797

J. Appl. Cryst. (1992). 25, 797-799

Determination of the Debye--Waller Factor of Molybdenum by Powder Neutron Diffraction

BY J. BASHIR, N. M. BUTT, M. NASIR KHAN AND Q. H. KHAN

Pakistan Institute of Nuclear Science and Technology, PO Nilore, Islamabad, Pakistan

AND ZHANG BAISHENG, YANG JILIAN, DING YONGFAN AND YE CHUNTUNG

China Institute of Atomic Energy, PO Box 275(30), Beijing, People's Republic of China

(Received 17 January 1992; accepted 7 July 1992)

Abstract

The Debye-Waller factor of molybdenum has been determined by the powder neutron diffraction tech- nique with a double-axis neutron diffractometer. The B value after correction for thermal diffuse scattering (TDS) was 0.25 (1) A 2, which corresponds to a Debye temperature 6) of 385 (7) K. These values are in good agreement with the values determined by others using X-ray diffraction and inelastic neutron scattering tech- niques.

Introduction

Molybdenum is a notable refractory metal. Outstand- ing characteristics of molybdenum are its high melting point (2883 K) and very small thermal expansion coefficient. Low thermal expansion is useful for a number of applications such as glass sealing. The resistivity of molybdenum is relatively small com- pared with conventional alloys and increases rapidly with temperature. This property makes it very suitable for use in high-temperature furnaces. Therefore, the study of its thermophysical properties is of consider- able interest. One such property is the Debye-Waller thermal parameter.

The Debye-Waller parameter B is the exponent of the temperature factor of the solid, which describes the temperature dependence of the intensity of neu- trons or X-rays scattered from a crystal lattice. B is related to the mean square amplitude of atomic vibra- tion (U 2) and the Debye temperature O through the relations

B = 8n2(u 2) (1)

= (6h2/mK)(T/O2)[tp(x) + x/4], (2)

where K is Boltzmann's constant and the other sym- bols have their usual significance (James, 1967).

Neutron and X-ray diffraction measurements pro- vide an accurate method of determining the thermal parameter of materials. However, with X-ray powder diffraction measurements there are severe problems

0021-8898/92/060797-03506.00

due to the effects of preferred orientation, porosity, surface roughness and line broadening (Paakkari, Suortti & Inkinen, 1970). All of these problems can be eliminated by use of the powder neutron diffraction technique.

The purpose of the present work was to obtain information about the thermal parameters of molyb- denum by measuring neutron diffraction intensities from a powder sample and to compare the results with those obtained by X-ray diffraction and inelastic neu- tron scattering.

Thermal diffuse scattering

In order to determine accurately the Debye-Waller factor, one must correct the diffracted inensities for the effect of thermal diffuse scattering (TDS). If I o is the true Bragg intensity and ~Io is the TDS contribu- tion to the intensity, then the total scattering intensity is

I = Io(1 + 0¢). (3)

The TDS contribution to the intensity, e, can be calculated using the relation

o~ = (8n/B)(Q2qmax/mVz)(~ i Ei(q)cos 2 o~i(q)/V2 1 (4)

(Willis, 1969), where the angle brackets, ( ) , indicate the average over all the q vectors contributing to TDS and the symbols are those defined by Willis (1969). This expression can be simplified at high temperatures (T > O) by replacing the mode energy Ei(q) with KT and assuming that the modes are acoustic and iso- tropic, with longitudinal and transverse velocities Vt and V,, respectively. It was further assumed that the volume seen by the detector is the sphere of radius qmax centered on the appropriate reciprocal-lattice point. In this case,

o~ = (8n/9)(Q2qmaxKT/mvz)[(1/V 2) + (2/V2)]. (5)

However, in the present experiment T < O (O -~ 390 K) and the assumption that the mode energy is KT is

© 1992 International Union of Crystallography

798 DETERMINATION OF DEBYE-WALLER FACTOR

no longer valid. The average mode energy in this case is (Schoening, 1969)

<E,> = K T ¢ , (6)

where ~, = x/(½ + 1/e x - 1) with x = 19/T. Equation (4) then becomes

= (8rc/9)(Q2qmaxKT~,/mVz)[(1/V 2) + (2/V2)]. (7)

If c¢ is sufficiently small to justify the replacement of 1 + ~ by C, the effect of including TDS in the estimate of the Bragg intensity is to decrease artificially the overall B value. This decrease, AB, is given by

ct = 2AB sin 2 0/22 = (Q2/8n2)AB, (8) . . . . . . . .

o r

AB = (64na/9)(qmaxKT~'/mVz)[(1/V 2) + (2/V2)]. (9)

Willis (1969) has also pointed out that the TDS correction to the intensities is required only when the neutron velocity is greater than the sound velocity in the material and that, to a first approximation, there is no correction for the scattering of slower-than- sound neutrons.

Experimental

The neutron diffraction experiment was performed on a powder sample of molybdenum using a double-axis diffractometer at the 15 MW heavy-water research reactor at the China Institute of Atomic Energy, Beijin~. Monochromatic neutrons of wavelength 1.184 A were obtained by the scattering of the primary beam from (002) planes of a pyrolytic graphite monochromator. Molybdenum powder of 99.9% purity (particle size 50~tm) was used to fill a thin-walled cylindrical vanadium container of diam- eter 1 cm and length 5cm. The intensity data were collected over an angular range of 20 from 26 to 131 ° with a step of 0.15 °. All measurements were made at room temperature (290 K).

Results and discussion

The powder diffraction data were refined by the Rietveld method (Rietveld, 1969). The observed (lobs) and calculated (Ica~c) intensities of the individual peaks are given in Table 1. The final R (nuclear) factor ( = 1 0 0 ~ l F o ~ s 1/cF2alcl/~ 2 -- Fobs) was 1.70% which indicates good reliability of data.

In order to correct the B factor for the effect of TDS, it is necessary to determine first whether the neutrons are slower or faster than sound in the solid. The neutron velocity corresponding to the neutron wavelength of 1.184/~ was 3.34 km s- 1. The velocities calculated for the one longitudinal mode and the two acoustic modes were 6.72 and 3.28 km s - t , respec- tively. For the calculation of the sound velocities,

Table 1. Observed and calculated neutron diffraction intensities for molybdenum at room temperature; c¢ is

the percentage T D S correction

hkl ~bs ~al¢ ~ (%)

110 133388 132060 0.19 200 34560 34021 0.37 211 92834 91840 0.56 220 34827 35059 0.74 310 59337 58006 0.93 222 17348 16744 1.11 321 91518 89316 1.30 400 10753 10410 1.48

411, 330 59077 58794 1.66 420 38146 38312 1.85 332 41601 39174 2.03

elastic constants from Huntington (1958) were used. Thus, the neutrons are slower than sound for the longitudinal modes and faster for the transverse modes. As pointed out earlier, the TDS correction is zero if the sound velocity exceeds the neutron velocity, so it was evaluated for the transverse modes only. Accordingly, (9) becomes

AB = (64~3/9)(KTqmaxd//mVz)[(2/V2)]. (10)

The TDS contribution to the Debye-Waller factor, AB, was 0.02 A 2 when calculated by this method. The TDS contribution to the diffraction intensities, ~, varied from 0.19% for the 110 reflection to 2.03% for the 332 reflection. The ~ values for the reflections examined at room temperature, calculated in the spherical approximation with AB = 0.02 A 2, are listed in Table 1.

The present TDS-corrected B value is 0.25 (1)A 2 and corresponds to a Debye temperature 19 of 385 (7) K, in agreement with X-ray diffraction results (Paakkari, 1974; Korsunskii, Genkin & Vigdorchik, 1977). This B value is also in agreement with the recommended value of Butt, Bashir, Willis & Heger (1988) [B = 0.25 (2)A2], which they obtained after carefully evaluating the experimental data available in the literature. Merisalo & Paakkari (1973) calculated the Debye temperature from the force constants given by Walker & Egelstaff (1969). They obtained a Debye temperature of 399 K (B = 0.24/~2). Powell, Martel & Woods (1977) reported 19 = 378K ( B = 0.26A 2) which they obtained b y fitting a general tensor Born-von Karman model to the experimentally determined phonon frequencies of Woods & Chen (1964) and Powell, Martel & Woods (1968). Both of these values are in essential agreement with our results.

In summary, the present experimental values are in good agreement with the values obtained by X-ray diffraction techniques and agree to within 4% with those calculated from data obtained by inelastic neutron scattering.

J. BASHIR et al. 799

One of the authors (JB) is grateful to Professor Sun Zunxun, Director, Institute of Atomic Energy, Beijing, for the hospitality shown to him during his visit to the Institute.

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