1. Introduction

Virtually all physical science is concerned at some stage with comparing experimental data with theoretical predictions. Although no theories are ever fully verifiable, one nearly always wants to find that theoretical model, from the limited set of possible models under consideration, which best represents the data, which allows the underlying phenomena to be better understood, and, if possible, which allows prediction outside the range of the original measurements. In the relatively early stages of investigation of a given domain, one usually does not know which of several theoretical or empirical models is likely to be most appropriate. This state of affairs is particularly likely to occur when the physical situation being studied is too complex to allow a tractable theoretical idealization, which is still sufficiently close to the experimental situation, to be accurate. Many-body interaction problems, such as that of determining the exact equation of state of a solid or liquid, fall in this category.

The problem of model discrimination is made difficult by the presence of random and systematic errors in the data. In the present paper, it is assumed that systematic error in the data is absent or at least negligible compared with other error. Systematic error can still be generated, of course, by the choice of an inappropriate model [1],' and a question of considerable importance is: Under what conditions is it possible to discriminate adequately between several more-or-less appropriate models, or equations? In the present paper, we shall be concerned with typical synthetic equationof-state data generated without significant error of any

An invited paper.

**Present address: Texas Instruments Incorporated, P.O. Box 5936, Dallas, Texas 75222. Figures in brackets indicate the literature references at the end of this paper.

kind, reserving a detailed discussion of the effect of random errors to a later paper. It will be shown that by using such exact "data" we can investigate what sort of discrimination is possible between various equations of state in practical cases where measurements are of limited precision.

In real life, experimental data have only limited accuracy and precision and always extend only over a limited range of the variables involved. This state of affairs suggests intuitively that one will be unable to discriminate adequately between two or more analytical models which are sufficiently close together in their predictions for the range considered. We are here concerned with ways of making this intuition quantitative at least for the specific equations considered here. Since better discrimination may sometimes appear possible than is actually the case, just because of the presence of more or less random errors which happen to fall in a particular way, it is important to consider exact data before data with random errors.

Although all that is often required of an equation of state, or more generally, a mathematical model of experimental results, is that it serve adequately as an interpolation and smoothing device for the data, the problem of model discrimination is usually still present even in this case. Unless the first model fitted passes all tests of adequacy, more than one model must be examined and a choice of available models made. The present paper discusses some general methods of model discrimination with specific illustrations taken from the equation of state field. Here we are concerned additionally with the task of estimating physically significant parameters of the material which led to the data in question.

Two somewhat different situations frequently arise in the equation of state area. Often one starts with no,

or only crude, knowledge of the underlying parameters of the material under investigation. These parameters are then determined by fitting various equations of state to P-V data, usually by least squares techniques [1]. The most appropriate, or "best fit" equation will usually be that which leads to minimum estimated standard deviations of the fitted data points and of the parameters. The values of the parameters obtained from this fit are taken to be the best available estimates

of the unknown material parameters. In general, how ever, such values will not usually be good estimates unless the choice of model is appropriate for the data and leads to randomly distributed, essentially stochastically independent residuals, and the fitting procedure itself leads to negligibly biased parameter estimates.

Sometimes one is able to obtain estimates of the parameters by other means than from fitting of direct P-V measurements. Now becoming popular for this purpose is the method of ultrasonic velocity measurements under pressure [2, 3], pioneered by Lazarus [4]. Once having parameter estimates available for a certain material, one can, given the appropriate equation of state, calculate volume values for a range of pressure. Of course, with a limited number of parameters available, as is always the case, calculated volumes can generally only be expected to remain reasonably accurate over a limited range of pressure. The nub of the problem here is usually in knowing what equation of state to use (and how far to trust it). Sometimes the determination of the best equation may be made by comparing ultrasonically derived parameters with those obtained from a least squares fit of direct P-V data for the same material.

In either approach, one eventually obtains a set of parameters believed to be appropriate for the material under investigation. Although in actual practice these parameters will always be uncertain to some degree, it is nevertheless useful to ask, as a limiting case, how well one can distinguish between various equations of state when the parameters are actually exact (or are so considered) but when available P-V data are of limited precision. Some answers to this question are discussed later for eight different equations of state of some current interest.

One of the important purposes of the present work is to point out that uniqueness is a limit seldom achieved in practice. Frequently an experimenter chooses a model to represent data of given range with the implication or statement that the chosen model is "best" or "most applicable" without realizing or investigating sufficiently to find that other different models are equally applicable for the given data.

Although the present analysis is concerned with discrimination between eight specific equations of state and thus involves quantitative results only for these equations, we expect that the results will also apply at least qualitatively to other not-too-different equations. More importantly, perhaps, the present discrimination methods and general approach can and should be applied to any experimental situation where it is important to establish one or more adequate

mathematical representations of the data or, better, of the underlying process which led to the predictable part of the data.

2. Equations of State Considered

The material parameters with which we shall be concerned, all for isothermal conditions, are the specific volume, Vo, at a given reference pressure Po; the bulk modulus at P=Po, Ko=-Vo(aP/V) | P=P and various pressure derivatives of the bulk modulus, K, also evaluated at P=Po. For simplicity, let p=P—Po; then V=Vo at p=0. Now K = n = (ǝK/JP)|p=0, and Kő = (ə2K/P2) | p=0. The symbol has been introduced to simplify subsequent equations; it is dimensionless. It is also useful to introduce the further dimensionless quantity = KoKő. Finally, define the dimensionless pressure variable z = p/K。 and the dimensionless density variable x = p/po= Vo/V.

Barsch and Chang [3] have recently given values for the parameters of CsI at 25 °C, plus temperature derivatives of these parameters. The quantity Vo may be calculated from x-ray measurements of the lattice constant. Other parameters such as Ko, Kó, and K | were obtained from ultrasonic measurements. Using the Barsch and Chang results, we have calculated the values of Vo, Ko, n and which then apply to Csl at 150 °C, an arbitrary choice of temperature. These values, as used in our computer studies, have 14 figure accuracy and may be considered the accurate values of some hypothetical material close to CsI at 150 °C. I Of course as applied to CsI itself, only a few places in each parameter value are significant. To five figures, the parameter values Vo= 1.0184, Ko = 1.0503 × 102 kbar, ʼn ≈ 6.0382, and ↓ = −6.9897. Here we have taken Po=0 and Vo at 25 °C as unity. Thus, all volumes used here are reduced specific volumes and are dimensionless. The original Barsch and Chang 25 °C values are V。=1, K。=118.9±0.6 kbar, n=5.86±0.11 and Kő=-0.052±0.002 kbar-1. These results lead to Y = −6.2 at 25 °C.

are:

We shall be interested here in comparisons of, or discrimination between, eight different equations frequently employed in equation of state studies [1, 3]. We have adopted the approach of Barsch and Chang of designating the ordinary Murnaghan equation as the first-order Murnaghan equation (ME1), and the equa tion previously [1] termed the second-order equation (SOE) as the second-order Murnaghan equation (ME2). There are several forms of this latter equation, depending upon the values of n2 and V; here only one of these forms is pertinent. All eight of the equations are given in the form z= f(x) in table 1, which also lists acronyms for each equation. Some, but not all of them, may be expressed in inverse form, with x as an explicit function of z. Note that three of the equations are "first-order" in the sense of Barsch and Chang [3]. They involve n=3 parameters: Vo, Ko, and n. The other five "secondorder" equations involve in addition. Finally, table 1 includes values of K = (K/ǝP) p→x

Form

n

f(x) x = p/po = Vo/V

3

3

η

The Keane equation only applies when n2 <¥ <0, conditions satisfied for the present parameter values. Although all of the equations must become poor models for sufficiently high z, failure is particularly evident for the UTE and ME2. The volume predicted by the UTE goes through zero at the finite z value of (n+1)-1[exp (n + 1) −1]. For the present form of the ME2, K' = (aK/aP)=0 at z=-n/ and V=0 at z=2/[(n2-2¥) 1/2 — n]. The 3SE also suffers from the disadvantage that it predicts zero volume at finite pressure. All the other equations require infinite z to produce zero volume.

The equations of table 1 are discussed in greater detail elsewhere [1, 3]. Although most of them have some macroscopic or phenomenological theoretical justification, here they may simply be regarded as empirical equations likely to be of some value in the P-V area.

In order to examine differences in the predictions of the various models, we have, for a given set of p or z values, calculated corresponding dimensionless V values, using in each equation the same 14-figure parameter values already mentioned. The V values were calculated using 14 figures, by iteration when necessary, with a resulting 13-figure or better accuracy. Finally, differences between Vvalues of each possible pair of equations were calculated for each z value. The differences obtained for p=11 kbar, or z = 0.1047, are

listed in table 2, all multiplied by 104 for convenience. The AV's shown are formed by taking the V of one of the equations listed in the left column and subtracting from it the V calculated using one of the equations in the top row. Since the ME, -3SE AV value is largest of all, the ME, yields the largest and the 3SE the smallest value for this value of z. Similarly, we see that the BE2 and KE volume predictions are closest together here.

2

In addition, in figures 1 to 5 we have plotted AV versus z for a variety of equation pairs. The boxed equation name is the equation from whose V values those of the equations named on the curves are subtracted. These five figures contain AV curves for most, but not quite all, of the possible pairs of equations. Curves have not been duplicated. Thus (VBE2-VBE1) appears in figure 3 for BE, but not its negative in figure 5 for BE. Negative values are indicated by using dashed lines.

max

Clearly, AV curves for all pairs not involving the UTE, 3SE, or ME2 will eventually reach a maximum, with AVm <1, as z increases, then decrease toward zero since both V's become arbitrarily small as z→ ∞. As the figures show, the situation is different for the ME, even within the present range. Since the parameter values used here lead to V <0 for z≥ 1.85, AV values which involve ME volumes become arbitrarily large in magnitude as z increases beyond this point. Clearly, the ME2 cannot be a useful model all the way to the point where it predicts zero or negative volumes. Nevertheless, it may be useful for a range ending sufficiently far below this point.

2

Of what value are the results shown in figures 1 to 5? They are of considerable value because they show how well the various equations of state considered here may be discriminated under the best possible conditions. Suppose, for example, that we wish to discriminate between the KE and other equations and are able to measure volume only up to z=0.1. Further, suppose that errors in p are negligible compared to

those in V. Figure 2 then shows that to distinguish the 3SE from the KE in the range 0z≤0.1, experimentally determined V values must be known to about one part in 104, or to four decimal places, near z ~0.1. Even less uncertainty would be required for a smaller range. The BE2 and KE cannot be reliably distinguished without a precision of about three parts in 106 near z=0.1 and higher precision for smaller z. Clearly, if the above precision has not been achieved, there would be no point in attempting to discriminate between the equation pairs discussed for the data in question. Barsch and Chang [3] have discriminated between the BE2 and KE for a situation where AVIVO 3 × 10-3 or more and have concluded that the BE2 was much better for their purposes than the KE. The present figure 2 results indicate that such discrimination is actually not significant with such precision in AV, for the present set of parameter values, over a pressure range from zero up to at least 200 kbar.

=

2

There are two reasons why we consider that the present curves represent the best possible discrimination. First, there are always some random errors in the determination of pressure values. To first order, we may take the expected or "controlled" pressure values as exact and consider that the actual pressure errors are incorporated as additional random errors in the volume values. It is then this total volume error which must be used in determining whether the curves allow equation discrimination within certain range of z. When parameter values are available, as from ultrasonic measurements, they may be used in several equations of state to calculate exact volumes over a given z range. These volumes may then be directly compared with a set obtained by direct measurement. Clearly, if the total errors in the latter set are not appreciably smaller (over most or all of the z range) than the AV's obtained with various equation pairs, no valid discrimination is possible. Even so, one of the several equations among which discrimination is impossible for the given z range may be far superior to the others for extrapolation beyond this range. Although all eight equations of figures 1 to 5 are indistinguishable for AV data of no better than 10-3 precision in the range 0≤ 0.1, clearly there are important differences between the predictions of the various equations for this same precision level at say z=1.5.

When an independently measured set of parameter values is unavailable, parameter value estimates must be obtained by fitting a model to the available data by some such procedure as least squares. Each different model fitted will then yield a different set of estimated parameter values. If AV values are obtained for a pair of models, using in each model the specific parameter values determined from a least squares fit of the data for the given model and range (case A), then the adjustment of the parameter values associated with the least squares procedure will generally lead to an appreciably different set of ▲V values than would have been obtained had the same parameter value set been used in each equation (case B). If the fits of the two equations for case A are sufficiently good, the corresponding A values may nearly all be much smaller than those