∫° Nonlinear Capillary Oscillation of a Charged Drop

GASES AND LIQUIDS

Nonlinear Capillary Oscillation of a Charged Drop D. F. Belonozhko and A. I. Grigor’ev Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia E-mail: [email protected] Received May 18, 1999; in ﬁnal form October 14, 1999

Abstract—In the quadratic approximation with respect to the amplitudes of capillary oscillation and velocity field of the liquid moving inside a charged drop of a perfectly conducting fluid, it is shown that the liquid drop oscillates about a weakly prolate form. This refines the result obtained in the linear theory developed by Lord Rayleigh, who predicted oscillation about a spherical form. The extent of elongation is proportional to the initial amplitude of the principal mode and increases with the intrinsic charge carried by the drop. An estimate is obtained for the characteristic time of instability development for a critically charged drop. © 2000 MAIK “Nauka/Interperiodica”.

Studies of capillary oscillation and stability of a 2. Consider a drop of an inviscid, perfectly conduct- charged drop are motivated by numerous applications ing liquid of density ρ characterized by the surface ten- [1]. The problem was reviewed in [1Ð7] and references sion γ and carrying a charge Q. The linear theory of therein. However, most theoretical analyses are based capillary oscillation postulates that there exists a time on the linearized system of fluid-dynamics equations. moment that can be treated as t = 0, when the drop Only recent studies have captured the nonlinear nature geometry is described by the second mode of linear of the phenomenon and provided essentially new infor- capillary oscillation of a small finite amplitude ε and mation about the mechanism of instability of a highly the velocity field is zero. The initial drop volume is charged drop and its capillary oscillation [2Ð7]. equal to that of a spherical drop of radius R. The problem is to determine the axially symmetric oscillations 1. A charged spherical drop with an intrinsic charge of the drop in the case of an axially symmetric potential Q greater than a certain critical value becomes unsta- velocity field in the drop. ble, because the electrical repulsive forces exceed the We use the dimensionless variables in which ρ ≡ 1, surface tension forces. At the close of the nineteenth γ ≡ ≡ century, Lord Rayleigh developed a linear model of this 1, and R 1. The mathematical model of the prob- instability [8]. Since then, studies of the instability of a lem is then written in spherical coordinates with the ori- highly charged drop and its generalizations have grown gin at the center of the drop as follows: into a broad area of research having important technical ∆ψ ==0; U — ⋅ ψ, (1) applications. ∆Φ ==0; EÐ—⋅Φ, (2) From the perspective of classical physics, the thermal motion of molecules gives rise to drop oscillations r ∞: |—Φ| 0, (3) of amplitude ~kT/γ about the equilibrium spherical r = 0: |—ψ| < ∞, (4) form, where k is Boltzmann’s constant, T is the absolute temperature, and γ is the surface tension of the liquid. r=1+ξΘ(),t Each mode is characterized by a specific critical value 1∂Φ  Ð------∫------dS =Q; Sr(),,Θϕ ≡0≤≤Θπ (5) of surface charge, and a supercritical charge leads to the 4π°∂n  onset of instability. The stability of the drop with 0≤ϕ<2π, respect to its intrinsic charge Q is generally character- ≡ 2 πγ 3 ized by the so-called Rayleigh parameter W Q /4 R , r==1+ξΘ(),t: Φ const, (6) where R is the drop radius. The second mode is the most unstable, and the corresponding critical value of ∂ψ ∂ξ 1 ∂ξ ∂ψ ------= ------+ ------, (7) the Rayleigh parameter is W = 4. 2∂Θ∂Θ c ∂r∂tr

In this paper, we solve Rayleigh’s problem in the 2 quadratic approximation with respect to the amplitudes ∂ψ 1 2 E ∆ᑠÐ ------Ð--- ()— ⋅ ψ +ᑠ==ᑠγ;ᑠ------, (8) of the velocity ﬁeld and surface perturbation and deter- ∂t 2 E E 8π mine the geometric form about which a subcritically ξ ε ()Θ charged liquid drop oscillates. t==0: r 1 ++* P2 cos , (9)

1002 BELONOZHKO, GRIGOR’EV In the zeroth approximation, the nontrivial relations 1 2 4 P (µ) = --(3µ Ð 1), dv = --π; are (2), (3), (5), and (8). They contain the electric 2 2 ∫ 3 potential, the capillary pressure, and the constant com- V (10) ponent of the difference of pressures inside and outside  ≤ ≤ ε ε ( (Θ)) the drop. Problem (1)Ð(11) reduces to one for the static 0 r 1 + + P2 cos , Φ ψ ≡ ξ ≡ V(r, Θ, t) ≡  * field 0(r), with the known functions 0 0 and 0 0: 0 ≤ Θ ≤ π, 0 ≤ ϕ < 2π, 2Φ Φ d 0 d 0 ψ = 0. (11) ------= 0, E1 = Ð------er, dr2 dr This is a boundary value problem with the free surface r = 1 + ξ(Θ, t) and three unknown functions: the 2π π Φ potential Φ = Φ(r, Θ, t) of the electric field outside the 1 d 0 (Θ) Θ ϕ ψ ψ Θ Ð----π-- ∫ ∫------ sin d d = Q, (16) drop, the potential = (r, , t) of the velocity field 4 dr r = 1 inside the drop, and the surface deviation ξ = ξ(Θ, t) 0 0 from a regular sphere with r = 1. Φ ∞ d 0 The potentials Φ and ψ satisfy the Laplace equa- r : ------0, tions (1) and (2) under the boundary and initial condi- dr tions (3)Ð(8) and (9)Ð(11), respectively. where er is the unit radial vector. Both kinematic and dynamic boundary conditions Φ The solution of the problem is 0 = Q/r. Using this for the velocity field U, (7) and (8), contain nonlinear solution and formulas (1B), (2C), and (5D)Ð(10D) from terms. Condition (4) implies that U is bounded at the ∆ᑠ the Appendices, the first- and second order problems center of the drop. In (8), is the difference between are easily formulated. The first order problem is written the constant pressure components inside and outside as the drop, ᑠ is the electric pressure on the drop surface E ∆ψ ⋅ ψ [9, 10], and ᑠγ is the capillary pressure under the 1 = 0, U1 = — 1, (17) curved drop surface [11]. ∆Φ = 0, E = Ð— ⋅ Φ , (18) It follows from (3) that the electric field E must be 1 1 1 ∞ | Φ | bounded at infinity. The continuity of the tangential r : — 1 0, (19) components of E(r, t) at the charged surface of a con- | ψ | ∞ ductor means that the surface is equipotential (see (6)) r = 0: — 1 < , (20) [10]. The condition for the normal components of E(r, t) 2π π at the interface is written in (5) in an integral form, Φ 1 d 1 (Θ) Θ ϕ where n is the outward unit normal to the drop surface. r = 1: Ð----π-- ∫ ∫------ sin d d = 0, (21) ξ 4 dr The quantity ∗ in (9) is determined by (10), which 0 0 means that the volumes of the initial drop and a spher- Φ Ð ξ Q = 0, (22) ical drop of unit radius are equal. 1 1 3. In dimensionless variables, the amplitude ε of an ∂ψ ∂ξ ------1 = ------1, (23) initial deviation from spherical form is measured in ∂r ∂t units of the spherical drop radius. We treat ε as a small parameter and seek a solution to the problem as a series ∂ψ ∂Φ ε 1 Q 1 ξ ξ ∆ ξ expansion in integer powers of , omitting the terms of ---∂------+ ----π-- ---∂------+ 2 1Q = 2 1 + Ω 1, (24) order higher than two: t 4 r 3 ξ ε ( (Θ)) Φ Φ Φ Φ (ε ) t = 0: 1 = P2 cos ; = 0 + 1 + 2 + O , (12) 1 2 (25) ψ ψ ψ (ε3) P (cos(Θ)) = --(3cos (Θ) Ð 1), = 1 + 2 + O , (13) 2 2 ξ ξ ξ ξ (ε3) ψ r = 1 + , = 1 + 2 + O , (14) 1 = 0. (26) Φ ∼ ψ ∼ ξ ∼ O(εm); m = 1, 2; Φ = O(1). (15) The problem for the second-order quantities is writ- m m m 0 ten as follows: The functions Φ, ψ, and ξ are assumed to be of the ∆ψ ⋅ ψ same order as their partial derivatives. Under these 2 = 0, U2 = — 2, (27) assumptions, since the Laplace operator is linear, we ∆Φ ⋅ Φ can use the expansions obtained in the Appendices to 2 = 0, E2 = Ð— 2, (28) split the problem (1)–(11) into problems of zeroth, first, ∞ | Φ | Φ Φ r : — á 2 0, (29) and second order [11] for the seven functions 0, m, ψ ξ | ψ | ∞ m, and m, where m = 1 and 2. r = 0: — á 2 < , (30)

TECHNICAL PHYSICS Vol. 45 No. 8 2000 NONLINEAR CAPILLARY OSCILLATION OF A CHARGED DROP 1003

π π − Θ 2 2 m(m + 1)Pm(cos )); and formulate the Cauchy prob- 1 dΦ ∂Φ ∂Φ r = 1: Ð------------2 + 2ξ ------1 + ξ ------1 lem for a homogeneous system of ordinary differential π ∫ ∫ 1 ∂ 1 2 4  dr r ∂r equations for the unknown functions Cm(t), Fm(t), and 0 0 (31) Z (t). By ﬁnding these unknowns and substituting them ∂ξ ∂Φ m 1 1 Θ Θ ϕ into (37)Ð(39), we can write the stable solutions to the Ð -∂---Θ------∂---Θ---- sin d d = 0, ﬁrst-order problem as ∂Φ ξ = εcos(ω t)P (cosΘ), (40) Φ ξ ξ 1 ξ2 1 2 2 2 Ð 2Q = Ð 1------+ 1 Q, (32) ∂r εω ψ = Ð------2 sin(ω t)P (cosΘ), (41) ∂ψ ∂2ψ ∂ξ ∂ξ ∂ψ 1 2 2 2 ------2 + ξ ------1 = ------2 + ------1 ------1, (33) ∂r 1 2 ∂t ∂Θ ∂Θ Φ ε (ω ) ( Θ) ∂r 1 = Qcos 2t P2 cos , (42)

∂ψ ∂Φ 2 2 Q 2 ξ ξ ∆ ξ Q ---∂------+ ----π-- ---∂------+ 2 2Q = 2 2 + Ω 2 ω = 8 Ð 2W, W = ------< 4. (43) t 4 r 2 4π 1∂ψ  2 1∂ψ  2 ∂ ∂ψ  Substituting (40)Ð(43) and the obvious equalities Ð ------1 Ð ------1 Ð ξ ------1 2 ∂r  2 ∂Θ  1∂r ∂t  2 1 2 18 (34) P (cosΘ) = -- + --P (cosΘ) + -----P (cosΘ), 2 ∂Φ 2 2 4 2 Q Q 1 5 7 35 Ð 2ξ Ð 2ξ ∆Ωξ + 5ξ ------+ 2ξ ------1 1 1 14π 14π ∂r ∂ ( Θ) 2  P2 cos  6 6 ( Θ) 72 ( Θ) 2 2 2 ------ = -- + --P2 cos Ð -----P4 cos Q ∂ Φ 1 ∂Φ  1 ∂Φ  ∂Θ 5 7 35 Ð ξ ------1 + ------1 Ð ------1 , 1 π 2 π ∂  π ∂Θ  4 ∂r 8 r 8 into (31)Ð(34), we obtain

2 π π ε 2 Φ t = 0: ξ = Ð----, (35) 1 d 2 2 5 r = 1: Ð------sinΘdΘdϕ = 0, (31') 4π ∫ ∫ dr  ψ 0 0 2 = 0, (36) Φ ξ ε2 2 (ω ) where ∆Ω is the angular part of the Laplace operator, 2 + 2Q = 2 Qcos 2t with r = 1. The functions ψ and Φ are determined 1 1 1 2P (cosΘ) 18P (cosΘ) (32') from (17)Ð(20) in the following general form: × -- + ------2------+ ------4------, 5 7 35 ∞ ψ = C rmP (cosΘ), C = C (t), (37) ∂ψ ∂ξ 1 ∑ m m m m 2 2 ε2ω ( ω ) ------Ð ------= 2 sin 2 2t m = 0 ∂r ∂t (33') ∞ ( Θ) ( Θ) F 1 P2 cos 27P4 cos Φ = F + ∑ ------m--- P (cosΘ), × Ð -- Ð ------+ ------, 1 m + 1 m (38) 5 14 35 m = 0 r ( ) ( ) ∂ψ ∂Φ Fm = Fm t , F = F t . 2 Q 2 ------+ ------+ 2ξ Q Ð 2ξ Ð ∆Ωξ ξ ξ Θ ∂t 4π ∂r 2 2 2 At any moment, the deviation 1 = 1( , t) of the drop surface is a single-valued continuous function of ω 2 ω2 cosΘ; therefore, it can be represented as a series in 2 2 9 2 Ð 22W + 40 2 = ε cos (ω t)------Ð ------terms of the orthogonal Legendre polynomials 2 20 4 Θ {Pm(cos )}: (34') 15ω 2 Ð 56W + 80 ω 2 ∞  2 (ω ) 2 2 ( Θ) +  cos 2t ------Ð ------ P2 cos ξ ( ) ( Θ) 28 4 2 = ∑ Zm t Pm cos . (39) m = 0 2 2  2 9(2ω Ð 21W + 20) ω  + cos (ω t)------2------Ð -----2- P (cosΘ). To solve problem (17)Ð(26), we have to substitute  2 28 4  4 (37)Ð(39) into (17)Ð(26); use the orthogonality of the Θ Legendre polynomials, i.e., the fact that {Pm(cos )} The system including (27)Ð(30), (31')Ð(34'), (35), are the eigenfunctions of the angular part of the Laplace and (36) is the reformulated second-order problem. It ∆ Θ operator in spherical coordinates ( ΩPm(cos ) = can be solved in the same way as the ﬁrst-order one.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1004 BELONOZHKO, GRIGOR’EV

4. The solution obtained by substituting the result- Rayleigh parameter W approaches the critical value Wc ing ﬁrst- and second-order terms into (12)Ð(14) is ω 2 = 4 for the principal-mode instability, 2 0 in ε2 (44)Ð(46) and the asymptotic character of the solution r = 1 Ð ---- cos2 (ω t) + εcos(ω t)P (cosΘ) 5 2 2 2 is violated. This fact determines the domain of uniform validity of the solutions written out here. The case of a ε2 highly charged drop with W 4 was considered in + ------(χ Ð (χ + χ )cos(ω t) + χ cos(2ω t)) [2]. Formula (44) is identical with the solutions found ω 2 1 1 2 2 2 2 2 (44) by using the Lindstedt–Poincaré and multiple-scales methods [11]. 18 2 × P (cosΘ) + -----ε (χ Ð (χ + χ )cos(ω t) 2 35 3 3 4 4 5. Solution (44) has some interesting features that have not been noted in [2Ð6]. χ ( ω ) ) ( Θ) + 4 cos 2 2t P4 cos , Equation (44) of the oscillating surface contains ( Θ) terms nonperiodic in time. The surface associated with 1 P2 cos Φ = ±2 πW -- + εcos(ω t)------these terms is written as r 2 3 r ε2 ( Θ) ε4 ( Θ) r = 1 + AP2 cos + BP4 cos , 2 * ε (47) + ------(χ + K Ð (χ + χ )cos(ω t) 44 Ð 5W 18 36 Ð 5W ω 2 1 1 2 2 A = ------, B = ------2 28(4 Ð W) 3512(6 Ð W) P (cosΘ) and can be considered as the surface about which the + (χ + K)cos(2ω t) )----2------(45) 2 2 3 drop oscillates. r The equation of the surface of a prolate spheroid β 18ε2(χ (χ χ ) (ω ) with a small eccentricity can be approximated by a + ----- 3 + 1 Ð 3 + 4 cos 4t Θ 35 series in {Pm(cos )}. In the quadratic approximation with respect to β, this equation has the form P (cosΘ) + (χ + 1)cos(2ω t) )----4------, 2 4 2 5  1 2 bβ r r = b 1 + --β + ------P (cos(Θ)),  6  3 2 2 εω 2 ε χ + χ ψ = Ð ------2 sin(ω t)r P (cosΘ) + ------1------2 sin(ω t) where b is the semiminor axis of the spheroid. Such a 2 2 2 ω 2 2 2 2 spheroid is the best approximation of the surface r∗ if

 2  1 2 bβ 2 --+ Gsin(2ω t) r P (cosΘ) (46) b 1 + --β = 1, ------= ε A. 2  2  6  3 χ χ These equations can be used to ﬁnd the eccentricity 18ε2 3 + 4 (ω ) ( ω ) 4 ( Θ) + ----- ------ω------sin 4t + Lsin 2 2t  r P4 cos , and semiaxes of the spheroid approximating the surface 35 4 deﬁned by (47). Denoting the semimajor axis of this 44 Ð 5W 23W Ð 116 36 Ð 5W spheroid by a, we have χ ≡ ------, χ ≡ ------, χ ≡ ------, 1 14 2 42 3 ω 2 β ε( (ε3) (ε3) 4 = 3A + O , b = 1 + O , 23A 2 44 Ð 5W (48) 12 + W Q2 a = 1 + ε ------+ O(ε ), A = ------. χ ≡ ------, W ≡ ------, 0 ≤ W Ӷ 4, 2 28(4 Ð W) 4 4(10 Ð W) 4π According to (48), even an uncharged drop with the 2(26 Ð 5W) (9 Ð 2W) 2(4 Ð W) initial disturbance ~P (cosΘ) oscillates about a spher- G ≡ ------, L ≡ ------, 2 21 2(10 Ð W) oid of eccentricity β = 1.1ε, which is close to the dimensionless amplitude ε of the initial disturbance. It 2ω 2 is clear that the eccentricity of a spheroid about which K ≡ ------2-, ω2 ≡ 2(4 Ð W), ω 2 ≡ 12(6 Ð W). 7 2 4 a drop of a viscous liquid oscillates must exponentially decrease as the energy of the initial deformation dissi- The sign of the potential in (45) corresponds to the pates. In the case of a perfect liquid drop, the energy of sign of the drop charge. In the expressions for Φ and ψ, an initial deformation is merely distributed between the the terms that depend on time only are omitted. interacting modes. The solution given by (44)Ð(46) is uniformly valid Curves 1 and 2 in Fig. 1 represent the surfaces (47) for W Ӷ 4, because it does not contain any secular and (48) with ε = 0.3 for a charged drop at W = 3. terms or terms with small denominators. When the Numerical calculations reveal that when ε < 0.3, the

TECHNICAL PHYSICS Vol. 45 No. 8 2000 NONLINEAR CAPILLARY OSCILLATION OF A CHARGED DROP 1005 ellipsoid described by (48) well approximates the sur- 4 face defined by (47) about which the drop oscillates. The accuracy of the approximation improves with decreasing ε. The influence of W on the accuracy of the approximation of a nonoscillating drop surface (47) by 3 the spheroid defined by (48) with ε = 0.3 becomes noticeable starting from W ≈ 3. As ε decreases, the value of W above which the approximation in (48) fails 3 1 2 4 4 2 1 3 increases, approaching the limit W = 4. Curves 3 and 4 in Fig. 1 show the contours of the –1 0 1 r(θ) limit deformations of an oscillating drop. The contour that is the most prolate along the polar axis (curve 3) represents the geometry of the drop at t = T2m (with m = ≈ 0, 1, 2, …), where T2 2.22 is the period of the second mode of capillary oscillation. At the moments T2/2 + T2m, the drop has the most oblate geometry, represented by curve 4. Fig. 1. Contours of possible forms taken by the drop. The relative elongation of the semimajor axis of the spheroid defined by (48) is described by the function δ δ(ε, W) = 1.5ε2(44 Ð 5W)/28(4 Ð W). 1.0 4 The graphs of this function are shown in Fig. 2 for 3 various values of ε. It is clear from this figure that the eccentricity of the spheroid defined by (48), i.e., the 0.8 extent of elongation about which the drop oscillates, rapidly increases with W. 6. Even though solution (44) has a nonasymptotic 0.6 character when W ≈ 4, it is interesting to compare its behavior to that of the first-order solution for W 4 2 as W approaches the critical value. Indeed, in the linear approximation with respect to ε, the surface equation 0.4 (ε, , Θ, ) ε (ω ) ( (Θ)) rs W t = 1 + cos 2t P2 cos has the following limit form as W 4: 0.2 (ε, , Θ, ) ε ( Θ) 1 lim rs W t = 1 + P2 cos , (49) W → 4 0 1 2 3 4 W where rs represents the equation of the drop surface in the first approximation with respect to ε. Fig. 2. Dependence of the relative elongation δ(ε, W) of the In this approximation, the drop has a prolate semimajor axis of the spheroid (48) on W at ε = (1) 0.1, nonoscillating form when W = 4. When the Rayleigh (2) 0.2, (3) 0.03, and (4) 0.4. parameter W slightly exceeds the critical value Wc = 4, this form elongates exponentially in time; when W < 4, the drop oscillates stably. rable to the preceding term in a characteristic time t ~ O(εÐ1/2) and exceeding it as t increases further, as indi- When the second-order terms are taken into account cated by the last term in (50). in (44), we have the limit Water drops of about a millimeter in diameter can be ε2 (ε, , Θ, ) ε ( (Θ)) treated as low-viscous and well conducting [1]. For a lim rs W t = 1 + P2 cos Ð ---- W → 4 5 water drop (ρ = 1 g/cm3) of radius R ~ 10Ð1 cm, capil- (50) Ð8 218 2 12 2 2 lary oscillations with an amplitude ~10 cm are char- + ε ----- sin (t 6)P (cosΘ) + -----ε t P (cosΘ). ε Ð7 35 4 7 2 acterized by a dimensionless amplitude ~ 10 . Accordingly, the dimensionless time t ~ εÐ1/2 ~ 3 × 103. In the first approximation, a small left neighborhood In the dimensionless variables used here, the time unit of W = 4 is a domain of stability; in the second-order is t∗ = ((ρR3)/γ)1/2 ~ 4 × 10Ð3 s. The corresponding phys- approximation, it is a domain where the asymptotic expansion is not valid. At nearly critical W 4, the ical time equals t1 = tt∗ ~ 10 s, which agrees with an second term in (44) grows with time, becoming compa- estimate obtained in [12] by using another method in

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1006 BELONOZHKO, GRIGOR’EV the spheroidal approximation for an unstable drop. This 2 2 Ð1/2 ∂nΘ 1 ∂ ξ 1  ∂ξ is the time interval within which the expansion (50) can ------= Ð------1 + ------∂Θ 2 2∂Θ be used to describe the evolution of the surface geome- r ∂Θ r try after the drop is critically charged. From physical (4A) 1  ∂ξ 2 ∂2ξ 1  ∂ξ 2 Ð3/2 considerations, a critically charged drop must be unsta- + ------1 + ------. 3∂Θ 2 2∂Θ ble. Therefore, the value tf obtained can be interpreted r ∂Θ r as an estimate for the time interval from the moment when the drop is charged to the moment when the drop The problem under consideration is solved in the becomes stable by losing a part of its charge. approximation in which the following asymptotic estimates and expansions with respect to the small dimen- 7. The nonlinear analysis of capillary oscillations of sionless amplitude ε of the initial drop deformation are a charged drop performed to the second-order terms valid: inclusive reveals that the time-averaged surface form of an oscillating, subcritically charged drop of a perfectly ∂ξ ∂2ξ conducting liquid is well approximated by the spheroi- ξ ∼ ------∼ ------∼ Oε, r = 1 + ξ, dal surface deﬁned by (48). The eccentricity of the ∂Θ ∂Θ2 approximating spheroid is proportional to the initial 1 2 3 1 2 3 deviation amplitude and considerably increases the -- = 1 Ð ξ + ξ + O(ε ), ---- = 1 Ð 2ξ + 3ξ + O(ε ), r 2 intrinsic charge of the drop. The relative elongation of r (5A) the spheroid compared to the radius of an equivalent 1 2 3 perfectly spherical drop is proportional to the ampli- ξ ξ (ε ) ----3 = 1 Ð 3 + 6 + O . tude squared of the initial deviation, increasing with the r intrinsic charge. By virtue of (5A), (1A)Ð(4A) can be rewritten as A water drop one millimeter in diameter carrying follows: the critical charge changes its form according to (50) 2 within an interval of about ten seconds, which is easy to 1 ∂ξ 3 n = 1 Ð ------+ O(ε ), (6A) observe experimentally; therefore, the effect can be r 2∂Θ detected.

∂ξ ∂ξ 3 nΘ = Ð ------+ ξ------+ O(ε ), (7A) APPENDIX ∂Θ ∂Θ A. Asymptotic Expansions of the Components ∂ ∂ξ 2 nr   (ε3) of the Normal to the Drop Free Surface --∂----- = ∂----Θ--- + O , (8A) and Their Derivatives r

2 2 At a moment t, the geometry of an oscillating drop ∂nΘ ∂ ξ ∂ ξ 3 surface obeys the following equation written in spheri------= Ð ------+ ξ------+ O(ε ). (9A) ∂Θ ∂Θ2 ∂Θ2 cal coordinates with unit basis vectors er, eΘ, and eϕ: F(r, Θ, t) = 0, f (r, Θ, t) = r Ð 1 Ð ξ(r, Θ, t). B. The Asymptotic Form of the Capillary Pressure Using the well-known relation under the Curved Drop Surface —F(r, Θ, t) For a given mean liquid surface curvature H and sur- n = ------, —F(r, Θ, t) face tension coefﬁcient γ, the Laplace pressure distribution at the surface is determined as ᑠγ = 2Hγ. The value we obtain the components of the normal vector n to this of H can be calculated as 2H = divn. In dimensionless surface and their derivatives with respect to the spheri- variables such that γ ≡ 1, we have ᑠγ = divn and cal coordinates: 2 2 Ð1/2 1 ∂(r A ) 1 (∂AΘ sinΘ) 1 ∂Aϕ 1  ∂ξ div A = ------r--- + ------+ ------. r = 1: n = 1 + ------, nϕ ≡ 0, (1A) 2 ∂ Θ ∂Θ Θ ∂ϕ r 2∂Θ r rsin rsin r r Since the problem is axially symmetric (nϕ ≡ 0), it 1 1  ∂ξ 2 ∂ξ nΘ = Ð-- 1 + ------, (2A) holds that r 2∂Θ ∂Θ r ∂ ∂ 2nr nr 1 nΘ nΘ ᑠγ = ------+ ------+ ------+ ----- cotΘ. ∂n 1 1  ∂ξ 2 Ð3/2 ∂ξ 2 r ∂r r ∂Θ r ------r = ---- 1 + ------, (3A) ∂ 3 2∂Θ ∂Θ r r r Substituting (5A)Ð(9A) into the last expression, we

TECHNICAL PHYSICS Vol. 45 No. 8 2000 NONLINEAR CAPILLARY OSCILLATION OF A CHARGED DROP 1007 have The resulting equation for ξ∗ has the roots Ðε2/5 and ∂2ξ ∂ξ ε2/5 Ð 1. Setting ξ∗ equal to the one for which the initial ᑠγ = 2 Ð 2ξ Ð ------Ð ------cotΘ ∂Θ2 ∂θ condition (9) can be written as an asymptotic expansion in ε, we rewrite (9) and (10) in a simpler form: ∂2ξ ∂ξ + 2ξ ξ + ------+ ------cotΘ + O(ε3). ε2 2 ε ( Θ) ∂Θ ∂θ t = 0: r = 1 + P2 cos Ð ----. (2C) 5 The unknown function ξ is sought in the form ξ = ξ ξ ε3 ξ ε ξ ε2 1 + 2 + O( ), 1 ~ O( ), 2 ~ O( ). Hence, D. Expansions of Potentials and Related Quantities 3 in the Vicinity of an Unperturbed Drop Surface ᑠγ = 2 Ð 2ξ Ð ∆Ωξ + 2ξ (ξ + ∆Ωξ ) + O(ε ), 1 1 1 1 1 The electrostatic potential Φ on the surface r = 1 + ∂2 ∂ (1B) ξ can be expressed in terms of Φ and its partial deriva- ∆Ω = ------+ ------cotΘ, 2 ∂Θ tives on the unit sphere surface with the required accu- ∂Θ racy as where ∆Ω is the angular part of the Laplace operator in r = 1 + ξ: spherical coordinates with r = 1. Φ = Φ(r, Θ, t) = Φ(1 + ξ, Θ, t) = Φ(1, Θ, t) 2 ∂Φ 1∂ Φ 2 3 C. Asymptotic Expansion of Initial Conditions + ------(1, Θ, t)ξ + ------(1, Θ, t)ξ + O(ε ). ∂ 2 r 2 ∂r Let us derive the asymptotic expansion of integral (10) over the drop volume at the initial moment. We Using (12), (14), and (15), we rewrite this expres- have sion as ξ ε ( Θ) r = 1 + ξ: r = 1 + * + P2 cos , ∂Φ 2π π r π Φ Φ Φ ξ 0 = 0 + 1 + 1------4 2 2π 3 r = 1  ∂r  --π = dv = r sinΘdrdΘdϕ = ------r sinΘdΘ r = 1 (1D) 3 ∫ ∫ ∫∫ 3 ∫ V 0 0 0 0 2 ∂Φ ∂Φ 1 2∂ Φ 3 + Φ + ξ ------0 + ξ ------1 + --ξ ------0 + O(ε ). 1 1 2 2 ∂ 1 ∂ 1 2 π π r r 2 ∂r 2 3 µ 2 [ ξ ε (µ)]3 µ r = 1 = ------∫ r d = ------∫ 1 + + P2 d , 3 3 * Now, the value of Φ on the surface r = 1 + ξ equals Ð1 Ð1 (1C) the sum of terms deﬁned on the surface r = 1 with the 1 ε 3 required accuracy. 2 = (1 + ξ )3 1 + ------P (µ) dµ, * ∫ 1 + ξ 2 Applying a similar method and using (6A) and (7A), Ð1 * we ﬁnd

1 r = 1 + ξ: ε ( ξ )3 3 (µ) 2 = 1 + ∫ 1 + ------P2 -  ∂—Φ  * 1 + ξ Φ = Φ + Φ + ξ ------0 * — — 0 r = 1 — 1 1 ∂  Ð1 r r = 1 2 3 (2D) 3ε 2 3 ∂ Φ ∂ Φ ∂2 Φ + ------P (µ) + O(ε ) dµ — 0 — 1 1 2 — 0 2 2 + —Φ + ξ ------+ ξ ------+ --ξ ------(1 + ξ ) 2 2 ∂r 1 ∂r 2 1 ∂ 2 * r r = 1 or (ε3) (ε3) + erO + eΘO , 2 3 3 ε 3 2 2 = 2(1 + ξ ) 1 + ------+ O(ε ), ∂Φ ∂Φ nΘ ∂Φ 1 ∂ξ * 2 5(1 + ξ ) ------= nr--∂----- + ------∂---Θ--- = 1 Ð --∂----Θ--- * ∂n r r 2 ε2 1/3 2 2 3 ( ξ ) 3 (ε3) ∂Φ ∂ Φ ξ ∂ Φ 1 = 1 +  1 + ------+ O  × ------+ ξ------+ ------* 5(1 + ξ )2 ∂r ∂ 2 2 ∂ 3 * r r r = 1 ε2 ≈ ( ξ ) 2 ∂ξ ∂ξ 1 + 1 + ------. + (1 Ð ξ + ξ ) Ð ------+ ξ------* 5(1 + ξ )2 ∂Θ ∂Θ *

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1008 BELONOZHKO, GRIGOR’EV

2 2 3 ∂Φ ∂ Φ ξ ∂ Φ 3 ∂Φ  ∂Φ ∂Φ ∂Φ × ------+ ξ------+ ------+ O(ε ), 1 2 ξ 1 ∂Θ ∂d∂Θ 2 2 ------dS = Ð Q + ------+ ------+ 2 1------∂r ∂Θ ∂n  ∂r ∂r ∂r (3D) 2 (7D) ∂Φ ∂Φ ∂Φ ∂ Φ ∂ξ ∂Φ 2 ------= ------0 + ------1 + ξ ------0 Ð ------1 ------0 ∂Φ ∂ξ ∂Φ  1 2 ξ 1 1 1 Θ ϕ ∂n ∂r ∂r ∂ ∂Θ ∂Θ + 1------Ð ------d d . r ∂ 2 ∂Θ ∂Θ  r r = 1 2 2 ∂ Φ ∂ ξ ∂Φ ∂Φ The expansions of the velocity potential ψ and its + ξ ------0 Ð ------2 ------0 + ------2 2 ∂ 2 ∂Θ ∂Θ ∂r partial derivatives on the drop surface are constructed r by analogy with (1D), and we have ∂2Φ ξ2 ∂3Φ ∂ξ 2∂Φ 1  ∂ψ 3 + ξ ------1 + ----1 ------0 + ------1 ------0 r = 1 + ξ: ψ = ψ + ψ + ξ ------1 + O(ε ), (8D) 1 2 3  ∂Θ ∂Θ 1 2 1 ∂ ∂r 2 ∂r 2 r r = 1

∂ξ ∂Φ ∂ξ ∂2Φ ∂ξ ∂Φ ∂ψ ∂ψ ∂ψ ∂2ψ 1 1 ξ 1 0 ξ 1 0 (ε3) 1 2 ξ 1 (ε3) Ð ------Ð 1------+ 2 1------+ O . ------= ------+ ------+ 1------2--- + O , (9D) ∂Θ ∂Θ ∂Θ ∂r∂Θ ∂Θ ∂Θ r = 1 ∂r ∂r ∂r ∂ r r = 1 For S{(r, Θ, ϕ) ≡ r = 1 + ξ(Θ, t); 0 ≤ Θ < π; 0 ≤ ϕ < ∂ξ ∂ψ ∂ξ ∂ψ 2π}, we have 1 1 (ε3) ∂----Θ---∂----Θ--- = -∂---Θ-----∂----Θ---- + O , r = 1 r2 sinΘ r2 sinΘ (10D) dS = ------dΘdϕ = ------dΘdϕ ∂ψ ∂ψ cosγ (e ⋅ n) ( ⋅ ψ)2  1  1 (ε3) r — = --∂------ + -∂----Θ---- + O . r r = 1 r = 1 ( ξ ξ2) Θ 1 + 2 + sin Θ ϕ = ------2------d d , 1 ∂ξ 3 (4D) 1 Ð ------+ O(ε ) REFERENCES 2∂Θ 1. A. I. Grigor’ev and S. O. Shiryaeva, Izv. Akad. Nauk, 2  2 1 ∂ξ  Mekh. Zhidk. Gaza, No. 3, 3 (1994). dS = 1 + 2ξ + 2ξ + ξ + ------sinΘdΘdϕ,  1 2 1 2∂Θ  2. J. A. Tsamopoulos and R. A. Brawn, J. Fluid Mech. 147, 373 (1984). where cosγ is the cosine of the angle between the radial 3. R. Natarayan and R. A. Brawn, Proc. R. Soc. London, unit vector er and the normal n to the drop surface. The Ser. A 410, 209 (1987). zeroth-order terms in (1)Ð(11) are easily determined by 4. W. A. Pelecasis and J. A. Tsamopoulos, Phys. Fluids A using (1D)Ð(4D), (1B), and (12)Ð(15). The complete 2, 1328 (1990). mathematical formulation of the zeroth-order problem Φ 5. J. Q. Feng and K. V. Beard, J. Fluid Mech. 227, 429 is given by (16). Using its solution, 0 = Q/r, we can (1991). simplify (1D)Ð(4D) and write out the following 6. T. G. Wong, A. V. Anilkumar, and C. P. Lee, J. Fluid approximations: Mech. 308, 1 (1996). r = 1 + ξ: 7. Z. C. Feng, J. Fluid Mech. 333, 1 (1997). Φ (Φ ξ ) 8. Lord Rayleigh (J. W. Strett), Philos. Mag. 14, 184 = Q + 1 Ð Q 1 r = 1 (5D) (1882). ∂Φ 9. L. D. Landau and E. M. Lifshitz, Course of Theoretical + Φ Ð Qξ + ξ ------1 + ξ2Q + O(ξ3), 2 2 1 ∂r 1 Physics, Vol. 8: Electrodynamics of Continuous Media r = 1 (Nauka, Moscow, 1982; Pergamon, New York, 1984). 2 ( Φ)2 2 ∂Φ 10. L. D. Landau and E. M. Lifshitz, Hydrodynamics (Nauka, ᑠ E — Q Q  1 ξ  Moscow, 1986). E = ----π-- = ------π------= ----π-- Ð ----π-----∂------+ 2 1Q 8 8 4 4 r r = 1 11. A. Nayfeh, Introduction to Perturbation Techniques 2 (Wiley, New York, 1981; Mir, Moscow, 1984). Q ∂Φ ∂ Φ ∂Φ 2 Ð ------2 + 2ξ Q + ξ Q------1 Ð 2ξ ------1 Ð 5ξ (6D) 12. S. O. Shiryaeva, A. I. Grigor’ev, and I. D. Grigor’eva, π 2 1 2 1 ∂ 1 4 ∂r ∂r r Zh. Tekh. Fiz. 65 (9), 39 (1995) [Tech. Phys. 40, 885 r = 1 (1995)]. ∂Φ 2 ∂Φ 2 1 1 1 1 (ε3) + ----π-- ---∂------+ ----π-- --∂---Θ---- + O , 8 r r = 1 8 r = 1 Translated by V. Gursky

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1009Ð1013. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 53Ð57. Original Russian Text Copyright © 2000 by Veresov, Grigorenko, Udovichenko.

GAS DISCHARGES, PLASMA

Electron Heat Transport in the Magnetic Filter of a Volume Plasma-Based Source of HÐ/DÐ Ions O. L. Veresov, S. V. Grigorenko, and S. Yu. Udovichenko Efremov Research Institute of Electrophysical Apparatus, St. Petersburg, 189631 Russia Received June 22, 1999

Abstract—A time-independent one-dimensional model of the electron energy balance in the region of the magnetic filter of a volume plasma-based ion source is justified. The local electron energy balance equation and the steady density profiles of the plasma components are used to determine the transverse (with respect to the magnetic field) electron temperature profile, which is found to agree well with the experimental profile. The temperature profile obtained analytically is then used to refine the particle balance in a plasma with two ion species and, accordingly, to find the optimum conditions for the formation of an HÐ/DÐ beam and for extracting the beam from the source. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION tionality coefficient was varied over a broad range and was determined by bringing the calculated profile of In [1], we proposed a model describing the transport of a plasma with two ion species across the magnetic the electron temperature as a function of the magnetic field in a steady volume plasma-based source of nega- flux into coincidence with the relevant experimental tive ions which is intended to inject ion beams into profile. The profile derived by Haas et al. [5] to cyclotron accelerators from the outside. Profiles of the describe the spatial variation of the electron tempera- electric field and of the densities of the plasma compo- ture along the magnetic filter is also semiempirical. nents obtained with this model allowed us to determine The constants characterizing this profile were found the optimum conditions for the formation of an HÐ/DÐ from the corresponding experimental profiles of the beam. The calculated plasma parameters were found to electron density, electron temperature, and plasma agree qualitatively with the experimental data obtained potential. In the theoretical model proposed in [5], the in multipole two-chamber ion sources with a magnetic steady-state electron plasma density across the mag- filter adjacent to the plasma electrode [1, 2]. However, netic field was described by an exponentially decreas- a qualitative comparison between the numerical and ing one-dimensional profile and the electron tempera- experimental results turned out to be incorrect, ture was assumed to be a two-dimensional function of because, in simulations, we specified the experimen- the coordinates. However, a comparison between the tally measured mean electron temperature in the first chamber (where the plasma is created) and neglected electron-energy relaxation length and the sizes of the the electron temperature variations in the second cham- chamber of the ion source under consideration shows ber (in the region of the magnetic filter), whereas the that electron heat conduction equalizes the electron experimental data show that the electron temperature temperature along the magnetic field. Consequently, depends on the magnetic and electric fields, as well as the electron temperature should be treated as a one- on some other plasma parameters in the magnetic filter dimensional function of the coordinate transverse to region. The plasma electrons along the magnetic filter the magnetic field. may be cooled markedly, which leads to a more efficient generation of negative ions via the dissociative In this paper, we apply the electron energy balance attachment of slow electrons to the excited gas mole- equation and the density profiles of the plasma compo- cules. nents [1] in order to derive a steady-state transverse (with respect to the magnetic field) electron tempera- The cooling of plasma electrons in the transverse magnetic field of a volume plasma-based ion source ture profile in a multipole two-chamber ion source. was studied in [3, 4]. However, the formula derived in Then, we use the temperature profile obtained analyti- those papers to describe the electron temperature vari- cally in order to refine the particle balance in a plasma ations is semiempirical, because the authors assumed with two ion species and, accordingly, to find the opti- that the electron heat flux through the magnetic filter mum condition for the formation of an HÐ/DÐ beam and is proportional to the electron flux itself. The propor- for extracting the beam from the source.

1010 VERESOV et al. JUSTIFICATION FOR THE USE In equation (1), the directed electron velocity is OF A ONE-DIMENSIONAL ELECTRON ENERGY defined as BALANCE EQUATION ue = ue|| + ued + uet, We analyze electron heat transport in a magnetic ( ) field using the classical scheme of a multicusp two- —|| neT e ue|| = be||Ee|| Ð De||------, chamber ion source described in [1, 2] as an example. neT e The second chamber (the region of the magnetic filter) ( ) (3) is a cylinder of radius R and length L . A plasma elec- —⊥ neT e f uet = be⊥E⊥ Ð De⊥------, trode with an exit hole is positioned at the end of the neT e second chamber and is aimed at extracting ion beams [ × ( )] [ × ] h — neT e from the source. The longitudinal profile of the mag- ued = bed E h + Ded ------, netic field, which is directed along the z-axis of a cylin- neT e der, is bell-shaped. We neglect the magnetic field non- where h is the unit vector along the magnetic field and uniformity and set the magnetic field B equal to its ue||, uet, and ued are the directed-velocity components mean value along a magnetic filter of length Lf in order along and across the magnetic field, respectively. L f for the magnetic flux ∫ Bdz to be conserved. We put In a strong magnetic field, the electron transport 0 ν the origin of the coordinates at the entrance to the sec- coefficients have the form be|| = e/me e0, De|| = Tebe||/e, 2 ond chamber, so that the region where the plasma is b = e/m ω , D = T b /e, b ⊥ = eν /m ω , and created (the first chamber) corresponds to the negative ed e Be ed e ed e e0 e Be values of the ordinate z. De⊥ = be⊥Te/e. The electric field E⊥ is driven by a current flowing The time-independent electron energy balance equation in the second chamber is generally three- through the magnetic filter when the plasma electrode dimensional. Under the condition that the energy of the is held at a positive bias potential relative to the walls directed motion is much lower than the thermal energy, of both the first and second chambers of the source. In the central region of the second chamber, the conditions 2 Ӷ ӷ ӷ me ue /2 Te, the energy balance equation has the E⊥ E|| and —⊥ne —||ne are satisfied for a plasma with form [6] two ion species such that the density of negative ions is comparable with the electron density. The electric field 2 2 Ӷ --div q + (n u )—T + --n T div u causes negative ions with the temperature Ti Te to 3 e e e e 3 e e e accumulate near the chamber axis, thereby creating an (1) ion–ion plasma with a weak field E|| ~ Ti/e in the axial 2me ν + ------neT e e0 = 0, region [7]. Since this field is not strong enough to con- m0 fine the electrons, the radial electron density ne(r) is nearly constant. An electronÐion plasma with a strong where qe is the electron heat flux, ue is the directed elec- ambipolar field E|| ~ T /e occupies the region near the tron velocity, n is the electron density, T is the electron e e e chamber wall; moreover, the plasma potential drops temperature, me is the mass of an electron, m0 is the ν preferentially in a narrow charged sheath in the imme- mass of a gas molecule, and e0 is the averaged (over diate vicinity of the wall. Consequently, far from the the relative velocities) rate of elastic collisions between wall of the second chamber, we can neglect the contri- electrons and molecules. bution to equation (1) of the velocity component ue|| in Let us show that the electron temperature is a one- comparison with the contribution of uet. dimensional function of the coordinates Te(z). In a In (1), the electron heat flux has the form weakly ionized plasma, the electron-energy relaxation lengths along and across the magnetic field are equal qe = qe|| + qet + qed, to [7] 5 1/2 ν qe|| = neT ebe||E|| Ð --ne De||—||T e, 10 Te m0 e0 2 λ|| = ------, λ⊥ = λ||------, (2)  3 2 ν  ω me e0 Be [ ( )] qet = ÐgTe neT ebe⊥E⊥ + —⊥ neT e De⊥ (4) where ω = eB/m c is the electron gyrofrequency. Be e 5 5  Ð --n D ⊥—⊥T = Ð g n T u Ð -- + g n D ⊥—⊥T , Estimates made for sources of the type under con- 2 e e e Te e e et 2 Te e e e λ ӷ λ sideration suggest that || R and ⊥ < Lf. Conse- quently, electron heat conduction equalizes the electron 5 q = Ð n T b [h × E] + --D [h × —⊥(n T )], temperature along the magnetic field, and the local ed e e ed 2 ed e e energy balance across the magnetic field can be ν ∂ν ∂ described by equation (1). where gTe = (Te/ e0)( e0/ Te) is the thermal diffusivity.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ELECTRON HEAT TRANSPORT IN THE MAGNETIC FILTER 1011

≤ σ β = S n , γ = S n ( ) , n is the atomic density, For Te 3 eV, the cross section e0 for elastic elec- AD H DA H2 v'' H tronÐmolecule collisions is independent of the electron n (v ) is the density of the excited gas molecules, SAD temperature [8]. In this case, the collision frequency is H2 '' 〈σ 〉 proportional to the electron velocity, and, under the = v AD is the rate of associative detachment of elec- assumption that the electrons obey a Maxwellian veloc- trons from negative ions in their collisions with 〈σ 〉 ity distribution, the effective collision frequency has the H atoms, SDA = v DA is the rate of dissociative attach- ν σ π 1/2 ment of electrons to the excited molecules H (v''), form [6] e0 = (4/3) e0 n vTe, where vTe = (8Te/ me) 2 H2 〈σ 〉 SMN = v MN is the ionÐion recombination rate, SED = is the electron thermal velocity and nH is the density 〈σ 〉 2 v ED is the rate of electron detachment from negative 〈σ 〉 of the gas molecules. Accordingly, the thermal diffusiv- ions in electronÐnegative-ion collisions, SIZ = v IZ is ity is equal to g = 1/2. Te the rate of ionization of H atoms by electrons, E0 is the Repeating the arguments regarding the longitudinal electric field of a current-carrying plasma at the ν ω 2 and transverse components of the directed velocity, we entrance to the filter, and b± = e ±0/2m± B± are the can also neglect the electron heat flux qe|| along the mobilities of positive and negative ions. magnetic field in comparison with the cross-field flux q ⊥. The components u and q of the electron velocity Equation (5) was derived in the approximation e ed ed γ Ӷ Ӷ and electron heat flux do not contribute to equation (1), d /dx SEDdnÐ/dx and dln(SED)/dx dln(nÐ)/dx. The because, for a uniform magnetic field, we have ∇[h × density of the excited molecules is determined from the ∇ × balance between their production and losses: E], [h —(neTe)] = 0. Hence, far from the wall of the second chamber of the ion source, the electron energy n n S = n ( )ν /bR, (7) balance equation (1) is one-dimensional and the H2 ef EV H2 v'' H2 directed electron velocity and electron heat flux are where n is the density of fast electrons emitted from described by the above expressions for uet(z) and qet(z), ef respectively. the heated cathode into the first chamber, S and ν EV H2 are the production rate of the excited molecules and their mean directed velocity, and b = 5Ð10 is the num- PROFILES OF THE ELECTRON TEMPERATURE ber of collisions of the excited molecule with the wall AND OF THE DENSITIES OF THE PLASMA that still do not change the excited state of the mole- COMPONENTS ALONG THE MAGNETIC FILTER cule. To determine the electron temperature from equa- Estimates show that the loss rates of the excited gas tion (1) requires a knowledge of the steady profiles of molecules on the walls of both the first and second the electron density and electric field in a current-carry- chambers are higher than the rates of their quenching in ing plasma. Plasma transport across a strong magnetic the processes of dissociative attachment (SDA) and ion- field is governed by heavy plasma ions rather than by ization (SIZ) in the source plasma. plasma electrons. Using the one-dimensional model α α developed in [1] to describe the transport of a plasma In a first approximation, we assume that 1 and 2 with two ion species across the magnetic field and tak- are independent of both the electron temperature and ing into account the condition that the plasma is quasi- the x-coordinate. Then, we obtain from (5) and (6) the neutral, we can obtain the following equations for the steady profile of the electron plasma density: densities of negative ions nÐ(x) and electrons ne(x) in a Ð Ð α Ð1 source of H /D ions: *( ) (α ) 2( (α )) ne x = exp 1 x 1 Ð α----- 1 Ð exp 1 x , (8) 1 * dnÐ α *2 α ------+ 2nÐ Ð 1nÐ* = 0, (5) dx where we introduce the notation ne* (x) = ne(x)/ne0 and β ne0 = ne(x = 0). ne = --nÐ. (6) α Ӷ γ Accordingly, for 1x 1, the electric field profile in a current-carrying plasma is given by the expression Here, x = z/L , n* = n /n , n = n (x = 0), f Ð Ð Ð0 Ð0 Ð ( ) ( ) ≈ [ (α α ) ] E⊥ x = E0/nÐ* x E0 1 + 2 Ð 1 x . (9) 2  β γ SMN L f b+ β ϕ ϕ α ϕ α = 1 + ------1 + ---- 1 + -- , Here, E0 = 2( 0 Ð e)/(2 + 2)Lf, where 0 is the plasma 2  γ β S b E b  γ ϕ ED + 0 Ð potential in the first chamber and e is the potential of the plasma electrode. S n L b β α γ IZ H f +  In the model proposed here to describe the electron 1 = ------1 + ---- 1 + -γ- , SED nÐ0 b+E0 bÐ heat transport in the second chamber of a volume

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1012 VERESOV et al. plasma-based ion source, equation (1) reduces to the This equation can be transformed into the ﬁrst-order following equation for the electron temperature: differential equation

2 2 2 d y Ady dy dE⊥* dy 2 2D Cy ------+ -------- Ð D + 0.33E⊥*------+ 0.29------= ------y Ð ------, (13) 2 y dx ydx dx dx 2A + 1 ( α )2( ) dx 1 + 2 x A + 1 ( ) 2 ( ) in which the second term on the right-hand side is a  dy d ln ne* d ln ne* (10) + 0.11E⊥* + 1.52----- ------+ 0.29y------small correction to the first term. Solving equation (13) dx dx dx2 by the method of successive approximations yields ( ) 2 x dln ne*  ( ≥ ) ( ) + 0.11y ------= 0, y x 0 = y0 x + C1------α------, (14)  dx  1 + 2 x where y (x) is defined in (11) and C = C[(2D/(2A + where y = T (x)/T , T = T (x = 0), E⊥* = E⊥eL /T , A = 0 1 e e0 e0 e f e0 −1/2 α 1)) + 3/2 2]/2(A + 1). 1.64, and D = 0.57ω 2 m 2 L 2 /m T . Be e f 0 e0 For x ≤ 1, we can neglect the second derivative of the Ignoring the electric field E⊥* and the derivative function y in comparison with the leading-order term D in equation (10) and apply the method of successive * 1 dne /dx, we obtain the following solution to equation approximations to solve the equation (10): dy 2 ( ) 1/2 1/2 ----- ( ) T e x 1 2D A  y0 x = ------= 1 Ð ------x . (11) dx Te0 2 2A + 1 (15) eE L ≤ 0 f [ (α α ) ]dy An analysis of equation (10) shows that, for x 1, + 0.33------1 + 2 Ð 1 x ----- Ð D1 y = 0, the electric field of a current-carrying plasma contrib- T e0 dx utes to the electron temperature only in the region where D = D Ð 0.29(α Ð α )eE L /T . where this field is maximum. At small values of the 1 2 1 0 f e0 argument x ≥ 0, it is important to take into account the In this case, the electron temperature profile has the electron density profile. form α Ӷ For 1x 1, we can use expression (8) to obtain 1D  1/2 1/2 from (10) the following equation for T at x ≥ 0: y(x ≤ 1) = 1 Ð ------1 x e 2 A  2 2 (16) d y dy 2 Ð2 y------+ A ----- Ð Dy + Cy (1 + α x) = 0, (12) 0.08eE0 L f [ (α α ) ]2 2 dx 2 Ð -----(--α------α------)------1 + 2 Ð 1 x . dx A 2 Ð 1 T e0 α α α where C = 0.38( 2 Ð 1) 2. We compared the calculated and experimental profiles Te(x) for the following parameters of the plasma and of the ion source [2]: B = 100 G, L ≈ 5 cm, R = Te, V f 5 cm, n = 1014 cmÐ3, n = 1013 cmÐ3, n ≈ 9 × 2.0 H2 H e0 1 12 Ð3 ≈ ϕ ≈ ϕ ≈ 10 cm , Te0 2 eV, 0 4 eV, and e 1.25 eV. Fig- ure 1 shows the experimental (curve 1) electron tem- 1.5 3 perature profile and the analytic (curve 2) profile Te(x) 2 calculated from (11). In a first approximation, we α α assume that the parameters 1 and 2 are independent 1.0 of both the electron temperature and the x-coordinate. We bring the profile n (x) calculated from (8) into coin- 4 e cidence with the experimental electron density profile 0.5 α ≈ α ≈ (see Fig. 2) to obtain 2 3.3 and 1 0.3. Substituting these parameter values into formulas (14) and (16), we determine the corrections to the electron temperature profile (11), which are represented in Fig. 1 by curves 3 0 0.2 0.4 0.6 0.8 1 and 4, respectively. x = z/L f Note that, for the above parameters of the ion- source plasma, the conditions for the electron energy Fig. 1. Electron temperature profiles along the source axis in λ ӷ λ the region of the magnetic filter: (1) experimental profile, relaxation || R and 1 < Lf and the condition (2) profile calculated from (11), (3) corrected profile (14), 2 Ӷ and (4) corrected profile (16). me ue /2 Te are all well satisfied. The directed electron

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ELECTRON HEAT TRANSPORT IN THE MAGNETIC FILTER 1013

n , cm–3 region adjacent to the plasma electrode. By controlling e this profile, it is possible to form ion beams with the 10 maximum current and to extract them from the source. The above plasma parameters are representative of 8 ion sources of the type under consideration. Under the 1 conditions prevailing in these sources, taking into 6 account the electron-density and electric-field profiles yields a small correction to the electron temperature 2 profile in the region of the magnetic filter. Specifically, 4 the variables ne(x) and Te(x) in the electron energy balance equation become independent of each other. The 2 electron density is determined from the model of the transport of heavy plasma components across the magnetic field, under the assumption that the plasma is 0.2 0.4 0.6 0.8 1.0 quasi-neutral. In this case, electron temperature varia- x = z/L tions may substantially change the plasma density pro- f file. Fig. 2. Electron density profiles in the magnetic filter: If the plasma electric field is strong (with an (1) experimental profile and (2) profile calculated from (8). ϕ increased potential 0 in the first chamber and a ϕ decreased potential e of the plasma electrode), then velocity should be estimated from the maximum com- the relation between the electron-density and electron- ponent ued in expression (3). temperature profiles becomes closer. An increase in the plasma electric field raises the intensity of the electron By virtue of the relationship β/γ ≈ 1.5, our model of heat flux q (4), which is directed toward the first cham- the electron heat transport across the magnetic field far et from the wall of the second chamber is one-dimen- ber (where the plasma is created) and is governed by sional with a fairly high degree of confidence. When the the directed electron velocity. As a result, the plasma associative detachment of electrons from negative ions electrons are cooled more efficiently in the magnetic β γ ӷ ӷ filter region adjacent to the plasma electrode. is enhanced (i.e., when / 1 and ne nÐ), the elec- tronÐion plasma expands from the wall toward the chamber axis and occupies most of the volume of the REFERENCES ion source [7]. In this case, the electron density becomes two-dimensional and the one-dimensional 1. O. L. Veresov, S. V. Grigorenko, and S. Yu. Udovi- approach used here fails. chenko, Zh. Tekh. Fiz. 70 (4), 111 (2000) [Tech. Phys. 45, 236 (2000)]. 2. K. Jayamanna, M. McDonald, D. H. Yuan, and CONCLUSION P. W. Schmor, in Proceedings of the European Particle The one-dimensional model proposed here to Accelerator Conference, EPAC-90, Nice, 1990, p. 647. describe the electron energy balance in the region of the 3. M. Ogasawara, T. Yamakawa, F. Sato, and Y. Okumura, magnetic filter of a volume plasma-based ion source in Proceedings of the International Symposium on the allowed us to determine the electron temperature pro- Production and Neutralization of Negative Ion Beams, file along the source axis. The analytic profile agrees Brookhaven, 1990, p. 596. well with the experimental one and makes it possible to 4. A. J. T. Holmes in Proceedings of the International Sym- refine the rates of the elementary processes that are sen- posium on the Production and Neutralization of Nega- sitive to the electron temperature and govern the equi- tive Ion, Brookhaven, 1992, p. 101. librium density profiles (6) and (8) of the plasma com- 5. F. A. Haas, L. M. Lea, and A. J. T. Holmes, J. Phys. D 24, ponents. 1541 (1991). In a second approximation, equation (5) for the den- 6. V. E. Golant, A. P. Zhilinskii, and I. E. Sakharov, Funda- sity of negative ions should be solved with the coordi- mentals of Plasma Physics (Atomizdat, Moscow, 1977; α α Wiley, New York, 1980). nate-dependent coefficients 2(x) and 1(x). The solution to this equation, 7. V. A. Rozhanskiœ and L. D. Tsendin, Collision Transfer in a Partially Ionized Plasma (Énergoatomizdat, Mos-  x  Ð1 cow, 1988). n* = F(x) α (x)F(x)dx + 1 , Ð ∫ 2  8. J. P. Shkarofsky, T. W. Johnston, and M. P. Bachynski, 0 The Particle Kinetics of Plasmas (Addison-Wesley, Reading, 1966; Atomizdat, Moscow, 1969). where F(x) = exp x α (x)dx, then enables us to refine ∫0 1 the density profile of negative ions in the magnetic filter Translated by O. E. Khadin

TECHNICAL PHYSICS Vol. 45 No. 8 2000

SOLIDS

Simulation of the Melting and Crystallization Processes in Monocrystalline Silicon Exposed to Nanosecond Laser Radiation S. P. Zhvavyœ Institute of Electronics, Belarussian Academy of Sciences, Minsk, 220090 Belarus Received October 12, 1998; in ﬁnal form, July 26, 1999.

Abstract—Numerical simulation of the melting and crystallization processes of monocrystalline silicon exposed to the nanosecond radiation of a ruby laser was carried out with the kinetics of the phase transformations accounted for on the basis of Kolmogorov equations. A two-dimensional mechanism of nucleation and growth of the new phase was invoked to describe the phase transitions. It was shown that the temporal dependences of monocrystal overheating and liquid phase supercooling in the melting and crystallization stages, respectively, are nonmonotonic and determined by the kinetics of the phase transitions. The maximum values of the overheating and supercooling were ~100 K. © 2000 MAIK “Nauka/Interperiodica”.

The melting and recrystallization processes of on the basis of the one-dimensional thermal conductiv- monocrystalline silicon initiated by nanosecond laser ity equation pulses have been studied in many papers (see, for ∂T ∂ ∂T example, [1Ð4]). Generally, the Stephan problem is ρcT()------= kxT(), ------solved to clarify the main relationships of the effect of ∂t ∂ x ∂x laser irradiation. This approach is warranted for pro- (1) ∂ϕ ∂Ψ cesses deviating only slightly from equilibrium condi- + SxT()ρ, ÐL------Ð ------tions. However, as follows from experimental studies ∂t ∂t [5Ð7] on irradiation of semiconductor surfaces by nano- and picosecond laser pulses, phase transitions with the boundary and initial conditions occur far from equilibrium. In [8, 9], the simulation of ∂T ()∞, ------==0, Tx t T0, laser annealing of amorphous silicon layers takes into ∂x account the nonequilibrium character of the processes x = 0 (2) (), and is based on a consideration of the phase state of the Txt=0 =T0, irradiated sample cell as a function of enthalpy and the ρ elapsed time before nucleation of the new phase. where is the density, c(T) is the specific heat capacity, Another approach [10] is based on solving the Stephan k(x, T) is the coefficient of thermal conductivity, L is the problem for the nonlinear dependence of the phase latent heat of phase transition, and T0 is the initial tem- boundary velocity on temperature. However, problems perature. involving the kinetics of the new phase formation have The heat source term S(x, t) in (1) describes the heat rarely been dealt with in these studies. evolved due to absorption of the laser radiation: This paper presents a model of melting and crystal- () (), ()α(), WT lization of monocrystalline silicon irradiated by a nano- Sxt = 1ÐR xT------τ- second ruby laser, which takes into account the kinetics i of the phase transformations using Kolmogorov equa- x (3) tions [11Ð13]. We previously used a similar approach in ×αexp Ð,∫()x',Tdx' a numerical simulation of laser annealing of amorphous silicon [14, 15], in which the crystallization process of 0 a highly supercooled melt is controlled by a three- where R and α(x, T) are the reflection and absorption τ dimensional growth mechanism of already available coefficients, respectively, and W and 1 are the energy nuclei. Here, it is assumed that both melting and crys- density and duration of the laser pulse, respectively. tallization occur as a result of homogeneous nucleation The two last terms on the right-hand side of Eq. (1) through two-dimensional layer-by-layer growth [11, describe the capacities of heat sinks and sources in 12, 16Ð18]. melting and crystallization of silicon. Here, ϕ(x, t) is Variation of the temperature of monocrystalline sil- the fraction of the melt formed at point x by the time t icon irradiated by nanosecond laser pulses is described after melting begins and Ψ(x, t) is the fraction of crys-

SIMULATION OF THE MELTING AND CRYSTALLIZATION PROCESSES 1015 tallized melt at point x by the time t after crystallization T, K begins; in this case, the condition ϕ(x, t) + Ψ(x, t) + γ(x, 2200 t) = 1 should be fulﬁlled, where γ(x, t) is the fraction of monocrystal not melted at point x by the time t. In the theory of phase transitions, the fraction of new phase 2 formed is expressed in terms of the nucleation rate J(t) 2000 and the growth rate V(t) [11, 12]: 1 t t n   1800 ϕ(x, t) = 1 Ð expÐβ∫ J(τ) ∫V(t')dt' dτ, (4)   τ Tm t1 1600 where t1 is the time of the start of nucleation at point x and β is the form factor. The function J(t) is determined by the nucleation mechanism. In this study, the kinetics of melting and crystallization is considered in the 1400 50 100 150 200 250 300 framework of the layer-by-layer growth model [11, 13]; t, ns i. e., it is assumed that the growth of the new phase proceeds as ongrowth of successive layers. The formation Fig. 1. Temporal dependence of the temperature of silicon of each layer proceeds by two-dimensional growth of surface at a radiation power density of W = (1) 1.5 and nuclei (the exponent n = 2 in (4)), and the nuclei of a (2) 2 J/cm2. new ith layer can emerge only in the crystallized regions of the preceding (i + 1)th layer. In this case, the d, µm expression for the nucleation rate has the form [11, 12] 0.4 2 kT  U   πaσ T  J(t) = N------exp Ð------exp Ð------m-- , (5) 2 h  kT  kT∆T  0.3

2 where N = N0 f(x, t), N0 is the density of atoms per cm at the interface, and f(x, t) = Ψ(x + a, t) + γ(x + a, t) = 0.2 ϕ 1 Ð (x + a, t) is the fraction of monocrystalline phase 1 of the preceding layer where crystallization centers of 0.1 the next layer can be formed. During melting, centers of the liquid phase can arise only in the crystalline regions of the layer and, in this 0 ϕ 50 100 150 200 250 300 case, f(x, t) = 1 Ð (x, t) [12]; U is the activation energy t, ns for the transition of an atom through the phase boundary; a is the interatomic distance (the monolayer Fig. 2. Temporal dependence of the silicon melting depth. height); σ is the surface energy of the phase boundary; ϕ = 0.01 (solid curve), ϕ = 0.99 (dashed curve); (1, 2) same ∆ ∆ T = T Ð Tm for melting; and T = Tm Ð T for crystalli- as in Fig. 1. zation. The expression used for the growth rate has the form τ with i = 70 ns. The values of the silicon parameters [13] used in solving the problem are listed in the table. ∆ ( ) kT  U   L* T In Fig. 1, the temporal dependences of the tempera- V t = a------expÐ------ 1 Ð expÐ------ , (6) h kT kTm T ture of a monocrystalline silicon surface are shown for two values of the energy density, W = 1.5 and 2 J/cm2. where L* is the melting heat per atom. It is seen that at the early stage of heating, a narrow In the two-phase (transition) region, which includes peak is observed on the temperature curve. As follows molten and crystalline silicon, the parameters of the from the calculations, this peak appears as the silicon problem are deﬁned as follows [14]: begins to melt and corresponds to overheating of the ( , ) ϕ( , ) ( , ) [ ϕ( , )] ( , ) surface layer. The overheating before melting starts is a x t = x t al x t + 1 Ð x t as x t , (7) as high as ∆T ≈ 100 K (Fig. 3), both at W = 2 J/cm2 and where the indices l and s refer to the liquid and crystal- W = 1.5 J/cm2. In the time interval ∆t < 1 ns in a near- line phases, respectively. surface layer ∆x ≈ 0.075 µm thick (Figs. 2, 5), nuclei of Equation (1), in combination with (2)Ð(7), was the liquid phase arise, which at ∆T ≈ 100 K start to grow solved numerically by a sweep method. The shape of at a high rate (Fig. 4). Because of the large latent heat L 2 π τ the laser pulse was speciﬁed by the function sin ( t/2 i) of the phase transition in silicon, formation of the melt-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1016 ZHVAVYŒ et al.

(T – Tm), K V, m/s 15 100 2 1 10 50 5 2 1 0 0

–50 –5

–100 –10 –15 50 100 150 200 250 300 50 100 150 200 250 300 , ns t t, ns

Fig. 3. Temporal dependence of overheating and supercool- Fig. 4. Temporal dependence of the velocity of advance of ing of silicon at the boundaries of the two-phase region ϕ = the boundaries ϕ = 0.01 (solid curve) and ϕ = 0.99 (dashed 0.01 (solid curve) and ϕ = 0.99 (dashed curve) ((1, 2) same curve) of the two-phase region at W = (1) 1.5 and as in Fig. 1). (2) 2 J/cm2.

ϕ T, K 1850 1.0 4 5 6 7 8 9 4 1800 3 0.8 3 1750 1 2 0.6 2 1700 T 0.4 m 1650

0.2 1600 1 1550 0 0.1 0.2 0.3 0.4 0 0.1 0.2 µ x, m x, µm

Fig. 5. Depth variation of the silicon liquid phase fraction at W = 2 J/cm2 at various times t = (1) 59, (2) 61, (3) 62, (4) 64, Fig. 6. Calculated temperature profiles at W = 2 J/cm2 for (5) 65, (6) 68, (7) 80, (8) 100, and (9) 133 ns. time t = (1) 59, (2) 61, (3) 62, (4) 64 ns. ing nuclei and their growth causes a reduction in over- layer forms. Thus, at the initial stage of melting of heating at the forward boundary of the two-phase monocrystalline silicon, in time ∆t ≈ 5Ð6 ns, a molten region down to ∆T ≈ 10 K (Fig. 3), cooling of the adja- layer ∆x ≈ 0.07Ð0.08 µm thick with a fairly narrow cent crystalline regions (Figs. 5, 6), and a significant transition region ∆x ≈ 0.015 µm is formed at the surface reduction in the propagation velocity of the forward (Figs. 2, 5). boundary down to V ≈ 2 m/s (Fig. 4).1 As the forward boundary slowly propagates into the sample bulk Further heating of silicon by laser radiation results (Fig. 5, curves 1Ð5), an increase in the melt fraction at in renewed increases in overheating and the velocity of the surface and formation of the rear boundary of the forward boundary movement, both of which attain their transition region occur; i.e., a continuous liquid phase maximum values at this stage (Figs. 3, 4). Thus, at W = 2 ∆ | ≈ | ≈ 2 J/cm , T ϕ = 0.01 80 K and V ϕ = 0.01 9 m/s. At the 1 The condition used to define the position of the rear boundary of rear boundary, the overheating is somewhat higher, the two-phase (transition) region is that the melting (crystalliza- ∆ | ≈ T ϕ = 0.99 105 K, and the velocity of the boundary tion) process is considered finished if ϕ = 0.99 (Ψ = 0.99) [8]. The | | position of the forward boundary is defined by the condition ϕ = movement V ϕ = 0.99 becomes equal to V ϕ = 0.01, remain- 0.01 (Ψ = 0.01). ing so until completion of the melting process. As the

TECHNICAL PHYSICS Vol. 45 No. 8 2000 SIMULATION OF THE MELTING AND CRYSTALLIZATION PROCESSES 1017

ψ T, K 1.0 1700 10 Tm 0.8 5 4 9 8 7 6 5 4 1650 3 0.6 3 0.4 1 1600 2 0.2 2 1 0 0.1 0.2 0.3 0.4 1550 x, µm 0 0.1 0.2 0.3 0.4 0.5 x, µm Fig. 7. Depth variation of the fraction of crystallized silicon at W = 2 J/cm2 for time t = (1) 136, (2) 139, (3) 142, (4) 145, Fig. 8. Calculated temperature profiles at W = 2 J/cm2 for (5) 150, (6) 201, (7) 243, (8) 251, (9) 259, and (10) 268 ns. time t = (1) 136, (2) 139, (3) 142, (4) 145, and (5) 150 ns. melt moves deeper into the sample bulk and the influx to ∆x ≈ 0.075 µm (Fig. 2), since the emerging nuclei of of light energy decreases, the overheating and the prop- the new layer are subjected to greater supercooling and, agation rate gradually decrease and, by the time the consequently, their nucleation and growth rates are laser radiation is ended, the ingress of melt into the higher than in the preceding layer. As more heat is semiconductor bulk ceases and the two-phase region released from growth of the crystalline phase, the tem- remains immobile for ∆t ≈ 20 ns (Figs. 2, 4). For this perature distribution becomes uniform (Fig. 8) and period of time, as a result of heat drain to the sample supercooling at the forward boundary (Ψ = 0.01) and volume, which is no longer compensated by the laser the rear boundary (Ψ = 0.99) of the transition region radiation, the overheating vanishes completely and the drops to ~4 K and ~18 K, respectively. The velocity of melt becomes supercooled by ∆T ≈ 100 K (Fig. 3) in the advance of the boundaries decreases to 3Ð4 m/s. The vicinity of the transition region and by 80 K (Fig. 1) at advance of the crystallization region toward the sample the surface. With the onset of, and increase in, super- surface is accompanied by a minor temperature | cooling, the formation and growth of the nuclei of the increase (of ~2 K); and the velocities V Ψ = 0.99 and | crystalline phase begins (Fig. 7, curve 8). The heat V Ψ = 0.01 drop from ~3 to 2.2 m/s and from 4 to 3 m/s, release in the silicon crystallization process causes an respectively. Only at the final stage, when the thickness increase in temperature and the emergence of a peak in of the transition layer becomes less than 0.05 µm (Fig. 2), the temperature profile within the two-phase region do the supercooling at the rear boundary and the veloc- | (Fig. 8, curves 2Ð4). The temperature gradient pro- ity V Ψ = 0.99 increase sharply (Figs. 3, 4). The reason is duced in the melt at the forward boundary of the two- that the heat sources contained in a narrow layer with phase region (Ψ = 0.01) causes spreading of the transi- ψ > 0.5 are insufficient to offset the heat which is being tion region toward the surface, increasing its thickness removed to the sample bulk.

Silicon parameter values Parameters Crystalline Si Molten Si ρ 3 × Ð3 , g/cm 2.328 2.53Ð0.152 10 (TÐTm) [19] c, J/g K 0.844 + 1.18 × 10Ð4T Ð 1.55 × 104TÐ2 [19] 1.04 L, J/g 1787 [19] 1521 8.97 k, W/cm K ------, T < 1200 K, ------, T ≥ 1200 K [20] 0.585 [19] T1.226 T0.5 R 0.35 0.72 α, cmÐ1 1578exp(T/493) [21] 106 [20] U, eV 1.22 [22] σ, erg/cm2 300

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1018 ZHVAVYŒ et al.

Thus, the temporal variations of the overheating of 9. V. Yu. Balandin, A. V. Dvurechenskiœ, and L. N. Aleksan- drov, Poverkhnost, No. 1, 53 (1986). crystalline silicon and supercooling of the melt proceed > > nonmonotonically and depend on the phase transfor- 10. R. Cerny, R. Sas> ik, I. Lukes, and V. Chab, Phys. Rev. B mation kinetics. Maximum overheating and supercool- 44, 4097 (1991). ing values, both equal to ~100 K, are attained at the 11. L. N. Aleksandrov, Crystallization and Recrystallization early stages of melting and crystallization, respectively. Kinetics of Semiconductor Films (Nauka, Novosibirsk, Formation of a continuous ﬁlm of melt at the surface of 1985). monocrystalline silicon takes 5Ð6 ns. The average 12. V. Z. Belen’kiœ, GeometricÐProbabilistic Models of velocity of advance of the two-phase region is ~8Ð9 m/s Crystallization (Nauka, Moscow, 1989). for melting and ~3Ð4 m/s for crystallization. 13. V. P. Skripov and V. P. Koverda, Spontaneous Crystalli- zation of Supercooled Liquids (Nauka, Moscow, 1984). 14. S. P. Zhvavyœ, Zh. Prikl. Spektrosk. 50, 589 (1989). REFERENCES 15. S. P. Zhvavyœ and O. L. Sadovskaya, Poverkhnost, 1. R. F. Wood and G. E. Giles, Phys. Rev. B 23, 2923 No. 11, 101 (1990). (1981). 16. Ya. I. Frenkel’, Kinetic Theory of Liquids (Nauka, Lenin- 2. D. H. Lowndes, R. F. Wood, and J. Narayan, Phys. Rev. grad, 1975; Clarendon, Oxford, 1946). Lett. 52, 561 (1984). 17. M. Vollmer, Kinetik der Phasenbildung (Steinkopff, 3. V. A. Pilipovich, V. L. Malevich, G. D. Ivlev, and Dresden, 1939; Nauka, Moscow, 1986). V. V. Zhidkov, Inzh.-Fiz. Zh. 48, 306 (1985). 18. L. N. Aleksandrov, Formation Kinetics and Structures of 4. G. M. Gusakov, A. A. Komarnitskii, and A. S. Em, Phys. Solid Layers (Nauka, Novosibirsk, 1972). Status Solidi A 107, 261 (1988). 19. A. R. Regel’ and V. M. Glazov, Physical Properties of 5. G. E. Jellison, Jr., D. H. Lowndes, D. N. Mashburn, and Electronic Melts (Nauka, Moscow, 1980). R. F. Wood, Phys. Rev. B 34, 2407 (1986). 20. A. E. Bell, RCA Rev. 40, 295 (1979). 6. M. Yu. Aver’yanova, S. Yu. Karpov, Yu. V. Koval’chuk, et al., Pis’ma Zh. Tekh. Fiz. 12 (18), 1119 (1986) [Sov. 21. M. Toulemonde, S. Unamuno, R. Heddache, et al., Appl. Tech. Phys. Lett. 12, 462 (1986)]. Phys. A 36, 31 (1985). 7. A. G. Cullis, H. C. Weber, N. G. Chew, et al., Phys. Rev. 22. G. Andra, H.-D. Geiler, G. Gotz, et al., Phys. Status Lett. 49, 219 (1982). Solidi A 74, 511 (1982). 8. R. F. Wood and G. A. Geist, Phys. Rev. B 34, 2606 (1986). Translated by M. Lebedev

TECHNICAL PHYSICS Vol. 45 No. 8 2000

SOLIDS

The Determination of the Model Interaction Potential Parameters from a Comparison of Experimental and Calculated Ion Ranges in an Amorphous Substance E. G. Sheœkin Research Enterprise for Hypersonic Systems, St. Petersburg, 196066 Russia Received November 13, 1998

Abstract—Parameters of the model atomic interaction potential are suggested to be determined by comparing the experimental and analytical values of projective ion ranges. The parameters were found for the interaction of Bi, Pb, Au, Yb, Er, Eu, Cs, Xe, Sn, Rb, Kr, Ga, and Cu ions with carbon and boron atoms. © 2000 MAIK “Nauka/Interperiodica”.

The proper selection of atomic interaction potential where f(X) = X[(1 + X)ln(1 + 1/X) Ð 1] and is of great importance in analysis and numerical simulation of low- and medium-energy ion motion in a sub- β2 ()ε stance. The interaction potential must adequately char- X = ------3/2 3/4 . acterize elastic scattering while being easy to apply. In 4()ε + βε [1], a model interaction potential that basically satisﬁes ε these requirements was proposed. It is written in the Figure 1 shows the stopping powers of ions sn( ) form of screened Coulomb potential calculated from (3) for three values of β. It follows that the stopping power depends on β only slightly for ε > 2. 21 Ð r / a , r ≤ a , ε ≤ () Z1Z2e Hence, results for ion energies 2 are the most appro- Vr =------ (1) priate if β is determined by comparing the theory and r 0, ra>, experiment. where Z1 and Z2 are the charges of an ion nucleus and Projective ranges will be estimated using results target atom, respectively; r is the distance between col- obtained in [2]. In that work, elastic scattering of ions liding particles; and a is the screening radius. is treated in terms of potential (1) and inelastic scatter- It is assumed that the screening radius depends on ing is considered within the approximation of continu- the ion energy as εÐ1/4 β aa= TF / . (2) sn This relationship provides an adequate description of the experimentally found stopping power of ions 0.5 in elastic collisions throughout the energy range. Here, 2/3 2/3 1/2 aTF = 0.8853a0/(Z1 + Z2 ) , e is the charge of 0.4 an electron, and a0 is the Bohr radius. The reduced ε ε energy is related to the ion energy E as = 1 2 0.3 Em2aTF/(Z1Z2e (m1 + m2)), where m1 and m2 are the 2 masses of an ion and target atom, respectively. The 3 β parameter appearing in (2) is viewed as an adjusting 0.2 parameter. In this work, we will try to estimate β for speciﬁc ionÐtarget pairs by comparing theoretical and experi- 0.1 mental values of projective ion ranges in a substance. For interaction potential (1) with screening radius (2), the stopping power of ions is given by [1] 0 10–3 10–2 10–1 100 101 ε ε ()ε ()()ε sn =------2fX , (3) Fig. 1. Stopping power vs. energy for model potential (1) β with screening radius (2) at β = (1) 0.5, (2) 0.6, and (3) 0.7.

1020 SHEŒKIN et al.

ε ε 2 + Q ous deceleration with stopping power sn( ) = k , b0 = ------γ---(------)-----, where k is a dimensionless parameter. In the calcula- Q + 1 + m2/m1 /2 tions, we will take into account the threshold character ε ε of ion deceleration. This means that an ion will stop if µ(ε , ε ) 1  m2 ''  m2 ' ε ε '' ' = -- 1 Ð ------ -ε--- + 1 + ------ ε---- , its energy becomes less than the threshold energy th. 2 m1 ' m1 '' The latter is related to the energy of atom displacement 1 Ed [3]. In view of these assumptions, the projective ion --- Ð 1 ε 1 ε'' Q range Rp( ) can be represented in the form [2] p (ε ε'') = ------, ε'' ≤ ε, e εQ ε  ˜ ˜ R (ε) = R p(ε) Ð µ (ε)R p(ε ), p th th γX(ε'')[X(ε'') + 1] N (ε ε ) ------(4) pn '' ' = 2 ˜ (ε) λ (ε) εi/2 ε''[γX(ε'') + (1 Ð ε'/ε'')] R p = 0 ∑ bi , i = 0 1 at (1 Ð γ )ε'' ≤ ε' ≤ ε'', ×  where 0 at ε > ε'', ε' < (1 Ð γ )ε''. β2 ε λ (ε) = ------, Let us briefly explain the physical meaning of the 0 π 2 ( ) ε ε n aTH 1 + Q/2 functions involved in (6). The function pe( '') defines the probability density of the ion energy being γ β2 γ n is the target atom density, Q = k , = 4m1m2/(m1 + changed from ε to ε" under inelastic deceleration, 2 µ ε ε m2) , and th is the mean direction cosine of ion motion pn( '' ') defines that of the ion energy being ε ε ε µ ε ε at the instant its energy becomes less than th. changed from " to ' under elastic collision, and ( '', ') ε ≤ is the scattering angle cosine of an elastically colliding At 2, we can put N = 6 in representation (4); in ion when its energy changes from ε" to ε'. The value of this case, the relative error of the ion range calculation ε is the extreme of the energy interval in which pro- will not exceed 0.5%, according to [2]. In [2], µ was max th jective ranges are calculated. calculated by the Monte Carlo method. In this work, we suggest an expression approximating the results of [2]: According to (4)Ð(6), for an ionÐtarget pair, the projective range depends on β, which characterizes elastic 2 Ð1  m m   0.25 scattering of ions, and k, which describes inelastic µ (ε) = 1 + 0.16 + 1.6------2 + 3.65 ------2 ε . (5) th  m m   deceleration of ions. Assuming these parameters to be 1 1 unknown, we will try to find them by contrasting exper- The use of (5) instead of the Monte Carlo expression imental and theoretical values of the projective ranges. for µ introduces additional relative and absolute errors As a measure of discrepancy, we will take residue th β into the projective ranges of no more than 0.1% and S( , k): 0.5 Å, respectively, throughout the energy range. N exp R exp(ε ) Ð R (β, k, ε ) 2 The coefficients b in (4) are found by solving the set (β, ) p i p i i S k = ∑ ------σ------, (7) i of linear equations with N variables i = 1 N exp ε σ where R p ( i) is an experimental projective range, i is F + b ψ , = 0, j = 1…N, (6) j ∑ i i j the standard deviation of the measured projective i = 1 ε ranges at an energy i, Nexp is the number of data points, β ε where and Rp( , k, i) is a theoretical projective range [unlike β ε (4), the parameters and k enter in explicit form]. max The parameters β and k will be found from the min- (λ ( (λ ) λ ))( (λ εi/2) λ εi/2) ε Fi = ∫ 0 + b0 L 0 Ð 0 L 0 Ð 0 d , imum condition for the residue S(β, k). Experimental 0 values of projective ranges will be taken from [4]. The σ σ ε value of i in [4] is defined as i = max(14 Å; max exp ε ψ ( (λ εi/2) λ εi/2)( (λ ε j/2) λ ε j/2) ε 0.05R p ( i)). i, j = ∫ L 0 Ð 0 L 0 Ð 0 d , In order that the errors involved in β and k be deter- 0 mined simultaneously with the calculation of these ε ε'' parameters, we will make the following assumptions. ( ) (ε ε ) (ε )µ(ε , ε ) Let the discrepancy between calculated and experimen- L f = ∫ pe '' ∫ f ' '' ' tal results be random. Also, let random discrepancies 0 0 between experimental and theoretical ranges of ions × (ε ε ) ε ε ε pn '' ' d 'd '', with an energy i be distributed by the normal law with

TECHNICAL PHYSICS Vol. 45 No. 8 2000 THE DETERMINATION OF THE MODEL INTERACTION 1021

2 β σ S( opt, k)/S0.95 the zero expectation and a dispersion i . Then [5], S in (7) is a random quantity obeying the chi-square (χ2) distribution with the number of degrees of freedom 1.0 1 (Nexp Ð 1). The probability density for this distribution is 0.8 ( ) ( ) 1 Nexp Ð 3 /2 ÐS/2 p S = ----(------)------S e , (8) Nexp Ð 1 /2Γ(( ) ) 2 Nexp Ð 1 /2 0.6 2 where Γ(x) is gamma function. The expectation of the random quantity S is (N Ð 1). exp 0.4 The probability that the random quantity S lies in the ≤ ≤ 3 interval 0 S SP is given by

SP 0.2 ∫ p(S)dS = P. (9) 4 0 0 0.02 0.04 0.06 0.08 0.10 The tolerance ranges for β and k are determined β ≤ k from the inequality S( , k)/SP 1. Given P, SP is found from (9). Fig. 2. Normalized residue vs. k. IonÐtarget pairs are β (1) GaÐC, (2) RbÐB, (3) AuÐC, and (4) CuÐC. Figure 2 shows S( opt, k)/SP vs. k curves for different β ionÐtarget pairs at P = 0.95. The value of opt is found β from the residue minimum for a given k and also opt β depends on k. opt vs. k curves for the pairs in Fig. 2 are 0.72 displayed in Fig. 3. β 0.70 From Fig. 2, it follows that S( opt, k) for CuÐC, AuÐ C, and RbÐB pairs is within the tolerance range S0.95 for 0.68 k in the interval of 0 ≤ k ≤ 0.1. For AuÐC, the projective ranges were measured for 0.0127 ≤ ε ≤ 0.127. In this 0.66 4 energy interval, ions are decelerated largely because of elastic collisions with target atoms; inelastic decelera- 0.64 1 β tion can be ignored. As a result, S( opt, k)/SP is virtually β 0.62 k-independent for this pair. For GaÐC, S( opt, k)/SP has a minimum, which is more distinct here than in the 0.60 other pairs. For this pair, experimental data were obtained for energies 0.142 ≤ ε ≤ 2.12, which are larger 0.58 2 than for the other pairs. Accordingly, inelastic scattering processes are most signiﬁcant in this case. As fol- 0.56 β 3 lows from Fig. 3, opt grows with k almost linearly. The β 0.54 weakest opt(k) dependence is observed for AuÐC. 0 0.02 0.04 0.06 0.08 0.10 Hence, it can be expected that β for this pair will be k determined with the greatest accuracy. β The k values obtained from the experimental data in Fig. 3. opt vs. k. IonÐtarget pairs are (1) GaÐC, (2) RbÐB, Figs. 2 and 3 (ε ≤ 2) are rough estimates. The exact (3) AuÐC, and (4) CuÐC. evaluation of β from these results thus requires that k be calculated with independent experimental or theoretical data. The inequality S(β , k)/S < 1 holds in a wide Analysis [4] of 19 ionÐtarget pairs similar to that illus- opt P trated in Figs. 2 and 3 showed that, under such condi- range of k. Hence, k can be determined with a not too ξ high accuracy. Therefore, we will take advantage of the tions, (10) provides the best estimate of k at = 1. The frequently used Lindhard’s expression (see, e.g., [6, 7]) associated value of k will be designated as k*. For example, k* = 0.0573 and 0.0541 for the GaÐC and RbÐ 0.0793Z 1/2Z 1/2 (m + m )3/2 B pairs (Fig. 2), respectively. The corresponding values k = ξ------1------2------1------2------. (10) 3/4 3/2 1/2 of S(β , k*)/S , as follows from Fig. 2, deviate from (Z 2/3 + Z 2/3) m m opt P 1 2 1 2 the smallest ones insigniﬁcantly. Hereafter, the param- It is assumed that ξ varies in the [1, 2] interval. In eter β will be found from the minimum condition for β β ≤ (10), m1 and m2 are expressed in atomic mass units. S( , k*)/SP. The inequality S( , k*)/SP 1 will be used

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1022 SHEŒKIN et al.

β S( , k*)/S0.95 Rp, A 1.2 800

1.0 600 3 2 1 0.8

0.6 400

0.4 200 0.2

0 0.52 0.56 0.60 0.64 0.68 0.72 0 50 100 150 200 β E, keV

Fig. 5. Energy dependence of the projective range of Au Fig. 4. Normalized residue vs. β. IonÐtarget pairs are (1) ions in carbon. Error bars, experiment [4]. Calculation by GaÐC, (2) RbÐB, and (3) Au–C. Circles denote the conﬁ- (4)Ð(6) at β = 0.556, 0.537, and 0.575 is represented by con- dence interval for β. tinuous, dashed, and dotted curves, respectively. to estimate the tolerance range of β. Figure 4 depicts lated values of R and ∆R listed in the table are in good β β p p normalized residue S( , k*)/SP vs. curves for three agreement with the experimental data. The maximum ∆ ionÐtarget pairs at P = 0.95. They look like quadratic absolute discrepancy Rp between them is 61 Å parabolas and have distinct minima whose positions (300-keV Cs ions in B). The associated relative error is specify the desired parameter β. The intersections of 23% for this case. Recall that β was estimated using the β the curves with the S( , k*)/SP = 1 level are treated as experimental data for projective ranges Rp alone. The β ∆ extremes of the tolerance ranges of for the corre- coincidence of the calculated and experimental Rp sponding pair. For example, for GaÐC and AuÐC, the values supports the validity of β’s for the ionÐtarget allowable values of β fall into the ranges 0.637 ≤ β ≤ pairs considered. In addition, this fact strengthens the 0.696 and 0.537 ≤ β ≤ 0.575, respectively, with a prob- selection of potential (1) for simulating ion motion in a ability P = 0.95. The relative error thus determined in β substance. for the different ionÐtarget pairs varies from 3.5 to 5.5%. Since this error is estimated approximately, we The calculated results are well approximated by the will set it equal to 5% on average. Figure 5 compares relationship the experimental projective ranges of Au ions in carbon and theoretical values calculated by (4)Ð(6) for three ∆R m 0.846 β ------p = ------2 ------. (11) 's. It is seen that all data points fall into the interval R m ε bounded by the extreme curves for β. At β = 0.556, p 1 1 + 0.473 β which corresponds to the minimum of S( , k*)/SP, the ∆ calculation and experiment coincide within the experi- From (11), the relative error Rp/Rp is no more than mental error. 2% for ions with m1/m2 > 5 and energies between 0.01 < ε < 2.5. This relationship can be used for esti- The table lists the values of β for 19 ionÐtarget pairs. ∆ mating Rp when Rp is calculated [for example, with With regard for the estimated error, they are rounded (4)Ð(6)] or known from experiment. off to two signiﬁcant digits. As follows from the table, the values of β are close to each other and lie in the To conclude, a method for determining the parame- interval 0.53 ≤ β ≤ 0.59, except for Ga and Cu ions. ters of the model interaction potential is suggested. It is These values were employed to calculate ion ranges by based on comparing experimental and calculated pro- the Monte Carlo method using the algorithm described jective ion ranges in amorphous substances. The poten- in [1]. In these calculations, the projective ion ranges Rp tial parameters estimated for 19 ionÐtarget pairs were ∆ and their standard deviations Rp were determined. The applied to calculate ion ranges by the Monte Carlo 5 number of histories was 10 . According to [1], the rela- method. Both the projective ranges Rp and standard ∆ tive error in this case is no more than 0.3%. The calcu- deviations Rp are in good agreement with the experi-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 THE DETERMINATION OF THE MODEL INTERACTION 1023

Experimental ion ranges vs. those calculated by the Monte Carlo method Experiment Monte Carlo Experiment Monte Carlo E, E, Ion Target β [4] calculation Ion Target β [4] calculation keV keV ∆ ∆ ∆ ∆ Rp, Å Rp, Å Rp, Å Rp, Å Rp, Å Rp, Å Rp, Å Rp, Å Bi B 0.54 20 180 30 177 32 Cs B 0.55 20 165 45 164 36 50 285 60 302 53 50 285 65 290 61 100 440 90 459 78 100 450 110 465 93 300 1050 170 965 153 300 1180 262 1126 201 C 0.53 15 140 27 146 28 C 0.57 20 170 43 176 41 40 245 37 256 48 50 290 69 309 68 80 390 60 386 70 100 490 105 491 104 150 615 115 573 101 200 820 152 824 164 Pb B 0.55 20 175 30 183 33 Xe C 0.55 20 150 30 165 38 50 310 70 312 55 50 290 60 290 64 100 450 100 474 81 100 480 100 464 99 300 1050 200 997 157 300 1200 230 1108 210 C 0.56 20 205 44 190 37 Sn C 0.58 30 235 45 228 54 50 315 60 322 60 50 310 65 316 73 100 495 91 488 88 100 515 100 509 112 200 790 137 763 132 300 1300 260 1241 241 Au B 0.57 20 200 50 193 36 Rb B 0.58 20 170 45 172 44 50 330 70 328 59 50 325 80 325 79 100 470 90 501 87 100 565 150 564 127 300 1100 172 1056 171 300 1550 320 1607 311 C 0.56 20 197 25 187 37 C 0.59 30 210 70 233 62 50 315 47 318 61 50 330 90 332 85 100 460 80 484 89 100 590 160 568 136 150 640 121 627 112 200 1077 270 1057 230 Yb B 0.56 20 180 40 180 36 Kr C 0.58 30 206 60 227 61 50 310 60 310 59 50 320 90 326 84 100 480 90 479 88 100 610 155 562 135 300 1100 190 1053 176 150 870 220 805 183 C 0.55 20 176 35 175 36 Ga C 0.67 20 216 52 222 65 50 295 59 300 60 50 415 110 423 115 100 490 95 463 89 100 730 200 740 185 200 800 150 742 137 300 2000 500 2111 442 Er C 0.58 10 135 48 129 28 Cu C 0.68 30 280 90 301 88 50 310 90 329 67 50 430 130 440 122 75 421 95 421 84 100 785 215 783 200 100 500 105 506 99 200 1547 400 1509 342 Eu C 0.55 30 220 45 215 46 50 320 64 293 62 100 458 90 458 93 200 729 140 752 144

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1024 SHEŒKIN et al. mental data. This counts in favor of the selected model 4. M. Grande, F. C. Zawislak, D. Fink, and M. Behar, Nucl. potential and supports the validity of its parameters. Instrum. Methods Phys. Res., Sect. B 61, 282 (1991). 5. D. J. Hudson, Statistics. Lectures on Elementary Statis- tics and Probability (Geneva, 1964; Mir, Moscow, REFERENCES 1967). 6. M. A. Kumakhov and F. F. Komarov, Energy Loss and 1. E. G. Sheœkin, Zh. Tekh. Fiz. 69 (5), 1 (1999) [Tech. Ranges of Ions in Solids (Belarus. Gos. Univ., Minsk, Phys. 44, 481 (1999)]. 1979). 2. E. G. Sheœkin, Zh. Tekh. Fiz. 69 (2), 93 (1999) [Tech. 7. H. Muramatsu, H. Ishii, E. Tanaka, et al., Nucl. Instrum. Phys. 44, 218 (1999)]. Methods Phys. Res., Sect. B 134, 126 (1998). 3. E. G. Sheœkin, Zh. Tekh. Fiz. 67 (10), 16 (1997) [Tech. Phys. 42, 1128 (1997)]. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 8 2000

SOLIDS

Theoretical and Experimental Study of Photoacoustic and ElectronÐAcoustic Effects in Solids with Internal Stresses K. L. Muratikov and A. L. Glazov Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received August 27, 1999

Abstract—Results of a theoretical and experimental study of photoacoustic and electronÐacoustic effects in solids with internal stresses are presented. In the theoretical part, an approach to describing these effects on the basis of a generalized concept of thermoelastic energy of a solid with internal stresses and the nonlinear Mur- naghan model for the elastic part of its energy is developed. The results of studying objects with internal stresses in the context of an integrated experimental approach incorporating the techniques of photodeﬂection and thermal-wave and photoacoustic microscopy with piezoelectric recording of the signal are reported. It is shown that a similar approach allows one to detect the arrangement of the strained surface areas of the object and to evaluate the extent to which its thermal and thermoelastic parameters are affected by internal stresses. The results of applying this approach to a study of Vickers indentations in silicon nitride ceramics are reported. © 2000 MAIK “Nauka/Interperiodica”.

Internal stresses can radically change the properties eration of the thermoelastic mechanism of generating of materials [1]. In this connection, the development of acoustic vibrations in solid-state objects. An important methods for detecting them in different materials has advantage of thermoelastic generation of sound is its received considerable attention. At present, a number of versatility [31]. In this connection, the photoacoustic methods are being used to tackle this problem. Among and electronÐacoustic methods for detecting internal these are primarily the optical method [2], the ultra- stresses may basically be regarded as belonging to a sonic technique [3], X-ray [4] and neutron [5, 6] dif- small group of general-purpose methods that make it fraction, magnetic measurements [7, 8], Raman spec- possible to detect internal mechanical stresses in troscopy [9, 10], mechanoluminescence [11], detection objects of different origin. of thermal radiation from the absorption of ultrasonic A body of experimental data in support of this pos- oscillations by the object being studied [12], and the sibility has been already obtained. The experimental methods based on holographic interferometry [13Ð15]. results obtained so far support the possibility of using These methods have shown their high efficiency in the photoacoustic and electronÐacoustic effects to solving problems of detecting internal stresses in dif- detect internal stresses in metals [16, 17, 20, 21, 28, 30] ferent types of objects. At the same time, serious and in ceramics [17Ð19, 22Ð25, 27]. Theoretical mod- restrictions are inherent in most of these methods due els of the possible influence of internal stresses on pho- to the origin of the corresponding physical processes. toacoustic and electronÐacoustic signals have also been The only exceptions are the last two methods, which suggested. A model for generating photoacoustic and are based on fairly general physical principles and can electronÐacoustic signals was proposed [20]; this be applied to a wide range of objects. However, rela- model explicitly relates the dependence of these signals tively moderate spatial resolution is inherent in these on stresses to the dependence of thermal material methods. In this connection, much attention has parameters on these stresses. The photoacoustic and the recently been given to studying the possibility of using electron acoustic effects were analyzed [26, 32] in the the photoacoustic [16Ð26] and electronÐacoustic [27Ð context of a nonlinear mechanical model of a solid with 30] effects to detect mechanical stresses in solids. An regard for the possible influence of internal stresses on important point to note when using either of these the thermoelastic component of the energy of the solid. methods is the conversion of the optical-radiation or This model yields correct estimates for the nonlinear electron-beam energy to thermal energy with its subse- mechanical and acoustic parameters of solids and quent transformation into acoustic energy owing to the makes it possible to explain the photoacoustic and elec- thermoelastic effect. Therefore, when considering the tronÐacoustic effects in ceramics with internal stresses, photoacoustic and electronÐacoustic effects, we may in which a profound influence of internal stresses on disregard the details of the interaction of the optical thermal parameters has not been observed [22Ð25]. In radiation and the electron beam with the material. In this connection, the prime objective of this work is the fact, we may restrict ourselves in both cases to consid- further theoretical and experimental study of photoa-

1026 MURATIKOV, GLAZOV coustic and electronÐacoustic effects in solids with In this model, the thermoelastic energy density is deter- internal stresses. mined by the expression An experimental investigation of the influence of 2 3 (λ µ)I1 µ ( )I1 internal stresses on the photoacoustic and electronÐ W E = + 2 ---- Ð 2 I2 + l + 2m ---- acoustic effects presents serious difficulties. They are 2 3 (2) mainly related to the need to monitor a large number of Ð 2mI1 I2 + nI3, different object characteristics during the experiments. This includes, first of all, independent monitoring of where λ and µ are Lamé coefficients; l, m, and n are the the thermal, elastic, and thermoelastic parameters of Murnaghan constants; I1 = ukk; the object. Furthermore, in these experiments, informa- 1[( )2 ] tion about the surface relief of the object is found to be I2 = -- ukk Ð ulmulm ; extremely useful, since it allows detection of strained 2 areas of the sample and, thereby, monitoring of the and 1 3 1 3 arrangement of areas with internal stresses. I = -- u u u Ð --u u u + --(u ) . 3 3 ik il kl 2 ik ik ll 2 ll We turn to a more detailed consideration of the subject. We begin by analyzing the theory of the phenom- Knowledge of the energy densities of a solid makes ena being considered. In [20, 26, 32, 33], the basic con- it possible to set up the equation of motion for the com- cepts of the theory of photoacoustic and electronÐ ponents of the body. In the context of nonlinear acoustic effects in solids with internal stresses were mechanics [37], this equation can be written as generally formulated. In addition, the possible depen- ∂ dence of the thermoelastic energy of the solid on strain Pik ρ ∆ --∂------= 0 uúúi, (3) was taken into account [26, 32, 33]. This dependence xk was chosen on the basis of theoretical analysis of the ∂ influence of strain on the thermal expansion coefficient Xi where Pik = ∂------tkm is the PiolaÐKirchhoff tensor; tkm is of the solids [34]. At the same time, from general con- xm siderations, the dependence of the thermoelastic energy the stress tensor related to the internal energy of the on strain can be represented in a somewhat more gen- ∂ ∂ body W = WE + WT by the relation tkm = W/ ukm; Xi are eral form taking into account the possible influence of coordinates of points of the strained body; x are coor- a change in the volume of the body under strain on this i dinates of points of the body in the initial unstrained energy. Such general representation of the thermoelas- ρ tic energy is used in this study. According to this state; and 0 is the density of the body in the initial approach, the density of the thermoelastic energy of the state. body can be represented as (correct to first order terms Note that the coordinates of points of the strained of the strain tensor stemming from the influence of and unstrained body are related by the equation optical radiation or an electron beam) ( , , , ) ( , , ) Xi x1 x2 x3 t = xi + Ui x1 x2 x3 (4) W = Ðγ (u Ð U )∆T, (1) ∆ ( , , , ) T ik ik ik + ui x1 x2 x3 t , γ γ β δ β γ where ik = 0[(1 + 0Ull) ik + 1Uik], 0 is the coefficient where Ui are components of the vector of the initial β ∆ of thermoelastic coupling for an unstrained body, 0 strain in the body; ui are components of the vector of β and 1 are coefficients determining the dependence of the strain caused by the effect of optical radiation or an thermoelastic coupling on the initial distortion, electron beam on the body and superimposed on the initial strain. ∂ ∂ ∂ ∂ 1 ui uk ul ul Further description of the problem depends on the uik = --∂------+ -∂------+ -∂------∂------ 2 xk xi xi xk chosen system of coordinates xi or Xi, which are referred to as the Lagrangian and Eulerian coordinates, is the tensor of overall strain in the body, Uik is the ten- respectively [38]. In the context of the current problem, ∆ sor of initial strain in the body, T = T Ð T0, and T0 is Eq. (3) is written in the Lagrangian representation; the ambient temperature. therefore, further consideration will be carried out in β this representation. Further solution of the problem Note that, for 0 = 0, Eq. (1) transforms into the depends on the mutual arrangement of the optical radi- expression for the thermoelastic energy density used in β β ation or electron beam and the object. In this study, for [26, 32, 33] while, for 0 = 1 = 0, it transforms into the the sake of definiteness, the geometry of the problem expression for the thermoelastic energy density of an will be considered as shown in Fig. 1. isotropic solid without internal stresses [35]. In its general formulation, the problem of acoustic- The Murnaghan model can be used to determine the vibration generation by a non-steady-state optical or thermoelastic energy density of a strained solid, taking electron beam in the context of thermoelasticity theory into account nonlinear effects under deformation [36]. goes beyond the above-formulated statements, which

TECHNICAL PHYSICS Vol. 45 No. 8 2000 THEORETICAL AND EXPERIMENTAL STUDY 1027 should be supplemented with an equation for non- 1 steady-state thermal-flux propagation in a solid. Basi- cally, such an equation can be derived by the method used, for instance, in [35]. However, in [35], this equation was derived in the context of linear elastic theory. In nonlinear elastic theory, the difference between the parameters of strained and unstrained objects should be U(r) taken into account more precisely. Therefore, we will ∆u(r, ) dwell briefly on deducing the thermal conductivity r t equation in the context of nonlinear elastic theory. According to [35], the rate of heat generation or R absorption in a unit volume is connected with the 0 2 entropy density S by the relationship T∂S/∂t. Therefore, for an arbitrary volume element of the strained body, the following equation is valid: Fig. 1. Geometry of sample arrangement: (1) is the exciting radiation or electron beam; (2) is the sample. ∂S dVT----- = Ð dV div Q + dVw, (5) ∫ ∂t ∫ ∫ where q is the vector of thermal-flux density per unit where Q is the vector of the thermal-flux density in the area of the unstrained body. solid with initial strains; w is the energy density The components of the heat flux q in a solid with released in the body owing to the effect of external initial strains can be specified in the form [39] sources (optical radiation or electron beam). ∂ The entropy density of a strained body with regard T qi = ÐKik -∂------, (10) to the impact of optical or electron-beam excitation can xk be written as where Kik is thermal-conductivity tensor of the body ( , , ) ( , ) ∆ ( , , ) S R T t = S0 U T + S R T t , (6) with initial strains. where the vectors R and U are defined by the compo- It should be noted that, according to the field theo- nents (X , X , X ) and (U , U , U ), respectively; S (U, ries of nonlinear mechanics [20, 39], the thermal-con- 1 2 3 1 2 3 0 ductivity tensor K generally has the structure T) is the entropy density of the body with initial strain; ik ∆ ( δ ) and S(R, T, t) is the variation of the entropy density of Kik = K0 ik + K1Uik + K2 IilUlk , (11) the body as a result of an external excitation. In its thermodynamic sense, the density of ther- where the quantities K0, K1, and K2 should be consid- moelastic energy of the body (1) represents the varia- ered as scalar functions of the three invariants of the ini- tion of the free energy density ∆F of the body. There- tial strain tensor, temperature, and convolution of its fore, the entropy variation ∆S is given by first derivatives with respect to coordinates with the initial strain tensor [20, 39]. ∂∆F ∂W ∆S = Ð------= Ð------T- = γ ∆u , (7) Using equations (5)Ð(10), as well as the outlined ∂T ∂T ik ik rules, we can derive the thermal conductivity equation for a body with initial strain in the starting coordinate ∆ where uik = uik Ð Uik. system, i.e., in the Lagrangian representation. On the

The entropy density S0 of the body with initial strain basis of the stated results, this equation can be obtained and given volume is related to temperature by the from (5) in the form expression ∂∆ ρ ∂∆ ρ T γ 0[( β )δ β ] uik ρ 0cv ---∂------+ 0T--ρ--- 1 + 0U pp ik + 1Uik -----∂------S0 = Cv lnT, (8) t t (12) ∂ ∂∆ ρ where Cv is heat capacity of the body with initial strain.  T 0 = ∂ Kik --∂------ + --ρ---W. To transform the coordinates in equation (5) into xi xk those of an unstrained body, we can use the relationship ρ ρ Note that all quantities in Eq. (12) are assumed to be dV = 0dV0 [38], where dV0 is a volume element of the expressed in the Lagrangian coordinates corresponding body in the unstrained state. In addition, it is necessary to the position of points in the unstrained body. The to use the relation [39] between integrals over the vol- influence of strains on the heat capacity of the body is ume for thermal fluxes in strained and unstrained bod- usually insignificant and is discussed elsewhere [40]. In ies the second term on the left-hand side of equation (12), ∆ the strains ui in this study are considered to be small. ∫dV div Q = ∫dV 0 div q, (9) Therefore, restricting ourselves to quantities of the first

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1028 MURATIKOV, GLAZOV ≅ order of smallness in (12), we may assume that T T0. V(ω) ρ Density in Eq. (12) can be determined (correct to the β β terms of the first order of smallness in the initial strains (ω) 1 + 0U pp + 1U33 (14) = V 0 ------, ρ ≅ ρ 3 3/2 in the body) from the relationship (1 Ð Ull) 0. If, in 1 + A [1 + 2lU + (4m' + n')U ] Eq. (12), we set β = β = 0, ρ = ρ , and K = Kδ , this pp 33 0 1 0 ik ik ω equation transforms into the well-known equation of where V0( ) is the photoacoustic piezoelectric signal thermal conductivity for isotropic bodies [35]. from the sample in the absence of initial strains [41], The second term on the left-hand side of Eq. (12) l m n describes dilatation processes in a solid during defor- l' = ------, m' = ------, n' = ------, ρ 2 ρ 2 ρ 2 mation. They are vital in the process of emission of 0cl 0cl 0cl infrared radiation by solids under deformation [12] but µ usually exert little influence on photoacoustic and elec- 3K = 4 and cl = ------ρ------. tronÐacoustic effects in solids. Therefore, when consid- 3 0 ering the photoacoustic and electronÐacoustic pro- Equation (14) shows that the dependence of the pho- cesses in solids, this term is usually ignored. toacoustic piezoelectric signal on internal stresses According to the formulated conditions, the deter- stems from the nonlinear elastic and inelastic properties mination of heat fluxes is generally an intricate mathe- of the medium. In deducing Eq. (14), no special matical problem. A further simplification can be carried assumptions were made about the nature of the interac- out on the basis of additional experimental data con- tion of radiation with the material and, hence, this result cerning the behavior of thermal properties of a material is also valid for an electronÐacoustic signal, provided it subjected to mechanical stresses. In [22Ð25], several is due to the thermoelastic mechanism. Note that, for photothermal methods were used to show that residual β 1 = 0, expression (14) transforms into the correspond- stresses only slightly affect the thermophysical proper- ing result for a photoacoustic signal obtained in ties of silicon nitride ceramics. [26, 32, 33]. In this study, investigations of the influence of inter- In the experimental part of this work, silicon nitride nal stresses on the photoacoustic effect are restricted ceramics were studied. The key feature of the experi- solely to silicon nitride ceramics. Therefore, according mental approach is the use of several photothermal and to experimental results obtained in [22Ð25], we assume photoacoustic techniques in combination with the opti- that the appearance of internal stresses in the sample cal deflection method [42]. This approach makes it pos- does not result in a noticeable variation of the thermal sible to measure independently the thermal, elastic, and parameters. In addition, we assume that the exciting thermoelastic parameters of the sample. In turn, the radiation is time-modulated by a harmonic law and the optical deflection method is very useful for determining sample surface is irradiated uniformly. Then, if radia- the surface relief of the sample, in particular, the position is absorbed in the near-surface region of the sam- tions of plastically deformed areas. ple, the non-steady-state component of temperature inside the sample will be defined by the expression We studied experimentally the influence of internal strains on the thermal and elastic parameters of samples σ ω ∆T(z, t) = ∆T eÐ z + i t, (13) of silicon nitride ceramics that were indented using the s Vickers method. The Vickers indentation method is one where σ = (1 + i) ω/2κ , κ is thermal diffusivity of the of the most reliable and reproducible methods for ∆ inducing stresses and cracks in samples [43]. The gen- sample, Ts is amplitude of the temperature oscillations ω eral structure of the region formed in the ceramics in at the sample surface, and is the circular frequency of the vicinity of the Vickers indentation site is shown in modulation of the exciting radiation. Fig. 2. Results of the photodeflection and photoreflec- Knowledge of the temperature distribution in the tion measurements, as well as the obtained thermal- sample allows displacements of the sample surface wave images of Vickers indentations in silicon nitride under the action of exciting radiation to be determined ceramics were considered in detail previously [22Ð25]. from Eq. (3) and the photoacoustic signal to be found Therefore, in this paper, only the optical deflection with a piezoelectric recording method. A detailed images of regions in the vicinity of Vickers indentations scheme for these calculations was described elsewhere are presented. A representative example of such images [26, 32, 33]. Therefore, passing over the details of the is shown in Fig. 3. According to these images, the max- calculations, we immediately present the net result for imum surface deformation was attained in the immedi- a photoacoustic signal with piezoelectric signal record- ate region of plastic deformation in the vicinity of an ing for the case when the thermoelastic energy of the indentation site. For instance, the surface deformation sample is defined by Eq. (1). Then, for the photoacous- amounted to 3Ð4 µm for an indentation load of 196 N. tic signal V(ω) in a sample with uniform initial strains As for the surface-relief changes in the ceramics away (i) (i) Ui = A xi (A are arbitrary coefficients characterizing from the indentation site, they are related to surface the strain in different directions), we obtain the follow- strain resulting from the formation of lateral surface ing result: cracks in the ceramics during Vickers indentation [43].

TECHNICAL PHYSICS Vol. 45 No. 8 2000 THEORETICAL AND EXPERIMENTAL STUDY 1029

(a) (b)

1 1

2 4

Fig. 2. Schematic representation of the impression and crack system in ceramics: (1) are subsurface horizontal cracks, (2) is the region of plastic deformation, (3) are radial cracks, (4) are subsurface medial cracks; (a) is the top view and (b) is the cross section. (‡)

(b)

Fig. 4. Region of silicon nitride ceramics near a Vickers indentation. The image was obtained by the photoacoustic method with a piezoelectric recording of the signal. Load is Fig. 3. Optical deﬂection image of the region of silicon 98 N. Modulation frequency of exciting radiation is 113 kHz. nitride ceramics near a Vickers indentation. Load is 98 N. (a) is an old indentation; the image size is 240 × 320 µm. (b) The image size is 500 × 500 µm. is a new indentation; the image size is 380 × 400 µm.

In this study, comprehensive investigations of the At the same time, a noticeable difference in the indentations were carried out by the photoacoustic structures of the old and new indentations should be technique with a piezoelectric signal recording method. noted. This difference becomes especially pronounced In particular, we studied both old indentations about if we superpose corresponding photoacoustic and opti- ﬁve years of age and new ones that were made several cal deﬂection images represented in the form of con- days before the measurements. In Fig. 4, images of tour plots. Examples of such superposed images are both old and new indentations obtained by the photo- shown in Fig. 5. It can be seen that, for new indenta- acoustic technique are presented. First, we note that these images are very similar in structure to those tions, the regions with maximum values of photoacous- obtained for Vickers-indented ceramics by the elec- tic signals are positioned near the tips of radial cracks. tronÐacoustic method [27]. This result supports the the- For old indentations, the regions with maximum values oretical conclusion that the photoacoustic and elec- of photoacoustic signals are appreciably closer to their tronÐacoustic signals are formed by the same mecha- centers and correspond approximately to the intersec- nisms. tion site of radial and subsurface lateral cracks. We also

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1030 MURATIKOV, GLAZOV

µm (a) note that the maximum values of photoacoustic signals 300 are somewhat higher in the latter case.

250 These features are presumably explained by the fact that gradual stress relaxation occurs with time in the vicinity of radial cracks. At the same time, near the 200 intersection site of radial and subsurface lateral cracks, the latter gradually develop and approach the surface. 150 This is consistent with the well-known result for silicon nitride ceramics: under the effect of stresses, the 100 motion of dislocations does not occur in its volume; at the same time, a noticeable motion of the dislocations 50 over the fracture surfaces can be observed [44]. To analyze this subject in greater detail, we studied 0 50 100 150 200 µm the dependence of the maximum value of the photoa- µm (b) coustic signal (PAS) near the tips of both types of 400 cracks on the indentation load. The results of these measurements are shown in Fig. 6. They show that, as 350 the load increases, the photoacoustic signal at first grows and then gradually levels off. This dependence of 300 the photoacoustic signal may be related to the fact that an increase in the crack length promotes an increase in 250 stresses at its tips [45]. Therefore, under a certain indentation load, the stress at the tips of the cracks 200 reaches a limiting value corresponding to the ultimate strength of the material. After that, a further increase in stresses becomes impossible and the photoacoustic sig- 150 nal levels off. 100 Expression (14) and the data shown in Fig. 6 allow estimation of the upper limit values of the parameters β β 50 0, l, l, m, and n. It should be taken into account that the maximum possible strains in present-day ceramics that do not result in their failure meet the condition 0 50 100 150 200 250 300 350 µm ≤ Umax 1% [46]. Then, in order to explain the variation in the photoacoustic signal at a level of 10% according Fig. 5. Superposition of two images of a silicon nitride β ≅ β ≅ ceramic region in the vicinity of a Vickers indentation; the to Eq. (14), we should assume that 0 1 10 and l, images were obtained by the photoacoustic and optical tech- ≤ ρ 2 niques and are shown as contour plots. (a) and (b) are the m, and n 10 0 cl . Theoretical estimation of the coef- same as in Fig. 4. ficient β [34] yielded a value on the order of unity. However, only metals were considered [34]. As for the PAS, arb. units quantities l, m, and n, the values obtained from estima- 0.6 tions with Eq. (14) are in the correct range of the Mur- naghan parameters [36]. At the same time, Eq. (14) makes it possible to explain qualitatively the difference 0.4 in the magnitudes of photoacoustic signals observed near the tips of the radial and subsurface lateral cracks. For subsurface lateral cracks, the principal component of the strain tensor that contributes to the photoacoustic 0.2 signal is the component U33. For radial cracks, the principal strains near the crack tips are related to U11 and U22. The strains U11 and U22 also result in strains of the 0 100 200 U33 type, but they are related by the Poisson ratio. The Load, N Poisson ratio for silicon nitride ceramics is approximately equal to 0.26 [44]. Therefore, according to Fig. 6. Dependence of the maximum value of the photoa- expression (14), the strains near the tips of radial cracks coustic signal (PAS) obtained by a piezoelectric recording on the load value. Modulation frequency of exciting radia- exert somewhat less influence on the photoacoustic sig- tion is 98 kHz. (᭹) correspond to old Vickers indentations. nal compared with the case of the subsurface lateral (᭡) correspond to new Vickers indentations. cracks.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 THEORETICAL AND EXPERIMENTAL STUDY 1031

Thus, the theoretical and experimental results 23. K. L. Muratikov, A. L. Glazov, D. N. Rose, and J. E. Du- reported here show that the photoacoustic and elec- mar, Pis’ma Zh. Tekh. Fiz. 24 (21), 40 (1998) [Tech. tronÐacoustic effects in solids with internal stresses can Phys. Lett. 24, 846 (1998)]. be described in the context of the nonlinear theory of 24. K. L. Muratikov, A. L. Glazov, D. N. Rose, and J. E. Du- elasticity and thermoelasticity. An important advantage mar, in Proceedings of the 10th International Confer- of the photoacoustic and electronÐacoustic methods for ence “Photoacoustic and Photothermal Phenomena,” detecting internal stresses is their versatility, since, 10 ICPPP, Rome, 1998; AIP Conf. Proc. 463, 187 according to the presented theory, their internal-stress (1998). sensitivity is based on the general linear elastic proper- 25. K. L. Muratikov, A. L. Glazov, D. N. Rose, and J. E. Du- mar, in Proceedings of the 3rd International Congress on ties of solids. Thermal Stresses, Thermal Stresses’99, Cracow, 1999, p. 669. REFERENCES 26. K. L. Muratikov, Pis’ma Zh. Tekh. Fiz. 24 (13), 82 1. I. A. Birger, Residual Stresses (MAShGIZ, Moscow, (1998) [Tech. Phys. Lett. 24, 536 (1998)]. 1963). 27. J. H. Cantrell, M. Qian, M. V. Ravichandran, and 2. A. Ya. Aleksandrov and M. Kh. Akhmetayanov, Pola- K. W. Knowles, Appl. Phys. Lett. 57, 1870 (1990). rization Optical Methods in the Mechanics of Deform- 28. Q. R. Yin, B. Y. Zhang, Y. Yang, et al., Prog. Nat. Sci., able Bodies (Nauka, Moscow, 1974). Suppl. 6, 115 (1996). 3. Y. H. Pao and W. Sachse, Phys. Acoust. 17, 61 (1984). 29. F. Jiang, S. Kojima, B. Zhang, and Q. Yin, Jpn. J. Appl. Phys., Part 1 37, 3128 (1998). 4. B. Eigenmann, B. Scholtes, and E. Macherauch, Mater. Sci. Eng. A 118, 1 (1989). 30. Y. Gillet and C. Bissieux, in Proceedings of the 10th International Conference “Photoacoustic and Photo- 5. M. R. Daymond, M. A. M. Bourke, R. B. von Dreele, thermal Phenomena,” 10 CPPP, Rome, 1998, p. 131. et al., J. Appl. Phys. 82, 1554 (1997). 31. L. M. Lyamshev, Radiation Acoustics (Nauka, Moscow, 6. M. Ceretti, C. Braham, J. L. Lebrun, et al., Exp. Tech. 20 1996). (3), 14 (1996). 32. K. L. Muratikov, in Proceedings of the 10th Interna- 7. É. S. Gorkunov and M. V. Tartachnaya, Zavod. Lab. 59 tional Conference “Photoacoustic and Photothermal (7), 22 (1993). Phenomena,” 10 CPPP, Rome, 1998, pp. 478Ð480. 8. C. C. H. Lo, F. Tang, Y. Shi, et al., J. Appl. Phys. 85, Part 33. K. L. Muratikov, Zh. Tekh. Fiz. 69 (7), 59 (1999) [Tech. 2A, 4595 (1999). Phys. 44, 792 (1999)]. 9. T. Iwaoka, S. Yokogama, and Y. Osaka, Jpn. J. Appl. 34. R. I. Garber and I. A. Gindin, Fiz. Tverd. Tela (Lenin- Phys. 24, 112 (1984). grad) 3, 176 (1961) [Sov. Phys. Solid State 3, 127 10. M. Bowden and D. J. Gardiner, Appl. Spectrosc. 51, (1961)]. 1405 (1997). 35. L. D. Landau and E. M. Lifshitz, The Theory of Elasti- 11. C. N. Xu, T. Watanabe, M. Akiyama, and X. G. Zheng, city (Nauka, Moscow, 1987). Appl. Phys. Lett. 74, 2414 (1999). 36. A. I. Lur’e, Nonlinear Theory of Elasticity (Nauka, Mos- cow, 1980). 12. A. K. Wong, R. Jones, and J. G. Sparrow, J. Phys. Chem. Solids 48, 749 (1987). 37. T. Tokuoka and M. Saito, J. Acoust. Soc. Am. 45, 1241 (1969). 13. G. N. Cheryshev, A. L. Popov, V. M. Kozintsev, and 38. G. I. Petrashen’, Propagation of Waves in Anisotropic I. I. Ponomarev, Residual Stresses in Deformable Solids Elastic Media (Nauka, Leningrad, 1980). (Nauka, Moscow, 1996). 39. A. Green and J. Adkins, Large Elastic Deformation and 14. M. J. Pechersky, R. F. Miller, and C. S. Vikram, Opt. Non-Linear Continuum Mechanics (Clarendon, Oxford, Eng. 34, 2964 (1995). 1960; Mir, Moscow, 1965). 15. C. S. Vikram, M. J. Pechersky, C. Feng, and D. Engel- 40. K. Y. Kim, Phys. Rev. B 54, 6245 (1996). haupt, Exp. Tech. 20 (6), 27 (1996). 41. W. Jackson and N. M. Amer, J. Appl. Phys. 51, 3343 16. M. Kasai and T. Sawada, Springer Ser. Opt. Sci. 62, 33 (1980). (1990). 42. M. Rosete-Aquilar and R. Díaz-Uribe, Appl. Opt. 32, 17. R. M. Burbelo, A. L. Gulyaev, L. I. Robur, et al., J. Phys. 4690 (1993). (Paris) 4, C7-311 (1994). 43. R. F. Cook and G. M. Pharr, J. Am. Ceram. Soc. 73, 787 18. D. N. Rose, D. C. Bryk, G. Arutunian, et al., J. Phys. (1990). (Paris) 4, C7-599 (1994). 44. D. G. Miller, C. A. Andersson, S. C. Singhal, et al., 19. H. Zhang, S. Gissinger, G. Weides, and U. Netzekmann, AMMRC CTR 76-32 (Watertown, Mass., 1976), Vol. IV. J. Phys. (Paris) 4, C7-603 (1994). 45. L. I. Sedov, A Course in Continuum Mechanics (Nauka, 20. M. Qian, Chin. J. Acoust. 14 (2), 97 (1995). Moscow, 1970; Wolters-Noordhoff, Groningen, 1971Ð 1972), Vol. 2. 21. R. M. Burbelo and M. K. Zhabitenko, Prog. Nat. Sci., Suppl. 6, 720 (1996). 46. T. J. Mackin and T. E. Purcell, Exp. Tech. 20 (2), 15 (1996). 22. K. L. Muratikov, A. L. Glazov, D. N. Rose, et al., Pis’ma Zh. Tekh. Fiz. 23 (5), 44 (1997) [Tech. Phys. Lett. 23, 188 (1997)]. Translated by M. Lebedev

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1032Ð1041. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 77Ð86. Original Russian Text Copyright © 2000 by Gorbachev, Zatevakhin, Krzhizhanovskaya, Shveœgert.

SOLID-STATE ELECTRONICS

Special Features of the Growth of Hydrogenated Amorphous Silicon in PECVD Reactors Yu. E. Gorbachev, M. A. Zatevakhin, V. V. Krzhizhanovskaya, and V. A. Shveœgert Institute for High-Performance Computing & Data Bases, St. Petersburg, 198005 Russia E-mail: [email protected] Received February 10, 1999; in ﬁnal form, October 27, 1999.

Abstract—A model has been developed to calculate the growth parameters of silicon ﬁlms in diode- and triode- type PECVD reactors and to analyze the factors affecting the deposition of silicon-containing radicals. Mech- anisms of the effect of diluting silane with molecular hydrogen on the ﬁlm growth process have been explained. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION In the framework of the model proposed in [3], the authors carried out a computational investigation of the One of the most widespread methods of growing physical and chemical processes occurring in a purely hydrogenated amorphous silicon films is the technol- silane plasma of a diode reactor and performed a ogy based on deposition from the gas phase of silicon- detailed analysis of the role of various components and containing radicals produced by the decomposition of the effect of the system parameters on film growth for silane by an RF discharge plasma in PECVD reactors. major operating regimes. It was found that, at low pres- At present, various types of such systems are used. In sures, molecular hydrogen accumulates in the chamber, one of these, the diode system, the substrate serves as so that silane ceases to be the only carrier component. one of the electrodes. In another system, belonging to a In addition, in practice, a technology based on diluting numerous class of systems with remote plasma and silane with hydrogen is often used, which saves silane usually called a triode system, the substrate is placed and makes the production of silicon films ecologically outside the plasma discharge (PECVD reactors). This system was proposed and thoroughly investigated in cleaner. Under these conditions, the model of [3] is [1]. By increasing the growth rate and film quality inadequate. through optimizing the parameters of PECVD reactors, In this study, a model that takes into account the significant technological advantages can be attained. changes in silane concentration during film growth is Therefore, various models are being proposed, which, presented. Transport due to diffusion was determined with differing degrees of detail, describe the processes using, instead of fixed coefficients, those calculated by in the growth chamber and allow improvements to be Wilke’s formula representing a first approximation of made in the reactor design. diffusion in a multicomponent medium. It has been The first comprehensive model of the growth pro- shown that further elaboration of the description of dif- cess of hydrogenated amorphous silicon film was pro- fusion processes does not improve the accuracy of the posed in [2]. calculations of the component fluxes onto the film surface. Calculations of the plasma discharge were carried A rather simple and effective model developed later out using a fluidic model [5] affording a means of cal- [3] is suitable for making prompt estimates of all basic culating electron densities. Expressions have also been process parameters with sufficient accuracy. Apart from obtained for the rate constants of reactions initiated by silane (SiH4), 12 major chemical components of the electron impact in silaneÐhydrogen mixtures. Hence, mixture are included in this model: SiHn, n = 1Ð3; H; the processes in reactors could be studied in a wider H2; Si2Hn, n = 3Ð6; Si3H8; Si2 H*6 ; and Si2 H**6 (aster- range of parameters and the processes taking place in isks denote the excited electronic state). Recently, a silane diluted with molecular hydrogen could be simu- similar model but with a different set of chemical com- lated. The proposed model describes the processes in ponents and a much more elaborate presentation of the triode-type reactors with the discharge zone some dis- electronic subsystem was proposed [4]. The authors tance away from the substrate. Finally, in this work, the carried out quite detailed measurements of the system effect of the reactor volume, which in almost every real parameters and provided the experimental data neces- installation does not coincide with the discharge zone sary for verifying the models. volume, has been accounted for in a correct manner.

SPECIAL FEATURES OF THE GROWTH 1033

MATHEMATICAL MODEL dependence in such a way as to circumvent calculations In the proposed model of a PECVD reactor, the gas- of the corresponding electron temperatures, it was sug- phase processes in its chamber are described by one- gested in [3] to use the following relation between the dimensional equations of chemical kinetics, in which, reaction constants kr and the ionization constants ki: as estimated in [3], only transfer processes by diffusion ⑀ ⑀ 0 ( 0) r/ i are taken into account: kr/kr = ki/ki , (4) ∂n ∂  ∂n  where ⑀ and ⑀ are the thresholds for the chemical reac------i = D ------i + F . (1) r i ∂t ∂x i ∂x  i tion and ionization, respectively [11], and the super- script “0” refers to certain conditions for which the val- Here, ni is the number density of the ith component and ues of the constants are known (below, these conditions Fi is the source term describing chemical transforma- are called “basic”). tions (for more detail, refer to [3]). The effective diffu- Then, assuming that the plasma as a whole is neu- sion coefficient of the ith component Di is calculated by tral, that the withdrawal rate of electrons is equal to the Wilke’s formula [6], commonly used in practical calcu- + lations of multicomponent gas mixtures: withdrawal rate of ions (mainly SiH2 ions), and that Ð1 the carrier mixture consists of silane and molecular Nk hydrogen, the following expression can be obtained ( ) using formula (2): Di = n Ð ni ∑ nk/Dki , (2) k = 1, 0 k ≠ 1 k n nSiH ----i = -----t ------4 0 0 nSiH where n is the total number density of the mixture. ki nt 4 The ambipolar diffusion coefficient D is calculated (5) ki ( 0 0 0 0 ) 0 2 nSiH /DSiH SiH + nH /DH SiH l  by the formula of the molecular kinetic theory of gases × ------4------4------2------2------2------2------. [7] with use of the LennardÐJones 6Ð12 potential (n /D + n /D )  l  SiH4 SiH4 SiH2 H2 H2 SiH2 3 µ 2 0 T /2 ki cm D = 0.002628------. (3) Here, nt and nt are the numerical densities of the mix- ki σ 2 Ω (1.1)*( ) s p ki ki T k*i ture under present and basic conditions, respectively. µ For the basic conditions, a silaneÐhydrogen mixture is Here, ki = mkmi/(mk + mi) is the effective mass of the chosen with partial hydrogen and silane pressures kth and ith species, m is the molecular weight of the ith i 0 0 0 σ σ σ pH = pSiH = 0.125 torr at the temperature T = 520 K species, ki = ( k + i)/2 is the effective collision diam- 2 4 0 eter, T * = kT/ε is the characteristic temperature, ε = and the interelectrode separation l = 2.5 cm, the reason ki ki ki being that the constants for these conditions were deter- ε ε k i is a parameter of the potential for intermolecular mined in [2]. interaction (potential well depth), k is the Boltzmann Ω (1.1)* Ω constant, and ki (T k*i ) is a reduced -integral of Table 1. Parameters of the LennardÐJones potential collisions for mass transfer normalized by the Ω-inte- σ ε Component i, Å i/k, K gral of the model of hard spheres. Values of the T k*i function for a wide range of characteristic temperatures SiH4 4.084 207.6 SiH 3.943 170.3 T * can be found in monograph [7]; in the calculations, 3 ki SiH 3.803 133.1 approximate formulas given in [8] were used. Values of 2 σ ε SiH 3.662 95.8 the LennardÐJones potential parameters i and i bor- rowed from [9] are given in Table 1. H 2.5 30.0 In [10], experimental determinations of the ambipo- H2 2.915 59.7 lar diffusion coefficients of silyl in silane and molecular Si2H6 4.828 301.3 hydrogen were carried out and found to be equal to * 140 ± 30 and 580 ± 140 cm2/s, respectively, at the tem- Si2H6 4.828 301.3 perature 320 K and pressure 1 torr. Under these condi- Si H** 4.828 301.3 tions, formula (3) gives the values of 111 and 536 cm2/s, 2 6 respectively, in good agreement with measured data. Si2H5 4.717 306.9 To finalize the formulation of the problem, we need Si2H4 4.601 312.6 to know the dependence of the rate constants of chem- Si2H3 4.494 318.2 ical reactions initiated by an electron beam on pressure Si H 5.562 331.2 p and the interelectrode separation l. To define this 3 8

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1034 GORBACHEV et al.

Table 2. List of chemical reactions and reaction constants magnitude: Reac- ∂n s c Constants*, i i i tion Reactions ÐDi------= ------ni. (6) cm3/s ∂x 2(2 Ð s ) no. i Here, ∂/∂x is a normal derivative at the outside of the R1 SiH + e SiH + H 3.000 × 10Ð11 4 3 π × Ð10 surface, ci = 2 2kT/ mi is the thermal velocity, and si R2 SiH4 + e SiH2 + 2H 1.500 10 × Ð12 is the deposition coefficient of the ith species. Values of R3 SiH4 + e SiH + H + H2 9.340 10 these coefficients are given in [2]: si = 0.15 for SiH3 and R4 SiH + e SiH + H 7.190 × 10Ð12 4 2 2 Si2H5; si = 1 for SiH, SiH2, Si2H3, and Si2H4; and si = 0 × Ð12 R5 H2 + e 2H 4.490 10 for all other components. The electron density in the × Ð10 R6 Si2H6 + e SiH3 + SiH2 + H 3.720 10 interelectrode gap is calculated in the fluidic approxi- R7 Si H + e Si H + 2H 3.700 × 10Ð11 mation using a model described in detail in [5]. In the 2 6 2 4 simulations, it is assumed that the hot electrons causing × Ð12 R8 SiH4 + H SiH3 + H2 2.530 10 dissociation are available only in the interelectrode gap. R9 SiH + SiH Si H* 1.000 × 10Ð11 Away from the gap, the energy of electrons diffusing 4 2 2 6 from the discharge zone drops exponentially and × Ð12 R10 SiH4 + SiH Si2H3 + H2 1.700 10 becomes insufficient for initiating chemical reactions. × Ð12 R11 SiH4 + SiH Si2H5 2.500 10 In this sense, a technological system in which the sub- R12 SiH + Si H SiH + Si H 5.000 × 10Ð13 strate is outside the discharge zone can be considered as 4 2 5 3 2 6 a system with remote plasma. R13 SiH + Si H Si H 1.000 × 10Ð10 4 2 4 3 8 For numerical solution of this problem, a method × Ð10 R14 SiH3 + H SiH2 + H2 1.000 10 described in [3] was employed. × Ð10 R15 SiH3 + SiH3 SiH4 + SiH2 1.500 10 Basic chemical reactions and their rate constants are R16 SiH + SiH Si H** 1.000 × 10Ð11 listed in Table 2. In addition, silane and disilane pyrol- 3 3 2 6 ysis reactions are taken into consideration, which, as is × Ð10 R17 SiH3 + Si2H5 SiH4 + Si2H4 1.000 10 proved by the calculations, make an insignificant con- × Ð11 R18 SiH3 + Si2H5 Si3H8 1.000 10 tribution to the total chemical balance. × Ð12 R19 SiH3 + Si2H6 SiH4 + Si2H5 3.270 10 × Ð13 R20 SiH2 + H SiH + H2 7.960 10 2. ANALYTICAL RESULTS × Ð10 R21 SiH2 + Si2H6 Si3H8 1.200 10 Asymptotic analysis based on the difference of scale × Ð12 R22 Si2H3 + H2 Si2H5 1.700 10 of different processes gave some analytical formulas R23 Si H + H + SiH + SiH 1.000 × 10Ð10 and an explanation of the interrelationship between 2 4 2 4 2 conditions in the chamber and various process parame- × Ð10 R24 Si2H5 + H Si2H4 + H2 1.000 10 ters. In [3], such relationships are given only for the × Ð12 R25 Si2H6 + H SiH4 + SiH3 7.160 10 diode system, whereas in this work, such an analysis × Ð11 R26 Si2H6 + H Si2H5 + H2 1.430 10 has been made for the triode system as well. The entire calculation area 0 < x < L was divided into two zones: R27 Si H* Si H + H 5.000 × 106 sÐ1 2 6 2 4 2 0 < x < l, where the density ne of electrons initiating the R28 Si H* + M Si H + M 1.000 × 10Ð10 chemical reactions was taken equal to that in the inter- 2 6 2 6 electrode gap, and l < x < L, where plasma is penetrat- (M, a species colliding with Si H* ) ing, but the energy of its electrons is not enough to ini- 2 6 tiate the reactions. It is assumed that in the latter zone × 7 Ð1 R29 Si2H6** SiH4 + SiH2 2.300 10 s of size d = L Ð l, only monomolecular reactions or reactions involving molecular collisions are proceeding. ** × 7 Ð1 R30 Si2H6 SiH4 + H2 2.300 10 s Estimates of basic processes similar to those in [3] × Ð11 R31 Si3H8 + H Si2H5 + SiH4 2.170 10 produced the following simplified equation for the × Ð11 behavior of silyl: R32 Si3H8 + SiH3 Si4H9 + H2 1.000 10 0 2 2 * Data are given for the interelectrode separation l = 2.5 cm, partial DSiH d nSiH /dx = 2k8nHnSiH Ð k32nSiH nSi H , (7) pressures of molecular hydrogen and silane p0 = 0.125 torr, and 3 3 4 3 3 8 temperature T 0 = 520 K. where ki is the constant of the ith reaction (i is the reaction number in Table 2); condition (6) with a deposition coefficient s = 0.15 is taken as the boundary condi- Other finalizing relationships for system (1) are SiH3 boundary conditions (3) accounting for a concentration tions for this equation. jump near the deposition surface, which, under the con- A corresponding analysis for atomic hydrogen, ditions specified above, can be of considerable which is the major initiator of silyl production, leads to

TECHNICAL PHYSICS Vol. 45 No. 8 2000 SPECIAL FEATURES OF THE GROWTH 1035 the following balance equation in the discharge region Because the cycle in not closed, silylyl is withdrawn 0 < x < l: through a reaction with an effective constant keff = ≅ 2 2 k9k13 n /(k23 n + k13 n ) k9/(n /n + 1). The ÐD d n /dx = 2k n n Ð k n n . (8) SiH4 H2 SiH4 H2 SiH4 H H 2 e SiH4 8 SiH4 H ≅ last approximate equality is a consequence of k13 k23. Outside the discharge region at l < x < L the behavior Therefore, the behavior of silylyl can be described by of atomic hydrogen is described by the same equation, the equations except that the first term on the right-hand side should 2 2 be omitted (n = 0); i.e., hydrogen gets outside the dis- ÐDSiH d nSiH /dx = k2nenSiH Ð kefnSiH nSiH , e 2 2 4 2 4 (12) charge zone only due to diffusion. Solving these equa- 0 < x < l, tions with boundary conditions (6) in the absence of deposition (s = 0) and with the condition of smooth 2 2 2 H ÐD d n /dx = k n Ð k n n , joining of solutions at the boundary of the zones (at SiH2 SiH2 15 SiH3 ef SiH2 SiH4 (13) x = l), we get an analytical expression for the atomic l < x < L. hydrogen concentration profile: By solving these equations with corresponding ( ) 2k2ne sinh d/LH x  boundary conditions, an analytical expression for the n = ------1 Ð ------cosh------, H k  sinh(L/L ) L  (9) flux of silylyl to the surface has been obtained, but, in 8 H H view of its clumsiness, we do not present it here. 0 < x < l, ( ) 2k2ne sinh l/LH L Ð x < < 3. RESULTS AND DISCUSSION nH = ------(------) cosh------, l x L. (10) k8 sinh L/LH LH (1) Comparison with experiment. Comparison with experimental data in the literature is difficult, because Substitution of expressions (9) and (10) into equa- the set of parameters describing the system available tion (7) yields an analytical dependence on the system there is far from complete. For this reason, we chose the parameters of the flow of silyl towards the surface: results given in [4] for comparison. In this work, exhaustive information on the experimental conditions 2k n n L 2 2 e SiH4 3 2 l LH 1 Γ = ------2sinh ------+ ------and results of numerical calculations are described. SiH3 sinh(L/L ) 2L 2 2 sinh(L/L ) 3 3 L3 Ð LH H The experimental installation was a chamber with a volume of 101 and contained two electrodes of radius 8 cm located 2.7 cm apart. For this configuration, the ×  l L L l d  cosh----- sinh------Ð cosh----- sinh------Ð sinh------ , (11) ratio of the reaction volume to the total volume of the L3 LH L3 LH LH chamber is Rν = 0.054 (for more details, see Section 3.5). L = D /(k n ), L = D /(k n ). The experiments were carried out with a mixture of H H 8 SiH4 3 SiH3 32 Si3H8 silane and hydrogen in equal molar concentrations with a flow rate equal to 3.6 l/h under normal atmospheric Here, LH and L3 are reaction-diffusion lengths for atomic hydrogen and silyl, respectively, introduced in conditions; the temperature in the reactor was 400 K. [3] and defined as the distance which a diffusing spe- A program for computing parameters of the RF dis- cies will travel before entering into a chemical reaction. charge between two plane-parallel electrodes was used For l L (d 0), expression (11) describes a to calculate the electron densities [5]. The main prob- diode system. lem was to obtain a good estimate of the discharge power. In [4], from a comparison of the self-bias poten- To conclude this section, we consider the behavior tial computed in simulations of argon plasma based on of silylyl. In the discharge zone, silylyl is produced corresponding experiments, it was determined that the mainly via the decomposition of silane by electron power directed to the discharge amounted to 50% of the impact, reaction R2, and through reaction R23, total power. The same conclusion was made from a whereas, outside this zone, it is produced in reactions comparison of the calculated concentrations of H2 and R15 and, again, R23. Withdrawal of silylyl through SiH in the hydrogenÐsilane plasma with experimental reactions in both cases depends on its reaction with 4 data. Therefore, in calculations performed for the sug- silane (R9). The Si2H4 radical is an intermediate prod- 2 uct of the cycle of fast chemical reactions R9, R27, and gested model, we used the power value of 0.05 W/cm R23; therefore, its concentration is close to the equilib- proposed in [4]. Figure 1 shows the dependence of the film growth rium value n e = k n n /(k n + k n ) [3]. Si2H4 9 SiH2 SiH4 23 H2 13 SiH4 rate on the discharge frequency at a constant pressure of If it were not for reaction R13 diverting part of the sili- 16 Pa in the reactor. It is seen that the results are in good con atoms to higher silanes, reactions R23 and R9 agreement with experimental data up to a frequency of would compensate one another and the only mecha- 30 MHz. At frequencies higher than 30 MHz, the cal- nism for silylyl to escape from the zone would be its culated curve tends to level off. The same tendency, diffusion to the surface for deposition and pumping out. although more gradual, was noted in [4]. At the same

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1036 GORBACHEV et al.

Rd, nm/s P, Pa 0.5 10 2

8 0.4 1

0.3 6

1 3 2 4 0.2 3 4 5 6 2 0.1

0 20 40 60 80 0 20 40 60 80 F, MHz F, MHz Fig. 2. Partial pressures of molecular hydrogen (1, 3, 5) and Fig. 1. Film growth rate as a function of discharge frequency silane (2, 4, 6) as a function of discharge frequency. at a constant pressure in the chamber p = 16 Pa. (1) Experi- (1, 3) Experiment [4], (2, 4) calculation [4], and (5, 6) this ment [4], (2) calculation [4], and (3) this work. work. time, in [4], the experimental dependence of the growth istic of real experimental setups were chosen [1, 2]: the rate on frequency was approximately linear. To explain temperature in the reaction chamber was taken equal to this effect, additional investigations are needed; their 520 K, the pressure was 0.25 torr, and the interelectrode possible directions have been proposed in [4]. distance was l = L = 2.5 cm. Under these conditions, the From the calculations of the partial pressures of reactor parameters corresponded to a characteristic molecular hydrogen and silane as functions of the dis- pumping time of τ = 1s at a pressure of 1 torr (for more charge frequency given in Fig. 2, it is seen that the data details on this characteristic, see Section 3.4). The cal- obtained using the approach developed in this study are culations were carried out for the pressure range 0.02 < in better agreement with experiment than the calcula- p < 1 torr. All the results given in this section refer to tion results obtained in [4]. the case of pure silane as the carrier. (2) Speciﬁc features of chemical kinetics at low One of main distinctions of the considered pro- pressures. With the model described in Section 1, cal- cesses is a considerable rise in the concentration of culations are possible of the processes at low pressures, atomic hydrogen with decreasing pressure. There are a at which silane decomposition becomes noticeable and number of reasons for this. First of all, as seen from for- the concentration of molecular hydrogen can no longer mula (9), the concentration of atomic hydrogen in the be considered small. Below, we discuss in detail the particular case of homogeneous discharge is given by qualitative differences of these processes from those formula taking place at high pressures. n = 2k n /k ; (15) Dependence of the electron density in the discharge H 2 e 8 zone on the process parameters was described by a for- i.e., it depends only on the electron density. This formula [3] derived for the case of constant discharge mula is also in good agreement with the calculation power results. At the same time, as seen from formulas (4), 0 0 0 (5), and (14), the electron density and the constant k n ( p l) = n p l / p l, (14) 2 e SiH4 e SiH4 SiH4 rise rapidly with decreasing silane pressure, so that the 0 concentration of atomic hydrogen increases by nearly where ne is the electron density under basic conditions, ﬁve orders of magnitude as the pressure in the chamber taken equal to 5 × 108 cmÐ3. This formula is true for decreases from 1 to 0.02 torr and the relative silane con- pressures that are not very low, but it was used for the centration decreases to 0.3 as a result of its decomposi- qualitative analysis in the entire range of pressures. For tion. The increase in atomic hydrogen concentration other key parameters of the problem, values character- causes qualitative changes in various processes.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 SPECIAL FEATURES OF THE GROWTH 1037

The behavior of other components with variations in reaction chamber begins. Other parts of this classifica- pressure does not change compared with the description remain qualitatively unchanged (see details in [3]). tion in [3], except for the fact that the concentration of (3) Effect of diffusion coefficients. The use of for- Si2H6 drops at pressures below 0.1 torr. The same effect mula (2) in calculating the diffusion coefficients does is observed for Si3H8. This is due to the appreciable not significantly change the main results (such as the drop in silane concentration and the increase in atomic growth rate and film composition) compared with cal- hydrogen concentration, the main reactants determin- culations in which these coefficients were assumed to ing the formation and decomposition of these compo- be equal to the coefficients of ambipolar diffusion of nents. The increase in atomic hydrogen concentration the respective components in silane, as was done in [3]. also affects the balance and role of different reactions This result appears to be quite evident at high pressures, in silyl decomposition. Thus, at low pressures, a notice- where the concentration of molecular hydrogen is able growth in the role of the decomposition process in rather low and the diffusion coefficients calculated by the silyl balance is observed; at 0.02 torr, it becomes formula (2) are close to the corresponding ambipolar dominant and prevails over deposition. Note, however, coefficients. However, at low pressures, these values that this result is entirely a property of the model, are distinctly different. because the use of formula (14) at very low pressures This fact suggests that the concentration profiles of gives values of the electron density that are too high. the depositing components adjust in such a way as to At pressures below about 0.08 torr in silyl decompo- maintain the productionÐdeposition balance, irrespec- sition, the reaction SiH4 + Si3H8 Si4H9 + H2 gives tive of the particular values of the diffusion coefficients (the rate of chemical decomposition of silyl and silylyl way as the dominant process to reaction SiH3 + at low pressures is small). SiH3 SiH4 + SiH2; and at pressures of about 0.02 torr, to the reaction between silyl and atomic For a more detailed investigation of this issue, cal- hydrogen, which begins to make a noticeable contribu- culations using the above approach were carried out for tion to the production of silyl. As before, silyl forma- a mixture of silane and molecular hydrogen at equal tion is completely dominated by the reaction between partial pressures and constant electron density 5 × silane and atomic hydrogen. 108 cmÐ3. In this case, the influence of the diffusion Contributions of individual reactions to the silylyl coefficients is more pronounced. For the same purpose, balance remain largely unchanged, but its production at a simplified formula was used to calculate reaction rate low pressures increases so much that most of the silylyl constants as functions of pressure [3]: ⑀ ⑀ is deposited instead of being decomposed. Because of k l0 p0 2 r/ i all these changes, the contribution of silylyl to film ----r = ------. 0  lp  growth, which, with decreasing pressure, increases at a kr greater rate than that of silyl, becomes dominant at a pressure of 0.02 torr. This should cause appreciable Under these conditions, the diffusion coefficients modifications in the film structure. calculated by Wilke’s formula differ from the ambipolar diffusion coefficients by a factor of 1.6Ð1.7. Still, at Taking into account silane decomposition at low low pressures where the silylÐsilylyl balance is domi- pressures revealed further details of the division [3] of nated by deposition, the fluxes of these components to all components into three groups: stationary, nonsta- the substrate are practically coincident in the two cases, tionary, and quasi-stationary. It was found that SiH can despite distinctly different concentration profiles. At be considered strictly stationary only at high pressures. high pressures, silylyl is spent mainly through reactions In addition, at high pressures, stationary components in the bulk and the values of fluxes obtained with the also include silylyl, the balance of which is maintained use of constant ambipolar diffusion coefficients are by two reactions with silane: SiH4 + e SiH2 + 2H + e somewhat less than those given by Wilke’s formula; and SiH + SiH Si H* , with the latter reaction however, this difference is a mere 10% for silyl and 4 2 2 6 20% for silylyl, which is much less than the difference keeping the concentrations of Si2 H*6 and Si2H4 (which in the coefficients, and the contribution of silylyl to film growth under these pressures is insignificant. is produced via decomposition of Si2 H*6 ) stationary. At low pressures, these last two components go over to the To check this, a series of calculations was carried quasi-stationary group because of the appreciable out for varying values of the diffusion coefficients. In decomposition of silane, resulting in an increase in one of these, the diffusion coefficient for silyl was electron density and reaction rate constants. This is assumed to be twice as large. The film growth rate was especially noticeable in the behavior of atomic hydro- almost unchanged, although at a pressure of 1 torr, the gen, whose concentration at high pressure is stationary silyl concentration dropped by a factor of 1.5. and adequately described by formula (15). At low pres- This fact is explained as follows. Because the dom- sures, the deviation from stationarity caused by these inant contribution to film growth comes from the flux of processes makes atomic hydrogen nonstationary and silyl, the growth rate can be analyzed using expression appreciable accumulation of this component in the (11). At d = 0, this expression is significantly simpli-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1038 GORBACHEV et al. fied, reducing to Γ = 2k n n L tanh(L /2L ). For SiH3 2 e SiH4 3 3 with a decreasing flow rate, the film growth rate increased by 30%. This was due to an increase in both real values of the diffusion coefficient, L/2L3 < 0.7 throughout the range of pressures studied and, there- the electron density and the reaction rate constants with Γ a decreasing partial concentration of silane. The rela- fore, the dependence of on L3 is weak. An SiH3 tive contribution of silylyl increased by about the same increase in the diffusion coefficient only reduces the amount. This was caused by an increase in the concen- dependence of the flux of silyl on its value. Thus, silyl tration of molecular hydrogen and, subsequently, a deposition is virtually unaffected by diffusion pro- greater contribution to the recovery of the silylyl con- cesses. centration from the cyclic reaction SiH4 + SiH2 A similar result was also obtained for silylyl. The Si H* Si H + H SiH + SiH . twofold increase in the silylyl diffusion coefficient did 2 6 2 4 2 4 2 not produce any significant changes. At low pressures, (5) The effect of chamber volume. The problem con- the silylyl concentration decreased, but the modified sidered above is closely related to the effect of the ratio profiles ensured exactly the same value of its flux. At of the reaction volume to the entire chamber volume, higher pressures, the silylyl balance is completely con- because circulation of the components throughout the trolled by chemical reactions; therefore, its concentra- volume can change the course of the processes. tion (as well as concentrations of all other components) To investigate this aspect in a one-dimensional case, remained the same, but the flux increased somewhat the following approach was suggested: the area under (by 37% at p = 1 torr). However, at this pressure, the study was expanded by adding a region R such that the contribution from SiH2 to film growth is quite small and ratio Rν would have been equal to the total chamber vol- cannot change the final result. ume. The calculations for a fixed parameter τ and vari- It can be concluded from the above that further able Rν gave estimates of the effect of circulation. The specifying the description of diffusion transport cannot results described below were obtained for a total cham- significantly change the results, especially in the case ber volume of 2 l and τ = 0.26 s; other parameters were of a pure silane plasma. the same as before. (4) Effect of the flow rate. The flow rate of the mix- Changing the parameter Rν from 1 to 0.2 resulted in ture through the working reactor volume is one of the an insignificant (about 25%) increase in the rates of important factors affecting the film growth parameters, film growth and silyl deposition. This increase was and, at the same time, it can be easily varied. In [3], an caused by increased concentrations of silane and silyl in approach that allows one to take into account the effect the reaction volume because of the influx of SiH4 from of the flow rate using a simple one-dimensional model the chamber volume. At the same time, silylyl deposition was suggested. In this approach, the initial steady-state remained nearly constant, because some increase in its problem is replaced with a nonstationary problem, production due to the reaction of silylyl decomposition τ which is to be solved for a time period equal to the (R15) was offset by reaction SiH4 + SiH2 and deposition characteristic time of transit of the mixture in the reac- occurs only from a thin near-wall layer, where the con- tor working volume. Decomposition of a considerable centration of silyl is low. Figures 3Ð5 illustrate some dis- part of silane occurring at low flow rates and a signifi- tinctive features of these processes. cant increase in the molecular hydrogen concentration In Fig. 3, profiles of typical representatives of the render the model suggested in [3] inapplicable for three classes of components are shown: nondepositing studying the role of the flow rate in a relatively wide and slowly reacting (H2), nondepositing and rapidly range of this parameter. reacting (H), and depositing and rapidly reacting Using the model presented in this study, the effect of (SiH2). It is seen that the concentrations of the latter the flow rate has been analyzed for the example of pure two components in the reaction volume are practically silane for τ ranging from 0.13 to 2.6 s, which, at a pres- independent of the chamber volume. They are deter- sure of 0.25 torr, corresponds to a change in the cham- mined by reactions proceeding in the discharge zone, ber volume from 1 to 20 l for the fixed flow rate of and circulation has little effect on them. The concentra- silane of 5 l/h at atmospheric pressure. Values of the tion of molecular hydrogen, on the contrary, is constant other parameters were the same as in the previous sec- throughout the chamber volume and is appreciably tions, and the electron density was calculated by for- reduced by circulation. mula (14). The small reaction-diffusion length for silylyl with As in [3], variation of the flow rate was found to respect to the chamber dimensions is the reason for the affect, first of all, the concentration of Si2H6: it weak influence of the chamber volume on the balance increased by an order of magnitude as the flow rate was and flows of this radical (Figs. 4, 5). Only some reduced by the same factor. At the same time, the con- increase in its production due to a higher silane concen- centration of Si3H8 increased by a factor of about 5. On tration can be noted; this increase is compensated by its the whole, variation of the flow rate corresponding to more intensive (for the same reason) decomposition, so the considered range of τ caused only insignificant vari- that flows onto the substrate turn out to be about the ations of the characteristics of the growing film. Thus, same.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 SPECIAL FEATURES OF THE GROWTH 1039

–3 13 –3 –1 C, cm B(SiH2), 10 cm s 1014

7.5 1012 SiN2 H F H2 5.0 A 10 10 SiH2 –D H F H2 A 2.5 –D 108

0 106

–2.5 104 0 5 10 x, cm 0 5 10 x, cm

Fig. 3. Concentration proﬁles of silylyl and molecular and Fig. 4. Silylyl balance at different values of the ratio of reac- atomic hydrogen at different values of the ratio of reaction tion volume to the total chamber volume Rν: F, production; volume to the total chamber volume Rν equal to 0.2 (curves) A, decomposition; D, diffusion; Rν = 0.2 (curves) and and 1 (dots). 1 (dots).

(6) The effect of diluting silane with molecular fact that with an increasing concentration of H2 and a hydrogen. In real equipment, a mixture of silane and decreasing concentration of silane, the fraction of sily- molecular hydrogen is widely used. To evaluate the lyl being deposited rises significantly. At the same time, effect of silane dilution, calculations have been carried the intensity of the above-mentioned cyclic reaction out for a silane content in this mixture varying from 100 rises, leading to the recovery of the SiH2 concentration. to 20%, the other parameters having the same values as before. The electron density was determined using an 12 –2 –1 RF discharge model [5]. Calculations for a discharge f(SiH2), 10 cm s power of 0.025 W/cm2 have shown that, in these condi- 10.0 tions, the average electron density rises nearly linearly from 4.3 × 108 to 6.7 × 108 cmÐ3 as the silane content in 7.5 the mixture is reduced. Rv = 0.2 Rv = 1 As the silane content at constant pressure is reduced, 5.0 the diffusion coefficients of the components, with the exception of H2, increase by a factor of 2Ð3. Variation 2.5 of the concentrations of all reaction products under these conditions is shown in Fig. 6. The film growth 0 rate rises, and the relative contribution of silylyl to overall deposition becomes larger (Fig. 7). The growth rate rises, despite falling silane concentration, because –2.5 of an increase in the total production of depositing components due to higher rate constants of reactions –5.0 initiated by electron impact (see expressions (4) and (5)) and increased electron density. In addition, the net –7.5 result of lowering the concentration and the simultaneous increase in the diffusion coefficients is that the –10.0 greater part of the produced silyl is deposited; this is 0 5 10 x, cm true for other depositing components, except Si2H3 and Si2H4, which are effectively decomposed by molecular Fig. 5. Silylyl fluxes at different values of the ratio of the hydrogen. The behavior of silylyl is explained by the reaction volume to the total chamber volume Rν.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1040 GORBACHEV et al.

C/Ctot (7) Specific features of the triode system. All the 100 results described above were obtained for a diode system in which the discharge zone is located between two 10–1 electrodes, one of which serves as a substrate for the growing film. One alternative technology is the so- SiH –2 3 called triode system, in which the discharge zone is 10 SiH2 SiH separated from the substrate. Formulation of the 10–3 H respective problem has been described in Sections 1 H2 and 2. 10–4 Si2H6 Si2H5 In order to investigate the effect of the separation of 10–5 Si2H4 electrodes from the substrate, calculations of film Si2H3 growth from a pure silane plasma were carried out for Si H 10–6 3 8 the same problem parameters as previously used (with SiH4 the exception of parameter L = 5 cm, with parameter l/L 10–7 varying from 0.4 to 1). The electron densities determined using the RF discharge model [5] were varied 10–8 from 6.8 × 108 to 4.9 × 108 cmÐ3. The calculation results 0.2 0.4 0.6 0.8 1.0 obtained using the above analytical expressions for the P /P SiH4 tot silyl flux (11) and the concentration profile of atomic hydrogen (9), (10) are in good agreement with numeri- Fig. 6. Concentrations of components as a function of the cal simulation results, as seen in Figs. 8 and 9, respec- silane fraction in the initial silaneÐhydrogen mixture at con- tively. stant pressure. As a result of varying the parameter l/L in the range specified above, the film growth rate (at a constant dis- Also worth noting is the considerable reduction in charge power) decreased by only half. The cause of the the Si H concentration with dilution (Fig. 6). The rea- slow decrease in the growth rate with increasing sepa- 3 8 ration between the electrode and the substrate is the son is that alongside the drop in production by the reac- increase in electron densities and rate constants of reaction SiH4 + Si2H4, it is decomposed via the reaction tions initiated by electron impact as the interelectrode with atomic hydrogen, whose concentration rises at a separation l is reduced, which partially compensates high rate. The result is a drastic drop in Si4H9 produc- decomposition of silicon-containing components out- tion, and the greater part of the silicon produced in the side the discharge zone. In addition, a drastic reduction reaction is deposited. in the contribution to the film growth from silylyl

Rd, A/min 2 Rd, A/min 10 140 R 101 120 tot SiH3 SiH Rd tot 100 2 100 SiH3 SiH3, (11) 80 SiH2 60 10–1

40 10–2 20

0 10–3 0.2 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 P /P SiH4 tot l/L

Fig. 7. Film growth rate and contributions of silyl and silylyl Fig. 8. Film growth rate and contributions of silyl and silylyl to it as a function of the fraction of silane in the initial as a function of the ratio of discharge zone to the total reac- silaneÐhydrogen mixture at constant pressure. tor zone. Numerical and analytical solutions (formula (11)).

TECHNICAL PHYSICS Vol. 45 No. 8 2000 SPECIAL FEATURES OF THE GROWTH 1041

10 –3 CH, 10 cm It has been shown that the widely used technique of 4.0 diluting silane with molecular hydrogen both increases the growth rate and reduces production of higher silanes, making this technology more economical and ecologically clean. Moreover, dilution increases the 3.0 contribution of silylyl to film growth, appreciably affecting its properties. Analysis of a widely used reactor system has been 2.0 carried out. Numerical simulation has shown that the effective decomposition of silylyl outside the discharge zone reduces its contribution to film growth as the substrate is displaced away from the discharge. 1.0 The analytical expressions obtained for silyl and silylyl fluxes and for the profile of atomic hydrogen closely approximate the results of numerical computations and can be used for making the corresponding 0 1 2 3 4 5 estimations. x, cm

Fig. 9. Atomic hydrogen profiles for the triode reactor sys- ACKNOWLEDGMENTS tem (l/L = 0.4): solid curve, analytical solution; dashed curve, numerical solution. This work was partly supported by the Russian Foundation for Basic Research, project no. 97-01- 00233 and INTAS, project no. 96-0235. should be noted, starting as soon as the electrode is moved away from the substrate (Fig. 8). This is especially remarkable in view of the fact that at the same REFERENCES time, its concentration and total production are rising. 1. O. A. Golikova, Fiz. Tekh. Poluprovodn. (Leningrad) 25, Two main reasons for this effect can be indicated. The 1517 (1991) [Sov. Phys. Semicond. 25, 915 (1991)]. first is that the reaction-diffusion length of silylyl is 2. M. J. Kushner, J. Appl. Phys. 63, 2532 (1988). much less compared with silyl; therefore, the bulk 3. Yu. E. Gorbachev, M. A. Zatevakhin, and I. D. Kagano- decomposition of silylyl is more efficient. The second vich, Zh. Tekh. Fiz. 66 (12), 89 (1996) [Tech. Phys. 41, reason is the very different mechanisms of their pro- 1247 (1996)]. duction. The major source of silyl is the reaction 4. G. J. Nienhuis, W. J. Goedheer, et al., J. Appl. Phys. 82, between silane and atomic hydrogen, which, as seen in 2060 (1997). Fig. 9, diffuses quite intensively beyond the discharge 5. M. I. Zhilyaev, V. A. Shveœgert, and I. V. Shveœgert, Prikl. zone. On the other hand, the main contribution to sily- Mekh. Tekh. Fiz. 35 (1), 13 (1994). lyl production comes from the reaction between silane 6. C. R. Wilke, Chem. Eng. Prog. 46 (2), 95 (1950). molecules themselves; therefore, production of SiH 2 7. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Mole- outside the discharge is small. cular Theory of Gases and Liquids (Wiley, New York, 1954; Inostrannaya Literatura, Moscow, 1961). CONCLUSIONS 8. I. P. Ginzburg, Friction and Heat Transfer in Moving Gas Mixtures (Leningrad. Univ., Leningrad, 1975). In this work, numerical investigations of the growth of hydrogenated amorphous silicon films have been 9. J. Perrin and O. Leroy, Contrib. Plasma Phys. 36 (3), 3 (1996). carried out under various conditions in the growth chamber. 10. N. Itabashi, K. Kamo, and N. Nishawaki, Jpn. J. Appl. Phys. 28, 325 (1989). Comparison with experimental data has demonstrated the ability of the model to predict the film 11. J. Perrin, J. Phys. D 26, 1662 (1993). growth rate and concentrations of individual components with reasonable accuracy. Translated by B. Kalinin

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1042Ð1044. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 87Ð90. Original Russian Text Copyright © 2000 by Baraban, Konorov, Malyavka, Troshikhin.

OPTICS, QUANTUM ELECTRONICS

Electroluminescence of Ion-Implanted SiÐSiO2 Structures A. P. Baraban, P. P. Konorov, L. V. Malyavka, and A. G. Troshikhin St. Petersburg State University, Institute of Physics, Universitetskaya nab., St. Petersburg, 198904 Russia Received October 6, 1999

Abstract—Speciﬁc features of the electroluminescence of ion-implanted (Ar ion implantation in oxide layer bulk) and ion-synthesized (SIMOX technology) SiÐSiO2 structures were studied. The electroluminescence from the electrolyte-insulator-semiconductor system was registered in the 250Ð800 nm range at room temperature. It has been found that implantation increases the concentration of centers already present in the oxide layer bulk and creates new luminescent centers. The nature and the models of the centers are discussed. © 2000 MAIK “Nauka/Interperiodica”.

Lately, ion implantation in solid-state structures has implanted into the silicon bulk at 650°C, followed by been widely used both in fundamental and applied annealing for 6 hours at 1320°C and the etching away research. This is due to a range of opportunities offered of the outer silicon layer, which resulted in the forma- by implantation, which produces controllable transfor- tion of a 390-nm-thick silicon dioxide layer. Type 3 mations of both the structure and atomic and electronic structures were prepared by the thermal oxidation of properties of solids by way of introducing selected ions KÉF-5 (100) silicon in a humid oxygen ambient at to a specified depth (depending on the impinging ion 950°C and the subsequent implantation of argon ions. energy). This allows the type and level of doping in the The ion energy, 130 keV, was chosen with a view to semiconductor near-surface region to change, to form having the maximum density of implanted ions in the submerged oxide layers in the bulk of silicon (SIMOX middle of the oxide layer, and the doses were in the technology), and to significantly modify the electro- 1013Ð3.2 × 1017 cmÐ2 range. Some of these structures physical properties of dielectric layers at the semicon- were subjected to a fast thermal (radiative) annealing ductors surface [1]. One of the main problems involved (FTA) at temperatures of 500Ð1100°C for 10 s. in the utilization of ion implantation is to determine the properties and nature of the defects introduced in the Figure 1 shows EL spectra of type 1 structures pre- process. The development of a fast, nondestructive pro- pared with the use of different technologies. Earlier, cess control method is an important fundamental and characteristic emission bands at energies 1.9, 2.3, 2.7, practical task. Electroluminescence (EL) is a highly 3.3, 3.8, and 4.6 eV, corresponding to various types of reliable method of studying SiÐSiO2 structures. It defects located in the oxide layer and at the SiÐSiO2 detects the presence and identifies the types of defects interface, were identified in this spectrum [2]. It has in the oxide layer, as well as their concentration and spatial distribution, by measuring the spectral distribu- L, arb. units tion and intensity of characteristic bands [2]. The purpose of this work is to study the specific fea- 1 tures of EL in ion-implanted and ion-synthesized SiÐ 6 SiO2 structures and apply EL to the study of defects 2 produced by implantation. The EL in the 250Ð800 nm range was registered 4 from an electrolyteÐinsulatorÐsemiconductor system 2 [2], with the sensitivity significantly enhanced due to utilization of a field electrode transparent in this spec- 1 2 tral range (1 pH water solution of Na2SO4). The measurements were carried out in a photon-counting regime at 293 K. In this paper, three types of SiÐSiO structures were 2 032 4 5 6 investigated. Type 1 structures were produced by the hν, eV thermal oxidation of silicon (grade KDB-10 (100)) in accordance with the usual technologies. Type 2 struc- Fig. 1. EL spectra of standard SiÐSiO2 structures produced tures were made by SIMOX technology. Oxygen ions by thermal oxidation of KDB-10 (100) silicon: (1) in water of energy 190 keV and dose 1.8 × 1018 cmÐ2 were vapor at 950°C; (2) in dried oxygen at 1100°C.

ELECTROLUMINESCENCE OF ION-IMPLANTED SiÐSiO2 STRUCTURES 1043 been shown that the luminescence centers responsible L, arb. units for the red EL band at 1.9 eV are silanol groups local- 12 ized in the outer SiO2 layer and that their concentration and spatial distribution correlate with those of the electron capture centers [2]. Three-coordinated silicon localized near the SiÐSiO2 interface is the center 8 responsible for the EL band at 2.3 eV, which is due to intracenter electron transitions in silicon atoms induced by hot electrons created in the oxide layer [2]. Lumi- 4 nescence centers responsible for emission in the ultraviolet (UV) spectral range are localized near the SiÐ SiO2 interface [2], but the nature of these centers in these particular structures has not yet been conclusively 0 ascertained. 2 3 4 5 The EL band at 2.7 eV is the most interesting. In hν, eV nonimplanted structures, it is observed only in the range of electric fields producing impact ionization of Fig. 2. EL spectra of SiÐSiO2 structures formed using the SiO2 matrix. Its localization is not fixed, being coin- SIMOX technology. cident with the position of the impact ionization probability maximum in SiO2 (determined by the electric L, arb. units L, arb. units field strength in the oxide layer), where the concentra- 4 tion of dangling SiÐO bonds is a maximum. 1.8 The EL spectrum of type 2 structures is presented in 6 Fig. 2. The band at 1.9 eV is not observed in this spec- 1 trum, indicating the absence of silanol groups in the oxide layer. This is explained by the specific technol- 1.2 4 ogy of these structures, which eliminates penetration of 2 the fragments of water molecules (hydrogen, hydroxyl groups) into the oxide layer. 0.6 At the same time, an intense band at 2.7 eV is 2 observed in the EL spectrum for electric fields not causing impact ionization. Its intensity is practically not 3 affected by etching half the oxide layer away. This 0 0 means that the centers that are responsible for this band are mainly localized near the SiÐSiO2 interface. The 2 3 4 5 energy position and the root-mean-square variance of hν, eV this band (0.35 ± 0.05 eV) are the same as in nonimplanted structures. An emission band at 4.4 ± 0.1 eV Fig. 3. EL spectra of argon-implanted SiÐSiO2 structures. occurs in the UV range of the spectrum, which is well D = (1) 1013; (2) 1014; (3) 1016; (4) 3.2 × 1017 cmÐ2. fitted by the Gaussian distribution function with a root- mean-square variance of 0.4 ± 0.1 eV. ther increase of the dose to 1017 cmÐ2 causes a drop in EL spectra of type 3 structures are presented in the intensity of these bands, which is succeeded by Fig. 3. There are three well-resolved EL bands in the × 17 Ð2 spectrum. Two of the bands have the same energy posi- rapid growth at a dose of 3.2 10 cm . The ratio of tions at 1.9 and 2.7 eV and fit the same Gaussian distri- intensities of the bands at 2.7 and 4.4 eV stays constant. butions as in the initial structures, but the EL band at To determine the localization area of the lumines- 2.7 eV is excited by fields lower than those causing the cence centers, we monitored the intensity variations of impact ionization in the oxide layer. As in type 2 struc- the bands at 1.9, 2.7, and 4.4 eV resulting from the pro- tures, one EL band in the UV range is observed at gressive etching away of the oxide layer. It has been energy 4.4 eV. It fits the Gaussian function with the found that the centers emitting at 1.9 eV are localized same variance. in the outer region of the oxide layers, as in the nonim- It has been found that the intensity of the EL band at planted structures. The localization area of the centers 1.9 eV grows with an increase in the implantation dose broadens with the implantation dose. Centers emitting for doses up to 1014 cmÐ2, levels off at 1015 cmÐ2, and at 2.7 and 4.4 eV are located mainly in the 30Ð140 nm drops down as the dose is further increased. The EL region from the SiÐSiO2 interphase boundary. Their bands at 2.7 and 4.4 eV are seen in spectra of the struc- distribution maximum is closer to the Si boundary than tures implanted with doses 1013 cmÐ2 and higher. A fur- that of the implanted argon ions. Increase of the dose

TECHNICAL PHYSICS Vol. 45 No. 8 2000

1044 BARABAN et al. causes the smearing of the spatial distribution of lumi- average distance between neighboring implantation- nescence centers and the shifting of its maximum induced defects accompanied by structural changes in towards the Si boundary. The intensities of all EL bands the oxide layer due to the change in the silicon-oxygen observed decrease as a result of FTA as the anneal tem- bond angle, the formation of =Si=Si=-type defects in perature is increased. the SiO2 bulk, and so on. The drastic enhancement of To reveal the nature of the defects responsible for these EL bands at the highest doses suggests the the EL at 2.7 and 4.4 eV, let us discuss the processes renewed generation of defects like two-coordinated sil- occurring in the silicon oxide layers during argon ion icon, which this time takes place in a dielectric layer implantation. The bombarding argon ions lose their whose structure and properties differ from those of the energy in the oxide layer through interactions with its initial SiO2 matrix. This argumentation is supported by atomic and electronic subsystems, thereby generating the nonmonotonic behavior of the intensity of a photo- the structural defects of the SiO2 matrix and electron- luminescence band at 2.7 eV with an increasing dose of hole pairs, respectively. silicon implanted in the oxide layer, as well as with an increasing concentration of excess silicon in SiO2 films In the outer region of the oxide layer, approximately [5, 6]. equal amounts of energy are dissipated due to interactions with the atomic and electronic subsystems. In the The abovementioned peculiarities of the EL band at bulk of the oxide layer, the more probable channel of 1.9 eV (due to silanol groups in the oxide layer) display energy dissipation for the bombarding argon atoms is the considerable transformation of the atomic structure the interaction with the atomic subsystem. Therefore, in the outer portion of the oxide layer as a result of ion the greatest amount of dangling SiÐO bonds occurs implantation. The growth of the concentration and near the maximum of the distribution of implanted localization area of the silanol groups is attributed to argon ions, and the Si and O atoms are found displaced the generation of these defects in the retardation tracks deeper into the oxide layer. The estimated displacement of bombarding argon ions due to the breaking of the SiÐO from the distribution maximum of implanted argon bonds and their subsequent capture by hydrogen and/or ions is 80Ð170 nm for oxygen atoms and 30Ð70 nm for by hydroxyl groups localized in the oxide layer or dif- Si atoms. As a consequence, two nonstoichiometric fusing from the ambient. This process results in the growth of the concentration of electron traps in the regions of SiOx with x > 2 and x < 2 are formed in the bulk of the oxide layer [3]. The region rich in oxygen outer region of the oxide layer and depends on the (x > 2) is located closer to the boundary with silicon argon implantation dose. due to the larger displacement of oxygen atoms com- In the case of the SiÐSiO2 structures produced by pared with that of Si atoms. SIMOX technology, the EL bands at 2.7 eV and 4.4 eV In the oxygen-deficient region of SiO , defects of are also attributed to defects of the two-coordinated sil- 2 icon (=S :) type. However, these defects in such struc- the two-coordinated silicon atom (O2 = Si:) type are tures are due to the formation of silicon clusters near formed; these we consider to be the EL centers respon- the SiÐSiO interface during the fabrication process. sible for the 2.7- and 4.4-eV bands. Such defects are 2 generated in ion implantation as two of the SiÐO bonds Thus, EL provides a quick and efficient (a spectrum in a silicon-oxygen tetrahedron become broken and the can be taken in less than 10 minutes) means of studying broken bonds spatially separated because of the greater ion-implanted SiÐSiO2 structures and getting informa- inward displacements of oxygen atoms in the oxide tion about their structural and electrophysical proper- layer. According to data in the literature [4], in defects ties. of this type, two radiative transitions are possible with energies 2.7 and 4.4 eV, their excitation energy being REFERENCES equal to approximately 5 eV. These luminescence centers are excited due to interaction with hot electrons 1. H. Ryssel and I. Ruge, Ionenimplantation (Teubner, whose mean kinetic energy in the electric fields of the Stuttgart, 1978; Nauka, Moscow, 1983). 2. A. P. Baraban, V. V. Bulavinov, and P. P. Konorov, Elec- range used for exciting EL in SiO2 is just around 5 eV. The decrease in the intensity of the 2.7- and 4.4-eV tronics of SiO2 Layers on Silicon (Leningrad. Gos. Univ., Leningrad, 1988). bands caused by exposure to external factors (FTA) is explained by the partial restoration of the broken bonds 3. B. Garido, J. Samitier, S. Bota, et al., J. Non-Cryst. in the oxide layer bulk and a decrease in the concentra- Solids 187, 101 (1995). tion of two-coordinated silicon. The nonmonotonic 4. L. N. Skuja, A. N. Streletsky, and A. B. Pakovich, Solid behavior of the intensity of 2.7- and 4.4-eV EL bands State Commun. 50, 1069 (1984). with an implantation dose is of special interest. This 5. W. Skorupa, R. A. Yankov, et al., Appl. Phys. Lett. 68, fact indicates that the variation of the concentration of 2410 (1996). =Si-type defects is nonmonotonic. The lowering of the 6. D. J. DiMaria, J. R. Kirtley, et al., J. Appl. Phys. 56, 401 concentration of these defects with an increasing dose (1984). is apparently due to the partial restoration of the broken bonds because of the considerable reduction of the Translated by S. Egorov

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1045Ð1050. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 91Ð96. Original Russian Text Copyright © 2000 by Parygin, Vershubskiœ, Kholostov.

ACOUSTICS, ACOUSTOELECTRONICS

Acousto-Optical Filtering Using Short Acoustic Trains V. N. Parygin, A. V. Vershubskiœ, and K. A. Kholostov Moscow State University, Vorob’evy gory, Moscow, 119899 Russia Received June 7, 1999

Abstract—The problem of controlling the characteristics of collinear acousto-optical filters is studied experimentally. It is shown that the use of an acoustic pulse in the form of a step-function signal makes it possible to considerably reduce the side lobes of the instrument function of a collinear acousto-optical filter. The changes in the shape of the transmission curve that occur as a result of the variations in the number of acoustic pulses simultaneously propagating in the crystal and in their duration are investigated. An experimental study of a spectrum consisting of two close lines is performed using a tunable collinear acousto-optical filter. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION THEORY OF THE COLLINEAR ACOUSTO- OPTICAL INTERACTION Tunable collinear acousto-optical filters are among the most promising optoelectronic devices. These fil- The propagation of acoustic trains in a crystal is ters have a narrow transmission band that can be elec- accompanied by an elastic strain wave described by the tronically tuned within an octave [1Ð3]. In the litera- strain tensor Slma(x, y, z, t). The strain wave changes the ture, two types of acousto-optical filters are described: refractive indices of the medium. This change is related collinear and noncollinear ones [4, 5]. Collinear filters, to the elasto-optical effect described by the tensor pjklm. which usually have a narrow transmission band, are The change in the components of the permittivity ten- characterized by higher selectivity, and this property is sor of the medium under the effect of the acoustic field important for the spectral analysis of optical radiation ∆ε 2 2 3 has the form jk = ÐN N ∑ p S . Here, Nj and for the channel multiplexing purposes. j k lm, = 1 jklm lm and Nk are the main refractive indices of the medium, In our recent publications [6Ð9], it was shown theo- and j, k, l, and m are the coordinate indices. retically that, in collinear acousto-optical filters, an The diffraction of light by sound is described by the electronic tuning of not only the central frequency but wave equation also the width of the transmission band and the shape of the transmission curve is possible when the filter is 1 ∂ 2 1 ∂ 2 rotrot E + ------εˆ E =Ð------∆εˆ()aE , (1) controlled by a pulsed acoustic signal rather than by a 2 ∂ 2 0 2 ∂ 2 continuous one. In this case, the duration of the control c t c t pulse determines the transmission band, and the form where E(x, y, z, t) is the electric vector of the light wave; of the pulsed signal governs the shape of the transmis- εˆ sion curve of the collinear filter. Our previous experi- 0 is the permittivity of the medium in the absence of ∆ε ε ments [10] demonstrated the possibility of controlling sound; ˆ is the variation of ˆ0 in the presence of the characteristics of a collinear filter by using single sound, this variation being proportional to the acoustic acoustic trains. strain; and a(x, y, z, t) is the distribution of the acoustic strain in the medium. The latter quantity can be repre- If we use a sequence of short acoustic trains with a sented in the form relatively small distance between them (less than the crystal length), several acoustic pulses may simulta- axyzt(),,, =aWxyz(),, Vxt(), 0 (2) neously propagate in the cell. In this case, the transfer × []()Ω characteristic of the cell has the form of a series of sev- expjKxÐ t + c.c., eral narrow peaks. The spacing between the peaks where a0 is the amplitude of the wave at the cell input depends on the number of peaks in the crystal, and the (at x = 0), and K and Ω are the wave number and the fre- number of peaks depends on the duration of each indi- quency of the acoustic train, respectively. In the general vidual pulse. case, the functions W(x, y, z) and V(x, t) describe the In this paper, we describe the experimental study of spatial distribution of the amplitude and the time enve- the dependence of the characteristics of a collinear lope of the trains, respectively. acousto-optical filter on the form and duration of suc- It should be noted that, for light beams of finite cessive acoustic trains. dimensions, we have rotrotE ≠ ∇2E, because graddivE

1046 PARYGIN et al. cannot be considered as zero even in an isotropic ( , , , ) ( , ) { 2 } Ud ky kz x t = fd x t exp jxky /2kd medium. (7) × { ( 2 2) 2( ) } It is known that, in the case of a collinear diffraction, exp Ð ky + kz rd x /4 . the polarization of the diffracted light beam is orthogonal to the polarization of light incident on the acousto- Here, ft(x, t) and fd(x, t) are the light beam amplitudes optical cell. Therefore, in the region of the interaction depending on x and t and taken along the axis at ky = of light and sound, it is quite natural to represent the kz = 0; ri(x) (where i = t, d) are the slowly varying radii light beam as a sum of the transmitted and diffracted of the Gaussian beams. According to our previous cal- beams with orthogonal polarizations: culations [7], the variations of the radii with the coordi- ( , , , ) [ ( ω )] nate are negligibly small. In addition, the radii of the E = etEt x y z t exp j kt x Ð tt (3) incident rt and diffracted rd light beams are related as ( , , , ) [ ( ω )] + ed Ed x y z t exp j kd x Ð dt , 2 2 rd = rt/ 1 + rt /R . where et and ed are the polarization directions of the transmitted and diffracted beams, respectively; ω and Let us substitute expressions (6) and (7) into equa- t tions (4) and (5). The integrals over dK and dK in the ω are the frequencies of the transmitted and diffracted y z d right-hand members of equations (4) and (5) can be cal- light beams; and Et(x, y, z, t) and Ed(x, y, z, t) are their culated analytically. Neglecting the variations of the amplitudes slowly varying along the x-axis. light beam radii, we obtain a system of two first-order We substitute the vector E determined by expression differential equations describing the collinear diffrac- (3) into equation (1). We use the diffraction condition tion of light by Gaussian acoustic trains in the case of a ω ω Ω ω d = t + and equate the amplitudes of exp{j tt} and strong acousto-optical interaction: ω ∂2 ∂ 2 exp{j dt}. Neglecting the quantities Et/ x and ∂ { η} ∂2 ∂ 2 ∂ ∂ ∂ ∂ f d exp Ð jx Ed/ x , as well as Et/ t and Ed/ e, we apply a two------= Ð jq f (x, t)V(x, t)------, (8) ∂x 1 t ( ) ρ 2 dimensional Fourier transform to both sides of equation 1 Ð jDx + t (2) in the yz plane. Then, omitting the mathematical ∂ f exp{ jxη} transformations analogous to those described in [6Ð9] ------t = Ð jq f (x, t)V(x, t)------, (9) and taking into account the orthogonality of the polar- ∂x 2 d ( ) ρ 2 1 + jDx + d izations et and ed, we arrive at a system of scalar equa- ρ tions for the spectra of the transmitted and diffracted where i = ri/R. light Ut and Ud: The use of finite beams in describing the diffraction ∂ 2 of light by sound makes it possible to determine the Ud ky ( η ) ( , ) efficiency of the diffraction not through the ratio of the j--∂------+ ------Ud = q1 exp Ð j x V x t x 2kd (4) power densities of incident and diffracted light but through the ratio of the luminous fluxes in the diffracted × ( , , ) ( , , , ) ∫∫ A K y Kz x Ut ky + K y kz + Kz x t dK ydKz, and incident light beams, as it is always done in the experiment. The luminous flux in a light beam can be ∂U k 2 calculated by using either the integral of the squared j------t + -----z--U = q exp( jηx)V(x, t) A*(K , K , x) ∂x 2k t 2 ∫∫ y z magnitude of the light field distribution over the beam t (5) cross-section or the integral of the squared magnitude × ( , , , ) Ud ky Ð K y kz Ð Kz x t dK ydKz. of the Fourier spectrum of the field over the angular coordinates ky and kz (the Parseval theorem). ∆ε 2 ∆ε 2 η Here, q1 = kd(ed ˆ et)/nd ; q2 = kt(et ˆ ed)/nt ; = kt + The luminous flux at the input of the acousto-optical K Ð kd is the parameter of detuning; ky, kz and Ky, Kz cell is determined by the formula P = 0.5 exp{Ð(k 2 + are the transverse components of the wave vectors of 0 ∫∫ y 2 2 light and sound, respectively; and A(Ky, Kz, x) = kz )rt /2}dkydkz, because ft(0) = 1; at the cell output, π 2 − 2 2 2 R exp{ (K y + Kz )R (1 Ð jDx)/4} is the Fourier spec- it is calculated by the formula P = trum of the function W(x, y, z) for the case of a Gaussian 2 2 2 0.5∫∫ f d (L) f d* (L)exp{Ð(ky + kz )rd /2}dkydkz, where distribution of the acoustic pulse amplitude in the yz plane. In the latter expression, R denotes the transverse expression (8) should be used for fd(L). The ratio P/P0 dimensions of the train at x = 0 and t = 0, D = 2ζ/KR is characterizes the efficiency of the acousto-optical dif- the train divergence in the transverse directions, and ζ fraction, and this quantity can be directly determined characterizes the transverse anisotropic spread. We rep- from the experimental data. resent the functions Ut and Ud in the form ( , , , ) ( , ) { 2 } EXPERIMENT Ut ky kz x t = f t x t exp jxkz /2kt (6) In our experimental studies, we used a collinear × { ( 2 2) 2( ) } exp Ð ky + kz rt x /4 , acousto-optical filter made on the basis of a CaMoO4

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ACOUSTO-OPTICAL FILTERING USING SHORT ACOUSTIC TRAINS 1047 crystal of length L = 4 cm. The time of the acoustic pulse propagation through the crystal was L/v = 1 2 3 11.6 µs. The acoustic wave was excited in the crystal by a piezoelectric transducer, which converted the electric energy of the generator to the energy of the acoustic 4 wave. For our experiments, we used a specially designed 5 generator that was capable of producing signals with an arbitrary envelope, i.e., arbitrarily shaped pulses. The envelope of the signal is formed by setting the value of 6 7 8 the signal amplitude every 0.1 µs; i.e., the envelope has µ the form of a set of rectangles of duration 0.1 s each. Fig. 1. Block diagram of the generator of arbitrarily shaped In this way, one can specify any form and duration of acoustic pulses: (1) a com-nopra interface, (2) a processor, the acoustic train, as well as the entire sequence of (3) memory (ROM), (4) memory (RAM), (5) a D/A con- trains. The maximum duration of a pulse produced by verter, (6) a high-frequency oscillator, (7) a multiplier, and the generator is 200 µs, and the minimum duration is (8) an amplifier. about 0.5 µs. The rate of the pulse series repetitions may vary from 0.5 to 1 kHz. After the amplification, the a/a0 output power reaches a level of 3 W. 1 The block diagram of the generator is shown in Fig. 1. The high-frequency unit generates the fundamental frequency 35Ð48 MHz, and this is precisely the sound frequency at which the diffraction of light takes place. The envelope is formed by the following units: the read-only memory (ROM), the processor, the random-access memory (RAM), and the digital-to-analog 0 t t t t t t (D/A) converter. The initial pulse or pulse series was 1 2 3 4 5 modeled on a PC and saved in the text format. Then, the Fig. 2. Time envelope of an acoustic pulse in the form of a envelope was sent through the corresponding port to the step-function signal. generator processor and to the ROM. After this, the communication between the computer and the generator was cut off. When the generator was turned off, the filter. This improvement is most pronounced for the information on the pulse form was stored in the ROM. pulse lengths somewhat less than the crystal length, Within several seconds after the termination of the data because in this case the transmission band is practically transfer from the PC to the generator, or after turning on not broadened, while the side lobes are already notice- the generator, the processor supplies the data on the ably suppressed. On the other hand, with a further pulse form to the RAM from which the digital data are decrease in the acoustic train duration, the filter band- sent to the D/A converter. As a result of the described width increases. This fact can be used in practice for process, the desired envelope of the signal is formed. controlling the characteristic of the collinear filter, but Then, by means of a multiplier, the high-frequency a strong suppression of the side lobes can be achieved oscillation of frequency 35Ð48 MHz is modulated by only with smooth control pulses. the envelope. The resulting pulse (or series of pulses) is We experimentally studied the level of the side lobes supplied to an amplifier and amplified to the necessary of the instrument function of a collinear filter con- amplitude. trolled by an acoustic pulse in the form of a step-func- In common practice, to obtain acoustic trains of tion signal. It was found that, in the case of an optimal finite length, rectangular pulses are used; for such choice of the form of the step-function signal, the level trains, the transfer characteristic has the form of the of the side lobes is considerably reduced. For example, function sinc2(x). For rectangular pulses, the level of if an acoustic train with the time envelope shown in the side lobes does not depend on the pulse duration Fig. 2 is used, its form can be described by the expres- and is as high as 5% of the central maximum (in the sion weak-interaction approximation). It should be noted ( , ) t Ð x/v 1 that, in contrast to rectangular pulses for which the V x t = A rect------Ð -- level of the side lobes is constant, for Gaussian pulses, t5 2 this quantity varies from zero (for short pulses, τ Ӷ L/v) t Ð x/v Ð t1 1 t Ð x/v Ð t2 1 to the level corresponding to rectangular pulses (for + B rect------Ð -- + C rect------Ð -- , long pulses, τ > L/v). t4 Ð t1 2 t3 Ð t2 2 The possibility of considerably reducing the side where the constants A, B, and C determine the width lobes allows one to improve the characteristics of the and the height of the steps.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1048 PARYGIN et al.

I/I0 I/I0 1.0 1.0

0.5 0.5

0 0 42.5 43.0 43.5 44.0 44.5 f, MHz 43.0 43.3 43.6 43.9 44.2 f, MHz Fig. 3. Instrument function of a filter controlled by two acoustic pulses of duration 3 µs. The solid line corresponds Fig. 4. Instrument function of a filter controlled by two to the distance between the pulses 6 µs; the dotted line acoustic pulses of duration 3 µs with the distance between shows the instrument function of a filter with a single pulse them 6 µs. The solid line corresponds to the theory, and the of duration 1 µs. dashed line shows the experimental data.

I/I0 I/I0 1.0 1.0

0.5 0.5

0 0 42 43 44 45 f, MHz 42 43 44 45 f, MHz

Fig. 5. Theoretical dependence of the transmission factor on the sound frequency for an acousto-optical filter in the case of five acoustic trains being present inside the crystal. The Fig. 6. Experimental instrument function of a filter con- distance between the trains is 2 µs; the train duration is trolled by five acoustic pulses of duration 1 µs with the dis- (solid line) 2 and (dashed line) 1 µs. tance 2 µs between them.

When uniform rectangular pulses are used and the trains of the same duration are present in the crystal, condition of weak interaction is fulfilled, the levels of beats are observed (the solid line in Fig. 3). The calcu- the first, second, and third side lobes are 4.7, 2.7, and lation is performed for the pulses of duration 1 µs with 1.0%, respectively. With the use of a step-function sig- the distance between them 6 µs. The envelope width nal shown in Fig. 2, it was possible to suppress the side and, correspondingly, the number of maxima are deter- lobes down to a level of 0.7%. mined by the duration of the short pulse. The use of In the case of the generation of short acoustic trains acoustic trains of longer duration leads to a narrowing (with the duration less than 3Ð4 µs), several pulses may of the envelope of the filter characteristic and to a simultaneously occur within the crystal. In such a situ- decrease in the number of maxima. This was observed in the experiment (Fig. 4) with the trains of duration ation, it is of interest to study the dependence of the µ µ shape of the transmission curve on the pulse duration 3 s with a distance of 6 s between them. and the number of pulses that simultaneously propagate With a further increase in the number of trains in the in the crystal (the number of pulses depends on the dis- crystal, the envelope of the transmission characteristic tance between them). Below, we will consider a is retained, because it is determined exclusively by the sequence of acoustic trains with a Gaussian time enve- duration of the pulses forming the sequence (the shorter lope. the acoustic train, the broader the envelope). As the For a single short acoustic train, we obtain a broad number of pulses increases, the beats take the form transmission band with a Gaussian characteristic with- shown in Fig. 5. The transmission function tends to a out any side lobes (the dotted line in Fig. 3). When two set of isolated narrow peaks whose width is determined

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ACOUSTO-OPTICAL FILTERING USING SHORT ACOUSTIC TRAINS 1049

f, MHz I/I0 1.0 1.0

0.6 0.5

0.2 0 1 2 ∆τ, µs 42.0 42.5 43.0 43.5 44.0 f, MHz

Fig. 8. Study of the optical spectrum consisting of two lines Fig. 7. Dependence of the frequency interval between the by an acousto-optical filter controlled by the pulses of dif- peaks of the comb, ∆f, on the pulse-repetition interval ∆τ. ferent duration: τ = (solid line) 2 and (dashed line) 12 µs. by the size of the crystal, i.e., by the length of interac- The results of the experiment are shown by the tion in the case of a continuous-wave operation. dashed line in Fig. 8. One can see that the acousto-opti- Figure 6 presents the experimental shape of the cal filter well resolves the spectral lines. The solid line shows the same spectrum studied by using short acous- transmission curve for a series of five pulses of duration τ Ӷ 1 µs spaced at 2 µs intervals. One can see that the tic pulses with L/v. In this case, the filter bandwidth experimental data agree well with the theoretical ones. exceeds the distance between the spectral lines, and the Thus, using a series of short trains whose individual resolution of the system is insufficient for observing the frequency bands are very wide, it is possible to create a two separate lines of the spectrum. This experiment filter with a comb-shaped transmission curve whose confirms the fact that the filter bandwidth increases peaks are characterized by bandwidths about those of with a decrease in the control pulse length. long pulses or even a continuous-wave signal. For a spectrum consisting of two lines, there still is an interesting possibility to determine the exact dis- As one can see from Figs. 5 and 6, in the case of a tance between the lines even with the use of short simultaneous propagation of several acoustic trains in pulses. When we use a single short pulse, we cannot the crystal, the transmission band of the filter has the resolve two close lines of the spectrum because of the form of a set of equidistant peaks. Figure 7 shows the broad transmission band. However, using a series of theoretical dependence of the frequency interval short pulses, we can tune the system in such a way that between the peaks of the comb on the time interval the peaks of the filter characteristic will coincide with between the acoustic pulses. This dependence is a the lines of the spectrum. In this case, the spectral char- hyperbolic one. It is evident that the interval between acteristic of the filter remains a periodic one for the the acoustic pulses determines the number of pulses optical spectrum consisting of two lines. simultaneously propagating in the crystal. It should be noted that the frequency interval between the peaks depends on the time interval between the acoustic trains CONCLUSION rather than on the number of pulses inside the crystal. Therefore, we obtain a continuous and smooth depen- The experimental studies described above showed dence rather than a step function. Correlating the exper- that the use of acoustic trains of finite length for con- imental dependence shown in Fig. 6 with the calibra- trolling the characteristics of a collinear acousto-opti- tion dependence given in Fig. 7, we obtain the interval cal filter is of great interest, because it provides a pos- between the acoustic pulses 2 µs, which is in complete sibility to vary the characteristics of the filter over wide agreement with the experiment. limits. The filter bandwidth can be increased by using short pulses. At the same time, the level of the side An important application of tunable acousto-optical lobes of the transmission function depends on the form filters is in the measurements of optical spectra. In our of the acoustic train and its duration. In the case of a experiments, we studied a spectrum consisting of two Gaussian (or close to it) acoustic train, the side maxima close lines that differed by 91 Å. The spectrum was are absent when the pulse duration is less than the time obtained by simultaneously supplying the optical sig- of the pulse propagation through the crystal τ = L/v. As λ µ nals from a HeÐNe laser ( 1 = 0.6328 m) and a semi- the pulse duration increases, the level of the side lobes λ µ conductor laser ( 2 = 0.6419 m) to the filter input. The tends to the value characteristic of a rectangular pulse. measurements were performed by using long acoustic By using pulses in the form of a step-function signal pulses (τ ≈ L/v). (with adequately chosen parameters), it is possible to

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1050 PARYGIN et al. suppress the side lobes down to a level of 0.7%. The use 4. A. Korpel, Acousto-Optics (Marcel Dekker, New York, of a series of Gaussian pulses allows one to obtain a ﬁl- 1988; Mir, Moscow, 1993). ter characteristic that has the form of a set of equidistant 5. A. P. Goutzoulis and D. R. Pape, Design and Fabrication peaks with a broad envelope. of Acousto-Optic Devices (Marcel Dekker, New York, An acousto-optical cell can also be used as a spec- 1989). trometer with a varying resolution, the latter being var- 6. V. N. Parygin and A. V. Vershubskiœ, Vestn. Mosk. Univ., ied by changing the pulse duration. Interesting possibil- Ser. 3: Fiz., Astron. 39 (1), 28 (1998). ities for spectrum identiﬁcation are offered by the use 7. V. N. Parygin, A. V. Vershubskiœ, and Yu. G. Rezvov, Opt. of a series of acoustic trains simultaneously propagat- Spektrosk. 84, 1005 (1998) [Opt. Spectrosc. 84, 911 ing in the crystal. (1998)]. 8. V. N. Parygin and A. V. Vershubskiœ, Akust. Zh. 44, 615 (1998) [Acoust. Phys. 44, 529 (1998)]. REFERENCES 9. V. N. Parygin and A. V. Vershubskiœ, Radiotekh. Élek- 1. L. G. Magdich, Izv. Akad. Nauk SSSR, Ser. Fiz. 44, 1683 tron. (Moscow) 43, 1369 (1998). (1980). 10. V. N. Parygin, A. V. Vershubskiœ, and K. A. Kholostov, 2. A. Harris, S. Mieh, and R. Fiegelson, Appl. Phys. Lett. Zh. Tekh. Fiz. 69 (12), 76 (1999) [Tech. Phys. 44, 1467 17, 223 (1970). (1999)]. 3. Jieping Xu and R. Stroud, Acousto-Optic Devices (Wiley, New York, 1992). Translated by E. Golyamina

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1051Ð1053. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 97Ð99. Original Russian Text Copyright © 2000 by Bazhukov, Gol’chevskiœ, Kotov.

RADIOPHYSICS

Calculation of the Relaxation Time from the Frequency Spectra of Ferrites K. Yu. Bazhukov, Yu. V. Gol’chevskiœ, and L. N. Kotov Syktyvkar State University, Syktyvkar, 167001 Komi Republic, Russia E-mail: [email protected] Received July 27, 1999

Abstract—A method for determining the relaxation times of the magnetic moments of ferrites from inverse Fourier transforms is suggested. The effect of the alternating external magnetic ﬁeld strength on the relaxation times was studied. It was found that the dispersion and absorption ranges in the magnetic spectra are associated with changes in the relaxation times of magnetic moments of ferromagnets. © 2000 MAIK “Nauka/Interperi- odica”.

DETERMINATION OF RELAXATION TIME [µ''(ω = 0) = 0 and µ''(ω ∞) = 0], and the third one µ ω ∞ Relaxation time is a key parameter of ferromagnets has a physical meaning [ '( ) = 0]. They also that specifies their frequency properties, such as the fre- help to solve the problem associated with the broaden- quency curve of the permeability (its slope), the width ing of the natural ferromagnetic resonance (FMR) of the absorption range, etc. The relaxation time τ is tra- peak. This effect is related to domain wall motion at zero static fields and makes the evaluation of τ difficult. ditionally found from the width of the peak of the imag- τ inary part µ'' of the permeability: τ = 2π/∆ω, where ∆ω At such fields, can be determined only in ferrites for is the half-peak width of µ''. This method is applied to which the contribution from wall motion is small in the saturation-magnetized magnets (single-domain crys- FMR frequency range. With the above restrictions, the tals and polycrystals). The most pressing problem is approximating curves eliminate this contribution. however to find relaxation times at small and vanishing Real µ'(ω) and imaginary µ''(ω) curves found in exper- µ ω − ω magnetizing fields, since ferromagnets are most fre- iments were approximated as '( ) = A1exp( B1/ ) + ω µ ω ω ωn quently applied just in this range of magnetization. For A2 exp(ÐB2/ ) and ''( ) = A exp(ÐB ), where n is a most of ferromagnets, experimental data for µ'' can be positive rational number. These expressions are fairly obtained only at lower-than-resonance frequencies pre- simple and provide a good fit to experimental data. sumably because of large losses (defined as µ''/µ' ratio, Approximating and experimental curves for several where µ' is the real part of the permeability) at higher samples are shown in Figs. 1Ð4. values. Therefore, the need for a method that can overcome these difficulties is obvious. In this work, the relaxation time τ of the magnetic OBJECTS OF INVESTIGATION moments of ferrites is derived from inverse Fourier Using the inverse transform method, we calculated transformation applied to the frequency dependence of the relaxation time for the magnetic subsystem of man- the permeability +∞ µ' 1 µ()t = ------∫ ()µ ' ()ω Ðiµ''()ω exp()ωiωtd.(1) 1800 2π Ð∞ 1600 For inverse Fourier transformation, the integrand 1400 must be defined in (–∞, +∞) and the functions µ'(ω) 1200 and µ''(ω), in [0, +∞). We assume µ' and µ" to be even in order to define them in the (–∞, +∞) range. The 1000 inverse Fourier transform is then a real time function 800 µ(t) [1]. The relaxation time τ of magnetic moments is 600 µ defined as a time during which (t) becomes e times 400 smaller than µ(t = 0). To apply this method, µ' and µ ω 200 ''( ) must be taken as continuous functions approxi- 104 105 106 ω, Hz mating experimental data. Inverse Fourier transformation imposes constraints on the approximating func- Fig. 1. µ' vs. frequency for the (211) single-crystal MnZ tions. Two of the constraints make integration easier spinel. h0 = 1 mOe. Symbols denote data points.

1052 BAZHUKOV et al.

µ'' little to the total permeability at high frequencies [5]. 700 A small contribution from wall displacement must improve the accuracy of determination of the true mag- 600 netic moment relaxation times, not the effective times, which are severely affected by wall motion. Moreover, 500 along with the suggested method, the conventional 400 approach to determining the relaxation time is also applicable in this case. 300 Calculations were carried out on (1) 200 Mn0.54Zn0.36Fe2.09O4 (111), (110), (211), and (100) spinel single crystals (the ring (torus) plane was parallel 100 to the plane of orientation) subjected to different alter- 0 nating magnetic fields h (1, 7, and 20 mOe) and (2) 7 0 104 105 106 10 Mn Zn Fe O spinel polycrystals with a porosity ω, Hz 0.63 0.29 2.08 4 of 0.003, mean grain size 20.2 µm, and field of anisotropy H = 0.04 Oe. In the latter case, h equaled 1 and Fig. 2. µ" vs. frequency for the (211) single-crystal MnZ A 0 spinel. h = 1 mOe. 2 mOe. Experimental dependences µ(ω) were taken 0 from [6, 7]. µ' 10000 COMPARISON OF THE METHODS FOR τ DETERMINATION 8000 The relaxation times obtained by inverse Fourier 6000 transformation lie near 10Ð7 s, while those derived from 4000 the half-peak width are in the 10Ð6Ð10Ð7 s range. In [3], the magnetic moment relaxation times in MnZ spinel 2000 were estimated at 10Ð7Ð10Ð8 s; hence, the inverse Fou- 0 Ð1 0 1ω rier transform gives the more accurate value. This is 10 10 10 , MHz explained as follows. In calculations using the inverse µ µ Fourier transform, both the real, ', and the imaginary, Fig. 3. ' vs. frequency for the polycrystals. h0 = 20 mOe. µ", parts of the permeability are used. Therefore, due to a greater number of data points, the accuracy of calcu- µ" lation is improved. A frequency dependence of µ' at 5000 high frequencies is easier to predict, since its slope, as a rule, is represented more comprehensively. Hence, µ' 4000 can be approximated with a much better accuracy than µ" (Figs. 1, 2). 3000 When the relaxation time was calculated with the conventional method, data points only for µ" were 2000 used. Note that the component µ' was approximated at frequencies from 104 to 107 Hz. Conversely, in the case 1000 µ of ", data at h0 = 1 mOe were obtained only for the ascending branch (in the 104Ð106 Hz range, Fig. 2). The 0 µ 10Ð1 100 101 ω, Hz further behavior of " was predicted very roughly from the conditions µ"(0) = 0 and µ"(∞) = 0. µ µ Fig. 4. " vs. frequency for the polycrystals. h0 = 20 mOe. From experimental data for ", the position of its peak remains unclear; hence, the height of the peak may be in great error. For the peak value of µ", we took ganeseÐzinc spinel (MnZ spinel) from experimental the highest-frequency data point. As follows from cal- frequency spectra µ(ω). A specific feature of MnZ culations, such an assumption is usually valid and the spinel is that the range of dispersion and the absorption relaxation times estimated by both methods are close to peak in the µ'' curve are due largely to magnetization each other. rotation [2Ð4], i.e., to FMR. Since MnZ spinel has low Thus, for the experimental data considered, inverse fields of anisotropy HA (below 1 Oe), FMR and domain Fourier transformation is the method of choice, since it wall resonance in them are observed in the 0.1Ð10 MHz provides more accurate values of the relaxation times and 1Ð10 kHz ranges, respectively [4]. Hence, they add of ferrites.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 CALCULATION OF THE RELAXATION TIME 1053

For polycrystals, the relaxation times obtained by traditional method is even preferred, since inverse Fou- both methods coincide. Here, we used experimental rier transformation requires data preprocessing. data for a wide frequency range and the peak of µ" is We can conclude that the relaxation time is indepen- distinct. In this case, the conventional method gives dent of the mutual arrangement of the crystallographic fairly accurate values of τ (related approximations are axes and alternating field in the single-domain ferrites. shown in Figs. 3, 4). The field amplitude affects the relaxation time only in the polycrystals. It can be suggested that frequency DISCUSSION characteristics of many radio engineering devices based on polycrystalline ferrites can easily be con- For the single crystals, the relaxation times calcu- trolled by varying the amplitude of an external alternat- lated from the experimental permeability data were ing field. found to be independent of the crystallographic orientation and alternating external field amplitude h0 within 1Ð20 mOe. The times remain constant at a level of 3 × REFERENCES Ð7 10 s. For polycrystals, the relaxation time at small h0’s 1. E. C. Titchmarsh, Introduction to the Theory of Fourier is 4 × 10Ð7 s, i.e., roughly equal to that of the single Integrals (Clarendon Press, Oxford, 1948, 2nd ed.; Gos- tekhizdat, Moscow, 1948). crystals. As h0 rises, the situation changes drastically. At h = 20 mOe, the magnetic moment relaxation time 2. S. Chikasumi, The Physics of Ferromagnetism. Mag- 0 netic Characteristics and Engineering Applications increases about twofold. Unlike the single crystals, the (Syokabo, Tokyo, 1984; Mir, Moscow, 1987). structure of the polycrystals undergoes substantial 3. S. Krupicka, Physik der Ferrite und der verwandten modification (damage). In this case, the vector of mag- magnetischen Oxide (Academia, Praha, 1973; Mir, Mos- netization may slowly return to the equilibrium posi- cow, 1976). tion, causing the relaxation time to increase. Similar 4. J. Smit and H. P. J. Wijn, Ferrites (Wiley, New York, results were obtained in [8Ð10]. It was found in these 1959; Inostrannaya Literatura, Moscow, 1969). works that the relaxation time increases as the energy 5. A. G. Gurevich, Magnetic Resonance in Ferrites and delivered to the magnetic subsystem grows. In [9], Antiferromagnets (Nauka, Moscow, 1973). nickel ferrite was studied; and in [10], nickelÐchro- 6. S. G. Abarenkova, S. A. Kochnov, I. V. Saenko, et al., mium alloys. It is likely that such an energy depen- Élektron. Tekh., Ser. 6: Mater. 8 (264), 28 (1991). dence of the relaxation time is typical of all spinel-like 7. M. A. Kharinskaya and S. G. Abarenkova, Élektron. ferrites within a certain range of field amplitudes. Tekh., Ser. 6: Mater. 3 (248), 23 (1990). 8. D. Park, Phys. Rev. 97, 60 (1955). CONCLUSIONS 9. L. Torres, M. Zazo, and J. Iniguez, IEEE Trans. Magn. If the relaxation time is difficult to estimate with the 29, 3434 (1993). traditional method, inverse Fourier transformation can 10. F. I. Stetsenko, Fiz. Tverd. Tela (St. Petersburg) 37, 598 be used. When the frequency range is such that the (1995) [Phys. Solid State 37, 326 (1995)]. vicinity of the imaginary component peak is fully covered, both methods give similar results. In this case, the Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 8 2000

ELECTRON AND ION BEAMS, ACCELERATORS

Electron Acceleration by Gaussian Electromagnetic Beam in a Stationary Magnetic Field V. P. Milantiev and Ya. N. Shaar Peoples' Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198 Russia E-mail: [email protected] Received June 22, 1999

Abstract—Relativistic electron motion in the electromagnetic Gaussian beam that propagates along a stationary magnetic field is studied. It is shown that, if the cyclotron resonance conditions are initially satisfied, electrons can be efficiently accelerated over a relatively small interval at a slightly lower rate than in a plane accelerating wave. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION ASSUMPTIONS AND BASIC EQUATIONS Among various charged particle acceleration mech- In the paraxial approximation, laser radiation can be anisms, the cyclotron autoresonance mechanism dis- represented as a Gaussian beam (GB) [9] that propa- covered by Kolomenskiœ and Lebedev [1] and, indepen- gates along a stationary magnetic field aligned with the z-axis: dently, by Davydovskiœ [2] is of great interest. This mechanism provides a high acceleration rate and suffi- E =()EcosΘ,,Ð0EsinΘ .(1) ciently low radiation losses [3]. A variety of designs of microwave [4Ð6] and laser [7, 8] electron accelerators Here, based on the cyclotron autoresonance have been pro- Θω ψ posed. At the same time, strictly speaking, the cyclo- =Ðt++kz k , tron autoresonance, being a relativistic effect, exists kψ= r2D/a2()1+D2Ð arctanD, only in the plane transverse electromagnetic wave trav- (1a) ()2Ð1/2 {}2 2()2 eling at the speed of light in vacuum along a stationary EE= 11+D exp Ðr /a 1 + D magnetic field if electrons are in the cyclotron reso- ≡ E fxyz(),, , nance with the wave at the initial moment of time. If 1 these conditions are violated, various techniques can be 2 ≡ 2 where D = 2z/ka z/zq; zq = ka /2 is the Rayleigh used to support the forced synchronism [3]. It was length; ω is the wave frequency; k = ω/c is the wave shown [7] that electrons can be accelerated to number in vacuum; a is the radius of the beam’s waist, extremely high energies in the field of high-power laser 22 radiation at a very high rate with very small radiative i.e., its minimal radius at z = 0; and r = x+y is the losses. However, these results were obtained under the transverse coordinate. assumption that the laser radiation has the form of a The magnetic components of the GB field can be plane wave. Actually, this assumption is usually found from Maxwell’s induction equation: invalid. In many cases, laser and microwave radiation (),, can be considered in the quasi-optical approximation as B=BxByBz, (2) a Gaussian beam. Clearly, in this case, the cyclotron where autoresonance conditions are a fortiori violated. There- fore, the electron energy does not necessarily grow ()Θ Θ Bx=E1Gcos + fQsin ,(2a) monotonically. In this paper, we show that, despite this ()ΘΘ circumstance, particles in the Gaussian beam can also By=ÐE1Gsin Ð fQcos ,(2b) acquire a significant energy over a short acceleration interval. A high acceleration rate can be achieved by 2fE ------1 shaping the guiding magnetic field to fit an appropriate Bz= Ð 2 2 () (2c) profile or by optimizing the parameters of the electron ka 1 + D injection and of the Gaussian beam. × ()()ΘxDÐ y sin + ()ΘyD+ x cos .

ELECTRON ACCELERATION BY GAUSSIAN ELECTROMAGNETIC BEAM 1055

Here, dPz εP⊥{ Θ Θ } ------= Ð ------fQcos + + Gsin + , (7) ∂kψ ∂f dτ γ Q = 1 + -∂------, G = Ð∂------. kz kz dγ ε ----- = Ð--P⊥ f cosΘ , (8) The phase velocity of the GB is dτ γ +

ω dZ Pz dX dY v = ------, ------= -----, ------≈ 0, ------≈ 0. (9) ph k(1 + ψ') dτ γ dτ dτ τ ω ε ω where Here, = t; R = kr; P = p/m0c, and = eE1/m0c are the dimensionless parameters and variables; m is the 2 2 2 2 0 dkψ 1 r (1 Ð D ) Ð a (1 + D ) electron rest mass; ω = eB /m c is the classical fre- kψ'(z) ≡ ------= ------. 0 0 0 dz 2 ( 2)2 quency of the electron cyclotron rotation in the guiding a zq 1 + D Ω ω ω magnetic field; 0 = 0/ ; the electron charge is Ðe, At a distance of about the Rayleigh length, D ≈ 1, so 2 2 ψ ≈ Ӷ where e > 0; and γ = 1 + P + P⊥ is the relativistic that '/k (1/kzq) 1 when r < a. Therefore, z factor (dimensionless electron energy). In the case of a  ψ' plane electromagnetic wave, Q = 1 and function f = 1 v ≈ c 1 Ð ---- > c. ph  k  such that G = 0. Then, system (5)Ð(9) yields the integral γ of motion Ð Pz = Y = const, which coincides with the Ω That is, the phase velocity is higher than the velocity cyclotron resonance condition at Y = 0. This is the of light. This means that the cyclotron autoresonance autoresonance [3]. If the electromagnetic field has the cannot exist in the GB. form of GB (1) and (2), according to (5)Ð(9), the cyclo- In order to extract the cyclotron rotation of a parti- tron resonance occurs when cle, we represent its momentum vector as γ ≈ Ω Ð PzQ 0. (10) ( Θ Θ ) p = pxez + p⊥ ex cos 0 + ey sin 0 . (3) Since the phase velocity of the GB is higher than the velocity of light, the electron cyclotron resonance con- Here, ex, ey, and ez are the unit vectors of the Cartesian dition, imposed at the initial time moment, is not pre- coordinate system; pz and p⊥ are the momentum com- served during the electron motion. This effect occurs ponents, respectively, along and perpendicular to the Θ because (10) is not an integral of motion. guiding magnetic field; and 0 is the phase of the particle cyclotron rotary motion in this field. The phase of field (1), as it acts on the electron, is governed by the NUMERICAL RESULTS equation It is difficult to derive an analytical solution to sys- dΘ dr tem (5)Ð(9). Therefore, we solved it numerically by the ------= Ð ω + -----∇Θ. (4) RungeÐKutta method. The motion of electrons injected dt dt at Z ≤ 0 was studied in the region x2 + y2 < a2. The Equations of electron motion in the GB field com- cyclotron resonance condition (10) was assumed to be bined with equation (4) constitute a two-period (or two- fulfilled exactly at the initial moment of time. This frequency) system, which contains oscillating factors requirement imposes rather stringent constraints on the Θ Θ Θ ± Θ domain of parameters Ω and Q . In fact, initially, the with phases , 0, and their combinations 0. In 0 0 the region of the electron cyclotron resonance, Θ + transverse momentum component must comply with Θ ≡ Θ the relationship 0 + is a slowly varying quantity. When the oscillation frequency is high and the magnetic field is strong, (γ Ω )2 Θ Θ Θ Θ 2 2 0 Ð 0 phases , 0, and Ð 0 must be regarded as rapidly ⊥ = γ Ð 1 Ð ------≥ 0, P 0 0 2 (11) varying quantities. Smoothing over rapidly varying Q0 phases [10] in the electron cyclotron resonance region Ω yields which yields the constraint on parameters 0 and Q0 mentioned above dP⊥ ε P ------= Ð -- f {γ Ð P Q}cosΘ + ε-----zGsinΘ , (5) Ω 2 2 ≥ dτ γ z + γ + 0 + Q0 1. (12) Θ Ω γ We will study the motion of electrons that satisfy d + 0 Ð + PzQ ε f {γ } Θ initial condition (10) and have an initial energy of ------= ------+ ------Ð PzQ sin + dτ γ P⊥γ 25 MeV or higher. The electron motion will be studied (6) P within a rather small 100-cm-long interval. This inter- ε z Θ val is chosen because it is necessary to find the optimal + ------γ--Gcos +, P⊥ parameters that provide a high electron acceleration

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1056 MILANTIEV, SHAAR

γ γ 600 A 5000 B 500 B 4000 400 C 3000 300 D 2000 A 200

100 1000

0 50 100 0 50 100 Z, cm Z, cm

Fig. 1. Electron energy γ on the acceleration interval versus GB width (parameter q = a2k2) for microwave radiation Fig. 2. Electron energy in the laser GB and stationary mag- λ Ω γ ε ( = 1 mm and 0 = 1) at 0 = 50 and = 1: (A) plane wave netic ﬁeld B = 100 kG at ε = 1 for (A) γ = 50 and λ = 10 µm 5 4 × 3 z γ λ 0 µ and the GB at q = 10 and the GB at q = (B) 10 , (C) 5 10 , (CO2 laser) and for (B) 0 = 500 and = 1 m (neodymium and (D) 103. glass laser).

γ 3000 γ A 500 2500 A B 400 B 2000 C 300 1500 C 1000 200

500 100

0 50 100 0 50 100 Z, cm Z, cm

Fig. 3. Energy of electrons accelerated in the GB under the Fig. 4. Energy of electrons injected at different points of λ Ω approximate initial cyclotron resonance conditions at λ = plane z0 = 0 (beam’s waist) versus z at = 1 mm, 0 = 1, 10 µm (CO laser), Ω = 0.01, γ = 50, and ε = 1: (A) plane 2 0 0 γ ε 2 2 ≡ ≤ wave with the exact cyclotron resonance; (B) the GB with 0 = 50, and = 1: (A) x + y r 0.1a; (B) r = 0.5a; and C = 0.01, 0.00985, and 0.0101; and (C) the GB with C = 0.011. (C) r = a. rate with the Rayleigh length being noticeably longer The acceleration rate significantly increases with the than the acceleration interval. radiation power. µ Figure 1 shows the electron energy on this interval At shorter wavelengths (less than 20 m, laser radiation) and the same guiding magnetic field, Ω is very as a function of the GB width (parameter q = a2k2) for 0 λ Ω small; hence, by virtue of constraint (12), the initial microwave radiation ( = 1 mm and 0 = 1). It can be cyclotron resonance conditions (10) cannot be satisfied. seen that the particle acceleration rate increases with Therefore, when the particles are accelerated in the 5 γ Ω the beam width at a constant wavelength. At q = 10 , the laser field, we impose the initial condition 0 Ð Pz0 = 0, energy grows as fast as in the plane wave. Under these which corresponds to the electron cyclotron resonance conditions, the electron energy can rise by more than an in the plane wave. The results are presented in Fig. 2. It order of magnitude. The acceleration process involves can be seen that, similarly to the plane wave [7], the GB all electrons regardless of the spread in the initial can also efficiently accelerate electrons on the consid- phases, which govern the electron motion at the initial ered interval, though at a slower rate. The initial GB stage; the electron energy increases by a factor of 7Ð8. width was specified such that the beam intensity

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ELECTRON ACCELERATION BY GAUSSIAN ELECTROMAGNETIC BEAM 1057 decreased by a factor of no more than e at a distance z = stationary magnetic field despite the fact that the initial 150 cm from the beam’s waist. Figure 2 plots the elec- cyclotron resonance between the wave and particle is tron energy as a function of z in a stationary magnetic violated during the particle motion. All particles are ε γ λ µ field Bz = 100 kG at = 1 for (A) 0 = 50 and = 10 m involved in the acceleration process regardless of their γ λ µ (CO2 laser) and for (B) 0 = 500 and = 1 m (neody- initial phases. mium glass laser). We studied the motion of a separate electron, Figure 3 shows the electron energy as a function of thereby ignoring the effect of the intrinsic field of the z when, at the initial moment of time, the cyclotron res- beam and the back effect of the beam on the accelerat- γ ing electromagnetic wave. This approximation is valid onance condition is satisfied approximately ( 0 Ð pz0 = const ≡ C ≠ Ω ) rather than exactly. The energy of elec- when the beam concentration (and current) is suffi- 0 ciently low. The single-particle model of interaction trons accelerated by the plane wave with the initial between the electron beam and the electromagnetic cyclotron resonance condition that is satisfied exactly is wave was shown to be valid if the beam current is lower plotted for reference. It can be seen that such particles than a certain tolerable value J(kA) Ӷ 8εωR /c, where can be accelerated on the 100-cm-long interval of our b Ω Rb is the beam radius [8]. As for the radiation losses, interest if the difference between C and 0 is small. If C is significantly higher than Ω , the energy oscillates their effect is of very small significance even at very 0 high electron energies [7]. rather than grows monotonically. Figure 4 shows the energy of electrons injected at ACKNOWLEDGMENTS different points of plane z0 = 0 (beam’s waist): (A) 2 2 This work was supported by the project “Funda- x + Y ≡ r ≤ 0.1a; (B) r = 0.5a; and (C) r = a. All mental Studies in Russian Universities.” remaining parameters of the GB and electron are the same as above. As shown in the figure, the acceleration rate decreases as the injection point moves away from REFERENCES the GB center because the GB energy decreases. Thus, 1. A. A. Kolomenskiœ and A. N. Lebedev, Dokl. Akad. all particles of a given cross section of the electron Nauk SSSR 145, 1259 (1962) [Sov. Phys. Dokl. 7, 745 beam are accelerated, though at different rates. (1963)]; Zh. Éksp. Teor. Fiz. 44, 259 (1963) [Sov. Phys. JETP 17, 177 (1963)]. In order to validate our calculations, we compared 2. V. Ya. Davydovskiœ, Zh. Éksp. Teor. Fiz. 43, 886 (1962) our solution to system (5)Ð(9) for the plane wave with [Sov. Phys. JETP 16, 629 (1963)]. results given in [7, 11]; the agreement was perfect. Let 3. V. P. Milant’ev, Usp. Fiz. Nauk 167, 3 (1997) [Phys. Usp. us focus on the results presented in [7], which predict 40, 1 (1997)]. that relativistic electrons can be accelerated in a high- 4. A. A. Vorobev, A. N. Didenko, A. P. Ishkov, et al., At. power laser plane wave to extremely high energies at a Énerg. 22 (1), 3 (1967). high rate if the electron cyclotron autoresonance is 5. H. R. Jory and A. W. Trivelpiece, J. Appl. Phys. 39, 3053 maintained. At first sight, this statement contradicts the (1968). conclusion [1, 3] that the acceleration rate decreases 6. E. T. Protasevich, Zh. Tekh. Fiz. 65 (6), 133 (1995) with increasing energy as 1/ γ . However, it can easily [Tech. Phys. 40, 1 (1995)]. be seen that the energy is also proportional to the wave 7. A. Loeb and L. Friedland, Phys. Rev. A 33, 1828 (1986). frequency and the dimensionless parameter ε. There- 8. A. Loeb and L. Friedland, Phys. Lett. A 129 (5/6), 329 fore, increasing the energy and acceleration rate with (1988). the power and frequency of the laser radiation at a given 9. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukho- ε is justified. rukov, The Theory of Waves (Nauka, Moscow, 1990). 10. V. P. Milant’ev, Zh. Tekh. Fiz. 64 (6), 166 (1994) [Tech. Phys. 39, 608 (1994)]. CONCLUSION 11. C. S. Roberts and S. J. Buchsbaum, Phys. Rev. A 135, Our calculations show that an electron beam can be 381. accelerated at a sufficiently high rate in the electromagnetic Gaussian beam that propagates along an external Translated by A.D. Khzmalyan

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1058Ð1062. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 104Ð107. Original Russian Text Copyright © 2000 by Ovsyannikova, Fishkova.

ELECTRON AND ION BEAMS, ACCELERATORS

Six-Electrode Deflectron L. P. Ovsyannikova and T. Ya. Fishkova Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia E-mail: [email protected] Received July 20, 1999

Abstract—In the two-dimensional approximation, the potential of a six-electrode deflectron (which was previously proposed by the authors) is obtained in a closed form. The field nonuniformity is calculated for this deflectron. The field distributions are computed on the axes of finite-length deflectrons using certain analytical formulas. In deflectrons, the axial trajectory of the beam can be horizontally and vertically deflected by a maximum angle which depends on the structure geometry. The nonlinearity of this deflection is calculated. The results are compared to those obtained for four- and eight-electrode deflectrons. © 2000 MAIK “Nauka/Inter- periodica”.

π In order to create a scan on a sample, electrostatic and vertical deflections, are, respectively, K1x = 23 / π omnidirectional deflection systems with spatially and K1y = 3/ . Therefore, equal deflections in both superposed deflection centers (so-called deflectrons) directions are provided by the feeding voltages that are are used in scan electron microscopes, electron-beam tubes, high- and low-energy electron diffractometers; coupled by the relationship Vx/Vy = 3 /2. in the techniques of secondary-ion and atom mass spec- In a closed form, the potential distribution of the six- troscopy; etc. Deflectrons do not focus beams of electrode deflectron that involves the corrected third charged particles (in the first-order approximation). harmonics has the form The main requirement imposed on these structures is Φ()6 that the field in the operation region should be as uni- π{()[]()()2 2 form as possible. = 1/ Vx + 1/2V y arctan 3+y/1Ðx Ðy Electrostatic deflectrons are most often designed as (2) ()[]()()2 2 cylinders or cones which are cut along generatrices into + V x Ð 1/2V y arctan 3xyÐ /1Ðx Ðy an even number of fragments [1, 2], having the forms []}()2 2 of planar electrodes placed on the sides of rectangular + V y arctan 2y/1Ðx Ðy . (square) boxes [3, 4] and cut planar capacitors [5, 6]. Here and below, the x and y coordinates are expressed In [7], we proposed a six-electrode deflectron with in terms of cylinder radius R. The field intensity com- identical angular dimensions of electrodes equal to π/3 ponents of this deflectron are (assuming that the gaps between the electrodes are infi- () ()π {()[()2 2 nitely small). The electrodes are located on a cylinder Ex 6 =Ð1/ R Vx+ 1/2Vy 31+x Ðy (cone) surface. The cross section of the deflectron is ] []()2 22()2() shown in Fig. 1. Inside an infinitely long cylinder that +2xy /1ÐxÐy +3xy+ +VxÐ 1/2Vy is cut along the generatrices, the potential distribution 2 2 2 22 can be represented in Cartesian coordinates as a series. ×[]31()+x Ðy Ð 2xy /1[()ÐxÐy When feeding voltages are applied as we propose ()2] []}()2 222 (Fig. 1), the series coefficients of the potential for the +3xyÐ +4Vyxy/1ÐxÐy +4y, six-electrode deflectron are expressed as follows: () ()π {() Ey6=Ð1/ R Vx+ 1/2V y π ()[]()π K()2n Ð 1 x= 4/ /2nÐ1sin 2nÐ1/3 , 2 ×[]1Ðx2++y223xy /1[()Ðx2Ðy2 K()2n Ð 1 y (1) 2 = 4/π/2()nÐ1{}b+ ()1Ðbcos[]()π2nÐ1/3 .+3()xy+ ] (3) ()[]22 [()2 22 These expressions imply that K3x = 0 always and ÐVxÐ 1/2Vy 1Ðx+ yÐ 23xy /1ÐxÐy K = 0 only at b = 0.5. The six-electrode deflectron 3y 2 provides a more uniform field than a standard four- +3()]xyÐ electrode deflectron. According to (1), the higher coef- []}()2 2 []( 2 222 ficients, which govern the sensitivities of the horizontal +2Vy 1Ðx +y /1ÐxÐy +4y.

y Expressions (3) imply that, when Vx/Vy = 3 /2, the Vy field intensity components on the axis are Ex(x = y = 0) = Ey(x = y = 0) = Eh, where Eh is the intensity of the corresponding uniform field. –Vx + bVy Vx + bVy Figure 2 demonstrates field nonuniformity ∆E nor- R malized to the intensity of the uniform field versus the distance to the axis. These results are calculated for the 0 x six-electrode deflectron using (3) (curves 2) and compared to the field nonuniformities of the elementary –Vx – bVy Vx – bVy four- and the more complicated eight-electrode [2] deflectrons with the electrode angular dimensions of –V π/2 (curves 1) and π/4 (curves 3). Specifying the value y of the field nonuniformity necessary for solving a problem under consideration, one can find the maximum Fig. 1. The cross section of the six-electrode deflectron. possible distance between the trajectory and the structure axis and, finally, determine the deflection angle. ∆E/E , % ∆ h Thus, when E/Eh = 0.1%, the distance from the axis must not exceed 0.02R, 0.17R, and 0.025R for the four-, six-, and eight-electrode deflectrons, respectively. We 4 1 2 should note that the difference in the distances is small 2 for the two latter deflectrons and, in most cases, the six- electrode deflectron is not inferior to the eight-elec- 3 trode one. Below, this circumstance is demonstrated by analyzing the trajectories and determining the nonlin- 2 earities of deflection through equal angles. For arbitrary deflectrons, the field distribution along 1 the axis (in the proximity of the axis) coincides with the field of a planar capacitor when the deflectrons have 0 equal lengths and interelectrode distances. In the case 0.2 0.4 0.6 when the deflectron is located in the free space, the field x/R distribution is calculated using the TEO program [8]. –1 The field intensities are calculated on the longitudinal axes of deflectrons of different lengths and normalized to the field intensity AEh observed at the center (see –2 Fig. 3a). Coefficient A as a function of deflectron length is shown in Fig. 3b. Taking into account the data given –3 3 in Fig. 3, we have chosen an analytical formula for the 3 field distribution. For short deflectrons (1 ≤ l/R ≤ 2), it 1 2 has the form –4 ( ) 3 ( ) E z = AEh cos Bz/R , (4) Fig. 2. The field nonuniformity of electrostatic deflectrons: where B = 0.9 Ð 0.2l/R. (solid curves) deflection in the x-axis direction, (dash-and- The origin z = 0 coincides with the deflectron center. dot curves) deflection in the y-axis direction, and (dashed In Fig. 3a, the field distribution calculated by (4) is curves) the diagonal deflection. marked with crosses. For long deflectrons (l/R > 3), A = 1. In this case, one can notice a section with Eh = of deflectrons are determined using (4)Ð(6). For the const, whose length depends on the structure length. short and long structures, the effective lengths are, For this section, we have found the empirical formula respectively, π/(2B) z0/R = l/R Ð 2.8. (5) 3 In addition, long deflectrons are characterized by an L = 2 ∫ cos (Bz/R)dz = 4R/(3B) (7) edge field that is virtually independent of the deflectron 0 length. This field can be represented in a simple form as and 2 [ ( )] π Ek = Eh cos z/ 2R . (6) 2 ( ) In Fig. 3a, the edge field calculated by (6) is marked L = z0 + 2∫ cos z/2R dz = l + 0.3R. (8) with dots surrounded by circles. The effective lengths 0

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1060 OVSYANNIKOVA, FISHKOVA

δ E/AEh i, % 1.0 . (a) . 15 0.8 . 6 (a) 10 . 5 1 0.6 . 5 2 4 . 3 . 0.4 0 2 . 10 20 30 40 α, deg 0.2 1 2 . –5 . 0 1 2 3 4 5 6 z/R –10 3 A –15 1.0 δ i', % (b) (b) 15 0.9 1 10 2

0.8 5

0 0.7 10 20 30 40 α, deg

0 1 2 3 4 l/R –5 2

Fig. 3. (a) The field intensity of deflectrons of different –10 3 length [l/R: (1) 1, (2) 2, (3) 3, (4) 4, (5) 6, and (6) 8] and (b) the coefficient governing the field intensity at the center of the structure. –15

The trajectories of the charged-particle beam in the Fig. 4. (a) The coordinate and (b) angle deflection nonlin- deflection structures are computed using the DEF pro- earity of the axial beam trajectory at the exit of deflectrons of the length l = 2R: (1) the four-, (2) the six-, and (3) the gram, which utilizes the MathCAD system of com- eight-electrode deflectrons. (Solid curves) deflections in the puter-aided mathematical calculations. In this program, x- and (dashed curves) y-axis directions. each second-order equation is reduced to a system of two first-order differential equations. The error in the Ð6 Using calculations of the parameters of the axial tra- solution is 10 . δ jectory, we find the nonlinearity of the coordinate ( i) The axial trajectories of the beams in the short and and slope angle (δ' ) deflections observed at the exit of long six-electrode deflectrons are calculated by the i DEF program. The fields of these deflectrons are spec- the field region by using the following formulas: ified by (3)Ð(6). The coordinates and slope angles δ δ' ' ' obtained at the exit of the field region are compared i = ri/rih Ð 1; i = ri /rih Ð 1, (9) with the calculations performed using a rectangular ' model with the effective lengths found from (7) and (8). where ri and ri are the distance from the axis and the The comparison shows that the difference observed for slope angle of the axial beam trajectory at the exit of the short deflectrons reaches 100% and, for long deflec- field region, respectively, and rih and ri'h are the same trons, this difference does not exceed 10%. This fact parameters for a uniform field. means that the longer the deflectrons, the less the beam parameters are affected by the form of the edge field Then, the deflection nonlinearity on the object is and, starting with a certain length, the variation of the ∆ δ δ λ ∆ δ radial field becomes a determining factor. = i' + i' , ' = i', (10)

TECHNICAL PHYSICS Vol. 45 No. 8 2000 SIX-ELECTRODE DEFLECTRON 1061

δ i, % electrode deflectron whose length is equal to the diam- (a) eter of its aperture (curves 2). Figure 5 shows the same 4 parameters calculated using the rectangular field model 1 for the deflectron with the length equal to the doubled aperture diameter. The data characterizing the coupling 2 of the efficiency of the deflection structure with the 2 deflection angle, as well as with the coordinate of the axial trajectory at the exit of the field region, are sum- 0 marized in the table. The efficiency of the deflection 10 20 30 α, deg structure is governed by the ratio between the main 3 feeding electrode voltages (±V) and the acceleration –2 2 Φ potential ( 0). Note that Vy = V and Vx = 3 /2V. δ From Figs. 4 and 5, one can see that the long and i', % (b) short deflectrons exhibit a substantially different behavior of the deflection nonlinearity. For the long 1 4 2 deflectron, the deflection-angle dependence corresponds to the field nonuniformity of the infinitely long deflectron observed when the distance from its longitu- 2 dinal axis increases. This nonuniformity mainly governs the deflection nonlinearity, which is virtually independent of the edge field. For the short deflectron, the 0 α, deg edge-field effect is essential. At deflection angles which 10 20 30 do not exceed 25°, the deflection nonlinearity of the 3 long deflectron is less than that of the short one by a –2 factor of 2Ð4. 2 We should note that, when the deflection angle exceeds 25Ð30° in the short deflectron, the deflection –4 nonlinearity abruptly varies and, hence, the shape of the spot is distorted due to a considerable difference Fig. 5. (a) The coordinate and (b) angle deflection nonlin- between the axial and edge beam trajectories. There- earity of the axial beam trajectory at the exit of deflectrons fore, it is inexpedient to use these modes in precision of the length l = 4R: (1) the four-, (2) six-, and (3) eight-elec- instruments such as scanning electron microscopes. trode deflectrons. (Solid curves) deflections in the x- and (dashed curves) y-axis directions. The parameters of four- (curves 1) and eight-electrode (curves 3) deflectrons are compared in Figs. 4 and 5. These parameters are calculated using the DEF pro- where λ is the distance from the exit of the field region gram in the case when the field distribution along the to the object. longitudinal axis is identical to the corresponding field Figure 4 demonstrates the nonlinearities of the coor- distribution of the six-electrode deflectron. As dinate (a) and angle (b) deflections for the short six- expected, the four-electrode structure exhibits the max-

Table l/R = 2 l/R = 4

Φ α0 x /R α0 α0 x /R α0 V/ 0 x i y yi/R x i y yi/R 0.05 3.5 0.192 3.5 0.192 5.9 0.225 5.9 0.225 0.1 7.0 0.386 7.0 0.386 11.6 0.449 11.8 0.450 0.2 14.1 0.779 14.3 0.782 21.7 0.891 23.1 0.907 0.25 26.2 1.100 28.8 1.140 0.3 21.2 1.183 21.8 1.202 0.4 26.8 1.577 28.0 1.627 0.5 30.5 1.918 32.4 2.004 0.6 32.8 2.196 35.0 2.320 0.8 35.5 2.630 38.4 2.840 1.0 37.1 2.972 41.1 3.330

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1062 OVSYANNIKOVA, FISHKOVA imum nonlinearity. For example, if we assume that δ = 3. L. P. Ovsyannikova and T. Ya. Fishkova, Zh. Tekh. Fiz. 1%, the deflection angle must not exceed 5°. In this sit- 56, 1348 (1986) [Sov. Phys. Tech. Phys. 31, 791 (1986)]. uation, the beam can be deflected by an angle of up to 4. L. P. Ovsyannikova and T. Ya. Fishkova, USSR Inven- 8° in the short six-electrode and in the eight-electrode tor’s Certificate No. 1365179, Byull. Izobret., No. 1 deflectrons and up to 18° and 15°, respectively, in the (1988). long eight-electrode and the six-electrode deflectrons. 5. L. P. Ovsyannikova, T. Ya. Fishkova, and V. V. Moseev, USSR Inventor’s Certificate No. 1557603, Byull. Izo- Thus, in certain cases, the proposed six-electrode bret., No. 14 (1990). deflectron with the corrected third harmonics in the 6. L. P. Ovsyannikova and T. Ya. Fishkova, Zh. Tekh. Fiz. expansion of the potential is superior because its 58, 1176 (1988) [Sov. Phys. Tech. Phys. 33, 692 (1988)]. parameters are close to those of the eight-electrode 7. T. Ya. Fishkova, A. A. Shaporenko, L. P. Ovsyannikova, deflectron and the design is simpler. et al., RF Inventor’s Certificate No. 1729247, Byull. Izo- bret., No. 23 (1993). 8. L. P. Ovsyannikova, S. V. Pasovets, and T. Ya. Fishkova, REFERENCES Zh. Tekh. Fiz. 62 (12), 171 (1992) [Sov. Phys. Tech. 1. B. É. Bonshtedt, USSR Inventor’s Certificate No. 143479, Phys. 37, 1215 (1992)]. Byull. Izobret., No. 24 (1961). 2. J. Kelly, Adv. Electron. Electron Phys. 43, 116 (1977). Translated by Efimova

TECHNICAL PHYSICS Vol. 45 No. 8 2000

SURFACES, ELECTRON AND ION EMISSION

Emission of Charged Clusters during Metal Sputtering by Ions V. I. Matveev Department of Thermal Physics, Academy of Sciences of Uzbekistan, ul. Karatal 28, Tashkent, 700135 Uzbekistan E-mail: [email protected] Received May 24, 1999

Abstract—A method for simulating processes of metal sputtering by ion bombardment in the form of large neutral and charged clusters with a number of atoms N ≥ 5 based on simple physical assumptions and in fair agreement with experiment is suggested. As an example, the ionization degrees and ionization coefﬁcients, as well as the relative cluster yields, are calculated as a function of the number of atoms in clusters of different metals (Ag, Nb, and Ta) bombarded by singly charged Ar+1 and AuÐ1 ions. A ﬂuctuation mechanism of charge state formation for large clusters, which describes the dependence of the charge state distributions on cluster size and target temperature, is developed. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION state of the sputtering products in the form of clusters, it is convenient to use the following more specific char- Sputtering of solids by ion bombardment plays an acteristics: the ionization coefficient important role in many fields of science and technology. This is primarily associated with technological J Q κ Q = ------N (1) applications in micro- and nanoelectronics and space N Q = 0 and fission technologies. The number of papers devoted J N to both the application and fundamental studies of sput- and the ionization degree of clusters with number of tering has increased considerably in recent years (see, atoms N for example, recent reviews [1Ð5] and the literature cited therein). A theoretical description and calcula- Q η Q J N tions of sputtering processes are extremely difficult, N = ------, (2) first of all because of the many-particle nature of the J N problem both at the stage of ion penetration into a solid Q and at the stage of formation of sputtering products, where J N is the flow of clusters escaping a surface con- which consist not only of single target atoms but also of sisting of N atoms and having a charge Q = 0, ±1, polyatomic particles, i.e., clusters. Sputtering processes ± Σ Q involving single target atoms are usually described [1] 2, … and JN = Q J N is the total flow of neutral and using the so-called collision cascade sputtering mecha- charged clusters of N atoms. nism [6]. Sputtering mechanisms in the form of two or A large number of both experimental and theoretical more bonded target atoms are still the subject of discus- works (see, for example, [9]) are devoted to studies of sion [1, 2, 5], since they describe the formation of large charge state formation for monatomic particles sput- clusters inadequately and differ considerably from the tered or scattered at a metal surface. The mechanism of cluster formation mechanisms in gas and plasma. At the charge state formation for polyatomic particles has present time, hopes of performing calculations based been studied much less extensively, both theoretically on “first principles” are connected with computer sim- and experimentally. Notably, in [10], the ionization ulations using molecular dynamics methods [1, 2, 5] κ +1 (see also the calculations in [7, 8]). However, these cal- coefficient N was found to depend on the number of culations are technically complicated, especially as the κ +1 atoms in a cluster: N increases drastically with number of atoms in the cluster increases, and are hard ≥ to reproduce in other investigations. The formation of increasing N, saturating at N 5, so that variations of κ +1 the charge state of the sputtered surface material is also N with a further increase in N are negligible. From a complicated problem. The ionization degree η = JQ/J this observation, a conclusion was made about the uni- is usually used as a quantitative characteristic of the versal character of the power law established empiri- charge state, where JQ is the flow of particles of charge cally for the relative yield of large clusters, both neutral Σ Q escaping the surface and J = QJQ is the total flow of and charged. In [11], an attempt based on semiempiri- escaping particles. However, to characterize the charge cal estimates of the degree of cluster excitation was

1064 MATVEEV undertaken to account for the formation of singly conduction band of the metal and the atoms of the clus- charged positive clusters as they leave the metal by sug- ter is possible. When the distance between the cluster gesting that the thermionic emission of an electron and the metal exceeds ξ, the electron exchange process from the cluster occurs as it leaves the metal. However, is stopped nonadiabatically. Below, when speaking of agreement with experimental results could only be electrons in the cluster, we mean only valance electrons reached by assuming the existence of clusters with and we call the set of corresponding states the cluster extremely high temperatures of ~1 eV. Moreover, the conduction band. We assume that an exchange process formation mechanisms for clusters with charges other between the conduction bands of the cluster and the than +1, including negative clusters, are not clear. In metal is possible. Then, the average number of elec- this paper, we have suggested a mechanism for ion trons nτ occupying the energy level ετ according to the bombardment sputtering of a metal in the form of neu- ε µ Θ tral and charged clusters with number of atoms N ≥ 5. Fermi distribution is equal to nτ = {exp[( τ Ð )/ ] + The mechanism is based on simple physical assump- 1}Ð1, where Θ is the temperature and µ is the chemical tions and is in good agreement with experimental ∆ 2 results. potential. Let us denote by nτ the rms deviation of the occupation numbers nτ from their equilibrium values 2 2 FORMULATION OF THE PROBLEM nτ ; then, ∆nτ = (nτ Ð nτ) = nτ (1 Ð nτ ) [15]. Obvi- Σ The process of charge state formation is an integral ously, the average number of electrons is Ne = τ nτ . part of the sputtering mechanism. Our consideration The number of electrons in the cluster conduction band largely relies on the suggestion made earlier in many 2 works [1] that large clusters leave a solid as an integral is N ; therefore, by definition, ∆N 2 = (N Ð N ) = block of atoms. This assumption was further developed e e e e Σ ∆ 2 in a sputtering model proposed in [12, 13]. According τ nτ . A cluster with Ne electrons in the conduction to [12, 13], the probability of events corresponding to the correlated motion of a block of N atoms with total band will be neutral if Ne = Ne , where Ne is the aver- momentum k can be expressed as age number of electrons in the cluster conduction band, equal to the number of atoms N in the cluster multiplied Wn by the valence γ (more precisely, by the number of elec- N trons a neutral metal atom gives to the conduction  β  k  2   β  k  2  γ expÐ------q Ð ----  Ð expÐ------q + ----  band). Thus, the cluster charge is Qe = (Ne Ð N )e, ប2 N ប2 N where e is the electron charge.  n0   n0  (3) = ------, k β Further calculations by these equations require 4q------N ប2 information about the cluster’s electron structure and n0 cannot be performed in the general case. However, assuming that the cluster size is large enough and the where β = 1/(2mω/ប); ω is the characteristic oscillation electron states are quasi-continuous, summation over frequency of the target atoms, m is the mass of the tar- the electron states can be replaced, in the standard way, get atoms, and q has the meaning of the average by integration over the band by the rule given in [15] momentum received by a metal atom at the early development stage of a collision cascade. ∞ 3/2 ε (ε ) (ε) 1/2Vme ε The probability (3) is the result of summing over all ∑ f τ = ∫ f 2 ------d , vibrational excited states of the cluster up to some prin- π2ប3 τ 0 cipal quantum number n0 when the energy accumulated in excited oscillations is sufficient to destroy the cluster. where m is the mass of the conduction band electron ≈ ∆ បω e This can be realized at n0 /( ) when the oscillation and V is the cluster volume. energy of all the oscillators (atoms in the cluster) becomes sufficient to remove one atom from the poten- Thus, tial well of depth ∆ describing the bonds between target ∞ atoms. 3/2 [(ε µ) Θ] ∆ 2 1/2Vme ε ε exp τ Ð / Ne = 2 -----2------3--∫d ------2- π ប {exp[(ετ Ð µ)/Θ] + 1} Model of Charge State Formation 0 ∞ (4) 3/2 Let us define the charge state of a block of N atoms. 1/2Vme expz As in the statistical derivation of the SahaÐLangmuir = 2 ------∫ dzΘ Θz + µ------. π2ប3 {expz + 1}2 equation [14], we will assume that as the cluster moves Ðµ/Θ a certain distance ξ (called critical) away from the metal surface, an exchange of electrons between the At temperatures less than the degeneracy tempera-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 EMISSION OF CHARGED CLUSTERS DURING METAL SPUTTERING BY IONS 1065 µ Θ ӷ ture, i.e., at / 1, of the contacting surface SN between the block of N 3/2 atoms and the metal. Let us assume that the contacting 2 1/2Vme surface is a hemisphere with its center on the initial ∆N ≈ 2 ------µΘ, e π2ប3 metal surface before sputtering. Its radius is obviously connected with the number of atoms in the cluster where the chemical potential of a degenerate Fermi gas 1 -- 3 with the number of particles Ne in a volume V is [15] 3 RN = N----π----- , (9) 2 2 d 2 -- -- ប2 3 µ ( π2)3 Ne = 3 ----------- . where d is the number of atoms in a unit volume. 2me V Then, the binding energy of a neutral cluster, which Thus, the rms deviation of the cluster charge from is proportional to SN, is ( γ ) 2 the equilibrium value Qe = Ne Ð N e = 0 is equal -- 2 2 3 -- -- 1 0 2  3  3 3 to: U = σS = σ2πR = σ ------N = δN , (10) N N N 2πd 1 1 -- -- 2 3 2 2 2 2 3 m ΘN  where δ obviously means the surface binding energy of (∆ ) ∆ e -----e QN = e Ne = e -----4 ------2---  , -- ប V a neutral cluster per atom in the cluster. π3 If a cluster carries away a charge Q , an energy U of (5) e c 1 2 interaction with the image charge should be added to -- -- 1 2 m Θ 3 -- 0 Q 3 e V γ 3 U N to obtain its binding energy with the metal U N . V = -----4 ------2------- N. -- ប N The energy of interaction can be expressed as π3 Qe U = ------, (11) Probabilities PN(Q) for values of Q can be deter- c 4χ mined from the standard equation for the probability of fluctuations where χ is some effective distance at which the overlap- ping of the wave functions of electrons of the cluster  2  ( ) 1 1 Q and the metal conduction bands vanishes as the cluster PN Q = ------expÐ------, (6) moves away. Therefore, we shall assume that χ is con- DN  2(∆ )2  2 QN nected with the work function ϕ of the target metal as where the normalization factor DN is determined by ប ± ± χ = ------. (12) summing over all possible values of Q = 0, 1, 2, … φ and is equal to 2me Thus, the energy binding the cluster of charge Q  2  e 1 Q and the metal can be expressed as DN = ∑ expÐ------. (7) 2 2  (∆ )  2 Q QN -- Q δ 3 U N = N + Uc. (13) Thus, to obtain the probability W Q for a cluster with N A cluster that before escaping (before overcoming N atoms and charge Qe to leave the metal, it is necessary Q to multiply the probability WN (see formula (3)) by the binding energy U N ) was imparted the momentum PN(Q): k will then move with the kinetic energy TN equal to Q ( ) 2 W N = W N PN Q . (8) k σ T N = ------Ð SN Ð Uc. (14) Equation (8) describes the probability for a cluster 2mN of N atoms to escape the metal if the cluster’s kinetic Taking TN = 0, we find from (14) the minimum energy is sufficient to break the bonds between the clus- momentum kQN the cluster needs to overcome the bind- ter of N atoms and other atoms of the metal. If the clus- Q ter is neutral, this energy is proportional to the square ing energy U N :

1 In principle, the fact that the equilibrium cluster charge is equal to k = k 2 + 2mNU , (15) zero follows from the assumption that the Fermi levels in the clus- QN 0N c ter and the metal coincide; if this is not the case, asymmetry between positively and negatively charged clusters will be 2 In the strict sense, χ is defined only to within an order of magni- observed. The corresponding changes to the resulting formulas tude, but we put the sign of equality in the definition of χ so as not can easily be made. to introduce extra fitting parameters.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1066 MATVEEV

1 1 5 5 ------( )2 ν ν [ σ ]2 [ δ]2 6 6 where k = 2mNT N + k0N + Uc , k0N = k01N , = k0N = 2mN SN = 2m N = k01 N , (16) β 2 δ ∆ 5/6, and the fact that k01 /n0 = / is taken into 1 -- account. δ 2 where k01 = (2m ) is the minimum momentum per We deﬁne the ionization degree (2) as atom in a cluster necessary to eject a neutral cluster. Q η Q W N Now, to obtain the escape probability of a whole N = ------, (20) cluster of N atoms, it is necessary to integrate the prob- W N ability formula (8) over all possible values of k, under where | | the condition that k > kQN and k is directed outside, i.e., is found within the solid angle of 2π: Q W N = ∑W N ∞ Q Q Q 3 π 2 ( ) W N = ∫ W N d k = ∫ 2 k dkW N PN Q . (17) Q is W N summed over all possible values of Q = 0, ±1, > > k k0N k k0N ±2, …. The ionization coefficient, which according to To integrate (17), we can write the probability in the (1) is the ratio of the number of clusters with charge Q form in which the integrand represents the distribution to the number of neutral clusters (for a given cluster size N), can be deﬁned as of the kinetic energy TN of the clusters; thus, we write (14) in the form Q κ Q W N N = ------Q--- -=-- --0. (21) 1 2 2 W T = ------(k Ð k Ð U ). (18) N N 2mN 0N c

By substituting the variables in (17), we obtain the RESULTS final expression for the escape probability of a cluster Our consideration apparently cannot be used to of N atoms with charge Q describe the sputtering of single atoms or small clusters, while comparison with experimental results has ∞ led to the conclusion [12, 13] that the model is valid for Q δ ÐN π ( ) q k clusters containing a certain minimum number of W N = 2 ∫kmNdT N PN Q 4------∆--- k01 Nk01 atoms (N ≥ 5). In experiments, the relative probability 0 of the yield of clusters with different numbers of atoms  δ 2  is usually measured. Thus, to compare it with experi- ×  q k  expÐ∆---------Ð ------  (19) mental data, the probability given by (19) should first  k01 Nk01  be divided by the probability of escape of a cluster with N = 5 (to be exact, any value in the range N ≥ 5 can be N  δ 2  chosen, but N = 5 is more convenient in this case);  q k  experimental results are normalized in a similar way. Ð expÐ∆---------+ ------  ,  k01 Nk01  Also, any arbitrary units can be defined to meet particular needs. Values of the binding energy δ (eV) and the variable δ ប Values of energy (eV) [16] and variable parameter q (at.u.) parameter q (atomic units = me = e = 1) for different for various ionÐtarget combinations ionÐtarget combinations are given in the table. In all Binding energy cases listed in the table, good agreement with experi- Incident ion Target q, au δ, eV [16] ment is observed. In the calculations, we kept the number of fitting parameters to a minimum and assumed Ar+ (5 keV) Ta 8.1 550 ∆ = δ. In general, analysis of the direct consequences AuÐ (6 keV) Ta 8.1 550 from Eq. (19) is difficult; thus, we shall consider the Ar+ (5 keV) Nb 7.47 380 results of numerical simulations and experiments pre- AuÐ (6 keV) Nb 7.47 310 sented in Figs. 1Ð8, which appropriately illustrate the general situation. Ar+ (5 keV) Ag 2.96 170 The simplest characteristics of the charge state dis- Note: Some differences in the values of the fitting parameter q from those we used earlier [12, 13] for tantalum and nio- tribution of clusters with a given number of atoms N is bium can be explained by the fact that in [12, 13], charge the ionization coefficient κ Q equal to the ratio of the state formation was not considered and that for tantalum, the N value of the sublimation energy was taken from a different number of clusters with charge Q ≠ 0 to the number of source. neutral clusters of the same size N. The dependence of

TECHNICAL PHYSICS Vol. 45 No. 8 2000 EMISSION OF CHARGED CLUSTERS DURING METAL SPUTTERING BY IONS 1067

κ Q κ 1 N N 0.6 100 0.5 κ1 10 10–2 0.4 κ1 10–4 0.3 5 0.2 10–6 κ2 10 0.1 κ2 10–8 5 0 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 12 14 N Θ, K Fig. 2. The dependence of the single ionization coefﬁcient Fig. 1. The dependence of the coefﬁcients of single and dou- on the number of atoms in Ag clusters: dots, experiment ble ionization of clusters on target temperature Θ. [10]; dashed line, calculation.

ηQ the ionization coefficients of Ta clusters with N = 5 and N 10 atoms as a function of target temperature Θ is given 1.0 in Fig. 1, illustrating the following general conclusions: η0 (1) The charge state varies with target temperature, and 0.8 5 the ionization coefficient increases with increasing temperature. (2) The larger the cluster charge, the more rarely that cluster occurs; for example, the number of 0.6 clusters with charge 2 is, as a rule, much less than those 0.4 Q with charge 1. (3) Larger clusters are more strongly Ση5 ionized; for example, κ 1 > κ 1 and κ 2 > κ 2 . 10 5 10 5 1 0.2 η5 The dependence of the ionization coefficient for Ag clusters on cluster size N at a target temperature Θ = 0 700 K is shown by the dashed line in Fig. 2. An impor- 0 500 1000 1500 2000 2500 3000 tant feature is the tendency to saturation of the ioniza- Θ, K tion coefficients with increasing cluster size. Qualita- tively similar behavior was observed in experiments Fig. 3. The ionization degree of Ta clusters as a function of 3 [10], whose results are also presented in Fig. 2. Since ηQ target temperature. N = 5; ∑ 5 , the ionization degree an ionization coefficient describing only charged clus- summed over all cluster charges corresponding to the total ters cannot account for the behavior of neutral clusters, relative number of clusters with arbitrary, but nonzero, Q η ηQ η0 it is necessary to introduce the ionization degree N as charge, i.e., ∑ 5 = 1 Ð 5 . the ratio of the number of clusters (of size N) with charge Q = 0, ±1, ±2, … to the total number of clusters of the same size N. The ionization degree shows charge ηQ redistribution between clusters of a given size; for N example, a rise in the number of charged clusters is 1.0 accompanied by a corresponding fall in the number of neutral clusters. This behavior of the ionization degree 0.8 0 is shown in Fig. 3 (for N = 5) and Fig. 4 (for N = 10). η10 Q Q Q 0.6 The relative yields Y = W N /W5 for singly charged Q N Ση10 1 0 (Y N ) and neutral (Y N ) clusters of TaN and NbN as a 0.4 1 function of the number of their atoms are shown in η10 Figs. 5Ð8. Tantalum and niobium targets were sputtered 0.2 by singly charged AuÐ1 ions at an energy of 6 keV and Ar+1 ions at an energy of 5 keV at target temperatures 0 0 500 1000 1500 2000 2500 3000 3 To be precise, in these experiments, the target temperature was Θ, K not registered, despite its possible dependence on laser irradiation. Fig. 4. Same as in Fig. 3 for N = 10.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1068 MATVEEV

1 0 YN ...... YN ......

101 101

100 100

10–1 10–1

10–2 10–2

3 5 7 10 3 5 7 10 N N 1 +1 Fig. 5. The relative yield Y of Ta singly charged clus- N N 0 0 ters as a function of N. Bombardment by AuÐ1 ions, Θ = Fig. 6. The relative yield Y N of TaN neutral clusters as a 1 +1 Θ 2273 K. Solid line, calculated Y values; heavy dots, function of N. Bombardment by Ar ions, = 300 K. Solid N 0 experiment [17, 18]; dashed line, calculated mass spectrum line, calculated values of Y N ; dots, experiment [19]; dashed 0 of neutral Y clusters; dotted line, power law curve [19] 1 N line, calculated mass spectrum of singly charged Y N clus- normalized to N = 5, i.e., values of the function NÐ8.5/5Ð8.5. ters; dotted line, see Fig. 5.

1 Y0 YN ...... N ......

101 101

100 100

10–1 10–1

10–2 10–2

3 5 7 10 3 5 7 10 N N

1 +1 Fig. 7. The relative yield Y N of NbN singly charged clus- Ð1 Θ 0 0 ters as a function of N. Bombardment by Au ions, = Fig. 8. The relative yield Y N of NbN neutral clusters as a 1 +1 Θ 2273 K. Solid line, calculated values of Y N ; dots, experi- function of N. Bombardment by Ar ions, = 300 K. Solid ment [20]; dashed line, calculated mass spectrum of neutral 0 line, calculated values of Y N ; dots, experiment [19]; dashed 0 Y N clusters; dotted line, power law curve [19] normalized line, calculated mass spectrum of singly charged clusters Ð8.1 Ð8.1 1 to N = 5, i.e., values of the function N /5 . Y N ; dotted line, see Fig. 7.

Θ = 2273 K and Θ = 300 K, respectively. The power Thus, only on the assumption that a cluster escapes law curve [19] normalized to a cluster with 5 atoms, as a whole can the ﬂuctuation mechanism of cluster i.e., values of the functions NÐ8.5/5Ð8.5 for tantalum and charge state formation be developed and the depen- N−8.1/5Ð8.1 for niobium, is also plotted in the ﬁgures for dence of the charge state distribution on cluster size and comparison. It is worth noting that mass spectra of the target temperature be described. It is known that it is neutral clusters are weakly dependent on temperature, technically much simpler to register charged particles whereas mass spectra of singly charged clusters are than neutral ones. Charge state formation processes, appreciably affected by target temperature but i.e., emission processes of charged and neutral parti- approach the mass spectra of the neutral clusters as the cles, are often interrelated. Therefore, experimental temperature increases. results for charged particles provide an indirect way to

TECHNICAL PHYSICS Vol. 45 No. 8 2000 EMISSION OF CHARGED CLUSTERS DURING METAL SPUTTERING BY IONS 1069 recover data for neutral particles and thus allow one to 11. V. Kh. Ferleger, M. B. Medvedeva, and I. A. Wojcie- simplify the experimental setup. chowski, Nucl. Instrum. Methods Phys. Res. B 125, 214 (1997). 12. V. I. Matveev and P. K. Khabibullaev, Dokl. Akad. Nauk REFERENCES 362, 191 (1998). 1. Sputtering of Solids: Fundamentals and Applications, 13. V. I. Matveev, S. F. Belykh, and I. V. Verevkin, Zh. Tekh. Collection of Articles, Ed. by E. S. Mashkova (Mir, Mos- Fiz. 69 (3), 64 (1999) [Tech. Phys. 44, 323 (1999)]. cow, 1989). 14. L. N. Dobretsov and M. V. Gomoyunova, Emission Elec- 2. H. H. Andersen, K. Dan. Vidensk. Selsk. Mat.-Fys. tronics (Nauka, Moscow, 1966). Medd. 43, 127 (1993). 3. H. M. Urbassek and W. O. Hofer, K. Dan. Vidensk. 15. L. D. Landau and E. M. Lifshitz, Statistical Physics Selsk. Mat.-Fys. Medd. 43, 97 (1993). (Nauka, Moscow, 1976; Pergamon, Oxford, 1980), Part 1. 4. I. A. Baranov, Yu. V. Martynenko, S. O. Tsepelevich, and Yu. N. Yavlinskiœ, Usp. Fiz. Nauk 156, 478 (1988) [Sov. 16. C. Kittel, Introduction to Solid State Physics (Wiley, Phys. Usp. 31, 1015 (1988)]. New York, 1976; Nauka, Moscow, 1978). 5. G. Betz and K. Wien, Int. J. Mass Spectrom. Ion Pro- 17. S. P. Belykh, U. Kh. Rasulov, A. V. Samartsev, and cesses 140, 1 (1994). I. V. Veryovkin, Nucl. Instrum. Methods Phys. Res. B 6. P. Sigmund, Phys. Rev. 184, 383 (1969). 136, 773 (1998). 7. A. Wucher and B. Y. Garrison, J. Chem. Phys. 105, 5999 18. S. F. Belykh, U. Kh. Rasulov, A. V. Samartsev, et al., (1996). Mikrochim. Acta., Suppl. 15, 379 (1998). 8. Th. J. Colla, H. M. Urbassek, A. Wucher, et al., Nucl. 19. A. Wucher and W. Wahl, Nucl. Instrum. Methods Phys. Instrum. Methods Phys. Res. 143, 284 (1998). Res. B 115, 581 (1996). 9. M. L. Yu, in Topics of Applied Physics Sputtering by Par- 20. S. F. Belykh, B. Habets, U. Kh. Rasulov, et al., in ticle Bombardment III, Ed. by R. Behrisch and K. Witt- Abstract Book of SIMS Europe, Muenster, 1998, p. 5. maack (Springer-Verlag, New York, 1991), pp. 91Ð160. 10. W. Wahl and A. Wucher, Nucl. Instrum. Methods Phys. Res. 94, 36 (1994). Translated by M. Astrov

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1070Ð1074. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 114Ð117. Original Russian Text Copyright © 2000 by Lobastov, Lysan, Peshekhonov, Smirichinskiœ.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Coordinate Correction for Parallax for Gamma-Ray Beams Detected by Counters Based on Thin-Film Drift Tubes S. P. Lobastov, V. M. Lysan, V. D. Peshekhonov, and V. I. Smirichinskiœ Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia E-mail: [email protected] Received June 1, 1999

Abstract—A model was put forward for finding the angle of incidence of a narrow gamma-ray beam and refining its coordinate when the beam is detected by cathode-readout counters based on thin-film drift tubes. The model can be used in solving small-angle scattering problems, which are widely met in X-ray diffraction analysis. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION not orthogonal to the anode wire may be resolved. A similar problem appears, for instance, in structure anal- The determination of neutral particle coordinates ysis related to small-angle scattering; in this case, the with the use of gas-filled detectors based on thin-film xenon mixture under excessive pressure is more appro- drift tubes is always complicated by parallax. If the par- priate. ticle path is deflected from the plane normal to the anode wire, the performance of these detectors degrades in comparison with the usual case. The spatial RESULTS AND DISCUSSION resolution of the detector can be improved by introduc- Model ing a correction for parallax [1] on condition that the beam coordinate distribution in the detector is given by It follows from experimental data (Figs. 2a, 2b) that, the Gaussian function. if a narrow beam is strongly deflected from the normal, the coordinate distribution of gamma quanta cannot be This paper deals with an experimental check of the represented by a broadened and shifted Gaussian func- parallax-correction method and extends it to the case tion. To treat the data, we assumed, in contrast to [1], when a linear position cathode-readout detector on the that, for each gamma quantum absorbed in the detector, base of thin-film drift tubes is used [2, 3]. the coordinate distribution obeys a Gaussian curve, and the parameter σ of this distribution is the spatial resolu- EXPERIMENTAL 1 A special collimator (Fig. 1) with orthogonal slit 3 3.6 mm of width 40 µm and inclined slit 4 of width 80 µm was designed for investigations. The detector [2] was purged by a Ar : CH (80 : 20) or Xe : CH (80 : 20) gas 4 4 2 mixture under normal pressure. It was irradiated by a 14 mm 55Fe source through slits 3 and 4 simultaneously. The 4 collimator can move up and down over a distance of 3 0.3 mm 1 mm with a precision of better than 20 µm. Beam orthogonality was thought to be provided if collimator Xe/CH4 movement did not change the coordinate of the peak leaving slit 3 (within the accuracy defined by the spatial 5 resolution of the detector). 10 mm Typical coordinate distributions after passing both slits for the argon and xenon mixtures are presented in 6 Fig. 2. It is evident that a gas mixture with a great absorption factor is preferable. In this case, two or more Fig. 1. Schematic of experiment: 1, 55Fe source; 2, collima- nearby peaks from narrow gamma-ray beams that are tor; 3 and 4, slits; 5, anode; and 6, strips.

COORDINATE CORRECTION FOR PARALLAX FOR GAMMA-RAY BEAMS 1071 γ γ γ N Φ Φ (a) 2000

1500 n

1000 a

500 Φ

γ (b) Φ 2500

2000

1500 Fig. 3. Geometry of the experiment.

1000 γ 500

0 0.1 0.2 0.3 0.4 0.5 Y x, cm dV Fig. 2. Typical coordinate distributions of the number of X R x0 α gamma quanta for (a) argon and (b) xenon gas mixtures (one bin along the x-axis equals 50 µm). xm Z tion of the detector. We assume also that σ is constant d/2 for each single event, being independent of the drift α path of the primary electron cloud toward the anode wire. Physically, this means that we neglect the diffusion of the drifting primary electron cloud. Fig. 4. Schematic of the process. The geometry of the experiment is shown in Fig. 3. The situation when the gamma ﬂux vector, the normal to the plane of the ﬂux, and the anode-wire axis are N x2/ndf 11.73/11 noncomplanar was not considered. According to our 600 Constant 563.1 ± 0.01 assumption, the differential probability that the event Mean 0.4478 ± 0.00006 cm Sigma 45.29 ± 0.95 µm will occur at a point x is given by the normal distribu- 500 tion Entries 7097 Mean 0.4474 cm 400 RMS 56.18 µm ∼ ( ( )2 σ2) dPg exp Ð x Ð xm /2 , (1) 300 where xm is the projection of the point where a gamma quantum is absorbed (Fig. 4). 200 The probability that the gamma quantum will be absorbed in the gas mixture is given by 100

dP ∼ exp(Ðµs), (2) 0 a 0.43 0.44 0.45 0.46 0.47 x, cm where µ is the absorption factor of the gas mixture in the detector and s is the gamma quantum range in the Fig. 5. Fitting by the Gaussian function (one bin along the mixture. x-axis equals 10 µm).

TECHNICAL PHYSICS Vol. 45 No. 8 2000

1072 LOBASTOV et al. ± ° α ° ± N χ2/ndf 87.9/46 ﬁnd x0 = 0.2292 0.0011 cm and (90 Ð ) = 13.20 x0 0.2292 cm 0.12°. The latter value agrees well with the measured (90° – α) 13.20° angle (13.25° ± 0.15°) of the used collimator. The ﬁt- 600 ting results are presented in Fig. 6.

400 CONCLUSIONS Mathematical model (4) for coordinate distribution 200 in a cathode-readout detector based on thin-film drift tubes is presented. The model takes into account the detector and incident beam geometries. It gives a good 0.15 0.20 0.25 0.30 0.35 fit to the experimental data and can be employed in x, cm experiments with space-separated gamma-ray beams, for instance, in X-ray diffraction analysis. Fig. 6. Fitting by model (4) (one bin along the x-axis equals 50 µm). A method [see (5)] is suggested that not only significantly refines the coordinate x0 but also finds the parallax-related angular coordinate α for a narrow gamma These processes are mutually independent; then, the beam from the shape of the peak in its coordinate dis- total differential probability that the event will be tribution. detected at the anode, including the probability that the gamma quantum is absorbed in a volume dV, takes the Experimental coordinate distributions were obtained form for the designed detector (Fig. 2). The angular coordinate of a narrow beam of gamma quanta, which are dP(x) ∼ exp(Ðµs)exp(Ð(x Ð x )2/2σ2)dV. (3) absorbed for the most part through photoeffect, was m first obtained from these data. Integrating (3) over the region where gamma quanta meet the gas mixture, we obtain the total probability The method can be employed for creating planar that the event will be detected at a point x: detectors for synchrotron beam imaging [4].

2 2 x + d/(2 sinα) R R Ð z The use of numerical integration (see Appendix) 0 will inevitably increase the computational time when ( ) ∼ P x ∫ dx0' ∫ dz ∫ dy the model becomes more complicated to improve the ( α) (4) α x0 Ð d/ 2 sin ÐR 2 2 accuracy of obtaining the coordinates x0 and (for Ð R Ð z instance, if diffusion or a more intricated incident beam × { ( µ ) ( ( )2 σ2)} exp Ð s exp Ð x Ð xm /2 , geometry is considered). Clearly, the greater the computational resource, the less the computational time.1 where

[( 2 2)1/2 ] α α s = R Ð z Ð y /sin , xm = x0' + ycot , APPENDIX x0 is the intersection point of the anode wire and mid- The main difficulties in taking the integral in (4), plane of a gamma flux (by definition, the midplane is parallel to the collimator slit), and d is the gamma flux ( α) 2 2 x0 + d/ 2 sin R R Ð z width (we neglect the beam divergence). ( ) ∼ P x ∫ dx0' ∫ dz ∫ dy Thus, the model is specified by the function P(x, x0, σ α x Ð d/(2 sinα) ÐR 2 2 , ). 0 Ð R Ð z × { ( µ ) ( ( )2 σ2)} exp Ð s exp Ð x Ð xm /2 , Fitting The parameter σ is determined as follows: in the are as follows: (1) This triple integral cannot be taken absence of parallax (α = 90°), the peak leaving slit 3 analytically and (2) Immediate numerical integration (Fig. 2b) is approximated with expression (4). This requires much computational time because of the three- gives σ ≈ 50 µm. For this experimental distribution, the dimensional domain of integration. ≈ µ root-mean-square deviation is RMS 56 m (Fig. 5). If However, (4) can be analytically integrated succes- this peak is approximated by a normal Gaussian func- ' tion within the range where a good correlation with the sively for the variables y and x0 . On simplification, we σ ≈ µ experiment is observed, we obtain G 45 m. 1 In this work, an AMD-K-6-II (333 MHz/32-Mb RAM) computer Now, putting σ = 50 µm, we apply this model to was used. The time to compute the data presented in Fig. 6 was approximate the data for inclined slit 4 (Fig. 2b) and 25 min.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 COORDINATE CORRECTION FOR PARALLAX FOR GAMMA-RAY BEAMS 1073 obtain

R/2  2x Ð 2x Ð 2 R2 Ð z2 cot(α) + d cosec(a) , , σ, α ≡  ( 2 2µ (α))  0  I(x x0 ) ∫ dz Ð 2 + exp 2 R Ð z cosec Erf ------  2 2σ  ÐR/2

Ð 2x + 2x + 2 R2 Ð z2 cot(α) + d cosec(a) + exp(2 R2 Ð z2µcosec(α))Erf------0------  2 2σ 

2 2 2  µsec(α)(2x Ð 2x Ð d cosec(α) + µσ sec(α))  + exp(2 R Ð z µcosec(α)) exp ------0------  2  

 Ð 2x + 2x Ð 2 R2 Ð z2 cot(α) + d cosec(α) Ð 2µσ2 sec(α) × Erf------0------   2 2σ 

Ð 2x + 2x + 2 R2 Ð z2 cot(α) + d cosec(α) Ð 2µσ2 sec(α)  Ð Erf------0------   σ   2 2 (5) 2 µsec (α)(2x Ð 2x + d cosec(α) + µσ sec(α)) + exp ------0------ 2 

 2x Ð 2x Ð 2 R2 Ð z2 cot(α) + d cosec(α) + 2µσ2 sec(α) × ÐErf------0------   2 2σ 

2x Ð 2x + 2 R2 Ð z2 cot(α) + d cosec(α) + 2µσ2 sec(α)   + Erf------0------    2 2σ   

Ð 2x + 2x Ð 2 R2 Ð z2 cot(α) + d cosec(α) + Erfc------0------  2 2σ 

2 2 2x Ð 2x + 2 R Ð z cot(α) + d cosec(α)   1 + Erfc------0------  sin(α) ------. σ 2 2  2 2    exp(2 R Ð z µcosec(α))µ

It only remains for us to integrate over z numeri- where the integration of the numerator and denomina- cally.2 The normalization factor is defined by tor is performed numerically with an accuracy of five significant digits. 1 K = --∞------. Because of the complexity of mathematical model (6) (7), the Newton method was initially used to minimize ( , , σ, α) ∫ dxI x x0 the χ2 functional, which provides a rapid convergence Ð∞ with the minimum number of iterations. However, this In the final form, the model is given by numerical method does not give an explicit functional I(x, x , σ, α) α σ ( , , σ, α) 0 ≡ dependence of the parameters {x0min, min, min}, mini- p x x0 = --∞------KI, (7) mizing the functional, on experimental values {Ni, xi}. ( , , σ, α) ∫ dxI x x0 This makes the accurate estimation of errors (standard Ð∞ deviations) introduced into the obtained parameters more difficult. Therefore, model (7) was first interpo- 2 Here, the domain of integration over z differs from that in general lated in x and the parameters in the vicinity of {x0min, expression (4), because actually the collimator (window) is R = α α 0.5 mm wide throughout the tubes. min, min} by a third-degree polynomial. Then the min-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1074 LOBASTOV et al. imum of the χ2 functional is searched using polynomial REFERENCES regression [5]. This method provides an explicit depen- 1. S. Yu. Grachev and V. D. Peshekhonov, Communication α σ dence of {x0min, min, min} on {Ni, xi}, resulting in a JINR, No. E13-96-83 (1996). more accurate estimation of errors in the sought param- 2. V. N. Bychkov et al., Nucl. Exp. Tech. 41, 315 (1998). 3. R. A. Astabatyan et al., Communication JINR, No. P13- eters. 98-271 (1998). 4. V. N. Bychkov et al., Nucl. Instrum. Methods Phys. Res. A 367, 276 (1995). ACKNOWLEDGMENT 5. J. R. Taylor, An Introduction to Error Analysis (Univ. Science Books, California, 1982; Mir, Moscow, 1985). We thank Yu.K. Potrebennikov for helpful consulta- tions. Translated by B.A. Malyukov

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1075Ð1080. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 118Ð124. Original Russian Text Copyright © 2000 by Randoshkin, Gusev, Kozlov, Neustroev.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Advantages of Anhysteretic Magnetooptic Films in Nondestructive Testing V. V. Randoshkin1, M. Yu. Gusev2, Yu. F. Kozlov2, and N. S. Neustroev2 1 Magnitooptoelektronika Laboratory, Institute of General Physics, Ogarev Mordovian State University, Russian Academy of Sciences, Bol’shevistskaya ul. 68, Saransk, 430000 Russia 2 Malinin Materials Science Research Institute, Moscow, 103460 Russia Received August 3, 1999

Abstract—The potentialities provided by bismuth-containing garnet ferrite ﬁlms with uniaxial anisotropy and easy-plane anisotropy for the visualization of spatially nonuniform magnetic ﬁelds in magnetooptic nondestructive testing are compared. © 2000 MAIK “Nauka/Interperiodica”.

Nondestructive testing and several physically simi- In the films of type II not subjected to an external lar techniques designed for other purposes [1Ð26] can magnetic field, the magnetization vectors lie in the film exploit the magnetooptic visualization of spatially non- plane even if the film splits up into domains of opposite uniform magnetic fields [1, 8, 27, 28]. Two types of bis- polarities. Therefore, the plane of polarization of light muth-containing single-crystal garnet ferrite films propagating normally to the film does not rotate. The (BcSCGFFs) with giant Faraday rotation [1, 29] are plane of polarization may rotate only in the case of used: with uniaxial magnetic anisotropy (type I) and Bloch walls. Bloch lines separating DW parts where the with easy-plane magnetic anisotropy (type II). For plane of polarization rotates in the opposite directions BcSCGFFs of type I, an inhomogeneous magnetic field can be visualized. can be judged from the domain configuration, while for The influence of a time-invariable but space-inde- the films of type II, the surface distribution of the angle pendent magnetic field on the films of type I depends on between the magnetization and film plane bears neces- the field intensity and gradient. A sufficiently intense sary information. alternating magnetic field with the spatial period considerably exceeding the equilibrium size of domains In the absence of an external magnetic field, the splits up the film into large oppositely polarized films of type I split up into oppositely polarized domains with smooth walls. These films visualize the domains and the areas covered by the domains of either lines of equal intensity H = 0. In particular, a gradient polarity are roughly equal. The difference between the magnetic field H = βx that can be produced by two per- areas increases with coercive force H , which is usually b c manent magnets with the C-shaped section forms pla- 0.1–1 Oe for these films. Owing to coercive force, the nar straight DWs. If then a constant bias field H is films of type I exhibit hysteresis at magnetization rever- b sal under a bias field H applied normally to the film. applied perpendicular to the film surface, the DWs will b move to another equilibrium position, again going In the films of type I, the plane of polarization in the through the points where the total magnetic field equals neighboring domains rotates in the opposite direction zero. In other words, lines of equal intensity H = ÐHb Θ by the same angle F, which does not depend on the are visualized. By varying the bias field, one can visualize lines of equal intensity for any H. bias magnetic field. As Hb increases, the area occupied by domains with the magnetization aligned with the Lines of equal intensity can be visualized even if an field grows at the expense of unfavorably magnetized intense magnetic field varies very smoothly (the gradi- domains up to the complete magnetization of the film. ent is small). In this case, the border between two oppo- The motion of domain walls (DWs) accounts for this sitely magnetized domains is no longer a planar DW, process. By changing the angle between the analyzer but a transient region of strip domains. The smaller the and polarizer in a polarizing microscope, one can com- gradient, the wider this transient region. In particular, a β pletely suppress monochromatic light passing through gradient magnetic field Hb = x induces a “comb” of the domains of a specific polarity. If white light is used, strip domains. It is significant that the curves envelop- domains become differently colored, the color depending the transient region go through points where the ing on the angle ϕ between the optical axes of the polar- external magnetic field H is equal to the saturation field ± izer and analyzer. When the polarizer and analyzer are of the film Hs. If then a bias field is applied to the film, crossed (ϕ = 0), domains have the same color and DWs these two curves will shift. The monodomain areas of appear dark. one polarity will expand; and those of the other, shrink.

1076 RANDOSHKIN et al.

ϕ The new lines of equal intensity will relate to H = Hb + F, deg Hs and H = Hb Ð Hs, respectively. Note that, in both cases, high-resolution magnetic ﬁeld mapping requires 5 a ﬁne measurement step.

In the case of a low magnetic field (less than Hs), monodomain areas in the films of type I are absent and field nonuniformity can be judged by the local ratio of the areas occupied by domains with different polarity. Hs It is evident that, in this case, the spatial resolution –2000 –1000 1000 2000 sharply drops. To remedy the situation, the bias field H⊥, Oe should be chosen such that the total magnetic field exceeds the saturation field. As for mapping of a nonuniform magnetic field that consists of weak variable and intense constant components, it is advisable to can- –5 cel out the constant component using an external bias field. Fig. 1. Hysteresis loop typical of films of type II magnetized The most problematic is the visualization of an perpendicularly to the film surface. alternating close-to-saturation magnetic field whose period coincides on the order of magnitude with the equilibrium period of domains in the films of type I. In netization vector, their sensitivity is considerably this case, the spatial period of the magnetic field may be higher than that of the films of type I. For this reason, a misestimated several times [30]. range of magnetic fields to be visualized may be broad and accommodated to applications. Experiments show Magnetization reversal in the films of type II sub- that the films of type II are suitable for magnetooptic jected to a bias field does not cause hysteresis (Fig. 1), nondestructive testing at magnetic fields within the since this phenomenon is not inherent in the rotation of −8 5 magnetization. If even a relatively weak magnetic field 10 Ð10 Oe range. is applied, the magnetization vectors come out of the When a magnetooptic signal from the films of type II film plane, their direction being the same for domains is photometrically recorded with a linear measuring of different polarity. Consequently, the plane of polar- channel, the output is defined by the Malus law Θ ization rotates. As the bias field grows, F increases 2 (ϕ Θ ) almost linearly until the film is magnetized to satura- U = U0 sin Ð F , tion and the magnetization vectors come out at right where U0 is the input signal. angles to the film surface (Fig. 1). With a further The linearity of the transfer characteristic can con- increase in Hs, the Faraday rotation angle remains considerably be increased by applying the two-channel dif- stant. ferential measurement mode, which means the auto- When a nonuniform magnetic field is applied to the matic subtraction of two images of the same visual field films of type II and white light is used for illuminating, that are obtained at two different angles between the the films change their color throughout the surface polarizer and analyzer axes (+ϕ and Ðϕ). In this case, (for monochromatic light, the transmitted intensity the output signal looks like changes). Being proportional to the normal component ϕ Θ of the magnetization, the local angle of Faraday rota- U = U0 sin2 sin2 F. tion depends on the corresponding component of the This form ensures the high linearity of conversion and external magnetic field. It is significant that, with the simplifies the calibration of results in absolute mag- films of type II (in contrast to those of type I), mapping netic units. of a nonuniform magnetic field does not require the The experimental setup, built around a polarizing application and variation of an additional external mag- microscope, was connected to a personal computer netic field. The films of type II provide information through a CCD video camera. The surface of a visual- about the field nonuniformity via the distribution of the izing film was covered first by a mirror coating and then Faraday rotation angle over the film surface. by a protective layer. The thus-prepared film and a bias- The films of type I are magnetized to saturation ing coil were placed on the microscope stage. The when the total magnetic field reaches Hs, which is less source of a magnetic field to be studied was placed near than but comparable to the saturation magnetization the surface of the visualizing film. π 4 Ms. In the films of type II, saturation occurs when the The most obvious application of magnetooptic visu- total magnetic field equals the field of magnetic anisot- alization of nonuniform magnetic fields is the nonde- ropy HK, which can be both much less and much more structive control of magnetically hard products, for π than 4 Ms. Since the films of type II respond to a weak example, magnetic-recording media or permanent external magnetic field by a slight rotation of the mag- magnets.

TECHNICAL PHYSICS Vol. 45 No. 8 2000

ADVANTAGES OF ANHYSTERETIC MAGNETOOPTIC FILMS 1077

(‡) (‡)

(b)

(c)

Fig. 2. Magnetooptic images of a (a) sound record on a magnetic tape, (b) VHS-tape record, and (c) R-DAT digital record visualized with the use of the ﬁlm of type II. (c)

Fig. 3. Magnetooptic images of records on a (a) hard mag- Fig. 2a shows the black-and-white magnetooptic netic disk, (b) flexible magnetic disk, and (c) metallic tape of a “black box” that are visualized with the use of the film image of a recorded tape. A colored image obtained of type II. with the experimental setup consists of a number of colored fringes (brown, yellow, and various tints of green) of different width with diffused borders against various brown tints). A video tape track is about 30 µm a yellow-green background (unfortunately, on a black- wide; and the recording period, about 5 µm. and-white picture, a significant part of information on a Fig. 2c shows the magnetooptic image of a tape with sample under study is lost). This means that an analog a digital sound record made in the R-DAT standard. The record is visualized. In experiments, the dynamic range record shows up as narrow alternate yellow-brown was found to be no less than 56 dB (this value depends fringes of different tints and widths. Also shown are only on the capability of metrological equipment). timing tracks (wide and narrow yellow and green Note that each of two tape tracks is about 450 µm wide. fringes of equal width). The periods of timing pulses were 170, 40, and 30 µm. Fig. 2b shows the magnetooptic image of a recorded Fig. 3a shows the magnetooptic image of a record VHS tape (the color picture is a set of narrow fringes of and labels on a hard magnetic disk. The width of the

TECHNICAL PHYSICS Vol. 45 No. 8 2000

1078 RANDOSHKIN et al.

(‡) (b)

Fig. 4. Magnetooptic images of (a) the permanent magnet surface and (b) a part of laser-printed text that are visualized with the use of the film of type II. visualized lines is less than 0.7 µm, which is unattain- the magnet surface. Dark intersecting fringes with light able when the films of type I are used. spots represent aggregates of particles with another Fig. 3b shows the magnetooptic image of a record structure or saturation magnetization. In the color orig- on a faulty flexible disk. Long curved lines are disk sur- inal, these spots are yellow with a green border (in face defects, while horizontal lines, which form vertical black-and-white images, the contrast between green tracks, depict records. Sharp local variations of the and dark brown parts is lost). Figure 4a indicates that track width and location, which are caused by start-stop the magnetooptic visualization of spatially nonuniform faults of the recording head, are noteworthy. In particu- magnetic fields may be helpful in the phase analysis of lar, the track width varies stepwise by about 50 µm. various materials. This means that the visualization of magnetic records Figure 4b shows the magnetooptic image of a laser- discovers the faults of a recording head. printed text (the text is not seen because of the mirror The films of type II are indispensable in restoring coating). The need for reading such a text appears in the corrupted or lost information written on the metallic case of a latent marking [33], specifically, when a text tape of a “black box.” As seen from Fig. 3c, distinct is written with a “magnetic paint” on a nonmagnetic black and white fringes alternate with low-contrast background of the same color. A text and an image may lines, which result from incomplete deletion of the pre- contain fragments that are drawn with both magnetic vious record (color contrast gives additional informa- and ordinary paints of the same color and grade into tion). Moreover, the noncoincidence between the tracks one another (a $100 banknote is an example). In this of the previous and subsequent records results in the case, the use of magnetooptic visualization of spatially presence of the previous one as bright dots at the track nonuniform magnetic fields is appropriate both for edges. developing and controlling printing processes and for revealing counterfeit banknotes. Fig. 4 shows the black-and-white magnetooptic image from the surface of a barium permanent magnet, The main difference between magnetically soft and which is used for producing a bias magnetic field in magnetically hard materials is that, in the normal state, bubble technology [31, 32]. The image is brown to the former do not produce stray fields, since the mag- green in the color image. In particular, narrow vertical netic flux closes inside a magnet. Stray fields, however, dark fringes on the light background correspond to arise if a magnetically soft sample is duly magnetized dark-brown ones on the orange background. These (magnetically “illuminated”). If the field required for fringes seem to be associated with polishing defects on this purpose is significantly lower than the field of

TECHNICAL PHYSICS Vol. 45 No. 8 2000

ADVANTAGES OF ANHYSTERETIC MAGNETOOPTIC FILMS 1079

dark brown colors in the color magnetooptic image. This variation of contrast suggests that the magnetic properties of the sample are nonuniform. “Magnetic illumination,” as applied to BcSCGF films, seems to be promising for the control of magnetic head manufacturing and positioning. By way of example, Fig. 5b shows the magnetooptic image of the magnetic path and head gap for hard disk recording. The irregularity and asymmetry of the image, specifically beard-like domains near the magnetic path, point to head faults and misarrangement. Fig. 5c shows the magnetooptic image of a 256-kbit (‡) bubble chip expander. Control elements of the bubble chip are made of magnetically soft permalloy. The stray fields, which are produced at the edges of the permalloy elements by “magnetic illumination,” are visualized by using the film of type II. As a consequence, these elements become visible. The high sensitivity of the BcSCGFFs is demonstrated by the fact that 3-µm-wide domains are seen in the color image of the bubble film (saturation magneti- π zation is 4 Ms = 320 Gs). Recall that this film and BcSCGFF are separated by the several protective layers and the mirror layer (in the black-and-white image, this faint contrast is lost). In other words, the films of type II (b) enable the magnetization control of various magnetic substances that do not possess magnetooptic properties. The magnetooptic images presented in Figs. 2Ð5 by no means exhaust the capabilities of the films of type II in the magnetooptic visualization of nonuniform magnetic fields. Yet, they give an idea of the range of the intensities and characteristic spatial dimensions of nonuniform magnetic fields that can be visualized with the films of type II. It is obvious that the use of the same BcSCGFF to solve many problems, even if possible, is unjustified. For every specific problem, it is advanta- (c) geous to select the film with an optimal set of properties. This can be done because of the unique possibility of modifying the BcSCGFF composition. In particular, Fig. 5. Magnetooptic images of the (a) edge of a magnetically soft material, (b) magnetic path and gap of a record the presence of three cation interstices of different size head for hard disk recording, and (c) bubble chip expander. allows more than half of the known chemical elements Visualization with the use of the magnetically illuminated to be incorporated into the film, thus providing a variety film of type II. of its properties [34Ð37]. uniaxial anisotropy for the films of type I, then the REFERENCES domain structure of a defect-free magnetically soft 1. V. V. Randoshkin and A. Ya. Chervonenkis, Applied material remains unchanged. Conversely, the rear- Magnetooptics (Énergoatomizdat, Moscow, 1990). rangement of domains indicates the presence of 2. N. F. Kubrakov, A. Ya. Chervonenkis, G. Ya. Merkulova, defects. With the films of type II, defects show them- et al., Zh. Tekh. Fiz. 54, 1163 (1984) [Sov. Phys.—Tech. selves through the local variation in the Faraday rota- Phys. 29, 662 (1984)]. tion angle. 3. A. Ya. Chervonenkis, N. F. Kubrakov, N. G. Svenskiœ, Figure 5a shows the magnetooptic image of the edge and T. P. Kisileva, USSR Inventor’s Certificate of a piece of a magnetically soft material. Stray fields No. 1072095, Byull. Izobret., No. 5 (1984). were observed under the constant field 150 Oe. They 4. N. F. Kubrakov, A. Ya. Chervonenkis, and M. V. Kash- arise at the edges and in the vicinity of defects and are cheev, Zh. Tekh. Fiz. 56, 1215 (1986) [Sov. Phys.— visualized with a BcSCGF film. The areas of different Tech. Phys. 31, 714 (1986)]. contrast in Fig. 5a correspond to yellow, green, and 5. N. F. Kubrakov, Proc. SPIE 1126, 85 (1989).

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1080 RANDOSHKIN et al.

6. A. V. Antonov, M. V. Gusev, E. I. Il’yashenko, and “New Magnetic Materials for Microelectronics,” Mos- L. S. Vomov, in Program and Abstract of the Interna- cow, 1996, pp. 182Ð183. tional Symposium on Magnetooptics, Kharkov, 1991, 23. A. A. Aœrapetov, V. L. Gribkov, A. P. Gubarev, et al., RF p. 70. Patent No. 2060491, Byull. Izobret., No. 14 (1997). 7. V. L. Gribkov, N. N. Kirukhin, V. A. Liskov, et al., in 24. V. V. Randoshkin, RF Patent No. 2047170, Byull. Izo- Advances in Magnetooptics II, Proceedings of the 2nd bret., No. 23 (1997). International Symposium on Magnetooptics; Fiz. Nizk. 25. V. V. Randoshkin, RF Patent No. 2092832, Byull. Izo- Temp., Suppl. S1 18, 429 (1992). bret., No. 28 (1997). 8. N. F. Kubrakov, Single-Crystal Garnet Ferrite Films and 26. R. M. Grechishkin, M. Yu. Gusev, N. S. Neustroev, et al., Their Application (Nauka, Moscow, 1992), pp. 136Ð164. in Proceedings of the XVI International Seminar “New 9. V. V. Randoshkin and M. V. Logunov, Pis’ma Zh. Tekh. Magnetic Materials for Microelectronics”, Moscow, Fiz. 19 (6), 62 (1993) [Tech. Phys. Lett. 19, 182 (1993)]. 1998, pp. 445Ð446. 10. V. V. Randoshkin and V. N. Dudorov, Pis’ma Zh. Tekh. 27. B. S. Vvedenskiœ, F. V. Lisovskiœ, and A. Ya. Chervonen- Fiz. 19 (22), 79 (1993) [Tech. Phys. Lett. 19, 731 kis, Tekh. Kino Telev., No. 6, 11 (1978). (1993)]. 28. A. Ya. Chervonenkis and N. F. Kubrakov, Pis’ma Zh. 11. V. V. Randoshkin and M. V. Logunov, Pis’ma Zh. Tekh. Tekh. Fiz. 8, 696 (1982) [Sov. Tech. Phys. Lett. 8, 303 Fiz. 20 (5), 17 (1994) [Tech. Phys. Lett. 20, 182 (1994)]. (1982)]. 12. A. A. Aœrapetov, V. L. Gribkov, A. P. Gubarev, et al., 29. A. K. Zvezdin and V. A. Kotov, Magnetooptics of Thin International Patent Appl. PCT/RU92/00210 (1999), Films (Nauka, Moscow, 1988). Bull. PCT, No. WO 93/11427. 30. S. V. Gerus, S. V. Lisovskiœ, and E. G. Mansvetova, Mik- 13. M. V. Logunov and V. V. Randoshkin, USSR Patent roélektronika 10, 506 (1981). No. 1813217, Byull. Izobret., No. 16 (1993). 31. A. H. Eschenfelder, Magnetic Bubble Technology (Sprin- ger-Verlag, Berlin, 1980; Mir, Moscow, 1983). 14. M. V. Logunov and V. V. Randoshkin, USSR Inventor’s Certiﬁcate No. 1824619, Byull. Izobret., No. 24 (1993). 32. Bubble-Based Elements and Devices: Reference Book, Ed. by N. N. Evtikhiev and B. N. Naumov (Radio i 15. M. V. Logunov and V. V. Randoshkin, RF Patent Svyaz’, Moscow, 1987). No. 2017182, Byull. Izobret., No. 14 (1994). 33. A. Ya. Chervonenkis, A. P. Gubarev, N. N. Kiryukhin, et 16. A. A. Aœrapetov, V. L. Gribkov, V. A. Lyskov, et al., RF al., in Proceedings of the XV All-Russia Seminar “New Patent No. 2002247, Byull. Izobret., Nos. 39Ð40 (1993). Magnetic Materials for Microelectronics,” Moscow, 17. A. A. Aœrapetov, V. L. Gribkov, V. A. Lyskov, et al., RF 1996, pp. 182Ð183. Patent No. 2011187, Byull. Izobret., No. 7 (1994). 34. V. V. Randoshkin, Single-Crystal Garnet Ferrite Films 18. V. L. Gribkov, V. V. Randoshkin, and A. Ya. Chervonen- and Their Application (Nauka, Moscow, 1992), kis, RF Patent No. 2022365, Byull. Izobret., No. 20 pp. 49−107. (1994). 35. A. M. Balbashov and A. Ya. Chervonenkis, Magnetic 19. V. V. Randoshkin, Defektoskopiya, No. 11, 34 (1994). Materials for Microelectronics (Énergiya, Moscow, 20. M. V. Logunov and V. V. Randoshkin, RF Patent 1979). No. 2047170, Byull. Izobret., No. 30 (1995). 36. V. V. Randoshkin and A. Ya. Chervonenkis, in Radioelec- tronics: The Current Status and Prospects (NIIÉIR, 21. R. M. Grechishkin, V. Yu. Goosev, S. E. Il’yashenko, and Moscow, 1985), Vol. 2, pp. 71Ð78. N. S. Neustroev, J. Magn. Magn. Mater. 157/158, 305 (1996). 37. V. V. Randoshkin, Defektoskopiya, No. 6, 58 (1997). 22. A. Ya. Chervonenkis, A. P. Gubarev, N. N. Kiryukhin, et al., in Proceedings of the XV All-Russia Seminar Translated by A. Sidorova

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 1081Ð1084. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 125Ð128. Original Russian Text Copyright © 2000 by Vasil’eva, D’yakonova, Erofeev, Lapushkina.

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Local Plasma Parameters and Integral Characteristics of a Magnetogasdynamic Channel under Conditions of Ionization Instability R. V. Vasil’eva, E. A. D’yakonova, A. V. Erofeev, and T. A. Lapushkina Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received July 16, 1999

Abstract—Results are presented from an experimental investigation of the onset of ionization instability in a disk-shaped Faraday magnetogasdynamic channel attached to a shock tube. The experiments were carried out in a pure inert gas (xenon) without alkaline additives. A relation is found between the integral plasma characteristics of a nonequilibrium magnetogasdynamic channel and the local parameters of a plasma that is unstable against the ionization instability. Mechanisms for amplifying perturbations and increasing the effective conductivity are revealed. It is concluded that these effects stem mainly from the features of three-body recombination in rare gases. © 2000 MAIK “Nauka/Interperiodica”.

The goal of this paper is to develop a self-consistent plasma moves along the channel. An increase in the physical model that accounts for the effect of an magnetic induction results in an increase in these increase in the effective conductivity during the onset parameters. of the ionization instability in pure inert gases. The model is based on the measurements of the local The experimental results show that the ratio plasma parameters and exploration of the structure of between the electron temperature Te and the tempera- plasma irregularities. ture of the heavy plasma component Tg attains Te/Tg = 4. α The experimental facility (see [1, 2]) consists of a At B = 0, the plasma is in the recombination state ( > α shock tube with a disk-shaped 32-cm-diameter and eq), whereas with selective heating of electrons in an 1-cm-high magnetogasdynamic (MGD) channel. The induced electric field, the degree of ionization α is magnetic induction is up to 1.4 T. Experiments were below the equilibrium value (both in the initial state and carried out in xenon at an incident shock wave Mach in the presence of fluctuations); i.e., in the MGD mode, α α number of 6.9 and initial pressure of 26 torr. At the the plasma is being ionized ( < eq). The degree of entrance to the disk channel, plasma parameters were ionization is high enough (α > 10Ð4) for the electron × 3 ρ as follows: r = 0.04 m, u0 = 1.27 10 m/s, 0 = velocity distribution to be considered Maxwellian. 3 α × −4 0.45 kg/m , Ta0 = 2600 K, Te0 = 3100 K, 0 = 2.6 10 , A characteristic feature of the plasma is that an increase and M = 2.45. A circular Faraday current and radial in the magnetic field leads to an increase in the degree Hall field were induced in the disk channel. The chan- of ionization; as a result, the ratio between the electronÐ nel operated either in a short-circuit Faraday channel neutral and electronÐion collision frequencies, mode or with a narrow insert shaped as a sector with ν ν three pairs of electrodes for connecting load resistors. ea ==naceQea, ei neceQei The load coefficient was varied within the 0 < k < 0.2 range. changes. Here, na and ne are the densities of atoms and The methods for determining the parameters of the ions, respectively; ce is the average electron velocity; gas-dynamic stream; measuring the effective plasma and Qea and Qei are the cross sections (averaged over conductivity, the Hall parameter, the electron density the Maxwellian distribution) for the electron momen- and temperature, and the value of the magnetic induc- tum transfer in collisions with atoms and in Coulomb tion Bcr that is critical for the onset of the ionization collisions, respectively. Under our experimental condi- ν ν instability; and recording glow irregularities are tions, we have 0.2 < ei/ ea < 1. In this case, the colli- described in [1Ð5]. sion frequency depends not only on the electron tem- The main results of the previous studies [1Ð3] used perature and atom density, but also on the electron den- in this paper are the following. At B > Bcr, the effective sity. The main distinguishing feature of a rare-gas plasma conductivity, the average electron density, and plasma is a fairly low recombination coefficient Kr. the level of the electron density fluctuations grow as the This feature stems from a specific structure of energy

1082 VASIL’EVA et al.

Te, K Figure 1 presents the typical time behavior of the local plasma parameters at a fixed point of the channel 10000 (r = 0.09 m) during the onset of the ionization instability. In the experiment, we measured the electron tem- 8000 perature and density. Both the conductivity and the Hall parameter were deduced from the measured values of 6000 Te and ne and the known density of atoms (at B = 0, na = 1024 mÐ3). Low-frequency fluctuations with a duration n , 1021 m–3 µ e of 20Ð50 s are seen in the dependences Te(t) and ne(t). There, fluctuations are caused by plasma processes. 4 Higher frequency oscillations with a period of less than 5 µs are likely attributed to a photomultiplier noise. For 2 this reason, below, we only consider low-frequency oscillations. Note that the intervals with higher (lower) 0 values of Te correspond to the intervals with higher σ, 102 S/m (lower) values of the electron density and conductivity. The Hall parameter varies in the opposite phase with 8 the conductivity, because Coulomb collisions play an important role under given conditions and the momen- 6 tum transfer rate increases with an increase in the elec- 4 tron density. Based on the above dependences, we can 〈 〉 〈 〉 〈σ〉 〈β〉 find the average values Te , ne , , and . However, 2 they do not clarify how the fluctuations with respect to β the average values are related to those with respect to the initial, unperturbed plasma parameters. Therefore, 2 the role of positive and negative perturbations in the build-up of oscillations should be revealed. 1 The irregularities are oriented in space in a definite manner. This was found by a frame-by-frame imaging 0 200 300 400 t, µs of the plasma glow [2]. The glowing irregularities turned out to be shaped as spokes inclined at an angle ° Fig. 1. Time evolutions of the measured electron tempera- of 20 with respect to the azimuthal direction. The ture and density and the calculated conductivity and Hall propagation velocity of the spokes was on the order of parameter at B = 1 T and r = 0.09 m. the flow velocity as if the they were “frozen” in the flow. As a spoke moves, the brightness of its glow y y increases. On average, there are two spokes in the chan- u nel simultaneously. They arise with definite time intervals. At radii much greater than the initial radius, the x glowing spokes can be approximately represented as Bz j' strips in rectangular coordinates (Fig. 2). In the disk k ' geometry, the ϕ and r directions correspond to the y and E' E j' x directions, respectively. Figure 2 displays the coordinate system and the vectors of the initial current j and θ 0 x δ the initial electric field E0* in the plasma. Here, tan = 1 2 β, and θ is the angle between the current and the normal to the sheet plane (π/2 < θ < π). We denote the changes δ 3 4 σ σ σ in the main plasma parameters as j' = j Ð j0 , ' = Ð 0, β β β ' = Ð 0, T e' = Te Ð Te0, and ne' = ne Ð ne0 (subscript 0 j0 stands for the parameter values in the unperturbed surrounding medium). In [6, 7], it was shown that, in an Fig. 2. Schematic diagram of current and field fluctuations unbounded plasma, the fluctuations of the current and under conditions of the ionization instability. electric field are related to the fluctuations of conductivity and the Hall parameter as follows: levels. Thus, at Te = 8000 K, the value of Kr for alkali × Ð39 6 σ' metals equals 5 10 m /s, whereas for rare gases, j' = J (sinθ Ð βcosθ)----- + J β'cosθ, (1) × Ð41 6 0 σ 0 Kr = 5 10 m /s. 0

TECHNICAL PHYSICS Vol. 45 No. 8 2000 LOCAL PLASMA PARAMETERS AND INTEGRAL CHARACTERISTICS 1083

(a) (b) t1 t2 A

Te' A B Te'

0 0

ne' ne'

0 0 σ' σ'

0 0 β' β'

0 0

0 1 2 3 4 k 0 1 2 3 4 k

Fig. 3. Illustration of the mechanism for amplifying positive electron density perturbations.

2 which adds to the initial current, thus providing an 1 + β σ' J ±E' = Ð ------0-J cosθ----- + ----0-(sinθ + β cosθ)β'. (2) additional Joule heating and increasing the electron σ 2 0 σ σ 0 0 0 0 temperature and density. Therefore, positive perturbations of Te and ne are increasing, whereas negative ones Here, the plus sign is used if the vectors E and k are of have no chance to rise. Indeed, let us assume that a neg- the same direction and the minus sign is used in the ative fluctuation of ne has occurred. This would lead to opposite case. a decrease in the conductivity and, according to for- Therefore, positive fluctuations of σ and negative mula (1), a decrease in the current, which, in turn, fluctuations of β lead to an increase in the electric field would result in a decrease in the Joule heating and Te, and a fluctuation current in the direction of j0. The per- but not in a further drop in ne. turbations of the main plasma parameters along the normal to the observed irregularities are shown sche- During the time interval between t1 and t2, the matically in Fig. 3. The numerals stand for certain plasma volume A moves along the channel from region 1Ð2 to region 3Ð4 (Fig. 3b). During this time, positive dimensionless distances along the normal correspond- σ ing to those in Fig. 2. Let us trace the evolution of pos- perturbations of Te, ne, and and negative perturbations itive and negative fluctuations of the electron density of β further develop and a new fluctuation arises in region 1Ð2. and temperature. Let the fluctuation of Te shown in Fig. 3a occur at the instant t1. According to the ioniza- In a bounded plasma, the structure of fluctuation tion kinetics [8], the higher the electron temperature, currents will be different compared to that in Fig. 2. the higher the electron density in region A (under our The current distributions in a Faraday channel with per- experimental conditions, the characteristic ionization fectly sectionalized electrodes were calculated in [6, 7]. × Ð5 time is about (1Ð5) 10 s). In region B, where the It was shown in those papers that the Faraday current electron temperature is decreased, the electron density flowing along the isolines of the electron density closes decreases only slightly with respect to the initial value mainly via the electrodes and partly inside the plasma. because of the low rate of three-body recombination in All of this influences the effective values of the conduc- rare gases (under our experimental conditions, the char- tivity and the Hall parameter. Here, we define the effec- acteristic recombination time is about 10Ð1 s). Accord- σ tive conductivity eff and the effective Hall parameter ingly, in region A, the conductivity increases, whereas β as follows: in region B, it remains almost the same. The change in eff the Hall parameter may be opposite in phase to the σ = 〈 jϕ〉/(uB), (3) change in the conductivity σ', as shown in Fig. 3a. eff A positive change in σ and a negative change in β in β 〈 〉 ( ) region A result in an increase in the fluctuation current, eff = Er / uB , (4)

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1084 VASIL’EVA et al.

σ, S/m Figure 5 presents the average and effective values of the Hall parameter. Their decrease with an increase in 900 the magnetic induction is caused by an increase in the electron density and a subsequent increase in the Cou- lomb collision frequency and the average momentum 〈σ〉 transfer rate. As a result, the Hall parameter decreases in both the irregularities and the surrounding medium. 600 β β As is seen from Fig. 5, the values of and eff are nearly the same. According to theoretical predictions, this can occur if the irregularity direction is close to that of the σ eff initial current [7]. Under our experimental conditions, 〈β〉 β 300 which correspond to = eff, irregularities in the shape of the spokes are only slightly inclined to the azimuthal direction, the instability develops at moderate values of β ≈ 1–2, the ﬂuctuations in the Hall parameter attain 40%, and the momentum transfer rate depends on the plasma parameters in a complicated way. 0 0.4 0.8 Bcr, T The results obtained can be summarized as follows: Fig. 4. Dependences of the average conductivity 〈σ〉 (ᮀ) and (i) a relation between the local plasma parameters and σ ᭝ effective conductivity eff ( ) on the magnetic induction. the integral characteristics of an MGD channel is established, (ii) mechanisms for amplifying perturbations β 〈β〉 and increasing the effective conductivity under condi- eff, tions of the ionization instability are revealed, and (iii) 3 R = 0.096 m it is shown that these effects stem mainly from the features of three-body recombination in rare gases.

〈β〉 REFERENCES 2 1. R. V. Vasil’eva, A. L. Genkin, V. L. Goryachev, et al., Low-Temperature Nonequilibrium-Ionization Plasma of β Inert Gases and MHD Generators (Ioffe Physicotechni- eff cal Inst., St. Petersburg, 1991), p. 206. 2. T. A. Lapushkina, R. V. Vasil’eva, A. V. Erofeev, and A. D. Zuev, Zh. Tekh. Fiz. 67 (12), 12 (1997) [Tech. 1 0 0.5 1.0 B, T Phys. 42, 1381 (1997)]. 3. R. V. Vasil’eva, A. V. Erofeev, D. N. Mirshanov, and Fig. 5. Dependences of the average (᭞) and effective (᭛) T. A. Alekseeva, Zh. Tekh. Fiz. 59 (7), 27 (1989) [Sov. values of the Hall parameter on the magnetic induction. Phys. Tech. Phys. 34, 728 (1989)]. 4. T. A. Lapushkina, E. A. D’yakonova, and R. V. Vasil’eva, Pis’ma Zh. Tekh. Fiz. 24 (2), 58 (1998) [Tech. Phys. where 〈jϕ〉 is the azimuth current density at the load Lett. 24, 66 (1998)]. coefﬁcient k 0. 5. A. V. Erofeev, R. V. Vasil’eva, A. D. Zuev, et al., in Pro- In essence, the values of σ and β are the integral ceedings of the 12th International Conference on MHD eff eff Electrical Power Generation, Japan, 1996, vol. 1, p. 74. characteristics of the MGD channel. 6. A. V. Nedospasov and V. D. Khait, Principles of Physical The dependences of the average and effective con- Processes in Low-Temperature Plasma Devices (Éner- ductivities on the magnetic induction are shown in goatomizdat, Moscow, 1991). 〈σ〉 Fig. 4. Being close to each other at B < Bcr, both and 7. L. A. Vulis, A. L. Genkin, and B. A. Fomenko, The The- σ eff increase with an increase in the magnetic induction. ory and Analysis of Magnetogasdynamic Flows in Chan- σ nels (Atomizdat, Moscow, 1971). This is caused by the increase in both 0 (due to selective electron heating) and positive perturbations of σ 8. L. M. Biberman, V. S. Vorob’ev, and I. T. Yakubov, (due to the onset of the ionization instability). At the Kinetics of Nonequilibrium Low-Temperature Plasmas highest value of the magnetic induction, the effective (Nauka, Moscow, 1982; Consultants Bureau, New York, conductivity is somewhat less than the average conduc- 1987). σ 〈σ〉 tivity ( eff/ = 0.7), which is due to a partial closing of ﬂuctuation currents inside the plasma. Translated by N. N. Ustinovskiœ

TECHNICAL PHYSICS Vol. 45 No. 8 2000

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Free-Running Emerald Laser V. V. Antsiferov Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia E-mail: [email protected] Received July 20, 1999

Abstract—The spectral and energy characteristics of an emerald laser are studied experimentally. Its performance is compared with that of an alexandrite laser. For free-running oscillation under normal conditions, undamped pulsating behavior of output intensity is observed, as is the case with other lasing media containing Cr ions. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION electronÐphonon interaction. Compared with alexan- 4 4 drite, emerald has a more complex crystal structure and Lasing due to the T2 A2 electronic vibrational a weaker crystal field, the latter being indicated by the transitions in Cr3+ ions of emerald was first reported in position of the U band. The crystal field of emerald is [1]. Next were studies of tunable emerald lasers, which D = 1600 cmÐ1 (for alexandrite, D = 1740 cmÐ1). 4 4 − q q used the T2 A4 transition and laser pumping [2 4] or the R-lines of Cr3+ [4, 5]. Energy characteristics of a In emerald, strong ultraviolet (UV) absorption flashlamp-pumped laser with a flux-grown crystal were arises at shorter wavelengths, than in alexandrite, presented in [6]. namely in the 300-nm band for flux-grown crystals and in the 360-nm band for hydrothermally grown ones; in Emerald, or chromium-doped beryl Cr3+ : the latter case, there are additional bands in the 380Ð Be3Al2(SiO3)6, is a negative uniaxial crystal with 450 nm region. Short-wave absorption of emerald pro- refractive indexes n0 = 1.58 and ne = 1.575. The crystal ceeds mainly from impurities, especially from iron. is green when the concentration of Cr3+ is 0.01Ð1%. The iron content is 0.001 wt % in flux-grown crystals The melting point is 1470°C, which is 400°C lower and 0.1 wt % in hydrothermally grown ones. than that of alexandrite. The thermal conductivity of 3+ Ð1 Ð1 In luminescence spectra of Cr ions of emerald emerald is 0.04 W cm deg , being nearly six times (Fig. 2), the most noticeable component is the wide less than that of alexandrite. Emerald crystals are grown by the hydrothermal or flux method. The latter α, cmÐ1 η yields crystals with a higher optical quality and lower (a) impurity content. The nonselective loss of hydrother- 3 mally grown crystals is of the order of 0.1 cmÐ1. 3 0.6 4 4 In an emerald laser, lasing due to the T2 A2 1 electronic vibrational transitions occurs in the 700Ð 2 850 nm range. The energy gap between the 4T and 2E 0.3 2 1 2 levels of Cr3+ is 400 cmÐ1, being half as high as that of alexandrite. At room temperature, the lifetime of the excited state of Cr3+ is 65 µs and the transition cross section σ is 3.3 × 10Ð20 cm2. 3 (b) Optical absorption spectra of emerald (Fig. 1) are typical of Cr3+ ions enclosed by the octahedral config- 2 1 uration of oxygen ions. The wide bands in the blue and 1 red (Y, U) regions represent the allowed transitions 4 4 4 4 2 A2 T1 and A2 T2, respectively. The triplet levels are split by the trigonal component of the crystal 0.3 0.4 0.5 0.6 0.7λ, µm field, hence the differences in the π and σ components. The narrow absorption lines at 681 and 684 nm stem Fig. 1. Optical properties of (a) flux-grown and (b) hydro- 4 2 thermally grown emerald crystals at T = 300 K: the absorp- from the spin-forbidden transitions A2 E (the R1 tance α for (1) E || C and (2) E ⊥ C and (3) the luminescence and R2 lines). The fine structure of the U band is due to quantum yield η vs. wavelength.

1086 ANTSIFEROV

I, arb. units The absolute quantum yield of Cr3+ luminescence is 1 0.7 for flux-grown emerald crystals and of the order of 0.01 for hydrothermally grown ones. The yield is flat in the absorption region of chromium ions, decreasing at 3 wavelengths shorter than 380 nm. This testifies that Cr3+ ions are responsible for the short-wave absorption. 2 2 Emerald lasers have received only marginal applica- 1 tion. It is difficult to produce emerald crystals of adequate size, since their growth rate is an order of magnitude less than that of alexandrite crystals. Another 0.6 0.7 0.8 0.9 λ, µm demerit is the toxicity of beryllium. Fig. 2. Emerald luminescence intensity vs. wavelength at T = 300 K for (1) E || C or (2) E ⊥ C. EXPERIMENTAL SETUP peak at 770 nm, corresponding to the 4T 4A tran- 2 2 In this study, we tested a 3 × 35-mm flux-grown sition. The R and R lines are not so strong as in alex- 1 2 emerald crystal [7] with an active volume V of andrite because of the smaller energy gap, ∆E, between a 2 4 0.21 cm3 and a Cr3+ concentration of 0.7 wt %. The end the levels E and T2. In emerald, the levels are near thermal equilibrium even at room temperature (kT = faces of the crystal were beveled (the bevel was one 208 cmÐ1) and the metastable level 2E is intensely degree) and supplied with antireflection coatings. The 4 depleted via the short-lived level T2. In alexandrite, laser was pumped by an ISP-250 flashlamp in a mono- this occurs at higher temperatures, since its ∆E is as block quartz clarifier. The UV pump radiation was cut large as 800 cmÐ1. off by a liquid filter. The energy characteristics were

Ð3 E /V , J cmÐ3 Eout/Va, J cm E out a Et (a) t (a) 1 1 0.2 3 0.13 3 1.0 1.0 0.12 4 4 0.1 0.11 0.5 0.5 2 2 0.10 0 0.2 0.4 0 40 80 T , °C Ð3 2 Ð3 T Eout/Va, J cm Eout/Va, J cm (b) (b) 1 1.6 2 1.2 1.0

0.8 0.5 2 0.4 1 0 0.2 0.4 E , k J 0 0.5 1.0 L, m p

Fig. 4. (a) (1, 3) Output energy density Eout/Va and (2, 4) Fig. 3. (a) (1, 3) Output energy density Eout/Va and (2, 4) threshold pump energy E vs. output-mirror transmittance t ° threshold pump energy Et vs. crystal temperature T for Ep = T2 for Ep = 0.4 kJ, L = 0.4 m, and T = 70 C. (b) Output 0.4 kJ and L = 0.4 m and (b) output energy density vs. cavity energy density vs. pump energy E for L = 0.4 m and T = ° p length L for Ep = 0.4 kJ and T = 70 C. The emerald laser is 70°C. The emerald laser is represented by curves 1 and 2 in represented by curves 1 and 2 in panel (a) and curve 1 in panel (a) and curve 1 in panel (b). The alexandrite laser is panel (b). The alexandrite laser is represented by curves 3 represented by curves 3 and 4 in panel (a) and curve 2 in and 4 in panel (a) and curve 2 in panel (b). panel (b).

TECHNICAL PHYSICS Vol. 45 No. 8 2000 FREE-RUNNING EMERALD LASER 1087 compared with those of an alexandrite laser [8Ð10] operated under similar conditions. Lasing was examined with a photodiode and an oscilloscope. The output spectrum was recorded with an STÉ-1 spectrograph, and the output energy was Ep measured with an IMO-2 meter.

OUTPUT-ENERGY AND SPECTRUM λ CHARACTERISTICS

Figure 3 shows the output-energy density Eout/Va Fig. 5. TEM mode spectra of the emerald laser at T = ° mnq against (a) the crystal temperature T or (b) cavity length 20 C for pump energies Ep = 3, 4, 6, and 8 Et (from the top L. The emerald laser is represented by curves 1 and 2 in down). Fig. 3a and curve 1 in Fig. 3b. The alexandrite laser is represented by curves 3 and 4 in Fig. 3a and curve 2 in The laser was tuned by means of a dispersive cavity Fig. 3b. It is seen that the output of the emerald laser is with three prisms made of TF-5 glass. The total angular much less sensitive to temperature variation (Fig. 3a). dispersion was about 3′/nm. The tuning range was 710Ð The reason is that, in emerald, the energy gap between 830 nm, the output wavelength being stable within ~1 nm. 2 4 the metastable E and upper T2 levels is much narrower For the TEMooq and TEMmnq modes of the emerald and the latter becomes populated even at room temper- laser, its output always has the form of undamped oscil- ature. The dependence on cavity length is stronger for lation, as is the case with other lasing media containing the emerald laser (Fig. 3b) due to the much lower ther- Cr ions. The evolution of the output spectrum depended mal conductivity of the medium. As is known, a posi- on the physical state of the emerald crystal and was tive spherical thermal lens arises in lasing solids under affected by the spurious longitudinal-mode selection in flashlamp pumping, which is equivalent to replacing a the cavity, as in the case of the alexandrite laser. With flat cavity with a spherical one. The focal length of the partial cutoff of UV pump radiation, the spectral evolu- thermal lens is much smaller in emerald. That is why tion changed materially. the cavity of an emerald laser becomes unstable at a smaller length, hence a more prominent and sharper fall in the output energy. REFERENCES We also compared the dependences of the output 1. M. L. Shand and J. C. Walling, IEEE J. Quantum Elec- tron. 18, 1829 (1982). energy on the transmittance T2 of the output mirror at the 2. J. Buchert and R. R. Alfano, Laser Focus, No. 9, 117 pump energy Ep = 0.4 kJ (Fig. 4a). The emerald laser was found to attain its maximum output energy at the higher (1983). 3. M. L. Shand and S. T. Lai, IEEE J. Quantum Electron. T2. The threshold pump energy Et of the emerald laser was much lower throughout the range of T . 20, 105 (1984). 2 4. J. Buchert, A. Katz, and R. R. Alfano, IEEE J. Quantum The output energy was also studied as a function of Electron. 19, 1477 (1983). Ep at optimal T2’s (Fig. 4b). A nonlinear rise was 5. Z. Hasan, S. T. Keany, and N. B. Manson, J. Phys. C 19, observed for both lasers, except for small values of Ep. 6381 (1986). The output energy of the emerald laser increased more 6. V. S. Gulev, A. P. Eliseev, V. P. Solntsev, et al., Kvan- steeply at modest Ep’s and saturated more rapidly at tovaya Élektron. (Moscow) 14, 1990 (1987). high Ep’s, so that the output energy densities of the 7. V. V. Antsiferov and G. I. Smirnov, Preprint No. 98-97, lasers at Ep = 0.5 kJ were nearly equal. This stems from IYaF Sib. Otd. RAN (Budker Institute of Nuclear Physics, more severe strains in the emerald crystal. Siberian Division, Russian Academy of Sciences, Novosibirsk, 1998). For the emerald laser, the spectra of the TEM mnq 8. V. V. Antsiferov, V. A. Kalyagin, G. V. Khaburzaniya, and modes (Fig. 5) were virtually the same as those for the A. V. Sharpov, Preprint No. 89-1, SFTI (Ioffe Physicote- alexandrite laser [9]. The lasers exhibited a fine discrete chnical Institute, Russian Academy of Sciences, St. spectral structure. This stems from spurious longitudi- Petersburg, 1989). nal-mode selection, no matter how weak it is, imparted 9. V. V. Antsiferov, A. I. Alimpiev, E. V. Ivanov, and by the beveled and coated end faces. At room tempera- G. V. Khaburzaniya, Zh. Tekh. Fiz. 62 (3), 9 (1992) ture and moderate values of Ep, the emerald laser emit- [Sov. Phys. Tech. Phys. 37, 239 (1992)]. ted in a wide wavelength band with the maximum at 10. V. V. Antsiferov, E. V. Ivanov, and G. I. Smirnov, Preprint 770 nm. As Ep grew, the output spectrum broadened No. 93-107, IYaF Sib. Otd. RAN (Budker Institute of predominantly toward shorter wavelengths. An eight- Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 1993). fold increase in Ep above the threshold resulted in an output bandwidth of about 20 nm. The bandwidth was an almost linear function of Ep. Translated by A. A. Sharshakov

TECHNICAL PHYSICS Vol. 45 No. 8 2000

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Adiabatic Compression of Matter by a Shell V. V. Prut Russian Research Center Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia E-mail: [email protected] Received August 4, 1999

Abstract—The problem of the spherical compression of condensed matter by a shell was solved in an incompressible medium approximation. Parameter values at the inner boundary of the shell are deﬁned by the solution of a self-modeling problem. At collapse, the velocity and kinetic energy of the shell behave asymptotically. © 2000 MAIK “Nauka/Interperiodica”.

Matter compression by a spherical shell is consid- For γ = const, the self-modeling equations are split: ered under the assumption that the shell compressibility we solve the first equation and then find ξ using quadra- is much less than that of the matter. Such problems are ture. In the general case γ ≠ const, the equations must of interest in the production and metrology of high be solved jointly. The self-modeling problem specifies energy densities, in particular, in treating target com- the piston path in such a way that the entire mass is pression during inertial nuclear fusion. Let us simplify isoentropically compressed to a point, i.e., to collapse. the problem by assuming the shell to be incompress- The self-modeling compression of the finite plasma ible: dρ/dt = 0. Then, shell motion is described by the mass by a piston (on condition that the equation of the ργ ordinary differential equation perfect gas (plasma) state is p = p0 ) was considered, ∂ e.g., in [1Ð12]. In this work, we suggest an equation of ρ 2 2 u1 11 1()2 2 h2r1u1 + r1 ------∂ - ---- Ð ---- +--- u 2 Ð u 1 = Ðp2+p1;state of real gas. Namely, if the specific volume V 0, t r1r 2 2 the thermodynamic functions depend on the properties π of perfect degenerate nonrelativistic electron gas. For 224()3 3 r1u1==r2u2;------r 2 Ð r 1 Vh, the normal density, they depend on two experimental 3 ∂ ∂ ρ parameters: bulk modulus B = ÐV( p/ V)s and exponent where h is the shell density; Vh, its initial volume; r, γ ∂ ∂ = Ð( lnB/ lnV)s. Cold pressure is introduced in the radii; u, velocities; and p, the pressure at the shell form boundaries. Subscripts 1 and 2 refer to the inner and α outer boundaries, respectively. All parameters are taken ()ÐaV 5/3 Ða as dimensionless with respect to the matter being com- pp=0e/V Ð e , pressed. 5/3 where p0 = (2/5)cF(Z/V0) and Z is the atomic number. Parameter values at the inner boundary of the shell The constant c = (3π2)2/3ប2/2m enters into the Fermi are defined by the solution of a self-modeling problem. F e ε 2/3 The associated equations are derived from the mass and energy according to the expression F = cF ne. momentum conservation equations at constant entropy. A self-modeling variable is represented as ξ = t/r, and We will use conventional designations: specific vol- ξ ume V, which is dimensionless relative to V0; dimen- self-modeling functions will be denoted as U = u and ρ C = cξ, where u is the mass velocity and c is the sound sionless density = 1/V; and parameters p and B, adiabatic velocity. Time is counted backward from t = dimensionless relative to B0. Hence, at V = 1, the 1 (initial state) to t = 0 (collapse). Then, in terms of the dimensionless sound velocity c equals 1. α independent variable C, the self-modeling mass and The values of a and were calculated from B0 and γ momentum conservation equations have the form 0 obtained from shock-wave and static measurements. γ 2 2 In the former, B0 and 0 are determined from the depen- ∂lnU/∂lnC=()() 1 Ð U ÐνC/H, ≈ dence of the shock wave velocity on mass D c0 + D1u; γ ∂ξln/∂lnC=()() 1 Ð U ÐC2/H, at V = 1, = 4D1 Ð 1. The value of B0 varies between ~1 kbar (for hydrogen, its isotopes, and helium) and where H = (1 Ð U)(1 Ð ηU) Ð C2 and η = (ν Ð 1)(γ Ð several megabars, and 3 ≤ γ ≤ 7. Calculations of a and 1)/2 + 1. Here, ν = 1, 2, and 3 for planar, cylindrical, α indicated that the interpolation formula for p(V) is γ and spherical space geometry, respectively, and = well realizable throughout the range of B0. Figure 1 γ(V) is defined by the equation of state. Acceleration is shows the dimensionless dependences of B and γ on V ∂ ∂ ∂ ξ ∂ ξ expressed as u1/ t = (dU/dC Ð U ln / C)/( dt/dC). for hydrogen. The transition from the initial value of γ

Fig. 1. γ and bulk modulus B vs. V.

u r 103 1.2 r 2 102 1.0 u1 1 0.8 10 0 0.6 10 r1 –1 u2 0.4 10 0.2 10–2 10–3 (a) (b) W E 7 1011 10 108 103 E 105 2 W2 Ek 2 10 –1 10 E1 W1 10–1 10–4 10–5 10–4 10–2 100 10–4 10–2 100 (c) t (d)

Fig. 2. Time dependences of (a) r, (b) u, (c) W, and (d) E for the inner and outer boundaries. Panel (d) also shows the variation of the shell kinetic energy Ek. to its final value is observed at ρ ≅ 10. The energy (per parameter that is determined from the condition ξ = 1 unit mass) is determined from the relationship p = at N. The piston path is found from the equation −∂ ∂ Ӷ ∂ ∂ ξ ( e/ V)z with a normalization e(1) = 0. At V 1, the lnt/ ln = 1 Ð U. 2 2 γ Ð 1 2 dimensionless sound velocity is c = c∞ ρ . π The power is defined as W = 4 r2 u2p2, the kinetic energy of the shell is The desired integral curve must start and end on the r (C, U) plane at singular points N = (1, 0) (node) and S = 2 2 ξ ξ ξ 2 4 2 (Cs, Us) (saddle). At S, = 0, and at N, = 1. At = 0 πρ u πρ  1 1 Ek = 4 h ∫ ----r dr = 2 hr1 u1 ---- Ð ---- , or 1, V = 0 or 1, respectively. The parameters of singular 2 r1 r2 points defined by γ(V) are calculated for both V’s. The r1 desired integral curve U(C) comes out of the saddle S and its internal energy Ei = 0 because of incompress- and enters the node N along the separatrix. The param- ibility. eters of S (having the subscript s) are U = 2/(ν(γ Ð 1) + 2) s The problem involves two dimensionless parame- ν γ ν γ ∆ ρ ρ and Cs = ( Ð 1)/( ( Ð 1) + 2). At small = C Ð Cs, ters: h and Vh. In the above calculations, we put h = π the curve leaves the saddle, following the analytical for- 10 and Vh = 4 /3, which is the initial volume of the mat- ξ Ω∆ω ω ν η mula = , where = ( Ð 1)k/( (2Us Ð 1) Ð 1 Ð k), ter. Hence, r2 = 1 at collapse. Qualitatively, these quan- k is the slope of the separatrix, and Ω is a variable tities do not have an effect on the results. Figure 2

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1090 PRUT ∞ ∞ shows the obtained relationships. A number of com- E2 and W2 , since the total energy of the pression features deserve attention. Near collapse, r1 = collapsing matter E ∞. U U U Ð 1 (γ ) ζ s ζ s s ζ ν1/4 + 1 /2 ( t) and u1 = Us t , where = c∞/ Cs . 3U 3U Ð 1 REFERENCES ≈ ζ s s 2 ≈ Then, u2 Us t /r2 (r2 const). The exponent ν γ ν γ 1. K. P. Stanyukovich, Unsteady Motions of Continuous 3Us Ð 1 = (4 Ð ( Ð 1))/( ( Ð 1) + 2) and changes sign Medium (Nauka, Moscow, 1971). γ ν when u = 1 + 4/ . This means that, near collapse, the 2. N. V. Zmitrenko and S. P. Kurdyumov, Prikl. Mekh. velocity of the outer boundary of the shell may increase Tekh. Fiz., No. 1, 3 (1977). γ γ (for > u), decrease, or remain unchanged. From 3. M. A. Anufrieva and A. P. Mikhaœlov, Differ. Urav. 19, Fig. 2, it follows that u2 ﬁrst grows and then, starting 483 (1983). ≈ × Ð3 Ӷ with t 5 10 , drops. Also, near collapse (r1 r2), 4. A. F. Sidorov, Prikl. Mat. Mekh. 55, 779 (1991). 3 2 2 5Us Ð 2 2 5. R. E. Kidder, Nucl. Fusion 14, 53 (1974). E = 2πρ r u = 2πρ ζ5 U t /r . The exponent k h 1 1 h s 2 6. M. M. Basko, Nucl. Fusion 35, 87 (1995). 5U Ð 2 = (6 Ð 2ν(γ Ð 1))/(ν(γ Ð 1) + 2) at γ = 1 + 3/ν s k 7. S. K. Zhdanov and B. A. Trubnikov, Pis’ma Zh. Éksp. also changes sign. Finally, the kinetic energy near col- γ γ Teor. Fiz. 21, 371 (1975) [JETP Lett. 21, 169 (1975)]. lapse may increase (at > k), decrease, or remain con- γ γ γ γ γ 8. Ya. M. Kazhdan, Prikl. Mekh. Tekh. Fiz., No. 1, 23 stant. Since h < n, there exists a range k < < u where (1977). the velocity of the outer boundary decreases and the 9. I. E. Zababakhin and V. A. Simonenko, Prikl. Mat. kinetic energy grows. This is explained by velocity and Mekh. 42, 573 (1978). accelerated-mass redistributions in the shell. However, γ γ 10. S. I. Anisimov and N. A. Inogamov, Prikl. Mekh. Tekh. for physically ultimate values, = 5/3 < k = 2; there- Fiz., No. 4, 20 (1980). fore, at collapse, both the outer boundary velocity u2 11. A. M. Svalov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. and the kinetic energy of the shell Ek tend to zero. Gaza, No. 3, 171 (1982). Figure 2 implies that the difference E2 Ð E1 due to the 12. A. N. Kraœko and N. I. Tillyaeva, Teploﬁz. Vys. Temp. pressure drop p2 Ð p1 and power difference W2 Ð W1 36, 120 (1998). contributes to the kinetic energy Ek = E2 Ð E1. At t < −3 ∝ 1/2 10 , Ek slowly drops: Ek t . It is clear, however, that Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 8 2000

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Electroviscous Effect in an Alternating Electric Field A. A. Ostapenko Radiophysics Research Institute, St. Petersburg State University, St. Petersburg, 198904 Russia Received October 28, 1998; in ﬁnal form, October 20, 1999

Abstract—Experimental dependence of the relative dynamic viscosities of liquid dielectrics on the applied alternating voltage amplitude and frequency is presented. The voltage frequency was varied from 20 Hz to 2 kHz. The variation of viscosity can be attributed to a change in the liquid structure, with ionÐmolecule complexes centered at the ions produced as a result of charge injection from an electrode. A change in the injection regime leads to a change in viscosity. © 2000 MAIK “Nauka/Interperiodica”.

In previous publications, it was pointed out that an up in the vicinity of the electrodes. The change in the electroviscous effect is observed in electric fields that regime of charge production due to a change in the fre- are transverse to the dielectric flow [1, 2]. The change quency of the external electric field can affect the rate in viscosity was attributed to momentum transfer due to of charge buildup or completely suppress it. Within the ion electrophoresis, the development of electrohydro- framework of the conventional model of the electrode dynamic flows, and the formation of ionÐdipole clus- double layer, injection can take place if a charge at least ters near electrodes [3]. In some studies, it was noted equal to the total double-layer charge has accumulated: that the electroviscous effect is not observed in nonpo- Φ lar liquids, but it was observed in the presence of small qC= dl , (1) amounts of polar additives [1]. The addition of a polar Φ where Cdl is the capacity of the double layer and is additive to a nonpolar liquid can substantially affect the the voltage drop across the double layer. A rough esti- conductivity of the medium. Indeed, it was noted in a mate shows that the double-layer capacity is number of studies (e.g., see [1, 2]) that conductivity ~10 µF/cm2, and the voltage drop is ~1 V for media of affects the intensity of the electroviscous effect. Apart the kind considered here [4]. In an alternating field, from polar additives, an important factor that affects the such a double layer is discharged and charged during a intensity of the electroviscous effect is the presence of half-period of every voltage cycle. Therefore, for injec- an injecting electrode. The process referred to as charge tion to take place, a charge equal to q must build up in injection into a medium was not studied specifically, the layer during at least a quarter-period of the applied and only the consequences of charge injection from an voltage; i.e., electrode were discussed. It was shown in [2] that charge injection plays a very important role in the elec- T/2 troviscous effect as manifested in constant electric qit≤ ∫ ()dt, (2) fields. The production of ions and ionÐmolecule complexes can change the structure of a liquid dielectric T/4 σ π σ medium and considerably affect the viscosity observed where i(t) = E0sin2 ft, is the conductivity of the in an electric field. The regime of charge injection from liquid, E0 is the amplitude of the external field intensity, an electrode can be changed by applying an alternating and f is its frequency. This yields an expression for the electric field to the electrode gap. critical frequency above which the required charge can- In this paper, the behavior of the observed viscosity not build up in the double layer: of polar liquid dielectrics as a function of the frequency σ Umax of the external alternating electric field is discussed. It f cr = ------π - , (3) should be noted that the effect of the frequency of the 2 d applied field on the viscosity of a liquid dielectric has where Umax is the amplitude of the applied voltage and previously been studied only at 50 and 1000 Hz [1], and d is the electrode-gap width. the frequency-dependent behavior has remained When f > fcr, charge injection does not take place, unclear. In other papers, the existence of this effect was and vice versa. The setup used in the experiment was ruled out completely. To this day, no acceptable expla- described in a previous paper [2]: a channel of rectan- nation has been given for these controversial results. gular cross section, 20 mm long and 3.5 mm wide, had For processes that take place near the electrodes top and bottom walls made of metal plates that served (electrochemical reactions) responsible for charge as electrodes. The electrode gap (channel height) was injection to take place, it is necessary that charge build 200 µm. The passage time of the liquid flow between

1092 OSTAPENKO

∆η/η ∆η/η (a) (b) 1.0 5 5

4 4 3 3 0.5 2 2

0.2 1 1

100 200 300 400 500 600 700 100 200 300 400 500 600 700 f, Hz

Fig. 1. Relative change in viscosity versus the frequency of applied alternating voltage for (a) acetone and (b) nitrobenzene at (1) 10, (2) 20, (3) 30, (4) 40, and (5) 50 kV/cm. marks on the capillary walls was measured by a photo- was measured as a function of the frequency of the detector within 0.1 s. The alternating voltage generated applied voltage at different amplitudes of the voltage by a special power supply was applied to the electrodes for various liquids. All of the curves obtained for the of the experimental cell. The error of frequency mea- polar liquids involved portions characterized by similar surement did not exceed 3%. The length of the segment behavior (see Figs. 1 and 2). At low frequencies, as the upstream of the capillary cell was adjusted so that the frequency is increased, the relative viscosity drops, flow was steady along the entire cell. Since dynamic reaches a minimum, and then rapidly increases. With a viscosity can be represented as η = A∆P/∆Q, where A further increase in frequency, viscosity exhibits a slow is a calibration constant and ∆P is the pressure drop decline. For example, the relative viscosity of acetone ∆ required to keep the flow rate at Q = V0/t (V0 is the vol- the relative viscosity at 400 Hz was equal to that ume of flowing liquid), the change in the observed vis- observed at ~60 Hz when the voltage amplitude was cosity is ∆η/η = (η Ð η)/η = (t Ð t)/t, where η is the held at 10 kV, whereas the minimum relative viscosity el el el ∆η η viscosity observed when voltage is applied to the cell, ( / )min increased with the applied voltage. In partic- ∆η η tel is the time required for the liquid volume to flow ular, the value of ( / )min at E = 40 kV/cm is three through the passage in the presence of the field, and t is times higher than that observed at E = 10 kV/cm for all the time required for the liquid volume to flow through curves (see Figs. 1 and 2). the passage in the absence of the field. The minima of the curves discussed here may be The polar liquids under study had high permittivi- associated with the critical frequencies for the charge ties (ε ~ 36Ð20), and decane (with ε ≈ 1.2Ð1.4) was buildup in the electrode double layer. Indeed, compar- ∆η η used as a reference nonpolar liquid. These liquids are ing the frequencies corresponding to ( / )min with the typical liquid dielectrics. The change in viscosity (i.e., critical frequency given by (3), one finds that they close the time required for a liquid to pass through the cell) (see table). It should be noted that the critical frequen-

Table Nitrobenzene Nitromethane Acetone Decane E, kV/cm f , Hz f , Hz f , Hz f , Hz f , Hz min f , Hz min f , Hz min f , Hz min cr (experiment) cr (experiment) cr (experiment) cr (experiment) 10 16 12 80 60 10 10 1.6 × 10Ð6 Ð 20 32 25 160 120 20 15 3.2 × 10Ð6 Ð 30 48 42 240 180 32 26 4.8 × 10Ð6 Ð 40 64 60 320 260 40 35 6.4 × 10Ð6 Ð 50 80 75 400 350 53 48 8.0 × 10Ð6 Ð

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ELECTROVISCOUS EFFECT IN AN ALTERNATING ELECTRIC FIELD 1093

∆η/η age amplitude is increased, the value of ∆η/η increases 4 by a factor of less than 1.5. It should be noted that when 1.0 3 a constant voltage was applied to a ﬂow cell containing a 0.9 nonpolar liquid, the value of ∆η/η also exhibited a relatively slow increase as the voltage was increased [2]. 0.8 The table compares the values of the critical fre- 0.7 quency fcr calculated by (3) with fmin exp measured in the experiment. The table demonstrates that the frequen- 0.6 cies corresponding to the minima of curves in Figs. 1 0.5 and 2 are in good agreement with that predicted by (3), whereas experiments with decane did not reveal any 0.4 2 critical frequency. According to (3), the critical frequency for decane at the ﬁeld intensity 10 kV/cm 0.3 would lie at ~10Ð2 Hz, i.e., in the far low-frequency 0.2 domain, which is also consistent with experimental results. 0.1 1 After the injection was “switched off,” the observed viscosity slightly increased with frequency and then 200 600 1000 1400 1800 f, Hz slowly declined. These trends can be explained by effects of the bulk conductance of the liquid due to the Fig. 2. Relative change in viscosity versus the frequency of molecular dissociation of the additive. applied voltage for nitromethane at (1) 10, (2) 20, (3) 30, and (4) 40 kV/cm. Thus, the studies described here have shown that the presence of injected space charge in a liquid medium ∆η η × –2 plays an important role in the mechanism of the electro- / 10 viscous effect. When the injection is prevented either 6 by isolating the electrode from the liquid [2] or by 5 changing the frequency of the applied voltage results in a decrease in relative viscosity. This may occur 4 because the concentration of ionÐmolecule complexes 3 decreases with injection intensity, i.e., because the for- 2 2 mation of an ionÐmolecule complex is centered at an injected charge. An increase in the size of a structural 1 1 element of a liquid would then lead to an increase in 0 viscosity and, under certain conditions, to manifesta- 100 200 300 400 500 600 700 f, Hz tions of other electrohydrodynamic effects.

Fig. 3. Relative change in viscosity versus frequency for decane at (1) 20 and (2) 40 kV/cm. REFERENCES 1. P. E. Sokolov and S. L. Sosinskiœ, Dokl. Akad. Nauk cies measured for all liquids used in this study lie SSSR 4, 1037 (1939). within 10Ð500 Hz (see table), i.e., in a domain of rela- 2. A. A. Ostapenko, Zh. Tekh. Fiz. 68 (1), 40 (1998) [Tech. tively low frequencies. For a nonpolar liquid such as Phys. 43, 35 (1998)]. decane, no minimum of this type was observed in the 3. Yu. M. Rychkov, V. A. Liopa, et al., Élektron. Obrab. frequency range explored here (see Fig. 3). The relative Mater., No. 5, 34 (1994). viscosity of decane decreases with increasing fre- 4. B. Gosse, J. Electroanal. Chem. Interfacial Electrochem. quency. In particular, as the frequency of the applied 61, 265 (1975). voltage is reduced by a factor of one hundred, the relative viscosity decreases by ~30%. As the applied volt- Translated by A.S. Betev

TECHNICAL PHYSICS Vol. 45 No. 8 2000

BRIEF COMMUNICATIONS

The Behavior of Higher Manganese Silicide Single Crystals under Electrochemical Treatment F. Yu. Solomkin Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received October 21, 1999

Abstract—For higher manganese silicide single crystals of certain geometry and superstructure orientation, an unusual metallization process is observed. This anomaly can be associated either with bulk processes similar to thermogalvanic effects or with the carrier redistribution in the near-surface regions that is due to the high thermoelectric power anisotropy and complex crystal structure. © 2000 MAIK “Nauka/Interperiodica”.

The use of electrochemical techniques in the tech- This observation seems to be an anomaly from the nology of high-temperature thermoelectric devices standpoint of conventional electrochemistry. During makes it possible to integrate electrochemical cleaning the electrochemical application of metal films (cathode (chemical activation) of the surface and subsequent polarization), preferential deposition on edges, corners, deposition of metal contacts into a continuous fabrica- and surface ridges, not etching, is common. tion process [1]. Also, it allows researchers to study the distributions of heat fluxes and electric currents. This effect is most prominent when the C-axis makes an angle of 45° with the largest side of the par- Higher manganese silicide (HMS) is viewed as the allelepiped (see figure). When the C-axis deviates from best candidate for high-temperature generators and this direction, the island length on the side faces may sensors that operate in a wide spectral range. This mate- change and the process on the end side may be com- rial has the tetragonal crystal structure, which consists pleted with the formation of the oval or dumb-bell. The of loosely bonded manganese and silicon sublattices. shape of the end face metallization strongly depends on The spacings of the sublattices along the tetragonal the parallelepiped dimensions. The two metal rings C-axis are generally incommensurable. Their incom- appeared on the end face when a : b = 1 : 3 and speci- mensurability and the existence of many crystal struc- mens were sufficiently long in the current passing tures in the narrow range of compositions (MnSe1.71Ð1.75) direction. The effect was absent for short specimens. cause the precipitation of manganese monosilicide on planes normal to the C-axis [2]. It can be inferred that the residual metallization pat- terns on the end side (rings) and side surfaces are due When HMS polycrystals are subjected to electro- to bulk processes similar to thermogalvanic effects [4]. chemical metallization (nickel plating), no anomalies It appears that, when the carriers move along stream- (from the electrochemistry viewpoint) are observed. lines, their paths twist and eventually two parallel flows Ohmic contacts with a resistivity of 10Ð5Ð10Ð6 Ω cm2 are produced where the carries spiral. This becomes are produced [3]. possible because of the specific superstructure orientation, specimen geometry, and temperature distribution. The electrochemical deposition of nickel on HMS The situation looks as if the carriers were drawn in two single crystals in the form of a parallelepiped whose parallel-mounted “solenoids” instead of being spent on end face is dipped into an electrolyte proceeds in sev- metal reduction at cathode polarization. Eventually, eral stages. Within the initial five seconds, the blanket metal reduction on the surface changes to its dissolu- metallization of the electrolyte-covered surface takes tion, causing an oval, dumb-bell, spots, and rings to place. Then, the metal begins to dissolve at the speci- appear successively on the end face. On the side sur- men corners and edges, forming metal islands on the faces, the tendency to complete metal etching off is side faces. The islands extend toward the end face (see observed. figure). On the end face, the dissolving metal film first takes the oval shape, then its central part narrows to An alternative explanation is the redistribution of form a dumb-bell, and eventually two metal spots the carriers in the near-surface region because of the remain. If the process continues further, the central part high thermopower anisotropy and complex crystal of the spots also dissolves and two metal rings are left structure. Cathodic and anodic areas forming on the on the surface. The same is reproduced on another surface begin to act in parallel. As the specimen tem- specimen in the same electrolyte. perature changes during the electrochemical reaction,

THE BEHAVIOR OF HIGHER MANGANESE SILICIDE SINGLE CRYSTALS 1095

C C 90° 45° 90° α b

Anomalous electrochemical metallization of HMS single crystals with the certain superstructure orientation and different geometry. so do the mutual arrangement and shapes of the areas REFERENCES with different polarization. In this way, the residual 1. M. I. Fedorov, V. K. Zaœtsev, F. Yu. Solomkin, and metal ﬁlm is patterned. M. V. Vedernikov, Pis’ma Zh. Tekh. Fiz. 27 (15), 64 (1997) [Tech. Phys. Lett. 27, 602 (1997)]. Thus, an anomalous metallization process is 2. V. K. Zaitsev, in Handbook of Thermoelectrics, Ed. by observed in HMS single crystals of the speciﬁc geome- D. M. Rowe (CRC Press, New York, 1995), pp. 299Ð309. try and superstructure orientation. This anomaly can be 3. F. Yu. Solomkin, RF Patent No. 2009571, Byull. Izobret., associated either with bulk processes similar to ther- No. 5 (1994). mogalvanic effects or with the carrier redistribution in 4. L. S. Stil’bans, The Physics of Semiconductors (Sov. the near-surface regions that is due to the high thermo- Radio, Moscow, 1967). electric power anisotropy and complex crystal structure. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 8 2000

BRIEF COMMUNICATIONS

Generation and Amplification of Electromagnetic Waves by an Annular Electron Beam in a Radial Electric Field in Free Space Yu. V. Kirichenko National Science Center, Kharkov Institute of Physics and Technology, Akademicheskaya ul. 1, Kharkov, 310108 Ukraine Received November 22, 1999

Abstract—Mechanisms for the generation and amplification of electromagnetic waves by a thin-walled annular beam of electrons rotating in a radial electric field in free space are studied theoretically. It is shown that electromagnetic waves can be generated and amplified under the Cherenkov resonance conditions. The frequencies and growth rates of the generated waves are determined, and the propagation characteristics and amplification coefficients of the amplified waves are found. © 2000 MAIK “Nauka/Interperiodica”.

The theory of the generation and amplification of where kϕ = m/rÐ is the azimuthal component of the wave electromagnetic waves by annular electron beams in vector, c is the speed of light, and m = |m|. waveguides (in particular, plasma-filled waveguides) is widely covered in the literature (see, e.g., [1, 2]). Under Inequalities (1) allow us to apply the potential certain conditions (such as a small beam diameter in approximation [3, 4]. Using the method described by comparison with the waveguide diameter, the field falls Dolgopolov et al. [5], we can show that, in the case of off exponentially in the radial direction with distance a thin layer such that from the beam, and the wavelengths are short), the δ Ӷ waveguide walls have essentially no effect on the dis- r+ Ð rÐ = r rÐ, (2) persion properties of electromagnetic waves. On the other hand, the problem of the generation and amplifi- the electrostatic potential Φ treated in a linear approxi- cation of waves by a thin-walled annular beam of elec- mation under the Cherenkov resonance conditions sat- trons moving simultaneously in the azimuthal and axial isfies the boundary conditions directions in free space is of interest in its own right. Φ Φ We introduce the cylindrical coordinates (r, ϕ, z) Φ Φ d d κΦ r ==r , ------Ð ------r , (3) + Ð r Ð and consider an unbounded (along the z-axis) cylindri- dr + dr r Ð cal electron layer in which the electrons rotate in the azimuthal direction. Let us assume that a positively where charged metal rod with radius a, linear charge density r Q, and a high but finite conductivity σ is located at the 2+ 2 2 2 m2 2VϕÐΩ()rr coordinate axis. The electrons are held on circular equi- κ= ------d r Ω ()r ------, (4) 2ω2∫ 2 Ω2()2 librium orbits by the radial electrostatic field of the rod, rÐm2Vϕ+ rr r Ð F0(r) = 2Q/r. We neglect the constant magnetic and electric self-fields of the electron layer and assume that Ω2 π 2 ω ω (r) = 4 e n(r)/me, m = Ð mVϕ/rÐ Ð kzVz, Vϕ and Vz the perturbations of the electromagnetic field, electron are the azimuthal and axial components of the unper- density, and electron velocity all depend on the coordi- ϕ ϕ ω turbed electron velocity, and Ðe < 0 and me are the nates z and and time t as exp[i(m + kzz Ð t)], where ≠ charge and the mass of an electron. m 0 is an integer, kz is the projection of the wave vector onto the z-axis, and ω is the frequency. We treat the Formulas (3) and (4) were derived under the Cher- problem in the hydrodynamic approximation. The enkov resonance condition unperturbed electron density n(r) is assumed to be non- ω Ӎ zero in the layer between the surfaces r = rÐ and r = r+. m 0 (5) We also assume that the following conditions are satisfied: and with allowance for the fact that, by virtue of conditions (1), the quantity kz/kϕ is small. Taking (2) into ӷ ω ӷ ω Ӷ ()1/2 Φ Φ kz ---- , k ϕ ---- , k z r Ð 2 m , (1) account, we match (r) and d (r)/dr at the boundaries c c between the rod and the layer to obtain the dispersion

1063-7842/00/4508-1096 $20.00 © 2000 MAIK “Nauka/Interperiodica” GENERATION AND AMPLIFICATION OF ELECTROMAGNETIC WAVES 1097 relation In addition, dispersion relation (6) enables us to r solve the problem of the amplification of an electro- η + 2 Ω2 2 2 m m 1 2 2V ϕ Ð (r)r magnetic wave travelling along the z-axis by an annular ω = Ð------(1 Ð iωδσ)---- drΩ (r)------, (6) m 2m r ∫ 2 2 2 electron beam under the Cherenkov resonance condi- η Ð 2V ϕ + Ω (r)r ω rÐ tion (5). To do this, we must set Im( ) = 0 in equation (6), in which case we have Im(k ) ≠ 0 and |Im(k )| Ӷ η η η2m δ πση z z where = rÐ/a, m = ( Ð 1)/2, σ = 1/(4 m), and Re(kz). Under condition (8), equation (6) gives the fol- |ωδσ| Ӷ 1. lowing expression for the amplification coefficient | | The parameter δσ accounts for energy losses in the Im(kz) : Ω 2 1/2 rod. We introduce the averaged quantity : η 1/2 2 Ω 2 1/2 Ω m m 2V ϕ Ð rÐ δr r Im(k ) = ------------ ----- ; (15) 2 Ð2 2 + 2 2 2 z  2m  2   V z η  2 2 rÐ 2 2V ϕ Ð Ω r 1 2 2V ϕ Ð Ω (r)r 2V ϕ + Ω r Ω ------Ð-- = ----- drΩ (r)------. (7) Ð 2 ΩÐ2 2 δr∫ 2 Ω2 2 2V ϕ + rÐ 2V ϕ + (r)r under condition (11), we obtain rÐ We start by solving the problem of the excitation of ωΩδ η 1/2 σm m electromagnetic waves. To do this, we must set Im(k ) = Im(k ) = ------z z 2V  η2m  0 in (6), which then implies that Im(ω) ≠ 0 and z |Im(ω)| Ӷ Re(ω). Under the condition 2 1/2 (16)  2Ω 2  1/2 rÐ Ð 2V ϕ δr 2 2 2 × ------ ----- . > Ω 2   2V ϕ rÐ (8)  2Ω 2 rÐ rÐ + 2V ϕ equation (6) yields The amplification coefficient (15) is much larger 1/2 mV ϕ δ | | (ω)  r δ  than (16). Under condition (11), the coefficient Im(kz) Re = ------+ kzV z + 0----- σ , (9) rÐ rÐ is governed by the energy losses in the rod and is proportional to the frequency of the amplified wave. 2 1/2 1/2 2 2 1/2 mη  2V ϕ Ð Ω r ∂r Note that, according to the first inequality in (1), the (ω) Ω m ------Ð- Im = ------2--m-- 2 ----- ; (10) magnetic and electric fields are both exponentially η  2 Ω 2 rÐ 2V ϕ + rÐ decreasing in the radial direction. Consequently, during under the condition the generation or amplification of a wave, an infinite annular beam in free space emits no electromagnetic 2 2 2 waves, thereby keeping its energy unchanged. In con- 2V ϕ < r Ω , (11) Ð trast, a generated or amplified wave will be emitted by equation (6) yields a semi-infinite annular beam (bounded, e.g., by the ∂ 1, 2 coordinate z0) through its end into a half-space z > z0. (ω) mV ϕ  r  Re = ------+ kzV z Ð 0-----  , (12) rÐ rÐ ACKNOWLEDGMENT η 1/2 (ω) 1δ ΩmV ϕ  m m Im = -- σ ------+ kzV z ------2--m-- I am grateful to V.V. Dolgopolov for fruitful discus- 2 rÐ η sions and valuable remarks. 2 1/2 (13)  2 2  1/2 r Ω Ð 2V ϕ δr × ---Ð------ 2 ----- . REFERENCES  2 2 rÐ r Ω + 2V ϕ Ð 1. Ya. B. Faœnberg, Fiz. Plazmy 11, 1398 (1985) [Sov. J. We can see that, under condition (8), which indi- Plasma Phys. 11, 803 (1985)]. cates that the angular electron velocity is high and the 2. A. N. Kondratenko, Plasma Waveguides (Atomizdat, electron density is low, the growth rate is considerably Moscow, 1976). higher than that under the opposite condition (11), 3. R. C. Davidson, Theory of Nonneutral Plasmas (Ben- according to which the instability is triggered by the jamin, New York, 1974; Mir, Moscow, 1978). dissipation of the wave energy in the rod. Note that the 4. V. V. Dolgopolov, Yu. V. Kirichenko, Yu. F. Lonin, and growth rate (10) increases with m as m1/2 , while the I. F. Kharchenko, Zh. Tekh. Fiz. 68 (8), 91 (1998) [Tech. growth rate (13) is exponentially decreasing. However, Phys. 43, 959 (1998)]. we must keep in mind that, by virtue of inequalities (1), 5. V. V. Dolgopolov, M. V. Dolgopolov, and Yu. V. Kiri- the positive integer m is limited by the condition chenko, Izv. Vyssh. Uchebn. Zaved. Radioelektron. 40 (12), 16 (1997). k rc m Ӷ ---z------. (14) V ϕ Translated by O. E. Khadin

TECHNICAL PHYSICS Vol. 45 No. 8 2000

THEORETICAL AND MATHEMATICAL PHYSICS

Analytical Solution to the Skin-Effect Problem for an Arbitrary Accommodation Coefficient of Electron Tangential Momentum A. V. Latyshev and A. A. Yushkanov Moscow Pedagogical University, ul. Radio 10a, Moscow, 107005 Russia E-mail: [email protected] Received February 23, 1999

Abstract—An analytical solution to the boundary-value problem of an electric field and electrons in a metal- filled half-space is obtained for arbitrary values of the tangential-momentum accommodation coefficient. The frequency of an external electromagnetic field directed tangentially to the surface is allowed to take on complex values. Both the normal and anomalous skin effects are considered. In the latter case, the low- and high-frequency limits are examined. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION problem were addressed in [5, 6]. Studies [7, 8] of high- frequency processes in metals highlighted the effi- The skin-effect problem has been solved analyti- ciency of the Case method [9], which expands a solu- cally for the specular and diffuse reflections of elec- tion in the singular generalized eigenfunctions of the trons from a metal surface. The former case corre- associated characteristic equation. In [10, 11], diffuse sponds to zero accommodation of electron tangential boundary conditions were applied to an electron momentum; and the latter, to total accommodation (the plasma subjected to an electromagnetic field perpen- tangential-momentum accommodation coefficient q is dicular to the surface. In contrast to the WienerÐHopf 0 or 1, respectively). For intermediate q’s, analytical method [3–6], Case’s method yields explicit expres- solutions have not yet been found. The aim of this study sions for discrete modes of the solution. As emphasized is to fill the gap. by some authors (see, e.g., [12]), under certain condi- For conventional boundary conditions, the behavior tions, such modes define to the largest extent the prop- of gases or electrons near the surface defies analytical erties of both the electromagnetic field and conduction description if the specular reflection coefficient α takes electrons. Furthermore, the Case method provides the on arbitrary values. However, modified boundary con- deepest understanding of different types of skin effect. ditions were developed in the kinetic theory of gases [1] That is why it is followed here. α so as to allow for an arbitrary q (q = 1 Ð for the spec- In this study, we apply the expansion in singular ular or diffuse case). They make the boundary-value eigenfunctions to finding the impedance at 0 ≤ q ≤ 1. In problems solvable in an analytical form (e.g., for the particular, formulas for diffuse (q = 1) and specular isothermal or thermal slip of gas, etc.). We are going to (q = 0) boundary conditions are derived. The normal extend this approach to electrons in metals. and anomalous skin effects are considered. In the latter The tangential momentum accommodation coeffi- case, the high- and low-frequency limits are explored. cient q is the ratio of the tangential momentum flux of For the anomalous skin effect, a relation is revealed electrons reflected from the surface to that of electrons between the macroscopic response (impedance) of a impinging on the surface: metal to an external field and the discrete spectrum. The Ð1 beauty of the Case method is that it dispenses with intu- 3 3 itive ideas, such as the concept of inefficiency [1, 13]. q =∫v nv τ fd vv∫nvτdv. () () + Ð FORMULATION OF THE PROBLEM v v Here, n and τ are, respectively, the normal and tan- We restrict ourselves to the case of a spherical Fermi gential components of the electron velocity. The plus surface and small oscillation frequencies, neglecting v v and minus signs correspond to n > 0 and n < 0, displacement current. The external field is assumed to respectively. The surface is assumed to be planar. be weak, which allows us to view the problem in a lin- An analytical solution to the problem of the anoma- ear context. Consider a metallic medium occupying a lous skin effect in a half-space was first obtained by the half-space. Let us introduce Cartesian coordinates so WienerÐHopf method [2Ð4]. Generalizations of the that the origin lies on the surface, the x-axis is perpen-

956 LATYSHEV, YUSHKANOV dicular to the surface and is directed into the metal, the ing electrons: y-axis is aligned with the electric field E(x)exp(αt), and 0 the z-axis is perpendicular to the magnetic field 2 H(x)exp(αt). (1 Ð q) ∫ (1 Ð µ' )ψ(0, µ')µ'dµ' After the time variable is separated, the equations Ð1 for electric field and electrons in the metal have the 1 form [2] = Ð ∫(1 Ð µ'2)ψ(0, µ')µ'dµ'. ∂ f ∂ f 0 v ------1 + (ν + α) f (x, v ) = ev E(x)------0, x ∂x 1 y ∂ε Here, q is the tangential-momentum accommodation ( ) ( ) coefficient, i.e., the probability of electron reflection. E'' x = Aj x . The last two conditions imply

Here, f0 is the Fermi distribution of electrons; f1 is the 1 α ε v ν 1 Ð q 2 correction to f0; f = f0 + f1exp( t); , e, m, , and are d = Ð4------∫ (1 Ð µ' )ψ(0, µ')µ'dµ'. (3) the electron energy, charge, mass, velocity, and colli- q sion rate, respectively; j is the current density; and A = Ð1 4παcÐ2. The current density is given by

3 EIGENFUNCTIONS  m  3 j(x) = Ð2e ------v f (x, v )d v . 2πប ∫ y 1 Let us separate the variables according to

ψη(x, µ) = exp(Ðx/η), eη(x) = exp(Ðx/η)E(η), Let us introduce the notation where η is the spectral parameter. We thus arrive at the ( ) ( ) α ν e x = E x /E0, w0 = 1 + / , characteristic system (τv ) µ v v x' = x/ F , = x/ F, (η µ)Φ(η, µ) Ð1η (η) Ð = kw0 E , eE f = v δ(v Ð v )ψ, k = Ð------0-----, (η) β Ð2η (η) β Ð2 1 y F ν E = w0 n , b = w0 , 2m v F ev 6 α where β = ------F------. ν2πប3 2 1 c E0 n(η) = ∫ (1 Ð µ2)Φ(η, µ)dµ. Here, vF is the electron velocity on the Fermi surface, Ð1 E0 is the electric field amplitude on the surface of the η metal, and δ(x) is the delta function. In what follows, Eliminating E( ) from the last two equations results instead of x', we will write x. Then, the equations in the characteristic equation become 1 3 (η Ð µ)Φ(η, µ) = --aη n(η), ∂ψ 2 µ------+ w ψ(x, µ) = ke(x), ∂x 0 where 1 (1) 2 5 2 α e v 1 e''(x) = β ∫ (1 Ð µ' )ψ(x, µ')dµ'. a = Ða ------, a = ------F-----, ν = --. 0 3 0 3 2 τ Ð1 (ν + α) πmប c The conditions on the surface and away from it are If η ∈ (Ð1, 1) and n(η) ≡ 1, the characteristic equa- assumed to have the form tion immediately yields its continuum eigenfunctions [14]: ψ(0, µ) = d, 0 < µ < 1; 1 3 1 λ(η) ψ(∞, µ) < µ < Φ(η, µ) = --aη P------+ ------δ(η Ð µ). = 0, Ð1 0; (2) η µ 2 2 Ð 1 Ð η e(0) = 1, e(∞) = 0, Here, PxÐ1 denotes a distribution, i.e., the principal where d relates to the mean velocity of the electrons value of the integral of xÐ1; δ(x) is the Dirac function; reflected from the surface. and λ(z) is the dispersion function of the problem, Suppose that the momentum flux of reflected elec- λ( ) 2 2( 2)λ ( ) trons equals 1 Ð q times the momentum flux of imping- z = 1 Ð az + az 1 Ð z c z ,

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ANALYTICAL SOLUTION TO THE SKIN-EFFECT PROBLEM 957 where of integration. In view of the behavior of κ = κ(δ) for δ ∈ ∆+ ∆Ð 1 ( ), we take the function 1 dτ κ λ (z) = 1 + ------X(z) = zÐ expV(z), c 2 ∫ τ Ð z Ð1 1 1 dτ is the Case dispersion function [9]. V(z) = ------[lnG(τ) Ð 2πiκ]------. 2πi∫ τ Ð z By definition, the discrete spectrum of the charac- 0 teristic equation comprises the zeros of the dispersion Explicit expressions for the zeros of the dispersion function that lie outside the cut [Ð1, 1]. The structure of function can be obtained from its factorization formu- the spectrum can be found by the technique developed las (presented without derivation): in [15]. First, we will consider the homogeneous Rie- 2 2 2 2 2 + mann boundary-value problem ω(z) = --(η Ð z )(η Ð z )X(z)X(Ðz), z ∈ ∆ , 3 0 1 X+(µ) = G(µ)XÐ(µ), 0 < µ < 1, (4) 2 2 2 Ð ω(z) = --(η Ð z )X(z)X(Ðz), z ∈ ∆ . where 3 0 G(µ) = λ+(µ)/λÐ(µ) = ω+(µ)/ωÐ(µ). The eigenfunctions associated with the discrete spectrum at hand are as follows: Here, Φ(±η , µ) ±1 η 3 1 ω±(µ) λ± δ (µ) (δ ± (µ)) k = --a k ±----η------µ---, = /a = 1 Ð p + i 2 q , 2 k Ð δ = 1/a = δ + δ , (±η ) η 2 ( , ) 1 2 E k = b k k = 0 1 . and We skip over the boundary regime, since this topic has been carefully treated in the context of the Rayleigh (µ) µ2[ ( µ2)λ (µ)] p = 1 Ð 1 Ð c , problem [17]. In [17], the continuity of the distribution function and its integral characteristics in the plane of π 3 2 q(µ) = --µ (1 Ð µ ). complex frequency were also demonstrated for the case 2 when the index of the Rayleigh problem varies in steps. Let Θ(µ) denote the regular branch of argG(µ) such Let us find explicit expressions for the zeros of the that Θ(0) = 0. Let us find the index [16] of problem (4). dispersion function in two limiting cases: |δ| Ӷ 1 and We have |δ| ӷ 1. If |δ| Ӷ 1, we are dealing with the anomalous skin κ 1 [Θ(µ)] 1 [Θ(µ)] 1 [ ω( )] ± 3 = ------(0.1) = ------(Ð1.1) = ------arg z γ , effect. Then, ω = δ ± iπµ /2. If δ is in the second quad- 2π 4π 4π η rant, the discrete spectrum consists of two zeros: 1 = where γ is a contour that runs clockwise around the cut δ π η η 3 δ π rexp(iarg /3) + i /6 and 2 = Ð 1, where r = 2 / . [Ð1, 1] and does not enclose the zeros of the dispersion η ᑬη π The zero 1 is in the first quadrant ( 1 > 0), since < function. δ π η δ arg < /2. The zero 2 is in the third quadrant. If is Let N and P respectively denote the total numbers of in the third quadrant, the spectrum also consists of two the zeros and poles of ω(z) outside the cut [Ð1, 1]. As η δ π π η η zeros: 1 = rexp(iarg /3 + i /6 + i2 /3) and 2 = Ð 1. follows from the last equality and the principle of argu- η π κ The zero 1 is in the second quadrant now, since Ð < ment [16], the index of problem (4) is = (N Ð P)/2. In δ π δ δ the δ plane, consider the region arg < Ð /2. Note that, if is real and negative (arg = π), the dispersion function has two purely imaginary η η η δ ∆+ {δ δ δ < δ < , δ < (δ )} zeros: 0 = ri and 1 = Ð 0. On the other hand, if is a = = 1 + i 2; 0 1 1 2 q 1 . η π purely imaginary number, then 0 = rexp(i /3) and Ð η η δ π η π η Let ∆ and γ respectively denote the exterior and 1 = Ð 0 for arg = /2 and 0 = rexp(Ði /3) and 1 = 0 η δ π boundary of δ ∈ ∆+. As in [15], it can be shown that, if Ð 0 for arg = Ð /2. δ ∈ ∆+, κ = 1 and hence N = 4, since P = 2 for all δ ∈ ∆±. Let δ be in the right half-plane. In this case, the dis- Also, if δ ∈ ∆Ð, κ = 0 and hence N = 2. That P = 2 fol- persion function has four zeros, two of which, lows from the expansion η [ ( δ π π )] ( , ) k = rexp i arg /3 + /6 + 2 k/3 k = 0 1 , 2 2 2 ᑬη ᑬη ω(z) = Ð --z + δ Ð ----- + o(1), z ∞. (5) lie in the upper half-plane (with 0 > 0 and 1 < 0) 3 15 η η η η and the other zeros are 2 = Ð 1 and 3 = Ð 0. For problem (4), let us select a solution that is non- In the context of the anomalous skin effect, we sin- vanishing and bounded at the extremes of the interval gle out two important cases: the high- and low-fre-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 958 LATYSHEV, YUSHKANOV quency conditions. Under the low-frequency condi- At x = 0, we obtain tions, the ﬁeld frequency is much lower than the elec- |α| Ӷ ν ν|α| Ӷ 1 3 1 3 tron collision rate: and also a0. 1 η 1 η A(η) λ(µ) A ------k----- + a------dη + ------A(µ) = d, Furthermore, w = 1 + α/ν = 1 and δ = Ð(ν + α)3/αa = ∑ k η µ ∫ η µ 2 0 0 2 k Ð 2 Ð 1 Ð µ (8) ν3 α ν3 ω ±η k = 0 0 Ð / a0 = Ð i/ a0. We then have two zeros: 0, η π < µ < where 0 = rexp(Ði /3). Under the high-frequency con- 0 1, ditions (which are close to an oscillatory regime), the 1 ﬁeld frequency is much higher than the electron colli- η2 η2 η2 (η) η sion rate: |α| ӷ ν. This situation occurs at low temper- A0 + A1 + ∫ A d = 1/b. (9) atures and low impurity concentrations. Here, w0 = 0 ατ ωτ δ α2 = Ði and = Ð /a0. The dispersion function has It is seen that (8) is a complete singular integral α ω δ ω2 four zeros, since = Ði , so that = /a0 > 0. The equation with the Cauchy kernel. To prove this, substi- expressions for the zeros are as in the above. tute expansion (6) into boundary condition (3) for d. We solve equation (8) by the CarlemanÐVekua regulariza- The case |δ| ӷ 1 corresponds to the normal skin tion [15]. The approach is built around an explicit solu- effect. Now the dispersion function has two zeros. As tion of the characteristic equation. follows from (5), the expression for them is Let us introduce the auxiliary function

η [ ( δ π )] δ ( , ) 1 3 k = rexp i arg /2 + k , r = 3 /2 k = 0 1 . 1 η A(η) N(z) = ------dη. 2∫ η Ð z If δ lies in the negative portion of the real axis, then 0 ᑬη δ η δ 0 > 0 always. If < 0, then 0 = ri. If > 0, then the Using the boundary values of N(z) and λ(z) reduces ±η zeros are real: 0. equation (8) to the nonhomogeneous Riemann boundary-value problem

EXPANDING THE GENERAL SOLUTION 1 η 3 λ+(µ) +(µ) 1 Ak k IN EIGENFUNCTIONS N Ð d Ð -- ∑ µ------η---- 2 Ð k k = 0 Let us represent the general solution of system (1) in 1 η 3 terms of the discrete and continuum eigenfunctions in λÐ(µ) Ð(µ) 1 Ak k such a way as to satisfy boundary conditions (2) (away = N Ð d Ð -- ∑ µ------η---- , 2 Ð k from the surface). The expansions read as follows: k = 0 0 < µ < 1. 1 ψ( , µ) 1 η 3 1  x  Now recall the results for problem (4) and reduce x = ∑ --a k η------µ--expÐw0η----- 2 k Ð k the last equation to the problem of finding an analytic k = 0 function from its zero jump over a cut: 1 (6)  x --- Φ(η, µ) (η) η 1 3 + ∫ expÐw0  A d , + + 1 A η η X (µ) N (µ) Ð d Ð -- ∑ ------k------k-- 0 µ η 2 Ð k k = 0 1 1 2 x 3 ( ) η   Ð Ð 1 A η e x = ∑ Akb k expÐw0η----- = X (µ) N (µ) Ð d Ð -- ∑ ------k------k-- , k µ η k = 0 2 Ð k k = 0 1 (7) < µ <  x η2 (η) η 0 1. + b∫ expÐw0η--- A d . 0 Its general solution is 1 3 1 1 η A(η) Ð1 ck Here, Ak (k = 0, 1) are unknown coefficients associated N(z) = d + ------+ X(z) ------, (10) Ð ∑ η ∑ η with the discrete spectrum, with A = 0 for δ ∈ ∆ ; 2 z Ð k z Ð k 1 k = 0 k = 0 A(η) is an unknown function associated with the continuous spectrum (continuous expansion coefficient); where c0 and c1 are arbitrary constants. ᑬ η ᑬ and w0/ k > 0 (k = 0, 1), with w0 > 0. Eliminating poles from (10) gives

1 3 Based on the boundary conditions, let us determine c = Ð--A η X(η ) (k = 0, 1). the expansion coefﬁcients in (6) and (7). Let δ ∈ ∆+. k 2 k k k

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ANALYTICAL SOLUTION TO THE SKIN-EFFECT PROBLEM 959

The condition N(∞) = 0 implies In view of (11), substituting (13) into (9) gives

d + c0 + c1 = 0. (11) 2c0b  1 1   1  Ð -----(------)-η---- Ð -η---- + 2bd1 + -η------(------) = 1. To the auxiliary function N(z), we apply the X 0 0 1 1 X 0 Sokhotskiœ formula. Then we substitute (10) into the resultant expression and determine the continuum Now substitute expansion (6) into boundary condi- expansion coefficient: tion (3). In doing so, we will take into account that the 1 first moments of the continuum and discrete eigenfunc- 3  1 1  ck πiµ A(µ) = ------Ð ------. (12) tions are respectively expressed as  + Ð  ∑ µ η (µ) (µ) Ð k X X k = 0 1 2 2 3 Using (12), we will calculate the integral in (9). (1 Ð µ )F(η, µ)µdµ = η Ð --aη , First, notice that ∫ 3 Ð1 1 1 2c η2 A(η)dη = ∑ ------k [J(η ) Ð J(0)], 1 ∫ η k 2 2 3 k (1 Ð µ )F(η , µ)µdµ = η Ð --aη (k = 0, 1). 0 k = 0 ∫ k k 3 k where Ð1 1 We thus obtain the equation µ (η ) 1 1 1 d ( , ) J k = ------∫ ------Ð ------k = 0 1 . 2πi +(µ) Ð(µ) µ Ð η q 2 X X k d------+ R Ð --aR = 0. 0 4(1 Ð q) 1 3 3 η The integral appearing in the formula for J( k) is calculated via the following integral expression (pre- Due to (13) and (11), we finally arrive at sented without derivation): 2c0  1 1   X'(0) 1 1  1 R = ------Ð ------Ð ----- Ð ----- 1 1 1 1 dµ 1 ( )η η   ( ) η η  ------= z Ð V + ------Ð ------, X 0 0 1 X 0 0 1 ( ) 1 π ∫ + Ð µ X z 2 i X (µ) X (µ) Ð z 0 2d  X'(0) 1  Ð -η------(------)-----(------)- Ð -η---- , where 1 X 0 X 0 1 (η η ) (η ) 1 R3 = 2c0 1 Ð 0 + 2d 1 Ð V 1 . 1 [ (τ) π ] τ V 1 = Ð------∫ lnG Ð 2 i d . 2πi The resultant system of equations uniquely deter- 0 mines the parameters d and c0 of general solution (10). η η Ð1 η It is now clear that J( k) = X( k) = k + V1. Hence, This completes the proof of expansions (6) and (7) for δ ∈ ∆+. (η ) ( ) 1 η 1 J k Ð J 0 = -----(--η------)- Ð k Ð -----(------). X k X 0 Furthermore, J'(0) = ÐX'(0)/X2(0) Ð 1. Thus, the inte- EXACT IMPEDANCE FORMULAS gral in (9) is First, consider the case δ ∈ ∆+. Differentiating (7) 1 1 for electric field gives η2 (η) η 2ck 1 η 1 ∫ A d = ∑ --η------(--η------)- Ð k Ð -----(------) , k X k X 0 1 1 0 k = 0 e'(0) = Ðbw ∑ A η + ηA(η)dη . which implies 0 k k ∫ k = 0 0 1 2c k η 1  According to the above notation, e'(0) = Ðbw R , where R2 = Ð ∑ --η----- k + -----(------) , (13) 0 1 k X 0 k = 0 R1 is determined from (13). To calculate the impedance, where we start from the formula Z = Ae(0)/e'(0). Since e(0) = 1 [see (2)], we have Z = A/e'(0). With the notation 1 1 ηl (η) η η l ( , , ) ( ) 1 Rl = ∫ A d + ∑ Ak k l = 1 2 3 . (η , η ) X' 0 1 g 0 1 = -----(------)- Ð -η---- Ð -η----. 0 k = 0 X 0 0 1

TECHNICAL PHYSICS Vol. 45 No. 8 2000 960 LATYSHEV, YUSHKANOV the desired exact expression is η η ( ) ( ) (η η ( ) η η ) A g Ð 1/ 0 1 X 0 + q/8 1 Ð q Ð 2a 0 1 X 0 + 0 + 1 Ð V 1 /3 Z = ------[------(------)------(--η------η------)------]------. (14) w0 g q/8 1 Ð q Ð 2a 0 + 1 Ð V 1 /3 Ð 2a/3

With q = 0, formula (14) becomes η η ( ) (η η ( ) η η ) 3A g Ð 1/ 0 1 X 0 Ð 2a 0 1 X 0 + 0 + 1 Ð V 1 /3 Z = Ð------(--η------η------)------. (15) 2aw0 g 0 + 1 Ð V 1 + 1

As q 1 (diffuse boundary conditions), the impedance tends to A (η , η )Ð1 Z = -----g 0 1 . w0

Now look at the case δ ∈ ∆Ð. The same reasoning as above leads to the exact formula (η ) η ( ) ( ) (η ( ) η ) A g 0 + 1/ 0 X 0 + q/8 1 Ð q + 2a 0 X 0 Ð 0 + V 1 /3 Z = ------(---η------)--[------(------)------(---η------)------]------, (16) w0 g 0 q/8 1 Ð q Ð 2a 0 Ð V 1 /3 Ð 2a/3

η η where g( 0) = X'(0)/X(0) Ð 1/ 0. Furthermore, For (16), consider the limits q = 0 and q 1, 1 1 πµ3(1 Ð µ2)dµ which refer to the zero accommodation of electron tan- V(0) = -- arctan------, gential momentum and to diffuse boundary conditions, π∫ 2(δ Ð p(µ)) µ respectively. With q = 0, the impedance is 0 1 πµ3( µ2) 3A 1 1 Ð µ Z = Ð------V 1 = Ð--∫ arctan------d . 2aw0 π 2(δ Ð p(µ)) (17) 0 g(η ) + 1/η X(0) + 2a(η X(0) + η Ð V )/3 × ------0------0------0------0------1------. For large |δ|’s, these yield g(η )(η Ð V ) + 1 0 0 1 1 1 2 2 1 With q 1, V(0) = ----- µ (1 Ð µ )dµ = ------, 2δ∫ 15δ A (η )Ð1 0 Z = -----g 0 . (18) w0 1 1 3 2 1 V = Ð----- µ (1 Ð µ )dµ = Ð------. 1 2δ∫ 24δ LIMITING CASES 0 δ ∈ ∆Ð δ Then consider the case q = 0. Based on (17), we We begin with the normal skin effect. If ( < obtain the same formula: Z = ÐAη /w . With an arbi- 0), then 0 0 trary q, formula (16) gives 1 3 2 η X'(0) 1 πµ (1 Ð µ )dµ A 0 2(1 Ð q) ------= V'(0) = -- arctan------. Z = Ð------1 Ð ------. π∫ (δ (µ)) 2 δ( δ ) X(0) 2 Ð p µ w0 9 q + 1 Ð q 0 Now, we proceed to the anomalous skin effect. First, For large |δ|’s, this yields consider the low-frequency limit. It is characterized by α ν ≈ δ δ δ η π 1 w0 = 1 + / 1, = Ði 0 ( 0 > 0), 0 = rexp(Ði /3), ( ) 1 µ( µ2) µ 1 V' 0 = ----δ-∫ 1 Ð d = ----δ-. and r = 3 2 δ /π . Let us employ the following asymp- 2 8 ± 0 totics for ω(z): ω (µ) = δ ± iπµ3/2. Hence, Since the normal skin effect is characterized by δ < 0, 1 1 δ + iπτ3/2dτ η δ η ≈ η V'(0) = ------ln------. 0 = Ðri, and r = 3 /2 , we have V'(0) Ð 1/ 0 Ð1/ 0. π ∫ 3 2 η 2 i δ Ð iπτ /2 τ Consequently, if q 1, then Z = ÐA 0/w0 [see (18)]. 0

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ANALYTICAL SOLUTION TO THE SKIN-EFFECT PROBLEM 961

After the change of the variable, we have V'(0) = equality V'/(3r), where 1 πµ3( µ2) ∞ 1 1 Ð µ V 1 = Ð--∫ arctan------d 1 Ð4/3 1 Ð x 3 π 2(δ Ð p(µ)) V' = ------x ln------dx = i------(1 + i 3). 2πi∫ 1 + x 2 0 0 implies π η Consequently, V'(0) = V'exp(Ði /3)/(3 0) = 1 η η ≈ 1 ( 2) 1 i/( 3 0) = V 0' / 0, where V 0' = i/ 3 . Finding the V 1 --∫ 1 Ð x dx = --. δ 2 3 asymptotics of the expressions for V(0) and V1 as 0 0 and using (16), we arrive at To calculate V'(0), we use the following asymptotic ω ω± µ δ ± πµ3 A η representation of the function : ( ) = i /2. Z = ------0-----, Then, we have w0 V 0' Ð 1 1 1 πµ3 dµ or Z = ÐAr/(2w0). V ' (0) = -- arctan------. 0 π∫ δ 2 Turn to the high-frequency limit. It is characterized 2 µ ω ν δ 0 by w0 = Ði / and > 0. Furthermore, there are four η π η πµ2 δ µ 3 δ π discrete modes, for which 0 = rexp(Ði /6) and 1 = Let x denote /(2 ) = x, so that = 2 / . Then rexp(Ði5π/6), where r = 3 2δ/π . ∞ 11 dx 3 π 1 3 Ð1 V ' (0) = ---- arctanx------= ----- ln 2 + -- Ð -- + ------. Let us expand X(z) as X(z) = (z Ð 1) expV0(z), 0 π∫ 4/3 π   r x r 2 2 3 where 0

1 3 2 In short, V ' (0) = V ' /r, where V ' = 3[ln 2 + π(−1/2 + 1 πτ (1 Ð τ ) dτ 0 0 0 V (0) = -- arctan------. 0 π∫ 2(δ Ð p(τ))τ Ð z 3 /3)/2]/π. Thus, the impedance for diffuse boundary 0 conditions is Notice that V is bounded if δ 0. Furthermore, 1 A 2δ/π Z = ------. η η ( ) δ w 0 1 X 0 = 3 /2, 0 V 0' Ð i X'(0)/X(0) = V ' (0) + 1, η η = Ðr2. It can easily be proven that this formula applies in 0 0 1 the general case as well [see (15)]. The asymptotics for ω(z) enable us to ﬁnd

1 3 CONCLUSIONS 1 πτ dτ V ' (0) = -- arctan------. 0 π∫ 2δ τ3 We developed a novel technique to obtain an analyt- 0 ical solution to the classical skin-effect problem in a generalized form. According to our approach, the fre- Hence, V 0' (0) = V 0' /r, where quency of the external electric field may take on complex values and the accommodation coefficient of elec- 1 tron tangential momentum q may lie anywhere between 1 Ð4/3 3 1 1 3 V ' = -- x arctanxdx = -- -- ln2 Ð -- + ------. 0 and 1. The technique consists in expanding the solu- 0 π∫ 2 π 2 3 0 tion to the boundary-value problem in the singular generalized eigenfunctions of the associated characteristic Due to (14) [or (15)], the impedance is equation. This enabled us to investigate the problem A r comprehensively. In particular, we derived explicit Z = ------. expressions for the discrete modes of the solution and w 0 V 0' Ð 1 constructed the frequency region D+ such that the prob- The same formula applies to diffuse boundary con- lem has four discrete solutions if the frequency lies in D+ and two discrete solutions if the frequency is outside ditions as well. Let us find V(0), V'(0), and V1. The above factorization of the dispersion function yields D+. The D+ region boundary represents the line of critical frequencies: crossing the boundary of D+ from the δ 2η2η 2 ( ( )) inside changes the structure of the solution. = -- 0 1 exp 2V 0 , 3 Remarkably, the singular-eigenfunction method enabled us to reduce the problem to a complete singular so that X(0) = i(π/4)2/3 3 δÐ1/6. We will demonstrate integral equation with the Cauchy kernel rather than to δ µ ≈ µ3 that V1 is bounded if 0. Since p( ) , the the characteristic equation. This seems to be the first

TECHNICAL PHYSICS Vol. 45 No. 8 2000 962 LATYSHEV, YUSHKANOV result of its kind in transport theory. The skin-effect 9. K. M. Case and P. F. Zweifel, Linear Transport Theory problem is thus opened up to the CarlemanÐVekua reg- (Addison-Wesley, Reading, Mass., 1967; Mir, Moscow, ularization based on an explicit solution of the charac- 1972). teristic equation. 10. A. V. Latyshev and A. A. Yushkanov, Zh. Vychisl. Mat. Mat. Fiz. 33, 600 (1993). REFERENCES 11. A. V. Latyshev and A. A. Yushkanov, Poverkhnost, No. 2, 25 (1993). 1. C. Cercignani, Theory and Application of the Boltzmann 12. V. M. Gokhfel’d, M. A. Gulyanski , M. I. Kaganov, and Equation (American Elsevier, New York, 1975; Mir, œ A. G. Plyavenek, Zh. Éksp. Teor. Fiz. 89, 985 (1985) Moscow, 1978). [Sov. Phys. JETP 62, 566 (1985)]. 2. A. A. Abrikosov, Introduction to the Theory of Normal Metals (Nauka, Moscow, 1972). 13. J. M. Ziman, Electrons and Phonons (Clarendon, Oxford, 1960; Inostrannaya Literatura, Moscow, 1962). 3. G. E. H. Reuter and E. H. Sondheimer, Proc. R. Soc. London, Ser. A 195, 336 (1948). 14. V. S. Vladimirov, Generalized Functions in Mathemati- 4. R. B. Dingle, Physica (Amsterdam) 19, 311 (1953). cal Physics (Nauka, Moscow, 1976). 5. P. M. Platzman and P. A. Wolff, in Solid State Physics, 15. A. V. Latyshev and A. A. Yushkanov, Teor. Mat. Fiz. 92, Suppl. 13 (Academic, New York, 1973; Mir, Moscow, 127 (1992). 1975). 16. F. D. Gakhov, Boundary-Value Problems (Nauka, Mos- 6. I. F. Voloshin, V. G. Skobov, L. M. Fisher, and cow, 1977, 3rd ed.; Addison-Wesley, Reading, Mass., A. S. Chernov, Zh. Éksp. Teor. Fiz. 80, 183 (1981) [Sov. 1966). Phys. JETP 53, 92 (1981)]. 17. A. V. Latyshev and A. A. Yushkanov, Teor. Mat. Fiz. 116, 7. A. V. Latyshev, A. G. Lesskis, and A. A. Yushkanov, 305 (1998). Teor. Mat. Fiz. 90, 179 (1992). 8. A. V. Latyshev and A. A. Yushkanov, Zh. Vychisl. Mat. Mat. Fiz. 33, 259 (1993). Translated by A. A. Sharshakov

TECHNICAL PHYSICS Vol. 45 No. 8 2000

Technical Physics, Vol. 45, No. 8, 2000, pp. 963Ð970. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 8Ð15. Original Russian Text Copyright © 2000 by Indeœtsev, Sergeev, Litvin.

THEORETICAL AND MATHEMATICAL PHYSICS

Resonance Vibrations of Elastic Waveguides with Inertial Inclusions D. A. Indeœtsev, A. D. Sergeev, and S. S. Litvin Institute of Problems in Machine Science, Russian Academy of Sciences, Vasil’evskiœ Ostrov, Bol’shoœ pr. 61, St. Petersburg, 199178 Russia Received March 2, 1999

Abstract—The problem of resonance oscillations of inertial inclusions in contact with elastic waveguides has triggered a number of theoretical investigations. It was shown [1Ð3] that related phenomena may be treated by solving the spectral problem posed for a differential equation that is defined in an infinitely long interval. For specific waveguide and inclusion parameters, a composite system that includes interacting objects with lumped and distributed parameters may have a mixed (continuous and line) eigenfrequency spectrum. The line spectrum may be observed both before and after the boundary frequency. It was noted [3, 4] that the presence of an isolated lumped inertial element causes the line eigenfrequency spectrum, which extends to the boundary frequency. So-called trap oscillations are responsible for this spectrum. However, little is yet known about these effects, which hinders their effective use in practice. First, conditions for trap oscillations should be generalized for the case of multielement inclusions in various infinite waveguides. Second, the effect of edge conditions on the line spectrum in a semi-infinite waveguide calls for in-depth investigation. The solution to these problems would formulate proper ways of tackling engineering challenges associated with the interaction of a railway track with high-speed rolling stock [5]. Issues discussed in this paper are also concerned with object character- ization from analysis of its eigenfrequency spectrum. In recent years, this technique has gained wide acceptance in crystallography and other fields of science and technology as a promising tool for the acquisition and pro- cessing of data on the internal structure of an object. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION This may explain the wide use of the BernoulliÐ Euler equation of a beam on a Winkler foundation in Any extended constructions represent objects with a studying the interaction of a railway track with rolling complex internal structure. Such objects integrate, nat- stock [5] or the wave equation in the physics of crystal urally or artificially, members of highly different den- lattices [6]. sity, modulus of rigidity, viscosity, etc. that deform However, the selection of a load-carrying contin- simultaneously. One basic member of such objects is a uum model imposes stringent restrictions on model- so-called elastic inertial load-carrying continuum. refining lumped members. Specifically, if the contin- Other members are mounted on, or placed into, it. A uum is a priori assumed to have string properties, railÐcross tie array is an example of an artificial struc- moment-type interactions of the continuum with any ture; crystals are natural objects of this kind. surrounding inertial and inertialess members must be immediately excluded from consideration. This The analysis efficiency in solving applied or exper- strongly restricts the elaboration of the mathematical imental problems is closely related to the justified model. It may so happen that this model will be impos- selection of a real object (physical) model. Then, the sible to improve by adding any finite number of lumped physical model is supplemented by an adequate mathe- members. matical model of the object. A model for the load-car- It is known that eigenfrequency spectra bear much rying continuum is of greatest importance. This model information on a real object’s properties, particularly, should be based on previous service experience or its internal structure. The agreement between the real experiments with the given object. It may happen that, object spectrum and its mathematical model is a neces- in experiment, various models of the inertial continuum sary condition for the validity of any physical theory. predict similar qualitative and quantitative (accurate to We, however, consider the dynamics of an object with an experimental error) results. In this case, designers an unknown, but a priori, complex internal structure first consider the traditional inertial continuum model for which only its eigenfrequency spectra are known (which is the simplest in terms of mathematical from experiments. In this case, the construction of a description). Then, this model is replenished by iner- mathematical model with a spectrally similar operator tial, elastic, and other lumped-parameter members nec- leaves many “dark spots.” It remains unclear how to essary for the rigorous description of subsequent exper- integrate its elementary inertial links into macrostruc- iments. tures.

964 INDEŒTSEV et al. In this work, the behavior of the eigenfrequency cides with that of the cross section at the point of fixing. spectra of complex objects was derived by rigorously Cases 1 and 2 refer to a so-called momentless contact solving formally stated problems of mathematical between the elastic line and inclusions. Case 3 means a physics. The discovered effects provide a greater possi- moment-type contact interaction. Note that a moment- bility to experimentally test the adequacy of available type lineÐinclusion contact is possible for a beam but mathematical models of real objects. The discussed impossible for a string. Various limiting processes dealt features are expected to show up in experiments with with in the above problems allow the determination of any systems whose dynamics is simulated by the above trap frequencies for a wide variety of edge conditions. equations. This gives a chance to trace a correlation between the Consider the problem of free vibrations of an infi- appearance (or disappearance) of some trap spectrum nite elastic inertial line on a Winkler foundation. It is and many physical factors, such as the elastic proper- known that this problem involves a continuous eigen- ties of the loaded line; type of degrees of freedom of an frequency spectrum. This spectrum corresponds to individual, purely inertial inclusion; type of lineÐinclu- eigenmodes in the form of propagating undamped sion contact; and inclusion size. waves. Such a spectrum is called continuous and lies For steady-state vibrations of two-element point above the boundary frequency. A Winkler classical inclusions of masses m1 and m2, the amplitudes of their foundation implies that a system has a single boundary transverse (relative to the equilibrium position) dis- frequency when interacting with infinite (or semi-infi- placements will be denoted as W1 and W2. For two rigid nite) one-dimensional elastic inertial continua, such Ψ Ψ that their behavior is described by the string equation, bodies, rotations 1 and 2 are added. The position of BernoulliÐEuler beam equation, or Timoshenko beam a single-element inclusion (rigid body) is specified by ω the transverse displacement of its center of mass W and equation. The boundary frequency b depends on the Ψ linear mass density of an elastic line ρ and modulus of a rotation . The rest of the parameters are designated as follows: s, longitudinal Lagrange coordinate of an ω ρ rigidity of the Winkler foundation k; that is, b = k/ . elastic-line cross section; ()' = ∂/∂s; w = w(s), trans- It was shown [1, 2] that, if lumped elastic inertial inclu- verse displacement amplitude of a cross section with a sions are embedded in these infinite or semi-infinite coordinate s; c2 = T/ρ, velocity of a transverse distur- systems, the latter, along with the continuous spectrum, bance in a string (T is the string tension); β4 = C/ρ, elas- may exhibit line eigenfrequency spectra. These spectra tic parameter of a BernoulliÐEuler beam (C is its flex- correspond to localized, or trap, vibrations. It was ural rigidity); and N1 and N2 are force amplitudes acting found, in particular, that, for strings, the line spectra of on the elastic line at points of contact with inclusions. studied inclusions always lie below the boundary frequency. In the case of beams (for certain combinations of their elastic and inertial properties and those of the INTERACTION OF STRING AND BEAM inclusions), these spectra may appear above the bound- WAVEGUIDES WITH TWO POINT MASSES ary frequency. Some of the previous conclusions, being valid as applied to specific situations, however need, Consider how trap spectra depend on inclusion correction upon generalizing obtained results. This parameters when a purely inertial inclusion is in refers to the sufficient condition for the existence of line momentless contact with an elastic inertial line. For fre- ω spectra and the effect of edge conditions on their behav- quencies below the boundary frequency b, the steady- ior when inclusion parameters vary according to the state vibration amplitude of an elastic line (string) or intrinsic properties of the elastic inertial line. BernoulliÐEuler beam that lies on a Winkler foundation In this work, we find trap vibration spectra in elastic and has a two-point moment-type contact with an inclu- systems of infinite length like a string or BernoulliÐ sion is given by Euler beam lying on a Winkler foundation. The systems have purely inertial inclusions that lack intrinsic vibra- N N λ2 ------1--δ( ) ------2--δ( ) tory dynamics. In this case, the line spectrum frequen- w'' Ð w = Ð 2 s Ð 2 s Ð L , ω ρc ρc cies lie below the boundary frequency b. As inclusions, the following objects were considered: (1) two ω 2 ω2 λ2 b Ð > material points of masses m1 and m2 spaced at an inter- = ------0, string, c2 val L, (2) a perfectly rigid body of mass m and moment (1) of inertia J (with respect to the center-of-mass position) IV λ4 N1 δ( ) N2 δ( ) that is momentlessly fixed on an elastic line at two w + 4 w = ------4 s + ------4 s Ð L , points spaced at an interval L, and (3) two perfectly ρβ ρβ rigid bodies of masses m1 and m2 and moments of iner- ω 2 ω2 4 b Ð tia J1 and J2 (with respect to the center-of-mass posi- λ = ------> 0, beam. tions) that are fixed on an elastic line in such a way that 4β4 the center-of-mass displacement coincides with that of the point of fixing and the center-of-mass rotation coin- The vibration amplitudes of an inclusion with two

TECHNICAL PHYSICS Vol. 45 No. 8 2000

RESONANCE VIBRATIONS OF ELASTIC WAVEGUIDES 965 masses m and m satisfy conditions L 1 2 (a) (c) 2 2 m2 m ω W = N , m ω W = N , m m 1 1 1 2 2 2 (2) m1 1 2 ( ) ( ) w 0 = W1, w L = W2. For the string, the solution of (1) bounded at infinity has the form (b) (d) λs Ðλ(L Ð s) m2 N e N e m ------1------+ -----2------, Ð∞ < s ≤ 0; 2 λρ 2 λρ 2 2 c 2 c m1 m1 λ λ( ) N eÐ s N eÐ L Ð s w(s) = -----1------+ -----2------, 0 ≤ s ≤ L; (3) λρ 2 λρ 2 2 c 2 c Fig. 1. Symmetric and asymmetric localized vibrations in Ðλs λ(L Ð s) (a, b) string and (c, d) beam waveguides with two-point N e N e inclusions. Shape “b” disappears when the interinclusion -----1------+ -----2------, L ≤ s < ∞. 2 2 distance is less than L∗. 2λρc 2λρc A similar solution for the BernoulliÐEuler beam is for the string and λs Ð2λ L N1e ω 2 ( I, II ( λ )) ------(cosλs Ð sinλs) m1m2 I, II 1 Ð e 1 + sin2 I, II L 3 4 ------8λ ρβ λ 3 ρβ4( ) 4 I, II m1 + m2 λ(s Ð L) (6) N2e λ ( λ( ) λ( )) Ð2 I, II L + ------cos s Ð L Ð sin s Ð L , 4m m (1 Ð e (1 + sin2λ , L)) 8λ3ρβ4 = 1 +− 1 Ð ------1------2------I----II------(m + m )2 Ð∞ < s ≤ 0; 1 2 Ðλs for the beam. N e ------1------(sinλs + cosλs) With the minus sign in (5) and (6), we find the λ3ρβ4 8 ω 2 eigenfrequency I of conventionally symmetric ( ) λ(s Ð L) w s = N e (4) waveguide vibrations (Figs. 1a, 1c); the plus sign gives + -----2------(cosλ(s Ð L) Ð sinλ(s Ð L)), 3 4 ω 2 8λ ρβ the eigenfrequency II of conventionally asymmetric 0 ≤ s ≤ L; waveguide vibrations (Figs. 1b, 1d) (if m1 = m2, vibrations become symmetric or asymmetric in the strict λ N eÐ s sense). ------1------(sinλs + cosλs) 8λ3ρβ4 Consider limiting cases for (5) and (6). If L 0, we have λ( ) N eÐ s Ð L 2 ( λ( ) λ( )) ( )ω 2 ω 2 + ------sin s Ð L + cos s Ð L , m1 + m2 I m1m2 L II 8λ3ρβ4 ------= 1, ------= 1 2 2 ρ 2( ) 2ρc ω Ð ω c m1 + m2 L ≤ s < ∞. b I In both cases, the vibration amplitude exponentially for the string and drops with distance from the lineÐinclusion contact 2 (m + m )ω region; hence, waveguide vibrations are localized near ------1------2------I------= 1, the inclusions. ρβ(ω 2 ω 2)3/4 2 2 b Ð I Matching the string or beam equation [equations (7) ω 2 (1)] with the inclusion equation [expression (2)] yields m1m2 L II ------= 1 two expressions for the spectral parameter ω2: ρβ4( )(ω 2 ω 2 )1/2 2 m1 + m2 b Ð II 2 m m ω , Ð2λ , L ------1------2------I---II------(1 Ð e I II ) for the beam. λ ρ 2( ) I, II c m1 + m2 If one of the masses (e.g., m1) in (5) is taken infi- (5) nitely large, one comes to the frequency equation of λ ( Ð2 I, II L) vibration for an infinite string that has a point inertial − 4m1m2 1 Ð e = 1 + 1 Ð ------. inclusion of mass m placed at a distance L from a ( )2 2 m1 + m2 hinged waveguide point with the coordinate s = 0. It is

TECHNICAL PHYSICS Vol. 45 No. 8 2000 966 INDEŒTSEV et al.

(a) L* (b) string with a single hinged point and one-mass inclusion. ∞ m2 ∞ m2 m1 = m1 = As follows from (7) and (9) for a string waveguide, there should exist a distance between inertial inclusions such that the realness and positiveness conditions for the square of the eigenfrequencies are violated. In this case, the associated eigenfrequency spectrum becomes Fig. 2. Localized vibrations of a point inertial inclusion in a complex, the existence criteria for localized undamped (a) semi-infinite string waveguide and (b) beam waveguide with hinged support. Shape “a” disappears when the dis- vibrations responsible for the given eigenfrequency are tance between the inclusion and waveguide boundary is less violated, and this localized vibration mode disappears than L*. (Figs. 1b, 2a). However, more thorough analysis is needed to gen- easy to check that results thus obtained completely eralize the conditions necessary for such a phenomenon ω coincide with those for a semi-infinite string with a to occur. The drift of II beyond the boundary fre- point inertial inclusion of mass m2 placed at a distance L quency with decreasing distance between the masses or from the rigidly fixed beginning of the waveguide s = 0. to the fixed support (Fig. 2b) is not predicted from the If m1 tends to infinity in the beam waveguide problem above results for a BernoulliÐEuler beam with similar [that is, in (6)], we arrive at the frequency equation of inclusions. vibration for an infinite beam with a point inertial inclusion of mass m2 placed at a distance L from a hinged waveguide point with the coordinate s = 0 (but not for INTERACTIONS OF STRING AND BEAM a semi-infinite beam!). Thus, we have WAVEGUIDES WITH A RIGID BODY UNDER MOMENTLESS TWO-POINT CONTACT 2 2 m ω Ð2λ L ω = 0, ------2------II--(1 Ð e II ) = 1 Consider elastic lines momentlessly interacting I λ ρ 2 II c with a single rigid body. Recall that its location is specified by the center-of-mass position and a rotation about for the string and a certain axis. For a rigid inclusion in momentless sym- 2 metric contact with an elastic line, conditions (2) are m ω Ð2λ L ω 2 = 0, ------2------II----(1 Ð e II (1 + sin2λ L)) = 1 (8) replaced by I λ 3 ρβ4 II 8 II 2 2 L L mω W = N + N , Jω Ψ = Ð ---N + ---N , for the beam. 0 1 2 0 2 1 2 2 (11) At L 0 in (8), we obtain the limiting equations Ψ Ψ L 0 L 0 2 ( ) ( ) m ω L w 0 = W0 Ð ------, w L = W0 + ------. ω 2 = 0, -----2------II----- = 1 2 2 I ρ 2 c From the latter, we come to the equations for fre- for the string and quencies of symmetric (subscript I) and asymmetric (subscript II) elastic-line vibrations: m ω 2 L ω2 = 0, ------2------II------= 1 (9) I 1/2 Ðλ L 2 Ðλ L 2β2(ω 2 Ð ω 2 ) mω2(1 + e I ) Jω (1 Ð e II ) b II ------I------= 1, ------II------= 1 λ ρ 2 λ ρ 2 2 for the beam. 4 I c II c L ω The condition I = 0 means that the system moves as a rigid unit. Since the hinged point with the coordi- for the string and nate s = 0 makes the infinitely large mass m immobile, 1 2 Ðλ L the zero frequency correlates with the system at rest in mω (1 + e I (cosλ L + sinλ L)) ------I------I------I------= 1, this case. 3 4 16λ ρβ If the distance L between the masses is less than I λ (12) 2 Ð II L 2 ω ( ( λ λ )) ρ ( ) 2 J II 1 Ð e sin II L + cos II L c m1 + m2 ρc ------= 1 ≈ ------≈ ------3 4 2 L 2 , L = L m = ∞ 2, (10) λ ρβ * ω * 1 ω 4 II L m1m2 b m2 b equations (7) and (9) for the string give real values of for the beam. the trap frequency ω that no longer satisfy the condi- II Localized vibrations of a string with a fixed rigid tion ω < ω . The former equation in (10) gives L for II b * body are represented in Fig. 3a. Similar vibrations of a a string with a two-mass inclusion; and the latter, for a beam are shown in Figs. 3c and 3d.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 RESONANCE VIBRATIONS OF ELASTIC WAVEGUIDES 967

If m ∞, (12) can be recast as (a) (c) m, J m, J Ðλ L Jω 2 (1 Ð e II ) ω 2 = 0, ------II------= 1 I λ ρ 2 2 II c L for the string and

ω2 = 0, (b) (d) I m, J m, J Ðλ L Jω 2 (1 Ð e II (sinλ L + cosλ L)) (13) ------II------II------II------= 1 λ 3 ρβ4 2 4 II L for the beam. Relationships (13) yield the frequency equations for a rigid solid that has a fixed center of mass Fig. 3. Symmetric and asymmetric localized vibrations in and is in two-point contact with an elastic line. For (a, b) string and (c, d) beam waveguides in momentless contact with a rigid body. Shape “b” is disappearing. J ∞, we obtain λ 2 Ð I L mω (1 + e ) 2 quency of localized vibrations of a string with two ------I------= 1, ω = 0 λ ρ 2 II purely mass inclusions disappears: 4 I c 2 2 2 2 ρc L Jω for the string and ω > ω ≈ ------, ⇒ L ≈ ------b-. (16) b II J ** ρ 2 Ðλ L c mω 2(1 + e I (cosλ L + sinλ L)) ------I------I------I------= 1, As was demonstrated, the effect of the disappear- λ 3ρβ4 16 I (14) ance or conservation of the higher trap spectrum immediately depends on the type of elastic inertial contin- ω2 II = 0 uum. Also, this effect is not a mere consequence of a rising number of degrees of freedom of a single inclu- for the beam. These relationships are frequency equa- sion. Nor is it directly associated with types of elastic tions for vibration of a rigid body that momentlessly lineÐinclusion contact. interacts with an elastic line at two points without rotation. To confirm the last statement, we will present expressions for trap spectra of free vibrations in the If m and J in (12) jointly tend to infinity, we obtain case of a purely inertial inclusion that has two intrinsic ω 2 ω 2 I = 0 and II = 0 for both beam and string. This is a degrees of freedom and is in moment-type contact with well-known result, indicating that string and beam sys- a BernoulliÐEuler beam. These are line spectra of free tems with two fixed hinged supports do not have trans- vibrations of a system consisting of a rigid body (bod- verse vibration frequencies lying below the boundary ies) fixed at its center of mass and lying on a BernoulliÐ frequency of line spectra. Euler beam supported by a Winkler foundation: If L 0 in (12), we obtain β3mω 2 βJω2 ------I------= 1, ------II------= 1. (17) ω 2 ω2 ρ(ω 2 ω 2)3/4 ρ(ω 2 ω 2 )1/4 m I J II 2 b Ð I 2 2 b Ð II ------= 1, ------2---- = 1 ρ ω 2 ω 2 ρc L 2 c b Ð I Free vibrations at these frequencies are shown in Figs. 4aÐ4c. for the string and Putting m = ∞ in (17), we arrive at the frequency mω 2 equation for vibrations of an infinite beam hinged at a ------I------= 1, point with the coordinate s = 0. This point coincides ρβ(ω 2 ω 2)3/4 2 2 b Ð I with the center of mass of a perfectly rigid body inter- (15) acting with a waveguide and having a moment of iner- Jω2 tia J: ------II------= 1 2ρβ2 L ω 2 Ð ω 2 ρ(ω 2 ω 2 )1/4 b II ω 2 ω 2 2 2 b Ð II I = 0, II = ------. (18) for the beam. Jβ Thus, in this case, the beam does not have, but the Putting J = ∞ in (17), we arrive at the frequency string has, the limiting (maximum) distance between equation for vibrations of an infinite beam supported at contact points. As this distance grows, the higher fre- a point with the coordinate s = 0. This support is a slid-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 968 INDEŒTSEV et al. The solution of problem (20)–(21) bounded at infin- (a) , (b) , m J m J ity and localized near the contacts with the inclusions is given by λ e s ------(A cosλs + D sinλs), λ3β4 1 1 (c) m, J m, J (d) 4 m, J Ð∞ < s ≤ 0; m, J λ eÐ s ------(Pcosλs + Qsinλs) 4λ3β4 ( ) λ(s Ð L) (22) w s = e Fig. 4. Symmetric and asymmetric localized vibrations in a + ------(H cosλ(s Ð L) + K sinλ(s Ð L)), beam waveguide in point moment-type contact with (a, b) λ3β4 one- and (c, d) two-element rigid inclusions. 4 0 ≤ s ≤ L; ing attachment of mass m: Ðλ(s Ð L) e ( λ( ) λ( )) ------3------4-- B2 cos s Ð L + S2 sin s Ð L , ρ(ω 2 ω 2)3/4 4λ β ω 2 2 2 b Ð I ω2 I = ------, II = 0. (19) ≤ < ∞ mβ3 L s , where A , D , P, Q, H, K, B , and S are constants. It is easy to notice that the localized vibration mode 1 1 2 2 disappears in none of the situations considered, as The joining conditions give the set of equations opposed to the previous cases for the string. m ω2W m ω2W -----1------1 = P, -----2------2 = H, 2ρβ4 2ρβ4 INTERACTION OF A BEAM WAVEGUIDE (23) WITH TWO SOLIDS BEING IN MOMENT POINT λJ ω2Ψ λJ ω2Ψ 1 1 = + ------2------2 = H + K. CONTACT WITH AN ELASTIC INERTIAL ------4------Ð P Q, 4 CONTINUUM ρβ ρβ To elucidate the effect of interest in an infinite beam Let both rigid bodies (two-element inclusion) be system, we will determine steady-state vibration fre- identical. Then, the frequencies of symmetric and quencies of a beam lying on a Winkler foundation and asymmetric vibrations are found independently. Since being in point moment contact with two rigid inclu- the localized vibrations may disappear with decreasing sions. We assume the frequencies to be below the interelement distance, the behavior of these spectra at ω boundary frequency b. Divide the infinite region occu- L 0 seems to be the most interesting. pied by the elastic line into three sections: those on the We will find the symmetric vibration frequencies by Ψ Ψ Ψ left and on the right of both inclusions and the region putting W1 = W2 = Ws and 1 = Ð 2 = s. Then, at between them. For each of the sections, we write the L 0, one obtains from set (23) elastic line equations and edge conditions:  mω2  2 2 λL ------Ð 1 λW + Ψ = 0, ω Ð ω  3 4  s s wIV + 4λ4w = 0, λ4 = -----b------> 0, 4λ ρβ 4β4 (24) ω2 (20) λ λ LJ  Ψ w(0) = W , w(L) = W , L Ws + ------Ð 1 s = 0. 1 2 2ρβ4 w'(0) = Ψ , w'(L) = Ψ , 1 2 For the free symmetric vibrations of a beam as well as the joining conditions for the partial (section) waveguide with two rigid inclusions, the frequencies solutions: obtained from the existence condition for the nontrivial solution of (24) are m ω2W = ρβ4w'''(0) Ð ρβ4w'''(0), 1 1 + Ð 4 λ3 ρβ4 2 2 2ρβ 4 sII ω = 0, ω = ------+ ------. (25) ω2Ψ ρβ4 ( ) ρβ4 ( ) sI sII J1 1 = Ð w+'' 0 + wÐ'' 0 , JL m (21) The symmetric vibrations are depicted in Fig. 4c. ω2 ρβ4 '''( ) ρβ4 '''( ) m2 W2 = w+ L Ð wÐ L , The presence of zero frequency among the roots of the characteristic equation merits attention. As is ω2Ψ ρβ4 ( ) ρβ4 ( ) J2 2 = Ð w+'' L + wÐ'' L . known, the zero frequency implies that a system moves

TECHNICAL PHYSICS Vol. 45 No. 8 2000 RESONANCE VIBRATIONS OF ELASTIC WAVEGUIDES 969 as a rigid unit. However, in our infinite system, unlim- results in the same effect: the upper frequency of the ited displacements of the inertial inclusions are impos- line spectrum disappears on exceeding the boundary sible because of the Winkler foundation. This apparent frequency: conflict is due to the fact that the frequencies were calculated using the limiting equations obtained from (23) ρβ2 ω 2 Ð ω 2 ρβ4 ω2 = ------b------I-, ω2 = ------. (29) at L 0. The frequency equation coincident with the I m L II J L first equation in (17) yields a more accurate value of 2 b 2 b ω 2 . For the refined frequency, system vibrations are With (23), one can obtain frequency spectra of free sI localized vibrations for variously posed two-body shown in Fig. 4a. problems: the centers of mass of both bodies are fixed, For asymmetric vibrations, the frequencies are neither one can rotate, the center of mass of one body is Ψ Ψ Ψ found by putting W1 = ÐW2 = Wa and 1 = 2 = a. At fixed and the other cannot rotate, etc. In each of these L 0, we obtain from (23) problems, the localized vibration modes of the beam waveguide disappear. In fact, for this to take place, it is 2 ω2 L m  λ λ Ψ sufficient that at least one of the inertial elements of a ------Ð 1 2 Wa Ð L a = 0, 8λρβ4 two-element inertial inclusion interacting with a Ber- (26) noulliÐEuler beam have the inertia of a rigid solid and ω2 λ LJ  λ Ψ that the beamÐinclusion interaction be of moment-type Ð 2 Wa + ------Ð 1 L a = 0. 2ρβ4 character. The frequencies of free asymmetric vibrations of the beam waveguide with two rigid inclusions are as fol- DISCUSSION lows: The string and beam waveguides considered in this work are partial mathematical models. They, however, ρβ4 λ ρβ4 ω 2 ω 2 2 8 aII to some degree of approximation account for the possi- aI = 0, aII = ------+ ------. (27) JL mL2 ble behavior of real objects under service or in experiments. The disappearance of one line spectrum when The corresponding asymmetric vibrations are given the parameters of an inertial inclusion are varied seems in Fig. 4d. Like the symmetric vibration spectrum, they to be rather intriguing in this respect. Since the eigen- possess the zero eigenfrequency. It is associated with frequencies of a linear system become resonant in the the rotation of both inertial inclusions with equal con- presence of an external harmonic action on the system, stant velocities. The reason for its appearance is the this effect can be used for thoroughly examining the same as in the previous case. The exact value of ω2 is internal structure of a continuum with massive inertial aI inclusions. The very discovery of this effect is an indi- derived from the frequency equation coincident with cation that an object is of an essentially discreteÐcon- the second expression in (17). The shape of the vibra- tinuous nature. However, distributed members, lumped tions for this case is shown in Fig. 4b. inclusions, and interactions between them need to be The appearance of both zero frequencies is very identified. If information on any of these components is important from the physical viewpoint. This means available, the effect of the appearance or disappearance that, as the interinclusion distance diminishes, the of the trap mode can help to clarify the properties of the approximate frequency equations become “insensitive” others. As follows from the aforesaid, a new trap reso- to the presence of the Winkler foundation. Under such nance frequency appears or disappears if variations of conditions, these equations predict that the entire sys- the geometric and inertial inclusion parameters are spe- tem will behave as a rigid unit. cially matched to the properties of all the components For high-frequency components of the trap spectra, of complex systems (comprising elastic inertial contin- approximate frequency equations (25) and (27) give the uum, noninertial elastic continuum, and purely inertial same interinclusion distance, namely, inclusions). The new mode is lost when the disturbance energy is transferred from the near-inclusion region to 2ρβ4 L ≈ ------, (28) the elastic inertial continuum and disappears when the * ω 2 energy is confined in the vicinity of the inclusion [1Ð3]. J b With very simple mechanical models, it was demon- at which these spectra disappear when exceeding the strated that the spectra of two closely related systems boundary frequency. may qualitatively behave in a fundamentally different Now we proceed with the limiting case when a rigid way when their parameters are varied similarly. It is body interacts with a semi-infinite beam fixed at its noteworthy that the qualitative distinction due to radi- beginning. We put the mass and moment of inertia of cally differing internal properties of objects is observed the body fixed at the point s = 0 tending to infinity. only in a rather narrow frequency range. This is of spe- Ψ Then, W1 = 0 and 1 = 0. Under such conditions, the cial significance for systems where the variations occur frequency equation derived from (23) at L ∞ in a natural manner; i.e., the parameters vary during

TECHNICAL PHYSICS Vol. 45 No. 8 2000 970 INDEŒTSEV et al. operation. Speciﬁcally, the case in point is transport ples are the upper part of a railway track or crystals, problems. In fact, such a situation may arise when a where a model problem is hard to solve analytically. train approaches some railway irregularity (bridge, via- duct, or switch). In these cases, the joint railwayÐtrack dynamics inevitably necessitates the solution to prob- REFERENCES lems like those considered in this work. Obviously, 1. V. A. Babeshko, B. V. Glushkov, and N. F. Vinchenko, some of them are still more complicated because of the Dynamics of Inhomogeneous Linearly Elastic Media need for taking into account the continuously varying (Nauka, Moscow, 1989). interinclusion distance. For this reason, they cannot be 2. A. K. Abramyan, V. L. Andreev, and D. A. Indeœtsev, treated rigorously. Here, the role of tests, which allow Model. Mekh. 6, 34 (1992). the qualitative prediction and correct interpretation of 3. A. K. Abramyan, V. V. Alekseev, and D. A. Indeœtsev, Zh. results, increases. Tekh. Fiz. 68 (3), 15 (1998) [Tech. Phys. 43, 278 Analytical methods can be applied to one-dimen- (1998)]. sional elastic inertial (string and beam) and noninertial 4. G. G. Denisov, E. K. Kugusheva, and V. V. Novikov, (Winkler foundation) continua with inclusions having Prikl. Mat. Mekh. 49, 691 (1985). no more than two elements. Only in these cases can we 5. G. M. Shakhunyants, Railway Track (Transport, Mos- ﬁnd analytical solutions to a number of like problems cow, 1987). for each of the elastic lines and compare them. This is, 6. L. Brillouin and M. Parodi, Wave Propagation in Peri- however, a rarely encountered exception. In our opin- odic Structures (Dover, New York, 1953; Inostrannaya ion, the value of our results is that they provide a better Literatura, Moscow, 1959). insight into similar phenomena in two- and three- dimensional continua with a complex structure. Exam- Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 45 No. 8 2000

THEORETICAL AND MATHEMATICAL PHYSICS

A High-Gradient Accelerator Based on a Faster-than-Light Radiation Source Yu. N. Lazarev and P. V. Petrov All-Russia Research Institute of Technical Physics, Russian Federal Nuclear Center Snezhinsk, Chelyabinsk oblast, 456770 Russia Received March 23, 1999

Abstract—A wide-band microwave generator using a faster-than-light source is proposed to be used as a charged particle accelerator. According to theoretical estimates, an electric ﬁeld amplitude as high as ~1011 V/m or more can be attained at the focus of a paraboloidal emitting surface with a focal parameter of ~1 m. These estimates are supported by numerical calculations. The schematic diagram of such an accelerator is suggested. © 2000 MAIK “Nauka/Interperiodica”.

ε INTRODUCTION where f is the energy of electrons emitted from the anode of the microwave source placed on the parabo- Wide-band microwave generators based on a faster- loid surface. than-light (FTL) emission source are very promising for applications [1]. In particular, emission directivity in Thus, these generators is provided by the shape of the emitting ε r E ∼ -----f- ----0 . (4) surface, and the direction of emission coincides with that e⑄ ⑄ of the specular reflection of emission-inducing radiation. Thus, the focusing of FTL emission presents no prob- If a particle of charge e and mass M is accelerated lems. This raises the question of whether the electric by this field, it gains a momentum p, field intensity at the focus can be increased to the point ⑄ ε r where it can be used for accelerating charged particles. peE∼∼------f ----0 , (5) c c ⑄ Consider a plane laser beam propagating along the and energy ε , axis of a paraboloid. Let the paraboloid consist of radi- a ating elements similar to those described in [1]. Let r p also the inner surface of the paraboloid be exposed to ε ----0 , ------> 1, f ⑄ Mc the laser beam (Fig. 1). ε ∼ a 2 (6) ε r 2 p Electromagnetic radiation generated by such an ------f - ----0 , ------< 1. 2 ⑄ FTL emission source must be focused at the paraboloid Mc Mc focus and must greatly increase the field intensity. The ε 2 3 ⑄ electromagnetic energy fluxes near the paraboloid sur- The values of f ~ 10 Ð10 keV, ~ 0.1Ð1 cm, and face and at the focus should coincide: 2 2 ∼ 2⑄2 E0 r0 E . (1) 1 2 3 Hence, the electric field at the focus can be estimated as

∼ r0 EE0---⑄- , (2) where E is the electric field at the paraboloid focus, E0 4 is the electric field near the paraboloid surface, r is the 6 0 5 paraboloid radius in the focal plane, and 2π⑄ is the 7 electromagnetic radiation wavelength. The field amplitude near the paraboloid surface can readily be determined using the results obtained in [1]: Fig. 1. Schematic diagram of an FTL paraboloid source: (1) ε photons, (2) emitted electrons, (3) photoemitter, (4) laser ∼ f E 0 ------, (3) radiation front, (5) paraboloid surface, (6) accelerating e ⑄ charge; and (7) electromagnetic wave.

972 LAZAREV, PETROV

⑄ 2 3 r0/ ~ 10 Ð10 seem to be attainable. Thus, it is hoped mally to the paraboloid surface, i.e., that the energy of the accelerated particle may run to about 1 GeV with a comparatively small-size accelera-  z  j(t, z, ρ) = j⊥ t + --, z, ρ n, tor (several meters). Since the acceleration length is  c  about ⑄, the accelerating gradient can be as high as ~10 GeV/m or even more. Rigorous treatment of the 1  ρ  (9) problem using analytical and numerical methods sup- n = ------Ð----; 1, ρ 2 r0 ports this result.     1 + ---- r0

ANALYTICAL CALCULATIONS we obtain the expression for the ﬁeld component along the paraboloid axis Electron acceleration. The analytical expression for 1 dv' α ∂ j⊥ the intensity of an electric ﬁeld generated by an FTL E = Ð------sin(α)cos ------, (10) z 2∫   ∂ pulse on the inner surface of the paraboloid, c r 2 t ρ2 r where α is the angle between the axis OZ and vector r' z (ρ) = ------Ð ----0, (Fig. 2). p 2r 2 0 In our case, the dipole approximation can be can be obtained with the retarded potentials in the applied: Lorentz gauge: 2π ω ϕ k ω d ω1 + τ z ω Ez = E0 ∫ ------∫ d ------f  + -- ρ r Ð r'  2π 1 Ð ω ⑄ t Ð ------ 0 ω c 0 ϕ( ) ν r = ∫d '------ρ r Ð r' ω2 (ϕ) (11) (7) + ⑄-- 1 Ð cos  ,  r Ð r'  jt Ð ------ π c 2 r0P0 A(r) = dν'------. E0 = ------. ∫ r Ð r' ⑄2 ω Considering that Here, = z'/z' + r0, z' is the coordinate of the vector r', ρ ρ (z0, 0) and (zk, k) are the coordinates of the extreme 1∂A points of the paraboloid, ω = z /z + r , ω = z /z + r , E = Ð ∇ϕ Ð ------(8) 0 0 0 0 k k k 0 c ∂t z and ρ are the cylindrical coordinates of the vector r, P0 is the amplitude of the dipole moment surface den- and assuming that the emission current is directed nor- sity, ( ) 1 zk Ð z' τ = ----- Ð ------, X T p cT p

j Tp is the characteristic time of variation of the dipole ⑄ moment, = cTp, r0 is the paraboloid radius in the focal r' plane, r – r' d 2 r0 f (x) = η(x)-----F(x), F(x) = 2.17x exp(Ðx) (12) z0 O α z zk dx 1 r z' Z is a dimensionless function speciﬁed for a pumping source that provides a linear growth of the radiation intensity, and η(x) is the Heaviside function. At the paraboloid axis, ω 2 k 1 + ω  z  E = E dω------f τ + -- ω . (13) z 0 ∫ 1 Ð ω  ⑄  ω 0 ω Fig. 2. Geometry used in computations: (1) focus and (2) The parameter varies from Ð1 to +1 over the paraboloid surface. paraboloid surface. It follows from the expression for

TECHNICAL PHYSICS Vol. 45 No. 8 2000 A HIGH-GRADIENT ACCELERATOR 973 ω Ez that the phase velocity of the field is c/ > c, because It follows that ωdz dτ dz c eE (1 + ω) ------+ ----- = 0 ⇒ ----- = ---. (14) p (τ) = p + ------0⑄------(ω Ð ω ) ⑄ dt dt ω z i 2 k 0 dt c (1 Ð ω) (20) Obviously, the closer the velocity of particles to the × F((1 Ð ω)(τ Ð τ )), phase velocity of the field, the higher the acceleration i |ω| ≈ ≡ (τ ) efficiency. Therefore, truncated paraboloids with 1 pi pz i . are of particular interest. Paraboloids with ω ~ 1 are preferable because of the factor (1 + ω)/(1 Ð ω) in the Expression (16) for the field component Ez was integrand for Ez. The physical meaning of this factor is obtained under the assumption that clear: the dipole radiation field is maximum in the plane z (ω ω ) < perpendicular to the dipole axis, whereas the dipole -- k Ð 0 1. (21) does not emit along its axis. When ω Ð1, the angle ⑄ between the dipole axis and direction toward the focus tends to zero; when ω 1, this angle tends to π/2. Hence, within this approximation, Only paraboloids with ω , ω ≈ 1 are therefore consid- (τ τ )(ω ω ) < k 0 Ð i k Ð 0 1. (22) ered in further discussions. For the points with coordinates (z, ρ = 0) lying on This inequality defines the time interval where the the paraboloid axis in the vicinity of the focus, above approximation is valid. The function F(τ) has a broad maximum at τ = 2. Therefore, the energy of the z τ (ω ω ) < accelerated electron increases until max: ⑄-- k Ð 0 1, (15) 2 the expression for E at the paraboloid axis can further τ Ð τ ≈ ------. (23) z max i 1 Ð ω be simplified to According to (22), the following inequality should  z  1 + ω E ≅ E f τ + -- ω ------(ω Ð ω ). (16) be met: z 0  ⑄  1 Ð ω k 0 2 (ω ω ) < ω ω ------k Ð 0 1. (24) If k is far from unity ( k < 0.95), the following 1 Ð ω equation is valid: τ Otherwise, max should be determined from (22). 1 + ω 1 + ω ω + ω Thus, it can be assumed that F = F = 1, and the ------≈ ------, ω = -----k------0 . (17) max 1 Ð ω 1 Ð ω 2 momentum of the accelerated relativistic electron can attain the value p equal to ω max Actually, the values of k close to unity are of no practical significance, because the required power of eE (1 + ω) p = p + ------0⑄------(ω Ð ω ). (25) the light source is inversely proportional to the sine of max i 2 k 0 c (1 Ð ω) the glancing angle. When ω 1, the sine decreases, whereas the power increases as 1/ 1 Ð ω . In addition, The quantity a significant difference between ω and ω (very long k 0 ω z Λ Λ paraboloid) results in nonuniform illumination and 1 + (ω ω )  0,  ------2 k Ð 0 = 2K------ ----, makes the wavelength ⑄ dependent on the coordinates (1 Ð ω) r0 r0 r0 (26) of the emission point. Λ = z Ð z Consider the process of electron acceleration. For a k 0 relativistic electron (the electron velocity is close to the is the effective length of the paraboloid, which varies velocity of light, v ≈ c), the equation of motion is writ- over a wide range. The value of K attains its maximum ≈ Λ ≈ ten as (K 1) at 0.5r0. For paraboloids of moderate length Λ z = z + c(t Ð t ) = z + ⑄(τ Ð τ ). (18) ~ (1Ð5)r0, which are of practical significance, we e i i i i have Using this equation, we obtain the formula for the Λ momentum of the accelerated electron: z0,  > K------ 0.5. (27) r0 r0 d p 1 + ω ------z = eE ------(ω Ð ω ) f ((1 Ð ω)(τ Ð τ )), dt 01 Ð ω k 0 i Assuming K = 0.5, we obtain (19) z 2πeΛP τ = Ð---k ω. p ≈ p + ------0 . (28) i ⑄ max i c⑄

TECHNICAL PHYSICS Vol. 45 No. 8 2000 974 LAZAREV, PETROV ε Substituting the current coordinate of the proton Since P = ------f--- , 0 2πe into the expression for the field phase, we obtain τ znω τ ziω ωv (τ τ ) p p λ ε + ---- = + --- + --- Ð i ----max----- ≈ ------i-- + ------f----. (29) ⑄ ⑄ c ⑄ 2 mec mec m c (34) e  v  z  = 1 Ð ---ω (τ Ð τ ) ≈ τ Ð τ ---i = Ðτ .  c  i i ⑄ i ε p Λ If ------f---- = γ Ð 1 ~ 1, ------i-- ~ 1, and --- ӷ 1, then 2 f ⑄ Now, the solution of the equation for proton m c mec e momentum can easily be obtained: Λ ε eE ω pmax f ≈ 0⑄1 + (ω ω ) (τ τ ) ------≈ ------; (30) P Pi + ------k Ð 0 F Ð i . (35) m c ⑄ 2 c 1 Ð ω e mec It follows that that is, P r m  P  ----max------∼ ----0 -----e(γ Ð 1) ------i- Ӷ 1 (36) Λ ⑄ f   (γ ) ≈ (γ ) Mc M Mc Ð 1 max -⑄-- f Ð 1 . (31) ⑄ × 3 γ ≈ If r0/ ~ 2 10 and f Ð 1 1, Hence, if Λ/⑄ ~ 103, the energy of the accelerated P ----max------≈ 0.5. (37) electron is approximately 1 GeV. So far, the accelera- Mc tion of a relativistic electron has been considered. If an electron is nonrelativistic but the condition In this case, a proton can be accelerated to about 100 MeV. Λ ---(1 Ð ω) ӷ 1, (32) ⑄ ANTICIPATED DESIGN AND PERFORMANCE OF THE ACCELERATOR is met, all the estimates obtained above remain valid in the order of magnitude. If an accelerated particle is out From the obtained results, the possible design and of phase with the accelerating field (this is true for non- performance of the accelerator can be envisioned. relativistic velocities v Ӷ c), the time of acceleration Electron accelerator. The condition Λ/⑄ = 103 spec- decreases in proportion to 1 Ð ω . However, if condition ifies the accelerator size. The value of ⑄ ≈ 0.2 cm seems to be attainable. In this case, the accelerating gap L (32) is met, the electron soon becomes relativistic, γ ≈ whereupon acceleration proceeds as described above. should be approximately 0.2 cm at f 2. The parabo- Λ ω ω loid length should be = 2 m. If k = 0.8 and 0 = 0.5, Proton acceleration. Protons and electrons have the paraboloid radius in the focal plane should be r0 = opposite electric charges. Therefore, when the acceler- ≈ 2 τ 2/3 m and the paraboloid surface area, S 22 m . The ating field reverses at > 2, the directions of the proton amount of energy accumulated in the accelerating gap velocity, phase velocity of the accelerating field, and near the paraboloid surface is easily estimated at accelerating force will coincide. The proton mass is approximately 1840 times the mass of an electron. (γ )2 f Ð 1 Therefore, it is reasonable to suggest that the proton U ∼ 116------S(J) ≈ 13(J). (38) L remains nonrelativistic throughout the acceleration process. We cannot say in advance how the proton posi- The output power of a laser with the light quantum ε ≅ tion in space will vary with time. However, this infor- energy k 2 eV that provides electron emission from mation is not necessary for making the required esti- the paraboloid surface (quantum efficiency Y ≈ 0.2) can mates. It is sufficient to obtain the formal solution of the also be easily assessed: equation of motion for the coordinate z (t): n 3 -- 2 τ ε (γ Ð 1) ∼ × 7----k ------f------( ) ≈ × 10 v v (τ) J 1.4 10 2 S W 7 10 (W). (39) z = z + dτ'---⑄ = z + (τ Ð τ )------⑄, Y L n i ∫ c i i c τ i (33) If the pulse duration is approximately 10 ps, the ≈ v (τ) laser energy is 0.4 J. ------Ӷ 1, The schematic diagram of the accelerator is shown c in Fig. 3. Obviously, a device accelerating particles to energies lower than 1 GeV would have smaller dimen- where v = v (τ) is the mean velocity of the particle. sions. Let us estimate the energy acquired by an elec-

TECHNICAL PHYSICS Vol. 45 No. 8 2000 A HIGH-GRADIENT ACCELERATOR 975

3 focus is approximately 27 cm and Λ = 20 cm. Assum- 2 ing, as above, that ⑄ = 0.2 cm, we ﬁnd that all the parameters of the accelerator should be decreased a 1 hundred times in comparison with the case considered 6 above: S ≈ 0.22 m2, J ~ 7 × 108 W, and U ~ 130 J. The 4 5 ε energy a acquired by an electron upon acceleration decreases only tenfold: ε ~ 100 MeV. The total length 7 a of the accelerator, including a laser and an optical system, is approximately 1 m. The injection of particles into such a system can be performed in two ways: (1) synchronously with the generation of a laser pulse (in this case, particles are accelerated in the vicinity of the Fig. 3. Schematic diagram of the accelerator: (1) paraboloi- paraboloid focus) or (2) prior to focusing the electro- dal mirrors of optical system, (2) conical mirrors, (3) parab- magnetic pulse (a dipole layer of low-energy electrons oloidal microwave source, (4) laser, (5) injector, (6) focus, and (7) beam outlet. is produced over the injecting surface, and the electromagnetic pulse captures as many electrons from this tron in such an accelerator with a characteristic size layer as it can accelerate). of 1 m. Proton accelerator. The energy of a proton attains ≈ ω ω ≈ γ ≈ ⑄ If r0 6.7 cm, k = 0.8, and 0 = 0.5, the distance 1 GeV at Pmax 1.8Mc. Assuming that f 3, = ω ω between the extreme point of the paraboloid and its 0.3 cm, k = 0.67, and 0 = 0, we obtain, using starting Z, cm 120 Ez, arb. units t = 5 ns Current

100

60 1 Focus r0 = 100 cm

40 2

0 50 100 150 ρ, cm

Fig. 4. Electric ﬁeld level lines: (1) paraboloid surface and (2) electric current.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 976 LAZAREV, PETROV

Ez, arb. units 1.0

1 2

0.5

–0.5 5.4 5.5 5.6 5.7 5.8 5.9 T, ns ρ Fig. 5. Electric ﬁeld at the paraboloid focus ( = 0, z = zf, and T0 = 0.05 ns): (1) analytical solution and (2) numerical solution. , arb. units Ez (a)

1.0 Paraboloid surface

Z/r 0 1.0 ρ/r 1.0 0 0 –0.2 Ez, arb. units (b)

1.0

Z/r 0 1.0 ρ/r 1.0 0 0 –0.2

Fig. 6. Spatial distribution of the electric component Ez of the electromagnetic wave focused onto the paraboloid focus. t/T0 = (a) 6 and (b) 0.8.

≈ 3⑄ ≈ equation (9), hence, r0 10 . Therefore, the paraboloid radius r0 P r 3 m; the paraboloid length, 6 m; and the maximum ----max------≈ 1.8 × 10Ð3----0; (40) Mc ⑄ radius, 6.7 m. A 100-MeV accelerator should have the

TECHNICAL PHYSICS Vol. 45 No. 8 2000 A HIGH-GRADIENT ACCELERATOR 977

Ez, arb. units 0.8

ρ 0.6 _ = 0 z/λ = 1.33 T = 0.05 ns _ 0 z/λ = 0.66 r0 = 100 cm _ ξ ξ 0.4 z/λ = 0 min = 0, max = 0.35 _ z/λ = 0.66 0.2

–0.2

–0.4

–0.6 –2 0 2 4 6 8 10 12 t/T0

Fig. 7. Time dependence of Ez in the vicinity of the paraboloid focus.

Ez, arb. units 0.8 t/T0 = 0.86

ρ = 0 0.6 t/T0 = 0.54 T0 = 0.05 ns t/T0 = 2 r0 = 100 cm ξ ξ min = 0, max = 0.35 0.4

t/T0 = 0 0.2

–0.2

–0.4 _ –13.33 –6.66 0 6.66 13.33 20 z/λ

Fig. 8. Spatial distribution of Ez in the vicinity of the paraboloid focus at various time instants.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 978 LAZAREV, PETROV

Eρ, arb. units Ez, arb. units 0.6 0.4 0.3 0.4 0.2 0.2 0.1 0 0 –0.1 –0.2 –0.2 –0.3 –0.4 –0.4 –5 0 5 10 t/T0 –5 0 5 10 t/T0

Hϕ, arb. units 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 –5 0 5 10 t/T0

Fig. 9. Time dependence of the electromagnetic ﬁeld components in the focal region: ρ/λ = 0.666; z/λ = (dashed line) 2.666, (solid line) 0, and (dash-and-dot line) 1.333.

≈ δ following dimensions: r0 0.7 m; length, 1.5 m; and where is the Dirac function. maximum radius, ≈1.6 m. The mesh size was taken to be ∆z/⑄ = ∆x/⑄ = 0.166.

RESULTS OF NUMERICAL Simulated results for the propagation of an electro- CALCULATIONS magnetic wave induced by an FTL pulse on the paraboloid surface are presented at different time instants in The computation of the spaceÐtime distribution of Figs. 4Ð6. It can be seen that the amplitude of the local- the electric field near the paraboloid focus was per- ized electromagnetic pulse increases as the focus is formed using a two-dimensional electromagnetic code. An FTL current pulse induced by radiation in an inner approached. For comparison, an analytic curve for ρ electric field intensity Ez is shown in Fig. 5. layer (dl = 0.25 cm) of the paraboloid (zp( ), 0 < z/r0 < 0.9, the focal distance r0 = 100 cm) was simulated using Calculations showed that a strong electric field is the expression for electric current density generated in the focal region of size տ⑄ (Figs. 7Ð9). The  z Ð z field direction remains invariable for the time interval j(t, z, ρ) = f t Ð ---k------δ(z (ρ))n, ≅2T , during which the field can be used for accelerat-  c  p 0 ing charged particles. It should be noted that the radial component of the electric field is much smaller than the 1  ρ  (41) n = ------Ð----; 1, longitudinal component and equals zero at the parabo-  ρ 2 r0  loid axis. In addition, the magnetic field upon accelera- 1 + ---- r0 tion is directed so that the Lorentz force, acting on the electron beam, presses it against the paraboloid axis t 2 Ð----- (i.e., the beam contracts). Thus, the accelerator can be ( )  t  T0  t  × Ð9 f t = ----- e ----- , T0 = 0.5 10 s, (42) expected to provide a small angular spread of acceler- T 0 T0 ated particles.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 A HIGH-GRADIENT ACCELERATOR 979

CONCLUSION present no problems if FTL microwave sources them- The obtained results show that the use of FTL emis- selves are developed. sion sources for accelerating charged particles is feasi- ble. Accelerators based on FTL microwave sources are expected to accelerate charged particles to energies of REFERENCES several hundred or even several thousand MeV, while 1. Yu. N. Lazarev and P. V. Petrov, Pis’ma Zh. Éksp. Teor. being signiﬁcantly smaller in size than today’s linear Fiz. 60 (9), 625 (1994) [JETP Lett. 60, 634 (1994)]. accelerators. The requirements for the performance of the accelerator components do not seem to be unique. Therefore, the production of such accelerators should Translated by Chamorovskiœ

TECHNICAL PHYSICS Vol. 45 No. 8 2000

ATOMS, SPECTRA, AND RADIATION

Emission of Particles from a Magnetic Trap Yu. G. Pavlenko Advanced Education and Science Center, Moscow State University, Moscow, 121357 Russia Received May 11, 1999

Abstract—Three types of magnetic traps that provide three-dimensional potential wells for neutral atoms and ferromagnetic particles are considered. The possibility of forming directed particle beams is discussed. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION EQUATIONS FOR PARTICLE MOTION AND MAGNETIC MOMENT Ions that are cooled to ultralow temperatures and The energy of the particle–magnetic-field interac- localized in electromagnetic traps open many possibil- µ ities for investigating quantum jumps, the quantum tion has the form U(t, x) = ÐmB, where m = g BS is the mean magnetic-moment operator, S is the mean effec- properties of photon statistics, optical bistability, and µ ប × the problems standing in the way of constructing fre- tive spin, g is the Landé factor, and B = e /2me = 5.787 quency standards. Experiments with ensembles of par- 10Ð5 eV/T is the Bohr magneton [9]. For a neutron and µ µ − ticles made it possible to observe the formation of proton, we have g B gn, p N, where gn = 3.826, µ × Ð8 ordered structures and phenomena similar to phase gp = 5.586, and N = 3.15 10 eV/T is the nuclear transitions [1, 2]. In connection with this, it becomes magneton. The magnetic moment of a ferromagnetic relevant to analyze the problems of the localization, particle is equal to µ = MV, where M is the volume den- accumulation, and confinement of neutral particles in sity of the magnetic moment, V is the volume of a par- 7 3 magnetic traps [1, 3Ð5]. Another aspect of this area of ticle, and Mmax ~ 10 (J/T cm ). research is associated with the possibility of emitting The equation of motion for the center of mass of an particles from a magnetic trap and implementing a atom in a quasi-uniform magnetic field is a conse- device that was called an atomic laser [6]. The escape quence of the Ehrenfest equation of particles can be driven by one or another triggering mechanism [7, 8]. 2 2 µ ∂ ∂ md x/dt = g BSn Bn/ x + mg0, (1) The motion of particles with a nonzero magnetic moment is described by a set of six nonlinear equations where g0 is the free-fall acceleration. for the coordinates of the center of mass and for the Equation (1) is also valid for a ferromagnetic parti- components of the magnetic moment. Here, we pro- cle [10]. The vector S satisfies the equation pose a new approach to investigating the dynamics of ⋅ the magnetic moment of a particle in a magnetic field. dS/dt = W S, (2) The dynamic equations for the magnetic moment are where W = ÐγB and γ = gµ /ប. represented in Hamiltonian form with the fundamental B Poisson bracket of complex “coordinates” and 2 2 Equation (2) has an obvious first integral S (t) = S0 . “momenta.” The solution obtained is a canonical trans- Note that, in electron systems, the cyclotron frequency formation that drives the new Hamiltonian to zero. We is equal to ν [GHz] = eB/(2πm ) = 27.9922B [T]. consider several configurations of a constant axisym- e metric nonuniform magnetic field that provide three- We consider particle motion in a magnetic field that dimensional potential wells for trapping cooled parti- is a superposition of the constant axisymmetric field cles in a bounded region. A resonant RF pulse triggers Bs(x), the constant uniform field B0 = (0, 0, b0), and the the spin-flip transition to the state of infinite particle RF magnetic field motion. The question then arises of how to pass over to () ()ω,,ω () the state of semi-infinite motion. We show that an axi- B~ t = bpcos t Ð0b p sin t ft, (3) symmetric field configuration provides the possibility where we introduced the function f(t) = 0 for t < t0 and of emitting particles in the form of a directed beam. In t > t + τ and f(t) = 1 for t ≤ t ≤ t + τ. order to implement this effect, three ring currents that 0 0 0 ρ ϕ produce a symmetric magnetic trap should be supple- The vector potential of the magnetic field, As( , , mented with an additional ring coil. z) = (0, Aϕ, 0), satisfies the conditions divB = 0 and

1063-7842/00/4508-0980 $20.00 © 2000 MAIK “Nauka/Interperiodica” EMISSION OF PARTICLES FROM A MAGNETIC TRAP 981 curlB = 0. In cylindrical coordinates, the component mean spin is equal to 〈S〉 = CB/B with C = S(0)B/B. The Aϕ = A(ρ, z) can be represented as a series [11]: thermally equilibrium states are characterized by C > 0. As the temperature decreases, we have C S . 3 0 A(ρ, z) = ρb (z)/2 Ð ρ b''(z)/16 Moreover, if C = S , then S(t) = S B/B, in which case s s (4) 0 0 …( )n + 1 [( ) ](ρ )2n Ð 1 (2n Ð 2)( ) the total energy of a particle can be written as E = + Ð1 1/ n Ð 1 !n! /2 bs z , 2 µ µ µ mv /2 Ð B(x) Ð mg0x, where = g BS0. ρ 2 2 1/2 where = (x + y ) and bs(z) is an arbitrary function. Substituting S = CB/B into (1) results in the equa- From (4), we find the constant nonuniform magnetic tion field: 2 2 µ ∂ ∂ md x/dt = g BC B/ x + mg0. (9) ( ) ≈ [ ( )( ' '''ρ2 ) Bs x Ð x/2 bs Ð bs /8 , Let the origin of the coordinates be at the axis of the (5) Ð(y/2)(b' Ð b'''ρ2/8), b Ð (ρ2/4)b'' ]. magnetic configuration, and let this axis coincide with s s s s the z-axis directed vertically upward. In order to deter- The lines of the constant magnetic field B(x) = B0 + mine the conditions under which the particles move in B (x) are described by ρ[A(ρ, z) + ρb /2] = const. Let us a bounded spatial region (|z| < L/2, ρ < L), we introduce s 0 | | ӷ | | consider some configurations of a nonuniform mag- the function bz = b0 + bs(z) and assume that bz Bs1 , | | ≈ ρ2 2 netic field. Bs2 , in which case we have B(x) bz + ( /8)[(bz' ) /bz Ð (i) In a hyperbolic configuration, the magnetic field '' intrinsically vanishes at the z-axis and increases when 2bz ]. From (9), we obtain the set of equations away from it, so that we can set 2 2 md z/dt 2 3 ( ) 2 (10) bs z = a1z + a2z /2 + a3z /6. (6) µ [ (ρ )( )] = g BC bz' + /4 bz'bz''/bz Ð bz''' Ð mg0, Here, the first term describes the quadrupole field 2 2 ( µ )[( )2 ] and the second terms defines the mirror field. md x/dt = g BC/4 bz' /bz Ð 2bz'' x, (11) (ii) A quasi-periodic magnetic field can be specified π λ 2 2 2 through the function bs(z) = bcoskz, where k = 2 / md y/dt = (gµ C/4)[(b' ) /b Ð 2b'']y. (12) with λ the spatial period, in which case we have B z z z In the paraxial approximation ρ Ӷ L, such that the B (x) ≈ [(bkx/2)sinkz, s ρ ρ2| | Ӷ (7) values of satisfy the condition bz'bz'' /bz Ð bz''' ( ) , ( 2ρ2 ) ] bky/2 sinkz b 1 + k /4 coskz . | | 4 bz' , equation (10) can be written as (iii) For a configuration produced by a ring current- carrying coil of radius a, centered at the z-axis and md2z/dt2 = Ð∂W(z)/∂z, (13) µ 3 lying in the z = h plane, we can set bs(z) = 0Ia/(2R ), µ 2 2 2 where the function W(z) = Ðg BCbz(z) + mg0z plays the where R = a + (z Ð h) , I is the current magnitude, and role of the potential energy. µ = 4π × 10Ð7 H/A2 is the permeability of the vacuum. 0 Equation (13) has an obvious first integral ( )2 ( ) ( ) ( )[ ( )] EQUATION OF PARTICLE MOTION dz/dt = G z , G z = 2/m E3 Ð W z , (14) IN A MAGNETIC TRAP where E3 is the total energy of the longitudinal particle Equations (1) and (2) yield a law according to which motion. Under the condition b'' > (b' )2/2b , equations the total energy of a particle changes as z z z (11) and (12) describe a harmonic oscillator with a µ ∂ dE/dt = Ðg BS Bt/dt, (8) time-varying frequency. 2 µ where E = mv /2 Ð g BS(t)Bt(t, x) Ð mg0x and Bt(t, x) = B0 + Bs(x) + B~(t) is the total magnetic field. EMISSION OF PARTICLES We start by solving equations (1) and (2) in the time FROM A MAGNETIC TRAP ≤ ≤ interval 0 t t0, assuming that B(x) = B0 + Bs(x). For cold particles moving in a magnetic trap, we The spatial and spin variables change with the charac- have C S0, in which case the energy of the particleÐ ω µ 1/2 Ω γ teristic frequencies a = ( B''/m) and = B, where magnetic-field interaction via the particle magnetic 2 ω Ӷ Ω moment is U(t, x) ÐµB, where µ = gµ S . Let the B'' ~ 200 T/m and B ~ 1 T. Since a , we can B 0 neglect the contribution of the rapidly oscillating com- RF magnetic field (3), which is rotating around the z- ponents of the vector S to equation (1) and switch to axis, be also present in the system, so that the total mag- slowly varying spatial variables. This procedure is netic field is Bt(t, x) = Bs(x) + B0 + B~(t). We assume ӷ | | equivalent to the replacement of S by its mean value. In that the condition b0 bs holds. A resonant pulse of τ π Ω the Appendix, we solve equation (2) and show that the the RF magnetic field with the duration = / p, where

TECHNICAL PHYSICS Vol. 45 No. 8 2000 982 PAVLENKO Ω γ p = bp, triggers the spin-flip transition S0 ÐS0 at ment of the particle away from the axis is equal ω γ 2 the frequency = b0 [12]. In fact, according to (A.14), to g /ω . the solution to the dynamic equations for the magnetic 0 12 τ moment on the time scales t > t0 + has the form S(t) = ÐS0B/B, where S0 = S(t0)B/B. For a particle moving in Emission of Particles from the Trap an alternating electric field, law (8), according to which After the interaction between a particle and a reso- the total particle energy changes, reduces to dE/dt = − µ ∆ nant RF pulse, the potential energy of the particle and g BSdB~(t)/dt, which yields the total energy E = its total energy are equal to W(z) W (z) = µb (z) + µ τ f z 2 b0 acquired by the particle over the time interval . mg z and E = E + 2µb . On the time scales t ≥ t + τ, ≥ τ 0 3f 3 0 0 On the time scales t t0 + , the energy of a particle equation (14) takes the form interacting with a constant magnetic field becomes µ ( )2 ( ) ( ) ( )[ ( )] Uf(t, x) = B. Under the action of the RF pulse, the con- dz/dt = G f z , G f z = 2/m E3 f Ð W f z . (16) ditions for the particle to move in a bounded region fail to hold and the particle escapes from the trapping Since the equation Gf(z) = 0 has the single real root region. z = z0, there remains only one turning point, so that the z0-particles moving in the z-direction leave the trap in the positive direction along the z-axis. The time at PARTICLE IN AN AXISYMMETRIC which the particles escape from the potential well can MAGNETIC TRAP be estimated from (16) as follows: Localization of a Particle ∼ σ 1/4 tax 2/ f D , In (6), we set a3 = Ða30 < 0 and a1 > 0, in which case the potential energy of a particle is equal to W(z) = where − µ 2 3 g BC(b0 + a1z + a2z /2 Ð a30z /6) + mg0z. The equation σ 2 = µa /3m, dW/dz = 0 implies that the potential energy has two t 30 local extremes, a maximum and a minimum, at the 4 [ 2 ( )( µ)] D = 3 z0 Ð 2a2z0/a30 Ð 2/a30 a1 + mg0/ . ± 2 points with the coordinates e2, 1 = [a2 (a2 + 2a30[a1 Ð The characteristic time scale tr on which the particle µ 1/2 mg0/g BC]) ]/a30, such that W(e1) > W(e2). If E3 is in beam diverges in the radial direction can be estimated µ µ the W(e2) < E3 < W(e1) range, then the particle oscillates from equations (15) with the replacement Ðg BC : in the potential well between two turning points z2 and µ 2 Ð1/2 Ӷ tr ~ 2[( /m)(a1 /b0 Ð 2a2)] . If tax tr, the particles z1, such that e1 < z1 < e2 < z3. In the well region, we have G(z) ≥ 0. The zeros of the function G(z) are such that occupy the region stretched out along the z-axis, forming a beam extended in the positive direction along the z3 > z2 > z1. Substituting the function G(z) in the form σ2 σ2 µ z-axis. G(z) = Ð (z Ð z1)(z Ð z2)(z Ð z3) with = g BCa30/3m into (14), we obtain a solution to equation (10): PARTICLE IN A QUASIPERIODIC ( ) ( ) 2[ω ( ) , ξ] z t = z3 Ð z3 Ð z2 sn z t + T /2 , MAGNETIC TRAP ω ξ ω σ Inserting b (z) = b + bcoskz into (10)Ð(12), we where sn[ z(t + T)/2, ] is an elliptic sine, z = (z3 Ð z 0 1/2 ξ2 2 ω obtain the following equations in the paraxial approxi- z1) , = (z3 Ð z2)/(z3 Ð z1), z3 Ð z(0) = (z3 Ð z2)sn [ zT/2, ξ ω mation: ], z(t) is a periodic function with the period 4K/ z, and K(ξ) is the complete elliptic integral. 2 2 ω 2 d kz/dt + z sinkz = Ðkg0, (17) Now, we analyze particle motion in the x- and y- 2 2 (ω 2 )[( ) 2 ] directions by examining the simplest situation, when d x/dt Ð z /2 b/2b0 sin kz + coskz x = 0, (18) the inequalities b ӷ |b | and (b' )2/b Ð 2b'' ≈ a 2 /b Ð 0 s z z z 1 0 2 2 (ω 2 )[( ) 2 ] d y/dt Ð z /2 b/2b0 sin kz + coskz y = 0, (19) 2a2 < 0 are assumed to be satisfied in the region where the particle moves along the z-axis. In this case, equa- ω 2 µ 2 tions (11) and (12) reduce to where z = g BCk b/m. We can draw an analogy between (17) and the equa- 2 2 ω 2 2 2 ω 2 d x/dt + 12 x = 0, d y/dt + 12 y = 0, (15) tion of motion of a pendulum with a constant force 2 2 2 moment. If kg < ω , the coordinates z of the equilib- where ω = (gµ C/4m)[2a Ð a /b ]. 0 z eq 12 B 2 1 0 rium points can be obtained from the equation The projection of the particle trajectory onto the ω 2 (x, y) plane is an ellipse, so that the particle moves in a z sinkzeq = Ðkg0 with coskzeq > 0. The potential energy µ bounded spatial region. Note that, if the magnetic axis is equal to W(z) = Ðg BC(b0 + bcoskz) + mg0z. In the is horizontal, then the equilibrium vertical displace- vicinity of an equilibrium point, we have z = zeq +

TECHNICAL PHYSICS Vol. 45 No. 8 2000 EMISSION OF PARTICLES FROM A MAGNETIC TRAP 983 ω α Acos( 0t + ). In other words, a particle oscillates with Equations (21)Ð(23) give the coordinates of the ω ω 2 ω 2 equilibrium point: z = 0, x = 2mg /[gµ Cb'' (0)], and frequency 0 such that 0 = z coskzeq. In this case, eq eq 0 B s equations (18) and (19) belong to the class of Hill equa- y = 0, b'' (0) = 3gµ C[Ia(a2 Ð 4h2)(a2 + h2)7/2 + I /2a 4 ]. Ӷ eq s B 0 0 tions. If kA 1, equations (18) and (19) reduce to the | | ӷ 2 2 If the potential well is sufficiently deep, bs(h) Mathieu equation d u/ds + (p + 2qcos2s)u = 0, where | | s = 2(ω t + α) and bs(0) , then the parameters of the magnetic system 0 should satisfy the inequality 2 p = Ð(1/8)[1 + bsin kz /(2b coskz )], 2 2 2 Ð3/2 2 2 Ð3/2 eq 0 eq (I/a )[1 + (1 + 4h /a ) Ð 2(1 + h /a ) ] ( )( ) Ð3/2 q = Ð A/8 Ð tankzeq + bsinkzeq/b0 . ӷ 2[ ( 2 2) ] I0/a0 1 Ð 1 + h /a0 . The theory of Mathieu functions implies that, in the ' parameter plane (p, q), there are regions in which the On the other hand, since bs (0) = 0, a particle will solutions are either bounded or unbounded [13, 14]. In move in a bounded region in the vicinity of the plane the plane (p, q), the solutions to equations (18) and (19) z = 0 in the in the x- and y-directions only if the inequal- are bounded in the first stability domain, which is ity bs'' (0) > 0 holds. Consequently, the particles can located to the right of the curve p (q) = Ðq2/2 + c0 generally be localized if the function bs(z) satisfies the 4 7q /128 + … and is bounded by the curve ps1(q) = 1 + conditions 2b'' Ð (b' )2/b > 0 and |b (h)| > |b (0)|. 2 s s 0 s s q Ð q /8 + … for q < 0 and by the curve pc1(q) = 1 Ð q Ð q2/8 + … for q > 0 [13]. Emission of Particles from the Trap In order to ensure the directed motion of the parti- PARTICLE IN A SPHERICAL SEXTUPOLE cles after the resonant pulse has come to an end, it suf- MAGNETIC TRAP fices to switch on an additional coil, with current Ic and Localization of a Particle radius r, located in the plane z = ÐH such that H > h. After the particle spin reverses direction, the potential The magnetic trap implemented experimentally by (c) µ Paul and his collaborators [3] and aimed at confining energy of the particle becomes W (z) = (b0 + bs + bc), µ 3 2 2 2 neutrons was produced by the magnetic field of three where bc(z) = 0Icr/(2R ), R = r + (z + H) , and bs(z) is ring current-carrying coils placed at the intersections of defined in (20). The magnetic field of the additional coil a spherical surface with the planes z = ±h and z = 0. The plays the role of a magnetic mirror that reflects particles equatorial current was chosen to flow in the direction in the positive direction along the z-axis. To make the opposite to the polar currents, in which case the two beam radially nondivergent, the additional coil should 2 points corresponding to the equilibrium states are those satisfy the condition 2(bs'' + bc'' ) Ð (bs' + bc' ) /b0 < 0. located symmetrically with respect to the equatorial plane (z ≠ 0). This circumstance complicates the extrac- tion of neutrons from the trap. ACKNOWLEDGMENTS We consider the motion of particles in a magnetic This work was supported in part by the Russian field produced by three unidirectional ring currents. To Foundation for Basic Research, project no. 97-01- do this, we treat (4) with 00957. b (z) = Ð (µ Ia/2)[1/R 3 + 1/R 3] Ð µ I a /2R 3, (20) s 0 1 2 0 0 0 0 APPENDIX where R 2 = a2 + (z + h)2, R 2 = a2 + (z Ð h)2, h > a/2, Equation (2) is derived by averaging the Heisenberg 1 2 equations of motion for the spin operator over the 2 2 2 and R0 = a0 + z . Let the z-axis be directed horizon- superposition of states of a quasiclassical wave packet. tally, and let the x-axis be directed vertically down- Consequently, equation (2) is neither Lagrangian nor ward. From (10)Ð(12), we obtain the following equa- Hamiltonian; it also cannot be derived variationally. tions in the paraxial approximation: But it can be made Hamiltonian with the help of the method that was developed by Schwinger, who found md2z/dt2 = ÐdW/dz, (21) the relation between the spin operator and the conju- gate “creation” and “annihilation” operators, which can 2 md2 x/dt2 = (gµ C/4)[(b' ) /b Ð 2b'']x + mg , (22) be introduced when examining two harmonic oscilla- B s 0 s 0 tions [15]. 2 We introduce a spinor-column Ψ with two complex md2 y/dt2 = (gµ C/4)[(b' ) /b Ð 2b'']y, (23) B s 0 s Ψ+ elements a1 and a2, such that = (a1* , a2* ). We µ ӷ σ where W(z) = Ðg BC[b0 + bs(z)] and b0 bs. denote the Pauli matrices by k (k = 1, 2, 3) and define

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Ψ+σ Ψ Λ the components Sk = (1/2) k of the vector S matrix mµ = [um(µ)] are the eigenvectors of the matrix through the relationships Bs/2B, which coincide with (A.4) to within a phase factor [16]: S = (a*a + a*a )/2, S = (a*a Ð a*a )/2i, 1 1 2 2 1 2 1 2 2 1 (Θ ) ( ϕ ) (A.1) u1(1) = cos /2 exp Ði /2 , ( * * ) S3 = a1 a1 Ð a2 a2 /2. (Θ ) ( ϕ ) u1(2) = Ðsin /2 exp Ði /2 , (A.5) We also introduce the “coordinates” and (Θ ) ( ϕ ) u2(1) = sin /2 exp i /2 , * “momenta” qk = ak and pk = iak (k = 1, 2) with the fun- (Θ ) ( ϕ ) δ u2(2) = cos /2 exp i /2 . damental Poisson bracket (PB) [qi, pk] = ik. The rela- ε tionship [S1, S2] = ijkSk for the PB puts equation (2) in The CT generated by the function F2 has the form Hamiltonian form dS/dt = [S, H] with the Hamiltonian (qα' = ∂F /∂ pα' , p = ∂F /∂q ) (q = Λ α qα' , p = H = WS. 2 m 2 m n n m Λ + + Λ + We consider the dynamic equations for the magnetic ( )µm pµ ). Since ( )αk = [uk(α)]*, we have moment in a constant nonuniform magnetic ﬁeld B0 + [ ] Bs(x). Along the phase trajectories of the system, we qn = un(α)qα' , pm = um(µ) * pµ' . (A.6) γ have W(t) = Ð B(t), where B(t) = B0 + Bs(x(t)). In terms of canonical variables, the Hamiltonian has the form Consequently, in the new variables, the Hamiltonian H'(q', p', t) = (H + ∂F /∂t) is equal to H' = H' + h' with (A.2) 2 0 H = ( 1/2 ) [ a * a Ω + a*a Ω + (a*a Ð a*a )Ω ], ( , , ) 1 2 Ð 2 1 + 1 1 2 2 3 H 0' q ' p ' t = Ð i H µ' α p µ' q α '', (A.7) Ω Ω ± Ω where ± = 1 i 2. h'(q', p', t) = ωµα pµ' qα' ,

The dynamic equations dak/dt = [ak, H] (k = 1, 2) Λ + Λ λ δ ω Λ + Λ become where Hµ' α = ( )µm Hmn nα = Ð µ µα, µα(t) = µk kα. ' dak/dt = ÐiHknan, (A.3) The CT puts the Hamiltonian H0 (q', p', t) in the diago- Ω Ω Ω nal form H' (q', p', t) = Ðλ c* c or where H11 = 3/2, H12 = Ð/2, H21 = +/2, and H22 = 0 n n n −Ω /2. 3 Ω( 2 2) λ H0' = Ð c1 Ð c2 /2. (A.8) We begin by solving the eigenvalue problem vn = λ ÐHnkvk. From the equation det(H + I) = 0, we obtain We assume that, along the phase trajectories of the λ ±λ λ Ω Ω the eigenvalues 1, 2 = 0 and 0 = /2, where = system, the magnetic field satisfies the condition 2 2 2 |ω | Ӷ Ω µ α (Ω + Ω + Ω )1/2. Substituting λ = λ into the µα ( , = 1, 2), in which case we have H'(q', p', 1 2 3 1, 2 ≈ λ eigenvalue problem yields the orthonormal eigenvec- t) Ð ncn* cn. The solution to the equations generated tors vn(1) and vn(2): by the Hamiltonian H' is a CT ck bk such that c1 = η η 1/2 b1exp(i /2) and c2 = b2exp(Ði /2) pass over to b1 = v ( ) = [(Ω Ð Ω )/2Ω] , 1/2 φ 1/2 φ 1 1 3 (S0 + C) exp(i /2) and b2 = (S0 Ð C) exp(Ði /2), Ω [ Ω(Ω Ω )]1/2 η Ω φ v 1(2) = Ð/ 2 Ð 3 , where (t) = ∫ (t)dt and C and are integration con- (A.4) Ω [ Ω(Ω Ω )]1/2 stants. The general solution to equations (A.3) has the v 2(1) = Ð +/ 2 Ð 3 , Λ Λ form an = nαcα. Inserting an = nαcα into (A.1) gives [(Ω Ω ) Ω]1/2 α v 2(2) = Ð 3 /2 . Sn = Rn(α) Sα' , where the real vectors Rn(α) ( = 1, 2, 3) We can parameterize the eigenvectors by introduc- are defined as Θ ϕ ing the angles and for the vector B in spherical R ( ) = B B /BB , R ( ) = ÐB /B , R ( ) = B /B, ϕ 1 1 1 3 12 1 2 2 12 1 3 1 coordinates through the relationships B1 = B12cos and ϕ Θ Θ B2 = B12sin , where B12 = Bsin , and B3 = Bcos , R2(1) = B2 B3/BB12, R2(2) = B1/B12, R2(3) = B2 /B, 2 2 1/2 where B12 = (B1 + B2 ) . R3(1) = ÐB12/B, R3(2) = 0, R3(3) = B3/B. (A.9) We represent Hamiltonian (A.2) in terms of the coordinates and momenta H = ÐiHmnpmqn and make the The components of the vector Sα' are determined by canonical transformation (CT) qk = ak, pk = iak* relationships (A.1) with the replacement an cn: ' ' * qk = ck, pk = ick generated by the function F2(q, p', ( 2 2)1/2 (η φ) S1' = S0 Ð C cos + , Λ+ t) = ( )µk pµ' qk, which depends on the old coordinates (A.10) ( 2 2)1/2 (η φ) and new momenta. Here, the columns of the unitary S2' = Ð S0 Ð C sin + , S3' = C.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 EMISSION OF PARTICLES FROM A MAGNETIC TRAP 985

The relationship BS = BC implies that, in a quasi- equal to uniform magnetic field, the projection C = BS/B of the ( ) [Ω ( )] ω vector S onto the tangent to a magnetic field line is an S1 t = S0 sin p t Ð t0 sin t, adiabatic invariant. The mean spin is equal to 〈S〉 = ( ) [Ω ( )] ω S2 t = S0 sin p t Ð t0 cos t, (A.14) R C = BC/B. 3 ( ) Ω ( ) S3 t = S0 cos p t Ð t0 . Note that the CT ak ck defines a transition to the If an RF field is nonzero over the time interval t Ð new basis vectors nk nk' (k = 1, 2, 3), in which the τ Ω τ π ≥ t0 = such that p = , then, on the time scales t t0 + τ magnetic field B is directed along the unit vector n3' . In , we have S1(t) = 0, S2(t) = 0, and S3(t) = ÐS0. In other Ð1 π fact, the matrix Eik = (R )ik describes the operations of words, the resonant “ -pulse” reverses the direction of ϕ ϕ ϕ Θ the magnetic moment. We conclude with the following rotating by the Eulerian angles 1 = , 2 = , and ϕ two comments. 3 = 0 about the 3-2-3 axes, respectively [17]. With allowance for (A.9), we represent the solution to equa- (i) We write equation (2) in tensor form, dSi/dt = ε Ω tion (2) as AikSk, where Aik = ijk j(t) is an antisymmetric tensor Ω Ω Ω with the elements A21 = 3, A32 = 1, and A13 = 2. In ( Θ Θ) ϕ ϕ σ S1 = S1' cos + S3' sin cos Ð S2' sin , this case, the equation Vi = AijVj has the eigenvectors

S = (S ' cosΘ + S' sinΘ)sinϕ + S' cosϕ, V(1) = (R(1) + iR(2))/ 2 , V(2) = [V(1)]*, and V(3) = R(3), 2 1 3 2 (A.11) σ Ω σ Ω which refer to the eigenvalues 1 = Ði , 2 = i , and Θ Θ σ S3 = Ð S1' sin + S3' cos . 3 = 0. The vectors V(α) compose the orthogonal basis δ [V(α)]*V(β) = αβ [8]. Now, we consider the dynamics of the magnetic (ii) The functional moment of a particle in the magnetic field Bt = B0 + Bs(x) + B~(t). The total Hamiltonian of the dynamic ( , , ) I = dtL ak* ak t equations for the magnetic moment has the form Ht = ∫ H + h, where the Hamiltonian H is defined in (A.2) and satisfies the EulerÐLagrange equation ( )[ * Ω ( ω ) * Ω ( ω )] h = Ð 1/2 a1 a2 p exp i t + a2 a1 p exp Ði t , , ( )[ ] L(ak* ak) = i/2 andan*/dt Ð an*dan/dt ≥ (A.12) t t0, + H(a*, a , t), Ω γ k k with p = bp. which is in Hamiltonian form [18]. A significant advan- ӷ | | Ω ≈ Ω γ We set b0 bs and 0 = b0. Under the reso- tage of the Hamiltonian formalism is the possibility of ω Ω nance condition = 0, the CT ak ck bk con- using new methods for integrating the canonical equa- verts the Hamiltonian to the form tions of motion [18Ð20]. Introducing the functional makes it possible to apply direct variational methods of (Ω )( ) the BubnovÐGalerkin type. Ht' = Ð p/2 b1*b2 + b2*b1 . (A.13)

Taking into account (A.9) or (A.11), we obtain the REFERENCES components of the vector S(t): 1. P. É. Toshek, Usp. Fiz. Nauk 158, 450 (1989). 2. K. N. Drabovich, Usp. Fiz. Nauk 158, 499 (1989) [Sov. ( ω ) S1 = Re b1*b2 exp Ði t , Phys. Usp. 32, 622 (1989)]. 3. W. Paul, Usp. Fiz. Nauk 160 (12), 109 (1990). ( ω ) (A1a) S2 = Im b1*b2 exp Ði t , 4. D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). 5. W. Petrich, M. E. Anderson, J. R. Ensher, et al., Phys. ( ) S3 = b1*b1 Ð b2*b2 /2. Rev. 74, 3352 (1955). 6. M. Holland, K. Burnett, C. Gardiner, et al., Phys. Rev. A The equations generated by Hamiltonian (A.13) and 54, 1757 (1996). supplemented with the boundary conditions S1(t0) = 0, 7. A. M. Gazman, M. Moore, and P. Meyste, Phys. Rev. A S2(t0) = 0, and S3(t0) = S0 have the solution 54, 977 (1996). 8. V. I. Yukalov, Laser Phys. 7, 998 (1997). ( )1/2 Ω ( ) b1 = i 2S0 cos p t Ð t0 /2, 9. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990). ( )1/2 Ω ( ) b2 = Ð 2S0 sin p t Ð t0 /2. 10. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Nauka, Moscow, 1982; Pergamon, Therefore, the components of the vector S(t) are New York, 1984).

TECHNICAL PHYSICS Vol. 45 No. 8 2000 986 PAVLENKO

11. W. Glaser, Grundlagen der Elektronenoptik (Springer- 17. J. Mathews and R. L. Walker, Mathematical Methods of Verlag, Vienna, 1952; Gostekhizdat, Moscow, 1957). Physics (Benjamin, Menlo Park, 1970; Atomizdat, Mos- 12. C. P. Slichter, Principles of Magnetic Resonance cow, 1972). (Springer-Verlag, Berlin, 1990; Mir, Moscow, 1967). 18. Yu. G. Pavlenko, Hamiltonian Methods in Electrody- 13. N. W. McLachlan, Theory and Application of Mathieu namics and Quantum Mechanics (Mosk. Gos. Univ., Functions (Clarendon, Oxford, 1947; Inostrannaya Lit- Moscow, 1988). eratura, Moscow, 1953). 19. A. M. Perelomov, Generalized Coherent States and 14. P. M. Morse and H. Feshbach, Methods of Theoretical Their Applications (Nauka, Moscow, 1987; Springer- Physics (McGraw-Hill, New York, 1953; Inostrannaya Verlag, New York, 1986). Literatura, Moscow, 1958), Vol. 1. 15. D. C. Mattis, The Theory of Magnetism (Harper and 20. A. M. Perelomov, Integrable Systems of Classical Row, New York, 1965; Mir, Moscow, 1967). Mechanics and Lie Algebra (Nauka, Moscow, 1990). 16. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Nauka, Moscow, 1980; Per- gamon, Oxford, 1982). Translated by G. V. Shepekina

TECHNICAL PHYSICS Vol. 45 No. 8 2000

GASES AND LIQUIDS

Nonlinear Development of Capillary Waves in a Viscous Liquid Jet Yu. G. Chesnokov St. Petersburg State Institute of Technology (Technical University), St. Petersburg, 198013 Russia Received March 9, 1999; in ﬁnal form, September 14, 1999

Abstract—The one-dimensional approximate equations describing the dynamics of a Newtonian viscous ﬂuid are used to analyze the nonlinear development of capillary waves in a jet. It is shown that the size of satellite droplets resulting from a nonuniform jet breakup decreases with the Reynolds number at a constant wavenumber. The satellite-droplet formation ceases at a certain value of the Reynolds number, which depends on the wavenumber and initial perturbation amplitude. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION solution is represented as a series in a small perturbation amplitude. In the other, the equations of fluid A liquid jet issuing out of a nozzle can break up into mechanics supplemented with appropriate boundary droplets as the amplitude of a perturbation is increased conditions are integrated numerically. by capillary forces. The perturbation may appear on the jet surface or arise from fluctuations of pressure, flow A number of studies were focused on the problem of rate, or physical characteristics of the fluid. This phe- nonlinear perturbation development in a cylindrical nomenon is employed in a variety of technical devices, inviscid jet [3Ð9]. In particular, it was shown that non- such as printers, chemical apparatus, etc. [1]. For this linear interaction between perturbations can result in reason, capillary waves on the surface of a liquid jet nonuniform breakup of a liquid jet into drops. Rela- have been studied over many years, both experimen- tively small droplets, so-called satellites, can form tally and theoretically. This subject has been discussed between relatively large, so-called main, drops. in a number of literature surveys [1, 2], and there is no Lafrance calculated the sizes of main drops and satel- need for reviewing it in detail here. lites as depending on the wavenumber and found that Lord Rayleigh established that a liquid jet is unsta- his results were in good agreement with experimental ble with respect to a sinusoidal disturbance whose data [8]. wavelength is longer than the circumference of the jet Direct numerical simulations of liquid-jet breakup cross section. The instability is driven by capillary were reported in [10Ð16], where both inviscid [13, 14] forces. When the jet radius locally decreases, capillary and viscous [10Ð12, 15, 16] liquid jets were considered. forces give rise to a corresponding local pressure In [15], a very detailed analysis of the effects of the increase. Vice versa, at the location where the jet radius Reynolds number, wavenumber, and initial perturba- increases, the pressure at the surface decreases. As a tion amplitude on the process of jet breakup into drop- result, the liquid flows from the regions where the jet lets was presented. However, numerical integration of contracts to the regions where it expands, and the per- the equations of fluid mechanics is not a universal turbation tends to increase with time elapsed. Solving method. The high computational costs make it impossi- the linearized equations of fluid mechanics, Rayleigh ble to do without the assumption of longitudinally peri- obtained an equation relating the growth rate of the per- odic flow or take into account the initial velocity pro- turbation amplitude to the perturbation wavelength. file. For this reason, many theoretical analyses of the The time interval from the initial moment of perturba- capillary instability of a liquid jet are based on the one- tion to the moment when the jet breaks up into droplets dimensional approximate equations derived by assum- predicted by the linear theory was found to be in fair ing that the transverse length scale of the flow is much agreement with experimental results. smaller than its longitudinal length scale. The principal drawback of the linear theory lies in This approach was developed, for example, in the fact that the linearized equations used in the theory [17−26]. These studies include analyses based on per- become inapplicable as the perturbation amplitude turbation methods [17Ð22] and numerical simulations grows. Accordingly, the results based on the theory are [23Ð26]. Eggers obtained a self-similar solution to the valid only for the relatively short interval where the per- one-dimensional equations describing the behavior of a turbation amplitude remains small. Two different liquid jet just before and after its breakup [27, 28]. The approaches can be used to allow for nonlinear interac- stability of this solution with respect to small perturbation between disturbances. In one approach, the desired tions was analyzed in [29]. In [30], it was shown that

988 CHESNOKOV there exists a countable set of self-similar solutions of STATEMENT OF THE PROBLEM this type, the Eggers solution being the most stable with In this paper, the capillary stability of a cylindrical respect to perturbations. jet of an incompressible Newtonian liquid is analyzed. Different authors considered different variants of Lord Rayleigh showed that the breakup of a jet into approximate one-dimensional equations. Systematic droplets is caused by the growth of axially symmetric perturbations. Therefore, perturbations of this particu- derivation of such equations was considered, for exam- lar form are considered in the present analysis. The ple, in [31, 32], where the results obtained by solving fluid velocity in the jet is assumed to be sufficiently various approximate equations were compared with the high, so that the effect of gravity on the breakup of a exact solution to Rayleigh’s problem. In [32], a proce- liquid jet driven by capillary forces is negligible. dure was developed for constructing approximate equa- It is convenient to seek a solution in terms of dimen- tions that are valid to an arbitrary order with respect to sionless variables. Denote the jet radius by a, the liquid the small parameter defined as the ratio of the trans- density by ρ, its viscosity by µ, and the surface tension verse and longitudinal length scales of the flow. It was at the interface between the liquid jet and the ambient shown that the equations derived therein can be used as gas by σ. Then, the jet radius a can be used as a refer- a basis for constructing a solution to the stability prob- ρ 3 σ 1/2 ence length; t0 = ( a / ) , as a reference time; v0 = lem for a cylindrical viscous liquid jet that is consistent σ ρ 1/2 σ with Rayleigh’s solution up to terms proportional to the ( / a) , as a reference velocity; and p0 = /a, as a ref- wavenumber squared. erence pressure. Consider the cylindrical coordinate system (r, Θ, z) with z-axis aligned with the symmetry In this paper, the approximate one-dimensional axis. Denote the axial and radial velocity components equations derived in [32] are used to solve the problem by u and v, respectively, and pressure by p. The govern- of nonlinear development of perturbations in a cylindri- ing equations for a liquid jet flow include the continuity cal jet of a viscous liquid. A solution to a similar prob- equation lem that does not rely on approximate one-dimensional v v + --- + u = 0, equations has been found only for the inviscid liquid r r z jet. As in [32], the temporal stability of a liquid jet is analyzed. This means that an infinite jet is considered the axial NavierÐStokes equation in a coordinate system moving with the liquid. The spa- 1 1( ) tial stability of a viscous liquid jet, i.e., the stability of ut + v ur + uuz = Ð pz + ------rur r + uzz , a semi-infinite jet under given initial conditions at the Re r nozzle outlet cross section, was analyzed in [19Ð21] by and the radial NavierÐStokes equation invoking the so-called Cosserat equations. These equations, as well as other variants of approximate one- 1 1( ) v v t + v v r + uv z = Ð pr + ------rv r r Ð ---- + v zz . dimensional equations, can be employed under the Re r r2 assumption that the transverse length scale of the flow is small compared to its longitudinal length scale. This Here, t is time and the Reynolds number Re is defined condition holds for long-wavelength perturbations as when the corresponding wavenumber is small. It Re = (ρaσ)1/2/µ. should be noted that the terms of leading order with respect to this parameter in the Cosserat equations, In these equations, the subscripts denote derivatives ∂ ∂ which contain the liquid viscosity, are not consistent with respect to the corresponding variables: vr = v/ r, with the exact equations [31]. This makes it impossible etc. The free surface of the jet is prescribed by the equa- to analyze the effect of viscosity on the development of tion perturbations within the framework of these equations. r = η(z, t). An analysis of nonlinear interaction between capillary waves on the surface of a cylindrical jet of a viscous liq- On the surface, a kinematic boundary condition η η η uid based on the equations proposed in [32] can be used t + u z = v at r = both to examine the effect of viscous friction forces on the development of capillary waves and to determine to and two dynamic boundary conditions are set. One of what extent these equations are valid for describing the these dictates that the tangential component of viscous nonlinear interaction between perturbations. To accom- stress vanish on the jet surface: plish the latter task, one should compare the results 2v η + (u + v )(1 Ð η 2) Ð 2u η = 0 at r = η. obtained in this study with solutions to the problem of r z r z z z z nonlinear development of capillary waves in inviscid By the other dynamic condition, the normal compo- σ liquid jets that can be found in literature, as well as with nent of viscous stress has a jump equal to (1/R1 + the results obtained by solving a similar problem 1/R2) across the surface, where R1 and R2 are the longi- numerically for viscous liquid jets. tudinal and transverse principal curvature radii of the

TECHNICAL PHYSICS Vol. 45 No. 8 2000

NONLINEAR DEVELOPMENT OF CAPILLARY WAVES 989 jet surface, respectively. In dimensionless form, it can transverse and longitudinal coordinates r and z. The be written as resulting equations would then involve a small parameter δ equal to the ratio of the reference lengths. The 1 [ ( )η ] 1 1 dimensionless parameter Re is assumed to be a quantity p Ð ------2v r Ð ur + v z z = pa + ----- + -----. Re R1 R2 on the order of unity (while δ 0). If the variables η, u, v, and p are sought in the form of series, Here, pa is the ambient pressure, assumed to be constant. In the cylindrical coordinate system, the principal ( ) curvature radii are calculated as η = ∑ δ2mη 2m (z, t), η m ≥ 0 1 1 1 zz ----- = ------, ----- = Ð------. 2 1/2 R 2 3/2 R1 η(1 + η ) 2 (1 + η ) ( ) , z z u = ∑ δ2 n + m r2nun 2m(z, t), The problem formulated here has a solution of the n, m ≥ 0 form η 2(n + m) 2n n, 2m u = v = 0, = 1, p = pa + 1. (1) v = δr ∑ δ r v (z, t), These formulas describe the motion of the liquid in n, m ≥ 0 a cylindrical jet in a coordinate system tied to the liq- ( ) , uid. This solution is not valid in the near field of a jet, p Ð p = δ2 n + m r2n pn 2m(z, t), where viscous relaxation of the velocity profile of the a ∑ nozzle flow takes place. According to some estimates n, m ≥ 0 (e.g., see [6]), the near-field length is small compared to then equations for the coefficients of these expansions the distance at which the capillary breakup of the jet can be obtained by substituting the series into the equa- occurs. For this reason, almost all studies of the capil- tions of fluid mechanics and boundary conditions and lary stability of liquid jets are focused on the analysis equating the coefficients of like powers of r and δ. In of the particular solution written out above with respect [32], equations of this kind were used to analyze the to small perturbations. stability of the flow described by (1) in the linear approximation with respect to the initial perturbation amplitude. At the initial moment, a sinusoidal perturba- ONE-DIMENSIONAL EQUATIONS tion, e.g., the jet-surface perturbation OF THE DYNAMICS OF A CAPILLARY JET 2 Calculation of nonlinear interactions between per- η = 1 + εcos(kz) + O(ε ) at t = 0, turbations in a viscous liquid jet is associated with cer- ε tain difficulties. When a solution is sought in the form was introduced, where is a small parameter and k is of a series in powers of a small parameter (perturbation the wavenumber. amplitude), it turns out that explicit analytical solutions When the analysis is restricted to the leading terms of the equations describing the liquid motion in the jet of asymptotic expansions, i.e., to the equations for η(0) cannot be obtained in the second approximation with and u0, 0, a perturbation with an arbitrary wavenumber respect to the small parameter. However, one can con- is found to be unstable. This is inconsistent with Ray- struct an approximate solution based on the assumption leigh’s exact results, which show that only the perturba- that the length scale of the longitudinal (axial) profiles tions with k < 1 are unstable. In the second approxima- of flow variables is much greater than the length scale tion with respect to δ, only long-wavelength perturba- of their cross-sectional profiles. Approximate one- tions are unstable. If the wavenumber k is greater than dimensional equations that are valid under this condi- a certain value depending on Re, the flow is unstable. tion have been considered by many authors. In [32], a The approximate expression for the rate of perturbation technique was developed for constructing equations of growth agrees with Rayleigh’s results up to terms of this type that are valid to an arbitrary order with respect order k2. to the small parameter defined as the ratio of transverse to longitudinal length scale of the flow. Actually, these In this paper, a somewhat different approach is equations are quite accurate when the perturbation applied. Consider the new variables wavelength is sufficiently long; i.e., the corresponding (0) 2 (2) 4 φ = η + δ η + O(δ ), wavenumber is small. Here, the derivation of the , , approximate equations is briefly outlined in order to w = u0 0 + δ2u0 2 + O(δ4). introduce the notation and assumptions that underlie the theoretical analysis. The equations for the coefficients η(0), η(2), u0, 0, and When the equations of fluid mechanics and bound- u0, 2 in the expansions were written out in [32]. Multi- ary conditions are written in a dimensionless form, dif- plying the equations for η(2) and u0, 2 by δ2; summing ferent reference lengths should be introduced for the the resulting relations with the equations for η(0) and

TECHNICAL PHYSICS Vol. 45 No. 8 2000 990 CHESNOKOV u0, 0, respectively; and dropping the terms of order δ4, equations and initial conditions for the first and second one obtains the equations for φ and w: approximations. In the linear approximation in ε, φ 1 1 φ φ 1φ 1 [φ3 φ φ2 1t + --w1z + -----w1zzzz = 0, (6) t + zw + -- wz + ----- wzz + 10 z wzz 2 16 2 16 (2) (φ φ2 φ 2φ) ] 3 1 + 6 zz + 3 z wz = 0, w = ------w + --w + φ + φ , (7) 1t Re 1zz 8 1zzzz 1z 1zzz 3 2 2 w + ww = --(φ w w + 2φφ w ) φ = cos(kz), (8) t z 8 z zz z z 1 φ φ 1t = 0 at t = 0. (9) 3 2 zwz 1φ2 5φφ + ------wzz + ------+ -- wzzzz + -- zwzzz φ Re φ 8 4 The equations for 2 and w2 are 3 (3) 1 1 1 5 2 5 1φ φ φ φ  φφ φ  φφ φ φ z  2t + --w2z + -----w2zzz + 1zw1 + -- 1w1z + 2 z + -- z  wzz +  zzz + -- z zz + ----φ--- wz 2 16 2 2 2 2 (10) φ φ φ 1 ( φ φ φ ) 3 + ----- 3 1w1zzz + 10 1zw1z + 6 1zzw1z = 0, + ----z + φ + ------z-----z--z. 16 φ2 zzz 4 φ 3 3 1 w + w w = --w w + ------w + --w + 2φ w These equations are obtained without introducing 2t 1 1z 8 1z 1zz Re 2zz 8 2zzzz 1z 1z different reference lengths for r and z; i.e., δ ≡ 1 here. This approach is advantageous in that the necessary 1 5 + --φ w + --φ w + 2φ w + φ w (11) algebra is somewhat simplified. The presentation below 4 1 1zzzz 4 1z 1zz 1zz 1zz 1zzz 1z shows that, in contrast to [32], the boundary that sepa- 3 rates the stable and unstable perturbations calculated in + φ + φ Ð 2φ φ + --φ φ . the linear approximation with respect to ε is identical 2z 2zzz 1 1z 4 1z 1zz with that predicted by Rayleigh’s theory; i.e., the flow is unstable when k < 1. The initial conditions for these equations are 1 φ = Ð--, (12) 2 4 APPROXIMATE EQUATIONS φ FOR PERTURBATION AMPLITUDES: THE FIRST 2t = 0 at t = 0. (13) APPROXIMATION A solution to equations (6) and (7) subject to initial Assume that the jet surface is perturbed at the initial conditions (8) and (9) can be sought in the form moment t = 0, while the flow velocity remains unper- φ = h (t)cosh(ikz), w = i f (t)sinh(ikz). turbed. Then, equations (2) and (3) must be supple- 1 1 1 1 mented with the following initial conditions: Here, i is the imaginary unit. Define 3 ε2 k k φ = 1 + εcos(kz) Ð ---- + O(ε4), (4) A(k) = -- Ð -----. 4 2 16 For h and f , one obtains the following expressions: φ 1 1 t = 0 at t = 0. (5) ω t ω t h (t) = α e 1 + β e 2 , Here, ε is a small parameter (initial perturbation ampli- 1 0 0 ω ω 1 1t 2t tude). A derivation of (4) can be found in [3]. It is based f (t) = ---(α ω e + β ω e ). on the condition that the jet volume is conserved. 1 A 0 1 0 2 A solution to the formulated problem is sought in the ω ω following form: Here, 1(k) and 2(k) are the roots of the quadratic equation 2 3 φ = 1 + εφ + ε φ + O(ε ), 2 6kA(k) 2 1 2 ω + ------ω Ð k(1 Ð k )A(k) = 0, (14) Re w = εw + ε2w + O(ε3). 1 2 α β and the coefficients 0 and 0 are calculated as Substituting these expansions into equations (2) and ω ω (3) and the initial conditions and collecting the coeffi- α 2 β 1 0 = ω------ω-----, 0 = ω------ω-----. cients of like powers of ε on both sides, one obtains the 2 Ð 1 1 Ð 2

TECHNICAL PHYSICS Vol. 45 No. 8 2000 NONLINEAR DEVELOPMENT OF CAPILLARY WAVES 991

In the linear approximation, the exact solution to the ( ) 3 2 problem of capillary stability of a viscous liquid jet was B1 = kA 2k 1 + --k  ; obtained by Rayleigh. It can be used to find a relation 8 for calculating the perturbation growth rate ω. As k3 6k2 k2 A(2k) shown in [32], this relation can be written as B = 3 Ð ------; B = ------1 Ð ------. 2 A(k) 3 Re   A(k) 2 4 16 2 F(k) k (2F(k) Ð 1) 2k ω ------+ ω------+ ------(F(k) Ð F(k )) 2 Re 2 1 The expressions for g2, h2, and f2 are Re (15) 1 2 1 2 2 = --k (1 Ð k ), g2 = Ð--h1 , 2 4 2ω t 2ω t (ω + ω )t ω t ω t 2 1 2 1 2 3 4 2 ω h2 = c1e + c2e + c3e + c4e + c5e , where k1 = k + Re; F(k) = kI0(k)/I1(k); and I0(k) and I1(k) are the first- and second-order modified Bessel 2 α B 2ω t functions of the first kind, respectively. 1 ω  0 2 1 f 2 = -----(------) 12c1 Ð ------ e Representing F(k) as a series in powers of k and A 2k 2 retaining only the leading two terms, one obtains 2 β B 2ω t ω  0 2 2 2 + 22c2 Ð ------ e k 4 2 F(k) = 2 + ---- + O(k ). 4 α β (ω ω )  B  1 + 2 t + (ω + ω ) c Ð ----0------0------2 e By using this relation, the sum of the second and 1 2  3 2  third terms on the left-hand side of (15) is transformed into ω ω ω 3t ω 4t --+ 3c4e + 4c5e . k2(2F(k) Ð 1) 2k4 ω------+ ------(F(k) Ð F(k )) Re 2 1 ω ω ω ω Re Here, 3 = 1(2k); 4 = 2(2k); and the coefficients ci 2 (i = 1, …, 5) are calculated as 3k 6 = ------ω + O(k ). Re α 2  ω 2  c = -----0- B ------1------Ð 1 + B ω 2 + B ω , 1 Ω 1 2 2 1 3 1 On the other hand, 1 2A (k)  ( ) I1 k ( ) ( 5) ------= A k + O k . 2  2  I (k) β ω 2 0 c = -----0- B ------2------Ð 1 + B ω + B ω , 2 Ω 1 2 2 2 3 2 Thus, equation (4) for small k can be derived from 2 2A (k)  (15) by neglecting terms of order k6. Calculations show α β  ω ω  that the exact and approximate solutions are in good c = ----0------0 2B ------1------2--- Ð 1 3 Ω 1 2  agreement, even at relatively large k. For example, at 3 2A (k) Re = 10 and k = 0.9, the exact and approximate values of ω are 0.17547 and 0.17555, respectively. 1 2 + --B (ω + ω ) + B (ω + ω ) , 2 2 1 2 3 1 2

SECOND APPROXIMATION (2ω Ð ω )c + (2ω Ð ω )c + (ω + ω Ð ω )c c = ------1------4------1------2------4------2------1------2------4------3-, In the second approximation, a solution to equations 4 ω Ð ω (10) and (11) subject to initial conditions (12) and (13) 4 3 can be sought in the form (2ω Ð ω )c + (2ω Ð ω )c + (ω + ω Ð ω )c c = ------1------3------1------2------3------2------1------2------3--l------3-. φ ( ) ( ) ( ) 5 ω ω 2 = h2 t cosh 2ikz + g2 t , 3 Ð 4 ( ) ( ) w2 = i f 2 t sinh 2ikz . The second approximation of the exact solution (i.e., the solution obtained without approximating the To write formulas for h2, g2, and f2 in a more com- governing equations) is not known for a viscous liquid. pact form, the following notation is introduced: However, the results obtained can be compared with the ω 2 exact solution for an inviscid liquid. In this case, 2 = Ω = 4ω Ð 2(ω Ð ω )ω + ω ω , i = 1, 2; −ω ω ω i i 3 4 i 3 4 1, 4 = Ð 3 and the first two terms in the expansions of ω in powers of k are identical with the exact solu- Ω (ω ω )2 (ω ω )(ω ω ) ω ω i 3 = 1 + 2 Ð 3 + 4 1 + 2 + 3 4; tion, so that the exact and approximate solutions differ

TECHNICAL PHYSICS Vol. 45 No. 8 2000 992 CHESNOKOV by a quantity on the order of k6. The leading two terms terms. Nevertheless, the derived formulas can be used in the expansion of c3 in powers of k are given by to obtain realistic estimates for the quantities of interest here. It is well known that the time to breakup of a liq- 1 63 2 4 c = -- + -----k + O(k ). uid jet into droplets can be quite accurately predicted 3 2 32 not only by the weakly nonlinear theory, but also by the linear theory. The geometry of the jet at the moment of A comparison with the corresponding term obtained its breakup can be quite accurately calculated by means in [5] shows that this expansion agrees with the exact of perturbation theory as well. This is demonstrated by one up to terms of order k2. However, in the expressions Ω comparing the results of a calculation of the size of the for i (i = 1, 2), which have the form main and satellite droplets depending on the wavenumber with experimental results [8]. Ω = 4ω 2 Ð ω 2 i i 3 Figure 1 shows that, at each value of the Reynolds in the approximation considered, the leading terms in number, the time to jet breakup first decreases as the the expansions of the first and second terms in powers wavenumber is increased and then reaches a minimum of k are mutually canceled. For this reason, only the and increases. As expected, the effect of the Reynolds leading terms turn out to be correct both in the expan- number increases with wavenumber. This is explained Ω by the increase in the spatial derivatives of fluid veloc- sions of i and in the expansions of ci (i = 1, 2). None- theless, the results of calculations of the jet based on the ity toward shorter wavelengths, which enhances the exact and approximate relations are in good agreement. effect of viscosity on the flow behavior. The location of the minimum of the time to jet breakup plotted versus wavenumber shifts leftwards with increasing Re. EFFECTS OF WAVENUMBER AND REYNOLDS A similar trend is predicted by the linear theory. Calcu- NUMBER ON THE DEVELOPMENT lations (not represented in this figure) show that the OF CAPILLARY WAVES time to jet breakup strongly depends on the initial perturbation amplitude. The graphs shown here are plotted The relations obtained above can be used to exam- for the initial perturbation amplitude ε = 0.01. ine the effect of viscosity on the evolution of perturbations. The strongest effect of fluid viscosity is on the Figure 2 shows that the Reynolds-number depen- duration of the interval from the initial moment of per- dence of the time to breakup is weak in a wide range of turbation to the moment when the liquid jet breaks up Re. Only at relatively low Re does the time to breakup into droplets. To find this quantity, the following steeply increase. In this domain, the instability devel- method was applied. With the Reynolds number and the ops very slowly and can be described by the results initial perturbation amplitude held constant, the coordi- obtained here even better. nates of points of the liquid jet surface were calculated The jet geometry at times close to the moment of at various times. The moment at which the radial coor- breakup into droplets strongly depends on the wave- dinate of one of the points vanishes was assumed to be number. The jet radius plotted versus longitudinal coor- the desired quantity. It should be pointed out here that, dinate for long-wavelength perturbations has a second generally, the formulas obtained above are not applica- maximum, starting from a certain moment (see Fig. 3). ble at such moments. At the moment of jet breakup, the This phenomenon is not observed for short-wavelength neglected terms in the expansions in powers of the perturbations (Fig. 4). The appearance of the second small parameter are of the same order as the retained maximum is an indication of satellite-droplet formation

t t 60 80

50 70 60 40 50 40 30 3 2 30 1 0.2 0.4 0.6 0.8 k 2 4 6 8 10 Re Fig. 1. Time to jet breakup versus wavenumber for various Fig. 2. Time to jet breakup versus Reynolds number for values of the Reynolds number: Re = (1) ∞; (2) 10; (3) 5. k = 0.4.

TECHNICAL PHYSICS Vol. 45 No. 8 2000 NONLINEAR DEVELOPMENT OF CAPILLARY WAVES 993 η η

2.0 1.50 1.25 1.5 1.00 1.0 0.75 1 1 2 2 0.5 3 0.50 3 4 4 5 0.25 5 2.5 5.0 7.5 10.0 12.5 15.0 z 2 4 6 8 z

Fig. 3. Jet shape at Re = 10 and k = 0.4 for t = (1) 18, (2) 19, Fig. 4. Jet shape at Re = 10 and k = 0.75 for t = (1) 14, (2) 15, (3) 20, (4) 21, and (5) 21.9305. (3) 16, (4) 17, and (5) 18.149.

η η

2.0 2.0

1.5 1.5

1.0 1.0 1 5 1 0.5 4 0.5 2 4 2 3 3 2.5 5.0 7.5 10.0 12.5 15.0 z 1 2 3 4 5 6 ζ

Fig. 5. Jet shape at the moment of breakup for k = 0.4 and Fig. 6. Jet shape at the moment of breakup for Re = 10 and Re = (1) 100, (2) 5, (3) 3, and (4) 0.3. k = (1) 0.1, (2) 0.3, (3) 0.45, (4) 0.525, and (5) 0.8. between the main droplets. As the Reynolds number is length. Note that the results presented here are consis- increased, the magnitude of the second maximum in the tent with the numerical results reported in [15]. graph of jet radius versus longitudinal coordinate at the moment of jet breakup decreases (Fig. 5), and the point REFERENCES where the jet radius vanishes shifts toward the midpoint 1. Monodispersion of Substances: Principles and Applica- of the segment bounded by the two principal maxi- tion, Ed. by E. V. Ametistov, V. V. Blazhenkov, A. K. Go- mums. At a certain value of Re, the second maximum rodov, et al. (Énergoizdat, Moscow, 1991). disappears completely. This means that the size of sat- 2. D. B. Bogy, Annu. Rev. Fluid Mech. 11, 207 (1971). ellite droplets decreases with increasing Re, and the jet 3. M.-C. Yuen, J. Fluid Mech. 33, 151 (1968). breakup ceases at a certain Re. The ﬂuid viscosity sup- 4. D. P. Wang, J. Fluid Mech. 34, 299 (1968). presses the development of the second harmonic, and 5. A. H. Nayfeh, Phys. Fluids 13, 841 (1970). satellite droplets do not form. Figure 6 shows the 6. M. P. Markova and V. Ya. Shkadov, Izv. Akad. Nauk dependence of the jet radius on ζ = kz for various k. It SSSR, Mekh. Zhidk. Gaza, No. 3, 30 (1972). demonstrates that, as the wavenumber increases, the 7. T. Kakutani, Y. Inoue, and T. Kan, J. Phys. Soc. Jpn. 38, point where the jet radius vanishes at the moment of jet 529 (1974). breakup shifts toward the center, while the jet radius at 8. P. Lafrance, Phys. Fluids 18, 428 (1975). the midpoint of the segment bounded by the two prin- 9. K. C. Chaudhary and L. G. Redekopp, J. Fluid Mech. 96, cipal maximums decreases. Therefore, the size of satel- 257 (1980). lite droplets decreases with the perturbation wave- 10. J. E. Fromm, IBM J. Res. Dev. 28, 322 (1984).

TECHNICAL PHYSICS Vol. 45 No. 8 2000 994 CHESNOKOV

11. D. W. Bousﬁeld, R. Keunings, G. Marrucci, and 23. V. E. Epikhin and V. Ya. Shkadov, Izv. Akad. Nauk, M. M. Denn, J. Non-Newtonian Fluid Mech. 21 (1), 79 Mekh. Zhidk. Gaza, No. 2, 12 (1993). (1986). 24. V. V. Blazhenkov, A. F. Ginevskiœ, V. F. Gunbin, et al., 12. F. Shokoohi and H. G. Elrod, J. Comput. Phys. 71, 324 Izv. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 54 (1993). (1987). 25. N. A. Razumovskiœ, Zh. Tekh. Fiz. 63 (9), 26 (1993) 13. N. N. Mansour and T. S. Lundgren, Phys. Fluids A 2, [Tech. Phys. 38, 752 (1993)]. 1141 (1990). 26. J. Eggers and T. F. Dupont, J. Fluid Mech. 262, 205 14. V. V. Vladimirov and V. N. Gorshkov, Zh. Tekh. Fiz. 60 (1994). (11), 197 (1990) [Sov. Phys. Tech. Phys. 35, 1349 (1990)]. 27. J. Eggers, Phys. Rev. Lett. 71, 3458 (1993). 15. N. Ashgriz and F. Mashayek, J. Fluid Mech. 291, 163 28. J. Eggers, Phys. Fluids 7, 941 (1995). (1995). 29. M. P. Brenner, X. D. Shi, and S. R. Nagel, Phys. Rev. 16. H. Huynh, N. Ashgriz, and F. Mashaek, J. Fluid Mech. Lett. 73, 3391 (1994). 320, 185 (1996). 30. M. P. Brenner, J. R. Lister, and H. A. Stone, Phys. Fluids 17. H. C. Lee, IBM J. Res. Dev. 18, 364 (1974). 8, 2827 (1996). 18. W. Y. Pimbley, IBM J. Res. Dev. 20, 148 (1976). 31. F. J. García and A. Castellanos, Phys. Fluids 6, 2676 19. D. B. Bogy, Phys. Fluids 21, 190 (1978). (1994). 20. D. B. Bogy, Trans. ASME, Ser. E 45, 469 (1978). 32. S. E. Bechtel, C. D. Carlson, and M. G. Forest, Phys. 21. D. B. Bogy, IBM J. Res. Dev. 23, 87 (1979). Fluids 7, 2956 (1995). 22. V. V. Blazhenkov, A. F. Ginevskiœ, V. F. Gunbin, and A. S. Dmitriev, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 53 (1988). Translated by A. S. Betev

TECHNICAL PHYSICS Vol. 45 No. 8 2000

GASES AND LIQUIDS

On Capillary Motion of a Viscoelastic Liquid with a Charged Free Surface S. O. Shiryaeva and O. A. Grigor’ev Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia E-mail: [email protected] Received May 18, 1999; in ﬁnal form, October 14, 1999.

Abstract—The structure of the capillary–relaxation motion spectrum in a liquid with a charged free surface has been investigated taking into account the viscosity relaxation effect. On the basis of numerical analysis of the dispersion equation for the wave motion in a viscoelastic incompressible liquid, it is shown that for a given wave number the range of characteristic relaxation times in which relaxation-type wave motion exists is limited and expands with increasing wave number. The growth rate of instability of the charged liquid surface markedly depends on the characteristic relaxation time and increases with its growth; in liquids with elastic properties, the energy dissipation rate of capillary motion is enhanced. At a surface charge density that is supercritical for the onset of TonksÐFrenkel instability, both purely gravitational waves and waves of a relaxational nature exist. © 2000 MAIK “Nauka/Interperiodica”.

INTRODUCTION uous medium model on the basis of hydrodynamics equations for a viscous liquid (as was done in [1Ð2, The problem of capillary motion in a liquid exhibit- 4−7]), on the assumption that the viscoelastic proper- ing elastic properties is of interest in connection with ties of a liquid can be accounted for by introducing numerous scientific and technological applications, and complex viscosity through the Maxwell formula [9] for this reason it has often been a focus of attention (see, for example, [1Ð7] and the references therein). νν=()1ÐiωtÐ1, Nevertheless, many issues remain unclear because of 0 * the known awkwardness of the theoretical solution, the representing a Fourier image of the exponential varia- variety of possible ways of choosing nondimensional tion of the viscosity of a viscoelastic liquid with time. parameters in the boundary problem, and the large ν In this expression, 0 is a coefficient of kinematic vis- number and complicated structure of nondimensional cosity at zero frequency, ω is the complex frequency, t∗ parameters arising in the final solution [1–7]. is the characteristic time of viscosity relaxation, and i is The essence of the problem is that under a suffi- the imaginary unit. ciently brief (t ≤ 10Ð5 s) external impact even non-New- tonian liquids show elastic properties: they are first The analysis in [1–7] of the influence of the viscos- deformed elastically, and, after the impact ends, the ity relaxation effect on the relationships governing cap- residual shear stresses persists, relaxing in time t ~ illary motion in a liquid with a charged free surface is −5 mainly qualitative, as it was performed either by 10 s [8] and setting the liquid in motion. This effect asymptotic methods [1Ð5, 7] or by numerical methods manifests itself in capillary wave motion, because, at a µ capable of establishing only qualitative relationships wavelength of ~10 m, the wave period is already com- [5, 6]. In the latter case, ways are sought of first deriv- parable with the characteristic relaxation time of the ing a nondimensional dispersion equation for use in elastic stress. As shown in [1Ð7], taking account of the numerical calculations. In the numerical analysis in elastic properties of a liquid leads to an appreciable [5, 6], the frequency and the enhancement and damping complication of the capillary motion spectrum, result- rates of capillary motion in a liquid were converted ing in limitation of the capillary wave spectrum and using either the wave motion frequency in an ideal liq- an increase in the wave energy dissipation rate due to uid with a charged free surface or the characteristic the formation of high-frequency phonon-type wave damping rate of the capillary waves. In both cases, the motions. objective was to diminish the number of nondimen- In the consideration below, the entire analysis (in sional physical parameters describing the capillary contrast to [3], where the dispersion equation for the motion of the liquid in the system considered. As a vari- wave motion of a viscoelastic liquid was derived in the able argument of the sought-for complex frequencies, a framework of a microstructure approach and statistical complex parameter was used, which depended on the mechanics methods) will be performed using a contin- wave number, the capillary pressure, and the pressure

996 SHIRYAEVA, GRIGOR’EV exerted by an electrostatic field on the free surface of n(t—)U + t(n—)U = 0, (6) the liquid, that is, ultimately, on the physical characteristics of the liquid, namely, its density, capillary con- Ð P(U) + ρgζ + 2ρνn(n—)U Ð P (ζ) + Pσ(ζ) = 0, (7) stant, surface tension, coefficient of viscosity, and the E surface density of the electrical charge. From this anal- Φ ysis, it is hard to find out just how the characteristics of = 0, (8) capillary motion in a liquid depend on parameters such Φ as wave number k or surface density κ of the electrical z = b: = V. (9) charge. Our purpose is to overcome this limitation. In the above expressions, n and t are the normal and 1. The problem will be solved by calculating the tangential unit vectors relative to the free liquid surface, spectrum of capillary waves on the charged flat surface respectively; P(U) is the pressure within the liquid due of an ideally conductive liquid of infinite depth border- to the capillary motion of the liquid and is of the first ρ ing a vacuum. The liquid, which has density , viscos- order of smallness in ζ; and P (ζ) and Pσ(ζ) are addi- ν σ E ity , and surface tension coefficient , is exposed to tional pressures on the free liquid surface related to the gravitational field g and electrostatic field E0 (the sur- electrical forces and the surface tension forces, respec- face density of the charge induced by field E0 on the tively, both of them resulting from the perturbation unperturbed free surface of the liquid is connected to E0 ζ(x, t) = Aexp(ikx Ð iωt) of the equilibrium flat surface πκ by the well-known relation E0 = 4 . The strength of of the liquid caused by the capillary wave motion and ζ the electrostatic field E0 near the liquid surface is deter- of the first order of smallness in [11, 12], mined by the potential difference between the free sur- 2 face of the liquid with zero potential and a flat counter- ∂ ζ Ð1 2 Pσ(ζ) = Ðσ------, P (ζ) = 4πε κ kζ. (10) electrode positioned parallel to the unperturbed flat sur- ∂ 2 E face of the liquid at distance b and with a potential x Φ = V. As the liquid is viscous, to describe the flows in it, Let us choose a Cartesian coordinate system with we divide the velocity field U = U(r, t) into two compo- || the z-axis directed vertically upward, nz Ðg (nz is a unit nents in accordance with the Helmholtz theorem: the vector of coordinate z), and x-axis along the propaga- potential component (with the velocity field potential tion direction of a flat capillary wave (~exp(ikx Ð iωt)). ψ(r, t)) and the vortex component (described by the Let us also assume that the plane z = 0 coincides with stream function ϕ(r, t). Then the expression for pres- the unperturbed free surface of the liquid. The function sure field P(U) in the liquid can be written in the form ζ ζ ω (x, t) = 0exp(ikx Ð i t) describes a small perturbation of the equilibrium flat surface of the liquid caused by ∂ψ P(U) = Ð ρ------Ð ρgζ. (11) thermal capillary wave motion with a very small ampli- ∂t ζ σ 1/2 tude ( 0 ~ (kT/ ) ), where k is the Boltzmann constant, T is the absolute temperature, and U(r, t) is the velocity 2. We will be seeking a solution of problem (1)Ð(6) field of the liquid motion caused by the free surface in the following form [12]: perturbation ζ(x, t) and having the same order of small- ( , , ) ( ( ) ( )) ness [10]. Ux x z t = ikBexp Ðkz Ð lCexp Ðlz Let us derive the spectrum of the capillary waves in × exp(ikx Ð iωt), a liquid for the given conditions. The mathematical formulation of the problem includes the linearized U (x, z, t) = (Ð kBexp(Ðkz) + ikCexp(Ðlz)) NavierÐStokes equation for an incompressible liquid; z the incompressibility condition; the Laplace equation × exp(ikx Ð iωt), for the electrical field potential near the liquid surface; and the corresponding boundary conditions l2 = k2 Ð iωνÐ1. ∂U 1 ------= Ð --—P(U) + ν∆U + g, (1) Here A, B, and C are constants and l is the characteristic ∂t ρ linear scale of the spatial variation of the vortex component of the velocity field. div U = 0, (2) We follow the reasoning in [12] and add an extra ∆Φ = 0, E = Ð—Φ, (3) term to the dynamic boundary condition for the normal component of the stress tensor to take into account the z = Ð∞: U = 0, (4) electric field pressure. Then, expressing the viscosity as ν ν ω a function of frequency = 0/(1 Ð i t∗) in accordance ∂ξ(x, t) z = 0: Ð------+ U = 0, (5) with the Maxwell formula, we obtain the dispersion ∂t z relationship characterizing the capillary motion of a

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ON CAPILLARY MOTION OF A VISCOELASTIC LIQUID 997

ImY ImY 2.5 5.0 x 2.5 5.0 x 0 0 1 2 7 8 3 4 7 –5 5 8 6 –5 6 ReY ReY 4

5 7 5

1 7 0 2.5 5.0 x 0 1 2.5 5.0 x 7

–5 –5 7 4

Fig. 1. Nondimensional real ReY(X) and imaginary ImY(X) frequency components as functions of the nondimensional wave number X calculated for β = 1, W = 0, and τ = 0.11. Fig. 2. Same as in Fig. 1 for τ = 0.17 and W = 0. viscoelastic liquid with a charged free surface in the face stability of the liquid with respect to its own dimensional form charge; and the capillaryÐgravitational wave with the ν 2 ν2 4 wave number k at the liquid surface sustains instability  2ik  4 0 k iω(1 Ð iωt ) ω + ------0------+ ------1 Ð ------*---- at W > (k + kÐ1) [13, 14].  ( ω ) 2 2 1 Ð i t (1 Ð iωt ) νk * * 3. Variations of the real ReY = ReY(X) and imaginary ImY = ImY(X) components of nondimensional fre- k( ρ σ 2 πκ2 ) = ρ-- g + k Ð 4 k . quency Y on nondimensional wave number X calculated numerically using Eq. (12) for various character- Introducing the nondimensional variables istic relaxation times τ and the TonksÐFrenkel ω 2 parameter W are presented in Figs. 1Ð8. σ ρ a X = ka, a = / g, Y = ---ν------, 0 The numerical calculations show (Figs. 1, 2) that at W = 0 (with no charge at the free surface of the liquid) t ν σa 4πκ2a the viscosity relaxation effect leads to the emergence of τ = --*------0, β = ------, W = ------, 2 2 σ a ρν a periodic relaxation motion (branch 4), as well as two 0 aperiodic relaxation motions (branches 5, 6). Branches we obtain 1Ð3 correspond to the capillaryÐgravitational liquid motions arising in nonviscous liquids as well. When the 2 2 4 iY(1 Ð iYτ) [Y(1 Ð iYτ) + 2iX ] + 4X 1 Ð ------parameter τ increases to 0.17 (Fig. 2), the aperiodic X2 (12) motion 3 vanishes and curves 1 and 4 combine into a β [ 2 ]( τ)2 single capillaryÐrelaxation periodic motion 7; an aperi- = X 1 + X Ð WX 1 Ð iY , odic motion 8 also arises as curves 2 and 5 combine. where a is the capillary constant of the liquid; W is the A further increase in τ does not change the general TonksÐFrenkel parameter characterizing the free-sur- qualitative picture of the capillary motions; however, it

TECHNICAL PHYSICS Vol. 45 No. 8 2000 998 SHIRYAEVA, GRIGOR’EV

ImY ImY

2 1

2 2 0 2.5 5.0 x 1 0 3 2.5 5.0 x 4 1 3 4 –2 5 5 6 6 –1

ReY ReY

4 2.5 4 5

1 1 0 0 2.5 5.0 x 1 2.5 5.0 x 1

–5 4 –2.5 4

Fig. 3. Same as in Fig. 1 for τ = 0.3 and W = 3. Fig. 4. Same as in Fig. 1 for τ = 1 and W = 3. decreases the frequency of the wave motions described region X > 1 of the capillary waves; in addition, the ape- by branch 7 and the damping rates of all branches. riodically damping motions 3 and 5 vanish. The presence of an electrostatic ﬁeld near a liquid Thus, when the parameter W is subcritical for the surface (due to the surface charge) with subcritical onset of TonksÐFrenkel instability, gravitational, capil- strength in the sense of the TonksÐFrenkel instability lary, and relaxation waves exist; at supercritical param- (W = 1), while not affecting the frequency of the relax- eter W, the capillary waves are unstable in a limited ation oscillations, decreases the capillary wave fre- range of wave numbers. At the same time, the gravita- quencies, and at larger τ (τ = 0.3), branch 1 combines tional waves (k Ӷ 1), as well as the relaxational waves, with 4 and branch 2 with 5. remain and the spectra of the wave numbers character- At a supercritical strength of the electrostatic ﬁeld izing these wave motions are contiguous (Figs. 3Ð5). (W = 3, see Figs. 3, 4), the locus corresponding to curve Investigation of the dependence of the instability 1 describing capillaryÐgravitational waves shrinks con- growth rates (the section of branch 2 at ImY > 0 in siderably and curve 2 corresponding to capillaryÐgrav- Figs. 3Ð5) on nondimensional wave number X for vari- itational aperiodic motions penetrates into the half- ous values of characteristic relaxation time τ of the plane ImY > 0, which is evidence of the onset of aperi- elastic stresses in a liquid was carried out in [13] using odic instability of the surface perturbations in the cor- the same dimensionless parameters as in the present responding range of wave numbers k. study. It was found that this dependence is very strong Increasing the parameter τ does not change the and that the growth rate increase reaches 100% of the spectrum of the waves, which became unstable because growth rate value at W = 6 when τ varies from 0.11 to 1, of supercritical charge; however, it causes an increase in contrast to the conclusions based on qualitative in the growth rates of unstable motions and decreases investigations made in [5, 6], where even such a strong the damping rates of stable motions. effect was obscured by the complicated form of the For a highly supercritical TonksÐFrenkel parameter dimensionless parameters and arguments used. W = 6 (Fig. 5), expansion of the spectrum of the waves The numerical investigation of the growth rate of experiencing instability is observed (branch 2) both unstable capillary motions in a liquid as a function of into region X < 1 of the gravitational waves and into nondimensional characteristic time τ of the viscosity

TECHNICAL PHYSICS Vol. 45 No. 8 2000 ON CAPILLARY MOTION OF A VISCOELASTIC LIQUID 999

ImY ImY 2 3

1.5 2.5 2 1 0 5 10 x 6 1 –1.5 4

ReY 0 5 10 τ 7.5 Fig. 6. Plot of the nondimensional growth rate of the instability vs. the nondimensional characteristic time τ of the viscosity relaxation. (1) X = 1 and W = 3; (2) X = 1 and W = 6; 4 (3) X = 5 and W = 6.

1 0 ImY 1 5 10 x 2.5 4 2

–7.5 0 1 5 10 W Fig. 5. Same as in Fig. 1 for τ = 1 and W = 6. 3 4 relaxation carried out for various wave numbers X and 5 various W show (Fig. 6) that the larger W and X are, the –2.5 faster the growth rates increase with τ. 6 The dependence of the components ReY = ReY(W) ReY and ImY = ImY(W) of the nondimensional frequency Y on the TonksÐFrenkel parameter W calculated for X = 1 and different values of τ shows that taking into account 4 the viscosity relaxation effect with τ = 0.1, in compari- 1.0 1 son with the purely gravitationalÐcapillary wave motions described by branches 1Ð3, leads to the onset of three aperiodic relaxation motions 4Ð6, one of which 0 (4) becomes periodic when τ is increased to 0.3 (Fig. 7). 5 10 W A further increase in characteristic relaxation time τ leads to convergence of the relaxation (branch 4) and 1 4 capillaryÐgravitational (branch 1) periodic motions, as –1.0 well as of aperiodic motions 2 and 5, accompanied by the formation of compound capillaryÐrelaxation motions. The real ReY = ReY(τ) and imaginary ImY = ImY(τ) Fig. 7. Nondimensional real ReY(W) and imaginary ImY(W) frequency components as functions of the TonksÐFrenkel components of nondimensional frequency Y as a func- τ tion of nondimensional characteristic time τ of the vis- parameter W for X = 1 and = 0.3. cosity relaxation calculated for Y = 1 and W = 3 are shown in Fig. 8. Branch 1 describes variation of the Fig. 8, within a limited range of τ values. Branches 3Ð instability growth rate with τ: the growth rate slowly 5 describe the aperiodic relaxation motions in a liquid. increases with increasing τ. Branch 2 describes the Fig. 8 and numerical calculations carried out for other periodic relaxation motions existing, according to values of W show that the periodic relaxation motions

TECHNICAL PHYSICS Vol. 45 No. 8 2000 1000 SHIRYAEVA, GRIGOR’EV

ImY parameter W and does not depend on the characteristic 1 time of viscosity relaxation, although the wave number 5 10 τ of the most unstable wave at W = const slowly increases 0 with increasing τ. 3 4 2 At a ﬁxed value of the wave number k = const, the 6 5 range of characteristic relaxation times τ in which periodic solutions exist is limited; however, it expands with –2.5 increasing wave number k. For large enough characteristic relaxation times τ, 7 the branches of the capillaryÐgravitational and relaxation waves combine into a single compound motion existing at any wave number k, including those corresponding to purely gravitational waves (k 0). ReY As the characteristic relaxation time τ decreases, the damping rates of relaxation-type capillary motions in 2 the liquid rapidly increase. 0.5

REFERENCES 0 τ 1. Yu. A. Bykovskiœ, É. A. Manykin, P. P. Poluéktov, et al., 5 10 Zh. Tekh. Fiz. 46 (11), 2211 (1976) [Sov. Phys. Tech. Phys. 21, 1302 (1976)]. –0.5 2. G. A. Levachova, É. A. Manykin, and P. P. Poluéktov, 2 Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 17 (1985). 3. C. F. Tejero and M. Baus, Mol. Phys. 54, 1307 (1985). 4. I. N. Aliev, Magn. Gidrodin., No. 2, 78 (1987). Fig. 8. Nondimensional real ReY(τ) and imaginary ImY(τ) frequency components as functions of the nondimensional 5. O. A. Grigor’ev and S. O. Shiryaeva, Izv. Akad. Nauk, characteristic time τ of the viscosity relaxation for X = 1 and Mekh. Zhidk. Gaza, No. 1, 98 (1996). W = 3. 6. S. O. Shiryaeva, O. A. Grigor’ev, M. I. Munichev, and A. I. Grigor’ev, Zh. Tekh. Fiz. 66 (10), 47 (1996) [Tech. exist within a limited interval of τ values whose extent Phys. 41, 997 (1996)]. is inversely proportional to the TonksÐFrenkel parame- 7. S. I. Bastrukov and I. V. Molodtsova, Dokl. Akad. Nauk ter W. As the wavelength increases, the range of τ in 350, 321 (1996) [Phys. Dokl. 41, 388 (1996)]. which the relaxation oscillations exist expands and the 8. B. B. Badmaev, Ch. S. Laœdabon, B. V. Deryagin, and instability growth rate of the capillary waves increases. U. V. Bazaron, Dokl. Akad. Nauk SSSR 322, 307 (1992). CONCLUSIONS 9. J. Frenkel, Kinetic Theory of Liquids (Nauka, Leningrad, 1975; Clarendon, Oxford, 1946). In summarizing the above consideration of the viscosity relaxation inﬂuence on the relationships in the 10. Ya. I. Frenkel’, Zh. Éksp. Teor. Fiz. 6, 348 (1936). capillary motions of a liquid with a charged free sur- 11. L. D. Landau and E. M. Lifshitz, Course of Theoretical face, let us note the following: Physics, Vol. 8: Electrodynamics of Continuous Media The instability growth rate for the branch of capil- (Nauka, Moscow, 1982; Pergamon, New York, 1984). lary motions unstable with respect to the surface charge 12. V. G. Levich, Physicochemical Hydrodynamics (Fizmat- strongly depends on the characteristic time of viscosity giz, Moscow, 1959). relaxation and on the surface charge density. For sufﬁ- 13. S. O. Shiryaeva and O. A. Grigor’ev, Pis’ma Zh. Tekh. ciently large surface charge densities (parameter W), Fiz. 25 (2), 1 (1999) [Tech. Phys. Lett. 25, 41 (1999)]. the instability growth rate increases appreciably with 14. A. I. Grigor’ev, S. O. Shiryaeva, V. A. Koromyslov, and an increase in the nondimensional characteristic relax- τ D. F. Belonozhko, Zh. Tekh. Fiz. 67 (8), 27 (1997) [Tech. ation time . Phys. 42, 877 (1997)]. The range of wave numbers in which surface instability with respect to the surface charge is observed is governed only by the nondimensional TonksÐFrenkel Translated by N. Mende

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