On the fourth-order anharmonic equation of state of solids

J. Phys.Ckem. Solids Pergamon Press 1970. Vol. 3 I, pp. 2003-2016.

ON THE FOURTH-ORDER EQUATION OF STATE ~rnoot”~ohe~

Printed in Great Britain.

ANHARMONIC OF SOLIDS;‘:

LEON TIIOMSEN Geological Observatory of Columbia University. Palisades. N .Y. 10964. U.S.A.

[Received 2 t Aprif I969: in revisedfclrm8 September 1969) Abstract-The non-uniqueness of the definition of finite ‘strain’ is demonstrated to lead to signifi~nt ambiSuit~s in the equation of state of elastic solids. The ambig~ties are resolved by the rigorous derivation (due to Leibfried and Ludwig) of the Mie-Griineisen equation. Closed expressions for &,(V) and r(F) are presented which contain no assumptions additional to those already present in the simple M-G equation. Numerical equations of state for NaCl and MgO are presented which satisfy all ultrasonic and shock-compression data. Comparison with previous tabulations for NaCi reveals small but significant discrepancies. 1. FINITE STRAIN

a well-known rest&[ 1,2] of continuum mechanics that the theory of finite strain may be written equiv~ent~y in terms of any of an infinite number of different definitions of the ‘strain’. Far the case of an elastic solid (i.e. one possessing a free-energy function which is uniquely defined in terms of deformation and temperature), this result implies, specifically, that the stress-drain relation may be written equivalently in an infinite number of ways. As examples, if the strain e is defined implicitly in terms of the deformation R-R, by IT Is

Cartesian axes. e is the finite strain tensor defined by (1). u is the (homogeneous) stress at the point R. 1 is the unit 3 x 3 tensor, (I), = Z&. F is the ~elmholtz free energy, or ‘elastic energy’ function. 11 is the symmetric finite strain tensor defined in relation toe by qii = * feij f eji -t etie,+).

Equation (2) is exact, i.e. independent of the size of e. Zt is derived by, e.g. Leibfried and Ludwig [ 31 and by Murnaghan [4]. Alternatively, the strain f can be defined implicitly by R-R, = fR

then the stress-strain

relation is

and then the stress-&ram

Here R is a vector to a point in the deformed lattice, & is the vector to the same point in the undeformed reference state, denoted in this section by a sub-zero. The components of both are referred to the same set of *~~ont-~obe~y tion No. 1424.

Geological

Observatory

Contribu-

(3)

(4)

relation is

with the symmetric tensor E defined in relation tofby Elj S 4 t .& +f$ -fmjfmi ) -

(6)

Equation (5) is afso exact, though its generality is restricted to special situations, including the impo~ant cases of isotropic bodies, and of 2003

2004

L. THOMSEN

pure strains unaccompanied by rotations. It is derived by, e.g. Mu~aghan [S]. Not directly involved in the proofs of (2) and (5) are the relations between strain and volume change: *= ldet (1+2r))]

p=_

F ( av > .

Equations (7) and (8), however, do not reduce to the same form, but rather to

(7)

(74

*= jdet (1-2~)) V is, of course, specific volume. The difference between these two definitions of strain (1) and (4) is clearly in the choice of the characteristic lengths used as coefficients of the strain on the RHS of the two definitions. In the first case. the characteristic lengths are taken from the reference state; in the second. from the deformed state. Because of this essential difference, the strains e and q are called Lagrangiun or ~ut~rju~ definitions of strain; the strains f and E are called Euierian or spatiaf. These, and other de~nitions of strain can be, and have been, derived with more generality than is needed here. An elegant. and exhaustive. discussion of finite strain is given by Truesdell[ 11; a more limited, pedagogic treatment is given by Thomsen[6]. Although (2) and (5) are exact. their usefulness is limited by the need for expressions for the free energy as a function of the strain. In general, the physical description of the free energy in terms of interatomic forces. etc. will impose requirements on the choice of a definition of strain[7]. Also, any approximations made in F will affect the results of the choice of a definition of strain. Hence, the appfication of (2) and (5) to* real solids will not be exact, and further, will depend critically upon the choice of a definition of strain. Different choices lead to non-equivalent equations of state. As an illustration of this. consider the special case of hydrostatic pressure P on an isotropic body. Both (2) and (5) reduce to

v.

=-

213

(> v

(8b)

71m

The fact that V, appears in the denominator of (7a) and in the numerator of (8a) is an expression of the essential difference between the Lagrangian and the Eulerian definitions of strain. If the free energy is assumed to be of the form i, F=z Aiq” (10) i=l

with i, = 3, then (9) and (7a) give the thirdorder Lagrangian

isotherm

P=$Ko(y”3-y-1’3)[1-$K;(y-2’3-1)]

(11)

where y is the volume ratio

vo y=-=P* V

PII

(12)

Here the coefficients Ai have been evaluated in terms of the boundary value measurements at the reference state, taken here to be defined by P = 0, T = T,, = room temperature. K is the isothermal incompressibility: K’ is (dK/aP),: p is density. This equation appears in Murnaghan’s 195 1 monograph [4], p. 71. As is clear from the symmetry apparent in (2) and (5), or in (9), (7a) and @a), an assumption equally as plausible (at this point in the argument) as ( 10) is

ON THE

FOURTH-ORDER

ANHARMONIC

2005

EQUATION

jm F = x &e j=l

(13)

with j, = 3. Equations (9) and (8a) then lead to the third-order Euierian isotherm P =#Ko(y”3_

y5’3)(1+2(K;-4)(y3’3-1)). (14)

This equation has achieved prominence in the literature as the ~ir~h-Murn~~ (B-M) equation of state through its application by Birch[&l l] and others to problems of finite compression in the interior of the earth. It was apparently first written by Murnaghan[S] in an early exposition of the thoughts which led eventually to his 195 1 treatise. The difference between (11) and (14) is an increasing function of y, becoming serious for y 2 l-2. The divergence is illustrated in Fig. 1, where curves for (11) and (14) are shown, together with the Lagrangian and Eulerian fourth-order isotherms corresponding to i, = 4, in (10) and to j, = 4 in (13). respectively. Also shown for comparison is a curve of the Murnaghan equation (12) p+

; (YP-l)

(15)

wherep = K;. The fourth-order curves require a knowledge of K;j = (~zK/L8Pz)T,~.=,, which is not measurable directly. Later sections of this paper develop a way of determi~ng this quantity indirectly. Figure 1 is drawn for a material with properties similar to NaCl. Since the fourth-order Lagrangian isotherm is considered the most important, Fig. l(b) shows the deviations of the others from this. The main point of Fig. 1 is to demonstrate that the ambiguity mentioned above is real, and serious for large compressions. The ambiguity is not resolvable with concepts of continuum mechanics, but requires a physical description of F in terms of an atomistic theory.

a P/Ko

.-I

& P -I

Fig. 1. (a) The upper set of curves is isotherm equations of state for third (3L) and fourth (4L) order Lagrangian formulas. for third (3E) and fourth (4Ef order Eulerian formulas. and for the Mumarrhan formula (MI. (b) The lower two sets of curves de&be the @&entape) departures of 3U, 3E, 4E, and M from 4L. in two ways: (1) as pressure differences at constant density, as functions of density (vertical scale), and (2) as density differences at constant pressure, as functions of pressure (horizontal scale). 2. LATTICE DYNAMICS-THE ENERGY

POTE.NTJAL

As Knopoff has pointed out[7], each model of atomic interaction leads to a de~nition of strain approp~ate to that model, and thence to an equation of state appropriate to that model. The purpose of this paper is to identify the definition of finite strain appropriate to that model of atomic interaction which underlies the theory of lattice dynamics, and to examine the consequences that follow. Only centrosymmetric lattices are considered. The model is defined by the assumption that the instantaneous lattice potential energy at time t may be approximated by a Taylor expansion in the atomic displacements:

2006

L. THOMSEN

where the super-tilde denotes evaluation in the fixed reference state. The reference state may be chosen arbitrarily, so long as the approximation implied by the truncation in (16) and (17) is not violated. It is convenient here to choose that state as the one where & is at a minimum (hence $J: = 0). This

The notation here is that of Leibfried and Ludwig[3~; in fact, it is fairly said that most of the ideas presented here are implicit, and some are explicit, in that excellent review. Briefly qf is the i component of the displacement from equilibrium of the pth atom in the unit cell which is located by m. The Coupling Parameter of first order, Q,z”is the indicated partial derivative of Qi>,evaluated at the equilibrium configuration (1 * . R * f . ) and so is a function of volume. The sums (over repeated indices) cover all lattice sites. The termination of the expansion at the fourth order is important, when thermal effects are signi~cant. for reasons that appear presently. It is desired to express (I 6) in terms of an expansion about a fixed reference position. The assumption of (16) as written requires, for example, that the first term be expandabie as &(. . .R..

.) =&(.

. .ff..

.)

x(R:-ii:)(RI--ii~)(R4--iiq,i--. A I .J J (17)

configuration, (. . - $3 . * .), of course would be that assumed by the crystal if it were harmonic. That it is not realized in an actual crystal except under unusual circumstances is of no concern here. It will be referred to as the ‘rest’ state. The coefficients $Ftfetc., are, of lj course, constants. The strain is introduced here for the first time as a simplifying notation. Using the simple Lagrangian definition (l),

R;--;=e;;k; I II r I

(18)

and assuming the strain to be homogeneous throughout the crystal, @,,is

The other terms of (I 6) are also expandable in this way, but they do not appear in the equation of state, and so are not treated here. It is a simple matter to convert the expansion (I 9) in e to the equivalent expansion in q appropriate to the stress-strain relation (2) and the rotational invariance of $,,

ON THE FOURTH-ORDER

ANHARMONIC

where

c,,

r*

z

BaeiiaEkz -

(214 1 CiJklmn =

= v

@lb) -

the 6:‘:

@ and to higher-order

This is’:ec&sd,

Though the description of the Ciiklin terms of the ~~~~ kF is easy[3,6], it is not given ri j 1 here. Again, the i&l etc., are constants. Turning now to the Eulerian definition of strain, it is clear that an equally valid representation of the static potential energy is:

The important point is that here, unlike in (19), the polynomial coefficients of the strain components are not constants, but depend upon the equilibrium configuration (* * - R - . *) imposed by the external stress. Therefore an Eulerian analog in E to to equation 20, i.e. a strain polynomiaI with constant coefficients is not possible. The conclusion follows that the Taylor expansion of G(t) in the atomic displacements from equilib~um, as in (161, requires the Taylor expansion of #Q( * * * R - - a) in the Lagrangian strain, as in (20). Of course, it is possible to write

terms.

according to (8b), E =

where

Wa)

and so forth. The (constants) &k[, etc. would then be related, in some complicated way, to

8% ar)iia~kZa~rnn

2007

EQUATlON

q$.0(?j2).

tw

Hence the third term on the right of (23) already contains terms in v,P. Since, as will be discussed below, it is important that the expansion (16) be truncated at the fourthorder terms, the expansion (23) is not permitted. Consideration of this point was excluded by postulate from recent work by Thomsen and Anderson [33,34]; these were discussions within the theoretical framework of Eulerian strain. The present conclusion is that the Lagrangian strain, rather than the Eulerian, is more fund~ental to the equation of state of solids wherever thermal effects are important. 3. LATTICE DYNAMICS-THE

FREE ENERGY

It

is well known that, in the harmonic approximation, the Helmholtz free energy is F=#ao+F,

(24)

where #@may be taken as the first two terms of (20), and the vibrational cont~bution is Fs=C

F-l-kTln

[

1 -eXP(-$)I}

WI

k

where the summation extends over all wave vectors k of all branches. The eigenfrequencies w(k) are second derivatives of the potential energy Qt; the spectral average is [3] (245)

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2008

Since, in this approximation,

the 4;

= $ 5

are constants, the w are constant, and F, is’; function of T only. The equation of state is then P=--.

d+o

(27)

In the fourth-order theory, the free energy is[3] F = $o+Fs+Fanh (28) where 4. is the quartic polynomial (20). F, is of the same form (25) as in the harmonic theory, but now the frequencies are quadratic functions of the strain r). In fact, the entire term F, must be considered to be a function of T, and a quadratic function of ~[3,6]. Incorporating the ‘Griineisen approximation’ whereby strain derivatives of the logarithms of the o are all assumed to be equal it isl3.61

and A is its variation with the strain: 1 a21n<;i2 hiikl = --2 a7)U&jkl-’

(33)

The specific anharmonic contribution Fanh is given in Section 10 of Leibfried and Ludwig[3]; it need not be reproduced here. As for F, in the previous (harmonic) approximation, Fanh is, in the fourth-order approximation, a function of T alone. Hence it will not enter into the thermal equation of state (2) but only into the caloric equation of state, i.e. Cv and T-derivatives thereof. The thermal equation of state which follows from (28) was first written over half a century ago[ 12, 131. In its simplest form, it is called the Mie-Griineisen (M-G) equation of state:

P = Pstatie + Pthermal

(34)

or more explicitly, d4o

P=--+yy.

x ( & - TCv)I wm.

(29)

Other variations [ 13, 141 on the exact form of this approximation yield the same results in the present context. Here U, is the vibrational contribution to the internal energy, evaluated at the rest state; in the fourth-order approximation, it is 1 exp (G/kT) C, is the corresponding

- 1

u/8

(35)

As explicitly defined by (20) and (29), it is referred to in this paper as the fourth-order anharmonic equation of state. It is convenient to restrict the generality of the treatment to those crystal classes whose strain response to the hydrostatic pressure is a diagonal tensor. (This includes al1 classes but triclinic and monoclinic.) For these cases, from (2), (28), PO), (29),

(30)

specific heat:

& = z ,($)‘(,,&,k*_

l)-Ze”“/kT

rti is the tensor form of the Griineisen meter, in the rest state: lalno2

(31) para-

(32)

Here the notation of Voigt[ 141 is adopted. It

ON THE

FOURTH-ORDER

is a simple matter to express the constants appearing here in terms of quantities measured in the laboratory. However, before this is done, a number of remarks of a general nature are in order. Of basic importance is the concept of a self-consistent approximation. The present development specifically does not purport to be an exact theory, but only an approximate one. It is the requirement of internal consistency within a well-defined approximation that produces the explicit functional form (36). This equation contains no approximations which are not already present in the simple form (35) of the M-G equation of state. These approximations are well known; the two most important bear some discussion. In order that the ‘thermal pressure’ term of (34-36) be linear in US, and hence linear in T at high T, it is necessary that the specific anharmonic term, Fan,,, in (28) depend only on T, and hence vanish upon operation with (a/N),. For F,,, to be thus constant in strain, it is necessary that 4;: be constant

ANHARMONIC

2009

EQUATION 4. CUBIC

CRYSTALS

The fourth-order anharmonic equation of state (36) contains a number of constants, all of which are definable in terms of laboratory measurements. While this is true for all crystal classes (except, of course, monoclinic and triclinic), the cubic crystals offer the simplest and most instructive cases, and are the only ones considered in the following. For these, the strain components are equal, and from (7a) (37) where a is the edge of the unit cube. The incompressibility K can be written (38) analogous quantities are defined by

(384

rjkl

in strain, i.e. that the Coupling Parameters of Jiftth-order vanish as in (16) and (20). Thus the linearity in U, of the thermal pressure is tied inextricably to the quartic expression in r) of the static pressure. The fourth-order theory thus defined describes the thermal dependence of the lattice ‘constants’ and of the elastic ‘constants’. It is not competent to describe the mixed derivative a2K/aPaT lo; for that a higher-order theory is required[6]. Entirely separate from, and additional to, this approximation is the Griineisen approximation, introducing w2. The simplification thereby provided is apparently crucial; little progress can be made without it. Moreover, it becomes rigorous in the high- and the lowtemperature regimes [ 31. However. by replacing a spectrum of functions with a single function, it thereby reduces the utility of such detailed information as is available. In particular, the role of the measured ‘reststrahlen’ frequency in this development is not clear.

(38b)

A =

$h*. 1

For this (cubic) symmetry, yij

also[3]

=p&j.

(39)

The equation of state becomes

It is easy to describe the six unknowns, 6, K,

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L. THOMSEN

9, I‘, A, A, in terms of the six measurements, a07 KoS, cyo, K& (aK/aT)pjo. Kg, respectively, at the ‘zero state’, P = 0, T = To = room temperature[Q. In this way, an extended portion of the P(T, V) surface of the solid is described by ‘extrapolation’ from the zero point. The value of, in fact, the necessity for such an approach (as opposed to earlier ‘curve-fitting’ procedures) has been emphasized by Anderson[ 15, 161 and others. Accordingly the constants are found by operating on (40) to yield the following iterative set: (4Oa)

-+1:‘+ [h-y”(l-*)]90) Ko%la,

K= [

(4W

I - 3r~o+~~o”-~(A-32)

a,Ko(K;-i)+$$)

h =-;$ UO

I

P

]+y”

ljkl

(404

(4Oe) KoK;+

KQ(K;+

1) -;

course, evaluation of (30) and (3 1) requires the use of a characteristic temperature & appropriate to the rest state rather than to the zero state [3,6]. The techniques of modern laboratories are sufficient in general to measure directly all required quantities except for KG. For soft materials (with small values of K,), even this measurement may be possible [30], however it is not necessarily useful with (40). This is because for these soft materials, the interatomic forces are weak, and the displacements may become large. so that, strictly, the fourthorder anharmonic approximation is insufficient. This insufficiency is demonstrated by the fact that for these soft materials it is often also possible to measure the higherorder effects d2Kl~P8T # 0. and F,m SK/ aT2 f 0. The adequate description of” such materials will require a higher order theory [6]. It is still possible. however, to determine the parameter A, and the corresponding K;i by indirect means. The quantities A and A are both of fourth order, that is to say, they are different linear combinations of the fourthorder Coupling Parameters (b E$‘. In certain

1

(4Of)

where alpha is (~~~~T)~~~ and KS is -V(aPI av),.

This is a closed system, provided that 0, and cV are known, and that all ‘measurements’ are actually realizable. The expressions (3Q) and (3 1) for the internal vibrational energy and the specific heat require assumptions on the spectrum of eigenfrequencies; Leibfried and Ludwig131 have estimated that the use of the Debye spectrum for this purpose incurs errors smaller than those due to the errors of me~~ement of a,. Kos, etc. Of

cases of high symmetry, and assuming such restrictions as nearest-neighbor, centralforce interaction, the quantity A can be calculated from A[6]. Because of the approximations involved, however, this calculation can only be considered to provide the order of magnitude of A. The parameter A can, however, be found exactly. if it is as&dined that the fou~h-order anharmonic approximation is sufficient to describe the solid under conditions of shockwave compression. This assumption has, in the past. been universally adopted, with the M-G equation of state being used, along with various approximations, to interpret the Hugoniot in terms of an isothermr321. The assumption is not always justified, as shown below. The topic of shock-wave data will require some discussion in order to free it from a number of encumbe~ng hypotheses;

ON THE FOURTH-ORDER

to this end treated next.

the

Griineisen

parameter

is

5. THE GRijNEISEN PARAMETER

A simple differentiation the well-known ‘thermal’ Griineisen parameter,

of (35) yields formula for the

VCYK -Y=-

cv .

(41)

Because the volume de~ndences of (Y and Cr have not been available, researchers have turned to assumptions on the nature of the vibrational spectrum in order to obtain expressions for y(V) [17-l 91. Knopoff and Shapiro[20] have given an excellent discussion of the difficulties engendered by such assumptions. However, no such hypotheses are necessary in the present development. With the definition

(of which (32) is the special case at zero strain) the Griineisen parameter for a cubic crystal is y(V) = (;)2’3(T+3*q)

(43)

ANHARMONIC

EQUATION

2011

with the constants + and X defined by (4Oc, e). Here. all the modes of vibration, both acoustic and optic, both low-frequency and high, are correctly represented by the measured values of CQ and (X/a?),. There are no assumptions in (43) beyond those already present in the simple M-G equations (35). The Griineisen parameter is shown graphically in Fig. 2(a), using the data of Table 1, for a number of cubic materials of interest. The calculation has also been done for noncubic crystals, in an informal way, i.e. through the literal application of (43) and (7a) to these materials. The results are graphed in Fig. 2(b) for several materials of geologic interest. with previous work[33,34] Comparison based on Eulerian strain emphasizes the importance of the choice of Lagrangian strain. It is to be noted that the coefficient of oi,/v in (40) contains a term additional to those of (43). It arises from the variation of U, with compression, i.e. in this approximation,

The strain dependence of the thermal expansivity is most easily found, not from (41), but by differentiating (40) with P held constant.

Fig. Z(a). The Griineisen parameter, normalized to its value at P = 0, T = TO. The basic reference va1ue.y. is listed in Table 1.

L. THOMSEN

2012

Fig, 2(b). The Griineisen parameter, according to (5.2) normalized basic reference value y is &ted in Table t.

to yo. The

Table 1. Parameters for the equation of state (40) calculated from equations 4O(a-f) with the zero pressure data from the indicated source Mg@

N&I

(A)

R. f kbar) 3 I’

55705’ 285.52 I -566” .5*OY

h

0359~

2,

it &I

4.1773 1733.8

1.752 4.51 1.355

327.F

SpineI”

Garner

8.052 2101-2 1.08 4.2 1.29

11.485 1876.6 1.I3 5-28 1.10

4.5168’ 144.8’

Zincitefi

966.V

CSI

1*535x 5‘369 I+617 29.29 118.0g

Forsterite

Hematite6

Corundum”

Quartz@

;

1357.3’0 6400’

2143.2 6.672

2629.7 6.312

419.0 4.808

1440-8 3.617

r r h

5-25’0 0~81~

0.49 6.01 O-32

O-65 4.72 I-59

I *OF

I*93 4.63 2.69

I.35 4.03 1-52

‘National Bureau of Standards Circular 539, i-9, Washington, D.C. (1953-1960). “SLAGLE 0. D. and McKINSTRY H. A..J. uppf. P&s. 38.437 t $967). SRUBfN T., JOHNSTON H. L. and ALTMAN H. W., J. phys. C&em. 65, 6.5 (1961). 4BARTELS R. A. and SCHUELE D. E., J. Phys. Chem. Solids 26,537 ( 1965). “cf. THOMSEN L.. ref. [6]. BAWDERSON 0. L., SCHREIBER E.. LIEBERMANN R. C. and SOGA N., Rev. Geophys. 6,491 (1968). %LAGLE 0. D. and MCKINSTRY H. A., J. appl. Phyhys.38,4S l(1967). s.IOHNSON J. W., AGRON P. A. and BREDIG M. A.. J. Am. them. Sot. 77, 2734 ( 1955). %ARSCH G. R. and CHANG Z. P.. paper presented at Ssnrp. opt t&e &ccurcrte ~~a~a~l~r~za~i~tt of the Ii&h

Pressure

(1968). ‘~KU~~AZAWA M.. unpublished data.

Emironrnent,

Guithershurg,

M~~l~~d

ON THE FOURTH-ORDER

The result is

ANHARMONIC

Substitution fourth-order

A--yz--y_ f31nCV a In T I>

-37

(45)

Pff=-$

EQUATION

2013

of (46) ef seq into (40) gives the Hugoniot:

HH

+Y(V,)[sp,(g-

1)-V+%].

(49)

where a(P=O,T)

3

=A KV

zi

2

a(P = 0, T)

c:,(T).

(454

This function CX(I/, T) decreases rapidly with V, difficulties with the fourth-order approximation may be expected near its vanishing point. In order to evaluate (45) properly, it is necessary to know P; hence the problem of determining A is now discussed. 6. HUGONIOT INTERPRETATION

Through the assumption that the fourthorder anharmonic theory is sufficient under conditions of shock-wave compression, the parameter A can be evaluated. The evaluation starts with the locus of Hugoniot points in the P-V plane, where the temperature is defined implicitly by

u, =

UO+fPH(VO--

V,).

(4)

The correction from Hugoniot to adiabat to isotherm is unambiguous, since y(Y) is well defined. However, it is logically more straightforward to introduce the Hugoniot condition (46) directly into (40). The internal energy U is composed of two parts

u(v, T) =4o(V) +,U,W, T)

(47)

and in the fourth-order theory, U, is defined by (44) and (30). $J,,is, from (20), &,(v)

= Ri;l

v,2+vrq3 [ .

+ (--3j4&/4 4!

1 .

(4)

When evaluated at a single Hugoniot point, (PH, V,) , this equation contains only one unknown, A, which is thereby determined. The equation can then be used to generate the entire locus of Hugoniot points. Further, the same constants can be used in (40) to generate the isotherms, adiabats, etc. This method of Hugoniot interpretation is equivalent in concept to more complicated schemes [2 l-231 ; it has, however, the advantage of internal consistency. It shares with these other methods the assumption of the fourth-order approximation; as discussed below, this assumption is not always justified. Only partial checks on the assumption are possible within the conceptual framework developed so far. The Hugoniot generated from (49) does, in fact, reproduce the measured Hugoniot within experimental error (for all minerals tested), but this is perhaps to be expected, because of the fitting procedure used. Another measure of the validity of this assumption is introduced in the next section. 7. EQUATIONS OF STATE OF NaCl and MgO

The calculation of the equations of state of NaCl and MgO was made for reasons of mutual comparison, Weaver et al.[26] statically compressed a powdered mixture of NaCl and MgO and simultaneously measured the lattice constant of each. Hence equation (40), when applied to this simultaneous compression data, should give the same calculated pressure for both solids. The constants for NaCl in Table 1 were used in conjunction with the shock-wave data of Fritz ef a/.[241 to determine hNaCL= 23. The constants for Mgo were used with the

2014

L. THOMSEN

shock-wave data of Carter et al. to determine AMgO= 22 &4. The postulation of a simple strength effect (cf.[22]) in MgO was not required to fit the Hugoniot data. The uncertainty in hMMgO, which results from scatter in the Hugoniot points, is not important for calculations below P = 300 kb. The (analytically) calculated isotherms are given in Tabfe 2 and compared with those given by Weaver et al.; the agreement is good. The advantages of the present theory over that of Weaver et al., and of Decker[25,28], include: (1) its internal consistency, (2) its easy extension to the description of arbitrarily complex crystals, and (3) its lack of limiting hypotheses (for example, no central-foroe theory can describe the elasticity of NaCl completely, as is known from measurements on the individual elastic moduli. On the other hand, the present theory is easily extendable to the description of the elastic moduli; such a paper is now in preparation.) It is an interesting exercise to calculate that part, Pii, of the pressure which is due to the quartic term in the strain, with coefficient A. It is plotted, as a fraction of P, in Fig. 3. Because of the larger compression of NaCl at a given pressure, PA is considerably greater for this mineral than for MgO. In fact. by 300 kbar, P,, for NaCl is too great to be realistically considered a perturbation term. A serious doubt may be entertained as to the sufficiency of the present fourth-order theory

j 0

&&f%G-j too

200

300

Fig. 3. The contribution of the fourth-order term, P.t, tothe equation of state (40) for NaCl and MgO.

in this case. The fact that the present calculations do not indicate such an insufficiency is not at present well understood. The sign of the second derivative I(: has been the subject of some speculation [ 11.16, 121 and at least one measurement [30], the consensus being that it is generally small and negative. With the parameter A determined by the Hugoniot for NaCl and MgO, iu;l is given directly by (4Of). It is, for NaCl, &J = -3.79; for MgO, Kli;,” = -3 t 4. Thus the present work is in accord with the current consensus. 8. DISCUSSION

The fourth-order anharmonic (thermal) equation of state is given by (36) and (40). Especially for cubic minerals, there is no Table 2. Sirn~~ta~e~u~ c~rnp~e~s~o~ of NaCl particular di~culty. Of the six parameters and MgO at 298°K. P[26] is from Weaver which characterize the equation, five are et al. 1261, in kb routinely measured at zero pressure; the sixth is easify obtained from the Hugoniot. The NaCl M&J Grtineisen parameter is unambiguously P P VW, VW0 p [261 Pi261 defined by (43). 28+-5 2s*1 0.983 29 The suficiency, however, of the fourth0.911 28 115%6 o-940 122-+2 114 0,764 124 order theory is another matter entirely. As 142-r-7 0.92% 1.54 15423 141 0.735 Chang and Barsch[30] have remarked, the 0.907 196*7 199k-3 194 0.700 198 series (20) converges slowly. It seems reason211-‘8 0902 207 0.692 208 21123 23228 0.895 227 228 232rt4 O-679 able to expect that the fourth-order approxi2.515% 0.888 24% 0,671 240 246r4 mation will begin to break down before 0*8%4 265-+-9 2625~4 0.662 254 259 v/v, = 0%. This is in accord with an estimate

ON THE FOURTH-ORDER

[31] that melting (and hence breakdown of the approximations occurs when the vibrational amplitude approaches a&O. These compressions will occur near P/K, - 4. Thus, for a soft material with weak interatomic forces, and hence a small incompressibility, the range of validity of the fourth-order theory should be restricted. For a harder material, the range will be correspondingly extended; for most oxides and silicates the theory is expected to be adequate below their points of transition to higher density phases. It can, of course, also be applied to the high pressure phases, if the relevant data are available. Eventually, one may hope to be able to understand in this way the equation of state of the homogeneous lower mantle of the earth. For these harder materials, however, the Griineisen approximation may lead to trouble. This is because, by virtue of their hardness, 0, is greater than room temperature, and the normal modes of vibration are not all saturated. The approximation may be drastic, and may be reflected in false values of the constants, leading perhaps to effects such as the slight divergence of the isotherms in Table 2 at intermediate compressions. It may be preferable, for these materials, to evaluate R, etc. with measurements at high temperature rather than at To. In this case, however, the intrinsic errors of measurement will be higher, and it may be that no advantage accrues. The situation should be separately evaluated in every case.

Acknowledgements-The author is thoroughfy indebted to 0. L. Anderson for his continued guidance and encouragement. I also benefited greatly from many discussions with R. C. Liebermann, and from a critical correspondence with T. Takahashi. In a more specific sense. I am grateful to Leon Knopoff and to G. E. DuvaIl for forcefully pointing out to me, at a time when I was not prepared to believe it, the ambiguity in the theory of finite strain. Financial support was provided through National Aeronautics and Space Administration contract NSG-445 and through Air Force Office of Scientific Research contract F44620-68-C-~79.

ANHARMONI~

EQUATION

2015

WRENCH 1. TRUESDELL C. and NOLL W.. Hnndb. Phys., Vol. 111/3. Springer, Berlin (1965). 2. TRUESDELL C.. The Mechanical Foundations of Elasticity and Fluid Dynamics. Gordon and Breach, New York (1966). G. and LUDWIG W., Solid State 3. LEIBFRIED Physics, Voi. 12, v. 275. Academic Press, New Y&k(1961). ^ 4. MURNAGHAN F. D., Finite Deformation of an Elastic Solid. Wiley, New York (195 1). 5. MURNAGHAN i?. D.,Am.J. &ath..59,235 (1937). 6. THOMSEN L., Ph.D. Thesis, Columbia University, New York (1969). 7. KNOPOFF L.,J. geophys. Res. &X,2929 (1963). 8. BIRCH F.,J. appl, Phys. 9,279 (1938). 9. BIRCH F., Bull. seis. Sot. Am. 29.463 (f 9393. 10. BIRCH F., Phys. Rev. 71,809 (1947). 11. BIRCH F.,J. ieophys. Res. 57,227 (1952). E.-Ann. Phvs. 39.257 (19121. 12. GRtiNEISEN i., Handb. Phys., Gal. i0, p, 1. 13. GRUNEISEN Springer, Berlin (1926). Also, in translation, NASA Republication RE 2-18-59W. 14. BORN M. and HUANG K., Dynamical Theory of Crystal Lattices. Oxford University Press, London (1954). 0. L., J. Phys. Chem. Solids 27, 15. ANDERSON 547 (1966). 0. L., Phvs. Earth Planet inter. I, 16. ANDERSON 169(1968). 17. SLATER J. C., Introduction to Chem~ral Physics. McG~w-Hiit, New York (19391. J. s. and MACDONALD D. K. C., 18. DUGDALE Phys. Reu. 89,832 (1953). L. V., Soviet Phys. Usp. 8, 52 19. ALT’SCHULER (1965). L. and SHAPIRO J. N., J. geophys. 20. KNOPOFF Res. 74, 1439 (1969). H. and KANAMORI H., J. geophys. 21. TAKEUCHI Res. 71,3985 (1966). D. L. and RING22. AHRENS T. J., ANDERSON WOOD A. E., Rev. Geoph., 7(4), 667 (1969). L., J. geophys. 23. SHAPIRO J. N. and KNOPOFF Res. 74,143s (1969). 24. FRITZ J. N., MARSH S. P., CARTER W. J., and MCQUEEN R. G., paper presented at Symp. on the Accuraie Characterization of the High Pressure Envjronment, Gaithersburg, Maryland (1968). 25. DECKER D. L., unpublished. 26. WEAVER J. S., TAKAHASHI T. and BASSETT W. A., paper presented at Symp. on the Accurate Characterization of the High Pressure Environment, Gaithersburg, Maryland (1968). 27. CARTER W. J., MARSH S. P., FRITZ J. N. and MCQUEEN R. G., paper presented at Symo. on the Accurate Characterizdtion of the High Pressure Environment, Gaithersbum. Maryland (1969). 28. DECKER D. L., J. appl. whys. 38,157 (1965j. 29. SCHWARTZSCHILD M., Structure and Evolution of the Stars. Princeton University Press, Princeton (1958).

2016 30. CHANG Z. z&r. 19,1381 31. GILVARRY 32. RICE M. H.,

L. THOMSEN

P. and BARSCH G. R., Phys. Rev. (1967). J. J., Phys. Rev. 102,308 (195.5). MCQUEEN R. G. and WALSH J. M.. Solid State Physics, Vol. 6, p. 1. Academic Press, New York (1958).

33. THOMSEN L. and ANDERSON 0. L., J. geaphys. Res. 74,98 1 (1969). 34. THOMSEN L. and ANDERSON 0. L., paper presented at Symp. on the Accurate Characterization of the High Pressure Environment, Gaithersburg,

Naryland( 1968).