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Calculation of the intermolecular dispersion potential for NH3-He couples
Volume 6. number 6
CALCULATION
15 September 1970
CHEMlCAL PHYSICS LETTERS
OF
THE
INTERMOLECULAR FOR
Grarpe de Physiwe
NH3 -He
DISPEMSON
POTENTIAL
COUPLES
J. BONAMY and D. ROBERT MoLBculaire, Facult@ des Sciences, 25, Besan~on, Fence
Received 8 June 1970 Revised manuscript received 25 June 1970 An ezpression for the dispersion potential.describing the interaction between NH3 and He mafecuhs is derived and an explicit calculztion of the molecular constants is performed. A comparison between ffie different contributions appearing in the intermolecular potential shows that the term in R-7 of ffie dispersion potential should be preponderant.
1. Recent studies have pointed out the necessity to take into account terms of order R-7 for the molecinteraction in pairs of polar linear molecules and spherical. molecules in order to reach a better understanding of spectral properties and more especially the properties of absorption and spectral-line broadening in these media [l-5].
ular dispersion
Such a potential term, previously introduced by Herman [6] and generalized by Duckingham [7], has been the subject of a qu~tttative study for the molecular constants of hydracid-rare gas mixtures [8,9], Recently [IO], collision broadening of the NH3 inversion spectrum by helium was investigated theoreticaliy considering only the R-6 dispersion contribution and dipole-induced dipole interaction The results appear to be about 50% smaller than the measured values. Thus, it seems interesting to check if interactions of higher order, such as dispersion inR-7 may have some effects upon the linewidth of NH3 lines. The calculation of such a term for NH3 interacting with He is the purpose of the present paper. 2. EXPRESSION OF THE R-7 DISPERSION TERM The dispersion interaction between a polar molecule theory in the form [6,8, II]:
and an atom has been obtained fram the London
where i refers to the ith orbital of the atom (here He) and j to the jth orbital of the polar molecule (here NH3); Rj is the distance between the nucleus of i and the cefiters of gravity of j, ej is the angle between Ri and the symmetry axis of the ammonia molecule, and C&’ and y&Zdepend on the molecular structures, The ammonia molecule has ten outer electrons. It was pointed out ]12] that its electronic structure could be described in terms of equivalent orbit& which are generalLy of a more localized character than molecular orbital& This Ieads to a simpler picture for the structure of the motecute. Ifi ammonia, there are one lone-pair equivalent orbital x(l).dire$ed along the axis of the moleoute and three bond-equivalent orbitals x(bi) (i = 1,2,3) directed along each NH-bond. They may be written as linear combinations of atomic orbitals cf,113,141. The coordinate system is shown in fig. 1. The interactions of the nitrogen fs-electrons are neglected [14]; SO that eaeh equivalent orbital is doubly occupied_ The electron density is given by: .
.
591
-Pi& 1. ‘Thepositive2 axis extendsoutwards from the plane of the paper, ‘The N atom is at the origin of the coo&in&? system. The plane of the three H atoms is
E’fg. 2. Cl and C&i are the centroids of orbitak XI and Xbi, C.M. is the center of mass of the molecute. IZtand &i are the distances between the centers of gravity of orbitals 1 of NH3 and i of He and $ of NH3 and i of He. R is the distance between the centers of mass of the interacting molecules, 6 is the angle between Rand the symmetry axis.
parallel to andabove theXY plane intersecting the positive. Z axis. HI is located in the 3’2 plane. (Y and JI of eq. (3) _are respectively t&angles between one of the NHbonds and the symmetry axis and between the projections pf NH& or.-yg;n theXY plane and the
. P
=.ztxd + x(b,?
i + x0%-&l f xoq2!,
(21
with: d
= COW1
w;%)
-
sinel&!N;2pz)
;
- k8inebSfIUYcOS@(N;2py) + ksinehsm@sin$@(N;2px); x(b3) 1 IrWH&)
+ k cosab$(N;2s) + k sineb COSo&(N;2p,)
- h sineb sincecos+# (N;2py) - hsinebsino sin+=%(N;2px).
(31 The values of the coefficients X, P, ~1 and eb, determined ZrornorthogonaLityconditions, are given in [13] and the atomic functions used are Slater orbitals centered on an * or N nucleus with a screening constant equal to 1.95 for nitrogen. From eq. (l), the interznolecular7 nteraction energy is a sum of energies associated with pairs of interacting orbitals, Let us Consfder the atom N and the atom He as origins of the coordinate systems~(x,y,z) and @‘,y’,z’) (fig. 2a). Let CIbe the distance between N and the center of mass of NH3 (a = O.OSS&_ After transfort+.ion from (.I$ B+}to (R, #) variables (figs, 2a and Zb) [S, S] the following expressions are obtained for u2J[cf. eq. (l)] according to the nature of equkmlent orbitals *:
._ * The index i of t?q. (If Jas beep omitted in the subsequent oalcniationa~ since xv&consider The sammation wiI& be made over j wbioh takes the foui.‘&bels I, bl, bz, b3.,
only one orbit+
for-fle;
; 4$l’b
Ico& coso! +ainB since cosq)3 *3ay b (cos@ coscy tsins sina! cos& icosm - cotgo sinol cos 0) 6
_ +‘b,
The &ressions for ub2 and ub2 are derived from that of ubl by changing rp respectiveLy into* upf- (2v/3) and yl + (4rf3). Here d$?l,-d,?b, dil)l anddp)? are defined throu h the same reelationships as in 181. D)j , dlR;i may be carried out using the vatues of the quanThe nnm rical#alculations of Cf,, r$, andd ,, olecule defined in the charge center of orbitafs and simply retit@ pj’ 9 ,>, til), ($ for thjNR6 9%, lated to results o [ 131 and of {xi (= {zi )-for He [ 15j_ The summation over the four pairs of interacting orbital9 leads to the folloting expression of the dispersion potential: r
where
cbfd(f)b _dfl)b sin3cr .l. 1 6
A; =
1’
(10)
The numerkal value of the constant c6 may be calcubited from eq. (26) of [6] and eqs, (2) to (6; of this paper. It is to be noted that the coefficient C6 may also be expressed from tile usual constantsof the Lennard-Jones potential by 4& or from perturbation theory [7] by:
fil) whore Ux, U2 and al, 1 and%
9
are, resp&veIy,
the ionization potentials and poiar&biIities
of maLecules
3. RRSULTS AND IXSCUSSION Using eqs. (1) and (41 to (101 and numerical results of [13,15-l?] .
one obWns the foIlowing values: 593
. .
‘.
1
:
,
I..
.
.
.
.
. ‘
.
.
_Volume 6, number 6
CHEMICAL
PHYSICS
LETTERS
..
-.
16 September
1670
! if)
= 6.234A;
= -0.014A;
$l’
A; = - 0.810 x 1O-32 cm4;
C6 = 26 x 16-”
erg‘&‘;
is = -6.687.
(12) The accuracy of such numerical values obtained through the.v:rriational method has been previously discussed [S, 81. TO date and to the authors’ knowledge, no experimental determination of the parameter C6. has been made, .so that a comparison is only possible between the value obtained above and the one derived from eq_ (11). This last one is C6 = 9 x 10-66 erg cm6. Although C6 differs by a factor of 3 from the result of the present calculations, it is to be noted that the perturbation method generally leads to values for the parameter smaller than the experimental ones [183. It should be noted that the expression (5) contains an additional term which depends upon the angle cp which denotes the rotation of the moIecule around its symmetry axis. This term characterizes the symmetric-top molecules. Furth_e&more, it may be seen that the expression for the dispersion potential obtained in eq. (5) has the same angular dependence aS the expression derived by Aigrot et aL 1191 who extended Buckingham’s calculations particularly to the case of symmetric-top molecules. However, no quantitative evaluation of molecular constants was obtained by these authors. A comparison between the different contributicns appearing in the intermolecular potential may now be made for the case of interactions NH3-He, taking into account only the angular parts of each of them. The expression for the intermolecular potential (up to terms in R-7) of Buckingham may be written as a sum of electrostatic, induction and dispersion potentials: 24 = Ue!ec
+ Uind
+
Udisp-
In the case considered: Uelec = 6 ;
Uind ’ Udipole-induced
The expressions “dipole-induced
dipole + Ziquadrupole-induced
dipole ;
(1) (2) UdiSp = “disp + Udisp’
for the terms above are:
dipole = -0.67 x 10-(j” @cos2a
~$1;~ = 3.5 x i0-6ow6c0.52a
;
; u quadl-upole-in&ced
u(‘) =- 37 x lo-68 disp
r7cOsa
dipole
= O-25 X lo_68 Rm7 COS36;
+ 26 x lo-68R-7~0~3e
- 13 x 10mtj8R-7 sin38 cos.3 cp, where the numerical values of the coefficients, derived from [lS] and [17], are given in E.S. cgs units. It is clear that, for a variation of A between 3 and 7A, the first value being close to the kinetictheory diameter and the second one being the mean distance between molecu!es for a pressure of 1 atmosphere, and a temperature of 20°C, the ~(2) -part of the dispersion potential is numerically important. Moreover, it is to be noted that this co%%bution contains two terms, odd in the angle-function COSa, and larger in magnitude than the ~u&rupo!e_in&ced &pole-P art of the potential, the only one to be considered till now. It has been shown [ZO] that such terms, oc’:din co&, play a preponderant part for the purpose of evaluating linewidths. An application of these calculations to the problem of rotational-lines broadening of ammonia is in progress. Actually, the presence of molecular hydrogen, ammonia and methane in the atmosphere of Jupiter has been definiteIy_established. That of helium seems very probable,- although to date no measurement has confirmed it [21]. The abundance of hydrogen and heHum would amount to 99% of the total, that of NH3 and CH4 to less than 0.1% [21,22]. A detailed study of the formation-of the NH3 lines under Jovian conditions seems therefore interesting in connection with the present results. The half-widths of the NHg-lines depend upon the states involved, the composition of the broadening gas and the temperature. The perturbing mechanism is dif(where the dipole-dipole interaction is predominant) from that of ferent for ammonia-self-broadeni n% as shovrn in this ,papcr) or by H2 (diljoi_eiquadrupole). This leads to _ .broadening.by He (dispersi&in Ra different dependence of the .linewiilths on quantum states and temperature. One may hope that such a study would give informations about physical properties of Jupiter’s atmosphere [23]. . I, __ . 594
Vohune 6, nmmber 6
.
The tiutbors would like:to thank &ofessor manuscript.
;lJ D. Rcbart
.and L. C&&y.
E. c;alatry for stimulating discussions
IS September 1970
and for reading the
Chem. Phys. Letters 1 (1967) 399: 1 (1968) 526.
[2] M. GiraudandD. Robert, Chem.Phys.Letters2 (1968)551.
C. G.Gray, J. Chem. Phys. 50 (1969) 549; Ph. D. thesis, University of Toronto 0967). M. Girazzd, D. Robert and L. Galatry, J. Chem. P&s., to be p~iished. R. Ii. Tipping and R. M.Herman, 3. f&ant. Spectry. Radiative Transfer, ta be publbhed~ R.M.Eersnan, 3.Chem. Phys. 44 (l966) 1346. A. D. Buckit~gham, Advan. Chem. Pbys. 12 (1967) 107; 3. Chem. Phys. 48 (1968) 3827. D. Robert, C. Giraidet and L. Galatry, Chem. Phys. Letters 3 (1969) 102. C. Girardet, D. Robert and L. Galatry, Compt. Rend. Acad. Sci. (Paris) 270 (1970) B481. J. 9. Murphy &nd J. E;Boggs. J. Chem. Phys. 50 (1969) 3320. F. London, J. Phy$. Chem. 46 0942) 305. J-H. van V&ok and A.Sherman, Rev.Mod.Phys. 7 (lS35) 167. A. B. F,Duncan and 3. A,PopIe, Trans. Faraday See. 49 (1963) 217. A.B. F.Duncan, .J. Chem. Phys. 27 (l957) 423. D. Ft.Hartree and W. Hartree, Proc. Roy. Sot. A366 4938) 450. H.Margenau and N. R. Kes’.xr, Theory of intermolecular forces (Pergmnon Press, New York, 1969) ch. 2. J. 0. Birschfalder, C. F; Curtiss and R. B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1967). C. Girardet and D. Robert, J. Mol. Struct., to be published. M. Aigrot, Y. F. Le Men and L. GaLatry, Ann. Sci. Univ. Besangon (1970). P. W. Anderson, Phys. @v. 80 (1960) 511. M. B. i%fcEIroy, J. Atmospheric Sci. 26 (1969) 798. 3. S. Hogan, S, I. fiasool and T. Encrenaz, J. Atmospheric Sci. 26 (1969} 898. D. P. Cruikshank and A.B. Binder, Astrophys. Space Sci. 3 @969) 347.
CALCULATION
15 September 1970
CHEMlCAL PHYSICS LETTERS
OF
THE
INTERMOLECULAR FOR
Grarpe de Physiwe
NH3 -He
DISPEMSON
POTENTIAL
COUPLES
J. BONAMY and D. ROBERT MoLBculaire, Facult@ des Sciences, 25, Besan~on, Fence
Received 8 June 1970 Revised manuscript received 25 June 1970 An ezpression for the dispersion potential.describing the interaction between NH3 and He mafecuhs is derived and an explicit calculztion of the molecular constants is performed. A comparison between ffie different contributions appearing in the intermolecular potential shows that the term in R-7 of ffie dispersion potential should be preponderant.
1. Recent studies have pointed out the necessity to take into account terms of order R-7 for the molecinteraction in pairs of polar linear molecules and spherical. molecules in order to reach a better understanding of spectral properties and more especially the properties of absorption and spectral-line broadening in these media [l-5].
ular dispersion
Such a potential term, previously introduced by Herman [6] and generalized by Duckingham [7], has been the subject of a qu~tttative study for the molecular constants of hydracid-rare gas mixtures [8,9], Recently [IO], collision broadening of the NH3 inversion spectrum by helium was investigated theoreticaliy considering only the R-6 dispersion contribution and dipole-induced dipole interaction The results appear to be about 50% smaller than the measured values. Thus, it seems interesting to check if interactions of higher order, such as dispersion inR-7 may have some effects upon the linewidth of NH3 lines. The calculation of such a term for NH3 interacting with He is the purpose of the present paper. 2. EXPRESSION OF THE R-7 DISPERSION TERM The dispersion interaction between a polar molecule theory in the form [6,8, II]:
and an atom has been obtained fram the London
where i refers to the ith orbital of the atom (here He) and j to the jth orbital of the polar molecule (here NH3); Rj is the distance between the nucleus of i and the cefiters of gravity of j, ej is the angle between Ri and the symmetry axis of the ammonia molecule, and C&’ and y&Zdepend on the molecular structures, The ammonia molecule has ten outer electrons. It was pointed out ]12] that its electronic structure could be described in terms of equivalent orbit& which are generalLy of a more localized character than molecular orbital& This Ieads to a simpler picture for the structure of the motecute. Ifi ammonia, there are one lone-pair equivalent orbital x(l).dire$ed along the axis of the moleoute and three bond-equivalent orbitals x(bi) (i = 1,2,3) directed along each NH-bond. They may be written as linear combinations of atomic orbitals cf,113,141. The coordinate system is shown in fig. 1. The interactions of the nitrogen fs-electrons are neglected [14]; SO that eaeh equivalent orbital is doubly occupied_ The electron density is given by: .
.
591
-Pi& 1. ‘Thepositive2 axis extendsoutwards from the plane of the paper, ‘The N atom is at the origin of the coo&in&? system. The plane of the three H atoms is
E’fg. 2. Cl and C&i are the centroids of orbitak XI and Xbi, C.M. is the center of mass of the molecute. IZtand &i are the distances between the centers of gravity of orbitals 1 of NH3 and i of He and $ of NH3 and i of He. R is the distance between the centers of mass of the interacting molecules, 6 is the angle between Rand the symmetry axis.
parallel to andabove theXY plane intersecting the positive. Z axis. HI is located in the 3’2 plane. (Y and JI of eq. (3) _are respectively t&angles between one of the NHbonds and the symmetry axis and between the projections pf NH& or.-yg;n theXY plane and the
. P
=.ztxd + x(b,?
i + x0%-&l f xoq2!,
(21
with: d
= COW1
w;%)
-
sinel&!N;2pz)
;
- k8inebSfIUYcOS@(N;2py) + ksinehsm@sin$@(N;2px); x(b3) 1 IrWH&)
+ k cosab$(N;2s) + k sineb COSo&(N;2p,)
- h sineb sincecos+# (N;2py) - hsinebsino sin+=%(N;2px).
(31 The values of the coefficients X, P, ~1 and eb, determined ZrornorthogonaLityconditions, are given in [13] and the atomic functions used are Slater orbitals centered on an * or N nucleus with a screening constant equal to 1.95 for nitrogen. From eq. (l), the interznolecular7 nteraction energy is a sum of energies associated with pairs of interacting orbitals, Let us Consfder the atom N and the atom He as origins of the coordinate systems~(x,y,z) and @‘,y’,z’) (fig. 2a). Let CIbe the distance between N and the center of mass of NH3 (a = O.OSS&_ After transfort+.ion from (.I$ B+}to (R, #) variables (figs, 2a and Zb) [S, S] the following expressions are obtained for u2J[cf. eq. (l)] according to the nature of equkmlent orbitals *:
._ * The index i of t?q. (If Jas beep omitted in the subsequent oalcniationa~ since xv&consider The sammation wiI& be made over j wbioh takes the foui.‘&bels I, bl, bz, b3.,
only one orbit+
for-fle;
; 4$l’b
Ico& coso! +ainB since cosq)3 *3ay b (cos@ coscy tsins sina! cos& icosm - cotgo sinol cos 0) 6
_ +‘b,
The &ressions for ub2 and ub2 are derived from that of ubl by changing rp respectiveLy into* upf- (2v/3) and yl + (4rf3). Here d$?l,-d,?b, dil)l anddp)? are defined throu h the same reelationships as in 181. D)j , dlR;i may be carried out using the vatues of the quanThe nnm rical#alculations of Cf,, r$, andd ,, olecule defined in the charge center of orbitafs and simply retit@ pj’ 9 ,>, til), ($ for thjNR6 9%, lated to results o [ 131 and of {xi (= {zi )-for He [ 15j_ The summation over the four pairs of interacting orbital9 leads to the folloting expression of the dispersion potential: r
where
cbfd(f)b _dfl)b sin3cr .l. 1 6
A; =
1’
(10)
The numerkal value of the constant c6 may be calcubited from eq. (26) of [6] and eqs, (2) to (6; of this paper. It is to be noted that the coefficient C6 may also be expressed from tile usual constantsof the Lennard-Jones potential by 4& or from perturbation theory [7] by:
fil) whore Ux, U2 and al, 1 and%
9
are, resp&veIy,
the ionization potentials and poiar&biIities
of maLecules
3. RRSULTS AND IXSCUSSION Using eqs. (1) and (41 to (101 and numerical results of [13,15-l?] .
one obWns the foIlowing values: 593
. .
‘.
1
:
,
I..
.
.
.
.
. ‘
.
.
_Volume 6, number 6
CHEMICAL
PHYSICS
LETTERS
..
-.
16 September
1670
! if)
= 6.234A;
= -0.014A;
$l’
A; = - 0.810 x 1O-32 cm4;
C6 = 26 x 16-”
erg‘&‘;
is = -6.687.
(12) The accuracy of such numerical values obtained through the.v:rriational method has been previously discussed [S, 81. TO date and to the authors’ knowledge, no experimental determination of the parameter C6. has been made, .so that a comparison is only possible between the value obtained above and the one derived from eq_ (11). This last one is C6 = 9 x 10-66 erg cm6. Although C6 differs by a factor of 3 from the result of the present calculations, it is to be noted that the perturbation method generally leads to values for the parameter smaller than the experimental ones [183. It should be noted that the expression (5) contains an additional term which depends upon the angle cp which denotes the rotation of the moIecule around its symmetry axis. This term characterizes the symmetric-top molecules. Furth_e&more, it may be seen that the expression for the dispersion potential obtained in eq. (5) has the same angular dependence aS the expression derived by Aigrot et aL 1191 who extended Buckingham’s calculations particularly to the case of symmetric-top molecules. However, no quantitative evaluation of molecular constants was obtained by these authors. A comparison between the different contributicns appearing in the intermolecular potential may now be made for the case of interactions NH3-He, taking into account only the angular parts of each of them. The expression for the intermolecular potential (up to terms in R-7) of Buckingham may be written as a sum of electrostatic, induction and dispersion potentials: 24 = Ue!ec
+ Uind
+
Udisp-
In the case considered: Uelec = 6 ;
Uind ’ Udipole-induced
The expressions “dipole-induced
dipole + Ziquadrupole-induced
dipole ;
(1) (2) UdiSp = “disp + Udisp’
for the terms above are:
dipole = -0.67 x 10-(j” @cos2a
~$1;~ = 3.5 x i0-6ow6c0.52a
;
; u quadl-upole-in&ced
u(‘) =- 37 x lo-68 disp
r7cOsa
dipole
= O-25 X lo_68 Rm7 COS36;
+ 26 x lo-68R-7~0~3e
- 13 x 10mtj8R-7 sin38 cos.3 cp, where the numerical values of the coefficients, derived from [lS] and [17], are given in E.S. cgs units. It is clear that, for a variation of A between 3 and 7A, the first value being close to the kinetictheory diameter and the second one being the mean distance between molecu!es for a pressure of 1 atmosphere, and a temperature of 20°C, the ~(2) -part of the dispersion potential is numerically important. Moreover, it is to be noted that this co%%bution contains two terms, odd in the angle-function COSa, and larger in magnitude than the ~u&rupo!e_in&ced &pole-P art of the potential, the only one to be considered till now. It has been shown [ZO] that such terms, oc’:din co&, play a preponderant part for the purpose of evaluating linewidths. An application of these calculations to the problem of rotational-lines broadening of ammonia is in progress. Actually, the presence of molecular hydrogen, ammonia and methane in the atmosphere of Jupiter has been definiteIy_established. That of helium seems very probable,- although to date no measurement has confirmed it [21]. The abundance of hydrogen and heHum would amount to 99% of the total, that of NH3 and CH4 to less than 0.1% [21,22]. A detailed study of the formation-of the NH3 lines under Jovian conditions seems therefore interesting in connection with the present results. The half-widths of the NHg-lines depend upon the states involved, the composition of the broadening gas and the temperature. The perturbing mechanism is dif(where the dipole-dipole interaction is predominant) from that of ferent for ammonia-self-broadeni n% as shovrn in this ,papcr) or by H2 (diljoi_eiquadrupole). This leads to _ .broadening.by He (dispersi&in Ra different dependence of the .linewiilths on quantum states and temperature. One may hope that such a study would give informations about physical properties of Jupiter’s atmosphere [23]. . I, __ . 594
Vohune 6, nmmber 6
.
The tiutbors would like:to thank &ofessor manuscript.
;lJ D. Rcbart
.and L. C&&y.
E. c;alatry for stimulating discussions
IS September 1970
and for reading the
Chem. Phys. Letters 1 (1967) 399: 1 (1968) 526.
[2] M. GiraudandD. Robert, Chem.Phys.Letters2 (1968)551.
C. G.Gray, J. Chem. Phys. 50 (1969) 549; Ph. D. thesis, University of Toronto 0967). M. Girazzd, D. Robert and L. Galatry, J. Chem. P&s., to be p~iished. R. Ii. Tipping and R. M.Herman, 3. f&ant. Spectry. Radiative Transfer, ta be publbhed~ R.M.Eersnan, 3.Chem. Phys. 44 (l966) 1346. A. D. Buckit~gham, Advan. Chem. Pbys. 12 (1967) 107; 3. Chem. Phys. 48 (1968) 3827. D. Robert, C. Giraidet and L. Galatry, Chem. Phys. Letters 3 (1969) 102. C. Girardet, D. Robert and L. Galatry, Compt. Rend. Acad. Sci. (Paris) 270 (1970) B481. J. 9. Murphy &nd J. E;Boggs. J. Chem. Phys. 50 (1969) 3320. F. London, J. Phy$. Chem. 46 0942) 305. J-H. van V&ok and A.Sherman, Rev.Mod.Phys. 7 (lS35) 167. A. B. F,Duncan and 3. A,PopIe, Trans. Faraday See. 49 (1963) 217. A.B. F.Duncan, .J. Chem. Phys. 27 (l957) 423. D. Ft.Hartree and W. Hartree, Proc. Roy. Sot. A366 4938) 450. H.Margenau and N. R. Kes’.xr, Theory of intermolecular forces (Pergmnon Press, New York, 1969) ch. 2. J. 0. Birschfalder, C. F; Curtiss and R. B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1967). C. Girardet and D. Robert, J. Mol. Struct., to be published. M. Aigrot, Y. F. Le Men and L. GaLatry, Ann. Sci. Univ. Besangon (1970). P. W. Anderson, Phys. @v. 80 (1960) 511. M. B. i%fcEIroy, J. Atmospheric Sci. 26 (1969) 798. 3. S. Hogan, S, I. fiasool and T. Encrenaz, J. Atmospheric Sci. 26 (1969} 898. D. P. Cruikshank and A.B. Binder, Astrophys. Space Sci. 3 @969) 347.