Physica 94B (1978) 319-330 © North-Holland Publishing Company

CRYSTAL EQUILIBRIUM AND LATTICE VIBRATIONAL PROPERTIES OF THE MATERIALS OF BCC STRUCTURES

O. P. GUPTA* and M. P. HEMKAR Physics Department, Allahabad University, Allahabad, India

Received 14 November 1977

A lattice dynamical model which assumes short range pairwise forces effective up to second nearest neighbours and electron-ion interaction on the lines of Bhatia, is considered in the study of the lattice vibrational properties of bcc metals. The authors consider an appropriate value of the screening parameter. The volume force is averaged over the whole Wigner- Seitz sphere. The ionic lattice is in equilibrium in a medium of electrons. The present theory has been satisfactorily applied to compute the dispersion curves, Debye characteristic temperatures, and temperature dependence of the Debye-Waller factors of chromium, molybdenum and tungsten.

1. Introduction

During the last two decades several attempts [1-5] have been made to study the lattice dynamics of cubic metals using realistic models on first principle approach. But, even today the task remains formidable. Normally, phonon spectra have been analyzed on the lines of the Born-IC~irmLrl [6] theory of lattice vibrations. This requires a large number of force constants to be used as independent parameters [7, 8] for interpreting satisfactory results. Proper physical interpretation cannot be given to these force constants and are incapable of describing the actual interaction in metals. Therefore, attempts [9-12] have been made to develop elastic force models in which the nature and magnitude of the forces between bare electrons and ions have been reasonably explained. These have been excellently reviewed by Joshi and Rajagopal [13]. In the models of de Launay [9], Sharma and Joshi [11], and Krebs [12] the crystal lattice is in equilibrium under a central force alone. An equilibrium condition on the model solid has not been imposed by these authors. Furthermore, they assume that in equilibrium the derivatives of the potential energy for each set of neighbouring ions (i.e. o~) are individually zero; in addition to this they also neglect electronic pressure.

* Also at Physics Department, J. Christian College, Allahabad, India.

In this context Thomas [14] stressed that the energy terms and hence their first derivatives cannot be set separately equal to zero in an equilibrium configura- tion and he emphasized that elastic constant theory only applies to a solid in equilibrium and that an equilibrium condition must be imposed explicitly for empirical models for lattice cohesion. Thus, in equi- librium the presence due to the ionic potential must be balanced by the pressure due to the electron gas.

In the present paper we propose a phenomenological model which considers the ion-ion interaction through second neighbour pair potential [15] to the electron- ion interaction on the lines of Bhatia using a realistic value of the screening parameter. The volume depend- ent force due to the compressibility of the electron gas and its interaction with ions has been averaged over the whole Wigner-Seitz sphere. The electron-ion interaction part of the dynamical matrix in Bhatia's approach is rigorous, based on sound physical grounds, and is easy to compute as evidenced by the published work [16, 17].

The present scheme has been utilized to compute the phonon dispersion, Debye characteristic tempera- ture, and temperature dependence of the Debye- Waller factor of chromium (Cr), molybdenum (Mo), and tungsten (W).

Chromium (24), molybdenum (42) and tungsten (74) belong to group VI-B of the periodic table, where

319

320 O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties of the materials of bcc structures

the number given in the parentheses are their atomic numbers. These transition elements crystallize in the bcc space lattice at room temperature with one atom per unit cell. They have incompletely filled (n - 1)d shells, where n is the principal quantum number, i.e. 4 for Cr, 5 for Mo, and 6 for W. Stringer and Rosenfield [18] have shown that the activation energy for the diffusion of oxygen atoms increases with absolute melting point in Cr, Mo, and W. Zener [19], who made a formal attempt, envisaged that, with the number of outer electrons per atom being six in group VI metals, the ground state electronic configuration in the free atom, and possibly in the metallic state would be (n - 1) dSns for Cr, Mo, and W. The five d-electrons would just half fill the d shell with all their spins aligned in one direction in accordance with Hund's rules. These facts are responsible for the broadening of ns, np, and (n - 1)d electron states of the free atom into overlapping bands. As a result of this the electrons concerned are found in the hybrid (spd) states giving rise to multiple chemical valence (Z). However, on the basis of effective chemical valence, which determines the number of electrons that may be treated as nearly free, we have taken Z = 3, 6, and 6 for Cr, Mo, and W, respectively. Besides these, the phonon spectra for chromium [20, 21], molybdenum [22, 23] and tungsten [24] display common features.

2. Theory

The system in which the vibrations take place consists of point ions having their equilibrium positions at the lattice points and on the electron gas surround- ing the ions. The assumption of point ions is valid for all cubic metals [5, 25] excepting group I-B metals. The phonon frequencies in the harmonic approxima- tion are given by the secular equation

[D12(q) - Mc°2,i ~ 121 = 0,

Following Maradudin et al. [15], the ion-ion contribution to the dynamical matrix for bcc structure is given by the following expressions:

where

D12(q ) = Dil~i(q) + Dil~e(q)

Dil-i i (q) = ~(20tl + ~1) (1 -- CLC2C3)

+ 4fl2 $2 + 4a 2 (S 2 + $2), (3)

" - i S D112 (q) = ~ 1 -- °tl)S1S2C3' (4)

where al , °t2,/31, and r2 are the pair potential force constants, S i and C i (i = 1,2,3) are respectively sin (naki) and cos (rraki), qi = 2nki, and a is the lattice constant. Here we have conf'med the ion-ion interac- tion up to second nearest neighbours only as the bare electron-ion interaction reduces the effect of the long range Coulomb interaction to short range forces between ions [2]. The merit of this procedure has very recently been enumerated by Upadhyay [26], who has shown that the central pairwise potential [15] stands on sound theoretical grounds and the angular force models produce no additional results and create confusion; these models need not be retained for lattice dynamical calculations. Furthermore, Bertoni et al. [27] have also shown that the central pairwise interaction between ions gives a good description of the lattice dynamics of cubic metals.

The modified form of the electronic part of the dynamical matrix using Bhatia's approach is

l~2e(q) Keq lq2~ - i 02(qro).

The screening parameter k c is given by

k2c = k (e) r(t) ,

(5)

(6)

where

(1) kc(P) = 0 . 3 5 3 m \ao! kF' (7)

(2)

Here D12(q ) are the elements of the dynamical matrix, q is the wave vector confined to first Brillouin zone, and M is the ionic mass.

and

f ( t ) = ½ + 1 - t 2 1 + t , ' 2 t I n I ~ " (8)

with t = k/2k F. Here a 0 is the Bohr radius, k F the

O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties o f the materials o f bcc structures 3 21

Fermi surface wave vector, and (m*/m) is the effective electronic mass to consider band structure effects [28]. The function f(t) of eq. (6) has been included as the screening parameter kc(P ) in a high density electron gas and is k dependent [29], kc(P ) has been borrowed from Pines [30], ~2 is the crystal volume, and the function G2(qro), which was first given by Laval [31] for averaging the volume dependent force due to the compressibility of the electron gas and its interaction with ions, is computed by an average over the whole Wigner-Seitz sphere and used successfully [1 i, 12], is given by

G2(x) [3(sinx- xc°sx)] 2, (9)

where x = qr 0 = qre/Z~(Raimes [32] ). Here r 0 is the interelectronic spacing, r e the radius of the Wiguer- Seitz sphere, and Z the chemical valence.

The potential energy U of the crystal is the sum of the potential energies Ui( = Z/U}), and U e correspond- ing to ion-ion and electron-ion interactions, respec- tively. The lattice equilibrium will be determined by vanishing of the total pressure independent of direc- tion, i.e. (aUi/aI2)a + pe = 0, where pe = aUefi)~2, the electronic pressure, is assumed to be isotropic. Writing ~ = 1/rj(dUi/dr/) andP e = 3/5 Ke for free electrons [9, 10], the equilibrium condition assumes the following form:

a 1 + a 2 + (3/lO)aKe = O. (10)

Large number of solid state phenomena such as neutron scattering, MSssbauer effect, electrical con- ductivity, and melting of crystals are governed by an exponential factor e -2w, often called the Debye- Waller (DW) factor. The DW factor is assumed to consist of two parts: (a) the phonon dependent part W' which is obtained from the knowledge of the whole vibration spectra, and (b) the second part W" is the contribution of the central part of the Bfillouin zone, corresponding to q = 0 and is evaluated in the Debye approximation.

(a) Following James [33] and a recent paper of one of us [34], we can write

8~r2~ tsin0~ 2 E 1 coth~_coq,_/l (11) 2W' = 3ran ~ X I q,i coq,/ \2--k-~sT]'

where m is the mass of the atom, N the total number of unit ceils in the crystal, coq ] the angular frequency of phonon wave vector q wit~ polarization ], eq i the polarization vector of the (q,]) lattice mode, ~i the Planck's constant divided by 2~r, K B the Boltzmann constant, T the absolute temperature, 0 the glancing angle, and X the wave length of the incident radiation and the summation extends over all the normal vibra- tions of the crystal.

(b) Since all the modes are acoustic for a monatomic lattice, co is zero when q = 0. Following Buyers and Smith [35], we can avoid this singularity in the sum by replacing the contribution to the sum at q = 0 with an integration of the energy term [E/4zr2v 2 (of the mode of lattice vibrations)] :

E =/ico[{exp (lico/kBT) - 1} -1 -4- ½] (12)

over a sphere of volume equal to 20 -3 of the Brillouin zone near q = 0. It is assumed that the frequency is given in this region by the average velocity times the wave vector. The zero phonon contribution W" is then written as

=~m {D(CL) + 2D(CT)} ' (13)

where

1 D ( C ) =

2~f kmax [{ ~co. } -1 0 47r21iCkdk exp (k--~) - 1

+ ~] , (14)

Here C L is the average velocity for the longitudinal phonons and C T is that for the transverse phonons in this region, kma x is the radius of the sphere of integra- tion and is given by

4zt k3 - 7 max = 20-3 (volume of the Brillouin zone)

or

1 ( x , l ~ (for bcc metals). kmax - 20a \~ffl

(15)

D(C) can be reduced to

322 O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties of the materials of bcc structures

0 et - 1] + ~Px,

where

4rr 2 k B T kma x . 41r2~ kma x C P = , x =

C 2 kBT

(16) aCll = ~(2Otl +/31) + 2/32 + aKe,

aC12 = ~(2/31 - 4or I - 3ot2) + aKe,

aC44 = ~(2ot 1 +/31) + 2ot 2,

mrr2v2 = ~(2ot 1 +/31).

(17)

and

2n'hkC

kBT

The value of the integral contained in the expression for D(C) can be substituted from standard tables. The DW factor W has been calculated from eqs. (11) and (13).

3. Numerical computation

The three equations of elastic constants and one equilibrium condition, eq. (10), together with one zone boundary frequency in the [100] direction, form the necessary set of equations to determine the five parameters of the theory. The remaining four equations are as follows:

Experimental values of the elastic constants and other parameters used in the calculations are summarized in table I. We have taken effective electron mass ratio (m*/m) to be one for Cr, Mo, and W as has been done earlier by Animalu [5]. This approximation implies that these metals have free electron-like band struc- tures. It is the mechanism of the d band hybridizing with the s - p band which decreases m* for transition metals. The calculated values of the force constants are listed in table II. These force constants are used to calculate frequency versus wave vector dispersion relations along the three symmetry directions [100], [110], and [111] for Cr, Mo, and W, determined from the solution of the secular eq. (1). The frequency dis- tribution of the normal modes of vibrations have been computed by Blackmann's root sampling technique [36]. For this purpose it is essential to determine the characteristic frequencies at a suitable number of points in an irreducible section of the first Brillouin

Table I Physical constants for metals used in calculations

Metal Density Lattice Elastic constants Source Phonon Source (gcm -3) constant frequency

(A) C 11 C 12 C44 ~'T (in units of 1011 dyn cm -2) (100)

(1012 Hz)

Chromium 7.19 2.8792 35.00 6.78 10.08 ref. 46 7.83 ref. 21 Molybdenum 10.22 3.1468 44.077 17.243 12.165 ref. 44 5.10 ref. 22 Tungsten 19.1 3.1650 52.327 20.453 16.072 ref. 44 5.56 ref. 24

Table II Calculated force constants

Metal a I a2 0 1 132 aKe (All in the units of 103 dyn cm -1 )

Chromium -0.1581 1.4541 39.4995 39.4989 -4.32 Molybdenum -11.0545 8.8802 52.7654 55.3582 7.2476 Tungsten -4.4952 2.6074 77.3494 58.3239 6.2927

O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties of the materials of bcc structures 323

zone. We have considered an evenly distributed mesh of 8000 wave vectors in the first Brillouin zone. From the Born cyclic boundary condition and consideration of lattice symmetry, these 8000 points reduce to 256 non-equivalent points including the origin lying within ~ th part of the Brillouin zone and weighted according to the number of similar points associated with them. The 24 000 frequencies corresponding to 8000 points ha the zone were obtained from the solutions of the secular determinant (1). The lattice specific heat at constant volume has been computed at different temperatures from the equation.

m C o = k B E(x) g(co) doo, (18)

b

where x = h w l k B T and fo g(¢o)dco = 3N0; k a and N O are the Boltzmann constant and number of atoms per mol, respectively. This procedure has been used at moderate temperatures down to ® 10, asg(6o), computed by sampling technique, cannot be very accurate in the low frequency range due to coarseness of the mess. For temperatures below O 10 down to about 2 - 3 K, where the lower frequencies have a predominant influence, the modified Houston's method as elaborated by Betts et al. [37] has been used. The frequency distribution can be written

p=l (19)

where ~2 is the solid angle ha the wave vector space and the summation over p is on the three modes of lattice vibrations for each q. The use ofeq. (19) ha (18) gives

qmstx V C v = k B (2r0 3 ~ j" .f E(x)q 2 dqd~.

P 0

(20)

In evaluating C o from (20), the integration over q was performed numerically and the angular integration was carried out by a modified Houston's spherical six-term integration procedure. The six direction of q used are: [100], [110], [111], [210], [211],and [221] and their weights are given by

aft J = 1081080 [117603IA + 76544IB + 17496Ic

+ 381251I D + 311040I E + 177147IF]. (21)

We have compared the theoretical results with the experimental data on C o (after correction for Cp - C o and the electronic specific heat) in terms of the equiva- lent Debye temperature O D.

The DW factor in terms of a temperature parameter Y may be written

Y = lOgl0e (X/sin O)2(2WTo -- 2WT),

where WTo and W T are the DW exponents at tempera- ture T O and T, respectively.

4. Results and discussion

4.1. Dispersion relations

M~511er and Mackintosh [20] have measured phonon dispersion curves of chromium by the neutron scatter- hag technique. Owing to the relative sparseness of data these dispersion curves did not receive much attention; however, an analysis ha terms of the fourth neighbour tensor force model of Begbie and Born [38] has been reported for the first time by Feldman [39]. Shaw and Muhlestein [21] have measured detailed phonon dispersion curves of chromium at room tem- perature by means of thermal neutron scattering. These phonon dispersion relations show four regions of anomalous behaviour. Our theoretical calculated dis- persion curves have been plotted in fig. 1 together with these experimental data [20, 21]. The overall agreement is good, particularly for low values of the wave vectors, but starts deteriorating as the wave vector increases. However, the experimental data of these workers [20, 21] differ considerably for large wave- vector values. It is also worthwhile to mention here that for chromium the elastic constant C44 is greater than C12, in contrast to the elastic constants of all the other bcc and fcc crystals for which C12 > C44. This unusual behaviour gives rise to a subtractive con- tribution of the electron gas when the compressibility factor Ke is parametefized in terms of the elastic constants of the cubic crystal (C12 - C44 = 2.2 Ke).

324 O. 1>. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties o f the materials of bcc structures

40

tJ "I"

(> >- 0 Z ld

,,t

mE

P H I1 N

[3ooj • O

O ~ 0

P

[ f l o ] i I I / . 4

T - - T T - -

£ ~ H 1

i s

r

T ', I 6

i t , : ~ "I"4

P R F . . . ~ N T C A k C U L A T | O N S ~ , ~ E}I, PF.. R ~ I r H T A 1- pO|N' I ' ,~ ~, ~ ' i -I

e x a , M~t.'-ER ~,.), M~-~wrosH ', ~ \ " - ~ - 7 - " - i ' ., J

1 ,' \ ." /

, .~ .~ • ~ t - 0 " $ " ~ " 4 . a 0 " 0 - t ' 2 ' a "k

~. L ' i ~ U C E , , b W A V E V E C T O ~ C O O R..I) | N / i T E . " ~

Fig. 1. Phonon dispersion curves of chromium along the symmetry directions.

.a"

A negative 2.2 Ke which affects the longitudinal frequencies alone, means that the electronic contribu- tion to the total interaction which ought to be additive is subtractive, and this results in the overall low values of the computed longitudinal frequencies in all the symmetry directions.

Neutron inelastic scattering of normal modes of vibration in molybdenum at room temperature has been made by many workers [22, 23, 40]. Woods and Chen [22] have measured the phonon dispersion relations in Mo by neutron spectroscopy for wave vectors in the three symmetry directions. Powell et al. [23] have reported extensive measurements of phonon dispersion curves at 296 K by using one phonon scatter- ing of thermal neutrons. Recently Walker and Egelstaff [40] have studied the phonon frequencies for wave vectors in a (110) plane at room temperature. The measured frequency wave vector results are in general agreement with major symmetry axis data of Woods and Chen, but show further anomalous behaviour in

several regions in this plane. The data of Woods and Chen, and PoweU et al. are plotted in fig. 2 together with the calculated values. The general agreement between theory and experiment is fair except at the zone boundary in the [110] direction, and in the region ~ = 0.5-1.0 in the [111] direction.

The calculated dispersion curves of tungsten are shown in fig. 3 together with the experimental points of Chen [41], and Larose and Brockhouse [42] measured at room temperature, using the methods of constant Q or constant E, as appropriate [43], with the use of triple-axis neutron spectrometry. The theory exhibits good agreement with the experimental data. Ultrasonic measurements [44] show that tungsten is remarkably isotropic within 1%; however, this behaviour is not confined to the long wavelength phonon domain as shown by the near degeneracy of the T 1 and T 2 branches.

Furthermore, the frequency where vector dispersion curves of Cr, Mo, and W have been compared with the

O. P. Gupta and M. 1). Hemkar/Crystal equilibrium and latHce vibrational properties o f the materials o f bcc structures

P

(o E(oo] E((f] : [~(o]

el

0 Z ~d ~T

U.

r /

f

/P ~ I I i ¢1 I I / /" , • ~ !~.~ I. o

<: T i ~ i II

\ . l

. . . . . /~ NI i'-1 ~ i . U ~ I ~ S F . N T CA I.- CUL#~ T I0 HS

F..'IL P F- RI IvlE N T/M- P O I H T 5 % 0 1 ~ , W(X)/)$ nH.D THEN A n-t- pO~,IELL el" (ft..

325

i I "l

/ J

li

N

/ / / / ]

L

I

\

O " 2 • 6 "~ "~, 4-o "S -6 "4 "~ o .4 "a

I~.~ 1) U C $. .b W A V f . V E c T o R - ~

Fig. 2. P h o n o n d i spers ion curves o f m o l y b d e n u m a long the s y m m e t r y d i rec t ions .

"3 "4 "5"

recent published phonon spectra of Animalu [5] by using new transition-metal model potential (TMMP) which includes s - d hybridization, in nearly free electron second order perturbation theory. The computed curves of Animalu are indicated by dotted lines in figs. 1-3. Except at the zone boundary in the [100] direction for Cr, Mo, and W, and at the zone boundary in the [111] direction for Cr and Mo, the results obtained with our model exhibit a compara- tively better agreement with the experiments. In addition to this the phonon spectra calculated by Animalu tend to disagree in detail with the experi- mental points. Animalu, himself, thought this to be due to some important factor left in the local TMMP approximation used in his calculation. Apart from this, the theory of Animalu is very complicated in mathematical formalism and involves great computa- tional labour and the results obtained by this theory are of the same order of magnitude as that given by our model with simple mathematical design and with less computational labour.

4.2. Specific heat

The specific heat at constant volume C o per gram atom has been calculated in the usual manner by numerical integration. In the present case the calcula- tion for each temperature was made separately using the input data appropriate to that temperature. The lattice parameter values at various temperatures were taken from Pearson [45] and its linear interpolated and extrapolated values. The experimental values of the temperature variation of the elastic constants, together with the linear interpolated and extrapolated values, have been borrowed from Bolef and de Klerk [46] for Cr; up to room temperature from Featherston and Neighbours [44] and above from Dickinson and Armstrong [47] for Mo; and up to room temperature from Featherston and Neighbours [44] and above from Lowrie and Gonas [48] for W.

The experimental data for specific heats C o for these metals are summarized in table III. Lattice specific heats have been estimated after deducting the

326 O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties of the materials of bcc structures

I -

U

~J

I l L

P H i i

Doo]

1 ~ P

' i,, ' I ' ' ' E t t l ] : [1 i o ] I

L~

L /" J

. . . . . A H I I ' I & I ~ U

, i

~ 1 I , /

~ j

"~RF... SF. N T C R L r - U LAT4 ON e. F."~,P¢ R I M E N T & L " P O I N T S ;

.-.-. coNsr~,Nr t l t , ~ o z t a~b 8 R o c K . o u s t "L CONS' r i l l , IT _ QJ

• t.~ C O N S T A N T F." 1 C H I : N ! c o . s v ~ N x Q J

H C O N S T A N T ~ . I c o M I ~ O N ~I31NlI~ IN JtlbOYF_ ~[ C O N S T R N T Q_j-rwo R E F E R E N C E S

1 = 1

-a "4 .6 "~ q-o

J

/ f

~ /

• ~ "4 ',9. 0 " 0 - t " ,2 . ,L .z~ -.S"

R E-.b U c E~ WAVE VECTOR COOR~I NA "rE

Fig. 3. Phonon dispersion curves of tungsten along the symmetry directions.

Table III Experimental specific heat data and their temperature range

Metal Temperature range Source (K)

Chromium 55-100 ref. 49 50-100 ref. 50 10-270 ref. 51 30-260 Heiniger's estimated

values (ref. 57)

Molybdenum 1-77 ref. 58 20-200 ref. 59 16-274 ref. 60

Tungsten 1-77 ref. 58 20-200 ref. 59 26-91 ref. 61

1 0 0 - 2 0 0 ref. 62 20-300 ref. 63

contribution of electronic specific heat (C e = "IT). The coefficients of electronic specific heat 7 used for this purpose are collected in table IV. Experimental and theoretical data for the specific heats are converted into Debye temperature O.

Fig. 4 shows the equivalent Debye O---T curve for Cr together with the extracted lattice vibrational contribution to the heat capacity C L, where C v = C L + Ce, from the experimental measurements of Simon

Table IV Electronic 2 specific heat coefficient (~cal/K g atom)

Metal 7 Source

Chromium 36 ref. 39 Molybdenum 500 reL 59 Tungsten 270 ref. 59

O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties of the materials of bcc structures 327

g O 0

$ 5 O

4OOl

I i , J

t & PR£gEPIT CALCULATION i E.XPP. RIPIF..N'rttL P O I N T S : A A • ChU~lOS AND FRRNT-O~'INI O O O HF.INI6rR'~S EETIPq/~TF-D VAbUEs l i e d .,~llViON IkNb I~UHE.tqANN

OAoA, AA£1 ANgER.SON

d . ~ , " , & & " & & A A & • • A A A • A & & , t

N ~ o o o o 0 0 0 0 0

"o ' ,io ,ho ~o ~0 t00

TF-. M pF... RA "~'u g & (K)

Fig. 4. Debye temperature of chromium as a function of temperature. The cross (X) and closed circle (*) represent the Debye ® values determined from the T = 0 K (extrapolated from T = 77 K) and T = 298 K elastic constants of Bolef and de Klerk (ref. 46).

and Ruhemann [49] , Anderson [50], and Clusius and Franzosini [51 ]. Theoretical studies of many workers [52-56] reveal that the conventional treatment of the electronic specific heat C e = ~,T shows inadequacy for chromium. As elaborated by Feldman [39], Heiniger's [57] estimated values have also been employed in the present study. The theoretical O - T curve shows a better overall fit with Heiniger's estima- ted values when compared to experimental data. In fig. 5 we plot the Debye O-Tcurve for Me together with the lattice heat capacity data of Horowitz and Daunt [58], Clusius and Fronzosini [59], and Simon and Zeidler [60]. A reasonably good agreement between theory and experiments is observed. For W, the experimental lattice heat capacity data of Horowitz and Daunt [58], Clusius and Franzosini [59] , Lange [61] , Zwikker [62] , and Kelly [63] have been plotted in fig. 6 together with our theoretical values. The data of Kelly have been transformed into lattice Debye temperatures into two categories, A and B, using two different values of electronic specific heat coefficients (details are given by Myers [64] ). The agreement between theory and experiments is good. A closer examination of figs. 4 - 6 indicates that the

5~

41.0

kr.a

t PKESI~NT C RLC ULATION$ ~.XPERIMEPITAL, P O I N T S :

~. A • C,b~) 61U.S RN) P/~INI~OSlN! + ..I- ~ SIMON ~bl~ 7.1r|.DbER • D • HOROW|TI[, AIdJb ~flUAIT

4m + & +

+ ~ • . .

TEI ' IP£RATUftE. (K)

Fig. 5. Debye temperature of molybdenum as a function of temperature.

328 O. P. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties of the materials of bcc structures

I PRESENT CALCULATION F. ~ P E. Ri I~ f.N TAI.- "POINTS;

~ k A & CL.U$1US &NI) FRANZOSlNI I # + , L~NSt

,,~,L o o o Z ~ K K E R - |~eQ • e r a 14o~.o~ltx~: ~.N.b I)AUNr

~°~o ~, ~o ,-', A, T F... PI I:'F._R K x U R ~ ( , K )

Fig. 6. Debye temperature o f tungsten as a func t ion o f temperature.

Debye O attains its highest value in the temperature range 0 -10 K, then decreases monotonically up to about 80-100 K temperature range, and afterwards it remains virtually temperature independent. This variation of the Debye O values with temperature for all the metals studied follows the general prediction given by Black•ann [65].

4.3. Debye - Waller factor

The reliable integrated X-ray intensity data for a single crystal of chromium have been reported by Wilson et al. [66] in the temperature range 100-520 K. The data were corrected for thermal diffuse scattering as well as changes in the lattice parameters. These authors have expressed their results in terms of the characteristic I)ebye temperature. For comparison we have calculated the quantity (X/sin 0) 2 log e (IT/ITo) as a function of temperature from the corresponding values of Debye temperatures. Ilyina and Kristkaya [67] have also reported the measurements of the X-ray Debye temperature of chromium but their measurements are confined to room temperature only. Reference temperature T O has been taken as 298 K.

The effect of temperature on the intensities of the X-ray reflections for molybdenum crystals was studied experimentally by Korsunskii [68] and the observa-

tions were taken in the temperature limit 100 < T < 400 K. He measured the intensity ratio 1291/I T of the crystals for the (232) and (322) reflection planes, respectively. The measured values of (X/sin 0) 2 lOgl0 × (IT/1T o) for the (232) and (322) planes have been plotted against temperature with reference temperature T0 = 291 K.

We display our calculated values of Y for Cr, Me, and W in fig. 7 together with the measurements of Wilson et al., Korsunskii, and Geshko. Our theoretical Y values are in satisfactory agreement with the experi- mental findings. It is worthwhile to mention here that the contribution of the central part of Brillouin zone is about 5-6% for all the three metals and its contri- bution gave a good fit of our theoretical curves with the experiments. The present study offers a satisfactory explanation of the observed temperature variation of the X-ray intensities of Bragg reflections up to a certain temperature. However, the marked discrepancy may be attributed to the neglect of lattice expansion [70] and other anharmonic effects [71-76] . Anharmonic contributions vary mostly as the square of the absolute temperature and become predominant at high tem- peratures. Furthermore, the frequencies decrease due to thermal expansion at elevated temperatures. This effect depends on the Griineisen parameter, which is

O';l

u.4

0"0

-O-1

-G-a.

O'S

\ - \

T£HP£ RATUR E (,K) 4do J~* ~ A * ' ' ' ' ' 400 JoO 4~ See 6o0 "~O 11100 q~<~

I I N ~ . " . . .

--.Y

EXPERIMENTAL POINTS: \ ~ "V. & & ~ I L , SON e~t ~ . \ , A ~ " " " ' . \ - -

. o . . . , u . \ ..... - - - - THEORETICAL CURVF- ~ V'""

I~_ _m (~3:~) !~1-&1N -I I~'(1 ! S UN & ~( [ I \ . ~ "4" "" • * ~ . ) ~ . , . j . . . . . . . . . . . \ ~ ' . "

T U N ; S T F N (,T,~ = a'~+K) \ ~ "'. :~THeO~[rlellt. euRv~ " \ \ ~ +

I [XP£RII " f£NTR~ POINTS" % ; ~. "0.+) ~ , , ~ . .~ .~o . \

\ \

\

Fig. 7. Debye-WaUer factors of chromium, molybdenum and tungsten as a function of temperature.

O. 1'. Gupta and M. P. Hemkar/Crystal equilibrium and lattice vibrational properties o f the materials o f bcc structures 329

temperature dependent. Several attempts [77-79] have been made to estimate this effect in terms of the Griineisen parameter, but a detailed discussion has not been possible. In addition to this our results are undoubtedly superior, and will eclipse those obtained earlier by many workers [80-84] since the schemes employed by these authors suffer from serious draw- backs of internal equilibrium, long range screening of the conduction electrons, and contribution of the central part o f the Brillouin zone. Hence much reliance cannot be bestowed upon their computed DW factors.

The discrepancy between theoretical results and experimental data can be attributed to the neglect of temperature variation of vibrational frequencies [85], neglect of cubic and higher order terms in the expan- sion of crystal potential, and to the anisotropy of the Fermi surface of Cr [86, 88] , Mo [88, 89] and W [88-91 ]. The incomplete d-electronic shells results in many unusual properties [92] particularly the large cohesive energy [93] which is expected to make an important contribution to the forces governing the lattice vibrational properties of transition metals. Interatomic forces in transition metals are of long range nature [94, 95] as inferred by the studies of the inelastic scattering of slow neutrons. Thus, we expect to improve our results further by considering more distant neighbours.

5. Concluding remarks

In spite of these drawbacks, short range models are worth attention for several reasons. First, the Fermi surface effects are presumably to a large extent strongly directional, and hence part of the observed long range effects are likely to be specific to limited ranges of directions rather than typical o f the crystal generally. Secondly, the effective long range forces are in certain metals comparatively weak due to their cancellation or to low density o f conduction electrons. Thirdly, in short range models coupled with the elastic constants some effects of atoms beyond those formally considered are always included in effects o f the latter through the correspondence equations. Simple models based on general properties o f the crystals could, furthermore, serve as a standard, representing smooth behaviour, in studies of various lattice vibrational properties of metals.

However, from obtained agreement with experi- ments, we conclude that the present lattice dynamical model accounts quite well for the vibrational proper- ties of Cr, Mo, and W.

Acknowledgement

The authors gratefully acknowledge the computer facilities made available to him at the Indian Institute of Technology, Kanpur.

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