The Reliability of Thermal Analysis of Cross-Country Gas Pipelines B

Technical Physics, Vol. 47, No. 4, 2002, pp. 375Ð379. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 1Ð5. Original Russian Text Copyright © 2002 by Kudryashov, Litvinenko, Serdyukov.

THEORETICAL AND MATHEMATICAL PHYSICS

The Reliability of Thermal Analysis of Cross-Country Gas Pipelines B. B. Kudryashov*, V. S. Litvinenko*, and S. G. Serdyukov** * St. Petersburg State Mining Institute (Technical University), Vtoraya liniya 21, St. Petersburg, 199026 Russia ** Lentransgaz Ltd. Received July 26, 2001; in ﬁnal form, October 10, 2001

Abstract—In the designs of gas pipelines that must be laid on the bottom of north and south seas, the reliability of analytical results for the gas temperature in the pipelines is questionable. The reason is that different authors take into account various factors inﬂuencing the temperature conditions in the pipelines. © 2002 MAIK “Nauka/Interperiodica”.

Today’s gas pipeline is a very complex and expen- projected to be laid in deep-water layers of the Barents, sive structure that is designed for long-term trouble- Black, and Caspian seas. Therefore, Shukhov’s formula free exploitation. The design and construction of any can no longer adequately describe the temperature dis- gas pipeline is based on extensive engineering work tribution. The authors of handbook [1] suggest the dif- that takes into account multiple factors, such as preset ferential equation capacity; mechanical strength; reliability; stability p Ð p against technogenic and environmental effects, includ- ÐkπDt()Ð t dx = Gc dt+ Gc Dh------1 -2dx ing natural calamities; and environmental safety. In r p p l addition, the pipeline material must withstand mechan- (2) ∆z v 2 ical wear and corrosion for a long time, the design must + Gg------dx+ Gd ------, be accident-proof, etc. l 2 The validity of the available design methods, specif- which, as they argue, includes all the factors influenc- ically, hydraulic (aerodynamic) methods, is beyond ing the gas temperature in the pipeline. Ignoring the gas question in most cases, since it has been repeatedly velocity variation (the last term on the right-hand side) checked during the long-term exploitation of the exten- and integrating Eq. (2), they come to an equation for the sive gas-pipe networks. However, the reliability of the gas temperature at any point of the pipeline: analysis of the gas temperature distribution in pipelines is questionable, because different authors variously tt= + ()t Ð t eÐax take into account factors influencing the temperature r 0 r conditions in gas pipelines. p Ð p 1 Ð eÐax ∆z1 Ð eÐax (3) Ð Dh------1 -2 ------Ð g------; Over many years, the temperature distribution in gas l a l c pa and oil pipelines has been estimated with well-known kπD Shukhov’s formula a = ------, kπD Gcp Ð------x ()Gcp tt= r + t0 Ð tr e , (1) where Dh is the JouleÐThomson coefficient for natural ° 5 gas, C/10 Pa; p1 and p2 are the gas pressures at the where t and t0 are the instantaneous and initial gas tem- beginning and at the end of the pipeline, Pa; ∆z is a rise ° peratures, respectively, C; tr is the environmental in the level, m; and l is the total length of the pipeline, m. (rock) temperature, °C; k is the coefficient of heat trans- 2 ° However, both initial equation (2) and Eq. (3), fer through the pipeline wall, W/(m C); D is the inner which is used in calculations, lack the term that takes diameter of the pipeline, m; G is the mass flow rate of into account the work of friction of the gas flow. Note the gas, kg/s; cp is the specific heat of the gas at constant ° that the authors of [1] contradict themselves. In the sec- pressure, J/(kg C); and x is the distance from the begin- tion devoted to the JouleÐThomson effect upon throt- ning of the pipeline. tling [1, p. 11], they argue that “…the work spent to Currently, gas pipelines many thousands of kilome- overcome obstacles… is converted to heat and is ters long pass through permafrost, cross huge water received by the substance…adiabatic throttling of obstacles, and have large level drops. New pipelines are gases, vapors, and liquids is an isoenthalpic process,

376 KUDRYASHOV et al. h = idem.” On the other hand [1, p. 16], it is argued that are intended to prevent negative temperatures that are “in the absence of external heat exchange, the tempera- expected at the end of the pipeline. ture of the gas being transported changes due to only “Throttling is a method of expanding a gas at con- the JouleÐThomson effect and a change in the position stant enthalpy when it passes through a throttle (that is, of the center of gravity of the flow.” through a local resistance, such as an aperture, nozzle, While analyzing the throttling of natural gas, Rus- pipe connection, valve, cock, tube constriction, etc.). sian engineers usually take the JouleÐThomson coeffi- The expansion, in this case, is attended by a tempera- cient in the range Dh = 3Ð5 K/MPa. A particular value ture change because the energy is spent to overcome the depends on the pressure and temperature. The value of internal molecular forces of mutual attraction” [2, p. 87]. Dh can be found by various techniques according to Upon the adiabatic throttling of oil in the near-bottom specific conditions [1–3]. We will find it with the for- region of a well, the oil temperature grows, while the mula for adiabatic expansion, taking the formula for gas temperature drops. This makes it possible to evalu- gas adiabatic expansion as the basis: ate the collecting properties of a pool and determine the k Ð 1 structure of strata [5]. In industrial condensed gas ------0 - k T2 p2 0 equipment, the throttling effect is used for cooling nat------= ----- . (4) ural gas in order to separate out water and readily con- T p 1 1 densing hydrocarbons [1]. In all these and similar The parameters of the improved version of the Blue cases, formula (4) for the JouleÐThomson coefficient is Flow project [4] are the following: rate of gas delivery valid and can be applied for calculations under specific simultaneously through two pipelines is 16 billion m3 conditions, since it holds only for sudden adiabatic gas pear year; the length and outer diameter of the pipelines expansion and this effect can by no means be “diffused” are 380 km and 24′′, respectively; the thickness of the throughout the many-kilometer pipeline. pipeline wall is 31.8 mm; the pipelines must be laid on Actually, when gas transport through a horizontal the Black Sea bottom at a depth of 2150 m; the methane pipeline lying deep in a rock is steady, the process is pressure at the beginning and end of the pipeline is similar to the adiabatic gas flow with friction, because 251.5 and 130.8 bar (abs.); the adiabatic exponent for the specific heat of rocks is high and their thermal con- methane (97.5% in natural gas) is k0 = 1.314; the initial ductivity is low. During the transport, the heat from the gas temperature at the Beregovaya station is t = 50°C or gas-heated pipeline wall returns to the gas and the gas T1 = 273 + 50 = 323 K (according to the estimates made in the pipeline cools down by converting the potential by Italian engineers [4]); the compressor discharge energy of pressure to the kinetic energy of the gas pressure is p1 = 251.5 bar; and the pressure at the Turk- (without allowance for external heat exchange). The ish coast is p2 = 130.8 bar. Then, the temperature at the degree of cooling is defined by the difference between end of the pipeline should be the initial and final velocities squared. In this case, the k Ð 1 gas temperature drop can be found, e.g., from the Ale------0 - 1.314Ð 1 k ------ksandrov formula [6] p 0 130.8 1.314 T ==T -----2 323 ------=276.28 K. 2 1  2 2 p1 251.5 v v ∆ k0 Ð 1 2 Ð 1 ∆ T = ------, (5) The temperature drop will be T = 323Ð276.28 = k0 2R 46.72 K, and the pressure drop will be ∆p = 251.5Ð 130.8 = 120.7 bar = 120.7 × 105 Pa. Accordingly, the where v1 and v2 are the cross-section-averaged veloci- JouleÐThomson coefficient in this case is ties of compressible gas at the beginning and end of the pipeline, m/s; k is the adiabatic exponent (for methane, ∆T 46.72 0 Dh ==------=3.87 K/MPa, k = 1.314); and R is the gas constant (for methane, R = ∆ 5 0 p 120.7× 10 500 J/(kg K)). which virtually coincides with the midpoint of the For the parameters of the Blue Flow version listed range Dh = 3Ð5 K/MPa. above, v1 = 4.27 m/s and v1 = 6.94 m/s. Then, accord- We do not know the methods of thermal analysis ing to formula (5), the gas temperature drop is less than applied by the Italian engineers in the Blue Flow 0.01°C. This value is convincingly confirmed by expe- project. It is noteworthy, however, that with the calcu- rience of many years in drilling wells with air (gas) lated value Dh = 3.87 K/MPa substituted into Eq. (3), blow and measuring the temperature distribution in which ignores the work of flow friction, we do obtain them [7Ð10]. Early in the development of this technol- ° the gas temperature at the Turkish coast t2 = Ð10 C. ogy, it was argued in many, mostly American, publica- That is why the Saipem corporation recommended [4] tions that blow with gas is in many respects a much to set up the last 6.4 km of the double pipeline (near the more promising approach than irrigation but the tem- Turkish coast) on supporting piles (the cost of this perature at the well bottom drops down to Ð40°C project is estimated at $ 78.85 million) or to increase because of the JouleÐThomson effect. Note that Amer- the diameter of one of the pipelines from 24 to 26′′ (an ican drilling engineers used formula (4) for sudden adi- extra cost of $ 115.02 million). Both recommendations abatic gas expansion.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE RELIABILITY OF THERMAL ANALYSIS 377

Consider a typical situation (Fig. 1). Upon rotor drilling with air blow, the ascending air flow velocity at the well bottom must be no less than v2 = 20 m/s. This value will be taken in subsequent calculations. Let the pressure in the drill pipe string near the cone drill bit be p1 = 8 bar and in the annular channel, 4 bar abs. Then, if the cross-sectional area of the inner channel of the v2 drill pipe string is equal to that of the annular channel of the well, the air velocity near the drill bit will be v = 1 T 10 m/s. Let the compressed air temperature near the 2 ° drill bit be t1 = 20 C or T1 = 293 K. The adiabatic expo- v1 p nent for air is k0 = 1.41. Under these conditions, for- 2 mula (4) for the air temperature at the bottom of the annular channel yields T1 1.41Ð 1 ------1.41 4 p1 T ===293 --- 239.5 KÐ 33.5°C, 2 8 ∆ and the temperature drop is T = T1 Ð T2 = 293Ð239.5 = 53.5°C. However, experience of drilling with blow suggests that this technology may encounter some difficulties associated with the insufficient cooling (overheating) of rock-crushing tools (drill bits). For this reason, drilling Fig. 1. Schematic representation of air blow of the well bottom. by diamond bits with blow has been found to be inap- propriate. One of the authors showed [8, 10] that the temperature must be calculated with formula (5), which sudden adiabatic flow of bottled air. Let the initial yields a reduction of the temperature at the well bottom by parameter values remain the same: the pressure and the temperature of the air inside the bottle are p1 = 8 bar abs 2 2 1.41Ð 1 20 Ð 10 and T1 = 293 K, respectively, and let the external pres- ∆T ==------0.15°C 1.41 2× 287 sure be p2 = 4 bar abs. It is known that if the bottle pressure exceeds the external one by a factor of more than at the same initial parameters and the gas constant for 1.86, the air velocity equals the velocity of sound; that air R = 287 J/(kg K). ≈ is, v2 330 m/s [12]. (At lower pressure drops, the Subsequently, it was found that the early bench Saint-Venant principle comes into action.) By defini- experiments (February, 1951) carried out by specialists tion, the bottle air velocity is v1 = 0 m/s. Formula (5) from Hughes Tool Co. [11] gave identical results. yields an air temperature reduction upon flowing out of In a 2-ft (≈0.6 m)-deep well of diameter 8 3/4′′ the bottle (≈220 mm), a new three-extension OSC jet drill bit of diameter 6 1/4′′ (≈160 mm) designed for drilling with 1.41Ð 1 3302 Ð 02 ∆T ==------55.2°C. air blow was installed without rotation (the diameter of 1.41 2× 287 each of the extensions was 5/16′′ (≈8 mm). Air circu- lated along the drill pipe string of diameter 2′′ Then, the temperature at the exit from the bottle is (≈50 mm) through the extensions at a flow rate of T = T Ð ∆T = 293 Ð 55.2 = 237.8 K = Ð35.2°C, which 350 ft3/min (≈10 m3/min) and at a temperature of 77°F 2 1 almost equals that predicted from formula (4) for sud- (25°C). The resulting pressure drop was 80 psi den adiabatic expansion. (≈5.5 bar abs). Under these conditions, the calculation by formula (4) for the adiabatic expansion of air yields In our opinion, if starting equation (2) would take the temperature drop ∆T = 116.5 K and the temperature into account heat release due to gas flow friction at the drill bit end t = Ð91.5°C. At the same time, the (instead of the unjustified inclusion of the Joule–Thom- measurements performed after the steady-state circula- son effect) and would not ignore the gas velocity variation had been established (in 17Ð30 min) showed that tion (the fourth term on the right-hand side), it would the temperature drop is only 9°F (5°C). However, even yield the same cooling in the pipeline as the Aleksan- this minor drop was explained by the absorption of the drov formula (provided that external heat exchange is heat of vaporization when the air is humidified, enter- absent) because cooling may be associated only with ing the well. the potential-to-kinetic energy conversion under these The general applicability (and, hence, validity) of conditions. At the same time, a change in the kinetic Aleksandrov formula (5) can be demonstrated with the energy can also be neglected, since this change is very

TECHNICAL PHYSICS Vol. 47 No. 4 2002 378 KUDRYASHOV et al.

0 km 99.5 km 224.6 km 325.6 km and i is the dimensionless hydraulic gradient, which is 123 4 found from the expression ∆p i = ------ρ -, (7) G = 246.5 kg/s g avl ∆ ∆ where p is the pressure drop in the pipeline ( p = pin Ð ρ pI = 49.8 bar pII = 46.4 bar pII = 40.4 bar pIII = 29.7 bar pf), Pa, and av is the average gas density in the pipe- tI = 40°C tII = 12.2°C tII = 10.0°C tIII = 8.0°C line, kg/m3. In the calculations, we used the actual mass flow 34.1 km rate of the gas G = 246.5 kg/s and the inner diameter of G6 = 6.53 kg/s the pipeline D = 1.195 m. The gas being transported is methane (accounts for 97.5% in natural gas), for which Fig. 2. Segment of the TorzhokÐMinskÐIvatsevichi 3 pipe- R = 500 J/(kg K). The overcompression factor is z = line (3rd branch) between the Rzhev and Orsha CSs ° (1, Rzhev CS; 2, Kh. Zhirki CS; 3, Smolensk CS; and 0.91, and the specific heat is cp = 2220 J/(kg C) (at an 4, Orsha CS). The daily average absolute pressure and tem- average temperature t = 20°C). The gas consumption at perature for the gas being transported are indicated. Sub- a rate G0 = 6.53 kg/s between the compressor stations scripts I, II, and III refer to the initial (output), intermediate, and final (input) values, respectively. Note: 1 bar = 105 Pa. at Rzhev and Kh. Zhirki is neglected. The rock temperature around the pipeline is taken to be constant, tr = 5°C at the end of October, throughout the segment con- small; hence, the gas cooling is also very small (on the sidered. Based on tentative estimates for external heat order of several hundredths of a degree). exchange, the coefficient of heat transfer from the gas 2 ° Let us compare the calculated values of the effects to the rock is set equal to k = 4.3 W/(m C). considered with the daily average (October 22, 2000) The calculations were performed by formula (6) temperature of the gas transported through the for two variants: (1) with regard for the mechanical TorzhokÐMinskÐIvatsevichi 3 pipeline on the segment work of friction of the gas flow (i = 0.021) and without between the compressor stations (CSs) in Rzhev and taking into account the JouleÐThomson effect (Dh = 0) Orsha. This 325.6-km-long segment is schematically and (2) in view of the JouleÐThomson effect (Dh = depicted in Fig. 2, where the subscripts I, II, and III 0.4 °C/105 Pa) and without taking into account the refer to initial (output), intermediate, and final (input) work of friction (i = 0). In both cases, external heat gas temperatures and pressures, respectively. The data exchange was equally taken into consideration [k = were provided by Lentransgas Ltd. 4.3 W/(m2 °C)]. Ignoring a level drop (∆z = 0), we supplement the The results of the calculations are summarized in the right-hand side of expression (3) by the term allowing table, where the data obtained by Shukhov’s formula (1) for the mechanical work of friction of the gas flow. for the same initial conditions are shown for compari- Then, the temperature distribution along the pipeline is son. given by Both analytical distributions (with external heat exchange equally taken into account) tend to the con- Ggi ∆p Gcp tt= r + ------Ð Dh------stant temperature value faster than the actual (mea- kπD l kπD sured) gas temperature distribution along the pipeline. kπD (6) Ð------This can be explained, for example, by the variable rock Ggi ∆p Gcp Gv p + t Ð t Ð ------+ Dh------e . temperature along the 325-km-long pipeline. However, 0 r kπD l kπD the agreement between the measured and analytical data is of minor concern for us. It is interesting to con- Here, g is the gravitational acceleration (g = 9.81 m/s2) trast the calculated gas temperature distributions in the pipeline when the work of friction is taken into account Comparison of the measured and analytical values of the nat- and the JouleÐThomson effect is not, and vice versa. In ural gas temperature along the path Rzhev CSÐOrsha CS, °C the former case, the gas temperature at the final point is nearly coincident with its average value on October 22, Compressor stations Rzhev Kh. Zhirki Smolensk Orsha 2000. In the latter case, the gas temperature calculated for the last two compressor stations (without pumping- Measurements 40 12.20 10.00 8.00 over) turned out to be lower than the rock temperature. Calculated data 40 9.83 8.18 8.14 This can be explained only by incorrectly taking into (i = 0.021, Dh = 0) account the JouleÐThomson effect especially when the Calculated data 40 6.06 4.21 4.16 work of friction is neglected. ° (Dh = 0.4 C/bar, i = 0) The gas velocities calculated at the beginning and at Data calculated by 40 6.86 5.05 5.00 the end of the segment under consideration were found Shukhov’s formula to be v1 = 3.34 m/s and v2 = 5.61 m/s for the actual tem-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE RELIABILITY OF THERMAL ANALYSIS 379 peratures and pressures. The temperature drop due to of thermodynamics and higher mathematics the partic- the potential-to-kinetic energy conversion was as low ipants demonstrated. Yet, some fundamental physical as ∆t = 5 × 10Ð3 °C, as estimated by formula (5). laws should be remembered. One of them states that the Note that the associated calculation made by giant compressor energy spent on gas transfer over sev- Shukhov’s formula (1) for the same initial conditions eral hundreds of kilometers at a rate of several millions gave the temperature value lying between those of cubic meters per day is eventually dissipated as heat, obtained in the above two variants, since it includes nei- and this fact cannot be ignored. ther the mechanical work of friction nor the JouleÐ All the above reasoning seems to be valid even with- Thomson effect. At the end of the 325-km-long pipeline out regard for the time factor, which has never been (the Orsha CS), the gas temperature found by considered in the design and analysis of the pipelines. Shukhov’s formula is equal to the rock temperature. For the recently built pipelines, however, this factor is However, the structure of this formula is such that it of great importance and will be considered in subse- cannot yield a lower temperature. For many decades, quent publications. this fact satisfied theorists and engineers engaged in the The authors admit that the reasoning and calcula- gas extraction industry. Formula (6) for the first variant tions reported in this article may make little difference (Dh = 0) also cannot yield temperatures below the rock for some of those taking part in the Blue Flow project. temperature, since it takes into account the work of fric- Anyway, they make anybody think and try to solve the tion of the gas flow. Nearly the same results were problem touched upon analytically and experimentally. obtained for a number of segments in the pipeline network exploited by Lentransgas Ltd. In the history of exploitation of pipelines passing REFERENCES through the Arctic permafrost and through unfrozen 1. B. P. Parshakov, R. N. Bikchentaœ, and B. A. Romanov, rocks in the European part of Russia and in Western Thermodynamics and Heat Transfer (in Technological Europe up to Greece and Portugal, engineers from Len- Processes of Oil and Gas Industry): Textbook for Insti- transgas Ltd. have never encountered any problems tutes of Higher Education (Nedra, Moscow, 1987). associated with rock freezing, frost-induced rock heav- 2. A. I. Gritsenko, Z. S. Aliev, O. M. Ermilov, et al., Man- ing, etc., except for a drastic drop in the gas pressure at the ual on Study of Wells (Nauka, Moscow, 1995). distributing stations of gas-transporting networks [13]. 3. Tables of Physical Quantities: Handbook, Ed. by In summary, our conclusions, as applied to the cal- I. K. Kikoin (Atomizdat, Moscow, 1976). culations within the Blue Flow project, are the follow- 4. Saipem. Gazprom-SNAM. Project “Blue Stream”: Busi- ness and Technical Application, June 30, 1999 (with sup- ing [14]. For the gas pipeline passing on the Black Sea plements). bottom from Russia to Turkey at a minimal water temperature t = +9°C, its temperature at the Turkish coast 5. Sh. K. Gimatudinov and A. I. Shirkovskiœ, Physics of Oil by no means can be negative. Thus, there is no reason and Gas Seam (Nedra, Moscow, 1982). for extensive discussion [4] on the danger of mechani- 6. V. L. Aleksandrov, Technical Hydromechanics (Gos- cal damage to the pipeline at the Turkish coast because tekhizdat, Moscow, 1946). of rock freezing and a fortiori for the “scientific” justi- 7. B. B. Kudryashov, Zap. Leningr. Gos. Inst. im. G. V. Ple- fication of extremely questionable proposals on khanova 62 (2), 43 (1969). “improving” the initial Blue Flow project, which costs 8. B. B. Kudryashov and A. I. Kirsanov, Boring of Explor- $ 79Ð115 million. atory Wells with Air (Nedra, Moscow, 1990). 9. B. B. Kudryashov, V. K. Chistyakov, and V. S. Lit- On November 23, 2000, the coordination meeting vinenko, Well-Boring under Aggregative State Change on gas transport from the Shtokman condensed gas of Rocks (Nedra, Leningrad, 1991). deposit was held at the St. Petersburg State Mining 10. B. B. Kudryashov and A. M. Yakovlev, Drilling in the Institute. Specialists from research institutes and uni- Permafrost (Oxonion Press, New Delhi, 1990). versities, as well as the administration and the leading engineers from the Gasprom joint-stock venture, were 11. Laboratory Report (Hughes Tool Company, 1951). present at the meeting (a total of 31 persons). Various 12. B. B. Kudryashov, Prom. Energ., No. 3, 14 (1962). and often mutually exclusive ideas concerning whether 13. S. G. Serdyukov, Author’s Candidate’s Dissertation (St. or not the mechanical work of friction and the JouleÐ Petersburg, 1998). Thomson effect should be taken into account in the 14. B. B. Kudryashov and A. V. Kozlov, Preliminary Report analysis of the temperature distribution along the pipe- on Reliability Estimate of Engineering Solutions on line were put forward. It was even suggested to use the Project “Blue Stream” (St. Petersburg, 2000). NavierÐStokes equations to solve this relatively simple problem. We greatly appreciate the intimate knowledge Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

THEORETICAL AND MATHEMATICAL PHYSICS

On the Frenkel Problem of Equivalent Steady Currents in a Sphere R. Z. Muratov Moscow State Mining University, Moscow, 117935 Russia Received July 16, 2001

Abstract—The Frenkel problem of substituting the 3D system of steady currents given in one of two concentric spherical regions by an equivalent system of currents (i.e., by that inducing the same external magnetic ﬁeld) that is distributed over the surface of the other region is considered. A method of multipole moments providing the direct solution (without calculating the ﬁelds) of the problem is described. The case of currents with the density components represented by cubic polynomials of the Cartesian coordinates is considered as an example. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION In this paper, we propose a solution to the simple problem of equivalent steady currents where the “orig- Dissimilar systems of sources are referred to as inal” currents are specified in the volume of a sphere of equivalent (or adequate) if they, being located inside a radius a and the equivalent currents are looked for on a finite space, induce the same fields outside the space. concentric sphere of radius R. The problem is consid- These may be equivalent distributions of gravitating ered in Cartesian (rather than in spherical) coordinates masses, electric charges, and, finally, steady electric in order to subsequently extend the approach used to currents. The characteristic feature of the last-named geometrically more general problems. The solution of case is the vector nature of the sources. the problem, with the Cartesian components of the It is well known that a charged sphere with a uni- original current density represented by polynomial form volume charge density and a concentric sphere of functions of the Cartesian coordinates, is obtained by arbitrary radius with a uniform surface charge density the method of multipole moments. As in the case of sca- induce the same external electrostatic fields coinciding lar sources, this method makes it possible to avoid the with that of a point charge located at the center of the cumbersome procedure of calculating the fields. spheres if the total charge is equal in all the three cases. These are examples of scalar equivalent sources. For a magnetostatic field, such a simple example is absent MULTIPOLE MOMENTS METHOD since there are no magnetic charges in the nature. As is well known, in a simply connected spatial The problem of finding the source distributions region that is an outer region with respect to steady equivalent to a given distribution is not new. In the case electric currents, one can deal with the pseudoscalar of scalar sources (charges), the theoretical statement of potential of the induced magnetic field (hereafter, mag- the problem and the discussion of general results were netic potential) and its multipole expansion [5] ∞ given in [1, p. 103; 2, p. 524]. It is worth noting that in l the general statement, the problem of equivalent ()Ð2 1 Φ()r = ∑ ------m … ∇ …∇ ---. (1) sources has an infinite set of solutions, as also follows ()2l ! i1 il i1 il r from the example given above. The statement of practi- l = 1 cal interest is that in which the distribution of sources is The quantities m … appearing in (1) are the com- given in a volume V (or on a closed surface S) and it is i1 il required to find the equivalent distribution of sources ponents of the lth-rank tensor, which constitute the on another surface S. This provides the uniqueness of magnetic multipole moment of the lth order. In what the solution. The method for solving such scalar prob- follows, we will distinguish the magnetic multipoles ()j lems in the case of ellipsoidal spatial regions was m … of the bulk currents with the density j(r) and the i1 il reported in [3, 4]. As for the vector problem of equiva- ()i magnetic multipoles m … of the surface currents with lent steady currents, the author has not encountered an i1 il example of its consideration. However, the solution of the surface density i(r). These multipole moments are such a problem is useful for refining the geomagnetic given by dynamo mechanism, estimating currents induced on superconducting shields, and tackling some other ()j 1 []∇ mi …i = ------∫ rj i …i dV, (2) issues of practical magnetostatics. 1 l ()l + 1 c 1 l

ON THE FRENKEL PROBLEM OF EQUIVALENT STEADY CURRENTS 381

()i 1 filled with currents is represented by either a sphere or m … = ------[]∇ri … dS. (3) i1 il ()l + 1 c°∫ i1 il its surface and the current density components (bulk or surface) are polynomial functions (of degree L > 0) of Here, the bracketed vectors mean the vector product. Cartesian coordinates, sum (1) involves a finite number Expressions (2) and (3) involve the irreducible sym- of terms, since all magnetic multipoles of the (L + 1)th metrical tensor d … (r), which is defined as the prod- or higher orders vanish. Here, we assume that the origin i1 il is placed at the center of the sphere. uct of the components of the vector operator [6] Now we turn to the general statement of the problem Dˆ = 2rr()∇ Ð r2∇ + r (4) and the method to solve it. Assume that an electric cur- (the vectors in parentheses mean the scalar product). rent with the Cartesian density components represented The product operates on unity; i.e., by L-degree polynomials l m n d … ()r = Dˆ i …Dˆ i ⋅ 1. (5) jx = ∑ Almnx y z , i1 il i l 1 ≤ lmnL++≤ 2 2 2 In particular, dx = x, dxx = 2x Ð y Ð z , dxy = 3xy, 2 2 2 l m n dxyy = 3x(4y Ð x Ð z ), and dxyz = 15xyz, so that jy = ∑ Blmnx y z , (11) 1 ≤ lmnL++≤ 1 () mx = ------∫ yjz Ð zjy dV; (6) 2c l m n jz = ∑ Clmnx y z 2 () 1 ≤ lmnL++≤ mxx = ---∫xyjz Ð zjy dV, (7) c is specified in the volume of a sphere with radius a.2 1 2 2 m = --- {}xzj Ð yzj + ()y Ð x j dV; (8) In (11), the coefficients A, B, and C of the polynomi- xy c∫ x y z als given are related by expressions following from the current steadiness condition 3 { ()2 2 2 mxyy = ------∫ 10xyzjx + x + z Ð 4y zjy 4c (9) divj = 0 (12) ()2 2 2 } +4y Ð 11x Ð z yjz dV, and the boundary condition on the sphere surface

15 2 2 2 2 j = 0. (13) m = ------{()z Ð y xj + ()x Ð z yj n S xyz 4c∫ x y It is required to find the current on the surface of a ()2 2 } 1 1 + y Ð x zjz dV. (10) concentric sphere of the radius R that is adequate to (11). The pattern of calculation is as follows. First, we Note that, owing to the special “normalization” [5] calculate all the independent components of the mag- of the magnetic moments, expansion (1) is identical to ()j netic multipoles m … whose rank is in the range 1 ≤ the multipole expansion of the electrostatic potential i1 il with only one exception: in sum (1), the term corre- l ≤ L. The total number of such components is sponding to l = 0 (magnetic charge) is absent. L For our purposes, two well-known properties of ()M ()() expansion (1) are essential. First, this expansion is uni- N L ==∑ 2l + 1 LL+ 2 . (14) versal, i.e., is applicable to arbitrary distributions of l = 1 currents if they occupy a finite spatial region. In partic- As it was mentioned, any Cartesian component of ular, the expansion is independent on what kind of cur- the equivalent surface current has also to be an L-degree rents (bulk or surface) induces the magnetic field. polynomial. Recall that the degree of a polynomial on Hence, if different systems of currents have the same the surface of a sphere multipole moments, these systems are adequate. For- mally, this implies the coincidence of the infinite num- x2 ++y2 z2 = R2 (15) ber of the magnetic multipole components, for series (1) is infinite. is that of the polynomial resulting from the initial one The second property of the multipole expansion by eliminating [in view of (15)] the even powers of any allows one to use the above universality just in the geo- Cartesian coordinate (e.g., z) and collecting the similar metric configuration considered. Indeed, if a region terms. These polynomials may be represented in various equivalent forms (see [3] for details). Hereafter, we 1 Hereafter, in the case of several relations (equations) that differ will write them in the symmetric (keeping the equiva- from each other by either the circular permutation or interchanging of the coordinates, we write only one relation (equation). It is 2 good to bear in mind that, in the case of interchanging, the In (11), we have taken into account that the coefficients before the pseudovariables change their sign. summands with x0y0z0 are equal to zero.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 382 MURATOV lence of the Cartesian directions) form: this requirement adds to (19) L ()L = 1 ()2 α i j k NS ==N L + 2 (21) ix = ∑ ∑ ijkx y z , lL= ijk++ = l homogeneous equations. L The condition of surface current steadiness β i j k jy = ∑ ∑ ijkx y z , (16) lL= ijk++ = l divi = 0 (22) L decreases the polynomial degree by unity, adding j = ∑ ∑ γ xi y jzk. z ijk ()L Ð 1 2 lL= ijk++ = l Ndiv ==N L (23) Thus, each of the components of the surface current more homogeneous equations for the coefficients. is the sum of homogeneous polynomials of the (L Ð 1)th and Lth degrees with the coefficients to be found. Since ()M Thus, the number of equations N L + NS + Ndiv ()H the number of the coefficients N L of a homogeneous always exceeds the number of unknowns N by unity. It L-degree polynomial is can be shown that one of the homogeneous equations is a linear combination of the rest. ()H 1 N = ---()L + 1 ()L + 2 , L 2 EXAMPLE OF ADEQUATE CURRENTS the number of the coefficients in each of the polynomials in (16) is In order to illustrate the method, we consider the case when the components of currents (11) and (16) are ()L ()H ()H ()2 N ==N L + N L Ð 1 L + 1 . (17) cubic polynomials. As was noted, this case generally involves 48 unknown coefficients of polynomials (16). The number of unknowns in our problem is defined Fortunately, as in the case of scalar sources [3], the cal- by the total number of coefficients in all three polyno- culations are substantially simplified, since the result- mials in (16): ing system of 48 linear algebraic equations splits into () N ==3N L 3()L + 1 2. (18) eight independent subsystems. Moreover, as in the case of scalar sources [3], it is sufficient to find the analytical In particular, the number of unknowns is 48 for solutions of four subsystems only because of the sym- L = 3, whereas, in the similar problem with scalar metry of the problem (providing the equivalence of the sources, the number of unknowns is three times Cartesian directions). The remaining solutions may be smaller. obtained by circular permutations in those found previ- Let us calculate the number of available linear alge- ously. braic equations needed to find the coefficients of the According to the aforesaid, instead of considering polynomials in (16). First of all, these are the inhomo- all fifteen independent components of the magnetic geneous equations arising as a result of equating the multipoles m, mij, and mijk, we can restrict our analysis corresponding independent components of the mag- to, e.g., the following ones: (1) m , m , and m ; (2) m netic multipole tensors (of rank from one to L) due to x xyy xzz xx and myy; (3) mxy; and (4) mxyz. These components are currents (11) and (16). Since any symmetric lth-rank split into four groups, each having an independent sub- irreducible tensor, such as that of the magnetic multi- system of equations for the polynomial coefficients. pole of the same rank, has 2l + 1 independent compo- () () () nents, the number of the resulting inhomogeneous Equating the components m i , m i , and m i cal- equations turns out to be x xyy xzz culated by formulas (6) and (9)3 to the corresponding L () () () ()M ()()2 components of the bulk currents m j , m j , and m j N L ==∑ 2l + 1 L + 1 Ð 1. (19) x xyy xzz l = 1 and completing the set of inhomogeneous equations obtained by necessary homogeneous equations result- The requirement that the normal component of the ing from (20) and (22), we find the following closed surface current density vanish on the surface of sphe- subsystem of seven equations (see Footnote 2): re (15), () γ γ γ β β β 15c ()j in ==ri S xix ++yiy ziz = 0, (20) 3 030 ++210 012 Ð 3 003 Ð 021 Ð 201 = ------mx , S 2πR6 increases the degree of the polynomials describing the current by unity, as is seen from (20). Consequently, 3 With jdV replaced by idS.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 ON THE FRENKEL PROBLEM OF EQUIVALENT STEADY CURRENTS 383 α β β β 〈 〉 〈 〉 5 111 ++3 003 Ð 4 021 201 and A011 = 0. Hereafter, the angular brackets … mean the sum of three terms of the circular permuta- γ γ γ 35c ()j tion. +12 030 Ð 11 210 Ð 012 = ------8mxyy, 2πR The solution of the second subsystem of equations α β γ α β γ α β γ (for the unknowns 011, 101, and 110) has the form 111 ++2 021 2 012 == 0,111 ++201 210 0, 7 β γ α 5c ()()j ()j a 003 +0,012 = 011 ==------mxx + 2myy ------A011. (30) 8πR6 7R6 where () () Equating the quadrupole moments m i to m j  xy xy ()j 2π 5 1 2 γ gives the value of the coefficient 002 and leads us to the mx = ------a C010 Ð B001 Ð ---a 15c  7 third subsystem of four equations for the remaining α β γ γ (24) unknown coefficients 101, 011, 200, and 020:  ---× ()B Ð4C + B Ð 4C , α β γ γ 15c ()j 201 210 021 012 101 Ð 011 Ð22 200 + 020 = ------mxy ,  4πR6 π α γ α β ()j 8 9()101 +0,200 ==101 +0,011 mxyy = ------a 4B201 Ð 14B021 + C210 Ð 11C012 . (25) 315c where Expressions (24) and (25) are the result of integra- ()j 8π 7 tion by formulas (6) and (9) and the subsequent simpli- mxy = ------a A101, (31) fication in view of the relationships 35c and C = 0, A = ÐB = ÐC = C . The unknown A ++2B 2C = 0, 002 101 011 200 020 111 021 012 coefficients are

A111 +0,B201 Ð B003 + C201 Ð C012 = γ 002 = 0, (32)

B021 Ð B003 +0,C030 Ð C012 = 7 5c ()j a α ==γ ------m =------A . (33) 2 2 101 020 π 6 xy 6 101 B001 +++a B003 C010 a C012 = 0 8 R 7R between the coefficients A, B, and C, which follow from ()i Equating the magnetic octupole components mxyz (12) and (13). ()j The solution of the first subsystem of the equations and mxyz calculated by (10) to each other gives zero α β β β γ γ values for α , β , and γ and results in the fourth (for the unknowns 111, 201, 021, 003, 210, 030, and 300 030 003 γ subsystem of six equations 012) has the form 7c ()j 7c ()j ()j 〈〉α α ------α = ------()m Ð m , (26) 102 Ð 120 = 8mxyz, 111 8 xyy xzz π 12πR 2 R β +0,γ ==α +0β c ()j 7 ()j 210 201 120 210 γ ==Ðβ ------3m + ------m , (27) 030 021 6x 2 xyy α α β β γ 4πR 6R for the unknown coefficients 120, 102, 210, 020, 201, γ and 021. Here, c ()j 7 ()j 7 ()j γ = ------3m + ------m Ð ------m . (28) π 210 6x 2 xyy 2 xzz ()j 4 9 〈〉 4πR 2R 6R mxyz = ------a A102 , (34) 63c Similarly, equating the corresponding quadrupole () () () () and components m i and m i found by (7) to m j and m j xx yy xx yy 〈〉A = 0, in view of (20) and (22) yields three equations forming 100 β the second subsystem: A120 Ð0,A102 ++210 Ð2B012 C003 Ð C021 Ð C201 =

15c ()j γ β 〈〉α A300 Ð0,A102 + C003 Ð C201 = 110 Ð 101 ==------mxx , 011 0, 8πR6 3A300 ++B210 C201 = 0, where, according to (7), A +++a2 A C a2C = 0, π 100 102 001 201 ()j 8 7() mxx = ------a C110 Ð B101 , (29) 2 105c C001 +0.a C003 =

TECHNICAL PHYSICS Vol. 47 No. 4 2002 384 MURATOV The calculations yield responsible only for its own components of the mag- α netic 2L field tensor. 300 = 0, (35) (2) Consider the case when a sphere of radius a with 9 7c ()j a given currents j(r) is concentrically placed into a spher- α ==------m ------〈〉A . (36) 102 π 8 xyz 8 102 ical cavity of radius R > a that is in the bulk of a super- 12 R 27R conductor. It is required to find the currents i'(r) Thus, the problem is solved, since the other coeffi- induced on the surface of the cavity. If the problem of cients are found by the circular permutation or inter- the surface currents i(r) adequate to the given bulk ones changing of the coordinates in formulas (26)Ð(28), j(r) that is discussed in this paper is solved, the problem (30), (32), (33), (35), and (36). regarding the superconductor is solved as well. The example considered involves the currents corre- Namely, sponding to the degrees L = 2 and 1 of the polynomials i'()r = Ðir(). as special cases. In particular, the simplest adequate currents with the components in the form of first-degree polynomials may be represented as REFERENCES jx ==0, jy B001z, jz =ÐB001 y; (37) 1. Ya. I. Frenkel, Electrodynamics (General Theory of 5 5 Electricity): Collection of Selected Works (Akad. Nauk a a SSSR, Moscow, 1956), Vol. 1. ix ==0, iy ------B001z, iz =Ð------B001 y. (38) 5R4 5R4 2. Ya. I. Frenkel, Electrodynamics (ONTI, Leningrad, 1935), Vol. 2. 3. R. Z. Muratov, Astron. Zh. 69, 604 (1992) [Sov. Astron. CONCLUSION 36, 306 (1992)]. We complete the paper with two remarks. 4. R. Z. Muratov, Zh. Tekh. Fiz. 67 (4), 1 (1997) [Tech. (1) Splitting the set of the algebraic equations for the Phys. 42, 325 (1997)]. polynomial coefficients of the surface currents into 5. B. V. Medvedev, Principles of Theoretical Physics three (L = 1), seven (L = 2), or eight (L ≥ 3) independent (Nauka, Moscow, 1977). subsystems simplifies the process of finding the solu- 6. S. P. Efimov, Teor. Mat. Fiz. 39 (2), 219 (1979). tion. At the same time, this means that, for a fixed L, there exists the corresponding number (3, 7, or 8) of independent systems of currents, each of which is Translated by M. Fofanov

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 385Ð388. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 11Ð14. Original Russian Text Copyright © 2002 by Perevezentsev, Svirina, Ugol’nikov.

GASES AND LIQUIDS

Model of Local Melting of Grain Boundaries Containing Impurity Atom Clusters V. N. Perevezentsev, Yu. V. Svirina, and A. Yu. Ugol’nikov Lobachevsky State University, pr. Gagarina 23, Nizhni Novgorod, 603600 Russia Received January 16, 2001; in ﬁnal form, October 2, 2001

Abstract—A model describing the initial stages of the local melting of grain boundaries containing impurity atom clusters is developed. The local melting process is considered as the formation of liquid zones whose geometry and number depend on the initial concentration and thermodynamic characteristics of the impurity atoms, as well as on the material parameters and temperature. The evolution of the liquidÐsolid interface structure with increasing temperature is discussed. The dependences of the liquidÐsolid interface parameters on temperature are obtained at various initial concentrations and thermodynamic characteristics of the impurity atoms. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION The aim of this study is to elaborate a model of grain-boundary local melting and to analyze the effect It is known that some microcrystalline aluminum of the concentration and thermodynamic characteristics alloys and composites are prone to superplastic defor- of impurity atoms segregating at the boundaries. It is mation at anomalously high rates (up to 103 sÐ1) [1Ð7] also of interest to see how the material parameters at near-solidus temperatures. The study of high-rate affect the structure of the boundaries at the initial stages superplasticity has shown that this effect is caused by of the local melting. The application of this model for the local melting of the grain boundaries. Experimental the description of high-rate superplasticity will be the investigations into the local melting of grain boundaries subject of subsequent publications. using differential scanning calorimetry, transmission electron microscopy, and Auger spectroscopy have revealed the following features of this phenomenon [6]: MODEL OF LIQUIDÐSOLID STRUCTURE (1) the local melting of the boundaries is closely related OF GRAIN BOUNDARY to the aggregation of impurity atoms at the boundaries (in particular, silicon and magnesium clusters at grain Let us suppose that, at the initial stage of the local boundaries and interfaces play a vital role in aluminum melting, a grain boundary can be represented as a solid δ alloys), (2) according to the phase and chemical com- layer of thickness 0 that contains impurity atoms. position of the alloy, the local melting temperature may ӷ v v Their concentration in this layer is C0 C0 , where C0 be different from the solidus temperature by several is the volume impurity concentration. It can be assumed degrees to tens of degrees, (3) both liquid and solid that, early in the local melting, liquid islands (droplets) zones are observed at the boundaries under local melt- appear and their number increases with temperature. ing conditions, and (4) the thickness of the liquid layer Therefore, we will view the liquidÐsolid structure of the does not exceed 30 mm at the final stages of the local grain boundary as a solid layer of the grain-boundary melting. Efforts to theoretically describe the liquidÐ phase with uniformly distributed liquid droplets. Dur- solid structure of the grain boundaries with regard for ing the local melting, the impurity is redistributed the features listed, as well as the application of the the- among the liquid and solid zones of the boundary. Let ory to the problem of high-rate superplasticity, was L b undertaken in [8, 9]. A model of the boundary structure Ci and Ci be the concentrations of impurity atoms in at the final stages of the local melting was proposed in the liquid and solid zones of the boundary, respectively. [8]. It treats a grain boundary as a liquid layer contain- To describe the grain boundary structure, it is necessary ing solid streaks. The main disadvantage of this model to specify parameters characterizing the equilibrium is the assumption that the liquidÐsolid interface is flat. geometry of the liquid droplets, the number of the liq- A more realistic model that considers conditions of uid zones, and the concentration of impurity atoms in phase and chemical equilibrium at the liquidÐsolid the liquid and solid zones of the boundary. Generally, interfaces was developed in [9]. An approach put for- the geometry of the liquid zones in phase equilibrium ward in [9] was applied to analyze the temperature can be represented as two spherical segments with a δ δ δ dependence of the superplastic deformation rate at the radius of curvature R, height y = ( Ð 0)/2 ( is the thick- initial stages of the local melting [10]. ness of the liquid zones), and base radius rL. These seg-

386 PEREVEZENTSEV et al.

2r atoms in the course of melting. In the limiting case 1 L where the outflow of the impurity atoms from the δ δ0 boundary into the grain body can be neglected in the local melting process, this condition becomes b L L 2 C δ = C δ ()1 Ð α ++C δ α C 2V n , (5) R L/SB L/G 0 0 i 0 L i 0 L i L L π 2 where VL = (1/3) y (3R Ð y) is the volume of the spher- Fig. 1. LiquidÐsolid interface structure: 1, grain I; 2, grain II. ical segment, nL is the number of the liquid droplets per α π 2 unit area of the grain boundary, and L = nL rL is the δ ments are separated by a cylinder of height 0 (Fig. 1). fraction of the liquid phase. In order to determine the parameters describing the The first term on the right-hand side of Eq. (5) is the geometry of the liquid zones (y, R, and rL), we consider number of the impurity atoms per unit area of the solid the balance of forces acting at the interfaces between zones at the boundary, and the second and the third ones the liquid and solid phases, namely, at the interface describe the number of the impurity atoms in the liquid between the grain body and the liquid zone, L/G, and at phase. By solving the system of Eqs. (1) and (3)Ð(5), that between the liquid zone and the solid grain-bound- one finds the analytical dependences of the geometrical ary phase, L/SB. The balance condition at the L/G inter- parameters n , y, R, and r on CL , Cb , and T. face can be written in the form L L i i The temperature dependences of the parameters nL, γ ∆ ()L v 2 L/S/Rq+ T/T = Ci Ð C0 kT, (1) y, R, and rL can be found from those of the impurity ∆ L b where T = Tm Ð T (Tm is the absolute melting point), atom concentrations Ci and Ci . To this end, we will λ ρ q = m is the heat of fusion at constant pressure, and take advantage of two more additional relationships. γ L/S is the energy per unit surface area of the liquidÐ The first of them is the condition of chemical equilib- solid interface. rium between the impurity atoms in the liquid and solid The first term on the left-hand side of Eq. (1) grain-boundary phases: describes the surface pressure and the second one rep- ΨL + kTln CLa3 = Ψb + kTln Cba3, (6) resents the configurational pressure related to a change i i i i in the bulk free energy due to the phase transition. The where ΨL and Ψb are the chemical potentials of the term on the right-hand side of Eq. (1) is the osmotic i i impurity in the liquid and solid phases, respectively, pressure of the impurity atoms. The balance condition and a is the interatomic spacing. at the L/SB interface has the form γ ∆ This condition implies that impurity diffusion L/S/rL + q T/T between the liquid and solid phases is absent in equilib- (2) +2()γ cosϕ Ð γ /δ = ()CL Ð Cb kT, rium. The second relationship can be written on L/S b 0 i i assumption that thermodynamic equilibrium exists and γ where b is the grain boundary surface energy (surface the free energy of the system remains unchanged (∆F = 0) tension). during the local melting. The expression for ∆F can be The third term on the left-hand side of Eq. (2) is the written in the form pressure acting on the L/SB interface due to the surface ∆ ∆ ∆ ∆ F ==Fγ ++Fm Gi 0, (7) tension at the L/G interfaces and the L/SB interface, and ∆ ϕ is the angle between the L/G interface and its refer- where Fγ is the change in the surface energy during ∆ ence (undisturbed) plane (Fig. 1). In view of the rela- the local melting, Fm is the change in the bulk energy, ∆ tionship cosϕ = 1 Ð y/R, Eq. (2) can be written as and Gi is the change in the chemical potential of the γ /r ++q∆T/T ∆γ/δ impurity atoms. L/S L 0 ∆ ∆ (3) In turn, the expressions for Fγ and Fm per unit γ δ ()L b Ð2 L/S y/ 0 RC= i Ð Ci kT, area of the boundary are written as where ∆γ = 2γ Ð γ . ∆ π γ π δ γ γ π 2 L/S b Fγ = 4 RynL L/S + 2 rL 0nL L/S Ð b nLrL, (8) The parameters y, R, and r are related by the rela- L ∆ ()∆ L tionship Fm = q T/TCÐ i kT 2V LnL (9) 2 () ()δ∆ ()L b π 2 ()b δ rL = y 2RyÐ . (4) + q T/TCÐ i Ð Ci kT 0 nLrL + C0 Ð Ci kT 0. To describe the structure of the liquidÐsolid inter- The first and second terms on the right-hand side of face, it is also necessary to determine the number of liq- Eq. (8) describe the increase in the surface energy of the uid droplets nL per unit area of the grain boundary. We system when the liquid droplets appear. The third term supplement the system of Eqs. (1), (3), and (4) by the stands for the decrease in the surface energy of the grain condition of conservation of the number of the impurity boundary due to the disappearance of the solid zones

TECHNICAL PHYSICS Vol. 47 No. 4 2002 MODEL OF LOCAL MELTING 387

2 rL/a nLa 24 0.005 (a) (a) 0.18 20 0.004

16 0.003 0.10 0.16 0.12 12 0.14 0.002 0.16 0.18 0.14 8 0.001 0.12 0.10 4 0 25 3.0

2.5 (b) (b) 2.0 20 0.01 2.5 2.0

15 1.0 3.0

1.5 1.5 10

1.0 5 0.00 0.80 0.84 0.88 0.92 0.96 1.00 0.80 0.84 0.88 0.92 0.96 1.00 T/Tm T/Tm Fig. 2. Temperature dependence of the liquid droplet radius (a) for different initial concentrations of the impurity atoms 3 C0' = C0a (indicated by figures at the curves) and (b) for Fig. 3. Temperature dependence of the number of the liquid different values of the thermodynamic characteristic ∆Ψ' = droplets per unit area of the boundary. (a) Values of C0' ∆Ψ /kTm of the impurity atoms (indicated by figures at the (indicated by figures at the curves) and (b) values of ∆Ψ' curves). (indicated by figures at the curves).

L during the local melting. On the right-hand side of (9), the boundary structure parameters (nL, y, R, rL, C0 , and the ﬁrst term describes the change in the energy during b the melting of the grain body, the second term is the Ci ) to be obtained under the local melting conditions γ γ δ change in the energy when the grain boundary phase for given material parameters ( L/S, b, q, and 0), melts, and the third term is the change in the chemical parameters characterizing the energy of the impurity potential of the grain boundary atoms due to the impu- atoms in the liquid and solid grain-boundary phases rity redistribution during the local melting. ΨL Ψb (i and i ), and initial concentration of the impurity The expression for the change in the chemical atoms. potential of the impurity atoms has the form NUMERICAL ANALYSIS OF BOUNDARY ∆ δ ()∆Ψ ()L Gi = ÐC0 0 + kTln C0/Ci , (10) STRUCTURE PARAMETERS δ where C0 0 is the amount of the impurity per unit area The numerical simulation was carried out for the ∆Ψ Ψb ΨL model alloy with the following typical values of the of the boundary and = i Ð i . 3 δ γ 2 parameters: qa ~ 1.5kTm, 0 ~ 2a, ba ~ 0.3kTm, and γ 2 System (1), (3)Ð(7) is a self-consistent system of L/Sa ~ 0.2kTm. As follows from the analysis, the tem- equations that enables the temperature dependences of perature dependences of the radius of the liquid drop-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 388 PEREVEZENTSEV et al. α ≈ aL mal ﬂuctuations. It should be noted that, at L 0.6, the 0.8 overlap of the liquid zones can be expected and the model becomes inapplicable. In this case, the grain (a) 0.16 0.14 boundary structure should be considered as a liquid layer with solid islands [9]. The variation of the mate- 0.6 γ 2 3 rial parameters ba and qa does not affect the temperature dependence of the boundary parameters but con- 0.18 siderably changes the melting point: the process starts 0.4 earlier in the materials with a lower heat of fusion qa3 γ 2 and a higher surface energy ba of the boundary.

0.2 CONCLUSION 0.12 A model describing the initial stages of the local 0.10 melting of grain boundaries containing impurity clus- 0 ters was developed. 0.8 The system of equations relating the liquidÐsolid interface parameters (geometry, dimensions and num- (b) ber of the liquid zones, and impurity content in the liquid and solid zones at the boundary) to temperature, ini- 0.6 tial concentration, and thermodynamic characteristics of the impurity atoms were derived. The numerical analysis of the model suggested 0.4 shows that the local melting starts (stable liquid droplets appear) at a finite fraction of the liquid phase. The fraction of the liquid phase invariably increases mono- 2.5 2.0 tonically with temperature, whereas the temperature 0.2 3.0 dependences of the size and number of liquid zones vary according to the initial concentration and thermo- 1.0 1.5 dynamic characteristics of the impurity. 0 0.80 0.84 0.88 0.92 0.96 1.00 ACKNOWLEDGMENTS T/Tm This work was supported by the Russian Foundation Fig. 4. Temperature dependence of the surface fraction of for Basic Research (project no. 00-02-16546). the liquid phase: (a) the same as in Fig. 2a and (b) the same as in Fig. 2b. REFERENCES 1. T. G. Nieh, C. A. Henshall, and J. Wadsworth, Scr. Met- lets rL (Fig. 2a) and of the number of the liquid droplets all. 19, 1375 (1985). nL per unit area of the boundary (Fig. 3a) are nonmono- 2. K. Higashi, Mater. Sci. Forum 170Ð172, 131 (1994). ∆Ψ 3 tonic at fixed = 1kTm. For fixed C0a = 0.1, the run 3. L. Koike, M. Mabuchi, and K. Higashi, Acta Metall. Mater. 43, 199 (1995). of the temperature dependences of rL and nL depends on the value of ∆Ψ (Figs. 2b, 3b). At small ∆Ψ (1kT and 4. K. Higashi, M. Mabuchi, and T. G. Langdon, ISIJ Int. 36, m 1423 (1996). 1.5kTm), these dependences are also nonmonotonic. At ∆Ψ 5. K. Higashi, T. G. Nieh, and J. Wadsworth, Acta Metall. high , rL and nL vary monotonically throughout the Mater. 43, 3275 (1995). temperature range considered. At the same time, for all 6. K. Higashi, Mater. Sci. Forum 243Ð245, 267 (1997). 3 ∆Ψ the values of C0a and , the liquid phase fraction 7. M. Mabuchi, H. Iwasaki, and K. Higashi, Acta Mater. 46, (Fig. 4) monotonically increases with temperature. The 5335 (1998). analysis shows that the temperature at which the liquid 8. V. N. Perevezentsev, Mater. Sci. Forum 243Ð245, 297 phase appears considerably depends on the initial (1997). impurity concentration at the boundary and on its 9. V. N. Perevezentsev, K. Higashi, and J. V. Svirina, Mater. energy characteristics. The local melting point T0 is Sci. Forum 304Ð306, 217 (1999). higher the lower the initial impurity concentration. It is 10. V. N. Perevezentsev, T. G. Langdon, and J. V. Svirina, in noteworthy that, in the model, the beginning of local Proceedings of the International Conference “Current melting (the appearance of stable liquid droplets) is Status of Theory and Practice of Superplasticity in Mate- characterized by the finite value of the liquid phase rials,” Ufa, 2000, p. 21. fraction. At temperatures T < Ti0, there are no stable droplets and the liquid phase can exist only due to ther- Translated by M. Lebedev

TECHNICAL PHYSICS Vol. 47 No. 4 2002

GASES AND LIQUIDS

Nonlinear Vibrations of a Charged Drop Due to the Initial Excitation of Neighboring Modes S. O. Shiryaeva Demidov State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia e-mail: [email protected] Received May 31, 2001

Abstract—The asymptotic analysis of the nonlinear vibrations of a charged drop that are induced by a multi- mode initial deformation of its equilibrium shape is performed. It is shown that when two, three, or several neighboring modes are present in the initial deformation spectrum, the mode with the number one (translational mode) appears in the second-order mode spectrum. The excitation of the translational mode follows from the requirement of center-of-mass immobility and causes the dipole components (which are absent in the linear analysis) to appear in the spectra of the acoustic and electromagnetic radiation of the charged drop. © 2002 MAIK “Nauka/Interperiodica”.

(1) The capillary vibration and the stability of a drop. Our aim is to find the spectrum of arising capil- charged drop are of considerable interest in a number of lary oscillations of a drop (its shape) for t > 0. areas of science and technology where charged drops Since the initial perturbation of the surface is axi- play an important part (see, e.g., [1Ð3] and Refs. symmetric and small, we assume that the shape of the therein). Because of much interest in a charged drop as drop remains axisymmetric at any time instant. Then, in a physical object, most related physical problems for- the polar coordinate system with the origin at the center mulated in the framework of linear models have already of mass of the drop, the equation describing the surface been solved. A large number of papers devoted to the of the drop in dimensionless variables where R = ρ = nonlinear analysis has appeared in recent years [4Ð13]. σ = 1 has the form This analysis allows one to gain much more information on the object. In view of the cumbersome calcula- r()Θ, t = 1 + ξΘ(), t , ξ Ӷ 1. (1) tions associated with the nonlinear problems, many The flow inside the drop is assumed to be potential. aspects of the nonlinear vibration of a charged drop ψ either have not been considered or are poorly under- In this case, the flow field in the drop, V(r, t) = — (r, stood. The latter concerns the so-called translational t), is defined only by the velocity potential function ψ(r, t). instability of drops and bubbles, which shows up when two neighboring modes are present in the spectrum of The set of equations for drop evolution consists of initially excited modes [14]. In accordance with [14], the Laplace equations for the field velocity potential the center of mass of a translationally unstable drop ψ(r, t) and electrostatic potential Φ(r, t) undergoing nonlinear oscillations acquires a transla- ∆ψ()r, t = 0, (2) tional velocity. This statement seems to be incorrect, since it is in contradiction with the well-known law of ∆Φ()r, t = 0 (3) theoretical mechanics: no motions inside a closed system can result in the motion of its center of mass. (hereafter, ∆ is the Laplacian operator) and boundary Therefore, the problem of nonlinear capillary vibra- conditions tions of a charged drop when the initial deformation of ψ(), the equilibrium spherical shape of a drop is induced by r 0: r t 0, (4) the excitation of neighboring modes is of interest. r ∞: Φ()r, t 0, (5) (2) Consider the time evolution of the shape of a ∂ξ ∂ψ 1 ∂ξ ∂ψ drop of an ideal incompressible perfectly conducting r ==1 + ξΘ(), t : ------Ð ------, (6) ∂ 2 ∂Θ∂Θ liquid with the density ρ and the surface tension coeffi- t ∂r r cient σ. We assume that the drop, having the charge Q ∂ψ 1 2 1 2 and radius R, is in a vacuum. At the time instant t = 0, ∆p Ð ------Ð ---()—ψ + ------()—Φ = — ⋅ n, (7) the equilibrium spherical shape of the drop undergoes ∂t 2 8π an axisymmetric perturbation of fixed amplitude, the Ψ()Φ,,Θ () perturbation being much smaller than the radius of the r t = s t . (8)

390 SHIRYAEVA We also introduce the condition of center-of-mass In Eqs. (3), (5), (8), and (9), it is assumed that the immobility and the condition of conservation of the electric charge is distributed over the surface and is in total charge and volume of the drop: equilibrium at any instant of time. This assumption is valid if the characteristic time of the charge redistribu- 1 Ð------∫()n ⋅ —Φ ds = Q, tion is much smaller than the characteristic hydrody- 4π° (9) namic time of surface vibration S Sr= []= 1 + ξΘ(), t , 0 ≤≤Θπ, 0 ≤≤φ 2π , ρR3 1/2 ε ε λ Ӷ ------* 0 σ , 2 4 ∫r drsinΘdΘdφ = ---π, 3 (10) where λ is the resistivity of the liquid and ε∗ is the per- v mittivity. v = []0 ≤≤r 1 + ξΘ(), t , 0 ≤≤Θπ, 0 ≤≤φ 2π , For a drop of distilled water (λ = 104 Ω m, ε∗ = 3 Θ Θ φ 80.08, ρ = 103 kg/m3, σ = 72.8 × 10Ð3 N/m, and ε = ∫err drsin d d = 0, 0 (11) 8.85 × 10Ð12 F/m), the left-hand side of this inequality is v two orders of magnitude smaller than the right-hand v = []0 ≤≤r 1 + ξΘ(), t , 0 ≤≤Θπ, 0 ≤≤φ 2π . one at R = 1 mm. Note that conditions (10) and (11) have to be satis- (3) We will take advantage of method of multiple fied at any instant of time including the initial one. scales [15] to solve the problem up to second-order Therefore, at t = 0, these conditions define the ampli- infinitesimals in ε. We expand the unknown functions tudes of the zero and first modes in the expansion of the ξ(Θ, t), ψ(r, t), and Φ(r, t) in the power series in the initial perturbation of the equilibrium spherical shape small parameter ε, and consider the series as functions of the drop ξ(Θ) in Legendre polynomials. In other of various time scales (rather than merely of time t) ε ≡ εm words, the zero and first modes cannot have arbitrary defined via the small parameter : Tm t. Then, amplitudes. Their amplitudes depend on the shape of ∞ the initial deformation (perturbation). () ξΘ(), t = εmξ m ()Θ,,,,T T T … ; The initial conditions are set as the initial deforma- ∑ 0 1 2 tion of the equilibrium spherical shape of the drop and m = 0 as the zero initial velocity of the surface: ∞ ψ(), εmψ()m (),,,,,Θ … ξΘ() ξ ()µ ξ ()µ ε ()µ r t = ∑ r T 0 T 1 T 2 ; (14) t ==0: 0P0 ++1P1 ∑ hiPi , m = 0 i ∈ Ξ (12) ∞ ∂ξ Φ(), εmΦ()m (),,,,,Θ … ∑ h ==1, ------0, r t = ∑ r T 0 T 1 T 2 . i ∂t i ∈ Ξ m = 0 where Ξ is the set of the numbers of initially excited Using standard techniques involved in the method modes and µ ≡ cosΘ. of multiple scales (see, e.g., [13] for details), we find ∆ the expression for the time evolution of the surface In (6)Ð(12), p is the difference in the constant pres- shape: sures inside and outside of the drop in equilibrium; n, Φ unit normal vector to the surface defined by (1); s(t),  ()Θ, ≈ ε ()1 () ()µ potential that is constant over the surface of the drop; r t 1 + ∑ Mi t Pi ε, amplitude of the initial perturbation of the surface  µ i ∈ Ξ shape; Pi( ), ith-order Legendre polynomials; hi, coefficient specifying the partial contribution of the ith ∞ ξ ε2 ()2 () ()µ ()ε3 vibration mode to the total initial perturbation; and 0 + ∑ Mn t Pn + O ; ξ  and 1, constants that are determined from conditions n = 0 (10) and (11) at the initial time instant and are equal to ()1 () ()ω Mi t = hi cos it ; 2 ξ ≈ ε2 hi ()ε3 0 Ð ∑ ------+ O , ()2i + 1 ()2 1 hi i ∈ Ξ M ()t = Ð--- ∑ ------()12+ cos()ω t ; (13) 0 2 ()2i + 1 i i ∈ Ξ ξ ≈ ε2 9ihi Ð 1hi ()ε3 1 Ð ∑ ------()()- + O . 2i Ð 1 1i + 1 () 9ih h i ∈ Ξ M 2 ()t = Ð∑ ------i Ð 1 i -cos()ω t cos() ωt ; 1 ()2i Ð 1 ()2i + 1 i i Ð 1 up to second-order infinitesimals in ε. i ∈ Ξ

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() ≥ M 2 = []N ()t Ð N ()0 cos() ωt , n ≥ 2; center of mass is at rest at the origin for t 0 because n n n n its displacement induced by the excitation of two neighboring modes is compensated. Note that the trans- 1 ()+ N ()t = --- ∑ ∑ h h {λ cos()()ω + ω t lational instability of a bubble due to the excitation of n 2 i j ijn i j ∈ Ξ ∈ Ξ two neighboring surface vibrational modes discussed in i j [14] is associated with incorrect initial conditions. λ()Ð ()}()ω ω + ijn cos i Ð j t ; Namely, the statement of the problem in [14] does not require that the center-of-mass system initially be at 2 2 Q rest. In other words, in [14], the velocity of the center ω ≡ nn()Ð 1 []()n + 2 Ð W ; W ≡ ------; n 4π of mass of the drop is other than zero with respect to the laboratory system of coordinates at t = 0. λ()± ≡ []ωγ ± ω ω η []2 ()ω ± ω 2 Ð1 ijn ijn i j ijn n Ð i j ; The time dependences of the amplitudes of the modes excited by the intermode interaction in the sec- 2 ε γ ≡ K ω ()niÐ1+ + 2njj[]()+ 1 Ð 1 --- ond-order approximation with respect to are shown in ijn ijn i Figs. 1 and 2 for various initial deformations of the spherical shape. In Fig. 1, the initial deformation is W + []ji()+ 1 Ð i()2i Ð72n + + 3 n----- defined by the superposition of two modes of equal 2 amplitudes; in Fig. 2, by that of three modes of the same amplitude. According to these data, the amplitude of 1 2 W + α ---ω +;n----- the fundamental mode has the highest rate of rise pre- ijn i i 2 sumably because its excitation energy is the least. According to Figs. 1aÐ1c, when the initial deformation n 1n of the equilibrium shape is due to two various modes, η ≡ K --- Ð1i + + α --- 1 + ----- ; ijn ijn2 ijn i 2 j the rate of rise of the fundamental mode amplitude increases with increasing number of initially excited 2 K ≡ []C000 ; modes. For three initially excited modes, the tendency ijn ijn is the same. It seems likely that, for ε = const, the α ≡ Ð ii()+ 1 jj()+ 1 C000CÐ110; energy injected into the vibrating system rapidly ijn ijn ijn increases as the numbers of modes responsible for the 0, if ijn++= 2g + 1, initial deformation grow.  where g is an integer; The number of modes excited in the second-order  ()gnÐ approximation depends on the fact that the coefficients 000 ≡ Ð1 2n + 1g! ()± ()± Cijn ------λ λ ()giÐ !()gjÐ !()gnÐ ! iin and jjn in (15) are nonzero only for even n in the closed intervals [0, 2i] and [0, 2j], respectively, whereas  1/2 ()2g Ð 2i !2()g Ð 2 j !2()g Ð 2n ! ()± ()± × ------, λ and λ are other than zero for n from the closed  ()2g + 1 ! ijn jin interval [|i Ð j|, (i + j)] such that n + i + j is even. There- if i + j + n = 2g (g is an integer); fore, when odd and even modes (with numbers k and p) are excited simultaneously, their interaction will gener- Ð110 ≡ Cijn 2n + 1n! ate all even modes from the closed interval [0; max(2k, 2p)] and all odd modes from the interval [|k Ð p|, (k + ()ijn+ Ð !ii()+ 1 1/2 × ------p)]. In the case of the initial excitation of either two ()nij+ Ð !()niÐ + j !()ijn+++1 ! jj()+ 1 (15) even or two odd modes, only even modes from the closed interval [0; max(2k, 2p)] are excited in the sec- ()Ð1 i ++1 z()iz+ Ð 1 !()njz++Ð1! ond-order approximation. × ∑------. z!()izÐ1+ !()nzÐ !()jnÐ + z Ð 1 ! z (4) Let us consider in greater detail whether the cen- The summation in the last expression is over all inte- ter of charge of a drop is displaced under nonlinear gers z yielding nonnegative values of the expressions vibration of its surface. The radius vector of the center 000 Ð110 of charge Rq is given by under the factorial sign. Note that Cijn and Cijn are the ClebschÐGordan coefficients [16], which are nonzero only if the subscripts satisfy the conditions |i Ð j| ≤ 1 1 ()⋅ Φ Rq ==----∫rdQ Ð------∫ n — rdS n ≤ (i + j) and i + j + n = 2g (g is an integer). Q 4πQ S S From (15) it follows that, if two neighboring vibration modes (even and odd) are present in the mode ()⋅ Φ 1 n — 3()Θ, Ω spectrum describing the initial surface perturbation, the = Ð ------π -∫ ------()r t erd , 4 Q ner amplitude of the first mode is nonzero. In this case, the Ω

TECHNICAL PHYSICS Vol. 47 No. 4 2002 392 SHIRYAEVA

(2) where Ω ≡ [0 ≤ Θ ≤ π; 0 ≤ φ ≤ 2π], r(Θ, t) is given by Mn (a) (15); and the parentheses stand for the scalar product of 2 3 the vectors.

The radial unit vector er of the spherical coordinate 0.02 system is related to the unit vectors of the Cartesian system as 3 Θφ Θφ Θ 4 4 er = ex sin cos + ey sin sin + ez cos . 5 5 1 6 6 Let us express the components of the vector Rq in 0 the Cartesian coordinates through the spherical func- 1 2 3 t 1 tions Ym (Θ, φ) in view of the relations 0 n

4π 0 cosΘ = ------Y ()Θφ, ; 3 1 –0.02 2π 1 Ð1 sinΘφcos = Ð ------[]Y ()Θφ, + Y ()Θφ, , 3 1 1 (b) π Θφ 2 []1()Θφ, Ð1()Θφ, 0.02 2 sin sin = i ------Y1 Ð Y1 . 5 3 Then, the components of the vector Rq will have the 4 6 form 4 3 7 ()⋅ Φ 1 []()Θ, 3 n — 1 8 Rqx = ------∫ r t ------() 0 π ner 26QΩ 1 8 6 1 7 2 t 0 × []Y1()Θϕ, + YÐ1()Θϕ, dΩ, 5 1 1 ()⋅ Φ i []()Θ, 3 n — Rqy = Ð------∫ r t ------() (16) 3 26πQ ner –0.02 Ω × []1()Θϕ, Ð1()Θϕ, Ω Y1 Ð Y1 d , 0.04 (c) ()⋅ Φ 1 []()Θ, 3 n — 0()ΩΘϕ, Rqz = ------∫ r t ------()Y1 d . π ner 23QΩ In view of (1) and expansions (14), the integrand in 0.02 2 4 (16) can be represented by the power series in the small 8 8 parameter ε up to terms of the order of O(ε3): 6 6 4 8 10 ()1 3()n ⋅ —Φ ∂Φ ()1 6 []r()Θ, t ------≈ Ð Q + ε ------Ð Qξ 12 12 ()ne ∂r 0 r r = 1 1 2 t 0 ∂Φ()2 ∂2Φ()1 ∂Φ()1 ε2 ξ()2 ξ()1 8 10 + ------Ð Q + ------3+ ------∂r ∂r2 ∂r ∂Φ()1 ∂ξ()1 –0.02 Ð ------+ O()ε3 . ∂Θ ∂Θ r = 1 Fig. 1. Time dependences of the dimensionless amplitudes ()2 Taking into account the form of the functions Mn of various modes of the capillary vibrations of a ξ(1)(Θ, t) and ξ(2)(Θ, t) in (15), as well as the potential charged drop that appear in the second-order approximation Φ(r, t) found by solving the boundary problem given due to interaction between modes at W = 3.9 and ε = 1, and the initial deformation in the linear approximation with by (3), (5), (8), and (9), namely, ε ε µ µ ε µ µ respect to : (a) [P2( ) + P3( )]/2, (b) [P3( ) + P4( )]/2, ()1 ()1 Ð()i + 1 ε µ µ Φ ()r,,Θ t = QM()t r P ()µ , and (c) [P4( ) + P6( )]/2. The numbers at the curves ∑ i i denote the mode number. i = Ξ

TECHNICAL PHYSICS Vol. 47 No. 4 2002 NONLINEAR VIBRATIONS OF A CHARGED DROP 393

∞ M(2) Φ()2 (),,Θ ()2 () n (a) 3 r t = QM∑ n t n = 1 0.02 2 4 ()1 () ()1 ()Ð()n + 1 ()µ 4 + ∑ ∑ iKijnMi t M j t r Pn , ∈ Ξ ∈ Ξ i j 4 5 6 we represent the integrand in (16) as an expansion in 6 3 6 Legendre polynomials: 8 1 7 8 0 3()n ⋅ —Φ 1 7 2 3 t []()Θ, ≈ ()µ 0 1 7 r t ------() ÐQP0 ner 5

ε () ()ω ()µ 5 Ð Qi∑ + 2 hi cos it Pi i ∈ Ξ ∞ –0.02 ε2 {()δ ()2 () Ð Qn∑ + 2 Ð n, 0 Mn t (17) (b) n = 0

Ð ∑ ∑ [()()i Ð 1 ()i + 1 Ð in()+ 1 Ð δ , K 4 n 0 ijn 0.02 i ∈ Ξ j ∈ Ξ 2 () 5 α ] 1 () ()1 ()} ()µ ()ε3 4 Ð ijn Mi t M j t Pn + O . 6 3 8 8 In the derivation of (17), the following expansions 5 10 1 coming from the ClebschÐGordan expansion [16] have 10 0 been used: 9 0.5 1.0 1.5 9 t 0 1 ∞ 10 8 9 8 7 6 P ()µ P ()µ = ∑ K P ()µ , 7 i j ijn n 7 n = 0 –0.02 6 ∂ ∂ ∞ Pi P j α ()µ ------∂Θ------∂Θ = ∑ ijnPn , 3 n = 0 ε µ µ α Fig. 2. The same as in Fig. 1 for (a) [P2( ) + P3( ) + where the coefficients Kijn and ijn are defined in (15). µ ε µ µ P4( )]/3 and (b) [P3( ) + P4( ) + P5]/3. Substituting (17) into (16), we turn from the Leg- endre polynomials to the spherical functions [16] Its displacement along the Oz axis is described by π ()µ 4 0()Θφ, P j = ------Y j ()2 j + 1  ≈ ε2 ()2 () 1 [α (()() Rqz M1 t + --- ∑ ∑ ij1 Ð i Ð 1 i + 1 taking into account the orthonormality condition for the  3 i ∈ Ξ j ∈ Ξ spherical functions (19)  ) ] ()1 () ()1 () ()ε3 m()Θφ, []k ()Θφ, Ω δ δ ∑ Ð2i Kij1 Mi t M j t  + O . ∫Y j Yn *d = jn, mk, . (18)  Ω Then, in the case of axisymmetric (with respect to Recall that, according to definitions introduced in α the z axis) vibrations of the surface, when the perturba- (15), the coefficients ijn and Kijn are expressed via the tion of its equilibrium spherical shape ξ = xi(Θ, t) is ClebschÐGordan coefficients and, hence, are nonzero independent of the angle φ the integrand in (17) is rep- only if i + j + n = 2g (g is an integer) and |i Ð j| ≤ n ≤ 0 (i + j). In our case, n = 1, so that resented by the series in spherical functions Y j with the zero superscript and, in view of (18), the center of charge in the xy plane is not displaced: ij+2= g Ð 1, (20)

Rqx ==Rqy 0. ijÐ1≤≤ij+ . (21)

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Rqz approximation with respect to the surface perturbation:

0.002 2 2 6ii()Ð 2 R ≈ ε R ∑ ------h h cos()ω t qz ()2i Ð 1 ()2i + 1 i i Ð 1 i 2 i ∈ Ξ 3 () 0.001 ×ω()≡ ε2 3iiÐ 2 1 cos i Ð 1t R ∑ ------hihi Ð 1 3 ()2i Ð 1 ()2i + 1 (24) i ∈ Ξ [][]()ω ω []()ω ω 0 × cos i + i Ð 1 t + cos i Ð i Ð 1 t , 2 4 t σ 2 ω2 ≡ ()()Q 1 i ------iiÐ 1 i + 2 Ð.------–0.001 1 ρR3 4πσR3

3 The plot Rqz = Rqz(t) for several numbers of initially excited neighboring modes is shown in Fig. 3. –0.002 2 (5) The presence of center-of-charge oscillations causes the drop to emit dipole electromagnetic radia- Fig. 3. Dimensionless displacement of the center of charge ε tion. Consider the case when two neighboring modes i of a drop Rqz vs. dimensionless time t for W = 3.9 and = 0.1. In the linear approximation with respect to ε, the initial and i Ð 1 are initially excited. The intensity Ie of the perturbation of the equilibrium spherical shape has the form electromagnetic radiation from a single drop has the ε µ µ ε µ µ (1) [P2( ) + P3( )]/2, (2) [P3( ) + P4( )]/2, and form [17] ε µ µ (3) [P3( ) + P4( )]/3. 4 { 2()ω ω 4 Ie = ------dω + ω i + i + 1 3c3 i i + 1 The right-hand inequality in (21) is always satisfied, 2 4 Ξ + dω ω ()ω Ð ω }, since, in the spectrum of the vibrational modes , the i + 1 Ð i i + 1 i zero and first modes are absent. The left-hand inequal- ≤ where c is the velocity of light in a vacuum, dω is the ity in (21) implies that j = i or i Ð 1 for j i and j = i + j 1 for j > i. Consequently, the index j should take the val- dipole moment of the drop with its center of charge ω ues {i Ð 1; i; i + 1} at a fixed i. However, the value j = i oscillating with a frequency j. contradicts requirement (20). Thus, the excitation spec- According to (24), in our situation, we have trum for a given i should also contain the (i + 1)th or ≡≡ (i Ð 1)th mode (i.e., a neighbor of the ith mode) in order dω + ω dω Ð ω QRqz, λ i i + 1 i + 1 i for the coefficients ij1 and Kij1 to be nonzero. In this case, the center of charge will be displaced along the z where Rqz is the peak value of Rqz. axis. Hence,

Using the explicit form of the ClebschÐGordan 4 2 2 4 4 δ I = ------Q ()R {}()ω + ω + ()ω Ð ω . (25) coefficients [16] and the Kronecker symbol m, k, one e 3 qz i i + 1 i + 1 i readily finds 3c In view of (24), expression (25) evaluates the order 3i 3()i + 1 of magnitude of the intensity of the background (noise) K = δ , ------+ δ , ------, ij1 jiÐ 1()2i Ð 1 ()2i + 1 ji+ 1()2i + 1 ()2i + 3 (22) electromagnetic radiation from various (natural and α δ ()() δ () artificial) liquid-drop systems, such as clouds of con- ij1 = ji, Ð 1 i Ð 1 i + 1 Kij1 + ji, + 1ii+ 2 Kij1. vection. In [18], the intensity of the electromagnetic radiation from a cumulus cloud was estimated in the Substituting (22) into (19) and reindexing, we framework of linear analysis. The physical model pro- reduce (19) to the more compact form posed in [18] considers the total multipole (starting from quadrupole) radiation from large, heavily charged hydrometeors (drops of R = 1 mm) freely falling in the ≈ ε2 ()2 () 3i ()1 () ()1 () Rqz M1 t + ∑ ------Mi t Mi Ð 1 t . (23) cloud and coagulating with smaller charged drops of ()2i + 1 µ i ∈ Ξ radius 10 m. The total intensity of the multipole electromagnetic radiation (except for the dipole radiation, Finally, taking into account the expressions for the which is absent in the linear analysis) of a cloud with a Ð2 ()1 ()2 diameter of 5 km, I = 3 × 10 W, found in [18] is essen- coefficients Mi and Mi in (15) and turning to tially overestimated. This is because the maximum dimensional variables, we find the displacement of the admissible charges of hydrometeors rather than their center of charge along the z axis in the second-order mean values (which are several orders of magnitude

TECHNICAL PHYSICS Vol. 47 No. 4 2002 NONLINEAR VIBRATIONS OF A CHARGED DROP 395

(2) smaller) were used in calculations and the concentra- M1 tion of hydrometeors of the given size was overstated. 0.002 Let us estimate the intensity of the background 1 dipole electromagnetic radiation with (24) and (25) 1 when the displacement of the center of charge is caused 0.001 by the excitation of only two neighboring modes with 3 i = 100 and i + 1 = 101. Note that, in this case, the sum 2 in (24), which determines the amplitude of the center- 3 of-charge displacement R , is reduced to the single 0 qz 3 2 4 t term with i = 101. The estimation is performed for a drop with the mean radius R = 30 µm and mean charge 1 Q = 2.5 × 105 C.G.S.E. The concentration of such drops –0.001 in a cumulus cloud is n ≈ 103 cmÐ3 according to refer- 2 ε2 2 ence data [19]. We also put = 0.1, h100 = h101 = 0.5, σ = 73 dyn/cm, and ρ = 1 g/cm3. Then, for a cloud with –0.002 a diameter of 5 km (as in [18]), the intensity of the electromagnetic background induced by the motion of the ()2 Fig. 4. Dimensionless amplitude M of the ﬁrst mode center of charge (with the excitation of the translational 1 (n = 1) of the drop surface oscillations vs. dimensionless mode) of the drops when their surface vibrates because time t. Curves 1Ð3 show the data corresponding to the of physical microprocesses inside the cloud (coagula- approximations in Fig. 3. tion with ﬁner drops, evaporation, condensation, as well as hydrodynamic and electrical interaction with I of the dipole acoustic radiation from the drop has the neighboring drops) is equal to I ≈ 10Ð6 W. The radiation s form [20] frequency in this case is of the order of megahertz. 2 2 3πρ v R 2 2 If we had assumed that neighboring modes in the I ≡ ------* -{()ω + ω Uω ω s 2 i i + 1 i + i + 1 range from k to k + m are excited (instead of two 2V (26) modes), the integral intensity of the radiation would 2 2 + ()ω Ð ω Uω ω }, have increased in proportion to m. i + 1 i i + 1 Ð i

(6) From expression (15), describing the shape of a where Uj is the peak velocity of the surface motion at vibrating drop at any instant of time, it is seen that, if the jth vibration frequency. two modes with sequential numbers are present in the Assuming that the translational mode (i = 1) appears mode spectrum specifying the shape of the initial per- because of the presence of two neighboring modes turbation, the mode with n = 1 corresponding to the (with numbers i and i + 1) in the spectrum of modes translational motion appears in the second-order vibra- responsible for the initial deformation, we ﬁnd, accord- tional spectrum. The appearance of this mode is the ing to (15), result of the requirement for the center of mass of a 2 9ihi Ð 1hi Uω ω ≈ ε R------h h (),ω + ω (27) vibrating drop to be at rest [see (11)]. The amplitude of i + i + 1 ()2i Ð 1 ()2i + 1 i i Ð 1 i i + 1 the mode with n = 1 for several numbers of initially excited neighboring modes is plotted in Fig. 4. 2 9ihi Ð 1hi Uω ω ≈ ε R------h h ().ω Ð ω (28) i Ð i + 1 ()()i i Ð 1 i i + 1 In the case of a compressible environment (instead 2i Ð 1 2i + 1 of a vacuum) with the density ρ∗, kinematic viscosity Substituting (27) and (28) into (26), we ﬁnally come v, and velocity of sound V, the mode with n = 1 due to to nonlinear vibrations of the drop results in the genera- 2 2 2 ω 3πρ v R 2 9ih h tion of dipole acoustic waves with the frequencies ( i + I ≡ ------* -ε R------i Ð 1 i -h h ω ω ω s 2 ()()i i Ð 1 i + 1) and ( i Ð i + 1). Note that the linear analysis of 2V 2i Ð 1 2i + 1 (29) the acoustic radiation from vibrating drops gives multi- ×ω{}()ω 4 ()ω ω 4 pole radiation (starting with quarupole one related to i + i + 1 + i + 1 Ð i . the fundamental mode n = 2). The dipole acoustic radi- Let us estimate the intensity of the dipole acoustic ation associated with the drop is absent in the linear radiation related to the translational mode in the spec- analysis. If the drop radius R is of the same order of trum of capillary surface vibrations for a rain drop with ω 1/2 magnitude as (v/ i) , the expression for the intensity R = 250 µm when the second and third modes are

TECHNICAL PHYSICS Vol. 47 No. 4 2002 396 SHIRYAEVA excited. We put σ = 73 dyn/cm, ρ = 1 g/cm3, ρ∗ = 1.3 × 4. J. A. Tsamopoulos and R. A. Brown, J. Fluid Mech. 147, 373 (1984). 10Ð3 g/cm3, V = 3.3 × 104 cm/s, W = 1, ε = 0.1, v = 2 5. J. A. Tsamopoulos, T. R. Akylas, and R. A. Brown, Proc. 0.15 cm /s, and h2 = h3 = 0.5. From (29), it is easy to R. Soc. London, Ser. A 401, 67 (1985). estimate the intensity of the dipole acoustic radiation of 6. O. A. Basaran and L. E. Scriven, Phys. Fluids A 1, 795 ≈ Ð13 a single drop: Is 10 erg/s. The integral power den- (1989). sity of the acoustic radiation at frequencies of tens of 7. R. Natarayan and R. A. Brown, Proc. R. Soc. London, kilohertz is ≈28 dB/km3 at the boundary of the rain- Ser. A 410, 209 (1987). filled space. Such a density corresponds to the sound 8. N. A. Pelekasis, J. A. Tsamopoulos, and G. D. Manolis, intensity of human speech. Phys. Fluids A 2, 1328 (1990). 9. Z. Feng, J. Fluid Mech. 333, 1 (1997). CONCLUSION 10. S. O. Shiryaeva, A. I. Grigor’ev, and D. F. Belonozhko, Pis’ma Zh. Tekh. Fiz. 26 (19), 16 (2000) [Tech. Phys. When the spectrum of modes responsible for the ini- Lett. 26, 857 (2000)]. tial perturbation of the equilibrium spherical shape of 11. S. O. Shiryaeva, Pis’ma Zh. Tekh. Fiz. 26 (22), 76 (2000) the drop has two or more neighboring vibrations, the [Tech. Phys. Lett. 26, 1016 (2000)]. translational mode appears, as follows from asymptoti- 12. D. F. Belonozhko and A. I. Grigor’ev, Zh. Tekh. Fiz. 70 cal calculations up to second-order infinitesimals. The (8), 45 (2000) [Tech. Phys. 45, 1001 (2000)]. translational mode is responsible for dipole acoustic 13. S. O. Shiryaeva, Zh. Tekh. Fiz. 71 (2), 27 (2001) [Tech. radiation from the drops. In the case of a charged drop, Phys. 46, 158 (2001)]. this causes dipole electromagnetic emission. These 14. Z. C. Feng and L. G. Leal, Phys. Fluids 7, 1325 (1995). effects may be of significant importance in analyzing physical phenomena taking place in multiphase liquid- 15. A.-H. Nayfeh, Perturbation Methods (Wiley, New York, drop systems of both artificial and natural origin: 1973; Mir, Moscow, 1976). clouds, fog, or rain. 16. D. A. Varshalovich, A. N. Moskalev, and V. K. Kherson- skii, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975; World Scientific, Singapore, 1988). ACKNOWLEDGMENTS 17. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2: The Classical Theory of Fields (Nauka, This work was supported by the Foundation of the Moscow, 1967; Pergamon, Oxford, 1971). President of the Russian Federation (project no. 00-15- 9925). 18. V. I. Kalechits, I. E. Nakhutin, and P. P. Poluéktov, Dokl. Akad. Nauk SSSR 262, 1344 (1982). 19. Clouds and Cloud Atmosphere: Handbook, Ed. by REFERENCES I. P. Mazin, A. Kh. Khrgian, and I. M. Imyanitov (Gidrometeoizdat, Leningrad, 1989). 1. A. I. Grigor’ev and S. O. Shiryaeva, Izv. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 3 (1994). 20. L. D. Landau and E. M. Lifshitz, Course of Theoretical 2. A. I. Grigor’ev, Zh. Tekh. Fiz. 70 (5), 22 (2000) [Tech. Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, Phys. 45, 543 (2000)]. 1986; Pergamon, New York, 1987). 3. D. F. Belonozhko and A. I. Grigor’ev, Elektrokhim. Obrab. Met., No. 4, 17 (2000). Translated by M.S. Fofanov

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Technical Physics, Vol. 47, No. 4, 2002, pp. 397Ð405. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 23Ð31. Original Russian Text Copyright © 2002 by Lapushkina, Bobashev, Vasil’eva, Erofeev, Ponyaev, Sakharov, Van Wie.

GAS DISCHARGES, PLASMA

Influence of Electric and Magnetic Fields on the Shock Wave Configuration at the Diffuser Inlet T. A. Lapushkina*, S. V. Bobashev*, R. V.Vasil’eva*, A. V. Erofeev*, S. A. Ponyaev*, V. A. Sakharov*, and D. Van Wie** * Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] ** Johns Hopkins University, Laurel, Maryland, 20723, USA Received September 20, 2001

Abstract—The possibility is investigated of influencing the shock wave configuration in a xenon plasma flow at the inlet of a supersonic diffuser by applying electric and magnetic fields. Flow patterns resulting from interaction of the plasma with external fields in the entire diffusor volume and its different sections are compared. The patterns are obtained by the Schlieren method using two recording regimes: individual frames or a succession of frames. The study focuses on the normal shock wave formation process under strong MHD interaction over the whole diffuser volume. Basic factors affecting the plasma flow velocity in the diffuser under externally applied fields are compared, namely, the ponderomotive force and the Joule heating of the gas by the electric field, which decelerate the supersonic flow, and the heat removal to the external electric circuit producing the opposite effect. It has been shown that the external fields are most effective if applied to the inlet part of the diffuser, while the flow in the diffuser section, where there is a large density of dissipative structures, is not readily responsive to external factors. It is suggested that the measure of response can be estimated by the energy that goes to the shock wave formation as a result of the flow interaction with the diffuser walls. © 2002 MAIK “Nauka/Interperiodica”.

FORMULATION OF THE PROBLEM applied external fields were chosen in such a way as to The present work is a continuation of a series of have the flow of a strong interaction type in the diffuser investigations [1, 2] initiated by the AJAX project [3, 4] as a whole and of a weak interaction type in its different and aimed at studying the effect of external factors on parts. The strong interaction, according to the authors' the shock wave patterns arising in a diffuser with com- classification [2], is that producing a normal shock plete internal flow compression. The main part of the wave in the diffuser due to MHD deceleration of the diffuser is a linearly converging duct formed by two flow. The formation mechanism of this shock wave is a wedge-shaped walls. The shock wave configuration separate problem also considered in this paper. consists of two attached shock fronts interacting with each other in the diffuser. An external electric field is EXPERIMENTAL applied in such a way that it enhances the current induced by the magnetic field. The flow is formed both A schematic of the experimental setup was as a result of the MHD interaction proper and due to described in [1, 2]. The setup consists of a vacuum test heating of the working gas by the external electric field. section integrated with a shock tube. The test section Both these factors produce the same effect, that is, they includes a supersonic nozzle accelerating the plasma decelerate the supersonic flow and accelerate the sub- flow to M0 = 4.3 and a diffuser with a set of electrodes sonic one. The experiment was carried out under condi- connected as in a segmented Faraday generator. tions in which the effects of these two factors were A pulsed magnetic field with a magnetic induction of comparable in strength. up to 1.5 T is produced by Helmholtz coils 30 cm in The main objective of this work was to trace the diameter. A slug of the shock-heated gas becomes stag- changes in the shock wave configuration caused by the nant at the end of the tube facing the test section, heats external fields and identify the processes and diffuser up to a stagnation temperature, and flows through the sections mainly responsible for changes in the flow pat- inlet slit into the nozzle and then the diffuser. As a tern by comparing flow patterns obtained with the working medium, a xenon plasma is utilized. The initial external factors applied over the whole diffuser volume xenon pressure in the low-pressure part of the shock or its particular sections and when only the electric field tube is 20 torr and the Mach number of the incident or both the electric and magnetic fields are applied. To shock wave is M = 8.3. Design parameters at the dif- ρ 3 solve this problem, the conditions in the diffuser under fuser inlet are as follows: gas density 0 = 0.0985 kg/m ;

398 LAPUSHKINA et al.

(‡) regime without external fields, as well as (b) a scheme of discontinuities formed. The picture clearly shows input shock fronts a and their interaction point located ± at a distance of Xc = (44 1.5) mm from the diffuser inlet. This characteristic length will be used in what follows for evaluating the influence of the external fields on the flow structure. The results of the reflection of the attached shocks fronts from one another are the shock fronts (b). It should be noted that, in this case, the near- wall layer (d) registered poorly and its position in the explanatory scheme is indicated rather arbitrarily. Additional discontinuities (c) seen in the picture arise (b) from the roughness of the diffuser walls. d In this setup, because of a large voltage drop across the thick near-wall layers, potential differences had to c be applied between the electrodes using an external source of electric power in order to achieve current den- a b sities required for MHD interaction. Application of the c external voltage was implemented by discharging long lines in a circuit consisting of a plasma gap between Ω d pairs of electrodes and a load resistance of 0.1 . Across pairs of electrodes connected to the long lines, the same voltages of 110 V were applied. The Fig. 1. A Schlieren pattern of the flow in the diffuser and a Ohm law for a circuit including a plasma gap between schematic diagram of gasdynamic nonuniformities without external fields. two electrodes, a load resistor, an MHD generator with the electromotive force ε = uBh, and a source of external voltage V has the form concentration of heavy particles n = 4.5 × 1023 mÐ3; () 0 V+ uBh = IReff + RL , (1) × 3 flow velocity u0 = 1.6 10 m/s; plasma conductivity σ µ where V is the voltage applied to the electrodes; u is the 0 = 800 S/m; plasma efflux duration tA = 350 s; Rey- ρ µ 6 flow velocity; B is the magnetic induction; h is the dis- nolds number Re = ( 0u0l)/ 0 = 10 , where l is the dif- µ tance between the electrodes in a pair; I is the current; fuser length and 0 is the coefficient of gas viscosity; Reff is the effective internal plasma resistance including and the initial magnitude of the Stuart parameter is a resistance of the flow core and a resistance of the near- St = (σB2)/(ρ u ) = 5 mÐ1 at B = 1 T. 0 0 0 wall layer; and RL is the load resistance. For visualization of the shock wave configuration, We studied the flow interaction with external fields two modifications of the Schlieren method are used. in three (overlapping) regions (see Fig. 2) by passing The first modification, described in [5], produces a sin- the current between appropriate electrodes. As seen in gle high-resolution flow pattern at a specified moment Fig. 2, with connection I, the voltage is applied to pairs in time from the beginning of efflux. As an illuminant, of electrodes in the diffuser from the third to the sev- an OGM-20 ruby laser is used with an exposure of enth. In this case, the interaction occurs in the whole 30 ns. In the second method, the development of the diffuser volume. The interaction zone length is roughly shock wave configuration throughout the efflux process 90 mm. With connection II, the voltage is applied only is recorded as a succession of frames. To this end, the to electrode pairs from the fourth to the seventh, so that illuminating and recording units of the Schlieren opti- the diffuser inlet is excluded from the interaction zone; cal system were modified. In this regime a Podmoshen- the interaction zone length in this case is about 70 mm. skiœ light source capable of maintaining a steady illumi- With connection III, the voltage is applied to the third nation of high intensity for 500 µs was used as an illu- pair of electrodes, thus confining the interaction to the minant. For recording we used a VSK-5 high-speed diffuser inlet; the interaction zone length is approxi- cinecamera, allowing one to take up to 130 frames at mately 20 mm. µ intervals from 4 to 25 s. For integrating the cinecam- Figure 3 demonstrates current density distributions era with the Schlierenoptics, the viewing-focusing sys- along the duct for the three schemes of applying the tem of the camera was modified to provide illumination external voltage in the case of MHD interaction and in of the entire frame by the probing light beam and accu- zero external magnetic field. The current density was rate focusing of the subject under study on the record- determined as the ratio of the current measured to the ing film. area of the electrode and the interelectrode gap. The Figure 1 shows a Schlieren pattern (a) of a flow in zero of the abscissa corresponds to the front of the dif- the diffuser obtained in the single-frame recording fuser. In the case of connections types I and II, the aver-

TECHNICAL PHYSICS Vol. 47 No. 4 2002

INFLUENCE OF ELECTRIC AND MAGNETIC FIELDS 399 age current densities are plotted at the midpoints of the I electrodes; for connection III, the current density is plotted over the whole width of the electrode. As seen III II in the plots, the electromotive force induced by the magnetic ﬁeld, contrary to expectations, did not increase the current density. To all appearances, the reasons are the Hall effect arising because of imperfect u sectioning of the electrodes and the increase of the near-wall layer width as a result of MHD interaction. Using IÐV characteristics, we evaluated the magnitudes ≈ Ω 1 5 6 7 of Reff. At B = 0 Reff 0.15 and at B = 1.3 T Reff = 0.2Ð 234 Ω 0.4 . Accordingly, the load factor k = RL/(RL + Reff) varies in the range from 0.2Ð0.4. 0 20406080X, mm Field of vision

COMPARATIVE ESTIMATIONS Fig. 2. Schematic of the channel with marked interaction OF THE INFLUENCE OF ELECTRIC zones for three types of connection to the external voltage. AND MAGNETIC FIELDS ON THE FLOW PARAMETERS j, 105A/m2 Elementary impacts on the flow are those of force and energy. The principle force on the flow is exerted by (a) 8 a ponderomotive force F = jB. The power developed by 6 the ponderomotive force is N = juB. 1 4 The energy impact has two components. The first component is the removal to the external electric circuit 2 0 of an electric power N2 = kjuB generated through MHD interaction. The second component is the heat supplied 10 from an external source of a power N3 = (1 Ð k)j V/h. 8 The work done by the ponderomotive force and the heat (b) supplied decelerate the supersonic flow, while the heat 6 removal accelerates it. To evaluate and compare the 4 actions of these factors on the flow parameters, we use 2 a set of equations [6] describing an MHD interaction. 0 An expression for variation of the flow Mach number M along the duct has the form 5 4 2 2 dMln 2 (c) ()M Ð 1 ------= Ð[]2 + ()γ Ð 1 M 3 dx 2 d ln()ρu γ jBu 1 × ------Ð ------(2) dx 2 ρu a –2 0 2 4 6 8 X, 10–2 m ()γ Ð 1 ()1 + γ M2 juB jV Ð ------k------+.()1 Ð k ------2 ρu hρu Fig. 3. Current density distribution along the duct. Connec- a tions: (a) I; (b) II; (c) III. B = 0 (triangles) and 1.3 T (circles). This equation can be expressed in the form of a reversal condition for the effect of external factors on Using Eq. (3), one can compare the effects of exter- subsonic and supersonic flows [6], nal factors on the flow. For example, it is possible to

2 determine at what values of the ratio between the exter- 2 dM ()M Ð 1 ------= ∑r N , (3) nal electric field and an electric field induced by the dx k k magnetic field a particular factor—gas heating or pon- k deromotive force—is predominant in decelerating the where Nk are the types of external factors and rk are flow. The condition of equality of the actions of these coefficients corresponding to them; here we consider two factors has the form the factors of the aforementioned types N , N , and N 2 1 2 3 uBh ()γ Ð 1 ()11 + γ M ()Ð k and add to them the geometrical factors N = 4 N1r1 = N3r3 or ------= ------2 -.(4) (dln(ρu))/dx. V γ []2 + ()γ Ð 1 M

TECHNICAL PHYSICS Vol. 47 No. 4 2002 400 LAPUSHKINA et al.

M the calculation data, leads to a drastic decrease in the Mach number over the interaction zone. This may 4.5 cause, as shown in [7], the formation of a shock wave of MHD deceleration, which transforms a supersonic 4.0 flow into a subsonic one. However, the variation of the 3.5 gas parameters along the diffuser axis is hardly an adequate flow characteristic for the diffuser as a whole. An 3.0 answer to the question of how the flow pattern changes under the influence of external fields should be 2.5 obtained from experiment.

2.0 INTERACTION WITH EXTERNAL FIELDS 1.5 IN THE WHOLE DIFFUSER VOLUME: CONNECTION I 1.0 0 0.05 0.10 x, m Figure 5 demonstrates flow patterns in the diffuser Inlet Diffusor obtained by the Schlieren method in the single-frame recording regime, as well as schematic diagrams of the shock wave structure. Figure 5.1 shows a flow structure Fig. 4. Variation of the flow Mach number along the axis of formed under the action of only the electric field. the gasdynamic path under the action of magnetic and elec- Clearly seen here are the attached shocks a, shocks b tric fields. B = 0, V = 0 (solid line); B = 1.3 T, V = 0 (dashed line); B = 0, V = 110 V (dotted line); B = 1.3 T, V = 110 V resulting from interaction between the attached shocks, (dotted-dashed line). and shocks f representing reflections of shocks b from the walls or the near-wall layers d. The near-wall layers can be clearly distinguished in the Schlieren patterns. It Consider a case of k = 0. For flows with large Mach is seen that the layer thickness increases downstream numbers (M > 5) (uBh)/V ≈ 1, this means that, at and is considerably larger than it would be without (uBh)/V > 1, the dominant factor in the flow decelera- passing the electric current (compare with Fig. 1). The tion is the ponderomotive force, and, at (uBh)/V < 1, the near-wall layer produces a shock wave l clearly seen by major role is played by the Joule heating caused by the the lower wall. Gasdynamic discontinuity e we identify external electric field. As the Mach number decreases, as a contact surface formed in the region beyond the the role of the ponderomotive force increases; for attached shocks because the gas flow at the diffuser example, for (uBh)/V > 0.4, it is dominant at M = 1, inlet is diverging and therefore the gas passing through becoming more so at still lower Mach numbers. As seen different parts of the attached shock front has different from (2), the relative role of the ponderomotive force densities, Mach numbers, and incidence angles. All this grows with increasing load factor, and at k 1 the results in a large density gradient, which can be seen in flow deceleration is mainly determined by the MHD the density field calculated in [8]. Probably, the electric interaction in the flow. field enhances this gradient, producing the contact sur- Figure 4 shows variation of the calculated Mach face e. In comparison with the flow pattern without number along the axis of the gasdynamic path under the external fields (see Fig. 1), the basic distinction of the action of magnetic and electric fields. The interaction flow pattern observed with an applied external electric zone occupies the length of the diffuser from the inlet field is that in the latter case the slope angles of the attached shock fronts increase, the shock front intersec- to a point where the attached shock fronts intersect. The ± calculation results are obtained by solving a set of equation point Xc shifts nearer the diffuser inlet, Xc = (37 tions describing variation of the flow velocity, the state 1.5) mm, and the points of the shock front reflection variables, and the Mach number and by using the Ohm from the diffuser walls become closer to the inlet. Thus, law [6] for conditions close to those of the present it is possible to conclude that the flow pattern formed in experiment. In the calculation, it was assumed that the electric field is similar to that emerging under a X = 0 corresponds to the critical cross section; T = weak MHD interaction as described in the numerical cr simulation of the process [8] and in the experiment [2]. 7700 K; ρ = 2.2 kg/m3; and a = 940 m/s, where the cr cr Fig. 5.2a shows Schlieren flow pattern at a fixed subscript acr refers to the critical cross section parameters. For conductivity in the interaction zone, we used moment of time, and Fig. 5.2b shows a scheme of the the effective conductivity of σ = 150 S/m obtained in basic gasdynamic nonuniformities observed in the dif- eff fuser under the joint action of the electric and magnetic the experiment. The load factor was k = 0.3. fields. It is seen that the flow pattern is appreciably dif- In Fig. 4, it is seen that the electric and magnetic ferent. First of all, attention is attracted by a highly fields singly applied affect the flow noticeably and developed near-wall layer, especially near the upper decrease the Mach number by about 25%. Simulta- wall, where the cathodes are located. Radical changes neous application of the external fields, as seen from occurred in the configuration of the inlet shock fronts.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 INFLUENCE OF ELECTRIC AND MAGNETIC FIELDS 401

(a) (a)

(b) (b) d d e n a b f a g b m e l n d d (5.1) (5.2)

Fig. 5. Connection I. (a) Schlieren patterns and (b) schematic diagrams of the ﬂow. B = (5.1) 0 and (5.2) 1.3 T.

At the place where oblique shock fronts intersected like a normal one at the core of the flow, the reflection before, now we see a triple shock configuration consist- remains regular. Then, at the moment t3, the regular ing of a normal shock g in the flow core and two oblique reflection of the shock fronts a from one another trans- shocks a and b. Between them one can also distinguish forms into a Mach reflection. The Mach wave g is discontinuities n; however, it is very difficult to con- formed with the reflected shock front b attached to it. clude if they are the same contact surfaces e or shock Thus, a triple configuration arises (see Fig. 5.2b). It waves due to the near-wall layer. Also, we have not should be noted that such a process of flow formation been able so far to identify discontinuity m, which can was unexpected. Rather, it was believed that the highest be seen by the upper wall. On the whole, the flow can pressure will build up at the end of the interaction zone be characterized as a typical case of strong interaction and the compression waves will produce an upstream with a normal shock front of MHD deceleration formed shock wave changing the inlet shock wave configura- in the flow core; however, the flow pattern is much tion. The experiment has revealed quite a different way complicated by the highly developed near-wall layer. of the formation of the MHD deceleration shock front. Dynamics of the formation process of the MHD deceleration shock front is illustrated in Fig. 6, which INTERACTION IN VARIOUS DIFFUSER shows a selection of Schlieren patterns obtained with SECTIONS: CONNECTION II the help of the high-speed cinecamera. The exposure To find a way of a more effective influence on the µ time is 2 s, and the time interval between the frames is shock wave configuration, let us consider how it 5.7 µs. Frames 2, 6, and 13 show how the gas enters the changes when an electric current is passed through var- diffuser. Seen in frame 22 are the already developed ious diffuser sections. Figure 8 demonstrates the flow near-wall layer and a second shock front near the upper structure for connection II. An external voltage is wall, which is probably caused by the near-wall layer. applied to all of the electrodes except the first pair. The subsequent frames demonstrate that all the changes A Schlieren pattern obtained with only the applications observed in the flow happen to the attached shock of electric field is presented in Fig. 8.1a. A schematic waves. Schematically, this transformation is shown in diagram of the shock fronts observed is given in Fig. 7. At some moment of time t1, a regular reflection Fig. 8.1b. It is seen that the slope angles of the attached of the attached shock fronts from each other takes shock fronts did not change much as compared to place. The pattern changes with time, and at the Fig. 1, and the distance Xc remained practically the moment t2, the lower shock front becomes noticeably same. This is evidence that, in this case, the current concave, while the shape of the upper shock front is dif- flowing through the plasma has little effect on the ficult to judge because it is distorted by the highly attached shock fronts a formed at the diffuser inlet. The developed near-wall layer. However, despite the con- positions of shock fronts b changed only slightly; their cavity, which makes the oblique shock front look more slope angles increased to such an extent that shock

TECHNICAL PHYSICS Vol. 47 No. 4 2002 402 LAPUSHKINA et al.

240

642

13 44

22 46

32 49

34 51

Fig. 6. Connection I. Frame-by-frame recording of Schlieren flow patterns. The frames are identified by their numbers. fronts f reflected from the near-wall layer became visi- Application of the magnetic field in addition to the ble. The current caused a noticeable growth of the near- electric field did not cause noticeable shifts of the wall layer starting near the first pair of electrodes in this attached shock fronts (see Figs. 8.2a and 8.2b), the only connection. observable effect being that their slight bending of con-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 INFLUENCE OF ELECTRIC AND MAGNETIC FIELDS 403 tact surface e became more clearly seen. In addition, close to the still more developed near-wall layer, gasdynamic discontinuity m was observed. t1

CONNECTION III When an external voltage was applied only to the first pair of electrodes located near the inlet diffuser edges (connection III), a shock wave configuration arose, which can be seen in Fig. 9. Passing the electric current without applying the magnetic field (Figs. 9.1a t2 and 9.1b) noticeably changed the slope of the shock fronts, and their intersection point shifted toward the ± diffuser inlet, Xc = (36 1.5) mm, and a marked growth of the near-wall layer was observed beginning right at the diffuser inlet. With an onset of the MHD interaction (Figs. 9.2a and 9.2b), no further change of Xc occurred; however, the attached shock fronts became slightly concave, and the wide near-wall layers could be clearly t3 a g b d seen, the upper layer being appreciably thicker than the lower one. Contact surface e was clearly discernible. Many more details of the gasdynamic structure were brought out in an experiment carried out with a gas Fig. 7. Schematic diagram of the MHD deceleration shock whose density was twice as high. These experimental formation. results are shown in Figs. 10a and 10b. It is seen that, under the joint action of the electric and magnetic fields, changes in the shape of the attached shock fronts for connection II and 0.04 for connection III. StII is are accompanied by additional disturbances in the almost an order of magnitude larger than StIII, which is vicinity of the attached shock fronts. evidence of a stronger MHD interaction in the second duct section. Contributions to the flow deceleration of the plasma DISCUSSION OF THE RESULTS heating by the electric field for the two connections can be compared if one evaluates the ratio of the amount of The experimental data show that connection II, in Joule heat, spite of creating a more extensive interaction region, has a weaker influence on the shock wave configuration ()jLV in the diffuser than connection III. Let us consider all Qc = 1 Ð k ------, the factors affecting the flow under applied external uh fields. This will allow us to select a diffuser section for 〈ρ 2 〉 more effective local impact on the inlet shock fronts. to the flow enthalpy H = (u + CpT) averaged over ∆ the time t = L/u, that is, the quantity P = Qc/H. Here, To compare quantitatively the energy expended for Cp is the specific heat of the gas at constant pressure. variation of the flow parameters with the two connec- Using the calculated data for the velocity and tempera- tion types, II and III, let us evaluate the work of the pon- ture of the downstream flow, we obtain the following deromotive force and the energy spent for heating the values of the parameter P for the two interaction zones: gas by the external electric field. Energy pick-up, PII = 0.6 and PIII = 0.07. It is seen that the effect on the according to estimates, plays a minor role, and there- flow of the Joule heating, like that of the ponderomotive fore it is not considered here. The work of the ponder- force, should be greater if the external voltage is omotive force over the length L is equal to A = jBL and applied using connection II. However, comparison of was determined from the experimental data. As a mea- the flow fields in regions II and III (see Figs. 8 and 9) sure of the influence of this work on the flow, the Stuart shows that in region II, where the applied impact is parameter St = jBL/〈ρu2〉 was used, which is defined as greater, changes in the positions of the shock fronts are the ratio between the work of the ponderomotive force substantially less than in region III, where a weaker and twice the average kinetic energy of the flow in the influence significantly changes the slope and shape of diffuser volume over the length L at V = 0 and B = 0. the attached shock fronts. Thus, external voltage affects From the kinetic energy estimated on the basis of cal- the attached shock fronts more significantly if applied culation data, we obtain Stuart parameter values of 0.3 to the diffuser inlet section.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 404 LAPUSHKINA et al.

(a) (a)

(b) d (b) d

a b f a b d (8.1) l d (9.1) (a) (a)

(b) (b) d l d e m b a f a b f e l e d (8.2) d (9.2)

Fig. 8. Connection II. (a) Schlieren patterns and (b) sche- Fig. 9. Connection III. (a) Schlieren patterns and (b) schematic diagrams of the ﬂow. B = (8.1) 0 and (8.2) 1.3 T matic diagrams of the ﬂow. B = (9.1) 0 and (9.2) 1.3 T.

In connection to this, differences in the flow struc- region. As the experiment has shown, the system of ture in regions II and III should be considered. Only shock fronts in region II is less responsive to external half of the attached shock fronts are found in region III, influences; that is, to change the flow structure shocks, most of the flow volume here being occupied by a con- a stronger impact is necessary than in the case of continuous flow. In region II, where the density of dissipa- tinuous flow. It appears natural to relate the observed tive structures is greater, the attached shock fronts per- unresponsiveness of the shocks to the energy taken sist; here they are reflected from one another, and prac- from the flow for their formation in the course of inter- tically all the reflected shock fronts are found in this action with the diffuser walls. The irreversible energy

TECHNICAL PHYSICS Vol. 47 No. 4 2002 INFLUENCE OF ELECTRIC AND MAGNETIC FIELDS 405

(a) CONCLUSION The major results of this study are as follows. It has been found that the near-wall layer has a strong influence on the flow pattern in the diffuser. If the interaction is strong, a shock wave of MHD deceleration is produced due to transition from the regular reflection of the attached shock fronts to the Mach reflection. It has been ascertained that, for effective control of the attached shock fronts, the external impacts should be applied to the diffuser inlet section. The flow containing shock waves is no less responsive to externally applied fac- (b) tors. It is suggested that this effect is associated with the d amount of energy that goes to the shock wave formation as a result of the flow interaction with the diffuser walls. l r a b f ACKNOWLEDGMENTS The authors express their sincere gratitude to l r B.G. Zhukov for helpful discussions. The study was d performed with the financial support of the ISTC.

REFERENCES Fig. 10. A Schlieren pattern of the flow and a schematic diagram of gasdynamic discontinuities under the action of 1. S. V. Bobashev, R. V. Vasil’eva, E. A. D’yakonova, et al., ρ electric and magnetic fields at a gas density of 0 = Pis’ma Zh. Tekh. Fiz. 27 (2), 63 (2001) [Tech. Phys. 0.196 g/m3. Lett. 27, 71 (2001)]. 2. S. V. Bobashev, E. A. D’yakonova, A. V. Erofeev, et al., in Proceedings of the 2nd Workshop on MagnetoÐ loss Q to a shock is given by the change of the entropy PlasmaÐAerodynamics in Aerospace Applications, Mos- S across the shock is as follows: cow, 2000, p. 64. 3. V. L. Fraœshtadt, A. L. Kuranov, and E. G. Sheœkin, Zh. QTS==∫ d 〈〉∆T S, Tekh. Fiz. 68 (11), 43 (1998) [Tech. Phys. 43, 1309 (1998)]. 4. E. P. Gurijanov and P. T. Harada, AIAA Pap., No. 96- ργ (5) ∆ p2 1 4609 (1996); in Proceedings of the 7th Aerospace Planes SS==2 Ð S1 Cv ln------γ , and Hypersonic Technology Conference, Norfolk, 1996. p ρ 1 2 5. S. V. Bobashev, E. A. D’yakonova, A. V. Erofeev, et al., AIAA Pap., No. 2000-2647 (2000). where indices 1 and 2 denote parameters in front of and 6. L. A. Vulis, A. L. Genkin, and V. A. Fomenko, Theory behind the shock, respectively. and Calculation of Magnetohydrodynamic Stream in Quantitatively, this energy loss is about two orders Channels (Atomizdat, Moscow, 1971). of magnitude less than the total flow enthalpy. A ques- 7. R. V. Vasil’eva, A. V. Erofeev, et al., Zh. Tekh. Fiz. 57 (2), tion arises whether the ratio of the energy lost to all 251 (1987) [Sov. Phys. Tech. Phys. 32, 150 (1987)]. shocks in the interaction zone and the energy supplied 8. Yu. P. Golovachev, Yu. A. Kurakin, A. A. Schmidt, and by the external fields can serve as a measure of the D. M. Van Wie, AIAA Pap., No. 2001-2883 (2001). response of the shocks to the external factors aimed at changing their positions. Translated by N. Mende

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 406Ð409. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 32Ð35. Original Russian Text Copyright © 2002 by Shuaibov, Dashchenko, Shevera.

GAS DISCHARGES, PLASMA

Optical Characteristics of the Steady-State Plasma of a Longitudinal Discharge in a He/CFC-12 Mixture A. K. Shuaibov, A. I. Dashchenko, and I. V. Shevera Uzhhorod National University, Uzhhorod, 88000 Ukraine e-mail: [email protected] Received September 28, 2001

Abstract—The emission from the plasma of a contracted longitudinal dc discharge in a He/CF2Cl2 mixture in the wavelength range 130Ð300 nm is investigated. It is shown that the discharge plasma emits within the range 150Ð260 nm. The emission consists mainly of the broad bands of Cl2 molecules and single-charged chlorine ions. The pressure and composition of the working mixtures, the discharge current, and the time during which the emitter can operate on a single gas ﬁll are optimized to attain the best characteristics of UV and VUV radiation. The results obtained are of interest for developing a steady-state source of VUV and UV radiation to be applied in microelectronics, photochemistry, and medicine. © 2002 MAIK “Nauka/Interperiodica”.

A dc glow discharge in the mixtures of Xe, Kr, and applied to develop a dc emission source operating on a Ar with chlorine has been successively used as a source He/CF2Cl2 mixture. of VUV and UV emission from rare gas chlorides with In this paper, we present the results of the studies of an average radiation power of 3Ð20 W and efficiency of VUV and UV emission from a longitudinal discharge 8Ð23% [1Ð3]. In [2], it was also found that He/Cl mix- 2 in a low-pressure He/CF2Cl2 mixture. tures can be employed in low-pressure electric-dis- A longitudinal dc discharge was ignited in a quartz charge lamps, which operate on the Cl (D'ÐA') 258-nm 2 tube with an inner diameter of 5 mm and interelectrode band and whose output parameters are close to those of distance of 100 mm. Cylindrical electrodes made of Ni the corresponding excimer lamps. An advantage of foil were placed in the discharge tube with open ends. such a UV source is that its working mixture does not The discharge tube was housed in a 10-l buffer chamber contain expensive rare gases like Xe or Kr. This pro- connected to a 0.5-m-long vacuum monochromator via vides the possibility of creating a lamp with a longitu- a LiF window. The open end of the discharge tube was dinal circulation of the working mixture, where the in front of the entrance slit of the vacuum monochroma- problem of the gas mixture lifetime becomes unimpor- tor. The spectrometer was calibrated in the wavelength tant. Lamps of this kind are characterized by the ele- range 165Ð350 nm with the help of continuous hydro- vated output radiation powers and efficiencies. In [4], gen emission; then, the curve obtained was extrapolated we optimized the optical characteristics of a compact to λ = 130 nm. The system for recording the character- emitter operating on a He/Cl2 mixture pumped by a lon- istics of the longitudinal discharge was identical to that gitudinal dc discharge. It was found that the emission described in [3, 4]. spectrum of this lamp can be extended to 195Ð200 nm In experiments, we studied the currentÐvoltage 1Σ 1Π by operating on the Cl2( Ð u) bands and using an characteristics, the power deposited in the discharge outer lamp envelope transparent in the wavelength plasma, the emission spectra in the 130- to 300-nm range λ ≤ 200 nm. Some medical and industrial appli- wavelength range, the intensities of the emission bands cations (sterilization of medical instrumentation, food as functions of the pressure and composition of the conservation, etc.) require powerful sources of short- working mixture, the dependences of the maximum wavelength radiation in the range 170Ð220 nm. Thus, in intensities in different emission bands on the discharge current, and the emitter lifetime. The emission band order to further extend the emission spectrum to the intensity was determined from the area under the corre- VUV spectral range, it is of importance to continue the sponding curve on the chart recorder paper with allow- experiments of [4] with another halogen-containing ance for the relative spectral sensitivity of the vacuum agent (CFC-12), whose dissociation products emit in spectrometer. At a CFC-12 pressure higher than the range 150Ð180 nm. It was found in [5, 6] that the 0.1 kPa, a longitudinal dc discharge in pure CFC-12 emission from the plasma of a transverse volume dis- and He/CFC-12 mixtures becomes contracted. As the charge in He/Cl2 and Ar/CF2Cl2 mixtures contains CFC-12 partial pressure and the total pressure of the VUV radiation from the products of dissociation of He/CFC-12 mixture increases, the diameter of the chlorine and CFC-12 molecules, which can also be plasma filament separated from the inner wall of a dis-

OPTICAL CHARACTERISTICS OF THE STEADY-STATE PLASMA 407 charge tube decreases from 2 to 1 mm. Although the Uch, kV discharge contraction leads to a decrease in the area of the working surface of the cylindrical emitter, it is still 4 (a) of interest from the point of view of increasing the time during which the emitter can operate on a single gas fill. The reason is that the service life of chlorine-containing 3 devices strongly depends on the coefficient of chlorine diffusion into glass, which significantly increases with 3 the temperature of the discharge tube [7]. A contracted 2 4 longitudinal discharge is the simplest emitter, in which 2 the plasma does not contact the glass envelope; thus, in 0 1 this respect, it is similar to an excimer lamp with microwave pumping [8]. The ignition of a usual dc glow dis- P, W charge in CFC-containing mixtures at CFC partial pres- (b) 2 sures higher than 100 Pa is questionable. In [9], such a 60 3 discharge was ignited in CCl vapor, but only when the 4 1 gas medium was excited with a fast longitudinal ioniza- 40 tion wave. The ionÐion plasma of a contracted discharge in electronegative gases significantly differs 20 from the common electronÐion plasma [10, 11]; this fact attracts additional interest to studying these discharges. In the short-wavelength spectral range, the 0 5 10 15 20 25 30 optical characteristics of an ionÐion plasma formed of Ich, mA CFC-12 molecules were not investigated, which impedes the development of steady-state sources of Fig. 1. (a) CurrentÐvoltage characteristics of a longitudinal VUVÐUV radiation. dc discharge in the He/CF2Cl2 = (1) 1.33/0.13, (2) 1.33/0.40, (3) 1.33/0.67, and (4) 6.70/0.40-kPa mix- Figure 1a presents the currentÐvoltage characteris- tures, and (b) the power deposited in plasma vs. discharge tics of a longitudinal discharge in a He/CFC-12 mix- current in the He/CF2Cl2 = (1) 1.33/0.13, (2) 1.33/0.67, and (3) 6.70/0.40-kPa mixtures. ture. At a discharge current of Ich = 1Ð15 mA, the discharge voltage (Uch) is inversely proportional to the discharge current, which is characteristic of the subnormal 156–162 nm Cl+* ≥ ** mode of discharge operation [12]. At Ich 15 mA, the 200 nm Cl2 longitudinal discharge switches to the normal mode, in 258 nm Cl2(D'–A') which Uch changes only slightly as the current increases. The increase in the CFC-12 content and the pressure of the He/CFC-12 mixture leads to a sharp increase in the discharge ignition voltage and the quasi- steady discharge voltage Uch in the normal mode (Fig. 1a). The power deposited in the discharge plasma increases with the total pressure of the mixture and the 150 180 210 240 270 λ, nm CFC-12 content and attains 30Ð65 W (Fig. 1b). Fig. 2. VUVÐUV emission spectrum from the steady-state The plasma emission spectrum corrected with electric-discharge plasma in a He/CF Cl mixture. allowance for the relative sensitivity of the recording 2 2 system is shown in Fig. 2. As in the case of a longitudinal discharge in a He/Cl2 mixture [4], the 200- and specific features of the plasma of a contracted longitu- 258-nm emission from chlorine molecules prevails. dinal discharge. ** Moreover, against the background of the Cl2 VUV The dependences of the VUV and UV emission emission bands in the 140- to 170-nm range, there are intensities on the composition of a He/CF2Cl2 mixture intense spectral lines of single-charged chlorine ions. are shown in Fig. 3. At low pressures of the mixture, the The identification of this linear spectrum is shown in optimum partial pressure of CFC-12 is in the range 0.3Ð the table. A group of unresolved Cl+* spectral lines 0.5 kPa. Further increase in the CFC-12 content leads forms the shortest wavelength part of the spectrum with to a decrease in the intensities of the majority of the λ 156Ð162 nm. In the plasma of a transverse discharge emission bands and the total intensity in the range 130Ð in Ar/CF2Cl2 mixture, no emission from chlorine ions 280 nm. The emission from chlorine ions is the most in the wavelength range 155Ð165 nm was recorded [6], sensitive to the CFC-12 content in the mixture (Fig. 3a). which is possibly related to the involvement of helium As for the dependence of the emission intensities on the in the production of the excited chlorine ions and the helium partial pressure, there are a narrow peak at

TECHNICAL PHYSICS Vol. 47 No. 4 2002

408 SHUAIBOV et al.

J, arb. units (a) tral range 130Ð280 nm also increases with the helium partial pressure. The optimum partial pressure of 100 helium in the mixture is within 5Ð6 kPa. Such behavior of the dependence of VUV and UV emission intensities 80 on P(He) in the ionÐion plasma of a contracted dis- 4 charge can be related to the significant contribution of 60 energy transfer from the metastable atoms and mole- 1 cules of helium to the molecules and main dissociation + 40 3 products of CFC-12 in the plasma (Cl2, Cl , etc.), as 2 well as to the efficient recombination of the Cl+ and ClÐ 20 ions toward the production of Cl (D') and Cl** mole- 100 300 500 P(CF Cl ), Pa 2 2 2 2 cules. (b) An increase in the discharge current in the range 2Ð 140 4' 30 mA enhances the emission intensity from chlorine molecules and ions (Fig. 4). For the shortest wavelength fraction of the emission spectrum, the rate at 100 which the intensity grows as Ich increases is nearly twice as high as that for chlorine molecules. The increase in the helium partial pressure from 0.67 to 60 1' 3' 4.00 kPa also increases the rate of the intensity growth 2' in the shortest wavelength part of the spectrum by a fac- 20 tor of approximately 1.5. When the discharge tube is 0 2 4 6 P(He), kPa cooled only with the working mixture of the contracted discharge under study, no saturation of the VUV and ≤ Fig. 3. Emission intensities from a longitudinal discharge in UV emission intensities is observed at the currents Ich He/CF2Cl2 mixtures vs. (a) the CFC-12 partial pressure at 30 mA. In the emitter based on a glow discharge in an P(He) = 1.33 kPa and (b) the helium partial pressure at Ar/Cl2 mixture (which is the working mixture of the P(CF2Cl2) = 0.40 kPa in the ranges (1, 1') 130Ð175, (2, 2') 175Ð215, and (3, 3') 215Ð280 nm and (4, 4') the total emis- lampsoperating on ArCl 175-nm and Cl2(D'ÐA') sion intensity in the 130- to 280-nm spectral range. 258-nm band system), under similar pumping condi- ≥ tions, the emission intensity dropped already at Ich 22 mA. P(He) = 1.3Ð1.4 kPa and a broad peak at P(He) = 5Ð 6 kPa in the spectral range where the emission from The resource characteristics of the VUVÐUV emitter pumped with a contracted longitudinal discharge Cl+* ions prevails. The emission from the Cl (D'ÐA') 2 and operating in the gas-static regime with the use of a 10l 258-nm band has a broad peak at P(He) = 4 kPa, and the buffer volume are seen in Fig. 5. Each curve in Fig. 5 intensity of the Cl2* emission band with a maximum at was obtained with a fresh He/CFC-12 = 1.33/0.4-kPa λ = 200 nm slightly increases with increasing P(He). At mixture at Ich = 20 mA. The lifetime (defined as the low helium pressures, the VUV emission from chlorine time it takes for the emission intensity to fall by a factor ions prevails. The total emission intensity in the spec- of 2) of an emitter operating on either λ = 162 or

J, arb. units 1.0 J, arb. units 1' 1.0 1 2' 3' 0.5 0.5 2 3 3 0 2 1 0 10 20 30 0510 15 20 25 t, min Ich, mA

Fig. 4. Maximum emission intensities at the wavelengths λ = (1, 1') 162, (2, 2') 200, and (3, 3') 258 nm vs. discharge Fig. 5. Maximum emission intensities at the wavelengths current in the He/CF2Cl2 = (1Ð3) 0.67/0.40-kPa and (1'Ð3') λ = (1) 258, (2) 200, and (3) 162 nm vs. discharge run time 4.00/0.40-kPa mixtures. in a He/CF2Cl2 mixture.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 OPTICAL CHARACTERISTICS OF THE STEADY-STATE PLASMA 409

Constituents of the VUV emission from chlorine ions in the The results of studying the optical characteristics of plasma of a He/CF2Cl2 mixture a contracted longitudinal discharge in He/CF2Cl2 mixtures can be summarized as follows. At a deposited λ, nm Elow, eV Eup, eV Transition power of 20Ð60 W, the discharge acts as a source of Cl II radiation in the 130- to 280-nm spectral range. The radiation is generated due to the spontaneous decay of 156.5 13.67 21.60 50 5 3d D01234,,,, Ð5f F1, 2, 3, 45 the dissociation products of CFC-12 molecules and incorporates the Cl (D'ÐA') 258-nm band, the chlorine 156.6 11.70 19.61 3p5 30P Ð4p'' 3S 2 0 1 emission continuum with a maximum at λ = 200 nm, + 160.4(5) 14.85 22.58 3d 30D Ð4f ' 3G and the emission from Cl * ions in the range 130Ð 23, 3, 4 175 nm. The optimum partial pressures of CFC-12 and 161.1 15.71 23.41 4s' 30D Ð5p'' 3D helium lie within the ranges 0.3Ð0.5 and 5Ð6 kPa, 2 2 respectively. The increase in the discharge current from 161.9 13.67 21.33 3d 50D Ð6p 5P 2 to 30 mA enhances the emission intensity throughout 3 2 the entire VUVÐUV spectral range with no tendency to 161.9 13.67 21.33 3d 50D Ð6p 5P saturation. The longest emitter lifetime is achieved 4 3 when operating on the 200-nm band. The total emission ≤ 173.1 14.85 22.01 3d 30D Ð5p' 3D power attains 1.5 W with an efficiency of 3%. The 3 3 active medium of the emitter is the ionÐion plasma; 178.7 11.65 18.59 3p5 30P Ð4p' 3P hence, it is of interest to diagnose this plasma with the 1 1 aim of studying the mechanisms for the production of the 178.9 15.65 22.58 3d' 30F Ð4f ' 3G excited chlorine ions and molecules and to perform the 3 4 relevant numerical simulations. It is promising to use the 179.2 11.65 18.57 3p5 30P Ð4p' 3P VUVÐUV emitter under study in a repetitive mode with 1 2 the pumping by a fast longitudinal ionization wave (nano- 30 3 second longitudinal low-pressure discharge). 179.3 15.68 22.59 3d' F4 Ð4f ' G5 179.4 15.68 22.59 30 3 3d' F4 Ð4f ' H5 REFERENCES 5 3 0 3 179.7 11.70 18.59 3p P Ð4p' P1 1. A. P. Golovitskiœ and S. V. Lebedev, Opt. Spektrosk. 82, 251 (1997) [Opt. Spectrosc. 82, 227 (1997)]. 50 5 185.7 13.67 20.35 3d D01234,,,, Ð4f F1, 2, 3, 45 2. M. I. Lomaev, A. N. Panchenko, É. A. Sosnin, and V. F. Tarasenko, Zh. Tekh. Fiz. 68 (2), 64 (1998) [Tech. 5 30 3 188.7 11.58 18.14 3p P2 Ð4p' D2 Phys. 43, 192 (1998)]. 3. A. K. Shuaibov, A. I. Dashchenko, and I. V. Shevera, Zh. 5 30 3 192.3 11.70 18.14 3p P0 Ð4p' D1 Tekh. Fiz. 71 (8), 121 (2001) [Tech. Phys. 46, 1049 (2001)]. 5 50 5 193.7 13.38 19.77 4s S2 Ð5p P3 4. A. K. Shuaibov, A. I. Dashchenko, and I. V. Shevera, Teplofiz. Vys. Temp. 39, 833 (2001). 30 1 199.6 16.39 22.60 3d' G4 Ð4f ' H5 5. A. K. Shuaibov, Zh. Tekh. Fiz. 70 (10), 117 (2000) [Tech. Phys. 45, 1346 (2000)]. 30 3 199.7 16.39 22.60 3d' G5 Ð4f ' H6 6. A. K. Shuaibov, L. L. Shimon, A. I. Dashchenko, and I. V. Shevera, Teplofiz. Vys. Temp. 38, 386 (2000). 30 3 199.8 16.39 22.59 3d' G3 Ð4f ' H4 7. V. I. Svettsov, A. L. Kupriyanovskaya, and A. B. Mary- shev, Zh. Prikl. Spektrosk. 35, 205 (1981). 30 3 199.9 16.39 22.59 3d' G4 Ð4f ' H5 8. A. P. Golovitskiœ, Pis’ma Zh. Tekh. Fiz. 24 (6), 63 (1998) [Tech. Phys. Lett. 24, 233 (1998)]. 9. L. M. Vasilyak, S. V. Kostyuchenko, A. V. Krasnochub, 258 nm does not exceed 15 min, whereas for the and M. E. Kuz’menko, Zh. Prikl. Spektrosk. 65, 302 200-nm band, it is longer than 60 min. Thus, in the gas- (1998). 10. Yu. I. Bychkov, S. L. Gorokhov, and A. G. Yastremskiœ, static regime, the emission from the Cl2** 200-nm band λ Kvantovaya Élektron. (Moscow) 30, 733 (2000). is of most interest. For an emitter operating on = 11. A. A. Kudryavtsev, A. L. Kuranov, V. G. Mishakov, et al., 258-nm line or on the chlorine ion transitions, it is rea- Zh. Tekh. Fiz. 71 (3), 29 (2001) [Tech. Phys. 46, 299 sonable to use the regime of slow circulation of the (2001)]. working mixture in the discharge tube. The He/CF2Cl2 12. Yu. P. Raizer, Gas Discharge Physics (Nauka, Moscow, flow rate may be within the range 0.3Ð1.0 l/min. The 1987; Springer-Verlag, Berlin, 1991). average output power in the 130- to 280-nm range does not exceed 1Ð1.5 W with an efficiency of 2Ð3%. Translated by N. Ustinovskiœ

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 410Ð414. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 36Ð40. Original Russian Text Copyright © 2002 by Shadrov, Tagirov, Boltushkin.

SOLIDS

Structure Characteristics and Magnetization Reversal of High-Coercivity Cobalt-Based Alloy Films V. G. Shadrov, R. I. Tagirov, and A. V. Boltushkin Institute of Solid-State and Semiconductor Physics, Belarussian Academy of Sciences, ul. Brovki 17, Minsk, 220072 Belarus Received June 14, 2001

Abstract—The structure of “continuous” Co–W and Co–Ni–W alloy ﬁlms, as well as Co-containing metallic oxide heterogeneous ﬁlms, covering the aluminum surface is studied in dependence of intergranular magnetic interaction and magnetization reversal processes by taking the angular dependences of the coercive force and irreversible susceptibility and also from δM curves. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION METHODS OF FILM PREPARATION AND CHARACTERIZATION In recent years, many researchers studying magnetic media have been concentrated on Co-based high-coer- Films of CoÐW alloy were obtained by electro- civity alloy films [1, 2]. The magnetic parameters of chemically depositing from sulfateÐchloride electro- these films vary in wide limits, making them suitable lytes on copper substrates. The electrolyte composi- for both longitudinal and vertical data writing. To date, tions were close to those described in [7, 9]. The depo- however, the processes of magnetization reversal and sition conditions were as follows: pH 6.6, current density 10 mA/cm2, and electrolyte temperature 18Ð intergranular magnetic interaction, which are of great ° scientific and practical interest, have been studied in 50 C. The CoÐ10 wt % NiÐ15 wt % W films were films condensed in a vacuum [3–5]. In the films obtained under somewhat different conditions: pH 3.5Ð 5.3, current density 55 mA/cm2, and electrolyte tem- obtained electrochemically, these processes are poorly ° understood [6, 7], although these films admit a wider perature 18Ð55 C. Co-containing metallic oxide het- control of their properties. For example, the electro- erostructures were obtained by electrochemical deposi- chemical filling of anodic oxide films (AOFs) covering tion into pores that were formed on the AD-1 aluminum the aluminum surface by a magnetic material makes it surface by anodizing in a sulfuric acid electrolyte under possible to easily control the magnetic separation of conditions reported in [10]. Some of the samples were ° Ð5 needle-like crystals and their diameter by varying the annealed at 240 C under a pressure of 10 torr. The geometric parameters of the AOF cellular structure [8]. structure of the films was studied with a DRON-2 diffrac- Also, the electrochemically obtained films are less tometer, EVM-100 LM and JEOL-100CX electron micro- expensive. scopes; magnetic characteristics, with an automated vibration magnetometer under fields of up to 15 kOe. In this work, we seek a correlation between the The difference in the crystal structure of the films structure of “continuous” Co–W and Co–Ni–W alloy causes differences in the processes of intergranular films, as well as Co-containing metallic oxide heteroge- magnetic interaction and magnetization reversal. neous films, covering the aluminum surface and the Unfortunately, methods for the direct observation of processes of intergranular magnetic interaction and magnetization reversal are of limited application (for magnetization reversal. In the Co–W films, the mag- example, Lorentz microscopy applies only to free thin netic separation of the grains is less pronounced and is films); therefore, indirect methods are widely used. due to higher concentrations of crystal defects, foreign Among them are those using the angular dependences impurities and compounds, etc., in the intergranular of the coercive force, remanent magnetization, rota- layers, hence, a reduced magnetic moment of the lay- tional hysteresis, and losses due to magnetization rever- ers. By varying the deposition conditions and/or pos- sal [3Ð6]. telectrolysis treatment of the Co–W films, one can If in the crystal consisting of isolated grains magne- change their structure parameters: grain size and shape, tization reversal is due to incoherent rotation, the angu- texture and texture uniformity, width of the boundary lar dependence of the coercive force is given by [11] regions. On the one hand, this affects the magnetic properties of the grains and their magnetic separation; 1.08SÐ2()1Ð 1.08SÐ2 on the other, this makes experimental data difficult to Hc = H0------, (1) Ð2 2 2 Ð2 1/2 interpret. []()1Ð 1.08S Ð sin ϕ()1Ð 1.16S

STRUCTURE CHARACTERISTICS AND MAGNETIZATION REVERSAL 411

Preparation conditions, composition, texture, and magnetic properties of the Co–W and Co–Ni–W ﬁlms (h = 1 µm)

° Sample no. T, C W, wt %/pH Texture Hc, Oe Hr, Oe Hr' , Oe CoÐW 1* 18 15/6.6 [001] 639 843 848 2 18 15/6.6 [001] 175 230 490 3 29 15/6.6 tr. [001] 536 628 634 4 40 16/6.6 [100] 391 422 583 5** 19 15/6.6 [001] 168 227 374 CoNiÐW 6 19 /3.5 tr. [001] 146 245 239 7 19 /4.5 [001] 233 292 307 8 19 /5.3 [001] 460 478 541 9 47 /5.3 [001] 555 635 694 * Thickness 0.05 mm; ** annealed sample 2.

where S = R/R0, R is the grain radius, and R0 is the char- STRUCTURE EXAMINATIONS acteristic radius depending on the exchange interaction As follows from the X-ray and electron microscopy energy and saturation magnetization. data, the Co–W films deposited on the copper sub- The character of magnetic interaction can be deter- strates consist of individual grains, which represent the mined, for example, by finding the δM(H) functions solid solution based on hcp cobalt [9]. At an electrolyte temperature of 20°C, first the transition layer of thick- χd and measuring the irreversible susceptibilities irr and ness about 50 nm forms. It consists of randomly ori- χr ented equiaxial grains separated by layers with an irr [5]. They are defined as increased concentration of defects, inclusions, hydrogen, and Co(OH) hydroxide. As the film thickens, δ () ()()() 2 MH = Id H Ð,I∞ Ð 2Ir H columnar grains of diameter 40Ð50 nm with the [001] texture start to form. χd χr irr ==dId/dH, irr dIr/dH, During the growth of the columnar grains, their active surface is gradually filled by adsorbed impurity where I∞ is the remanent magnetization after the satu- particles and hydroxide sol particles, which are from time to time captured by moving growth steps. As a rating field has been switched off; Id(H) and Ir(H) are the remanent magnetizations after switching off the sat- result, the columns contain zones with an increased dis- urating field H when a negative field is applied to the tortion of the crystal lattice, which separate more perfect zones. With a rise in the electrolyte temperature, sample in the state I∞ and when a positive field is the relative adsorption of impurities and hydroxides applied to the demagnetized sample, respectively; and decreases and the column growth rate rises. Eventually, Hr and Hr' are the fields at which Id(H) = 0 and Ir()Hr' =blocks extended in the column growth direction appear 0.5I∞. [9]. The compositions of the films, their texture, and magnetic parameters vs. electrolyte temperature and According to [12], for noninteracting single-domain pH are listed in the table. As the electrolyte temperature ° particles, the relationships δM(H) = 0 and H = H' are is raised to 32 C, the axis in the Co–W films changes r r the orientation from the hcp lattice to the film plane and valid. In the case of magnetostatic (dipoleÐdipole) δ platelet-like grains appear. Simultaneously, the total interaction between single-domain particles, M(H) < 0 amount of the impurities drops. At an electrolyte tem- and Hr > Hr' . If the grains are divided into domains perature of 40Ð45°C, all the grains become platelet- and/or in the case of exchange interaction between the like. δ grains, M(H) > 0 and Hr = Hr' . In the Co–Ni–W films, the dependence of the texture on the electrolyte temperature is weaker: the [001] The type of interaction can also be judged from the texture changes to the [100] texture at an electrolyte χ ° position of the irr(H) peaks: in the absence of interac- temperature of 47 C. The texture is also more stable tion, the peaks coincide; for dipoleÐdipole interaction, against pH variations: in the range 3.5 < pH < 5.3, the χd χr texture remains the same, [001]. One might expect that, the peak of irr (H) is shifted relative to that of irr (H) with pH varying in the so wide limits, the impurity con- toward weaker fields; and for exchange interaction, the centration will considerably change: for example, the former peak lies in a higher field range. impurity concentration in electrodeposited Co films

TECHNICAL PHYSICS Vol. 47 No. 4 2002 412 SHADROV et al.

Hc, Oe the saturation magnetization in Co–W films is lower than in pure cobalt, the critical size will increase. At a 1500 1 S = 1.43 large energy of uniaxial magnetic anisotropy, which may be as high as 1016 erg/cm2, individual grains can be 1250 viewed as permanent magnets. If the energy of anisot- 2 ropy appreciably exceeds the energy of magnetostatic 1000 1.38 interaction between the grains, the film represents an ensemble of isolated noninteracting particles; then, 1000 3 δ 1.42 according to [5], M(H) = 0, Hr = Hr' , and the angular 750 dependence of the coercive force is given by expres- 4 sion (1). 1500 The measurements on the Co-containing AOFs, 1.30 consisting of individual needle-like grains separated 1250 5 from each other, confirm the above conclusion: at the early stage, the angular dependences are coincident 1500 with the calculated ones if S is taken to be 1.43, 1.38, and 1.42 for curves 1Ð3, respectively (Fig. 1). With 1250 regard for the finite sizes of cobalt particles (in calculations performed in [11], infinitely long cylinders are 1000 considered) and the dispersion of easy-magnetization axis and the anisotropy field, the coincidence will be 750 even better (the associated correction has been made for uniaxial cobalt–chromium films with parameters close 03060 90 to those of the films studied in this work [15]). Note that ϕ, deg the values of Hc1 for the films with different pore diameters differ in accordance with theoretical predictions Fig. 1. Coercive force vs. magnetization reversal angle. [8]. With this circumstance taken into account, the Continuous lines, calculation; o, data points. Curve 5 was constructed according to the displacement theory [14]. associated values of S virtually coincide. This can be 1, Co film with pores of diameter 18 nm; 2, annealed sample explained by the lack of adequate experimental data 3; 3, Co film with pores of diameter 30 nm; 4, CoÐ20 wt % (specifically, for the contribution from the crystallo- W film with the [001] texture; and 5, CoÐ20 wt % W film graphic component of the magnetic anisotropy) and with the [100] texture. also by the fact that our model is to some extent conventional. At the same time, an increase in Hc1 in the changes from 2.3 × 1015 cmÐ3 at pH 1.7 to 1.4 × 1017 cmÐ3 annealed samples is attended by a decrease in S; in at pH 5.7 [13]. other words, the refinement of the needle-like crystal- lite structure and the related growth of anisotropy can Co films consisting of individual needle-like grains be associated with the enhanced effect of rotation in of diameter (AOF pore diameter) 18 and 30 nm and magnetization reversal. This is also confirmed by a length ≈1 µm have the hcp lattice with the randomly decrease in the rotational hysteresis integral in the oriented c axis (the I002/I100 X-ray intensity ratio is 2 to annealed structures [16]. 3). The annealing of the films of both types obtained Incoherent rotation may also be responsible for under the above conditions affects the geometric magnetization reversal in the hcp Co–W films, which parameters of the columnar (cellular) structure insignif- have the [001] texture and columnar grains. The exper- icantly. iment (Fig. 1, curve 4) coincides with the calculation at least up to 30°Ð40° if S is set equal to 1.30. At higher angles of magnetization reversal, the experimental and MAGNETIC MEASUREMENTS analytical curves diverge considerably, indicating the AND DISCUSSION major contribution from another mechanism of magne- The “continuous” Co-based structures and AOFs tization reversal, namely, domain wall displacement studied in this work consist of grains separated by non- [4, 7]. magnetic layers (boundaries) or those with a decreased As follows from the aforesaid, magnetization rever- magnetization. The sizes of the grains (needles of diam- sal in the Co and CoÐ20 wt % W films deposited at eter 18Ð30 nm on the AOF surface and columns of room temperature ([001] texture) is described well by diameter 40–50 nm in the Co–W films) suggest that the the model of separated uniaxial grains. If, however, the grains are almost single-domain. Such an assumption energy of uniaxial magnetic anisotropy decreases or the seems to be quite reasonable: the critical grain size cor- saturation magnetization grows, the effect of magneto- responding to purely single-domain cobalt has been static interaction becomes more pronounced. Such a estimated at 30 nm [14]. If it is taken into account that situation is observed in the CoÐW and CoÐNiÐ15 wt %

TECHNICAL PHYSICS Vol. 47 No. 4 2002 STRUCTURE CHARACTERISTICS AND MAGNETIZATION REVERSAL 413

W films, where the magnetic moment lies in the film δM, arb. units plane in spite of the columnar morphology and [001] 0.2 texture. This means that the energy of the demagnetiz- 1 ing field exceeds the magnetostatic energy; in this case, the film cannot be represented as an ensemble of noninteracting particles. The experimental data suggest the –0.2 5 presence of magnetostatic interaction, since δM < 0 and ' Hr < Hr for most of the films (see table and Fig. 2). 3 It should also be stressed that the columns, while –0.6 having a rather perfect [001] texture, consist of several 4 blocks slightly misoriented relative to each other. These blocks are separated by zones where the crystal lattice 2 is highly distorted [9]. Thus, the blocks inside the col- –1.0 umns can be viewed as single-domain magnetostati- 0 1000 2000 3000 cally interacting particles. The validity of this assump- H, Oe tion is confirmed by the measurements of the magnetic characteristics of samples 2 (CoÐW) and 8 (CoÐNiÐW): Fig. 2. Field dependences of δM taken from the CoÐW δ films. Curve numbers correspond to sample numbers in the for both M(H) < 0. The same measurements taken table. from sample 1 (CoÐW), which, like sample 2 (CoÐW), was obtained at room temperature, indirectly support the presence of magnetostatic interaction between the x, arb. units blocks: at room temperature, magnetostatic interaction in sample 1 is almost absent. The reason for the absence 2.0 of magnetostatic interaction in sample 1 is the absence of the block structure, since the thickness of this sample 1 coincides with that of the transition layer, consisting of equilibrium grains. 1.5 More evidence in favor of this assumption is the reduced interaction in the annealed Co–W films (h = 2 1 µm) with the [001] texture (Fig. 2, curve 5). This 1.0 reduction is due to the poorer magnetic separation between the substructure elements because of a decrease in the concentration of crystal defects and their redistribution. As a result, the contribution of the 0.5 substructure elements to the resulting magnetostatic interaction drops (with the intergranular interaction remaining almost unchanged). As the electrolyte temperature grows, the adsorption 0 1000 2000 H, Oe of the hydroxides decreases and the probability of the hydroxides being captured by the growing crystal lat- Fig. 3. Field dependences of the irreversible susceptibility tice drops. Simultaneously, the [001] texture changes to for the Co–W film (sample 3): (1) χd and (2) χr. the [100] texture. Because of this, the impurities are pushed out to the grain boundaries and the lattice in the columns becomes more perfect. Eventually, the blocks (the peaks change places). Possibly, at an electrolyte in the grains disappear and their contribution to the temperature of 29°C, the grains not only are refined but magnetostatic interaction vanishes (CoÐW sample 3). also grow. As a result, grains coarser than purely single- Under these conditions, there is no reason to assume a domain grains may appear. Simultaneously, magneto- considerable magnetostatic interaction between the static interaction between the blocks is partially grains, since the grain boundaries remain sufficiently retained and that between the grains takes place. wide. It is also necessary to take into account that the [001] texture axis, along which the energy of magnetic A further rise in the electrolyte temperature causes anisotropy is the highest, may change the orientation new structures to appear. The columns are replaced by and lie in the film plane. platelets, and the grain size grows. The impurities are Note that the situation in sample 3 appears to be removed from the grains, and the relative volume of the more complicated. From the irreversible susceptibility grain boundaries shrinks. As a consequence, the mag- curves (Fig. 3), it follows that exchange interaction is netic separation of the grains degrades, and the magne- observed in weak fields (the peak of χr lies to the left of tostatic interaction grows (samples 4, CoÐW, and 9, that of χd) and magnetostatic interaction, in high fields CoÐNiÐW).

TECHNICAL PHYSICS Vol. 47 No. 4 2002 414 SHADROV et al. Studying the Co–Ni–W films deposited at different REFERENCES pH, we could clarify the effect of grain size and the 1. K. E. Johnson, C. M. Mate, J. A. Mertz, et al., IBM J. magnetic separation of the grains on magnetization Res. Dev. 40, 511 (1996). reversal. The decrease in pH from 5.3 to 3.5 drastically 2. P. J. Grundy, J. Phys. D 31, 2975 (1998). (by one to two orders of magnitude) reduces the impu- 3. J. C. Lodder and L. Cheng-Zhang, J. Magn. Magn. rity concentration in the films [13]. This affects the Mater. 74, 74 (1988). grain boundary thickness and the grain size: the grain boundaries become thinner and the grains, “purer” and 4. R. Ranjan, J. S. Gau, and N. Amin, J. Magn. Magn. coarser. As a result, magnetostatic interaction weakens Mater. 89, 38 (1990). 5. P. I. Mayo, K. O’Grady, R. W. Chantrell, et al., J. Magn. and exchange interaction appears (Hr' < Hr, sample 6). Magn. Mater. 95, 109 (1991). 6. U. Admon, M. P. Dariel, E. Grunbaum, and J. C. Lodder, J. Appl. Phys. 66, 316 (1989). CONCLUSIONS 7. V. G. Shadrow, R. I. Tagirov, A. V. Boltushkin, and (1) Upon the electrodeposition of cobalt into pores N. N. Kozich, J. Magn. Magn. Mater. 118, 165 (1993). in anodized aluminum, films consisting of needle-like 8. D. AlMawlawi, N. Coombs, and M. Moskovits, J. Appl. ferromagnetic particles form. The particles are close to Phys. 70, 4421 (1991). single-domain grains by size, and magnetization rever- 9. A. V. Boltushkin, V. G. Shadrov, T. A. Tochitskiœ, and sal in them is accomplished by incoherent rotation. Zh. P. Apkhipenko, Elektrokhimiya 9, 1105 (1990). (2) Continuous Co–W and Co–Ni–W films depos- 10. V. G. Shadrov, A. V. Boltushkin, L. B. Sosnovskaya, ited at room temperature consist of grains divided into et al., Metally, No. 2, 120 (1999). blocks, which are presumably responsible for the mag- 11. S. Shtrikman and D. Treves, J. Phys. Radium 20, 286 netostatic interaction observed. (1959). (3) The coarsening of the grains and a decrease in 12. E. P. Wohlfarth, J. Appl. Phys. 29, 595 (1958). the concentration of adsorbed impurities, which take 13. S. Nakahara and S. Mahajan, J. Electrochem. Soc. 127, place when the electrolyte temperature rises and pH 283 (1980). declines, violates the conditions under which the grains 14. S. V. Vonsovskiœ, Magnetism (Nauka, Moscow, 1971; are single-domain. This shows up as magnetostatic Wiley, New York, 1974). interaction between the grains. 15. J. Nakamura and S. Iwasaki, IEEE Trans. Magn. 23, 153 (1987). 16. A. V. Boltushkin, V. G. Shadrov, N. N. Kozich, et al., ACKNOWLEDGMENTS Vestsi Akad. Navuk Belarusi, Ser. Fiz.-Mat. Navuk, This work was supported by the Belarussian Foun- No. 1, 47 (1993). dation for Basic Research (grant no. T99-107) and by the NATO grant HTECH LG, no. 940656. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

SOLIDS

Formalization of Models of a Deformable Polycrystalline Material in Terms of Mesomechanics V. V. Ostashev and O. D. Shevchenko Pskov Polytechnical Institute (Branch of St. Petersburg State Technical University), Pskov, 180680 Russia Received June 15, 2001

Abstract—The stage of formalization is necessary for constructing an adequate mathematical model of any phenomenon. By the example of a deformable polycrystalline material, it is shown that the formalization algorithm is based on a closed set of deﬁnitions, which represents the material under study as an open nonlinear dynamic structurally stable hierarchical (multilevel) dissipative self-organizing information system. Formaliza- tion principles, as applied to models of plastic deformation under static loading, are experimentally veriﬁed with polycrystalline fcc materials (MO copper and 08Kh18N10T austenitic steel). © 2002 MAIK “Nauka/Interperiodica”.

GENERAL STATEMENTS sions and are separated from each other by boundaries. The description of plastic deformation in a poly- The evolution of the system depends on the dynamics crystalline material on the mesoscopic level requires of mesodefects, their mechanical characteristics, and the qualitative formalization of their mathematical the interaction within the representative volume. The models and imposes certain demands on the rigor of the interaction between mobile mesodefects results in the terminology used and basic definitions. In the general formation of dissipative structures, as a rule, of the cor- case, a formalization algorithm must map the physical relation type; however, the functional ordering also model onto the mathematical one and vice versa. favors the formation of geometrical structures characteristic of the mesoscopic level. This means that meso- It is known that the “hottest” issues in mesomechan- defects containing both the shear and rotational compo- ics are geometrical image of a deformation-induced nents of the strain make possible the displacement of defect (hereafter, deformation defect), structural levels volume structure elements of different scale (subgrains, of deformation, dissipative structures, and representa- grains, their conglomerates, or extended blocks) in a tive volume. The physical mechanisms of plastic defor- deformable solid. mation on the mesoscopic level have been considered by the authors of mesomechanics [1, 2]. In the general case, they are as follows. PRINCIPLES OF FORMALIZATION (1) The basic mechanism of deformation is initial glide, which always results in initial material rotation. Thus, models of physical mesomechanics are based All other deformation mechanisms are accommodation on the description of nonlinear interactions between ones, which provide the relaxation of the field of rota- large-scale deformation defects called mesodefects, tional moments acting on a structural deformation ele- and the deformable material refers to open nonlinear ment from the ambient material. dynamic structurally stable, hierarchical dissipative, (2) The accommodation mechanisms of deforma- self-organizing information systems. tion are realized through secondary fluxes of defects An open system always suggests mass transfer dur- and can cause both the material rotation (multiple ing the deformation process, as well as energy and glide) and the crystallographic rotation of a structural information exchange through the fluxes of deforma- element of deformation (grain-boundary glide, grain tion defects. If the intensity of these fluxes is suffi- migration, or fragmentation). ciently high, ordered spaceÐtime structures arise. The (3) The natural relation between shear and rotation system captures the transfer fluxes even if they are leads to the fact that a translational rotational vortex somewhat structured, transforms and organizes them, (TRV) of different scale rather than shear is an elemen- and endows with its own spaceÐtime structure. In tary act of plastic deformation in mesomechanics. essence, this is the way in which a deformable material Within the framework of this approach, a deform- becomes structured, i.e., self-organizes. One can con- able polycrystalline material is treated on the mesos- clude that a deformable crystalline material adapts to copic level as an ensemble of interacting elements— loading conditions by varying the extent of its open- mesodefects, which have certain geometrical dimen- ness. As an open system, a material can quantitatively

416 OSTASHEV, SHEVCHENKO be characterized with the equation of entropy balance sented as a set U of structural levels and a set R of links (relations) between them. A pair of sets U and R is ds ds dse ----- = ------i + ------, (1) divided into lower level subsystems whose dynamics is dt dt dt defined by the interaction between mesodefects in the where the terms stand for the rate of entropy production representative volume. In this representation, systems in an open system, the rate of entropy production in the of the previous level are mesodefects for next-level sys- system due to internal irreversible processes, and the tems; i.e., the distribution of mesodefects over the lev- rate of entropy exchange with the environment, respec- els is mapped by the set of embedded sets: tively. ds /dt is always positive by definition, while i () () () dse /dt may be both positive and negative. U j R j U j + 1 R j + 1 U j + 2 R j + 2 SS. (2) The nonlinear behavior (nonlinearity) of a polycrystalline material in response to an external action shows Open nonlinear nonequilibrium systems, in which up in its integral properties and includes a variety of new structures may appear through the dissipation possible stationary states. A dynamic system is defined (scattering) of the flux entering the system and propa- as a process or object whose initial state is specified by gating in it, refer to self-organizing systems. Basic pre- a set of parameters and for which there always exists an mises to self-organization in dissipative systems are: an operator that describes the behavior of the system in energy or information flux from the outside due to the time and space. openness of the system; the nonequilibrium state of the Nonlinear dynamic systems differ from strictly system, enhancing fluctuations to the point where chaos determinate ones in that their behavior is sometimes creates order; and the presence of potentiality, which akin to random. Such a phenomenon is called dynamic allows stability exchange between the structures inside. chaos. Dynamic chaos results from internal interactions The emergence of dissipative structures during defor- in the system rather than as a consequence of random mation follows from general thermodynamic princi- external actions. In other words, it is due to the intrinsic ples. dynamics of a nonlinear system. The chaos (as an The representation of a deformable polycrystalline intrinsic property of the system) arises under almost material by an information system is based on the idea any deformation conditions. However, it may go unno- ticed if the range of parameters where it manifests itself that matter and energy are the products of the process is extremely narrow, the time interval of its manifesta- of information transmission and storage [3, 4]. Without tion is very extended, or it is shaded by more vigorous pretending to the strict definition of the term “informa- processes. tion,” we point to its three most general properties: information cannot exist without the interaction of A polycrystalline material as a structurally stable objects (in our case, deformation defects), information system is characterized by the stability of the parame- is not lost by any of them during the interaction, and the ters of its geometrical structure and by the invariance of interaction leads to the formation of a new complicated the functional relations between the dynamic and statis- system in which new information properties not found tical components of the system. Obviously, any in the initial systems may emerge. dynamic solution and any ordering in the structure is a fluctuation from the viewpoint of the statistical Thus, as applied to our problem, any interaction approach. Thus, the instability of dynamic states means between the deformation defects during which one of the stability of statistical ones. The reverse is also true: them acquires information and the other does not loose the stability of dynamic solutions means the absence of it is called information interaction. Information interac- fluctuation relaxation and, therefore, the instability of tions are asymmetric. We can distinguish conceptually the statistical component. coupled and transmitted informations. Coupled, or By a multilevel hierarchical system, we mean a set structural, information C(S) is defined by a set of statis- of structures with different levels of hierarchy. Correct tical data on the state of mesodefects for each structural structurization of nonlinear systems is one of the most level. Coupled information is a measure of the com- complicated problems in simulating plastic micros- plexity of the material structure under deformation: trains. It is based on the Leibniz principle of identifica- () () tion of state indistinguishability. As applied to our prob- CS = Ð f 1 log f 1 + f 2 log f 2 + f n log f n lem, this principle is stated as follows: upon originating (3) and interacting, deformation defects that define a struc- = Ð ∑ f k log f k, tural level must have properties so much alike that the defects are indistinguishable when being analytically where fk are the parameters of the frequency distribu- processed with a given method. Three scale levels are tion of the deformation mode for mesodefects at a conventionally identified: macrolevel, mesolevel, and structural level considered. microlevel. The hierarchical structure of the mesolevel can in its turn be subdivided into structural levels as fol- Transmitted information I(S) is represented as the lows. A deformable polycrystalline material is repre- map of lower level coupled information onto a structure

TECHNICAL PHYSICS Vol. 47 No. 4 2002

FORMALIZATION OF MODELS OF A DEFORMABLE POLYCRYSTALLINE MATERIAL 417 of higher hierarchical level and is defined as conven- linear, shear, and rotational modes of four discrete tional information strains under the conditions of the factorial experiment. Each of the periodograms characterizes the number of () [ ()ε ()ε ()ε ()ε IS = Ð ∑ f k log f k Ð ∑ f kω log f kω harmonics of certain wavelength and amplitude in the (4) spectrum of corresponding strain. The phase diagram is ()ε ()]ε Ð ∑ f kγ log f kγ , plotted in the M operatorÐN operator coordinates; the bifurcation diagram, in the plastic microstrainÐperiod where ∑ f (ε)(log f ε) is the coupled information at coordinates. The method of sliding frequency window k k was used to find bifurcations present in the phase dia- ε ε a structural level considered, and ∑ f kω ( )(log f kω ) gram and boundaries between domains of attraction for different attractors [6]. and ∑ f γ (ε)(log f γ ε) is the conventional informa- k k From the periodograms of the random functions tion due to the linear plastic deformation, which depend ε (x |e ) , the synchronization coefficient for the set of on the shear and rotational deformation components. ij i j i deformation defects associated with mesh sizes from 10 to 120 µm was calculated [7, 8]. EXPERIMENTAL BASIS Statistical processing included the search for the The experimental validation of the formalization plastic microstrain distribution, the calculation of the principles discussed in this study is based on the inves- autocorrelation functions, as well as tests for stationar- ε | tigation into the field of displacements of the nodes of ity and ergodicity of the random functions ij(xi ej)i. a dividing network applied directly on a deformable sample. A mesh of the network 10 µm in size is the geometrical image of a mesodefect on a lower structural DISCUSSION level. More stable structural levels are studied by Given a set of periodograms, the state of a polycrys- selecting a set of model materials (such as MO copper) talline material strained can be characterized on each of undergoing static deformation under different condi- the levels as the dynamic evolution of the frequency tions (grain size, rate of loading, operating length of the distribution. It is assumed that the material as a nonlin- sample, and rigidity of the tester). We designed a 24 fac- ear system has a spectrum of unrealized but realizable torial experiment to find samples with the highest plas- stationary states. The analysis of the periodograms of ticity, those with the highest strength, and those having the translational and rotational deformation modes a set of desired characteristics. The field of displace- shows that the frequency spectrum varies during the β ments is transformed into the field of distortions ij = deformation process: spectral harmonics appear and/or ε ω ij + ij by numerical differentiation. For the plane case, disappear, which is characteristic of only nonlinear sys- β all components of the tensor ij for the symmetric and tems (Fig. 1). Two, convergent and divergent, stages of antisymmetric components are calculated by the self-organization can be discriminated from the peri- Cauchy formulas. odograms. Analytically, the latter is associated with an increase in the number of harmonics in the spectrum; For a network mesh with the size i, the linear, shear, the former, with a decrease in this number. and rotational components of the plastic microstrains determined over an ensemble of mesodefects within the The joint analysis of the phase diagrams for linear, ε | length x are given by some random functions ij(xi ej)i, rotational, and shear modes in the material as a where i = 10, 20, …, 120 µm and j is the average strain. dynamic system shows that dissipative structures The distinctive feature of plastic microstrains on the develop in a certain sequence. They show up as bifurca- mesoscopic level is their oscillating character and non- tions of different codimensionality along with regular uniform spatial distribution. They can be considered as and chaotic attractors. A qualitative modification of the relatively periodic processes that satisfy the general phase portrait is represented by a bifurcation; an estab- definition [5] lished motion, by an attractor. In these terms, for a poly- ε []ε() ()≤ ε ε > i xTx+ Ð i x , 0. (5) On a given structural level, the process is described 0.916 by a number of dynamic components Yi(x) in form (1) and random quantities R(x) characterizing relations between mesodefects inside the level: 1 () () () 0.026 Hx = ∑Yi x + Rx. (6) n Using such a representation and the resonance Fig. 1. Periodogram of linear plastic microstrains. The search method we calculated the periodograms for the material is MO copper with an average strain of 2%.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 418 OSTASHEV, SHEVCHENKO

2.466 The coordinated activity of the whole complex system of interacting deformation defects on each of the structural levels is provided by synchronization, which means the most general case of establishing certain frequency relations between individual deformation defects and groups of defects. A quantitative measure N of synchronization is the synchronization coefficient (SC) [2]. The SC is calculated from the periodograms of linear plastic microstrains for a set of deformation defects identified with mesh sizes from 10 to 20 µm. From the variation of the SC, one can divide deformation defects into three groups and specify the hierar- –3.024 chy of structural levels: IV, the level of intragrain plas- –2.391 M 2.782 tic strains with defect sizes of 10 and 20 µm; III, the level of intergrain plastic strains with defect sizes of 40 Fig. 2. Phase diagram for the shear deformation mode. The µ material is MO copper with an average strain of 2%. and 60 m; and II, the level of interaction between groups of grains (80 and 120 µm) (Fig. 3). The classification by SC illustrates the spaceÐtime development of η, % plastic deformation. Synchronization, or the occur- 80 rence of coherence, is a mechanism of self-organiza- IV tion. 60 III Generally speaking, it is stable self-organization processes which are central to the concept of a dissipa- 40 tive structure in synergetics. The universality of this II concept implies the possibility of geometrical charac- 20 teristics being replaced by their information and numerical analogs. ∆ 0 2 4 6 8 , % Thus, the concepts of synergetics offer scope for a unified approach to formalizing all known models of Fig. 3. Graphical identification of the structural levels from plastic deformation in polycrystals on the mesoscopic the synchronization coefficient η vs. deformation ∆ dependence. The defect size is IV, 10Ð20; III, 40Ð60; and II, 80Ð level. 120 µm. REFERENCES crystalline material deformed under different condi- 1. V. E. Panin, Physical Mesotechnology and Computer- tions, one can indicate general rules and distinguishing Aided Design of Materials (Nauka, Novosibirsk, 1995), features. Vols. I, II. A material deformed can be represented as a struc- 2. V. A. Likhachev, V. E. Panin, and E. É. Zosimchuk, Cooperative Deformation Processes and Localization of turally stable system; i.e., the topological structure of Deformation (Naukova Dumka, Kiev, 1989). the phase diagram is stable in a range of strains up to some critical value at which one of the deformation 3. V. V. Rybin, High Plastic Strains and Metal Failure modes bifurcates. (Metallurgiya, Moscow, 1986). 4. S. Ya. Berkovich, Cellular Automata as a Model of Real- The stability of dissipative, for example, high-plas- ity: Search for New Representations of Physical and ticity structures is maintained owing to the strictly Informational Processes (Mosk. Gos. Univ., Moscow, dynamic shearÐrotationÐshear process, so that the sta- 1993). ble limit cycle persists on the phase diagram in one of 5. M. G. Serebryannikov and A. A. Pervozvanskiœ, Detec- the deformation modes (shear or rotational) at any tion of Latent Periodicities (Nauka, Moscow, 1965). instant (Fig. 2). 6. O. D. Shevchenko, V. V. Ostashev, and V. K. Fedyukin, Tr. Pskov. Politekh. Inst., No. 2, 142 (1998). From the topological structure of the phase dia- 7. V. V. Ostashev and O. D. Shevchenko, in Proceedings of grams on the different levels, one can judge the fulfill- the XXXV Workshop “Topical Problems of Strength,” ment of the scale invariance principle and, hence, the Pskov, 1999, Part II, p. 433. stability of the dissipative structures. 8. V. V. Ostashev and O. D. Shevchenko, Pis’ma Zh. Tekh. The considerable difference between the representa- Fiz. 24 (16), 50 (1998) [Tech. Phys. Lett. 24, 645 (1998)]. tive volumes under the conditions of high plasticity and high strength indicates that plastic deformation develops more steadily. Translated by D. Zhukhovitskiœ

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 419Ð425. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 46Ð52. Original Russian Text Copyright © 2002 by Smolyar, A. Eremin, V. Eremin.

SOLIDS

The Distribution of Liberated Energy and Injected Charge under Normal Incidence of a Fast Electron Beam on a Target V. A. Smolyar, A. V. Eremin, and V. V. Eremin Volgograd State Technical University, pr. Lenina 28, Volgograd, 400131 Russia e-mail: [email protected] Received July 23, 2001

Abstract—Formulas for the distributions of the liberated energy and injected charge are derived within the diffusion model of the kinetic equation for an electron beam normally incident on the target. The theory does not involve empirical adjustable parameters. The calculation of the liberated energy for the case of a planar directional electron source in an inﬁnite medium (C, Al, Sn, and Pb) correlates well with the Spencer data, obtained from the exact solution of the Bethe equation, which is also used as the basis in the diffusion model. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION where v is the electron velocity, s is the path length, N is the volume atom concentration, and σ is the differ- As follows from [1], the diffusion approximation of ential scattering cross section. the kinetic equation can be used as a basis for constructing a backscattering and penetration model for elec- The problem of an electron beam striking a target trons incident on a target. Unlike the starting Bethe dif- should be complemented by an electron source located fusion model [2] and semiempirical models based on it on the target surface and emitting electrons along the z [3], our theory does not involve adjustable parameters axis, which is aligned with the inner normal n to the needed for backscattering coefficients to reasonably surface: agree with associated experimental data. The diffusion S()r,,W s = S δ()δx ()δy ()δz ()δn Ð W ()s , (2) model has been extended up to final kinetic expressions 0 for the backscattering and penetration of electrons and where S0 is the emission rate of the source, that is, the supplements, along with other analytical approaches number of electrons emitted in unit time. intended to simplify the kinetic equation [4], the avail- Having penetrated into the target, the beam is scat- able software suites developed for Monte Carlo numer- tered and becomes isotropic with time. When the direc- ical simulation (such as the ETRAN program [5]) or for tional electron anisotropy becomes small, the diffusion numerical simulation with semiempirical algorithms approximation is applied. With this approximation, (the EDMULT program [6]). The model suggested in only the first two terms can be left in the expansion of [1] is a mathematically closed diffusion approximation f(r, W, t) in Legendre polynomials: of the kinetic equation. Therefore, the good fit of the analytical backscattering coefficients to the experimen- 1 f ()r,,W s = ------()f ()r, s + 3W ⋅ Jr(), s , (3) tal values allows us to hope that the analytical energy 4π and injected charge distributions will not require empirical adjustable parameters as well. where F()r, s = ∫ f ()r,,W s dW BASIC STATEMENTS OF THE DIFFUSION MODEL OF THE KINETIC EQUATION is the electron density and In the Bethe designations, the kinetic equation for Jr(), s = Wf ()r,,W s dW the electron density f(r, W, t), where r is the coordinate ∫ space, W covers motion directions, and t is a time is the electron flow density vector. instant uniquely related to a path s, has the form [2] Eventually, kinetic equation (1) is reduced to the dif- ∂f ∂f fusion equation ----- ==------ÐW ⋅ ∇f + ∫dW'Nσ()v, W' ⋅ W ∂s v∂t (1) ∂F λ()s ------= Ð------∆F (4) × []f ()r,,W t Ð f ()r,,W t , ∂s 3

420 SMOLYAR et al. and the equation for the diffusion flow As the total diffusion length, we take the mean displacement of stopped electrons emitted by the planar λ()s J = Ð------∇F. (5) source, which is inserted into an infinite scattering 3 medium. The displacement is counted normally to the Here, λ is the transport length defined by the integral source plane. The formula for the mean displacement of such electrons as a function of residual range was found π by Lewis [7]: 1 πΘ()Θ()()Θ σΘ(), v () λ------()= 2 ∫ sin d 1 Ð cos N s , (6) s R0 d + 1 0 〈〉zr() = ------()1 Ð r , d + 1 where Θ is the scattering angle. At the interface, there is no electron flow from the where R0 = s(E0, 0) is the initial range of incident elec- λ free space into the target other than incident flow (2), so trons, r = R(E)/R0 is the residual range, and d = R0 / . that With this expression, one can locate an isotropic f ()r,,W s = 0, W ⋅ n > 0, s > 0. (7) source that replaces the incident flow, i.e., find the total diffusion length: Condition (7) cannot be met in diffusion approxima- R tion (4); however, it can be satisfied having been direc- z ≡ 〈〉zr()= 0 = ------0 -. (11) tion-averaged in the target. Substituting (4) and (5) into d d + 1 (3) and averaging the electron flow inside the target over angles, we come to the boundary condition According to statement (ii) of the diffusion model, the path length ξ of diffusing electrons and the total 2λ∂F Fxy(),,,0 s = ------()xy,,,0 s . (8) path length of electrons that is counted from the point 3 ∂z where the electrons enter the target [formula (9)] relate as Thus, the basic statements of the diffusion model are as follows. ζ R0 Ð zd (), = ------sE0 E . (i) Kinetic equation (1) is replaced by Eqs. (4) and R0 (5), which suggests that the model is based on the diffusion approximation. Thus, the diffusion model of the problem of incident (ii) Incident flow (2) is changed to an isotropic electron flow of unit intensity (Eqs. (1) and (2)) is math- source, which is placed at a total diffusion depth z and ematically reduced to the equation of diffusion from d δ source: emits electrons with an energy E0 that have an initial range shorter by zd. ∂F λζ() ------= Ð ------∆F + δ()δx ()δy ()δζzzÐ (), (12) (iii) Additional condition (7) is imposed at the ∂ζ 3 d boundary with the free space. It implies the absence of the diffusion flux from the free space into the target. which has the boundary conditions 1 λζ()∂F ---Fxy(),,,0 ζ = ------()xy,,,0 ζ . (13) ANALYTICAL SOLUTION TO THE PROBLEM 2 3 ∂z OF AN ELECTRON BEAM INCIDENT ON A SEMI-INFINITE TARGET The problem stated by Eqs. (12) and (13) can be solved numerically with standard techniques. To ana- Starting kinetic equation (1) assumes continuous lytically solve this problem, we introduce a new vari- deceleration; therefore, the total path of an electron in able— the “age” of electrons, the target is obtained by integrating the reciprocal of the mean electron energy loss per unit path: ζ τζ() 1 λζ()ζ E = ---∫ d (14) 0 3 (), dE 0 sE0 E = ∫ ------〈〉ε()-. (9) E —and average the diffusion coefficient in boundary E condition (13). Then, diffusion equation (12) and By definition, boundary condition (13) will take the form ε max ∂ F ∆ δ()δ()δ()δτ() 〈〉ε() ε()ε, ε ------= F + x y zzÐ d , (15) E = ∫ winel. E d , (10) ∂τ 0 2 ∂F where w (E, ε) is the number of electron inelastic col- Fxy(),,,0 τ = --- 〈〉λ ------()xy,,,0 τ (16) inel. 3 ∂z lisions per unit length with an energy loss ε per colli- ε sion and max is the maximal energy loss. and have constant coefficients. Here, 〈λ〉 is age-aver-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE DISTRIBUTION OF LIBERATED ENERGY 421 aged transport length: b τ 0.6 0 〈〉λ 1 λτ()τ = τ---- ∫ d , 0.5 0 0 τ 0.4 where 0 is the age of the stopped electrons that is calculated by formula (14), where the upper limit of the 0.3 integral equals R0 Ð zd. The solution of problem (15), (16) yields the density 0.2 of the electrons with the age τ:  ()2 0.1 (),,,τ 1 zzÐ d Fxyz = ------expÐ------2 πτ 4τ 0 20406080100 ()2 Z zz+ d ()2τ () + expÐ------τ - Ð aaexp + az+ zd (17) 4 Fig. 1. (+) Analytical and (o) experimental [9] values of the backscattering coefficients for 40-keV electrons normally  2 2 2 × τ zz+ d 1 x + y incident on the target. erfca + ------------expÐ------, 2 τ  2 πτ 4τ where a = 2〈λ〉/3. and ξ(E) = |dE/dt| is the mean energy lost by electrons The density of the remaining electrons calculated by with an energy E per second. formula (17) and multiplied by the electron charge rep- With regard for the fact that resents the distribution of the injected charge in the tar- (),,, (),,,() get for the unit intensity of the incident beam. For the FxyzEdE= FxyztE dt, beam intensity S0 electrons per second, we obtain the we obtain density of the charge injected in unit time: E0 dQ() x,, y z dW ------= S eF() x,,, y z τ , (18) ------= FxyztE(),,,()dE, (21) dV 0 0 dV ∫ 0 where e is the electron charge and dV is an element of the volume. instead of (20), where F(x, y, z, t) is the solution of kinetic equation (1), which was integrated over all The backscattering coefficient is found by subtract- directions and has form (17) in the diffusion approxi- ing the density of the stopped electrons [formula (17)] mation. integrated over the target volume from unity. The formula for the backscattering coefficient thus obtained Until now, we have not used any scattering features. has the form For this reason, the aforesaid equally refers to electrons, positrons, and light ions. To obtain numerical z b = erfc ------d - results, it is necessary to specify the elastic and inelastic τ scattering cross sections in formula (6) for transport length 2 0 (19) and in formula (10) for energy loss per unit path length for 2 z given target material and energy of penetrating particles. Ð exp()a τ + az erfc a τ + ------d - . 0 d 0 τ 2 0 RESULTS AND DISCUSSION Now, we will calculate the density of the beam energy liberated in the target (for brevity, the liberated The target material and the electron energy enter energy density), which is designated as dW(x, y, z)/dV. into the diffusion model through the transport length By definition, the liberated energy density is equal to and mean energy loss per unit length, rather than the energy lost by all electrons in unit volume per sec- through the elastic and inelastic scattering cross second. Therefore, tions as in (1). The transport length and the mean energy loss per unit length were calculated with Spen- E0 cer standard formulas [8]: dW ------= FxyzE()ξ,,, ()E dE, (20) ∫ ()2 dV 1 π 2 ()T + 1 0 ------2= re NZ+ 1 ------λ()T T 2()T + 2 2 (22) where F(x, y, z, E)dE is the number of electrons with an energy in the interval (E Ð dE, E + dE) per unit volume × []ln()11/+ η Ð 1/() 1 + η ,

TECHNICAL PHYSICS Vol. 47 No. 4 2002 422 SMOLYAR et al.

dW/d(z/R0) 4 (a) (b)

2 1 1 1 2 2 0 4 (c) (d) 1 3 1

2 2 2

0 0.5 1.0 0 0.5 1.0 z/R0

Fig. 2. Liberated energy distribution for a planar directional source of electrons with an energy E0 placed into an inﬁnite scattering medium (dashed lines, exact solutions [12]; continuous lines 1, diffusion model) and into a semi-inﬁnite target (continuous curves 2, diffusion model). The linear energy density dW/d(z/R0), corresponding to the unit intensity of the source, is converted to dimensional units via the factor E0/R0. (a) C, (b) Al, (c) Sn, and (d) Pb. E0 = (a, b) 25, (c) 50, and (d) 100 keV.

()2 electrons on the target with experimental data obtained 〈〉ε() π 2 T + 1 []() T = 4 re NZ------()ln1.166T/IZ , (23) in [9]. The calculations were made for solid elements TT+ 2 from lithium to uranium, whose densities are given in where T = E/mc2 is the energy in units of the rest energy [10]. From Fig. 1, it follows that the diffusion model gives fairly exact values of the backscattering coeffi- of an electron, re is the classical radius of an electron, N is the number of atoms in unit volume, cient. However, it cannot confirm the dependence of the backscattering coefficient on the atomic number of the η = 1.75× 10Ð5Z2/3/[]TT()+ 2 target, which was found in [11]. is the parameter of nucleus screening by atomic elec- The discrepancy between the model calculations of trons, and the liberated energy and the actual values may be checked with the US NBS tables [12], which list the 13.6Z, Z < 10 Spencer data [8] for the liberated energy density distri- ()  IZ = Ð1.19 butions in graphite, aluminum, tin, and lead. This com- ()9.76+ 58.8Z Z, Z ≥ 10 parison is of interest also due to the fact that the Bethe kinetic equation is used as basic in the diffusion model is the mean ionization potential of target atoms (in elec- suggested. Spencer proceeds from Bethe kinetic equa- tronvolts). tion (1) and solves the problem of energy liberation for Formulas (22) and (23) are valid for energies from a monoenergetic electron source placed into an infinite I(Z) to 1 MeV; i.e., in the interval where energy losses scattering medium. Its theory does not involve adjust- are defined by the ionization and excitation of atoms able parameters, and the energy liberation is expressed and radiation losses can be neglected. through the double spatial-range moments of the coef- Figure 1 compares the backscattering coefficients ficients of the f(r, W, t) expansion in Legendre polyno- calculated by formula (19) for the normal incidence of mials. Spencer directly looks for the liberated energy

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE DISTRIBUTION OF LIBERATED ENERGY 423

dW/dV dQ/dV 10 2.0 (a) 1 (b) 1 9 1.8 2 8 1.6 3 7 1.4

6 1.2

5 2 1.0 4 4 0.8

3 0.6 3 2 0.4 4 1 0.2 5 5 0 0.5 1.0 1.5 0 0.5 1.0 1.5 z/R0

Fig. 3. Distributions of the (a) energy density and (b) injected charge in a semi-infinite aluminum target for a normally incident sharply focused electron beam with an energy E0 = 25 keV. The distance r from the beam axis is expressed in units of the initial range R0: r/R0 = (1) 0.05, (2) 0.1, (3) 0.25, (4) 0.4, and (5) 0.7. The energy density dW/dV is converted to dimensional units via the 3 3 factor E0/R0 . The injected charge density dQ/dV is converted to dimensional units via the factor e/R0 , where e is the electron charge. Both values correspond to the unit intensity of the source. density, not the solution of the kinetic equation. Our of the energy liberated from a wide monoenergetic calculations of the stopping power, total trajectory electron beam, is compared with the Spencer calcula- range, and elastic collision transport length performed tions [12]. Work [12] tabulates the distribution of the for aluminum at an electron energy of 25 keV using the linear density of the energy liberated when a planar cross sections available from the database of the Ioffe electron source placed into an infinite medium unidi- Physicotechnical Institute, Russian Academy of Sci- rectionally emits monoenergetic electrons normally to ences [13], virtually coincide with the values listed in the source plane. The discrepancy between our calcula- the Spencer tables [12]. tions for a semi-infinite target and the tabulated energy In Fig. 2, the liberated energy density calculated by values for an infinite medium can partially be explained (21) and integrated over the coordinates transverse to by the fact that the electrons penetrating into an infinite the z axis, that is, the distribution of the linear density medium experience multiple scattering and may find

Table 〈 〉 Element E0, keV bbE Eb /E0 W/E0 Ws/E0 C 25 0.073 0.051 0.703 0.949 0.972 Al 25 0.159 0.118 0.741 0.882 0.923 Sn 50 0.415 0.343 0.824 0.656 0.787 Pb 100 0.512 0.437 0.852 0.563 0.725 Note: b, backscattering coefﬁcient; b , coefﬁcient of backward dissipated energy (the ratio of the backscattered electron energy to the E 〈 〉 energy of the incident electron beam); Eb /E0, ratio of the mean energy of a backscattered electron to the energy of an incident electron; W/E0, ratio of the energy absorbed by the target per incident electron to the energy of an incident electron (this work); and Ws/E0, the same taken from the Spencer tables [12].

TECHNICAL PHYSICS Vol. 47 No. 4 2002 424 SMOLYAR et al.

dQ/dV

1.5

1.0

0.5

0 1.0 0.8 1.0 0.8 0.6 0.6 0.4 z/R0 0.4 r/R 0.2 0.2 0 0

Fig. 4. Injected charge density distribution in a semi-infinite aluminum target for a normally incident sharply focused electron beam with an energy E0 = 25 keV. The z axis is directed normally to the surface from the point of beam incidence. r is the distance to the z axis. The injected charge density dQ/dV, corresponding to the unit intensity of the source, is converted to dimensional units via 3 the factor e/R0 , where e is the electron charge and R0 is the initial electron range. themselves on the left of the source, which emits to the this depth. Here, the discrepancy between the diffusion right along the z axis, and then again pass to the right model and physical reality shows up most vividly. If the half of the space. Such transitions may occur repeat- beam is wide and planar, the discrepancy decreases, yet edly; consequently, in an infinite medium, the energy remaining appreciable, as follows from the curves in density on the right of the source plane must be larger Figs. 2 and 3. The injected charge distribution is, in than that near a semi-infinite target subjected to elec- essence, the distribution of the stopped electrons. That tron bombardment. The diffusion model does show that is why the location and the geometry of the source, be the energy density near the surface bombarded is lower it an incident beam or an isotropic source located at the than that in an infinite medium, as follows from Fig. 2. diffusion depth, is of minor significance. Figure 4 dem- The table shows the backscattering coefficients and the onstrates the detailed three-dimensional injected fractions of the backward dissipated energy obtained by charge distribution obtained by formula (18). It is seen integrating over the space of energy density and that this distribution is much smoother than the energy injected charge (see Fig. 2). distribution. The larger the atomic number of an element and the smaller the electron energy compared with the initial Thus, the model suggested in this work is a mathe- energy, the smaller the discrepancy between the diffu- matically closed diffusion approximation of the kinetic sion model and the rigorous solutions. This feature of equation that describes the problem of an electron beam the diffusion model is also observed in the liberated incident on a target. The model does not require adjust- energy distributions shown in Fig. 2. able parameters to bring the backscattering coefficients, It can be expected that the injected charge distribu- the liberated energy distribution, and, apparently, the tion simulated within the diffusion model must be injected charge distribution into coincidence to experi- closer to reality than the liberated energy distribution, mental data. since the latter characteristic is more integral. As follows from Fig. 3, the distributions of the injected charge and liberated energy for the case of a sharply focused ACKNOWLEDGMENTS beam incident normally to the target surface (formulas (17) and (18)) greatly diverge. A sharp peak of the The authors thank the developers of the electron energy density at the depth of total diffusion is due to scattering archive (Ioffe Physicotechnical Institute, the fact that, according to the diffusion model, an inci- Russian Academy of Sciences) for submission of the dent beam is replaced by an isotropic electron source at database.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE DISTRIBUTION OF LIBERATED ENERGY 425

REFERENCES 9. A. Ya. Vyatskin, A. N. Kabanov, B. N. Smirnov, and V. V. Trunev, Radiotekh. Élektron. (Moscow) 21, 895 1. V. A. Smolyar and A. V. Eremin, Radiotekh. Élektron. (1976). (Moscow) 46 (5), 599 (2001). 2. H. Bethe, M. E. Rose, and L. P. Smith, Proc. Am. Philos. 10. Tables of Physical Quantities: Handbook, Ed. by Soc. 78 (4), 573 (1938). I. K. Kikoin (Atomizdat, Moscow, 1976). 3. K. Kanaya and S. Okayama, J. Appl. Phys. 5, 43 (1972). 11. Yu. D. Kornyushkin, Zh. Tekh. Fiz. 69 (6), 40 (1999) [Tech. Phys. 44, 645 (1999)]. 4. L. A. Bakaleœnikov, S. G. Konnikov, K. Yu. Pogrebitskiœ, et al., Zh. Tekh. Fiz. 64 (4), 9 (1994) [Tech. Phys. 39, 354 12. L. V. Spencer, Energy Dissipation of Fast Electrons (1994)]. (National Bureau of Standards, Washington, 1959), 5. S. M. Seitzer, Appl. Radiat. Isot. 42, 917 (1991). Monograph 1. 6. T. Tabata et al., Radiat. Phys. Chem. 35, 821 (1990). 13. http://www.ioffe.rssi.ru/ES. 7. H. W. Lewis, Phys. Rev. 78, 526 (1950). 8. L. V. Spencer, Phys. Rev. 98, 1597 (1955). Translated by V. Isaakyan

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SOLID-STATE ELECTRONICS

Conditions for the Production of a Single Conducting Nanostructure by Electroforming V. M. Mordvintsev and S. E. Kudryavtsev Institute of Microelectronics and Informatics, Russian Academy of Sciences, ul. Universitetskaya 21, Yaroslavl, 150007 Russia e-mail: [email protected] Received January 22, 2001

Abstract—The process of electroforming (the production of a carboniferous conducting medium when the current flows through an organic material under a high electric field) in open sandwichlike structures with an insulating gap several tens of nanometers in width is considered. It is shown experimentally that there are factors that both favor (external ballast resistor) and prevent (local spreading resistance and the presence of the initial conductivity in the insulating gap) the production of a single conducting element between the electrodes. A simple model of the process in terms of the equivalent electric circuit is proposed. The model helps to find the trade- off between these factors and to construct an IÐV diagram, which exhibits a region within which a single conducting nanostructure can be electroformed. An expression that relates the minimum permissible resistance of the nanostructure to its geometric parameters is derived. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION memory. In this structure, electroforming greatly differs in comparison with that in the AlÐAl2O3ÐW struc- Electroforming in metalÐinsulatorÐmetal (MIM) ture. Here, the carboniferous conducting medium tends structures [1, 2] has been studied for a long time. The to form along the entire perimeter of the insulating gap analysis of a huge body of experimental data (including simultaneously. This makes it difficult to form the sin- those obtained recently) shows the following [3]. Elec- gle conducting nanostructure. The basic factors respon- troforming can be considered as a self-organization sible for such behavior are the high resistivity of the sil- process in the nanometer insulating gap in the carbon- icon electrode and the appreciable initial conductivity iferous conducting medium between the metallic elec- of the insulating gap surface. In this paper, we discuss trodes of the MIM structure. The carboniferous mechanisms of the processes and study conditions medium is produced in an organic material coming under which it is possible to fabricate the single con- from the outside (from the gas phase or, e.g., from a ducting nanostructures even in the presence of these photoresist film deposited on the structure) whose con- unfavorable factors. ductivity changes when the organic molecules are destroyed by an electron impact as a result of current passage. EXPERIMENTAL RESULTS AND MECHANISMS In [4], a so-called MIM nanodiode with a carbonif- OF THE PROCESSES erous active medium was designed. It is based on an It was shown [5] that the presence of a series ballast open sandwichlike MDM (metalÐdielectricÐmetal) resistor Rb during the electroforming process is favor- structure where an insulating gap exposed to an organic able to the formation of the single conducting element material is formed at the end face of the dielectric film (nanostructure) in the insulating gap of an AlÐAl2O3ÐW with a thickness of several tens of nanometers by open sandwichlike structure where the dielectric is sev- locally etching two upper layers of a conventional sand- eral tens of nanometers wide. The physical mechanism wichlike MDM structure. Electroforming in open sand- of the process is as follows. A high-value (no less than wichlike AlÐAl2O3ÐW structures with an oxide thick- several megaohms) series-connected resistance pro- ness of 20Ð40 nm [5] showed that, under certain condi- vides strong degenerative feedback between the current tions, the carboniferous conducting medium forms only through and voltage U across the structure when the at one site of the end face perimeter over a length com- supply voltage V gradually increases. When the current parable to the insulating gap width. This fact indicates increases because of a rise in the conductivity of the that a single conducting structure with nanometer sizes insulating gap, the voltage across the gap drastically in all three dimensions arises. drops. As the field strength in the gap increases, some From the practical point of view, the design of the (first) nanotip on the cathode surface (tungsten) starts to MIM nanodiode in the form of an open sandwichlike emit, leading to the formation of the carboniferous con- SiÐSiO2ÐW structure [6] is promising for nonvolatile ducting medium. As soon as this takes place, conditions

CONDITIONS FOR THE PRODUCTION 427 for emission from other regions worsen, causing the ity is too high, it is impossible to produce a single con- single conducting element to appear. Its growth is ducting nanostructure even in structures based on low- accompanied with an increase in the conductivity, cur- resistivity silicon (with a resistivity of 0.1 Ω cm), as fol- rent, and, hence, voltage drop across the ballast resis- lows from experiments. tance. Such a process lasts until the voltage U across the MIM structure (at a given Rb) decreases to the threshold The presence of the appreciable current before electroforming allows one to determine the threshold volt- value Uth (close to 3 V). The formation of the carboniferous conducting medium under these conditions age Uth, which initiates the formation of the carbonifer- becomes impossible (see below), and the growth of the ous conducting medium from organic molecules. This nanostructure, as well as the increase in its conductiv- voltage switches the electroformed structure from the ity, stop. When Rb = 0, a local electric breakdown devel- low- to the high-resistivity state. Usually, this voltage is ops in the AlÐAl2O3ÐW structure. about 3 V [2]. Unlike AlÐAl2O3ÐW structures [5], here it is not associated with the voltage required to induce In the open sandwichlike SiÐSiO2ÐW structures made on p-type boron-doped silicon wafers with differ- the current inside the insulating gap. The time depen- ent resistivities, electroforming shows a number of fea- dences of the current at fixed voltages U across the SiÐ tures [6]. First, the gap has an appreciable initial (i.e., SiO2ÐW structure are shown in Fig. 1. The smooth ini- existing before the electroforming) conductivity, which tial current is observed when U is less than some is proportional to the structure perimeter. This conduc- threshold value Usf. At U > Usf, specific current jumps tivity is presumably associated with the chemical etch- with the gradual increase in its mean value are seen, ing of silicon oxide (setup and fabrication technology which means the beginning of electroforming. The pro- are detailed in [6]). Second, the forming can be carried cess starts with a time delay, which is the smaller, the out without the ballast resistance. In this case, no elec- higher U. The value of Usf is in the range between 4 and trical breakdown is observed, unlike the AlÐAl2O3ÐW 5 V and is almost independent of the SiO2 film thick- structures, and the current in the highly conducting ness in the range from 10 to 40 nm. The fact that it is state is several orders of magnitude larger. This means somewhat higher than the typical value 3 V implies a that the electroforming takes place over the entire gap; voltage drop across some structure elements. The value i.e., the conducting medium forms along the whole of Usf is also independent of the radius of curvature of exposed perimeter of the structure. the tungsten electrode edge (the radius can be varied by These features can be explained by the high resistiv- varying tungsten film etching conditions), which seems ity of the silicon. Because of this, a large local resistance Rs to the carrier flow toward the silicon electrode is series-connected to every conducting nanostructure 0 160 320 480 t, s arising in the insulating gap. According to estimates Ω [6], Rs is on the order of 1 M for silicon with a resis- 0 (a) tivity of 10 Ω cm; therefore, the spreading resistance may substantially influence all the processes. In particular, it prevents local electrical breakdowns. Moreover, 0 the voltage drop across the local spreading resistance (b) decreases the voltage across a forming conducting element, leaving the total voltage across the gap 0 unchanged. The growth of the corresponding nanostructure slows down and ceases when the threshold voltage is attained. At the same time, new conducting (c) elements may originate in other regions of the gap. In other words, the local resistance series-connected to a conducting nanostructure stimulates electroforming over the entire gap (parallel electroforming) and sup- presses the formation of a single nanostructure. The validity of this statement was confirmed in experiments with SiÐSiO ÐW structures based on high-resistivity 2 1 µA silicon (with a resistivity of 10 Ω cm). All attempts to produce a single conducting nanostructure under these conditions failed: in all the cases, parallel forming was J developed. The initial conductivity distributed over the insulat- Fig. 1. Time dependences of the current through open sandwichlike SiÐSiO2–W structures at fixed voltages. The struc- ing gap also favors parallel electroforming, because the tures are covered by a resist layer. The resistivity of the sil- Ω electron flows initiate the formation of the conducting icon is 10 cm, and the SiO2 layer thickness is 11.4 nm. medium at many sites simultaneously. If this conductiv- U = (a) 3.6, (b) 4.0, and (c) 4.4 V.

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428 MORDVINTSEV, KUDRYAVTSEV

E in an appropriate energy position. It can tunnel to the (a) autodecay repulsive term of the molecule (Fig. 2b) if the voltage U applied to the nanometer gap of width H is somewhat higher than the potential barrier Φ (or V0 ϕ somewhat lower than the work function 0). Since a σ typical value of the work function is 4.5Ð5.0 eV, the ϕ 0 Φ threshold voltage Usf must be close to the physical 1 ϕ parameter Uth, which is somewhat lower than 0 and is U = 0 usually about 3 V. We will find the same value of Uth F when electrons tunnel through a lower barrier and then are accelerated by the electric field up to a necessary 22kinetic energy. The same mechanism apparently defines the threshold voltage for the modification of the solid surface V0 (i.e., for molecular bond destruction) in a scanning tun- E (b) neling microscope. Usually, this voltage is about 3Ð4 V [10Ð12]. This mechanism is also responsible for the Φ specific peak appearing in the quasi-static IÐV characteristics of electroformed MIM structures near 3Ð4 V [2]. The appearance of the peak at a decreased voltage F σ can be explained by the fact that the formation of the C carboniferous conducting medium (hence, the decrease ϕ U 0 in the insulating gap width and the change in the dielectric composition in the gap) takes place only at higher- than-threshold voltages.

Note that the threshold voltage Uth defined by the H A mechanism described above could not be found at the early stage of electroforming in the AlÐAl2O3ÐW structures, because the initial conductivity here is lacking and the formation of the carboniferous conducting Fig. 2. Simplified potential diagrams of the MIM structure medium is limited by the current, which passes only at illustrating the mechanism responsible for the appearance the voltage (electric field) providing electron emission of the threshold voltage Usf (3Ð4 V) during the electroform- from the cathode. The value of Usf varied from 5 to ing. The qualitative dependence of the cross section of dis- 15 V [5] depending on the tungsten electrode geometry. sociative electron attachment to an adsorbed molecule on the electron kinetic energy is shown. 1, adsorbed organic In the SiÐSiO2ÐW structures, the current passes over insulator; 2, metal; C, cathode; and A, anode. the open face end even at very low voltages and does not control the formation of the conducting medium. Here, the process is limited by low electron energies. At to be quite natural unlike the forming in the AlÐAl2O3Ð higher voltages U, the value of Usf (close to Uth) can W structures. reliably be detected. Thus, we can argue that the initial smooth current is Thus, we can suggest that the primary effect of the the prerequisite for the electroforming process (i.e., for silicon substrate resistivity ρ on the electroforming pro- the formation of the carboniferous conducting cess in the SiÐSiO2ÐW structures is that it governs the medium), which starts when the voltage U exceeds the spreading resistance. As ρ decreases, the situation threshold value Usf. The aforesaid is in good agreement comes close to that in the MIM structures. Experimen- with the model discussed in [7]. The carboniferous con- tal data completely confirm this statement. Figure 3 ducting medium forms as a result of the destruction of shows the currentÐvoltage characteristics taken in the organic molecules in the flow of slow electrons due to process of electroforming for two structures with close dissociative electron attachment. Having captured an thicknesses of silicon oxide and greatly differing (by electron, the molecule passes to the autodecay repulsive two orders of magnitude) resistivities of the silicon term of a negative ion [8]. The effective cross section σ wafers. As is known [7], a decrease in the current with of this process resonantly depends on the electron increasing voltage is due to the partial burnout of the kinetic energy E (Fig. 2) and has a peak, which can be carboniferous conducting medium. In the low-resistiv- very close to E = 0 [9]. For the molecules adsorbed on ity silicon, the averaged current maximum is one order the solid surface, which occurs in our system, the curve of magnitude lower and shifts to lower voltages and the σ(E) is somewhat shifted downward from the vacuum current essentially oscillates. The distinctions from the level V0, as qualitatively shown in Fig. 2a. The dissoci- case of the high-resistivity silicon are caused just by the ation of the molecule is possible only if the electron is decrease in the spreading resistance, because of which

TECHNICAL PHYSICS Vol. 47 No. 4 2002 CONDITIONS FOR THE PRODUCTION 429 parallel forming is weakly pronounced: separate cur- J, µA rent peaks are the results of the forming in one or sev- (a) eral regions of the insulating gap where the conducting 500 medium forms and burns out before it forms in other 400 regions. Therefore, the current amplitudes are separated in time and do not add up as in Fig. 3a. 300 For low-resistivity silicon, the conditions under 200 which a single conducting nanostructure forms [6], as 100 in the AlÐAl2O3ÐW MIM structures, can readily be provided by using a high-value stair-step ballast resistor. 0 800 MODEL (b) 700 Thus, there are several factors variously influencing the formation of the single conducting structure during 600 electroforming in real SiÐSiO2ÐW structures. Condi- 200 tions under which they are counterbalanced are of great interest. Strictly speaking, the formation of the con- 100 ducting medium and the increase in the current should be described by a system of differential equations that 062 4 8 10 U, V constitute a difficult mathematical problem. However, it is possible to describe such objects in terms of the Fig. 3. Quasi-static currentÐvoltage characteristics of open simplified equivalent circuit, which takes into account sandwichlike SiÐSiO2ÐW structures during electroforming basic features of the process. for the first stage of voltage increase without the ballast resistance. The structures are covered by a resist layer. The Electroforming is usually carried out under quasi- exposed perimeter length is 64 µm. The rate of change of steady-state conditions, i.e., when the voltage varies voltage is 0.05 V/s. The resistivity of the silicon is (a) 10 and Ω slowly. Therefore, the averaged (not instantaneous) (b) 0.1 cm. The SiO2 layer thickness is (a) 16 and state of the conducting medium inside the insulating (b) 17 nm. gap is of interest. Then, as the first approximation, one can use the simplest equivalent circuit, including resis- V tors only. Such an equivalent circuit for a real metalÐ insulatorÐsemiconductor structure is shown in Fig. 4. l J Rb 1 34 The external ballast resistance Rb is series-connected to the entire structure. The local spreading resistance Rs is 5 series-connected to each of the regions of insulating gap 3 where the carboniferous conducting medium may a form (nanotips 4 on cathode surface 1). The initial con- H R0 ductivity distributed over the insulating gap is simulated by the resistance R0. The spreading resistance Rs Rs Rs series-connected to R0 is ignored because the current spreads over a large area. The supply voltage V increases gradually (which is typical of electroform- 2 ing), and the voltage U across the insulating gap is generally a part of the total voltage V. Fig. 4. Insulating gap of the metalÐinsulatorÐsemiconductor Let us assume first that the initial conductivity of the structure and the equivalent electric circuit during the elec- structure is negligible; i.e., R approaches infinity. It is troforming. 1, metallic cathode; 2, semiconductor anode; 0 3, insulating gap; 4, nanospikes on cathode surface; and clear that the origination (field emission from a ran- 5, conducting nanostructure (carboniferous conducting domly appeared nanospike and the subsequent forma- medium). tion of the carboniferous conducting medium) of the first conducting element with increasing V does not at which other conducting elements do not originate as depend on Rb and Rs values and is caused by the nonuniform emission from the insulating gap perimeter. yet. Since, in real experiments, the emission is always non- The appearance of the first conducting element is uniform, single nanostructure 5 begins to grow. How- characterized by the growth voltage V = Ug1 = Ug cor- ever, this is accompanied by an increase in the conduc- responding to the emission initiated at the “weakest” tivity and in the current through the structure. As a site. As the nanostructure grows and the current result, the voltage across the resistances drops. There- increases, the voltage V is redistributed between the fore, it is of interest to find the maximum conductivity insulating gap and the resistances Rb and Rs. If we have

TECHNICAL PHYSICS Vol. 47 No. 4 2002 430 MORDVINTSEV, KUDRYAVTSEV

J J III Jcr II

Jcr IV 1

J0 I 2

Uth Ug Vcr V Uth[R0/(R0 + Rs)] Ug Vcr V Fig. 5. CurrentÐvoltage diagram showing the region (dashed) of existence of the single conducting nanostructure in the case of the zero initial conductivity. Fig. 6. The same as in Fig. 5 in the general case.

only the first nanostructure, the total current J flows ratio Ucr/Jcr and is given by through it, so that the voltage UON across the nanostruc- Uth ture is given by rcr = ------Rs. (7) Ug Ð Uth U = VJRÐ ()+ R . (1) ON b s This resistance depends neither on the external volt- Since the current through all other regions of the age V nor on the external ballast resistance Rb. Since the insulating gap is absent, the voltage UOFF across them is conductivity of the nanostructure is defined (on U = VJRÐ . (2) assumption that the resistivity of the conducting OFF b medium is constant) by its geometric characteristics, If UOFF < Ugi, where i = 2, 3, …, and Ugi are the such as h = H Ð a and l (Fig. 4), condition (7) is the lim- growth voltages for the second and subsequent ele- itation imposed on the sizes a and l of the single con- ments (the voltages form a monotonically increasing ducting nanostructure. The slopes of lines 1 and 2 in sequence), other conducting elements do not arise. Fig. 5 are 1/(Rb + Rs) and 1/Rb, respectively. Hence, in Assuming that the conditions for the emission initiation the special case Rs = 0, corresponding to the MIM struc- are everywhere similar, we can take the stronger ine- ture, the lines in the diagram become parallel, the criti- quality cal point disappears, and the conductivity of the single U < U , (3) nanostructure can reach any value. OFF g In region II (Fig. 5), other conducting elements can- i.e., assume that the process is characterized by one not arise but, in contrast to region I, the growth of the value Ug = Ug1, which is the least in the sequence. first nanostructure is also impossible. In region III, The originated nanostructure can grow through the other conducting elements can arise and the growth of extension of the carboniferous conducting medium if the first nanostructure is impossible. Finally, in region IV, > both processes can take place. UON Uth. (4) When the structure has an appreciable initial con- The diagram obtained from Eqs. (1)Ð(4) is shown in ductivity, i.e., R0 is finite, we must consider two addi- Fig. 5. The single conducting nanostructure can exist tional effects. First, the growth voltage Ug decreases only in the dashed region. The coordinates of the criti- because the current distributed over the insulating gap cal point (point of intersection) are obtained from the initiates the formation of the carboniferous conducting solution of Eqs. (1) and (2): medium. Second, R0 inhibits other processes. The first effect facilitates the formation of all the conducting ele- ()Rb V cr = Ug + Ug Ð Uth ----- , (5) ments, while the second one hampers the appearance of Rs even the first of them. U Ð U It should be emphasized that the growth voltage Ug J = ------g th-. (6) cr R is not a physical constant. It depends on the insulating s gap width, cathode geometry on the nanometer scale, Let for given Rb and Rs the maximum voltage V at distribution of the initial current along the insulating which the single conducting nanostructure still exists gap, and rate of rise of the voltage. However, Ug can be Vcr. The conductivity of the nanostructure can grow easily be found experimentally from the current jump until the current exceeds Jcr. At the critical point, the when voltage gradually increases; therefore, it has a resistance of the nanostructure is determined as the certain value for a specific structure.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 CONDITIONS FOR THE PRODUCTION 431

Prior to the origination of the first conducting ele- Our analytical results are in good agreement with ment, the total current flows through R0. Since it arises the experimental data presented above. The single con- when the voltage across the gap reaches the value Ug, ducting nanostructure forms if the electroforming the following inequality holds: parameters remain within the dashed region in Fig. 6. U J > ------g = J . (8) CONCLUSION R 0 0 Our experimental results have shown that there are Once the first conducting element has arisen, the factors both favoring (external ballast resistance) and following equations become valid: preventing (local spreading resistance and initial conductivity) the production of a single conducting ele- VJR= b ++UJ1 Rs, (9) ment by electroforming open sandwichlike structures UJ+ 1 Rs ==J2 R0, JJ1 + J2. with an insulating gap several tens of nanometers in width. A simple model using the equivalent electric cir- Here, J1 is the current through the conducting nanostruc- cuit allows us to find the trade-off between these factors ture in the “on” state, J2 is the current through the resis- and construct a currentÐvoltage diagram that has a tance R0, and U is the voltage across the insulating gap. region within which the single conducting nanostruc- A solution of system (9) in view of inequality (4), ture can be reliably electroformed. The conductivity, as which is the condition for the growth of the nanostruc- well as the sizes of the conducting element, are basi- ture, yields the inequality cally limited from above. The expression for the R extreme value of the nanostructure resistance was < 1 b derived. J ------V Ð ------Uth . (10) Rs R0 Rs + R0 Rb + ------Rs + R0 ACKNOWLEDGMENTS Finally, the condition when none of the conducting This work was supported by the State Scientific and elements but the first one can originate is derived from Technical Subprogram “Promising Technologies and Eqs. (2) and (3) and has, as before, the form of the ine- Devices for Micro- and Nanoelectronics” of the Minis- quality try of Science of the Russian Federation. 1 1 J > ----- V Ð ----- U . (11) REFERENCES R R g b b 1. G. Dearnley, A. M. Stoneham, and D. V. Morgan, Rep. For the growth of the single conducting nanostruc- Prog. Phys. 33, 1129 (1970). ture to proceed, inequalities (8), (10), and (11) must be 2. H. Pagnia and N. Sotnik, Phys. Status Solidi A 108 (1), fulfilled simultaneously (dashed region in Fig. 6). This 11 (1988). region is similar to that in Fig. 5 except for the fact that 3. K. A. Valiev, V. L. Levin, and V. M. Mordvintsev, Zh. the current J0 is its lower boundary. The critical point, Tekh. Fiz. 67 (11), 39 (1997) [Tech. Phys. 42, 1275 where straight lines (10) and (11) intersect, has the (1997)]. coordinates 4. K. A. Valiev, S. E. Kudryavtsev, V. L. Levin, et al., Mik- roélektronika 26, 3 (1997). Rb Rs ++Rb R0 Rs R0 Rb Ucr = ------Ug Ð ----- Uth, (12) 5. V. M. Mordvintsev, S. E. Kudryavtsev, and V. L. Levin, Rs R0 Rs Zh. Tekh. Fiz. 68 (11), 85 (1998) [Tech. Phys. 43, 1350 (1998)]. Rs + R0 1 Jcr = ------Ug Ð ----- Uth. (13) 6. V. M. Mordvintsev and S. E. Kudryavtsev, Mikroélek- Rs R0 Rs tronika (in press). 7. V. M. Mordvintsev and V. L. Levin, Zh. Tekh. Fiz. 64 As before, the critical current Jcr depends neither on V nor on R . Hence, if the initial conductivity is so large (12), 88 (1994) [Tech. Phys. 39, 1249 (1994)]. b 8. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 147, that J0 > Jcr, the single nanostructure cannot form in 459 (1985) [Sov. Phys. Usp. 28, 956 (1985)]. principle for any values of the external parameters. The 9. J. P. Johnson, L. G. Christophorou, and J. G. Carter, minimum resistance of the single conducting nano- J. Chem. Phys. 67, 2196 (1977). structure, which corresponds to its maximum size, is 10. P. N. Luskinovich, V. D. Frolov, A. E. Shavykin, et al., generally given by the expression Pis’ma Zh. Éksp. Teor. Fiz. 62, 868 (1995) [JETP Lett. Rs 62, 881 (1995)]. Uth + Ug----- 11. E. F. Golov, G. M. Mikhaœlov, A. N. Red’kin, et al., Mik- R0 rcr = ------Rs. roélektronika 27, 97 (1998). Ug Ð Uth 12. R. M. Penner, M. J. Heben, and N. S. Lewis, Appl. Phys. This expression and Eqs. (12) and (13) turn to cor- Lett. 58, 1389 (1991). responding formulas for the case when the initial conductivity is absent if R0 approaches infinity. Translated by M. Astrov

TECHNICAL PHYSICS Vol. 47 No. 4 2002

SOLID-STATE ELECTRONICS

Negative Crystals of Silicon Carbide V. A. Karachinov Ya. Mudryœ State University, Sankt-Peterburgskaya ul. 41, Novgorod, 173003 Russia Received May 10, 2001

Abstract—Systems of negative silicon carbide crystals are classiﬁed and studied by experimental methods. The crystal structure and morphology forming during growth, etching, and erosion are discussed. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION RESULTS AND DISCUSSION The problem of implementing quantum-size effects By negative crystal (NC), one means a faceted cav- resulting in nanostructures with a single electronic sys- ity inside a positive crystal that is usually filled with a tem, as well as in nanocomposites, stimulates the mother medium (solution or gas). NCs are viewed as search for new materials and methods of their fabrica- three-dimensional crystal defects [6, 10]. It is necessary tion [1]. New materials are frequently prepared by pro- to distinguish opened NCs, those communicating with ducing a regular or quasi-regular set of voids. In a semi- a crystallization medium (etch pits or striation), and conductor crystal, such voids can be pores made by closed NCs (inclusion of the mother medium, foreign etching; in composites, natural spaces between the balls inclusions, etc.). NCs with an artificially created sur- [2]. Among wide-gap semiconductors, silicon carbide face (faceted or unfaceted) will be considered as is of particular interest for nanoelectronics. Its proper- pseudo-NCs. To date, a large number of NCs has been ties have already found application in microwave elec- identified in SiC crystals. They can be classified by tronics, optoelectronics, and power electronics [3]. their specific features. However, a set of voids in SiC crystals may be used in areas untraditional for this material, such as ultrasonics, 1. NC Classification micromechanics, chemistry, thermal electronics, and even jewellery [3Ð5]. Generally, the problem here is to Experimental data show that various SiC crystalli- controllably produce a system of negative crystals in zation conditions, as well as various treatments of SiC SiC and to study their properties [6, 7]. crystals, such as etching, mechanical processing, and erosion, may lead to the formation of different NCs and This work is devoted to the classification and exper- even NC systems (see table). imental investigation of such systems. It is known that there is a relation between SiC crystallization and decrystallization; namely, nucleation mechanisms, as well as growth kinetics and mecha- EXPERIMENTAL TECHNIQUES nisms, have common features [7]. Under certain condi- The objects of experiment were 6H-SiC single crys- tions, this may result in the simultaneous growth of tals grown in a vacuum or argon at growth temperatures positive and negative crystals. Together with individual T = 2100Ð3000 K with the conventional Lely and NCs, NC systems may form, where NCs are distributed g randomly or along particular crystallographic direc- LÉTI (the abbreviation from Leningradskiœ Élek- tions. Lely- and LÉTI-grown SiC positive crystals may troTekhnicheskiœ Institut) techniques [8]. The concen- contain NCs distributed along the directions perpendic- × 17 tration of uncompensated donors was NdÐNa = 5 10 Ð ular to the (0001) basis planes or along the oblique × 18 Ð3 × 4 × 5 10 cm ; the dislocation density, ND = 1 10 Ð1 directions, NCs of variable (arbitrary) orientation, and 107 cmÐ2. The electrical erosion processing of the SiC those growing parallel to the (0001) basis planes. Typi- crystal was carried out with the well-known sewing cal examples of crystallographically oriented NCs are method and by filamentary electrode cutting [4, 9] on shown in Fig. 1. In spite of high SiC crystallization tem- both commercial and laboratory equipment. The struc- peratures, NCs grew by the solidÐliquidÐvapor mecha- ture and the morphology of the crystals were examined nism [10]. Silicon or impurities (metallic solvents), by the Lang X-ray diffraction method and anisotropic which may fall into the reaction zone from the graphite chemical etching in a KOH solution at T = 450Ð600°C, fixtures or be specially introduced into the vapor source as well as in a metallurgical microscope and a BS-340 [16], serve as the liquid phase. SiC positive crystals scanning electron microscope (SEM) operating in the obtained by the Lely method grow in an unbounded secondary electron mode. space, as a result of which they take the form of thin

NEGATIVE CRYSTALS OF SILICON CARBIDE 433

Systems of negative crystals in 6H-SiC

Crystallographic NC production methods NC shape NC system orientation, localization

1 SiC sublimation growth [8, 11] Holes (pits) Irregular Growth faces, arbitrary Disks Unfaceted cavities (pores) Quasi-regular [0001] Prisms Pyramids Irregular Arbitrary Pseudo-NC of ordinary shape Regular [0001] 2 SiC etching [11Ð13]: chemical, Hexagonal pits Quasi-regular (0001) electrochemical Circular ﬂat-bottomed pits Irregular thermal, by molten metals or Filamentary (whiskers) Irregular silicon, through mask (stencil) Unfaceted cavities (pores) Quasi-regular Pseudo-NC (pits) Regular

3 SiC erosion [9, 14, 15]: Pits Quasi-regular (0001), (1010), (112 0) electrical, beam, mechanical, ultrasonic Filamentary " Pseudo-NC of regular and Regular Any irregular shape platelets. In such crystals, NCs are usually oriented 2. Quasi-regular Systems of NCs along the direction of maximal growth rate ([1120 ]). (i) Etch pits. The chemical etching of SiC crystals Fig. 1a is illustrative in the sense that it demonstrates in salt melts and oxide melts, as well as in alkaline solu- the growth of NCs in the direction parallel to the (0001) tions, readily generate etch pits on the (0001)Si face. basis plane through the development of a filamentary The sizes and shapes of the etch pits depend largely on NC. Such a mechanism is consistent with the conven- growth crystal defects and etching conditions [12]. Ear- tional model of negative crystal evolution [7]. lier [11, 12], it has been shown that small hexagonal The use of seeds and also the limited growth space etch pits uniquely correspond to edge and screw dislo- in the LÉTI method (sublimation profiling [11]) pro- cations; circular flat-bottomed pits, to dislocation loops vide the oriented growth of the positive and negative due to vacancies; and large hexagonal pits, to growth crystals. Much lower crystallization temperatures (Tg = NCs. Of most interest for us are hexagonal etch pits 1700Ð2000°C for the growth in a vacuum) compared related to upright dislocations (UDs). This is because with the Lely method raise the liquid phase formation these dislocations readily form when SiC crystals grow probability, particularly at the early stage of growth, in the [0001] direction. As follows from the X-ray dif- hence, a high density of NCs extended in the [0001] fraction data, the dislocation lines are straight (Fig. 2) direction. Such an NC system may also form at crystal- and are aligned primarily with the C axis. In the crystals lization temperatures above 2100°C, where the density grown at temperatures below 2300°C, the UD density of “upright” dislocations, along which voids (usually exceeds ~105 cmÐ2 and varies along the crystal insignif- decorated by graphite) are produced, is high. High icantly (Fig. 2b). UDs form a rather stable system; how- growth rates of the positive crystals and the depletion of ever, well-known methods, such as complex profiling, the SiC vapor source intensify this process. Figure 1b isovalent doping, and the use of polytype gates (layers), shows the traces of solidified melt inside the NC. may provide the control of the UD inheritance process and form quasi-regular dislocation (hence, etch-pit) Unfortunately, the fact that NCs forming in SiC dur- ensembles [11]. ing growth have different shapes and sizes are today treated as process errors [17]. Only regular systems of The cooling rate after the crystallization has a great NCs in SiC are of practical value. However, they are effect on the UD distribution over the crystal cross sec- difficult to produce during crystallization, since the tion. It affects the temperature field and, consequently, growth of NCs has a random character and has been thermoelastic stresses generated by standard disloca- poorly studied. The exception is quasi-regular sets of tion distributions. Fig. 3a shows the experimentally etch pits and regular systems of pseudonegative crys- found distribution of the etch pit density along the tals. Some of their features are discussed below. radius of the SiC crystal. At a cooling rate of ≈970 K/h

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434 KARACHINOV

– [1120]

(‡) 1.5 mm (‡) 1 mm –2 ND, cm 107 (b)

1 106

105

104 2 012 3 l, mm

150 µm Fig. 2. Dislocation structure of the 6H-SiC crystals (LÉTI (b) ≅ ° method, Tg 2100 C). (a) UD pattern obtained by the Lang method from the (0012) plane, MoKa1 radiation; (b) varia- Fig. 1. Growth negative crystals. (a) Lely method, growth in tion of the etch pit (dislocation) density along the crystal. the [0001] direction: (1) hexagonal prism and (2) filamentary NC; (b) LÉTI method, growth in the [0001] direction: (1) hexagonal prism and (2) traces of melt crystallized on with the flat undistorted growth front, the pyramid tip the cavity bottom. was symmetric relative to the base. This observation is corroborated by the X-ray topographic data, which indicate that the UD lines in the crystals studied cross or lower, one can observe the weak tendency to a the (0001) growth face largely at right angles. V-shaped distribution. The etch pit distribution markedly changes starting from a cooling rate of ≈1650 K/h (ii) Breakdown figures. It is known that chemical or higher. It becomes W-shaped with more or less dis- etching under strong electric fields generates a set of tinct boundaries at the center and at the edges of the pits that are inclined relative to the basic position. For quartz crystals, for example, this inclination may reach crystal (Fig. 3b). Since the cooling rate is finite, the ≈ ° temperature over the cross section has no time to equal- 10 [12]. It is also known that a discharge generated ize, which generates stresses depending also on the by a strong electric field can produce (without using crystal size (Fig. 3c). chemicals) specific figures on the SiC crystal surface. The figures may appear as usual erosion holes or as a The SEM examination of the etch pit structure on set of negative filaments known as EFT defects [14]. the silicon faces indicates the following (Fig. 4). It is clearly seen that the pits have the form of a pyramid In the case of individual discharges, a quasi-regular with a regular hexagonal base. Because of a difference set of EFT defects usually forms. It, however, may con- between the normal and tangential components of the tain ordered groups of extended (coupled) defects SiC dissolution rate at the sites where dislocations (chains or mosaic). Spikes oriented in the 〈〉1120 emerge on the surface, the terrace structure forms. No directions are dominant coupling elements (bridges) asymmetry of the etch pits and the base facet was here. The morphology of EFT defects was refined with revealed. In most of the pits observed in the crystals the SEM (Fig. 5) and was found to be a combined result

TECHNICAL PHYSICS Vol. 47 No. 4 2002

NEGATIVE CRYSTALS OF SILICON CARBIDE 435

–2 ND, cm 106 (a)

105 1 2

4 10 µ 0 500 1000 1500 Rk, m

8 µm

Fig. 4. SEM micrograph of the hexagonal etch pit structure on the (0001)Si face. LÉTI growth method.

(b) 1 mm

z, mm (c) 2 ×104 2.05 × 4 2.56 10 2375 K × 4 1 3.07 10 4 4 4.10×10 2380 K ×104 5.12 50 µm 4 (‡) 3 ×10 2385 K 6.15 × 4 8.20 10 × 5 1.00 10 2 2390 K 1.23×105 2395 K 1.64×105 1 2 2.05×105 2399 K 2.46×105 8 6 4 2 0 2 4 6 8 1 r, mm

Fig. 3. Effect of the cooling rate on the dislocation structure. µ (a) Experimental radial distribution of the etch pit density, (b) 8 m cooling rate is (1) ≈1650 and (2) ≈970 K/h; (b) natural (0001)Si face (etch pit pattern), cooling rate is ≈1650 K/h; and (c) calculated distributions of the temperature and ther- Fig. 5. SEM micrograph of the EFT defect structure on the moelastic stresses in the crystal (ﬁgures on the right are (0001)Si face in SiC. (a) General view: (1) spikes and stresses in N/m2). (2) hole; (b) growth step: (1) spike and (2) step.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 436 KARACHINOV

3 V, mm /min (a) 2 2 1 1

041 2 3 5 6 Ep, J (b)

1 2 0.2 (‡) 1 mm – 0.1 [1120]

00.4 0.8 1.2 h, mm

Fig. 7. Growth parameters of cylindrical pseudo-NCs obtained by the sewing method in transformer oil. (a) Experimental growth rate vs. electric pulse power: growth in the (1) [0001]Si direction and (2) [0001]C direction; (b) growth rate variation along the pseudo-NC: (1) tubular electrode and (2) cylindrical electrode (steel).

spikes are shown in Fig. 5b. The steps remained on the surface even after the long-term etching of the crystals in a boiling aqueous solution of KOH. (b) 1 mm (iii) Regular sets of pseudo-NCs. The sewing principle, which is widely used in the size profiling of mate- Fig. 6. (a) Pseudonegative crystals: (a) hollow crystal grown rials [19], may serve as a basis for methods that produce ≅ ° in argon at Tg 2100 C and (b) crystal obtained by erosion regular sets of voids—pseudonegative crystals—in in water (filamentary brass electrode of diameter d ≈ both single-crystal and polycrystalline SiC. Figure 6a 100 µm). demonstrates a pseudo-NC in the form of a hollow crystal obtained by the self-sewing of a positive single of erosion and partial surface breakdown of the crystal. crystal during its growth from the vapor. By its formation conditions, this defect can be classi- Erosion methods, such as electrical erosion in liquid fied as a negative breakdown figure [18]. It has the insulators, offer wide applicability and are easy to 6mmm point symmetry group. The effects of orienta- implement. Note that the early pseudo-NC obtained in tion and pinching of electron avalanches spreading over SiC was threaded. Figure 6b shows a regular NC sys- the SiC facet during the discharge and acting as tem; Fig. 7, typical growth parameters for a cylindrical extended heat sources produce a system of oriented pseudo-NC. The sewing depth is traditionally a param- cavities (spikes). Careful examination shows that the eter of great interest. Experiments with bulk single spikes oriented in the [1120 ] and [1010 ] directions crystals and layer structures (sets of platelets) demon- differ in shape and size. Sharp-front spikes are oriented strated that the NC growth rate monotonically mostly in the former direction. This is consistent with decreases as a rule. This is presumably explained by an the general idea that the crystallographic direction increase in the evacuation length for SiC erosion products [19]. influences SiC evaporation channels with the participation of the silicon liquid phase [7]. Along with ordinary It is known that the basic geometrical characteristics spikes, recuperative partition-divided spikes also of SiC pseudo-NCs are directly related to the shape of appear. As a rule, recuperative spikes branch. High tem- the profiling electrode. The profiling of a material hav- perature gradients and the presence of the silicon-based ing a high erosion resistance at an effective diameter liquid phase favor the epitaxial growth of SiC. Typical d ≤ 1 mm is a challenge. Therefore, the use of a set of growth steps of height from 0.5 to 1.0 µm around the positive filamentary crystals seems to be promising for

TECHNICAL PHYSICS Vol. 47 No. 4 2002 NEGATIVE CRYSTALS OF SILICON CARBIDE 437 solving this problem. However, this statement calls for 8. V. I. Levin, Yu. M. Tairov, M. G. Travadzhyan, and further experimental verification. V. F. Tsvetkov, Izv. Akad. Nauk SSSR, Neorg. Mater. 14, 1062 (1978). 9. V. A. Karachinov, Kristallografiya 43, 1097 (1998) CONCLUSION [Crystallogr. Rep. 43, 1038 (1998)]. Special technological actions on SiC matrix crystals 10. R. S. Wagner, J. Cryst. Growth 3, 159 (1968). may form crystallographically oriented systems of neg- 11. V. A. Karachinov, Author’s Abstrct of Candidate’s Dis- ative crystals in them. The growth of NCs during the sertation (Leningrad, 1985). electrical erosion of SiC is accompanied by the epitax- 12. K. Sangwal, Etching of Crystals. Theory, Experiment ial growth of a positive crystal layer. One can produce and Application (Elsevier, Amsterdam, 1987; Mir, Mos- regular systems of pseudo-NCs and carry out the size cow, 1990). profiling of SiC with any degree of complexity. 13. A. G. Ostroumov, M. I. Abaev, and M. I. Karklina, Izv. Akad. Nauk SSSR, Neorg. Mater. 15, 1497 (1979). REFERENCES 14. V. A. Karachinov, in Proceedings of the 3rd Interna- tional Conference “Crystals: Growth, Properties, Real 1. V. N. Bogomolov, D. A. Kurdyukov, A. V. Prokof’ev, Structure, Application” (VNIISIMS, Aleksandrov, et al., Pis’ma Zh. Éksp. Teor. Fiz. 63, 496 (1996) [JETP 1997), Vol. 2, p. 154. Lett. 63, 520 (1996)]. 15. V. G. Ral’chenko, V. I. Konov, A. A. Smolin, et al., in 2. M. E. Kompan and N. Yu. Shibanov, Fiz. Tekh. Polupro- Proceedings of the International Workshop “Silicon vodn. (St. Petersburg) 29, 1859 (1995) [Semiconductors Carbide and Related Materials” (Novg. Gos. Univ., 29, 971 (1995)]. Novgorod, 1997), p. 41. 3. P. A. Ivanov and V. E. Chelnokov, Fiz. Tekh. Polupro- 16. V. A. Il’in, V. A. Karachinov, Yu. M. Tairov, and vodn. (St. Petersburg) 29, 1921 (1995) [Semiconductors V. F. Tsvetkov, Pis’ma Zh. Tekh. Fiz. 11, 749 (1985) 29, 1003 (1995)]. [Sov. Tech. Phys. Lett. 11, 312 (1985)]. 4. V. A. Karachinov, in Proceedings of the 3rd Interna- 17. B. A. Kirillov, Sb. Nauchn. Tr. S-Peterb. Gos. Élek- tional Conference “Crystals: Growth, Properties, Real trotekh. Univ., No. 488, 87 (1995). Structure, Application” (VNIISIMS, Aleksandrov, 1997), Vol. 2, p. 240. 18. Progress in Dielectrics, Ed. by J. B. Birks and J. H. Schulman (Wiley, New York, 1959; GÉI, Moscow, 5. Yu. V. Deryugin, Single Crystals of Corundum in Jewel- 1962). lery Industry (Mashinostroenie, Leningrad, 1984). 19. N. K. Mitskevich and I. G. Nekrashevich, Electroerosion 6. V. A. Mokievskiœ, Crystal Morphology (Nedra, Lenin- Machining of Metals (Nauka i Tekhnika, Minsk, 1988). grad, 1983). 7. E. I. Givargizov, Growth of Filamentary and Scaly Crys- tals from Vapor (Nauka, Moscow, 1977). Translated by V. Isaakyan

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Technical Physics, Vol. 47, No. 4, 2002, pp. 438Ð443. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 66Ð71. Original Russian Text Copyright © 2002 by Babichev, Kozlovskiy, Romanov, Sharan.

SOLID-STATE ELECTRONICS

A Silicon Stress-Sensitive Unijunction Transistor G. G. Babichev, S. I. Kozlovskiy, V. A. Romanov, and N. N. Sharan Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine e-mail: [email protected] Received May 14, 2001

Abstract—The performance of a silicon stress-sensitive unijunction transistor is investigated. The transistor is classed as a stress-sensitive semiconductor lateral bipolar device with an S-type input (emitter) IÐV characteristic. The optimal layout of the device and its basic parameters are determined. The device can serve as a basis for designing relaxation oscillators with the physically integrated function of mechanical stress-to-signal frequency conversion at the output. © 2002 MAIK “Nauka/Interperiodica”.

A unijunction transistor (or double-base diode) is a ter current Ie = Is (Is is the saturation current) passes semiconductor device with one pÐn junction and two through the pÐn junction. This current is due to the volt- η η base contacts. Its static (S-shaped) IÐV characteristic age Ubb , where = le/lx is the voltage transfer coeffi- has a region with negative differential resistances. cient, or the internal coefficient of voltage division, and Because of this, unijunction transistors are used in l is the spacing between the emitter and the base elec- pulse generators, sawtooth generators, threshold e trode B1. We assume that the geometric sizes of the devices, and converters. Circuits based on unijunction emitter are small compared with the base region. As the transistors are simpler and more reliable than those negative potential difference U grows, the junction built around diode-connected and bipolar transistors e [1Ð3]. Therefore, they are promising for transducers of becomes directly biased and the emitter starts to inject physical quantities. For example, transducers where the nonequilibrium electrons, which drift to the base elec- output signal frequency depends on magnetic field trode B1 in the sweeping field. Because of this, the strength have been designed on the basis of magnetic- resistance Reb1 of the first base decreases and the direct field-sensitive unijunction transistors (unijunction bias of the emitter junction increases further. This pro- magnetotransistors) [2, 4]. cess is of an avalanche character and passes the transis- In this work, we consider the design and basic char- tor to the on state. The input (emitter) IÐV characteristic acteristics of a unijunction strain-sensitive transistor, of the unijunction transistor is schematically shown in around which transducers of mechanical quantities, pri- Fig. 2. It has two points of practical value: the peak marily pressure transducers with the frequency output, (Up, Ip) and the valley (Uv, Iv) of the voltage, where the can be built. It is known [1, 2, 5] that frequency-output transducers can be readily interfaced with digital sys- + tems and offer a high noise immunity because fre- p l quency-modulated signals are weakly sensitive to inter- y ference and crosstalk. B2 Unijunction transistors can be made by conven- E p-Si tional methods of integrated technology and be placed lx + le on the planar side of a profiled silicon membrane of EE n B1 type (having two rigid central areas) [6]. This mem- + [100] brane converts a load uniformly distributed over its sur- p [110] face to uniaxial elastic compressiveÐtensile strain in σ that its part where the transistor is placed. The layout, [010] electrical circuit, and crystallographic orientation of the + + unijunction transistor are shown in Fig. 1. In the absence of elastic strain, the transistor oper- U U ates like a conventional unijunction transistor [1Ð3]. e bb With a potential difference Ubb applied to the base con- Fig. 1. Layout, connection diagram, and crystallographic tacts and in the absence of a voltage between the emitter orientation of the unijunction stress-sensitive transistor. B1 and the base electrode B1 (Ue = 0), a small inverse emit- and B2, base electrodes; E, emitter.

A SILICON STRESS-SENSITIVE UNIJUNCTION TRANSISTOR 439 ∂ ∂ derivative Ue/ Ie vanishes: Ie ∂U ∂2U ------e()I = I = 0 for ------e()I = I < 0, (1) ∂I e p ∂ 2 e p e Ie I ∂U ∂2U v ------e()I = I = 0 for ------e()I = I > 0. (2) ∂I e v ∂ 2 e v e Ie

The parameters Up, Ip and Uv , Iv are accordingly referred to as peak (switch-on) voltage and current and valley (saturation) voltage and current, respectively [1, 2]. Ip In the case of uniaxial elastic strain in the crystallo- Uv Up Ue graphic direction [110], the initially isotropic mobility µ p of minority carriers (holes) in the base region of a Fig. 2. Input IÐV characteristic of the unijunction stress-sen- strain-sensitive transistor becomes anisotropic. There- sitive transistor. fore, the holes, drifting in the longitudinal sweeping field of the base (in the x direction; Fig. 1), shift to the Let the base of the unijunction stress-sensitive tran- transverse direction, and the transverse potential differ- sistor be bounded by the segments ence Vy appears. The strain-induced transverse potential << << << difference modulates the injection from the emitter pÐn 0 xlx,0yly,0zlz. (5) junction and, hence, the voltage Up of switching the unijunction stress-sensitive transistor to the high-con- We will consider the unijunction stress-sensitive ductivity state. transistor operating under two limiting conditions: low The operation of a unijunction stress-sensitive tran- and high injection levels in the first base. In the former sistor is formally the same as the operation of a unijunc- case, the emitter characteristic of the device is given by tion magnetotransistor, in which the Hall field modu- () T Ie () lates the emitter injection. This effect has been named Ue Ie = --- ln1 + ---- + Ie Reb1 Ie injection modulation [7]. Note that the operation of e Is (6) early unijunction magnetotransistors was based on []() ()Ð1 ()σ another, so-called deflection effect [2, 4]. Here, a mag- + Ubb 1 + Reb2 Ie /Reb1 Ie + V y /2. netic field modulates the resistance of the first base, Here, T is temperature in energy units, e is the charge of deflecting emitter-injected nonequilibrium carriers an electron, and R is the resistance of the second base toward the lateral surfaces of the base, which have a low eb2 or high surface recombination rate. (i.e., of the area between the emitter and the base electrode B2). As follows from (6), the elastic strain modu- A unijunction stress-sensitive transistor can be char- lates the emitter injection by the transverse potential acterized through the absolute and relative strain sensi- difference Y (σ). For the uniform strain, the transverse tivities in terms of switching-on voltage (S and S ) y AP RP potential difference is expressed as [8] and saturation (valley) voltage (SAV and SRV): ()σ ()σ (),, ∂ V y = a UbbFle lx ly , (7) U p()v SAP() AV = ------, ∂σ σ → 0 where F(l , l , l ) is a geometric factor and (3) e x y ∂ ∞ 1 U p()v n SRP() RV = ------. 4 []11Ð ()Ð U () ∂σ (),, p v σ → 0 Fle lx ly = ----- ∑ ------π2 n2 Here, σ is the mechanical stress in the base region of the n = 1 (8) stress-sensitive transistor. By writing σ 0, it is πnl πnl × sin ------e tanh ------y . meant that the above parameters are determined at l 2l small σ. In order to optimize the layout and the geome- x x try of the unijunction stress-sensitive transistor, we will In view of (7), the emitter characteristic of the analyze the emitter characteristic Ue(Ie) and evaluate device takes the form the parameters Up, Ip, Uv, and Iv. The analysis will be performed in the low uniform strain approximation [6, 8], T Ie (), σ --- () i.e., when the anisotropy parameter is small (|a| Ӷ 1): Ue Ie = ln1 + ---- + Ie Reb1 Ie e Is ()σ Πσ a = 44 /2, (4) (9) R ()I a()σ Π + U ------eb1 e - + ------Fl(),,l l . where 44 is the shear piezoresistive coefficient bb []() () e x y Reb1 Ie + Reb2 Ie 2 for p-Si.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 440 BABICHEV et al. Let us analyze expression (9). According to experi- hole current density and means the absence of the hole mental data [1], the resistance of the second base Reb1 component at the boundary of the emitter pÐn junction. can be assumed to be independent of the emitter current The solution of problem (16)Ð(18) has the form in the first approximation: () () Ie Lxsinh /L R ()I ≅ R ()I = 0 = R . (10) px = p0 + ------()(). (19) eb2 e eb2 e eb2 eDnSe b + 1 cosh le/L At the same time, the dependence Reb1(Ie) is gov- Since the injection is low, the emitter current in erned by a number of factors [1, 2], such as base geom- expression (19) is limited: etry, boundary conditions at the contacts, nonequilib- ≤ ≤ I Ӷ eD S ()b + 1 p /Lltanh()/L . (20) rium carrier distribution along the base segment 0 x le, e n e 0 e etc. For the steady-state case, the set of equations for The resistance of the first base is found by integra- the distribution of nonequilibrium carrier concentration tion: in the quasi-neutrality approximation has the form le 1 ρ ---div()j +0,RGÐ = (11) () 0 dx p 0 Reb1 Ie = -----∫[]------()∆(), , (21) e Sb 1 + b + 1 pxIe / p0 0 ()≡ () () div j div jn + jp ==0, curl E 0, (12) ρ where 0 is the base resistivity in the absence of injec- µ ∇ µ ∇ tion and S is cross-sectional area of the base. Integrat- jp ==e p pE Ð eDp p, jn e nnE Ð eDn n, (13) 0 ing (21) yields ∆ nnÐ 0 ==ppÐ 0 p. (14) () () Reb1 Ie = 0 L Here, n and p are the concentrations of electrons and Reb1 Ie = ------()2 holes, respectively; n0 and p0 are their equilibrium val- le 1 + PIe µ µ ues; n, Dn and p, Dp are the mobilities and the diffu- (22) 1 + []PI()+ 1 + PI()2 tanh()l /2L sion coefficients of electrons and holes, respectively; jn × ln------e e e , and jp are the densities of the electron and hole currents, []() ()2 () 1 + PIe Ð 1 + PIe tanh le/2L respectively; R and G0 are the rates of volume recombination and generation, respectively; and E is electric where P(Ie) = Ie/I0 and I0 = eDSep0 cosh( le/L)/L. field. Eliminating E from Eqs. (13), we obtain the To find the peak voltage U by formula (9), one must expression for the hole current density p determine, along with the dependence Reb1(Ie) and the j j ≅ ------Ð eD∇p, (15) current Ip, the geometric factor F(le, lx, ly), which is p b + 1 shown in Fig. 3 as a function of the base size and the ≅ µ µ emitter position. Its peak value is observed at lx ly and where D = 2bDp/(b + 1) and b = n/ p. le = lx/2. These relationships for lx, ly, and le correspond We assume that the nonequilibrium carrier distribu- to the case of uniform (constant) strain within the tran- tion in the base is quasi-one-dimensional and take into sistor base. The transverse potential difference V (σ) account the boundedness of the base in the yz plane by y τ and the geometric factor for the nonuniform case are introducing the effective lifetime eff and the associated given in [8, 13]. τ diffusion length L = D eff [10, 11]. According to [12], To simplify calculations, we will take advantage of ӷ the effective value of L in our case (lx, ly lz) is limited the following experimental observation [1]: in a wide ≈ by the base thickness: L lz. Substituting expression (15) range of Ie, the dependence of the resistance of the first for the hole current density into Eq. (11), we find, under base on the emitter current can be approximated by the the above assumptions, the equation for the hole con- expression [2, 4] centration: () () Reb1 Ie = 0 2 Reb1 Ie = ------ξ , (23) ∂ p ppÐ 0 1 + Ie ------2 Ð0.------2 = (16) ∂x L where ξ is an approximating factor.

The boundary conditions are taken in the form The Reb1 vs. emitter current dependence calculated by formulas (22) and (23) is shown in Fig. 4. The curve ∂p I ------()xl= = ------e -, (17) is seen to be well approximated by expression (23) with ∂ e () ξ × 4 Ð1 x eSe Db+ 1 = 1.11 10 A . With formula (22), the curve Reb1(Ie) () was constructed for the silicon transistor base region px= 0 = p0, (18) × × × × µ measuring lx ly lz = 100 100 10 m and silicon 2 Ð1 where Se is the surface area of the emitter junction. electrophysical parameters (at 300 K) Dn = 35 cm s , 2 Ð1 × 15 Ð3 × 10 Ð3 Condition (17) follows from expression (14) for the Dp = 13 cm s , p0 = 5 10 cm , ni = 1.4 10 cm .

TECHNICAL PHYSICS Vol. 47 No. 4 2002 A SILICON STRESS-SENSITIVE UNIJUNCTION TRANSISTOR 441

Ω F(le, lx, ly) Rebl, 0.8 1 1250* 2 * 0.6 3 * 1200 * * 0.4 * * 1150 * * 0.2 * * 024 6 8 10 0 0.5 1.0 1.5 2.0 µ Ie, A ly/lx Fig. 4. Resistance of the ﬁrst base vs. emitter current. Con- Fig. 3. Geometrical factor vs. transistor base size. The emit- tinuous curve, calculation by formula (22); symbols, ter position le/lx = (1) 0.5, (2) 0.3, and (3) 0.8. approximation by formula (23).

, V An expression for the current peak is found by sub- Up 3 stituting expression (23) into expression (9) for the input characteristic and equating the ﬁrst derivative of 4.5 the emitter characteristic ((expression (1)) to zero. At low injection levels (I ξ Ӷ 1), the expression for the 4.0 e 2 current peak Ip is 3.5 ≅ T Ip ------[]η()ξη ()-. (24) 3.0 2 Ubb 1 Ð Ð Reb1 Ie = 0 1 2.5 Given Ip, the peak voltage can be easily found from –0.06 –0.04 –0.02 0 0.02 0.04 0.06 a (9). Figure 5 shows the analytic dependence of the peak voltage on the anisotropy parameter a. The curve Up(a) Fig. 5. Peak voltage vs. anisotropy parameter. Ubb = (1) 4, is linear, which is of importance for applications. (2) 6, and (3) 8 V. According to (3) and (9), the absolute strain sensitivity in terms of switching-on voltage, S , does not AP U , V depend on the emitter current: p –10 –1 5 SRP, 10 Pa Π 44 S = ------U Fl(),,l l , (25) 4.0 AP 4 bb e x y 4 12 Π × Ð9 Ð1 ≅ 3 with 44 = 1.4 10 Pa and Ubb = 4 V, we have SAP 3.5 10Ð9 V/Pa. 2 The relative strain sensitivity in terms of switching- 3.0 on voltage SRP vs. potential difference Ubb is shown in 1 Fig. 6. As Ubb grows, SRP tends to saturation (curve 1), 2 3 4 5 6 7 8 while S linearly increases (curve 2). AP Ubb, V Because of the shortening effect under high injec- ξ ӷ tion levels (Ie 1), the transverse potential difference σ ≈ Fig. 6. (1) Relative strain sensitivity in terms of switching- Vy in (6) can be neglected, Vy( ) 0, so that the emitter on voltage and (2) switching-on (peak) voltage vs. potential characteristics of the stress-sensitive transistor takes the difference Ubb. form Substituting approximating dependence (23) into () T Ie ()() Ue Ie = --- ln1 + ---- + Ie + Ibb Reb1 It , (26) formulas (26) and (2), we obtain the valley current e Is 2[]U ξη Ð R ()I = 0 ()1 Ð η where I = U /(R + R ) ≅ U /R is the interbase ≅ bb eb1 e bb bb eb2 eb1 bb eb2 Iv ------2 -. (27) current. Tξ ()1 Ð η

TECHNICAL PHYSICS Vol. 47 No. 4 2002 442 BABICHEV et al.

– obey the inequality l T R Rb1 a > ------e - ≅ 0.1. (29) eLUv Since the maximal value of the anisotropy parameter is somewhat smaller than a = 0.06 in our case (which C corresponds to an elastic stress of 86 MPa), the effect of Rb2 mechanical stress on Uv and Iv will be negligible. U0 Let us briefly consider the principal electrical circuit + of mechanical quantity (pressure, force, and acceleration) transducers based on unijunction stress-sensitive Fig. 7. Simple multivibrator based on the unijunction stress- sensitive transistor. transistors. Like related devices built around conventional unijunction transistors [1Ð3, 15], the strain-sensitive transducers also admit any of standard operating Frequency, kHz conditions: amplification, generation of harmonic and 5.5 relaxation oscillations, switching, detection, etc. Uni- 5.0 1 junction stress-sensitive transistors are best suited to be used in switches and relaxation oscillators. 2 4.5 Figure 7 shows the circuit of the simplest multivi- 4.0 brator based on a unijunction stress-sensitive transistor. The oscillation frequency of the multivibrator depends 3.5 on the elastic mechanical stress in the base of the tran- 3 sistor. In the steady-state operating mode, the oscilla- 3.0 4 tion period of the multivibrator is given by [1Ð4] 0 20 40 60 80 σ, MPa ()σ Uv Ð U0 T = RCln ------()σ . (30) Up Ð U0 Fig. 8. Oscillation frequency of the multivibrator in Fig. 7 vs. elastic uniaxial mechanic (1, 3) compressive and (2, 4) The output signal frequency of the multivibrator vs. tensile stress in the base of the stress-sensitive transistor. elastic mechanical stress in the base of the unijunction Ð3 Ubb = (1, 2) 6 and (3, 4) 4 V. RC = 10 s, U0 = 10 V. stress-sensitive transistor is depicted in Fig. 8. Thus, we have obtained the expressions for peak The valley voltage Uv is easy to find by substituting (switching) and valley currents (voltages) in the input IÐV characteristic of the silicon unijunction stress-sen- Iv into expression (26) for the emitter characteristic. sitive transistor. Their dependence on the mechanical When the injection level in the stress-sensitive stress in the base, potential difference between the base device is high, the uniaxial elastic strain modulates the electrodes, and base geometry have been evaluated. resistance Reb1 of the first base, as with unijunction magnetotransistors [2, 4], by virtue of the deflection REFERENCES effect. Because of the strain-induced anisotropy of the hole mobility, nonequilibrium holes are deflected 1. I. G. Nedoluzhko and E. F. Sergienko, Unijunction Tran- sistors (Énergiya, Moscow, 1974). toward either the nearest or farthest (relative to the emitter) lateral side of the base (Fig. 1). As a conse- 2. I. M. Vikulin and V. I. Stafeev, Physics of Semiconductor Devices (Radio i Svyaz’, Moscow, 1990). quence of the deflection effect, the valley voltage and current may depend on the elastic strain. The necessary 3. S. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981; Mir, Moscow, 1984), Vol. 1. condition for such a dependence is an excess of the transverse field E over the diffusion one [9, 10, 14]: 4. I. M. Vikulin, L. F. Vikulina, and V. I. Stafeev, Galvano- y magnetic Devices (Radio i Svyaz’, Moscow, 1983). T 5. S. V. Gumenyuk and B. I. Podlepetskiœ, Zarubezhn. Éle- E ≅ aE > ------, (28) y x eL ktron. Tekh. 12 (343), 3 (1989). 6. G. G. Babichev, I. P. Zhad’ko, S. I. Kozlovskiy, et al., Zh. where E is the sweeping field. Tekh. Fiz. 64 (9), 84 (1994) [Tech. Phys. 39, 908 x (1994)]. According to inequality (28), anisotropy parameter 7. H. P. Baltes and R. S. Popovic, Proc. IEEE 74, 1107 values at which the drift field exceeds the diffusion one (1986).

TECHNICAL PHYSICS Vol. 47 No. 4 2002 A SILICON STRESS-SENSITIVE UNIJUNCTION TRANSISTOR 443

8. I. I. Boœko, I. P. Zhad’ko, S. I. Kozlovskiy, and V. A. Ro- 13. I. P. Zhadko, G. G. Babichev, S. I. Kozlovskiy, et al., manov, Optoélektron. Poluprovodn. Tekh. 27, 94 (1994). Sens. Actuators 90, 89 (2001). 9. I. I. Boœko and V. A. Romanov, Fiz. Tekh. Poluprovodn. 14. Z. S. Gribnikov and R. N. Litovskiœ, Fiz. Tekh. Polupro- (Leningrad) 11, 817 (1977) [Sov. Phys. Semicond. 11, vodn. (Leningrad) 14, 675 (1980) [Sov. Phys. Semicond. 481 (1977)]. 14, 397 (1980)]. 10. Z. S. Gribnikov, G. I. Lomova, and V. A. Romanov, Phys. Status Solidi 28, 815 (1968). 15. S. A. Garyainov and I. D. Abezgauz, Negative Resis- 11. M. A. Lampert and P. Mark, Current Injection in Solids tance Semiconductor Devices (Énergiya, Moscow, (Academic, New York, 1970; Mir, Moscow, 1973). 1967). 12. G. G. Babichev, V. N. Guz’, I. P. Zhad’ko, et al., Fiz. Tekh. Poluprovodn. (St. Petersburg) 26, 1244 (1992) [Sov. Phys. Semicond. 26, 694 (1992)]. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 444Ð447. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 72Ð75. Original Russian Text Copyright © 2002 by Ryabukho, Ul’yanov, Minenkova.

OPTICS, QUANTUM ELECTRONICS

Formation of a Quasi-Regular Interference Pattern in a Speckle-Modulated Laser Beam Reflected from a Transparent Optically Nonuniform Layer V. P. Ryabukho, S. S. Ul’yanov, and I. F. Minenkova Saratov State University, Moskovskaya ul. 155, Saratov, 410026 Russia Institute of Problems of Precision Mechanics and Control, Russian Academy of Sciences, Rabochaya ul. 24, Saratov, 410028 Russia e-mail: [email protected] Received June 18, 2001

Abstract—Conditions for the formation of a quasi-regular interference pattern in the speckle-modulated radiation reﬂected from a rough transparent layer irradiated by a focused laser beam are found. A laser method for determining the optical thickness of the transparent coating deposited onto the rough surface is proposed. © 2002 MAIK “Nauka/Interperiodica”.

α α Popov [1] and Kiedron [2] considered an optical where I1(, z) and I2(, z) are the angular distributions method for measuring the thickness of relatively thin of the intensities in the beams reflected from the smooth transparent plates and layers with optically smooth sur- (top) and rough (bottom) surfaces of the layer, respec- faces. The method is based on the interference of opti- tively; α (α, β) is the direction of reflection; ϕ(α ) is a cal waves reflected by both sides of the object at various random spatialÐangular distribution of the phase of the angles of incidence of a laser beam. The spacing of iso- ∆ψ α clinic fringes in this case depends on the optical thick- speckle-modulated field; and ( ) is the determinate ness of the plate or the layer reflecting the beam. Priez- phase difference introduced by a transparent layer with zhev et al. [3] discussed the use of a similar method the mean thickness h and the refractive index n. employing a focused laser beam for determining the In the plane of incidence (α, β = 0), the determinate thickness of the cornea. In the above papers, the layers phase difference is written as [5] studied had optically smooth surfaces, which do not π introduce irregular phase perturbations into the waves ∆ψ() α 4 h 2 2 α reflected. If at least one surface of the layer is rough, a = ------λ n Ð.sin (2) speckle structure, disturbing the regular interference It follows from Eq. (1) that the resulting interference pattern, forms in the laser beam reflected [4]. The purpose of this work is to consider conditions for the formation of a quasi-regular interference pattern in the speckle-modulated field reflected from an optically nonuniform transparent layer. Also, we will discuss a α 4 technique for measuring the local thickness of a rough 0 transparent layer that is based on the precise focusing of 1 the incident laser beam on the scattering surface of the 5 layer. Let us consider a focused laser beam striking a transparent layer with the rough bottom surface – h (Fig. 1). The field reflected represents the superposition 2 of the laser beam with the uniform wave front and the speckle-modulated scattered beam. For the diffraction 3 field away from the focal spot on the layer surface, the angular distribution of the intensity in the resulting reflected field is given by Fig. 1. Incidence of a focused laser beam on an optically nonuniform transparent layer and the reflection of uniform ()α, ()α, ()α, ()α, ()α, and speckle-modulated beams. 1, Incident Gaussian laser I z = I1 z ++I2 z 2 I1 z I2 z beam; 2, transparent layer with the rough bottom surface; (1) 3, substrate; 4, reflected beam with the uniform wave front; ×ϕαcos[]()+ ∆ψα(), and 5, speckle-modulated reflected beam.

FORMATION OF A QUASI-REGULAR INTERFERENCE PATTERN 445

(‡) (b)

Fig. 2. Development of quasi-regular interference fringes against the background of the laser speckle structure as the beam waist approaches the rough surface of a thin transparent layer with an optical thickness nh ≈ 1.6 × 30 µm. (a, b) transverse size ε of the Λ ε ≈ Λ ε Λ a speckles is smaller than the spacing a of the interference fringes, (c) α α, and (d) α > α. pattern in the beam reflected is qualitatively specified Zernicke theorem for the speckle field [6, 7]: by the relationship between the phase distribution ε ≅ ()λα()π α parameters ϕ(α ) and ∆ψ(α ). The determinate compo- α cos 0 / wcos , (4) ∆ψ α α nent of the phase difference ( ) shows up in the where 0 is the angle of incidence of the axial beam spatial distribution of the intensity of the resulting (Fig. 1) and w is the radius of the Gaussian beam on the speckle-modulated field as regular interference fringes scattering surface of the layer. The coefficient α α if the spatial (angular) rate of change of the phase dif- cos 0/cos takes into account the variation in the ference ∂∆ψ(α)/∂α is substantially greater than that of speckle size away from the optical axis of the reflected ∂ϕ α ∂α α ≠ α the random phase difference ( )/ in the speckle- laser beam ( 0). ∆ψ α modulated field. The fastest variation of ( ) takes The speckle size is increased and, consequently, place in the plane of incidence. For this plane, the condition (3) is satisfied if the laser beam waist lies on formal condition for regular fringe appearance is writ- the rough surface of the layer. In this case, w in Eq. (4) ten as is minimized. Figure 2 illustrates the effect of the deter- Ð1 minate component of the phase difference ∆ψ()α on εα >Λ2π∂∆ψα[]()()/∂α = α()α , (3) the interference pattern in the reflected laser beam. where εα is the mean angular size of speckles in the Quasi-regular interference fringes develop against the field reflected from the rough surface and Λα(α) is the background of the speckle structure as the waist of the angular spacing of regular interference fringes incident beam (with angle of incidence of about 45°) expected. approaches the scattering surface of the layer. Figure 2d In other words, it is necessary that the transverse shows the interferogram corresponding to the mini- size of the speckles in the light scattered by the rough mum size of the laser spot on the rough surface and, surface be larger than the spacing of the interference accordingly, to the maximum size of the speckles in the fringes expected. For a Gaussian laser beam, the trans- beam reflected. verse size of speckles in the plane of incidence can be In this case, the angular spacing Λα of the almost estimated by the expression based on the Van CittertÐ parallel interference fringes is given by an expression

TECHNICAL PHYSICS Vol. 47 No. 4 2002 446 RYABUKHO et al.

Λα, rad Λα, rad 0.5 1.00

0.4 0.08

0.3 0.06 0.04 0.2 1 2 0.02 0.1 3 0 10 20 30 40 50 60 – 04020 60 80 90 h , µm α, ° Λ Fig. 3. Angular spacing a of interference fringes versus the Fig. 4. Angular spacing of interference fringes versus the angle of reflection α of laser radiation (λ ≈ 0.63 µm) from thickness of the lacquer coating applied on the rough metal α ≈ the surface of the transparent layer with the refractive index surface. Squares, data points; continuous line, theory ( 0 n = 1.5 and the thickness h = (1) 10, (2) 20, and (3) 50 µm. 45°, λ ≈ 0.63 µm, and n ≈ 1.577). derived from Eqs. (2) and (3): ture of the field reflected. One can observe a rather large number of quasi-regular interference fringes in the λ 2 2 α cross section of the reflected beam if the angular aper- Λ ()α n Ð sin α = ------. (5) ture θ = 2λ/πw of the laser beam reflected from the h sin2α 0 smooth surface of the layer is larger than the angular Figure 3 shows the variation of the fringe spacing spacing Λα of the fringes. Using Eq. (6), we obtain an with the angle of reflection α in the plane of incidence additional condition for the observation of the fringes for transparent layers of various optical thickness nh . when the focused laser beam is incident at an angle of The spacing tends to infinity at normal (α = 0) and graz- α ≈ 45°: α ≈ ° ing incidence ( 90 ). In the former case, interference πw rings centered on the normal to the layer surface are h ≥ ------0 n2 Ð.0.5 (7) observed in the reflected beam. Almost straight fringes 2 with the least spacing appear when the angles of inci- One can use this relationship for estimating the mini- ° dence are close to 45 . It is seen from Fig. 3 that in a mum layer thickness h at which the regular interfer- wide angular range centered at 45°, the spacing of the fringes varies insignificantly. The fringes are equis- ence pattern still can be observed in the reflected laser beam at a given numerical aperture NA of the focusing paced and run parallel to each other. For given n, the ≅ λ constancy of the spacing allows the simple calculation objective. For example, if NA = 0.2, w0 /NA, and n = 1.5, we find h ≅ 6 µm. The maximum value of h of the layer thickness h if the value of Λα is measured: min min that can be measured in practice is limited by the reso- λ 2 lution of the detectors and analyzers of the interference h = ------n Ð.0.5 (6) Λ Λα pattern with the minimum spacing min. Figure 4 shows experimental and theoretical varia- Λ One can determine the value of α from several fringes tions of the angular spacing of the interference fringes observed in a rather wide angular aperture of the laser with the thickness of the transparent lacquer coating beam reflected (Fig. 2d). (n = 1.577) covering the rough metal surface. The angle α ≈ ° In experiments, we used samples with randomly of incidence of the focused laser beam was 0 45 . irregular interfaces, in particular, transparent lacquer The theoretical curve Λα(h) was calculated by Eq. (6). layers of various thickness applied onto the rough metal We also determined the optical thickness ∆ = hh of the surface. The refractive index of the lacquer (n = 1.577) lacquer layer (Fig. 4) by the method of low-coherence was determined by the immersion technique [8]. The interferometry [9, 10] using a MII-4 Linnik interferom- laser beam was focused on the bottom rough surface of eter equipped with a source of white light. The error of the layer with microobjectives with NA = 0.1Ð0.2. We measurement was no greater than 3 µm. pre-expanded the laser beam with an auxiliary microobjective to completely fill the aperture of the Figure 5 compares the thickness h of the lacquer focusing objective. This makes it possible to diminish layers measured by the methods of focused laser beam the radius of the beam waist w0 and to increase the and low-coherence interferometry. Note that the use of transverse size εα of the speckles and the angular aper- both these methods, as well as of low-coherence inter-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 FORMATION OF A QUASI-REGULAR INTERFERENCE PATTERN 447

– ACKNOWLEDGMENTS h , µm This work was supported in part by the Russian 60 Foundation for Basic Research (grant no. 00-15- 96667), the Program “Leading Scientific Schools of the 50 Russian Federation,” and the U.S. Civilian Research 40 and Development Foundation for the Independent 30 States of the Former Soviet Union (CRDF) (grant no. REC-006). 20 10 REFERENCES 02010 30 40 50 60 1. Yu. N. Popov, Prib. Tekh. Éksp., No. 3, 237 (1978). – h , µm 2. P. W. Kiedron, Proc. SPIE 62, 103 (1986). 3. A. V. Priezzhev, V. V. Tuchin, and L. P. Shubochkin, Fig. 5. Comparison of the lacquer thickness values mea- Laser Diagnostics in Biology and Medicine (Nauka, sured by low-coherence (horizontal axis) and laser (vertical Moscow, 1989). axis) interferometry. 4. M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979; Mir, Moscow, 1980). 5. M. Born and E. Wolf, Principles of Optics (Pergamon, ferometry and confocal microscopy [11Ð13], enables Oxford, 1969; Nauka, Moscow, 1973). one to independently determine the geometrical thick- 6. J. W. Goodman, in Laser Speckle and Related Phenom- ness h of the layer and its refractive index n by jointly ena, Ed. by I. Dainty (Springer-Verlag, Berlin, 1975), ∆ pp. 9Ð75. solving equations (6) and = nh . From these equa- 7. J. W. Goodman, Statistical Optics (Wiley, New York, 2 tions, n and h can be approximated as n ≈ (Λα∆/λ) + 1985; Mir, Moscow, 1988). 2 8. N. M. Melankholin, Methods for Studying Optical Prop- 0.25 and h ≈ (λ∆/Λα) if the optical thickness ∆ is suf- erties of Crystals (Nauka, Moscow, 1970). ficiently large. 9. D. A. Usanov, O. N. Kurenkova, A. V. Skripal’, and V. B. Feklistov, USSR Inventor’s Certificate No. Thus, the focusing of the laser beam with a small 1772627, Byull. Izobret., No. 40 (1992). beam waist and, accordingly, a large angular aperture 10. D. A. Usanov and A. V. Skripal’, Television-Controlled (divergence) on the scattering surface of a transparent Measuring Microscopy (Saratovskiœ Univ., Saratov, layer makes it possible to observe quasi-regular inter- 1996). ference fringes in the speckle-modulated radiation 11. T. Fukano and I. Yamaguchi, Opt. Lett. 21, 1942 (1996). reflected. From their spacing, one can measure the local 12. M. Haruna, M. Ohmi, T. Mitsuyama, et al., Opt. Lett. 23, thickness of the layer on the rough surface. The method 966 (1998). presented can be used for the contactless optical moni- 13. K. Yoden, M. Ohmi, Y. Ohnishi, et al., Opt. Rev. 7, 402 toring of the thickness of transparent coatings (e.g., (2000). protective lacquer coatings in microelectronics) applied on optically rough surfaces. Translated by A. Chikishev

TECHNICAL PHYSICS Vol. 47 No. 4 2002

ELECTRON AND ION BEAMS, ACCELERATORS

Cerenkov Radiation Fields in an Inhomogeneously Filled Circular Waveguide A. S. Vardanyan and G. G. Oksuzyan Yerevan Physics Institute, Yerevan, 375036 Armenia e-mail: [email protected] Received August 28, 2000; in ﬁnal form, September 17, 2001

Abstract—Cerenkov radiation ﬁelds excited in a partially ﬁlled circular waveguide with a channel in which an electron beam propagates are studied. The surface of the channel is covered by a thin quasi-conducting layer to remove the electrostatic charge due to the beam passage. The effect of the thickness of the cylindrical dielectric layer is investigated. The thin quasi-conducting layer is shown to attenuate the Cerenkov waves generated in the lossless medium only slightly. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION βc. The charge distribution along each bunch is Gauss- The Cerenkov radiation produced by a single elec- ian: f(ξ) = (πξ2 )Ð1/2exp(Ðξ2/ξ2 ); in the cross section, tron bunch and by a periodic train of bunches in a cir- the bunches are homogeneous. cular waveguide loaded by a lossless cylindrical lay- ered dielectric has been studied in [1]. In particular, the The waveguide is filled with a dispersed dielectric cross-sectional distribution of the longitudinal compo- having an empty channel of radius b. A thin conducting δ nent of the electric field produced by a periodic train of film of thickness = b Ð b1 (b1 > r0) is applied on the bunches was shown to be quasi-stepwise, remaining interior surface of the channel (Fig. 1). quasi-constant in gaps between the dielectric layers. ε ε The permittivity of the dielectric insert is = 1 (in This behavior is observed because, when the bunches the region b ≤ r ≤ a). In the channel, ε = ε = 1. In the follow periodically, the single-mode field in the ch 3 conducting layer, the permittivity is complex-valued: waveguide is set up at the bunch repetition rate, i.e., ε ε ε with a wavelength much greater than the spacing 2 = ' + i ''. between the dielectric layers. The waveguide radius is We will seek solutions for the potential ϕ in the chosen so that this mode alone is excited (this radius cylindrical coordinate system (r, ϕ, z) in the form will be referred to as the resonance radius), with the ∞ field intensity remaining sufficiently high. In fact, as ()ω ω ϕ Φω(), 2I1 kr0 φ  ()ω early as in 1940, Mandelstam, and in 1947, Ginzburg = ∫ r ------p---- expi---- z Ð vt d , (1) kr0 v v and Frank [2], demonstrated that, in a channel with a Ð∞ radius smaller than the wavelength, Cerenkov radiation is generated as in a continuum. Such a field pattern [3−5] makes the Cerenkov radiation mechanism promising for the acceleration of charged particle bunches. This paper discusses the problem of removing the electrostatic charge accumulated on the surface of the a δ channel when the electron beam passes through it. This charge can be removed by applying a thin conducting b layer whose parameters are chosen so as to minimize the attenuation. ε Expressions for the fields are obtained here by the 1 method developed in [1]. ε 2 ε 3 CERENKOV RADIATION IN THE PRESENCE OF A CONDUCTING LAYER

Let a periodic train of N identical cylindrical Fig. 1. Waveguide cross section: a is the waveguide radius, bunches of radius r0, charge q, and spatial period d pass b is the channel radius, and δ is the thickness of the conduct- along the z axis of a circular waveguide at a speed v = ing layer.

CERENKOV RADIATION FIELDS 449 where Let the medium that ﬁlls the waveguide meet the ε µ β2 Φω(), r Cerenkov condition 1 1 > 1. In this case, the coefﬁ- cient a is given by  q ------[]K ()αkr + I ()kr 0 < rb≤ Ð δ ε πv 0 0 α ()2 ()Ð1  = K1 kb1 + -----kK0 kb1 D , (5a)  fs q []η ()γ() δ ≤≤ (2) = ------π ε K0 sr + I0 sr b Ð rb  v 2 where   q []Θ ()ξ() ≤≤ ε ------K0 Ðipr + I0 Ðipr bra. ()2 () πv ε DI= 1 kb1 Ð -----kI0 kb1 ,  1 fs Here, I ()χsb + K ()sb ω ω f = ------0 1 0 1 , (5b) k ==---- 1 Ð;β2 s ---- 1 Ð;ε β2 I ()χsb + K ()sb v v 2 1 1 1 1 ω pΨ ε I ()sb Ð sΨ ε I ()sb ε β2 χ ------1 2 1 2 1 0 - p = ---- 1 Ð;1 = Ψ ε () Ψ ε (). v p 1 2K1 sb + s 2 1K0 sb

I0, I1, and K0 are the modified Bessel functions; and Without going into details of calculating the coeffi- η γ Θ ξ N ω ω cients , , , and , we give the final formulas for the ω n n φ ---- = ∑ φ ------exp i------()z Ð vt , (3) combinations of the functions I0 and K0 that enter into pv v v expressions (2): n = 1 where the function ε k I ()χsr + K ()sr ηK ()γsr +I ()sr = Ð------2 ------0 0 DÐ1, ω 0 0 2 ()χ() Ði----ξ s pI0 sb1 + K0 sb1 ω Ð1 v (5c) φ ---- = ()f ()ξξ d f ()ξ e dξ v ∫ ∫ Θ ()ξ() K0 Ðipr + I0 Ðipr describes the radiation of a single bunch and summa- ε ()χ()Ψ () (5d) tion is performed over positive and negative frequen- 1k I0 sb + K0 sb 1 r Ð1 = Ð ------()χ()Ψ------()-D , cies. b1 pI0 sb1 + K0 sb1 1 b α η γ Θ ξ The coefficients , , , , and in (2) are deter- Ψ Ψ mined from the following boundary conditions for the where 1 and 2 are the Abel functions Fourier components of the fields and potentials: Ψ () ()() ()() 1 b = J0 pb N0 pa Ð J0 pa N0 pb , (5e) 2 2 E ω ==E ω or()ϕ 1 Ð β ω ()ϕ1 Ð ε β ω z 3 z 2 3 2 2 Ψ = J ()pb N ()pa Ð J ()pa N ()pb . (5f) () 2 1 0 0 1 at rb= 1 ; Integration over frequency in (5) is performed by the ∂ϕ ∂ϕ ω3 ω2 method of residues on the complex plane ω as Hϕω = Hϕω or ------= ε ------()at rb= ; 3 2 ∂r 2 ∂r 1 described in [1]. The poles are calculated from the equation D = 0. Taking into account that solutions of ()ϕε β2 ()ϕε β2 Ezω1 = Ezω2 or 1 Ð 1 ω1 = 1 Ð 2 ω2 (4) this equation are complex-valued, we obtain the following expressions for the longitudinal electric field com- ()at rb= ; ponent (for the three regions of the waveguide): ∂ϕ ∂ϕ ω1 ω2 2 Hϕω = Hϕω or ε ------= ε ------()at rb= ; 2q 1 Ð β I ()χsb + K ()sb Ψ ()r 1 2 1 ∂r 2 ∂r E = Ð------∑------0 0 ------1 - z1 b v I ()χsb + K ()sb Ψ ()b ϕ () 1 λ 0 1 0 1 1 Ezω1 ==0or ω1 0 at ra= . (6a) ()ω 2I1 kr0 dD The longitudinal electric field component Ez in the × ------F ---- , kr v ------ω waveguide can be written as 0 d ωω= λ ∞ q()1 Ð β2ε 2 ()χ () Φω(), 2q 1 Ð β I0 sr + K0 sr Ez = Ð------Re ∫ r E = Ð------∑------v z2 ()χ() b1v I0 sb1 + K0 sb1 Ð∞ (5) λ (6b) ω () i----()z Ð vt ()ω 2I1 kr0 ω v 2I1 kr0 dD × ------φ ---- e iωdω. × ------F ---- , p kr v ------ω kr0 v 0 d ωω= λ

TECHNICAL PHYSICS Vol. 47 No. 4 2002 450 VARDANYAN, OKSUZYAN mode regime sets up at a given frequency (mode) that 2q 1 Ð β2 I ()sr 2I ()kr E = Ð------∑------0 ------1 0 is equal to or is a multiple of (l ≠ 1) the bunch repetition z3 b v I ()sb kr 1 λ 0 1 0 rate. When n = 1 and l = 1, the fundamental mode grows (6c) (at ω = 2πv/d) and we have ω × F ---- dD , v ------ω ω 2 2 d ωω= λ  w πξ w F ------1 = 2Ð1expÐ ------1 ------Ð ------1 z Ð vt pv ω 2 v where 1 d

2 2 ω Ð w 2 w w1d (7a) 2Ðexp ------ξ Ð ---- z Ð vt 1 Ð expÐN------2 2 v v ω w ξ ω 4v × ------cos------1z Ð vt + ------1 . F---- = ------v 2v v wd ωd 2wd w1d  1 Ð exp Ð------cos------+ exp Ð------1 Ð expÐ------v v v v When w = Imω 0,  ωwξ2 wd ×  v cos----z Ð t + ------Ð expÐ------2 2  v 2v v ω ω ξ ω 1 1 1() limF p------= 2N exp Ð------cos------z Ð vt . w → 0 v 4v 2 v ωwξ2 wNd × cos---- z Ð vt + ------Ð exp Ð------(7) v 2v v Further, to clarify the effect of the conducting layer on Cerenkov radiation, we will consider the field in a very thin (in terms of wavelength) cylindrical dielectric ωwξ2 wd layer coaxial with the waveguide. Having studied the × cos---- z Ð vt + ------Ð Nd Ð exp Ð------v 2v v field in this layer, we will turn to the effect of a thin conducting layer applied on the interior surface of the channel. ω ξ2  × w () cos----z Ð vt + ------Ð N Ð 1 d . v 2v  LOSSLESS DIELECTRIC CYLINDRICAL INSERT Here, N is the number of bunches in the train, Reω IN THE WAVEGUIDE ω, and Imω w. In the conducting layer, the real part The study of Cerenkov radiation in a cylindrical of the permittivity is taken to be equal to 1. dielectric layer is of interest for the waveguide technol- The radiation is maximum when the fields produced ogy and also helps to make some useful estimates. The by individual bunches add in phase. If the thickness of formulas obtained in [1] can be used to develop an algo- the conducting layer and the radius of the channel are rithm for numerically analyzing the effect of layers given, the waveguide radius can be chosen so as to sat- with various thicknesses, in particular, very thin layers. isfy the condition for in-phase radiation of the bunches In order to perform a comprehensive analysis of the ω π ( nl/2v)d = l (l = 1, 2, …). In this case, the single- field, we calculated the resonance radius of the waveguide (at the bunch repetition rate f = 3 GHz and the permittivity ε = 2.1), a = 3.8 cm. With this value, the Table main condition for the Cerenkov radiation to have a dis- l λ, cm δ, cm |E |, kV/m crete spectrum at harmonics of the bunch repetition fre- 1 quency, i.e., the condition for synchronism between the 1 10.0 3.1 48.490 charge motion and the phase velocity of the wave gen- ω γ 2 5.0 1.4 42.230 erated (vph = / = v), can be satisfied by choosing the appropriate radius and thickness of the dielectric pipe. 3 3.33 0.81 33.750 It was found that, when the dielectric cylinder is very 4 2.5 0.65 24.950 thin, the particle velocity can be equal to the phase 5 2.0 0.4 17.200 velocity of the wave only for modes with very large 6 1.66 0.3 11.150 numbers (for very short wavelengths). In other words, with a thin dielectric insert, the structure becomes a 7 1.43 0.25 6.850 below-cutoff waveguide at longer wavelengths (smaller 8 1.25 0.2 4.050 mode numbers). It should be noted here that, at very 9 1.11 0.17 2.350 short wavelengths, the condition β2ε > 1 is violated 10 1.0 0.14 1.400 because the permittivity tends to 1. At a wavelength λ ≈ 11 0.91 0.12 1.300 10 cm, the waveguide with a cylindrical insert less than 1 cm thick is evanescent. For the second harmonic of 12 0.83 0.1 0.130 the bunch repetition frequency (λ ≈ 5 cm) to propagate

TECHNICAL PHYSICS Vol. 47 No. 4 2002 CERENKOV RADIATION FIELDS 451

Ez, MV/m Ez, MV/m 30 (a) 20 30 (z – vt), cm 10 0 20 –20 –15 –10 –5 –10 –15 10 –20 0 123r, cm Fig. 2. Ez component of the ﬁeld produced at the axis of the channel by a train of N = 10000 bunches versus parameter δ Å (z Ð vt). 35 MV/m = 70

(b) in the waveguide, the thickness must be 0.5 cm or less; for the fifth harmonic (λ ≈ 2 cm), ≈1 cm; and so on. The 10 MV/m field amplitude decreases with increasing harmonic number. In our calculations, the waveguide radius a = 3.8 cm and the channel radius b = 0.7 cm were constant and the thickness δ was taken such that this waveguide radius (a = 3.8 cm) was resonant for various harmonics I ε'' = 103 ε = 2.1 of the bunch repetition frequency. b – δ b The table lists the electric field amplitude for the fundamental mode versus thickness of the dielectric Fig. 3. Radial distributions of the field component Ez over cylinder. The data were calculated by formulas the cross section (z Ð vt) = 0 in the (a) waveguide and obtained in [1] for a single bunch (N = 1). As follows (b) conducting layer I. N = 10000 bunches. from the table, the amplitude of the fundamental mode rapidly decreases with decreasing thickness δ. In par- δ , kV/m ticular, at = 0.05 cm, the amplitude |E1| is as small as Ez 0.07 V/m; therefore, the generation of the Cerenkov 20 waves in a thin (≈70 Å thick) conducting layer can be 15 neglected and the real part of the permittivity can be assumed to be equal to 1 (ε' = 1). 10 5

INTENSITY OF THE CERENKOV RADIATION 0 1234 (QUASI-CONDUCTING LAYER) r, cm The thin conducting layer may significantly decrease the field amplitude and reduce the efficiency Fig. 4. Radial field distribution of eleven higher modes over the cross section (z Ð vt) = Ð2.5 cm at N = 10000 bunches of the Cerenkov radiation mechanism in the two-beam (at the axis, the field is zero). acceleration scheme to zero. In fact, calculations by formulas (6) and (7) for a 10-Å-thick high-conductivity (ε'' ≈ 107) layer show that the field attenuation is intol- erably high. At the same time, to remove the static FN charge accumulated on the surface of the channel, it is 3000 not necessary to use a good conductor: a quasi-conduct- 2500 ing material would suffice for this purpose. 2000 Figure 2 shows the field produced by 10 000 bun- 1500 ches, each comprising 3 × 109 electrons, at the 1000 waveguide axis for a 70-Å-thick layer and the permit- 500 tivities ε' = 1 and ε'' = 103. The field was calculated by formulas (6a) and (7). Estimates made for eleven 0 5000 10 000 15 000 20 000 25 000 modes show that only the fundamental mode is N enhanced in the periodic train of bunches and the quasi- monochromatic wave with an amplitude of ≈35 MV/m Fig. 5. Interference factor FN versus number of bunches. sets up, as illustrated in Fig. 2. The curve rapidly saturates.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 452 VARDANYAN, OKSUZYAN Thus, the 70-Å-thick quasi-conducting layer does the complex permittivity, the attenuation due to the not prevent the generation of a sufficiently strong field, dielectric loss will primarily be observed. while removing the static charge. The Cerenkov radiation in a circular dielectric- Figure 3a plots the cross-sectional field distribution loaded waveguide with an axial channel covered by a in the presence of the 70-Å-thick quasi-conducting high-conductivity layer has been considered. Even if layer. Inside the layer, the field decreases more rapidly this layer is very thin (≈10 Å), it introduces a strong than in the dielectric; in the channel, it is quasi-uniform attenuation into the system, making it low-efficient. At as before. For illustration, Fig. 3b represents the field the same time, a low-conductivity layer can provide distribution in the layer on an enlarged (by a factor of both a small attenuation and the effective removal of 106) scale. the static charge from the wall. Figure 4 shows the radial distribution of the total Finally, in a waveguide with high losses (nontrans- field from 11 higher modes at the zero phase of the first parent medium), the condition for Cerenkov radiation mode (z Ð vt = Ð2.5 cm). The field amplitude is less than (synchronism between the phase velocity of the wave 0.1% of the fundamental mode amplitude (20 kV/m generated and the velocity of the emitting charge) and 35 MV/m) in this case. becomes uncertain, because the phase velocity itself In Fig. 5, the interference gain factor turns out to be complex. Therefore, the effect of the real part of the permittivity should be neglected, especially w d 1 Ð exp ÐN------1 because, for a very thin (in terms of wavelength) dielec- v tric layer, ε 1 at high frequencies and the condition FN = ------for Cerenkov radiation is violated as we noted above. w d 1 Ð exp Ð------1 v ACKNOWLEDGMENTS is plotted against N for the case when the fields produced by N bunches add in phase. As is seen, the lossy We are grateful to É.D. Gazazyan for the valuable film causes the curve to saturate. At N = 3000 bunches, discussions. This work was supported by the INTAS the gain is ≈70%; at N = 10 000, ≈99%. (grant no. A-087).

CONCLUSION REFERENCES The channel through which electron bunches pass 1. A. S. Vardanyan, Izv. Akad. Nauk Arm., Fiz. 34, 323 insignificantly decreases the field intensity at the bunch (1999). repetition frequency when the wavelength of the radia- 2. V. L. Ginzburg and I. M. Frank, Dokl. Akad. Nauk SSSR tion (≈10 cm) is much greater than the channel diameter 56, 699 (1947). (1.4 cm). At the fundamental harmonic, the longitudi- 3. L. D. Landau and E. M. Lifshitz, Course of Theoretical nal electric field component is almost constant across Physics, Vol. 8: Electrodynamics of Continuous Media the channel. (Nauka, Moscow, 1982; Pergamon, New York, 1984). The study of the Cerenkov radiation in the thin 4. B. M. Bolotovskiœ, Usp. Fiz. Nauk 75, 295 (1961) [Sov. cylindrical layer has shown that Cerenkov waves are Phys. Usp. 4, 781 (1962)]. efficiently generated at very high harmonics of the fun- 5. É. D. Gazazyan and É. M. Laziev, Izv. Akad. Nauk Arm. damental frequency and that the dielectric layer exerts SSR, Ser. Fiz.-Mat. Nauk 16 (2), 79 (1963). a minor effect on the Cerenkov radiation at the repetition frequency of the electron bunches. Therefore, with Translated by A. Khzmalyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

SURFACES, ELECTRON AND ION EMISSION

The Laser-Induced Formation of Superhard Structures and Phase Transformations of Carbon in the Near-Surface Layer of Cast Iron G. I. Kozlov Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow, 117526 Russia Received May 10, 2001; in ﬁnal form, August 20, 2001

Abstract—Superhard structures with a microhardness of (2–5) × 1010 Pa formed in the near-surface layer of gray iron covered by a thin copper inductor and processed by laser radiation are experimentally studied. The copper favors the formation of globular (or even spherical) carbon and stimulates the saturation of the near-surface layer of iron by carbon. The elemental analysis is performed, and the Raman spectrum from carbon globules is recorded. Based on this spectrum and the spectra from other carbon structures, the author argues that the laser processing converts the graphite of the near-surface layer to pyrolytic carbon, which offers a high hardness. A possible mechanism behind this process is discussed. © 2002 MAIK “Nauka/Interperiodica”.

Laser quenching is a most effective way of harden- intensity on the surface. The combination of the multi- ing the metal and alloys surfaces. This process is highly beam CO2 laser and the long-focus lens provided the nonequilibrium because the characteristic time of crys- uniform distribution of the laser intensity over the cen- tal structure rearrangement in metals and alloys sub- tral part of the beam in the focal plane and a natural jected to laser processing is comparable to the heating decline of the intensity to the beam periphery. and cooling times. Physically, the hardening of metals To trigger the mechanism inducing the phase trans- and alloys implies the production of structures making formations, a thin copper layer was applied on the iron plastic deformation (or, in other words, the motion of surface immediately before the processing. Copper as dislocations) difficult. Basically, this can be achieved in an inductor was taken because it, like γ-Fe, has the fcc different ways, e.g., by producing phase inhomogene- lattice and is therefore expected to strongly influence ities in the crystal structure of metals and alloys, the phase transitions, the structure and properties of decreasing the grain size, or fabricating harder and various constituents, and eventually the physical and more perfect crystal structures. mechanical behavior of the iron surface. After a number In this work, which elaborates upon the studies of trials, we succeeded in developing the effective tech- [1, 2], an intriguing phenomenon is reported: the nique for applying a thin copper layer on iron with a induced formation of superhard structures in the near- special paste. The paste consists of copper powder and surface layer of gray iron when the material is covered the emulsion readily absorbing the radiation of a CO2 by a thin copper inductor and then laser-processed. The laser. It is easy to apply on the surface, and its thickness term “induction” here means that the crystal structure in these experiments was 100Ð150 µm. Once the paste of the inductor controls (during cooling) the postirradi- had been dried, the surface was irradiated. The radia- ation phase transformations in near-surface layers of tion power was kept constant at 5.5 kW, and the radia- metals and alloys being in contact with the inductor. tion intensity was selected in such a way that the thickness of the molten iron was small and the ironÐcopper mixing was insignificant. With this beam power, the EXPERIMENTAL SETUP AND METHODS above condition was met when the focal spot diameter OF INVESTIGATION was about 8 mm and the scan rate over the surface, The structure and phase transformations taking 0.67 cm/s. place in the near-surface layer of gray iron under the After the processing, the copper layer on the surface laser irradiation were studied with the special setup was removed by milling to a depth of 0.5 mm (first step) incorporating a cw multibeam 6-kW CO2 laser with dif- and 0.9 mm (second step). Then, a 10-µm-thick layer fusion cooling [3], a focusing system, and a special was chemically etched off to minimize the effect of the control device that provides scanning over the surface milling-deformed zone. The milling exposed the spec- being processed. As a focusing means, a NaCl lens with ular matrix within which superhard structures formed a focal distance of 1.1 m was used. The lens could move during the laser processing, as will be evidenced by fur- along the laser beam axis, thus making it possible to ther examination. We studied the phase composition, vary the size of the focal spot and, hence, the radiation microhardness, and microstructure on the surface of the

454 KOZLOV steps and on the end faces of specially prepared speci- vides the microhardness in the range of (0.8Ð1.1) × mens. 1010 Pa [4]. The phase and elemental compositions, as well as The superhard structures formed by the laser pro- the microstructure, were studied by X-ray diffraction cessing of the iron surface covered by the thin copper (monochromatic FeKα radiation) and in a CAMSCAN film is undeniably related to the inducing effect of cop- CS44C-100S scanning electron microscope equipped per on the mechanism and the kinetics of postirradia- with a LINC ISIS-L200D energy dispersion analyzer. tion phase transformations (i.e., those proceeding dur- The size of the excitation zone was about 0.3 mm2. The ing cooling). It is important to identify structure com- microhardness distribution across the laser radiation ponents responsible for the so high microhardness track was found with a PMT-3 hardness meter at an values. For this purpose, we performed the phase com- indentation load of 0.1 kg. The microhardness was also position analysis. measured in separate regions with various optical con- In this analysis, the diffraction pattern from the area trast. processed was compared with that from the unproc- α essed region, which incorporated largely -Fe, Fe3C, and carbon. The pattern taken from the first step in the MICROHARDNESS, PHASE COMPOSITION, specular matrix indicates that here the material consists AND MICROSTRUCTURE OF THE LASER of ferrite, austenite, cementite, and carbon. All reflec- IRRADIATION ZONE tions are strongly diffuse in comparison with those for the unprocessed region, indicating the disperse struc- The microhardness of the matrix was naturally of ture of these components. In the range of angles corre- primary interest early in this study. Its distribution sponding to the martensite doublet, the split of the (0.25-mm pitch) across the hardened zone at the first reflection is faint. In deeper layers (second step), the step of the specular matrix is shown in Fig. 1. The dis- intensity of martensite doublet reflection grows and the tribution is nonuniform: in places, the microhardness is reflections become narrower. From these results, we as high as (2Ð5) × 1010 Pa. This means that superhard can conclude, with surprise, that the hardening of gray structure components have formed. The highest value, iron is not related to the formation of martensite. It 5.15 × 1010 Pa, was observed within a regularly shaped seems that the inducing effect of the copper is in that it light gray region of size about 20 µm. In an optical acts as an austenite-stabilizing agent, reducing the aus- microscope, one can distinguish several structure com- tenite transformation temperature and thus greatly ponents differing in color, shape, and hardness. We refining pearlite, which forms during cooling. How- failed to measure the microhardness of most dark glob- ever, grain refining is only one reason for iron harden- ular precipitates (as will be referred to later), because ing by laser irradiation under our experimental condi- they do not give optical contrast (the indentations tions. We failed to determine structure components against the dark background are not seen). In deeper responsible for the so high microhardness values with layers (second step), the microhardness values were the help of the phase analysis. It was established, how- lower; however, here too, superhard regions, (1.3Ð ever, that martensite forms in small amounts and is not 1.5) × 1010 Pa, were observed. Such high values cannot responsible for the laser-induced hardening of induc- be associated with laser thermal hardening, which pro- tor-covered iron. The microstructure of the specimen was examined 10 on its end face. The optical image of the specular zone H, 10 Pa is shown in Fig. 2, where the boundary of the molten 10 zone is distinctly seen. Within this zone, the surface 2 layer of the iron is heavily saturated by carbon. Along 8 with black carbon precipitates, dark gray needles (pre- 1 sumably bainite) and light ferrite regions with round- 6 ended dispersed cementite precipitates are observed. The microstructure varies with distance from the sur- 4 face. The concentration of the graphite precipitates decreases, and lamellar pearlite becomes a dominant 2 component. The number of the light ferrite regions with cementite precipitates inside also decreases. 0 0 1 2 3 4 5 6 More detailed information on the near-surface d, mm microstructure of the iron in the matrix zone was derived with the electron microscopic pattern taken Fig. 1. Microhardness H across the hardened zone (d is the from the central region of the laser spot (Fig. 3). Here, width of the zone). A cast iron specimen with a laser irradi- ≈ µ ation track is shown. The composition, microstructure, and three layers can be distinguished: a 200- m-thick dark microhardness were studied on the (1) end face and (2) on near-surface layer with the dispersed structure, a the surface. ≈100-µm-thick transition layer with a minor concentra-

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THE LASER-INDUCED FORMATION OF SUPERHARD STRUCTURES 455

1 1

500 µm

Fig. 2. Microstructure of the specular matrix zone. 200 µm (1) Boundary of the molten zone. tion of precipitates, and a light layer with carbon in the Fig. 3. Microstructure of the iron surface layer across its globular and lamellar forms. As follows from the ele- depth after laser processing. mental analysis, the dark layer consists mainly of carbon and copper, while the transition layer includes mainly iron and carbon. At the point marked by the cross in Fig. 2, the element concentrations are as follows: carbon, 37.82%; iron, 58.56%; silicon, 2.88%; and copper, 0.74%. Thus, having studied the phase composition and the microstructure of the matrix, we conclude that the inducing effect of the copper is two- fold: the considerable enrichment of the surface by carbon and the stabilization of austenite. The latter appreciably reduces the austenite transformation temperature and, accordingly, reﬁnes structures resulting during cooling (including carbides and pearlite).

THE TRANSFORMATION OF GRAPHITE INTO PYROLYTIC CARBON DURING LASER (‡) PROCESSING The question remains to be answered as to which structure constituents are responsible for the high microhardness values. The only way to clarify the situation seems to be the in-depth analysis of the iron surface microstructure. The micrographs taken both before and after the laser irradiation (Fig. 4) distinctly show that the molten zone contains a large amount of globular carbon precipitates. Note first of all that, during the crystallization of the iron from the melt, the graphite nucleates at one site and then grows, taking on the form of heavily curved flakes. Therefore, in the plane of the metallographic section (Fig. 4b), graphite in iron is usually seen as straight and curved lamellas, which are different sections of the graphite flakes. After the laser µ irradiation of the iron surface, the carbon structure in (b) 100 m the near-surface layer radically changes, as follows from Fig. 4a. The initial variously shaped lamellas virtually transform into globules, some of which having a Fig. 4. Microstructure of the iron surface layer (×100) nearly perfect spherical shape (Fig. 5). (a) after and (b) before laser processing.

TECHNICAL PHYSICS Vol. 47 No. 4 2002

456 KOZLOV elemental analysis showed that some globules consist of nearly pure graphite, while others contain silicon (to 7.1%), chromium (4.5%), iron (1.1%), and oxygen (20.7%) impurities. Such a difference in the globule composition is likely to be associated with the dynamics of the phase transformations and diffusion processes during the laser irradiation. The concentration of the first three impurities drops with irradiation time. As for the oxygen, it is by far adsorbed on carbon from air after the metallographic section has been prepared and the globules have been exposed. Thus, the globules can be thought of as carbon structures forming from the melt. The structure of the globular carbon was judged 25 µm from the Raman spectra for carbon inclusions in the irradiation zone at the first step in the specular matrix. Unlike the Raman spectra for pyrolytic graphite and Fig. 5. Microstructure of the laser-processed zone with diamond, of which narrow peaks at 1580 and 1332 cmÐ1 spherical carbon globules (×400). are typical, the spectrum of the globular carbon (Fig. 6) has a specific two-humped structure. This circumstance I, arb. units and also the widths of the peaks indicate that the laser processing of the copper-covered gray iron surface (a) 1580 cm–1 –1 causes a strong disorder in the structure of the graphite 1350 cm contained in the near-surface layer of the gray iron. As a result, long-range order, typical of crystal structures, practically disappears, and the molecular carbon structures become fragmented and chaotic; that is, carbon becomes amorphous. The widths of the peaks in the Raman spectrum imply that the particles are small, 50Ð 80 nm in size. It was of interest to compare the Raman spectrum of the globular carbon with that of CVD pyrocarbon [5]. The latter spectrum is shown in Fig. 6b. The surprising 1200 1400 1600 thing is that these spectra, which were taken from the structures prepared under entirely different conditions, 1350 cm–1 1580 cm–1 are almost identical. One therefore can conclude that (b) the laser processing of the copper-covered iron converts the near-surface graphite to pyrocarbon, offering a high hardness.

GRAPHITE-TO-PYROCARBON CONVERSION MECHANISM A mechanism of graphite-to-pyrocarbon conversion deserves special discussion. It can be suppose that, over a period of time after the laser irradiation, the near-surface carbon is in the liquid state. This supposition is 1200 1400 1600 1800 based on the analysis of the iron microstructure in the ∆v, cm–1 irradiation zone and explains the shrinkage of the carbon into globules and the subsequent formation of the Fig. 6. Raman spectrum intensity vs. Raman line shift ∆v: amorphous pyrocarbon. In other words, it may be con- (a) carbon inclusions in the laser irradiation zone and ceived that the amorphous pyrocarbon is the super- (b) pyrocarbon. cooled liquid phase passed to the condensed state. It is known that if the cooling rate of some alloys is sufﬁ- ciently high, the viscosity of the liquid metal increases Next, it is necessary to elucidate the structure and to such a degree that crystallization centers have no composition of the globular carbon. Whether it is time to arise and the metal solidiﬁes into an amorphous graphite or has some another structure? To answer this structure with short-range order. If short-range order question, we performed appropriate investigations. The extends to 1.3Ð1.8 nm, such a structure is considered to

TECHNICAL PHYSICS Vol. 47 No. 4 2002

THE LASER-INDUCED FORMATION OF SUPERHARD STRUCTURES 457 be amorphous [4]. If its characteristic size is between 2 associated literature data indicate that the crystalliza- and 10 nm, the structure is fine-grained. Since dislocation of the copperÐiron system from the melt after laser tions cannot glide over slip systems in amorphous processing causes stresses as high as 4 × 108 Pa due to alloys, they usually offer a very high strength. the considerable difference in the temperature coeffi- Until recently, it was however believed that the liq- cients of linear expansion. So high stresses may, in uid phase of graphite (for example, due to laser pro- principle, favor the transition of graphite to the liquid cessing) can form and exist only under high pressures. state. It is easy to show that high-pressure conditions may The question of whether or not liquid graphite exists occur in our experiments. First, we note that the tem- under similar conditions has arisen many times but still perature and phase stresses induced by the laser pro- remains unresolved. It has been believed until recently cessing of the copper-covered iron in the surface and that graphite heated to temperatures above 3800 K subsurface layers of the metal develop in a complex under the atmospheric pressure vaporizes without pass- manner. After the irradiation, the surface layer cools ing to the liquid phase. However, if the pressure is and shrinks last. When shrinking, it acts on the subsur- increased to 107 Pa, graphite passes to the liquid (mol- face layer, causing compressive temperature stresses in ten) state. Indeed, the signs of thermal reflowing were the latter, while the surface is under tensile stresses. If detected in experiments on laser heating of graphite at the phase transformations in the subsurface layer cause pressures p > 107 Pa. It was supposed [6] that the graph- an increase in the specific volume (for example, mar- 7 tensitic transformations), the compressive stresses will ite reflowing also takes place at p < 10 Pa. In this case, inevitably grow because of the phase component. Anal- however, the reflowing signs cannot be detected, since ysis of the temperature and phase stresses arising upon the film of the melt is very thin and quickly evaporates laser processing of copper-covered cast iron is a com- after the heating. The parameters of the triple (solidÐ plex problem, which calls for special consideration. liquidÐvapor) point for carbon were determined by extrapolating the curves of phase equilibrium for Of interest is to estimate temperature stresses aris- graphite melting and evaporation [6]. They were found ing in the iron when the iron–copper system solidifies 5 to be pT = 10 Pa and TT = 4000 K. These values consid- by cooling from the laser processing temperature to 7 room temperature. The reason for the stresses is the erably differ from those established earlier (pT = 10 Pa, great difference between the thermal expansion coeffi- TT = 5000 K). cients of the metals. For copper and iron plates of Unfortunately, the possibility of graphite-to-car- length l in the free state, the difference in the linear byne phase transition during graphite heating was not expansion coefficients would cause a difference ∆l in considered in [6]. Therefore, it is not quite clear to α α the linear dimension. Let 1 and 2 be the thermal which solid and liquid phases the triple point found cor- expansion coefficients of copper and iron, respectively, responds. One can assume that the above-mentioned and T1 and T0 be the melting point of copper and the considerable spread in the triple point parameters is normal temperature. Then, the value of ∆l is given by associated with a change in the phase composition. The ∆ ()α()α change in the phase composition may also be the reason llT= 1 Ð T 0 2 Ð 1 . (1) for the dependence of the melting point of “graphite” Under experimental conditions, the linear sizes will on the heating time of the specimen, which is observed not of course differ because the copper and iron layers in experiments carried out by various researchers. are fused into each other. From the phase diagram of carbon (Fig. 7) sug- Using the Hooke law and taking into account the gested by Whittaker [7] and complemented by the triple fact that in equilibrium compressive stresses in the iron point parameters obtained in [6], it follows that at tem- must be balanced by tensile stresses in the copper, we peratures above 2600 K graphite transforms into car- arrive at the approximate expression for the compres- byne; hence, no graphite liquid can exist. The diagram σ sive stress c in the near-surface iron layer: also shows that carbyne is stable between 2600 and 3800 K in a wide pressure range up to the diamond for- σ E1E2 ()α()α mation pressure. The triple point parameters for car- c = Ð------T 1 Ð T 0 2 Ð 1 , (2) E1 + E2 byne somewhat differ from those reported above: pT = × 4 2 10 Pa and TT = 3800 K. Hence, at temperatures where E1 and E2 are the respective elastic moduli of above 3800 K, carbyne passes to the liquid state. It was copper and iron. reported [7] that carbyne liquid is transparent and col- Substituting the numerical values of the quantities orless, as well as has a low emissivity [7]. This corre- involved in Eq. (2), we find that stresses arising in the lates with the comment [8] that films of molten carbon iron under the experimental conditions may reach 4 × were “sufficiently transparent.” One can therefore con- 108 Pa. This value is consistent with the literature data clude that the triple point parameters obtained in [6] [4] obtained by directly measuring temperature stresses may also refer to carbyne. Thus, carbyne, not graphite, due to laser processing. Thus, the simplified one- liquid must exist! Similarly, at still higher pressures, dimensional estimate of the possible stresses and the one might expect the existence of diamond liquid at the

TECHNICAL PHYSICS Vol. 47 No. 4 2002 458 KOZLOV

p, Pa far exceeds that of graphite. The density of various car- 1 × 9 byne modiﬁcations varies between 2.68 and 12 10 3.43 g/cm3, while that of graphite is no more than 6×109 2.25 g/cm3. The formation of triple bonds indicates that 3 liquid carbyne must consist of polyine molecules of ≡ type (ÐC CÐ)p, which are stable at high temperatures. 2 Thus, based on the body of the data reported, we can 1 × 105 argue that when the cast iron surface is subjected to laser processing, the graphite of the surface layer heats 2 × 104 up, takes the globular shape, and turns to carbyne (or carbyne liquid if the temperature is sufﬁciently high). 4 5 During the subsequent fast free cooling of the surface layer, the carbyne decomposes, turning to opaque pyro- 4 × 10–1 carbon. In summary, we stress that the above effects are of a general character. The laser processing of various metalÐalloy systems provides extreme experimental 2600 3800 T, K conditions and offers considerable scope for the control of phase transformations. Researchers are given an Fig. 7. PÐT phase diagram of carbon suggested by Whit- opportunity to modify the surface and improve its phys- taker and complemented by the triple point (+) found in [6]. ical and mechanical parameters, as well as develop (1) Diamond; (2) carbyne; (3) liquid; (4) graphite; and materials with unique properties. (5) vapor.

ACKNOWLEDGMENTS boundary of the diamond region! Of course, this rea- The author is indebted to Prof. V.T. Bublik and soning is valid if the Whittaker phase diagram is true, T.B. Sagalova for the fruitful discussions and assis- rather than purely hypothetical, and the carbyne region tance. does exist. This work was partially supported by the Russian It should be noted that the existence of carbyne is Foundation for Basic Research (grant no. 00-01- presently virtually beyond question. It has been found 00212). in the natural form and is produced as thin films in the laboratory conditions. From the phase diagram given, it follows that carbyne can be obtained from carbon if REFERENCES one, first, reaches the region where carbyne is thermo- 1. G. I. Kozlov, Pis’ma Zh. Tekh. Fiz. 25 (24), 61 (1999) dynamically stable and, second, retains it during cool- [Tech. Phys. Lett. 25, 997 (1999)]. ing to the normal conditions. In other words, it is nec- 2. G. I. Kozlov, Pis’ma Zh. Tekh. Fiz. 26 (11), 84 (2000) essary that quenching be fast, since, if the temperature [Tech. Phys. Lett. 26, 490 (2000)]. is reduced smoothly, reverse processes take place and 3. G. I. Kozlov and V. A. Kuznetsov, Kvantovaya Élektron. carbyne readily decomposes, turning to carbon. The (Moscow) 16, 1360 (1989). ratio of the forward and back reaction rates (i.e., the 4. A. G. Grigor’yants and A. N. Safonov, Principles of rates of carbyne formation and decomposition) is Laser Thermostrengthening of Alloys (Vysshaya Shkola, 1 : 500 [7]. For this reason, carbyne has been obtained Moscow, 1988). only as thin films to date. As for the molecular structure 5. K. J. Huttinger, Chem. Vap. Deposition 4, 151 (1998). of carbyne, we note that the dissociation of graphite at 6. É. I. Asinovskiœ, A. V. Kirillin, and A. V. Kostanovskiœ, temperatures above 2600 K implies the breaking of the Teplofiz. Vys. Temp. 35, 716 (1997). single bonds and the production of triple-bond chains in 7. A. G. Whittaker, Nature 276, 695 (1978); Science 200, the form of long molecules. These chains may vari- 763 (1978). ously combine to form the hexagonal structure of one 8. É. I. Asinovskiœ, A. V. Kirillin, A. V. Kostanovskiœ, and of the carbyne’s modification. V. E. Fortov, Teplofiz. Vys. Temp. 36, 740 (1998). As a result of the triple bond formation, atoms in the chain move closer together, so that the carbyne density Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 459Ð464. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 88Ð93. Original Russian Text Copyright © 2002 by Popok, Azarko, Khaibullin.

SURFACES, ELECTRON AND ION EMISSION

The Effect of High Implant Doses and High Ion Current Densities on Polyimide Film Properties V. N. Popok*, I. I. Azarko*, and R. I. Khaibullin** * Belarussian State University, Minsk, 220050 Belarus e-mail: [email protected] ** Kazan Physicotechnical Institute, Russian Academy of Sciences, Kazan, 420029 Russia e-mail: [email protected] Received September 17, 2001

Abstract—Ar+ and Ar2+ ions with energies of 40 and 80 keV are implanted into thin polyimide films. The implant doses and the ion current densities are varied in a wide range between 2.5 × 1014 and 1.5 × 1017 cmÐ2 and between 1 and 16 µA/cm2, respectively. The effect of the implantation parameters on the electrical, paramagnetic, and optical properties of the ion-modified near-surface polymer layer is studied. It is shown that the radiation-stimulated thermolysis of polyimide and its chemical constitution are responsible for a monotonic growth of the electrical conductivity of the layer with increasing ion current at a given implant dose. When the ion current density is fixed, the conductivity grows stepwise with implant dose, whereas the concentration of paramagnetic centers and the optical transmission of the modified layer decrease. The dependences observed are treated within a model of the structural reconfiguration of the polymer carbonized phase formed during the implantation. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION radiation hardness and thermal stability. On the other hand, an increase in the ion current density at a fixed In recent years, conductive polymers have become a dose raises the conductivity of the implanted polymer popular micro- and optoelectronic material in both sci- layer [9]. Thus, the optimization of the ion implantation entific research and applications [1]. The reason is the conditions for the formation of conductive layers with advances in the synthesis of new conductive polymers desired electrical properties seems to be a topical prob- and, hence, their extended potentialities in these areas lem in ion-beam modification of polymers. of technology. Polymer films with the metallic type of In this work, we study the effect of argon ion conduction have found application as corrosion inhibi- implantation on the electrical, paramagnetic, and opti- tors, compact capacitors, and antistatic coatings for cal properties of polyimide. Polyimide is a representa- photographic films and monitor screens. Moreover, the tive of the most thermally stable polymers and allows methods of synthesizing polyconjugated systems the ion current density to be varied in wide limits up to (which have evolved from the early 1970s) have made µ 2 it possible to produce polymers of n- or p-conductivity 16 A/cm . To date, such a high value of the ion current by doping. These materials are used in fabricating density, as applied to implantation into polymers, has organic electronic components, such as light-emitting not been reported in the literature. The use of argon ions diodes, transistors, solar cells, storage batteries, etc. eliminates the doping effect of impurities and makes it [1, 2]. possible to study the purely radiation effect of ions on the polymer. In addition, argon implantation into poly- A promising way for producing conductive polyimide under other process conditions (ion beam param- mers is ion implantation [3Ð5]. As a result of irradia- eters) has been extensively studied [9Ð11], which tion, a nanostructured system of carbon atoms with allows us to comparatively analyze surface modifica- conjugated bonds is formed in the implanted layer of a tions. polymer [6]. The conductivity of the implanted layer monotonically rises with dose and may vary within 10 to 15 orders of magnitude [3, 7, 8]. The use of high EXPERIMENTAL implantation doses to produce high-conductivity poly- The object of investigation was 40-µm-thick polymer layers implies that the process is carried out at imide films of density 1.43 g/cm3 (Fig. 1a). The advan- increased ion current densities, which cuts the irradia- tage of polyimide is that it does not exhibit the transition time and improves the process efficiency. However, tion to the viscoelastic state with increasing tempera- a high ion current inevitably means a high power being ture. The softening temperature depends on the released in the layer irradiated; therefore, this parame- concentration of imide groups (degree of imidization) ter is critical for organic materials because of their poor s: at s = 1, this temperature exceeds 500 K. The glass

460 POPOK et al.

(a)

O O NON O O n

(b) N . . O .. N N . O O N .. . N . N O O

Fig. 1. (a) Chemical formula of a polyimide elementary unit and (b) polymer structure transformation upon implantation. transition temperature is 590 K [12], so that polyimide silver paste applied on the sample surface. Optical can be considered as fairly stable against radiation- transmission spectra from the implanted films were induced thermal destruction. The implantation of Ar+ recorded in the visible range with a Hitachi-330 spec- and Ar2+ ions was carried out with an ILU-3 ion-beam trophotometer at room temperature. accelerator at room temperature at a residual pressure of 10Ð5 torr. The process parameters were the following: the energy 40 and 80 keV, respectively; doses D = 2.5 × ION-BEAM MODIFICATION OF POLYMER STRUCTURE 1014Ð1.5 × 1017 cmÐ2; and the ion current density j = 1Ð 16 µA/cm2. To avoid sample breakdown and overheat- To better understand results obtained in this work, ing during the implantation, the samples were placed we will first briefly consider the concept of polymer into a special cassette that provided efficient ion charge structure modification that has been generally accepted drainage from the polymer and tight contact between to date [4, 6Ð8]. Because of the electronic and nuclear the films and the metal substrate, which was cooled by stopping of the ions, the molecular structure of the running water. The films implanted were visually free polymer is damaged along the trajectories of the ions. of any serious thermal damages (charring, surface cor- Note that unlike inorganic materials, the breakage of rugation, etc.) under all the implantation conditions. chemical bonds in polymers takes place not only However, at some critical value of the ion current den- because of nuclear collisions but also the electronic sity (18 µA/cm2 at a dose of 1.0 × 1016 cmÐ2 in our excitation [16]. The effect of electron stopping on the case), local fused spots and blisters were detected on polymer structure is most pronounced when the ions the sample surface. are implanted in compounds with heteroatom-containing functional groups (polyimide in our case). The Electron paramagnetic resonance (EPR) studies destruction of the electron shells of atoms and the were carried out with a Varian E112 spectrometer in the breakage of chemical bonds are selective processes, X band (9.3 GHz) at room temperature. The concentra- aimed primarily at breaking weaker bridge bonds tion NPC of paramagnetic centers in the implanted layer between macromolecules and at transforming func- was calculated by the conventional technique [13] with tional chemical groups with a high degree of electron regard for the thickness of the radiation-damaged layer. delocalization. However, the core of the polymer dam- According to the calculations by the SRIM-2000 com- aged region (latent track), which has the highest con- puter program [14] and to Rutherford backscattering centration of radiation-induced defects, forms largely data [15], the thickness of the polymer layer modified as a result of collisions with nuclei. For the doses used is 80 ± 10 and 150 ± 10 nm for 40- and 80-keV argon in this work, the latent tracks overlap [4]. This, as well ions, respectively. The sheet resistivity of the polymer as developed cascades of the electronic excitation and samples modified was measured by the standard two- ionization of a polymer (so-called penumbras [17]), point probe technique [8] with an E7-14 emittance causes the considerable fragmentation of polymer meter. The linear contacts were made of fine-grained chains, followed by radiation-induced cross-linking.

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THE EFFECT OF HIGH IMPLANT DOSES 461

During the stopping of the ions in polymers, the layer ρ, Ω cm irradiated loses highly volatile components and the 109 latent tracks serve as channels through which gases diffuse to the polymer surface. As a result of outgassing, 108 the near-surface layer becomes enriched by carbon 7 atoms and carbon clusters form. The nucleation of car- 10 bon nanoclusters in polymers upon implantation has 6 been confirmed by neutron scattering [18], electron 10 microscopy [19], optical spectroscopy [20], and EPR 105 studies [21]. Today, this fact is considered to be commonly accepted. As the implantation dose increases, 104 the clusters usually grow, interlink, and coagulate up to the formation of a quasi-continuous carbonized layer 024 6 8 10 12 14 16 [22]. The type of conduction in this layer may vary µ 2 from semiconducting to metallic, depending on the j, A/cm irradiation conditions. Fig. 2. Resistivity vs. ion current density j for the samples irradiated by 40-keV Ar+ ions with a constant dose of 1 × 16 Ð2 ELECTRICAL, OPTICAL, AND PARAMAGNETIC 10 cm . PROPERTIES OF POLYIMIDE VS. ION CURRENT T, % DENSITY 40 Figures 2Ð4 show the electrical, optical, and para- 1 magnetic properties of the polyimide layer irradiated as 2 functions of the ion current density j at a fixed dose of 30 4 1.0 × 1016 cmÐ2 of singly charged argon ions Ar+. It follows from Figs. 2 and 3 that, as j grows, the resistivity 8 ρ linearly decreases and the optical transmission of the 20 modified layer monotonically declines. A rise in the 12 conductivity with growing j is generally consistent with 16 data obtained in [9]. However, it is impossible to com- 10 pare the absolute resistivity values, since in [9] they are given in units Ω/ᮀ. In all polyimide samples studied, the singlet isotropic EPR line with a g factor of 0 2.0025 ± 0.0005, which is close to the g factor of a free 460 480 500 520 540 560 580 600 electron, is observed. This EPR signal is related to λ, nm unpaired π electrons, entering into the electron system of the carbon clusters, which are produced during the Fig. 3. Optical transmission curves for the same experimental conditions as in Fig. 2. Figures at the curves show the ion implantation [21]. Figure 4 demonstrates the effect of 2 the ion current density on the EPR signal parameters. It current density values in µA/cm . is seen that N monotonically drops, while the line- PC ∆ 20 3 width ∆H increases with growing j. H, G NPC, 10 spin/cm Thus, as the ion current density increases at a fixed 4.50 3.0 dose, the conductivity grows, the transmission and the concentration of paramagnetic centers decrease, and 3.75 2.5 the EPR line broadens. This implies the structure modification of the polyimide carbonized phase under the 3.00 2.0 conditions of a growing energy effect. As the ion current grows, the composition and the structure of the sur- 2.25 1.5 face layer are affected not only by radiation-induced phenomena (fractionation, cross-linking, atom dis- 1.50 1.0 placements, etc.) but also by the local heating of the polymer near the latent track due to the transfer of the 0.75 0.5 ion kinetic energy to atoms of the polymer matrix (thermal spike effect) [23]. As a result of the implantation, 0 024 6 8 10 12 14 16 unified radiation-induced thermolysis takes place. The µ 2 characteristic time of this process after the ionÐtarget j, A/cm Ð12 Ð10 collision is 10 Ð10 s, and the space around the track Fig. 4. EPR linewidth and concentration of paramagnetic heats up to temperatures above 1000 K [24]. It is known centers vs. ion current density. The experimental conditions that, upon implantation into polyimide, ether links are the same as in Fig. 2.

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462 POPOK et al.

ρ, Ω cm between aromatic rings degrade ﬁrst, which is accom- 109 panied by the release of hydrogen molecules and carbon oxide [8, 25]. During polymer thermolysis, imide 108 2 groups may convert to imine and amide groups [11]. As the ion current density increases, so does the power dis- 107 sipated in the polymer. Therefore, it is logical to assume 1 that extended polycondensed aromatic structures may 106 form (Fig. 1b) by analogy with the structure transfor- 5 mations in polyimide under pyrolysis (1000Ð1200 K) 10 [26]. The growth of the carbonized phase with increas- 104 ing j, which is accompanied by the development of systems of conjugated links, is responsible for the growth 103 of the conductivity of the layer irradiated, as well as for 1015 1016 1017 the decrease in its optical transmission and in the num- D, cm–2 ber of paramagnetic centers. Fig. 5. Dose dependences of the resistivity for the samples irradiated by (1) singly charged (E = 40 keV, j = 8 µA/cm2) and 2 POLYIMIDE PARAMETERS (2) doubly charged (E = 80 keV, j = 4 µA/cm ) argon ions. VS. IMPLANT DOSE

20 3 NPC, 10 spin/cm Figures 5Ð8 show that the experimentally found dependences of the electrical and paramagnetic param- 2.0 2 eters of the polyimide on the implant dose at a constant ion beam power P = Ej = 0.32 W for the singly charged 1 2 1.5 (E = 40 keV, j = 8 µA/cm ) and doubly charged (E = 80 keV, j = 4 µA/cm2) argon ions are nonmonotonic. For doses between 5.0 × 1014Ð2.5 × 1016 cmÐ2, we 1.0 ρ observe a slight decline in (Fig. 5); a rise in NPC (Fig. 6); an almost constant ∆H (2.5Ð3.0 G for Ar+ and 0.5 1.5Ð2.0 G for Ar2+); and a decrease in the optical transmission, which tends to saturation (Fig. 7). For doses 0 × 16 Ð2 14 15 16 17 above 2.5 10 cm , the conductivity rises stepwise 10 10 10 10 + 2+ –2 for the samples irradiated by both Ar and Ar ions. D, cm This rise is accompanied by a drastic drop of the optical Fig. 6. Dose dependences of the paramagnetic center con- transmission, a decrease in the concentration of para- centration for the same experimental conditions as in Fig. 5. magnetic centers (Figs. 5Ð7), and a broadening of the EPR line (to 4.0 G for Ar+ and 2.5 G for Ar2+ ions). Sim- T, % ilar dose dependences of the EPR spectrum parameters 60 were observed in polyimide irradiated by argon ions 1 that had energies close to those mentioned above (40Ð 50 90 keV) [10]. 2 40 These experimental dependences of the polyimide parameters on the implantation dose are easy to under- 3 stand within the above concept of implantation-induced 30 6 4 structure modiﬁcation of polymers. Namely, the 20 5 increase in the concentration of paramagnetic centers 7 with dose in the interval 5.0 × 1014Ð2.5 × 1016 cmÐ2 9 10 8 (Fig. 6) indicates the growth of the carbon clusters in the polymer irradiated. The constancy of the EPR line- 0 width in this interval implies the absence of qualitative 460 480 500 520 540 560 580 600 changes in the electron subsystem of the carbonized λ, nm phase. The evidence for this fact is that ρ for these doses varies only slightly (Fig. 5). The reduction of the Fig. 7. Optical transmission curves for the samples irradi- optical transmission with increasing implant dose also ated by Ar+ ions (40 keV, j = 8 µA/cm2). The dose D = implies the growth of the carbon clusters, which are (1) 5 × 1014, (2) 1 × 1015, (3) 2.5 × 1015, (4) 5 × 1015, basic absorbing centers in the polymer irradiated. The (5) 1 × 1016, (6) 2.5 × 1016, (7) 5 × 1016, (8) 7.5 × 1016, and tendency of the transmission to saturate (curves 4Ð6 in (9) 1.5 × 1017 cmÐ2. Fig. 7) indicates that the processes of radiation-induced

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE EFFECT OF HIGH IMPLANT DOSES 463 cross-linking and carbon cluster agglomeration are T, % coming to an end. 30 1 As has been mentioned, the resistivity, as well as the 25 paramagnetic and optical parameters of the samples, 2 16 Ð2 vary stepwise at doses >2.5 × 10 cm . This can be 20 explained by the transition of the carbonized phase, which is characterized by the presence of extended car- 15 bon cluster agglomerates and conjugated macrorings in the polymer-like structure, to the phase of amorphous 10 carbon or graphite-like material with individual polymer chain fragments. Such a conclusion correlates well 5 with the results obtained in [10], where it has been con- cluded from the temperature dependence of the con- 0 ductivity that the conduction type in Ar+- and N+-irradi- 460 480 500 520 540 560 580 600 ated polyimide changes from semiconducting to metal- λ, nm lic at doses exceeding 3 × 1016 cmÐ2, that is, when the phase change is expected. A similar percolation transi- Fig. 8. Transmission curves for polyimide irradiated as in tion has been also observed in other polymers, for Figs. 5 and 6. The dose is 1 × 1016 cmÐ2. example, after the low-current (0.1 µA/cm2) implanta- × 16 Ð2 tion of light ions with doses higher than 5 10 cm layer modified grows in comparison with the irradiation [22]. Therefore, we can conclude that the phase transi- by Ar2+ ions. tions observed are dose-related, rather than current- related, processes. CONCLUSION Thus, the variations of the EPR spectrum parameters and the resistivity correlate, leading us to conclu- In this article, we report a complex study on the sion that the electrical and paramagnetic properties of electrophysical parameters of polyimide films as func- ion-irradiated organic semiconductors are closely tions of argon ion implantation conditions. It is shown related. that, as the ion current density increases from 1 to 16 µA/cm2 at a high fixed dose (1.0 × 1016 cmÐ2), the However, the electrical, paramagnetic, and optical conductivity of polyimide grows by five orders of mag- parameters of the samples irradiated by singly charged nitude without signs of radiation- or temperature- argon ions differ from the parameters observed on the induced destruction of the polymer. This allows us to samples irradiated by doubly charged ions (Figs. 5, 6, conclude that high-current implantation is an efficient 8). One reason may be that the effects of electronic practical means of controlling the conductivity of this deceleration on polymer structure modification differ polymer. The variation of the electrical parameters of for ions with differing charge and energy. Another rea- polyimide irradiated correlates with that of the para- son is the enhanced effect of radiation-stimulated ther- magnetic and optical characteristics. Based on this fact, molysis. As was noted, the effect of electron stopping the idea of polymer structure modification and carbon- on the molecular structure of polyimide is considerable ized phase formation under the action of high-power because of weak ether links between aromatic rings [8]. ion beams and due to the polyimide specific chemical When being decelerated, 80-keV Ar2+ ions cause the structure is put forward. It is found that the electrophys- significant radiation-induced ionization of the polymer, ical parameters of the polymer films depend on the dose which, in turn, produces much more radical changes in stepwise when it exceeds a certain value (~2.5 × its structure than in the case of the singly charged ions. 1016 cmÐ2 in our case). Such behavior finds an explana- This is reflected by higher values of NPC (Fig. 6) and tion within our model of carbonized phase restructur- lower values of the optical transmission (Fig. 8). The ing. It is shown that the change in the ion charge at the conductivity also drops (Fig. 5) because of the increase constant ion beam power causes only a quantitative dis- in the concentration of radiation-induced defects. It is crepancy between the polymer parameters, because not improbable, however, that the decrease in the opti- they produce different effects of ionization and radia- cal transmission for the case of Ar2+ ions is due to the tion-induced thermolysis; qualitatively, however, the thickening of the modified polyimide layer because of dose dependences of the polyimide parameters remain an increase in the projected ion range (see the section the same. EXPERIMENTAL). When Ar+ ions are implanted into polymers, a higher value of j favors thermolysis effects, which, as was noted, lead to the restructuring of the car- ACKNOWLEDGMENTS bonized phase; consequently, the concentration of para- The authors are indebted to I.A. Karpovich for the magnetic centers decreases and the conductivity of the assistance in the electrical measurements.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 464 POPOK et al.

R.I. Khaibullin thanks the Russian Foundation for 12. C. Feger, M. M. Khojasteh, and J. E. McGrath, Polyim- Basic Research (grant no. 99-03-32548) and the Lead- ides, Chemistry and Characterization (Elsevier, Amster- ing Scientific Russian Schools Organization (grant dam, 1989). no. 00-15-96615) for the financial support. 13. J. E. Wertz and J. R. Bolton, Electronic Spin Resonance: Elementary Theory and Practical Applications (McGraw-Hill, New York, 1972; Mir, Moscow, 1975). REFERENCES 14. J. F. Ziegler, in The Sopping and Range of Ions in Matter 1. The 2000 Nobel Prize in Chemistry (for the Discovery (SRIM-2000), www.research.ibm.com/ionbeams. and Development of Conductive Polymers), Press 15. V. N. Popok, I. I. Azarko, R. I. Khaibullin, et al., Appl. Release, www.nobel.se/chemistry/laureates/2000/press. Phys. A (in press). html. 16. J. Davenas, X. L. Xu, G. Boiteux, et al., Nucl. Instrum. 2. C. K. Chiang, C. R. Fisher, Y. W. Park, et al., Phys. Rev. Methods Phys. Res. B 39, 754 (1989). Lett. 39, 1098 (1977). 17. J. L. Magee and A. Chattejee, J. Phys. Chem. 84, 3529 3. T. Venkatesan, L. Calcagno, B. S. Elman, et al., in Ion (1980). Beam Modification of Insulators, Ed. by P. Mazzoldi and 18. D. Fink, K. Ibel, P. Goppelt, et al., Nucl. Instrum. Meth- G. W. Arnold (Elsevier, Amsterdam, 1987), pp. 301Ð379. ods Phys. Res. B 46, 342 (1990). 4. G. Marletta and F. Iacona, in Materials and Processes 19. G. R. Rao, Z. L. Wang, and E. H. Lee, J. Mater. Res. 8, for Surface and Interface Engineering, Ed. by Y. Pauleau 927 (1993). (Kluwer, Dordrecht, 1995), pp. 597Ð640. 20. I. P. Kozlov, V. B. Odzhaev, I. A. Karpovich, et al., Zh. 5. R. E. Gied, M. G. Moss, J. Kaufmann, et al., in Electrical Prikl. Spektrosk. 65, 377 (1998); J. Appl. Spectr. 65, 390 and Optical Polymer Systems, Ed. by D. L. Wise, (1998). G. E. Wnek, D. J. Trantolo, et al. (Marcel Dekker, New 21. I. P. Kozlov, V. B. Odzhaev, V. N. Popok, et al., Zh. Prikl. York, 1998), pp. 1011Ð1030. Spektrosk. 65, 562 (1998); J. Appl. Spectr. 65, 583 6. V. N. Popok, Poverkhnost, No. 6, 103 (1998); Surf. (1998). Investigation 14, 843 (1999). 22. V. N. Popok, I. A. Karpovich, V. B. Odzhaev, et al., Nucl. 7. E. H. Lee, Nucl. Instrum. Methods Phys. Res. B 151, 29 Instrum. Methods Phys. Res. B 48, 1106 (1999). (1999). 23. A. Miotello and R. Kelly, Nucl. Instrum. Methods Phys. 8. V. B. Odzhaev, I. P. Kozlov, V. N. Popok, et al., Ion Res. B 122, 458 (1997). Implantation of Polymers (Belaruss. Gos. Univ., Minsk, 1998). 24. H. De Cicco, G. Saint-Martin, M. Alurralde, et al., Nucl. Instrum. Methods Phys. Res. B 173, 455 (2001). > 9. M. Iwaki, Nucl. Instrum. Methods Phys. Res. B 175Ð > > 177, 368 (2001). 25. V. S vorc ik, R. Endrs t, V. Rybka, et al., Eur. Polym. J. 31, 189 (1995). 10. A. N. Aleshin, A. V. Gribanov, A. V. Dobrodumov, et al., Fiz. Tverd. Tela (Leningrad) 31 (1), 12 (1989) [Sov. 26. C. Z. Hu, K. L. De Vris, and J. D. Andrade, Polymer 28, Phys. Solid State 31, 6 (1989)]. 663 (1987). 11. A. De Bonis, A. Bearzotti, and G. Marletta, Nucl. Instrum. Methods Phys. Res. B 151, 101 (1999). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 465Ð469. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 94Ð98. Original Russian Text Copyright © 2002 by Takibaev, Spitsyna, Potrebenikov, Yushkov, Kopzhasarova.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

The Operation of High-Speed Emergency Protection Based on Uranium Hexafluoride Zh. S. Takibaev*, S. A. Spitsyna**, G. K. Potrebenikov**, A. V. Yushkov**, and K. A. Kopzhasarova** * Kazakh National Nuclear Center, Almaty, 480082 Kazakhstan e-mail: [email protected] ** Al-Farabi State University of Kazakhstan, ul. Vinogradova 95, Almaty, 480012 Kazakhstan Received March 17, 1999; in ﬁnal form, April 12, 2001

Abstract—The dynamic properties of a gasÐliquid one-time direct emergency protection device used in nuclear power plants are studied using computer experiments within a model suggested. In the case of reactivity emergency, an absorbent is introduced into the reactor core automatically without human intervention and any external control. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION The actuator is a simple pneumatic relay, which In modern reactor designs, the internal safety of a responds when the pressure in the sensing element nuclear reactor is the issue of fundamental importance. exceeds a preset value. When introduced, the absorbent This concept of internal safety means that, using nega- is kept outside the core. If the actuator comes into tive feedbacks, a reactor can be stabilized or shut down action, the metallic bellows moves upward, thereby if there is a real heavy accident hazard. Direct emer- allowing the absorbent to enter into the working gency protection (DEP) functioning in the reactor core volume. unattended and responding to a deviation of any con- The leak tightness and the accuracy of the relay trollable parameter from a preset value is an example. [5, 6] specify the operating pressure, which varies Such devices must feature a high speed, i.e., quickly between 0.12 and 0.18 MPa in automated pneumatic sense an impermissible change in a controllable param- devices [7]. The sensitivity threshold of pneumatic eter and quickly perform a safety action, i.e., to intro- relays is usually 20 to 40% lower. Let the sensitivity duce an efficient neutron absorbent. threshold pth for the relay of the DEP element actuator In a number of works [1Ð4], it has been suggested to be 105 Pa. The bellows (in the pneumatic relay), which improve the DEP speed by introducing compounds activates the control valve and works in tension, may be having a large thermal-neutron fission cross section replaced by a piston. However, the tight joint of the bel- into the sensitive elements. Such DEP devices sense the lows to the sensor body better prevents fissioner primary sign of reactivity emergency (increased neu- leakage. tron flux density), and the resulting temperature effects are used by the control means of the safety system to It has been suggested [1, 2] to use gaseous UF6 as a provide the delivery of a neutron adsorbent to the reac- sensitive medium in DEP devices. At typical intrareac- tor core. tor temperatures, this compound can be considered as In this work, we consider a version of gasÐliquid perfect gas [8]. In addition, its thermophysical proper- DEP built around a one-time element. The sensitive ties are well understood [9Ð11]. medium here is a gas containing a fissile component. 235 When uranium in UF6 is enriched by the U iso- The actuator responds when the pressure in the volume tope, the volume concentration of fissile nuclei in the exceeds a preset value and provides the delivery of a sensing element is given by liquid neutron absorbent into the working volume of the DEP element located in the reactor core. ρ χ γ N A U 5 = ------µ -, (1) U GASÐLIQUID DEP ELEMENT ρ For the DEP element to be placed in the reactor core, where NA is the Avogadro number, U is the UF6 den- µ χ its shape and sizes must be compatible with those of sity, U is the UF6 molecular mass, and is the enrich- standard reactor components: fuel element, rod guide ment (hereafter, the subscript 5 refers to 235U and the tube in the control and protection system (CPS), or fuel subscript U, to UF6; the molecular mass is calculated assembly. without allowance for enrichment).

466 TAKIBAEV et al.

(a) where t is the mean temperature of the fuel element, 4 Θ ρ τ t, K p × 10 , Pa is the coolant temperature, a = qv/ 'C, and 0 = 850 10.5 ξRCρ'/2. t For a rod-type sheath-free fuel element of radius R, th 10.0 ρ pth density ', and speciﬁc heat capacity C, the thermal 800 9.5 resistance ξ is related to the thermal conductivity λ of the fuel kernel material as ξ = R/4λ. For a sheathed fuel 9.0 ξ λ δ λ δ 750 element, = R/4 + / 1, where is the sheath thickness and λ is the thermal conductivity. This relation- t 8.5 1 0 p0 ship holds if heat transfer from the sheath to the coolant 700 8.0 λ ӷ λ 0 0.05 0 0.02 0.04 0.06 0.08 is perfect. Usually, 1 , so that the second term in the expression for ξ is neglected in the estimates. Two (b) geometric parameters (r, the sensing element radius; 850 R1, outer radius of the DEP element sheath) are used in 10.0 ρ τ ξ ρ p the dynamics simulation: a = qv/ UCU, 0 = R1CU U/2, tth th ξ λ τ and = r/4 U. In the steady state ( = 0), the mean tem- 800 Θ τ 9.5 perature of the sensing element is t0 = + a 0. The corresponding pressure p0 in the volume of the sensing ele- 750 t 9.0 p ment should not exceed the pressure pth, at which the 0 0 DEP actuator comes into action: 700 ρ 0 0.05 0 0.05 U Rm < p0 = ------µ -t0 pth, (4) (c) U

10 where Rm is the gas constant. 900 tth pth Since pth is given in specifications for actuators, the range of the UF density is bounded from above by the 9 6 ρ µ Θ 5 Θ 800 inequality U < Upth/Rm . At pth = 10 Pa and = ρ 3 579 K, U < 7250 kg/m . The DEP element operates at µ ρ 8 the temperature tth = Upth/ URm, which must be t0 p0 700 attained at the threshold neutron flux density nth. The 0 0.05 0.10 0 0.05 0.10 temperature growth lags behind the growth of the neu- τ, s tron flux. The thermal delay of the DEP element τ depends on its time constant 0, which, in turn, depends Fig. 1. Mean temperature and pressure in the sensing on the UF6 density in the sensing medium. For a given medium (element) at ρ = 0.9β and τ = 0. (a) ρ = 5000 kg/m3 ρ U Upth = p0tth/t0. χ ρ 3 χ and = 0.045, (b) U = 5000 kg/m and = 0.055, and Let the DEP element be placed into the rod guide ρ 3 χ (c) U = 4500 kg/m and = 0.05. tube of a 1000 VVÉR water-moderated water-cooled × Ð11 power reactor. Then, for nth = 1.4 at Ef = 3.14 10 J σ × Ð28 2 The sensing element (medium) can be viewed as a [13], f = 580.2 10 m . In this case, the mean neu- Φ 18 Ð1 Ð2 volume source with an energy release volume density tron flux density is 0 = 10 s m , the coolant temperature in the core of the 1000 VVÉR reactor is Θ = Σ Φ qv = Ef f , (2) 579 K [14], r = 0.004 m, and R1 = 0.0063 m. If the tem- τ perature dependences of the parameters a and 0 are Φ where Ef is the energy release per fission event, = taken into account upon solving Eq. (3), it can be solved Φ Φ only numerically. 0n, n is the relative neutron flux density, 0 is the neu- Σ γ σ Σ tron flux density in the steady state, f = 5 f, and f and With a positive reactivity introduced, the DEP ele- σ are the respective thermal-neutron macro- and ment behavior can be described by solving heat con- f duction equation (3) jointly with the set of equations of microscopic fission cross sections for 235U nuclei. point kinetics [12] The quasi-stationary heat conduction equation for dn ρβÐ ------==------n +1,Σλ C , n()τ = 0 an infinite cylinder is given by [12] dτ l i i Θ β β dt Ð t dCi i λ τ i ,,… -----τ = ------τ - + an, (3) ------τ- = ---- n ÐiCi, Ci() = 0 = ------λ , i = 1 6, (5) d 0 d l l i

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE OPERATION OF HIGH-SPEED EMERGENCY PROTECTION 467 Θ dt Ð t ()τ l, m (a) (b) -----τ ==------τ - + an, t = 0 t0, d 0 1 Θ τ 3 3 1 3 where t0 is a solution of the equation t Ð = a(t) 0(t). For the positive reactivity ρ = 0.9β, the solutions of 2 2 set (5) at the initial time instant (τ = 0) are depicted in 2 Fig. 1a, where the mean temperature and pressure in the ρ 3 sensing medium are shown for U = 5000 kg/m and 1 χ = 0.45. The dashed lines parallel to the abscissa axis indicate the values corresponding to the steady-state conditions and to the instant of actuator operation. The 0 0.5 1.0 0 0.5 1.0 time delay of the DEP element actuator is several tens τ of milliseconds, starting from the instant of reactor run- , s away. Note that this time for the automated emergency Fig. 2. DEP control member level in the reactor core after protection of the reactor may be as high as several hun- ε the actuator operation. (a) P0 = 1.5 MPa; = (1) 0.1, dred of milliseconds [15]. To further cut the DEP ele- ε (2) 0.15, and (3) 0.2. (b) = 0.1; P0 = (1) 1.4, (2) 1.8, and ment time delay, one can increase the enrichment of (3) 2.2 MPa. 235 UF6 by the U isotope. From Fig. 2b, it follows that this can be done if the temperature and the pressure in nels, respectively. The coefficients are given by the sensing medium are elevated. ξ 1 = 64/Re1, RESPONSE TIME OF DEP ξ = 64F(Θ)/Re. Here, Liquid neutron absorbents are usually used for eliminating excess reactivity, for example, in boron systems F(Θ) = (1 Ð Θ)2/[1 + Θ2 + (1 Ð Θ)2/lnΘ] of VVÉR reactor control [16]. With boron dissolved in Θ = r/R, the moderator [17], one can control only small reactivity variations. It has been suggested [15] to apply CPSs and Re1 and Re are the Reynolds numbers for the circu- where liquid absorbents circulate through hermetically lar and annular channels, respectively [21]. Substitut- ξ ξ sealed pipelines in special channels made in the core. ing 1 and into (6), we obtain ∆ η 2 A strong absorbent of thermal neutrons is gadolin- P1 = 8 lv1/R , σ × Ð24 2 2 2 ium ( a = 4.6 10 m ), for example, in the form ∆P = 8ηlvF(Θ)/[R (1 Ð Θ )], ρ 3 Gd(NO3)3 [18]. With a density Gd > 3000 kg/m , the where η is the dynamic viscosity. The Poiseuille formu- macroscopic thermal-neutron absorption cross section las for the circular and annular channels are obtained by Ð2 of Gd(NO3)3 exceeds 250 cm . Then, the DEP element expressing the velocities through the friction drags: working volume filled with this material can be consid- v = R2∆P /8vl, ered as a black absorbing rod [19] introduced into the 1 1 reactor core. In terms of hydraulics, a neutron absor- v = R2(1 Ð Θ2)∆P/8ηhF(Θ). bent container with an inner radius R can be represented For a DEP element with R = 0.0059 m, the reactor as a circular channel with a hydraulic diameter dl = 2R core height is H = 3.66 m. Let r = 0.004 m, as before. η ρ and the working volume of the absorbent, as an annular At the coolant temperature, = 0.085 Pa s and Gd = channel with a hydraulic diameter d = 2(R Ð r), where r 3430 kg/m3. The absorbent filling the working volume is the inner radius of the annular channel. The absor- throughout the core depth fills the container to a level bent flow is laminar and becomes steady in areas com- L = (1 Ð Θ2)H. It produces a hydrostatic head of ρgL. parable in size to the hydraulic diameters. The friction This head of the absorbent in the container makes it dif- drags for circular and annular channels of length l and ficult to provide a necessary rate of its delivery to the h, respectively, are [20] working volume of the DEP element. The rate of delivery can be increased if an extra pressure in the container ρ v 2 ρ v 2 ∆ ξ l Gd 1 ∆ ξhGd is provided by a compressible gas. P1 ==1-----------, P ---------, (6) d1 2 d 2 Let P0 be the initial gas pressure in the container. The gas layer thickness will be expressed through the ε τ where v1 and v are the cross-section-averaged veloci- core height: L0 Ð L = H. Within a time after the actu- ties of the steady flow in the circular and annular chan- ator has come into action, a part of the absorbent enters ξ ξ nels, respectively, and 1 and are the coefficients of from the container into the working volume, filling it to hydraulic resistance for the circular and annular chan- a level h; accordingly, its level in the container will

TECHNICAL PHYSICS Vol. 47 No. 4 2002 468 TAKIBAEV et al. decrease from the initial level L to a level l = L Ð (1 Ð Since the instantaneous ﬂow rate of an incompress- 2 Θ )h. Then, ible liquid is the same in any cross section, v1 = (1 Ð Θ2 Θ2)v. Hence, we have P = P0(L0 Ð L)/(L0 Ð l) = P0(L0 Ð L)/[L0 Ð L + (1 Ð )h].

P εR2 v () 0 h = ------2 . (7) 8η()1 Ð Θ2 H[]ε + ()1 Ð Θ2 h/H {}11Ð []Ð F()/1Θ ()Ð Θ2 h/H

Treating v(h) as the rate of displacement of the different initial thicknesses ε of the gas layer and vari- absorbent level in the working volume, we come to the ous initial gas pressures P0 in the container. It is seen equation for the depth to which the DEP control mem- that the actuator can be introduced into the reactor core ber descends after the actuator operation: for a time comparable to that for which the rods of automated emergency protection are introduced when they dh ------v ()h , h()τ = 0 = 0. (8) remedy a reactivity accident. dτ Figure 2 demonstrates the solutions of Eq. (8) for REACTIVITY ACCIDENT SIMULATION n ρ 25 In a reactivity accident, the DEP element behavior is 0.04 3 described by the set of equations including the equa- (a) 3 tions of point kinetics and Eqs. (3) and (8): 20 2 dn ρδρÐ ()h Ð β 0.02 ------==------n +1,Σλ C , n()τ = 0 15 dτ l i i 1 2 β β 10 dCi i λ ()τ i ,,… 02 ------τ = ---- n ÐiCi, Ci = 0 = ------λ , i =1 6, τ, s d l l i 1 (9) 5 dt Θ Ð t ----- = ------+an, t()τ = 0 = t , t()ττ≥ = t , nth dτ l 0 th th 0 dh 15 v ()ττ≤ ()ττ> τ 3 -----τ- = (),h h th = 0, h th + 1, = H, (b) d τ where th is the time instant at which the actuator comes τ 10 2 into action and 1 th is the time for which the absorbent is introduced into the DEP element working volume. The DEP system supplements the automated emer- 1 gency protection of the reactor [22]. In [14], the thresh- 5 old neutron ﬂux density is taken to be nth = 1.2; we put nth = 1.4. nth A reactivity accident is simulated by introducing the 0 function ρ(t) that speciﬁes the positive reactivity 10 growth in the equations of point kinetics. If the positive ρ τ βτ τ ≥ (c) 3 reactivity grows with a constant rate, ( ) = vβ ( 0), 8 where vβ is the rate of positive reactivity growth in terms of the total fraction β of delayed neutrons, it is 6 2 Fig. 3. Remedy of the reactivity accident due to the growth of the positive reactivity with different rates: v1 = (1) 2, 4 1 Ð1 ε v2 = (2) 4, and v3 = (3) 6 s . The DEP parameters are = ρ 3 χ τ 0.2 and U = 5000 kg/m . (a) P0 = 1 MPa; = 0.05; 1 th = 2 n τ th 1.08 s; and th = (1) 0.4, (2) 0.24, (3) 0.18 s. (b) P0 = 2 MPa; χ τ τ = 0.05; 1 th = 0.54 s; and th = (1) 0.4, (2) 0.24, and χ τ τ (3) 0.18 s. (c) P0 = 2 MPa; = 0.06; 1 th = 0.54 s; and th = 0 0.1 0.2 0.3 0.4 0.5 τ, s (1) 0.35, (2) 0.22, and (3) 0.16 s.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 THE OPERATION OF HIGH-SPEED EMERGENCY PROTECTION 469 reasonable to limit the reactivity: 6. M. D. Lemberg, Relay Systems Pneumo-Automation (Énergiya, Moscow, 1968). ρ β ≤≤ττ  0v β t,0 max 7. V. N. Dmitriev and V. G. Gradetskiœ, Foundations of ρτ()=  (10) Pneumo-Automation (Mashinostroenie, Moscow, 1979). ρ v βτ ττ>  0 β max, max. 8. G. I. Verkhivker, S. D. Tetel’baum, and G. P. Konyaeva, τ At. Énerg. 24, 158 (1968). At fixed max, to small vβ, there corresponds a slow 9. A. G. Morachevskiœ and I. B. Sladkov, Physicochemical growth of a relatively small reactivity; at high vβ, a Properties of Molecular Inorganic Compounds τ large reactivity grows with a high rate. We put max = 1 s. (Khimiya, Moscow, 1987). Figure 3 demonstrates the solutions of Eqs. (9) 10. V. V. Malyshev, in Thermophysical Properties of Gases when a reactivity accident is simulated by different (Nauka, Moscow, 1976), pp. 97Ð105. growth rates of a positive reactivity. From Fig. 3a, it is 11. S. S. Bakulin, S. A. Ulybin, and E. P. Zherdyaev, in Ther- seen that the gasÐliquid DEP operates the faster, the mophysical Properties of Gases (Nauka, Moscow, more dangerous is the reactor runway. Figure 3b shows 1976), pp. 40Ð45. the solutions of Eqs. (9) at a high initial pressure in the 12. I. A. Kuznetsov, Emergency and Transient Processes in neutron absorbent container. The speed of the gasÐliq- Fast Reactors (Énergoatomizdat, Moscow, 1987). uid DEP rises because the time of absorbent introduc- 13. B. S. Petukhov, L. G. Genin, and S. A. Kovalev, Heat tion into the core decreases. Figure 3c shows the solu- Transfer in Nuclear Power Plants (Énergoatomizdat, Moscow, 1986). tions of Eqs. (9) for the case where the gasÐliquid DEP spends a lesser power to remedy the reactor runaway 14. B. A. Dement’ev, Nuclear Power Reactors (Énergoat- omizdat, Moscow, 1990). because of a greater enrichment of the uranium in the sensing medium. Here, the DEP speed increases 15. I. Ya. Emel’yanov, P. A. Gavrilov, and B. N. Seliverstov, Control and Safety of Nuclear Power Plants (Atomizdat, because the delay time of the actuator is smaller. Moscow, 1975). The results of reactivity accident simulation imply 16. V. A. Sidorenko, Safety Problems of Water-Moderated that the emergency protection system considered offers Water-Cooled Power Reactor Works (Énergoatomizdat, a high speed. Moscow, 1977). 17. G. G. Bartolomeœ, G. A. Bat’, V. D. Baœbakov, and M. S. Alkhutov, Foundations of Theory and Computing REFERENCES Methods of Nuclear Power Reactors (Énergoatomizdat, 1. Zh. S. Takibaev, Preprint No. 3-96, NYaTs RK (Almaty, Moscow, 1989). 1996). 18. J. M. Harrer, Nuclear Reactor Control Engineering (Van 2. Zh. S. Takibaev, Izv. Akad. Nauk Resp. Kaz., Ser. Fiz.- Nostrand, Princeton, 1963; Atomizdat, Moscow, 1967). Mat., No. 2, 54 (1997). 19. B. A. Dement’ev, Kinetics and Regulation of Nuclear Reactors (Énergoatomizdat, Moscow, 1986). 3. J. S. Takibaev, in Proceedings of Fujii-e Symposium “Various Methods to Operate and Control Processes in 20. P. A. Kirillov, Yu. S. Yur’ev, and V. P. Bobkov, Handbook an Active Zone in Condition of Extraordinary Situa- of Heat-hydraulic Calculations (Nuclear Reactors, Heat tion,” Tokyo, 1996, p. 225. Exchanger, Steam Generators) (Énergoatomizdat, Mos- cow, 1990). 4. Zh. S. Takibaev, in Proceedings of the International Sci- entific and Practical Conference “Nuclear Power Engi- 21. B. S. Petukhov, Heat Transfer and Resistance in Lami- neering in Kazakhstan: Advanced Aspects,” Aktau, nar Fluid Flow in Pipes (Énergiya, Moscow, 1967). Kazakhstan, 1996, p. 71. 22. R. R. Ionaœtis and N. L. Shvedov, At. Tekh. Rubezhom, No. 1, 10 (1988). 5. I. S. Balakirev and A. É. Sofiev, Application of Pneumo- and Hydro-Automation Facilities in Chemical Industry (Khimiya, Moscow, 1973). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 470Ð473. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 99Ð102. Original Russian Text Copyright © 2002 by Belyaev, Drokin, Shabanov, Shepov.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

High-Frequency Dielectric Spectra from Liquid Crystals of Series nCB and nOCB B. A. Belyaev, N. A. Drokin, V. F. Shabanov, and V. N. Shepov Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received July 5, 2001

Abstract—The design of a resonant frequency-tunable high-sensitivity microstrip sensor is suggested. The permittivity dispersion of liquid crystals of two homologic series, alkylcyanobiphenyls (7CB and 8CB) and alkyloxycyanobiphenyls (7OCB and 8OCB), is studied at frequencies of 100Ð900 MHz. The dielectric spectra are shown to be the sum of the Debye relaxation and dielectric resonances observed at f ≈ 160, 280, 360, 450, 550, and 650 MHz. The dielectric resonances are present in the spectra of all the samples in both the nematic and isotropic phase. The substitution of an oxygen atom (series nOCB) for a carbon atom (series nCB) in liquid crystal molecules has a minor effect on the dielectric resonance frequencies but changes the resonance intensities and splits some of the resonance lines. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION and type of liquid-crystal ordering (nematic or smec- tic). At the same time, the intensity of this extra region The ColeÐCole diagrams are very suitable for eval- strongly varies with temperature and homologic type. uating the deviation of liquid crystal dielectric spectra from the Debye frequency dependence of the dielectric In this work, we further refine the method of mea- constants [1, 2]. This requires measurements to be suring the dielectric dispersion with microstrip reso- made at several widely separated frequencies in the nant sensors and study the dielectric spectra of two high-frequency and microwave ranges. The deviations types of liquid crystals: n-alkylcyanobiphenyls (7CB are usually described by a set of Debye relaxation and 8CB) and n-alkyloxycyanobiphenyls (7OCB and regions. However, because of the lack of adequate 8OCB) at frequencies from 100 to 900 MHz. Also, we experimental data, the reason for a number of spectral compare these spectra and discuss the effect of oxygen features remains unclear. In [3], the dielectric spectra in (nOCB series), which substitutes carbon (nCB series) a wide frequency range (100 kHzÐ10 GHz) were stud- in liquid crystal molecules, on the relaxation and reso- ied by time-domain spectroscopy. It was shown that the nance characteristics of the dielectric spectra. dielectric spectrum taken from liquid crystals of the nCB series shows individual faint non-Debye disper- MICROSTRIP INSTRUMENT SENSOR sion regions at frequencies of 39, 225, and 550 MHz for AND EXPERIMENTAL METHODS 7CB and 34.3, 225, and 600 MHz for 8CB. It is of interest that these regions are observed in both the meso- High-sensitivity frequency-tunable (in discrete morphic state and the isotropic phase. steps) resonance sensors built around microstrip structures have proved themselves a promising tool for mea- The existence of extra low-intensity narrow-band suring the dispersion in the decimeter range [6]. Such dispersion regions was also reported in [4], where the sensors operate with small amounts (volumes) of mate- dielectric spectra of liquid crystals placed in porous rials tested and take the dielectric spectra by measuring media were studied. In that work, the dispersion fea- the real and imaginary components of the permittivity tures of 5CB and 8CB crystals were observed for the with a small frequency step. perpendicular component of the permittivity in the For taking the dielectric spectra from the liquid nematic phase at 100 MHz. crystals, we used the modified design of the microstrip In [5], the dispersion region of the permittivity was resonant sensor [6] (Fig. 1). The sensor substrate is a studied in nCB liquid crystals at frequencies from 50 to 15 × 30-mm alumina plate with a permittivity ε = 9.6 500 MHz with special microstrip resonant sensors. Dis- and a thickness h = 1 mm. The lower plated side of the tinct extra dispersion regions imposed on the conven- substrate is soldered to the metallic base of the sensor, tional Debye relaxation pattern were found. It is worthy which serves as a screen. On the upper side of the sub- to note that the frequency of the most intense region at strate, microstrip conductors forming a half-wave rect- 300 MHz does not depend on a specific representative angular microstrip resonant loop are patterned by from this homologic series, as well as on temperature chemical etching. Input and output microstrip lines

HIGH-FREQUENCY DIELECTRIC SPECTRA 471

(A and B) are connected to the opposite sides of this loop through coupling capacitors Cc. A capacitive measuring interdigitation Cx with a spacing of 0.15 mm is placed at the antinode of the microwave field of the resonator. Three breaks to connect lumped elements con- C trolling the operation of the sensor were etched in the x microstrip conductor at the maximum of the microwave AB field. The resonant frequency of the sensor can be varied in wide limits. This is achieved by inserting either Cc L Cc calibrated inductances L, which lower the frequency of C1 C2 the sensor, or an electrically controlled capacitor (var- actor V), which raises the frequency, into the central break of the microstrip conductor. In this way, the fre- V quency of the sensor can be increased from 100 to R1 R2 R3 R4 900 MHz in 20 MHz steps. The capacitors C1 and C2 and resistors R1ÐR4 serve to isolate the controlling circuits and to apply a dc orienting electric field to the Fig. 1. Microstrip resonant sensor. sample. In the nematic phase, the long axis of the molecules rotate toward the microwave field when a voltage of ≈25 V is applied to the sample. In this case, the ε||' ε'' longitudinal component ε||( f ) of the permittivity is 18 6 measured. The transverse component ε⊥( f ) is measured 10 MHz by applying a permanent magnetic field H = 2500 Oe 3 perpendicularly to the microwave electric field. 600 MHz 1.2 MHz 0 A 0.20-mm-thick liquid crystal sample was placed 3159 ε' directly on the interdigitation, which was bounded by a 10 glass rim, and covered by a Teflon film to prevent con- tamination. Identical expendable measuring cells were made for either liquid crystal homologue by a specially developed precise lithography technique that offers 2 high reproducibility. The cells were connected to the 67 8 9 mutual resonance loop. Such an approach provides a log(f ) quick replacement of the samples and also prevents them from being contaminated during measurements, ε which inevitably occurs when a single measuring cell is Fig. 2. Frequency dependence of ||' ( f ) in semilogarithmi- used. cal coordinates for 7CB. The inset shows the ColeÐCole diagram. The real component of the permittivity was calculated from the difference in the resonance frequencies of the sensors with and without the sample. The imagi- quency dependence of the permittivity with the Debye nary component was calculated from the change in the relaxation formula. By way of example, Fig. 2 demon- loaded Q factor of the resonator when the liquid crystal strates the dielectric spectrum ε||' ( f ) for the 7CB sam- was placed in it. The amplitudeÐfrequency responses of ple. The inset here shows the ColeÐCole diagram with the microstrip sensors were recorded with an R4-37 data points. In the right-hand side of this diagram, the automated meter of complex transfer coefficients. For data points are fairly well described by a semicircle. In the nematic samples, the cell temperature was kept at the left-hand side, however, the measurements mark- T = (T Ð 4) K, where T is the temperature of the n Ð i n Ð i edly deviate from the semi-circle both to the left and to nematic-to-isotropic phase transition, for their dielec- ε tric spectra to be convenient to compare. For the sam- the right. As follows from the dispersion curve ||' ( f ) in ples in the isotropic phase, the temperature was kept at Fig. 2, such a deviation occurs because the dielectric T = (Tn Ð i + 4) K. The absolute accuracy of temperature spectrum contains faint yet well distinguishable non- maintenance was ∆T = ±0.1 K or higher. Debye dispersion regions at f > 100 MHz. Figure 3 shows the high-frequency part of the RESULTS dielectric spectrum for the real, ε||' ( f ), and imaginary,

Along with taking the high-frequency spectra, we ε||'' ( f ), components of the permittivity of the 7CB liq- measured the permittivity at some frequencies in the uid crystal sample. The dashed curve corresponds to the megahertz range. It was necessary for providing the desired accuracy in numerically approximating the fre- Debye approximation of ε||' ( f ) with a single relaxation

TECHNICAL PHYSICS Vol. 47 No. 4 2002

472 BELYAEV et al.

Ð8 ε'|| ε''|| time τ = 3.25 × 10 s. The extra dispersion regions with

ε||'' ( f ) peaks at f ≈ 160, 280, 360, 450, 550, and 5 650 MHz. Several hardly distinguishable peaks at 1 higher frequencies up to 900 MHz are also observed. 0.45 Comparing curves 1 and 2, one can see that the peaks of ε||'' ( f ) in Fig. 3 coincide with the points where the . dispersion curve ε||' ( f ) meets the dashed curve, which 2 corresponds to the Debye approximation. With the narrow extra dispersion regions, such behavior of ε||' ( f )

3 0.05 and ε||'' ( f ) is typical of resonance processes. These may 50 150 250 350 450 550 650 f, MHz be microwave-ﬁeld-induced intramolecular vibrations of mobile molecular fragments. A phenomenological dielectric spectrum can be described by the sum of the Fig. 3. High-frequency spectra (1) ε||' ( f ) and (2) ε||'' ( f ) for Debye relaxation and dielectric resonances: 7CB sample. ()ε ε 0' Ð ∞' 1 ε'()ω Ð ε∞' = ------+ ---∑∆ε ω2τ2 2 i 1 + i ε' (1) R 2 2 0.4 1 + ω ()ωω+ g 1 Ð ω ()ωωÐ g × ------0i 0i i +,------0i 0i i 1 ()ωω2 2 ()ωω2 2 1 + + 0i gi 1 + Ð 0i gi 0.2 ε ε where 0' and ∞' are the real components of the static τ π and high-frequency permittivities, = 1/2 fD is the 0 relaxation time (fD is the Debye resonance frequency), ∆ε ω is the resonance intensity, 0 is the resonance frequency, g is the damping factor, i is the resonance num- –0.2 ber, and ω = 2πf. For more detailed comparison of the frequencies 0.4 and heights of the resonance peaks, we proceeded as 2 follows. From the spectrum ε||' ( f ) measured, the 0.2 numerically approximated Debye relaxation contribu- ε tion was subtracted. The “resonance spectra” 'R ( f ) thus calculated for the 7CB, 7OCB, and 8OCB samples 0 are given in Fig. 4 for the nematic phase. Even slight distortions of the lines are discovered, and the behavior of the resonance frequencies and peak intensities is –0.2 clearly seen. Note that the resonance spectrum for 8CB differs from that for 7CB insigniﬁcantly and is omitted 0.4 in Fig. 4. It follows from the spectra that the typical res- 3 onance frequencies determined from Fig. 3 (f ≈ 160, 280, 360, 450, 550, and 650 MHz) do not depend on the 0.2 mesogene type for both nCB and nOCB series. How- ever, the presence of oxygen in an alkyloxycyanobiphe- nyl molecule (7OCB and 8OCB) between the rigid core 0 and loose HÐCÐH alkyl groups increases the intensity of the resonance peaks and even causes some of them to split. –0.2 100 300 500 700 In the isotropic phase, the shape of the spectra f, MHz remains virtually unchanged for both series, but the resonance line intensity increases almost twice. Such behavior was also observed in [5, 7] for 5CB and 6CB ε ε Fig. 4. Resonance spectra 'R obtained from ||' ( f ) for samples. Comparing the Debye relaxation parameters (1) 7CB, (2) 7OCB, and (3) 8OCB samples. obtained by approximation, one can see that the substi-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 HIGH-FREQUENCY DIELECTRIC SPECTRA 473 tution of an oxygen atom for a carbon atom in liquid tals has a minor effect on the dielectric resonance fre- crystal molecules strongly affects the Debye parame- quencies but changes the resonance intensities and ters: the relaxation time decreases, the high-frequency causes some of the resonance line to split. component ε||' (∞) of the permittivity grows, and the dielectric anisotropy increases. The numerical values of ACKNOWLEDGMENTS these parameters for the liquid crystals studied are very close to those obtained in [1, 2, 8]. This work was supported by the Russian Foundation for Basic Research (grant no. 00-03-32206). It was shown [8] that an angle β between the dipole moment of an alkyl group and the axis of symmetry of Shepov greatly appreciates the financial support the molecule increases with n in the nCB series and still from the expert group in the young scientist competi- stronger in the nOCB series. In other words, as the tion (Siberian Division, Russian Academy of Sciences) length of alkyl groups in the nCB and nOCB series devoted to Akademician Lavrent’ev’s 100th birthday. increases, so does the normal component of the total dipole moment of the molecules. It seems likely that the REFERENCES change in the resonance line intensity and the split of the lines are also associated with an increase in the nor- 1. D. Lippens, J. P. Parneix, and A. Chapoton, J. Phys. mal component of the dipole moment of the molecules. (Paris) 38, 1465 (1977). Of course, this assumption needs experimental verifica- 2. J. M. Wacrenier, C. Druon, and D. Lippens, Mol. Phys. tion; however, it should be taken into account in study- 43, 97 (1981). ing the dielectric spectra considered in this work. 3. T. K. Bose, B. Campbell, S. Yagihara, and J. Thoen, Phys. Rev. A 36, 5767 (1987). 4. G. P. Sinha and F. M. Aliev, Phys. Rev. E 58, 2001 CONCLUSION (1998). With special microstrip frequency-tunable resonant 5. B. A. Belyaev, N. A. Drokin, V. F. Shabanov, and sensors, we studied the high-frequency dielectric spec- V. N. Shepov, Fiz. Tverd. Tela (St. Petersburg) 42, 956 tra of alkylcyanobiphenyl (7CB and 8CB) and alkylox- (2000) [Phys. Solid State 42, 987 (2000)]. ycyanobiphenyl (7OCB and 8OCB) liquid crystals at 6. B. A. Belyaev, N. A. Drokin, and V. N. Shepov, Zh. Tekh. frequencies between 100 and 900 MHz. The dielectric Fiz. 65 (2), 189 (1995) [Tech. Phys. 40, 216 (1995)]. spectra of each of the crystals are described by the sum 7. B. A. Belyaev, N. A. Drokin, V. F. Shabanov, and of the Debye relaxation and dielectric resonances at f ≈ V. N. Shepov, Pis’ma Zh. Éksp. Teor. Fiz. 66, 251 (1997) 160, 280, 360, 450, 550, and 650 MHz. For the longitu- [JETP Lett. 66, 271 (1997)]. dinal component ε||' ( f ) of the permittivity, the relax- 8. S. Urban, B. Gestblom, and A. Wurflinger, Mol. Cryst. ation is characterized by a single relaxation time for Liq. Cryst. 331, 1973 (1999). each of the crystals. By comparing the dielectric spectra, we conclude that oxygen in the nOCB liquid crys- Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Generation of Magnetic Field by Detonation Waves S. D. Gilev and A. M. Trubachev Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 15, Novosibirsk, 630090 Russia e-mail: [email protected] Received July 30, 2001

Abstract—The generation of a magnetic field by a system of detonation waves in a condensed explosive is reported. The convergence of the detonation waves, which exhibit a high conductivity in the chemical reaction zone, increases the magnetic field at the axis of the system. The fact of magnetic field generation is demonstrated experimentally. Features of the new method of magnetic cumulation are discussed. A simple compression model that qualitatively agrees with experimental data is proposed. © 2002 MAIK “Nauka/Inter- periodica”.

INTRODUCTION and the subsequent expansion of the detonation products. The peak conductivity in the chemical reaction Magnetic flux cumulation by a conducting shell set zone (≈1 mm) was measured to be ≈250 (Ω cm)Ð1 and in motion by detonation products of high chemical ≈ Ω Ð1 explosives (HEs) has been used to obtain extrahigh 30 ( cm) outside it. magnetic fields up to 2 × 103 T and energy densities on the order of 106 J/cm3 [1Ð5]. Difficulties associated HIGH-CONDUCTIVITY ZONE with this method are conducting boundary instability, OF DETONATION PRODUCTS conducting layer overheating, and complexity of the system that initiates the magnetic flux. The shock-wave The peak electrical conductivity of the TNT detona- magnetic cumulation method [6Ð9] largely eliminates tion products [15] is the highest among HEs. However, these problems. In this method, a material placed into it is insufficient for magnetic cumulation. The conduc- the compression region becomes a good conductor tivity can be increased by introducing additives (a metal under the action of a shock wave. A system of shock or semiconductor powder) into HEs. The metallic addi- waves converging towards the axis traps and com- tive is inherently conductive. The semiconductor addi- presses the magnetic flux. The strongest magnetic fields tive is initially nonconducting but acquires good con- produced by small-size shock-wave generators with an ducting properties (i.e., passes to the metallic state) HE charge of about 0.5 kg were about 400 T [7]. under the action of a shock wave due to the thermal excitation of the semiconductor subjected to strong In this paper, it is proposed to compress the mag- compression. netic flux and generate strong magnetic fields with the We measured the electrical conductivity of the deto- help of detonation waves in condensed explosives. The nation products of HEÐmetal mixtures. The experiment conducting layer of the detonation products acts as a was similar to that described in [15]. The shunt was of piston, which compresses the magnetic flux. This tech- the same width as the space occupied by the HE. It was nique retains many advantages of the shock-wave demonstrated that increasing the weight content of the method and still does not require an additional power metal in the mixture to ≈50% augments the detonation supply. The efficiency of the new magnetic cumulation product conductivity by more than three orders of mag- method depends on the conductivity of the detonation nitude compared with HEs without the additive. The products, which must be high enough. maximum content of the conducting additive in the The conductivity of the detonation products from mixture is limited because its too great amount impairs most condensed HEs is σ ~ 1 (Ω cm)Ð1 [10Ð13]. Early the detonation power. For mixtures with linear dimen- measurements, however, were erroneous: it has been sions of about 1 cm, the resistance of the detonation shown [14] that for trinitrotoluene (TNT) detonation products may be of several milliohms. products, σ ≈ 25 (Ω cm)Ð1. A new measuring procedure Figure 1 shows the conductivity profile of the that significantly improves the time resolution and cyclotetramethylenetetramine (HMX)/Al = 60/40 mix- raises the upper limit of conductivity measurement has ture in a detonation wave. As follows from the figure, been proposed [15]. When applied to cast TNT, this the detonation conductivity profile is highly nonuni- procedure revealed the complex nature of detonation form. The maximal conductivity is observed in the conduction associated with the chemical reaction zone ≈2-mm-thick zone adjacent to the detonation front.

GENERATION OF MAGNETIC FIELD BY DETONATION WAVES 475

This high-conductivity zone travels through the HE σ, 103(Ω cm)–1 together with the wave front and is essentially coinci- 2.0 dent with the chemical reaction zone. The aluminum oxidation reaction and the expansion of the detonation products rapidly decrease the conductivity. Such a spe- 1.5 cific object (a high-conductivity zone propagating at a ≈ speed of 5 km/s) is of interest for high-power electro- 1.0 magnetic systems. In particular, the high-conductivity zone of the HE chemical reaction can be used for com- pressing the magnetic flux and generating a magnitude 0.5 field. 0 –1 0 1 2 3 4 5 DETONATION GENERATOR x, mm OF MAGNETIC FIELD Fig. 1. Conductivity profile of the HMX/Al = 60/40 mixture Figure 2 is the cross-sectional view of a detonation in a detonation wave. x = 0 is the position of the detonation magnetic field generator. The generator comprises front. 12 initiators 1, equispaced in a circle. The initiators det- onate auxiliary HE 2. The central region of the generator (62 mm in diameter and 50 mm in height) is filled with operating HE 3 (HMX/Al = 60/40 mixture of density ≈1.4 g/cm3). The detonation waves excited in the operating HE converge to the axis, at which inductive 1 pickup 4 is placed. 234 5 The degree of convergence of the detonation waves to the axis was estimated by detecting the optical radiation from the generator surface with the help of a VFU streak camera operating in a frame-by-frame mode. Figure 3 shows a sequence of photographs with the operating region magnified to 150 mm. The time increases from top to bottom and from left to right. The Fig. 2. Detonation generator of magnetic field: 1, initiation interval between frames is 1.6 µs. The maximum lumi- points; 2, boost explosive; 3, operating explosive; 4, induc- nance corresponds to the region near the detonation tive pickup; and 5, shell. front. The efficiency of wave initiation and the degree of convergence can be evaluated as satisfactory. dimensional Maxwell equations and the Ohm law yield Prior to the detonation, a magnetic field of 2 T was the system of equations created in the operating space. To this end, a bank of ∂B capacitors was discharged through external Helmholtz ------= Ðµ j, (1) coils. The current through the coils was monitored with ∂r 0 a reference inductive gage; the signal from measuring pickup 4, with an S9-27 oscilloscope. 1∂()rE ∂B ------∂ - = Ð------∂ , (2) Figure 4a shows an integrated record of the voltage r r t from the measuring pickup. This signal is proportional σ() to the magnetic field at the pickup location. The mag- j = EuB+ , (3) netic field at the axis of the system increases as the det- where B is the magnetic induction, j is the electric cur- onation waves in the operating HE converge. This rent density, E is the electric field, and u is the mass means that a conducting plug, which compresses the velocity. The magnetic field at the front of the wave is magnetic flux, arises in the HE. In this experiment, the continuous (the front is infinitely thin and, therefore, magnetic field intensity grew by a factor of 2.2. does not disturb the current). Hence, the electric field is also continuous. Equations (1)Ð(3) and the continuity condition for the electric field lead to the relationship ANALYSIS OF THE DETONATION GENERATOR PERFORMANCE r ∂ 1 ∂ --- B = uB + ------B , (4) 2------∂ µ σ------∂ Consider a converging cylindrical detonation wave t rr= f 0 r rr= f propagating in a transverse magnetic field. Let the HE conductivity change in a stepwise manner from zero to which is valid at the wave front. Here, rf is the current σ at the infinitely thin front of the wave. The one- radius of the front. If σ ∞ and the mass velocity u

TECHNICAL PHYSICS Vol. 47 No. 4 2002

476 GILEV, TRUBACHEV

Fig. 3. Convergence of detonation waves (a sequence of photographs) in the generator.

V (a) ln(B/B0) (b) 80 0.8 790 790790

60 0.6

40 0.4

20 0.2

06462 66 68 70 0 0.5 1.0 1.5 2.0 t, µs ln[1/(l – t/T)]

Fig. 4. Magnetic field generation by a detonation wave in the condensed explosive. (a) Record of the inductive pickup (the signal is proportional to the magnetic field at the axis); the arrows show the time instants when the conducting layer is formed in the explosive and when the wave arrives at the pickup. (b) Experimental data processing; the slope of the curve determines the effective magnetic field growth index α. and the wave velocity D are constant, the following where α is an effective exponent (α < u/D at finite σ and relationship can easily be obtained from (4): α u/D as σ ∞). n 2---- D If formula (5) is valid, the representation of experi- Bt() R0 ------= ----- . mental data in the coordinates (ln(B/B0) and ln[1/(1 Ð B0 r f t/T)]) (Fig. 4b) must yield a straight line (t is the time Here, B is the initial magnetic field and R is the initial from the beginning of magnetic compression and T is 0 0 the total compression time). From the slope of this line, radius of the region occupied by the HE. This formula α was obtained in [6] for the shock-wave compression of one determines the exponent 2 . In Fig. 4b, the time T the magnetic field. The u/D ratio depends on the com- was calculated from the diameter of the region occu- pressibility of the material and tends to unity in highly pied by the HE and the detonation wave velocity D porous substances. In the case of a detonation genera- found in other experiments. As can be seen from the fig- tor, the value of u/D can be varied in narrow limits. For ure, the curve is close to a straight line, which indicates the validity of formula (5). As a result, the average normal ChapmanÐJouguet detonation of condensed α ≈ explosives, u/D ≈ 0.25 [16]. The second term on the exponent thus found is 0.24. This value is close to right-hand side of Eq. (4) is negative; hence, the ohmic u/D typical of condensed HE detonation. loss decreases the rate of field rise. Let the field be com- The experiments show that the increase in the mag- pressed by a uniform conducting layer with a finite con- netic field significantly depends on the detonation ductivity that moves at a constant velocity. A numerical regime. For example, for the damped detonation, the solution to this model problem shows that the magnetic field was observed to increase by a factor of ≈8. This field at the axis can be described by the formula effect occurs because the compression (and the effec- 2α tive exponent α) increases when the detonation wave Bt() R0 ------= ----- , (5) transforms into the shock wave. The use of damped det- B0 r f onation also extends the operating life of the generator.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 GENERATION OF MAGNETIC FIELD BY DETONATION WAVES 477

A thorough analysis of the initial stage of the cumula- ties to cumulation systems (energy independence, flex- tion process reveals the presence of overcompressed ibility, and the ability to dynamically change the com- detonation, which occurs when the mixture is initiated pression geometry). by a high booster explosive. This effect in Fig. 4b shows up in the larger slope of the curve at the beginning of the process. Thus, the transient detonation ACKNOWLEDGMENTS regimes affect the magnetic cumulation efficiency. We are grateful to A.M. Ryabchun for developing a program that simulates magnetic flux compression by a conducting layer. DISCUSSION This work was supported by the Russian Foundation Our experiments have shown that detonation waves for Basic Research (grant no. 99-02-16807). can be used for generating a magnetic field. The efficiency of the detonation generator is limited by two factors. First, the ratio u/D for a detonation wave is rela- REFERENCES tively small. This leads to high convection losses of the 1. C. M. Fowler, W. B. Garn, and R. S. Caird, J. Appl. Phys. magnetic flux through the wave front. Second, the field 31, 588 (1960). is compressed by the conducting region of nonzero 2. A. D. Sakharov, Usp. Fiz. Nauk 88, 725 (1966) [Sov. resistance. In our experiments, the magnetic Reynolds Phys. Usp. 9, 294 (1966)]. µ σ number of the conducting region Rem = 0 ux0 (x0 is the 3. H. Knoepfel, Pulsed High Magnetic Fields (North-Hol- ≈ land, Amsterdam, 1970; Mir, Moscow, 1972). dimension of the region) was estimated as Rem 0.4. HEs with a higher conductivity of the detonation prod- 4. A. S. Lagutin and V. I. Ozhogin, Pulsed High Magnetic ucts and with a wider chemical reaction zone can pro- Fields in Physical Experiment (Énergoatomizdat, Mos- vide better results. cow, 1988). 5. Megagauss and Megampere Pulsed Technology and An advantage of the detonation method of magnetic Applications: Proceedings of the 7th International Con- cumulation is that it extends the potentialities of the ference on Generation of Megagauss Magnetic Fields device. The use of the detonation wave gives more free- and Related Experiments (VNIIÉF, Sarov, 1997), dom in choosing the compression geometry and allows Vols. 1, 2. for its transformation in the cumulation process. This is 6. S. D. Gilev and A. M. Trubachev, Pis’ma Zh. Tekh. Fiz. impossible to achieve in the method of compression 8, 914 (1982) [Sov. Tech. Phys. Lett. 8, 396 (1982)]. with a metallic shell [1, 2] and is difficult to realize in 7. E. I. Bichenkov, S. D. Gilev, A. M. Ryabchun, et al., the shock-wave compression method [6Ð9], where rar- Prikl. Mekh. Tekh. Fiz., No. 3, 15 (1987). efaction waves become significant because of the edge 8. K. Nagayama, Appl. Phys. Lett. 38, 109 (1981). effects. No fundamental limitations are imposed on the 9. K. Nagayama and T. Mashimo, J. Appl. Phys. 61, 4730 size of the initial field-generating areas. The magnetic (1987). field generator can also integrate different approaches 10. A. A. Brish, M. S. Tarasov, and V. A. Tsukerman, Zh. to magnetic cumulation. Éksp. Teor. Fiz. 37, 1543 (1959) [Sov. Phys. JETP 10, Note that, since the output signal depends on the 1095 (1960)]. state of the detonation products, experiments with a 11. R. L. Jameson, S. J. Lukasik, and B. J. Pernick, J. Appl. detonation generator may be useful for studying the Phys. 35, 714 (1964). detonation process. An increase in the field signifi- 12. A. P. Ershov, P. I. Zubov, and L. A. Luk’yanchokov, Fiz. cantly depends on the transient detonation regimes. Goreniya Vzryva 10, 864 (1974). This means that the magnetic cumulation principle can 13. A. G. Antipenko, A. N. Dremin, and V. V. Yakushev, be the basis for the noninvasive diagnostics of detona- Dokl. Akad. Nauk SSSR 225, 1086 (1975). tion processes. 14. A. P. Ershov, N. P. Satonkina, O. A. Dibirov, et al., Fiz. Goreniya Vzryva 36 (5), 97 (2000). 15. S. D. Gilev and A. M. Trubachev, Zh. Tekh. Fiz. 71 (9), CONCLUSION 123 (2001) [Tech. Phys. 46, 1185 (2001)]. The method of generating a magnetic field by a sys- 16. Physics of Explosion, Ed. by K. P. Stanyukovich (Nauka, tem of detonation waves in a condensed HE is shown to Moscow, 1975). be efficient. Unlike shock wave generation, detonation is a self-sustained process, which imparts new proper- Translated by A. Khzmalyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 478Ð483. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 107Ð112. Original Russian Text Copyright © 2002 by Noskov, Malinovskiœ, Cooke, Wright, Schwab. EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Modeling of Discharges in Spatially Charged Dielectrics M. D. Noskov*, A. S. Malinovskiœ*, C. M. Cooke**, K. A. Wright**, and A. J. Schwab*** * Tomsk Polytechnical University, Tomsk, 634034 Russia e-mail: [email protected] ** Massachusetts Institute of Technology, MA 02139-4307 Cambridge, USA e-mail: [email protected] *** University of Karlsruhe, 76128 Karlsruhe, Germany e-mail: [email protected] Received August 2, 2001

Abstract—A stochastic–deterministic model of a discharge induced by space charge in dielectric is developed. The spatiotemporal and current characteristics of the discharge are simulated numerically. The simulation results are compared with the experimental data on discharges in polymethylmethacrylate samples charged by means of an electron beam. The interrelation among the growth of the current-carrying channels, energy transfer, and energy release during the discharge is discussed. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION and breakdown strength [12] on the discharge development. However, the interrelation between the spa- In low-conductivity solid dielectrics, long-lived tiotemporal and current characteristics of the discharge space charges are possible to produce. A rather high structure cannot be adequately described without local space charge density in dielectric can be achieved, allowance for the dynamics of the charge propagation e.g., with the help of moderately intense electron beams along the discharge channels and the change in the [1Ð4]. The electric field generated by space charge can channel conductance. With this in mind, stochasticÐ provoke a discharge between the charged region and the deterministic models were proposed [13, 14], in which dielectric surface. The growth of the discharge channels the discharge channel growth and the charge propaga- results in the formation of branching and twisting sto- tion were described self-consistently. In our study, this chastic discharge structures. The use of an electron approach is developed and employed to describe a dis- beam allows one to control the amount and spatial discharge induced by space charge. The results from com- tribution of space charge. This makes it possible to puter simulations of the discharge structure are com- evaluate the effect of the electric charge distribution pared with the experimental data obtained when study- inside the dielectric on the spatiotemporal and current ing discharges in Plexiglass (polymethylmethacrylate). characteristics of the discharge structure [5, 6]. It is reasonable to use numerical simulations to quantitatively describe a discharge induced by space EXPERIMENTAL SETUP charge. The greatest success in describing the stochastic growth and branching of the discharge channels was Space charge was produced by an electron beam achieved using a fractal model of dielectric breakdown accelerated up to an energy of 3 MeV with the help of proposed in [7]. In this model, the growth of the chan- a Van de Graaff generator. The electron beam was inci- nel was assumed to depend randomly on the electric dent on a rectangular 12.7-mm-thick polymethyl- field and the probability of the channel formation was methacrylate sample (Fig. 1a). At such a sample thick- assumed to be proportional to a certain degree of the ness, 3-MeV electrons produced a 2-mm-thick charged local electric field. The electric field was calculated layer in the middle of the sample. The value and spatial from the Laplace equation, and the discharge structure distribution of space charge were controlled by chang- was regarded as an electrode continuation. In [8], the ing the beam intensity and masking the sample surface. model was generalized by introducing the threshold An electron beam with a current of several microam- electric field for the channel growth, assuming that the peres produced space charge with a density on the order voltage drop along the channels is constant. Various of 1 µC/cm2. The discharge was ignited when the tip of modifications of the fractal model of breakdown in a grounded needle touched the sample lying on a dielectric were used to describe the formation of den- grounded surface (Fig. 1b). The current flowing dritic and bushy structures [8, 9] and the influence of through the needle during the discharge was measured the nonuniform permittivity [9, 10], conductivity [11], with the help of a Rogowski coil and an oscillograph.

MODELING OF DISCHARGES IN SPATIALLY CHARGED DIELECTRICS 479

STOCHASTICÐDETERMINISTIC MODEL (a) OF THE DISCHARGE DEVELOPMENT The model describes the growth of the discharge 1 channels, the charge propagation along the channels, the dynamics of the electric field, and the change in the 2 channel conductance. The discharge channel grows due to the destruction of dielectric and the onset of the conduction phase. Up to now, no quantitative theory of the 3 formation of the conducting channels has been developed. This is related to both the complicated stochastic behavior of the channel growth and the great number of (b) physicochemical processes involved (heating, crack- 4 ing, gas release, dissociation, ionization, recombination, etc.) that occur in a strong electric field and lead to 5 the onset of the conduction phase in dielectric. Since the phase transition occurs due to the electric field 6 energy, one can attempt to describe the discharge structure growth by relating the probability of the conduct- 78 ing channel formation to the local density of the electric field energy. The form of this stochastic dependence Fig. 1. Schematic of (a) the formation of space charge in can be found either by analyzing physicochemical pro- dielectric and (b) the discharge ignition: (1) electron beam, (2) mask, (3) charged layer, (4) oscillograph, (5) Rogowski cesses in dielectric or from experimental data on the coil, (6) needle igniting the discharge, (7) discharge chan- discharge development. In our model, a stepwise nels, and (8) grounded surface. dependence of the growth probability on the local energy density is used as the first approximation. Inside the discharge structure, the probability density ω for the set such that, in accordance with the experimental con- discharge channel to grow along the n direction is ditions, the potential on the sample surface be zero. It assumed to be proportional to the squared projection En was assumed that the charge density changed only on of the local electric field onto this direction, provided the sample side that contacted the grounded surface. At that the projection magnitude exceeds a certain critical every instant, the calculated charge distribution on this value Ec, surface was adjusted so that its potential remained zero. The charge propagation along the discharge channel ωαΘ()2 = En Ð Ec En, (1) is described by Ohm’s law α where is the growth rate constant, Ec is the critical I = γ E , (3) Θ l electric field for the discharge channel growth, and (x) γ is the step function (Θ(x) = 1 for x > 0 and Θ(x) = 0 for where I is the current, is the channel conductance per x ≤ 0). unit length (the product of the conductivity by the channel cross-sectional area), and E is the projection of the The electric field potential is calculated as a super- l position of the fields created by the charges located electric field onto the channel direction. along the discharge structure L, on the sample surface S, During the discharge, the channel conductance per and inside the dielectric volume V, unit length changes due to the channel widening, ionization, recombination, electron attachment to the λ σ ρ ϕ dl dS dV channel wall and gas molecules, etc. These processes = ∫------πεε - ++∫∫------πεε - ∫∫∫------πεε -, (2) 4 0r 4 0r 4 0r depend on both the energy released in the channel due L S V to Joule heating and the energy dissipated into the sur- ε ε where is the dielectric constant; 0 is the permittivity rounding medium. Hence, it is reasonable, as the first of free space; λ, σ, and ρ are the linear, surface, and approximation, to use the following equation for the space charge densities, respectively; and r is the dis- time evolution of the conductance per unit length: tance between the observation point and the point dγ 2 where the charge is located. ------= χγE Ð ξγ, (4) dt l The densities of the charges located along the discharge channels and on the dielectric surface alter dur- where χ and ξ are the parameters governing the ing the discharge. It is necessary to take into account increase and decrease in the channel conductance, the surface charges because, when producing the space respectively. charge inside the dielectric, the ions of the opposite The first term on the right-hand side of Eq. (4) sign are deposited onto the sample surface. Moreover, relates the growth of the channel conductance per unit the sample lies on the grounded surface. In modeling, length to Joule heating. It can be regarded as an ana- the initial distribution of the surface charge density was logue of the RompeÐWeizel formula for the conduc-

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(a) (b) per unit length according to Eq. (4). The simulations were stopped when the discharge structure growth terminated and the channel conductance dropped to zero. α Thus, the model incorporates five parameters: Ec, , χ ξ γ , , and 0, which, together with a sample shape and the magnitude and spatial distribution of space charge, determine the spatiotemporal and current characteristics of the discharge. The physical meaning of the model parameters is quite clear. The critical electric field Ec is related to the existence of the threshold field for the formation of a conducting channel in dielectric. An increase in Ec reduces the dendricity of the discharge structure. The growth rate constant α relates the probability of the channel formation to the local density of the electric field energy. An increase in α stimulates the discharge structure growth. The parameter χ gov- Fig. 2. (a) Experimental and (b) computed pictures of the erning the increase in the conductance per unit length discharge structure in the space charge region of a sample relates the rate at which the conductance grows to the 90 × 135 mm in size for a charge density of 0.8 µC/cm2 at Joule heating power in the channel. The parameter ξ the sample top. governing the reduction in the conductance per unit length determines the rate at which the conductance ξ tance of a spark channel [15] expressed in the differen- decreases (the quantity reciprocal of can be treated as tial form. The second term describes the decrease in the the characteristic time during which the channel remains in the conducting state after the energy release channel conductance per unit length caused by energy χ ξ dissipation. The conductance per unit length of a newly terminates). An increase in and decrease in enlarge created channel is set equal to γ . the discharge channel conductance per unit length and 0 reduce the voltage drop required to sustain the channel Based on the above model, we developed a three- in the conducting state. This is accompanied by an dimensional numerical code, which enabled computer increase in the electric field at the ends of the channels simulations of discharges in spatially charged dielec- and the accelerated growth of the discharge structure. γ trics. In most of the numerical models of the discharge The initial conductance per unit length 0 characterizes structure growth (see, e.g., [7Ð13]), a rectangular lattice the processes leading to the formation of conducting γ was used to calculate the electric field and to describe channels in dielectric. A decrease in 0 decelerates the the channel growth. The possible growth directions conductance growth in the newly created channels but were limited by the lattice edges and diagonals. In our hardly affects the conductance of the existing channels. model, a nonlattice algorithm is used, which allows us The model parameters can be determined by analyzing to avoid this limitation. The discharge structure is rep- either the processes in dielectric and the discharge resented as a dendrite chain of point charges set at a dis- channels or the experimental data on the discharge tance d from each other. The d value determines the development. The parameter values used in our study α × Ð6 2 2 × 7 χ × minimum spatial scale in modeling. The smaller d, the ( = 7 10 m /V s, Ec = 5 10 V/m, = 4 −5 2 ξ 8 Ð1 γ × Ð9 more detailed the description of the discharge structure. 10 S m /J, = 10 s , and 0 = 2 10 S m) were However, the reduction in d significantly increases the determined by comparing the simulation results with computation time. Here, we set d = 2 mm. The electric the experimental data on discharges in polymethyl- field was calculated using the charge method [16], methacrylate. based on the principle of superposition of the fields created by individual charges. In computer simulations, the time step was set at ∆t = 2 × 10Ð9 s. At every time RESULTS AND DISCUSSION step, the following procedures were performed in succession: (i) the calculation of the field using Eq. (2) The growth of the discharge structure begins with with allowance for change in the charge surface den- the formation of one or a few channels that start at the sity; (ii) the simulation of the discharge structure needle tip initiating the discharge. The channels form growth by adding new points to either the existing only within the charged layer inside the dielectric sam- structure or the initiating needle according to the distri- ple. The shape of the region occupied by the discharge bution of the probability density (1); (iii) the calcula- structure corresponds to the shape of the space charge tion of the change in the charge distribution along the distribution. If a mask is used and the charge is discharge structure in accordance with the charge con- implanted only in one-half of the sample, then the dis- servation law and conduction equation (3); and (iv) the charge structure grows only within this zone and stops calculation of the change in the channel conductance at its edge (Fig. 2). If the charge is implanted in a belt

TECHNICAL PHYSICS Vol. 47 No. 4 2002 MODELING OF DISCHARGES IN SPATIALLY CHARGED DIELECTRICS 481

(a) I, A 60 (a) 50 40 30 20 10 (b) 0 γ, 10–4 S m L, mm 2.0 (b) 140 120 1.5 1 100 80 1.0 60 2 40 Fig. 3. (a) Experimental and (b) computed pictures of the 0.5 discharge structure growing along the charged sinusoidal 20 belt on a sample 80 × 160 mm in size for a charge density 2 0 of 0.8 µC/cm inside the belt. 0 0.2 0.4 0.6 0.8 t, µs shaped like a sinusoid, then the discharge structure Fig. 4. Time evolution of the discharge in a uniformly grows within this belt (Fig. 3). charged sample: (a) the simulated (solid line) and measured (dashed line) discharge currents and (b) the discharge struc- The growth rate of the discharge structure and the ture length (1) and the maximum conductance per unit current ﬂowing through the needle depend on the mag- length in the channels (2). nitude and spatial distribution of space charge in dielectric. The higher the charge density, the faster the dis- I, A charge structure growth and the higher the maximum 100 (a) current. If the sample is charged uniformly, then, as the discharge develops, the current decreases. Figure 4a 80 presents the simulated and measured time evolutions of the discharge current for a uniformly charged sample 60 50 × 140 mm in size (the charge density is 0.6 µC/cm2). The growth rate of the discharge structure decreases 40 with time. The simulated time evolution of the discharge structure length L (the distance from the needle 20 tip to the structure farthest point) is shown in Fig. 4b (curve 1). The channel conductance per unit length 0 decreases simultaneously with the deceleration of the γ, 10–4 S m L, mm structure growth (Fig. 4b, curve 2). When the space 3.0 (b) 140 charge is distributed nonuniformly, the discharge characteristics become correlated with the distribution of 2.5 120 the charge density in the channel regions. Figure 5a pre- 1 100 sents the time evolution of the current in a 50 × 140 mm 2.0 × 80 sample, in which ﬁve transverse 50 24 mm belts with 1.5 2 a charge density of 0.8 µC/cm are separated by 5-mm- 2 60 1.0 wide uncharged belts. The current maxima and minima 40 correspond to the moments when the channels pass 0.5 through the charged and uncharged belts, respectively. 20 When the channels pass through the uncharged belts, 0 the growth rate of the discharge structure and the chan- 0 0.2 0.4 0.6 0.8 µ nel conductance per unit length also decrease (Fig 5b). t, s The results obtained show that the charge transport Fig. 5. Same as in Fig. 4 for a nonuniformly charged along the channels and the energy released in the chan- sample.

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(a) ends. Hence, for the voltage drop along the channel to be insignificant, the discharge structure should be well 1 conducting. To sustain the discharge channel in the conducting state, the energy should be permanently 2 released in the channel. The electrostatic electric field created by the space charge in dielectric serves as an 3 energy source. The growth of the discharge structure ensures the conversion of the field energy into the energy of Joule heating. Since the energy density distri- 4 bution correlates with the space charge density distribution, the discharge channels tend to occupy the entire 5 charged region. If there is an uncharged belt, the channels can cross it due to the field created by the charges 6 located at the opposite side of the belt. However, in this case, the released power decreases, which leads to a (b) decrease in the conductance, current, and growth rate of the channels. If the uncharged belt is sufficiently wide, 1 the channel conductance can drop to zero and the discharge structure stops growing. A similar effect was 2 observed in both simulations and experiments. The lengthening of the discharge structure leads to a 3 decrease in the power released per unit length of the channel. Hence, the structure growth is accompanied 4 by a decrease in the channel conductance per unit length. This results in the reduction of the electric field at the channel ends and the deceleration of the growth. 5 The latter leads to a decrease in the power released in the channel, which, in turn, causes a further decrease in 6 the channel conductance. As a result, the conductance drops to zero and the discharge structure stops growing. Fig. 6. Discharge structures in samples with different The higher the space charge density in dielectric, the ρ ρ charge densities: (a) experimental results for n = / 0 = higher the energy density of the electrostatic field. (1) 1, (2) 1.5, (3) 2.0, (4) 2.5, (5) 3.0, and (6) 3.5 and (b) simu- Hence, the dimensions of the discharge structure lation results for the charge density equal to (1) 0.25, (2) 0.38, (3) 0.51, (4) 0.63, (5) 0.76, and (6) 0.88 µC/cm2. should increase with the charge density. This conclusion agrees with the results of simulations and experiments with a discharge in a sample 42 × 350 mm in size L, mm at various charge densities. The discharge structure starts to grow when the charge density reaches a certain 300 ρ critical value 0. Increasing the charge density in a step- ρ wise manner with a step of 0.5 0 results in a significant 200 enlargement of the discharge structure (Fig. 6). The dependence of the structure length on the charge den- ρ 100 sity (normalized to 0) is shown in Fig. 7. A comparison of the experimental data with the sim- 012 3 4 ulation results shows that the model adequately ρ ρ / 0 describes the main features of a discharge induced by space charge in dielectric. This allows us to conclude Fig. 7. Measured (crosses) and simulated (circles) discharge that the main assumptions underlying the model reflect structure length vs. the reduced space charge density. the real physical processes governing the development of the discharge. Since the model is rather general, it can also be applied to describe discharges in various nels vary self-consistently with the growth of the dis- materials under different conditions. The results charge channels, which results in the formation of the obtained can be used to study the discharge phenomena discharge structure in a spatially charged dielectric. The in dielectric components of the apparatus that are necessary condition for the channel growth is the pres- exposed to charged particle beams (e.g., accelerators, ence of a sufficiently high electric field at the channel space crafts, etc.).

TECHNICAL PHYSICS Vol. 47 No. 4 2002 MODELING OF DISCHARGES IN SPATIALLY CHARGED DIELECTRICS 483

CONCLUSION 3. J. Furuta, E. Hiraoka, and A. Okamoto, J. Appl. Phys. 37, 1873 (1966). A numerical model of a discharge induced by space 4. B. Gross and S. V. Nablo, J. Appl. Phys. 38, 2272 (1967). charge in dielectric is proposed. The model is based on 5. C. M. Cooke, E. R. Williams, and K. A. Wright, in Pro- both a stochastic approach to describing the growth of ceedings of the IEEE International Symposium on Elec- the discharge channels and the use of deterministic trical Insulation, Philadelphia, 1982, p. 95. equations for calculating the dynamics of the electric 6. C. M. Cooke, E. R. Williams, and K. A. Wright, in Pro- ﬁeld and charge transport. The three-dimensional ceedings of the International Conference on Properties numerical model allows one to perform computer sim- and Applications of Dielectric Materials, Xian, China, ulations of the spatiotemporal and current characteris- 1985, Vol. 2, p. 1. tics of the discharge. A quantitative agreement between 7. L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Phys. the simulation results and the experimental data on the Rev. Lett. 52, 1033 (1984). discharge development in polymethylmethacrylate 8. H. J. Wiesmann and H. R. Zeller, J. Appl. Phys. 60, 1770 samples charged with the help of electron beams indi- (1986). cates the reliability of the model. The results obtained 9. V. R. Kukhta, V. V. Lopatin, and M. D. Noskov, Zh. Tekh. show that the charge transport and the energy released Fiz. 65 (2), 63 (1995) [Tech. Phys. 40, 150 (1995)]. in the channels vary self-consistently with the growth 10. O. S. Geﬂe, A. V. Demin, V. R. Kukhta, et al., Élek- of the conducting channels, which results in the forma- trichestvo, No. 7, 61 (1994). tion of the discharge structure in a spatially charged 11. D. I. Karpov, V. V. Lopatin, and M. D. Noskov, Élek- dielectric. trichestvo, No. 7, 59 (1995). 12. P. J. J. Sweeney, L. A. Dissado, and J. M. Cooper, J. Phys. D 25, 113 (1992). ACKNOWLEDGMENTS 13. V. Lopatin, M. D. Noskov, R. Badent, et al., IEEE Trans. This work was supported by the Deutsche Fors- Dielectr. Electr. Insul. 5, 250 (1998). chungsgemeinschaft and the Russian Foundation for 14. M. D. Noskov and A. S. Malinovskiœ, in Proceedings of the VI International Conference “Modern Problems of Basic Research. Electrophysics and Electrohydrodynamics of Fluids,” St. Petersburg, 2000, p. 159. REFERENCES 15. R. Rompe and W. Weizel, Z. Phys. B 122, 9 (1944). 16. H. Singer, H. Steinbigler, and P. Weiss, IEEE Trans. 1. B. Gross and K. A. Wright, Phys. Rev. 114, 725 (1959). Power Appar. Syst. 93, 1660 (1974). 2. H. Lackner, I. Kohlberg, and S. V. Nablo, J. Appl. Phys. 36, 2064 (1965). Translated by N. Ustinovskiœ

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Technical Physics, Vol. 47, No. 4, 2002, pp. 484Ð490. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 113Ð119. Original Russian Text Copyright © 2002 by Gall’, Rut’kov, Tontegode.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Carbon Diffusion between the Volume and Surface of (100) Molybdenum N. R. Gall’, E. V. Rut’kov, and A. Ya. Tontegode Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia Received September 18, 2001

Abstract—The diffusion of carbon atoms between the volume and the surface of (100) molybdenum is directly studied at temperatures between 1400 and 2000 K (i.e., at process temperatures) for the first time. The balance of carbon atoms in the system is determined. The difference in the activation energies of carbon dissolution and ∆ precipitation, E = Es1 Ð E1s, is found for the case when the diffusion fluxes of dissolved and precipitated carbon atoms are in equilibrium. This difference defines the enrichment of the surface by carbon relative to the bulk. The experimentally found activation energy of carbon dissolution is Es1 = 3.9 eV. The activation energy of carbon precipitation is estimated at E1s = 1.9 eV. The latter value is close to the energy of bulk diffusion of carbon in molybdenum. © 2002 MAIK “Nauka/Interperiodica”.

Particle diffusion in solids has been the subject of with carbon adsorption on the molybdenum surface at much investigation [1Ð4]. However, atom diffusion temperatures between 300 and 2000 K in a wide range between the volume and the bulk of a metal has escaped of surface and bulk carbon concentrations have been the attention of researchers. This kind of diffusion is of traced from the very beginning of the adsorption to the importance in physical and chemical processes taking complete carbonization (Mo2C) across the tape in place on heated solid surfaces, because relations [7, 8]. between the dissolution and precipitation of impurities must be taken into consideration in this case. Fields where diffusion between surface and bulk plays a great EXPERIMENTAL part are heterogeneous catalysis, metallurgy, semicon- The experiments were carried out under superhigh ductor electronics, corrosion of metals, etc. Yet, we are vacuum conditions (P ≤ 1 × 10Ð10 torr) in a high-resolu- familiar only with one work devoted to this issue [5], ∆ ≤ where the dissolution of carbon on the (100)W surface tion ( E/E 0.1%) Auger spectrometer equipped with a prism energy analyzer [9]. Directly heated textured was studied on a quantitative basis. The authors of [5] × × have established the extremely intriguing fact: the molybdenum tapes measuring 1 0.02 40 mm with energy of activation of carbon dissolution is high, the work function constant over the tape (eϕ = 4.45 eV) 5.0 eV, and far exceeds the energy of activation of bulk were used as substrates. The work function uniformity carbon diffusion in tungsten, 2.55 eV. This means that, was verified with a mass spectrometer by thermionic even knowing the activation energy of bulk diffusion, emission methods and by surface ionization of Na and one cannot comprehensively describe the transport of K atoms [10]. The surface and bulk of the tape were impurities between the bulk and surface of different purified by ac heating to T = 2200 K in the oxygen atmosphere (P ~ 10Ð5 torr) and in superhigh vacuum. metals. Moreover, in some cases of practical signifi- O2 cance, surface-to-bulk diffusion may control the overall After the purification, only Auger peaks of molybde- rate of impurity or dopant diffusion in metals. It is num were observed. During the purification, the tapes therefore necessary to find the energy parameters of were so textured that the (100) face was exposed. The surface-to-bulk and bulk-to-surface transport in solids accuracy of substrate orientation was as high as ≈99.5%, from direct experiments and tabulate the findings in rel- as follows from the X-ray diffraction data. The mean evant reference books together with the energy of bulk grain size in the tape measured in a scanning tunnel diffusion. microscope was ≈20 µm. Both our and literature data Unfortunately, work [5] has not been elaborated [11] indicate that such a treatment of metallic samples upon. In review [6], where gas adsorption, desorption, can effectively remove nonmetal impurities (C, S, N, O, and reactions on metal surfaces are considered, diffu- P, etc.) from both the surface and the bulk. sion over the surface, rather than normally to it, is dis- Temperatures above 900 K were measured with an cussed. The aim of this article is to study the diffusion optical micropyrometer; below 900 K, by extrapolating of carbon atoms between the surface and the bulk in the dependence of the temperature on tape heating cur- thin textured molybdenum tapes. Processes associated rent. The temperature nonuniformity over the tape was

CARBON DIFFUSION BETWEEN THE VOLUME AND SURFACE 485 less than 10 K within 40 mm. Note that the Auger spec- Auger signal intensity from carbon, arb. units tra were taken directly from the heavily heated (up to 2100 K) tapes. This is of fundamental importance in 30 IC studying equilibrium high-and medium-temperature processes occurring between surface and bulk with the participation of carbon atoms, because excess carbon does not precipitate on the surface in this case (unlike 20 AES [5], where it precipitated upon cooling). Carbon was applied on both the working and back surface of the tape from a calibrated carbon atom Mo 10 C source [12]. At room temperature, the sticking coeffi- C cient of carbon was assumed to equal unity [13]. The surface carbon concentration NC, that is, the amount of carbon per 1 cm2 of the surface, was estimated as 20 40 60 NC = v Ct, Annealing time, s v where C is the calibrated atom flux density and t is the Fig. 1. Auger signal from carbon on the working molybde- deposition time. num surface vs. time of annealing at 1450 K. Carbon with a × 16 Ð2 Note that carbon desorbs from the refractory metal concentration NC = 1 10 cm is deposited on the back surfaces only at temperatures above 2000 K [14]. side of the tape at 300 K. Saturation corresponds to the for- × 15 Ð2 mation of surface carbide with NC = 1 10 cm . RESULTS The use of the calibrated flux makes it possible to (1) Carbon Distribution between the Surface find the carbon distribution between the surface and the and Bulk of the Tape bulk of the molybdenum tape (see table). To do this, we If the tape covered by carbon on its working side to applied carbon in amounts corresponding to NC on × 16 Ð2 the concentration NC = 1 10 cm at 300 K is heated either of the sides (at 300 K) and drove it in the tape by to 1450 K, a major portion of the carbon will dissolve heating to 1600 K. Then, the tape was slowly (for sev- in the bulk. The carbon remaining on the surface with a eral minutes) cooled to room temperature and Auger × 15 Ð2 concentration Ns = 1 10 cm forms surface molyb- peaks of carbon were recorded. The table lists the car- denum carbide MoC [7]. If we deposit the same amount bon concentration on the working side (or back side, of carbon (1 × 1016 cmÐ2) on the back side of the tape at since the concentrations are the same) at 300 K and the 300 K and then heat up the tape to 1450 K, Auger sig- amount of carbon that passed into the bulk from 1 cm2 nals of carbon will be observed after a time. The peak of the surface, Nb = NC Ð 2Ns. Since the carbon is uni- is the highest at t > 40 s corresponding to surface car- formly distributed over the molybdenum tape, one can × 15 Ð2 bide with Ns = 1 10 cm (Fig. 1). Thus, irrespective calculate its concentration in one layer of interstitials: of the side on which the carbon is applied to a concen- × 16 Ð2 tration NC = 1 10 cm at 300 K, the carbon concen- N = N /m, × 1 b tration on both sides turns out to be equal, Ns = 1 1015 cmÐ2, after heating the tape to 1450 K and the car- × 16 × 15 × 15 Ð2 Carbon distribution between the surface and the bulk of bon in an amount NC = 1 10 Ð2 10 = 8 10 cm dissolves in the bulk of the molybdenum tape. It fol- molybdenum at different doses NC µ lows that bulk diffusion in the 20- m-thick tape pro- N N N N ceeds with a high rate at 1450 K. C s b 1 × 14 × 14 × 14 × 9 We will show that our results for bulk diffusion are 5 10 <1 10 ~4 10 3 10 consistent with those available in the literature. The 1 × 1015 2.5 × 1014 5 × 1014 3.4 × 109 coefficient of bulk diffusion of carbon in molybdenum 3 × 1015 7 × 1014 1.6 × 1015 1.1 × 1010 is given by [15] 4.5 × 1015 9 × 1014 2.7 × 1015 2 × 1010 D[]cm2/s = 3.4× 10Ð2 exp{}Ð1.78[] eV 11 600/T . 7.5 × 1015 1 × 1015 5.5 × 1015 4 × 1010 × 16 × 15 × 16 × 11 For unidirectional diffusion, the diffusion length λ 8 10 1 10 7.8 10 5.5 10 varies with time as λ = (2Dt)1/2. Putting λ = h = 20 µm, Note: Ns, surface concentration of carbon (on both sides); Nb = we find the time it takes for carbon to diffuse from one NC Ð 2Ns, its bulk concentration; N1, carbon concentration ≈ in the first subsurface layer. The concentrations are given in side of the tape to the other at 1450 K: t 90 s, which atoms/cm2, and the last two quantities are calculated under is in good agreement with our experimental data the assumption that carbon is uniformly dissolved in the (Fig. 1). bulk of the substrate.

TECHNICAL PHYSICS Vol. 47 No. 4 2002 486 GALL’ et al.

5 where m = 2h/d ~ 1.4 × 10 is the number of layers of that Es1 appreciably exceeds E0 for all surface carbon × 15 Ð2 interstitials in the tape and d = 3.06 Å is the lattice con- coatings on molybdenum if Ns < 1 10 cm . stant in the [100] direction [16]. The factor 2 takes As was mentioned, the great difference between E account of the fact that the unit cell of the bcc lattice has s1 two equivalent layers of interstitials that are parallel to and E0 has been reported in [5] for the (100)WÐC sys- the (100) face. tem and in our work [18] for the (100)TaÐC system. We believe that this effect is typical of many MeÐC sys- The calculated value of N1 is also given in the table. tems. At small concentrations of the carbon deposited, NC < 5 × 1014 cmÐ2, its major portion is in the bulk, so that N < 1 × 1014 cmÐ2. For N = 1 × 1014 cmÐ2, the amount (3) Diffusion of Surface Carbon into the Metal s C × 15 Ð2 × at NC > 1 10 cm of the carbon grows both on the surface (to Ns = 2.5 14 Ð2 × 14 Ð2 10 cm ) and in the bulk (Nb = 5 10 cm ). At still Experiments show that surface carbide MoC offers × 15 Ð2 good thermal stability. The Auger signal of the carbon higher NC, NC = 7.5 10 cm , the surface concentra- × 15 Ð2 taken directly from the heated surface carbide tion becomes maximal, Ns = Ns max = 1 10 cm (that decreases only at T > 1400 K. We were interested in the is, MoC carbide forms on the surface), while the bulk kinetics of the dissolution of the surface carbon in the × 15 Ð2 concentration of the carbon reaches Nb = 5.5 10 cm . metal at concentrations higher than 1 × 1015 cmÐ2. For It is of interest that a further rise in NC by one order of this purpose, pure molybdenum was covered by a car- × 16 Ð2 magnitude, NC = 8 10 cm , does not increase the bon film with a concentration of 2 × 1016 cmÐ2 at 300 K. surface concentration of the carbon: surface carbide Then, heating the sample stepwise, we followed the × 15 Ð2 MoC with NC = 1 10 cm is observed on both sides. dissolution of the surface carbon using the Auger technique. Of great interest is the fact that the excess carbon (i.e., the amount of carbon exceeding that needed for (2) Carbon Diffusion into the Metal the formation of the surface carbide) completely dis- × 15 Ð2 at Small NC < 1 10 cm solves at 900Ð1000 K, while the surface carbide Carbon impurity was carefully removed from the remains intact up to T = 1400 K. If carbon of concen- × 15 Ð2 tape, and carbon was applied on the tape to the concentration Ns = 5 10 cm is applied on the previously × 14 Ð2 formed surface carbide MoC at 300 K, the excess car- tration NC < 5 10 cm at 300 K. Then, the tape temperature was stepwise increased and the kinetics of carbon also dissolves in the tape upon heating to 900Ð bon dissolution in the bulk of the metal was examined 1000 K. with Auger electron spectroscopy. It turned out that up It appears that there exist two types of adsorption to T ≈ 1250 K, the surface carbon did not dissolve. The sites on the (100)Mo surface, which greatly differ in the dissolution became noticeable at T = 1350 K, and its energy of dissolution activation. The surface carbide characteristic time was t ≈ 1 min. To find the kinetic originates at sites with strong bonds, and carbon atoms parameters that characterize the diffusion of the surface are arranged in “depressions” between four molybde- carbon into the bulk, we will take advantage of the num atoms. When all such sites are occupied (which results obtained in [17], where the similar problem was corresponds to a surface carbon concentration of 1 × solved. A decrease in the surface carbon concentration, 1015 cmÐ2), sites with weak bonds come into play and dNs, for a time dt is related to the dissolution flux vs1 the temperature threshold of marked carbon diffusion τ ≈ and the mean lifetime s1 of the carbon adatoms to dis- into the metal shifts by 400 K toward lower tempera- solution as tures. For T = 900 K and the dissolution time t ≈ 1 min, we estimated the activation energy E of the dissolu- τ s1 dN ==Ðv s1dt ÐNdt/ s1. (1) tion of the carbon occupying the weakly bonding sites. Integrating (1) when the initial surface concentra- Using the method described in the previous section and τ Ð13 ≈ again putting 0 = 10 s, we found Es1 2.5 eV. tion N0 drops to N for the dissolution time t yields Now let us discuss the nature of the weakly bonding {}τ NN= 0 exp Ðt/ s1 . (2) sites. It is easy to evaluate that the area under all carbon atoms in the surface carbide is insignificant. Indeed, if From (2), one can estimate the energy of activation the diameter of a carbon atom is d = 1.4 Å, the total area E of carbon diffusion into the molybdenum volume: s1 occupied by the carbon atoms entering into MoC {}[]()τ π 2 × 15 2 ≈ × 15 2 Es1 = kTln tln N0/N / 0 . (3) amounts to S = ( d /4) 10 Å 1.6 10 Å per 1 cm2 or ≈16% of the total surface area of the metal. τ Ð13 Putting T = 1350 K and 0 = 10 s, we obtain Es1 = Therefore, if the carbon of the surface carbide occupies 3.9 eV from the initial portion of the dissolution curves. all depressions between four surface atoms of molybde- Such a value of this energy for (100)Mo is unexpect- num (from these depressions, the carbon diffuses into edly too high, exceeding the energy of activation of the bulk with the higher energy of activation, E ≈ ∆ ≈ s1 bulk diffusion, E0 = 1.78 eV [15], by E 2 eV. Note 3.9 eV), carbon atoms covering the surface carbide are

TECHNICAL PHYSICS Vol. 47 No. 4 2002 CARBON DIFFUSION BETWEEN THE VOLUME AND SURFACE 487 adsorbed around the strongly bonding sites, from which Coverage they diffuse into the metal with the lower energy of acti- Θ ≈ C vation, Es1 2.5 eV. Note that the strong dependence of Y the activation energy of carbon dissolution on the car- 1.0 bon concentration on the molybdenum surface, which is observed in our work, is discovered for the first time. 0.8 It seems likely that such a dependence is typical of 3 other metalÐcarbon systems and of other adsorbates. In 1 2 particular, different adsorption sites have been also dis- 0.6 covered in (100)WÐSi [19] and (100)MoÐSi [20] systems: the silicon of surface silicide remains on the sur- 0.4 face up to high temperatures (T = 1400Ð1500 K), whereas the silicon applied on the surface silicide 0.2 X readily dissolves in the metal at 900Ð1000 K. 1200 1400 1600 1800 2000 (4) Equilibrium Diffusion of Carbon Atoms Annealing temperature, K between the Surface and the Bulk Fig. 2. Equilibrium coverage of the (100)Mo surface by car- Consider the results of experiments performed bon vs. substrate temperature. The concentration of carbon under equilibrium conditions, i.e., when the carbon atoms in the layer of interstitials is (1) 1.0 × 1011, (2) 2 × 1011, and (3) 4 × 1011 cmÐ2. Θ = 1 corresponds to the carbon atom flux from the surface, vs1, equals the flux from the × 15 Ð2 concentration in the surface carbide (Ns = 1 10 cm ). nearest subsurface layer to the surface, v1s. It is of interest which information on the (100)MoÐC system can be gained from these experiments. Carbon with a concen- face (1014Ð1015 cmÐ2) and in the near-surface layer of × 16 Ð2 11 Ð2 tration NC = 1.5 10 cm was applied on the working interstitials (~10 cm ) in equilibrium stands out. This side of the tape at 300 K. Then, it was driven in the may occur if the rate of carbon precipitation far exceeds Ӷ metal by heating to 2000 K. After cooling to 300 K, that of dissolution; that is, E1s Es1 may be expected in both sides of the tape exhibited the surface carbide with this case. × 15 Ð2 N = 1.5 10 cm , while the concentration of the dis- × 16 Ð2 2 If another portion of carbon with NC = 1.5 10 cm solved carbon in the bulk per 1 cm was found to be is now deposited on the working side at 300 K and then × 15 Ð2 Nb = 13 10 cm . In other words, the carbon concen- driven in the metal by heating to 2000 K, we will obtain tration per layer of interstitials was a new equilibrium curve Θ = f(T) (Fig. 2, curve 2). Since for each given T = const the diffusion flux of the N ==N /m 13× 1015/1.4× 105 ≈ 110× 11 cmÐ2. 1 b carbon to the surface v1s increases (because the bulk Next, the tape temperature was increased stepwise concentration of the carbon rises), so does the equilib- from 300 to 2000 K, and the Auger signal directly from rium concentration Ns and the curve as a whole shifts the heavily heated sample was recorded. The tempera- toward higher temperatures, retaining its shape (Fig. 2). ture dependence of the carbon concentration on the sur- Curve 3 in Fig. 2 corresponds to a new, still greater face normalized to the carbon concentration in the sur- increase in the bulk concentration of the carbon. × 15 Ð2 Θ face carbide with Ns = 1 10 cm ( = 1) is shown Let us write expressions for the dissolution, vs1, and in Fig. 2 (curve 1). Up to 1400 K, the surface concen- precipitation, v1s, fluxes when the surface and bulk of tration of the carbon remains constant and then the car- the tape exchange particles [4]: bon of the surface carbide starts to dissolve. At 1620 K, v = N*()1 Ð N /N W half the surface carbon is in the dissolved state. The s1 s 1 1m s1 (4) bulk concentration of the carbon is low (N ≈ 1011 cmÐ2), (){} 1 = Ns 1 Ð N1/N1m Cs1exp ÐEs1/kT , so that this portion of carbon makes a minor contribu- {}() tion to the Auger signal. With a further increase in the v 1s = N1W1s = N1C1s exp ÐE1s/kT 1 Ð Ns/Nsm . (5) temperature, the surface concentration of the carbon ≈ × 13 Ð2 Here, Ns and Nsm are the instantaneous and highest pos- continues to fall (Ns 5 10 cm at 1900 K). The curve Θ = f(T) is well reproduced with a temperature sible surface concentrations of the atoms, respectively; rise or fall. Also, no time changes in the surface concen- N1 is the atom concentration in the layer of interstitials tration were observed. The steady states in our experi- that is the closest to the surface; N1m is the concentra- ments are apparently associated with equilibrium con- tion corresponding to the limiting solubility; Ws1 and ditions, under which the surfaceÐbulk mass transfer of W1s are the dissolution and precipitation probabilities, the carbon is absent; that is, the oppositely directed dif- respectively; Es1 and E1s are the energies of dissolution fusion fluxes become equal to each other V1s = Vs1. The and precipitation, respectively; and Cs1 and C1s are pre- great difference in the carbon concentration on the sur- exponentials. The factor (1 Ð N1/N1m) in Eq. (4) reflects

TECHNICAL PHYSICS Vol. 47 No. 4 2002 488 GALL’ et al.

face carbon entirely dissolves in platinum if N1 < E N (T). At N = N (T), the dissolution of the surface s1 Surface 1m 1 1m carbon completely ceases (vs1 = 0) and newly deposited Bulk carbon is accumulated in the layer adsorbed. It is inter- E1s esting that if the carbon occupies even a small fraction, ~10Ð5, of interstitials in the platinum, its diffusion into E the bulk is terminated. The results for the (100)MoÐC 0 Ӷ system were processed for N1 N1m; hence, the factor

(1 Ð N1/N1m ) could be neglected. Fig. 3. Carbon atom energy on the surface and in the bulk as a function of distance normal to the surface. The mass transfer between the surface and bulk of a metal speciﬁes the energy of dissolution activation Es1 ∆ and the energy of precipitation activation E1s. In gen- E, eV eral, both may differ from E , the energy of activation 2.0 0 of bulk diffusion (Fig. 3). We will try to determine the ∆ difference E = Es1 Ð E1s in the activation energies from the carbon equilibrium distribution in the molybdenum tapes with various carbon content. To do this, we cut the family of the curves in Fig. 2 at Ns = const (X section). 1.9 Then, as follows from Eqs. (4) and (5) for the case v1s = vs1 and Ns = const, () {}∆ ln N1 T = ln NsC1s/Cs1 Ð E/kT. (6)

From the slope of the straight line lnN1(T) = f(1/kT), ∆ 1.8 we found E for each of the sections Ns = const. In the surface coverage interval 0 < Θ < 0.5, ∆E = const = 2.0 ± 0.1 eV. At Θ > 0.5, ∆E slightly diminishes, reach- ing 1.8 eV at Θ = 0.95 (Fig. 4). Knowing ∆E, one finds from (6) that Cs1/C1s = 380. Of interest is the fact that at Θ < 0.5 (Fig. 2), the sur- 1.7 Θ face coverage at T = const is directly proportional to the 0.2 0.4 0.6 0.8 1.0 C Coverage amount of the carbon dissolved in the bulk (Y section), of which the constancy of ∆E in this coverage range is ∆ Θ an indication. For Θ > 0.5, ∆E drops and the surface Fig. 4. Dependence E = Es1 Ð E1s = f( ) for the C/(100)Mo adsorption system when the diffusion fluxes coverage grows much more slowly compared with the from and to the surface are in equilibrium. Θ = 1 corre- bulk concentration of the carbon (Fig. 2). It should be × 15 C Ð2 sponds to surface carbide with Ns = 1 10 cm . stressed that the data in Fig. 4 refer to surface carbon × 15 Ð2 Θ concentrations Ns < 1 10 cm ( < 1), when the carbon is in the adsorbed layer and enters into the carbide, the fact that the dissolution flux of the carbon into the ∆ metal vanishes when the limiting solubility of the car- so that Es1 involved in the relationship E = Es1 Ð E1s is bon in the metal is reached. the energy of dissolution activation for the carbon in the surface carbide. The factor (1 Ð Ns /Nsm) in Eq. (5), the so-called Let us consider to what extent the value ∆E = 2.0 eV Langmuir factor, calls for special discussion. The Θ found for < 0.5 (Es1 = 3.9 eV in this case) is adequate. appearance of this factor in this equation is basically ∆ We have E1s = Es1 Ð E = 3.9 Ð 2.0 = 1.9 eV, which is due to historical reasons. Its primary goal is to show close to the energy of activation of carbon bulk diffu- that the surface can receive a limited number of ada- sion in molybdenum. Consequently, in our MoÐC sys- toms (no more than Nsm). As applied to precipitation, its tem, the activation energy of carbon precipitation from meaning is not quite clear; it is anyway obvious that Nsm the nearest subsurface layer is close to the activation is much greater than the concentration of the carbon in energy of bulk diffusion E0 = 1.78 eV [15]. × 15 Ð2 the surface carbide, 1 10 cm . Therefore, if the car- The question arises as to why the surface coverage bon concentration is equal to or lower than that in the × 15 Ð2 surface carbide, this factor can be ignored in calcula- equals Ns = 1 10 cm (MoC surface carbide), being tions, since it is close to unity and its contribution is independent of the temperature and the carbon concen- negligible. tration in the bulk of the molybdenum at T < 1400 K (Fig. 2). It might be expected that at T < 1400 K, the When studying the interaction of carbon atoms with bulk diffusion is “frozen,” preventing the carbon from the heated (111)Pt surface, we found [21] that the sur- being accumulated in the adsorbed layer over the sur-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 CARBON DIFFUSION BETWEEN THE VOLUME AND SURFACE 489 face carbide. This, however, contradicts both to the Carbon Auger-signal, arb. units experimental data (carbon atoms rapidly leave the sur- 40 I face for the bulk through the surface carbide at lower C temperatures, 900Ð1000 K) and to the calculation (as 30 3 follows from the available literature data [15], freezing 2 1 the diffusion processes is out of the question). On the 20 other hand, extrapolating the known data for the limiting solubility of carbon in molybdenum at T > 1800 K 10 [22] to lower temperatures, one finds that the volume concentrations of carbon found experimentally are close to the limiting values at 900Ð1400 K. Direct 100 200 experiments have shown that the deposition of carbon Time of annealing, s at 1000 K results in the formation of bulk molybdenum carbide when the carbon interplanar concentration Fig. 5. Temperature dependence of the Auger signal inten- × ≈ × 11 Ð2 ≈ sity from carbon for the carbonized Mo tape (NC = 5 reaches N1 7 10 cm , or 0.07 at. %. 1016 cmÐ2): (1) 1100, (2) 1160, and (3) 1210 K. Prior to The closeness of the carbon concentration to the sol- annealing, the surface carbide was removed in the oxygen ubility limit may alter the kinetic and energy parame- atmosphere. IC = 7 corresponds to residual carbon on the ters of the bulk diffusion of carbon in molybdenum. surface. Such effects are known and have been reported in the literature [23]. For example, we demonstrated with the using expression (5) for the flux, we determined the C/(10Ð10)Re system [24] that the activation energy of activation energy of carbon precipitation for three tem- the bulk diffusion of carbon significantly grows when peratures, which was found to be E = E = 2.5 ± approaching the saturated ReÐC solid solution and the 0 1s 0.2 eV. This value seems to be close to the activation diffusion is frozen at T < 1100 K. In pure rhenium, how- energy of the bulk diffusion. Knowing E and the value ever, carbon quickly migrates in the bulk of the metal at 1s × 11 Ð2 T = 800Ð900 K. One can expect that the activation N1 = 5 10 cm , one can estimate from (5) the pre- × 12 Ð1 energy of the bulk diffusion of carbon in the MoÐC sys- exponential: C1s = 2 10 s . tem at T < 1400 K exceeds E0 = 1.78 eV, since the bulk Thus, our assumption that the diffusion is frozen at concentration of the carbon dissolved approaches the T < 1400 K is invalid. Therefore, the constancy of the solubility limit. coverage in this temperature range can reasonably be To clarify the situation, we carried out the following associated with the equalization of the potential barri- ers for carbon dissolution in and precipitation from the experiment. First, the solid solution of carbon in ∆ molybdenum corresponding to curve 1 in Fig. 2 was bulk.]; that is, E 0. produced. Then, the temperature of the tape covered by the MoC surface carbide was lowered to the room value DISCUSSION and oxygen was adsorbed nearly to saturation (P ≈ O2 The scarcity of reliable literature data on transport 1 × 10Ð7 torr, t ≈ 30 s). Once the tape had been heated to processes at the surfaceÐvolume interface in metals can 1000 K, we observed the simultaneous decrease in the be related to several reasons. First, to obtain quantita- Auger peaks for oxygen and carbon. Since oxygen tive results, it is necessary to know the exact concentra- remains on the surface of pure carbon-free molybde- tion of dissolved carbon in a metal and carefully vary it, num up to high temperatures, T ≥ 1800 K, the simulta- since even trace amounts of impurities (~10Ð3 at. %) neous reduction of the intensity of the Auger peaks for may drastically influence experimental findings. In our O and C can be explained by the production of CO or work, this difficulty was obviated by properly selecting CO2 molecules and their low-temperature desorption. samples and carbon deposition methods. The use of In this way, we markedly decreased the surface concen- textured metallic tapes makes it possible to easily and tration of carbon from 1 × 1015 to 2 × 1014 cmÐ2 without effectively remove impurities, including carbon, from changing the carbon concentration in the volume. Then, the sample and, in addition, quickly reach uniform we raised the temperature to 1100Ð1200 K and equilibrium between the surface carbon and carbon in observed the precipitation of the carbon on the surface the bulk. In terms of homogeneity and orientation accu- up to the formation of the MoC surface carbide. The racy, the surface of the tape approaches the single-crys- early portions of the precipitation curves taken by the tal surface; however, the fabrication of single-crystal AES method are shown in Fig. 5. First, it is seen that the tapes is as yet a technological challenge. The use of the bulk diffusion is not frozen. Second, the early portions absolutely calibrated flux of carbon atoms allows us to are linear, indicating that the carbon precipitation flux completely cover the sample and evaluate the impurity is constant (v1s = const); in other words, the depletion particle balance in the system. For 40-mm-long tapes of the near-surface region is compensated for by the and a carbon deposition area of 30 mm, the migration rapid delivery of the carbon from the interior. Third, of carbon to the cold edges can be neglected. The feasi-

TECHNICAL PHYSICS Vol. 47 No. 4 2002 490 GALL’ et al. bility of carbon deposition on both sides of the tape is 4. A. Ya. Tontegode, Zh. Tekh. Fiz. 43, 1042 (1973) [Sov. also of great importance. This often made it possible to Phys. Tech. Phys. 18, 657 (1973)]. verify physical concepts put forward. 5. K. J. Rawlings, S. D. Foulias, and B. J. Hopkins, Surf. Surface carbon detection was the basic issue. It was Sci. 109, 513 (1981). found that the Auger spectra of adsorbed carbon taken 6. S. J. Lombard and A. T. Bell, Surf. Sci. Rep. 13, 1 directly from the sample heated up to 2000 K provides (1991). a better insight into the physics of the processes, since 7. A. Ya. Tontegode, M. M. Usufov, E. V. Rut’kov, and the high-temperature state of the surface is affected by N. R. Gall’, Zh. Tekh. Fiz. 62 (10), 148 (1992) [Sov. Phys. Tech. Phys. 37, 1038 (1992)]. high-rate precipitation when the heating current is switched off. 8. E. V. Rut’kov, A. Ya. Tontegode, M. M. Usufov, and N. R. Gall, Appl. Surf. Sci. 78, 179 (1994). 9. V. N. Ageev, E. V. Rut’kov, A. Ya. Tontegode, and CONCLUSION N. A. Kholin, Fiz. Tverd. Tela (Leningrad) 23, 2248 (1981) [Sov. Phys. Solid State 23, 1315 (1981)]. The direct study of high-temperature carbon diffu- 10. É. Ya. Zandberg, E. V. Rut’kov, and A. Ya. Tontegode, sion between the (100) surface and the bulk of molyb- Zh. Tekh. Fiz. 46, 2610 (1976) [Sov. Phys. Tech. Phys. denum was performed for the ﬁrst time. The kinetic 21, 1541 (1976)]. parameters of carbon transport from the surface (disso- 11. B. M. Shepilevskiœ and V. G. Glebovskiœ, Poverkhnost, lution) and to the surface (precipitation) were deter- No. 7, 26 (1982). mined. The nature of thermally stable molybdenum 12. N. R. Gall, E. V. Rut’kov, A. Ya. Tontegode, et al., Chem. carbide on the surface at T < 1400 K was discussed. It Vap. Deposition 5, 72 (1997). was inferred that its thermal stability is apparently asso- 13. J. H. Leck, Pressure Measurement in Vacuum Systems ciated with the equalization of the activation energies (Chapman and Hall, London, 1964), pp. 70Ð100. for both processes. It should be taken into account that 14. A. Ya. Tontegode, Prog. Surf. Sci. 38, 201 (1991). a considerable difference between the activation ener- 15. P. S. Rudman, Trans. Metall. Soc. AIME 239, 1949 gies of bulk diffusion (≈1.8 eV) and dissolution (1967). ≈ ( 3.9 eV) may cause great (by many orders of magni- 16. C. Kittel, Introduction to Solid State Physics (Wiley, tude) differences in the rates of diffusion processes. New York, 1975; Nauka, Moscow, 1978). This is of importance in metallurgy, heterogeneous 17. A. Ya. Tontegode and F. K. Yusifov, Poverkhnost, No. 4, catalysis, and other applications. These values, along 20 (1994). with the kinetic parameters for bulk diffusion, must 18. N. R. Gall, E. V. Rut’kov, and A. Ya. Tontegode, Surf. undeniably be tabulated in reference books on metal Sci. 472, 187 (2001). physics. 19. V. N. Ageev, E. Yu. Afanas’eva, N. R. Gall’, et al., Pov- erkhnost, No. 5, 7 (1987). ACKNOWLEDGMENTS 20. N. R. Gall, E. V. Rut’kov, A. Ya. Tontegode, and M. M. Usifov, Phys. Low-Dimens. Semicond. Struct. This work was supported by the Program “Atomic 9/10, 137 (2000). Surface Structures” of the Ministry of Science of the 21. E. V. Rut’kov and A. Ya. Tontegode, Fiz. Tverd. Tela (St. Russian Federation (project no. 4.6.99). Petersburg) 38, 635 (1996) [Phys. Solid State 38, 351 (1996)]. 22. E. Fromm and E. Gebhardt, Gase und Kohlenstoff in REFERENCES Metallen (Springer-Verlag, Berlin, 1976; Metallurgiya, 1. R. M. Barrer, Diffusion in and Through Solids (Cam- Moscow, 1980). bridge Univ. Press, Cambridge, 1941). 23. G. Schulze, Metallphysik (Akademic, Berlin, 1967; Mir, 2. P. G. Shewmon, Diffusion in Solids (McGraw-Hill, New Moscow, 1971). York, 1963; Metallurgiya, Moscow, 1966). 24. N. R. Gall, S. N. Mikhailov, E. V. Rut’kov, and A. Ya. Tontegode, Surf. Sci. 191, 185 (1987). 3. M. P. Seach, Practical Surface Analysis by Auger and X-ray Photoelectron Spectroscopy (Wiley, London, 1983), pp. 273Ð317. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 491Ð494. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 120Ð123. Original Russian Text Copyright © 2002 by Belyaev, Rubets, Nuzhdin, Kalinkin.

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Effect of Thermodiffusion on Perfection of Crystal Structures Formed by Condensation from Vapor Phase A. P. Belyaev, V. P. Rubets, M. Yu. Nuzhdin, and I. P. Kalinkin St. Petersburg State Technological Institute, St. Petersburg, 198013 Russia e-mail: [email protected] Received February 5, 2001

Abstract—Results are reported of an investigation of the effect of nonuniform synthesis conditions stimulating thermodiffusion on the perfection of the crystalline structure of cadmium telluride films synthesized in vacuum by condensation from the vapor phase. Results of technological, geometrical, electorgraphical, and electron- microscopic studies are given. It is shown that the positive effect of the nonuniform conditions on the perfection of the crystalline structure formed has a thresholdlike character. In the case where thermodiffusion takes place, the substrate temperature has an unusual effect on the crystalline perfection of the film obtained. Against expectations, raising the substrate temperature tends to increase the disorder in the structure. It has been found that thermodiffusion affects the duration of the Ostwald ripening phase and therefore provides a means of controlling the structure formation processes. The experimental data are shown to agree with the theory of the film formation processes. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION using a MII-4 microinterferometer accurate to ± µ The latest years have seen an increased interest of 0.03 m. Structural studies were made with an ÉMR- researchers and technologists in nonstandard condi- 100 electronograph and a PÉM-100 electron micro- tions of film synthesis [1, 2]. One of these is obviously scope. All structural data presented below refer to the synthesis in a graded temperature field. It is reported [3] central area of the films. that such fields improve the crystalline perfection. However, subsequent studies of the film growth mech- EXPERIMENTAL RESULTS anisms under noniniform conditions revealed that their influence has a thresholdlike character and is beneficial The effect of the graded temperature field on the only under certain regimes. Given below are new perfection of the crystalline structure formed was stud- research results on the vapor-phase growth processes of ied with the use of technological, geometrical, and cadmium telluride films in a graded temperature field, structural methods. The main results are given in which show the threshold character of the nonuniform Figs. 1Ð3. conditions and are consistent with the theory of film Figure 1 shows the effect on the film structure per- growth. fection of the substrate temperature and the temperature field. Electronograms in Figs. 1a and 1b demonstrate degradation of the crystalline perfection of the SAMPLES AND EXPERIMENTAL films at lower temperatures under uniform conditions. TECHNIQUE Comparing Figs. 1a and 1b, it can be seen that while at Film formation processes both under uniform (same a temperature of Ts = 523 K (Fig. 1a), the film structure temperature at all substrate points) and nonuniform (in is close to epitaxial; at Ts = 473 K (Fig. 1b), it is already a graded temperature field) conditions were studied on polycrystalline. Electronograms in Figs. 1cÐ1e refer to cadmium telluride films synthesized on muscovite sub- nonuniform conditions. The electronograms in strates. The film thickness for different samples did not Figs. 1c–1e were obtained from films synthesized at µ exceed 1 m. The films were synthesized by a quasi- temperatures Ts of 473, 490, and 523 K, respectively. It closed (hot wall) method [3] under a vacuum of ≈10Ð3 Pa is seen in these electronograms that the influence of the at a substance flux incident onto a substrate of Φ = temperature gradient on crystalline perfection can be 1.32 × 1016 cmÐ2 sÐ1. The temperature gradient of the both beneficial and detrimental. The effect is positive at thermal field along the substrate (nonuniform synthesis low substrate temperatures. In the absence of the tem- conditions) was produced by heating its central part and perature field at Ts = 473 K, the films obtained are poly- cooling the periphery [3]. The substrate was heated by crystalline (Fig. 1b), whereas in the opposite case the a heater at the center of the substrate holder. The tem- structure formed is close to epitaxial (Fig. 1c). The pos- perature was measured by chromelÐalumel thermocou- itive effect is a function of the substrate temperature. ples. Film thickness and its uniformity were measured From Figs. 1c (473 K) and 1d (490 K), it is seen that at

492 BELYAEV et al.

(‡) (b) (c)

(d) (e)

Fig. 1. Electronograms from cadmium telluride films synthesized under uniform conditions (a, b) and under temperature gradient (cÐe) at substrate temperatures of Ts = 523 (a, e), 473 (b, c), and 490 K (d). a higher temperature there is less ordering in the struc- the radial coordinate x. Curve 1 refers to growth under ture. uniform conditions at Ts = 473 K; curves 2 and 3 refer The effect of nonuniform temperature is negative at to growth in a nonuniform temperature field at Ts = 523 high temperatures, as seen in the electronograms in and 473 K, respectively. The figure clearly demon- Figs. 1a and 1e. At Ts = 523 K, the films grown with a strates the effect of the nonuniform temperature field on uniform temperature profile are close to epitaxial the growth rate, which is lower in the central area and (Fig. 1a) and have a significantly worse structure if higher at the periphery. The growth rate and its nonuni- grown under a temperature gradient (Fig. 1e). formity in the radial direction depend on the substrate Figure 2 demonstrates the effect of the temperature temperature, both being higher at elevated tempera- field on uniformity of the growth-rate distribution along tures. The effect of the temperature field on surface mor- vc, nm/s phology is seen in Fig. 3. Films in Figs. 3a and 3b were 8.5 grown at Ts = 523 K, and the film in Fig. 3c at 473 K. Electron micrographs in Figs. 3a and 3c show the sur- 8.0 faces of films synthesized under nonuniform conditions and in Fig. 3b, under uniform conditions. From the 7.5 electron-microscopic studies, it follows that with the 7.0 temperature field the size of film crystallites is smaller 1 if the substrate temperature is high. At lower substrate 6.5 temperatures under nonuniform growth conditions, the 3 6.0 crystallites grow to a larger size. 5.5 2 RESULTS AND DISCUSSION 0 0.2 0.4 0.6 0.8 1.0 r, arb. units According to current concepts, the process of film synthesis from a vapor phase on the surface of a solid Fig. 2. Dependence on the coordinate of the growth rate of consists of several stages. These are nucleation induced layers of cadmium telluride (the x axis originates at the sub- by fluctuations, the Ostwald ripening (OR) or coales- strate center). Curve 1 refers to growth under uniform con- cence, and merging into a continuous film. Perfection ditions at Ts = 473 K; curves 2 and 3 refer to growth in a temperature field at Ts = 523 and 473 K, respectively. of the crystal structure formed is usually determined by

TECHNICAL PHYSICS Vol. 47 No. 4 2002

EFFECT OF THERMODIFFUSION ON PERFECTION OF CRYSTAL STRUCTURES 493

(‡) (b) (c)

Fig. 3. Surface morphology of cadmium telluride films grown in uniform conditions (b) and with the temperature gradient (a, c) at × substrate temperatures Ts = 523 (a, b) and 473 K (c) ( 2000). the OR stage [4]. At this stage, by means of the gener- from the source in all the above experiments was the alized diffusion field, there takes place a correlated same; therefore, in what follows we consider only the redistribution of the atoms of the condensed substance sinks of a substance. between disperse particles (DP) formed, as a result of In the experiment in question, as shown in [3], the nucleation and formation of correlated chemical bonds sinks of a substance are formed by a nonuniform tem- (incorporation of atoms into the crystal lattice of DPs). perature field. Their capacity can be estimated from the Hence, it can be supposed that the deterioration of the equation [6] crystal structure due to the temperature field at high substrate temperatures (Figs. 1a, 1e), as well as the k i = ÐρD ∇c + ----T-∇T . (2) higher perfection obtained in the case of relatively low T temperatures (Figs. 2b, 2c), occur at the OR stage. In this connection, let us consider the effect of tempera- Here, i is the substance flux due to thermodiffusion, ρ is ture on OR in a nonuniform temperature field. the effective density of adatoms on the surface, D is the coefficient of the surface diffusion of atoms, ∆c is the The growth rate of a DP in a generalized field is adatom concentration, ∇T is the temperature gradient, affected by other DPs and depends on the concentration k is the thermodiffusion ratio, and Dk is the coeffi- of the components in the system and on temperature. In T T particular, if the DP growth rate is limited by incorpo- cient of thermodiffusion. ration, the expression for the growth rate has the According to [3], at low temperature, sinks of this form [5] kind lengthen the OR stage and thus serve to make the film crystal structure more perfect. At higher tempera- σβ 2 ψ ()αΘΘ () 2 V m 1 R tures, as testified by the experimental results under dis- ϑ = ------1Ð , (1) cussion, the effect of thermodiffusion becomes nega- k TR R B cr tive. The reason can now be easily seen from Eqs. (1) where σ is the surface tension, β is the unit boundary and (2). Increasing the temperature decreases both the flux onto a DP, V is the volume of an atom in the solid DP growth rate v, which ensures the correlated rear- m rangement of atoms in a DP at the OR stage, and the phase, ψ (Θ) and α(Θ) are the parameters accounting 1 capacitance of substance sinks affecting the OR process for the DP shape, kB is the Boltzmann constant, T is the duration. This conclusion is further supported by the temperature, R is the DP radius, and Rcr is the critical decrease of the size of crystallites (Figs. 3a, 3c) and radius of a DP. increase of the film growth rate (Fig. 2). The concentration of components on the substrate In addition, this conclusion helps to explain the neg- surface (entering (1) implicitly through Rcr) is deter- ative influence of the sinks on the film structure perfec- mined by the substance fluxes from/to sources/sinks tion in comparison with uniform conditions. As men- and their initial distribution function over the size tioned above, at high temperatures, when under uni- space, which, in addition to temperature, are again form conditions the films growth is epitaxial, the determined by the substance sources/sinks. The flux introduction of thermodiffusion flows are found to

TECHNICAL PHYSICS Vol. 47 No. 4 2002

494 BELYAEV et al. degrade the crystalline structure of the film (compare ripening stage and compensating for the negative effect Figs. 1a and 1e). Evidently, the reason is that, in the of the sinks on the incorporation process. presence of thermodiffusion sinks, the atoms at the OR (3) During the normal layer-by-layer film growth stage do not have enough time to be incorporated into from the vapor phase with nondecaying sources of sub- the crystal lattice of a DP and be involved into second- stance, both nucleation due to fluctuations and the Ost- ary nucleation, which, naturally, impairs the crystalline wald ripening take place. perfection of the growing film (the effect predicted theoretically in [1]). Additional evidence of this is provided by morphology studies (Figs. 3a, 3b). Sizes of the ACKNOWLEDGMENTS crystallites in films grown in the presence of sinks hap- We are grateful to S.A. Kukushkin for helpful sug- pen to be considerably less than in films grown at the gestions and discussions. This work was supported by same substrate temperature but under uniform condi- the Russian Foundation for Basic Research, project no. tions. Therefore, although the theory distinguishes 99-03-32676. three stages in the process of first-kind phase transformation on the surface of a solid, these stages may, in fact, be concurrent. REFERENCES 1. D. A. Grigor’ev and S. A. Kukushkin, Zh. Tekh. Fiz. 68 (7), 111 (1998) [Tech. Phys. 43, 846 (1998)]. CONCLUSIONS 2. A. P. Belyaev, V. P. Rubets, and I. P. Kalinkin, Zh. Tekh. From the results presented above, the following con- Fiz. 71 (4), 133 (2001) [Tech. Phys. 46, 495 (2001)]. clusions can be made. 3. A. P. Belyaev, V. P. Rubets, M. Yu. Nuzhdin, and I. P. Kalinkin, Fiz. Tverd. Tela (St. Petersburg) 43, 745 (1) Thermodiffusion sinks, spontaneous or inten- (2001) [Phys. Solid State 43, 778 (2001)]. tional, occurring during vapor-phase growth on the sur- 4. S. A. Kukushkin and A. V. Osipov, Usp. Fiz. Nauk 168, face of films having nondecaying sources of a sub- 1083 (1998) [Phys. Usp. 41, 983 (1998)]. stance can have totally different effects on the crystal- 5. S. A. Kukushkin, Fiz. Tverd. Tela (St. Petersburg) 35, line perfection of the growing film. 1582 (1993) [Phys. Solid State 35, 797 (1993)]. (2) Weak thermodiffusion sinks interfere with the 6. L. D. Landau and E. M. Lifshitz, Course of Theoretical incorporation of atoms into the crystal lattice and pro- Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, duce disorder in the crystalline film growing on the sub- 1986; Pergamon, New York, 1987). strate. Higher capacity sinks stimulate formation of a perfect crystalline structure by extending the Ostwald Translated by B. Kalinin

TECHNICAL PHYSICS Vol. 47 No. 4 2002

Technical Physics, Vol. 47, No. 4, 2002, pp. 495Ð496. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 4, 2002, pp. 124Ð125. Original Russian Text Copyright © 2002 by Korytchenko, Volkolupov, Krasnogolovets, Ostrizhnoœ, Chumakov. BRIEF COMMUNICATIONS

Shock Wave Strength and Discharge Energy in Gases K. V. Korytchenko, Yu. Ya. Volkolupov, M. A. Krasnogolovets, M. A. Ostrizhnoœ, and V. I. Chumakov Uskoritel’ Research Complex, Kharkov, 61108 Ukraine e-mail: [email protected] Received April 2, 2001; in ﬁnal form, September 11, 2001

It is demonstrated that taking advantage of the parameters ensuring the formation of the shock wave dynamic properties of a gas stream and applying the under the conditions outlined above and estimated for 5 technique for directional shock wave formation, one P0 = 10 Pa and T0 = 293 K were the following: the tem- can generate an intense shock wave at low gas dis- perature T = 3840 K, the velocity D = 2935 m/s, the charge energies. In our experiments, we used the setup pressure P = 100 × 105 Pa, and the density ρ /ρ = 6.76. schematically shown in the figure. It operates as fol- 1 1 2 lows. Cooled air from an external source is forced The necessary condition for the shock wave genera- through inlet 5 made in conductor 3 and then through tion in denotation tubes is the laminarity of the flow in hole 7 into cavity 4 of insulator 1. The dimensions of any outflow mode. As is known from experiments, with outlet 6 in the conductor, inlet 5, and holes 7 are chosen disturbances near the inlet eliminated, the flow remains so that the pressure in cavity 4 of insulator 1 is higher laminar up to the Reynolds number R = 105, where R = than the ambient pressure. The pressure smoothly drops ν Umaxd/2 , Umax is the velocity of the fluid at the tube to the ambient value in and outside channel 6. The gas axis, d is the diameter of the tube, and ν is the fluid flow velocity in channel 6 cannot exceed the critical kinetic viscosity. In our case, Umax corresponds to the velocity of sound C∗2. Thus, the velocity of disturbance velocity D of the shock wave front and d, to the diame- propagation in the gas flow varies from C0 to C∗2 (C0 is ter of the outlet. The maximal heating temperature T in the velocity of sound in the stagnant gas flow) [1]. the insulator cavity can be determined by equating the maximal velocity of the air flow heated to the critical Closing switch S causes the capacitive energy stor- velocity of sound C∗ : age to discharge. Part of the discharge energy is 2 released as the kinetic energy, which raises the pressure 2 ()γ 2 in cavity 4 and gives rise to a compression wave propa- C 2M + 1 D M()γ + 1 T ==------* ------, gating along channel 6. The incoming gas pressure pre- 2γR 2γR vents the gas from flowing towards channel 5. Since the rate of vibrational-to-kinetic energy conversion is where M is the molar mass of the gas in kg/mol and R is directly proportional to the gas pressure, deactivation the gas constant. processes in the cavity lead to a further rise in the pressure and faster increasing-amplitude compression waves begin to propagate in the somewhat heated gas. 1 This results in the formation of a shock wave. This pro- 2 4 cess is similar to the shock wave formation in detona- 5 tion tubes [2]. Based on the theory of vibrational deactivation of gas molecules [3], a technique for evaluating the design parameters of the setup described above has been elaborated. It takes into account shock wave formation conditions in detonation tubes and the gas properties. 3 6 7 The design of the setup extends the time of shock wave formation, which allows the more efficient use of C the discharge energy through an increase in the duration of the discharge and also due to the fact that the discharge is initiated in the higher-pressure region. The Setup for the shock wave formation by a gas discharge.

496 KORYTCHENKO et al. The discharge duration is found from the time t of REFERENCES shock wave formation: 1. L. D. Landau and E. M. Lifshitz, Course of Theoretical l l Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, t = ------Ð ------, 1986; Pergamon, New York, 1987). C01 C02 2. F. A. Baum, K. L. Stanyukovich, and B. I. Shekhter, where l is the outlet length and C01 and C02 are the Physics of Explosion (Fizmatgiz, Moscow, 1959). velocities of sound in the stagnant ﬂow under the initial conditions and in the gas heated by the electrical dis- 3. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock charge, respectively. Waves and High-Temperature Hydrodynamic Phenom- ena (Nauka, Moscow, 1963; Academic, New York, For example, if d = 0.001 m and l = 0.0015 m, we 1966). ﬁnd that R ≈ 104, T = 11100 K, and t = 4 × 10Ð5 s. For comparison, the time of shock wave formation in open air under normal initial conditions is no less than 10Ð5 s. Translated by A. Sidorova-Biryukova

TECHNICAL PHYSICS Vol. 47 No. 4 2002

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On the Surface States in Magnesia and Baria A. P. Savintsev1 and A. I. Temrokov1, 2 1 Kabardino-Balkar State University, ul. Chernyshevskogo 173, Nalchik, 360004 Russia e-mail: [email protected] 2Research Institute of Applied Mathematics and Automation, Russian Academy of Sciences, Nalchik, 360000 Russia Received May 18, 2001

Abstract—The band gap narrowing due to the Tamm states, as determined from the spectroscopy data for magnesia and baria single crystals, are compared with analytical results for the (100) face of these crystals. © 2002 MAIK “Nauka/Interperiodica”.

It is well known that the interface between a solid The results of calculation are compared with UV and a vacuum or another medium is usually character- spectroscopy data for BaO and MgO single crystals ized by specific energy states, so-called surface states. with a thickness of 0.38 and 1.2 mm, respectively [2]. When the surface disturbance is sufficiently high, The oxides have the crystal lattice of rock-salt type with bound surface states, such as Tamm states [1], may the (100) cleavage plane, which features the smallest σ appear at the surface. In a number of cases, surface [2, 6]. states merge with surface energy bands. As a rule, the energy distribution of surface states is difficult to reli- As follows from Fig. 2, the transmission band edge ably establish. It is commonly supposed that, in ionic in the BaO and MgO crystals is not coincident with the and partially ionic crystals, there exist acceptor surface fundamental absorption edge Ea. The fundamental ± bands below the conduction band and donor surface absorption edge Ec Ð Ev is 4.0 0.2 and 6 eV in BaO bands above the valence band. It is also assumed that and MgO, respectively [2]. The transmission band edge the surface bands in these crystals merge with the conduction and valence bands (Fig. 1). Therefore, the presence of surface (Tamm) states in dielectric crystals may alter the band gap at their surface. We compare the pre- Ec dicted effect of Tamm states on the band structure in E1 BaO and MgO crystals with spectroscopic data. There is not much data today on the surface properties of BaO E2 and MgO compounds, having the rock-salt structure Ev [1, 2]. The band gap narrowing due to Tamm states (∆E) at Fig. 1. Bands of ionic surface states [1]: Ec, conduction the dielectric surface was calculated for the (100) crys- band edge; Ev, valence band edge; E1, acceptor surface- ∆ state band edge; and E , donor surface-state band edge. tal face. According to Fig. 1, E = Ec + E2 Ð Ev Ð E1. 2 The quantum-mechanical calculation of the band gap narrowing for the BaO (100) face gives ∆E = 0.6 eV [3]. K, % 80 In the quasi-classical approximation, the band gap narrowing due to surface states is related to the face 60 specific surface energy σ [3, 4] as 40 12 ∆E = 4σ/n, 20 where n is the number of particles per unit face surface 0 area. 3.0 3.5 4.0 4.5 5.0 5.5. 6.0 E, eV Setting σ = 957 mJ/m2 for the MgO (100) face and the lattice constant to be equal to 0.43 nm [5], we find Fig. 2. Transmission of (1) BaO and (2) MgO single crystals that ∆E = 1.11 eV. versus photon energy.

498 SAVINTSEV, TEMROKOV ± ± E1 Ð E2 is 3.5 0.05 and 4.55 0.10 eV in BaO and REFERENCES MgO, respectively. From these values, the band gap 1. S. Davison and J. Levine, Surface States (Academic, narrowing London, 1970; Mir, Moscow, 1973). 2. A. A. Vorob’ev, Physical Properties of Ionic Crystalline ∆ EE==c + E2 Ð Ev Ð E1 Ea Ð Et Dielectrics (Tomskiœ Univ., Tomsk, 1960), Vol. 1. 3. T. M. Taova, A. I. Temrokov, and A. Yu. Kishukov, Effect is estimated at 0.5 ± 0.2 and 1.45 ± 0.10 eV for BaO and of High-Power Energy Fluxes on Substance (Inst. Vys. MgO, respectively. Temp. Akad. Nauk, Moscow, 1992), pp. 66Ð77. 4. A. P. Savintsev and A. I. Temrokov, in Proceedings of the Thus, we conclude that (1) the energy difference 15th International Conference “Equations of State,” between the fundamental absorption edge and the Elbrus, 2000, p. 25. transmission band edge can serve as a measure of the 5. A. I. Temrokov, Doctoral Dissertation, Leningrad, Len- band gap narrowing due to surface states and (2) the ingrad Gos. Univ., 1982. band gap narrowing calculated for the (100) face of 6. V. D. Kuznetsov, Surface Energy of Solids (GITTL, baria and magnesia is in good agreement with the trans- Moscow, 1954). mission vs. photon energy spectra of these single crystals. Translated by A. Sidorova-Biryukova

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A Relationship for the Surface Impedance at the Skin Effect for an Arbitrary Kernel of Integral A. I. Spitsyn Kharkov State Technical University of Radio Engineering, Kharkov, 433053 Ukraine Received June 21, 2001

Abstract—For the case of the skin effect in metals and an arbitrary function J(R) entering into the nonlocal relation between the current density and the electric ﬁeld, general distributions of the electric ﬁeld and surface impedance are found. The results obtained are applicable to both normal and superconducting metals and are represented as the Fourier transform of the function J(R). © 2002 MAIK “Nauka/Interperiodica”.

In determining the surface impedance of both nor- density vector j and the vector potential A: mal metals and superconductors, one uses the nonlocal ∂A relation between the current density j and the electric E ==Ð------ÐiωA, field E at the skin effect, which generally has the form ∂t [1, 3] (4) RR()⋅ A j = ÐC ------JR()dV. ()⋅ A∫ 4 () RR E () R jr0 = CE∫------JRdV, (1) V R4 V The coefficients CE and CA relate as where integration is performed over the whole space, ω CA = i CE. (5) R = r Ð r0 is the difference between the radius vectors of the point of integration and the point at which the We also assume that the function J(R) has the current density is determined, and (…) is the scalar ∞ product. asymptotics as R : α We set J(0) = 1 and represent the proportionality JR()∼ lÐR/ ,Reα > 0. (6) coefficients CE for a harmonically time-varying field (~liωt) in the form For normal metals, α = ξ; for superconductors in the α ξ stationary case, = F [5]. 3 C = ------, (2) Consider a metal bounded by a planar surface S and E π ωµ ξδ2 4 i 0 assume that E = E(z). Next, introduce the rectangular coordinate system with its origin on the plane (see fig- where δ is the complex penetration depth, which equals ure) and with the z axis directed inward to the metal. Let δ c.l if the local limit exists [4]. Relationship (2) contains the parameter

∞ O x ξ = ∫ JR()dR, (3) ϕ ρ 0 r which, in the case of normal metals, is equal to l/(1 + 0 r iωτ), where l is the mean free path of a conduction elec- z0 tron and τ is the relaxation time [2]. For superconductors (in the stationary case), the y R parameter ξ coincides with the coherence length introduced into the BCS theory [3]. z In the literature, instead of representation (1), one may encounter the relationship between the current Figure.

500 SPITSYN the x axis be aligned with the vector E. Then, from rela- Eq. (13), we find tionship (1), j = j = 0 and x z () 2 ˜ () ωµ Ω˜ () Ð2E'0 Ð0.k E k Ð i 0 E k = (15) 2 x E () ˜ ˜ jx ==jCE∫------4-JRdV, Ex =E. (7) Here, E (k) and ΩE (k) are the Fourier transforms of the R Ω V functions E(z) and E(z): Now, introduce the cylindrical coordinate system +∞ +∞ related to the initial one (see figure): ˜ () ()ikz0 Ω˜ () Ω ()ikz0 E k = ∫ Ez0 e dz0, E k = ∫ E z0 e dz0. ρϕ ρϕ x ==cos , y sin , Ð∞ Ð∞ 2 ρ2 ()2 ρ ρ ϕ From (15), R ==+ zzÐ 0 , dV d d dz. | | 2E'0() After introducing the variable u = R/ z Ð z0 , relationship E˜ ()k = Ð------, (16) 2 ωµ Ω˜ () (7) can be written in the form k + i 0 E k ∞ + which, after the inverse Fourier transformation, leads to () Ω()() jz0 = ∫ E zzÐ 0 Ezdz, (8) the solution Ð∞ ∞ 2E'0() coskz dk ∞ Ez()= Ð------0 . (17)  0 π ∫ 2 ˜ Ω () π 1 1 () k + iωµ ΩE()k E z = CE∫--- Ð ----- Juz 0 0 u u3 (9) 1 Ω˜ π []() () From relationship (9) for E (k), we have = CE EJ1 z Ð EJ3 z , ∞ where Ω˜ () π 1 1 ˜ k E k = CE∫----- Ð ----- J--- du, (18) ∞ u2 u4 u () 1 () Juz EJn z = ∫------du. (10) un ˜ 1 where J(k) is the Fourier transform of the function J(|z|). With regard for relation (4) between j and A, expression (8) is replaced by The surface impedance is given by ∞ +∞ iωµ E()0 2iωµ dk () Ω()() Z ==Ð------0 ------0 ------, (19) jz0 = Ð ∫ A zzÐ 0 Azdz. (11) E'0() π ∫ 2 ωµ Ω˜ () k + i 0 E k Ð∞ 0 δ ωµ In the relation between j and A through the plane and the complex penetration length c = Z/(i 0) has Ω kernel of integral A, we use, as is customary, the minus the form sign before the integral. It is evident that ∞ 2 dk Ω = iωΩ . (12) δ = ------. (20) A E c π∫ 2 ˜ k + iωµ ΩE()k The corresponding equation for the electric field E 0 0 inside the metal, In general, the skin depth is defined by the expres- +∞ sion [4] 2 d E ωµ Ω ()() δ δ ------2- Ð0,i 0 ∫ E zzÐ 0 Ezdz = (13) s = Re c. (21) dz0 ∞ Ð Now, we will show that the choice of the coefficient will be solved for the case of the mirror reflection of CE in form (2) is in agreement with nonlocal relation- electrons from the boundary. In this case, the solution is Ω ship (1). Since the characteristic distance of E(z) vari- symmetric about the plane S [4]: ation is on the order of |ξ| and E(z) varies over a dis- δ Ez()= Ez()Ð . (14) tance on the order of the skin depth s, then, in the non- 0 0 |ξ| Ӷ |δ | δ δ local limit c.l ( c = c.l), the electric field in Setting the boundary conditions in the form Eq. (8) can be factored out from the integral:

d +∞ ----- Ez() ==E'0(), E()∞ 0 dz z = 0 ()≈Ω() Ω() ˜ ()() jz0 Ez0 ∫ E zzÐ 0 dz = E 0 Ez0 . and applying the Fourier transformation to the terms of Ð∞

TECHNICAL PHYSICS Vol. 47 No. 4 2002 A RELATIONSHIP FOR THE SURFACE IMPEDANCE 501 ξ Using this relation in Eq. (13), we obtain with the associated relationships for a and n given in [4, 8]. The terms of series (24) are real and have the 2 1 δ = ------. (22) same form as in the Pippard theory of superconductiv- c.l ωµ Ω˜ () i 0 E 0 ity but different values of the parameter ξ. From (18), it follows that 4 REFERENCES Ω˜ ()0 = ---πC ξ; (23) E 3 E 1. D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958). consequently, the choice of the coefﬁcient C in form 2. G. E. H. Reuter and E. H. Sondheimer, Proc. R. Soc. E London, Ser. A 195, 336 (1948). (2) corresponds to the limit |ξ| Ӷ |δ |. c.l 3. R. Meservey and B. B. Schwartz, Superconductivity To conclude, we give the expressions for the func- (Marcel Dekker, New York, 1969), pp. 17Ð191. tion J(R) in different cases. When calculating the sur- 4. F. F. Mende and A. I. Spitsyn, Surface Impedance of face impedance of normal metals [6], one uses the func- Superconductors (Naukova Dumka, Kiev, 1985). tion J(R) from the ReuterÐSondheimer theory: J(R) = exp(ÐR/ξ), where ξ = l/(1 + iωτ). In the case of the Pip- 5. J. Halbritter, Z. Phys. 243, 201 (1971). pard kernel in the phenomenological theory of super- 6. A. I. Spitsyn, Radiotekh. Élektron. (Moscow) 38, 2152 conductivity, J(R) = exp(ÐR/ξ) at the coherence length (1993). ξ ξ = p > 0 [7]. In the microscopic theory of supercon- 7. A. B. Pippard, Proc. R. Soc. London, Ser. A 216, 547 ductivity (the stationary case), (1953). ∞ 8. A. I. Spitsyn, Zh. Tekh. Fiz. 64 (4), 68 (1994) [Tech. 4a exp()ÐR/ξ ()Tl, Phys. 39, 385 (1994)]. JR()= ------∑ ------n -, (24) π ()2 2 πtanh------n = 0 12+ n + 1 a 2a Translated by Yu. Vishnyakov

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The Possibility of Gaining Information on the Crystallographic Orientation of Cylindrical Samples from Surface-Impedance Measurements at Microwave Frequencies A. I. Spitsyn Kharkov State Technical University of Radio Engineering, Kharkov, 433053 Ukraine Received June 21, 2001

Abstract—From measurements of the Q factors or frequency shifts of H1 polarized modes in a coaxial cavity information on the crystallographic orientation of a single-crystal sample of which the cavity inner cylinder is made can be derived. The choice of H1 modes in the coaxial cavity and its sizes at microwave frequencies is optimized. It is found that the determination of the crystallographic orientation of a uniaxial crystal at the H121 mode is much more accurate than that at E1 modes. © 2002 MAIK “Nauka/Interperiodica”.

Measurements of the mean surface impedance Z = any of the E or H modes with the zeroth subscript and then, using the determined value of the partial Q factor R + i X of single crystals at microwave frequencies θ provide a possibility of gaining information on the for the cylindrical surface, find the angle . For this pur- crystallographic orientation of samples with the aniso- pose, it is more appropriate to use the TEM or E0 mode tropic surface impedance [1]. In [1], on the assumption of the coaxial cavity. In the figure, curve 1 represents θ θ of the local tensor relation between the electric field and the ratio ZTEM( )/Z01 as a function of . This ratio is the current density, we calculated the mean surface equal to that of the partial Q factors for the cylindrical impedance and its components for cylindrical samples. surface of the sample, Q(π/2)/Q(θ), or to the ratio of the The calculation was performed for the E0 and E1 modes frequency shifts in the coaxial cavity with respect to the (for example, of a cylindrical or coaxial cavity) with the frequency of a cavity with the perfectly conducting first subscripts 0 and 1 for the case of a uniaxial crystal. walls (the shifts are due to the cylindrical surface), ∆ω θ ∆ω π γ However, the corresponding results for the H modes are ( )/ ( /2). The ratio = Z01/Z03 was taken to be more appropriate for determining the crystallographic orientation. In this study, particular numerical results θ π θ θ θ are presented for the H1 mode in the coaxial cavity TEM( /2)/( TEM( )), QH(0)/(QH( )) where the coaxial is made of a sample under investigation. 1.4 In the expansion of the Fourier component of the Zxx tensor in the polar angle ϕ on the cylindrical of a uniax- 1 ial crystal, the coefficient a2 defines surface the value of 2 Z - for the H mode and has the form [1] H 1 1.3 1 2 a = ---()θZ Ð Z sin , (1) 2 2 01 03 where Z03 and Z01 are the surface impedances with the 1.2 resistivities equal to the principal values of the resistiv- ρ ρ ity tensor, 3 and 1, in the directions parallel and perpendicular to the C principal axis of the uniaxial crys- 3 tal, respectively, and θ is the angle between the cylin- 1.1 drical axis Z and the C axis. Note that the ratio a2/ZH for the H1 mode may amount to several tens of percent, whereas the corresponding ratio for the E modes is as θ 1 1.0 low as several percent. 0.4 0.8 1.2 π/2 In determining the angle θ for a given sample, one can measure the Q factor of the TEM mode or that of Figure.

THE POSSIBILITY OF GAINING INFORMATION 503 equal to 1/2 . This dependence is specified by the C axis lies. Thus, the direction of the C axis of a uniaxial cylindrical crystal is defined by the two directions. sample material. The figure (curves 2 and 3) shows the θ dependences of The direction of the C axis can be determined by the ratio Q(0)/Q (θ) for two polarized modes H in measuring the Q factors (or frequency shifts) of the two 1, 2 121 the above coaxial cavity when the ratio Z /Z = 1/2 . orthogonal polarized H1 modes in the coaxial cavity. 01 03 From [1], it follows that the mean surface impedance of In a coaxial cavity, there always exist irregularities the cylindrical surface in the case of the H modes is rep- (holes, pins, etc.), which, together with anisotropy, resented as affect the resulting directions of polarized modes. If the orientations of polarized modes are specified in the α ZH = Zzz + v Zxx, (2) main by irregularities, there additionally appears the that is, depends on both the mean component of the sur- possibility of changing the directions of mode polarizations in the single-crystal sample, for example, by rotat- face impedance tensor Zzz and Zxx . Since the maxi- ing the sample about the z axis in the presence of the mum difference between Zzz and ZTEM is much less irregularities. In this case, the ratio of the partial Q factors of the polarized modes is [1, 2] than between Zxx and Z0 , where Z0 is the mean sur- () face impedance of the cylinder measured for the Q 0 Z0 ± 1 a2 ϕ α Ӷ ------()θ = ------cos2 0, (4) H0 mode, it is desirable that /v 1. For the H1ml mode, Q12, Z01 2Z01 2 ϕ α πlR 1 R where 0 is the angle between the polarization direction ---- ==------2 ------, p -----1. (3) v h 2µ4 () R of one of the polarized modes and the projection of the p 1m p 2 ϕ π θ crystal C axis. At 0 = 0 or /2, the ratio Q(0)/Q1, 2( ) Here, R1 and R2 are the inner and outer radii of the coax- for one of the polarized modes will be maximal and for µ the other, minimal. ial cavity, respectively, and 1m is the mth root of the equation Attaining the maximum and minimum Q factors of the polarized modes by rotating the cylindrical sample ()µ ()µ ()µ ()µ J1' 1m N1' p 1m Ð0,J1' p 1m N1' 1m = about the z axis, one can find the plane containing the C axis from the directions of their polarizations. Note that where J1 and N1 are the Bessel and Neumann functions, the values of Q1 and Q2 for the resulting polarized respectively. modes can also be used for the independent determina- In order that the ratio α/v be small, it is necessary to tion of the angle θ. take the mode subscript l = 1. If the cavity sizes are such π ≤ Such a procedure of gaining information on the that R2/h 1, the mode subscript is m = 2, and p = 1/2 crystallographic orientation of cylindrical samples is µ (hence, the root 12 = 6.565), this ratio in the 3-cm also applicable in the general case of triaxial crystals. Ð3 range will be ~10 . In this case, the term αZzz in relationship (2) can be neglected and it is possible to con- REFERENCES sider that ZH , in our case, depends only on Zxx . 1. A. I. Spitsyn, Zh. Tekh. Fiz. 64 (11), 105 (1994) [Tech. As was shown in [2], in the absence of irregularities Phys. 39, 1146 (1994)]. in a coaxial cavity, the anisotropy is the main factor that 2. A. I. Spitsyn, Available from the State Scientific and influences the orientation of the polarized modes. Technical Library of Ukraine No. 653-Uk (Khar’kov, Therefore, the resulting polarized modes, as in the case 1996), p. 97. of planar samples [3, 4], will be oriented in such a way 3. F. F. Mende, A. I. Spitsyn, A. V. Skugarevski, and that one of them has the polarization direction along the L. A. Maslova, Cryogenics 25 (2), 92 (1985). projection of the C axis onto the plane perpendicular to 4. F. F. Mende, A. I. Spitsyn, and N. N. Dubrov, Zh. Tekh. the cylindrical axis of symmetry, and the other is ori- Fiz. 50, 1609 (1980) [Sov. Phys. Tech. Phys. 25, 938 ented perpendicularly to this projection. Consequently, (1980)]. knowing the polarization directions of the steady-state polarized modes, one can find the plane where the Translated by Yu. Vishnyakov

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The Structure and Fatigue Life of Titanium Alloys Processed by Electrical Pulses S. V. Loskutov and V. V. Levitin Zaporozh’e State Technical University, Zaporozh’e, 69060 Ukraine e-mail: [email protected] Received August 14, 2001

Abstract—The structure and properties of VT3-1 titanium alloy processed by electrical pulses are investigated. The electric pulse processing improves the fatigue life of titanium specimens, as follows from high-cycle fatigue tests. © 2002 MAIK “Nauka/Interperiodica”.

The effect of high-power current pulses on the metal ing in an ultrasonic field, as well as by vibration treat- structure was considered in [1Ð4]. It was shown that the ment. In the process of vibration treatment, the speci- electric current of density on the order of 100 MA/m2 mens were fixed in a cell containing ceramic granules. not only thermally affects the metal crystal lattice but The cell was put in vibration motion, and the vibration also has a specific (electroplastic) effect on lattice time and amplitude were varied. defects. For instance, the fatigue life of steels was To process the specimens with unipolar current extended after they had been processed with electric pulses, we used the discharge of a capacitor bank from current pulses [1]. The same processing seems to be a T1220 press generating electrohydraulic pulses. The promising for titanium alloys as well. current through the specimen was controlled by varying VT3-1 and VT8 titanium alloys, containing α and the length and the thickness of a fine copper wire to be β phases, offer high operating characteristics and espe- exploaded. For the most part, the specimens were pro- cially high strength. The reason is the metastability of cessed with current pulses of height ≈170 MA/m2 and the α and particularly β modifications, as well as the duration of 10 µs. The heating temperature was deter- complex kinetics of the phase transformations, which mined with a ChromelÐCopel thermocouple elastically are accompanied by the formation of new metastable pressed against the middle of the specimen. It is obvi- phases [5]. These transformations proceed in local vol- ous that the temperature thus measured is somewhat umes, and the kinetics of these processes is related to underestimated. During the processing with current the type and concentrations of alloying elements. The pulses, the temperature was varied in the range between phase composition and the phase transformation kinet- 80 and 250°C. Changes in the surface state of the spec- ics depend on the thermal treatment of the titanium imens were estimated by taking the contact potential alloys, the value and duration of dynamic and static difference (CPD) distribution. loads, surface plastic strains, etc. Different hardening treatments cause high (up to 850 MPa) residual com- The specimens were investigated before and after pression stresses in the surface layer of parts made of the current pulse processing with a DRON-3M comput- titanium alloys. The residual stresses change the prop- erized diffractometer [6]. The CoKβ monochromatic erties of the material that influence the fatigue strength. radiation was employed. Residual macrostresses were Experience suggests that the application of high-den- measured with the 2θÐsin2ψ method, and microstrains sity current pulses as the final stage of treatment may and coherent scattering block sizes were approximately bring parts subjected to load cycling into a more energy found from the diffraction peak broadening. favorable near-equilibrium state. This effect is associ- A small piezoelectric shaker [7] was employed in ated with the enhanced diffusion mobility of atoms in the high-cycle fatigue tests. The vibration frequency the vicinity of lattice defects because of preferred heat and the dynamic load were varied, and the number of evolution around them. cycles to failure was noted. The critical load at which Our aim was to perform X-ray diffraction investiga- the specimens fail the test base (the prescribed number tion into structure changes in the titanium alloys after of cycles was 2 × 106) was looked for all types of electric pulse processing and to reveal its effect on the mechanical processing. Each batch comprised six spec- high-cycle fatigue life. imens mechanically processed under identical condi- Test specimens (shovels) made of VT3-1 titanium tions. Then, the specimens were subjected to the elec- alloy had the working part measuring 2 × 10 × 60 mm. tric pulse processing and tested at this critical load to The surface was cold-hardened with small balls vibrat- estimate their lifetime. For each of the batches, the

THE STRUCTURE AND FATIGUE LIFE 505 mean number of cycles to failure is listed in table. Devi- –U, mV ations from the mean were within 30% for all of the specimens. 160 Oxide films colored from light yellow to brown 1 appear on the surface of the specimens after the electric 120 pulse processing. However, the CPD profile (or the 2 energy relief of the surface) changes insignificantly, as follows from the figure, where the shift of the CPD dis- 80 tribution is shown. Near the clamp (l = 22 mm), where 3 the specimen cross section is the largest and the current 40 passes, the change in the CPD distribution is also minor. Within the operating part, the CPD tends to grow 4 (the electron work function increases), which is typical of oxidizing titanium alloy surfaces. 0510 15 l, mm To find residual stresses in strained metals, one CPD distribution over the specimen surface: (1) initial, selects X-ray reflections with Bragg angles close to π/2. (2) in 48 h, (3) after single-pulse processing (8 kV, heating The accuracy of this method depends on strain-induced to 250°C), and (4) in 48 h after the pulse. lattice oriented microdistortions, which may violate the linear dependence of the interplanar spacing d on sin2ψ, ψ ing of the specimens, we carried out a number of exper- where is the angle between the normal to the speci- iments. The specimens were heated by direct current to men surface and the normal to a reflecting crystallo- ° ° graphic plane. It was found that the plane stress state of 350 C and annealed at 650 C for 3 h in a vacuum (the × Ð5 the surface layer is improved after the current pulse residual pressure was 2 10 torr or below). 2 processing (the plots in the 2θÐsin ψ method become In accordance with the X-ray diffraction data, in more linear) [8]. Presumably, this means that the defect these specimens, the intensity of the diffraction peaks distribution in the surface layer becomes more uniform grows insignificantly. The phase composition remains and the plane stress state sets up. almost unchanged. However, the residual macros- The short processing time and relatively low heating tresses become tensile, the microstrains are reduced, temperatures raise the question regarding a mechanism and mosaic blocks increase in size. The heating of the of structure transformations when high-power current specimens to 350°C in air results in the formation of an pulses disturb the crystal lattice. To reveal structural oxide film similar to that formed during the electric changes caused by the dc heating and vacuum anneal- pulse processing.

Characteristics of the titanium alloy structure and the fatigue test results Mean number N Batch Residual macros- Microstrains Block size Processing of cycles to failure, number tresses σ, MPa ε, 10Ð3 D, nm ×106 08 Mechanical grinding and annealing Ð82 2.5 17 0.1 The same with subsequent current pulse Ð103 1.7 19 0.3 application (t = 250°C) 07 Mechanical grinding and annealing Ð37 2.8 18 0.5 The same with subsequent current pulse Ð38 2.5 16 1.2 application (t = 250°C) 04 Ultrasonic processing with balls 1.9 mm Ð530 1.6 13 1.3 in diameter, 3 min The same with subsequent current pulse Ð560 1.5 14 1.8 application (t = 80°C) 05 Vibration grinding and polishing, 30 min Ð590 1.2 12 1.8 The same with subsequent current pulse Ð360 2.0 20 2.6 application (t = 225°C) 06 Vibrational hardening with ceramic Ð320 0.9 14 2.7 granules 1.9 mm in diameter The same with subsequent current pulse Ð285 2.4 22 5.0 application (t = 250°C)

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506 LOSKUTOV, LEVITIN After the vacuum annealing, the diffraction peak power current pulse undergo structure modifications intensity grows, the resolution of closely spaced peaks that are distinguished from those appeared after the from different phases is improved, and the residual similar dc heating or after the vacuum annealing. The macrostresses relax. For the initially stressed surfaces, plane stress state of the surface layer after the current the growth of microstrains and mosaic blocks after the pulse processing is improved (the plots in the 2θÐsin2ψ annealing is more significant than in the case of the method become more linear). As a result of the electric stress-free surfaces. In the specimens subjected to the pulse processing, the fatigue life of the titanium alloy ultrasonic hardening and vibration grinding, the specimens surface-hardened by different procedures annealing reduces the spread of the structure parame- rises. ters from specimen to specimen and macro- and microstresses become nearly equal in all the specimens. The measurements and the test results are given in REFERENCES the table. The electric pulse processing is seen to extend 1. L. B. Zuev, O. V. Sosnin, S. F. Podboronnikov, et al., Zh. the fatigue life of the specimens subjected to the differ- Tekh. Fiz. 70 (3), 24 (2000) [Tech. Phys. 45, 309 ent pretreatments. In the case of the surface-hardened (2000)]. specimens, the electric pulse processing reduces the 2. A. I. Pinchuk and S. L. Shavreœ, Fiz. Tverd. Tela (St. residual macrostresses. However, the residual micros- Petersburg) 43, 1416 (2001) [Phys. Solid State 43, 1476 tresses rise and mosaic blocks somewhat increase. (2001)]. The X-ray diffraction data also point to the fact that 3. N. E. Kir’yanchev, O. A. Troitskiœ, and S. A. Klevtsur, the titanium alloys processed by electric pulses change Probl. Prochn., No. 5, 101 (1983). their phase composition. The diffraction peaks (200)β 4. N. V. Dubovitskaya, S. M. Zakharov, and L. N. Larikov, Fiz. Khim. Obrab. Mater., No. 3, 128 (1980). and (110)β alter to the greatest extent. The intensity of the former grows, which may be explained by the 5. M. A. D’yakova, E. A. L’vova, and S. Z. Khabliev, Fiz. α'' β low-temperature transformation [9]. Met. Metalloved. 44, 1254 (1977). As follows from the high-cycle fatigue tests, the 6. B. A. Serpetskiœ, V. V. Levitin, S. V. Loskutov, and lifetime of the specimens subjected to a single current V. K. Man’ko, Zavod. Lab. 64 (3), 28 (1998). pulse (when the residual compression stresses are 7. S. V. Loskutov, V. V. Levitin, V. K. Man’ko, et al., Zavod. reduced by 100Ð150 MPa) increases two or three times Lab. 65 (7), 43 (1999). at the same levels of the dynamic load. Conceivably, the 8. D. M. Vasil’ev and V. V. Trofimov, Zavod. Lab. 50 (7), 20 partial reduction of the residual stresses is attended by (1984). changes in the structure that improve the fatigue 9. E. A. L’vova, M. A. D’yakova, and S. Z. Khabliev, Izv. strength of the specimens. Akad. Nauk SSSR, Met., No. 1, 154 (1979). From the experimental data obtained, we can conclude that the titanium specimens subjected to a high- Translated by B. Malyukov

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Detection and Measurement of the Current of Ribbon-Shaped Electron Beams at a Particular Section of a Converging Emitter System V. P. Narkhinov Department of Physical Problems, Presidium of Buryat Scientiﬁc Center, Siberian Division, Russian Academy of Sciences, Ulan-Ude, 670047 Russia e-mail: [email protected] Received August 22, 2001

Abstract—A method and an associated device for directly measuring the current distribution over 28 radially converging electron beams are considered. The operation of the device, which can easily be installed in the coaxial system of a gas-discharge electron source, is described. The short-focus radial beams are found to feature a relatively high current uniformity. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION anode rings 5 and collector electrode 6. The collector A number of design solutions are employed in gas- has 28 1.2-mm-diam. holes 7 and is arranged so that the discharge electron injectors in order to form beams centers of the holes and of 28 cells 3 coincide. propagating along a certain direction. Coaxially The design of the device is illustrated in Fig. 2. arranged electron sources can produce both diverging 60-mm-long 5-mm-diam. Faraday cup 1, enclosed in [1] and radially converging [2] charged particle beams. protective housing 2, is fixed to holder 3 by a stud and Annular plasma emitters with a radially converging nut. Conducting copper bushing 5 between dielectric beam, which are used for pumping lasers, significantly holder 3 and intermediate shaft 4 provides a sliding increase the lasing efficiency and power because the contact between pressure terminal 6 and rotating Fara- electron beam energy is uniformly applied to the gas- day cup 1. Shaft 4 (intermediate bushing) is fitted to the eous mixture [2]. shaft of synchronous motor 7. Holder 8 of pressure ter- The radially converging beams are also used in elec- minal 6 and the SD-54 synchronous motor are mounted tronÐion processing of materials and for curing lac- on table 9 (aluminum disk), which is installed in a quers and resins on wires and cables. For a radially con- recess made in support insulator 11. When the device is verging electron beam to efficiently act on an object, its energized through wires 10 and vacuum-tight connec- azimuthal power density distribution should be as uni- tors, the Faraday cup starts rotating (n = 2.24 rpm) form as possible. inside collector electrode 6 (Fig. 1). Electron beams are complex subjects of study. Some of the electrons penetrating through the small Therefore, there are no universal methods that can com- holes are captured by the Faraday cup and produce prehensively characterize any beam, although a vast electric signals, which are measured. variety of well-established diagnostics tools exist [3]. This paper proposes a technique for evaluating the EXPERIMENTAL RESULTS azimuthal current uniformity of converging electron AND DISCUSSION beams over a given section of the transport path. The currents of the electrons captured are proportional to the current densities near the holes: DESCRIPTION OF THE EXPERIMENTAL Ir() SETUP Jr()= ------, S Figure 1a shows gas-discharge electron source 1, which produces 28 ribbon-shaped radially converging where I(r) is the current through the hole and S is the electron beams (with viewing lid 2 removed); Fig. 1b, cross-sectional area of the hole. the device developed for conducting the experiment. The measurement error due to the error δ in measur- Electrons produced by a gas-discharge plasma, which ing the currents is given by is localized in 28 elementary Penning cells 3, are δ extracted through adjustable slit 4 by applying a voltage ∆J = ---; from a high-voltage rectifier across the gap between S

508 NARKHINOV

3 2 8 1 6 5 1 9

3 4 4

5 10 (‡) 7 (‡)

6 2

7 (b)

Fig. 1. Plasma electron source with the built-in experimental device for the diagnostics of beam currents uniformity of: (1) plasma electron source, (2) viewing lid, (3) Penning gas-discharge cells, (4) emission slit, (5) anode rings, (6) collector electrode, and (7) holes.

(b) i.e., the error in determining the current density depends on the absolute error δ in measuring the cur- Fig. 2. (a) Design and (b) installation of the device with one rent of the electrons that pass through a hole of cross- Faraday cup: (1) Faraday cup, (2) protective insulating housing, (3) dielectric holder, (4) intermediate bushing sectional area S. (shaft), (5) conducting bushing, (6) pressure terminal, (7) electric motor, (8) pressure terminal holder, (9) table, Our estimates showed that the scatter in the currents (10) wires, and (11) insulating support. is relatively small (no more than 5% including the measurement error). The average density of the azimuthal electron current from all the 28 radially converging the rotating Faraday cup, one of the most accurate beams was 10Ð5 A/cm2. instruments, measured the currents repeatedly. In the next series of experiments, the cross section The well-known traveling-slit, perforated-chamber, of the ribbon beams was varied from 15 to 40 mm2 by vibrating-probe [3], and rotating-probe [4] techniques are inapplicable to this problem, because they cannot changing the height of the exit window of the emission handle a large number of beams simultaneously. Our slit in 0.5-mm steps with the beam width kept constant method for detecting and measuring the current distri- (10 mm). It turned out that the current uniformity is bution over the variable cross sections of 28 radially independent of the beam cross section. The current den- converging ribbon electron beams can be used for ﬁnd- sities calculated in six experiments were constant ing actual currents along the circumference of a within one tenth. 140-mm-diam. cylindrical body being processed. The experimental results can be considered quite The use of a single rotating Faraday cup greatly sim- reliable, because the electron emission was stable and pliﬁes measurements in a multibeam radially converg-

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DETECTION AND MEASUREMENT OF THE CURRENT 509 ing system. Due to its simple design, the device is easy ties for the improvement and modiﬁcation of both the to fabricate and maintain, which favors a further devel- method and the device. opment of the method for charged particle beam diagnostics. REFERENCES Varying the diameter of the collector electrode and the radial position of the Faraday cup with the remain- 1. A. M. Efremov, B. M. Koval’chuk, Yu. E. Kreœndel’, ing components kept unchanged, one can measure the et al., Prib. Tekh. Éksp., No. 1, 167 (1987). current distribution in any cross section of the electron 2. S. P. Bugaev, L. G. Vintizenko, V. I. Gushenets, et al., in beams. When the anodeÐcollector spacing is longer Proceedings of the VII All-Union Workshop on High- than 10 mm, one should supply the source with an Current Electronics, Tomsk, 1988, Part II, p. 174. accelerating electrode and apply the above technique, 3. S. I. Molokovskiœ and A. D. Sushkov, Intense Electron placing the perforated collector coaxially with the and Ion Beams (Énergoatomizdat, Moscow, 1991). accelerating electrode. 4. O. K. Nazarenko, V. E. Lokshin, and K. S. Akop’yants, By displacing the centers of the holes from the beam Élektron. Obrab. Mater., No. 1, 87 (1979). axes, one can ﬁnd the current density distribution over the beam cross section. Thus, there are vast potentiali- Translated by A. Khzmalyan

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A Change in the Physical State of a Nonequilibrium Blood Plasma Protein Film in Patients with Carcinoma E. Rapis Tel-Aviv University, Ramat-Aviv, 64239 Israel Received September 24, 2001

Abstract—Pronounced changes in the physical phase and in the phase transition dynamics of proteins in drying blood plasma are observed. The morphologies (topologies) of these nonequilibrium ﬁlms in donors and in patients with various types of metastatic carcinoma qualitatively differ by the process of protein ﬁlm self- assembly and by symmetry type. New types of defects and solid crystals appear, liquid crystals persist for a long time, etc. The microscopic examination of the drying protein plasma topology can be used for diagnosing metastatic carcinoma. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION are direct signs of biological self-organization. If so, the breaking of protein self-organization at the cell The functionality and the structure of material in level should be considered as the basic stage in the both organic and inorganic nature depend basically on pathogenesis of malignant tumors. However, the study its physical state (phase) and phase transition features. of the cancer pathology is being concentrated largely Yet, a correlation between the formation of functioning on the cell morphology, immunology issues, search for protein at the supramolecular (cell) level and its phase oncogenes, etc. For example, there is evidence for bio- transitions under various condensation conditions (for chemical, colloidal, and other changes in the blood example, the solÐgel transition) has not been ade- indices [9Ð12], as well as for the appearance of quately studied to date, although phase transitions in glutamic acid and its protein constituents in blood (Sav- biological systems specify the protein behavior. They ina and Tuev, 1998). However, protein and its self-orga- attend the most important dynamic phenomena in a liv- nization are being investigated at the molecular and ing organism, such as cell and nucleus division; protein atomic levels (the study of molecular chain spiraliza- synthesis; fermentative, immune, and locomotive processes; etc. [1]. Thus, this offers scope for devising pro- tion [13] is an example). It appears that a distinct ten- tein phase tests. dency in studying the pathology of malignant tumors (specifically carcinomas) in terms of protein self-orga- Our early experiments [2Ð5] have clearly demon- nization at the supramolecular level is absent. strated the possibility of in vitro observing the phase transition dynamics in the proteinÐwater system. These To consider carcinoma from this standpoint using observations have been compared with the in vivo data for self-organization reported in [2Ð5], we per- behavior of the drying blood protein plasma, which is formed a special investigation. normally the formation of the “protos” protein modification. Upon the drying, the biological self-organization of the protein film with the specific geometry (topology) and different symmetry types and scales takes place (Fig. 1). However, these data have not been used to compare the physical state (phase transitions, etc.) of normally and pathologically self-organized protein in medicine and biology. However, it is well known that the growth of malignant cells breaks the order of symmetry at the cell level. In the absence of ordered film growth, the space distribution of the cells changes, which eventually causes the generalization of the process and the death of the organism [6Ð8]. It is of interest to trace whether these pathological changes are related to self- organization in biological systems. It is believed that the formation, division, and multiplication of cells, as well as their systematic arrangement in space and time, Fig. 1. Typical protein symmetry.

A CHANGE IN THE PHYSICAL STATE 511

MATERIALS AND METHODS Our aim was to visualize the liquidÐsolid transition in the fresh plasma separated out from patient’s blood prior to experiments and in the blood plasma kept in a refrigerator at +1.0°C for three days. The plasmas were placed on a solid substrate in equal amounts (for details, see [2Ð5]). The objects of investigation were plasma samples taken from five patients (45 samples) with metastatic cancer (mammary gland carcinoma, neck-of-womb carcinoma, and lungs carcinoma) (group 1). They were compared with blood samples taken from 20 healthy donors (45 samples) (group 2) (Fig. 2). To date, we performed only phenomenological studies; however, even the qualitative data are rather con- vincing. The morphologies (topologies) of the drying blood films for the two groups distinctly differ in a number of physical parameters: phase transition, symmetry type and scale, protein phase, etc. In group 2, the disturbances showed up as irregular division into protein blocks (films) with symmetry and similarity violation and increased cell nuclei. The central part of the cell (nucleus) becomes too large compared with the irregular movable (fluxible) edge, at which a great array of asymmetric defects is observed (Fig. 3). The contours of the defects and films only tend to Fig. 2. Micrograph of the drying blood plasma of a healthy flex; there are no regular parallel flexed lines or films donor. entering into each other (Fig. 2). The films lose the regular orbital (spiral) form, looking like leaves or warts with irregular edges and hyperfine structures inside. sionality, fractality, nucleation, and adhesion to the sub- Small randomly arranged tapered protrusions are fre- strate; (3) the equilibrium solid-crystal phase, appear- quently observed. The most surprising thing is the clus- ing under nonequilibrium conditions of condensation. tering of well-faceted cubic crystals, which usually do This means that we established changes in the blood not form under nonequilibrium conditions. They are as plasma proteins taking place in patients with cancer if slipped over straight thin filaments, which are parallel when the protein condenses during self-assembling to each other. With time (in several months), the crys- (phase transitions). tals form large agglomerates, retaining the regular This is because our approach makes it possible to shape. determine, from morphological (topological) patterns, Another surprising thing was that these crystals normal protein self-organization with certain symme- appeared near the protein films (in an optical micro- tries and scales in a complex system, such as the blood scope, one half of the film had usual black color; the plasma. Such behavior of protein is based on its prop- other, with crystal clusters seen inside, had an irregular erty to compete in activity with other plasma ingredi- shape and was colored white). ents. It follows from our results that the protein concentration in the blood plasma of a healthy person provides the regular morphology of a drying protein film DISCUSSION (Fig. 2). Thus, we succeeded to visualize the profound The profound qualitative changes in the physical changes in the phase transition and the phase state of properties of the blood plasma proteins in patients with blood plasma proteins in patients with metastatic can- metastatic carcinoma highlight a new aspect of the cer. Under nonequilibrium conditions, instead of the problem: extracellular pathological change in the pro- solid-state modification “protos” [2–5], identified in the teins with violation of self-organization (upon phase plasma of the donors, we revealed the coexistence of transitions), which alters their phase state. three protein states: (1) the long-lived (up to six to eight Therefore, the need for detailed investigation into months) liquid-crystal phase with a very high mobility the phases and symmetries of the solid modification of (“fluidity”); in this case, the great disturbance of optical the protein, as well as of its liquid-crystal phase, for effects (hence, of the optical axis orientation) was various forms of cancer is obvious. The investigation observed; (2) the strongly modified protos state with should be performed by medical men in cooperation the violation of symmetry, self-similarity, tridimen- with physicians engaged in different field of physics

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512 RAPIS

(‡) (b) (c)

Fig. 3. Micrograph of the drying blood plasma of a patient with cancer. using contemporary biological and physical technolo- 2. E. G. Rapis and G. Yu. Gasanova, Zh. Tekh. Fiz. 61 (4), gies. This problem becomes of primary importance in 158 (1991) [Sov. Phys. Tech. Phys. 36, 406 (1991)]. the treatment of cancer, since little is known about 3. E. G. Rapis, Pis’ma Zh. Tekh. Fiz. 21 (9), 13 (1995) chemical and morphological variations in a cancer cell [Tech. Phys. Lett. 21, 321 (1995)]. and also about changes in the physical phases of blood 4. E. G. Rapis, Pis’ma Zh. Tekh. Fiz. 23 (7), 29 (1997) plasma proteins. It is necessary to ﬁnd reasons for those [Tech. Phys. Lett. 23, 263 (1997)]. physical changes found experimentally. They can 5. E. G. Rapis, Zh. Tekh. Fiz. 70 (1), 121 (2000) [Tech. hardly be ignored in studying the pathogenesis of Phys. 45, 121 (2000)]. malignant transformations. 6. D. S. Coffey, Nat. Med. 4, 1342 (1998). Even our pioneering observations, as yet statisti- 7. V. Backman et al., Nature 406 (6), 36 (2000). cally unreliable, must attract the attention of the scien- 8. T. Goetze and J. Briokman, Biophys. J. 61, 109 (1992). tiﬁc society. 9. A. B. Classman and E. Jones, Ann. Clin. Lab. Sci. 24 (1), 1 (1994). 10. G. Buccheri, D. Ferringino, C. Ginardi, et al., Eur. J. ACKNOWLEDGMENTS Cancer 33, 50 (1977). I sincerely thank Profs. M. Amus’ya, A. Arel’, 11. B. Ya. Gurvits and K. G. Korotkov, Consciousness and E. Braudo, V. Buravtsev, V. Volkov, A. Zaikin, Physical Reality. JSSN. No. 1029-4716, Vol. 1, No. 1 M. Klinger, L. Manevich, S. Moiseev, Yu. Noeman, (Publishing Company Folium, 1998), pp. 84Ð90. I. Prigogine, and M. Safro for the fruitful discussions, 12. J. A. Forrester, E. J. Ambrose, and M. G. P. Stoker, valuable comments, and encouragement. Nature 201, 945 (1964). 13. P. G. Wolynes and W. A. Eaton, Phys. World 12 (9), 39 (1999). REFERENCES 1. B. M. Alberts et al., Mol. Biol. Cell 3, 1 (1992). Translated by V. Isaakyan

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