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INTERACTION-INDUCED-LIGHT-SCATTERING AS
A PROBE OF MANY-BODY CORRELATIONS IN
FLUIDS
F. Barocchi
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C9, supplément au n012, Tome 46, décembre 1985 page C9-123
INTERACTION-INDUCED-LIGHT-SCATTERING AS A P R O B E OF MANY-BODY CORRELATIONS IN FLUIDS
F. Barocchi
Dipartimento d i F i s i c a , U n i v e r s i t à d i Firenze, Largo E.Fermi, 2 Firenze, ItaL2J
Abstract - The spectrm of the light which is scattered by a fluid in the low frequency region (betweenw5 a n d ~ 3 0 0 cm-') can be related to the Fourier transfom of the correlation function between polarizability distortions cau- sed by the interactions amonc the mycroscopic constituents of the system. If only interactions between pairs of molecules are considered inportant it can be shown that the above sentioned correlation function reduces to an average of pair polarizability distortions perfomed with respect to a particular kind of four-points space-time distribution function. In particular the zero- th moment of the spectrun is related with the two, three and four particles static correlation function, while the second Koment of the spectrum is rela- ted with the two and three particles correlation functions
The intensity of the light which, in a depolarized geonetry, is scattered by a fluid of isotropic molecules is directly related to the polarizability anisotropies which are induced by the interactions among the elenentary constituents of the systen. The spectral distribution of this Depolarized-Interaction-Induced-Light-Scatterin (DILS) has been object of investigation for several years in the past both in fluids of iso- tropic and anisotropic nolecules /1/. hong the others, noble Eases fluids are of particular interest since in this case the polarizability anisotropies are related only with the translational dynamics, therefore, the DILS spectra can be more easily interpreted. The typical shape of DILS spectra is an exponential behaviour which goes from few c n l up to 200-300 cm-' with an intensity decay of several order of xa- gnitude. For example in the low density Ar-on gas at roon temperature the intensity varies a factor 10-~ between 5 and 300 cm-' /2/. This exponential behaviour makes very difficoult to measure these spectra in the far wing.
In recent years, the careful measurernent of the DILS absolute intensity and lineshape in Argon, Kripton and Xenon /3/, together with the possibility of perforninc very reliable theoretical calculation of spectra of interacting pairs of atons / 4 / , led to the determination of the induction mechanisr? which accounts in those systems for the absolute spectral behaviour over the entire neasured frequency range /3,5/. Since the interaction induced pair polarizability anisotropy for the noble gases Argon, Kripton
C9-124 J O U R N A L DE PHYSIQUE
and Xenon is now known, one can proceed and try to use this knowledge in order to understand the relevance of nany-body corrclation for DILS spectra in dense fluids and then possibly use DILS as a probe of these nany-body correlations.
In order to accomplish insight in this problem let us first examin the behaviour of the pair polarizability anisotropy p(.r) as a function of the pair distance r
,
in the previously mentioned noble gases. Fig.1 shows, as a typical example, the beha- viour off(x) for Argon where its experinentally deternined form is:/Z,6/and x is the reduced length x = P/r,,
d o =
1.674 i3 is the polarizability of the Argon atom,rm
= 3.7631
is the distance of the minimum of the Banker et al. poten- tial of Argon /7/, A*= - 6 2,
B*= 850, xo = .138.Fig.1
-
Behaviour of the pair-polarizability anisotropy of Argon as a function of the reduced distance x.The behaviour ofg(x) shows a strong decay versus x such thatp(~) at r = 3 r m is only few % of its value at r =rm. This fact indicates that the induction mechanisr?. which generates DILS is confined at short range of interatomic distances and there- fore will probe, mostly, correlations among atoms within the first two-three nearest neighbourhood shells.
If we assume that also in high density fluids the DILS spectrum arises only from pair polarizability anisotropy correlations (this hypotesis has been verified as it will be shown in the following) the cross section can be written /8/:
do-
-
=
fko
J P ~ V
I(w)
where
and
with the summations (i,j,k,l) performed over the !.I particles of the system in the volume V and the induced pair polarizability tensor is yiven by
where
ki.
(t) is the vector distance between atons i and j at time t.In order $0 compare experimental results and theoretical calculations it is useful to refer to the first two spectral monents of 1 (w) which are the ones measured with good precision. The even moments of I(w) are defined as usually:
From Eqs. ( 3 ) , (4), (5) the expressions for
M
and 1q2 can be calculated and result / 9 / :where m is the mass of the atoms and
'V
is the vector gradient in the 3N-dimensional space.
Experimental values for and in Argon at roon temperature have been measpred as a function of density between 10 and 500 Amagat. Also their theoretical values have been calculated by neans of :.lolecular Dynamics coquter simlation /1,8/ assuming that only pair polarizability of f o m (1) and pair potential are important.
JOURNAL DE PHYSIQUE
0.00 2.00 4.00 ( 1 0 2 1 0.00 2.00 4.00 110 21 DENSITT IRCPGRTI DENSITT [AERGRTI
Fig.2
-
Density behaviour of a) Mo/? and b) Pl2/? in Argon at room temperature. The dots with bars are the experimental data parts, the dots are the conputer simulation calculations.their dependence from two-three- and four-particles distribution functions. It is easy to show that for a monoatornic fluid:
Mo = X0(2) + Mo(3) + M0(4)
where :
and
M2 = M2 (2) + :t2(3)
2
In Mo and M2, given by the previous expressions, the factors N /2, FJ3 and ~ ~ corre / 4
from the number of pairs, triplets and quadruplets which contributes when N>,Zl. Eq.(9) and (13) also show that in
Ib
there is a contribution of pairs, triplets and quadruplets while :42 is detemined only by pairs and triplets. This happens because the velocity correlation,in the contribution of the quadruplets to M2,is identically zero / 8 / . If we take into account the isotropicity property of an homogeneous fluid, by using Eq. (5) and perforning a change of variables,:
'
$
(2) and X2 (3) can be written:A5
-9
here
QL3
is the angle between the directionsCl2
and V13 at tine t = 0.As we have explained in the preceding part @(ij) is known fron low density DILS
experiments, g(V12) is the pair distribution function of the fluid so that at any density and temperature can be derived fron neasurements by either neutron or X-ray scattering and also calculated by neans of molecular dynamics simulation. This means that M2(2) can be derived fron sources others than DILS, therefore measurements of
J O U R N A L DE PHYSIQUE
function of the fluid.
REFERENCES
/1/ See for exanple the various papers, and references therein, reported in: Procee- ding of the Nato Advanced Research Workshop on "Phenornena Induced by Intermolecular Interactions", Bonas 1983, published by Plenuir, New York (198 5)
.
/2/ Barocchi,F., Zoppi,M., in Proceedings of the LXXV School of Physics E.Fermi, Varenna, 1978
,
Edited by North Holland, Amsterdam (1980)./3/ Barocchi,F., Zoppi,M., Fromhold,L. and Proffit,i.I., Can.J.of Physics
g
(1981) 1473./4/ Fromhold L., Adv.Chem.Phys. (1981) 1
/5/ Barocchi,F., Zoppi,M., Bafile,U. and Magli,R., Chem.Phys.Lett.
9
(1983) 135. /6/ Bafile,U., Magli,R., Barocchi,F., Zoppi,:4. and Fromhold,L., M01.Phys. (1933) 1149./7/ Bobetic,M.V., Barker,J.A., Phys.Rev.B 2 (1970) 4149
/8/ Vershneya,D., Shirron,S. F., ~itovitz,~:~.
,
Zoppi ,M.,
Barocchi ,F.,
Phys .P.ev.Ag
(1981) 77.Zoppi,M., Barocchi,F., Litovitz,T.A., Newmann,ll., Vershneya,D., Can.J.of Phys.
