Solid State Communications, Vol. 54, No. 2, pp. 163-166, 1985. Printed in Great Britain.

CRYSTAL DYNAMICS OF NIOBIUM

O.P. Gupta

Physics Department, J. Christian College, Allahabad, India

(Received 25 October 1984 by G.S. Zhdanov)

0038-1098/85 $3.00 + .00 Pergamon Press Ltd.

The crystal dynamics of niobium is studied employing a two parameter model. The model satisfies the Cauchy relation and the symmetry require- ments of the lattice, and the lattice is in equilibrium without recourse to external forces. The agreement between theory and the experiments is fairly good.

1. INTRODUCTION

NIOBIUM CRYSTALLIZES in the bcc space lattice at room temperature with one atom per unit cell. Zener [1], who made a formal attempt, envisioned that, the number of outer electrons per atom being five for group-V metals; the ground state electronic configu- ration in the free atom and possibly in the metallic state would be 4d45s or 4d35s 2. These are responsible for broadening of 5s, 5p and 4d 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). But on the basis of effective chemical valence, which determines the number of electrons which may be treated as nearly free, Z has been taken as 5. This choice is consistent with the atomic spectroscopic data [2] and the systematics of the d-band width [3].

In the present communication, calculations of the phonon dispersion, frequency spectrum, temperature variation of Debye temperature and Debye-Waller (DW) factor of niobium are reported employing a two parameter model. The model satisfies the Cauchy relation and the symmetry requirements of the lattice, and the lattice is in equilibrium without recourse to external forces. The theoretical results are compared with the available experimental data.

2. THEORY

Following Gupta and Kharoo [4], the contribution to the dynamical matrix from the three-body force con- stants for a bcc crystal, may be written as

Dei(g) = 4 KeaS~C~C~ , (la)

e Dii(g ) = 4 KeaSiSi CiCjC~ . (1 b)

Here C i = cos (aqi), S~ = sin (aq~), ql(i = x, y, z) are the cartesian components of the phonon wave-vector q, a the semi-lattice constant and K e the bulk modulus of the electron gas. It may be noted that the electronic con- tribution to the dynamical matrix is invariant with

respect to a reciprocal lattice vector, which is not the case with other electron-gas models [5].

Retaining the two-body central interaction between the ions in the general form ~(r) and the dynamical matrix corresponding to this central interaction can be expressed in terms of the first (~') and second (~b") derivatives of this interaction potential. The contri- bution to the dynamical matrix due to the central inter- action upto nearest neighbours can be written in a straightforward way as

D~i(q ) = (8/3)(~b" + 2 x 3t/2a-~b')(1 - -GCiCk) , (2a)

C 1 Dij(q ) = (8/3)(~b"-- 3-r a -1 ¢')SiSjC~. (2b)

The phonon frequencies can be obtained by solving the secular equation

det ID--mw2II = 0, (3)

where D is the total dynamical matrix, i.e., sum of the dynamical matrix due to the two-body and effective three-body interactions (equations (1) and (2)), m is the mass of the ion, ¢o the phonon frequency and I is the 3 x 3 unit matrix. In the long-wavelength limit (i.e., for q~0) , the force constants involved in the dynamical matrix of a bcc lattice can be related to the measured elastic constants:

Clt = (1/3)a -I ¢" + (2/3)3 -~-a-2 ¢' + Ke, (4a)

C12 = (1/3)a-l dP"--(4/3)3-~a-2 ¢' + Ke, (4b)

C44 = (1/3)a -1 ¢ " + (2/3)3 -~a-2 ~'. (4c)

For equilibrium, the first derivative of the potential energy U(= U c + U e) of the crystal must vanish, i.e.,

(dUe/dUt)a __pc = O, (5)

where __pc = dUe/d fl, U e is the potential energy of the crystal corresponding to a central interaction, ~2 the atomic volume and U e the electronic energy. The funda- mental work of Kagan and co.workers [6] provides an exact relation between compressibility and susceptibility

163

164 CRYSTAL DYNAMICS OF NIOBIUM

of the electron gas, namely (equation (2.16) of Brovman et al. [6] )

K~ = n '2 /H(0) , (6)

where n' is the charge density and II(0) is the value of the susceptibility function in the long-wavelength limit. Because the total amount of charge remains constant, the bulk modulus of the electron gas can be written as

K e = -- I2(dpe/dI2) = n ' (dpe /dn ' ) . (7)

Equations (6) and (7) yield

dpe /dn ' = n'/rI(0). (8)

Integrating this we obtain (assuming II(O) independent of electron density)

P~ = n '= /2 I I (0 ) . (9)

Comparison of equations (6) and (9) gives

pe = ½K e (10)

The equilibrium condition, equation (5), with the aid of equation (10) takes the form

K~ = 2 x 31na-2~b'. (1 1)

The condition (11) and the two equations for the elastic constants (4a) and (4b) form the set of equations necessary to determine the unknown force constants (~", a -1 ¢' and aKe).

The contribution to the Cauchy deficiency (CD) in the present scheme can be written as

C1~ --C44 = Ke -- 2 x 3~/2a-~¢'. (12)

The equilibrium condition (11) implies that equation (12) can be written as

C12--Can = O, (13)

i.e., the CD is zero. Thus the present model satisfies the Cauchy relation.

> -

taJ

3. NUMERICAL COMPUTATION

The input parameters used in the calculations are Cxl = 24.6, Ct2 = 13.4 (both in units o f 10 l° Nm -2 [7]) and a = 0 . 1 6 5 0 4 n m . The calculated force con- stants are ¢" = 47.86161, a -1 ¢' = 16.00802 and aKe =

18.48447 (all in units of Nm -1 ).These force constants are used to calculate the frequency versus wave-vector dispersion relations along the three principal symmetry directions for niobium, determined from the solution of the secular equation (3) along these directions. The frequency spectrum, lattice specific heat and DW factor are calculated as described earlier [8].

4. RESULTS AND DISCUSSION

The theoretical results for phonon dispersion curves

p r , ~ , , p ,

~ool

i i 0.2 0.4 0.6 0.8 1.0 0'.8 0',6 0'.4 012

Vol. 54, No. 2

H ~ F P ~.'~ P Z~ N

I

i i i i 0 0.1 0.2 0.3 0.4. 0.5

REDUCED WAVE VECTOR

Fig. 1. Phonon dispersion curves of niobium along the principal symmetry directions. Theoretical curves:

- - Experimental points: o Sharp [9], • Powell et al. [ 1 0 ] .

1.3

1.2

1.1

1.o

0.9

o.s

~" [).7 ".~ ~: 0.8

o 5

0.4

0.3

0.2

0.1

0 0

i :

Ei ,~- "i ~i ",

. I : . 1 ': ( !

\%+';~'n"4 " I n

2) (THz)

Fig. 2. The frequency spectrum of niobium. Theoretical curve: - - Experimental measurements of: . . . . . . Sharp [9], - - - - Powell et al. [10].

along with the experimental values [9, 10] are displayed in Fig. 1. The overall agreement between theory and experiments is reasonably good except L modes in [100] direction for ~" = 0.55 to 0.97, Tmodes in [111] direction for ~" = 0.60 to 0.80 and T2 modes in [110] direction for ~" = 0.10 to 0.35.

Vol. 54, No. 2 CRYSTAL DYNAMICS OF NIOBIUM 165

300

29(3

280

270 %

260.

25O

240

0 O 0 0 0 0 -

... oO/I - -

?,/o 6)

5'0 I;o 3oo T(K)

Fig. 3. Debye temperature of niobium as a function of temperature. Theoretical curve: Experimental measurements of (a) neutron data: . . . . . . Sharp [9],

Powell et al. [10], (b) calorimetric data: o Clusiu et al. [11], • Morin and Malta [12], (c)elastic data: Z~Hubbell and Brotzen [13].

300 20001 ~ . oooI--.. \

,.,; ' ~'. ~,'-~_, T(K) o ,oo

-1000 ' ~ \.,.....

-2000 ~ "x x " -3000 ~ " " x \

-4000 \ ""

Fig. 4. Debye-Waller factor of niobium as a function of temperature. Theoretical curve: Experimental measurements of (a) neutron data: . . . . Powell et al. [10], (b) X-ray data: • Eihenhaver et al. [17], e Linkoaho [18] . . . . . . . Korsunskii et al. [19].

For the first time, theoretical results for frequency spectrum, temperature variation of Debye temperature O c and DW factor are presented. Our theoretical values for frequency spectrum compare well with the experi- ments [9, 10] in Fig. 2. The theoretical results for lattice specific heat expressed in terms of O c are depicted in Fig.' 3 along with the neutron [9, 10], calorimetric [11, 12] and elastic [13] measurements. The calorimetric data [11, 12] are corrected for Cp --Co and electronic specific heat in the usual manner [8]. For this purpose the coefficient of electronic specific heat has been taken as 7.79 mJ mo1-1 K -2 from Kittel [14]. The elastic Debye temperatures have been obtained from the measured elastic constants [13] using rapid evalu- ation method [15]. The general behaviour and the

position of minimum of these experiments [9-13] and of the present theory show closer resemblance. The variation of theoretical ®c values with T follows the general prediction of Blackman [16]. In Fig. 4, the calculated values of DW factor agree satisfactorily with the X-ray data [17-19] while the neutron measure- ments [I0] show large deviations.

Calculations of phonon frequencies of niobium have been reported by some workers [2, 20] using first prin- ciples in second-order perturbation theory. The approxi- mations [21 ] and shortcoming of zero CD [6] in the first principles approach [2, 20], preclude the possibility of its use in studying the metals. Finnis [22] has shown that the dynamical matrix theory in second order is not suitable for the calculation of the phonon spectrum of metals. If third- and higher-order terms are taken, one gets an expression for the CD even by imposing the equilibrium condition [6]. Therefore, attempts [5, 8, 21] have been made to develop elastic force models in which the nature and the magnitude of the forces between bare electrons and ions have been accounted reasonably. The crystal lattice is in equilibrium under central forces alone in these models [5] as the equili- brium condition on the model solid has not been imposed by their authors. These models [5, 8, 21] con- tained four or five parameters which were fitted to elastic constants and one or two experimental fre- quencies. All these models [5, 8, 21] suffer from the shortcoming of zero CD; therefore these models cannot be justified in the calculations of force parameters from data on elastic constants, nor should force constants so obtained be used in the dynamical matrix to calculate phonon frequencies. Furthermore, if the number of force constants becomes inordinately large, the model only possesses the characteristics of an interpolation formula, and good agreement under these circumstances represents only the internal consistency of the measurements.

From an overall study of these analyses it is evident that the present two-parameter model, though simple, satisfies the symmetry requirements, the internal force equilibrium condition of the lattice and the Cauchy relation, and provides fairly good crystal dynamics of niobium.

REFERENCES

1. ZenerC.,Phys. Rev. 81,440 (1951). 2. Animalu A.O.E., Phys. Rev. 138, 3542, 3555

(1973). 3. Anirnalu A.O.E., Phys. Rev. B10, 4964 (1974). 4. Gupta O.P. & Kharoo H.L., J. Chem. Phys. 74,

3577 (1981). 5. Pal S. & Gupta R.P., J. Phys. Soc. Jpn. 21, 2208

(1966); Bose G., Gupta H.C. & Tripathi B.B., J. Phys. F2, 426 (1972); Bahpai R.P. &

166 CRYSTAL DYNAMICS OF NIOBIUM Vol. 54, No. 2

Neelkandan K., Nuovo Cimento B7, 177 (1972); Singh V.P., Kharoo H.L., Prakash J. & Pathak L.P., Acta Phys. Aca. ScL Hung. 39, 37 (1975); Brescansin L.M., Padial N.T. & Shukla M.M., Nuovo Cimento B34, 103 (1976); Goel A., Mohan S. & Dayal B.,Physica B96,229 (1979).

6. Brovman E.G. & Kagan YU, Soviet Physics-JETP 30, 721 (1970); ibid. Dynamical Properties of Solids, Vol. 1, ed. Horton G.K. & Maradudin A.A., North-Holland: Amsterdam (1974); Brovrnan E.G., Kagan YU & Kholas A., Soy. Phys.-JETP 30, 883 (1970).

7. BolefD.I,J. Appl. Phys. 32, 100 (1961). 8. Gupta O.P., D. Phil. Thesis AUahabad University,

Allahabad (1978); Gupta O.P. & Hemkar M.P., Nuovo Cimento B45,255 (1978).

9. Sharp E.I.,J. Phys. C2, 421 (1969). 10. 'Powell B.M., Martel P. & Woods A.D.B., Can. J.

Phys. 55, 1601 (1977). 11. Clusius K., Franzosini P. & Piesbergen V.,

Z. Naturforsch. 15a, 728 (1960). 12. Morin F.J. & Maita J.P., Phys. Rev. 129, 1115

(1963)~

13. Hubbell W.C. & Brotzen F.R., J. Appl. Phys. 43, 3306 (1972).

14. Kittel C., Introduction to Solid State Physics, vth. Edition, Wiley: New York (1976) p. 167.

15. Overton JR. W.C. & Schuch A.F., Los Alamos Scientific Laboratory Report LA-3615, December (1966).

16. BlackmanM.,Rep. Progr. Phys. 8, 11 (1941). 17. Eiherthaver C.M., Pelah I., Hughes D.J. &

Palevsky H., Phys. Rev. 109, 1046 (1958). 18. Linkoaho M.V.,Phil. Mag. 23, 191 (1971). 19. Korsunskii M.I., Genkin YA. E. & Vigdorchuk L.I.,

Soy. Phys. -SolidState 13,761 (1971). 20. Sinha S.K. & Harmon B.N., Phys. Rev. Lett. 35,

1515 (1975); Allen P.B., Phys. Rev. BI6, 5139 (1977); Varma C~VI. & Weber W., Phys. Rev. B19, 6142 (1979); Singh R.S., Gupta H.C. & Tripathi B.B., J. Phys. Soc. JprL 51, 111 (1982);Ho K.M., Fu C.L. & Harmon B,N., Phys. Rev. B29, 1575 (1984).

21. Gupta O.P., Acta Phys. Pol. A64, 269 (1983); J. Phys. Soc. Jpn. 53, 2575 (1984) and references cited therein.

22. Finnis M.W.,J. Phys. F4, 1645 (1974).