Thermal vibrations in BeO

JOURNAL OF NUCLEAR MATERIALS

14 (1964) 275-281 @NORTH-HOLLANDPUBLISHING CO., AMSTERDAM

THERMAL

VIBRATIONS

A. W. PRYOR

IN Be0

and T. M. SABINE

Materials Division, AA EC Research Establishment, Lucas Heights. Azcsdralia

Neutron diffraction patterns, taken on fine-grained polycrystalline samples of Be0 at various temperatures from room temperature to 900” C, were analyzed to determine the structure parameter and the nuclear vibration amplitudes. The structure parameter was found to be 0.3778f0.001 at room temperatures increasing slowly to 0.3796~0.091 at 909” C. The vibration amplitudes were

correlated with specific heat results by assuming a simple model for the vibration spectrum. The temperature parameters at room temperature were found to be B. = 0.27f0.06 As and B, = O.SSjO.06 Al. The Debye parameter determined by diffraction was confirmed by observations of the coherent inelastic cross sections for 6 A neutrons at temperatures up to 2000” K.

1. Introduction

the scattering factors, by measuring the intensity of a given line as a function of temperature. It is likely, however, that 8, will vary with temperature, and a determination of the vibration amplitudes, and hence of @,, at one temperature can be achieved only if the scattering factors are known with high accuracy. This is why neutron diffraction studies, where the scattering factors are independent of the Bragg angle, are so valuable. Kuleshov et a1.l) have recently determined the vibration amplitudes in Be0 at room temperature from neutron powder diffraction patterns. In performing these measurements on Be0 there are three major difficulties. First, the Debye temperature is very high (- 1150’ K), and at room temperature the reduction of the intensities of the diffraction spectra due to the thermal vibrations is therefore quite small. Secondly, single crystals of Be0 are so nearly perfect that extinction becomes important for crystal dimensions greater than about 20,~; this means that the measurements must be made on fine-grained specimens. Third, the space group of Be0 is such that there are no reflections which come solely from the atoms of one species; this necessitates a detailed computer analysis of all available diffraction lines.

The object of these experiments was to determine the amplitudes of vibration of the Be and 0 nuclei in beryllium oxide as a function of temperature. There are two reasons for doing this. First, as a study in itself, it is of interest to see if the vibration amplitudes, directly determined by diffraction methods, agree with the expected values calculated from the specific heat of the solid. Second, it is an almost indispensable preliminary to a study of the electronic states of the two atoms by X-ray methods. In the analysis of X-ray diffraction intensities an incorrect choice of the atomic scattering factors may still give low residual errors, since the effect of the mistaken scattering factors may be compensated by incorrect temperature factors, and it is only if the temperature factors are known independently that one can determine whether the chosen scattering factors are actually appropriate, If it could be assumed that the vibration amplitudes obey a Debye theory function with temperature with one constant parameter, f3,, the Debye temperature for diffraction, then this parameter could be determined from X-ray intensity data, independent of uncertainties in

IV.

STRUCTURE

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PHYSICAL

PROPERTIES

A. W.

PRYOR AND

T. M. SABINE

2. Experimental

Methods

The powder patterns were taken with a neutron spectrometer installed on the high-flux reactor HIFAR. This spectrometer had the valuable property of unusually high resolution at large Bragg angles “). Even so there is considerable overlapping of lines in the typical pattern shown in fig. 1. The samples were contained in a thin-walled tube and were heated by passing a heavy current along this tube. Below 500’ C the tube was vanadium and above 500” C stainless steel. Even though the tube was operated in an argon atmosphere, retained by aluminium foil windows, the problem of oxidation of the tube limited the temperature of measurements to less than 1000’ C. The temperature was measured by a thermocouple in contact with the specimen. The specimens were cold-pressed and sintered, of mean grain size 1 to 2/t. The spectrometer was controlled by a monitor counter in the incident monochromatic beam. After the accumulation of a preset number of monitor counts the total detector count was printed and the counter moved 3 minutes to the next position. The integrated intensity was determined by adding the total number of counts in the line or group of lines and subtracting the background level as determined on each side of the line. A complete diffraction pattern was taken in 40 or 120 hours. The time was always adequate for a statistical accuracy better than 1 per cent but owing to various unknown experimental factors, the residual error in the results was usually of the order of 2 to 4 per cent. No correction was made for the component of the inelastic scattering which peaks at the Bragg position, or for multiple scattering in the specimen. Both effects are thought to be of minor significance. 3. Results

arid Method of Refinement

The intensity data, Ior,,, were converted to structure factor data by applying the usual adjustments for multiplicity and Lorentz factor. When one intensity figure was obtained for a

THERMAL

VIBRATIONS

group of lines it was divided amongst them in the ratio of the calculated intensities. When proceeding through several cycles of data refinement this allotment was made afresh on each cycle using the values of intensities calculated on the basis of the parameters determined on the previous cycle. The observed structure factors were analyzed by means of a full-matrix least-mean-squares

271

IN BE0

B, was 60-60 per cent, and between each of them and the scale factor 30-40 per cent. Correlations between z and the other parameters were about 10 per cent. Eleven separate sets of data were analyzed, taken at temperatures between 27’ C and 9.lP C. The values of B, and B, are shown in fig. 2. (The lines drawn through these ‘results were computed as described below). ,

I

I

I .

I

I

Zoo

400

I

100

600 TEMPERATURE

Fig. 2. Temperature

factor

(W
refinement programme *). Weighting factors were allotted in accordance with the statistical procedure described by Evans 4). Each set of data was subjected to three refinement cycles to determine the best values of four parameters: the scale factor, the structure parameter z, and the oxygen and beryllium vibration amplitudes B, and B,. The correlation between B, and

(-K

Iwo

lzw

)

for the 0 and Be nuclei. The solid lines are calculated spectra described in the text.

from

As for the z-parameter, from a line drawn through all results it was deduced that z = 0.3778 fO.001 at 27’ C and increases to 0.3796f0.001 at 900” C. This means that the vertical bonds are about 0.6 per cent longer than the nearhorizontal bonds at 27” C and about 0.8 per cent longer at 900” C. X-ray evidence has shown that it is very IV.

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278

A.

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difficult to prepare compacted Be0 specimens completely free from preferred orientation. The preferred orientation in Be0 is such that the c-direction in the crystallites tends to lie normal to the pressing direction. To determine the effect of a small amount of preferred orientation on the structural and thermal parameters, a set of observed intensities was arithmetically adjusted to simulate various degrees of preferred orientation, and re-analyzed. It was found that up to 20 per cent preferred orientation, the isotropic B values were not altered but the standard deviation increased. A systematic increase in the z-parameter was found but the values lay within one standard deviation. 4. Discussion

of the Vibration

Parameters

The values of (M:) and (II!), the mean square vibration amplitudes of the 0 and Be nuclei, are given directly by the analysis of the diffraction data since B, = 8n2(a&. In this discussion we shall assume a simple model for the vibration spectra and show that this model can account satisfactorily for the observations of the vibration amplitudes, and also for the available data on specific heat. The use of this model allows us to quote the values of B, and B, at room temperature with more precision than the actual room temperature observations, considered alone, would justify. Assuming that the vibrations are isotropic, Blackman ‘j) proves the following formula relating the vibration amplitudes to the vibration spectrum, G(y) :

where the sum is over the g particles in the unit cell, G(v) is normalized to 1, Y is the mode frequency, x = hv/kT where h and k are the Planck and Boltzmann constants and kTq(x) is the average energy of an harmonic oscillator, that is, r](x) = x/2 coth 42. Since. BeO. contains two molecules per unit cell there are 12N modes of vibration in an assembly containing N unit cells. Of these, 3N are “acoustic” modes and 9N are “optic”

T.

M.

SABINE

modes 6). We assume that the acoustic modes have constant velocity and may therefore be described by a Debye frequency spectrum with maximum frequency vn, and that the optic modes all occur at a single frequency, vs. Using this simple “Debye-Einstein” model it is possible to compute the equivalent Debye temperatures for diffraction, 8,, and for specific heat, 8,) and so to compare them with the values derived, respectively, from our observations of vibration amplitudes and from available specific heat data. (The details of these computations are given by several authors, for example Blackman ‘)). This comparison is shown in fig. 3. For the two parameters vn and vE of the Debye-Einstein model we have assumed hv, = 0.057 eV and hv, = 0.089 eV. These two values were chosen to give a good fit to the specific heat results. The values of Bc were derived from the specific heat data of Victor and Douglas 8) and Kelley “). The observed values of C, were adjusted by means of the formula: C, = C,(l-33yorT), where the Gruneisen constant y was calculated as 2.20 from the compressibility data of Weir lo) and the linear expansion coefficient a was taken from Collins rl). In deriving 0, from C, at high temperatures, an additional correction was made for anharmonicity. Following Peierls 6), the harmonic value was related to the actual value by: C; = C,(l--AT). A value of A = 3 x lOA ‘K-l gave constant values for 0, in the range 300-700” K. The decrease in 8, at higher temperatures may be due to small errors in the C,-C, correction. The values of 8, at 0” K, calculated from the elastic constants given by Nolle Ia) using the averaging formulae of Wolcott l*), and measured by Aslanian and Weil Ia), are also shown in fig. 3. The values of 8, derived from the observed values of B, and B, are shown with error bars corresponding to one standard deviation.

THERMAL

VIBRATIONS

The values of 19, and 8, calculated from the Debye-Einstein model have been corrected for the expansion of the lattice using the formula: B(T, V,) -vo y B(T, V,) = ( v, 1 *

IN BE0

279

atoms move together, that is, (mus) = m”imn

(0~:) = m”ImB (ui),

and that for the optic modes they vibrate in opposition, that is,

They are shown as solid lines in fig. 3. The assumed value of kv, is in satisfactory agreement with the value that Sinclair 15) deduced from his observations of the neutron

(mN2) = m, (24~) = mB (21;). The solid lines shown in fig. 2 were calculated in this way.

\

\ ‘.

90(Elastic

0

Constants)

200

600

400 Ta?~~mturc

Fig.

3. Debye

temperatures

Km

UOC

1000

--OK

for specific heat (0,) and for diffraction (f&J versus temperature. calculated from the model described in the text.

scattering law for polycrysta.lline BeO, that the optic modes peak at 0.086 eV. The infra-red dispersion frequency occurs at 12,~la) which is equivalent to 0.103 eV. With further assumptions it is possible to calculate the individual vibration amplitudes from the Debye-Einstein model. We assume that for the acoustic modes, the Be and 0

The solid lines are

It can be stated confidently that, at room temperature, the value of 8, is 1120f100° K. Values outside this range could not be reconciled with the values of 8, at higher temperatures, or with the values of 8, calculated from the specific heat data, or with Sinclair’s observation of hv, . From this value of 8,, together with the ratio of B, to BB which is calculated IV.

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A. W.

280

PRYOR

AND T.

M. SABINE

from the model and supported by the experimental evidence at high temperature, the following values may be calculated: B, = 0.27f0.06 x lo-l6 cm2 B, = 0.35&0.06x lo-l6 cm2.

Kuleshov et al.‘), from an experiment identical to that described here, report an average value for the B-factor of both atoms of 0.92f 0.02 x lo-l6 cm2 at room temperature, from which they quote the Debye temperature, 8,, as 602&13’ C. They claim that this value of 6, agrees well with chemical, thermal, and mechanical data on BeO. In view of the results obtained in this investigation, the statement is inexplicable. 5. The Coherent Inelastic

Section

Neutron Cross

TEMPERATURE

(OK )

Fig. 4. The coherent inelastic cross section for 6 A neutrons versus temperature. The solid line is calculated from the Marshall-Stuart formulae using values of 19~ from fig. 3 and average masses and cross sections, as described in the text.

that the vibrational modes are all acoustic modes with a Debye spectrum, then it is implied that (zoo) = (zc& and one may consider that the scattering arises from the Be and 0 nuclei vibrating together with a total scattering cross section of (bo+urJ, where these (T’Sapply to the individual nuclei, and an average mass, h& = $(mo+m,). Applying the MarshallStuart formulae with these figures, and the values of 8, established in section 4, one calculates a value of bikes which is 5 to 15 per cent below the observed value. This discrepancy is largely removed if, while retaining the Debye spectrum hypothesis, one chooses the average mass and cross section in the calculations in such a way as to take account of the difference in (N”,) and (z&J. Since (T~$‘:~ is proportional to (1 -e-2M), and assuming, in the spirit of the incoherent approximation, that the contributions of the nuclear species are additive, one obtains the following expression for M:

An additional property of solids, which is very sensitive to the presence of thermal vibrations, is the inelastic neutron cross section which is conveniently measured at neutron wavelengths beyond the maximum which can be scattered by Bragg reflection (4.68 A for BeO). These measurements were therefore made in the temperature range 77-2000’ K, using a monochromatic beam of 6 A neutrons from equipment previously described 17). The results, shown in fig. 4, agree very closely with those reported by Begum, Rao, and Umakantha 18) in the range below 1000” K. In agreement with those authors it was found that absorption and smallangle scattering amounted to about 0.150 barns. These measurements, though comparatively easy to take, are difficult to interpret. The complex formulae involved have recently been presented in a form convenient for computation by Marshall and Stuart 19). Their equations for the coherent inelastic cross section ai:& apply ar(1-e-2M) = ao(l-e2Mo)+o,(l-e2MB), rigorously only for a monatomic cubic solid, The “incoherent approximation” is used to- where ur is the appropriate scattering cross section for the molecule and M contains the gether with the assumption of a Debye vibration Suitable spectrum. Marshall and Stuart point out 20) appropriate average mass, Mav. that the appropriate Debye parameter is the values of ur and MBv may be calculated by 8, obtained from diffraction studies. equating the first two terms of the power series. If the calculation is confined to the hypothesis The adjustment of ‘TTfrom (uo-&) is less than

281

DISCUSSION

2 per cent and the adjustment of M,, from Q(mo+mB) is 6 to 8 per cent. (This is similar to the averaging procedure adopted by Begum et ~1.1~) except that they assume also that (u2) is inversely proportional to the mass). Using these values in the Marshall-Stuart formulae, and the values of 8, shown in fig. 3, extrapolated to higher temperatures where necessary, the calculated curve shown in fig. 4 is obtained. The agreement is reasonable at low temperatures and good at high temperatures. This procedure for applying the MarshallStuart formulae to a diatomic molecule is a simplification of a very complex problem, but the results may be taken as a confirmation of the vibration amplitudes observed by diffraction methods, and as an indication that they continue to behave in the same manner up to temperatures of 2000” K. 6. Conclusions The temperature factor parameters at room temperatures were found to be B, = 0.27kO.06 A2 and B,,= 0.35f0.06 A2 corresponding to a Debye temperature for diffraction, 8,, of 1120’ K. 7. Acknowledgments

The authors acknowledge with gratitude that J. D. Browne made a considerable contribution to these experiments by supervising the operation of the spectrometer and the initial processing of the results. The furnace for the diffraction measurements was built by D. H. Cato. The samples were supplied by K. D. Reeve of the Ceramics Group. We are grateful also for much assistance in the numerical computations. The least mean squares

refinement program was written for the most part by Suzanne Hogg; J. P.Pollard contributed the matrix solution sub-program, and N. W. Bennett supplied a sub-routine for the function y(t).. We are grateful, too, for the helpful cooperation of members of the Computer Operating Group. Following the oral presentation of the paper at Newport we have incorporated several suggestions by Dr. B. Dawson, Dr. G. K. White and Prof. E. 0. Hall. References 1) E.

M. Kuleshov, G. C. Sadukov and 2. A. Sokolova, Zh. Fiz. Khim. (Moscow) 36 (1963) 1369 x) T. M. Sabine and J. D. Browne, Acta Cry&. 16 (1963) 436 *) H. Lipson and W. Co&ran, The Determination of Crystal Structures (G. Bell and Sons, 1953) ‘) H. T. Evans, Acta Cryst. 14 (1961) 689 ‘) M. Blackman, Acta Cryst. 9 (1966) 734 *) R. E. Peierls, Quantum Theory of Solids (Oxford University Press, 1056) ‘) M. Blackman, Handbuch der Physik, Vol. VII (Berlin, 1066) O) A. C. Victor and T. B. Douglas, J. Res. N.B.S. 67A (1963) 326 ‘) K. K. Kelley. J. A. Chem. Sot. 41 (1939) 1217 lo) C. E. Weir, J. Res. N.B.S. 56(4) (1966) 187 rr) C. G. Collins, Radiation effects in BeO. This Conference, p. 69 I*) H. Nolle, An assessment of the effect of radiation induced lattice growth on the strength of BeO. Unpublished ia) N. M. Wolcott, J. Chem. Phys. 31 (1969) 536 i4) J. Aslanian and L. Weil, Cryogenics (p. 36, March 1963) Is) B. N. Sinclair, Proc. IAEA Conf. on Inelastic Neutron Scattering (Chalk River, Vol. 2, p. 199, October 1062) ia) H. W. Newkirk, private communication i7) T. M. Sabine, A. W. Pryor and B. S. Hickman, Phil. Mag. 8 (1063) 43 **) R. J. Begum, L. Mahdav Rao and N. Umakantha, J. Phys. Chem. Solids 23 (1062) 1747 i@) W. Marshall and R. N. Stuart, Lawrence Radn. Lab. Rept. UCRL-6568 (1959) *O) W. Marshall and R. N. Stuart, Lawrence Radn. Lab. Rept. UCRL-6112 (1989)

DISCUSSION B. S. HICKMAN (AAEC) Dr. Austerman is to be congratulated for developing the molybdate flux method to a stage where such beautiful

single crystals can be produced on a reasonably large scale. Also it should be recorded that many of us are very grateful to Dr. Austerman and the USAEC for making crystals available for our experiments. IV.

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