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A first-principles calculation of lattice expansion
Volume 94A, number 2
PHYSICS LETTERS
28 February 1983
A FIRST-PRINCIPLES CALCULATION OF LATTICE EXPANSION ~ M. BERARD and R. HARRIS
McGill University, Ernest Rutherford Physics Bldg., 3600 University Street, Montreal, Quebec, Canada H3A 2T8 Received 5 May 1982 Revised manuscript received 6 December 1982
Using the RPA we have calculated the phonon modes for a model of solid argon without using the usual harmonic approximation for the interatomic pair potential. The procedure permits a determination of the coefficient of expansion without recourse to additional parameters or further approximations: agreement with experimental values is most satisfactory.
The calculation of phonons for a harmonic solid in the random phase approximation (RPA) is a wellknown technique in the theory o f quantum crystals [1,2]. Less well known is the result that, in the RPA, the harmonic phonon frequencies are exactly those of the corresponding classical eigenmodes [ 3 - 5 ] . Even less well known is the corresponding anharmonic problem, where calculation of the phonon modes as a function of lattice spacing permits a calculation of the coefficient o f linear expansion. It is the purpose of this communication to report results of such a calculation for solid argon, using as input only the known Lennard-Jones interatomic pair potential: the calculation represents a first step in our application of the RPA to solids which are strongly anharmonic. When tunneling from one lattice site to another is neglected, the dispersion relation ~o/¢ for the phonon modes of a periodic crystal becomes [ 1 ] 2 - (h~k) 2 ] ~ a ' [~a~,
-
a~,£~, E~o ~ , ~ , ( ~ ) ~ ,
= 0,
3"7
where the indices a a ' and ?3" label Hartree basis states on different lattice sites. In the harmonic approximaWork supported by the NSERC of Canada and the FCAC programme of the Government of Quebec. 0 031-9163/83/0000--0000/$ 03.00 © 1983 North-Holland
tion, these states would correspond to Einstein-like oscillators localized on the respective sites. The other parameters are defined as: ~ a a , = ~2/ - f z i , ,
fia,=fa-
fi, ,
£2/ = eigenenergy a on site i ,
fi=exp(-~2i/kBT), t
~ota' = ½((%" [SPi[O~) + (°ti[SPilOti) ) , where 6 Pi is the infinitesimal change in the density matrix [ 1], and the parameter o ( - k ) is the Fourier transform of the inter-site pair potential. In the general case of a three-dimensional crystalline solid with anharmonic pair potentials, the dispersion relation must be obtained numerically. No explicit harmonic approximation needs to be made and thus the method represents an advance over the "quasi harmonic" approximation which is commonly used in the literature [6]. For the present application, the phonon frequencies can be obtained as a function of the lattice spacing a [which enters in the matrix elements v ( - k ) ] , and of the temperature T (which enters in the factors faa, ). In the usual way, the free energy can then be calculated as a function of a and T, and minimized to give a(T) and the coefficient of lattice expansion ~ a/~ T[p. To illustrate the method, we have applied it to fcc solid argon, using the Lennard-Jones 6 - 1 2 pair potential as given, f o r example, by Kittel [7]. The potential 89
Volume 94A, nmnbcr 2
PHYSICS LETTERS
was expanded to fourth order about the known zero temperature equilibrium position for pairs of atoms that were fifth nearest neighbours or closer, and a basis set of 10 harmonic oscillator functions per atom used to self-consistently determine the Hartree basis states. For simplicity, the calculation was limited to temperatures well below the characteristic separation of the Hartree states, which, in temperature units, is of order of the Debye temperature. In this limit, for a given lattice spacing, the phonon frequencies are not temperature dependent, and the method resembles the quasiharmonic approximation. The general expression for the free energy is given by the analogue ofeq. (63) of ref. [1]
F(a, T) "~ 9xf2 NaoB(a - ao) 2 + k a T ~ In (1 - exp [-hc%(a)/k B T] }, k?, written with the zero point energy absorbed in the first term. This formulation is possible because the(anharmonic) phonons (with wavevector k and polarization ?,) are the explicit eigenstates of the system. At zero temperature, numerical evaluation of the first term permits the determination of the binding energy EO, the equilibrium spacing a 0 and the bulk modulus B 0. Our values of these quantities show characteristic quantum corrections [8] to the classical values which in all cases improve the agreement with experiment. At finite temperatures, the second term contributes a characteristic temperature dependence. When the temperature is much less than the Debye temperature only the low frequency phonons contribute and since in this region their dispersion relation is found to be very linear, the summation gives a term proportional to T 4. Thus the coefficient of lattice expansion is given by:
1 3a _ 2rr2kB(kBT) 37 a aT
90
15B0(hc) 3
'
28 February 1983
with the average phonon velocity c defined in terms of the individual velocities ckx as
c3
12rr
~2
c~x(g2k ) '
where fZk represents the angular coordinates of the wave-vector k. 3' is Gruneisen's constant, defined by: 7-
Y~kh"[kxCkh Y~k~.CkX ,
")'gx-
- V ~C°kh ('Okh 3V '
where V is the volume. Numerical calculations were performed using six representative phonon directions, and gave a value of a = 0.49 X 10 -7 K -3, with 7 = 2.68. This value of 7 compares favourably with the classical value of around 2.8 [9], and the value of a is in quantitative agreement with the low temperature data for solid argon [10] particularly in view of the large uncertainties in this data. We conclude, therefore, that the method is capable of accurate results, at least at low temperatures and for weak anharmonicity. We intend, therefore, to investigate its application to higher temperatures and to situa. tions with stronger anharmonicity.
References [1 ] D.R. Fredkia and N.R. Werthamer, Phys. Rev. 138A (1965) 1527. [2] N.S. Gillis and N.R. Werthamer, Phys. Rev. 167 (1968) 607. [3] L.G. Caron, J. Math. Phys. 14 (1973) 839. [4] W. Brenig, Z. Phys. 171 (1963) 60. [5] G. Meissner, Z. Phys. 205 (1967) 249. [6] G. Liebfried and W. Ludwig, Solid State Phys. 12 (1961) 276. [7] C. Kittel, Introduction to solid state physics (5th Ed.) (Wiley, New York, 1976). [8] N. Bernades, Phys. Rev. 120 (1960) 1927. [9] G.K. Horton and J.W. Leech, Proc. Phys. Soc. 82 (1963) 816. [101 D.L. Pollack, Rev. Mod. Phys. 36 (1964) 748.
PHYSICS LETTERS
28 February 1983
A FIRST-PRINCIPLES CALCULATION OF LATTICE EXPANSION ~ M. BERARD and R. HARRIS
McGill University, Ernest Rutherford Physics Bldg., 3600 University Street, Montreal, Quebec, Canada H3A 2T8 Received 5 May 1982 Revised manuscript received 6 December 1982
Using the RPA we have calculated the phonon modes for a model of solid argon without using the usual harmonic approximation for the interatomic pair potential. The procedure permits a determination of the coefficient of expansion without recourse to additional parameters or further approximations: agreement with experimental values is most satisfactory.
The calculation of phonons for a harmonic solid in the random phase approximation (RPA) is a wellknown technique in the theory o f quantum crystals [1,2]. Less well known is the result that, in the RPA, the harmonic phonon frequencies are exactly those of the corresponding classical eigenmodes [ 3 - 5 ] . Even less well known is the corresponding anharmonic problem, where calculation of the phonon modes as a function of lattice spacing permits a calculation of the coefficient o f linear expansion. It is the purpose of this communication to report results of such a calculation for solid argon, using as input only the known Lennard-Jones interatomic pair potential: the calculation represents a first step in our application of the RPA to solids which are strongly anharmonic. When tunneling from one lattice site to another is neglected, the dispersion relation ~o/¢ for the phonon modes of a periodic crystal becomes [ 1 ] 2 - (h~k) 2 ] ~ a ' [~a~,
-
a~,£~, E~o ~ , ~ , ( ~ ) ~ ,
= 0,
3"7
where the indices a a ' and ?3" label Hartree basis states on different lattice sites. In the harmonic approximaWork supported by the NSERC of Canada and the FCAC programme of the Government of Quebec. 0 031-9163/83/0000--0000/$ 03.00 © 1983 North-Holland
tion, these states would correspond to Einstein-like oscillators localized on the respective sites. The other parameters are defined as: ~ a a , = ~2/ - f z i , ,
fia,=fa-
fi, ,
£2/ = eigenenergy a on site i ,
fi=exp(-~2i/kBT), t
~ota' = ½((%" [SPi[O~) + (°ti[SPilOti) ) , where 6 Pi is the infinitesimal change in the density matrix [ 1], and the parameter o ( - k ) is the Fourier transform of the inter-site pair potential. In the general case of a three-dimensional crystalline solid with anharmonic pair potentials, the dispersion relation must be obtained numerically. No explicit harmonic approximation needs to be made and thus the method represents an advance over the "quasi harmonic" approximation which is commonly used in the literature [6]. For the present application, the phonon frequencies can be obtained as a function of the lattice spacing a [which enters in the matrix elements v ( - k ) ] , and of the temperature T (which enters in the factors faa, ). In the usual way, the free energy can then be calculated as a function of a and T, and minimized to give a(T) and the coefficient of lattice expansion ~ a/~ T[p. To illustrate the method, we have applied it to fcc solid argon, using the Lennard-Jones 6 - 1 2 pair potential as given, f o r example, by Kittel [7]. The potential 89
Volume 94A, nmnbcr 2
PHYSICS LETTERS
was expanded to fourth order about the known zero temperature equilibrium position for pairs of atoms that were fifth nearest neighbours or closer, and a basis set of 10 harmonic oscillator functions per atom used to self-consistently determine the Hartree basis states. For simplicity, the calculation was limited to temperatures well below the characteristic separation of the Hartree states, which, in temperature units, is of order of the Debye temperature. In this limit, for a given lattice spacing, the phonon frequencies are not temperature dependent, and the method resembles the quasiharmonic approximation. The general expression for the free energy is given by the analogue ofeq. (63) of ref. [1]
F(a, T) "~ 9xf2 NaoB(a - ao) 2 + k a T ~ In (1 - exp [-hc%(a)/k B T] }, k?, written with the zero point energy absorbed in the first term. This formulation is possible because the(anharmonic) phonons (with wavevector k and polarization ?,) are the explicit eigenstates of the system. At zero temperature, numerical evaluation of the first term permits the determination of the binding energy EO, the equilibrium spacing a 0 and the bulk modulus B 0. Our values of these quantities show characteristic quantum corrections [8] to the classical values which in all cases improve the agreement with experiment. At finite temperatures, the second term contributes a characteristic temperature dependence. When the temperature is much less than the Debye temperature only the low frequency phonons contribute and since in this region their dispersion relation is found to be very linear, the summation gives a term proportional to T 4. Thus the coefficient of lattice expansion is given by:
1 3a _ 2rr2kB(kBT) 37 a aT
90
15B0(hc) 3
'
28 February 1983
with the average phonon velocity c defined in terms of the individual velocities ckx as
c3
12rr
~2
c~x(g2k ) '
where fZk represents the angular coordinates of the wave-vector k. 3' is Gruneisen's constant, defined by: 7-
Y~kh"[kxCkh Y~k~.CkX ,
")'gx-
- V ~C°kh ('Okh 3V '
where V is the volume. Numerical calculations were performed using six representative phonon directions, and gave a value of a = 0.49 X 10 -7 K -3, with 7 = 2.68. This value of 7 compares favourably with the classical value of around 2.8 [9], and the value of a is in quantitative agreement with the low temperature data for solid argon [10] particularly in view of the large uncertainties in this data. We conclude, therefore, that the method is capable of accurate results, at least at low temperatures and for weak anharmonicity. We intend, therefore, to investigate its application to higher temperatures and to situa. tions with stronger anharmonicity.
References [1 ] D.R. Fredkia and N.R. Werthamer, Phys. Rev. 138A (1965) 1527. [2] N.S. Gillis and N.R. Werthamer, Phys. Rev. 167 (1968) 607. [3] L.G. Caron, J. Math. Phys. 14 (1973) 839. [4] W. Brenig, Z. Phys. 171 (1963) 60. [5] G. Meissner, Z. Phys. 205 (1967) 249. [6] G. Liebfried and W. Ludwig, Solid State Phys. 12 (1961) 276. [7] C. Kittel, Introduction to solid state physics (5th Ed.) (Wiley, New York, 1976). [8] N. Bernades, Phys. Rev. 120 (1960) 1927. [9] G.K. Horton and J.W. Leech, Proc. Phys. Soc. 82 (1963) 816. [101 D.L. Pollack, Rev. Mod. Phys. 36 (1964) 748.