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

## 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 [1, 2, 3, 4, 5]) and are, therefore, equally suitable for the mentioned asymptotic regions.

The isochoric heat capacities of diamond, graphite and vitreous ${\text{SiO}}_{2}$, discussed in Sect. 2, are special cases of the broken power-law density (1). In the case of diamond, ${C}_{V}(T)$ interpolates between the constant Dulong–Petit high-temperature limit and the Debye ${T}^{3}$ 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, ${C}_{V}(T)$ interpolates between the linear low-temperature slope of the degenerate electron gas and the classical Dulong–Petit regime, and in the case of the ${\text{SiO}}_{2}$ 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 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 [25, 26, 27]. In the case of copper, the available experimental pressure range (up to 450 GPa [26]) will be extended with data sets obtained from DFT calculations [28]. 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

### 2.1 Diamond

Diamond | Graphite | ${\text{v - SiO}}_{2}$ | |
---|---|---|---|

${b}_{0}$ | $1.8605\times {10}^{-7}$ | $1.4843\times {10}^{-5}$ | $1.0618\times {10}^{-4}$ |

${b}_{1}[\text{K}]$ | 67.435 | 0.60277 | 2.6677 |

${b}_{2}[\text{K}]$ | 282.02 | 40.466 | 10.585 |

${b}_{3}[\text{K}]$ | – | 562.35 | 247.30 |

${\beta}_{0}$ | 3 | 1 | 1.22 |

${\beta}_{1}$ | 1.2496 | 1.6796 | 4.3172 |

${\beta}_{2}$ | ${\beta}_{0}+{\beta}_{1}$ | 1.3235 | 4.0656 |

${\beta}_{3}$ | – | ${\beta}_{0}+{\beta}_{1}-{\beta}_{2}$ | ${\beta}_{0}+{\beta}_{1}-{\beta}_{2}$ |

${\eta}_{1}$ | 0.30536 | 0.69130 | 2.7558 |

${\eta}_{2}$ | 2.1816 | 1.1347 | 1.7243 |

${\eta}_{3}$ | – | 0.56727 | 0.93528 |

${\chi}^{2}$ | 0.144 | 0.0539 | 0.0635 |

dof | 117–5 | 114–8 | 78–8 |

SE | 0.070 | 0.070 | 0.431 |

$1-{R}^{2}$ | $2.41\times {10}^{-4}$ | $5.55\times {10}^{-5}$ | $5.16\times {10}^{-4}$ |

The asymptotic limits of the Debye function are $D(x>>1)\sim \phantom{\rule{thickmathspace}{0ex}}{\pi}^{4}/(15{x}^{3})$ and $D(x<<1)\sim 1/3$, so that ${C}_{D}(T\to 0)\sim \phantom{\rule{thickmathspace}{0ex}}{c}_{D}{T}^{3}$ and ${C}_{D}(T\to \mathrm{\infty})\sim 3{n}_{a/m}R$, where ${n}_{a/m}$ denotes the number of atoms per molecule. The units are ${c}_{D}[{\text{J/(K}}^{4}\text{mol)]}$ and ${C}_{D}[\text{J}/(\text{K mol)]}$ as above. The amplitude ${c}_{D}$ is taken from the least-squares fit of ${C}_{V}(T)$ in (3), ${c}_{D}={b}_{0}$, 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 ${c}_{D}$ 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.

### 2.2 Graphite

The units are ${\alpha}_{\mathrm{\perp}}[1/\text{K}]$ and $T[\text{K}]$. The expansion ${\alpha}_{||}$ in the basal plane is negligible compared with ${\alpha}_{\mathrm{\perp}}$. 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, ${C}_{V}\approx {C}_{P}$ in the temperature range of the available data points.

There is no indication of a ${T}^{3}$ slope anywhere to be seen in the data set in Fig. 2. Therefore, we choose the amplitude ${c}_{D}$ defining the Debye temperature in a way that the low-temperature ${T}^{3}$ slope of the Debye curve ${C}_{D}(T)$ [stated in (4)] cuts through the inflection point of the empirical heat capacity ${C}_{V}(T)$, see Fig. 4. This gives a Debye temperature of $\theta =438.7\text{K}$ [with ${c}_{D}=2.303\times {10}^{-5}\phantom{\rule{thinmathspace}{0ex}}\text{J}/({\text{K}}^{4}\phantom{\rule{thinmathspace}{0ex}}\text{mol})$ in (4)]. A different choice of $\theta $ would just shift the ${T}^{3}$ 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 $, except at very high temperature; see also Ref. [41] 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 ${C}_{V}^{}(T)$ in (7) (with fitting parameters in Table 1) according to Eq. (5).