- Email: [email protected]
Mean field theory of melting
Solid State Communications,VoI. 15, pp. 677—681, 1974.
Pergamon Press.
Printed in Great Britain
MEAN FIELD THEORY OF MELTING H. Fukuyama and P.M. Platzman Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A. (Received 21 February 1974 byA.G. Chynoweth) A simply microscopic theory of melting is presented and applied to derive Lindemans law for alkali metals and to estimate the melting density and temperature of a Coulomb solid.
HISTORICALLY: the theory of melting has been characterized by its lack of quantitativeness, In 19101 L.indeman suggested that a solid would melt when the mean squared deviation of the atoms in a solid exceeded a certain empirical fraction of the normal lattice separation. Since the physical assumptions underlying Lindeman’s law are reasonable and its predictions roughly correct it is only natural to look for the underlying microscopic theory which justifies and extends such a simple description. Unfortunately most attempts in this direction have either been unsuccessful or at best qualitative in their approach.2
The transverse mode instability to be discussed in this paper may be considered a superheating transition. The true melting temperature may of course, driven by a completely different mechanism, occur at a lower temperature. We cannot prove that the amount of superheating is small. However, by considering several explicit examples we will be able to make it plausible that such a theory does indeed contain a simple physical picture of melting. Qualitatively we expect the thermal motion of long wavelength phonons to destroy the solid. Moreover, we know a posteriori that, even near the melting point, the fractional displacement of the atoms in most solids is still quite small.6 Thus, we should be able to describe most solids right up to their first order transition by some type of quasi-harmonic description of the vibration spectrum. The SCHA is such a theory. It is based on the observation that if an atom in a crystal is executing vibrational motion it is not proper to treat this motion in an approximation where the other atoms are held stationary. The effective forces on an atom arise from a thermodynamic average of the restoring forces for arbitrary configurations of the atoms.
In 1968 A. Kugler3 suggested that the so-called Self-Consistent Harmonic Approximation (SCHA) to the phonon spectrum of a solid4 might be applied to the problem of the stability of a Wigner lattice.5 His work was primarily numerical and restricted entirely to the zero temperature three dimensional Coulomb solid. In this work we will show analytically how such a microscopic theory predicts a sudden transition from a simple metallic solid with a well defined transverse sound velocity to a new state (liquid!!) where no self-consistent transverse mode exists. Since melting is a first order transition between liquid and solid phases a true theory of melting should of necessity consider both phases. The Lindeman approach to melting, like the current approach is a one phase theory. It only treats the solid and says nothing about the liquid. Such theones evolve from an intuitive sense of what must be occurring. Physically
In the SCHA the normal mode frequencies (WkX) of a group of particles of mass m interacting via a two particle potential V(~q) are, 2
— —
1 V * ~ e~(4~) ~~(k) (1
—
e
)X (V~V~ V (r, 7)>
we know that liquids do not support transverse modes so that it is reasonable to assume that an instability of the transverse modes signals the onset of melting.
(1)
The position ‘~is the sum of the equilibrium position ~ and a displacement which is to be averaged over 677
678
MEAN FIELD THEORY OF MELTING
the phonon distribution at temperature T. This type of mean field theory quite accurately describes the
I. CLASSICAL SOLID
low temperature4 phonon spectra of a variety of anharmonic solids. The average over the oscillators is easily carried out and, in terms of the fourier coefficients Pq7 of the potential, equation (1) may be written as, ~
=
(~(k) q)(c~(k) q) X e~~ -
(eik ‘R
1) Vq exp [—D(q, R)].
—
(2)
The effects of self consistency are contained in, 2 (1— cos k’~R)wk’x’ D(q,R) = ~ ) (c~(k’) ‘q) X coth (I3~k’x’/2). (3) In three dimension D(q,1~)approaches a constant value at largeR, and even for the smallest R(R 0) contamed in equation (2) it is a good approximation to set (for a cubic material)8 D(q,R)
q2(u2)/3
D(q).
In many solids a Debye approximation (WkX = C0k) to the phonon spectrum is a crude but qualitatively accurate description. At high temperature (‘~T hwk) it yields, /2 2\/ KT \ + 2(~)~ (8) D(q) = ~ ~‘
where C~(C,)is the isotropic temperature dependent transverse (longitudal) sound velocity and a is the effective lattice constant for this isotropic model
q,R ~
X
Vol. 15, No.4
(4)
i.e. n = ir/6a3. Near melting (as we shall see) D is about one for q = K 1 (2ir/a) (the smallest reciprocal lettice vector) so that the sum in equation (6) is accurately approximated by a single term i.e. (~,t)2 = (K et(k))2 {vK~~Vk+K}e~. ~
-
(9)
(v) nearest
In most solids the sum over reciprocal lattice vectors in equation (6) converges rapidly and the leading term in the sum (in the absence of self-consistency effects) is an excellent approximation9 to the low temperature phonon frequency i.e. (w((T))i
=
(~~2 =
exp~—D~K 1)). (10)
w~(0)
Using equation (4) the sum over R in equation (2) yields m(w~)2 = Nk2v,~+ N ~ {[(k + K~)-e’(k)] vk+K~, —
C0,
2/C~ = = exp When C~ C1 (—12T/mC2)or Cfor all Tequation (10) becomes, C T = mC~Rexp 12R) (11) (—
v*O 1Kv
-
e’(k)J 2 VK }
(5)
and =
N
~
(K~ et(k)2)fvk+ ‘cv -
—
VK}
(6)
v~O
for the longitudal (1) and transverse (t) modes. In these equations the effective self-consistent potential, vq
=
vq
exp(—q2(u2)/3).
(7)
The essential feature of equations (5) and (6) is that the longitudal mode equation involves a summation over all reciprocal lattice vectors while the transverse mode equation contains only non zero reciprocal lattice vectors. In many situations it is indeed the case that the behavior of longitudinal mode is dominated by the first term in equation (5). It is also true that this (Ky = 0) term is least affected by self consistency requirements [seeequation (7)] . Since we cannot solve equations (4)—(7) explicitly, we will demonstrate that they predict a transition by considering two simplified cases.
with R = T/mC2 .~oSince the function on the right hand side of equation (11) has a maximum it will have no solution for real positive C2 when T
~‘
Tm
=
mC~/12e = 0.031mC~.
(12)
At Tm the velocity is still finite but has softened to a point where, C2(Tm)/Co~ = e~. (13) For the alkali metals6 Lindeman’s law may be written as, T% = 0.039 mCg. (14) These results are in remarkable agreement. In equation (14) C~is the velocity determined from the low temperature Debye temperature. In the alkali metals the Debye temperature is only weakly dependent on temperature (of the order of ten per cent). The crudeness of the approximations, in particular the characterization of the phonon spectrum by a single parameter, precludes any sensible discussion of such questions as the density (temperature)
Vol. 15, No.4
MEAN FIELD THEORY OF MELTING
dependence of C~in both equation (13) and equation (14) predicted softening, in the alkali metals, of the transverse velocity [equation (13)] would be intersting to observe. We know of no such experiments near the melting point. The physics of the situation described by equation (11) is clear. As the temperature increases the mean square deviation the potential in an exponential way increases, and the sound velocity softens decreases. The decreasing sound velocity in turn, exponentially feeds back on the potential until a temperature is reached, where the potential runs away from the velocity and no transverse mode exists. We have assumed that both transverse and longitudal velocities soften simultaneously. We would expect that a self-consistent solution would show that the longitudal mode is less temperature dependent. In order to estimate the importance of this effect we consider the opposite extreme where only the transverse mode is temperature dependent and the longitudal velocity retains its zero temperature value (C, — C 0). In this case a similar procedure yields a Tm 0.039mC~in perfect but obviously fortuitous agreement with the empirical Lindeman law. The essential point is not the exact numerical agreement but the rough qualitative agreement. II. COULOMB SOLID Under suitable conditions, a set ofN negatively charged electrons immersed in a uniform positive background is constant expected a,todensity crystallize a BCC lattice (lattice n = into 2/a3).5 At low temperature and high (low) density the properties of the gas (solid) have been extensively investigated.”2 In this quantum limit estimates of the melting density based on a variety of models give a spread of r~ (r 2r 3 = me 0, where 4irr~/3 n’) values from S to 100011,13 At high temperatures where the system is 2 suggests treated classically, numerical simulation’ that crystallization occurs when I’~ e2 /r 0 (K T) is between 100 and 175. The remainder of this paper will be devoted to a discussion of the melting of such a Coulomb solid in both quantum and classical regimes within the framework of the SCHA. For a BCC lattice one can show that near melting equation (6) is, to a fraction of one per cent, accurately
67
represented by the sum over the twelve nearest neighbor reciprocal lattice vectors of the form — ~ [1,1,0] K0 [1,1,0]. K, a In this case we may expand ~k+K, —vjç1 ask ~ 0 to —
obtain an explicit expression ~orthe anisotropic transverse sound velocity, 2 = (4irne2 fm)S (15) c’
where S depends on k and ex(k). For k in the [1, 0, 0] direction the two transverse modes are degenerate and S 2exp (—2K~,y) (16) 1~= 4K~y with y (u2)/3. The S obtained by averaging over direction is SAy = 2/(5K~ X—27K~+ 8(K~ 7)2 )exp (—2Kg y) (17) Because of the self consistency (7 * 0) and the anisotropy of the spectrum equation (15) is still a complicated set of coupled equations. In order to obtain a value of r~at melting we will make four quite different approximations. (A)We take k along the [1,0,0] direction and evaluate ~ by artificially requiring the transverse sound mode to be isotropic and linear over the entire zone, while the longitudal mode (plasmon) is assumed to be dispersionless and independent of self-consistenc) requirements. In this case 0.92 2.81 IC0’1 K~y= ~~P’I~+ ~i7i ~ (18) 2 /ma. Substituting equation (18) and with ‘C~=(16) e into equation (l5)we obtain r~ 14 equation at melting (see Table 1). (B) We evaluate ~ in the same way as above but use the average velocity everywhere i.e. equation (15) with S given in equation (17). In this case we find an r~ 19. (C) We include some of the effects of the anisotropy of the phonon spectrum into the calculation of 7, by writing K~7= 0.92 + 5.35 /CCARR\ (19) vrvi~ r~ r~ C / The quantity CCARR 0.608 C0 is the value of the transverse sound velocity along ~I, 0,01 in the pure
680
MEAN FIELD THEORY OF MELTING
Vol. 15, No.4
(no SCHA) Coulomb solid as calculated by Carr.’°The numerical form of equation (19) requires K~7 to be equal to its value in the pure Coulomb solid i.e.10 K~7= 6.27/r 2 when C= CCARR. The implicit 1” is that the fluctuations, in this low assumption then, temperature regime, all scale like C’. Using equation (19) and equation (15) we arrive at an r 1 = 18.
Table 1. The values ofr~(quantum regime) and [‘~ (classical regime) for several approximate solutions of the mean field equation (15) (A) (B)(C)(D) r~ 14 19 18 19
(D) We take into account the dispersion of the plasmon in the scheme (C) by evaluating the first term in equation (5) using a value of 7 determined from equation (16) at melting. This changes 0.92 in equation 1.36, yielding r~= 19.
Since the value of r~and [‘~~are roughly constant independent of our approximation we can have some confidence that such numbers are characteristic of the predictions of our simple theory. A modified theory could of course lead to quite different values of r~. However, the value of r~obtained here fits in well with the scatter of values obtained by other authors, while the value of r0 is in disagreement with the numerical results obtained in reference (11). In order to determine whether the simple theory of melting presented here does in fact contain the essential physics of melting it will be important to examine in numerical detail the predictions of the SCHA as they apply for example to the alkali metals. Here the experimental data will provide us with ultimate test of the theory.
All these estimates are close to one another and in rather remarkable agreement with the extensive numerical calculations (r~ 22) presented in reference (3).14
In the classical regime ~ T ~> hwkx the low lying transverse modes are so heavily weighted that the plasmon branch plays practically no role i.e. 2) (22) K~y = (3.28)(KT/mC Using equation (22) it is straightforward to repeat our results for models A, B and D. We are unable to fill in column C in Table 1 since in the classical regime there has been no calculation of(u2) analogous to Maradudins.10 The results for r 0 are shown in the second row of Table 1.
—~
~‘o
______
16
25
16
Acknowledgements We would like to thank T.R. Brown, C.C.Grimes and P.C. Hohenberg for many —
useful discussions. Several comments by R. Kubo and C. Herring are also acknowledged.
REFERENCES 1.
LINDEMAN F., z.f Phys. 11,609, (1910).
2. 3.
BROUT R.H.,Phase Transitions, Benjamin, New York (1965). IDA Y.,Phys. Rev. 187 951 (1969). KUGLER A., Ann. Phys. 53, 133 (1969).
4.
KOHLER T.R, Phys. Rev. 165 942 (1968); WERTHAMER N.R., Theory of Lattice Dynamics of Rare Gas Crystals in Rare Gas Solids (Edited by KLEIN M.C. and VENABLES J.) Academic Press, New York (1973).
5. 6.
WIGNER E.P., Phys. Rev. 46, 1002 (1934), and Trans. Faraday Soc. 34, 678 (1938) PINES D., Elementary Excitations in Solids, Benjamin, New York (1963).
7.
The use of a well defined fourier transform of a two body potential to characterize the phonon specturm is most sensibly applied to simple metals. It is well known that in such materials, for example Na, [see COHRAN V~ in Lattice Dynamics (Edited by WALLIS R.F.) Pergamon Press] a rather smooth and well defined Fourier transform accurately represents all of the detailed inelastic neutron data. The situation is not as clear for the rare gas solids where hard cores play an important role.
8.
Such an argument fails in two dimensions where D(q,R) lnR at large R. We will have more to say on this matter in a later publication. We have estimated such terms for the case of Na and find that the first term gives 90% of the velocity.
9.
-+
Vol. 15, No.4
MEAN FIELD THEORY OF MELTING
681
10.
Reference (2) wrote down an equation similar to equation (11), of this paper, on the basis of intuitive arguments.
11.
COLDWELL-HORSTALL R.A. and MARADUDIN A.A.,J. Math. Phys. 1,395 (1960), CARRW.J.,Phys. Rev. 122 1437 (1961). DeWETTE F.W., Phys~Rev. 135, A287 (1964). BRUSH S.G., SAHLIN H.L. and TELLER E., J. Chem. Phys. 452102(1966). VAN HORN H.M., Phys. Rev. Lett. 28A 706 (1969), HANSEN J.P., Phys. Rev. Lett. 41A, 213 (1972).
12. 13.
MO1’T N., Phil. Mag. 6287 (1961); VAN HORN H.M., Phys. Rev. 111,442 (1958).
14.
In addition to calculating the melting density of the Coulomb solid within the SCHA Kugler attempted to estimate by means of a rather crude and uncontrolled type of perturbation theory, the effects of anharmonicity not contained in the SCHA. He found that for the Coulomb solid such effects became important at much higher values of r3 i.e. that deviations from SCHA could possibly play an important role in determining the melting density.
Une théorie microscopique simple de Ia fusion est presentée et appliquée a Ia demonstration de Ia loi de Lindeman eta 1 estimation de la densité et de la temperature d’un solide de Coulomb au point de fusion.
Pergamon Press.
Printed in Great Britain
MEAN FIELD THEORY OF MELTING H. Fukuyama and P.M. Platzman Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A. (Received 21 February 1974 byA.G. Chynoweth) A simply microscopic theory of melting is presented and applied to derive Lindemans law for alkali metals and to estimate the melting density and temperature of a Coulomb solid.
HISTORICALLY: the theory of melting has been characterized by its lack of quantitativeness, In 19101 L.indeman suggested that a solid would melt when the mean squared deviation of the atoms in a solid exceeded a certain empirical fraction of the normal lattice separation. Since the physical assumptions underlying Lindeman’s law are reasonable and its predictions roughly correct it is only natural to look for the underlying microscopic theory which justifies and extends such a simple description. Unfortunately most attempts in this direction have either been unsuccessful or at best qualitative in their approach.2
The transverse mode instability to be discussed in this paper may be considered a superheating transition. The true melting temperature may of course, driven by a completely different mechanism, occur at a lower temperature. We cannot prove that the amount of superheating is small. However, by considering several explicit examples we will be able to make it plausible that such a theory does indeed contain a simple physical picture of melting. Qualitatively we expect the thermal motion of long wavelength phonons to destroy the solid. Moreover, we know a posteriori that, even near the melting point, the fractional displacement of the atoms in most solids is still quite small.6 Thus, we should be able to describe most solids right up to their first order transition by some type of quasi-harmonic description of the vibration spectrum. The SCHA is such a theory. It is based on the observation that if an atom in a crystal is executing vibrational motion it is not proper to treat this motion in an approximation where the other atoms are held stationary. The effective forces on an atom arise from a thermodynamic average of the restoring forces for arbitrary configurations of the atoms.
In 1968 A. Kugler3 suggested that the so-called Self-Consistent Harmonic Approximation (SCHA) to the phonon spectrum of a solid4 might be applied to the problem of the stability of a Wigner lattice.5 His work was primarily numerical and restricted entirely to the zero temperature three dimensional Coulomb solid. In this work we will show analytically how such a microscopic theory predicts a sudden transition from a simple metallic solid with a well defined transverse sound velocity to a new state (liquid!!) where no self-consistent transverse mode exists. Since melting is a first order transition between liquid and solid phases a true theory of melting should of necessity consider both phases. The Lindeman approach to melting, like the current approach is a one phase theory. It only treats the solid and says nothing about the liquid. Such theones evolve from an intuitive sense of what must be occurring. Physically
In the SCHA the normal mode frequencies (WkX) of a group of particles of mass m interacting via a two particle potential V(~q) are, 2
— —
1 V * ~ e~(4~) ~~(k) (1
—
e
)X (V~V~ V (r, 7)>
we know that liquids do not support transverse modes so that it is reasonable to assume that an instability of the transverse modes signals the onset of melting.
(1)
The position ‘~is the sum of the equilibrium position ~ and a displacement which is to be averaged over 677
678
MEAN FIELD THEORY OF MELTING
the phonon distribution at temperature T. This type of mean field theory quite accurately describes the
I. CLASSICAL SOLID
low temperature4 phonon spectra of a variety of anharmonic solids. The average over the oscillators is easily carried out and, in terms of the fourier coefficients Pq7 of the potential, equation (1) may be written as, ~
=
(~(k) q)(c~(k) q) X e~~ -
(eik ‘R
1) Vq exp [—D(q, R)].
—
(2)
The effects of self consistency are contained in, 2 (1— cos k’~R)wk’x’ D(q,R) = ~ ) (c~(k’) ‘q) X coth (I3~k’x’/2). (3) In three dimension D(q,1~)approaches a constant value at largeR, and even for the smallest R(R 0) contamed in equation (2) it is a good approximation to set (for a cubic material)8 D(q,R)
q2(u2)/3
D(q).
In many solids a Debye approximation (WkX = C0k) to the phonon spectrum is a crude but qualitatively accurate description. At high temperature (‘~T hwk) it yields, /2 2\/ KT \ + 2(~)~ (8) D(q) = ~ ~‘
where C~(C,)is the isotropic temperature dependent transverse (longitudal) sound velocity and a is the effective lattice constant for this isotropic model
q,R ~
X
Vol. 15, No.4
(4)
i.e. n = ir/6a3. Near melting (as we shall see) D is about one for q = K 1 (2ir/a) (the smallest reciprocal lettice vector) so that the sum in equation (6) is accurately approximated by a single term i.e. (~,t)2 = (K et(k))2 {vK~~Vk+K}e~. ~
-
(9)
(v) nearest
In most solids the sum over reciprocal lattice vectors in equation (6) converges rapidly and the leading term in the sum (in the absence of self-consistency effects) is an excellent approximation9 to the low temperature phonon frequency i.e. (w((T))i
=
(~~2 =
exp~—D~K 1)). (10)
w~(0)
Using equation (4) the sum over R in equation (2) yields m(w~)2 = Nk2v,~+ N ~ {[(k + K~)-e’(k)] vk+K~, —
C0,
2/C~ = = exp When C~ C1 (—12T/mC2)or Cfor all Tequation (10) becomes, C T = mC~Rexp 12R) (11) (—
v*O 1Kv
-
e’(k)J 2 VK }
(5)
and =
N
~
(K~ et(k)2)fvk+ ‘cv -
—
VK}
(6)
v~O
for the longitudal (1) and transverse (t) modes. In these equations the effective self-consistent potential, vq
=
vq
exp(—q2(u2)/3).
(7)
The essential feature of equations (5) and (6) is that the longitudal mode equation involves a summation over all reciprocal lattice vectors while the transverse mode equation contains only non zero reciprocal lattice vectors. In many situations it is indeed the case that the behavior of longitudinal mode is dominated by the first term in equation (5). It is also true that this (Ky = 0) term is least affected by self consistency requirements [seeequation (7)] . Since we cannot solve equations (4)—(7) explicitly, we will demonstrate that they predict a transition by considering two simplified cases.
with R = T/mC2 .~oSince the function on the right hand side of equation (11) has a maximum it will have no solution for real positive C2 when T
~‘
Tm
=
mC~/12e = 0.031mC~.
(12)
At Tm the velocity is still finite but has softened to a point where, C2(Tm)/Co~ = e~. (13) For the alkali metals6 Lindeman’s law may be written as, T% = 0.039 mCg. (14) These results are in remarkable agreement. In equation (14) C~is the velocity determined from the low temperature Debye temperature. In the alkali metals the Debye temperature is only weakly dependent on temperature (of the order of ten per cent). The crudeness of the approximations, in particular the characterization of the phonon spectrum by a single parameter, precludes any sensible discussion of such questions as the density (temperature)
Vol. 15, No.4
MEAN FIELD THEORY OF MELTING
dependence of C~in both equation (13) and equation (14) predicted softening, in the alkali metals, of the transverse velocity [equation (13)] would be intersting to observe. We know of no such experiments near the melting point. The physics of the situation described by equation (11) is clear. As the temperature increases the mean square deviation the potential in an exponential way increases, and the sound velocity softens decreases. The decreasing sound velocity in turn, exponentially feeds back on the potential until a temperature is reached, where the potential runs away from the velocity and no transverse mode exists. We have assumed that both transverse and longitudal velocities soften simultaneously. We would expect that a self-consistent solution would show that the longitudal mode is less temperature dependent. In order to estimate the importance of this effect we consider the opposite extreme where only the transverse mode is temperature dependent and the longitudal velocity retains its zero temperature value (C, — C 0). In this case a similar procedure yields a Tm 0.039mC~in perfect but obviously fortuitous agreement with the empirical Lindeman law. The essential point is not the exact numerical agreement but the rough qualitative agreement. II. COULOMB SOLID Under suitable conditions, a set ofN negatively charged electrons immersed in a uniform positive background is constant expected a,todensity crystallize a BCC lattice (lattice n = into 2/a3).5 At low temperature and high (low) density the properties of the gas (solid) have been extensively investigated.”2 In this quantum limit estimates of the melting density based on a variety of models give a spread of r~ (r 2r 3 = me 0, where 4irr~/3 n’) values from S to 100011,13 At high temperatures where the system is 2 suggests treated classically, numerical simulation’ that crystallization occurs when I’~ e2 /r 0 (K T) is between 100 and 175. The remainder of this paper will be devoted to a discussion of the melting of such a Coulomb solid in both quantum and classical regimes within the framework of the SCHA. For a BCC lattice one can show that near melting equation (6) is, to a fraction of one per cent, accurately
67
represented by the sum over the twelve nearest neighbor reciprocal lattice vectors of the form — ~ [1,1,0] K0 [1,1,0]. K, a In this case we may expand ~k+K, —vjç1 ask ~ 0 to —
obtain an explicit expression ~orthe anisotropic transverse sound velocity, 2 = (4irne2 fm)S (15) c’
where S depends on k and ex(k). For k in the [1, 0, 0] direction the two transverse modes are degenerate and S 2exp (—2K~,y) (16) 1~= 4K~y with y (u2)/3. The S obtained by averaging over direction is SAy = 2/(5K~ X—27K~+ 8(K~ 7)2 )exp (—2Kg y) (17) Because of the self consistency (7 * 0) and the anisotropy of the spectrum equation (15) is still a complicated set of coupled equations. In order to obtain a value of r~at melting we will make four quite different approximations. (A)We take k along the [1,0,0] direction and evaluate ~ by artificially requiring the transverse sound mode to be isotropic and linear over the entire zone, while the longitudal mode (plasmon) is assumed to be dispersionless and independent of self-consistenc) requirements. In this case 0.92 2.81 IC0’1 K~y= ~~P’I~+ ~i7i ~ (18) 2 /ma. Substituting equation (18) and with ‘C~=(16) e into equation (l5)we obtain r~ 14 equation at melting (see Table 1). (B) We evaluate ~ in the same way as above but use the average velocity everywhere i.e. equation (15) with S given in equation (17). In this case we find an r~ 19. (C) We include some of the effects of the anisotropy of the phonon spectrum into the calculation of 7, by writing K~7= 0.92 + 5.35 /CCARR\ (19) vrvi~ r~ r~ C / The quantity CCARR 0.608 C0 is the value of the transverse sound velocity along ~I, 0,01 in the pure
680
MEAN FIELD THEORY OF MELTING
Vol. 15, No.4
(no SCHA) Coulomb solid as calculated by Carr.’°The numerical form of equation (19) requires K~7 to be equal to its value in the pure Coulomb solid i.e.10 K~7= 6.27/r 2 when C= CCARR. The implicit 1” is that the fluctuations, in this low assumption then, temperature regime, all scale like C’. Using equation (19) and equation (15) we arrive at an r 1 = 18.
Table 1. The values ofr~(quantum regime) and [‘~ (classical regime) for several approximate solutions of the mean field equation (15) (A) (B)(C)(D) r~ 14 19 18 19
(D) We take into account the dispersion of the plasmon in the scheme (C) by evaluating the first term in equation (5) using a value of 7 determined from equation (16) at melting. This changes 0.92 in equation 1.36, yielding r~= 19.
Since the value of r~and [‘~~are roughly constant independent of our approximation we can have some confidence that such numbers are characteristic of the predictions of our simple theory. A modified theory could of course lead to quite different values of r~. However, the value of r~obtained here fits in well with the scatter of values obtained by other authors, while the value of r0 is in disagreement with the numerical results obtained in reference (11). In order to determine whether the simple theory of melting presented here does in fact contain the essential physics of melting it will be important to examine in numerical detail the predictions of the SCHA as they apply for example to the alkali metals. Here the experimental data will provide us with ultimate test of the theory.
All these estimates are close to one another and in rather remarkable agreement with the extensive numerical calculations (r~ 22) presented in reference (3).14
In the classical regime ~ T ~> hwkx the low lying transverse modes are so heavily weighted that the plasmon branch plays practically no role i.e. 2) (22) K~y = (3.28)(KT/mC Using equation (22) it is straightforward to repeat our results for models A, B and D. We are unable to fill in column C in Table 1 since in the classical regime there has been no calculation of(u2) analogous to Maradudins.10 The results for r 0 are shown in the second row of Table 1.
—~
~‘o
______
16
25
16
Acknowledgements We would like to thank T.R. Brown, C.C.Grimes and P.C. Hohenberg for many —
useful discussions. Several comments by R. Kubo and C. Herring are also acknowledged.
REFERENCES 1.
LINDEMAN F., z.f Phys. 11,609, (1910).
2. 3.
BROUT R.H.,Phase Transitions, Benjamin, New York (1965). IDA Y.,Phys. Rev. 187 951 (1969). KUGLER A., Ann. Phys. 53, 133 (1969).
4.
KOHLER T.R, Phys. Rev. 165 942 (1968); WERTHAMER N.R., Theory of Lattice Dynamics of Rare Gas Crystals in Rare Gas Solids (Edited by KLEIN M.C. and VENABLES J.) Academic Press, New York (1973).
5. 6.
WIGNER E.P., Phys. Rev. 46, 1002 (1934), and Trans. Faraday Soc. 34, 678 (1938) PINES D., Elementary Excitations in Solids, Benjamin, New York (1963).
7.
The use of a well defined fourier transform of a two body potential to characterize the phonon specturm is most sensibly applied to simple metals. It is well known that in such materials, for example Na, [see COHRAN V~ in Lattice Dynamics (Edited by WALLIS R.F.) Pergamon Press] a rather smooth and well defined Fourier transform accurately represents all of the detailed inelastic neutron data. The situation is not as clear for the rare gas solids where hard cores play an important role.
8.
Such an argument fails in two dimensions where D(q,R) lnR at large R. We will have more to say on this matter in a later publication. We have estimated such terms for the case of Na and find that the first term gives 90% of the velocity.
9.
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Vol. 15, No.4
MEAN FIELD THEORY OF MELTING
681
10.
Reference (2) wrote down an equation similar to equation (11), of this paper, on the basis of intuitive arguments.
11.
COLDWELL-HORSTALL R.A. and MARADUDIN A.A.,J. Math. Phys. 1,395 (1960), CARRW.J.,Phys. Rev. 122 1437 (1961). DeWETTE F.W., Phys~Rev. 135, A287 (1964). BRUSH S.G., SAHLIN H.L. and TELLER E., J. Chem. Phys. 452102(1966). VAN HORN H.M., Phys. Rev. Lett. 28A 706 (1969), HANSEN J.P., Phys. Rev. Lett. 41A, 213 (1972).
12. 13.
MO1’T N., Phil. Mag. 6287 (1961); VAN HORN H.M., Phys. Rev. 111,442 (1958).
14.
In addition to calculating the melting density of the Coulomb solid within the SCHA Kugler attempted to estimate by means of a rather crude and uncontrolled type of perturbation theory, the effects of anharmonicity not contained in the SCHA. He found that for the Coulomb solid such effects became important at much higher values of r3 i.e. that deviations from SCHA could possibly play an important role in determining the melting density.
Une théorie microscopique simple de Ia fusion est presentée et appliquée a Ia demonstration de Ia loi de Lindeman eta 1 estimation de la densité et de la temperature d’un solide de Coulomb au point de fusion.