Elastic Constants and Coefficients of Thermal Expansion of Laves Phase Cr 2 X (X=Hf, Nb, Ta, Zr) Alloys
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I
Pl
( fv ) j=
-.
I
_ _)V
0
where p is the pressure and B is the bulk modulus. Nonzero value of CTE results from the anharmonicity of the crystal, i.e., the volume dependence of phonon frequencies. We use a Debye model to KK8.29.1 Mat. Res. Soc. Symp. Proc. Vol. 552 0 1999 Materials Research Society
describe the vibrating lattice. Within the Debye model, (ap/aT)v is equal to Ycv, where 'Yis the Gruineisen parameter given by
(lno
/=_
D)
(2)
a(In V)
and c, is the constant-volume specific heat. In Eq.(2), o%is the Debye frequency. Therefore, oxcan be written in a simple form: o=
(3)
C. 3B(T)
In Eq.(3), the bulk modulus B(T) consists of B0 and 8B. B0 is the bulk modulus at T=0 and 8B includes the anharmonic contribution due to volume expansion and phonon contribution at finite temperature. Using the relation kBED= riO%, y becomes a(InOo) Y=
OV aeO VQ(n)
(4)
where the Debye temperature OD is given by OD=(1b/kB)(67t 2N/V) 11 3Vm. Here, vm denotes the mean sound velocity of the propagation velocities v 1, v2, v3 (one longitudinal and two transverse) of elastic waves and vi's are the solutions of the cubic equation (Christoffel equation) in v2 [7]: Ak
(5)
1=P 0,
where Fik's are the Christoffel stiffnesses and p is the mass density. The six Christoffel stiffnesses are given in terms of elastic constants. RESULTS AND DISCUSSIONS Elastic constants and CTE
We performed total-energy calculations using the full-potential linearized augmented planewave (FLAPW) method within the local-density approximation (LDA). The FLAPW method solves the local-density-functional equations without any shape approximation to the potential or charge density. The atomic positions are relaxed by calculating Hellmann-Feynman forces acting on the atoms. First, we obtained the theoretical equilibrium lattice constants. For bcc Cr, a lattice constant of 2.79 A was obtained which is in fair agreement with the experimental value of 2.88 A. The theoretical lattice constants of Laves phases Cr2X are 6.947 A (X=Hf), 6.822 A (X=Nb), 6.809 A (X=Ta), and 6.990 A (X=Zr), which are in good agreement (within 2-3%) with the experimental lattice constants 7.157 [8], 6.991 [2], 6.985 [4], 7.208 [8] A, respectively. The smaller theoretical lattice constants are expected since LDA underestimates the lattice constant. Table I. Calculated elastic constants (in units of GPa), bulk modulus B0 (GPa). Debye temperature OD (K), y at experimental volumes for Cr and Cr 2X (X = Hf, Nb, Ta, Zr).
I]C Cr(antiferro) Cr(para) CrHf CrNb CrTa CrZr
373 393 225 250 281 201
C12
CB
0
65 122 126 170 173 126
81 68 58 58 73 46
168 212 159 197 209 151
KK8.29.2
o)
y
545 509 311t 354 334 332
1.10 2.02 1.39 1.44 1.70 0.91
To determine the Debye temperatures, we evaluated the elastic constants. Similarly in Ref. [9], three independent elastic constants C 11, C12, and C4 in the cubic structure were determined from three relations (i) U=(Cll-C 12 ) e2 for e1=-e 2 -e, (ii) U=(C11 +C12)e2 for e1=e 2-e, and (iii) U=(1/2)C44e 2 for e6=e, where ei's are the strain
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