- Email: [email protected]
Depolarization of light scattering by atomic and molecular solutions with strongly anisotropic translational-orientational fluctuations
Received 2 k&r 1971
T~~~~o~-o~en~ti~n~
fiuctuations are’showMo raise &lower the depolarktion
of Iight scattered by so-
lutions ac~rding to the structure of the ne+r&der regioq of anisotropic molecules. In solutions,of atoms, iranslati~nal ~u~~~tio~~ induci: anistitropic scattering &ready described in the bi-molecular radial cotrelations approximaCon; in accordance with the Latest experimexk, this scattering undergoes ti increase or decrease depending on ternary radial correlations. A general expression fox the effective cCpticaianisotropy of the nekrder region in sofutions is applied to the dilute case permking, in a,novel manner, determination of both the v&e anb s&n of the anisotropy for isolate4 &lute mokcules.
.‘l. Introductkln Ikpokization of l&t scattered by an inter&ion-less gas is known to be due solely. to the intrinsic anisotropy of the isolated molecule [l] , In real gases, liquids [2] an6 real solutions [3 3 , the depolarizing agent generally resides in an effective optical anisotropy of the region of near ordering, where various statistical fluctuations take place. On neglecting internal interference as well as frequency and spatial dispersion, this effective anisotropy can be detied,:fdr multi-component systems, as f3,4] :
where we have, ikroduded ‘rhe deviation’tensor of’&tical ino1e~uhr.rpolarizability: ..
:
(_
Above, rn@@ is the electric‘dipole moment’iuduced ti mole&e p of-species i (immersed in a medium with a. :’ XC@ number of’decdes N= r;iNi) by the electric field E of the incident light wave; Se@is KxonccKer’s unit. tens&, the summation is ovei recurring indices.& and p, and ( 1 stands for’statistical averaging iiz the prese&&f ,.. _,. ,.“.I. \ ... :‘, ti0ik.h correlatiork Thk deviation te,ritir (2) is defmed to vanishfoi isolated atok’and is&opi&.iy poia&ble’molecbIes i.e. when a line& vector function RI@? = ani” holds; wbe$$Pi, dencjtes the, SC&~ op.&al po~a~ab~~‘.of n$ecule p of --
-..
_.
;_,,
~~~~~~‘~o~‘~~‘~~~~of _.:. .. :
~
,.. :
fhysicj
,.‘:.
‘.
A;Mi~~~~.ULiiYetSiiY,Poma;i,P~~~., . .I.
Volume 10, number 5
CHEMICAL PHYSICS LETTERS
I September 1971
species i. The deviation tensor (2) and consequently the effective anisotropy (I) differ from zero only if: ,_(i) the.moIecuIes are intrlnsicaliy anisotropic in the ground state, (ii) if isotropically pohu-izable atoms or molecules interact subject to translational Kirkwood-Yvon fluctuations,
or’if the molecular interactions perturb their intrinsic polarizabilities [2] in accordance with the Jansen-Mazur [S] ,madel, and elec(iii) if the atom s or molecules are subject to non-linear poiarizability in the rime- and ~ati~~~uctua~g tric field of n~i~bour~g molecules [3,6] .
2. Angular correlations For optically inactive molecules with intrinsic anisotropy given by the linear polarizability tensor a$;?, the deviation tensor (1) becomes simply, in the absence of molecular fields, D$‘(O)
= a$)
- Q. SLIP,
(3)
whence the effective anisotropy (1) can be written in the fomr of an expansion:
where
is the optical ~i~~o~ of the isolated molecule of component i at molar fractionfi. The parameter Je describes angular correlations betyeen nroiectdes of species r’andi; for simpkity, it for the case of axially-syinmetric molecules (when $1~ 02; # ~$2 and ri = L&$- ayi) [3] : JA=P
if
2v
rr
(3 cos2 e,yq)-
l)g’?‘(~ ‘I
T )dT dT
P’4
P
4’
we adduce
(6)
g!?)(r ,T ) denoting the binary correlation function for molecules p and q of species i andi at configurations rP zd P w%h their symmetry axes subtending the angle 6$4); p = N/V is the number densiv of molecules in the me&m of volume Y. The expansion (4) applied to a two-component solution yields [3] :
whence the anisotropy of the solution ii found to be an additive property only-as long as ang&r correlations are absent. If present between like molecules (@I and.&) or between ones of different species (&>, angular corielations cause a deviation from addivity in I’; proportional to the square of the molar fractions. Numericaily, the angular-correlation.parameterS ,$ can be determined experimentally in pure liquids [7-91 and solutions [lO--131 and calculated ~eore~c~y 4or weak correlations [2,3] and for ‘he very strong angdfar correlations existing in benzene and nitrobenzene’ [4]. In the two l.imiting cases fl4j of,mofecuIes correlated (i) into parallel pairs @amtropism, J$ = z) and partners (diatropism, .I$’ = -z/2), one cg3 gain information ,(ii) into pairs 0 f rru~tually perperdicuh regarding the number z of.nearest neighbdurs. ‘.
where r7@61) =‘-$(3?&
rPsB-r”,,&,,,) is the tensor of interaction bepeen dipoles induce; in moiecu~es~and
: ,s al distancerp~.
Insertion oieq. (8) into eq. {I) yields, in addition to the a~satro~y (4), yet tiother rinisotropy due to transiaiional and o~enta~oR~ quotations:, ,‘.
Bpi +nd Bqi denoting the anglesbetween the symmetry axes of mole~les p and q (of speciesi andj) and the vector rpbq,respectively. On considering’theform of the co~~~a~anfa~tor~~~~~.~~(i2> wz note that the contibution (XI) is &n-zero anly,if correlations of the orientational kind are pre,sentand that this co#ribution is positive or neg$ive ackording : to tihether the mol!cules tend to com$ne into roughly pamtropic or’dia@opicpa@ [4] .‘. ; l%e anisotropy Ij?j?j/k is given explicit@in the apptkdix.
The second approbation wil.ibe&r&d&d h&e on& after neglec@g i&&& &isotrop~ i,e; whin the die1 ation(3) y&&hes;in eq. c&jthe sole ~~n~ibutio~to the effective arG.&opi (f~isth&n d& 6 puiely translational : fhyiuations:
CHEMICAL’PHYSICS LETTERS
Voiume’lO,number 5
L September 1971
involving the bi-molecular radial correlation factor JR = ij
21 j-jr-6 V
(2)’ pq gfj
(Tp’Tq)drpdrq. )
which is subject to calculation uslng~Ki.rkwoodls rigid-sphere model 161 and the general LeMard-Jones potential [15]. The next contribution, that with three-molecule correIations, is (neglecting anisotropy) of the form: = 6(ojc7&+oi$
I’;,,
ak+af “i+)J$
,
(15)
with the ternary radial correlation parameter of the form: Jtk
=T
P2 .JlJ
wpq*‘4d2-r2PQ r2qs ] r5+ P4 qs
g!!)(r Jlk
7
P? 4’
7
)dr dr dr
s.
P
4
s ’
g${r,, rq, rs) denoting the ternary correlation function for three moleculesp, ~7and s of species r’,j and k at con,’I figurations rp, rq and TV. The ternary correlation parameter (16) deserves particular attention since it can be positive or negative, where; as the parameter (14) is always positive. When the vectors ‘pg. and rqqsare approximately parallel (as+mmetric structure of nearest ordering), Jzk is positive, raising the effective anisotropy. If rpq is almost perpendicular on fqs or R is negative thus lowering the effective anisubtending the angle n/3 (symmetric structure of nearest ordering), Jiik sotropy. The latter case is close to reality (especially in atomic substances) since with increasing density the structure of near order tends to become more and more symmetrical, and effective anisotropy has to decrease. TkA conclusion is confiied by the latest measurements of the spectrum of the depolarized light scattered by liquefied [ 16-181 and highly compressed [ 191 rare gases.
5. Dilute solutions By combining eqs. (4) and (9) we obtain a general expression for the effective anisotropy of muki-component systems:
where the first term expresses the additivity of anisotropies of the is.o!ated molecules of the components in mixture. The higher tel?ns defme deviations from addivity in the effective anisotropy arising from bi-moIecular, trimolecular and multi-molecular correlations of the radial and angular kinds. On applying expansion (17) to dilute solutions’as done by us in the Kerreffect theory of multi-component systems. [20], one obtains for inftitely high dilution: CT; ( ~f"o:~, )
;-
1+ z$ I';l+r:ll +...)t2(r:i-~~1)t3(~12-~:ll)f...)i_o, fq(
(18)
:
where f = f2is the molar fraction of solute_ If the solvent (index 1) consists of intrindcally non-anisotropic molecules (7I = 0), eq. (Is) yiehls with regard .to the previous results (6), (lo), (13) and (15),
519
Volume 10, number 5
+ 1g
b#$
1
CHEMICAL PHYSICS LZM-ERS
+a2@12
- 34
J;J
1 f-+0.
..
septe+er
1971
(19)
* p2/p1, when considering ca&s in which a1 ? a2 the effects of pure redistribution = &lJg, cancel out mutually and we obtain by eq. (19), for‘the optical anisotropy of a’solute molecule at infinite dilu-
Isinw J&/.$~
tion,
(20) Consequen~y, in determinations of valnes of the optical anisotropy of axially symmetric molecules from measurements of light scattering by ‘infinitely dilute solutions, an important’role belongs to solvent-solute molecular transiationaI--orientationd.cross fluctuations whcse positive or negative contribution, to 7:_ is considerable. Moreover, obviously, eq. (20) can provide a new method for determining not only the value but also the sign of the optical anisotropyl.of axially symmetric molecules, since J& can be positive or negative 143. In fact, as.confiYied by the latest experiments, anisotropic molecules whik continually regrouping themselves in the fluctuational region of near ordering, at the same ti..e.reorient themselves with respect to each other causing an increase or decrease in the depolarization of scattered light. In atomic and molecular solutions lacking intrinsic optical anisotropy, the effective anisotropy induced by fluctuations in bi-molecular regrouping can undergo a lowering or an enhancement under the influence of multi-molecular regroupings in the near or&r region.
Acknowledgements The author wishes to thank most haarttiy Professor P. Bothorel and Dr. J.R. Lalanne for their kind interest
and for~discussing certain aspects of this paper and K. Flatau, for the translation into English.
Appendix ~Si.ncethe polarizability clz)=
tensor for molecules with axial symmetry (unit vector S) is
(aPi- f7Pi) 6,@ + yPi SF” SF’), ,
we get with regard to eqs. (1) and (8) (in an approximation
(A-1) linear in the interaction tensor T):
VoInme 10, number .5
CHEMICAL PHYSICS LETTERS
f September 1971
In bi-mofcciilar correlation approximation we obtain.from eq. (A.2) expressions (4), (6) and (9)~(12). In the case of a solution where the molecules of component 1 have zero intrinsic anisotropy and those of com-ponent 2 are axially symmetric eq. (A.2) leads to the contribution
describing =W
ternary correlations between Iwo isotropic molecules and one anisotropic one. rL% 12 and %A 112 defme the influence of a solvent on the optical anisotropy of mote&es
References fl] H.A.Stuart, Mstek~fstruktur (Sprirkger, IBerlin, 1967). f2] S.KieIich, J. Chem. Phys. 46 (1967) 4090 and references therein. [3] S.Kietich, Acta Phys. Polcn. 19 (1960) 573; 20 (1961) 83; 33 (1968) 63: J. Phys. (Pa.&) 29 (1966) 619. f4] S.Kiel.ieh, J.R.Lalanne and F.B.Martin, Proc. Conf. on Light Scattering, Paris (1971). [S] L.Jansen and P.Mazur, Physica 21 (195.5) 193,208. [6] SXieiich, Chetn. Phys. Letters 2 (1968) 112. [71 G.I)ezeLic, J. Chem. Phys 45 (1966) 185. f 8 j ,BXasprowicz and SXieLich, Acta Phys. Pobn. 33 (1968) 495. (91 J.R.Lalanne? 3. Phys. (Paris) 30 (1969) 643. 1101 S.Kieiich and M.Surma, Poznari Sot. Friends Sci;, Mathematical and Naturai Sci f I (1962) lS3; M.Surma, Acts Phys. Polon. 25 (1954) 485. f 111 BFechncr, Acta Phys. Poion. 36 (1969) 297. Cl21 CSuch, C.Cleinent and P.Bothorei, Compt. Rend. Acad, Sci. (Paris) 271C (1970) 225; : CSuch, ThBx, Bordeaux (1970). (131 HZ-Lucas am! D.A.Jackson, Mol. Phys. 20 (t971) 801. -. (141 J.A.Prins,and W.prins, Physica 23 (1957) 253. [ 151 S.Kielich, Chem. Phys. Letters 7 (1970) 347. [ 161 J.P.McTaye; P.A.Fleury and D.B.I+Prd, Whys. Rev. 188 (1969) 303. [ 171 W.S.Gorna.l, H.E.Howard-Lock and B.P.Stoicheff,‘Fhys. Rev. IA (1970) 1288, [18] B.SimicGlavaski, D-A-Jackson and I.G.Powles, Phys. Letters 34A (1971) 25.5. [19] M.TNt%au, G.C.Tabisq B,Oksengvrn and B.Vodar, J. C&ant. Spectry. Radiative Transfer 10 (1970) 8sY; J.P.McTague and G.Birnbaum. Phys. Rev. fA (1471) 1376. 12Oi S.Kietich, Mot Phys. 5 {.I963) 49.. ‘:
in: soh~tion.
T~~~~o~-o~en~ti~n~
fiuctuations are’showMo raise &lower the depolarktion
of Iight scattered by so-
lutions ac~rding to the structure of the ne+r&der regioq of anisotropic molecules. In solutions,of atoms, iranslati~nal ~u~~~tio~~ induci: anistitropic scattering &ready described in the bi-molecular radial cotrelations approximaCon; in accordance with the Latest experimexk, this scattering undergoes ti increase or decrease depending on ternary radial correlations. A general expression fox the effective cCpticaianisotropy of the nekrder region in sofutions is applied to the dilute case permking, in a,novel manner, determination of both the v&e anb s&n of the anisotropy for isolate4 &lute mokcules.
.‘l. Introductkln Ikpokization of l&t scattered by an inter&ion-less gas is known to be due solely. to the intrinsic anisotropy of the isolated molecule [l] , In real gases, liquids [2] an6 real solutions [3 3 , the depolarizing agent generally resides in an effective optical anisotropy of the region of near ordering, where various statistical fluctuations take place. On neglecting internal interference as well as frequency and spatial dispersion, this effective anisotropy can be detied,:fdr multi-component systems, as f3,4] :
where we have, ikroduded ‘rhe deviation’tensor of’&tical ino1e~uhr.rpolarizability: ..
:
(_
Above, rn@@ is the electric‘dipole moment’iuduced ti mole&e p of-species i (immersed in a medium with a. :’ XC@ number of’decdes N= r;iNi) by the electric field E of the incident light wave; Se@is KxonccKer’s unit. tens&, the summation is ovei recurring indices.& and p, and ( 1 stands for’statistical averaging iiz the prese&&f ,.. _,. ,.“.I. \ ... :‘, ti0ik.h correlatiork Thk deviation te,ritir (2) is defmed to vanishfoi isolated atok’and is&opi&.iy poia&ble’molecbIes i.e. when a line& vector function RI@? = ani” holds; wbe$$Pi, dencjtes the, SC&~ op.&al po~a~ab~~‘.of n$ecule p of --
-..
_.
;_,,
~~~~~~‘~o~‘~~‘~~~~of _.:. .. :
~
,.. :
fhysicj
,.‘:.
‘.
A;Mi~~~~.ULiiYetSiiY,Poma;i,P~~~., . .I.
Volume 10, number 5
CHEMICAL PHYSICS LETTERS
I September 1971
species i. The deviation tensor (2) and consequently the effective anisotropy (I) differ from zero only if: ,_(i) the.moIecuIes are intrlnsicaliy anisotropic in the ground state, (ii) if isotropically pohu-izable atoms or molecules interact subject to translational Kirkwood-Yvon fluctuations,
or’if the molecular interactions perturb their intrinsic polarizabilities [2] in accordance with the Jansen-Mazur [S] ,madel, and elec(iii) if the atom s or molecules are subject to non-linear poiarizability in the rime- and ~ati~~~uctua~g tric field of n~i~bour~g molecules [3,6] .
2. Angular correlations For optically inactive molecules with intrinsic anisotropy given by the linear polarizability tensor a$;?, the deviation tensor (1) becomes simply, in the absence of molecular fields, D$‘(O)
= a$)
- Q. SLIP,
(3)
whence the effective anisotropy (1) can be written in the fomr of an expansion:
where
is the optical ~i~~o~ of the isolated molecule of component i at molar fractionfi. The parameter Je describes angular correlations betyeen nroiectdes of species r’andi; for simpkity, it for the case of axially-syinmetric molecules (when $1~ 02; # ~$2 and ri = L&$- ayi) [3] : JA=P
if
2v
rr
(3 cos2 e,yq)-
l)g’?‘(~ ‘I
T )dT dT
P’4
P
4’
we adduce
(6)
g!?)(r ,T ) denoting the binary correlation function for molecules p and q of species i andi at configurations rP zd P w%h their symmetry axes subtending the angle 6$4); p = N/V is the number densiv of molecules in the me&m of volume Y. The expansion (4) applied to a two-component solution yields [3] :
whence the anisotropy of the solution ii found to be an additive property only-as long as ang&r correlations are absent. If present between like molecules (@I and.&) or between ones of different species (&>, angular corielations cause a deviation from addivity in I’; proportional to the square of the molar fractions. Numericaily, the angular-correlation.parameterS ,$ can be determined experimentally in pure liquids [7-91 and solutions [lO--131 and calculated ~eore~c~y 4or weak correlations [2,3] and for ‘he very strong angdfar correlations existing in benzene and nitrobenzene’ [4]. In the two l.imiting cases fl4j of,mofecuIes correlated (i) into parallel pairs @amtropism, J$ = z) and partners (diatropism, .I$’ = -z/2), one cg3 gain information ,(ii) into pairs 0 f rru~tually perperdicuh regarding the number z of.nearest neighbdurs. ‘.
where r7@61) =‘-$(3?&
rPsB-r”,,&,,,) is the tensor of interaction bepeen dipoles induce; in moiecu~es~and
: ,s al distancerp~.
Insertion oieq. (8) into eq. {I) yields, in addition to the a~satro~y (4), yet tiother rinisotropy due to transiaiional and o~enta~oR~ quotations:, ,‘.
Bpi +nd Bqi denoting the anglesbetween the symmetry axes of mole~les p and q (of speciesi andj) and the vector rpbq,respectively. On considering’theform of the co~~~a~anfa~tor~~~~~.~~(i2> wz note that the contibution (XI) is &n-zero anly,if correlations of the orientational kind are pre,sentand that this co#ribution is positive or neg$ive ackording : to tihether the mol!cules tend to com$ne into roughly pamtropic or’dia@opicpa@ [4] .‘. ; l%e anisotropy Ij?j?j/k is given explicit@in the apptkdix.
The second approbation wil.ibe&r&d&d h&e on& after neglec@g i&&& &isotrop~ i,e; whin the die1 ation(3) y&&hes;in eq. c&jthe sole ~~n~ibutio~to the effective arG.&opi (f~isth&n d& 6 puiely translational : fhyiuations:
CHEMICAL’PHYSICS LETTERS
Voiume’lO,number 5
L September 1971
involving the bi-molecular radial correlation factor JR = ij
21 j-jr-6 V
(2)’ pq gfj
(Tp’Tq)drpdrq. )
which is subject to calculation uslng~Ki.rkwoodls rigid-sphere model 161 and the general LeMard-Jones potential [15]. The next contribution, that with three-molecule correIations, is (neglecting anisotropy) of the form: = 6(ojc7&+oi$
I’;,,
ak+af “i+)J$
,
(15)
with the ternary radial correlation parameter of the form: Jtk
=T
P2 .JlJ
wpq*‘4d2-r2PQ r2qs ] r5+ P4 qs
g!!)(r Jlk
7
P? 4’
7
)dr dr dr
s.
P
4
s ’
g${r,, rq, rs) denoting the ternary correlation function for three moleculesp, ~7and s of species r’,j and k at con,’I figurations rp, rq and TV. The ternary correlation parameter (16) deserves particular attention since it can be positive or negative, where; as the parameter (14) is always positive. When the vectors ‘pg. and rqqsare approximately parallel (as+mmetric structure of nearest ordering), Jzk is positive, raising the effective anisotropy. If rpq is almost perpendicular on fqs or R is negative thus lowering the effective anisubtending the angle n/3 (symmetric structure of nearest ordering), Jiik sotropy. The latter case is close to reality (especially in atomic substances) since with increasing density the structure of near order tends to become more and more symmetrical, and effective anisotropy has to decrease. TkA conclusion is confiied by the latest measurements of the spectrum of the depolarized light scattered by liquefied [ 16-181 and highly compressed [ 191 rare gases.
5. Dilute solutions By combining eqs. (4) and (9) we obtain a general expression for the effective anisotropy of muki-component systems:
where the first term expresses the additivity of anisotropies of the is.o!ated molecules of the components in mixture. The higher tel?ns defme deviations from addivity in the effective anisotropy arising from bi-moIecular, trimolecular and multi-molecular correlations of the radial and angular kinds. On applying expansion (17) to dilute solutions’as done by us in the Kerreffect theory of multi-component systems. [20], one obtains for inftitely high dilution: CT; ( ~f"o:~, )
;-
1+ z$ I';l+r:ll +...)t2(r:i-~~1)t3(~12-~:ll)f...)i_o, fq(
(18)
:
where f = f2is the molar fraction of solute_ If the solvent (index 1) consists of intrindcally non-anisotropic molecules (7I = 0), eq. (Is) yiehls with regard .to the previous results (6), (lo), (13) and (15),
519
Volume 10, number 5
+ 1g
b#$
1
CHEMICAL PHYSICS LZM-ERS
+a2@12
- 34
J;J
1 f-+0.
..
septe+er
1971
(19)
* p2/p1, when considering ca&s in which a1 ? a2 the effects of pure redistribution = &lJg, cancel out mutually and we obtain by eq. (19), for‘the optical anisotropy of a’solute molecule at infinite dilu-
Isinw J&/.$~
tion,
(20) Consequen~y, in determinations of valnes of the optical anisotropy of axially symmetric molecules from measurements of light scattering by ‘infinitely dilute solutions, an important’role belongs to solvent-solute molecular transiationaI--orientationd.cross fluctuations whcse positive or negative contribution, to 7:_ is considerable. Moreover, obviously, eq. (20) can provide a new method for determining not only the value but also the sign of the optical anisotropyl.of axially symmetric molecules, since J& can be positive or negative 143. In fact, as.confiYied by the latest experiments, anisotropic molecules whik continually regrouping themselves in the fluctuational region of near ordering, at the same ti..e.reorient themselves with respect to each other causing an increase or decrease in the depolarization of scattered light. In atomic and molecular solutions lacking intrinsic optical anisotropy, the effective anisotropy induced by fluctuations in bi-molecular regrouping can undergo a lowering or an enhancement under the influence of multi-molecular regroupings in the near or&r region.
Acknowledgements The author wishes to thank most haarttiy Professor P. Bothorel and Dr. J.R. Lalanne for their kind interest
and for~discussing certain aspects of this paper and K. Flatau, for the translation into English.
Appendix ~Si.ncethe polarizability clz)=
tensor for molecules with axial symmetry (unit vector S) is
(aPi- f7Pi) 6,@ + yPi SF” SF’), ,
we get with regard to eqs. (1) and (8) (in an approximation
(A-1) linear in the interaction tensor T):
VoInme 10, number .5
CHEMICAL PHYSICS LETTERS
f September 1971
In bi-mofcciilar correlation approximation we obtain.from eq. (A.2) expressions (4), (6) and (9)~(12). In the case of a solution where the molecules of component 1 have zero intrinsic anisotropy and those of com-ponent 2 are axially symmetric eq. (A.2) leads to the contribution
describing =W
ternary correlations between Iwo isotropic molecules and one anisotropic one. rL% 12 and %A 112 defme the influence of a solvent on the optical anisotropy of mote&es
References fl] H.A.Stuart, Mstek~fstruktur (Sprirkger, IBerlin, 1967). f2] S.KieIich, J. Chem. Phys. 46 (1967) 4090 and references therein. [3] S.Kietich, Acta Phys. Polcn. 19 (1960) 573; 20 (1961) 83; 33 (1968) 63: J. Phys. (Pa.&) 29 (1966) 619. f4] S.Kiel.ieh, J.R.Lalanne and F.B.Martin, Proc. Conf. on Light Scattering, Paris (1971). [S] L.Jansen and P.Mazur, Physica 21 (195.5) 193,208. [6] SXieiich, Chetn. Phys. Letters 2 (1968) 112. [71 G.I)ezeLic, J. Chem. Phys 45 (1966) 185. f 8 j ,BXasprowicz and SXieLich, Acta Phys. Pobn. 33 (1968) 495. (91 J.R.Lalanne? 3. Phys. (Paris) 30 (1969) 643. 1101 S.Kieiich and M.Surma, Poznari Sot. Friends Sci;, Mathematical and Naturai Sci f I (1962) lS3; M.Surma, Acts Phys. Polon. 25 (1954) 485. f 111 BFechncr, Acta Phys. Poion. 36 (1969) 297. Cl21 CSuch, C.Cleinent and P.Bothorei, Compt. Rend. Acad, Sci. (Paris) 271C (1970) 225; : CSuch, ThBx, Bordeaux (1970). (131 HZ-Lucas am! D.A.Jackson, Mol. Phys. 20 (t971) 801. -. (141 J.A.Prins,and W.prins, Physica 23 (1957) 253. [ 151 S.Kielich, Chem. Phys. Letters 7 (1970) 347. [ 161 J.P.McTaye; P.A.Fleury and D.B.I+Prd, Whys. Rev. 188 (1969) 303. [ 171 W.S.Gorna.l, H.E.Howard-Lock and B.P.Stoicheff,‘Fhys. Rev. IA (1970) 1288, [18] B.SimicGlavaski, D-A-Jackson and I.G.Powles, Phys. Letters 34A (1971) 25.5. [19] M.TNt%au, G.C.Tabisq B,Oksengvrn and B.Vodar, J. C&ant. Spectry. Radiative Transfer 10 (1970) 8sY; J.P.McTague and G.Birnbaum. Phys. Rev. fA (1471) 1376. 12Oi S.Kietich, Mot Phys. 5 {.I963) 49.. ‘:
in: soh~tion.