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Thermal contributions to the elastic constants of sodium
522
TECHNICAL
8. BRBBRICX R. F., Bull. Am. Phys. Sot. 11,222 (1966); STRAUSS A. J. and BRIZUUCK R. F., Bull. Am. Phys. Sac. 11, 222 (1966); &IKBIUCK R. F., J. Phys. Chem. Solids 27,149s (1966). 9. KLMENSP. G., Phys. Reu. 119,507 (1960). 10. DAMOND. H., J. AppI. Phys. 37,318l (1966).
J. Phys. Chem. So&Es Vol. 28, pp. 522-523.
Thermal
NOTES
subscripts, b and c are lattice parameters associated with the other two principal axes, and 711 is an average of the quantities -[(u/v,)(d~~/&)] where the vf are the normal mode frequencies. Although pI1 and its derivatives are diflkult to calculate, one can make a crude estimate of the contribution of equation (I) if one assumes that under hydrostatic conditions
contributions to the, elastic COnaallb of sodium (Received 18 July 1966)
that a large discrepancy exists between the theoretical predictions of the logarithmic derivative, A, of the quantity c’ = (C,,--&J/2 for sodium, and the value determined experimentally by DANIELS.(~) The theoretical prediction, however, was based on a calculation at O’K, and DANIELS’~)data are taken at room temperature. Since A = A,+& where A, is the thermal contribution to A it seems possible that the thermal contribution to C.” may account for the discrepancy. We shall consider the logarithmic derivative
which implies 3xdy N--
IT HASbeen observed by BROOKSQ)
A =-
dlnC’ d In x
X = 1.043
where x = v/v,, v is the specific volume, vo the specific volume at zero temperature and pressure, and l-043 is the value of x at room temperature and pressure. DANXEL~(~’had obtained the value A = -243, whereas BROOKS(~)has shown that the theoretically predicted A, can be, at most, A o = -1.97 if temperature effects are excluded. It can be shown that for temperature, T, greater than the Debye theta, the theoretical therma contribution Cr.’ to C’ for a b.c.c. monatomic solid is given by
x
[u(fg)b,c-b(Jg),,I (1)
where 6, is the density at O”K and zero pressure, n is the number of atoms per gram, k is the Boltzmann constant, a is a lattice parameter associated with a principal axis and the 11
Y dx where y is the normal Grtineisen parameter, Since, under hydrostatic conditions, we must have equation (1) becomes, with this 711= Y, assumption
G’ N-
- 96,pkTy
(2)
2
where the prime denotes the derivative of y w.r.t.b It follows from this that the thermal contribution, A,, to the logarithmic derivative of C” is given by
A
3fy” r = $[l
(3)
+(cs’fc,‘)l
where C,’ is the nonthermal con~ibu~on to c’, The quantities y” and y’ can be estimated by use of the two mode approximation’3’ for y, i.e.
6
2
dlnx
6
2
dlnx
1 I
(4)
where C4,0 is computed according to the theory set forth by Fucr-rs(*) and C,,O is computed essentially by means of the equation of state set forth by BROOI&~) (corrected for ion core interactions), with its parameters adjusted so as to reproduce the high pressure low temperature experimental data of BEECROFTand SWENSON@)on sodium. By this procedure one can calculate at x = l-043, y’ N 0.23, y” N 0‘85, y’;“y’ 1: 3.7. The ratio Conic=’ can be
TECHNICAL
from the data of DIEDERICH and TRIVIfrom which one finds Cc’&’ 2! -5.14. Substitution of these numbers into equation (3) yields AT 1: -0.9, which is of the right order of magnitude to account for the aforementioned discrepancy. It seems worthwhile to repeat that this calculation is crude. It is, at best, an order of ~nitude computation which tends to show the possible importance of thermal contributions to A.
estimated SONNO
U.S. Naval Ordnance Laboratory h%xr Spring, Wwyland
D. JOHN PASTINE 20910
ReJerences 1. 2. 3. 4.
2: 7.
B~ooxs H.,
Trans. Met&.
Sot.
AIM&‘,
22f,
546-558 (1963). DANIEU W. B., Pbys. Reu. 119,1246-1252 (1960). PASTINBD. J., Phys. Rev. 138, A767-A770 (1964). FUCHS K., Proc. R. Sot. A153, 622-629 (1936); A157,444-450 (1936). BROOKSH., Supplt. Nuovo Cim. 7, 207-244 (1958). BEECROFT R. I. and SWENSONC. A., J. Phys. Chem. l&329-344 (1960). DIEDERICHMARY E. and TRIV~SONNOJ., J. Phys.
Chem. So&& 27,637-642
J. Phyr. Chem. Solids
(1965).
Vol. 28, pp. 523-525.
Atomic parameters for the interpretation of ESR and diamagnetic susceptibility data (Received 24 August 1966)
WE HARE calculated new values of certain atomic parameters which can be used in the interpretation of ESR and diamagnetic susceptibility data. (a) Parameters for ESR calcukations Table 1 gives values of the atomic parameters aa (for the s-wave function of the valence shell) and (for the p-wave function of the valence shell) where : =
* I 0
F”P,~(Y)
dr
(14
and
For P,(r), which is the value of the radial wave function at a distance r (atomic units) from the
523
NOTES
nucleus, we have used the self-consistent solutions of the Hartree-Fock-Slater equations obtained by HERMAN and SKILLMAN. These have corrections for exchange effects. The integration of equation (la) was carried out by the method of quadrature using a digital computer, while the limiting value of equation (1 b) was obtained from an extrapolation of (P~(~)/~)2 at small values of I. This extrapolation took into account the curvature of P,(Y) with Y (i.e. it was not simply a linear extrapolation). The values of the isotropic and anisotropic hyperflne coupling factors (which we shall call A0 and B, respectively) which correspond to the above atomic parameters are also given in Table 1 for the isotopes which are indicated. These coupling factors have been calculated from: A, = (8~g~~~3~)~2(0) MC/S B. = (2gj+/5h)
(r -3 > MC/S
The quantities in these expressions have their usual meaningsc2) We find that the wave functions of Herman and Skilhnan in general lead to larger values of and #02(0) than those calculated from earlier wave functions (see, for example, the values tabulated by MORTON.@)) , viz. we obtain values of 2.002 a.u. and l-692 au. respectively for the Herman-Skillman and the Roothaan functions. (b) Parameters for diamagnetic susceptibility calculations Table 2 gives values of the classical diamagnetic susceptibility of certain ions and atoms which are of interest in the interpretation of diamagnetic
TECHNICAL
8. BRBBRICX R. F., Bull. Am. Phys. Sot. 11,222 (1966); STRAUSS A. J. and BRIZUUCK R. F., Bull. Am. Phys. Sac. 11, 222 (1966); &IKBIUCK R. F., J. Phys. Chem. Solids 27,149s (1966). 9. KLMENSP. G., Phys. Reu. 119,507 (1960). 10. DAMOND. H., J. AppI. Phys. 37,318l (1966).
J. Phys. Chem. So&Es Vol. 28, pp. 522-523.
Thermal
NOTES
subscripts, b and c are lattice parameters associated with the other two principal axes, and 711 is an average of the quantities -[(u/v,)(d~~/&)] where the vf are the normal mode frequencies. Although pI1 and its derivatives are diflkult to calculate, one can make a crude estimate of the contribution of equation (I) if one assumes that under hydrostatic conditions
contributions to the, elastic COnaallb of sodium (Received 18 July 1966)
that a large discrepancy exists between the theoretical predictions of the logarithmic derivative, A, of the quantity c’ = (C,,--&J/2 for sodium, and the value determined experimentally by DANIELS.(~) The theoretical prediction, however, was based on a calculation at O’K, and DANIELS’~)data are taken at room temperature. Since A = A,+& where A, is the thermal contribution to A it seems possible that the thermal contribution to C.” may account for the discrepancy. We shall consider the logarithmic derivative
which implies 3xdy N--
IT HASbeen observed by BROOKSQ)
A =-
dlnC’ d In x
X = 1.043
where x = v/v,, v is the specific volume, vo the specific volume at zero temperature and pressure, and l-043 is the value of x at room temperature and pressure. DANXEL~(~’had obtained the value A = -243, whereas BROOKS(~)has shown that the theoretically predicted A, can be, at most, A o = -1.97 if temperature effects are excluded. It can be shown that for temperature, T, greater than the Debye theta, the theoretical therma contribution Cr.’ to C’ for a b.c.c. monatomic solid is given by
x
[u(fg)b,c-b(Jg),,I (1)
where 6, is the density at O”K and zero pressure, n is the number of atoms per gram, k is the Boltzmann constant, a is a lattice parameter associated with a principal axis and the 11
Y dx where y is the normal Grtineisen parameter, Since, under hydrostatic conditions, we must have equation (1) becomes, with this 711= Y, assumption
G’ N-
- 96,pkTy
(2)
2
where the prime denotes the derivative of y w.r.t.b It follows from this that the thermal contribution, A,, to the logarithmic derivative of C” is given by
A
3fy” r = $[l
(3)
+(cs’fc,‘)l
where C,’ is the nonthermal con~ibu~on to c’, The quantities y” and y’ can be estimated by use of the two mode approximation’3’ for y, i.e.
6
2
dlnx
6
2
dlnx
1 I
(4)
where C4,0 is computed according to the theory set forth by Fucr-rs(*) and C,,O is computed essentially by means of the equation of state set forth by BROOI&~) (corrected for ion core interactions), with its parameters adjusted so as to reproduce the high pressure low temperature experimental data of BEECROFTand SWENSON@)on sodium. By this procedure one can calculate at x = l-043, y’ N 0.23, y” N 0‘85, y’;“y’ 1: 3.7. The ratio Conic=’ can be
TECHNICAL
from the data of DIEDERICH and TRIVIfrom which one finds Cc’&’ 2! -5.14. Substitution of these numbers into equation (3) yields AT 1: -0.9, which is of the right order of magnitude to account for the aforementioned discrepancy. It seems worthwhile to repeat that this calculation is crude. It is, at best, an order of ~nitude computation which tends to show the possible importance of thermal contributions to A.
estimated SONNO
U.S. Naval Ordnance Laboratory h%xr Spring, Wwyland
D. JOHN PASTINE 20910
ReJerences 1. 2. 3. 4.
2: 7.
B~ooxs H.,
Trans. Met&.
Sot.
AIM&‘,
22f,
546-558 (1963). DANIEU W. B., Pbys. Reu. 119,1246-1252 (1960). PASTINBD. J., Phys. Rev. 138, A767-A770 (1964). FUCHS K., Proc. R. Sot. A153, 622-629 (1936); A157,444-450 (1936). BROOKSH., Supplt. Nuovo Cim. 7, 207-244 (1958). BEECROFT R. I. and SWENSONC. A., J. Phys. Chem. l&329-344 (1960). DIEDERICHMARY E. and TRIV~SONNOJ., J. Phys.
Chem. So&& 27,637-642
J. Phyr. Chem. Solids
(1965).
Vol. 28, pp. 523-525.
Atomic parameters for the interpretation of ESR and diamagnetic susceptibility data (Received 24 August 1966)
WE HARE calculated new values of certain atomic parameters which can be used in the interpretation of ESR and diamagnetic susceptibility data. (a) Parameters for ESR calcukations Table 1 gives values of the atomic parameters aa (for the s-wave function of the valence shell) and
* I 0
F”P,~(Y)
dr
(14
and
For P,(r), which is the value of the radial wave function at a distance r (atomic units) from the
523
NOTES
nucleus, we have used the self-consistent solutions of the Hartree-Fock-Slater equations obtained by HERMAN and SKILLMAN. These have corrections for exchange effects. The integration of equation (la) was carried out by the method of quadrature using a digital computer, while the limiting value of equation (1 b) was obtained from an extrapolation of (P~(~)/~)2 at small values of I. This extrapolation took into account the curvature of P,(Y) with Y (i.e. it was not simply a linear extrapolation). The values of the isotropic and anisotropic hyperflne coupling factors (which we shall call A0 and B, respectively) which correspond to the above atomic parameters are also given in Table 1 for the isotopes which are indicated. These coupling factors have been calculated from: A, = (8~g~~~3~)~2(0) MC/S B. = (2gj+/5h)
(r -3 > MC/S
The quantities in these expressions have their usual meaningsc2) We find that the wave functions of Herman and Skilhnan in general lead to larger values of