Flow birefringence in binary para hydrogen-noble gas mixtures

Physica 130A (1985) 49&504 North-Holland, Amsterdam

FLOW BHUZFRINGENCE

IN BINARY PARA HYDROGEN-NOBLE GAS MIXTURES

H. VAN

HOUTEN,*

F. BAAS,

Huygens

P.M.J.

MARIE’

and J.J.M.

Laboratorium der Rijksuniversiteit, The Netherlands

Received

26 October

BEENAKKER

LAden,

1984

Experimental flow birefringence data for para hydrogen-noble presented. The consistency with available data on depolarized discussed. The results disagree with recent close coupled calculations

gas mixtures Rayleigh line by McCourt.

at 293 K are broadening is

1. Introduction The intermolecular

pair potential

(polyatomic) gases. An information about the the potential energy phenomena associated

determines

the bulk

properties

of dilute

analysis of these properties can therefore yield useful potential. Of special interest is the anisotropic part of surface. Especially sensitive to this part are the with angular momentum polarizations. Examples are

the field effects on transport properties and optical phenomena such as flow birefringence (FBR) and depolarized Rayleigh line broadening (DPR). Since straightforward inversion of the experimental data is not possible, in practice

the experimental

potential sections

surface on the level of effective collision cross sections. may be calculated numerically from the potential surface

results

are compared

with calculations

based

on a

Such cross employing

scattering theory. A relatively simple case is the interaction between a diatomic molecule and a noble gas atom. At present for the various modifications of the hydrogen molecules (exact) close coupled calculations are possible, while approximate methods have to be used for heavier molecules. The bulk properties of gases may be expressed in terms of the effective cross sections with the help of kinetic theory. It must be recalled, however, that these expressions can only be derived under certain assumptions. In order to get * Present ‘Present Netherlands.

address: address:

Philips Research Laboratories, Eindhoven, The Netherlands. F.O.M. -Institute for Atomic and Molecular Physics, Amsterdam,

037%4371/85/$03.30 (North-Holland

0

Physics

Elsevier Publishing

Science

Publishers

Division)

B.V.

The

FLOW BIREFRINGENCE IN BINARY MIXTURES

491

manageable expressions it is, for example, in general necessary to work with cross sections averaged over the rotational states. In many cases a single moment description works quite well, but unknown scalar factors will occur in such a description (see ref. 1). This complicates a comparison with scattering calculations. An exception to this situation is furnished by systems with only a single rotational level excited (e.g. para hydrogen at T = 300 K, where 98% of the molecules are in j = 0 or j = 2 states). From a theoretical point of view the most simple systems are thus binary mixtures of gases of single level molecules and noble gas atoms. Ab initio potential energy surfaces are now available for the interaction between modifications of hydrogen molecules and noble gas atoms2-5), and close coupled calculations of many cross sections have recently been performed6*‘). In this paper we will limit the discussion to para hydrogen-noble gas systems. Experimental information on such systems is in general difficult to obtain, since the small non-sphericity of the hydrogen molecules leads to quite small effects. A study of the viscomagnetic effec?“), which depends quadratically on the small production cross section for the z-type polarization produced in viscous flow, does at present not seem likely to yield sufficiently accurate data for para hydrogen-noble gas systems. This polarization can also be studied by optical methods, however. Since linear molecules are optically anisotropic the polarization will result in an anisotropic dielectric tensor. The medium will thus become birefrigent”13). This effect is called flow birefringence (FBR). Since it depends linearly on the strength of the polarization, and therefore linearly on the production cross section for the z polarization, the effect is still measurable for para hydrogen-noble gas mixtures. In this paper measurements of flow birefringence for such systems are presented. Flow birefringence data for N,and HD-noble gas mixtures have been reported earlier by Baas et al.14). It should be noted that birefringence in pH,-noble gas mixtures is sensitive to both the pH,-pH, and the pH,-noble gas interaction. Although quantities determined solely by the pH,-noble gas interaction may be extracted from the measurements by an extrapolation procedure (see section 4) the information content of the data can only be used to the full if scattering theory calculations are performed on both the pH,-pH2 and the pH,-noble gas interaction. The complete concentration dependence of flow birefringence can then be calculated and compared with the experimental results. 2. Theory The single moment kinetic theory of flow birefringence12) (FBR) was extended to binary mixtures by Kohler and Halbritter in 197413). As pointed out in

492

H. VAN HOUTEN

ref. 1, this approach overlap below)

integral

has in general

between

the tensor

and the one produced

may be concentration

et al.

to be modified relevant

in viscous

dependent15).

by the introduction

for flow birefringence

flow. In a mixture

For single

this overlap

level systems

of an

(see eq. (2) integral

this complication

does not arise, however. More

recently

a multilevel

treatment

was given by Liu and McCourt”).

It is shown in the appendix that for single level systems both theories lead to the same results, as is to be expected. Here only the final expression for the concentration dependence of the flow birefringence coefficient is given in the notation we use for the single moment description. In FBR experiments the observable is the anisotropic part of the dielectric tensor F’. A constitutive relation defines the flow birefringence coefficient p: T=

-2pE,

(1)

where n denotes the symmetric traceless part of a tensor and u the mean mass velocity of the mixture. In general a second contribution to F exists, due to differences fringence),

between the velocities of the various components (diffusio-birebut this contribution vanishes for the geometry of our apparatus’“)

(see section tensor iP3):

3). For

a description

of FBR

the

relevant

angular

momentum

(2) with ( )A,Oan equilibrium average with respect to the one particle distribution function for species A molecules; J* has the eigenvalue j(j + 1). The anisotropit part of the dielectric tensor is proportional to the non-equilibrium average of this tensor. The resulting expression for the flow birefringence coefficient as a function of the mole fractions xA and xn in terms of effective collision cross sections reads13) (with xn = 1 - xA, and xA the polyatomic gas mole fraction):

B=

XAEA[xAy,G(02JA),, (~,)I.4

fi

+ xn~(02)A),,]-

FLOW BIREFRINGENCE IN BINARY MIXTURES

493

with yi = {2m,/(m, + ma)}“* (note the different definition of yi in ref. 16) and (?I&0 = (SkTln;u,)“* with pAB the reduced mass of the pair of molecules A-B. Here d = [x,y&(201A),

+ x,6(20

IA),aI

~[x,G,(201B), + x,y,~(20P),,l-

~&J*(;~~)AB

3

(4)

and

(5) with ((Y”- a’) the polarizability anisotropy (S.I. units). When xA approaches zero the ratio p(x,)/x, reaches a constant value:

(6) with xAB given by

(7) From eq. (6) it follows that p(x,)/x, for xA = 0 is determined by the ratio between production and decay cross sections G($I~),/G(O2IA),. Finally, for xA = 1 the expression for the single component gas is recovered %U4A P(1) = LL(v,>,,V/z ~(2OlA),~(O2IA),,

(8) .

3. Experiment The experimental set-up used to measure flow birefringence is an improved version of the one developed by Baas et al. 14.17).A cylindrical Couette flow cell is employed, with inner cylinder rotating. The effect to be measured is very small (typical value for the refractive index difference is lo-l4 for pure pH,), so that long measuring times were necessary (typically 10 hours for a single data point). Hence special attention was given to the stability of the optical set-up.

TABLEI Experimental results for the concentration dependence of the flow birefringence coefficient para hydrogen-noble gas mixtures at 293 K (xa is the para hydrogen mole fraction).

p&-Ne

p&-He P(h) XA

0.870 0.778 0.755 0.616 0.604 0.523 0.393 0.349 0.292 0.217 0.145 0.083

P(l) 0.99

0.96 0.99 0.94 0.96 0.89 0.80 0.74 0.69 0.58 0.37 0.26

W&r

pHrAr

a-A)

xA

P(h)

P(1)

0.850 0.700 0.687 0.555 0.546 0.335 0.209 0.105

xA 0.895 0.825 0.713 0.649 0.617 0.490 0.465 0.422 0.302 0.201 0.152 0.108

0.98 0.98 0.99 0.89 0.85 0.70 0.46 0.24

PHz-Xe

P(x4)

xA

P(1) 0.96 0.93 0.8.5 0.79 0.76 0.62 0.59 0.55 0.46 0.35 0.26 0.19

0.846 0.704 0.616 0.514 0.376 0.259 0.180

for

P(1) 0.85 0.67 0.60 0.51 034 0.30 0.20

PM

xA 0.882 0.748 0.636 o.s13 0.338 0.204

P(l) 0.82 0.64 0.52 0.41 0.23 0.10

1.2

0.6

0.4

0.2 PlXnr $111

I 0

I

0

XA

I

I

a2

QL

I

0.6

I 0.8

d

1s

*

Fig. 1. p(x.&p(l) versus the para hydrogen mole fraction XA for pHrHe at 293 K. The estimated error is kO.05 for each data point. The drawn line reflects a two parameter fit of the theoretical curve to the measurements.

FLOW

BIREFRINGENCE

IN BINARY

MIXTURES

495

Measurements of the concentration dependent flow birefringence coefficient were made alternately for a pH,-noble gas mixture and for pure pH, in order to discriminate against sensitivity drift. The ratio p(xA)/p(l) was thus obtained. Frequent checks on the occurrence of spurious effects were made by performing measurements on a pure noble gas, for which no effect should be observed. The pressure of the gas mixtures was chosen sufficiently high in order to avoid pressure dependent corrections associated with Knudsen effects. Measurements have been performed for the systems pH,-He, -Ne, -Ar, -Kr and -Xe at 293 K. The results for /3(x&p(l) are listed in table I. As a typical example the experimental data for pH,-He have been plotted as a function of x, in fig. 1. The curve drawn through these points is a fit of the theoretical curve derived from eqs. (3)-(8) with two adjustable parameters R, and R, (see section 5). No values for R, and R, are given here, however, since the shape of the curve was found to be rather insensitive to R, and R, separately.

4. Data analysis In fig. 2 the data points for pH,-He have been plotted as /?(x,)lx,l3(1) versus xA. Similar plots have also been made for the other systems investigated. In this way extrapolation to x, = 0 (and multiplication by p(1)) directly yields the

Fig. 2. p(x,)/x~p(l) versus the para hydrogen extrapolated value at XA = 0 is given in table II.

mole

fraction

XA for

pHrHe

at 293 K. The

496

H. VAN HOUTEN

et al.

TABLE II Extrapolated results from the concentration dependence of flow birefringence (see text). Estimated error 20%. ,im P(XA) __ m-0 x.&P(l)

System

$L-He

3.6 2.7 2.0 1.0 0.47

pHTNe p&-Ar pE-Kr p&Xe

quantity

given

in eq. (6). The results

are listed in table

II. The estimated

error

is 20%. The value of the pure pH, flow birefringence coefficient at 293 K is p(l) = 0.29 x 10m16s (see ref. IS), while the polarizability anisotropy at the wavelength the FBR 0.35 x 10d40F m* 19).

experiments

were

carried

out

(632.8 nm) is (a”-

LY-) =

Values for the ratio G($/~)AB/G(O~jA)AB of production and decay cross obtained from the extrapolated results for sections may be lim __,,fi(~~)/~~j3(1) listed in table II, provided eA and xAB are known (see eqs. (5) to (7)). For this purpose the cross section @20(B),, has been calculated from the pure noble

gas shear

viscosity

coefficient

qB according

to

kT G’201B)BB

= (uBJ,rlB

’

with (oB)” = (16kT/n-m,)“2. (An analogous Since no sufficiently accurate experimental

expression is valid for G(20(A),,.) data are available for the viscosity

TABLE III The cross sections G(20/AjAA and &(20(B) aa at 293 K from experimental viscosity data”‘) further viscosity cross sections calculated from potentials at 293 K (see text). System

%‘~~B)BB

Potential

(10~” mZ)

p&-He p&-He pb-Ne p&--k p&-Kr p&Xe

18.4 18.4 18.4 18.4 18.4 18.4

11.7 11.7 16.7 32.6 42.4 58.3

HFD-B MSV(GH)l HF’D-B BC3(6-8)

23.7 24.2 31.1 44.9

-2.84 -3.08 - I .40 -0.98

13.3 13.7 4.36 3.20

and

FLOW

BIREFRINGENCE

IN BINARY

MIXTURES

497

cross sections G(201A), and G($It),, values calculated from various potentials by McCourt have been used instead (see section 6). The viscosity cross sections are listed in table III. For Kr and Xe such calculations have not been performed. The experimental results thus obtained for the ratio are listed in table IV. These results will be compared with ~(%&JWO2lA), calculations in section 6. TABLE IV Comparison between the theoretical cross section ratio &$&J@5(021A)Aa calculated from potential surfaces by McCou&‘) (T= 293 K) with experimental (FBR) values. The estimated experimental uncertainty is 20% (notation of single moment treatment; see text). System

Potential

p&-He

HFD-B MSV(GH) HID-B BC3(6-8)

pHTNe pHz--Ar

5. Consistency

Theory

0.049 0.061 0.106 0.067

1

with data on depolarized

Experiment

0.35 0.67 0.88 0.60

0.14 0.091 0.12 0.11

0.058 0.058 0.070 0.094

Rayleigh line broadening

From eqs. (3) and (8) it can be seen that -except for the viscosity cross sections -only two unknown parameters occur in p(x,)/p(l): the ratios

R

=

’

UP%), G(O2jA),

’

(10)

and

(11) If pH2 may indeed be approximated by a single level system at 293 K, the single moment treatment used here should be realistic. The relevant decay cross sections G(O21A), and G(O2/A), then are known from depolarized Rayleigh line broadening experiments (DPR). The ratio R, (see eq. (10)) can thus be obtained directly from the DPR data, so that eq. (3) (together with eq. (8)) can be fitted to the measured concentration dependence of p(xA)/p(l) with only R, as adaptable parameter (see eq. (11)). From the DPR data quoted in

498

H. VAN HOUTEN

ref. 21 it is found 293 K. The results Viscosity section

cross 6 have

indistinguishable

that

I?, = 0.90 for pH,-He

for pH,-He

sections been

et al

are plotted

calculated used.

curves.)

(The

from two

The values

and R, = 2.35 for pH,-Ne.

and

in fig. 3 and for pH,-Ne the

potential

surfaces found

The agreement

R, = 1.50 for pH,-Ne surfaces

for pH,-He

in fig. 4.

discussed

yield

From WXL4 tive error

in

practically

for R, are R, = 1.17 for pH,-He between

the measurements

and the

curves obtained in this way is not as good as one might hope, but discrepancies are still within the range permitted by the experimental certainties. theoretical

at

One may thus conclude that the internal consistency description of DPR and FBR is not contradicted by these

the un-

of the results.

the values found for R, in principle G($;),, can be determined since can be taken from VME measurements for pure pH,. The cumulawould

be rather

big, however.

I

p

I

I

I

H,-He

x.4 )

I

I

I

I

0.2

0.L

0.6

na

Fig. 3. Combined FBR-DPR consistency test for pHrHe. The curve reflects a single parameter fit of eq. (3) (and eq. (8)) to the flow birefringence measurements with RI = 0.90 from depolarized Rayleigh experiments and R2 as adaptable parameter (Rz = 1.17).

FLOW BIREFRINGENCE

I

IN BINARY MIXTURES

I

I

499

I

pH,-Ne

0.6

I

I

I

I

0.2

04

0.6

a6

xA Fig. 4. Combined FBR-DPR consistency test for pHrNe. The curve reflects a single parameter fit of eq. (3) (and eq. (8)) to the flow birefringence measurements with RI = 1.50 from depolarized Rayleigh experiments and R2 as adaptable parameter (R2 = 2.35).

6. Comparison

with close coupled calculations

The only available full quantum mechanical (close coupled) calculations (for T = 293 K) of the relevant cross sections for the pH,-He, -Ne and -Ar

interactions are the recent results obtained by McCourt6). The calculations were done for several potential surfaces. (In the case of pH,-Ar interpolated values from ref. 22 have been used.) For pH,He two potentials have been considered: the Hartree-Fock damped dispersion, type B (HFD-B) of Rodwell and Scoles4) and the Morse-Spline fitted Van der Waals (MSV(GH)l) potential of Shafer and Gordon’). The HFD-B potential was also used for pH,-Ne, while for pH,-Ar the BC3(6-8) potential by Le Roy and Carley’) was considered (see also ref. 22). McCourt did not perform calculations for the pH,-Kr and -Xe systems. For the cross sections related to the pH,-pH, interactions no data are available. As a consequence only the initial slopes of the concentration

500

H. VAN HOUTEN

dependence and decay

of ,f3(~,)//3(1) cross sections

IV. It is evident values

can be tested

directly.

for the pH,-noble

that no agreement

The results

gas interaction

between

for the ratio of these cross sections

et al.

the calculated

~(~~~),,/~(O~~A),,

for the production are listed

in table

and the measured is found,

except

for pH,-Ar. With additional cross section the full curves can be calculated.

data from the viscomagnetic According to ref. 9 6($2),,

effect for pure pH, = 0.010 x 10Y2”m’

and G(O2lA),, = 0.49 X IO-” m2 at 293 K. Even the resulting curve for pH?-Ar, where the ratio of production and decay cross sections is in reasonable agreement with theory, shows large discrepancies with the measurements (see fig. 5). Apart from pointing to a discrepancy between experiment and the calculated results based on the HFD-B, MSV(GH)l and BC3(6-8) surfaces, this clearly demonstrates that by a comparison between theory and experiment of the cross section ratio 63(!$~)~/(5(02lA),. alone valuable information is lost.

,_ l.i

I

I

I

I

I

I

0.2

0.1

0.6

I 0.8

PbAr I_ 1.0

0.6

XA

Fig. been and been

I

1.0

5. p(x~)/p(l) versus the para hydrogen mole fraction XA for pHrAr at 293 K. The curve has obtained from eq. (3) (and eq. (8)). V’tscomagnetic effect data for the pHrpH2 cross sections calculations based upon the potential by Le Roy and Carley for the pHrAr interaction have used.

FLOW BIREFRINGENCE IN BINARY MIXTURES

501

Only a comparison of the full curve with the calculated results allows a realistic test of a potential surface. Summarizing, it may be concluded that the calculations by McCourt are in disagreement with our measurements of the concentration dependence of flow birefringence.

Appendix

The multilevel mixtures

and single moment

theories of flow birefringence in binary

In this appendix the results of the multilevel treatment of flow birefringence developed by Liu and McCourt16) are summarized. It will be shown that the results of this theory reduce to those of the single moment theory by Kohler and Halbritte?) in the case of a single level system. This appendix will also serve to define the various notations used in this paper. For a discussion of the scattering theory expressions for the cross sections we refer to ref. 16 (the notation employed there deviates in some cases from the one used here). First some conventions about the notation of effective cross sections should be mentioned. The cross sections are essentially matrix elements of the linearized Waldmann-Snider collision operator with respect to normalized basis functions in the velocity (IV) and angular momentum (J) space. The basis functions f#@ consist of an irreducible tensor of rank p in W and of rank q in J. Here i is a species label (i = A, B). Scalar functions of W* and 5’ may also be present. In this paper, however, the W* dependence will be disregarded (in ref. 1 it has been shown that this is realistic for the polarization produced in viscous flow). It has already been mentioned that the single moment description of flow birefringence13) should be modified to take the scalar factor of the polarization produced in viscous flow properly into account. For a single level system this complication plays no role, however. In the multilevel approach no scalar factor depending on J* is needed, since use is made of a complete set of second rank irreducible angular momentum tensors, given in spherical tensor notation by OZjlA _ -

4,

C (-l)i-mfVS(A,-i,, -2,)ljm,> (jmil

,

with (1::) a 3-j symbol. These tensors are normalized according to

(A.11

H. VAN HOUTEN

502

with pi the fractional

occupation

et al.

of a j, m state

(pi = pj/(2j + 1)). (Note

with T!‘(j) in ref. 16.) 4, 02,jJAcorresponds While in the single moment approach by Kohler cross sections

are used, in the multilevel

theory

with elements

labelled

number.

arrays denoted

are one

by the j quantum

dimensional,

by G(zj];),.

example G”2’(02)A), proach the following

for example

that

et al.13) only j-averaged

arrays

of cross sections

The production

EJ(‘)(~(‘$~~, which

occur,

cross section

has components

The decay cross section arrays are two dimensional, for with components denoted by G(~$l~),. In this apresult for p as a function of the mole fractions xA and xB

is obtained16): d(l)

XAEA __. P= ~ (v,),A

[~,y,@~~(02(A),,

+ x&+~)(O~]A)~~]-

. pf2’

viii

. ’ b,(Y,~(20kh~

- G,<:gi;)A,) + x,~,(20iB),,l

(A.3) with d as in eq. (4) .sA as in eq. (5) and yi = {2m,/(m, + m2)}“2 (note a factor d/z difference with the definition employed by Liu et al.“)). Here the dots indicate contraction over j-labels and [ 1-i denotes inversion with respect to the basis formed by the tensors c$‘~,“* (see eq. (A.l)). Finally d(l) is a row vector with components dj equal to the reduced matrix elements of the tensor relevant for flow birefringence

40i’* (see eq. (2)):

(A.4)

and pC2)is a tensor with components the result for the flow birefringence

pij = 6,,pj. In the high dilution coefficient becomes

!!!?.[@*‘((Q(A),]-’ with xAB as in eq. (7). In the case of a single level system

. p(2). @l)($l;)AB

eq. (A.5) simplifies

limit (x, + 0)

,

considerably:

(A3

FLOW BIREFRINGENCE IN BINARY MIXTURES

lim

XA+O

p(x,)= XA

'A

_xm(p,)l/2

hB)LT~2 ’

G(%il%L3 G(O2,ilA)m *

503

64.6)

Here j is the quantum number of the excited level. For this single level case the production and decay cross section arrays occurring in the multilevel theory’6.z) can be seen to be related to the cross sections used in the single moment approach according to (A.7) and G(O2, jlA)-

= G(O2lA),

.

(A.8)

Similar relations hold for the cross sections representing the A-A interaction. It is clear, therefore, that eqs. (A.6) and (6) are essentially the same in this case. This also applies to eqs. (A.3) and (3) which describe the full concentration dependence of p. In the single level case both treatments are thus equivalent. In the case that a few levels are excited, however, the multilevel approach has to be used rather than the single moment theory. Finally a word is in order regarding the notation of the various cross sections. The close coupled calculations by McCourt are presented in terms of thermally averaged cross sections (see e.g. ref. 22). The relation with the cross sections used in this section is16)

WZ 2lN, = G&,2)),

(A-9)

and (A. 10)

The pure pH, cross sections have not yet been calculated. The viscosity cross sections are also presented in a different way in ref. 22. These cross sections are related to the ones used here by6) Q(20lB),

= (2)’

EJ~)+ G,‘,“,

(A.ll) (A. 12) (A. 13)

H. VAN HOUTEN

504

et al.

Acknowledgements The coupled

authors

wish to thank

calculations

This work tee1 Onderzoek

Prof.

for the pH,-He,

is part of the research der Materie

F.R. -Ne

McCourt and -Ar

program

(Foundation

for performing systems

of the Stichting

for Fundamental

ter) and was made possible by financial support Organisatie voor Zuiver-Wetenschappelijk Onderzoek tion for the Advancement

the close

at 293 K. voor FundamenResearch

on Mat-

from the Nederlandse (Netherlands Organiza-

of Pure Research).

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) IS) 16) 17) 18) 19) 20) 21) 22) 23)

E. Mazur, H. van Houten and J.J.M. Beenakker. Physica 13OA (1985) SOS, this volume, R. Shafer and R.G. Gordon, J. Chem. Phys. 58 (1973) 5422. W. Meyer, P.C. Hariharan and W. Kutzelnigg, J. Chem. Phys. 73 (1980) 1880. W.R. Rodwell and G. Stoles, J. Phys. Chem. 86 (1982) 1053. R.J. Le Roy and J.S. Carley, Adv. Chem. Phys. 42 (1980) 3.53. F.R. McCourt, University of Waterloo, Ontario, Canada. private communication. W.E. Kiihler and J. Schaefer, Physica IU)A (1983) 185. P.G. van Ditzhuyzen, B.J. Thijsse, L.K. van der Meij. L.J.F. Hermans and H.F.P. Knaap, Physica 88A (1977) 53. P.G. van Ditzhuyzen, L.J.F. Hermans and H.F.P. Knaap, Physica 88A (1977) 452. H. H&man, F.G. van Kuik, K.W. Walstra, H.F.P. Knaap and J.J.M. Beenakker. Physica 57 (1972) 501. A.L.J. Burgmans, P.G. van Ditzhuyzen and H.F.P. Knaap, 2. Naturforsch. 2Sa (1973) 849. S. Hess, Springer Tracts in Mod. Phys. 54 (1970) 136. W.E. Kohler and J. Halbritter, Physica 74 (1974) 294. F. Baas, J.N. Breunese and H.F.P. Knaap, Physica 88A (1977) 34. E. Mazur, Thesis, University of Leiden. lY81, chap. IV. W.-K. Liu and F.R. McCourt, J. Phys. Chem. 87 (1983) 2Y23. F. Baas, J.N. Breunese, H.F.P. Knaap and J.J.M. Beenakker, Physica 88A (1977) I. H. van Houten and J.J.M. Beenakker, Physica 13OA (1985) 465. this volume. N.J. Bridge and A.D. Buckingham, Proc. Roy. Sot. (London) Ser. A295 (1966) 334. Data Book, vol. II, Thermophysical Properties Research Center, ed. (Purdue University, Lafayette, Indiana, 1966). P.W. Hermans, L.J.F. Hermans and J.J.M. Beenakker. Physica 122A (1983) 173. J.M. Hutson and F.R. McCourt, J. Chem. Phys. 80 (1984) 113.5. W.-K. Liu and F.R. McCourt, Chem. Phys. Lett. 62 (1979) 489.