2.2. Single-Phase Region

All of the thermodynamic properties of the homoge neous fluid in the one-phase region can be obtained . directly from the free-energy F(p, T). We assume that no higher order discontinuities exist anywhere in this region including the critical isochore x=*. This implies that the expression for μ(p, T) and p(p, T) in the (x, t) plane exist and are everywhere convergent for large R< x < ∞.

In general, the isothermal compressibility KT is given by

with the thermodynamic stability [5] requiring that Kr be everywhere non-negative. The dominant thermodynamic behavior near the critical isochore (x large) is given by the leading term in this expression (s = 1) and is like that found from simple scaling. As the critical point is approached along the coexistence curve within the single-phase region, the compressibility diverges as t' where y' = ẞ(8 − 1), provided Mi ( − xo) # 0.

The thermodynamic behavior of the specific heat in the homogeneous phase is given by

with o(x)=0 (neglecting the smooth contribution from Fo(p, T)). Since p1(x) = p.μ1(x), the lowest possible contribution to the specific heat must come from an s 2 term in this expression. Here in the single phase, just as in the two-phase region, the available experimental evidence suggests that the pressure term (82p/8T)p. is much larger than the chemical potential term [12]. About the critical isochore above Te, the specific heat varies as -". where a 2-ẞ(d+s−1), with s determined by the first nonvanishing term in the limit as xx. (The anomaly in the specific heat along the critical isochore is also given by Fr ta, where a=2-ẞ(8+r) and F, denotes the limit of ƒ (x) · x −B(ô+r (1) as x→ ∞.] Such structure admits the possibility of different values for the critical indices a and a'. If, for example, the leading s' = 2 were finite in the two-phase region and then vanish above T. at large x [see eqs 15, 17], s

would be greater than 2 and thus the difference |aa|ẞss' could be greater than zero.

3. Discussion

The extended formulation of thermodynamic scaling presented here provides a quite general description for the critical region of a fluid. The formalism proceeds beyond the usual lowest order asymptotic behavior and accommodates both the chemical potential and pressure as equivalent physical variables. Within the description it is possible to note contributions from the various higher ordered terms (s> 1) and of the two quantities p, μ to the various physical anomalies. We found, for example, that the difference between the potential and pressure variables is not only reflected in the shape of the coexistence curve (as determined by the value of x1), but also enters into the leading behavior of the specific heat.

We also expect extended scaling to be valid beyond the asymptotically small range of simple scaling, with contributions from the additional terms affecting the determination of the values of the critical indices away from the acutal critical point. For example, the experimentally determined shape of the coexistence curve B* would be greater or less than the true ẞ, the modification depending upon both the sign and magnitude of the additional terms of eq (1).

In the rather special cases where 1/B is an even integer, the formulation recovers the classical thermodynamic description of a van der Waals system. For B=2, this is just the familiar parabolic coexistence curve of the mean-field result, with the usual rectilinear diameter p=pe+ pet.

As mentioned before, any comparison of these results to the physical features must remain somewhat qualitative. This is because the formulation is essentially analytic everywhere and must therefore suffer the same defects as all "classical" equations of state. However, we believe the utility of this analysis lies in the display of certain features which must eventually be contained in any complete thermodynamic description of the critical region.

The author wishes to thank J. M. H. Levelt-Sengers and Melville S. Green for discussions involving this work.

4. References

[1] Widom. B., J. Chem. Phys. 43, 3898 (1965).

[2] Kadanoff, L., et. al., Rev. Mod. Phys. 39, 395 (1967). See also Fisher, M. E., Repts. Prog. Phys. XXX, 615 (1967).

[3] Domb, C., and Hunter. D. L... Proc. Phys. Soc. 86, 1147 (1965) Compare their series in the Ising model to our eq 2.

[4] Vicentini-Missoni. Levelt Sengers, J. H. M., and Green, M. S.. J. Res. Nat. Bur. Stand. (U.S.), 73A (Phys. and Chem.). 563 (1969): also Phys. Rev. Letters 18, 1113 (1967). [5] A complete discussion of the thermodynamic properties of simple scaling is given by Griffiths, R. B., Phys. Rev. 158. 176 (1967).

1. Introduction

The dependence np1/n*3 is known to be a good approximation for simple spectra [2, 3, 4]. The change in n* corresponding to the fine structure splitting is thus approximately constant for a series with structure proportional to up (np P and snp 3P2-0 intervals) [2. 4]. Observations of unperturbed series of this type usually show An* to be constant within 2 percent, except for the lowest series member or two. The constants in (1) were determined by graphically fitting the data from such series in ten spectra,2 indicated by stars after the element symbols in table 1. The series members that determined the value of ζωη for each of these spectra are listed under the heading "np Data." These ten spectra range in atomic number from 11 to 81, with an average disagreement of 6 percent between the observed value of n* and

*3

Figures in brackets indicate the literature references at the end of this paper.

It is notable that the data for eight of these spectra are from accurate observations made in Lund, Sweden, during the past 15 years.

the value from (1). The maximum disagreement is 12 percent.

3

The other data in the table are less reliable. Most configurations of the type (core) np with the core including a partly filled p or d subshell appear to be distorted by perturbations that are significant compared to up. The data were usually taken from the lowest such configuration, to minimize the relative error in S. The n* values for these lowest configurations are systematically larger than the limiting constant values (from eq (1)) by~ 15 percent. At least part of this effect is understood [3, 1]. Part of the remaining scatter in these values is probably due to configuration interactions neglected in the calculations. In some spectra there may also be uncertainty in obtaining properly corresponding values of 5, and n* [1].

It seems likely that the formula values of n (table 1) are correct to~ 15 percent for (core) np configurations (n greater than any principal quantum number in the core) in all atoms Z = 10 to 90. This accuracy must of course not be expected if perturbation has affected the value of n *3 by more than a few percent. Any other large percentage deviations from the formula are suspect and probably unphysical. In many cases rather complete and detailed calculations would be necessary to obtain values of , by the levelfitting method as accurate as the formula values. It would thus be advantageous in some instances to fix this parameter.

Some examples of these various points may be inferred from table 2, where values of obtained by different methods for several rare-gas configurations are compared. The smaller values of ,,, as in Ne, are

* Only a representative selection of the data available from complex spectra is given in the table. The un values listed for the first few elements following hydrogen are not accurate, since other interactions that contribute significantly to the term intervals were not allowed for. The results and discussion are not applicable for Z ≤ 5.