Chapter 3. The complex behaviour of metals under severe thermal and stress

Transcription

1 Chapter 3 SECOND ORDER ELASTIC CONSTANTS, THIRD ORDER ELASTIC CONSTANTS, PRESSURE DERIVATIVES OF SECOND ORDER ELASTIC CONSTANTS AND THE LOW TEMPERATURE THERMAL EXPANSION OF D-TIN 3.1 Introduction The complex behaviour of metals under severe thermal and stress environments is influenced markedly by their elastic properties. Indeed the response of a material tc~ an applied stress is always determined by the elastic constants and their pressure derivatives. Furthermore, both the mechanical and thermodynamic equation;$ of state of a metal that contains dislocations require detailed knowledge of the single crystal elastic constants for calculation of elastic fiee energy density [l-41. Thermal expansion and elastic properties of tin were investigated earlier by several authors [5-81, but they present either partial results or results obtained by the applications of 'not very precise' methods of measurements. Moreover, majority of them were performed with polycrystalline samples. We have made here an attempt to calculate the complete set of second order elastic constants and their pressure derivatives, third order elastic constants and the low temperature thermal expansion of tetragonal P-tin

2 3.2 Second-order elastic Constants Second-order elastic constants and their pressure derivatives provide insight into the nature of binding forces between atoms since they are represented by the derivatives of the internal energy of the crystal. Interactions upto fifth neighbours are considered here. The volume of the unit cell is ($1), where a is the lattice constant in the basal plane and p is the axial ratio. The position co-ordinates of the two non-equivalent atoms in the basis are The position co-ordinates of the various neighbours of the (p) atom in the basis cell are glven in tables 3.1 to 3.5.

3 Table 3.1 Position co-ordinates of first neighbours of the (3 atom

4 Table 3.2 Position co-ordinates of second neighbours of the (3 atom Table 3.3 Position co-ordinates of third neighbours of the

5 Table 3.4 Position co-ordinates of fourth neighbours of the 0 atom

6 Table 3.5 Position co-ordinates of fifth neighbours of the are given by The components of the interatomic vectors in a homogeneous deformation Here E,,, the deibrmation parameter is related to the macroscopic Lagranyian strams rl,, by

7 ~! [,., E + E.. +CEkiEk 1 " 2 " JI k and Wi are the componer~ts of the internal displacements and are replaced by the - relative internal dtsplacements W by the relation The potential energy is expanded in powers of the changes in the squares of the vector distance R (K) as where k2 and k3 are the second and brd order potential parameters respectively given by configuration k, = IS zero as the derivati~d are taken in the equilibrium (7J(r ),=.

8 - The Internal d~splacements W, are obtained in terms of the Lagrangian -- - strains by min~m~sing the strain energy with respect to W. in terms of the Lagrangian strains parameters to the first order are obtained as Substituting in the expression for strain energy per unit volume of the undefomed state glver~ in equation (1.17) of Chapter I and comparing the resulting expression with the lattice energy density in equation (1.33) of chapter I, we get the expression for second order elastic constants of P-tin as 11 kywhere C - I>

9 Here a is the lattice parameter and p is the axial ratio of the crystal p-tin. k2 is the second-order parameter characterising two body interactions has been obtained by substituting the value of CII = (73.5 GPa) measured by Mason and Bomlnel [9] in equation (3.6a) and the k2 thus obtained is given in Table 3.7. This value of kz has been used in equations (3.6) to obtain the second-order elastic constants of P-tin The values of second-order elastic constants for P-tin thus obtaned are ylven m table 3.6 along with other measured values. Table 3.6 Second-order elastic constants of b-tin (in G Pa) along with the measured values cij - Present work - Experimental Values - Ref [9] Ref [lo] c C c c

10 3.3 Third-order elastic C:onstants The non-lmear elastic properties of a solid can be expressed in terms of the third-order elastic constants which quantify the coefficients of the cubic term in the expansion of strain energy density and is the leading term in the vibrantional anharmonicity of the long wavelength acoustic phonons. We derive here the expression for the third-order elastic constants of the P-tin using the method of homogenous deformation [11,12]. There are twelve non- vanishmg elastic constants in thrd-orderthe case [I31 whose structure is A tetragonal 4/mmm 'The expression for the third-order elastic constants have been obtained by comparing the strain energy density derived from equation (1.17) of Chapter I with the lattice energy density derived from continuum model approximation given m equation (1.33) of Chapter I. The expressions thus obtained for thud-order elastic constants of P-tin are

11 (1' where B = k3-- and C == kz,, P' P LI kz i.r the second order potential parameter whose value is given in table 3.7 k3 the thud-order potential parameter has been obtained by substituting the value of CI1l= -410 GPa measured by Swartz et al [8] in equation (37a). The value of k3 thus calculated is given table 3.7. These values of kz and k3 have been used in equations (3.7) to obtain all the hrd-order elastic constants of P-tin. The values of third-order elastic constants thus

12 obtained for P-tin are collected in table 3.8 along with the experimental values of Swartz et a1 [8]. Table 3.7 Values of Potential Parameters k2 and kj of P-tin Table 3.8 Third-order elastic constants of b-tin (in -GPa) along with the experimental r-- values Cilk Present work Experimental Values Ref [8]

13 3.4 Pressure derivatives of the Second-order elastic constants We lnvestlgate here the effect of pressure on the second-order elastic constants of 13-tin using finite strain elasticity theory [14]. The values of second- order and third order elasttc constants given in table 3.6 and 3.8 respectively are substituted in equat~ons (1 48) of Chapter 1 to get the pressure derivatives of the second-order elastic constant of P-tin The values of pressure derivatives thus obtained are collected in table 3.9alony with other reported values. Table 3.9 Pressure derivatives of the Second- order elastic constants of P-tin dc.. u - dp dc,, dp, 5 dp 5 dp dc',, dp, dc,, dp, - dc', dp,- Present work ;!.30,- Experimental values Ref [8] Reported values Ref

14 3.5 Low temperature thermal expansion of p-tin the acous1:ic wave velocities and Gruneisen function depend on the direction of propagaticln of the elastic waves. The second and third order elastic constants given in tables 3.6 and 3.8respectively are used to obtain the Gruneisen functions y 'i (6) and y " J (6) for different acoustic modes at intervals of 10" ranging froin 0" to 90' using equations (2.23) of Chapter 11. The results of wave velocities obtained from equation (2.21) of Chapter 11 and the GPs for corresponding elastic wave velocities at different angles 0 are summarised in Table 3.10

15 Table 3.10 Generalised GPs for elastic waves propagating at different angles Oto the crystal axis in the p-tin Crystal The variation of the generalised Gruneisen function y ', (0) an d y "., (0) for different angles 8 to the c-axis of the crystal P-tin are shown in figures 3.1 and 3.2 respectively

16 Figure 3.1 Variation of the generalised Gruneisen parameters y/ as a function of angle " - 8 to the c-axis of p-tin 8 in degrees

17 Figure 3.2 variation of the generalised Gruneisen mete" Y angle 0 to the c-axis of p-tin : as a function Of 0 in degrees

18 The low temperature limits y~(0) and ~ ~(0) are obtained from equations (2.25) and (2.26 )of Chapter I1 by numerical integration procedure using the data given in table 3.10 Since the solid angle of the cone of semi-vertical angle is proportional to Sin 8, the values of yj' x-~': and y, ' x.'', have been multiplied by Sin 8 and have been summed over all values of e. Thus the low temperature limits y~(0) and ~ ~(0) have been evaluated using equation (2.25) and (2.26 ) for P-tin and are given in Table The elastic compliances S11, Slz, SI~ and S33 and isothermal compreissibility x,, are evaluated using equations (2.30) of Chapter 11 These values of St, and X is, along with the values of y~(0) and ~ ~(0) given in Table 3.11 are substituted in equation (2.29) of Chapter I1 to calculate the Brugger gammas ylbr(0) and yl IB'(~). The values thus obtained for ylbr(3) and y 1 "(0) are presented in Table The low temperature limit of the volume lattice thermal expansion y~ for P-tin is obtained using equation (2.31) of Chapter I1 by substituting the values of yb(0) and y ibr(0) fiom Table The value of y~ thus obtained is given in Table 3.1 1

19 Table 3.11 The values of Cruneisen functions y~(0) and yl(0), y~u'(~) and 71 lbr(0) and YL for P-tin 3.6 Results and Discussion Since P-tin possesses tetragonal symmetry it exhlbits six second-order elastic constants. The values of second order elastic constants of p-tin obtained in the present work are collected in table 3.6 along with the experimental values Q of Mason and Bomrnel [9] and Bridgman [lo]. The elastic: constant C33 which fi corresponds to the wave propagation along the c-axis of the crystal is in reasonable agreement with the experimental values. Measurements of elastic constants except that along crystallographic axes may lead to imprecise results due to the uncertanty in the orientation of single crystals. This is, in part, a reason for the deviation of the present second-order elastic constants from the corresponding experimental values. Even the experimental values are found to differ among themselves.

20 The third order elastic constants obtained in the present work are given in table 3.8 along wlth the experimental results of Swartz et al [S]. All the third order elastic constants are: negative and are generally one order of magnitude higher than the second order elastic constants. Swartz et al [8] obtained thxd order elastic constants of a-tin from measurements of the hydrostatic and uniaxial stress dependence of sound velocity on single crystals. The third order elastic constant C333 IS higher in magnitude than all other values. This shows a greater anharmonicity along W-axid the crystal. The pressure derivatives of the second-order elastic constants of a-tin obtained in the present work are shown in table 3.9. The results obtained in the present work are of the same order as that of the other reported values. The low temperature limit of the Gruneisen function obtained in the present work for p-tin is given in table An important feature of the Gruneisen functions of P-tin is that all the mode values are positive. Figure 3.1 shows the vanation of Gruneisen function y,' for three acoustic branches of the elastic waves as a function of angle 0, which the direction of propagation makes with the c-axis of the crystal. The transverse acoustic mode of GP y : assumes a minimum value 1.05 along the c-axis of P- tin, while it acquues a maximum at 0=60. The transverse acoustic mode yi assumes a minimum along c-axis and a maximum value1.63 at 0= 60' to the c- axis of the clystal. The longitudinal acoustic mode y : assumes minimum value

21 of 0 at angle 8 = 40" whle it extubits a maximum of 0.83 both along c-axis and perpendicular to c-axis. Figure 3.2 shows the variation of,y: ' with angle 8 in P- tin. The transverse acoustic rnode y " has a maximum of 7.81 at asigle 0 = 60" l and a minimum of 1 41 perpendicular to c-axis of P-tin. The transverse acoustic mode y: ' assumes a maximum value of 1.28 along c-axis of the crystal and steadily decreased upto 0.36 perpendicular to the direction of c-axis. The longitudinal acoustic mode of'gruneisen b tion yi' exhibits a maximum value along a direction perpendicular to c-axis of the crystal. Y ''assumes a minimum 3 value of 0.53 at angle 0 = 60". The low temperature limit of the volume lattice thermal expansion y, = 0.83 indicates that the thermal expansion of P-tin is positive down to absolute zero.

22 References [1] G. Kern, G. Kresse and J. Hafner, Phys. Rev. B. 59,8551 (1999). [2] E.L. Peltzer-y- Blanca. A. Svane, N.E. Christensen, C.O. Rodriguez, O.M. Cappanrim and M.S. hloreno, Phys. Rev. B 48, (1993). [3] N.E. Chnstensen and M. Methfossel, Phys. Rev. B, 48,5797 (1993). [4] H.Olijnyk, Phys Rev. H.,46,6589 (1992). [5] M.Wolcyrz, R. Kubiak and S. Maccejwslu, Phys. Stat. Sol.(b) 107, 245 (1981). [6j G.K. Whlte, Thermal Expansion 8, edited by Thomas A. Hahn, Plenum, [7] T.H.K. Bamon, J.G. Collins and G.K. White, Adv. Phys., 29,609 (1980). [8j K.D. Swartz, W.B. Chua, and C.Eibaum, Phys. Rev.B 6,426 (1972). [9] W.P. Mason and H.E. Bornmel J. Amt. Soc. Am. 28, 930 (1956) [lo] P.W. Bridgeman Proc Nat. Acad. Soc. US. 10,411 (1924) [ll] M. Born and K. Huang "Dynamcal Theory of Crystal Lattices", Oxford, London, [12] R. Srinivasan, Phys. Rev. 144, 620 (1966). [I31 S. Bhagavantam, 'Crystal Symmetry and Physical Properties' Academic Press, New York, [14] R. Ramji Rao and A. Padrnaja, J. Appl. Phys. 62,440 (1987).