PHYSICAL REVIEW B VOLUME 57, NUMBER 6 1 FEBRUARY 1998-II Hybrid beat in nuclear forward scattering of synchrotron radiation Yu. V. Shvyd'ko II. Institut fu¨r Experimentalphysik, Universita¨t Hamburg, D-22761 Hamburg, Germany U. van Bu¨rck, W. Potzel, and P. Schindelmann Physik-Department E15, Technische Universita¨t Mu¨nchen, D-85748 Garching, Germany E. Gerdau and O. Leupold II. Institut fu¨r Experimentalphysik, Universita¨t Hamburg, D-22761 Hamburg, Germany J. Metge European Synchrotron Radiation Facility, Boi te Postale 220, F-38043 Grenoble, France H. D. Ru¨ter II. Institut fu¨r Experimentalphysik, Universita¨t Hamburg, D-22761 Hamburg, Germany G. V. Smirnov R.R.C. ``Kurchatov Institute,'' 123182 Moscow, Russia Received 7 July 1997 Asymmetric nuclear-resonance broadening, as originating, e.g., from magnetic-hyperfine-field distributions in magnetic alloys, has strong effects on the time evolution of nuclear forward scattering of synchrotron radiation. In thin samples of an Invar alloy, resonance broadening and the resulting dephasing in time cause a fast decay of the coherent scattering signal. In thick samples, the asymmetry of the broadening strongly affects dynamical scattering. Quantum beat and dynamical beat blend into a fast hybrid beat with thickness dependent period and field distribution sensitive modulation. S0163-1829 98 01405-2 I. INTRODUCTION gets yields dynamical beats DB's in the time response of NFS.7­10 The coherent combination of these two beat phe- The advent of intense pulsed sources of synchrotron ra- nomena makes the time dependence of NFS both complex in diation SR made the observation of pure nuclear Bragg structure and rich in information. scattering of SR Ref. 1 possible providing the basis for This happens already for rather simple systems, where the Mo¨ssbauer spectroscopy in the time domain for a review nuclei at their sites experience definite values of the see, e.g., Refs. 2,3 . But only the introduction of meV- magnetic-hyperfine fields, electric field gradients, and elec- resolution x-ray monochromators made the observation of tron densities. Nowadays, however, interest has shifted to- nuclear forward scattering NFS of SR feasible4 and has wards more complex systems, which are characterized by thus opened a broad field of applications for time-domain distributions of the hyperfine-field parameters. Such distribu- spectroscopy. tions are usually directly connected with the nature of the In conventional Mo¨ssbauer spectroscopy the dependence sample under study, as for example magnetic-field distribu- of nuclear absorption on the energy of the incident radiation tions in intermetallic alloys, but they might also be produced is measured. The recorded signal in this case presents the by the experimental technique, e.g., by pressure gradients incoherent sum of the spectral components of the transmitted hardly avoidable in high-pressure cells. radiation. In time-domain spectroscopy, by contrast, the scat- The present work was aimed at studying the influence of tering spectrum of nuclei excited by a pulse of white SR is inhomogeneous nuclear-resonance broadening on the time measured in the time domain, where the response is formed dependence of NFS. We were able to reveal pronounced ef- by the coherent sum of the spectral components of the scat- fects due to resonance broadening. In particular, due to an tered radiation. This results in important interference effects asymmetry of the broadening, DB and QB blend into a fast specific for time-domain Mo¨ssbauer spectroscopy. hybrid beat. As an example we investigated the effects aris- Interference of radiation scattered by nuclear resonances ing from a distribution of hyperfine fields in magnetic Invar of different energies, arising, e.g., due to hyperfine interac- alloys. Since this was, to our knowledge, the first study in tions, leads to quantum beats QB's .5,6 Henceforth we shall this field, we were primarily interested in revealing and un- refer to the interference of this type as inter-resonance inter- derstanding the basic effects, rather than details of our actual ference. In optically thick targets, multiple nuclear scattering samples. makes the spectral components below and above resonance In Sec. II basic aspects of the NFS theory are presented. to dominate.3 The interference of these two groups of com- In particular, the influence of inhomogeneous nuclear- ponents intraresonance interference in optically thick tar- resonance broadening on the time dependence of NFS is dis- 0163-1829/98/57 6 /3552 10 /$15.00 57 3552 © 1998 The American Physical Society 57 HYBRID BEAT IN NUCLEAR FORWARD SCATTERING . . . 3553 cussed. In Sec. III the measurement procedure and the where ( . . . . . . ) is the 3 j symbol, N0 is the number of resonant samples under study are described. The results of the mea- nuclei per unit volume, wj is a weight factor of the nuclear surements of NFS by thin and thick samples of iron and site j, f LM is the Lamb-Mo¨ssbauer factor and is the inter- Invar are given in Sec. IV and discussed in Sec. V. nal conversion coefficient, and hs is the magnetic polariza- tion vector of the radiation. In defining the polarization fac- II. BASIC ASPECTS OF NFS THEORY tor Psm M we have assumed in accordance with our A. General formulas experimental conditions that n k holds at each nuclear site. Two types of solutions of the wave equation 1 are of inter- The propagation of a radiation field E(r,t) through a reso- est for us. nant medium in form of a plate of thickness L) can be The first one is the solution in energy space which de- described in space and time by the wave equation9,11,12 scribes the transmission of monochromatic incident radiation with energy E through the sample plate. The boundary con- dEs , dition for the radiation amplitude at the entrance surface is d d Ks Es , , 1 then given by Es(0,t) E s, where E s is the amplitude of the incident radiation. The solution of Eq. 1 is therefore of the E E steady-state type and the transmission amplitude Rs(E) Ks s l l exp i , 2 l Es(L,t)/Es(0,t) equals 0 2 z/ s Lsin ; 0t/ . 3 Rs E exp i l 0 . 9 Here Es( , ) is the s-polarization component of the ampli- l E El i 0/2 tude of the radiation field E(r,t) E( ,t)exp ikr iEt/ which varies slowly compared to the variation of the expo- The second type of solutions of interest are time re- nent; z is the spatial coordinate in the direction of the internal sponses of the nuclear-resonance ensemble onto the instan- normal to the entrance surface; the index l m M, j is taneous excitation by a short radiation pulse. The boundary used to numerate both the different nuclear transitions be- condition in this case reads Es(0,t) E s (t). The general tween ground and excited states and different nuclear sites; solution of the wave equation can be presented as a power is the angle between incident beam and surface. For sim- series of :12 plicity we assume that the nuclei experience only magnetic hyperfine interactions and that the magnetic quantization axis p Es , As s , 10 n is perpendicular to the wave vector k at each nuclear site. 0 p 1 p! Ap Under these assumptions the energy of the nuclear transition l is given by with the functions Asp( ) determined according to the follow- ing recursion relations: El E0 mM Bj , As0 E s , 11 mM Bj m g /Ig M e /Ie Bj , 4 s where E A E sKs , 12 0 is the energy of the nuclear transition between the 1 ground and excited state in the absence of hyperfine interac- tions, g,e are nuclear magnetic moments, Ig,e are nuclear s s spins and m,M are magnetic quantum numbers in the ground Ap d Ks Ap 1 , 13 and excited states, respectively, Bj is the value of the mag- netic hyperfine field at the nuclear site j. which one obtains by inserting Eq. 10 into Eq. 1 and The value taking into account the boundary condition. (t) is the unit step function. 1 We shall present examples of a few particular solutions. s s l 4 kLgl 5 First we consider the case of a thin nuclear resonance scat- terer, i.e., when s 1. In this case the series in Eq. 10 may is a partial effective resonance thickness parameter for the l be cut off after the term with p 1. Then using Eqs. 12 , 2 specific transition l and polarization s. Here the solution for the radiation field at the exit surface of the sample, i.e., when z L, at times t 0 reads gs Ig 1 Ie 2 s l g j 3P , 6 m m M M m M i Es L,t exp 0 s 2 t lexp l Elt . 14 4 2I 1 g RN0 e 1 j k wjf LM ; R k2 2 2I Thus it is seen that the nuclear system responds at the fre- g 1 1 , 7 quencies of the excited nuclear transitions. The time spec- nhs 2, m M 0, trum, which is I(t) Es(L,t) 2, contains a sum of sinusoidal functions, producing periodic modulations of the signal at Psm M 1 8 the difference frequencies (El El )/ . These modulations 2 1 nhs 2 , m M 1, have been called quantum beats.5,6 3554 YU. V. SHVYD'KO et al. 57 In the next example we consider a sample of arbitrary thickness, however, with an unsplit nuclear resonance, i.e., El E0 for all l. In this case the summation over l in Eq. 2 yields a single and polarization independent effective reso- nance thickness value s l,s l , which corresponds to T/4, where T is the effective thickness used in conventional Mo¨ssbauer spectroscopy. All coefficients in Eq. 10 can then be calculated explicitly Ap( ) ( )p p 1/(p 1)!. The summation of the series in Eq. 10 results for t 0) in i E L,t exp 0 E0t 2 t J1 2 . 15 This expression, where J1 is the Bessel function of first kind and order one, was first obtained in Ref. 8. The Bessel modu- lation of the nuclear response to the prompt excitation is an effect of multiple coherent nuclear scattering, which can be observed in samples of significant thickness,4,10 and which is now often referred to as dynamical beat. A related Bessel modulation of the nuclear signal in time due to sample thick- ness was also observed with Mo¨ssbauer radioactive sources by using delayed coincidence7 and fast shutter9 techniques. In thick samples ( 1) the Bessel function describes a faster decay as compared to the natural decay exp( 0t/ ) in case of a thin sample see Eq. 14 . This effect is often referred to as coherent speed-up of the decay in NFS.8,9,13,10 If the nuclear transition energies El are well separated then even in case of a thick sample one can use an approxi- mate solution9 i s Es L,t s 0 FIG. 1. Time evolution of NFS and magnetic hyperfine field l exp . l Elt 2 t J1 2 l s profile inset for a nuclear scatterer equivalent to a magnetized l 16 57Fe foil of 3.5 m thickness, calculated for different field distri- butions: definite value of magnetic field a , symmetrical distribu- which comprises both QB's and DB's. tions of Lorentzian shape b , and of Gaussian shape c . In the general case the hyperfine splitting is not zero, the transition energies are not well separated, and the sample is field distributions, or it can be of dynamic origin due, e.g., to not thin, then the time response can be calculated only nu- relaxation, diffusion, etc. These cases can be described with merically. The results of the numerical calculations pre- the same equation 1 , however, with the kernel Ks( ) aver- sented in this paper were performed by using Eqs. 1 ­ 13 aged over the corresponding statistical distributions.12 implemented in the computer program MOTIF.14 We give two simple examples. Let us first consider a model, where at different nuclear sites the magnetic hyper- B. Application to inhomogeneous nuclear-resonance fine field takes different values, and these values have a broadening Lorentzian distribution around B with a full width at half maximum given by B. We shall replace the weights in Eq. An example for NFS in the case of a nonbroadened 7 by the distribution function nuclear resonance is given in Fig. 1 a . The figure shows a calculated time spectrum I(t) of NFS by a magnetized iron B 1 sample under conditions where the two equivalent m M w b 0 nuclear transitions are excited. It is assumed that the 2 b B 2 B/2 2 magnetic hyperfine field has a definite value of about 33 T and replace the summation over the site number j in Eq. 2 see inset . The time spectrum reflects the inter-resonance by integration over b between infinite limits. Performing the interference between the scattered radiation components by a integration we obtain fast QB of period 8 ns. The intraresonance interference leads to a pronounced DB modulation with minima at 77 and E EmM 260 ns. This case corresponds to the approximation given in Ks smMexp i 2 1 mM , 17 Eq. 16 . mM 0 Which changes in the NFS time spectra can be expected if the nuclear resonances experience inhomogeneous broaden- EmM E0 mM B ; mM mM B / 0 . ing? The answer certainly depends on the origin of the The kernel 17 looks just the same as the original one Eq. broadening. It can be of static origin due to, e.g., hyperfine- 2 with the only exception that the exponential damping 57 HYBRID BEAT IN NUCLEAR FORWARD SCATTERING . . . 3555 constant is now (1 mM)/2 instead of 1/2 . This means that all solutions for the time dependences of NFS, such as Eqs. 14 ­ 16 , are also valid in this case with the only modifica- tion that the exponential damping factor 0 should be re- placed by (1 mM) 0. In agreement with Refs. 15­17 one can thus conclude that a Lorentzian field distribution does not change the positions of the QB and DB minima. The essential influence of the Lorentzian inhomogeneous broad- ening is a faster decay of the NFS signal. These features become obvious from a comparison of Figs. 1 a and 1 b . By contrast, a Gaussian distribution of the magnetic hy- perfine fields 1 b B 2 w b exp B 2 2 B 2 leads to more significant changes in the kernel: E E 2 2 Ks s mM mM mMexp i . 18 mM 0 2 2 The solution of the wave equation with this kernel Fig. 1 c yields the same QB modulation as in Figs. 1 a and 1 b . However, the DB modulation is now changed drastically. In particular, the DB minima are now shifted to earlier times. These two simple examples clearly demonstrate that the time dependence of NFS and in particular the DB and QB are very sensitive to the detailed form of the inhomoge- neous broadening. Experimental studies of this question and further theoretical discussion are presented in the following sections. III. SAMPLES AND MEASUREMENTS An Invar alloy of composition Fe FIG. 2. Mo¨ssbauer spectra of a 1 m iron metal foil a and of 65Ni35 , enriched to 95% in 57Fe, was used as sample material. Such alloys are ferro- Invar foils of 0.7 m b , 6 m c and 40 m, d thickness. The foils were mounted in a weak magnetic field perpendicular to the magnetic, and are considered in Mo¨ssbauer spectroscopy as beam direction. The solid lines are guides to the eye based on the classical, but still disputed18­20 examples for predominantly superposition of six independent Lorentzians. inhomogeneous resonance broadening due to static magnetic hyperfine field distributions. At room temperature they ex- hibit a broadening of several natural linewidths ture at the bending magnet beam-line F4 Ref. 22 and at 0.18 Two different foils of 10 and 40 m thickness were prepared. A liquid He temperature in a cryostat at the wiggler beam-line piece of the 10 m foil was used later to roll foils of 6 and BW4.23 The SR was monochromatized to 6 meV by a high- 1 m thickness. Reference measurements were performed resolution monochromator.24 The SR transmitted through the with foils of pure iron metal, enriched to 95% in 57Fe, of sample was recorded by an avalanche photo diode.25 The corresponding thicknesses. samples were placed in magnetic fields, which aligned the The foils were roughly characterized by conventional hyperfine fields perpendicular to the beam. Due to the linear Mo¨ssbauer spectroscopy at room temperature, using unpolar- polarization of SR magnetic-field vector perpendicular to ized radiation from an intense 57CoRh source. Figure 2 de- the plane of the storage ring and the pure M1 character of picts the spectra of a 1 m iron reference foil, and of Invar the 14.4 keV nuclear resonance of 57Fe, either the two foils of 0.7, 6, and 40 m thickness as used in the experi- nuclear transitions with M m 0 were excited when the ments. The foils were placed in an external magnetic field of external magnetic field was oriented vertically, or the four 0.1 T in order to orient the hyperfine fields perpendicular nuclear transitions with M m 1 when the field was ori- to the beam direction. The thinnest Invar foil exhibits a mag- ented horizontally . At room temperature a field of 0.13 T netic hyperfine splitting, which is 80% of that of the iron produced by a permanent magnet was applied, and in the foil, and a resonance broadening, which is 8 0 for the cryostat at liquid He temperature a field of 1 T was provided M m 0 transitions. In all spectra except the one in Fig. by superconducting coils. 2 b broadening effects due to resonance absorption in opti- In the experiments, at first NFS by iron foils was mea- cally thick samples prevail. sured in order to demonstrate as a reference the characteristic Studies of the time dependence of NFS by these samples21 patterns produced by interresonance and intraresonance in- were performed at HASYLAB Hamburg at room tempera- terference for a material without inhomogeneous broadening. 3556 YU. V. SHVYD'KO et al. 57 FIG. 3. Time evolution of NFS of SR for a 1 m iron metal foil a and a 1.2 m Invar foil b in a vertical magnetic field of 0.13 T. The solid lines are fits using the NFS theory. FIG. 4. Time evolution of NFS of SR for a 1 m iron metal foil a and a 1.2 m Invar foil b in a horizontal magnetic field of The same measurements were then performed using the In- 0.13 T. The solid lines are fits using the NFS theory. The dashed var foils as an example for a material with inhomogeneous lines are guidelines to the eye, pointing out the 100 ns modulation broadening. of the fast beat, starting high low at time zero for the even un- even numbered beats, respectively. IV. RESULTS A more complicated beat pattern was observed when the A. Thin samples samples were placed in a horizontal magnetic field. In this geometry the four nuclear transitions with M m 1 are As discussed in Sec. II, inhomogeneous broadening is ex- excited. The QB pattern observed in case of the thin iron foil pected to lead to a faster decay of the NFS signal. In order to is depicted in Fig. 4 a . We shall discuss it in Sec. V A. The distinguish this effect from the coherent speed-up, which corresponding time dependence for the 1.2 m Invar foil is arises in optically thick samples, it was important to use shown in Fig. 4 b . Although the QB patterns are more com- samples as thin as could be tolerated for intensity reasons. plicated, compared to that in Fig. 3, they show the same Figure 3 a shows the time dependence of NFS by a 1 m basic difference: a faster decay of the signal in the time spec- iron foil placed in a vertical magnetic field. Only the two trum of Invar despite comparable effective resonance thick- transitions with M m 0 were excited, yielding a QB with nesses. a single period of 2 / E 14 ns which is defined by the separation E 63.3 B. Thick samples 0 between the energies EmM of the transitions excited. This separation corresponds to a hyper- The next step was to measure the NFS from thick foils to fine field B 33 T at the iron nuclei. Because of the finite reveal the influence of the inhomogeneous broadening on the sample thickness, the initial decay is about three times faster combination of interresonance and intraresonance interfer- than natural as discussed previously,10 and the first minimum ence. In a comprehensive study, the time dependences of of the DB occurs at 270 ns as expected for a sample with NFS of Invar foils of different thicknesses and of the corre- effective thickness parameter s sponding iron foils were measured for the two cases of trans- mM 1.95 corresponding to a Mo¨ssbauer thickness T 7.8 for each M m 0 transition. verse magnetization. Because of better evidence and greater simplicity, only the measurements in a vertical magnetic The time evolution of NFS by an Invar foil of thickness field, where the transitions with M m 0 are excited alone, 1.2 m, which has an effective thickness parameter smM will be shown in the following. 1.4 is shown in Fig. 3 b . Although the effective thickness As an example for the results obtained with the iron is less than for the iron foil, the decay is much faster. It is samples, Fig. 5 a shows the time dependence of NFS by a approximately exponential with a decay constant 6.5 times foil of 9 m thickness. The fast QB is modulated by a larger than natural, and the signal is definitely lost after 150 pronounced slow DB, as previously reported.26 This case ns. The period of the QB is almost 16 ns, which is larger than provides an example, where the approximate solution Eq. for iron metal, in agreement with the smaller hyperfine split- 16 holds. A strong initial speed-up of the decay leads im- ting observed in the Mo¨ssbauer spectra compare Fig. 2 . mediately into the first DB minimum at 26 ns. The period 57 HYBRID BEAT IN NUCLEAR FORWARD SCATTERING . . . 3557 FIG. 5. Time evolution of NFS of SR for thicker foils in vertical magnetic fields. Iron metal foil of 9 m thickness at 4 K in a field of 1 T a and Invar foil of 10 m thickness at room temperature in a field of 0.13 T b . The solid lines are fits using the NFS theory. of the DB increases with time,8,3 and so the later DB minima FIG. 6. Time evolution of NFS of SR for a 40 m Invar foil in appear with increasing distances at times 90, 190, and 320 a vertical magnetic field of 0.13 T, mounted at right angle to the ns. beam a and inclined by 45° b and 58° c , yielding thick- By contrast, the time dependence of NFS of a correspond- nesses of 40, 57, and 76 m, respectively. The solid lines are fits ing Invar foil of 10 m thickness Fig. 5 b showed three using the NFS theory. unexpected features: 1 The apparent QB has a shorter pe- riod 14 ns than measured for the 1.2 m Invar foil 16 was measured from a thinner Invar foil of 6 m thickness, ns , 2 no modulation by a DB can be recognized, 3 in- stead the apparent QB shows a perceptible high-low modu- which should yield well separated and relatively broad DB lation. minima. Figure 7 a shows the time evolution of NFS by this In order to study these surprising features in more detail, foil. Again, the DB modulation, which should have a first the time dependence of NFS of the 40 m Invar foil was minimum at 70 ns, is completely missing. measured Fig. 6 a . All three anomalies listed above are At that time model considerations and fit results indicated, observed again. The apparent QB has now a period of 12 ns that the absence of the DB modulation might be connected only, corresponding to a hyperfine splitting larger than in with an asymmetry of the field distribution, which is typical iron metal, and much larger than given by the Mo¨ssbauer for Invar at room temperature.18 At low temperatures, by spectrum of this foil Fig. 2 d . contrast, the field distribution had been found to be much This compression of the apparent QB was observed so far more symmetrical.18 In order to check the influence of dif- in this study for foils of different thickness. In order to ex- ferent resonance shapes, the NFS of the same 6 m Invar foil clude any influences of metallurgical differences between was then measured at 4 K Fig. 7 b . Indeed, the time de- various foils, the same foil was measured with the effective pendence now reveals a DB envelope with rather pro- thickness being increased by tilting around a vertical axis. nounced minima around 50 and 150 ns. This way the conditions for nuclear excitation stayed the same only the transitions with M m 0 were excited , but the effective foil thickness was increased to 57 and 76 m, V. INTERPRETATION AND DISCUSSION respectively. The measured time dependences are shown in Figs. 6 b and 6 c . The apparent QB is still more com- A. Time spectra in the absence of inhomogeneous broadening pressed, with the period decreasing to a value of only 11 ns. In cases where two equivalent nuclear transitions with A QB of this period would correspond to an inner magnetic M m 0 are excited, e.g., in the case of Figs. 3 a and 5 a , field of more than 40 T. the time spectra of NFS are very easy to ``read'' since they The absence of the DB could in principle be explained by contain a QB pattern with a single frequency and a very thickness inhomogeneities which are known to smear out the distinct DB modulation which depends on the sample thick- sharp DB minima. In order to exclude this effect, the NFS ness. Already in cases when four nuclear transitions with 3558 YU. V. SHVYD'KO et al. 57 beats, starting with zero at time zero, we can easily follow the 100 ns modulation of the even beats and the corre- sponding antiphase modulation of the uneven beats compare guidelines in Fig. 4 a . At later times, however, dynamical effects make the scat- tering by the outer transitions largely vanish in a time win- dow centered at 360 ns, which is the first minimum of the DB pattern for the outer transitions. In this time window the scattering from the inner transitions prevails, which is char- acterized by a slow QB with a period of 50 ns. The domi- nance of the scattering by the inner transitions, however, is only temporary: At still later times the contribution of the outer transitions would recover and the fast beat would dominate again. B. Faster decay caused by resonance broadening The time dependences of NFS by the thin Invar foil com- pare Figs. 3 b and 4 b show much faster decays than those of the thin iron metal foil compare Figs. 3 a and 4 a . In particular, the Invar NFS signal is definitely lost after times of 300, 150, and 100 ns for the inner, middle, and outer pairs of transitions, respectively. This demonstrates in agree- ment with the results of Sec. II B, that the faster decay in case of the thin Invar foil is primarily due to inhomogeneous nuclear resonance broadening. In this case the radiation com- ponents are spread in frequency in a range wider than the FIG. 7. Time evolution of NFS of SR for a 6 m Invar foil at natural linewidth, and therefore they run out of phase faster room temperature in a vertical magnetic field of 0.13 T a and at than in a natural lifetime. 4 K in a vertical magnetic field of 1 T b . The solid lines are fits using the NFS theory. The insets show the field distributions used A similar, faster decay of the NFS signal due to dephasing for the fit. can be observed in case of dynamic inhomogeneous reso- nance broadening, for instance in cases of diffusion28 or M m 1 are excited, the interpretation becomes less evi- relaxation.29 This raises the question, whether Mo¨ssbauer dent. Therefore we shall describe more precisely the relevant spectroscopy in time domain offers possibilities to distin- time spectrum shown in Fig. 4 a . For this discussion we guish between dynamic and static origins of inhomogeneous shall use the approximation of a thin nuclear resonance scat- nuclear resonance broadening. In conventional Mo¨ssbauer terer. According to Eq. 14 we can write the expression for spectroscopy a distinction between different origins of inho- the NFS intensity mogeneous broadening is not straightforward. It requires ei- ther measurements using the selective excitation double 2 Mo¨ssbauer method SEDM 30 or a special analysis of the I t exp 0t/ cos2 t/2 line shapes.19 It seems, that in time-domain spectroscopy the 3cos t/2 cos t/2 answer is not straightforward as well. 1 9cos2 t/2 . C. Hybrid beat caused by asymmetry of the resonance broadening In this expression the first term corresponds to the interfer- ence of the two outer transitions in 57Fe separated in energy It was possible to fit all NFS time evolutions14 of the by , the third term arises from the interference of the two Invar foils by assuming asymmetric magnetic field distribu- inner transitions with energy separation , and the second tions. Such distributions are typical for Invar alloys.18 Fol- term describes the interference of the outer and the inner lowing Ref. 18 we have modeled appropriate distributions by transitions.27 In accordance with Eqs. 5 ­ 8 the ratios of the superposition of three Gaussians of different weights Wi , the thickness parameters s positions Bi , and widths Bi : mM are 3:1:1:3. In the time window from 0 to 270 ns Fig. 4 a a fast beat with a period of 8.1 ns is dominant, which corresponds to the Wi Bj Bi 2 first term in the above expression. The third term can be wj exp . neglected. The second term, however, introduces an interest- i 1,2,3 Bi 2 2 Bi 2 ing modulation of the fast beat. Similar to the case of nuclear Bragg diffraction,27 the interference of the four M m The weights Wi were normalized, so that jwj 1. Typi- 1 transitions causes also in NFS a high-low pattern with cally we used about 40 different nuclear sites in the simula- changing order. The order depends on the factor cos( t/2), tions. The fit parameters Wi ,Bi , Bi were optimized.31 As which changes sign every 51 ns. When we number the fast an example, the resulting profiles used for the fit of the 6 m 57 HYBRID BEAT IN NUCLEAR FORWARD SCATTERING . . . 3559 FIG. 8. Energy dependence of the NFS intensity in case of an optically thick scatterer with two resonance lines separated by a large hyperfine splitting. Due to multiple scattering, each resonance FIG. 9. Time evolution of NFS left , energy dependence of shows the typical double-hump structure. The hyperfine splitting is NFS right and field profile center for different magnetic field given by distributions: definite value of magnetic field a , symmetrical dis- Q , the separation of the two humps by D a . This situation would yield a time evolution of NFS characterized by a tributions of Lorentzian shape b and of Gaussian shape c , fast QB with frequency slightly asymmetrical distribution corresponding to Invar at 4 K d , Q , modulated by a DB with a frequency decreasing in time of order of and strongly asymmetrical distribution e based on Invar at room D . b When the two inner humps are canceled, a hybrid beat with frequency temperature. Q D is expected. Invar foil at room temperature and at 4 K are shown as insets asymmetrical oscillator distribution is just such a mecha- in Figs. 7 a and 7 b , respectively. nism. Figure 9 shows the effect of resonance broadening on The important question is, why an asymmetric nuclear the two double-hump structures and on the corresponding resonance broadening causes such drastic effects in NFS. An time evolution of NFS. In Fig. 9 a a definite value was answer can be found if details of the energy spectrum of NFS assumed for the magnetic field, as in the case of iron metal are considered. In NFS of SR in thick samples, multiple for instance. The resulting double-hump structures are well scattering leads in the energy dependence of the forward shaped, and in the time spectrum a fast QB is modulated by scattered intensity to a so called double-hump structure at a pronounced DB. In Figs. 9 b and 9 c symmetrical distri- each resonance transition.3 Figure 8 shows a typical NFS butions of Lorentzian shape and of Gaussian shape, respec- energy spectrum 1 Rs(E) 2 resulting from the excitation tively, were assumed. The general double-hump character of two nuclear transitions at E E0 Q/2. Rs(E) is calcu- and the resulting strong modulation of the time spectrum by lated according to Eq. 9 . The separation Q of the centers a DB remain essentially unchanged. In addition, we want to of the two double humps determines the QB arising from point out that the interference of neighboring oscillators in an inter-resonance interference, and the separation D of the asymmetrical oscillator distribution is a more general phe- humps, which increases with sample thickness, causes the nomenon, which is, e.g., also observed in x-ray physics.32 DB. If we now imagine a mechanism, that would just cancel A fundamental change, however, occurs, when an asym- the inner ones of each pair of humps, we immediately obtain metry is introduced to the field distribution. In case of asym- an explanation for the anomalous features observed in case metric broadening, with oscillator densities which decrease of the thick Invar foils compare Fig. 8 b : DB and QB slower in the region between the lines than outside of them, cease to exist separately, and instead blend into a new hybrid the inner humps are much stronger affected by destructive beat, which originates from the interference of the radiation interference than the outer ones see Fig. 9 d . In the case of components belonging to the two outer humps, with a beat a pronounced asymmetry, they can even be completely can- frequency given approximately by the sum of Q and D . celled see Fig. 9 e . These hump structures then yield NFS Thus the DB would disappear, and the period of the new time evolutions with less pronounced or finally completely hybrid beat would decrease with increasing sample thick- missing DB modulations Figs. 9 d and 9 e . Note that the ness, as observed in the experiment. field distribution of Fig. 9 d is actually the one used for the Indeed, interference of neighboring oscillators in an fit of the NFS by the 6 m Invar foil measured at 4 K 3560 YU. V. SHVYD'KO et al. 57 compare Fig. 7 b . The field distribution of Fig. 9 e is the depicted in Fig. 8 b , however, with much smaller separation one used for the fit of the NFS of the same foil measured at of the outer humps, yielding a slow beat. Such a slow hybrid room temperature compare Fig. 7 a , however with the po- beat is quite common for NFS by thick samples, which ex- sitions Bi of the three Gaussians being scaled so that the hibit, e.g., quadrupole splitting. The slow hybrid beat is pres- highest position corresponds to the one of Figs. 9 a ­9 d . ently under study. The resulting hybrid beat was found to be extremely sen- sitive to details of the field distribution. The rather smooth VI. CONCLUSION slope observed in Fig. 7 a seems to be accidental, in general the hybrid beat is modulated by a high-low pattern as ob- In the time evolution of NFS by thin Invar foils a fast served in Fig. 5 b and Fig. 6. Such a high-low pattern arises, decay of the scattering signal was observed. It originates whenever the beat of two dominating radiation components from dephasing in time, which is caused by inhomogeneous is disturbed by additional frequency components. Thus the nuclear-resonance broadening. In the case of thick Invar high-low modulation reveals details of the frequency spec- foils, a strong influence of the asymmetry of the resonance trum of the scattered radiation. This frequency spectrum is broadening on the time response was observed, which in determined by the interference of the scattering from the os- particular leads to a blend of QB and DB into a fast hybrid cillator distribution, which reflects the actual field distribu- beat. tion in the sample. The fast hybrid beat is typical for large hyperfine split- It should be noted, that already in the case of a thin foil tings in case of asymmetric magnetic field distributions. see Fig. 3 b the same interference effects lead to a sharp Since it has a shorter period than the original QB, it would QB, in spite of a pronounced asymmetry of the oscillator yield erroneous values for the hyperfine splitting, if it is mis- distribution compare the room-temperature field distribution interpreted as a pure QB. This again demonstrates that mul- depicted in Fig. 7 a . Also in the kinematical scattering by tiple scattering in case of a coherent scattering channel such thin foils destructive interference largely suppresses the con- as NFS plays an important role. The fast hybrid beat revealed tribution of the oscillators at the inner slopes of their energy itself as being very sensitive to the particular shapes of the distribution. This leads in the energy dependence of the scat- magnetic field distribution in Invar. In the present paper the tering to peaks, which are sharper, more symmetrical and inhomogeneous broadening was modeled with static mag- slightly shifted outwards as compared to the corresponding netic field distributions, where the nuclei of the ensemble strongly asymmetrical dips of the absorption spectra. Such experience definite, but slightly different magnetic fields. an energy dependence of the scattering yields a sharp QB with slightly increased frequency. ACKNOWLEDGMENTS Noteworthy, a similar, but slow hybrid between QB's and This work was funded by the Bundesministerium fu¨r Bil- DB's arises already in the absence of resonance broadening, dung, Wissenschaft, Forschung und Technologie under Con- when the hyperfine splitting is so small, that it is in the order tract Nos. 05 5WOAAI/643WOA and 05 643GUA. U.v.B. of the energy separation of the double humps. In this situa- and G.V.S. would like to thank S. L. Ruby, J. Arthur, A. Q. tion the interference of the forward scattering amplitudes is R. Baron, G. S. Brown, and A. I. Chumakov for hospitality destructive in between the split resonances, and constructive and collaboration in the preliminary experiments at the outside. This leads again to a two-hump structure as the one SSRL. 1 E. Gerdau, R. Ru¨ffer, H. Winkler, W. Tolksdorf, C. P. Klages, 9 Yu. V. Shvyd'ko, S. L. Popov, and G. V. Smirnov, Pis'ma Zh. and J. P. Hannon, Phys. Rev. Lett. 54, 835 1985 . E´ksp. Teor. Fiz. 53, 217 1991 JETP Lett. 53, 231 1991 ; J. 2 E. Gerdau and U. van Bu¨rck, in Resonant Anomalous X-ray Scat- Phys.: Condens. Matter 5, 1557 1993 ; 5, 7047 1993 . 10 tering, Theory and Applications, edited by G. Materlik, C. J. U. van Bu¨rck, D. P. Siddons, J. B. Hastings, U. Bergmann, and R. Sparks, and K. Fischer North-Holland, Amsterdam, 1994 , p. Hollatz, Phys. Rev. B 46, 6207 1992 . 11 589. Yu. V. Shvyd'ko, T. 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Stewart, Hyperfine Interact. 42, 1157 20 M. Dube´, P. R. L. Heron, and D. G. Rancourt, J. Magn. Magn. 1988 . Mater. 147, 122 1995 ; M.-Z. Dang, M. Dube´, and D. G. Ran- 31 Actually, the optimal fit parameters for the time dependences of court, ibid. 147, 133 1995 . NFS by different foils varied slightly. This can easily be ex- 21 Preliminary studies were performed at the wiggler beamline 10-2 plained by minor metallurgical changes introduced due to differ- of the Stanford Synchrotron Radiation Laboratory. However, the ent heat treatment in the preparation process, in particular by the quality of the data was not sufficient to allow an interpretation at annealing in the course of rolling. The thicknesses of the foils that time. 22 were sufficiently homogeneous so that in all fits averaging over R. Ru¨ffer, D. Giesenberg, H. D. Ru¨ter, R. Hollatz, E. Gerdau, J. Metge, K. Ruth, W. Sturhahn, M. Grote, and R. Ro¨hlsberger, thickness variations proved to be unnecessary. 32 Hyperfine Interact. 58, 2467 1990 . In x-ray physics, the peculiar shape of the refractive index near an 23 O. Leupold, E. Gerdau, H. D. Ru¨ter, W. Meyer-Klaucke, A. X. atomic edge is caused by asymmetric resonance broadening R. Trautwein, and H. Winkler unpublished . W. James, The Optical Principles of the Diffraction of X-rays 24 The high resolution monochromator with channel-cut Si 4 2 2 G. Bell and Sons, London, 1958 , Chap. IV.1. k . Near an and Si 12 2 2 crystals in the nested geometry was designed by atomic edge, the atomic oscillator distribution, as obtained by E. Gerdau, R. Ru¨ffer, and H. D. Ru¨ter based on the proposal of x-ray absorption measurements, is extremely asymmetric, re- T. Ishikawa, Y. Yoda, K. Izumi, C. K. Suzuki, X. W. Zhang, M. sembling a saw-tooth distribution. For each oscillator of this Ando, and S. Kikuta, Rev. Sci. Instrum. 63, 1015 1992 . distribution, the scattering amplitude is negative above and posi- 25 A. Q. R. Baron, Nucl. Instrum. Methods Phys. Res. A 352, 665 tive below resonance. Within the distribution, the scattering am- 1995 . plitudes above the individual resonances are almost cancelled by 26 S. Kikuta, in Resonant Anomalous X-ray Scattering, edited by G. corresponding amplitudes of opposite sign belonging to neigh- Materlik, C. J. Sparks, and K. Fischer North-Holland, Amster- boring resonances of slightly higher energy. As a result, only a dam, 1994 , p. 635. small positive amplitude below the atomic edge remains. The 27 U. van Bu¨rck, R. L. Mo¨ssbauer, E. Gerdau, R. Ru¨ffer, R. Hollatz, essential point is, that the real part of resonant scattering, which G. V. Smirnov, and J. P. Hannon, Phys. Rev. Lett. 59, 355 changes sign at resonance, is extremely sensitive to the particu- 1987 . lar distribution of oscillators.