PHYSICAL REVIEW B VOLUME 57, NUMBER 9 1 MARCH 1998-I Exact Bragg backscattering of x rays Yu. V. Shvyd'ko, E. Gerdau, J. Ja¨schke, O. Leupold, M. Lucht, and H. D. Ru¨ter II. Institut fu¨r Experimentalphysik, Universita¨t Hamburg, D-22761 Hamburg, Germany Received 27 October 1997 Exact 180° Bragg scattering of pulsed synchrotron radiation was observed by using a semitransparent detector and a time-of-flight technique. The angular dependences of Bragg scattering of monochromatic 14.413 keV x rays with only 0.5 eV bandwidth were studied by utilizing the (1 3 4 28) reflection of an Al2O3 crystal at different temperatures. By heating the crystal first the exact backscattering shows up achieving maximum intensity and 1.7 mrad angular width at 372.40 K. By increasing the temperature it develops into the usual Bragg reflection with much narrower angular profile. The measured dependences are in agreement with the theory. The fit is sensitive to a 5 10 9 Å variation of interplanar distance. S0163-1829 98 08609-3 Bragg scattering of x rays with angles close to 180° is a pulses and secondly to count only resonant photons in the 0.5 well established experimental technique. It is successfully eV band. The beam divergence was 20 rad in the vertical applied for monochromatization of hard x rays to meV and (z,x) and 80 rad in the horizontal (z,y) plane. sub-meV bandwidths.1­8 Despite this fact the properties of Sapphire single crystals, instead of Si or Ge which are backscattering remain experimentally not fully studied. So standard in x-ray crystal optics, were used as backscattering far, to our knowledge, in no experiment were x rays ob- mirrors. Several reasons motivated this choice. Bragg's law served which were reflected from a crystal exactly opposite 2dhkl sin B hc/E for x-rays with energy E scattered from to the direction of the incident beam.9 It is due to this fact crystal planes (hkl) with interplanar distance dhkl reduces in that up to now applications involving exact backscattering, the case of backscattering ( B /2) to EB hc/2dhkl . Ev- e.g., Fabry-Perot10,11 and other types of x-ray backscattering ery reflection (hkl) has its Bragg energy EB . However, be- interferometers are not realized. Our purpose was to observe cause of the cubic symmetry of Si and Ge many reflections exact Bragg backscattering of x rays, to study its angular and have the same dhkl and thus the same Bragg energy. There- energy dependences, to observe its transition to the usual fore the number of matching energies is low: once per Bragg scattering and to test the validity of the dynamical 500­250 eV in the range 10­25 keV. Furthermore, pure ex- theory of x-ray diffraction under these extreme conditions. act backscattering in crystals with such a high symmetry is Observation of scattering at exactly 180° has an obvious not possible due to multiple beam Bragg diffraction,10 and experimental difficulty. The x-ray source or the detector is the reflectivity for hard x rays with E 25 keV decreases rapidly because of the relatively low Debye temperatures. blocking the reflected or the incident x rays, respectively. To By contrast crystals with lower symmetry like hexagonal overcome this problem we have applied a semitransparent Al detector with a good time resolution and made use of the 2O 3 with high Debye temperature and low photoabsorp- tion suit much better for backscattering. Sapphire single pulsed structure of synchrotron radiation. The time-of-flight crystals allow exact backscattering at least once per 15 eV 2L/c to the crystal and back to the detector separated by in the range 10­25 keV and even more often for harder x a distance L was used to distinguish between the incident rays. By heating or cooling Al and reflected radiation pulses, see Fig. 1. 2O 3 for not more than 100 K from room temperature one can fulfill the backscattering Our experiment (L 9.9 m and 66 ns was performed condition for any x-ray energy above 10 keV. E.g., for the at the wiggler beam line BW4 at HASYLAB Hamburg . The DORIS-III positron storage ring was operated in the five-bunch mode producing radiation pulses from the wiggler every 192 ns ( 200 ps duration . A double-crystal Si 111 monochromator provided radiation with 3 eV bandwidth at the energy of the nuclear resonance in 57Fe 14.413 keV . The 57Fe nuclei in an iron foil of 10 m thickness enriched in 57Fe to 95% were excited resonantly by the incident beam. They emit 14.413 keV photons predominantly in the forward direction12,5 within a narrow energy band of less than 0.5 eV width and with an average delay of 40 ns. Only these delayed resonant quanta are counted to make use of the ra- FIG. 1. Scheme of the backscattering experiment; M is the diation with 0.5 eV bandwidth. As a semitransparent detec- Si 111 monochromator shown rotated by 90° about z axis ; D is tor served an avalanche photodiode with a 100 m thick the semitransparent x-ray detector; F is the 10 m 57Fe foil; V is sensitive Si wafer.13 Its time resolution was 1 ns. This time the 8 m vacuum tube; S is the Al2O3 single crystal in the oven; resolution was thus used twice in the experiment. Firstly, to is the angle between the wave vector k of the incident radiation and distinguish between the incident and reflected radiation the scattering vector H. 0163-1829/98/57 9 /4968 4 /$15.00 57 4968 © 1998 The American Physical Society 57 BRIEF REPORTS 4969 FIG. 2. Angular dependence of the exact Bragg scattering of the 14.413 keV x rays with 3 eV bandwidth from an Al2O3 crystal measured at different angular deviations from normal incidence to the (1 3 4 28) reflecting planes. The detector was not moved. Therefore the width of the rocking curve is given by the angular acceptance of the detector. FIG. 3. The temperature dependences of Bragg scattering of 14.413 keV 57Fe energy the (1 3 4 28) back reflection in resonant 14.413 keV photons with 0.5 eV bandwidth from an sapphire can be predicted at 380 K with angular acceptance Al of 1.3 mrad, energy bandwidth of 6.2 meV, and 88% reflec- 2O 3 crystal measured at different deivations from normal in- cidence to the (1 3 4 28) reflecting planes. Solid lines are fits with tivity the crystal data of Refs. 14,15 were used . Lorentzians. The width of the curve at 0.2 mrad is 100 mK In our experiments sapphire crystals in form of a disk 15 equivalent to 10.0 meV x-ray bandwidth . mm in diameter and 1 mm thick cut perpendicular to the c axis were employed.16 The crystal was installed in an oven Bragg reflection. Nevertheless the width of the Bragg reflec- on a four-circle diffractometer at the end of the vacuum tube, tion at 2.78 mrad still has the large value of 0.24 mrad. see Fig. 1. At first the crystal c axis was oriented parallel to The reflectivity of 14.413 keV nuclear resonant radiation the incident beam by detecting the exact back reflection was measured to be 64%, close to the theoretical value for a 0 0 0 30 of x rays with an energy E 14.315 keV. Then the perfect crystal, at the exact 180° Bragg scattering condition. (1 3 4 28) planes were set perpendicular to the incident The angular dependences were fitted by using the dynami- beam by detecting the exact back reflection (1 3 4 28) of cal theory of x-ray diffraction. The principles underlying the the incident 14.413 keV x rays with the broad 3 eV band. theory see, e.g., Refs. 17,18 are valid for backscattering The angular position of the crystal at which it reflects these x too.1,19,20 The solution for the angular and energy depen- rays back into the detector with maximum intensity was dences of Bragg scattering is expressed through the scatter- taken as the reference point 0 for the exact backscatter- ing amplitudes 0 , and h , crystal thickness l , asymmetry ing, see Fig. 2. As a next step the temperature of the crystal parameter b and the parameter a (H2 2Hk)/k2 which is a was scanned to find the temperature region at which the function of the energy E and the angle of incidence 14.413 keV resonant quanta in the 0.5 eV band are re- /2 of the x rays. The total reflectivity in a nonab- flected. sorbing ( 0, h 0) and semi-infinite crystal occurs within the Figure 3 shows such temperature scans recorded at differ- interval (a ,a ) where ent angular deviations of x rays from normal incidence to the (1 3 4 28) planes. The temperature in the oven was con- a 2 h / b 0 1/b 1 . 1 trolled with a relative accuracy of 1 mK. At 0.2 mrad the maximum of reflectivity is achieved at T We have found that the following presentation for a: R 372.40 0.01 K. The width of the reflection curve is 100 mK cor- responding to 10.0 meV energy width. This is more than the E a 4 B EB expected 6.2 meV. With increasing the temperature E E sin 2 where maximum reflectivity is reached increases propor- can be used which is valid for any value of the angle tional to 2. This square dependence which is observed 0 . However, the theoretical approaches of Refs. 17 only for backscattering has a remarkable consequence, and 18 use the approximation a 2( namely an extraordinarily large angular acceptance as dem- B )sin 2 B which fails in the backscattering region. As it was shown in Ref. 20 onstrated below. another approximation is applicable for backscattering Figure 4 shows the angular dependences of Bragg scatter- ( 1): ing of 14.413 keV resonant radiation measured at different temperatures T of the same Al 2O3 crystal. Below the refer- a 2 2 2 E E ence temperature at T T T B /EB . 3 R 0 the Bragg scattering scarcely takes place. Approaching TR the Bragg scattering Figure 5 is a kind of DuMond diagram modified for back- builds up with the maximum at 0 and reaches the maxi- scattering note the 2 dependence , i.e., the spectral- mum angular width full width at half maximum of 1.7 angular region of total reflection which is depicted according mrad at T 0. With increase of the temperature above TR to Eqs. 1 , 3 for the particular case b 1. The diagram the maximum reflectivity deviates from 0, the rocking helps to see that our experimental results Figs. 3,4 qualita- curve narrows and gradually develops into the conventional tively agree with the main theoretical predictions of Refs. 4970 BRIEF REPORTS 57 FIG. 5. Spectral-angular region of total reflection of the Bragg backscattering. Note that the abscissa scales with 2. lines in Fig. 4. By taking crystal defects more precisely into account one could certainly describe better the wings of the angular distributions. It is remarkable that the fit turned out to be sensitive to as small as a 5 10 9 Å variation of the average interplanar distance. The temperature dependence obtained from this fit is dhkl dR(1 6.944 10 6 T/K) where dR 0.430 108 Å. By using this dependence one ob- tains for the variation of the energy of backscattered x rays with crystal temperature a value of dE/dT 0.100 eV/K. The following data were used in the calculations: E 14.4132 keV, 0 ( 78.395 i0.289) 10 7, h ( 4.036 i0.129) 10 7. In summary, exact 180° Bragg scattering of synchrotron radiation was observed by using a semitransparent detector and the time-of-flight technique. The angular dependences of Bragg scattering of resonant 14.413 keV 0.5 eV FIG. 4. The angular dependences of Bragg scattering of mono- bandwidth photons were studied at different temperatures chromatic 14.413 keV x rays with 0.5 eV bandwidth measured at of Al different temperatures T of an Al 2O3 crystals. It was observed that by heating the 2O 3 crystal utilizing the crystal first the exact backscattering shows up with an (1 3 4 28) reflection. is the angular deviation from normal in- extremely broad angular width. By increasing the tempera- cidence of the x rays to the (1 3 4 28) reflecting planes. T T ture it develops into the usual Bragg reflection with much TR where TR 372.40 K. Solid lines are the fits using the dynami- cal theory of Bragg scattering. narrower angular profile. The results are in a good agreement with the dynamical theory of Bragg scattering. The fit 1,19,20. It also shows that the largest angular acceptance of is sensitive to a 5 10 9 Å variation of the interplanar 2 h and simultaneously the highest relative energy reso- distance. lution h are achievable in the exact backscattering geom- We have found by extended simulations that for x rays of etry. The fits of Fig. 4 were performed by using the results of any energy in the range 10­70 keV noncubic and hard single Refs. 17 and 18 but with the nonapproximated parameter a crystals like sapphire allow backscattering with significant of Eq. 2 . reflectivity at a temperature in the range of 200­450 K. This We have made certain assumptions to take into account may open a broader field of applications of Bragg back- the observed broadening of the temperature profiles and an- scattering in x ray optics with enhanced luminosity and high gular curves compared with the theoretical widths for an energy resolution. Backscattering mirrors for x ray interfer- ideal crystal. It is supposed that the broadening is caused by ometers and resonators would be possible. Especially attrac- a variation of the interplanar distance dhkl with crystal depth. tive is the fact that the longitudinal coherence length of As the real distribution is not known, it is simply assumed nuclear resonant photons is in the order of meters. Also for that at some depth the atomic planes are shifted by half of any Mo¨ssbauer transition in the mentioned energy range not the radiation wavelength relative to the ideal undisturbed po- only one reflection can be found to be used for a meV or sition. Accordingly the coherent response of the rest of the sub-meV energy-resolution backscattering monochromator. crystal does not contribute significantly to the scattering. The This will allow observation of new nuclear excitations and crystal thus has a smaller effective thickness. We have ascer- promote further the Mo¨ssbauer spectroscopy in time tained that the angular and energy dependences calculated domain21,5 as well as the phonon spectroscopy by means of for the crystal with a thickness l 70 m may describe inelastic nuclear scattering.22­24 Backscattering experiments satisfactorily our experimental results as it is shown by solid with sapphire crystals of better quality are in progress. 57 BRIEF REPORTS 4971 The authors are indebted to M. Gerken, K. Geske, D. Knicker Universita¨t Dortmund for their help in preparation Giesenberg, B. Lohl, J. Weber, H.-C. Wille, to Professor Dr. and performance of the experiment. This work has been W. Guse from the Mineralogisch-Petrologisches Institut funded by the Bundesministerium fu¨r Bildung, Forschung Universita¨t Hamburg and to Professor Dr. Kiendl and R. und Technologie under Contract No. 05 643GUA1. 1 K. Kohra and T. Matsushita, Z. Naturforsch. A 27, 484 1972 . 13 A. Q. R. Baron, Nucl. Instrum. Methods Phys. Res. 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