Experimental determination of the equation of state of ... - Springer Link

Journal of Low TemperaturePhysics, VoL 34, Nos. 3/4, 1979

Experimental Determination ot the Equation ot State ot Solid Hydrogen and Deuterium at High Pressures* A. Driessen, J. A. de Waal, and Isaac F. Silvera N a t u u r k u n d i g Laboratorium, Universiteit van A m s t e r d a m , A m s t e r d a m , The N e t h e r l a n d s

(Received July 6, 1978)

Isochores of solid hydrogen and deuterium have been measured for melting line pressures up to 2000 bar. These are corrected to correspond to para-H2 and ortho-D2. The 4.2 K isotherm of H2 has been determined and compared to the extrapolated low-pressure isochore of Anderson and Swenson (AS). Deviations have led us to reanalyze the A S data. The 4.2 K isotherm and the isochores are used, with the aid of a Mie-Griineisen analysis, to determine P - V - T data for these solids up to 25 kbar. An analysis is presented which enables a determination of P - V - T - C ~ for all values of C1, the ortho-para concentration. Results, including the Debye temperature, Griineisen constant, and bulk modulus, are presented both in tabulated and graphical form. We also present what we feel to be the best of the 4.2 K, zero-pressure molar volumes of disordered H2 and De as a function of C1. Our measurements show no indication of the premelting phase transition reported in recent Russian literature and a possible explanation is presented.

1. I N T R O D U C T I O N Solid hydrogen and deuterium have been the objective of intense study for the past decades. At ultra high pressure (~0.1 • 10 6 bar) interest has centered on the insulator-metal transition (see, e.g., Ref. 1); at lower pressures (0-25 kbar), which is the experimental region we have studied, 2 interest has focused on molecular interactions, dynamics, and crystalline phases o~ the o r t h o - p a r a species of the hydrogens. For a number of years studies were centered on the zero-pressure solid at low temperatures (see Ref. 3 for a review). However, as the understanding of this has advanced, there has been a trend toward studies as a function of density, which *Supported in part by the Stichting FOM. 255 0022-2291/79/0200-0255503,00/0

t~ 1979PlenumPublishingCorporation

256

A. Driessen, J. A. de Waal, and Isaac F. $ilvera

generally require knowledge of the equation of state (EOS), as pressure is the measured parameter. Surprisingly little experimental information is available on the E O S of the solid hydrogens. Available m e a s u r e m e n t s tend to localize on the melting lines, the 4.2 K isotherms, and the molar volume at zero pressure. Simon e t al. 4 measured the melting line to 5 kbar in 1929. Mills and Grilly 5 repeated these m e a s u r e m e n t s with greater accuracy to a m a x i m u m pressure of 3.5 kbar. The accuracy was further refined by Goodwin and R o d e r 6 at lower pressures. Molar volumes on the melting line have been determined by D w y e r e t a / . 7"* to pressures of 412 bar..The 4.2 K isotherms of H2 and D2 were first measured by Stewart 9 in 1956 from 2 to 20 kbar and were repeated and extended by Anderson and Swenson 1~ (AS) in 1974 from 0.5 to 25 kbar. D u r a n a and McTague jl also recently determined the 4.2 K isotherm, to 5 kbar, but for para-H2 rather than normal H2 (0.75 ortho-0.25 para) used in the earlier measurements. T h e r e is also a large group of m e a s u r e m e n t s dealing with the determination of the molar volume at zero pressure. These m e a s u r e m e n t s have been made by thermodynamic techniques and x-ray, electron, and neutron diffraction and will be discussed in a later section. The m e a s u r e m e n t s that we report here are of the isochores of H2 and D2 to pressures of - 2 kbar (at the melting line). Molar volumes have not been measured, but determined by correlation with existing data at the melting line and 4.2 K. The measured isochores have been accurately fit by integration of the specific heat, using a M i e - G r / i n e i s e n - D e b y e model similar to Spain and Segall's treatment for solid helium.~2 Because hydrogen is a q u a n t u m solid, it is highly compressible. As a result, the weak anisotropic electric q u a d r u p o l e - q u a d r u p o l e ( E Q Q ) interaction has an important effect on the molar volume at low temperatures and cannot be neglected (a typical quadrupolar pressure of - 2 0 bar corresponds to an - 1% volume change in H2 at zero pressure). This effect depends on the o r t h o - p a r a concentration, as the molecules in the J = 1 rotational state (ortho-H2 or para-D2) bear E Q moments, while spherically shaped J = 0 molecules (para-H2 and ortho-D2) do not. The hydrogens are almost never in an equilibrium o r t h o - p a r a concentration, as conversion rates are rather slow. Since the quadrupolar pressures are nonnegligible, the equation of state must also be labeled by the o r t h o - p a r a concentration. The lattice specific heat model is extended to include the contribution due to the anisotropic intermolecular E Q Q interaction. This enables a determination of the E O S for H2 and D2 up to 2 kbar as a function of o r t h o - p a r a concentration. We then use our model to extrapolate the EOS to --25 kbar. The extrapolations are shown to be quite accurate by comparison *We use volume corrections given by Younglove. 8

Equation of State of Solid Hydrogen and Deuterium

257

with other data at certain P V T points. The model also enables a determination of the high-temperature values of the Debye temperatures and the Gr/.ineisen fitting constants. Using our data, we can, for the first time, experimentally determine the 4.2 K isotherm from 0 to 340 bar. We find disagreement with the AS extrapolation and reanalyze their raw data combined with our data to redetermine the 4.2 K isotherm to 25 kbar. Finally, the molar volume at zero pressure has been determined and compared to existing literature values. We present what we feel are the best available values as a function of o r t h o - p a r a concentration. The paper is organized as follows. In the next two sections we discuss our experimental system and procedures. In Section 4 we discuss the theory used in the analyses of our results. The determination of the volumes of the isochores is handled in Section 5. The E O S is presented in Section 6 and 7. The paper is concluded with a discussion of the phases of the solid in Section 8. 2. E X P E R I M E N T A L

SYSTEM

The basic experimental technique in the measurement of an isochore is to seal a sample of known molar volume into a cell such that it has a uniform density and temperature and to measure the pressure as a function of temperature (see, e.g., Ref. 13). In practice one generally makes quasiisochoric measurements as the cell volume changes due to thermal expansion and mechanical distortion of the cell walls. Corrections can be applied to the results for these two effects. The latter effect is generally used to measure the pressure with a transducer. Our cell is shown in Fig. 1. The cell was made of solution-annealed BeCu 25, selected because of its resistance to hydrogen enbrittlement. After machining, the cell was hardened by a heat treatment of 3 h at 350~

~

\

thermometer indium \ ~-,\-.,\\\., \,. ,,., ~

L

60m.m

D

Fig. 1. The berylium copper isochore cell.

258

A. Driessen, J. A. de Waal, and Isaac F. Silvera

to give a hardness of 380 Vickers. The cell has a nominal sample volume of 1.2 cm 3. The design of the cell represented a compromise between maxim u m pressure and sensitivity. Although our pressure-generating apparatus can achieve - 7 kbar, we decided on a limit of - 2 kbar in favor of accuracy and sensitivity. The cell wails were 1.5 m m thick and could withstand a m a x i m u m theoretical pressure of 2.5 kbar before plastic deformation would begin. The calculated pressure dependence of the volume, ( 1 / V ) 0 V/aP, is 4.7 x 10-6/bar at r o o m temperature and 4.2 x 10-6/bar at 77 K. The pressure of the sample was measured with resistance strain gauges (see, e.g., Ref. 14) connected in a full Wheatstone bridge. The strain gauges ( B L H Electronics Inc., FSM 25-35-SG) have a polymide glass-reinforced carrier and were attached with B L H P L D 700 single-component polymide adhesive, which can be used in the t e m p e r a t u r e range 0 - 7 0 0 K. The two active gauges were attached to the thin wall at the position of m a x i m u m strain and the two passive, t e m p e r a t u r e compensating gauges were attached at the lower, thick wall (Fig. 2). Interconnections between the strain gauges were made with low-resistance copper wire. The bridge was connected to an external ac measuring bridge with e v e n o h m wire. Bridge m e a s u r e m e n t s were reproducible to - 2 / z strain. Pressure calibration was p e r f o r m e d on a dead weight system of the Van der Waals L a b o r a t o r y of the University of A m s t e r d a m at three temperatures: 293, 77, and 26 K. Helium was used as a pressure transmitting medium. The accuracy of the calibration points was 0.3 bar; a sensitivity of 1.1/x strain/bar was determined which was insensitive to temperature in the range 77-26 K. The zero-pressure shift in this range was 5 # strain. Relative pressure changes of 0.1 bar were easily detectable. During m e a s u r e m e n t s a hysteresis of m a x i m u m 100/x strain was observed. The source, which was not discovered until after all m e a s u r e m e n t s were completed, was in the passive gauges. Unfortunately, these were located near the Bridgman seal and were sensitive to the forces acting on the indium packing, which evidently possessed hysteresis. Nevertheless, pressures produced in the same manner could be reproduced to within a few bars. the corrections for the hysteresis were the ultimate limitation to the accuracy of the pressure measurement. The isochore cell was connected to the pressure generating system via a 1 m m O D x 0.18 m m ID stainless steel (ss) capillary as shown in Fig. 2. This capillary was centered in an ss tube which could be evacuated or filled with exchange gas. A heating wire wound around the capillary prevented plugging during filling of the cell. A high-pressure valve at the top of the cryostat separated the cell from the pressure generation system during measurement. The dead volume of the capillary and valve was 0.04 cm 3. The cell was suspended from the capillary, inside of two bi'ass cans which were immersed in a liquid helium bath. The space between the two cans could be filled with


259

Vacuum or e-exchange gas lines I__ Capillary-heater

~H J

Liquid Helium

~

_High-pressure capillory

~

Heater

A :S-0Tsure Passive strain gauges

~

_~Ge-Thermometer I

/

J

Fig. 2. The isochore cell mounted in the cryogenic environment. helium gas for heat exchange or evacuated for thermal isolation. The inner cart always contained helium gas for thermal exchange, providing for a uniform t e m p e r a t u r e of the BeCu 25 cell. The cell was heated by warming the wall of the inner brass can with a resistance wire heater. The t e m p e r a t u r e could be controlled to 0.05 K and was determined with a calibrated g e r m a n i u m resistance t h e r m o m e t e r with an accuracy better than 0.1 K over the measuring region. A pair of C u - C o thermocouples were used to monitor the cell gradient, which was always negligible during measurements. The pressure generating system can compress a few liters of H2, D2, or other condensable gases to a m a x i m u m pressure of 7 kbar. 15 This is done by

260


condensing the gas into a high-pressure cylindrical gas-oil separator which can be cooled with liquid helium. After condensation, the separator is warmed and the gas is compressed to high pressure by driving a piston with oil pressure, such that the volume of the gas side is reduced. The compressed gas is cleaned of residual oil by a cryogenic trap before entering the measuring cell. High-purity normal H2 (3/4 ortho, 1/4 para) and normal D2 (1/3 para, 2/3 ortho) gas was used. Before condensation of a sample, the separator, high-pressure capillaries, and cell were well flushed, first with helium and tffen with H2 or D2. The cell and measuring system was checked for accuracy by reproducing the melting line of helium 5 and found to be satisfactory. 3. EXPERIMENTAL

PROCEDURE

From these and other pressure experiments with H2 and D2, we have found it necessary to use elaborate procedures and great care to load the cell in a reliable manner. To fill the cell, the space between the capillary and its enclosing ss tube is evacuated and up to 4 W of power is dissipated in the capillary heater. The sample is introduced into the cell at a pressure and temperature in the liquid phase, such as point A in Fig. 3. The capillary heater is then turned off and the capillary is cooled by introducing He exchange gas in its surrounding tube. This seals the capillary with solid H2 at 4.2 K. The two main experimental difficulties are (1) unwanted freezing (plugging) of the sample in the capillary, which prevents filling of the cell, and (2) creep of solid HE in the capillary 16"17(after it is intentionally frozen to seal the cell), resulting in a change of the molar volume. The high pressure valve at the top of the cryostat is closed as a further precaution and the cell is then slowly cooled. The trajectory in Fig. 3 is followed, first entering the melting line at point B, where both solid and liquid phases coexist, then departing to enter the solid state isochore at C, and finally arriving at point D at 4.2 K. Isochores could be measured with either increasing or decreasing temperature. A point on an isochore was measured by regulating the temperature and waiting approximately 5 rain for equilibrium, for which we used the criterion that changes in the pressure reading were less than 0.2/z strain in 1 min. As we were restricted to a maximum pressure of 2 kbar, point D was limited to 1.15 kbar for H2 and 1 kbar for D2 if we used an A B C D trajectory. To reach higher density solid isochores we attempted to fill the cell along trajectory EF, but the capillary would always block before F was reached. Thus we filled the cell using a trajectory of the type HIJK. If the external valve was open and no exchange gas was introduced in the capillary isolation tube, then upon warming from K to J we could see a sudden drop of pressure


N

0

--

2Kbor Limit

261

7"

K

A

[

--

..~..

.

.

.

.

.

.

.

P~.(V.Tf)

IPo,lV

I

42

T

~

Fig. 3. Diagram showing various trajectories used in measuring isochores

(see text), and variables in Eqs. (19) and (20).

in the pressure generating system due to a small flow into the cell, and we could j u m p to the isochore LM. This effect was reproducible, but preventable by cooling the capillary to 4.2 K. Several isochores were m e a s u r e d for H~ and D2 to fill the P - T plane with nearly equally spaced curves as seen in Figs. 4 and 5. A few isochores were measured twice with different samples to check for reproducibility and possible inclusions of impurities. Within experimental error, all isochores were reproducible. Normally, we measured points on the isochores with 3-5 deg t e m p e r a t u r e spacings. However, as we expected some irregularities due to the r e p o r t e d negative thermal expansion of C o o k e t aL 16 or a structural phase change, ~s-21 we have made m e a s u r e m e n t s in steps smaller than 1 deg kelvin and made slow, continuous runs on an x - y recorder, both with decreasing and increasing temperature. We saw no indications of these irregularities, except in runs in which our sample loading and capillary sealing procedures were faulty. Thus a trajectory such as I J K - ~ J L , if not

262


2000

r

I

~

I

I

t

~

l

J

l

t

i

t

I

i

i

i

l

I

Para-Hydrogen

V=17.029 1500 V=12500 L. a aC~

V=17974 L. 03

~, lOOO &

V=18.414

melting line of para H2

Liquid V= 19.299 solid 500

V=20.165

V=21,17."

0

~'o present data, corrected for EQO interoction o Jarvis-Meyer, para H2 - - cal.cutated [sochores

10

20 30 40 50 Temperature [K] Fig. 4. Isochores of solid hydrogen, corrected to correspond to the para species. Data points on the melting line and in the liquid phase have been omitted, as well as points below 8 K. The solid lines are fits with the Mie-Grfineisen theory.

carefully m o n i t o r e d , could be i n t e r p r e t e d as an isochoric increase in pressure with d e c r e a s i n g t e m p e r a t u r e , similar to the result of C o o k e t al. 16 I n the p r e s e n t work, m e a s u r e m e n t s in H2 show a d r o p in p r e s s u r e at lower t e m p e r a t u r e s , b u t this is d u e to s h o r t - r a n g e o r d e r i n g of the E Q m o m e n t s as s e e n earlier by Jarvis e t al. 22


2000

,

,

,

i

I

i

,

~

i

I

'

i

i

,

I

'

i

,

i

I

i

263

I

O r t h o - Deuterium

V= 15.898 1500 V= 16.237 L.

~1ooo

V=16.792r

L-

..melting line 3f normal-D-

V=1711 5 V=17.407 solid

500

/

liquid

V=18.05z, / [] data points, c o r r e c t for EQQ interaction - - calculated Isochor v=19030 V=19861 0

10

20 30 Temperature [K]

/.0

50

Fig. 5. Isochores of solid deuterium, corrected to correspond to the ortho species. Data points on the melting line and in the liquid phase have been omitted, as well as points below 8 K. The solid lines are fits with the Mie-Grtineisen theory. T h e raw d a t a r e q u i r e d several corrections, m o s t of which were a p p l i e d iteratively: 1. T o o b t a i n true isochores from quasiisochores we used a relative v o l u m e c h a n g e of 4.1 x 1 0 - 6 / b a r in the t e m p e r a t u r e r e g i o n of interest. W e always took the c o r r e c t i o n r e f e r e n c e v o l u m e at the i n t e r s e c t i o n of the solid isochore a n d the m e l t i n g line.

264


2. The strain gauges exhibited a large, but reproducible hysteresis. For every isochore we have one or more accurately measured points on the melting line for which the pressure is known fairly precisely, s'6 It was possible to quantitatively describe the behavior of the hysteresis. The pressure could be determined with an error of 3 bar + 0.06 Ap, where A p is the difference between the calibration point at the melting line and the pressure point in question. 3. The thermal expansion of the vessel has been neglected, as the total expansion between 0 and 50 K is less than 0.05% 4. Below - 8 K the zero point of the strain gauges changes by an a m o u n t equivalent to 5 bar, as determined with the e m p t y cell. Correction for this would have been quite complicated. Since these points were not important for the analysis, we have retained only points measured above IOK. 5. The isochores were corrected to represent pure para-H2 and orthoD2. The correction scheme will be discussed later. Virgin gas samples were always normal; however, some catalytic conversion took place during condensation in the separator a s well as continual intrinsic conversion in the cell, which takes place at rates of 2 - 5 % / h in H2, 23 depending on the density, and much m o r e slowly in D2. Cycled gas from the cell could be sampled and analyzed by means of R a m a n scattering 24 to determine the o r t h o - p a r a content. One such H2 sample had a value of 0.49 ortho-H2 after being studied for a week. Consideration of catalytic conversion and the t i m e t e m p e r a t u r e - d e n s i t y history of samples led us to the value of 0 . 5 + 0 . 1 5 ortho-H2 concentration and 0 . 2 5 + 0 . 0 5 para-D2 concentration. These values were used in the correction to give the pure J = 0 isochores and the uncertainties are one of the largest sources of error. The molar volume of isochores of H2 with solid pressures less than - 3 4 0 bar was taken from the corrected data of D w y e r e t al., 7 who measured the molar volume on the melting line. For higher pressures we used the 4.2 K data of AS, which we have corrected in a manner to be discussed later. Our data measured to 10 K was easily extrapolated to 4.2 K. For D2 we used the AS isotherm, to which we applied a small correction for the para content. 4. T H E O R E T I C A L

CONSIDERATIONS

4.1. Mie-Griineisen

Picture

The analysis (see, e.g., Ref. 24) of the E O S is based on the Helmholtz free energy, which can be separated into a z e r o - t e m p e r a t u r e part F0 and an incremental thermal part F*,

F(V~ T~ C1) ~-~Fo(V ,

Cx)+F*(V, T,

C~)

(1)

Equation Of State o[ Solid Hydrogen and Deuterium

265

where V is the molar volume and C1 the concentration of the E Q bearing species (molecules in the Y = 1 rotational state); F * = 0 at T = 0. For our purposes, the only important terms in the free energy are the lattice (translational) and E Q Q (orientational) contributions. We assume these to be additive, so that we can write F=

E

F,~

(2)

,~=L,O

where L is the lattice contribution and C) the E Q Q contribition. The assumption of additivity is based on the weak coupling between translational and rotational motions 26 and on the very weak o r t h o - p a r a concentration d e p e n d e n c e of the isotropic part of the intermolecular interaction 27 which enters directly into the dynamical matrix of the phonons. As a result FL( V, T, Ca) is independent of Ca. The pressure P and bulk modulus B are determined f r o m the thermodynamic relations P= --(OF/aV)T B = -- V(OP/O V ) T

(3) (4)

F r o m Eq. (2) we have

P = Z P~(v, T, G ) = Z Pod(V, C 0 + E P* (V, T, C1)

(5)

with analogous expressions for the bulk modulus. P* is called the thermal pressure, Po the quadrupolar pressure, etc. In the following, if the Ca dependence is dropped, i.e., if we write P*(V, T), we imply Ca = 0 or that there is no Ca dependence. We have found that a Mie-Griineisen picture as used by Spain and Segall a2 for helium can be extended to Hz and D2 to calculate the equation of state. The thermal pressure can be given in terms of a characteristic t e m p e r a t u r e 0~ and Griineisen constant 7~: P* ( V, T, C1) = [y~( V)/V]

U* ( V, T, C1)

(6)

Here

yo(v)=

- d In o~(v) dtn v

(7)

and U* (V, T, C1) is the t e m p e r a t u r e - d e p e n d e n t part of the internal energy

u * ( v, T, Ca) =

(

Cw, ( V, x, Ca ) dx

(8)

=0

where Cv is the specific heat at constant volume. For the total pressure we

266

A. Driessen, J. A. de Waai, and Isaac F. Silvera

have

r~ e(v, r, c,)= Po.(V,c,)+2 --y-

ff:o Cv,~(V, x, C1) dx

(9)

4.2. Lattice Contribution For the lattice contribution we use the D e b y e approximation to calculate CvL :

9R xae x CvL(V, T ) = --x 3 JrXl o ~ (e x -

(lO)

1)2 dx

where R = Nokn is the gas constant and xo = Oo/T, Oo being the D e b y e temperature. Equations (9) and (10) yield

fo

x3

P * ( V , T ) - y L ( V ) 9NokB T 4 ~' _ _ dx V 03(V) ex- 1

(lla)

where the reduction of the double integral to a single integral is discussed in Ref. 12. Combining with Eq. (7), we get the differential-integral equation

P*L(V, T) - dOo(V) 9Nok~ T4 dV 04(V)

~0x ~ _ x3 _

dx

(llb)

e*-I

where OD(V) is unknown.

4.3. EQQ Contribution The second contribution to the thermal pressure comes from the anisotropic interactions, which are dominated by the attractive E Q Q interaction in the density ranges considered here. This interaction is described by a traceless tensor (see, e.g., Ref. 28); as a result the energy center of gravity is unshifted when the electric quadrupole m o m e n t Q ~ 0. For T much greater than the m a x i m u m splitting AEo, all levels are equally populated and P o ( V , T = ~ , C,) = 0. The E Q Q coupling p a r a m e t e r is F = 6e2Q2/25R

s

(12)

where R is the distance between the interacting molecules. For nearest neighbor pairs in H2 at zero pressure F -~ 0.84 K (experimental) and A E o 10F = 8.4 K; thus Po will be effective for all temperatures in the solid state


267

and will be largest at low temperatures. Since the E Q Q energy is lowered by decreasing R, Po will be a negative pressure, tending to decrease the lattic parameter or molar volume. At a given separation, F is almost identical for H2 and D2, as the mass difference has very little effect on the interaction potential. From Eq, (5),

Po(V, T, C,) ~- Poo(V, C1)+P~)(V, T, C1)

(13)

Using Po(V, oo, 6"i)= 0, we can write Po(V, T, C , ) = - P ~ ( V ,

oo, C,)+P*(V, T, C1)

(14)

From Eq. (9) we find oo

(15) where Yo = - d In F/d In V = - ( V / F ) O F / O V = 5/3, as has been shown by Jarvis et al. 22 Cvo has been measured by several experimenters, 29-31 and Berlinsky and Harris 32 have derived an expression for it which gives a reasonably good fit to experimental data for kT/F ~> 8. This expression and the details of the evaluation of Eq. (15) are given in the Appendix. The results are most easily given as P o V / T vs. the dimensionless parameter F/T, where F~2(V ) = FD2(V) = 154 V -s/3

(16)

with V given in cm3/mole and F in degrees kelvin. A sufficient number of points is given in Table I or Fig. 6 to enable easy interpolation. As an example, at a molar volume of 23 cm3/mole, FH2 = 0.575 cm -1. For C1 = 1 we find P Q ~ - 2 1 b a r at 1 0 K and ~ - 1 2 b a r at 20 K. Our calculations are compared in Fig. 6 to the data of Jarvis e t al. 22 and others and give a reasonable representation. The theory is only valid above the orientational ordering temperature 3 Tr which is 2.8 K for H2 and 3.8 K for Dz at zero pressure and C 1 = 1 and scales a s V - 5 / 3 . 33 For C1 = 1, the region of validity is T/F ~> 8, as given by Berlinsky and Harris; apparently it can be used at lower temperatures without great error, in particular for (71 < 1. We have assumed that the volume dependence of Po is obtained by using Eq. (16) in the expression for Cv~ Eq. (15).

268

A. Driessen, J. A. de Waai, and Isaac F. Silvera

Values of

TABLE i Determined by Evaluating Eq. (15) or Eq. (A3) as a Function of G(V, T, C1)= 154 V-5/3/T for Selected Values of C1 '~

-PoV/T

G (v, 7")]• o . 0 5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49

0 0 0 0 0.1 0.1 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.7 0.7 0.8 0.9 0.9 1 1.1 1.2 1.3 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.8 2.9 3 3.1 3.2 3.3 3.4 3.6 3.7

0.15

0.25

0.33

0.40

0.50

0.60

0.65

0.75

0 0.1 0.2 0.3 0.5 0.7 1 1.2 1.5 1.9 2.2 2.6 3 3.5 3.9 4.4 4.9 5.4 5.9 6.5 7 7.6 8.2 8.8 9.4 10 10.6 11.2 11.9 12.5 13.2 13.8 14.5 15.1 15.8 16.5 17.2 17.9 18.5 19.2 19.9 20.6 21.3 22 22.7 23.5 24.2 24.9 25.6

0.1 0.2 0.5 0.9 1.4 1.9 2.6 3.3 4.2 5 6 7 8.1 9.2 10.3 11.5 12.7 14 15.3 16.6 17.9 19.3 20.6 22 23.4 24.9 26.3 27.7 29.2 30.6 32.1 33.6 35 36.5 38 39.5 41 42.4 43.9 45.4 46.9 48.4 49.9 51.4 52.9 54.4 55.9 57.4 58.9

0.1 0.4 0.9 1.6 2.4 3.4 4.6 5.8 7.2 8.7 10.3 12 13.8 15.6 17.5 19.5 21.5 23.5 25.6 27.7 29.8 32 34.1 36.3 38.5 40.7 43 45.2 47.4 49.7 51.9 54.1 56.4 58.6 60.9 63.1 65.4 67.6 69.9 72.1 74.3 76.6 78.8 81 83.2 85.5 87.7 89.9 92.1

0.1 0.6 1.3 2.2 3.4 4.9 6.5 8.3 10.2 12.3 14.5 16.9 19.3 21.8 24.4 27 29.7 32.5 35.2 38 40.9 43.7 46.6 49.5 52.4 55.2 58.1 61.1 64 66.9 69.8 72.7 75.6 78.5 81.4 84.2 87.1 90 92.8 95.7 98.6 101 104 107 110 113 115 118 121

0.2 0.9 2 3.5 5.3 7.5 9.9 12.6 15.6 18.7 22 25.4 28.9 32.6 36.3 40.1 43.9 47.8 51.7 55.6 59.6 63.5 67.5 71.5 75.4 79.4 83.4 87.3 91.3 95.2 99.2 103 107 111 115 119 122 126 130 134 138 141 145 149 153 156 160 164 167

0.3 1.3 2.9 5 7.6 10.6 14.1 17.8 21.9 26.1 30.6 35.2 40 44.9 49.8 54.8 59.9 65 /70.1 75.2 80.3 85.4 90.6 95.7 101 106 111 116 121 126 131 136 141 146 150 155 160 165 169 174 179 183 188 193 197 202 206 211 215

0.4 1.5 3.4 5.8 8.9 12.4 16.4 20.7 25.3 30.2 35.3 40.6 46 51.5 57.1 62.8 68.5 74.2 79.9 85.6 91.3 97.1 103 108 114 120 125 131 136 142 147 153 158 163 169 174 179 185 190 195 200 205 210 215 220 225 230 235 240

0.5 2 4.4 7.7 11.7 16.3 21.4 27 32.9 39.1 45.6 52.2 58.9 65.8 72.7 79.7 86.6 93.6 101 108 114 121 128 135 142 148 155 162 168 175 181 188 194 200 206 213 219 225 231 237 243

0.80

0.90

1.00

0.6 0.7 2.3 2.9 5 6.3 8.7 10.9 13.2 16.5 18.4 22.9 24.1 30 30.3 37.6 37 45.6 43.9 54 51 62.6 58.3 71.3 65.8 80.2 73.3 89.1 80.9 98.1 88.5 107 96.2 116 104 125 111 134 119 142 126 151 134 160 141 168 149 177 156 185 163 193 170 201 177 209 184 217 191 225 198 233 205 241 212 248 219 256 225 263 232 271 238 245

0.9 3.6 7.8 13.4 20.1 27.9 36.4 45.4 55 64.8 74.9 85.1 95.4 106 116 126 137 147 157 167 176 186 196 205 214 224 233 242 250 259 268 276 285

Equation of State of Solid Hydrogen and Deuterium Table I. Continued

G (v, T~N U 0 . 0 5 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0,58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

3.8 3.9 4.1 4.2 4.3 4.4 4.6 4.7 4.8 4.9 5.1 5.2 5.3 5.5 5.6 5.7 5.9 6 6.1 6.3 6.4 6.6 6.7 6.8 7 7.1 7.3 7.4 7.5 7.7 7.8 8 8.1 8.3 8.4 8.6 8.7 8.9 9 9.2 9.3 9.4 9.6 9.7 9.9 I0.1 10.2 10.4 10.5 10.7 I0.8

0.15

0.25

0.33

26.3 27 27.7 28.5 29.2 29.9 30.6 31.3 32.1 32.8 33.5 34.2 35 35.7 36.4 37.2 37.9 38.6 39.4 40.1 40.8 41.5 42.3 43 43.7 44.5 45.2 45.9 46.7 47.4 48.1 48.9 49.6 50.3 51.1 51.8 52.5 53.3 54 54.7 55.5 56.2 56.9 57.7 58.4 59.1 59.9 60.6 61.3 62 62.8

60.4 61.9 63.4 64.9 66.3 67.8 69.3 70.8 72.3 73.8 75.2 76.7 78.2 79.7 81.1 82.6 84.1 85.6 87 88.5 89.9 91.4 92.9 94.3 95.8 97.2 98.7 100 102 103 104 106 107 109 110 112 113 114 116 117 119 120 122 123 124 126 127 129 130 131 I33

94.3 96.5 98.7 101 103 105 107 110 112 114 116 118 120 122 125 127 129 131 133 135 137 139 141 143 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198

0.40

124 127 129 132 135 138 140 143 146

0.50

171 174 178 182 185 189

0.60

220 224 228 233

"Po is in bars, V in cm3/mole, and T in degrees kelvin.

269

270

A. Driessen, $. A. de Waal, and Isaac F. Silvera

250

~200 ,.....

"6

E E15C

(3

t.-.--

~100

13_o I

50

0 0

0.1

0.2

l-'lkT

0.3

O./-.,

0.5

Fig, 6. Graphical display of quadrupolar pressure vs. the quadrupolar interaction constant, both parametrized with temperature. A comparison is made to experiment.

For pure J -----] H 2 and D 2 , below critical temperatures Tc of 2.8 K and 3.8 K, 34 at P = 0, respectively, a phase transition takes place to the orientationally ordered Pa3 structure, with an incremental decrease in Po and a concomitant decrease in volume. Although the order-disorder phase line has not been studied in detail, recent experiments 33 to - 5 kbar indicate that the critical temperature scales as V -5/3 as expected for E Q Q interactions. At concentration C~< 1 the transition temperature is reduced and apparently goes to zero* for concentrations Ct = 0.55 for V = V0, with the critical concentration going to Ca -~ 0.4 for higher densities. For T-

0~ 0

500

~

1000 PaL(V) [bar]

1500

2000I

Fig. 9. Straight line fits to Eq. (20). in this case, an uncertainty of + 1 5 % in C1 would correspond to about • 10 bar error. In order to average out errors on a given isochore we looked for an empirical relationship connecting all of the isochores. We found a linear relationship to exist between the z e r o - t e m p e r a t u r e pressure P0(V) and the thermal pressure where the solid (s) isochore intersects the melting line, P * ( V , Ts), or the thermal pressure where the liquid (f) isoch.ore intersects the melting line, P* (V, Tt), all shown in Fig. 3. These data are plotted in Figs. 8 and 9 (along with similar data for helium for comparison) and can be fit to straight lines:

P*(V, Ts) = P*(Vo, Ts)+asPoL(V)

(19)

P*(V, Ti)= P*(Vo, Tr)+afPoL(V)

(20)


274

T A B L E I! Constants Found in the Fit to Eqs. (19) and (20)

Species Ha Dz He

P~ ( Vo, Ts), bar 24.0 90.7 - 12.7

PL( Vo, Tf),

oq

rms deviation, bar

bar

af

rms deviation, bar

0.1194 0.2065 0.0704

4.3 13.4 3.0

251.4 597.5 42.4

0.4782 0.4576 0.2329

1.5 36.8 1.9

where V0 is the molar volume for Po(V) = 0. A small extrapolation of the data to T = 0 was done with the D e b y e model. P* (V, Ti) was determined by the change of slope in the data points at the melting line. For P*(V, Ts), more accuracy could be obtained by using the Debye model to extrapolate the highest point on the solid isochore to the melting line; where the Debye model was applied, one iteration was used in the volumes. Additional data (triangles in Fig. 9) could be obtained by combining the liquid density measurements of Weber 37 with the corrected AS isotherm. An unpublished low-pressure isochore in para-H2 of Jarvis and Meyer* has also been used. Equations (19) and (20) were fit with a weighted least squares computer program and the results are given in Table II. The larger rms deviations for Dz result from the hysteresis error, which increases with the differences in pressure of the solid isochore from that of the calibration point on the melting line. This is 2-3 times larger for D2. 5.2. Molar V o l u m e s

5.2.1. General The objective is now to assign a volume to each of the isochores. For para-H2 this can be done for pressures up to - 3 4 0 bar by use of the solid molar volume at the melting line Vs determined by Dwyer etal. 7'8 For higher pressures we can use the 4.2 K isotherm of AS, hut with some modification. In their measurements, values are given of P(V/V'0, 4.2 K, normal) up to 25 kbar. In the piston displacement method, volume is measured relative to a reference volume V, at pressure /9,. AS could not take P, = 0 due to difficulties with frictional forces. At P = P~, their stated absolute accuracy in Vr was +0.5%, much larger t h a n the relative error. To achieve a higher precision they extrapolated from P - - 5 0 0 bar to 0 bar so that they could express their results in terms of V0 (instead of Vr), which supposedly was known to a higher precision than +0.5%, giving an improved absolute *We thank H. Meyer for providing us with these data and for permitting us to use them.

Equation

.4[

of

~

i

State

t

I

ot

Solid

I

J

Hydrogen and Deuterium

r

p

i

r

,

f

i

275

[

J

[

I

CI=.75

.2

C~

error bars of AS I 9

I

I

I

J

20

I

I

I

I

I

15

L

I

I

10

V [cc/rno[e]

Fig. 10. A plot showing the percent change in volume in Ha relative to para-n2 for several values of ortho concentration. AV is arbitrarily taken as zero at 20 cm3/mole. The crossed area is the relative error band of AS. For Ca -- 340 bar. T h e differences in the isochores are not a result of c o m p a r i n g o u r p a r a - H 2 to their n-H2 (their samples identified as n-H2 m a y have had Ca = 0.5 r a t h e r than 0.7538) as they m e a s u r e d on b o t h n-H2 and p - H z and f o u n d no difference b e t w e e n the two within their relative experimental accuracy of 0 . 1 - 0 . 3 % in V/Vo (there is, however, a difference in the i s o t h e r m as the reference v o l u m e ff'o differs for n- and p-H2.). W e d e c i d e d to reinterpret the AS data. We r e c o v e r e d their c o m p r e s sion data for p - H z f r o m their Table II and their deviation plot (Fig. 2), giving us P vs. V/Vo for P ~ > 5 0 0 b a r . To this we a p p e n d o u r i s o t h e r m for

276


P < 3 4 0 bar. The two data sets are then fit to a Birch relation in which we take the AS V0 as an additional variable (we found ~'o = 23.36 cm3/mole). The actual zero-pressure volume V0 was also varied to give best agreement with the data of Dwyer et al., yielding V0 = 23.14 cm3/mole. We then used this Birch relation to determine the volumes of our isochores (see Fig. 19, Section 7). The fact that AS could not distinguish between n-H2 and p-H2 in their compression data is not inconsistent with our remark that the E Q Q forces are important. This is demonstrated by Fig. 10, where we plot the calculated A V~ V due to Po for selected values of C 1 a s a function of V. The crossed area indicates the 0.1-0.3% relative error band of AS; their stated absolute error was of order 0.5%. Evidently within their experimental error, the systematic effects due to the E Q Q forces would not be visible for samples with Ca ~