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Is depolarized light scattered from simple liquids mainly double-scattered?
Volume 64, number 3
IS DEPOLARIZED
CHEBIIC4L
LIGHT
SCAmERED
PHYSICS LETTERS
FROlM SIMPLE
LIQUIDS
15 July 1979
MAINLY
T. KEYES, Branka M. LADANYI SterIing Chemistry Laboratoqp. YaIe 0% ersity. New Haven. Connecticut 06520.
DOUBLE-SCATTERED?
USA
and Paul A. MADDEN i’Xe Uni~~ersiQChemical Laboratory, Cambridge 032 IEW. UK Received 8 January
1979; in final form 30 April 1979
The multiple-scattering series for depohrized light scattering from simple liquids is investigated. It is found that the leading term in the serif%, due to doubIe-double scattering, is probably not large compared to double-triple (DT) scattering contributions_ Existing theories, which neglect DT scattering, have concluded that the experimental data cannot be explained using the dipole-induced dipole (DID) model for the pair polarizability. It is suggested that p theory nhich includes DT scatterins will not lead to rejection of the DID model.
I _ Introduction The earliest theories [l] of the depolarized light scattering intensity (Z) for fluids composed of spherical particies were based upon the dipole-induced dipole (DID) model and upon the assumption that only the two-body (“double scattering”) term in the interaction-induced polarizability is important, that is Z=ZDo whereZDO is the* double-double depolarized intensity_ A quantitative test of this theory against experiment [2] requires an accurate two-body potential for the low-density gas or the two-, three- and four-body distnbution functions for the liquid- The most recent experimental work on the gas phase [3,4] indicates that the model predicts reasonably well (up to 20% difference) the integrated intensity and that earlier studies had contradicted this result because they had underestimated the experimental intensity at very low frequency_ Alder et al. [S] have calculated ZDD for hard-sphere and Lennard-Jones fluids representative of argon. They found that the calculated intensity was an order of magnitude greater than the experimental value [6] at the triple-point density and temperature, and attributed this disagreement to failure of the DID model for the pair polarizabilityMany quantum-mechanical calculations [7] of the pair polarizability have been performed_ Most show a substantial reduction in the anisotropy below the DID value at physically important separations. However. some of errors [Sj and neglect of correlation. Dacre [9] performed a very accurate calculation on He2 in which both these effects are included; the deviations the ‘ab initio studies can be criticised on the grounds of basis-set superposition
he finds from DID are not large at the important separations_ Clarke et al. [IO] made a critical study of the semianalytical extensions of the DID model and have shown how many of the corrections to DID expected at short range cancel for the inert gases so that the departures from DID are small [ 111, that is, consistent with the 20% disagreement between theory and experiment in the gas phase but not with the factor of 10 indicated by the comparison in the liquid_ With the exception of the comparisorL of the calculated and experimental liquid-state intensity, the above survey indicates that the DID model of the pair polarizability anisotropy is an excellent first approximation. It is our goal to explain the liquid-state experiments within the DID modeI; this note represents our first step towards this goal.
479
VoIume64, number 3
CHEMICAL PHYSICS LE-i-i-ERS
15 July 1979
The obviotrs alternative to a breakdown of the DID model is a breakdown of the approximation Z = ZDD_It is possible to write a series for Z, Z=ZDE)fZ~=fZ=~f-ZTTf---,
01
where T stands for tripIe scattering (and involves the induced polarizabihty of 3 molecules). Each term in the series is expected to be smaher than the preceding term by a factor of order o/03, where Q!is the poIarimbiIity and u the collision diameter. For the inert Bses [ I2], 0.06 > a/u3 > 0.03, so the multiple scattering series might appear to be rapidIy convergent. However, for Iiquids, ZD~ is very much smaller than rough estimates would indicate. ZDt, is composed of two-- three- and four-particfe terms: ZDi) =zgA *z&
+zg,
_
(9
For dilute gases, ZDDa Z$h, but for Iiquids,in the DID model, Alder et aI. [S] have shown that ZDDis typically only two or three percenr of Z$$ due to a rather strikbag cancellation between ZD’22, +Z#& {+) and Z$?h f-)_ In fact, ZDD/ZF& w o/133 and, thus, there is HOreason to believe that the multipIe-scattering series converges rapidlyIt is dear then, that any theory of Iiquid scattering which considers ZD~ alone is on shaky ground, and that, at Ieast, the next terms Z,, f Z, must be investigated; we begin that task in the next section_ The current state of affairs for simple systems is reminiscent of that for depoIarized scattering from Liquidscomposed of non-spherical moIecuIes in the recent past_ There, single scattering (S) gives depoiarization, so Z=Zss’Zs,
+ZDs i- I.“_
(3)
Theories 1131 based on the approximation Z ==Zss predicted intensities larger (- 2X) than experimental values. This led to various tamperings [I 3J with the true theoretical expressions for ZsS_ It was recently found [ 141 that ISI, -f Z*S is not small compared to Zss_ and that a theory with I= Zss -t I&-, f $S is in good agreement with experiment. So there exists a strong precedent for investigating more than the first nonzero term in the muttipfescattering s&ies, ZSSrepresents the scattering by an assembly of non-spherical moIecuIes each of which experiences the isotropic Lorentz fi_eIdin a dielectric continuum [ 141, while adding Zs~ + ZD~introduces anisotropy into the tieId due to the shape of the moIecuIe. It is mainfy the effect of this anisotropy that reduces the theoretical intensity_ Since this picture has been so fruitfu1 in the study of scattering from non-spherical molecufes, it is desirable to use it in simple systems.. Double scattering from simple particlesmay be thought of, rougIdy, as singIe scattering from interacting pairs of “transient diatomi&_ Insofar as this picture is true, we feel that a good theory shouId treat the interacting pair as residing in a pair-shaped cavity in a dielectric. We use this idea as a guide in examining the muitiple-scattering series for simpIe systems_ We shalt fo.muIare the idea more precisely in a future publication-
2. Double-doubIe and double-tripl
scattering
The mu~tipIe-scatte~g series has been written down by several authors f 15, Itif; we use the expressions in ref[16] _The expression for Z is represented by a series of diagrams_ Particles are denoted by dots. A iz%e represents a component of the ‘%ut” dipole tensor T&(T) (T’(r) = 0, r < a, a * 0), referred to laboratory-fixed axes. A tbfck Em? represents the same component of the product of two dipole tensors, i.e.
Note that when 1 = 3 this term is the one usually considered as a contribution to the pair poIar&abifity varying as rr26; as such it woufd be simpIe to incorporate into a computer sinudation. In this context we note that this sec-
ond-order DID term is not the only rrT6 contribution to the pair polar&ability; the dispersion contribution is also important, particuiarIy for ii&t atoms [ 173 _ Since our purpose is to compare with first-order DID calculations we 480
Volume 64, number 3
CHEMICAL PHYSICS LETTERS
15 July 1979
shall not formally distinguish between this and other triple-scattering terms For an n-particle diagram, integration over all 3n coordinates with the n-particle distribution plied_ We then have, omitting gecmetrical factors, I,,
= or4p’[(122 + 2)/3]4(2
I:: + 4p ,+
P’:
function is im-
:) 1
(41
where fz is the refractive index and p the density. It is well known that the two- and four-particle terms resemble the contribution to 139 of one and two transient “diatomics”, but no simple interpretation has been given to the three-particle term. The idea that shaped cavities are important for depoiarization yields a clear physical interpretation for @A _ Let us consider evaluating that term by first performing the integral over particle 3. This is basically the integral of a dipole tensor over all space but the region excluded by particles 1 + 3, - the shaped cavity formed by those particles - through the three-particle distribution function_ An analogous calculation gives (approximately) the correction, due to shape, of the electric field within a shaped cavity_ The remaining 1-Z dipole tensor is just a transient diatomic, so 1gA is a “cross term” between (a) a transient diatomic, responding with that part of its polarizability (%) which does not depend on interactions to an anisotropic field, and (b) a transient diatomic responding to the applied fieid with its anisotropic (pair property) polarizabilityThus, it makes sense that ig& is negative and causes major cancellation at liquid densities_ It contains the bulk of the shaped-cavity effects, and so should, in analo,v to the nonspherical case, reduce the intensity_ Other cavity contributions to I are possible, and we must look for them in f,,T t- 1m_ It is not bard to show that
(5) where the terms are grouped for convenience of interpretation_ The first bracket,contains the interference between the rzT6 and r$3 terms in the pair polarizability. Note that the two-, three- and four-particle terms appear in the same proportion as in fDD _The functionsto be averaged in these terms are very similar to those in ID,-, and we should thus expect substantial cancellation between them. The second bracket of (5) contains terms which modify the value of one of the dipole propagators from the vacuum value T’ towards the value appropriate to propagation in the dielectric medium around a scattering pairConsider the first term. It is an analogue of c: in ID, in which, whilst propagating between 1 and 2, one of the dipoIe fields is rescattered
(Yp
by particle 3_ It is instructive to write this diagram as
b3=ap/drl
?
*2
dr3 Tap(r~a) T&(rt3)
Trp(r32)C1g(3)(r1
,r2,r3)
-g(2)(r12)1
+g(*)(r12))
(6)
where gc3) and g(‘) are the two- and three-particle distribution functions_ The first term now has to have the position of particle S correlated with either 1 or 2. The remaining term arises from the free integration of the product of dipole tensors over the position of particle 3; it gives the “renormalization” of the dipole tensor, to first order in an, due to propagation of the dipole field through a uniform medium with dielectric constant n2 where (n2 - l)/(n2 + 2) = (4x/3) op. The correlated term gives the correction to this dielectric constant due to the local structure around 1 and 2. Proceeding in this way we may rewrite the second bracket of (5) as 481.
CHEMICAL. PHYSICS LETTERS
VoIume 64. number 3
15 July 1979
(7) This shows how the “renomAised”
terms are highly cancelling;
we expect that the ‘korrelated”
terms have the
same property_ It is now tempting to construct a fully renormalised theory by separating out the uniform-medium terms in (‘7) and associating them with IDD_ It is easy to sum these terms to infinite order in cup, in fact -(4ir/3)(up appears here as the f&t-order term in the expansion of [( 1 - ap 4~/3)(1 + ocp &r/3)]-1 _ It would also be straightforward
to “correct”
the simulation
results, by multiplication
by the square of this factor. We shall not follow
this
1 The last two terms in (5) are the true cavity corrections_ For 2u repeat the argument given for 1g&- noting that ‘u possesses two 1-2 dipole tensors_ The integral over particle 3 stilI generates the shaped-cavity correction ?o the field. Now, however, that field is tensor-multiplied by one of the l-2 dipoIe tensors, which corresponds to a morecule responding to the cavity field with its anisotropic polarizability- The result multiplies another l--2
approach
for reasons which will become
cIear below.
dipole tensor, so that the diagram represents a cross term between (a) a transient diatomic
responding
to the shapedevity
field with its anisotropic
polarizability,
and
(b) a traasient diatomic responding to the applied field with its anisotropic polarizability_ Thus b is Iogicalry grouped with 1gA _ Our overall picture then, is that the buIk of the shaped-cavity corrections to IDD is already contained in @A, These terms reduce the predicted intensity by introducing an even more but smaller terms are present in I,,. complete cancellation
of the positive IDD terms_
We have calculated Ig+ and 1g+ for hard spheres at p = 0.6 and 09. corresponding to the dense gas and liquid respectiveIy, using the Kirkwood superposition approximation for g (3) and obtaining the required g@)‘s from the Monte Garlo ca1cuIation.s of Barker and Henderson [IS] _ The results are referred to 1g& for convenience and are collected in table I_ We also give therein values of ID&g& obtained by interpolating the values of Alder et al- [5] Since we have not calcuIated alI the diagrams in (5) it is important to avoid faIse conclusions which might arise by ftiIure to anticipate cancellations; for this reason we substitute into (1 j and (5), retain uncalculated terms and obtain for the reduced density p of 09 @!&
= 0.024
i- (a/03)
+ (ar/03)
- 0-630~
1240
- 3358
t4p’i+p3
p + p* ; 0
1
/ 6
f)+@,03)
(-3358p+@
k]_
(8)
TabIe L GIculated doubk-triple
scattering d&rams and the total doubIe-double
densities of 0.6 and 0.9. CoIumn 2 has units of O3 -___-
--
__-_-
to \‘? ($A
-
-----
-__-
p = 09
O-620
-3358
--0_315
O-024
p = 0.6
0.565
-3.195
-0.304
0.094
a) Obtained by interpolation amoq
482
intensity r&the
the hard-sphere resuIts in ref_ I.51 .
= 2 S> ). for reduced
Volume 64, number 3
CHEhlICAL PHYS:CS LETTERS
15 July 1979
It is clear that most of the terms in Z,, are at least comparable in magnitude to what has previously been regarded as “the” intensity (the first term in (8)). In the first correction, arising from the second-order DID term in the pair polarizability, we find, as anticipated, from the analogy with I DD, that the two- and three-particle terms are of opposite sign (and we expect the four-particle term to be positive)_ However, in ZDn it is found that 2Zg& = The implication of ou: results for these terms is that the degree of cancellation of the r-ii6 term in Z8X = -2Z&_ the pair polarizability is much smaller than that of the rG3 term alone. This suggests that the simple extension of the computer simulation to include this extra term in the pair polarizability might significantly improve agreement with experiment _ The values contained in the second bracket of(S) show why renormalisation of the dipole tensors is inadvisable in this problem_ From (6) we see that p
/Q=
( >)comlatM/
I:-4X/3=-0.315,
which shows that the correlated term almost cancels the renormalisable uniform-medium term. In physical temrs, it appears that at the ‘12 distances important in Zgh the medium between 1 and 2 is closer to a vacuum than a medium of dielectric constant !z2_ The largest overall correction to ZDD comes from the cavity term, which gives a substantial reduction of intensity below the first-order DID level. We see no reason why the other term on the last line of (8) should nearly cancel this term.
3. Discussion Hopefully, it is evident that we do not purport to have obtained the DT intensity_ We have simply shown that the two- and three-body DT i- TD terms are larger than ZDr, in liquids- Thus, no conclusions should be drawn about the pair polarizability by assuming I = ZDD_ How, then, stands the theory of depolarized scattering from liquids of simple particles? In our opinion, two possibilities exist. The multiple-scattering series isordered by the small parameter o/as_ However_ the near-perfect cancellation of the three parts of ZDD a&o defines, in a sense, another small parameter, denoted X, with X = a/03_ We thus have, in relative units, ZOD = A,-
‘DT = cd)
iv,
I,
= (cY/u3)2?t~ ,
(9
where fl and y are unknown, pending the answer of the crucial question: does any further near-perfect cancellation exist in the multiple-scattering series_ The two possibilities are: and the DID model is questionable; (l)fl= l;thenZ*ZDD, (2) fi x 0; then ZDD z ZDT %, In, unless 7 behaves very oddly, Z = ZI,D + ZTD f ZDT, and the DID model may be successful. Case (1) has been assumed by all authors so far, but its v-ii Sty does depend on having ZDT x X2, which ~BS not been shown. Careful evaluation of the full ID= is now of first priority, but our guess is that case (2) holds. Near-perfect cancellation only occurs for a good reason, which, in ZDD, is the shaped-cavity correction (Zg& = -92X) to scattering in vacua (ZgA + Z$$J = 2X). We have separated and discussed those terms in ID-P for which we expect similar cancellation and do not see any reason to expect cancellation of the remainder. We think it likely that a theory of type (2) will agree with experiment_
References Ill W-M- Gelbart, Advan Chem. Phys. 26 (1974) I_ 383
Volume 64, number 3
CHPMICAL. PHYSICS LETTERS
15 July 1979
[2] GE_ Tabisz, in: hfolecufar Spectroscopy, Vat 6, Specialist Periodical Reports, eds. RF_ Barrow and D.A. Long (The Chemical Society, London, 1979). [3] L Frommhold, K-H_ Hong and M_H_ Proftitt, hfoL Phys. 35 (1978) 665,691. [4] F_ Barrochi and hf. Zoppi, Phys_ Letters 66A (1978) 99; 69A (1978) 187; D.P. Shelton and GE_ Tabisz, to be published[S] BJ_ Alder, JJ. Weiss and HI._ Strauss, Pbys_ Rev_ A7 (1973) 281_ [6] P-k Ffeury and J_P_bfcTague, Opt_ Commun l(l969) 164; W-S_ Gornalf. HE_ Howard-Lock and BP_ Stoicheff, Phys. Rev. Al (1970) 1288; KS_ Gabefnick and H.L Strauss, J_ Chem_ Phys_ 49 (1968) 2334. [7] T-K_ Lfm, B. Lfndcrand R_A_ Kmmhout, J. C’hem_Phyr 52 (1970) 3831; A-D. Buckfngham and R.S. Watts, DfoL Phys. 26 (1973) 7; E_F_ O’Brien, V_P_ Gutschick, V_ McKay and J_P_McTague, Phys_ Rev. A 8 (1973) 690; PJ_ Fortune, P-R Certain and L._W_Bruch, Chem_ Phyr Letters 27 (1974) 233; PJ_ Fortune and P-R_ Certain, J. Chem_ Phys. 61(1974) 2620; P_R Certain and PJ_ Fortune, J. Chem Phys. 55 (1971) 5818; W_ Meyer, Chem Phyr 17 (1976) 27; N_S_ Osthrnd and D-L hfcrriticld, Chem Phys_ Letters 39 (1976) 612; J-W. Kressand JJ_ Kozak, J_ Chem Phys_ 66 (;977) 4516; P. Laffemand, DJ. David and B. Bigot, hfol. Phys. 27 (1974) 1029; Rk Harris, D-F. Helfer and W-M_ GcIbart, J_ Chem Phys_ 61 (1974) 3854; D.F. HeIfcr, R.A. Hanisand W.&f. Gclbart, J. Chem. Pbys. 62 (1975) 1947; D-W_ Oxtoby and W.M. Gelbart, Mot Phys- 30 (1975) 535_ 181 N_S_ Ostfund and D-L._ MenitieId, Chem_ Phys. Letters 39 (1976) 612_ i9] P-D. Dacre, bfol Phys. 36 (1978i 541_ [lo] K.L. Cfarke~P_A_ hfadden and A-D_ Buckingham, Mol. Phys_ 36 (1978) 301. [ II] K-L. Cfarke, Ph.D. Thesfs, Univcrsfty of Cambridge (1978). [I21 Landolt-Bornstein, Zahfenwerte und Funktionen, 11-Band 8 Teil (Springer, Berlin, 1962). ]13] DJ. Counou,Trans_ Faraday Sot_ 65 (1969) 2654; S_ Kieffch, J_ Chem Phys_ 46 (1967) 4090; J_ Phys_ C 33 (1968) 191. [14] B.Sf_ Laianyi and T_ Keyes, bfol. Phyn 33 (1977) 1063,124?; T_ Keyes and B_M_ Lndanyi, hfoL Phyr 33 (1977) 1099,1271_ [15] B.U_ FekIcrhof, Physica 76 (1974) 486; H-hfJ. Boots, D_ Bcdeaux and P. hfazur, Physica A79 (1975) 397; E Hynne. bfoL Phys- 34 (1977) 681; B.Af. Ladanyi and T_ Keyes, bfot Phys- 31 (1976) 1685. 1161 B.Bf. Ladanyi andT_ Keyes, hfof. Phys. 34 (1977) 1643. 1171 kD_ Bucki@um and K_L._CIxke, Chem Phys. Letters 57 (1978) 321_ [IS] J-4 Barker and D_ Henderson, fbfoL Phys_ 21<1971) 187_
484
IS DEPOLARIZED
CHEBIIC4L
LIGHT
SCAmERED
PHYSICS LETTERS
FROlM SIMPLE
LIQUIDS
15 July 1979
MAINLY
T. KEYES, Branka M. LADANYI SterIing Chemistry Laboratoqp. YaIe 0% ersity. New Haven. Connecticut 06520.
DOUBLE-SCATTERED?
USA
and Paul A. MADDEN i’Xe Uni~~ersiQChemical Laboratory, Cambridge 032 IEW. UK Received 8 January
1979; in final form 30 April 1979
The multiple-scattering series for depohrized light scattering from simple liquids is investigated. It is found that the leading term in the serif%, due to doubIe-double scattering, is probably not large compared to double-triple (DT) scattering contributions_ Existing theories, which neglect DT scattering, have concluded that the experimental data cannot be explained using the dipole-induced dipole (DID) model for the pair polarizability. It is suggested that p theory nhich includes DT scatterins will not lead to rejection of the DID model.
I _ Introduction The earliest theories [l] of the depolarized light scattering intensity (Z) for fluids composed of spherical particies were based upon the dipole-induced dipole (DID) model and upon the assumption that only the two-body (“double scattering”) term in the interaction-induced polarizability is important, that is Z=ZDo whereZDO is the* double-double depolarized intensity_ A quantitative test of this theory against experiment [2] requires an accurate two-body potential for the low-density gas or the two-, three- and four-body distnbution functions for the liquid- The most recent experimental work on the gas phase [3,4] indicates that the model predicts reasonably well (up to 20% difference) the integrated intensity and that earlier studies had contradicted this result because they had underestimated the experimental intensity at very low frequency_ Alder et al. [S] have calculated ZDD for hard-sphere and Lennard-Jones fluids representative of argon. They found that the calculated intensity was an order of magnitude greater than the experimental value [6] at the triple-point density and temperature, and attributed this disagreement to failure of the DID model for the pair polarizabilityMany quantum-mechanical calculations [7] of the pair polarizability have been performed_ Most show a substantial reduction in the anisotropy below the DID value at physically important separations. However. some of errors [Sj and neglect of correlation. Dacre [9] performed a very accurate calculation on He2 in which both these effects are included; the deviations the ‘ab initio studies can be criticised on the grounds of basis-set superposition
he finds from DID are not large at the important separations_ Clarke et al. [IO] made a critical study of the semianalytical extensions of the DID model and have shown how many of the corrections to DID expected at short range cancel for the inert gases so that the departures from DID are small [ 111, that is, consistent with the 20% disagreement between theory and experiment in the gas phase but not with the factor of 10 indicated by the comparison in the liquid_ With the exception of the comparisorL of the calculated and experimental liquid-state intensity, the above survey indicates that the DID model of the pair polarizability anisotropy is an excellent first approximation. It is our goal to explain the liquid-state experiments within the DID modeI; this note represents our first step towards this goal.
479
VoIume64, number 3
CHEMICAL PHYSICS LE-i-i-ERS
15 July 1979
The obviotrs alternative to a breakdown of the DID model is a breakdown of the approximation Z = ZDD_It is possible to write a series for Z, Z=ZDE)fZ~=fZ=~f-ZTTf---,
01
where T stands for tripIe scattering (and involves the induced polarizabihty of 3 molecules). Each term in the series is expected to be smaher than the preceding term by a factor of order o/03, where Q!is the poIarimbiIity and u the collision diameter. For the inert Bses [ I2], 0.06 > a/u3 > 0.03, so the multiple scattering series might appear to be rapidIy convergent. However, for Iiquids, ZD~ is very much smaller than rough estimates would indicate. ZDt, is composed of two-- three- and four-particfe terms: ZDi) =zgA *z&
+zg,
_
(9
For dilute gases, ZDDa Z$h, but for Iiquids,in the DID model, Alder et aI. [S] have shown that ZDDis typically only two or three percenr of Z$$ due to a rather strikbag cancellation between ZD’22, +Z#& {+) and Z$?h f-)_ In fact, ZDD/ZF& w o/133 and, thus, there is HOreason to believe that the multipIe-scattering series converges rapidlyIt is dear then, that any theory of Iiquid scattering which considers ZD~ alone is on shaky ground, and that, at Ieast, the next terms Z,, f Z, must be investigated; we begin that task in the next section_ The current state of affairs for simple systems is reminiscent of that for depoIarized scattering from Liquidscomposed of non-spherical moIecuIes in the recent past_ There, single scattering (S) gives depoiarization, so Z=Zss’Zs,
+ZDs i- I.“_
(3)
Theories 1131 based on the approximation Z ==Zss predicted intensities larger (- 2X) than experimental values. This led to various tamperings [I 3J with the true theoretical expressions for ZsS_ It was recently found [ 141 that ISI, -f Z*S is not small compared to Zss_ and that a theory with I= Zss -t I&-, f $S is in good agreement with experiment. So there exists a strong precedent for investigating more than the first nonzero term in the muttipfescattering s&ies, ZSSrepresents the scattering by an assembly of non-spherical moIecuIes each of which experiences the isotropic Lorentz fi_eIdin a dielectric continuum [ 141, while adding Zs~ + ZD~introduces anisotropy into the tieId due to the shape of the moIecuIe. It is mainfy the effect of this anisotropy that reduces the theoretical intensity_ Since this picture has been so fruitfu1 in the study of scattering from non-spherical molecufes, it is desirable to use it in simple systems.. Double scattering from simple particlesmay be thought of, rougIdy, as singIe scattering from interacting pairs of “transient diatomi&_ Insofar as this picture is true, we feel that a good theory shouId treat the interacting pair as residing in a pair-shaped cavity in a dielectric. We use this idea as a guide in examining the muitiple-scattering series for simpIe systems_ We shalt fo.muIare the idea more precisely in a future publication-
2. Double-doubIe and double-tripl
scattering
The mu~tipIe-scatte~g series has been written down by several authors f 15, Itif; we use the expressions in ref[16] _The expression for Z is represented by a series of diagrams_ Particles are denoted by dots. A iz%e represents a component of the ‘%ut” dipole tensor T&(T) (T’(r) = 0, r < a, a * 0), referred to laboratory-fixed axes. A tbfck Em? represents the same component of the product of two dipole tensors, i.e.
Note that when 1 = 3 this term is the one usually considered as a contribution to the pair poIar&abifity varying as rr26; as such it woufd be simpIe to incorporate into a computer sinudation. In this context we note that this sec-
ond-order DID term is not the only rrT6 contribution to the pair polar&ability; the dispersion contribution is also important, particuiarIy for ii&t atoms [ 173 _ Since our purpose is to compare with first-order DID calculations we 480
Volume 64, number 3
CHEMICAL PHYSICS LETTERS
15 July 1979
shall not formally distinguish between this and other triple-scattering terms For an n-particle diagram, integration over all 3n coordinates with the n-particle distribution plied_ We then have, omitting gecmetrical factors, I,,
= or4p’[(122 + 2)/3]4(2
I:: + 4p ,+
P’:
function is im-
:) 1
(41
where fz is the refractive index and p the density. It is well known that the two- and four-particle terms resemble the contribution to 139 of one and two transient “diatomics”, but no simple interpretation has been given to the three-particle term. The idea that shaped cavities are important for depoiarization yields a clear physical interpretation for @A _ Let us consider evaluating that term by first performing the integral over particle 3. This is basically the integral of a dipole tensor over all space but the region excluded by particles 1 + 3, - the shaped cavity formed by those particles - through the three-particle distribution function_ An analogous calculation gives (approximately) the correction, due to shape, of the electric field within a shaped cavity_ The remaining 1-Z dipole tensor is just a transient diatomic, so 1gA is a “cross term” between (a) a transient diatomic, responding with that part of its polarizability (%) which does not depend on interactions to an anisotropic field, and (b) a transient diatomic responding to the applied fieid with its anisotropic (pair property) polarizabilityThus, it makes sense that ig& is negative and causes major cancellation at liquid densities_ It contains the bulk of the shaped-cavity effects, and so should, in analo,v to the nonspherical case, reduce the intensity_ Other cavity contributions to I are possible, and we must look for them in f,,T t- 1m_ It is not bard to show that
(5) where the terms are grouped for convenience of interpretation_ The first bracket,contains the interference between the rzT6 and r$3 terms in the pair polarizability. Note that the two-, three- and four-particle terms appear in the same proportion as in fDD _The functionsto be averaged in these terms are very similar to those in ID,-, and we should thus expect substantial cancellation between them. The second bracket of (5) contains terms which modify the value of one of the dipole propagators from the vacuum value T’ towards the value appropriate to propagation in the dielectric medium around a scattering pairConsider the first term. It is an analogue of c: in ID, in which, whilst propagating between 1 and 2, one of the dipoIe fields is rescattered
(Yp
by particle 3_ It is instructive to write this diagram as
b3=ap/drl
?
*2
dr3 Tap(r~a) T&(rt3)
Trp(r32)C1g(3)(r1
,r2,r3)
-g(2)(r12)1
+g(*)(r12))
(6)
where gc3) and g(‘) are the two- and three-particle distribution functions_ The first term now has to have the position of particle S correlated with either 1 or 2. The remaining term arises from the free integration of the product of dipole tensors over the position of particle 3; it gives the “renormalization” of the dipole tensor, to first order in an, due to propagation of the dipole field through a uniform medium with dielectric constant n2 where (n2 - l)/(n2 + 2) = (4x/3) op. The correlated term gives the correction to this dielectric constant due to the local structure around 1 and 2. Proceeding in this way we may rewrite the second bracket of (5) as 481.
CHEMICAL. PHYSICS LETTERS
VoIume 64. number 3
15 July 1979
(7) This shows how the “renomAised”
terms are highly cancelling;
we expect that the ‘korrelated”
terms have the
same property_ It is now tempting to construct a fully renormalised theory by separating out the uniform-medium terms in (‘7) and associating them with IDD_ It is easy to sum these terms to infinite order in cup, in fact -(4ir/3)(up appears here as the f&t-order term in the expansion of [( 1 - ap 4~/3)(1 + ocp &r/3)]-1 _ It would also be straightforward
to “correct”
the simulation
results, by multiplication
by the square of this factor. We shall not follow
this
1 The last two terms in (5) are the true cavity corrections_ For 2u repeat the argument given for 1g&- noting that ‘u possesses two 1-2 dipole tensors_ The integral over particle 3 stilI generates the shaped-cavity correction ?o the field. Now, however, that field is tensor-multiplied by one of the l-2 dipoIe tensors, which corresponds to a morecule responding to the cavity field with its anisotropic polarizability- The result multiplies another l--2
approach
for reasons which will become
cIear below.
dipole tensor, so that the diagram represents a cross term between (a) a transient diatomic
responding
to the shapedevity
field with its anisotropic
polarizability,
and
(b) a traasient diatomic responding to the applied field with its anisotropic polarizability_ Thus b is Iogicalry grouped with 1gA _ Our overall picture then, is that the buIk of the shaped-cavity corrections to IDD is already contained in @A, These terms reduce the predicted intensity by introducing an even more but smaller terms are present in I,,. complete cancellation
of the positive IDD terms_
We have calculated Ig+ and 1g+ for hard spheres at p = 0.6 and 09. corresponding to the dense gas and liquid respectiveIy, using the Kirkwood superposition approximation for g (3) and obtaining the required g@)‘s from the Monte Garlo ca1cuIation.s of Barker and Henderson [IS] _ The results are referred to 1g& for convenience and are collected in table I_ We also give therein values of ID&g& obtained by interpolating the values of Alder et al- [5] Since we have not calcuIated alI the diagrams in (5) it is important to avoid faIse conclusions which might arise by ftiIure to anticipate cancellations; for this reason we substitute into (1 j and (5), retain uncalculated terms and obtain for the reduced density p of 09 @!&
= 0.024
i- (a/03)
+ (ar/03)
- 0-630~
1240
- 3358
t4p’i+p3
p + p* ; 0
1
/ 6
f)+@,03)
(-3358p+@
k]_
(8)
TabIe L GIculated doubk-triple
scattering d&rams and the total doubIe-double
densities of 0.6 and 0.9. CoIumn 2 has units of O3 -___-
--
__-_-
to \‘? ($A
-
-----
-__-
p = 09
O-620
-3358
--0_315
O-024
p = 0.6
0.565
-3.195
-0.304
0.094
a) Obtained by interpolation amoq
482
intensity r&the
the hard-sphere resuIts in ref_ I.51 .
= 2 S> ). for reduced
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CHEhlICAL PHYS:CS LETTERS
15 July 1979
It is clear that most of the terms in Z,, are at least comparable in magnitude to what has previously been regarded as “the” intensity (the first term in (8)). In the first correction, arising from the second-order DID term in the pair polarizability, we find, as anticipated, from the analogy with I DD, that the two- and three-particle terms are of opposite sign (and we expect the four-particle term to be positive)_ However, in ZDn it is found that 2Zg& = The implication of ou: results for these terms is that the degree of cancellation of the r-ii6 term in Z8X = -2Z&_ the pair polarizability is much smaller than that of the rG3 term alone. This suggests that the simple extension of the computer simulation to include this extra term in the pair polarizability might significantly improve agreement with experiment _ The values contained in the second bracket of(S) show why renormalisation of the dipole tensors is inadvisable in this problem_ From (6) we see that p
/Q=
( >)comlatM/
I:-4X/3=-0.315,
which shows that the correlated term almost cancels the renormalisable uniform-medium term. In physical temrs, it appears that at the ‘12 distances important in Zgh the medium between 1 and 2 is closer to a vacuum than a medium of dielectric constant !z2_ The largest overall correction to ZDD comes from the cavity term, which gives a substantial reduction of intensity below the first-order DID level. We see no reason why the other term on the last line of (8) should nearly cancel this term.
3. Discussion Hopefully, it is evident that we do not purport to have obtained the DT intensity_ We have simply shown that the two- and three-body DT i- TD terms are larger than ZDr, in liquids- Thus, no conclusions should be drawn about the pair polarizability by assuming I = ZDD_ How, then, stands the theory of depolarized scattering from liquids of simple particles? In our opinion, two possibilities exist. The multiple-scattering series isordered by the small parameter o/as_ However_ the near-perfect cancellation of the three parts of ZDD a&o defines, in a sense, another small parameter, denoted X, with X = a/03_ We thus have, in relative units, ZOD = A,-
‘DT = cd)
iv,
I,
= (cY/u3)2?t~ ,
(9
where fl and y are unknown, pending the answer of the crucial question: does any further near-perfect cancellation exist in the multiple-scattering series_ The two possibilities are: and the DID model is questionable; (l)fl= l;thenZ*ZDD, (2) fi x 0; then ZDD z ZDT %, In, unless 7 behaves very oddly, Z = ZI,D + ZTD f ZDT, and the DID model may be successful. Case (1) has been assumed by all authors so far, but its v-ii Sty does depend on having ZDT x X2, which ~BS not been shown. Careful evaluation of the full ID= is now of first priority, but our guess is that case (2) holds. Near-perfect cancellation only occurs for a good reason, which, in ZDD, is the shaped-cavity correction (Zg& = -92X) to scattering in vacua (ZgA + Z$$J = 2X). We have separated and discussed those terms in ID-P for which we expect similar cancellation and do not see any reason to expect cancellation of the remainder. We think it likely that a theory of type (2) will agree with experiment_
References Ill W-M- Gelbart, Advan Chem. Phys. 26 (1974) I_ 383
Volume 64, number 3
CHPMICAL. PHYSICS LETTERS
15 July 1979
[2] GE_ Tabisz, in: hfolecufar Spectroscopy, Vat 6, Specialist Periodical Reports, eds. RF_ Barrow and D.A. Long (The Chemical Society, London, 1979). [3] L Frommhold, K-H_ Hong and M_H_ Proftitt, hfoL Phys. 35 (1978) 665,691. [4] F_ Barrochi and hf. Zoppi, Phys_ Letters 66A (1978) 99; 69A (1978) 187; D.P. Shelton and GE_ Tabisz, to be published[S] BJ_ Alder, JJ. Weiss and HI._ Strauss, Pbys_ Rev_ A7 (1973) 281_ [6] P-k Ffeury and J_P_bfcTague, Opt_ Commun l(l969) 164; W-S_ Gornalf. HE_ Howard-Lock and BP_ Stoicheff, Phys. Rev. Al (1970) 1288; KS_ Gabefnick and H.L Strauss, J_ Chem_ Phys_ 49 (1968) 2334. [7] T-K_ Lfm, B. Lfndcrand R_A_ Kmmhout, J. C’hem_Phyr 52 (1970) 3831; A-D. Buckfngham and R.S. Watts, DfoL Phys. 26 (1973) 7; E_F_ O’Brien, V_P_ Gutschick, V_ McKay and J_P_McTague, Phys_ Rev. A 8 (1973) 690; PJ_ Fortune, P-R Certain and L._W_Bruch, Chem_ Phyr Letters 27 (1974) 233; PJ_ Fortune and P-R_ Certain, J. Chem_ Phys. 61(1974) 2620; P_R Certain and PJ_ Fortune, J. Chem Phys. 55 (1971) 5818; W_ Meyer, Chem Phyr 17 (1976) 27; N_S_ Osthrnd and D-L hfcrriticld, Chem Phys_ Letters 39 (1976) 612; J-W. Kressand JJ_ Kozak, J_ Chem Phys_ 66 (;977) 4516; P. Laffemand, DJ. David and B. Bigot, hfol. Phys. 27 (1974) 1029; Rk Harris, D-F. Helfer and W-M_ GcIbart, J_ Chem Phys_ 61 (1974) 3854; D.F. HeIfcr, R.A. Hanisand W.&f. Gclbart, J. Chem. Pbys. 62 (1975) 1947; D-W_ Oxtoby and W.M. Gelbart, Mot Phys- 30 (1975) 535_ 181 N_S_ Ostfund and D-L._ MenitieId, Chem_ Phys. Letters 39 (1976) 612_ i9] P-D. Dacre, bfol Phys. 36 (1978i 541_ [lo] K.L. Cfarke~P_A_ hfadden and A-D_ Buckingham, Mol. Phys_ 36 (1978) 301. [ II] K-L. Cfarke, Ph.D. Thesfs, Univcrsfty of Cambridge (1978). [I21 Landolt-Bornstein, Zahfenwerte und Funktionen, 11-Band 8 Teil (Springer, Berlin, 1962). ]13] DJ. Counou,Trans_ Faraday Sot_ 65 (1969) 2654; S_ Kieffch, J_ Chem Phys_ 46 (1967) 4090; J_ Phys_ C 33 (1968) 191. [14] B.Sf_ Laianyi and T_ Keyes, bfol. Phyn 33 (1977) 1063,124?; T_ Keyes and B_M_ Lndanyi, hfoL Phyr 33 (1977) 1099,1271_ [15] B.U_ FekIcrhof, Physica 76 (1974) 486; H-hfJ. Boots, D_ Bcdeaux and P. hfazur, Physica A79 (1975) 397; E Hynne. bfoL Phys- 34 (1977) 681; B.Af. Ladanyi and T_ Keyes, bfot Phys- 31 (1976) 1685. 1161 B.Bf. Ladanyi andT_ Keyes, hfof. Phys. 34 (1977) 1643. 1171 kD_ Bucki@um and K_L._CIxke, Chem Phys. Letters 57 (1978) 321_ [IS] J-4 Barker and D_ Henderson, fbfoL Phys_ 21<1971) 187_
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