Physica 122B (1983) 23f5-240 North-Holland Publishing Company

THERMAL PROPERTIES OF LEAD

O.P. GUPTA Physics Department, J. Christian College Allahabad, India

Received 22 February 1983

Calculation of the temperature dependence of equivalent Debye temperature 6’ and Griineisen parameter of lead is

performed using a lattice dynamical model. The model considers short range pairwise forces effective upto second

neighbours, and an improved electron-ion interaction on the lines of Bhatia. The ionic lattice is in equilibrium in a medium

of electrons. The theoretical results are found to be in fairly good agreement with the experimental values.

1. Introduction

Quite recently, we have investigated the lattice dynamics and the Debye-Waller factor of lead [l, 21 using a realistic lattice dynamical model. The model considers the short range pair-wise forces effective upto second neighbors and an improved electron-ion interaction on the lines of Bhatia [3]. The ionic lattice is in equilibrium in a medium of electrons. The model has been des- cribed earlier [l]. The model has been employed successfully to explain the lattice dynamics and thermal properties of many cubic metals [4-91.

The present model has been utilized to com- pute the equivalent Debye temperature and Grtineisen parameter of lead as a function of temperature.

2. Theory

2.1. Specific heat

The lattice specific heat at constant volume can be written as [5]

where x = hvlKT, K the Boltzmann constant, 0 the solid angle in the wave vector space, the Einstein specific heat function E(x) = x2eX/(eX - l)*, and other synbols have their usual meanings [5].

2.2. Griineisen parameter

The microscopic Griineisen parameter y4,j can be written as [lo, 111

Yq.i = - d log Wr,j d log P dlogPdlog V > T’ (2)

and the thermodynamic Griineisen function is then given by [lo, 111

(3)

where symbols have their usual meanings [lo, 111.

3. Numerical computation

3.1. Specific heat

The frequency distribution function has been calculated by Blackmann’s root sampling tech- nique [12] for a discrete subdivision in wave vector space so as to get a grid of 8000 equally spaced points inside the first Brillouin zone. After obtaining the frequency distribution, lat- tice specific heat at constant volume can be computed from eq. (1). The sampling technique was used at moderate temperatures down to O/10 (0: Debye temperature) and below O/10,

0378-4363/83/0000-0000/$03.00 @ 1983 North-Holland

O.P. Gupta / Thermal properties of lead 237

Cv has been calculated from the Houston 's method as elaborated by Horton and Schiff [13].

3.2. Griineisen parameter

In eq. (3) the integration over q is performed numerically and the integration over J2 is carried out by employing the modified Houston 's spherical six term integration procedure as developed by Betts et al. [14]. The six directions used for the computation are [100], [110], [111], [210], [211], and [221]. The microscopic Gr/ineisen parameters yq.j of eq. (2) are com- puted in terms of observed values of adiabatic elastic constants Cij and their pressure deriva- tives dCddP, using phonon dispersion relations computed from the solutions of secular deter- minant of the present scheme [1]. To account in detail for the behaviour of y, models must be used in which the interatomic forces are as far as possible like those actually present in specific solids, a fact pointed out by Barron [15] also. Keeping the long range nature [16] of forces of Pb in mind, we use here a four parameter force model [1] considering interatomic interactions to be effective upto second neighbors. These force constants are related to three elastic constants of the crystal and one zone boundary (ZB) frequency. Hence to compute ycj one must know the pressure derivative of both elastic constants and the ZB frequency. As the pressure derivative of ZB frequency is unknown, the four

Table I Pressure derivatives of elastic constants for lead

dCuldP dC12/dP dC44/dP Source

296 K 5.94 5.33 2.06 Ref. [19] 195 K 5.82 5.26 1.97

I 4.5753 3.541 1.86 Ref. [20] II 4.4746 3.5826 2.02 Ref. [21]

4.1577 2.4832 1.4561

parameters (A1, A 2 , B h B2) have been reduced to three (A1, B1, B2) by taking a suitable ratio of A1/Az which gives a better fit of our phonon dispersion curves with experiments. The tem- perature dependent values of elastic constants have been borrowed from Waldorf and Alers [17] in the temperature range (60-300K) and their linear extrapolated and interpolated values for small interval in the temperature range (1- 300K). The inter- and extrapolated values of lattice parameter in the temperature range (1- 300 K) have been borrowed from Pearson [18]. Other input parameters are the same as quoted earlier [1]. Pressure derivatives of elastic con- stants used in the present study are taken from Miller and Schuele [19], Suzuki [20], and Mathur and Gupta [21]. These values are listed in table I.

4. Results and discussion

In fig. 1, we display our theoretical results of

115 I I I I I I I I I

105 ~) ..I..I.. ........

85 --.~ o d . . . . . '~_'.,d'"

75 I I I I I I I I I

0 20 40 60 80 100 120 140 160 180 200

T ( K )

Fig. 1. Equivalent Debye tempera ture of lead as a function of temperature . Theoretical curves: input data 80 K , - . . . . . . . . . input data 100 K . . . . . . . . . . input data 300K. Exper imental points: A Meads et al., + NBS Tables, • Hoeven and Keesom, • Horowitz et al., O Stedman et al.

238 O.P. Gupta / Thermal properties of lead

specific heat at constant volume expressed in terms of equivalent Debye temperature , ~9 c as a function of temperature . The theoretical results are expressed by solid, dashed, and dotted curves which have been obtained by fitting input data [1] appropr ia te at 80, 100, and 300K, respec- tively. The experimental measurements [22-26] are also plotted in fig. 1. The nature and the position of the minimum of these curves are similar. The solid curve shows reasonably good agreement with the experiments. Though the position of the minimum shown by our theoreti- cal curves matches with that of the experimental points yet the experimental values are a little higher than ours.

The computed values of microscopic Griineisen parameters yq, j for lead are shown in fig. 2 as a function of reduced wave vectors along the principal symmetry directions [100], [110], and [111]. The microscopic Griineisen parameters provide a picture for the relative contribution of longitudinal and transverse modes of lattice vibrations to the effective value of Griineisen parameter . For comparison, we have also displayed the values borrowed from avail- able experimental measurements [19, 27-29]. It is evident from the fig. 2 that the agreement be- tween the theoretical yq,/ curves and the experimental points is reasonably good in spite of the fact that observed values of Brockhouse et

P J

0 .2

X K ~: P ~ . L . - i I i i i , • , • , , _

1

, i [1107, , I , 0 1 1 ; ~ I • 4 .6 .8 " .8 .6 .t, .2 0 .1 .2 .3.4 -5

REDUCED WAVE VECTOR

Fig. 2. Microscopic Griineisen parameter for lead along the principal symmetry directions [100], [110], and [111] at room temperature. L and T stand for longitudinal and transverse modes. Theoretical curves: - - Miller and Schuele, - ....... Suzuki I, -.-.-.-. Suzuki II, -..-..- Mathur and Gupta. Experimental points: V!I' Brockhouse et al., F-10I Miller and Schuele, A • Lechner and Quittner, • McWhan et al.

al. [27], and Lechner and Quit tner [28] show large scatter at certain points. The uncertainty in the results of Brockhouse et al. [27] in [111] direction is probably of the same order of mag- nitude as in [100] direction [28]. The yq,j curves derived from the experimental pressure deriva- tives of elastic constants of Miller and Schuele [19] show comparat ively a good agreement with the experiments as compared to those derived f rom the d C J d P values of Suzuki [20], and Mathur and Gupta [21]. Further, in the micro- scopic theory of the equation of state of solids, y is a weighted average of yq.j [15]. The weighting factors are the contributions (Co)q.j of all the modes to Co. In the limit of classically high temperatures , where (Cv)q,j is equal to the Boltzmann constant K for all the modes, y is the simple average of the yq, j. In the case of Pb this should be true already at room tempera ture because of its low Debye tempera ture (O = 105 K) [30]. It can be seen in fig. 2 that at the Brillouin zone boundaries in [100], [110], and [111] directions, the yq, j values are larger for transverse than for the corresponding longitu- dinal modes.

For the first time, the results of calculation of t empera ture variation of the Griineisen paramete r of lead are presented in fig. 3 in the tempera ture range (1-300K). For comparison, we have also displayed the values derived f rom experimental measurements of thermal expan- sion [19,31-36] and specific heat data. The sources of experimental thermal expansion and the specific heat data used in the evaluation of Griineisen parameters are listed in table II. It is evident f rom fig. 3 that the general shape of the theoretical and experimental y - T curves is similar in spite of the fact that the observed values show large scatter at certain points parti- cularly at lower temperatures . In fig. 3, the theory is in fair agreement with the observations at the intermediate temperatures down to ~9/5. It corroborates Barron 's [15] prediction that y remains constant to a few percent down to about 0/3, and below this tempera ture the uncertainty in 3/is relatively large; however, y does decrease with falling tempera ture below about 0/5. It is worth noting (fig. 3) that the y - T curve derived

O.P. Gupta I Thermal properties of lead 239

3 " 2 | ~ = 1 / 5 K I I I I I I l l m u I I I I I II II I I

2.9 ~ . . . . . . . . . . . . . . . .

~.~ • "......~ .,~ 2.61--- .. ~ I " , , , , ~ - r t : : l -~UO3 .

/

2.0

, 7 ,oo , t , , 1 5 10 50 100 200 300

T(K) -

Fig. 3. Griineisen parameter of lead as a function of temperature in the temperature range (1-300 K). Theoretical curves: - - Miller and Schuele, - . . . . . . . Suzuki I, - .- .- .- . Suzuki II, - . . - . . - Mathur and Gupta. Experimental points: + Miller and Schuele, • White, O Channing and Weintrouh, [] Rubin et al., A Nix and MacNair, • Johnston, • Dheer and Surange.

from the experimental pressure derivatives of elastic constants of Miller and Schuele [19] show comparatively good agreement with the experi- ments as compared to those derived from the theoretical dCJdP values of Suzuki [20], and Mathur and Gupta [21]. It may be emphasized, therefore, that a theoretical investigation of Griineisen parameter is possible when we con- sider a correct temperature dependence of dCu/dP values. Barron [15] predicts for a fcc lattice value of 0,3 for the difference between the high and tow temperature limits for nearest neighbor central interaction and a decrease of this value when higher neighbors also interact. Following this statement, one concludes that more distant neighbors should be taken into account in order to improve the results.

Table II The experimental thermal expansion and specific heat data for lead

Thermal expansion data Specific heat data

Temperature limit Source Temperature limit Source (K) (K)

3-20 [191 80-300 [311 15--130 [221 80-300 [32] 4.2-10 [33] 3-11 [251 3-11 [35]

80-300 [34] 15-300 [22] 15-270 [36]

However, it emerges from the present study that the lattice dynamical model [1] provides a reasonable description of the temperature varia- tion of equivalent Debye temperature and Grfineisen parameter of lead.

Acknowledgements

We wish to express our sincere gratitude to Professor S.M. Shapiro for mailing a preprint. Thanks are also due to Dr. M.P. Hemkar for useful discussion about figure one.

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