VOLUME 77, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 2 DECEMBER 1996 Transverse X-Ray Coherence in Nuclear Scattering of Synchrotron Radiation A. Q. R. Baron, A. I. Chumakov, H. F. Grünsteudel, H. Grünsteudel,* L. Niesen, and R. Rüffer European Synchrotron Radiation Facility (ESRF), BP 220, F-38043 Grenoble Cedex, France (Received 17 July 1996) The time response of nuclei excited by pulsed synchrotron radiation is affected by correlations in the nuclear response transverse to the beam direction. The coherent addition of the radiation scattered by nuclei having a distribution of Doppler shifts accelerates the decay in the forward scattering from a rotating foil of 57Fe stainless steel. A model based on Huygens's construction shows good agreement with the data, allowing estimation of the source size or transverse coherence length. Implications for spectroscopic experiments using nuclear forward scattering are discussed. [S0031-9007(96)01697-3] PACS numbers: 76.80.+y, 42.25.Hz, 78.70.Ck Conclusions about microscopic properties are fre- Here, as distinct from previous work, we focus on the quently drawn from measurements of macroscopic effects of transverse coherence: how the responses of samples. This is true for most experiments using x-ray nuclei in a sample separated perpendicular to the beam probes since x-ray sources are generally too weak (and the combine in forward scattering. Interest in the effects relevant cross sections too small) to directly investigate of transverse coherence, including speckle measurements microscopic samples. It is then necessary to properly [8,9] and phase contrast microscopy [10,11] is increasing relate the macroscopic measurement to the microscopic with the advent of higher brilliance x-ray sources. With response. For this purpose, one identifies the limits of nuclear scattering, one may combine a spectroscopic mea- incoherent and coherent scattering [1]. Most x-ray spec- surement with structural information from transverse co- troscopic measurements (e.g., XAFS, x-ray fluorescence) herence. Aside from its basic physical interest, transverse measure incoherent scattering processes which are essen- coherence is of practical importance for NFS measure- tially local. Structural measurements (e.g., diffraction), ments on thin samples exhibiting domain structure [12]. in contrast, typically investigate coherent scattering from A priori, one might expect there to be two limits: a "well- extended portions of the sample. mixed" sample where the response of the nuclei should be This paper explores coherence in a relatively new added coherently and a "well-separated" case where the re- type of spectroscopy: nuclear forward scattering (NFS) sponses would add incoherently. This paper investigates of synchrotron radiation [2,3]. NFS experiments might the border between these limits. be considered the time-domain analog of conventional The essential concept for this experiment is that we frequency-domain Mössbauer transmission experiments. introduce a uniform spatial gradient in the velocity of They examine the time response of nuclei excited by nuclei in a sample, resulting in a corresponding spatial a pulse of synchrotron radiation, preserving the in-line gradient in the Doppler shift in their response (absorption/ source-sample-detector arrangement of the transmission emission) frequencies. This gradient is carefully arranged experiment. However, where the transmission experiment so that all nuclei along the path of an infinitesimal cross measures the absorption of a beam passing through a section beam respond at the same frequency, and also sample, the NFS measures the coherently re-emitted so that transverse displacement of this beam will shift wave field. Reviews of nuclear scattering of synchrotron that frequency. Noting that the NFS time response of a radiation may be found in [4] and [5], while some aspects moving foil is independent of its absolute velocity, the of coherence in such measurements are discussed in [6]. gradient introduced in this fashion will only affect the Coherence in NFS is immediately interesting and perti- measured time response if there is a coherent addition nent for a sample having different nuclear environments. of the scattering from transversely separated portions of If these environments result in different nuclear response the sample. In particular, the coherent addition of the frequencies, then coherent combination of their impulse re- scattering from nuclei with a distribution of Doppler sponses will give corresponding quantum beats in the time shifts, essentially a broadening of the frequency response, response, while incoherent combination will not. Previous is expected to result in a faster decay of the impulse work has investigated several limits. NFS from metallic response. This acceleration of the decay is the signal we 57Fe samples shows beats between the transitions to dif- look for and observe. ferent (Zeeman split) ground states [2,3]. Similarly, two This work was done at the nuclear resonance beam samples placed in succession, one after the other, in the line [13] of the European Synchrotron Radiation Facility x-ray beam (or in the arms of an amplitude splitting inter- (ESRF). The synchrotron provided short 120 ps ferometer [7]) give a time response in NFS that is a coher- pulses of x rays every 2.8 ms. The bandwidth of ent combination of the responses of the individual samples. the 14.4 keV radiation was reduced to 3 eV using a 4808 0031-9007 96 77(23) 4808(4)$10.00 © 1996 The American Physical Society VOLUME 77, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 2 DECEMBER 1996 Si 111 monochromator and then to 6 meV using "high z0 is the location of the point source, zd is the location resolution monochromator" [13­15]. The radiation was of a point detector, and zs is the source point of the allowed to fall onto a stainless steel (SS) foil (4.5 mg cm2) secondary waves in the sample (see Fig. 1). Ws zs is 95% enriched in 57Fe, oriented at 45± to the beam path. the field amplitude at the sample. The first exponential The foil was mounted on a dc motor (horizontal axis accounts for the Doppler shift in the nuclear response perpendicular to the foil surface) run at rotation rates up to frequency with height [18]. t0 is the natural lifetime of 100 Hz. An avalanche diode detector [16] measured the the nucleus (t0 141 ns for 57Fe), and A is the vertical NFS time response (see Fig. 1). Slits limiting the vertical distance necessary to shift the nuclear response frequency aperture of the system were placed immediately (10 cm) by one natural linewidth. One has A y0 2pN sin u after the foil and just (4 cm) in front of the detector. The where u is the angle between the rotation axis and the slits were measured with the x-ray beam to be 15 mm beam direction, y0 is the Doppler velocity corresponding high, full width at half maximum (FWHM). They were to a natural linewidth, and N is the rotation frequency. larger than the beam size in the horizontal. With the Taking y0 97 mm sec (appropriate for the 14.4 keV slits in place, count rates were 10 photons s in the time resonance of 57Fe) and u p 4, A mm 22 N Hz . window of about 10­150 ns after the synchrotron pulse w in Eq. (1) is the (relative) geometric phase associated ( 10 mA storage ring current). with the different path lengths for different points of the The NFS from the SS foil at rest is shown in Fig. 2(a). sample. Taking S to be the distance from the source plane This is well fit using an exponential decay modulated by a to the sample plane and D the distance from the sample Bessel function, as expected for the NFS from a single line plane to the detector (see Fig. 1), one has k 2p l nuclear response [3,17]. As the rotation rate is increased µ µ [Fig. 2(b)­2(d)], the decay becomes much faster. This z z k 1 1 w z 0 d 0, zs, zd 2k 1 1 1 z2s. (2) demonstrates interference between transversely separated S D 2 S D parts of the beam. In contrast, the response measured at The term linear in zs is kept in the usual Fraunhofer limit high rotation rate, without slits, is indistinguishable from [19]. For our experimental geometry, terms of higher that measured at rest [Fig. 2(a)]. One also notes that, order than second are negligible. within statistics, the location of the Bessel minimum (at One must integrate the intensity over the finite source t 55 ns) was not affected by rotation. and detector sizes. For a detector of spatial acceptance In order to understand these results we develop a model given by Wd, and a source intensity distribution W0, the based on Huygens's construction. For a point source, the response observed by a pointlike detector is the integral over the secondary waves emitted by the nuclei in the sample. Taking the impulse response of the foil at rest to be G0 t , integration over the foil yields the time- dependent field amplitude at the detector Z E z0, zd, t ~ G0 t dzse2izst At0eiw z0,zs,zd Ws zs . (1) FIG. 2. Forward scattered time response for (a) the foil at rest FIG. 1. Experimental setup. The perspective drawing shows (with slits in place) or at 95 Hz without slits. (b)­(d) show the general orientation while the cross section shows quantities the response at higher rotation rates. (b)­(d) are normalized in used in the text. The source-sample distance was S 41 m. the same incident rate and counting time as the response at rest The detector was D 2.5 m downstream of the sample. in (a). 4809 VOLUME 77, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 2 DECEMBER 1996 measured time response is lifetime is modified by the detector, source, and sample Z Z sizes, as one integrates over the different phasing of the I t ~ dz0dzdjE z0, zd, t j2W0 z0 Wd zd . (3) response pattern. Large detectors, sources, and samples blur the interference pattern leading to measurement of an Momentarily, we illustrate some of the essential con- incoherent response. cepts of this problem by considering a time-domain two The measured time responses, divided by the fit to the slit interference pattern. If the rotating foil is masked by time response at zero rotation rate, are shown in Fig. 3. two small slits, having separation L, and, for simplicity, The solid lines are calculations using Eq. (3), assuming negligible height, the intensity measured by a point detec- slits of 15 mm height limit the illumination of the sample tor at zd for a point source at z0 is found from Eq. (1) by and detector. In order to obtain good agreement with taking WS zs d zs 1 L 2 1 d zs 2 L 2 , µ the data, it was necessary to assume a source size of z z I z 0 d about 290 mm FWHM s0 120 mm . In contrast, 0, zd, t ~ jG0 t j2 1 1 cos Vt 1 kL 1 kL , S D measurements of the electron beam emittance suggest the (4) vertical source size should be about 55 mm FWHM [22]. where V L At The difference is explained by imperfect transport of the 0 is the frequency difference between the response at the 1L 2 and 2L 2 locations of the slits. x-ray beam from the source to the sample. In general, this The intensity variation with detector position (for t could be the result of windows and filters installed on the fixed) is a two-slit spatial interference pattern with angular beam line [11,23]. However, recent measurements [24] period l L. The intensity contrast is a direct indication of the fringes from a boron fiber [10] are in agreement of the coherence of the scattering from the two slits with the results measured here and show that the dominant [20]. If either the source or the detector subtend angles contribution to the increase in the source size is due to the larger than l L, the contrast will be reduced and the slit high resolution monochromator. responses become incoherent. The same effect occurs in Figure 4 demonstrates the effect of shifting the detector the time response. The two geometric terms in the cosine out of the direct line of source and sample (i.e., to finite determine the initial phase of the temporal beat pattern, zd). As expected from Eq. (5), there is a shift in the peak which then oscillates with frequency V 2p. Large of the time response. This shift (and the good agreement detector or source sizes lead to integration over differently with theory) is further confirmation of the interpretation phased beat patterns and incoherent superposition of presented here. One notes that this not a slowing down responses. This qualitatively demonstrates the importance of the NFS response, which shows an accelerated decay of the slit in front of the detector and small source size. after the peak is reached. Returning to the rotating foil calculation, an analytic We define an effective coherence length Lc as the mini- form for the response may be derived if all distributions mum transverse separation between parts of the sample are assumed to be Gaussian (where the size of the source, necessary to ensure that their responses add incoherently. the beam on the sample, and the detector are given by s Investigation of Eqs. (5) and (6), or extension of Eq. (4), 0, s gives Lc l 2pa. In the limit that both the detector s, and sd, respectively). This gives [21] µ 1 1 t z 2 I z d d, t . 0 ~ jG0 t j2 exp 2 1 . 2a2 kA t0 D (5) The quantity zd allows for a possible offset of the center of the detector from the direct line of the source and sample. a is the measured divergence of the radiation from the foil at rest: µ s2 s2 s s 2 1 a2 0 1 d 1 s 1 s 1 . (6) S2 D2 D S 4k2s2s The various terms in a are identified as the effects of finite source, detector and sample sizes, and the Fraunhofer diffraction term, respectively. The foil responds with a reduced lifetime and a shift in the time response. Both may be qualitatively explained by the two-slit example. The shift in the time response with detector position is directly analogous to the above mentioned shift in phase of the beat pattern with detector position. Here, however, only the initial part FIG. 3. Measured time responses divided by the fit to the response at rest. Solid lines are calculations based on Eq. (3). of the "beat" pattern is visible before the continuum of The vertical scale is the intensity relative to response of the foil excited frequencies washes it out. Likewise, the observed at rest. 4810 VOLUME 77, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 2 DECEMBER 1996 Permanent address: Nuclear Solid State Physics, Materials Science Center, University of Groningen, N’enborgh 4, 9747 AG Groningen, Netherlands. [1] See, e.g., S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, edited by R. K. Adair et al., International Series of Monographs on Physics (Clarendon Press, Oxford, 1994), Vol. 1. [2] J. B. Hastings et al., Phys. Rev. Lett. 66, 770 (1991). FIG. 4. Effect of moving the detector off the direct source- [3] U. van Bürck et al., Phys. Rev. B 46, 6207 (1992). sample axis. The phasing of the scattered wave shifts the peak [4] E. Gerdau and U. van Bürck, in Resonant Anomalous of the response to later times. Normalization as in Fig. 3. X-Ray Scattering. Theory and Applications, edited by G. Materlik, C. J. Sparks, and K. Fischer (Elsevier, New and sample are small, this definition corresponds with York, 1994), pp. 589­608. the usual one (see, e.g., [20]). However, as the effects [5] G. V. Smirnov, Hyperfine Interact. 97/98, 551 (1996). [6] U. van Bürck and G. V. Smirnov, Hyperfine Interact. 90, of sample and detector sizes are not negligible for most 313 (1994). NFS measurements, we explicitly include them here. In [7] K. Izumi et al., Jpn. J. Appl. Phys. 34, 4258 (1995). fact, typical running conditions at ESRF (without the small [8] M. Sutton et al., Nature (London) 352, 608 (1991). slits used here) will be dominated by the sd 0.5 mm [9] S. Brauer et al., Phys. Rev. Lett. 74, 2010 (1995). detector acceptance at a distance of D 1 m. One then [10] A. Snigirev et al., Rev. Sci. Instrum. 66, 5486 (1995). expects incoherent addition of the responses for parts of [11] P. Cloetens et al., J. Phys. D 29, 133 (1996). the sample separated transversely by more than Lc [12] H. F. Grünsteudel et al., Hyperfine Interact. C 1, 509 300 Å. However, bulk samples exhibiting structure of this (1996). scale transverse to the x-ray beam may be expected to [13] R. Rüffer and A. I. Chumakov, Hyperfine Interact. 97/98, have similar structure parallel to the beam direction [25]. 589 (1996). This should result in coherent combination of the various [14] T. Ishikawa et al., Rev. Sci. Instrum. 63, 1015 (1992). [15] T. S. Toellner, T. Mooney, S. Shastri, and E. E. Alp, responses until the domain size begins to approach the in Optics for High-Brightness Synchrotron Beamlines, sample thickness, in which case incoherent addition would edited by J. Arthur SPIE Proceedings Vol. 1740 (SPIE- begin to dominate. International Society for Optical Engineering, Bellingham, An application of this work would be to change WA, 1992), p. 218. effective coherence length to investigate the scale of [16] A. Q. R. Baron, Nucl. Instrum. Methods Phys. Res., variation of the nuclear response in a sample. With Sect. A 353, 665 (1994). the slits used here, Lc 3 mm, but one could hope to [17] Y. Kagan, A. M. Afanas'ev, and V. G. Kohn, J. Phys. C increase this length a factor of 2 or more, by increasing 12, 615 (1979). the sample-detector separation or shrinking slit sizes. [18] The velocity along the direction of x-ray propagation Removal of the detector to infinity makes this an angular depends only on the height above the (horizontal) plane resolved measurement, so one expects to see similar containing the x-ray beam and the rotation axis. [19] M. Born and E. Wolf, Principles of Optics (Pergamon changes in the time response by employing a crystal Press, New York, 1980), pp. 382­386. analyzer after the sample [26]. Recent work shows this [20] L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965). is the case [27], and such an analyzer might provide an [21] The integral over the exponential of a second order easier method of investigating the sample length scales via polynomial with complex coefficients may be found in the time response than the slits used here, which reduce I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, the count rate. Finally, while the velocity correlation Series and Products (Academic Press, Inc., New York, introduced in this work provides a good test case, more 1980), p. 485, Eq. 3.923. generally, the nuclear scattering amplitude is sensitive to [22] J. M. Filhol (private communication). hyperfine fields (magnetic dipole and electric quadrapole) [23] A. Snigirev, I. Snigireva, V. G. Kohn, and S. M. and even to chemical binding (isomer shifts), so one may Kuznetsov, Nucl. Instrum. Methods Phys. Res., Sect. A investigate length correlations in these quantities as well. 370, 634 (1996). [24] A. Snigirev, I. Snigireva, C. Raven, and M. Drakopoulos A. B. would like to thank P. Cloetens for discussing this performed these measurements. work with him. We also thank the ESRF staff for helping [25] Some structured samples, especially multilayers in a to make experiments possible on ID18, particularly, grazing incidence specular reflection geometry, might J. Ejton and Z. Hubert. We thank E. Gerdau for the loan provide exceptions. of the SS foil. H. G. acknowledges the support of the [26] This is similar to a small angle scattering geometry. In Deutsche Forschungsgemeinschaft. fact, the angular distribution of nuclear scattering gives information about magnetic correlation lengths in iron samples. Yu. V. Shvyd'ko et al., in Phys. Rev. B (to be *On leave from Medizinische Universität zu Lübeck, published). Institut für Physik, Ratzeburger Allee 160, D-23538 [27] A. Q. R. Baron et al. (unpublished); and also Lübeck, Germany. R. Röhlsberger (private communication). 4811