Effect of the Ion-Beam-Induced Getters on the Parameters of Neutron-Irradiated Gaas Heterostructures S

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 645Ð646. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 1Ð5. Original Russian Text Copyright © 2000 by Obolenskiœ, Skupov.

Effect of the Ion-Beam-Induced Getters on the Parameters of Neutron-Irradiated GaAs Heterostructures S. V. Obolenskiœ and V. D. Skupov Nizhegorodskiœ State University, Nizhniœ Novgorod, Russia Received February 28, 2000

Abstract—It was found that the negative inﬂuence of neutron irradiation on the parameters of Schottky-gate ﬁeld-effect transistors based on epitaxial GaAs heterostructures can be markedly reduced by preliminarily implanting the heterostructures with Ar ions from the side of a substrate. The effect is explained by the far- range gettering of impurities and defects from the active transistor regions in the course of the neutron irradiation, which suppresses the formation of irradiation-induced deep energy levels. © 2000 MAIK “Nauka/Inter- periodica”.

Theoretically, the electrical and functional charac- tors were exposed to a pulsed neutron radiation with an teristics of discrete devices and integrations based on average particle energy of 1 MeV, a pulse duration of GaAs must exhibit a higher stability with respect to the less than 1 ms, and a maximum fluence of 1015 cmÐ2 per action of high-energy corpuscular and photon beams as pulse. Changes in the parameters of transistors were compared to the properties of silicon-based analogs [1]. judged by their currentÐvoltage (IÐV) characteristics There is some experimental evidence that confirms this measured on an L2-56 instrument. In addition, the hypothesis, but a high level of defectness in single crys- parameters of transistors were calculated within the tals and epitaxial layers of GaAs frequently levels off framework of a theoretical model [3] using the values the advantage of GaAs over Ge. of electron gas parameters in the active layers of analo- Recently [2], we demonstrated that bombardment of gous transistors determined in [2]. the substrate in epitaxial GaAs heterostructures at room The results of our measurements demonstrated that temperature with medium-energy ions from the side of the character of degradation of the IÐV curves of the a substrate produces a long-range effect improving the neutron-irradiated transistors was qualitatively the parameters of Schottky-gate field-effect transistors same for the control samples (not implanted with based on these heterostructures. The character of the argon) and the pretreated structures with getters formed changes in the parameters of transistors suggested that by the ion bombardment (Fig. 1). However, the rate of the ion bombardment reduced the concentration of degradation of the latter structures was only about half crystallographic defects detrimentally affecting the that of the control samples with the same neutron flu- mobility and concentration of charge carriers in the ence (Fig. 1a). As seen from Fig. 1b, the neutron irradi- active layers of transistor. Therefore, it would be natu- ation led to an increasing channel resistance in the tran- ral to expect that this pretreatment may also improve sistors of both groups, but the ion-beam-induced getter- the radiation stability of devices based on the ion-bom- ing markedly reduced this effect. Moreover, the barded heterostructures. The purpose of this work was pretreated devices showed no residual conductivity in to study the long-range gettering effect of the prelimi- the sourceÐdrain chain at large negative voltages on the nary ion-beam treatment on the electrical characteris- gate (Fig. 1b). This conductivity is determined by resis- tics of GaAs transistor heterostructures subject to sub- tances of the buffer layer and the substrate at their epi- sequent neutron irradiation. taxial contact boundary [4]. The absence of the residual + Ð conductivity confirms the fact [2] that the concentration The experiments were performed on n nn transis- Ð tors with a gate width of 0.5 µm. The structure design gradient of the donor distribution profile at the nn and preparation technology were described elsewhere interface increases upon ion bombardment from the [2]. Before cutting into separate transistor structures, substrate side. As a result, the drain current becomes a the sample wafers with transistors were thinned to linear function of the gate potential in the cut-off volt- 100 µm and irradiated from the substrate side with age region. The linearity was retained upon neutron 90-keV argon ions to a total ion dose of 5 × 1015 cmÐ2 irradiation of the ion-beam-gettered transistor struc- at an ion beam current density not exceeding tures, although the cut-off voltage of these transistors (as well as that of the control samples) somewhat decreased. 0.5 µA/cm2. Only half of the wafer surface was exposed to the ion beam, while the other half was shielded by a The preliminary gettering markedly decreases the titanium mask. Upon cutting, the nonpackaged transis- rate of the neutron-fluence-dependent degradation of

646 OBOLENSKIŒ, SKUPOV

Id, mA S, mA/V 250 100 (a) (a) 200 1 75 2 2 150 2' 50 1 100 25 1' 50 0 2.5 5.0 7.5 10 0 2 4 6 V , V d Kp, dB Id, mA 200 6.0 (b) 160 (b) 1 4.5 120 2 2' 1 2 80 3.0

40 1.5 1' 0 –3 –2 –1 0 Vg, V 0 2 4 6 8 10 Fig. 1. The plots of drain current I (a) versus drain voltage d D × 1015 cm–2 Vd at a gate voltage Vg = 0 V and (b) versus the gate voltage n at Vd = 5 V for the GaAs field-effect transistors (1, 2) before × and (1', 2') after neutron irradiation to a fluence of Dn = 1 Fig. 2. The plots of (a) the IÐV curve slope S and (b) power 15 Ð2 10 cm : (1, 1') control samples (not implanted with gain coefficient Kp versus neutron fluence for the GaAs argon); (2, 2') samples gettered by the preliminary ion-beam field-effect transistors based on (1) control (not implanted treatment; solid lines show the results of model calculations. with argon) and (2) ion-beam-gettered heterostructures. the transistor IÐV curve slope and the power gain factor the ion-beam-induced increase in the stability of GaAs (Fig. 2). Note a good coincidence of the experimental field-effect transistors with respect to subsequent neu- data and the values calculated by the model [3] using tron irradiation. the parameters from [2] for the transistors before and after neutron irradiation. This coincidence, on the one hand, confirms validity of the model concepts [3] and, REFERENCES on the other hand, proves the possibility for effectively 1. R. Zulig, in VLSI Electronics, Vol. 10: Surface and Inter- increasing the radiation stability of GaAs transistors by face Effects in VLSI; Vol. 11: GaAs Microelectronics, ed. means of ion-beam-induced gettering. by N. Einspruch (Academic, New York, 1985; Mir, Mos- In our opinion, the above results indicate that the cow, 1988). preliminary ion irradiation accelerates the rearrange- 2. S. V. Obolenskiœ, V. D. Skupov, and A. G. Fefelov, ment of the components of the impurityÐdefect compo- Pis’ma Zh. Tekh. Fiz. 25 (16), 50 (1999) [Tech. Phys. sition in each layer of the transistor heterostructure and Lett. 25, 655 (1999)]. boundary regions. This leads, in particular, to a 3. N. V. Demarina and S. V. Obolenskiœ, Microelectron. decrease in the concentration of background impurities Reliab. 39, 1247 (1999). and antistructural defects responsible for the deep level 4. M. Shur, GaAs Devices and Circuits (Plenum, New formation in the course of neutron irradiation [5]. York, 1987; Mir, Moscow, 1991). These impurities and defects are gettered by extended 5. V. A. Novikov and V. V. Peshev, Fiz. Tekh. Poluprovodn. defects of the dislocation type and by the free surface. (St. Petersburg) 32 (4), 411 (1998) [Semiconductors 32, We believe that the rearrangement process is initiated 366 (1998)]. and driven by elastic waves generated in the ion stop- 6. P. V. Pavlov, Yu. A. Semin, V. D. Skupov, and D. I. Te- ping region and at the inner boundaries of a multilayer tel’baum, Fiz. Tekh. Poluprovodn. (Leningrad) 20 (5), structure irradiated from the substrate side [2, 6]. 503 (1986) [Sov. Phys. Semicond. 20, 315 (1986)]. Future investigations with the use of deep-level spectroscopy have to specify the processes responsible for Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 647Ð649. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 6Ð11. Original Russian Text Copyright © 2000 by M. Ruvinskiœ, Freik, B. Ruvinskiœ, Prokopiv.

Mechanisms of Formation and Charge States of Intrinsic Defects in Lead Telluride Films M. A. Ruvinskiœ, D. M. Freik, B. M. Ruvinskiœ, and V. V. Prokopiv Physicochemical Institute, Stefanik Carpathian University, Ivano-Frankivsk, Ukraine Received January 11, 2000

Abstract—A crystallochemical model of the vapor-phase epitaxy of lead telluride ﬁlms is suggested. The model is based on the simultaneous formation of singly- and doubly-charged Frenkel defects in the cationic sublattice. The results of numerical calculations based on this model are in good agreement with the known experimental data. © 2000 MAIK “Nauka/Interperiodica”.

1. It is well known [1Ð4] that compositional changes Equation (1) characterizes equilibrium in the within the homogeneity range of lead chalcogenides sourceÐvapor system during the decomposition of lead provide a control of the electric properties of the ﬁlms telluride in the evaporator at the evaporation tempera- of these compounds, including the type of conductivity ture Te. Reactions (2)–(5) deﬁne the vaporÐcondensate and the charge carrier concentration. However, there is equilibrium responsible for the formation of intrinsic still no commonly accepted opinion about the prevalent defects at the substrate temperature Ts. type of intrinsic defects in AIVBVI compounds and their charge state [4Ð7]. The intrinsic conductivity and the ionization of Frenkel defects are described by Eqs. (6) and (7)Ð(10): 2. In order to interpret the dependence of the charge Ð + carrier concentration on various growth parameters “0” e + h , Ki(Ts) = np; (6)

(including the partial pressure PTe of tellurium vapors 2 Ð + ' Ð + in the growth zone [8]) in lead telluride ﬁlms grown “0” VPb + Pbi , KF (Ts) = [VPb ][Pbi ]; (7) from the vapor phase by the hot-wall method, we sug- 2Ð 2+ 2Ð 2+ gest the ﬁrst model describing disorder in the metal “0” VPb + Pbi , KF'' (Ts) = [VPb ][Pbi ]; (8) sublattice by the Frenkel mechanism with the simulta- () neous formation of singly- and doubly-charged defects Ð 2+ Ð e Ð 2+ “0” VPb + Pbi + e , KF (Ts) = [VPb ][Pbi ]n;(9) 2+ + (interstitial lead ions, Pbi and Pbi , and lead vacan- 2Ð + + 2Ð Ð “0” VPb + Pbi + h , cies, VPb and VPb ). 3. Using the methods suggested in [9], the sourceÐ ()h 2Ð + K (Ts) = [V ][Pb ]p; (10) vapor and vaporÐcondensate equilibria can be F Pb i ∆ described by the following quasichemical reactions: Kj = K0jexp(Ð H/kT). PbTeS PbV +()1/2 Te V , K ()T = PP 1/2 ; (1) The general condition of electroneutrality has the 2 PbTe e Pb Te2 V + Ð []+Ð1 Pb Pbi +Pbe , KPb' , V()Ts = inPPb ; (2) Table

V Ð 0 + K ∆H, eV () Reaction constant 0 1/2 Te2 V Pb ++ Te Te h , Ð Ð1/2 (3) 3/2 × 18 K' , ()T = []V pP ; KPbTe, Pa 1.4 10 3.51 Te V s Pb Te2 Ð6 Ð1 × 30 KPb,' V , cm Pa 5.5 10 Ð1.01 V 2+ Ð Pb Pbi + 2e , Ð6 Ð1/2 K ' , , cm Pa 1.2 × 1038 0.25 (4) Te2 V () []2+ 2Ð1 KPb'' , V Ts = Pbi nPPb ; Ð9 Ð1 × 51 KPb,'' V , cm Pa 1.9 10 Ð0.87 ()1/2 Te V V 2 ++Te 0 2h+, 2 Pb Te Ð9 Ð1/2 59 K '' , , cm Pa 1.0 × 10 0.39 2Ð 2Ð1/2 (5) Te2 V K'' , ()T = []V pP. Te2 V s Pb Te2

log , cm–3 1/2 Ð1 nH B = 1 + K' P (K K' , ) , F Te2 PbTe Pb V 1018 1 Ð1/2 Ð1/2 C = K + K K' , P , D = 2K K'' , P . i PbTe Pb V Te2 PbTe Pb V Te2 The charge carrier concentration determined from the Hall effect nH is Ð1 17 10 nH = n Ð p = n Ð Kin . (13) p n Using this model, it is also possible to determine the concentration of charged defects.

4. The intrinsic conductivity constant Ki was deter- 2 mined with due regard for the temperature dependence 16 10 of the bandgap Eg(T) and the effective mass of the den- 10–3 10–2 10–1 100 sity of states m(T) [10] as ( ) ( ) logp , Pa Ki T = NC NV exp ÐEg/kT , (14) Te2 Fig. 1. The concentration of charge carriers in lead telluride where films as function of the partial pressure of tellurium vapors: ( πប2)3/2 (1) experiment [8]; (2) calculation by Eqs. (12) and (13). NC = NV = 2 mkT/2 , The substrates were the (111) cleavages of BaF2 crystals (T = 833 K; T = 653 K). e s m(T) = g 3/2K 1/30.048m T/300,  C 0 ( ≥ ) – + –3 2– 2+ –3 gC = 4, K = m||/m⊥ = 9 T 300 K ; log([VPb ], [Pbi ]), cm log([VPb ], [Pbi ]), cm 1020 1020.51 × Ð4( ) ( ) ( ≥ ) Eg = 0.217 + 4.5 10 T Ð 77 eV T 77 K . 4 The KF' constant was determined from the equilib- 1019 3 rium condition 2 Ð1 K' (T ) = K (T )K' , (T )K' , (T )K (T ). (15) F s PbTe s Pb V s Te2 V s i s 1018 1 For the ideal vapor, we have ( ) ( )( )3/2 KPbTe T s = KPbTe T e T s/T e . 1017 1020.49 10–3 10–2 10–1 100 The reaction constants KPbTe, KPb' , V , KTe' , V , KPb'' , V , logp , Pa 2 Te2 and K'' , were taken from [2, 11] (see table). Te2 V Fig. 2. Calculated concentrations of charged defects in lead telluride films as functions of the partial pressure of tellu- 5. The experimental and the calculated dependences Ð + 2+ 2Ð of the defect and charge carrier concentrations in PbTe rium vapors: (1) [V ], (2) [Pb ], (3) [Pb ], (4) [V ] Pb i i Pb films as functions of the partial pressure P of tellu- (T = 833 K; T = 653 K). Te2 e s rium vapors in the growth zone are shown in Figs. 1 and 2. It is seen from Fig. 2 that the concentrations of form 2+ 2Ð doubly-charged defects [Pbi ] and [VPb ] in the films + 2+ Ð 2Ð are considerably higher than the concentrations of sin- p + [Pbi ] + 2[Pbi ] = n + [VPb ] + 2[VPb ]. (11) gly-charged defects. This is consistent with the qualita- According to Eqs. (1)Ð(11), the carrier concentra- tive tendency to the prevailing formation [12, 13] of tion n can be determined in terms of the equilibrium multiply charged vacancies and interstitials capable of constants of the quasichemical reactions and pressure multiple ionization in the presence of rather shallow acceptor and donor levels [12, 13]. However, because P using the equation Te2 of partial compensation of the doubly-charged defects, an important role in the change of the charge carrier An4 + Bn3 Ð Cn Ð D = 0, (12) concentration in PbTe films (dependent on various growth factors) is also played by singly-charged where + Ð defects—Pbi and VPb (Fig. 2). Upon increasing the 1/2 Ð1 A = 2K'' , K' P (K K' , K' , K ) , tellurium-vapor pressure P , the electron concentra- Te2 V F Te2 PbTe Pb V Te2 V i Te2

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 MECHANISMS OF FORMATION AND CHARGE STATES 649 tion first decreases, then the conductivity changes from 4. N. J. Parada and G. W. Pratt, Phys. Rev. Lett. 22, 180 the n- to the p-type and the hole concentration increases (1969). (Fig. 1). This corresponds to an increase of the concen- 5. M. Schenk, H. Berger, and A. Kimakov, Cryst. Res. tration of singly-charged acceptor lead vacancies [V Ð ] Technol. 23 (1), 77 (1988). Pb 6. D. M. Zayachuk and V. A. Shenderovs’kiœ, Ukr. Fiz. Zh. and a decrease of the concentration of singly-charged 36 (11), 1692 (1991). + donor lead interstitials [Pbi ] (Fig. 1). 7. D. M. Freik, V. V. Prokopiv, A. V. Nych, et al., Mater. Sci. Thus, it can be stated that the equilibrium state of Eng. B 48, 226 (1997). the defect subsystem in PbTe films is rather compli- 8. A. Lopez-Otero, Appl. Phys. Lett. 26 (8), 470 (1975). cated. The experimental results can be satisfactorily 9. F. A. Kröger, The Chemistry of Imperfect Crystals interpreted only assuming that singly- and doubly- (Wiley, New York, 1964; Mir, Moscow, 1969). charged defects simultaneously exist in the cationic 10. T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor sublattice. Opto-Electronics (Butterworths, London, 1973; Mir, Moscow, 1976). 11. A. M. Gas’kov, V. P. Zlomanov, and A. V. Novoselova, REFERENCES Izv. Akad. Nauk SSSR, Neorg. Mater. 15 (8), 1476 1. N. Kh. Abrikosov and L. E. Shelimova, Semiconductor (1979). IV VI Materials Based on A B Compounds (Nauka, Mos- 12. F. A. Kröger, J. Phys. Chem. Solids 26, 1717 (1965). cow, 1975). 13. V. L. Vinetskiœ and G. A. Kholodar’, Statistical Interac- 2. V. P. Zlomanov, PÐTÐX Diagrams of Two-Component tion of Electrons and Holes in Semiconductors (Naukova Systems (Mosk. Gos. Univ., Moscow, 1980). Dumka, Kiev, 1969). 3. Yu. I. Ravich, B. A. Efimova, and I. A. Smirnov, Semi- conducting Lead Chalcogenides (Nauka, Moscow, 1968; Plenum, New York, 1970). Translated by L. Man

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 650Ð653. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 12Ð20. Original Russian Text Copyright © 2000 by Ginzburg, Novozhilova, Rozental’, Sergeev.

Longitudinal Self-Focusing of an Electron Bunch under Coherent Emission Conditions N. S. Ginzburg, Yu. V. Novozhilova, R. M. Rozental’, and A. S. Sergeev Institute of Applied Physics, Russian Academy of Sciences, Nizhniœ Novgorod, 603600 Russia Received March 10, 2000

Abstract—The longitudinal motion of an electron bunch with a length much smaller than its wavelength is studied theoretically under coherent emission conditions. It is demonstrated that the bunch can be focused under the action of a longitudinal component of the ponderomotive force of the radiation ﬁelds. This mechanism is operative when a bunch moves parallel to metallic surfaces partly reducing the space-charge forces. © 2000 MAIK “Nauka/Interperiodica”.

1. The creation of short electron bunches stable with boundaries of the layer. The width of the layer is respect to Coulomb repulsion is a task of practical sig- denoted by b. All of the electrons have the same initial nificance [1–3]. This study demonstrates that certain phase of cyclotron gyration, so that the transverse progress toward solving the problem can be made by momentum of an electron can be represented as p+ = ω ω γ using the ponderomotive force exerted by radiation px + ipy = p⊥exp(i Ht), where H = eH0/mc 0 is the γ fields. Specifically, a transverse oscillation velocity gyromagnetic frequency with 0 being the initial value should be imparted to the electrons in a bunch so that of the relativistic factor. Let us divide the layer into a the radiation fields thus produced would focus the finite number of sheets of charge (macroelectrons) dif- bunch in the longitudinal direction. The oscillatory fering in the longitudinal coordinate z', so as to analyze motion may represent a cyclotron gyration of particles the dynamics of the layer. For each sheet of charge, the in a uniform longitudinal magnetic field. Alternatively, surface current density is j = j + ij = j exp(iω t), the oscillations may be caused by an external electro- + x y 0 H where j = σv⊥ with σ being the surface charge density magnetic field or a nonuniform magnetostatic field 0 and v⊥ = p⊥/mγ, the transverse velocity. The field gen- (such as that in an undulator). erated by a sheet is a circularly polarized plane wave The ponderomotive interaction between two particles oscillating in phase is attractive if the distance ()ω E+ ==Ex + iEy E0exp i Ht (1) between particles is shorter than half of the radiation wavelength [4]. Naturally, the ponderomotive force with the amplitude depends on the magnitude of the transverse oscillation velocity. However, even at relativistic velocities, the π (), 2 (), force is weaker than the Coulomb repulsion of the par- E0zt =Ð------j 0 z ' tzzÐÐ/'c,(2) ticles. It is therefore necessary to partly neutralize the c action of the space charge. This can be achieved if the where j (z', t Ð |z Ð z'|/c) is the surface current density bunch moves parallel to metallic surfaces. 0 amplitude taken at a retarded time instant The present study addresses the longitudinal electron focusing under the conditions of cyclotron radia- j()z',tzzÐÐ/'c = σν⊥()z', t exp()ÐiωzzÐ/'c.(3) tion emission. First, we analyze the simplest one- 0 H dimensional model that represents the electron ensem- In Eq. (3), the characteristic time of the bunch ble as a layer extending to infinity along both of the focusing is assumed to be much greater than the time transverse coordinate axes, assuming that the space- lag b/c across the layer; in other words, the longitudinal charge forces are partly reduced. Then we proceed to a motion is assumed to be weakly relativistic. In a plane bunch of electrons gyrating in a uniform magnetic field wave, the magnetic field is H+ = Hx + iHy = iE+. Accord- between two conducting planes. Based on the particle- ingly, the longitudinal ponderomotive force with which in-cell simulation with a two-dimensional version of one sheet of charge acts upon a unit area of another the KARAT code, we demonstrate that the bunch can sheet that lies at the distance |z Ð z'| is expressed as be contracted in the longitudinal direction. 2. Consider a one-dimensional model. Let there be pond 2 2 F =Ð()2πσ v⊥()zt, v⊥()z',t/c a layer of electrons gyrating in a uniform magnetic field z (4) ×()()()ω (H0 = H0z0) that is directed perpendicularly to the sgn zzÐ'cos zzÐ' H/c.

LONGITUDINAL SELF-FOCUSING OF AN ELECTRON BUNCH 651

Fz force surpasses the ponderomotive force (Fig. 1), the coul former must be partly neutralized so that longitudinal Fz focusing is possible. This can be achieved by introducing an ion background or metallic surfaces parallel to pond which the bunch would move. Fz Suppose the space-charge force is partly reduced, the degree of the reduction being evaluated by the fac- –π –π/2 π/2 π tor r < 1. Let us describe the motion in terms of the – ' Fpond Z Z Lagrangian variables, with which the position of each coul z α τ Fz plane of charge (macroparticle) Z (Z0, ) and its velocity are functions of the time τ and the initial coordinate Z0 (it is assumed that the transverse electron motion is relativistic and the longitudinal motion is weakly rela- Fig. 1. The interaction of macroelectrons in a one-dimen- tivistic.) Then the equations of motion for the macro- sional model: the longitudinal components of the Coulomb and ponderomotive forces vs. the distance between sheets of electrons are as follows: charge. 2 d Zα  ∂a*  ⊥ ------= Rei p α  pond 2 ∂Z Z = Z Figure 1 demonstrates that F changes its sign. dτ α z (5a) pond B Note that Fz is an attractive force if the distance is + Ir sgn(Zα Ð Zα )dZ , less than half the wavelength. Also shown in Fig. 1 is ∫ ' 0α' the longitudinal component of the Coulomb force 0 coul πσ2 d p⊥ Fz = 2 sgn(z Ð z'), which is a repulsive force for α 2 2 ------= i p⊥ (1/ 1 Ð β⊥ + p⊥ Ð 1) Ð a, (5b) any distance between the planes. Since the Coulomb dτ α 0 α

Zα (a) 0.5

–0.5

0 400 800 1.0 – 2 (b) p⊥

– 0.5 P × 104

0 400 800 τ

Fig. 2. Time variation of (a) the longitudinal coordinates of macroelectrons and (b) the mean-square transverse momentum and the Ð4 2 radiation power at I = 10 , β⊥ = 0.8, B = 0.6, and r = 0.7.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 652 GINZBURG et al.

x, cm dition for neutralization of the Coulomb repulsion by (a) the radiative attraction: 2 0.4 r<β⊥. (6) t = 0 ns Figure 2a shows the trajectories of macroelectrons H0 0.2 in the case where condition (6) is met at the initial moment of time τ. Figure 2b depicts the radiation | |2 | |2 power W = a Z = 0 + a Z = B and the mean-square trans- 0 –0.5 0 0.5 z, cm verse momentum of the macroparticles for the same parameters. Comparing the two figure panels with each other shows that effective coherent emission from the (b) (e) layer with a pronounced longitudinal self-focusing takes place for τ ≤ 800. As τ increases further, the self- focusing changes to expansion of the layer. The reason is that the transverse momentum decreases during the t = 3 ns emission to reduce the ponderomotive force which is proportional to the squared momentum. 3. For a two-dimensional model, the longitudinal self-focusing was demonstrated using a particle-in-cell (c) (f) simulation with the help of the KARAT code that directly embodies Maxwell’s equations together with the equations of motion. The bunch was assumed to move between two metallic planes reducing the Cou- t = 4.5 ns lomb repulsion (Fig. 3). Figure 3a shows the initial configuration of the bunch with a density of 1010 cmÐ3 and the dimensions 1 and 0.5 mm along the z- and the (d) (g) x-axis, respectively. The electrons gyrate under the action of a uniform magnetic field oriented along the z-axis and having a magnetic flux density of 3 T. The initial energy of the electrons is 1 MeV. Figures 3bÐ3d t = 6 ns show how the bunch expands due to the Coulomb interaction, the emission fields being ignored. Figures 3eÐ3g present configurations of the bunch experiencing the Fig. 3. Particle-in-cell simulation of longitudinal focusing: focusing action of the radiation at various time instants. (a) initial configuration of a bunch; (b, c) evolution of the Comparing the figures suggests that the bunch under- bunch with neglect of the emission; (d, e) evolution of the bunch under the action of the radiation field. The initial den- goes contraction during an initial stage of the process, sity of the bunch is 1010 cmÐ3, the electron energy is 1 MeV, accompanied by the emission from oscillating elec- and the magnetic flux density is 3 T. trons. Then, the radiation power and the mean transverse velocity decrease (Fig. 4) to attenuate the ponderomotive force. This results in expansion of the bunch. where 4. It should be emphasized that the above analysis a(Z, τ) applies to a bunch moving as a whole with a transla- B tional velocity v||0 and that the process has been studied 2 2 (5c) in the intrinsic frame of reference K' moving with the = I p⊥ exp(Ði Z Ð Zα )/ 1 Ð β⊥ + p⊥ dZ ∫ α' ' 0 α' 0α' unperturbed translational velocity. For the laboratory 0 frame of reference K, the Lorentz transformations give the following expressions of the electron transverse is the radiation amplitude. Equations (5) employ the γ ω velocity, energy, and concentration, respectively: v⊥ = following dimensionless quantities: a = eE0/(mc 0 H), ω τ ω γ ω γ γ γ γ γ γ β2 Z = Hz/c, = Ht, p⊥ = p⊥/(mc 0), B = b H/c, I = v⊥' ||, = ' ||, and n = n' ||, where ( || = 1/ 1Ð ||0 ). For example, at v|| = 0.98c, the bunch dealt with in Section 3 ω 2 /2ω 2 , and ω = 4πe2n/mγ , where ω denotes 0 p H p 0 p has an energy of 7.5 MeV, the pitch factor β⊥/β|| = 0.2, the relativistic plasma frequency. and a concentration of 5 × 1010 cmÐ3 (the current den- According to Eq. (5c), for a given charge per unit sity being ~250 A/cm2). Note that the Lorentz (relativ- area, the radiation amplitude attains its maximum a = istic) time dilation must be allowed for as well: if the IB when the layer width is much smaller than the wave- time to focus ∆τ' is 3 ns in the intrinsic frame, then the length: B Ӷ 1. Then, Eq. (5a) gives the following con- corresponding value in the laboratory frame is ∆τ =

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 LONGITUDINAL SELF-FOCUSING OF AN ELECTRON BUNCH 653

2 P, W (p⊥/mc) sider, for example, the motion inside a capacitor. The 8 7.8 process obeys Eqs. (5a) and (5c), where the amplitude of the transverse momentum is a constant proportional to the amplitude of the capacitor field. Remarkably, the self-focusing and coherent emission in this model system will continue indefinitely, provided condition (6) is 6 met. Also interesting is the case where the oscillations are excited by a traveling electromagnetic pump wave. Then, the longitudinal self-focusing and coherent emission (scattering) must be accompanied by acceleration of the bunch as a whole due to the longitudinal momentum imparted by the pump wave. An opposite effect 4 7.6 must occur when a short bunch moves in a variable undulator field. Then, the bunch would be retarded as a whole and the self-focusing would provide for the coherent emission. 2 This study was supported by the Russian Founda- 0 2 4 tion for Basic Research, project no. 98-02-17308. t, ns Fig. 4. Time variation of the mean-square transverse REFERENCES momentum and the radiation power. The values of parameters are as in Fig. 3. 1. Ya. B. Faœnberg, A. K. Berezin, V. A. Balakirev, et al., in Relativistic High-Frequency Electronics, ed. by A. V. Gaponov-Grekhov (Inst. Prikl. Fiz. Ross. Akad. Nauk, Nizhniœ Novgorod, 1993), Vol. 7, p. 104. ∆τ'γ|| ≈ 15 ns. During this time, the bunch having a translational velocity close to the velocity of light will 2. P. Chen, J. M. Dawson, R. Huff, and T. Katsouleas, Phys. Rev. Lett. 54 (7), 692 (1985). travel ~4.5 m. 3. J. B. Rosenzweig, Phys. Rev. Lett. 58 (6), 555 (1987). 5. In conclusion, note that there must be no decrease 4. N. S. Ginzburg and A. S. Sergeev, Pis’ma Zh. Éksp. Teor. in the transverse oscillation velocity during coherent Fiz. 54 (8), 445 (1991) [JETP Lett. 54, 446 (1991)]. emission if the electrons perform forced oscillations, being exposed to external time-dependent fields. Con- Translated by A. Sharshakov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 654Ð655. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 21Ð25. Original Russian Text Copyright © 2000 by Gorbunov, Stacewicz.

Photo EMF Observed upon the Laser Resonance Excitation of Sodium Vapors N. A. Gorbunov* and T. Stacewicz** * Institute of Physics, St. Petersburg State University, St. Petersburg, Russia ** Institute of Experimental Physics, Warsaw University, Hoza 69, 00-689 Warsaw, Poland Received March 16, 2000

Abstract—A method of the direct light-to-electric energy conversion in photoplasma is proposed. The photo emf was observed in experiments on the pulsed laser excitation of sodium vapors. It is demonstrated that the separation of charged particles in the plasma is caused by the inhomogeneous distribution of resonance-excited atoms in the direction of optical excitation. © 2000 MAIK “Nauka/Interperiodica”.

Optical radiation can be detected using the light- We have generated a photo emf by resonance pulsed induced changes of conductivity in semiconductors [1] laser irradiation of sodium vapors. A glass tube with the and in the electric discharge plasma (optogalvanic vapors had a length of L = 70 mm and an inner diameter effect) [2]. The effect of the photo emf generation in of d = 20 mm. Cylindrical kovar electrodes welded at semiconductors was also comprehensively studied and the ends of the tube measured the electric current. Sap- found wide practical applications in the light-to-elec- phire windows on the electrode edges allowed for the tric energy converters. The phenomenon of the photo input of the laser beam. A heater controlled the density emf generation in plasma is not as well studied. of the sodium vapors. A potential difference between points in the illuminated gas medium arises in two stages: first, the forma- A pulsed tunable laser with the spectral half-width tion of charged particles (photionization); second, sep- of about 0.5 Å provided the resonance excitation of aration of these charges in space. sodium vapors (resonance transitions 3SÐ3P1/2, 3/2). The The light-induced plasma formation in metal vapors energy of the laser pulse amounted to 20 mJ at a pulse is extensively studied both theoretically and experi- duration of about 7 ns and a repetition rate of 10 Hz. An HP54220 A digital oscilloscope measured the voltage mentally (see, for example, [3Ð5]). Resonance absorp- Ω tion of the incident light quanta leads to maximum ion- U across the measuring resistor R = 50 connected in ization of the medium. Effective absorption of the res- series with the gas cell and a testing constant voltage onance quanta is followed by ionization mainly caused (ϕ) source. by the processes involving electrons heated in super- Figure 1a shows the plots of voltage U versus time elastic collisions with the optically excited atoms. at [Na] = 3 × 1014 cmÐ3 and ϕ = 0. Curve 1 refers to the The studies of mechanisms of the photoinduced case when the wavelength corresponding to the maxi- potential difference generation were stimulated by mum laser power is centered at the sodium 3S1/2Ð3P3/2 interest in the direct light-to-electric energy conversion absorption band. Consider the qualitative interpretation in photoplasma. Dunning and Palmer [6] proposed a method of solar energy conversion in photoplasma of this phenomenon. In the case under consideration, based on the MHD transformation. The laser energy the laser power density is insufficient for homogeneous conversion into electricity in a thermoemission con- optical saturation of the resonance transition over the entire cell length. The light energy is absorbed within a verter employing a CO2 laser for the plasma initiation were experimentally studied in [7, 8]. The mechanism few millimeters near the input window. This leads to of charge separation upon resonance optical excitation inhomogeneous distribution of the excited atoms along is less elaborate. Brandenberg [9] observed a photo emf the cell axis and, consequently, to inhomogeneous in the microwave discharge in krypton upon illuminat- plasma formation. Plasma emission mainly concen- ing the cell perpendicularly to its axis by laser light trates in a narrow layer near the input window. An with a wavelength of 785.5 nm (1s3Ð2p3 transition). excess of positive ions related to the diffusion outflow A mechanism of the photo emf generation in this sys- of more mobile electrons gives rise to a positive poten- tem (the potential difference was about 0.1 V at a laser tial in this region. The resulting ambipolar potential dif- ∆φ ≈ power of about 1 mW) is related to violation of the dis- ference (Te/e)/ln(n1/n2) depends on the electron charge symmetry caused by the light-induced changes temperature (Te) and the charged particle densities n1 of the Kr metastable state (1s3) density. and n2 in both ends of the cell [10].

PHOTO EMF OBSERVED UPON THE LASER RESONANCE 655

U, V age pulses. The voltage U reaches its maximum value 1.2 0.2 µs after termination of the exciting laser pulse. This (a) coincides with the characteristic time of relaxation of 1.0 the electron temperature Te caused by collisions with 1 helium atoms. The consequent deionization regime ter- minates the growth of the charged particles concentra- 0.8 tion in the plasma. Shortening of the photo emf decay time in comparison with the case of pure sodium is 0.6 explained by increasing rates of the Te relaxation and the resonance radiation transfer caused by the buffer 0.4 2 gas influence. An external field leads to increasing load current, 0.2 provided the electron drift direction in this field coincides with the ambipolar diffusion direction (Fig. 1b, 0 curve 1). In the opposite case (Fig. 1b, curves 3 and 4), the load current depends on the difference between the photo emf and the bias voltage. The maximum photo emf mea- 1.0 ∆ϕ sured by the compensation method was max ~ 3 V. (b) 0.8 1 ϕ = +0.5 V Thus, the inhomogeneous photo-excitation of atoms 2 ϕ = 0 V in a gas-filled cell leads to generation of the photo emf. ϕ 0.6 3 = –0.5 V This effect is similar to the Dember emf generation 1 4 ϕ = –1.5 V upon inhomogeneous excitation of a uniform semiconductor [1]. In contrast to the case of semiconductors, 0.4 2 the photo emf in plasma exhibits a resonance character with respect to the excitation wavelength. 0.2 3 The authors are grateful to L. D. Tsendin for helpful discussions. 0 4 This work was supported by the KBN grant –0.2 (Poland).

0 1 2 3 4 5 REFERENCES t, µs 1. V. L. Bonch-Bruevich and S. G. Kalashnikov, The Phys- Fig. 1. The plots of voltage versus time measured in a cell ics of Semiconductors (Nauka, Moscow, 1990). ﬁlled with (a) sodium vapors and (b) sodiumÐhelium mix- 2. V. N. Ochkin, N. G. Preobrazhenskiœ, and N. Ya. Sha- ture. See the test for explanation. parev, Optogalvanic Effect in Ionized Gases (Nauka, Moscow, 1991). 3. I. M. Beterov, A. V. Eletskiœ, and B. M. Smirnov, Usp. The value of the photocurrent in the external circuit Fiz. Nauk 155 (2), 265 (1988) [Sov. Phys. Usp. 31, 535 depends on the relationship between the load resistance (1988)]. and the internal resistance of the photo emf source. It is 4. V. A. Kas’yanov and A. N. Starostin, in Chemistry of seen that the photocurrent reaches its maximum value Plasma, ed. by B. M. Smirnov (Énergoatomizdat, Mos- in ~1 µs after the laser pulse termination, which coin- cow, 1990), Vol. 16, pp. 67Ð97. cides with the time instant when the electron density 5. N. N. Bezuglov, A. N. Klyucharev, and T. Stasevich, Opt. reaches a maximum [5]. The subsequent decay is Spektrosk. 77 (3), 342 (1994) [Opt. Spectrosc. 77, 304 caused by the Te relaxation and spatial blurring of the (1994)]. primary ionization zone due to the resonance radiation 6. G. J. Dunning and A. J. Palmer, J. Appl. Phys. 52 (12), transfer. A shift of the laser wavelength by ~1 Å toward 7086 (1981). the absorption band wing leads to decreasing photo emf 7. R. W. Thompson, E. J. Manista, and D. L. Alger, Appl. (Fig. 1a, curve 2). This was explained by a more homo- Phys. Lett. 32 (10), 610 (1978). geneous excitation of the sodium vapors along the laser 8. E. J. Britt, N. S. Rasor, G. Lee, and K. W. Billman, Appl. beam, as conﬁrmed by observations of the plasma Phys. Lett. 33 (5), 385 (1978). emission. 9. J. R. Brandenberg, Phys. Rev. A 36 (1), 76 (1987). Figure 1b shows the plots of voltage versus time for 10. V. A. Rozhanskiœ and L. D. Tsendin, Collisional Transfer a mixture of a sodium vapors ([Na] ≈ 1015 cmÐ3) and in a Partially Ionized Plasma (Énergoatomizdat, Mos- helium ([He] = 7 × 1017 cmÐ3) at various constant bias cow, 1988). voltages. In contrast to the case of pure sodium and ϕ = 0, we observe shortening of the photoinduced volt- Translated by A. Chikishev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 656Ð658. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 26Ð31. Original Russian Text Copyright © 2000 by Kal’yanov.

Transients in an Autostochastic Oscillator with Delayed Feedback É. V. Kal’yanov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow oblast, Russia Received February 4, 2000

Abstract—A numerical analysis of the turn-on transients in an autostochastic oscillator with delayed feedback is reported. Dependence of the transients on the intensity of chaotic initial conditions is studied. It is demonstrated that the chaotic oscillations may switch from one basin of attraction to another. © 2000 MAIK “Nauka/Interperiodica”.

Delayed-feedback oscillators (DFOs) are widely amplifier nonlinearity. The equations are as follows: employed as models in various areas of scientific inquiry [1Ð5]. Nevertheless, the dynamics of DFOs is xú = y, still poorly understood due to its complicated nature. In 2 2 2 yú +()ω/Q y +ωx = ω {}B[]()1 Ð z /δ[]xt()Ðτ Ðz, particular, little is known about the transients observed under chaotic conditions at large feedback delays. The δzú+z=xt()Ðτ, (1) processes must differ from those occurring in oscillators with a zero delay. Chaos is the factor that seriously where x, y, and z are the time-dependent variables; the ω complicates the transients in DFOs. On the other hand, dot denotes differentiation with respect to the time t; the understanding of chaotic transients and ways to and Q are the natural frequency and the Q-factor of the second-order filter, respectively; δ is the time constant reduce them is necessary in some cases, e.g., when τ using autostochastic DFOs for communications based of the first-order filter; is the feedback delay; and B is on deterministic chaos [6]. the gain parameter. It is pertinent to note that the very concept of tran- In contrast to the zero-delay case [8], the initial con- sient in its conventional sense seems to apply to reg- ditions cannot be considered independent of time (constant) when dealing with the turn-on transients in ular operation only. A chaotic oscillation can hardly DFOs. Circulating in the loop, an initial disturbance (in be treated as a steady process because its state varies a real system, arising from noise oscillations of various continuously. In fact, the chaotic operation of a DFO origin) keeps acting upon the DFO in a random fashion. may be considered as a kind of “transient” state [7]. It is therefore necessary to specify initial conditions in On the other hand, there is a noticeable growth in the the form of time-dependent noise (or noise-like) oscil- oscillation magnitude once a chaotic regime has lations. To obtain reproducible results, it is expedient to been turned on, just as in a regular operation mode. construct the initial conditions from a solution to a sys- Furthermore, the maximum fluctuations of the cha- tem of equations describing systems with chaotic otic oscillation eventually become bounded from behavior. In particular, it is convenient to use the cha- above so that the oscillation reaches some kind of otic solutions to relatively simple nonlinear Rössler saturation. It is therefore possible to define the equations: buildup time of a chaotic oscillation under certain conditions. This paper presents the results of simu- uú=ÐvÐw, lating the turn-on transients accompanying the exci- vú=u+αv, (2) tation of chaotic oscillations in a DFO described by fairly general delay equations. wú=Ðβw++uvσ, The DFO under study is a closed loop comprised of where u, v, and w are the time-dependent variables, a nonlinear amplifier, first- and a second-order filters, whereas α, β, and σ are constants [9]. At α = 0.3, β = a delay line, and a differentiating circuit. The simula- 8.5, and σ = 0.4, the solutions to Eqs. (2) are chaotic tion is based on a set of nonlinear differenceÐdifferen- with a funnel-shaped attractor [1]; the spectrum has no tial equations using a third-order approximation of the regular components.

TRANSIENTS IN AN AUTOSTOCHASTIC OSCILLATOR 657

x (a) S, dB (a) 0 –10 –3

–30 x (b)

0 –50

(b) –3

–10

x (c) –30 0

–3 –50 0 120 t 0 1 2 3 ω

Fig. 1. Waveforms of growing oscillations for different intensities of the initial chaotic conditions γ = 0.000001 (a); Fig. 2. Power spectrum densities for different intensities of 0.001 (b); 0.1 (c). the initial chaotic conditions γ = 0.000001 (a); 0.001 (b).

In connection with the above considerations, the ini- escent time, its length being denoted by T. As the γ tial conditions for x, y, and z in Eqs. (1) are determined value increases, the quiescent time drops: T = 80 for γ = by γu(t), γv(t), and γw(t), respectively, where u(t), v(t), 0.0001 and T = 45 for γ = 0.001 (Fig. 1b). In fact, T is and w(t) constitute a solution to Eqs. (2) with constants virtually zero for γ = 0.1 (Fig. 1c). Thus, one should indicated above. The coefficient γ makes it possible to raise γ so as to reduce the quiescent time and increase control the intensity of the chaotic oscillations repre- the buildup rate of a chaotic oscillation. senting the initial conditions for system (1). To elimi- Remarkably, the power spectrum density distribu- nate transients in the solution to Eqs. (2), the initial con- tion in a “steady-state” oscillation regime (set in after ditions are set as follows: u(0) = 5.536877, v(0) = ∈ τ γ − the time t [0, 2400] for = 9.4) depends on for a 1.617695, and w(0) = 0.1127826. These values of u, v, given τ. By taking various values of γ, we discovered and w are attained after a sufficiently long time from the two different shapes of the spectrum S, which are beginning of oscillation. The numerical analysis shown in Fig. 2. The spectrum in Fig. 2a is obtained for employed a fourth-order RungeÐKutta method with a γ = 0.000001 and that in Fig. 2b, for γ = 0.001 (the delay step size of 0.05. The parameters of Eqs. (1) were fixed τ ω δ being fixed at 9.4). Note that the low-frequency part at the following values: Q = 1, B = 2.9, = 1, and = 0.1. has more power in the latter case. For γ = 0.8, the spec- The results of numerically solving Eqs. (1) with the trum is as in Fig. 2b, whereas for γ = 0.0001, 0.01, or initial conditions specially selected as described above even 1, the spectrum is as in Fig. 2a. support our expectation that oscillations in the DFO A change in the spectrum of oscillations in the cha- arise with a lag increasing with τ. This phenomenon otic mode observed in Fig. 2b suggests that the oscilla- occurs in both regular and chaotic operation modes. tions may go to another basin of attraction (correspond- Now consider the effect of changes in γ. Figure 1 ing to lower frequencies). In other words, the intensity shows the excitation of chaotic oscillations for three of the chaotic initial conditions may determine the values of γ at a delay of τ = 9.4. If γ is very small (γ = basin of attraction where a chaotic oscillation will be 0.000001, see Fig. 1a), then the oscillation reaches an excited. appreciable magnitude only after a sufficiently long To sum up, this study has demonstrated that the time (t ∈ [0, 140]). This interval may be called the qui- turn-on transients in an autostochastic DFO may take a

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 658 KAL’YANOV considerable time that increases with the feedback 3. V. Ya. Kislov, Radiotekh. Élektron. (Moscow) 38 (10), delay. To reduce the transient time, one should increase 1783 (1993). the magnitude of the chaotic initial conditions. In prac- 4. É. V. Kal’yanov, Pis’ma Zh. Tekh. Fiz. 21 (16), 71 (1995) tice, this can be achieved with an ampliﬁer possessing [Tech. Phys. Lett. 21, 665 (1995)]. a high level of internal noise. On the other hand, it should be borne in mind that the chaotic oscillation 5. A. Yu. Loskutov, A. V. Mushenkov, A. I. Odintsov, et al., Kvantovaya Élektron. (Moscow) 29 (2), 127 (1999). may switch from one basin of attraction to another. This study was supported by the Russian Founda- 6. L. M. Pecora, T. L. Caroll, G. A. Johnson, et al., Chaos 7 tion for Basic Research (project no. 98-02-16722). (4), 520 (1997). 7. E. A. Kotyrev and L. E. Pliss, Radiotekh. Élektron. (Moscow) 10 (9), 1628 (1965). REFERENCES 8. A. O. Maksimov and E. V. Sosedko, Pis’ma Zh. Tekh. 1. Yu. I. Neimark and P. S. Landa, Stochastic and Chaotic Fiz. 25 (17), 1 (1999) [Tech. Phys. Lett. 25, 675 (1999)]. Oscillations (Nauka, Moscow, 1987; Kluwer, Dordrecht, 1992). 9. O. E. Rossler, Phys. Lett. A 57 (5), 397 (1976). 2. L. Glass and M. Mackey, From Clocks to Chaos. The Rhythms of Life (Princeton Univ. Press, Princeton, 1988; Mir, Moscow, 1991). Translated by A. Sharshakov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 659Ð661. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 32Ð38. Original Russian Text Copyright © 2000 by Bazhenova, Bormotova, Golub, Novikov, Shcherbak.

Effect of Shock Waves Outgoing from Partly Blocked Channel upon an Obstacle T. V. Bazhenova, T. A. Bormotova, V. V. Golub, S. A. Novikov, and S. B. Shcherbak Institute of Thermal Physics of Extremal States, Department of the Institute of High Temperatures, Russian Academy of Sciences, Moscow, Russia Received January 18, 2000

Abstract—Partial channel blocking affects the relative contributions of shock waves and concurrent jet ﬂow. This results in decreasing or increasing the dynamic shock-wave action upon an obstacle, depending on the duration of action and the Mach number. © 2000 MAIK “Nauka/Interperiodica”.

Description of the effect of shock waves outgoing flange (mounted perpendicularly to the tube walls) was from a partly blocked channel upon an obstacle is a arranged within the visual field of plane-parallel win- complex problem of gas dynamics. For large Mach dows of a pressure chamber. A flat obstacle could be numbers of the shock wave, the waveÐobstacle interac- mounted at a variable distance from the shock-wave tion is determined by the shock-wave action, while tube edge. The flow patterns were visualized by an small M values imply a dominating role of the cocur- IAB-451 shadowing instrument and registered with a rent jet flow. Intermediate cases feature an interplay of high-speed VSK-5 camera. the two processes. We have studied the effect of partial The corresponding nonautomodel diffraction prob- channel blocking on the relative contributions of two lem was numerically modeled by solving the Euler flow components. Partial blocking of the channel may equations using the Godunov method with second- enhance the diffracted shock wave action upon the order accuracy. A difference scheme, based on the inte- obstacle because the initial pressure drop in the wave gral form of the conservation laws, was constructed by increases reaching a level intermediate between the val- the finite volume method. We have employed the first- ues for the incident and reflected waves. order-accuracy monotonic schemes in terms of the The action of a weak diffracted shock wave outgo- Godunov and StegerÐWarming spatial coordinates, the ing from a round open channel upon an obstacle was related second-order schemes, and the ChakravartiÐ experimentally studied in by Panov et al. [1]. Serova [2] Harten schemes constructed using a similar approach. performed a numerical calculation for the early stage of The second-order schemes were constructed using the this process and obtained an isobar field coinciding total variation diminishing (TVD) concept. The bound- with the experimental configuration of the incident and ary conditions were selected in accordance with the reflected waves. The effect of a weak diffracted shock system geometry, taking into account the possibility of wave with an obstacle was also studied in the context of partial channel blocking and mounting an obstacle at a finding the optimum shape for a car muffler [3, 4]. Yu certain distance from the tube edge. The field of flow and Gronig [5] showed that attenuation of a diffracted parameters was calculated for a given Mach number of shock wave outgoing from an axisymmetric channel the incident shock wave, taking into account partial can be decreased by placing a coaxial cylindrical insert reflection of the shock wave from the diaphragm. The at the tube output. It was recommended to use this out- results of calculations were represented in dimension- put channel geometry to enhance the shock wave action less coordinates normalized to the parameters (density ρ upon processed surfaces. The effect of partial blocking 0 and pressure p0) of unperturbed gas in the pressure of the channel output upon the shock waveÐobstacle chamber. Distances are expressed in units of the tube interaction was not studied. diameter d. The dimensionless time t is related to the Our experiments were performed in a shock-wave τ p current time τ by the formula t = ------0 . tube with a 40 × 40 cm2 cross section connected to a d ρ cylindrical vacuum chamber with a diameter of 80 cm 0 and a length of 120 cm. The shock-wave tube end was We have obtained the Toepler patterns describing closed with a flange carrying a channel with a circular diffraction of the shock waves with 1.15 < M0 < 5 and cross section (diameter d = 20 mm; length l = 100 mm) their interaction with the obstacle. The experimental situated inside the tube. The tube edge could be addi- data describing propagation of the diffracted wave front tionally shielded with a diaphragm having a 0.5d hole along the symmetry axis was compared to the results of in the middle. The shock-wave tube edge with the numerical calculations, where the shock-wave front

660 BAZHENOVA et al.

P/P0 (a strong shock wave with supersonic flow behind). 80 The distance between the channel edge and the obstacle was varied from 0.2d to 1.5d. (a) Figure 1 shows time variation of the pressure at the center of the obstacle (plate) mounted at a distance of 60 0.5d from the channel edge. Data are presented for a shock wave with M0 = 3 or 1.15 outgoing from the open channel or a channel blocked with a diaphragm (hole diameter, 0.5d). For M0 = 3 (Fig. 1a), the diffracted shock wave produces an initial pressure jump followed 40 by a rapid drop, which is similar to the reflection of a 2 spherical shock wave. When a boundary between the outgoing flow and the flow of the unperturbed gas behind the diffracted wave comes to the obstacle, the 20 1 pressure exhibits the second increase. Then, a quasistationary state is established at the center of the plate. As seen, the second pressure pulse is insignificant as compared to the pressure level observed at the instant of the 0 shock wave reflection. With the diaphragm installed, 2 4 6 the pressure at the obstacle decreases but the character of pressure variation remains the same as for the open P/P0 channel. This result agrees with our previous data [6] 2.00 showing that a reflecting nozzle mounted at the edge of (b) a channel decreases the pressure at the obstacle. When the same obstacle is exposed to a weak shock wave (Fig. 1b), the diaphragm favors an increase in the amplitude and duration of the shock wave action. Here, 1.60 1 in contrast to the case of a diffracted wave outgoing from the open channel, the pressure on the obstacle mostly increases at a later stage of the interaction featuring retardation of the cocurrent flow rather than upon the diffracted wave reflection. Similar patterns of pres- 2 sure variation were observed when the obstacle was 1.20 mounted at other distances from the channel edge. In order to compare the effects of shock waves outgoing from an open and partly blocked channel, we have calculated the ratio of pressure pulse intensities developed in the two cases at various time instants 0.80 (Fig. 2). The pulse intensity ratio varies with time in the 0 2 4 6 8 10 same manner for all distances from the channel edge to t obstacle (from 0.2d to 1.5d). In the first stage of the shock waveÐobstacle interaction, the shock wave out- Fig. 1. Time variation of the relative pressure produced by a going from a partly blocked channel produces a pres- shock wave with M0 = 3 (a) and 1.15 (b) at the obstacle mounted at a distance of 0.5d in front of the channel output: sure pulse 15Ð20% smaller as compared to that in the (1) channel partly blocked by a diaphragm with a hole diam- case of open channel. Subsequently, the pressure pulse eter of 0.5d; (2) open channel (d is the channel diameter). of a shock wave outgoing from a partly blocked channel increases and becomes greater than that of a similar wave from the open channel. Upon reaching a quasista- position was determined as the coordinate of the point tionary state, the excess pressure reaches 25%. with maximum density gradient (this very point is reg- The results of our experiments and calculations istered on the experimental Toepler patterns). The showed that the partial channel blocking may either results of numerical calculations showed a satisfactory decrease or increase the dynamic action of shock waves coincidence with the experimental data. We have also upon the obstacle, depending on the duration of action compared the time variation of the pressure at the and the Mach number. When the Mach number is small, obstacle, determined by numerical calculations, with the shock wave dominates in the first stage of interac- the data measured for the shock waves outgoing from the tion and the cocurrent jet wave, in the second stage. open or partly blocked channel with M0 = 1.15 (a weak When a diaphragm with the 0.5d hole is mounted at the shock wave with subsonic flow behind) and M0 = 3 tube edge, the shock wave exhibits partial reflection

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

EFFECT OF SHOCK WAVES OUTGOING 661

I/I0 from the open channel at the same distance. In the fol- 1.3 lowing, the reflected part of the shock wave significantly affects parameters of the flow behind the wave front. This perturbation is transferred toward the sym- 4 metry axis (with the velocity of sound) and in the direc- 1.2 tion of the primary shock wave propagation (with the 2 flow velocity). This results in increasing pressure at the 3 flat obstacle mounted in front of the channel output. As 1.1 1 a result, the pressure pulse at the obstacle produced by the shock wave coming from the partly blocked channel exceeds the level observed for the wave outgoing from the open channel. 1.0 This phenomenon can be used to control the shock- wave pressure pulse developed at the obstacle by changing the channel output geometry. 0.9 The work was partly supported by the Russian Foundation for Basic Research.

0.8 0 4 8 12 16 REFERENCES t 1. B. Yu. Panov, A. I. Starshinov, and E. A. Ugryumov, Gazodin. Teploobmen 1, 108 (1970). Fig. 2. Time variation of the ratio of pressure pulse intensi- 2. V. D. Serova, Gazodin. Teploobmen 6, 121 (1981). ties for the shock waves with M0 = 1.15 outgoing from the partly blocked channel (I) (diaphragm hole diameter 0.5d) 3. K. Phan and J. Stollery, in Proceedings of the 14th Inter- national Symposium on Shock Tubes and Shock Waves, and the open channel (I0). The pressure was determined at a ﬂat obstacle spaced by (1) 0.2d, (2) 0.5d, (3) 1.0d, and Sydney, 1983, pp. 519Ð525. (4) 1.5d from the channel output edge. 4. S. Matsumura, O. Onodera, and K. Takayama, in Pro- ceedings of the 19th International Symposium on Shock Waves, Marseille, 1993, ed. by R. Brun and L. Z. Dumi- from the diaphragm and the central part is diffracted trescu (Springer Verlag, Berlin, 1995), Vol. 3, into the open space. A distance of 1.0d to the obstacle pp. 367−372. is equivalent to two diameters of hole in the diaphragm. 5. Q. Yu and H. Gronig, Shock Waves 6 (5), 249 (1996). Attenuation of the diffracted shock wave intensity is 6. T. V. Bazhenova, V. V. Golub, T. A. Bormotova, et al., calibrated by the hole diameter. In this ﬁrst stage, the Izv. Akad. Nauk, Mekh. Zhidk. Gaza, No. 4, 110 (1999). intensity of the shock wave outgoing from the partly blocked channel is smaller than the intensity of a wave Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 662Ð665. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 39Ð47. Original Russian Text Copyright © 2000 by Biryulin, Evlampieva, Melenevskaya, Bocharov, Zgonnik, Ryumtsev.

Transformation of the Electronic Structure of Fullerene C60 Caused by Complex Formation in N-Methylpyrrolidone Solutions Yu. F. Biryulin, N. P. Evlampieva, E. Yu. Melenevskaya, V. N. Bocharov, V. N. Zgonnik, and E. I. Ryumtsev Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, Russia Institute of Physics, St. Petersburg State University, St. Petersburg, Russia Institute of Macromolecular Compounds, Russian Academy of Sciences, St. Petersburg, 199004 Russia Received March 29, 2000

Abstract—Evolution of the electronic absorption and photoluminescence spectra of the fullerene C60 solutions with various concentrations in a polar solvent (N-methylpyrrolidone) was studied. Comparison with the electrooptical properties of these solutions shows that a slow (associative) fullereneÐsolvent interaction mechanism is operative at large (nearly saturating) concentrations (1 × 10Ð3 g/cm3), whereas the solutions of relatively low concentration (less than 5 × 10Ð4 g/cm3) exhibit fast complex formation between fullerene and the solvent molecules. The latter interaction signiﬁcantly changes the electronic structure of fullerene. © 2000 MAIK “Nauka/Interperiodica”.

A specific property of fullerene C60 is its ability tical properties (birefringence in the electric field) for to form complexes of the donorÐacceptor type with various concentrations and storage times of the C60 various organic and inorganic electron donors in sol- solutions in MP. vents [1]. It was recently demonstrated that fullerene C60 exhibits the donorÐacceptor interaction not only with the known complex-forming substances (e.g., tet- EXPERIMENTAL rafulvalenes [2] and calixarenes [3]) but also with the The electronic absorption spectra of C60 solutions in conventional organic solvents to form complexes with MP were measured in the 220Ð900 nm spectral range a certain number of the solvent molecules [1, 4, 5]. on a Specord M40 spectrophotometer employing Masin et al. [4, 5] used NMR spectroscopy to deter- 2-mm- and 1-cm-thick cells. mine the parameters of the crystal lattice of the C60 á A modulated argon laser (wavelength, 488 nm; 4C6H6 and C60 á C6H5(CH3) complexes obtained by modulation frequency, 135 Hz, power, less than slow evaporation of the C60 solutions in benzene and 100 mW) excited the photoluminescence (PL) of the toluene, respectively, at room temperature. solutions. Detection system used an FEU-62 wideband Based on the results presented below, we conclude photomultiplier tube (detection range, 280Ð1200 nm) and a lock-in amplifier. that fullerene C60 forms complexes with the molecules of one more solvent: polar cyclic organic compound Electrooptical experiments used a pulsed electric N-methylpyrrolidone. field with a rectangular pulse duration of 40 µs. A pho- Recent hydrodynamic and electrooptical studies [6] toelectric detection system employed a low-frequency of C solutions in N-methyl-pyrrolidone (MP) and modulation of the elliptically polarized light to measure 60 the electric-field-induced birefringence of the sample N-methylmorpholine revealed changes in the proper- solutions [7]. ties of these solutions caused by slowly developing aggregation process and showed a special role of the The spectral and electrooptical measurements were solvents in this process. The purpose of this work was carried out at room temperature. to study in further detail the interaction of MP as the The experiments were performed with commercial solvent with fullerene C60 by optical methods. fullerene C60 of the 99 mass % grade produced by the We measured and analyzed the electronic absorp- “Fullerenovye Tekhnologii” company (Fullerene Tech- nologies Ltd., Russia). tion and photoluminescence spectra of C60 solutions in MP prepared various methods and traced evolution of The initial solutions were prepared by dissolving the spectra with time. The changes of the spectral char- C60 in MP for two days to a maximum concentration of × Ð2 3 acteristics were compared to changes of the electroop- c1 = 0.1 10 g/cm [6]. Then, a part of the solution

TRANSFORMATION OF THE ELECTRONIC STRUCTURE 663 with the initial concentration c1 was diluted with MP to I, a.u. 1.72 eV obtain a series of solutions for the optical studies with 1.0 1.65 eV (a) the concentrations c2 = 0.8c1, c3 = 0.6c1, c4 = 0.5c1, c5 = 0.2c , and c = 0.1c . 0.9 1.72 eV 1 6 1 0.8 0.7 RESULTS AND DISCUSSION 0.6 1.65 eV 0.5 1.72 eV 3 The PL studies of the C60 solutions in MP at various 1.67 eV 0.4 concentrations and storage times revealed a sharp vari- 2 ation of the PL characteristics at certain concentrations, 0.3 which were indicative of changes in the electronic 0.2 1 structure of fullerene. 0.1 0.60 0.65 0.70 0.75 0.80 0.85 Figures 1a and 1b show the pattern of changes in the × λ, µm PL spectra at the concentrations c1 and c6 = 0.01 10−2 g/cm3, respectively, over a time interval of I, a.u. 38 days. The concentrated solution (Fig. 1a) exhibits no 1.0 significant changes in shape of the PL spectrum after a (b) one-month storage, but shows an approximately three- 0.8 3 fold increase in the quantum efficiency (radiative recombination yield). The PL spectrum of the diluted 0.6 solution (c1/10) shows substantial qualitative changes. The spectrum measured immediately or two days after 0.4 preparation exhibits a maximum at 1.65 eV and resem- 2 bles a PL spectrum of the fullerene C60 solution in tol- 0.2 uene [8]. A fifteen-day storage leads to a several-fold 1 increase in the PL intensity, a substantial broadening of 0 the spectrum, and a high-energy shift of its maximum 0.50 0.55 0.60 0.65 0.70 0.75 0.80 by 0.4Ð0.5 eV (Fig. 1b). The two weeks that follow the λ, µm sharp transformation bring virtually no changes in the spectrum shape. Fig. 1. Time evolution of the PL spectra of the C60 solutions in MP with the concentrations (a) 0.1 × 10Ð2 g/cm3 and Thus, the results of the PL studies show that the time Ð2 3 evolution of the intermolecular interactions in the (b) 0.01 × 10 g/cm : the spectra are measured (1) 2, (2) 15, fullerene solutions in MP substantially depends on the and (3) 38 days after preparation. solution concentration. A conventional associative mechanism dominates at high fullerene concentrations: that the C60 absorption maximum completely disap- in the first days, the PL spectrum exhibits virtually no pears in three days. In contrast, solutions prepared from differences from the spectrum of the molecular ≤ the initial solution of lower concentration (c 0.5c1) fullerene [8] and then some slow changes proceed in exhibit a faster evolution of the absorption spectra: time with increasing intermolecular interactions changes similar to those shown in Fig. 2 occur within (Fig. 1a). A different, rather fast (taking from several several hours. days to several hours) process causes drastic changes in the electronic structure of fullerene in the diluted solu- The C60 solutions in MP demonstrate rather unusual tion (c = 0.1c1, Fig. 1b) and gives rise to the recombi- spectral changes. The formation of the C60 solvates in nation transitions that substantially differ from those various solvents normally leads to a high-energy shift typical of isolated C60. of the UV absorption edge by approximately 0.1 eV as compared to the absorption of individual C60 [10]. The We studied optical absorption of the C60 solutions in MP in the UV range at a concentration of ~(0.005Ð character of the UV absorption spectra of the C60 solu- 0.006) × 10Ð2 g/cm3. A characteristic C absorption tions in MP does not comply with the conventional sol- 60 vation of C by solvent molecules and indicates a tran- band at 330 nm [9] shows up only in the spectra of solu- 60 tions prepared immediately prior to the measurement sition of C60 in this solvent into a state with a new elec- and then slowly “disappears” with time. The rate of the tronic structure of the optical transitions. The solutions of high (c ~ 0.1 × 10Ð2 g/cm3) and low (c ≤ 0.05 × spectral changes depends on the concentration of the − initial stock solution from which the sample was pre- 10 2 g/cm3) concentrations exhibit slow and very fast pared. Figure 2 shows evolution of the UV absorption fullerene state transformations, respectively. spectrum of a C60 solution in MP with a concentration Qualitative changes of the fullerene interaction with × Ð2 3 of 0.005 10 g/cm prepared by diluting the c1 solu- MP molecules during serial dilution are additionally tion immediately before the measurements. It is seen confirmed by experimental results on the pulsed-

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 664 BIRYULIN et al.

1 2

åP 0 300 400 500 600 700 λ, nm

× Ð2 3 Fig. 2. Time evolution of the UV absorption spectra of a C60 solution in MP with the concentration 0.005 10 g/cm : the spectra were measured (1) immediately after dilution of the initial solution with the concentration 0.1 × 10Ð2 g/cm3; (2) 4 hours later; (3) three days later. (MP) absorption spectrum of the solvent.

electric-field-induced birefringence of the C60 solutions exhibits additional weak absorption bands peaked at in MP. 435 and 460 nm similarly to the C60 complex with Figure 3 shows the change in sign of the electrooptical poly(vinylpyrrolidone)—a polymeric analog of MP [10]. × Ð2 3 effect at a concentration of (0.05Ð0.06) 10 g/cm , Summarizing the results of the studies of the C60 which proves modification of the solution properties. solutions in MP we arrive at the following conclusions High symmetry of the C60 molecule suggests its on the character of the fullerene-solvent interaction and optical isotropy. Then the observed negative electroop- its kinetics upon serial dilution. tical effect (the absolute birefringence being smaller (1) MP molecules are strongly associated with C60 than that of the solvent) can be explained by the fact molecules at a near-saturation concentration of about that a part of the solvent molecules bound to (or associated with) C60 molecules are not involved into the electric-field-induced orientation. The change of sign of the K × 1010, cm–5 g–1(300 V)–2 electrooptical effect points to different involvement of the polar solvent molecules into the field-induced ori- 10 entation at a concentration lower and higher than 0.05 × Ð2 3 10 g/cm . The positive effect at a concentration of 5 about 0.05 × 10Ð2 g/cm3 can be related to the formation of a polar asymmetric complex of C60 with MP mole- 0 cules. The orientation of these complexes in the electric 0.02 0.04 0.06 0.08 field accounts for the larger absolute value of the electrooptical effect in comparison with that in the case of –5 pure MP. × 2 3 Figure 4 compares the optical absorption spectra of c 10 , g/cm the solutions of C60 and the C60ÐMP complex (obtained Fig. 3. Plots of the specific (per fullerene unit mass) elec- by vacuum evaporation of MP) in methylene chloride. trooptical constant K versus C60 concentration measured The spectrum of the complex retains the main spectral during serial dilution of the C60 solution in MP. The curves features of the C60 absorption in the UV range and belong to two independently prepared samples.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 TRANSFORMATION OF THE ELECTRONIC STRUCTURE 665

100 2a 80 2

60 1 40

1a 20 ~ ~

0 ~ ~ ~ ~ 200 300 400 600 800 λ, nm

Fig. 4. The optical absorption spectra of (1) C60 and (2) C60ÐMP complex solutions in methylene chloride; curves (1a) and (2a) show the long-wavelength parts of the corresponding spectra scaled by the factors 5 and 10, respectively.

0.1 × 10Ð2 g/cm3. The character of the fullereneÐMP REFERENCES interaction changes with time and depends on the initial 1. D. V. Konarev and R. N. Lyubovskaya, Usp. Khim. 68 solution concentration. (1), 23 (1999). 2. A. Izuoka, T. Tachikawa, T. Sugawara, et al., J. Chem. (2) Serial dilution of the solutions down to a concen- Soc. Chem. Commun. 1472 (1995). Ð2 3 tration of (0.05Ð0.06) × 10 g/cm leads to sharp 3. T. Suzuki, K. Nakashima, and S. Shinkai, Chem. Lett. changes of their spectral, optical, and electrooptical 699 (1994). characteristics, which indicates the transition of C60 4. F. Masin, A.-S. Grell, I. Messari, and G. Gusman, Solid into the state with a new electronic structure, presum- State Commun. 106 (1), 59 (1998). ably due to a fullerene complex formation with MP. 5. A.-S. Grell, I. Messari, P. Pirotte, et al., in Molecular The PL spectra of the solutions exhibit a high-energy Nanostructures: Proceedings of the International Win- shift of the maximum by ~0.4Ð0.5 eV, while bands terschool on Electronic Properties of Novel Materials, 1998, ed. by H. Kuzmany [AIP Conf. Proc. 442, 28 characteristic of C60 disappear from the electronic (1999)]. absorption spectra. 6. N. P. Evlampieva, S. G. Polushin, N. P. Lavrenko, et al., Zh. Fiz. Khim. 74 (7), 1282 (2000). (3) The C ÐMP complex extracted from solution 60 7. V. N. Tsvetkov, in Rigid-Chain Polymer Molecules and dissolved in methylene chloride retains all features (Nauka, Leningrad, 1986), Ch. 7. of the C60 absorption spectrum and exhibits additional 8. Yu. F. Biryulin, L. V. Vinogradova, and V. N. Zgonnik, peaks at 435 and 460 nm typical of the C60Ð Pis’ma Zh. Tekh. Fiz. 24 (22), 71 (1998) [Tech. Phys. poly(vinylpyrrolidone) complex. Hence, C60 molecules Lett. 24, 896 (1998)]. interact with the heterocycles of these compounds in 9. V. N. Bezmel’nitsyn, A. V. Eletskiœ, and M. V. Okun’, the same way. Usp. Fiz. Nauk 168 (11), 1195 (1998) [Phys. Usp. 41, 1091 (1998)]. This work was supported by the Russian Interdisci- 10. V. N. Zgonnik, L. V. Vinogradova, E. Yu. Melenevskaya, plinary ScientiﬁcÐTechnical Program “Fullerenes and et al., Zh. Prikl. Khim. 70 (9), 1538 (1997). Atomic Clusters” (project no. 98076) and the Russian Federal Program “Integration” (project no. 7-326.38). Translated by A. Chikishev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 666Ð667. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 48Ð52. Original Russian Text Copyright © 2000 by Morozov, Bugrova, Lipatov, Kharchevnikov.

Medium-Frequency (f < 1 MHz) Oscillations in a Quadrupole Electric-Discharge Galathea Trap (“String-Bag”) A. I. Morozov, A. I. Bugrova, A. S. Lipatov, and V. K. Kharchevnikov Kurchatov Institute of Atomic Energy, State Scientiﬁc Center of the Russian Federation, Moscow, Russia Moscow Institute of Radio Engineering, Electronics, and Automatics, Moscow, Russia Received March 22, 2000

Abstract—Two oscillation modes (f1 ~ 25 kHz, f2 ~ 250 kHz) experimentally observed in an electric-discharge galathea trap of the quadrupole type (“string-bag”) are described and the nature of these oscillations is discussed. © 2000 MAIK “Nauka/Interperiodica”.

1. Previously [1Ð3], we have described a quadrupole tively ejected from the trap. This hypothesis is con- (“string-bag”) variant of the multipole electric-dis- firmed by the following facts. charge galathea trap (EDT-M). For convenience, Fig. 1 reproduces a schematic diagram of this device and (a) The frequency f1 is close to the inverse time of shows a radial distribution of the magnetic field z-com- 1 〈σ 〉 ponent in the plane z = 0. ion accumulation in the trap: f1 ~ --τ- = n0 V ~ 30 kHz, The discharge and plasma parameters of the “string- × 12 Ð3 where n0 ~ 3 10 cm is the xenon concentration in bag” trap reported in [1–3] represented time-averaged the working chamber and 〈σV〉 ~ 10Ð8 cm3/s. The pres- characteristics, while oscillations possible in the sys- ence of a tube supplying xenon weakly changes the dis- tem were only briefly mentioned in [4]. Below, we will present experimental data on the medium-frequency charge parameters as compared to those in the system (f < 1 MHz) oscillations observed in a quadrupole trap without tube operating under the same gas pressure. operating in the “barrier” mode (this regime, realized (b) The ion and electron currents to the probes by placing a cathode in the region of zero magnetic exhibit synchronous oscillations with the relative field, was previously referred to as a “discharge with ~ || mantle” [1, 2]) and suggest the most probable interpre- amplitudes (J /J ), being close on the experimental tation. accuracy scale of ~30% (Fig. 3). The experiments were performed using an EDT-M device of the “string-bag” type operated in the follow- Hz, Oe ing regime: working gas, xenon; gas flow rate mú = (a)3 160 (b) 2 mg/s; discharge voltage Up = 200 V; discharge current Jp = 200 mA; barrier field strength H1 = 20 Oe; residual air pressure, ~2 × 10Ð4 Torr. Under these conditions, the 120 plasma was characterized by a maximum electron tem- 1 2 1 perature of Te max ~ 20 eV, a maximum charge density of z 80 × 10 Ð3 nmax ~ 9 10 cm , and a minimum potential well |ϕ | depth of min ~ 50 V. r 40 2. Oscillograms of the discharge current J ~ reveal 1 0 the presence of quite regular oscillations (Fig. 2) with the frequencies f ~ 25 kHz, f ~ 250 kHz. H1 1 2 –40 3. Oscillations with the frequency f1 represent, as 12 16 20 24 suggested previously [3], a “reset” wave process pre- r, cm venting the trap from overfilling in the course of contin- Fig. 1. (a) Schematic diagram of the quadrupole electric- uous ionization of supplied gas. The accumulated discharge trap: (1) magnetic coils; (2) cathode; (3) xenon plasma reaches a threshold of the magnetohydrody- inlet tube; (b) radial distribution of the magnetic field Hz namic instability (Ohkawa’s surface [5]) and is convec- (r, z = 0); H1 is the barrier field.

MEDIUM-FREQUENCY (f < 1 MHz) OSCILLATIONS 667

(a) (a) 10 µS µ 10 mV 50 mV 20 S

(b) 5 µS 5 mV (b) µ 20 mV 5 S

Fig. 2. Oscillograms of the discharge current oscillations: (a) 20-µs scale; (b) 5-µs scale with the low-frequency component suppressed.

(c) The relative amplitude of oscillations of the (d) 5 µS probe currents increases when the probes are moved 5 mV from the plasma column center to the periphery. (d) The ion current minima measured with a probe placed inside the plasma volume coincide with the maxima of ion current ejected from the trap, the latter measured by a probe oriented along the radius.

4. Direct experiments with oriented probes showed Fig. 3. Oscillograms of the (a, b) ion and (c, d) electron that the flux of ions ejected from the trap has both radial probe current oscillations: (a, c) 20-µs scale; (b, d) 5-µs and azimuthal velocity components. The latter compo- scale with the low-frequency component suppressed. nent is apparently related to an azimuthal electric field, which was experimentally detected and explained by The work was supported by the Ministry of Atomic azimuthal dependence of the reset wave field. Energy of the Russian Federation.

5. As for the oscillations with the frequency f2, there are rather few experimental data on this phenomenon. REFERENCES However, taking into account a regular character of 1. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and these oscillations, their electron and ion energy scale V. K. Kharchevnikov, Pis’ma Zh. Tekh. Fiz. 18 (8), 1 (1992) [Sov. Tech. Phys. Lett. 18, 242 (1992)]. (Te ~ Ti ~ 20 eV), and the transverse size of the plasma volume, we may suggest that these oscillations repre- 2. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and sent an ion-acoustic volume “resonance.” Indeed, the V. K. Kharchevnikov, Fiz. Plazmy 19 (12), 1411 (1993) wavelength of these oscillations can be calculated by [Plasma Phys. Rep. 19, 741 (1993)]. the formula 3. A. I. Morozov and V. V. Savel’ev, Usp. Fiz. Nauk 168 (11), 1153 (1998) [Phys. Usp. 41, 1049 (1998)]. 4. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and c k(T + T ) 1 λ = ----3- = ------e------i------≈ 3 cm, V. K. Kharchevnikov, in Proceedings of the XXVI f 2 M f 2 Zvenigorod Conference on Plasma Physics and CTR, Zvenigorod, 1999, Vol. 18, p. 83. × Ð22 where Te = Ti = 20 eV, M ~ 2 10 g is the xenon ion 5. H. G. Voorhies and T. Ohkawa, Phys. Fluids 11 (7), 1572 λ (1968). mass, and f2 = 200 kHz. This value, being close to the effective transverse size of the plasma volume, is in our opinion a strong evidence for the above suggestion. Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 668Ð670. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 53Ð57. Original Russian Text Copyright © 2000 by Mursalov, Bodyagin, Vikhrov.

Calculation of Correlations in the Surface Structure of Solids S. M. Mursalov, N. V. Bodyagin, and S. P. Vikhrov Ryazan State Radioengineering Academy, Ryazan, Russia Received November 16, 1999

Abstract—Methods for the calculation of correlations in the surface structure of solids are analyzed. A new approach is proposed based on determination of the mean reciprocal information for two-dimensional systems. The approach is applied to interpretation of the results of experimental investigation of the surface of amorphized hydrogenated silicon. © 2000 MAIK “Nauka/Interperiodica”.

The existing methods used for investigations of the information for a given pair of points by integrating order in solid materials and the mechanisms and over Z2: dynamics of growth processes have significant drawbacks, and the task of studying these systems is still far (), (), (), PAB z1 z2 from being solved [1]. This situation necessitates the IAB = ∫ PAB z1 z2 log ------() ()d z 1 d z 2 . (2) PAz1PBz2 development of alternative approaches. A possible vari- 2 ant is offered by the methods of nonlinear dynamics, Z since a substance in the course of solidification can be Thus, MRI is a function defined on D2, adopting considered as a nonlinear self-organizing system [2, 3]. non-negative values. Let us consider the case when D is a rectangular An important characteristic of nonlinear systems is domain and a random function is the surface relief of a their mean reciprocal information (MRI). It is this material studied. We assume that the surface properties information—rather than distribution functions, auto- are invariant relative to the origin of the frame; this correlation function, and Fourier spectra—that charac- implies that we will study various convolutions of the terizes correlations in chaotic nonlinear systems. More- I(A, B) function. over, the search of local MRI minimum is an important stage in the procedure of embedding and calculating Let R be a domain of non-negative real values and the system attractor dimensionality [4]. L = L(r) be the MRI defined on R as function of the distance on D determined as a convolution of I(A, B) over The reciprocal information is determined as the circles with a radius r. The physical meaning of L is the amount of information about the random function value average amount of information which the known func- at point A that becomes known when we know the tion value at a given point carries about the values at a function value at point B. Let D be the domain of defi- distance r. The L(r) value is calculated using methods nition of a random function and Z, the codomain; PX, of numerical integration, by constructing multidimen- the probability distribution density at a point X (as a sional histograms on D × D × R, followed by summing function on Z); and PXY, the joint probability distribu- over equal distances. tion density at the points X and Y (as function on Z2). Figure 1 shows typical shapes of L(r) for various The reciprocal information IAB(z1, z2) for a given pair of two-dimensional objects. Here, curve 1 refers to a sig- the known and predicted values is calculated by the fol- nal of the type A(x, y) = sin(wx) + cos (wy); curve 2, to lowing formula: the surface of amorphized hydrogenated silicon (a-Si:H) film; and curve 3, to a random two-dimen- P()z , z sional noise. I ()z , z = log------AB 1 2 . (1) AB 1 2 P()zP()z The a-Si:H films were deposited from a glow dis- A1B 2 charge in 100% silane (discharge parameters: power, Here and below, the logarithm is binary and the values 50 mW/cm2; pressure, 70 Pa; silane flow rate, are expressed in bits. 200 cm3/s) onto substrates heated to various temperatures. The MRI were calculated using the sample sur- According to Eq. (1), the reciprocal information is face profile measured by the method of atomic force defined as a function on Z 2 and has non-negative values microscopy. The profile height was determined at dis- ≥ (since PAB PAPB). The mean reciprocal information is crete points and measured from a certain level consid- determined as the mean expected value of reciprocal ered as the zero level.

CALCULATION OF CORRELATIONS IN THE SURFACE STRUCTURE OF SOLIDS 669

L(r)

2 1

Fig. 1. The typical shapes of L(r) curves for various two- dimensional objects (see the text for explanation).

L(r)

1 2

r Fig. 2. The shapes of normalized L(r) curves for the same two- dimensional objects as in Fig. 1.

The shapes of MRI curves presented in Fig. 1 poses curve 3). Here, the normalized MRI is determined as serious questions. In particular, curve 1 exhibits poorly pronounced periodicity, while curve 3 (noise) reveals r L (r) = ------L(r). (3) no expected random uniform signal. A reason for these N log(1 + r) discrepancies is specific to the systems with dimension- alities greater than unity: the quality of statistics is significantly different for various distances. Indeed, this Figure 2 shows the plots of normalized MRI calcu- quality is directly determined by the size of a sample lated for various two-dimensional structures under the set. For the two-dimensional case, this size is directly same initial conditions as those used in Fig. 1. Appar- proportional to the circle length with a radius equal to ently, the normalized MRI adequately reflects spatial the argument (2πr). correlations in the systems studied. Indeed, the MRI for A(x, y) = sin(wx) + cos(wy) exhibits a clearly pro- In the case of studying a real sample, this effect is nounced sinusoidal shape. The MRI of a random noise, manifested by the incorrect selection of the MRI mini- as expected, is uniform. The MRI for an a-Si:H sample mum or its absence, which results in an attractor being surface has a distinct minimum not coinciding with that poorly “developed” in space. observed in Fig. 1. The distances for which the MRI function exhibits maxima are approximately equal to The variable quality of statistics can be correctly the size of inhomogeneities (islands) on the a-Si:H sur- taken into account using the MRI normalization calcu- face. The same pattern is observed for a-Si:H samples lated by Eq. (2). The normalization factor can be repre- deposited at different substrate temperatures. The char- log(1 + r) sented by the quantity L (r) = ------describing the acter of the MRI variation is highly sensitive toward the 3 r conditions of film deposition. A more detailed analysis MRI plot for a random two-dimensional noise (Fig. 1, of the physical meaning of the MRI curve and the

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 670 MURSALOV et al. effects of technological parameters on this function is a 2. G. Nicolis and I. Prigogine, Exploring Complexity (Free- subject of special investigation. man, New York, 1989; Mir, Moscow, 1990). Thus, the proposed approach based on the MRI cal- 3. A. A. Aivazov, N. V. Bodyagin, and S. P. Vikhrov, in culation can provide for an adequate description of correlations in spatially distributed two-dimensional Materials Research Society USA, Spring Meeting Pro- systems. ceedings, 1996, Vol. 420, pp. 145Ð151. 4. H. D. I. Abarbanel, R. Brown, J. J. Sidorovich, and REFERENCES L. S. Tsimiring, Rev. Mod. Phys. 65 (4), 1331 (1993). 1. A. A. Aœvazov, B. G. Budagyan, S. P. Vikhrov, and A. I. Popov, Disordered Semiconductors: Learning Aid (Mosk. Énerg. Inst., Moscow, 1995). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 671Ð674. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 58Ð64. Original Russian Text Copyright © 2000 by Pavlov, Anishchenko.

Dynamic Characteristics of Chaotic Processes Determined from Point Process Analysis A. N. Pavlov and V. S. Anishchenko Saratov State University, Saratov, Russia e-mail: [email protected] Received April 4, 2000

Abstract—The capability of calculating the highest Lyapunov index during analysis of the so-called point processes [1] is analyzed. Two mathematical models describing the generation of pulses by receptor neurons are considered and the conditions are established for which the dynamic characteristics of chaotic oscillations, determined from the output sequence of pulses, are retained during linear transformations of the neuron input signal. © 2000 MAIK “Nauka/Interperiodica”.

The study of information processing in living organ- Within the framework of the IF model, the S(t) signal is isms is a currently important task of natural sciences. frequently represented by a function of variables of the Solving this task encounters a large variety of particular small-scale dynamic system. A set of times Ti corre- problems, including the question of how are data coded sponding to the moments of spike generation (Fig. 1a) by the nerve cells. Each cell (receptor neuron) is essen- is determined from the equation tially a threshold device transforming a complex input signal S(t) into a sequence of identical pulses (spikes) Ti+1 registered at the output (Fig. 1a). Since the shape of the () θ ∫Stdt==, Ii Ti+1ÐTi, (1) output pulses is independent of the external factors, all information about the input signal S(t) must be con- Ti verted into the length of time intervals between output θ where is the threshold level and Ii are the time inter- pulses—interspike intervals (ISIs) [2, 3]. vals (IF ISIs). The integral value is reset to zero on In how much detail can the input signal be charac- reaching the threshold. terized upon analysis of the output sequence of spikes? The TC model introduces a threshold level Θ deter- In recent years, this problem has drawn the attention of Θ researchers in connection with the problem of dynamic mining the equation of a secant plane S = , where S(t) system (DS) reconstruction. In order to apply recon- is considered as a DS coordinate. The spike production struction methods to analysis of the point processes corresponds to the time instants when the threshold (where the information is carried in the form of times level is crossed by the signal S(t) in one direction (e.g., of various events), it is necessary to answer the question upward). From the standpoint of the DS theory, the formulated by Sauer [1]: can the output ISI sequence time intervals between spikes (TC ISIs) represent the determine the state of a system if the input signal is times of phase trajectory return to the secant plane. deterministic and generated by the DS with small-scale In this work, we attempted to answer the question as dynamics? to how the threshold level and the ISI sequence struc- An answer to this question was also originally given ture affect the results of the reconstruction. The conver- by Sauer [1], according to which an ISI set can be con- sion of a continuous input signal into a sequence of sidered as points of a new coordinate of state, in using spikes is a nonlinear transformation. Moreover, this which it is possible to characterize the small-scale transformation is accompanied by a partial loss of dynamics at the input. Then, Sauer [4] proved the information about the external factor (in particular, embedding theorem for time intervals, thus extending about the signal shape in the TC model). Can we still rigorous mathematical formalism developed by Tak- calculate characteristics of the input signal using the ens [5] to the case of point processes. The possibilities ISI sequence and what are the necessary conditions of reconstruction were investigated by numerical meth- providing for this possibility? ods in [6Ð9]. The study was focused on calculating the highest λ By now, various models have been developed to Lyapunov index 1, which is apparently the most infor- describe the process of spike generation, including the mative invariant of a complex dynamic process. Below, λ rather popular (and biologically justiﬁed) “integrate- we will discuss the conditions under which the 1 value and-ﬁre” (IF) and “threshold crossing” (TC) models [8]. can be determined from analysis of a point process.

672 PAVLOV, ANISHCHENKO

(a) S(t)

neuron

Ti Ti + 1 kS, Sint ω– H, ω 1.8 int (b) 1.14 (c)

0.6 0.98 0 t 40 10 t 140

Fig. 1. (a) A schematic diagram illustrating the input signal conversion by a receptor neuron. (b) A linear transformation of the input 1 signal: solid curve, θ-- S(t); dashed curve, Sint(t) interpolation; points, 1/Ii(Ti) values. (c) The values of instantaneous frequency ωH (Ti) (according to Hilbert) corresponding to the threshold level crossing (black circles connected by dashed line) and the π ω 2 /Ii(Ti) values (open circles connected by solid line representing the int(t) function).

Let us first consider the IF model. It was demon- Rössler system in a chaotic mode: S(t) = x(t) + C, θ = 35, strated [6] that, in the high-frequency approximation, C = 40. an IF ISI sequence represents a nonlinear transformation of the input signal Raising the threshold level leads to an increase in the average time interval I and, hence, a decrease in the ≈ θ ( ) λ Ii /Si, Si = S Ti . (2) accuracy of Eq. (2). Figure 2b shows the results of 1 calculation depending on the selection of constant C for Since the highest Lyapunov index is invariant with the same test system (variation of the C value is equiv- λ respect to nonlinear transformations, the 1 value cal- alent to shifting the threshold level). As seen, the index culated for the Ii sequence must coincide with the remains virtually unchanged for C > 30. This corre- results of calculation using the S(t) signal. Our sponds to I < T0/5ÐT0/4, where T0 is the base period of approach to calculating the Lyapunov index is essen- oscillations of the x(t) signal (cf. Fig. 2c). For smaller tially as follows [10, 11]. Once the IF ISI sequence is C values, the size of the time window occupied by the known, we may use Eq. (2) to obtain the sequence vector of state is greater than the signal correlation time, which hinders reliable determination of the 1 ≈ 1 ( ) dynamic characteristics of the external action (input --- θ--Si = kS T i (3) Ii signal) [8]. representing the values of the input signal multiplied by The DS reconstruction from a sequence of recovery times presents a more complicated problem. A possible a certain constant k at the fixed time instants Ti. In order to pass to a signal with uniform time scale, the points approach to solving this task, proposed in our previous 1/I (T ) are interpolated by a smooth function S (t), for work [9], is as follows. First, a transition is performed i i int from the time intervals I to the points ω(T ) = 2π/I cor- example, by a cubic spline. This restores the linear i i i responding to the instantaneous frequency values aver- transformation of the input signal to a certain approxi- ω mation: S (t) ≈ kS(t) (Fig. 1b). Consequently, the S (t) aged over the Ii intervals. Then, the (Ti) points are int int interpolated by a smooth function (a cubic spline) signal would retain both geometric and dynamic char- ω acteristics of the attractor corresponding to the input int(t) to pass to a signal with the uniform time scale signal (external action). Evidently, this scheme is only used in the attractor reconstruction. Using the resulting valid within a certain approximation. However (see time dependence, it is possible to describe behavior of λ H Fig. 2a), the 1 value calculated from the Sint(t) signal the average instantaneous frequency ω (t) (Fig. 1c), by a method described in [12] coincides with the results while the reconstructed attractor retains the dynamic λ of the 1 calculation proceeding directly from S(t). characteristics of chaotic oscillations in S(t) (Fig. 2d). By analogy with [6], the input signal was represented [For the TC model, the S(t) signal was represented by by a linear transformation of the first coordinate of a the first coordinate x(t) of the Rössler system.]

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 DYNAMIC CHARACTERISTICS OF CHAOTIC PROCESSES 673

0.10 0.12 (a) (b)

λ λ 1 1

0.06 1 τ 2 0 C 100

10 0.10 (c) (d)

– λ I 1

0.06 0 C 100 2 τ 9

0.12 60 (e) (f)

λ – 1 I

0.02 0 θ 16 0 θ 18

Fig. 2. (a, d) Plots of the highest Lyapunov index versus the delay time τ calculated for various dimensions of the embedding space using the (a) IF ISI and (d) TC ISI sequences for a Rössler system representing a source of chaotic oscillations (dashed lines show λ Θ the 1 values calculated using the given system of equations); (b, e) plots of the highest Lyapunov index versus threshold level calculated for the (b) IF and (e) TC models (in the former case, the threshold variation was modeled by equivalent change in the input signal shift C, see the text); (c, f) plots of the average time interval I versus threshold for the (c) IF (Θ modeled by C) and (f) TC models. The dynamic characteristics of the input chaotic oscillations can be determined provided that I does not exceed the characteristic time Tc indicated by the dashed line.

By analogy with the IF model, we have studied in the input set geometry [8], we may still calculate the detail dependence of the quality of reconstruction on dynamic characteristics using the ISI sequence. [Natu- the selection of the threshold level Θ. A shift of this rally, in speaking of the retained characteristics, we threshold has a clear physical meaning. Indeed, assume imply the results of approximate numerical experi- that we have changed the input signal amplitude. From ments rather than of the rigorous mathematical calcula- the standpoint of the DS theory, this would affect nei- tion: the characteristics can be evaluated to within ± ther the geometry nor dynamics of the chaotic oscilla- Ϸ 10%.] tions. At the same time, a change in the amplitude sig- The main conclusions from our investigation are as nificantly modifies the structure of the TC ISI output follows. The dynamic characteristics of a signal from sequence (i.e., the ISI distribution function and the small-scale dynamic system entering the neuron input recovery time transformation). These changes are so can be determined from the output ISI sequence, pro- significant that it was previously considered impossible vided that the average time interval does not exceed the to estimate characteristics of the chaotic signal at large characteristic time scale Tc. The time scales may be dif- Θ (small-amplitude input signals) [7, 8]. However, as ferent for various mathematical models of spike gener- seen from Fig. 2e, the highest Lyapunov index is inde- ation: for the IF model, the Tc value does not exceed the time required for the correlation function to attain the pendent of the threshold, provided that I does not first zero (for signals with clearly pronounced base fre- exceed a characteristic time scale Tc of the chaotic quency in the spectrum, this corresponds to a quarter of ≈ oscillations (in our case, the time of predictability Tc the base period); for the TC model, the characteristic λ 1/ 1 [13]). Therefore, despite the inability to estimate time scale is markedly greater and equals approxi-

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 674 PAVLOV, ANISHCHENKO mately to the predictability time. In this work, we pre- (American Mathematical Society, Providence, 1997), sented the results of calculations performed for the Vol. 11, pp. 63Ð75. Rössler system. The results were conﬁrmed by data of 5. F. Takens, in Dynamical Systems and Turbulence. Lec- a series of experiments performed with various sources ture Notes in Mathematics (Springer-Verlag, Berlin, of chaotic oscillations. Thus, the dynamic characteris- 1981), pp. 366Ð381. tics of random oscillations determined from the output 6. D. M. Racicot and A. Longtin, Physica D (Amsterdam) ISI sequence are retained upon linear transformations 104, 184 (1997). of the neuron input signal. For the restrictions formu- 7. R. Hegger and H. Kantz, Europhys. Lett. 38, 267 (1997). lated above, the accuracy of determining these charac- 8. R. Castro and T. Sauer, Phys. Rev. 55, 287 (1997). teristics is independent of the structure of the output 9. N. B. Janson, A. N. Pavlov, A. B. Neiman, and sequence of spikes. V. S. Anishchenko, Phys. Rev. E 58, R4 (1998). One of the authors (Pavlov) gratefully acknowl- 10. A. N. Pavlov, O. V. Sosnovtseva, E. Mosekilde, and edges the support from INTAS Foundation (grant V. S. Anishchenko, Phys. Rev. E 61 (2000) (in press). no. YSF 99-4050). The work was also partly supported 11. A. N. Pavlov, E. Mosekilde, and V. S. Anishchenko, in by a grant from the Royal Society of London. Stochaos: Stochastic and Chaotic Dynamics in the Lakes, ed. by D. S. Broomhead, E. A. Luchinskaya, P. V. E. McClintock, and T. Mullin (American Institute REFERENCES of Physics, Melville, 2000), pp. 611Ð616. 1. T. Sauer, Phys. Rev. Lett. 72, 3911 (1994). 12. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, 2. A. Longtin, A. Bulsara, and F. Moss, Phys. Rev. Lett. 67, Physica D (Amsterdam) 16, 285 (1985). 656 (1991). 13. J. Brindley and T. Kapitaniak, in Chaotic, Fractal, and 3. D. Pierson and F. Moss, Phys. Rev. Lett. 75, 2124 Nonlinear Signal Processing, ed. by R. A. Katz (Ameri- (1995). can Institute of Physics, Woodbury, 1996), pp. 605Ð611. 4. T. Sauer, in Nonlinear Dynamics and Tine Series, ed. by C. Culter and Kaplan, Fields Institute Communications Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 675Ð677. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 65Ð71. Original Russian Text Copyright © 2000 by Ivanov, Skvortsov.

The Laser-Induced Surface Evaporation A. Yu. Ivanov and G. E. Skvortsov St. Petersburg State University, St. Petersburg, 199164 Russia Received March 24, 2000

Abstract—A special mode of laser action—the surface evaporation—has been experimentally studied on a large number of various materials. Characteristic features of the pure evaporation mode have been established, and some possible applications of the method for modifying surfaces and clean-cutting of biopolymers are indicated. © 2000 MAIK “Nauka/Interperiodica”.

τ Interaction of a laser radiation with solid surfaces is the excitation relaxation time 2, and the process dura- τ considered in numerous publications (see review arti- tion 3. cles [1Ð3]). Despite the enormous amount of related data, some modes of this strongly nonequilibrium pro- The modes of the laser-induced processes are deter- cess have not been studied as yet. One such phenome- mined by a set of acting factors g and the corresponding non is the anomalous scattering considered in the structural factors s(g) [5]. In accordance with the char- recent publication [4]. acteristics indicated above, this set of acting factors for a pulsed laser includes the following quantities: Below, we describe a special type of the laser sur- ν λ δ face treatment—the pure evaporation mode—underly- g Ω Λ t G ====------[]: ----ν , ----λ- ; T 1 ----τ - , ing a number of technological operations. This mode is sg s s 1 also of great interest as a typical example of pulsed res- (1) τ∆ onance-like modes. 2 t T2==τ---- , T 3 -----τ ; We will present the results of the experimental study 1 3 of the laser-induced surface evaporation for various wε j ωδt materials (metals, dielectrics, and biopolymers). Based s W====----- , w s τ---- , J ---- , E ------ε ; (2) on the qualitative analysis of this mode, we suggest a ws 1 j s s set of quantities to describe this mode and the experi- wδt h mental conditions of its implementation. Some possible e==------j / v , v =----- , (3) applications of the mode are also indicated. ah δ t I. Consider the modes of surface treatment with the where v is the velocity of the evaporation front. aid of a pulsed laser in terms of the laws of processes If the necessary data on the structure are lacking, it occurring in the corresponding systems [5]. is expedient to use some special (critical) values of the The effect of a pulsed laser is characterized by the acting factors gc corresponding to transitions between following quantities: the frequency ν and the wave- various essentially different modes: length λ of the laser radiation; the pulse duration δt and () ν ν τ τ ε ε , … the pulse repetition period ∆t; the power w [W] and the sg gc: s c, 1 1c, s c .(4) intensity (power density) j [W/cm2] of the laser radia- Usually, the dynamic structural transitions take tion; jú = w/a, where a is the area of the illuminated spot place in the vicinity of such particular values. on the surface. At present, several particular modes of laser- The object is a surface described by the following induced processes on the surface are known. The most set of structural factors: the coefficient of radiation intense of these modes is the anomalous scattering [4] τ λ Ð8 3 absorption k, the characteristic times s and lengths s, taking place under the conditions δt ~ 10 s, 10 < j < ε 7 the binding energy s of a particle on the surface, and 10 . The second specific mode, which is considered in the characteristic intensity js of the absorbed radiation. this article, is the pure laser-induced evaporation imple- All these structural factors depend on the radiation fre- mented under the conditions δt ~ 10Ð8, 108 < j < 1010. At quency and other quantities characterizing the laser j > 1012, the latter mode is transformed into a specific radiation indicated above. shock-wave mode described elsewhere [1]. The laser-induced effect is determined by the pene- In accordance with the above data, the threshold val- α 3 8 12 tration depth h, the area a of the irradiated region, ues are taken to be jc1 = 10 , jc2 = 10 , and jc3 = 10 .

676 IVANOV, SKVORTSOV A number of specific modes can be obtained by nephrite, malachite, ruby, marble, etc.), plastics (texto- combining the appropriate values of criteria (1)Ð(3) lite, organic glass, foam plastic, etc.), rubber, and Ω, Λ Ѥ Ѥ ѥ biopolymers (paper, wood, fabrics, leather). The root- 1, J 1, T 1 (5) mean-square deviation of the surface roughness of for the threshold values indicated above and for other these materials ranged from 0.007 to 100 µm. known threshold values. Thus, more than thirty specific The laser power in the experiments varied from 104 modes can be obtained for the above three threshold to 106. The beam spot at the normal beam incidence values of j with due regard for the frequency resonance was usually a = 3.1 × 10Ð6 cm2. The absorption coeffi- Ω = 1 and the relaxation “resonance” T ≈ 1. Some of these modes are described in [6]. cient k (averaged over the experiments) had the value II. Now consider the conditions necessary for imple- 0.5. In this case, j = 0.5w/a = 1.6 × (109Ð1010). The mentation of the pure evaporation mode, which pro- evaporation depth under the action of one pulse ranged vides intense material evaporation without changing within (1Ð4) × 10Ð2 cm2. the chemical composition in the irradiated region. This Using the above parameters, we obtained the fol- mode should necessarily be pulsed and resonance-like. lowing basic values for metals and biopolymers (aver- The first requirement should be met because, in most aged over the materials of one type). materials, the radiation with intensity exceeding 108 For metals: within the time δt > 10Ð7 would initiate combustion [7]. w = 2 × 105, h = 3 × 10Ð2, v = 1.5 × 106, The second requirement—the resonance-like char- (6) acter with respect to intensity—is imposed by the fol- j = 3.2 × 1010, e = 2.2 × 104; lowing limitations. At j < 107, the evaporation for time δ Ð8 ≅ ε ρ × 4 ε t ~ 10 is rather weak because of a low energy e Al: = e/ Al = 0.85 10 J/g, s = 1.14 eV . δ λ × Ð4 ρε ε ≅ 4 j t/ = 2 10 j < s0, where s0 10 J/g is the tabulated average specific energy of evaporation. The j val- For wood: 10 9 ues exceeding 10 lead to combustion. At j ~ 10 , w = 2 × 104, h = 10Ð2, v = 5 × 105, absorption should attain the maximum value [8]. Thus, (7) in the general case, the evaporation mode is imple- j = 3.2 × 109, e = 6.4 × 103; ε = 1.1 × 104. mented within the range 108 < j < 1010. For each mate- It should be noted that the energy of a particle rial, the range of intense pure evaporation is rather nar- detachment in the laser-induced evaporation mode is row, and the process is of the resonance-like nature. smaller than the specific evaporation heat (1.2 × 104) by The evaporation mode is more effective when evap- a factor of 1.4. For biopolymers, the sublimation heat oration is provided by a series of pulses. In this case, the was not considered at all. material evaporated by the first pulse must go away so that the new pulse would directly interacts with the Being compared with the well-known thermal “open” surface rather than ionize the vapor. A number (combustion) mode, the evaporation mode is character- of experiments showed that the most appropriate pulse ized by a number of distinctive features, namely: repetition frequency is 50Ð100 Hz. The velocity of 1. Evaporation occurs without melting, i.e., directly laser-beam scanning of the surface was chosen in the from the solid phase. range (2Ð4) mm/s. For a laser spot size of ~10Ð3 cm, the 2. The width and the depth of cutting are smaller number of acting pulses varied from three to six. than those provided by conventional methods, whereas To provide for the clean-cutting of biopolymers, it is the quality of cutting is higher. important to use a laser wavelength close to the wave- 3. No oxides or other compositional changes were λ ≈ length of bound-water absorption s 540 nm. The detected in the evaporation regions (the exceptions are condition of frequency (energy) resonance was satis- titanium, sulfur, and phosphorus). fied by selecting laser radiation with the wavelength 4. All the cutting parameters (width, depth, and the λ = 530 nm. In most experiments, we used this laser rate of cutting) depend on the material and its rough- radiation and varied only its power. ness and can be controlled by a number of methods. III. Now consider the experimental results. The 5. Precise layer-by-layer removal of a material is experiments were performed with the use of a solid- possible, including the “erasing” of a text printed on state Nd3+ : YAG laser (LTIP4-5 type) with the wave- paper. length 530 or 1060 nm and a power of 106 W operating 6. The quality of cutting is higher the more homo- in the pulse-periodic mode with δt = 2 × 10Ð8 and ∆t = geneous the material is and is inversely proportional to 0.01Ð0.02 s. the integrated reflection coefficient and the surface We studied the surfaces of various materials—met- roughness. als (Al, Zn, Cu, Fe Ti, Mg), alloys, stainless steel, Dur- 7. Among a large number of readily evaporated alumin, brass, dielectrics (C, S, P), various glasses, materials, the exceptions are titanium, sulfur, and phos- concrete, minerals (agate, jasper, lasurite, fluorite, phorus, whose vapors are ignited almost simulta-

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 THE LASER-INDUCED SURFACE EVAPORATION 677 neously with the beginning of evaporation. Our inition, the laser evaporation mode is a resonance-like attempts at obtaining high-quality glass cuts also failed: mode with respect to the logarithmic intensity G = 9 in the laser-induced evaporation mode, glasses start log( j /js), js = 10 . The effective mode for each material breaking in an irregular manner several hours upon is also of the resonance nature and should be refined cutting. experimentally. 8. Usually, laser evaporation involving the interac- The laser evaporation mode finds numerous practi- tion of a laser beam with the vapor plasma (without cal applications. This treatment provides for the evapo- contact with the surface) is accompanied by sound gen- rative modification (cleaning, polishing, restoration) of eration. various surfaces, precise clean-cutting of thin layers, With an increase of the deposited energy (radiation and high-quality perforating. Another promising appli- dose), the evaporation mode is changed to the well- cation is the use of the vapors formed during laser irra- known thermal mode. diation under the appropriate condensation conditions IV. We should like to make some concluding for obtaining various clusters, including nanoclusters. remarks concerning the experiments and assess the It is also possible to use this mode for obtaining various technological possibilities provided by the laser evapo- structural fullerene-like structures [10]. ration mode. Within the law of mode sequence [5], mode 3 (con- REFERENCES sidered above) is intermediate between mode 2 (anomalous scattering [4]) and thermal-evaporation mode 4 1. S. I. Anisimov, A. M. Prokhorov, and V. E. Fortov, Usp. preceding mode 5 described elsewhere [1Ð3]. Fiz. Nauk 142 (3), 395 (1984) [Sov. Phys. Usp. 27, 181 Each of the above modes is characterized by the (1984)]. basic and accompanying dynamic structural transi- 2. G. Chapter, Rev. Mod. Phys. 59 (3), 119 (1987). tions. Thus, mode 2 provides for the formation of a sur- 3. S. I. Anisimov, Ya. R. Imas, and Yu. V. Khadyko, Effect face electromagnetic structure. Mode 3 is determined of High-Power Radiations on Metals (Moscow, 1970). mainly by the evaporation (sublimation) transition. 4. A. Yu. Ivanov, Pis’ma Zh. Tekh. Fiz. 25 (16), 29 (1999) Mode 4 gives rise to the evaporation and ionization [Tech. Phys. Lett. 25, 645 (1999)]. transitions, etc. 5. G. E. Skvortsov, Pis’ma Zh. Tekh. Fiz. 23 (6), 85 (1997) Since each structural transition necessarily includes [Tech. Phys. Lett. 23, 246 (1997)]; Pis’ma Zh. Tekh. Fiz. an anomalous stage [9], the detailed study of each of 23 (7), 23 (1997) [Tech. Phys. Lett. 23, 261 (1997)]; Pis’ma Zh. Tekh. Fiz. 23 (10), 17 (1997) [Tech. Phys. these modes reveals some anomalies. In particular, for Lett. 23, 383 (1997)]. the laser evaporation mode, such an anomaly consists in the generation of sound, indicating a nonstationary 6. V. M. Akulin and N. V. Karlov, Intense Resonance Inter- (pulsating) character of the nonequilibrium evaporation. actions in Quantum Electronics (Moscow, 1987). 7. Yu. I. Dymshits, Zh. Tekh. Fiz. 47 (3), 532 (1977) [Sov. An important property of the evaporation mode is Phys. Tech. Phys. 22, 322 (1977)]. the resonance-like character. It is expedient to deter- 8. G. G. Vilenskaya and I. V. Nemchinov, Dokl. Akad. mine a measure of this character using the relationship Nauk SSSR 186 (5), 1048 (1969) [Sov. Phys. Dokl. 14, 1 g 560 (1969)]. G = ------, G = ----, (8) r 2 ( )2 g 9. G. E. Skvortsov, Pis’ma Zh. Tekh. Fiz. 25 (1), 81 (1999) 1 + r± 1 Ð G c [Tech. Phys. Lett. 25, 35 (1999)]. ѥ where r± is the resonance degree at g gc. 10. J. M. Hunter, J. L. Fyc, and M. F. Jarrold, J. Chem. Phys. A mode can be considered as resonance when 3, 1785 (1993). Eq. (8) describes a rather narrow pulse, for example, if the conditions r± > 2 are satisfied. According to this def- Translated by L. Man

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 678Ð681. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 72Ð79. Original Russian Text Copyright © 2000 by Rodnyœ, Mikhrin, Mishin.

Measurement of Scintillator Parameters Using Pulsed X-ray Excitation P. A. Rodnyœ, S. B. Mikhrin, and A. N. Mishin St. Petersburg State Technical University, St. Petersburg, 195251 Russia Received February 26, 2000

Abstract—A setup for measurement of the major scintillator parameters, including decay time constants, light yield, and emission spectrum, has been developed. It consists of three main components: a small-size source of short X-ray pulses, a cryostat, and a registration system. The ultimate parameters characterizing the source are as follows: pulse duration, ~0.5 ns; repetition rate, 100 kHz; X-ray tube voltage, 30 kV; and current amplitude, 0.5 A. The registration system allows the measurements to be carried out in the 200Ð800 nm spectral range over the time intervals from 0.5 ns to 50 µs with resolution not worse than 0.1 ns. © 2000 MAIK “Nauka/Interperi- odica”.

Measurements of the main scintillator parameters came into use that are capable of producing stable- such as decay time constants, light yield, and emission amplitude pulses with a ≥10 ns duration [4]. spectrum conventionally employ excitation by gamma- quanta or by particles (electrons, protons, neutrons, Short X-ray pulses can be obtained from an XRT etc.) [1]. Recently, pulsed X-ray radiation has also been equipped with a laser-excited photocathode [5]. Oper- used to study the luminescence and scintillation prop- ating at an anode voltage of 30 kV, such a source can erties of wide-gap crystals [2]. This technique has obvi- produce pulses with a ~100 ps duration. A signiﬁcant disadvantage of this kind of sources is a very low ous advantages. In comparison to gamma-quanta and ≤ µ heavy particles (α, n), X-ray pulses provide a more uni- amplitude of the XRT anode current ( 50 A), with the form and stable excitation of the bulk of a sample, result that the X-ray pulse energy absorbed by a sample which leads to a higher stability of the parameters of only slightly exceeds the energy of a single gamma- 60 emitted light pulses. Excitation by electrons results in quantum from a Co source. charging of sample surface and requires the object In this paper, we describe a source with a three-elec- under study to be placed in vacuum. These problems do trode XRT (30 kV, 0.5 A), which is capable of produc- not arise when X-ray pulses are used. Looking for new ing subnanosecond X-ray pulses at a repetition rate of scintillator materials, the researchers often have to deal up to 100 kHz. The source comprises the two main with imperfect small-size samples, and intense X-ray parts: an RTI2-0.05 three-electrode XRT and a modula- pulses prove to be an indispensable tool in this situation. tor (Fig. 1, blocks 2 and 3). The modulator, which con- Experimental techniques based upon pulsed X-ray radi- sists of a GaAs LEDÐthyristor optocouple, delivers a ation are simple and reliable, which is especially valu- control pulse with an amplitude of up to 290 V and a able when a large series of specimens have to be exam- full width at half maximum (FWHM) of ~0.5 ns into a ined [1, 2]. 50 Ω load [6]. In the present layout, the modulator is Scintillators are widely used in high-energy physics, triggered by pulses from a G5-56 generator 1 (pulse positron emission tomography, computer-aided X-ray amplitude, 10 V; pulse duration, 100 ns; repetition rate, tomography, and other areas where high event count 100 kHz) and energized by a power supply unit 6. It is rates are encountered. Consequently, “fast” scintillators important to note that the compact semiconductor mod- with nanosecond and subnanosecond emission times ulator is connected to the XRT gridÐcathode assembly are to be employed. Only a limited number of such via a coaxial joint and forms a single block with the latter; this makes it possible to accomplish subnanosec- scintillators are now available (BaF2, ZnO : Ga), and new materials capable of producing short scintillations ond commutation [6]. The high voltage (up to 30 kV) is are being actively searched for. Thus, there is a need in supplied to the XRT from source 4 with control block 5. short (subnanosecond) exciting pulses. Bias supply unit 7 provides a constant negative voltage of 100Ð250 V at the XRT grid, which can also change Commercially available pulsed sources with two- the X-ray pulse duration to a small extent. The layout electrode X-ray tubes (XRTs) [3] have a number of described is capable of producing short (0.5 ns) pulses drawbacks: poor stability of the X-ray pulse parameters, of the anode current and the corresponding X-ray low repetition rate, and a high anode voltage (~106 V). pulses. The latter are slightly broadened (by ~30%) Fairly recently, three-electrode (controllable) XRTs with respect to the gridÐcathode current pulses, whose

MEASUREMENT OF SCINTILLATOR PARAMETERS 679

6 4 5

1 2 3 7

19 X-ray 8 20

22 ......

9 11 ...... 18 21

10 12 14 13

16 17 15

Fig. 1. Block diagram of the setup for measuring the scintillator parameters under pulsed X-ray excitation: (1) pulse generator; (2) modulator; (3) pulsed X-ray source; (4) high-voltage power supply unit; (5) control block; (6) modulator power supply unit; (7) bias source unit; (8) decoupling transformer; (9) START channel time-locking circuit; (10) time-to-amplitude converter; (11) frequency meter; (12) STOP channel time-locking circuit; (13) ampliﬁer; (14) photomultiplier tube; (15) photomultiplier power supply unit; (16) analog-to-digital converter; (17) IBM PC; (18) cryostat; (19) beryllium cryostat window; (20) quartz cryostat window; (21) sample; (22) monochromator.

FWHM duration varies from 0.6 to 1 ns, depending on high-voltage (anode) elements, and the radiation shield the bias voltage. Instability of the X-ray photon flux at assembly around the working beam. These components the XRT axis is ~1% at an anode-current instability are described in the literature [4, 7]. Note that, at equal of 0.5%. operating voltages (30 kV), the maximum value of the The source, comprising the XRT and the modulator, anode current (0.5 A) in an XRT with thermionic cath- is designed as a cylinder 15 cm in length and 10 cm in ode (RTI2-0.05) is 104 times higher than the corre- diameter (including the lead shield). This unit can be sponding value obtained in the tube with photocathode easily mounted together with a photomultiplier tube [5]. In the rated operating mode, the tungsten anode of (PMT) on an optical bench and matched with a cryostat RTI2-0.05 emits up to 107 X-ray photons per pulse with in an experimental setup. The power consumption by a mean energy of ~17 keV. the source (excluding the G5-56 generator) amounts to The sample under study is placed in a steel cryostat 18 50 W. When the tube is producing 0.5-A 1-ns pulses at (Fig. 1). In the region of the X-ray beam passage, the repetition rate of 100 kHz, the average anode current cryostat wall thickness equals 2 cm, which is sufficient equals ~25 µA, which is much lower than the maxi- to absorb the incident radiation completely. To prevent mum allowed average current of 500 µA. The 30 kV irradiation of the PMT detector by X-rays, the axes of XRT operating voltage is also lower than the rated level the cryostat entrance beryllium window 19 (for the (50 kV). To record a kinetic curve or a scintillator emis- X-ray beam) and exit quartz window 20 (for the light sion spectrum, the data are accumulated for several flux) are arranged at a right angle to each other. The minutes. Thus, dissipative losses at the XRT anode in cryostat design allows the sample orientation with the course of the measurements are not high, and the respect to these axes to be varied so as to optimize col- unit does not require any special cooling. lection of the emitted light. The sample temperature Some of the components of this compact X-ray can be varied from 80 to 600 K under a vacuum of 3 × source are virtually identical to those used in the famil- 10Ð3 Torr; however, most of the measurements with new iar 10 ns pulse sources, including the high-voltage sup- materials are performed at room temperature without ply unit, the control block, the insulation system for the evacuation of the cryostat volume.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

680 RODNYŒ et al.

N electronics in the time-domain measurements is less than 100 ps. 3 10 (a) Let us consider how does the setup work. Synchro- nization pulses from the G5-56 generator 1 are passed to modulator 2, the subnanosecond output pulses of which open the tube of the X-ray source 3. Simulta- 102 neously, a constant bias voltage, which determines the anode current amplitude, is applied to the XRT grid from source unit 7. The high-voltage power supply unit 4, controlled by block 5, provides up to 30 kV at 1 the XRT anode. Modulator 2 is powered by a stabilized 10 500-V source 6. The X-ray pulse enters cryostat 18 through the beryllium window 19 and hits the sample 21. The light emitted by the sample leaves the cryostat 0 10 20 30 40 through the quartz window 20 and, after being dis- t, ns persed by monochromator 22, enters the photomultiplier 14 supplied with high voltage from source 15. N A special small-size monochromator provides for the 104 time-resolved emission spectra recording over a (b) 200−800 nm range. The monochromator step motor is controlled by computer 17. 3 10 In the study of samples with low light yield, the monochromator is not used and the sample emission 102 goes directly to the PMT. The PMT output pulses are enhanced by amplifier 13 and passed through the STOP TLC unit 12 to enter the time-to-amplitude converter 101 (TAC) 10. In the other channel, the synchronization pulses are passed through the decoupling transformer 8 and the START TLC unit 9 to TAC 10. The amplitude 100 of the TAC output pulse is measured by a 12-bit analog- 0 10 20 30 40 50 to-digital converter 16, and the value obtained is stored t, µs by the computer 17 in a data file. Subsequent processing of the experimental data is carried out using the Fig. 2. Luminescence kinetics of (a) BaF2 and (b) Gd2O2S : Pr measured using pulsed X-ray excitation. N is the number of common application program packages. The frequency pulses counted. meter 11 is used in the layout adjustment. The light yield is determined by referencing the output signal obtained from the sample under study to the signal The sample (Fig. 1, 21) mounted on a rod in the cry- from a standard scintillator such as CsI : Tl. ostat, is located at a distance of 6 cm from the X-ray This scheme operates stably when the load (i.e., the beam entrance window. It can be estimated that an 2 PMT single-electron pulse detection rate) does not energy deposited by each X-ray pulse in a 1-cm -area exceed 10% of the excitation pulse repetition rate. At sample (thick enough to completely absorb the incident higher loads, two-photon events become important, X-ray radiation) is ~109 eV. Thus, in its action on the which distorts the measurement results. To illustrate the sample, such a short X-ray pulse is essentially equiva- performance of the setup described, we present the lent to a gamma-quantum of ~109 eV energy. Possess- recorded luminescence kinetics of the well-known ing an energy efficiency of ~1%, the scintillator will BaF2 and Gd2O2S : Pr scintillators (Fig. 2). The decay emit ~3 × 106 optical photons per pulse. time constants characterizing the fast and slow components in the BaF2 emission (Fig. 2a), determined from A system used for registration of the emission from these data, are 0.88 ± 0.02 and 850 ± 5 ns, respectively. crystals under study, built within the CAMAC standard, Both values agree well with those found in the litera- is based on the STARTÐSTOP technique widely ture [10]. employed in physical experiments [8, 9]. The time- locking circuit (TLC) 12 provides for synchronizing The luminescence kinetics of Gd2O2S : Pr shown in the STOP channel to a definite point at the front edge of Fig. 2b exhibits, in contrast to the case of BaF2, a sin- the single-electron pulses arriving from the PMT, gle-exponent decay with a time constant of 3.05 ± which eliminates the effect of pulse amplitude fluctua- 0.05 µs. tions on the time resolution. Requirements to the These experimental results demonstrate that, with START TLC unit 9, operating with standard triggering the setup described, it is possible to carry out wide- pulses, are less stringent. The net error introduced by dynamic-range measurements of the luminescence

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 MEASUREMENT OF SCINTILLATOR PARAMETERS 681 kinetics over a large time interval (from 0.5 ns to 50 µs) 6. P. A. Rodnyœ, B. R. Namozov, A. V. Rozhkov, and with a time resolution of at least 0.1 ns. A. G. Ryzhkov, RF Patent No. 2054739, 1996. 7. V. G. Guk, A. A. Lobanov, and N. N. Skachkov, Élek- tron. Tekh., Ser.: Élektrovak. Gazorazryad. Prib. 2 (133), REFERENCES 37 (1991). 1. L. V. Viktorov, V. M. Skorikov, V. M. Zhukov, and B. V. Shul’gin, Izv. Akad. Nauk SSSR, Neorg. Mater. 8. W. W. Moses, Nucl. Instrum. Methods Phys. Res. A 336, 27, 1699 (1991). 253 (1993). 2. P. A. Rodnyœ, Appar. Metod. Rentgen. Anal. 39, 152 9. F. L. Gerchikov, V. A. Gudovskikh, N. T. Danil’chenko, (1988). et al., Pis’ma Zh. Tekh. Fiz. 6 (15), 911 (1980) [Sov. 3. S. P. Vavilov, Pulsed X-ray Techniques (Énergiya, Mos- Tech. Phys. Lett. 6, 394 (1980)]. cow, 1981). 10. M. Laval, M. Moszynski, R. Alemand, et al., Nucl. 4. F. L. Gerchikov, Controlled X-ray Radiation in Instru- Instrum. Methods 206, 169 (1983). ment Building (Énergoizdat, Moscow, 1987). 5. S. E. Derenzo, W. W. Moses, S. C. Blankerspoor, et al., IEEE Trans. Nucl. Sci. 41, 629 (1994). Translated by M. Skorikov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 682Ð685. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 80Ð87. Original Russian Text Copyright © 2000 by Akimov, Babenko.

Field and Input Admittance of a Vertical Magnetic Dipole Placed above a Plane Grid Screen V. P. Akimov and L. A. Babenko St. Petersburg State Technical University, St. Petersburg, 195251 Russia Received March 23, 2000

Abstract—A solution describing the radiation ﬁeld of a vertical magnetic dipole placed above a plane grid screen is presented. It is shown that both the solution and the related computational procedure can be substantially simpliﬁed using methods of virtual images and averaged boundary conditions. The obtained results can be used to calculate the parameters of electromagnetic screens involving metal grids. © 2000 MAIK “Nauka/Interperiodica”.

In order to obtain a unidirectional radiation from the Upon applying the Fourier transform in the XY- simplest antennas (such as dipoles and loops) and plane to Eqs. (1) and (2), we obtain arrays of such antennas, solid or grid metal screens are usually applied. In the case of a solid screen, the m ˜ '' γ 2 ˜ iI Lδ ()≥ antenna characteristics can be very simply calculated M1 + M1 = ------µ ω- zhÐ ,z0, (3) using the mirror image method. Grid screens are not 0 perfectly reﬂecting surfaces; therefore, in this case, the ˜ γ2 ˜ ≤ image method is inapplicable. M2'' +0,M2 = z 0, (4)

To calculate the radiation characteristics of antennas 2 d M˜ placed above grid screens, numerical methods are usu- where M˜ '' = ------, γ2 = k2 Ð α2 Ð β2, α and β are the ally applied [1]. Such methods are rather time-consum- dz2 ing even for modern computing facilities. However, variables of the Fourier transform, and ᑣmγ < 0. these problems can be solved using methods of averaged boundary conditions [2] and virtual images [3], The structure of equations (3) and (4) coincides with which substantially simplify the computational proce- that of equations describing a Z-directed transmission dure. This approach was successfully used [4, 5] to cal- line with the propagation constant γ. Solutions to these culate the fields of vertical and horizontal electric equations can be written as dipoles placed above a plane metal grid. m γ γ Proceeding to the solution of the magnetic problem, ˜ Ði z I L Ði zhÐ ≥ M 1 =Ce Ð0,------γωµ- e ,z (5) consider Fig. 1, showing a vertical magnetic dipole of 2 0 length L with dipole current Im placed at height h above ˜iγz the surface of a plane wire grid (I) with square mesh. M2=De ,z≤0. (6) The mesh size is a Ӷ λ (where λ is the radiation wave- Ӷ The constants C and D are determined from the bound- length), the wire radius is r0 a, and the contacts between wires are perfect. For such parameters, the ary conditions on the grid surface representing the con- grid can be treated as an effective solid surface with tinuity condition for the tangent components of the averaged boundary conditions [2]. In order to deter- electric field intensity and the averaged boundary con- mine fields in the upper and bottom half-spaces, intro- dition: duce component Mz = M of the magnetic Hertz vector Eτ=Eτ,(7) satisfying the Helmholtz equation 12c=0

2 2 i m 1 ()∇ + k M()r= ------I L δ ()rzÐh,z≥ 0, (1) Eτ=iηκ jτ +------∇∇()jτ ,(8) 1 ωµ 0 12 0 2k z=0 ()∇2 2 () ≤ + k M2r= 0, z 0, (2) µ η 0π κ a a where = -----ε = 120 , = λ--- ln------π - is the grid where k2 = ω2ε µ , ε and µ are the permittivity and 0 2 r 0 0 0 0 0 × permeability of free space, r are the coordinates of the parameter, and jτ = z0 (H2 Ð H1) is the averaged cur- δ observation point, and (r Ð z0h) is the Dirac delta. rent density on the grid surface.

Taking into account that vectors E and H are related Z to the Hertz vector via relationships r ωµ ∇ × 2 ∇(∇ ⋅ ) ImL E = Ði 0 M, H = k M + M , h boundary conditions (7) and (8) can be written in terms of the Fourier images M˜ as θ I M˜ = M˜ , (9a) 1 2 X η ˜ κ( ˜ ' ˜ ' ) M1 = ω-----µ---- M1 Ð M2 , (9b) 0 –h dM˜ where M˜ ' = ------. Fig. 1. Magnetic dipole above the grid screen. dz

Using relationships (5), (6), and (9), we obtain Taking into account formulas (13) and (14), we can write expression (10) in the form k Ðiγh 2iγκ Ðiγh C = ÐA------e , D = A------e , γκ γκ ∞ k k + 2i k + 2i   i------s   ˜  Ðiγ z Ð h  ik 2κ Ðγs  Ðiγ (z + h) m M1 = A e + ∫----κ--e e ds e I L   2   (15) A = Ð----γ---ω----µ----. 0 2 0 ˜ 0 ˜ 1 = M1 + M1 , ˜ ˜ Now the Fourier images M1 and M2 can be written ˜ 0 ˜ 1 as where M1 and M1 are the Fourier images of the Hertz vectors corresponding to the dipole and grid current, ˜ Ðiγ z Ð h Ðiγ (z + h) M1 = A(e + R(γ )e ), z ≥ 0, (10) respectively. In the physical space, the Hertz vector of the dipole ˜ iγ (z Ð h) M2 = A(1 + R(γ )e ), z ≤ 0, (11) ﬁeld is determined by the expression

m (γ ) k 0( ) iI L ( ) R = Ð------. (12) M1 r = Ð----π---µ------ω---G r Ð z0h , (16) k + 2iγκ 4 0

Note that the first and the second terms in formula (10) eÐikd where G(r Ð z h) = ------is the free-space Green’s func- describe the incident and reflected wave, respectively, 0 d and R(γ) is essentially the reflection coefficient. In order to pass from the Fourier images to functions tion. By analogy with formula (16), we can represent 1 defined in the physical space, we should perform the M1 as inverse Fourier transform. We may choose, however, ∞ k another way and derive an expression for the Hertz vec- m i------s tor corresponding to the grid currents by considering 1( ) iI L ik 2κ ( ( )) M1 r = Ð----π---µ------ω------κ--∫e G r Ð z0 Ð h + is ds,(17) some dummy source (image) similarly to the case of 4 0 2 the perfectly conducting plane. 0 γ where r is the position vector determining coordinates Assume that R( ) is the Laplace transform of func- of the observation point and (Ðh + is) are the coordi- tion f(s), that is, nates on the dummy source simulating the effect of the ∞ grid surface. Expression (17) can be treated as the Hertz Ðγ vector of a virtual source placed in the complex space. R(γ ) = ∫ f (s)e ds. (13) This source is a semi-infinite linear magnetic dipole 0 extended along the s-axis from 0 to ∞. The current of this source depends on the coordinate s on the imagi- From relationships (12) and (13), it follows that nary axis and equals

k m k i------s i----κ--s ( ) ik 2κ m'( ) iI L ik 2 f s = ------e . (14) I s = Ð----π---µ------ω------κ--e . (18) 2κ 4 0 2

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 684 AKIMOV, BABENKO

∆ ∆ Y' = Y/Y0 mirror-symmetric (relative to the grid plane) point; that 0.2 is, we come to the conventional mirror image method. 3 Thus, in order to determine the radiation field of a a vertical magnetic dipole placed above the grid (for z > ---- = 100 r 0), we should add the field of the real dipole to that of 0 its virtual image (18). The field in the bottom half-space 4 1 can be determined in accordance with relationship (11). If the observation point is located in the upper half- space (z > 0) at a large distance from both the dipole 2 and the grid surface, then, passing to corresponding 0 limits, we obtain a well-known relationship for the reflection coefficient of a plane TE wave from a grid with square mesh,

TE 1 R = Ð------. 1 + i2κcosθ Using the above technique and the method of induced magnetomotive forces, we can derive a relationship for determination of the relative variation of –0.2 input admittance of the magnetic dipole related to the 0 1 2 presence of grid screen: h/λ ∞ ∆Y 3 Ði4πh/λ  1 ∆ λ ∆ ------= Ð------e ------Fig. 2. The plots of (1) Re( Y') for a/ = 0.001; (2) Re( Y') πκ ∫ 3 λ ∆ λ ∆ Y0 4 2π(2h/λ Ð iτ) for a/ = 0.25; (3) Im( Y') for a/ = 0.001; and (4) Im( Y') 0 for a/λ = 0.25. (19) i  π(i/κ Ð 2)τ  τ + ------2- e d , Green’s function in relationship (17) has the form (2h/λ Ð iτ) 

G(x, y, z, s) π ε λ 2 0 λ 2 where is the wavelength and Y0 = ------(L/ ) is the 2 2 2 3 µ exp(Ðik x + y + (z + h Ð is) ) eÐikd 0 = ------= ------; radiation conductance of the magnetic dipole placed in 2 2 2 d x + y + (z + h Ð is) the free space. Formula (19) is well suited for calculations, since that is, it involves the complex distance ﬁrst introduced the integral is fast-convergent owing to the rapidly low- by Wait in transformation of the well-known Sommer- ering magnitude of the integrand. Figure 2 presents the feld solution for a source located near an interface. real and imaginary parts of the dipole input admittance as functions of h/λ for two values of a/λ. The integral appearing in expression (17) converges if, calculating it for a given complex distance d, we As is well known, a physical implementation of the choose the branch with ᑣmd < 0. Consider two limit- magnetic dipole is a loop of small radius b with electric e m ing cases: (i) the grid is absent and (ii) the grid trans- current I . The magnetic current I on the dipole can be forms to the perfectly conducting surface. If we assume expressed in terms of the loop current as [6] that r /a 0 as a result of decreasing radius of the m e 0 I = iωµ I S/L, grid wire, then evidently κ ∞, which corresponds 0 to “disappearance” of the grid surface. In this case, we where S = πb2 is the loop area. 1 ∆ obtain from Eq. (17) that M1 (r) 0; that is, the Variation Zp in the loop input impedance related to dipole is situated in the free space. If the grid density is the effect of grid screen can be expressed in terms large, that is, if 1/2κ ∞, then, performing partial of ∆Y: integration in formula (17), we obtain that ω2µ 2 2 ∆ ∆ ∆ 0 S ∆ Z p Y Z p = ------2------Y or ------= ------, m L R0 p Y0 1( ) iI L ( ) M1 r = ----π---µ------ω---G r + z0h . 4 0 ω2µ 2 2 0 S where R0p = Y0 ------2------is the input resistance of the This expression corresponds to the Hertz vector of a L magnetic dipole with opposite current placed at the loop placed in the free space [6].

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 FIELD AND INPUT ADMITTANCE 685

In conclusion, it should be noted that the obtained 3. I. V. Lindell, Methods for Electromagnetic Field Analy- results can be used to calculate the parameters of grid sis (Clarendon, Oxford, 1992). electromagnetic screens designed for the protection of 4. I. V. Lindell, V. P. Akimov, and E. Alanen, IEEE Trans. physical installations and objects. Electromagn. Compat. EMC-28 (2), 107 (1986). 5. V. P. Akimov, in Waves and Diffraction-90 (Akad. Nauk REFERENCES SSSR, 1990), Vol. 3, pp. 101Ð104. 1. J. R. Wait, Antenna Theory (McGraw-Hill, New York, 6. G. T. Markov, Antennas (Gosénergoizdat, Moscow, 1969). 1960). 2. M. I. Kontorovich, M. I. Astrakhan, V. P. Akimov, and G. A. Fersman, Electrodynamics of Network Structures (Radio i Svyaz’, Moscow, 1987). Translated by A. Kondrat’ev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 686Ð688. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 88Ð93. Original Russian Text Copyright © 2000 by Morozov, Bugrova, Lipatov, Kharchevnikov, Kozintseva.

The Barrier Operation Mode of an Electric-Discharge Galathea Trap A. I. Morozov, A. I. Bugrova, A. S. Lipatov, V. K. Kharchevnikov, and M. V. Kozintseva Kurchatov Institute of Atomic Energy, State Scientiﬁc Center of the Russian Federation, Moscow, Russia Moscow Institute of Radio Engineering, Electronics, and Automatics, Moscow, Russia Received March 22, 2000

Abstract—Experimental plasma parameters in an electric-discharge galathea multipole trap operating in a barrier mode (“discharge with mantle”) at a magnetic ﬁeld strength below 20 Oe are reported. The local plasma τ parameters exhibit saturation with increasing cathode emission current. The plasma energy lifetime E and the “barrier” coefﬁcient β are estimated. © 2000 MAIK “Nauka/Interperiodica”.

Electric-discharge devices of a new type, called the The magnetic field strength became zero at a point multipole electric-discharge galathea traps (EDT-M), with the coordinates r0 = 17 cm, z = 0, and a maximum were suggested by Morozov and Bugrova. The first barrier field was observed at r1 = 21.7 cm, z = 0. The experimental results were obtained on a model device discharge was generated between an incandescent cath- with quadrupole poloidal magnetic field created by two ode placed in the zero-field position and the vacuum rings carrying currents passing in the same direction [1]. chamber walls. The gas was supplied via a tube to the It was found that the system may operate in two modes region near the cathode. With the usual anode voltage featuring the discharge with “plasmid” or with “man- (U = 200 V) applied in the absence of magnetic field, a tle.” Since the plasma parameters under otherwise “gas-jet” discharge was initiated between the cathode equal conditions were higher in the former mode, the and the gas inlet tube with the discharge current J2 close discharge “with plasmid” was paid more attention both to the cathode emission current [6]. During the subse- in [1] and in subsequent publications [2Ð4]. quent experiments, the cathode emission current was Later, it was understood that the character of the normalized to the J2 value. The measurements were plasma configuration was determined primarily by the typically conducted at J2 = 200 mA. cathode position [5]. When the cathode is situated in the region of zero magnetic field, the system operates in Once the magnetic field was switched on, a blue the regime with “mantle,” whereby the plasma (specif- glow was observed all around the rings, showing virtu- ically, electrons generating an electric field keeping ally no visible attenuation with increasing distance ions in the plasma region) is held in the trap by a mag- from the cathode in the azimuthal direction. The mea- netic barrier (Fig. 1a). Should the cathode be situated in surements were carried out in a steady-state regime at a a nonzero field, a discharge configuration with “plas- barrier field strength of H1 = 20 Oe. This field held only mid” is observed (Fig. 1b) in which electrons are held electrons, whereas ions were kept by the electric field mostly by the magnetic plugs. In this regime, only elec- of electrons. Selection of the field strength was deter- trons penetrating through the plugs may reach the mined by the wish to avoid overheating the coils in the region of capture (to leave this region shortly after). course of long-term operation. Figure 1 shows the computer-simulated particle tra- A. Integral characteristics. In the barrier operation jectories illustrating the dynamics of electron motion in mode, the discharge current J was virtually indepen- the two regimes. In accordance with the pattern of elec- 1 dent of the barrier field strength H1, being determined tron motion, the regime depicted in Fig. 1a was called [5] primarily by the cathode emission current (i.e., by the the “barrier” mode (B-mode) and that in Fig. 1b, the J value). As the gas supply rate grew from zero to a “plug” mode (P-mode). 2 certain level mú , the discharge current increased from Below, we report on the first results obtained for the * EDT-M system operation in the B-mode. The experi- zero to J1 ~ J2. ments were performed on a quadrupole EDT-M device B. Radial variation of the local plasma parameters. of the “string-bag” type described in [1]. The trap was ϕ The values of n, Te, and the ionization potential as operated with the working gas (xenon) flowing at a rate functions of r were measured with the aid of probes. of mú = 2 mg/s under conditions of constant pumping, The results of these measurements revealed a (gener- the residual air pressure in the chamber being ~(2Ð3) × ally natural) tendency of the above quantities to satura- 10−4 Torr. tion with increasing cathode emission current. The sat-

THE BARRIER OPERATION MODE 687

r, cm (a) (b) 24

16 2 1 1 14

12 2

0 2 4 6 8 10 0 2 4 6 8 10 Z, cm Fig. 1. Computer-simulated electron trajectories in an EDT-M trap operated in the (a) barrier and (b) plug modes: (1) cross section of plasma-surrounded coils; (2) cathode.

≈ uration was reached for J2* 200 and 300 mA in the P- kTe, eV and B-modes, respectively. 30 Figure 2 (dashed curves) shows the curves of n(r), ϕ 20 Te(r), and (r) in the plane z = 0 observed in the P-mode (U = 200 V; J = 200 mA; mú ≈ 2 mg/s) and (solid 1 10 ≈ curves) B-mode (U = 200 V; J1 = 300 mA; mú 2 mg/s). In the plug mode (see dashed curves), the plasma con- δ ≈ centrates within a narrow ( r ~ 2- to 3-cm) region at r ϕ 13.5 cm, where ions are kept in a potential well with a , V |ϕ | × 10 Ð3 depth of min ~ 70 eV and Te ~ 30 eV, n ~ 5 10 cm –20 at the center. In the barrier mode (solid curves), the –40 δ Ϸ plasma occupies a wider ( r ~ 15 cm) region at r –60 17.5 cm with a potential well depth of |ϕ | ~ 40 eV min –80 and T ~15 eV, n ~ 8 × 1010 cmÐ3 at the center. e max max –100 C. (r, z)-distribution of plasma parameters. We have n, 1010 cm–3 ϕ measured the values of n(r, z), Te(r, z), and (r, z) in the ≈ 9 barrier mode (U = 200 V, J1 = 300 mA, and mú 2 mg/s) with the aid of probes and determined the regions of 7 approximately equal parameters (Fig 3). As seen, the 5 × maximum parameters (Te max ~ 20 eV; nmax ~ 9 3 10 Ð3 |ϕ | 10 cm ; min max = 50 eV) are reached at the points 1 displaced from the plane z = 0 (where ions are kept in a × potential well with a depth of Te, 0 = 15 eV, ne, 0 = 8 12 14 16 18 20 22 24 26 10 Ð3 |ϕ | r, cm 10 cm , and min 0 = 40 eV). Fig. 2. Distribution of the plasma parameters in the central This circumstance, together with non-equipotential plane (z = 0) of an EDT-M trap operated in the (solid curves) character of the magnetic force lines, indicates that no barrier and (dashed curves) plug modes.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 688 MOROZOV et al.

r, cm be determined by different methods. The maximum β ∞ local value is ( loc)max = . A more adequate character- (a) (b) β 24 istic is provided by the barrier coefﬁcient 0 deﬁned by 2 the formula 22 III 2 8πnk(T + T ) β = ------e------i--. (1) 20 III II 0 2 I I H1 18 II The effective ion temperature can be estimated from 16 the known ϕ(r) and n(r) values by means of an “abel- 1 1 ization” procedure. Under “weak” assumptions, we 14 ≈ |ϕ | may assume kTi 1/3 min . Substituting Te = 20 eV, n = × 10 Ð3 |ϕ | 12 9 10 cm , and min = 50 eV into formula (1), we β ≈ obtain 0 0.3. 10 (b) The characteristic time of keeping the plasma τ energy (energy lifetime) E is evaluated by the formula 0 2 4 6 8 10 0 2 4 6 8 10 3 Z, cm --nκ(T + T )dV r, cm ∫2 e i τ = ------. (2) E J U 24 (c) 1 For the data of Fig. 3, this formula yields τ ≈ 30 µs, 22 E 2 which is close to the value calculated by classical for- 20 III mula. I The authors are grateful to V.V. Savel’ev for com- 18 II puter simulation of the electron trajectories. The work 16 was supported by the Ministry of Atomic Industry of 1 the Russian Federation. 14 12 REFERENCES 1. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and 10 V. K. Kharchevnikov, Pis’ma Zh. Tekh. Fiz. 18 (8), 1 (1992) [Sov. Tech. Phys. Lett. 18, 242 (1992)]. 0 2 4 6 8 10 2. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and Z, cm V. K. Kharchevnikov, Pis’ma Zh. Tekh. Fiz. 18 (24), 54 (1992) [Sov. Tech. Phys. Lett. 18, 815 (1992)]. Fig. 3. Distribution of the plasma parameters (a) ne, (b) Te, 3. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and ϕ and (c) over the (r, z) plane in the barrier operation mode: V. K. Kharchevnikov, Fiz. Plazmy 19, 1411 (1993) (a) I, 8.5 × 1010 cmÐ3 < n < 9 × 1010 cmÐ3; II, 8 × [Plasma Phys. Rep. 19, 741 (1993)]. 10 −3 10 Ð3 10 Ð3 10 cm < n < 8.5 × 10 cm ; III, 1 × 10 cm < n < 4. A. I. Morozov and V. V. Savel’ev, Usp. Fiz. Nauk 168 × 10 Ð3 8 10 cm ; (b) I, 15 eV < kTe < 20 eV; II, 5 eV < kTe < (11), 1153 (1998) [Phys. Usp. 41, 1049 (1998)]. |ϕ| 15 eV; III, kTe < 5 eV; (c) I, 45 eV < < 50 eV; II, 40 eV < 5. A. M. Bishaev, A. I. Bugrova, M. V. Kozintseva, et al., in |ϕ| < 45 eV; III, 10 eV < |ϕ| < 40 eV. Proceedings of the XXVI Zvenigorod Conference on Plasma Physics and CTR, Zvenigorod, 1999, p. 82. 6. A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and “Maxwellization” takes place because of a relatively V. K. Kharchevnikov, Pis’ma Zh. Tekh. Fiz. 17 (19), 29 short particle lifetime on the trap. (1991) [Sov. Tech. Phys. Lett. 17, 691 (1991)]. β τ β D. Estimates of and E. (a) The parameter for a quadrupole EDT-M device of the “string-bag” type can Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 689Ð693. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 94Ð102. Original Russian Text Copyright © 2000 by Reznikov, Polekhovskiœ.

Amorphous Shungite Carbon: A Natural Medium for the Formation of Fullerenes V. A. Reznikov and Yu. S. Polekhovskiœ St. Petersburg State University, St. Petersburg, 199164 Russia Received March 6, 2000

Abstract—A comparative analysis of data on the density, porosity, and intermolecular space of high-carbon shungites, graphite, glassy carbon, and C60 fullerite gave an estimate of the fullerene content in the shungite samples which agrees with the values obtained by electrochemical and polar solvent extraction methods. A low yield of fullerenes in the extracts obtained with nonpolar solvents is explained by the high polarity and large adsorption energy of fullerenes and related compounds. © 2000 MAIK “Nauka/Interperiodica”.

π + Ð The methods of C60 and C70 fullerene synthesis -states (C e ). Resonance excitation of a solvated C60 developed by the beginning of 1990s suggested that molecule in solution [11] must involve transitions into + these compounds may be present in carbon-containing states with the electron affinity energy EaC evaluated rocks. Indeed, shortly after the publication of reliable 0 [12] as 3EaC = 3.81 eV. Taking into account the vibra- data on the physicochemical properties of C60 and C70, Ð1 tional transitions in C60 (272, 496, and 776 cm [11, these fullerenes were found in high-carbon shungites + π [1, 2]. These shungites still remain the only natural 13]), this EaC value corresponds to the energy of the - objects known to contain aromatic carbon-containing band maximum in the spectra of solutions. According to the quantum-chemical calculations [14], the energy molecules. Up to now, no investigations were under- π taken to determine the concentration, molecular composition of the -band maximum for a non-associated position, and distribution of fullerenes and/or their C60 molecule was evaluated as 3.43 eV, which is derivatives in shungites (Sh) with various structures of 65 meV below the electron work function [15]. The difference coincides, to within 3 meV, with the energy of the carbon-containing substance (Csh). This informa- “breathing” oscillations of the highly symmetric C tion is of value for elucidating the nature of Csh and 60 identifying the component responsible for the medical molecule. Note also that 3.435 eV is the middle (arith- 0 and ecological properties of Sh. metic mean) value between EaC = 1.27 eV [10] and the Filippov et al. [3] pointed out that there must be a ionization potential of C60 (5.6 eV), which confirms the + relationship between the properties of shungite carbon above C60 model in the form of mutually bound C ions π Csh, its fullerene-like structure, and the content of Cn surrounded by a cloud of -electrons. fullerenes. Yushkin [4] suggested that a common mechanism may be responsible for the formation of C and The non-associated C60 molecule admits the anal- sh ogy between their delocalized π-electrons and the sÐd Cn. These hypotheses are justified provided that the concentration of fullerenes is related to the C structure electron states of small-size icosahedral metallic parti- sh cles [9, 13]. For example, the ionization potential and and correlated, at least in certain types of Sh, with the the surface plasmon energy of Ag [16] virtually coin- content of completely amorphized carbon phase [3, 5], 13 the latter being the most probable initial material for the cide with the corresponding values for C60, while the C synthesis. maximum of the plasma absorption band of Ag6 (consid- n ered as an elementary particle with metal bonds [17]) Previously [6], we have presented evidence of a coincides with the calculated energy of the π-band commercially significant C content (about 1%) in n maximum of the non-associated C . A difference Sh-3. The purpose of this work was to perform a com- 60 parative analysis of some macrophysical properties of between C60 and small metallic particles consists in (i) the form of the surface electron density distribution Csh, its closest structural analog—glassy carbon (GC) and (ii) the possibility of shape changing in the C60 car- [7, 8], and C60 fullerene, with a view to estimating the limiting content of C and elucidating a model of meta- bon unit cell. The scattering of electromagnetic waves n on these objects is determined by collective oscillations morphism in the shungite carbon structure. of the conduction electrons [18], while the plasma oscil- For a carbon unit cell size of 1.0 Å [9] and a covalent lations divide into πÐσ- and π-states [9, 15, 19]. The 0 atomic radius of C = 0.77 Å [10], carbon atoms in C60 π-states being localized upon adsorption on an electro- + can be considered as C ions surrounded by delocalized neutral surface, the C60 molecule (as a solid particle)

690 REZNIKOV, POLEKHOVSKIŒ of molecules. Therefore, the interaction between low- energy states and π-states of the acceptor molecule must be a fundamental energy characteristic of a C60 crystal. For the calculated mean energy of 3.435 eV, an equidistant value with respect to EaC60 is 1.865 eV corresponding to the bandgap of C60 fullerite [9, 13]. This implies that the 2C60 dimer is an element of self-simi- 4 larity in the fullerite structure, which may pass as a whole into solution to perform as the self-similarity 3 element in the corresponding fractal aggregates [13]. 2 In a non-associated C60 molecule, the electron shell 1 thickness can be considered as corresponding to the 300 400 500 nm thermodynamic equilibrium between the orbital and covalent radii of carbon atoms (1.39 Å). In this case, the Fig. 1. Typical spectra of C60: (1, 2) luminescence (curve 2 ratio of the volume occupied by a carbon atom to the illustrates a time variation of the emission spectrum); Х volume of a C60 molecule is 0.78. The same value is (3, 4) luminescence excitation at (3) 410 and (4) 497 nm. ρ obtained for the density ratio of C60 fullerite ( = 1.697 g/cm3) and graphite (Х2.25 g/cm3) calculated loses its metallic properties. In this case, an excited C60 taking into account a difference in CÐC bond lengths particle with electron-acceptor properties must contain [5, 13]. For an equivalent surface filling of the entire ρ a bound electron-hole pair. Indeed, the luminescence C60 volume, the density ( ) of fullerite increases to ρ spectra of C60 absorbed in a microporous glass or reach the value for graphite. Therefore, the volume of adsorbed on the surface of polymer molecules exhibit the interlayer space in graphite is equivalent to the bands peaked at 365, 368, and 410, 433 nm shifted intermolecular volume in fullerite. A special feature of from the 361 and 407 nm bands in accordance with the the C60 structure is the presence of pentahedra uni- frequencies of the characteristic vibrational states of formly distributed over the carbon cell surface, C60 (Fig. 1). whereby each carbon atom is shared by two hexagons and one pentagon. This implies that the pentahedra are The thermal stability of fullerenes, together with formed as a result of self-organization in the excited C60 high electronegativity and polarizability of C60 [15, 19], molecules. allows this molecule to be considered as an adatom of large radius possessing the properties of elements of the A collective character of the self-organization of carbon molecules into an aromatic structure is indi- VII group. For example, C60 is, like iodine, soluble in nonpolar solvents and has the values of ionization cated by coinciding energies of formation of the C60 molecules and πÐσ-plasmons [13, 14]. It was also energy and Ea close to those of At [10]. pointed out [13, 14] that the synthesis of fullerenes Thus, C60 may exhibit, depending on the environ- from C-clusters is an effective process provided that an ment, both the individual molecular properties and the energy released upon the contact of clusters amounts to properties of metallic or semiconducting particles, or a few eV, which is just the energy characteristic of the even halogen atoms. This accounts for the ability of C60 electron affinity of linear carbon molecules [21]. There- to form compounds with various types of chemical fore, the excited amorphized graphite-like clusters and bonds, for the large energies of C60 adsorption on elec- chain molecules in a Csh structure may form aromatic tron-donor surfaces, and for the effective solid-state fullerene-like molecules. This is confirmed by a pyro- interaction of C60 with ionic and/or molecular com- lytic synthesis of fullerenes from chain hydrocarbons pounds [20]. under conditions of weak interaction with environment A 10% decrease in the intermolecular distance in in an oxygen-free atmosphere. C60 crystals as compared to the interlayer spacing in The samples of Sh-1 and GC exhibit, in addition to graphite (3.35–3.36 Å) suggests that the πÐπ exchange similar X-ray-amorphous structures and coinciding interaction in this system is possible as well. A high features in the 400Ð1600 cmÐ1 (see Fig. 2, curves 1 and 2) mobility of C60 molecules near the position of equilib- range of the IR absorption spectra, close values of the rium in the crystal lattice [13] allows us to assume that, strength characteristics, thermal expansion coefficients, when the carbon cell shifts from the equilibrium posi- heat capacity, and electric conductivity [5]. On the tion, one of the neighboring molecules may act as a other hand, the thermal conductivity values differ donor, and the other, as an acceptor of electron density. 6−7 times. The low thermal conductivity of Sh-1 as A shift of the π-states of the donor molecule to a low- compared to that of GC, with close values of most mac- energy level of the acceptor molecule corresponds to rophysical characteristics, implies that the interphase EaC60 = 2.65 eV [13]. This model, in which the donor boundaries may contain a considerable amount of heat- and acceptor are equivalent, can be applied to any pair scattering centers. A middle (arithmetic mean) value

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 AMORPHOUS SHUNGITE CARBON 691 between the densities of graphite and C60 fullerite at 290 K is 0.02 g/cm3 greater than the density of GC. This difference is related to a fully amorphized phase present in the GC structure [5], the density of which may be 1 comparable with that of soot (ρ = 2.1Ð2.18 g/cm3). Thus, the GC structure can be considered as a superposition of graphite-like packets (GLPs) with ultimate geome- 2 tries of bent hexagonal layers. The ratio of limiting values of the interplanar spacing in GLPs (3.65 and 3.41 Å for Csh with globular and sheet morphology, respectively [3]) coincides with the density ratio of the corresponding types of Sh-1 (1.83Ð 3 1.96 g/cm3), which indicates that these packets are the main structural units of Csh. A 2Ð7% increase in the interlayer spacing in GLP with respect to that in crystalline graphite corresponds to a decrease in the density 4 down to ρ = 2.1Ð2.2 g/cm3. This estimate does not take into account a possible change in the density within 2−4% due to a difference in interatomic distances 1500 1250 1000 750 500 between the Csh and graphite structures, since the bend- cm ing and small dimensions of GLPs necessarily imply the presence of carbon vacancies. For a C structure Fig. 2. IR absorption spectra: (1) glassy carbon; (2) shungite sh (Sh-1); (3) C60 in KCl matrix; (4) solid amorphized porosity of 10Ð12% [5], the average density of GLPs C60-methylpyrrolidone phase. correspond to the ρ value of Sh-1. A middle value between the densities of GLPs and C60 fullerite (1.9−1.95 g/cm3) falls in the range of ρ values of Sh-1. absence of any pronounced GLP orientation. An evi- For an average C-globule size of about 10 nm, the free dence for the glassy structure of Csh is provided by the volume of C-globules coincides with that of C60 mole- coincidence of the Sh-1 density with that (ρ = cules provided that the globules form a four-layer 1.9 g/cm3) of the melt of a mixture comprising chain packet, which accounts for the fullerene-like structure and aromatic C-clusters [24]. In this context, the glob- of Csh [3, 4, 22]. A smaller number of features in the IR ular structure can be considered as resulting from an spectrum of Sh-1 in comparison with the spectrum of ordered mass rearrangement within a closed volume, GC (see Fig. 2, curves 1 and 2) is indicative of a higher beginning with nucleation of the graphite-like networks degree of order in the shungite carbon structure. in the field of uniformly distributed catalytically active ρ 3 centers. A decrease in the number of crystallization The density of C60 was estimated as = 2.03 g/cm [19]. Determined within the framework of the centers corresponds to an increase in the size of GLPs fullerene-like model, the interglobular porosity must and, hence, of the C-globules. fall within 30Ð35%, while the total porosity must be In this model, the odd number of layers in the packet about 40%. A discrepancy between this estimate and is a consequence of the primary monolayer nucleation; the value (10Ð12%) presented above is eliminated if we however, this does not exclude the possible nucleation assume that (i) a soot-like phase and/or Cn carbon phase of two layers and an even number of layers in the is present in the inter- and intraglobular space and (ii) packet. Thus, GLPs can be considered as the elements the GLPs are for the most part intercalated [8]. The lat- of self-similarity in the fractal structure of Csh. Assum- ter assumption, like the hypothesis of a fullerene-like ing the precipitation-like accumulation of C-globules, the Csh structure, is based upon the coincidence of the main fractal dimensionality must fall within 2.3Ð2.5 D [25]. spectral features observed in the 400Ð1500 cmÐ5 range The interglobular porosity of this structure must be not for Sh-1 and C60 dimers in a KCl matrix [20] (Fig. 2, lower than 20%. In our opinion, it is the total porosity curves 2 and 3). The IR spectrum of Sh-1 displays a level of Csh that is the main evidence in favor of the Raman band of graphite at 1575Ð1580 cmÐ1; the fact recrystallization mechanism. The crystallization model that this band is allowed in the IR spectrum is also of Csh formation is confirmed by a filament (whisker) related to the intercalation of monovalent electron- crystal structure of some shungite carbon samples [26]. donor carbon atoms [23]. Within the framework of the vaporÐliquidÐcrystal Considering GLPs as the walls of globules cannot mechanism [27], the filament crystal growth begins in explain the fact that the number of layers in the GLPs is a quasiliquid phase supplied with the mobile C-clus- odd before and even after the heat treatment of Sh sam- ters, which makes possible the fullerene-like structure ples [3]. A fractal character of the Csh structure implies formation [28]. The growth of filament crystals require a mixed type of the interaction between GLPs and the a sufficient free volume and a small number of primary

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 692 REZNIKOV, POLEKHOVSKIŒ crystallization centers. The recrystallization process ature maxima are attributed to changes in the vibra- may not involve some part of the C-clusters. Thus, the tional mobility of C60 and C70 molecules, and the high- main difference of the GLP formation in Sh is the temperature maxima, to changes in the structure of C60 closed space in which the process occurs, while the free fullerite with a large admixture of C70 [32]. volume formation provides conditions for the second- The calculated estimates of the content of C in Sh ary recrystallization of weakly bound C-clusters into n the final aromatic structures. are confirmed by data on the extraction of thermoacti- vated Sh-1 samples with polar solvents capable of The fact that the ρ values of GC and Sh-1 can be forming compounds containing donorÐacceptor bonds obtained within the framework of the same GLP model with Cn. The samples obtained upon the thermal implies that a difference between the calculated and removal of hydrocarbons and solvent from the measured densities of Sh-1 (0.03Ð0.07%) is related to ρ extracted bitumous phase exhibited crystallization of the presence of either a C-phase with = 1.6Ð carbonaceous particles with a cubic and/or dendrite 1.66 g/cm3 characteristic of the GC samples upon a crystal habit. The X-ray diffraction data allowed these high-temperature treatment [5] or fullerenes. The use of particles to be identified with C60Ð70. The spectral fea- a macrophysical value for evaluating the total density tures of the extracted carbon particles and their solu- of statistically distributed C molecules or their small- n tions also corresponded to the C60Ð70 structure. The esti- size aggregates is possible because the inter-pore space mated mass of the isolated crystalline particles varied in Csh is definitely smaller than the intermolecular vol- within 1.5Ð2.0 wt % of the Sh-1 sample, depending on ume in fullerite and in the adsorbed state of Cn. The the extraction conditions. The experimental data were existing notions about the shungite substance forma- indicative of the possible technological efficiency of tion [3], as well as the presence of organic compounds, using natural fullerenes extracted from shungites in microorganisms, and water in the shungite samples, are medical and ecological applications. inconsistent with the thermal mechanism of low-density C phase formation. sh REFERENCES As follows from the above considerations, the concentration of C in Sh samples with most disordered 1. P. R. Buseck, S. J. Tsipurski, and R. Hettich, Science n 257, 215 (1992). structure is estimated at 3Ð5 wt % of C . This estimate sh 2. S. V. Kholodkevich, A. V. Bekhrenev, V. K. Donchenko, is confirmed by observation of a resolved band at et al., Dokl. Akad. Nauk 330 (3), 340 (1993). Ð1 1180 cm in the IR spectrum of Sh-1 (see Fig. 2, 3. M. M. Filippov, A. I. Golubev, P. V. Medvedev, et al., curve 2), which is a characteristic band in the spectrum Organic Substances of Shungite-Bearing Rocks of Kare- of C60. A similar band is observed in the spectrum of a lia (Petrozavodsk, 1994). solid amorphized C60Ðmethylpyrrolidone phase (see 4. N. P. Yushkin, Dokl. Akad. Nauk 337 (6), 800 (1994). Fig. 2, curve 4). At the same time, the spectra of this 5. Shungites of Karelia and Ways to Their Complex Use, phase and Sh-1 contain, in contrast to the spectrum of ed. by V. A. Sokolov and Yu. K. Kalinin (Petrozavodsk, C60ÐKCl samples [29], no characteristic bands in the 1976). region of 1420Ð1430 cmÐ1, which is most probably 6. V. A. Reznikov, Yu. S. Polekhovskiœ, and V. E. Kholmo- explained by the interaction of π-states in the major gorov, in Carbonaceous Formations in Geologic History, Abstracts of Papers (Petrozavodsk, 1998), pp. 71Ð72. part of Cn with the matrix components. The common features observed in the IR spectra of Sh-1, GC, 7. Yu. K. Kalinin, Zap. Vses. Mineral. OÐva 119 (5), 1 − (1990). C60 KClx, and ultradisperse graphite particles [30] reflect the presence of like structural groups of carbon 8. S. V. Kholodkevich and V. V. Pobochiœ, Pis’ma Zh. Tekh. atoms forming bent hexagonal layers [31]. Fiz. 20 (3), 22 (1994) [Tech. Phys. Lett. 20, 99 (1994)]. 9. S. V. Kozyrev and V. V. Rotkin, Fiz. Tekh. Poluprovodn. According to the above estimates, there are 30Ð (St. Petersburg) 27 (9), 1409 (1993) [Semiconductors 40 Cn molecules per globule, which allows the associa- 27, 777 (1993)]. tive arrangement of these molecules in the free space of 10. Properties of Inorganic Compounds, ed. by A. I. Efimov Csh. This hypothesis is consistent with the calorimetric (Khimiya, Leningrad, 1983). effects observed in the temperature intervals near 11. V. P. Belousov, V. P. Budtov, O. B. Danilov, et al., Opt. 190−200, 235Ð245, and 280 K. For the samples of Sh-3 Zh. 64 (12), 3 (1997) [J. Opt. Technol. 64, 1081 (1997)]. dehydrated in vacuum for 24 h at 280 K, the high-tem- 12. V. A. Voll, Fiz. Tekh. Poluprovodn. (St. Petersburg) 29 perature maximum is observed 5Ð7 K below, and the (10), 2071 (1995) [Semiconductors 29, 1081 (1995)]. low-temperature maximum, 5Ð7 K above the corre- 13. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 165 sponding peaks for Sh-1. The high-temperature max- (9), 977 (1995) [Phys. Usp. 38, 935 (1995)]. ima are usually registered during the first DSC runs in 14. A. L. Chistyakov and I. V. Stankevich, Izv. Akad. Nauk, the heating modes. These features allow us to assign the Ser. Khim., No. 9, 1680 (1995). observed calorimetric features to processes in the 15. V. V. Afrosimov, A. A. Basalaev, and M. N. Panov, Zh. ordered molecular aggregates with various relative con- Tekh. Fiz. 66 (5), 10 (1996) [Tech. Phys. 41, 412 centrations of C60 and C70 molecules. The low-temper- (1996)].

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 AMORPHOUS SHUNGITE CARBON 693

16. Yu. I. Petrov, Physics of Small Particles (Nauka, Mos- 25. J. Freder, Fractals (Plenum, New York, 1988; Mir, Mos- cow, 1982). cow, 1991). 17. V. A. Reznikov and A. V. Struts, Opt. Spektrosk. 73 (2), 26. V. V. Kovalevski, A. N. Saphronov, and Ju. A. Markov- 355 (1992) [Opt. Spectrosc. 73, 207 (1992)]. ski, Mol. Mater. 8, 21 (1996). 18. A. V. Klyuchnik, Yu. E. Lozovik, and A. V. Solodov, Zh. 27. E. I. Givargizov, Growth of Filament and Platelike Crys- Tekh. Fiz. 65 (6), 203 (1995) [Tech. Phys. 40, 633 tals from Vapors (Moscow, 1977). (1995)]. 28. V. A. Reznikov, T. É. Kekhva, and B. T. Plachenov, 19. Yu. M. Shul’ga and A. S. Lobach, Fiz. Tverd. Tela Pis’ma Zh. Tekh. Fiz. 16 (22), 1 (1990) [Sov. Tech. Phys. (St. Petersburg) 35 (4), 1092 (1993) [Phys. Solid State Lett. 16, 847 (1990)]. 35, 557 (1993)]. 29. A. A. Sukhanov and V. A. Reznikov, Pis’ma Zh. Tekh. 20. V. A. Reznikov and A. A. Sukhanov, Pis’ma Zh. Tekh. Fiz. 25 (9), 56 (1999) [Tech. Phys. Lett. 25, 363 (1999)]. Fiz. 25 (8), 45 (1999) [Tech. Phys. Lett. 25, 313 (1999)]. 30. Yu. S. Polekhovskiœ and V. A. Reznikov, in Ore Forma- 21. L. N. Sidorov and O. V. Boltalina, Soros. Obrazov. Zh. tion and Localization in Earth’s Crust (St. Petersburg 11, 35 (1997). State. Univ., St. Petersburg, 1999), pp. 123Ð147. 22. V. V. Kovalevskiœ, Zh. Neorg. Khim. 39 (1), 31 (1994). 31. V. M. Loktev, Fiz. Nizk. Temp. 18 (3), 217 (1992) [Sov. J. Low Temp. Phys. 18, 149 (1992)]. 23. V. I. Ivanov-Omskiœ, A. A. Andreev, and G. S. Frolova, Fiz. Tekh. Poluprovodn. (St. Petersburg) 33 (5), 608 32. K. Kniaz, J. E. Fisher, L. A. Girifallo, et al., Solid State (1999) [Semiconductors 33, 569 (1999)]. Commun. 96, 739 (1995). 24. V. A. Polukhin and E. A. Kibanova, Zh. Fiz. Khim. 73 (3), 494 (1999). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 694Ð697. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 15, 2000, pp. 103Ð110. Original Russian Text Copyright © 2000 by Ginzburg, Zotova, Sergeev, Rozental’, Yalandin.

Pulsed EHF Superradiance Due to the Stimulated Scattering of a High-Power Pump Wave by a Counterpropagating Electron Bunch N. S. Ginzburg, I. V. Zotova, A. S. Sergeev, R. M. Rozental’, and M. I. Yalandin Institute of Applied Physics, Russian Academy of Sciences, Nizhniœ Novgorod, 603600 Russia Institute of Electrophysics, Ural Division, Russian Academy of Sciences, Yekaterinburg, Russia e-mail: [email protected] Received February 4, 2000

Abstract—Stimulated scattering of a pump wave by an extended electron bunch counterpropagating with a relativistic translational velocity is studied theoretically. It is demonstrated that the realization of a superradiance regime ensures coherent emission of a short electromagnetic pulse with a frequency much higher than the pump frequency. © 2000 MAIK “Nauka/Interperiodica”.

1. In [1Ð5], pulsed superradiance (SR) due to differ- the scattered H-waves are represented in terms of the ent elementary mechanisms was theoretically studied. vector potentials: The process is essentially the coherent emission of an electromagnetic pulse by an electron bunch as a result A , ()zt, i()ω, thÐ , z of electron clustering. Although the bunch extends over is []∇ ψ , is is Ais, =Re------⊥ is, z0 e ,(1) several tens of wavelengths, the different parts emit in k⊥is, phase because the group velocity of the wave differs from the translational velocity of electrons so that the where the subscripts “i” and “s” refer to the incident wave slips relative to the particles. Subnanosecond SR and the scattered wave, respectively; Ai, s(z, t) are pulses have been experimentally generated in the ψ microwave region [6Ð9] using the stimulated emission slowly varying wave amplitudes; i, s = φ inis, obtained by different methods including the cyclotron, J(k⊥ r)e are the membrane functions of a Cherenkov, and undulator mechanisms. In connection, nis, is, round waveguide; k⊥ , h are the transverse and lon- it is interesting to explore the possibility of producing is, i, s ultrashort pulses during the stimulated Compton scat- gitudinal wavenumbers, respectively; and ni, s are the tering of a high-power pump wave by a counterpropa- azimuthal numbers. Assume that the condition of com- gating electron bunch. At relativistic electron veloci- bination-frequency wave matching is fulfilled: ties, the frequency of the scattered radiation must be much higher (due to the Doppler effect) than that of the ωÐ hν|| ≈Ωω Ð h ν|| =.(2) pump wave. s s i i This paper presents a theoretical analysis of the SR At the same time, we suppose that the incident and effect due to stimulated scattering from an electron scattered waves are well out of the cyclotron resonance bunch propagating along a uniform magnetic field in a with the electrons: smooth waveguide. Two simulation models are consid- ωνω ӷ π ered. The first model rests on the method of an averaged is,Ðhis,|| ÐHT2,(3) ponderomotive force used to describe the motion of ω γ electrons in the field of two electromagnetic waves. The where H = eH0/m0c 0 is the gyromagnetic frequency, second model employs direct numerical simulation γ ν22 1/2 ν using the particle-in-cell KARAT code. 0 = (1 Ð 0 /c ) is the relativistic mass factor, || is the longitudinal electron velocity, and T is the characteris- 2. Let us examine the scattering of a pump wave by tic interaction time. an annular electron bunch of radius Rb and length b. The bunch propagates along a uniform transport Assume that As(z, t) is a slowly varying function of magnetic field H0 = H0z0 in a smooth circular the longitudinal coordinate z and the time t and that the waveguide of radius R. The fields of the incident and pump wave amplitude is fixed. Then, the stimulated

PULSED EHF SUPERRADIANCE DUE TO THE STIMULATED SCATTERING 695 scattering of the pump wave by the bunch under condi- f, GHz tions (2) and (3) obeys the equations 250 2π  ∂ 1 ∂  1 Ðiθ 200 + ------a = if (t Ð z/ν||)k a GI-- e dθ , ∂z v ∂t s c i π ∫ 0 gr 150 0 (4) ω 2 s  ∂ 1 ∂  2 iθ ---- θ µ { * } ω 100  +  = kc Im asai G*e , i ∂z ν||∂t 50 γ 2 where ai, s = eAi, s/2m0 0c are the normalized wave θ ω amplitudes; = ct Ð hcz is the time-dependent phase angle of electrons with respect to the combination-fre- –10 0 10 20 30 40 ω ω ω ω quency wave; kc = c/c, c = s Ð i is the combination h, 1/cm frequency; h = h Ð h is the longitudinal wavenumber c s i Fig. 1. Dispersion relation for the waveguide and the elec- of the combination-frequency wave; tron beam.

Ω J (k⊥ R )J (k⊥ R ) ni + 1 i b ns + 1 s b , MW G = -β---------Ω------ω------P ||  + H 20 15 J (k⊥ R )J (k⊥ R )  ( )φ ni Ð 1 i b ns Ð 1 s b i ns Ð ni + ------Ω------ω------e Ð H  10

µ γ Ð2βÐ3 5 is the electronÐwave coupling coefficient; = 0 || is ν the clustering parameter; gr is the group velocity of the 0 scattered wave; 240 260 280 300 320 340 360 380 t, ps eJ0 1 I = ------Fig. 2. The plot of scattered power vs. time. 3 γ 2 m0c 2 0hskc R Ns is the current parameter; J0 is the beam current; Ns = wave with a power of up to 60 MW. A waveguide trans- 2 2 (1 Ð n /ν2)J (ν) is the scattered-wave parameter with former converts the output wave into an H11 pump wave s ns that is transmitted to a waveguide carrying a subnano- ν = k⊥ R being a root of the J' (x) = 0 function; and f(x) s ns second electron bunch from the other accelerator. The is an unperturbed profile of the bunch. Introducing new transmission line comprises two quasioptical mirrors. ζ τ β independent variables = Ckc z, = (1/ gr Ð The parameters of the forthcoming experiment are as −1 follows: bunch current, 200 A; particle energy, 250 keV; 1/β||) Cω (t Ð z/ν||) brings Eqs. (4) to the form c pulse duration, 250 ps; bunch radius, 0.2 cm; transport π 2 2 magnetic field, 13 kOe; waveguide radius, 0.3 cm. The ∂ ∂ θ ∂ θ θ a a (τ) i Ði θ { i } scattering is expected to occur in a wave with the same ----- + ----- = f -- ∫ e d 0 ------= Im ae (5) ∂ζ ∂τ π ∂ζ2 structure as that of the pump wave. 0 with the initial and boundary conditions The diameter of the waveguide for the scattering θ θ θ should be close to the cut-off value for the pump wave. a τ = a0, ζ = 0 + rcos 0, = 0 = 0 This decreases the group velocity of the H11 pump wave ∂θ θ ∈ [ , π ) so that its transverse electric field becomes stronger for ------= 0, 0 0 2 , ∂ζ a given power flux. (That is why the E01 wave is con- ζ = 0 verted into the H11 wave.) As a result, electrons acquire µ 2 µ | |2 1/3 where a = as ai* G*/C , C = ( I aiG ) is the gain, a higher oscillation velocity. For example, if the and the parameter r describes a small initial modulation waveguide radius is reduced from 0.4 to 0.3 cm, then β i of the bunch density. the group velocity gr of the pump wave drops to about The simulation was carried out for the values of β i ≈ parameters close to those selected for the forthcoming gr 0.5 so that the electron oscillation velocity rises experiment with two synchronized RADAN accelera- by a factor almost as large as 1.5 to reach β⊥ ≈ 0.13. The tors [7Ð9] producing beams with nanosecond and sub- dispersion relation plotted in Fig. 1 shows that the fre- nanosecond pulse durations. One of the beams is used quency of the scattered wave is about 170 GHz for the in a backward-wave oscillator to generate a 38-GHz E01 parameters selected. Figure 2 presents the results of

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 696 GINZBURG et al.

px /m0c

0.4

–0.4

0 10 20 30 z, cm

Fig. 3. Phase portrait of an electron bunch oscillating in a pump wave field. simulation based on Eqs. (5). The scattered power is P, MW shown as a function of time for an interaction space 4.5 length of 20 cm and a bunch duration of 250 ps. It is (a) seen that duration of the scattered pulse is of the order of 100 ps and the peak power is 17 MW. 3. The possibility of superradiance due to the stim- 3.0 ulated wave scattering was also supported by the particle-in-cell simulation using the KARAT code. The numerical experiment was carried out within the framework of a simplified plane two-dimensional model. The 1.5 process was assumed to take place in a planar waveguide with a 38-GHz H01 pump wave. The computation was performed for a pumping power density of 100Ð150 MW/cm, a beam current of 200Ð500 A/cm, an 0 electron energy of 200 keV, a transport magnetic field 1.6 2.0 2.4 2.8 β of 12 kOe, and a pump wave group velocity ( gr) of t, ns about 0.5. These values correspond to those used with the first model. By contrast, the bunch duration was (b) increased to 700 ps. Figure 3 presents the phase portrait of a bunch oscillating in a pump field. Figure 4 displays the power waveform of the pulse and the spectrum of scattered , a.u. f radiation. It is seen that the spectrum is centered at S 150 GHz and the peak power is about 1.5 MW. Also note that the peak power is lower than that obtained with the first model. The reason is that the second model is more realistic: it takes into account the scatter in electron positions, the effect of the space charge, etc. 100 200 300 To sum up, both models confirm the possibility of f, GHz superradiance due to the stimulated wave scattering. In Fig. 4. Pulsed scattered radiation: (a) power waveform for conclusion, note that the mechanism concerned works z = 0; (b) power density spectrum. The data are obtained by in the submillimeter band as well. Specifically, if the a plane-model particle-in-cell simulation using the KARAT bunch energy is raised to 300 keV and pump frequency code.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 PULSED EHF SUPERRADIANCE DUE TO THE STIMULATED SCATTERING 697 is raised to 70 GHz, then the frequency of the scattered 4. V. V. Zheleznyakov, V. V. Kocharovskiœ, and Vl. V. Ko- radiation will fall in the 400Ð500 GHz range. charovskiœ, Izv. Vyssh. Uchebn. Zaved., Radioﬁz. 29 (9), This study was supported by the Russian Foun- 1095 (1986). dation for Basic Research (project no. 98-02-17308) 5. N. S. Ginzburg, Yu. V. Novozhilova, and A. S. Sergeev, and by the State Interdisciplinary Scientiﬁc-Techno- Pis’ma Zh. Tekh. Fiz. 22 (3), 39 (1996) [Tech. Phys. logical Program, “Microwave Physics” section, project Lett. 22, 359 (1996)]. no. 1.13. 6. N. S. Ginzburg, A. S. Sergeev, I. V. Zotova, et al., Phys. Rev. Lett. 78 (12), 2365 (1997). 7. D. A. Jaroszynski, P. Chaix, and N. Piovella, Phys. Rev. REFERENCES Lett. 78 (9), 1699 (1997). 1. R. Bonifacio, N. Piovella, and B. W. J. McNeil, Phys. 8. N. S. Ginsburg, A. S. Sergeev, Yu. V. Novozhilova, et al., Rev. A 44, 3441 (1991). Nucl. Instrum. Methods Phys. Res. A 393, 352 (1997). 2. N. S. Ginzburg, Pis’ma Zh. Tekh. Fiz. 14 (5), 440 (1988) 9. N. S. Ginzburg, Yu. V. Novozhilova, A. S. Sergeev, et al., [Sov. Tech. Phys. Lett. 14, 197 (1988)]. Phys. Rev. E 60, 3297 (1999). 3. N. S. Ginzburg, I. V. Zotova, and A. S. Sergeev, Pis’ma Zh. Éksp. Teor. Fiz. 60 (7), 501 (1994) [JETP Lett. 60, 513 (1994)]. Translated by A. Sharshakov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 698Ð700. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 1Ð7. Original Russian Text Copyright © 2000 by Antonov, Buznikov, Rakhmanov.

Remagnetization of a Circularly Anisotropic Amorphous Wire in an Alternating Magnetic Field A. S. Antonov, N. A. Buznikov, and A. L. Rakhmanov Institute of Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] Received February 17, 2000

Abstract—Remagnetization of an amorphous ferromagnetic wire with circular anisotropy in an alternating longitudinal magnetic field was theoretically studied within the framework of a quasistationary approximation. A frequency spectrum of the emf generated in a probing coil wound on the wire was determined, and analytical expressions describing the dependence of the emf amplitude on the alternating magnetic field amplitude H0 and the constant magnetic bias field value He were derived. The results can be applied to the development of weak magnetic field sensors. © 2000 MAIK “Nauka/Interperiodica”.

The interest of researchers in studying magnetically effect in the wire is small and the alternating magnetic soft amorphous wires is related to their unique physical field is homogeneously distributed in the wire cross- properties and good prospects for various technological section. The maximum frequency for which this applications. Investigations performed in the past approximation is valid can be estimated in the follow- decade showed evidence that the high-frequency imped- ing way. A product of the specific conductivity of the ance of these wires may exhibit a very large change in amorphous film σ by the characteristic magnetic per- response to weak external magnetic fields, which was µ˜ 19 Ð1 called the giant magnetic impedance effect [1Ð4]. This meability usually does not exceed 10 s [6]. Then, effect is now extensively investigated in the context of the skin layer thickness δ = c(2πσµ˜ ω)Ð1/2 in a wire with obtaining highly sensitive magnetic field sensors as d ≅ 10 µm will exceed the diameter for the frequencies well as magnetic recording media and devices. As for f = ω/2π below 10 MHz. the magnetic field sensors, of considerable interest is the study of nonlinear remagnetization processes in As is well known, the distribution of the easy aniso- amorphous ferromagnetic wires [5]. Indeed, use of the tropic magnetization axes in amorphous ferromagnetic nonlinear effects may eliminate some undesirable wires is controlled for the most part by the magneto- phenomena such as hysteresis, metastable domain striction effect [5]. The magnetic properties of non- states, etc. spun amorphous wires with negative magnetostriction In this work, we have theoretically studied the constants are descried within the framework of a model remagnetization of an amorphous ferromagnetic wire assuming the presence of two regions in the wire: a cen- with circular anisotropy in an alternating longitudinal tral region (core) with homogeneous longitudinal mag- magnetic field. A frequency spectrum of the emf gener- netization and an outer shell with a circular direction of ated in a probing coil wound on the wire was deter- the magnetic anisotropy [1, 2, 5, 7Ð9]. In a wire with mined, and it was established that all harmonics are d ≅ 10 µm, the core size is small and the contribution of present in the spectrum provided that the amplitude of this region to the total free energy of the wire can be the alternating magnetic field exceeds a certain thresh- neglected. For the sake of simplicity, we will addition- old level. The results confirmed a high sensitivity of the ally assume that the wire is single-domain, which is harmonic amplitudes vk (k is the harmonic number) with respect to the constant magnetic bias field. valid for sufficiently short samples (with a length not exceeding several centimeters) [5, 10, 11] in the shell of Consider an amorphous ferromagnetic wire with a which no equilibrium domain structure can be realized diameter d in an alternating longitudinal magnetic field because of the low magnetostriction constant. ˜ H = He + H , where He is the constant magnetic bias ˜ ω Under these assumptions, the free energy of the wire field and H = H0sin( t). The harmonic alternating is the sum of two terms representing the energies of component H˜ produces a change in the wire magneti- magnetic anisotropy and the magnetic moment in the zation with time and, in accordance with the Faraday applied longitudinal field. Since the alternating field law, generates a circular emf Vϕ in a coil wound around frequency is sufficiently small, the remagnetization the wire. We will assume for simplicity that the skin process can be described within the framework of a

REMAGNETIZATION OF A CIRCULARLY ANISOTROPIC AMORPHOUS WIRE 699 quasistationary approximation [12, 13], where the free π 2 ω vkc/ d NM energy density U has the following form: 4 ( ) 2 θ ( ˜ ) θ U = MHa/2 sin Ð M He + H sin . (1) Here, M is the saturation magnetization, H is the aniso- θ a 3 tropic field, and is the angle between the magnetic 3 moment vector and the easy axis (circular direction). The emf Vϕ generated in the probing coil is given by the formula 2 2 d/2 2 2 1 8π N dMz π d N dMz V ϕ = Ð------rdr = Ð------, (2) c ∫ dt c dt 0 1 4 2 θ where N is the number of turns and Mz = Msin is the longitudinal magnetization component. The latter quantity can be determined from the condition of min- 0 imum free energy: 1 10 ( )[ (ω )] ˜ < H0/Ha  M/Ha He + H0 sin t , He + H Ha, Mz =  (3) Fig. 1. The plots of harmonic amplitudes v versus alternat- ±M, H + H˜ > H . k e a ing magnetic field amplitude H0 calculated for He/Ha = 0.5 As can be seen from Eqs. (2) and (3), small ampli- for k = 1 (1); 2 (2); 3 (3); 4 (4). tudes of the alternating field H0 and the anisotropic field (Ha < He) lead to a linear response signal in the coil, Figure 1 shows the plots of the first (k ≤ 4) harmonic containing only the first harmonic. For H0 > Ha Ð He, amplitudes as functions of the alternating field ampli- the frequency spectrum of the emf signal will contain tude H0 calculated for a fixed magnetic bias field He by all harmonics. Introducing dimensionless harmonic formulas (4) and (5). As seen, for H0 > Ha Ð He, the fre- π 2 ω amplitudes vk = cvk/ d NM into Eqs. (2) and (3) and quency spectrum of the emf signal displays all harmon- performing simple transformations, we obtain ics. As the H0 value increases, the odd harmonic ampli- v ( ) tudes grow and the even harmonic amplitudes pass 1 = H0/Ha through a maximum and decay. As can also readily be × [τ τ (τ ) (τ ) (τ ) (τ )] 2 Ð 1 + sin 2 cos 2 Ð sin 1 cos 1 , shown using formulas (4) and (5), the dimensionless amplitudes of all odd harmonics asymptotically tend to {( )τ } {( )τ } H0 sin k + 1 2 Ð sin k + 1 1 a constant value (vk = 4), while those of the even har- v k = ------Ha k + 1 monics decrease proportionally to 8kHa/H0. Figure 2 shows the plots of the first harmonic ampli- sin{(k Ð 1)τ } Ð sin{(k Ð 1)τ } + ------2------1---- , tudes versus the magnetic bias field He calculated for a k Ð 1 fixed amplitude of the alternating field H0. As seen from these curves, the first harmonic dominates in the k = 3, 5, …, (4) region of small He values. The contributions due to {( )τ } {( )τ } higher harmonics become significant when H H0 cos k + 1 2 Ð cos k + 1 1 e v k = ------increases to a sufficiently high level; however, the sig- Ha k + 1 nal in the coil vanishes for He Ð H0 = Ha since no remag- {( )τ } {( )τ } netization takes place in the wire. Note that some har- cos k Ð 1 2 Ð cos k Ð 1 1 + ------, monics are zero at intermediate values of the magnetic k Ð 1 bias field amplitude (Fig. 2), which reflects a change in k = 2, 4, … , the phase of these harmonics by π. For a nonlinear remagnetization regime to be estab- where lished, the longitudinal magnetic field amplitude must π < exceed the anisotropic field strength in the wire during Ð /2, H0 Ð He Ha, τ =  some part of the cycle of the alternating magnetic field 1 {( ) } > Ð arcsin Ha + He /H0 , H0 Ð He Ha, variation. In this case, all harmonics will appear in the (5) frequency spectrum of the emf generated in the probing π < coil. As seen from Fig. 2, the harmonic amplitudes are  /2, H0 + He Ha, τ =  highly sensitive with respect to the constant magnetic 2 {( ) } > arcsin Ha Ð He /H0 , H0 + He Ha. bias amplitude He, which is an important characteristic

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 700 ANTONOV et al.

π 2 ω REFERENCES vkc/ d NM 4 1. K. Mohri, T. Kohzawa, K. Kawashima, et al., IEEE 1 Trans. Magn. 28 (5), 3150 (1992). 2 2. R. S. Beach and A. E. Berkowitz, Appl. Phys. Lett. 64 4 (26), 3652 (1994). 3 3. K. V. Rao, F. B. Humphrey, and J. L. Costa-Kramer, J. Appl. Phys. 76 (10), 6204 (1994). 4. J. Velazquez, M. Vazquez, D.-X. Chen, and A. Hernando, Phys. Rev. B 50 (22), 16737 (1994). 2 5. M. Vazquez and A. Hernando, J. Phys. D 29 (4), 939 3 (1996). 6. N. A. Usov, A. S. Antonov, and A. N. Lagar’kov, 1 J. Magn. Magn. Mater. 185 (1Ð2), 159 (1998). 7. F. B. Humphrey, K. Mohri, J. Yamasaki, et al., in Pro- ceedings of International Symposium on Magnetic Prop- erties of Amorphous Materials (Elsevier, Amsterdam, 0 1987), p. 110. 2 4 6 8. K. Mohri, F. B. Humphrey, K. Kawashima, et al., IEEE He/Ha Trans. Magn. 26 (5), 1789 (1990). 9. L. V. Panina, K. Morhi, T. Uchiyama, et al., IEEE Trans. Magn. 31 (2), 1249 (1995). Fig. 2. The plots of harmonic amplitudes vk versus magnetic bias field value He calculated for H0/Ha = 5 and k = 1 (1); 10. N. A. Usov, A. S. Antonov, A. M. Dykhne, and 2 (2); 3 (3); 4 (4). A. N. Lagar’kov, Élektrichestvo, No. 2, 55 (1998). 11. A. S. Antonov, A. N. Lagar’kov, and I. T. Yakubov, Zh. Tekh. Fiz. 69 (3), 58 (1999) [Tech. Phys. 44, 317 for weak magnetic field sensors; for example, let us (1999)]. estimate the sensitivity of the second harmonic ampli- 12. R. S. Beach, N. Smith, C. L. Platt, et al., Appl. Phys. ∂ ∂ | Lett. 68 (19), 2753 (1996). tude | V2/ He for a typical cobalt-based microscopic wire [14]. Assuming d = 10 µm, M = 500 G, f = 5 MHz, 13. A. S. Antonov, N. A. Buznikov, I. T. Iakubov, et al., in H = 0.5 Oe, and H /H = 5, N = 100, we obtain Proceedings of the Moscow International Symposium on a 0 a Magnetism (Mosk. Gos. Univ., Moscow, 1999), Part 2, ∂ ∂ | ≈ Ð1 | V2/ He 10 V/Oe. In conclusion, it should be noted p. 252. that amplitudes of all the first harmonics are of the same 14. M. Vazquez, J. M. Garcia-Beneytez, J. M. Garcia, et al., order of magnitude. For this reason, the phenomenon of in Proceedings of the Moscow International Symposium nonlinear remagnetization of an amorphous wire con- on Magnetism (Mosk. Gos. Univ., Moscow, 1999), sidered above may be of interest from the standpoint of Part 1, p. 259. developing magnetic-field-controlled frequency converters. Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 701Ð704. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 8Ð16. Original Russian Text Copyright © 2000 by Ginzburg, Sergeev, Peskov, Arzhannikov, Sinitskiœ.

Planar Free-Electron Lasers with Combined 1D/2D Bragg Mirror Resonators: A Theoretical Study N. S. Ginzburg, A. S. Sergeev, N. Yu. Peskov, A. V. Arzhannikov, and S. L. Sinitskiœ Institute of Applied Physics, Russian Academy of Sciences, Nizhniœ Novgorod, 603600 Russia Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, Russia Received February 21, 2000

Abstract—The operation of a free-electron laser with a combined Bragg mirror resonator is studied theoretically. The resonator comprises a pair of planar Bragg reﬂectors with a two- and a one-dimensional relief pattern, respectively, and is closed in the transverse direction so as to ensure unidirectional outcoupling. It is demonstrated that this design provides for a possibility of obtaining spatially coherent radiation from a sheet electron stream with the cross size exceeding the wavelength by several orders of magnitude. © 2000 MAIK “Nauka/Interperiodica”.

1. Using sheet electron streams in free-electron pilot FELs with 2D distributed feedback. We think that lasers (FELs) seems to be a promising way for increas- effort should be made to design resonators closed in the ing output power, whereby the interaction space is transverse direction. developed in a transverse direction. It is now conceivable that a gigawatt output power level can be studied This paper analyzes the operation of a sheet beam in the millimeter wave band, since high-current relativ- FEL with a resonator comprising two dissimilar Bragg istic electron beams with a width of up to 150 cm and a reflectors bearing 2D and 1D relief patterns (Fig. 1). power of up to 50 GW are already available [1, 2]. The Situated at the cathode (injector) end, the 2D reflector main problem is the synchronization of waves from dif- synchronizes the radiation in the transverse direction. ferent parts of the stream. A two-dimensional (2D) dis- At the collector end, it is sufficient to place a conven- tributed feedback has been suggested to solve the task tional 1D reflector to close the feedback loop. Both [3, 4]. The feedback is effected by a 2D Bragg resona- reflectors are closed in the transverse direction, but tor comprised of a planar waveguide section with the modest losses (e.g., ohmic) are still imparted to the inner surface having a double periodic relief pattern, input reflector for the sake of lasing stability. These the translation vectors being noncollinear. In contrast to losses have insignificant effect on the energy balance, a one-dimensional (1D) resonator with a corrugated provided that optimum parameters are selected. This inner surface, four partial waves are coupled in the 2D means that the reflection coefficient of the output case, two of which propagate with the stream and in the reflector is not too large and the radiation is amplified opposite direction and the other two travel along the predominantly after the input reflector. Thus, the power transverse axis, thus synchronizing radiation in the is mostly emitted by the electron stream in the longitu- transverse direction. Reported in [4, 5], the analysis of dinal direction and little is dissipated or scattered. the dynamics of such FELs showed that a steady-state 2. Let the input reflector have the following 2D single-frequency operation can be achieved with relief pattern: almost any stream width and that an increase in the system width results solely in longer transients, provided aa= cos()h z cos()h x , (1) the optimum parameters are selected. On the other 2 2 2 hand, the simplest configurations [3–5] suffer from the requirement that the system must be open in the trans- where the pattern height 2a2 is assumed to be small. verse direction. As a result, the power has to be coupled The pattern insures the coupling and mutual scattering out in two directions, whereas only unidirectional out- of four partial waves coupling is required in most cases. In [6], a way to ([Ꮽ Ðihz Ꮽ ihz Ꮾ Ðihx ensure this was suggested whereby the resonator EE=0Re +e ++Ðe +e should be supplied with additional edge Bragg struc- (2) Ꮾihx ] iωt ) tures defecting the transverse power flows by 90° +Ðee , toward the electron stream. In our opinion, this resonator configuration is too complicated to be employed in provided the propagation constant h satisfies the Bragg

702 GINZBURG et al.

y x z

lx ν ||

l2 l0 l1

Fig. 1. Conﬁguration of a planar FEL with a combined Bragg mirror resonator driven by a sheet relativistic electron beam. The input and the output reﬂector have a 2D and a 1D relief pattern, respectively.

resonance condition [3]: of the electron beam; β|| is the translational electron β velocity; gr is the group velocity of the waves; h ≈ h2. (3)

2 2 1/3 In Eqs. (2) and (3), h = 2 π/d , with d being the  eI0 λ µk  2 2 2 C = ------Ꮽ Ꮾ  3 πγ  period of the pattern; ±(x, z, t) and ±(x, z, t) are mc 8 a0 slowly varying functions; and ω = 2π f = h2 c is the

Bragg frequency, which is chosen to serve as a carrier is the Pierce gain parameter; I0 is the beam current per frequency. Allowing for excitation of the synchronous α Ꮽ unit width; a0 is the cavity height (see Fig. 1); 2 is the wave + by the electron stream, the mutual scattering wave coupling parameter for the 2D pattern, which is of the partial waves by the pattern (1) is described by proportional to a [9]; and F(x) is the electron stream the following system of equations [7]: 2 density proﬁle (see Fig. 1). ∂ ∂  β Ð1  α ( ) ( ) ∂ + gr ∂τ A+ + i 2 B+ + BÐ = F X J, Z η, %

 ∂ Ð1 ∂  20 Ð β A + iα (B + B ) = 0, (4) ∂Z gr ∂τ Ð 2 + Ð

 ∂ Ð1 ∂  15 ± β B± + iα (A + A ) = 0. ∂X gr ∂τ 2 + Ð 1 2 3 The boundary conditions for Eqs. (4) are 10 ( ) A+ Z = 0 = 0, B+ + RBÐ X = 0 = 0, (5) 5 ( ) BÐ + RB+ = 0, X = Lx where R ≤ 1 is the reflection coefficient of the cavity side walls for the B± waves. The dimensionless quanti- 0 40 80 120 160 200 ties in Eqs. (4) are defined as follows: t, ns Fig. 2. Simulated turn-on transients in a combined1D/2D Z = zCω/c, X = xCω/c, L , = l , Cω/c, x z x z Bragg mirror FEL: the electronic efficiency vs. time at a τ = tCω; cavity width lx of (1) 20, (2) 70, or (3) 140 cm. Other param- α eter values are as follows: 2 = (1) 0.08, (2) 0.04, or Ꮽ Ꮾ κµ γ ω 2 κ ≈ β β Ð1 α Ð1 (A±, B±) = ( ±, ±)e / mc C ; ⊥/ || is the (3) 0.03 cm ; 1 = (1) 0.1 or (2, 3) 0.03 cm ; f = 75 GHz; µ ≈ γÐ2 × Ð3 beamÐwave coupling parameter; is the inertial C = 6 10 ; R = 0.95; l2 = 18 cm; l0 = 30 cm; l1 = 10 cm; bunching parameter [8]; γ is the relativistic mass factor and ∆ = Ð1.5.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 PLANAR FREE-ELECTRON LASERS 703

Like the smooth portion of the resonator, the output 1D reflector deals with two partial waves only: ([Ꮽ Ðihz Ꮽ ihz] iωt) Ꮽ E = E0 Re +e + Ðe e . (6) +, kV/cm 60 These waves travel parallel and antiparallel to the 140 translational beam velocity. The 1D Bragg pattern is 30 defined as 105 ( ) 0 70 a = a1 cos h1z , (7) 60 x, cm 40 35 where h = 2π/d (with 2a and d being the height and 20 1 1 1 1 z, cm 0 the period of the pattern, respectively). The waves are scattered if Ꮽ –, kV/cm 60 h ≈ h1/2 . (8) 140 The scattering is described by the equations 30 105  ∂ Ð1 ∂  + β A + iα A = F(X)J, 0 70 ∂Z gr ∂τ + 1 Ð 60 x, cm (9) 40 35 ∂ ∂ z, cm 20  β Ð1  α 0 Ð gr AÐ + i 1 A+ = 0, AÐ = 0, ∂ ∂τ Z = Lz Z Ꮾ +, kV/cm α where 1 is the wave coupling parameter for the 1D 60 pattern [9] and lz = l2 + l0 + l1 is the total cavity length (Fig. 1). In the smooth portion of the resonator, the 30 amplification of the wave Ꮽ obeys an equation similar + 0 to Eqs. (9) with a zero wave coupling parameter. The 140 partial amplitudes Ꮽ± must be considered continuous at 105 the boundaries between the resonator portions. 70 According to [8], the averaged particle motion in x, cm 0 Ꮽ the field of synchronous wave + is described by the 35 20 equation 40 0 60 z, cm ∂ ∂ 2 θ  β Ð1  θ ( i )  + ||  = Re A+e (10) Fig. 3. Spatial distributions of the partial wave amplitudes ∂Z ∂τ Ꮽ Ꮾ ± and + under steady-state FEL operation conditions. subject to the boundary conditions The parameter values are the same as for curve 3 in Fig. 2. θ = θ ∈ [0, 2π ), Z = 0 0 3. We carried out a simulation based on Eqs. (4), (9),  ∂ Ð1 ∂  (11) and (10). The results are presented in Figs. 2 and 3. The + β|| θ = ∆, ∂Z ∂τ Z = 0 parameter values were close to those of 4-mm-band FEL experiments conducted at the Institute of Nuclear θ ω where = t Ð hz Ð hwz is the instantaneous phase of Physics using the U-2 and U-3 accelerators [10, 11]. electrons with respect to the synchronous wave, ∆ = Specifically, we set the input (2D) reflector length at 18 cm, the output (1D) reflector length at 10 cm, the (ω Ð hν|| Ð h v||)/ωC is the initial frequency detuning w smooth portion length at 30 cm, and the cavity height at from wiggler synchronism at the carrier frequency, a = 1 cm. The beam current per unit width was h = 2π/d , and d is the period of the wiggler. Having 0 w w w 1 kA/cm; the particle energy, 1 MeV; the wiggler solved Eq. (10), one finds the electron current J = period, 4 cm; and the particle oscillation velocity in the 2π Ðiθ 1 θ wiggler field, β⊥ ≈ 0.2. Accordingly, the Pierce gain --∫ e d 0 that excites the synchronous wave, this π 0 parameter was about C ≈ 6 × 10Ð3. term entering into Eqs. (4) and (9). Note that Eqs. (4), (9), and (10) are written under the assumption that both Figure 2 depicts the electronic efficiency as a func- the 1D and 2D Bragg resonance conditions (3) and (8) tion of time for different widths of the interaction are met. This is obviously possible when h = h /2, space. As seen, a steady-state FEL operation is possi- 2 1 ble. Note that the width was increased while simulta- which in turn is satisfied if d2 = 2 d1. neously decreasing the wave coupling parameter, the

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 704 GINZBURG et al. other dimensions being fixed. To obtain stable opera- REFERENCES tion, a certain loss was imparted to the resonator for the 1. A. V. Arzhannikov, V. S. Nikolaev, S. L. Sinitsky, and Ꮾ± waves by setting the reflection coefficients of the M. V. Yushkov, J. Appl. Phys. 72 (4), 1657 (1992). input reflector at about 0.95 for the waves. This could 2. A. V. Arzhannikov, V. B. Bobylev, V. S. Nikolaev, et al., be implemented by depositing an absorber onto the in Proceedings of the 10th International Conference on cavity side walls. High-Power Particle Beams, San Diego, 1994, Vol. 1, p. 136. Figure 3 presents the spatial distributions of the par- Ꮽ Ꮾ 3. N. S. Ginzburg, N. Yu. Peskov, and A. S. Sergeev, Pis’ma tial wave amplitudes + and ± under steady-state Zh. Tekh. Fiz. 18 (9), 23 (1992) [Sov. Tech. Phys. Lett. conditions for the case where lx = 140 cm and the sheet 18, 285 (1992)]. relativistic electron beam occupies about 80% of the 4. N. S. Ginsburg, N. Yu. Peskov, A. S. Sergeev, et al., Nucl. Ꮽ cavity width. In the beam region, + is seen to be Instrum. Methods Phys. Res. A 358, 189 (1995). almost uniform in the transverse direction. This ensures 5. N. S. Ginzburg, N. Yu. Peskov, and A. S. Sergeev, Opt. that all of the stream parts contribute the same power to Commun. 112, 151 (1994). the radiation and that the average efficiency is suffi- 6. N. S. Ginzburg, N. Yu. Peskov, and A. S. Sergeev, ciently high. Estimates of the partial power flows at the Radiotekh. Élektron. (Moscow) 40 (3), 401 (1995). resonator ends (Fig. 3) indicate that the values of the 7. N. S. Ginzburg, N. Yu. Peskov, A. S. Sergeev, et al., Phys. reflection coefficients mentioned in the preceding para- Rev. E 160 (1), 935 (1999). graph have little or no effect on the energy balance, the 8. V. L. Bratman, G. G. Denisov, N. S. Ginzburg, and resultant losses being within 1Ð2%. Under these condi- M. I. Petelin, IEEE J. Quantum Electron. QE-19 (3), tions, the maximum output power of the FEL may 282 (1983). reach 5Ð10 GW. 9. N. F. Kovalev, I. M. Orlova, and M. I. Petelin, Izv. Vyssh. To sum up, the above analysis has demonstrated that Uchebn. Zaved., Radiofiz. 11 (5), 783 (1968). using FELs with a combined Bragg resonator compris- 10. M. A. Agafonov, A. Arzhannikov, N. S. Ginzburg, et al., ing two dissimilar planar reflectors with 2D and 1D IEEE Trans. Plasma Sci. 26 (3), 531 (1998). relief patterns opens the way to obtaining a spatially 11. N. V. Agarin, A. V. Arzhannikov, V. B. Bobylev, et al., in coherent radiation from a sheet electron stream whose Abstracts of the 21st International FEL Conference, width exceeds the wavelength by several orders of Hamburg, 1999, p. Mo-O-03. magnitude. The resonator is closed in the transverse direction, thus ensuring unidirectional outcoupling. Translated by A. Sharshakov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 705Ð706. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 17Ð21. Original Russian Text Copyright © 2000 by Razumov, Tumarkin.

Electrical Properties of Thin BaxSr1 Ð xTiO3 Films for Microwave Applications S. V. Razumov and A. V. Tumarkin St. Petersburg State Electrotechnical University, St. Petersburg, Russia Received April 12, 2000

In recent years, ferroelectric BaxSr1 Ð xTiO3 (BSTO) tering of 76-mm-diam ceramic targets of various com- films used in numerous applications have attracted the positions. The sputtered material was deposited either attention of researchers as possible materials for micro- in pure oxygen or in a 50% O2 + 50% Ar gas mixture at wave devices [1, 2]. This interest is based on the strong a total gas pressure of 8 Pa. The synthesis temperature dependence of the permittivity of BSTO films on the was varied from 700 to 900°C. The RF discharge power applied electric field strength, reliable operation of the was varied from 120 to 360 W. The electrical properties BSTO-based microwave semiconductor devices at a of the synthesized films were studied using planar field strength of up to ~40 V/µm and above [3], and a capacitor structures with 0.35-µm-thick copper elec- relatively low dielectric loss tangent of these materials trodes and an interelectrode gap width of ~7 µm. The in the microwave frequency range [4]. These properties electrical characteristics were measured at a frequency make the BSTO films promising materials for the of 1 GHz and a voltage U applied to the capacitors var- microwave devices such as varactors, phase-rotating ied from 0 to 300 V (E = 0Ð40 V/µm). devices, and tunable filters operating at room temperature [5]. Figures 1a and 1b show the plots of the control abil- From the standpoint of physical properties in the ity factor K = Cmax(U)/Cmin(U) versus the synthesis microwave frequency range, the most promising sys- temperature for the films obtained by sputtering from tems are offered by the BSTO films deposited onto sap- Ba0.8Sr0.2TiO3 and Ba0.5Sr0.5TiO3 targets onto sapphire phire and, especially, polycore substrates. However, and polycore substrates. Figures 2a and 2b present the now there is a need for ferroelectric films to combine a temperature dependences of the dielectric loss tangent sufficiently strong field dependence of the permittivity for the same films. As seen from these data, both the with low microwave energy losses (at least, with a control ability and the microwave losses of the films are room-temperature loss tangent tan δ < 0.01 at 1 GHz). determined to a considerable extent by the content of Creating such BSTO films presents a quite difficult Ba in the deposit. In particular, the films produced by task. sputtering the Ba0.8Sr0.2TiO3 target have a control ability factor of up to 2.5 at rather high losses (tan δ = 0.04 One of the most acceptable ways to obtain the BSTO films with perfect structure and high physical at U = 0 and above), which makes this material ineffec- properties is offered by the ion-plasma deposition tech- tive in the microwave frequency range. niques. However, fabrication of the BSTO films with More promising results were obtained for the films the properties acceptable for the microwave applica- prepared from the Ba0.5Sr0.5TiO3 target. These films tions by the method of RF magnetron sputtering exhibit a clear tendency of both the control ability fac- requires establishing a correlation between the main tor and the dielectric loss tangent to increase with the technological parameters of the film deposition process synthesis temperature (Figs. 1a and 1b). However, the (including the synthesis temperature, working gas mix- losses can be reduced by replacing pure oxygen with ture composition, and target composition) and the electrical properties of the resulting BSTO films. the O2ÐAr mixture, albeit at the expense of the control ability factor reduced to ~1.5. Apparently, the problem The purpose of this work was to study the properties of obtaining BSTO films combining good control abil- of BSTO films obtained on single- and polycrystalline ity and low losses can be solved by compromising sapphire substrates by the RF magnetron sputtering of between the target composition (with the Ba and Sr a ceramic target. contents not exceeding 0.5/0.5), a sufficiently high syn- The sample films were prepared on a Leybold Z-400 thesis temperature, and the working gas mixture com- technological magnetron system by the “on-axis” sput- position.

706 RAZUMOV, TUMARKIN

δ Cmax/Cmin tan (U = 0) 3.5 (a) 0.09 (a) 0.5 Ba (O) 0.08 3.0 0.8 Ba (O) 0.5 Ba (Ar + O) 0.07 0.06 2.5 0.05 2.0 0.04 0.03 0.5 Ba (O) 1.5 0.02 0.8 Ba (O) 0.01 0.5 Ba (Ar + O) 1.0 700 750 800 850 900 950 0 700 750 800 850 900 950 3.5 0.09 (b) 0.08 0.5 Ba (O + Ar) 3.0 0.5 Ba (O + Ar) 0.5 Ba (O) (b) 0.5 Ba (O) 0.07 0.8 Ba (O) 0.8 Ba (O) 2.5 0.06 0.05 2.0 0.04 0.03 1.5 0.02 0.01 1.0 0 650 700 750 800 850 900 950 650 700 750 800 850 900 950 T, °C T, °C Fig. 1. Plots of the control ability factor versus temperature Fig. 2. Plots of the dielectric loss tangent versus temperature for various BSTO ﬁlms deposited onto (a) sapphire and for various BSTO ﬁlms deposited onto (a) sapphire and (b) polycore substrates. (b) polycore substrates.

Investigations of the film sample structure by X-ray of the synthesis of BSTO film combining a high control diffraction showed that the films synthesized in the ability and low energy losses in the microwave fre- indicated temperature range contain blocks with the quency range. orientations (111), (110), and (100), their ratio depending on the synthesis temperature. We may suggest that a multiphase character of the films accounts for the REFERENCES rather high level of RF losses, although we failed to find 1. J. M. Ponds, S. W. Kirchoefer, W. Chang, et al., Integr. any strict correlation between changes in the phase Ferroelectr. 22, 317 (1998). composition and the RF losses. 2. F. Miranda, F. W. van Keuls, R. R. Romanofsky, and All the above considerations equally apply to the G. Subramanyam, Integr. Ferroelectr. 22, 269 (1998). BSTO films deposited onto sapphire and polycore sub- 3. Jaemo Im, O. Auciello, P. K. Baumann, et al., Appl. strates, since no significant differences between the Phys. Lett. 76, 625 (2000). electrical properties of these films were observed. 4. J. D. Baniecki, R. B. Laibowitz, T. W. Shaw, et al., Appl. Thus, we have demonstrated the possibility of con- Phys. Lett. 72, 498 (1998). trolling the electrical properties of BSTO films grown 5. A. B. Kozyrev, A. V. Ivanov, T. B. Samoilova, et al., by method of RF magnetron sputtering on both single- Integr. Ferroelectr. 24, 297 (1999). and polycrystalline Al2O3 substrates. The obtained results determine the direction for further investigation Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 707Ð709. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 22Ð29. Original Russian Text Copyright © 2000 by Ganzherli, Denisyuk, Konop, Maurer.

A Thin-Layer Bichromated Gelatin for Holography Sensitive in the Red Spectral Range N. M. Ganzherli, Yu. N. Denisyuk, S. P. Konop, and I. A. Maurer Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia Received December 16, 1999

Abstract—A variant of thin-layer optically sensitive material for holographic applications, based on a bichromated gelatin with glycerol and Methylene Blue dye additives, is suggested. The material is sensitive in the red spectral range, possesses the property of self-development, and provides for the possibility of the real- time reconstruction of holographic images. The bichromated gelatin is prepared using potassium bichromate. © 2000 MAIK “Nauka/Interperiodica”.

Gelatin with chromate additives has been used as The development of information recorded in a BG a medium for hologram recording since 1968 [1]. Pre- layer implies that water, one of the necessary compo- viously, bichromated gelatin (BG) was known as a nents, has to be present in the composition at an amount medium sensitive in the blue-green and ultraviolet sufficient to reveal the recorded image. Using sandwich spectral ranges. The task of expanding the spectral sen- structures with a BG layer confined in a closed volume sitivity range to the red, so as to provide for the holo- between two glasses allows a certain amount of water graphic information recording using HeÐNe laser radi- to be retained in the layer upon gelation. In this case, a ation, is still important. Solving some practical tasks hologram is registered immediately in the water-satu- requires holographic media in the form of thin layers rated layer. (with thicknesses from 100 µm to a few millimeters). Previously [10Ð12], we have developed a thick- Another requirement is the obtaining of self-develop- layer optically sensitive material based on a bichro- ing layers, which would allow for the real-time image mated gelatin for the blue-green spectral range. The reconstruction. material was in the form of a 1Ð3 mm thick gel cast between two glasses. A certain amount of water According to [2], the intrinsic sensitivity of a retained in the layer allowed a real-time registration of bichromated gelatin in the red spectral range is very the holograms. An obvious disadvantage of this thick- 2 µ small (about 15Ð150 J/cm for 10Ð30 m thick layers). layer system was the limited lifetime of the holograms. In order to increase the spectral range, the chromate- In order to increase the water content in the BG layer, it containing colloidal systems are doped with dyes pos- was suggested to introduce a certain amount of glycerol sessing maximum absorption in the desired region. The to the emulsion composition [13, 14]. Glycerol may main problem encountered in selecting such a sensi- play a manyfold role in self-developing BG layers. tizer is related to the low solubility of many dyes in the presence of ammonium bichromate. For example, In addition to providing for the self-development Methylene Blue tends to precipitate in the aqueous process, this component increases the optical sensitiv- solutions of ammonium bichromate. However, most ity and extends it toward the longwave spectral region. works devoted to the red sensitization of BG layers Moreover, glycerol acts as a plasticizer and increases were performed with Methylene Blue (MB) [3Ð9]. The the amount of water molecules capable of developing experimental data indicate that successful operation of the recorded image due to the presence of hydrogen the BG + MB system is possible provided that the solu- bonds. Self-developing bichromated gelatin layers containing up to 95% glycerol (relative to the dry gelatin tion pH is maintained on a level no lower than 9.0. weight) and having a thickness of 5Ð10 µm upon drying The first publication concerning the use of BG lay- were described in [15]. It was established that an opti- ers in a real-time operation mode not involving addi- mum content of free water providing a maximum sen- tional photochemical processing appeared in 1984 [9]. sitivity is reached in compositions containing 90Ð95% According to the data reported, undeveloped BG layers glycerol (relative to the dry gelatin weight). with a diffraction efficiency (DE) of 0.1Ð0.7% can be Diffraction efficiency is the main holographic used for the optical data processing at an exposure of parameter characterizing a recording medium. Previ- 200 mJ/cm2. It was also pointed out that holding the ously [16], we studied the holographic characteristics BG layers for several hours in a humid atmosphere of glycerol-containing layers with MB additives, in leads to some increase in the DE level. which the BG was obtained using ammonium bichro-

708 GANZHERLI et al.

η, % and allowed to stand for 2 h at 25°C. To this solution was added glycerol (100% of the dry gelatin weight) 30 and the mixture was treated for another 2 h at 40°C. 1 Finally, potassium bichromate (at an amount of 20, 40, 20 or 60% of the dry gelatin weight) and a necessary 2 amount of a 0.5% MB solution (1 ml MB per 100 ml of the mixture) were added and the mixture was cast 10 3 inside metal rings of certain thickness on glass substrates. Upon a 24-h gelation, during which the layer 2.5 5.0 7.5 10 J/cm2 thickness somewhat decreased, the samples were ready for recording holograms. Fig. 1. The typical plots of diffraction efﬁciency η versus exposure for the holograms of two plane waves recorded in We have studied the holographic characteristics of a symmetric scheme using an incident radiation power den- the sample BG layers upon recording the holograms of sity of 54 mW/cm2 in BG layers with various content of two plane waves in a symmetric scheme using a HeÐNe potassium bichromate (relative to the dry gelatin weight): laser operating at a wavelength of 0.63 µm. The conver- (1) 60; (2) 40; (3) 20%. gence angle of the interfering beams was varied from 15° to 35°. We have also varied a power density of the mate.The sample layers with a thickness of 2 mm were recording radiation, the potassium bichromate content prepared by in the form of gels between two glass (20Ð60% of the dry gelatin weight), and the MB con- plates. The MB precipitation was prevented by adding centration. The holographic characteristics of the emul- ammonia to pH 9. As long as ammonia did not evapo- sions were measured 24 h after casting. The layer thick- rate from the glass-covered layers, MB did not precipi- nesses varied from 0.6 to 2.2 mm. tate. The sample holograms of two plane waves were recorded in a symmetric scheme using a HeÐNe laser. Figure 1 shows the typical plots of DE versus expo- The beam convergence angle corresponded to a spatial sure measured for the holograms studied. The optimum potassium bichromate concentration in layers with a frequency of 100 mmÐ1. thickness on the order of 1.3 mm was 60% (relative to We have analyzed the DE as a function of exposure the dry gelatin weight), for which the a maximum DE for various component concentrations. In particular, it level of 29% was achieved. Increasing the potassium was found that increasing the MB content leads to an µ bichromate content above 60% led to a decrease in the increase in the absorption at 0.63 m and a decrease in DE value. The same effect was produced by increasing the optical transmission, which results in a drop of the the MB concentration, which was explained by a drop DE of holograms. At the same time, increasing the con- in the sample layer transmission. In the course of dry- centration of ammonium bichromate to 50% (relative ing, the sensitivity of sample layers gradually to the dry gelatin weight) increases the optical sensitiv- decreased and the DE level of holograms dropped. We ity and the maximum achievable DE level of the 2 have determined the DE of the holograms from two recorded gratings (DE reached 15% at a 10 mW/cm plane waves as a function of the spatial frequency of power density of the incident radiation). recorded gratings. It was established that the maximum In some applications, it would be convenient to use DE gradually decreases with increasing spatial fre- layers with the open surface rather than sandwich struc- quency, amounting to 26, 23, 18, 9, and 2% for the tures conﬁned between glass plates. However, as noted interfering beam convergence angles 17° (470 mmÐ1), above, MB begins to precipitate as soon as pH 20° (550 mmÐ1), 25° (685 mmÐ1), 30° (820 mmÐ1), and decreases below 9.0. In the ammonium bichromate ° Ð1 based layers described above, the pH was maintained 35 (955 mm ), respectively. by adding certain amounts of ammonia and keeping the We have also studied the angular selectivity of samples closed upon gel casting. We have replaced three-dimensional (3D) holograms with respect to vari- ammonium bichromate with potassium bichromate, the ation of the hologram reading angle. Figure 2a shows other composition components and preparation fea- the corresponding DE plot for a hologram measured in tures remaining the same. Ammonia was not added and a 1.37 mm thick layer by varying the angle of incidence the sample layers were open both during and after cast- of the reading beam at various time instants after the ing. In this way, we have succeeded in obtaining self- hologram recording. As seen, the DE level decreases developing layers sensitive in the red spectral range. with time as a result of degradation of the recorded Both preparation and casting of these layers are much holographic grating, which is explained by changes in like the technology of obtaining self-developing BG layers sensitive in the blue range. Upon drying, the structure of the gel-like medium containing a large layer thickness varied from 100 µm to 2.2 mm. amount of glycerol. The sample layers were prepared using the follow- Previously [17], a simple expression was obtained ing procedure. A 10% gelatin solution was prepared that relates the angular selectivity δα of the 3D holo-

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

A THIN-LAYER BICHROMATED GELATIN FOR HOLOGRAPHY 709

η, % η, % 30 (a) 20 (b) 4

15 2 20 5 3 1 10

1 10 2 3 5

–2 0 2 4 –8 –6 –4 –2 0 2 4 6 8 10 A, min of arc A, min of arc Fig. 2. The plots of diffraction efﬁciency η versus hologram reading angle A measured (a) at various times upon recording [t = 2 (1); 40 (2); 80 min (3)] in a 1.37 mm thick BG layer and (b) for BG layers of various thicknesses [d = 610 (1); 730 (2); 1020 (3); 1720 (4); 2140 µm (5)]. gram to the hologram thickness d, light wavelength λ, 2. C. Solano, R. A. Lessard, and P. C. Roberge, Appl. Opt. and the interfering beam convergence angle α: 24 (8), 1189 (1985). δα λ α. 3. T. Kubota, T. Ose, M. Sosaki, and K. Honda, Appl. Opt. = /d 15 (2), 556 (1976). Upon substituting the experimental values of δα = 4. T. Kubota and T. Ose, Appl. Opt. 18 (15), 2538 (1979). 5′ = 0.0014 rad, α = 15° = 0.259 rad, and λ = 0.63 µm, 5. R. Changkakoti, S. S. C. Babu, and S. V. Pappu, Appl. we obtain an estimate of the sample thickness d = Opt. 27 (2), 324 (1988). 1.73 mm. The table below presents the experimental δα η 6. R. Changkakoti and S. V. Pappu, Appl. Opt. 28, 340 values of and the maximum DE value max for BG (1989). layers of various thicknesses d (Fig. 2b) and the effec- 7. J. Blyth, Appl. Opt. 30, 1598 (1991). tive thicknesses d" calculated using the above formula: 8. K. Wang, L. Guo, J. Zhu, et al., Appl. Opt. 37 (2), 326 (1998). δα , min 13.5 9 6.5 5 4 3.5 9. S. Colixto and R. Lessard, Appl. Opt. 23 (8), 1989 η (1984). max, % 7.5 14.5 16 29 21 10 µ 10. Yu. N. Denisyuk, N. M. Ganzherli, and I. A. Maurer, d, m 610 730 1020 1370 1720 2140 Pis’ma Zh. Tekh. Fiz. 21 (17), 51 (1995) [Tech. Phys. d'', µm 630 930 1310 1730 2200 2420 Lett. 21, 703 (1995)]. 11. Yu. N. Denisyuk, N. M. Ganzherli, and I. A. Maurer, Taking into account the approximate character of Proc. SPIE 2688, 42 (1996). the theoretical expression for the angular selectivity 12. Yu. N. Denisyuk, N. M. Ganzherli, and I. A. Maurer, and uncertainty of the experimental δα and α values, Opt. Spektrosk. 83 (2), 320 (1997) [Opt. Spectrosc. 83, the agreement between measured and calculated thick- 320 (1997)]. nesses can be considered as quite satisfactory. The pro- 13. V. P. Sherstyuk, A. N. Malov, S. M. Maloletov, and posed material, sensitive in the red spectral range, V. V. Kalinkin, Proc. SPIE 1238, 218 (1989). allows a real-time reconstruction of the recorded holo- 14. Yu. N. Vygovskiœ, S. P. Konop, A. N. Malov, and graphic image and, hence, can be used for investigating S. N. Malov, Laser Phys. 8 (4), 901 (1998). features of the hologram recording. The bichromated 15. S. P. Konop, A. G. Konstantinova, and A. N. Malov, gelatin is cheap, simple in preparation, and ensures Proc. SPIE 2969, 274 (1996). good reproducibility of results. The next task is to 16. Yu. N. Vygovskiœ, P. A. Draboturin, A. G. Konop, et al., increase the lifetime of recorded holograms. in Laser Application in Science and Technique (Inst. Fiz. ILT Sib. Otd. Ross. Akad. Nauk, Irkutsk, 1997), This work was supported by the Russian Foundation − for Basic Research, project no. 99-02-18481. pp. 149 159. 17. Yu. N. Denisyuk, Zh. Tekh. Fiz. 60 (6), 59 (1990) [Sov. Phys. Tech. Phys. 35, 669 (1990)]. REFERENCES 1. T. A. Shankoff, Appl. Opt. 7 (10), 2101 (1968). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 710Ð712. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 30Ð34. Original Russian Text Copyright © 2000 by Denisov, Novoselov, Tkachenko.

Dissociation of Nitrogen Oxides under the Action of Pulsed Electron Beams G. V. Denisov, Yu. N. Novoselov, and R. M. Tkachenko Institute of Electrophysics, Ural Division, Russian Academy of Sciences, Ekaterinburg, Russia e-mail: [email protected] Received March 23, 2000

Abstract—The decomposition of nitrogen oxides NOx in model gas mixtures ionized by microsecond pulsed electron beams was studied. The main decomposition mechanism is the dissociation of oxides caused by the interaction with atomic nitrogen. The ﬁnal decomposition products are molecular oxygen and nitrogen. © 2000 MAIK “Nauka/Interperiodica”.

Smoky waste gases of thermal power plants contain The experiments were conducted on a setup com- a considerable proportion of toxic nitrogen oxides (NO prising an electron accelerator; a plasmachemical reac- and NO2). The task of purification of the waste gases tion chamber; and a system providing preparation, from toxic impurities has stimulated extensive investi- admission, and monitoring of the gas mixture studied. gations into the possibility of decomposing these impu- The accelerator (analogous to that described in [6]) rities by electrophysical methods. These methods are formed a pulsed electron beam with the following based on the laws of chemical reactions in gases ion- parameters: beam cross section, 10 × 100 cm; electron ized under the action of various electric discharges and energy, 200 keV; pulse duration (FWHM), 5 µs; beam electron beams. In particular, these tasks were solved current density behind the output foil, 4.5 × 10Ð3 A/cm2. by using continuous-mode accelerators forming elec- The electron beam entered the reaction chamber with tron beams with a current density from 10Ð9 to as volume of 12 dm3. 10−5 A/cm2 [1Ð3]. This electron-beam treatment pro- A model gas mixture was prepared from pure vides for a high degree (98–100%) of purification from nitrogen oxides, with the specific energy spent for the molecular nitrogen (N2), oxygen (O2), and nitric oxide removal of one toxic molecule being 15Ð20 eV [4]. (NO) in a special mixer and then admitted to the plasmachemical reaction chamber, which was preliminarily A mechanism for nitrogen oxide removal with the evacuated and doubly washed with pure nitrogen. The aid of continuous electron beams is sufficiently well gas mixture composition was as follows: nitrogen, 80Ð studied (see, e.g., [5]). In the presence of water vapor, 90%; oxygen, 0Ð20%, and nitric oxide, 100Ð1000 ppm. the electron-beam ionization leads to the formation of The qualitative and quantitative composition of the mixture was monitored using a TESTO-350 gas ana- free radicals of the Oá, OHá, and O2Há types. These radicals react with nitrogen oxides to form nitric acid. lyzer. The error of the concentration determination did Upon adding ammonia, the acid forms an ammonium not exceed 3%. Upon entering the plasmachemical chamber, the gas mixture was forced to flow over a salt (NH4NO3) in the form of a solid powder trapped by filters. This purification mechanism, involving the for- closed loop during the entire experimental cycle. Dur- mation of some product that has to be trapped and ing circulation of the mixture without electron-beam removed, implies a significant complication of the puri- irradiation, the NO component interacted with oxygen fication technology. Indeed, additional equipment and exhibited the natural chemical oxidation to NO2. (besides the electron accelerator and reaction chamber) After some time, when a dynamic equilibrium between is necessary for the supply of ammonia and the removal nitrogen oxides was established, the gas mixture was of powdered ammonium salts. treated with the electron beam. The treatment consisted in irradiating the gas by series of 50 electron-beam The elemental composition of nitrogen oxides coin- pulses. cides with that of pure air. In connection with this, the most economically efficient purification system is Typical plots of the concentrations of nitrogen apparently that based on the nitrogen oxide conversion oxides [NO], [NO2], and [NOx] (where [NOx] = [NO] + directly into molecular oxygen and nitrogen. The pur- [NO2]) versus the total number N of the electron-beam pose of this work was to study the possibility of this pulses are depicted in the figure. Curves 1, 2, and 4 conversion using a gas mixture modeling typical smoky were obtained for a gas mixture with an oxygen content waste gases. of 10%, and curve 3 refers to a mixture free of oxygen.

DISSOCIATION OF NITROGEN OXIDES 711

As seen, all the curves reflect gradual decrease in the [NOi], ppm corresponding oxide concentration with the total number of electron-beam pulses. This pattern differs from that reported [7] for mixture with the NO content above 500 1200 ppm, where decay in the NO2 concentration with increasing N was preceded by an initial growth stage. 400 1 Apparently, irradiation of a mixture containing a small amount of nitric oxide is either not accompanied by the 300 oxidation to NO2 or the amount of NO2 formed is below 2 the sensitivity level of the measuring instrument. It 200 3 should be noted that experiments with oxygen-free gas 4 mixtures containing small amounts of NO showed a 100 decrease in the NO concentration without any NO2 formation. 0 100 200 300 According to the commonly accepted notions about N, pulses the free-radical mechanism of NOx oxidation [5], a decrease in the concentration of nitrogen oxides in the Plots of the concentration of nitrogen oxides [NOi] (i = 1, 2, x) mixture must be accompanied by the formation of versus the number of electron-beam pulses for gas mixtures (1, 2, 4) containing 10% of oxygen and (3) free of oxygen: nitric acid as the final reaction product; however, our (1) NOx; (2, 3) NO; (4) NO2. Solid lines connect the exper- experiments revealed the presence of nitric acid in very imental points, dashed lines show the results of calculations. small, virtually trace amounts. This result is evidence that the mechanism of nitrogen oxide removal is not restricted to the oxidation process. Moreover, the elec- concentration of nitrogen oxides are as follows: tron-beam treatment of the oxygen-free mixture (see figure, curve 3) was also accompanied by a decrease in N + NO N2 + O, (1) the NO content. N + NO 2NO, (2) A mechanism alternative to the free-radical nitrogen 2 oxide reaction with the formation of nitric acid as the N + NO2 N2 + 2O, (3) final product is offered by the NOx dissociation yielding atomic species, followed by the formation of molecular nitrogen and oxygen. As is known, the energy of an O + NO2 = O2 + NO. (4) electron beam injected into a nitrogenÐoxygen mixture is spent not only for ionization and excitation of the The atomic oxygen and nitrogen form the correspond- mixture component but for the nitrogen dissociation as ing molecular species. The results of calculation of the well. A calculation of the electron beam energy distri- decreasing concentrations of nitrogen oxides according bution in air performed by the Monte-Carlo method to Eqs. (1)–(4) are depicted in the figure by dashed showed that the dissociation of molecular nitrogen by lines. The calculations were performed using the reac- the direct electron impact mechanism takes about 2.5% tion rate constants from [8]. As seen, Eqs. (1)Ð(4) rather of the total beam energy. Estimates showed that the satisfactorily describe the experimental dependences. concentration of atomic nitrogen formed upon the dis- Thus, we have demonstrated that purification of the sociation of N2 molecules caused by the direct impact smoky waste gases from nitrogen oxides by using of high-energy electrons under our experimental condi- pulsed electron beams is based primarily on the mech- tions does not exceed 1012 cmÐ3. The concentration of anism of NOx dissociation rather than oxidation. The atomic nitrogen in the gas mixture may significantly dissociation reactions are most effective in the gas mix- increase at the expense of dissociative recombination of tures with a small oxygen content. The process of elim- + the N2 ions. Our calculation showed that about 33.5% inating nitrogen oxides from the gas mixtures can be of the beam energy is spent the formation of these ions. described as follows. The electron-beam irradiation of The corresponding concentration of molecular nitrogen + a gas mixture leads to the formation of N2 ions, which 13 Ð3 ions is ~10 cm . The dissociative recombination of exhibit dissociative recombination resulting in the for- + N2 leads to the formation of atomic nitrogen with a mation of a sufficiently high concentration of atomic concentration of ~8 × 1013 cmÐ3. nitrogen. The interaction of atomic nitrogen with nitrogen oxides leads to their dissociation with the eventual The subsequent reactions involving atomic nitrogen formation of molecular oxygen and nitrogen. This and impurity molecules and leading to a decrease in the mechanism of the toxic impurity elimination can be

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

712 DENISOV et al. used to develop a wasteless electrophysical technology 4. J. C. Person and D. O. Ham, Radiat. Phys. Chem. 31 for the purification of smoky waste gases. (1−3), 1 (1988). 5. A. A. Valuev, A. S. Kaklyugin, G. É. Norman, et al., This work was supported by the International Scien- Teplofiz. Vys. Temp. 28 (5), 995 (1990). tific-Technological Center, grant no. 271. 6. K. A. Garusov, D. L. Kuznetsov, Yu. N. Novoselov, and V. V. Uvarin, Prib. Tekh. Éksp. 3, 180 (1992). 7. G. V. Denisov, Yu. N. Novoselov, and R. M. Tkachenko, REFERENCES Pis’ma Zh. Tekh. Fiz. 24 (4), 52 (1998) [Tech. Phys. Lett. 24, 146 (1998)]. 1. S. Masuda, Pure Appl. Chem. 60 (5), 727 (1988). 8. V. N. Kondrat’ev, Rate Constants of Vapor-Phase Reac- 2. E. L. Neau, IEEE Trans. Plasma Sci. 22 (1), 2 (1994). tions (Nauka, Moscow, 1970). 3. N. Frank and S. Hirano, Radiat. Phys. Chem. 35, 416 (1990). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 713Ð715. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 35Ð40. Original Russian Text Copyright © 2000 by Lyamshev.

Optico-Acoustic Characterization of the Fractal Structure of Radiation from a Laser with Unstable Resonator M. L. Lyamshev Research Center of Wave Phenomena, Institute of General Physics, Russian Academy of Sciences, Moscow, 117942 Russia e-mail: [email protected] Received March 28, 2000

Abstract—The thermooptical excitation of sound in a liquid by ﬂuctuating laser radiation with sine-modulated intensity is studied theoretically. The intensity is randomly distributed over the beam cross section. It is assumed that the random processes are homogeneous and that the spatial spectrum of intensity ﬂuctuations obeys a power fractal law. Possibilities for the optico-acoustic characterization of the fractal radiation structure of a laser with unstable resonator are discussed. © 2000 MAIK “Nauka/Interperiodica”.

Chaotic operation of lasers has attracted much atten- According to [7], the laser-induced thermooptical tion in recent years. For example, Loskutov et al. [1] sound excitation obeys the equation studied the routes to chaos and the properties of chaotic 2 κmω oscillation in an unstable resonator of a nonuniformly ()µ∆ + k = i------A µ Ixy(), exp()Ðµz. (1) pumped fast-flow laser. A scheme of information chaos Cp using an optical communications channel based on syn- Here, p is the acoustic pressure; κ is the coefficient of chronized chaotic lasers with random pumping was volumetric expansion; C is the heat capacity; µ is the considered in [2]. p optical absorption coefficient; A is the transmission Key features of a nonlinear dynamic system can be coefficient of the interface; m is the degree of intensity described in terms of scaling and the correlation dimen- modulation; I(x, y) is the intensity distribution over the sion or fractal dimension [3]. The scaling may occur in beam cross section at the interface; and k = ω/c, with c the emission harmonics of a plasma produced by being the speed of sound in the liquid. We assume that intense laser radiation [4]. It was also demonstrated that A = 1. The time-dependent factor exp(Ðiωt) is then the modal structure of the radiation from a laser with omitted. unstable resonator has a fractal character [5]. For a slit- The solution to Eq. (1) can be represented in the aperture laser, the output intensity distribution was form [7] found to have a fractal dimension of D = 1.6. For a circular-aperture laser, the fractal dimension was evalu- κωm ated at D = 1.3. pr()= i------µ ∫ Ix()',y'exp()Ðµz' Cp (2) It was interesting to find out whether it is possible to Ω examine the fractal structure of radiation of a laser with × p˜ ()x',,y' z' ⁄xyz,, dx'dy'dz', unstable resonator by optico-acoustic techniques. This study addresses the thermooptical excitation of sound where p˜ (r'/r) is the solution to the boundary-value in a liquid by laser radiation with sine-modulated inten- problem of the diffraction of the field from a point sity. The intensity fluctuations of the radiation are dis- source situated at the point r where the field p(r) is to tributed in a random fractal fashion over the beam cross be determined. Let us consider p(r) in the Fraunhofer section. The effect of spatial and temporal laser inten- region. Then, we have sity fluctuations on the excitation of sound in a liquid exp()ikr has already been studied in [6], but the charter of distri- p˜ ()r' ⁄ r = ------{exp[]Ði ()α x ' ++β y ' γ z ' bution was not specified. 4πr (3) Consider a laser beam incident on the interface Ð exp[]Ði()αx' + βy' Ð γz' }, between the atmosphere and a liquid which occupy the α2 β2 γ2 2 2 2 2 1/2 upper and the lower half-space, respectively. Let us where + + = k and r = (x + y + z ) . define the Cartesian coordinates (x, y, z) in such a way Let the laser intensity distribution be a random func- 〈 〉 that the origin lies on the interface and the z-axis is tion of the type I(x, y) = I0f(x, y), where f(x, y) = 0. directed downward along the beam. The absorbed radi- Furthermore, assume that the random processes are ation produces thermal sources of sound in the liquid. homogeneous.

1063-7850/00/2608-0713 $20.00 © 2000 MAIK “Nauka/Interperiodica” 714 LYAMSHEV Upon substituting Eq. (3) into Eq. (2) and integrat- Substituting Eqs. (7) and (8) into Eq. (4) and inte- ing the result with respect to z, we arrive at the follow- grating the result, we obtain ing expression for the mean-square acoustic pressure 2 2 2 2 2 2 κ ω m 1 µ γ 〈 p(r) 2〉 : 〈 p(r) 〉 = ------I ση G(α), (9) 2 π2 2 2 2 2 0 0 C p 4 r (µ + γ ) 2 2 2 2 2 2 κ ω m 1 µ γ 2 η α 〈 p(r) 〉 = ------I σ where 0 is the laser spot size on the interface and G( ) 2 π2 2 2 2 2 0 C p 4 r (µ + γ ) is the spectral density of intensity fluctuations. The lat- (4) ter quantity is expressed as × B(ξ, η)exp[Ði(αξ + βη)]dξdη, ∫∫  1 ξ η Γ ν + --  2 ξ ξ η 〈 〉 (α) 0 where B( , ) = f(x', y')f(x'', y'') is the normalized cor- G = ------1 . (10) relation function of intensity fluctuations; ξ = |x' Ð x''|; πΓ(ν) ν + -- ( α2ξ 2) 2 η = |y' Ðy''|; and σ is the area of the laser spot on the 1 + 0 interface. Strictly speaking, the integration with respect αξ α ξ η If 0 > 1, then G( ) obeys a power law (like any sta- to and is confined to the irradiated interface region. tistical fractal): However, if B(ξ, η) rapidly decreases within the beam ∞ Ð(2ν + 1) cross section and B( ) = 0, then the integration can be G(α)αξ > ∼ α . (11) extended to the interval from Ð∞ to +∞. 0 1 The properties of statistical fractals are often Now, consider the case of a circular aperture. Here, expressed in terms of structural functions (correlation Eq. (4) becomes functions) and their spectra. Remarkably, the functions 2 2 2 2 2 2 κ ω m 1 µ γ 2 represent power laws. This follows from the scale 〈 p(r) 〉 = ------I πa G(k⊥),(12) 2 π2 2 2 2 2 0 invariance of fractals [8]. C p 4 r (µ + γ ) In the context of statistical fractals related to the where wave motion, an important characteristic is the power fluctuation spectrum having the form +∞ δ G(k⊥) = B(ρ)exp(Ðik⊥ρ)dρ, (13) G(q) ~ q , (5) ∫ Ð∞ where q is the wavenumber of spatial fluctuations. For objects with a fractal surface, the exponent δ is k⊥ is the horizontal component of the wave vector k, 2 2 2 expressed as k⊥ = α + β , ρ = |ρ' Ðρ''|, and a is the laser beam spot δ = D Ð2d, (6) radius on the interface. Let us write the correlation function B(ρ) in the where D is the fractal dimension and d is the dimension ξ ρ ξ ρ of the embedding space. form of Eq. (8), where is replaced by and 0, by 0, the symbol ρ denoting the correlation radius of inten- Let us find the mean-square acoustic pressure in the 0 case of a slit aperture extending along the x-axis. For sity fluctuations. this system, we can write For the spectral density G(k⊥) defined by Eq. (13), (ξ, η) (ξ) (η) we have B = B1 B2 , (7) ρ2 η ≈ Γ(ν + 1) 0 where B2( ) 1 (since the intensity fluctuations can be G(k⊥) = ------. (14) πΓ(ν) 2 2 ν + 1 considered totally correlated in the transverse direc- (1 + k⊥ ρ ) tion). For the longitudinal direction, the normalized 0 ρ correlation function is expressed as follows [9]: If k⊥ 0 > 1, then ν Ð2(ν + 1) 1  ξ   ξ  G(k⊥) ∼ k⊥ . (15) B(ξ) = ------Kν ---- , (8) ν Ð 1  ξ   ξ  2 Γ(ν) 0 0 To compute the acoustic field in the liquid, one has to evaluate ν for each particular case. Let us set the where Γ(ν) is the gamma function, Kν(ξ/ξ ) is the mod- 0 embedding space dimension d = 2 according to the ified Bessel function of the second kind (also known as ξ numerical experiment reported in [5]. Then, Eqs. (5), the Macdonald function), and 0 is the longitudinal cor- (6), (11), and (15) imply that ν = 0.7 for the slit aperture relation radius of the intensity fluctuations. Note that ν ν and = 0.35 for the circular aperture. This computation B(0) = 1, B(∞) = 0, and B(ξ)ξ < ξ ~ (ξ/ξ ) . This means 0 0 also uses the fact that D = 1.6 and D = 1.3 for the two that the correlation function has a power form and, cases, respectively [5]. hence, can be used for describing the fractal structure of It follows from Eqs. (9)Ð(11) and (12)Ð(15) that the laser intensity fluctuations. fractal structure of the radiation produced by a laser

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 OPTICO-ACOUSTIC CHARACTERIZATION OF THE FRACTAL STRUCTURE 715 with unstable resonator can be examined by optico- REFERENCES acoustic techniques. Indeed, assume that µ Ӷ k, θξ ӷ θρ ӷ 1. A. Yu. Loskutov, A. V. Mushenkov, A. I. Odintsov, et al., ksin 0 >1, and kl 1 or that ksin 0 > 1 and ka 1, Kvantovaya Élektron. (Moscow) 29 (2), 127 (1999). where l is the length of the slit aperture and a is the laser 2. A. P. Napartovich and A. G. Sukharev, Kvantovaya Éle- beam spot radius on the interface. Then, the mean- ktron. (Moscow) 25 (1), 85 (1998). square sound pressure at a point of interest in the 3. F. C. Moon, Chaotic and Fractal Dynamics (Wiley, New x0z-plane York, 1992).

2 δ δ 4. V. P. Silin, Kvantovaya Élektron. (Moscow) 26 (1), 11 〈 p(r) 〉 ∼ Cq ∼ k , (16) (1999). 5. G. P. Karman and J. P. Woerdman, Opt. Lett. 23 (24), where C is a constant determined by the parameters of 1909 (1998). the problem. 6. F. V. Bunkin, Selected Works (Nauka, Moscow, 1999). The exponent is expressed as 7. L. M. Lyamshev, Thermooptical Laser Sound Genera- tion (Nauka, Moscow, 1989). log 〈 p(r) 2〉 δ = ------. (17) 8. M. Schroeder, Fractals, Chaos, Power Laws (Freeman, logk New York, 1990). 9. V. I. Tatarskiœ, Theory of Fluctuational Phenomena dur- Varying the modulation frequency, one obtains the ing the Sound Propagation in Turbulent Atmosphere dependence δ = ϕ(k) and hence D ~ ϕ(k). (Akad. Nauk SSSR, Moscow, 1959). This study was funded by the Russian Foundation for Basic Research, project no. 99-02-16334. Translated by A. Sharshakov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 716Ð717. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 41Ð45. Original Russian Text Copyright © 2000 by Kartashov, Popov.

Thermal Fluctuation Electromagnetic Field in the Medium as the Driving Force of Its Magnetosensitivity Yu. A. Kartashov and I. V. Popov North-West Correspondence Polytechnic Institute, St. Petersburg Received July 23, 1999

Abstract—The so-called kT problem relating the magnetosensitivity and thermal oscillations of a condensed medium is solved. An external magnetic field acting upon a free charged particle occurring in a thermal fluctuation electromagnetic field inside the condensed medium gives rise to a rotational moment having the same direction for charged particles of the same sign. The rotational moment is proportional to the thermal energy kT and can substantially increase under the cyclotron resonance condition. © 2000 MAIK “Nauka/Inter- periodica”.

t Theories interpreting the magnetosensitivity of ˜ q t q V = ------exp Ð τ-- +i ------∫Btd' condensed media [1], including the biological ones [2], m0 m 0 encounter a stumbling block known as the kT prob- 0 ∞ lem. The problem is that the potential energy of a t × ˜()ωω ω t' q magnetosensitive particle (molecule, atom, proton, ∫ e d exp it'++--τ i ------∫Btd '' dt(2) m 0 etc.) in a magnetic field (MF) is very small in compar- Ð∞ 0 ison with its thermal energy, even in the case of rather t strong MFs. ˜ t q + V 0 exp Ð τ-- Ði ------∫Btd', m 0 This work attempts to solve this problem based on 0 the concept of a thermal electromagnetic field (EMF) where the z axis of the Cartesian coordinates is parallel considered as the driving force of the charged particle ˜ ˜ ˜ motion. It is demonstrated that an external magnetic to B0; V = Vx + iVy, V 0 is the value of V at t = 0; and field gives rise to a rotational moment the direction of ˜ ∞ ˜ ω ω ω E = Ex + iEy = ∫ ∞ e ( )exp(i t)d . which is determined by the particle charge sign. This Ð phenomenon is analogous to the effect of a weak elec- Consider a periodic MF. For t ӷ τ, the solution (2) tric field (EF) on charged particles in a conducting can be rewritten as medium, adding a small drift velocity against the back- ∞ ground of very large thermal velocities. ˜ Ӎ q ()νΩ V ------∑b n b ν* exp Ði t m0 The motion of a particle (molecule, atom, proton, ν,∞n =Ð etc.) with the charge q and the mass m in an EMF is ∞ (3) 0 []()ω Ω described by the equation ×˜()ωexp i +n tω ∫ e------τ ()ωΩ Ω- d , 1/ +i ++c n ∞ dV m Ð m ------=Ð ------0 V +q VB×+qB, (1) 0 dt τ where τ where V is the particle velocity, is its relaxation time ∞ t ˜ ()Ω q ˜ in a given medium, B = B0 + B (B0 and B are the con- ∑b n exp in t = exp i------∫ B d t ' , m0 stant and alternating components of the magnetic Ð∞ 0 induction, respectively, B || B˜), and E is the equilib- 0 Ω = (q/m )B is the cyclotron frequency, and Ω is the rium thermal fluctuation EF strength inside the c 0 0 MF oscillation frequency. medium. The ensemble-averaged z-component of a rota- In the spectral representation, a solution to Eq. (1) is tional moment due the Coulomb forces is given by the as follows: expression

( × ) ( ) network), thus changing its structure. Since the Mz = r qE z = q xEy Ð yEx (4) moment is proportional to kT, the MF effect on the = (q/2i)(ρ˜ *E˜ Ð ρ˜ E˜ *), structure increases with the temperature (until the structure is destroyed by the fluctuation oscillations). ˜ Thus, there is no sense in the kT problem since there is where r is the particle radius-vector, ρ˜ = x + iy = ∫V dt, no contradiction between thermal oscillations and mag- and the asterisk denotes complex conjugation. netosensitivity of the medium. On the contrary, the magnetosensitivity is related to thermal oscillations. Averaging in Eq. (4) with an allowance of the fluc- tuationÐdissipation theorem [3] yields the expression Note that the rotational moment under consideration presumably accounts for the phenomenon of water 2 (ω) (ω ) (ω)δ(ω ω ) magnetization [1] upon its motion in MF, since Mz e˜* e˜ ' = --gE Ð ' , 3 decreases the turbulence microscale to change the adsorption and other properties of water. The phenom- where enon of the “water memory” [5] is possibly related to the effect of the rotational moments on a system exhib- (ω) θ(ω, ) π 2τ gE = 3m0 T / q , iting the Bjerrum defects. Since life on Earth appeared and evolved under the influence of the geomagnetic θ(ω, T) = បω/2 + បω/[exp(បω/kT) Ð 1] field, the Mz variation under the conditions of a hypo- is the mean quantum oscillator energy. In order to sim- magnetic field may substantially perturb the organic plify the mathematical transformations, below we life structure [6]. បω Ӷ assume that kT. Then, the rotational moment can The allowance of an amplitude-modulated MF in be represented as Eqs. (2) and (4) yields a cyclotron resonance at the ∞ modulation frequency in agreement with the experi- 2kT exp[Ði(n Ð ν)Ωt] mental observations [7].  * Mz = Ð----τ----- ∑ bn bν------τ------(---Ω------ν---Ω-----)--  1/ Ð i c + ν, n = Ð∞ It is clear that a thermal fluctuation MF also induces (5) a rotational moment in the medium. In the general case, ∞ exp[i(n Ð ν)Ωt]  this moment is proportional to kT in a higher power Ð ∑ b bν*------. than that in Eq. (5). n 1/τ + i(Ω + νΩ) ν, ∞ c  n = Ð Note, finally, that a rotational moment acts upon a In particular, in the case when only the stationary MF is charged particle in a magnetic field regardless of the δ δ nature of forces causing oscillations of the particle. operative (bn = n0, where n0 is the Kronecker symbol), expression (5) is reduced to 2kTΩ τ REFERENCES M = Ð------c-----. (6) z Ω 2τ2 1 + c 1. V. I. Klassen, Magnetization of Aqueous Systems (Khimiya, Moscow, 1982). Note that the sign in Eqs. (5) and (6) indicates that 2. V. V. Novikov and M. N. Zhadin, Biofizika 39 (1), 45 the direction of Mz for the positively charged particles (1994). is opposite to that of B0. The above expressions show 3. S. M. Rytov, Introduction to Statistical Radiophysics that either a periodic (at the frequency of the external (Nauka, Moscow, 1966). MF) or stationary (when the alternating MF component 4. A. R. Liboff, S. D. Smith, and B. R. McLeod, in Mecha- is absent) rotational moment Mz proportional to the nistic Approaches to Interactions of Electric and Elec- thermal energy kT acts upon a free charged particle. It tromagnetic Fields with Living System (Plenum, New York, 1987), p. 109. follows from Eq. (5) that Mz substantially increases if the condition 5. M. V. Berezin, R. R. Lyapin, and A. M. Saletskiœ, Pre- print No. 21, MGU Fiz. Fak. (Moscow State University, Ω Ω c + n = 0 (7) Physical School, Moscow, 1988). 6. V. I. Kopanev and A. V. Shakula, Effect of Hypomagnetic is met and the relaxation time is sufficiently large τ ӷ Ω Field on Biological Objects (Nauka, Leningrad, 1985). 1/ c. Expression (7) represents a condition of the experimentally observed cyclotron resonance [4]. 7. W. R. Adey, Proc. IEEE 68 (1), 119 (1980).

The moment Mz generates rotational moments in the bond network of the medium (e.g., in the H-bond Translated by A. Chikishev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 718Ð720. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 46Ð51. Original Russian Text Copyright © 2000 by Kolmakov.

Conical Waves Producing Longitudinal Power Flows I. A. Kolmakov Received September 29, 1999

Abstract—A conical electromagnetic wave converging to its axis is studied theoretically. It is demonstrated that the wave produces an intense self-accelerating flow of energy (momentum). Conical waves may find various applications such as in the acceleration of particles, the generation of high-power pencil beams, the transformation of simultaneously written spatial data into a time waveform, the measurement of flow rate, etc. It is noted that conical waves could set the stage for the manifestation of unknown properties of matter. © 2000 MAIK “Nauka/Interperiodica”.

This paper concerns an unexplored object of wave ical surface of the CW, the magnitudes E and H are science: a conical wave (CW) converging to its axis. maximum. For the sake of simplicity, assume that E The gradient structure of such waves causes a progres- and H vary in a sinusoidal fashion in the direction of the θ sive self-compression of the field near the axis (from inward normal ln and that 0 is small. the vertex to the base of the cone). In general, the process is accompanied by a motion accelerating in the Consider the focusing of the CW. While a CW longitudinal direction. This property enables one to equiphase surface is at a distance from the origin such produce an intense pencil flow of energy (momentum) that the energy flow produced by the compression has by creating an appropriate distribution of electric cur- no effect, the velocity of the surface in the direction of ln equals that of light, c; the velocity along the 0Z-axis rent or acoustic pressure over the radiator aperture. θ is v0Z = c/sin 0; and the velocity of the intersection CWs may be acoustic or electromagnetic waves. It point of the radial plane and the equiphase surface is is conceivable that there may be CWs of some other θ v0R = c /cos 0. The “gradient” compression (see below) nature. The CW radiator has the shape of a truncated of the interior CW field arises in the vicinity of the ori- cone that emits in the direction of its inward normal (so gin and then spreads to distant CW regions. This adds as to focus the CW). It is necessary that all points of the to the velocity of the energy flow (mainly) along the 0Z- radiator surface emit simultaneously so that the wave ≥ axis: vZ = v0Z = const at Z = 0, whereas vZ vZ0 at Z > vector be orthogonal to the CW front. In acoustics, this 0. Thus, the focusing of a CW is accompanied by the is achieved with electroacoustic (piezoceramic, etc.) formation of a longitudinal self-accelerating flow of radiators and the CWs can be visualized by the shadow or interference methods (see, e.g., [1, 2]). In the case of electromagnetic waves, the requirement of simulta- 1 neous emission calls for a specially designed radiator, j, E and there seems to be no visualization techniques other r than indirect ones. Nevertheless, theoretical studies revealed that there are general laws governing both K acoustic and electromagnetic CWs. This fact may facil- 2(max) itate research into the general aspect as well as the elec- S tromagnetic aspect of the problem. E 1 θ The present study concentrates on electromagnetic 0 S ln CWs, the notation being borrowed from [3]. It is perti- H nent to note that the velocity of an energy flow near the 0 Sz H z axis, which is dealt with in what follows, has nothing in common with the phase or group velocity of light. The energy flow arises because a CW acts as a conical plunger that both focuses and accelerates the wave energy in the direction of the CW axis. The figure sketches (1) a radiator and (2) a CW at the instant when the vertex of the cone is at the origin, the angle between θ the generatrix and the axis being denoted by 0. The arrows indicate the directions of the current density j, the fields E and H, and the wave vector k. On the con- Figure.

CONICAL WAVES PRODUCING LONGITUDINAL POWER FLOWS 719 energy (momentum) near the 0Z-axis, with the flow It is seen that Eqs. (1)–(3) completely define the CW velocity growing toward the 0Z-axis. A CW must have field before the compression stage. a gradient structure for the flow to be efficient, e.g., to The second part of the problem is solved with the have a sufficient duration. Specifically, the field must help of the energy and momentum conservation laws: increase in going from the origin along the 0Z-axis and ∂ k ∂ k from the surface of field maximum (the conical surface T i / X = 0. Let us integrate the corresponding equa- in figure) toward the 0Z-axis. tions over the region bounded by the conical surface, taking into account that, during the focusing, the region Now, we outline how to solve the problem of a lon- contracts with a speed equal to c. For example, S in a gitudinal CW field. The first is to find a CW field in the Z three-dimensional space is found as follows: form of a harmonic wave traveling in the direction of ln. This stage of evolution goes on before the interior CW Ð1 Sú dV(t) = {σ d f + R σ ϕd f ϕ + σ d f }. (4) region begins contracting. It is characterized by the ∫ Z °∫ ZR R Z ZZ Z ↑↓ relations S1 ln and W1 = S1c, where S1 is Poynting’s Deriving a solution obtained from Eq. (4), we have to vector and W1 is the energy density. Then, the field integrate it using Eqs. (2) with respect to time subject compression is described in terms of S and the energy to initial condition (3) for t = 0. Applying the mean- flow velocity v with the initial condition in the form of value theorem, we arrive at expressions for S and W . Below, we present the frag- 1 1 2( )  θ ments of a solution procedure to substantiate the above 3cEl 0 ln cos2 0 S˜ = ------------[1 Ð cos[2k(l Ð L )]] ideas and reproduce the necessary expressions. Z 8πsinθ 2 n n 0  12kLn Solving the first part of the problem reduces to treat- (5) ing a CW wave equation derived from Maxwell’s equa-  2 2 3 3 4 4   tions in the cylindrical coordinates. The solution has 1 [ ] c t [ ] c t [ ] c t [ ] + ------ct b1 + ------b2 + ------b3 + ------b4  , the form r2z 2 3 2   ( ) Ð1 { ( )} where El = El 0 ln Ln exp Ðik Ln Ð ln + ct ; z  a  a2  1  [b ] ------Ð r 1 + ----2 ---- l Ð1 1 =   Ð  n + ------2-----0------2 , Hϕ = E (0)cos2θ l L cosθ 2 2 ( θ ) l 0 n n (1) 0 4k r 2cos 0 ×{( )Ð1 [ ] [ ]} 2kLn sin k(ln Ð Ln Ð ct) Ð cos k(ln Ð Ln Ð ct) , ( 0) [ ] 3 r Ð r  a2 6 b2 = ------θ------1 + ---- Ð ------θ---- where l designates the direction along and parallel to zsin 0 2 sin 0 the cone; L are current values of the coordinates of the n 1 1 3a  1  points in the interior region of the CW, counted in the Ð ------1 + ------l + ------θ  θ  n 2 0  cos 0 2 zsin 0 θ direction of ln; and ln are the values of Ln on the bound- 16k r cos 0 ary (i.e., on the cone surface). The components of the Maxwell’s tension tensor are as follows: l  + 1 + ------n------, 0 θ  σ ( π)Ð1( 2 2 2 ) 4r cos 0 ZZ = 8 EZ Ð ER Ð Hϕ ;

Ð1 2 2 2 3  1 a2 σ = (8π) (E Ð E Ð Hϕ ); [ ] RR R Z b3 = Ð------θ----1 + ------θ---- + ---- , zsin 0 cos 0 2 σ σ ( π)Ð1 (2) RZ = ZR = 4 EREZ; 2 3l  Ð1 2 2 2 [b ] = a k 1 Ð ------n----- cosθ , σϕϕ = (8π) (Hϕ Ð E Ð E ); 4 1  θ  0 R Z zsin 0 σ ϕ, σϕ , σ ϕ, σϕ = 0; R R Z Z 0 < ( θ ) θ 2 θ r r, a1 = 1 + cos2 0 cos2 0, a2 = cos 2 0. where ER, EZ, and Hϕ are described by Eqs. (1) and (2). ˜ Ð2 The field compression and the energy density are given Equation (5) indicates that SZ is proportional to r Ð1θ by the formulas and sin 0. In the context of a linear approximation with respect to t, the flow density rapidly rises with θ {( )Ð1 ( ) 2 ( )} S1 = 2N1ccos2 0 4kLn sin 2X Ð cos X ; time. Taking into account nonlinear terms reveals that the growth slows down to some extent after a time. The { 2 θ 2 ( ) W1 = N1 2cos 0 cos X (3) reason is that the integration procedure was linearized Ð2 and, more importantly, that the amplitude was assumed + (2k ) cos2θ [sin(X) Ð 4kL cos(X)]sin(X) }, Ln 0 n to be uniform over the radiator surface. If one aims to produce an accelerating flow or at least to maintain a 2 π Ð1 where N1 = El (0)ln(8 Ln) and (X) = (ln Ð Ln Ð ct). steady velocity, then the radiator field must increase

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 720 KOLMAKOV with Z (according to a certain law), as mentioned eration over conventional approaches [4] has been above. demonstrated above.) The technique presented here ˜ could also serve to generate high-power pencil beams The expression for Sϕ shows that the flow rotates of electromagnetic radiation. Furthermore, it suggests about the 0Z-axis. The rotation becomes more and the idea for a novel device that transforms data from the ˜ more intense as r approaches zero. The component SR form of a spatial distribution (written simultaneously Ð1θ ˜ ӷ ˜ onto a CW equiphase surface) into the form of a time- varies proportionally to cos 0 so that SZ SR for dependent quantity. CWs offer a promising means of ˜ small r. During the compression, SR first rises then producing the echo effects [5], including a change in the radiation direction. Other potential areas of applica- drops, decreasing more rapidly than does S˜ . A more Z tion are flowmeters [6], medicine, and some other tech- detailed description of CWs can be achieved with the nologies. Finally, CWs could set the stage for the man- help of numerical methods, whose algorithm is evident ifestation of yet unknown properties or types of matter. from the preceding. The velocity of the energy flow in n ˜ Ð1 the 0Z-direction, defined by v Z = SZ W , is expressed as follows: REFERENCES 1. S. A. Abrukov, Shadow and Interference Methods for the n 2( )( π ˜ θ )Ð1 v Z = cEl 0 8 W sin 0 Investigation of Optical Inhomogeneities (Kazan. Univ., Kazan, 1962). × { 3 ( 2)Ð1 θ 2 θ (6) lnr z 6kLn sin 0 cos 0 2. D. Holder and R. North, Schlieren Methods (H. M. Sta- tionary Off., London, 1963; Mir, Moscow, 1966). + 2rl { f (t)}sin2 θ cosÐ1 θ }. n 0 0 3. L. D. Landau and E. M. Lifshitz, The Classical Theory Here, f(t) denotes the expression in braces appearing in of Fields (Nauka, Moscow, 1973; Pergamon, Oxford, 1975). Eq. (4). The expression for W˜ is omitted because it is n 4. A. N. Matveev, Electrodynamics and Relativity Theory rather cumbersome. It follows from Eq. (6) that v z (Vysshaya Shkola, Moscow, 1964). grows with time, since the denominator increases at a 5. I. A. Kolmakov and N. N. Antonov, Fiz. Plazmy 18 (10), lower rate than does the numerator; all the comments 1372 (1992) [Sov. J. Plasma Phys. 18, 708 (1992)]. on Eq. (5) apply to v n as well. 6. I. A. Kolmakov, Pis’ma Zh. Tekh. Fiz. 25 (5), 53 (1999) Z [Tech. Phys. Lett. 25, 189 (1999)]. In conclusion, converging CWs may find applications in many areas. For example, they could be used for particle acceleration. (The advantages of CW accel- Translated by A. Sharshakov

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 721Ð722. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 52Ð56. Original Russian Text Copyright © 2000 by Kolyada.

Pulsed High-Current Electron Beams Formed Without Vacuum Yu. E. Kolyada Azov State Technical University, Mariupol, Russia Received November 30, 1999; in ﬁnal form, March 10, 2000

Abstract—Microsecond pulsed electron beams were experimentally obtained under atmospheric pressure conditions in a high-current arc discharge channel of a plasma accelerator. © 2000 MAIK “Nauka/Interperiodica”.

Some applications involving electron beams, such as niently presented as electron-beam processing, radiation-induced polymer- E 3 Z ization, modification of the surface properties of various -----c = 3.88 × 10 --- , (3) materials, encounter the problem of extracting the beams P I into atmosphere. This is achieved by using either a thin metal foil separating the vacuum chamber of an acceler- where the parameters Ec/P and I are measured ator from the atmosphere or a complicated scheme of in V/(cm Torr) and eV, respectively; in air, the inelastic differential pumping. However, good prospects in this loss factor can be taken equal to I = 15Ð80 eV. respect are offered by methods of direct electron beam In contrast to previous works dealing with nanosec- production under atmospheric conditions. ond pulsed beams, this work was devoted to experimental investigation of the microsecond electron The formation of high-energy electrons by dis- beams formed under the action of additional high-volt- charge in air was experimentally studied in [1Ð4], age pulses in a high-current arc discharge channel of an where the electron beam production was accompanied edge plasma accelerator operating under atmospheric by the X-ray emission registered in the initial stage of pressure conditions. the spark discharge development. The electron beam Figure 1 shows a schematic diagram of the experi- pulse duration was of the order of 10 ns in the range mental system. The plasma accelerator (PU) depicted of kV voltages. The beam formation under these condi- in the right-hand part of the scheme comprises a thick- tions was governed by the electron runaway effect [5, 6]. wall dielectric tube (wall thickness, ~1 cm; internal For a nonrelativistic electron, this phenomenon takes place provided that the retarding force is smaller than the electrostatic force of the external (accelerating) ⌳100 kV field. The retarding force is related to ionization losses and described by the well-known BetheÐBloch formula: 1 125 kV P2 4 2 neZ 2 ⑀ F()⑀ = Ð------0 - ln------, (1) πε2 I C1 8 0 R where e is the electron charge, ⑀ is the kinetic energy of P1 L electron, n0 is the concentration of gas molecules, Z is the atomic number, I is the average energy of inelastic R 5 kV ε losses, and 0 is the dielectric constant. The critical C2 electric field strength, above which the acceleration C1 (and runaway) of electrons takes place, is given by the formula

3 e n0Z Fig. 1. A schematic diagram of the experimental system: Ec = ------. (2) 4πε22.72I (1) plasma accelerator; (2) electron beam; C2, capacitor 0 bank (charge voltage, 5 kV; high-voltage start-up pulse amplitude, 100 keV); L, decoupling capacitor; C1, R, P1, According to [6], this relationship can be conve- P2, Marx generator (charging voltage, 125 kV).

722 KOLYADA

a The high-voltage pulse acting upon the discharge gap of the plasma accelerator (occurring at this time instant at a reduced pressure) produced a pulsed electron beam. Figure 2 shows typical oscillograms of the (a) discharge current of the Marx generator (measured with a Rogowsky coil), (b) voltage applied to the accelerating b gap, (c) X-ray radiation measured by a crystal detector with multiplier phototube, and (d) microwave signal c detected by a horn antenna in a 3Ð10 cm wavelength range. The oscillograph beam was swept at a rate of 5 µs/div with a sensitivity of 1 kA/div (for current) and 125 kV/div (for voltage). An analysis of the oscillograms shows that a microsecond pulsed electron beam is formed in the system with a beam current of ~1 kA d and an electron energy of ~200 keV. The electron beam production is preceded by a high-power arc discharge in the accelerator channel, which heats the emitter (rod electrode) to a temperature close to the melting point. This probably accounts for excitation of the autother- moelectron emission. Fig. 2. Typical oscillograms of the (a) discharge current of Estimates of the Ec/P ratio by formula (3) suggest the Marx generator, (b) voltage applied to the accelerating that the electron runaway effect can be manifested gap, (c) X-ray radiation pulse, and (d) microwave pulse in a under the experimental conditions studied. Indeed, for 3Ð10 cm wavelength range. The oscillograph beam was a gas pressure in the discharge channel decreasing to swept at a rate of 5 µs/div with a sensitivity of 1 kA/div (for − current) and 125 kV/div (for voltage). 1 5 Torr during the plasma accelerator operation and an average electric ﬁeld strength of ~20 V/cm in the gap, we obtain E/p ~ (0.4Ð2) × 104 V/(cm Torr). In air, × diameter, 8 mm; length, 40 cm) with two (rod and ring) calculation by formula (3) yields Ec/P = (0.8Ð3.5) electrodes spaced by ~10 cm. The electric circuit was 103 V/(cm Torr), which is markedly lower than the level supplied from a capacitor bank with a capacitance C2 = achieved in experiment. × Ð3 1.5 10 F operating at a working voltage of 5 kV. The Thus, the results of our investigation demonstrate design and operation of a similar accelerator scheme the possibility of obtaining microsecond pulsed elec- were described in detail [7]. A discharge between electron beams formed in a high-current arc discharge trodes was initiated by high-voltage (~100 kV) start-up channel without vacuum. pulses with a base width of ~5 µs. The inductor coil The author is grateful to Ya.B. Fa nberg for fruitful with L = 0.3 mH prevented the starting pulses from pen- œ discussions and useful comments on the experiment etrating into the capacitor bank. Upon the interelec- planning and interpretation. trode gap breakdown, the capacitor bank produced a high-power discharge pulse with a duration of ~1.4 ms, whereby not less than 60% of the stored energy was REFERENCES converted into the plasma bunch energy. 1. Yu. L. Stankevich and N. S. Kalinin, Dokl. Akad. Nauk A characteristic feature of this accelerator was that SSSR 177 (1), 72 (1967) [Sov. Phys. Dokl. 12, 1042 the high-power discharge formation and the dense (1968)]. plasma bunch injection into the ambient medium 2. R. C. Noggle, E. P. Kriger, and I. R. Way Land, J. Appl. account for a pressure increase of up to 100 atm in the Phys. 39 (10), 4746 (1968). discharge channel. This pulse is followed by a depres- 3. L. V. Tarasova, L. N. Khudyakova, T. V. Loœko, and sion-type wave, whereby the pressure at the pulse end V. A. Tsukerman, Zh. Tekh. Fiz. 44 (2), 564 (1974) [Sov. may drop markedly below the atmospheric pressure Phys. Tech. Phys. 19, 352 (1974)]. level (1Ð5 Torr). At this instant, a negative high-voltage 4. P. A. Bokhan and G. V. Kolbychev, Zh. Tekh. Fiz. 51 (9), pulse was applied to the interelectrode gap from a two- 1823 (1981) [Sov. Phys. Tech. Phys. 26, 1057 (1981)]. stage Marx generator (see the left part of Fig. 1). The 5. A. V. Gurevich, Zh. Éksp. Teor. Fiz. 39 (5), 1296 (1960) generator circuit parameters are as follows: C1 = [Sov. Phys. JETP 12, 904 (1960)]. 0.64 µF, R = 48 kΩ (P1 and P2 are the start-up and 6. Yu. D. Korolev and G. A. Mesyats, The Physics of Pulse decoupling discharge gaps, respectively). The genera- Breakdown in Gases (Nauka, Moscow, 1991). tor circuits charged by a voltage of 125 kV provided for 7. I. A. Glebov and F. G. Rutberg, High-Power Plasma a voltage pulse of ≤250 kV in the load (interelectrode Generators (Énergoatomizdat, Moscow, 1985). gap). The inductance L prevented the starting pulses from penetrating into the circuit of capacitor bank C2. Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 723Ð725. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 57Ð62. Original Russian Text Copyright © 2000 by Grinyaev, Chertova.

Dynamic Theory of Defects and Creep in Solids Yu. V. Grinyaev and N. V. Chertova Institute of Strength Physics and Materials Science, Siberian Division, Russian Academy of Sciences, Tomsk, 634055 Russia Received December 20, 1999

Abstract—The dynamic equations of the ﬁeld theory of defects are used to analyze the creep characteristics. A number of interesting results are obtained, of which some are consistent with the well-known experimental data, whereas others are quite new. In particular, the existence of a critical value of the applied stress and an unstable steady-state creep rate are established, which separate the stable and unstable creep modes and can be of great practical importance. © 2000 MAIK “Nauka/Interperiodica”.

It is well known that a deformable solid is a compli- study the creep phenomenon. In the above expressions, cated hierarchic system consisting of a large number of α is the dislocation-density tensor, Vext is the velocity of interacting structural elements that are considered on elastic displacements, ρ is the density of the medium, η different scales [1]. The behavior of various hierarchic is the viscosity coefficient, B and S are the theory con- systems is a subject of synergetics [2], where various stants, and δ is the Kronecker symbol. The signs (×) systems are studied on the micro-, meso-, and mac- and (á) indicate the vector and the scalar multiplication, rolevels. On the microscopic level, individual structural × elements are considered by setting their positions, the sign (. ) corresponds to the vector multiplication velocities, and the laws governing their interactions. On with respect to the first indices and the scalar multipli- the mesoscopic level, the variables describing an cation with respect to the second indices. Equation (2) is obtained under the condition that defects are uni- ensemble of interacting structural elements are intro- α duced. Finally, in order to describe the system on the formly distributed, so that the , I field strengths are macroscopic level, the initial mesoscopic level is used independent of the coordinates. According to [7], such to develop the methods which would provide the pre- a situation can be observed at the material yield stress, diction of macroscopic structures. In what follows, we when single defects are randomly distributed over the consider the characteristics of deformation in solids material and cannot form any spatial structure. Since during creep within the framework of the field theory of many of the experimental data on creep were obtained defects [3, 4] representing a mesoscopic description of under tensile tests on rods [8Ð9], we consider here an equation of the uniaxial deformation. In terms of the the system. According to various physical theories [5, 6], η τ η the elementary creep processes in solids at moderate dimensionless quantities v = Ð(B/ )I11, = ( /B)t, and η2 σ temperatures are provided mainly by dislocation S = (B/ ) 11, this equation has the form motion. Using the dynamic equations of the field theory of defects, dv 2 ------= v /2 Ð v + S, (3) × τ B()∇ ⋅ I = ÐB()α . I Ð ρVext, d ∂α ∇α⋅ ==0, ∇ ⋅ I ------, where v is the plastic deformation rate. At S = const ∂t (which corresponds to creep under a constant stress), we may use the stationarity condition ∂I δ2 (1) S∇α⋅ =Ð B----- ÐS αα⋅ Ð --- α ∂t 2 v 2/2 Ð v +0S = (4) δ2 ext ÐBI⋅ IÐ --- I ÐηIÐ σ, 2 to determine two rates of the steady-state creep: we arrive at the equation v1==p112+ Ð,Sv2 ==q 112Ð Ð.S(5) ∂I δ2 ext B----- +++0,BI⋅ IÐ --- I ηIσ=(2) ∂t 2 Analyzing the plot (4) and the phase portrait (3), we see that the stationary state q is stable, whereas the state p relating the velocity I of the defect flow to the exter- is unstable. If the controlling parameter S approaches nally induced stresses σext, thus making it possible to the value 1/2, the stable and the unstable stationary

724 GRINYAEV, CHERTOVA v (a) Figures 1a and 1b show the strain rate curves illustrating the evolution at S = 0.2. At short times, the form τ 0.4 of the function v( ) is determined by the initial value v0 that can vary within three ranges: 0 < v0 < q, q < v0 < p, < v > τ 0.3 q 0 p and v0 > p. Figure 1a shows the v( ) curves obtained at τ v0 from the two first ranges. At v0 > pv( ), the curve has a singularity (the denominator of the expression goes to 0.2 v = q zero, Fig. 1b), whence the time to the system failure v0 < q (when the strain rate becomes inﬁnite) is determined as 0.1 τ [ ( )] [( ) ( )] 1 = 2/ p Ð q ln v 0 Ð q / v 0 Ð p . At S > S∗, a solution to Eq. (3) can be written as 0 1 2 3 τ ν α2 ν2 + α2 v (τ) + (b) = ------ν------+ ------ν------ 20 (8)  cos(ατ/2)  × ------, 10 cos(ατ/2) Ð v /αsin(ατ/2) α 0 where 2 = p Ð q and v = 1 Ð v0. Obviously, the lifetime 2 4 6 8 10 τ of a real system is limited by the condition –10 cos(ατ/2) Ð (v /α)sin(ατ/2) = 0, (9) –20 from which it follows that the time to failure equals τ ( α) (α v) –30 2 = 2/ arctan / . Usually, the experimental data on creep are repre- Fig. 1. Evolution of the creep rate at S < S∗: (a) v0 < p and sented in the form of a creep curve characterizing the (b) v0 > p. change in deformation with time. Within the present approach, the corresponding dependences are obtained by integrating Eqs. (7) and (8) with respect to time. At ∈ S < S∗, the creep curve is described by the relationship ε(τ) ε τ 12 = 0 + q (10) ( ) [( ) ( ) (( )τ )] + 2ln p Ð q / p Ð v 0 Ð q Ð v 0 exp p Ð q /2 , 9 S > S* and, at S > S∗, it has the form ε(τ) = ε + τ 6 0 (11) Ð 2ln cos(ατ/2) Ð (v /α)sin(ατ/2) . S < S* 3 Figure 2 shows the curves of stable and unstable creep obtained at S = 0.15, v0 = 0.7 and S = 0.55, v0 = ε 0.8, with 0 = 0.0002 in both cases. 0 5 10 15 20 τ The creep analysis based on the equation describing the evolution of a ﬂow of defects showed that the char- Fig. 2. Typical creep curves. acter of the creep process is essentially dependent on the applied load S and the initial deformation rate v0. At a constant stress, the range of the stable creep is limited states approach one another and coincide at by the conditions 0 < S < S∗ and 0 < v0 < p, where S∗ S* = 1/2 , (6) is the critical stress and p is the unstable stationary deformation rate. The stress S allows us to introduce a whereas at S > 1/2, they both simultaneously disappear. ∗ At S < S∗, the solution to Eq. (3) has the following limit of the stable creep determined by the material constants σ* = η2/2B. Beyond the stable creep limit form: τ τ (v0 > p or S > S∗), the time to system failure ( 1, 2) [(v ) (v )] [( )τ ] (τ) p Ð q 0 Ð p / 0 Ð q exp p Ð q /2 decreases with an increase of the external load and the v = ------[---(------)----(------)---]------[--(------)---τ------]--. (7) 1 Ð v 0 Ð p / v 0 Ð q exp p Ð q /2 initial deformation rate.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 DYNAMIC THEORY OF DEFECTS AND CREEP IN SOLIDS 725

The expressions obtained for the creep rate describe REFERENCES the well-known monotonic increase of the rate with 1. V. E. Panin, V. A. Likhachev, and Yu. V. Grinyaev, Struc- stresses [8] and the fact that q(S = 0) = 0, which is also tural Levels of Deformation in Solids (Nauka, Novosi- taken into account in constructing the phenomenologi- birsk, 1985). cal expressions for this quantity [9]. The typical creep curve has three regions. In the ﬁrst region, the deforma- 2. H. Haken, Advanced Synergetics: Instability Hierarchies tion rate gradually decreases down to the minimum of Self-Organizing Systems and Devices (Springer-Ver- lag, New York, 1983; Mir, Moscow, 1985). value. In the second segment, the deformation rate remains constant. Finally, in the third region, the defor- 3. Yu. V. Grinyaev and V. E. Panin, Dokl. Akad. Nauk 353 mation rate increases again until the process is ended (1), 37 (1997) [Phys. Dokl. 42, 108 (1997)]. with the specimen failure. In our case, the curve ε(τ) at 4. Yu. V. Grinyaev, N. V. Chertova, and V. E. Panin, Zh. S < S∗ has only the stages of non-steady-state and Tekh. Fiz. 68 (9), 134 (1998) [Tech. Phys. 43, 1128 steady-state creep, whereas the third segment is not (1998)]. described by this expression. However, the typical 5. A. H. Cottrell, Dislocations and Plastic Flow in Crystals experimental creep curves were obtained under con- (Clarendon Press, Oxford, 1953; Metallurgizdat, Mos- stant loading conditions. According to [8], in the case cow, 1958). of constant stresses considered here, no creep accelera- 6. V. M. Rozenberg, Creep in Metals (Metallurgiya, Mos- tion can be observed until the very moment of specimen cow, 1967). failure. The dependence ε(τ) at S > S∗ is described 7. N. A. Koneva and É. V. Kozlov, Izv. Vyssh. Uchebn. in [9], where it is also shown that the creep curve does Zaved., Fiz., No. 2, 89 (1990). not necessarily contain a region of decreasing deforma- 8. L. N. Kachanov, Theory of Creep (Fizmatgiz, Moscow, tion rate: upon a short period of deformation at a con- 1960). stant rate, the creep rate starts increasing. In fact, the 9. Yu. N. Rabotnov, Mechanics of Deformed Solids whole diagram reduces to the third segment alone. The (Nauka, Moscow, 1979). results obtained in our study indicate that at different values of the applied stress, all the three creep modes can be observed on the same specimen. Translated by L. Man

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 726Ð728. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 63Ð67. Original Russian Text Copyright © 2000 by Abramov, Novik.

A Single-Electron Transistor Model Based on a Numerical Solution of the Poisson Equation I. I. Abramov and E. G. Novik State University of Informatics and Radio Engineering, Minsk, Belarus Received December 25, 1998; in ﬁnal form, October 25, 1999

Abstract—A numerical model of a single-electron metal transistor is proposed based on a solution of the Pois- son equation. The model provides a good agreement with experimental data on the currentÐvoltage character-

At present, single-electron transistors (SETs) are con- Here, I1 and I1 are the direct and reverse currents ventionally simulated using semiclassical models [1]. passing through I the drain tunneling junction, respec- A disadvantage in common for these models is that fit- ting to the experimental data is performed using param- tively; I2 and I2 are the corresponding currents in the eters—capacitances and resistances of the tunneling source tunneling junction; and ρ(n) is the probability of junctions—that cannot be calculated. As a result, the finding n excess charge carriers on the “island” (see models do not adequately reflect the structure design Fig. 1). and physical properties of the SET materials, which hinder prediction of the electrical characteristics of The values of voltage drops across the tunneling final devices. junctions are determined taking into account the results of calculation of the electrostatic potential distribution The purpose of this work was to elaborate a model ϕ and the contact potential differences in the structure based on the geometric and physical parameters of studied. Resistances of the tunneling junctions can be SETs. calculated using well-known relationships [4, 5]. An analysis of published data showed that most Assuming that the magnetic field produces a negli- types of SETs can be described using a structural gibly small effect on the transistor operation, the elec- scheme depicted in Fig. 1 [2]. Below, we will also study trostatic potential distribution ϕ can be determined the SET structures of this kind. Within the framework from the Poisson equation of the semiclassical approach, the currentÐvoltage ∇ε∇ϕ characteristics of SET structures can be calculated = ÐqM, (3) using a master equation [3]. According to this, currents ε passing the source and drain tunneling junctions obey where is the dielectric permittivity and qM is the bulk

the following relationship: I x ∂ρ()∂ ()()ρ() AB n /t=[I1nÐ1+I2nÐ1]nÐ1

[]ρ()()() (1) D +I1n+1+I2n+1n+1 S

2 2 I I

[]ρ() () () []ρ() () () ÐI1n+I1n nÐI2n+I2n n. D C

The total current through the SET structure is

2 I G +∞

ρ()()() ()

I=∑n I1nÐI1n

∞ 2 n=Ð I (2) +∞ y ρ()()() () = ∑n I2nÐI2n . Fig. 1. Schematic diagram representing the proposed SET n=Ð∞ model: D, drain; I, “island”; S, source, G, gate.

A SINGLE-ELECTRON TRANSISTOR MODEL BASED ON A NUMERICAL SOLUTION 727

I, pA 0.8 1 0.6 2 3

0.4

0.2

0 0.1 0.2 0.3 0.4

Vd, V Fig. 2. Comparison of the experimental and calculated currentÐ voltage characteristics: (1) experimental data; (2) proposed model; (3) MOSES program. density of charge. Below, we also assume electroneu- computer facilities versus 20 min for the MOSES pro- trality of the plasma at the “island” (except for the gram on a Pentium 200. excess free charge carriers) and neglect charges in the Let us analyze the factors responsible for the devia- dielectric. Therefore, qM is determined only by the aver- tion from experiment observed for the results of calcu- age density of excess charge carriers immediately on lations using both our numerical model and the MOSES the “island.” Under these assumptions, Eq. (3) can be model (based on the Monte-Carlo method) [10]. In the solved in the active SET region between the source, proposed model, the electrical characteristics are cal- drain, and gate (the two-dimensional region ABCD in culated using a two-dimensional potential distribu- Fig. 1). The boundary conditions are represented by the tion in the structure determined by solving the Pois- Dirichlet conditions at the contacts and the Neumann son equation. In the well-known models [1, 10], the conditions at the free boundaries. potential distribution is set according to a one-dimensional case, irrespective of the contact potential dif- The finite-difference approximation of the Poisson ferences. Therefore, our model provides for a better equation (3) was formulated within the framework of agreement with experiment than the above models. the TikhonovÐSamarskiœ approach [6]. All values were Still, small deviation from the experiment can be normalized using coefficients introduced in [6]. The explained by neglect of the parasitic effects in our system of linear algebraic equations obtained upon the model. Note, however, that these models are also not finite-difference approximation was solved using a taken into account in the known semiclassical matrix trial method [7]. models. The proposed semiclassical two-dimensional Thus, we have developed a new semiclassical two- numerical SET model was used as a base for a special dimensional numerical model of a metal SET based on computational program written in PASCAL adapted to solving the Poisson equation. The model provides good agreement with experimental data on the currentÐvolt- an IBM PC/AT and included into a program package age characteristics of SETs and can be used for predict- NANODEV intended for the modeling of nanoelec- ing electrical characteristics of these devices. tronic devices [8]. Using this program, we have calculated the currentÐvoltage characteristics of a real metal The work was pertly supported by the Scientific- SET [9] based on the Ti/TiOx/Ti tunneling junctions Technological Programs of the Republic of Belarus on operating at a nonzero temperature (T = 103 K). As Informatics, Low-Dimensional Systems, and Nano- seen from the data presented in Fig. 2, the results of cal- electronics. culations performed using the proposed model agree well with the experimental data. For comparison, Fig. 2 REFERENCES shows a curve calculated by the MOSES program [10]. Note that the time required for calculating a currentÐ 1. Single Charge Tunneling: Coulomb Blockade Phenom- voltage characteristic is 3 min for our program and ena in Nanostructures, ed. by H. Grabert and

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

728 ABRAMOV, NOVIK

M. H. Devoret (Plenum, New York, 1992), NATO ASI 7. A. A. Samarskiœ and E. S. Nikolaev, Methods of Solving Ser., Ser. B: Phys., Vol. 294. Network Equations (Nauka, Moscow, 1978). 2. I. I. Abramov and E. G. Novik, Izv. Belarus. Inzh. Akad., 8. I. I. Abramov, I. A. Goncharenko, and E. G. Novik, No. 2 (6/2), 4 (1998). Pis’ma Zh. Tekh. Fiz. 24 (8), 16 (1998) [Tech. Phys. 3. K. K. Likharev, IEEE Trans. Magn. MAG-23 (2), 1142 Lett. 24, 293 (1998)]. (1987). 9. K. Matsumoto, M. Ishii, K. Segawa, et al., Appl. Phys. 4. J. G. Simmons, J. Appl. Phys. 34 (6), 1793 (1963). Lett. 68 (1), 34 (1996). 5. H. Fukui, M. Fujishima, and K. Hoh, Jpn. J. Appl. Phys., 10. R. H. Chen and K. K. Likharev, Appl. Phys. Lett. 72 (1), Part 1 36 (6B), 4147 (1997). 61 (1998). 6. I. I. Abramov and V. V. Kharitonov, Numerical Modeling of the Elements of Integrated Circuits (Vysshaya Shkola, Minsk, 1990). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 729Ð733. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 68Ð76. Original Russian Text Copyright © 2000 by Demidov, Kalinikos.

Green’s Tensor of the Maxwell Equations for a Planar MetalÐDielectricÐFerromagnetÐDielectricÐMetal Sandwich Structure V. E. Demidov and B. A. Kalinikos St. Petersburg State Electrotechnical University, St.Petersburg, 197376 Russia e-mail: [email protected] Received April 5, 2000

Abstract—Expressions for the dyadic Green’s functions of the Maxwell equations for a planar metalÐdielec- tricÐferromagnetÐdielectricÐmetal sandwich structure are derived taking into account the electromagnetic retardation. © 2000 MAIK “Nauka/Interperiodica”.

Spin waves propagating in ferrite films and layered the Maxwell equations. In the first approach, expres- structures are widely applied in electrically tuned sions for components of the permeability tensor are devices used for the processing of microwave signals [1]. obtained from the magnetization equation and are used Among these devices are filters, resonators, delay lines, in the solution of the Maxwell equations. In the second suppressors of weak signals, convolvers, etc. Usually, approach, the dyadic Green’s functions of the Maxwell the dispersion characteristics of spin waves propagat- equations accounting for boundary conditions are first ing in ferromagnetic films and layered structures based obtained and then used in solving the magnetization on these films are calculated using the magnetostatic equation. As was shown in [2], the second approach approximation, that is, neglecting the electromagnetic (the method of dyadic Green’s functions or Green’s retardation. This substantially simplifies the form of tensor) is more convenient, especially, in solving prob- dispersion equations and, in some cases, makes it pos- lems of the spectrum of spin waves with account of sible to write these equations in an explicit form (see, both the dipoleÐdipole and exchange interaction. In for example, [2, 3]). Such an approach is justified when particular, the problem of the spectrum of spin waves the permittivity of both the ferromagnetic film and the and the problem of linear excitation of such waves in fer- ambient space is small and the structure operates at romagnetic films and layered structures were solved moderate frequencies (below 10 GHz). Among the dis- using this method [2, 3]. Another important advantage advantages of the magnetostatic approximation are of the method of the dyadic Green’s functions is that the substantial errors arising in attempts to calculate the first steps of calculations do not require linearization of parameters of such films in the millimeter wave band as the magnetization equation. This makes the second well as in the case when the film is in contact with approach more suited to solve problems related to non- objects having a large permittivity. linear spin waves (see, for example, [5]). In recent years, interest has revived in creating The dyadic Green’s functions for the metalÐdielec- devices for the microwave signal processing based on tricÐferromagnetÐdielectricÐmetal (MDFDM) planar ferroelectric materials with a large permittivity con- sandwich structures were obtained previously [2]. trolled by an applied static electric field (see, for exam- However, the derivation was based on the solution of a ple, [4]). Advantages of such devices are the high speed magnetostatic equation system, which does not allow and small energy required for the electrical tuning. one to use these solutions for description of the struc- A similar mechanism can also be proposed to control tures involving layers with large permittivity, in partic- dispersion characteristics of spin waves in structures ular, ferroelectric films. involving contacting ferromagnetic and ferroelectric The purpose of this work is to derive expressions for layers. In order to evaluate parameters of these struc- the dyadic Green’s functions (Green’s tensor) of a pla- tures, it is necessary to develop a theory of spin waves nar MDFDM sandwich structure on the basis of solu- taking into account the electromagnetic retardation. tion of the complete system of Maxwell equations In the theory of normal spin oscillations and waves involving vortex fields. propagating in ferromagnetic films and layered struc- Consider a plane-parallel layer structure infinitely tures based on these films, two approaches are usually extended in the YOZ plane. The structure consists of an applied. These approaches differ in methods of simul- isotropic ferromagnetic film of thickness L separated taneous integration of the magnetization equation and on both sides from perfectly conducting screens by

730 DEMIDOV, KALINIKOS

ε a a

k Z ε z L L

ε b b

Geometry of the layered structure. dielectric layers of thickness a and b, respectively. The remaining components can be expressed in terms of ferroelectric film is described by variable magnetiza- these known values. ⑀ tion m and permittivity L, and the dielectric layers are Let us choose components h and E as primary ⑀ ⑀ x z described by permittivities a and b, respectively (see functions, since these components satisfy boundary figure). The origin of coordinate system is placed on the conditions on the metal surfaces. These functions are middle plane of the ferromagnetic film. determined from two second-order differential equa- Let us relate waves of the magnetic and electric tions, ω ω field, h = hk(x)exp(i( t Ð kzz)) and E = Ek(x)exp(i( t Ð 2 2 ∂ x 2 x 2 x ∂ x ∂ z kzz)) to the waves of variable magnetization m = ------h Ð γ h = Ð k m Ð ------m + ik m , (2) ω ∂ 2 k i k 0i k ∂ 2 k z∂x k mk(x)exp(i( t Ð kzz)). These relationships can be x x obtained from the complete system of Maxwell equa- ∂2 ∂ tions ------Ez Ð γ 2E z = iωµ m y, (3) 2 k i k 0∂ k ω⑀ ⑀ ∂x x rot h = i 0 rE, γ γ γ γ rot E = Ðiωµ (h + m), where the parameter i equals a, b, or L, respectively, 0 (1) for each of the structure layers: div h = Ðdiv m, γ 2 = k 2 Ð k 2 , k 2 = ω2⑀ µ ⑀ , div E = 0 L z 0L 0L 0 0 L γ 2 2 2 2 ω2⑀ µ ⑀ in the following form: a = kz Ð k0a , k0a = 0 0 a, L/2 γ 2 2 2 2 ω2⑀ µ ⑀ b = kz Ð k0b , k0b = 0 0 b. h (x) = Gˆ h(x, x'; k )m (x')dx', k ∫ z k The remaining components are connected with the pri- ÐL/2 mary ones via expressions L/2 ωµ ( ) ˆ ( , ) ( ) E y = Ð------0(h x + m x), Ek x = ∫ GE x x'; kz mk x' dx', k k k kz ÐL/2 z 1 ∂ y z h = Ð ------E Ð m , where Gˆ h (x, x'; k ) and Gˆ E (x, x'; k ) are the dyadic k ωµ ∂ k k z z i 0 x Green’s functions of Eqs. (1). 2 It can be easily shown that system (1) splits into two y iω⑀ ⑀ ∂ z k y h = ------0-----i E + ---0---i m , independent equation systems for the components (Ex, k γ 2 ∂x k γ 2 k hy, Ez) and (hx, Ey, hz) of the electromagnetic field, i i respectively. The first triplet of the field components x kz y corresponds to transverse magnetic (TM) waves, Ek = ω-----⑀-----⑀---hk . whereas the second triplet corresponds to the transverse 0 i electric (TE) waves. It would suffice to know one of the Solving equations (2) and (3) in each layer, we components involved in each triplet, because the obtain expressions for each of six components of the

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 GREEN’S TENSOR OF THE MAXWELL EQUATIONS 731 (γ ) (γ ) ≥ electromagnetic field as functions of the variable mag- ωµ k sinh L xb sinh L xa' x x' yx 0 z  netization. These expressions involve constants that can GE = -γ------(---γ------) L sinh Ld  (γ ) (γ ) < be obtained upon satisfying boundary conditions on the sinh L xa sinh L xb' x x' metal surfaces and layer interfaces. ωµ k Ð ------0-----z{sinh(γ x' )T G (x) + sinh(γ x' )T G (x)}, Separating contributions from different components γ N L a b 3 L b a 4 of the variable magnetization into corresponding com- L ponents of the electromagnetic field, we obtain the final (γ ) (γ ) ≥ expressions for the Green’s functions. In the ferromag- iωµ sinh L xb sinh L xa' x x' yz 0  netic layer, these expressions take the following form: GE = ------(---γ------)- sinh Ld  (γ ) (γ ) < sinh L xa sinh L xb' x x' 2 xx kz iωµ G = Ð δ(x Ð x') Ð ------0{ (γ ) ( ) (γ ) ( )} h γ (γ ) Ð ------cosh L xa' T bG3 x + cosh L xb' T aG4 x , L sinh Ld N  (γ ) (γ ' ) ≥ sinh L xb sinh L xa x x' (γ ) (γ ) ≥ ×  iωµ sinh L xb cosh L xa' x x' zy 0  sinh(γ x )sinh(γ x' ) x < x' GE = Ð------(---γ------)- L a L b sinh Ld  (γ ) (γ ) < sinh L xa cosh L xb' x x' 2 iωµ kz { (γ ) ( ) (γ ) ( )} + ------0{cosh(γ x' )T yG y(x) + cosh(γ x' )T yG y(x)}. + γ------sinh L xa' T bG3 x + sinh L xb' T aG4 x , y L a b 3 L b a 4 L N N

(γ ) (γ ) ≥ For layer a, ik sinh L xb cosh L xa' x x' G xz = Ð------z------ h (γ ) 2 sinh Ld  (γ ) (γ ) < (γ ( )) sinh L xa cosh L xb' x x' xx kz sinh a x Ð a Ð L/2 Gh = Ðγ------(--γ------)------a N cosh aa ikz{ (γ ) ( ) (γ ) ( )} + ------cosh L xa' T bG3 x + cosh L xb' T aG4 x , × ( (γ ) (γ ) ) N sinh L xa' Db + sinh L xb' T a ,

γ (γ ( ))  (γ ) (γ ) ≥ xz ikz L sinh a x Ð a Ð L/2 zx ikz cosh L xb sinh L xa' x x' G = ------ Gh = Ð--γ------(--γ------)------h (γ ) a N cosh aa sinh Ld  (γ ) (γ ) < cosh L xa sinh L xb' x x' × (cosh(γ x' )D + cosh(γ x' )T ), ik L a b L b a Ð ------z{sinh(γ x' )T G (x) + sinh(γ x' )T G (x)}, N L a b 1 L b a 2 (γ ( )) zx ikz cosh a x Ð a Ð L/2 Gh = ------(---γ------)------N cosh aa γ cosh(γ x )cosh(γ x' ) x ≥ x' zz ------L------ L b L a Gh = Ð (γ ) × ( (γ ' ) (γ ' ) ) sinh Ld  (γ ) (γ ) < sinh L xa Db + sinh L xb T a , cosh L xa cosh L xb' x x' γ (γ ( )) γ zz L cosh a x Ð a Ð L/2 + ----L{cosh(γ x' )T G (x) + cosh(γ x' )T G (x)}, Gh = Ð------(---γ------)------N L a b 1 L b a 2 N cosh aa × ( (γ ' ) (γ ' ) ) 2 (γ ) (γ ) ≥ cosh L xa Db + cosh L xb T a , k cosh L xb cosh L xa' x x' yy 0L  Gh = -γ------(---γ------) L sinh Ld  (γ ) (γ ) < 2 cosh L xa cosh L xb' x x' k cosh(γ (x Ð a Ð L/2)) G yy = ------0--L------a------2 h γ y cosh(γ a) k L N a Ð ------0--L---{cosh(γ x' )T yG y(x) + cosh(γ x' )T yG y(x)}, γ y L a b 1 L b a 2 L N × ( (γ ) y (γ ) y ) cosh L xa' Db + cosh L xb' T a , ωµ  (γ ) (γ ) ≥ xy 0kz cosh L xb cosh L xa' x x' ωµ k ⑀ γ cosh(γ (x Ð a Ð L/2)) G = ------ xy 0 z L a a E γ (γ ) GE = ------L sinh Ld  (γ ) (γ ) < γ y ⑀ γ cosh(γ a) cosh L xa cosh L xb' x x' a N a L a ωµ k Ð ------0-----z{cosh(γ x' )T yG y(x) + cosh(γ x' )T yG y(x)}, × (cosh(γ x' )D y + cosh(γ x' )T y ), γ y L a b 1 L b a 2 L a b L b a L N

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 732 DEMIDOV, KALINIKOS ωµ (γ ( )) ωµ γ (γ ( )) yx 0kz sinh a x Ð a Ð L/2 yz i 0 L sinh b x + b + L/2 GE = ---γ------(--γ------)------GE = Ð----γ------(---γ------)------a N cosh aa b N cosh bb × ( (γ ) (γ ) ) × ( (γ ) (γ ) ) sinh L xa' Db + sinh L xb' T a , cosh L xa' T b Ð cosh L xb' Da ,

ωµ γ (γ ( )) zy iωµ ⑀ γ sinh(γ (x + b + L/2)) yz i 0 L sinh a x Ð a Ð L/2 G = ------0 ----L------b ------b------GE = ----γ------(--γ------)------E y ⑀ γ cosh(γ b) a N cosh aa N b L b × ( (γ ) (γ ) ) × ( (γ ) y (γ ) y ) cosh L xa' Db + cosh L xb' T a , cosh L xa' T b Ð cosh L xb' Da .

iωµ ⑀ γ sinh(γ (x Ð a Ð L/2)) The remaining components of the dyadic Green’s func- G zy = Ð------0 ----L------a ------a------tions are identically zero. E y ⑀ γ (γ ) N a L cosh aa The above formulas involve the following designa- tions: × ( (γ ' ) y (γ ' ) y ) cosh L xa Db + cosh L xb T a . ( ) (γ ( )) G1 x = cosh L x + L/2 For layer b, γ L (γ ) (γ ( )) + -γ--- tanh bb sinh L x + L/2 , 2 (γ ( )) b xx kz sinh b x + b + L/2 Gh = γ------(---γ------)------( ) (γ ( )) b N cosh bb G2 x = Ðcosh L x Ð L/2 γ × ( (γ ' ) (γ ' ) ) ----L (γ ) (γ ( )) sinh L xa T b Ð sinh L xb Da , + γ tanh aa sinh L x Ð L/2 , a ik γ sinh(γ (x + b + L/2)) ( ) (γ ( )) xz z L b G3 x = sinh L x + L/2 Gh = --γ------(---γ------)------b N cosh bb γ L (γ ) (γ ( )) + -γ--- tanh bb cosh L x + L/2 , × ( (γ ) (γ ) ) b cosh L xa' T b Ð cosh L xb' Da , G (x) = Ðsinh(γ (x Ð L/2)) ik cosh(γ (x + b + L/2)) 4 L G zx = Ð------z ------b------γ h N cosh(γ b) L (γ ) (γ ( )) b + -γ--- tanh aa cosh L x Ð L/2 , a × ( (γ ) (γ ) ) sinh L xa' T b Ð sinh L xb' Da , (γ ) N = sinh Ld γ cosh(γ (x + b + L/2)) 2 zz L b   γ  Gh = ------(--γ------)------× sinh(γ L) 1 + ------L----tanh(γ a)tanh(γ b) N cosh bb L  γ γ a b   a b × (cosh(γ x' )T Ð cosh(γ x' )D ), L a b L b a γ γ   (γ ) L (γ ) L (γ )  + cosh L L -γ--- tanh bb + -γ--- tanh aa  , k 2 cosh(γ (x + b + L/2)) b a  G yy = Ð------0--L------b------h y (γ ) γ N cosh bb γ L (γ ) L (γ ) (γ ) Ta = sinh La Ð -γ--- tanh bb cosh La , b × (cosh(γ x' )T y Ð cosh(γ x' )D y ), L a b L b a γ T = Ð sinh(γ b) + ----L tanh(γ a)cosh(γ b), ωµ k ⑀ γ cosh(γ (x + b + L/2)) b L γ a L G xy = Ð------0-----z ----L------b ------b------a E γ y ⑀ γ cosh(γ b) b N b L b γ (γ ) L (γ ) (γ ) Da = sinh L(a + L) + -γ--- tanh aa cosh L(a + L) , × ( (γ ) y (γ ) y ) a cosh L xa' T b Ð cosh L xb' Da , γ ωµ (γ ( )) D = sinh(γ (b + L)) + ----L tanh(γ b)cosh(γ (b + L)), yx 0kz sinh b x + b + L/2 b L γ b L b GE = Ð---γ------(---γ------)------b N cosh bb xa = x + a + L/2, xb = x Ð b Ð L/2, × ( (γ ) (γ ) ) sinh L xa' T b Ð sinh L xb' Da , d = a + b + L.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 GREEN’S TENSOR OF THE MAXWELL EQUATIONS 733

y y y y REFERENCES Elements Gi , N , T i , and Di can be obtained from elements Gi, N, Ti, and Di using the following changes: 1. J. D. Adam, D. M. Back, K. M. S. V. Bandara, et al., Physics of Thin Films. Thin Films for Advanced Elec- γ ⑀ γ γ ⑀ γ L L a L L b tronic Devices (Academic, New York, 1991). -γ--- ⑀------γ----, -γ--- ⑀------γ----. a a L b b L 2. B. A. Kalinikos, Izv. Vyssh. Uchebn. Zaved., Fiz. 24 (8), 42 (1981). The obtained expressions for components of the dyadic Green’s functions can be used in the solution of 3. V. F. Dmitriev and B. A. Kalinikos, Izv. Vyssh. Uchebn. a wide spectrum of problems coupled with the investi- Zaved., Fiz. 31 (11), 24 (1988). gation of both linear and nonlinear spin waves propa- 4. M. J. Lancaster, J. Powell, and A. Porch, Supercond. Sci. gating in layered MDFDM structures of an arbitrary Technol. 11, 1323 (1998). transverse geometry. 5. M. P. Kostylev, B. A. Kalinikos, and H. Doetsh, J. Magn. This work was supported by the Russian Foundation Magn. Mater. 145, 93 (1995). for Basic Research (project no. 99-02-16370) and the Ministry of General and Professional Education of the Russian Federation (project no. 97-8.3-13). Translated by A. Kondrat’ev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 734Ð736. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 77Ð82. Original Russian Text Copyright © 2000 by Sil’nikov, Mikhaœlin, Petrov, Meshcheryakov, Ermolaev, Kochetova, Usherenko.

Effect of the Shock-Wave Treatment on the Dynamic Strength of 38KhN3MFA Steel M. V. Sil’nikov, A. I. Mikhaœlin, A. V. Petrov, Yu. I. Meshcheryakov, V. A. Ermolaev, N. S. Kochetova, and S. M. Usherenko Special Materials Research and Production Corporation, St. Petersburg, Russia Pulsed Processing Research and Production Corporation, Minsk, Belarus Received October 19, 1999; in ﬁnal form, March 17, 2000

Abstract—Effect of the shock-wave treatment on the dynamic strength of a 38KhN3MFA grade steel was studied in two different regimes—shock wave propagation and superdeep incorporation of ultrafine powder. In the first case, an increase in the dynamic strength is observed, related primarily to a decrease in the proportion of hard and brittle domains in the material as a result of the shock-wave action. The superdeep ultrafine powder incorporation produced no significant effects on the dynamic properties of steel. © 2000 MAIK “Nauka/Inter- periodica”.

The effects of shock waves on the properties of met- structureless zone of amorphized material, a region of als and alloys have always attracted the attention of sci- strong fragmentation, and a domain of cellular disloca- entists and technologists [1Ð4]. From the fundamental tion structure. It was reported that the shock-wave treat- standpoint, this is a way to probe materials in the range ment using a microparticle jet stream can modify the of extremely high pressures and deformation rates, properties of materials, in particular, markedly increase while researches are mostly interested in obtaining the wear resistance of cutting instruments [6]. Simo- materials with high mechanical and technological nenko et al. [7] proposed a model of particle capture properties not achievable for the other methods of pro- and entraining by the shock-wave front, but the mecha- cessing. Although a considerable amount of experi- nism of superdeep penetration is still unclear. The two mental and theoretical information has been gained to types of treatment were specially selected so as to allow the present concerning the effects of shock-wave action us to separate the effect of the shock wave, accompany- on the properties of solids, any new data in the field are ing the superdeep particle penetration, from a change in still of great value. the material structure and properties related to the The purpose of this work was to study the effects of incorporation of powder in the material studied. two preliminary shock-wave treatment regimes on the The dynamic strength was studied with the aid of a dynamic strength of a structural alloyed steel of the setup comprising a pneumatic light-gas gun and a dif- 38KhN3MFA grade. The experiments were performed ferential laser interferometer. Nonplanarity of the on cylindrical specimens with a diameter of 52 mm and striker and target at the moment of collision did not a height of 10 mm, this shape being necessary for sub- exceed 10Ð3 rad, and the collision velocity was 300Ð sequent investigation of the dynamic mechanical prop- 350 m/s. The interference signals were processed as erties of steel. In the first treatment regime, an explo- described in [8] to obtain the target free surface velocity sion-initiated shock wave propagated through the spec- V and the velocity dispersion ∆V as functions of time. imens. In the second case, the same specimens were These data were used to determine the split velocity subjected to a triple joint action of a shock wave and a and split strength of the target material. high-velocity (3000 m/s) dense (1Ð4 g/cm3) jet stream As seen from the data presented in Fig. 1, the split of a mixture of ultrafine SiC and Ni powders. The time velocity V = Vmax Ð Vmin of the 38KhN3MFA steel of material exposure to the particle flux was 100 µs. increased from 187 to 235 m/s for the specimens The background pressure in both cases was 10 GPa. treated by the first method and to 248 m/s for the spec- As is known [5, 6], the exposure of a metal to such imens subjected to the combined action of the shock a high-velocity jet stream results in penetration of the wave and ultrafine powder. The split strength increased ultrafine particles to a very high material depth (up to a from 4.30 to 5.41 and 5.70 GPa. Thus, the main contri- few millimeters). This leads to the formation of a mod- bution to the dynamic strengthening is related primarily ified composite layer representing a viscous matrix of to the shock wave action. the base material with incorporated solid powder parti- In order to elucidate factors responsible for the cles, featuring the tracks of particle propagation. The increase in the dynamic strength of the 38KhN3MFA tracks exhibit a complicated morphology, comprising a steel, we have studied the material structure by optical

EFFECT OF THE SHOCK-WAVE TREATMENT ON THE DYNAMIC STRENGTH 735

V; ∆V, m/s 400 (a) (b) (c) 350

300 1 1 1

250 V V 200

150 2 100

50 2 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 t, µs

Fig. 1. Time variation of (1) the free surface velocity V and (2) the velocity dispersion ∆V for 38KhN3MFA steel specimens (a) in the initial state, (b) upon the shock-wave treatment and (c) upon the joint action of shock wave and dispersed particle incorporation.

(a) (b)

Fig. 2. Micrographs showing the morphology of 38KhN3MFA steel specimens (a, d) in the initial state, (b) upon the shock-wave treatment, and (c) upon the joint action of shock waves and incorporated ultrafine particles. The section plane is (a, b, c) parallel and (d) perpendicular to the rolling direction (magnification, ×100). microscopy and measured the micro- and macrohard- direction, the structure of steel in both initial and ness at various points in the specimen cross section. treated states comprises the alternation of light and Micrographs presented in Fig. 2 show that the region of dark regions with dimensions corresponding to the split fracture in the initial specimens contains a consid- band separation in Figs. 2aÐ2c. This fact is most clearly erable number of secondary cracks. The fracture sur- manifested in the initial specimen (Fig. 2d). No notice- faces also show an inhomogeneous band structure, able traces of the superdeep penetration of ultrafine which is also most pronounced in the initial state. particles were observed. The bands are parallel to the rolling direction of a The microhardness of steel was markedly different material from which the specimens were made in the light and dark regions, amounting to 473 and (Fig. 2c). Along the axis perpendicular to the rolling 321 HV50, respectively. Probably, the light bands cor-

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

736 SIL’NIKOV et al. respond to the regions of either increased carbon con- 2. J. Rinehart and J. Pearson, Behavior of Materials under tent or localized rolling deformation. In specimens Impulsive Loads (Pergamon, Oxford, 1954; Inostran- upon the shock-wave treatment by both methods, the naya Literatura, Moscow, 1958). microhardness of bands remained the same, but the 3. A. A. Deribas, The Physics of Explosion Welding and Hardening (Nauka, Novosibirsk, 1980). proportion of hard regions markedly decreased. As a 4. P. Yu. Pashkov and Z. M. Gelunova, The Effect of Shock result, the macrohardness of treated specimens Waves on Hardened Steels (Volgograd, 1969). decreased to 42 HRC (for both methods) against 5. V. G. Gorobtsov, K. I. Kozorezov, and S. M. Usherenko, 45 HRC in the initial state. Thus, the increase in the Poroshk. Metall. (Minsk) 6 (1982). dynamic strength of the 38KhN3MFA steel upon the 6. K. I. Kozorezov, V. N. Maksimenko, and S. M. Usher- shock-wave treatment of the type studied is due to a enko, in Selected Topics in Modern Mechanics (1981), decrease in the proportion of hardest (and, probably, Part 1. most brittle) component of the initially inhomogeneous 7. V. A. Simonenko, N. A. Skorkin, and V. V. Bashurov, Fiz. material. Goreniya Vzryva 27 (4), 46 (1991). 8. Yu. I. Meshcheryakov and A. K. Divakov, Preprint No. 25, Leningr. Filial Inst. im. A. A. Blagonravova Akad. Nauk SSSR (Leningrad Branch of the Blagonra- REFERENCES vov Inst. Acad. Sci. USSR). 1. J. Rinehart and J. Pearson, Explosive Working of Metals (Pergamon, Oxford, 1963; Mir, Moscow, 1966). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 737Ð739. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 83Ð88. Original Russian Text Copyright © 2000 by Dubinov, Kozlova.

The Energy-Analyzing Properties of a Solenoid with a Screw Magnetic Axis A. E. Dubinov and N. V. Kozlova Sarov Physical Engineering Institute, Russian Federal Nuclear Center, Sarov, Russia Received March 31, 2000

Abstract—The shape of a beam autograph in an electron energy analyzer with a screw magnetic axis was theoretically analyzed in a single-particle approximation. It is established that the autograph represents a ﬂat smooth spiral involving inward with increasing energy and possessing self-tangency points. The dynamic range of the beam spectrum variation in the energy analyzer based on a screw-magnetic-axis solenoid is markedly greater as compared to that in an analogous system with toroidal magnetic ﬁeld. © 2000 MAIK “Nauka/Inter- periodica”.

At present, magnetostatic energy analyzers [1, 2] measured in the pulsed high-current direct-action are the principal means of investigating the characteris- accelerators [3, 4] exhibit no low-energy components, tics of high-current beams of charged particles. Of most although the simplest analysis of the operation of these interest are the energy analyzers based on solenoids accelerators indicates that low-energy electrons must conjugated with the magnetic systems of particle accel- significantly contribute to the total energy spectrum. In erators. other words, electrons with small energies are reflected by the magnetic field and cannot reach the photo- Previously [3, 4], we described an electron spec- graphic plate. Electrons with very large energies strike trometer based on implementing a toroidal solenoid. the analyzer chamber walls and are absorbed, thus also The principle of the electron beam energy determina- not reaching the plate. tion in such a device is as follows. First, the analyzed part is cut from a beam with the aid of a thick dia- Second, the autographs obtained on a photographic phragm with a small hole. Then, the extracted part is plate in a toroidal spectrometer exhibit nondifferentia- passed along the curvilinear solenoid axis. As is known, ble features of the reversal point type, hindering the electrons drifting in such a magnetic field deviate from analysis of the spectrum. the axis [5], the deviation being dependent on the lon- Below, we consider a different configuration of gitudinal energy of electrons entering into the solenoid. solenoid for the energy analyzer in which the magnetic Placing a transverse indicator plate (e.g., a photo- axis has the form of a screw line (Fig. 1). This solenoid graphic plate) at the solenoid output gives the so-called was considered previously [6] as the prototype of a “beam autograph”—a curve every point of which cor- thermonuclear trap with curvilinear magnetic axis. responds to a certain energy. By analyzing of the intensity of blackening (optical density) at various points of The spatial distribution of magnetic field in a sole- the autograph, we may determine the energy spectrum noid with the screw magnetic axis with a radius r0 = a/2 of electrons. can be approximately described in the cylindrical coordinates [6]: However, it was found that the magnetic field configuration in this spectrometer has several disadvan- tages. First, solenoids with a toroidal magnetic field 2 possess a rather narrow dynamic range for the energy 1ε 3 ε 2 r θ --- B 0 1 + ------2 sin measurements, limited from both above and below. 2 8 a These limitations are related to a special character of B r 2 the charged particle motion in this magnetic field, 1 1 2 r Bϕ =--- ε B 1 + --- ε ----- cosθ, 0 2 which is well known from the analysis of similar 2 8 a motions in the toroidal thermonuclear traps. The limi- Bz   tation from below is related to the existence of an 2 r B 1 Ð ε ------cosθ energy range in which the particles are “locally 02a confined” and travel by the so-called “banana” trajectories [5]. The limitation from above becomes significant θ ϕ α α ε ε when the Larmor diameter exceeds the toroidal sole- where = Ð z, = /a, and , a, and B0 are constant noid coil radius. Indeed, the energy spectra of electrons quantities.

738 DUBINOV, KOZLOVA charge to mass ratio) and γ is the Lorentz factor (invariant in the given problem). The above equations of motion were solved by the z RungeÐKutta method; the error of the calculation was determined using the law of the total electron energy ( 2 2 2 ) conservation v r + v ϕ + v z = const . Fig. 1. Schematic diagram of a solenoid with a screw magnetic axis. Figure 2 shows an example of an autograph for a beam injected into solenoid along the z-axis, which was calculated for the following parameters: magnetic field The shape of a beam autograph on a photographic ε B0 = 0.1 T; = 0.1; electron transit distance L = 0.3 m; plate can be calculated using a single-particle approxi- screw magnetic axis radius r = 0.1 m. mation. The equations of motion of a relativistic elec- 0 tron can be written in the following form [7]: It was found that the beam autographs have the shape of a smooth spiral involving inward, not possessing any nondifferentiable features hindering the data  2   v ϕ  processing. However, the spiral exhibits a self-tangency   gr + ------v r  r  point (Fig. 2b, point A) in the vicinity of which the d     ---- v  = , beam energy spectrum processing encounters certain dt ϕ  v rv ϕ    gϕ Ð ------difficulties. As is readily seen, the dynamic energy     γ ∞ v z  r  range is limited neither from above (for , the   curve in Fig. 2 involves toward the central point of the gz spiral autograph) nor from below (we failed to find “banana” trajectories even for an electron energy of where 10 keV). Thus, the results of our calculations demonstrated     advantages of the energy analyzer with a screw mag-  gr   v ϕ Bz Ð v z Bϕ  ηγ Ð1 netic axis, which possess a broader dynamic range for  gϕ  =  v B Ð v B  ,    z r r z  the energy measurements as compared to the analyzer  gz   v r Bϕ Ð v ϕ Br  with toroidal magnetic field. However, the autograph still contains points complicating the data processing. It should be also noted that the solenoid with a screw η = e/m is the specific charge of electron (electron magnetic axis is smaller than its toroidal counterpart.

r, m (a) 90 y, m 120 60 0.002 0.008 (b) 150 0.006 30 0.004 y 0.002 ϕ 180 0 , deg 0 A 0 x

210 330

–0.002 240 300 –0.002 0 0.002 270 x, m

Fig. 2. An example of the beam autograph (a) plotted in the polar coordinates (the electron entrance point with the coordinates r = 0.01 m and ϕ = 0° is indicated by the black circle) and (b) depicted on a greater scale in the local (x, y) coordinates (black circles corresponding to the energy varied from 100 to 300 keV at 2 keV point; open circle corresponds to 100 keV).

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 THE ENERGY-ANALYZING PROPERTIES OF A SOLENOID 739

REFERENCES (Russian Federal Nuclear CenterÐInstitute of Experi- mental Physics, Sarov, 1998), p. 232. 1. A. I. Gerasimov, E. G. Dubinov, and B. G. Kudasov, Prib. Tekh. Éksp., No. 3, 31 (1971). 5. Yu. N. Dnestrovskiœ and D. P. Kostomarov, Numerical 2. N. I. Tarantin, Magnetic Static Analyzers of Charged Simulation of Plasmas (Nauka, Moscow, 1993). Particles (Énergoatomizdat, Moscow, 1986). 6. A. V. Georgievskij, V. E. Ziser, V. V. Nemov, et al., Nucl. 3. A. E. Dubinov, N. V. Minashkin, V. D. Selemir, et al., in Fusion 14 (1), 79 (1974). Proceedings of 9th IEEE International Pulsed Power 7. A. S. Roshal’, Modeling of Charged Particle Beams Conference, Albuquerque, 1993, p. 708. (Atomizdat, Moscow, 1979). 4. N. V. Stepanov, V. E. Vatrunin, and V. D. Selemir, in Investigations into Plasma Physics: A Collection of Sci- entiﬁc Works, ed. by V. D. Selemir and A. E. Dubinov Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 740Ð743. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 89Ð96. Original Russian Text Copyright © 2000 by Volyar, Fadeeva.

Nonparaxial Gaussian Beams: 2. Splitting of the Singularity Lines and the Optical Magnus Effect A. V. Volyar and T. A. Fadeeva Simferopol State University, Simferopol, Ukraine Received November 23, 1999

Abstract—The Poynting vector field lines for the lowest modes of a nonparaxial Gaussian beam exhibit a number of loops and rings in the vicinity of the phase singularity lines (Airy’s fringes), with negative energy fluxes present inside these loops and rings. The positions of these fluxes are nonsymmetric with respect to rotation about the optical axis. This asymmetry leads to a local splitting of the phase singularity lines, after which the beam cross section transforms from circular to elliptic. The asymmetry of the beam cross section can be eliminated by considering a superposition of circularly polarized even and odd modes. However, this approach only uniformly redistributes the negative energy fluxes in the azimuthal direction of the cross section, rather than completely eliminates these fluxes. Any small perturbation of the resulting symmetric beam gives rise to a unique phenomenon—the optical Magnus effect in the free space, whereby the beam intensity pattern rotates upon changing the circular polarization from right to left. This effect implies the presence of a spinÐorbit coupling in the nonparaxial Gaussian beam propagating in the free space. © 2000 MAIK “Nauka/Interperiodica”.

In the first part of this work,1 we demonstrated that ization direction. It might be expected that the above a linearly polarized nonparaxial low-order Gaussian phenomena will not accompany a circularly polarized beam may exist in the form of four modes, representing nonparaxial Gaussian beam possessing no special even (LPev) and odd (LPod) waves with predominant x- directions of this kind. or y-polarization. We have suggested that the linear The purpose of this work was to study the vector polarization of modes is the factor accounting for a properties of circularly polarized nonparaxial Gaussian change in symmetry of the beam cross section in the beams propagating in the free space in the vicinity of region of transition from circular to elliptic shape (or the focal plane. vice versa) and, hence, for the vector field separation 1. Let us consider a monochromatized low-order with respect to parity (in the general case, this assump- single-mode beam (l = 0) propagating in free space in tion is really valid for unconfined waves subject to the z-axis direction, representing a solution to the Max- strong focusing [1, 2]). It should also be noted that the well equations. We will also require that this beam parity and polarization characteristics of a nonparaxial would transform into a paraxial Gaussian beam upon Ӷ beam are subject to strict formal relationships. In par- going to the paraxial approximation (kz0 1, where z0 ticular, the y-electric polarization corresponds to an π λ is the Rayleigh length and k0 = 2 / is the wavenumber even mode and the x-magnetic polarization, to an odd in vacuum) [2]. The Cartesian field components of this mode. For the even modes, the electric field vector is beam were tabulated in the first part of this work. We parallel to the large semiaxis of the elliptic cross sec- will use these components to form a circularly polar- tion, while the same condition holds in the odd modes ized (CP) field in the cylindrical {r, ϕ, z} coordinate for the magnetic field vector. ()σ ⇒ system by the following scheme: CPev LP(ey) + In additions, it was established that the axial energy σ ()σ ⇒ σ σ ± flux near the focal plane may acquire negative values in i LP(ex) and CPod LP(hy) + i LP(hx) (here, = 1 the vicinity of the ring-shaped phase dislocations is the spirality factor determining the direction of rota- (Airy’s fringes). The position of Airy’s fringes, strictly tion of the polarization vector): confined by the focal plane in the single-mode states, CP : e = σ()ziz+ Fexp()iσϕ , can change for the mode superpositions of the LPev ± ev r 0 1 od ∂ iLP types, where the (+) and (Ð) signs correspond to F0 eϕ ==iz()+ iz Fexp()iσϕ ,eσ------exp()i σϕ , the position of fringes in front of or behind the focal 0 1 z ∂r plane. These unique phenomena, occurring in the free 2 space, were related to the mutual asymmetry of electric h==()G+rFexp()iσϕ ,hϕ iσG exp()iσϕ , and magnetic field and the presence of a special polar- r12 1 ∂ F1 1 Tech. Phys. Lett. 26 (7), 573 (2000). h= Ð()ziz+ ------exp()i σϕ ; (1) z0 ∂r

NONPARAXIAL GAUSSIAN BEAMS: 2. SPLITTING OF THE SINGULARITY LINES 741

(σ) (σ) ( 2 ) ( σϕ) (σ, κ) ⇒ κ κ ± CPod : er = G1 + r F2 exp i , the scheme CP CP(ev) + i CP(od) (where = 1 σ ( σϕ) is the orbital index): eϕ = i G1 exp i , {σ( ) κ( 2 )} ( σϕ) ∂ er = z + iz0 F1 + i G1 + r F2 exp i , ( ) F1 ( σϕ) ez = Ð z + iz0 ------exp i . σ{σ( ) κ } ( σϕ) ∂r eϕ = i z + iz0 F1 + i G1 exp i , σ( ) ( σϕ) ∂ hr = Ð z + iz0 F1 exp i , e = Ðiσ-----{(z + iz )F + iκσF }exp(iσϕ), (3) z ∂r 0 1 0 ( ) ( σϕ) hϕ = Ði z + iz0 F1 exp i , {σ( ) κ( 2 )} ( σϕ) hr = Ð z + iz0 F1 + i G1 + r F2 exp i , ∂F h = Ðσ------0 exp(iσϕ). (2) κσ{σ( ) κ } ( σϕ) z ∂r hϕ = z + iz0 F1 + i G1 exp i , ∂ σκ {( ) κσ } ( σϕ) For brevity, Eqs. (1) and (2) are written in the CGS sys- hz = Ði ----- z + iz0 F1 + i F0 exp i . (4) tem of units, with wavenumbers expressed in the units ∂r of k = 2π/λ. Now, the electric and magnetic fields are almost sym- An analysis of these expressions shows that imple- metric. In order to render these fields fully symmetric mentation of the circularly polarized basis and cylindri- from the standpoint of the paraxial approximation, let cal coordinates cannot solve the problem of mutual us take into account that the second term in parentheses symmetry of the electric and magnetic fields of the sin- in the expressions for er and hr is smaller than the first ӷ 2 gle-mode beam. In order to provide for the further sym- term by two orders of magnitude (G1 r F2), the rela- metrization of the fields, we will use a superposition of tionship being valid even under a weak condition of ≥ even (1) and odd (2) modes constructed according to kz0 1. This term reflects the deviation of the field

y (a) x (b) 10 10

0 0

–10 –10 –10 0 10 –10 0 10 z z (c) (d) y y 10 10

5 5

–10 –5 0 5 10 x –10 –5 0 5 10 x

–5 –5

–10 –10

Fig. 1. Maps of the Poynting vector ﬁeld lines for the even LP(ey) mode in the (a) x = 0 and (b) y = 0 planes and maps of the singu- π λ larity lines Pz(x, y, z) = 0 of the same mode for (c) kz0 = 1 and (d) kz0 = 5 (all lengths expressed in units of k = 2 / ).

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 742 VOLYAR, FADEEVA

y y –10 0 10 –10 0 10 –10 –10

0.3

0.2 0.02 1 0.3 0.2 0 0

0.3

10 10 (a) x (b) x z z –10 0 10 –10 0 10 –10 –10 0.3 0.2 0.01 0.3 0.2 0.01

0.03 0.44 0.1 0.02 0.38 0 0

10 10 (c) x (d) x σ κ Fig. 2. Maps of the intensity levels for a superposition of the circularly polarized CP( , ) mode and a plane monochromatic wave α σ with the amplitude a0 = 1 propagating along the optical axis ( = 0) in the (a, b) xy and (c, d) xz planes for (a, c) = 1 and σ − (b, d) = 1 (kz0 = 1). polarization from the circular type. In the paraxial the negative energy fluxes in the z-axis direction. In approximation, the electric and magnetic fields must Fig. 1a, the field lines in the vicinity of Airy’s fringes in × satisfy a relationship of the type ht = zˆ et. Then, the x = 0 plane exhibit a singularity of the cusp type (no Eqs. (3) and (4) imply that negative energy flux). At the same time, the ring and loop lines of the Poynting vector field in the orthogonal σκ = 1. (5) plane y = 0 (Fig. 1b) are indicative of the presence of negative energy fluxes. The rigid relationship between the “spin” and “orbital” characteristics of the vector field is indicative of the The main part of the light flux is concentrated within presence of a spinÐorbit coupling in the nonparaxial a region at the focal plane with the coordinates r < z0 beam. In the following paper, we will attempt to strictly and is projected onto the observation plane as a family prove this statement in the case of nonparaxial optical of hyperbolas. In this region, lying in the vicinity of the vortices. Here, we will restrict the consideration to an optical axis and corresponding to the paraxial approxi- illustrative interpretation of this phenomenon. It should mation, the field lines represent a family of straight be noted that we have checked the validity of the con- generators of a one-sheet hyperboloid of rotation [4]. dition σκ = –1 for the fields described by Eqs. (3) and Note that the loop and ring trajectories are indicative of (4) by computer simulation. It was found that these the presence of the sinÐorbital coupling in the non- fields, albeit not representing travelling waves, form a paraxial beam [5]. A comparison of the map of field special state of the “light droplet” type mentioned in the lines in Fig. 1a to that reported for the wave energy flux first part of this work. behind a high-aperture microobjective lens [2, Fig. 4.21] Figures 1a and 1b show maps of the Poynting vector shows good coincidence of the results obtained by different methods. field lines for the even LP(ey) mode (see table in the first part of this work) in the x = 0 and y = 0 planes, respec- 2. A remarkable feature is offered by inhomoge- tively. Note the lack of symmetry in the field lines with neous distribution of the negative energy fluxes and the respect to the π/2 rotation. This asymmetry is related to related splitting of singularity lines. This effect resem-

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 NONPARAXIAL GAUSSIAN BEAMS: 2. SPLITTING OF THE SINGULARITY LINES 743 bles the phenomenon of multiplicity of the atomic a comparison of Figs. 2a and 2b indicates that the cen- energy levels arising due to the influence of a micropar- tral spot, in which the major field intensity is concen- ticle spin. Figures 1c and 1d show families of the sin- trated, exhibits a transverse shift at a distance of d ∝ λ/2 π gularity lines (Pz = 0) of the LP(ey) mode, on which at the expense of rotation by upon switching the cir- both real and imaginary parts of the transverse electric cular polarization direction. It is important to note that, and magnetic field components simultaneously turn simultaneously with the transverse shift, the beam node zero. The singularity lines of the LP(ex) mode will be exhibits a longitudinal shift by approximately the same merely rotated by π/2. Splitting of the singularity lines distance (Figs. 2c, 2d). It would apparently be natural leads to a unique phenomenon—the optical Magnus to relate the observed transverse shift to a difference in effect for a single-mode beam in the free space. Let us the propagation velocities of beams with orthogonal consider this effect in some more detail. circular polarizations along the optical axis. However, The presence of the spinÐorbit coupling implies a this assumption is erroneous because the longitudinal difference in the beam wavefunction response to the Poynting vector component ∫Pz dS , which is responsi- right-hand (σ = 1) and left-hand (σ = Ð1) circular polarizations. However, the fields (3) and (4) are symmetric ble for the group velocity, is the same for both orthog- with respect to rotation about the optical axis. In order onal polarizations. to break the external symmetry, let us introduce a small The authors are grateful to V.G. Shvedov for fruitful perturbation in the form of a plane wave with the ampli- discussion. α tude a0 , propagating at the angle relative to the z-axis. The results of the small perturbation action upon the single-mode beams with the right-hand and left-hand REFERENCES circular polarizations in the focal plane are illustrated 1. M. Born and É. Wolf, Principles of Optics (Pergamon, in Fig. 2. As seen from the intensity Pz distribution in Oxford, 1969; Nauka, Moscow 1973). the focal plane z = 0, the polarization switching from 2. S. Solimeno, B. Crosignani, and P. Di Porto, Diffraction σ = 1 to σ = Ð1 results in the whole pattern rotation by and Confinement of Optical Radiation (Academic, the π angle. Orlando, 1986; Mir, Moscow, 1989). The phenomenon of rotation of the speckle pattern 3. E. Abramochkin and V. Volostnikov, Opt. Commun. 125, in the field of radiation of an optical fiber upon chang- 302 (1996). ing the circular polarization direction is known as the 4. A. V. Volyar, T. A. Fadeeva, and V. G. Shvedov, Pis’ma optical Magnus effect [6]. This phenomenon is based Zh. Tekh. Fiz. 25 (5), 87 (1999) [Tech. Phys. Lett. 25, upon the fundamental principles of interaction of radi- 203 (1999)]. ation and medium with a gradient of the index of refrac- 5. A. V. Volyar, V. Z. Zhilaœtis, and V. G. Shvedov, Opt. Spe- tion. Our case presents a unique manifestation of the ktrosk. 86 (4), 664 (1999) [Opt. Spectrosc. 86, 593 Magnus effect in the free space. Apparently, the role of (1999)]. the refractive index gradient is played by the gradient of 6. A. V. Volyar and T. A. Fadeeva, Pis’ma Zh. Tekh. Fiz. 23 the nonparaxial single-mode beam intensity. The possi- (23), 59 (1997) [Tech. Phys. Lett. 23, 927 (1997)]. bility for observation of the transverse beam displace- 7. N. B. Baranova, A. Yu. Savchenko, and B. Ya. ment by a distance of the order of half-wavelength in Zel’dovich, Pis’ma Zh. Éksp. Teor. Fiz. 59 (4), 216 the focal plane of the lens was indicated by the results (1994) [JETP Lett. 59, 232 (1994)]. of approximate calculations [7]. Sadykov [8] substanti- ated the transverse shift of the focal spot within the 8. N. R. Sadykov, Opt. Spektosk. 84 (2), 293 (1998) [Opt. framework of a geometric-optical approach. Spectrosc. 84, 250 (1998)]. At first glance, the transverse beam shift is not related to rotation of the intensity pattern. Nevertheless, Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 744Ð746. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 97Ð102. Original Russian Text Copyright © 2000 by Raevskiœ, Reznichenko, Smotrakov, Eremkin, Malitskaya, Kuznetsova, Shilkina.

A New Phase Transition in Sodium Niobate I. P. Raevskiœ, L. A. Reznichenko, V. G. Smotrakov, V. V. Eremkin, M. A. Malitskaya, E. M. Kuznetsova, and L. A. Shilkina Research Institute of Physics, Rostov State University, Rostov-on-Don, 344090 Russia e-mail: [email protected] Received March 28, 2000

ε Abstract—The temperature dependences of the dielectric constant of sodium niobate single crystals NaNbO3 and sodium niobate-based (Na,Li)NbO3 and (Na,K)NbO3 solid solutions with low content of the second component have been studied. A weak anomaly observed in ε(T) indicates a new phase transition in the vicinity of ° 150 C in NaNbO3. The introduction of lithium or potassium into NaNbO3 crystals increases the temperature of this transition. The X-ray powder diffraction data suggest that the new transition takes place between two rhombohedral phases. © 2000 MAIK “Nauka/Interperiodica”.

A large number of phase transitions (at present, their were applied onto the (001) faces of the grown crystals. number amounts to six [1]) provided by the rotation of The content of LiNbO3 (1.5Ð2.5 mol %) and KNbO3 oxygen octahedra and by ordered ion displacements (2Ð3 mol %) in the (Na,Li)NbO3 and (Na,K)NbO3 crys- have attracted the attention of both theoreticians and tals was evaluated by comparing their structural param- experimenters to the antiferroelectric sodium niobate eters and the phase transition temperatures with the crystals (NaNbO3) with a perovskite structure for about experimental values obtained for the ceramics and data forty years. The studies of sodium niobate and sodium reported for the corresponding solid solution crystals niobate-based solid solutions are also of great impor- [12Ð14]. tance since they make the basis or are the components Figure 1 shows the ε(T) curves for sodium niobate of various functional materials possessing piezoelec- and (Na,Li)NbO3 and (Na,K)NbO3 crystals. In addition tric, electrooptical, and pyroelectric properties and of to the well-known anomalies in the regions of 240Ð270 materials used in capacitors, posistors, etc. [2Ð5]. and 350Ð360°C (corresponding to the well-known Although it was stated that sodium niobate undergoes phase transitions) [1, 12Ð14], all the curves show weak no phase transitions in the temperature range from Ð50 ° ° anomalies in the vicinity of 150 C (for sodium niobate to 350 C [1], the temperature dependences of the elec- crystals) and 175Ð185°C (for the crystals of solid solu- trophysical parameters of sodium niobate and sodium tions). In fact, these anomalies have already been niobate-based solid solutions often showed weak observed in a large number of crystals, including those anomalies [6Ð10] which could be associated, in partic- synthesized under various conditions. Moreover, the ular, with some unknown (e.g., electric-ﬁeld-induced temperatures of all the anomalies on the ε(T) curves or impurity-related) phase transitions. changed in a regular manner with the Li and K content Below, we describe the detailed study of dielectric in the solid solutions. Thus, it was possible to assume and structural properties of sodium niobate and some that the anomaly in ε(T) in the vicinity of 150°C for sodium niobate-based solid solutions in the tempera- sodium niobate was associated with a certain phase ture range from 20 to 400°C. Since most of the phase transition. It should also be emphasized that the analy- transitions in sodium niobate are accompanied by very sis of published data on the properties of sodium nio- weak anomalies in the behavior of the dielectric con- bate crystals and ceramics shows that weak anomalies stant ε [1], our studies were performed on single crys- of ε(T) in the vicinity of 150°C have been repeatedly tals providing much better observation of the anomalies observed in various experiments [7Ð10]; however, they ε in than the corresponding ceramics. NaNbO3 single have been ignored and received no interpretation crystals and (Na,Li)NbO3 and (Na,K)NbO3 solid solu- because of their low values in comparison with the tions with a low content of the second component were features corresponding to the well-known phase tran- obtained by the method of mass crystallization in the sitions. Na2CO3ÐNb2O5ÐB2O3 system [11] as colorless trans- Despite the fact that the anomalies in ε(T) revealed parent 2-mm-long plates with a thickness ranging in our study are rather strongly smeared in comparison within 0.2Ð0.3 mm. The dielectric constant was mea- with anomalies associated with the well-known phase sured during continuous heating or cooling at a rate of transitions, the temperatures of all these anomalies 2Ð3 K/min with the aid of a P5083 ac bridge in the were almost independent of the measuring ﬁeld fre- 1−100 kHz frequency range. The Aquadag electrodes quency in the 1Ð100 kHz range. As is seen from Fig. 1,

A NEW PHASE TRANSITION IN SODIUM NIOBATE 745 the anomalies in ε(T) corresponding to the well-known ε ε phase transitions are characterized by a pronounced 1600 350 1 temperature hysteresis (for the sake of clarity, the ε(T) curve obtained in the cooling mode is shown only for 2 the (Na,Li)NbO3 crystals), whereas the temperature positions of the newly revealed weak anomalies in ε(T) 3 are almost independent of the mode: heating or cooling. 230 3 The results obtained lead to the assumption that the 1200 2 character of the phase transition in the vicinity of 150°C in sodium niobate is close to a second-order 1 phase transition. 110 The studies of the grown crystals in the polarized 100 140 180 220 light showed that propagation of the phase fronts or 800 T, °C changes in the extinction pattern take place only at temperatures in the 240Ð270 and 350Ð360°C regions, corresponding to the well-known phase transitions. Simi- 2 lar results were obtained earlier for Na0.98Li0.02NbO3 crystals [12] together with a change in the slope of the temperature dependence of birefringence in the vicinity 400 ° of 150 C indicating a possible phase transition. 1 The temperature dependences of the structural 2 parameters of NaNbO3 and Na0.975Li0.025NbO3 powders synthesized by the method of solid-phase reactions 3 showed no changes in the symmetry of the perovskite structure in the temperature range from 25 to 330Ð 0 100 200 300 400 350°C. In the temperature range from 150 to 200°C (where the anomalies in ε(T) were observed), only T, °C insigniﬁcant changes in the slope of the temperature Fig. 1. Dielectric constant as a function of temperature, ε(T), dependences of the unit cell parameters a, b, and c and for (1) NaNbO crystals and (2) (Na,Li)NbO and β 3 3 the angle of the rhombohedrally distorted unit cell (3) (Na,K)NbO3 solid solutions with low content of the sec- were recorded. In the vicinity of 270°C, (corresponding ond component measured in the heating (solid lines) and cooling (dashed lines) modes at a frequency of 1 kHz. to the phase transition in Na0.975Li0.025NbO3 accompanied by a pronounced anomaly in ε(T)), only rather weak changes in the slope of the temperature depen- a, b, c, nm dences of the unit cell parameters were observed. The results obtained are consistent with the structural data 0.394 a = c (a) for Na0.98Li0.02NbO3 powders [13] and lead to the 1 assumption that the phase transitions in sodium niobate 0.392 2 and the sodium niobate-based solid solutions (indicated by weak anomalies in ε(T) in the vicinity of 150Ð 200°C) take place between two rhombohedral phases. 0.390 b Thus, the dielectric, optical, and structural studies 0.388 and the analysis of the corresponding published data 0 100 200 300 400 500 indicate that, in addition to the six well-known phase transitions in NaNbO3, one more phase transition β, deg occurs in the vicinity of 150°C which is close to a sec- 90.4 ond-order phase transition and is assumed to take place (b) between two antiferroelectric rhombohedral phases. 90.3 The existence of the phase transition in NaNbO3 and 1 sodium niobate-based solid solutions in the 150Ð200°C 90.2 temperature range (which falls within the working 2 interval of numerous functional sodium niobate-based 90.1 materials) can produce a considerable effect on the 0 100 200 300 400 temperature and temporal stability of various characteristics of these materials. This fact should necessarily be T, °C taken into account in designing new sodium niobate- Fig. 2. Temperature dependences of the unit cell parameters based materials for various purposes. In particular, it is of a rhombohedrally distorted perovskite-type structure of important to reﬁne the corresponding temperatureÐ (1) NaNbO3 and (2) Na0.975Li0.025NbO3 powders.

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 746 RAEVSKIŒ et al. composition phase diagrams of NaNbO3-based solid amic Materials: Optimization of Search (Paik, Rostov- solutions and the theoretical models which describe the on-Don, 1995). sequence of phase transitions in NaNbO3. 6. V. A. Isupov, Izv. Akad. Nauk SSSR, Ser. Fiz. 22 (12), 1504 (1958). The study was supported by the Russian Foundation 7. N. N. Kraœnik, Fiz. Tverd. Tela (Leningrad) 2 (4), 685 for Basic Research, project no. 99-02-17575. (1960) [Sov. Phys. Solid State 2, 633 (1960)]. 8. L. Pardo, P. Duran-Martin, J. P. Mercurio, et al., J. Phys. REFERENCES Chem. Solids 58 (9), 1335 (1997). 9. K. Konieczny, W. Smiga, and C. Kus, Ferroelectrics 190, 1. G. A. Smolenskiœ, V. A. Bokov, V. A. Isupov, et al., The 131 (1997). Physics of Ferroelectric Phenomena (Nauka, Leningrad, 10. K. Konieczny, Mater. Sci. Eng. B 60, 124 (1999). 1985). 11. V. G. Smotrakov, I. P. Raevskiœ, M. A. Malitskaya, et al., 2. N. F. Borelly and M. Layton, IEEE Trans. Electron Neorg. Mater. 16 (5), 1065 (1980). Devices 16 (6), 511 (1969). 12. A. Sadel, R. Von der Mühll, and J. Ravez, Mater. Res. 3. P. Wang, T. Bai, Z. Chen, and M. Liu, J. Huazhong Univ. Bull. 18, 45 (1983). Sci. Technol. 16 (5), 123 (1988). 13. A. Sadel, R. Von der Mühll, J. Ravez, et al., Solid State 4. I. P. Raevskiœ, L. A. Reznichenko, and M. A. Malitskaya, Commun. 44 (3), 345 (1982). Pis’ma Zh. Tekh. Fiz. 26 (3), 6 (2000) [Tech. Phys. Lett. 14. M. Badurski and K. Stroz, J. Cryst. Growth 46, 274 26, 93 (2000)]. (1979). 5. A. Ya. Dantsiger, O. N. Razumovskaya, L. A. Rezni- chenko, and S. I. Dudkina, High-Efﬁciency Piezocer- Translated by L. Man

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000

Technical Physics Letters, Vol. 26, No. 8, 2000, pp. 747Ð750. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 26, No. 16, 2000, pp. 103Ð110. Original Russian Text Copyright © 2000 by Chirkov, Ageev.

On the Possibility of Observing the AharonovÐBohm Effect under Nonstationary Potential Conditions A. G. Chirkov* and A. N. Ageev** * St. Petersburg State Technical University, St. Petersburg, Russia ** Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia Received February 26, 2000

Abstract—Based on a theoretical analysis of the nonstationary excitation of a solenoid, it is demonstrated for the ﬁrst time that vector potentials exist in the space outside the solenoid which can, in the general case, provide conditions for observation of the AharonovÐBohm effect. © 2000 MAIK “Nauka/Interperiodica”.

Since the first descriptions of the AharonovÐBohm As is known [12], the zero-field potential can be rep- effect (ABE) [1Ð3], numerous theoretical and experi- resented in the following form: mental works have been devoted to elucidating the 0 0 1∂χ nature of this phenomenon (see, e.g., [4, 5]). An origi- A ==grad χ, ϕ Ð ------, (2) nal approach to observing this effect, not involving c ∂t electron interactions with either magnetic or electric χ fields, consisted in realizing the situation where elec- where is a differentiable function. trons would be moving in a region free of such fields. Note that the zero-field potentials are responsible These conditions are most simply realized in the sta- for the ABE manifestations in all the works published tionary regime; this is just the case that was studied so far dealing with the stationary case. This is related until the beginning of 1990s. On the present-day level primarily to the circumstance that the ABE is observed of technology, this regime was almost exhaustively in the “pure” form, whereby the shift of the interference studied in the experiments of Tonomura et al. [6]. bands is determined by the potentials rather than by the fields [13, 14]. On going to the nonstationary case, the Eventually, researchers turned to elucidating the question naturally arises as to whether a situation may possibility of the ABE manifestations in the case of take place whereby the zero-field potentials are created nonstationary fields [7–10]. This expansion of the field in a certain space region? In this context, the first point of observation is very promising from the standpoint of in analysis of the possibility of the ABE manifestations experimental investigations, which may provide a in the case of nonstationary fields must consist in deter- deeper insight into the ABE nature. On the other hand, mining the zero-field potentials. To our knowledge, this interpretation of the ABE observed under these condi- problem was not considered in the works published to tions presents a more complicated task. date. As seen from the relationships determining fields With a view to elucidating the ABE nature, the elec- through potentials, tromagnetic potentials are conveniently represented as Ð1 the sums of two terms: B==rot A, E Ðc ∂A/∂t Ð grad ϕ, (3) ϕ0 ≠ AA==f+ A0and ϕ ϕf + ϕ0, (1) the zero-field potentials exist only provided that 0, which implies that the Coulomb (radiation) gauge is where the superscript “f” indicates the potentials inapplicable. describing nonzero electromagnetic fields (“field” Let us consider the MaxwellÐLorentz equations potentials) and superscript “0” refers to the potentials describing, in the conventional notation, the electro- that, albeit corresponding to a solution of the Maxwell magnetic field in a vacuum: equations with boundary conditions, do not directly 4π 1 ∂ E produce electromagnetic fields (“zero-field” or div E ==4πρ, rot B ------j + ------. (4) “excess” potentials). It should be noted that functions c c ∂ t analogous to the zero-field potentials have been used for a long time in the mathematical physics. In particu- Substituting expressions (1) and (3) into Eq. (4), we lar, a zero-field vector potential appears (and is obtain equations determining the potentials: uniquely determined) in the problem of vector recon- f 0 f1∂Λ 0 1 ∂Λ struction from the given rotor and divergence a finite ϕ+++------ϕ ------= Ð 4πρ, (5a) domain [11]. c∂t c ∂t

748 CHIRKOV, AGEEV

f f 0 0 4π The nonzero vector potential components Aρ and Aα A Ð grad Λ + A Ð grad Λ = Ð------j, (5b) c have the following form (here and below, the harmonic time dependence is omitted) [16]: where

2 1 ∂ ( ) (α α ) ( , ) ∆ Aρ = ∫ jα r' sin Ð ' G r r' dV', (9a) = Ð ----2 ------2, c ∂t V ∂ϕf Λf f 1 = div A + ------. ( ) (α α ) ( , ) c ∂t Aα = ∫ jα r' cos Ð ' G r r' dV', (9b) Let us search for the Af and ϕf potentials in the form V of solutions to inhomogeneous equations: π i (2) | | where G(r, r') = Ð---- H0 (k r Ð r' ) is the Green func- ϕf = Ð 4πρ, (6a) c (2) tion of the Helmholtz equation [16], H0 is the Hankel f 4π A = Ð------j. (6b) function, and k = ω/c. Integrals entering into Eqs. (9) c are readily calculated using an addition formula for the Hankel functions [16]: Now, the terms containing zero-field potentials van- ish identically by virtue of definition (2) and the Max- (2)( ρ2 2 ρ (α α )) well equations reduce to the Lorentz condition H0 k + R Ð 2 Rcos Ð ' f +∞ (2)( ) ( ρ) ρ < (10) f 1∂ϕ (α α )Hm kR Jm k , R, div A + ------= 0, (7) Ðim Ð '  ∂ = ∑ e ( ) c t  ( ) 2 ( , ρ) ρ > m = Ð∞ Jm kR Hm k , R. involving only the field potentials rather than complete potentials. This makes superfluous the Strutton con- As a result, we obtain dition π2 0 2 i I0 R Ðinα 0 1∂ϕ 1 ∂ χ Aα = Ð------e div A + ------≡ ∆χ Ð ------= 0. c ∂ 2 2 (11a) c t c ∂t ( ) ( ) 2 ( ) ( ρ) 2 ( ) ( ρ) ρ < Hn + 1 kR Jn + 1 k + Hn Ð 1 kR Jn Ð 1 k , R, Thus, within the framework of the scheme adopted ×   ( ) (2) ( ρ) ( ) (2) ( ρ) ρ > for solving the MaxwellÐLorentz equations, Eq. (7) is a Jn + 1 kR Hn + 1 k + Jn Ð 1 kR Hn Ð 1 k , R; necessary part of the problem rather than merely a condition facilitating the solution. In the case when solu- π2 I0 R Ðinα tions of the nonlinear Eqs. (6) are expressed in terms of Aρ = Ð------e retarding potentials, the Lorentz condition (7) becomes 2c (11b) the identical relation [15] and the remaining part of sys- (2) (2) H (kR)J (kρ) + H (kR)J (kρ), ρ < R, tem (5) is satisfied by the zero-field potentials. There- ×  n + 1 n + 1 n Ð 1 n Ð 1 ( ) ( ) fore, the nonzero field can be described by potentials J (kR)H 2 (kρ) Ð J (kR)H 2 (kρ), ρ > R. satisfying Eqs. (6) with a necessary condition (7). n + 1 n + 1 n Ð 1 n Ð 1 An example of a system featuring nonstationary For n = 0 (i.e., in the absence of angular modulation), zero-field potentials, presenting generalization of the Eqs. (11) yield the following relationships: stationary magnetostatic ABE, is offered by the nonsta- ( ) tionary excitation of a cylindrical solenoid of infinite π2  2 ( ) ( ρ) ρ < 2i I0 R H1 kR J1 k , R, length with negligibly thin walls. In a cylindrical coor- Aα = Ð------ ρ α (2) dinate system ( , , z) with the origin on the solenoid c J (kR)H (kρ), ρ > R (12) axis and the z-axis coinciding with the solenoid axis, 1 1 the volume current distribution is described by the fol- and Aρ = 0. lowing equations: In the stationary case (ω 0), these relationships (ρ, α, ) δ(ρ ) ( α ω ) jα z = I0 Ð R expi Ð n + t , lead to the known formulas Aα = Jρ/cR and Aα = JR/cρ (8) (ρ > R), where J = 2πRI is the solenoid wall current per jρ = jz = 0, 0 unit length). The only nonzero magnetic field compo- where R is the solenoid coil radius and ω is the cyclic nent corresponding to potentials (12) has the following frequency of the current. form:

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 ON THE POSSIBILITY OF OBSERVING THE AHARONOVÐBOHM EFFECT 749

(a) Alternating current f Aα ( ) π2  2 ( ) ( ρ) ρ < 2 2i I0 Rk H1 kR J0 k , R, Bz = Ð------ (13) c  ( ) (2)( ρ) ρ > J1 kR H0 k , R; (b) Stationary case 1 4πI B = ------0 (ρ < R), B = 0 (ρ > R). (14) z c z 0 In accordance with Eqs. (1), let us divide into two parts the total vector potential in the space outside the solenoid coil so as to separate a term with zero rotor in Eq. (12). This can be done in a unique mode as follows: –1 ρ π2 2i I0 R (2) Aα = Ð------J (kR)H (kρ) c 1 1 –2 (15) π ( ) 5 10 4 I0 R J1 kR f 0 ρ + ------≡ Aα + Aα , k kc ρ f f 0 The field potential Aα calculated as function of the distance where Aα is the field potential component, Aα is the from solenoid kρ at three time instants corresponding to ω π π ρ ω π (2) ρ t = /2 [solid line, J1(k )], t = /4 [dashed line, zero-field potential component, and H1 (k ) = ( ) π π ρ π ρ 2  ω 2 ρ π ρ 0.25 - J1(k ) Ð Y1(k ) Ð ----ρ- ], and t = 0 [dotted line, H1 (k ) Ð 2i/ k . Separating the real parts of the k potential components in Eq. (15), we obtain  2  − -πY (kρ) + ----- ]. Calculated for kr = W = 1.  1 kρ  f π ( ρ) ω Re Aα = W J1 k sin t -  (16a) Note that, according to Eq. (15), there is a certain relationship between the solenoid coil radius and the 2  π ( ρ) ω  wavelength of the electromagnetic field, determined by Ð ---ρ-- + Y1 k cos t , k  the roots of the equation J1(kR) = 0, for which the fields outside the solenoid are zero. Under these nonstation- 0 2 ary conditions, however, the zero-field potentials are Re Aα = W----- cosωt, (16b) kρ also vanishing (a system of the closed waveguide type). The zero-field potentials are also vanishing for all n ≠ 0. where In the general case, there are three situations in π ( ) 2 I0 RJ1 kR which the ABE can be observed in a nonstationary W = ------c regime. These cases correspond to the de Broglie electron waves interacting with (i) zero-field potentials, and Y1 is the Neumann function. (ii) both zero-field and field potentials, and (iii) field f potentials only. The third case was considered in [7, 17] The plots of Aα calculated according to Eqs. (16) at but still requires a deeper analysis. The ABE observed various time instants are depicted in the figure. An anal- in the case of the de Broglie electron waves interacting ysis of these curves indicates that situations are possi- with the field potentials is sometimes referred to as the ble when electrons, travelling by certain trajectories AharonovÐBohm quasi-effect. and emitted at definite time moments, occur under the action of zero-field potentials only. Thus, we have demonstrated that zero-field poten- f 0 tials may exist in the space outside a solenoid excited in In the stationary case, Aα = 0 and Aα = JR/cρ. Pro- a nonstationary regime, which is a necessary but insuf- ceeding from the condition that the electric field is ficient condition for manifestation of the AharonovÐ absent, we obtain the following expression for the sca- Bohm effect. lar zero-field potential: π ϕ0 4 iI0 R ( )α REFERENCES = Ð------J1 kR , (17) c 1. W. Franz, Verh. Dtsch. Phys. Ges. 20, 65 (1939). where α is the azimuthal angle. In the stationary case, 2. W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. London, ϕ0 = 0. Sect. B 62, 8 (1949).

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000 750 CHIRKOV, AGEEV

3. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). 11. A. N. Tikhonov and A. A. Samarskiœ, Equations of Math- ematical Physics (Nauka, Moscow, 1953; Pergamon, 4. S. Olariu and I. I. Popesku, Rev. Mod. Phys. 57, 339 Oxford, 1964). (1985). 12. M. G. Savin, Lorentz Gauge Problems in Anisotropic 5. M. Peskin and A. Tonomura, Lect. Notes Phys. 340, 115 Media (Moscow, 1979). (1989). 13. D. H. Kobe, Ann. Phys. (N. Y.) 123, 381 (1979). 6. A. Tonomura et al., Phys. Rev. Lett. 56, 792 (1986). 14. T. H. Boyer, Phys. Rev. D 8, 1679 (1973). 7. B. Lee, E. Yin, T. K. Gustafson, and R. Chiao, Phys. 15. I. E. Tamm, The Principles of Electricity Theory (Nauka, Rev. A 45, 4319 (1992). Moscow, 1966). 16. G. T. Markov and A. F. Chaplin, Excitation of Electro- 8. A. N. Ageev, Yu. M. Voronin, I. P. Demenchonok, and magnetic Waves (Moscow, 1967). Yu. V. Chentsov, Pis’ma Zh. Tekh. Fiz. 22 (4), 25 (1996) 17. A. N. Ageev, S. Yu. Davydov, and A. G. Chirkov, Pis’ma [Tech. Phys. Lett. 22, 144 (1996)]. Zh. Tekh. Fiz. 26 (2000) (in press) [Tech. Phys. Lett. 26, 9. G. Afanasiev and M. Nelhiebel, Phys. Scr. 54, 417 392 (2000)]. (1996). 10. A. Vourdas, Phys. Rev. A 56, 2408 (1997). Translated by P. Pozdeev

TECHNICAL PHYSICS LETTERS Vol. 26 No. 8 2000