VOLUME 79, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 OCTOBER 1997 Quasielastic Scattering of Synchrotron Radiation by Time Domain Interferometry A. Q. R. Baron,1 H. Franz,2 A. Meyer,1,2 R. Rüffer,1 A. I. Chumakov,1 E. Burkel,3 and W. Petry2 1European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France 2Physik-Department, E13, Technische Universität München, D-85747 Garching, Germany 3Universität Rostock, FB Physik, D-18051 Rostock, Germany (Received 27 May 1997) We use synchrotron radiation and time resolved x-ray detection to measure structural relaxations of glycerol [C3H5(OH)3] having time scales of 30 to 200 ns at 1.5 Å21 momentum transfer. Foils containing 57Fe (14.4 keV nuclear resonance, 141 ns lifetime) are placed before and after the nonresonant sample, and a small difference 70 MHz is established in their nuclear response frequencies. Quasielastic scattering from the sample perturbs the 70 MHz quantum beat pattern of the nuclear scattering. A simple model relates the perturbation to the dynamic structure factor of the sample. [S0031-9007(97)04191-4] PACS numbers: 61.10.­i, 61.20.Lc, 76.80.+y The motion of atoms and molecules on angstrom length (possibly disturbing the system) or a more complicated and scales and few ns time scales is a subject of great interest, much lower count rate Rayleigh scattering of Mössbauer providing the possibility to gain insight into many physi- radiation (RSMR) measurement [10] must be performed. cal processes, including diffusion [1], glass transitions, and RSMR uses a narrow line source before the sample and a the motions of complex biological molecules and polymers narrow line absorber after the sample to directly measure [2,3]. Here we introduce a method of measuring quasielas- the frequency distribution of the scattered radiation. We tic x-ray scattering from samples having relaxations on note that there has been one attempt to do an RSMR time scales of 15 ns to more than 150 ns (energy scales measurement using synchrotron radiation [11]. of 50 to ,5 neV) and at large momentum transfers, in We describe this technique as time domain interferom- principle up to 14 Å21 (this work is at 1.5 Å21). The es- etry. The essence of most interferometry measurements sential idea of this technique is to set up a temporal in- is to combine the wave scattered from an object with a terference pattern in the nuclear scattering from two foils reference wave. Usually, one measures the spatial inten- containing 57Fe (14.4 keV nuclear resonance, 141 ns life- sity distribution and thus obtains information about the time) and then see how it is modified by quasielastic scat- spatial characteristics of the sample. Here we measure a tering from a sample placed between the two foils. temporal intensity distribution (quantum beats), and ob- Previously, scattering experiments at large momentum tain information about the temporal characteristics of the transfers and sub-meV resolution have largely been the scattering from the sample. We use two single line nu- domain of neutron work. The spin echo technique can clear scatterers (57Fe containing foils) chosen to have dif- measure relaxation on few ns (or shorter) time scales ferent response frequencies (see Fig. 1). After excitation (energy transfers of 100 neV or more) and momentum by a pulse of synchrotron radiation, the nuclear scattering transfers of a few Å21 [4]. However, high resolution (long from them will show quantum beats (our interference pat- time scales) and large momentum transfer are difficult to tern), as the radiation reemitted by the two foils at slightly achieve simultaneously. Light scattering techniques [5] different frequencies goes in and out of phase. If one can achieve neV energy resolution (or better), but are places a sample between the two foils, from which the limited to low momentum transfers. scattering changes on a time scale of the order of the In some special cases, when the sample contains a quantum beat period, it will perturb the beats. In essence, suitable nucleus with a low-lying excited state, it is also after the impulse excitation, the wave from the second foil possible to do x-ray (or g-ray) measurements with few acts as a reference, allowing the sample-induced modula- neV resolution. The Mössbauer effect allows one to take tions of the wave from the first foil to be measured [12]. advantage of the intrinsically small linewidths of nuclear The amplitude of the quantum beats may be related resonances, e.g., 5 neV for the 14.4 keV resonance of 57Fe, to the dynamic structure factor of the sample using to investigate samples either in conventional frequency- a semiclassical model. We take the responses of the domain absorption experiments [6], or, more recently, two 57Fe foils after impulse excitation at t 0 to be in time domain measurements [7]. Both frequency [8] G1 t e2i v01V t and G2 t e2iv0t, where 1 and 2 indicate and time domain [9] methods have been applied to the the foils before and after the sample, respectively. v0 is study of diffusion. However, these techniques are usually the nuclear response frequency, and V is the frequency limited in the momentum transfer to exactly that of the difference in the response of the two foils (which may absorbed photon (7.3 Å21 for 57Fe). If the sample does not be introduced artificially via Doppler shift). G1 and G2 contain a suitable isotope, then either one must be implaned are slowly varying functions of time (in the limit of thin 0031-9007 97 79(15) 2823(4)$10.00 © 1997 The American Physical Society 2823 VOLUME 79, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 OCTOBER 1997 is essentially the Debye-Waller factor [14]. One has I q, t ~ jG t j2 1 1 fqe q cos Vt Z 3 dv Sn q, v eivt , FIG. 1. Experimental setup. 57Fe stainless steel foils were where n indicates normalization (division) by the integral placed before and after the sample, with the first mounted of the dynamic structure factor over frequency. on a Mössbauer drive in constant velocity mode. Detectors Two effects are clear from the above equation. First, measured both the transmitted radiation and that scattered by if the amount of quasielastic scattering is small relative the glycerol into its structure factor maximum. to the total scattering fqe ø 1 , then the amplitude of foils, jG the beats will be small, and the measured response will 1j2 and jG2j2 will be exponential decays with the natural lifetime, 141 ns for 57Fe). be essentially the single foil response. Second, if there A pulse of synchrotron radiation excites foil 1, then is significant quasielastic scattering, then the decay of is scattered by the sample with some finite momentum the quantum beats (the reduction of the contrast of the transfer, q, and excites foil 2. Considering only single beats with time) will directly give the Fourier transform scattering from the sample (Born approximation limit), the of the dynamic structure factor. In particular, structural ratio of the field scattered by the sample to the incident field relaxation around a glass transition is frequently modeled is proportional to the integral over the electron density r using a Kohlrausch function [15], giving of the sample, with appropriate phasing for the momentum I q, t ~ jG t j2 1 1 fqe q cos Vt e2 t tk b , transfer. The time dependence of the scattered field (after impulse excitation at t 0) is where tk and b are the (q-dependent) parameters describ- Z ing the quasielastic scattering from the sample. E Note that no time domain effects are visible if one, m q, t ~ G1 t e2i v01V t rm r, t eiq?r dr or both, of the foils are removed. Within the above Z plane wave analysis, this is due to the average over 1 G2 t e2iv0t rm r, t 0 eiq?r dr . sample microstates. However, in light of recent x-ray The first term is the field from foil 1 after scattering intensity fluctuation measurements [16], we examine this from the sample, while the second is the field from foil 2 in more detail. In particular, if one could observe the (the reference wave). The subscript m indicates that this x-ray speckle pattern (essentially the exact diffraction expression is for some particular microstate of the system pattern of a microstate of the system) from the sample, and its subsequent evolution. We neglect the influence then one could relate temporal correlations in the speckle of the wave from foil 1 on the emission from foil 2, intensity of the sample dynamics. However, this typically which leads to a convolution term which is small if the requires that the incident beam have a size comparable to frequency shift, V, is much larger than the width of the its transverse coherence length (some few microns) and lines used in the experiment. The measured intensity that the detector size be similarly reduced (to that of a will be the absolute square of E single speckle) resulting in a severe drop in flux. Thus, m, averaged over sample configurations, since the experiment is performed over while x-ray correlation experiments have been extended many successive synchrotron radiation pulses, which are to times of 0.1 ms [17], reduction to ms scales will not correlated with the sample microstate. Taking (for be difficult even at third generation synchrotron radiation simplicity) foils 1 and 2 to be identical G sources. In addition, the bandwidth used to gain sufficient 1 G2 G , flux to achieve event the 0.1 ms resolution limits the range I q, t ~ jG t j2 S q, t 0 1 S q, t cos Vt , of momentum transfer to less than 1022 Å21. where S q, t , the intermediate scattering function, is pro- Our experiment was performed on the Nuclear Reso- portional to the Fourier transform of the dynamic structure nance Beamline [18] of the European Synchrotron Ra- factor [13], diation Facility (ESRF). The storage ring was run in Z Z 16 bunch mode, providing x-ray pulses of about 100 ps S q, t dr dr0 eiq? r2r0 rm r, t rm r0, t 0 duration every 176 ns. The incident radiation was tuned Z to the 14.4 keV nuclear resonance in 57Fe. The flux was ~ dv S q, v eivt. about 109 photons sec in a 6 meV bandwidth, at an aver- age storage ring current of 70 mA. The beam size was We take fqe to be the ratio of the scattering within our about 0.4 3 1.5 mm2. experimental time window to the total scattering [where the In order to demonstrate the feasibility of this technique, total scattering is the integral of S q, v over frequency, we choose glycerol as our sample because it has been or S q, t 0 ]. fqe includes both elastic scattering and the subject of many previous investigations. Here, we quasielastic scattering with time scales greater than about reference a tiny fraction of the recent work, hoping 10 ns (energy scales less than or the order of 75 neV) and that the interested reader will look up the references in 2824 VOLUME 79, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 OCTOBER 1997 the papers. Previous work includes measurements where resonant isotopes were introduced into the sample [19], RSMR experiments [20], and neutron scattering [21]. In particular, our results may be compared to RSMR measurements by Elwenspoek et al. [22]. The first foil [stainless steel (SS) 95% enriched in 57Fe] was mounted about 30 cm upstream of the sam- ple on a Mössbauer drive moved at a constant velocity of 6.3 mm sec 64.9G0 (see Fig. 1). A second SS foil (nearly identical to the first) was placed, at rest, 60 mm downstream of the sample and intercepted both x rays transmitted directly through the glycerol sample and those scattered at finite angles. Time resolving avalanche pho- todiode (APD) detectors were placed downstream of the FIG. 2. Forward scattering from (a) foil 1 alone and (b) both second foil. A small APD was used to measure the for- foils in series, with foil 1 on the Mössbauer drive at constant ward scattering from the two foils while a larger one, velocity (temperature at 200 K). Lines are fits (see text). 10 3 10 mm2 [23], was placed near the sample in the structure factor maximum for scattering from the glycerol introduced from the constant velocity motion of the drive q 1.5 Å21 or 2u 12± . The acceptances of the de- is clear. Note that we used a nitrogen bath cryostat to tectors, as seen from the sample, were about 2 3 1025 avoid introducing vibrations (e.g., by the compressor of and 2.4 3 1022 sr, respectively. A baffle was installed to a closed cycle refrigerator). The fit in the figure uses prevent the detector in the structure factor maximum from the parameters found from measurements of the two foils seeing scattering from the direct beam hitting the second individually and a constant velocity shift corresponding to foil. In addition, Al (120 mm) placed in front of this detec- 64.9G0. In addition, for good agreement, we included a tor and a high discriminator threshold ensured that back- slight broadening of 0.3G0 which is probably caused by ground events from 6.4 keV Fe K x-ray fluorescence were imperfect drive motion or residual effects of vibrations. not detected (e.g., proceeding from internal conversion in This broadening and the thickness distributions in the foils the second SS foil). Count rates in this detector were typi- account for the slight decrease in beat amplitude at later cally 2 s21 in the window of 15­150 ns after the prompt times. The forward time response was constant. pulse. Prompt rates were 105 Hz. Count rates in the for- Figure 3 shows the change in the time response in the ward detector were about 2 orders of magnitude larger. scattering at the structure factor maximum of the glycerol The glycerol sample, 5 mm thick, was mounted in with temperature. One immediately notes that the contrast a cryostat with thin kapton windows. Removal of the in the beats decreases with increasing temperature. In par- cryostat from the path of the direct beam showed the ticular, at 263 K, the beat contrast at later times is remark- background (due, e.g., to air scattering along the beam ably reduced from that at early times, corresponding to the path) in the detector in the structure factor maximum expectations for quasielastic scattering. In addition, the was about 3%. The contribution from the thin kapton amount of inelastic scattering increases with temperature, windows of the cryostat was estimated to be an additional leading to an overall, time independent, reduction in the 2%. Both backgrounds were ignored in the data analysis. beats. (Measurements were also made without foil 2 in Figure 2(a) shows the time dependence of the nuclear place. These always showed just the time response of the forward scattering (see [7]) from foil 1, without foil 2 first foil [Fig. 2(a)], independent of the glycerol tempera- in place. One observes the expected exponential decay ture, as expected from the model above.) modulated by a Bessel function from multiple nuclear The data were well fit using a stretched exponential scattering [24]. The minimum at t 75 ns is not very for S q, t . The quality of the data was not sufficient to sharp, indicating the SS foils have a thickness distribution. fit all parameters (b, tk, and fqe) independently so we A good fit to the time response was found taking chose b 0.7 as was found from neutron measurements an average thickness of T 27.3 (T the number of [21] and allowed the tk and fqe to vary. Fitting gave absorption lengths at resonance) and a distribution of tk . 5 ms, fqe 0.93 6 0.06 at 200 K, tk 180 6 sample thickness of 615% over the area of the x-ray 70 ns, fqe 0.87 6 0.05 at 250 K, and tk 34 6 8 ns, beam. This thickness corresponds to a linewidth 6G0 fqe 0.79 6 0.05 at 263 K. (At 276 K, the data quality in a conventional Mössbauer transmission experiment did not allow unique determination of the parameters- (neglecting source width). The time response of foil 2 is here we show the data are consistent with tk 12 ns, nearly identical to that of foil 1, with an average thickness fqe 0.7.) The results (200­263 K) agree nicely with of T 26.2 and a distribution of 617%. the RSMR work of Elwenspoek et al. [22]. Figure 2(b) shows the time dependence of the forward We have demonstrated a new technique that may be scattering in the direct beam passing through the two used to do time domain measurements of quasielastic foils. The quantum beat at 70 MHz (13.6 ns period) scattering with large momentum transfers. In comparison 2825 VOLUME 79, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 OCTOBER 1997 H. F., A. M., and W. P. acknowledge the support of the BMBF (Project No. 05 643WOB), as does E. B. (Project No. 05 643HRA5). [1] K. S. Singwi and A. Sjölander, Phys. Rev. 119, 863 (1960); 120, 1093 (1960). A recent review is G. Vogl, in Mössbauer Spectroscopy Applied to Magnetism and Materials Science, edited by G. B. Long and F. Grandjean (Plenum Press, New York, 1996), Vol. 2, p. 85. [2] See, e.g., articles in Dynamics of Disordered Materials, edited by D. Richter, A. J. Dianoux, W. Petry, and J. Teixeira (Springer-Verlag, Berlin, 1989), Vol. 37. [3] A recent review is G. U. Nienhaus and F. Parak, Hy- perfine Interact. 90, 243 (1994). See also F. Parak and H. Fraunfelder, Physica (Amsterdam) 201A, 332 (1993). [4] See, e.g., articles in Neutron Spin Echo, edited by F. Mezei (Springer-Verlag, Berlin, 1980), Vol. 128. [5] See, e.g., articles in Dynamic Light Scattering, edited by R. Pecora (Plenum Press, New York, 1985). [6] R. L. Mössbauer, Z. Phys. 151, 124 (1958). [7] Experimental papers include E. Gerdau et al., Phys. Rev. Lett. 54, 835 (1985); E. Gerdau et al., Phys. Rev. Lett. 57, 1141 (1986); J. B. Hastings et al., Phys. Rev. Lett. 66, 770 (1991). A recent review is G. V. Smirnov, Hyperfine Interact. 97/98, 551 (1996). FIG. 3. Scattering from glycerol measured in structure factor [8] G. Vogl, Hyperfine Interact. 53, 197 (1990), and refer- maximum at different temperatures. Lines are fits (see text). ences therein. to other synchrotron based work, this technique has ac- [9] B. Sepiol et al., Phys. Rev. Lett. 76, 3220 (1996); cess to a unique range of momentum and time scales. The A. Meyer et al., Z. Phys. B 103, 479 (1997). time scales here are much longer than the ps time scales [10] D. C. Champeney, Rep. Prog. Phys. 42, 1017 (1979). of even the highest resolution inelastic x-ray scattering [11] J. Z. Tischler et al., in Resonant Anomalous X-Ray measurements with either crystal analyzers [25] or nu- Scattering, edited by G. Materlik et al. (Elsevier Science B.V., Malente, Germany, 1994), p. 647. clear absorption techniques [26] and much shorter than the [12] Conceptually, this is similar to an optical heterodyne 0.1 ms scales that have been reached by speckle mea- technique. H. Z. Cummins et al., Phys. Rev. Lett. 12, surements [17]. Neutron scattering measurements provide 150 (1964). more direct competition with this technique, allowing [13] Correlations and response functions are summarized in access to similar (or slightly shorter) time scales and S. W. Lovesey, Theory of Neutron Scattering from Con- momentum transfers. However, they have difficulty si- densed Matter (Clarendon Press, Oxford, 1994), Vol. 1. multaneously achieving both high energy resolution and This formalism is usually traced back to L. van Hove, large momentum transfer, and they may suffer from inco- Phys. Rev. 95, 249 (1954). herent scattering backgrounds. The last is very different [14] In the event that the sample has significant excitations from this work where the dominant channel (Thomson outside the bandwidth of the incident radiation (6 meV charge scattering) is entirely coherent. for this work) there will be a correction. [15] R. Kohlrausch, Ann. Phys. (Leipzig) 91, 56 (1854). This work is similar to the RSMR technique, and might [16] S. Brauer et al., Phys. Rev. Lett. 74, 2010 (1995). be thought of as a time domain analog to that method. [17] T. Thun-Albrecht et al., Phys. Rev. Lett. 77, 5437 (1996). However, aside from being a direct time domain measure- [18] R. Rüffer and A. I. Chumakov, Hyperfine Interact. 97/98, ment which might add insight in some cases, this method 589 (1996). seems to more easily access longer time scales, which are [19] G. U. Nienhaus et al., Phys. Rev. B 43, 3345 (1991). sometimes obscured by the broad linewidths of the sources [20] K. Ruebenbauer et al., Phys. Rev. B 49, 15 607 (1994). and absorbers used in RSMR. Also, the inherent bril- [21] J. Wuttke et al., J. Chem. Phys. 105, 5177 (1996). liance of synchrotron radiation permits one to do experi- [22] M. Elwenspoek, M. Soltwisch, and D. Quitmann, Mol. ments with small 1 mm2 samples and with modest or Phys. 35, 1221 (1978). even extreme (mrad) collimation, such as investigation of [23] A. Q. R. Baron, Nucl. Instrum. Methods Phys. Res., quasielastic scattering in the neighborhood of Bragg peaks. Sect. A 353, 665 (1994). [24] Y. Kagan et al., J. Phys. C 12, 615 (1979). The authors thank G. V. Smirnov for useful discus- [25] E. Burkel et al., Europhys. Lett. 3, 957 (1987); F. Sette sions. We thank E. Gerdau for loaning us the SS et al., Phys. Rev. Lett. 75, 850 (1995). foils and J. Peisl for the bath cryostat. A. B thanks [26] M. Seto et al., Phys. Rev. Lett. 74, 3828­3831 (1995); F. Comin for facilitating the completion of this paper. W. Sturhahn et al., Phys. Rev. Lett. 74, 3832 (1995). 2826