X-ray acoustic resonance in real crystal

The influence of crystal defects and ultrasound absorption on X-ray diffraction under the condition of X-ray acoustic resonance are theoretically investigated. It is shown that the rocking curve and integral intensity of diffraction are very sensitive to defect's type and concentration. 1 INTRODUCTLON In the first papers [I, 21 devoted to the X-ray acoustic resonance the most fruitful1 results were obtained in case of dynarnical X-ray scattering on thick ideal crystal (pa > 6, y-linear absorption coefficient, a-thickness). It is well known that in that case a wave field corresponded to Bloch function (branch of dispersion surface, DS) with strong absorption can be neglected in comparison with Bloch function with weak absorption on the exit surface of the crystal. As result the rocking curve has not pendellosung oscillations (Bonmann effect). In vibrating crystals anomalous transparence of X-ray (Borrmann effect) is disrupted for plane waves with such deviation from Bragg's law that the condition of X-ray acoustic resonance can be satysfied : nKs = Re (6K) (1) where n = 1 , 2 ...number of the phonon taking part in the diffraction, 6Kdistance between two sheets of the DS along the phonon wave vector Ks direction. Rocking curve has minims for these deviations (see fig. 1). Convenient experimental scheme is shown on fig. 2. -2.0 -1 .O 0.0 1 .o p,deviation f rom Bragg's law Figure 1 Figure 2 Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1994229 JOURNAL DE PHYSIQUE IV 2 CRYSTAL DEFECTS For simplicity we restrict ourselves in what follows to considering symmetric X-ray Laue diffraction with a Bragg angle 0 and transversal ultrasonic standing wave. Taking account of multiple reflections of the ultrasonic wave from exit and entrance surfaces and absorption (with coefficient a) one can approximate displacement [3] in ultrasonic wave as : us = wcos (Ksz), w = wo (A1 coswt + iA2sinot) (2) where w ultrasound frequence, ttime, wamplitude of the atom displacement in ultrasonic wave, Ksis perpendicular to w, zcoordinate along normal to the entrance surface : We start by deriving a general expression for diffracted wave and then we estimate the influence of the defects. All distances are measured in what follows in units NTC, A-extinction length. We shall look for the solution of Takagi-Taupin's equations ( ~ h = xh-) in real crystal : aEo/aso-iexp(ihu)Eh = 0, aEh/ash-iexp(-ihu)Eg = 0 (3) where we took into account atom's displacement due to ultrasonic vibration us and due to defects Udef, U = Us + Udef. It is convenient [4] for considered crystal with low defect concentration to treate the coherent and diffuse part of the diffracted wave separately : In what follows we shall be interested in coherent intensity [4], EC , because diffuse part is relatively small. Omiting the diffuse part and substituting static Debye-Waller kctor I2 = instead of exp(st: ihudef) allows us to use solution for ideal vibrating crystal with renormed extinction length A' = NE. As result the form of the DS and the resonance condition are changed. In summary, the influence of the defects on the diffraction arises from ultrasound absorption (amplitude w and absorption coefficient a) and from renormed extinction length. For example, minima positions on rocking curve of defect crystal are determined by condition (1) and consequently by extinction length and E but their depths are determined by amplitude w and consequently by ultrasound absorption. 3 PERTURBATION THEORY We shall look for the solution in the form of an expansion : Eg (x,z) = jm-oo (ipxJsin0) Eg @,z) dp, g-0,h (5) where parameter p is correlated with deviation from Bragg's angle 60, p = 60 sin2B(Nh) and Eg (p,z) is sum of the two Bloch's waves with excitation amplitude Qm (z), m = 1,2. Em (r) = (Pm (2) ( E o ~ @) + Ehm (P) exp(-ihr)I exp(i 4 Qz) (6) where Q = 6412 = dp2 + (l+iq)2, Eom, Ehm are solutions of the Takagi-Taupin's equations for unvibrating crystal. One can derive the closed set of equations related on functions $2, $1 : vla$l/az = iKs (hw) sinKsz (exp(-iGQz)Q2-@1, v2a(P2/& = iKs (hw) sinKsz (exp(iGQz)$l-$2 where v1,2 = 2Q/(kQ-p), hdiffraction vector. According I.R. Entin [I] for case K = 6Q, one should omit a fast oscillating part in [6] and the set of the equations reduces to the simple form : 2via$i/az = Ks (hw) expli(-K~ + 6Q)z1$2, 2v2a$2/aZ = -Ks (hw) exp[-i(-Ks + 6Q)z]$l Using perturbation theory and neglecting the amplitude of strong absorbed wave $'2 in comparison with $1 I.R. Entin obtained : The developed perturbation theory was valid only in case of a rather small ultrasonic amplitude,(hw)2/Im(~~)<where Im(GQ) dynamical absorption coefficient {Im(GQ) = 0.15).4 EXACT SOLUTIONExact solution [5] consists of four terms corresponding to four sheets of the DS : $m j=12= C fmjexp[hmjz], m = 1,2(10)wherehmj = i/2 [(-1)mG + ( 1 j 4 ~ 2+~ 2 ~ ~ 2 ~ ( h w ) ( l+ iq)2/4~2](1 1)Additional spliting between new sheets of DS are proportional Bessel's function of the first orderJl(hw). We note here that formula (11) describes diffraction on crystal with high displacement amplitude(hw 2 1) and reduces to the well known result [I] in the limiting case when hw << 1. As result we candefine w with high accuracy.Theoretically and experimentally it is known that the integral intensity of the diffraction (ID) onvibrating crystal goes monotonically beyond kinematic limit when hw increases. On the basis of tHe exacttheory anomalous dependance (minimum) of the IID on ultrasonic amplitude was explained. It was foundthat minimum depth depends on static DW factor and increases with annealing [6].Summarising, we can say that X-ray resonance can be considered as a very accurate (= 10-6)method for investigation of the defect structure through extinction length determination and throughultrasonic absorption.REFERENCES1 I.R. Entin, Phys. Stat. Sol. (B) 2 (1978) 575.2 I.R. Entin, V.I. Khrupa, Acta Cryst. (A) 46 (1990) Sup.435.3 J.C. Miller, D.I. Bolef, J. Appl. Phys. 39 (1968) 4589.4 A.M. Polyakov, F.N. Chukhovskii, D.I. Piskunov, Jurnal experimentalnoi i teoreticheskoi fiziki.99 (1991) 589 (in Russian).5V.L. Nosik, Krystallographia 38 (1993) 44-54 (in Russian).6 V.I. Khmpa, Metallofizika 13 (1991) 49 (in Russian).