Non-Metals
5.3. ROCKSALT STRUCTURE 1. Alkali Halides
Most of the alkali halides crystallize in the 6-coordinated cubic rocksalt structure;
cations and anions each form a fcc lattice, and together occupy the sites of a simple cubic lattice. At high pressures the crystals undergo a first order transition to the more compact 8-coordinated cesium chloride structure, where cations and anions each form a simple cubic lattice, and together occupy the sites of a bec lattice.
Transition pressures increase from a few GPa for Rb halides to tens of GPa for Na halides; they have been predicted by ab initio electronic computations for several of the salts in good agreement with experiment [Sim98]. The remaining halides, CsCI, CsBr and CsI, have the cesium chloride structure at atmospheric pressure, at least up to room temperature.
Qualitative differences in the lattice dynamics and thermal properties of rocksalt halides may be expected to reflect differences in the ratios of atomic masses and of ionic radii, and also in the nature of the short-range interatomic potential. The differences in ionic radii are illustrated in Fig. 5.3, which compares LiF with RbBr and CsF using values for the Pauling radii and nearest neighbor distances taken from Born and Huang, see [Bor54, p. 18]. It is clear that the rocksalt structure can comfortably accommodate a wide range of ratios of ionic radii. The ionic masses do not directly affect the elasticity, but do affect the vibrational spectrum and hence the heat capacity and thermal expansion. In contrast, the nature of the interatomic potential affects all three properties.
Elasticity. We consider first the volume and the elasticity. Table 5.2 shows how the molar volume V increases with increasing size of each ion. At the same time the bulk modulus B decreases, so that the product VB remains roughly constant throughout the table. The shear stiffness for transverse waves polarized perpendicular to {100} planes, C44, also decreases with the size of either ion, but more drastically
IS8 ChapterS
(a) LiF
(b) RbBr
(c) CsF
Fig. 5.3. Schematic view of a (100) plane in (a) LiF, (b) RbBr and (c) esF.
when the cation is changed. The ratio C44/ B thus depends mainly on the type of alkali ion; and so also does its pressure derivative, which becomes negative for the heavier cations. This has important consequences at low temperatures, affecting the heat capacity and particularly the thermal expansion. Our main source of elastic data for this table are from [Lew67, Har79] and compilations by Hearmon [Hea66, Hea69].
Heat Capacity. The first definitive measurements of Cp by Clusius et al. [Clu49]
were followed by extensive work at the National Research Council (Ottawa) on K and Na halides [Bar57b]. The latter achieved sufficiently high accuracy (error bars of ±O.I%) for the Taylor expansion in Eq. (2.87) to be fitted to the data below 8/25, which gave the higher order TS and T7 terms as well as the T3 term and hence the corresponding terms in w4 and w6 for the frequency distribution. The T3 term gave values for 8~ in good agreement with best elastic values.
At higher temperatures the accuracy was sufficient to allow extrapolation of (8C )2 versus I/T2 to give the hannonic high temperature limit 8£ for several salts (see Table 5.2 and Section 2.6.2). For other salts in Table 5.2, Eq. (2.76) was used to derive 8£ from values of
<
w2>
computed from the DD3N models of Hardy and Karo [Har79].Plots of8C(T) show a characteristic pattern, as in Fig. 5.4 (see [Bar57b, p. 485]), with a minimum near 8D / 15 and an increase towards 8ao at higher temperatures; the
Non-Metals 159 Table 5.2. Data for alkali halides of NaCI structure. Most are from [Bar80]
with elastic data from [Lew67]. Asterisk. indicates use of room temperature data, .1 use of elastic data, and m use of
<
tAl'].>
[Har79]fall-off in 8C at still higher temperatures for the potassium halides is an anharmonic effect, due to a small positive contribution to C v. The shapes of these curves depend both on the mass ratio and on the type of alkali ion. 80 depends through the elastic wave velocities on the density, and hence on the average mass (ml +m2)/2, whereas to a rough approximation 8£ depends on the reduced mass [Bla42]:
C2
1(1 1)
(800 ) oc-
-+-2 ml m2
For crystals with similar interionic forces, the ratio
eo /
8£ thus becomes proportional to~, where TJ = (ml-m2)/(ml +m2)' To a good approximation this factor accounts for changes in 8 0/8£ as the halide ion is changed. But in addition, the lowering of the ratio C44 / B with increasing atomic number of the alkali ion affects the weighting of the low-lying transverse modes, producing a relative lowering of 80, so that the mass-corrected ratio depends chiefly on the cation. For example, KCI and RbBr both have ionic mass ratios close to 'unity but different ratios 80/8£.Thermal Expansion. At room temperature linear expansion coefficients lie mainly in the narrow range 36 to 45 x 10-6 K-1• This is a consequence of a similarly narrow range (1.4-1.7) for the Griineisen function at T '" 8D for these salts (Fig. 5.5), since VB varies little. Values for the fluorides are lower. There is a clear trend for a273 to increase with the size of the halide ion, and except for CsF
160 ChapterS
'1'/8.
Fig. 5.4. Reduced plot of
e
C against T for potassium halides [Bar57b].to decrease with the size of the alkali ion. Sources of a values in Table 5.2 are the reviews [Bar80, Mer73) and the book by Krishnan et at. [Kri79).
At low temperatures "I decreases markedly (excepting for the lithium salts). This is another dimensionless property that depends primarily on the alkali ion (Fig. 5.5), as can be seen also from the values of ')'0 given in Table 5.2; these correlate well with the change in the alkali ion and agree within limits of measurement with values of "lei calculated from the pressure dependence of the ultrasonic velocities. In particular, transverse modes propagating in {I OO} planes and polarized normal to the planes do not bring into play the strong force-constant of the nearest neighbor pair potential, and so tend to have low frequencies and small or negative gammas because of the tension effect (Section 2.6.3). In the acoustic limit these are governed by C44, which has a negative pressure dependence for both the potassium and rubidium halides;
and this plays a major part in lowering the average 'YO. For example, RbCI has dC44/dP = -0.6, which gives a value of about -1.3 for the associated mode 'Y.
The weighted average over all the low frequency modes of RbBr gives 'Yo' ~ -0.05, agreeing well with
"Ib
h = -0.03.By contrast, for the three Cs halides which have the CsCI structure 'Y ~ 2.0 over the whole range from 2 to 300 K (see review [Bar80, p. 661]).
5.3.2. Other Crystals of Rocksalt Structure
Numerous other solids crystallize in the rock salt structure, including the alkaline earth oxides (MgO is discussed in Section 5.6) and many compounds of the transition and post-transition metals, including PbS, PbSe, PbTe, SnTe, etc. For many of
Non-Metals
these there are heat capacity data in Vol. 5 of Thermophysical Properties of Matter [Tou70b] and thermal expansion data in Vol. 13 of the same Series [Tou77]. They are not generally of major cryogenic interest.