Applied Physics A

, 126:102

# Caloric and isothermal equations of state of solids: empirical modeling with multiply broken power-law densities

• Roman Tomaschitz
• Roman Tomaschitz
• 1
1. 1.ViennaAustria
Article

## Abstract

Empirical equations of state (EoSs) are developed for solids, applicable over extended temperature and pressure ranges. The EoSs are modeled as multiply broken power laws, in closed form without the use of ascending series expansions; their general analytic structure is explained and specific examples are studied. The caloric EoS is put to test with two carbon allotropes, diamond and graphite, as well as vitreous silica. To this end, least-squares fits of broken power-law densities are performed to heat capacity data covering several logarithmic decades in temperature, the high- and low-temperature regimes and especially the intermediate temperature range where the Debye theory is of limited accuracy. The analytic fits of the heat capacities are then temperature integrated to obtain the entropy and caloric EoS, i.e. the internal energy. Multiply broken power laws are also employed to model the isothermal EoSs of metals (Al, Cu, Mo, Ta, Au, W, Pt) at ambient temperature, over a pressure range up to several hundred GPa. In the case of copper, the empirical pressure range is extended into the TPa interval with data points from DFT calculations. For each metal, the parameters defining the isothermal EoS (i.e. the density–pressure relation) are inferred by nonlinear regression. The analytic pressure dependence of the compression modulus of each metal is obtained as well, over the full data range.

## Keywords

Multi-parameter equation of state (EoS) Caloric EoS of carbon allotropes Specific heat of vitreous silica Thermal EoS and compression modulus of metals High-pressure regime Multiply broken power laws

## 1 Introduction

The aim of this paper is to develop analytic equations of state (EoSs) for solids which can reproduce empirical data sets covering several orders in temperature and pressure, including the extended crossovers between the asymptotic low and high pressure and temperature regimes. The proposed EoSs are multiply broken power laws, which do not involve truncated series expansions in density, pressure or temperature (frequently used in empirical EoSs, cf. e.g. the reviews [, , , , ]) and are, therefore, equally suitable for the mentioned asymptotic regions.

At first, we study caloric EoSs, i.e. the temperature dependence of the internal energy, obtained by integration of the isochoric heat capacity. We work out three specific examples, obtaining closed analytic expressions for the heat capacities of diamond, graphite and vitreous silica by least-squares regression. The isochoric heat capacities are structured as multiply broken power laws [, , ],
${C}_{V}\left(T\right)={b}_{0}{T}^{{\beta }_{0}}\prod _{k=1}^{n}{\left(1+\left(T/{b}_{k}{\right)}^{{\beta }_{k}/|{\eta }_{k}|}\right)}^{{\eta }_{k}}$
(1)
with positive amplitudes ${b}_{0}$$b_{0}$ and ${b}_{k}$$b_{k}$, $k=1,\dots ,n$$k = 1, \ldots ,n$, and ${b}_{k}<<{b}_{k+1}$$b_{k} < < b_{k + 1}$. The exponents ${\beta }_{k}$$\beta_{k}$, $k=1,\dots ,n$$k = 1, \ldots ,n$ are positive, ${\beta }_{0}$$\beta_{0}$ is real, and the exponents ${\eta }_{k}$$\eta_{k}$ can be positive or negative. ${C}_{V}\left(T\right)$$C_{V} (T)$ is analytic on the positive real axis and consists of $n+1$$n + 1$ approximate power-law segments, $\propto {T}^{{\beta }_{0}}$$\propto T^{{\beta_{0} }}$, ${T}^{{\beta }_{0}+{\beta }_{1}\text{sign}\left({\eta }_{1}\right)},\dots ,{T}^{{\beta }_{0}+{\sum }_{k=1}^{n}{\beta }_{k}\text{sign}\left({\eta }_{k}\right)}$$T^{{\beta_{0} + \beta_{1} {\text{sign}}(\eta_{1} )}} , \ldots ,T^{{\beta_{0} + \sum\nolimits_{k = 1}^{n} {\beta_{k} {\text{sign}}(\eta_{k} )} }}$ in the intervals $T<<{b}_{1}$$T < < b_{1}$, ${b}_{1}<$b_{1} < < T < < b_{2} , \ldots ,b_{n} < < T$, respectively. The amplitudes ${b}_{k}$$b_{k}$ define the break points between the power-law segments and the exponents ${\eta }_{k}$$\eta_{k}$ determine the extent of the transitional regions. Power laws in log–log plots appear as approximately straight segments, and the exponents ${\eta }_{k}$$\eta_{k}$ determine the extent of the transitional regions between the power-law segments.

The isochoric heat capacities of diamond, graphite and vitreous ${\text{SiO}}_{2}$${\text{SiO}}_{ 2}$, discussed in Sect. 2, are special cases of the broken power-law density (1). In the case of diamond, ${C}_{V}\left(T\right)$$C_{V} (T)$ interpolates between the constant Dulong–Petit high-temperature limit and the Debye ${T}^{3}$$T^{3}$ low-temperature slope. In the intermediate temperature range, the broken power law (1) is sufficiently adaptable to model deviations from the Debye theory such as the boson peaks of glasses. In the case of graphite, ${C}_{V}\left(T\right)$$C_{V} (T)$ interpolates between the linear low-temperature slope of the degenerate electron gas and the classical Dulong–Petit regime, and in the case of the ${\text{SiO}}_{2}$${\text{SiO}}_{2}$ glass, the linear low-temperature slope is replaced by a power law with non-integer index close to one, generated by a fermionic two-level system. In the latter two examples, a phononic cubic temperature slope does not emerge in the empirical data sets, as the low-temperature regime is dominated by Fermi systems.

In recent years, new phenomenological EoSs have been proposed to model density–pressure data sets obtained from Hugoniot shock compression of, for instance, alkali metals [, ], alkaline earths [], aluminum [, , ], iron [], copper [], tin [] and tungsten [], to mention but a few. In the second part of this paper, multiply broken power laws are employed as isothermal EoSs of metals, with emphasis on the high-pressure (GPa and TPa) regime. In this case, the pressure dependence of the mass density $\rho$$\rho$ has the analytic shape
$\rho \left(P\right)={\rho }_{0}\prod _{k=1}^{n}{\left(1+P/{b}_{k}\right)}^{\phantom{\rule{thinmathspace}{0ex}}{\eta }_{k}}$
(2)
with positive amplitudes ${\rho }_{0}$$\rho_{0}$, ${b}_{k}$$b_{k}$, ${b}_{k}<<{b}_{k+1}$$b_{k} < < b_{k + 1}$, and real exponents ${\eta }_{k}$$\eta_{k}$, $k=1,\dots ,n$$k = 1, \ldots ,n$. These parameters are temperature dependent, but will be treated as constants in the examples studied here, at ambient temperature of 300 K. Density (2) consists of $n+1$$n + 1$ power-law segments, $\propto 1$$\propto 1$, ${P}^{{\eta }_{1}},\dots ,{P}^{{\sum }_{k=1}^{n}{\eta }_{k}}$$P^{{\eta_{1} }} , \ldots ,P^{{\sum\nolimits_{k = 1}^{n} {\eta_{k} } }}$, in the intervals $P<<{b}_{1}$$P < < b_{1}$, ${b}_{1}<$b_{1} < < P < < b_{2} , \ldots ,b_{n} < < P$. In contrast to the broken power law (1), we have put ${\beta }_{0}=0$$\beta_{0} = 0$ and ${\beta }_{k}=|{\eta }_{k}|$$\beta_{k} = \left| {\eta_{k} } \right|$ in (2), so that $\rho \left(P\right)$$\rho (P)$ is analytic at $P=0$$P = 0$, which is essential to model the compression modulus in the low-pressure range consistent with ultrasonic measurements. When discussing the compression modulus in Sect. 3, we will also need the logarithmic derivative of density (2), ${\rho }^{\mathrm{\prime }}\left(P\right)/\rho \left(P\right)={\sum }_{k=1}^{n}{\eta }_{k}/\left(P+{b}_{k}\right)$$\rho^{\prime}(P)/\rho (P) = \sum\nolimits_{k = 1}^{n} {\eta_{k} /(P + b_{k} )}$, to show how the low-pressure limit of EoS (2) is related to the Murnaghan EoS [, , , , , ].

In Sect. 3, the EoS (2) will be put to test by performing least-squares fits to high-pressure data sets of Al, Cu, Mo, Ta, Au, W, and Pt, which cover an extended pressure range, from ambient (zero) pressure to several hundred GPa [, , ]. In the case of copper, the available experimental pressure range (up to 450 GPa []) will be extended with data sets obtained from DFT calculations []. The least-squares fit of the EoS of copper then covers pressures reaching 60 TPa. In Sect. 4, we present our conclusions.

## 2 Heat capacity, caloric EoS, and entropy of diamond, graphite and vitreous silica

In this section, the modeling of empirical heat capacity data by multiply broken power-law densities is discussed. Two carbon allotropes, diamond and graphite, and a glass, ${\text{v - SiO}}_{2}$${\text{v - SiO}}_{ 2}$, are studied as specific examples. Explicit formulas for the molar isochoric heat capacities are obtained by nonlinear least-squares regression, cf. Figs. 1, 2 and 3. Comparisons with the Debye theory are depicted in Fig. 4. The temperature variation of the internal energy and entropy variables obtained from the regressed heat capacities is shown in Fig. 5. Alternative attempts to find caloric EoSs for diamond and graphite are discussed in Refs. [, , , ] and references therein.

### 2.1 Diamond

We perform a ${\chi }_{}^{2}$$\chi_{{}}^{2}$ fit to the available data sets [, , ] of the diamond heat capacity, depicted in the double-logarithmic plot in Fig. 1, by employing a broken power law of type (1) as isochoric heat capacity,
${C}_{V}^{}\left(T\right)={b}_{0}{T}^{3}\left(1+\left(T/{b}_{1}{\right)}^{{\beta }_{1}/{\eta }_{1}}{\right)}^{{\eta }_{1}}\frac{1}{\left(1+\left(T/{b}_{2}{\right)}^{{\beta }_{2}/{\eta }_{2}}{\right)}^{{\eta }_{2}}}$
(3)
with amplitudes ${b}_{1}<<{b}_{2}$$b_{1} < < b_{2}$. The exponents ${\beta }_{i}$$\beta_{i}$, ${\eta }_{i}$$\eta_{i}$, $i=1,2$$i = 1,2$ are positive, and ${\beta }_{2}=3+{\beta }_{1}$$\beta_{2} = 3 + \beta_{1}$, so that the classical Dulong–Petit limit ${C}_{V}\left(T\to \mathrm{\infty }\right)\sim 3R$$C_{V} (T \to \infty )\sim 3R$ is recovered, with gas constant $R = 8.314{\text{ J/(K mol)}}$. This limit also requires amplitudes in (3) related by ${b}_{0}=3R{b}_{1}^{{\beta }_{1}}/{b}_{2}^{{\beta }_{2}}$$b_{0} = 3Rb_{1}^{{\beta_{1} }} /b_{2}^{{\beta_{2} }}$. The low-temperature limit is ${C}_{V}\left(T\to 0\right)\sim \phantom{\rule{thickmathspace}{0ex}}{b}_{0}{T}^{3}$$C_{V} (T \to 0)\sim \;b_{0} T^{3}$; the units used are ${C}_{V}^{}\left[\text{J}/\left(\text{K mol)}\right]$$C_{V}^{{}} [{\text{J}}/({\text{K mol)}}]$, $b_{0} [ {\text{J/(K}}^{4} {\text{ mol)]}}$, and ${b}_{i}\left[\text{K}\right]$$b_{i} [{\text{K}}]$.
Broken power laws composed of multiple factors $\left(1+\left(T/{b}_{i}{\right)}^{{\beta }_{i}/|{\eta }_{i}|}{\right)}^{{\eta }_{i}}$$(1 + (T/b_{i} )^{{\beta_{i} /\left| {\eta_{i} } \right|}} )^{{\eta_{i} }}$ are quite efficient for data sets stretching over several decades in temperature, as demonstrated in the subsequent examples. In (3), we have three successive power laws, $\propto {T}^{3}$$\propto T^{3}$, ${T}^{3+{\beta }_{1}}$$T^{{3 + \beta_{1} }}$, ${T}^{3+{\beta }_{1}-{\beta }_{2}}$$T^{{3 + \beta_{1} - \beta_{2} }}$, in the intervals $T<<{b}_{1}$$T < < b_{1}$, ${b}_{1}<$b_{1} < < T < < b_{2}$ and ${b}_{2}<$b_{2} < < T$, respectively, which appear as approximately straight segments in log–log plots, and the exponents ${\eta }_{1}$$\eta_{1}$, ${\eta }_{2}$$\eta_{2}$ determine the transitional regions around the break points ${b}_{1}$$b_{1}$ and ${b}_{2}$$b_{2}$. The parameters ${b}_{1}$$b_{1}$, ${b}_{2}$$b_{2}$, ${\eta }_{1}$$\eta_{1}$, ${\eta }_{2}$$\eta_{2}$ and ${\beta }_{1}$$\beta_{1}$ obtained from the least-squares fit are recorded in Table 1. In contrast to the Debye heat capacity, cf. (4), the heat capacity of diamond exhibits an inflection point in the crossover region between the low- and high-temperature regimes, see Fig. 1; the tangent of ${C}_{V}$$C_{V}$ at the inflection point is shown as green dotted line depicting the power law $\propto {T}^{3.547}$$\propto T^{ 3. 5 4 7}$.
Table 1

Fitting parameters of the molar isochoric heat capacity ${C}_{V}\left(T\right)$$C_{V} (T)$ of diamond, graphite and vitreous silica, see Figs. 1, 2 and 3

Diamond

Graphite

${\text{v - SiO}}_{2}$${\text{v - SiO}}_{ 2}$

${b}_{0}$$b_{0}$

$1.8605×{10}^{-7}$$1.8605 \times 10^{ - 7}$

$1.4843×{10}^{-5}$$1. 4 8 4 3\times 10^{ - 5}$

$1.0618×{10}^{-4}$$1. 0 6 1 8\times 10^{ - 4}$

${b}_{1}\left[\text{K}\right]$$b_{1} [{\text{K}}]$

67.435

0.60277

2.6677

${b}_{2}\left[\text{K}\right]$$b_{2} [{\text{K}}]$

282.02

40.466

10.585

${b}_{3}\left[\text{K}\right]$$b_{3} [{\text{K}}]$

562.35

247.30

${\beta }_{0}$$\beta_{0}$

3

1

1.22

${\beta }_{1}$$\beta_{1}$

1.2496

1.6796

4.3172

${\beta }_{2}$$\beta_{2}$

${\beta }_{0}+{\beta }_{1}$$\beta_{0} + \beta_{1}$

1.3235

4.0656

${\beta }_{3}$$\beta_{3}$

${\beta }_{0}+{\beta }_{1}-{\beta }_{2}$$\beta_{0} + \beta_{1} - \beta_{2}$

${\beta }_{0}+{\beta }_{1}-{\beta }_{2}$$\beta_{0} + \beta_{1} - \beta_{2}$

${\eta }_{1}$$\eta_{1}$

0.30536

0.69130

2.7558

${\eta }_{2}$$\eta_{2}$

2.1816

1.1347

1.7243

${\eta }_{3}$$\eta_{3}$

0.56727

0.93528

${\chi }^{2}$$\chi^{2}$

0.144

0.0539

0.0635

dof

117–5

114–8

78–8

SE

0.070

0.070

0.431

$1-{R}^{2}$$1 - R^{2}$

$2.41×{10}^{-4}$$2.41 \times 10^{ - 4}$

$5.55×{10}^{-5}$$5.55 \times 10^{ - 5}$

$5.16×{10}^{-4}$$5.16 \times 10^{ - 4}$

The listed parameters (amplitudes ${b}_{i}$$b_{i}$, exponents ${\beta }_{i}$$\beta_{i}$, ${\eta }_{i}$$\eta_{i}$) define the multiply broken power laws (3) (for diamond), (7) (graphite) and (8) (vitreous ${\text{SiO}}_{2}$${\text{SiO}}_{2}$) representing the empirical heat capacities. Some of these parameters are interrelated, cf. Sect. 2. ${b}_{0}$$b_{0}$ is in units of $\text{J}/\left({\text{K}}^{1+{\beta }_{0}}\phantom{\rule{thinmathspace}{0ex}}\text{mol}\right)$$\text{J}/(\text{K}^{1+{\beta}_0} \,\text{mol})$. Also recorded are the minimum of the least-squares functional ${\chi }^{2}={\sum }_{i=1}^{N}\left({C}_{V}\left({T}_{i}\right)-{C}_{Vi}{\right)}^{2}/{C}_{Vi}^{2}$$\chi^{2} = \sum\nolimits_{i = 1}^{N} {(C_{V} (T_{i} ) - C_{Vi} )^{2} /C_{Vi}^{2} }$ and the degrees of freedom (dof: number $N$$N$ of data points $\left({T}_{i},{C}_{Vi}\right)$$(T_{i} ,C_{Vi} )$ minus number of independent fitting parameters). The standard error of the fits, $\text{SE}=\left({\sum }_{i=1}^{N}\left({C}_{V}\left({T}_{i}\right)-{C}_{Vi}{\right)}^{2}/N{\right)}^{1/2}$${\text{SE}} = (\sum\nolimits_{i = 1}^{N} {(C_{V} (T_{i} ) - C_{Vi} )^{2} /N)^{1/2} }$, and the coefficient of determination, ${R}^{2}=1-{\sum }_{i=1}^{N}\left({C}_{V}\left({T}_{i}\right)-{C}_{Vi}{\right)}^{2}/\left(N{\sigma }^{2}\right)$$R^{2} = 1 - \sum\nolimits_{i = 1}^{N} {(C_{V} (T_{i} ) - C_{Vi} )^{2} } /(N\sigma^{2} )$, with sample variance ${\sigma }^{2}={\sum }_{i=1}^{N}\left({C}_{Vi}-{\overline{C}}_{V}{\right)}^{2}/N$$\sigma^{2} = \sum\nolimits_{i = 1}^{N} {(C_{Vi} - \bar{C}_{V} )^{2} /N}$ and mean ${\overline{C}}_{V}={\sum }_{i=1}^{N}{C}_{Vi}/N$$\bar{C}_{V} = \sum\nolimits_{i = 1}^{N} {C_{Vi} } /N$, are listed as well

In Figs. 1 and 4, we have also indicated the Debye approximation,
$\begin{array}{rl}{C}_{D}\left(T\right)& =9{n}_{a/m}R\left[4D\left(\theta /T\right)-\frac{\theta /T}{{\text{e}}^{\theta /T}-1}\right],\phantom{\rule{1em}{0ex}}D\left(x\right):=\frac{1}{{x}^{3}}\underset{0}{\overset{x}{\int }}\frac{{y}^{3}\text{d}y}{{\text{e}}^{y}-1},\\ \theta & ={\left(\frac{12}{5}{\pi }^{4}\frac{{n}_{a/m}R}{{c}_{D}}\right)}^{1/3}.\end{array}$
(4)

The asymptotic limits of the Debye function are $D\left(x>>1\right)\sim \phantom{\rule{thickmathspace}{0ex}}{\pi }^{4}/\left(15{x}^{3}\right)$$D(x > > 1)\sim \;\pi^{4} /(15x^{3} )$ and $D\left(x<<1\right)\sim 1/3$$D(x < < 1)\sim 1/3$, so that ${C}_{D}\left(T\to 0\right)\sim \phantom{\rule{thickmathspace}{0ex}}{c}_{D}{T}^{3}$$C_{D} (T \to 0)\sim \;c_{D} T^{3}$ and ${C}_{D}\left(T\to \mathrm{\infty }\right)\sim 3{n}_{a/m}R$$C_{D} (T \to \infty )\sim 3n_{a/m} R$, where ${n}_{a/m}$$n_{a/m}$ denotes the number of atoms per molecule. The units are $c_{D} [ {\text{J/(K}}^{4} {\text{ mol)]}}$ and ${C}_{D}\left[\text{J}/\left(\text{K mol)]}$$C_{D} [ {\text{J}}/({\text{K mol)]}}$ as above. The amplitude ${c}_{D}$$c_{D}$ is taken from the least-squares fit of ${C}_{V}\left(T\right)$$C_{V} (T)$ in (3), ${c}_{D}={b}_{0}$$c_{D} = b_{0}$, cf. Table 1, to recover the cubic low-temperature slope. Accordingly, the Debye temperature (4) of diamond is $\theta = 2186{\text{ K}}$. The low-temperature amplitude ${c}_{D}$$c_{D}$ is the only adjustable parameter of the Debye heat capacity, so that it is not surprising that the Debye approximation becomes inaccurate in the crossover region.

The temperature dependence of the caloric EoS (molar internal energy with zero-point energy subtracted) and molar entropy of diamond,
(5)
is shown in Fig. 5, calculated by substituting ${C}_{V}\left(T\right)$$C_{V} (T)$ in (3) with fitting parameters recorded in Table 1.

### 2.2 Graphite

The heat capacity data of graphite tabulated in the experimental papers [, , , ] refer to the isobaric heat capacity ${C}_{P}$$C_{P}$. The conversion of isobaric (at ambient pressure) to isochoric heat capacities ${C}_{V}$$C_{V}$ is done with the approximate formula ${C}_{V}\approx {C}_{P}/\left(1+0.526\phantom{\rule{thinmathspace}{0ex}}T{\alpha }_{\mathrm{\perp }}\left(T\right)\right)$$C_{V} \approx C_{P} /(1 + 0.526 \, T\alpha_{ \bot } (T))$, where $0.526$$0.526$ is the dimensionless Grüneisen constant of graphite [] and ${\alpha }_{\mathrm{\perp }}$$\alpha_{ \bot }$ the thermal expansion coefficient perpendicular to the basal plane, cf. Refs. [, , ],
$\begin{array}{rl}& {\alpha }_{\mathrm{\perp }}\left(0\le T\le 80\right)=5.35×{10}^{-9}{T}^{2}-3.755×{10}^{-11}{T}^{3},\\ & \begin{array}{rl}{\alpha }_{\mathrm{\perp }}\left(80\le T\le 273\right)& =2.435×{10}^{-7}T-7.69×{10}^{-10}{T}^{2}\\ & \phantom{\rule{1em}{0ex}}+8.875×{10}^{-13}{T}^{3},\end{array}\\ & {\alpha }_{\mathrm{\perp }}\left(273\le T\le 1100\right)=2.722×{10}^{-5}+3.05×{10}^{-9}\left(T-273\right),\\ & {\alpha }_{\mathrm{\perp }}\left(1100\le T\le 3000\right)=2.975×{10}^{-5}+9.604×{10}^{-9}\left(T-1100\right).\end{array}$
(6)

The units are ${\alpha }_{\mathrm{\perp }}\left[1/\text{K}\right]$$\alpha_{ \bot } [1/{\text{K}}]$ and $T\left[\text{K}\right]$$T[{\text{K}}]$. The expansion ${\alpha }_{||}$$\alpha_{||}$ in the basal plane is negligible compared with ${\alpha }_{\mathrm{\perp }}$$\alpha_{ \bot }$. In the case of diamond, cf. Sect. 2.1, the conversion to isochoric heat capacities has already been done in the experimental papers. In the case of vitreous silica, cf. Sect. 2.3, ${C}_{V}\approx {C}_{P}$$C_{V} \approx C_{P}$ in the temperature range of the available data points.

The heat capacity of graphite has an electronic and a phonon component, cf. Fig. 2; at low temperature, the linear electronic heat capacity overpowers the phonon component. For the least-squares fit depicted in Fig. 2, we use a broken power law similarly structured as in (3):
$\begin{array}{rl}{C}_{V}^{}\left(T\right)& ={b}_{0}T\left(1+\left(T/{b}_{1}{\right)}^{{\beta }_{1}/{\eta }_{1}}{\right)}^{{\eta }_{1}}\\ & \phantom{\rule{1em}{0ex}}×\frac{1}{\left(1+\left(T/{b}_{2}{\right)}^{{\beta }_{2}/{\eta }_{2}}{\right)}^{{\eta }_{2}}}\frac{1}{\left(1+\left(T/{b}_{3}{\right)}^{{\beta }_{3}/{\eta }_{3}}{\right)}^{{\eta }_{3}}},\end{array}$
(7)
where the factors are ordered by increasing magnitude of the amplitudes, ${b}_{1}<<{b}_{2}<<{b}_{3}$$b_{1} < < b_{2} < < b_{3}$, and the exponents ${\beta }_{i}$$\beta_{i}$, ${\eta }_{i}$$\eta_{i}$, $i=1,2,3$$i = 1,2,3$, are positive. We put ${\beta }_{3}=1+{\beta }_{1}-{\beta }_{2}$$\beta_{3} = 1 + \beta_{1} - \beta_{2}$ so that ${C}_{V}\left(T\to \mathrm{\infty }\right)\sim 3R$$C_{V} (T \to \infty )\sim 3R$, which also requires amplitudes related by $b_{0} [ {\text{J/(K}}^{2} {\text{ mol)]}} = 3Rb_{1}^{{\beta_{1} }} /(b_{2}^{{\beta_{2} }} b_{3}^{{\beta_{3} }} )$. The fitting parameters are ${b}_{i}$$b_{i}$, ${\eta }_{i}$$\eta_{i}$, $i=1,2,3$$i = 1,2,3$, and ${\beta }_{1}$$\beta_{1}$, ${\beta }_{2}$$\beta_{2}$, cf. Table 1. The tangent at the inflection point of ${C}_{V}^{}\left(T\right)\left[\text{J}/\left(\text{K mol)]}$$C_{V}^{{}} (T) [ {\text{J}}/({\text{K mol)]}}$ is depicted as dotted green line $\propto {T}^{2.586}$$\propto T^{ 2. 5 8 6}$ in Fig. 2, and the electronic low-temperature asymptote ${C}_{V}^{}\left(T\right)\sim \phantom{\rule{thickmathspace}{0ex}}{b}_{0}T$$C_{V}^{{}} (T)\sim \;b_{0} T$ is also shown in this figure.

There is no indication of a ${T}^{3}$$T^{3}$ slope anywhere to be seen in the data set in Fig. 2. Therefore, we choose the amplitude ${c}_{D}$$c_{D}$ defining the Debye temperature in a way that the low-temperature ${T}^{3}$$T^{3}$ slope of the Debye curve ${C}_{D}\left(T\right)$$C_{D} (T)$ [stated in (4)] cuts through the inflection point of the empirical heat capacity ${C}_{V}\left(T\right)$$C_{V} (T)$, see Fig. 4. This gives a Debye temperature of $\theta = 438.7{\text{ K}}$ [with ${c}_{D}=2.303×{10}^{-5}\phantom{\rule{thinmathspace}{0ex}}\text{J}/\left({\text{K}}^{4}\phantom{\rule{thinmathspace}{0ex}}\text{mol}\right)$$c_{D} = 2.303 \times 10^{ - 5} \,\text{J}/(\text{K}^{4}\,\text{mol})$ in (4)]. A different choice of $\theta$$\theta$ would just shift the ${T}^{3}$$T^{3}$ slope parallel to the depicted slope. It is evident from Fig. 4 that the standard Debye approximation (4) cannot give a reasonable fit to the heat capacity of graphite, irrespective of the choice of $\theta$$\theta$, except at very high temperature; see also Ref. [] and references therein for modifications of the Debye theory regarding graphite. The internal energy and entropy functions of graphite are shown in Fig. 5, obtained by integrating the empirical ${C}_{V}^{}\left(T\right)$$C_{V}^{{}} (T)$ in (7) (with fitting parameters in Table 1) according to Eq. (5).

### 2.3 Vitreous SiO2${\text{SiO}}_{2}$${\text{SiO}}_{2}$

For the heat capacity of vitreous silica [, , , , ], we use a broken power law similar to that of graphite:
$\begin{array}{rl}{C}_{V}^{}\left(T\right)& ={b}_{0}{T}^{{\beta }_{0}}\left(1+\left(T/{b}_{1}{\right)}^{{\beta }_{1}/{\eta }_{1}}{\right)}^{{\eta }_{1}}\\ & \phantom{\rule{1em}{0ex}}×\frac{1}{\left(1+\left(T/{b}_{2}{\right)}^{{\beta }_{2}/{\eta }_{2}}{\right)}^{{\eta }_{2}}}\frac{1}{\left(1+\left(T/{b}_{3}{\right)}^{{\beta }_{3}/{\eta }_{3}}{\right)}^{{\eta }_{3}}},\end{array}$
(8)
where the positive amplitudes ${b}_{i}\left[\text{K}\right]$$b_{i} [{\text{K}}]$ are ordered by increasing magnitude, ${b}_{1}<<{b}_{2}<<{b}_{3}$$b_{1} < < b_{2} < < b_{3}$. The exponents ${\beta }_{i}$$\beta_{i}$ and ${\eta }_{i}$$\eta_{i}$, $i=1,2,3$$i = 1,2,3$, are positive, and we put ${\beta }_{3}={\beta }_{0}+{\beta }_{1}-{\beta }_{2}$$\beta_{3} = \beta_{0} + \beta_{1} - \beta_{2}$ so that ${C}_{V}\left(T\to \mathrm{\infty }\right)\sim 9R$$C_{V} (T \to \infty )\sim 9R$, which also requires $b_{0} [ {\text{J/(K}}^{{1 + \beta_{0} }} {\text{ mol)]}} = 9Rb_{1}^{{\beta_{1} }} /(b_{2}^{{\beta_{2} }} b_{3}^{{\beta_{3} }} )$. The data points in Fig. 3 refer to the total heat capacity of the phonons and the fermionic two-level system which dominates the phonon heat capacity of the glass at low temperature [, ]. The low-temperature heat capacity of the two-level system is slightly different from linear, with power-law exponent of