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Polarization energies and quasi-free electron states of rare gas crystals
Volume 6, number 3
CHEMICAL PHYSICS LETTERS
POLARIZATION AND
QUASI-FREE
ELECTRON
I
August i9iO
ENERGIES S-TATES
OF
RARE
GAS
CRYSTALS
L. E. LYONS and M. G. SCEATS Department of Chemistry, Univmsity of Quaenshzd, St.Lucin, Queensland 4067, AustmEia
Received Revised
manuscript
16 March received
1970 28 April
1970
Crystal polarization energies are calculated at several temperatures several DreSSUreS for He. These results are used to estimate energies from spktroscopic data for Xe, Kr. and Ar.
Crystal polarization energies P have been calculated for molecular crystals [l-5] using a classical multipole theory [6,7]. The charge distribution causing the polarization is assumed to correspond with the centre of mass of an atom and is thus related to the electronic polarization energies associated with a ‘hole’ left behind after photoionization or to a charge carrier localized on a molecular site between ‘hops’ [8]. Calculation of P allows estimation of: a) an r-dependent potential function [9] for the interaction of an electron with its hole in the static approximation of an electronic polaron theory [lo, 111; b) the energy of the quasi-free electron state [12] if the band gap is known, and hence the
stability of the quasi-free electron state; c) the photoelectric threshold for the case of narrow valence bands. For solid xenon, krypton, argon and neon previously P has been calculated by Fowler [9] using the first order approximation of Mott and Littleton [13] in which the energy of interaction between point-dipoles and the parent ion was evaluated, the first sheath of molecules being considered separately. However, the values cf the lattice parameter chosen by FowIer [9] for his calculation for Xe and Ne were much less than the measured values * near O°K. For Kr, the value chosen was greater than measured values [14] at the triple point. The only realistic calculation was that for Ar, the lattice param-
Kr.
eter used corresponding
Ar and Ne and at
to experiment
at 20°K
P41-
Jn this paper the polarization energies of Xe, Kr, Ar and Ne are presented at several temxratures using realistic lattice parameters and the polarization energy of He is presented at several pressures. The calculations, performed by computer, are based on a classical multipoIe theory [6,7] using both solid and gas phase polarizabilities. The results are used to estimate the energy of the quasi-free electron state. For centrosymmetric molecules P mav be expanded in a power series of the electric field F and its gradient F [4], thus P = -$(+Y&FB”N’Fv’)
where c+p.,,is the second order polarizability tensor; ypy6E the hyperpotarizability; Qby the quadrupole moment; and p, y, 6, E denote particular components. Here YN is the distance of the Nth atom from the ion, and the sumrnatiorr is over all atoms in the lattice. The electric field F acting on atom N is given by F(rhi,
* For a review of the solid state of heavy rare gases, see ref. [14].
for Xe.
of the quasi-free electron staate
= -+N
.
(21
rN 217
‘,,‘,,
.’
.I’
.
,.
bdc
1.73
1.73 -0.176
0.204
0.2008 0.2008
-0.173 -0.173
0.1998
6.1917
1.6052
1 MJ52
1.6137
1.6137
2.40
2.436
3.90
3.973
3.993
a)
a)
-0.170 -0.170
0.003 0.003
-0.192
-0.649
-1.17
-1.15
-1.14
-1.11
-1.28
-1.26
-1.39
-1.37
-1.30
-0.174
0.004
0.014
0.15
0.23
0.14
0.21
0.33
0.32
0.49
0.47
0.42
(2)
-0.173
-0.196
-0.662
-1.32
-1.37
-1.28
-1.32
-1.61
-1.58
-1.88
-1.84
-1.72
(eV
P
Solid phase polarlzabilities
0.003
-0.196
-0.669
-0.61
-1.20
-1 .I6
-1.15
-1.12
-1.31
-1.27
-1.42
-1.37
-1.30
results
0.003
0.004
0.016
0.04
0.16
6.23
0.15
0.22
0.35
0.33
0.52
0.46
0.42
Calculated
Table 1 energies of rare gas solids
ref. (241, except for He result from ref. [25].
-0.176
0.204
-0.200
29
0.204
100
-0.705
-0.65
-1.24
-1.39
-1.19
-1.33
-1.66
-1.60
-1.94
-1. .86
-1.72
29.7
0.204
0.392
1.63
1.63
1.63
1.63
2,465
2.465
4.01
4.01
4.01
Polarization
Gas phase polwizabilities
1255
i) Extrapolated value. b) From ref. [14], .’‘c) From ref. 1221; “a) C~cu’atecl a6 in text using data from
,’
‘.,
1.73
hcp
16
hcp
fee
1
1
” Helium
,;., ,.
._
20
hCP
20
1
1
64 20
hCF
ZCC
1
1
54
20
fee
1
1
1
1
Press. (c;.nD . ,
fee
54
20
fee
fee
77
I fee
fee
146
Phase
.
Temp.. (oK)
fee
_’
‘,
‘;;
:. .’ ‘ Neon
.,_
,,-
: -,Argon
Krypton ,,
‘.
‘.
Crystal
,.
,/’
‘,
‘. Xenon
.‘,‘_
m’“,
w
k,
O°K
<
O°K
20°K
>117’K
c
Effective temp. h)
-0.68
-1.10
-1.10
-1.32
&)
Fowler’a results (Ref. [SJ)
iI
2
2
;
k?
%
;
2
B
G3
x1
ifi
m .
i
Volume 6, number 3
CHEMICALPHYSICSLETTERS
The effects of quadrupoles are negligible for molecular crystals 143. The first term in eq. (1)
gives rise to two interactions, the ion-induced dipole interaction Pm and the induced dipoleinduced dipole interaction PBB. Therefore, neglecting qua&mole moments, P=P~+PDD-I-PH,
(3)
where PH arises from the hyperpolarizability. For Ar, PH = -0.004 eV when y is 0.7 x 10-36 esu from Kerr constant data [15] and the summation is over twelve nearest neighbours*. Hence and PH is negligible. The summations for P PDD were carried out by computer [4,5 F. The input data were lattice constants and polarizabilities. If lattice polaron deformations [16] are neglected then the dimensions of the lattice around the ion are those of the bulk crystal. Xe, Kr, Ar and Ne crystallize in an fee unit cell [14], but argon has an hcp phase which sometimes coexists with the fee phase [17]. Helium crystallizes in three possible phases, a-hcp [18,19] at low pressure, /3-fee [ZO] at high pressure and y-bee [21] at low pressure. In the calculations, the polarizability was assumed isotropic for the hcp phases. As a first approximation, the polarizabilities were those evaluated for the gas phase [22,23]. As a second approximation those evaluated for the solid phase were from refractive index data For completely ionized states or for polarons with large orbits, the static dielectric constant is required to calculate the polarizability. The refractive indices of solid Xe, Kr and Ar have been measured as a function of wavelength and temperature [24]. The data were extrapolated to infinite wavelength using a Cauchy dispersion formula strictly true only for gases but often applied to solids [6]. The polarizability was then calculated using the Lorentz-Lorenz relationship. For temperatures outside the region covered by the refractive index data, a was extrapolated. For He, the solid phase‘polarizability used was theoretically calculated [25] on the statistical atom model. Because Pm converged slowly, it was caIculated for ail molecules within 3, 6 and 9 unit cells of the ion and then graphically extrapolated to infinity, except for He which has a small-polarizability wnere the tabuiated result is for three unit cells. PBB rapidly converged and so the interactions of each induced dipole with its nearest neighbours were evaluated for all molecules ,within 3 unit cells of the ion. f This is acceptable becausePHhas (see ref; [3]). _ -.
: :. -.
an?-- 6.de&ndence
1 August 1970
Calculated values of Pm, PDD and P are shown in table I for several temperatures and pressures. The results of Fowler [9] are shown for comparisonf, and although the results cannot be strictly compared, they were in fair ag-reement. The only case where the results can be directly compared with those of Fowler is for Ar at 20°K, where the two values agree to within 0.01 eV. As the temperature is increased or the pressure decreased, P changes towards zero. Polarons for the rare gas crystals derived from the highest energy filled valence band have been shown to be static polarons [9]_ The potential function [9] for interaction of an electron and a hole for the static electronic p&u-on [lo, 111 is given by V(r) = -e2/r
+ e2/7-(1 - l/c)
Ly=-
f 2P I
2eP (E -l)e2’
(41
(61
We can estimate at what distances V(r) becomes hydrogenic (viz. -e2/er) using values of Pandc.
The photoelectric threshold 1C from the filled valence bands of small width is given by IC =IG+P,
where ZG is the ionization potential for the corresponding free-atom level. The band gap 18 is given by 18 =zC + VO =lG+Pi
v9,
(7)
where VO is the energy of the quasi-free electron state [12]. For the rare gas solids ;3 can be determined experimentally from the convergence of the energy levels of X-ray Wannier excitons or Table 2 The energy of the quasi-free electron stite in solid rare gases Temperature r90
P + Vo lev)
Xenon Krypton KryptOll
20 20 70
-1.95a)
-1.39
-1.70 a) -l.‘iab)
-1.28 -I_27c)
-0.56 -a.42 -0.46
Argon
80
-1.45b).
-1.10 ct
-0.35
Medium
VO cevl
a) From X&y excitcn data. r&f. E261. b) From Xe doped matrices, ref. [ZSl: c) Estimated from values given in table 1. t Yowler uses the term “self energy- instead of “pol_arization energy*. ,_ ., -. 219 :
-.
[l - e-op]
where E is the dielectric constant and LYthe damping parameter given by
-.
-_.._
_
Volume 6. number.3
CHEMICAL PHYSICS LETTER8
Wannier excitons of doped crystals, Using calculated values of ZJ and experimental values of ZS and ZG, Vo was ,estimated (table 2), Because Vo is negative the quasi-free electron state is electronically stable. Fro is seen-to approach zero with decreasing polarizability and lattice parameter. The authors wish to thank Mr. 3. Angus for helpful discussions. The USAF, Office of Scientific Research,- Directorate of Chemical Sciences -(Gram No. AFCSR-68-1561) and the Australian Research Grants Committee are thanked for support of the programme of which this work is part.
REFERENCES [I] L.E. Lyons. J. Chem.Soc. (1957) 5001. [2] L.E. Lyons and 3. C. Mackie, Proc. Chem.Soc. (1962) 71. [3] J. C. Mackie, Ph. D. thesis, University of Sydney (1963). [4] M. Batley. Ph. 13. thesis, University of Queensland (1966). [S] L. J. Johnston, Ph. D. thesis, University of Queensland (1969). [6] C. J. F. Bottcher. Theory of electric polarization ’(Elsevier, Amsterdam. 1952). [7] A. D. Buckingham. Quart. Rev. (London) 13 0959) [8] %. Gtaeserand R. S. Berry. J. Chem. Phys. 44 (1966) 3797. [9] W.B. Fowler. Phys.Rev. 151 0966) 657.
[I-O] H. Haken and W. Schottky, 2. Physik. Chcm. 16 (1958) 218. [ll] Y. Toyozawa. Progr. Theoret. Phys. (Kyoto) 12 '(1954) 422. [12] B. R& and J. Jorlner. Chem. Phys. Letters 4
(l969) 155. [13] N. F. Mott and M. J. Littleton, Trans. Faraday Sot. 34 0.938) 485.
[14] G. L. Pollack, Rev. Mod. Phys. 36 (1964) 748. [15] A. D. Buckingham. Trans. Faraday Sot. 52 0956) 1035. [16] T. D. Lee, F. E. Low and D. Pines. Phys. Rev. 90 (1953) 297. [17] C. S-Barrett and L. Meyer, J. Chem. Phys. 41 (1964) 1078. [18] D. G. Henshaw. Phys. Rev. 109 (1958) 328. [19] J. Dugdale and F. E. Simon, Proc. Roy. Sot. A218 (1953;. 291. [%O]R. L. Mills and A. F. Schuch, Phys. Rev. Letters 6 (l961) 263. 1211J. H. Vignos and H. A. Fairbank, Phys. Rev. Letters 6 (1961) 265; A. F.Schuch and R. L. Mills. Phys. Rev. Letters 8 (1962) 469; E. R.Grilly and R. L. Mills, Ann. Phys. 18 (1962) 250; G.Ahlers, Phys.Rev. 135 (1964) AlO. [22] G. A. Cook, Argon, helium and the rare gases (Interscience. New York, 1961). [23] T.Kihara, Advan. Chem. Phys. 1 (1958) 267. (2.41A. C.Sinnock and B. L. Smith, Phys. Rev. 181 Q969) 1297. [25] C.A.Ten Seldam and S.R.de Groot. Physica 18 (1952) 905. [2S] R. Haensel, G. Keirel, P. Schrieber and G. Kunz. to be published. 1271 G. Baldini, Phys. Rev. 137 (1965) A608.
<.-
.220‘
1 August 1970
: :
t ‘- r
:
:
_-
..
: :. Y ._,._-
_’
_>. .. -_ .’
. .,
CHEMICAL PHYSICS LETTERS
POLARIZATION AND
QUASI-FREE
ELECTRON
I
August i9iO
ENERGIES S-TATES
OF
RARE
GAS
CRYSTALS
L. E. LYONS and M. G. SCEATS Department of Chemistry, Univmsity of Quaenshzd, St.Lucin, Queensland 4067, AustmEia
Received Revised
manuscript
16 March received
1970 28 April
1970
Crystal polarization energies are calculated at several temperatures several DreSSUreS for He. These results are used to estimate energies from spktroscopic data for Xe, Kr. and Ar.
Crystal polarization energies P have been calculated for molecular crystals [l-5] using a classical multipole theory [6,7]. The charge distribution causing the polarization is assumed to correspond with the centre of mass of an atom and is thus related to the electronic polarization energies associated with a ‘hole’ left behind after photoionization or to a charge carrier localized on a molecular site between ‘hops’ [8]. Calculation of P allows estimation of: a) an r-dependent potential function [9] for the interaction of an electron with its hole in the static approximation of an electronic polaron theory [lo, 111; b) the energy of the quasi-free electron state [12] if the band gap is known, and hence the
stability of the quasi-free electron state; c) the photoelectric threshold for the case of narrow valence bands. For solid xenon, krypton, argon and neon previously P has been calculated by Fowler [9] using the first order approximation of Mott and Littleton [13] in which the energy of interaction between point-dipoles and the parent ion was evaluated, the first sheath of molecules being considered separately. However, the values cf the lattice parameter chosen by FowIer [9] for his calculation for Xe and Ne were much less than the measured values * near O°K. For Kr, the value chosen was greater than measured values [14] at the triple point. The only realistic calculation was that for Ar, the lattice param-
Kr.
eter used corresponding
Ar and Ne and at
to experiment
at 20°K
P41-
Jn this paper the polarization energies of Xe, Kr, Ar and Ne are presented at several temxratures using realistic lattice parameters and the polarization energy of He is presented at several pressures. The calculations, performed by computer, are based on a classical multipoIe theory [6,7] using both solid and gas phase polarizabilities. The results are used to estimate the energy of the quasi-free electron state. For centrosymmetric molecules P mav be expanded in a power series of the electric field F and its gradient F [4], thus P = -$(+Y&FB”N’Fv’)
where c+p.,,is the second order polarizability tensor; ypy6E the hyperpotarizability; Qby the quadrupole moment; and p, y, 6, E denote particular components. Here YN is the distance of the Nth atom from the ion, and the sumrnatiorr is over all atoms in the lattice. The electric field F acting on atom N is given by F(rhi,
* For a review of the solid state of heavy rare gases, see ref. [14].
for Xe.
of the quasi-free electron staate
= -+N
.
(21
rN 217
‘,,‘,,
.’
.I’
.
,.
bdc
1.73
1.73 -0.176
0.204
0.2008 0.2008
-0.173 -0.173
0.1998
6.1917
1.6052
1 MJ52
1.6137
1.6137
2.40
2.436
3.90
3.973
3.993
a)
a)
-0.170 -0.170
0.003 0.003
-0.192
-0.649
-1.17
-1.15
-1.14
-1.11
-1.28
-1.26
-1.39
-1.37
-1.30
-0.174
0.004
0.014
0.15
0.23
0.14
0.21
0.33
0.32
0.49
0.47
0.42
(2)
-0.173
-0.196
-0.662
-1.32
-1.37
-1.28
-1.32
-1.61
-1.58
-1.88
-1.84
-1.72
(eV
P
Solid phase polarlzabilities
0.003
-0.196
-0.669
-0.61
-1.20
-1 .I6
-1.15
-1.12
-1.31
-1.27
-1.42
-1.37
-1.30
results
0.003
0.004
0.016
0.04
0.16
6.23
0.15
0.22
0.35
0.33
0.52
0.46
0.42
Calculated
Table 1 energies of rare gas solids
ref. (241, except for He result from ref. [25].
-0.176
0.204
-0.200
29
0.204
100
-0.705
-0.65
-1.24
-1.39
-1.19
-1.33
-1.66
-1.60
-1.94
-1. .86
-1.72
29.7
0.204
0.392
1.63
1.63
1.63
1.63
2,465
2.465
4.01
4.01
4.01
Polarization
Gas phase polwizabilities
1255
i) Extrapolated value. b) From ref. [14], .’‘c) From ref. 1221; “a) C~cu’atecl a6 in text using data from
,’
‘.,
1.73
hcp
16
hcp
fee
1
1
” Helium
,;., ,.
._
20
hCP
20
1
1
64 20
hCF
ZCC
1
1
54
20
fee
1
1
1
1
Press. (c;.nD . ,
fee
54
20
fee
fee
77
I fee
fee
146
Phase
.
Temp.. (oK)
fee
_’
‘,
‘;;
:. .’ ‘ Neon
.,_
,,-
: -,Argon
Krypton ,,
‘.
‘.
Crystal
,.
,/’
‘,
‘. Xenon
.‘,‘_
m’“,
w
k,
O°K
<
O°K
20°K
>117’K
c
Effective temp. h)
-0.68
-1.10
-1.10
-1.32
&)
Fowler’a results (Ref. [SJ)
iI
2
2
;
k?
%
;
2
B
G3
x1
ifi
m .
i
Volume 6, number 3
CHEMICALPHYSICSLETTERS
The effects of quadrupoles are negligible for molecular crystals 143. The first term in eq. (1)
gives rise to two interactions, the ion-induced dipole interaction Pm and the induced dipoleinduced dipole interaction PBB. Therefore, neglecting qua&mole moments, P=P~+PDD-I-PH,
(3)
where PH arises from the hyperpolarizability. For Ar, PH = -0.004 eV when y is 0.7 x 10-36 esu from Kerr constant data [15] and the summation is over twelve nearest neighbours*. Hence and PH is negligible. The summations for P PDD were carried out by computer [4,5 F. The input data were lattice constants and polarizabilities. If lattice polaron deformations [16] are neglected then the dimensions of the lattice around the ion are those of the bulk crystal. Xe, Kr, Ar and Ne crystallize in an fee unit cell [14], but argon has an hcp phase which sometimes coexists with the fee phase [17]. Helium crystallizes in three possible phases, a-hcp [18,19] at low pressure, /3-fee [ZO] at high pressure and y-bee [21] at low pressure. In the calculations, the polarizability was assumed isotropic for the hcp phases. As a first approximation, the polarizabilities were those evaluated for the gas phase [22,23]. As a second approximation those evaluated for the solid phase were from refractive index data For completely ionized states or for polarons with large orbits, the static dielectric constant is required to calculate the polarizability. The refractive indices of solid Xe, Kr and Ar have been measured as a function of wavelength and temperature [24]. The data were extrapolated to infinite wavelength using a Cauchy dispersion formula strictly true only for gases but often applied to solids [6]. The polarizability was then calculated using the Lorentz-Lorenz relationship. For temperatures outside the region covered by the refractive index data, a was extrapolated. For He, the solid phase‘polarizability used was theoretically calculated [25] on the statistical atom model. Because Pm converged slowly, it was caIculated for ail molecules within 3, 6 and 9 unit cells of the ion and then graphically extrapolated to infinity, except for He which has a small-polarizability wnere the tabuiated result is for three unit cells. PBB rapidly converged and so the interactions of each induced dipole with its nearest neighbours were evaluated for all molecules ,within 3 unit cells of the ion. f This is acceptable becausePHhas (see ref; [3]). _ -.
: :. -.
an?-- 6.de&ndence
1 August 1970
Calculated values of Pm, PDD and P are shown in table I for several temperatures and pressures. The results of Fowler [9] are shown for comparisonf, and although the results cannot be strictly compared, they were in fair ag-reement. The only case where the results can be directly compared with those of Fowler is for Ar at 20°K, where the two values agree to within 0.01 eV. As the temperature is increased or the pressure decreased, P changes towards zero. Polarons for the rare gas crystals derived from the highest energy filled valence band have been shown to be static polarons [9]_ The potential function [9] for interaction of an electron and a hole for the static electronic p&u-on [lo, 111 is given by V(r) = -e2/r
+ e2/7-(1 - l/c)
Ly=-
f 2P I
2eP (E -l)e2’
(41
(61
We can estimate at what distances V(r) becomes hydrogenic (viz. -e2/er) using values of Pandc.
The photoelectric threshold 1C from the filled valence bands of small width is given by IC =IG+P,
where ZG is the ionization potential for the corresponding free-atom level. The band gap 18 is given by 18 =zC + VO =lG+Pi
v9,
(7)
where VO is the energy of the quasi-free electron state [12]. For the rare gas solids ;3 can be determined experimentally from the convergence of the energy levels of X-ray Wannier excitons or Table 2 The energy of the quasi-free electron stite in solid rare gases Temperature r90
P + Vo lev)
Xenon Krypton KryptOll
20 20 70
-1.95a)
-1.39
-1.70 a) -l.‘iab)
-1.28 -I_27c)
-0.56 -a.42 -0.46
Argon
80
-1.45b).
-1.10 ct
-0.35
Medium
VO cevl
a) From X&y excitcn data. r&f. E261. b) From Xe doped matrices, ref. [ZSl: c) Estimated from values given in table 1. t Yowler uses the term “self energy- instead of “pol_arization energy*. ,_ ., -. 219 :
-.
[l - e-op]
where E is the dielectric constant and LYthe damping parameter given by
-.
-_.._
_
Volume 6. number.3
CHEMICAL PHYSICS LETTER8
Wannier excitons of doped crystals, Using calculated values of ZJ and experimental values of ZS and ZG, Vo was ,estimated (table 2), Because Vo is negative the quasi-free electron state is electronically stable. Fro is seen-to approach zero with decreasing polarizability and lattice parameter. The authors wish to thank Mr. 3. Angus for helpful discussions. The USAF, Office of Scientific Research,- Directorate of Chemical Sciences -(Gram No. AFCSR-68-1561) and the Australian Research Grants Committee are thanked for support of the programme of which this work is part.
REFERENCES [I] L.E. Lyons. J. Chem.Soc. (1957) 5001. [2] L.E. Lyons and 3. C. Mackie, Proc. Chem.Soc. (1962) 71. [3] J. C. Mackie, Ph. D. thesis, University of Sydney (1963). [4] M. Batley. Ph. 13. thesis, University of Queensland (1966). [S] L. J. Johnston, Ph. D. thesis, University of Queensland (1969). [6] C. J. F. Bottcher. Theory of electric polarization ’(Elsevier, Amsterdam. 1952). [7] A. D. Buckingham. Quart. Rev. (London) 13 0959) [8] %. Gtaeserand R. S. Berry. J. Chem. Phys. 44 (1966) 3797. [9] W.B. Fowler. Phys.Rev. 151 0966) 657.
[I-O] H. Haken and W. Schottky, 2. Physik. Chcm. 16 (1958) 218. [ll] Y. Toyozawa. Progr. Theoret. Phys. (Kyoto) 12 '(1954) 422. [12] B. R& and J. Jorlner. Chem. Phys. Letters 4
(l969) 155. [13] N. F. Mott and M. J. Littleton, Trans. Faraday Sot. 34 0.938) 485.
[14] G. L. Pollack, Rev. Mod. Phys. 36 (1964) 748. [15] A. D. Buckingham. Trans. Faraday Sot. 52 0956) 1035. [16] T. D. Lee, F. E. Low and D. Pines. Phys. Rev. 90 (1953) 297. [17] C. S-Barrett and L. Meyer, J. Chem. Phys. 41 (1964) 1078. [18] D. G. Henshaw. Phys. Rev. 109 (1958) 328. [19] J. Dugdale and F. E. Simon, Proc. Roy. Sot. A218 (1953;. 291. [%O]R. L. Mills and A. F. Schuch, Phys. Rev. Letters 6 (l961) 263. 1211J. H. Vignos and H. A. Fairbank, Phys. Rev. Letters 6 (1961) 265; A. F.Schuch and R. L. Mills. Phys. Rev. Letters 8 (1962) 469; E. R.Grilly and R. L. Mills, Ann. Phys. 18 (1962) 250; G.Ahlers, Phys.Rev. 135 (1964) AlO. [22] G. A. Cook, Argon, helium and the rare gases (Interscience. New York, 1961). [23] T.Kihara, Advan. Chem. Phys. 1 (1958) 267. (2.41A. C.Sinnock and B. L. Smith, Phys. Rev. 181 Q969) 1297. [25] C.A.Ten Seldam and S.R.de Groot. Physica 18 (1952) 905. [2S] R. Haensel, G. Keirel, P. Schrieber and G. Kunz. to be published. 1271 G. Baldini, Phys. Rev. 137 (1965) A608.
<.-
.220‘
1 August 1970
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