Here, UM is the classical static-lattice energy, CM is the Madelung constant (Table 2.4), Uq accounts for the zero-point quantum vibrations, and ul belongs to the family of average phonon frequency moments up = Ks)ph K. (2.102)
From Eq. (2.97), one obtains the internal energy
and the heat capacity
As a rule, the main contribution into the internal energy comes from the classical static-lattice energy. The Madelung constant CM, which determines this energy, is remarkably close to 0.9 for all lattice types. 4 Let us recall that the value 0.9 is predicted by the ion-sphere model (§2.3.2b). Therefore, the ion-sphere expression UM = -0.9 Nn Z2e2/ai is sufficiently accurate in the ion liquid and solid, as long as the ions are strongly coupled.
The low-order frequency moments of bcc, fcc, and hcp crystals are very similar (see Table 2.4). According to Eq. (2.96), u2 = 3 for any OCP lattice. Many thermodynamic and kinetic properties of Coulomb crystals are remarkably independent of the crystal type and are similar to those in a strongly coupled Coulomb liquid. For instance, the transport properties of degenerate electrons which scatter off ions are almost the same in the liquid and crystal phases (Baiko et al., 1998). The difference between the liquid and crystal and between different crystal types is really tiny (also see §§ 2.3.4 and 2.4.6).
4For the bcc Coulomb lattice, this constant was first calculated by Fuchs (1935).
Table 2.4. Parameters of Coulomb crystals: Madelung constant Cm, frequency moments un and l = (ln(wfcs/wpi))ph (Baiko et al., 2001b).
bcc 0.895 929 255 682 12.972 2.79855 0.5113875 0.25031 -0.831298
fcc 0.895 873 615195 12.143 2.71982 0.5131940 0.24984 -0.817908
hcp 0.895 838120459 12.015 2.7026 0.51333 0.24984 -0.81597
2.3.3 d Analytic formulae for thermodynamic functions
Classical Coulomb crystal. Asymptotically, at r ^ one has Ui ~ UM, where UM is given by Eq. (2.101). A thermal correction
to the internal energy of the OCP at r » 1 is very small. The term 3NnkBT comes from the harmonic lattice vibrations and Uanharm represents anharmonic corrections discussed in § 2.3.3 e.
In the classical regime (T > Tpi), one has huks ^ kBT, and Eq. (2.97) reduces to
with the term
presented in Table 2.4. Subtracting the ideal-gas free energy (2.71) from Fi given by Eq. (2.106), one obtains the excess free energy of ions in the harmonic approximation,
Strong quantum limit. Now consider the regime of T ^ Tpi. In this case, only a small central part of the Brillouin zone, where zks < 1, contributes actually to the averaging in Eq. (2.97). In addition, the contribution from the optical (longitudinal) phonons (s = 3) is much smaller than that from the acoustic ones (s = 1, 2). Carr (1961) evaluated CVi and obtained CVi = (NnkB(T/Tpi)3, where ( is a numerical constant (its accurate value is given on p. 83). This yields
Because of the rapid decrease of the T-dependent terms at low T, the ion heat capacity becomes smaller than the electron one in the strong quantum limit, tp ^ 0 (§ 2.4.6).
Harmonic Coulomb crystal at arbitrary temperature. At intermediate temperatures T ~ Tpi the asymptotic expressions (2.106) and (2.109) become inaccurate and thermodynamic quantities should be evaluated numerically. For estimates, however, one can use a simple model (Chabrier et al., 1992) that treats two acoustic modes as degenerate Debye modes with uks = atupik/qBZ and the longitudinal mode as an Einstein mode with uks = alupi. We shall call it D2E ("2 Debye + 1 Einstein modes") model. In this model
3 (in(i _ e-zks))ph = _3 Ds(atn) + 2 ln(1 - e-an) + ln(1 - e-ain),
and n rx tn
is a Debye function. We can propose the interpolation formulae for D1 and o(x2)] and large-x asymptotes,
D3 which are accurate within 7 parts in 104 and reproduce the small-x [up to
1(X) ~ 1 + 1 x + Hz x2 + 0.00139x4 + (6/n2) Ri(x) ' (. )
~ 1 + 8 x + 3290 x2 + 0.01656 x3-2 + 5.9 x 10"5 x6 + (5/n4) R3(x) '
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