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Equation of state of cubic solids; some generalizations
J. Phys. Chem. Solids
Pergamon Press 1968. Vol. 29, pp. 1337-1348.
EQUATION
Printed in Great Britain.
OF STATE OF CUBIC GENERALIZATIONS’r
SOLIDS;
SOME
C. A. SWENSON lnstitute for Atomic Research and Department of Physics, Iowa State University, Ames, Iowa 50010, U.S.A. (Received
26 December
1967)
Abstract-The equations of state (pressure-volume-temperature relationships) for a number of cubic solids can be approximated-at high temperature (T > 0,) by a temperature-independent isothermal bulk modulus BT which varies linearly with the applied pressure. This is demonstrated by an analysis of both piston-displacement and ultrasonic data, as well as by an elementary calculation based on the Mie-Griineisen equation of state. These generalizations also should be valid for ST at low temperatures to within one or two percent except for the inert gas solids. The high temperature equation of state which follows from this analysis contains the following features: (a) the Griineisen constant y is a temperature-independent linear function of volume, (b) ET is a linear function of temperature at constant pressure, (c) (a(B,-BB,)/aT)v=PB~ and (.3(B,-&)/#‘)T=-PyT, where p is the volume thermal expansion coefficient. These relationships should hold as a first approximation for most cubic solids. INTRODUCTION
of state (pressure-volumetemperature) data often are required to express high pressure data (such as resistivity or magnetization) in terms of a volume dependence. In many instances, the limited equation of state data which exist for a given substance must be extrapolated well beyond their range of validity. These data might include only one pressure-volume isotherm at room temperature, a limited number of elastic constant or thermal expansion measurements as a function of temperature, etc. The equation of state ddta which exist for NaCl are reasonably extensive, and a semitheoretical interpolation and extrapolation of these data have been suggested as a basis for pressure standardization [ 11. Similarly detailed primary data do not exist for most other solids although an analysis of various types of data suggests correlations which can form the basis for elementary and reasonably reliable EQUATION
extrapolations of their thermodynamic functions [2]. The basic description of a solid can be expressed in terms of the Helmholtz free energy F(T, V) = U,(V) + F*(T,
= U,(V)+U*(T,V)-TS(T,V)(l) where UI,(V) is the cohesive energy at absolute zero, U* is the temperature-dependent internal energy, and S is the entropy. Both of these latter terms, and indeed all terms indicated with an * in this paper, approach zero at T = 0. The equation of state and the bulk modulus can be obtained as P(T,V)
=-(aF/W),=P,(V)-i-P*(T,V)
(2) and B(T, V) = - V(aPlaV), = &(I/)+B,*(T,
tWork was performed in the Ames Laboratory of the U.S. Atomic Energy Commission. Contribution No. 2225.
V)
= + V(a2F/a~)T v).
(3)
By using these equations we assume that in a theoretical treatment the ground state and 1337
C. A. SWENSON
1338
excited state contributions to the energy are independent and additive. We also might assume as a logical extension that the energy contributions due to various systems in a solid, such as magnetic spins, free electrons, lattice vibrations, etc., are independent and additive. The following discussion will be restricted to lattice properties and will apply most reliably only at temperatures which are greater (in principle, much greater) than the Debye temperature 0,. In this limit, the vibrational properties of solids are essentially classical, and the specific heat at constant volume is given by the Dulong and Petit value, CV = 3R. Thermal contributions P* and B* will be shown to be relatively small. Complications due to anharmonic and defect contributions will not be considered, even though they can be of great importance under certain conditions. An implicit restriction to cubic solids will be made although the basis for the restriction is not clear. The first part of this paper suggests semiquantitative guidelines which are to be used for extrapolation if adequate data do not exist. These guidelines should not be used for more than a qualitative test of the reliability of experimental data. The second part of this paper will consider the equation of state at lower temperatures and will make plausible the conclusions reached in the first part and their extrapolation to low temperatures. EQUATION OF STATE AT HIGH TEMPERATURES
Basic relationships (1) The Mie-Griineisen equation of state is obtained by using a characteristic temperature to represent the volume dependence of F*[F*/T = @*(O(V)/T)]. Equation (2) then becomes P(T,V)
=Po(V)+y,(V)U*/V
(4)
with yU = - d In 0/d In V. This type of relationship is useful when discussing lattice properties, and the Debye model represents a special
case. In practice, yU as defined by equation (4) will be a slowly varying function of both temperature and volume and will approach a constant value at high temperatures. The isobaric change in volume of a solid with temperature (the thermal expansion) can be related to the increase of P* with increasing temperature. At zero pressure, the volume expansion is given by P*( T, V) = -P,(V). The definition of the bulk modulus (equation (3)) can be combined with this relationship and equation (4) to calculate small volume changes as AV/V, = (V(T) - V,,)/V, = P*/B&, = yJJ*IBJ.
(5)
Equation (4) is not particularly useful since the various thermodynamic quantities appearing in it are not easily measured. The Griineisen relationship, P = yCJBJ
(6)
follows from equation (4) if y is a function of volume only, and the mathematical identity @pIaT) V= P&
(7)
is used. Here, p = (aln V/aT), is the volume thermal expansion coefficient and CV is the heat capacity at constant volume. Our simple model (equation (4)) assumes that y is temperature-independent, while more complex calculations (such as those by Fritz and Brugger[3]) show that y as defined by equation (6) should be different from yU as defined by equation (4) at intermediate temperatures and that the two should be equal and independent of temperature only for temperatures greater than 015 to e/2. This is confirmed by experiment [3,4]. Both BT and V are slowly varying functions of temperature and at all temperatures (for most solids) the temperature dependences of /3 and C, are very similar. In the classical or high temperature limit,
EQUATION
OF STATE
Cv = 3R, independent of volume. CV cannot be determined directly since experimental data are taken at constant pressure. The resulting temperature-dependent specific heat at constant pressure C, is related to CV by the following expression [5];
CP = C,( 1+ fly T)
(8)
where y is defined operationally by equation (6). A similar relationship exists between the adiabatic bulk modulus BS (which is determined from ultrasonic measurements) and the isothermal bulk modulus BT; Bs=B,(l+PyT).
(9)
Equations (7-9) can be used to express the definition of y (equation (6)) in two alternate and useful ways as follows: Y = P&V/C,
(6a)
and (aP/aT)”
= r(C,/V).
(6b)
(2) Anderson [6,7] demonstrates in a recent paper that most pressure-volume isotherms can be fitted by the Mumaghan relationship P(T, I’) = BTo(T, I’,)/n)((I’&‘)n--
OF CUBIC
(10) in lieu of direct P-V measurements[6]. One must, of course, convert the adiabatic data obtained from the ultrasonic measurements to isothermal values using equation (9) and a relationship between ( dBs/tIP)T and (dB,/dP),. This latter relationship is quite complex[8], but it can be simplified considerably for high temperature data as we will demonstrate below. (3) Considerable evidence exists that at high temperatures the isothermal bulk modulus is solely a function of volume[2]. This evidence comes from two sources. First, relatively crude P-V-T measurements on the alkali metals Li[9], Na[lO], and K[l I] and on solid xenon[ 121 can be represented well by this approximation and equation (10). These data cover wide relative pressure (to 20 kbars) and temperature (20°K to the melting point) ranges, and BT( T, V) values are reliable to approximately 3 per cent. Second, as Lazarus pointed out [ 131, ultrasonic data which are obtained as a function of temperature at atmospheric pressure and as a function of pressure at constant temperature can be combined to evaluate the explicit temperature dependence of various elastic (C)as follows: (aln C/aT)p
1) (10)
where n = (aB,/dP), is a dimensionless constant (called BI, by Anderson) which is a characteristic of the solid involved and usually has a value between 3 and 6. BTo and V, are the zero pressure values of the isothermal bulk modulus and molar volume, respectively. This relationship holds remarkably well for a great number of substances[6]. Although equation ( 10) cannot be justified on theoretical grounds, it is probably the most useful single relationship for the extrapolation of P-V data to higher pressures. Indeed, Anderson suggests that values of BTO and n as obtained in relatively low pressure high precision elastic constant measurements be used in equation
1339
SOLIDS
= (aln C/dT), - (aln C/dP),(aP/aT), = (aln C/aT)“-
(aln C/aP)&Bp (11)
When ultrasonic measurements are analyzed in this manner, they give most directly (aln BJdT) “. A correction must be applied to obtain (aln BT/aT) v Table 1 gives magnitudes for (aln BT/dT) v for 16 cubic solids for which both pressure and temperature-dependent ultrasonic data exist near room temperature. The listing is not complete, and no ‘selection’ of data was made. The primary data which were used in these calculations are not given but will be found in the references indicated. The method which
C. A. SWENSON
1340
was used to convert from the adiabatic to the isothermal temperature derivative will be described in the following section. For almost all cases, the magnitude of (alnB,/aT), is less than 10m4OK-‘. This suggests that BT will vary by less than 1 per cent in a 100 deg. temperature interval. While an effect of this order of magnitude can be measured easily with modern techniques, it is effectively zero for the extrapolations of the equation of state which are considered in this paper. In making comparisons of this type it is desirable that both the temperature and pressure derivatives be obtained for the same sample and preferably in the same laboratory.
This is the case for very few of the substances listed in Table 1. This discussion of experimental and elementary theoretical aspects of the equation of state of solids suggests the following postulates which will be used to establish guidelines for the extrapolation of P-V data. B,(V)
(aB,/aT),
= BT(V,,) +nP
(12aj
= BT(F/O) (v,/Vn.
(12bj
= 0
(12cj
T > 0,; CV = 3R and y is independent
(12dj
Table 1. Summary of various 300°K thermodynamic parameters for a number of cubic solids. In most instances values of 8, y and p, as well as of the derivative (alnB,/aT), are given in (or can be deduced from) data given in the references. (aln B,/aT) V was calculated as described in the text. Where two references are given, the first involves zero pressure data and the second high pressure data
( lO-4 “K-l)
cu &
AU Al Fe Ta NaCl KC1 LiF NaF NaI TlBr CaF, BaF, Na K Xe’“’ Li’fJI
345 227 162 430 477 260 320 235 740 490 164 132 508 300 160 95 64
2.0 2.4 2.9 2.35 1.6 1.6 1.56 1.45 1.6“ 1.4 1.71 2.3 1.90 1.6 1.1 1.1 2.5
0.495 0.576 0.423 0.70 0.35 0.20 1.21 1.10 1.4 1.1 1.36 1.53 0.57 0.55 2.0 2.5 10”’
390
0.8
1.34
(“‘Piston-displacement @‘165”K.
of T.
+1+x + 1.62 + 0.96 + 1.3 + 1.0 -0.4 + 1.5 + 1.0 + 1.3 + 1.6 + 1.2 +2.2 + 0.34 +0.06 + 1.1 +5.16 -
data only.
-0.1 +0.3 -0.2 -0.2 +0.4 -0.7 -0.3 -0.5 -0.3 +0.1 -1.1 -0.3 -0.5 - 1.2 +0.9 +2.8 0 0
114,151 [14,151 [14,15] [16, 171 [18, 191 [20] [211 [211 [221 [221 [231 [241 1251 [25] [261 ~271 [I21 [91
EQUATION
OF STATE OF CUBlC SOLIDS
Here, equations (12a) and (12b) are a direct of the assumption of the consequence Murnaghan relationship equation (10). In his original formulation and discussion, Anderson made no comment on the temperaturedependence of n [6]. Our postulate that the isothermal bulk modulus is independent of temperature at constant volume (( 12~) above) implies that IZ must be independent of temperature as well as pressure or volume. This is verified by experiment for NaCl and KC1[21] and for CaF, and BaF,[25] and is approximately correct for copper[28]. Data for CaF, exist[29] which disagree with those of Wong and Schuele[25] in that they show a significant temperature dependence of the pressure derivatives. The reason for the discrepancy is not understood.
Conclusions The above postulates lead to a number of general statements which should apply as a first approximation to a description of the equation of state of solids in the classical region. These statements include first a rather general recipe for the extrapolation of a single P-V isotherm in both pressure and temperature, and then suggestions as to the volume dependence of y, the isobaric temperature dependence of p and BT and relatively simple relationships between the pressure and temperature derivatives of Bs and BP These ideas will be considered in detail in the following sections. The extrapcilation of P-V data in both T and P. Equation (12~) states effectively that all P-V isotherms have the same slope for a given molar (or specific) volume. As will be shown in the second half of this paper, the error involved in applying this statement to all temperatures from absolute zero upwards is of the order of one or 2 per cent of BT at most if the inert gas solids are not included. Hence, if one P-I’ isotherm is known at a temperature To, any other P-V isotherm at a temperature
1341
T can be generated by translating this isotherm parallel to the pressure axis by an amount equal to Al’* = P*(T) -P* (T,) (equation (2)). The important variable here is the molar volume; these statements do not apply to the relative volume changes V/V,, or Av/F,,. As we will show in a following section, P* does not depend on volume within our approximations (equations (12c, d)). The original P-V isotherm which is used can be one which has been determined directly by experiment or which has been constructed from ultrasonic measurements of BT( T, P = 0) and n = (aB,/aP), using equation (10). In any event, equation (10) will be useful to extrapolate this isotherm to other molar volumes. AP* can be determined as the difference between the pressures necessary to obtain a given molar volume at T as compared with To (equation (2)). This implies that P* ( T) and hence the equation of state can be determined immediately from a single P-V isotherm and the zero pressure volume thermal expansion. This, for instance, is the procedure which was used to generate an equation of state for indium metal from Bridgman’s room temperature isotherms and measured zero pressure volume thermal expansions [2,30]. The method outlined in the preceding paragraphs will give (by equations (12b-c)) an estimate of BT( T, P = 0). If data exist for the temperature dependence of BT at zero pressure, it is preferable to use these in equation (10) along with a value for n to determine P-V isotherms rather than to assume equation (12~). Isotherms which are calculated from equation (10) are as precise as the value of BTO which are used, anddepend on the value of n only in second and higher orders. At the other extreme, a minimum of expansion and bulk modulus data exist for a solid. An estimate of BT and n can be made from Anderson’s correlations [6] and at least in the classical limit the volume expansion can be estimated from equation (5). The volume dependence of y. Equation (12~) implies the following;
1342
C. A. SWENSON
-(aBJaT)”
= V(azP/aTaV) = v(a/av)
(aP/aT)”
= 0. Hence, (MlaT), is independent of volume. Also, an elementary thermodynamic relationship
(a2PlaT2)V = T-l(aC,lav)T
(13)
can be used to show that in the high temperature limit (where C, = 3R) (aP/aT),is independent of temperature also. An immediate and useful consequence is that
since the thermal pressure is equivalent to a negative pressure applied to the solid. All of the quantities multiplying (T - T,,) on the right-hand side are constants, independent of temperature and volume, so at atmospheric pressure, BT should to a first approximation be linear function of the temperature. This result suggests that it is more meaningful to plot BT as a function of temperature than the compressibility ( kT = l/B,). Equation (14) can be combined with equation (16) to predict similarly the temperature dependence of p for a classical solid: P(T, P = 0) = (PBT)w=o/[BT(TO, -n(rlWR(T-
fiBT = (aP/aT) v = constant, independent of T,Vfor T > 0,. (14) Equation (14) can be checked experimentally and has been found to hold to within 3 per cent for the high temperature P-V-T data for Li[9], Na[lO] and K[ll], and for solid xenon [ 121. The constancy of BT also is evident in the atmospheric pressure data for Cu near room temperature (as collected by Overton
m. Equation (14) can be combined with equation (6b) to establish that (in this limit where C, is a constant) the Griineisen constant is a linear function of the volume. This follows directly from rewriting equation (6b) as: y/V = (aP/aT) &2, = constant.
(15)
There is no direct way of verifying this prediction, which also has been used in the analysis of shock-wave data[3 11. The isobaric temperature-dependence and BT. The temperature-dependence
of /I
of BT at atmospheric (or zero) pressure can be obtained from equations (12a) and (4) as:
B,(T,P= O)-BT(To,P= 0) =-nP*(T,l') =-n(rC,W)(T-To)
(16)
P = 0) To)]. (17)
Equation (17) can be verified for NaCl since this solid satisfies our postulates, and thermal expansion data exist for it up to the melting point [32]. Indeed, the temperature dependence of the thermal expansion is of basic interest since vacancy effects should appear as an anomalous increase in p with increasing temperature, and equation (17) should give a first approximation to ‘normal’ behavior. Room temperature values of the various thermodynamic quantities for NaCl have been used together with equation (17) to calculate the crosses in Fig. 1. The agreement with experiment over a factor of 3 in absolute temperature and almost 2 in (Y= p/3 is quite remarkable. The complete P-V-T relationship for NaCl at these high temperatures could be calculated using equations (10) and (16) and this result. Relationships
between
the
derivatives
of
BT and B,. Ultrasonic measurements of elastic constants give directly the adiabatic bulk modulus Bs, while the isothermal modulus is needed to establish an equation of state. The relationship between these (equation (9)) can be expressed as Bs - BT = B&yT
= yzCvT/V.
(18)
EQUATION
1343
OF STATE OF CUBIC SOLIDS
In practice, (a In C,/a In T)v is of the order of (a In &/a In T)P c O-1 for T > f3, if a Debye model is used. For practical purposes near room temperature (to within 10 per cent or so) (a In B&WV = (&I&)
(a lnB&Wv-Py. (19a)
I
d
I
xEQUATION I7 (CALCULATED) 4..al(il’ 100
’ ’ ’ ’ ” ” ’ ’ ’ ’ 300 500 700 TEMPERATURE “‘2
1
Fig. 1. A comparison between the linear expansion coefficient a = p/3 for NaCl as calculated from equation (17) and as measured[32]. Only room temperature, low pressure data were used in the extrapolation.
Over-ton has outlined a procedure for converting temperature and pressure derivatives, of Bs to the same derivatives for B&3]. This procedure is quite complex and requires detailed data. It can be simplified considerably by using the ideas expressed in the preceding section and by making use of combinations of thermodynamic quantities which are slowly varying with temperature or pressure. The relationship between (dB,/aT) v and is obtained as follows. Both (a&/W v experimentally and theoretically, y is found to be dependent on volume only for, roughly, T > O-2 &[3,4]. Hence, using the right-handside of equation (18), (aBslaT)v(aNaT), = YGIV + (WV) (aCv/aT)v = PBg 11+ (a In CL& In T),]. (19)
This relationship was used to obtain the derivatives given in Table 1. The shortened version of equation (19a) (by the arguments of the preceding section) is valid only if (a In BT/ aT),= 0. The fact that this derivative is negative in almost every case when equation (19a) is used to analyze the data implies that we have ignored the actual temperature dependence of Cv (or an explicit temperature dependence of y). The pressure derivatives are more difficult to calculate since the volume dependence of y must be known. Because y is temperatureindependent for T > 0.2 0,, we will assume (perhaps incorrectly) that the high temperature linear dependence of y on volume (equation (15)) holds in this region also. Then, from equation (18),
=-P-yT[l+(alnCv/alnV)T].
(20)
But from equations (6b) and (13), if y/V= constant, independent of T, (alnCv/~lnV)T=y(alnCv/alnT)v
(21)
and (a&lap) r = (aBs/aP)T +PyT[l
+y(a
In C,/a In T),].
(22) Since (dB,/aP)T is of the order of 3 to 5, and /3yT< O-1 for most solids161, the correction is of the order of 2 per cent or less. The sign of the correction depends on the sign of y (or the sign of p), and for a ‘normal’ solid (p > O)(aB,/aP), < (aBTIaP)T. This is
1344
C. A. SWENSON
physically reasonable since BS > &in general, and the entropy must decrease with increasing pressure (BS + BT) for a solid with positive thermal expansion. The quantity (a In C& In T) Vwhich appears in both equation (19) and equation (22) can be evaluated for a Debye solid. This derivative varies from+O.l at T=& to+0*38 at T= &/2 to +3 for very low temperatures. Anharmonic effects will tend to increase these effects for a real solid for T > &,, but in most cases the uncertainties in the differences (equations (19), (22)) will likely be masked by experimental uncertainties. If adequate experimental data exist for determining explicitly the temperature and volume dependence of both CV and y so that precise corrections can be made, the actual interest will be in these dependences, not in the corrections.
V,, is the molar volume of the solid at absolute zero and zero pressure. This calculation has been made for three typical solids (Cu, NaCl,.K), and the results are shown as the BT( V,,) curves in Figs. 2-4. 1460
1440
I
I
I
I
I
I
,
,
,
,
,
,
150
200
1
-
TEMPERATURE DEPENDENCE OF BT AT CONSTANT VOLUME
The foregoing discussion has been based largely on the assumption that BT is a function of volume only and, hence, that (aB,/W), is independent of temperature. This assumption is based in part on the data in Table 1 and on experimental results for the equation of state of the alkali metals [9-l I] and solid xenon [ 121. A direct measurement of BT as a function of temperature at constant volume would be of considerable interest; this is, unfortunately, not possible experimentally. One can, however, use equation (12b), the thermal expansion of a solid and the assumption that (aB,/aP), = n is constant to correct to constant volume conditions ultrasonic data which have been taken as a function of temperature at constant (zero or atmospheric) pressure. Hence, one can obtain the temperature dependence of the thermal contribution to the bulk modulus as: BT*(T,~~)=BT(T,V~)--B~(T=O,~~) =B~(T,P=O)(V(T,P=O)/V,,)” -B&T
= 0, I’,,).
(23)
23ot 0
SO
100
TEMPERATURE,
250
4
300
OK
Fig. 2. The temperature dependence of the elastic constants for copper at atmospheric pressure and at constant volume. The curve for B, was from equation (23) while an equivalent relationship was used for Bs and the shear constants C’ = (C,, - C,,)/2 and C,. The slopes of the constant volume curves at 300°K are identical with those which would be calculated from equation (11).
The sources of experimental data for these figures are indicated in Table 2. The isobaric temperature dependences of the isothermal and adiabatic bulk moduli and of the shear constants are shown also. The constant volume temperature dependences of Bs, C44, and C’ = (C,, - Cl,)/2 can be calculated in much the same fashion, with II being replaced in each case by (aIn c/aIn V),. Theresults of these calculations are given also in Figs. 2-4. The assumption that the pressure derivatives
EQUATION
OF STATE
OF CUBIC
1345
SOLIDS
42 ?
L 0
I
50
I
too
1
1
L
1
150
200
2!50
300
I
TEHPERATURE:K TEMPERATURE.-K
Fig. 3. The temperature dependence of the elastic constants for NaCl at atmospheric pressure and at constant volume calculated as for Fig. 2. The calculated slope for C, at constant volume and 300°K is shown; the slopes for all of the other moduli agree with those calculated from equation (I 1).
are temperature-independent for these elastic constants is of questionable validity although the results of Bartels and Schuele show little change in these derivatives for NaCl and KC1 between 295 and 195”K[21]. The behavior of BT at constant volume is qualitatively different from that which is found for Bs and the shear moduli. As the temperature increases from absolute zero, BT( V,,) first decreases slightly before becoming relatively constant for both Cu and NaCl, with B,*(3OO“K, V,) = -O-005& and-O-03& respectively for these two. The data for potassium metal are such that BT* (T, V,,) is essentially zero at all temperatures. The linear temperature dependence of BS, CM, and C’ at constant volume and high temperatures can be interpreted in terms of anharmonic effects which presumably are small in the case of B,[33-351.
Fig. 4. The temperature dependence of the elastic constants for potassium metal at atmospheric pressure and at constant volume calculated as for Fig. 2. The slopes of the constant volume curves (equation (11)) are shown where they do not agree with those calculated here. The behavior of Br(T, V,,) for potassium appears to be anomalous above 250°K.
Similar calculations for BT can be made for any solid for which reliable ultrasonic data exist over a range of temperature and pressure. Table 2 presents the result of a number of such calculations. The various input data appear in the second through fifth columns, and BT* (3OO”K, V,,) is given in the seventh column. In general, B,* is small and negative and is of the order of 1 or 2 per cent of B,. These results (Table 2) can be understood qualitatively in terms of the Mie-Gri_ineisen equation of state (equation (4)) and the Debye approximation. The high temperature form of the internal energy of a Debye solid is given by U* (T, V) = 3RT - (9/8)R&.
In this limit P* and BT* become (2-4)):
(24)
(equations
C. A. SWENSON
1346 P*(T,
V) =
(r/V)
(3RT-
(9/8)RB,)
(25)
and BT* (T, V) = - (al’*/13 In V), = - (9/8) (R&~/V,).
(26)
The definition y = - (d In O/d In V) and the linear dependence of y on V have been used in these calculations. V, is the volume of one gram-atomic weight, since equation (24) refers to a solid with 3No(N0 = Avagadro’s number) degrees of freedom. Both y and 8 should be high temperatures values (so-called yrn and 0, ). The right-hand side of equation (26) and data given in Tables 1 and 2 can be used to calculate the values of Br* which are shown in the last column in Table 2. The agreement in both sign and magnitude is good and is well
within experimental uncertainties in most instances. As was mentioned earlier, the ideal situation would involve both temperature and pressure measurements on the same crystal in the same laboratory, and this is seldom the case. The significance of these small, negative values for BT* can be understood as follows. At absolute zero the equation of state and, hence, BT contain contributions due both to the static lattice and to the zero point motion, while at high temperatures (in the classical region) the zero point motion effects must disappear. Thus, this apparent softening of the solid with increasing temperature is due directly to the disappearance of the zero point energy in the classical limit. In our use of the quasi-harmonic approximation (equation (4)) we have lumped all anharmonic effects into
Table 2. Summary of absolute zero and 300°K isothermal bulk moduli and of the data used to calculate BT* (V,, 300°K). References are indicated for the ultrasonic data (the first reference indicates zero pressure, the second high pressure data), while AVIV0 was obtained or deduced from the references, Corruccini and Gniewek[36], Collins and White[4], or (for the alkali halides) White [37] B,(VJ
BT(P = 0,300”K) (kbars, lo+’ d/cm2)
CU
As AU Al Fe Ta NACI LiF KC1 NaF NaI TiBr CaF, Na K xe’*j Li@)
1420 1087 1803 793.8 1731 1942 273.9
1332 996 1667 723.2 1640 1901 233.9
g&J 700 514.3 178.6 257 882 882 74 37 30 123
664 173.2 630 454.0 149.7 201 817 825 61.6 30 15”” 112
(a) Normalized.
&* AVIV0 ( l0-2)
(asslaP),
(cms/ztom)
(kbars)
BT*
(es. (23Meq. (26))
References
1.0 1.3 1.0 1.3 0.63 0.45 2.5
5.59 6.18 6.43 5.22 5.97 3.19 5.27
7.1 10.6 10.4 10.0 7.4 10.9 13.2
-8 -12 -33 -25 -29 -14 -9
-18 -12 -12 -22 -16 -6 -6
138, 151 114,151 [14,151 [17,t61 118,191 [39,201 [40,211
1.4 2.4
5.14 5.34
5.0 18.3
-30 -1
-31 -3
2.3 3.0 4.0 097
5.18 5.4 6.5 4.91 6.0 ;:“97
7.5 20 19 8.0
-3 -2.1 0 -20 -8 -2 +0.5 --1*[email protected]) 0
[4r’;2:‘1 r421 [40,221 143,231 1241 [251 [291 [26,441 r27,451
;:: 1O+P’ 3.4
6. l’d’ 3.6(@
22.7 43.3 38.5 12.7
-12 -2.2 -3.5 -21 -0.7 -0.3 -0.8 -2
EQUATION
OF STATE OF CUBIC SOLIDS
the Griineisen constant, and anharmonic contributions to BT do not appear. It is remarkable that they are not observed experimentally. The shear constants appear to be much more sensitive to these anharmonic effects. No attempt will be made to evaluate critically the input data to Table 2. Inconsistencies exist for LiF while earlier inconsistencies for NaI have been resolved by the publication of more recent ultrasonic data. The high pressure data for CaF, due to different laboratories are in disagreement at room temperature by 20 per cent in the pressure derivatives of the bulk moduli, and indeed, one of these sets of data[29] appears to contradict many of the ideas presented in this paper. A careful reevaluation of these results would appear to be in order. The ultrasonic data for sodium and potassium (see Table 2) can be interpreted as being consistent with the high-pressure piston-displacement results[9, lo] although in the case of sodium some uncertainty exists. B,( T,P = 0) behaves strangely for tantalum [39], and (in contrast to all other solids, the behavior of which is similar to that in Figs. 2-4) this modulus actually decreases with increasing temperature at constant volume near room temperature [20]. The original suggestion that the isothermal bulk modulus is a function of volume only[2] arose from the piston-displacement pressurevolume-temperature measurements on the alkali metals Na[9], K[lO] and Lilll]. Similar data for solid xenon did not quite fit into this pattern and required a value of ET* = 1.6 kbars to reproduce the data (Table 2) [ 121. The calculated values of BT* (of the order of 0.01 B,) for the alkali metals are sufficiently small so that they could not be observed with this relative crude technique. On the other hand, the much greater relative importance of zero point effects for the lightly bound inert gas solids makes this contribution to BT of greater importance. Neon represents an extreme example. Here, B,(l’,,) = 11.2 kbars, while from equation (26) (using 8 = 75”K,
1347
y = 3, V,, = 14 cm3/mole) BT* = 4.7 kbars [46]. It is obvious that zero point energy effects will not be small and softening of the solid should occur as the temperature is increased which is even greater than one would expect from the volume dependence of the bulk modulus (n - 6). This, indeed, is what is observed. Similar, although less dramatic, considerations apply to argon[47] and krypton[48]. One certainly cannot assume that BT is a function of volume only for these solids even to a first approximation. sum&Y
To a first approximation, the equation of state of cubic solids can be represented by a temperature-independent isothermal bulk modulus which is a linear function of the pressure. If zero pressure thermal expansions are not known, these can be calculated from the above using the Mie-Griineisen equation of state and a Griineisen constant y which is linearly dependent on the volume. In the next approximation, BT is found to decrease slightly at constant volume with increasing temperature at low temperatures due to the disappearance of zero point energy effects, although it is relatively temperature-independent for temperatures greater than the Debye temperature f3,. Similar considerations do not apply to the adiabatic bulk modulus Bs or the shear constants where anharmonic effects play a large role at the higher temperatures. The lighter inert gas solids represent an exception to the above generalizations. In this case, the zero point energy is of much greater relative importance in determining the low temperature thermal properties for these solids. Acknowledgements-The author is indebted to Dr. C. S. Smith for discussions over a number of years which assisted in clarifying the ideas presented in this paper. REFERENCES 1. DECKER D. L., J. appl. Phys. 36; 157 (1965); ibid., 37,5012 (1967). 2. SWENSON C. A., Thermal Contribution to the Equation of State of Solids, USAEC Rept. IS-870
1348
3. 4. 5. 6.
7.
8. 9. 10. 11.
C. A. SWENSON
(1964). The present paper presents a simplified treatment of the ideas advanced in this report. In many instances more recent data have been used in extending the analysis. BRUGGER K. and FRITZ T. C., Phys. Rev. 157, 524(1967). COLLINS J. G. and WHITE G. K., Progress in Low Temperature Physics (Edited by Gorter) Vol. IV, p. 450. North-Holland, Amsterdam (1964). See, for instance, ZEMANSKY M. W., Heat and Thermodynamics, 4th Edn. McGraw-Hill, New York (1957). ANDERSON 0. L., J. Phys. Chem. Solids 27,547 (1966). This paper defines B,’ = (aB,/aP),, and discusses equations of state in terms of this quantity. We have chosen to restrict the arguments to (M&P),. The difference is small in most cases. MACDONALD J. ROSS, Rev. mod. Phys. 38,699 (1966). This paper defines I&,’= (M,v/aP),, and posed equations of state of the Murnaghan type. For our purposes, the differences are small. OVERTON W. C.,J. them. Phys. 37,716 (1962). SWENSON C. A., J. Phys. Chem. Solids 27, 33 (1966). BEECROFT R. 1. and SWENSON C. A., J. Phys. Chem. Solids 18,329 (1961). MONFORT C. E. and SWENSON C. A., J. Phys. Chem. Solids 26,291
12. PACKARD
Chem. Solids 24,1405 (1963). 13. LAZARUS D., Phys. Rev. 76,545
18. RAYNE
J. A. and CHANDRASEKHAR B. S., Phys. Rev. 122,1714(1961). 19. ROTTER C. A. and SMITH C. S., J. Phys. Chem. (1966).
20. CHECHILE R. A. and SMITH Communication ( 1967). 21. BARTELS R. A. and SCHUELE 22. MILLER
Phys. Chem. Solids 27,637
C. S., Private D. E., J. Phys.
(1965).
R. A. and SMITH C. S., J. Phys. Chem.
Solids 25, 1279 ( 1964).
23. ROBERTS R. W. and SMITH C. S.. Private Communication (1967). 24. MORSE G. E. and LAWSON A. W.. J. Phvs. Chem. Solids 28, 939 (1967). The data given in Table 2 above differ slightly from those given in Table 3 of this reference due to a computational error in producing Table 3. The actual data (Tables 1 and 2) are correct. We are indebted to Dr. Morse for correspondence on this point.
(1966).
W. R. and TRIVISONNO J*, J Phys. Chem. Solids 26,273 (1965). 28. SALAMA K. and ALERS G. A., Phys. Rev. 161, 673 (1967). 29. HO PAUL S. and RUOFF A. L., Phys. Rev. 161, 864 (1967).
30. MONFORT
C. E. and SWENSON
C. A., J. Phys.
Chem. Solids 26,623 ( 1965). 31. ALDER B. J., Solids Under
Pressure (Edited by Warshauer and Paul). D. 385. McGraw-Hill. New York(1963). ” 32. ENCK F. D. and DOMMEL .I. G., J. appl. Phys. 36,839
(1965).
33. HIKI Y., THOMAS J. F. and GRANATO A. V., Phys. Reb. 153,764 (1967). 34. PASTINE D. J., J. Phys. Chem. Solids 28, 522 (1967).
35 SLAGLE
0. D. and McKINSTRY H. A., J. appl. (1967). These authors interpret the total linear dependence of the elastic constants at constant pressure as an anharmonic effect. We have chosen to eliminate the explicit volume dependence as is done in references[33] and [34]. CORRUCCINI R. J. and GNIEWEK, Nat. Bur. Stand. Monogr. 29 (1961). WHITE G. K., Proc. R. Sot. A286.204 (1965). ALERS G. A. and THOMPSON D. O., J. uppl. Phys. 32,283 (196 1). FEATHERSTONE F. H. and NEIGHBOURS J. R., Phvs.
(1949).
327 (1964).
28, 1225 (1967). The pressure
27. MARQUANDT
C. A., J. Phys.
J. R. and ALERS G. A., Phys. Rev. 14. NEIGHBOURS 111,707 (1958). 15. DANIELS W. B. and SMITH C. S., Phys. Rev. 111, 713 (1958). 16. SCHMUNK R. E. and SMITH C. S., J. Phys. Chem. Solids 9, 100 (1959). 17. KOMM G. N. and ALERS G. A., J. uppl. Phys. 35,
Chem. Solids 26,537
D. E., J. Phys. Chem. dependence of the elastic constants of CaF, (as well as of BaF,) have been determined at 195”i(‘also as an extension of this work (C. Wong and D. E. Schuele, Private Communication). 26. DIEDERICH M. E. and TRIVISONNO J., J. Solids
(1965).
J. R. and SWENSON
Solids 27,267
25. WONG C. and SCHUELE,
36. 37. 38. 39.
38
437
Phys. Rev. 130,1324(1963). 40. LEWIS J. T., LEHOCZKY J. T. and C. V., Phys. Rev. 161,877 (I 967). 41. NORWOOD M. H. and SQUIRE C. F., 84,758 (1958). 42. BRISCOE C. V. and SQUIRE C. F., 106, 1175 (1957). The room temperature
BRISCOE Phys. Rev., Phys.
Rev.
data given in this paper disagree with those given in reference 1221. We have normalized the data to give the same B,( T, P = 0) value at room temperature. 43. CLAYTOR R. N. and MARSHALL B. J., Phys. Rev. 120,332 (1960). 44. DANIEk W: B., ihys. Rev. 119,1246 (1960). Chem. 45. SMITH P. A. and SMITH C. S.. J. Phvs. . ~~ Solids 26,279 (1965).
46. BATCHELDER D. N., LOSEE D. L. and SIMMONS R. 0.. Phys. Rev. 162,767 (1967). 47. PETERSON 0. G., BATCHELDER D.. N. and SIMMONS R. 0.. Phvs. Rev. 150.703 (1966). 48. URVAS A. O., L&SEED. L. and SIMMONS R. O., J. Phys. Chem. Solids 2& 2269 (1967).
Pergamon Press 1968. Vol. 29, pp. 1337-1348.
EQUATION
Printed in Great Britain.
OF STATE OF CUBIC GENERALIZATIONS’r
SOLIDS;
SOME
C. A. SWENSON lnstitute for Atomic Research and Department of Physics, Iowa State University, Ames, Iowa 50010, U.S.A. (Received
26 December
1967)
Abstract-The equations of state (pressure-volume-temperature relationships) for a number of cubic solids can be approximated-at high temperature (T > 0,) by a temperature-independent isothermal bulk modulus BT which varies linearly with the applied pressure. This is demonstrated by an analysis of both piston-displacement and ultrasonic data, as well as by an elementary calculation based on the Mie-Griineisen equation of state. These generalizations also should be valid for ST at low temperatures to within one or two percent except for the inert gas solids. The high temperature equation of state which follows from this analysis contains the following features: (a) the Griineisen constant y is a temperature-independent linear function of volume, (b) ET is a linear function of temperature at constant pressure, (c) (a(B,-BB,)/aT)v=PB~ and (.3(B,-&)/#‘)T=-PyT, where p is the volume thermal expansion coefficient. These relationships should hold as a first approximation for most cubic solids. INTRODUCTION
of state (pressure-volumetemperature) data often are required to express high pressure data (such as resistivity or magnetization) in terms of a volume dependence. In many instances, the limited equation of state data which exist for a given substance must be extrapolated well beyond their range of validity. These data might include only one pressure-volume isotherm at room temperature, a limited number of elastic constant or thermal expansion measurements as a function of temperature, etc. The equation of state ddta which exist for NaCl are reasonably extensive, and a semitheoretical interpolation and extrapolation of these data have been suggested as a basis for pressure standardization [ 11. Similarly detailed primary data do not exist for most other solids although an analysis of various types of data suggests correlations which can form the basis for elementary and reasonably reliable EQUATION
extrapolations of their thermodynamic functions [2]. The basic description of a solid can be expressed in terms of the Helmholtz free energy F(T, V) = U,(V) + F*(T,
= U,(V)+U*(T,V)-TS(T,V)(l) where UI,(V) is the cohesive energy at absolute zero, U* is the temperature-dependent internal energy, and S is the entropy. Both of these latter terms, and indeed all terms indicated with an * in this paper, approach zero at T = 0. The equation of state and the bulk modulus can be obtained as P(T,V)
=-(aF/W),=P,(V)-i-P*(T,V)
(2) and B(T, V) = - V(aPlaV), = &(I/)+B,*(T,
tWork was performed in the Ames Laboratory of the U.S. Atomic Energy Commission. Contribution No. 2225.
V)
= + V(a2F/a~)T v).
(3)
By using these equations we assume that in a theoretical treatment the ground state and 1337
C. A. SWENSON
1338
excited state contributions to the energy are independent and additive. We also might assume as a logical extension that the energy contributions due to various systems in a solid, such as magnetic spins, free electrons, lattice vibrations, etc., are independent and additive. The following discussion will be restricted to lattice properties and will apply most reliably only at temperatures which are greater (in principle, much greater) than the Debye temperature 0,. In this limit, the vibrational properties of solids are essentially classical, and the specific heat at constant volume is given by the Dulong and Petit value, CV = 3R. Thermal contributions P* and B* will be shown to be relatively small. Complications due to anharmonic and defect contributions will not be considered, even though they can be of great importance under certain conditions. An implicit restriction to cubic solids will be made although the basis for the restriction is not clear. The first part of this paper suggests semiquantitative guidelines which are to be used for extrapolation if adequate data do not exist. These guidelines should not be used for more than a qualitative test of the reliability of experimental data. The second part of this paper will consider the equation of state at lower temperatures and will make plausible the conclusions reached in the first part and their extrapolation to low temperatures. EQUATION OF STATE AT HIGH TEMPERATURES
Basic relationships (1) The Mie-Griineisen equation of state is obtained by using a characteristic temperature to represent the volume dependence of F*[F*/T = @*(O(V)/T)]. Equation (2) then becomes P(T,V)
=Po(V)+y,(V)U*/V
(4)
with yU = - d In 0/d In V. This type of relationship is useful when discussing lattice properties, and the Debye model represents a special
case. In practice, yU as defined by equation (4) will be a slowly varying function of both temperature and volume and will approach a constant value at high temperatures. The isobaric change in volume of a solid with temperature (the thermal expansion) can be related to the increase of P* with increasing temperature. At zero pressure, the volume expansion is given by P*( T, V) = -P,(V). The definition of the bulk modulus (equation (3)) can be combined with this relationship and equation (4) to calculate small volume changes as AV/V, = (V(T) - V,,)/V, = P*/B&, = yJJ*IBJ.
(5)
Equation (4) is not particularly useful since the various thermodynamic quantities appearing in it are not easily measured. The Griineisen relationship, P = yCJBJ
(6)
follows from equation (4) if y is a function of volume only, and the mathematical identity @pIaT) V= P&
(7)
is used. Here, p = (aln V/aT), is the volume thermal expansion coefficient and CV is the heat capacity at constant volume. Our simple model (equation (4)) assumes that y is temperature-independent, while more complex calculations (such as those by Fritz and Brugger[3]) show that y as defined by equation (6) should be different from yU as defined by equation (4) at intermediate temperatures and that the two should be equal and independent of temperature only for temperatures greater than 015 to e/2. This is confirmed by experiment [3,4]. Both BT and V are slowly varying functions of temperature and at all temperatures (for most solids) the temperature dependences of /3 and C, are very similar. In the classical or high temperature limit,
EQUATION
OF STATE
Cv = 3R, independent of volume. CV cannot be determined directly since experimental data are taken at constant pressure. The resulting temperature-dependent specific heat at constant pressure C, is related to CV by the following expression [5];
CP = C,( 1+ fly T)
(8)
where y is defined operationally by equation (6). A similar relationship exists between the adiabatic bulk modulus BS (which is determined from ultrasonic measurements) and the isothermal bulk modulus BT; Bs=B,(l+PyT).
(9)
Equations (7-9) can be used to express the definition of y (equation (6)) in two alternate and useful ways as follows: Y = P&V/C,
(6a)
and (aP/aT)”
= r(C,/V).
(6b)
(2) Anderson [6,7] demonstrates in a recent paper that most pressure-volume isotherms can be fitted by the Mumaghan relationship P(T, I’) = BTo(T, I’,)/n)((I’&‘)n--
OF CUBIC
(10) in lieu of direct P-V measurements[6]. One must, of course, convert the adiabatic data obtained from the ultrasonic measurements to isothermal values using equation (9) and a relationship between ( dBs/tIP)T and (dB,/dP),. This latter relationship is quite complex[8], but it can be simplified considerably for high temperature data as we will demonstrate below. (3) Considerable evidence exists that at high temperatures the isothermal bulk modulus is solely a function of volume[2]. This evidence comes from two sources. First, relatively crude P-V-T measurements on the alkali metals Li[9], Na[lO], and K[l I] and on solid xenon[ 121 can be represented well by this approximation and equation (10). These data cover wide relative pressure (to 20 kbars) and temperature (20°K to the melting point) ranges, and BT( T, V) values are reliable to approximately 3 per cent. Second, as Lazarus pointed out [ 131, ultrasonic data which are obtained as a function of temperature at atmospheric pressure and as a function of pressure at constant temperature can be combined to evaluate the explicit temperature dependence of various elastic (C)as follows: (aln C/aT)p
1) (10)
where n = (aB,/dP), is a dimensionless constant (called BI, by Anderson) which is a characteristic of the solid involved and usually has a value between 3 and 6. BTo and V, are the zero pressure values of the isothermal bulk modulus and molar volume, respectively. This relationship holds remarkably well for a great number of substances[6]. Although equation ( 10) cannot be justified on theoretical grounds, it is probably the most useful single relationship for the extrapolation of P-V data to higher pressures. Indeed, Anderson suggests that values of BTO and n as obtained in relatively low pressure high precision elastic constant measurements be used in equation
1339
SOLIDS
= (aln C/dT), - (aln C/dP),(aP/aT), = (aln C/aT)“-
(aln C/aP)&Bp (11)
When ultrasonic measurements are analyzed in this manner, they give most directly (aln BJdT) “. A correction must be applied to obtain (aln BT/aT) v Table 1 gives magnitudes for (aln BT/dT) v for 16 cubic solids for which both pressure and temperature-dependent ultrasonic data exist near room temperature. The listing is not complete, and no ‘selection’ of data was made. The primary data which were used in these calculations are not given but will be found in the references indicated. The method which
C. A. SWENSON
1340
was used to convert from the adiabatic to the isothermal temperature derivative will be described in the following section. For almost all cases, the magnitude of (alnB,/aT), is less than 10m4OK-‘. This suggests that BT will vary by less than 1 per cent in a 100 deg. temperature interval. While an effect of this order of magnitude can be measured easily with modern techniques, it is effectively zero for the extrapolations of the equation of state which are considered in this paper. In making comparisons of this type it is desirable that both the temperature and pressure derivatives be obtained for the same sample and preferably in the same laboratory.
This is the case for very few of the substances listed in Table 1. This discussion of experimental and elementary theoretical aspects of the equation of state of solids suggests the following postulates which will be used to establish guidelines for the extrapolation of P-V data. B,(V)
(aB,/aT),
= BT(V,,) +nP
(12aj
= BT(F/O) (v,/Vn.
(12bj
= 0
(12cj
T > 0,; CV = 3R and y is independent
(12dj
Table 1. Summary of various 300°K thermodynamic parameters for a number of cubic solids. In most instances values of 8, y and p, as well as of the derivative (alnB,/aT), are given in (or can be deduced from) data given in the references. (aln B,/aT) V was calculated as described in the text. Where two references are given, the first involves zero pressure data and the second high pressure data
( lO-4 “K-l)
cu &
AU Al Fe Ta NaCl KC1 LiF NaF NaI TlBr CaF, BaF, Na K Xe’“’ Li’fJI
345 227 162 430 477 260 320 235 740 490 164 132 508 300 160 95 64
2.0 2.4 2.9 2.35 1.6 1.6 1.56 1.45 1.6“ 1.4 1.71 2.3 1.90 1.6 1.1 1.1 2.5
0.495 0.576 0.423 0.70 0.35 0.20 1.21 1.10 1.4 1.1 1.36 1.53 0.57 0.55 2.0 2.5 10”’
390
0.8
1.34
(“‘Piston-displacement @‘165”K.
of T.
+1+x + 1.62 + 0.96 + 1.3 + 1.0 -0.4 + 1.5 + 1.0 + 1.3 + 1.6 + 1.2 +2.2 + 0.34 +0.06 + 1.1 +5.16 -
data only.
-0.1 +0.3 -0.2 -0.2 +0.4 -0.7 -0.3 -0.5 -0.3 +0.1 -1.1 -0.3 -0.5 - 1.2 +0.9 +2.8 0 0
114,151 [14,151 [14,15] [16, 171 [18, 191 [20] [211 [211 [221 [221 [231 [241 1251 [25] [261 ~271 [I21 [91
EQUATION
OF STATE OF CUBlC SOLIDS
Here, equations (12a) and (12b) are a direct of the assumption of the consequence Murnaghan relationship equation (10). In his original formulation and discussion, Anderson made no comment on the temperaturedependence of n [6]. Our postulate that the isothermal bulk modulus is independent of temperature at constant volume (( 12~) above) implies that IZ must be independent of temperature as well as pressure or volume. This is verified by experiment for NaCl and KC1[21] and for CaF, and BaF,[25] and is approximately correct for copper[28]. Data for CaF, exist[29] which disagree with those of Wong and Schuele[25] in that they show a significant temperature dependence of the pressure derivatives. The reason for the discrepancy is not understood.
Conclusions The above postulates lead to a number of general statements which should apply as a first approximation to a description of the equation of state of solids in the classical region. These statements include first a rather general recipe for the extrapolation of a single P-V isotherm in both pressure and temperature, and then suggestions as to the volume dependence of y, the isobaric temperature dependence of p and BT and relatively simple relationships between the pressure and temperature derivatives of Bs and BP These ideas will be considered in detail in the following sections. The extrapcilation of P-V data in both T and P. Equation (12~) states effectively that all P-V isotherms have the same slope for a given molar (or specific) volume. As will be shown in the second half of this paper, the error involved in applying this statement to all temperatures from absolute zero upwards is of the order of one or 2 per cent of BT at most if the inert gas solids are not included. Hence, if one P-I’ isotherm is known at a temperature To, any other P-V isotherm at a temperature
1341
T can be generated by translating this isotherm parallel to the pressure axis by an amount equal to Al’* = P*(T) -P* (T,) (equation (2)). The important variable here is the molar volume; these statements do not apply to the relative volume changes V/V,, or Av/F,,. As we will show in a following section, P* does not depend on volume within our approximations (equations (12c, d)). The original P-V isotherm which is used can be one which has been determined directly by experiment or which has been constructed from ultrasonic measurements of BT( T, P = 0) and n = (aB,/aP), using equation (10). In any event, equation (10) will be useful to extrapolate this isotherm to other molar volumes. AP* can be determined as the difference between the pressures necessary to obtain a given molar volume at T as compared with To (equation (2)). This implies that P* ( T) and hence the equation of state can be determined immediately from a single P-V isotherm and the zero pressure volume thermal expansion. This, for instance, is the procedure which was used to generate an equation of state for indium metal from Bridgman’s room temperature isotherms and measured zero pressure volume thermal expansions [2,30]. The method outlined in the preceding paragraphs will give (by equations (12b-c)) an estimate of BT( T, P = 0). If data exist for the temperature dependence of BT at zero pressure, it is preferable to use these in equation (10) along with a value for n to determine P-V isotherms rather than to assume equation (12~). Isotherms which are calculated from equation (10) are as precise as the value of BTO which are used, anddepend on the value of n only in second and higher orders. At the other extreme, a minimum of expansion and bulk modulus data exist for a solid. An estimate of BT and n can be made from Anderson’s correlations [6] and at least in the classical limit the volume expansion can be estimated from equation (5). The volume dependence of y. Equation (12~) implies the following;
1342
C. A. SWENSON
-(aBJaT)”
= V(azP/aTaV) = v(a/av)
(aP/aT)”
= 0. Hence, (MlaT), is independent of volume. Also, an elementary thermodynamic relationship
(a2PlaT2)V = T-l(aC,lav)T
(13)
can be used to show that in the high temperature limit (where C, = 3R) (aP/aT),is independent of temperature also. An immediate and useful consequence is that
since the thermal pressure is equivalent to a negative pressure applied to the solid. All of the quantities multiplying (T - T,,) on the right-hand side are constants, independent of temperature and volume, so at atmospheric pressure, BT should to a first approximation be linear function of the temperature. This result suggests that it is more meaningful to plot BT as a function of temperature than the compressibility ( kT = l/B,). Equation (14) can be combined with equation (16) to predict similarly the temperature dependence of p for a classical solid: P(T, P = 0) = (PBT)w=o/[BT(TO, -n(rlWR(T-
fiBT = (aP/aT) v = constant, independent of T,Vfor T > 0,. (14) Equation (14) can be checked experimentally and has been found to hold to within 3 per cent for the high temperature P-V-T data for Li[9], Na[lO] and K[ll], and for solid xenon [ 121. The constancy of BT also is evident in the atmospheric pressure data for Cu near room temperature (as collected by Overton
m. Equation (14) can be combined with equation (6b) to establish that (in this limit where C, is a constant) the Griineisen constant is a linear function of the volume. This follows directly from rewriting equation (6b) as: y/V = (aP/aT) &2, = constant.
(15)
There is no direct way of verifying this prediction, which also has been used in the analysis of shock-wave data[3 11. The isobaric temperature-dependence and BT. The temperature-dependence
of /I
of BT at atmospheric (or zero) pressure can be obtained from equations (12a) and (4) as:
B,(T,P= O)-BT(To,P= 0) =-nP*(T,l') =-n(rC,W)(T-To)
(16)
P = 0) To)]. (17)
Equation (17) can be verified for NaCl since this solid satisfies our postulates, and thermal expansion data exist for it up to the melting point [32]. Indeed, the temperature dependence of the thermal expansion is of basic interest since vacancy effects should appear as an anomalous increase in p with increasing temperature, and equation (17) should give a first approximation to ‘normal’ behavior. Room temperature values of the various thermodynamic quantities for NaCl have been used together with equation (17) to calculate the crosses in Fig. 1. The agreement with experiment over a factor of 3 in absolute temperature and almost 2 in (Y= p/3 is quite remarkable. The complete P-V-T relationship for NaCl at these high temperatures could be calculated using equations (10) and (16) and this result. Relationships
between
the
derivatives
of
BT and B,. Ultrasonic measurements of elastic constants give directly the adiabatic bulk modulus Bs, while the isothermal modulus is needed to establish an equation of state. The relationship between these (equation (9)) can be expressed as Bs - BT = B&yT
= yzCvT/V.
(18)
EQUATION
1343
OF STATE OF CUBIC SOLIDS
In practice, (a In C,/a In T)v is of the order of (a In &/a In T)P c O-1 for T > f3, if a Debye model is used. For practical purposes near room temperature (to within 10 per cent or so) (a In B&WV = (&I&)
(a lnB&Wv-Py. (19a)
I
d
I
xEQUATION I7 (CALCULATED) 4..al(il’ 100
’ ’ ’ ’ ” ” ’ ’ ’ ’ 300 500 700 TEMPERATURE “‘2
1
Fig. 1. A comparison between the linear expansion coefficient a = p/3 for NaCl as calculated from equation (17) and as measured[32]. Only room temperature, low pressure data were used in the extrapolation.
Over-ton has outlined a procedure for converting temperature and pressure derivatives, of Bs to the same derivatives for B&3]. This procedure is quite complex and requires detailed data. It can be simplified considerably by using the ideas expressed in the preceding section and by making use of combinations of thermodynamic quantities which are slowly varying with temperature or pressure. The relationship between (dB,/aT) v and is obtained as follows. Both (a&/W v experimentally and theoretically, y is found to be dependent on volume only for, roughly, T > O-2 &[3,4]. Hence, using the right-handside of equation (18), (aBslaT)v(aNaT), = YGIV + (WV) (aCv/aT)v = PBg 11+ (a In CL& In T),]. (19)
This relationship was used to obtain the derivatives given in Table 1. The shortened version of equation (19a) (by the arguments of the preceding section) is valid only if (a In BT/ aT),= 0. The fact that this derivative is negative in almost every case when equation (19a) is used to analyze the data implies that we have ignored the actual temperature dependence of Cv (or an explicit temperature dependence of y). The pressure derivatives are more difficult to calculate since the volume dependence of y must be known. Because y is temperatureindependent for T > 0.2 0,, we will assume (perhaps incorrectly) that the high temperature linear dependence of y on volume (equation (15)) holds in this region also. Then, from equation (18),
=-P-yT[l+(alnCv/alnV)T].
(20)
But from equations (6b) and (13), if y/V= constant, independent of T, (alnCv/~lnV)T=y(alnCv/alnT)v
(21)
and (a&lap) r = (aBs/aP)T +PyT[l
+y(a
In C,/a In T),].
(22) Since (dB,/aP)T is of the order of 3 to 5, and /3yT< O-1 for most solids161, the correction is of the order of 2 per cent or less. The sign of the correction depends on the sign of y (or the sign of p), and for a ‘normal’ solid (p > O)(aB,/aP), < (aBTIaP)T. This is
1344
C. A. SWENSON
physically reasonable since BS > &in general, and the entropy must decrease with increasing pressure (BS + BT) for a solid with positive thermal expansion. The quantity (a In C& In T) Vwhich appears in both equation (19) and equation (22) can be evaluated for a Debye solid. This derivative varies from+O.l at T=& to+0*38 at T= &/2 to +3 for very low temperatures. Anharmonic effects will tend to increase these effects for a real solid for T > &,, but in most cases the uncertainties in the differences (equations (19), (22)) will likely be masked by experimental uncertainties. If adequate experimental data exist for determining explicitly the temperature and volume dependence of both CV and y so that precise corrections can be made, the actual interest will be in these dependences, not in the corrections.
V,, is the molar volume of the solid at absolute zero and zero pressure. This calculation has been made for three typical solids (Cu, NaCl,.K), and the results are shown as the BT( V,,) curves in Figs. 2-4. 1460
1440
I
I
I
I
I
I
,
,
,
,
,
,
150
200
1
-
TEMPERATURE DEPENDENCE OF BT AT CONSTANT VOLUME
The foregoing discussion has been based largely on the assumption that BT is a function of volume only and, hence, that (aB,/W), is independent of temperature. This assumption is based in part on the data in Table 1 and on experimental results for the equation of state of the alkali metals [9-l I] and solid xenon [ 121. A direct measurement of BT as a function of temperature at constant volume would be of considerable interest; this is, unfortunately, not possible experimentally. One can, however, use equation (12b), the thermal expansion of a solid and the assumption that (aB,/aP), = n is constant to correct to constant volume conditions ultrasonic data which have been taken as a function of temperature at constant (zero or atmospheric) pressure. Hence, one can obtain the temperature dependence of the thermal contribution to the bulk modulus as: BT*(T,~~)=BT(T,V~)--B~(T=O,~~) =B~(T,P=O)(V(T,P=O)/V,,)” -B&T
= 0, I’,,).
(23)
23ot 0
SO
100
TEMPERATURE,
250
4
300
OK
Fig. 2. The temperature dependence of the elastic constants for copper at atmospheric pressure and at constant volume. The curve for B, was from equation (23) while an equivalent relationship was used for Bs and the shear constants C’ = (C,, - C,,)/2 and C,. The slopes of the constant volume curves at 300°K are identical with those which would be calculated from equation (11).
The sources of experimental data for these figures are indicated in Table 2. The isobaric temperature dependences of the isothermal and adiabatic bulk moduli and of the shear constants are shown also. The constant volume temperature dependences of Bs, C44, and C’ = (C,, - Cl,)/2 can be calculated in much the same fashion, with II being replaced in each case by (aIn c/aIn V),. Theresults of these calculations are given also in Figs. 2-4. The assumption that the pressure derivatives
EQUATION
OF STATE
OF CUBIC
1345
SOLIDS
42 ?
L 0
I
50
I
too
1
1
L
1
150
200
2!50
300
I
TEHPERATURE:K TEMPERATURE.-K
Fig. 3. The temperature dependence of the elastic constants for NaCl at atmospheric pressure and at constant volume calculated as for Fig. 2. The calculated slope for C, at constant volume and 300°K is shown; the slopes for all of the other moduli agree with those calculated from equation (I 1).
are temperature-independent for these elastic constants is of questionable validity although the results of Bartels and Schuele show little change in these derivatives for NaCl and KC1 between 295 and 195”K[21]. The behavior of BT at constant volume is qualitatively different from that which is found for Bs and the shear moduli. As the temperature increases from absolute zero, BT( V,,) first decreases slightly before becoming relatively constant for both Cu and NaCl, with B,*(3OO“K, V,) = -O-005& and-O-03& respectively for these two. The data for potassium metal are such that BT* (T, V,,) is essentially zero at all temperatures. The linear temperature dependence of BS, CM, and C’ at constant volume and high temperatures can be interpreted in terms of anharmonic effects which presumably are small in the case of B,[33-351.
Fig. 4. The temperature dependence of the elastic constants for potassium metal at atmospheric pressure and at constant volume calculated as for Fig. 2. The slopes of the constant volume curves (equation (11)) are shown where they do not agree with those calculated here. The behavior of Br(T, V,,) for potassium appears to be anomalous above 250°K.
Similar calculations for BT can be made for any solid for which reliable ultrasonic data exist over a range of temperature and pressure. Table 2 presents the result of a number of such calculations. The various input data appear in the second through fifth columns, and BT* (3OO”K, V,,) is given in the seventh column. In general, B,* is small and negative and is of the order of 1 or 2 per cent of B,. These results (Table 2) can be understood qualitatively in terms of the Mie-Gri_ineisen equation of state (equation (4)) and the Debye approximation. The high temperature form of the internal energy of a Debye solid is given by U* (T, V) = 3RT - (9/8)R&.
In this limit P* and BT* become (2-4)):
(24)
(equations
C. A. SWENSON
1346 P*(T,
V) =
(r/V)
(3RT-
(9/8)RB,)
(25)
and BT* (T, V) = - (al’*/13 In V), = - (9/8) (R&~/V,).
(26)
The definition y = - (d In O/d In V) and the linear dependence of y on V have been used in these calculations. V, is the volume of one gram-atomic weight, since equation (24) refers to a solid with 3No(N0 = Avagadro’s number) degrees of freedom. Both y and 8 should be high temperatures values (so-called yrn and 0, ). The right-hand side of equation (26) and data given in Tables 1 and 2 can be used to calculate the values of Br* which are shown in the last column in Table 2. The agreement in both sign and magnitude is good and is well
within experimental uncertainties in most instances. As was mentioned earlier, the ideal situation would involve both temperature and pressure measurements on the same crystal in the same laboratory, and this is seldom the case. The significance of these small, negative values for BT* can be understood as follows. At absolute zero the equation of state and, hence, BT contain contributions due both to the static lattice and to the zero point motion, while at high temperatures (in the classical region) the zero point motion effects must disappear. Thus, this apparent softening of the solid with increasing temperature is due directly to the disappearance of the zero point energy in the classical limit. In our use of the quasi-harmonic approximation (equation (4)) we have lumped all anharmonic effects into
Table 2. Summary of absolute zero and 300°K isothermal bulk moduli and of the data used to calculate BT* (V,, 300°K). References are indicated for the ultrasonic data (the first reference indicates zero pressure, the second high pressure data), while AVIV0 was obtained or deduced from the references, Corruccini and Gniewek[36], Collins and White[4], or (for the alkali halides) White [37] B,(VJ
BT(P = 0,300”K) (kbars, lo+’ d/cm2)
CU
As AU Al Fe Ta NACI LiF KC1 NaF NaI TiBr CaF, Na K xe’*j Li@)
1420 1087 1803 793.8 1731 1942 273.9
1332 996 1667 723.2 1640 1901 233.9
g&J 700 514.3 178.6 257 882 882 74 37 30 123
664 173.2 630 454.0 149.7 201 817 825 61.6 30 15”” 112
(a) Normalized.
&* AVIV0 ( l0-2)
(asslaP),
(cms/ztom)
(kbars)
BT*
(es. (23Meq. (26))
References
1.0 1.3 1.0 1.3 0.63 0.45 2.5
5.59 6.18 6.43 5.22 5.97 3.19 5.27
7.1 10.6 10.4 10.0 7.4 10.9 13.2
-8 -12 -33 -25 -29 -14 -9
-18 -12 -12 -22 -16 -6 -6
138, 151 114,151 [14,151 [17,t61 118,191 [39,201 [40,211
1.4 2.4
5.14 5.34
5.0 18.3
-30 -1
-31 -3
2.3 3.0 4.0 097
5.18 5.4 6.5 4.91 6.0 ;:“97
7.5 20 19 8.0
-3 -2.1 0 -20 -8 -2 +0.5 --1*[email protected]) 0
[4r’;2:‘1 r421 [40,221 143,231 1241 [251 [291 [26,441 r27,451
;:: 1O+P’ 3.4
6. l’d’ 3.6(@
22.7 43.3 38.5 12.7
-12 -2.2 -3.5 -21 -0.7 -0.3 -0.8 -2
EQUATION
OF STATE OF CUBIC SOLIDS
the Griineisen constant, and anharmonic contributions to BT do not appear. It is remarkable that they are not observed experimentally. The shear constants appear to be much more sensitive to these anharmonic effects. No attempt will be made to evaluate critically the input data to Table 2. Inconsistencies exist for LiF while earlier inconsistencies for NaI have been resolved by the publication of more recent ultrasonic data. The high pressure data for CaF, due to different laboratories are in disagreement at room temperature by 20 per cent in the pressure derivatives of the bulk moduli, and indeed, one of these sets of data[29] appears to contradict many of the ideas presented in this paper. A careful reevaluation of these results would appear to be in order. The ultrasonic data for sodium and potassium (see Table 2) can be interpreted as being consistent with the high-pressure piston-displacement results[9, lo] although in the case of sodium some uncertainty exists. B,( T,P = 0) behaves strangely for tantalum [39], and (in contrast to all other solids, the behavior of which is similar to that in Figs. 2-4) this modulus actually decreases with increasing temperature at constant volume near room temperature [20]. The original suggestion that the isothermal bulk modulus is a function of volume only[2] arose from the piston-displacement pressurevolume-temperature measurements on the alkali metals Na[9], K[lO] and Lilll]. Similar data for solid xenon did not quite fit into this pattern and required a value of ET* = 1.6 kbars to reproduce the data (Table 2) [ 121. The calculated values of BT* (of the order of 0.01 B,) for the alkali metals are sufficiently small so that they could not be observed with this relative crude technique. On the other hand, the much greater relative importance of zero point effects for the lightly bound inert gas solids makes this contribution to BT of greater importance. Neon represents an extreme example. Here, B,(l’,,) = 11.2 kbars, while from equation (26) (using 8 = 75”K,
1347
y = 3, V,, = 14 cm3/mole) BT* = 4.7 kbars [46]. It is obvious that zero point energy effects will not be small and softening of the solid should occur as the temperature is increased which is even greater than one would expect from the volume dependence of the bulk modulus (n - 6). This, indeed, is what is observed. Similar, although less dramatic, considerations apply to argon[47] and krypton[48]. One certainly cannot assume that BT is a function of volume only for these solids even to a first approximation. sum&Y
To a first approximation, the equation of state of cubic solids can be represented by a temperature-independent isothermal bulk modulus which is a linear function of the pressure. If zero pressure thermal expansions are not known, these can be calculated from the above using the Mie-Griineisen equation of state and a Griineisen constant y which is linearly dependent on the volume. In the next approximation, BT is found to decrease slightly at constant volume with increasing temperature at low temperatures due to the disappearance of zero point energy effects, although it is relatively temperature-independent for temperatures greater than the Debye temperature f3,. Similar considerations do not apply to the adiabatic bulk modulus Bs or the shear constants where anharmonic effects play a large role at the higher temperatures. The lighter inert gas solids represent an exception to the above generalizations. In this case, the zero point energy is of much greater relative importance in determining the low temperature thermal properties for these solids. Acknowledgements-The author is indebted to Dr. C. S. Smith for discussions over a number of years which assisted in clarifying the ideas presented in this paper. REFERENCES 1. DECKER D. L., J. appl. Phys. 36; 157 (1965); ibid., 37,5012 (1967). 2. SWENSON C. A., Thermal Contribution to the Equation of State of Solids, USAEC Rept. IS-870
1348
3. 4. 5. 6.
7.
8. 9. 10. 11.
C. A. SWENSON
(1964). The present paper presents a simplified treatment of the ideas advanced in this report. In many instances more recent data have been used in extending the analysis. BRUGGER K. and FRITZ T. C., Phys. Rev. 157, 524(1967). COLLINS J. G. and WHITE G. K., Progress in Low Temperature Physics (Edited by Gorter) Vol. IV, p. 450. North-Holland, Amsterdam (1964). See, for instance, ZEMANSKY M. W., Heat and Thermodynamics, 4th Edn. McGraw-Hill, New York (1957). ANDERSON 0. L., J. Phys. Chem. Solids 27,547 (1966). This paper defines B,’ = (aB,/aP),, and discusses equations of state in terms of this quantity. We have chosen to restrict the arguments to (M&P),. The difference is small in most cases. MACDONALD J. ROSS, Rev. mod. Phys. 38,699 (1966). This paper defines I&,’= (M,v/aP),, and posed equations of state of the Murnaghan type. For our purposes, the differences are small. OVERTON W. C.,J. them. Phys. 37,716 (1962). SWENSON C. A., J. Phys. Chem. Solids 27, 33 (1966). BEECROFT R. 1. and SWENSON C. A., J. Phys. Chem. Solids 18,329 (1961). MONFORT C. E. and SWENSON C. A., J. Phys. Chem. Solids 26,291
12. PACKARD
Chem. Solids 24,1405 (1963). 13. LAZARUS D., Phys. Rev. 76,545
18. RAYNE
J. A. and CHANDRASEKHAR B. S., Phys. Rev. 122,1714(1961). 19. ROTTER C. A. and SMITH C. S., J. Phys. Chem. (1966).
20. CHECHILE R. A. and SMITH Communication ( 1967). 21. BARTELS R. A. and SCHUELE 22. MILLER
Phys. Chem. Solids 27,637
C. S., Private D. E., J. Phys.
(1965).
R. A. and SMITH C. S., J. Phys. Chem.
Solids 25, 1279 ( 1964).
23. ROBERTS R. W. and SMITH C. S.. Private Communication (1967). 24. MORSE G. E. and LAWSON A. W.. J. Phvs. Chem. Solids 28, 939 (1967). The data given in Table 2 above differ slightly from those given in Table 3 of this reference due to a computational error in producing Table 3. The actual data (Tables 1 and 2) are correct. We are indebted to Dr. Morse for correspondence on this point.
(1966).
W. R. and TRIVISONNO J*, J Phys. Chem. Solids 26,273 (1965). 28. SALAMA K. and ALERS G. A., Phys. Rev. 161, 673 (1967). 29. HO PAUL S. and RUOFF A. L., Phys. Rev. 161, 864 (1967).
30. MONFORT
C. E. and SWENSON
C. A., J. Phys.
Chem. Solids 26,623 ( 1965). 31. ALDER B. J., Solids Under
Pressure (Edited by Warshauer and Paul). D. 385. McGraw-Hill. New York(1963). ” 32. ENCK F. D. and DOMMEL .I. G., J. appl. Phys. 36,839
(1965).
33. HIKI Y., THOMAS J. F. and GRANATO A. V., Phys. Reb. 153,764 (1967). 34. PASTINE D. J., J. Phys. Chem. Solids 28, 522 (1967).
35 SLAGLE
0. D. and McKINSTRY H. A., J. appl. (1967). These authors interpret the total linear dependence of the elastic constants at constant pressure as an anharmonic effect. We have chosen to eliminate the explicit volume dependence as is done in references[33] and [34]. CORRUCCINI R. J. and GNIEWEK, Nat. Bur. Stand. Monogr. 29 (1961). WHITE G. K., Proc. R. Sot. A286.204 (1965). ALERS G. A. and THOMPSON D. O., J. uppl. Phys. 32,283 (196 1). FEATHERSTONE F. H. and NEIGHBOURS J. R., Phvs.
(1949).
327 (1964).
28, 1225 (1967). The pressure
27. MARQUANDT
C. A., J. Phys.
J. R. and ALERS G. A., Phys. Rev. 14. NEIGHBOURS 111,707 (1958). 15. DANIELS W. B. and SMITH C. S., Phys. Rev. 111, 713 (1958). 16. SCHMUNK R. E. and SMITH C. S., J. Phys. Chem. Solids 9, 100 (1959). 17. KOMM G. N. and ALERS G. A., J. uppl. Phys. 35,
Chem. Solids 26,537
D. E., J. Phys. Chem. dependence of the elastic constants of CaF, (as well as of BaF,) have been determined at 195”i(‘also as an extension of this work (C. Wong and D. E. Schuele, Private Communication). 26. DIEDERICH M. E. and TRIVISONNO J., J. Solids
(1965).
J. R. and SWENSON
Solids 27,267
25. WONG C. and SCHUELE,
36. 37. 38. 39.
38
437
Phys. Rev. 130,1324(1963). 40. LEWIS J. T., LEHOCZKY J. T. and C. V., Phys. Rev. 161,877 (I 967). 41. NORWOOD M. H. and SQUIRE C. F., 84,758 (1958). 42. BRISCOE C. V. and SQUIRE C. F., 106, 1175 (1957). The room temperature
BRISCOE Phys. Rev., Phys.
Rev.
data given in this paper disagree with those given in reference 1221. We have normalized the data to give the same B,( T, P = 0) value at room temperature. 43. CLAYTOR R. N. and MARSHALL B. J., Phys. Rev. 120,332 (1960). 44. DANIEk W: B., ihys. Rev. 119,1246 (1960). Chem. 45. SMITH P. A. and SMITH C. S.. J. Phvs. . ~~ Solids 26,279 (1965).
46. BATCHELDER D. N., LOSEE D. L. and SIMMONS R. 0.. Phys. Rev. 162,767 (1967). 47. PETERSON 0. G., BATCHELDER D.. N. and SIMMONS R. 0.. Phvs. Rev. 150.703 (1966). 48. URVAS A. O., L&SEED. L. and SIMMONS R. O., J. Phys. Chem. Solids 2& 2269 (1967).