Non-Metals
5.2. RARE GAS SOLIDS
The history of the discovery of the rare gas elements, and of fifty years of progress in unravelling their physical properties, is well told by George Horton in the opening chapter of the two volumes of Rare Gas Solids [Kle76, Kle77]. Both theory and experiment are covered comprehensively in succeeding chapters.
All these solids are weakly bound by van der Waals interatomic forces, which can be represented fairly closely by pair potentials of similar shape [Bar76a], scaled by parameters
eo
for energy and ro for distance. For the heavier rare gases this leads to a law of corresponding states, so that they have similar properties when scaled to units derived from these two parameters and the atomic mass (see, for example [Row69]).But departures from this simple law of corresponding states become larger for the lighter rare gas solids, as the ratio of the zero point vibrational energy to the binding energy of the interatomic potential increases from about 0.03 for Xe to 0.25 for Ne and nearly unity for He. This ratio depends on the de Boer dimensionless parameter
A*
= hhjmeor'6
(5.1)tabulated in Table 5.1 [deB49]. In helium the zero point energy is so large that even at the lowest temperatures solidification occurs only under pressure of at least 25 bar, and even then the vibrations are strongly anharmonic. For this reason we consider first the heavier rare gas solids.
5.2.1. Ne, Ar, Kr and Xe
Purely central forces would give the hcp structure, but small many-body effects cause all these crystals to be fcc [Nie76]. Because the bonding is weak they have low melting points and large expansion coefficients, making pure strain-free single crystals difficult to produce and presenting a challenge to the experimentalist. Many measurements have been made on samples which were condensed in metal or glass cylinders and thereby severely strained, and so not truly representative. Thermal and elastic properties are reviewed by P. Korpiun and E. Luscher in the second volume of Rare Gas Solids [Kle77], and high pressure thermodynamic data by C. A. Swenson in the same volume. Data for Ne, Ar, Kr and Xe from these two sources are given in Table 5.1, together with values for the two helium isotopes (see Section 5.2.2 and the review by H. R. Glyde in [Kle76]).
Figure 5.1 illustrates the temperature dependence of 6c for Ar [Bea61]. For Kr, Xe and Ne the graphs are very similar in shape to that for Ar. The reduced values of 6c (T)/60 versus T /60 are quantitatively similar for all four, with a shallow
Non-Metals ISS Table 5.1. Data for rare gas solids. A· is the de Boer parameter,
T, Is the triple point and • denotes solid phase under pressure of 3.3 MPa eHe) and 2.5 MPa (4He) [K1e76, K1e77]
Property 3He 4He Ne Ar Kr Xe
A* 3.08 2.67 0.593 0.186 0.103 0.063
T,(K) 24.556 83.806 115.763 161.391
AJ(g/cm3) 0.123* 0.190* 1.507 1.771 3.093 3.781 Yo (cm3/mol) 24.3* 21.0· 13.4 22.6 27.1 34.7
B~(GPa) 0.019 0.027 1.12 2.9 3.5 3.7
eo(K) 15-18* 26* 74 92 72 64
10 2.9* 2.6 2.7 2.6 2.5
minimum near 00/10. The rise in 0c (T, Vo) at high temperatures indicates that the anharmonic contribution to Cy is negative, as often seen in a close-packed crystal.
Thermal expansion measurements by X-rays and capacitance dilatometry and also equation of state data (see [Bar80, p. 654]) lead to limiting values of ')'0 ~ 2.6 and values in the range of 2.7 to 2.9 at higher temperatures (Fig. 5.2).
The most accurate elastic data appear to be those derived from Brillouin scattering or from the limiting slopes of phonon dispersion curves obtained by inelastic neutron scattering (see [Kor77]). At low temperatures departures from the Cauchy relation
Cl2 = C44 are small, as expected for solids dominated by pair potentials (Section 2.8.6); The crystals are elastically anisotropic, with 2C44/(Cll - Cl2) ~ 2.4 as T --} 0 [Kor77].
95 - 0 - V" feT)
,...,
90@
b 8580~ ________ ~~ ________ ~~ ________ ~ __________ ~
o
10 20 30 40T(OK)
Fig. 5.1. Variation of €Ie (T) for Ar [Bea61).
IS6 3.0
2.8
2.6
2.4
o
205.2.2. HeUum
Kr
40 60
T (K)
80
Fig. 5.2. y(T) for Nc, Ar, Kr and Xc [BarSo, p. 655).
ChapterS
100
Helium gas, at least at low pressures, may be close to 'perfection' as far as the equation of state is concerned, but the liquid and solid states present a complex picture which is still being unravelled and would take a volume to cover in detail.
The weak binding forces and large zero point energy prevent solidification under normal pressure (see phase diagram in Fig. 4.7): the more common isotope 4He liquefies under the standard pressure of 100 kPa (1 bar) at 4.21 K, and 3He at 3.19 K. The minimum pressure to produce solid 4He increases from about 2.5 MPa (25 bar) near absolute zero to about 8 MPa near 3 K. Even under these pressures the volume is considerably larger than the equilibrium volume of the static lattice, and gives a negative static lattice compressibility; it is the zero-point vibrational energy that stabilizes the structure at these volumes. But although the quasi-harmonic approximation thus breaks down completely, the heat capacity and thermal expansion behave similarly to other solids, with a renormalized effective vibrational spectrum (Section 2.6).
The thermodynamic properties above 1 K are well established. The solid phase is hcp at moderate pressures, except for a narrow range along the melting curve between 1 and 2 K where it is bec. For 3He the solidification pressures are higher (Fig. 4.7 and Table 5.1), and there is a more extended bec phase region at lower pressures. Both isotopes transform from hcp to fcc at high T and P.
Because of the large compressibility of solid helium, there have been many measurements of both heat capacity and (ap / aT)v as functions of volume and temperature, which lead to values of@C(V, T) and 'Y(V, T). For 4He, 'Yo increases
Non-Metals 157 from about 2.5 at a molar volume V = 17.5 cm3 to 2.9 at V = 21 cm3, indicating that 'Y/V is fairly constant and q = dln'Y/dln V ~ 1 (e.g., [Swe77]).
Dugdale and Franck [Dug64] made extensive measurements of Cv of solid (and fluid) 3He and 4He from 3 K up to the melting points at different densities. Curves of aC versus V measured near T = a /20 show parallel behavior of the two isotopes, each giving 'Y
=
-dIna/dIn V=
2.4. At much lower temperatures ('" mK) the nuclear spin of 3He gives rise to magnetic properties in both the bec and fcc phases.These, and also the properties of 3He-4He mixtures, are described in Solid Helium Three [Dob94].
Theoretical treatment of these highly anharmonic crystals is reviewed extensively in [Gly76]. More general references are the monographs by Wilks [Wil67], Keller [KeI69], Dobbs [Dob94], and Pobell [pob96].
5.3. ROCKSALT STRUCTURE