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Isotherms of the rare gas solids
ISOTHERMS OF THE RARE GAS SOLIDS FRANCIS BIRCH Hoffman Laboratory, Harvard University, Cambridge, MA 02138,U.S.A. (Received 10May 1976;accepted 163une 1976) Ahstraet-The experiment data of Anderson, Fugate and Swenson for the rare gas solids are recast in a dimensionless form which exhibits the close approach to the Lennard-Jones form except at the highest temperatures, and also in a form derivedfrom the theory of finite elasticity.The latter is particularly useful when ultrasonic data furnish initial values of the modulus of compression. It also has the merit that no o priori assumptions concerning interatomic potentials are implicit, except that the strain-energy may be expanded in powers of the strain.
KzK,‘=
The extensive experimental data for the rare gas solids given by Anderson, Fugate and Swenson[ll and Anderson and Swenson[2] offer an opportunity for testing various equations of state at large compressions. These solids, all Uing cubic symmetry, are among the most compressible; the maximum pressure of the experiments, 20 kilobars (kbars) suffices to reduce the volumes to about threequarters or less of the initial volumes. The authors have represented their measurements by isotherms suggested by Lennard-Jones form: P = ZA, v-”
Cna,X(n t l)n’a, -xn’a,li(n
f#J V@= Ca, /(n - 1)
(2)
These relations are valid for any isotherm having the form (1). The number of independent a. is evidently always one less than the number used in the formula for P.
The Lennard-Jones formula is a special case with only a, and a;, of which only one is independent, since a3 = - a, = - K,/2. Again, the relations are:
(0 d = ( V,K,/4) ( - Y-* + Y-‘/2)
where the n are odd integers, beginning with n = 3, V is molar volume, A,, are functions of temperature. Most of the isotherms require 3 terms, with n = 3,5 and 7. For a few an A, term is added, and for several, A, and AJ suffice. Standard deviations from these smooth functions are of the order, 0.02cm3/mole, or roughly 0.1% of the volumes. The authors point out the pronounced similarity of shape of the isotherms, and the possibility of a reduced equation of state for Ar, Kr and Xe. While (1) is convenient for discussion of variations at constant volume, the isotherms may be represented to advantage by ~n~oducing the coefficients a, = A, (TM V,“, where V, is the molar volume at P = 0 and temperature T. This eliminates large factors of powers of ten appearing in the A,. The a, have the dimension of pressure and are simply related to the physical parameters. Let K denote the bulk modulus, defined by K = - V(~~/~V)=, K’ its first pressure derivative, K” its second pressure derivative; the subscript zero denotes the values of these quantities at P = 0. Let y = V(T)/ V,( T), and 4(T) the Helmholtz potential. We have then the explicit relations: P= Ba.y-”
O=Za,
K = Cna.y-”
K. = %a,, K; = Hn2a./Ko
K” = [Pna,y-‘“‘“H(n t I)nzu~y~(nc2’ - ~n%.y-‘“+“2(n f
(Znn,y-‘“+“)’
+ l)na.y-‘“‘2’1
&I = - V,K,/g
P=fK~2)(-y-3ty-S)
K’ = (25 - 9y2)1(5- 3y*) k&K” = (120y6)/(3y2 - 5)3
I(;=8 K,K;=
- 15.
(3)
Table 1 shows the a. calculated from the Refs. [l, 21, and the dimensionless ratios, 2a,/Ko; the bulk modulus and derivatives, at P = 0, as calculated from (2), are given in Table 2. The close approach to the simple LennardJones relations is notable for all temperatures except near the melting temperatures, where an As term was found necessary. Otherwise, K; is near 8, and KoKi near -15. The values differ slightly from those of Table 1 of Anderson and Swenson because of rounding-off errors and possibly somewhat different procedures. When the energy or the pressure is represented by only two powers of the volume, K; and K&l: are determined by the exponents alone. For example, if we write P=an(-yTm
I(’ = ~n’a”y-‘“+“s~:na~y-‘“+”
+ l)na,
+y-“)
(4)
then KL = m + n, and K,KZ = - mn. The relation for K; is equivalent to the well-known relation of Gruneisen [3], who, however, expressed it in terms of powers of r in the energy, instead of powers of V in the pressure. When further terms are added, these simple relations are no longer exact, but the exponents continue to dominate except for certain critical ratios of the a,. These data for the rare gas solids may also be 175
176
FRANCISBIRCH Table 1. Reduced coefficients: II. = A. / V,’ kilobars T,“K -a, a, Ne
4.2 13.5 19.9 4.2 20 40 60 71 4.2 20 40 60 77 90 100 110 4.2 20 40 60 80 100 120 140 159
Ar
Kr
\Xe
5.694 5.258 3.475 15.841 15.053 12.348 8.912 3.984 18.478 16.423 14.168 12.447 IO.189 8.575 7.605 6.834 19.761 19.475 17.699 16.011 13.528 12.219 II.245 9.039 2.755
5.900 5.382 2.606 17.365 16.329 12.927 8.533 -0.142 20.256 17.223 14.491 12.616 9.679 7.760 6.734 5.985 21.384 21.197 18.880 16.907 13.526 12.282 11.243 8.461 -3.614
Dimensionless a7
-0.206 -0.123 1.076 -0.207 -1.528 -1.274 -0.584 +0.377 t5.197 -1.069 -1.782 -0.792 -0.321 -0.164 +0.511 to.817 to.876 to.852 - I.623 - 1.724 -1.184 -0.900
+0.580 t8.112
Bulk modulus and its pressure derivatives, at P = 0 From P = Ca.y-” From finite strain T,“K K, K; -K,,K; K, K; -K,K: kbars kbars
Ar
Kr
Xe
4.2 13.5 19.9 4.2 20 40 60 17 4.2 20 40 60 77 90 100 110 4.2 20 40 60 80 100 120 140 159
10.97 10.27 8.28 28.60 27.57 23.50 18.57 14.10 33.37 31.30 27.71 24.60 21.40 18.79 16.99 15.39 36.28 35.50 33.01 30.21 27.05 24.57 22.48 19.24 14.77
7.85 7.90 8.44 7.58 7.63 7.80 8.16 9.69 7.57 7.79 7.91 7.94 8.19 8.35 8.41 8.44 1.64 7.61 7.71 7.79 8.00 8.00 8.00 8.24 9.56
14.9 14.9 16.8 14.8 14.8 14.8 15.2 26.7 14.7 14.8 14.9 14.9 15.2 15.5 15.6 15.6 14.8 14.8 14.8 15.1 15.0 15.0 15.0 15.3 24.6
10.97 7.82 IO.31 7.79 8.26 8.50 28.44 7.73 27.61 7.61 23.39 7.87 18.58 8.05 13.94 9.69 33.23 7.69 31.49 7.67 21.79 7.82 24.19 7.75 21.48 8.05 18.92 8.08 17.18 8.04 15.57 8.07 36.26 7.66 35.45 7.66 32.92 7.77 30.13 7.81 27.02 7.97 24.68 1.87 22.46 7.98 19.36 8.01 14.61 10.08
13.8 12.5 17.5 17.5 14.8 15.4 11.7 33.3 16.9 12.9 13.0 11.4 11.5 9.0 7.5 7.4 15.3 15.8 15.5 15.3 13.6 12.1 13.6 10.0 38.0
represented in a form, independent of assumptions concerning atomic interactions, derived from the theory of finite elastic strain [4,5]. An isotherm may be written:
P = 3Koj(l+2f)5’*(1 t af+bfZ+.
.).
-1.743
-2a,lK, 2a,lK, 2a,lK, 2a,lK, 1.038 1.0236 0.840 1.108 1.092 I.051 0.960 0.565 1.107 1.049 1.023 1.012 0.952 0.913 0.895 0.888 1.089 1.097 1.072 1.060 1.000 1.000 1.000 0.940 0.373
1.075 1.048 0.630 1.214 1.185 1.100 0.919 -0.020 1.214 1.101 I.046 1.026 0.905 0.826 0.793 0.778 1.179 1.194 I.144 I.119 1.000 1.000 1.000 0.880 -0.489
-0.038 -0.024 to.260 -0.050 -0.107 -0.092 -0.050
to.041 to.737 -0.152 -0.107 -0.051 -0.023 -0.013 +0.048 to.087 to.103 to.111 -0.089 -0.097 -0.072 -0.060
to.060 +I.099
-0.236
Here f is the hydrostatic strain, taken positive and related to volume according to V/ V,= (1t 2f)-3’2; Vo, K,,, a and b are functions of temperature only. Also, K/, = 4 t 2a/3; K{=(9K,)-'(66 -60 -4aZ-35). Given P and V/V,by experiment, we form for each pair of values the ratio
Table 2.
Ne
a9
(5)
F(f) = P[3f(l+ 2f)“‘]-’ = K,(l t of+ bf t. .). (6) F(f) is frequently a nearly linear function of f, with a relatively small range of variation even for large compressions; a quadratic in f is usually adequate to represent experimental compressions. This corresponds to a fourth-order expansion of C$in the Eulerian strain. Equation (6), written in terms of V/V,, instead of h has been successfully used to represent the measurements for solid hydrogen and deuterium [6,7] and to obtain K, for neon [ 11.In all cases, a quadratic for F(f), found by least squares, reproduces the pressures to 0.01 kbars or better; these pressures were found from (1) and the a. of Table 1, and have thus been smoothed in a particular way; application of (6) to the original points would perhaps be more informative. The values of K.and K;I found with (5) are generally close to those found from (2), but systematic differences in Kbdemonstrate the slightly different shapes of the isotherms. Expansions in powers of the pressure are frequently used to represent the variation of ultrasonic effective constants; for example, K may be expanded in a Taylor’s series:
= K,[l + K;P/K,tKoK;(P/Ko)*/2t.].(7)
Isothermsof the rare gas solids
177
With only the quadratic term usually available, this must be restricted to small values of P/K@. With the coefficients of Table 2 for neon, (7) gives K = 0 for P = 13 kbars, which is within the experimental range. Integrations of (7) to give pressure-volume relations must also be restricted to small compressions. It is generally advantageous to insert ultrasonic values, with suitable thermodynamic corrections, in (5) to give isotherms of wide range[l], though Kg is probably still best determined from large compressions. A similar restriction to small compressions applies to the Murnaghan equation [9]:
or crystal structure: the range for Kb is approximately from 4 to 8, for KoKb: from perhaps -4 to -15, the latter pertaining to the rare gas solids. Within a group of solids having a common structural feature, such as the closed shells of the rare gas solids, to a crude first approximation only K,, varies, and P/K, is represented by nearly the same equation for all except neon. The virtual equivalence of (2) and (5) demonstrates the compatibility of the atomic and macroscopic approaches; (5) has the advantage of easy incorporation of ultrasonic data, and a more straightforward statistical procedure for the reduction of pressure-volume measurements.
P = (K,/K;) [(V/ VJ“6 - 11.
REFERENCES
(8)
This is derived from the assumption that K’ = Kb, independent of pressure. That this is not valid is shown by (2) and by the calculations of K’ at 17kbars in Table 1 of Anderson and Swenson[2]. For V/V,, = 0.8, (8) gives pressures which are about 20% too high for the rare gas solids, with increasing discrepancies at larger compressions. It is a striking characteristic of inorganic crystalline solids that, despite a variation of K. by several thousand-fold, the dimensionless quantities, Kb and K,K{ remain within narrow bounds, regardless of composition
JPCS VOL. 38 NO. 2-F
1.
2. 3. 4. 5. 6. 7. 8.
9.
Anderson M. S., Fugate R. Q. and Swenson C. A., .J. Low Temp. Phys. 10, 345 (1973). Anderson M. S. and Swenson C. A., I. Phys. Chem. Solids 36, 145 (1975). Gruneisen E., Ann. Physik 39, 257 (1912). Birch F., Phys. Rev. 71, 809 (1947). Birch F., J. Geophys. Res. 57, 227 (1952). Stewart J. W., J. Phys. Chem. Solids 1, 146 (1956). Anderson M. S. and Swenson C. A., Phys. Rev. BlO, 5184 (1974). Barsch G. R. and Chang Z. P., in Accurate Characterization of the High-Pressure Environment (Edited by E. C. Lloyd), p. 173. Nat. Bur. Standards Special Publ. 326 (1971). Murnaghan F. D., Proc. Not. Acad. Sci. 30, 244 (1944).
KzK,‘=
The extensive experimental data for the rare gas solids given by Anderson, Fugate and Swenson[ll and Anderson and Swenson[2] offer an opportunity for testing various equations of state at large compressions. These solids, all Uing cubic symmetry, are among the most compressible; the maximum pressure of the experiments, 20 kilobars (kbars) suffices to reduce the volumes to about threequarters or less of the initial volumes. The authors have represented their measurements by isotherms suggested by Lennard-Jones form: P = ZA, v-”
Cna,X(n t l)n’a, -xn’a,li(n
f#J V@= Ca, /(n - 1)
(2)
These relations are valid for any isotherm having the form (1). The number of independent a. is evidently always one less than the number used in the formula for P.
The Lennard-Jones formula is a special case with only a, and a;, of which only one is independent, since a3 = - a, = - K,/2. Again, the relations are:
(0 d = ( V,K,/4) ( - Y-* + Y-‘/2)
where the n are odd integers, beginning with n = 3, V is molar volume, A,, are functions of temperature. Most of the isotherms require 3 terms, with n = 3,5 and 7. For a few an A, term is added, and for several, A, and AJ suffice. Standard deviations from these smooth functions are of the order, 0.02cm3/mole, or roughly 0.1% of the volumes. The authors point out the pronounced similarity of shape of the isotherms, and the possibility of a reduced equation of state for Ar, Kr and Xe. While (1) is convenient for discussion of variations at constant volume, the isotherms may be represented to advantage by ~n~oducing the coefficients a, = A, (TM V,“, where V, is the molar volume at P = 0 and temperature T. This eliminates large factors of powers of ten appearing in the A,. The a, have the dimension of pressure and are simply related to the physical parameters. Let K denote the bulk modulus, defined by K = - V(~~/~V)=, K’ its first pressure derivative, K” its second pressure derivative; the subscript zero denotes the values of these quantities at P = 0. Let y = V(T)/ V,( T), and 4(T) the Helmholtz potential. We have then the explicit relations: P= Ba.y-”
O=Za,
K = Cna.y-”
K. = %a,, K; = Hn2a./Ko
K” = [Pna,y-‘“‘“H(n t I)nzu~y~(nc2’ - ~n%.y-‘“+“2(n f
(Znn,y-‘“+“)’
+ l)na.y-‘“‘2’1
&I = - V,K,/g
P=fK~2)(-y-3ty-S)
K’ = (25 - 9y2)1(5- 3y*) k&K” = (120y6)/(3y2 - 5)3
I(;=8 K,K;=
- 15.
(3)
Table 1 shows the a. calculated from the Refs. [l, 21, and the dimensionless ratios, 2a,/Ko; the bulk modulus and derivatives, at P = 0, as calculated from (2), are given in Table 2. The close approach to the simple LennardJones relations is notable for all temperatures except near the melting temperatures, where an As term was found necessary. Otherwise, K; is near 8, and KoKi near -15. The values differ slightly from those of Table 1 of Anderson and Swenson because of rounding-off errors and possibly somewhat different procedures. When the energy or the pressure is represented by only two powers of the volume, K; and K&l: are determined by the exponents alone. For example, if we write P=an(-yTm
I(’ = ~n’a”y-‘“+“s~:na~y-‘“+”
+ l)na,
+y-“)
(4)
then KL = m + n, and K,KZ = - mn. The relation for K; is equivalent to the well-known relation of Gruneisen [3], who, however, expressed it in terms of powers of r in the energy, instead of powers of V in the pressure. When further terms are added, these simple relations are no longer exact, but the exponents continue to dominate except for certain critical ratios of the a,. These data for the rare gas solids may also be 175
176
FRANCISBIRCH Table 1. Reduced coefficients: II. = A. / V,’ kilobars T,“K -a, a, Ne
4.2 13.5 19.9 4.2 20 40 60 71 4.2 20 40 60 77 90 100 110 4.2 20 40 60 80 100 120 140 159
Ar
Kr
\Xe
5.694 5.258 3.475 15.841 15.053 12.348 8.912 3.984 18.478 16.423 14.168 12.447 IO.189 8.575 7.605 6.834 19.761 19.475 17.699 16.011 13.528 12.219 II.245 9.039 2.755
5.900 5.382 2.606 17.365 16.329 12.927 8.533 -0.142 20.256 17.223 14.491 12.616 9.679 7.760 6.734 5.985 21.384 21.197 18.880 16.907 13.526 12.282 11.243 8.461 -3.614
Dimensionless a7
-0.206 -0.123 1.076 -0.207 -1.528 -1.274 -0.584 +0.377 t5.197 -1.069 -1.782 -0.792 -0.321 -0.164 +0.511 to.817 to.876 to.852 - I.623 - 1.724 -1.184 -0.900
+0.580 t8.112
Bulk modulus and its pressure derivatives, at P = 0 From P = Ca.y-” From finite strain T,“K K, K; -K,,K; K, K; -K,K: kbars kbars
Ar
Kr
Xe
4.2 13.5 19.9 4.2 20 40 60 17 4.2 20 40 60 77 90 100 110 4.2 20 40 60 80 100 120 140 159
10.97 10.27 8.28 28.60 27.57 23.50 18.57 14.10 33.37 31.30 27.71 24.60 21.40 18.79 16.99 15.39 36.28 35.50 33.01 30.21 27.05 24.57 22.48 19.24 14.77
7.85 7.90 8.44 7.58 7.63 7.80 8.16 9.69 7.57 7.79 7.91 7.94 8.19 8.35 8.41 8.44 1.64 7.61 7.71 7.79 8.00 8.00 8.00 8.24 9.56
14.9 14.9 16.8 14.8 14.8 14.8 15.2 26.7 14.7 14.8 14.9 14.9 15.2 15.5 15.6 15.6 14.8 14.8 14.8 15.1 15.0 15.0 15.0 15.3 24.6
10.97 7.82 IO.31 7.79 8.26 8.50 28.44 7.73 27.61 7.61 23.39 7.87 18.58 8.05 13.94 9.69 33.23 7.69 31.49 7.67 21.79 7.82 24.19 7.75 21.48 8.05 18.92 8.08 17.18 8.04 15.57 8.07 36.26 7.66 35.45 7.66 32.92 7.77 30.13 7.81 27.02 7.97 24.68 1.87 22.46 7.98 19.36 8.01 14.61 10.08
13.8 12.5 17.5 17.5 14.8 15.4 11.7 33.3 16.9 12.9 13.0 11.4 11.5 9.0 7.5 7.4 15.3 15.8 15.5 15.3 13.6 12.1 13.6 10.0 38.0
represented in a form, independent of assumptions concerning atomic interactions, derived from the theory of finite elastic strain [4,5]. An isotherm may be written:
P = 3Koj(l+2f)5’*(1 t af+bfZ+.
.).
-1.743
-2a,lK, 2a,lK, 2a,lK, 2a,lK, 1.038 1.0236 0.840 1.108 1.092 I.051 0.960 0.565 1.107 1.049 1.023 1.012 0.952 0.913 0.895 0.888 1.089 1.097 1.072 1.060 1.000 1.000 1.000 0.940 0.373
1.075 1.048 0.630 1.214 1.185 1.100 0.919 -0.020 1.214 1.101 I.046 1.026 0.905 0.826 0.793 0.778 1.179 1.194 I.144 I.119 1.000 1.000 1.000 0.880 -0.489
-0.038 -0.024 to.260 -0.050 -0.107 -0.092 -0.050
to.041 to.737 -0.152 -0.107 -0.051 -0.023 -0.013 +0.048 to.087 to.103 to.111 -0.089 -0.097 -0.072 -0.060
to.060 +I.099
-0.236
Here f is the hydrostatic strain, taken positive and related to volume according to V/ V,= (1t 2f)-3’2; Vo, K,,, a and b are functions of temperature only. Also, K/, = 4 t 2a/3; K{=(9K,)-'(66 -60 -4aZ-35). Given P and V/V,by experiment, we form for each pair of values the ratio
Table 2.
Ne
a9
(5)
F(f) = P[3f(l+ 2f)“‘]-’ = K,(l t of+ bf t. .). (6) F(f) is frequently a nearly linear function of f, with a relatively small range of variation even for large compressions; a quadratic in f is usually adequate to represent experimental compressions. This corresponds to a fourth-order expansion of C$in the Eulerian strain. Equation (6), written in terms of V/V,, instead of h has been successfully used to represent the measurements for solid hydrogen and deuterium [6,7] and to obtain K, for neon [ 11.In all cases, a quadratic for F(f), found by least squares, reproduces the pressures to 0.01 kbars or better; these pressures were found from (1) and the a. of Table 1, and have thus been smoothed in a particular way; application of (6) to the original points would perhaps be more informative. The values of K.and K;I found with (5) are generally close to those found from (2), but systematic differences in Kbdemonstrate the slightly different shapes of the isotherms. Expansions in powers of the pressure are frequently used to represent the variation of ultrasonic effective constants; for example, K may be expanded in a Taylor’s series:
= K,[l + K;P/K,tKoK;(P/Ko)*/2t.].(7)
Isothermsof the rare gas solids
177
With only the quadratic term usually available, this must be restricted to small values of P/K@. With the coefficients of Table 2 for neon, (7) gives K = 0 for P = 13 kbars, which is within the experimental range. Integrations of (7) to give pressure-volume relations must also be restricted to small compressions. It is generally advantageous to insert ultrasonic values, with suitable thermodynamic corrections, in (5) to give isotherms of wide range[l], though Kg is probably still best determined from large compressions. A similar restriction to small compressions applies to the Murnaghan equation [9]:
or crystal structure: the range for Kb is approximately from 4 to 8, for KoKb: from perhaps -4 to -15, the latter pertaining to the rare gas solids. Within a group of solids having a common structural feature, such as the closed shells of the rare gas solids, to a crude first approximation only K,, varies, and P/K, is represented by nearly the same equation for all except neon. The virtual equivalence of (2) and (5) demonstrates the compatibility of the atomic and macroscopic approaches; (5) has the advantage of easy incorporation of ultrasonic data, and a more straightforward statistical procedure for the reduction of pressure-volume measurements.
P = (K,/K;) [(V/ VJ“6 - 11.
REFERENCES
(8)
This is derived from the assumption that K’ = Kb, independent of pressure. That this is not valid is shown by (2) and by the calculations of K’ at 17kbars in Table 1 of Anderson and Swenson[2]. For V/V,, = 0.8, (8) gives pressures which are about 20% too high for the rare gas solids, with increasing discrepancies at larger compressions. It is a striking characteristic of inorganic crystalline solids that, despite a variation of K. by several thousand-fold, the dimensionless quantities, Kb and K,K{ remain within narrow bounds, regardless of composition
JPCS VOL. 38 NO. 2-F
1.
2. 3. 4. 5. 6. 7. 8.
9.
Anderson M. S., Fugate R. Q. and Swenson C. A., .J. Low Temp. Phys. 10, 345 (1973). Anderson M. S. and Swenson C. A., I. Phys. Chem. Solids 36, 145 (1975). Gruneisen E., Ann. Physik 39, 257 (1912). Birch F., Phys. Rev. 71, 809 (1947). Birch F., J. Geophys. Res. 57, 227 (1952). Stewart J. W., J. Phys. Chem. Solids 1, 146 (1956). Anderson M. S. and Swenson C. A., Phys. Rev. BlO, 5184 (1974). Barsch G. R. and Chang Z. P., in Accurate Characterization of the High-Pressure Environment (Edited by E. C. Lloyd), p. 173. Nat. Bur. Standards Special Publ. 326 (1971). Murnaghan F. D., Proc. Not. Acad. Sci. 30, 244 (1944).