PHYSICAL RE VIE% 8 VOLUME 13, NUMBER 7 1 APRI L 1976 Effective pair interaction and the Van der Waals constants for second-layer helium atoms in bilayer films A. D. Novaco* Lafayette College, Easton, Pennsylvania 18042 (Received 16 October 1975) The effects of surface normal vibration on the effective pair interaction for 'He atoms in the second layer of a bilayer has been calculated. The single-particle potential for atoms in the second layer included pair interactions between the second-layer 'He atom and fixed carbon atoms in the graphite plus the pair interaction between the second-layer He atom and an oscillating first layer. The shift in the pair interaction between the second-layer atoms is about 10 — 15% of the "bare" interaction. Rough estimates were made of the effect upon the Van der Waals constants for the second-layer imperfect gas and these estimates were compared to empirical values. INTRODUCTION The experimental results for helium monolayer films are often interpreted in terms of theoretical calculations for matter in hvo dimensions. The second layer of a helium bilayer film has also been analyzed using these models. The physisorbed film does, of course, differ from a truly two-di- mensional system since the adatoms oscillate in the z or surface-normal direction. Reducing this quasi- two- dimensional problem to a two- dimen- sional one requires the replacement of the "bare" pair interaction v by an effective interaction p», this interaction being the average of e over the z- wise motion of the adatom pa. ir. In a recent pub- lication, it was shown that the difference between v and e» has a negligible effect on the properties of the first-layer helium liquid. ' However, an empirical virial-coefficient analysis of data for low-temperature helium adsorption isotherms5 indicated that this effect is important for the sec- ond layer. This analysis found that the van der Waals parameters a and b differed from the cal- culations for a two-dimensional imperfect quantal gas. The two-dimensional calculations for 'He predict (for T& 1 K) a second virial coefficient which (for the purposes of this calculation) can be fitted with little error by the van der Waals form B(T) = b — a/kT, with b = 8 A and a/0 = 53 A K. The empirical analysis found b = 7. 3 A and a/k = 32 A K for temperatures near 4 K. The dis- crepancy was attributed (in part) to the z-wise os- cillations of the adatoms in the second layer. This shift in a and 5 leads to a predicted depression of the liquid-gas transition temperature T, of about 50fp. The ca. lculation presented here is one of z}» using z-wise probability distributions calcu- lated for a helium atom in the second layer of a bilayer film. The calculated shift in the two- body interaction is then used to estimate the shifts in the a and b coefficients. These calcula- tions support the hypothesis that the large-ampli- tude, anharmonic oscillations of the second layer are responsible for much of the depression of the liquid-gas critical temperature which has been ob- served in these second-layer films, ' CALCULATION OF v~ D The calculation of v» involves averaging g. over the z-wise motion of a pair of adatoms using a probability density P(z) = IM(z) I for each adatom. The intera, ction v», where OO v, n(r} = dz, J dz, P(z,} wco 00 x V{[r'+ (z, — z, )']"')P{z, ), is then a function of x, where x is in the in-plane separation between the adatom pair. The function M(z) is the ground-state wave function for a single He atom in a substrate potential well U~(z) where the interaction Uo(z) is the average of the crystal- line substrate field, U(r, z) over a unit cell. The field U(r, z) is obtained by summing two-body inter- actions between the second-layer helium atom and both carbon atoms and first-layer helium atoms. The effect of the z-wise motion of the helium atoms in the first layer on U(r, z) is included. The wave function M(z) was calculated by using an expansion in an orthonormal basis set M "(z) using methods described in Refs. 3 and 8. The probability densi- ty P(z) is plotted in Fig. 1 for (a) the second layer using the Uo(z) in Ref. 8, (b) the second layer us- ing a U, (z) generated by ignoring the z-wise os- cillations of the first layer, and (c) the first lay- er. Table I contains the values of the average dis- placement of the helium atoms from the graphite surface plane and the rms deviation from the aver- age value for each P(z). The "bare" interaction used to calculate g~» was the I ennard- Jones 6-12 interaction with so= 10. 22 K and 0 = 2. 556 A. 9 vzn interactions for the various P(r) functions are plotted in Fig. 2. The "bare" interaction is es- sentially identical to curve c. The v» given by