Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited

Tomaschitz, R.

doi:10.1107/S2053273321005507

research papers

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ISSN: 2053-2733

Volume 77| Part 5| September 2021| Pages 420-432

https://doi-org.cc/10.1107/S2053273321005507

Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited

Roman Tomaschitz ^a ^*

^aSechsschimmelgasse 1/21-22, Vienna, 1090, Austria
^*Correspondence e-mail: tom@geminga.org

Edited by A. Altomare, Institute of Crystallography - CNR, Bari, Italy (Received 23 March 2021; accepted 26 May 2021; online 29 July 2021)

A phenomenological model of anisotropic lattice vibrations is proposed, using a temperature-dependent spectral cutoff and varying Debye temperatures for the vibrational normal components. The internal lattice energy, entropy and Debye–Waller B factors of non-cubic elemental crystals are derived. The formalism developed is non-perturbative, based on temperature-dependent linear dispersion relations for the normal modes. The Debye temperatures of the vibrational normal components differ in anisotropic crystals; their temperature dependence and the varying spectral cutoff can be inferred from the experimental lattice heat capacity and B factors by least-squares regression. The zero-point internal energy of the phonons is related to the low-temperature limits of the mean-squared vibrational amplitudes of the lattice measured by X-ray and γ-ray diffraction. A specific example is discussed, the thermodynamic variables of the hexagonal close-packed zinc structure, including the temperature evolution of the B factors of zinc. In this case, the lattice vibrations are partitioned into axial and basal normal components, which admit largely differing B factors and Debye temperatures. The second-order B factors defining the non-Gaussian contribution to the Debye–Waller damping factors of zinc are obtained as well. Anharmonicity of the oscillator potential and deviations from the uniform phonon frequency distribution of the Debye theory are modeled effectively by the temperature dependence of the spectral cutoff and Debye temperatures.

Keywords: anisotropic lattice vibrations; thermodynamic functions; Debye–Waller factors; non-cubic crystals; temperature-dependent spectral cutoff; directional Debye temperatures; effective phonon speed; oscillator mass; heat capacity; zero-point energy.

1. Introduction

The aim is to develop an effective theory of lattice vibrations in anisotropic crystals that can accurately reproduce the empirical thermodynamic functions and temperature evolution of Debye–Waller B factors. This is motivated by the fact that the standard Debye theory usually fails to model the extended phonon peak in the lattice heat capacity of non-cubic crystals, which emerges in the crossover region between the low- and high-temperature regimes. Moreover, the Debye temperatures inferred from X-ray diffraction measurements of the mean-squared atomic displacements differ from those obtained from the low-temperature heat capacity, resulting in different X-ray and caloric Debye temperatures (cf. e.g. Gopal, 1966 ; Butt et al., 1988 ; Peng et al., 1996 , 2004 $[Peng, L.-M., Dudarev, S. L. & Whelan, M. J. (2004). High-Energy Electron Diffraction and Microscopy. Oxford: Oxford University Press.]$ ), which should be identical in a self-consistent theory of lattice vibrations. The linear high-temperature scaling of the B factors predicted by the Debye theory is also not shown empirically, underestimating the measured factors (cf. e.g. Killean & Lisher, 1975 ; Martin & O'Connor, 1978a ,b ; Shepard et al., 1998 ; Malica & Dal Corso, 2019 ).

A frequently used method to model empirical deviations from the Debye theory is to employ a temperature-dependent Debye temperature $[\theta (T)]$ in the thermodynamic functions, to be determined from measured heat capacities or Debye–Waller B factors (cf. e.g. Barron & Munn, 1967b ; Martin, 1968 ; Skelton & Katz, 1968 ). As is the case with constant Debye temperatures, the $[\theta (T)]$ functions obtained from caloric and diffraction experiments differ. Moreover, a phonon partition function with varying Debye temperature does not define an equilibrium system, since the equilibrium condition $[\partial S/\partial U = 1/T]$ on the internal-energy derivative of entropy is violated once the Debye temperature becomes temperature dependent (Tomaschitz, 2020a ). Nevertheless, the idea of using a varying Debye temperature to account for deviations from the standard Debye theory is attractive because of its relative technical simplicity, avoiding perturbative expansions. Alternative attempts to adapt the Debye theory to reality include the addition of anharmonic terms to the oscillator potential (cf. e.g. Merisalo & Larsen, 1977 , 1979 ; Field, 1983 ; Rossmanith, 1984 ; Kumpat & Rossmanith, 1990 ) and/or a modification of the continuous and uniform phonon density of states assumed in the Debye theory (cf. e.g. Gopal, 1966; Meissner et al., 1978 ).

Here, we will discuss thermal vibrations of the hexagonal close-packed zinc structure (Altmann & Bradley, 1965 ), as an example of an anisotropic elemental crystal. The equilibrium relation $[\partial S/\partial U = 1/T]$ can be maintained with varying Debye temperatures $[{\theta _j}(T)]$ for the normal vibrations, provided that the spectral cutoff $[\Lambda (T)]$ of the thermodynamic functions is also allowed to vary with temperature. (In anisotropic crystals, the vibrational normal components admit different Debye temperatures labeled by j, which coincide in the case of cubic monatomic crystals.) The empirical B factors and heat capacity can be described with the same set of functions $[{\theta _j}(T)]$ and $[\Lambda (T)]$ so that X-ray and caloric Debye temperatures coincide. The starting point in Section 2.1 will be a partition of the atomic oscillations into vibrational normal components, each component being described by a temperature-dependent Debye temperature and spectral cutoff. The varying spectral cutoff and Debye temperature of each normal component can be extracted from empirical heat capacity data by least-squares regression and from the zero-temperature limits of measured B factors.

Once the directional Debye temperatures and the spectral cutoff are known, one can calculate the effective phonon speed determining the dispersion relation of each normal component, which suffices to calculate the thermodynamic functions. The temperature evolution of the effective oscillator mass of the normal vibrations can also be determined from the empirical B factors. We will explain the general formalism for anisotropic monatomic crystals and work out a specific example, the thermodynamic functions and B factors of zinc.

This paper is organized as follows. In Section 2.1, we discuss the lattice internal energy, entropy and the mean-squared vibrational amplitudes, the partitioning of atomic vibrations into normal components, and the partial energies and entropies and B factors of the normal vibrations.

In Section 2.2, the temperature-dependent spectral cutoff and Debye temperatures of the vibrational normal components are derived. The partial oscillator density and effective phonon speed of each normal component are inferred from the respective Debye temperature and spectral cutoff. In this section and the Appendix, we explain the regression of the varying Debye temperatures and spectral cutoff from heat capacity data and empirical B factors.

In Section 3, we consider Debye–Waller B factors of anisotropic elemental crystals defined by the mean-squared amplitudes of the orthogonal normal vibrations. The zero-point lattice energy can be obtained from the zero-temperature limits of three B factors measured by X-ray diffraction and from the amplitude of the cubic low-temperature scaling of the isochoric heat capacity. The reasoning in Sections 2 and 3 applies to anisotropic elemental crystals in general.

In Section 4, the thermodynamic variables and mean-squared vibrational amplitudes of the zinc structure are studied. Heat capacity measurements of zinc continuously extend from the low-temperature regime up to the melting point (Seidel & Keesom, 1958 ; Phillips, 1958 ; Eichenauer & Schulze, 1959 ; Garland & Silverman, 1961 ; Zimmerman & Crane, 1962 ; Martin, 1968, 1969 ; Cetas et al., 1969 ; Mizutani, 1971 ; Grønvold & Stølen, 2002 ), converted from isobaric to isochoric values (Barron & Munn, 1967b; Arblaster, 2018 ), and B factors of vibrations in the basal plane and along the hexagonal axis have also been measured over a wide temperature range [see Skelton & Katz (1968), Albanese et al. (1976 ), Pathak & Desai (1981 ) and earlier measurements reviewed by Barron & Munn (1967a ) and Rossmanith (1977 )]. We model the lattice heat capacity of the zinc structure with a multiply broken power-law density (Tomaschitz, 2020a,b , 2021b ,c ) and perform a least-squares fit to the isochoric data sets, subtracting the electronic heat capacity (cf. Section 4.1).

Analytic least-squares fits are also performed to the measured B factors of zinc, up to the melting point (cf. Section 4.2). The amplitude of the low-temperature lattice heat capacity and the zero-temperature limits of the regressed B factors allow us to estimate the zero-point internal energy of the zinc structure (cf. Section 4.3). Integrations of the regressed lattice heat capacity give the thermal components of the internal energy and the entropy (cf. Section 4.6). Based on these empirical functions and the zero-temperature limits of the axial and basal B factors of zinc, one can determine the varying Debye temperatures of normal vibrations orthogonal and parallel to the basal plane as well as the temperature-dependent spectral cutoff (cf. Sections 4.4 and 4.5). Using this input, the effective phonon speed defining the dispersion relations of the normal vibrations can be calculated (cf. Section 4.7). The effective oscillator mass of the normal vibrations is inferred from the temperature evolution of the regressed B factors (cf. Section 4.8). Second-order B factors defining the non-Gaussian correction of the Debye–Waller damping factors are derived in Section 4.9. Section 5 contains the conclusions.

2. Varying Debye temperatures and spectral cutoffs of non-cubic elemental crystals

2.1. Lattice internal energy, entropy and mean-squared vibrational amplitudes

We consider anisotropic (i.e. non-cubic) monatomic crystals and split the molar internal energy of the lattice vibrations into three components, corresponding to atomic normal vibrations, $[U = \sum_{j = 1}^3 {{U_j}}]$ ,

$[\eqalignno{&{U_j}(T) = {{4\pi } \over {{{(2\pi)}^3}}}R{{{\Lambda ^3}(T)} \over {n_{{\rm{at}}}}}T\left\{D\left[{{{\theta _j}(T)} \over T}\right] + {1 \over 8}{{{\theta _j}(T)} \over T}\right\},&\cr &D(x): = {1 \over {x^3}}\int\limits_0^x {{t^3 \,{\rm d}t} \over {\exp (t) - 1}},&(1)}]$

where R = 8.314 J (K mol)⁻¹ is the gas constant, the $[{\theta _j}(T)]$ are varying Debye temperatures, $[\Lambda (T)]$ (cm⁻¹) is the temperature-dependent spectral cutoff and $[n_{{\rm{at}}}]$ (cm⁻³) the atomic density (Tomaschitz, 2020a, 2021a ). The energy components U_j(T) are in units of J mol⁻¹. D(x) denotes the Debye function, with asymptotic limits $[D(x \to 0)\sim 1/3]$ and $[D(x \to \infty)]$ $[\sim {\pi ^4}/(15{x^3})]$ . The coordinate system defining the normal oscillations can be arbitrarily chosen; in the case of zinc, one will conveniently choose the hexagonal axis and two perpendicular axes in the basal plane. The standard Debye internal energy is recovered by assuming a constant Debye temperature $[{\theta _j}(T) = {\theta _{\rm{D}}}]$ and constant spectral cutoff, $[\Lambda (T) = {\Lambda _{\rm{D}}} = {(6{\pi ^2}n_{{\rm{at}}})^{1/3}}]$ . The spectral representation of the partial energies (1) reads

$[\eqalignno{{{{U_j}} \over V}& = {{4\pi } \over {{{(2\pi)}^3}}}\int\limits_0^{\Lambda (T)} \hbar \omega _{j,{\bf{k}}}(T)&\cr &\quad\times\left\{{1 \over {\exp[\hbar \omega _{j,{\bf{k}}}(T)/(k_{\rm{B}}T)] - 1}} + {1 \over 2}\right\}{k^2}\,{\rm{d}}k, &(2)}]$

where the temperature-dependent linear dispersion relations $[\omega _{j,{\bf{k}}} = c_{{\rm eff},j}(T)k]$ are determined by the effective phonon speed $[c_{{\rm eff},j} = k_{\rm{B}}\theta _j(T)/[\hbar \Lambda (T)]]$ of the normal vibrations (labeled by subscript j = 1,2,3). The conversion of the specific energy densities U_j/V in (2) to molar quantities (1) is done by multiplying (2) with $[V/{\rm{mol}}]$ and using $[n_{{\rm{at}}} = N_{\rm{A}}{\rm{mol}}/V]$ and $[R = N_{\rm{A}}k_{\rm{B}}]$ .

The entropy can likewise be decomposed into normal components $[S = \sum_j S_j]$ , with partial entropies

$[\eqalignno{{S_j}(T)& = {{4\pi } \over {{{(2\pi)}^3}}}R{{{\Lambda ^3}(T)} \over {n_{{\rm{at}}}}}&\cr &\quad\times\left({4 \over 3}D\left[{{{\theta _j}(T)} \over T}\right] - {1 \over 3}\log \left\{1 - \exp[ - \theta _j(T)/T]\right\}\right). &(3)}]$

The temperature-dependent effective oscillator density is $[n_{{\rm{oscil}}}(T)]$ = $[[4\pi /(2\pi)^3]\Lambda ^3(T)]$ , with partial densities $[n_{{\rm{oscil}},j}]$ = $[ n_{{\rm{oscil}}}(T)/3]$ generating the energy and entropy components in (1) and (3). The partition function is assembled as $[k_{\rm{B}}\log Z = S - U/T]$ and the Helmholtz free energy as $[F = - k_{\rm{B}}T\log Z]$ .

The mean-squared displacement $[\langle u_j^2(T)\rangle ]$ (cm²) of the normal oscillators vibrating parallel to coordinate axes labeled j reads (Tomaschitz, 2020a)

$[\eqalignno{&\langle u_j^2(T)\rangle = {3 \over {{m_{{\rm{eff}},j}}(T)}}{{{\hbar ^2}T} \over {{k_{\rm{B}}}\theta _j^2(T)}}\left\{{D_1}\left[{{{\theta _j}(T)} \over T}\right] + {1 \over 4}{{{\theta _j}(T)} \over T}\right\},&\cr & {D_1}(x): = {1 \over x}\int\limits_0^x {{t\,{\rm d}t} \over {\exp(t) - 1}},&(4)}]$

where $[m_{{\rm{eff}},j}(T)]$ is the temperature-dependent effective oscillator mass. The corresponding Debye–Waller B factors are $[{B_j} = 8{\pi ^2}\langle u_j^2(T)\rangle]$ . The asymptotic limits of the Debye function in (4) are $[{D_1}(x \to 0)\sim 1]$ and $[{D_1}(x \to \infty)\sim {\pi ^2}/(6x)]$ . A derivation of (1)–(4) based on an effective field theory for the normal oscillations can be found in Tomaschitz (2021a). Here, the focus is on the practical application to zinc.

2.2. Temperature dependence of the Debye temperatures and spectral cutoff

The lattice internal energy and entropy are related to the lattice heat capacity C_V(T) [in units of J (K mol)⁻¹] by $[U(T) = \int_0^T {C_V}(T)\,{\rm d}T + {U_0}]$ and $[S(T) = \int_0^T {C_V}(T)/T\,{\rm{d}}T]$ , respectively, where the zero-point internal energy U₀ is an integration constant. In Section 4.1, an analytic representation of C_V(T) will be obtained from a least-squares fit of a multiply broken power-law density to heat capacity data of zinc. The analytic fit can then be integrated as indicated to find the internal energy and entropy. The zero-point energy U₀ will be determined from X-ray and γ-ray diffraction measurements of Debye–Waller B factors (cf. Section 4.2).

We specialize the Debye temperatures of the normal oscillations in (1)–(4) as $[{\theta _j}(T) = {\hat c_j}\theta (T)]$ , where the $[{\hat c_j}]$ are positive temperature-independent constants. This ansatz is sufficiently general to model anisotropic vibrations of elemental crystals. The effective phonon speed of the normal oscillations is $[c_{{\rm{eff}},j} = {\hat c_j}c_{{\rm{eff}}}]$ , with $[c_{{\rm{eff}}} = k_{\rm{B}}\theta (T)/[\hbar \Lambda (T)]]$ [cf. after (2)].

Once U(T) and S(T) are extracted from the empirical heat capacity and B factors, the varying Debye temperature $[\theta (T)]$ can be found by combining equations (1) and (3),

$[{3 \over 4}{{TS(T)} \over {U(T)}} = \Delta (d), \Delta (d): = {{\sum_j \{4D({{\hat c}_j}d) - \log [1 - \exp( - {{\hat c}_j}d)]\} } \over {\sum_j [4D({{\hat c}_j}d) + {{\hat c}_j}d/2]}}\eqno(5)]$

where we substituted $[{\theta _j} = {\hat c_j}\theta (T)]$ and used the shortcut $[d = \theta (T)/T]$ . In the Appendix, we will show that the Δ function (5) is monotonously decreasing on the positive real axis, from infinity to zero, for any choice of positive constants $[{\hat c_j}]$ . It will also be necessary to explicitly invert $[\Delta (d)]$ in order to calculate $[\theta (T)]$ by solving (5), $[\theta (T) = T{\Delta ^{ - 1}}({{3 \over 4}}TS/U)]$ . The constants $[{\hat c_j}]$ in (5) can be obtained from zero-temperature limits of measured B factors (cf. Sections 3 and 4), so that the numerical inversion of $[\Delta (d)]$ in (7) needs to be done only once; a practical inversion method is outlined in the Appendix, cf. after (30).

The spectral cutoff $[\Lambda (T)]$ in (2) is found by solving $[U = \sum_j {{U_j}}]$ in (1) for $[\Lambda (T)]$ , with $[{\theta _j} = {\hat c_j}\theta (T)]$ substituted for the directional Debye temperatures. By expressing $[\theta (T)]$ and $[\Lambda (T)]$ as functions of entropy and internal energy, the equilibrium condition $[\partial S/\partial U = 1/T]$ mentioned in Section 1 is preserved (Tomaschitz, 2020a). U(T) and S(T) are empirical functions, obtained from the regressed heat capacity and the low-temperature limits of the regressed B factors, as exemplified with zinc in Section 4.4.

3. Low-temperature limits of the Debye–Waller B factors and zero-point energy of anisotropic lattice vibrations

At low temperature, the lattice heat capacity scales as $[{C_V}(T \to 0)\sim {c_{V0}}{T^3}]$ , with amplitude c_V0 obtained from a least-squares fit (cf. Section 4.1), so that the internal energy converges to its zero-point limit, $[U(T \to 0)\sim {U_0}]$ , and the lattice entropy scales as $[S(T \to 0)\sim {c_{V0}}{T^3}/3]$ (cf. the beginning of Section 2.2). The low-temperature limit of $[\Delta (d)]$ in (5) reads [cf. (30)]

$[\Delta (d \to \infty)\sim {{8{\pi ^4}} \over {15}}{1 \over {{d^4}}}{{\sum_j {1/\hat c_j^3} } \over {\sum_j {\hat c_j} }}. \eqno(6)]$

Substitution of the stated limits into (5) gives the zero-temperature limit of the Debye temperature $[\theta (T)]$ ,

$[{\theta _{T = 0}} = \left({{32{\pi ^4}} \over {15}}{{{U_0}} \over {{c_{V0}}}}{{\sum_j {\hat c}_j^{ - 3}} \over {\sum_j {\hat c}_j}} \right)^{1/4}. \eqno(7)]$

The zero-temperature limit of the B factors (4) reads

$[{B_{j,T = 0}} = 8{\pi ^2}{3 \over {{m_{{\rm{eff}},j}}(0)}}{1 \over 4}{{{\hbar ^2}} \over {{k_{\rm{B}}}{{\hat c}_j}{\theta _{T = 0}}}}. \eqno(8)]$

Here and in (6), we used the asymptotic expansions of the Debye functions D(x) and D₁(x) [cf. after (1) and (4) and (30)]. The zero-temperature limit of the effective oscillator mass is the atomic mass, $[{m_{{\rm{eff}},j}}(T = 0) = m]$ . Equation (8) defines a system of three equations, labeled by j = 1,2,3, that can be solved for $[{\theta _{T = 0}}]$ , $[{\hat c_2}]$ and $[{\hat c_3}]$ :

$[{\theta _{T = 0}} = 6{\pi ^2}{{{\hbar ^2}} \over {{k_{\rm{B}}}}}{1 \over {{{\hat c}_1}{B_{1,T = 0}}m}},\, {\hat c_j} = {\hat c_1}{{{B_{1,T = 0}}} \over {{B_{j,T = 0}}}}.\eqno(9)]$

We substitute this into (7) and solve for U₀ to find the zero-point internal energy,

$[{U_0} = {{15} \over {32{\pi ^4}}}\left({{{\hbar ^2}} \over {{k_{\rm{B}}}}}6{\pi ^2}\right)^4{{{c_{V0}}} \over {{m^4}}}{{\sum_j {1/{B_{j,T = 0}}} } \over {\sum_j {B_{j,T = 0}^3} }}, \eqno(10)]$

where c_V0 is the empirical amplitude of the low-temperature lattice heat capacity, $[{C_V}\sim {c_{V0}}{T^3}]$ .

One of the temperature-independent constants $[{\hat c_j}]$ defining the Debye temperatures $[{\theta _j}(T) = {\hat c_j}\theta (T)]$ , cf. Section 2.2, can be absorbed into $[\theta (T)]$ , so that we can put $[{\hat c_1} = 1]$ from the outset. The remaining constants $[{\hat c_{j = 2,3}}]$ defining the directional Debye temperatures $[{\theta _j}]$ can be inferred from the zero-temperature limits B_{j,T = 0} of the empirical B factors by way of (9).

4. Thermodynamic functions and Debye–Waller factors of zinc

4.1. Regressed lattice heat capacity of the zinc structure

We perform a least-squares fit to the empirical heat capacity of zinc (Eichenauer & Schulze, 1959; Martin, 1968, 1969); see also Seidel & Keesom (1958), Phillips (1958), Garland & Silverman (1961), Zimmerman & Crane (1962), Cetas et al. (1969), Mizutani (1971) and Grønvold & Stølen (2002) for experimental data, converted to isochoric values and depicted in the double-logarithmic plot in Fig. 1. The isochoric heat capacity $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ can be split into a linear component stemming from the degenerate electron gas and a multiply broken power law (Tomaschitz, 2017 , 2020c , 2021b,c) for the lattice heat capacity C_V,

$[\eqalignno{C_{V,{\rm{e}} + {\rm{L}}}(T)& = b_{{\rm{elec}}}T + C_V(T),&\cr C_V(T)& = b_0 T^3 [1 + (T/{b_1})^{\beta _1/\eta _1}]^{\eta _1}&\cr &\quad\times{1 \over {[1 + (T/b_2)^{\beta _2/\eta _2}]^{\eta _2}}}{1 \over {[1 + (T/b_3)^{\beta _3/\eta _3}]^{\eta _3}}}. &(11)}]$

The amplitudes $[b_{{\rm{elec}}}]$ , b₀ and $[b_1 \,\lt\, {b_2} \,\lt\, {b_3}]$ and the exponents $[{\beta _i}]$ and $[{\eta _i}]$ are positive and related by $[b_3 = [3Rb_1^{\beta _1}/(b_2^{\beta _2}b_0)]^{1/\beta _3}]$ and $[{\beta _3} = 3 + {\beta _1} - {\beta _2}]$ , so that the classical Dulong–Petit limit $[{C_V}(T \to \infty)\sim 3R]$ is recovered, with gas constant R = 8.314 J (K mol)⁻¹. The isochoric data points in Fig. 1 include the electronic component of the heat capacity, and the least-squares fit is done with $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ . The independent fitting parameters $[b_{{\rm{elec}}}]$ , b_0,1,2, $[{\beta _{1,2}}]$ , $[{\eta _{1,2,3}}]$ are recorded in Table 1. The lattice heat capacity C_V(T) of the zinc structure is obtained by subtraction of the electronic contribution $[b_{{\rm{elec}}}T]$ from the regressed $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ and has the low-temperature limit $[C_V(T \to 0)\sim {b_0}{T^3}]$ . The units used are C_V [J (K mol)⁻¹], $[b_{{\rm{elec}}}]$ [J (K² mol)⁻¹], b₀ [J (K⁴ mol)⁻¹] and b_i (K). The linear electronic power law in (11) is exponentially cut off at high temperature (classical regime), but in the temperature range below the melting point we do not need to consider this, since even the linear power law is dwarfed by the lattice heat capacity except, of course, in the low-temperature regime (see Fig. 1).

Table 1
Fitting parameters of the isochoric heat capacity $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ (electronic and lattice heat capacity) of zinc

The recorded amplitudes b_i and exponents $[{\beta _i}]$ , $[{\eta _i}]$ define the multiply broken power-law density (11) used for the fit of the heat capacity in Fig. 1. (Some of the parameters are interrelated, e.g. $[{\beta _3} = 3 + {\beta _1} - {\beta _2}]$ , cf. Section 4.1.) $[{b_{{\rm{elec}}}}]$ is the amplitude of the linear electronic power law included in the $[{\chi ^2}]$ fit, cf. (11). The listed $[{\chi ^2}]$ is the minimum of the least-squares functional $[{\chi ^2} = \sum_{i = 1}^N [C({T_i}) - {C_i}]^2/C_i^2]$ , where C is a shortcut for $[C_{V,{\rm{e}} + {\rm{L}}}]$ ; the degrees of freedom [d.o.f.: number N of data points (T_i,C_i) minus number of independent fitting parameters] are also recorded. The standard error SE = $[\{\sum_{i = 1}^N [C({T_i}) - {C_i}]^2/N\}^{1/2}]$ of the fit is 0.0589. The coefficient of determination, $[{R^2} = 1 - \sum_{i = 1}^N [C({T_i}) - {C_i}]^2 /(N{\sigma ^2})]$ , with sample variance $[{\sigma ^2} = \sum_{i = 1}^N ({C_i} - \bar C)^2/N]$ and mean $[\bar C = \sum_{i = 1}^N {C_i} /N]$ , is estimated as $[1 - {R^2} = 3.88 \times {10^{ - 5}}]$ .

$[{b_{{\rm{elec}}}}]$ [J (K² mol)⁻¹]	b₀ [J (K⁴ mol)⁻¹]	b₁ (K)	b₂ (K)	b₃ (K)	$[{\beta _1}]$	$[{\beta _2}]$	$[{\eta _1}]$	$[{\eta _2}]$	$[{\eta _3}]$	$[{\chi ^2}]$	d.o.f.
6.3899 × 10⁻⁴	6.2314 × 10⁻⁵	7.4222	11.615	53.290	5.6567	6.6810	1.9078	2.7421	1.0039	0.0545	67 − 9

Figure 1
Isochoric heat capacity of zinc. Data points from Martin (1968

[Martin, D. L. (1968). Phys. Rev. 167, 640-651.]

, 1969

) (circles) and Eichenauer & Schulze (1959

[Eichenauer, W. & Schulze, M. (1959). Z. Naturforsch. A, 14, 28-32.]

) (squares). The least-squares $[\chi^2]$ fit (solid blue curve) is performed with $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ in (11)

(combined electronic and lattice heat capacity), the fitting parameters are stated in Table 1

. The classical 3R Dulong–Petit limit is indicated by the black dotted horizontal line. The lattice heat capacity C_V(T) of the zinc structure [red solid curve depicting the multiply broken power-law density in (11)

] has a cubic low-temperature slope and is obtained from the least-squares fit $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ (solid blue curve) by subtracting the linear electronic low-temperature heat capacity $[{b_{{\rm{elec}}}}T]$ (blue dotted line), cf. (11)

and Table 1

. C_V(T) has an inflection point (black diamond) in log–log representation at $[{T_{{\rm{inflec}}}} ]$ = 9.085 K, $[{C_V}({T_{{\rm{inflec}}}})]$ = 0.0986 J mol⁻¹; the inflection tangent $[ \propto T^{4.223}]$ is depicted as a red dotted line. The Debye approximation $[{C_{V,{\rm{D}}}}(T)]$ (black dashed curve) with constant Debye temperature $[{\theta _{\rm{D}}}]$ = 314.79 K, cf. (12)

, admits the same low- and high-temperature limits as C_V(T) but does not have a log–log inflection point and largely deviates from the data points in the crossover regime. The maximum of the relative deviation $[({C_V} - {C_{V,{\rm{D}}}})/{C_V}]$ of the Debye approximation from the empirical heat capacity occurs at 18.5 K, where $[{C_V}/{C_{V,{\rm{D}}}} = 3.66]$ and C_V = 1.44 J (K mol)⁻¹. The difference $[{C_V} - {C_{V,{\rm{D}}}}]$ is listed in Table 2

, for selected temperatures. The temperature range is cut off at the melting point of zinc, at 692.73 K. Residuals of the $[\chi ^2]$ fit of $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ are shown in the lower panel.

In contrast to the Debye heat capacity, the empirical lattice heat capacity of the zinc structure has an inflection point in log–log representation (cf. Fig. 1); the tangent of C_V(T) at the inflection point is depicted as a red dotted line $[ \propto T^{4.223}]$ , at $[T_{{\rm{inflec}}}]$ = 9.085 K, $[{C_V}(T_{{\rm{inflec}}})]$ = 0.0986 J mol⁻¹. $[C_{V,{\rm{e}} + {\rm{L}}}(T)]$ and C_V(T) are plotted in Fig. 1 up to the melting point of zinc at 692.73 K.

Also shown in Fig. 1 is the Debye approximation,

$[\eqalignno{{C_{V,{\rm{D}}}}(T) &= 9R\left[4D({\theta _{\rm{D}}}/T) - {{{\theta _{\rm{D}}}/T} \over {{\exp(\theta _{\rm{D}}/T) - 1}}}\right],&\cr {\theta _{\rm{D}}}& = \left({{12} \over 5}{\pi ^4}{R \over {{b_0}}}\right)^{1/3},&(12)}]$

with Debye function D(d) in (1). The constant Debye temperature of zinc is $[{\theta _{\rm{D}}}]$ = 314.79 K, obtained from the low-temperature amplitude b₀ in (11) and Table 1. The constant effective phonon speed is $[{c_{{\rm{eff,D}}}} = {k_{\rm{B}}}{\theta _{\rm{D}}}/(\hbar {\Lambda _{\rm{D}}})]$ , cf. after (2), with cutoff factor $[{\Lambda _{\rm{D}}} = {(6{\pi ^2}n_{{\rm{at}}})^{1/3}}]$ , cf. after (1), so that $[{c_{{\rm{eff,D}}}}]$ = 2.62 km s⁻¹, based on the atomic density $[n_{{\rm{at}}}]$ = 6.5702 × 10²² cm⁻³. The zero-point energy in the Debye model is $[{U_{0,{\rm{D}}}} = 9R{\theta _{\rm{D}}}/8]$ , cf. (1) and (30), which gives $[{U_{0,{\rm{D}}}} ]$ = 2944.5 J mol⁻¹ for zinc, incompatible with the zero-temperature limits of the measured B factors (cf. Section 4.3). The empirical heat capacity of zinc has an extended phonon peak at around 20 K (cf. Fig. 1), which is not reproduced by the Debye approximation; the deviation $[{C_V}(T) - {C_{V,{\rm{D}}}}(T)]$ is listed in Table 2, for a few selected temperatures.

Table 2
Isochoric heat capacity C_V(T), thermal energy $[{U_{{\rm{therm}}}}(T)]$ and entropy S(T) of the zinc structure at selected temperatures below the melting point

The listed values of C_V(T) are calculated from the regressed broken power law (11) with parameters in Table 1. $[{U_{{\rm{therm}}}}]$ and S are obtained by integrating C_V(T), cf. Section 2.2. Also recorded is the deviation of the Debye approximations $[{C_{V,{\rm{D}}}}(T)]$ , $[{U_{{\rm{therm,D}}}}(T)]$ , $[{S_{\rm{D}}}(T)]$ from the empirical values. $[{C_{V,{\rm{D}}}}]$ is calculated via (12), $[{U_{{\rm{therm,D}}}}]$ and $[{S_{\rm{D}}}]$ by integrating $[{C_{V,{\rm{D}}}}]$ , see Sections 2.2, 4.1 and 4.6. The maximal relative deviations are indicated in the captions of Figs. 1 and 4.

T (K)	C_V [J (K mol)⁻¹]	$[ {C_V} - C_{V,{\rm{D}}}]$ [J (K mol)⁻¹]	$[{U_{{\rm{therm}}}}]$ (J mol⁻¹)	$[{U_{{\rm{therm}}}} - {U_{{\rm{therm,D}}}}]$ (J mol⁻¹)	S [J (K mol)⁻¹]	$[S - {S_{\rm{D}}}]$ [J (K mol)⁻¹]
5	0.009276	1.487 × 10⁻³	0.01069	9.551 × 10⁻⁴	0.002814	2.174 × 10⁻⁴
10	0.1478	0.08547	0.2887	0.1329	0.03636	0.01559
25	3.237	2.268	20.94	14.86	1.087	0.7633
50	10.86	4.856	199.9	113.9	5.741	3.403
75	16.05	4.222	541.9	230.3	11.22	5.299
100	19.06	3.141	984.2	322.0	16.29	6.362
200	23.16	1.050	3151	507.1	31.15	7.728
298	24.10	0.4961	5476	578.0	40.60	8.024
400	24.47	0.2787	7956	616.0	47.75	8.135
500	24.63	0.1787	10411	638.3	53.23	8.185
600	24.73	0.1239	12880	653.2	57.73	8.212
692.7	24.78	0.09267	15175	663.1	61.29	8.228

4.2. B factors of lattice vibrations parallel and perpendicular to the hexagonal axis

The Debye–Waller B factor of zinc defined by vibrations along the principal axis perpendicular to the basal plane is denoted by $[{B_{{\rm{axial}}}} = 8{\pi ^2}\langle u_{{\rm{axial}}}^2(T)\rangle]$ and the B factor of vibrations parallel to the basal plane by $[{B_{{\rm{basal}}}} = 8{\pi ^2}\langle u_{{\rm{basal}}}^2(T)\rangle]$ , cf. (4). Thus, relating to the notation of Sections 2 and 3, $[{B_1} =: {B_{{\rm{axial}}}}]$ , $[{B_2} = {B_3} =: {B_{{\rm{basal}}}}]$ , and analogous identifications are made for other quantities such as basal and axial Debye temperatures, $[{\theta _1} =: {\theta _{{\rm{axial}}}}]$ and $[{\theta _2} = {\theta _3} =: {\theta _{{\rm{basal}}}}]$ . We use X-ray and Mössbauer γ-ray diffraction data of the $[{B_{{\rm{axial}}}}]$ and $[{B_{{\rm{basal}}}}]$ factors (Skelton & Katz, 1968; Albanese et al., 1976; Pathak & Desai, 1981) and perform least-squares fits with the fit function $[{B_j}(T) = 8{\pi ^2}\langle u_j^2(T)\rangle]$ , where j can take the values `axial' and `basal', and

$[\langle u_j^2(T)\rangle = {3 \over m}{{{\hbar ^2}T} \over {{k_{\rm{B}}}\theta _{{\rm{X}},j}^2}}\left[{D_1}(\theta _{{\rm{X}},j}/T) + {1 \over 4}{{\theta _{{\rm{X}},j}} \over T}\right] + {m_{2,j}}{T^2} + {m_{3,j}}{T^3}. \eqno(13)]$

A similar fit function was used by Shepard et al. (1998) for the B factors of alkali halides; see also Martin & O'Connor (1978b). In equation (13), m = 65.38 u is the atomic mass of zinc, and the Debye function D₁(d) is defined in (4). The fitting parameters $[\theta _{{\rm{X}},j}]$ , m_2,j and m_3,j are recorded in Table 3, the subscript j labeling axial and basal components as defined above. Debye temperatures inferred from X-ray (or γ-ray) diffraction are marked with a subscript X. The regressed B factors $[{B_{{\rm{axial}}}}]$ and $[{B_{{\rm{basal}}}}]$ are depicted in Fig. 2. We also note $[3\hbar^2/({\rm u}k_{\rm B})]$ = 145.526 Å²K, to be substituted in (13), which gives $[\langle u_j^2(T)\rangle]$ (Å²) in terms of m (u), $[{\theta _{{\rm{X}},j}}]$ (K), T (K), m_2,j (Å² K⁻²) and m_3,j (Å² K⁻³).

Table 3
Fitting parameters of the Debye–Waller factors B_j(T) of zinc in Fig. 2

The index j takes the values `axial' and 'basal,' referring to normal vibrations along the hexagonal axis and in the basal plane, cf. Section 4.2. The least-squares fits are performed with B_j(T) in (13), depending on the listed parameters $[{\theta _{{\rm{X,}}j}}]$ , m_2,j and m_3,j. Also recorded are the zero-temperature limits of $[{B_{{\rm{axial}}}}]$ and $[{B_{{\rm{basal}}}}]$ . The goodness-of-fit parameters $[{\chi ^2}/{\rm{d.o.f.}}]$ , SE and R² are defined as in the caption of Table 1, with obvious replacements.

j	$[{\theta _{{\rm{X,}}j}}]$ (K)	m_2,j (Å² K⁻²)	m_3,j (Å² K⁻³)	B_j(T = 0) (Å²)	$[{\chi ^2}]$	d.o.f.	SE	R²
Axial	157.44	−7.1973 × 10⁻⁸	1.8786 × 10⁻¹⁰	0.2791	0.0321	35 − 3	0.110	0.995
Basal	262.05	1.3034 × 10⁻⁸	0	0.1677	0.00301	21 − 2	0.0142	0.9994

Figure 2
Debye–Waller B factors of zinc. Data points from Skelton & Katz (1968

[Skelton, E. F. & Katz, J. L. (1968). Phys. Rev. 171, 801-808.]

) (squares) and Pathak & Desai (1981

[Pathak, P. D. & Desai, R. J. (1981). Phys. Status Solidi A, 64, 741-745.]

) (circles) were obtained by X-ray diffraction. Mössbauer γ-ray diffraction data (diamonds) are from Albanese et al. (1976

[Albanese, G., Deriu, A. & Ghezzi, C. (1976). Acta Cryst. A32, 904-909.]

). The B factors $[{B_{{\rm{axial}}}}(T)]$ (red solid curve) and $[{B_{{\rm{basal}}}}(T)]$ (blue solid curve) are defined by the mean-squared atomic displacements $[\langle u_{{\rm{axial}}}^2(T)\rangle]$ (parallel to the hexagonal axis) and $[\langle u_{{\rm{basal}}}^2(T)\rangle]$ in the basal plane, cf. Section 4.2

. The black dotted curve depicts the averaged factor $[({B_{{\rm{axial}}}} + 2{B_{{\rm{basal}}}})/3]$ . The least-squares regression is performed with the analytic fit function B_j(T) in (13)

; the fitting parameters of the axial and basal B factors are listed in Table 3

. The red and blue dashed curves show the Debye approximations, obtained by dropping the quadratic and cubic terms in (13)

. The regressed $[{B_{{\rm{axial}}}}(T)]$ , $[{B_{{\rm{basal}}}}(T)]$ and the varying Debye temperatures in Fig. 5

determine the effective oscillator masses $[{m_{{\rm{eff, axial}}}}(T)]$ and $[{m_{{\rm{eff, basal}}}}(T)]$ of normal vibrations orthogonal and parallel to the basal plane, cf. (19)

and Fig. 8

. The zero-point internal energy (see Fig. 4

) is inferred from the zero-temperature limits of the B factors, cf. (14)

and Table 3

. Residuals of the least-squares fits are shown in the lower panels.

If the quadratic and cubic terms in (13) are dropped, we obtain the B factors of the Debye approximation [with constant $[{\theta _j} = \theta _{{\rm{X}},j}]$ and $[{m_{{\rm{eff}},j}} = m]$ in (4)], which are also depicted in Fig. 2. The low-temperature limits of the mean-squared vibrations $[\langle u_j^2(T)\rangle]$ in (4) (with constant mass and Debye temperature) are constant, and the high-temperature limits are linear in temperature since $[{D_1}(x \to 0)\sim 1]$ . The B_j(T) in (13) (with $[j = {\rm{axial}},{\rm{basal}}]$ ) cover the full temperature range up to the melting point; the zero-temperature limits B_j(0) are listed in Table 3.

In Fig. 3, we plot the Index functions $[{\rm Index}[{B_j}(T)]]$ = $[T{B'_j}(T)/{B_j}(T)]$ of the axial and basal B factors, together with the Index of their Debye approximations. The Index functions quantify the log–log slope of the B-factor curves in Fig. 2. That is, a tangent to the B-factor plots in Fig. 2 represents a power law $[ \propto {T^\alpha }]$ whose exponent $[\alpha]$ is the slope of the tangent line (in the log–log coordinates of Fig. 2) given by the Index function. Thus, if the temperature dependence of the B factor can be approximated in an interval by a power law, then the Index curve is nearly constant in this interval, with a value close to the power-law exponent.

Figure 3
Index functions of the B factors of zinc. Depicted are the log–log slopes $[{\rm{Index[}}{B_j}(T)] = TB'_j(T)/{B_j}(T)]$ of the B_j(T) factors ( $[j = {\rm{axial}},{\rm{basal}}]$ ) in Fig. 2

. The red/blue solid curves show the slope of the axial/basal B factors at the respective temperature, and the red/blue dashed curves are the slopes of their Debye approximations, cf. the end of Section 4.2

. The black dotted curve depicts the Index of the averaged B factor in Fig. 2

. Intervals in which the Index curves are constant imply power-law scaling with the respective Index as exponent. Below 10 K, the B factors are constant. The depicted temperature range is cut off at the melting point (in all figures). Close to the melting point at 692.73 K, the Debye approximations scale linearly with temperature, whereas the slopes of the empirical B factors increase with temperature, reaching 2 (axial) and 1.2 (basal) at the melting point. The inclusion of a cubic term in the axial B factor (13)

substantially improves the $[{\chi ^2}]$ of the fit (in contrast to the basal B factor, cf. Table 3

); the slope below the melting point does not exceed 2 despite the cubic term.

4.3. Zero-point internal energy of the zinc structure

The zero-point energy of the lattice vibrations is calculated from the zero-temperature limits of the B factors via (10),

$[{U_0}\,({\rm J\ mol}^{-1}) = 3.27662 \times {10^{11}}{{{c_{V0}}} \over {{m^4}}}{{1/{B_{{\rm{axial}}}}(0) + 2/{B_{{\rm{basal}}}}(0)} \over {B_{{\rm{axial}}}^3(0) + 2B_{{\rm{basal}}}^3(0)}}. \eqno(14)]$

The B factors at T = 0 are listed in Table 3. As for units, c_V0 [J (K⁴ mol)⁻¹], B_j (Å²), m (u), and we used $[{\hbar ^2}/({\rm{u}}{k_{\rm{B}}})]$ = 48.50875 Å²K to obtain the numerical factor in (14). The constant c_V0 is the amplitude of the low-temperature heat capacity, cf. the beginning of Section 3, to be identified with the fitting parameter b₀ in (11) and Table 1. The zero-point energy of zinc consistent with the measured B factors is thus U₀ = 556.3 J mol⁻¹.

The zero-point energy of the Debye approximation, cf. after (12), is by a factor $[{U_{0,{\rm{D}}}}/{U_0} = 5.29]$ larger than U₀ obtained from B-factor measurements. While the zero-point lattice energy is not accessible by caloric measurement of energy differences, the thermal energy $[{U_{{\rm{therm}}}}(T) = U(T) - {U_0}]$ can be obtained empirically by integrating the measured heat capacity, cf. Section 2.2, and a plot thereof is depicted in Fig. 4. The maximal relative deviation of $[{U_{{\rm{therm}}}}(T)]$ from the Debye approximation $[{U_{{\rm{therm,D}}}}(T) = {U_{\rm{D}}}(T) - {U_{0,{\rm{D}}}}]$ happens at 23.45 K and is substantial, $[{U_{{\rm{therm}}}}/{U_{{\rm{therm,D}}}} = 3.46]$ at this temperature (see also the captions to Figs. 1 and 4 and Section 4.6). In view of this, the large $[{U_{0,{\rm{D}}}}/{U_0}]$ ratio is not surprising. The zero-point internal energy stems from the second term in (1) and (2) and is finite because of the finite spectral cutoff. The zero-temperature limit of the spectral cutoff $[\Lambda (T)]$ is smaller than the constant spectral cutoff $[{\Lambda _{\rm{D}}}]$ of the Debye theory (cf. Section 4.1 and Fig. 6 in Section 4.5); therefore U₀ is smaller than the Debye zero-point energy [cf. the spectral representation (2) of U(T)].

Figure 4
Internal energy $[U(T) = \int_0^T {{C_V}\,{\rm{d}}T + {U_0}}]$ and entropy S(T) = $[\int_0^T {({C_V}/T)\,{\rm{d}}T}]$ of the zinc structure. The regressed lattice heat capacity C_V(T) in the integrands is stated in (11)

, with fitting parameters in Table 1

, see Fig. 1

. The lattice energy U(T), including the zero-point energy U₀ = 556.3 J mol⁻¹ [extracted from the zero-temperature limits of the measured Debye–Waller B factors, cf. (14)

and Table 3

], is depicted as a red solid curve, its thermal component $[{U_{{\rm{therm}}}}(T) = \int_0^T {{C_V}\,{\rm{d}}T}]$ as a green solid curve, and the lattice entropy as a blue solid curve. At high temperature, $[U(T) \propto T]$ , and the entropy diverges logarithmically. The low-temperature slopes are $[{U_{{\rm{therm}}}}(T) \propto {T^4}]$ and $[S(T) \propto {T^3}]$ , cf. Section 3

. The black dashed curves show the Debye approximations $[{U_{\rm{D}}}(T)]$ , $[{U_{{\rm{therm,D}}}}(T)]$ and $[{S_{\rm{D}}}(T)]$ , based on the Debye heat capacity $[{C_{V,{\rm{D}}}}]$ in (12)

(with $[{\theta _{\rm{D}}}]$ = 314.79 K). The Debye zero-point energy of zinc, $[{U_{0,{\rm{D}}}}]$ = 2944.5 J mol⁻¹, see after (12)

, is by a factor $[{U_{0,{\rm{D}}}}/{U_0} = 5.29]$ larger than the zero-point energy consistent with the measured B factors. The phonon peak in the second logarithmic temperature decade in Fig. 1

is also clearly visible in $[{U_{{\rm{therm}}}}(T)]$ and S(T). The largest relative deviation $[({U_{{\rm{therm}}}} - {U_{{\rm{therm,D}}}})/{U_{{\rm{therm}}}}]$ of the Debye approximation from the empirical thermal energy happens at 23.45 K, where $[{U_{{\rm{therm}}}}/{U_{{\rm{therm,D}}}} = 3.46]$ and $[{U_{{\rm{therm}}}}]$ = 16.3 J mol⁻¹. For entropy, the maximum of $[(S - {S_{\rm{D}}})/S]$ occurs at 24.82 K, where $[S/{S_{\rm{D}}} = 3.35]$ and S = 1.06 J (K mol)⁻¹. The differences $[{U_{{\rm{therm}}}} - {U_{{\rm{therm,D}}}}]$ and $[S - {S_{\rm{D}}}]$ at selected temperatures are listed in Table 2

4.4. Varying axial and basal Debye temperatures

The directional Debye temperatures $[{\theta _j}(T) = {\hat c_j}\theta (T)]$ (labeled j = 1,2,3 in Section 2.2) of the zinc structure are denoted by $[{\theta _1} = {\theta _{{\rm{axial}}}}(T)]$ and $[{\theta _2} = {\theta _3} = {\theta _{{\rm{basal}}}}(T)]$ as in Section 4.2. The constants $[{\hat c_j}]$ can be extracted from the zero-temperature limits of the B factors, cf. (9): $[{\hat c_1} = {\hat c_{{\rm{axial}}}} = 1]$ and $[{\hat c_2} = {\hat c_3}]$ = $[{\hat c_{{\rm{basal}}}}]$ = $[{B_{{\rm{axial}}}}(0)/{B_{{\rm{basal}}}}(0)]$ = 1.664 (see Table 3).

Thus, $[{\theta _{{\rm{axial}}}}(T) = \theta (T)]$ and $[{\theta _{{\rm{basal}}}}(T) = {\hat c_{{\rm{basal}}}}\theta (T)]$ , where $[\theta (T)]$ is found by solving (5). The Δ function (5) of zinc reads [cf. (23)]

$[\eqalignno{\Delta (d) &=1- (2\log [1 - \exp( - d)] + d&\cr &\quad + 2\{2\log [1 - \exp( - {\hat c}_{{\rm{basal}}}d)] + {\hat c}_{{\rm{basal}}}d\})&\cr &\quad /\{ 8D(d) + d + 2[8D({{\hat c}_{{\rm{basal}}}}d) + {{\hat c}_{{\rm{basal}}}}d]\}, &(15)}]$

with $[d = \theta (T)/T]$ and $[{\hat c_{{\rm{basal}}}}]$ stated above. $[\theta (T)]$ is calculated as indicated in Section 2.2, $[\theta (T) = T{\Delta ^{ - 1}}[{{3 \over 4}}TS(T)/U(T)]]$ . The inversion $[{\Delta ^{ - 1}}]$ of (15) is sketched in the Appendix, cf. after (30). The lattice internal energy $[U(T) = \int_0^T {{C_V}\,{\rm{d}}T} + {U_0}]$ and entropy $[S(T) = \int_0^T {{C_V}/T\,{\rm{d}}T}]$ are empirical input functions, inferred from the regressed lattice heat capacity C_V(T) [cf. (11) and Table 1] and depicted in Fig. 4. The zero-point internal energy of zinc is U₀ = 556.3 J mol⁻¹ (cf. Section 4.3). The Debye temperatures $[{\theta _{{\rm{axial}}}}(T) = \theta (T)]$ and $[{\theta _{{\rm{basal}}}}(T)]$ = $[{\hat c_{{\rm{basal}}}}\theta (T)]$ are plotted in Fig. 5; their zero-temperature limits coincide with the fitting parameters $[\theta _{{\rm{X}},j}]$ of the B factors in Table 3, $[{\theta _{{\rm{axial}}}}(0) = {\theta _{{\rm{X}},{\rm{axial}}}}]$ , $[{\theta _{{\rm{basal}}}}(0) = {\theta _{{\rm{X}},{\rm{basal}}}}]$ , according to (8) and (13) [where $[{D_1}(x \to \infty)\sim {\pi ^2}/(6x)]$ ].

Figure 5
Varying Debye temperatures $[{\theta _{{\rm{axial}}}}(T)]$ and $[{\theta _{{\rm{basal}}}}(T)]$ of zinc, referring to normal vibrations along the hexagonal axis and in the basal plane. $[{\theta _{{\rm{axial}}}}]$ and $[{\theta _{{\rm{basal}}}}]$ are calculated from the temperature variation of the internal energy and entropy depicted in Fig. 4

and from the zero-temperature limits of the Debye–Waller B factors in Fig. 2

and Table 3

, cf. Section 4.4

. The Debye temperatures converge to constant limit values at low and high temperature, cf. Section 3

, but the constant high-temperature limits are not attained within the depicted temperature range of the solid phase. The horizontal dashed straight line indicates the constant Debye temperature $[{\theta _{\rm{D}}}]$ = 314.79 K, cf. after (12)

, used in the Debye approximations in Figs. 1

and 4

Constant axial and basal Debye temperatures for zinc were already introduced by Grüneisen & Goens (1924 ). The plots of the temperature-dependent caloric and X-ray Debye temperatures of zinc reported by Barron & Munn (1967b), Martin (1968) and Skelton & Katz (1968) (which largely differ from one another) are not comparable with the ones in Fig. 5, since they were derived with a constant (temperature-independent) spectral cutoff. Also, these authors do not use directional Debye temperatures, but calculate one temperature-dependent Debye temperature $[\theta _{\rm{D}}^{}(T)]$ either by comparing the Debye heat capacity (12) with the empirical heat capacity curve, or by comparing the averaged experimental B factor $[({B_{{\rm{axial}}}} + 2{B_{{\rm{basal}}}})/3]$ with the isotropic Debye B factor obtained by dropping the quadratic and cubic terms in (13) and replacing $[\theta _{{\rm{X}},j}]$ by $[\theta _{\rm{D}}(T)]$ . As mentioned, the $[\theta _{\rm{D}}(T)]$ curves obtained by these two methods substantially differ, cf. Skelton & Katz (1968), and a temperature-dependent Debye temperature without simultaneous variation of the spectral cutoff also violates a basic equilibrium condition (cf. Sections 1 and 2.2).

4.5. Temperature variation of the spectral cutoff

In contrast to the Debye temperatures $[{\theta _{{\rm{axial}}}}(T)]$ and $[{\theta _{{\rm{basal}}}}(T)]$ , the cutoff factor $[\Lambda (T)]$ of the internal energy (1) and entropy (3) is the same for normal vibrations orthogonal and parallel to the hexagonal axis. $[\Lambda (T)]$ is calculated as explained in Section 2.2,

$[\eqalignno{&\Lambda (T) = &\cr &\left\{{{(2\pi)^3} \over {4\pi }}{1 \over {RT}}{{n_{{\rm{at}}}U(T)} \over {D(d) + d/8 + 2[D({\hat c}_{{\rm{basal}}}d) + {\hat c}_{{\rm{basal}}}d/8]}}\right\}^{1/3}, &\cr &&(16)}]$

with U(T) [see after (15)] and $[d = \theta (T)/T]$ and $[{\hat c_{{\rm{basal}}}}]$ in Section 4.4. The rescaled dimensionless cutoff factor $[h(T): = \Lambda (T)/n_{{\rm{at}}}^{1/3}]$ is plotted in Fig. 6, where $[n_{{\rm{at}}} ]$ = 6.5702 × 10²² cm⁻³ is the atomic density of zinc. h(T) increases moderately with temperature, from 2.49 at zero to 3.77 at the melting point (692.7 K). Like the Debye temperatures, the cutoff depends on the zero-temperature limits of the B factors, which enter via the zero-point energy (14) and $[{\hat c_{{\rm{basal}}}}]$ . The temperature dependence of $[\Lambda (T)]$ compensates for the variation of $[\theta (T)]$ , so that the equilibrium condition $[\partial S/\partial U = 1/T]$ is preserved (Tomaschitz, 2020a).

Figure 6
Spectral cutoff in the thermodynamic variables of zinc. The spectral cutoff $[\Lambda (T) = n_{{\rm{at}}}^{1/3}h(T)]$ , $[n_{{\rm{at}}}^{1/3}]$ = 0.4035 Å⁻¹, of the internal energy, cf. (1)

and (2)

, depends on the atomic density of zinc, $[n_{{\rm{at}}}]$ = 6.5702 × 10²² cm⁻³, and a dimensionless cutoff factor h(T) (red solid curve), cf. Section 4.5

. In contrast to the axial and basal Debye temperatures of zinc depicted in Fig. 5

, the spectral cutoff is the same for normal vibrations orthogonal and parallel to the basal plane. h(T) is calculated from the lattice internal energy U(T) of the zinc structure, cf. Fig. 4

, and the Debye temperatures $[{\theta _{{\rm{axial}}}}(T)]$ and $[{\theta _{{\rm{basal}}}}(T)]$ by way of (16)

. Both the low- and high-temperature limits of h(T) are constant. At high temperature, h(T) converges to the constant cutoff factor of the Debye approximation, $[{h_{\rm{D}}} = {(6{\pi ^2})^{1/3}}]$ , cf. after (12)

, indicated here as a black dashed straight line. (This convergence is not visible in the temperature range shown in this figure, as it happens above the melting point.) In the low-temperature regime, h(T) depends, like $[{\theta _{{\rm{axial}}}}]$ and $[{\theta _{{\rm{basal}}}}]$ , on the zero-point internal energy U₀, cf. Fig. 4

4.6. Internal energy and entropy components

The molar internal energy of the zinc structure can be decomposed into partial energies of normal vibrations orthogonal and parallel to the basal plane, U(T) = $[{U_{{\rm{axial}}}} + 2{U_{{\rm{basal}}}}]$ , cf. (1), where

$[{U_{{\rm{axial}}}}(T) = {{4\pi } \over {{{(2\pi)}^3}}}R{{{\Lambda ^3}(T)} \over {n_{{\rm{at}}}}}T\left\{D[{\theta _{{\rm{axial}}}}(T)/T] + {1 \over 8}{{{\theta _{{\rm{axial}}}}(T)} \over T}\right\}, \eqno(17)]$

and analogously for $[{U_{{\rm{basal}}}}(T)]$ . The partial energies $[{U_{{\rm{axial}}}}]$ and $[{U_{{\rm{basal}}}}]$ depend on the spectral cutoff $[\Lambda (T)]$ and the Debye temperatures $[{\theta _{{\rm{axial}}}}(T)]$ and $[{\theta _{{\rm{basal}}}}(T)]$ , respectively, calculated in Sections 4.4 and 4.5. An analogous decomposition holds for the lattice entropy, $[S(T) = {S_{{\rm{axial}}}} + 2{S_{{\rm{basal}}}}]$ , cf. (3), with axial entropy component

$[\eqalignno{{S_{{\rm{axial}}}}(T)& = {{4\pi } \over {{{(2\pi)}^3}}}R{{{\Lambda ^3}(T)} \over {n_{{\rm{at}}}}}&\cr &\times\left({4 \over 3}D[{\theta _{{\rm{axial}}}}(T)/T] - {1 \over 3}\log \{1 - \exp[ - \theta _{{\rm{axial}}}(T)/T]\}\right), &\cr &&(18)}]$

and analogously for the basal component $[{S_{{\rm{basal}}}}(T)]$ . U(T) and S(T) coincide with the empirical functions obtained from integrations of the regressed heat capacity [see after (15)] and from the zero-temperature limits of the B factors determining the zero-point energy U₀, cf. Section 4.3. The temperature evolution of U(T), its thermal component $[{U_{{\rm{therm}}}} = U(T) - {U_0}]$ and the entropy S(T) are plotted in Fig. 4, together with the respective Debye approximations $[{U_{\rm{D}}}(T)]$ , $[{U_{{\rm{therm,D}}}}]$ = $[{U_{\rm{D}}}(T) - {U_{0,{\rm{D}}}}]$ and $[{S_{\rm{D}}}(T)]$ . The latter are calculated by integrating the Debye heat capacity $[{C_{V,{\rm{D}}}}(T)]$ in (12) with constant Debye temperature $[{\theta _{\rm{D}}}]$ = 314.79 K and spectral cutoff $[{\Lambda _{\rm{D}}} = {(6{\pi ^2}n_{{\rm{at}}})^{1/3}}]$ , cf. Section 2.2. The differences $[{U_{{\rm{therm}}}}(T) - {U_{{\rm{therm,D}}}}(T)]$ and $[S(T) - {S_{\rm{D}}}(T)]$ are recorded in Table 2, at selected temperatures, and the maximal relative deviation of the Debye theory from the empirical $[{U_{{\rm{therm}}}}(T)]$ and S(T) is indicated in the caption of Fig. 4 (and is clearly visible in this figure).

4.7. Effective phonon speed and dispersion relations of axial and basal normal vibrations

The axial and basal components of the effective phonon speed are $[{c_{{\rm{eff, axial}}}} = {c_{{\rm{eff}}}}(T)]$ and $[{c_{{\rm{eff, basal}}}} = {\hat c}_{{\rm{basal}}}{c_{{\rm{eff}}}}(T)]$ (orthogonal and parallel to the basal plane, respectively), where $[{c_{{\rm{eff}}}}(T) = {k_{\rm{B}}}\theta (T)/[\hbar \Lambda (T)]]$ , cf. Section 2.2. The Debye temperature $[\theta (T)]$ and the zero-temperature ratio $[{\hat c_{{\rm{basal}}}} = {B_{{\rm{axial}}}}(0)/{B_{{\rm{basal}}}}(0) = 1.664]$ were obtained in Section 4.4 and the spectral cutoff $[\Lambda (T)]$ in Section 4.5. The atomic density of zinc is $[n_{{\rm{at}}}]$ = 6.5702 × 10²² cm⁻³ (based on a mass density of 7.133 g cm⁻³ and a molar mass of 65.38 g mol⁻¹), so that $[c_{{\rm{eff}}}(T)]$ (cm s⁻¹) = 3244.5 θ(T) (K)/h(T), with dimensionless cutoff factor $[h(T) = \Lambda (T)/n_{{\rm{at}}}^{1/3}]$ , cf. Section 4.5. The temperature-dependent axial and basal dispersion relations are $[\omega = c_{{\rm{eff, axial}}}(T)k]$ and $[\omega = c_{{\rm{eff, basal}}}(T)k]$ , cf. after (2). Instead of anharmonic perturbations of the oscillator potential, we use a temperature-dependent spectral cutoff and varying Debye temperatures in the spectral representation (2), so that the dispersion relations stay linear. The effective phonon velocities $[{c_{{\rm{eff, axial}}}}(T)]$ and $[{c_{{\rm{eff, basal}}}}(T)]$ are plotted in Fig. 7.

Figure 7
Effective phonon speed of zinc defining the temperature-dependent dispersion relations of axial and basal normal vibrations, cf. Section 4.7

. $[{c_{{\rm{eff, axial}}}}(T)]$ refers to vibrations along the hexagonal axis and $[{c_{{\rm{eff, basal}}}}(T)]$ to vibrations in the basal plane. The constant effective phonon speed of the Debye theory, $[c_{{\rm{eff,D}}} ]$ = 2.62 km s⁻¹, cf. after (12)

, is indicated by the black dashed horizontal line. Since $[c_{{\rm{eff, axial}}}(T) \propto {\theta _{{\rm{axial}}}}(T)/\Lambda (T)]$ and analogously for $[c_{{\rm{eff, basal}}}]$ , cf. Section 4.7

, the phonon speed converges to finite limit values at low and high temperature; the constant high-temperature limits are not attained within the depicted temperature range below the melting point. For comparison, measured transversal/longitudinal sound velocities in zinc vary between 2.0 and 4.2 km s⁻¹ (Ledbetter, 1977

[Ledbetter, H. M. (1977). J. Phys. Chem. Ref. Data, 6, 1181-1203.]

4.8. Temperature variation of the effective oscillator mass of zinc

We substitute the analytic fits (13) of the mean-squared atomic displacement $[\langle u_j^2(T)\rangle]$ into (4) and solve for $[m_{{\rm eff },j}(T)]$ , with index j taking the values `axial' and `basal'. The oscillator mass of normal vibrations along the hexagonal axis is found as

$[\eqalignno{{m_{{\rm{eff, axial}}}}(T)({\rm{u}})& = {{145.526} \over {\langle u_{{\rm{axial}}}^2(T)\rangle (\AA^2)}}{T \over {\theta _{{\rm{axial}}}^2(T)}}&\cr &\quad\times\left\{{D_1}[{\theta _{{\rm{axial}}}}(T)/T] + {1 \over 4}{{{\theta _{{\rm{axial}}}}(T)} \over T}\right\}, &(19)}]$

and $[{m_{{\rm{eff, basal}}}}(T)]$ , the oscillator mass of basal oscillations, reads analogously. $[\langle u_{{\rm{axial}}}^2(T)\rangle]$ and $[\langle u_{{\rm{basal}}}^2(T)\rangle]$ denote the least-squares fits (13) to the empirical mean-squared vibrations, and $[3\hbar^2/({\rm u}k_{\rm B})]$ = 145.526 Å²K in (4). $[{\theta _{{\rm{axial}}}}(T)]$ and $[{\theta _{{\rm{basal}}}}(T)]$ are the Debye temperatures calculated in Section 4.4. At zero temperature, the effective oscillator mass coincides with the atomic mass, $[{m_{{\rm{eff, axial}}}}(0) = {m_{{\rm{eff, basal}}}}(0) = m]$ , cf. Section 3. The temperature variation of $[{m_{{\rm{eff, axial}}}}(T)]$ and $[{m_{{\rm{eff, basal}}}}(T)]$ is depicted in Fig. 8.

Figure 8
Effective oscillator mass of zinc. The curves $[{m_{{\rm{eff, axial}}}}(T)]$ and $[{m_{{\rm{eff, basal}}}}(T)]$ show the temperature-dependent effective oscillator mass of normal vibrations parallel and orthogonal to the hexagonal axis, respectively, cf. Section 4.8

. The mass functions $[{m_{{\rm{eff, axial}}}}(T)]$ , $[{m_{{\rm{eff, basal}}}}(T)]$ are calculated from the regressed B factors $[{B_{{\rm{axial}}}}(T)]$ , $[{B_{{\rm{basal}}}}(T)]$ in Fig. 2

and Debye temperatures $[{\theta _{{\rm{axial}}}}(T)]$ , $[{\theta _{{\rm{basal}}}}(T)]$ in Fig. 5

, by way of (19)

. The horizontal dashed line indicates the atomic mass of zinc, which is the zero-temperature limit of $[{m_{{\rm{eff, axial}}}}(T)]$ and $[{m_{{\rm{eff, basal}}}}(T)]$ . The temperature range is cut off at the melting point.

4.9. Second-order B factors of zinc

Non-Gaussian contributions to the Debye–Waller (diffraction intensity attenuation) factors $[\exp( - 2{M_j})]$ are defined by higher-order terms in the ascending series expansion $[{M_j} = [{Q^2}/{(4\pi)^2}]({B_j} + {B_{2,j}}{Q^2} + \ldots)]$ , where Q² is the squared diffraction vector (Wolfe & Goodman, 1969 ; Day et al., 1995 , 1996 ; Shepard et al., 1998, 2000 ; Wang et al., 2017 ), and the subscript j labels the vibrational normal components. In the case of zinc, j takes the values `axial' and 'basal,' cf. Section 4.2, and the diffraction vector is directed accordingly, so that we do not need to take cross-correlations into account. The leading order of M_j is defined by the B factor $[{B_j}(T) = 8{\pi ^2}\langle u_j^2(T)\rangle]$ , cf. (4) and Section 4.2. The second-order coefficient B_2,j(T) depends on the vibrational amplitude averages $[\langle u_j^2(T)\rangle]$ and $[\langle u_j^4(T)\rangle]$ via the cumulant $[{\kappa _j} = 3\langle u_j^2(T)\rangle ^2 - \langle u_j^4(T)\rangle]$ , $[{B_{2,j}} = 2{\pi ^2}{\kappa _j}(T)/3]$ . This cumulant can be expressed in terms of the Debye function D₁ (Tomaschitz, 2021a),

$[\eqalignno{{B_{2,j}}(T)(\AA^4)& = {{2{\pi ^2}} \over 3}\left[{{145.526} \over {{m_{{\rm{eff}},j}}(T)}}{T \over {\theta _j^2(T)}}\right]^2&\cr &\quad\times \left[D_1^2({d_j}) + {1 \over 2}{d_j}D_1({d_j}) + {{d_j^2} \over 8}\right], &(20)}]$

where we use the shortcut $[{d_j}(T) = {\theta _j}(T)/T]$ and $[3\hbar^2/({\rm u}k_{\rm B})]$ = 145.526 Å²K. $[{m_{{\rm{eff}},j}}(T)]$ is the effective mass (19) in atomic units, the Debye function D₁ is defined in (4), and $[j = {\rm{axial}},{\rm{basal}}]$ . The varying Debye temperatures $[{\theta _j}(T)]$ of zinc are calculated in Section 4.4. The Debye approximation of B_2,j(T) in (20) is obtained by replacing $[{m_{{\rm{eff}},j}}(T)]$ by the atomic mass and $[{\theta _j}(T)]$ by the constant directional Debye temperatures $[\theta _{{\rm{X}},j}]$ in Table 3; see also Section 4.2, where the Debye approximation of the B_j(T) factors is obtained in the same way.

The second-order B factors (20) are related to the B_j(T) factors (4) by

$[{B_{2,j}}(T) = {1 \over {96{\pi ^2}}}B_j^2(T)\left[1 + {{d_j^2} \over {16D_1^2({d_j}) + 8{d_j}{D_1}({d_j}) + d_j^2}}\right], \eqno(21)]$

which is obtained by substituting the effective mass (19) into (20). The temperature evolution of the second-order factors $[{B_{2,{\rm{axial}}}}(T)]$ and $[{B_{2,{\rm{basal}}}}(T)]$ of zinc is depicted in Fig. 9, together with their Debye approximations. The Index functions $[{\rm Index}[{B_{2,j}}(T)] = TB'_{2,j}(T)/{B_{2,j}}(T)]$ [cf. the end of Section 4.2 and equation (31) in the Appendix] of the second-order factors are plotted in Fig. 10, representing the log–log slope of the B_2,j(T) curves in Fig. 9.

Figure 9
Second-order B factors of zinc. Depicted are the axial (red solid curve) and basal (blue solid curve) second-order B factors quantifying the non-Gaussian component of the Debye–Waller damping factors, cf. Section 4.9

. The solid curves are plots of B_2,j(T) in (21)

. The averaged (isotropized) second-order B factor $[({B_{{\rm{2, axial}}}} + 2{B_{{\rm{2,basal}}}})/3]$ is indicated by the black dotted curve. The dashed curves are the Debye approximations of $[{B_{2,{\rm{axial}}}}]$ and $[{B_{2,{\rm{basal}}}}]$ , cf. after (20)

, which admit quadratic power-law scaling close to the melting point. See also Fig. 10

depicting the temperature dependence of the log–log slope of these factors.

Figure 10
Index functions of the second-order B factors of zinc. The log–log slope of the axial and basal second-order B factors in Fig. 9

is here depicted by the Index curves $[{\rm{Index}}[{B_{2,j}}(T)] = TB'_{2,j}(T)/{B_{2,j}}(T)]$ , cf. Section 4.9

. The dashed curves show the slope of the Debye approximations in Fig. 9

, and the black dotted curve depicts the Index function of the averaged second-order B factor in Fig. 9

. At high temperature, the Indices are by a factor of two larger than the slopes of the Gaussian B_j(T) factors in Fig. 3

, because of the quadratic B_j(T) dependence of the B_2,j(T) factors, cf. (21)

and (31)

. [The second temperature-dependent factor in (21)

varies only moderately and decreases monotonously, being close to 2 at low temperature and close to 1 at high temperature.] Above 200 K, the slope of the axial Debye approximation (red dashed curve) is nearly constant and approximately 2, whereas the slope of $[{B_{2,{\rm{ axial}}}}(T)]$ (red solid curve) varies between 2 and 4 in the 200–693 K interval.

5. Conclusion

We outlined a practical method to describe anisotropic lattice vibrations in elemental crystals, focusing on a specific example, the hexagonal close-packed zinc structure. The experimental lattice heat capacity, internal energy and equilibrium entropy as well as the temperature evolution of the measured B factors of zinc can self-consistently be modeled, from the low-temperature regime up to the melting point, by employing the varying Debye temperatures $[{\theta _{{\rm{axial}}}}(T)]$ , $[{\theta _{{\rm{basal}}}}(T)]$ (cf. Section 4.4), the temperature-dependent spectral cutoff $[\Lambda (T)]$ (cf. Section 4.5) and the effective oscillator masses $[{m_{{\rm{eff, axial}}}}(T)]$ , $[{m_{{\rm{eff, basal}}}}(T)]$ (cf. Section 4.8).

In Sections 2 and 3, we sketched the general formalism for anisotropic monatomic crystals. The atomic vibrations are partitioned into normal components described by a temperature-dependent spectral cutoff $[\Lambda (T)]$ and directional Debye temperatures $[{\theta _j}(T)]$ , the index j labeling three vibrational normal components. (In the case of zinc, the two components in the basal plane are identical, and normal vibrations are labeled as axial and basal accordingly, cf. Section 4.) In contrast to cubic monatomic crystals, the Debye temperatures of the normal components differ in anisotropic crystals. By the way, in compound crystals, the spectral cutoffs of the normal vibrations of different atomic species also differ in general (Tomaschitz, 2021a).

The varying spectral cutoff and Debye temperatures can be extracted from the experimental lattice heat capacity and low-temperature Debye–Waller B factors, as explained in Sections 2.2 and 3 and illustrated with zinc in Section 4. The effective phonon speed $[{c_{{\rm{eff}},j}}(T)]$ defining the dispersion relations of the normal vibrations is determined by the Debye temperatures $[{\theta _j}(T)]$ and spectral cutoff $[\Lambda (T)]$ , cf. Sections 2.2 and 4.7. The effective oscillator density $[{n_{{\rm{oscil}},j}}(T)]$ of the vibrational components is calculated from the spectral cutoff, cf. Section 2.1, and the effective oscillator mass $[{m_{{\rm{eff}},j}}(T)]$ from the respective Debye–Waller factor B_j(T), cf. Section 4.8.

We have chosen to model empirical heat capacity and B-factor data of anisotropic crystals by way of a temperature-dependent spectral cutoff, temperature-dependent directional Debye temperatures and effective oscillator masses. Apart from these temperature dependencies and the use of directional quantities in the case of anisotropy, there are no further changes made in the Debye theory. The harmonic oscillator potential, resulting in linear (but temperature-dependent) dispersion relations for the normal vibrations, and the uniform density of phonon states of the Debye theory are retained, which makes the described non-perturbative formalism possible.

Several other methods to account for the deviations of the Debye theory from empirical data are reviewed by Gopal (1966). Anharmonic perturbations of the oscillator potential and nonlinear phonon dispersion curves have been used by Merisalo & Larsen (1977, 1979), Field (1983), Malica & Dal Corso (2019), Ulian & Valdrè (2019 ). A discrete spectral component (Einstein terms) can be added to the Debye heat capacity at optical frequencies (cf. Meissner et al., 1978; Hoser & Madsen, 2016 , 2017 ; Sovago et al., 2020 ). A non-uniform phonon density of states was used by Tewari & Silotia (1990 ) for zinc and Malica & Dal Corso (2019) for several other monatomic crystals.

The phonon peaks occurring in the lattice heat capacity of anisotropic crystals in the crossover region between the low-temperature and classical regimes largely deviate from the Debye heat capacity as exemplified by zinc in Fig. 1 and graphite (Tomaschitz, 2020b). The onset of the phonon peak is indicated by an inflection point in the log–log slope of the heat capacity, cf. Fig. 1.

When modeling heat capacity data, the preference of experimentalists seems to be the Debye theory in combination with a temperature-dependent Debye temperature, cf. e.g. Goetsch et al. (2012 ), Li et al. (2017 ) and references therein. In this case, the spectral cutoff has also to vary with temperature in order to satisfy the equilibrium condition $[\partial S/\partial U = 1/T]$ , which would otherwise be violated by a varying Debye temperature (Tomaschitz, 2020a). This also holds true for anisotropic crystals: since the Debye temperature $[{\theta _j}(T)]$ of each vibrational normal component varies with temperature, the spectral cutoff $[\Lambda (T)]$ has to vary as well to ensure that the internal-energy derivative of entropy coincides with the inverse temperature. The above condition on the entropy derivative is preserved by relating $[{\theta _j}(T)]$ and $[\Lambda (T)]$ to the internal energy and entropy, cf. Section 2.2.

In contrast to the Debye theory, where the zero-point energy of the phonons is already determined by the constant Debye temperature, cf. Section 4.1, the zero-point energy U₀ emerges as a free constant in the temperature-dependent spectral cutoff $[\Lambda (T)]$ and Debye temperatures $[{\theta _j}(T)]$ , cf. Section 2.2. The Debye–Waller factors B_j(T) also depend on U₀ by way of the Debye temperatures $[{\theta _j}(T)]$ , especially in the low-temperature regime, cf. Section 3. Therefore, the zero-point energy U₀, which is calorically not measurable, can be inferred from the zero-temperature limits of the measured B factors, as explained in Section 3 for anisotropic crystals and specifically for zinc in Section 4.3. Finally, and again in contrast to the Debye theory, there is no need to distinguish between caloric and X-ray Debye temperatures, as the varying directional Debye temperatures $[{\theta _j}(T)]$ extracted from the empirical heat capacity are identical to those used in the modeling of the Debye–Waller B factors.

APPENDIX A

Invertibility of the Δ function, and logarithmic derivative of second-order B factors

A1. Invertibility of Δ(d)

We demonstrate that the function $[\Delta (d)]$ in (5),

$[\eqalignno{\Delta (d)& = {{\sum_j {\chi _j}\{4D(d_j) - \log [1 - \exp( - d_j)]\}} \over {\sum_j {{\chi _j}[4D({d_j}) + {d_j}/2]} }},&\cr D(d)&: = {1 \over {d^3}}\int\limits_0^d {{{{x^3}\,{\rm{d}}x} \over {{\exp(x)} - 1}}},&(22)}]$

with $[{d_j} = {\hat c_j}d]$ , is strictly monotonously decreasing on the positive real axis, for any choice of positive constants $[{\hat c_j}]$ and $[{\chi _j}]$ . Here, we consider a more general function than $[\Delta (d)]$ stated in (5) where $[{\chi _j} = 1]$ and j = 1,2,3. Equation (22) is the generalization of $[\Delta (d)]$ in (5) for compound crystals (Tomaschitz, 2021a), where the constants $[{\chi _j}]$ depend on the atomic composition and the summation in (22) is over $[j = 1,\ldots,3N]$ , with N atomic species in the formula unit. [ $[\Delta (d)]$ in (5) applies to elemental crystals.]

We will show that $[\Delta '(d) \,\lt\, 0]$ for $[d \,\gt\, 0]$ , which means that $[\Delta (d)]$ is invertible on the positive real axis, as required for the calculation of the Debye temperatures in Sections 2.2 and 4.4. Before differentiating, it is convenient to eliminate D(d_j) in the numerator of (22),

$[\Delta (d) = 1 - {{\sum_j {{\chi _j}\{2\log [1 - \exp( - d_j)] + {d_j}\}} } \over {\sum_j {{\chi _j}[8D({d_j}) + {d_j}]} }}. \eqno(23)]$

Using the derivative of the Debye function in (22),

$[D'(d) = - {3 \over d}D(d) + {1 \over {\exp(d) - 1}}, \eqno(24)]$

we find that $[\Delta '(d) \,\lt\, 0]$ is equivalent to $[\sum_{j,k} {{{\hat c}_j}{\chi _j}{\chi _k}{f_{jk}}} \,\gt\, 0]$ , with

$[\eqalignno{{f_{jk}}&: = \left[{2 \over {\exp({d_j}) - 1}} + 1\right][8D({d_k}) + {d_k}]&\cr &\quad+ \left[{{24} \over {{d_j}}}D({d_j}) - {8 \over {{\exp({d_j})} - 1}} - 1\right]&\cr &\quad\times \{2\log [1 - \exp( - d_k)] + {d_k}\},&(25)}]$

where $[{d_i} = {\hat c_i}d]$ . The coefficients f_jk are independent of the positive constants $[{\hat c_j}]$ and $[{\chi _j}]$ . It suffices to show that $[{f_{jk}} \,\gt\, 0]$ . To this end, we rearrange the f_jk,

$[\eqalignno{{f_{jk}}& = 2\left[1 + {2 \over {{\exp({d_j}}) - 1}}\right]\{4D({d_k}) - \log [1 - \exp( - d_k)]\}&\cr &\quad + {6 \over {{d_j}}}\left[4D({d_j}) - {{{d_j}} \over {\exp({d_j}) - 1}}\right]\{2\log [1 - \exp( - d_k)] + {d_k}\},&\cr &&(26)}]$

and also note two identities obtained by integration by parts,

$[D(d) - \log [1 - \exp( - d)] = - {3 \over {d^3}}\int\limits_0^d \log [1 - \exp( - x)]{x^2}\,{\rm{d}}x \,\gt\, 0, \eqno(27)]$

with Debye function D(d) in (22), and

$[4D(d) - {d \over {\exp(d) - 1}} = {1 \over {{d^3}}}\int\limits_0^d {{{x^4}\exp(x)\,{\rm{d}}x} \over {[\exp(x) - 1]^2}} \,\gt\, 0. \eqno(28)]$

Because of the positivity of the expressions (27) and (28), it is evident that the condition $[{f_{jk}} \,\gt\, 0]$ with f_jk in (26) is equivalent to the inequality $[\psi ({d_j}) + \varphi ({d_k}) \,\gt\, 0]$ , where

$[\eqalignno{\psi ({d_j})&: = {{{d_j}} \over 3}{{1 + 2/[\exp({d_j}) - 1]} \over {4D({d_j}) - {d_j}/[\exp({d_j}) - 1]}},&\cr \varphi ({d_k})&: = {{2\log [1 - \exp( - d_k)] + {d_k}} \over {4D({d_k}) - \log [1 - \exp( - d_k)]}}.&(29)}]$

We consider $[\psi ({d_j})]$ and $[\varphi ({d_k})]$ as two functions of independent variables d_j and d_k on the positive real axis. $[\psi ({d_j})]$ is positive for $[{d_j} \ge 0]$ and attains its minimum at d_j = 0, $[\psi (0) = 2]$ . By the way, the asymptotic limits of the Debye function in (22) are

$[D(d) = {1 \over 3} - {d \over 8} + {{{d^2}} \over {60}} - {\rm{O}}(d_{}^4), D(d) = {{{\pi ^4}} \over {15}}{1 \over {{d^3}}} + {\rm{O}}[\exp( - d)].\eqno(30)]$

The numerator of $[\varphi ({d_k})]$ in (29) has a zero at $[{\tilde d_k}]$ = $[- \log \{[3 - (5)^{1/2}]/2\} \simeq 0.96]$ . $[\varphi ({d_k})]$ is positive for $[{d_k} \,\gt\, {\tilde d_k}]$ . In the range $[0 \le {d_k} \,\lt\, {\tilde d_k}]$ , $[\varphi ({d_k})]$ is negative and attains its minimum at d_k = 0, $[\varphi (0) = - 2]$ . Both functions $[\psi ({d_j})]$ and $[\varphi ({d_k})]$ in (29) increase monotonously on the positive real axis. Thus the inequality $[\psi ({d_j}) + \varphi ({d_k}) \,\gt\, 0]$ holds true for independent variables d_j and d_k on the positive real axis, and equality is only attained at d_j = d_k = 0. Since $[\psi ({d_j}) + \varphi ({d_k}) \,\gt\, 0]$ is equivalent to $[{f_{jk}} \,\gt\, 0]$ , cf. (25) and (26), the derivative of $[\Delta (d)]$ in (22) and (23) is negative along the positive real axis, irrespective of the choice of positive constants $[{\hat c_j}]$ and $[{\chi _j}]$ , so that $[\Delta (d)]$ is invertible.

As for the practical inversion of $[\Delta (d)]$ in (5) [or (15) in the case of zinc], the asymptotic limits of $[\Delta (d)]$ [calculated by substituting (30)] can be inverted analytically. The inversion in the crossover region (typically stretching over two or three logarithmic decades) is done numerically. First, we determine the numerical values of the parameters $[{\hat c_j}]$ in (5) from the zero-temperature limits of the empirical Debye–Waller B factors, cf. (9) and Section 4.4, so that $[\Delta (d)]$ only depends on the variable d. In the crossover interval, we choose a descending sequence of closely spaced and logarithmically equidistributed points p_n and generate a table of function values $[[{p_n},\Delta ({p_n})]]$ . The inverse of $[\Delta (d)]$ in the crossover region is approximated by a cubic polynomial interpolation of the function values defined by the inverted table $[[\Delta ({p_n}),{p_n}]]$ .

A2. Logarithmic derivative of B_2,j(T)

The numerical evaluation of the Index function [see after (21) and Fig. 10] of the second-order B factors B_2,j(T) in (21) is done with the logarithmic derivative

$[\eqalignno{{{{B'_{2,j}}(T)} \over {{B_{2,j}}(T)}} &= 2{{{B'_j}(T)} \over {{B_j}(T)}} + {{{d_j}{d'_j}(T)} \over {8D_1^2({d_j}) + 4{d_j}D_1^{}({d_j}) + d_j^2}}&\cr &\quad \times \Bigl\{1 + {{4{D_1}({d_j}) + {d_j}} \over {16D_1^2({d_j}) + 8{d_j}D_1^{}({d_j}) + d_j^2}}&\cr &\quad\times\Bigl[4{D_1}({d_j}) - {d_j} - {{4{d_j}} \over {\exp({d_j}) - 1}}\Bigr]\Bigr\},&(31)}]$

obtained by using the identity $[{D'_1}(x) ]$ = $[- {D_1}(x)/x]$ + $[1/[\exp(x) - 1]]$ for the Debye function D₁(x) in (4). Here, $[{d_j} = {\hat c_j}d]$ , with $[d = \theta (T)/T]$ , as in (20) and (22). [We also note $[{d_j}(T) = {\theta _j}(T)/T]$ with $[{\theta _j} = {\hat c_j}\theta (T)]$ , cf. Sections 2.2 and 4.9.] $[{B'_j}(T)]$ is the derivative of the regressed B factors in (13), and the Debye temperatures $[{\theta _j}(T)]$ of zinc are calculated in Section 4.4.

The derivative $[{d'_j}(T) = {\hat c_j}d'(T)]$ in (31) is found by differentiating the equation $[\Delta (d) = {{3 \over 4}}TS/U]$ in (5), with the Δ function of zinc in (15),

$[d'(T) = {3 \over 4}\left[{{TS(T)} \over {U(T)}}\right]'{1 \over {\Delta '(d)}}. \eqno(32)]$

The internal energy U(T) and entropy S(T) are calculated from the regressed heat capacity and the zero-temperature limits of the B_j(T) factors, cf. Section 4.4. The derivatives in (32) can be done analytically, using the integral representations of U(T) and S(T) in Section 4.4 and also the identity (24) for [D'(d)] when differentiating $[\Delta (d)]$ in (15). This way to obtain the derivative [d'(T)] [or $[\theta '(T)]$ ] is preferable to direct numerical differentiation of $[d(T) = {\Delta ^{ - 1}}({{3 \over 4}}TS/U)]$ , since the inverted Δ function is approximated by an interpolated function as pointed out above.

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Volume 77| Part 5| September 2021| Pages 420-432

https://doi-org.cc/10.1107/S2053273321005507

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited

1. Introduction

2. Varying Debye temperatures and spectral cutoffs of non-cubic elemental crystals

2.1. Lattice internal energy, entropy and mean-squared vibrational amplitudes

2.2. Temperature dependence of the Debye temperatures and spectral cutoff

3. Low-temperature limits of the Debye–Waller B factors and zero-point energy of anisotropic lattice vibrations

4. Thermodynamic functions and Debye–Waller factors of zinc

4.1. Regressed lattice heat capacity of the zinc structure

4.2. B factors of lattice vibrations parallel and perpendicular to the hexagonal axis

4.3. Zero-point internal energy of the zinc structure

4.4. Varying axial and basal Debye temperatures

4.5. Temperature variation of the spectral cutoff

4.6. Internal energy and entropy components

4.7. Effective phonon speed and dispersion relations of axial and basal normal vibrations

4.8. Temperature variation of the effective oscillator mass of zinc

4.9. Second-order B factors of zinc

5. Conclusion

APPENDIX A

Invertibility of the Δ function, and logarithmic derivative of second-order B factors

A1. Invertibility of Δ(d)

A2. Logarithmic derivative of B2,j(T)

References

research papers

A2. Logarithmic derivative of B_2,j(T)