Equations of state

The scientific reference for the equation-of-state framework adopted by Quantas is the review and revalidation by Angel, Gonzalez-Platas, and Alvaro, EosFit7c and a Fortran module (library) for equation of state calculations [1]. That work reviews the connection between linear elasticity and equations of state, derives the implemented formulae, and discusses their ranges of validity. The complete citation guidance is collected in Citing Quantas.

An equation of state (EOS) describes how a structural quantity changes with pressure, temperature, or both. For a volume EOS, the fundamental variables are pressure \(P\), volume \(V\), and temperature \(T\). Quantas also applies the same formalism to linear quantities, such as lattice parameters, through the conventions described at the end of this chapter.

The presentation follows the order used by Angel et al. (2014):

1. pressure–volume equations at fixed temperature;

2. volume–temperature equations at fixed pressure;

3. pressure–volume–temperature equations obtained by coupling compression and thermal expansion.

The equations below define the scientific models. Parameter estimation, uncertainty treatment, diagnostics, and command-line options belong to the EOS fitting: implementation and workflow workflow rather than to this chapter.

Reference state and notation

At a selected reference temperature \(T_{\mathrm{ref}}\) and zero pressure, let

\[V_0 = V(P=0,T_{\mathrm{ref}}),\]

\[K_0 = -V\left(\frac{\partial P}{\partial V}\right)_{T, P=0,T=T_{\mathrm{ref}}},\]

with pressure derivatives

\[K'_0 = \left(\frac{\partial K}{\partial P}\right)_{P=0,T=T_{\mathrm{ref}}}, \qquad K''_0 = \left(\frac{\partial^2 K}{\partial P^2}\right)_{P=0,T=T_{\mathrm{ref}}}.\]

Pressure is positive under compression. \(K_0\) has the same unit as pressure, \(K'_0\) is dimensionless, and \(K''_0\) has units of inverse pressure.

The order assigned to an EOS identifies which terms of its defining expansion are retained. A lower-order equation fixes or implies one or more derivatives of the bulk modulus; a higher-order equation introduces additional freedom, but that freedom is useful only when the data span a sufficient pressure or volume range.

Pressure–volume equations of state

An isothermal P–V equation extends infinitesimal linear elasticity to finite compression. Linear elasticity defines the bulk modulus at one state, but it does not uniquely determine how \(K\) changes over a finite pressure interval. Different EOS families therefore embody different assumptions about the evolution of strain energy or bulk modulus.

Murnaghan equation

The Murnaghan EOS assumes a bulk modulus that varies linearly with pressure,

\[K(P)=K_0+K'_0P.\]

Integration gives

\[P(V)=\frac{K_0}{K'_0} \left[ \left(\frac{V_0}{V}\right)^{K'_0}-1 \right],\]

or, equivalently,

\[V(P)=V_0 \left(1+\frac{K'_0P}{K_0}\right)^{-1/K'_0}.\]

Advantages

• It is algebraically simple and exactly invertible between \(P\) and \(V\).

• Its parameters have an immediate physical interpretation.

• It is useful for modest compressions and for applications that repeatedly require both \(P(V)\) and \(V(P)\).

Limitations

• The assumption implies \(K''=0\) at every pressure.

• Most solids exhibit a small negative \(K''_0\), so the equation becomes progressively less realistic as compression increases.

• Angel et al. note that it is normally reliable only over relatively small compressions, approximately up to \(V/V_0\simeq0.9\) for many solids.

Tait equation

The modified Tait EOS is an invertible generalization of Murnaghan that allows non-zero curvature in \(K(P)\). In the parameterization used by Angel et al., define

\[a=\frac{1+K'_0}{1+K'_0+K_0K''_0},\]

\[b=\frac{K'_0}{K_0}-\frac{K''_0}{1+K'_0},\]

\[c=\frac{1+K'_0+K_0K''_0} {(K'_0)^2+K'_0-K_0K''_0}.\]

The pressure form is

\[P(V)=\frac{1}{b} \left\{ \left[ \frac{V/V_0+a-1}{a} \right]^{-1/c}-1 \right\},\]

and the inverse relation is

\[V(P)=V_0\left[1-a\left(1-(1+bP)^{-c}\right)\right].\]

Quantas supports second-, third-, and fourth-order Tait forms. The third-order form uses the common truncation \(K''_0=-K'_0/K_0\); the fourth-order form allows \(K''_0\) to remain independent.

Advantages

• It retains the practical invertibility of Murnaghan.

• It represents the observed curvature of the bulk modulus more realistically.

• Angel et al. report that Tait and Birch–Murnaghan commonly yield mutually consistent parameters for ordinary solid P–V datasets.

Limitations

• It is primarily an empirical compressional representation rather than an expansion derived from a specific finite-strain energy.

• Its auxiliary parameters \(a\), \(b\), and \(c\) can become singular for non-physical combinations of \(K_0\), \(K'_0\), and \(K''_0\).

• A fourth-order form is not automatically preferable: \(K''_0\) is often weakly constrained and strongly correlated with the lower derivatives.

Birch–Murnaghan equation

The Birch–Murnaghan EOS is derived by expanding the strain energy in the Eulerian finite strain

\[f_E=\frac{1}{2} \left[ \left(\frac{V_0}{V}\right)^{2/3}-1 \right].\]

To fourth order in the strain expansion, the pressure is

\[P=3K_0f_E(1+2f_E)^{5/2} \left\{ 1+\frac{3}{2}(K'_0-4)f_E +\frac{3}{2} \left[ K_0K''_0+(K'_0-4)(K'_0-3)+\frac{35}{9} \right]f_E^2 \right\}.\]

The normalized pressure

\[F_E=\frac{P}{3f_E(1+2f_E)^{5/2}}\]

is a polynomial in \(f_E\), which makes an \(F_E\)–\(f_E\) plot a useful diagnostic of EOS order.

The supported truncations are:

• second order: \(K'_0=4\);

• third order: \(K'_0\) is independent and \(K''_0\) is implied by

  \[K''_0=-\frac{1}{K_0} \left[(3-K'_0)(4-K'_0)+\frac{35}{9}\right];\]

• fourth order: \(K''_0\) is independent.

Advantages

• It is grounded in finite-strain theory and has a clear hierarchy of orders.

• It is widely used for solids at moderate compression.

• The normalized-pressure representation provides a direct visual test of whether higher-order curvature is supported.

• It is also naturally connected to integrated energy–volume equations used in first-principles calculations.

Limitations

• A finite-order Eulerian expansion is not expected to remain accurate under extreme compression; Angel et al. identify approximately \(V/V_0<0.6\) as a regime where finite-strain EOS can become inadequate for many solids.

• Higher-order terms require a sufficiently broad and precise dataset.

• Apparent improvement from a fourth-order fit may reflect parameter correlation rather than physically resolved curvature.

Natural-strain or Poirier–Tarantola equation

The natural-strain EOS uses the Hencky strain

\[f_N=\frac{1}{3}\ln\left(\frac{V_0}{V}\right),\]

chosen here to be positive on compression. Its fourth-order pressure form is

\[P=3K_0\frac{V_0}{V}f_N \left\{ 1+\frac{3}{2}(K'_0-2)f_N +\frac{3}{2} \left[ 1+K_0K''_0+(K'_0-2)+(K'_0-2)^2 \right]f_N^2 \right\}.\]

The corresponding lower-order forms imply:

• second order: \(K'_0=2\);

• third order:

  \[K''_0=-\frac{1}{K_0} \left[1+(K'_0-2)+(K'_0-2)^2\right];\]

• fourth order: \(K''_0\) is independent.

Advantages

• It provides a mathematically consistent alternative finite-strain measure.

• It can be useful when the natural strain produces a simpler normalized relation for a particular material or pressure interval.

• Like Birch–Murnaghan, it offers a systematic order hierarchy.

Limitations

• The implied \(K''_0\) of the third-order form is commonly more negative in magnitude than the Birch–Murnaghan value.

• Angel et al. observe that this often gives a poorer description of ordinary solid P–V data.

• Agreement or disagreement with Birch–Murnaghan is dataset dependent and should be interpreted as model sensitivity, not merely as a statistical ranking.

Vinet equation

The Vinet EOS was derived from a generalized interatomic potential rather than from a finite-order strain-energy polynomial. Define

\[f_V=1-\left(\frac{V}{V_0}\right)^{1/3}, \qquad \eta=\frac{3}{2}(K'_0-1).\]

Then

\[P=3K_0\frac{f_V}{(1-f_V)^2}\exp(\eta f_V).\]

The standard form contains \(V_0\), \(K_0\), and \(K'_0\) and is conventionally called third order. Quantas also provides a second-order representation with the implied value \(K'_0=1\).

Advantages

• It is designed to behave more realistically than finite-strain polynomials at very large compression.

• It is compact and often effective over broad pressure ranges.

• Its normalized-pressure form can be used to assess deviations from the assumed exponential behavior.

Limitations

• There is no compelling theoretical truncation to a lower-order Vinet EOS; the second-order form is mainly a constrained representation.

• A model intended for high compression is not necessarily the best choice for a narrow low-pressure dataset.

• The ordinary three-parameter form fixes the implied \(K''_0\); resolving additional curvature requires a different extended model.

Choosing a P–V family

No single P–V EOS is universally superior. A scientifically defensible choice considers the compression range, the physical origin of the data, and whether the dataset genuinely resolves higher derivatives.

P–V model summary

Family	Main idea	Principal strength	Principal caution
Murnaghan	Linear \(K(P)\)	Simple and invertible	\(K''=0\); limited compression range
Tait	Invertible nonlinear \(K(P)\)	Flexible and practical	Empirical auxiliary parameters; high-order correlation
Birch–Murnaghan	Eulerian finite-strain expansion	Standard solid-state EOS and clear order diagnostics	Finite-order expansion degrades at extreme compression
Poirier–Tarantola	Natural/Hencky strain expansion	Alternative finite-strain description	Often implies stronger curvature and poorer ordinary P–V fits
Vinet	Generalized interatomic-potential form	Good behavior at high compression	Lower-order truncation has limited theoretical basis

Volume–temperature equations of state

At fixed pressure, the volume thermal-expansion coefficient is

\[\alpha(T)=\frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P.\]

Integration gives

\[V(T)=V_{\mathrm{ref}} \exp\left[ \int_{T_{\mathrm{ref}}}^{T}\alpha(t)\,\mathrm dt \right].\]

Thermodynamics requires \(\alpha\rightarrow0\) and \(\partial\alpha/\partial T\rightarrow0\) as \(T\rightarrow0\) for a stable non-degenerate solid. Simple empirical expressions often describe a restricted high-temperature interval very well without satisfying this low-temperature limit. Conversely, forms designed for low-temperature saturation may become unsuitable when extrapolated far above their intended range.

Berman equation

Let \(\Delta T=T-T_{\mathrm{ref}}\). The Berman form used by Quantas is

\[V(T)=V_{\mathrm{ref}} \left[1+\alpha_0\Delta T +\frac{1}{2}\alpha_1\Delta T^2\right].\]

The exact expansion coefficient of this truncated volume expression is

\[\alpha(T)= \frac{\alpha_0+\alpha_1\Delta T} {1+\alpha_0\Delta T+\frac{1}{2}\alpha_1\Delta T^2}.\]

Advantages

• It is simple, transparent, and effective for many moderate- or high-temperature datasets.

• The quadratic term captures smoothly varying non-linear expansion.

• \(V(T_{\mathrm{ref}})=V_{\mathrm{ref}}\) and \(\alpha(T_{\mathrm{ref}})=\alpha_0\) exactly.

Limitations

• It does not enforce the third-law low-temperature limit.

• \(\alpha_1\) is a coefficient of the volume polynomial, not exactly \((\partial\alpha/\partial T)_{T_{\mathrm{ref}}}\).

• The bracketed volume factor must remain positive; broad extrapolation can become non-physical.

Fei equation

The Fei expansion coefficient is

\[\alpha(T)=\alpha_0+\alpha_1T+\alpha_2T^{-2}.\]

Integration yields

\[V(T)=V_{\mathrm{ref}}\exp\left[ \alpha_0(T-T_{\mathrm{ref}}) +\frac{1}{2}\alpha_1(T^2-T_{\mathrm{ref}}^2) -\alpha_2 \left(\frac{1}{T}-\frac{1}{T_{\mathrm{ref}}}\right) \right].\]

The simplified form sets \(\alpha_2=0\).

Advantages

• The fitted coefficients describe the expansion function directly.

• They are defined against absolute temperature and are independent of the chosen reference temperature.

• The linear form \(\alpha_0+\alpha_1T\) is useful for broad high-temperature trends.

Limitations

• The inverse-square term diverges as \(T\rightarrow0\).

• Even the simplified form does not force \(\alpha(0)=0\).

• It should therefore be regarded as a finite-temperature representation, not a universal low-temperature law.

Modified Holland–Powell equation

The modified Pawley–Redfern–Holland expression adopted by EosFit7 and Quantas is

\[V(T)=V_{\mathrm{ref}} \left\{ 1+\alpha_0(T-T_{\mathrm{ref}}) -2(10\alpha_0+\alpha_1) \left(\sqrt{T}-\sqrt{T_{\mathrm{ref}}}\right) \right\}.\]

Setting the residual \(\alpha_1=0\) gives the simplified Holland–Powell form.

Advantages

• It is compact and can describe the approach toward approximately constant expansion at high temperature.

• The simplified form is useful for lower-resolution datasets with few independently resolvable thermal parameters.

• It is widely used in thermodynamic databases.

Limitations

• It is not a low-temperature EOS. Below

  \[T_{\mathrm{limit}}= \left(\frac{10\alpha_0+\alpha_1}{\alpha_0}\right)^2,\]

  the predicted expansion becomes negative for the usual positive \(\alpha_0\) case. The simplified form has \(T_{\mathrm{limit}}=100\ \mathrm K\).

• The coefficient \(\alpha_0\) is not generally the exact expansion at \(T_{\mathrm{ref}}\).

Salje equation

The Salje form explicitly represents low-temperature saturation:

\[V(T)=\left[ V_0^{1/3} +p_1\theta_{\mathrm{sat}} \left( \coth\frac{\theta_{\mathrm{sat}}}{T}-1 \right) \right]^3.\]

It satisfies

\[V(0)=V_0, \qquad \alpha(0)=0.\]

Advantages

• It has the correct zero-temperature saturation of thermal expansion.

• It is particularly suitable for low-temperature datasets.

• The saturation scale \(\theta_{\mathrm{sat}}\) has a clear qualitative role.

Limitations

• Above a few times \(\theta_{\mathrm{sat}}\), the predicted expansion becomes nearly temperature independent.

• That behavior is inconsistent with the approximately linear increase of \(\alpha(T)\) observed at high temperature for many solids.

• It should not be used as an unrestricted high-temperature extrapolation.

Kroll form of Holland–Powell

The Kroll–Holland–Powell form relates thermal expansion to an Einstein oscillator while retaining parameters at a chosen reference temperature. Let

\[\xi_0= \frac{(\theta_E/T_{\mathrm{ref}})^2 \exp(\theta_E/T_{\mathrm{ref}})} {[\exp(\theta_E/T_{\mathrm{ref}})-1]^2},\]

\[A(T)=\alpha_{\mathrm{ref}}\frac{\theta_E}{\xi_0} \left[ \frac{1}{\exp(\theta_E/T)-1} -\frac{1}{\exp(\theta_E/T_{\mathrm{ref}})-1} \right],\]

and, with \(k=K'_0\),

\[B=-\frac{1}{k(k+2)}.\]

The volume can be written

\[V(T)=V_{\mathrm{ref}} \left\{ -k+(1+k) \left[ 1-\frac{k(k+2)}{k+1}A(T) \right]^B \right\}.\]

Advantages

• The Einstein function gives low-temperature saturation while retaining a realistic high-temperature trend.

• The reference identities \(V(T_{\mathrm{ref}})=V_{\mathrm{ref}}\) and \(\alpha(T_{\mathrm{ref}})=\alpha_{\mathrm{ref}}\) are exact.

• It offers a physically motivated bridge between purely empirical thermal expansion and lattice-vibrational models.

Limitations

• It is more complex than the simple polynomial forms.

• \(\theta_E\) is frequently weakly constrained by ordinary V–T data.

• Its robustness depends on the availability of a credible compressional \(K'_0\) and on remaining within a physically admissible temperature interval.

Choosing a V–T family

The temperature interval is often the decisive criterion.

V–T model summary

Family	Intended behavior	Principal strength	Principal caution
Berman	Smooth polynomial expansion	Simple and effective over restricted ranges	No low-temperature saturation
Fei	Direct polynomial/inverse-square \(\alpha(T)\)	Reference-independent expansion coefficients	Divergent or non-zero low-temperature limit
Modified Holland–Powell	Approximately constant high-temperature expansion	Compact database-oriented form	Negative expansion below its limiting temperature
Salje	Low-temperature saturation	Correct \(\alpha(0)=0\)	Unrealistic unrestricted high-temperature behavior
Kroll–Holland–Powell	Einstein-based low/high-temperature crossover	More physical global temperature dependence	Extra parameters may be weakly constrained

Pressure–volume–temperature equations of state

A P–V–T EOS combines an isothermal compressional EOS with a thermal model. The central thermodynamic issue is not only the increase of the zero-pressure volume \(V_{0T}\), but also the change of the zero-pressure bulk modulus \(K_{0T}\) with temperature. Angel et al. discuss three principal coupling strategies: a linear temperature dependence of \(K_0\), an Anderson–Gruneisen relation, and thermal pressure.

Linear variation of the zero-pressure bulk modulus

The simplest model is

\[K_{0T}=K_0+ \left(\frac{\partial K_0}{\partial T}\right) (T-T_{\mathrm{ref}}),\]

combined with a selected V–T equation for \(V_{0T}\) and a selected P–V family evaluated with those temperature-dependent reference values.

Advantages

• It is transparent and introduces only one direct temperature derivative.

• In combination with non-linear thermal expansion and an isothermal EOS with non-zero \(K''_0\), it contains the principal second derivatives of volume with respect to pressure and temperature.

• It is often adequate over a restricted experimental temperature range.

Limitations

• A constant \(\partial K_0/\partial T\) is a local approximation.

• Extrapolation can drive \(K_{0T}\) toward zero or negative values.

• Hellfrich and Connolly showed that this coupling can predict non-physical negative thermal expansion at moderate pressure for many materials.

Anderson–Gruneisen coupling

The Anderson–Gruneisen parameter \(\delta\) relates the temperature variation of the zero-pressure bulk modulus to thermal expansion [2]:

\[K_{0T}=K_0 \exp\left[ -\delta \int_{T_{\mathrm{ref}}}^{T}\alpha(t)\,\mathrm dt \right].\]

Using the zero-pressure thermal volume,

\[K_{0T}=K_0 \left(\frac{V_0}{V_{0T}}\right)^{\delta}.\]

Advantages

• It couples thermal softening directly to expansion instead of imposing a constant temperature derivative.

• It generally behaves more smoothly under pressure and avoids some of the negative-expansion pathologies of linear coupling.

• The approximation \(\delta\simeq K'_0\) can provide a first estimate when independent information is limited.

Limitations

• Treating \(\delta\) as constant is itself an approximation.

• \(\delta\simeq K'_0\) is not a thermodynamic identity.

• Broad extrapolation can fail when the pressure or temperature dependence of \(\delta\) becomes important.

Holland–Powell Einstein thermal pressure

The thermal-pressure approach writes

\[P(V,T)=P(V,T_{\mathrm{ref}})+\Delta P_{\mathrm{th}}(T).\]

The reference term is any selected isothermal P–V EOS. In the Holland–Powell Einstein model,

\[\Delta P_{\mathrm{th}}(T)= \alpha_{\mathrm{ref}}K_0 \frac{\theta_E}{\xi_0} \left[ \frac{1}{\exp(\theta_E/T)-1} -\frac{1}{\exp(\theta_E/T_{\mathrm{ref}})-1} \right].\]

The thermal pressure vanishes on the reference isotherm.

Advantages

• It gives a direct physical interpretation: heating at fixed volume creates an additional pressure.

• The Einstein function produces low-temperature saturation and an approximately linear high-temperature trend.

• The formulation avoids several pathologies associated with prescribing a constant \(\partial K_0/\partial T\).

Limitations

• In this form, thermal pressure depends only on temperature, not explicitly on volume.

• The single Einstein frequency is an effective representation of the vibrational spectrum.

• \(\theta_E\) may be poorly determined unless the temperature range is broad and precise.

Mie–Gruneisen–Debye thermal pressure

Quantas also includes a Mie–Gruneisen–Debye (MGD) coupling as an extension beyond the model set reviewed in detail by Angel et al. (2014). MGD is a quasi-harmonic thermal-pressure EOS in which the vibrational internal energy is represented by a Debye spectrum [3]. In compact form,

\[P(V,T)=P(V,T_{\mathrm{ref}}) +\frac{\gamma(V)}{V} \left[U_D(V,T)-U_D(V,T_{\mathrm{ref}})\right],\]

with the appropriate unit and atom-count normalization.

The full Quantas form uses

\[\gamma(V)=\gamma_0\left(\frac{V}{V_0}\right)^q,\]

and

\[\gamma(V)=-\frac{\partial\ln\theta_D}{\partial\ln V}.\]

For \(q\ne0\), integration gives

\[\theta_D(V)=\theta_{D0} \exp\left[ \frac{\gamma_0-\gamma(V)}{q} \right].\]

The Debye thermal term uses

\[D_3(x)=\frac{3}{x^3} \int_0^x\frac{t^3}{\exp(t)-1}\,\mathrm dt.\]

Quantas also represents the explicitly named EosFit q-compromise approximation, in which \(\theta_D\) and \(\gamma/V\) are held constant.

Advantages

• It introduces explicit volume dependence into the thermal pressure.

• It connects compression, the Gruneisen parameter, and lattice-vibrational energy within a quasi-harmonic framework.

• It is well suited to broad P–V–T applications when the material is reasonably represented by an effective Debye spectrum.

Limitations

• A single Debye temperature cannot reproduce all details of a real phonon density of states, especially at low temperature.

• \(\theta_{D0}\), \(\gamma_0\), and \(q\) can be strongly correlated unless the dataset spans a broad P–T–V domain.

• Intrinsic anharmonicity at fixed volume, electronic excitations, magnetic effects, and phase transitions are outside the basic MGD model.

• The physical normalization must be consistent with the chosen cell or molar volume.

Choosing a P–V–T coupling

P–V–T coupling summary

Coupling	Main assumption	Principal strength	Principal caution
Linear \(K_0(T)\)	Constant \(\partial K_0/\partial T\)	Minimal and transparent	Local model; can become non-physical on extrapolation
Anderson–Gruneisen	\(K_0\) scales with thermal volume	Smooth pressure–temperature coupling	Constant \(\delta\) is approximate
Einstein thermal pressure	Temperature-only oscillator pressure	Physical low/high-temperature crossover	No explicit volume dependence of thermal pressure
Mie–Gruneisen–Debye	Debye energy with volume-dependent \(\gamma\) and \(\theta_D\)	Broad quasi-harmonic P–V–T description	Effective-spectrum approximation and correlated parameters

Linear equations of state

The same EOS families can describe a lattice parameter or another linear quantity \(x\), provided the finite-strain conventions remain consistent with the volume EOS. Following Angel et al., Quantas uses the auxiliary quantity

\[q=x^3\]

inside the volume-like equation. The physical linear expansion and compressibility are then

\[\alpha_x=\frac{1}{x} \left(\frac{\partial x}{\partial T}\right)_P, \qquad \beta_x=-\frac{1}{x} \left(\frac{\partial x}{\partial P}\right)_T.\]

For three orthogonal principal directions,

\[\alpha_1+\alpha_2+\alpha_3=\alpha_V, \qquad \beta_1+\beta_2+\beta_3=\frac{1}{K}.\]

A linear modulus is

\[M_x=\frac{1}{\beta_x}.\]

For an isotropic or cubic material, \(M_x=3K\). The cubed-length construction therefore preserves consistency between volume and linear elasticity, but the numerical moduli reported for a linear EOS are physical linear moduli, not volume-like auxiliary values.

Scientific interpretation and model selection

An EOS is not only a curve through data. Its derivatives determine bulk modulus, expansion, and pressure–temperature response, so model choice should be based on physical scope as well as residual size.

When comparing equations, consider:

• whether the sampled compression or temperature interval matches the intended domain of the model;

• whether an additional EOS order is supported by independent curvature rather than parameter correlation;

• whether the reference state lies inside or near the observed domain;

• whether extrapolated volumes, moduli, expansion coefficients, Debye temperatures, and thermal pressures remain physically admissible;

• whether conclusions are stable when another scientifically plausible EOS family is used.

A statistically converged representation can still be scientifically inappropriate outside its calibration range. Phase transitions, elastic or vibrational instabilities, intrinsic anharmonicity, electronic transitions, and changes in bonding cannot in general be repaired by increasing the order of a single-phase EOS.

What Quantas provides

Within the standalone EOS workflow, Quantas evaluates and analyzes:

• Murnaghan P–V;

• Birch–Murnaghan P–V, orders 2–4;

• natural-strain/Poirier–Tarantola P–V, orders 2–4;

• Vinet P–V, orders 2–3;

• Tait P–V, orders 2–4;

• Berman, Fei, modified Holland–Powell, Salje, and Kroll–Holland–Powell V–T models;

• linear, Anderson–Gruneisen, Einstein thermal-pressure, and MGD P–V–T couplings;

• volume and linear forms where the underlying scientific convention is applicable.

The workflow pages explain how these models are selected and assessed; the present chapter defines the physical assumptions behind them.

Bibliographic references

[1]

Ross J. Angel, Javier Gonzalez-Platas, and Matteo Alvaro. “EosFit7c and a Fortran module (library) for equation of state calculations”. Zeitschrift fuer Kristallographie, 229, 405-419 (2014). DOI: 10.1515/zkri-2013-1711.

[2]

Orson L. Anderson. Equations of State of Solids for Geophysics and Ceramic Science. Oxford University Press, New York (1995).

[3]

Lars Stixrude and Carolina Lithgow-Bertelloni. “Thermodynamics of mantle minerals - I. Physical properties”. Geophysical Journal International, 162, 610-632 (2005). DOI: 10.1111/j.1365-246X.2005.02642.x.