VOLUME 76, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 1996 Time Domain Study of 57Fe Diffusion using Nuclear Forward Scattering of Synchrotron Radiation B. Sepiol,1 A. Meyer,2,3 G. Vogl,1 R. Rüffer,3 A. I. Chumakov,3 and A. Q. R. Baron3 1Institut für Festkörperphysik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria 2Physik-Departement, Technische Universität München, D-85747 Garching, Germany 3European Synchrotron Radiation Facility, F-38043 Grenoble, France (Received 27 November 1995) Diffusion of 57Fe in the intermetallic alloy Fe3Si was investigated using nuclear forward scattering of synchrotron radiation parallel to the [113] crystal direction with the aim to demonstrate the feasibility of a diffusion study in a crystalline solid. The jumps between different sites on a lattice, corresponding to a finite residence time on one and the same lattice site, cause a loss of coherence of scattered synchrotron radiation. This gives rise to a decay of the forward scattered intensity which is faster than under static conditions. From the time dependence of the decay diffusivities of iron are derived. [S0031-9007(96)00019-1] PACS numbers: 76.80.+y, 66.30.Fq In the past, two methods have been applied for studying is well known from QMS studies [5,7]. These studies the diffusion mechanism in solids on an atomistic scale have proven that Fe atoms which occupy sites on three and in space and time. These are quasielastic Mössbauer sublattices (Fig. 1) jump between nearest neighbor iron spectroscopy (QMS) and quasielastic neutron scattering sites remaining at each sublattice site for different resi- (QNS) [1]. In the case of QMS the diffusion manifests dence times. In the Mössbauer spectrum this leads to a itself through the broadening and change in shape of sum of three different Lorentzians with different broaden- the resonance line(s). For free long-range diffusion on a ings, weights of Lorentzians, and broadenings depending Bravais lattice it is a simple broadening of one Lorentzian on crystal orientation. In the [113] direction the spectrum shaped line. In the case of a non-Bravais lattice it can be a is composed of two lines only: one unbroadened and one superposition of Lorentzians, their number corresponding diffusionally broadened line. to the number of sublattices [2]. A single crystal of the Fe-Si alloy was grown by the Here we present the first measurement of solid state dif- Bridgman technique with a composition 75.5 at. % Fe fusion using nuclear resonance scattering of synchrotron (natural 57Fe abundance), 24.5 at. % Si. Slices of about radiation (SR) [3,4]. This technique permits studies di- 10 mm diameter were cut by a wire saw with the surface rectly in the time domain, whereas QMS and QNS studies parallel to the (113) plane. Two samples of about 24 and are performed in the energy domain. 15 mm final thickness were prepared. The choice of the investigated material is most im- In order to compare most directly we measured both portant for such a feasibility study. It should be a ma- conventional Mössbauer spectroscopy and nuclear for- terial well known and already thoroughly investigated ward scattering of SR. Conventional Mössbauer absorp- with QMS and tracer diffusion. The diffusion mechanism tion spectra were registered with the 15 mm sample and should be easy from the Mössbauer point of view. All 57Co in Rh as a source (Fig. 2) and in the same furnace these requirements are well fulfilled by the intermetallic which was afterwards used for the synchrotron experi- alloy Fe3Si as argued in the following. ment. Column 2 of Table I gives the line broadening (a) Fe3Si crystallizes in a simple structure (cubic, D03 of the broad line Gd which yields directly the residence consisting of three iron sites and one silicon site in a prim- time on a particular lattice site with respect to the natural itive cell) and is perfectly ordered up to the melting point (Tm 1500 K). (b) Single crystals of Fe3Si are stable during the high-temperature measurement. They can be grown, oriented, cut, and polished up to the required final thickness. (c) Fe3Si shows extremely fast diffusion of the iron atoms, the fastest of all iron intermetallics with high iron content found up to now. Thus diffusion phenomena can be observed at low temperature (about 900 K) which reduces technical problems, as has been demonstrated by QMS [5] and confirmed by tracer diffusion studies [6]. FIG. 1. D0 An essential advantage of Fe 3 structure of Fe3Si (2 8 of elementary cell). The 3Si for a feasibility test iron atoms occupy the sublattices a1, a2, and g, the silicon is that the diffusion mechanism of iron in this structure atoms the sublattice b. 3220 0031-9007 96 76(17) 3220(4)$10.00 © 1996 The American Physical Society VOLUME 76, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 1996 were counted by a fast avalanche photodiode (APD) detector [12,13]. In the present work because of limited experimental time, low count rate from the unenriched sample, and overload of the detector caused by the prompt pulse, measurements are reliable from 30 ns after the pulse is on. In principle the short-time limit is determined by the synchrotron radiation pulse length ( 100 ps) and the detector resolution ( 100 ps FWHM). The detector tail after the prompt pulse [14], from available APD's, indicates that with measurements beginning about 2 ns after the pulse, at the cost of seriously reduced count rate, the signal of delayed forward scattering will be above the noise. Forward scattered SR was registered with two different sample thicknesses, one with the 24 mm sample and another one with both samples packed together (total thickness 39 mm) which leads to an increase in the delayed count rate. Depending on the synchrotron ring current, sample thickness, and temperature, the count rates were between 2 and 30 delayed counts s. This resulted in a measuring time of 1 h at the highest temperature. The constant background of the APD diode was 0.05 counts s. Figure 3 shows time spectra-forward scattered inten- sity as a function of time after the SR pulse-for four dif- FIG. 2. Conventional Mössbauer absorption spectra of Fe ferent temperatures and two sample thicknesses: 24 mm 3Si measured in the [113] direction at (a) 876 K, (b) 916 K, [Figs. 3(a) and 3(d)] and 39 mm [Fig. 3(b) and 3(c)]. At (c) 959 K, and (d) 1013 K. At and above 916 K two 876 K an exponential decay of the intensity appears which Lorentzian lines are observable, one of which is strongly becomes steeper with increasing temperature. As will be broadened. shown the increase in the slope is an effect of faster diffu- sion at higher temperatures. A second decay shows up at longer times. linewidth G0 of the 14.4 keV Mössbauer level. From the In the following we propose an interpretation. With- residence time the diffusivity (column 3 of Table I) fol- out diffusion the time dependence of forward scattered lows directly via the Einstein-Smoluchowski equation [8]. intensity as a function of sample thickness z and time t is The experiments with nuclear resonance scattering of described by the following relation [15­17]: SR were carried out at the nuclear resonance beam line of the ESRF (for details see [9]). The storage ring L p operated in 16-bunch mode providing short pulses of IFS z, t ~ exp 2qt J2 Lt , (1) t 1 x rays every 176 ns. The radiation from the undulator source, optimized for the 14.4 keV transition in iron, was with q 1. Here t t t0 with t0 the natural lifetime of the excited state (141 ns), L s filtered by a double Si(111) reflection followed by a high 0fLMnx z the effec- resolution nested monochromator [9,10]. The delayed tive thickness where s0 is the nuclear absorption cross events, resulting from the nuclear forward scattering [11], section (2.56 3 10222 m2), fLM the Lamb-Mössbauer factor derived from Ref. [18] (fLM 0.42 at 876 K), n the number of iron atoms per unit volume (5.66 3 1022 cm23 at 700 K), x the isotope abundance of 57Fe TABLE I. Line broadening of the broad line Gd G0 from (0.021), z the thickness of the sample, and J1 the first or- the Mössbauer (QMS) data (Fig. 2), q from forward scattering der Bessel function. of synchrotron radiation (SR) (Fig. 3) and the corresponding diffusivities D Diffusive jumps between different sites lead to a loss QMS and DSR. Only statistical errors are given. of coherence of the radiation and therefore an accelerated T (K) Gd G0 DQMS m2 s21 q DSR m2 s21 decay; as a consequence q will be greater than 1 [17,19]. 876 1.2(4) 2.0 6 3 10214 2.2(3) 2.1 5 3 10214 q is connected to the diffusional line broadening of one 916 4.1(4) 6.8 7 3 10214 4.8(3) 6.1 5 3 10214 resonance line in conventional QMS in the following way: 959 10.6(9) 1.7 2 3 10213 8.0(8) 1.1 2 3 10213 Gd 1013 35(2) 5.8 3 3 10213 10.2(8) 1.5 2 3 10213 q 1 1 . (2) G0 3221 VOLUME 76, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 1996 sionally accelerated decay of forward scattered intensity (Fig. 3). Column 5 of Table I gives the diffusivities D de- duced from the forward scattered intensity of SR. Up to T 959 K the diffusivities agree sufficiently well with the values deduced from conventional QMS and mean times confirmed by tracer measurements of samples with similar Fe-Si composition [6]. Indeed the diffusivity is extremely high compared to other alloys at comparable temperatures, e.g., 1.1 3 10213 m2 s21 at 959 K. Notice that at 876 K diffusion is too slow to lead to a two-step exponential decay in the accessible time window, whereas at 1013 K the short-time decay is so fast that its reli- able determination is problematic, the tabulated value of 1.5 3 10213 m2 s21 being definitely too low. The reason is certainly the lack of data below 30 ns. Opening the time window down to 2 ns would extend the range of dif- fusivities up to 10212 m2 s21 (about 50G0 in conventional 57Fe QMS). We regard this as the upper limit of diffusiv- ities accessible to SR. The exponential decay in forward scattered intensity dominating at longer times (e.g., above about 80 ns at 959 K) corresponds to the narrow line in Fig. 2 which is not broadened through diffusion. We used samples with two different thicknesses; therefore, the difference FIG. 3. Nuclear forward scattering of synchrotron radiation in the slopes of Figs. 3(c) and 3(d) is due to the different at the same sample as in Fig. 2 at (a) 876 K, (b) 916 K, effective thicknesses L. (c) 959 K, and (d) 1013 K. The line through the data is a We finally note that a more exact fit of the diffusional fit according to Eq. (1). Above 916 K a two-step exponential decay is clearly visible. The two straight lines indicate the decay of the forward scattered intensity can be accom- asymptotic behavior of the decay at short and long times. plished with an 57Fe enriched sample. In that case one leaves the asymptotic region and, for longer times, regis- ters the first minimum of the Bessel function. Then both For diffusion on a non-Bravais lattice as in the present values, q and the effective thickness L, can be fitted in- case of Fe3Si and in the [113] direction we find two dependently, whereas for our nonenriched sample the ef- Lorentzians in the conventional QMS spectra; therefore, fective thickness L had to be fixed. A detailed theory we expect two different q values corresponding to a of nuclear forward scattering of SR in the case of diffu- two-step exponential decay in Eq. (1). Equation (1) sion in crystalline lattices going beyond the asymptotic ap- contains time dependence in a twofold way: via the proach of the present paper and accounting for different exponential connected with diffusion, exp 2qt , and via crystal directions will be the subject of a more extensive a part connected with the effective thickness L. In a first paper [20]. approximation we split the exponential part into a sum In conclusion, we state that it is possible to follow of two exponentials, one in the limit of short-time decay diffusion by nuclear resonance scattering of SR, i.e., corresponding to the broad QMS line and one for long Mössbauer spectroscopy in the time domain, in complete times. We fitted the spectra of Fig. 3 with the effective analogy to conventional quasielastic Mössbauer spectro- thickness L due to the change of the sample thickness z copy (QMS) or quasielastic neutron scattering (QNS), and to the temperature dependence of the fLM factor. The both in the energy domain. The method and its results fitting procedure took account of the constant background appear even more straightforward since the decay of co- of the APD diode and of delayed quanta from the previous herence is caused in a direct and appealing way by the pulse. The values of q for the short-time decay are given stochastic motion of the jumping atom, whereas conven- in column 4 of Table I. tional QMS and QNS register diffusion by following the At temperatures where both the diffusional line broad- time-energy Fourier transform of the diffusion process. ening in Fig. 2 and the short-time behavior in Fig. 3 can Compared to the problematic standard task in QMS and be determined with sufficient reliability (between 876 and QNS of separating various differently broadened lines, the 959 K) and at identical temperatures the ratio Gd G0 is separation of different exponential decay rates of the in- nearly the same whether determined from conventional tensity of scattered synchrotron radiation is more direct. QMS (Fig. 2) or derived via Eq. (2) from the diffu- This advantage of the SR method compared to QMS and 3222 VOLUME 76, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 1996 to QNS can be exploited in the case of diffusion investi- [8] For a D03 structure D 124a2 t, with the Fe3Si lattice gations of complicated non-Bravais lattices. constant a 5.72 Å at 770 K and the residence time Further advantages are as follows: with a SR beam t 6 ¯h Gd. size of less than 1 mm2 at the sample position diffusion [9] R. Rüffer and A. I. Chumakov, Hyperfine Interact. 97­98, investigations in tiny crystals and recrystallized foils 509 (1996). should be possible. Furthermore, a divergence of the [10] T. Ishikawa, Y. Yoda, K. Izumi, C. K. Suzuku, X. W. x-ray beam in the mrad range will enable measurements Zhang, M. Ando, and S. Kikuta, Rev. Sci. Instrum. 63, of diffusional line broadening with considerably reduced 1015 (1992). [11] J. B. Hastings, D. P. Siddons, U. van Bürck, R. Holland, "smearing" of the crystal orientation. and U. Bergmann, Phys. Rev. Lett. 66, 770 (1991). The authors appreciate valuable discussions with V.G. [12] S. Kishimoto, Rev. Sci. Instrum. 63, 824 (1992). Kohn. This work was supported by grants from the [13] A. Q. R. Baron and S. L. Ruby, Nucl. Instrum. Methods Austrian FWF (Project No. S5601) and from the German Phys. Res., Sect. A 343, 517 (1994); A. Q. R. Baron, Nucl. BMBWFT (Project No. 05 643WOB). Instrum. Methods Phys. Res., Sect. A 352, 665 (1995). [14] S. Kishimoto, Nucl. Instrum. Methods Phys. Res., Sect A 351, 554 (1994). [15] F. J. Lynch, R. E. Holland, and M. Hamermesh, Phys. Rev. [1] G. Vogl, Hyperfine Interact. 53, 197 (1990); K. Rueben- 120, 513 (1960). bauer, J. G. Mullen, G. U. Nienhaus, and G. Schupp, Phys. [16] Yu. Kagan, A. M. Afanasev, and V. G. Kohn, J. Phys. C Rev. B 49, 15 607 (1994); B. Sepiol, Defect Diffus. Forum 12, 615 (1979). 125­126, 1 (1995). [17] G. V. Smirnov and V. G. Kohn, Phys. Rev. B 52, 3356 [2] O. G. Randl, B. Sepiol, G. Vogl, R. Feldwisch, and (1995); G. V. Smirnov, Kurchatov Institute, Moscow, K. Schroeder, Phys. Rev. B 49, 8768 (1994). Report No. IAE-5907 9, 1995. [3] E. Gerdau, R. Rüffer, H. Winkler, W. Tolksdorf, C. P. [18] O. G. Randl, G. Vogl, W. Petry, B. Hennion, B. Sepiol, Klages, and J. P. Hannon, Phys. Rev. Lett. 54, 835 (1985). and K. Nembach, J. Phys. Condens. Matter 7, 5983 (1995). [4] See, e.g., E. Gerdau and U. van Bürck, in Resonant [19] In the paper by G. V. Smirnov and V. G. Kohn [17] Anomalous X-Ray Scattering. Theory and Applications, the case of free diffusion in a dispersive medium was edited by G. Materlik, C. J. Sparks, and K. Fischer calculated. For diffusional broadening of one Lorentzian (Elsevier, New York, 1994), p. 589. line it was shown that relation (1) holds. In our case we [5] B. Sepiol and G. Vogl, Phys. Rev. Lett. 71, 731 (1993). may regard this as the asymptotic limit of diffusion in a [6] A. Gude, Doktorarbeit, Universität Münster, 1995. crystal lattice for short jump lengths. [7] B. Sepiol and G. Vogl, Hyperfine Interact. 95, 149 (1995). [20] A. Meyer et al. (to be published). 3223