PHYSICAL REVIEW B VOLUME 57, NUMBER 17 1 MAY 1998-I Diffusion in a crystal lattice with nuclear resonant scattering of synchrotron radiation B. Sepiol Institut fu¨r Materialphysik der Universita¨t Wien, Strudlhofgasse 4, A-1090 Wien, Austria A. Meyer Physik-Department E13, Technische Universita¨t Mu¨nchen, D-85747 Garching, Germany and European Synchrotron Radiation Facility, F-38043 Grenoble, France G. Vogl Institut fu¨r Materialphysik der Universita¨t Wien, Strudlhofgasse 4, A-1090 Wien, Austria H. Franz Physik-Department E13, Technische Universita¨t Mu¨nchen, D-85747 Garching, Germany R. Ru¨ffer European Synchrotron Radiation Facility, F-38043 Grenoble, France Received 9 October 1997 We report on a method for probing the elementary diffusion jumps in crystalline lattices on an atomistic scale. The method makes use of synchrotron radiation coherently scattered in the forward direction after nuclear resonant excitation. The decay of forward-scattered radiation is faster ``diffusionally accelerated'' when atoms move on the time scale of the excited-state lifetime because of a loss of coherence. The accelera- tion of the decay rate differs for different crystal orientations relative to the beam; thus providing information not only about the rates but also about the directions of the diffusion jumps. As a first application we studied the diffusion of 57Fe in the intermetallic alloy Fe3Si parallel to the 111 and 113 crystal directions yielding the diffusion mechanism of iron and its diffusion coefficient. From a comparison with conventional quasielas- tic Mo¨ssbauer spectroscopy the advantages of the method are deduced. S0163-1829 98 07317-2 I. INTRODUCTION turbed scattering process. From this ``diffusionally acceler- ated decay,'' details on the diffusion process can be derived. After the original proposal by Singwi and Sjo¨lander on We will in the following use the abbreviation NFS for how to study diffusion in solids with Mo¨ssbauer nuclear forward scattering. spectroscopy1 and the experiments on crystalline solids by The choice of an appropriate system is decisive for a Knauer and Mullen2 that method has been applied with study that is intended to demonstrate the capacities and limi- growing success for studying the diffusion mechanism in tations of this method for investigating diffusion. It should solids on an atomistic scale.3 Since the method makes use of be a well-known material already thoroughly investigated quasielastic energy broadening in analogy to quasielastic with QMS and the tracer diffusion method. The effects to be neutron scattering QNS it is called quasielastic Mo¨ssbauer expected from diffusion should be of reasonable size under spectroscopy QMS . In the case of QNS, diffusion manifests easily manageable experimental conditions. itself through the energy broadening of the scattered inten- All these requirements are well fulfilled by the ordered sity; in the case of QMS, through the change in shape of the intermetallic alloy Fe3Si as argued in the following. a nuclear resonance line s . Fe3Si crystallizes in a simple structure cubic D03, consist- Here we present the first full report on a measurement of ing of three iron sites and one silicon site in a primitive cell diffusion in a single crystal using the new technique of and is perfectly ordered up to the melting point Tm 1500 nuclear resonant scattering of synchrotron radiation4,5 SR K . b Single crystals of Fe 3Si are stable during high- for a review, see Ref. 6 . A preliminary letter containing temperature measurements. They can be grown, oriented, cut part of the experimental results has been published earlier.7 and polished up to the required final thickness. c Fe3Si Recently nuclear resonant scattering of SR has been applied shows extremely fast diffusion of the iron atoms, the fastest to the study of dynamics around the glass transition.8 of all iron intermetallics with high iron content found up to The nuclear resonant scattering technique enables studies now. Thus diffusion phenomena can be observed at low tem- in the time domain, whereas QMS and QNS studies perform perature about 900 K , which reduces technical problems. in the energy domain. The general idea using nuclear reso- QMS Refs. 9,10 has proven that Fe atoms which occupy nant scattering of synchrotron radiation is that the collective sites on three sublattices Fig. 1 jump between nearest- coherent state during the scattering process is destroyed by neighbor iron sites on different sublattices remaining on each diffusion, which leads to a faster decay of the scattered in- sublattice for different residence times. Tracer diffusion tensity in the forward direction with respect to an undis- studies11 have confirmed the diffusivities determined from 0163-1829/98/57 17 /10433 7 /$15.00 57 10 433 © 1998 The American Physical Society 10 434 SEPIOL, MEYER, VOGL, FRANZ, AND RU¨FFER 57 range 17° to 17° degrees relative to the furnace axis. Temperature is controlled by a chromel-alumel thermo- couple with a relative accuracy of 1 K. The absolute tem- perature was calibrated via the Curie temperature of the sample known from a differential scanning calorimeter analysis . For the measurements at the synchrotron and the Mo¨ssbauer measurements, as well, the furnace was mounted on a goniometer head permitting a change in the orientation of the samples relative to the direction of the beam. The samples were encapsulated between two beryllium oxide disks, each one 0.9 mm thick, granting more than 40% transparency for the 14.4 keV radiation. The encapsulation was necessary to avoid inflection of the samples during the measurements at high temperatures. FIG. 1. Elementary cell of the D03 structure of Fe3Si. The iron atoms occupy the sublattices 1, 2 shaded circles , and black C. Measurement of nuclear forward scattering circles , the silicon atoms the sublattice open circles . of synchrotron radiation NFS... the QMS measurements and extended the determination of The experiments with nuclear forward scattering of SR diffusivities to a wider temperature range than accessible to were carried out at the nuclear resonance beamline of the QMS. ESRF for details see Ref. 12 . The storage ring operated in The aim of this paper is to explore the possibilities of 16-bunch mode providing short pulses of x rays duration using this new method of studying diffusion, and to study the 100 ps every 176 ns. The radiation from the undulator sensitivity limits of the method. In Sec. II sample preparation source, optimized for the 14.4 keV transition in iron, was and high-temperature measurements with NFS and with filtered by a double Si 111 reflection followed by mono- QMS are described. In Sec. III we deal first with the method chromatization in a nested high resolution mono- of calculation of NFS spectra in the presence of diffusion. chromator.12,13 The delayed events, resulting from NFS,14 We briefly describe in general form the calculation of the were counted by a fast avalanche photo diode APD self-correlation function and specify it for the D0 detector.15,16 Because of overload of the detector from the 3 structure. We then discuss the particular case of Fe prompt synchrotron pulse, data taken during the first 20 ns 3Si. In Sec. IV we interpret NFS spectra on the basis of the theory. In Sec. V we after each pulse have to be discarded.17 In principle, the du- account for the fact that due to the high resolution of the ration of this ``lost time'' is determined by the SR pulse NFS method, with respect to both geometry and time, it is length and the detector resolution ( 100 ps FWHM , but necessary to include relaxation due to jumps of iron atoms due to the detector overload the lost time is considerably between lattice sites with different isomer shifts. Section VI longer. contains conclusions and an outlook. Depending on crystal direction and temperature, the count rates were between 90 for the 113 crystal direction parallel II. EXPERIMENT to the synchrotron beam and 140 111 direction delayed counts per second at 827 K, and between 3 and 60 delayed A. Sample counts per second at 967 K. This resulted in measuring times Two single crystals of the Fe-Si alloy were grown by the between 5 and 60 min. The constant background of the APD Bridgman technique with the compositions a 75.5 at. % Fe, diode was 0.02 cps and could be neglected in the spectra 24.5 at. % Si and b 74 at. % Fe, 26 at. % Si, respectively. fitting. The ingots were 99.99% iron Johnson Matthey and 99.999% silicon Goodfellow . Sample a contained natural D. Mo¨ssbauer measurements iron concentration of 57Fe 2.1 at. % , sample b was en- In order to be able to recognize most directly the advan- riched 5.1 at. % 57Fe . Preliminary results on sample a tages and possible drawbacks of NFS compared to QMS, we have already been reported.7 measured with both techniques on identical samples. For the For sample a two slices of about 10 mm diameter were Mo¨ssbauer absorption measurements we used a 57Co in Rh cut by a wire saw with their surfaces parallel to the 113 source and the same furnace that was afterwards used for the plane. After cutting the samples were ground to final thick- synchrotron experiment. The high-temperature Mo¨ssbauer nesses of 24 m and 15 m, respectively. For sample b spectra for the sample a have been shown in Fig. 2 of Ref. one slice oriented with its surface parallel to the 335 plane 7. Within the resolution of the Mo¨ssbauer method the high- was produced final thickness 21 m . The thickness of the temperature spectra of sample b are identical with those of slices was constant within about 1 m. sample a . B. Furnace III. DIFFUSION: PRINCIPLES OF NUCLEAR RESONANT SCATTERING AND MO¨SSBAUER SPECTROSCOPY The vacuum furnace was produced by CIGNUS Cracow, Poland . It was originally designed for Mo¨ssbauer measure- A. General outline of the theory of quasielastic methods ments at temperatures up to about 1100 K. The distance be- for studying diffusion tween entrance and exit beryllium windows ( 15 mm is The original theory of Singwi and Sjo¨lander1 deals with only 40 mm. These dimensions enable measurements in the continuous diffusion, whereas Chudley and Elliott shortly 57 DIFFUSION IN A CRYSTAL LATTICE WITH NUCLEAR . . . 10 435 later described jump diffusion on an empty Bravais lattice,18 (Q, ) mirrors all dynamical effects on QMS, QNS, and i.e., an empty lattice with all sites equivalent. The action of NFS spectra: In QMS the real part of (Q, ) called S(Q, ) diffusion is contained in what Randl,19 following the termi- in earlier papers19 is the essential factor of the absorption nology of neutron scattering, calls the intermediate scattering cross section and of the emission probability and in QNS of function I(Q,t) named momentum-time correlation func- the scattering cross section. tion by Smirnov and Kohn20,21 and marked Fs(Q,t) by them . I(Q,t) is the Fourier transform from space into mo- B. Diffusion in Fe3Si mentum of the diffusional part of the space-time self- correlation function G(r,t), In the present paper we study diffusion in an ordered al- loy, namely Fe3Si, which crystallizes in cubic D03 structure Fig. 1 . The D03 superlattice is built up of four interpen- I Q,t dr exp iQr...G r,t , 1 etrating fcc sublattices with three of them two named , also called A and C in the literature and one or B) occu- where Q is the momentum transfer to the interacting radia- pied by iron atoms and the fourth ( or D) by silicon. The tion. In the case of diffusion G(r,t) is the solution of the sublattice can be divided into two parts with different sym- diffusion equation for the given system. metry of and nearest neighbors NN's . Iron atoms on For diffusion on non-Bravais lattices the intermediate -sublattice sites have four iron NN's on -sublattice sites, scattering function can be written in the following way:22,19 whereas iron atoms on -sublattice sites have eight iron NN's on sublattice sites. In earlier Mo¨ssbauer work9 it has I Q,t w been proven that a simple diffusion model with NN jumps of p Q exp p Q t/2 , 2 p iron atoms between and iron sublattice sites describes where diffusion in the stoichiometric composition. The jumps of p(Q)/2 is the pth negative eigenvalue of the jump matrix A and silicon atoms are much slower and decoupled from the iron jumps.11 For a stoichiometric D03 structure the jump matrix A Eq. w p 2 p Q cibi . 3 4 has the following form:9 i wp(Q) is the weight of the component in the measured spec- 1 2 E E* tra corresponding to the pth eigenvalue, with ci the probabil- A E* 1 0 . 7 ity of finding the atom on the ith sublattice and bp i the ith E 0 1 component of the pth eigenvector of the jump matrix. The 1 elements of the jump matrix A are given by Here is the jump rate of the iron atom from a site on one sublattice into a vacancy on a distinct NN site on the 1 1 sublattice and vice versa and E is a function of the structure A k i j n exp iQlij ij . 4 of the jump lattice, ji* ji k j ij Here each site of the ith sublattice is surrounded by n E cos Q i j sites xd cos Qyd cos Qzd of the jth sublattice with the kth site at a vector distance lkij . isin Qxd sin Qyd sin Qzd , 8 1 i j is defined as the jump rate from a site of symmetry i to any nearest-neighbor site of symmetry j. Generally, for a with Qx , Qy , and Qz the components of the vector of trans- nonequal occupation of different sublattices due to, e.g., lat- ferred momentum ( Q 7.3 Å 1 for 57Fe and d a/4, tice disorder, the matrix A is not Hermitian and must be where a is the Fe 3Si lattice constant (a 5.655 Å at room transformed by a similarity transformation into Hermitian temperature and 5.726 Å at 967 K . Because we assumed form.19 The elements of the jump matrix A specify the vari- only NN iron jumps the matrix elements A23 and A32 corre- ous allowed jumps for the diffusing atom and the accompa- sponding to jumps between both sublattices are zero. The nying jump rates. matrix says that in general there will be three components in The momentum-energy correlation function called the the spectra with three different eigenvalues p/2 for QMS universal resonance function by Smirnov and Kohn20,21 corresponding to three diffusionally broadened lines and (Q, ) is calculated by the Laplace transformation, corresponding weights.9 With Q parallel to special crystal directions, the number of components can be less than three: there are only two Q, 0 2 dt exp i t 0t/2 I Q,t , 5 components when Q is parallel to the 113 crystal direction 0 and even only one for the 111 direction. and further on from Eq. 2 , IV. DIFFUSION: EXPERIMENTAL RESULTS w AND THEIR INTERPRETATION Q, i p Q 0/2 p i 0 p Q .../2 , 6 As can be derived from what has been said before, in where for QMS and NFS, 0 is the energy width of the QMS diffusion leads to a simple broadening of one or more excited nuclear level and p(Q... is the energy width Lorentzian-shaped lines.19 The above-described way of de- FWHM caused by diffusion leading for QMS to ``diffu- termining diffusional line broadening was applied in Refs. 9, sional line broadening.'' 10, and 19 to QMS and QNS. 10 436 SEPIOL, MEYER, VOGL, FRANZ, AND RU¨FFER 57 On the basis of the same theory, by calculating the Fou- rier integral over (Q, ), Smirnov and Kohn20,21 have de- termined the time dependence of the amplitude of the electric field E(t,z) for NFS transmitted through the sample of thick- ness z. Physically speaking that means considering the inter- ference of all forward scattered components, d E t,z E0 z 2 exp i t exp 12L Q, , 9 where L 0fLMn z is the effective sample thickness with 0 the nuclear absorption cross section (2.56 10 22 m2), f LM the Lamb-Mo¨ssbauer factor, n the number of iron atoms per unit volume, and the 57Fe isotope abundance. The function E0(z) is determined by the intensity of the SR within the frequency band of the monochromator system re- duced by electronic absorption in the sample. The intensity of NFS is the square modulus of E(t,z). The essential result of the calculation is the following: When the time between diffusive jumps becomes compa- rable to or even shorter than the natural lifetime of the nucleus ( 0 141 ns , the coherence will be destroyed. This will lead to an accelerated decay rate of the coherently for- ward scattered intensity. We now turn to our special case: diffusion in Fe3Si. In a thin target, the initial time response of the diffusionally ac- celerated decay in NFS is exponential. As has been argued in Ref. 7 for the explanation of NFS spectra one for Bravais lattices or several exponential decays for non-Bravais lat- FIG. 2. Nuclear forward scattering of SR from the Fe3Si sample tices are expected, with their number equal to the number of b with the synchrotron beam parallel to the directions 111 left the corresponding absorption lines in Mo¨ssbauer spectros- and 113 right . The full lines are fits according to the theory of copy. Sec. V. Note the nearly exponential decay along 111 and the more This approximation was applied in the earlier short complicated decay along 113 at elevated temperatures. communication7 on diffusion in Fe3Si studied by NFS. There 5.1%, the effective sample thickness can be calculated to the time-dependent spectra measured in the 113 direction lie between 8.0 at 827 K and 7.0 at 967 K with about 10% were fitted with a sum of two exponentials: one decay with error. natural lifetime corresponding to the unbroadened Mo¨ss- Figure 2 shows time spectra of forward-scattered intensity bauer line with natural linewidth 0, the other faster decay as a function of time after the SR pulse for both sample ``diffusionally accelerated'' with slope and relative contri- orientations and at two and five different temperatures, re- bution determined from a free fit corresponding to the broad spectively. Figure 3 shows NFS spectra with the beam par- line in Ref. 9. The validity of this approximation, called also allel to three different crystal directions all at 967 K this was a kinematical approximation for a thin sample was confirmed the highest measuring temperature . The strong dependence in the Ref. 21. Each line in the Mo¨ssbauer spectrum gives of the spectrum shape both on temperature and on the direc- actually rise to one exponential decay of intensity, but as tion is clearly visible. The lines in Figs. 2 and 3 are fitted NFS is a coherent scattering method, interference terms be- with the complete theory described further down in Sec. V. tween the different resonances contribute to the total inten- We eventually, however, ran into a problem with the cal- sity. culations based on Eqs. 6 and 9 . As can be seen from Fig. As has already shortly been reported in Ref. 7, the fits 4, where we show NFS spectra in the 113 direction, at made use of the known values of sample thickness z and sufficiently high temperature shown here is 939 K the spec- Debye temperature of Fe3Si, D 369 K, known from Mo¨ss- trum can be well described with weights wp 0.9 and 0.1 bauer and phonon measurements,23 and yielded consistency Eq. 3 for the fast and the slow components in agreement with the atomistic diffusion model put up from QMS.9 From with a model of jumps between the - and -Fe sites.24 For p the iron diffusivity was calculated. lower temperatures, however, shown here is 885 K, it is evi- Sample b was measured in two orientations, namely, (i) dent that the model described above does not fit dashed with the 113 and (ii) the 111 crystal directions parallel to line . Fitting weights wp and different isomer-shift values for the synchrotron beam one measurement was performed in both components p in Eq. 6 provides much better the 335 direction . From the Debye temperature, sample agreement with experiment full curve in Fig. 4 . The reason thickness z 21(1) m and 57Fe isotope enrichment of the effective change in weight wp(Q) of the fast compo- 57 DIFFUSION IN A CRYSTAL LATTICE WITH NUCLEAR . . . 10 437 from QMS with its poorer effective energy resolution due to the poor angular resolution of Mo¨ssbauer spectroscopy. In the following we seek to include the effects of relaxation of hyperfine interactions into our model. V. FLUCTUATING HYPERFINE INTERACTIONS AND DIFFUSION Different iron sites in intermetallic alloys feel different hyperfine interactions, resulting from the difference in their atomic neighborhood. In general this leads to an isomer shift due to Fe atoms at different sites or causes quadrupole split- ting due to a noncubic symmetry of the neighborhood or leads to magnetic splitting in a magnetic neighborhood. In the present case cubic, paramagnetic in the relevant tem- perature range only an isomer shift between the components from the and sites of Fe atoms can appear. At low temperatures where diffusion is very slow, in fact two lines appear in the Mo¨ssbauer spectra with proportion being 2:1 originating from the occupation of the and sublattices by 57Fe atoms independent of crystal orientation. At very high temperatures where only diffusion defines the spectrum shape, and relaxation effects are veiled by the large width of the lines, the diffusion theory of Sec. III suf- FIG. 3. NFS spectra of Fe3Si sample b with the synchrotron fices, the weights being given by the solution of Eq. 3 with beam along three different crystal directions taken at 967 K. Lines the jump matrix Eq. 7 . In particular, for the spectra in 113 are fits according to the theory of Sec. V. direction the weights of broad and narrow line are 0.9 and 0.1, respectively9,21 see Fig. 4, 939 K . nent decreases with decreasing temperature to about 0.7 at We now turn to the intermediate temperature region. 885 K must be sought in incomplete description of the ex- Here the hyperfine components start to fluctuate, if the atoms perimental results by the model used up to now. We seek the jump between lattice sites during the lifetime of the excited reason for that deficiency in the following: what has been nuclear state, but this fluctuation is still not veiled by the neglected is the relaxation of isomer shifts of various com- diffusional linewidth as is the case for very high tempera- ponents due to the mobility of the atoms. It is appealing that tures. the measured temperature dependence of weights is a relax- The theory of the combined effects of diffusion and fluc- ational effect. This effect was suggested25 but not evident tuating hyperfine interaction has been developed by Dattagupta26 and Dattagupta and Schroeder27 who adapted the Blume28 approach for incorporating diffusion effects to QMS. They get Q, 0 0 2 i k i 1 A 1 m cm, k,m 2 10 where 1 is a unit matrix, the term in parentheses is the matrix element (k,m) and k and m are sublattice indices. This method demands a matrix inversion but offers the possibility to introduce effects of the relaxation of hyperfine interac- tions. Dattagupta and Schroeder were interested in QMS and QNS and therefore calculated only the real part of (Q, ), which is essentially the absorption probability and the scat- tering cross section. The Dattagupta-Schroeder approach was developed in principle to include the effect of vacancies near the Mo¨ss- bauer atom. In diffusion in the intermetallic phase Fe FIG. 4. NFS spectra of Fe 3Si we 3Si sample b with the synchrotron neglect the influence of vacancies due to the negligibly low beam along 113 direction taken at 885 and 939 K. Dashed curves: fit with model of Secs. III and IV, without accounting for relaxation values29 of vacancy-induced hyperfine interactions in this al- and without the difference in isomer shift between the component loy. We rather consider here only one type of hyperfine in- corresponding to the unbroadened resonance and the faster diffu- teractions, namely, the monopole interaction responsible for sionally accelerated component. Full curves: isomer shift differ- the isomer shift differences due to different distributions of ence and contributions are fitted. Fe and Si atoms in the neighborhood of the 57Fe atoms, i.e., 10 438 SEPIOL, MEYER, VOGL, FRANZ, AND RU¨FFER 57 TABLE I. Sample thickness L, jump rate 1 and diffusivity D as a function of temperature and crystal orientation for sample b . Data as in Fig. 5. Values without errors were kept fixed in the fit. T K Direction L 1 106 s 1) D m2 s 1) 827 113 7.1 4 0.9 3 1.2(4) 10 14 858 113 6.2 4 3.8 5 5.1(7) 10 14 885 113 6.2 4 6.7 3 9.1(4) 10 14 913 113 6.7 13.8 8 1.9(1) 10 13 939 113 6.3 31 3 4.2(4) 10 13 967 113 6.0 48 5 6.6(7) 10 13 967 335 6.0 40 5 5.4(7) 10 13 827 111 6.9 4 0.3 3 0.4(4) 10 14 913 111 6.7 4 13.8 1.9 10 13 967 111 6.0 50 5 6.7(7) 10 13 FIG. 5. Diffusivities of Fe in Fe3Si. Full circles present our NFS on and sites. This is done, following Ref. 27 by adding result for sample b , 74 at. % Fe, 26 at. % Si. Diamonds, triangles, the isomer-shift matrix V to Eq. 10 . and hexagons, data from tracer diffusion studies by Gude and Mehrer, Ref. 11. Q, 0 0 thickness z taking into account its roughly estimated charac- 2 i k i 1 A iV 1 m cm. k,m 2 ter. Note that due to the method of the single-crystal foil 11 preparation grinding , the thickness of the foil is sufficiently The isomer-shift matrix can be added as a constant, since it is constant and to use a thickness distribution was not neces- independent of the nuclear variables of the excited and sary. ground states of the Mo¨ssbauer nucleus. For Fe For the sake of comparison with the result of other diffu- 3Si it has the following form: sion studies it is useful to calculate the diffusivities from the diffusional acceleration of the intensity decay. Diffusion 0 0 jumps with jump rates 1 i j in intermetallic compounds be- V 0 0 0 , 12 tween ith and jth sublattices with distance vectors lij con- tribute as partial diffusivities to the diffusion coefficient31 0 0 0 where is the isomer shift difference between and 1 2 1 iron sites absolute values of isomer shifts are not measured D 6 lij ij ci , 13 i, j in NFS . The value of is known from the RT-Mo¨ssbauer measurements and is practically temperature independent where ci are sublattice occupations. Taking into account that 0.16 1 mm s 1 . By substituting Eq. 12 into Eq. 11 and we have only one jump rate 1 due to the equal sublattice the result into Eq. 9 the shape of the forward-scattered occupations and with lij a 3/4, Eq. 13 can be simpli- intensity for diffusion in Fe 1 3Si is calculated. fied to the form D (1/24)a2 . The resulting diffusivities To fit the spectra the numerical procedures of matrix in- are given in Fig. 5 and Table I. They are in good agreement version and fast Fourier transformation FFT from the with data from literature QMS Ref. 32 and tracer NAG-Fortran Library Routine Document were applied. The diffusivity11 showing the reliability of the new NFS method number of fitted parameters was reduced from five to three for studying diffusion. compared to the fit with two Lorentzian lines. The only fitted In retrospect we have to admit that the influence of dif- parameters for diffusion in the D03 lattice are the scattering ferent isomer shifts from different iron sites and their relax- intensity at time zero which is in fact a trivial parameter ation could have been recognized in Mo¨ssbauer spectra as proportional to the measuring time and the synchrotron cur- well, again at moderate temperature. There line smearing ap- rent only , the sample thickness L and the jump rate, the peared around 900 K.25 At high temperatures the increasing latter being the only diffusional parameter. The isomer-shift diffusional broadening again veils the small differences in difference between iron positions on and sublattices was the isomer shift of the Mo¨ssbauer lines. The reason why held constant at 0.16 mm s 1. relaxation effects are more evident in NFS spectra than in As to be seen from Fig. 4, T 885 K, the full curve which QMS spectra is twofold: a QMS spectra are the result of a takes into account relaxation, gives a much better description convolution with the geometrical resolution the divergence of the continuous change of the time spectrum than the of rays for receiving reasonable statistics in reasonable dashed curve, which does not account for relaxation.30 All time was about 7°), which forbids an experimental resolu- solid lines in Figs. 2 and 3 were fitted with the above- tion of the small energy shifts, as, e.g., small isomer shifts described method, too. The fitted parameters are listed in and, in particular, their change through relaxation. The syn- Table I. The effective thicknesses of the sample are in good chrotron beam, in contrast, has nearly negligible divergence, agreement with the values approximated from the measured so that the problem with the geometrical resolution is absent. 57 DIFFUSION IN A CRYSTAL LATTICE WITH NUCLEAR . . . 10 439 b In QMS the Mo¨ssbauer source adds its linewidth to the There are a few definite advantages of the NFS method. spectrum, which is not the case for time-resolved NFS with a Diffusion investigations are orders of magnitude faster synchrotron radiation. than with classical Mo¨ssbauer spectroscopy QMS. This per- mits fast measurements even of samples that stand the high VI. CONCLUSIONS AND OUTLOOK temperatures only for short time. Even shorter measuring times will be possible by the help of an integral method In conclusion we state that it is possible to follow diffu- where all delayed counts irrespective of the delay time are sion in the time domain by observing the diffusional accel- counted as a function of the crystal orientation.34 b The eration of the decay rate of nuclear forward scattering NFS highly brilliant synchrotron beam with a size of less than 1 of SR, in analogy to conventional quasielastic Mo¨ssbauer mm2 at the sample position and a divergence in the rad spectrocopy QMS or quasielastic neutron scattering QNS , range permits measurements with considerably reduced both in the energy domain. The conclusions from the Mo¨ss- ``smearing'' of the crystal orientation. c The narrow beam bauer measurements on the jump mechanism are fully con- will enable diffusion investigations in tiny crystals and re- firmed: the jumps are between Fe NN sites on and sub- crystallized foils. lattices. NFS spectra can be fitted in a consistent way if one uses a ACKNOWLEDGEMENTS full relaxational-matrix formalism. Uncorrelated jumps be- tween NN sites are sufficient to fit the spectra. Attempts to The authors appreciate the most valuable discussions with include correlations between jumps based on the Monte V. G. Kohn and G. V. Smirnov and the continuous exchange Carlo simulations did not improve the fit quality. The prob- of ideas with W. Petry. They thank A. Q. R. Baron for help lem of correlation between diffusion jumps in a D03 lattice with the APD detector and A. I. Chumakov for advice with has been studied through calculations by Szabo´33 with the the experiments. This work was supported by grants from the result that assuming only nearest neighbor jumps provides Austrian FWF Project No. S5601 and from the German sufficient agreement with experimental results. BMBF Project No. 05 643WOB . 1 K. S. Singwi and A. Sjo¨lander, Phys. Rev. 119, 863 1960 ; 120, 19 O. G. Randl, B. 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