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Statistical Theory of Dynamical Diffraction of Mössbauer Radiation V

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Statistical Theory of Dynamical Diffraction of Mössbauer Radiation V Crystallography Reports, Vol. 47, No. 1, 2002, pp. 1–16. Translated from Kristallografiya, Vol. 47, No. 1, 2002, pp. 7–23. Original Russian Text Copyright © 2002 by Nosik. DIFFRACTION AND SCATTERING OF X-RAY AND SYNCHROTRON RADIATION Statistical Theory of Dynamical Diffraction of Mössbauer Radiation V. L. Nosik Shubnikov Institute of Crystallography, Russian Academy of Sciences, LeninskiÏ pr. 59, Moscow, 117333 Russia Received July 27, 2000; in final form, April 4, 2001 Abstract—The statistical theory of dynamical diffraction of Mössbauer radiation providing the account for the mutual influence of coherent and elastic diffusion scattering has been considered. The sources of elastic and inelastic diffuse scattering for a hematite crystal are analyzed. The theoretical results are compared with the res- onance nuclear scattering data obtained in the experiments performed at the Photon Factory in Tsukuba, Japan. © 2002 MAIK “Nauka/Interperiodica”. INTRODUCTION phase transitions, diffusion, and relaxation) and its high Soon after the discovery of the Mössbauer effect in intensity open new possibilities for conducting experi- 1958, the process of diffraction of Mössbauer radiation ments on the diffuse scattering of Mössbauer radiation from crystals with resonance nuclei was studied in and require the construction of a corresponding theory. detail for both ideal single crystals [1, 2] and polycrys- An important notion for further consideration is tals [3]. The coherence of Mössbauer radiation as well coherence (in application to both the electromagnetic as the interference of the resonance and electron scat- field of the incident beam and the field inside the crys- tering were established. It was also shown that the tal) and an ensemble of atoms in the crystal. Because of effects of dynamical diffraction (the anomalous trans- the high monochromaticity of Mössbauer radiation, the mission in the Laue diffraction, the total external reflec- coherence length l of the train of electromagnetic waves tion in the Bragg diffraction, etc.) should be studied on generated by an individual nucleus in the emission of a perfect crystals purposefully enriched with a resonance quantum is determined by the lifetime of the excited isotope. Thus, the suppression of inelastic channels of state (τ ≈ ΓÐ1) and, for Co57, is equal in vacuum to l = nuclear reactions was studied on almost perfect iso- cτ = 30 m, i.e., has a value considerably exceeding the tope-enriched lead [4], iron [5], FeBO [6], and hema- coherence length for the X-ray radiation from conven- 3 Γ ≈ tite [7] crystals. tional sources ( x 1 eV) with the same wavelength µ Recently, the widespread use of powerful synchro- (lx = 0.3 m). The synchrotron radiation used in exper- tron-radiation (SR) sources considerably increased the iments has an intermediate coherence length deter- interest in studying time-delayed signals of nuclear mined mainly by the energy resolution of the mono- radiation. The short (0.2 ns) SR pulses separated by chromator used. considerable time intervals (1–2 ns) provide the unique In crystals, the radiation is scattered by a set of possibility of separating the “fast” (elastic) electron and atoms forming a coherent ensemble if all the nuclei the “slow” (resonance) nuclear Mössbauer-radiation have the same scattering amplitude and are arranged (MR) response of the crystal. strictly periodically. However, in actual fact, it is not the Moreover, the polarization characteristics and the case and one has to single out a certain coherently scat- high SR intensity allow unique experiments to be con- tering volume ξ, which can be either larger or smaller ducted for pure nuclear scattering (yttrium-iron garnets than the coherent volume of the electromagnetic field Γ Γ [8], iron borate [9], hematite [10]) and interference of (~ c). In the final analysis, c is determined by the electron and nuclear scattering in hematite [11]. At dielectric constant of the medium [17], which makes its present, the theory and practice of the use of coherent calculation under diffraction conditions even more nucleus scattering are studied quite well [12–16]. complicated. Any deviations from the strict periodicity Because of a very narrow width of nuclear levels, in the arrangement of identical atoms give rise to dif- the MR experiments require the use of specific crys- fuse scattering with the intensity also dependent on the tals–monochromators with a high energy resolution volume in which it occurs. necessary for diminishing the contribution from elec- Moreover, in Bragg (reflection) diffraction, the key tron scattering and the special protection from vibra- parameter can be the penetration depth of the field into tions. The high MR sensitivity to the magnetic structure the crystal, which coincides with the extinction length and various dynamical excitations (phonons, structural Λ if the radiation is incident at the Bragg angle, 1063-7745/02/4701-0001 $22.00 © 2002 MAIK “Nauka/Interperiodica” 2 NOSIK whereas far from this angle, it coincides with the Considering MR diffraction, one has to distinguish absorption length µ –1 . between elastic and inelastic scattering. In elastic scat- a tering, a quantum is scattered in a random way, but its It should be emphasized that the region in which dif- energy remains constant and it can experience reso- fuse waves are formed can also be finite. Thus, in the nance scattering and participate in dynamical diffrac- case of thermal diffuse scattering (TDS), the largest tion. In inelastic scattering, e.g., by phonons, a quantum បω ≈ –3 contribution comes from acoustic phonons having the acquires an additional energy ( s 10 eV), which maximum wavelength in the given crystal, so that, in excludes its participation in dynamical scattering. This this case, the whole crystal can be considered as a scat- consideration of the incoherent part of the diffraction tering volume. However, one of the most attractive con- field of Mössbauer radiation drastically differs from cepts of the microscopic theory of phase transitions consideration in the case of X-rays [19]. attribute these transitions to instability of some optical oscillations, whose excitation frequency decreases anomalously with the approach to the transitions tem- Elastic Incoherent Processes ω | |1/2 perature 0 = T Ð Tc [18]. The condensation of “soft” Spin incoherence. A nucleus of an isotope in the phonons makes the lattice rearrangement energetically ground state possesses a total angular momentum J = ប µ favorable and, in this case, the volume of the restruc- Ig, where Ig is the nuclear spin m = g nIg. The projec- ω –1 tion of the magnetic moment onto the chosen quantiza- tured region increases proportionally to 0 . tion axis can have 2Ig + 1 values In the final analysis, the relationship of the distances , ,,… at which the coherent and incoherent waves are formed mI = –Ig –1Ig + Ig. (1) determines the contribution of the coherent and diffuse scattering into the total diffraction intensity. Since the nuclear magneton ប At present, attempts are being made to construct a µ e n = --------- (2) consistent statistical theory of dynamical diffraction mnc which would describe the interaction between the µ coherently and diffusely scattered radiations. It was is a thousand times less than the Bohr magneton B, the established that the corrections for diffuse scattering in magnetic order in the system of nuclear spins can be the analysis of the structure and properties of real crys- attained either in strong magnetic fields or at very low tals can considerably distort the ideal diffraction pattern temperatures (0.1 K). Since the Mössbauer line is very by imposing a certain basic limit on the accuracy of the narrow even in a relatively weak field of the atomic diffraction experiment. A high coherence of Mössbauer moment, only the nuclei with definite moment projec- radiation would allow one to approach this limit, tions can experience nuclear resonance. Statistically, although this would require that very complicated the number of such nuclei equals experiments be conducted. ()() β ≈ ) wm = 1/Z exp– mI M 1/(2Ig + 1 , Below, we make an attempt to construct a theory of (3) statistical diffraction of Mössbauer radiation for a well- ()β β–1 known hematite crystal such that it would consistently Z ==∑ exp – mI M , kT. take into account incoherent elastic scattering (i.e., mI scattering occurring without a change in the frequency 57 of the scattered radiation) and its effect on the coherent Thus, for the Fe isotope (Ig = 1/2), only a half of component. We also analyze inelastic-scattering chan- the nuclei experience resonance coherent scattering. nels (phonons, magnons, and critical fluctuations), Isotopic incoherence. The natural content of the which can make a considerable contribution to the resonance isotope is low (2.2%). Similar to binary solid intensity of diffuse scattering. solutions [20], the replacement of Fe56 by Fe57 in a We deliberately limit our consideration to a hematite hematite crystal results in small displacements of crystal, because it allows us to compare our results with neighboring atoms and changes the energy of their the data of the well-known diffraction experiments interaction leaving the symmetry of the crystal lattice made on imperfect crystals with a complex magnetic unchanged. structure. The isotope-enriched crystals grown from melt [21] with the c-axis being normal to their surface had the thickness T = 0.4 mm and the surface area 5 × 10 mm2. SOURCES OF DIFFUSE SCATTERING The detailed X-ray study of the crystals [21, 22] Consider possible sources of diffuse scattering of showed that they were slightly bent (with the curvature Mössbauer radiation. Incoherent (diffuse) scattering radius 8–64 m) and had growth sectors (domains). can be defined as a process in which some information The statistically complete description of any system about the phase shift during scattering from individual suggests the knowledge of all the moments of the polar- nuclei is lost. izability distribution. We restrict our consideration only CRYSTALLOGRAPHY REPORTS Vol.
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