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Planar channeled relativistic electrons and positrons in the field of resonant hypersonic wave
Nuclear Instruments and Methods in Physics Research B 201 (2003) 25–33 www.elsevier.com/locate/nimb
Planar channeled relativistic electrons and positrons in the field of resonant hypersonic wave L.Sh. Grigoryan a,*, A.H. Mkrtchyan a, H.F. Khachatryan a, V.U. Tonoyan a, W. Wagner b a
b
Institute of Applied Problems in Physics, 25 Hr. Nersessian Str., 375014 Yerevan, Armenia Institute of Nuclear and Hadron Physics, Forschungszentrum Rossendorf e.V., PF 510119, D-01314 Dresden, Germany Received 30 January 2002; received in revised form 4 June 2002
Abstract The wave function of a planar channeled relativistic particle (electron, positron) in a single crystal excited by longitudinal hypersonic vibrations (HVs) is determined. The obtained expression is valid for periodic (not necessarily harmonic) HV of desired profile and single crystals with an arbitrary periodic continuous potential. A revised formula for the wave number of HV that exert resonance influence on the state of a channeled particle was deduced to allow for non-linear effects due to the influence of HV. 2002 Elsevier Science B.V. All rights reserved. PACS: 61.85.þp; 62.65.þk Keywords: Quantum theory of channeling; Acoustic vibrations
1. Introduction The phenomenon of intense emission of electromagnetic radiation at the channeling of relativistic particles in a single crystal was predicted by Kumakhov [1]. The experiments that followed confirmed the predictions of theory that are important for applications: high spectral-angular distribution of the intensity of emitted radiation, the monochromaticity, high degree of polarization etc.
(see [1b,2–8]). It is expected that the intensity of radiation from a channeled particle may be essentially increased at an excitation of lattice vibrations by HV [9–17]. The standing (harmonic) acoustic wave that is excited along the direction of relativistic particle motion (OZ-axis) modulates the density of a single crystal according to the law q ¼ q0 þ Dq cosðks zÞ sinðxs tÞ q0 þ Dq cos ks z;
* Corresponding author. Tel.: +374-2-240577; fax: +374-2281861. E-mail addresses: [email protected], levonshg@yandex. ru (L.Sh. Grigoryan).
ð1:1Þ
where Dq ¼ Dq sin xs t0 is the modulation depth and t0 is the instance of relativistic particle incidence on the single crystal (see (2.1.13)). The
0168-583X/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 3 0 6 - X
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acoustic wave indirectly affects the channeled particle because the potential energy U ð~ r; qÞ U ð~ r; q0 þ Dq cos ks zÞ
ð1:2Þ
of particle is a parametric function of q. The relativistic particle moves in the field U ð~ r; qÞ U ð~ r; q0 þ Dq cos x tÞ;
ð1:3Þ
which oscillates at effective cyclic frequency c ð1:4Þ x ¼ cks ¼ xs xs vs (z c t and vs is the velocity of acoustic wave propagation). According to (1.3), the acoustic wave may exert strong influence on the transverse motion of channeled particle when vs x ¼ x0 ) xs ¼ x0 ð xmax Þ: ð1:5Þ c Here x0 is the frequency of oscillations that the negatively/positively charged particle makes about the atomic plane (between atomic planes) in case of planar channeling, or the frequency of its rotation about the chain of atoms (between chains of atoms) in case of axial channeling. As the influence of acoustic wave is determined by the periodical variation of the parameter q, it is to have parametric resonance behavior. The swinging of oscillations under parametric resonance may occur (see [18,19]), when the parameter varies according to arbitrary periodical law with the frequency xðsÞ ¼
xmax ; s
ð1:6Þ
where s ¼ 1; 2; 3; . . . is an integer and xmax is the maximum resonant frequency. Similarly, in the presence of resonant acoustic wave a channeled particle (electron, positron) may swing and the radiation intensity would increase. So, in the field of acoustic wave the channeled electrons with energies 10 6 E 6 100 MeV may provide an exceptionally versatile and inexpensive source of monochromatic, narrow directed X-rays with numerous and promising applications. Based on (1.5) and (1.6), one can expect that resonant influence of acoustic wave on an electron with energy within 10 6 E 6 100 MeV will be at frequencies xs =2p 5 GHz (the hypersound). The limits of such an action are determined by non-
linear effects in the wave function of particle caused by the hypersound. The present work deals with determination of the wave function of a channeled relativistic particle (electron, positron) in a single crystal with arbitrary periodic continuous potential excited by longitudinal periodic HV of desired profile. Nonlinear effects of the action of HV on channeled particle state have been taken into account. The wave function of the planar channeled particle in HV-excited single crystal is calculated in Section 2. In Section 3 a formula similar to (1.6) for HV exerting the resonant influence on the state of channeled particle is derived, as well as an expression for the wave function of channeled particle under conditions of exact resonance is obtained. The summary of results of the present paper is given in the last section.
2. The wave function of a planar channeled particle 2.1. The basic equations The wave function of a channeled particle is determined by equation [1b,5–8] ðE2 2EU m2 c4 þ c2 h2 DÞwð~ rÞ ¼ 0;
ð2:1:1Þ
if the inequality jU ð~ rÞj E
ð2:1:2Þ
is taken into account and the sufficiently small influence of particle spin is neglected, E is the total energy of relativistic particle. At its parallel motion close to atomic planes the particle is subjected to the influence of a ‘‘continuous’’ potential U ðx; y; zÞ ¼ U0 ðxÞ;
ð2:1:3Þ
that is produced by averaging U ð~ rÞ over a unit cell in the channeling plain (an area with linear di). We suppose that mensions of an order of 1 A OX-axis of the Cartesian coordinate system x, y, z is directed along the sense of transverse oscillations of the particle. The results obtained using the continuous potential U0 ðxÞ describe experimental data sufficiently well (see [1b,5–8], and (3.1.8) as well).
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
Similar to U ð~ rÞ the continuous potential is also a periodic function, U0 ðx þ dx Þ ¼ U0 ðxÞ;
ð2:1:4Þ
dx is the distance between atomic planes forming a channel. In the field U0 ðxÞ the particle makes two independent motions. First, it is in a uniform rectilinear motion along the channeling direction (OZ-axis) that is described by the plane wave, 1 i pffiffiffiffi exp pz z ; ð2:1:5Þ h lz where lz is the thickness of single crystal along the axis OZ and pz is the corresponding momentum. And, second, the particle executes transverse oscillations described by some wave function SðxÞ. The total wave function 1 i ð0Þ pz z : w ðx; zÞ ¼ pffiffiffiffi SðxÞ exp ð2:1:6Þ h lz The equation for SðxÞ may be obtained by substitution of (2.1.6) into (2.1.1) and one arrives at a Schrodinger-type equation [1b,5–8] h2 d 2 þ U ðxÞ SðxÞ ¼ eSðxÞ ð2:1:7Þ 0 2M dx2 with relativistic mass M ¼ E=c2 and the energy of transverse motion in the following form: 1 2 E c2 ðpz2 þ m2 c2 Þ e¼ 2E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) E ffi c pz2 þ m2 c2 þ e:
ð2:1:8Þ
In virtue of periodicity of U0 ðxÞ, the solutions of Eq. (2.1.7) are orthonormal Bloch functions Skl ðxÞ ¼ 1kl ðxÞ expðikxÞ;
ð2:1:9Þ
with 1kl ðx þ dx Þ ¼ 1kl ðxÞ and Z Sk0 l0 ðxÞSkl ðxÞ dx ¼ dk0 k dl0 l :
27
and l is a subscript for numeration of energy bands ekl . Here for convenience we avail of the scheme of reduced zones, because in the strongcoupling limit the energy bands of particle grow narrower (k ! 0) with following transition to discrete energy levels numbered by l subscript. Now consider the case when a longitudinal HV is excited along the channeling direction, i.e. along OZ-axis, owing to which the distance dz between identical nuclei along OZ-axis is modulated according to some law 1 dz ¼ d0z þ fþ ðks z xs tÞ þ f ðks z þ xs tÞ;
ð2:1:12Þ
where f are periodic functions describing the profiles of hypersonic waves propagating in the positive (þ) and negative ( ) directions of OZaxis, d0z is a corresponding distance in the absence of HV, ks ¼ 2p=ks and xs are the wave number and cyclic frequency of HV. In case of channeling 1 xs lz 2p c
ð2:1:13Þ
and therefore, the phase ks z xs t does not practically change [9–17] (the adiabatic approximation), and dz ¼ d0z þ gðks zÞ;
ð2:1:14Þ
where g ¼ fþ ðks z xs t0 Þ þ f ðks z þ xs t0 Þ
ð2:1:15Þ
is a periodic function meeting the conditions Z ks gðz þ ks Þ ¼ gðzÞ; gðzÞ dz ¼ 0 ð2:1:16Þ 0
(the second condition may be always approached by means of corresponding renormalization of d0z ). The potential energy of particle is a function of the parameter dz , U ð~ r; dz Þ ¼ U ð~ r; d0z Þ þ U ð1Þ ð~ rÞgðzÞ þ . . . ;
ð2:1:17Þ
where ð2:1:10Þ
In (2.1.9) hk is the quasi-momentum of a particle from the first Brillouin zone p p 6k 6 ð rÞ ð2:1:11Þ dx dx
U ð1Þ ð~ rÞ ¼
oU ð~ r; d0z Þ od0z
ð2:1:18Þ
1 In case of standing wave the density of a single crystal is modulated according to the law (1.1).
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describes the response of system to the variation of dz . After averaging over an elementary cell in the channeling plane U ðx; y; z; dz Þ ¼ U ðx; zÞ;
ð2:1:19Þ
where U ðx; zÞ ffi U0 ðxÞ þ U1 ðxÞgðzÞ
ð2:1:20Þ
and U1 ðxÞ ¼ U ð1Þ ðx; y; zÞ:
ð2:1:21Þ
Eq. (2.1.20) is obtained taking into account the relation gðzÞ ¼ gðzÞ as the averaging is over an elementary cell, the linear dimensions of which are much less than ks . After the substitution of (2.1.20) in (2.1.1) one has ½E2 2EðU0 þ U1 gÞ m2 c4 þ c2 h2 Dwð~ rÞ ffi 0: ð2:1:22Þ This equation describes the quantum state of channeled particle in the field of hypersonic wave. To simplify the calculations one would have specify the functions U0 ðxÞ, U1 ðxÞ and gðzÞ. But we shall not do that to make the best of an opportunity to model the real potential of a single crystal and real (not necessarily harmonic) lattice vibrations. The product U1 ðxÞgðzÞ in (2.1.22) indicates that the variables x and z are not separated any more. For this reason (2.1.6) is not a solution of (2.1.22). Nevertheless, one can expand the wave function wð~ rÞ in a series in orthogonal functions (2.1.9), X wðx; zÞ ¼ Ckl ðzÞSkl ðxÞ: ð2:1:23Þ kl
The coefficients Ckl are functions of z and have to be determined. If HV are cut out, then we shall have only one summand in (2.1.23), 1 i ð0Þ pz z Ckl ¼ dk0 k dl0 l pffiffiffiffi exp ð2:1:24Þ h lz and (2.1.23) passes into (2.1.6). After HV are cutin again, the summands with l 6¼ l0 will appear, in this case the appreciable contribution being made only by summands with jl l0 j 6 lm ;
ð2:1:25Þ
where lm is some (probably large) number. Below we shall assume that everywhere the summation is carried out with a finite number of summands that meet the condition (2.1.25). By substituting (2.1.23) into (2.1.22) and taking account of the orthogonality condition (see (2.1.10)), we obtain the following system of differential equations with periodic coefficients: h2 d2 E2 m2 c4 þ ekl Ckl ðzÞ 2M dz2 2E X þ gðzÞ all0 ðkÞCkl0 ðzÞ ffi 0; ð2:1:26Þ l0
for jkj < r (for the sake of brevity the case of k ¼ r is not considered), Z all0 ðkÞ ¼ 1kl ðxÞU1 ðxÞ1kl0 ðxÞ dx: ð2:1:27Þ Of course, the set (2.1.26) is to be supplemented with conditions of continuity of wave function and of its gradient on both the surfaces z ¼ 0 and z ¼ lz of the single crystal. Note that the solutions of Eq. (2.1.7) are determined with accuracy to a constant phase /, and, therefore, the substitution Skl ðxÞ ! S kl ðxÞ ¼ Skl ðxÞ expði/Þ
ð2:1:28Þ
is admissible. We shall try to eliminate this ambiguity. This aim in view the wave function 2 of particle prior to the incidence on crystal (z < 0) will be taken in the form wðx; z < 0Þ expðxpx0 þ zpz0 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 E ¼ c m2 c2 þ ðpx0 Þ þ ðpz0 Þ and a complex number Z i 0 xpx dx ¼ akl expðibkl Þ Skl ðxÞ exp h
ð2:1:29Þ
ð2:1:30Þ
with modulus akl and phase bkl will be considered. After substitution of (2.1.28) we have Z i 0 xpx dx ¼ akl exp½iðbkl /Þ: S kl ðxÞ exp h ð2:1:31Þ
2 To be correct, the part of wave function describing the motion of particle in the positive direction of OZ-axis.
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
To fix the phase it suffices to put the imaginary part of (2.1.31) equal to zero. As a result / ¼ bkl
ð2:1:32Þ
(in what follows we shall omit the line in S kl ). 2.2. Hypersonic super lattice Now continue the determination of Ckl and introduce a function uðzÞ defined as Ckl ¼
Ckl u Akl u: u
ð2:2:1Þ
If we substitute (2.2.1) into (2.1.26) subject to the requirement that the summands containing Akl cancel each other (this does not refer to summands containing the derivatives of Akl ). As a result we come to an equation h2 d2 þ a ðkÞgðzÞ uðzÞ ¼ vuðzÞ ð2:2:2Þ ll 2M dz2 (compare with (2.1.7)) with respect to u, v¼
E2 m2 c4 ekl 2E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
) E ¼ v þ ekl þ
2
ðv þ ekl Þ þ m2 c4 :
ð2:2:3Þ
When the amplitude of HV decreases (g ! 0), uðzÞ passes into (2.1.24) with the relation between ekl , pz and E, similar to that in (2.1.8). Eq. (2.2.2) is a Schrodinger-type equation for a particle with mass M in one-dimensional lattice with a basis translational vector ks~ nz and a periodic potential U ðzÞ ¼ all ðkÞgðzÞ;
ð2:2:4Þ
i.e. for particle in a lattice excited by HV. The energy spectrum of particle inside such a ‘‘super lattice’’ has a band structure (we omit relevant subscripts), and the wave function has a form of Bloch function, ukl ðzÞ ¼ nkl ðzÞ expðizKkl Þ; nkl ðz þ ks Þ ¼ nkl ðzÞ;
29
(2.2.2) and (2.2.3) is made in subscripts k and l). Unlike the case of transverse oscillations here it is convenient to use the scheme of expanded zones, when different energy bands vkl are located in Kkl space in different Brillouin zones. It is connected with the fact that the particle motion along OZaxis is semi-classical. Really, the condition of quasi-classical motion dg 3 ð2:2:6Þ pkl ðzÞ hM all ; dz where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cpkl ¼ c 2M½vkl all gðzÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ E2 m2 c4 2Eðvkl þ all gÞ is transformed as dg E2 ch all dz
ð2:2:7Þ
ð2:2:8Þ
and satisfied with high accuracy. So, for the wave propagating along positive (þ) or negative ( ) direction of OZ-axis Z 1 i z ukl pffiffiffiffiffiffiffiffiffiffiffiffi exp pkl ðzÞ dz ð2:2:9Þ h 0 pkl ðzÞ in the first approximation in h. Comparing (2.2.5) with (2.2.9), we come to Z ks 1 hKkl ffi pkl ðzÞ dz ð2:2:10Þ ks 0 and
Z z 1 i nkl ðzÞ pffiffiffiffiffiffiffiffiffiffiffiffi exp ð pkl hKkl Þ dz : h 0 pkl ðzÞ ð2:2:11Þ After substitution of (2.2.1) in (2.1.26) and use of (2.2.2) one comes to the following equation for Akl : ! 0 X u ukl0 h2 kl A00kl þ 2A0kl all0 Akl0 ffi0 þ gðzÞ 2M ukl ukl l0 6¼l
ð2:2:5Þ
ð2:2:12Þ
with the quasi-momentum hKkl (the allowance for the dependence on all and ekl followed from
(the prime at the function means a derivation with respect to z).
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2.3. Quickly convergent perturbation theory
b2 þ þ Q bn þ b ¼Q b1 þ Q Q
in powers of D. Substituting (2.3.11) into (2.3.6) and equating the quantities of the same order of smallness, we have
Now introduce the quantities Pkl ih ( Wll0
u0kl ; ukl
all0 g 0
ukl0 ukl
at l0 6¼ l; at l0 ¼ l;
the column 0 1 .. B . C C A B @ Akl A; .. .
ð2:3:1Þ
ð2:3:2Þ
as well as the matrices b kWll0 k; W
Pb kdll0 Pkl k:
ð2:3:3Þ
By making use of these definitions one can rewrite Eq. (2.2.12) in a compact form, h2 d 2 ih b d b A ¼ 0: P þ W ð2:3:4Þ 2M dz2 M dz If we make a substitution Z Z i b ðzÞ dz Að0Þ; AðzÞ exp Q h 0
ð2:3:12Þ
Since for energies of particle (positron, electron) E P 10 MeV, the parameter D is very small, and, consequently, the series in (2.3.11) is to converge quickly. Thus, the wave function of the channeled particle is X wðx; zÞ ¼ 1kl ðxÞnkl ðzÞ exp½iðxk þ zKkl Þ kl
X
i exp h
Z
z
b dz Q
0
Akl0 ð0Þ: ll0
ð2:3:13Þ
ð2:3:5Þ
1 b2 1 b b ih b 0 b ¼ 0: Q þ PQ Q þW ð2:3:6Þ 2M M 2M b ¼ 0, and therefore In the absence of HV Q
Similar expressions for special forms of functions U0 ðxÞ, U1 ðxÞ and gðzÞ have been derived in b ¼Q b 1. [13,15,17] in the approximation of Q
3. Resonance
ð2:3:7Þ
where j Fb j stands for the greatest modulo element of matrix Fb . This inequality follows from pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:3:8Þ j Pb j ffi max Pkl E2 =c2 m2 c2 (see (2.3.1) and (2.2.7), (2.2.9)), b j maxfjall0 gj; l 6¼ l0 g jW
b 1 ¼ M Pb 1 W b; Q b0 Q b 2 Þ; b 2 ¼ 1 Pb 1 ðih Q Q 1 1 2 b 3 ¼ 1 Pb 1 ðih Q b0 Q b 1Q b 2Q b2 Q b 1 Þ; Q 2 2 .. .
l0
we come to a matrix equation
b j jW b j cj Pb j E; cj Q
ð2:3:11Þ
3.1. The condition of resonance According to (2.2.5) (2.3.1) and (2.3.12) 8 i < h all0 ðkÞMFll0 ðk; zÞ b 1 Þ 0 ¼ exp½izðKkl0 Kkl Þ at l0 6¼ l; ðQ ll : 0 at l0 ¼ l;
ð2:3:9Þ
and due to the smallness of the parameter jU0 j jall0 gj c hks ; ; D max 1 ð2:3:10Þ l;l0 E E E for a relativistic particle. Having in view the estimates in Eq. (2.3.7) one can expand
ð3:1:1Þ where gðzÞnkl0 ðzÞ 0 expð izKkl Þ½nkl ðzÞexpizKkl X ¼ Fll0 ðk; nÞ expðinks zÞ
Fll0 ðk; zÞ ¼
n
ð3:1:2Þ
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
is a periodic function of z, and n ¼ 0; 1; 2; . . . As a result X b 1 Þ 0 ¼ i all0 ðkÞM Fll0 ðk; nÞ ðQ ll h n exp½izðnks þ Kkl0 Kkl Þ
at l0 6¼ l: ð3:1:3Þ
Generally speaking nks þ Kkl0 Kkl 6¼ 0
ð3:1:4Þ
and exponents in summands quickly oscillate. For this reason Z z b 1 ðzÞ dz ffi 0 ð3:1:5Þ Q 0
for macroscopic values 8 9 < = 2p : z max ks ; : nks þ Kkl0 Kkl ; In this case AðzÞ is a constant, Z i Zb AðzÞ ffi exp Q 1 ðzÞ dz Að0Þ ffi Að0Þ h 0
ð3:1:6Þ
ð3:1:7Þ
ð3:1:9Þ
and ali l0i Fli l0i ðk; ni Þ 6¼ 0;
ð3:1:10Þ
where i ¼ 1; 2; 3; . . . In this case there is a constant b c in the equality part Q 1 bc þ Q b v ðzÞ b 1 ðzÞ ¼ Q Q 1 1
0
Thus, we have for the whole series of perturbation theory b v ðzÞ; b ðzÞ ¼ Q bc þ Q Q
ð3:1:14Þ
where bc ¼ Q bc þ Q bc þ ... Q 1 2 and Z Z
b v ðzÞ dz ffi 0: Q
ð3:1:15Þ
Here AðzÞ ffi
(see (2.1.16)). Eq. (3.1.7) is violated if n ks þ Kkl0i Kkli ¼ 0
In analogy to (3.1.5) the integral of a variable part b v is equal to zero, Q 1 Z Z b v ðzÞ dz ffi 0: ð3:1:13Þ Q 1
0
and attest to the absence of the influence of HV on the channeled particle state. This implies, in particular, that the potential (2.1.20) may be replaced by the value averaged over the spatial period of hypersonic wave, Z ks 1 U ðx; zÞ ¼ U ðx; zÞ dz ¼ U0 ðxÞ ð3:1:8Þ ks 0
i
31
ð3:1:11Þ
the elements of which are i h Mali l0i Fli l0i ðni Þ at l ¼ li ; l0 ¼ l0i ; c b ð Q 1 Þll0 ¼ 0 in other cases: ð3:1:12Þ
i bc exp z Q Að0Þ: h
ð3:1:16Þ
b c 6¼ 0, then the wave function of particle is If Q strongly modulated by a weak hypersonic wave (g 0) with the period much exceeding ks (similar to the case of parametric resonance [18,19]). According to (2.2.5), hKkl is the quasi-momentum and Gs ¼ ni ks is the reciprocal vector of onedimensional super lattice excited by hypersound. Hence, one can regard (3.1.9) as a diffraction condition, Kkli Kkl0i ¼ Gs
ð3:1:17Þ
on the super lattice excited by hypersound. The introduction of a hypersonic wave adds momentum to the problem and momentum conservation (3.1.17) leads directly to the resonance condition ks ¼
ki s
ð3:1:18Þ
that is similar to (1.6) at xmax vs ki . Here s ¼ jni j ¼ 1; 2; 3 . . . and 1 ki ¼ Kkli Kkl0i ekli ekl0i ch
ð3:1:19Þ
is the largest value of wave number determined by transition frequency between l0i and li zones of
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transverse motion (see (2.1.2), (2.1.16), (2.2.7), (2.2.10)). The expressions for ki were derived in [13,15,17] for special forms of functions U0 ðxÞ, U1 ðxÞ and gðzÞ. From (3.1.18) the following conclusion is made, that is important for practical applications: the wave number of HV may take on series i ¼ 1; 2; 3; . . . of values, in case of which the hypersonic vibrations of lattice have resonant influence on the channeled particle. In each series, along with the largest value ks ¼ ki there are also fractional values of ki : ki =2, ki =3, . . . The matrix element all0 is determined by the wave function of transverse oscillations of the particle in the channel (see (2.1.27)), and the periodic function gðzÞ is determined by HV of a single crystal (see (2.1.15)). Both these quantities b c as multipliers (see (3.1.12) and (3.1.2)), enter Q 1 while the latter, in its turn, appears in the index of exponent in (3.1.16). It is clear that in the absence of correlation between HV of the single crystal and transverse oscillations of particle in the channel b c ¼ 0 (after averaging over an arbitrary initial Q 1 phase of HV), and the hypersound will not influence the channeled particle state, even if the condition (3.1.18) is observed. So, under conditions of resonance the correlation between the phase of HV and the phase of transverse oscillations of the particle is a necessary condition. 3.2. Wave function The simplification of expression (2.3.13) is possible in two cases: far from the resonance and at an exact resonance. In the first case (see [13,15,17]) the perturbation theory in terms of HV amplitudes is applicable. Below we shall consider the second case that is of practical interest. If þ bc ¼ Q bc Q ð3:2:1Þ is an Hermitian matrix, then it can be diagonalized 0 1 .. . 0 B C b 1 Q b ¼B bcX C ð3:2:2Þ X ql @ A .. 0 .
b is some unitary transformation and ql are real (X eigenvalues determined by the hypersound vibrations power). As a result X i 0 0 Akl ðzÞ ¼ Xll ðkÞ exp zql Bkl0 ; ð3:2:3Þ h l0 where Bkl ¼
X
X 1 ll0 Akl0 ð0Þ
ð3:2:4Þ
l0
are new integration constants. Thus, in the case (3.2.1) one has X wðx; zÞ ¼ Bkl Ukl ðx; zÞ; ð3:2:5Þ kl
where the functions X i Xl0 l ðkÞ1kl0 ðxÞ Ukl ðx; zÞ ¼ exp zql h l0 nkl0 ðzÞ exp½iðxk þ zKkl0 Þ
ð3:2:6Þ
are independent solutions of Eq. (2.1.22). In conformity with (2.2.11) nkl ðzÞ 6¼ const;
ð3:2:7Þ
b 1 matrix in (3.1.1) is not and for this reason Q Hermitian, b þ 6¼ Q b 1: Q 1
ð3:2:8Þ
So, generally speaking, the condition (3.2.1) is not satisfied. In this case instead of (3.2.5) one can use a more general formula X 1kl ðxÞnkl ðzÞ exp½iðxk þ zKkl Þ wðx; zÞ ¼ kl
X l0
i bc exp z Q h
Akl0 ð0Þ;
ð3:2:9Þ
ll0
which is obtained by substitution of (3.1.16) into (2.3.13).
4. Conclusions (a) The wave number of HV may take on series of values i ¼ 1; 2; 3 . . ., for which the resonance influence of longitudinal HV on the wave func-
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
(b)
(c)
(d)
(e)
tion of planar channeled particle will be the case. In each of these series, along with the largest value ks ¼ ki there are also the fractional values of ki : ki =2, ki =3, . . . For resonant ks , • weak HV strongly influence the particle state: the spatial oscillations of the wave function are modulated with a period that much exceeds the wavelength ks of hypersound; • the correlation of the phase of transverse oscillations of particle with that of HV determines the degree of impact of hypersound on the wave function. The wave function of channeled particle in case of exact resonance is determined by expression (3.2.9). The items (a), (b), (c) are valid for single crystals with an arbitrary continuum potential U0 ðxÞ, which is excited by periodic (not necessarily harmonic) HV of desired profile gðzÞ. One may arrive at conclusions analogous to (a), (b) for the case of axial channeling.
Acknowledgements The authors are thankful to the referee for valuable comments. The work has been supported in part by grant # 1361 from Ministry of Education and Science of the Republic of Armenia and grant # A-100.2 from International Science and Technology Center.
References [1] (a) M.A. Kumakhov, Phys. Lett. A 57 (1976) 17; (b) M.A. Kumakhov, Radiation of Channeled Particles in Crystals, Energoatomizdat, Moscow, 1986 (in Russian). [2] V.V. Beloshitsky, F.F. Komarov, Phys. Rep. 93 (1982) 117.
33
[3] J.U. Andersen, E. Bonderup, R.H. Pantell, Ann. Rev. Nucl. Part. Sci. 33 (1983) 453. [4] B.L. Berman, S. Datz, in: A.W. Saenz, H. Uberall (Eds.), Coherent Radiation Sources, Springer-Verlag, Berlin, 1985, p. 165. [5] V.A. Bazylev, N.K. Zhevago, Radiation of Fast Particles in Matter and in External Fields, Nauka, Moscow, 1987 (in Russian). [6] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, High Energy Electromagnetic Processes in Aligned Single Crystals, Nauka, Novosibirsk, 1989 (in Russian). [7] M.A. Kumakhov, R. Wedell, Radiation of Relativistic Light Particles during Interaction with Single Crystals, Spektrum Akademischer Verlag, Heidelberg, 1991. [8] A.I. Akhiezer, N.F. ShulÕga, High Energy Electrodynamics in Matter, Gordon and Breach, Luxemburg, 1996. [9] H. Ikezi, Y.R. Lin-Liu, T. Ohkawa, Phys. Rev. B 30 (1984) 1567. [10] A.R. Mkrtchyan, R.H. Gasparyan, R.G. Gabrielyan, Phys. Lett. A 115 (1986) 410. [11] A.R. Mkrtchyan, R.H. Gasparyan, R.G. Gabrielyan, A.G. Mkrtchyan, Phys. Lett. A 126 (1988) 528. [12] A.R. Mkrtchyan et al., Conversion potential of Armenia and ISTC programs, Proceedings of the International Seminar, Yerevan, 2–7 October 2000, Pt. 1, p. 165. [13] L.Sh. Grigoryan, A.R. Mkrtchyan, B.V. Khachatryan, H.F. Khachatryan, H. Prade, W. Wagner, Radiat. Effects Defects Solids 152 (2000) 269. [14] L.Sh. Grigoryan, A.H. Mkrtchyan, H.F. Khachatryan, R.P. Vardapetian, H. Prade, W. Wagner, Radiat. Effects Defects Solids 152 (2000) 225. [15] L.Sh. Grigoryan, A.R. Mkrtchyan, H.F. Khachatryan, A.H. Mkrtchyan, H. Prade, W. Wagner, Radiat. Effects Defects Solids 153 (2000) 13. [16] L.Sh. Grigoryan, A.R. Mkrtchyan, A.H. Mkrtchyan, H.F. Khachatryan, W. Wagner, M.A. Piestrup, Radiat. Effects Defects Solids 153 (2001) 221; 289; 307. [17] (a) L.Sh. Grigoryan, A.R. Mkrtchyan, A.H. Mkrtchyan, H.F. Khachatryan, H. Prade, W. Wagner, M.A. Piestrup, Nucl. Instr. and Meth. B 173 (2001) 132; (b) L.Sh. Grigoryan, A.R. Mkrtchyan, A.H. Mkrtchyan, H.F. Khachatryan, H. Prade, W. Wagner, M.A. Piestrup, Nucl. Instr. and Meth. B 173 (2001) 184. [18] L.D. Landau, E.M. Lifshitz, Mechanics, Nauka, Moscow, 1973 (in Russian). [19] A.M. Prokhorov (Ed.), Encyclopedic Dictionary of Physics, Sovetskaya Encyclopedia, Moscow, 1983, p. 520 (in Russian).
Planar channeled relativistic electrons and positrons in the field of resonant hypersonic wave L.Sh. Grigoryan a,*, A.H. Mkrtchyan a, H.F. Khachatryan a, V.U. Tonoyan a, W. Wagner b a
b
Institute of Applied Problems in Physics, 25 Hr. Nersessian Str., 375014 Yerevan, Armenia Institute of Nuclear and Hadron Physics, Forschungszentrum Rossendorf e.V., PF 510119, D-01314 Dresden, Germany Received 30 January 2002; received in revised form 4 June 2002
Abstract The wave function of a planar channeled relativistic particle (electron, positron) in a single crystal excited by longitudinal hypersonic vibrations (HVs) is determined. The obtained expression is valid for periodic (not necessarily harmonic) HV of desired profile and single crystals with an arbitrary periodic continuous potential. A revised formula for the wave number of HV that exert resonance influence on the state of a channeled particle was deduced to allow for non-linear effects due to the influence of HV. 2002 Elsevier Science B.V. All rights reserved. PACS: 61.85.þp; 62.65.þk Keywords: Quantum theory of channeling; Acoustic vibrations
1. Introduction The phenomenon of intense emission of electromagnetic radiation at the channeling of relativistic particles in a single crystal was predicted by Kumakhov [1]. The experiments that followed confirmed the predictions of theory that are important for applications: high spectral-angular distribution of the intensity of emitted radiation, the monochromaticity, high degree of polarization etc.
(see [1b,2–8]). It is expected that the intensity of radiation from a channeled particle may be essentially increased at an excitation of lattice vibrations by HV [9–17]. The standing (harmonic) acoustic wave that is excited along the direction of relativistic particle motion (OZ-axis) modulates the density of a single crystal according to the law q ¼ q0 þ Dq cosðks zÞ sinðxs tÞ q0 þ Dq cos ks z;
* Corresponding author. Tel.: +374-2-240577; fax: +374-2281861. E-mail addresses: [email protected], levonshg@yandex. ru (L.Sh. Grigoryan).
ð1:1Þ
where Dq ¼ Dq sin xs t0 is the modulation depth and t0 is the instance of relativistic particle incidence on the single crystal (see (2.1.13)). The
0168-583X/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 3 0 6 - X
26
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
acoustic wave indirectly affects the channeled particle because the potential energy U ð~ r; qÞ U ð~ r; q0 þ Dq cos ks zÞ
ð1:2Þ
of particle is a parametric function of q. The relativistic particle moves in the field U ð~ r; qÞ U ð~ r; q0 þ Dq cos x tÞ;
ð1:3Þ
which oscillates at effective cyclic frequency c ð1:4Þ x ¼ cks ¼ xs xs vs (z c t and vs is the velocity of acoustic wave propagation). According to (1.3), the acoustic wave may exert strong influence on the transverse motion of channeled particle when vs x ¼ x0 ) xs ¼ x0 ð xmax Þ: ð1:5Þ c Here x0 is the frequency of oscillations that the negatively/positively charged particle makes about the atomic plane (between atomic planes) in case of planar channeling, or the frequency of its rotation about the chain of atoms (between chains of atoms) in case of axial channeling. As the influence of acoustic wave is determined by the periodical variation of the parameter q, it is to have parametric resonance behavior. The swinging of oscillations under parametric resonance may occur (see [18,19]), when the parameter varies according to arbitrary periodical law with the frequency xðsÞ ¼
xmax ; s
ð1:6Þ
where s ¼ 1; 2; 3; . . . is an integer and xmax is the maximum resonant frequency. Similarly, in the presence of resonant acoustic wave a channeled particle (electron, positron) may swing and the radiation intensity would increase. So, in the field of acoustic wave the channeled electrons with energies 10 6 E 6 100 MeV may provide an exceptionally versatile and inexpensive source of monochromatic, narrow directed X-rays with numerous and promising applications. Based on (1.5) and (1.6), one can expect that resonant influence of acoustic wave on an electron with energy within 10 6 E 6 100 MeV will be at frequencies xs =2p 5 GHz (the hypersound). The limits of such an action are determined by non-
linear effects in the wave function of particle caused by the hypersound. The present work deals with determination of the wave function of a channeled relativistic particle (electron, positron) in a single crystal with arbitrary periodic continuous potential excited by longitudinal periodic HV of desired profile. Nonlinear effects of the action of HV on channeled particle state have been taken into account. The wave function of the planar channeled particle in HV-excited single crystal is calculated in Section 2. In Section 3 a formula similar to (1.6) for HV exerting the resonant influence on the state of channeled particle is derived, as well as an expression for the wave function of channeled particle under conditions of exact resonance is obtained. The summary of results of the present paper is given in the last section.
2. The wave function of a planar channeled particle 2.1. The basic equations The wave function of a channeled particle is determined by equation [1b,5–8] ðE2 2EU m2 c4 þ c2 h2 DÞwð~ rÞ ¼ 0;
ð2:1:1Þ
if the inequality jU ð~ rÞj E
ð2:1:2Þ
is taken into account and the sufficiently small influence of particle spin is neglected, E is the total energy of relativistic particle. At its parallel motion close to atomic planes the particle is subjected to the influence of a ‘‘continuous’’ potential U ðx; y; zÞ ¼ U0 ðxÞ;
ð2:1:3Þ
that is produced by averaging U ð~ rÞ over a unit cell in the channeling plain (an area with linear di). We suppose that mensions of an order of 1 A OX-axis of the Cartesian coordinate system x, y, z is directed along the sense of transverse oscillations of the particle. The results obtained using the continuous potential U0 ðxÞ describe experimental data sufficiently well (see [1b,5–8], and (3.1.8) as well).
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
Similar to U ð~ rÞ the continuous potential is also a periodic function, U0 ðx þ dx Þ ¼ U0 ðxÞ;
ð2:1:4Þ
dx is the distance between atomic planes forming a channel. In the field U0 ðxÞ the particle makes two independent motions. First, it is in a uniform rectilinear motion along the channeling direction (OZ-axis) that is described by the plane wave, 1 i pffiffiffiffi exp pz z ; ð2:1:5Þ h lz where lz is the thickness of single crystal along the axis OZ and pz is the corresponding momentum. And, second, the particle executes transverse oscillations described by some wave function SðxÞ. The total wave function 1 i ð0Þ pz z : w ðx; zÞ ¼ pffiffiffiffi SðxÞ exp ð2:1:6Þ h lz The equation for SðxÞ may be obtained by substitution of (2.1.6) into (2.1.1) and one arrives at a Schrodinger-type equation [1b,5–8] h2 d 2 þ U ðxÞ SðxÞ ¼ eSðxÞ ð2:1:7Þ 0 2M dx2 with relativistic mass M ¼ E=c2 and the energy of transverse motion in the following form: 1 2 E c2 ðpz2 þ m2 c2 Þ e¼ 2E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) E ffi c pz2 þ m2 c2 þ e:
ð2:1:8Þ
In virtue of periodicity of U0 ðxÞ, the solutions of Eq. (2.1.7) are orthonormal Bloch functions Skl ðxÞ ¼ 1kl ðxÞ expðikxÞ;
ð2:1:9Þ
with 1kl ðx þ dx Þ ¼ 1kl ðxÞ and Z Sk0 l0 ðxÞSkl ðxÞ dx ¼ dk0 k dl0 l :
27
and l is a subscript for numeration of energy bands ekl . Here for convenience we avail of the scheme of reduced zones, because in the strongcoupling limit the energy bands of particle grow narrower (k ! 0) with following transition to discrete energy levels numbered by l subscript. Now consider the case when a longitudinal HV is excited along the channeling direction, i.e. along OZ-axis, owing to which the distance dz between identical nuclei along OZ-axis is modulated according to some law 1 dz ¼ d0z þ fþ ðks z xs tÞ þ f ðks z þ xs tÞ;
ð2:1:12Þ
where f are periodic functions describing the profiles of hypersonic waves propagating in the positive (þ) and negative ( ) directions of OZaxis, d0z is a corresponding distance in the absence of HV, ks ¼ 2p=ks and xs are the wave number and cyclic frequency of HV. In case of channeling 1 xs lz 2p c
ð2:1:13Þ
and therefore, the phase ks z xs t does not practically change [9–17] (the adiabatic approximation), and dz ¼ d0z þ gðks zÞ;
ð2:1:14Þ
where g ¼ fþ ðks z xs t0 Þ þ f ðks z þ xs t0 Þ
ð2:1:15Þ
is a periodic function meeting the conditions Z ks gðz þ ks Þ ¼ gðzÞ; gðzÞ dz ¼ 0 ð2:1:16Þ 0
(the second condition may be always approached by means of corresponding renormalization of d0z ). The potential energy of particle is a function of the parameter dz , U ð~ r; dz Þ ¼ U ð~ r; d0z Þ þ U ð1Þ ð~ rÞgðzÞ þ . . . ;
ð2:1:17Þ
where ð2:1:10Þ
In (2.1.9) hk is the quasi-momentum of a particle from the first Brillouin zone p p 6k 6 ð rÞ ð2:1:11Þ dx dx
U ð1Þ ð~ rÞ ¼
oU ð~ r; d0z Þ od0z
ð2:1:18Þ
1 In case of standing wave the density of a single crystal is modulated according to the law (1.1).
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describes the response of system to the variation of dz . After averaging over an elementary cell in the channeling plane U ðx; y; z; dz Þ ¼ U ðx; zÞ;
ð2:1:19Þ
where U ðx; zÞ ffi U0 ðxÞ þ U1 ðxÞgðzÞ
ð2:1:20Þ
and U1 ðxÞ ¼ U ð1Þ ðx; y; zÞ:
ð2:1:21Þ
Eq. (2.1.20) is obtained taking into account the relation gðzÞ ¼ gðzÞ as the averaging is over an elementary cell, the linear dimensions of which are much less than ks . After the substitution of (2.1.20) in (2.1.1) one has ½E2 2EðU0 þ U1 gÞ m2 c4 þ c2 h2 Dwð~ rÞ ffi 0: ð2:1:22Þ This equation describes the quantum state of channeled particle in the field of hypersonic wave. To simplify the calculations one would have specify the functions U0 ðxÞ, U1 ðxÞ and gðzÞ. But we shall not do that to make the best of an opportunity to model the real potential of a single crystal and real (not necessarily harmonic) lattice vibrations. The product U1 ðxÞgðzÞ in (2.1.22) indicates that the variables x and z are not separated any more. For this reason (2.1.6) is not a solution of (2.1.22). Nevertheless, one can expand the wave function wð~ rÞ in a series in orthogonal functions (2.1.9), X wðx; zÞ ¼ Ckl ðzÞSkl ðxÞ: ð2:1:23Þ kl
The coefficients Ckl are functions of z and have to be determined. If HV are cut out, then we shall have only one summand in (2.1.23), 1 i ð0Þ pz z Ckl ¼ dk0 k dl0 l pffiffiffiffi exp ð2:1:24Þ h lz and (2.1.23) passes into (2.1.6). After HV are cutin again, the summands with l 6¼ l0 will appear, in this case the appreciable contribution being made only by summands with jl l0 j 6 lm ;
ð2:1:25Þ
where lm is some (probably large) number. Below we shall assume that everywhere the summation is carried out with a finite number of summands that meet the condition (2.1.25). By substituting (2.1.23) into (2.1.22) and taking account of the orthogonality condition (see (2.1.10)), we obtain the following system of differential equations with periodic coefficients: h2 d2 E2 m2 c4 þ ekl Ckl ðzÞ 2M dz2 2E X þ gðzÞ all0 ðkÞCkl0 ðzÞ ffi 0; ð2:1:26Þ l0
for jkj < r (for the sake of brevity the case of k ¼ r is not considered), Z all0 ðkÞ ¼ 1kl ðxÞU1 ðxÞ1kl0 ðxÞ dx: ð2:1:27Þ Of course, the set (2.1.26) is to be supplemented with conditions of continuity of wave function and of its gradient on both the surfaces z ¼ 0 and z ¼ lz of the single crystal. Note that the solutions of Eq. (2.1.7) are determined with accuracy to a constant phase /, and, therefore, the substitution Skl ðxÞ ! S kl ðxÞ ¼ Skl ðxÞ expði/Þ
ð2:1:28Þ
is admissible. We shall try to eliminate this ambiguity. This aim in view the wave function 2 of particle prior to the incidence on crystal (z < 0) will be taken in the form wðx; z < 0Þ expðxpx0 þ zpz0 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 E ¼ c m2 c2 þ ðpx0 Þ þ ðpz0 Þ and a complex number Z i 0 xpx dx ¼ akl expðibkl Þ Skl ðxÞ exp h
ð2:1:29Þ
ð2:1:30Þ
with modulus akl and phase bkl will be considered. After substitution of (2.1.28) we have Z i 0 xpx dx ¼ akl exp½iðbkl /Þ: S kl ðxÞ exp h ð2:1:31Þ
2 To be correct, the part of wave function describing the motion of particle in the positive direction of OZ-axis.
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
To fix the phase it suffices to put the imaginary part of (2.1.31) equal to zero. As a result / ¼ bkl
ð2:1:32Þ
(in what follows we shall omit the line in S kl ). 2.2. Hypersonic super lattice Now continue the determination of Ckl and introduce a function uðzÞ defined as Ckl ¼
Ckl u Akl u: u
ð2:2:1Þ
If we substitute (2.2.1) into (2.1.26) subject to the requirement that the summands containing Akl cancel each other (this does not refer to summands containing the derivatives of Akl ). As a result we come to an equation h2 d2 þ a ðkÞgðzÞ uðzÞ ¼ vuðzÞ ð2:2:2Þ ll 2M dz2 (compare with (2.1.7)) with respect to u, v¼
E2 m2 c4 ekl 2E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
) E ¼ v þ ekl þ
2
ðv þ ekl Þ þ m2 c4 :
ð2:2:3Þ
When the amplitude of HV decreases (g ! 0), uðzÞ passes into (2.1.24) with the relation between ekl , pz and E, similar to that in (2.1.8). Eq. (2.2.2) is a Schrodinger-type equation for a particle with mass M in one-dimensional lattice with a basis translational vector ks~ nz and a periodic potential U ðzÞ ¼ all ðkÞgðzÞ;
ð2:2:4Þ
i.e. for particle in a lattice excited by HV. The energy spectrum of particle inside such a ‘‘super lattice’’ has a band structure (we omit relevant subscripts), and the wave function has a form of Bloch function, ukl ðzÞ ¼ nkl ðzÞ expðizKkl Þ; nkl ðz þ ks Þ ¼ nkl ðzÞ;
29
(2.2.2) and (2.2.3) is made in subscripts k and l). Unlike the case of transverse oscillations here it is convenient to use the scheme of expanded zones, when different energy bands vkl are located in Kkl space in different Brillouin zones. It is connected with the fact that the particle motion along OZaxis is semi-classical. Really, the condition of quasi-classical motion dg 3 ð2:2:6Þ pkl ðzÞ hM all ; dz where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cpkl ¼ c 2M½vkl all gðzÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ E2 m2 c4 2Eðvkl þ all gÞ is transformed as dg E2 ch all dz
ð2:2:7Þ
ð2:2:8Þ
and satisfied with high accuracy. So, for the wave propagating along positive (þ) or negative ( ) direction of OZ-axis Z 1 i z ukl pffiffiffiffiffiffiffiffiffiffiffiffi exp pkl ðzÞ dz ð2:2:9Þ h 0 pkl ðzÞ in the first approximation in h. Comparing (2.2.5) with (2.2.9), we come to Z ks 1 hKkl ffi pkl ðzÞ dz ð2:2:10Þ ks 0 and
Z z 1 i nkl ðzÞ pffiffiffiffiffiffiffiffiffiffiffiffi exp ð pkl hKkl Þ dz : h 0 pkl ðzÞ ð2:2:11Þ After substitution of (2.2.1) in (2.1.26) and use of (2.2.2) one comes to the following equation for Akl : ! 0 X u ukl0 h2 kl A00kl þ 2A0kl all0 Akl0 ffi0 þ gðzÞ 2M ukl ukl l0 6¼l
ð2:2:5Þ
ð2:2:12Þ
with the quasi-momentum hKkl (the allowance for the dependence on all and ekl followed from
(the prime at the function means a derivation with respect to z).
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L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
2.3. Quickly convergent perturbation theory
b2 þ þ Q bn þ b ¼Q b1 þ Q Q
in powers of D. Substituting (2.3.11) into (2.3.6) and equating the quantities of the same order of smallness, we have
Now introduce the quantities Pkl ih ( Wll0
u0kl ; ukl
all0 g 0
ukl0 ukl
at l0 6¼ l; at l0 ¼ l;
the column 0 1 .. B . C C A B @ Akl A; .. .
ð2:3:1Þ
ð2:3:2Þ
as well as the matrices b kWll0 k; W
Pb kdll0 Pkl k:
ð2:3:3Þ
By making use of these definitions one can rewrite Eq. (2.2.12) in a compact form, h2 d 2 ih b d b A ¼ 0: P þ W ð2:3:4Þ 2M dz2 M dz If we make a substitution Z Z i b ðzÞ dz Að0Þ; AðzÞ exp Q h 0
ð2:3:12Þ
Since for energies of particle (positron, electron) E P 10 MeV, the parameter D is very small, and, consequently, the series in (2.3.11) is to converge quickly. Thus, the wave function of the channeled particle is X wðx; zÞ ¼ 1kl ðxÞnkl ðzÞ exp½iðxk þ zKkl Þ kl
X
i exp h
Z
z
b dz Q
0
Akl0 ð0Þ: ll0
ð2:3:13Þ
ð2:3:5Þ
1 b2 1 b b ih b 0 b ¼ 0: Q þ PQ Q þW ð2:3:6Þ 2M M 2M b ¼ 0, and therefore In the absence of HV Q
Similar expressions for special forms of functions U0 ðxÞ, U1 ðxÞ and gðzÞ have been derived in b ¼Q b 1. [13,15,17] in the approximation of Q
3. Resonance
ð2:3:7Þ
where j Fb j stands for the greatest modulo element of matrix Fb . This inequality follows from pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:3:8Þ j Pb j ffi max Pkl E2 =c2 m2 c2 (see (2.3.1) and (2.2.7), (2.2.9)), b j maxfjall0 gj; l 6¼ l0 g jW
b 1 ¼ M Pb 1 W b; Q b0 Q b 2 Þ; b 2 ¼ 1 Pb 1 ðih Q Q 1 1 2 b 3 ¼ 1 Pb 1 ðih Q b0 Q b 1Q b 2Q b2 Q b 1 Þ; Q 2 2 .. .
l0
we come to a matrix equation
b j jW b j cj Pb j E; cj Q
ð2:3:11Þ
3.1. The condition of resonance According to (2.2.5) (2.3.1) and (2.3.12) 8 i < h all0 ðkÞMFll0 ðk; zÞ b 1 Þ 0 ¼ exp½izðKkl0 Kkl Þ at l0 6¼ l; ðQ ll : 0 at l0 ¼ l;
ð2:3:9Þ
and due to the smallness of the parameter jU0 j jall0 gj c hks ; ; D max 1 ð2:3:10Þ l;l0 E E E for a relativistic particle. Having in view the estimates in Eq. (2.3.7) one can expand
ð3:1:1Þ where gðzÞnkl0 ðzÞ 0 expð izKkl Þ½nkl ðzÞexpizKkl X ¼ Fll0 ðk; nÞ expðinks zÞ
Fll0 ðk; zÞ ¼
n
ð3:1:2Þ
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
is a periodic function of z, and n ¼ 0; 1; 2; . . . As a result X b 1 Þ 0 ¼ i all0 ðkÞM Fll0 ðk; nÞ ðQ ll h n exp½izðnks þ Kkl0 Kkl Þ
at l0 6¼ l: ð3:1:3Þ
Generally speaking nks þ Kkl0 Kkl 6¼ 0
ð3:1:4Þ
and exponents in summands quickly oscillate. For this reason Z z b 1 ðzÞ dz ffi 0 ð3:1:5Þ Q 0
for macroscopic values 8 9 < = 2p : z max ks ; : nks þ Kkl0 Kkl ; In this case AðzÞ is a constant, Z i Zb AðzÞ ffi exp Q 1 ðzÞ dz Að0Þ ffi Að0Þ h 0
ð3:1:6Þ
ð3:1:7Þ
ð3:1:9Þ
and ali l0i Fli l0i ðk; ni Þ 6¼ 0;
ð3:1:10Þ
where i ¼ 1; 2; 3; . . . In this case there is a constant b c in the equality part Q 1 bc þ Q b v ðzÞ b 1 ðzÞ ¼ Q Q 1 1
0
Thus, we have for the whole series of perturbation theory b v ðzÞ; b ðzÞ ¼ Q bc þ Q Q
ð3:1:14Þ
where bc ¼ Q bc þ Q bc þ ... Q 1 2 and Z Z
b v ðzÞ dz ffi 0: Q
ð3:1:15Þ
Here AðzÞ ffi
(see (2.1.16)). Eq. (3.1.7) is violated if n ks þ Kkl0i Kkli ¼ 0
In analogy to (3.1.5) the integral of a variable part b v is equal to zero, Q 1 Z Z b v ðzÞ dz ffi 0: ð3:1:13Þ Q 1
0
and attest to the absence of the influence of HV on the channeled particle state. This implies, in particular, that the potential (2.1.20) may be replaced by the value averaged over the spatial period of hypersonic wave, Z ks 1 U ðx; zÞ ¼ U ðx; zÞ dz ¼ U0 ðxÞ ð3:1:8Þ ks 0
i
31
ð3:1:11Þ
the elements of which are i h Mali l0i Fli l0i ðni Þ at l ¼ li ; l0 ¼ l0i ; c b ð Q 1 Þll0 ¼ 0 in other cases: ð3:1:12Þ
i bc exp z Q Að0Þ: h
ð3:1:16Þ
b c 6¼ 0, then the wave function of particle is If Q strongly modulated by a weak hypersonic wave (g 0) with the period much exceeding ks (similar to the case of parametric resonance [18,19]). According to (2.2.5), hKkl is the quasi-momentum and Gs ¼ ni ks is the reciprocal vector of onedimensional super lattice excited by hypersound. Hence, one can regard (3.1.9) as a diffraction condition, Kkli Kkl0i ¼ Gs
ð3:1:17Þ
on the super lattice excited by hypersound. The introduction of a hypersonic wave adds momentum to the problem and momentum conservation (3.1.17) leads directly to the resonance condition ks ¼
ki s
ð3:1:18Þ
that is similar to (1.6) at xmax vs ki . Here s ¼ jni j ¼ 1; 2; 3 . . . and 1 ki ¼ Kkli Kkl0i ekli ekl0i ch
ð3:1:19Þ
is the largest value of wave number determined by transition frequency between l0i and li zones of
32
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
transverse motion (see (2.1.2), (2.1.16), (2.2.7), (2.2.10)). The expressions for ki were derived in [13,15,17] for special forms of functions U0 ðxÞ, U1 ðxÞ and gðzÞ. From (3.1.18) the following conclusion is made, that is important for practical applications: the wave number of HV may take on series i ¼ 1; 2; 3; . . . of values, in case of which the hypersonic vibrations of lattice have resonant influence on the channeled particle. In each series, along with the largest value ks ¼ ki there are also fractional values of ki : ki =2, ki =3, . . . The matrix element all0 is determined by the wave function of transverse oscillations of the particle in the channel (see (2.1.27)), and the periodic function gðzÞ is determined by HV of a single crystal (see (2.1.15)). Both these quantities b c as multipliers (see (3.1.12) and (3.1.2)), enter Q 1 while the latter, in its turn, appears in the index of exponent in (3.1.16). It is clear that in the absence of correlation between HV of the single crystal and transverse oscillations of particle in the channel b c ¼ 0 (after averaging over an arbitrary initial Q 1 phase of HV), and the hypersound will not influence the channeled particle state, even if the condition (3.1.18) is observed. So, under conditions of resonance the correlation between the phase of HV and the phase of transverse oscillations of the particle is a necessary condition. 3.2. Wave function The simplification of expression (2.3.13) is possible in two cases: far from the resonance and at an exact resonance. In the first case (see [13,15,17]) the perturbation theory in terms of HV amplitudes is applicable. Below we shall consider the second case that is of practical interest. If þ bc ¼ Q bc Q ð3:2:1Þ is an Hermitian matrix, then it can be diagonalized 0 1 .. . 0 B C b 1 Q b ¼B bcX C ð3:2:2Þ X ql @ A .. 0 .
b is some unitary transformation and ql are real (X eigenvalues determined by the hypersound vibrations power). As a result X i 0 0 Akl ðzÞ ¼ Xll ðkÞ exp zql Bkl0 ; ð3:2:3Þ h l0 where Bkl ¼
X
X 1 ll0 Akl0 ð0Þ
ð3:2:4Þ
l0
are new integration constants. Thus, in the case (3.2.1) one has X wðx; zÞ ¼ Bkl Ukl ðx; zÞ; ð3:2:5Þ kl
where the functions X i Xl0 l ðkÞ1kl0 ðxÞ Ukl ðx; zÞ ¼ exp zql h l0 nkl0 ðzÞ exp½iðxk þ zKkl0 Þ
ð3:2:6Þ
are independent solutions of Eq. (2.1.22). In conformity with (2.2.11) nkl ðzÞ 6¼ const;
ð3:2:7Þ
b 1 matrix in (3.1.1) is not and for this reason Q Hermitian, b þ 6¼ Q b 1: Q 1
ð3:2:8Þ
So, generally speaking, the condition (3.2.1) is not satisfied. In this case instead of (3.2.5) one can use a more general formula X 1kl ðxÞnkl ðzÞ exp½iðxk þ zKkl Þ wðx; zÞ ¼ kl
X l0
i bc exp z Q h
Akl0 ð0Þ;
ð3:2:9Þ
ll0
which is obtained by substitution of (3.1.16) into (2.3.13).
4. Conclusions (a) The wave number of HV may take on series of values i ¼ 1; 2; 3 . . ., for which the resonance influence of longitudinal HV on the wave func-
L.Sh. Grigoryan et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 25–33
(b)
(c)
(d)
(e)
tion of planar channeled particle will be the case. In each of these series, along with the largest value ks ¼ ki there are also the fractional values of ki : ki =2, ki =3, . . . For resonant ks , • weak HV strongly influence the particle state: the spatial oscillations of the wave function are modulated with a period that much exceeds the wavelength ks of hypersound; • the correlation of the phase of transverse oscillations of particle with that of HV determines the degree of impact of hypersound on the wave function. The wave function of channeled particle in case of exact resonance is determined by expression (3.2.9). The items (a), (b), (c) are valid for single crystals with an arbitrary continuum potential U0 ðxÞ, which is excited by periodic (not necessarily harmonic) HV of desired profile gðzÞ. One may arrive at conclusions analogous to (a), (b) for the case of axial channeling.
Acknowledgements The authors are thankful to the referee for valuable comments. The work has been supported in part by grant # 1361 from Ministry of Education and Science of the Republic of Armenia and grant # A-100.2 from International Science and Technology Center.
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