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On the theory of light scattering from fluids and polymers
Volume 19, number 1
ON THE
CHEMICAL PHYSICS LETTERS
‘.
THEORY
OF
LIGHT
SCATTERING Robert
FROM
I hlarch 1973
FLUIDS
AND
POLYMERS
A. HARRIS
Department of Chemistry, University of California, Berkeley. California 94 720, US.4 Received 18 September Revised manuscript We present a simpie quantum mechanical AU multiple scattering effects are obtained.
There has been a renewal in the theory scattering of light from gases and liquids, In this note we construct a hamiltonian may be atoms or molecules in a liquid, or interaction between the units takes place each unit, and (b) the interaction between statistical mechanical aspects of part (b).
derivation
1972
received 18 December
1972
of the theory of light scittering
from fluids and po!ymers.
of light scattering from fluids and polymers. In particular, the multiple and the scattering of light from optically active solutions [I, 21. theory of the scattering light off of the centers of N “units”. The units monomers on a polymer: In the Born-Oppenheimer approximation the at two levels: (a) the interaction between the electrons (and nuc!ei) of the center of the units. We consider “(a)“, and do not discuss the
Our paper is developed under the following approximations whose relaxation is discussed at the end.of our paper: (a) neglect overlap between units; (b) the size of the unit is small compared to wavelength of light; (c) assume the Born-Oppenheimer approximation; (d) assume that the frequency of the light is large compared to an excitation of the centers of the units, but small compared to an electronic excitation of the system;(e) neglect retardation effects;(f) assume the monomers are not optically active. The theory is only trivially modified by omitting assumptions (e) and (f); readily modified by relaxation of (but do-able) changed when (c) and (d) are removed. parts (a) and (b), and fundamentally For a fluid the full hamiltonian takes the form,
(1) The defmitions are the usual ones. p,(u) is the charge density operator relative to the center of charge of unit 0. pa is the electric dipole moment operator of center u. TN is the kinetic energy operator of the centers i&j.
The eigenvalue equation is, (HR$H~~)~R~)=(EK+E~~)IRM). Consider
a transition
between
(2) two states RM+
RIM’. The amplitude is
Volume 19, number 1 T(t)w_,R;&
CHEMICAL PHYSICS LETTERS
1 March 1973
=-(R’M’Iexp(-~~slTi)lRM).
(3)
Suppose we make a unitary transformation
on H which removes V. Eq. (3) becomes
T(t)mf_,~‘nf
IR.M> ,
= (R’M’I exp [i(S- f/t/E-S)]
(4)
where
and H = HR SHh? t V” -I-__. .
(6)
If we limit ouFelves to lowest order scattering processes, then v” is the crucial term and has the form
V= +ilV,Sl.
(7)
A totally straightforward application of assumptions(c)
+ (terms that make no contribution
and (d) yields a Y” which may be written explicitly as,,
to light scattering).
~~~(0) is the n arid b element of the adiabatic ground state electronic polarizability of the entire system:
where lg), e. and In>, en are the exact grcllnd and excited states and energies, respectively, of HM. It is a function of ali coordinates, CR,). xab(w) drops off as IR, - Rb I increases. Thus we may replace (8) by
v’ e kqG
Q&aiiE,(k)Ef(@
o% exp(-i(k-@(Ra+Rb)/2]
[l f i(~&)*(&-&,)/2
+ ...]e~(k).x.,(w),~-,(~). (IO)
Because we do not assume the dipole limit for the interaction betu-een monomers, we must calculate density7density susceptibility xob(t:d,a);
x&Jb)a
c
n
the
’ (glp,(Y)ln)(nIp6(v?lg) (nlPb(~Ig)(gJp,(V?ln)
j
0 t i6 - (En-fo)
which gives, xOb(w) upon “averaging”,
t
w +(en-eo)
(11)
I March 1973
CHEMICAL PHYSICS LETTERS
Volume 19, number 1
Under the assumptions given above, time dependent Hartree theory (or RPA approximation) provides an excellent description of the electronic states of the fluid [3]. In that case x,,(r, Y’,w) satisfies the equation, (13) where X&Y,r’, w) is the density susceptibility of the 0th unit in the Hartree approximation: this equation is nothing more than X
&
=
F,bX,W -
The dipole limit of
cC x,(4~T,;x&).
(14)
If we iterate eq. (1 I), and substitute the result into eq, (9) we find d3R c
S3(R-R,)
exp [-i(k-x)-R]
e~(k)*~~.ej;(~)
u +
X
Jd3R
id3ROz
s3(R-%)
X&QX#3~4)~4
Ir, - R-r,
+RI
-q(Z)
g3 (R+)
+ .,.
, I
exp[-i(k-%)(R+R)/2]
e:(K). 1 ... ld3rl
...d3rGPr
(15)
where !he origin and defiiition of each term may be obtained from (8) and (11). We see clearly the multiple scattering structure of the expansion. Upon substitution into the ordinary golden rule, summing over final translational states of the fluid, and averaging over initial translational states of the fluid one fimds the cross section for light scattering including all multiple scattering effects. The light scattering from a polymer solution is readily obtainable from eqs. (8) and (1 l), as exciton solutions of(l1). We have said nothing about the new states exp(-iS)lRM). It may be readily shown that they only renormahze the “old” states. Indeed, in the processes which we are considering, the “old” states are the proper ones to use. The generalization of time dependent Hartree theory to include photons is easy. In the Coulomb gauge one carries out time dependent Hartree theory between the photons and the units and between the units. This generalization leads properly to the retarded interaction between units. The expansion of the susceptibility in a power series in this retarded interaction leads directly to the conventional vector dependent multiple scattering fonnuIae. If the units are optically active, a minor generalization of the formalism presented here is necessary_ A little reflection shows that the electric dipole-magnetic dipole susceptibility which will now occur may readily be obtained simply by modifying the inhomogeneous term, and the r’ label in eq. (1 l), as the coupling of units to one another is the same. If the frequency of the incident light is near an electronic excitation, the complex susceptibility must be used. The result is resonance Raman theory in the R, degrees of freedom. In a sense Fourier transforms of FranckCondon factors appear rather than polarizabilities. Away from resonance the deviation from the adiabatic susceptibilities are readily found by moment expansions. In conclusion, we see that subject to a few restrictions, in the frequency region of interest, light interacts with the electronic polarizability of a fluid or polymer. If time dependent Hartree theory is used to evaluate the polarizability, a multiple scattering hamiltonian is naturally obtained. This hamiltonian may be used to describe the scattering of more than one beam of light off of a fluid or polymerr. Professor L. Jansen sent me B manuscript by Pasmanteret al. 141.WN.Iethere YCcertain things in common with my work, the fundamental method, nature of approximations, and usefulness of the results, are quite different.
t After this manuscript was submitted,
-51
Volume
19, number
1
CHEhiICAL PHYSICS LETTERS
I wish to thank my friends Virginia Weiss and William Gelbart for many fine discussions.
References [I] W.M. Gclbart, J. Chem. Phys. 57 (1972) 699, and references therein. [2] L. BIum and H.L. Frish, J. C’hem. Fhys. 52 (1970) 4379; L. Barron, J. Chem. Phys. 55 (1971) 2001, and references therein. [3] R.A. Harris, J. Chem. Phys. 43 (1965) 959; A.D. McLachlan, R.D. Gregory and hl.A. Ball, Mol. Phys. 7 (1964) 119; R.A. Harris, J. Chem. Phys. 47 (1967) 4487. [4] R.A. Pasmanter, R. Samson and A. Ben-Reuven, Chem. Phys. Letters 16 (1972) 470.
1 March 1973
ON THE
CHEMICAL PHYSICS LETTERS
‘.
THEORY
OF
LIGHT
SCATTERING Robert
FROM
I hlarch 1973
FLUIDS
AND
POLYMERS
A. HARRIS
Department of Chemistry, University of California, Berkeley. California 94 720, US.4 Received 18 September Revised manuscript We present a simpie quantum mechanical AU multiple scattering effects are obtained.
There has been a renewal in the theory scattering of light from gases and liquids, In this note we construct a hamiltonian may be atoms or molecules in a liquid, or interaction between the units takes place each unit, and (b) the interaction between statistical mechanical aspects of part (b).
derivation
1972
received 18 December
1972
of the theory of light scittering
from fluids and po!ymers.
of light scattering from fluids and polymers. In particular, the multiple and the scattering of light from optically active solutions [I, 21. theory of the scattering light off of the centers of N “units”. The units monomers on a polymer: In the Born-Oppenheimer approximation the at two levels: (a) the interaction between the electrons (and nuc!ei) of the center of the units. We consider “(a)“, and do not discuss the
Our paper is developed under the following approximations whose relaxation is discussed at the end.of our paper: (a) neglect overlap between units; (b) the size of the unit is small compared to wavelength of light; (c) assume the Born-Oppenheimer approximation; (d) assume that the frequency of the light is large compared to an excitation of the centers of the units, but small compared to an electronic excitation of the system;(e) neglect retardation effects;(f) assume the monomers are not optically active. The theory is only trivially modified by omitting assumptions (e) and (f); readily modified by relaxation of (but do-able) changed when (c) and (d) are removed. parts (a) and (b), and fundamentally For a fluid the full hamiltonian takes the form,
(1) The defmitions are the usual ones. p,(u) is the charge density operator relative to the center of charge of unit 0. pa is the electric dipole moment operator of center u. TN is the kinetic energy operator of the centers i&j.
The eigenvalue equation is, (HR$H~~)~R~)=(EK+E~~)IRM). Consider
a transition
between
(2) two states RM+
RIM’. The amplitude is
Volume 19, number 1 T(t)w_,R;&
CHEMICAL PHYSICS LETTERS
1 March 1973
=-(R’M’Iexp(-~~slTi)lRM).
(3)
Suppose we make a unitary transformation
on H which removes V. Eq. (3) becomes
T(t)mf_,~‘nf
IR.M> ,
= (R’M’I exp [i(S- f/t/E-S)]
(4)
where
and H = HR SHh? t V” -I-__. .
(6)
If we limit ouFelves to lowest order scattering processes, then v” is the crucial term and has the form
V= +ilV,Sl.
(7)
A totally straightforward application of assumptions(c)
+ (terms that make no contribution
and (d) yields a Y” which may be written explicitly as,,
to light scattering).
~~~(0) is the n arid b element of the adiabatic ground state electronic polarizability of the entire system:
where lg), e. and In>, en are the exact grcllnd and excited states and energies, respectively, of HM. It is a function of ali coordinates, CR,). xab(w) drops off as IR, - Rb I increases. Thus we may replace (8) by
v’ e kqG
Q&aiiE,(k)Ef(@
o% exp(-i(k-@(Ra+Rb)/2]
[l f i(~&)*(&-&,)/2
+ ...]e~(k).x.,(w),~-,(~). (IO)
Because we do not assume the dipole limit for the interaction betu-een monomers, we must calculate density7density susceptibility xob(t:d,a);
x&Jb)a
c
n
the
’ (glp,(Y)ln)(nIp6(v?lg) (nlPb(~Ig)(gJp,(V?ln)
j
0 t i6 - (En-fo)
which gives, xOb(w) upon “averaging”,
t
w +(en-eo)
(11)
I March 1973
CHEMICAL PHYSICS LETTERS
Volume 19, number 1
Under the assumptions given above, time dependent Hartree theory (or RPA approximation) provides an excellent description of the electronic states of the fluid [3]. In that case x,,(r, Y’,w) satisfies the equation, (13) where X&Y,r’, w) is the density susceptibility of the 0th unit in the Hartree approximation: this equation is nothing more than X
&
=
F,bX,W -
The dipole limit of
cC x,(4~T,;x&).
(14)
If we iterate eq. (1 I), and substitute the result into eq, (9) we find d3R c
S3(R-R,)
exp [-i(k-x)-R]
e~(k)*~~.ej;(~)
u +
X
Jd3R
id3ROz
s3(R-%)
X&QX#3~4)~4
Ir, - R-r,
+RI
-q(Z)
g3 (R+)
+ .,.
, I
exp[-i(k-%)(R+R)/2]
e:(K). 1 ... ld3rl
...d3rGPr
(15)
where !he origin and defiiition of each term may be obtained from (8) and (11). We see clearly the multiple scattering structure of the expansion. Upon substitution into the ordinary golden rule, summing over final translational states of the fluid, and averaging over initial translational states of the fluid one fimds the cross section for light scattering including all multiple scattering effects. The light scattering from a polymer solution is readily obtainable from eqs. (8) and (1 l), as exciton solutions of(l1). We have said nothing about the new states exp(-iS)lRM). It may be readily shown that they only renormahze the “old” states. Indeed, in the processes which we are considering, the “old” states are the proper ones to use. The generalization of time dependent Hartree theory to include photons is easy. In the Coulomb gauge one carries out time dependent Hartree theory between the photons and the units and between the units. This generalization leads properly to the retarded interaction between units. The expansion of the susceptibility in a power series in this retarded interaction leads directly to the conventional vector dependent multiple scattering fonnuIae. If the units are optically active, a minor generalization of the formalism presented here is necessary_ A little reflection shows that the electric dipole-magnetic dipole susceptibility which will now occur may readily be obtained simply by modifying the inhomogeneous term, and the r’ label in eq. (1 l), as the coupling of units to one another is the same. If the frequency of the incident light is near an electronic excitation, the complex susceptibility must be used. The result is resonance Raman theory in the R, degrees of freedom. In a sense Fourier transforms of FranckCondon factors appear rather than polarizabilities. Away from resonance the deviation from the adiabatic susceptibilities are readily found by moment expansions. In conclusion, we see that subject to a few restrictions, in the frequency region of interest, light interacts with the electronic polarizability of a fluid or polymer. If time dependent Hartree theory is used to evaluate the polarizability, a multiple scattering hamiltonian is naturally obtained. This hamiltonian may be used to describe the scattering of more than one beam of light off of a fluid or polymerr. Professor L. Jansen sent me B manuscript by Pasmanteret al. 141.WN.Iethere YCcertain things in common with my work, the fundamental method, nature of approximations, and usefulness of the results, are quite different.
t After this manuscript was submitted,
-51
Volume
19, number
1
CHEhiICAL PHYSICS LETTERS
I wish to thank my friends Virginia Weiss and William Gelbart for many fine discussions.
References [I] W.M. Gclbart, J. Chem. Phys. 57 (1972) 699, and references therein. [2] L. BIum and H.L. Frish, J. C’hem. Fhys. 52 (1970) 4379; L. Barron, J. Chem. Phys. 55 (1971) 2001, and references therein. [3] R.A. Harris, J. Chem. Phys. 43 (1965) 959; A.D. McLachlan, R.D. Gregory and hl.A. Ball, Mol. Phys. 7 (1964) 119; R.A. Harris, J. Chem. Phys. 47 (1967) 4487. [4] R.A. Pasmanter, R. Samson and A. Ben-Reuven, Chem. Phys. Letters 16 (1972) 470.
1 March 1973