Theory of Cyclotron Superradiance from a Moving Electron Bunch Under Group Synchronism Conditions N

THEORETICAL AND MATHEMATICAL PHYSICS

Theory of Cyclotron Superradiance from a Moving Electron Bunch under Group Synchronism Conditions N. S. Ginzburg, I. V. Zotova, and A. S. Sergeev Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603600 Russia Received August 5, 1999

Abstract—A theory is presented of cyclotron superradiance from an electron bunch rotating in a uniform magnetic ﬁeld and drifting at a velocity close to the group velocity of a wave propagating in a waveguide. It is shown that, in a comoving frame of reference, the bunch emits radiation at a frequency close to the cutoff frequency of the waveguide. Superradiance implies the azimuthal self-bunching of electrons, which is accompanied by coherent emission of the stored rotational energy in a short electromagnetic pulse. Linear and nonlinear stages of the process are analyzed. The growth rate of the superradiance instability is determined. It is shown that the maximum growth rate is attained under group synchronism conditions. The peak power and the characteristic duration of the cyclotron superradiance pulse are determined by numerical simulation. The characteristic features of the superradiance pulses are described in the comoving and laboratory frames. The results of theoretical analysis are compared with experimental data. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION electron bunch is close to the group velocity of the electromagnetic wave: Induced emission from a spatially localized electron ≈ ensemble (bunch) whose size significantly exceeds the V || V gr. (1) emission wavelength but is much smaller than the interaction scale length is one of the promising methods for The low rate at which the energy outflows from the generating ultrashort electromagnetic pulses. Emission electron resonator formed by the electron bunch allows from such a bunch can be considered a classical analog the maximum growth rate of the SR instability [16, 19]. of a quantum effect known as Dicke superradiance Condition (1) can be met, e.g., when the radiation [1−3], which is the emission of an electromagnetic propagates in a waveguide; in this case, the dispersion pulse with a duration shorter than the relaxation time of curves of the wave and of the electron flux are tangent the emitting atomic ensemble with population inver- (Fig. 1a).1 Note that, in the comoving frame, in which sion. In classical electrodynamics, superradiance (SR) the bunch as a whole is at rest, the regime of group syn- can be attributed to various mechanisms for induced chronism corresponds to the emission of radiation at a emission [4Ð21]; in particular, SR can be produced by frequency close to the cutoff frequency and, therefore, an electron bunch rotating in a uniform magnetic field exhibits several advantages typical of electronÐwave (cyclotron SR) [10Ð22]. interaction in gyrotrons, in which the operating mode is Cyclotron SR implies the azimuthal self-bunching also excited at the quasi-cutoff frequency [22]. In par- of electrons, which is accompanied by coherent emis- ticular, SR at the quasi-cutoff frequency is less sensitive sion of the stored rotational energy. The phase self- to the scatter in the electron bunch parameters, includ- locking is related to the relativistic dependence of the ing the longitudinal bunch spreading caused by both gyrofrequency on the particle energy [23] and is similar Coulomb repulsion and the spread in the initial electron to that occurring in cyclotron resonance masers (CRM). velocities. However, in CRMs, quasi-continuous electron beams Earlier, we reported on the first experimental obser- are used; i.e., the electrons that leave the interaction vations of cyclotron SR in the millimeter wavelength region are replaced by the electrons continuously range [20, 21]. In those experiments, we used electron injected from the cathode, thus providing the condi- bunches with a length of 5Ð7 cm, particle energy of up tions for steady-state generation. In contrast, SR is to 200Ð250 keV, and current of about 200Ð500 A. The pulsed in character and can be realized when each par- electron bunches propagated in a 30-cm smooth cylin- ticle stays in a moving or resting electron bunch for a drical waveguide placed in a uniform magnetic field. long time (ideally, for an infinitely long time). This fact We observed the generation of ultrashort (up to 400 ps) also accounts for the absence of an SR threshold [18]. electromagnetic pulses with the peak power higher than

Earlier, we showed [19] that the regime of group 1 Group synchronism condition (1) can also be met in the case synchronism is the most favorable for the observation when an electron bunch propagates in a dispersive medium, e.g., of cyclotron SR. In this regime, the drift velocity of the in a homogeneous plasma.

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ω ω' waveguide modes, yields the following expression for (a) (b) the transverse component of the magnetic ﬁeld:2

V || V ||V ' γ  [ , ] γ  ph H⊥ = ||H⊥ Ð ----- z0 E⊥  = ||H⊥1 Ð ------ 0. c c2 Therefore, the regime of group synchronism in the ω c ω ω laboratory frame (the dispersion curves are tangent) ω H' = c H corresponds to the emission of radiation at a quasi-cutoff frequency in the comoving frame (Fig. 1b). h h' Thus, in the K ' frame, the bunch electrons rotate in the magnetic field but the bunch as a whole is at rest. Fig. 1. Dispersion curves corresponding to the regime of The linear size of the bunch in the z' direction is b' = group synchronism in (a) laboratory frame and (b) comov- γ ± b ||0. The bunch radiates isotropically in the z' direc- ing frame. tions. Assuming that the transverse field structure of the emitted radiation coincides with that of one of the 200 kW. SR pulses were recorded only under condi- waveguide modes E⊥(r⊥), we represent the radiation tions close to the condition of group synchronism with field in the form various waveguide modes. Far from group synchro- E' = Re[E⊥(r⊥)A'(z', t')exp(iω t')], (2) nism, the generation of SR pulses was absent. Thus, the c ω experimental results proved that the regime of group where c is the carrier frequency coinciding with the synchronism is optimum for SR. cutoff frequency of the operating mode. According to This paper is devoted to a theoretical study of cyclo- the dispersion relations, the evolution of the axial field tron SR in the regime of group synchronism. A'(x', t') distribution is described by the inhomogeneous parabolic equation ∂2a ∂a 1. CYCLOTRON SUPERRADIANCE i------+ ------= 2if (Z')GJ. (3) IN THE COMOVING FRAME ∂Z'2 ∂τ'

1.1. Basic Equations 2π π βˆ Θ The transverse electric current J = 1/ ∫ +d 0 Let a planar bunch of cyclotron oscillators of length b 0 move in a cylindrical waveguide at a longitudinal exciting the electromagnetic field can be found from velocity close to the group velocity of an electromag- the equations of electron motion. Assuming that the netic wave, so that condition (1) is satisfied. In the lab- Ӷ oratory frame, the dispersion curve of the electromag- transverse electron velocity is nonrelativistic (V ⊥' c), Ð1 ω2 ω 2 1/2 these equations can be represented in the form of equa- netic wave (h = c ( Ð c ) ) and of the electron flux tions for nonisochronous oscillators, which are well- ω ω ω ( Ð hV|| = H) are tangent (Fig. 1a). Here, c is the known in the CRM theory: ω γ waveguide cutoff frequency and H = eH0/mc is the 2 relativistic gyrofrequency. ∂βˆ + ------+ iβˆ +( βˆ + Ð ∆ Ð 1) = ia. (4) We consider the emission from the electron bunch in ∂τ' the comoving frame K', in which the bunch as a whole Equation (4) describes azimuthal self-bunching of is at rest. The Lorentz transformation for the longitudi- electrons caused by the dependence of the gyrofre- nal wave number h' is quency on the electron energy. In Eqs. (3) and (4), we ω use the following dimensionless variables: the γ  V || γ  V ||  h' = ||h Ð ------ = ||h1 Ð ------ , βˆ c2 V c2 normalized transverse electron velocity + = ph ω β β β exp(i ct)( ' + i ' )'/ ⊥' ; ω x y 0 where Vph = /h is the phase velocity of the electromag- 2 2 Ð1/2 3 netic wave and γ|| = (1 Ð V|| /c ) . a = (2eA'/mcω β⊥' )J (R ω /c), c 0 m Ð 1 0 c

It is well known that, when an electromagnetic wave 2 2 Z' = z'β⊥' ω /c, τ' = t'β⊥' ω /2; propagates in a waveguide, the relationship VgrVph = c 0 c 0 c holds [23]. It is seen from the group synchronism condition (1) that the longitudinal wavenumber vanishes in 2 Similar considerations for transverse magnetic (TM) modes show the comoving frame. Similarly, the allowance for the that the transverse component of the electric ﬁeld vanishes in the comoving frame when condition (1) is satisﬁed. For this reason, relationship E⊥ = [H⊥, z0]Vph/c (where z0 is a unit vec- the interaction with TM modes is ineffective in the regime of tor), which is valid for transverse electric (TE) group synchronism.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 THEORY OF CYCLOTRON SUPERRADIANCE FROM A MOVING ELECTRON BUNCH 815 the detuning of the unperturbed cyclotron frequency where hˆ = (Ω + ∆)1/2 and χ = (Ω + ∆ Ð 4G(Ω Ð 1)/Ω2)1/2 from the cutoff frequency (in the comoving frame) ∆ = are the wavenumbers outside and inside the bunch, ω ω ω β 2 respectively. 2( ' Ð c)/ c ⊥' ; and the form factor H 0 First, we consider a relatively short electron bunch 2 2 Ӷ 1 eI 1 λ J (R ω /c) h'b' 1; (8) G = ------0------m----Ð---1------0------c------π 3 4 5 2 2 2 2 i.e., we assume that the bunch length is much smaller 4 mc β⊥ β|| γ || πR J (ν )(1 Ð m /ν ) 0 0 0 m n n than the wavelength of the waveguide mode λ' = 2π/h'. written under the assumption that an annular electron In this case, χB Ӷ 1 and χ/hˆ Ӷ 1, which makes it pos- bunch with the injection radius R0 radiates in a circular sible to reduce the characteristic equation (7) [19] to waveguide of radius R, where I0 is the total current in λ π ω π ν iΩ2 Ω + ∆ + 2GBΩ = 2GB. (9) the laboratory frame, = 2 c/ c = 2 R/ n, m is the azi- ν muthal index of the waveguide mode, and n is the In the regime of exact group synchronism ∆ = 0 nth root of the equation J' (ν) = 0. The function f(Z ') ω ω Ӷ m ( H' = c), assuming that the bunch is rarefied (G 1), describes the axial distribution of the electron density. we can neglect cyclotron absorption [the second term A uniform (correct to small fluctuations specified by on the left-hand side of Eq. (9)] and find an analytical the parameter r Ӷ 1) initial distribution of the electrons solution to Eq. (9). Among the solutions to Eq. (9), over cyclotron rotation phases yields the following ini- there is a unique solution, tial conditions for the set of equations (3) and (4): π Ω ( )2/5   = 2GB expi-- , (10) βˆ + τ = exp[i(Θ + rcosΘ )], 5 ' = 0 0 0 (5) Θ ∈ [ , π] 0 0 2 , a τ' = 0 = 0. corresponding to the excitation of a mode that grows in time (ImΩ < 0) and whose electromagnetic energy flux is directed to the bunch periphery (Rehˆ > 0). According 1.2. The Linear Stage of Cyclotron SR to (10), the growth rate of the SR instability is In the linear approximation, the bunch of cyclotron Ω ( )2/5 (π ) oscillators can be regarded as a finite-length active Im = 2GB sin /5 (11) medium that forms an active resonator exhibiting a dis- or, in the dimension variables, crete eigenmode spectrum. To determine the eigen- 2 modes, we assume that the electrons are distributed π  eI β⊥ λ ω 1ω    0 0 b uniformly in the bulk of a planar electron bunch of Im ' = -- csin-- ------3 ------3------3 2 5 mc β|| γ || πR β ω ∈ 0 0 length B = ⊥' b' c/c, so that f(Z ') = 1 at Z ' [ÐB/2, (12) 0 2 ( ω )  2/5 B/2]. The emission field can be presented as a(Z ', t') = Jm Ð 1 R0 c/c × ------ . a(Z ')exp(i∆τ' + iΩτ'), where 2 (ν )( 2 ν 2) Jm n 1 Ð m / n a(Z ') = C exp(iχZ ') + C exp(ÐiχZ ') 1 2 Note that, according to Eqs. (11) and (12), in spite in the bulk of the layer and of the fact that the bunch is limited in the longitudinal direction and there are energy losses due to emission of ( ) (− ˆ ) a Z ' = C3, 4 exp +ihZ ' radiation, the instability has no threshold, which is explained by an infinite lifetime of electron oscillators on the left (the upper sign) and on the right (the lower in the region of interaction with the electromagnetic sign) of the electron bunch. We linearize Eqs. (3) and field. (4) with allowance for the boundary conditions Due to a positive electron frequency shift 2/5  ∂a  Re Ω = (2GB) cos(π/5) > 0 , (13) {a}, ------ = 0, (6) ∂ B  Z ' Z' = ±--- the emission frequency exceeds the cutoff frequency 2 even at ∆ = 0. As a result, the real part of the longitudi- corresponding to the continuity of the transverse elec- nal wavenumber Rehˆ becomes positive. This means tric and magnetic fields at the boundaries of the elec- that the group velocity also becomes nonzero, which tron bunch. As a result, we obtain a characteristic equa- causes the energy outflow from the electron bunch. tion determining the complex eigenfrequencies Ω: Figure 2 shows the dimensionless growth rate ImΩ, ˆ χ 2 Ω ˆ ( χ ) h +  electron frequency shift Re , and real (Reh ) and exp Ð2i B = ------ , (7) hˆ Ð χ imaginary (Imhˆ ) parts of the wavenumber versus the

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ˇ Since, in the vicinity of the cutoff frequency, the Reh, Imh longitudinal wavenumber is h' ~ cÐ1 2ω (ω' Ð ω ) and

2 c c ˇ it follows from (10) and (11) that the frequency shift is |ω ω | ω of the same order as the growth rate ( ' Ð c ~ Im '), Reh we can rewrite inequality (8) in the form

2 b' Ӷ λ------1, (15) 1 cT' (a) where T' = (Imω')Ð1 is the characteristic growth time of the SR instability (inverse growth rate). Note that, under condition (15), the characteristic

equation (9) can be obtained by linearizing Eqs. (3) and 0 ˇ (4), in which we should set f(Z') = Bδ(Z'), where δ(Z') Imh is the delta function. An electron bunch of an arbitrary length exhibits an ReΩ, ImΩ infinite number of unstable modes that differ from each 0.3 other by the number of field oscillations along the longitudinal coordinate. Figure 3 shows the growth rate of the first symmetric and first antisymmetric modes ver- (b) 0.2 sus the parameter B at a constant number of particles in the bunch (constant bunch charge). It is seen that the first symmetric mode [its growth rate is given by (11) at Ӷ ReΩ B 1 has the maximum growth rate at any electron 0.1 bunch length. As the bunch length increases, the growth rates of the other modes get closer to the growth rate of 0 1 2 3 4 ∆ the first symmetric mode. 0 In the limiting case of an extended layer (B ӷ 1), the ImΩ characteristic equation (7) can be represented in the form –0.1 2 3 l π 2 Ω Ð ------Ω Ð 4GΩ + 4G = 0, (16) B2 Fig. 2. (a) The real (Rehˆ ) and imaginary (Imhˆ ) parts of the longitudinal wavenumber and (b) the electron frequency where l = 0, 2, 4, … for symmetric modes and l = 1, 3, shift ReΩ and the growth rate of the SR instability ImΩ ver- 5, … for antisymmetric modes. ∆ sus the detuning parameter for a short electron bunch at Ӷ GB = 0.01. The dashed curve shows to the space charge If the particle density is sufficiently low (G 1), the parameter q = 0.015. first order of the perturbation theory over the small parameter BÐ1 yields the following solution to this equation: detuning parameter ∆. It is seen that detuning from the cutoff frequency (i.e., the violation of the tangent 1/3  π 2π  regime) leads to a decreasing growth rate. If the detun- Ω = (4G) exp i-- + i------( j Ð 1) ,  3 3  (17) ing is positive (ω' > ω ), the instability develops at any H c where j = 1Ð3. arbitrarily large value of the parameter ∆. At ∆ ӷ 1, the asymptotics Expression (17) apparently coincides with a well- 1/2 Ð1/4 known relationship for the instability growth rate in an Ω = (GB) ∆ (1 Ð i), infinitely long bunch of nonisochronous oscillators [24]. (14) ∞ ˆ ∆1/2 ( )1/2∆Ð3/4( ) The limiting value of the growth rate in Fig. 3 at B h = + GB 1 Ð i /2, is determined by the root j = 3 in (17), corresponding corresponding to crossing of the dispersion curves [15], to the solution that grows in time. are valid. In the range of negative detunings (ω' < ω ), Figure 4 shows the growth rate versus the detuning H c parameter ∆ at a fixed total charge and two different the instability ceases at ∆ < ∆ú ≈ Ð0.3. lengths of the bunch. It is seen that the larger the bunch

TECHNICAL PHYSICS Vol. 45 No. 7 2000 THEORY OF CYCLOTRON SUPERRADIANCE FROM A MOVING ELECTRON BUNCH 817 length, the more pronounced the decrease in the growth |lmΩ| rate with detuning from the regime of group synchro- 0.3 nism.

1 1.3. The Nonlinear Stage of Cyclotron SR 0.2 We studied the nonlinear stage of the cyclotron SR 2 instability under the group synchronism conditions by numerically solving Eqs. (3) and (4) for the parameters close to the experimental ones (see above): the operating mode was TE21; the waveguide radius was about 0.1 0.5 cm; the electron injection radius was 0.25 cm; and β⊥/β|| ~ 1. The corresponding values of the dimensionless parameters were G = 0.12 and B = 10. Figure 5a shows the waveforms of the amplitude of the emission field at three values of the detunings ∆. It is seen that the radiation is emitted in the form of a short pulse. The 0 5 10 15 20 main part of the transverse oscillatory energy of elec- B trons is transformed into the radiation energy in a time Fig. 3. The absolute values of the growth rates of (1) the first of a few inverse growth rates. The characteristic dura- symmetric and (2) the first antisymmetric modes as function of the electromagnetic pulse is determined by the tions of the bunch length B for GB = 0.1. electron azimuthal phasing and dephasing times. Indeed, a comparison of Figs. 5a and 5b shows that the |lmΩ| duration of the electromagnetic pulse is almost equal to 0.3 the duration of the azimuthal current pulse. For β⊥' ~ 0.5, the duration the main SR pulse is about ten cyclotron periods. The additional maxima of the radiation 1 field in Fig. 5a are apparently related to the additional 0.2 maxima of the current, which are typical of the inertial azimuthal bunching of the particles. 2 According to the linear theory, the growth rate of the 0.1 SR instability is maximum at ∆ = 0. However, SR pulses can be generated at both positive and negative values of the detuning parameter ∆. Detuning from the 0 group synchronism condition decreases the growth rate ∆ and slightly lowers the SR peak power. –1 0 1 2 3 4 5

Fig. 4. The absolute value of the growth rate of the first sym- 1.4. Allowance for the Space Charge Field metric mode as a function of the detuning parameter ∆ for the fixed bunch charge GB = 0.1 and bunch lengths of B = The transverse Coulomb field arising in the course (1) 0.5 and (2) 10. of the azimuthal bunching of the particles may significantly influence the SR process. This field may give rise to a “negative” mass effect [25, 26] and sometimes electron can be found by averaging over the positions increase the growth rate of the SR instability. of all the other planes. As a result, the equation of motion (4) takes the form (see [26] for details) We consider the effect of the space charge within a model widely used in the gyrotron theory [22]. If the ∂βˆ 2 ------+- + iβˆ ( βˆ Ð ∆ Ð 1) radius and length of an annular electron bunch are sub- ∂τ' + + stantially larger than its thickness, we can assume that 2π (18) the space charge field has only a radial component and βˆ (τ , Θ ) βˆ (τ , ψ) 1 ψ + ' 0 Ð + ' that the field structure coincides with that of a planar = ia Ð iq----π-- ∫ d ------, 2 βˆ +(τ', Θ ) Ð βˆ +(τ', ψ) charged bunch. The bunch is formed of electrons whose 0 0 Larmor-orbit centers lie on one line and that have dif- 2 where q = 2(ω /β⊥' ω ) is the space charge parameter, ferent azimuthal positions with respect to the guiding b 0 H centers. The bunch can be presented as an ensemble of ω π charged planes consisting of electrons that fall into a b = 4 ene/m is the plasma frequency, and ne is the narrow interval of the initial azimuthal angle dψ and electron density. differ from each other by the position of the Larmor- Equation (18) and the excitation equation in its orbit center. Then the effective force acting on a given former representation (3) describe SR with allowance

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|a| given by 1/2 (a) h = (Ω + ∆˜ ) , 0.3 Ω 1/2 (19) χ Ω ∆˜ + 2q˜ Ð 1  1 2 =  + Ð 4G------ , (Ω Ð q˜ )2 Ð 4q˜ (q˜ Ð 1) 0.2 where q˜ = 2q/3π, ∆˜ = ∆ Ð 3q˜ , and only the first harmonic of the space charge field is taken into account. 0.1 3 We will analyze the effect of the space charge in the case of a relatively short electron bunch satisfying inequality (8), which means that the bunch length b is less 0 than the wavelength of the waveguide mode. At the J same time, we assume that the bunch length is larger 0.10 than the radius of electron cyclotron rotation and use (b) the right-hand side of (18) to write the expression for the space charge field. With the use of (19), the characteristic equation (7) transforms into 0.05 i Ω Ð ∆˜ {(Ω Ð q˜ )2 Ð 4q˜ (q˜ Ð 1)}+2GBΩ = 2GB. (20) The dashed line in Fig. 2 shows the growth rate of the SR instability as a function of ∆ with allowance for 0 10 20 30 τ the space charge field. It is seen that the space charge slightly increases the growth rate of the SR instability. Fig. 5. (a) The field amplitude at the edge of the electron This can be attributed to the “negative mass” effect, bunch versus time in the comoving frame at B = 10; G = 0.1; which causes the azimuthal bunching of electrons even and ∆ = (1) 0, (2) 1, and (3) Ð0.7. (b) The transverse electron in the absence of the radiation field [25]. Numerical current in the same cross section as a function of time at solution of Eqs. (3) and (18) in the case of a short elec- ∆ = 0. tron bunch also proves that allowance for the space charge field leads to an increase in the growth rate and |a| a slight increase in the peak amplitude of the radiation 0.03 field (Fig. 6).

2. CYCLOTRON SR IN THE LABORATORY 2 1 FRAME 0.02 It follows from the above analysis that, in the comoving frame K', the electron bunch radiates isotropically in both +z' and Ðz' directions along the waveguide axis. Since, in this frame, the radiation frequency is close to the cutoff frequency, the group veloc-

0.01 ities V gr' of the electromagnetic pulses emitted in both directions are rather low (their difference from zero is related to the electron frequency detuning, see Section 1.2). In the laboratory frame, the bunch moves with the

longitudinal velocity V|| > V gr' (under the experimental 0 20 40 60 80 100 τ conditions, V|| ~ 0.7c). If the bunch moves towards a detector, the radiation emitted in the +z' (forward) Fig. 6. The electric field amplitude in the comoving frame direction in K' frame affects the detector earlier than versus time (1) with and (2) without allowance for the space that emitted in the Ðz' (backward) direction. Indeed, it charge field in a short-bunch model at GB = 0.01 and q = 0.015. follows from the velocity addition law that, if the source (electron bunch) velocity is higher than the electromagnetic wave group velocity in the source frame, for the space charge field. The form of characteristic then, in the laboratory frame, both the radiation compo- equation (7) derived within the linear approximation is nents propagate in the direction of the source motion. also conserved, but, in the case under consideration, the The group velocity of the “forward” pulse in the labo- wavenumbers outside and inside the electron bunch are ratory frame is slightly higher and that of the “back-

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|a| length, which was experimentally demonstrated in 0.2 [20, 21]. (a) The radiation power at the detector that is at rest in the laboratory frame is determined by the vector product of the electric and magnetic fields on the trajectory of the detector motion in the (t', z') plane. These fields 0.1 can be obtained by the Lorentz transformation of the fields in the comoving frame. Integration over the waveguide cross section yields the following expression for the power in the laboratory frame: 2 5 2 2(ν )( 2 ν 2) 0 πm c πR J n 1 Ð m / n 8 6 P = ------γ || β⊥ 0.2 2 2 2 0 0 2 e λ J (R ω /c) m Ð 1 0 c (21) (b)  2 2  ∂a* × β|| a + β⊥ γ || (1 + β|| )Im a------.  0 0 0 0  ∂Z'  0.1 Note that (21) is written with allowance for the fact that the transverse component of the magnetic field is small in the K' frame.

CONCLUSIONS 0 10 20 30 τ Therefore, we have shown that cyclotron SR can be Fig. 7. The field amplitude recorded by a detector in the lab- used to generate short (about ten cyclotron oscillations) oratory frame as a function of time at ∆ = (a) 0 and (b) Ð0.4. electromagnetic pulses. The group synchronism regime appears to be the most favorable for cyclotron SR observation. A detuning from this regime leads to a ward” pulse is slightly lower than the longitudinal decrease in both the growth rate of the SR instability velocity of the electron bunch. Thus, the distance and the SR peak power. Note than this decrease is even between the two pulses increases with increasing more pronounced for positive detunings ∆ if we take observation length. into account the longitudinal dynamics of a real elec- The shape of the SR pulse at the detector can be tron bunch caused by the spread in the initial longitudi- obtained as follows. First, we calculate the field distri- nal velocities and longitudinal Coulomb repulsion. bution in the (z', t') plane and then determine the field These two factors cause relative displacement of elec- on the z' = ÐV||t' + const line, along which the detector trons in the comoving frame, which may result in sub- moves. Figure 7a shows the electric field amplitude at stantial suppression of the SR instability if the displace- the detector versus time calculated for the case of the ment is comparable with the wavelength of the exact group synchronism (∆ = 0). It is seen that the waveguide mode λ' = 2π/h'. Under the exact group syn- detector signal has a two-hump shape; i.e., the emitted chronism regime (∆ = 0), h' tends to zero, the wave- radiation consists of two pulses. The first and second length λ' of the waveguide mode becomes infinitely pulses are formed by photons emitted by the bunch in large, and the radiation is almost independent of the the K' frame in the positive and negative directions longitudinal displacement of electrons. However, as ∆ along the z'-axis, respectively. Due to the Doppler increases (i.e., the regime of group synchronism is vio- effect, the carrier frequency of the first pulse is higher lated), the longitudinal wavenumber increases and, than that of the second pulse. Hence, the first pulse accordingly, the wavelength of the waveguide mode turns out to be substantially shorter than the second decreases and tends to the vacuum wavelength λv' = pulse. The amplitude of the first pulse is larger, because 2πc/ω'. In this case, the same longitudinal displacement it arrives at the detector earlier than the second pulse can result in substantial suppression of SR. Experimen- and, thus, is less subjected to dispersion spreading. tal results [20, 21] demonstrate almost complete sup- Note that, for negative values of the detuning pression of the microwave signal as the detuning from parameter ∆, the difference between the group veloci- the exact group synchronism increases. ties becomes negligible. Hence, the signal detected in Finally, we estimate the parameters of cyclotron SR the laboratory frame at a given observation length looks pulses in the regime of exact group synchronism under like a single pulse (Fig. 7b). the conditions close to experimental (see the Introduc- Therefore, the SR pulse shape can be varied by tion). The pulse duration, which can be found from changing the observation length at a constant ∆ or/and Fig. 5, is approximately 300 ps and agrees well with the changing the detuning parameter at a given observation experimental data. The corresponding growth rate of

TECHNICAL PHYSICS Vol. 45 No. 7 2000 820 GINZBURG et al. the SR instability in the comoving frame is equal to 9. D. A. Jarosynski, P. Chaix, N. Piovella, et al., Phys. Rev. 10 Ð1 0.9 × 10 s . For the bunch drift velocity V|| = 0.5 s, the Lett. 78, 1699 (1997). distance corresponding to the e-fold increase in the 10. V. I. Kanavets and A. Yu. Stabinis, Vestn. Mosk. Gos. power in the laboratory frame is about 1.6 cm. In exper- Univ., Ser. Fiz. 14, 186 (1973). iments, such an increase was observed at a distance of 11. V. V. Zheleznyakov, V. V. Kocharovskiœ, and Vs. V. Ko- approximately 4 cm. Numerical results predict a peak charovskiœ, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 29, power of 8 MW, whereas the experimentally detected 1095 (1986). power amounted to 200Ð300 kW. Such a discrepancy 12. Yu. A. Il’inskiœ and N. S. Maslova, Zh. Éksp. Teor. Fiz. between the calculated and experimental values of the 94, 171 (1988) [Sov. Phys. JETP 67, 96 (1988)]. growth rate and the peak power can be attributed to the 13. L. A. Vaœnshteœn and A. I. Kleev, Dokl. Akad. Nauk presence of a spread in the initial electron energies and SSSR 311, 862 (1990) [Sov. Phys. Dokl. 35, 359 pitch angles, which are typical of real electron bunches. (1990)]. Note also that the emission power reported in [20, 21] 14. E. S. Mchedlova and D. I. Trubetskov, Pis’ma Zh. Tekh. should only be considered a lower estimate, because, at Fiz. 19 (24), 26 (1993) [Tech. Phys. Lett. 19, 784 present, the methods for recording individual subnano- (1993)]. second electromagnetic pulses require additional 15. N. S. Ginzburg, I. V. Zotova, and A. S. Sergeev, Pis’ma improvement. Zh. Tekh. Fiz. 15 (14), 83 (1989) [Sov. Tech. Phys. Lett. 15, 573 (1989)]. 16. N. S. Ginzburg and A.S. Sergeev, Zh. Tekh. Fiz. 60 (8), ACKNOWLEDGMENTS 40 (1990) [Sov. Phys. Tech. Phys. 35, 896 (1990)]. We are grateful to A. D. R. Phelps and M. I. Yalan- 17. N. S. Ginzburg and A. S. Sergeev, Zh. Éksp. Teor. Fiz. din for fruitful discussions. This work was supported by 99, 438 (1991) [Sov. Phys. JETP 72, 243 (1991)]. the Russian Foundation for Basic Research, project 18. N. S. Ginzburg and A. S. Sergeev, Fiz. Plazmy 17, 1318 no. 95-02-04791. (1991) [Sov. J. Plasma Phys. 17, 762 (1991)]. 19. N. S. Ginzburg, I. V. Zotova, and A. S. Sergeev, Pis’ma Zh. Éksp. Teor. Fiz. 60, 501 (1994) [JETP Lett. 60, 513 REFERENCES (1994)]. 1. R. H. Dicke, Phys. Rev. 99, 131 (1954). 20. N. S. Ginzburg, I. V. Zotova, A. S. Sergeev, et al., Pis’ma 2. A. V. Andreev, V. I. Emel’yanov, and Yu. A. Il’inskiœ, Zh. Éksp. Teor. Fiz. 63, 322 (1996) [JETP Lett. 63, 331 Cooperative Phenomena in Optics (Nauka, Moscow, (1996)]. 1988), p. 277. 21. N. S. Ginzburg, I. V. Zotova, A. S. Sergeev, et al., Phys. 3. V. V. Zheleznyakov, V. V. Kocharovskiœ, and Vs. V. Ko- Rev. Lett. 78, 2365 (1997). charovskiœ, Usp. Fiz. Nauk 159 (2), 193 (1989) [Sov. 22. A. V. Gaponov, M. I. Petelin, and V. K. Yulpatov, Izv. Phys. Usp. 32, 835 (1989)]. Vyssh. Uchebn. Zaved. Radiofiz. 28, 1538 (1975). 4. R. Bonifachio, C. Maroli, and N. Piovella, Opt. Com- mun. 68, 369 (1988). 23. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975; Inostrannaya Literatura, Moscow, 1965). 5. R. Bonifachio, N. Piovella, and B. W. J. McNeil, Phys. Rev. A 44, 3441 (1991). 24. V. V. Zheleznyakov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 3, 57 (1960). 6. R. Bonifachio, L. D. Salvo, L. Narducci, and E. D. An- gelo, Phys. Rev. A 150, 50 (1994). 25. V. L. Bratman and M. I. Petelin, Izv. Vyssh. Uchebn. 7. N. S. Ginzburg, Pis’ma Zh. Tekh. Fiz. 14, 440 (1988) Zaved. Radiofiz. 28, 1538 (1975). [Sov. Tech. Phys. Lett. 14, 197 (1988)]. 26. V. L. Bratman, Zh. Tekh. Fiz. 46, 2030 (1976) [Sov. 8. N. S. Ginzburg and A. S. Sergeev, Pis’ma Zh. Éksp. Teor. Phys. Tech. Phys. 21, 1188 (1976)]. Fiz. 54, 445 (1991) [JETP Lett. 54, 446 (1991)]. Translated by A. Chikishev

TECHNICAL PHYSICS Vol. 45 No. 7 2000

THEORETICAL AND MATHEMATICAL PHYSICS

The Classification of Anisotropic Stressed States in Spherical Bodies A. S. Baranov Pulkovo Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140 Russia Received April 19, 1999

Abstract—Tensor ﬁelds (for example, of pressure) with desired properties of symmetry under rotation were constructed in an explicit form. Tensor components represented as polynomials in Cartesian coordinates satisfy a biharmonic equation. Possible applications of the obtained results in various domains of technical physics are discussed. © 2000 MAIK “Nauka/Interperiodica”.

STATEMENT OF THE PROBLEM For the specific case of spherical bodies, the classification of tensor fields is naturally related to the sym- Harmonic and biharmonic functions are of frequent metry under rotation of the entire body, in contrast to use in technical physics. They are applied in the theo- the more general case [3], where it may seem somewhat ries of elasticity, plasticity, equilibrium figures in a formal. That is why we consider the spherical case rotating liquid, etc. An apparatus for expanding the func- separately. It requires a solution to the biharmonic tions over an appropriate basis depends on the body equation ∆∆U = 0 to meet representations of rotation shape. Most often, spherical functions are used [1]. group [6]. Such representations may be one- or two- Spheroidal functions are less known [2]. Spheroidal valued, but we are certainly interested in the former. bodies are usually characterized by harmonic func- Each representation is characterized by an index n, tions. In [3], we elaborated the biharmonic function which runs over 0, 1, 2, … values and combines 2n + 1 apparatus for spheroidal bodies to tackle a variety of states of an object into a unified set. Under any rota- problems when the bounding surface is close to sphe- tions, the states are linearly transformed into each roidal. Spherical bodies, however, can also be treated in other. One cannot get by with a lesser number of states; wider mathematical terms than is generally appreci- otherwise, new states will arise at some rotations. ated. A trivial example is the transformation of a single vec- Indeed, in technical physics, we handle not only tor under rotations: in this transformation, all three vec- scalar characteristics. The tensor character of pressure tor components are involved. An example of another and pressure-related parameters is perhaps the most sort is the transformation of scalar fields on a sphere. In prominent example. Such an anisotropic analog of this case, the standard basis of representation at fixed pressure is used in magnetometric prospecting for gas n’s is a set of 2n + 1 spherical harmonics with the same fields, as well as in exploration of bulk materials, inte- index [7]. rior of the Earth and other planets, etc. In star dynamics and the theory of plasma, the analog of pressure also In general, spinor, vector, and tensor fields on a exhibits more or less anisotropic properties [4]. The sphere have been studied by Gel’fand and coworkers, same is true for applications such as the study of equi- but the derived formulas are of a rather abstract charac- librium in round elastic bodies. The only difference ter. They are hard to apply to specific technical prob- here is that gravitational forces are replaced by external lems, especially if the harmonicity or biharmonicity forces applied to the surface [5]. Therefore, it is of condition is additionally set. Therefore, we will address interest to classify spatial distributions of anisotropic the problem in a somewhat different way. pressure inside a body in terms of symmetry under rotation. CONDITIONS OF EQUILIBRIUM Pressure tensor components will be assumed to be harmonic or biharmonic functions. Such a restriction Of basis states (or vectors) corresponding to a given n, often follows from additional physical conditions we shall consider the one which is the easiest to express [1, 3]. Then, a random pressure distribution that satis- in Cartesian coordinates. With scalar fields, experience fies the harmonicity or biharmonicity condition, as well suggests that this is the extreme “vector,” which, being as the condition of equilibrium inside the volume, can rotated about the polar axis through an angle Θ, be represented as a superposition of simpler distribu- acquires the factor exp(inΘ). In the case of scalar fields, tions that satisfy certain symmetry requirements. this vector is called sectorial harmonic.

822 BARANOV

Let m be the degree of polynomials describing pres- vanish when the operator Ix + iIy is applied, where Ix sure tensor components. A pressure tensor is known to and Iy are the infinitesimal operators of rotations about have the form of a symmetrical matrix the axes x and y, respectively (it is assumed that rotation, viewed from the positive ends of both horizontal   axes, proceeds in the same direction).  Pxx Pxy Pxz  ( , , )  P P P  Pxy = Pyx Pxz = Pzx Pyz = Pzy ,  yx yy yz  For the Ix + iIy operator, the table of action on the  P P P  tensor components alone is easily constructed in the zx zy zz same way as for rotations about the x-axis (with the where x, y, and z are Cartesian coordinates and z is the changed roles of coordinates). Eventually, six equa- polar axis. tions that relate the pressure tensor components are obtained [6]. On rearrangement (details are omitted), Since the component Pzz by itself does not change we come to the following set: when rotated about the polar axis, the azimuth depen- Θ dence must show up through the factor exp(in ) or, in ∂ ∂ ∂P Cartesian coordinates, the factor ϕ [ϕ = (x + iy)n],   ( ) zz ( ) n n Ð z∂ + i∂  Pzz + x + iy ---∂------Ð 2 Pxz + iPyz = 0, where the other factor is some polynomial in z and R2 = x y z 2 2 x + y . Hence, if we start with the component Pzz, it ϕ ∂P ∂P  should be sought in the form Pzz = c0 n (m = u), Pzz = (x + iy) ------x--z + i------y--z ϕ 2 2 ϕ  ∂ ∂  c1z n (m = n + 1), or Pzz = (c1R + c2z ) n (m = n + 2); z z for m < n, it is necessary to construct a similar polynomial and put Pzz = 0. It will be shown below that the  ∂ ∂  | | Ð z + i (P + iP ) Ð H = 0, case m Ð n > 2 is impossible. ∂x ∂y xz yz 1 When rotated about the polar axis, the quantity Φ = P + P , also a scalar, is transformed in a quite similar xx yy ∂Φ ∂Φ ∂Φ way but with other coefficients. We will pursue our ( )   ( ) x + iy ------Ð z------+ i------ + 2 Pxz + iPyz = 0, consideration as follows. When rotated about the polar ∂z ∂x ∂y axis, the components P and P by themselves trans- (1) xz yz ∂H  ∂ ∂  form following the same law as the components of a (x + iy)------1 Ð z + i H = 0, Θ ∂z ∂x ∂y 1 two-dimensional vector; that is, Pxz Pxzcos Ð Θ Θ Θ Pyzsin and Pyz Pxzsin + Pyzcos . For their ∂ ∂ ∂ complex combination, we have Pxz + iPyx ( ) H   ( ) Θ Θ Ð x + iy ------+ z + i  H Ð 4 Pxz Ð iPyz = 0, Fexp(i ) and Pyz Ð iPxz F1exp(i ), where F = Pyz + ∂z ∂x ∂y iPxz and F1 = Pyz Ð iPxz. In this way, the components Pxz and Pyz by them- ∂P ∂P  (x + iy) ------x--z Ð i------y--z selves transform. Here, account is taken of the fact that,  ∂z ∂z  under rotation, as was noted, a point in space to which they are related changes when the field as a whole  ∂ ∂  rotates. Hence, to find these components in a fixed Ð z + i (P Ð iP ) + 2P Ð Φ = 0. coordinate system, the factor exp(inΘ) must be ∂x ∂y xz yz zz included in the above formulas. Thus, unlike Pzz, the ± ϕ general expression Pxz iPyz must involve the factor n. Equations (1) are basic in our problem. Recall that The second factor should again be either a polynomial we are interested in only those tensor fields satisfying of the [m Ð (n ± 1)] degree in z and R2 or zero if the the mechanical equilibrium condition expression in the brackets is a negative. Finally, the components P , P , and P transform under rotation ∂P ∂P ∂P ∂P ∂P ∂P xx xy yy ------x--x + ------x--y + ------x--z = ------y--x + ------y--y + ------y--z about the polar axis in the same way as the components ∂x ∂y ∂z ∂x ∂y ∂z of a two-dimensional tensor, that is, in the same way as ∂P ∂P ∂P pair products of two different vectors. The combina- = ------z--x + ------z--y + ------z--z = 0, tions H = Pxx Ð Pyy Ð 2iPyx and H1 = Pxx Ð Pyy + 2iPyx ∂x ∂y ∂z show simple properties under rotation. Similar consid- ρ erations indicate that H and H1 must involve the factors where Pxx = pxx + V in the presence of self-gravitation; ϕ ϕ ± ≥ ρ n ± 2 and n Ð 2, respectively [if m Ð (n 2) 0]. Pxy = pxy, …; is the material density, which is assumed The previous argument referred to rotations about to be constant; pxx, pxy, … are the components of pres- the polar axis. However, they are insufficient to prove sure as such; and V(x, y, z) is the gravitational potential. the desired total symmetry of the field. A typical feature Naturally, P and p coincide in the absence of external of the extreme state under consideration is that it must forces.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 THE CLASSIFICATION OF ANISOTROPIC STRESSED STATES 823

After transformation, the equilibrium condition can Then, 2(n Ð 1)c1 + c2 = 0. For example, at c1 = 4 and − be written in the form [3] c2 = 8(n Ð 1), we have ∂ ∂Φ ∂Φ ∂ ∂ ( )ϕ ( ) ϕ ( )     Pxz Ð iPyz = 2 2n Ð 1 n Ð 1, Pxz = 2n Ð 1 z n Ð 1, 2 Pxz + iPyz + ------+ i------ Ð  Ð i  H1 = 0, ∂z ∂x ∂y ∂x ∂y ϕ Pyz = i(2n Ð 1)z n Ð 1, ∂ ∂Φ ∂Φ  ∂ ∂  Φ ( )ϕ 2 (P Ð iP ) + ------Ð i------+ + i H = 0, = Pxx + Pyy = Ð2 2 + 1/n n, ∂z xz yz  ∂x ∂y  ∂x ∂y (2) P = Ð(2n + 1 + 1/n)ϕ , ∂ ∂ zz n   ( ) ∂ + i∂  Pxz Ð iPyz [ 2 ( ) 2]ϕ x y H = 4R Ð 8 n Ð 1 z n Ð 2,

∂ 2 2  ∂ ∂  Pzz [ ( ) ]ϕ + Ð i (P + iP ) + 2------= 0. Pxx Ð Pyy = 2 R Ð 2 n Ð 1 z n Ð 2, ∂x ∂y xz yz ∂z P = i[R2 Ð 2(n Ð 1)z2]ϕ . In what follows, we will consider various indexes n xy n Ð 2 at a given degree m of the polynomials. The other solution will be essentially biharmonic rather than harmonic. For c1 = 4(n + 1) and c2 = 4(n + 2), we have DETERMINATION OF THE BASIC ϕ ϕ FUNCTIONS Pxz Ð iPyz = Ð4z n Ð 1, Pxz = Ð2z n Ð 1, ϕ The extreme case n = m + 2 is the easiest. In this Pyz = Ð2iz n Ð 1, case, as was noted, the tensor components are free from [ 2 2]ϕ the factors exp(inΘ), exp[i(n ± 1)Θ], and exp[i(n + Pxx Ð Pyy = 2(n + 1)R + 2(n + 3)z n Ð 2, Θ ϕ 2) ]. Then, H = c n remains as the only other-than- [( ) 2 ( ) 2]ϕ zero combination, so that, with the constant coefficient c Pxy = i n + 1 R + n + 3 z n Ð 2, properly selected (c = 4), we have Φ = P + P = Ð4ϕ , P = 0. ϕ ϕ ϕ xx yy n zz Pxx = n, Pyy = Ð n, Pxy = i n, Consider now the case n = m Ð 1. Only H1 vanishes ϕ Pxz = Pyz = Pzz = 0. immediately. Let Pzz = cz n Ð 1 (according to the general rule). Then, the first equation in (1) yields P + iP = Notice that this solution automatically satisfies both xz yz ϕ 3 2 ϕ the equilibrium and harmonicity conditions. Next, we (c/2) n. We can, however, take H = (c1z + c2zR ) n Ð 3. take n = m + 1. The factors exp(inΘ), exp[i(n + 1]Θ, and Substituting this equality into the last but one equation in Θ Φ 2 2 ϕ exp[i(n + 2) ] are impossible here, hence Pzz = = (1) yields Pxz Ð iPyz = (1/4)[Ðc2R + (2c2 Ð 3c1)z ] n Ð 2. ϕ Pxz + iPyz = H1 = 0. However, H = cz n Ð 1 and we obtain From the last equation in (1), we find Φ = [2c + ϕ ϕ Pxz Ð iPyz = Ð(c/4) n from the last but one equation in (1). 3(c2 Ð c1)/2]z n Ð 1. Finally, the third equation in (1) For example, at c = 8, relates the coefficients through the expression (c2 Ð Φ ϕ ϕ ϕ c1)/2 + c = 0. Eventually, we arrive at = (c2 Ð Pxz = Ð n, Pyz = Ði n, Pxx = 2z n Ð 1, ϕ c1)z n − 1/2. Taking into account the second equilibrium ϕ ϕ Pyy = Ð2z n Ð 1, Pxy = 2iz n Ð 1. condition in (2), one finds the unique relationship between the coefficients c and c : (n + 2)c Ð (n + Again, the equilibrium and harmonicity conditions 1 2 1 3)c2 = 0. We can take c1 = 4(n + 3), c2 = 4(n + 2), and are satisfied automatically. Now let us pass to the case c = 2. Then, m = n. The possibility of the factors exp[i(n + 1)Θ] and Θ ϕ exp[i(n + 2) ] is eliminated; that is, Pxz + iPyz = H1 = 0. Pxz + iPyz = n, 2 2 ϕ We can start with the component H = (c1R + c2z ) n Ð 2. P Ð iP = Ð[(n + 2)R2 + (n + 5)z2]ϕ , Then, the last but one equation in (1) yields Pxz Ð iPyz = xz yz n Ð 2 ϕ Φ (c1 Ð c2)z n Ð 1/2, and the last equation gives Ð 2Pzz = ϕ ϕ Φ Pxx + Pyy = Ð2z n Ð 1, (c1 Ð c2) n/2. The second equation in (2) yields = ϕ 2 2 (c2 Ð 3c1) n/(2n), so that Pzz = (1/4)[(c2 Ð 3c1)/n Ð (c1 Ð P Ð P = 2z[(n + 3)z + (n + 2)R ]ϕ , ϕ xx yy n Ð 3 c2)] n. [( ) 2 ( ) 2]ϕ ϕ In the previous calculation, we invoked equilibrium Pxy = iz n + 3 z + n + 2 R n Ð 3, Pzz = 2z n Ð 1. condition (2). The components remain harmonic or Finally, consider the last case n = m Ð 2. Symmetry biharmonic functions. The difference from the preced- considerations cannot immediately eliminate any of the ing cases is that we are dealing with two linearly inde- combinations. Let us start with the component P = pendent solutions. One of them is obtained if the har- zz 2 2 ϕ monicity condition is imposed on the components. (cR + cz ) n Ð 2. Then, the first equation in (1) yields

TECHNICAL PHYSICS Vol. 45 No. 7 2000 824 BARANOV

ϕ 4 Pxz + iPyz = (c Ð c)z n Ð 1. In addition, H = (c1z + It is easy to verify that, in all of the cases, the values 2 2 4 ϕ 2 2 2 c2R z + c3R ) n Ð 4. After substitution into the last but of the quadratic form Q = x Pxx + y Pyy + z Pzz + one equation of (1), we obtain 2xyPxy + 2xzPxz + 2yzPyz give a space field with a desired type of symmetry; in other words, at a given [( ) 2 ( ) 2]ϕ Pxz Ð iPyz = z c3 Ð c2/2 R + c2/2 Ð c1 z n Ð 3. r (= (x2 + y2 + z2)1/2), they are proportional to the spher- From the last equation of (1), it follows that ical harmonic of order n. Φ OPERATOR APPROACH  c  2 5  2 = c Ð ----2 + 2c R + --c Ð 3c Ð 2c + 2c z ϕ . It would be worthwhile to consider another  3 2  2 2 1 3  n Ð 2 approach to constructing found solutions that is based on the direct construction of the pressure matrix com- Return to the second equation in (1). It yields the ponents. It uses some initial scalar function ϕ and sym- ϕ n relationship H1 = (c Ð c) n. Finally, the third equation metry-preserving operations. in (1) relates the coefficients by the expression For n = m, we, as expected, have two variants of the c Ð c = c + c Ð c . (3) 2 ϕ 3 1 2 pressure tensor. The first is Pxx = (2J x Ð n Ð 1) n and ϕ Turn to the equilibrium condition again. The substi- Pxy = (JxJy + JyJx) n, where tution of the functions found above into Eqs. (2) yields ∂ ∂ ∂ ∂ ∂ ∂ relationships that are partly dependent on the previous J x = y∂ Ð z∂ , J y = z∂ Ð x∂ , Jz = x∂ Ð y∂ one and on each other. Two independent equations z y x z y x remain: are rotation operators [3]. ( )( ) The other components are easily constructed by 2 n + 1 c Ð c + 2c3 Ð c2 + 4c = 0, (4) analogy. ( ) ( ) 2 n + 2 c3 Ð nc2 + 4 n Ð 1 c = 0. The second variant for our case is As a result, we have found three relationships for ∂ϕ ∂ϕ ∂ϕ P = 2x------n Ð (n + 3)ϕ , P = x------n + y------n. five parameters; hence, two independent parameters are xx ∂x n xy ∂y ∂x actually available. These may be, for example, c1 and c2. Then, in particular, Consider the case n = m Ð 1. We have [( )( ) ] [( )( )] ∂U c = n + 4 n Ð 1 c + c2 / n + 1 n + 2 . P = 2(n + 4)J ------Ð 2xJ ∆U, xx x ∂x x Consider the biharmonicity condition. It is nontriv- ∂U ∂U ial only for the function H (the other functions under P = (n + 4)J ------+ J ------Ð (yJ + xJ )∆U, consideration become biharmonic automatically) and xy y ∂x x ∂y x y yields 3c1 + 2(n Ð 3)c2 + 4(n Ð 2)(n Ð 3)c3 = 0. 2ϕ where U = r n. Combining the obtained relationships with (3) and Finally, consider n = m Ð 2. Then, (4), we find, with an accuracy to an arbitrary common 2 coefficient, 2∂ U 2 ∂U 2 P = r ------Ð αx ∆U + βx------+ γU + δr ∆U, xx 2 ∂ ( )( ) 2 ∂x x c1 = 2 n Ð 3 n + 2 , c2 = 2n Ð 5n Ð 6 ∂2 β ∂ ∂ and then 2 U α ∆  U U Pxy = r ------Ð xy U + --y------+ x------ , 2 3 2 ∂x∂y 2 ∂x ∂y 2n Ð 2n Ð 3 2n Ð 5n + n + 6 c = Ð------, c = ------, 3 2(n Ð 2) 4(n Ð 2) where 2 2n3 Ð 5n2 Ð 3n + 12 2n + 5 2n + 9n + 8 c = ------. α = Ð------, β = Ð------, 4(n Ð 2) 2(n + 3) n + 3 It is easy to check that, for n = m + 2, any nonnega- 2n3 + 15n2 + 37n + 28 tive values of m are admissible. In the other cases, m γ = Ð------, δ = Ð1. 2(n + 3) should be bounded below; otherwise, fractional functions, rather than polynomials, would have been the In these formulas, the coefficients are such that the result. Specifically, at n = m + 1 or n = m, m should be pressure tensor components satisfy biharmonicity and ≥1 (for n = m, the only variant among those considered equilibrium conditions (2). Using the formulas, we per- in this work, i.e., the linear combination, remains). formed the necessary calculations to make sure that the Similarly, for n = m Ð 1 and n = m Ð 2, the restriction results obtained are in agreement with those reported in m ≥ 3 is imposed. the basic part of the paper (with an accuracy to multi-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 THE CLASSIFICATION OF ANISOTROPIC STRESSED STATES 825 plying the pressure tensor matrix by an arbitrary fac- The constructed stress fields can be useful for repre- tor). senting equilibrium and vibratory states in elastic and The agreement occurs for any n but, obviously, is plastic bodies largely of spherical or close-to-spherical most readily established for not too large n’s. shape. Certainly, in many applications, it will be neces- Notice the following easy-to-check feature of the sary to use a superposition of a finite or infinite number solutions found; namely, the biharmonic solutions for of found solutions. n = m, m Ð 1, and m Ð 2 subjected to the Laplace operator are transformed, with an accuracy to a constant REFERENCES factor, into the harmonic solutions for, respectively, n = m + 2, m + 1, and m (this m differing from the pre- 1. V. A. Antonov, E. I. Timoshkova, and K. V. Kholshevni- vious m). This feature can also serve as a means to kov, Introduction to the Newtonian Potential Theory check the solutions. (Nauka, Moscow, 1988). 2. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge Univ. Press, Cambridge, 1931; CONCLUSION Inostrannaya Literatura, Moscow, 1952). Thus, for each m, we can construct only six linearly 3. A. S. Baranov, Zh. Vychisl. Mat. Mat. Fiz. 37, 395 independent tensor fields satisfying the equilibrium and (1997). biharmonicity conditions for each of the components. 4. K. F. Ogorodnikov, Dynamics of Stellar Systems (Fiz- The six fields refer to five types of symmetry, the dual- matgiz, Moscow, 1958; Pergamon, Oxford, 1965). ity of the solutions being observed only at n = m. 5. A. E. H. Love, A Treatise on the Mathematical Theory of Recall that each of the symmetry types is (2n + 1)- Elasticity (Cambridge University Press, Cambridge, fold degenerate; that is, 2n + 1 linearly independent ten- 1927; ONTI, Moscow, 1935). sor fields result when the configurations considered 6. G. Ya. Lyubarskiœ, The Theory of Groups and Its Appli- rotate about different axes. Similar degeneracy of sca- cation (Fizmatgiz, Moscow, 1958). lar, vector, and other fields is frequently encountered in 7. N. Ya. Vilenkin, Special Functions and the Theory of applications (see, for example, [8]). The symmetry of Group Representations (Nauka, Moscow, 1965; Ameri- internal states is usually due to external conditions. can Mathematical Society, Providence, 1968). Naturally, the geometrical symmetry group is narrower 8. A. S. Baranov, Astron. Zh. 70, 223 (1993) [Astron. Rep. for spheroidal bodies [3]. It is, however, reasonable to 37, 117 (1993)]. expect that some deeper symmetry formally similar to the considered symmetry of the fields in spherical bodies will also be found for spheroidal bodies. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 7 2000

THEORETICAL AND MATHEMATICAL PHYSICS

Relativistic Particle in a Traveling Magnetic Field A. I. Toropova Advanced Education and Science Center, Moscow State University, Moscow, 121357 Russia E-mail: [email protected] Received April 27, 1999

Abstract—A Hamiltonian formalism is applied to derive an exact solution to the equation of motion of a charged particle in the electromagnetic field of a traveling current wave. The particle motion is studied in a monochromatic magnetic field and in the traveling jump-like front of the magnetic field, and the wave mechanism for betatron acceleration is analyzed. It is shown that, in each of these situations, a charged particle can be accelerated simultaneously in both the longitudinal and transverse directions. © 2000 MAIK “Nauka/Inter- periodica”.

INTRODUCTION µ µ µ where x = (ct, x, y, z); e()1 = (0, 1, 0, 0); e()2 = (0, 0, 1, In present-day relativistic mechanics, only a few 0); nµ = (1, 0, 0, 1); kµ = (ω/c)nµ; B(kx) is an arbitrary problems relevant to the motion of charged particles in function of the argument kx = ωt Ð ωz/c [4]; the metric the field of an electromagnetic wave excited in a real tensor is gµν = diag(1, Ð1, Ð1, Ð1); the scalar product of electrodynamic system are known to possess exact ana- µ two 4-vectors is defined as ab = a bµ = a0b0 Ð ab, so that lytic solutions. Pavlenko et al. [1] solved the equations 2 µ a ≡ a aµ; and the 4-potential satisfies the wave equa- of motion of a charged particle in the field of a TEM ν µ µ wave propagating in a quadrupole waveguide. This tion ∂ ∂νA = 0 and the Lorentz gauge ∂µA = 0. electrodynamic waveguide system, which was first pro- The electromagnetic field tensor has the form posed by V. Paul, made it possible to develop a high- µν µν ()[]()µν ()µν resolution mass spectrometer of nonrelativistic parti- F = t Bkx + e1x1Ð e2x fB'/2, cles [2, 3]. An analysis of the domains of stable particle µν µ ν µ ν µν µ ν µ ν µν motion showed that quadrupole waveguides can also be where t = e()2 e()1e()1e()2 , f = e()1k Ð ke()1 , 1 = used to focus relativistic particles, to separate them out µ ν µ ν by mass and specific charge, and to reduce the spread e()2k Ð ke()2 , and B' = dB/d(kx). in transverse velocities of the beam particles. The electric and magnetic fields are equal to In this paper, we derive exact analytic solutions to (),, the Hamiltonian equations of motion of a relativistic E=yk0B'/2 Ð0xk0B'/2 , (),, ω particle in the electromagnetic field of a traveling cur- B==xk0B'/2 yk0B'/2 B ,k0/c. rent wave in an axisymmetric electrodynamic system. The solutions obtained make it possible to investigate Obviously, we have EB = 0. Note that, in the case the stability domains and to determine the kinetic B = const, potential (1) defines a constant uniform mag- ω energy and longitudinal momentum of the particle. We netic field. The traveling magnetic field Bz = B( t Ð consider particle acceleration in a monochromatic ωz/c) initiates vortex electric and magnetic fields in the magnetic field and in the traveling jumplike front of the plane orthogonal to the symmetry axis of the system. magnetic field. We show that, in the regime of betatron The function B(kx) satisfies the natural boundary con- ∞ wave acceleration, a condition analogous to the well- ditions: B(kx) B1 for kx Ð and B(kx) B2 ∞ known 2 : 1 rule should hold. The solution method pro- for kx +, where B1 and B2 > B1 are positive con- posed here is based on the theory of canonical transfor- stants. mations and applies to any mechanism for field excitation. SOLUTION OF THE EQUATIONS OF MOTION ELECTROMAGNETIC FIELD We describe the particle trajectory in parametric OF AN ELECTRODYNAMIC SYSTEM µ µ form: x = x (τ), where τ is the intrinsic time. The The 4-potential of the electromagnetic field can be µ 4-velocity of the particle is xú = (ctú , xú ); the superior represented as dot indicates the derivative with respect to τ, so that we µ µ µ µ () []() () () 2 2 Ð1/2 A x = e2xe1Ðe1xe2Bkx/2, (1) have xú = γ(c, v), where γ = [1 Ð v /c ] . We solve the

1063-7842/00/4507-0826 $20.00 © 2000 MAIK “Nauka/Interperiodica” RELATIVISTIC PARTICLE IN A TRAVELING MAGNETIC FIELD 827 σ τ τ equations of motion of a charged particle in terms of the where ( ) = ku + kx0. Equations (10) and (11) are Hamiltonian formalism. The particle motion in the generated by the part of Hamiltonian (2) that is inde- electromagnetic field defined by the 4-potential (1) is pendent of the momentum components p0 and pz: described by the Hamiltonian [5] H = (1/2m)[ p 2 + p 2] + (e/2mc)[y p Ð x p ]B(σ) H(x, p) = Ð (1/2m)[ p Ð (e/c)A(x)]2 + mc2/2. (2) 12 x y x y 2 2 2 2 2 (12) µ ν µν + (e /8mc )(x + y )B (σ). Taking into account the relationship [x , p ] = Ðg for the fundamental Poisson bracket, we arrive at the The first integral of Eqs. (10) and (11) is the projec- equations tion of the generalized particle momentum onto the z- µ µ µ µ µ axis: xú = [x , H], mxú = p Ð (e/c)A , (3) ( ) ( )( 2 2) (σ) α µ Mz = m xyú Ð yxú + e/2c x + y B , púµ = [ pµ, H], púµ = (e/c)xúα∂A /∂x , (4)

µ in which case law (9) describing how the particle with the boundary conditions x (0) = (0, x0, y0, z0) and kinetic energy increases becomes µ xú (0) = uµ, where uµ = γ (c, v ) and γ = [1 Ð (v /c)2]−1/2. 0 0 0 0 τ ( 2ω 2)( 2 2)( 2) ω Equations (3) and (4) have three integrals of motion. dE/d = e /8mc x + y B ' Ð e Mz/mc. One of the integrals can be obtained by taking the scalar τ τ product (convolution) of Eq. (4) with the 4-vector kµ: From Eq. (8), we ﬁnd cdpz/d = dE/d , which allows kp = mku. Then, taking the convolution of Eq. (3) with us to draw the important conclusion that the longitudinal momentum of the particle increases simultaneously kµ yields the second integral of motion: kxú = ku, which with its energy. can also be written in terms of coordinates as Equations (10) and (11) can also be solved by carry- ctúÐ zú = nu. (5) ing out a sequence of canonical transformations (CTs).

We thus obtain the wave phase along the particle tra- First, we make the CT x, y, px, py x', y', px' , py' such τ jectory: kx = ku + kx0. The third integral of motion for that Eqs. (3) and (4) has the form xú2 = c2 or x = (1/2m)1/2(x' + y'), y = Ði(1/2m)1/2(x' Ð y'), ( ú)2 2 2 2 2 (13) ct Ð xú Ð yú Ð zú = c . (6) ( )1/2( ) ( )1/2( ) px = m/2 px' + py' , py = i m/2 px' Ð py' . Resolving Eqs. (5) and (6) in tú and zú, we obtain In the new variables, Hamiltonian (12) becomes ctú = nu/2 + (1/2nu)(c2 + xú2 + yú2), ( )( ) (7) H12' = px' py' + ie/2mc y' py' Ð x' px' B zú = Ð nu/2 + (1/2nu)(c2 + xú2 + yú2). (14) + (eB/2mc)2 x'y'. The integral of motion (5) is actually a consequence of Eqs. (3) and (4) taken with µ = 0.3: Now, we eliminate the second term in (14) by performing the CT mctú = p0, mzú = pz, ( ) ( ) ( β ) ' ( β ) ' pú0 = e/c k0 yxú Ð xyú B'/2, (8) x' = exp Ði /2 x1 , y' = exp i /2 x2 , ( ) ( ) púz = e/c k0 yxú Ð xyú B'/2. ' ( β ) ' ' ( β ) ' px = exp i /2 p1 , py = exp Ði /2 p2 , (15) Since the increment E = mc2 tú in the particle kinetic τ energy is governed by the vortex electric ﬁeld, we can β(τ) = (e/mc)∫dΘB σ(Θ) , write 0 τ τ ( ω )( ) dE/d = cxúE, dE/d = Ð e /c xyú Ð yxú B'/2. (9) which puts the Hamiltonian in the form Equations (3) and (4) taken with the integral of ( )2 motion (5) and µ = 1.2 can be reduced to the Hamilto- H12'' = p1' p2' + eB/2mc x1' x2' . (16) nian system The solution to the canonical equations ( ) (σ) mxú = px + c/c yB /2, (10) τ τ ( ) (σ) dx1' /d = p2' , dx2' /d = p1' , myú = py Ð c/c xB /2, ( ) (σ) ( ) (σ) τ ( )2 τ ( )2 púx = e/c yúB /2, púy = Ð e/c xúB /2, (11) d p1' /d = Ð eB/2mc x2' , d p2' /d = Ð eB/2mc x1'

TECHNICAL PHYSICS Vol. 45 No. 7 2000 828 TOROPOVA can be represented as With this Wronskian, the Hamiltonian transformed to the new coordinates using the above procedure, h = ( )1/2( ) x1' = 1/2 wa1 + w*a2* , ∂ ∂τ H12'' + F1/ , is identically zero. x' = (1/2)1/2(wa + w*a*), 2 2 1 (17) ( )1/2( ) p1' = 1/2 wú a2 + wú *a1* , PARTICLE MOTION IN THE TRAVELING JUMPLIKE FRONT OF THE MAGNETIC FIELD ( )1/2( ) p2' = 1/2 wú a1 + wú *a2* , Of particular importance is the magnetic ﬁeld (1), τ where a1 and a2 are constants. The function w is a for which the function complex solution to the oscillator equation 2(σ) ( )( 2 2) ( )( 2 2) B = 1/2 B1 + B2 + 1/2 B2 Ð B1 d2w/dτ2 + [eB(σ)/2mc]2w = 0 (18) × tanh[ p(σ Ð σ )] + [B /2chp(σ Ð σ )]2 ω 1/2 ω τ 0 0 0 with the initial condition w = (2/ 1) exp(Ði 1 /2) (where ω = eB /mc) for τ Ð∞. The Wronskian of 2 2 1 1 takes on the limiting values B and B and has a max- the two functions w and w* is independent of τ: wwú * Ð 1 2 2 2 2 wú w* = 2i. We substitute (15) and (17) into (13) to imum in the range B0 > B2 Ð B1 . In quantum mechan- arrive at the solution to the equations of motion (10) ics, this function is known as the Eckart potential [6]. ӷ ω and (11): The most interesting case here is pku 1, which corresponds to a sharp jump in the function B(σ). Setting ( ) [( ) ( β )] σ ≈ x = 1/ m Re wa1 + w*a2* exp Ði /2 , B0 = 0, we arrive at a function such that B( ) B1 for σ σ σ ≈ σ σ (19) < 0 and B( ) B2 for > 0. Since the relationship ( ) [( ) ( β )] σ Ð σ = ku(τ Ð τ ) with τ = (σ Ð kx )/ku > 0 holds y = 1/ m Im wa1 + w*a2* exp Ði /2 . 0 c c 0 0 along the trajectories of a particle, we can integrate the In order to determine the functions t(τ) and z(τ), we second-order differential equations following from τ τ insert (19) into (7). Below, we will be interested only in (10) and (11) over a small vicinity of the point = c in the solutions to Eq. (18) in the limit τ ∞, in which order to obtain the boundary conditions in the form of the function B(kx) approaches a constant value B . In ∆ ω 2 the incremental velocity components xú ( 2 Ð this case, we can use the asymptotic expressions ω τ ∆ ω ω τ 1)y(c)/2 and yú = Ð( 2 Ð 1)x( c)/2. In accordance ( ω )1/2[ ( ω τ ) ( ω )τ ] with (20), the solution to Eq. (18) can be written as w = 2/ 1 C1 exp Ði 2 /2 + C2 exp i 2 /2 , ω (20) ( ω )1/2 ( ω τ ) τ ≤ τ 2 = eB2/mc. w(1) = 2/ 1 exp Ði 1 /2 , c; (21a) Let us make two remarks. ( ω )1/2[ ( ω τ ) ( ω τ )] w(2) = 2/ 1 Dexp Ði 2 /2 + Sexp i 2 /2 , (i) The coefﬁcients C1 and C2 can be found from the τ ≥ τ , solutions known in quantum mechanics, because c asymptotics (20) are consistent with the problem of , ( , ) ( ω τ ± ω τ ) D S = D0 S0 exp Ð i 1 c/2 i 2 c/2 , particle scattering by a one-dimensional potential. (21b) , ( )( ± ω ω ) According to [6], the quantities 1/C1 and C2/C1 play the D0 S0 = 1/2 1 1/ 2 . role of the amplitudes of the forward and backward waves into which the wave that is incident on the poten- Let us set x0 = 0, y0 > 0, z0 = 0, and v0 = (v1, 0, 0). Then tial from the right is scattered. Let 1/p be the character- µ γ γ β ω τ σ xú (0) = (c 0, u1, 0, 0), u1 = 0v1, and = 1 follow. istic scale on which the function B( ) varies. Then, if τ this function changes adiabatically, |dB(σ)/dτ| Ӷ Substituting (21a) into (19) and setting = 0 gives σ ω Ð1/2 ω Ð1/2 pkuB( ), we can see that the ratio C2/C1 ~ (m 1) a1 = iR and (m 1) a2* = i(y0 Ð R) with R = −πω ω ≤ τ ≤ τ exp( 1/pku) is exponentially small [6]. u1/ 1. In the interval 0 c, the particle trajectory is τ ω τ τ (ii) Since the Wronskian of the linearly independent described by the equations x( ) = Rsin 1 , y( ) = ω τ τ solutions w and w* is equal to 2i, solution (17) is a Rcos 1 + (y0 Ð R), and z( ) = 0; i.e., the trajectory is a CT x' , p' x = a , p = ia* (n = 1, 2) whose gen- circle of radius R centered at (0, y0 Ð R, 0). Inserting n n n n n n (21b) into (19), we obtain the solution to Eqs. (10) and erating function depends on both the old and new vari- τ τ τ τ (11) in the interval > c: x( ) = Rex+ and y( ) = Imx+, ables [5]: τ where x+( ) = x + iy, which can also be written as p' = ∂F /∂x' , p = ∂F /∂x , n 1 n n 1 n (τ) [ ( ω τ ) ( )] x+ = i D0 Rexp Ði 1 c + S0 y0 Ð R ( )[ ' ' ( ' ' ) ] × [ ω τ τ ] [ ( ω τ ) ] F1 = 1/w* wú *x1 x2 Ð i 2 x1 x2 + x2 x1 + wx1 x2 . exp Ði 2( Ð c) +i S0 Rexp Ði 1 c + D0(y0 Ð R) .

TECHNICAL PHYSICS Vol. 45 No. 7 2000 RELATIVISTIC PARTICLE IN A TRAVELING MAGNETIC FIELD 829

ω τ η µ 2 Setting [D0Rexp(Ði 1 c) + S0(y0 Ð R)] = R2exp(Ði ) bounded by the curve c1(q) = 1 Ð q Ð q /8 + … in the µ yields the components of the 4-velocity: upper half-plane (q > 0) and by the curve c1(q) = µ − (τ) ω [ω (τ τ ) η] c1( q) in the lower half-plane (q < 0). xú = 2 R2 cos 2 Ð c + , (22) (τ) ω [ω (τ τ ) η] Of particular interest is the case of parametric reso- yú = Ð 2 R2 sin 2 Ð c + . nance, in which it may be expected that the energy and longitudinal momentum will increase significantly at 2 2 ú The kinetic energy T = E Ð mc (where E = mc t ), µ = 1, 4, 9, …. An analysis of the stability domains and the longitudinal momentum pz = mzú can be found allows us to conclude that charged particles can be from (7): accelerated in the regime of parametric resonance and can be separated out by specific charge. γ [(ω )2 (ω )2] ( γ ) ctú = c 0 + 2 R2 Ð 1 R / 2c 0 , (23) [(ω )2 (ω )2] ( γ ) zú = 2 R2 Ð 1 R / 2c 0 . BETATRON ACCELERATION REGIME This indicates that the particle moves along a spiral It is well known that particle losses can be prevented by a focusing magnetic field which decreases away trajectory of radius R2, with the axis lying at a distance | ω τ | from the axis of the system. We consider particle r2 = S0Rexp(Ði 1 c) + D0(y0 Ð R) from the z-axis. Note τ ω ω τ τ motion in a traveling nonuniform magnetic field that, for v1 = 0, we have xú ( ) = 2R2cos 2( Ð c), defined by the following components of the 4-potential τ ω ω τ τ yú ( ) = Ð 2R2sin 2( Ð c), R2 = S0y0, and r2 = D0y0. In in cylindrical coordinates: τ the other particular case, y0 = R, we obtain xú ( ) = ρ ω ω τ τ ω τ 2R2cos[ 2( Ð c) + 1 c], ( ρ) ρρ (ρ, ) A0 = Aρ = A z = 0 , Aϕ = 1/ ∫d B kx , (τ) ω [ω (τ τ ) ω τ ] yú = Ð 2 R2 sin 2 Ð c + 1 c , 0 ω ω R2 = D0 R, r2 = S0 R. where kx = t Ð z/c. From (23), we can see that, after the particle passes The Lagrangian describing the motion of a relativis- through the magnetic field front, its energy becomes tic particle can be written in terms of the intrinsic time higher. (in SI units) as [9] 2 2 2 L = (m/2)[ρú + ρ2φú + zú2 Ð c2tú ] PARTICLE MOTION IN A MONOCHROMATIC ρ FIELD + eφú ∫dρρB(ρ, ωt Ð ωz/c). We consider the function B(kx) = B0 + bcoskx. Then, from (18) we obtain the Hill equation 0 The EulerÐLagrange equations have the first inte- 2 τ2 ( )[Ω ω σ(τ)]2 σ τ σ τ τ d w/d + 1/4 0 + 0 cos w = 0, grals (5) and (6): kx = ( ) and ( ) = ku + kx0. We Ω = eB /mc, ω = eb/mc. solve the equations of motion by analyzing the acceler- 0 0 0 ation cycle on a cylindrical surface of constant radius. Setting Setting φú = Ω and ρ = R, we obtain the equations τ µ (Ω 2 ω 2 ) ( )2 2s = ku + kx0, = 0 + 0 /2 / 4ku , dE/dτ = Ð(eΩω/2π)dΦ/dσ, ω Ω ( )2 (ω )2 R q = 0 0/ 4ku , q2 = 0/8ku , (25) Φ(σ) = 2π∫dρρB(ρ, σ), we arrive at the Hill equation in the standard form: 0 2 2 (µ ) d w/ds + + 2qcos2s + 2q2 cos4s w = 0. (24) 0 = mΩ + eB(R, σ), (26)

The exponential index in (15) is equal to 2 mR dΩ/dτ + (e/2π)dΦ/dτ = 0, (27) β(τ) = Ω τ + (ω /ku)[sin(kuτ + kx ) Ð sin(kx )]. 0 0 0 0 2 τ2 ( Ωω π ) Φ σ Ӷ Ӷ md z/d = Ð e /2 c d /d , (28) In the case q2 q (or b 2B0), we arrive at the Mathieu equation [7, 8]. The theory of Mathieu func- where Φ is the total magnetic flux through a membrane tions implies that, in the plane of the parameters (µ, q), bounded by the particle orbit. there are regions that correspond to either bounded or From (26) and (27), we obtain the equation dΦ/dτ = unbounded solutions [7]. In the region of small µ and q 2πR2dB/dτ. We supplement this equation with the ini- in the (µ, q) plane, the solution to Eq. (24) is finite in tial conditions B(R, σ) = 0 and Φ(σ) = 0 at τ = 0 and the first stability domain, which is located to the right integrate it over the acceleration cycle. As a result, we µ 2 4 Φ σ of the curve c0(q) = Ðq /2 + 7q /128 + … and is arrive at an analogue of the “betatron rule”: ( m) =

TECHNICAL PHYSICS Vol. 45 No. 7 2000 830 TOROPOVA

π 2 σ σ τ Φ 2 R B(R, m), where m = ku m + kx0. Substituting REFERENCES and Ω into (25) and (28), we find the kinetic energy T = 2 1. Yu. G. Pavlenko, N. D. Naumov, and A. I. Toropova, Zh. E Ð mc and the z-component of the 4-velocity: Tekh. Fiz. 67 (7), 98 (1997) [Tech. Phys. 42, 809 (τ ) 2γ 2 2 2( , σ ) ( ) (1997)]. E m = mc 0 + e R B R m / 2mnu , 2. W. Paul, Usp. Fiz. Nauk 160 (12), 109 (1990). ( ) 2 2 2( , σ ) ( ) mzú = mzú 0 + e R B R m / 2mcnu . 3. Pradip K. Ghosh, Ion Traps, The International Series of Monographs on Physics (Clarendon, Oxford, 1995). We denote the maximum magnetic field strength by σ 4. I. A. Malkin and V. I. Man’ko, Dynamic Symmetries and B(R, m) = Bm and assume that zú(0) = 0. Then, we have γ Coherent States in Quantum Systems (Nauka, Moscow, 0 = 1, so that the particle kinetic energy at the end of 1979). τ 2 2 2 the acceleration cycle is T( m) = e R Bm /(2m). Since 5. Yu. G. Pavlenko, Hamiltonian Methods in Electrody- ecRBm = 300[R(m)Bm(T)] MeV, the kinetic energy namics and Quantum Mechanics (Mosk. Gos. Univ., τ τ Moscow, 1988). T( m) can be represented as T( m) = 2 2 2 6. A. I. Baz’, Ya. B. Zel’dovich, and A. M. Perelomov, Scat- 45[R (m)Bm (T)/mc (MeV)] GeV. For proton accelera- tering, Reactions and Decays in Nonrelativistic Quan- τ 2 2 tum Mechanics (Nauka, Moscow, 1971; Israel Program tion, we have Tp( m) = 45[R (m)Bm (T)] MeV; and, for τ for Scientific Translations, Jerusalem, 1966). electron acceleration, we have Te( m) = 7. P. M. Morse and H. Feshbach, Methods of Theoretical 2 2 90[R (m)Bm (T)] GeV. Note that, in the regime of con- Physics (McGraw-Hill, New York, 1953; Inostrannaya ventional betatron acceleration, the kinetic energy of a Literatura, Moscow, 1958), Vol. 1. τ 2 2 2 1/2 particle is equal to T( m) = [(mc ) + (ecRBm) ] Ð 8. N. W. McLachlan, Theory and Application of Mathieu 2 ≈ mc ecRBm. Functions (Clarendon, Oxford, 1947; Inostrannaya Lit- eratura, Moscow, 1953). 9. Yu. G. Pavlenko, Lectures on Theoretical Mechanics ACKNOWLEDGMENTS (Mosk. Gos. Univ., Moscow, 1991). I am grateful to Yu.G. Pavlenko for useful discussions. This work was supported by the Russian Founda- tion for Basic Research, project no. 97-01-00957. Translated by O.E. Khadin

TECHNICAL PHYSICS Vol. 45 No. 7 2000

ATOMS, SPECTRA, AND RADIATION

Electromagnetic Transitions between Rydberg States of a Hydrogen Atom: Violation of the Dipole Selection Rules in a Strong Field O. B. Prepelitsa Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau, 277028 Moldova Received November 30, 1998; in ﬁnal form, August 3, 1999

Abstract—One-quantum bound–bound transitions between high-excited states of a hydrogen atom are considered. Electron wave functions involving an electromagnetic field even in a zero-order approximation are constructed semiclassically. With these functions, it is shown that transitions accompanied by violation of the dipole selection rules are possible in strong fields. The probability of such transitions is a nonlinear function of electromagnetic field intensity. © 2000 MAIK “Nauka/Interperiodica”.

The probability of transitions between states of the external electromagnetic field and fall outside the hydrogen-like atoms under the action of an electromag- framework of perturbation theory. As we will show, the netic field has been studied in detail (e.g., [1–4]). Basi- dipole selection rules for the orbital quantum number cally, the available tables of oscillator forces [1] and the are violated in strong fields (the selection rules for the values of reduced matrix elements for the dipole magnetic quantum number holds true), and the proba- moment [2Ð4] make it possible to calculate the probabilities of one-photon transitions become nonlinear bilities of one-photon transitions between arbitrary functions of the electromagnetic field intensity. states of a hydrogen-like atom in the dipole approxima- Since high-excited states are semiclassical, an elec- tion. However, such calculations are valid for moder- tron in such a state is located basically in the vicinity of ately strong electromagnetic fields weakly perturbing its classical trajectory. Therefore, it is worthwhile to the corresponding atomic states. Therefore, stationary consider the motion of a classical electron in a wave wave functions for the Coulomb problem are used as field in more detail. It is well known that, if in the wave functions in the zero-order approximation [1, 5]. absence of an electromagnetic field, an electron moves As follows from classical treatment, such an approach in a path r(t), then in a field with strength E(t) and fre- is valid for electromagnetic field strengths bounded by quency ω and under the conditions the inequality min r()t ӷ max a()t , (2) E E Ӷ ------at , (1) 1 0 ()2 ω ӷ --- , (3) 2nif, T 2 5 ប4 where Eat = M e / is the atomic field strength; e and where M are the electron charge and mass, respectively; and () () e E t ni, f are the principal quantum numbers of the initial a t = ------2 , and final states. m ω In this paper, we consider the probability of one- and T is the period of the unperturbed electron motion, photon transitions between high-excited states of a it will move in the quasistationary path (see the figure) hydrogen atom with large orbital angular momenta. In r ' ()t = r ()t Ð a ()t . (4) contrast to the standard approach [1Ð4], quasistationary wave functions describing an electron in the field of Thus, the influence of a high-frequency wave is both the Coulomb potential and a high-frequency elec- basically reduced to the appearance of oscillations tromagnetic wave are used as wave functions in zero- about unperturbed electron trajectory r(t) (for more order approximation (we will further refine the condi- details, see [6, Section 30]). This trajectory formally tion on the field frequency). Taking into account the coincides with the electron trajectory in the absence of electromagnetic field in the zero-order wave function the external field in a noninertial reference frame with makes it possible to relax constraints on the strength of coordinates related to the initial ones by (4). Therefore,

832 PREPELITSA

z perturbation. The validity of dividing the Hamiltonian into the principal term H0 and perturbation Vint also follows from comparison of the magnitudes of (6) and (7) (they are treated classically here) in the trajectory r(t). Indeed, one can consider Vint as a small perturbation if it is much less than H0. Using the virial theorem, it is sufficient to ascertain that (7) is much less than the sec- y ond term in (6). Since the latter is proportional to 1/|r| at large distances, whereas (7) is proportional to 1/r2, one can easily see that (7) is small when (2) is satisfied. Since the Feynman paths located in the vicinity of a x classical electron trajectory make a major contribution to the evolution of a high-excited electron (due to the Figure. semiclassical nature of the state), the division of the Hamiltonian into the principal term H0 and perturbation Vint holds true from the quantum-mechanical point of we may semiclassically conclude that the electron wave view as well. Therefore, the solutions of the Shrödinger function in the wave field is approximately represented equation with the Hamiltonian H by the Coulomb wave function in the noninertial refer- 0 ence frame: ∂Ψ(r, t) iប------= H Ψ(r, t). (8)  t 2 2  ∂t 0 c i e A (τ) Ψ(r, t) = Ψ (r Ð a(t), t)expÐ-- dτ------ , (5) ប ∫ 2  2Mc  Ð∞ should be used as wave functions in the zero-order approximation. where A(t) is the vector potential of the external electromagnetic field in the dipole approximation, c is the Immediate substitution implies that function (5) sat- speed of light, and Ψc(r, t) is the Coulomb wave func- isfies Eq. (8), while Ψc(r, t) is the subject to the tion. Shrödinger equation in the Coulomb problem: Note that (5) is a state with indefinite energy and orbital angular momentum. However, it is convenient to ∂Ψc( , ) ˆ 2 2 ប r t  P e  Ψc( , ) characterize it by the number triple (nlm), which refers i ------= ------Ð ----- r t . to the corresponding Coulomb wave function. ∂t 2M r Relation (5) can be obtained more accurately. The Hamiltonian of an electron in both the Coulomb and Using well-known formulas of the Coulomb prob- electromagnetic fields has the form lem, we represent criteria of the applicability of (2) and (3) to the model in the form  e  2 ˆ -- ( ) P Ð A t  2 ω 2 c e E0  at 1 Ӷ H = ------Ð -----. ------------ ------1, (9) 2M r E ω 2 at 2  l  n 1 Ð 1 Ð -- The Hamiltonian of the system can be represented  n  as follows: ω H = H + V , at 1 0 int ------Ӷ 1, (10) ω 3  e  2 n Pˆ Ð --A(t) (6)  c  2 e where ω = Me4/ប3 is the atomic frequency and n and l H0 = ------Ð ------(-----)-, at 2M r Ð a t are the principal and orbital quantum numbers. e2 e2 V = ------Ð ----. (7) Since the interaction of an atom with the field is int r Ð a(t) r considered in the dipole approximation (vector poten- One can readily see that (4) is the solution of the tial A(t) is independent of coordinates), conditions (9) classical equations of motion resulting from Hamilto- and (10) should be complemented with a condition set- nian (6). Thus, precisely (6) basically determines the ting an upper bound on the field frequency: electron trajectory in the wave field. Therefore, expres- ω sion (6) is the Hamiltonian in the zero-order approxi- ---max r(t) Ӷ 1, mation, whereas (7) should be considered as a small c

TECHNICAL PHYSICS Vol. 45 No. 7 2000 ELECTROMAGNETIC TRANSITIONS BETWEEN RYDBERG STATES 833 or, alternatively, the integration variable and some transformations we obtain 2 2 2  l  ω e ∞ l n 1 + 1 Ð ------Ӷ 1. (11)     ω ប ( ) lm l (α ) n at c Aif t = ∑ ∑ C fi M fi 0 l = 1 m = Ðl The more the orbital quantum number l, the better (14)  i  inequalities (9) and (11) are satisfied, since this is the exp -- (ε Ð ε Ð mបω)t Ð 1 very case when the electron is located far from the ប f i  × ------, nucleus with overwhelming probability. Furthermore, ε Ð ε Ð mបω due to the semiclassical nature of the state that implies f i l ӷ 1 along with n ӷ 1, we will assume n, l ӷ 1 here- ( ) lm 2l + 1 ( )( ) l Ð m ! after. For a state with l/n ~ 1, conditions (9) and (11) C fi = ------π----- 2li + 1 2l f + 1 (------)--- can be represented in the form 4 l + m !     (15) ω 2 m l f l li l f l li E0  at 1 × P (0)    , ------Ӷ 1, (12) l  ω  2  Ðm m m   0 0 0  Eat n f i ∞ 2 l 2 e α 2 ω e l (α ) ( ) 0 ( ) n ------Ӷ 1. (13) M fj 0 = ∫drRn l r ------Rn l r , ω បc f f rl Ð 1 i i at 0 (16) Conditions (9)Ð(13) are consistent in a wide range α eE0 0 = ------, of parameters. Note that inequalities (9) and (12) Mω2 bounding the electromagnetic field strength essentially differ from similar inequality (1), which determines the where Pm (x) is the associated Legendre polynomial, applicability region of the standard perturbation theory. l Expressions (9) and (12) reflect the fact that the influ-  l l l  ence of the field on electrons increases with decreasing  f i  is the Wigner 3j-symbol, index i (f)   field frequency, so that, in the limiting case of ω = 0 m f m mi (stationary field), perturbation theory is no longer valid. means that the corresponding magnitude refers to the We consider the transition from the initial atomic initial (final) state, and Rnl(r) is the radial part of the Ψ Coulomb wave function. state i(r, t) characterized by quantum numbers (nilimi) Ψ to the final state f(r, t) with quantum numbers (nf lfmf). It is worth noting here that integral (16) can be cal- According to the above discussion, wave functions of culated analytically, but we do not present the result the considered states in the wave field have the form here because it is rather cumbersome (detailed calcula- (5). Operator Vint defined by (7) is the operator that tion of integrals of the form (16) is presented in the Ψ Ψ mathematical appendix in [5]). mixes states i(r, t) and f(r, t). Thus, the transition amplitude takes the form The properties of the Wigner 3j-symbols (e.g., [5]) and zeros of Legendre polynomials t ( ) i Ψ ( , ) Ψ ( , ) m  π Aif t = Ð-- ∫dt'∫dr *f r t' V int i r t' . P (0) ∼ cos (l + m )-- ប l  2 0 imply that transition amplitude (14) is nonzero if the We consider a circularly polarized electromagnetic following conditions are satisfied: field. We also assume that the atom is oriented in such a manner that the quantization axis is perpendicular to m f Ð mi = m, (17) the polarization plane of the external electromagnetic ≤ ≤ field. Choosing the coordinate frame where the quanti- li Ð l l f li + l; (18) zation axis is directed along the 0z-axis, we represent , , , … the electromagnetic field strength in the form l f + li + l = 2 p, p = 1 2 3 ; (19) ( ) ( ω ω ) , , , … E t = E0 ex cos t + ey sin t , l + m = 2 p', p' = 1 2 3 . (20) Expression (17) implies that the number of photons m where ex and ey are the unit vectors directed along 0x and 0y, respectively. coincides with the difference between the magnetic quantum numbers of the final and initial states. This Taking into account the explicit form of the Cou- corresponds to the well-known selection rule for mag- lomb wave functions Ψc(r, t), after a simple change of netic quantum numbers in dipole transitions under the

TECHNICAL PHYSICS Vol. 45 No. 7 2000 834 PREPELITSA action of circularly polarized radiation. Furthermore, We present some numerical estimations. For exam- we will restrict our consideration to one-photon transi- ple, for the ratio of matrix elements η = tions and, hence, take m = 1 in (14) and (15). In this 1 α 3 α η M fi (0)/M fi ( 0) (see (16)), we obtain ~ 10, where case, it follows from (20) that l should be an odd num- 1 α ber. Taking (19) into account, one can conclude that M fi ( 0) defines the amplitude of the ordinary dipole the difference between the orbital quantum numbers transition from the state ni = 10, li = 5, and mi = 0 to the of the final and initial states should be an odd number 3 α state nf = 30, lf = 6, and mf = 1; and M fi ( 0) determines as well: the amplitude of the forbidden dipole transition from , , , … the same initial state to the final state nf = 30, lf = 8, and l f Ð li = 2 p + 1, p = 0 1 2 . (21) × 5 ω 14 Ð1 mf = 1 for E0 = 9 10 V/cm, = 10 s . Thus, the probability of the forbidden dipole transition turns out Otherwise, the probability of transition turns out to 2 be zero. The term with the smallest l satisfying the pre- to be only 10 times less than the probability of the ordinary dipole transition in the perturbation theory. sented conditions makes the major contribution to the sum in (14). According to the summation rule for angu- The above consideration of boundÐbound transi- lar momentum (18), the smallest l is tions in a high-frequency electromagnetic field is an advance outside the framework of perturbation theory l = l f Ð li . (22) (compare criteria (12) and (1)). It became possible due to a more accurate account of the interaction of the On this basis, the transition amplitude takes the final atom with the wave field compared to a standard form approach, even at the beginning of solving the problem, i.e., in the zero-order wave functions (5). Note that similar wave functions for the states of continuous spec-  i (ε ε បω)  trum were considered in multiphoton ionization of expប-- f Ð i Ð t Ð 1 ( ) l1 l (α ) atoms [8Ð11]. This made it possible to fall outside the Aif t = C fi M fi 0 ------ε------ε------ប---ω------. (23) f Ð i Ð framework of perturbation theory and obtain the limiting case of atom ionization by a stationary field (it is Here, l is defined by (22) and the denominator in well known that perturbation theory is inapplicable to (23) is assumed to be nonzero. the case of a stationary field, since the radius of conver- gence for the corresponding series is equal to zero). ± In the case lf = li 1 (l = 1), the obtained expressions Note also that the obtained results are applicable not coincide with those describing one-photon transitions only to a hydrogen atom but to multielectron atoms as ӷ in ordinary perturbation theory when the transition well. Since for n 1, an electron is located on average operator is expressed in the “acceleration representa- far from the atomic residual, the influence of the latter can be taken into account by replacing n by n Ð δ , tion” (e.g., [7]). Such transitions are dipole-allowed in l where δ is the quantum defect. perturbation theory, and they are the most likely. How- l ever, as expression (21) shows, transitions between the The resonance case (ε Ð ε = បω) has remained | | f i states with l = lf Ð li = 2p + 1, where p = 1, 2, 3, …, are beyond consideration since it requires other mathemat- also possible. It is clear that such transitions are dipole- ical methods. The problem is complicated by the fact forbidden in the standard approach. that, in the resonance case, the standard two-level approximation is no longer valid for two reasons: the Since the transition probability is proportional to the electron quickly leaves the selected two-level system, intensity of the exciting field to the l power, as is seen and the wave packet spreads due to strong mixing of the from (16) and (23), the transition probability becomes states coupled by transitions accompanied by violation a nonlinear function of the field intensity for l ≥ 3. This of the dipole selection rules. From the classical point of fact essentially differs from the result of the standard view, this means that electron motion in the wave field perturbation theory where the probability of a one-pho- becomes stochastic (for more details, see [12]). ton transition is proportional to the intensity to the first power. Therefore, the observation of nonlinear dependence of the probability of the process on the intensity REFERENCES of the applied electromagnetic field can be an experi- 1. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of mental verification of the results obtained in this paper. One- and Two-Electron Atoms (Academic, New York, | | ӷ Note that nf Ð ni 1, as follows from both condition 1957; Fizmatgiz, Moscow, 1960). បω (10) and a natural assumption that quantum is of the 2. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, order of the energy level spacing between the initial and Quantum Electrodynamics (Nauka, Moscow, 1980; Per- final atomic states. gamon, Oxford, 1982).

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3. L. A. Bureeva, Astron. Zh. 45, 1215 (1968) [Sov. Astron. 9. A. I. Nikishov and V. I. Ritus, Zh. Éksp. Teor. Fiz. 50, 12, 962 (1968)]. 255 (1966) [Sov. Phys. JETP 23, 168 (1966)]. 4. S. P. Goreslavskiœ, N. B. Delone, and V. P. Kraœnov, Zh. 10. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. Éksp. Teor. Fiz. 82, 1789 (1982) [Sov. Phys. JETP 55, Éksp. Teor. Fiz. 50, 1391 (1966) [Sov. Phys. JETP 23, 1032 (1982)]. 922 (1966)]. 5. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: 11. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. Non-Relativistic Theory (Nauka, Moscow, 1989, 4th ed.; Éksp. Teor. Fiz. 51, 309 (1966) [Sov. Phys. JETP 24, 207 Pergamon, Oxford, 1977, 3rd ed.). (1966)]. 6. L. D. Landau and E. M. Lifshitz, Mechanics (Nauka, 12. N. B. Delone, V. P. Kraœnov, and D. L. Shepelyanskiœ, Moscow, 1973; Pergamon, Oxford, 1976). Usp. Fiz. Nauk 140, 355 (1983) [Sov. Phys. Usp. 26, 551 7. M. Ya. Amus’ya, The Photoelectric Effect in Atoms (1983)]. (Nauka, Moscow, 1987). 8. L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1964)]. Translated by M.S. Fofanov

TECHNICAL PHYSICS Vol. 45 No. 7 2000

ATOMS, SPECTRA, AND RADIATION

Positron Scattering by Phonons in Metals D. A. Gryaznykh All-Russia Research Institute of Technical Physics, Russian Federal Nuclear Center, Snezhinsk, Chelyabinsk oblast, 456770 Russia E-mail: [email protected] Received March 23, 1999; in ﬁnal form, October 26, 1999

Abstract—Processes involved in positronÐmatter interaction are studied. The stopping power and mean free path are calculated for positrons with an energy of about 1 eV, which are scattered mostly by phonons. The mean free path and range of positrons in tungsten, as well as the stopping power of tungsten, are calculated for positron energies between 0.025 and 10 eV. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION The kinetic equation for positron distribution function f(r, p, t), where p is the positron wave vector, is Low-energy positron sources are finding wide application in atomic physics, solid-state physics, physics ∂ ++vp()∇ F∇ f()rp,,t and chemistry of surfaces and thin films, materials ∂t r p technology, study of lattice defects, etc. One of the (1) methods for producing high-flux low-energy positron ∂ = Ð()λ +κ f+f()rp,,t. beams is the use of a charged particle accelerator. This ∂t b i method has been implemented, for example, in the s λ SPring-8 complex [1]. Superconductive wiggler mag- Here, b is the rate of annihilation of delocalized pro- nets installed in the electronÐpositron storage ring tons; κ is the rate of positron trapping by lattice defects; generate high-power synchrotron radiation with a pho- fi is the source function; and []s is the rate of scattering ton energy significantly higher than the pair-production threshold. The positrons thus produced are thermalized ∂ =dq[]R()qp, f()rq,,tÐR()pq, f()rp,,t ,(2) by passing through a decelerator. The process of ∂t∫ positron deceleration should be taken into account to s properly assess the thermalization efficiency and yield where R is the scattering amplitude. of slow positrons. The intensity of low-energy positron scattering by The ionization processes have been well studied conduction electrons can be calculated in terms of the both theoretically and experimentally [2], whereas the random-pulse approximation for electronÐpositron intricate lattice processes (collective excitations, inter- interaction [4]: action with holes and phonons) have not yet been sub- 2 2 ប2()2 jected to adequate experimental study. The goal of this , 1e a 0 δkpq+Ð Rel()pq = π------ប------π - ∫ d k------work was to calculate the parameters of positron scat- 4 k F 2me tering by phonons in tungsten. Calculations were made ប 2 q 2 ប 2 k 2 ប 2 p 2 using the existing theoretical models of positronÐ + ------Ð ------(3) phonon interaction and experimental data on the 2 m * 2 m e 2 m * positron energies in metals. ប 2 ()2 ប22 × 0 kpq+Ð , 0k, 1 Ðf F ------T fF------T , 2me 2me KINETIC EQUATION FOR POSITRONS where a0 is the Bohr radius, kF is the Fermi wave vector, Positron interaction with matter involves different 0 m* is the positron effective mass, and fF (E, T) = processes depending on the positron energy. High- Ð1 energy positrons induce ionization [2]; positrons with {exp[(E Ð EF)/kBT] + 1} . The random-pulse approxi- an energy higher than the plasmon-production thresh- mation can be applied if the positron energy is lower old may cause collective excitations [3]; positrons with than the ionization threshold (about 10 eV). lower energies may cause, production of electronÐhole A comprehensive review of the theory of positronÐ pairs; and those with an energy of about 1 eV experi- matter interaction and its applications to materials tech- ence mostly phonon scattering. nology is given in [5].

POSITRON SCATTERING BY PHONONS IN METALS 837

POSITRONÐPHONON INTERACTION Thermalized positrons are scattered mainly by longitudinal acoustic phonons. In the Debye approxima- p q tion with E(k) = បsk, the expression for scattering amplitude takes the form [2] q p γ 2  , [ 0 (ប , ) ]δ R ph(p q) = ------q Ð p  f B s q Ð p T + 1 -- π2  Fig. 1. Feynman diagrams for phonon emission and absorp- 4 tion. ប2 2 ប2 2 ×  p q ប  0 (ប , )δ (4) ------Ð ------Ð s q Ð p  + f B s q Ð p T 2m* 2m* where R1(p, q)dqdt is the probability that a positron will pass from the state with the wavenumber p to the state ប2 2 ប2 2  ± ×  q p ប  θ(ω ) with the wavenumber between q and q dq in a time dt. ------Ð ------Ð s q Ð p   D Ð s q Ð p , 2m* 2m*  Using Eq. (4) and removing delta functions, we obtain ω π 1/3 where s is the sound velocity, D = s(6 n) is the 2 2 0 (ប ∆ ) < < γ q ∆  f B s 0 + 1, qmin q p, 0 Ð1 0 Debye frequency, and f (E, T) = [exp(E/k T) Ð 1] is R1 = ------ (9) B B 2π pបs 0 (ប ∆ ) < < the BoseÐEinstein distribution function. The ﬁrst sum- f B s 0 , p q qmax, mand in equation (4) represents phonon emission; and where the second summand, phonon absorption. Feynman ប diagrams of the processes of phonon emission and ∆ = ------p2 Ð q2 . (10) absorption are given in Fig. 1. 0 2ms The positronÐphonon coupling constant in the Substituting (10) into (9), we come to deformation potential approximation is given by 2 γ q ប 2 2 2 E R = ------(q Ð p ) d 1 π 2 3 γ = ------, (5) 2 p4m s (2ρs)1/2 2 0  ប 2 2  where ρ is the density, E is the deformation potential f ------( p Ð q ) + 1 q < q < p, (11) d B 2m  min given by × 2 ∂ 0  ប 2 2  E+ ------( ) < < E = V------, (6) f B  q Ð p  , p q qmax. d ∂V 2m Now consider the energy distributions using the and E is the total energy of a crystal with a positron at + energy variable y = ប2p2/2mk T: the lowest energy level (k = 0). Experimental [6] and p B theoretical data [7] for the deformation potentials of γ 2 m(k T)3 several materials are given in the table. R (y , y ) = ------B------(y Ð y )2 y p q π 5 3 p q 2 ប s p For tungsten, E+ = Ð1.31 eV. It belongs to the same ≈ 0 (12) group as molybdenum. Therefore, Ed Ð11 eV.  f (y Ð y ) + 1, y < y < y , ×  B p q min q p  f 0 (y Ð y ), y < y < y . POSITRON RANGE B q p p q max The values of y and y are determined from the In Eq. (4), R depends only on the absolute values min max p = |p|, q = |q|, and ∆ = |p Ð q|. Therefore, Eq. (2) can conditions ≤ ∆ ≤ ∆ < ω be written as p Ð q p + q, s D (13) ∂ ( ) π and are equal to f p 2 ∆ ∆[ ( , , ∆) ( ) -----∂------= ------∫qdq d R q p f q t s p (7) 2  2ms 2myp,  ( , , ∆) ( ) ] ymin = maxy p + ------Ð s ------yp Ð yD , Ð R p q f p . kBT kBT (14) Integrating with respect to ∆ yields 2  2ms 2myp,  ymax = miny p + ------+ s ------yp + yD , ∂f ( p) kBT kBT ------= dq[R (q, p) f (q) Ð R ( p, q) f ( p)], (8) ∂ ∫ 1 1 បω t s where yD = ( D)/(kBT).

TECHNICAL PHYSICS Vol. 45 No. 7 2000 838 GRYAZNYKH

λ , cm Consider the energy range where positron backscat- ph ± 10–5 tering does not occur; i.e., xmin(max) = yp yD. This is the case when

T = 1000 K 2 > kBTyD 2ms y p ----------- + ------ . 2m s kBT The mean time between interactions is

2 ( )3 1 γ m kBT 1 –6 ------= R (y , y )dy = ------Ꮽ, (15) 10 τ ∫ y p q q π 5 3 T = 200 K ph 2 ប s p where 2 4 6 8 10 y E, eV D 3 Ꮽ 2 0 ( ) yD = 2 ∫ x f B x dx + -----. (16) Fig. 2. Mean free path of positrons in tungsten with regard 3 for positron scattering by phonons at T = 200Ð1000 K. 0 Therefore, the mean free path is 2 /ρ , eV/g cm 4 3 dE dx ប p 4π ប s E 100 λ = τ ------= ------, (17) ph ph m γ 2 ( )3 Ꮽ m kBT where E is the positron kinetic energy. The energy loss is given by 10 dy ------p = (y Ð y )R (y , y )dy dt ∫ q p y p q q 3 4 (18) γ 2 m(k T) y 1 = Ð ------B------D----- π 5 3 1 8 ប s p 0 2 4 6 8 10 and the stopping power by E, eV 2 (បω )4 dE m dy p γ m D Ð1 Ð------= Ð(k T)------= ------E . (19) B ប π 4 3 Fig. 3. Stopping power of tungsten with regard for positron dx p dt 16 ប s scattering by phonons at T = 300 K. Let us calculate the same parameters for phonons in tungsten. For tungsten (in the system of units c = 1), , cm Ltot |E | = 11 eV, ω = 3.534 × 1013 sÐ1, ρ = 19.3 g cmÐ3 = 0.008 d D 2.917 × 1065 eV sÐ3, s = 1.112 × 10Ð5 = 3.33 km sÐ1, and γ2 = 1.865 × 10Ð59 eV s3. 0.007 Putting T = 293 K, we obtain kBT = 0.0252 eV, yD = 0.9230, and Ꮽ = 0.8819. 0.006 The range under consideration is E > 4.58 eV. 0.005 Within this range, calculations yield ([Ep] = eV) τ = 1.219 × 10Ð14 E s, 0.004 ph p λ × Ð7 ph = 7.228 10 E p cm, (20) 0.003 dE Ð1 2 Ð1 0 2 4 6 8 10 Ð------= 372E eV cm g . E, eV ρdx

Fig. 4. Positron range in tungsten calculated with regard for To calculate the mean time between interactions and positron scattering by phonons at T = 300 K. the energy loss in the entire range, the following

TECHNICAL PHYSICS Vol. 45 No. 7 2000 POSITRON SCATTERING BY PHONONS IN METALS 839

Deformation potential Ed determined by Eq. (6) and positron where lifetime for several materials ( , បω ) exp th Emin = max E Ð s 2mE E Ð D , Material E , eV E , eV E⊥, eV τ, ps d d ប2ω 2 (25) ˜ D Al Ð6.7 Ð7.70 Ð4.41 166 E = ------. 2ms2 Cu Ð9.4 Ð9.45 Ð4.81 106 Ag Ð11 Ð9.48 Ð5.36 120 The temperature dependences of dE/dx and positron Mo Ð16 Ð14.3 Ð1.92 111 range Ltot are involved only in the expressions for sound W Ð1.31 100 velocity and Debye frequency. In the approximation used and within the temperature range under consider- Au Ð4.59 10 ation, these values can be regarded as temperature- independent. equations should be used: The mean free path of a positron in tungsten and the stopping power of tungsten were calculated for 2 ( )3  positron energies between 0.025 and 10 eV and temper- 1 γ m kBT 1 ------= ------Ᏺ (y Ð y ) -- atures ranging from 200 to 1000 K. A plot of the mean τ π 5 3 2 p min ph 2 ប s p free path vs. positron energy is presented in Fig. 2; the stopping power vs. energy, in Fig. 3; and the positron  range vs. energy, in Fig. 4. Ᏺ ( ) 1( )3 + 2 ymax Ð y p + -- y p Ð ymin , 3  (21) 3 CONCLUSION dy γ 2 m(k T) 1 ------p = ------B------ÐᏲ (y Ð y ) -- dt 2π ប5 3 p 3 p min Thermalized positrons are scattered mainly by s  acoustic phonons. The Debye approximation allows the phonon contribution to positron scattering to be esti-  Ᏺ ( ) 1( )4 mated. In this work, the mean free path of a positron in + 3 ymax Ð y p Ð -- y p Ð ymin , 4  tungsten and the stopping power of tungsten were calculated for the positron energies 0.025Ð10 eV and temperatures 200Ð1000 K. where

x n ACKNOWLEDGMENTS Ᏺ ( ) y dy n x = ∫------(------)------. exp y Ð 1 This work was supported by the International Sci- 0 ence and Technology Center, project no. 767. Assuming E ӷ ms2, we obtain

4 4 REFERENCES 4πប s E λ = ------, (22) ph γ 2 m 1. F. Ando, J. Synchrotron Radiat. S3, 201 (1996). 3Ᏺ E Ð Emin 1( ) 2T 2 ------+ -- E Ð E  T 3 min  2. A. Perkins and J. R. Carbotte, Phys. Rev. B: Condens. Matter 1, 101 (1970); W. R. Frensley, Rev. Mod. Phys. 2 ( )4 62, 745 (1990). dE γ m E Ð Emin Ð------= ------, (23) π 4 3 3. J. Oliva, Phys. Rev. B: Condens. Matter 21, 4909 (1980). dx 16 ប s E 4. E. R. Woll and J. P. Carbotte, Phys. Rev. 164, 985 (1967). E dx πប4 5. M. J. Puska and R. M. Niemenen, Rev. Mod. Phys. 66, Ltot = ∫------dE = ------841 (1994). dE γ 2m3s 0 6. E. Soinen, H. Huono, P. A. Huttunen, et al., Phys. Rev. B: E Condens. Matter 41, 6227 (1990). ln------, E < E ˜ , (24) 2ms2 7. O. V. Boev, M. J. Puska, and R. M. Niemenen, Phys. × Rev. B: Condens. Matter 36, 7786 (1987). E˜ m2s4 ln------+ ------, E > E˜ , 2ms2 ប4ω 4 ( 2 ˜ 4) D E Ð E Translated by K. Chamorovskiœ

TECHNICAL PHYSICS Vol. 45 No. 7 2000

Technical Physics, Vol. 45, No. 7, 2000, pp. 840Ð848. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 26Ð34. Original Russian Text Copyright © 2000 by Grigor’ev, Koromyslov, Shiryaeva.

GASES AND LIQUIDS

Instability of a Charged Spherical Viscous Drop Moving Relative to a Medium A. I. Grigor’ev, V. A. Koromyslov, and S. O. Shiryaeva Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia E-mail: [email protected] Received July 12, 1999

Abstract—It is shown that, as the velocity of the flow around a charged drop of viscous liquid increases the drop charge value critical for the occurrence of drop instability rapidly decreases. It is found that, for some domains of values of the charge, the ratio of densities of the media, and the ambient velocity, the even and odd modes of the drop capillary oscillations pairwise couple with each other, which represents drop vibrational instability against the tangential discontinuity of the velocity field at the drop surface. At medium velocities larger than those associated with such domains, the instability growth rates for odd modes exceed the incre- ments of even modes with smaller orders, which corresponds to the parachute-like deformation of the drop in the flow. © 2000 MAIK “Nauka/Interperiodica”.

ν INTRODUCTION kinematic viscosity 2 and has radius R and charge Q. The instability of a charged viscous drop moving Let us find the critical conditions for instability of the relative to a medium is manifested in a number of aca- drop capillary oscillations under these assumptions. We demic, engineering, and technological problems. The will use spherical coordinates with the origin at the center of the drop and employ the linear approximation in problem of stability against the tangential discontinuity ξ Θ of velocity field at the drop free surface arises in oper- the perturbation value ( , t) of the equilibrium spher- ation of various dripÐjet devices, in paint sputtering, ical surface of the drop that is caused by the capillary and in the atomization of liquid fuels [1Ð3]. This prob- wave motion of the liquid. The equation for the perturbed surface of the drop is taken in the form r(Θ, t) = lem is of considerable interest for studying thunder- ξ Θ storm electricity in the context of investigation of the R + ( , t). physical mechanism of lightning initiation: lightning The set of hydrodynamic equations modeling the may start with corona discharge in the neighborhood of capillary motion of the liquid in the system includes the a free-falling, large, water-bearing hailstone [4Ð6]. The Euler equation for the potential (curlV1 = 0) motion of instability of a drop either falling in the atmosphere or the medium and the NavierÐStokes equation for the moving with a constant speed in a medium with higher drop density was studied in a large number of papers [7]. ∂ Nevertheless, the conditions of occurrence of the insta- V1 1 ∇ ()2 1∇ ------∂ +--- V 1 = Ð ----ρ- P 1 , (1) bility against the tangential discontinuity of the veloc- t 2 1 ity field at the surface of a drop moving relative to a medium are still poorly understood. The same problem ∂ V 1 ------2 = Ð ----- ∇ P + ν ∆ V ; (2) in the case of a charged drop is almost unstudied. Insta- ∂ t ρ 2 2 2 bilities of two types must evidently occur in this case: 2 the instability of the drop against its charge [3, 8] and continuity equations the instability of the dropÐmedium interface against the tangential discontinuity of the velocity field, i.e., the div V j ==0, j 1; 2 (3) instability of the KelvinÐHelmholtz type [9, 10]. with the boundary conditions Therefore, it is of interest to study the conditions for occurrence of instability of a charged viscous-liquid r ==0: V 0, (4) drop moving with a constant speed in a medium. We 2 solve the problem treating the surrounding medium as ∂ξ 1 ∂ξ ∂ξ rR==:------V Ð --- V Θ ------, ------=V , (5) a perfect liquid. ∂t 1 r r 1 ∂Θ ∂t 2 r 1. Let a perfect incompressible dielectric liquid of ρ ε ()⋅∇ ()⋅∇ density 1 and dielectric constant move with the con- nr V2+0,tnV2=(6) stant velocity U relative to a spherical drop that consists ρ ρν()⋅∇ of a perfectly conducting liquid with density 2 and ÐP1ÐPσ+PE=Ð2P2+22n1nV2,(7)

INSTABILITY OF A CHARGED SPHERICAL VISCOUS DROP 841 ∞ ρ σ r : V1 = U; (8) unity: 2 = R = = 1. The characteristic scales corresponding to the new basis have the form the condition reflecting the invariance of the volumes of 3/2ρ 1/2σÐ1/2 Ð1/2ρ Ð1/2σ1/2 both media r* = R, t* = R 2 , U* = R 2 , Ð1σ 3/2σ1/2 ν 1/2ρ Ð1/2σ1/2 P* = R , Q* = R , * = R 2 . ξ(Θ, ϕ, t)dΩ = 0; (9) ∫ ν ≡ ν We denote the dimensionless quantities as 2 1 Ω ρ ≡ ρ and 1 . and the following condition meaning that the center of Taking into account the axial symmetry of the prob- mass of the system is fixed: lem, we will ignore the dependence of the quantities on the angle ϕ. This means that the velocity field has no azimuthal rotational component, which is of no inter- (Θ, ϕ, ) Ω ∫ t erd = 0. (10) est, because it does not couple with the potential Ω and rotational poloidal velocity components [11] and does not affect the drop stability. The velocity field V2 Hereafter, subscript 1 relates to parameters of the exter- in the drop is represented as the sum of two orthogonal nal medium; subscript 2 relates to the parameters of the fields [11]: drop; n and t are the unit vectors, respectively, normal ( , ) ∇Ψ ( , ) ∇ × (∇ × )Ψ ( , ) V2 r t = 1 r t + r 2 r t , (13) and tangent to the dropÐmedium interface; Vj(r, t) and Pj(r, t) are the velocity and pressure fields, respectively; where the first term defines the potential part of the Pσ is the following perturbation of the pressure of the velocity field and the second one is the rotational poloi- surface tension forces [9]: dal part. σ In accordance with [9, 10], the velocity field in the Pσ(ξ) = Ð-----(2 + ∆Ω)ξ, (11) medium in a neighborhood of the unperturbed surface R2 of the spherical drop is written in the form ∇ϕ ∇ϕ(0) where σ is the surface tension at the interface; ∆Ω is the V1 = + , angular part of the Laplacian in the spherical coordi- 3 (14) (0) ∇ϕ(0) R [ ( ⋅ ) ] nate system; PE is the following perturbation of the V1 = = Ð ------3n U n Ð U + U, electric field pressure related to the perturbation of the 2r3 drop surface [11]: Ψ Ψ ϕ where 1, 2, and are the first-order quantities. ∞ n Ψ Ψ ϕ 2 2 3. Scalar functions 1(r, t), 2(r, t), and (r, t) Q ( ) m(Θ, ϕ) Q ξ PE = ----π---ε-- ∑ ∑ n + 1 AnmYn Ð ----π---ε-- , appearing in the expression for the velocity fields of liq- 4 2 uid motion in the medium and in the drop, as well as the n = 0 m = Ðn ξ π 2π (12) perturbation (r, t), are sought in the form ξ(Θ, ϕ, ) m(Θ, ϕ) Θ Θ ϕ Anm = Q∫ ∫ t Yn sin d d , Ψ ( , ) Ψ ( ) (Θ) ( ) ( , ) 0 0 j r t = ∑ jn r Yn exp St j = 1 2 , n where Y m (Θ, ϕ) are the normalized spherical func- n ϕ( , ) ϕ ( ) (Θ) ( ) tions; and dΩ is the solid angle element. r t = ∑ n r Yn exp St ; (15) n 2 The term proportional to ~V 1 was retained in (1), ξ( , ) (Θ) ( ) because it includes the first- and second-order terms. r t = ∑ZnYn exp St . Conditions (9) and (10) set a lower bound on the spec- n trum of the system capillary oscillations [11]. We solve the problem (1)Ð(15) by using the scalar- 2. To simplify the solution procedure, it is appropri- ization method described in detail elsewhere [11] and ate to go over from the dimensional MTL basis, where applied to particular problems in [12Ð15]. Omitting inter- M, L, and T are the units of mass, length, and time, mediate calculations, we write the following final system respectively, to another, more convenient, basis reduc- of algebraic equations for unknown amplitudes Zn of the ing the number of the model parameters. In this basis, liquid capillary motion forming the contour of the per- ρ the units of the volume density 2 of the constituent turbed interface: substance of the drop, drop radius R, and surface ten- ρ 2 ρ {κ 2 ν ( ) sion σ are taken as the basic units and are equated to U KnZn Ð 2 Ð USLnZn Ð 1 + nS + 2 Fn x S

TECHNICAL PHYSICS Vol. 45 No. 7 2000 842 GRIGOR’EV et al.

ρ 2 γ } ρ ρ 2 viscosity ν and the frequency S as Ð U Mn + n Zn + USInZn + 1 + U JnZn + 2 = 0,

2 2νF (x)S ∼ 2νF S Ð 2ν νs, Q 9α β 9β α n n0 ≡ ------≡ n n Ð 1 ------n------n--+----1 W πε, Mn ------+ ( ), 1 4 2n 2 n + 2 F = --(n Ð 1)(2n + 1). n0 n 9α α (9n + 6)α K ≡ ------n------n---Ð---1 , L ≡ ------n-, n 2n n 2n(n + 1) Retaining in (16) the terms of no higher than the ﬁrst degree in ν and neglecting the coupling of modes, we (9n + 12)β 9β β obtain the dispersion equation I ≡ ------n-----, J ≡ ------n------n---+----1 , (16) n 2(n + 1)(n + 2) n 2(n + 2) κ 2 ν ρ 2 γ nS + 2 Fn0S Ð U Mn + n = 0. (17) ( ) ( )( ) α ≡ n n Ð 1 β ≡ n + 1 n + 2 n ------, n ------, Solutions to (17) are as follows: (2n Ð 1)(2n + 1) (2n + 1)(2n + 3)  nρ  Ð1 S ρ 1 S = Ðν(n Ð 1)(2n + 1) ------+ 1 x = --, κ ≡ ------+ --, γ ≡ (n Ð 1)[n + 2 Ð W], (n + 1)  ν n (n + 1) n n Ð1 2  nρ  ( ) 1( )( ) ± [n(n Ð 1)[W Ð (n + 2)] + nρU M ] ------+ 1 . Fn x = -- n Ð 1 2n + 1 n ( )  n n + 1 ( ) Ð1 Three different cases can be realized: 1( )2( ) x in x + -- n Ð 1 n + 1 1 Ð ------(-----)- , n 2in + 1 x (a) If where in(x) represents the spherical Bessel functions. 1 2 W < (n + 2) Ð ------ρU M , (18) In the limit of a nonviscous drop, the problem under (n Ð 1) n consideration transforms to the problem of motion of a ω perfect-liquid drop with constant velocity U in a perfect the quantity S determines the natural frequencies n of medium, which was solved in [10]. Since Fn(x) also the surface oscillations for the charged drop of low-vis- tends to zero at ν 0 [11], Eq. (16) reduces to the cosity liquid. These frequencies coincide with the nat- corresponding equation obtained in [10]. In the case ural frequencies of oscillations of the drop of perfect when ρ = 0, the problem transforms to the problem of liquid; and the allowance for viscosity results in the β capillary oscillations of a charged viscous drop in vac- appearance of the damping decrement n of the drop uum, which was solved in [11, 12], and Eq. (16) surface natural oscillations, which is proportional to the reduces to the equation derived in these papers. At U = 0, viscosity: we have the problem of capillary oscillations of a ρ Ð1 charged drop in a dielectric medium. This problem is β ± ω ≡ ν( )( ) n  ν S = Ð n i n Ð n Ð 1 2n + 1 ------+ 1 the limiting case of the problem studied in [15] at 1 = 0. (n + 1) Thus, the problem correctly reduces to simpler limiting Ð1 cases. 2  nρ  ± i [n(n Ð 1)[(n + 2) Ð W] Ð nρU M ] ------+ 1 . 4. In the limiting case of a drop of low-viscosity liq- n (n + 1)  uid, one can analytically study the effect of viscosity on the natural frequencies of capillary oscillations, their (b) If the condition opposite to (18) is valid, S δ damping rates, and the growth rate of instability. In this describes the rising rate n for instability of the charged case, x ӷ 1 and we use the following asymptotic expan- drop of low-viscosity liquid, sion of the spherical Bessel functions at large values of δ δ β ≡ ω β argument [11]: S = n = n0 Ð n n Ð n,

1 x 1 where δ = |ω | is the rising rate of instability of the i (x) ≈ -----e 1 + 0 -- , n0 n n x ӷ 1 2x x charged drop of perfect liquid. Then, the ratio of the Bessel functions has the (c) As in the case of perfect liquid [10], equality in asymptotic behavior (18) determines both the boundary separating stable and unstable solutions and the critical relation between ( ) 1 the charge of the drop and its velocity with respect to in x ≈ 1 + 0 -- , ------(-----)- x the medium. in + 1 x ӷ x 1 5. To analyze the case of a high-viscosity liquid, i.e., and the second term in the braces in (16) depends on the x 0, we use the following asymptotic expansion of

TECHNICAL PHYSICS Vol. 45 No. 7 2000 INSTABILITY OF A CHARGED SPHERICAL VISCOUS DROP 843 the spherical Bessel functions for small argument [11]: the approximation of high viscosity: ρ i (x) Ð1 2 n 2 ν ρ 2 γ x n ≈ 2 x An + ------S + BnS Ð n U Mn + n n = 0, ------(-----)- Ð 1 (------)- 1 Ð (------)--(------) (n + 1) 2in + 1 x 2n + 1 2n + 1 2n + 5 (19) 3 2 4 3(4n + 8n + 6n + 3) x A = ------, (20) ------… n 2 + 2 . (2n + 1) (2n + 5) (2n + 1)(2n + 5) (2n + 7) 2(n Ð 1)(2n2 + 4n + 3) Substituting (19) into (16), neglecting the coupling Bn = ------(------)------. of modes, and collecting the terms with the same pow- 2n + 1 ers in S, we obtain the following dispersion equation in Solutions to (20) have the form

Ð1 2 nρ 2 nρ S = Ð B ν ± B ν + 4 A + ------[n(n Ð 1)[W Ð (n Ð 2)] + nρU M ] A + ------. (21) n n n n + 1 n n n + 1

Let us analyze the dependence of these solutions on time solutions the parameter W. [ ( )[ ] ρ 2 ]ν Ð2 (a) If condition (18) is fulﬁlled and the radicand is n n Ð 1 n + 2 Ð W Ð n U Mn kp Ð1 (22) negative, the expression for S becomes complex. The Bn nρ ω = ----- A + ------. imaginary part of S determines the frequencies n of 4 n n + 1 the surface natural oscillations, and the absolute value β of the real part represents damping decrements n: Equation (22) determines the bifurcation points, i.e., ν β ± ω the values kp of viscosity for given drop charge Q and S = Ð n i n. oscillation mode order n such that the oscillation fre- ω Therefore, the drop surface undergoes damped quency n vanishes and, instead of one damping decre- oscillations. β β (1) β (2) ment n, two decrements, n and n , appear. The (b) If condition (18) is valid and viscosity ν is large numerical calculations with W = 0, ρ = 0, and U = 0 enough so that the radicand is positive, both roots of yield the following critical values of dimensionless vis- equation (21) are negative. Their absolute values deter- cosity at which the capillary oscillations vanish: ν ≈ ( ) ( ) mine damping decrements β 1 and β 2 of the pertur- 2.1, 2.66, 3.15, 3.57, 3.96, and 4.31 for n = 2, 3, 4, 5, 6, n n and 7, respectively. It is seen from (22) that, as the Ray- bation of the drop surface. The time dependence of the leigh parameter W and velocity U of the medium amplitude of the perturbation described by the Θ increase, the critical value of the dimensionless viscos- nth-order spherical function Yn( ) in (15) has the form ity ν decreases. The increase of the relative density of of the sum of two exponents the medium results in the increase of the critical value ν ( β (1) ) ( β (2) ) of the dimensionless viscosity for low velocity val- Zn = C1 exp Ð n t + C2 exp Ð n t . ues, i.e., when U ≤ 1, and leads to the decrease of the At small values of time t (t 0), the perturbation critical value for high velocity values. damping is characterized by the smaller decrement (c) If the condition opposite to (18) is valid and the β (1) β (2) roots of (21) are real and have opposite signs, the drop n , because the exponent with the larger value n surface is unstable, because solutions exponentially ris- vanishes faster. ing in time appear. The positive root determines the rate δ The condition that the radicand in (21) is equal to n of instability rising, which depends strongly on vis- zero separates the periodic and aperiodic, decaying in cosity as follows:

Ð1 2 nρ 2 nρ δ = B ν + 4 A + ------[n(n Ð 1)[W Ð (n + 2)] + nρU M ] Ð B ν A + ------. n n n n + 1 n b n n + 1

Therefore, as in the limiting cases of perfect and relation between the charge of the drop and its velocity low-viscosity liquids, the equality in (18) in the case of with respect to the medium. high-viscosity liquids describes both the boundary 6. Let us return to the general case of arbitrary vis- between stable and unstable solutions and the critical cosity. The condition necessary and sufﬁcient for the

TECHNICAL PHYSICS Vol. 45 No. 7 2000 844 GRIGOR’EV et al.

ReS This equality represents the dispersion equation 3 2 5 describing the spectrum of the drop capillary oscillations depending on the dimensionless physical param- 4 eters W, U, ρ, and ν. Variation in these parameters 10 changes the spectrum of capillary oscillations: at defi- 8 ρ 2 nite values of W, U, and , certain solutions Sn vanish 5 9 and become positive with further variation of the 2 6 7 parameters. Amplitudes of the corresponding capillary 2 3 4 4 5 0 3 oscillations increase exponentially in time; i.e., the drop becomes unstable and decomposes [3]. The zero 6 7 solutions to the dispersion equation appear under the 4 condition that the free term in equation (23) vanishes. –5 5 2 This condition determines the relation between the 5 6 7 10 10 charge and velocity of the drop that is critical for the 11 3 –10 11 drop instability occurrence, and it can be easily obtained by putting S = 0 in (23). The numerical calcu- 50 100 U lation for the resulting equation demonstrates that the desired critical relation between W and U differs little ImS from the analytical form obtained by neglecting the 7 coupling modes: 6 ( ) ( )Ð1ρ 2 Wn = n + 2 Ð n Ð 1 UMn . (24) 5 2 10 As ρU increases, the W parameter value critical for 4 occurrence of the drop instability against drop charge 3 rapidly decreases, which gives reason to revive the phys- 2 ical model of lightning initiation developed in [4Ð6]. 8 9 0 This model is based on the idea that the corona discharge lights in the neighborhood of a large melting 2 hailstone freely falling in a thundercloud. The results 3 obtained above enable us to derive correct numerical –10 4 estimates and to fit the model of discharge initiation to 5 real thundercloud parameters using measured values of charge, velocity of falling drops, and strength of elec- 6 tric field inside the cloud. 7 Successive-approximation numerical calculations for dispersion equation (23) reveals that the frequency 50 100 U S = S(U) of the drop capillary oscillations as a function of the velocity of the incident flow has the same quali- Fig. 1. Real ReS(U) and imaginary ImS(U) parts of the ν dimensionless frequency S of the drop capillary oscillations tative character for various modes at W = 0 and = 0.1; against dimensionless ambient velocity U for ρ = 10Ð3, ν = it is shown in Fig. 1 for the first six modes at ρ = 10Ð3. 0.1, and W = 0. Curves 2 to 7 display the respective modes. Curve 1 is absent, because the corresponding mode is responsible existence of nontrivial solutions to homogeneous sys- for the translational displacement of the drop [9]. Note tem (16) is that the determinant composed of the coef- that the second mode couples with the third; and the ficients of desired amplitudes Z be equal to zero: fourth mode couples with the fifth, which results in the n appearance of oscillatory solutions 8 and 9, respectively. The parts ReS > 0 of the curves ReS = ReS(U) χ ρ ρ 2 … 2 USI2 U J2 0 correspond to the rate of instability rising of relevant ρ χ ρ ρ 2 … modes of the drop capillary oscillations. Thus, for the Ð USL3 3 USI3 U J3 velocity region associated with curves 8 and 9, the 2 = 0, ρU K ÐρUSL χ ρUSI … vibrational instability of the drop takes place that is typ- 4 4 4 4 ical of the KelvinÐHelmholtz type instabilities [9, 10, ρ 2 ρ χ … (23) 0 U K5 Ð USL5 5 16]. Curves 10 and 11 correspond to the aperiodic rota- … … … … … tional polar motion of liquid. This motion does not affect the instability of the drop and was analyzed in 2 2 more detail in [13, 15]. χ = κ S + 2νF Ð ρU M + γ ; j j j j j It is of interest that, to the right of the velocity region j = 2, 3, 4, 5, …. where the second and third modes couple with one

TECHNICAL PHYSICS Vol. 45 No. 7 2000 INSTABILITY OF A CHARGED SPHERICAL VISCOUS DROP 845

ReS ReS 5.0 2 4 3

2.5 6 2 3 4 2 2 4 4 6 6 2 2 0 0 3 3 4 4 5 –2.5 5 6 6 –3 7 7 –5.0 7 3 5

5 10 U 2 4 ImS U ImS 15 7 7

6 6 2 5 5 3 10 7.5 4 4 3 3 4 3 5 2 2 5 2 4 6 4 4 0 2 3 4 0 2 3 2 2 5 4 4 2 4 3 3 3 5 4 –10 5 –7.5 4 3 6 5 2 7 6 5 10 U –15 7 2 4 U Fig. 2. The same as in Fig. 1, but for ρ = 0.1. The curves for polar motions are not shown. Fig. 3. The same as in Fig. 1, but for ρ = 1. another, the rising rate for aperiodic instability of the The above-mentioned difference between rising third mode is larger than that of the second mode. The rates for the second and third modes must be phenom- same is true for the fourth and fifth modes. This fact can enologically manifested in the pattern of the decompo- be explained on the basis of Eqs. (16) with neglect of sition of the unstable drop. If the second fundamental the coupling of modes. Indeed, the derivative of S 2 mode has the maximum instability rising rate at given n drop and medium densities and ambient velocity ρ 2 with respect to U determines the rate of Sn increase (velocity of the drop falling in the rest medium), the 3 with varying U and is proportional to nMn ~ n . It means unstable drop takes a shape close to the spheroid Θ that, even in the absence of the coupling of modes, the defined by the second Legendre polynomial P2(cos ) rising rate for the instability of high modes against and then decomposes as described in [8, 17]. If the third increasing ambient velocity increases with the order of mode has the maximum instability rising rate, the modes. An analysis of the classical KelvinÐHelmholtz unstable drop takes the parachute-like shape defined by Θ instability at the plane free surface of a liquid also the third Legendre polynomial P3(cos ) and decom- revealed the increase in the wave number of the most poses into a wealth of small, and several large, frag- unstable mode with the velocity of the displacement ments [7]. The vibrational instability of a free falling flow [9, 16]. drop was also observed in experiments [7].

TECHNICAL PHYSICS Vol. 45 No. 7 2000 846 GRIGOR’EV et al.

ReS ReS

3 2 5 4 3 2 5 4

7.5 7.5 7 8 7 8 10 6 9 10 6 9 2 2 4 2 3 5 6 7 0 34 5 6 7 0 4 3 6 6 5 7 6 7 7 4 4 2 2 –7.5 –7.5 5 5 11 3 3 12 50 100 U 50 100 U ImS ImS 7 7 10 6 10 6 5 5 4 4 10 3 3 8 2 8 9 10 0 0 9

–10 –10 50 100 U 50 U

Fig. 4. The same as in Fig. 1, but for the drop charge W = 3.5 subcritical in the Rayleigh sense. Curves 2Ð7 display respective modes. Curve 10 results from the coupling Fig. 5. The same as in Fig. 4, but for the drop charge W = 4.5 between modes 6 and 7. Curves 11 and 12 correspond to the supercritical in the Rayleigh sense. The curves for polar rotational polar motions of the liquid. motions are not shown.

Numerical calculations reveal that, as the medium face, as well as the spectrum of unstable oscillations, density increases, the critical value of dimensionless depend on the ratio of densities of the media. It is seen velocity U ensuring the instability of the nth mode from Figs. 2 and 3 that only the even modes are unsta- decreases and the variety of liquid motions slightly ble at ρ = 10Ð1, whereas both the even and the odd changes. The vibrational instability resulting from the modes are unstable at ρ = 1 and at smaller values of the coupling of the second and third modes takes place over velocities. As the velocity of the medium increases, the the entire region U > 12. The domain of the KelvinÐ amplitudes of all unstable modes ﬁrst increase aperiod- Helmholtz type instability is extended, and the frequen- ically and then, at larger velocities, the instability cies corresponding to given solutions increase. In all becomes vibrational. The many-valued functions in other respects, the curves S = S(U) are qualitatively Figs. 2 and 3 correspond to the aperiodically unstable similar to the curves in Fig. 1. branches. Figures 2 and 3 show the functions S = S(U) calcu- If the drop carries a charge subcritical in terms of the lated for ρ = 0.1 and 1, respectively, and demonstrate Rayleigh stability (see Fig. 4), the basic features of the that only the even modes are unstable. The damping function S = S(U) coincide with those described above. decrements of the odd modes of capillary motions in The charge presence only decreases the critical velocity the domain where these modes should be unstable rap- values at which the drop becomes unstable, which is in idly increase with the ambient velocity. agreement with the analytical result (24). The critical velocities at which various modes of the Figure 5 presents the pattern for the case when the drop capillary oscillations become unstable against the drop charge is slightly supercritical (W = 4.5) in terms tangential discontinuity of the velocity ﬁeld at drop sur- of the Rayleigh instability (the drop becomes unstable

TECHNICAL PHYSICS Vol. 45 No. 7 2000 INSTABILITY OF A CHARGED SPHERICAL VISCOUS DROP 847

ReS ReS 15 3

4 5 3 6 8 2 4 4 7 0 3 7.5 8 4 4 2 2 9 9 9 6 5 4 5 5 3 –3 6 6

7 7

0 50 100 U 0.5 1.0 ρ ImS ImS 0.1 7

6 8 9 9 7.5 5 0 4 9 3 0 8 4 –0.1 3 4 50 100 U 5 –7.5 Fig. 6. The same as in Fig. 4, but for the drop charge W = 10 6 highly supercritical in the Rayleigh sense. The curves for 7 polar motions are not shown. 0.5 1.0 ρ at W = 4 [3]). The function S = S(U) in this figure is sim- Fig. 7. Real ReS(ρ) and imaginary ImS(ρ) parts of the fre- ilar to that given in Fig. 4, but, as compared to Fig. 4, quency S of the drop capillary oscillations as functions of the ratio of densities of the medium and the drop for ν = 0.1, the critical velocity value at which the drop is unstable U = 1, and W = 4.5. Curves 2Ð7 display respective modes. in the KelvinÐHelmholtz sense is lower and the second Branches 8 and 9 are formed due to the coupling of modes. mode is initially unstable. The function S = S(U) for highly supercritical charge W = 10 is plotted in Fig. 6. In contrast to the represent the motions resulting from the coupling of cases considered above, the rising rate for the funda- modes. Curves 8 and 9 correspond to the vibrationally mental mode takes the minimum value as compared to unstable and oscillatory damped motions, respectively. the values of instability rising rates for higher modes whose instability is vibrational or aperiodic (depending CONCLUSIONS on velocity U) and determines the drop decomposition. It is of interest that nonsequential, i.e., third and sixth, A charged drop in a flow around it can decompose modes couple with one another, which results in the into highly charged drops when the drop charge is less formation of the branch of the vibrational instability. than that in the rest of the surrounding medium. This behavior is due to the composition of two instability Numerical calculations reveal that both decrements types: the instability of the drop free surface against the and frequencies of liquid capillary motions decrease tangential discontinuity of the velocity field and the only slightly with increasing ρ. Figure 7 shows the real instability against the drop charge. Depending on the and imaginary parts of the frequency as functions of the ratio of the densities of the drop and the medium, the ratio of the densities of the medium and the drop for the drop charge, and ambient velocity, the drop can decom- supercritical drop charge value W = 4.5. Curves 8 and 9 pose due to both aperiodic and vibrational instabilities.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 848 GRIGOR’EV et al.

Owing to the aperiodic instability at low ambient veloc- 7. A. L. Gonor and V. Ya. Rivkind, Itogi Nauki Tekh., Ser.: ities, the drop is deformed to a stretched spheroid and, Mekh. Zhidk. Gaza 17, 98 (1982). because of the electrostatic repulsion, decomposes 8. A. I. Grigor’ev and S. O. Shiryaeva, Zh. Tekh. Fiz. 61 either via the Rayleigh mechanism [8] or into several (3), 19 (1991) [Sov. Phys. Tech. Phys. 36, 258 (1991)]. fragments of comparable dimensions [17]. The aperi- 9. L. D. Landau and E. M. Lifshitz, Fluid Mechanics odic instability at high ambient velocities leads to the (Nauka, Moscow, 1986; Pergamon, Oxford, 1987). instability of odd modes; as a result, the drop is 10. A. I. Grigor’ev, V. A. Koromyslov, and S. O. Shiryaeva, deformed to a parachute-like shape and, due to the Zh. Tekh. Fiz. 69 (5), 7 (1999) [Tech. Phys. 44, 486 aerodynamic forces, decomposes into a wealth of (1999)]. small, and several large, drops. The results of the study 11. S. O. Shiryaeva, A. É. Lazaryants, et al., Preprint No. 27, are in qualitative agreement with experimental data [7]. IMRAN (Institute of Microelectronics, Russian Acad- emy of Sciences, Yaroslavl, 1994). 12. A. I. Grigor’ev and A. É. Lazaryants, Zh. Vychisl. Mat. REFERENCES Mat. Fiz. 32, 929 (1992). 13. S. O. Shiryaeva, M. I. Munichev, and A. I. Grigor’ev, Zh. 1. Monodisperse Substances: Principles and Application, Tekh. Fiz. 66 (7), 1 (1996) [Tech. Phys. 41, 635 (1996)]. Ed. by V. A. Grigor’ev (Énergoatomizdat, Moscow, 1991). 14. A. I. Grigor’ev and S. O. Shiryaeva, Izv. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 107 (1997). 2. S. I. Shevchenko, A. I. Grigor’ev, and S. O. Shiryaeva, Nauchn. Priborostr. 1 (4), 3 (1991). 15. A. I. Grigor’ev, S. O. Shiryaeva, and V. A. Koromyslov, Zh. Tekh. Fiz. 68 (9), 1 (1998) [Tech. Phys. 43, 1011 3. A. I. Grigor’ev and S. O. Shiryaeva, Izv. Akad. Nauk, (1998)]. Mekh. Zhidk. Gaza, No. 3, 3 (1994). 16. V. A. Grigor’ev and S. O. Shiryaeva, Zh. Tekh. Fiz. 66 4. V. M. Muchnik and Yu. S. Rud’ko, Tr. Ukr. Nauchno- (2), 23 (1996) [Tech. Phys. 41, 124 (1996)]. Issled. Gidrometeorol. Inst., No. 103, 96 (1971). 17. V. A. Koromyslov, A. I. Grigor’ev, and S. O. Shiryaeva, 5. V. A. Dyachuk and V. M. Muchnik, Dokl. Akad. Nauk Zh. Tekh. Fiz. 68 (8), 31 (1998) [Tech. Phys. 43, 904 SSSR 248, 60 (1979). (1998)]. 6. A. I. Grigor’ev and S. O. Shiryaeva, Phys. Scr. 54, 660 (1996). Translated by V. Gursky

TECHNICAL PHYSICS Vol. 45 No. 7 2000

SOLIDS

Phase-Transition Radiation and the Growth of a New Phase S. A. Sall’ and A. P. Smirnov Vavilov State Optical Institute, All-Russia Research Center, ul. Babushkina 36/1, St. Petersburg, 199034 Russia Received July 16, 1999

Abstract—It was shown that, under certain conditions, the growth of a new phase becomes much like a cooperative optical phenomenon during which the heat of phase transition evolves as a sequence of superradiance pulses (phase-transition radiation). The optical control of new-phase growth and the effect of optical properties of substances on the kinetics of phase transitions are considered. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION from the available phase-transition conceptions and is not considered in analysis of the transition kinetics. The phenomenon of strong near-infrared above- thermal radiation of boiling-up water has been described in [1]. The radiation intensity from the THE FORMATION OF PHASE-TRANSITION waterÐglass interface at a wavelength λ = 1.7Ð1.8 µm RADIATION PULSES is approximately two orders of magnitude greater than that of absolutely black body at 100°C. This effect Three stages of new-phase growth (fluctuation was related to the evolution of condensation heat as a nucleation, extension of nuclei, and coalescence) are nonequilibrium radiation, which was called phase- usually recognized. Here, the second stage is the con- transition radiation. A similar phenomenon also takes cern. Let us assume that, in a single-component super- place upon the boiling up of metals [2]. In metallurgy, cooled melt, nuclei grow to the point where the transi- the condensation and solidification of metal vapors tion heat and temperature take macroscopic values. For are known to be attended with a burst of radiation at metals, this is attained at r0 > 20 nm [5], where r0 is the frequencies that are considerably higher than that of nucleus radius. As chemical bonds form, each molecule the heat radiation peak at the phase transition temper- or atom relaxes from the melt-related state to the crys- ature. The bursts were also observed at the crystalliza- tal-related one. The former condition can be treated as tion of alkali halides and sapphire from the melt [3, 4]. excited in relation to the latter. The spectra of the above-thermal radiation were relatively wide (of the order of 1013 sÐ1) and free of emis- We estimate the probability of excitation energy sion lines and bands. The photon energy in the peak of being converted to light emission at phase transition. the spectra corresponded to the melting heat of the For a free molecule in the excited state, its optical life- substance per molecule. It was found that the intensity time (the longitudinal relaxation time) is equal to T1 = of the above-thermal radiation depends on melt cool- 10Ð7Ð10Ð8 s. For transitions in the near-infrared range at ing conditions, and the leading and trailing edges of T ~ 103 K, the nonradiative multiphonon relaxation the burst do not necessarily coincide with the begin- * Շ Ð9 ning and completion of crystallization. The radiant time in solids is T1 10 s [6]. Then, the probability Ӷ energy of the burst includes a significant part of the of light emission p ~ T 1* /T1 1. Therefore, most of the total radiant energy of a crystallizing melt. Note that transition energy is converted to heat. It is assumed here the spectrum of the phase-transition radiation of boil- that bonding does not change the position of a particle ing-up water [1] contains the fundamental absorption passing from the melt into the crystal. Hence, in accor- bands and the radiation peak is shifted to the short- dance with the FranckÐCondon principle, the optical wave spectral range. transition to the crystal ground state is allowed and the Although experimental evidence on phase-transi- probabilities of optical transitions to the excited states tion radiation is still poor, the mere fact of its occur- of the crystal are small. Actually, the particle position rence is surprising. This phenomenon does not follow may vary and the transition to the crystal ground state

850 SALL’, SMIRNOV may go through a number of intermediate states, for related to the dipole matrix element d of the transition which multiphonon relaxation is more essential [6]. as T = 3បc3/4|d|2 ω 3 . Then, the probability of light emission will be still 1 0 Ӷ lower. Phase transitions with p 1 will be called With no regard for transverse relaxation and in the nonradiative. adiabatic approximation, the dynamics of radiation Q ∆ ∆ For a radiative phase transition with p ~ 1, the and the inversion K of the system ( K = K Ð Kc, where time T1 of the optical transition between the melt and Kc is the number of particles passed to the ground state crystal ground states has to be less than or comparable of the crystal) are written as in [10]: to the nonradiative relaxation time T * . This is obtain- 1 π2 ω 2 dQ 4 Q c V d∆K 2Q able in a great ensemble of particles. The feasibility of ------= Ð------, ------= Ð------, (2) dt ω λ3 dt បω radiative phase transition in an ensemble of particles 3 0 0 was originally treated in terms of quantum electrodynamics even before the phenomenon of collective spon- ω2 taneous radiation (or superradiation) had been experi- where V is the domain volume, t is time, and c = − π| |2∆ ω ប mentally discovered [7, 8]. As is known, the phenome- 8 d K 0/ V is the square of the so-called cooper- non of superradiation is that a system of excited ative frequency of the medium. particles undergoes the optical transition to the lower state due to their interaction with each other through the The solution of system (2) has the form common radiation ﬁeld, the transition time being much ∆ [( ) τ] shorter than the radiative decay time of an individual K = ÐK tanh t Ð td /2 , (3) particle [9]. Next, the feasibility of radiative phase tran- 2 Ð1 Q = បω K/4τcosh [(t Ð t )(2τ) ]. sition will be considered in terms of the theory of 0 d superradiation, where the semiclassical approach is τ often employed [10]. The superradiation pulse duration = T1/K is K times shorter than that of the spontaneous radiation Let us have an ensemble of K (K ӷ 1) particles that from an individual particle. The maximum power Ӷ λ λ បω τ 2 surround nuclei in a melt area of size l , where is Qmax = 0K/4 , varying in proportion to K , is approx- the wavelength of radiation corresponding to the spe- imately K times higher than the initial spontaneous ciﬁc (per particle) transition heat. Also, let parameters radiation level Q0 and is achieved within the delay time * τ T2 and T 2 stand for the dephasing times of the wave td = ln[Qmax/Q0] or, in view of Eq. (1), functions of excited particles due to uniform and nonuniform line broadening (the parameter T is also T K 2 t = -----1 ln---. (4) referred to as the transverse relaxation time). For cubic d K 4 Ð10 crystals with homogeneous line broadening, T2 = 10 Ð 10Ð9 s; for noncubic and impure crystals with inhomo- In our case, the superradiation mode is realized only Ð11 Ð10 * Ð7 * Ð9 geneous broadening, T2* = 10 Ð10 s [6]. Thus, the if td < T2, T2 . Assuming that T1 = 10 s, T1 = 10 s, Ð10 Ӷ and T2 = 10 s and taking into account (4), we arrive condition T2* T2 < T 1* < T1 is met. We assume that the optical transitions of particles from the ground state at the requirement for the threshold number of particles տ 5 of the melt to that of the crystal are allowed. For super- Kt 10 for uniform-broadening media (cubic crys- radiation to take place, its pulse duration τ should be tals). This requirement is readily met even for one Ӷ λ much less than the dephasing times T or T * and the nucleus. The lumped model of superradiation (l ) in 2 2 media with nonuniform broadening seems to be photon transit time through the melt area of size l τ Ӷ τ unlikely [9]. In this case, phase-transition superradia- should be much less than ; that is, l/c , where c is tion may occur in a melt transparent at the frequency ω , the velocity of light (we assume that the refractive 0 index of the medium n ~ 1). The initial level Q of when the radiation propagates from one nucleus to 0 another practically without loss and superradiation noncoherent spontaneous radiation at a transition pulses are simultaneously generated from the whole frequency ω depends on the new-phase growth 0 melt area of size l տ λ. The radiation dynamics in this rate K/T 1* : system is similar, most of the energy being radiated along the greatest extension of the area. In the melts of Q ∼ pបω K/T * = បω K/T , (1) cubic metals having a high conductivity, the penetra- 0 0 1 0 1 ω tion depth of radiation with the frequency 0 is about where ប is Planck’s constant (thermal radiation from 102 nm [11]. Hence, if the internuclear distance is much the melt is neglected). The time T1 is conventionally greater than this value, each nucleus is able to indepen-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 PHASE-TRANSITION RADIATION AND THE GROWTH OF A NEW PHASE 851 dently generate superradiation pulses with a duration of crystallization heat results in the partial melting of τ Շ 10Ð12 s. nuclei, and levels for new superradiation pulses arise. Heat transfer in the melt becomes due to radiant heat conduction. This equalizes the temperature in the melt, NEW-PHASE COLLECTIVE GROWTH and the heat is quickly removed toward the crystallizer IN THE SUPERRADIATION MODE walls. At the stage of coalescence, when growing grains The nonradiative growth rate ν of a nucleus, which come closer together, polaritonic solitons and the effect is defined as the number of new-phase particles formed of self-induced transparency [10] may add to the rate of heat removal from the melt. in a unit time, equals K/T 1* . Therefore, there is a Because of the temperature spread ∆T of an actual ν * threshold value for this rate, t = Kt/T1 , that marks the phase transition [13], the integrated-over-time spec- ν ν beginning of the superradiative process. If < t, the trum of phase-transition radiation broadens by a value ν ν ∆ω ∆ ∆ ប ∆ new phase grows nonradiatively; if exceeds t, after a of i ~ Hf T/ TNA ( Hf is the transition heat per delay time td, a superradiation pulse is generated and a mole of a substance, T is the average temperature of nucleus grows up jumpwise. In this case, most of the transition, and NA is Avogadro’s number), which may transition heat is removed from the nucleus into the be far in excess of the linewidth ∆ω ≈ τÐ1 of an individ- ambient melt by radiation and the heating of the inter- ∆ω ≈ 13 Ð1 ual superradiation pulse (in [4], i 10 s ). More- face limits the growth rate to a lesser degree. Then, over, when passing through the medium, phase-transi- other superradiation pulses may form. In the case of tion radiation may broaden further due to nonlinear high-melting crystals, the thermal radiation from the optical effects [14]. melt is essential for the initiation of a superradiation pulse. The orientation ordering of the dipole moments As shown in [15], optical radiation is stimulatory to of molecules due to the reradiation field at the second- the formation of large-scale density fluctuations (dila- harmonic frequency may also contribute to the new- tons and compressons) and new-phase nuclei. Because phase growth rate [9]. of this, phase-transition radiation, acting on an area free of nuclei, makes for nucleation and fast growth of the The classical theory of new-phase growth suggests new phase over the medium. This point calls for special that the maximum growth rate is observed when, at investigation and is beyond the scope of this article. some temperature, an optimum relation between the ω supercooling of the melt and its viscosity is set. Actu- In melts transparent at the frequency 0, the super- ally, for the majority of substances, the crystal growth radiation pulses will appear from large regions if the rate turns out to be well above the theoretical value and, temperature distribution over them is rather uniform ∆ Շ ប τ∆ in addition, does not vary over a wide temperature ( T NA T/ Hf). When the parameter l/c becomes range [12]. This inconsistency is still left unexplained. comparable to τ, superradiation may appear as a train of Within the framework of our model, it finds a natural pulses with decreasing height (the oscillatory mode). explanation. Indeed, if the phase transition is radiative, At l/c > T2, superradiation may change to superlumi- the growth rate considerably increases and the temper- nescence (when Qmax ~ K) and the radiation becomes ature, supercooling, and viscosity at the interface are induced [10]. The duration of such pulses is of the order governed not by external conditions but by transport of of l/c, and light emission, in this case, carries away no radiation inside the crystal. more than half the energy stored by excited levels Consider bulk crystallization in a supercooled metal (induced amplification occurs at ∆K > 0). melt. Let the bulk of the melt have nuclei of radius r0 > The oscillation threshold is defined by the condition 20 nm. If the nonradiative growth rate of a nucleus or a β/α = 3λ2KT l/4πVT > 1, and the transition to the Ӷ λ ν 2 1 number of nuclei in an area of size l reaches t, then superluminescence mode is observed when λ2 τ π a delay time td later, a superradiation pulse forms. We 3 K l/8 VT1 > 10 [16]. Cooling a melt transparent to ω assume roughly that, if this pulse, acting on other the radiation at the frequency 0 generates a train of nuclei, reaches the level of Q0, they also start to super- superradiation pulses, which cause nuclei to grow Ð7 Ð9 5 radiate. At T1 = 10 s, T1* = 10 s, and K = 10 , the jumpwise with each pulse. value of Qmax will exceed Q0 approximately by a factor of 105. During a superradiation pulse, a nucleus with OPTICAL CONTROL OF NEW-PHASE r0 = 20 nm increases its radius by less than 1 nm. If non- GROWTH linear adsorption is neglected, the radiation flux density 2 If the melt is inside an optical resonator, the emerg- falls with distance r as (r0/r) exp(Ðkr), where k is the ω ing radiation increases and crystallization proceeds absorption factor of the melt at the frequency 0. Put- faster. For instance, if a superradiating medium with Ð1 ting r0 = 20 nm and k = 0.01 nm , we will find that the α = 10Ð3 and β = 10Ð1 is placed into a resonator with initial (driving) pulse has an effect up to the distance plane-parallel semitransparent mirrors with a reflectiv- a ≈ 500 nm. Then, the maximum interface velocity in ity ρ = 0.8, the radiation peak increases and the delay ≈ × 6 such a melt will be a/td 5 10 cm/s. The evolution of time is reduced by almost one order of magnitude [16].

TECHNICAL PHYSICS Vol. 45 No. 7 2000 852 SALL’, SMIRNOV

In this case, the pulse width decreases by a factor of of transition. In substances with large values of k and T2 1/lnρÐ1 ≈ 4.5. For media with nonuniform broadening, (that is, in metals with the cubic crystal lattice) or with the influence of the resonator on the radiation parame- low values of k and T 2* (that is, in noncubic crystals, ters is qualitatively similar; that is, the resonator can be such as Si, Ge, Al O , etc., which are transparent at the * 2 3 used to provide the superradiation condition td < T2, T 1 . frequency of transition), the amorphous phase is more On the contrary, the use of the resonator at superlumi- plausible. Finally, the amorphous phase readily forms nescence will result in a decrease in the radiation peak in substances with large k and low T * , that is, in dielec- intensity [16] and rate of crystallization. Notice that 2 generators of phase-transition radiation are radically trics with the noncubic crystal lattice (SiO2, B2O3, distinguished from those of coherent radiation, which GeO2, etc.), which are opaque at the transition fre- are known in quantum electronics; namely, in the quency. While amorphous substances of the first group former case, pumping can be performed by low-grade are unknown, the substances of the second and third heat (ordinary heating) rather than by electric current, groups have been obtained amorphous only under high-power gaseous-discharge lamp, etc. This may also superfast cooling of the melt or in the form of films, be essential in designing low-grade heat utilizers. small particles, and gels; the substances of the fourth group form glasses when the melt solidifies. The sub- The use of laser pulses or continuous infrared radi- stances of the fourth group crystallize from the melt, ation from a high-power source (laser, arc lamp, globar, solution, or vapor exclusively nonradiatively, that is, etc.) to promote superradiation is another way of con- particle by particle and comparatively slowly. Small trolling the crystallization rate. Optical control of the particles and thin films (water drops, metal and semi- crystallization rate seems to be rather intriguing in pro- metal films, carbon black particles, Al O , etc.) are ducing materials not only with desired properties but 2 3 also with desired grain size distribution (for instance, in fairly prone to vitrification or supercooling, because the devices of gradient optics). In this case, it should be requirement for Kt is not met here. taken into consideration that coarse grains will enhance Phase transitions occurring with energy absorption radiation scattering in the melt and the oscillatory mode (evaporation, melting, and polymorphous transforma- may eventually break down. tions) may likewise bear some resemblance to a coop- Optical control would be also appropriate for use in erative optical phenomenon, However, in this case, it is the case of vaporÐliquid transitions taking place in gas associated with energy collective absorption, or super- liquefiers and solidÐsolid transitions in the technology absorption. It is this phenomenon that can apparently of producing different crystal modifications. explain experiments where substances melted when exposed to Շ1-ps (subpicosecond) laser pulses. A considerable rise in the phase-transition rate due to the radiative processes should cut the lifetime of Our method of analyzing new phase growth can be metastable states such as supercooled vapor and liquid. extended to the more general case of multicomponent It is evident that a high crystallization rate associated systems, as well as to a number of related (in our opin- with the radiative phase transition inhibits the occur- ion) phenomena: chemical reactions (including self- rence of the vitreous state under rapid cooling of the propagating high-temperature synthesis), explosive melt. One can substantially reduce the radiative growth crystallization, polymorphous transformations, sonolu- rate or even break down the superradiation mode by minescence, explosive emission, etc. adding impurities that absorb the radiation at the fre- ω quency 0 or cause significant nonuniform broadening REFERENCES of a radiation line into the melt. This effect should be allowed for in technologies of glasses and Pyroceram- 1. W. R. Potter and J. G. Hoffman, Infrared Phys. 18, 265 like materials. (1968). 2. V. A. Burtsev, N. V. Kalinin, and A. V. Luchinskiœ, Elec- The complex crystal structure, polymorphism, a trical Explosion of Conductors and Its Application in great discrepancy between maxima in the temperature Electrophysical Equipment (Énergoatomizdat, Moscow, dependences of the nucleation rate and crystal growth 1990). rate, high viscosity of the melt, rapid cooling, and the 3. V. A. Tatarchenko, Kristallografiya 24, 408 (1979) [Sov. absence of foreign inclusions are all thought to cause Phys.ÐCrystallogr. 24, 238 (1979)]. the formation of amorphous solids. It follows from our 4. V. A. Tatarchenko and L. M. Umarov, Kristallografiya consideration that the parameters k and T2 or T 2* , as 25, 1311 (1980) [Sov. Phys.ÐCrystallogr. 25, 748 well as the system geometry, which specifies the (1980)]. parameters K, α, and β, play a key role in the formation 5. Yu. I. Petrov, The Physics of Ultimate Particles (Nauka, of amorphous substances. In many cases, they become Moscow, 1982). deciding. The amorphous phase is most difficult to 6. N. V. Karlov, Lectures on Quantum Electronics (Nauka, form in substances with low k and high T2, that is, in Moscow, 1988). salts with the cubic crystal lattice (NaCl, KCl, KJ, etc.), 7. M. E. Perel’man, Dokl. Akad. Nauk SSSR 203, 1030 which are transparent to the radiation at the frequency (1972).

TECHNICAL PHYSICS Vol. 45 No. 7 2000 PHASE-TRANSITION RADIATION AND THE GROWTH OF A NEW PHASE 853

8. N. Scribanowitz, I. P. Hermann, I. C. MacGillivray, and 13. A. P. Smirnov, in A Set of Singular Temperature Points M. S. Feld, Phys. Rev. Lett. 30, 309 (1973). in Solids (Nauka, Moscow, 1986), pp. 210Ð239. 9. A. V. Andreev, V. I. Emel’yanov, and Yu. A. Il’inskiœ, 14. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, The Cooperative Phenomena in Optics (Nauka, Moscow, Optics of Femtosecond Laser Pulses (Nauka, Moscow, 1988). 1988). 15. G. I. Grozovskiœ, N. V. Krasnogorskaya, and A. P. Smir- 10. V. V. Zheleznyakov, V. V. Kocharovskiœ, and Vl. V. Ko- nov, in Current Problems in Biosphere Research and charovskiœ, Usp. Fiz. Nauk 159 (2), 194 (1989) [Sov. Protection (Gidrometeoizdat, St. Petersburg, 1992), Phys.ÐUsp. 32, 835 (1989)]. Vol. 2, pp. 427Ð430. 11. N. I. Koroteev and I. L. Shumaœ, The Physics of High- 16. A. V. Andreev, Usp. Fiz. Nauk 160 (12), 1 (1991) [Sov. Power Lasing (Nauka, Moscow, 1991). Phys.ÐUsp. 33, 997 (1990)]. 12. M. Vollmer, Kinetik der Phasenbildung (Steinkopff, Dresden, 1939; Nauka, Moscow, 1986). Translated by B.A. Malyukov

TECHNICAL PHYSICS Vol. 45 No. 7 2000

SOLIDS

On the Turbulization of the Viscocrystalline Phase S. O. Gladkov Semenov Institute of Chemical Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, 117977 Russia Received October 13, 1997; in ﬁnal form, December 24, 1999

Abstract—The dependence of the mean size of dispersed phase particles on the physical parameters of a system (temperature, density, and sound velocity in a substance) was found. The generalized FokkerÐPlanck equation was used to calculate the particle size distribution. The obtained binary distribution function was proved to adequately describe a large array of experimental data in actual physical conditions. It was shown rigorously that the ﬁne-grain phase (powder) results when the viscocrystalline phase is subjected to shear loads. The shape of the distribution turned out to be independent of external actions, i.e., remained the same both on sedimentation and at pressure drop. © 2000 MAIK “Nauka/Interperiodica”.

Interest in the properties of fine-grain powders breaks. As a result, many fine pellets with a character- stems largely from the fact that many industrial tech- istic linear size R are produced. niques are intimately related to powder metallurgy. It is To describe this picture mathematically, we should particularly remarkable that theoretical findings are first elucidate how the shape of the pellet size distribu- quickly employed in practice in this field. The problem tion depends on R. For simplicity and without loss of discussed below also has a direct bearing on powder generality (as will be seen later), we will assume the technology, since its solution physically justifies a pellets to be spherical. number of processes and makes it possible to produce Initially, the pellets were absent. This means that the high-quality fine powder. distribution function at t = 0 (t is time) equals zero and, Let us have a viscocrystalline phase where the vis- hence, no equilibrium (or quasi-equilibrium) distribution function existed. It appeared at times on the order cous liquid and crystalline phase occupy 85Ð95 and 5Ð of τ, when thermostatic control due to phonons set in. δ Ð4 Ð6 15 vol.%, respectively. A very thin ( = 10 Ð10 cm) The situation when phonons do not interact and are free near-surface layer of this substance is subjected to a to move from boundary to boundary is known as Knud- shear load F (Fig. 1). Upon being stripped off, this layer sen situation [1]. This would have been the case if the is released to a thermostat and is converted into a pow- pellets had roughly equal sizes. Actually, however, the der. The first question immediately arises as to what dispersed phase derived is highly nonuniform and con- forces are behind the formation of the fine-grain phase. tains pellets of different sizes. Then, along with the To tackle the question, imagine the following situation Knudsen mechanism, one must also take into consider- (Fig. 2a). Let a ∆x-long thin layer of a viscocrystalline ation phonon interaction. Formally, this means replac- structure be dropped into a thermostat. Because of tem- ing the time τ by τ*, where 1/τ* = τ Ð1 + τ Ð1 . (Here, δ 0 1 perature fluctuations T in the thermostat, each piece of τ is used instead of τ.). The time τ can be given by the the structure is under its own temperature T + δT(x) that 0 1 differs from others. The characteristic range of x has a δ very small dimension R on the order of . As follows 1 from Fig. 2b, “icicles” that form under the action of the gravity force began to nonuniformly “freeze” from the F δ lower end. The freezing process is in no way related to 2 heat conduction (we are dealing with very short times!) but is due to the phonon transfer of the given local temperature by a distance on the order of R. The transfer τ time = R/cs, where cs is the sound velocity in the solid phase (for example, polipropylene or metal). The upper 3 τ portion (lower end) of an icicle thus cooled for a time δ Fig. 1. Fine powdering process: Vsh, shear rate; , removed begins to “pool” the as yet liquid lower portion (upper layer thickness; (1) shear stress; (2) viscocrystalline sub- end). This results in necking, and the neck eventually stance; and (3) thermostat.

ON THE TURBULIZATION OF THE VISCOCRYSTALLINE PHASE 855 formula (a) ∆x/V = ∆t Ն τ = R/c τ λ 3 sh s 1/ 1 = R , (1) where λ depends on the ambient temperature T.1 1 This function is readily determined if a specific V mechanism of phonon interaction is known (see, for sh example, [2]). Here, consideration must be given to one important point. In a “normal” crystalline substance, ∆x τ the time 1 is proportional to N/V, where N is the number of atoms in the lattice and V is the crystal volume. (b) Therefore, as applied to our case (highly nonequilibrium state), the assumption (as yet purely hypothetical) T(x) = T + δT(x) should be made that the core of a forming pellet has a fixed number of atoms (of the already crystalline structure!). Then, we can introduce into consideration a phonon gas and treat volume as a parameter that fluctu- ates. This is included in formula (1). It will be shown below that this assumption makes possible the exact ∆ determination of the distribution function with allow- x ance for a spread in particle sizes. Running ahead, we notice that the distribution function obtained by solving 2 the generalized FokkerÐPlanck equation has two peaks. 1 3 Thus, for the time τ*, pellets can grow up only to a certain (limiting) size. This circumstance is reflected in Fig. 2. (a) Origination of fine powder: ∆x, small displace- τ τ ∆ the macroscopic equation ment of viscocrystalline phase 1 for a time 0 ( 0 = x/cs) and R, radius of formed crystalline structure. (b) Early stage γ ( 〈 2〉 2) of icicle formation: (1) neck; (2) grain; and (3) finely dis- dR/dt = R R Ð R , (2) persed phase. where λ is a parameter (that will be discussed later) and 〈R〉 is the limiting radius of the pellets. In formulas (3)Ð(5), n is the normal to the spherical pellet surface, p = mu is the pellet momentum, m is the CALCULATION OF THE DISTRIBUTION pellet mass (m = ρν, where ρ is the pellet density and FUNCTION ν = 4πR3/3 is the pellet volume), and u is the pellet For our specific problem, the general form of a velocity. kinetic equation that takes into account gravity forces The operator Lp{f} describes the particle velocity or pressure drops must have the form distribution (see, e.g., [3, 4]). Its form is typical of the ∂ ∂ ( )∂ ∂ ∂ ∂ { } τ FokkerÐPlanck equation: f / t + dR/dt f / R + F f / p = Lp f + f / *, (3) { } ∂( ∂ ∂ ) ∂ where f = f(R, p, t) is a desired distribution function and Lp f = D f / p + kpf / p, (6) F is an external force acting on pellets. where k and D are coefficients. In the case of sedimentation (motion in a gravitational field), The solution of (3) should be sought only for the case ∂f/∂t = 0. In fact, for times less than τ, the classical F = mg Ð 6πηRu, (4) consideration fails and the problem must be treated as purely quantum. For times comparable to τ, the pellet and in the case of a pressure drop, distribution function has already been established and ( )π 2 only the steady-state solution should be found. In con- F = P1 Ð P2 R n. (5) nection with this, we put f = f1{u}f2{R} and, after sep- 1 Notice that ballistic (collisionless) phonon movement from aration of variables, obtain boundary to boundary is actually an n-fold process. Phonons [(∂ ∂ )γ ( 〈 2〉 2) Ð1 τ ] striking one boundary pass to it a part of their energy, reflect, f 2/ R R R Ð R f 2 Ð 1/ * reach another boundary, pass to it a part of the energy again, etc. (7) The process is repeated n times until the phonons are thermalized = f Ð1[L { f } Ð F∂ f /∂p]. and acquire the mean kinetic energy. Such phonons in combina- 1 p 1 1 tion are nothing else than a thermostat. As the pellet size R grows, phonon interaction comes into play. Generally, at a certain (criti- Equation (7) is solvable if the right- and left-hand τ cal) size Rcr, phonon collisions become competitive with the 0 sides equal some constant B. The simple mathematics mechanism (the condition τ ӷ τ is violated). In this case, the Ð1 Ð1 τ τ τ 1 0 τ τ τ formula 1/ * = 1/ 0 + 1/ 1 is valid. in view of the relationship 1/ * = 0 + 1 yields the

TECHNICAL PHYSICS Vol. 45 No. 7 2000 856 GLADKOV

f2(R) Two cases are possible: B < 0 (a < 0) and B > 0 (a) (a > 0). In the former case, the distribution function has either one maximum or two maxima and one minimum (Fig. 3a); in the latter, either one maximum and one minimum or two maxima and two minima (Fig. 3b). In 2 both cases, the distribution function reflects the actual physical situation dealt with in the process of powder production. Now we turn to the constant γ, involved in (1). The 1 formation of a crystalline structure is known [5] to be a highly nonequilibrium process. Under such conditions, crystallization formally depends on two temperatures: γ f2(R) cs/2 R R thermostat temperature T and crystallization tempera- (b) ture Tcr . As the temperatures are close to each other in our given case, the coefficient γ, which characterizes the rate of nucleus growth, depends on their difference. 3 The growth rate can be related only to the viscosity of the system. The latter is known [6, 7] to be described by the expression η η ( ∆ ( )) = 0 exp + /kB T Ð Tcr , (10) 4 ∆ η where is the energy gap and 0 is the viscosity at T = ∞. The nucleation rate (the rate at which a hydrodynamic viscous flux adheres to the basic crystalline η c /2γ 2 R R structure) must be inversely proportional to . The only s Rmin = 2BR /cs possible relation in this case has the form Fig. 3. (a) Particle size distribution function for B < [c + γ 2 ν s = u0 / , (11) λ〈 〉4 〈 〉 〈 〉 ε ε R ]/ R . (1) Peak at R1 max = R (1 Ð ) for = 0.25 + Ӷ where the kinematic viscosity ν and dynamic viscosity b/4(1 Ð a), B < 0, a < 0; (2) B < 0, b 1, a < 0, R1 max = η η νρ ρ 〈 〉 1/2 〈 〉 ε 〈 〉 are conventionally related as = ( is the density). 4 R b /3, R2 min = R (1 Ð ), and R3 min = 3 R /4. (b) Dis- λ〈 〉4 〈 〉 The velocity u0 is defined by the exchange interac- tribution function for B > [cs + R ]/ R : (3) B > 0, a > 0, 〈 〉 tion integral Jex and mean interatomic spacing a ; i.e., B Ӷ 1, R + 4〈R〉b1/2/3, and R = 〈R〉(1 Ð ε); (4) B > 1 max 2 min 〈 〉 ប 0, equation (9) has four real roots. In this case, an analytic u0 = Jex a / , (12) solution is impossible to find, and the distribution function vs. radius curve is shown schematically. where ប is Planck’s constant. Eventually, the desired growth rate of a microscopic nucleus is given by desired pellet size distribution function: γ = (J 〈a〉/ប)2/ν. (13) { } ÐB/γ ( 〈 〉 )µ( 〈 〉 )ν ex f 2 R = AR R + R R Ð R (8) This expression adequately covers real situations, × { ( γ ) (λ β)} exp Ð cs/ R Ð R/ , since it includes all the basic physical parameters needed for growth description at both the micro- and where A is a normalizing constant, which is found from macroscopic levels. the conservation-of-mass condition ∫ f 2 {R}dR = M µ (M is the total mass of one removed layer); = (cs + THE EFFECT OF A HIGH-FREQUENCY λ〈R〉4)/2γ〈R〉 + B/2γ; and ν = Ðµ. ELECTRIC FIELD ON THE FORMATION OF THE DISPERSED STRUCTURE f2{R} curves for cases of practical value are demonstrated in Fig. 3. Their extrema are found by solving the When the molten material is exposed to an ac elec- algebraic equation of the fourth order tric field whose frequency satisfies the inequality

4 3 2 ω ӷ K (14) x + ax + ax Ð ax + b = 0, (9) (K is the rate of a chemical reaction that speciﬁes the where exchange interaction), the exchange integral diminishes, affecting the crystallization process [see, e.g., 〈 〉 λ 〈 〉 3 λ 〈 〉 4 x = R/ R , a = B/ R , b = cs/ R . [8, 9]). This is associated with the fact that exchange

TECHNICAL PHYSICS Vol. 45 No. 7 2000 ON THE TURBULIZATION OF THE VISCOCRYSTALLINE PHASE 857 coupling sets in because the wave functions of elec- ACKNOWLEDGMENTS trons of neighboring atoms overlap. Since the electrons strongly interact with the ac field, the exchange energy This work was partially supported by the Russian becomes frequency-dependent. It is significant that the Foundation for Basic Research (grant no. 96-03- frequency range where this takes place is rather narrow 03237). and only specific frequencies of the field can influence the exchange energy. From the physical standpoint, condition (14) merely indicates that the chemical reac- REFERENCES tion has no time to be completed during the cycle of the 1. J. M. Ziman, Principles of the Theory of Solids (Cam- electric field; hence, exchange coupling weakens. bridge Univ. Press, London, 1972; Mir, Moscow, 1982). If the field begins to act from the instant that the 2. V. L. Gurevich, Kinetics of Phonon Systems (Nauka, entire substance represents a viscocrystalline structure, Moscow, 1980). it must act to the end of crystallization. Only then can 3. W. Coffey, M. Evans, and P. Grigolini, Molecular Diffu- one expect the realization of the above effect: dispersed sion and Spectra (Wiley, New York, 1984; Mir, Moscow, pellets become still finer (exchange interaction weak- 1987). ens); and their agglomerate, more homogeneous. 4. S. O. Gladkov, Phys. Rep. 182 (4-5), 211 (1989). As follows from the above formulas, the structure will be homogeneous if (1) the substances to be dis- 5. J. Frenkel, Z. Phys. 26, 117 (1924). persed are selected such that the sound velocity is as 6. H. Fogel, Z. Phys. 22, 645 (1921). low as possible, (2) the relaxation time τ is as long as 1 7. G. S. Fulcher, J. Am. Ceram. Soc. 8, 339 (1925). possible, (3) the melting (crystallization) point is as high as possible, and (4) the applied field frequency is 8. S. O. Gladkov, Chem. Phys. Lett. 174, 636 (1990). such that the exchange interaction does not show up 9. S. O. Gladkov, Phys. Lett. A 163 (5), 460 (1992). [8, 9]. The field frequency ω must be such that electron bonds have no time to form; that is, it must exceed K, where K is the chemical reaction rate. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 7 2000

OPTICS, QUANTUM ELECTRONICS

Short-Laser-Pulse-Induced Photoelectric Phenomena and Reorientation in Nematics Activated with Ionic Dyes S. V. Serak, A. A. Kovalev, and A. V. Agashkov Institute of Electronics, Belarussian Academy of Sciences, Minsk, 220090 Belarus E-mail: [email protected] Received March 16, 1999; in ﬁnal form, December 21, 1999

Abstract—The photoelectric phenomena and orientational nonlinearity induced by nanosecond laser pulses in planar layers of liquid crystals oriented by silicon oxide (SiO) and activated with polymethine dyes were investigated. These phenomena are due to the photogeneration of surface and bulk charges in the liquid crystal cell, their spatial distribution between the grating vector and the beam propagation directions, and the electrohydrodynamic instability. © 2000 MAIK “Nauka/Interperiodica”.

Previously [1, 2], we have observed the phenome- nying photoelectric phenomena were studied in this non of static grating recording in the planar layers of work. nematics oriented with silicon oxide and activated with dyes such as phthalocyanines, bisanthenes, and poly- methines used for the passive Q-switching of ruby LIQUID CRYSTAL SAMPLES lasers [3, 4] and laser wave front inversion [5, 6]. The The experiments were performed with planar ori- × gratings were recorded at a radiation power of 5 ented liquid crystals of two types. Samples of the first 106 W/cm2 and had a spatial frequency of not less than type had both conducting and orienting coatings, while 500 line/mm. They existed for a long time (several those of the second type had only the orienting coating. weeks) and could be erased either by heating liquid The orientation was performed by the oblique vacuum crystals above the temperature of transition to the iso- deposition of SiO layers with a thickness of about tropic state or by applying an electric voltage of about 300 Å. Director slope angle to the substrate surface var- 30 V. The lattices represented the regions of “domains,” ied within 1°Ð3°. Stoichiometry of the near-surface sil- where the director orientation deviated from the initial, icon oxide layer to a depth of 14 Å was studied by surrounded by disclinations [7, 8]. Study of the reorien- means of X-ray photoelectron spectroscopy using an tation in these regions by means of polarization micros- ESCA spectrometer (EMgKα1, 2 = 1253.6 eV). A maxi- copy and dynamic holography revealed a number of mum attributed to Si4+, characteristic of silicon dioxide, spectral features connected with the photorefractive was observed in the Si2p electron band. After the ion properties of dye-activated liquid crystals. sputter cleaning, an additional maximum shifted by The experimentally observed photorefraction in liq- 0.5 eV, which is characteristic of SiO, was observed as well. The ratio of oxygen to silicon was equal to uid crystals (LCs) [9,10] is of considerable interest due approximately 1.2. Such an inhomogeneous structure to the unusually large nonlinear response of the contains oxygen vacancies, which favor the formation medium. By now, various mechanisms for the photore- of local energy levels in the bandgap of the dielectric fraction phenomena observed in nematics activated SiO film. We must also note that the evaporated films with dyes [9,10], fullerene C60 [11], and discotic nem- exhibit some absorption at the ruby laser wavelength atics [12] have been proposed. A space charge field (λ = 694.3 nm), with the absorption coefficient equal to induced by the interference field of continuous wave α ≈ 1.5 cmÐ1. To study the conductivity, we used LC gas lasers at a small static voltage (1Ð2 V) can be cells with transparent evaporated electrodes of indium explained by diffusion, drift, or transport of the photo- oxide In O covered with a SiO film. generated charges or by anisotropy of the dielectric per- 2 3 mittivity and conductivity (KarrÐHelfrich effect). We used nematics with a small positive dielectric Another possible reason for the photorefraction can be anisotropy such as 5TsB (∆ε = 5) and a mixture of the photovoltaic effect in LC cells reported in [13, 14]. ZhK-440 (∆ε < 0) with up to 15 wt. % ZhK-497 (∆ε > 0) (NIOPIK, Moscow). Figure 1a shows the We have used nanosecond ruby laser emission structures of polymethine dyes PK-686 and PK-742 pulses to record orientational dynamic holograms of (Institute of Organic Chemistry, Kiev) used as the acti- high diffraction efficiency in nematics activated with vating agents. The absorbtion dichroism of PK-686 and ionic polymethine (cyanine) dyes [15]. The accompa- PK-742 additives in 5TsB was equal to 0.02 and 0.33;

SHORT-LASER-PULSE-INDUCED PHOTOELECTRIC PHENOMENA 859 and that in the ZhK-440+ZhK-497 mixture was 0.46 (a) and 0.69, respectively. The LC cell thickness was equal CH2 CH2 to 50 µm. CH2 CH2 C–CH=CH–CH=CH–CH=C + + INTRINSIC CONDUCTION OF LC ACTIVATED N N WITH IONIC POLYMETHINE DYES J– LCs activated with ionic dyes represent weak elec- R R trolyte solutions. In polar LC solvents, polymethine R=C6H5 –Pä-742 dyes dissociate into organic cations and iodine anions R=CH3 –Pä-686 (JÐ). In the stationary case, when the electric current density satisfies the condition dj(t)/dt = 0, the conduc- (b) U tivity is determined by the well-known equation [16] – + ± σ = e(µ+ + µÐ)n (1) (µ+ µÐ)[γ ( ) γ ]1/2 = e + D E n0/ R , where e is the carrier charge; µ± are the positive and negative ions mobilities; n = n+ = nÐ is the concentration of charge carriers; n0 is the impurity concentration; and γ γ D(E) and R are the constants of dissociation and γ recombination, respectively ( D depends on the field U' RH strength in strong electric fields). The conductivity measurements were performed using the scheme presented in Fig. 1b. The value of σ was determined as σ = dU'/[SR (U Ð U')], where d is Fig. 1. The structure of polymethine dye (a) and the scheme H of conductivity measurements in LCs activated with poly- the layer thickness, S is the sample cross section methine dyes (b). through which the current flows, RH is the load resistance, U is the source voltage, and U' is the voltmeter 9 –1 –1 reading. The perpendicular conductivity component σ⊥ × 10 , Ω cm , µA σ ⊥ I ⊥ (j L, L being the layer director) was measured dur- 9 –1 –1 10 σ⊥ × 10 , Ω cm 2.0 ing the study of planar layer. The parallel component δ|| was 1.5 times larger. Figure 2 shows a typical currentÐ 10 voltage characteristic of an LC activated with polyme- 1 3 thine dyes and the plots of conductivity versus voltage 8 1.6 across the sample for various dye concentrations. As is 5 2 seen, the current increases slowly and the conductivity remains virtually constant and equal to σ ≈ 10−9 Ω−1 cmÐ1 in the range of voltages from 0 to 1 V. 6 1' 1.2 Note for comparison that the intrinsic conductivity of 0 50 150 σ ≈ × Ð11 ΩÐ1 Ð1 –1 5TsB is 0 2 10 cm and that of the α, cm ZhK-440 + ZhK-497 mixture is approximately 5 × 1 0.8 10−11 ΩÐ1 cmÐ1. 4 Already at a small voltage (~0.1 V), the LC cell 3' exhibits the properties of a galvanic cell. This might be evidence of the electric double layer existence in the 2 0.4 vicinity of the electrode. According to the formula for 2' the Debye screening radius, the layer thickness is rD = 1' (Dε/4πσ)1/2, where D is the diffusion coefficient. For the molar concentrations of impurities typically used in 0 −4 Ð3 0 0.5 1.0 1.5 2.0 experiments (10 …10 mol/l), the screening radius is U, V usually smaller than 1 µm [17]. Taking this into account, we obtain the value of diffusion coefficient D = Ð6 2 Ð1 σ Ð9 ΩÐ1 Ð1 Fig. 2. A currentÐvoltage characteristic of the LC cell 10 cm s for the conductivity value = 10 cm . (dashed line) and the plots of (1Ð3) conductivity and (1'Ð3') σ 2 α Using the Einstein relationship D = kBT /ne for the photoconductivity versus absorbance and voltage U for (1, × 16 Ð3 α Ð1 × 16 Ð3 diffusion coefficient, we may estimate the concentra- 1') n0 = 3 10 cm ( = 14 cm ); (2, 2') 9.6 10 cm tion of carriers n = 1.4 × 1013 cmÐ3. It is also possible to (α = 48 cmÐ1); (3, 3') 3 × 1017 cmÐ3 (α = 154 cmÐ1).

TECHNICAL PHYSICS Vol. 45 No. 7 2000 860 SERAK et al.

α α –3 σ 1/2 I0exp(– d), J cm Onsager theory ~ exp(E) [18]. The conductivity of × 17 Ð3 2 4 6 8 10 12 14 a solution with the concentration n0 = 3 10 cm − 0.9 rises from 10 9 to 9 × 10Ð9 ΩÐ1 cmÐ1, and the concentra- 2 tion of carriers also grows by almost one order of mag- 3.0 × 14 Ð3 A nitude to reach n = 1.3 10 cm . At a large current,

µ 0.5

, the concentrational emf causes saturation of the cur- I rentÐvoltage characteristic. 2.5 3 0.1 1 0 50 150 PHOTOINDUCED CONDUCTION IN LIQUID 1 –1 – α, cm 2.0 CRYSTALS ACTIVATED WITH POLYMETHINE

cm DYES 1 –

Ω The conductivity of LC cells increased as a result of , 9 1.5 pulsed excitation by radiation of the ruby laser (pulse

10 3' duration, 60 ns). The radiation beam was expanded × with a lens to ﬁt the size of the sample, after which the ⊥

σ 2 2' 2 energy density was equal to ~0.1 J/cm . The photocur- 1.0 rent was measured by means of an oscillograph (Fig. 1b) using the maximum value in a pulse response. In the presence of a static electric field, the current 0.5 1 relaxed to its stationary value within a time period of 1' about 10 ms. Here we can see a steady growth of the photoconductivity with an increase in the solution concentration and the voltage across the electrodes (Fig. 2, 0.05 0.10 0.15 0.20 –2 curves 1'Ð3'), which makes these curves different from I0, J cm the initial ones (Fig. 2, curves 1Ð3). One of the possible reasons for the photoconductivity growth is the addi- Fig. 3. The plots of photoconductivity versus energy density tional dissociation of excited molecules into ions, at various α, cmÐ1: (1) 14; (2) 48; (3) 154. The inset shows which are distributed in the cell volume in accordance the plots of photocurrent versus absorbance in (1) the with formula ρ(z) = ρ exp(Ðαz), where α is the absorp- absence and (2) the presence of voltage across the cell elec- 0 trodes. U = 1.6 V. tion coefficient. Here, the z axis is parallel to the beam propagation and perpendicular to the layer director L and the substrate. This is confirmed by the growth of calculate the mobility of the carriers µ = σ/2en = 2 × conductivity with an increase in the absorption coeffi- 10−5 cm2 V−1 sÐ1. The mobilities and diffusion coefficient and radiation intensity (Fig. 3). For the cell with α Ð1 cients of ions are not the same because of a consider- = 154 cm , the number of carriers increases from × 14 × 14 Ð3 able difference in size (iodine ion radius is r ≈ 2.19 Å, 1.3 10 up to 2 10 cm . We may describe the jÐ dependence of the photoconductivity on the radiation ≈ while that of the organic cation is rOC+ 20 Å). A rela- intensity I and the absorption coefficient α under some tionship between the mobility of ions and the viscosity simplifying assumptions. The balance equation of the of a liquid insulator established by Walden [18] photoinduced ion production is as follows [9]: µχ π χ gives = e/6 r, where is the viscosity. This allows ∂ µ± ± n γ + Ð ( ) one to estimate and D using the ratio r /rOC+. ----- + n n = aI z , (2) jÐ ∂t R The final values are µÐ = 1.8 × 10Ð5 cm2 VÐ1 sÐ1, µ+ = γ ± 2 α α 2 × 10Ð6 cm2 VÐ1 sÐ1; DÐ = 9 × 10Ð7 cm2 sÐ1, and D+ = where R = D e /kBT, I = I0exp(Ð z), and a = a0 is a 10−7 cm2 sÐ1. Then the parameter ν = (D+ Ð DÐ)/(D+ + DÐ) constant characterizing the photodissociation effi- characterizing the relative mobility of ions is 0.8, which ciency. is considerably higher than the values in other photore- The dissociation of the excited dye molecules can fractive LCs [9, 10]. In practice, this value is slightly occur in the presence of states with large lifetimes, smaller than the estimate due to the different degrees of where the probability of collisions with molecules of cation and anion coupling with the molecules of sol- the crystal matrix is sufficiently large. On the other vent. hand, the intercombination conversion from a triplet level increases the temperature in a local interaction In contrast to the region of voltages below 1 V, region and also results in a growth of the degree of dis- where the current grows slowly, both conductivity and sociation. Study of the kinetics of interaction of pulsed current increase rapidly in the interval from 1 to 2 V. ruby laser radiation with polymethine dyes in LCs This is due to the growing role of dye ionization with showed that dye molecules radiate via a three-level increasing electric field strength in accordance with the scheme involving a metastable state with a lifetime of

TECHNICAL PHYSICS Vol. 45 No. 7 2000 SHORT-LASER-PULSE-INDUCED PHOTOELECTRIC PHENOMENA 861

~1.5 µs [19]. Thus, the main processes of ion recombi- induced in the bulk of the LC by the interference radia- nation and dissociation occur after termination of the tion field as a static lattice of domains. laser pulse. Then the balance equation (2) splits into The relaxation time of the induced charge was of the two parts: order of milliseconds. The Debye relaxation time of the τ ε πσ ε σ × ∂n volume charge is D = ⊥/4 ⊥. For ⊥ = 10 and ⊥ = 3 ----- = aI exp(Ðαz) for t < τ, (3) Ð9 ΩÐ1 Ð1 τ ∂t 0 10 cm , this formula gives D = 0.3 ms, which is smaller by one order of magnitude than the value ∂n + Ð observed in experiment. Thus, inhomogeneous distri------= +γ n n = 0 for t > τ, (4) ∂t R bution of the volume charge along the beam direction up to a depth exceeding the electric double layer thick- where τ is the pulse duration. ness arises as a result of the photoinduced unipolar The first equation gives the number of radiation- injection of charge carriers at the surface and the disso- α τ ciation of dye molecules. This leads to the appearance induced charge carriers n0' = aI0exp(Ð z) . The second of a photoinduced electric field Ez. Photogeneration of equation allows us to determine variation of the the volume charge also results in the electric double induced charge density with time after the laser pulse: layer destruction and more intensive growth of the cur- n' rent beginning with very small voltages, as can be seen n'(t) = ------0------. (5) in Fig. 2 (curves 1'Ð3'). γ n0' Rt + 1 γ ≈ × Ð13 3 Ð1 µ For the parameters R 5.7 10 cm s , t = 100 s DIFFRACTION GRATINGS IN LIQUID (the moment the maximum current amplitude is CRYSTALS ACTIVATED WITH POLYMETHINE ≈ × 14 Ð3 DYES reached), and n0' 0.7 10 cm (the maximum number of induced charge carriers under our experimental The orientational photorefractive nonlinearity in γ ≈ LCs induced by the pulsed radiation of the ruby laser conditions) we obtain the value n0' Rt 0.004, which is operating in the TEM00 mode (pulse energy E = 60 mJ, much less than unity. Taking into account Eqs. (1) and pulse duration τ ≈ 60 ns) was studied by means of (5), the photoinduced conductivity can be expressed as dynamic holography using the radiation of a HeÐNe σ'(t) ≈ e(µ+ + µÐ)a αI exp(Ðαz)z. (6) laser as a probe. The geometry of experiment in which 0 0 the direction of the lattice vector q coincides with that σ α α Here, the function '{ I0exp(Ð d)} is in fact close of the director L (see Fig. 4a) was selected from a num- to linear, which is illustrated in Fig. 3 (curves 1'Ð3'). ber of various possible geometries. If the polarization We observed the phenomenon of photocurrent exci- vectors of the pumping waves E1 and E2 and the prob- tation in the absence of any external static electric field. ing wave E3 are parallel to the director L, the diffraction This effect qualitatively differs from that discussed efficiency of the gratings is 2Ð3 times greater than in || ⊥ above. In the presence of an external field, the photo- the case of E1, 2 L, E3 L and an order of magnitude ⊥ current polarity follows the polarity of the applied volt- larger than that for E1, 2 L. The director is mainly age, while without the field, the polarity of the signal on reoriented in the xz plane. the load resistor RH is determined by the direction of the The radiation field generates charges inhomoge- incident light beam. The photocurrent flows from the neously distributed in space along the x-axis (with a illuminated electrode to the dark one (i.e., cations move grating constant Λ) and along the z-axis (Fig. 4b). In the from the bulk of the crystal toward the input electrode). absence of an external static electric field, a photoin- High-power laser radiation (~106 W/cm2) may give duced field Ez arises in the region of diffraction maxima rise to the injection of electrons from the electrode (this along the z-axis and the diffusion of induced charges is effect was observed experimentally) and to the genera- observed along the x-axis with a lattice relaxation time τ Λ2 π2 tion of charges on the surface of silicon oxide films. D = /4 D. Weak electric current accompanied by Combined thermal and field effects of the laser radia- the gradient of positive ion concentration along the tion will result in the accumulation of negative charge z direction might cause isotropic electrohydrodynamic at the film surface [20]. This process is favored by the instability. The mechanism of this instability formation existence of local energy levels in the bandgap of insu- under conditions of unipolar injection was considered lating SiO film, by the light absorption in the film, and by Felici [21]. Figure 4b shows a model describing the by the injection of charge carriers from the electrode. mechanism of instability in our case. Due to inhomoge- Recombination of the free charge carriers and cations neous charge density distribution, both in the laser of the solution at the relief film surface [15] will cause beam direction and along the substrate, an electrostatic δρ the formation of radical ions of dye molecules. These force F = Ez arising in the bulk of the layer will cause ions can settle as defects on the surface, which is one of the formation of two vortices over a length equal to the the possible reasons for the formation of disclinations, grating constant Λ. Rotation of the director as a result bounding the regions with different orientations of the hydrodynamic flow will have a period close to

TECHNICAL PHYSICS Vol. 45 No. 7 2000 862 SERAK et al.

(a) the cell thickness [22]. The threshold voltage of insta- Ð1 0 +1 bility of an LC with ∆ε > 0 may be lower than that of a electrohydrodynamic instability of the anisotropic type Ez E0 according to the KarrÐHelfrich model [16]. In the presence of a static electric field E0, the inter- z ference field of the laser radiation gives rise to the photoinduced field Ex (Fig. 4b) due to anisotropic conductivity (the conductivity is equal to σ|| along the x-axis and to σ⊥ along the z-axis. In accordance with the KarrÐ y Helfrich model, the field Ex is determined by the for- x mula [23] Ex (σ|| Ð σ⊥)cosΘsinΘ E1 E2 Ex, σ = Ð------2------2------E0, (7) σ|| cos Θ + σ⊥ sin Θ E3 Θ (b) where is the angle of director deviation in the xz plane. + Electrohydrodynamic instability of the anisotropic + + + + + ++ + + type arises if there is an initial disturbance of the direc- + + + + + + Ez E0 + Ð + tor orientation in the layer, which can be induced by an + Ð + isotropic instability. The rotation of the layer director is + + Ð enhanced when the direction of the static field E0 coin- + Ð + cides with the direction of the induced field Ez. The + Ð + period of sinusoidal deformation of the director remains the same and is close to the layer thickness. Ð Typical oscillograms of the first-order diffraction in Ex Ex the LC cells oriented with SiO without electrodes are presented in Fig. 5. Two types of the characteristic relaxation time are observed, the first being the thermal Fig. 4. Schematic diagrams showing (a) the interaction geometry and (b) a model of the electrohydrodynamic insta- τ ρ 2 Λ2 π2λ diffusion time T = 0 C p /4 T (equal to 0.5 ms, see bility formation in LCs activated with polymethine dyes. Fig. 5a) and the second, the orientational time, which is τ determined essentially by the charge diffusion D = Λ2/4π2D (Figs. 5bÐ5f). Typical values of the LC param- ρ 3 3 χ × Ð2 (a) (d) eters are as follows: 0 = 10 kg/m , = 7 10 kg/(m s), λ Ð6 2 Cp = 1500 J/(kg K), 1 = 0.16 J (K s m), D = 10 cm /s. For a grating constant of Λ = 40 µm the time constants τ τ T and D are equal to 0.5 ms and 400 ms, respectively, in good agreement with experiment. The orientational component is observed after vanishing of the thermal (b) (e) component and has a buildup period of 20 ms (a), while the relaxation time varies from 0.4 to 1 s. The time parameters of the holograms depend not only on the lattice constant and properties of LC media, but on the conditions of orientant deposition and the laser radia- (c) (f) tion intensity as well, which results in the formation of both dynamic and static gratings. A typical oscillogram of variation of the probing beam of a rectangular shape in the case of equilibrium of the hydrodynamic and orientational moments is shown in Fig. 5d. A grating constant Λ close to 40 µm is the optimum value for effec- div tive gratings (Fig. 6). The diffraction efficiency η grows with increasing radiation intensity and absorption coef- α 3 Fig. 5. Osscillograms of the first-order diffraction intensity ficient, reaching a maximum at I0 = 16 J/cm (Fig. 7a). α η for the probing beam of a HeÐNe laser in LC cells oriented With the further growth of I0, the value of drops as with (aÐe) silicon oxide and (f) polyimide. Time scale a result of convective motions in the bulk of the cell. (ms/div): (a) 0.1, (b) 5, (cÐf) 100; α = 14cmÐ1; (e) lattice relaxation in the presence of electric field U = 1.2 V (higher Heating of the liquid crystal does not affect the diffrac- amplitude) and in the absence of the field. tion efficiency but results in a decrease of the relaxation

TECHNICAL PHYSICS Vol. 45 No. 7 2000 SHORT-LASER-PULSE-INDUCED PHOTOELECTRIC PHENOMENA 863

η, % both η and conductivity. Here the value of η is approx- 3.0 imately three times greater than the value without bias voltage. At U = 1.4 V, which is close to the Fredericksz ≈ transition (for our LCs, the threshold voltage is UFr 2 V), η decreases, because the geometry of experiment 2.0 does not provide for the formation of electrohydrodynamic instability. The diffraction efficiency in the presence of electric field was equal to 10%. 1.0 As a result of comparative experiments involving nonionic dyes and other orienting substances, we may conclude that the reorientation described above has 10 20 30 40 50 60 remarkable features in the ionic compositions LC + dye Λ, µm with orienting SiO layers. Figure 5 compares orientational components for the LC cells with SiO and poly- Fig. 6. The plot of diffraction efficiency η versus grating imide orientants. The conductivity and the grating constant Λ. appear in the LC cells without a static electric field and are only enhanced by the external field. Apparently, we initially observe an isotropic electrohydrodynamic τ χΛ2 π2 ° time Θ = /4 K from 600 (T = 20 C) to 150 ms instability with a lower threshold and then the instabil- ° (T Ð Tis = 1 C), which corresponds to the reduction in ity of anisotropic character, which is similar to the opti- viscosity χ (in the isotropic phase, this time is 50 ms). cal KarrÐHelfrich effect in liquid crystals. Our experi- The maximum diffraction efficiency of the gratings was mental conditions provide for the observation of these equal to 3Ð4%. effects [22]. Indeed, the efficiency of gratings in planar layers is maximum when the grating constant Λ is close The efficiency of the gratings can be increased using to the cell thickness. In addition, the cells exhibit a pos- an additional static field. Typical oscillograms of the itive anisotropy of conductivity (σ|| ≈ 1.5σ⊥). Finally, diffraction beam intensity with and without voltage the effect in LCs with positive dielectric anisotropy across the cell electrodes are presented in Fig. 5e. Here, (∆ε > 0) is observed at voltages U < U . Dynamics of no relaxation of the cell to the initial state is observed Fr the orientational grating shows that reorientation because of a static grating formation. The character of occurs within a period of ~20 ms simultaneously with the diffraction efficiency variation as function of the α the photocurrent relaxation (10 ms) and the onset of absorbed radiation I0 is not altered in the presence of macroscopic mobility in the liquid crystal volume. the field (Fig. 7b). The results of our study reveal several subsequent stages in the variation of η with Thus, the photogeneration of surface and bulk increasing voltage. The efficiency does not change con- charges in the LC cell, the distribution of these charges siderably in the region below 0.6…0.8 V, where the in space in the beam direction and along the grating conductivity and also exhibits no significant variation vector, and the interaction of photoinduced fields with (Fig. 2). The range 0.6…1.3 V reveals the growth of the applied static field result in the phenomena of elec-

η, % η, % 5.0 5.0 (a) (b) 2' 4.0 4 4.0

3.0 3.0 3 2.0 2.0 1' 2 1.0 1.0 1 0 10 20 30 0 0.5 1.0 1.5 2.0 α –3 I0, J cm U, V

α α Ð1 Fig. 7. The plots of diffraction efﬁciency versus (a) parameter I0 and (b) voltage for = 14 (1), 48 (2, 1'), 154 cm (3, 4, 2') and U = 0 (1Ð3), 1.2 V (4).

TECHNICAL PHYSICS Vol. 45 No. 7 2000 864 SERAK et al. trohydrodynamic instability and effective reorientation 8. S. V. Serak and A. A. Kovalev, Mol. Cryst. Liq. Cryst. of the LC layer director. 320, 417 (1998). 9. E. V. Rudenko and A. V. Sukhov, Zh. Éksp. Teor. Fiz. 105, 1621 (1994) [JETP 78, 875 (1994)]. ACKNOWLEDGMENTS 10. I. C. Khoo, H. Li, and Y. Liang, Opt. Lett. 19, 1723 Authors gratefully acknowledge discussions with (1994). S.P. Zhvavyœ and N.A. Usov and the valuable assistance 11. I. C. Khoo, Mol. Cryst. Liq. Cryst. 282, 53 (1996). of E.A. Tyavlovskaya in the study of X-ray photoelec- 12. R. Macdonald, P. Meindl, G. Chilaya, and tron spectra of SiO ﬁlms. D. Sikharulidze, Mol. Cryst. Liq. Cryst. 320, 115 (1998). 13. S. Sato, Jpn. J. Appl. Phys. 20, 1989 (1981). The work was supported by the International 14. I. C. Khoo, B. D. Guenther, and S. Slussarenko, Mol. INTAS-Belarus foundation (grant no. 0635). Cryst. Liq. Cryst. 321, 419 (1998). 15. A. A. Kovalev and S. V. Serak, Proc. SPIE 3318, 327 REFERENCES (1998). 16. L. M. Blinov, Electro- and Magnetooptics of Liquid 1. V. A. Pilipovich, A. A. Kovalev, G. L. Nekrasov, and Crystals (Nauka, Moscow, 1978). S. V. Serak, Dokl. Akad. Nauk BSSR 22, 36 (1978). 17. P. de Genues, The Physics of Liquid Crystals (Claren- 2. G. L. Nekrasov, Yu. V. Razvin, and S. V. Serak, Optical don, Oxford, 1974; Mir, Moscow, 1977). Data-Processing Techniques, Ed. by V. A. Pilipovich 18. S. N. Koœkov, Physics of Dielectrics (Leningrad (Nauka i Tekhnika, Minsk, 1978). Politekh. Inst., Leningrad, 1967). 3. A. A. Kovalev, G. L. Nekrasov, and S. V. Serak, Mol. 19. A. A. Kovalev, G. L. Nekrasov, and S. V. Serak, Zh. Prikl. Cryst. Liq. Cryst. 193, 51 (1990). Spektrosk. 45, 400 (1986). 4. A. A. Kovalev and S. V. Serak, Zh. Prikl. Spektrosk. 52, 20. Handbook of Thin Film Technology, Ed. by L. I. Maissel 197 (1990). and R. Glang (McGraw-Hill, New York, 1970), Vol. 2. 5. A. A. Kovalev, S. V. Serak, G. L. Nekrasov, et al., Kvan- 21. N. J. Felici, Rev. Gen. Electr. 78, 717 (1969). tovaya Élektron. (Moscow) 22, 838 (1995). 22. S. A. Pikin, Structural Transformations in Liquid Crys- 6. A. A. Kovalev, S. V. Serak, and G. L. Nekrasov, Mol. tals (Nauka, Moscow, 1981). Cryst. Liq. Cryst. 320, 425 (1997). 23. W. Helfrich, J. Chem. Phys. 51, 4092 (1969). 7. G. L. Nekrasov and A. A. Kovalev, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. 3, 105 (1980). Translated by S. Egorov

TECHNICAL PHYSICS Vol. 45 No. 7 2000

OPTICS, QUANTUM ELECTRONICS

Analogs of the Brewster Effect and Total Internal Reflection for Cylindrical Waves M. A. Kaliteevskiœ and V. V. Nikolaev Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received July 6, 1999

Abstract—Analogs of the Brewster effect and total internal reflection were investigated for cylindrical waves passing through a cylindrical interface. It was found that, in the case of cylindrical geometry, the Brewster effect is also observed but weakens with decreasing moment of momentum of a cylindrical wave. Wave reflection from a small-radius cylindrical interface is analyzed. Asymptotic expressions for the reflection factor of a cylindrical wave are obtained when the radius of the interface tends to zero. It was found that, as the radius the interface decreases, the reflection factor of a cylindrical wave with a nonzero moment of momentum approaches unity from the left but does not reach this value. © 2000 MAIK “Nauka/Interperiodica”.

.INTRODUCTION ratio between the tangential electric field components The incidence of a plane light wave on the planar in the reflected and incident waves) for the E- and H- interface between two media with refraction indices n polarized cylindrical light waves scattered from a cylin- 1 drical interface of radius ρ is given by [7] and n2 seems to be the physical phenomenon that has () () been investigated in most detail [1, 2]. The Brewster n C 1 Ð n C 1 effect (the reflection factor is exactly equal to zero r = ------2 m2 1 m1 , (1) θ d ()2 ()2 when a p-polarized wave is incident at an angle BR that n1Cm1 Ð n2Cm2 satisfies the relationship tanθ = n /n ) and total BR 2 1 and internal reflection (the reflection factor is exactly equal ()1 ()1 to unity when n1 > n2 and the angle of incidence n /C Ð n /C θ 2 m2 1 m1 exceeds the critical value sin tot = n2/n1) are finding rd = ------()2 ()1- , (2) wide application in technology. In the last few years, n1/2Cm1 Ð n2/Cm2 designers have been trying to use cylindrical structures ()12, ()12, ' ρ ()12, ρ in various optoelectronic devices [3, 4]. Several theo- respectively. Here, Cml = Hm (nlK )/Hm (nlK ), retical papers are devoted to the propagation of cylin- ()12, ρ ω drical waves in laminated cylindrical structures and Hm (nlK ) is the Hankel function, K = /c, c is the through a cylindrical interface [5Ð7]. However, a num- speed of light, and the derivative sign means differenti- ber of intriguing features, such as an increase in the ation with respect to the whole argument and not just reflection factor when the radius of the interface is to ρ. small and the dependence of optical properties on the Figures 2 and 3 show the variations of the phase and light polarization, call for further investigation. In this squared absolute value of the reflection amplitude fac- paper, we try to contribute to the study of these effects. tor for the E-polarized cylindrical wave scattered from the interface between media with refraction indices of RESULTS AND DISCUSSION 1.0 and 3.0. In the figures, the three curves correspond to moments of momentum m = 0, 2, and 5. Figure 2 Consider a cylindrical light wave with a frequency ω illustrates the situation when the external medium is that propagates perpendicularly to the axis of structure optically denser, while in Fig. 3, the situation is symmetry (z-axis) and is incident on the cylindrical reversed. Figures 4 and 5 present the same for the H- interface between two media with refraction indices n1 polarized wave. (internal medium) and n2 (external medium). The In a cylindrical wave, Poynting’s vector is radially media are assumed to be infinite along the z-axis directed when m = 0. When the moment of momentum (Fig. 1). Cylindrical waves of arbitrary polarization can is nonzero, the angle Θ between Poynting’s vector and be represented as a superposition of E (electric field E the radius depends on ρ and satisfies the approximate is parallel to the axis of symmetry) and H (magnetic relationship field H is parallel to the axis of symmetry) waves [7]. Θ ρ The reflection amplitude factor rd (introduced as the sin =m/nK , (3)

866 KALITEEVSKIŒ, NIKOLAEV

(a) (b)

Θ Θ 1 2 Θ 1 Θ 2 ρ

ρ n1

n2 n1 n2

ρ π Fig. 1. Trajectory of energy transfer in a refracted cylindrical wave for = 0.55*(2 /K) and m = 5: (a) n1 = 3 and n2 = 1 and (b) n1 = 1 and n2 = 3.

π π/4 1 (a) (a) 0 2 3 2

ϕ ϕ 3 1 19π/20 –π 1 1 3 (b) (b) 2 3 2 R R

1 1

0 0.5 1.0 1.5 0 0.5 1.0 1.5 ρK/2π ρK/2π

Fig. 2. (a) Phase and (b) square of the absolute value of the reﬂection amplitude factor for the E-polarized scattered cylindrical wave vs. interface radius. n1 = 1, n2 = 3, and m = (1) 0, (2) 2, and (3) 5. Fig. 3. The same as in Fig. 2 for n1 = 3 and n2 = 1.

The azimuth component of Poynting’s vector being As ρ decreases, the angle Θ determined by (3) proportional to m/ρ. Within a small portion of the inter- increases. When n1 > n2 and a light wave comes from face, one could consider an incident cylindrical wave as a the internal medium, total internal reflection is ρ ρ Θ Θ plane wave incident on the plane interface at an angle Θ. observed if = tot and = tot. The critical value of ρ ρ This, however, may yield qualitatively false results, as the radius tot is determined by the formula K tot = demonstrated below. m/n2. When n1 < n2, total internal reflection must be absent. As ρ increases, the azimuth component of Poynt- Θ The above considerations are in conflict with the ing’s vector vanishes and angle approaches zero; i.e., following properties of cylindrical waves. First, using the incidence of a cylindrical wave on a cylindrical (1) interface of large radius is equivalent to the normal (1), one can show (taking into account that Cml is the (2) incidence of a plane wave on a planar interface [7]. complex conjugate to Cml ) that the reflection intensity Indeed (Figs. 2Ð5), for large ρ’s, the square of the factor does not depend on the sequence of layers, as in reflection factor magnitude tends to 25% and its phase the case of the normal incidence of a plane wave on a approaches zero or π, depending on medium alterna- planar interface (Figs. 2Ð5) [7]. Second, after the wave tion, for both polarizations. crosses the interface, Θ determined by (3) does not

TECHNICAL PHYSICS Vol. 45 No. 7 2000 ANALOGS OF THE BREWSTER EFFECT 867 π 10π/9 (a) (a) 1 ϕ ϕ 1 2 3 2 3 8π/9 –π/3 1 1 (b) 3 (b) 2 1 R 1 2 3 R

0 0.5 1.0 1.5 0 0.5 1.0 1.5 ρK/2π ρK/2π

Fig. 4. The same as in Fig. 2 for the H-polarized cylindrical Fig. 5. The same as in Fig. 3 for the H-polarized cylindrical wave. wave.

Θ depend on the refraction index of the layer from which to the Brewster angle BR. It is easy to check that this the wave comes. However, the angles Θ on both sides “Brewster radius” satisfies the relationship of the interface are formally related by Snell’s law [1, 2] [see (3)]. Kρ = m n Ð2 + n Ð2. (4) In fact, when the radius ρ of the interface decreases BR 1 2 and reaches a certain m-dependent characteristic value Figures 4 and 5 demonstrate that, when m ≠ 0, the (m is assumed to be nonzero), the reflection intensity dependences of the reflection factor for the H-polarized factor starts increasing, asymptotically approaching wave on interface radius have minima located close to unity from the left (see Figs. 2Ð5 and Appendix). This those predicted by (4). This fact enables us to argue that phenomenon, briefly described in [5], was called total these minima reflect the Brewster effect. In the minima, reflection at small radius. It was indicated that the − m + 1 the reflection factor is other than zero but the minimum reflection amplitude factor is equal to ( 1) i at a for m = 5 is deeper than for m = 2. This can be explained small (but nonzero!) radius. According to our calcula- if we take into account that the Brewster effect is a tions, the absolute value of the reflection factor, while result of the transversality of electromagnetic waves approaching unity as ρ tends to zero, always remains ρ [1, 2]. Hence, this effect must be weaker in the case of less than unity at any nonzero . The phase of the cylindrical waves, whose Poynting’s vector is locally reflection factor behaves in a quite different way, π transverse. As the moment of momentum m increases, approaching . A small variation of the reflection factor the Brewster radius grows, the curvature of the surface magnitude from unity may affect the observed optical decreases, the situation more and more resembles the properties of an actual cylindrical structure only plane case, and the minimum becomes deeper. slightly. However, being involved in the phase of the reflection factor, such an error radically changes the Our analysis demonstrates that, for cylindrical calculated spectrum of the optical eigenmodes in the structures, a phenomenon similar to the Brewster effect cylindrical system. is observed but, instead of total internal reflection, the Thus, total internal reflection is not observed for reflection factor grows at a small radius. cylindrical waves propagating perpendicularly to the axis of symmetry, which can be explained in simple physical terms.1 In the plane case, when total internal APPENDIX reflection occurs, light passes through the interface into the lower index medium, turns, and comes back to the Asymptotics of the Reflection Factor higher index medium. Such is not the case in cylindri- for the Small-Radius Interface cal systems, because the curvature of the interface exceeds that of the light “trajectory.” The reflection factor of the E-polarized wave scattered from the interface between media 1 and 2 is rep- For H-polarized waves, one can find, for every m, a ρ Θ resented by radius BR such that determined by (3) will be equal (1) (1) 1 n C Ð n C Note that the situation under consideration has nothing to do with ----2------m---2------1------m---1- light propagation in fibers, where total internal reflection takes r = (2) (1) . place for light waves propagating along the axis of symmetry. n1Cm1 Ð n2Cm2

TECHNICAL PHYSICS Vol. 45 No. 7 2000 868 KALITEEVSKIŒ, NIKOLAEV Replacing the Hankel functions by combinations of wave can be written in the form the Bessel (J ) and Weber (Y ) functions in the formula m m 2n G(n Kρ) (1, 2) r = Ð 1 Ð i------1------1------for Cm , one can use the approximate expressions ( ρ ρ ) n1F(n1K ) Ð n2F(n2K ) (( ρ )2 ρ ρ ) ' ' '  ' '  n1G(n1K ) + n1G(n1K )*n2G(n2K ) (1) Jm + iYm Ym Ym Jm Jm + 2------C = ------≅ ------+ i------Ð ------ 2 (9A) m 2 ( ( ρ) ( ρ)) Jm + iYm Y  Y  n2F n2K Ð n1F n1K m Ym m (1A) G 3 J' J Y' J 2 + O---  . + ----m------m-- Ð ----m------m--, F Y 2 Y 3 m m At m = 0, this yields for the reflection amplitude (r) and intensity (R) factors and iln(n /n ) lim (r) = ------1------2------(10A) ρ → π ( ) ( ) J' Ð iY' Y' Y' J J'  0 Ð iln n1/n2 C 2 = ----m------m- ≅ ----m-- Ð i----m------m- Ð ----m-- m 2 Jm Ð iYm Y  Y  and m Ym m (2A) 2 ( ) 2 ln n2/n1 Jm' Jm Ym' Jm lim (R) = ------, (11A) + ------Ð ------, ρ → 0 π2 2 ( ) 2 3 + ln n2/n1 Ym Ym respectively.2 as the argument of the functions approaches zero. This For m = 1, is valid, since, at small arguments, the Weber function tends to infinity, while the Bessel function approaches πn2 (1, 2) r = Ð 1 + i------1------zero (or unity when m = 0). The formula for Cml can ( 2 2) ( ρ) n1 Ð n2 ln K (1, 2) (12A) be written in the form Cm (x) = F(x) + iG(x). When π2(n 4 + n 2n 2) m = 0, F(x) and G(x) are defined by + ------1------1-----2------+ O(Kρ) ( 2 2)2 2 ( ρ) 2 n2 Ð n1 ln K 2i 1 F(x) = Re ------+ O(x), (3A) π + 2i(ln(x/2) + γ ) x and π2 2 2 n1 n2 (( ρ)2) 2i 1 R = 1 Ð ------+ O K . (13A) G(x) = Im ------+ O(x). (4A) ( 2 2)2 2 ( ρ) π + 2i(ln(x/2) + γ ) x n2 Ð n1 ln K For m ≥ 2, For m = 1, π( ) 2m m Ð 1 n1 ( ρ)2m Ð 2 1 r = Ð 1 + i------K F(x) = Ð -- Ð xlnx + O(x), (5A) (m Ð 1)!222m Ð 3(n 2 Ð n 2) x 2 1 π2( )2 ( 4m 2m 2m) (14A) m Ð 1 n1 + n1 n2 ( ρ)4m Ð 4 π π 3 5 + ------K G(x) = -- x + --(2ln(x/2) Ð 1 + 2γ )x + O(x ), (6A) (m Ð 1)!424m Ð 5 ( 2 2)2 2 4 n2 Ð n1 + O((Kρ)6m Ð 6) where γ is the Euler constant. and For m ≥ 2, π2( )2 2m 2m m Ð 1 n1 n2 ( ρ)4m Ð 4 1 1 3 R = 1 Ð ------K F(x) = Ð m -- + ------x + O(x ). (7A) ( ) 4 4m Ð 6( 2 2)2 (15A) x 2(m Ð 1) m Ð 1 ! 2 n2 Ð n1 + O((Kρ)8m Ð 8). π 2m Ð 1 2m + 1 G(x) = ------x + O(x ). (8A) The reflection amplitude factor of the H-polarized (m Ð 1)!222m Ð 1 2 For m = 0, one should prefer exact formula (1) rather than the The reflection amplitude factor of the E-polarized asymptotic approximation, which is too awkward.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 ANALOGS OF THE BREWSTER EFFECT 869 wave can be represented in the form For m ≥ 2, πn 2m + 1n Ð1 ( ρ) 1 2 ( ρ)2m 2n2 G n1k0 r = Ð 1 Ð i------K r = Ð 1 Ð i------(m Ð 1)!2m22m Ð 2(n 2 Ð n 2) ( Ð1 ( ρ) Ð1 ( ρ)) 2 1 n1 F n1k0 Ð n2 F n2k0 (21A) π2( 4m 2m 2m) 2 2 2 n1 + n1 n2 n1 n2 4m 6 ((nÐ1G(n Kρ)) + n Ð1G(n Kρ)*n Ð1G(n Kρ)) + ------(Kρ) + O((Kρ) ) + 2------2------1------1------1------2------2------( ) 4 2 4m Ð 3( 2 2)2 2 m Ð 1 ! m 2 n2 Ð n1 (n Ð1F(n Kρ) Ð n Ð1F(n Kρ)) 2 2 1 1 (16A) and G 2 π2 2m + 2 2m + 2 + O --- , n1 n2 4m  F  R = 1 Ð ------(Kρ) ( ) 4 2 4m Ð 4( 2 2)2 m Ð 1 ! m 2 n2 Ð n1 (22A) which, at m = 0, yields [9] 8m + O((Kρ) ). lim (r) = 1 (17A) ρ → 0 ACKNOWLEDGMENTS and This work was supported by the Russian Foundation for Basic Research and Scientiﬁc and Technical Pro- lim (R) = 1. (18A) gram “Nanostructures.” ρ → 0 REFERENCES For m = 1, 1. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1969; Nauka, Moscow 1973). 3 πn n 2. N. I. Kaliteevskiœ, Wave Optics (Nauka, Moscow, 1971). r = Ð 1 Ð i------1------2----(Kρ)2 ( 2 2) 3. T. Egrogan and D. G. Hall, J. Appl. Phys. 68, 1435 n2 Ð n1 (1990). (19A) 4. D. Libilloy, H. Benisty, C. Weisbuch, et al., Appl. Phys. π2 4 2( 2 2) n1 n2 n2 + n1 4 6 Lett. 73, 1314 (1998). + ------(Kρ) + O((Kρ) ) ( 2 2)2 5. Y. Jiang and J. Hacker, Appl. Opt. 33, 7431 (1994). 2 n2 Ð n1 6. A. A. Tovar and G. H. Clark, J. Opt. Soc. Am. A 14, 3333 (1997). and 7. V. V. Nikolaev, M. A. Kaliteevskiœ, and G. S. Soko- lovskiœ, Fiz. Tekh. Poluprovodn. (St. Petersburg) 33, 174 π2n 4n 4 (1999) [Semiconductors 33, 147 (1999)]. R = 1 Ð ------1------2----(Kρ)4 + O((Kρ)8). (20A) ( 2 2)2 n2 Ð n1 Translated by I. Eﬁmova

TECHNICAL PHYSICS Vol. 45 No. 7 2000

Technical Physics, Vol. 45, No. 7, 2000, pp. 870Ð877. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 57Ð64. Original Russian Text Copyright © 2000 by Demidov, Efremov, Ivkin, Ivonin, Petrov, Fortov.

OPTICS, QUANTUM ELECTRONICS

Evolution of the Glow of an Aerogel Irradiated with a High-Power Pulse Electron Beam B. A. Demidov, V. P. Efremov, M. V. Ivkin, I. A. Ivonin, V. A. Petrov, and V. E. Fortov Russian Research Centre, Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received July 6, 1999

Abstract—The evolution of the glow of the energy-release zone in porous transparent aerogel, with a density of 0.03Ð0.25 g/cm3, which is irradiated by a high-power pulse electron beam, is studied experimentally. In τ ≤ τ τ ≈ Ð6 τ ≈ × addition to a fast ( beam) and a luminescent ( 10 s) glow components, a slow glow component ( 2 10Ð5 s) has been revealed. The appearance of this slow component is associated with an aerogel rarefaction wave and its destruction (cracking) arising after its isochoric bulk heating by electron radiation, which may occur due to an electrostatic ﬁeld induced under irradiation. The discovered glow is utilized to visually determine the current position of the rarefaction wave front. The sound velocity measured as a function of the density of SiO2 aerogels with porosities of 10Ð100 allowed us to experimentally determine the percolation parameter of the aerogel equation of state. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION Upon impact compression of porous aerogels, the temperature behind the shock wave front with an Porous condensed media are widely used to solve amplitude of about 10 kbar is large enough to cause an numerous important problems in science and technol- intense glow. In this case, the glow front position virtu- ogy. They are promising materials for damping short- ally coincides with the position of the shock wave front, term impact loads, are used as radiation converters on which is used for determining the shock adiabat of the heavy-ion accelerators, and are included in targets for weakly ionized plasma that appears [13]. However, the inertial thermonuclear fusion [1, 2]. most interesting physical effects manifest themselves in studies of aerogels under pressures of 1 to 50 kbar, Shock-induced compression of porous substances when aerogels preserve the features of their internal has long been used in shock-wave physics for con- structure. In this loading range, an aerogel with a high structing equations of state [3, 4]. The effects of intense optical transparency is able to emit light in the visible energy fluxes on condensed substances are being inves- region under irradiation by an intense electron beam. tigated in many laboratories [5, 6], but the analysis of This allows one to use convenient optical diagnostics the action of pulse radiation fluxes on porous media is for obtaining information on the equation of state of an incomplete; the models for describing such media aerogel with a porosity that changes several times under isochoric heating and formation of shock waves under rarefaction [8]. A nonlinear self-consistent equa- under these conditions have not been considered. tion of state representing fractional properties of highly The interaction of a high-power high-current pulse porous materials was modeled in precisely this way. In particular, this equation allowed one to calculate the electron beam with a highly porous material (SiO2 aerogels) was studied in [7Ð9]. In these experiments, velocities of the flying-apart irradiated and rear aerogel the aerogel was a homogeneous medium, because the parts, which differ by two orders of magnitude, under typical irradiation and rarefaction time was sufficient irradiation with a high-power pulse electron beam [9]. for the pressure in the aerogel structural elements to be The objective of this study is to analyze the glow equalized. In experiments performed in [10], the inter- evolution along the beam radius and into the depth of action of high-power pulse laser radiation with an an aerogel irradiated by a high-power pulse electron “Agar-agar”-type (C14H18O7) low-density porous sub- beam, as well as to investigate the interaction of the stance (irregularly alternating cavities and particles of electron beam with targets of complex configurations various shapes and normal solid-state density) was ana- consisting of several aerogel layers of different densi- lyzed. Therefore, under pulse irradiation [10], “Agar- ties. agar” is a heterogeneous medium. The absorption of intense radiation fluxes, energy transfer mechanisms, This work also aims to study the sound velocity as a and hydrodynamic processes in homogeneous and het- function of the density of porous SiO2 aerogels with a Π ρ ρ erogeneous porous media have specific features and are porosity = 0/ Ð 1 in a porosity range of 10Ð100 in currently attractive to researchers [11, 12]. order to directly verify the power dependence of the

EVOLUTION OF THE GLOW OF AN AEROGEL IRRADIATED 871

Monochromator X-ray detector

Electron beam Aerogel

Aluminium Laser foil

Spall particle Aluminium detector foil

To interferometer Streak Camera camera

Fig. 1. Layout of the diagnostic equipment. sound velocity in an aerogel on its porosity and to sion spectrum under study; a spall particle detector for directly determine the percolation coefficient that can measuring the velocity of split-off fragments by using be used in the equation of the aerogel state [9]. the time-of-flight technique; an X-ray detector for reg- istering the instant of the interaction between the electron beam and aerogel; and cameras for recording the EXPERIMENTAL CONDITIONS trajectories of flying-away glowing fragments in the The behavior of transparent porous SiO2 aerogels integral regime. The velocity of the elastic disturbance with a density ρ = 0.03Ð0.25 g/cm3 under fast isochoric propagation in the aerogel was measured under modi- heating was studied experimentally on the Kal’mar fied experimental conditions. The anode aluminum foil electron accelerator operating with the following was replaced by a 4.5-mm-thick stainless steel disk parameters of the electron beam: the current is I = 10Ð with a diameter of 90 mm, to which an aerogel sample 15 kA, the electron energy is U = 270Ð300 keV, and the was attached by a special holder. A pulse electron beam current pulse FWHM duration is τ = 150 ns. Aerogel excited a short pressure pulse in the metallic anode, samples 5Ð34 mm thick with cross sections of 25Ð which was transmitted to the aerogel sample and prop- 50 mm were exposed to the electron beam. The sample agated in it as an elastic disturbance. The instant at under study was placed in a vacuum chamber of the which this disturbance reached the free aerogel surface Kal’mar accelerator behind the anode foil. Figure 1 was recorded by a differential laser interferometer. shows the arrangement of the diagnostic equipment. An electron beam with an effective diameter of 11Ð EXPERIMENTAL RESULTS 12 mm passed through the 10-µm-thick aluminum foil (the accelerator anode) and hit the sample positioned The time dependence of the aerogel glow along the immediately adjacent to the foil or at a 5-mm distance radius is shown in Fig. 2 presenting photochronograms from it. A 10-µm-thick aluminum foil set up on the rear of the interaction process between the electron beam 2 of the aerogel served to reflect a diagnostic laser beam. (j = 15 kA/cm , U = 300 keV) and aerogels of different The following diagnostic devices were used: an FÉR-7 densities. The entrance slit of the FÉR-7 streak camera streak camera with slit scanning that allowed us to was positioned perpendicular to the electron beam axis obtain a time-scanned aerogel glow intensity distribu- and, in both cases, was focused 2Ð3 cm from the irradi- tion in the radial and longitudinal directions; a differen- ated sample surface. tial laser interferometer (based on an LGN-215 laser) The dynamics of electron beam interaction with lay- with an interference period of 26 m/s that records the ered aerogel targets is illustrated by the photochrono- time of arrival and mass velocity of a disturbance at the grams in Fig. 3 (here, the FÉR-7 entrance slit is parallel rear surface of the aerogel; an MDR-2 monochromator to the electron beam axis). The photochronogram in with a special cassette extending the range of the emis- Fig. 3a refers to the action of the electron beam (j =

TECHNICAL PHYSICS Vol. 45 No. 7 2000

872 DEMIDOV et al.

mm 2.5 µs 9

–9 (‡)

2.5 µs 9

–9 (b)

Fig. 2. Time-dependent aerogel glow along the radius of the zone of the electron-beam energy release; ρ = 0.25 (a), 0.034 (b) g/cm3.

15 kA/cm2, U = 300 keV) on a four-layer target with Figure 5a shows aerogel samples with a density of ρ ρ 3 the following aerogel densities: 1 = 0.08, 2 = 0.15, 0.25 g/cm with a cross section of 25 × 25 mm and an ρ ρ 3 3 = 0.05, and 4 = 0.15 g/cm with the corresponding initial thickness of 22.5 mm after being exposed to the 2 thicknesses h1 = 4, h2 = 4, h3 = 4, and h4 = 3 mm. The electron beam (j = 15 kA/cm , U = 300 keV). As a layers are numbered from the front (irradiated) target result of the separation of a split-off plate, the sample surface. thickness decreased by 5 mm. For a low-intensity irra- 2 Figure 3b shows a photochronogram for the interac- diation mode (j = 8 kA/cm , U = 250 keV), irreversible tion of the electron beam (with the same parameters) effects occur in the bulk of a similar aerogel sample, with a three-layer target at a faster scan. The target con- resulting in its turbidity. However, the spalling effect is ρ 3 absent (Fig. 5b) and the sample thickness remains sists of the following aerogel layers: 1 = 0.05 g/cm , ρ 3 ρ unchanged. h1 = 10 mm; 2 = 0.25 g/cm , h2 = 2.5 mm; and 3 = 3 0.05 g/cm , h3 = 9 mm. The velocity of ﬂying-apart fragments at constant The photochronogram in Fig. 3c shows the interac- parameters of the electron beam (j = 15 kA/cm2, U = tion dynamics between the electron beam and a three- 290 keV) depends heavily on the aerogel thickness and layer target, in which the ﬁrst layer has the highest den- density. For example, the velocity of spalled fragments ρ 3 ρ 3 3 sity ( 1 = 0.25 g/cm , h1 = 5 mm; 2 = 0.034 g/cm , h2 = of an aerogel sample with a density of 0.16 g/cm 10 mm; and ρ = 0.25 g/cm3, h = 5 mm). changes from 500 m/s at a 5.5-mm sample thickness to 3 3 80 m/s at a 14.6-mm thickness. The studies performed with the MDR-2 monochromator have shown that the aerogel emission from the Figure 6 shows a set of interferograms characteriz- zone of the electron-beam energy release spans the ing the propagation velocity of elastic disturbances in entire visible spectrum region. For example, Fig. 4 aerogels of various densities under the same parameters shows a part of the emission spectrum for an aerogel of the electron beam producing pressure pulses in the with ρ = 0.034 g/cm3 exposed to the electron beam (j = metallic anode of the accelerator. The densities of the 2 ρ ρ 10 kA/cm , U = 270 keV). The slit of the monochroma- aerogels under study were as follows: 1 = 0.02, 2 = ρ ρ ρ ρ tor is adjusted to the region of the electron-beam energy 0.057, 3 = 0.076, 4 = 0.13, 5 = 0.16, and 6 = release. The bright lines in the upper part of the spec- 0.25 g/cm3. The sample thicknesses were h = 22, h = trum are the reference lines of a mercury lamp. 1 2 34, h3 = 30, h4 = 18, h5 = 25, and h6 = 26 mm, respec- The electron beam interacts with the aerogel and tively (1Ð6 in Fig. 6). excites a shock wave with a velocity reaching 500Ð 600 m/s in dense aerogels. Arriving at the free aerogel The results of processing the interferograms are pre- surface, the shock wave causes a spall destruction. sented in Fig. 7 in the form of a plot of the propagation

TECHNICAL PHYSICS Vol. 45 No. 7 2000 EVOLUTION OF THE GLOW OF AN AEROGEL IRRADIATED 873

7.5 µs mm 15

0 (‡)

24 mm 2.5 µs

0 (b)

18 mm 7.5 µs

0 (c)

Fig. 3. Time-dependent aerogel glow along the depth of multilayer targets: (a) four-layer target and (b, c) three-layer target.

τ ≈ Ð7 velocity of an elastic disturbance as a function of the Fig. 2a. The first glow stage lasts 1 10 s; and the aerogel density. durations of the second and third stages are 10Ð6 and 2 × 10Ð5 s, respectively. The duration of the first glow stage coincides with that of the electron-beam action, ANALYSIS OF EXPERIMENTAL causing a glow through the excitation of aerogel atoms RESULTS and molecules (mainly by secondary electrons) and The comparison of the photochronograms shown in subsequent reemission in the visible spectrum region Figs. 2a and 2b allows us to conclude that the aerogel [14, 15]. Changes in the intensity of this glow along the density appreciably affects the character of the glow in target radius show that the electron-beam current den- the region of the electron-beam energy release. In con- sity decreases from the center to the periphery, making trast to Fig. 2b, three stages of the aerogel glow with it possible to estimate the effective electron-beam significantly differing durations are clearly observed in diameter (11Ð12 mm).

TECHNICAL PHYSICS Vol. 45 No. 7 2000 874 DEMIDOV et al.

0.435 µm 0.491 µm

Fig. 4. Integrated spectrum of the glow in the blue-green wavelength region in the zone of the electron-beam energy release.

‡ b

Fig. 5. Aerogel samples exposed to an electron beam: (a) high-intensity and (b) low-intensity irradiation modes.

The glow induced by the direct action of the elec- density ρ ≤ 0.2 g/cm3 [8] have revealed a strong influ- tron beam on optically transparent dielectric media can ence of the aerogel electrization on the formation of the be divided into two stages, taking into account the zone of electron-beam energy release. In experiments degree of its origin and decay delays. The first, most with high-porosity aerogels, the internal electrostatic intense glow stage has virtually no time lag (τ ≈ 10Ð12Ð field reaches the maximum possible threshold field Ð10 ≈ 10 s) and is directly associated with cascade cooling Eth 70Ð80 kV/cm for high-energy secondary elec- of high-energy secondary electrons with energies ε ≈ trons with energies of the order of the optical phonon ε ≈ τ ≤ ε ≈ ω ε g 10Ð15 keV. The second stage, with a time lag energy (h Ð g). Therefore, a mass rarefaction 1 µs, is determined by the slow relaxation of the excited wave propagating into the aerogel (≈1 cm) may lead to atoms and molecules with excitation energies of about an increase in the electrostatic electron energy, with a ε g and is responsible for the luminescence effect. subsequent excitation and reemission of atoms in the The third stage of the aerogel glow revealed in our optical wavelength region. experiments has the longest time lag and is of the great- When a highly porous aerogel was irradiated with est interest. This glow propagates in the radial direction an electron beam (Fig. 2b), the FÉR-7 entrance slit was from the central region of the target to its periphery adjusted to the beginning of the energy-release zone (in with a velocity of 500Ð550 m/s. The measured velocity contrast to the experiment in Fig. 2a). Therefore, the is close to the maximum propagation velocity of the aerogel rarefaction in the slit zone began immediately region with a glow duration τ ≈ 20 µs from the zone of after termination of the irradiation, and no pronounced the electron-beam energy release to deep into the target separation of the second and third glow stages was [9]. Hence, this glow coincides with the pattern of the observed. aerogel dynamic rarefaction, with the mass compres- The photochronograms shown in Fig. 3 also confirm sion wave propagating deep into the aerogel. a great influence of the induced electrostatic field on the The high bulk electrization of the aerogel, which glow evolution during the interaction between the elec- arises under its irradiation by a pulse electron beam, is tron beam and the aerogel. Experiments with a depth- evidently responsible for the appearance of the slow inhomogeneous aerogel directly point to the existence glow component. Experiments on recording the profile of internal rarefaction waves due to inhomogeneities of the fast glow component in an aerogel with the initial causing a prolonged glow of the aerogels' interfaces,

TECHNICAL PHYSICS Vol. 45 No. 7 2000 EVOLUTION OF THE GLOW OF AN AEROGEL IRRADIATED 875 which are clearly seen in Fig. 3. The glow duration of 1 the aerogels' interfaces in Figs. 3a and 3b is longer than that of the interfaces in Fig. 3c. This is explained by the 200 µs first layer in the layered targets in Figs. 3a and 3b being a highly porous aerogel, which ensures a higher electrization at the interface of aerogels and, correspondingly, 2 a longer glow. Note that the difference in the interface glow durations for various layered targets shows only an insignificant contribution of the plasma in the accel- 100 µs erator diode chamber to the glow. The integrated aerogel-emission spectrum pre- 3 sented in Fig. 4 is actually continuous. In the short- wavelength region, a system of bands typical of molecular spectra is observed. Fairly intense isolated Si lines, 50 µs which could help diagnose macroscopic parameters of the aerogel, are absent. In order to extract quantitative information from spectroscopic data, further experi- 4 mental and theoretical investigations are necessary. Figure 5 shows spalling phenomena in porous 50 µs media, which are nontrivial for such media. It is clear that such phenomena have a threshold effect that depends on the medium porosity. 5 Figure 7 shows a linear dependence of the propagation velocity of an elastic disturbance in the aerogel on its density. This velocity can be considered as the sound 50 µs velocity Cs, at least for comparatively dense aerogels. According to the cluster model of the equation of 6 state for a porous medium [16], thermodynamic parameters depend on the density according to a power law. 25 µs In particular, the sound velocity in the aerogel can be ρ ρ (γ Ð 1)/2 represented by the dependence Csi = Csk( i/ k) , ρ ρ Fig. 6. Interferograms illustrating the propagation velocity where i and k are the densities of different porous of an elastic disturbance in aerogels of various densities. aerogel samples and γ is the percolation coefficient characterizing the aerogel structure. For porous metals, the power law for the sound velocity and the percola- V, m/s tion coefficient were determined in [17]. The revealed 400 linear dependence of the sound velocity in the aerogel on its density confirms a power dependence of the sound velocity in the aerogel on its porosity and allows 300 us to directly determine the percolation coefficient, which was found to be equal to 3. The measured coefficient was close to the value γ = 3.2 utilized in [9] in 200 the numerical hydrodynamic simulation of an aerogel exposed to intense irradiation. 100 NUMERICAL SIMULATION In order to determine the induced electrization of 0 0.1 0.2 ρ, g/cm3 highly porous aerogels, we numerically calculated the interaction of the electron beam from the Kal’mar Fig. 7. Elastic disturbance velocity in an aerogel as a func- accelerator with aerogels. tion of its density. When an electron beam is absorbed by an aerogel, strong electric fields arise in it, leading to its breakdown and a charge flow from the volume of the energy- requires an allowance for the high-energy conductivity release zone to the aerogel surface. These electric fields [14, 15] of dielectrics, which is brought about during may significantly distort the electron energy-release the absorption of electron radiation. profile [8, 9]. To correctly determine the effect of these The high-energy conductivity in wide-gap dielec- fields on the formation of the energy-release zone trics is determined by the fact that, when electrons are

TECHNICAL PHYSICS Vol. 45 No. 7 2000 876 DEMIDOV et al.

Ω smax, 1/( cm) Thus, the high-energy conductivity is 0.30 1.5ε 2 g σ ≈ e (ε)τ (ε) ε 0.25 ----- ∫ F ⊥ d , me { , ηω } max T 0 0.20 τ ≈ τ ≈ µω ηω ε ηω 1/2 where ⊥ d0 0(T/ 0)( / 0) is the relaxation ε 0.15 time of an electron pulse and F( ) is the nonequilibrium distribution function for high-energy electrons, which can be estimated from the energy balance 0.10 { ε ε ε } ε Qd0( ) + QT( ) Ð QE( ) F( ) 0.05 , [ (ε ε )3/2] = G(x t) 1 Ð /1.5 g , ≈ ε 0 0.5 1.0 1.5 2.0 where G(x, t) W(x, t)/1.5 g is the volumetric genera- Depth, cm tion rate for high-energy electrons and the expression in the square brackets approximately describes the den- Fig. 8. Profile of the maximum induced conductivity during sity of energy states. aerogel irradiation. The F(ε) singularity (zeroing of the expression in the braces) corresponds to the high-energy breakdown ε ≈ ε cooled to an energy 1.5 g of the order of magnitude that develops at the breakdown field Ebr [8], of the forbidden zone ε ≈ 10Ð20 eV, the energy loss g ≈ µ(ω )( ηω )1/2[ (ηω ε )1/2 rate abruptly falls (by a factor of 1000), because it Ebr 0/e me 0 1 + 0/ * becomes impossible to continue exciting electrons × ( 3 µ6) ] ( Π)1/3 ≤ ( Π)1/3 [ ] from the forbidden zone to the conduction band. The NT a / / 1 + 200/ 1 + kV/cm , main cooling channel [14, 15] for electrons with ener- ε ω ε ε which is several orders of magnitude lower than the gies in a range {T, h } < < 1.5 g is the interaction equilibrium value. Here, we take into account that with optical phonons and with electrons thermalized at (ε*/ηω ) < (ε /ηω ) ≈ µÐ2 and (1 + Π)1/3 is the correc- the temperature T with the energy loss rates 0 g 0 tion for the material porosity (discontinuity) Π. ≈ µω ω (ε ω )1/2 Qd0 0h 0 /h 0 The specific energy release of the electron beam as a function of the Lagrangian coordinate of the absorp- Q ≈ ω /µ5(N a3)hω (hω /ε)1/2, tion depth and irradiation time was calculated in the T 0 T 0 0 diffusion approximation [18] taking into account the respectively. quasistationary generation of electric fields, as well as ω 3 ≈ actual oscillograms of the current and voltage of the Here, 0 is the optical phonon frequency; (NT a ) Ð11 3/2 1/2 Kal’mar accelerator and the dependence of the electron 10 W{(x, t)T (x, t)} is the number of thermalized path depth on the electron energy taken from [19]. electrons in an elementary cell a3 determined by binary 3 Numerical calculations have shown that, at an irra- collisions; W(x, t) [J/(cm ns)] is the energy-release rate 2 of the electron beam at a depth x and moment t; and µ = diation power J0 = 15 kA/cm , U = 300 keV, the high- (m/M)1/4 ≈ 0.1 is the series expansion parameter [15]. energy breakdown intensity Ebr = (70Ð80) kV/cm is The electron heating power in the induced electrostatic reached at the moment of termination of the irradiation field E in the prebreakdown regime is evaluated as in almost the entire energy-release zone. In highly porous aerogels with a depth R ≈ 2 cm of the energy- Q ≈ eE(2e/m )1/2, release zone, the electrostatic energy becomes compa- E e rable with the energy of primary electrons, thus where me is the electron mass. strongly affecting the formation of the energy-release ε ω zone [8, 9] in an aerogel with a density of 0.03 g/cm3. With further electron cooling, < h 0, the emission of optical phonons becomes impossible and only the The calculated profile of the maximum conductivity, interaction of electrons with acoustic phonons and traps which is induced during irradiation, corresponding to (defects of the crystal lattice) is efficient. The latter this regime is shown in Fig. 8. effect depends heavily on the dielectric features and determines the slow charge relaxation after the irradia- CONCLUSIONS tion is completed [14, 15]. Since the experimentally measured time lag of the slow glow component in the (1) The spatial glow of an aerogel irradiated by a aerogel is rather large, we neglect this low-energy con- high-intensity pulse electron beam is studied experi- ductivity. mentally.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 EVOLUTION OF THE GLOW OF AN AEROGEL IRRADIATED 877

(2) Along with the classical glow of a transparent 4. A. I. Funtikov, Teplofiz. Vys. Temp. 36, 405 (1998). dielectric medium exposed to an electron beam, a slow 5. G. I. Kanel’, S. V. Razorenov, A. V. Utkin, and V. E. For- glow component (≈2 × 10Ð5 s) is revealed, which coin- tov, ImpactÐWave Phenomena in Condensed Media cides with the pattern of dynamic aerogel rarefaction— (Yanus-K, Moscow, 1996). with a compression wave propagating deep into the 6. I. I. Zalyubovskiœ, A. I. Kalinichenko, and V. T. Lazurik, aerogel. Introduction to Radiation Acoustics (Vishcha Shkola, (3) The origin of the slow glow component is evi- Kharkov, 1986). dently due to the significant volumetric electrization of 7. B. A. Demidov, V. P. Efremov, I. A. Ivonin, et al., Zh. a highly porous aerogel arising under its irradiation by Tekh. Fiz. 67 (11), 19 (1997) [Tech. Phys. 42, 1258 an electron beam. (1997)]. 8. B. A. Demidov, V. P. Efremov, I. A. Ivonin, et al., Zh. (4) Experiments with layered targets have con- Tekh. Fiz. 68 (10), 112 (1998) [Tech. Phys. 43, 1239 firmed the existence of internal rarefaction waves from (1998)]. aerogel inhomogeneities. 9. B. A. Demidov, V. P. Efremov, I. A. Ivonin, et al., Zh. (5) Direct experiments have confirmed a power Tekh. Fiz. 69 (12), 18 (1999) [Tech. Phys. 44, 1413 dependence of the sound velocity in the aerogel on its (1999)]. density and helped determine the percolation coeffi- 10. A. É. Bugrov, I. N. Burdonskiœ, V. V. Gavrilov, et al., Zh. cient, the value of which (γ = 3) agrees satisfactorily Éksp. Teor. Fiz. 111, 903 (1997) [JETP 84, 497 (1997)]. γ with the value = 3.2 utilized in previous calculations. 11. J. Lindl, Phys. Plasmas 2, 3933 (1995). 12. G. Yu. Gus’kov, N. V. Zmitrenko, and V. G. Rozanov, ACKNOWLEDGMENTS Pis’ma Zh. Éksp. Teor. Fiz. 66, 521 (1997) [JETP Lett. 66, 555 (1997)]. We are grateful to the staff members at the Institute 13. V. E. Fortov, A. S. Filimonov, V. K. Gryaznov, et al., in of Catalysis (Russian Academy of Sciences, Novosi- Proceedings of the International Conference on Physics birsk) for helping us prepare aerogel samples and to of Strong Coupled Plasmas, Ed. by W. D. Kraft (World B.A. Bryunetkin for helping us record the optical spec- Scientific, Singapore, 1996), pp. 317Ð321. trum of the aerogel emission. 14. D. I. Vaœsburd, High-Energy Solid-State Electronics This work was supported by the Russian Foundation (Novosibirsk, 1982). for Basic Research, project no. 97-02-16729a. 15. V. F. Gantmakher and I. B. Levinson, Scattering of Cur- rent Carriers in Metals and Semiconductors (Nauka, Moscow, 1984). REFERENCES 16. I. M. Sokolov, Usp. Fiz. Nauk 150 (2), 221 (1986) [Sov. 1. M. Burchell and R. Thomson, Bull. Am. Phys. Soc. 40, Phys. Usp. 29, 924 (1986)]. 1409 (1995). 17. Yu. A. Krysanov and S. A. Novikov, Prikl. Mekh. Tekh. 2. G. Yu. Gus’kov, N. V. Zmitrenko, and V. G. Rozanov, Fiz., No. 6, 57 (198). Pis’ma Zh. Éksp. Teor. Fiz. 108, 548 (1995) [JETP 81, 18. K. Kanaya and S. Okadama, J. Phys. D 5 (1), 43 (1972). 296 (1995)]. 19. V. F. Baranov, Electron Radiation Dosimetry (Atomiz- 3. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock dat, Moscow, 1974). Waves and High-Temperature Hydrodynamic Phenome- na (Nauka, Moscow, 1966, 2nd ed.; Academic, New York, 1966, 1967). Translated by A. Seferov

TECHNICAL PHYSICS Vol. 45 No. 7 2000

Technical Physics, Vol. 45, No. 7, 2000, pp. 878Ð882. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 65Ð69. Original Russian Text Copyright © 2000 by Kilosanidze, Kakichashvili.

OPTICS, QUANTUM ELECTRONICS

On the Theory of SpaceÐTime Polarization Holography B. N. Kilosanidze and E. Sh. Kakichashvili Institute of Cybernetics, Academy of Sciences of Georgia, Tbilisi, 380086 Georgia Received July 26, 1999

Abstract—Polarization holographic recording and the reconstruction of the ﬁeld of a nonstationary object wave are considered theoretically. Expressions for the nondiffracted beam, as well as the virtual and real images, formed by a spaceÐtime polarization hologram are analyzed. It is shown that, under certain conditions imposed on the isotropic, anisotropic, and gyrotropic responses of a polarization-sensitive medium, one can adequately reconstruct the space structure, time waveform, and polarization characteristics of the ﬁeld of a nonstationary object in the virtual image. © 2000 MAIK “Nauka/Interperiodica”.

Recently evolved so-called time holography, In (1), Eob(x0, y0, z0, t0) is the field immediately extending the holographic method to recording and behind the object. It is formed when a totally polarized ω reconstructing the time behavior of nonstationary wave monochromatic wave of frequency 0 with the Jones processes, has attracted considerable interest. This vector [11] direction represents a great heuristic contribution to ω completing the construction of holography fundamen- i 0z1 E0y E ==Eˆ exp Ð------,0≤ε ------≤ 1 (2) tals [1, 2]. The concept of holographic recording and 0 0x c i ε E reconstruction of nonstationary wave fields was first 0x put forward in [3]. It is based on the unambiguous rela- propagates along the z axis through a nonstationary tion between the time waveform of a nonstationary pro- object with the Jones matrix [11] cess and its frequency spectrum [4, 5]. A rigorous the- ,,, oretical substantiation of spaceÐtime holography for Mob()x0 y0 z0 t0 the scalar description of nonstationary waves was given in [6]. mˆ ()x ,,,y z t mˆ()x ,,,y z t = 11 0 0 0 0 12 0 0 0 0 . In this work, the developed theoretical approach is mˆ()x ,,,y z t mˆ()x ,,,y z t extended to the case when the state and degree of polar- 21 0 0 0 0 22 0 0 0 0 ization of nonstationary electromagnetic wave fields are treated rigorously. Earlier, in order to prove the fea- In the following, the properties of the nonstationary sibility of recording and reconstructing the state and object are assumed to be independent of the frequency ≠ ω degree of polarization of arbitrary electromagnetic of illuminating light; that is, mîj (x0, y0, z0, t0) f( 0). waves, the holographic method was modified for sta- Such a limitation, not being fundamental, makes it pos- tionary wave fields [7–9]. sible to substantially simplify subsequent calculations. We represent the field from a nonstationary object in Clearly, for actual media, the condition of object inde- the paraxial approximation of the Kirchhoff vector dif- pendence of the light frequency is an approximation fraction integral modified for the case of nonstationary similar to the “black screen” approximation used in wave fields [10]: [12, 13]. The nonstationary object depolarizes the illuminat- iω ing monochromatic wave, which is assumed to be ini- E ()xyz,,,ω ,t≈------E ()x ,,,y z t ob 2πc∫∫r ob 0 0 0 0 tially totally polarized. In the general case, the wave transmitted through the object is partially elliptically S0T0 (1) polarized. The modified Jones vector for the transmit- 1 ×expiω()ttÐ Ð --- r dt dS , ted wave immediately behind the object can be repre- 0 c0 0 sented in the form of partially coherent elliptically polarized orthogonal components [14]: ω where c is the velocity of light; is the frequency; (x0, y0, z0, t0) and (x, y, z, t) are the space and time coordi- 1 E ()x ,,,y z t = EˆM ()x ,,,y z t nates of object and observation points, respectively; r is ob 0 0 0 0 Ax ob 0 0 0 0 iε the distance between these points; S0 and T0 are the (3) ε space and time intervals occupied by the object; and ⊕ ˆ ,,, iω EByMob()x0 y0 z0 t0 expit0, dS0 = dx0dy0. 1

1063-7842/00/4507-0878 $20.00 © 2000 MAIK “Nauka/Interperiodica” ON THE THEORY OF SPACEÐTIME POLARIZATION HOLOGRAPHY 879 where ε = E /E = E /E ; 0 ≤ ε ≤ 1; ⊕ is the symbol Ay Ax Bx By  ε designating noncoherent summation of amplitudes (it × ω ( ) 1 i  expi t Ð t0 Ð --r dS0dt0   . was introduced in [14], where the Jones vectorÐmatrix c  1 method was formally extended to partially polarized light); Eˆ is the complex amplitude of a component of The intensity of the electric vector of the net wave is A described by the real part of expression (5) [16]: basis A; and Eˆ B is the complex amplitude of a compo- Re(EΣ) = pcosωt + qsinωt, (6) nent of basis B orthogonal and noncoherent to basis A. For the reference illuminating wave, we use a wave where the parameters p and q of the net ellipse are passed through an infinitely narrow time gate with a determined in terms of polarization ellipse components δ-shaped transmission. It is known that the interruption for the bases A and B according to the rules [14] ( ) ⊕ ( ) ⊕ of a wave train changes the wave frequency: an initially p = Re EΣ A Re EΣ B = pA pB, monochromatic train becomes nonmonochromatic (7) ( ) ⊕ ( ) ⊕ after passing the gate. In this case, according to the def- q = Im EΣ A Im EΣ B = qA qB. δ inition of the function, the passed wave has a contin- In order to record net wave (5), we use a polariza- uous spectrum, with the spectral density constant over tion-sensitive medium [17, 18]. We assume that both the entire frequency range [15]. In addition, the gate the recording medium and the nonstationary object are totally depolarizes an initially polarized wave. Then, spectrum-nonselective in the entire frequency range. the modified Jones vector of the reference wave can be represented as an orthogonal basis of elliptically polar- The photoanisotropy and photogyrotropy induced in ized components [14]: the photosensitive recording medium are related to the polarization characteristics of the inducing light via the  1  π iε relationship obtained in [19, 20]. This relationship con- E = E expiϕ ⊕ E expi Ψ Ð -- tains complex coefficients of the light-induced ellipti- or 0x iε 0x  2  1 (4) cal birefringence; and the vector photoresponse of the  1  polarization-sensitive medium is described by the func- × expiω t Ð --z νˆ  c  tions of isotropic, sˆ , anisotropic, L , and gyrotropic, ν ˆ G responses. In our work, we assume that the func- ε ϕ Ψ where = E0y/E0x and E0x, E0y and , are the respec- ν ν tions sˆ , ˆ L , and ˆ G do not depend on the frequency of tive amplitudes and initial phases of two mutually non- incident radiation. coherent components. The light-induced anisotropy and gyrotropy of the In polarization holographic recording, mutually polarization-sensitive medium can be described by coherent components of the orthogonal basis of the ref- Jones matrices [8, 11]. In [20], rules of constructing the erence and object waves independently interfere with Jones matrix for a polarization-sensitive medium were each other at corresponding frequencies and the result- formulated for the case of partially polarized inducing ing fields are summed up noncoherently (additively). radiation. With these rules and the relationship The net field in the hologram plane has the form obtained in [20], we come to the resulting expression for the Jones matrix:  , , , ϕ ω 1  EΣ(x y z t) = Eor + Eob = E0x expi expi t Ð --z  M M   c M = exp(Ð2iκdnˆ ) 11 12  , (8) 0   M21 M22 i ω where + ------Eˆ M (x , y , z , t ) 2πc ∫ ∫ r Ax ob 0 0 0 0 κ i d[ ( ) ( ) S0 T0 M11, 22 = 1 Ð ------sˆ I1 + I2 A + sˆ I1 + I2 B 2nˆ 0  ± νˆ cos2Θ (I Ð I ) ± νˆ cos2Θ (I Ð I ) ], × ω ( ) 1  1 L A 1 2 A L B 1 2 B expi t Ð t0 Ð --r dS0dt0   c  iε κ i d[ν Θ ( ) (5) M12, 21 = Ð------ˆ L sin2 A I1 Ð I2 A 2nˆ 0   π  1  ⊕ Ψ ω ν Θ ( ) − ν ( − ) E0x expi Ð -- expi t Ð --z + ˆ L sin2 B I1 Ð I2 B + i ˆ G I± Ð I+ A  2 c − ν ( − ) ] + i ˆ G I± Ð I+ B . κ π λ λ i ω In formula (8), = 2 / ; is the wavelength of the + ------Eˆ ByM (x , y , z , t ) initial illuminating wave; d is the thickness of the 2πc ∫ ∫ r ob 0 0 0 0 recording medium; nˆ is the complex refractive index S0 T0 0

TECHNICAL PHYSICS Vol. 45 No. 7 2000 880 KILOSANIDZE, KAKICHASHVILI of the medium in the initial (unilluminated) state; (I1 + and M+1 is the matrix responsible for the real image, I2)A and (I1 + I2)B are the ﬁrst Stokes parameters, (I1 Ð   I2)A and (I1 Ð I2)B are the second Stokes parameters, and κd (M ) (M ) M ≈ Ð------exp(Ð2iκdnˆ ) +1 11 +1 12  (12) (I± Ð I−) and (I± Ð I−) are the fourth Stokes parameters +1 π 0 + A + B 4 cnˆ 0  (M ) (M )  Θ Θ +1 21 +1 22 for A and B components, respectively; and A and B are the orientation angles (measured counterclockwise with matrix elements from the x axis) of the major axis of the polarization ellipse for A and B components, respectively. ω ( ) ˆ * [( ± ν )( * ε * ) Expressing the Stokes parameters appearing in (8) M+1 11, 22 = ∫ ∫∫ ---EAx sˆ ˆ L mˆ 11 Ð i m ˆ 12 r  in terms of the parameters pA, pB, qA, and qB [8], we S T Ω obtain the formula for the holographic matrix repre- 0 0 sented as the sum of three matrices in the entire fre- − + iε(sˆ + νˆ )(mˆ * Ð iεmˆ * ) ]E expiϕ + Eˆ B*y quency range: L 21 22 0x

M = M0 + MÐ1 + M+1. (9) × [( − ν ) ε ε ± ν ε ] sˆ + ˆ L (mˆ 22* Ð i mˆ 21* ) + i (sˆ ˆ L)(mˆ 12* Ð i m ˆ 11 * ) Here, M is the matrix responsible for the undiffracted 0 π  ω beam, × Ψ  ω 1  E0x expi Ð -- expÐi---zexpi t0 + --r 2  c c   iκdsˆ 2 2 1 0 M ≈ exp(Ð2iκdnˆ ) 1 Ð ------(1 + ε )E  ; (10) × ω 0 0 0x d dt0dS0, nˆ 0  0 1  M is the matrix responsible for the virtual image, ω Ð1 ( ) ˆ * [(ν ± ν )( * ε * ) M+1 12, 21 = ∫ ∫∫ ---EAx ˆ L ˆ G mˆ 21 Ð i mˆ 22 r    Ω κ (M ) (M ) S0 T0 ≈ d ( κ ) Ð1 11 Ð1 12  MÐ1 ----π------exp Ð2i dnˆ 0 (11) 4 cnˆ 0  ( ) ( )  MÐ1 21 MÐ1 22 ε(ν ± ν )( ε ) ] ϕ ˆ * + i ˆ L ˆ G mˆ 11* Ð i mˆ 12* E0x expi + EBy with matrix elements × [(ν ± ν ) ε ˆ L ˆ G (mˆ 12* Ð i mˆ 11) ω ( ) { ˆ [( ± ν )( ε ) MÐ1 11, 22 = ∫ ∫∫ --- EAx sˆ ˆ L mˆ 11 + i mˆ 12 r −  π  Ω + iε(νˆ + νˆ )(mˆ * Ð iεmˆ * ) ]E expi Ψ Ð --  S0 T0 L G 22 21 0x   2  ε( − ν )( ε ) ] ϕ ˆ Ð i sˆ + ˆ L mˆ 21 + i mˆ 22 E0x expÐi + EBy ω  1  × [(sˆ +− νˆ )(mˆ + iεmˆ ) Ð iε(sˆ ± νˆ )(mˆ + iεmˆ )] × expÐi---zexpiω t + --r dωdt dS . L 22 21 L 12 11 c  0 c  0 0  π  ω × E expÐi Ψ Ð -- expi---z In the above formulas, mˆ ≡ mˆ (x , y , z , t ) are the 0x  2 c ij ij 0 0 0 0  coordinate- and time-dependent elements of the two- dimensional matrix of the nonstationary object. In this  1  work, we do not analyze convolutions. Under certain × expÐiω t + -- r d ω d t dS ,  0 c  0 0 relationships between the response functions, namely, for ω sˆ = νˆ , νˆ = Ðνˆ , (13) (M ) = ---{Eˆ [(νˆ ± νˆ )(mˆ + iεmˆ ) L L G Ð1 12, 21 ∫ ∫∫ r Ax L G 21 22 Ω expressions (11) and (12) are simpliﬁed. It should be S0 T0 noted that conditions (13) are satisﬁed with a high ε ν − ν ε ] ϕ ˆ Ð i (ˆ L + ˆ G)(mˆ 11 + i mˆ 12) E0x expÐi + EBy accuracy for many polarization-sensitive media [8].

×[(ν − ν ) ε Then the matrices MÐ1 and M+1 are given by ˆ L + ˆ G (mˆ 12 + i mˆ 11) π  κdνˆ ω ε ν ± ν ε ] Ψ  M ≈ ------L-- exp(Ð2iκdnˆ ) ---M P Ð i (ˆ L ˆ G)(mˆ 22 + i mˆ 21) E0x expÐi Ð --  Ð1 π 0 ∫ ∫∫ ob 2 2 cnˆ 0 r  Ω S0 T0 (14) ω  1  1 × expi---zexpÐiω t + -- r d ω d t d S ; × expÐiω t + --(r Ð z) dωdt dS , c  0 c  0 0 0 c 0 0

TECHNICAL PHYSICS Vol. 45 No. 7 2000 ON THE THEORY OF SPACEÐTIME POLARIZATION HOLOGRAPHY 881

For the nondiffracted wave, we obtain κdνˆ ω ≈ L ( κ ) * M+1 Ð----π------exp Ð2i dnˆ 0 ∫ ∫∫ ---P*Mob 2 cnˆ 0 r iκdsˆ 2 2 Ω ≈ ( κ ) ------( ε ) S0 T0 (15) E0 exp Ð2i dnˆ 0 1 Ð 1 + E0x nˆ 0 × ω 1( ) ω expi t0 + -- r Ð z d dt0dS0.  1  π iε c × E expiϕ ⊕ E expi Ψ Ð -- (18) 0x iε 0x  2  1 Expressions (14) and (15) involve the object matrix  1  Mob and matrix P, which is × expiω t' Ð --z' ,  c     aˆ + ε2bˆ Ðiε(aˆ Ð bˆ )  and the virtual and real images are given, respectively, P =   , by  iε(aˆ Ð bˆ ) ε2aˆ + bˆ  iκdνˆ E (x', y', z', t') ≈ ------L---- exp(Ð2iκdnˆ )E 2 (1 + ε2) where Ð1 ( π )2 0 0x 2 c nˆ 0  π aˆ = Eˆ E expÐiϕ, bˆ = Eˆ E expÐi Ψ Ð -- 2 Ax 0x By 0x   ω  1 2 × ----- Eˆ M (x , y , z , t ) ∫∫ ∫∫ r'r Ax ob 0 0 0 0 iε Ω S S0 T0 (19) and P* and Mob* are Hermitian conjugates. iε ⊕ Eˆ M (x , y , z , t ) expiω Let us illuminate the obtained hologram by a recon- By ob 0 0 0 0  1 structing nonpolarized wave with complex amplitudes E' expiϕ' and E expiΨ' (ε' = E' /E' ) and a fre- 0x 0y 0y 0x 1 quency ω': × (t' Ð t ) Ð --(r' + r) dωdt dS dS 0 c 0 0  1   π iε' E = E' expiϕ' ⊕ E' expi Ψ' Ð -- and rec 0x iε' 0x  2  1  (16) κ νˆ , , , ≈ i d L ( κ ) 2 × ω  1  E+1(x' y' z' t') Ð------exp Ð2i dnˆ 0 E0x expi 't' Ð --z . ( π )2 c 2 c nˆ 0

The transmitted wave takes the form 2 ω  1 × ----- P*M* (x , y , z , t ) ∫∫ ∫∫ r'r A ob 0 0 0 0 iε i ω' ω' S S T Ω , , , 0 0 (20) E(x' y' z' t') = ----π-----∫---- MErec expÐi----r'dS, (17) 2 c r' c iε S ⊕ P*M* (x , y , z , t ) expiω B ob 0 0 0 0  1 where S is the hologram area and r' is the distance between a point on the hologram surface and the obser- 1 vation point. × (t' + t ) Ð --(r' Ð r + 2z) dωdt dS dS, 0 c 0 0 Sequentially substituting expressions (10), (14), and (15) into formula (17), we determine the nondiffracted where beam and the virtual and real images formed by the hologram. Now it is necessary to determine which π * ϕ * Ψ  wave should be used to reconstruct the object field in PA = expi P*, PB = expi Ð -- P*. the virtual image. Evidently, this requires the determi- 2 nation of the eigenvectors (and their related eigenvalues) of the matrix P. It has been found that, correct to a The integrals in formulas (19) and (20) will be taken constant factor, the eigenvectors of the matrix P are in the linear approximation for distances r and r' and Ω  1 iε infinitely large domains of integration S, S0, T0, and . and and the eigenvalues are (1 + ε2)aˆ and Ω iε  1 The integrals over the domains S and are essentially space and time δ functions. Calculations similar to (1 + ε2)bˆ . This implies that the reconstructing wave those performed in [6] lead to the final expressions for should be identical to the reference wave used in the images formed by the spaceÐtime polarization holo- recording. gram.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 882 KILOSANIDZE, KAKICHASHVILI

For the virtual image, we obtain from (19) at z' = z0 REFERENCES π κ νˆ 1. D. Gabor, Proc. R. Soc. London, Ser. A 197, 454 (1949). , , , ≈ 2 i d L ( κ ) 2 ε2 EÐ1(x' y' z' t') Ð------exp Ð2i dnˆ 0 E0x(1 + ) 2. Yu. N. Denisyuk, Dokl. Akad. Nauk SSSR 144, 1275 nˆ 0 (21) (1962) [Sov. Phys. Dokl. 7, 543 (1962)]. ε 3. V. A. Zubov, A. V. Kraœskiœ, and T. I. Kuznetsova, Pis’ma × ˆ , , ,  1 ⊕ ˆ , , , i  Zh. Éksp. Teor. Fiz. 13, 443 (1971) [JETP Lett. 13, 307 EAx Mob(x' y' z' t')  EByMob(x' y' z' t')  . iε 1 (1971)]. It follows from formula (21) that this expression, 4. V. A. Zuœkov, V. V. Samartsev, and R. G. Usmanov, correct to a constant factor, describes the complete Pis’ma Zh. Éksp. Teor. Fiz. 32, 293 (1980) [JETP Lett. reconstruction of both the spaceÐtime structure and 32, 270 (1980)]. polarization characteristics of the nonstationary object 5. P. M. Saari, R. K. Kaarli, and A. K. Rebane, Kvantovaya wave field. Élektron. (Moscow) 12, 672 (1985). 6. Sh. D. Kakichashvili and E. Sh. Kakichashvili, Pis’ma For the real image, we obtain from (20) at z' = 2z Ð z0 Zh. Tekh. Fiz. 24 (11), 76 (1998) [Tech. Phys. Lett. 24, π κ ν 446 (1998)]. 2 i d ˆ L 2 E (x', y', z', t') ≈ Ð------exp(Ð2iκdnˆ )E 7. Sh. D. Kakichashvili, Opt. Spektrosk. 33, 324 (1972). +1 nˆ 0 0x 0 8. Sh. D. Kakichashvili, Polarization Holography (Lenin- grad, 1989). × * *  , , , 2z   1 (22) PA Mob x' y' z' ----- Ð t'  ε 9. Sh. D. Kakichashvili and B. N. Kilosanidze, Zh. Tekh. c i Fiz. 67 (6), 136 (1997) [Tech. Phys. 42, 709 (1997)].  2z  iε 10. Sh. D. Kakichashvili, Pis’ma Zh. Tekh. Fiz. 20 (22), 78 ⊕ P*M* x', y', z', ----- Ð t' . B ob  c   1 (1994) [Tech. Phys. Lett. 20, 925 (1994)]. 11. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941). One can see from formula (22) that, at the distance 12. G. R. Kirchhoff, Selected Works (Moscow, 1988). z' = 2x Ð z0, the image with the pseudoscopic spatial 13. F. Kottler, Prog. Opt. 4, 283 (1965). structure of the object field is formed symmetrically to 14. Sh. D. Kakichashvili, Zh. Tekh. Fiz. 65 (7), 200 (1995) the virtual image about the hologram plane. It has an [Tech. Phys. 40, 743 (1995)]. inverted time waveform with a time delay due to pass- 15. A. A. Kharkevich, Spectra and Analysis (Moscow, ing the distance 2z = z' + z0 from the observation point 1962). to the real image. Its polarization state is transformed 16. M. Born and É. Wolf, Principles of Optics (Pergamon, according to the matrices PA* and PB* . Oxford, 1969; Nauka, Moscow, 1973). In conclusion, the capability of the polarization 17. F. Weigert, Verh. Dtsch. Phys. Ges. 21, 479 (1919). holographic method to reconstruct the spatial structure, 18. H. Zocher and K. Coper, Z. Phys. Chem. 132, 313 time waveform, and polarization state of the initial field (1928). of a nonstationary object ultimately extends the poten- 19. Sh. D. Kakichashvili, Opt. Spektrosk. 52, 317 (1982) tialities of the holographic method. [Opt. Spectrosc. 52, 191 (1982)]. 20. Sh. D. Kakichashvili and B. N. Kilosanidze, Pis’ma Zh. Tekh. Fiz. 21 (23), 6 (1995) [Tech. Phys. Lett. 21, 951 ACKNOWLEDGMENTS (1995)]. We are grateful to Sh.D. Kakichashvili for the statement of the problem and helpful discussions. Translated by A. Kondrat’ev

TECHNICAL PHYSICS Vol. 45 No. 7 2000

RADIOPHYSICS

Relativistic Microwave Devices with Postacceleration of an Electron Beam in the Interaction Space É. B. Abubakirov and A. V. Savel’ev Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603600 Russia E-mail: [email protected] Received December 1, 1998; in ﬁnal form, November 11, 1999

Abstract—We analyzed the possibility of improving the efficiency of microwave devices operating with relativistic electron beams in systems with particle postacceleration in the interaction space. The fundamental feature of this approach is the formation of the accelerating-potential profile with the inherent electric field of a high-current electron beam. It is shown that the use of the space-variant beam-potential sag helps raise the estimated efficiencies of relativistic Cherenkov TWT and BWO up to values of about 50%. © 2000 MAIK “Nauka/Interperiodica”.

Facilities with postacceleration of an electron beam where J is the beam current, ν is the electron velocity, in the interaction space are a promising version of pow- and R and r are the anode and electron-beam radii. erful RF devices. This kind of acceleration helps One can easily see that, as the ratio R/r changes, i.e., increase the average energy of previously formed elec- as the beam approaches or recedes from the transport- tron bunches, as in Reltron [1], simultaneously reduc- channel wall, the electron energy in the beam may vary ing the relative energy spread, and thus effectively within wide margins. This control of the beam position achieve a high output RF power. Applying an accelerat- can be attained by creating the required configuration ing electric field to the electrons captured by a synchro- of the lines of force of the static magnetic field focusing nous electromagnetic wave also provides the mode of the high-current electron beam. For instance, diverging their adiabatic deceleration [2, 3]. In this case, the elec- field lines (magnetic field decreasing along the chan- tron-beam potential energy in the accelerating static nel) maintain a constant electron acceleration. field is actually converted to RF radiation. Let us estimate the possibilities of controlling the particle energy under changes of the “beam-potential Note that, as a rule, electron postacceleration is per- sag.” The electron energy in the stationary beam is min- formed by introducing supplementary high-voltage 2 γ 1/3 electrodes which need intricate and bulky insulation imum Emin = mc (a Ð 1) when the current has the lim- and are unsuitable for transporting the electron beam. iting value for this transport channel At the same time, for high-current electron beams 3()γ 2/3 3/2 focused by a longitudinal magnetic field, the particle mc a Ð 1 J = ------, (2) energy is likely to be controlled by the beam-inherent lim 2eRln()/r fields. Powerful relativistic microwave devices are con- γ ventionally used when the electron currents account for where e and m are the electron charge and mass, a = 2 an appreciable fraction of the limiting vacuum value. 1 + eUa/mc , and Ua is the accelerating voltage. If the The inherent fields of this electron flux are rather electron energy in the initial beam state is close to the strong, and, therefore, the potential difference between minimum value, the possible relative increment in their the beam and the transport-channel wall ∆U is large. In γ 1/3 γ 1/3 kinetic energy is (eUa Ð Emin)/Emin = ( + 1). It is this connection, the particle energy in the beam E = a a ∆ evident that the particle energy in the transport chan- e(Ua Ð U) may noticeably differ from the peak energy nels can be increased three times even for slightly rela- determined by the anode (accelerating) voltage U . The γ a tivistic ( a 1) flows. This acceleration technique drop of potential ∆U depends on the beam current, its seems to be best suited to relativistic-electron beams, configuration, and position in the transport channel. For whose potential profile, defined by relation (1), is virtu- example, for a circular cross section of the drift channel ally unchanged during their interaction with the RF and a narrow tubular (circular) electron beam, we have field because of the weak energy dependence of the relativistic particle velocity. 2JRln()/r ∆U = ------, (1) Now we analyze the usability of electron-beam ν postacceleration for specific tasks. By way of example,

884 ABUBAKIROV, SAVEL’EV

g(ζ) η, % is the initial detuning of synchronism; I = γ 3 πω2 | α|2 α2 | |2 2 1.0 50 (0 eJ/ mP) Ez/ ; = Ez /2h P is the coupling resistance of electrons with the wave; P is the power 0.8 40 transferred by the wave; h is its longitudinal wave number; σ = 4γ T(eJ/mc3)/π(γ 2 Ð 1)3/2 is the coefficient of 30 0 0 0.6 space charge; T is the depression coefficient; e and m η are the electron charge and mass. 20 0.4 To give specific expressions to the dependences α(ζ) and σ(ζ), we assume that the electrodynamic sys- 0.2 10 tems of the TWT and BWO are circular corrugated α α χ χ σ g waveguides. In this case, = 0I0( r)/I0( r0); = σ χ 0 0 0T(r)/T(r0), where r0 is the initial beam radius; is the 0 1 2 3 4 5 6 7 8 9 10 transverse wave number of the synchronous harmonic; ζ T(r) = I0(pr)[I0(pR)K0(pr) Ð I0(pr)K0(pR)]/I0(pR); I0 and K0 are the modified zero-order Bessel function; and p = Fig. 1. The efficiency and the effective accelerating force versus the longitudinal coordinates in the TWT with elec- k/ γ 2 Ð 1 . Sign (Ð) in expressions (4) corresponds to tron postacceleration. 0 the case when the electrons and the energy in the wave (TWT) move in the same direction, and the sign (+) consider the relativistic Cherenkov traveling-wave tube corresponds to their counterpropagation (BWO). If (TWT) and the backward-wave oscillator (BWO). For the beam entering the interaction space is not modu- simplicity, we use a one-dimensional TWT (BWO) lated, the boundary conditions in (4) are written as fol- simulation in which the particle acceleration is taken lows: into account by means of an additional force that is ( ) ϑ( ) ϑ ∈ [ , π] w 0 = 1, 0 = 0 0 2 , dependent on the longitudinal coordinate. Electron tra- (5) ( ) ( ) (ζ ) ( ) jectories are assumed to coincide with the lines of the F 0 = F0 TWT or F k = 0 BWO , guiding magnetic field. This approximation is valid if ξ the radius and pitch of the Larmor electron spiral in the where k is the dimensionless length of the interaction magnetic field are small with respect to the characteris- space. tic scale of the focusing-field-strength variation. Under While simulating the TWT, the configuration of the these assumptions, the force acting on the electron that guiding-magnetic-field lines was selected so that the moves along the field line of force r(z) is defined with electrons were postaccelerated in the region where a relation (1): compact bunch of electrons had already been formed. For simplicity, the accelerating force was assumed to be dU 2eJ 1dr ζ G(z) = e------sinϕ ≈ ------. (3) constant in this region and a specific dependence r( ), dr ν r dz needed for evaluating α(ζ) and σ(ζ), was found from (3). Note that the beam radius should be varied ϕ Ӷ where = arctan(Hr /Hz) 1 and Hr and Hz are the adiabatically smoothly in order to avoid a significant vector components of the focusing-magnetic-field loss of electrons from the bunch produced. A solution strength. The interaction of an ultrarelativistic electron of Eqs. (4) is shown in Fig. 1 as the interaction effi- beam (γ = E/mc2 ӷ 1) with a synchronous wave is ciency versus the longitudinal coordinate described by a system of nonlinear equations [4], sup- 2π plemented with the accelerating force G(z): (ζ) ϑ ∫ w d 0  2π  dw  iϑ Ðiϑ  η 1 0 ------= ReαF + iσ e dϑ  e  + g(ζ), = 1 Ð ----π------ζ------. (6) dζ  ∫ 0 2   (ζ ) ζ 0 (4) 1 + ∫g ' d ' 2π 0 dϑ 1 dF iϑ ------= ----- Ð δ, ------= +−αI e dϑ . ζ 2 ζ ∫ 0 Simultaneously with the postacceleration, an d w d 0 increase in the beam radius causes the coupling coeffi- γ γ cients and the space charges to change (Fig. 2). In this Here, w = / 0 is the electron energy normalized by γ ν2 2 Ð1/2 case, the relative change in the kinetic energy of elec- the initial value; = (1 Ð /c ) is the relativistic trons due to postacceleration is ∆E/E = 0.75. The peak mass factor; ϑ = ωt Ð kz is the particle phase with 0 efficiency achieved in this case is nearly 50%. respect to the synchronous wave; F = 2eγ E /mcωα; 0 z In high-current devices, the space charge signifi- γ ω ζ γ 2 ω δ γ 2 g = 2 0G/mc ; = kz/2 0 ; k = /c; = 2 0 (h/k Ð 1) cantly affects the electron-bunching process in the

TECHNICAL PHYSICS Vol. 45 No. 7 2000 RELATIVISTIC MICROWAVE DEVICES WITH POSTACCELERATION 885

σ(ζ) α(ζ) δw 0.30 1.2 1.5 α 1.0 0.25 1.0 0.5

0.20 0.8 0

0.15 0.6 –0.5 σ –1.0 0.10 0.4 –1.5 0.05 0.2 0 1 2 3 ζ 0 1 2 3 4 5 6 7 8 9 10 ζ Fig. 3. Variations of the normalized energy δw = w(ζ) Ð ζ ∫ g (ζ')dζ' Ð 1 of electrons with various initial phases (solid Fig. 2. Variations in the coupling and space-charge coefﬁ- 0 cients in the TWT with postacceleration of the electron lines) and their average energy (dashed line) in the BWO beam. with electron postacceleration along the interaction space.

field of a synchronous wave, preventing the formation when implementing the postacceleration principle, the of a compact bunch. The advantage of the postaccel- configuration of the static magnetic field confining the eration scheme in question is that the beam broaden- electron beam corresponds to beam decompression, ing results in the attenuation of the space charge and which helps maintain the microwave-device efficiency reduction of the repulsive forces, favoring formation for finite values of the magnetic field [6]; therefore, of a dense electron bunch. Moreover, bringing the similar estimates are used. The longitudinal motion is Ð1 beam closer to the walls of the waveguiding system in assumed to govern until ν⊥ Ӷ cγ , where ν⊥ is the Cherenkov devices is accompanied by an increase in transverse electron velocity [4]. When a coaxial diode the synchronous-harmonic amplitude, which provides with magnetic insulation is used to form the electron more effective deceleration of the electron bunch by beam, those particles emitted by the cathode side have the wave. the maximum transverse velocity; their Larmor radius ν ω 2 The feasibility of beam postacceleration in the is defined by the relation R⊥0 = ⊥/ Hc = (mc /e)Ec /Hc, BWO was similarly analyzed. It is interesting to com- where ν⊥ corresponds to the drift velocity in the pare these results for the BWO with the coupling-resis- crossed radial electric field at the cathode side Ec and tance jump [5]. As in [5], the interaction space in the focusing magnetic field close to the cathode Hc. As the BWO with postacceleration can be functionally divided magnetic field changes, the peak-to-peak amplitude of into two parts. The electron bunching predominantly ≈ occurs in the part with reduced coupling resistance and transverse oscillations increases as R⊥ R⊥0 Hc/H ; relatively low electron energy; in the other BWO part, hence, ν⊥ ≈ ω R⊥ = (c/γ)E / H H . Therefore, the energy is taken up from the bunched flow. Analyzing H c c Ӷ the electron phase distribution, we see that the bunch is transverse velocities are small when Ec Hc H . not formed until ζ ≈ 1.5 and the average electron energy Naturally, this condition is more stringent than that slightly changes up to this moment (Fig. 3). The elec- with a homogeneous magnetic field which is equal to trons then reach the region of stronger coupling with the field at the cathode; however, the difference is only the field, where the bunch gives up most of its energy to the wave. In contrast to [5], the coupling of the elec- Hc H times. trons with the wave varies smoothly and is accompanied by the postacceleration of the electron bunch. Thus, the calculations corroborate the feasibility of When α(ζ )/α(0) = 3, ζ = 3.25, and ∆E/E = 1, this using particle postacceleration to increase the sensitiv- k k 0 ity of microwave devices with relativistic electron BWO has an efficiency of 48.2%. It is important that, beams. The advantage of the postacceleration scheme without the postacceleration, this drop of the coupling is that the longitudinal distribution of potential required resistance provides a significantly lower efficiency (18.4%) of the BWO with the given parameters. for its implementation is formed by the electron beam itself. In addition, the space charge of the beam Now we discuss the features in using the method decreases, and the coupling between electrons and the with finite magnetic fields, when particles may move wave increases during the interaction. However, it is across the magnetic-field lines of force. Note that, important that just imparting additional kinetic energy

TECHNICAL PHYSICS Vol. 45 No. 7 2000 886 ABUBAKIROV, SAVEL’EV to electrons, i.e., postacceleration, plays the decisive 4. N. F. Kovalev, M. I. Petelin, M. D. Raœzer, and role in raising the device sensitivity. A. V. Smorgonskiœ, Relativistic High-Frequency Elec- tronics (IPF AN SSSR, Gorki, 1979), p. 76. REFERENCES 5. N. F. Kovalev and V. I. Petrukhina, Élektronika SVCh, No. 7, 102 (1977). 1. R. B. Miller, W. F. McCullough, K. T. Lankaster, and C. A. Muehlenweg, IEEE Trans. Plasma Sci. 20, 332 6. E. B. Abubakirov, M. I. Fuchs, N. G. Kolganov, et al., in (1992). Proceedings of the 3rd Int. Workshop “Strong Micro- 2. E. D. Belyavskiœ, Radiotekh. Élektron. (Moscow) 16, wave in Plasmas,” Nizhni Novgorod, Institute of Applied 208 (1971). Physics, 1996, vol. 2, pp. 810Ð828. 3. N. S. Ginzburg, I. A. Man’kin, V. E. Polyak, et al., Rela- tivistic High-Frequency Electronics (IPF AN SSSR, Gorki, 1985), Vol. 7, p. 37. Translated by N. Goryacheva

TECHNICAL PHYSICS Vol. 45 No. 7 2000

ELECTRON AND ION BEAMS, ACCELERATORS

Dispersion Dilution of the Beam Emittance in Linear High-Energy Electron Accelerators V. M. Tsakanov Erevan Institute of Physics, Erevan, 375036 Armenia E-mail: [email protected] Received February 12, 1999

Abstract—Dispersion dilution of the beam emittance in a linear accelerator due to the initial uncorrelated spread of the particle energy in a bunch is considered. Both coherent beam oscillations and the case of the disturbed central trajectory and its local correction are treated. Emphasis is given to the analytic description of the dispersion dilution of the beam emittance. Exact analytic expressions derived for the emittance evolution along the accelerator are strengthened by the numerical simulation of particle tracks in the main linear accelerator of the future electronÐpositron collider. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION on the phase plane, resulting in BEDD. In this case, the Liouville theorem is invalid and the beam emittance is The achievement of ultimately small electron defined statistically as the root-mean-square (rms) (positron) beam emittances in the future electronÐ spread of particles over the phase plane of transverse positron collider is a key challenge in obtaining high oscillations. luminosity at the collision site [1Ð3]. In the main linear accelerator of the collider, where particles are acceler- BEDD has been treated from various standpoints ated from several GeV to several hundred GeV, the nat- [9Ð17]. Both the analytic characterization of the phe- ural emittance, in the ideal case, must shrink as much nomenon and numerical simulation of particle trajecto- as possible because of the adiabatic damping of trans- ries in a linear accelerator have been performed. Phys- verse particle oscillations. Actually, however, the tra- ically, BEDD in linear accelerators is well understood jectories of the particles are disturbed because of toler- (see, e.g., [9, 13, 15]); however, the unified approach to ances imposed on elements of the electronÐoptic sys- the problem that makes possible the rigorous analytic tem of the accelerator and transverse wake fields characterization of the beam emittance in the presence arising in accelerating sections [4, 5]. In addition, a of an initial spread of the particle energies is lacking in bunch has an energy spread both correlated with the the literature. Let us consider in detail two important particle longitudinal coordinate (because of the interac- issues. tion with the accelerating structure) and uncorrelated First, the two-particle model of bunch is obviously (because of beam preforming). The BalakinÐ inappropriate for the analytic description of BEDD NovokhatskyÐSmirnov method [6] allows one to sup- when the spread of the particles is large. Indeed, if the press an increase in the emittance due to the correlated relative phase incursion of betatron oscillations of a energy spread and wake fields within a bunch if so- nonequilibrium particle is 2π (one turn of the nonequi- called particle autophasing conditions are met [7, 8]. librium particle about the phase ellipse with respect to However, the remaining uncorrelated energy spread an equilibrium particle), the rms emittance goes to causes beam emittance dispersion dilution (BEDD). zero. In reality, however, as the particles spread, the This poses major problems for obtaining energetic entire phase ellipse of the transverse betatron oscilla- beams with ultimately small lateral and vertical emit- tions of the center of gravity of a beam becomes occu- tances. pied by particles with nonequilibrium energies. There- An increase in the emittance of a beam with an ini- fore, in the analytic description of BEDD, one should tial energy spread is associated both with coherent beta- rely on the actual energy distribution of the particles in tron oscillations of a beam with nonzero initial ampli- a bunch. tudes (deflection and angle of deflection) of its center of The second remark concerns BEDD numerical sim- gravity and with the disturbed central trajectory. The ulation when the central trajectory is disturbed and disturbance of the trajectory stems from quadrupole associated corrections are made. Note that lens misar- lens misarrangement. When passing through the focus- rangement can be taken into account only by specifying ing system of the accelerator, particles having different the offsets of the quadrupole lenses relative to the energies acquire a phase incursion of betatron oscilla- accelerator axis. The offsets are presented as a random tions and, if the central trajectory is nonzero, separate set of uncorrelated displacements with a known rms

888 TSAKANOV alignment precision. However, the disturbed central Note that the above formula is nothing else than the trajectory may significantly (by several orders of mag- cell chromaticity, which defines a relative change in the nitude) differ for two sets [18, 19]; hence, it is neces- phase incursion of betatron oscillations per period for a sary to average the BEDD effects over many sets of the nonequilibrium particle. The parameter α specifies the displacements of focusing components from the zero law of phase variation with particle acceleration µ trajectory. As shown in [20], by introducing the rms (energy). Thus, the phase incursion 1 during the first area of the disturbed machine phase ellipse, one can period and parameter α uniquely characterize the rigorously describe the rms disturbance of the central focusing systems of an accelerator and linear optics of trajectory (along the accelerator) to which particle an electron beam if it is assumed that lateral and verti- tracks converge when averaged over many sets of lens cal oscillations are uncoupled. Such an approach is misarrangements. Virtually, the analytic description applicable to all possible beam trajectories. As shown provides the exact averaging of the rms beam parame- below, the BEDD effect to a great extent depends on the ters over all possible trajectories of particles in a bunch. phase incursion at the beginning of an accelerator and In this work, we study the BEDD effect in linear its variation along the accelerator. It is significant [8] accelerators that is due to the uncorrelated initial parti- that, with the focusing system defined in such a way, the particle autophasing conditions are met and the cle energy spread in a bunch. First, we will consider α coherent betatron oscillations with the central trajec- exponent in formula (1) equals 0.5. tory disturbed because of quadrupole lens misarrange- The linearized equation for transverse motion of a ment. Then, the case when a dipole corrector in each of nonequilibrium particle in a linear accelerator where the quadrupole lenses compensates for central trajec- quadrupole lenses are displaced from the axis has the tory distortions by determining the position of the cen- form ter of gravity of a beam will be discussed. Results for γ ' ( δ)( ) FODO (F, focusing lens; O, open gap; D, defocusing x'' + -γ-- + K x 1 Ð x Ð xq = 0, (2) lens) periodicity cells for various phase incursion vs. particle energy dependences along the accelerator [8, where x is the transverse (vertical or lateral) displace- 21] will be presented. Analytic expressions for rms δ δ ment of the particle, x0 = x(0, 0) and x0' = x'(0, 0) are beam emittance are compared with simulations of par- δ δ γ γ ticle trajectories in the main accelerator to be used in the initial amplitudes, (z) = 0 0/ (z) is an accelera- the SBLC thermal linear collider (accelerating field fre- tion-dependent change in the uncorrelated relative δ ∆ δ Ӷ quency 3 GHz) and the TESLA superconducting linear energy deviation, 0 = E/E0 ( 0 1) is the initial collider (accelerating field frequency 1.3 GHz) [22]. uncorrelated energy deviation, Kx is the lens strength, and xq(z) = xqk are random displacements of the quadrupole lenses from the axis. The derivative is taken with EQUATIONS OF MOTION AND PHASE respect to the coordinate z along the accelerator axis. DISPERSION OF BETATRON OSCILLATIONS The general solution of the equation of motion is As usual, we assume that the focusing system of an represented as the sum of betatron oscillations xb and accelerator consists of many FODO cells (periods), the disturbed trajectory xd: each having two accelerating sections. Particles in a x(z, δ) = x (z, δ) + x (z, δ). (3) bunch experience the same acceleration with a constant b d gradient γ', so that the equilibrium energy of the parti- The explicit form of these solutions is found by γ γ γ cles varies linearly along the accelerator: (z) = 0 + 'z, applying Twiss matrix formalism [23Ð25]: γ γ where 0 and (z) are, respectively, the initial and γ 1/2 instantaneous Lorenz factors of an equilibrium particle ( , δ)  0 β( ) [µ( ) ϑ ] xb z = ab------z  cos z + b , (4) and z is the particle position γ. It is assumed that the γ(z) phase incursion of betatron oscillations per cell varies z with energy as [8] ( , δ) δ ( , ) ( ) xd z = ∫M12 z' z K x z' dz', (5) µ µ α n 1 (β β ) 1 0 tan----- = --Kn Lq n max Ð n min = tan-----gn , (1) 2 4 2 δ where the M element of the Twiss matrix for a non- µ 12 where n is the phase incursion in the nth cell, gn = equilibrium particle is given by γ /γ , γ = γ + (n Ð 1)∆γ is the energy of a particle enter- 0 n n 0 γ 1/2 ing the nth cell, ∆γ is the energy gain per FODO cell, δ , β β  (z') [µ µ ] ∂ ∂ M12(z' z) = (z') (z)-γ------ sin (z) Ð (z') . (6) Kn = ec( B/ x)/E is the normalized strength of quadru- (z) pole lenses, e is the electron charge, c is the speed of light, ∂B/∂x is the gradient of the lens magnetic field, E Here, β = β + ∆β and µ = µ + ∆µ are the amplitude β is the instantaneous energy of an equilibrium particle, function and phase of betatron oscillations for the par- and Lq is the quadrupole lens length. ticle, respectively; ∆β and ∆µ are their changes relative

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DISPERSION DILUTION OF THE BEAM EMITTANCE 889 β µ ϑ to the equilibrium values and ; and ab and b are the ∆v initial transverse deﬂection and angle of deﬂection (at 0.20 (a) 0.04 (b) the exit to the accelerator), respectively. It can be shown that ∆β and ∆µ are expressed through the parameters of 0.16 the focusing system as [23, 24] 0.03 0.12 z 0.02 ∆β δ β [ µ µ ] (z) = ∫ (z') (z')K x(z')sin 2 (z) Ð 2 (z') dz', (7) 0.08 0 0.04 0.01 z ∆µ δ β 2 [µ µ ] (z) = Ð∫ (z') (z')K x(z')sin (z) Ð (z') dz'. (8) 0 20 40 60 80 100 0 20 40 60 80 100 0 E, GeV

Thus, the mean change in the amplitude function in Fig. 1. Beam chromaticity vs. particle energy in the main linear accelerators can be ignored, unlike magnetic res- accelerator of the (a) SBLC (δ = 0.01) and (b) TESLA δ 0 onance accelerators [24]; on the other hand, the mean ( 0 = 0.014) colliders. increase in the phase incursion causes beam spreading on the phase plane and, hence, an increase in the rms beam emittance. The current emittance ε is defined sta- erator parameters, it is given by tistically as the rms spread of particles on the phase µ γ 1 α plane of transverse oscillations [26]: ∆µ = Ð2δ tan-----1 -----0----(1 Ð g ). (12) 0 2 ∆γ α ε = 〈∆x2〉 〈∆x'2〉 Ð 〈∆x∆x'〉 2, (9) An important parameter of the machine is its chromaticity ξ. For linear high-energy accelerators, it can where ∆x = x Ð x ; ∆x' = x' Ð x' , and x and x' are the be defined as a change in the number of betatron oscillations of a nonequilibrium particle with the rms energy position and angular coordinate of the center of gravity deviation σ⑀ from the equilibrium energy after a single of a beam. ξ ∆µ π transit of a bunch through the accelerator: = s/2 . Averaging is accomplished over all particles in a In linear accelerators, chromaticity is virtually respon- bunch. We are interested in the averaged (over the sible for beam spreading on the phase plane of trans- instantaneous phases of betatron oscillations) emit- verse betatron oscillations. The beam spread, as evi- tance along the accelerator. Then, ε = 〈∆x2〉/β if the dis- denced by the foregoing, depends on the initial rms persion of the betatron function is absent. energy deviation in the beam, energy gain per cell, The uncorrelated particle energy spread in a bunch phase incursion in the first cell, and a variation of phase is observed when the bunch is injected into the main incursion along the accelerator. linear accelerator. The spread is the consequence of the Figure 1 shows the growth of the chromaticity along beam forming history (storage ring and bunch com- the main accelerator in the SBLC thermal and TESLA pressor). If all of the particles have the same accelera- superconducting colliders with constant betatron phase tion in a given cross section of the bunch, the spread is incursions per period (α = 0) of π/2 and π/3, respec- inversely proportional to the energy of an equilibrium tively [22]. Notice that the small phase incursion per particle. Then, in the thin-lens approximation, the mean period of the focusing system and the relatively large change in the phase incursion for a nonequilibrium par- particle energy gain per period (~900 MeV) in the δ ticle that has the relative initial energy deviation 0 is TESLA design are key factors that specify the small given as the sum over periodicity cells: chromaticity of the machine. This, as demonstrated below, is significant for retaining the natural beam emittance in the main linear accelerator. 1 ∆µ = Ð--δ ∑g K L (β Ð β ). (10) 2 0 n n qn n max n min n COHERENT OSCILLATIONS For our representation of the focusing system, this OF THE BEAM formula is recast as To begin with, we will consider the dynamics of particles in a bunch with an uncorrelated initial Gaussian µ α energy spread of coherent betatron oscillations entering ∆µ δ 1 1 + 1∆γ = Ð2 0 tan-----∆-----γ-∑gn , (11) 2 into a linear accelerator (x0 and x0' are the initial trans- n verse amplitudes). The focusing system is assumed to which is the energy integral at a slow energy variation be perfectly arranged with respect to the accelerator along the accelerator (∆γ/γ Ӷ 1). In terms of the accel- axis. With the amplitude dispersion ignored, betatron

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∞ x'inv, mrad 2 2 2 σ ( ) (δ) 〈 ( , δ) ( )〉 , ϑ δ (a) (b) x z = ∫ P0 x z Ð x z aβ βd , (16) 4 Ð∞ δ 2 where P0( ) is the initial uncorrelated particle energy distribution. Let it be Gaussian with the rms devia- σ 0 tion ⑀. Averaging yields the expression that locates the cen- –2 ter of gravity of a beam in the accelerator: γ 1/2 ( ) β( ) 0  –4 x z = a0 z ------ γ(z) (17) × ( ∆µ 2 ) [µ( ) ϑ ] (c) (d) exp Ð s /2 cos z Ð 0 , 4 ∆µ where s is the mean shift of the betatron phase of a 2 nonequilibrium particle for the initial rms energy deviation σ⑀ [see (12)]. 0 The rms size of the bunch is found similarly: γ 2 γ 2 0 2 a0 0 2 –2 σ = ----σβ + ------{1 Ð [cos (µ Ð ϑ ) x γ 2 γ 0 (18) –4 ( µ ϑ ) ] ( ∆µ 2) } Ð cos 2 Ð 2 0 exp Ð s , – 40 – 20 0 20 40 – 40 – 20 0 20 40 where σβ is the initial rms transverse size of the bunch. xinv, mm The mean contribution to the increase in the rms Fig. 2. Phase ellipse splitting in the invariant phase plane emittance is determined by averaging the obtained (x, x') for beam coherent betatron oscillations in the main expression over the instantaneous phase of betatron accelerator of the SBLC. The phase ellipses of equilibrium- oscillations µ within (0, 2π). Ignoring the dispersion of energy particles (solid line) and those with higher (dashed the amplitude betatron function, we will have for the line) and lower (dotted line) energies are shown. E = (a) 3.15, (b) 25, (c) 100, and (d) 250 GeV. dispersion dilution of the bunch emittance: 2 γ a0 0 2 ∆ε = ε Ð εβ = ------[1 Ð exp(Ð∆µ )], (19) oscillations of a particle with a small initial relative 2 γ s δ energy deviation 0 from the equilibrium value [the ε ε term x in (3)] can be written as where β and are the natural and rms instantaneous b beam emittances. ( , δ) ( , δ) ( , δ) x z = xc z + xβ x , (13) 2 If it is taken into account that a0 is the area of the where initial central phase ellipse, the emittance dilution is limited by the half-area of the ellipse of coherent beta- γ 1/2  0  tron oscillations of the beam: x , β(z, δ) = a , β β(z)------c 0  γ (z) (14) 1γ 2 2 × [µ( ) ∆µ( , δ) ϑ ] ∆ε = ------0(γ x + 2α x x' + β x' ), (20) cos z + z Ð 0, β . max 2 γ x 0 x 0 0 x 0 ϑ α β γ Here, a0 and 0 define the initial position of the center where x, x, and x are the Twiss matrix parameters at of gravity of the bunch on the phase plane (x, x') and aβ the exit to the accelerator. ϑ and β define the initial position of the particle relative Note that the BEDD effect does not depend on the to the center of the bunch. natural emittance. At a high beam spread, the effect Then, the instantaneous center of gravity x and rms becomes a crucial factor if the vertical emittance is size σ = 〈∆x2〉1/2 of the bunch are defined as the means small, since the machine ellipse may considerably x exceed the beam natural emittance. Hereafter, it will be over energy spread and initial coordinates of the parti- supposed that the initial phase ellipse of coherent beta- cles in the bunch: tron oscillations coincides with the natural phase ∞ ellipse, or, in other words, that the initial amplitudes ( ) (δ) 〈 ( , δ)〉 δ (deflection and angle) of the center of gravity lie on the x z = ∫ P0 x z aβ, ϑβd , (15) 2 ε Ð∞ one-sigma contour (a0 = β0). Figure 2 shows the evo-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DISPERSION DILUTION OF THE BEAM EMITTANCE 891 lution of the normalized phase ellipses of particles with x'inv, mrad different energies in the main linear accelerator of the SBLC collider. As the particles are accelerated, the 4 ellipses in the phase plane are split so that low-energy particles are ahead of equilibrium particles in phase, 2 M while high-energy ones lag behind. Bunch particles E + ∆E tend to occupy the machine phase ellipse of coherent 0 betatron oscillations irrespectively of the initial oscilla- 0 tion phase (Fig. 3). Of importance here is that the center E – ∆E of gravity of the beam approaches the accelerator axis –2 0 E0 and the coherent betatron oscillation amplitude expo- A nentially drops with increasing BEDD, which is associ- –4 ated with Landau collisionless damping [5]. For an – 50 – 40 – 30 – 20 – 10 0 10 20 30 40 50 uncorrelated initial Gaussian particle energy spread in x , mm a bunch, the expression for Landau damping parameter inv follows from formulas (12) and (17): Fig. 3. Machine phase ellipse (M) and actual phase portrait (A) of a beam after particle spreading in the phase plane at µ γ 2 1 2 1 1 0 1 α beam coherent oscillations in the SBLC linear accelerator. α (z) = --∆µ = -- σ⑀ tan------(1 Ð g ) . (21) E = 250 GeV. L 2 s 8 2 ∆γ α ∆ε ε At small beam spreads (ξ Ӷ 1), the BEDD effect can / , % 40 be approximated as (a)

µ γ 2 30 α = 0 ∆ε 2 2 1 0  1 α 2 ------≈ 2σ⑀ tan ------(1 Ð g ) . (22) ε 2 ∆γ α2 20 The BEDD evolution along the main linear acceler- α = 0.5 ator in the SBLC and TESLA colliders [see (19)] are 10 shown in Fig. 4 (dashed line). The solid line stands for the BEDD numerical simulation in the accelerator. The 0 analysis and numerical simulation are seen to be in 3.0 good agreement. The same is also true for the variation (b) of the Landau damping parameter with particle acceler- 2.5 ation. Thus, the derived formulas can be thought of as 2.0 α the rigorous analytic description of uncorrelated BEED = 0 for the case of coherent betatron oscillations in linear 1.5 accelerators. Note that, if the autophasing condition 1.0 (α = 0.5) [8] is met, the uncorrelated BEDD is markedly suppressed, as follows from the data for the SBLC 0.5 design (Fig. 4, dotted line). It is also worthy to notice that the BEDD effect at coherent oscillations is revers- 0 10 20 30 40 50 60 70 80 90 100 ible. As shown in [17], an increase in the emittance can E, GeV be almost totally avoided if particles on the phase plane are redistributed in a dispersionless positive-chromatic- Fig. 4. Relative dilution of the rms emittance of a bunch at beam coherent oscillations. Solid line, particle tracing; ity arc that is placed in the injection stage. dashed line, analytic description. (a) SBLC (δ = 0.01) and δ 0 (b) TESLA ( 0 = 0.014). DISTURBANCE AND CORRECTION δ OF THE CENTRAL TRAJECTORY tory ( 0 = 0, x0 = 0, x0' = 0) can be represented in the form We now proceed to a study of the disturbance term in the equation of motion. Recall that disturbance is due β(z) to lens misarrangement. Let us ﬁrst consider the distur- x (z) = ------∑K L x β γ sin[µ(z) Ð µ(z )], d γ(z) k qk qk k k k bance of the central trajectory with regard for random k (23) lens offsets from the accelerator axis. In view of (3), (5), and (6), the partial solution (of the equation of where quantities with the subscript k refer to the corre- motion) that corresponds to the disturbed central trajec- sponding values in the kth quadrupole lens.

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A2, mm mrad aging the particle track over 100 or more sets of random 4 lens offsets. In the presence of an initial uncorrelated particle 3 energy spread in a bunch, nonequilibrium particles will 1 tend to occupy the disturbed phase ellipse of the accelerator. For a nonequilibrium particle with an initial 2 3 δ energy deviation 0, its trajectory will then be deﬁned by the expression 4 1 2 β(z) x (z, δ) = ------∑K L x β γ d γ(z) k qk qk k k (27) 0 25 50 75 100 125 150 175 200 k E, GeV × [µ µ ∆µ , ] , δ sin (z) Ð (zk) + (zk z) + xβ(z ), Fig. 5. Area variation for the rms disturbed machine ellipse along the accelerator (particle tracing). Averaging over where 100 sets of random lens displacements from the axis. The rms deviation is (1) 0.1 (SBLC) and (2) 0.5 mm (TESLA). γ µ γ α 0 1 1  k (3, 4) analytic description. ∆µ(z , z) = Ð2δ ------tan------1 Ð ---- (28) k 0∆γ 2 α  γ 

The random lens offsets are assumed to be mutually is the phase shift of the betatron oscillations for the uncorrelated; hence, cross terms make no contribution nonequilibrium particle under the action of the kth qua- to the rms displacement of the central trajectory drupole lens. 〈 〉 ≠ ( xqkxql = 0 at k l). As for free betatron oscillations, we introduce the rms disturbed instantaneous phase Assuming, as before, the Gaussian initial particle ellipse of the beam [20]: energy distribution and performing averaging over particle energies and positions, we will obtain the expres- γ 〈 2〉 α 〈 〉 β 〈 2〉 2 sion for the rms transverse beam size: x xd + 2 x xd xd' + x xd' = A . (24) π The area of the disturbed ellipse (divided by 2 ) is 2 1 2 β(z) 2 2 σ (z) = -- 〈 x 〉 ------∑K L β γ given by x 2 q γ(z) k qk k k k (29) 2 2 〈 x 〉 µ 2 2 q γ n × [1 Ð exp(Ð∆µˆ )] + σβ (z), A = 16------γ------∑ n tan----- k Lc (z) 2 n (25) ∆µ 2 where ˆ k is now the phase shift of the betatron oscil- 〈 x 〉 γ µ 1 γ  γ  2 Ð α = 16------q------0- tan-----1 ------0 ---- Ð 1 . lations for the nonequilibrium particle with the rms 2 ∆γ 2 2 Ð α γ γ  energy deviation. Lc 0 α β γ In deriving the rms beam size, averaging was also Here, x, x, and x are instantaneous Twiss matrix µ parameters and L is the FODO cell length. In deriving performed over the instantaneous phase (z) within c (0; 2π). On averaging, account was taken of the fact (25), we made use of the following relationships for the that the undisturbed trajectory averaged over the initial amplitude functions of a symmetric FODO cell [25]: amplitudes (aβ, ϑβ) goes to zero and the rms deviation µ is coincident with the instantaneous undisturbed rms β β 2Lc max + min = ------, KLq Lc = 4sin---. (26) 2 2 sinµ 2 beam size 〈xβ 〉 = σβ . Supposing that a single cell makes a negligible contribution to the phase shift Notice that the rms displacement of the lens and tol- (which is justiﬁed for ∆γ/γ Ӷ 1), the rms beam size can 2 erance aq are related through the relationship aq = be represented as the sum over the periodicity cells: 〈 2 〉 3 xq . As shown in [20], expression (25), which relates 2 α 2 〈 x 〉 β(z) µ γ  the area of the rms disturbed phase ellipse to the key σ ( ) q 1 0 γ x z = 8------γ---(-----)-tan-----∑γ---- n Lc z 2 n accelerator parameters, yields the exact rms distur- n (30) bance of the central trajectory for uncorrelated random × [ ( ∆µ 2)] σ 2 ( ) lens offsets. The variation of the disturbed phase ellipse 1 Ð exp Ð ˆ n + β z . area along the main accelerator in the SBLC and TESLA designs is depicted in Fig. 5. The rms phase Passing from the sum to the energy integral, one ellipse areas converge to the exact solution after aver- obtains the fairly good approximation for an increase in

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DISPERSION DILUTION OF THE BEAM EMITTANCE 893 the admittance (provided that the beam spread is ∆ε/ε small): 350 (a) ∆ε 〈 2〉 γ 3 300 ≈ xq σ2 0  ------32------⑀ ------ 250 εβ ε L ∆γ 0 c (31) 2 Ð α 200 3 µ 1  γ  × tan -----1 ------Ð 1 . 2 γ  150 2 4 Ð α 0 100 Note the very strong dependence of the BEDD on 50 the phase incursion per period of the focusing system µ ∆γ 0 1 and the energy gain per periodicity cell . Figure 6 compares the relative emittance increase along the 30 (b) main accelerator for the SBLC and TESLA colliders. 25 The rms beam sizes were averaged over 100 sets of lens offsets. Clearly, the central trajectory of the beam needs 20 correction. 15 Let us correct the central trajectory by measuring the position of the center of gravity of the beam and find 10 the rms BEDD. We assume that beam position monitors 5 are embedded in each of the quadrupole lenses with a misarrangement b relative to the center of the kth lens. k 0 10 20 30 40 50 60 70 80 90 100 Let the kth lens be randomly displaced by xqk from the E, GeV accelerator axis. The misarrangements of the accelerator components are assumed to be constant. Once the Fig. 6. BEDD in the case of the disturbed central trajectory center of gravity in the kth lens has been determined, (solid line). Dashed line, analytic curve. (a) SBLC (δ = µ π δ µ π 0 the trajectory of subsequent bunches is corrected 0.01, = /2) and (b) TESLA ( 0 = 0.014, = /3). toward the lens center by previous dipole correctors with a resolution d . In so doing, the central trajectory k As usual, we will assume that the additional disper- remains disturbed, but the disturbance does not grow sion function due to dipole correctors goes to zero at the along the accelerator. The center-of-gravity displace- end of the accelerator and does not contribute to the rms ment in the kth quadrupole lens after correction, xck, is beam emittance. Then, the solution of (34) using the given by Twiss matrix element M12 [see (6)] appears as x = x + b + d . (32) ck qk k k z γ ∆ δ 0 [ ] δ , Note that all of the quantities are random and mutu- x(z) = 0∫γ------K x(z') xc(z') Ð xq(z') M12(z' z)dz'.(35) ally uncorrelated. The rms deviation of the center of (z') gravity along the accelerator is expressed as 0 With the above correcting technique, the BEDD 〈 2〉 〈 2〉 〈 2 〉 〈 2 〉 xc = xq + xBPM + xRES , (33) exact value can be found if, for any given set of random quantities xqk, bk, and dk along the accelerator, the dis- 〈 2 〉1/2 where xq is the rms deviation of the lenses from the turbed central trajectory xc(z) is represented as a smooth function of coordinate z, F(z), with the exten- accelerator axis, 〈x 2 〉1/2 is the rms deviation of the BPM sion F(zk) = xck. The square of the relative displacement beam position monitors from the centers of the lenses, of a nonequilibrium particle averaged over instanta- 〈 2 〉1/2 neous phases and energy deviations is then given by and xRES is the rms resolution of the monitors. γ It is easy to check that, correct to the first order in 〈∆ 2 〉 1 〈 2 〉σ 2β 0 δ Ӷ x (z) = -- xB ⑀ (z)------energy deviation ( 0 1), the displacement of a non- 2 γ(z) equilibrium particle from the disturbed central trajec- (36) ∆ γ tory, x = xδ Ð xc, satisfies the equation × 0 2 2 (β β ) ∑γ----Kn Lqn n max + n min , γ n ∆ '∆ ( δ)∆ n x'' + -γ-- x' + K x 1 Ð x (34) 2 2 2 where the quantity 〈x 〉 = 〈x 〉 + 〈x 〉 depends = δK (x Ð x ) Ð δG (z), B BPM RES x c q x only on the arrangement accuracy of the correctors and ρ where Gx(z) = ecB(z)/E = 1/ s(z) is the instantaneous their resolution. curvature of the central trajectory disturbed by the Passing to integration in view of formulas (26) for dipole correctors. the amplitude functions, we come to the expression for

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∆ε/ε, % betatron oscillations, the disturbance of the central trajectory, and the correction of the disturbed trajectory 2.0 were investigated. The obtained expressions allow researchers to precisely estimate the beam quality and α 1.5 = 0 predict its behavior in high-energy linear accelerators, α = 0.5 as well as to optimize the focusing and correcting sys- 1.0 tems. Under the conditions of the energy-dependent betatron phase incursion, autophasing was shown to 0.5 considerably suppress the uncorrelated BEDD. SBLC 0 ACKNOWLEDGMENTS 6 The author thanks R. Brinkmann, J. Rossbach, and 5 R. Wanzenberg for the support and valuable remarks. α = 0 4 Special thanks to É. Gazazyan for the fruitful com- ments and discussion. 3 2 REFERENCES 1 TESLA 1. A. Skrinsky, in Proceedings of the 12th International 0 20 40 60 80 100 Conference on High Energy Accel., Fermilab, Batavia, E, GeV IL, 1983, pp. 104Ð108. 2. J. T. Seeman, SLAC-PUB-4886 (1989), p. 5. Fig. 7. BEDD in the linear accelerator for the corrected central trajectory. Averaging over 100 sets of monitor displace- 3. N. Holtcamp, in Proceedings of the 15th International ments from the lens centers and their resolutions (solid line). Conference on High Energy Accel., Hamburg, 1993, The rms deviations for the monitor displacements and reso- pp. 770Ð776. lutions are, respectively, 0.1 and 0.01 mm for the SBLC and 4. K. Bane, SLAC-PUB-4169 (1986), p. 44. 0.5 and 0.05 mm for the TESLA. Dashed line, analytic curve. 5. A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley, New York, 1993), p. 286. the relative dilution of the emittance along the acceler- 6. V. Balakin, A. Novokhatsky, and V. Smirnov, in Proceed- ator: ings of the 12th International Conference on High 2 α Energy Accel, Fermilab, Batavia, IL, 1983, pp. 119Ð120. ∆ε 2 〈 x 〉 γ µ 1 γ σ B 0 1  0 7. V. Balakin, Preprint No.88-100, IYaF (Budker Institute --ε---- = 8 ⑀ -ε------∆-----γ- tan-----α--- 1 Ð --γ-- . (37) 0 Lc 2 of Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 1988). Note that the correction of the central trajectory considerably diminishes its smearing even in compari- 8. V. M. Tsakanov, Phys. Rev. ST Accel. Beams 1, 041001 son with free betatron oscillations of a beam. Figure 7 (1998). shows the relative dilution of the beam lateral emittance 9. I. M. Kapchinskiœ, Usp. Fiz. Nauk 132, 639 (1980) [Sov. for the SBLC and TESLA designs after such a correc- Phys. Usp. 23 (4), 835 (1980)]. tion. Again, the results of particle tracing are averaged 10. I. M. Kapchinskiœ, The Theory of Linear Resonant over 100 sets of center-of-gravity random displace- Ampliﬁers (Énergoizdat, Moscow, 1982), p. 240. ments from the accelerator axis. Since the vertical emit- 11. Linear Accelerators of Ions, Ed. by B. P. Murin (Atom- tance is, as a rule, markedly smaller than the lateral, the izdat, Moscow, 1978), Vol. 1, p. 264. accuracy of arrangement of the lenses and correcting 12. V. Balakin, A. Novokhatsky, and V. Smirnov, in Proceed- system in the vertical plane must be much higher. With ings of the 12th International Conference on High the autophasing condition fulﬁlled (α = 0.5) and the Energy Accel., Fermilab, Batavia, IL, 1983, p. 121. same correcting system used, the uncorrelated BEDD 13. R. D. Ruth, SLAC-PUB-4436 (1987), p. 22. considerably decreases, as before (the dashed line for the SBLC design). 14. R. D. Ruth, in Proceedings of 1991 Part. Accel. Confer- ence, San Francisco, 1991, p. 2037. 15. T. Raubenheimer, SLAC-387 (1991), p. 111. CONCLUSION 16. M. Drevlak, M. Timm, and T. Weiland, in Proceedings of The exact analytic expressions for BEDD in high- the 18th International Lingc Conference, Geneva, 1996, energy linear accelerators were derived for the case p. 621. when the betatron oscillation phase randomly varies 17. V. M. Tsakanov, Nucl. Instrum. Methods Phys. Res. with the equilibrium particle energy. Free coherent A 421, 423 (1999).

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18. A. Mosnier and A. Zakharian, in Proceedings of the 4th 23. E. D. Courant and H. S. Snyder, Ann. Phys. (N.Y.) 3, 1 European Part. Accel. Conference, London, 1994, (1958). p. 1111. 24. M. Sands, The Physics of Electron Storage Rings, 19. M. Drevlac, DESY 95-225 (1995), p. 148. SLAC-121 (1971), p. 172. 20. V. M. Tsakanov, Zh. Tekh. Fiz. 69 (7), 95 (1999) [Tech. Phys. 44, 825 (1999)]. 25. J. Rossbach and P. Schmuser, in Proceedings of the CERN Accelerator School, CERN 94-01, 1994, p. 17. 21. G. Guignard, in Proceedings of 1993 Part. Accel. Con- ference, New York, APS, 1993, p. 3600. 26. J. Buon, in Proceedings of the CERN Accelerator 22. Conceptual Design of a 500 GeV e+ eÐ Linear Collider School, CERN 94-01, 1994, p. 89. with Integrated X-ray Laser Facility, Ed. by R. Brink- mann, G. Materlik, J. Rossbach, and A. Wagner (DESY, Hamburg, 1997), Vols. I and II, p. 275. Translated by V. Isaakyan

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Technical Physics, Vol. 45, No. 7, 2000, pp. 896Ð904. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 83Ð91. Original Russian Text Copyright © 2000 by Kalinin, Kozhevnikov, Lazerson, Aleksandrov, Zhelezovskiœ.

ELECTRON AND ION BEAMS, ACCELERATORS

Dynamical Chaos in a Charged-Particle Flow Produced by a Magnetron Injection Gun: Numerical Simulation and Experiment Yu. A. Kalinin, V. N. Kozhevnikov, A. G. Lazerson, G. I. Aleksandrov, and E. E. Zhelezovskiœ Chernyshevsky State University, Saratov, 410026 Russia E-mail: [email protected] Received March 4, 1998; in ﬁnal form, July 2, 1999

Abstract—A theoretical and experimental study of spatial and temporal current oscillations in a magnetron injection gun is presented. Basic features of the device operation are ascertained. Complicated system dynamics, namely transitions from regular to chaotic oscillations in response to changing the system parameters, is revealed. The conclusions are drawn based on the analysis of the computed electron trajectories and output- current waveforms and spectra. Experimentally, the spectra of the beam-current oscillations, noise-intensity spectral density, etc., are obtained. Strong broadband microwave oscillations of the output current are observed for a wide range of the lengths of the emitting and the nonemitting portions of the cathode. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION DESCRIPTION OF THE MODEL AND THE CONDITIONS OF THE NUMERICAL Complicated dynamical phenomena (particularly, SIMULATION dynamical chaos) in nonlinear oscillators have attracted considerable interest for many years due to their signif- A schematic of a MIG is shown in Fig. 1. It is seen icance for both basic research and various applications. that a real MIG is a fairly complicated device; there- Nevertheless, among a large number of papers devoted fore, the studying of the processes occurring in it is a to numerical and experimental investigations of chaos, rather intricate problem. Hence, when modeling the only few studies concern the dynamics of systems with dynamics of the electron flow in such a device, we will distributed parameters or many degrees of freedom. use a simplified model shown in Fig. 2. We consider a This fact stems from serious difficulties faced by any- plane-electrode diode in a magnetic field; the dimen- one who tries to construct realistic models of such sys- sions of the cathode and the anode in the x- and z-direc- tems under the conditions where they behave in a com- tions are assumed to be much larger than the interelec- plicated fashion. By now, a lot of papers have been trode spacing. Our numerical analysis of the electron devoted to the dynamical chaos of oscillations in elec- motion employs a version of a particle method (see, e.g., [5]) in which we allow for the forces acting on a tron flows without a magnetic field (O-type tubes); however, there are few papers in which analogous phenomena were studied in the presence of the crossed y electric and magnetic fields (M-type tubes). As early as in [1, 2], it was noted that M-type tubes can produce E 4 B intense internal noise, which makes them promising for 5 practical applications. The reason for an anomalously high noise level in M-type tubes is still poorly understood. For M-type 3 diodes and guns, an attempt to explain the internal noise in terms of the complicated dynamics of an elec- 12------z tron stream in crossed fields has been made in [3]. Here, we extend this approach to the magnetron injection gun Fig. 1. Schematic of the magnetron injection gun: (1) cathode (the dashed line shows the boundary of the emitting (MIG), which may be useful for designing high-power region), (2) injection region, (3) helical electron beam, microwave noise sources [4]. (4) control electrode, and (5) anode.

DYNAMICAL CHAOS IN A CHARGED-PARTICLE FLOW 897

y x be supplemented with the equation of motion along the z-axis: E úzú = Py, (1) where z and y are dimensionless coordinates and P is the ratio between the amplitudes of the longitudinal and transverse components of the electrostatic field. B z Computer simulations of the complicated electron dynamics in an MIG involved the following control Fig. 2. Plane-electrode diode (a simplified model of a real parameters: the emission current, the cathode length, device). the drift length (i.e., the length of the nonemitting portion of the cathode), and the parameter P = Es/Ea (the ratio between the longitudinal, Es, and transverse, Ea, particle in the y- and the z-directions only. The only y- components of the electric field). The fixed parameters component of the space charge field is taken into were the magnetic induction, the initial electron veloc- account. The general concept of the model was formu- ity, the interelectrode spacing, etc. Note that such lated in [5], and its application to the problem of parameters as the emission current, the cathode length, dynamical chaos was reported in [3]. Neglecting the and the drift length are also used in the models of a edge effects, the configuration under study implies the magnetic diode and an M-type gun. following structure of the fields. The alternating elec- The numerical simulation yielded graphic represen- tric field has the Ez component. In the z-direction, a tations of electron trajectories, waveforms of the output drawing field Ez is also applied, whose amplitude and induced currents, and spectral charts (Figs. 3Ð6). depends on y as Ez = Py and is constant along the z- direction. The magnetic field is uniform and has only one component B = B . Thus, the crossed static fields RESULTS OF NUMERICAL z 0 SIMULATIONS Ey and B0 govern the motion of emitted electrons in the (x, y) plane, whereas the drawing electric field controls Numerical simulations revealed transitions to the electron motion along the z-axis. intense chaotic electron motion, which resembles turbulence and manifests itself in chaotic noiselike oscil- With allowance for the above assumptions, the lations of the output current. The parameters of these equations of electron motion in a M-type MIG [3] must oscillations are very sensitive to the above control

S, dB –10 (a) (b)

–20

–30

–40

–50

–10 (c) (d)

–20

–30

–40

–50 0 10 0 10 f/f0

ω ω Fig. 3. Numerical simulation: the MIG output-current spectrum at a small emission current ( p/ c = 0.16) for the cathode length Lc = (a) 1, (b) 2, (c) 5, and (d) 10 mm.

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898 KALININ et al. 5.0 (f) (b) (d) is the distance between anode c Ð

a L 0 f / 2.5 f = (a, b) 0.01, (c, d) 0.25, and (e, f) 2.0; dB 0 P

, S 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 – – – – – – – – – – – – – – – (e) (a) (c) 1500 1000 , comp. steps T 500 0 a – c 0 0 0 L 1.0 0.5 1.0 0.5 1.0 0.5 Numerical simulation: (a, c, e) the electron trajectories and (b, d, f) output spectrum for . 4. Fig and the cathode.

TECHNICAL PHYSICS Vol. 45 No. 7 2000

DYNAMICAL CHAOS IN A CHARGED-PARTICLE FLOW 899

La–c 1.0 (a)

0.5

0 1.0 (b)

0.5

0 1.0 (c)

0.5

0 1.0 (d)

0.5

0 0 500 1000 1500 T, comp. steps

Fig. 5. Numerical simulation: electron trajectories for different values of the cathode length.

parameters: I0 (the maximum emission current), Lc (the increase in the cathode length or/and a decrease in the cathode length), Ldr (the drift length), and P. parameter P = Es/Ea. Figure 3 displays the normalized The scenario of the transition to chaos is very simi- noise spectral density (NSD) as a function of the fre- lar to those for a magnetic diode and an M-type gun [3] quency (normalized to the cyclotron frequency) for dif- but is quite different from those for systems with few ferent values of the cathode length. Figure 4 shows how degrees of freedom. the electron trajectories and NSD vary as the parameter P increases. For small values of the emission current, a transition to the chaotic regime of electron motion occurs and the Figure 3 demonstrates that an originally excited oscillations of the output current appear with an periodic oscillation becomes quasi-periodic and then

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900 KALININ et al.

A, dB Iind Ip (a) Ig (b) –10 2.832 0.61 –20 Iind 1.955 0.41 –30 –40 0.978 0.20 –50 0 2.5 5.0 0 500 1000 1500 f/f0 T, comp. steps

Iint Ip I (c) 2.837 p 0.627 Iint 1.954 0.416

0.977 0.208

0 500 1000 1500 T, comp. steps

A, dB Iind Ip (d) Ig (e) –10 2.738 0.61 –20 1.852 0.40 –30 Iind 0.20 –40 0.976 –50 0 2.5 5.0 0 500 1000 1500 f/f0 T, comp. steps

Fig. 6. Numerical simulation: (a, d) the output oscillation spectra and (b, c, e) waveforms of the gun current (Ig) and induced current (Iind) for I0 = (a, b) 1.442 (c) 1.443Ð1.449, and (d, e) 1.450 A. chaotic. Note that the energy of chaotic oscillations their behavior is transformed completely, and a kind of concentrates in the lower frequency region. The excita- fluid turbulence arises in a fraction of the electron flow; tion and development of this type of oscillation stem there is a drift motion of a jet, on which a disordered primarily from the oscillations of the beam boundary motion of each of the other individual jets is superim- due to the modulation of the time required for different posed. The spectrum of such oscillations is fairly wide, groups of electrons to reach this boundary. The beam and their intensity concentrates in the lower frequency boundary is formed by the groups of electrons (each region (Fig. 3d). As the emission current increases fur- group consisting of large number of particles) simulta- ther, chaos sets in at smaller drift lengths, appearing as neously occurring in this spatial region. Figure 5 shows a strong and broadband output current oscillations the evolution of the electron motion with an increase in (Fig. 6a). the cathode length. One can see the onset of chaotic The transition to chaos resemble Landau’s scenario oscillations of the beam boundary (Figs. 5a, 5b), the for the onset of turbulence. However, there is some dif- expansion of chaos (Fig. 5c), and the eventual estab- ference. At certain values of the parameters, the MIG lishment of turbulence in the entire interelectrode space produces a stationary current. As some parameter var- (Fig. 5d). ies, quasi-periodic (sometime, single-frequency) oscil- ω ω As the emission current increases ( p/ c > 0.5, lations are excited. As the parameter is varied further, ω ω where p and c are the plasma frequency and the the number of spectral components increases (the noise cyclotron frequency, respectively), the output current appears). The further variation in the parameter leads to exhibits oscillations due to the development of turbu- a decrease in the number of spectral components; then, lence in the electron flow (Fig. 5d). Starting from a cer- the noise level increases again, and so on. Finally, the tain value of the emission current, the electron trajecto- transition to the regime of strong turbulence occurs. We ries begin to drift as a whole in the longitudinal direc- also revealed that, at certain intermediate values of the tion under the action of an accumulated space charge, parameter, the oscillations of the MIG current may dis-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DYNAMICAL CHAOS IN A CHARGED-PARTICLE FLOW 901 appear completely so that the output current becomes S, A2/Hz time independent. This occurs in a narrow parameter 1 range. After some time, quasi-stationary oscillations 6 × 10–16 appear again; further, they may transform into turbu- × –16 lence and then terminate. Such a termination (quench- 5 10 ing) of oscillations may recur several times (Fig. 6). For 4 × 10–16 the ﬁxed parameters, we observed up to three quench- 3 × 10–16 ings. Under fully developed chaos, the NSD in an MIG 2 was found to exceed the spectral density of the Schot- 2 × 10–21 tky noise in electron guns by six to seven orders of magnitude (Fig. 7). 7.5 8.0 8.5 9.0 9.5 10.0 I , mA We also carried out the stability analysis of the solu- 0 tions for different ranges of the control-parameter val- Fig. 7. Numerical simulation: the noise spectral density ues. In particular, we studied the dependence of the nor- S vs. the maximum emission current I0 for (1) fully-devel- malized bandwidth on the step size (expressed in frac- oped chaos and (2) Schottky noise. tions of π) and on the number of the layers between any two successively extracted layers (Figs. 8, 9). It was f/f0 found that the solution is the most stable if the layers 30 are extracted at each step (or at least every ﬁfth or sixth 25 step) and if the step size (expressed in fractions of π) is 15Ð30 (an optimum value is 25). Such a large step size 20 was dictated by computational problems. 15 10 5 EXPERIMENTAL RESULTS 0 2 4 6 8 10 12 14 The experiments were carried out with a model MIG n schematically shown in Fig. 10. The device includes a Fig. 8. Numerical simulation: the normalized bandwidth conic cathode with a 1.5-mm-wide porous metallic f/f0 vs. the number n of layers between any two successively thermionic emitter, control electrodes, and an anode. withdrawn layers. The anodeÐcollector spacing is 3 mm. The angle of the cathode face with the axis is 15°. f/f0 The NSD in the MIG beam was measured with an 30 analyzer, which was a segment of a slow-wave helix attached to a shield by means of ceramic rods and 20 matched with an output coupler. Behind the analyzer, there was an electron collector (microwave probe), 10 which was connected to the output coupler via a match- 0 ing unit. The collector was designed as a segment of 1 11 21 31 41 51 N coaxial cable, which allowed us to measure the NSD.

The MIG was placed into a demountable vacuum Fig. 9. Numerical simulation: normalized bandwidth f/f0 assembly under continuous evacuation [6]. The mag- vs. the step size (expressed in fractions of π). netic field was produced by permanent magnets; the maximum magnetic field strength was 2000 Oe. 6 8 9 10 The setup scheme allowed us to move the magnetic 1 3 4 7 focusing system in both the longitudinal and transverse U U directions. The measurements were carried out in a y a pulsed regime (anode modulation). The other electrodes were connected to dc voltage sources. The signals from the probes (the helix and the collector) were 5 examined with an S4-60 spectrum analyzer, which 2 B operates in the range 200 MHz to 19 GHz, and with a high-frequency S1-74 spectrum analyzer. The analyz- ers were exploited in conjunction with high-Q filters Fig. 10. Schematic of the model gyrotron MIG: (1) cathode, (the passband being 2 to 4 MHz), which offer the 1- to (2) emitter, (3) control electrode, (4) anode, (5) electron beam, (6) slow-wave helix, (7) absorber, (8) output coupler, 2-GHz and 2- to 4-GHz tuning ranges; the detected out- (9) high-frequency probe (collector), and (10) inner conduc- put signals were recorded by an EPP-09 recorder. tor of the collector.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 902 KALININ et al.

S, A2/Hz 40 (a) (b) 30 20 10

40 (c) (d) 30 20 10

40 (e) (f) 30 20 10 0.2 0.5 0.8 1.2 1.5 1.8

40 (g) (h) 30 20 10

40 (i) (j) 30 20 10

(k) (l) 30 20 10 2.1 3.0 3.6 4.4 5.0 5.6 F, GHz

Fig. 11. Experiment: the oscillation spectra of the electron beam for (a, b, g, h) U/U0 = 1.0 and I/I0 = 1.0, (c, d, i, j) U/U0 = 0.57 and I/I0 = 0.43, and (e, f, k, l) U/U0 = 0.33 and I/I0 = 0.11. Panels (aÐf) refer to 0.2 < F < 2.0 GHz, and panels (gÐl) refer to 2.0 < F < 6.0 GHz.

Figure 11 presents typical spectra of stochastic the change in the heater voltage. Note that, by varying oscillations in the beam, measured in the 200 MHz to the heater voltage, we could change the structure of the 6 GHz range in different operating regimes. It is seen oscillation spectrum and vary the amplitude of oscilla- that the intensity of oscillations is the highest at lower tions in a wide range. frequencies (400Ð500 MHz). The higher frequency components grow with an increase in the accelerating Thus, chaotic oscillations may arise in a high-cur- voltage and the beam current. rent beam produced by an MIG. The chaos stems from the presence of virtual cathodes. The properties of the Figure 12 displays the chaotic spectra for different excited oscillations can be controlled via the voltages values of the beam current, the latter being varied via applied to the electrodes, the magnitude and distribu-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DYNAMICAL CHAOS IN A CHARGED-PARTICLE FLOW 903

S, A2/Hz 40 (a) (b) 30 20 10

40 (c) (d)

10 0.2 0.5 0.8 0.2 0.5 0.8 F, GHz

Fig. 12. Experiment: the oscillation spectra of the electron beam for different values of the heater voltage: (a) U/Uh = 0.39 with I/I0 = 0.11, (b) U/Uh = 0.47 with I/I0 = 0.35, (c) U/Uh = 0.72 with I/I0 = 0.71, and (d) U/Uh = 1.0 with I/I0 = 1.0. tion of the magnetic field, and the magnitude and har- taining of oscillations in the MIG beams and explain monic content of an external signal (single-frequency, why the measured noise level is higher than the theo- multifrequency, or noise-like signal) that comes from a retical one. Specifically, two factors should be high- microwave source or from the output of the slow-wave lighted. The first factor is the formation of a virtual helix through a feedback loop. cathode (a minimum of the potential) near the MIG cathode; the influence of this factor increases with an increase in the width of the emitting belt at the cath- DISCUSSION ode. The second factor is the formation of a magnetic Figure 13 shows the NSD at frequencies of f = 5 and mirror in the region where the magnetic field 10 GHz versus the normalized heater voltage Uh = Uh0, increases. A substantial radial component of the mag- where Uh0 is the heater voltage corresponding to the netic field in this region gives rise to a second virtual space-chargeÐlimited current. The figure also shows cathode. The chaotic oscillations in high-current MIG the spectral density S(Uh/Uh0) of Schottky noise. It is beams result from a decreased drift velocity of elec- seen that the intensity of oscillations at the output of the trons and a significant scatter in the beam electron anode is much higher than the intensity of Schottky noise (by six to seven orders of magnitude) and does not fall and even rises at the heater voltages higher S, A2/Hz than U . h0 10–20 Note that the variations in the heater voltage Uh and 1 –18 the accelerating voltage U0 in the experiment corre- 10 spond to the variations in the emission current I0 and 2 the parameter P in numerical simulations. Therefore, 10–16 we can point out a qualitative agreement between the behaviors of the experimental and computed spectra 10–14 when the relevant parameters are varied. We also note 3 that, in both the experiment and numerical simulations, 10–12 the NSD in the MIG current is higher by six to eight orders of magnitude than the intensity of Schottky noise. 0.8 0.9 1.0 1.1 1.2 U /U The measured dependences of the spectra of cha- h h0 otic current-density oscillations on the electric and Fig. 13. Experiment: the NSD at the output of the anode vs. magnetic parameters in MIG-based systems allowed the normalized heater voltage at frequencies of (1) 5 and us to reveal the mechanism for the excitation and sus- (2) 10 GHz; curve 3 refers to Schottky noise.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 904 KALININ et al. velocity as well as from oscillations of the parameters MIGs could serve as high-power sources of broadband of the virtual cathodes. signals.

REFERENCES CONCLUSIONS 1. G. Baneman, in Crossed-Field Microwave Devices, Ed. The results of numerical and experimental studies by E. Okress (Academic, New York, 1961; Innostran- show that, in an MIG, chaotic oscillations whose spec- naya Literatura, Moscow, 1961), Vol. 1. tral intensity is higher than the intensity of Schottky 2. B. L. Usherovich, Electronics Reviews, Ser.: Microwave noise can arise. The numerical and experimental results Electronics (1969), Vol. 7. are in good qualitative agreement. Thus, the theoretical 3. E. E. Zhelezovskiœ, A. G. Lazerson, and B. L. Usherov- model presented can adequately describe the processes ich, Pis’ma Zh. Tekh. Fiz. 21 (18), 12 (1995) [Tech. occurring in real devices. Quantitative discrepancies Phys. Lett. 21, 730 (1995)]. between the theory and the experiment (including a 4. E. E. Zhelezovskiœ, USSR Inventor’s Certiﬁcate higher noise level measured) may be attributed to the No. 788993 (Moscow, 1980). assumptions of the model (such as replacing a realistic 5. L. M. Lagranskiœ and B. L. Usherovich, Voprosy gun conﬁguration with a plane-electrode diode and Radioélektron., Ser. I: Élektron. 1, 3 (1964). neglecting the cycloidal motion of electrons). 6. Yu. A. Kalinin and A. D. Essin, Experimental Methods and Tools in Vacuum Microwave Electronics (Saratov, Finally, the results obtained allow us to infer that the 1991), Part 1. anomalously high noise level observed previously (see, e.g., [3]) has a dynamical nature. We think that such Translated by A.A. Sharshakov

TECHNICAL PHYSICS Vol. 45 No. 7 2000

Technical Physics, Vol. 45, No. 7, 2000, pp. 905Ð908. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 92Ð95. Original Russian Text Copyright © 2000 by Dymont, Samtsov, Nekrashevich.

SURFACES, ELECTRON AND ION EMISSION

Effect of Thermal Annealing on Spectral Properties of Electrodeposited Carbon Films V. P. Dymont, M. P. Samtsov, and E. M. Nekrashevich Institute of Solid-State Physics and Semiconductors, Academy of Sciences of Belarus, Minsk, 220072 Belarus E-mail: [email protected] Received April 7, 1999

Abstract—The effect of thermal annealing on properties of carbon ﬁlms deposited on nickel electrodes by the electrodeposition method was studied. It has been shown that annealing at a temperature of 300°C results in the formation of nanosize diamond clusters. With an increase in the annealing temperature, the size of diamond clusters diminishes. At an annealing temperature of 900°C, all of the carbon enters into reaction with nickel, thus forming nickel carbide. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION EXPERIMENTAL RESULTS A number of properties of diamond-like films A vacuum anneal of carbon films results in visually (transparency in the visible and infrared spectral detectable modifications of the surface. A film annealed ranges, chemical stability, high hardness and good heat at 300°C changes in color from light-brown to almost conduction) make them an attractive material for use in black and remains physically intact. Annealing at various technological fields. Characteristics of the dia- 600°C makes the film still darker: it becomes black and mond-like carbon films largely depend on the method glassy. After annealing at 900°C, the film becomes and conditions of growth. Earlier, we developed a new transparent. For all samples of the Raman scattering technique of producing diamond-like carbon films (RS) spectra, infrared (IR) spectra and spectra of Auger using the electrolytic process. Description of this electrons were taken. method and some properties of the obtained coatings are given in [1, 2]. In this work, results of studies of the (a) As-deposited film. An RS spectrum of the as- electrodeposited films annealed at various temperatures deposited film with a broad asymmetric band in the fre- by the methods of Raman scattering, infrared absorp- quency range 1200Ð1700 cmÐ1 is shown in Fig. 1. In tion, and Auger spectroscopy are presented. Fig. 2, an IR absorption spectrum of the same film fea-

SAMPLE PREPARATION AND MEASUREMENT Intensity, arb. units TECHNIQUES The films were prepared by the electrolytic method at a low (Ð40°C) temperature. As an electrolyte, acety- lene solution in liquid ammonia was used. The electrol- ysis was carried out at a voltage of 10 V for 15 h. The film deposition technique is described in more detail elsewhere [2]. Under these conditions, fairly thick (1Ð2 µm) smooth light-brown high-ohmic films were obtained. Properties of as-deposited films and G films annealed at 300, 600, and 900°C were studied. The annealing was carried out for 1 h in quartz D ampoules pumped out to pressures not higher than 10−4 Pa. Raman scattering spectra in the range of 1000Ð 1800 cmÐ1 at room temperature were registered using the RAMALOG-44 spectrometer (wavelength of incident radiation λ = 514.5 nm). IR spectra in the range 1000Ð4000 cmÐ1 were taken with the PERKIN- 1000 1200 1400 1600 1800 ELMER 180 instrument in reflection geometry at room Raman shift, cm–1 temperature. Auger spectra were registered with the PERKIN-ELMER PHI-660 spectrometer. Fig. 1. Raman spectrum of the as-deposited film.

906 DYMONT et al.

Transmittance, arb. units bands in the 1000Ð1800 cmÐ1 range disappear. Consid- erable changes are seen in the Auger spectra as well (Fig. 6). The lines due to nitrogen are missing and, besides, signiﬁcant changes are found in the shape of the low-energy part of the carbon peak (inset in Fig. 6).

DISCUSSION Raman spectra of the two known crystalline forms of carbon, graphite (sp2 hybridization), and diamond (sp3 hybridization) have been considered in detail in [3]. For monocrystalline and polycrystalline diamond, only one peak at a frequency of 1332 cmÐ1 is observed. The spectrum of monocrystallyne graphite also consists of a line in the 1590Ð1600 cmÐ1 range. The Raman 1000 1200 1400 1600 1800 –1 spectrum of polycrystalline graphite consists of two Wave numbers, cm bands. A G-line (graphite line) is localized around 1600 cmÐ1 and a D-line (disordered graphite) in the Fig. 2. IR absorption spectrum of the as-deposited film. vicinity of 1355 cmÐ1. The D-line in the scattering spectrum is due to the presence of small-sized crystallites of dN(E)/dE graphite which causes the violation of the selection rules. The ratio of integrated intensities of the D- and G-lines (ID/IG) is inversely proportional to the size of crystallites [3]. The Raman spectra of carbon films containing carbon in sp2 hybridization are similar to spectra of finely crystalline carbon [4]. The absence in the RS spectra of the diamond line, even in those cases N O Ni where X-ray diffraction analysis indicates the presence Ni of diamond, is usually explained by the fact that the Ni efficiency of Raman scattering for diamond is lower by C a factor of 55 than for graphite. Therefore, the quantity of the sp3-hybridized carbon is judged by indirect evi- 100 200 300 400 500 600 700 800 900 dence. Therefore, the spectral position of the G-line Kinetic energy, eV gives an indication of the ratio of carbon in sp2 and sp3 Fig. 3. Auger spectrum of the as-deposited film. hybridization states in a sample [5]. Contributions to the profile of the RS line of a carbon film may come from other vibrations related to sp3 hybridization car- turing two broad absorption bands peaking at 1580 and bon. For example, the spectrum may contain the fol- 1350 cmÐ1 is shown. In an Auger spectrum of the film lowing lines: a line due to disordered sp3-bonded car- in Fig. 3, besides the carbon peak, peaks due to nitro- bon at 1140 cmÐ1; a line due to distorted sp3-bonded gen, oxygen, and nickel are present. carbon at 1488 cmÐ1; and a line due to hexagonal mod- (b) Film annealed at 300°C. An RS spectrum of this ification of diamond (lonsdaleite) at 1305 cmÐ1 [6]. film is shown in Fig. 4. It is generally similar to the Thus, RS spectra of carbon films can be represented spectrum of the previous sample, except for some as a superposition of Gaussian curves, with peak posi- enhancement of the scattering intensity at lower fre- tions, integrated intensities, and halfwidths as fitting quencies. The IR spectrum of the annealed film differs parameters. By comparing the obtained results with only slightly from that of the as-deposited film. There published data, one can make judgments about the are also no significant differences in the Auger spectra types of chemical bonds in the compounds comprising of these samples. the films under study, their relative content, as well as (c) Film annealed at 600°C. In the Raman spectra of the degree of perfection of the crystalline structure. this film (Fig. 5), further enhancement of the relative The experimental spectra could be approximated in intensity of the scattering band in the vicinity of most cases by four Gaussian curves under the assump- 1350 cmÐ1 can be seen. The Auger and IR spectra of the tion that two of the curves are related to sp2- and two samples remain practically unchanged. more, to sp3-hybridized carbon. The spectrum of the (d) Film annealed at 900°C. Spectral characteristics film prior to annealing is best described with two of the film annealed at this temperature show consider- Gaussian curves centered at 1352 cmÐ1 (D-line) and able transformations. In the RS spectrum, the scattering 1568 cm−1 (G-line). This shift of the G-line from the

TECHNICAL PHYSICS Vol. 45 No. 7 2000

EFFECT OF THERMAL ANNEALING ON SPECTRAL PROPERTIES 907 asymptotic position for graphite at 1600 cmÐ1 towards Intensity, arb. units smaller wave numbers may be caused by the presence of sp3-hybridized carbon. According to [7], the position of the G-line at 1568 cmÐ1 corresponds to a 70%-content of sp3-hybridized carbon. The ratio of integrated intensities of the D- and G-lines is equal to 0.5. This value is characteristic of hydrogenated amorphous silicon films (usually in the range 0.3Ð3.0), and its change is due to the difference in the number and/or size of the graphite clusters. The halfwidth of the D- and G-lines G is equal to 180 cmÐ1. The same value is characteristic of carbon films containing small graphite clusters. 1 2 The presence in the IR absorption spectrum of D bands in the region from 1350 to 1600 cmÐ1 suggests that part of the carbon atoms has been replaced by nitrogen. This conclusion is based on an observation that, in the IR range, vibrations related to sp2-bonded carbon are inactive. Substitution of nitrogen for part of the carbon lifts this prohibition [8]. The presence of 1000 1200 1400 1600 1800 nitrogen in the film is confirmed by the nitrogen line in Raman shift, cm–1 the Auger spectra. Regrettably, qualitative determina- Fig. 4. Raman spectrum of the film annealed at 300°C. tion of nitrogen in the film from available data is not possible. It can only be hypothesized that the nitrogen Intensity, arb. units content does not exceed 5Ð10 at. %. It is highly probable that, at room temperature, the film consists of small graphite clusters doped with nitrogen and uniformly distributed in the matrix of amorphous hydrogenated carbon. After annealing at 300°C, the D- and G-components of the Raman scattering spectrum become narrower G and shift towards positions asymptotic for graphite (Fig. 4), which is evidence of the structural ordering of sp2-hybridized forms of carbon. The number and size 1 2 of the graphite clusters do not change because the ratio D of integrated intensities ID/IG remains equal to about 0.5. Besides, components related to the sp3-hybridized carbon appear in the film spectrum; a broad line centered at 1320 cmÐ1 (curve 1 in Figs. 4, 5), which can be related to finely dispersed diamond, and a line centered at 1440 cmÐ1 (curve 2 in Figs. 4, 5), which can be 3 1000 1200 1400 1600 1800 related to distorted sp -hybridized carbon [6]. The for- –1 mation of the diamond form of carbon in the films at Raman shift, cm atmospheric pressure does not run counter to the gener- ° ally accepted view of the stability of allotropic modifi- Fig. 5. Same as Fig. 4, at 600 C. cations of carbon (graphite and diamond) in the ultradisperse state. For example, according to estimates also remains equal to 0.5. Halfwidths of the lines D and Ӷ in [10], carbon clusters with size 10 nm at room tem- G decrease considerably, down to 90 cmÐ1 and 70 cmÐ1, perature and normal pressure are stable in diamond respectively. Also, the position of the G-line is close to modification, whereas the bulk material representing the asymptotic position for graphite. This indicates the the stable phase under these conditions is graphite. greater perfection of the structure of existing graphite The RS spectrum of the film annealed at 600°C is clusters. The observed shift of the position of the “dia- also composed of four Gaussian curves (Fig. 5). The mond” line (curve 1) towards smaller wave numbers lines centered at 1350 cmÐ1 (D) and 1593 cmÐ1 (G) can and the simultaneous increase of the integrated inten- be ascribed to polycrystalline graphite; the line cen- sity and the halfwidth of the line relating to the dis- tered at 1480 cmÐ1 (curve 2) to distorted sp3-hybridized torted sp3-hybridized carbon (curve 2) indicate that the carbon; and the line at 1280 cmÐ1 (curve 1) to finely dis- size of diamond clusters decreases as the temperature is persed diamond. The ratio (ID/IG) at this temperature raised.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 908 DYMONT et al.

dN(E)/dE Raman spectra suggests that annealing at 300°C, besides producing a higher degree of order in the film, causes the formation of nanosize diamond clusters. With a further increase of the annealing temperature, 252 260 the crystal structure of the graphite clusters becomes C more perfect, and the size of diamond clusters diminishes. At temperatures above 600°C, the interaction of carbon with nickel in the film bulk takes place. As a 230 250 270 E, eV result, all carbon reacts with nickel thus forming nickel carbide. The work was supported by the Foundation for O Ni Basic Research of the Republic of Belarus. One of the Ni authors acknowledges financial support from the Min- C Ni istry of Education of the Republic of Belarus. 100 200 300 400 500 600 700 800 900 Kinetic energy, eV REFERENCES 1. V. P. Novikov and V. P. Dymont, Pis’ma Zh. Tekh. Fiz. Fig. 6. Auger spectrum of the film annealed at 900°C. 22 (7), 39 (1996) [Tech. Phys. Lett. 22, 283 (1996)]. 2. V. P. Novikov and V. P. Dymont, Appl. Phys. Lett. 70, 200 (1997). It is very likely that during the annealing of the film 3. F. Tuinstra and J. L. Koenig, J. Chem. Phys. 53, 1126 at 900°C, the interaction of carbon with finely dis- (1970). persed nickel, present in the film bulk, takes place. As a 4. V. D. Andreev, T. A. Nachal’naya, Yu. I. Sozin, and result of this interaction, nickel carbide is formed. An V. A. Semenovich, Tekh. Sredstvo Svyazi, Ser. Tekhnol. Auger spectrum of such a film shown in the inset in Proizvodstva Oborudovanie, No. 4, 18 (1991). Fig. 6 is very much like that of the pure nickel carbide. 5. R. O. Dillon, J. A. Woollam, and V. Katkanat, Phys. Rev. B 29, 3482 (1984). CONCLUSIONS 6. L. C. Nistor, J. V. Landuyt, V. G. Ralchenko, et al., Dia- mond Relat. Mater., No. 6, 159 (1997). Analysis of the obtained data warrants a conclusion 7. S. Prawer, K. W. Nugent, Y. Lifshitz, et al., Diamond that the film that is not subjected to heat treatment con- Relat. Mater., No. 5, 433 (1996). sists of nanosize graphite clusters uniformly distributed 8. J. H. Kaufman, S. Metin, and Saperstein, Phys. Rev. B in a matrix of hydrogenated amorphous carbon in 39, 13053 (1989). which a part of the carbon atoms has been replaced by 9. M. M. Lacerda, D. F. Franceschini, F. L. Freire, et al., nitrogen atoms. It has been established that quite a large J. Vac. Sci. Technol. A 15, 1970 (1997). quantity of nickel and some oxygen are present in the film. Therefore, the electrolytic method under given 10. M. Y. Gamarnik, Nanostruct. Mater. 7, 651 (1996). conditions yielded films of rather complex composition containing 70Ð80 at. % of carbon. Analysis of the Translated by B. Kalinin

TECHNICAL PHYSICS Vol. 45 No. 7 2000

SURFACES, ELECTRON AND ION EMISSION

Deformation of the Contact Area and Adhesive Friction between a Scanning Frictional Microscope Probe and the Atomic-Flat Surface G. V. Dedkov and M. B. Dyshekov Kabardino-Balkar State University, ul. Chernyshevskogo 173, Nalchik, 360004 Russia E-mail: gv [email protected] Received November 22, 1999

Abstract—The determination of the equilibrium atomic structure of a nanotribocontact, formed by a hard probe to be viewed as a paraboloid of revolution and subjected to an external load, with the soft surface modeled by a set of parallel atomic planes is considered. Structural, energy, and load characteristics are calculated. In addition, dissipative static adhesive friction as a function of the normal load and the radius of probe curvature for the diamondÐgraphite system is derived. A number of approximations of the interatomic potentials is used. It is shown that an allowance for the deformation of the contact area causes the adhesive frictional force in the tensile (negative) load range to decrease. For positive loads in a range of 0Ð80 nN, the variation of the frictional force (when deformation is taken into account) depends on the radius of the probe curvature and the used approximation of the interaction potential. The dependence of friction on the radius of the probe curvature is close to a direct proportionality. The calculated results are compared with the available experimental data. © 2000 MAIK “Nauka/Interperiodica”.

The achievements of scanning probe microscopy of atoms of both the nanoprobe and surface; therefore, have stimulated the advent and development of the during the dynamical relaxation, the energy is novel promising fields of research in physics, such as “trapped” in the contact area, whereas it must be actu- nanotribology and nanolithography [1Ð3], that makes it ally dissipated into the bulk of the contacting bodies via necessary to detail the nanostructure friction mecha- vibrational modes. As a result, when loading a contact, nisms. According to present-day concepts, the fric- the response of the simulated system may significantly tional force involves deformation and adhesive compo- differ from that of an actual one. There are certain dif- nents, the latter being theoretically described in the ficulties in a qualitative determination of the contribu- most difficult manner. There are two main approaches tion of the “internal” and “external” frictions [9–11], to solving this problem. They are based on the macro- which are associated with energy dissipation into the scopic contact theory [4Ð6] and molecular-dynamics bulk of the rubbing bodies and governed by the dissipa- methods [7–11]. In the first case, the theory uses the tive component of lateral forces. Finally, the temporal values of elastic characteristics known at the mac- and velocity scales for the numerical experiments rolevel and a semiempirical relationship of Bowden (typical values are 1–1000 m/s) significantly differ and Tabor for the frictional force [4] from the scanning velocities in the probe microscopy (10Ð4Ð10Ð7 m/s). In [12], we suggested a “quasistatic” F = τA, adhesive friction model for the nanoprobe slipping τ along the atomically flat surface, in which each elemen- where is the shear strength and A is the actual contact tary slipping act (“microslip”) is accompanied by a area. Generally speaking, both quantities lose their drastic disruption of existing adhesion bonds between meaning on the atomic scale; therefore, the results atoms of the tip and surface and, on the other hand, by derived by means of the contact theory must be treated the avalanche-like process of creating new bonds. The critically. In this case, in addition, the physical mecha- residual potential atomic energy of disrupting and new nism of friction-related irreversible energy losses is not arising bonds is further transformed into the heat via revealed. Numerical simulations have shortcomings as vibrational modes. According to this model, we have well, since one has, first of all, to limit the number of the following expression for a dissipative static (veloc- particles involved in the process of the dynamical relax- ity-independent) adhesive frictional force ation that makes it difficult to evaluate the fraction of the energy dissipated into the heat. This is especially ∆ the case if a “thermal” model of the system is initially ∑ wm considered. In molecular-dynamics calculations, inevi- F = ------m - , (2) tably, it is necessary to consider only the finite number ∆x

910 DEDKOV, DYSHEKOV

W, eV (a) F 0 20

W1 10 W2

0 m = 0 R h h 1 h r 0 ΣW m = 1 h0 –10 m = 2

Z –20 W3 (b) 0 –30 r –5.0 –4.8 –4.6 –4.4 –4.2 –4.0 h1, A Z Fig. 2. The single contributions and the total contact energy Fig. 1. (a) Schematic diagram of the contact area between for a diamondÐgraphite system for the probe with R = 15 nm the tip of a probe microscope and the surface and (b) the as a function of the vertical shift of the top graphite shift directions of atoms of the top atomic plane of a mate- plane, h1. The height of the probe with respect to the non- rial due to deformation. shifted position of the top plane is h = Ð 0.2 nm.

∆ where wm is the change in energy of the mth bond due hard diamond probe and a (0001) graphite surface. The to a microslip and ∆x is the corresponding travel of the choice of this system is governed by simplicity (identi- probe. The initial position of the nanoprobe corre- cal atoms) and the possibility of applying the rather sponds to the minimum of the tribosystem energy. The simple approximations of interatomic potentials. In experiments performed with frictional microscopes addition, the graphite was used as a standard test-object demonstrate that a typical travel ∆x is approximately in nanoprobe microscopy. equal to the surface lattice constant independently of The diamond probe is assumed to have the form of dimensions of the contact area. In the framework of the a paraboloid of revolution with a radius of curvature R. model under consideration, this fact has a simple geo- It is oriented by the symmetry axis [111] along the nor- metric explanation, since the total number of the adhe- mal to the surface, which is modeled by a set of atomic sive bonds and the probeÐsurface interaction energy planes separated by a spacing h0. Figure 1a shows an quantitatively display the periodicity of the surface approximate deformation of the surface, when an exter- atomic structure for a lateral travel of the probe apex, nal normal force P is applied to the probe. The shift and its equilibrium position corresponds to the mini- directions of atoms of a single atomic plane (initial mum of the tribosystem energy and is separated by a atom sites are shown by the dashes) are given in Fig. 1b. spacing equal to the lattice period from other neighbor- We assume that the shape of the deformed atomic lay- ing positions. To overcome the corresponding potential ers replicates the axial probe symmetry and may be barrier, each microslip needs an “activation energy” defined by a model function z(r). In specific calcula- accumulated in the form of the elastic deformation tions, we used the functions energy in the cantileverÐprobeÐsurface system as the movable part of the microscope scans along the surface. b4 z(r) = ------, (3) In [12], we considered a simplified contact model 2R(r2 + b2a2m) regardless of the deformation of the contact area, which is justified only for fairly large distances of the probe where R is the radius of the probe curvature, m is the apex from the surface. In this study, we take the defor- serial number of the atomic plane (counting inward of mation into consideration by means of a geometric the sample), and a and b are the variational parameters. model allowing us to find the form of the contact area Equation (3) automatically takes into account the and the arrangement of atoms in it by minimizing the fact that the radius of curvature of the top atomic plane energy of the system. Further, we calculate the fric- at r = 0 is R and with rising m (at a > 1), the planes tional force using formula (2). The particular numerical become more and more “smooth.” The suggestion calculations are performed for the contact between a about the radius of curvature, however, is not basically

TECHNICAL PHYSICS Vol. 45 No. 7 2000

DEFORMATION OF THE CONTACT AREA 911 important. With allowance for the above suggestions, b the relation between the (x, y) coordinates of an atom 50 2 2 on the nondeformed plane and (x', y', z'( x' + y' )) on R = 15 nm the deformed one is given by equation (3), in which the 40 (x, y) coordinates are replaced by the “primed” ones, and the formulas R = 10 nm 2 2A r' 2 2 30 r = r' Ð ------, r = x + y , 2 2 3 (r' + B ) (4) R = 5 nm

2 2 20 r' = x' + y' , x/y = x'/y', where A = b4/2R and B = bam. Let us represent the contact energy as the sum of 10 three main contributions

W = W1 + W2 + W3, (5) 0 –8 –4 0 4 8 where W1 is the change in the energy of planes due to h, A deformation of the covalent bonds, W2 is the change in the coupling energy between planes, and W3 is the Fig. 3. The dependence of the parameter b on the height h of probeÐsurface coupling energy. the probe apex above the top nonshifted graphite plane for several radii of the probe curvature corresponding to the For the covalent carbon bond energy in the planes, energy minimum of the probeÐsurface system. Calculated we used the Morze approximation curves correspond to the potential 3. For positive h, the surface of graphite determined by equation (3) is “heaved” ε(d) upwards, tending to form a connective neck with the probe. (6) ( ( ( )) ( α( ))) = ÐU0 exp Ð2a d Ð d0 Ð 2exp Ð d Ð d0 , P, nN where d0 = 0.142 nm is the equilibrium bond length, α 120 U0 = 3.68 eV, and = 3.093/d0. The small bond deformation energy for this choice of constants is 35.2(d Ð 2 2 d0) /d0 eV in agreement with [13]. The probeÐsurface R = 15 nm coupling and the interaction of carbon atoms of differ- 80 ent planes were calculated in the approximation of the pairwise interactions using the CÐC potential evaluated R = 10 nm by means of the electron gas model [12] and with the LennardÐJones potential in the form 40 R = 5 nm ( ) ( Ð6 ( 6 ) Ð12) U r = ÐC6 r Ð r0 /2 r , (7) × Ð18 6 where C6 = 3.745 10 JA is the dispersive coupling 0 constant for carbon atoms in their nonvalent states and r0 = 3.81 A. A combined potential was also applied, which coincides with the electron gas model at r < –40 0.25 nm and smoothly changes to the potential (7) at –8 –4 0 4 8 r > 0.25 nm. In this case, both the short-range repulsion h, A and long-range attraction are taken into account more correctly. The corresponding approximations to the Fig. 4. The dependence of the normal load at the contact on potentials are numbered by the indices 1, 2, and 3. the apex height for several probe radii. The calculations cor- The equilibrium characteristics of the contact were respond to the potential 3. calculated by minimizing the energy W over the parameters a and b at ﬁxed values of R and h for each partic- energy (minimized over parameters a and b) with ular lateral position of the probe apex with respect to respect to h. The friction was evaluated with formula (2) at the surface (Fig. 1a). The value of h was assumed to be given values of a, b, and h. positive if the probe was shifted upwards with respect to the top nondeformed graphite plane; otherwise, it Figure 2 shows generic dependences of individual was negative. The load P of the contact and the normal contributions and the total contact energy on the verti- contact hardness were found by differentiation of the cal shift of the top atomic plane, h1, at the point r = 0

TECHNICAL PHYSICS Vol. 45 No. 7 2000 912 DEDKOV, DYSHEKOV

F, nN (Fig. 1a). In this case, the probe apex height (R = 40 15 nm) above the upper nondeformed graphite plane was equal to Ð0.2 nm. The parameters b and h1 are con- (a) 2 nected by the relationship b = 2R h1 . It is seen that 30 the dependence W3(h1) follows the dependence of the pairwise interaction potential r; i. e., it has a minimum, a sharp rise at h1 0, and a smoother increase with ∞ the zero asymptotic at h1 Ð . The sum of energies 20 W1 + W2, on the contrary, changes monotonically, since the deformation of atomic planes and their excess inter- 1 actions increase with the rise in b. As a result, the total energy W always possesses the pronounced minimum | | ≤ | | 10 if we take into account the limitation h h1 , at which the probe does not actually “puncture” the nearest plane shifted downwards. 0 The parameter a characterizes the degree of relaxation of the atomic layer displacements with the dis- 25 tance inward of the sample. The minimum over a is more ﬂattened and depends slightly on the probe radius (b) and distance h. In our evaluations, a = 1.054 was 20 obtained. The amplitude of the vertical plane shifts was decreased by a factor of 10 at m = 22. Figure 3 shows the dependences b(h) for the energy 15 2 minimum of the contact for different radii of the nanoprobe. In Fig. 4, the dependences of the normal load P 1 on h are given. The range of positive values of h corre- 10 sponds to the separation of the probe in its lifting from the surface. In Figs. 5a and 5b are shown the calculated friction-force dependences on the normal load for the 5 potentials 1 and 2 (corresponding to the approximations of the electron gas model and Lennard-Jones, respectively) with and without allowance for the defor- 0 mation of the contact area. For both cases, the radius of curvature was equal to 5 nm. The nonmonotonic char- 25 acter of the calculated points in the range of small loads is determined by the discreteness of an atomic struc- (c) ture. In this case, the dependence W(a, b) reveals a vari- 20 ety of fairly close minima, making it difﬁcult to determine the most probable structure, since even a small probe shift in the vertical direction may give rise to a nonmonotonic variation in the coupling energy. Thus, 15 as a whole, the approximation 1 leads to higher values of the friction force as compared with the approximation 2. In addition, the allowance for deformation of the 10 2 contact area causes the frictional force to increase. For the potential 1, this increase is several times in the 1 range of positive loads. For the potential 2, it lies in the 5 interval between 0 and 50% within a load range of 0 to 60 nN. For negative loads, the potential 2 with the allowance for the deformation gives somewhat smaller 0 values of the pulloff probe forces (in magnitude) than –20 0 20 40 60 80 in the opposite case. P, nN Figures 6a and 6b show the frictionÐload dependences for the nanoprobes with radii of 5, 10, and Fig. 5. The frictionÐload dependences for the potentials (a) 1, (b) 2, and (c) 3. The radius of the probe curvature is 15 nm using combined potential 3. A comparison of the 5 nm. Curves 1 and 2 are plotted without and with allowance corresponding curves in Figs. 6a and 6b for the probe for the contact deformation, respectively. with a radius of curvature of 5 nm shows that, for loads

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DEFORMATION OF THE CONTACT AREA 913

F, nN and 48 nm), diamond (probe)—tungsten carbide (the 60 radius of the probe curvature is 110 nm), and Pt R = 15 nm (probe)—mica (the radius of the probe curvature is (a) 140 nm). 50 In analyzing the dependences given in these studies, one can, as is in our calculations, note an irregular char- R = 10 nm 40 acter of the experimental points for small loads. These irregularities are more pronounced at a small probe radius when the discreteness of the structure is more 30 strongly exhibited. Note that in the absence of the contact area deformation, the theoretical dependences are smooth (compare a and b in Fig. 6). The results of our 20 R = 5 nm calculations at R = 10 nm and the data from [15] for − Si NbSe2 at R = 12 nm are close qualitatively. In the 10 quantitative respect, our values of frictional forces are two to three times greater. The discrepancy may be caused by a different kind of interacting atoms and, cor- 0 –40 0 40 80 120 respondingly, load characteristics of tribosystems, by an error of formula (2), and (or) experimental data. 60 The conclusion about the proportionality of the frictional force to the radius of the probe curvature is con- 50 (b) ﬁrmed by the comparison of the experimental data from R = 15nm [14, 15] with each other, since the friction for the Pt−mica contact (R = 140 nm) in [14] is higher by an 40 order of magnitude than that for the SiÐNbSe2 contact (R = 12 nm). In [16], however, the values of the fric- R = 10nm tional force are anomalously small for the large probe 30 radius (110 nm), and, therefore, too small for such a stiff contact shear stress value (238 MPa) calculated in 20 R = 5nm the DerjaguinÐMullerÐToporov contact approximation [6]. Since the probe proﬁle was uncontrolled, the results may be explained by the fact that the probe tip 10 in actuality was considerably smaller (provided that the friction was the same). It is possible that the radius of the tip curvature was about 10 nm. 0 –100 0 100 200 300 400 P, nN REFERENCES Fig. 6. The frictionÐload dependences for the probes of var- 1. C. M. Mate, G. M. McClelland, R. Erlandsson, and ious radii. The calculated curves are obtained for the poten- S. Chiang, Phys. Rev. Lett. 59, 1942 (1987). tial 3 in two cases: (a) without and (b) with allowance for 2. Fundamentals of Friction: Macroscopic and Micro- deformation of the contact area. scopic Processes, Ed. by G. L. Singer and H. M. Pollock (Cluwer, Dordrecht, 1991). 3. B. Bhushan, Tribol. Int. 28, 85 (1995). of 0Ð80 nN, the regard for deformation gives rise to 4. F. P. Bowden and D. F. Tabor, The Friction and Lubrica- somewhat lesser values of the frictional force than in tion of Solids (Clarendon Press, Oxford, 1964). the opposite case. Conversely, for a larger probe radius, 5. K. L. Johnson, K. Kendall, and A. D. Roberts, Proc. the frictional force is greater when deformation is taken R. Soc. London, Ser. A 324, 301 (1971). into account. The calculation results also indicate that 6. B. V. Derjaguin, V. M. Muller, and Y. P. Toporov, J. Col- the frictional force is approximately proportional to the loid Interface Sci. 53, 314 (1975). radius of probe curvature. 7. D. J. Maugis, J. Colloid Interface Sci. 150, 243 (1992). Unfortunately, the lack of information makes it 8. U. Landman, M. D. Luedtke, and M. W. Ribarsky, J. Vac. impossible to compare in detail our results with exper- Sci. Technol. 7, 2829 (1989). imental ones. Among the known studies devoted to 9. U. Landman and M. D. Luedtke, Fundamentals of Fric- measuring “dry” adhesive friction in a vacuum by tion: Macroscopic and Microscopic Processes, Ed. by means of a scanning frictional microscope, one can G. L. Singer and H. M. Pollock (Cluwer, Dordrecht, notice papers [14Ð16] in which the frictionÐload depen- 1991), p. 463. dences for a number of contacts are given. These are Si 10. A. V. Pokropivnyœ, V. V. Pokropivnyœ, and V. V. Sko- (probe)—NbSe2 (the radii of the probe curvature are 12 rokhod, Pis’ma Zh. Tekh. Fiz. 22 (2), 1 (1996) [Tech.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 914 DEDKOV, DYSHEKOV

Phys. Lett. 22, 46 (1996)]; V. V. Pokropivnyœ, V. V. Sko- 13. W. A. Harrison, Electronic Structure and the Properties rokhod, and A. V. Pokropivnyœ, Trenie i Iznos 17, 579 of Solids: The Physics of the Chemical Bond (Freeman, (1996). San Francisco, 1980; Mir, Moscow, 1983). 11. A. Buldum, S. Ciraci, and I. P. Batra, Phys. Rev. B 57, 14. R. W. Carpick, N. Agrait, D. F. Ogletree, and M. Salme- 2468 (1998). ron, Langmuir 12, 3334 (1996). 15. M. A. Lanz, S. J. O’Shea, M. E. Welland, and 12. G. V. Dedkov, in Proceedings of the 8th International K. L. Johnson, Phys. Rev. B 55, 10776 (1997). Conference on Tribology, NORDTRIB'98, Aarhus, Den- 16. M. Enashescu, R. J. A. van den Oetelaar, R. W. Carpick, mark, DTI Tribology Centre, 1998, Vol. 1, p. 47; Pis’ma et al., Phys. Rev. Lett. 81, 1877 (1998). Zh. Tekh. Fiz. 24 (19), 44 (1998) [Tech. Phys. Lett. 24, 766 (1998)]; Mater. Lett. 38, 360 (1999); Wear 232 (2), 145 (1999). Translated by Yu. Vishnyakov

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Technical Physics, Vol. 45, No. 7, 2000, pp. 915Ð921. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 102Ð108. Original Russian Text Copyright © 2000 by Vostrikov, Agarkov, Dubov.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Kinetics of Radiation Cooling of Fullerenes A. A. Vostrikov, A. A. Agarkov, and D. Yu. Dubov Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia E-mail: [email protected] Received March 10, 1999; in ﬁnal form June 29, 1999.

Abstract—Emission from fullerene molecules excited by means of electron impact in crossed beams under conditions of single collisions between electrons and C60 molecules in a kinetic energy range Ee from 25 to 100 eV was studied experimentally. Emission spectra in a wavelength range from 300 to 800 nm; the emission excitation functions and the temperature of emitting molecules as a function of Ee were measured with a reso- +* lution of 1.6Ð3.2 nm. The contribution to emission from ionized C60 molecules has been determined and data +* on the emissivity of the C60 ion have been obtained. It has been shown that the emission spectra can be well approximated with the spectral distribution of thermal emission from a black body (Planck’s formula), taking into account the lowering of emissivity for a small particle. The emission can be observed starting with electron ≈ energy of about 27 eV; the emission excitation function is of a nonresonant form, peaking at an energy of Ee 70 eV. As Ee is increased, the temperature of emitting particles rises and reaches its maximum value of 3100Ð ≈ 3200 K at Ee 47 eV. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION The discovery in 1985 of fullerenes possessing a unique stability with respect to fragmentation stimu- In previous investigations at our laboratory, emis- lated investigations of highly-excited fullerene mole- sion from microparticles of condensed matter (clusters cules in the gas phase [9Ð17], in particular of electron- of molecular gases (CO2)n, (N2O)n, (H2O)n, and (N2)n induced processes [10, 11, 14, 17]. It has been found formed in supersonic jets under excitation by means of that the primary excitation of fullerene molecules, like electron impact) was studied [1Ð3]. It was found that that of other complex molecules and clusters, initiates the energy of vibrational [1] and electronic [2, 3] exci- cascades of secondary processes, while the complex tations is quickly dissipated through the heating of the energy spectrum of the excited states (including collec- cluster. The rate of return from excited states back to tive excitations) facilitates the rapid dissipation of the the ground states rises dramatically with the cluster primary excitation and attainment of partial or com- size, and the emission spectrum corresponds to emis- plete thermalization. However, the basic distinction sion from electron-excited states of molecules ejected between fullerenes and most other microsystems lies in from the cluster. A similar picture was observed under the extreme stability of the carbon cage [8]. Because of the laser excitation of vibrational degrees of freedom of this high stability, fragmentation, which is the domi- SF6 molecules in clusters [4]. Thus, the ejection of mol- nant cooling channel of an excited “excessive” com- ecules in excited states and the evaporation are the plex system (it is monomolecular disintegration in the major cooling channels in weakly bound clusters [5], case of a molecule and evaporation of structural units in and the effect of radiation cooling is not observed. Still, the case of a cluster), is impeded by the relaxation of there is indirect evidence that radiation cooling in the excited fullerene. As a consequence, competing molecular systems plays an important part, for exam- (not evaporating) molecule cooling processes of com- ple, in gas-phase ion-molecular processes [6]. Corre- parable intensity come to the foreground, namely, elec- spondingly, ion-molecular processes in clusters can tron and photon emission (analogues of macroscopic also proceed with the participation of radiation cooling. processes of thermionic emission and thermal emis- Tightly bound clusters (clusters of less volatile sub- sion, respectively). For example, in [13, 14], a continu- stances) were found to display radiation cooling with a ous emission spectrum was registered in the visible continuous emission spectrum similar to Planck’s spec- range from a ﬂux of C60 molecules heated in various trum [7, 8]. The temperature of emitting clusters, as in ways. The molecules were either vaporized from a solid the case of tungsten clusters Wn with n > 200, could be sample by laser radiation [13] or excited by an electron as high as 3800 K [8]. The clusters were heated by impact [14]. means of an oxidation reaction (Nbn clusters in [7]) or In this study, thermal emission from fullerenes was by laser irradiation (Nbn, Wn, and Hfn clusters [8]). excited in crossing beams of C60 and low-energy elec-

916 VOSTRIKOV et al.

6 and, following spectral dispersion, it was registered PM with an FEU-79 photomultiplier (PM). The spectral 7 sensitivity of the system of optical diagnosis was cali- brated using a CI 10Ð300 lamp as a standard emitter. In order to determine the flux of photons I(λ), the registered spectra were normalized using a curve of relative η λ 1 DP MC spectral sensitivity /( ). The integrated intensity was registered by a photomultiplier, with the monochromator replaced by a ZS-8 broad-band filter (F). Under these conditions, the integrated spectral sensitivity 8 5 curve had a maximum at 500 nm and a halfwidth of 4 F 150 nm. The spatial resolution of the detecting system PM 3 (the size of the spot from which the emission was collected) was determined using a point radiation source. 2 +* 6 To isolate the contribution from C60 ions to the total emission, an electric field was applied to the Fig. 1. Schematic of the experimental setup. region of beam crossing which was produced by two deflecting plates (DP) placed along the axes of molecular and electron beams. With increasing electric field ξ trons. A continuous emission spectrum was observed strength between the plates, the time spent by the ions after excitation by means of electron impact in a wave- in the field of view diminished. Because of the magnetic collimation of electrons, there was no distortion length range from 300 to 800 nm (the results given here ξ were obtained with spectral resolution ∆λ = 3.2 nm; of the electron beam by the field . To verify this, we check measurements at ∆λ = 1.6 nm yielded identical measured in the same setup the emission due to short- + 2Σ+ λ spectra). The spectrum could be described by Planck’s lived states of N2 (B u , = 391 nm) and found that formula for emission from a sphere of much smaller Ӷ λ their emission intensity did not depend on the strength diameter than the emission wavelength, d [19]. of the pulling electric field. Generally speaking, contributions to emission may

EXPERIMENTAL TECHNIQUE also come from fragments, both neutral, C60* Ð 2n , and +* The procedure and the experimental setup for the charged, C60 Ð 2n . But the threshold for fragmentation generation of the molecular beam and excitation of C60 under electron impact suffers a signiﬁcant kinetic shift, by electrons was identical to the ones used in experi- being observed at 42Ð44 eV, and at Ee < 100 eV, the ments on the measurement of the absolute cross sec- fraction of fragments does not exceed 5Ð7% [10, 12]. Ð + tions of C60 and C60 [11, 17, 20]. For emission regis- Taking this into account and also considering the char- tration, the setup was equipped with an optical system. acter of I(Ee) plots obtained in this work, we conclude The schematic of the experiment is shown in Fig. 1. that the emission is due mainly to unfragmented fullerene molecules. The effusion molecular beam was formed by vapor- izing fullerenes from source 1 heated to T = 800 K and Contributions to emission from other sources was 0 eliminated in the following ways. The electron-induced intersected by an electron beam at an angle of 90°. The emission from the background gas and other possible beam of electrons emitted from oxide cathode 2 was sources was measured while the beam source aperture formed by a system of apertures 3Ð5, collimated with was closed with shutter 8. The contribution to the signal the magnetic ﬁeld (300 G) of magnets 6 and registered of thermal radiation from heated parts of the fullerene at collector 7 of the Faraday cup type. The current of source was eliminated by modulating the electron electrons did not exceed 60 µA and the density of mol- beam; it was chopped at a frequency of 80 Hz, and the ecules in the region of crossing of the beams was of the 10 -3 photomultiplier signal was measured in the lock mode order of 10 cm . In this way, a pairwise interaction of at the chopping frequency. electrons and molecules was ensured. The energy of electrons in the beam Ee varied in the range from 0 to 100 eV. RESULTS AND DISCUSSION The emission from the intersection region of the two In Fig. 2, typical spectra are shown (curves 1) of beams was collected at right angles to the electron emission, generated as a result of electron impact on the beam; the angle with the molecular beam direction was fullerene molecule in crossing beams at electron ener- 40°. A system of short-focus lenses imaged the emis- gies 66 (a) and 30 eV (b). In order to obtain these sion onto the entrance slit of an MUM monochromator curves, the contribution of the background gas was sub- (MC) (200Ð800 nm, reciprocal dispersion 3.2 nm/mm) tracted from the total PM signal (curve 2); then the dif-

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KINETICS OF RADIATION COOLING OF FULLERENES 917

I, real. units I, real. units; η, arb. units 2 60 10 60 (a) (b) 50 50 4 40 1 3 10 40 30 1 30 20 1 100 20 10 2 0 10 3

–10 10–1 0 300 400 500 600 700 800 300 400 500 600 700 λ, nm λ, nm

Fig. 2. Electron-induced radiation spectra. Dashed line is an approximation of the experimental data by formula (1).

ference signal (desired signal, curve 3) was corrected At T > 1500 K, the curve of Eν(T) is well approximated for the spectral sensitivity of the optical system by a linear function Eν(eV) ≅ 13.9 + 0.0143(T Ð 1500). (curve 4). Figure 3 shows the measured emission intensity of Let us compare this spectrum with the one that is the fullerene as a function of energy Ee. Here curve 1 is due to the thermal emission of a small spherical particle an integrated emission (measured in “ﬁlter + photomul- heated to a temperature T. The rate of photon emission λ λ ∆λ tiplier” scheme); curve 2 is the emission at a wave- Ith in the wavelength range [ , + ] from a heated length λ = 540 nm, ∆λ = 3.2 nm. Curve 3 in Fig. 3 is the body is described by Planck’s formula dependence of the temperature of emitting particles T(E ) obtained by ﬁtting the spectral data with formula I (λ, T) e th (1) using the least squares method. It can be seen that 4  hc   (1) as Ee is increased to about 47 eV, the temperature T = 2πcS∆λε(λ, T)/ λ exp ------Ð 1 ,  λkT  rises in proportion to Ee. At Ee > 47 eV, the temperature of emitting particles rises to a maximum value of T* ≈ where S is the emitting area and ε(λ, T) is the emissivity 3100Ð3200 K corresponding to the internal energy of ≅ of a solid. It follows from the emission theory for small the C60 molecule Eν 36 eV. Taking into account that particles [19] that the emissivity of a sphere of diameter d Ӷ λ is ε(λ) ∝ d/λ (because photon emission comes + I, I(C58), arb. units T, K from the bulk of the particles instead of their surface). 100 3500 The C60 molecule is spherically symmetrical and hol- ε λ ∝ λÐ1 3000 low; for this geometry, the function ( ) will 3 obviously be valid. The outer diameter of the electron 80 shell of a molecule is d ≈ 1 nm. Below we assume 1 2500 60 d 2000 ε(λ) = ε --, (2) 4 0λ 1500 40 2 ε where 0 is a numerical constant. 1000 20 In order to derive the internal energy Eν of a 500 fullerene molecule corresponding to the temperature of 0 0 emitting molecules, we calculated, following [13], the 20 30 40 50 60 70 80 90 100 vibrational energy in the approximation of harmonic Ee, eV intramolecular vibrations of C60. Contributions of all the 46 vibrational modes were summed up, taking into Fig. 3. Plots of the emission intensity (1, 2), the temperature account their degree of degeneration. The results of cal- + culations with the use of data on frequencies presented of emitting molecules (3) and the current of C58 fragments in [21] practically coincided with calculations in [12]. (4) [10] versus electron energy.

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µ I, arb. units; tr, s temperature dependence [9]. Note, at this point, that the 35 sum kT* + Uf, where Uf is the ionization potential of 30 C58, turned out to be close to the threshold energy Ee for the dissociative ionization [10, 12, 18]. 25 Shown in Fig. 3 (curve 4) is the dependence [10] of 2 20 the current due to C + ions generated as a result of the 3 58 15 +* 1 metastable fragmentation of C60 during the time inter- 10 val 7.7Ð31.2 µs lapsed after interaction with an electron ≈ for an initial thermal energy of the molecule Eν, 0 5 1' 2' 4.5 eV. It is seen that the threshold for the generation of 0 the fragments is indeed close to the energy E , at which –40 –30 –20 –10 0 10 20 30 40 e ξ, V/cm the temperature rise of emitters ceases, i.e., hotter molecules, apparently, rapidly loose energy through the evaporation of C . Fig. 4. Plots of the emission intensity (1, 2) and the calcu- 2 lated ion transit time (3) versus the electric field strength. Heating the C60 molecule causes not only its fragmentation but also the thermal emission of an electron, +, arb. units; , µs , K which is a faster process [9]. Therefore, the main emis- I tr T sion source at E ≥ 40 eV, as shown earlier by the 10 3200 e I +* present authors [22], is provided by the ionized C60 3000 molecules. In Fig. 4, the integrated intensity I is plotted as a function of the strength ξ and polarity of the elec- 2800 tric field between the deflecting plates at energies Ee = 40 (curve 1) and 65 eV (curve 2). Also shown in this 2600 12 +* figure is the time of stay of a C60 ion in the emission II 6 2400 region under observation (curve 3) calculated for an ion produced in the center of electron beam and having the 3 2200 0.01 starting velocity equal to the velocity of a C60 molecule 1 in the effusion beam. The asymmetry of the curves is 1 2000 0 1 2 related to the fact that the optical axis and the molecule 10 10 10 velocity are not collinear (the angle between them is ξ, V/cm 40°). Fig. 5. Experimental (᭺, ᭝) and calculated (dashed lines) It can be easily shown that, at strong pulling fields, plots of the emission intensity of molecules, ion transit time the time of stay of the ions in the region under observa- (I) and their ultimate temperature (II) against the electric tion t and, consequently, their emission intensities I+ field strength (values of ε are indicated at the curves). r 0 are proportional to ξÐ1/2. By plotting I(ξ) in coordinates I(ξ) Ð ξÐ1/2 and extrapolating to the origin of the coordi- the main emission source is the ionized molecule (ion- nates, we find that the contribution to emission from ization potential Ui = 7.6 eV [11, 18]) and that the ther- neutral particles does not exceed 16 and 4% for Ee = 40 mal energy of the C60 molecule in the effusion beam and 65 eV, respectively. These values are shown in ≅ Fig. 4 by horizontal lines 1' and 2'. Eν, 0 = Eν(T0) 4.6 eV, we obtain that the energy transferred to the molecule by an electron (Eν + Ui Ð Eν, 0) Data in Fig. 4 allow estimates to be made of the amounts to 39 eV. Thus, the primary and secondary +* emissivity of C60 . For this purpose, we subtracted electrons take away only 8 eV. The transfer of such a from the total emission I the contribution of neutral par- great amount of energy to the C60 molecule implies a ticles. In Fig. 5, the symbols correspond to data for the multielectron excitation process. ion emission I+(ξ) derived from the left-hand branches The existence of a limit for the temperature increase of curves 1 and 2 in Fig. 4. The calculated dependence + of time t on ξ is shown in Fig. 5, curve I. of the C * molecule can be explained by competition r 60 The emission at small ξ (corresponding to large val- between the radiation cooling of the molecule heated ues of time t ) does not vary in proportion to t . We by an electron and an alternative cooling process by r r relate this behavior of I+(ξ) to the radiation cooling of +* evaporation of C2 through the process C60 +* the C60 ion during its stay in the region under observa- +* C60 Ð 2n + nC2, the rate constant of which has a stronger tion. In order to derive the dependence of the emission

TECHNICAL PHYSICS Vol. 45 No. 7 2000 KINETICS OF RADIATION COOLING OF FULLERENES 919 intensity on the cooling rate, we integrate the emission I+, arb. units λ λ ε λ flux energy q = (hc/ )Ith with respect to . Taking ( ) 10 in the form of (2), we get, as a result of integration, an expression similar to the StephanÐBoltzmann formula 8 for a small particle 6 q = ε σ T 5 (3) 0 c 4 σ π 2 5 where c = 24.888 2 hc Sd(k/hc) . 2 Making use of 0 qdt = ÐCdT, (4) –3 –1 1 3 and integrating (4), we get variation with time of the rs, mm +* temperature of C60 Fig. 6. Transverse profiles of the electron-induced emission (E = 70 eV) from nitrogen ions (open circles) and fullerene Ð1/4 e T(t, T ) = T {1 + t[4ε σ T 4]/C} , (5) molecules (solid circles). Symbols show the experimental i i 0 c i results, and solid curves show the results of calculations. where Ti is the initial ion temperature. + 2Σ+ λ +* Substituting (5) into (1), we get an expression for nitrogen ions N2 (B u , = 391 nm) and C60 ions. the ion emission intensity I + as a function of the time The obtained distributions of the emission intensity are th + in transit t and the initial temperature T . The integral of shown in Fig. 6. Maxima of I curves in Fig. 6 have i been made equal. Note that in these measurements, + Ith (t) over the ion trajectory with account taken of the nitrogen was leaked in the chamber to a pressure of spatial variation of the sensitivity of the optical system 10−4 Pa. The lifetime of the state is B 2Σ+ ~ 0.1 µs; i.e., η(r) u + ∞ the thermal motion of the excited N2 ions during a + + η( ( )) radiative lifetime can be neglected. In this case, the N = ∫Ith r t dt (6) measured emission intensity is given to within a device 0 constant by the integral represents the number of photons registered by the opti- +* ( ) 3 η( ) ( ) cal system per one emitting C60 ion. As the rate of for- In rs = ∫ d r r Ð rs P r . (7) + * ξ P(r), η(r) ≠ 0 mation of C60 ions does not depend on the field , the + | | N (ξ) dependence to within a constant A coincides with Here, rs is the distance from the optical axis to the the measured signal I+(ξ). electron beam axis, and the integration is carried out over the region where both the excitation level P(r) and Depicted by a dashed line in Fig. 5 are the calcula- η + ξ λ the sensitivity of the registration system (r) are non- tion results for N ( ) at Ti = 3150 K and = 540 nm for ε zero. For a fullerene, taking into account the radiation several values of the constant 0. For illustration pur- cooling, we get poses, the calculated N+(ξ) curves have been normalized at their maxima in the region of large times tr . In + ξ ( ) 3 ( ) comparing the experimental data points for I ( ) with I f rs = ∫ d r0P r0 + + calculated N (ξ) curves, the differences between N and ( ) ≠ P r0 0 I+ values were minimized using the least squares (8) ε method with constants A and as fitting parameters. It × 3 η( ) ( ( )) was found that the best agreement with the model is ∫ d r r Ð rs Ith t r Ð r0 , ε ε observed at 0 = 5.7 for Ee = 40 eV (triangles) and 0 = η(r) ≠ 0 6 for Ee = 65 eV (open circles). The final temperature Tf where t(r Ð r ) = |r Ð r |/|v | is the time of ion transit and +* 0 0 0 of C60 ions exiting the region under observation for v0 is the starting velocity vector (measurements were ε ξ 0 = 6 is shown by curve II in Fig. 5. carried out at = 0). A qualitative confirmation of the temperature varia- + The emission profile of N2 calculated by formula (7) +* tion of the quantity T(C60 ) has been obtained from and the emission profile of the fullerene derived using ε measurements of the emitting region profiles along the formulas (1), (5), and (8) at 0 = 6 are shown in Fig. 6 direction perpendicular to the optical observation axis as solid curves. It can be seen that the shift of the and the axis of the electron beam for emission from fullerene emission profile relative to that of nitrogen is

TECHNICAL PHYSICS Vol. 45 No. 7 2000 920 VOSTRIKOV et al.

Emissivity of fullerenes Emitting particle Coefﬁcient ε Process considered References

* × Ð4 C60 0.5Ð1.2 10 Cooling of heated molecules in a beam [5] +* × Ð2 C60 Ð 2n 2Ð4.4 10 Metastable fragmentation [8] Ð* ~1.2 × 10Ð4 Thermionic emission [9] C60 +* 1.0Ð1.1 × 10Ð2 Optical emission This study C60

well described by our model of the radiation cooling C60 displays a resonant character typical of molecular + of C * . spectra [24] and showing that, in this place, the number 60 of optically active states is small. The continuous char- Previously [12, 15, 16], in studies of relaxation pro- acter of the electron-induced emission of highly excited cesses in a gaseous fullerene, estimates of emissivity fullerene molecules can possibly be explained by opti- have been made. However, in analyzing these data, the cal transitions between electron-excited states whose following points should be taken into consideration. ** Firstly, indirect estimates of the radiation cooling con- density is high [25]. The shape of the observed C60 stant are strongly inﬂuenced by the choice of the rate emission spectra indicates that the highly excited constants of competing processes, mechanisms of fullerene molecule is large enough for complete ther- which are not quite well understood. Secondly, in [12, mal equilibrium between emission modes to be 15, 16], experimental determinations were carried out achieved. of the loss rates of the internal energy q in different temperature ranges and the ε values were “recovered” under certain assumptions differing from those adopted CONCLUSIONS in this work. For example, the dependence of ε on λ In this work, the emission of fullerene molecules was ignored [12, 15], and it was assumed [16] that q ∝ excited by electron impact in crossing beams under 7.6 ε λ ∝ λ 3.6 Ð T or ( ) (d/ ) . In view of the above, we have conditions of single inelastic collisions C60 + e in the used the data of [12, 15, 16] for the rate of radiative range of collision energies from 25 to 100 eV was stud- energy loss q(T) and recovered the values of ε using ied, and the following results have been obtained. ε λ formulas (2) and (3). Values of for = 540 nm and Collisions between C and electrons produce emis- d = 1 nm are given in the table. 60 * +* Note that our values of ε are much higher than those sion from C60 molecules and C60 ions at electron of indirect estimations. This is probably due to the dif- energies starting from ≈27 eV. The major contribution + ference in temperatures (in [12, 16] the temperature of to the emission comes from C * ions, which form after particles did not exceed 1800 K), which can have a two- 60 fold effect. Firstly, as the temperature is increased, the the fast thermal emission of electron. + The emission excitation function I(E ) has been cooling rate of C * can rise because of the evaporation e 60 derived. The function has a nonresonant character and of C2. However, this effect appears to be not strong reaches a maximum at an energy E ≈ 70 eV. enough because the observed cooling rates are the same e at energies 40 and 65 eV, whereas fragmentation at It has been found that the spectrum of electron- 40 eV is practically zero (curve 4 in Fig. 3; see also induced emission of fullerene molecules is described [10, 12]). Secondly, the possibility that ε depends on T by Planck’s formula for radiation from small particles cannot be excluded. In our calculations above, this of sizes much smaller than the radiation wavelength. dependence was neglected because of a rather small The temperature of molecules determined from emission spectra rises in proportion to E as the electron ** e temperature drop observed during crossing by the C60 energy is increased to ≈47 eV and then stays constant at ion of the region under observation (curve II in Fig. 5). T ≈ 3150 K. Considering the distribution of tempera- The most complicated issue is the formation mech- tures of the molecules, the heating can be as high as ≈ anism of the Planck’s emission spectrum of a nanopar- Tm 3800 K. ticle. In ordinary media, the continuous thermal emis- +* sion spectrum is formed as a result of the transfer of The kinetics of radiation cooling of C60 has been resonant emission under conditions of photon reab- investigated and the emissivity value obtained: ε ≈ 10Ð2 sorption [23]. It is evident that because of the low den- at λ = 540 nm and d = 1 nm. The results of works [7, 8, sity of C60 molecules in the beam, the reabsorption does 13, 14] and of this study show that the method of exci- not take place. The absorption spectrum of unexcited tation of metal clusters and C60 molecules has no effect

TECHNICAL PHYSICS Vol. 45 No. 7 2000 KINETICS OF RADIATION COOLING OF FULLERENES 921 on the emission spectrum; metal particles and fullerene 11. A. A. Vostrikov, D. Yu. Dubov, and A. A. Agarkov, molecules produce Planck’s spectrum for thermal Pis’ma Zh. Tekh. Fiz. 21 (17), 73 (1995) [Tech. Phys. emission. This is probably the common property of Lett. 21, 715 (1995)]. tightly bound clusters and molecules. 12. E. Kolodney, B. Tsipinyuk, and A. Budrevich, J. Chem. Phys. 102, 9263 (1995). 13. R. Mitzner and E. E. B. Campbell, J. Chem. Phys. 103, ACKNOWLEDGMENTS 2445 (1995). The authors are grateful to S. V. Drozdov for his 14. A. A. Vostrikov, D. Yu. Dubov, and A. A. Agarkov, assistance. Pis’ma Zh. Éksp. Teor. Fiz. 63, 915 (1996) [JETP Lett. The work was supported by the Russian Foundation 63, 963 (1996)]. for Basic Research (grant nos. 97-02-17804 and 98-02- 15. K. Hansen and E. E. B. Campbell, J. Chem. Phys. 104, 17804) and a grant from the Ministry of Education of 5012 (1996). the Russian Federation. 16. J. U. Andersen, C. Brink, P. Hvelplund, et al., Phys. Rev. Lett. 77, 3991 (1996). 17. A. A. Vostrikov, D. Yu. Dubov, A. A. Agarkov, et al., REFERENCES Mol. Mater. 20, 255 (1998). 1. A. A. Vostrikov and S. G. Mironov, Chem. Phys. Lett. 18. C. Lifchitz, Mass Spectrom. Rev. 12, 261 (1993). 101, 583 (1983). 19. C. F. Bohren and D. R. Huffman, Absorption and Scat- 2. A. A. Vostrikov, D. Yu. Dubov, and V. P. Gilyova, tering of Light by Small Particles (Wiley, New York, Z. Phys. D 20, 205 (1991). 1983; Mir, Moscow, 1986). 3. A. A. Vostrikov and V. P. Gileva, Pis’ma Zh. Tekh. Fiz. 20. A. A. Vostrikov, D. Yu. Dubov, and A. A. Agarkov, 20 (15), 40 (1994) [Tech. Phys. Lett. 20, 625 (1994)]. Pis’ma Zh. Tekh. Fiz. 21 (13), 55 (1995) [Tech. Phys. 4. I. M. Beterov, Yu. V. Brzhazovskiœ, A. A. Vostrikov, et al., Lett. 21, 517 (1995)]. Kvantovaya Élektron. (Moscow) 7, 2443 (1980). 21. R. A. Jishi, R. M. Mirie, and M. S. Dresselhaus, Phys. 5. A. A. Vostrikov, D. Yu. Dubov, and V. P. Gileva, Zh. Rev. B 45, 13685 (1992). Tekh. Fiz. 59 (8), 52 (1989) [Sov. Phys. Tech. Phys. 34, 22. A. A. Agarkov, D. Yu. Dubov, and A. A. Vostrikov, in 872 (1989)]. Proceedings of the 17th International Symposium on 6. R. C. Dunbar, Int. J. Mass Spectrom. Ion Processes 100, Molecular Beams, France, 1997, pp. 399Ð402. 423 (1990). 23. Yu. K. Zemtsov, A. Yu. Sechin, A. N. Starostin, et al., Zh. 7. U. Frenzel, A. Roggenkamp, and D. Kreisle, Chem. Éksp. Teor. Fiz. 114, 135 (1998) [JETP 87, 76 (1998)]. Phys. Lett. 240, 109 (1995). 24. P. F. Coheur, M. Carleer, and R. Colin, J. Phys. B 29, 8. U. Frenzel, U. Hammer, H. Westje, et al., Z. Phys. D 40, 4987 (1996). 108 (1997). 25. W. A. Chupka and C. E. Klots, Int. J. Mass Spectrom. Ion 9. Z. Wan, J. F. Christian, Y. Basir, et al., J. Chem. Phys. 99, Processes 167Ð168, 595 (1997). 5858 (1993). 10. M. Foltin, M. Lezius, P. Scheier, et al., J. Chem. Phys. 98, 9624 (1993). Translated by B. Kalinin

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Discharge Leader Self-Organization on the Water Surface V. P. Belosheev Vavilov State Optical Institute, All-Russia Research Center, ul. Babushkina 36/1, St. Petersburg, 199034 Russia Received May 20, 1999

Abstract—Experimental studies on the self-organization of dissipative dynamic systems have shown that this property is inherent to the structure of the discharge leader. © 2000 MAIK “Nauka/Interperiodica”.

In recent years, interest in studying the self-organi- character of structuring and structure breakdown, as zation phenomenon in living and physical systems has well as the relation between the dissipated power and quickened. The latter are especially promising in this structuring. respect because of their simplicity and the availability of well-developed experimental methods and tools. At present, few systems where self-organization is known CONDITIONS FOR LOW STRUCTURING to take place are the objects of investigation. A wider LOW structuring is demonstrated more distinctly coverage of dissimilar systems would be of great value with the two-dimensional LOW [4]. The setup for for the in-depth analysis of the self-organization mech- LOW study (Fig. 1) is essentially a 1.5-cm-high cylin- anisms and properties. drical glass cuvette 1 of diameter 9 cm, two-thirds full In this work, the self-organization of the discharge of tap water 2 with a conductivity of ≅1 × 10Ð4 ΩÐ1 cmÐ1. leader on the water surface (LOW) was discovered. Cathode 3, an 0.05-cm-thick brass disk 7 cm in diame- This study is an extension of [1], where an arc discharge ter was placed concentrically to the cuvette 0.3 cm initiated on the water surface was used as a source of UV below the water surface. The end of a stainless steel radiation to deactivate microorganisms [2]. wire 7.5 × 10Ð3 cm in diameter served as anode 4. The When a cylindrical cathode was partially immersed anode was 0.1Ð0.3 cm above the water surface at the in water (vertically) and the anode tip was above the center of the cuvette. water surface at a distance of 1Ð3 cm from the cathode In experiments, we used 0.1-µF-storage capacitor 5, [1], the formation of the leader was found to shunt the which was charged to the initial voltage U0 = 3Ð6 kV. water resistance in the gap and cause nonlinear current A discharge was initiated by shortening air gap 6. feedback (between the gap voltage and resistance) to Oscillograms of the capacitor voltage Uc and discharge occur. In other words, if the LOW is initiated by a current i (Fig. 3) were obtained using resistive voltage capacitor, an R(t)C discharge arises in the circuit and an divider 7 and shunt 8. The discharge was photographed exponential current and voltage decay is not observed. with camera 9. The experimental conditions were Further studies of LOW [3, 4] under creeping dis- detailed in [3, 4]. charge conditions (Fig. 1) have shown that the LOW At the initial voltage U0 = 3Ð6 kV (positive anode 1 may be quasi-one- or quasi-two-dimensional depend- in Fig. 4), cone-shaped corona 3 was initiated in air ing on whether the cathode is one- or two-dimensional gap 2. As soon as the corona cone base reached the (Fig. 2). The leader core is driven by the potential dif- water surface, from three to six initial leaders 4 began ference between it and the water surface in front of it. The leader structure forms by repeatedly splitting the core because of its ﬂattening. The LOW structure thus obtained is fractal with the fractal dimension D = 0.96 9 and 1.85 for quasi-one- and quasi-two-dimensional dis- 6 charges, respectively. Their evolution is self-consistent 7 [4, 5]. 5 Here, we report additional information that provides 4 a deeper insight into the role of a corona in the leader 1 evolution. Based on these data and also in view of self- organization signs [3Ð5] found in nonequilibrium 8 dynamic dissipative systems [6, 7], we establish the 3 2 fact of LOW self-organization. The self-organization signs are, speciﬁcally, the spontaneous and threshold Fig. 1. Experimental setup.

DISCHARGE LEADER SELF-ORGANIZATION ON THE WATER SURFACE 923

Ω (‡) i, A; Uc, kV R, 3 (a) 30 600 20 6 400 1 10 3 200 2 0 10 20 30 40 t, µs Ω i, A; Uc, kV R, k 3 (b) 6 1 2 4 2 1 2 3 0 20 40 60 80 t, µs

Fig. 3. Time dependences (oscillograms) of the (1) capacitor voltage, (2) discharge current, and (3) discharge circuit resistance. U0 = (a) 6 and (b) 3 kV. 1 cm

+ 1 3 (b) – – + + + + +– + + 2 – + 5 + + – + + –+ + + + + + 3 3 – + + – + + + + + + + + + – 4 + + + + + + + + + – + 3 – + + + 6 – – –

– 1 cm Fig. 4. Corona (dashed lines) and contracted (continuous lines) discharges in the anodeÐwater gap. Pluses and Fig. 2. (a) Two-dimensional discharge and (b) the end of a minuses are charges on the plasma and water surfaces; one-dimensional discharge. arrows, current streams. to develop over it. The onset of the discharge in the gap expected, were absent in the photograph of the entire was indicated by the current passing through the dis- leader. According to [8], it was assumed that they are charge circuit. Its value increased from that typical of comparable in size with the core radius and remained coronas to i0 = 3 A (U0 = 6 kV) (Fig. 3, curve 2). The unresolved. However, in the photograph taken with a final value was set after the corona cone had touched greater magnification from a one-dimensional leader the water surface. During this stage (≈100 ns), cathode developed in the gap (Fig. 2b), filament-like formations layer 5 (Fig. 4), necessary for passing the conduction against the diffuse background are seen in the last cen- current, formed in the anodeÐwater gap near the water timeter of the channel on both of its sides. Away from and the initial corona was successively changed to the core, branches are observed. The filaments were glow, abnormal glow, and arc discharges 6. The current observed most distinctly in a microscope with a 16× value i0 equals U0/R0, where R0 is the discharge circuit magnification. The filament length increased from resistance, which includes the plasma resistance in the 150 µm at the core to 400 µm at 1 cm from it. The spac- anodeÐwater gap and the resistance of the water layer ing between the filaments was 100 and 30 µm, respec- between the cathode and corona cone. A further tively. The filament diameter was found to be 20Ð30 µm increase in the current is due to the LOW evolution, throughout the channel. The angle between the channel specifically, an increase in the leader area and water and filaments increases from 20° at the core of the layer under the leader structure. This water layer leader to 80° at its end. However, streamers ahead of defines the discharge circuit resistance at this stage of the leader were not observed either in the photograph or the discharge [4]. in the microscope. Here, it should be taken into consid- In [3], the structure of a quasi-one-dimensional eration that the photograph of the entire leader is iden- leader was represented as consisting of a channel, arms, tical to the original only at the end part of the stopped and branches. Streamers at the leader core, though leader. Away from the core, the image of the leader,

TECHNICAL PHYSICS Vol. 45 No. 7 2000

924 BELOSHEEV

ϕ, kV intact, since the channel potential near the anode is still 6 relatively high. However, as the capacitor and anode voltages drop, the leader core moves towards the anode 5 and the leader shortens. 4 1 Thus, the LOW evolution proceeds at two levels. It begins in the plasma of the corona cone near the water 3 2 surface (the microlevel), where initial leaders, early 2 macroelements, start to form (Fig. 4). However, the 3 evolution of the macrostructure through the movement 1 of the cores and their splitting are also provided by microprocesses taking place in the corona and cathode 1 2 3 4 5 layer. The appearance of the corona and cathode layer l, cm has a threshold character. Hence, LOW structuring also has the voltage threshold. Fig. 5. (1) Core potential, (2) water surface potential, and (3) coreÐwater potential difference vs. the leader length. LOW STRUCTURE AND POWER DISSIPATION including that observed in the microscope, is the con- IN THE DISCHARGE CIRCUIT tinuous superposition of its previous images. The other sign of true self-organization in dynamic With regard for data for pulse coronas [8, 9], one dissipative systems is the growth of dissipated power may suppose that the observed filaments are streamers during self-organization. As noted, the LOW structure arising when the moving core displays corona. develops from three to six initial leaders by repeatedly splitting their cores. The leader core is driven by the Coulomb force, which is proportional to the potential difference Each of the cores in the LOW structure moves between the core and water surface in front of it, U = through the interaction with the net field of polarization ϕ ϕ charges on the water surface and with elements sur- c Ð w (Fig. 5). In addition, the core motion is closely related to a plasma generated at the core front. It has rounding the water. Therefore, the leading cores at the been suggested [3] that the plasma is generated largely periphery of the structure move in the radial direction, in the cathode layer, the voltage drop across which is while those lagging behind and being at the center of 0.3Ð0.4 kV. Therefore, the core stops when ∆U = 0. the structure move according to the randomly directed local field of the structure. However, the observed corona of the core and the coincidence (within the experimental error) of the A combination of the determinate (radial) and ran- corona threshold voltage in air (2.3 kV [10]) with the dom movements of the core makes the structure spa- potential of the core at the instant it stops (2 kV) for any tially stochastic and fractal [5]. The fractal dimension of the initial voltages in the range 3Ð6 kV strongly sug- D of the structure does not depend on the number of ini- gests that the corona discharge significantly contributes tial leaders and on the initial voltage in the 4Ð6 kV to plasma generation. On the other hand, it follows range (the range being studied). The fractal dimension from the above that streamers would arise only ahead of of the structure is related to the full length L of its ele- D the moving core if its potential exceeds 2.3 kV. Hence ments in a given radius r; namely, L ~ r . In our case, the absence of streamers in front of the stopped core in the radius of the structure equals the length of the lead- the photograph. ers l, l ~ t1/2 [3, 4]; hence, L ~ tD/2. At the same time, the discharge circuit resistance is defined by that of the The major role of the core corona in the evolution of water layer under the structure, that is, depends on the the leader is also strengthened by the decrease in the 5 4 structure size and element density. Eventually, R ~ core velocity from (1Ð2) × 10 to (1Ð2) × 10 cm/s when ÐD/2 ≈ 1/L ~ t . As the structure develops, the voltage across the core potential drops to 3 kV (Fig. 5). This is also the capacitor reduces insignificantly (Fig. 3); therefore, the threshold potential for the evolution of the leader, D/2 the current in the circuit i = U0/R ~ t , and the power this threshold being of synergistic nature. The neces- 2 D/2 sary value of the anode potential is the sum of the dissipated in the discharge circuit is P ~ i R ~ t . That threshold potential for the corona discharge initiated is, the power dissipated in the circuit at the stage of cur- from the tip, 2.3 kV, and the threshold voltage drop rent growth is proportional to the full length of the across the cathode layer, 0.3Ð0.4 kV. The core moves structure elements. with a velocity of (1Ð2) × 104 cm/s (∆U ≥ 0.4 kV) also As stated above, when the corona ceases, the core because of plasma generation in the cathode layer. Only slows down, and when its potential becomes equal to when this layer breaks down (∆U ≤ 0.3 kV) does the that of the water, the core stops. The current ceases to core stop. Note that, when the leader core stops, the rest grow, the power balance in the channel is violated, the of the channel may continue to display corona and, plasma in the core breaks down, and the structure accordingly, the surrounding cathode layer may remain shrinks. This, in turn, causes the current, and hence,

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DISCHARGE LEADER SELF-ORGANIZATION ON THE WATER SURFACE 925 power in the discharge circuit to decrease further, distance away from the core; hence, the core must flat- etc. [3]. ten on its axis. One can argue, in this case, that the The analysis of the origination and evolution of the plasma formation (a cylindrical channel and a hemi- LOW structure, as well as its relation to the power dis- spheric core) cannot be dynamically stable. According sipated in the discharge circuit, allows us to argue, by to the type of coreÐelectrode coupling, the formation analogy with self-organization in nonequilibrium will necessarily be converted either to a streamer with dynamic dissipative systems [6, 7], that the LOW struc- a narrowing channel or to an expanding leader. There- ture evolution represents one more example of dissipa- fore, analysis of streamer or leader evolution within the tive system self-organization. cylindrical model will inevitably cause errors. Generally, self-organization in a system occurs STRUCTURE SELF-ORGANIZATION when it is necessary to enhance the energy flux through it from an external source with a growing gradient at Once the fact of self-organization is established, it the system boundaries. Under such conditions, a mech- makes sense to treat this phenomenon at length. We anism of energy transfer through the system changes to proceed from the assumption that system elements pro- a higher-rate one. In liquids, for example, heat conduc- vide necessary conditions for structuring and structure tion changes to Benard cellular convection, and in evolution. lasers, spontaneous lasing changes to induced lasing. In By elements of the well-developed LOW structure, our case, the system cannot enhance the energy flux by we will mean leader channels in the region from the locally increasing the process rate, since the water con- corona cone to the most remote (along the radius) core, ductivity remains unchanged. Here, the energy flux side arms emerging from the channels or larger arms to grows owing to the extensive development of the sys- the core most remote from them, and arms emerging tem in the form of LOW structuring. Remarkably, not from the last bifurcations together with their cores. The only LOW structure elements are produced but also the role of branches in LOW self-organization is insignifi- nonlinear interaction of the entire system with the cant, and they will be excluded from consideration. energy source to sustain structure instability is pro- A specific feature of LOW structure self-organization vided. is that these structure elements appear immediately It is known that positive feedback (PF) arising in a during LOW evolution and self-organization. system is a sign of its instability. In our case, it shows As noted, at the early stage of structuring, areas with up (at the stage of current growth) as a continuously an elevated charge density, which give rise to cores, maintained unambiguous correspondence between the appear in the boundary layer of the corona nonequilib- length of a leader and the current through it (lÐi corre- rium plasma. These areas move, because the charged spondence) [3], i.e., between the radius of the structure plasma is unstable in the electric field. When moving, and the current through it. This correspondence reflects the cores leave behind them a conducting plasma chan- the relation between the leader core potential and the nel, which transfers the corona cone potential to the anode (corona cone) potential. This relation sets in cores. The resulting coreÐchannel plasma formation is through the channel, whose conductivity depends on the initial macrostructure element, a leader nucleus. Its the passing current. Note that PF is inherent in leader further development implies core splitting and the evolution and can be considered as an immanent prop- emergence of side arms, which are new structure ele- erty in developing each of the structure elements. ments. As the number of structure elements grows and the system becomes unstable, self-organization at the As the cores are split, the number of structure ele- macrolevel starts. Here, the early channels become the ments increases and so does the charge interaction first (basic) ordering parameters. The side arms and between them. This interaction appears as negative branches of leaders set up the final hierarchy of order- feedback (NF), which is inherent in the entire structure. ing parameters in LOW structuring. It is obvious that the PF prevails early in structuring; Leader core splitting is certainly the key process in therefore, the number of structure elements doubles LOW structuring. In [5], core splitting was associated within 0.2 cm (Fig. 6). Then, NF comes into play, and with a like residual charge present in front of a core. As the rate of element production diminishes. Subse- the core moves, the charge builds up, so that the core quently, the partial contributions of NF and PF to the front flattens at points where it touches the channel overall process presumably vary, as follows from the envelope. It appears, however, that the residual charge varying number of elements (Fig. 6, curve 1). This may not be a decisive factor in core splitting. Indeed, a combined action continues until all structure elements streamer core is not split, as a rule, but retains its shape develop over the entire structure area and depends on (self-sharpens) [11]. This self-consistent process can be the element position in the structure. It is responsible explained by a decrease in the energy density in the for the competitive growth of the elements and their streamer channel with distance from the core; accord- selection by formation rate and, hence, length. Eventu- ingly, the rate of ionization drops, and the channel nar- ally, the structure becomes hierarchically self-similar, rows. On the contrary, a leader channel expands with i.e., fractal [5].

TECHNICAL PHYSICS Vol. 45 No. 7 2000 926 BELOSHEEV

n Current stratification observed in the cathode layer of a 50 3 glow discharge [10] and cathode spot ordering [12] are indirect evidence for this supposition. In this case, the 2 current structure in the cathode layer will modulate the 40 1 charge density in the plasma layer adjacent to it on the anode side (Fig. 4), and the crests of the charge density at the boundary of this layer will give rise to initial lead- 30 ers. In both mechanisms, the charge density varies in a 20 regular manner, including in the periphery of the layer, which provides the symmetric arrangement of initial leaders. Note that the symmetry is also a necessary con- 10 dition for the resulting structure to be electrically stable. The nucleation of initial leaders may follow the 0.5 1.0 1.5 third possible scenario. Since the lÐi correspondence, r, cm acting as PF, is universally present in the system, radial charge-density microfluctuations in the corona plasma Fig. 6. Number of elements vs. radius: (1) actual structure, layer at the water surface may be brought to the mac- (2) doubled number, and (3) averaged curve to calculate D. rolevel. The competition between (and eventually the selection of) macrofluctuations will cause radial splitting (contraction) of the current into several channels However, the spaceÐtime characteristics of the (Fig. 4), and the charge interaction between the macrof- structure also depend on other factors. Since each of the luctuations will provide the electrical stability of the elements is subjected to the determinate and random channels. fields and the element position specifies their combined Obviously, the above scenarios are not mutually action on it during the structure evolution, the LOW exclusive and may coexist during self-organization at structure exhibits time-variant stochasticity. On the the microlevel. The early stage of LOW structuring other hand, the nonstationarity of the energy source and calls for further investigation. It would be useful for the absence of limits for structuring make the system finding determinateness limits at the early stage of self- spatially irregular, unlike Benard cells and laser modes, organization in any system. In our opinion, the phe- i.e., space-stochastic. nomenon of LOW structuring offers a greater scope for In the foregoing, LOW self-organization after the experimentation than other physical systems. appearance of several initial leaders was considered. Our studies of LOW structuring and structure evolu- However, a mechanism of their appearance, i.e., pro- tion [3Ð5] can be summarized as follows. cesses in the plasma layer at the water surface, is also of great interest. This nonequilibrium plasma layer (1) LOW structuring exhibits all of the signs of self- forms when the corona cone base comes in contact with organization in nonequilibrium dynamic dissipative the water surface. The diameter of the layer is close to systems [6, 7]. that of the cone base (0.1Ð0.3 cm, depending on U0), (2) LOW self-organization proceeds in two stages. and its thickness is comparable to the core diameter, First, the radial charge-density structure forms in the ≈10Ð2 cm. As the current through the layer grows when disk-shaped layer of the nonequilibrium plasma near contracted along the anodeÐwater gap axis, so does the water surface; then, the LOW current channels (within ≈100 ns) the energy flux through the layer. This evolve from the charge density structure. may induce ionicÐacoustic vibrations in the plasma (3) The evolution and self-organization of the LOW layer. Charge density crests at the layer periphery may structure impose limits on the specific power (power initiate the nucleation of initial leaders. per element) dissipated in the system. This leads to a Another possible initiation mechanism is the occur- rise in the number of structure elements by splitting rence of a planar double electric layer at the instant and, hence, the total dissipated power; in other words, when the positively charged cone base touches the neg- the self-organization proceeds in the extensively devel- atively charged water surface (Fig. 4). Further, this oping system. layer serves as the cathode layer to provide a discharge (4) A structure element moves because of the move- in the anodeÐwater gap. Its diameter and thickness are ment of its core, which is driven by the source potential. 0.1Ð0.3 and ≈10Ð3 cm, respectively [10]. With regard The source potential is transferred to the core through for the planar geometry of the cathode layer, oppositely the channel. Hence, the cores and channels fulfil differ- moving electrons and ions in it, and a drastic increase ent functions during the process. in the energy flux when the current contracts, a current (5) The self-organization of the structure is gov- structure of Benard cell type may form in this layer. erned by a combined action of PF, the immanent prop-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DISCHARGE LEADER SELF-ORGANIZATION ON THE WATER SURFACE 927 erty of each of the elements, and NF, the immanent 4. V. P. Belosheev, Zh. Tekh. Fiz. 68 (11), 63 (1998) [Tech. property of the structure as a whole. This leads to the Phys. 43, 1329 (1998)]. competition between structure elements, their selec- 5. V. P. Belosheev, Zh. Tekh. Fiz. 69 (4), 35 (1999) [Tech. tion, and the fractality of the structure. In addition, Phys. 44, 381 (1999)]. dynamic self-consistency between the structure and 6. I. P. Prigozhin, Usp. Fiz. Nauk 131 (2), 189 (1980). power source is provided. 7. H. Haken, Synergetics: an Introduction (Springer-Ver- (6) The nonstationarity of the power source and the lag, Berlin, 1977; Mir, Moscow, 1980). unrestrictedness of LOW structuring are responsible 8. É. M. Bazelyan and I. M. Razhanskiœ, Spark Discharge for the spaceÐtime stochasticity of the structure. in Air (Nauka, Novosibirsk, 1988). (7) Based on items 1Ð4, LOW self-organization can 9. I. S. Stekol’nikov, The Nature of Long Spark (Akad. be considered as a universal evolutionary model. Nauk SSSR, Moscow, 1960), p. 55. 10. Yu. P. Raœzer, The Physics of Gas Discharge (Nauka, Moscow, 1987). REFERENCES 11. Yu. P. Raœzer and A. N. Simakov, Fiz. Plazmy 24, 754 1. V. P. Belosheev, Zh. Tekh. Fiz. 66 (8), 50 (1996) [Tech. (1998) [Plasma Phys. Rep. 24, 700 (1998)]. Phys. 41, 773 (1996)]. 12. Yu. D. Korolev and G. A. Mesyats, The Physics of Pulse 2. V. P. Belosheev, RF Patent No. 2042641 (May 14, 1992). Breakdown (Nauka, Moscow, 1991). 3. V. P. Belosheev, Zh. Tekh. Fiz. 68 (7), 44 (1998) [Tech. Phys. 43, 783 (1998)]. Translated by V. Isaakyan

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Technical Physics, Vol. 45, No. 7, 2000, pp. 928Ð930. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 115Ð117. Original Russian Text Copyright © 2000 by D’yachkov, Zhilyakov, Kostanovskiœ.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Melting of Aluminum Nitride at Atmospheric Nitrogen Pressure L. G. D’yachkov, L. A. Zhilyakov, and A. V. Kostanovskiœ Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow, 127412 Russia E-mail: [email protected] Received August 19, 1998; in ﬁnal form August 26, 1999

Abstract—Experimental results on the melting of aluminum nitride heated by an electric arc burning in a nitrogen atmosphere at a pressure of 0.2Ð0.3 MPa are presented. A qualitative explanation of the dissociation suppression mechanism under arc heating is proposed. It has been shown that the suppression is possible at atmospheric pressure due to the photoactivation of aluminum on the sample surface by resonant radiation of aluminum vapors present in the electric arc. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION surface layer having properties differing from those of Aluminum nitride (AlN) is a promising material the starting material and possessing high hardness. possessing a combination of properties valuable for X-ray phase analysis has shown that the dominant practical use [1]. Their full utilization is possible for phase (up to 99%) in the molten layer is a perfectly mono- or polycrystalline AlN. Dense crystalline alumi- crystallized aluminum nitride. Detailed description of num nitride can be obtained either by gas-phase growth the experiment as well as of the preceding studies has [2] or by crystallization from melt [3, 4]. The latter been given in [6]. method is more productive but its implementation is This paper considers the suppression mechanism of extremely difficult. Heating of aluminum nitride at AlN dissociation as well as the specific conditions of atmospheric pressure results in intensive dissociation the electric arc heating favoring the AlN melting. The and complete destruction of the sample (transition from possibility of melting aluminum nitride at atmospheric solid to gas phase) already at 2400Ð2700 K, which is pressure has been shown. ± much lower than its melting point Tm = 3070 100 K [5]. Therefore, to determine the melting parameters of aluminum nitride and produce it in polycrystalline DISSOCIATION SUPPRESSION form, growth from melt at elevated nitrogen pressure is MECHANISM used in order to suppress the thermal dissociation of Suppose that the surface of an AlN sample is heated AlN. For example, laser melting requires a nitrogen to melting point Tm. However, for the melting really to pressure not lower than 10 MPa [3], while in a con- occur, it is necessary that a dynamic equilibrium tainer melting, it is 0.5 MPa [4]. between the condensed aluminum nitride and the gas In [6] an experiment on heating a porous AlN phase products of its decomposition be established ceramics by an ac electric arc at nitrogen pressures 0.2Ð according to the reaction 0.3 MPa was carried out. The arc was initiated under AlNc Al + 1/2N (1) the AlN sample and the current was 60Ð80 A. Indica- 2 tions of aluminum nitride melting were observed. (c denotes the condensed state). Otherwise, if the par- The choice of the alternating current arc as a heater, tial pressures of nitrogen and aluminum vapors are too the porous ceramics as an AlN sample, and the scheme low, instead of melting, an intensive dissociation will of heating from the bottom is not accidental. The alter- take place. The constant of reaction (1) given by + 1/2 nating field confines Al ions [7] and having the arc K = P(Al)P (N2) (2) under the sample reduces the carry-over of aluminum vapor forming as a result of AlN dissociation from the at thermodynamic equilibrium can be expressed, for reaction zone by convection currents. However, heating example, in terms of constants of the reactions from the bottom poses the problem of melt confine- AlN(c) Al + N and N2 2 N , which can be ment. The porous sample structure through the capil- found in [8]; at T ≅ 3000 K, it is equal to 0.12 MPa3/2. lary enables a sufficient amount of molten AlN phase to This means that at a partial nitrogen pressure P(N2) of, be accumulated for unambiguous instrumental proof of say, 0.1 MPa, the equilibrium pressure of aluminum the fact of melting. In these conditions, samples have vapors should be 0.38 MPa, which is impossible since been obtained with a clearly distinguishable remelted the saturated vapor pressure of aluminum at this tem-

1063-7842/00/4507-0928 $20.00 © 2000 MAIK “Nauka/Interperiodica” MELTING OF ALUMINUM NITRIDE AT ATMOSPHERIC NITROGEN PRESSURE 929 perature is only 0.24 MPa [8]. If P(Al) = 0.24 MPa, the time dt and maintaining equilibrium in (3) is deter- equilibrium nitrogen pressure will be equal to P(N2) = mined by the expression 0.25 MPa as it follows from (2). P(Al*) dN = ------dsdt, (6) Thus, the aluminum nitride can be melted only at 2πMkT nitrogen pressures of no less than 0.25 MPa. In this m case, if P(N2) is not too high, the aluminum vapor where M is the mass of the aluminum atom. It is should be close to saturation, which is difﬁcult to pro- assumed that in the near-surface gas layer of a thickness vide technically since, in the absence of the hermeticity of the order of the atom free path, the temperature is of the heated volume, the vapor will intensively diffuse close to the surface temperature Tm. to colder regions and condense there. To reduce P(Al) by a factor of n while maintaining the equilibrium of If the occupation of the excited states is governed by process (1), it is necessary to increase the nitrogen pres- some nonthermal process and is in excess of that given sure by a factor of n2. Therefore, the melting of AlN can by (5), only the equilibrium in (3) (but not in (1)) is of be achieved only at pressures that are high enough. importance. In that case, the melting is possible at lower gas phase pressures. The conclusion made above is true for the condi- When a sample is heated by arc, the excitation of tions of thermal excitation of aluminum atoms. In the aluminum atoms at the surface can be caused by radia- process of AlN dissociation, excited aluminum atoms tion from the arc. Then, the condition for equilibrium in Al* are released, presumably to the ﬁrst excited state (3) and therefore, the possibility of melting, can be 4s(2S) with energy E* = 3.14 eV [9]. Therefore, the described as follows. After dissociation by (3), alumi- reaction that is actually taking place is num atoms pass to the ground state and escape to the arc plasma. The density of aluminum vapors in the arc

AlN(c) Al* + 1/2N2, (3) plasma increases until the intensity of radiation from the arc due to the aluminum resonant transition and the reaction constant can be written in the follow- 4s(2S) 3p(2P) becomes high enough for excitation ing form of such a number of aluminum atoms at the sample sur- 1/2 face that an equilibrium between the direct and reverse K* = P(Al*)P (N2), (4) reactions in (3) is established. To estimate the minimum required aluminum vapor density in the arc, the number where P(Al*) is the partial pressure of aluminum atoms 2 of resonant photons arriving at the sample surface in the 4s( S) state. should be no less than that given by (6). Expressing At T = 3000 K, the equilibrium pressure of excited P(Al*) from (4) and substituting into (6), we obtain that atoms is the flux density of resonant photons emitted by the arc should be no less than ( ) ( ) g*  E* K*(T ) P Al* = P Al Σ----(------) expÐ------ j = ------m------. (7) T kT (5) π ( , ) 2 MkT mP N2 T m × Ð6 ( ) = 1.77 10 P Al . At the same time, the flux density of aluminum atoms coming to the sample surface should be no less Here g* = 2 is the statistical weight of the excited state; Σ than the value given by (7), and all these atoms can be is the statistical sum of an atom, which at a specified considered to be in the ground state. For this to occur, temperature is actually equal to the statistical weight 2 the aluminum vapor pressure in the near-surface layer g = 6 of the ground state 3p( P). As a result, we obtain should be no less than the equilibrium partial pressure 3/2 K* = 210Pa ; that is, at atmospheric nitrogen pressure, of excited atoms at the melting temperature, which is P(Al*) = 0.7 Pa. given by equation (4). After dissociation by (3), aluminum atoms pass to The flux density of photons at an emission wave- the ground state and an equilibrium between the ground length λ is and excited states sets up. Only then, if excitation/de- 2 excitation of the atoms in the near-surface gas layers lN*A 2πe lNagf  E*  j = ------a------= ------exp Ð------, (8) results from collision processes, the direct and reverse a 4π mc λ2Σ( )  kT  reactions (3) (as well as (1)) may come to an equilib- Ta a rium making the melting possible. This will occur at the where A is the transition probability (Einstein coeffi- extremely high equilibrium gas phase pressures speci- * fied above, although the vast majority of aluminum cient); f is the oscillator strength; and Na and Na are atoms in the ground state are not involved in the pro- the densities of excited atoms and the total density of cesses (3). In this case, the number of excited alumi- aluminum atoms in the radiating plasma layer of thick- num atoms arriving at a sample surface element ds in ness l at a temperature Ta, respectively.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 930 D’YACHKOV et al. For the transition under consideration, f = 0.12 [10]. nitride and melt it by an electric arc in a nitrogen envi- To make an estimation, assume that the temperature Ta ronment at pressures close to the atmospheric pressure. and electron density ne in the arc are independent of the In this study, relying on the analysis of the dissociation aluminum vapor density; that is, the desired minimum mechanism, it has been shown that, even at aluminum density Na is low enough. We take Ta = 12000 K and vapor partial pressure in the arc of the order of 10Ð 17 Ð3 100 Pa, the conditions for the photoactivation of the ne = 10 cm [11]. Thus, it is assumed that the density 17 -3 number of Al atoms necessary to suppress the AlN dis- of aluminum ions is less than 10 cm . Assume also sociation are ensured and the melting of aluminum that the arc plasma is transparent for the resonant radi- nitride at atmospheric nitrogen pressure is made possi- ation; i.e., the characteristic transverse dimension of the ≈ ble. This is a result of aluminum photoactivation at the arc l can be taken equal to 0.5 cm. AlN sample surface at the melting temperature by res- Based on the assumptions made and equating (7) onant radiation of the aluminum vapor from the hot and (8), we obtain zone of the arc. It is important, therefore, that the arc be 15 positioned under the AlN sample; otherwise the con- 2 × 10 N ≈ ------, vective currents would carry aluminum vapors away a ( , ) P N2 T m from its surface and the necessary vapor density may not be attained. where the nitrogen pressure is measured in Pa and the Ð3 The method considered can also be successfully aluminum atom density, in cm . applied in melting other easily dissociating high-melt- From the Saha equation, it is easy to determine that ing-temperature nonmetallic nitrides, like nitrides of ≈ the aluminum ion density is Ni 30Na. At atmospheric boron, silicon, and others. ≈ × 12 Ð3 ≈ × nitrogen pressure, Na 6 10 cm and Ni 2 14 Ð3 17 Ð3 10 cm < 10 cm , as was assumed. In the conditions under consideration, the broadening of spectral REFERENCES lines is of the Stark type and the resonance line has a 1. G. V. Samsonov, Nitrides (Naukova Dumka, Kiev, Lorenz contour with a halfwidth ∆λ = 0.016 nm [12]. 1969). For the Lorenz contour, the adsorption coefficient at the 2. G. A. Slack and T. F. McNelly, J. Cryst. Growth 42, 560 center of the line is (1977). 3. A. V. Kostanovskiœ, Teplofiz. Vys. Temp. 32, 742 (1994). 2λ2 e Na f 4. A. V. Kostanovskii and A. V. Kirillin, in Proceedings of kλ = ------mc2∆λ the International Symposium on Nitrides, Saint-Malo, France, 1996, Vol. B28. and at nitrogen pressures approaching the atmospheric 5. V. L. Vinogradov, A. V. Kostanovskii, and A. V. Kirillin, pressure, the optical thickness kλl ~ 0.1; that is, for the High Temp.ÐHigh Press. 23, 685 (1991). resonance aluminum line, the arc plasma is transparent. 6. L. A. Zhilyakov and A. V. Kostanovskiœ, Teplofiz. Vys. Therefore, the assumptions made are valid. Temp. 37, 71 (1999). ≈ × At an aluminum vapor density Na + Ni 2 7. D. A. Zhilyakov, D. N. Gerasimov, and A. V. Kosta- 1014 cmÐ3, the respective partial pressure in the arc is novskiœ, Teplofiz. Vys. Temp. 35, 147 (1997). about 30 Pa. Taking into account that aluminum comes 8. L. V. Gurvich, I. V. Veœts, V. A. Medvedev, et al., Ther- from the AlN dissociation at the sample surface and the modynamic Properties of Substances (Nauka, Moscow, fact that, with heating from the bottom, convective cur- 1978Ð1982), Vols. 1Ð4. rents return the aluminum vapors to the sample surface, 9. L. A. Zhilyakov, A. V. Kostanovskiœ, and A. V. Kirillin, it can be assumed that, near the surface, the vapor pres- Dokl. Akad. Nauk SSSR 35, 606 (1994). sure will be not lower than in the arc. The latter exceeds 10. A. A. Radtsig and B. M. Smirnov, Reference Data on the equilibrium partial pressure of excited atoms at Atoms, Molecules, and Ions (Énergoatomizdat, Moscow, 1986; Springer-Verlag, Berlin, 1985). Tm = 3000 K, which is of the order of 1 Pa as follows from (5). Therefore, the second condition for suppress- 11. É. I. Asinovskiœ, A. V. Kirillin, and V. L. Nizovskiœ, Sus- ing the dissociation will be fulfilled as well. tained Electric Arcs and Their Application in Thermal Physical Experiment (Nauka, Moscow, 1992). 12. H. R. Griem, Spectral Line Broadening by Plasmas CONCLUSIONS (Academic Press, New York, 1974; Mir, Moscow, 1978). In [6], it has been experimentally shown that it is possible to suppress the dissociation of aluminum Translated by M. Lebedev

TECHNICAL PHYSICS Vol. 45 No. 7 2000

Technical Physics, Vol. 45, No. 7, 2000, pp. 931Ð936. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 7, 2000, pp. 118Ð124. Original Russian Text Copyright © 2000 by Aleksandrov, Balabas, Vershovskiœ, Pazgalev.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

A New Model of a Quantum Magnetometer: A Single-Cell CsÐK Tandem Based on Four-Quantum Resonance in 39K Atoms E. B. Aleksandrov, M. V. Balabas, A. K. Vershovskiœ, and A. S. Pazgalev Vavilov State Optical Institute, All-Russia Research Center, ul. Babushkina 36/1, St. Petersburg, 199034 Russia E-mail: [email protected] Received September 3, 1999

Abstract—A new model of a quantum magnetometer based on the principle of CsÐK tandem was developed and experimentally tested. The speciﬁc features of the magnetometer are a single absorption cell containing Cs and K vapors, digital synthesis of the potassium resonance frequency by multiplying the cesium resonance frequency by a conversion factor (the ratio of the atomic constants), and the use of four-quantum resonance in potassium atoms. Device readings were found to be insensitive (within 10 pT) to variations of the main operating parameters even if the variations go far beyond the normal service variability. © 2000 MAIK “Nauka/Inter- periodica”.

INTRODUCTION PRINCIPLE OF OPERATION Since the late 1950s, the absolute values of mag- In metrology, the problem of combining high accu- netic fields falling into the geophysical range have been racy with high speed is usually solved by integrating usually measured with optically pumped quantum two different measuring devices into a single system so magnetometers. Optically pumped magnetometers that the readings of a high-speed device are corrected (OPMs) are significantly more sensitive and faster than with a slower-speed but high-precision device. This conventional proton (precession) magnetometers principle is used, for example, in modern time-keeping [1−3]. In addition, the recently developed OPM models systems and was proposed to be applied in precise mea- provide a higher accuracy of measurement than the pro- surements of the magnetic induction magnitude [4]. ton instruments. Many OPM models of different mea- A magnetometer built around this principle is often suring accuracy, resolving power, speed, working called tandem. It consists of OPMs of two types: a spin range, power consumption, weight, dimensions, price, generator with the output frequency proportional to the etc., have been described in the literature. In view of a external magnetic induction (MxÐOPM) and a passive wide variety of OPM applications, it is hardly probable radio spectrometer where a feedback loop is employed that they will ever be covered by one unified multipur- for holding a selected resolved magnetic resonance line pose OPM design. Therefore, the development of new (Mz-OPM). The latter OPM provides a higher accuracy OPM models best suited to specific types of measure- of resonance frequency measurement, whereas ment remains a topical problem. Mx-OPM oscillates near the center of gravity of a group The goal of this work is to describe a new OPM of unresolved lines and, therefore, is prone to large sys- model intended largely for observatory measurements tematic errors. of the magnetic field magnitude in the geomagnetic range. The model is distinguished by high speed (0Ð In the early tandem model, the atomic vapor of the 87 100 Hz band) and high resolution (~10 pT/Hz1/2) in rubidium isotope Rb was used in both components. combination with the record-breaking long-term stabil- However, 87Rb proved to be a poor choice, because its ity (reading variations within 10 pT) and a strictly lin- magnetic resonance spectrum contains a group of lines ear dependence of the resonance frequency on mag- the spacing between which in the slightly disturbed netic induction. Although such good characteristics magnetic field of the Earth is only several times larger have previously been achieved in some other types of than their widths. A CsÐK tandem [5] (cesium is used OPMs, the model under consideration is the first to in the Mx component; and potassium, in the Mz compo- combine all of these advantages. Note also that, in the nent), developed many years after its Rb counterpart, is field of metrology, it is always worthwhile to develop a free from this disadvantage, because the magnetic res- device based on an essentially new approach even if its onance spectrum of potassium is completely resolved metrological performance is not superior to that of pre- in the whole range of the earth’s magnetic fields. This vious models. prevents the interference of adjacent spectral lines.

932 ALEKSANDROV et al.

magnetic field intensity H1, the four-quantum resonance width can be as small as several hertz, whereas the other resonance lines are significantly broadened. This allows the problem of searching and holding the required resonance line (the total number of resonance lines is 10) to be easily solved. An example of the potassium magnetic resonance spectrum calculated for the case of pumping with circularly polarized D1-line light is shown in Fig. 1. The curve represents the superposition of multiquantum resonances of multiplicity 1Ð 4 in a strong magnetic field H1. The underlying discrete –1000 0 1000 f – f0, Hz spectrum shows four single-quantum resonances in a weak field H1. The abscissa is f Ð f0, where f is the fre- Fig. 1. Calculated dependence of the pump light absorption quency of the radio-frequency (RF) alternating mag- by 39K vapors on the frequency f of RF field H (f). 1 netic field applied and f0 is the four-quantum resonance frequency. The continuous spectrum was calculated for The distinguishing features of the OPM tandem the static magnetic field 50 µT and the relatively high γ described in this work are as follows: (1) the absorption amplitude of the alternating magnetic field H1 ( H1 = Γ γ cell, containing a mixture of Cs and K vapors, is shared 200 0, where = 7 Hz/nT is the gyromagnetic ratio and Γ by the component magnetometers and (2) the potas- 0 is the width of the single-quantum resonance line). sium Mz-OPM uses the four-quantum (4Q) magnetic The (discrete) spectrum of slightly excited potassium resonance line corresponding to the transition between atoms is calculated for the weak alternating magnetic | 〉 ⇔ | 〉 γ Γ the sublevels F = 2, mF = 2 F = 2, mF = Ð2 . field H1 such that H1 = 0. This spectrum consists of ⇔ four almost equidistant lines corresponding to mF The common absorption cell by both magnetome- m + 1 transitions for F = 2. Their width is far less than ters eliminates the systematic error caused by a differ- F ence in the magnetic fields in two separate cells. In the distance between them and cannot be resolved in addition, this reduces the size of the device. The use of the scale of the figure. It is seen that the two spectra four-quantum magnetic resonance allows the device diverge significantly. That of potassium excited by the resolution to be increased many times. Another advan- strong magnetic field is characterized by the extraordi- tage is that the dependence of the four-quantum reso- narily narrow four-quantum resonance line and resonance frequency on the magnetic induction is strictly nance lines of multiplicity n = 1Ð3, broadened by the linear. As shown in [6, 7], the high-order resonance line magnetic field. n = 2F is prominent in the spectrum of n-quantum tran- The block diagram of the device is shown in Fig. 2. ∆ ∆ | | sitions like F = 0, mF = n . The frequency of this res- The detector of the magnetometer incorporates an evac- onance is almost independent of the alternating mag- uated spherical glass bulb 80 mm in diameter. The inner netic field intensity H1. Moreover, this resonance line is surface of the bulb is covered by a paraffin film. The the narrowest and has the highest intensity. In the case bulb has an extension containing a globule of the of potassium, the maximum value of the total angular cesiumÐpotassium alloy. Thus, at a temperature of momentum F equals 2. Therefore, the multiplicity of about 50°C, the densities of Cs and K vapors inside the the high-order resonance is 4. At optimum values of the bulb are kept close to each other.

Cs RF coils Modulation Ref. Fmod = 5Hz K CsK Cell PhD

KRF H Phase CsRF control Frequency PhD VCO synthesizer × Fout = Fin 2.002 Ref.

Frequency counter

Fig. 2. Simpliﬁed functional diagram of the CsÐ4QK tandem.

TECHNICAL PHYSICS Vol. 45 No. 7 2000

A NEW MODEL OF A QUANTUM MAGNETOMETER 933

The bulb is illuminated by beams of circularly METROLOGICAL PERFORMANCE polarized light from cesium- and potassium-discharge OF THE MAGNETOMETER lamps. Only the D1 lines of resonance doublets are The absolute values of the constants involved in used. The potassium light is aligned with the static equation (1) are known with an accuracy to ~10Ð7. Tak- magnetic field, whereas the cesium beam makes an 39 ° ing the values of gj and gi for the potassium isotopes K angle of ~45 with the field direction. After passing 41 through the cell, both beams are detected with silicon and K from [8]: photodiodes. The resulting photocurrents are amplified g = 2.00229421(24), and applied to the inputs of two phase detectors (PhDs). f The cesium spin generator is based on the principle of 39g = Ð1.14193489(12) × 10Ð4, self-tuning of the voltage-controlled oscillator (VCO) i frequency to the cesium resonance frequency. For this 41 ( ) × Ð4 purpose, the signal from the cesium beam photodetec- gi = Ð0.7790600 8 10 tor is applied to one of the inputs of the phase detector. µ and that of B/h from the CODATA Recommendations, The reference signal is generated by the VCO. If the (1997): frequency of the VCO, feeding the RF induction coil, is µ ( ) close to the cesium resonance frequency, the pump B/h = 13.99624677 94 Hz/nT, beam, having passed the bulb, becomes amplitude- modulated at the VCO frequency. The modulation we obtain for proportionality factor fK/H phase depends on how close the VCO frequency is to ( ) f K/H = 7.00466137 58 Hz/nT. (2) the resonance frequency. The signal from the VCO is applied to the second input of the phase detector. In Formally, the absolute accuracy of the tandem is turn, the VCO frequency is controlled by the output limited by an error involved in scaling factor fK/H (2). voltage of the phase detector. In this manner, the VCO However, in terms of reproducibility of device read- is tuned to the resonance frequency. ings, the absolute accuracy of the tandem can go beyond these limits. Equation (1) relates the output fre- The cesium resonance frequency is a rough measure quency of the magnetometer to the magnetic field of the external magnetic field intensity. However, its intensity through the atomic and fundamental constants value needs correction because of significant system- and, therefore, imposes no limitations on the reproduc- atic errors. The correction is performed by bringing an ibility. The limitations occur when the parametric additional loop for controlling the VCO frequency in dependence of the four-quantum resonance frequency the device circuit through the signal of four-quantum on the following factors is taken into account: the spec- potassium resonance. The doubled frequency of cesium tral composition and intensity of the pump light, the resonance deviates only by 0.1% from the four-quan- amplitude of the resonance-inducing alternating mag- tum potassium resonance frequency throughout the netic field, the density of potassium vapors, and the range of geomagnetic fields. This allows the potassium static magnetic field intensity. Magnetization of mag- resonance frequency to be synthesized by merely mul- netometer parts and an imperfect procedure for deter- tiplying the VCO frequency by a constant factor of mining the resonance peak position can also contribute 2.002395 …. Along with the frequency multiplication, to systematic errors. provision is made for a slow (at a frequency of 5 Hz) frequency modulation of the VCO signal. The signal In four-quantum resonance OPMs, the systematic induced by the potassium beam is synchronously errors are expected to be lower than in conventional sin- detected at the modulation frequency. This allows the gle-quantum resonance instruments because of a smaller resonance width and higher resolution. The res- error signal for correcting the VCO frequency (fVCO) to be obtained (provided that f lies within the cesium olution may, however, be increased at the expense of VCO reproducibility due to reduced pump light intensity. It is resonance line width). Thus, the output frequency well known that pumping causes optical shifts of the dynamics is measured using the fast cesium magnetom- resonance frequency. These are of two types [9]. One eter; then, the obtained data are corrected with regard occurs when RF coherence is transferred from the for the narrow potassium resonance frequency mea- atomic ground state to the excited state [10]. An optical sured with the slower potassium magnetometer. shift of this type is not observed in the case of four- The frequency fK of four-quantum resonance in quantum resonance, since the coherence cannot be potassium atoms depends on the magnetic field inten- transferred to the excited state at single-quantum opti- sity H as cal excitation. The second component of the optical shift is the Stark shift [11]. It is typical of four-quantum µ ( ) ( ) f K = H B g j + 3gi / 4h , (1) resonance but can be linearly decreased by reducing the pump light intensity or even completely eliminated by µ where B is the Bohr magneton; gj and gi are the elec- selecting the appropriate spectrum of pump light. tron and nuclear g factors for potassium, respectively; Experimental study of the long-term stability of the and h is Planck’s constant. magnetometer is extremely difficult because of the lack

TECHNICAL PHYSICS Vol. 45 No. 7 2000 934 ALEKSANDROV et al.

H – H0, pT(0.1s) 200

150

100

–50

–100

–150

–200 0 10 20 30 40 50 60 70 80 90 100 110 120 t, s

σ Fig. 3. CsÐK tandem resolution. H0 = 49574.500 nT, (0.1 s) = 18 pT.

H – H0, pT 200 150 100 50 0

I1 – 50 I1 I2 I1 I2 –100 –150 –200 –250 –300 0 100 200 300 400 500 600 t, s

Fig. 4. Effect of variations in the cesium lamp intensity on the readings of the Cs–K tandem (filled circles) and cesium magnetometer µ µ (open circles). H0 = 49574.470 nT, I1 = 3.0 A, and I2 = 1.5 A. of standards of the desired accuracy. Also, it is hard to experiments, however, the magnetometer sensitivity produce a sufficiently stable magnetic field. To tackle was limited by a number of other factors, including the the problem, we placed the magnetometer into the sta- magnetic field stability. Hence, the obtained value is bilized magnetic field and measured changes in its merely the lower limit of sensitivity. Figure 3, charac- readings induced by variations in pump light intensity, terizing the device resolution, shows the scatter in the RF-field strength, operating cell temperature, and device readings at a rate of 10 counts per second in a phase modulation parameters. These variations can be stabilized magnetic field of about 0.5 Oe. The disper- made sufficiently fast, so that the drift of the magnetic sion of counts is 18 pT. Averaging over one second field intensity can be neglected. decreases the dispersion almost tenfold. This means that the shot noise of the photocurrent does not limit the The magnetometer resolution is basically limited by device sensitivity (i.e., theoretically, the sensitivity can the performance of the cesium spin generator. In the be increased).

TECHNICAL PHYSICS Vol. 45 No. 7 2000 A NEW MODEL OF A QUANTUM MAGNETOMETER 935

H – H0, pT 200

150

100

50 I1 I2 I1 I2 0 I1 I2

– 50

–100

–150

–200

–250 –300 0 100 200 300 400 500 600 t, s

Fig. 5. Effect of variations in the potassium lamp intensity on the readings of the CsÐK tandem. H = 49574.470 nT, I = 7.6 µA, µ 0 1 and I2 = 4.6 A.

H – H0, pT 200 150 100 50 H1 H2 0 H1 H2 H1 – 50 –100 –150 –200 –250 –300 0 100 200 300 400 500 600 t, s

Fig. 6. Dependence of the Cs–K tandem readings on the amplitude of the RF ﬁeld inducing resonance in potassium atoms. H0 = 49574.470 nT and H2 = 2H1.

The effect of variations in the cesium lamp intensity on its readings was expected. It can be seen that they on OPM readings is shown in Fig. 4 (ﬁlled circles indi- remain unchanged within a random error of about cate the tandem readings; open circles, readings of the 10 pT (abrupt changing in the light intensity causes cesium magnetometer decoupled from the potassium transient overshoots). On the contrary, the readings of magnetometer). The intensity of the cesium lamp was the cesium spin generator alone drift with time and subjected to stepwise twofold variations (alternate two- change abruptly by approximately 0.2 nT in response to fold increase and decrease). For the tandem, no effect the changes in the intensity of the cesium lamp.

TECHNICAL PHYSICS Vol. 45 No. 7 2000 936 ALEKSANDROV et al.

H – H0, pT CONCLUSION 150 Testing of the CsÐK tandem supported its high metrological performance. It was found that, within the limits of about 10 pT, the device readings were virtually 100 independent of the pump light intensity, RF-field strength, and absorption cell temperature, even if changes in these parameters went far beyond the normal service variability. The most pronounced effect on 50 the tandem readings was found to be exerted by the variations in the RF field inducing four-quantum resonance in potassium atoms. In this connection, it is rec- ommended to use the instrument as a stationary mea- 0 suring device. In this case, the (normal) projection of the alternating magnetic field H1 on the direction of the 45 50 55 60 65 70 magnetic field H to be measured is relatively easy to , °C 0 T maintain at a constant level. Fig. 7. Dependence of the CsÐK tandem readings on the absorption cell temperature. H0 = 49573.870 nT. REFERENCES 1. E. B. Aleksandrov, Opt.-Mekh. PromÐst. 55 (12), 27 The variations in the potassium lamp intensity were (1988). expected to bring about small changes in the tandem 2. E. B. Alexandrov and V. A. Bonch-Bruevich, Opt. Eng. readings. The experimental curve is shown in Fig. 5. It 31, 711 (1992). is hard to perceive any systematic variation in the tandem readings (within the limits of 10 pT) against the 3. E. B. Aleksandrov, V. A. Bonch-Bruevich, and background of noise and drift, which is caused, in all N. N. Yakobson, Opt. Zh., No. 11, 17 (1993). probability, by the imperfect magnetic field stabilizer. 4. A. H. Allen and P. L. Bender, J. Geomagn. Geoelectr. 24 (1), 105 (1972). A more pronounced effect on the tandem readings is exerted by variations in the RF field inducing four- 5. E. Pulz, H.-J. Linthe, and A. Best, in Proceedings of the quantum resonance in potassium atoms (Fig. 6). It can VI Workshop on Geomagnetic Observatory Instruments, Data Acquisition, and Processing, Dourbes, Belgium, be seen that a 1.5-fold increase in the RF-field ampli- 1994, pp. 7Ð13. tude makes the device readings larger by 10Ð20 pT. As expected, the RF-field variations had no effect on the 6. E. B. Aleksandrov and A. S. Pazgalev, Opt. Spektrosk. device readings in the case of cesium resonance. 80, 534 (1996) [Opt. Spectrosc. 80, 473 (1996)]. Generally, OPM readings strongly depend on the 7. E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, Opt. absorption cell temperature. The temperature depen- Spektrosk. 82, 14 (1997) [Opt. Spectrosc. 82, 10 (1997)]. dence of the tandem readings is shown in Fig. 7. The 8. A. Beckmann, K. D. Boeklen, and D. Elke, Z. Phys. 270, device readings remain constant within 10 pT within an 173 (1974). uncommonly wide temperature range of 45Ð65°C. At 9. C. Cohen-Tannoudji and A. Kastler, Prog. Opt. 5, 3 higher temperatures, potassium resonance is not (1966). observed. 10. B. S. Mathur, H. Tang, and W. Happer, Phys. Rev. 171, High absolute accuracy of the new magnetometer is 11 (1968). especially difficult to verify by experiment. To do this 11. B. R. Bulos, A. Marshall, and W. Happer, Phys. Rev. A 4, requires a sufficiently stable and uniform magnetic 51 (1971). field and a high-precision reference instrument. In our 12. E. V. Blinov, R. A. Zhitnikov, and P. P. Kuleshov, Zh. experiments, a certified alkaliÐhelium magnetometer Tekh. Fiz. 49, 588 (1979) [Sov. Phys.ÐTech. Phys. 24, [12] was used for this purpose. Within the magnetic 336 (1979)]. induction range 30Ð60 µT, the discrepancy between the readings of the tandem and alkali-helium magnetometer did not exceed 0.5 nT. Translated by K. Chamorovskiœ

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The Threshold Current of a Balanced Double-Contact Low-Inductance Interferometer I. N. Askerzade Institute of Physics, Academy of Sciences of Azerbaijan, pr. Dzhavida 33, Baku, 370143 Azerbaijan E-mail: [email protected] Received May 13, 1999

Abstract—The dependence of the threshold current of a double-contact low-inductance interferometer on the capacitance of a Josephson contact and geometric inductance was found. © 2000 MAIK “Nauka/Interperiod- ica”.

The miniaturization of electrodes and striplines in structive data readout [5] and the need for using exter- superconducting electron devices is known to increase nal clocked power supplies in pulse-actuated circuits the circuit impedance. Therefore, the design of low-cur- [6, 7]. With this in mind, we evaluated the threshold rent switches is of great importance. Interference current in a balanced double-contact interferometer in devices, which offer higher sensitivity to a magnetic field the low-inductance approximation. than isolated Josephson contacts, seem to be the most It is known that, in the low-conductance limit, a bal- appropriate in this respect. Also, interference devices anced double-contact interferometer behaves as a sin- made by advanced submicron technology allow for a gle Josephson contact with a critical current considerable decrease in the contact capacitance [1]. ϕ ϕ 1 Ð 2 A double-contact interferometer is the simplest IM = 2Ic cos------device whose critical current depends on an applied 2 magnetic field [2]. Memory cells on isolated contacts ϕ ϕ ϕ and an effective phase = ( 1 + 2)/2. For closedness, with the use of double-contact interferometers were the equation for total magnetic flux, and remain to be the best candidates for superconducting computers. Double-contact interferometers are also ϕ ϕ Ð ϕ = ϕ Ð lsin-----e cosϕ , (1) the basis for high-speed parallel analog-to-digital con- 1 2 e 2 verters. A variety of nonlinear properties inherent in ϕ πΦ Φ double-contact interferometers makes possible the cre- where e = 2 e/ 0, must be added. ation of other pulse [3] and digital [4] devices. In [3], At low l’s (l Ӷ 1) and in view of (1), the supply cur- for example, a double-contact interferometer was used rent (in terms of contact critical current) within the as a short strobe driver. resistive model is given by In double-contact interferometers, switching ϕ ϕ Ie e ϕ l 2 e ϕ between various states strongly depends on the external ie ==----2cos----- sin + --- sin ----- sin2 . (2) β Ic 2 2 2 (supply) current Ie, the capacitance parameter of a Josephson contact, and the geometric inductance l. In deriving an analytic dependence of the threshold β π 2 Φ current on l and β, we ignore a similar dependence for Here, = 2 Ic RN C/ 0 (C is the capacitance; RN is the Φ the supply current, since it is negligible in comparison resistance in the normal state; and 0 is a magnetic flux Φ ប π Φ with the desired one. We will also ignore decay in the quantum, 0 = /2e), and l = 2 LIc/ 0. As was noted contacts, assuming βÐ1 to be a small parameter. With [2], there always exists the threshold current IQ of the these assumptions met, the transition free energy is interferometer (which depends on l, β, and critical cur- expressed as rent ratio Ic /Ic ) such that, at Ie > IQ, the system is 1 2  2 ϕ βϕú e switched into the resistive (R) state. At Ie < IQ, the sys- ()ϕ EE==c ------2+ cos----- 1 Ð cos tem passes to the superconducting (S) state.  2 2 In spite of the considerable progress into fabricating (3) Josephson-contact circuits with double-contact inter- l2 ϕ  +--- sin -----e ()12Ð cos ϕ, ferometers, a dependence of the threshold current IQ on 42  l and β still remains unclear. In our opinion, this prob- Φ π lem is also closely related to the problem of nonde- where Ec = 0Ic/2 .

938 ASKERZADE Resistive losses in energy per cycle (2π) are calcu- (5) in the small parameter l; eventually, we obtain lated by the formula 2 π I 4 1/2 lsin (ϕ /2)  2 Q (ϕ )  e  ------= ------cos e/2 1 Ð ------(--ϕ------)- . (8) ϕ(ϕ) ϕ 2Ic π β  6cos e/2  Wi = Ec ∫ ú d . (4) 0 It follows from (8) that the value of IQ/2Ic decreases ϕ The expression for ϕú (ϕ) is found from (3): at l = 0 and e = 0, as in the case of the recovery current for a single contact [2]. Along with the critical current, 1/2 ϕ the threshold current of a balanced double-contact ϕ 2 e( ϕ) interferometer turns out to be modulated by an external ú = β-- e Ð 2cos----- 1 Ð cos  2 magnetic field. With the inductance l fixed, the thresh- (5) old current vanishes at some flux Φ . This means that, 1/2 e ϕ  for I ≠ 0 and large β's, the system will always change l 2 e( ϕ) e Ð -- sin ----- 1 Ð cos2  , over to the R state. 4 2  where e = E/Ec. REFERENCES The infinite phase variation is possible only if e ≥ 1. K. K. Likharev, A. B. Zorin, and V. K. Semenov, New ϕ 4cos( e/2). At a fixed current, the energy delivered Horizons for Superconductor Electronics (Moscow, from the power source is found as 1988), Vol. 1.

2π 2. K. K. Likharev, Introduction to the Dynamics of Joseph- son Contacts (Nauka, Moscow, 1985). ϕ π We = Ec ∫ ied = 2 Ecie. (6) 3. P. Wolf, B. J. van Zeghbroeck, and U. Deutsch, IEEE 0 Trans. Magn. 21, 226 (1985). . 4. C. A. Hamilton and F. L. Lloyd, IEEE Trans. Magn. 19, The mean voltage (ϕ = ϑ) is determined by equat- 1259 (1983). ing energies (4) and (6). The supply current at which 5. R. E. Harris, C. A. Hamilton, and F. L. Lloyd, Appl. Josephson oscillation ceases is found from the limit of Phys. Lett. 35, 720 (1979). ϕ (5) at e = 4cos( e/2): 6. S. R. Whiteley, G. K. G. Hohenwarter, and S. M. Faris, 2π IEEE Trans. Magn. 23, 899 (1987). 1 7. I. N. Askerzade, Zh. Tekh. Fiz. 70 (1), 68 (2000) [Tech. i = ------ϕú dϕ. (7) e 2π ∫ Phys. 45, 66 (2000)]. 0 In this case, integral (4) is calculated by expanding Translated by V. Isaakyan

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Dependence of the Transverse Dynamics of the Plasma Electrons in the Ion Focus Regime on the Ratio of the Radius of a Relativistic Electron Beam to the Radius of the Ion Channel E. K. Kolesnikov and A. S. Manuœlov Smirnov Research Institute of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, 198904 Russia Received May 28, 1999

Abstract—A study is made of the effect of the magnetic self-ﬁeld of a relativistic electron beam propagating in the ion focus regime on the transverse dynamics of plasma electrons. For Gaussian radial proﬁles of the beam and the ion density in the channel, the maximum deviation of the plasma electrons from the axis of the beamÐ plasma system is determined as a function of the space-charge neutralization fraction, the ratio of the characteristic beam radius to the channel radius, and the net beam current. © 2000 MAIK “Nauka/Interperiodica”.

Recently, much attention has been devoted to the and ion channel, respectively) influences the transverse problem of the propagation of a relativistic electron collisionless dynamics of the plasma electrons in the beam (REB) in the ion focus regime (IFR) [1Ð8]. The process of charge neutralization at the beam front in the η IFR arises when the ion line density Ni in the plasma IFR. In considering situations with different values, channel is lower than the line beam density Nb. At the we also study how the escape of plasma electrons away entrance to a preformed plasma channel, the electric from the beam region is affected by the magnetic self- self-field of the beam front expels the plasma electrons field of the beam with a current in the range Ib = 5Ð from the channel, so that the ions remaining in the 50 kA. Note that the case η = 1 was examined by channel focus and guide the beam, thereby preventing Briggs and Yu [4] for a Bennet beam and a Bennet pro- its transverse dispersion. file of the ion density in the channel. The main onset condition of the IFR implies a suffi- We consider a paraxial monoenergetic REB guided ciently low pressure of the background gasÐplasma with a preformed plasma channel in the IFR. We direct medium. In this case, the electrons that are produced in the z-axis of the cylindrical coordinate system (r, θ, z) a preformed plasma channel or a channel created along the channel axis and assume that the electron and directly by the electron-beam ionization of a gas and ion densities in the channel obey Gaussian distribu- are pushed away from the beam path by the transverse tions, component of the collective electric field cause no addi- 2 tional ionization of the background plasma. This situa- Nbi, r n , = ------expÐ ------, (2) bi π 2 tion occurs under the condition Rbi,  R bi, λ ӷ R , (1) i b where the subscripts b and i stand for the parameters of λ where i is the characteristic scale length on which the the beam and ion channel, respectively, and Nb and Ni avalanche ionization develops and Rb is the characteris- are the line densities of the beam electrons and the ions tic beam radius. in the channel. For the space charge of an REB to be neutralized in We also assume that the z-component of the collec- the IFR, the transverse component of the electric field tive electric field, Ez, is small, the net beam current Ib is of the beam front should rapidly expel the background time independent, and the electron motion in the chan- electrons away from the beam path. However, the pro- nel is collisionless. Under all the above assumptions, cess of charge neutralization can also be affected by the the energy equation yields magnetic self-field of the beam, which may, to a sub- 2 dγ e v r 2Ib r stantial extent, prevent the background electrons from ------=------1 Ð expÐ ------leaving the beam region. dt β 3 r2 mc R b In this paper, assuming that the radial profiles of the (3) beam and the ion density in the channel are Gaussian in 2 η r shape, we investigate how the parameter = Rb/Ri Ðf1Ð expÐ ------, c2 (where Rb and Ri are the characteristic radii of the beam Ri

940 KOLESNIKOV, MANUŒLOV

rm/Rb rm/Rb 102 50 80 60 40 40 6 20 30 10 5 8 20 3 2 6 1 1 2 3 4 4 10 2

0 0.2 0.4 0.6 0.8 1.0 0 3 6 9 12 I , kA fc b

η η Fig. 1. Dependence of rm/Rb on fc for = Rb/Ri = (1) 5, Fig. 2. Dependence of rm/Rb on Ib for = (1) 1.5, (2) 1, and (2) 2, (3) 1.5, (4) 1, (5) 0.5, and (6) 0.1. (3) 0.5. where m and e are the mass and charge of an electron, field of the beam. With allowance for the relationship β = v /c, v and v are the radial component and the z r z v 2 v 2 z-component of the beam electron velocity, I is the net γ r γ 2 γ z b  ----- = Ð  ----- Ð 1, (7) beam current, c is the speed of light, fc = Ni/Nb is the c c γ effective space-charge neutralization fraction, is the from (5) we find Lorentz factor, and t is the time. v 2 Note that the right-hand side of (3) incorporates the γ r ˜ ᑬ( ) ᑬ( )  ----- = 2Ib Rb Ð f c Ri -- contribution of the radial component E of the electric c r (8) field of both the beam electrons and the ions in the 2 f 2 channel. Ð f I˜ ᑬ(R )ᑬ(R ) + ----c-I˜ ᑬ (R ) , c b b i 2 b i Assuming that the plasma electrons are initially immobile, from (3) we obtain ˜ ᑬ ξ ᑬ ξ where Ib = Ib/I A* and the notation ( ) = (r, r0, ) is introduced for brevity. I γ Ð 1 = ----b-[ᑬ(r, r , R ) Ð f ᑬ(r, r , R )], (4) The maximum deviation of a plasma electron from * 0 b c 0 i I A the axis of the beam–plasma system is defined as the distance r between the electron stopping point (v = 0) β γ m r where I A* = 17 [kA] is the Alfvén current for = 1 and and the axis and can be found from the equation ∞ µ µ µ r0 = r at t = 0, E1(z) = ∫ d exp(Ð )/ is the integral f ᑬ(R )[2 Ð f ᑬ(R )I˜ ] z ᑬ(R ) = ----c------i------c------i------b---. (9) exponential function, and b ( ᑬ( )˜ ) 2 1 Ð f c Ri Ib

2  2 2 Figure 1 shows the dependence of rm/Rb on fc for dif-  r  r0  r  η ᑬ( , , ξ)   ferent values of = Rb/Ri at Ib = 5 kA and r0/Rb = 0.1. r r0 = ln---- Ð E1 ----2 + E1----2 . (5) r0 ξ  ξ Note that a curve analogous to curve 4 was obtained by Briggs and Yu [4] for Bennet proﬁles nb and ni of the On the other hand, taking into account the paraxial beam and the ion density in the channel. Figure 2 shows η character of the beam, we turn to formula (2) and the the dependence of rm/Rb on Ib for = 0.5, 1, and 1.5 at z-component of the equation of motion of a plasma fc = 0.5. From Fig. 1, we can see that the divergence of electron to obtain the ion channel (η < 1) has a signiﬁcant impact on the escape of plasma electrons from the beam region. In γv I particular, for η = 0.1, the maximum deviation of the ------z = ----b-ᑬ(r, r , R ). (6) c I* 0 b plasma electrons from the system axis in the neutraliza- A Ӎ 2 tion phase (up to fc 0.6) is rm/Rb > 10 . For larger val- η η Note that the right-hand side of (6) incorporates the ues of the parameter ( > 1), in the range fc > 0.2, the contribution of the θ-component of the magnetic self- maximum deviation rm is much smaller. A comparison

TECHNICAL PHYSICS Vol. 45 No. 7 2000 DEPENDENCE OF THE TRANSVERSE DYNAMICS 941 of curves 1Ð3 with curve 4 (η = 1) confirms the quali- 3. J. Krall, K. Nguyen, and G. Joyce, Phys. Fluids B 1, tative conclusion that the beams with larger radii more 2099 (1989). efficiently prevent the escape of plasma electrons in the 4. R. J. Briggs and S. S. Yu, Rep. No. UCID-19399 (Law- radial direction. That is why, for the same value of fc, rence Livermore National Laboratory, Livermore, CA, curves 1Ð3 lie below curve 4. It is seen from Fig. 2 that 1982), p. 47. the magnetic self-field of the beam, Bθ, which is pro- 5. V. B. Vladyko and Yu. V. Rudyak, Fiz. Plazmy 19, 1444 (1993) [Plasma Phys. Rep. 19, 760 (1993)]. portional to the net beam current Ib, may, to a substantial extent, prevent the background electrons from leav- 6. E. K. Kolesnikov and A. S. Manuœlov, Radiotekh. Élek- ing the beam region. tron. (Moscow) 37, 694 (1992). 7. E. K. Kolesnikov and A. D. Savkin, Pis’ma Zh. Tekh. Fiz. 20 (1), 54 (1994) [Tech. Phys. Lett. 20, 26 (1994)]. REFERENCES 8. E. K. Kolesnikov and A. G. Zelenskiœ, Zh. Tekh. Fiz. 65 1. H. L. Buchanan, Phys. Fluids 30, 221 (1987). (5), 188 (1995) [Tech. Phys. 40, 510 (1995)]. 2. R. F. Fernsler, R. F. Hubbard, and S. P. Slinker, Phys. Fluids B 4, 4153 (1992). Translated by G.V. Shepekina

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Effect of Multiple Scattering and the External Magnetic Field on the Resistive Sausage Instability of a Relativistic Electron Beam E. K. Kolesnikov and A. S. Manuœlov Smirnov Research Institute of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, 198904 Russia Received May 28, 1999

Abstract—The problem is studied of how multiple Coulomb scattering and the external longitudinal magnetic ﬁeld affect the resistive sausage instability of a relativistic electron beam propagating in a gasÐplasma medium with nonzero ohmic conductivity. It is shown that these factors signiﬁcantly reduce the amplitude of the sausage mode. © 2000 MAIK “Nauka/Interperiodica”.

In recent years, much attention has been paid to the The transverse dynamics of such an REB is dynamics of relativistic electron beams (REBs) propa- described by the following equations for the doubled gating in gasÐplasma media [1Ð12]. Among the prob- root-mean-square beam radius R2 and the root-mean- lems associated with the guiding of REBs, of special square beam emittance E2 [1, 3, 5] interest is that of investigating the large-scale resistive instabilities of the beams. Along with the resistive fire- 2 2 2 2 ∂ R 2 U k R 4E4 P θ hose instability (the mode with the azimuthal wave ------++------c - = ------+ ------, (1) number m = 1), an important role is played by the resis- ∂z2 R 4 R3 R 3 tive sausage instability (the mode with the azimuthal wave number m = 0), which manifests itself as axisym- ∂ 2 3 ∂ 2 E α R U R σ2 metric perturbations of the beam radius. Physically, the ------=Ð2ph------+1ngR.(2) mechanism for the onset of the resistive sausage insta- ∂ z E ∂ z 2 bility (RSI) is governed by the phase delay between the eddy currents induced by these perturbations and the 〈〉22 oscillating component of the beam current density. The Here, U = kβr is the generalized perveance of the RSI of an REB was studied in a number of papers [3Ð beam (where the angle brackets stand for averaging 5, 9, 10], which, however, did not treat the often over the radial profile of the beam current density); 2 encountered effects of multiple Coulomb scattering of kβ is the squared betatron wavenumber of the beam the beam electrons by the atoms and molecules of the 2γ 3 background gas and did not take into account the exter- electrons; kc = eB0/ mc is the cyclotron wavenumber nal longitudinal magnetic field. of the beam electrons in a longitudinal magnetic field of strength B0 (where e and m are the charge and mass of In this paper, we use analytic methods to derive the γ θ perturbed beam radius as a function of the background an electron and is the Lorentz factor); Pθ is the -com- gas density and the strength of the external longitudinal ponent of the generalized momentum of the beam elec- α magnetic field. trons in the beam segment under consideration; ph is the phase-mixing coefficient of the beam electrons [8]; We consider an axisymmetric paraxial REB propa- σ 1 is the transport cross section for multiple scattering gating along the z-axis of the cylindrical coordinate of the beam electrons by the background gas atoms; system (r, θ, z) in a scattering gasÐplasma medium with σ πσ ӷ and ng is the density of gas atoms. a high ohmic conductivity such that 4 Rb/c 1, where Rb is the characteristic beam radius and c is the Assuming that, in the linear stage of the RSI, the speed of light. We assume that the beam is completely perturbed quantities are small (in particular, δR = R Ð Ӷ charge-neutralized and that the magnetic self-field of R0 1, where R0 is the doubled equilibrium beam the beam (or, in other words, the beam current) is neu- radius), from (1) and (2) we find tralized only partially, the degree of neutralization being fm. We also assume that the beam propagates 2 2 ∂δR4 U 0 2 2δU 4δE ------δ Rkδ = ------, along a steady-state external uniform magnetic field of ------2+++2 cR------3 (3) ∂ R0 strength B0. zR 0 R0

EFFECT OF MULTIPLE SCATTERING AND THE EXTERNAL MAGNETIC FIELD 943

2 α 3 2 δR/δR(0) ∂δE ph R0 U0 ∂ δR ------= Ð ------+ 2R δRσ n , (4) ∂ 2 0 1 g 1.0 z 4 E0 ∂z 0.6 where the zero subscript denotes unperturbed parame- 2 ter values. 0.2 1 0 Assuming that, during the RSI, the perturbations of –0.2 3 the beam radius are self-similar, we can obtain –0.6 δR δU = ------2Ψ(1 Ð f )U , (5) –1.0 R m 0 0 0 50 100 150 z, cm where ∞ Radial perturbation δR/δR(0) vs. z for (1) case (25), (2) case 2 (26), and (3) case (27). 4π 3 2 Ψ ------( ) = 2 ∫drr Jb r , (6) Ib 0 Laplace transform and assuming ∂δR/∂z(0) = 0, we Ib is the net beam current, and Jb(r) is the radial profile obtain of the beam current density. We will work under the condition ∞ δR(0) δR(z) = ------∫ dΩexp(ÐiΩz) Lz0 ӷ 2πi ------1, (7) Ð∞ Lz1 (14) (Ω + iα ᏾)Ω where × ------p--h------, (ΩΩ 2 Ω3 α ᏾Ω2 Λ ) 0 Ð Ð i ph Ð i sc R0 δR Lz0 = ∂------∂----, Lz1 = ∂----δ------∂----. (8) R0/ z R/ z Λ 2 ᏾ where i is the imaginary unit, sc = 4S/ R0 , and = Condition (7), which implies that, in the positive U /E . direction along the z-axis, the equilibrium root-mean- 0 0 square beam radius changes much more gradually than Clearly, the integral over Ω in (14) can be evaluated the perturbed beam radius δR, allows us to solve equa- by the method of residues. In the particular case with no ≤ Λ tions (3) and (4) over distances of z Lz0 by taking the scattering ( sc = 0), we readily obtain Laplace transform α ᏾ ∞ δ ( ) δ ( ) Ð ph z R z = R 0 exp------ ∆F = dzexp(iΩz)F(z), (9) 2 ∫ (15) α ᏾ 0 × ( Ψ ) ph ( Ψ ) cos z 1 + ------Ψ------sin z 1 , where F is a function of z. 2 1 In this way, equations (3) and (4) yield where Ω is defined by (13) and F(Ω) 0 ∆R = ------. (10) D(Ω) 2 2 1/2  2 α ᏾  Here, Ψ = Ω Ð -----p--h------. (16) 1  0 4  α ∂δ (Ω)  i phU0 R( ) Ωδ ( ) F = 1 + ----Ω------ ---∂------0 Ð i R 0 , (11) Expression (15) is a generalization of the analogous E0 z ≠ expression derived by Lee [5] to the case B0 0. From 2 2 iα ΩU 4iS (15), we can easily see that the processes of phase mix- (Ω) Ω Ω ph 0 ------D = 0 Ð Ð ------Ð 2, (12) ing partially suppress the RSI on the spatial scale Lph ~ E0 ΩR α 0 2E0/( phU0). σ where S = 2 1ng, U0 = Ib/IA, IA is the limiting Alfvén With allowance for multiple Coulomb scattering current, and Λ ≠ ( sc 0), the poles of the integrand in (14) can only be 4U found by solving the cubic equation Ω2 ------0( Ψ) 2 0 = 2 1 Ð f m + + kc (13) R 3 2 2 0 Ω + iα ᏾Ω Ð ΩΩ + iΛ = 0. (17) Ψ ph 0 sc with fm as the degree of current neutralization and as the form-factor defined in (6). Taking the inverse Again, applying the method of residues and per-

TECHNICAL PHYSICS Vol. 45 No. 7 2000 944 KOLESNIKOV, MANUŒLOV forming laborious manipulations, we obtain Λ × Ð3 Ð3 kc = 0, sc = 2.3 10 cm (27) δ ( ) δ ( ) [ ( ) ] 2 × Ð3 R z = R 0 exp Ð b/3 + T/2 z ------2--- under the conditions R0 = 1 cm and U0 = 2 10 . The (4A + 9T ) Λ × Ð3 Ð3 value sc = 2.3 10 cm in (27) corresponds to the  case in which the scattering gas is nitrogen at atmo- × ( Ψ ) ( ) spheric pressure, the beam particle energy being E =  2A + 2B Ð 3T 1 cos Az --  2 5 MeV. The value of kc in (26) corresponds to B0 = (18) 100 G at γ = 10. [2AΨ + 3T(A + B)]  + ------1------sin( Az)  From the figure, we can see that the external longi- A  tudinal magnetic field and the processes of multiple scattering and phase mixing all act to suppress the 4(T2 + bT/3 + 2b2/9) radial perturbations of the beam during the RSI. This + exp[Ð(b/3 Ð T)z]------, ( 2) conclusion agrees with the results of experiments, in 4A + 9T which no undamped axisymmetric perturbations of the where beam radius were observed. 1/3 1/3 T = (M Ð (P)1/2) + (M + (P)1/2) , (19) REFERENCES Λ sc α U0 1. E. P. Lee, Phys. Fluids 19, 60 (1976). M = T1 + ------, b = ph------, (20) 4 E0 2. E. P. Lee, Phys. Fluids 21, 1327 (1978). 3. E. P. Lee and R. K. Cooper, Part. Accel. 7, 83 (1976). 2 bΩ b3 Λ T = ------0 Ð ----7 + -----s--c, (21) 4. E. P. Lee and S. S. Yu, Rep. No. UCID-18330 (Lawrence 1 6 2 4 Livermore National Laboratory, Livermore, CA, 1979), p. 23. Ω4 2 5. E. P. Lee, Rep. No. UCID-18940 (Lawrence Livermore 0  2 b  P = ------Ω Ð ---- + Λ T , (22) National Laboratory, Livermore, CA, 1981), p. 34. 27  0 4  sc 1 6. R. F. Fernsler, R. F. Hubbard, and M. Lampe, J. Appl. 2 Phys. 75, 3278 (1994). Ω 2 b 3 2 A = 0 Ð ---- + --T , (23) 7. E. R. Nadezhdin and G. A. Sorokin, Fiz. Plazmy 14, 619 3 4 (1988) [Sov. J. Plasma Phys. 14, 365 (1988)]. 8. W. A. Barletta, E. P. Lee, and S. S. Yu, Nucl. Fusion 21, b T 2b T B = -- + --- ----- Ð --- . (24) 961 (1981). 3 2 3 2 9. M. Lampe and G. Joyce, Phys. Fluids 26, 3371 (1983). In order to illustrate the results obtained, we con- 10. E. K. Kolesnikov and A. S. Manuœlov, Pis’ma Zh. Tekh. sider an REB with a Bennet radial current-density pro- Fiz. 17 (3), 46 (1991) [Sov. Tech. Phys. Lett. 17, 96 α (1991)]. file truncated at r = Rb ( ph = 0.62). The figure shows δ 11. E. K. Kolesnikov and A. S. Manuœlov, Zh. Tekh. Fiz. 62 the z-profiles of R obtained from formula (18) for the (9), 55 (1992) [Sov. Phys. Tech. Phys. 37, 924 (1992)]. cases 12. E. K. Kolesnikov and A. S. Manuœlov, Zh. Tekh. Fiz. 67 Λ kc = 0, sc = 0, (25) (7), 108 (1997) [Tech. Phys. 42, 819 (1997)]. 2 × Ð3 Λ kc = 3.6 10 , sc = 0, (26) Translated by G.V. Shepekina

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Cooperative Mechanism of Self-Diffusion in Metals V. G. Chudinov Physicotechnical Institute, Ural Division, Russian Academy of Sciences, Izhevsk, 426001 Russia; E-mail: [email protected] Received June 1, 1999

Abstract—The atomic mechanism of self-diffusion in a face-centered cubic structure (exempliﬁed by aluminum) is simulated by the molecular-dynamics method without resorting to a priori information. No conﬁrma- tion of vacancy-jump mechanisms of diffusion has been obtained. The cooperative mechanism of diffusion suggested previously by Khait and Klinger and based on the assumption of possible local melting near vacancies (within oneÐtwo nearest coordination shells) appears to be valid. As a result, the preexponential factor proves to depend on the temperature and heat of melting, and the exponential factor depends on the heat of melting and the current temperature. © 2000 MAIK “Nauka/Interperiodica”.

1. Atomic diffusion in solids is one of the fundamen- means the following. A diffusion jump may occur when tal phenomena which is a base for understanding a lot a fluctuation in the atomic distribution appears around of effects. But, strange as it may seem, there is no uni- a vacancy. After relaxation of the given fluctuation, one fied point of view on the diffusion mechanisms that of the atoms surrounding the vacancy may occupy the were devoid of internal contradictions. The quantities vacancy position. entering into the equations for diffusion parameters A similar approach was developed, e.g., by Klin- become meaningful only when the atomic mechanism ger [4]. In his opinion, the temperature in some region of the process is known. As a rule, some conclusions on around a vacancy may exceed, due to a fluctuation, the these quantities can be made by measuring the energies melting (more correctly, quasimelting) temperature of controlling the process. For simple metals far from the the crystal. Upon the subsequent solidification, the melting point, such a parameter is the energy of vacancy may occupy another site of the restored crystal vacancy self-diffusion, which counts in favor (but by no lattice. It is very important that Klinger obtained an means proves) of the widely used model of diffusion expression for the diffusion coefficient as a function of through vacancy jumps. As other possible mechanisms, the temperature and heat of melting the following are usually listed: direct exchange, cyclic exchange (ring mechanism), simple interstitial mecha- λ ασ0 ()2/3 λ nism, interstitialcy mechanism, crowdion mechanism, 1ν 2 k m Ð ls VLn nm D = --- d exp------expÐ ------,(1) etc. [1]. “First-principles” calculations based on these 6 kTm kT models underestimate the diffusion rates by tens and even hundred of times and have a rather heuristic sig- where ν is the mean frequency of atomic vibrations; d nificance. The inconsistencies appearing in such an is the length of a diffusion jump; Tm, the melting tem- λ “individual” description of diffusion phenomena led perature; m, the heat of melting; n, the number of Frenkel’ [2] to the idea that in a self-consistent diffu- atoms of the second phase; σ 0 , the surface tension at sion theory, the interacting particles had to be consid- ls ered as a coherent system, like in the Debye theory of the liquidÐcrystal interface; VL, the volume per atom in heat capacity. the liquid phase; and k, the Boltzmann constant. Agreement with experiment for simple metals is An attempt to build such a theory was undertaken by obtained when n = 12Ð20, i.e., at local disordering Khait [3]. In his opinion, the application of the standard within 1Ð2 coordination shells. This agrees with the theory of fluctuations has been unsuccessful because of generalized coordinate relation for the self-diffusion the incorrect consideration of thermal-energy fluctua- energy E = 15λ [1]. tions as small corrections to the equilibrium distribu- m tion function in spite of the correct estimations of mac- In this work, we used the molecular-dynamics roscopic parameters. The thing is in that although the method (MDM) to qualitatively test the ideas devel- short-lived fluctuations that involve a limited number of oped in [1Ð4]. The method consists in a numerical solu- particles give a negligible contribution to the averaged tion of the Newton equations of motion when pairwise equilibrium parameters, they nevertheless may play an interaction potentials (PIP) are known. Its main advan- important part in some kinetic processes. In application tage is the minimum of physical information taken to diffusion through the vacancy mechanism, this a priori [5].

946 CHUDINOV

E, 10–13erg formed having the face-centered cubic (fcc) lattice and 1 containing ~1500 sites (one of which was vacant). At first, all of the atoms were given the same velocity (corresponding to a given temperature) with random direc- 0 tions. After that, no intervention to the system was made. After a period of ~10Ð12 s, an equilibrium state with a Maxwell distribution of velocities was reached. –1 The criterion of a jump was an atomic translation to 0.3 0.4 0.5 0.6 0.7 another WignerÐSeitz cell constructed for the ideal lat- nm tice. Obviously, some of the jumps were reversible and did not contribute to the diffusion. These jumps were Fig. 1. Pairwise interaction potential for aluminum. not taken into account. All the atoms were numbered and their positions were traced in time. The average jump time was ~10Ð13 s. About 100 jumps occurring It should be noted that the work requires a lot of according to a common scheme were observed. The computer time, since diffusion processes are very slow melting temperature for a given PIP was ~1100 K. One on the MDM scale. could hardly expect better agreement when using a 2. Several approaches are used in MDM to study model potential. We believe that the main purpose diffusion phenomena. when using the MDM should be the investigation of atomic mechanisms of diffusion (at least qualitatively) 1. Determining diffusion coefficients by calculating rather than fitting numeric results to experimental data. mean-square displacements 3. The temperature in the experiment was 900 K. [ (τ) ( )]2 This choice was accounted for by two reasons: (a) at ∑ x j Ð x j 0 lower temperatures, there were no jumps at all or their 1 D = ------j------, (2) numbers for the time periods used (10Ð11 s) were insuf- 6 τ ficient for reliable statistical analysis; (b) at higher tem- τ peratures, double and triple jumps took place along where xj( ) is jth coordinate of an atom at the time moment τ and 〈〉 stands for averaging over all atoms. with common single jumps, which led to deviations The method is integral and does not allow one to deter- from the Arrenius law. mine the atomic mechanism of diffusion. By assuming that an elementary act of vacancy dif- 2. Determining the diffusion barrier by “squeezing” fusion occurs as the overcoming of a potential barrier a diffusing atom through a fixed lens of atoms in the by one of the surrounding atoms and the subsequent direction toward a vacancy. relaxation of this fluctuation of thermal energy, we can 3. The same as in 2, but with atomic relaxation. expect the existence of some characteristic jump frequency and a uniform distribution of jumps in time. 4. The same as in 2, but with “shooting” the diffus- However, they proved to occur as series consisting of ing atom through the lens. 5Ð10 acts for 10Ð12 s. This agrees with the estimations In the last three methods, the model of diffusion is of the theory of short-lived large-energy fluctuations actually specified. The corresponding energies differ in (SLEF) [3, 4]. a ratio of 1 : 0.1 : 0.3, respectively. We propose a substantially different approach. At a Analysis of velocity vectors and kinetic and poten- fixed temperature, we construct a crystallite with this or tial energies of diffusing atoms revealed that there is that defect. Then, we let the atoms in the equilibrium neither increase in the velocity in the vacancy direction state perform free vibrations and jumps and fix their before a jump, nor a pronounced potential barrier to be coordinates in the computer memory after certain time overcome when moving away from the four-atomic periods. After executing an elementary diffusion act, a lens. return to the preceding time moment was made, and the Figure 2 displays the kinetic and potential energies process was repeated with a more detailed analysis of before and after a single diffusion act for two nearest- the parameters of atoms surrounding the defect (kinetic neighbor coordination shells around a vacancy. One and potential energies, velocity vectors of atoms in the can see that the kinetic energy increases by a factor of structure, etc.). Thus, no model assumptions on the ~4, but the potential energy varies within the range of mechanism of the diffusion act is done in the computer common fluctuations. The atomic structure around a experiment, except for specifying the interatomic inter- vacancy is typical of liquid. The vacancy location can- action. not be identified reliably. The fluctuation disappears for We used a PIP of the Morse type for aluminum cut ~10Ð12 s, and the vacancy may occupy any site of the at 0.7 nm (Fig. 1), which provided stability of the face- crystal lattice after the solidification of the “quasimol- centered cubic lattice, and an MDM20 computer code ten” region. So, a cooperative mechanism of diffusion [5]. A crystallite with cyclic boundary conditions was is realized, in which the number of atom jumps, includ-

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COOPERATIVE MECHANISM OF SELF-DIFFUSION IN METALS 947

arb. units erative mechanism of diffusion is preferable. It can be (a) interpreted as the quasimelting of the nearest surround- 6 ings of a vacancy due to rather small kinetic energy ﬂuctuations (with 20Ð100 atoms being involved in the 4 process). Then, diffusion according to a nonbarrier mechanism, close to the diffusion mechanism in liquid, 2 takes place; i.e., this is an anharmonic process in which 0 the peculiarities of interatomic interaction become a decisive factor [5]. , 103 K T (b) ACKNOWLEDGMENTS 4 The author would like to thank E.I. Salamatov for fruitful discussion and useful remarks. 2 REFERENCES 0 0 1 2 3 t, 10–13 s 1. B. S. Bokshteœn, Diffusion in Metals [in Russian] (Met- allurgiya, Moscow, 1978). Fig. 2. Dependence of (a) the kinetic and (b) the potential 2. Ya. D. Frenkel’, Introduction to the Theory of Metals energies (in temperature units) per atom near a vacancy (1st (Fizmatgiz, Moscow, 1958). coordination shell) before and after jump. The conditional moment of the jump occurrence is indicated by an arrow. 3. Y. L. Khait, Physica A (Amsterdam) 103, 1 (1980). 4. M. M. Klinger, Metalloﬁzika (Kiev) 6 (5), 11 (1984). 5. V. M. Dyadin and V. G. Chudinov, Available from ing nonreversible ones, is much more that in the tradi- VINITI, No. 1537-V 91 (1991). tional theory. 4. In conclusion, we would like to notice that the whole body of the results obtained suggest that a coop- Translated by M. Astrov

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On the Influence of Ionization Density in a Fast Charged Particle Track on the Light Output of Scintillations

V. K. Lyapidevskiœ and M. I. Ryazanov Moscow State Engineering Physics Institute (Technical University), Kashirskoe sh. 31, Moscow, 115409 Russia Received June 4, 1999; in ﬁnal form, September 6, 1999

Abstract—It is shown that ionization density due to a fast charged particle inﬂuences the light output of a scin- tillator through the relaxation transition of an electron from an impurity atom to a host-material ion. © 2000 MAIK “Nauka/Interperiodica”.

(1) In organic scintillators, regardless of their Both factors decrease the light output at the fundamen- nature, the ratio of light outputs due to α particles and tal frequency. electrons (α/β ratio) is known to be about 0.1 [1, 2]. The same has also been observed in inorganic scintilla- (4) Consider now how the ionization density inﬂu- tors [3]. The independence of the α/β ratio on the scin- ences the fraction of ionized impurities. An increase in tillation material and impurity type suggests that this the ionization density raises the average number of ions effect is related to general, rather than speciﬁc, proper- per unit volume and, thus, shortens the average interi- ties of scintillators. Hence, the explanation of the onic distance and the distance between host ions and α/β ratio uniqueness must rely on general scintillation impurity atoms. Eventually, the probability that a laws and mechanisms. A uniqueness mechanism for the bound impurity electron will be trapped by a host ion α/β ratio was not considered earlier; however, the effect increases. Consequently, the fraction of ionized impu- has been supposed to be associated with moleculeÐ rity grows with the ionization density and the light out- molecule and moleculeÐion interactions [4]. put drops. α (2) When exposed both to electrons and to parti- (5) The α/β ratio is the same for different scintilla- cles, organic scintillators in the pure state scintillate tors, because the energetically favorable capture of a weakly and the scintillation intensities are close to each α β bound impurity electron by a host atom may attend any other. This indicates that the / ratio depends on the relaxation process. Therefore, for a given ionization states taken by scintillation impurities under the action density, this capture has only a weak dependence on the of electrons and α particles. The effects fast electrons and heavy α particles have on a material differ in that a energy levels of excited impurity atoms and host-mate- heavy particle passing through the substance creates a rial properties. much higher ionization density than a light electron. It (6) It follows from the above that the qualitative fea- seems therefore natural to look for an explanation of α β tures of the light output vs. particle-induced ionization the unique / ratio through the effect of ionization dependence (the existence of the unique α/β ratio and density on impurities. the emission at other frequencies when the ionization (3) A host atom ionized by a moving particle may density rises) can be accounted for within a single appear near an impurity atom. Usually, the ionization mechanism. According to this mechanism, the capture potential of the latter is lower than that of the former. In of a bound impurity electron by a host ion affects the other words, the energy level of a valence impurity state of scintillation impurities. electron is above that of a host-atom electron; hence, the capture of a bound impurity electron by a host ion (7) It should be noticed that, when an insulator is is energetically favorable. Such a capture may occur at doped by impurities in which electrons are bound more any relaxation process in the excited state. A bound weakly than in host atoms (in particular, scintillation impurity electron may also be trapped by a host ion if impurities [5]), the number of displaced atoms in a an impurity atom was excited during scintillation. The heavy particle track decreases [6]. Thus, two effects ionization of an impurity atom changes the occupancies (loss in light output for heavy particles and atomic dis- of excited electron states and shifts their energy levels. placement in heavy particle tracks) result from the

ON THE INFLUENCE OF IONIZATION DENSITY 949 same process: the capture of a loosely bound impurity 3. V. V. Averkiev, Ya. A. Valbis, F. Kh. Grigoryan, V. K. Lya- electron by a host ion. pidevskiœ, et al., Luminescent Detectors and Transduc- ers of Ionizing Radiation (Nauka, Novosibirsk, 1985). 4. V. S. Vyazemskiœ, I. I. Lomonosov, A. N. Pisarevskiœ, ACKNOWLEDGMENTS et al., Scintillation Method in Radiometry (Atomizdat, This work was supported by the Russian Foundation Moscow, 1961). for Basic Research. 5. V. V. Averkiev, V. K. Lyapidevskiœ, and N. B. Khokhlov, Izv. Akad. Nauk SSSR, Ser. Fiz. 50, 568 (1986). REFERENCES 6. V. K. Lyapidevskiœ and M. I. Ryazanov, Pis’ma Zh. Tekh. Fiz. 23 (16), 51 (1997) [Tech. Phys. Lett. 23, 638 1. I. M. Rozman and S. F. Kilin, Usp. Fiz. Nauk 69, 469 (1997)]. (1959). 2. M. D. Galanin and Z. A. Chizhikova, Opt. Spektrosk. 48, 196 (1958). Translated by V. Isaakyan

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Simulation of an Early Stage of a Conductor’s Electrical Explosion S. I. Tkachenko Institute of Pulse Research and Engineering, National Academy of Sciences of Ukraine, Nikolaev, 327018 Ukraine Received August 4, 1999

Abstract—Phase-transformation waves that may arise in a conductor in the course of an electrical explosion are considered. An estimate allowing one to predict the possibility of occurrence of several phase transformations at the front of the single wave is proposed. Mathematical simulation results are presented. © 2000 MAIK “Nauka/Interperiodica”.

The problem of calculating the electrical explosion is formed. The descending slope of the peak corre- of the skin layer in a gap for the one-dimensional case sponds to a new phase transition. in Cartesian coordinates was considered in [1]. This paper is devoted to the simulation of the early stage of Figure 1b displays numerical calculation results the electrical explosion of cylindrical copper conduc- corresponding to instants of 17.1, 17.4, 17.7, and tors. We use a model similar to that employed in [2]; 17.8 ns. The bottom arrow indicates the solidÐliquid- i.e., we solve the one-dimensional set of magnetohy- metal transition. It can be seen that the region of this drodynamics equations supplemented with the circuit phase transition is virtually not displaced (during the equation; the electrical conduction in the solid and liq- time interval pointed out in the plots); therefore, the uid states is described by a semiempirical equation [3] first current wave is stationary. and in the plasma state, by an equation proposed in [4] The upper arrows indicate the transition of the metal for dense plasma. Following [5], we assume that at ρ/ρ 0 into the nonconducting state; in accordance with them, = 0.2 conduction vanishes completely. the front of the second current wave is located at the Variation of the distribution of both the local param- same instants. eters of the conductor (density, temperature, pressure, It is clear seen that the second current wave, which and electrical conductivity) and the integral ones (cur- “takes in” the rest of the current that has remained in rent, voltage, resistance, and wire radius) is caused by the molten conductor layers, moves with a velocity the propagation of the phase-transformation waves much higher than the first wave (v varies roughly from through the conductor [6Ð8]. Figure 1 displays curves 2 × 4 × 5 × 3 describing the current-density and temperature distri- 7 10 to 2 10 m/s while v1 ~ 5 10 m/s), catches butions over the radial coordinate of a copper wire (l = up the latter, joins it, and after that, the joint current wave propagates. Its further propagation gains specific 15 mm and r0 = 0.1 mm) at various instants at the following circuit parameters: L = 5 nH, C = 25 nF, and features of skin layer explosion [1] with the only distinction that, in an axially symmetric system, any pro- U0 = 40 kV. The curves are shown in two panels of Fig. 1 in order to separate an explosion stage specified cesses proceed more intensely (in accordance with [1] 4 by the propagation of only one current wave (Fig. 1a) v2 = 10 m/s). This phenomenon features the explo- caused by the redistribution of the current in the con- sions whose energy characteristics may cause fast heat- ductor due to melting wave propagation through it from ing of the molten layers of a conductor up to the tem- another stage in which entire loss of the conduction in perature of the next phase transition. the outer layers occurs (Fig. 1b). Such a process will be observed if a matter in some In Fig. 1a, numerical-calculation results for instants phase state (f) is heated up to a temperature at which the of 13, 16, and 17 ns are shown. The arrows near the next phase transition proceeds so fast that the both abscissa indicate solidÐmelt phase transition. It can be phase transitions may occur at a single wave front. In seen that their locations correspond to the trailing edge this case, the lifetime of the intermediate phase state is of the current wave. At the instant of 17 ns, another much less than the time necessary for the phase transi- phase transition occurs (it is indicated by the upper tion wave to travel a distance equal to the phase thick- arrow) at which the conduction entirely vanishes; at the ness. The lifetime of any phase state can be evaluated same time, one more peak in the current-density curve as the time necessary for heating the medium from the

SIMULATION OF AN EARLY STAGE 951

j, 1012A m–2 T, 103 K j, 1012A m–2 T, 103 K 15 16 15 17 13 17.7 6 (a) 25 (b)

10 20 10 4 15 17.8

5 10 5 2 17 17.4 5 17.1 0 0 0 0.05 0.10 0.15 0 0.04 0.08 0.12 r, mm

Fig. 1. Distributions of the current density (solid lines) and the temperature (dashed lines) over the radial coordinate at various instants of time (the instants expressed in nanoseconds are denoted by numbers near the curves). temperature of the preceding phase transition to that of the instant of 21 ns, does the next phase transition wave the next phase transition: start to move towards the center, causing the conduction to vanish (in accordance with [9], the phase transi- ∆ ρ ∼ 2τ σ c f T f j λ/ f . (1) tion wave may move only under a certain condition). When this takes place, the current wave moves together Here, cf is the specific heat of the medium at the phase with the melting wave crest. ρ σ state considered; f is the density; f is the electrical ∆ If the stored energy is not sufficient for the regime conductivity; j is the current density; T = T1f Ð T2f is the difference between the temperatures of the succes- described to be achieved (for example, for the same parameters and C = 20 nF), then only a single current sive phase transitions; and τλ is the phase lifetime wave is formed and travels similarly to the process which must be less than the time required for the phase- shown in Fig. 1a. In this regime, some conduction transition wave front to travel through the conductor τ remains in the expanded outer layers, which results in w = r0/u (here u is the velocity of the phase-transition nonzero current density there. It is only after the melt- wave front). ing wave reaches the central layers that a new phase If one uses the estimate for the phase-transition wave is formed in the outer layers which causes the front velocity proposed in [8], then, for superfast metal conduction to vanish; when this takes place, only τ Ӷ τ regimes at λ w, it is possible to write a single current-wave crest, which accompanies the ∆ wave of the first solidÐmelt phase transition, is clearly c f T Ӷ r0 seen. ----λ------δ----. (2) f j δ µσ ω Ð1/2 µ Here j = (0.5 f ) is the skin depth; is the mag- CONCLUSIONS netic permeability; and ω = IÐ1dI/dt. According to [9], the existence of the second current wave (in the case of The simulation results demonstrate that regimes of the ultrafast (nanosecond) electrical explosion of a con- explosion of a copper wire with r0 = 0.1 mm) is possible ω 7 Ð1 ductor differ from ones of the fast (microsecond) [2] for > 10 s ; it follows from (2) that a regime at explosion (from the standpoint of the phase transition which several phase transitions may proceed at the dynamics) in the relation between the running time of front of a single wave becomes possible at higher rates ω ӷ 10 Ð1 the wave of the material phase transition moving from of the current variation 10 s . The regime pre- the outer boundary to the center and the lifetime of the sented is the limiting one for the second condition phase state considered. In ultrafast regimes, the lifetime ω ≈ 10 Ð1 since, for the given parameters, 10 s . For this of intermediate phases may be much less than the time particular reason, the liquid metal at first occupies a of the phase boundary displacement from the conductor rather extended region ~r0 (it can be seen from the tem- periphery to its center; therefore, it becomes possible to perature curves, for example, at the moment of the sec- realize such regimes in which a phase wave catches up ond current wave formation) and then it practically dis- to the front of the preceding phase transition and, fur- appears when the waves merge into a single one at t ≈ ther on, several phase transitions occur at a single wave 17.8 ns. front. Exactly in this case (ω ӷ 1010 sÐ1), one can con- Further on, the melting wave first moves toward the ventionally assume that the conductor material trans- center and, only after it reaches the symmetry axis at forms from the condensed state immediately into the

TECHNICAL PHYSICS Vol. 45 No. 7 2000

952 TKACHENKO dense plasma. Probably, the products of such an explo- 5. S. N. Kolgatin, A. Ya. Polishchuk, and G. A. Shneerson, sion have a more uniform size. Teploﬁz. Vys. Temp. 31, 890 (1993). 6. F. D. Bennet, Phys. Fluids 8, 1425 (1965). REFERENCES 7. V. A. Burtsev, N. V. Kalinin, and A. V. Luchinskiœ, Elec- trical Explosion of Conductors and Its Application in 1. G. A. Shneerson, Zh. Tekh. Fiz. 43, 419 (1973) [Sov. Electrophysical Equipment (Énergoatomizdat, Moscow, Phys. Tech. Phys. 18, 268 (1973)]. 1990). 2. N. T. Kuskova, S. I. Tkachenko, and S. V. Koval, J. Phys. 8. N. I. Kuskova, Pis’ma Zh. Tekh. Fiz. 24 (14), 41 (1998) 9, 6175 (1997). [Tech. Phys. Lett. 24, 559 (1998)]. 3. H. Knoepfel, Pulsed High Magnetic Fields (North-Hol- 9. S. I. Tkachenko and N. I. Kuskova, J. Phys. 11, 2223 land, Amsterdam, 1970; Mir, Moscow, 1972). (1999). 4. A. Ya. Polishchuk, Doctoral Dissertation (Institute of High Temperatures Scientiﬁc Association, Russian Academy of Sciences, Moscow, 1991). Translated by N.P. Mende

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BRIEF COMMUNICATIONS

High-Frequency Boundary for the RayleighÐJeans Approximation in Microwave Radiometry Problems I. A. Semin Ryazan State Pedagogical University, Ryazan, 390000 Russia Received March 17, 1999; in ﬁnal form, December 24, 1999

Abstract—The Kirchhoff waveguide theorem provides an adequate description of an electrodynamic system of a high-sensitivity microwave radiometer with wideband detector. A transition from the general analytical representation of this theorem to a particular case is analyzed. This transition is equivalent to a transition from the Planck distribution to the RayleighÐJeans approximation. A simple relationship is derived that determines, for a preset accuracy, the high-frequency boundary for the above transition as a function of the temperature. © 2000 MAIK “Nauka/Interperiodica”.

The development of nonlinear superconducting ele- vides considerable simplification of the computation, ments led, on the one hand, to the creation of novel which makes this approach preferred to that based on devices with unique parameters [1] and, on the other the Planck distribution. However, this simplicity can hand, to the formulation of new radiophysical prob- sometimes be reached at the expense of lower preci- lems. This is true, in particular, for the so-called sion. In particular, Karlov and Chikhachev [8] were Josephson’s edge junctions representing one type of the among the first researchers pointing out that the Ray- nonlinear superconducting elements [2]. leighÐJeans approximation is inapplicable to the A theoretical basis for experimental investigations description of the radiation sources operating at ~4.2 K [2Ð4] of the detector properties of the edge junctions even in the millimeter wavelength range. was provided by [5, 6]. In particular, Zavaleev and The purpose of our work was to elaborate a criterion Likharev [6] calculated the detector characteristics of a for determining, for a preset accuracy, a high-frequency junction included into a wideband electrodynamic sys- boundary for the correct transition from the Planck dis- tem modeled by a long line. However, unsatisfactory tribution to the RayleighÐJeans approximation. agreement of the calculated characteristics with experimental results allows us to decline the above model in In this context, let us consider equation (1) for the favor of the Kirchhoff waveguide theorem [7]. average energy of a quantum oscillator. In the limit One term in the analytical expression of the Kirch- hν Ӷ kT, this expression coincides, to within a factor hoff theorem represents the average energy of a quan- describing the number of oscillators insignificant for tum oscillator described, to within the zero-oscillation the following analysis, with that provided by the Ray- energy, by the formula leighÐJeans approximation: 〈〉ε ν()()ν Ð1 〈〉ε P =h exph/kT Ð 1 ,(1) RJ = kT. (2) where h is the Planck constant, k is the Boltzmann constant, and T is absolute temperature. In order to obtain the desired criterion, let us compare expressions (1) and (2). Note that all ν > 0 and Equation (1) allows significant simplification in the T > 0 obey the inequality extremal cases when hν Ӷ kT and hν ӷ kT. From the standpoint of the Kirchhoff waveguide theorem appli- Ð1 cation, only the first case is of interest. Thus, the Kirch- kT> hν()exp()hν/kT Ð 1 ,(3) hoff theorem can be expressed in two forms: the general, corresponding to rigorous equation (1), and an which can be readily checked by expanding the expo- asymptotic, following from equation (1) in the limit of nent into a Maclaurin series. On introducing the nota- hν Ӷ kT. tion It should be noted that a transition from the general ν formulation of the Kirchhoff theorem to the asymptotic xh= /kT, (4) case is equivalent to the transition from the Planck distribution to that described by the RayleighÐJeans law we may rewrite inequality (3) in the following form:

(referred to below as the RayleighÐJeans approxima- Ð1 Ð1 tion). The RayleighÐJeans approximation usually pro- x > ()expx Ð 1 . (5)

954 SEMIN

Table quency νβ. On the other hand, by virtue of the properties of the β(x) function, this quantity is a measure of νβ, GHz β the maximum uncertainty appearing in the frequency T = 10 K T = 80 K T = 300 K interval (0, νβ) when the RayleighÐJeans approximation is used instead of the Planck distribution. In this 0.001 0.4212 3.370 12.64 context, the β value is expediently termed the error 0.005 2.087 16.70 62.66 level of the transition from the Planck distribution to 0.010 4.181 33.45 125.4 the RayleighÐJeans approximation and the νβ fre- 0.050 21.20 169.6 635.9 quency, the high-frequency boundary for the RayleighÐ Jeans approximation for the β uncertainty level. 0.100 43.16 345.3 1294 Solving equation (6) and taking into account formula (4), we obtain the following expression:

Introducing a coefficient β such that 0 ≤ β < 1, this νβ = x(β)kT/h, (9) expression can be transformed to an equation relative to x: which completes the analytical description. A quantitative pattern corresponding to equations (1 Ð β)(expx Ð 1) = x. (6) (6) and (9) is illustrated by data in the table. The coefficient β possesses the following properties. First, this quantity represents a methodological ACKNOWLEDGMENTS uncertainty caused by using the RayleighÐJeans approximation instead of the Planck distribution. The author is grateful to V.A. Il’in for creative sup- Indeed, using equations (1), (2), (4), and (6), we may port. express β as β ( 〈ε〉 〈ε〉 ) 〈ε〉 ∆ 〈ε〉 〈ε〉 REFERENCES = RJ Ð P / RJ = RJ/ RJ. (7) 1. V. P. Koshelets and G. A. Ovsyannikov, Zarubezh. Second, any permissible β value at a given fixed Radioélektron., No. 6, 31 (1983). temperature divides the frequency interval into two 2. A. L. Gudkov, V. A. Kulikov, V. N. Laptev, et al., Pis’ma parts, one of which is conveniently interpreted as the Zh. Tekh. Fiz. 12, 527 (1986). region of applicability of the RayleighÐJeans approxi- 3. A. L. Gudkov, V. A. Il’in, V. N. Laptev, et al., Pis’ma Zh. mation. Indeed, relationship (6) can be used to define β Tekh. Fiz. 14, 826 (1988). as function of x: 4. V. A. Il’in, V. G. Mirovskiœ, I. A. Semin, and V. S. Étkin, β = 1 Ð x/(expx Ð 1). (8) Radiotekhnika, No. 12, 17 (1988). 5. H. Kanter and F. L. Vernon, J. Appl. Phys. 43, 3174 A physical domain for the expression (8) is the (1972). interval (0, ∞), in which the function β(x) is a monoton- 6. V. P. Zavaleev and K. K. Likharev, Radiotekhn. Élektron. ically increasing function of x with the range (0, 1). 23, 1268 (1978). Therefore, an arbitrary ordinate in the plot of β(x) 7. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskiœ, Intro- determines a level uniquely corresponding to certain x. duction to Statistical Radiophysics. Random Fields Since the x and ν values obey a linear relationship (4) (Nauka, Moscow, 1978), Part. 2. at a fixed temperature T, any β level uniquely defines a 8. N. V. Karlov and B. M. Chikhachev, Radiotekhn. Élek- certain frequency νβ. tron. 4, 1047 (1959). Thus, on the one hand, the β value defined above represents a level determining the certain boundary fre- Translated by P. Pozdeev

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