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Ground-state energy of high-density helium
Volume 79A, number 5,6
PHYSICS LETTERS
27 October 1980
GROUND-STATE ENERGY OF HIGH-DENSITY HELIUM H. NIKI, H. NAGARA, H. MIYAGI and T. NAKAMURA Department of Material Physics, Faculty of EngineeringScience, Osaka University, Toyonaka, Japan 560 Received 17 July 1980
Ground-state energies of the high-density helium are calculated for fcc and bcc structures, up to fourth order in the structural expansion. The result manifests that bcc is much stabler than fcc in the region of high density.
A number of theoretical studies have been done for high pressure behavior of solid helium [1—5].The transition pressure from insulating to metallic phase was predicted to be ~95 Mbar [4]. The high-density helium is of astrophysical interest; its solubiity into
screened electron—ion interaction. Here the matrix element associated with the momentum transfer g of electrons is given by
metallic hydrogenof may be important internal structure some planets [6].for studying the All of the existing studies [1—4] assumed fcc as the ground-state structure. In this letter, however, we report that bcc helium is much more stable than fcc at high density and at zero temperature. Consider the system of atoms with atomic number Z. Generally the ground-state energy E of the considered system is written as
v1(g) = —(8/3ir)(ar )~g
EEeg+EM+Est.
(1)
Here Eeg denotes the energy of electron gas, EM the Madelung energy due to charge of atomic nucleus, and E5~the structure-dependent electronic energy, all per electron and in Rydberg units. Now, in the atomic units we introduce r5, which denotes the radius of sphere equivalent to volume per electron. Then Eeg is a function r5, being independent of structures. And 2/3aof r1 ~ E =Z M
M
~
‘
‘
‘
with a~the Madelung constant. We note that the aMs are identical to each other for bcc and fcc up to the four significant figuresbcc [7].and Then, Z the first energy difference between fcc for mustsmall come from E 5t.
Thus we look into E5~by using the structural expansion [8—il]. In ref. [11], E5~is expanded in the 428
ui&) =
u1(g)/e~(g,0),
2
(3) (4)
,
where g denotes the reciprocal lattice vector in units of the Fermi momentum (11 = 1), e~(g,0) the static Lindhard dielectric function, and a = (4/9iT)h/3. The relative stability of bcc to fcc can most simply be observed in the primary term of the expansion. It is given by 1 Iv 1 (g) 2 E2~~-g
v2(g)
[1—1/e~(g,O)]
,
(5)
where u2(g) is the electron—electron coupling and has a similar expression to u1(g). We notice here that e~(g,0) increases as g becomes shorter. For Z = 1 the shortest g-vectors are equal to 2.280 for bcc and to 2.2 16 for fcc. Here the numbers of shortest g-vectors (12 for bcc and 8 for fcc) are more effective than E than their length in producing E2 [bcc] lower 2 [fcc]. However the longer g-vectors contribute to E2 considerably and as resultants we have E2 [fcc] lower than E2 [bcc]. Let us now consider the case of Z = 2. Generally the relevantg-vectors are ob3. tamed from those g-vectors for Z = 1 by a scaling factorZ~I Then the shortest under consideration are fairly smaller than 2 for both structures, where the second shortestg-vectors are still larger than 2. Then the contributions from shortest g-vectors are most ef-
Volume 79A, number 5,6
PHYSICS LETTERS
fective in determining the relative stability. Thus we have E2 [bcc] lower than E2 [fcc], with difference
We utilize them to figure out E. The fourth-order
0.0084 at r5 = 1.0. Contributions from the higher order terms work to widen the energy separation produced by the primary term, at least up to fourth order. In fig. 1 we show our fourth order result by broken lines. The terms taken into account are in accord with ref. [11], and materials needed for evaluation are given in refs. [10—13].In the estimation the correlation energy of electron gas is taken into account in the Noziéres— Pines interpolation formula. An expansion has been proposed [10,11] in powers of eq. (3) with replacement of e~(g,0)by e(g, 0), the exact static dielectric function. This expansion seems to give us a more rapidly convergent series than that with the use of e~(g,0). Here the expanded terms are much more simplified. However, a fourth-order term arising from combined action between the vertex function and the self-energy still remains, though it was overlooked in refs. [8,9]. Accurate estimates of e(g, 0)are now availabl~in the region of rs ~2.0 [13]. AE2
E(Ry)
=
0.7
0.8
0.9
1.0
r5
27 October 1980
result thus obtained is shown in fig. i with solid lines. In the same figure, we also show the fcc result by Simcox and March [2] by a dotted line. That is near our result in third order. Thus the fourth-order result is remarkably different from the third order one. In our fourth-order result the energy separation between fcc and bcc is still large, in the region of r5 1.0: For both structures the energy curves must tend to the same limit, E —2.90 for free atom, as r5 increases. Ef. fect of the neglected terms including Hubbard’s H-graph [9] must be examined to improve the present result. In fig. 2 we show the resulting curves for pressure (P)versus molar volume (Vm). Those curves are compared with that by Zharkov et al. [5], which is based on the QSM method, namely a modified TFD method.
1~M_bar)
100
1.1
\\..
..,.,.
30
\\...
Zharkov et a
20
~.
-2.0
/
.....
~
N
bcc
Simcox-March
N
Sirnc~-March fcc
5 .
2
-2.5
I
Fig. 1. Ground-state energy (E) versus rs.
04
06
~
08
3) Vm (cm 1~0
Fig. 2. Curves for pressure (F) versus molar volume (Vm). The solid lines are the results obtained from the solid ones in fig. 1. The other curves from the broken ones in fig. 1 may be omitted.
429
Volume 79A, number 5,6
PHYSICS LETTERS
Notable deviation occurs at r5 ~ 0.95 which corre3. sponds ~0.64 cm NowtotheVm energy difference between fcc and bcc amounts to 0.1 per atom at r 5 = 1.0. This is quite a large figure compared with the zero-point energy difference, which is estimated from the compressibility and proves to be at most one order of magnitude smaller. There is no question for bcc helium to be stabler than fcc one, in the high density region. This result may change a previous estimation for the transition pressure to the metallic state [4]. We finally mention that the transition pressure for bcc is lower than that for fcc [14]. References [1] C.A. ten Seldom, Proc. Phys. Soc. A70 (1956) 97, 529.
430
27 October 1980
[2) L.N. Simcox (1962) 830. and N.H. March, Proc. Phys. Soc. 80 [3] D. Brust, Phys. Lett. 38A (1972) 157. [4] D. Brust, Phys. Lett. 40A (1972) 255.
[5] V.N. Z~rkov,V.P. Trubitsyn, l.A. Tsarevskiy and A. Makalkin, Izv. Earth Phys. 10 (1974) 7.
[6] D.J.
Stevenson, Phys. Rev. 12 (1975) 3999; Phys. Lett. 58A (1976) 282. [7] C.A. Sholl, Proc. Phys. Soc. London 92 (1967) 437. [8] E.G. Brovman, Yu Kagan and A. Kholas, Soy. Phys. JETP 34 (1972) 1300; 35 (1972) 783. [9] i. Hammerberg and N.W. Ashcroft, Phys. Rev. B9 (1974) 409. [10] T. Nakamura, H. Nagara and H. Miyagi, Prog. Theor. 63 (1980) 368. and T. Nakamura, Prog. Theor. [11] Phys. H. Miyagi, H. Nagara Phys. 63 (1980) 1509. [12] H. Miyagi and H. Nagara, Prog. Theor. Phys. 64 (1980) No. 3. [13] H. Miyagi, to be published. [14] J. Hama, private communication.
PHYSICS LETTERS
27 October 1980
GROUND-STATE ENERGY OF HIGH-DENSITY HELIUM H. NIKI, H. NAGARA, H. MIYAGI and T. NAKAMURA Department of Material Physics, Faculty of EngineeringScience, Osaka University, Toyonaka, Japan 560 Received 17 July 1980
Ground-state energies of the high-density helium are calculated for fcc and bcc structures, up to fourth order in the structural expansion. The result manifests that bcc is much stabler than fcc in the region of high density.
A number of theoretical studies have been done for high pressure behavior of solid helium [1—5].The transition pressure from insulating to metallic phase was predicted to be ~95 Mbar [4]. The high-density helium is of astrophysical interest; its solubiity into
screened electron—ion interaction. Here the matrix element associated with the momentum transfer g of electrons is given by
metallic hydrogenof may be important internal structure some planets [6].for studying the All of the existing studies [1—4] assumed fcc as the ground-state structure. In this letter, however, we report that bcc helium is much more stable than fcc at high density and at zero temperature. Consider the system of atoms with atomic number Z. Generally the ground-state energy E of the considered system is written as
v1(g) = —(8/3ir)(ar )~g
EEeg+EM+Est.
(1)
Here Eeg denotes the energy of electron gas, EM the Madelung energy due to charge of atomic nucleus, and E5~the structure-dependent electronic energy, all per electron and in Rydberg units. Now, in the atomic units we introduce r5, which denotes the radius of sphere equivalent to volume per electron. Then Eeg is a function r5, being independent of structures. And 2/3aof r1 ~ E =Z M
M
~
‘
‘
‘
with a~the Madelung constant. We note that the aMs are identical to each other for bcc and fcc up to the four significant figuresbcc [7].and Then, Z the first energy difference between fcc for mustsmall come from E 5t.
Thus we look into E5~by using the structural expansion [8—il]. In ref. [11], E5~is expanded in the 428
ui&) =
u1(g)/e~(g,0),
2
(3) (4)
,
where g denotes the reciprocal lattice vector in units of the Fermi momentum (11 = 1), e~(g,0) the static Lindhard dielectric function, and a = (4/9iT)h/3. The relative stability of bcc to fcc can most simply be observed in the primary term of the expansion. It is given by 1 Iv 1 (g) 2 E2~~-g
v2(g)
[1—1/e~(g,O)]
,
(5)
where u2(g) is the electron—electron coupling and has a similar expression to u1(g). We notice here that e~(g,0) increases as g becomes shorter. For Z = 1 the shortest g-vectors are equal to 2.280 for bcc and to 2.2 16 for fcc. Here the numbers of shortest g-vectors (12 for bcc and 8 for fcc) are more effective than E than their length in producing E2 [bcc] lower 2 [fcc]. However the longer g-vectors contribute to E2 considerably and as resultants we have E2 [fcc] lower than E2 [bcc]. Let us now consider the case of Z = 2. Generally the relevantg-vectors are ob3. tamed from those g-vectors for Z = 1 by a scaling factorZ~I Then the shortest under consideration are fairly smaller than 2 for both structures, where the second shortestg-vectors are still larger than 2. Then the contributions from shortest g-vectors are most ef-
Volume 79A, number 5,6
PHYSICS LETTERS
fective in determining the relative stability. Thus we have E2 [bcc] lower than E2 [fcc], with difference
We utilize them to figure out E. The fourth-order
0.0084 at r5 = 1.0. Contributions from the higher order terms work to widen the energy separation produced by the primary term, at least up to fourth order. In fig. 1 we show our fourth order result by broken lines. The terms taken into account are in accord with ref. [11], and materials needed for evaluation are given in refs. [10—13].In the estimation the correlation energy of electron gas is taken into account in the Noziéres— Pines interpolation formula. An expansion has been proposed [10,11] in powers of eq. (3) with replacement of e~(g,0)by e(g, 0), the exact static dielectric function. This expansion seems to give us a more rapidly convergent series than that with the use of e~(g,0). Here the expanded terms are much more simplified. However, a fourth-order term arising from combined action between the vertex function and the self-energy still remains, though it was overlooked in refs. [8,9]. Accurate estimates of e(g, 0)are now availabl~in the region of rs ~2.0 [13]. AE2
E(Ry)
=
0.7
0.8
0.9
1.0
r5
27 October 1980
result thus obtained is shown in fig. i with solid lines. In the same figure, we also show the fcc result by Simcox and March [2] by a dotted line. That is near our result in third order. Thus the fourth-order result is remarkably different from the third order one. In our fourth-order result the energy separation between fcc and bcc is still large, in the region of r5 1.0: For both structures the energy curves must tend to the same limit, E —2.90 for free atom, as r5 increases. Ef. fect of the neglected terms including Hubbard’s H-graph [9] must be examined to improve the present result. In fig. 2 we show the resulting curves for pressure (P)versus molar volume (Vm). Those curves are compared with that by Zharkov et al. [5], which is based on the QSM method, namely a modified TFD method.
1~M_bar)
100
1.1
\\..
..,.,.
30
\\...
Zharkov et a
20
~.
-2.0
/
.....
~
N
bcc
Simcox-March
N
Sirnc~-March fcc
5 .
2
-2.5
I
Fig. 1. Ground-state energy (E) versus rs.
04
06
~
08
3) Vm (cm 1~0
Fig. 2. Curves for pressure (F) versus molar volume (Vm). The solid lines are the results obtained from the solid ones in fig. 1. The other curves from the broken ones in fig. 1 may be omitted.
429
Volume 79A, number 5,6
PHYSICS LETTERS
Notable deviation occurs at r5 ~ 0.95 which corre3. sponds ~0.64 cm NowtotheVm energy difference between fcc and bcc amounts to 0.1 per atom at r 5 = 1.0. This is quite a large figure compared with the zero-point energy difference, which is estimated from the compressibility and proves to be at most one order of magnitude smaller. There is no question for bcc helium to be stabler than fcc one, in the high density region. This result may change a previous estimation for the transition pressure to the metallic state [4]. We finally mention that the transition pressure for bcc is lower than that for fcc [14]. References [1] C.A. ten Seldom, Proc. Phys. Soc. A70 (1956) 97, 529.
430
27 October 1980
[2) L.N. Simcox (1962) 830. and N.H. March, Proc. Phys. Soc. 80 [3] D. Brust, Phys. Lett. 38A (1972) 157. [4] D. Brust, Phys. Lett. 40A (1972) 255.
[5] V.N. Z~rkov,V.P. Trubitsyn, l.A. Tsarevskiy and A. Makalkin, Izv. Earth Phys. 10 (1974) 7.
[6] D.J.
Stevenson, Phys. Rev. 12 (1975) 3999; Phys. Lett. 58A (1976) 282. [7] C.A. Sholl, Proc. Phys. Soc. London 92 (1967) 437. [8] E.G. Brovman, Yu Kagan and A. Kholas, Soy. Phys. JETP 34 (1972) 1300; 35 (1972) 783. [9] i. Hammerberg and N.W. Ashcroft, Phys. Rev. B9 (1974) 409. [10] T. Nakamura, H. Nagara and H. Miyagi, Prog. Theor. 63 (1980) 368. and T. Nakamura, Prog. Theor. [11] Phys. H. Miyagi, H. Nagara Phys. 63 (1980) 1509. [12] H. Miyagi and H. Nagara, Prog. Theor. Phys. 64 (1980) No. 3. [13] H. Miyagi, to be published. [14] J. Hama, private communication.