Journal of Low Temperature Physics, Vol. 54, Nos. 5/6, 1984

The High-Pressure Equation of State of Solid Molecular Tritium

Altred Driessen* and Isaac F. Silverat

Natuurkundig Laboratorium der Universiteit van Amsterdam, Amsterdam, The Netherlands

(Received August 3, 1983)

We have calculated the equation of state for solid molecular tritium using a semiempirical approach, closely guided by experimental results for He and De. The T = OK isotherm is calculated from a self-consistent phonon model of the free energy. Temperature effects are treated by the Mie-Griineisen model. A tabulation of pressure, bulk modulus, and thermal expansion is given for a dense set of molar volumes, as a function of temperature up to P = 22 kbar.

1. INTRODUCTION

Tritium, the mass-three isotope of hydrogen, is of general interest in comparing the isotopes of hydrogen and of particular interest because of its use as a thermonuclear fuel. In the latter case, in certain approaches it is compressed to ultra high densities and temperatures required for nuclear burning. However, due to its radioactivity the number of experiments on the equation of state (EOS) of condensed tritium have been restricted.

The experimental and theoretical work on the EOS of T2 is closely connected to the Los Alamos Scientific Laboratory: In 1950 Hammel 1 applied the de Boer law of corresponding states z for quantum systems to Tz. One year later Grilly measured the densities 3 and the vapor pressure 4 of liquid T2. The most recent measurement, to our knowledge, on the EOS is the determination of the melting line by Mills and Grilly 5 in 1956. Theoretical calculations were done by Rogers and Brickwedde 6 and recently by Mills et al. 7 They, like Hammel, use the quantum mechanical principle of corresponding states.

All of the work given above deals only with the liquid state. With the progress in the knowledge of the hydrogen potential and the successful

*Present address: Department of Physics, Vrije Universiteit Amsterdam, The Netherlands. tPermanent address: Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts.

565 0022-2291/84/0300-0565503.50/0 © 1984 Plenum Publishing Corporation

566 Alfred Driessen and Isaac F. Silvera

application of the Mie-Griineisen picture to H 2 and D2, 8'9 w e are now able to give an EOS of T2 in the solid state with reasonable accuracy, in the following section we give a brief theoretical background. In Section 3 we calculate the T = 0 isotherm, using a potential due to Silvera and Goldman 1° (SG), and the experimental isotherms of H2 and D2 as given in Ref. 9. Section 4 deals with the properties at T ~ 0, and results are given in Section 5 and compared with other predictions. In the Appendix we give extended tables of the EOS. The results presented in the following are only valid for J = 0 tritium, where J is the rotational quantum number. The influence of J = 1 molecules on the EOS of the hydrogen isotopes is described in Refs. 8 and 11.

2. T H E O R E T I C A L B A C K G R O U N D

The starting point for the determination of the EOS is the pair potential ~b(R) of two hydrogen molecules with intermolecular distance R. To a good approximation this is believed to be identical for all three hydrogen isotopes. Many forms of these potentials have been proposed, lz which are partially based on a priori calculations or are found by fitting parameters of semiem- pirical analytical functions. For our purposes we find the semiempirical potential of Silvera and Goldman, 4~sG, to be satisfactory. It is given by 1°

with

~sG(R)=exp(a-flR-vR2)+f(R)( Y. CJRi+Cg/R 9) i=6,8,10

(1)

f(R) = exp [-(1.28Rm/R - 1)2], R < 1.28Rm

-- 1, R > 1.28Rm

Rm = 3.41 A and the coefficients (all in atomic units) are a = 1.713, fl = 1.5671, Y = 0.00993, C9 = 143.1, and C6 = -12 .14 , C8 = - 2 1 5 . 2 , and C10 = -4813 .9 as given by Meyer. 13

The SG potential, Eq. (1), is an effective pair potential for the solid, in which many-body effects in the low-density solid are accounted for by a repulsive long-range term, C9/R 9.

With the aid of the effective solid pair potential it is possible to calculate the Helmholtz free energy F(V, T), which we can split into a zero- temperature part F0(V) and an incremental part F*(V, T), 8

F ( V, T) = Fo(V) + F * ( V, T) '(2)

The pressure P, the bulk modulus B, and the thermal expansion coefficient

The High-Pressure Equation of State o[ Solid Molecular Tritium 567

are determined from the thermodynamic relations

P= - ( O F / O V ) 7 -

B = - V ( O P / 0 V ) 7-

A = (1/V)(OV/OT)p

(3)

(4)

(5)

Similar to the free energy, we can split these variables into zero- and nonzero-temperature parts. For the pressure we get, for example,

P( V, T) = Po(V) + P*( V, T) (6)

Po(V) represents the T = 0 isotherm. The second term in Eq. (6) is caused mainly by thermally excited phonons and can be calculated with the aid of the Mie-Griineisen picture and the Debye model. 8'14 Spain and Segal114 derived an equation for the thermal pressure, where the Debye temperature 0D(V) is the only parameter to be determined by the material in question,

Io P*(V, T) 3/(V) 9NokB T4 XD _ _ dx (7) V 03 (V) e ~ - I

where

d In OD(V) , / ( v ) = - (8)

d V

R = N o k B is the gas constant and XD = OD/T .

3. T H E T = 0 I S O T H E R M

The most crucial point in the EOS is the volume-pressure relation at fixed temperature. The influence of temperature is more easily calculated, because the effects are relatively small, or, in other words, isochores as shown in Fig. 1 are fairly fiat.

With the SG potential, Eq. (1), we have a powerful means for calculating properties of the hydrogen isotopes, as it was actually determined by fitting a self-consistent phonon calculation of the free energy to the T = 4.2 K isotherm and other thermodynamic data of solid D2. With this potential the T = 0 K isotherm of solid H2 could also be generated in good agreement with experiment. A recent demonstration of the reliability of this potential is the calculation of the melting temperature of hydrogen and deuterium at room temperature by Ross and Young. 15 For this calculation they used the SG potential, with results that compare well to the experimental values of Mao and Bell 16 and Diatschenko and C h u 17 for H2 and Mills et al.18 for D2.

To generate the T = 0 K isotherm of T2 we use the same potential and calculation, with mass as the only variable. This calculation is discussed in

5 6 8 A l f r e d Driessen and Isaac F . S i l v e r a

i ~ i ~ I i I J I I i I ~ ~ I ~ ~ i ~ I

.~= P[kbor) o D~isochores _ _ = ~ ° ~ 2

" o 1 1 ° 2 ; ' 1 v=10 m31mole o - I - ° - - - - - T Z - - - -

01- / d

15- ~ -- melting line

. . . . . .

, u V = 1 2 c m 3 / m o l e o ~ o _ ~ o . . . . ~ - o ~ / v - 0 O - - . 0 " - - - - - - ~ " A ~ ~ - '~ - ' -

V = 1 6 c m 3 / m o l e ~ "

50 100 150 200

Fig. 1. The P - T plane of H2, D2, and T 2 with some selected calculated isochores from the tables in the Appendix and Ref. 9. The melting lines shown are from the same tables; solid line: D2; dashed line: H2.

Ref. 10 and we use the same computer code. The free energy is calculated in the self-consistent phonon approximation as a function of density. Using the same SG potential for all three, differences in the free energy arise from the different contributions of the zero-point energy. From the free energy, it is easy to calculate the pressure by numerical differentiation, using Eq. (3). We note that the theoretical approximations are most valid for T2 due to its larger mass. In Fig. 2 we show the results of these calculations for the three isotopes and compare to experimental data for He and D2. For a more sensitive assessment of this approach, in Fig. 3 we present a deviation plot of Fig. 2. We show, as a function of volume, the pressure difference with respect to the experimental HE isotherm.

The theoretical isotherms of H E and D2 are in reasonable agreement with experiment, but a compensation of small deviations in the H E and D2 isotherms could improve the accuracy of the calculated T2 isotherm. For this we start with the free energy and assume

F ( V ) = Fst(V) + Fzp(V) (9)

where F, t(V) is the static energy and is independent of the isotope, and Fzp(V) is the dynamical term at T = 0, which is due to the zero-point

The High-Pressure Equation o| State of Solid Molecular Tritium 569

20

15

10

[cm3/mole]

--O0

~ o

I I I I

_ _ our results

o H21 o D 2 theory

T2~

5 10 15 20 25

Fig. 2. The isotherms at T --- 0. Solid lines: our best isotherms as used in the tables of the Appendix and Ref. 9; theoretical points are calculated from the Silvera--Goldman potential.

Fig. 3. Deviation plot of Fig. 2. We show, as a function of volume, the pressure difference of the isotherms with respect to our best H2 isotherm from Ref. 9 (drawn as a horizontal line at zero). Solid (T~) and dash-dot (D2) lines: our best isotherms as found in the tables of the Appendix and Ref. 9. Dashed line: pressure difference due to a 1% change in molar volume of H 2, to give an impression of the importance of the deviations. Theoretical points are calculated from the Sil- vera-Goldman potential.

~-°'~-°-o-'o-e o o o "~-H 2

Z~P[k bar] "~- ~ o

/

-0.5- O..o. "Q ~ '~ --I

o 0 2 ) t h e o r y ~ a ,. 1 _2 aZ2~ ~ " ° ' 4

2_5 20 t5 10

570 Alfred Driessen and Isaac F. Silvera

motion. To a good approximation / ] ~1/2 r & [ 02 .~%

where ~ is the pair potential for the solid, R is the intermolecular distance, and M is the molecular mass.

By differentiation [Eq. (3)], we find for the total pressure

P(V) = Pst(V) + (1/ M)l/2 g( V) (10)

where the only isotope-dependent parameter is the molecular mass M of the isotopes; M = 2, 4, and 6 for H2, D2, and T2, respectively; g(V) is a function dependent only on the molar volume. From Eq. (10) we get

PH2(V) = Pst(V) + g(V)

PD2(V) = Pst(V) + (1/2)1/2g(V) (11)

PT2(V) = P~t(V) + (1/3)~/2g(V)

Solving for the T2 isotherm, we get

1 1 -1

This isotherm is plotted in Fig. 3, solid line. We expect that with the calculations of the SG potential we can improve

the T2 isotherm, because the dynamical calculations of the zero-point energy are explicitly done. From Fig. 3 it can be seen that the SG isotherms seem to slightly underestimate the isotope effect, i.e., the function g(V) in Eqs. (10) and (11) is too small. We can make a correction for this. From Eq. (11) we have

g(V) = [PH2(V) --PD2( V)]( - 1/~/2) -1

= [PT2(V) --PD2( V)](1/x/3 - 1/x/2)-' (13)

A volume-dependent factor K can be determined from experiment, which is defined by g( V)e~p = K( V)g( V)s6,

K(V) =[PH2(V)exp-PD2(V)~xp][PH2(V)so--PD2(V)sG]-' (14)

With the considerations given above we can successively improve the T2 isotherm calculated with the SG potential PT~( V)SG.

First step: PT2(V) = P7:( V)sG. Second step: PT2(V) = PD~( V)exp + [PT~( V ) s o - PD~( V)SG]-

With this, we have taken the experimental DE isotherm as a reference. We

The High-Pressure Equation of State of Solid Molecular Tritium 571

TABLE 1 Coefficients for the Birch Relation, Eq. (16), for P-T2

Vo, cm3/mole B 1, bar B2, bar

18.771 6269.5 11,195.3

can rewrite this isotherm with Eq. (11)

PT2(V) = eo2( V)oxo + (1/ , /3 - 1/,/2)g( V)sG

Third step:

ev2(V) = PD~( V)oxp + K ( V ) ( 1 / , / 3 - 1/,/2)g( V)sc

= P D 2 ( V ) e x p + K ( V ) [ P T 2 ( V ) s G - P D 2 ( V ) s o ] (15)

where we have used Eqs. (14) and (11). It turns out that the corrected SG T2 isotherm [Eq. (15)] is identical

with the T2 isotherm equation (12) within the accuracy of the calculation. This implies that the isotherms calculated by Silvera and Goldman are in the form of Eq. (10).

For the generation of tables for the EOS, we need an analytical form of the T2 isotherm. For this we use a modified Birch relation 19'2°

P ( V ) = y5 ~ Bn(Y2_l ) . (16) t l = l

where Y = ( V o / V ) 1/3 and V0 is the zero-pressure volume. We take 10 (P, V) points calculated by Eq. (15) and determine the coefficients of the Birch relation (16). These coefficients are given in Table I.

4. P R O P E R T I E S F O R T # 0

For T ~ 0 K we can derive the thermodynamic properties with the aid of Eq. (7) if we have knowledge of the Debye temperature Or)(V) (kBOi) = hOD is the energy with the highest allowed phonon frequency in the Debye spectrum). One expects the following relation:

~OD(V) -- [k(V) /M] 1/2 (17)

where M is the molecular mass and k(V) some average force constant. In the case of isotopes, ignoring quantum crystal effects, which will become less important at high pressure, we expect the force constant to be identical for all species, so we get

COD(V) ~ (1 / M ) 1/2 (18)

572 Alfred Driessen and Isaac F. $ilvera

for V - - const, or

rOD(H2) : rOD(D2) : rOD(T2) = x/2:1 : ~/2--'~ (19)

In Fig. 4 we show 0D(V) as a function of molar volume for the three hydrogen isotopes. Relation (19) is in fairly good agreement with experi- ment. To show this, we multiply 0D(D2) with x/2. The resulting line (dashed

400

300

200

100

• l e D [ K ] I ! E i ~ , I ~ , v

t

v

. \~, ',\ J

) , . \ " "',',. H "~, \ t o ~ '.."x - ~ T2t"~., \ - -..~

~ . V [ c m 3 / m o l e ] I I I ; 1 I [ I I I I I I ~

10 15 20 24

Fig. 4. The Debye temperature 0 D as a function of molar volume for HE, DE, and T 2. Solid lines: our results as used in the tables of the Appendix and Ref. 9; dashed lines: 0 D of H2 and T 2 calculated from 0D(D2) with the aid of Eq. (19); triangles: results of Liebenberg et al. 2~ determined from volume measurements in the solid along the melting line; diamonds: high-temperature average of the results of Krause and Swenson. 22

The High-Pressure Equation ot State ot Solid Molecular Tritium 573

line in Fig. 4) follows closely the experimental 0D(H2), but seems to give a systematic underestimate of the isotope effect.

For a quantitative analysis a volume-dependent parameter k(V) can be determined from experiment by

0D(H2) = 0D(D2)(~/2-- 1)k(V)+ 0D(D2)

For the calculation of 0D(T2) we use a similar relation and get, with Eq. (19),

0D(T2) = 0o(D2)(x/2/3 -- 1)k(V) + 0D(D2)

---- 0D(D2)[1 + k( V)(x/2-~- 1)] (20)

For a tabulation of the EOS we need an analytical form for 0D(T2). We use the same expression as for H2 and D2,

0D(V) = exp (C1 + C2x + C3x 2) (21)

with x =In (V /V0) ; C1, C2, and Ca are coefficients to be determined by experiment. We take 10 values for 0D(T2) from Eq. (20) and determine by a least square fit the coefficients of Eq. (21). The results are given in Table II and Fig. 4. For completeness we show in Fig. 5 the Griineisen parameter y, Eq. (8), for all three isotopes.

The last information needed for the tabulation is the melting line. Mills and Grilly 5 made measurements up to 3.1 kbar. Goodwin 23 presents a slight correction on the basis of accurate data for p-H2. This melting line we use for the low-density range (V > 14 cm3/mole).

At higher density there are no experimental data available for T2. Li~benberg et al., 2~ however, measured the melting line for HE and DE up to 17 kbar. Diatschenko and Chu 17 extended the data for HE up tO 55 kbar and room temperature and got identical results. As the high-density melting lines of the two isotopes H2 and DE coincide within the accuracy of the measurements, we use, to a good approximation, the same melting line for T 2 a s for D 2 for V < 14 cm3/mole (see Fig. 1).

TABLE !I

Coefficients for the Debye Temperature , Eq. (21), for p -T z

C1 C2 C3 Vo, em a mole

4.4127 - 1 . 8 2 8 6 - 0 . 2 2 7 6 0 18.771

574 Alfred Driessen and Isaac F. Siivera

1

V [ c m 3 1 m o l e ]

0 I I I I I I I I f I ~ I I 10 15 20 2/.

Fig. 5. The Oriineisen parameter y as a function of molar volume for H 2, D 2, and T2. Solid lines: our results are used in the tables of the Appendix and Ref. 9; diamonds: experimental results of Krause and Swenson zz for Hz.

5. R E S U L T S A N D D I S C U S S I O N

We now have all the data available to calculate the complete EOS of solid Tz. We use the T = 0 K isotherm as determined in Section 3, and the P*, Eq. (7), with the 0D(V) and the melting line as given in Section 4. Because of the two different melting lines at low and high densities, we present the results for the EOS in the Appendix in two tables: V = 19.3 to 14 cm3/mole, low density, and V = 14 to 10 cm3/mole, high density. Figure 1 shows three isochores in the P - T plane. We expect the resulting error in volume to be less than 1% for the low-density data and 1.5% at high density.

There are no experimental data to compare with, and even theoretical predictions 1'6'7 deal mostly with the liquid state. The only comparison can be made at the triple point. The experimental data of Grilly 3 give for the liquid phase

Ttr ip = 20.62 K; Vtr ip = 22.05 cm3/mole

Hammel I predicts a volume difference for the liquid and solid phases at

The High-Pressure Equation of State of Solid Molecular Tritium 575

the triple point of 2.66 cm3/mole. The molar volume of the solid at the triple point should be 22 .05- 2.66 = 19.39 cm3/mole. This is in reasonable agreement with our value of 19.25 cm3/mole.

APPENDIX

In this Appendix we present tables of the EOS for para-T2 (Tables III and IV). We give for a dense set of volumes the pressure P, the bulk modulus B = - V OP/OV, and the thermal expansion A = - ( 1 / V ) OV/OT along an isochore as a function of the reduced temperature T~ Tms, where Tms is the melting temperature.

For the calculation of the thermal properties we use a temperature- independent Debye temperature 0D and Griineisen parameter y, which is given for each isochore. Although the accuracy of the listed values for the calculated thermodynamic variables exceeds the experimental limits by sometimes two or three orders in magnitude, we believe that our presenta- tion can be useful for interpolation and numerical calculation of other thermodynamic variables.

The values are given in the following units: volume, cm3/mole; T and 00, K; T/Tms and y, dimensionless; A, 10-6K-1; P and B, bar for the low-density tables, and kbar for the high-density tables. For some further remark about the use of these tables see Ref. 9.

TABLE IU

Equation of State of Para-Tritium (Low Densities)

T / T m = 0.0 0.2 0.4 0.6 0.7 0.8 0.9 1.0

V = 19.30 T = 0.00 0D= 78.39 P = -106.30

3 ' = 1.841 B = 3483.1 A = 0.0

V = 19.20 T = 0.00 12.54 0o = 79.15 P = -87.882 -67.212

3"= 1.839 B = 3606.7 3525.0 A = 0.0 1634.7

V= 19.10 T = 0.00 4.29 8.59 ~,2.88 8o = 79.91 P= -68.717 -68.406 -63.802 -46.474

3'= 1.836 B= 3733.9 3732.1 3711.1 3649.0 A = 0.0 77.8 600.7 1642.8

V= 19.00 T = 0.00 4.41 8.82 13.23 0 o= 80.68 P= -48.437 -43.456 -24.876 -8 .858

3"= 1.834 B= 3864.8 3862.3 3839.8 3774.3 A = 0.0 79.3 610.2 1658.2

V= 18.90 T= 0.00 4.53 9.05 13.58 O~= 81.47 P= -28.024 -27.659 -22.276 -2,384

3'= 1.832 B= 3999.7 3995.3 3971.0 3902.1 A = 0.0 80.9 619.5 1656.3

16.28 18.31 20.35 -56.66 -35.81 -11.68 3330.5 3291.6 3255.8 2814.5 3366.1 3866.5

14.63 16.72 18.82 20.91 -53.074 -34.810 -12.763 12,651

3485.4 3444.2 3484.5 3368.3 2228.3 2805.7 3341.9 3825.3

15.03 17.18 19.32 21.47 -31.413 -12.053 11.218 37.950

3607.8 3565.4 3525.0 3488.4 2227.2 2791.1 3310.9 3777.0

15.43 17.64 19.84 22.04 -8 ,858 11.634 36.166 64.252 3731.6 3688.2 3647.1 3610.3 2225.1 2775.6 3279.7 3728.9

15.84 18.10 20.36 22.63 14.629 36.291 62.120 91.596 3857.9 3813.4 3771.6 3734.5 2221.7 2758.7 3247.5 3680.5

576 Alfred Driessen and Isaac F. Silvera

TABLE Ill--continued

T/T m = 0.0 0.2 0.4 0.6 0.7 0.8 0.9 1.0

V =

0D=

3'=

V= 18.80 T= 0.00 0 o = 82.26 P= -6 .44

3'= 1.829 B= 4138.5 A = 0.0

V= 18.70 T= 0.00 0D= 83.07 P= 16.01

7 = 1.827 B= 4281.4 A = 0.0

18.60 T= 0.00 83.88 P= 39.36

W m

0D-----

3'=

4.64 9.29 -6 ,05 -0 .24

4136.5 4110.5 82.3 627.7

4.76 9.53 16.44 22.70

4277.8 4250,0 83.8 636.0

4.89 9.77 39.82 46.56

1.824 B = 4428.7 4423.5 4393.7 A m

V= 18.50 T= 0D= 84.71 P=

3 '= 1.822 B= A =

V = 18,40 T= 0.00 Oo= 85.55 P= 88.89

7 = 1.819 B= 4736.5 A = 0.0

18.30 T= 0.00 86.40 P= 115.14

0.0 85.2 643.8

0.00 5.01 10,02 63.64 64.14 71,39

4580.3 4571.5 4539.7 0.0 86.7 651.5

5.14 10.27 89.42 97.21

4737.0 4703,1 87.9 656.8

5.27 10.53 115.71 124.07

1.817 B= 4897.4 4891.3 4855.1 A = 0.0 89.4 664.1

V= 18.20 T= 0.00 5.40 10.80 0D= 87.27 P= 142.42 143.04 152.00

3'= 1.815 B= 5063.1 5060.7 5022.2 A = 0.0 90.6 669.4

V= 18.10 T= 0.00 5.53 11.06 0D= 88.14 P= 170.79 171.46 181.06

7 = 1.812 B= 5233.9 5228.1 5187.2 A = 0.0 92.0 675.4

V= 18.00 T= 0.00 5.67 11.22 OD= 89.03 P= 200.27 200,99 211.27

7 = 1.809 B= 5409.8 5406.5 5363.0 A = 0.0 93.2 680.0

V= 17.90 T= 0.00 5.81 11,61 OD= 89.94 P= 230.91 231.68 242.67

3'= 1.807 B= 5591.1 5584.4 5538.3 A = 0,0 94.5 684.8

V= 17.80 T= 0.00 5,95 11.89 0D= 90.85 P=

3'= 1.804 B= A =

V= 17.70 T= OD= 91.78 P=

3'= 1.802 B= A =

V = 17.60 T = 0D= 92.72 P =

y = 1.799 B = A =

13.93 16.25 18.57 20.90 23.22 21.03 39.07 61.94 89.11 120.01

4038,0 3992.3 3946.7 3904.2 3866.8 1659.0 2213.7 2736.8 3209.7 3626.4

14.29 16.67 19.06 21.44 23.82 45.40 64.52 88,64 117.18 149.55

4174.0 4126.7 4080.1 4037.0 3999.5 1661.9 2206.3 2715.7 3173.5 3574.9

14.66 17.10 19.55 21.99 24.43 70.78 91.00 116.41 146.37 180.24

4314.1 4265.4 4217.6 4174.1 4136.6 1663.2 2196.9 2692,9 3135.7 3521.8

15.03 17.54 20.04 22.55 25.05 97.18 118.55 145.29 176.70 212.13

4456.3 4406,2 4357.5 4313,5 4275.9 1663.9 2187.0 2669.6 3097.9 3469.6

15.41 17.98 20.55 23.12 25.69 124.64 147.21 175.32 208.22 245.23 4616.0 4564,4 45.14.8 4470.3 4432.5 1658.4 2169,1 2637.0 3050.0 3406.5

15.80 18.43 21.07 23.70 26,33 153.23 177.04 206.57 241,03 279,67 4764.2 4711.2 4660.7 4615.8 4578.1 1657.7 2157.9 2613,0 3012.4 3355.3

16.19 18.89 21.59 24,29 26.99 182.97 208.05 239.04 275.09 315,41 4927.4 4873.0 4821.5 4776.3 4738.5 1652.0 2140,4 2581.7 2966.9 3296.1

16,59 19.36 22.13 24.89 27.66 213.91 240.32 272.83 310.52 352.57 5088.5 5032.7 4980.5 4934,9 4897.4 1647.8 2125.0 2553,4 2925.3 3241.5

17.00 19.84 22,67 25.50 28.34 246.07 273.85 307.92 347.29 391,10 5260.5 5203.3 5150.3 5104.5 4067.0 1640.4 2105.8 2520.9 2879.2 3182.8

17,42 20.32 23.22 26.12 29.03 279.50 308.69 344.36 385.45 431.07 5~31,7 5373.1 5319.4 5273.2 5235.9

2836.0 3127,5

26,76 29.73 425.17 472.68 5450.4 5413.2 2791.2 3071.0

27.41 30.45 466,38 515.80 5641.3 5604.3 2741.9 3009.9

28.06 31.18 509.13 560.50 5828.0 5791.4 2697.1 2954.1

1633.5 2087,7 2490.2

17.84 20.81 23.79 262.76 263.58 275.33 314.30 344.97 382.29 5778.0 5770.6 5721.7 5611.0 551.1 5494,7

0.0 95.7 689.1 1625.4 2068.1 2458.1

0.00 6.09 12.18 18.27 21.32 24.36 295.85 296.73 309.27 350.46 382,64 421.67 5970.5 5970.6 5918.7 5804.0 5742.8 5687,7

0.0 96.8 691.9 1613.7 2044.4 2421.5

0.00 6.24 12.47 18.71 21.83 24.94 330.24 331.18 344.56 388,03 421.77 462.55 6169.0 6166.6 6111.9 5992.9 5930,4 5874.7

0.0 98.0 695.2 1603.7 2023.2 2388.2

The High-Pressure Equation of State of Solid Molecular Tritium 577

T A B L E l l ~ c o n t i n u e d

~ = 0.0 0.2 0.4 0.6 0.7 0.8 0.9 1.0

V = 17,50 T = 0.00 6.39 12.77 19.16 22.35 25.54 28.73 31.93 0D= 93.67 P = 365.97 366.98 381.24 427.13 462.48 505.08 553,59 606.99

y 1.797 B = 6373.6 6365,8 6307.9 6184.8 6121.2 6064.7 6018.1 5981.9 A = 0.0 99.2 698.7 1594.0 2002.5 2356.1 2653.6 2900.2

V = 17.40 T = 0.00 6.54 13.08 19,61 22.88 26.15 29.42 32.69 0D= 94.64 P = 403.09 404.17 419.38 467,79 504.84 549.31 599.84 655.33

7 = 1.794 B = 6584,5 6576.1 6515.0 6387.7 6322.8 6265.9 6219.4 6183.2 A = 0.0 100.4 701.4 1582.6 1980.0 2321.8 2608.0 2844.5

V = 17.30 T = 0.00 6.69 13.38 20.07 23.42 26.77 30.11 33.46 0D= 95.62 P = 441.67 442.82 459.00 509.97 548.72 595,09 647.65 705.25

y = 1.791 B = 6802.0 6794.0 6729.5 6597.9 6531.8 6474.3 6427.7 6392.4 A = 0.0 101,4 703.0 1569.1 1955,4 2285.8 2561.4 2787.8

V = 17.20 T = 0.00 6.85 13.70 20,56 23.98 27.41 30.83 34,26 0D= 96.62 P = 481.75 482.99 500.22 553.96 594.53 642,93 697.64 757,47

7 = 1.789 B = 70~6.2 7013.5 6945,4 6809.6 6742,5 6684.4 6637.9 6603.3 A = 0.0 102.7 705,7 1557.9 1933.4 2252.9 2518.0 2734.8

V = 17.10 T = 0,00 7.01 14.03 21.04 24.55 28.05 31.56 35.06 0o= 97.63 P = 523.39 524.71 543~4 599.58 641.99 692.43 749,30 81L39

y = 1.786 B = 7257.5 7247.0 7175.2 7035.2 6966.8 6908.4 6862.0 6827.7 A = 0.0 103.7 706.7 1543.2 1907.8 2216.3 2471.0 2678.9

V = 17.00 T = 0.00 71.17 14.35 21.52 25.11 28.70 32.28 35.87 8D= 98.66 P = 566.65 568.06 587.50 646.88 691.15 743.64 802.69 867,04

= 1.783 B = 7496.0 7489.0 7413,6 7269.3 7200,1 7141.3 7095,2 7061.4 A = 0.0 104.5 706.3 1526.7 1880.5 2178.2 2423.0 2622.0

V = 16.90 T = 0.00 7.34 14.68 22.03 25.70 29.37 33,04 36.71 0o = 99.70 P = 611.60 613,10 633.76 696.20 742,45 797.12 858.48 925.22

7 = 1.781 B = 7742,0 7737.2 7658.0 7509.5 7439.0 7379,9 7333.9 7301,0 A = 0.0 105.5 706.8 1511.5 1854.6 2141.8 2376.9 2567.5

V = 16.80 T = 0.00 7.51 15.02 22.53 26.29 30.04 33.80 37.55 OD= 100.76 P = 658.30 659.89 681.80 747,33 795.58 852.45 916.13 985.28

7 = 1.778 B = 7995.8 7983.3 7900.0 7747.0 7675.7 7616.2 7570.6 7538.1 A = 0.0 106.5 707.1 1496.7 1829,9 2107.3 2333,6 2516.2

V ~ 16.70 T = 0.00 7.69 15.37 23.06 26.90 30.75 34.59 38.43 OD= 101.84 P = 706.8 708.5 731.8 800.6 851.0 910.2 976.3 1048.0

7 = 1.775 B = 8257.6 8245,1 8157.6 8000.3 7297,7 7868.2 7822.8 7791,2 A = 0.0 107.5 707.2 1481.1 1804,1 2071.7 2289.1 2463.8

V = 16,60 T = 0.00 7.86 15.73 23.59 27.52 31,45 35.38 39.31 0D= 102.93 P = 757.2 759.0 783.7 855.9 908.4 969.9 1038.5 1112.8

7 = 1.773 B = 8527.8 8515.6 8423.8 8262.0 8188.7 8128.8 8083.6 8053.0 A = 0.0 108.4 706.2 1464.1 1777.1 2035.2 2244.0 2411.2

V = 16.50 T = 0.00 8.04 16.09 24.13 28.15 32.17 36.19 40.21 8D= 104.04 P = 809.6 811.5 837.6 913.3 968.0 1032.0 1103.1 1180.0

7 = 1.770 B = 8806.6 8791,5 8695.2 8529.3 8454.9 8395,2 8350.5 8320.5 A = 0.0 109.3 705.2 1447.2 1750.6 1999.5 2200.0 2360.2

V = 16.40 T = 0.00 8.23 16,45 24.68 28.79 32,91 37.02 41.13 8D= 105.16 P = 864.0 866.0 893.7 973.0 1030,0 1096.4 1170.2 1249.8

y = 1,767 B = 9094.3 9087.0 8986.3 815.9 8740.9 8680.7 8636.6 8607.4 A = 0.0 110.0 703.0 1428.3 1721.8 1961.5 2153.8 2306.9

V = 16.30 T = 0.00 8.41 16.83 25.24 29.45 33.66 37.86 42.07 Oo= 106.30 P = 920.5 922.7 951.9 1035.0 1094.4 1163.4 1239.9 1322.2

y = 1.764 B = 9391.3 9380.2 9274.6 9099.7 9023.7 8963.4 8920.0 8891.8 A = 0.0 110.8 701.4 1411.1 1695.4 1926.5 2111.1 2257.6

578 Al~ed Driessen and Isaac F. Silvera

TABLE Ill--continued

T / T m = 0.0 0.2 0.4 0.6 0.7 0.8 0.9 1.0

V = 16.20 T = 0.00 8.61 17.21 25.82 30.12 34,42 38.72 43.03 0D= 107.46 P = 979.2 981.6 1012.5 1099.5 1161.2 1232.9 1312.2 1397.4

7 = 1.762 B = 9697.9 9684.0 9573.5 9394.1 9317,4 9257.1 9213.9 9186.9 A = 0.0 111.6 699.4 1393.3 1668,6 1891.2 2068.5 2208.8

V = 16.10 T = 0.00 8.80 17.60 26.40 30.80 35.28 39.60 44.00 0D= 108.64 P = 1040.3 1042.8 1075.4 1166.4 1230.7 1305.1 1387.2 1475.4

7 = 1.759 B = 10014 9991 9875 9692 9614 9554 9512 9486 A = 0.0 112.5 697.5 1376.2 1642.8 1857.3 2027.7 2161.9

V = 16.00 T = 0,00 9.00 17.99 26.99 31,49 35.98 40,48 44.98 0D= 109.84 P = 1103.7 1106.3 1140.8 1235.9 1302.7 1379.8 1464,8 1555.9

7 = 1.756 B = 10341 10332 10212 10024 9946 9886 9844 9819

A = 0.0 112.9 693.0 1354.7 1612.1 1818.5 1981.7 2189.9

V = 15.90 T = 0.00 9.20 18.41 ~7.61 32.21 36.81 41.41 46.01 0D= 111.05 P = 1169.6 1172.4 1208.8 1308.3 1377.8 1457,9 1545.9 1640.2

7 = 1.753 B = 10679 10669 10543 10351 10272 10213 10171 10147 A = 0.0 113.7 690.4 1336.4 1585.3 1784.0 1940.4 2063.0

V = 15.80 T = 0.00 9.41 18.82 28.23 32.93 37.64 42.34 47.05 0D= 112.28 P = 1238.0 1241.0 1279.5 1383.5 1455.7 1538.7 1629.9 1727.3

y = 1.750 B = 11027 11008 10876 10679 10600 10540 10500 10477 A = 0.0 114.4 687.6 1318.5 1559.3 1750.7 1900.9 2018.3

V = 15.70 T = 0.00 9.62 19.23 28.85 33.66 38.46 43.27 48.08 0D= 113.54 P = 1309.2 1312.3 1352.9 1461.3 1536.3 1622.3 1716.6 1817,2

7 = 1.747 B = 11388 11386 11249 11048 10968 10909 10869 10848 A = 0.0 114.7 682.0 1296~2 1528.5 1712.4 1856.3 1968.4

V = 15.60 T = 0.00 9.84 19.68 29.51 34.43 39.35 44.27 49.19 0D= 114.81 P = 1383.1 1386.5 1429.4 1542.8 1620.8 1710.0 1807.6 1911.7 "7 = 1.744 B = 11760 11743 11600 11395 11314 11255 11217 11197

A = 0.0 115.6 679.7 1279.4 1504.1 1681.1 1819.2 1926.3

V = 15.50 T = 0.00 10.05 20.11 30.16 35.18 40.21 45.24 50.26 0D= 11~.11 P = 1460.0 1463.5 1508.6 1626.8 1707.8 1800.1 1901.0 2008.5

7 = 1.741 B = 12144 12124 11975 11766 11684 11625 11588 11569 A = 0.0 116.0 674.8 1259.3 1476.5 1646.9 1779.3 1881.9

V = 15.40 T = 0.00 10.27 20.55 30.82 35.96 41.10 46.24 51.37 0D= 117.42 P = 1539.9 1543.6 1591.1 1714.4 1798.3 1894.0 1998.3 2109.3

y = 1.738 B = 12541 12520 12365 12151 12070 12011 11975 11957 A = 0.0 116.4 670.0 1239.4 1449.1 1613.0 1740.0 1838.1

V = 15.30 T = 0.00 10.50 21.01 31.51 36.76 42.02 47.27 52.52 0D= 118.76 P = 1622.9 1626.9 1676.9 1805.5 1892.6 1991.7 2099.6 2214.2

7 = 1.735 B = 12952 12944 12782 12564 12482 12424 12389 12373 A = 0.0 116.8 664.5 1218.3 1420.5 1577.9 1699.5 1793.2

V = 15.20 T = 0,00 10.74 21.48 32.22 37.60 42.97 48.34 53.71 0D= 120.11 P = 1709.2 1713.5 1766.2 1900.3 1990.8 2093.5 2205.1 2323.5

y = 1.733 B = 13377 13358 13190 12968 12885 12838 12793 12779 A = 0.0 117.4 660.3 1199.9 1395.1 1546.5 1663.0 1752.6

V = 15.10 T = 0.00 10.97 21.94 32.91 38.40 43.89 49.37 54.86 0D= 121.49 P = 1799.0 1803.5 1858.8 1998.3 2092.0 2198,2 2313.3 2435.5

y = 1.730 B = 13815 13793 13619 13391 13309 13252 13219 13205 A = 0.0 117.6 654.1 1179:3 1367.8 1513.5 1625.3 1711.0

V = 15.00 T = 0.00 11.22 22.43 33.65 39.26 44.87 50.47 56.08 0D= 122.90 P = 1892.3 1897.0 1955.2 2100.7 2197.9 2307.8 2426.8 2553.0

7 = 1.726 B = 14269 14250 14069 13838 13755 13698 13666 13655 A = 0.0 117.9 648.5 1159.2 1341.0 1480.9 1588.0 1669.7

The High-Pressure Equation of State of Solid Molecular Tritium 579

TABLE Ill--continued

T~ T,,, -~ 0.0 0.2 0.4 0.6 0.7 0.8 0.9 1.0

V = 14.90 T = 0.00 11.46 22.92 34.38 40.11 45.85 51.58 57.31 0o= 124.32 P = 1989.3 1994.3 2055.5 2206.9 2307.6 2421.2 2544.2 2674.4

T= 1.723 B = 14739 14716 14528 14293 14210 14154 14123 14114 A = 0.0 118.2 642.5 1139.2 1314.5 1449.0 1551.6 1629.7

V = 14.80 T = 0.00 11.71 23.43 35.14 41.00 46,86 52.71 58.57 0D= 125.77 P = 2090.2 2095.5 2159.8 2317.4 2421.7 2539.3 2666.4 2800.7

7 = 1.720 B = 15524 15199 15003 14764 14681 14626 14597 14589 A = 0.0 118.4 636.6 1119.3 1288.4 1417.6 1516.0 1590.8

V = 14.70 T = 0.00 11.97 23.93 35.90 41.88 47.87 53.85 59.83 0v= 127.25 P = 2195.1 2200.7 2268.1 2432.0 2540.8 2661.6 2792.7 2931.2

7 = 1.717 B = 15726 15709 15507 15263 15180 15126 15098 15093 A = 0.0 118.4 629.4 1098.2 1261.2 1385.3 1479.4 1550.9

V = 14.60 T = 0.00 12.23 24.47 36.70 42.82 48.94 55.06 61.17 0D= 128.75 P = 2304,2 2310.1 2381.0 2551.7 2663.7 2789.5 2925.0 3068.0

7 = 1.714 B = 16246 16223 16013 15766 15683 15629 15603 15599 A = 0.0 118.8 623.6 1079.1 1236.1 1355.2 1445.5 1513.0

V = 14.50 T = 0.00 12.50 25.01 37.51 43.76 50.01 56.26 62.51 OD= 130.27 P = 2417.7 2424.0 2498.4 2675.9 2791.9 2921.9 3061.9 3209.3

T = 1.711 B = 16784 16742 16523 16272 16189 16136 16111 16110 A = 0.0 119.0 617.6 1060.4 1211.9 1326.6 1413.0 1478.2

V = 14.40 T = 0.00 12.78 25.56 38.33 44.72 51.11 57.50 63.89 0D= 131.82 P = 2535.7 2542.4 2620.5 2805.2 2925.3 3059.8 3204.3 3356.4

y = 1.708 B = 17340 17318 17092 16837 16754 16703 16688 16680 A = 0.0 119.0 610,1 1039.4 1185.1 1295.0 1377.6 1440.0

V = 14.30 T = 0.00 13.05 26.11 39.16 45 .69 52.22 58.74 65.27 OD= 133.40 P = 2658.6 2665.6 2747.5 2939.3 3063.6 3202.5 3351.6 3508.5

7 = 1.705 B = 17916 17888 17654 17394 17312 17262 17240 17242 A = 0.0 119.0 603,0 1019.7 1160.1 1265.6 1345.0 1404.6

V = 14.20 T = 0.00 13.34 26.67 40.01 46.68 53.35 60.02 66.69 % = 135.00 P = 2786.4 2793.8 2879.6 3078.9 3207.5 3351.0 3504,9 3666.6

T = 1.702 B = 18513 18487 18245 17981 17898 17849 17830 17835 A = 0.0 118.9 595.6 999.8 1134.9 1236.3 1312,2 1369.0

V = 14.10 T = 0.00 13.63 27.26 40.88 47.70 54.51 61.33 68.14 % = 136.63 P = 2919.4 2927.2 3017.2 3224.2 3357.3 3505.6 3664.4 3831.2

T = 1.698 B = 19131 19090 18840 18573 18490 18443 18425 18432 A = 0.0 119.0 588.7 980.9 1111.2 1208.4 1281.2 1335.6

580 Alfred Driessen and Isaac F. Silvera

~ o - ~ - ~ • ~ o o ~ ~ ~ o o ~

II ii II II ii i~ It ii ii ii 11 ii ir ii II ii ii ii Jl ii if ii ii ii ii ii iJ JI it ~F iI ii

n ii it ii Jl ii ~1 ii ii rl ii ii ii if ii ii ii fl i~ ii it ii rl II

The High-Pressure Equation o! State og Solid Molecular Tritium $81

m ~

~ 4

o ~

II II II II II il II 11 II II II II II II II II II II II II II II il II II II II II II n II tl II II II II

II II II II II II II II II II II 11 II II II II II II 11 II II U II II II U II

582 Alfred Driessen and Isaac F. Silvera

r~

II If JI II If IJ IJ Ir I~ II II II II fr II II fl ~1 II fl II tf II I[

The High-Pressure Equation ot State ot Solid Molecular Tritium 583

I I I I I I I L I B I I I I I I I I I I I I E I I I I I I ] l l I I J ~ l l l l I l J l l l l l i l l l l l l t i l l l l l l l I t 1 1 1 1 1 1

II ]1 II II II El II il II li II Lt II II II II II II II ~l II II ~1 II II II I]

5 8 4 A l f r e d D r i e s s e n a n d I s a a c F . $ i l v e r a

J

II II II II II ]1 II II II II II II II II II il II II II II II II II II II II II II

it ~l II It II II n u II iI u II II iI II tt I~ II ~l ii II

The High-Pressure Equation o[ State of Solid Molecular Tritium 585

ACKNOWLEDGMENTS

We thank Dr V. V. Goldman for discussions and assistance in running his self-consistent phonon program. The assistance of E. van der Poll with the computer codes for the fittings and. tabulations was of great aid. The financial support of the Stichting FOM is gratefully acknowledged.

REFERENCES

1. E. F. Hammel, J. Chem. Phys. 18, 228 (1950). 2. J. de Boer, Physica 14, 139 (1948). 3. E. R. Grilly, J. Am. Chem. Soc. 73, 5307 (1951). 4. E. R. Grilly, J. Am. Chem. Soc. 73, 843 (1951). 5. R. L. Mills and E. R. Grilly, Phys. Rev. 101, 1246 (1956). 6. J. D. Rogers and F. G. Brickwedde, J. Chem. Phys. 42, 2822 (1965). 7. R. L. Mills, D. H. Liebenberg, and J. C. Bronson, Appl. Phys. 49, 5502 (1978). 8. A. Driessen, J. A. de Waal, and I. F. Silvera, J. Low Temp. Phys. 34, 255 (1979). 9. A. Driessen, Thesis (1982), unpublished; A. Driessen and I. F. Silvera, 3". Low Temp.

Phys. 54, 361 (1984). 10. I. F. Silvera and V. V. Goldman, J. Chem. Phys. 69, 4209 (1978). 11. A. Driessen, Thesis (1982), unpublished; A. Driessen, I. F. Silvera, and E. van der Poll,

to be published. 12. I. F. Silvera, Rev. Mod. Phys. 52, 393 (1980). 13. W. Meyer, Chem. Phys. 17, 27 (1976). 14. I. L. Spain and S. Segall, Cryogenics 11, 26 (1971). 15. M. Ross and D. A. Young, Phys. Lett. A 78, 463 (1980). 16. H. K. Mao and P. M. Bell, Science 203, 1004 (1979). 17. V. Diatschenko and C. W. Chu, Science 212, 1393 (1981). 18. R. L. Mills, D. H. Liebenberg, J. C. Bronson, and L. C. Schmidt, Rev. Sci. Instrum. 51,

891 (1980). 19. F. Birch, J. Geophys. Res. 57, 227 (1952). 20. M. S. Anderson and C. A. Swenson, Phys. Rev. B 10, 5184 (1974). 21. D. H. Liebenberg, R. L. Mills, and J. C. Bronson, Phys. Rev. B 18, 4526 (1978). 22. J. K. Krause and C. A. Swenson, Phys. Rev. B 21, 2533 (1980). 23. R. D. Goodwin, Cryogenics 2, 353 (1962).