 1. Introduction
 2. Varying Debye temperatures and spectral cutoffs of noncubic elemental crystals
 3. Lowtemperature limits of the Debye–Waller B factors and zeropoint energy of anisotropic lattice vibrations
 4. Thermodynamic functions and Debye–Waller factors of zinc
 5. Conclusion
 A1. Invertibility of Δ(d)
 A2. Logarithmic derivative of B2,j(T)
 References
 1. Introduction
 2. Varying Debye temperatures and spectral cutoffs of noncubic elemental crystals
 3. Lowtemperature limits of the Debye–Waller B factors and zeropoint energy of anisotropic lattice vibrations
 4. Thermodynamic functions and Debye–Waller factors of zinc
 5. Conclusion
 A1. Invertibility of Δ(d)
 A2. Logarithmic derivative of B2,j(T)
 References
research papers
Thermodynamics of lattice vibrations in noncubic crystals: the zinc structure revisited
^{a}Sechsschimmelgasse 1/2122, Vienna, 1090, Austria
^{*}Correspondence email: tom@geminga.org
A phenomenological model of anisotropic lattice vibrations is proposed, using a temperaturedependent spectral cutoff and varying Debye temperatures for the vibrational normal components. The internal lattice energy, B factors of noncubic elemental crystals are derived. The formalism developed is nonperturbative, based on temperaturedependent 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 and B factors by leastsquares regression. The zeropoint internal energy of the phonons is related to the lowtemperature limits of the meansquared vibrational amplitudes of the lattice measured by Xray and γray diffraction. A specific example is discussed, the thermodynamic variables of the hexagonal closepacked 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 secondorder B factors defining the nonGaussian contribution to the Debye–Waller damping factors of zinc are obtained as well. Anharmonicity of the oscillator potential and deviations from the uniform phonon of the Debye theory are modeled effectively by the temperature dependence of the spectral cutoff and Debye temperatures.
and Debye–WallerKeywords: anisotropic lattice vibrations; thermodynamic functions; Debye–Waller factors; noncubic crystals; temperaturedependent spectral cutoff; directional Debye temperatures; effective phonon speed; oscillator mass; heat capacity; zeropoint 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 of noncubic crystals, which emerges in the crossover region between the low and hightemperature regimes. Moreover, the Debye temperatures inferred from Xray diffraction measurements of the meansquared atomic displacements differ from those obtained from the lowtemperature resulting in different Xray and caloric Debye temperatures (cf. e.g. Gopal, 1966; Butt et al., 1988; Peng et al., 1996, 2004), which should be identical in a selfconsistent theory of lattice vibrations. The linear hightemperature 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 temperaturedependent Debye temperature 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 functions obtained from caloric and diffraction experiments differ. Moreover, a phonon with varying Debye temperature does not define an equilibrium system, since the equilibrium condition on the internalenergy derivative of 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 assumed in the Debye theory (cf. e.g. Gopal, 1966; Meissner et al., 1978).
Here, we will discuss thermal vibrations of the hexagonal closepacked zinc structure (Altmann & Bradley, 1965), as an example of an anisotropic elemental crystal. The equilibrium relation can be maintained with varying Debye temperatures for the normal vibrations, provided that the spectral cutoff 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 can be described with the same set of functions and so that Xray 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 temperaturedependent Debye temperature and spectral cutoff. The varying spectral cutoff and Debye temperature of each normal component can be extracted from empirical data by leastsquares regression and from the zerotemperature 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, and the meansquared 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 temperaturedependent 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 data and empirical B factors.
In Section 3, we consider Debye–Waller B factors of anisotropic elemental crystals defined by the meansquared amplitudes of the orthogonal normal vibrations. The zeropoint lattice energy can be obtained from the zerotemperature limits of three B factors measured by Xray diffraction and from the amplitude of the cubic lowtemperature scaling of the isochoric The reasoning in Sections 2 and 3 applies to anisotropic elemental crystals in general.
In Section 4, the thermodynamic variables and meansquared vibrational amplitudes of the zinc structure are studied. measurements of zinc continuously extend from the lowtemperature 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 of the zinc structure with a multiply broken powerlaw density (Tomaschitz, 2020a,b, 2021b,c) and perform a leastsquares fit to the isochoric data sets, subtracting the electronic (cf. Section 4.1).
Analytic leastsquares fits are also performed to the measured B factors of zinc, up to the melting point (cf. Section 4.2). The amplitude of the lowtemperature lattice and the zerotemperature limits of the regressed B factors allow us to estimate the zeropoint internal energy of the zinc structure (cf. Section 4.3). Integrations of the regressed lattice give the thermal components of the internal energy and the (cf. Section 4.6). Based on these empirical functions and the zerotemperature 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 temperaturedependent 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). Secondorder B factors defining the nonGaussian 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 noncubic elemental crystals
2.1. Lattice internal energy, and meansquared vibrational amplitudes
We consider anisotropic (i.e. noncubic) monatomic crystals and split the molar internal energy of the lattice vibrations into three components, corresponding to atomic normal vibrations, ,
where R = 8.314 J (K mol)^{−1} is the the are varying Debye temperatures, (cm^{−1}) is the temperaturedependent spectral cutoff and (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 and . 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 and constant spectral cutoff, . The spectral representation of the partial energies (1) reads
where the temperaturedependent 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 and using and .
relations are determined by the effective phonon speed of the normal vibrations (labeled by subscriptThe
can likewise be decomposed into normal components , with partial entropiesThe temperaturedependent effective oscillator density is = , with partial densities = generating the energy and and (3). The is assembled as and the Helmholtz free energy as .
components in (1)The meansquared displacement (cm^{2}) of the normal oscillators vibrating parallel to coordinate axes labeled j reads (Tomaschitz, 2020a)
where is the temperaturedependent effective oscillator mass. The corresponding Debye–Waller B factors are . The asymptotic limits of the Debye function in (4) are and . 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 C_{V}(T) [in units of J (K mol)^{−1}] by and , respectively, where the zeropoint internal energy U_{0} is an integration constant. In Section 4.1, an analytic representation of C_{V}(T) will be obtained from a leastsquares fit of a multiply broken powerlaw density to data of zinc. The analytic fit can then be integrated as indicated to find the internal energy and The zeropoint energy U_{0} will be determined from Xray and γray diffraction measurements of Debye–Waller B factors (cf. Section 4.2).
are related to the latticeWe specialize the Debye temperatures of the normal oscillations in (1)–(4) as , where the are positive temperatureindependent constants. This ansatz is sufficiently general to model anisotropic vibrations of elemental crystals. The effective phonon speed of the normal oscillations is , with [cf. after (2)].
Once U(T) and S(T) are extracted from the empirical and B factors, the varying Debye temperature can be found by combining equations (1) and (3),
where we substituted and used the shortcut . 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 . It will also be necessary to explicitly invert in order to calculate by solving (5), . The constants in (5) can be obtained from zerotemperature limits of measured B factors (cf. Sections 3 and 4), so that the numerical inversion of in (7) needs to be done only once; a practical inversion method is outlined in the Appendix, cf. after (30).
The spectral cutoff in (2) is found by solving in (1) for , with substituted for the directional Debye temperatures. By expressing and as functions of and internal energy, the equilibrium condition mentioned in Section 1 is preserved (Tomaschitz, 2020a). U(T) and S(T) are empirical functions, obtained from the regressed and the lowtemperature limits of the regressed B factors, as exemplified with zinc in Section 4.4.
3. Lowtemperature limits of the Debye–Waller B factors and zeropoint energy of anisotropic lattice vibrations
At low temperature, the lattice c_{V0} obtained from a leastsquares fit (cf. Section 4.1), so that the internal energy converges to its zeropoint limit, , and the lattice scales as (cf. the beginning of Section 2.2). The lowtemperature limit of in (5) reads [cf. (30)]
scales as , with amplitudeSubstitution of the stated limits into (5) gives the zerotemperature limit of the Debye temperature ,
The zerotemperature limit of the B factors (4) reads
Here and in (6), we used the asymptotic expansions of the Debye functions D(x) and D_{1}(x) [cf. after (1) and (4) and (30)]. The zerotemperature limit of the effective oscillator mass is the . Equation (8) defines a system of three equations, labeled by j = 1,2,3, that can be solved for , and :
We substitute this into (7) and solve for U_{0} to find the zeropoint internal energy,
where c_{V0} is the empirical amplitude of the lowtemperature lattice .
One of the temperatureindependent constants defining the Debye temperatures , cf. Section 2.2, can be absorbed into , so that we can put from the outset. The remaining constants defining the directional Debye temperatures can be inferred from the zerotemperature 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 of the zinc structure
We perform a leastsquares fit to the empirical ; 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 doublelogarithmic plot in Fig. 1. The isochoric 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 C_{V},
of zinc (Eichenauer & Schulze, 1959The amplitudes , b_{0} and and the exponents and are positive and related by and , so that the classical Dulong–Petit limit is recovered, with R = 8.314 J (K mol)^{−1}. The isochoric data points in Fig. 1 include the electronic component of the and the leastsquares fit is done with . The independent fitting parameters , b_{0,1,2}, , are recorded in Table 1. The lattice C_{V}(T) of the zinc structure is obtained by subtraction of the electronic contribution from the regressed and has the lowtemperature limit . The units used are C_{V} [J (K mol)^{−1}], [J (K^{2} mol)^{−1}], b_{0} [J (K^{4} 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 except, of course, in the lowtemperature regime (see Fig. 1).

In contrast to the Debye cf. Fig. 1); the tangent of C_{V}(T) at the inflection point is depicted as a red dotted line , at = 9.085 K, = 0.0986 J mol^{−1}. and C_{V}(T) are plotted in Fig. 1 up to the melting point of zinc at 692.73 K.
the empirical lattice of the zinc structure has an inflection point in log–log representation (Also shown in Fig. 1 is the Debye approximation,
with Debye function D(d) in (1). The constant Debye temperature of zinc is = 314.79 K, obtained from the lowtemperature amplitude b_{0} in (11) and Table 1. The constant effective phonon speed is , cf. after (2), with cutoff factor , cf. after (1), so that = 2.62 km s^{−1}, based on the atomic density = 6.5702 × 10^{22} cm^{−3}. The zeropoint energy in the Debye model is , cf. (1) and (30), which gives = 2944.5 J mol^{−1} for zinc, incompatible with the zerotemperature limits of the measured B factors (cf. Section 4.3). The empirical of zinc has an extended phonon peak at around 20 K (cf. Fig. 1), which is not reproduced by the Debye approximation; the deviation is listed in Table 2, for a few selected temperatures.
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 and the B factor of vibrations parallel to the basal plane by , cf. (4). Thus, relating to the notation of Sections 2 and 3, , , and analogous identifications are made for other quantities such as basal and axial Debye temperatures, and . We use Xray and Mössbauer γray diffraction data of the and factors (Skelton & Katz, 1968; Albanese et al., 1976; Pathak & Desai, 1981) and perform leastsquares fits with the fit function , where j can take the values `axial' and `basal', and
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 of zinc, and the Debye function D_{1}(d) is defined in (4). The fitting parameters , 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 Xray (or γray) diffraction are marked with a subscript X. The regressed B factors and are depicted in Fig. 2. We also note = 145.526 Å^{2}K, to be substituted in (13), which gives (Å^{2}) in terms of m (u), (K), T (K), m_{2,j} (Å^{2} K^{−2}) and m_{3,j} (Å^{2} K^{−3}).

If the quadratic and cubic terms in (13) are dropped, we obtain the B factors of the Debye approximation [with constant and in (4)], which are also depicted in Fig. 2. The lowtemperature limits of the meansquared vibrations in (4) (with constant mass and Debye temperature) are constant, and the hightemperature limits are linear in temperature since . The B_{j}(T) in (13) (with ) cover the full temperature range up to the melting point; the zerotemperature limits B_{j}(0) are listed in Table 3.
In Fig. 3, we plot the Index functions = 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 Bfactor curves in Fig. 2. That is, a tangent to the Bfactor plots in Fig. 2 represents a power law whose exponent 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 powerlaw exponent.
4.3. Zeropoint internal energy of the zinc structure
The zeropoint energy of the lattice vibrations is calculated from the zerotemperature limits of the B factors via (10),
The B factors at T = 0 are listed in Table 3. As for units, c_{V0} [J (K^{4} mol)^{−1}], B_{j} (Å^{2}), m (u), and we used = 48.50875 Å^{2}K to obtain the numerical factor in (14). The constant c_{V0} is the amplitude of the lowtemperature cf. the beginning of Section 3, to be identified with the fitting parameter b_{0} in (11) and Table 1. The zeropoint energy of zinc consistent with the measured B factors is thus U_{0} = 556.3 J mol^{−1}.
The zeropoint energy of the Debye approximation, cf. after (12), is by a factor larger than U_{0} obtained from Bfactor measurements. While the zeropoint lattice energy is not accessible by caloric measurement of energy differences, the thermal energy can be obtained empirically by integrating the measured cf. Section 2.2, and a plot thereof is depicted in Fig. 4. The maximal relative deviation of from the Debye approximation happens at 23.45 K and is substantial, at this temperature (see also the captions to Figs. 1 and 4 and Section 4.6). In view of this, the large ratio is not surprising. The zeropoint internal energy stems from the second term in (1) and (2) and is finite because of the finite spectral cutoff. The zerotemperature limit of the spectral cutoff is smaller than the constant spectral cutoff of the Debye theory (cf. Section 4.1 and Fig. 6 in Section 4.5); therefore U_{0} is smaller than the Debye zeropoint energy [cf. the spectral representation (2) of U(T)].
4.4. Varying axial and basal Debye temperatures
The directional Debye temperatures (labeled j = 1,2,3 in Section 2.2) of the zinc structure are denoted by and as in Section 4.2. The constants can be extracted from the zerotemperature limits of the B factors, cf. (9): and = = = 1.664 (see Table 3).
Thus, and , where is found by solving (5). The Δ function (5) of zinc reads [cf. (23)]
with and stated above. is calculated as indicated in Section 2.2, . The inversion of (15) is sketched in the Appendix, cf. after (30). The lattice internal energy and are empirical input functions, inferred from the regressed lattice C_{V}(T) [cf. (11) and Table 1] and depicted in Fig. 4. The zeropoint internal energy of zinc is U_{0} = 556.3 J mol^{−1} (cf. Section 4.3). The Debye temperatures and = are plotted in Fig. 5; their zerotemperature limits coincide with the fitting parameters of the B factors in Table 3, , , according to (8) and (13) [where ].
 Figure 5 depicted in Fig. 4 
Constant axial and basal Debye temperatures for zinc were already introduced by Grüneisen & Goens (1924). The plots of the temperaturedependent caloric and Xray 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 (temperatureindependent) spectral cutoff. Also, these authors do not use directional Debye temperatures, but calculate one temperaturedependent Debye temperature either by comparing the Debye (12) with the empirical curve, or by comparing the averaged experimental B factor with the isotropic Debye B factor obtained by dropping the quadratic and cubic terms in (13) and replacing by . As mentioned, the curves obtained by these two methods substantially differ, cf. Skelton & Katz (1968), and a temperaturedependent 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 and , the cutoff factor of the internal energy (1) and (3) is the same for normal vibrations orthogonal and parallel to the hexagonal axis. is calculated as explained in Section 2.2,
with U(T) [see after (15)] and and in Section 4.4. The rescaled dimensionless cutoff factor is plotted in Fig. 6, where = 6.5702 × 10^{22} 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 zerotemperature limits of the B factors, which enter via the zeropoint energy (14) and . The temperature dependence of compensates for the variation of , so that the equilibrium condition is preserved (Tomaschitz, 2020a).
4.6. Internal energy and 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) = , cf. (1), where
and analogously for . The partial energies and depend on the spectral cutoff and the Debye temperatures and , respectively, calculated in Sections 4.4 and 4.5. An analogous decomposition holds for the lattice , cf. (3), with axial component
and analogously for the basal component . U(T) and S(T) coincide with the empirical functions obtained from integrations of the regressed [see after (15)] and from the zerotemperature limits of the B factors determining the zeropoint energy U_{0}, cf. Section 4.3. The temperature evolution of U(T), its thermal component and the S(T) are plotted in Fig. 4, together with the respective Debye approximations , = and . The latter are calculated by integrating the Debye in (12) with constant Debye temperature = 314.79 K and spectral cutoff , cf. Section 2.2. The differences and are recorded in Table 2, at selected temperatures, and the maximal relative deviation of the Debye theory from the empirical 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 and (orthogonal and parallel to the basal plane, respectively), where , cf. Section 2.2. The Debye temperature and the zerotemperature ratio were obtained in Section 4.4 and the spectral cutoff in Section 4.5. The atomic density of zinc is = 6.5702 × 10^{22} cm^{−3} (based on a of 7.133 g cm^{−3} and a molar mass of 65.38 g mol^{−1}), so that (cm s^{−1}) = 3244.5 θ(T) (K)/h(T), with dimensionless cutoff factor , cf. Section 4.5. The temperaturedependent axial and basal dispersion relations are and , cf. after (2). Instead of anharmonic perturbations of the oscillator potential, we use a temperaturedependent spectral cutoff and varying Debye temperatures in the spectral representation (2), so that the dispersion relations stay linear. The effective phonon velocities and are plotted in Fig. 7.
4.8. Temperature variation of the effective oscillator mass of zinc
We substitute the analytic fits (13) of the meansquared atomic displacement into (4) and solve for , with index j taking the values `axial' and `basal'. The oscillator mass of normal vibrations along the hexagonal axis is found as
and , the oscillator mass of basal oscillations, reads analogously. and denote the leastsquares fits (13) to the empirical meansquared vibrations, and = 145.526 Å^{2}K in (4). and are the Debye temperatures calculated in Section 4.4. At zero temperature, the effective oscillator mass coincides with the , cf. Section 3. The temperature variation of and is depicted in Fig. 8.
4.9. Secondorder B factors of zinc
NonGaussian contributions to the Debye–Waller (diffraction intensity attenuation) factors are defined by higherorder terms in the ascending series expansion , where Q^{2} 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 crosscorrelations into account. The leading order of M_{j} is defined by the B factor , cf. (4) and Section 4.2. The secondorder coefficient B_{2,j}(T) depends on the vibrational amplitude averages and via the cumulant , . This cumulant can be expressed in terms of the Debye function D_{1} (Tomaschitz, 2021a),
where we use the shortcut and = 145.526 Å^{2}K. is the effective mass (19) in the Debye function D_{1} is defined in (4), and . The varying Debye temperatures of zinc are calculated in Section 4.4. The Debye approximation of B_{2,j}(T) in (20) is obtained by replacing by the and by the constant directional Debye temperatures 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 secondorder B factors (20) are related to the B_{j}(T) factors (4) by
which is obtained by substituting the effective mass (19) into (20). The temperature evolution of the secondorder factors and of zinc is depicted in Fig. 9, together with their Debye approximations. The Index functions [cf. the end of Section 4.2 and equation (31) in the Appendix] of the secondorder factors are plotted in Fig. 10, representing the log–log slope of the B_{2,j}(T) curves in Fig. 9.
5. Conclusion
We outlined a practical method to describe anisotropic lattice vibrations in elemental crystals, focusing on a specific example, the hexagonal closepacked zinc structure. The experimental lattice B factors of zinc can selfconsistently be modeled, from the lowtemperature regime up to the melting point, by employing the varying Debye temperatures , (cf. Section 4.4), the temperaturedependent spectral cutoff (cf. Section 4.5) and the effective oscillator masses , (cf. Section 4.8).
internal energy and equilibrium as well as the temperature evolution of the measuredIn Sections 2 and 3, we sketched the general formalism for anisotropic monatomic crystals. The atomic vibrations are partitioned into normal components described by a temperaturedependent spectral cutoff and directional Debye temperatures , 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 B factors, as explained in Sections 2.2 and 3 and illustrated with zinc in Section 4. The effective phonon speed defining the dispersion relations of the normal vibrations is determined by the Debye temperatures and spectral cutoff , cf. Sections 2.2 and 4.7. The effective oscillator density of the vibrational components is calculated from the spectral cutoff, cf. Section 2.1, and the effective oscillator mass from the respective Debye–Waller factor B_{j}(T), cf. Section 4.8.
and lowtemperature Debye–WallerWe have chosen to model empirical Bfactor data of anisotropic crystals by way of a temperaturedependent spectral cutoff, temperaturedependent 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 temperaturedependent) dispersion relations for the normal vibrations, and the uniform density of phonon states of the Debye theory are retained, which makes the described nonperturbative formalism possible.
andSeveral 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 at optical frequencies (cf. Meissner et al., 1978; Hoser & Madsen, 2016, 2017; Sovago et al., 2020). A nonuniform phonon 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 and graphite (Tomaschitz, 2020b). The onset of the phonon peak is indicated by an inflection point in the log–log slope of the cf. Fig. 1.
of anisotropic crystals in the crossover region between the lowtemperature and classical regimes largely deviate from the Debye as exemplified by zinc in Fig. 1When modeling 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 , which would otherwise be violated by a varying Debye temperature (Tomaschitz, 2020a). This also holds true for anisotropic crystals: since the Debye temperature of each vibrational normal component varies with temperature, the spectral cutoff has to vary as well to ensure that the internalenergy derivative of coincides with the inverse temperature. The above condition on the derivative is preserved by relating and to the internal energy and cf. Section 2.2.
data, the preference of experimentalists seems to be the Debye theory in combination with a temperaturedependent Debye temperature,In contrast to the Debye theory, where the zeropoint energy of the phonons is already determined by the constant Debye temperature, cf. Section 4.1, the zeropoint energy U_{0} emerges as a free constant in the temperaturedependent spectral cutoff and Debye temperatures , cf. Section 2.2. The Debye–Waller factors B_{j}(T) also depend on U_{0} by way of the Debye temperatures , especially in the lowtemperature regime, cf. Section 3. Therefore, the zeropoint energy U_{0}, which is calorically not measurable, can be inferred from the zerotemperature 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 Xray Debye temperatures, as the varying directional Debye temperatures extracted from the empirical 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 secondorder B factors
A1. Invertibility of Δ(d)
We demonstrate that the function in (5),
with , is strictly monotonously decreasing on the positive real axis, for any choice of positive constants and . Here, we consider a more general function than stated in (5) where and j = 1,2,3. Equation (22) is the generalization of in (5) for compound crystals (Tomaschitz, 2021a), where the constants depend on the atomic composition and the summation in (22) is over , with N atomic species in the formula unit. [ in (5) applies to elemental crystals.]
We will show that for , which means that 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),
Using the derivative of the Debye function in (22),
we find that is equivalent to , with
where . The coefficients f_{jk} are independent of the positive constants and . It suffices to show that . To this end, we rearrange the f_{jk},
and also note two identities obtained by integration by parts,
with Debye function D(d) in (22), and
Because of the positivity of the expressions (27) and (28), it is evident that the condition with f_{jk} in (26) is equivalent to the inequality , where
We consider and as two functions of independent variables d_{j} and d_{k} on the positive real axis. is positive for and attains its minimum at d_{j} = 0, . By the way, the asymptotic limits of the Debye function in (22) are
The numerator of in (29) has a zero at = . is positive for . In the range , is negative and attains its minimum at d_{k} = 0, . Both functions and in (29) increase monotonously on the positive real axis. Thus the inequality 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 is equivalent to , cf. (25) and (26), the derivative of in (22) and (23) is negative along the positive real axis, irrespective of the choice of positive constants and , so that is invertible.
As for the practical inversion of in (5) [or (15) in the case of zinc], the asymptotic limits of [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 in (5) from the zerotemperature limits of the empirical Debye–Waller B factors, cf. (9) and Section 4.4, so that 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 . The inverse of in the crossover region is approximated by a cubic polynomial interpolation of the function values defined by the inverted table .
A2. Logarithmic derivative of B_{2,j}(T)
The numerical evaluation of the Index function [see after (21) and Fig. 10] of the secondorder B factors B_{2,j}(T) in (21) is done with the logarithmic derivative
obtained by using the identity = + for the Debye function D_{1}(x) in (4). Here, , with , as in (20) and (22). [We also note with , cf. Sections 2.2 and 4.9.] is the derivative of the regressed B factors in (13), and the Debye temperatures of zinc are calculated in Section 4.4.
The derivative in (31) is found by differentiating the equation in (5), with the Δ function of zinc in (15),
The internal energy U(T) and S(T) are calculated from the regressed and the zerotemperature 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 when differentiating in (15). This way to obtain the derivative [or ] is preferable to direct numerical differentiation of , since the inverted Δ function is approximated by an interpolated function as pointed out above.
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