Low temperature limit of Grüneisen's gamma of copper from third order elastic constants data

Volume 47A, number 5

PHYSICS LETTERS

22 April 1974

LOW TEMPERATURE LIMIT OF GRt)NEISEN'S GAMMA OF COPPER FROM THIRD ORDER ELASTIC CONSTANTS DATA R.R. RAO ~

Institute of Physicsof CondensedMatter, Universityof Geneva,Switzerland Received 4 February 1974 A simple procedure to calculate the low temperature limit of the Gr/Jneisen gamma in cubic crystals from third order elastic constants data is presented and is applied to the case of copper. In this paper a simple method is presented to calculate the individual mode gammas 7i for the elastic waves in copper from its third order elastic (TOE) constants data and hence the low temperature limit ~L is evaluated. The microscopic mode gamma is also designated as Grfineisen parameter (GP) and is defined by the equation - d l n coi 3'i - dln V

(1)

where w i is the frequency of the ith normal mode and V is the volume. The Griineisen gamma (3') is an average over the individual 3'i, the weighting factors being the amount of excitation of the mode. So the Grfneisen gamma is temperature dependent through the temperature variation of the weighting factors, the individual 3'i being considered independent of temperature. It follows that 3' should remain constant after all the modes are completely excited and this is called the high temperature limit 7H" Also the Griineisen gamma should be constant in the T3-region of temperature where one can apply the Debye theory. At very low temperatures, the number of normal modes excited in the ]th acoustic branch is proportional to v~ 3 (0, ¢), v] being the wave velocity. The low temperature limit of the Griineisen gamma is designated as 7L and is given by

N3= 1 f v~ 3 (0, ¢)d ~2

POW2tjj = ~-j AS NpN m uk kpm ]k, pm

(3)

Po is the density of the unstrained material; uj the components of the displacements from the strained state; and w, the natural velocity for wave propagation perpendicular to a plane whose natural normal is N. To the first order in the strain ASkpm is given by

ASk, pm =r~s (@],mk,rs + Cpm,rs~jk) ers (4)

+ ~q Gq,mk el"q + ~q Cpf,mq erq el~ are the deformation parameters; Cp/mk and C_I mk rs are the second order elastic (SOE) and t~ird order elastic (TOE) constants respectively. For a cubic crystal, subjected to a pressure P, there will be uniform volume strain e (negative) and we have exx = eyy = ezz = (--e/3), el/= 0 when i :/:/'. The secular equation determining po w2 = X is

X3-AX 2+BX-C=O

(5)

where A = Dxx + Dzz

B= DxxDxy + DyyDyz + DzzOxz

~3= 1 f 3"i(0' ¢) vT3 (0, ¢)dg2 ~L =

Thurston and Brugger [1] give the following equation for elastic wave propagation in a homogeneously deformed crystal:

(2)

Dxy Dyy

Dy z Dzz

DxxDxyDzI On leave from the Department of Physics, Indian Institute of Technology, Madras - 600036, India.

Dxz Dxx (6)

C= Dxy Dyy Dy z . Dxy Dyz Dzz

407

Volume 47A, n u m b e r 5

PHYSICS LETTERS Table 1

Calculated values of X i and Yi for the case 0 = 0 and vations 0-values for copper ( X i in arbitrary units)

0

XI

X2

X3

"gl

3'2

3"3

5°

16.81

7.38

7.56

2.18

2.14

2.22

15 °

18.35

5.82

7.56

2.23

2.05

2.22

25 °

20.12

4.05

7.56

2.27

1.78

2.22

35 °

21.40

2.77

7.56

2.29

1.38

2.22

45 °

21.86

2.31

7.56

2.29

1.14

2.22

55 °

21.40

2.77

7.56

2.29

1.38

2.22

65 °

20.12

4.05

7.56

2.27

1.78

2.22

75 °

18.35

5.82

7.56

2.23

2.05

2.22

85 °

16.81

7.38

7.56

2.18

2.14

2.22

~ A jk,pm S Np N n

HereDlk=

(7)

pm

is a function of e if the SOE and TOE constants of the material are known. The generalized GPs for elastic modes are given by

7 i = - ( 1 / 2 X i ) dXi/de 1 X2(dA/de)o - Xi (dB/de)o + (dC/de)o -

-

.

2<

.

.

.

.

.

3x 2

.

.

.

.

(8)

2 AoX , +

Here X i (i = 1, 2, 3) are the values of po w2 in the unstrained state for a given direction of propagation and the subscript 0 refers to the values of the constants A, B, C and their derivatives in the unstrained state. The SOE and TOE constants of copper measured by Hiki and Granato [2] at the room temperature are used in the present calculations, Salama and Alers [3] also measured the TOE constants of copper at room temperature which agree well with those of Hiki and Granato [2]. The direction of wave propagation is given in polar co-ordinates (0, 0); 0 is the angle which the

408

22 April 1974

wave-vector makes with the z-axis and 0 is the azimuthal angle. Table 1 gives the values of X i and 7i for waves propagating at different angles 0 to the Z-axis in the XZ plane. Tile low temperature limit 7 L is then calculated from the formula $3= I f 7i X~ 3/2 dg2 ")'L

x23__ 1 f X 7 3 / 2

de

and has tile value ~k = 1.73. This compares well with the experimental value 1.72 +- 0.03 obtained from the thermal expansion data of copper by Carr et al. [41. In the Debye approximation the high temperature limit is the mean value of ~'i over the elastic waves propagating in all directions and the present calculations give a value of 2.07 while the experimental 7H is 2.0. The agreement is surprisingly good. Thus the above method yields a precise value for YL and the computational labour is very much reduced unlike in the calculations of Sheard [5] or Collins [6] who used the continuum model combined with the experimental values of the pressure dependence of elastic constants. Sheard's value for ~L o f c o p p e r is 0,76 which is extremely low and Collin's value is 1.77 which is obtained by an approximate integration of the 7i's taking into account summation over six symmetry directions. The author wishes to thank Professor M. Peter for his kind encouragement during the work.

References ! l ] R.N. T h u r s t o n and K. Brugger, Phys. Rev. 133A (1964) 1604. [2] Y. Hiki and A.V. Granato, Phys. Rev. 144A (1966) 411. [3] K. Salama and G.A. Alers, Phys. Rev. 161A (1967) 673. [4] R.H. Carl, R.D. M c C a m m o n and G.K. White, Proc. Royal Soc. (London), A280 (1964) 72. [5] F.W. Sheard, Phil. Mag. 3 (1958) 1381. [6] ].G. Collins, Phil. Mag. 8 (1963) 323.