Volume 35, number 3 CHEMICAL PHYSICS LE TERS 15 September 1975

THERMAL NEUTRON SCA’I-TERING BY LIQUID ARGON

AND THE LNTERhfOLECULAR POTENTiAL

IVitold BROSTOW D~partemort de chimic, Liniwrsithde Afoorm&abl, Montr6al, QuPbec H3C 3 VI, Canada

Received 8 May 1975

Revised manuscripi received 18 June 1975

The most accurate neutron scattering data MIW available for a.rpon have been used to calcuhh the interatomic potcntiaL Comparison of results is made for two different procedures, and using severA approximations for each procedure.

Forces acting between molecules of gases, iiquids and solids are most often expressed in terms of inter- action potentials 11 as functions of molecular separa- tion R. We have quite a variety of ti(Rj equations,

starting from the simplest hard rods model, through Mie potentials [l] (often misnamed Lennard-Jones potentials) up to the function of Shavitt and BOYS

[2] containing an unlimited number of adjustable

parameters. Various starting points are possibIe for establishing z@?) functions. Quantum mechanical calculations now supply us with portions of potential energy curves, at least for noble gases [3]. Here we

are going to consider an approach which in principle supplies the entire curve [4] : calculation of u(R) from the ‘oinary radial distribution function g(R). We shall confine ourselves to argon, as it is for Ar that we are now making numerical calculations based on the

informational model of liquids [5] _ Moreover, it is for Ar that Yarnell et al. [6] have obtained possibly the most accurate g(R) data now available, resulting from thermal neutron scattering measurements at 85.0 K. In all our cakulatjons described below the data from ref. [6] are used.

The intermolecuiar potential u(R) may be related to g(R) through a rigorous expansion [4] in terms of the number density p =Iv/V:

a(R) = exp [-GYkTI I$o gi(R) pi

(1)

resuming pair-wise additivity of the configurational energy of the system, we may write

s,(R)= 1

9, (R) = A

z !+, + g,, +92x +g,, (4)

Consecutive terms in the third member of eq. (4) give a short-hand representation of the respective terms in the second member of (4). The cluster clia- grams above involve integrals over products of Mayer f functions:

z;:i(Rii> = exp [-~(Rij)/kT] - 1 . (5)

A line between a pair or points i and j Indicates the presence of the functionJT in the integrand; circles indicate coordinates held fiied in the integration.

Now some of the cluster integrals mzy be related to the in‘tensity of coherently scattered radiation such as X-rays or slow neutrons. Thus, considering the limit as R tends to zero, Jonah [7] has derived a formula, valid if the expansion (1) contains only the first two terms:

Here s is related to the scattering angIe and i(s) to the

387

Volume 3.5, number 3 CHEMICAL PHYSICS LETTERS 1.5 September i975

scattering intensity; they rare defined in ref. [4] or

ref. [6]. (Pings (41 uses tile same symbols as we do here, while Yarnell et al. (61 denote ours by Q and

our i(s) by S(Q) - 1.) Another relation between the scattering parame-

ters and some cluster integrals has been obtained by Pings [8,9]. Considering the difference between the

net radial distribution function giR)- 1 and the direct

correlation function c(R), he obtained

s - si’(s) _ = & o

JdW 1 + i(s) ‘Ii7 (sR)ds -2&R . (7)

According to pings [9], his equation should be valid

at densities for which Pv7 data can be adequately re-

presented in terms up to the fourth virial coefficient,

but negIecting fifth and higher virial coefficients.

Moreover, Ping [9] has also observed that by the definition of gz3 we have

Pz g23(R) = [P gl (R)]’ - (8)

The last result m;iy be uSed in conjunction with either (6) of (7). Tn.&n2 eq. (6) and remembering

that it has been based on the assumption

v(R)= 1 +pgJR), 6%

and also using (1) and (S), we obtain by pure a!gebra

P28*3W = 8-w __1 2 . 1 + J, @)/2&R 1 (10)

Alternatively, combining eq. (8) with (7) rather than with (6) and clearly neglecting the two terms of the

order of p2 in the first member of (7), Pings [9] has obtained

p2 g2#?) $(R)/(2Ti”pR)’ . (11)

Given the above expressions, we also consider, in

addition to (9d), other approximations to the func- tiony(R) as defined by (I). 'ihe Percus-Yevick ap- proximation [IO] is

AR) =go + pgl + p2 (&I f gz2) . Pb)

The convolution-hy_pernetkd chain approximation,

proposed independently by :;everal authors, is ti our notation

388

For a high density Mie 12-6 fluid Madden and Fi:!s

[i l] have found that (SC) is superior to (9b). For

lower densities, however, in spite of containing one term less, (9b) sometimes gives better results than (SC). Apparently cancellation of omitted diagrams plays a

role here. This had led Carley [13-l to sugest the formula

J,(R) =g(-J f Kl f A&l +&-zz f C&23) I (9d)

where cc is a flexible parameter, which can even be

considered as a function of R. In actual calculations, Cariey has treated cc as a constant, and he has found that its best value for a system of hard spheres is

equal to 0.27. We know that the hard sphere model,

in spite of its simplicity, gives valuable results for pure components as well as mixtures; essential aspects of

this approach are discussed in fieglewski’s monograph [13]. In our calculations based on (9d) we have thus

adopted cc = 0.27.

Terms of the order of p3 and higher in eq. (1) have been neglected in eqs. (9). Errors so introduced are

virtually unknown. We have no choice in the matter,

however, since there are no relations between the scattering intensity i(s) and gk for k > 2. Thus, in our

calculations we have used either (6) or (7) in conjunc-

tion with one of the eqs. (9). The resultingy(R) was substituted into (1) and rc(_K) extracted therefrom.

The necessary values ofg(R) have been obtained by

Yarnell et al. from their diffraction intensity data as

~~)-l=1~~~(s)sin(sR)ds. 2?i2pR 0

(12)

Essential results of our calculations are given in

table 1. Numbers in the first coIumn are for identifi-

cation only; E, represents the minimum potential

ener= value and R, the corresponding distance. For

comparison, we include values of E, and R, for u(R) functions proposed in :he literature; some of these have been listed in ref. [14] _ We include, among

others, the Bae potential; Bae has demonstrated [15] that his two-parameter potential gives better results than other two-parameters models, and comparable in accuracy to tfiree-parameter potentials. But it should be remembered that Bae values for Ar [l6] have been obtained from the second v-Sal coefficients only.

By “standard Jxnnard-Jones” we mear. the Me potential with m = 6 and 11 = lZS often used due toits simplicity, with the generally accepted characteristic

Volume 35, number 3 CHEMICAL PHYSICS LETTERS i5 September 1955

Table 1

Charnctcristic parameters for argon interaction potentialrc(R)

No. Equations Approximation cc Rm Cm/k (nn1) W)

1 (61, (5) (9a) 0 0.3728 -104.5 2 (7) (9b) 0 0.4560 -213.0 3 (7),(11) WI 0.27 0.4382 -147.0

4 (7),0 1) PC) . 0.3893 -111.5

Bat [ISI 0.3501 -173.6 Smith-Thakkar II [I 7 j 0.3617 -152.4 Kihara type, Barker et al. [ 1 S] 0.375 -142.1 multiparameter, Maitland-Smith 1191 0.375 -142.5 Parson-Siska-Lsc [20] 0.376 - 140.7 rz(R*)-6, hlaitland-Smith [21] 0.376 -142.1 Barker-Fisher-Watts [22] 0.3761 -142.1 standard iennard-Jones (23 1 0.382 -119.86 Mie, Lichtenthaler-Schifer 1241 0.382 -119.4

----

parameter values [23]. Lichtenthaler and Sckifer [24] have considered a Mie potential with WI = 6 and 11

adjustable, tut from the second virial coefficients

they have found FI = 12. In view of experimental and other errors, we do

not present full a(R) curves obtained. Now let us

consider the main resu!ts as shown in table 1. Case 1 is different from all the others; while it corresponds formally to the approximation (9s). the usual proce- dure of substituting an eq. (9) into (1) is not applied here. By using (6) in conjunction with (5) one ob-

tains [c(R) without the reference to g(R). This is an advantage, as the Fourier transform of scattering in- tensity eq. (12) is known to introduce some errors,

particularly for low scattering angles; this problem

has been discussed by Pings [4]. Case 1 furnishes us Hith a position R, of the potential enera minimum.

We have also somewhat modified case 1, by adding the term given by the r.h.s. of eq. (10). The results were virtually unchanged. In particular, the position

afR, remained the same within kO.0003 run, and

this might well be the true location of this minimum. As for the corresponding value of E, obtained in case 1, this is clearly influenced by neglecting higher terms in the expansion for y(R). Moreover, the depth of the attractive well is very sensitive to experimentA

errors in i(s). Mentine and Jones [25] have found that the attractive wall in U(R) for Na can be raised up to zero by LUI error in i(s) of only 0.006.

Cases 2-4 are based on eq. (7) instead of (6) We

notice that location ofR,, depends on the approxi- mation made. Also full U(R) curves show larger undu-

lations for values ofR >R, than the respective curve obtained in case 1. Apparently eq. (7) is more sensi- tive to experimental errors in diffractometric meas- urements than (6); this can be explained by the fact

that the denominator of the integrand in (7) represent

the square of the respective denominator in (6). The

well depth also depends on the approximation used.

In spite of experimental errors and neglecting p3 and higher terms in (I), there is one value of em which

agrees well with most of ihe potentials listed in the

second part of the table; it has been obtained for the Carley approximation (9d) and using the value of cc found for hard spheres.

Apparently, given the present accuracy of experi- mental data and the approximations involved, it is

not yet possible to Iccate the well depth properly. Inspection of potential parameters listed in table 1, however, provides us with an interesting observation:

I E, I decreases along with increasing R,. This is found when passing from the Bae potential through various functions to the _Mie potantial parameters found by

Jkhtenthaler and SchB’fer. Drawing the respective

E, versus R, curve, for R, = 0.373 nm we find Em/k = -147 f 5 K.

For further calculations, the parameter c, may be developed into a function of distance and possibly also,of density, in accordance with Carley’s suggestion

of improving moderate and high-density soluiions. It appears that the Jonah eq. (6) has not been used

before in actual calculations (cf. ref. [36]). We have found that eq. (6) is less sensitive to experimental errors than (7); given the current accuracy of thermal neutron and X-ray scattering data, we thus recommend

the use of (6). But it has to be remembered, that the Pings eq. (7) supplies us with more terms in the ex-

pansion (1); thus with an improvement of the ex-

perimental accuracy eq. (7) will become incre3singly more useful. We clearly need more scattering data, of accuracy at feast comparable to these in ref. [6], and in particular measured at low densities and high tem- peratures. When snch data wiil become available, an

important feature of eqs. (6) and (7) might become highly advantageous: both these equations have been derived without the assumption of pairwise additivity

of the intermolecular enerw.

389

Volume 35, ntimber 3 CHEMICAL PHYSICS LETTERS 15 September 1975

Sincere thanks are due to all those who have

commented upon the present work, in particular to Professor D.A. Sonah of Department of’ Mathematics, University of Sierra Lzane, Freetown, and to Dr. I. Shavitt of Battelle Coiumbus Ibboratories, Columbus,

Ohio.

1121 D.D. Carley, Phys. Rev. A10 (1974) 863.

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390