PHYSICAL REVIEW B VOLUME 57, NUMBER 2 1 JANUARY 1998-II Debye-Waller factors of alkali halides C. K. Shepard* and J. G. Mullen Purdue University, West Lafayette, Indiana 47907-1396 G. Schupp University of Missouri Research Reactor, Columbia, Missouri 65211 Received 8 August 1997 Using very-high-intensity ( 70 Ci 183Ta Mo¨ssbauer sources, we have measured the Debye-Waller factors DWF's of sodium chloride, potassium chloride, and potassium bromide single crystals for several of the (h00) and (nnn) Bragg reflections. We have used an approach which properly accounts for thermal expansion over the temperature range of our experiment, from 90 K to 900 K, about 100 K below the melting point of our crystals. We have found that a procedure used to analyze data by earlier workers leads to incorrect parameters in the Debye-Waller factor exponent, and our procedure does not require empirical parameters to account for the effects of thermal expansion. Additionally, we find three items of significance. Contrary to earlier results, we observe that the cations and the anions have identical DWF's in NaCl and also in KBr. We observe terms in the expansion of the DWF exponential which are quartic in the scattering wave vector Q in NaCl and KCl, with some evidence for a Q4 term in KBr. The size of the Q4 contribution is reported and varies with the direction of momentum transfer. We also observe that the Debye temperature and the coefficient of the anharmonic Q2 term also vary with the direction of momentum transfer. We believe our data are the definitive evidence for a nonspherical thermal cloud in a cubic crystal; the ions have a larger amplitude of oscillation in the h00 direction than in the nnn direction, contrary to the commonly held view of crystallographers that the most general form of the mean-square thermal motion is of an ellipsoid shape. S0163-1829 98 03602-9 I. INTRODUCTION peratures and for the lowest-order reflections, TDS is a small fraction of the total scattering and so a poor estimate of the The reduction in elastic scattering of photons by the ther- amount of TDS will not greatly affect the DWF. However, as mal motion of atoms in a crystal is characterized by the the temperature of the sample approaches the melting tem- Debye-Waller factor DWF . It is given by perature, the TDS can dominate the elastic scattering for 600 and higher-order reflections. In these cases, a good DWF e 2M eiQ *u l 2, 1 experimental measurement of TDS is required to avoid a systematic bias due to theoretical estimates of the TDS. where Q is the scattering wave vector and u l is the displace- Since the energy change in the scattered photon is small, ment from equilibrium of the lth ion in the basis. Its tem- 10 meV, standard x-ray techniques are unable to distin- perature dependence provides a method for examining the guish the elastic scattering from the inelastic scattering. This lattice dynamics of a crystal, including the anharmonic terms may be accomplished by using the high-energy resolution of of the lattice potential. The measurement of the Debye- the Mo¨ssbauer effect. Waller factor of single crystals, as a function of temperature, is complicated by thermal diffuse scattering and by thermal By using a Mo¨ssbauer -ray source, the TDS can be mea- expansion of the sample crystal. Thermal diffuse scattering sured experimentally and the elastic intensity can be ob- TDS is inelastic scattering due to lattice phonons. Since the tained. The resonant photons from the 46.5-keV Mo¨ssbauer DWF characterizes the elastic scattering, the inelastic scat- level in 183W produced by the decay of 183Ta have an tering must first be removed from the total scattering inten- energy width of 2.5 eV. A resonant photon which is inelas- sity. The thermal expansion of the crystal creates difficulties tically scattered by the crystal will have its energy shifted by in carrying out the experiments and in properly analyzing the an amount of the order of 10 meV, making it nonresonant data as the scattering vectors and the number of scattering with the 46.5-keV Mo¨ssbauer effect transition which is de- sites illuminated by the incident beam vary with temperature. tectable when it is analyzed by a Mo¨ssbauer absorber foil. The high-energy resolution of Mo¨ssbauer radiation allows us Since the energy width of the Mo¨ssbauer line is of the order to distinguish between elastic and inelastic scattering, and we of 10 3 times the phonon energy, the probability of a non- have carefully corrected for the effects of thermal expansion resonant photon being scattered into the resonant window for measured by earlier workers , allowing us to make very absorption is negligible. Any TDS in the scattering crystal, accurate measurements of the DWF for NaCl, KCl, and KBr. therefore, lowers the observed resonant fraction of the beam It is important to experimentally separate the elastic and on scattering. The recoilless fraction of the beam before scat- inelastic scattering from the sample crystal. To observe an- tering may be measured by Doppler-shifting the absorber in harmonic effects, we must make measurements at high tem- the beam before scattering from the crystal putting the ab- peratures and for higher-order Bragg reflections. At low tem- sorber between the source and the sample ; the recoilless 0163-1829/98/57 2 /889 9 /$15.00 57 889 © 1998 The American Physical Society 890 C. K. SHEPARD, J. G. MULLEN, AND G. SCHUPP 57 FIG. 1. Comparison of Mo¨ssbauer spectra. The upper spectrum FIG. 2. Rocking curves for potassium chloride at room tempera- is the full Mo¨ssbauer spectrum collected in this experiment. The ture. lower spectrum is a two-point, on- and off-resonance spectrum similar to those collected by earlier researchers. period of time, we have eliminated these common sources of systematic error from our experiments. fraction after scattering may be measured by moving the ab- sorber to a position between the sample and the detector. II. EXPERIMENTAL METHOD AND APPARATUS Measurement of Debye-Waller factors by this method has been hindered by the low intensity of typical Mo¨ssbauer The experiments on three alkali halides were carried out sources. The typical 100-mCi 57Co source requires very at MURR using the Mo¨ssbauer facility.7 The reactor pro- close geometry with large acceptance angles and uses only duces very intense 183Ta sources weekly. These sources are two points on the Mo¨ssbauer spectrum on and off reso- transferred into a stationary cask for the presently reported nance to determine the elastic fraction.1­6 Typically, the dis- Mo¨ssbauer diffraction studies. This instrument consists of a tance from source to detector is on the order of 25 cm, with stationary, heavily shielded source cask, a rotary stage for acceptance angles of several degrees, leading to cosine crystal scattering which accommodates either a furnace or a cryostat, an oscillating stage for the Mo¨ssbauer absorbers, smearing of the spectrum. These sources are widely used for and a solid-state photon detector. The sample stage and the other purposes because of their much greater energy resolu- detector table, which holds the detector and the oscillating tion, only 4.7 neV; however, the resolution of the 183Ta stage, are driven by computer, allowing control of and 2 sources is more than adequate to resolve the differences in the orientation of the sample crystal and the scattering energy due to phonon interactions and these sources can be angle . The distance between the source and the detector is manufactured with far greater photon intensity in the Mo¨ss- approximately 1.5 m. bauer transition. We have utilized these 183Ta supersources The -ray beam is collimated to 1 in. in height and 0.125 available at the University of Missouri Research Reactor in. in width before reaching the sample crystal. For the MURR to make accurate measurements of the DWF. The lowest-order reflections it is possible to collimate the beam increase in intensity of a factor of about 500 in Mo¨ssbauer to 0.0625 in. in width if needed. After scattering from the photons allows us to collect full Mo¨ssbauer spectra rather sample crystal, the beam passes through a 1 in. by 0.5 in. than two-point spectra see Fig. 1 and to collimate our beam opening before reaching the detector. The extra width allows to about 0.3°.7 In addition, it allows us to fit our spectra to the entire scattered beam to be detected. The scattered beam the true line shape of the transition, rather than assuming a is wider than the incident beam due to penetration into the Lorentzian shape, which leads to errors in the measured elas- sample crystal; we choose to accept the entire scattered beam tic scattering fraction in on- and off-resonance measurements so that thermal expansion corrections are simplified. and even erroneous Debye temperatures.8­14 As a source de- We used NaCl and KBr crystals fabricated by Bicron, Inc. cays, absorber nuclei build up in the source, and source reso- These crystals were grown and cut so that the desired Bragg nance self-absorption SRSA begins to distort the shape of planes were parallel to the crystal face. We obtained two the observed Mo¨ssbauer line. We used sources for only 1 crystals for each salt, one oriented on the h00 and one on week, during which SRSA is negligible. However, over the the nnn . These crystals were nominally 2.5 1.2 0.125 lifetime of a 57Co source, the observed source recoilless frac- in. The length of the crystals allowed the entire photon beam tion can change by more than 30%, changing the line shape to illuminate part of the crystal face and scatter in reflection and distorting the measurements of elastic scattering geometry. These crystals had rocking curves with a full fraction.15 By collecting full Mo¨ssbauer spectra, fitting them width at half maximum of less than 0.5°; a typical rocking to the true line shape, and using sources only for a short curve ( scan is shown in Fig. 2. Bicron also fabricated 57 DEBYE-WALLER FACTORS OF ALKALI HALIDES 891 KCl crystals, but these had large mosaic spreads; other KCl tions were much less sensitive to the alignment of the crystal crystals were obtained from the University of Utah. These in the beam. crystals were much smaller, so that the entire crystal was At each temperature we examined the rocking curves ( bathed by the photon beam. Some data were taken using the scan of each Bragg peak and collected a Mo¨ssbauer spec- larger crystals; the same results were obtained in each case. trum after scattering from each peak. In this way we mea- The temperature of the sample crystals was adjusted from sured the total intensity elastic plus inelastic of each reflec- about 90K to room temperature by using a liquid nitrogen tion and determined the elastic scattering fraction for each Dewar. The sample crystals, in a boron nitride holder, are reflection. We oscillated a resonant absorber, a natural tung- placed in an isothermal holder which is attached to a cold sten foil 2 mils thick, between the sample crystal and the finger. There is a thermal switch which may be evacuated to detector. A line shape analysis of each Mo¨ssbauer spectrum, inhibit heat transfer between the sample and the nitrogen based on an analytic expansion of the transmission bath when the sample is being heated above nitrogen tem- integral,8­12 determined the recoilless fraction of the photon peratures by small rod heaters. The temperature is controlled beam after scattering from the sample. The Mo¨ssbauer line by a thermocouple and measured by two platinum resistors shape depends on the thickness of the absorber, the SRSA in the source, the recoilless fraction of the source, the width of located on the isothermal holder. the transition, and the number of counts collected. The thick- Above room temperature the temperature of the sample ness of the absorber and the SRSA were taken as known crystals may be heated up to near their respective melting values from previous high-precision measurements.11,12 Ad- points by the use of a furnace. The sample crystals, in a ditionally, we have shown that, provided the actual absorber boron nitride holder, are placed in an isothermal copper thickness and SRSA are constant, the elastic scattering frac- holder, with 1-mil copper end windows for the beam to pass tion does not change for small inaccuracies in the thickness through, which is between two heating coils. There are five number of the absorber or the source. That is, using a slightly thermocouples used for temperature control and measure- inaccurate value for one of these parameters will affect the ment of temperature. At even the highest temperatures the recoilless fraction measured in both the before and after po- temperature variation was less than 3°, and usually the tem- sitions, but the elastic scattering fraction will be unaffected. perature was constant within 1°. The outside of the furnace We also took the width of the transition ( / , where is the is water cooled, and there are up to four tantalum heat shields lifetime of the excited state to be known, after confirming surrounding the sample, with windows cut out for beam that there was no instrumental broadening. In this manner, transmission. we reduced the number of parameters to avoid parameter We wanted to collect data as near to the melting points of correlation problems in fitting these spectra. the crystals as possible the melting points of the crystals are To measure the recoilless fraction of the photon beam 1074 K for NaCl, 1043 K for KCl, and 1007 K for KBr ; before scattering, we used a LiF calibration crystal. At room however, at temperatures near the melting point the alkali- temperature, the 200 Bragg plane of LiF is a nearly 100% halide crystals tended to evaporate excessively. This had two elastic scatterer of 46.5-keV rays16,17 and so we can deter- effects; the crystal gradually became thinner, moving the po- mine the recoilless fraction of the incident beam by scatter- sition of the crystal face away from the -ray beam, and ing from LiF. The ratio of the recoilless fraction after scat- depositing the evaporated material on the window foils, at- tering to the recoilless fraction before scattering is the elastic tenuating the intensity of the transmitted beam. To overcome scattering fraction, and this, when multiplied by the total this problem we reduced the maximum temperature to which scattered intensity, determines the elastic intensity, and thus we raised our crystals and we coated the crystals with col- the DWF. loidal graphite. The coating of carbon did not affect the dif- fraction peaks, but it did reduce the evaporation problem, III. EXPANSION OF THE DEBYE-WALLER FACTOR eliminating it completely below 850 K. At and above 850 K there was evidence of some small amount of evaporation The exponential of the Debye-Waller factor can easily be occurring; we monitored the drop in count rate caused by expanded in a power series in Q and T. For cubic crystals evaporated material depositing on the copper windows and with inversion symmetry, keeping only terms of order Q4 or corrected for this effect, but it did introduce additional un- less, the DWF exponential can be written quite simply. To certainty into these very highest data points. simplify calculations, we normalize our data to the intensity At each temperature it was necessary to align the sample at room temperature and take the natural logarithm, to deter- crystal in the gamma-ray beam. Due to the thermal expan- mine the exponential of the DWF. It is customary to divide sion of the crystal and furnace or cryostat, the position of the out the Q2 dependence of this exponential so that anhar- crystal relative to the beam changed slightly each time the monic effects may be more easily seen. sample temperature was changed. Aligning the crystal was A typical function to which previous investigators have fit accomplished by a hand-driven screw which translated the the integrated intensity measurements is given by18­20 crystal into and out of the beam. This alignment process introduced the largest, nonstatistical uncertainty into the Y YH 4 2 m2T2 m3T3 Q2m4T3 , 2 data. For the lowest-order reflections, this is very significant. For the higher-angle reflections, the actual changes in inten- where the rescaled elastic intensities are given by sity with temperature were large, so that a small variation is less important. Also, it was at low angles that the beam size 4 2 I Y ln . 3 was nearly the same as the crystal size; the high-angle reflec- Q2 I0 892 C. K. SHEPARD, J. G. MULLEN, AND G. SCHUPP 57 the intensities of the reflections are corrected for thermal expansion, m3 0.21 By using thermal expansion data22­24 to correct our data for the changes in Q, the Lorentz polariza- tion factor, and the number of scattering centers in the -ray beam, we eliminated m3. After eliminating m3 from the analysis by correcting for thermal expansion, we can fit our data to the function Y YH 4 2 m2T2 Q2m4T3 , 5 which is the same as Eq. 2 when m3 0. We first determine m4 by a direct comparison of reflections within a family, independently of the other parameters. By writing Eq. 5 for two different reflections with Y values given by Y1 and Y2, we can eliminate the harmonic and m2 terms if we take the difference between them. This gives us Y 2 2 1 Y 2 C1 C2 4 2Q1m4T3 4 2Q2m4T3, 6 where C1 and C2 are the constant terms in Eq. 4 for the two different reflections. The terms with m2 and D drop out because they do not depend on the magnitude of Q, and so they are the same for each reflection. From this it can be seen that a plot of (Y 2 2 1 Y 2)/ (4 )2(Q1 Q2) versus T3 will have a slope of m4. After determining m4 we fit for all other parameters simultaneously. FIG. 3. A plot of the Q4 contribution to the Debye-Waller fac- tor. The slope of each line corresponds to the m4 coefficient. Q1 IV. THERMAL-EXPANSION CORRECTIONS and Q2 are two different scattering wave vectors which have the same direction. We have investigated the use of the parameter m3, and we find that the results obtained when using it with intensity I and I0 are the integrated elastic intensities at the tempera- data not corrected for thermal expansion are in error. The ture T and at the reference temperature, respectively. The correction for the number of scattering centers is independent harmonic term YH is given by of Q , which means that the correction to the Y values de- pends on Q . The corrected Y values differ from the uncor- Y rected Y values by large amounts for low-order reflections, H C 12h2 T D , 4 mk 2 T while the difference is much smaller for the 600 reflections. B D This can be clearly seen in Fig. 4. Because this difference in where Y values varies dramatically with Q, a failure to properly account for these thermal-expansion effects makes it impos- 1 x zdz sible to correctly determine m x 4. x , The thermal-expansion corrections to the Y values from 0 ez 1 changes in Q , from the explicit dependence on Q and from C is a constant and a parameter in the fitting process, m is the change in the Lorentz polarization term which is due to the mean ionic mass, kB is the Boltzmann constant, and h is its explicit Q dependence, are straightforward. The change in the Planck constant. m2, m3, and m4 are the anharmonic the number of scattering centers is more complicated. Our terms of interest, and D is the Debye temperature. By direct crystals expand in three dimensions, length, depth, and comparison of parallel reflections, the value of m4 can be height. The expansion in depth of the crystal does not affect determined independently of the other parameters see Fig. the number of scattering sites, because the penetration depth 3 , but the others must be found by fitting the data for the of the rays increases as well. The net effect of this expan- remaining four simultaneously. This presents a problem be- sion is to broaden the scattered beam slightly. In these ex- cause all but C are highly correlated. This problem has been periments, the collimation at the detector was such that the ignored in the past by simply selecting a value for the Debye entire broadened beam was still accepted into the detector. temperature, ignoring m3 or carrying out some thermal ex- Consequently, the expansion in this dimension did not cause pansion corrections, and fitting only for m2 and m4, which is a change in the intensity of the scattered beam. determined as above. We reduced the problem greatly by The originally selected crystals are all taller than the correcting our measured intensities for the thermal expansion height of the beam, and so as they expand the number of of the sample crystals. Both m2 and m3 are associated with scattering sites decreases. Likewise, the crystals are longer thermal expansion. m2 contains a part from thermal expan- than the effective beam width on the face of the crystal, and sion, but other factors also contribute. m3 is wholly due to again as they expand in this direction, the number of scatter- the nonlinearity of thermal expansion with temperature. If ing sites decreases. These two combine to give a two- 57 DEBYE-WALLER FACTORS OF ALKALI HALIDES 893 FIG. 4. A plot of the rescaled elastic intensities Y(Q ,T) vs FIG. 5. A plot of Y(Q ,T) vs temperature for NaCl. The scale temperature for KBr showing the difference between the expansion for each curve is the same. corrected and uncorrected Y values. The uncorrected data are the diamonds; the corrected data are the pluses. due to the alignment difficulties discussed earlier. Addition- ally, notice that the error bars (1 statistical errors are much dimensional correction to the scattered intensity. However, larger on the low-order reflections than the high-order reflec- we also used two smaller crystals KCl , for which the crys- tions. This is a consequence of the definition of the Y's. The tal was smaller than the beam in both of these directions. In actual elastic intensities were found to greater precision in these cases, we made no correction for a change in the num- the case of the low-order reflections, but to determine the Y ber of scattering sites. The 111 and 200 reflections on the values we must divide by Q2. This reduces the uncertainty in larger crystals are potential problems. If the crystals are cor- the high-order Y's. rectly oriented, the crystals are longer than the beam; how- To fit our data to Eq. 5 , even after independently deter- ever, if the crystal is slightly misaligned, the beam may be mining m4, it was necessary to place constraints on the pa- wider than the length of the crystal. In this case, a one- rameters D and m2. The correlation between these param- dimensional correction would be more appropriate. This may eters was so large that fitting the data from a single reflection have been the case with NaCl. The uncertainties on these gave uncertainties greater than the value of the parameter. It low-angle data points are such that the fitted parameters are was essential to fit several reflections simultaneously. The not affected by which correction is made. However, the one- Debye temperature and m2 were thought to be fixed for a dimensional correction gives a result with a slightly smaller given crystal, not varying with Q .21,25,26 The Debye tempera- 2. ture was thought to vary as 1/ mass, from Eq. 4 and be- To make these thermal expansion corrections correctly, it cause the phonon spectra of these three crystals are almost is essential to know the variation of the lattice constants with temperature. The coefficient of thermal expansion is not con- stant at either low temperature or high temperature, and so using the room temperature value, as has been done in the past, leads to large errors. Since the lattice constants of all three of these alkali halides have been measured as a func- tion of temperature over the entire range of this experiment, we have used these experimental values instead. We used the fitting functions of Pathak et al. to calculate the lattice con- stants at each measured temperature.22­24 We then calculated Q(T) and the relative number of scattering sites illuminated using these lattice constants. V. RESULTS AND DISCUSSION We took our measured elastic scattering intensities, cor- rected them for thermal expansion, and converted them into Y values to more readily see anharmonic effects. The experi- mental Y values and the curves fitted to these points for NaCl, KCl, and KBr are shown in Figs. 5­7, respectively. The lowest-order reflections deviate most from the curves FIG. 6. A plot of Y(Q ,T) vs temperature for KCl. 894 C. K. SHEPARD, J. G. MULLEN, AND G. SCHUPP 57 FIG. 7. A plot of Y(Q ,T) vs temperature for KBr. FIG. 9. A residual plot for NaCl showing the better fit obtained when allowing D and m2 to be different for the two sets of reflec- identical, differing by only a scaling factor. We tested these tions. Each vertical division is 1 Å. assumptions to determine how best to constrain our param- eters. for each crystal varied according to 1/ mass, where the mass We discovered that it was not possible to fit the two dif- referred to here is the mean ionic mass of the alkali-halide ferent sets of reflections with the same Debye temperature crystal for each of the three crystals studied. The values for and m the two different directions were different, but both sets of 2. While this reduced the uncertainties in the param- eters, the fitted curves did not correspond to the data. This is reflections followed this relationship. Therefore, we made seen in Figs. 8 and 9, and this effect is even more pro- this a constraint in our fitting process. We fit all our (h00) nounced for KCl.27 When they are constrained to have the data together, allowing the m2 values to vary independently same from one crystal to another but requiring the Debye tempera- D and m2, the (h00) data and the (nnn) data move away from the fitted curve in opposite directions as the tem- tures to vary according to the reciprocal of the square root of perature increases. We can, however, constrain the mean ionic mass. We fit our (nnn) data in the same D and m2 to be the same for each reflection in a family that is, for the manner. Three important points were revealed by these 200 , 400 , and 600 reflections we can require that they analyses and are discussed below. be the same . This causes no problems in fitting the data. The first is that the Debye-Waller factor varies with crys- Within errors, the values for the Debye temperature found tal orientation in these cubic crystals. All three of the non- trivial parameters determined in our data fitting process, the Debye temperature and the two anharmonic constants m2 and m4, varied with crystal orientation. The difference in the Q4 contribution was not unexpected, but because these crys- tals all have cubic symmetry, it is believed that the Debye temperature and the Q2T2 term will be the same for the h00 and the nnn directions or, in fact, for any direction.21,25,26 The mean-square displacement of an ion is the same in any direction, and so the Debye temperature and m2 must be identical for the different directions. This is not the case. The coefficient m2 differs between the two direc- tions, tending to be larger in the nnn direction m2 7.6(5) 10 8 Å2/K2 in the nnn direction versus m2 1.7(4) 10 8 Å2/K2 in the h00 direction for KCl.... The Debye temperatures in the nnn direction were 14 5 % larger than in the h00 direction. A larger Debye tempera- ture implies a smaller mean-square displacement from equi- librium; a larger m2 implies a larger mean-square displace- ment from equilibrium, although this term is a lesser contribution to the mean-square displacement, especially at FIG. 8. A residual plot for NaCl showing the poor fit obtained low temperatures. when requiring These results indicate that at lower temperatures, the ions D and m2 to be the same for both sets of reflec- tions. Each vertical division is 1 Å. Notice that the (h00) points oscillate with a larger amplitude in the h00 direction than slope up while the (nnn) points slope down. in the nnn direction. Thus, the probability density for the 57 DEBYE-WALLER FACTORS OF ALKALI HALIDES 895 TABLE I. Debye-Waller factor parameters. The parenthetical quantity associated with each number is the error in the last figure of the measured quantity. When two digits are given in parentheses, e.g., D 283(18), we mean D 283 18. Crystal Direction D K m2 Å2/K2) m4 Å4/K3) NaCl (h00) 281 7 1.4 4 10 8 11.2 9 10 13 NaCl (nnn) 320 13 1.4 5 10 8 4.1 8 10 13 KCl (h00) 249 6 2.5 4 10 8 10 2 10 13 KCl (nnn) 283 18 8.0 5 10 8 3 3 10 13 KBr (h00) 197 5 7.3 5 10 8 1 2 10 13 KBr (nnn) 224 10 6.8 6 10 8 - ions is not spherically symmetric; at low temperatures the it suggests that there is such a dependence and that in the mean-square radius may change by as much as 30% from nnn direction it may be quite large. In the KCl samples, nnn to h00 . As the temperature is increased, m2 contrib- the nnn reflections also show no Q4 dependence within utes more, and so the oscillation of the ions becomes more errors m4 1(30) 10 14 Å4/K3 ; however, the h00 re- spherical. For KBr, where the values of m2 are nearly equal flections do show such a dependence m4 in the two directions, the asymmetry remains large up to high 1.1(2) 10 12 Å4/K3 . Both sets of reflections show a Q4 temperatures. For NaCl and KCl, where the values of m2 are dependence for the NaCl samples; however, the coefficient is very different in the two directions studied, the asymmetry significantly larger in the h00 direction h00, m4 1.2(1) decreases with increasing temperature. In general, our results 10 12 Å4/K3; nnn, m4 5.2(8) 10 13 Å4/K3.... These indicate that the thermal cloud of the ions is not spherically values are listed in Table I. symmetrical in these cubic crystals, and the asymmetry di- The third point is that the (nnn) reflections with all odd minishes with increasing temperature. indices and the (nnn) reflections with all even indices have The general assumption has been that in cubic crystals the same temperature dependence. If measurements of the like these, where there is inversion symmetry about every intensity for both even and odd ordered (nnn) reflections are ion, the vibration of the ions about their equilibrium points is made, it is possible to separate the scattering of the two types spherically symmetric. This is an unstated assumption in of ions, and measure individual DWF's for the ions. We can most discussions of Debye-Waller factors and most measure- write the scattered intensity for each type of reflection in two ments of it. It follows from the more general notion that the ways, once using an average DWF and once using an indi- thermal cloud is an ellipsoid, and in cubic crystals where vidual DWF for cations and anions separately, as Martin and x2 y2 z2 it follows that the thermal cloud must have O'Connor did.1 The total scattering can be written in terms spherical symmetry. Thus a single Debye temperature is of an average Debye-Waller factor as given for a material, independent of the direction of scatter- ing. Our results demonstrate that there is a difference in the Debye-Waller factors for different directions within a crystal Isum Ep f1 Q f2 Q e MS Q ,T 2 7 and that the thermal cloud is not spherically symmetric. The second point is that we do observe a Q4 contribution for the even-order reflections and to the DWF. Earlier measurements1,4,5 reported a Q4 contri- bution but reported coefficients varied by two orders of mag- I nitude, discrepancies that are so large as to shed doubt on diff Ep f 1 Q f2 Q e MD Q ,T 2 8 these earlier claims of a nonvanishing Q4 term. All the re- ported values are larger than our data indicate. In KCl we for the odd-order reflections. It can also be written in terms find m of the separate Debye-Waller factors as 4 1.1(2) 10 12 Å4/K3, while Martin and O'Connor1 report a value of m4 4(1) 10 12 Å4/K3 and Solt et al.5 report m4 1.5(6) 10 10 Å4/K3. These earlier Isum Ep f1 Q e M1 Q ,T f2 Q e M2 Q ,T 2 9 measurements were limited by a low source intensity, which forced them to make compromises in the data collection and for the even-order reflections and analysis, and it is unclear how, if at all, they accounted for thermal expansion. Our high-photon-intensity measurements allow us to draw IDIFF Ep f1 Q e M1 Q ,T f2 Q e M2 Q ,T 2 a definitive conclusion that there is a Q4 contribution at these 10 temperatures for NaCl and KCl. The evidence for such a Q4 dependence in the KBr samples is not compelling m4 for the odd-order reflections. In these equations E is an in- 4(2) 10 13 Å4/K3 for the h00 direction, and m4 tensity term which depends only on the incident intensity and 4(2) 10 12 Å4/K3 for the nnn direction, when m4 is p( ) is an angular-dependent term which takes into account determined by a simultaneous fit with m2 and D...; however, polarization and the volume of the crystal irradiated. Isum is- 896 C. K. SHEPARD, J. G. MULLEN, AND G. SCHUPP 57 ested in anharmonic effects. The amount and quality of data taken below the Debye temperature of each sample were insufficient to allow any conclusions to be drawn about the relative DWF's of the two ion types at low temperatures. If one considers the system in the light of the equipartition theorem as has Disatnik et al.,31 it becomes clear that the amplitudes of vibration for the two types of ions should be the same at high temperature. By equipartition, the average kinetic energies of the two ion types should be equal, and thus the average potential energies of the two should be equal. The potential energy for a particle on a spring is given by 12 ku2, and since the two ions share the same spring con- stant k, they will have the same mean-square displacement, whenever the temperature is high enough for the equiparti- tion theorem to hold. At low temperatures, where equiparti- tion would not be valid, the change in scattered intensity with temperature is too small to allow us to see any differ- ences between the scattering from the two types of ions in this experiment. FIG. 10. A plot of the rescaled single-ion elastic intensity YSI(Q ,T) vs temperature for the separate ion types in NaCl. The diamonds are the cation data and the pluses are the anion data. VI. CONCLUSION In conclusion, we have carefully collected Debye-Waller the intensity when the reflections from the two types of at- factor data for NaCl, KCl, and KBr, and corrected them for oms are in phase, and Idiff is the intensity when the reflec the thermal-expansion of each crystal. The thermal- tions from the two types of atoms are out of phase. For the expansion corrections are absolutely essential for a proper rocksalt structure, scattering from the (nnn) planes depends analysis of the data, and must be made with care. Accurate on the sum of the scattering from the alkali and halide ions lattice constant data over the entire temperature range are when n is even and the difference when n is odd, and so the needed, as the expansion of these crystals is complicated. DWF for the individual ions, e 2M1(Q ,T) and e 2M2(Q ,T), One must correct for three effects, the change in Q in the may be calculated from data from the (nnn) planes. Refer- definition of the rescaled elastic intensities the Y values de- encing all intensities to room temperature and using known fined in Eq. 3 , the change in Q in the Lorentz polarization atomic scattering factors f ,28 and our measured intensities for factor, and the number of scattering sites illuminated by the the (nnn) reflections, we fit all our (nnn) data simulta- gamma beam. neously to the individual Debye-Waller factors. We had data We find that the Debye temperatures and the anharmonic for the 111 , 222 , 333 , and 444 reflections for NaCl, coefficients vary with the mean ionic mass, such that the and the 111 and 222 reflections for KBr. The data for Debye temperature varies according to 1/ mean mass and NaCl clearly show that the different ions have the same tem- m2 increases with increasing mass. In the (h00) direction m4 perature dependence. For KBr the results are not as clear, decreases with increasing mean mass, though the functional due to the larger uncertainties in these measurements, but dependence is not apparent, while in the (nnn) direction our they are consistent with this hypothesis. Figure 10 shows the data do not indicate a systematic change in m4 with mean Y values for the single-ion scattering, YSI , for NaCl. mass. The reasons for the dependence of m4 on mass are not These results are in contrast to earlier measurements by yet understood. We find that there is a quartic term in the Martin and O'Connor,1 who claimed to have seen different DWF exponent (m4 is nonzero , but that it is much smaller temperature dependences for the individual ions. Their mea- than earlier reports had indicated. This dependence is obvi- surements, however, suffered from the 57Fe experimental ous in the cases of NaCl and KCl, but less convincing for drawbacks associated with inadequate photon intensities, and KBr because of our large experimental errors in this latter they did not correctly account for thermal expansion effects. case. Additionally, they did not directly determine the single-ion We have determined that the two types of ions in these scattering; they attempted to estimate what the scattering crystals have the same mean-square displacement from equi- would be in certain cases and used these estimates to deter- librium, as would be expected at high temperatures from the mine the single-ion scattering. Given the systematic errors equipartition theorem. This supports claims by Huiszoon and introduced in their experiments and the contrast in intensity Groenewegen29 and Jex et al.30 that, for temperatures greater between the all-odd and the all-even reflections, the differ- than the Debye temperature and in the harmonic approxima- ence between our results and their claims is not surprising. tion, the mean-square displacements of ions are independent The identical DWF's for the two ion types require that the of their masses. mean-square displacements of the ions be identical in the We have found that the Debye temperature and the coef- nnn direction. For temperatures above the Debye tempera- ficient m2 do vary with direction, indicating that the oscilla- ture, this is expected.25,29,30 The bulk of our measurements tions of the ions about their equilibrium positions are not were taken at high temperatures, as we were primarily inter- spherically symmetric. It is generally accepted that the ther- 57 DEBYE-WALLER FACTORS OF ALKALI HALIDES 897 mal cloud is isotropic for cubic crystals with inversion sym- ACKNOWLEDGMENTS metry about each atom. This experiment does not support This work was prepared with the support of the U.S. De- this belief. The data clearly show DWF's which are different partment of Energy, Grant No. DE-FG02-85 ER 45199 and in the two directions studied here, leading us to the conclu- NSF Grant No. DMR-9623684 and is taken from the Ph.D. sion that the ions have different amplitudes of oscillation in thesis work of Carmen K. Shepard of Purdue University. the two directions. 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