PAST -DJ 68-0061 Reprinted from THE PHYSICAL REVIEW, Vol. 166, No.3, 703-709, 15 February 1961

Printed in U. S. A.

Equation of State for Sodium Metal

D. JOHN PASTINE U. S. Naval Ordance Laboratory, Silver Spring, Maryland

(Received 26 July 1967 j revised manuscript received 2 October 1967)

The equation of state of solid sodium metal is derived in order to provide an additional tool for the calibration of very high-pressure devices. The derivation divides itself naturally into two parts. The first part contains a calculation of the OaK isotherm, which is based on the already well-developed theory of the alkali metals. In the second part, the thermal contribution to the pressure is determined. The calculations in this part include an evaluation of the Griineisen parameter and an estimate of anharmonic effects. The comparisons which are made between theory and experiment show the over-all agreement to be quite good.

I. INTRODUCTION

I T is the purpose of this work to present for possible use in the calibration of very high-pressure devices a complete (P, V,T) equation of state for solid sodium metal. The need for accurate equations of state in high-pressure calibration and the mechanics of their use have been elaborated elsewhere,l and to date only one other equation of statel has been developed for this purpose. As a basis for pressure calibration, sodium has two important advantages. The first is that it is highly compressible, a quality which implies that small errors in the measurement of volume will not lead to gross errors in the associated pressure. The second advantage is that sodium has consist~ntly shown itself to be quite amenable to theoretical treatment. This, of course, helps to inspire confidence in the results of theoretical calculation. Against these two advantages one must weigh the disadvantage that sodium is chemically quite active, a characteristic which will sometimes require it to be chemically insulated in use.

Theoretical treatments of various aspects of the equation of state of sodium abound, and this work is built in great part on the works of previous authors. Some contributors2-5 whose work may clearly be seen to relate to this one and who are not referenced else-where are given in Refs. 2-5.

Section II deals with the calculation of the two quantities fundamental to any equation of state. These are the specific energy along the OaK isotherm and the Grtineisen parameter 'Y. Since neither one of these quantities submits to exact treatment for real materials, approximations are used whenever necessary. Such approximations as are made are kept within the bounds demanded by reason and thermodynamic consistency.

In Sec. III, the predicted (P, V,T) relationships are presented for some specialized equations of state (isothermal and Hugoniot), and, where possible, com-

1 D. L. Decker, J. App!. Phys. 36, 157 (1965). J J. Bardeen, J. Chem. Phys. 6, 367 (1938). I P. Gombas, Die Statistisclte Theorie des Atoms ftnd lilre

Anwendungen (Springer-Verlag, Vienna, 1949). 4 K. F. Berggren and A. Froman, Quantum Chemistry Group,

Uppsala University, Uppsala, Sweden, Report No. A4443-411 , July, 1965 (unpublished).

i J. C. Raich and R. H. Good, J. Phys. Chem. Solids 26, 1061 (1965).

166

parison is made with experimental data. In general, the agreement between theory and experiment is quite favorable.

II. THEORY

A. List of Symbols and Description of Units

The symbols (those not defined in the text) of all the quantities to be considered in subsequent calcu-lations are listed below, together with their significance and the units in which they are calculated.

Symbol Significance Units

x Specific volume divided by the dimensionless specific volume at P = 0, T = 0

Po Pressure along the OaK isotherm kbar P. Pressure contribution arising from kbar

OaK lattice vibrations Do Density at P=O, T=O glcc 'Yo Value of the Griineisen parameter dimensionless

in the quasiharmonic approxi-mation

n Number of atoms per gram number/g k Boltzmann constant ergrK

k=1.38X1o-16 ElD Debye temperature OK e Magnitude of the electron charge statcoulombs

e=4.80223X1O-10 Ro Radius of a sphere of volume

equivalent to the volume per Bohr units and em

atom in the solid R"o Value of R" at P=O, T=O Bohr units and cm Vo .volume per atom cm-1l

B. Griineisen Equation

What is sought is an equilibrium relation between P, x, and T. The well-known Grlineisen equation provides such a relation, and that is

(1)

Equation (1) is easily derived from the precepts of statistical mechanics,6 and it is valid provided that T is large compared to €I D, and provided that the normal-mode frequencies are almost completely in-dependent of vibrational amplitude at constant volume. When the latter condition is not completely fulfilled, it can be shown quite generally that, if the anharmonic contribution to the free energy is small, then in the range ElD< T < 3ElD, the first-order correction to Eq. (1)

I J. E. Mayer and M. G. Mayer, StatisticaJ Mechanics Oohu Wiley & Sons, Inc., New York, 1961).

703

704 D. JOHN PASTINE 166

is such that7

(2)

in which 11"( is the product of T and a function of x only.

This correction 11"( to "(0 arises from first-order an-harmonic effects which are neglected in the quasi-harmonic (QH) approximation (i.e., the approximation which assumes the normal-mode frequencies to be functions of volume only and independent of amplitude at constant volume). In these calculations, it will be assumed that second-order anharmonic effects, at least in the ranges T::=; 30 D and x::=; 1.1, are completely negligible and that Eq. (2) is an accurate relation.

C. Cohesive Energy

Since Po= -oo(dip/ dx), ip(x) must be evaluated. Four separate contributions to ip(x) will be considered :

(1) ipE (x), which is the contribution due to the valence electrons;

(2) ipI (x), a repulsive contribution due to interactions between ion cores;

(3) ipW(x), arising from the attractive Van der Waals in teractions between ion cores;

(4) ipV(x), a contribution due to the OOK lattice vibrations.

All other contributions to ip(x), such as the one8 which arises from the volume variation of the core wave functions, will be considered either constant or negli-gible in our range of interest. Reasonable semitheo-retical expressions for ipI and ipw already exist,9-11 and these will be used here. The contribution cpI to ipI due to the interaction between one ion and one nearest neighbor, a distance r angstrom units away, is given asH

cpI= (1.25X1(j12) exp[(1.75-r)/ 0.345] erg. (3)

The analogous contribution cpw to ipw isll

cpW= (-2.5XlO-!2)/r6 erg. (4)

Considering only interactions between nearest neigh-bors, the expressions for ipI (x) and ip W (x) for the sodium bcc lattice are as follows:

ipI(X) = (130.92X109) Xexp[S.07(1-2.0611xI/8)] erg/ g, (5)

ipW(x)= (-0.1187X109)/ x2 erg/ g. (6)

ipv can be estimated using the Debye approxi-mation, so that

ipV(x) = (9/ 8)nk0D= (4.0651 X 106) 0 D (x) erg/ g, (7)

7 D. J. Pastine, in Proceedings of the IUTAM, Paris, 1967 (to be published).

8 H. Brooks, Nuovo Cimento 7, 207 (1958). g M. Born and ]. E. Mayer, Z. Physik 75, 1 (1932). 10]. E. Mayer and W. Helmholtz, Z. Physik 75, 19 (1932). 11 K. Fuchs, Proc. Roy. Soc. (London) A153, 622 (1936).

where for eD, the approximation12

eD(x)= e D(l)/x(1+'Yo)

is used, and eDl!) is taken asl3 152°K. By far the greatest contribution to ip(x) is given by ipE(X). ipE(X) may be considered the sum of four conceptually distinguishable contributions. These are as follows:

(1) ipoSE(X), the contribution (ground-state energy) which one would have if all valence electrons were in the ground state (i.e., not translating throughout the lattice) and interacted with nothing save their respec-tive ion cores;

(2) ipFE(X), a contribution (the Fermi energy) which occurs because the exclusion principle prohibits all the valence electrons from being in the same energy state;

(3) ipexE(x), a contribution (exchange energy) which results from interactions between electrons and de-generacies which occur because of their indistinguish-ability;

(4) ipcorrE(x), a contribution (correlation energy) which arises because valence-electron motions are somewhat correlated because of the mutual Coulomb in teractions.

It has been pointed out by Brooks8,14 that for the alkali metals (particularly sodium and potassium) and certain polyvalent metals, the term ipFE(X) can be calculated with reasonable accuracy in the free-electron approximation. Since this point of view is the result of rather exhaustive theoretical investigations and is supported in large part by experimentap5-17 evidence, it will be adopted here. The experimental support arises because, in keeping with the predictions of the free-electron approximation, the Fermi surfaces of sodium, potassium, rubidium, and cesium have been observed to be very nearly spherical, with the sphericity and Fermi momenta given by the free-electron values to within experimental error. From the theoretical point of view, it has been observed that deviations of the density of energy states at the Fermi surface from the free-electron value can be simply related!8 to deviations of the Fermi surface from sphericity. Moreover, the rather elaborate calculations of Ham!9 and Bienenstock and Brooks20 imply that the Fermi energy for sodium is even more free-electron-like than it is for the rest of the alkalis. Thus, from both the theoretical and experi-mental points of view, there is a strong indication that

12 See, for example, D. ]. Pastine, J. Appl. Phys. 35, 3407 (1964). This approximation should be reasonable, since the calculated 'Yo(x) is a very slowly varying function of volume.

13 D. L. Martin, Phys. Rev. 139, AlSO (1965). 14 H. Brooks, Trans. AIME 227, 546 (1963). 16 C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963). 16 H. J. Foster, P. Meijer, and V. Mielczarek, Phys. Rev.

139, A1849 (1965). 17 K. Okumura and 1. M. Templeton, Phil. Mag. 7, 1239

(1962) j 8, 889 (1963). 18 M. H. Cohen and V. Heine, Advan. Phys. 7, 395 (1958). 19 F. S. Ham, Phys. Rev. 128, 82 (1962) j 128, 2524 (1962). 10 A. Bienenstock and H. Brooks, Phys. Rev. 136, A787 (1964).

166 EQUATION OF STATE FOR SODIUM METAL 705

if>FE for sodium is given by the free-electron value21 to within 2 or 3%. Since it can be expected that the sodium valence electrons behave very much as if they were free, it is reasonable to use the free-electron approximation for the correlation and exchange energies as well. Therefore, following Brooks8.14 and Raimes,22 the following expressions (in atomic units) for the correlation, exchange, and Fermi energies per atom will be used:

IPexE = 0.284/ Ra

IPFE= 2.21/Ra2

Ry, (9)

Ry. (10)

Equation (8) is an approximation developed by Pines23

for the region of metallic valence-electron density. Equation (9) is the result24 of a first-order perturbation correction to the energy of a free-electron gas, in which the balance between the self-exchange and self-Coulomb energies has been taken into account. Since they are both approximations, neither Eq. (8) nor Eq. (9) is exact. Their contributions to the cohesion are suffi-ciently small, however, to make the inexactness of the sum IPcorrE+ IPexE unlikely to lead to much error. For example, at P= 0, T= 0, one can calculate25.2~ Rao= 3.93 Bohr units, at which point, by Eqs. (8) and (9), IPcorrE+lPexE=+0.003 Ry,27 One may compare this with the experimental cohesive energy per atom -0.0829 Ry.28 The relation Eq. (10) is well known, of course, and for perfectly free electrons it is exact. Again taking Rao= 3.93 Bohr units, one can write Eqs. (8), (9), and (10) in terms of x [i.e., X= (Ra/ RaO)3] and convert to cgs units to obtain

if>exE(x)+if>corrE(x) = 109[41.2058/x1/L 79.887+35.7213 Xln(3.9329x1/3)-5.8515x1/3] erg/g (11)

and if>FE (x)=109(81.511/x2/B) erg/g. (12)

The quantity if>GSE(x), which provides the greatest

21 Large apparent deviations from the free-electron value of the density of states at the Fermi surface have been experimentally observed for sodium and potassium (see Ref. 15) . The apparent density of states, however, is significantly affected by many electron and electron-phonon interactions. A brief discussion of these effects has been given by Brooks (Ref. 14).

22 S. Raimes, Phil. Mag. 43, 327 (1952). 23 D. Pines, Nuovo Cimento Suppl. 7, 329 (1958). Sj See, for example, Ref. 3, p. 22. U G. A. Sullivan and J. W. Weymouth, Phys. Rev. 136, A1141

(1964). 28 R. J. Corrucini andlJ. J. Gnieweck, Natl. Bur. Std. (U.S.)

Monograph No. 29 (1961). 27 Brooks (Ref. 14) has estimated a correction to the sum

tpccrrE and tp.".E for sodium based on calculated deviations of the valence-electron density at Ra from the free-electron values. When estimated in this way, the correction is rather large, because at R = Ra the valence-electron density can deviate by as much as 10% from the free-electron value. If, however, the correction is averaged over the entire atomic volume using typical sodium wave functions, it becomes entirely negligible.

28 D. R. Stull and G. C. Sinke, Thermodynamic Properties oj the Elements (American Chemical Society, Washington, D. C., 1956), Vol. 18.

contribution to if>E(X), will be calculated in the manner proposed by Hellmann and Kassatotschkin.29 This method requires that one select reasonable analytic forms (with adjustable constants) for both the zero-order valence-electron wave functions and the valence-electron-core interaction. The constants which appear in the wave function are determined by minimizing the valence-electron energies with respect to them, while the constants which appear in the interaction energy are determined by demanding agreement between theoretical and experimental determinations of specific energy levels. This technique (with some variations) has recently been used successfully by Szasz and McGinn30 to calculate the spectra of heavier atoms. When applied to the free alkali atom, the method enables one to predict with reasonable accuracy the energy change between the first excited sand p states, a change which is typically about 0.08 Ry. Now, as it happens, the change in the ground-state energy per atom IPGSE for sodium in the range of compression 0.4~ x~ 1.04 is only about31 0.05 Ry. This suggests that if the constants in the valence-electron-core-interaction term are adjusted so as to reproduce the solid-state valence-electron energy and some of its derivatives near P = 0, T = 0, there is a very good chance (provided the other contributions to if> are determined with reasonable accuracy) of predicting it accurately over a large range of compression. Moreover, one can hope to absorb in these constants some of the error in other approximations. So far as the zero-order wave function is concerned, it is certainly reasonable to take it to be constant. This is not at all a rash assump-tion. As a matter of fact, it was shown by Wigner and Seitz31 and Frohlich32 that the ground-state wave function 1/IGS for sodium shows considerable constancy between X= 1.1 and the point at which if>GSE has a minimum, which, for sodium, corresponds to an x of about 0.4. The ground-state wave functions do, of course, show some spatial variation, especially in the region of the core. This is because 1/IGS must have nodes which result from orthogonalization requirements. How-ever, it was observed by Hellman29 that the orthogonali-zation between valence and core wave functions causes the valence electron to behave as if it were acted upon by a repulsive potential (pseudopotential) in the core region. Hellman made use of this concept by choosing a form of the valence-electron-core interaction which has a minimum and is Coulombic in character at sufficiently large distances from the core. More re-cently,S3-35 the validity and accuracy of this approach

29 H. Hellmann and W. Kassatotschkin, Acta Physicochim. URSS 5, 23 (1936).

80 L. Szasz and G. McGinn, J. Chem. Phys. 42, 2363 (1965). 81 E. Wigner and F. Seitz, Phys. Rev. 43, 804 (1933) . 32 H. Frohlich, Proc. Roy. Soc. (London) A158, 97 (1937). 33 L. Kleinman and J. C. Philips, Phys. Rev. 116, 287 (1959) . 14 M. H. Cohen and V. Heine, Phys. Rev. 122, 1821 (1961) . 35 J. A. Hughes and J. Callaway, Phys. Rev. 136, A1390

(1964) .

706 D. JOHN PASTINE 166

have been much more thoroughly established. It has been shown, for example, that the valence-electron wave function which results when the proper repulsive potential is included in the Hamiltonian, has the same eigenvalues as the true valence-electron function, but inside the core it has no nodes. Because of this, it is advantageous to use the pseudopotential concept, since the associated nodeless (pseudo) wave functions for sodium will show greater constancy throughout the atomic volume than the true wave functions.

In accordance with Hellman, it will be assumed that the valence electron at a distance R from its respective core sees a potential vCR) given by

v(R)= - (e2/R)[1-Ae-KR]. (13)

This form is chosen because, aside from being physically quite reasonable, it is both simple and analytic. It is reasonable because, for large enough R, Eq. (13) reduces to Coulombic,36.37 and the repulsive term (to which CPGS E is not very sensitive35) provides a reason-able approximation to the true repulsion. Since the zero-order wave function is taken to be constant (i.e., 1/IGs*1/IGs= 1/ Va), with Eq. (13), the energy CPGSE (Ra) can be calculated quite simply by the relation

(14)

which, after integration, becomesB8

CPGS E = -3e2/ 2Ra+(3e2A/ Ra3K2) x[1- (1+KRa)e-KRaJ erg/ atom. (15)

Again taking Rao= 2.0809 A and substituting e=4.80223 X 10---10 esu, one can easily calculate il>GSE (x) from Eq. (IS) and obtain

il>GSE (x) = 109{ -435.273/ x1/3+ (201.043z/ x) X[1-(1+yx1/3) exp(-yx1l3)J}, (16)

where Z= (A / K2) X 1016 cm2, y=KRao, and the units of il>oSE(X) are erg/ g. Summing the contributions of Eqs. (5), (6), (7), (11), (12), and (16) provides a complete expression for iI>(x). The pressure Po along the OaK isotherm is then given by

Po= -ortlil>/dx . (17)

D. OaK Isotherm

Since accurate predictions of pressure are desired, It is sensible to evaluate the parameters z and y in Eq. (16) by requiring the theoretical values of Po to agree as well as possible with the best available experi-mental data. This has been done using the low-tempera-

36 The unlikelihood of appreciable deviation of the potential from Coulombic in the region between the core and Ra was first pointed out by Wigner and Seitz (Ref. 31) .

17 L. Szasz and G. McGinn, J. Chem. Phys. 45, 2898 (1966). as The resemblance of this relation to the one derived by

Frohlich (Ref. 32) is noteworthy.

22 .0 - THEORETICAL

20 .0 1 :: :~ p. (~b.r)~ 'EXPERIMENTA L 14 .0

12.0

"'"" '~ '-....."-10 .0

B.O

6 .0

•• 0

2.0

O.O'------'------'---_-'-__ .........::::.J 0 .800 0.850 0.900 0 .950 1.000

FIG. 1. Result of fitting the theoretical oaK isotherm to the experimental (Ref. 39) data.

ture experimental data of Beecroft and Swenson,s9 The results are shown in Fig. 1, with a more complete version of the theoretical isotherm given in Table I. The values of z and y which best reproduce the experi-mental data are Z= 0.4325 and y= 6.71. The fact that the theoretical sublimation energy at P 0= 0 (48.08X 109 erg/ g) is so close to the experimental value (47.314X 109

erg/ g) is a strong indication that the theory is generally sound and that the calculated value of il>osE is quite close to the true one. In order to inspire confidence in pressure values calculated beyond the experimental range, it is necessary to show that the approximation, Eq. (16), for il>osE (which is fit to the experimental data in the range 0.840:::; x:::; 1), even though it may not represent the true ground-state energy (because possible errors in the calculation of other quantities have been absorbed into it), will still provide a very good approximation to the total energy and pressure in some range of x

166 EQUATION OF STATE FOR SODIUM METAL 707

which is reasonably similar to, but not necessarily equal to, the true one. That this is probably the case can be shown in several ways. First, if any or all of the following possible corrections to the equations for Po are made, and the parameters z, yare readjusted so as to restore agreement with the Beecroft and Swenson data, the resulting isotherms do not deviate from the values in Table I by more than 1.5% at 100 kbar. The corrections are:

(1) the addition of ±10% to the sum ."E+corrE

+I+W+V or to any term in the sum; (2) the multiplication of FE by a factor ranging

from 0.95 to 1.25, to account for possible deviations from the free-electron approximation (density-of-states correction) ;

(3) the addition of the maximum correction40 to (x) which could arise as a result of the sphere ap-proximation;

(4) the inclusion of next nearest-neighbor inter-actions in the expressions for w and v.

Second, if z and yare found so as to obtain exact agreement with the reduced experimental data at Po=O and Po= 1.044 kbar (x=0.9869), the agreement with experiment at 18 times this range of pressure and 12 times the compression range is still within 4.8% in pressure. Third, if z and y are adjusted so that exact agreement with the experimental cohesive energy and lattice constant at x= 1 is obtained, the agreement at 19 kbar remains within 2.5% in pressure.

Finally, one can show that Eq. (15) is, in fact, a good approximation to

708 D. JOHN PASTINE 166

TABLE III. Calculated values of P., -Yo, and tJ.-y/T.

p. tJ.-y/T x (kbar) -Yo (10-( OK-I)

1.06 0.487 0.890 9.65 1.05 0.496 0.888 9.17 1.04 0.505 0.886 8.71 1.03 0.514 0.884 8.35 1.02 0.523 0.882 8.07 1.01 0.534 0.881 7.70 1.00 0.544 0.879 6.91 0.96 0.586 0.873 5.74 0.92 0.635 0.868 4.98 0.88 0.689 0.865 4.03 0.84 0.752 0.863 3.19 0.80 0.823 0.861 2.52 0.76 0.905 0.861 1.89 0.72 1.00 0.861 1.33 0.68 1.11 0.862 0.83 0.64 1.25 0.864 0.40 0.60 1.40 0.866 0.01

result that

1l'Y= 1.27550nkT[x'Yo" -'Yo' (d Inco/ d lnx)]/co, (20)

where the primes denote derivatives with respect to x. The approximations which lead to Eqs. (18) and

(20) are not nearly as groundless as they may seem in this skeletal description. As a matter of fact, they lead to such good agreement with experiment7 that there is little doubt that the calculated values of 'Yo and 1l'Y are accurate to well within ±20% for solid sodium in the ranges 0.58~ x~ 1.06 and 200oK~ T ~450oK. This accuracy is sufficient to calculate (in the above ranges of x and T) the thermal contribution to the pressure in sodium to within ±1 kbar at 100 kbar, so that at this pressure, the total accuracy of the isotherms should be within ±3% in pressure.

m. CALCULATED RESULTS A. Theoretical Isotherms

In Table III are listed values of p . , 'Yo, and ll'Y/T for various values of x. Since, according to Eq. (20), 1l'Y is linear in T, the isotherm at any temperature can be calculated using Tables I and III and Eq. 00, taking 50= 1.0128 glcc and n= 2.618X 1()22 atoms/g.

0.860 0.880 0.900 0.920 0 .940

18.0

14.0 I P (1(b or: 10.0 x

6 .0

2.0

0 .960 0.980 1.000 1.020 1.040 1.060

FIG. 2. Comparison of the theoretical 3000 K isotherm with experimental (Ref. 39) data.

In Fig. 2, the 3000 K isotherm is graphed and compared with the experimental data.39 The agreement can be seen to be quite reasonable. The theoretical isothermal-bulk moduli at 0, 300, and 349°K are 79.0, 67.7 and 64.3 kbar, respectively. These values compare well with the experimentaP9 values of 78.2, 65.3 (an average taken from three different sets of data), and 63.9 kbar, respectively.

The theoretical coefficient of volume thermal expan-sion at P= 0, T= 3000 K is 219X 10-6 OK-I, which again compares well with the experimental value25 210.41 ±5.65X 10-6 OK-I.

B. Hugoniot Equation of State

The theoretical Hugoniot pressure p" can be calcu-lated with the relation43

P 0- p.+ [50 ( 'Yo+ 1l'Y)/ x ][ch(Xi)-CPI (X) + 3nk Ti] ~= ,

1- [('Yo+ll'Y) / 2x] (Xi- X) (21)

where Xi is the value of x at which P,,=O, Ti is the initial temperature, and cpl=cp-cpV. Since 1l'Y depends on the temperature T" along the Hugoniot, T" was estimated by first taking 1l'Y= 0 in Eq. (21), calculating the pressure, and then determining the temperature from the relation

T,,=x(P,,-po+P .)/50('Yo+ll'Y)3nk. (22)

The temperatures from Eq. (22) were then used to calFulate 1l'Y along the Hugoniot, which, in turn, was used to recalculate p" by Eq. (21).

Since the difference between the initial values of p" (~th ~'Y=O) and the final values differed by less than 1 ~bar between 40 and 100 kbar, further iteration seemed unnecessary. The results of the calculation are given in Table IV, along with the reduced experimental results of Rice.44 The agreement is again quite good. In Fig. 3, ('Yo+ 1l'Y) , calculated (for T,,

166 EQUATION OF STATE FOR SODIUM METAL 709

to indicate that the Dugdale-MacDonald relation will provide a fairly good approximation to 'Y along the Hugoniot for the alkali metals. However, a much better approximation to the calculated 'Y (along the Hugoniot) is given by 'Y= 1.1xl/2, where 1.1 is the value of 'Y at x= 1.

C. Further Indications of Accuracy

Swenson's has observed that along the sodium 3000 K isotherm, 'Y (i.e., 'Yo+~'Y) is given very nearly by the relation

'Y='Y~, ~~

where 'Yl~1.17. Since the calculations in this section indicate that 'Y is an increasing function of T, then Eq. (23) should represent the lower limit of 'Y along the Hugoniot. If it were assumed that Eq. (23) is the correct relation for 'Y along the Hugoniot, then the agreement obtained with the Rice data at, say, x = 0.6517 and the prediction of Eq. (21) would be destroyed. In order to restore the agreement, Po would have to be increased above its calculated value by 2.71 kbar. Conversely, if it is assumed that'Y is every-where equal to its upper limit, 'Yl [in order for Eq. (21) to be finite where finite values of Ph are observed, 'Y must decrease with decreasing x], then the agreement with exper!ment at x= 0.6517 is again destroyed, and to restore It, Po would have to be decreased below its calculated value by 4.32 kbar. Since the terms to the right of Po in the numerator of Eq. (21) account for

1.20

1.00

0.90

0.80

DUG DALE AN D MAC DONALD

ALON G TH E HUGONIOT

ALONG THE 3000 K ISOTHERM

, ,

0.70 '-----1-__ -'-__ -l __ --.J 0 .65 0.75 0.85 0. 95 1.05

FIG. 3. Pr~diction of t;he Dugdale-MacDonald (Ref. 45) formula compared With theoretical values of 'Y along the Hugoniot and 3000 K isotherm.

,. C. A. Swenson (private communication).

eo

70

60

so

40

30

• 1 PI',, ·)

I BRIDGMAN (1945)

- THEORETICAL

FIG. 4. Comparison of the calculated 3000 K isotherm with Bridgman's 1945 data (Ref. 47) .

only a small part of the Hugoniot pressure at X= 0.65, and since they are relatively insensitive to changes in Po, then the above considerations indicate that at x=0.6517, Po should certainly be somewhere between 64.6 and 71.6 kbar (the calculated value of Po at x=0.6517 is 67.3 kbar).

Frohlich32 has shown that the minimum value of 'PGsE(Ra) should be such that Ra'PGSE(Ra)~-e2 = -23.061X10-2O (esu). In these calculations, 'PGSE has its minimum at Ra= 1.53 A, at which Ra'PGSE(Ra) = - 22.42X 10-20. Brookss has calculated the position of the minimum of 'PGSE(Ra), using the quantum-defect method. His theoretical evaluation puts the minimum at Ra= 1.58 A. The theoretical 3000 K isotherm calcu-lated in this work was found to be in good agreement with the 1945 date of Bridgman.47 A comparison is made in Fig. 4.

: Rice44 has observed that sodium should be in a ~olten state along the entire experimental Hugoniot. Ait first sight, this seems to imply that a valid com-parison between the theoretical Hugoniot for solid sodium and the experimental data cannot be made. There is evidence, however, that the Hugoniot for sO,dium should be essentially the same, whether or not a 'transition to the liquid state occurs. The evidence arises from the fact that the slope dU./du of the relationship between shock velocity U. anl particle velocity Up (the relationship which determines the Hugoniot) is very nearly the same whether it is (a) derived from low-pressure experimental data on solid sodium,43 (b) derived from Rice's data43 •44 (which pre-sumably deals with liquid sodium), or (c) determined by the Thomas-Fermi (T-F) theory,43 which is accurate at very high pressures (> 10 Mbar), and which does n.ot distinguish between solid and liquid states. Clearly, smce dU./du p , for either solid or liquid sodium should at high pressures, approach the TF value, a~d sinc~ the latter value is nearly the same as the low-pressure (s lid-state) and intermediate-pressure (liquid-state) values, then the fact of melting should have little effect on the sodium Hugoniot.

,7 P. W. Bridgman, Proc. Am. Acad. Arts Sci. 74, 425 (1945).