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Ground state surface tension of liquid 3He
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
GROUND STATE SURFACE TENSION OF LIQUID 3He G.D. BUCHAN Department of Natural Philosophy, University of Aberdeen, Aberdeen, Scotland
*
Received 12 July 1976 A new method for the energy of the fermion liquid in correlated basis function approximation is described. The method is applied to 3He, and extended to a successful variational calculation of the surface tension.
Recent years have seen several variational calculations (e.g. [1}) for the free surface of 4He, based on trial “modulated” wave functions of Jastrow form. Here we present an extension of this approach to 3He, using a correlated basis function (CBF) method. The calculation is made possible by recognition of a new technique for the fermion liquid energy, replacing the cluster series [2] by a closed expression. For the bulk liquid, following [21, we write ~F(rl,r
(1)
2,...,rN)Ii~(rl,...,rN)~,
(H)=E~+~— ~
k~_~N(N_1)I_hfdrlfdr2Vlu(rl2)fdr3N~2~Vl~.
(2)
Here ~ E~correspond to the exact ground state of the boson-type system, and ~ is a Slater determinant containing plane wave states {k~}.The form of the last (exchange)5)term Eexch in (~) derives from use ofisthe in that term. Its innermost integral realusual and Jastrow ansatz ~1i~=for ~ from the exp{~u(r~1)} may be substituted expression(with u(r) = —(b/r) ,
V
2
1n~(r1r2)=N(N—l)1’
fdr3
N(110)
V 1(~t~)+4)(r1r2) V1u(r12)+fdr3n~P(ri,r2,r3)V1u(r13)
(3) obtained by differentiating the 2-particle distribution function (I is the normalisation integral). Hence 2u(r)+ (Vu(r))2] + ~ p2ffdT Eexch T~ ~ drgF(r) [V/ 12 dr13 V1u(r12) V1u(r13)g~(r1,r2, r~), (4)
f
where g~ g~)are the radial distribution and 3He, 3-particle functions respectively. 4He and Bose correlation systems were described with the molecular dynamics In a simultaneous description of results of [3], with Lennard—Jones potential parameters (e = 11.21 K, a = 2.505 A) chosen to reproduce saturation energy and density of 4He. g~(r) was derived from the a-eq. of [1], and from it g~(r) via a cluster expansion [2]. Forg~the Kirkwood superposition approximation (KSA) was used. Our results EF —2.08 K/N, p = 0.015 1 A—3 (experiment —2.52 K/N, 0.0164 A—3) mark an improvement over previous similar calculations (e.g. [4]); and over the corresponding cluster series [2] result obtained by us (—1.87 K/N, 0.0152 A—3). Turning to the free surface, the calculation proceeds in the same spirit. Thus (with z normal to the surface); ,
~
exp{~rF(zi)},
and the fermion surface is coupled to a mass —3 boson surface (both with bulk density p *
(5)
0.0151 A—3),
Present address: Macaulay Institute for Soil Research, Craigiebuckler, Aberdeen, Scotland.
35
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
described by Consequently the surface tension ~ contains y~explicitly, plus remaining terms which account for FD statistics. The one-body factors in MF = fl exp{~tF(zl)} serve as “amplitudes” for the plane-wave orbitals in 0, and allow a variational approach to 7F• The second term in (2) cancels on subtracting the bulk ener~
gy to derive 7F• Making the Jastrow-type approximation =
H
H
exp{~u(r,1)}
1>1=1
exp{~tB(zl)}
(6)
1=1
is treated by standard methods [1] (which also yields t~(z))and the “statistics’ terms (7F _7B) may be reduced in a manner similar to (2) to (4) above. Expressing 7F — 7B in terms of the density profile flF(Z) requires elimination of tF via the equation 7B
VInF(rl)
_
nF(~l) Vi
[tF(r,)
+
tB(rl)} +fdr2 V,u(r,2)n~kr,,r2)+F(r1)
(7)
F(rl)(N/Jfdr2...NO,1I~,sMF)2Vl(0t0)fdr2f(rl,r2)I(N_1), the last term is awkward but may be shown [5] to vanish in thermodynamic limit. The proof hinges on the finite, short-ranged behaviour off(r1, r2). The final expression for 7F [5] is complicated but contains only flF(Z), t~(z),gf~(r1r2, r3) (forwhich KSA is used) and g~(r1,r2). For the latter we make the mean-density ansatz [1] ,
~
For both boson and fermion surfaces the profile
[1 +exp(j3z)1~
(8) 2, i3 = 0.75 A’ becomes 2, j3 = was result 7Brelatively = 0.18 erg cmerrors in the latter arise from 7F a=Monte-Carlo (0.15 ±0.02)evaluation erg cm of the 6-di(1.2 used. ±0.2)The A’boson for 3He. The large mensional integral containing g~(r 2) and r3). However the with experiment (0.152 erg cm the “thinning” effect of statistics on1,r2, ~urfacethickness are good both agreement clear. For comparison, in ref. [1] (with which our mass —3 boson calculation is closely allied) the result i3 = 1.0 A—’ is obtained for 4He. Thus while our mass —3 boson result above shows the expected dilation of surface thickness relative to 4He (due to decrease in mass), the effect of statistics is to cause the 3He interface to revert to a width comparable or smaller to that of 4He. This thinning effect is due to exchange. In a crude density-functional model it would be suggested by the relative forms of the boson “quantum-pressure term” n(z)
(h2/8m)J’dz(Vp)2/p
(9)
and its corresponding fermion form — the “Weizsacker term”, which is expression (9) multiplied by one-ninth (e.g. ref. [6]). This latter term expresses the “interaction” between FD statistics and a density gradient. A fuller discussion may be found in [5]. The absence in the above of any explicit reference to the zero-point energy of surface waves will be noted. There is a hidden controversy on this point in the literature: thus apparently successful calculations either omit this contribution entirely (e.g. [1]), or depend crucially on its inclusion (e.g. [7]). The author believes that the wave-function type calculation described above already implicitly contains the major part of this contribution [7], with only the very long-wavelength modes left to account for [8}. The effect is paralleled in bulk liquid calculations by the need to include the Reatto and Chester [9] term in u(r) to account for long-wavelength phonons. The resulting energy change is known to be only slight (e.g. [1, 3]). 36
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
Fuller details and a discussion of the above calculation will be published elsewhere. I am grateful to the Science Research Council for financial support; and also to the Computing Department of Aberdeen University. I am indebted to Dr. R.C. Clark for suggesting and supervising this line of research.
References [1] C.C. Chang and M. Cohen, Phys. Rev. A8 (1973) 1930. [2J F.Y. Wu and E. Feenberg, Phys. Rev. 128 (1962) 943. [31D. Schiff and L. Verlet, Phys. Rev. 160 (1967) 208. [41C.-W. Woo, Phys. Rev. 151 (1966) 138. [5]G.D. Buchan, Ph.D. thesis (1976), University of Aberdeen, unpublished. [6] K.A. Brueckner, J.R. Buchier, S. Jorna and R.J. Lombard, Phys. Rev. 171 (1968) 1188. [7] W.F. Saam and C. Ebner, Phys. Rev. Lett. 34 (1975) 253. [81 G.D. Buchan, M.Sc. thesis (1974), University of Aberdeen, unpublished. [9] L. Reatto and G.V. Chester, Phys. Rev. 155 (1967) 88.
37
PHYSICS LETTERS
1 November 1976
GROUND STATE SURFACE TENSION OF LIQUID 3He G.D. BUCHAN Department of Natural Philosophy, University of Aberdeen, Aberdeen, Scotland
*
Received 12 July 1976 A new method for the energy of the fermion liquid in correlated basis function approximation is described. The method is applied to 3He, and extended to a successful variational calculation of the surface tension.
Recent years have seen several variational calculations (e.g. [1}) for the free surface of 4He, based on trial “modulated” wave functions of Jastrow form. Here we present an extension of this approach to 3He, using a correlated basis function (CBF) method. The calculation is made possible by recognition of a new technique for the fermion liquid energy, replacing the cluster series [2] by a closed expression. For the bulk liquid, following [21, we write ~F(rl,r
(1)
2,...,rN)Ii~(rl,...,rN)~,
(H)=E~+~— ~
k~_~N(N_1)I_hfdrlfdr2Vlu(rl2)fdr3N~2~Vl~.
(2)
Here ~ E~correspond to the exact ground state of the boson-type system, and ~ is a Slater determinant containing plane wave states {k~}.The form of the last (exchange)5)term Eexch in (~) derives from use ofisthe in that term. Its innermost integral realusual and Jastrow ansatz ~1i~=for ~ from the exp{~u(r~1)} may be substituted expression(with u(r) = —(b/r) ,
V
2
1n~(r1r2)=N(N—l)1’
fdr3
N(110)
V 1(~t~)+4)(r1r2) V1u(r12)+fdr3n~P(ri,r2,r3)V1u(r13)
(3) obtained by differentiating the 2-particle distribution function (I is the normalisation integral). Hence 2u(r)+ (Vu(r))2] + ~ p2ffdT Eexch T~ ~ drgF(r) [V/ 12 dr13 V1u(r12) V1u(r13)g~(r1,r2, r~), (4)
f
where g~ g~)are the radial distribution and 3He, 3-particle functions respectively. 4He and Bose correlation systems were described with the molecular dynamics In a simultaneous description of results of [3], with Lennard—Jones potential parameters (e = 11.21 K, a = 2.505 A) chosen to reproduce saturation energy and density of 4He. g~(r) was derived from the a-eq. of [1], and from it g~(r) via a cluster expansion [2]. Forg~the Kirkwood superposition approximation (KSA) was used. Our results EF —2.08 K/N, p = 0.015 1 A—3 (experiment —2.52 K/N, 0.0164 A—3) mark an improvement over previous similar calculations (e.g. [4]); and over the corresponding cluster series [2] result obtained by us (—1.87 K/N, 0.0152 A—3). Turning to the free surface, the calculation proceeds in the same spirit. Thus (with z normal to the surface); ,
~
exp{~rF(zi)},
and the fermion surface is coupled to a mass —3 boson surface (both with bulk density p *
(5)
0.0151 A—3),
Present address: Macaulay Institute for Soil Research, Craigiebuckler, Aberdeen, Scotland.
35
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
described by Consequently the surface tension ~ contains y~explicitly, plus remaining terms which account for FD statistics. The one-body factors in MF = fl exp{~tF(zl)} serve as “amplitudes” for the plane-wave orbitals in 0, and allow a variational approach to 7F• The second term in (2) cancels on subtracting the bulk ener~
gy to derive 7F• Making the Jastrow-type approximation =
H
H
exp{~u(r,1)}
1>1=1
exp{~tB(zl)}
(6)
1=1
is treated by standard methods [1] (which also yields t~(z))and the “statistics’ terms (7F _7B) may be reduced in a manner similar to (2) to (4) above. Expressing 7F — 7B in terms of the density profile flF(Z) requires elimination of tF via the equation 7B
VInF(rl)
_
nF(~l) Vi
[tF(r,)
+
tB(rl)} +fdr2 V,u(r,2)n~kr,,r2)+F(r1)
(7)
F(rl)(N/Jfdr2...NO,1I~,sMF)2Vl(0t0)fdr2f(rl,r2)I(N_1), the last term is awkward but may be shown [5] to vanish in thermodynamic limit. The proof hinges on the finite, short-ranged behaviour off(r1, r2). The final expression for 7F [5] is complicated but contains only flF(Z), t~(z),gf~(r1r2, r3) (forwhich KSA is used) and g~(r1,r2). For the latter we make the mean-density ansatz [1] ,
~
For both boson and fermion surfaces the profile
[1 +exp(j3z)1~
(8) 2, i3 = 0.75 A’ becomes 2, j3 = was result 7Brelatively = 0.18 erg cmerrors in the latter arise from 7F a=Monte-Carlo (0.15 ±0.02)evaluation erg cm of the 6-di(1.2 used. ±0.2)The A’boson for 3He. The large mensional integral containing g~(r 2) and r3). However the with experiment (0.152 erg cm the “thinning” effect of statistics on1,r2, ~urfacethickness are good both agreement clear. For comparison, in ref. [1] (with which our mass —3 boson calculation is closely allied) the result i3 = 1.0 A—’ is obtained for 4He. Thus while our mass —3 boson result above shows the expected dilation of surface thickness relative to 4He (due to decrease in mass), the effect of statistics is to cause the 3He interface to revert to a width comparable or smaller to that of 4He. This thinning effect is due to exchange. In a crude density-functional model it would be suggested by the relative forms of the boson “quantum-pressure term” n(z)
(h2/8m)J’dz(Vp)2/p
(9)
and its corresponding fermion form — the “Weizsacker term”, which is expression (9) multiplied by one-ninth (e.g. ref. [6]). This latter term expresses the “interaction” between FD statistics and a density gradient. A fuller discussion may be found in [5]. The absence in the above of any explicit reference to the zero-point energy of surface waves will be noted. There is a hidden controversy on this point in the literature: thus apparently successful calculations either omit this contribution entirely (e.g. [1]), or depend crucially on its inclusion (e.g. [7]). The author believes that the wave-function type calculation described above already implicitly contains the major part of this contribution [7], with only the very long-wavelength modes left to account for [8}. The effect is paralleled in bulk liquid calculations by the need to include the Reatto and Chester [9] term in u(r) to account for long-wavelength phonons. The resulting energy change is known to be only slight (e.g. [1, 3]). 36
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
Fuller details and a discussion of the above calculation will be published elsewhere. I am grateful to the Science Research Council for financial support; and also to the Computing Department of Aberdeen University. I am indebted to Dr. R.C. Clark for suggesting and supervising this line of research.
References [1] C.C. Chang and M. Cohen, Phys. Rev. A8 (1973) 1930. [2J F.Y. Wu and E. Feenberg, Phys. Rev. 128 (1962) 943. [31D. Schiff and L. Verlet, Phys. Rev. 160 (1967) 208. [41C.-W. Woo, Phys. Rev. 151 (1966) 138. [5]G.D. Buchan, Ph.D. thesis (1976), University of Aberdeen, unpublished. [6] K.A. Brueckner, J.R. Buchier, S. Jorna and R.J. Lombard, Phys. Rev. 171 (1968) 1188. [7] W.F. Saam and C. Ebner, Phys. Rev. Lett. 34 (1975) 253. [81 G.D. Buchan, M.Sc. thesis (1974), University of Aberdeen, unpublished. [9] L. Reatto and G.V. Chester, Phys. Rev. 155 (1967) 88.
37