# A Simple Derivation of the Birch–Murnaghan Equations of State (EOSs) and Comparison with EOSs Derived from Other Definitions of Finite Strain

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Derivations of the Birch–Murnaghan Equation of State (EOS)

#### 2.1. Eulerian Finite Strain

#### 2.2. The Second-Order Birch–Murnaghan EOS

#### 2.3. The Third-Order Birch–Murnaghan EOS

#### 2.4. The Fourth-Order Birch–Murnaghan EOS

## 3. Equations of States from Other Finite Strain Definitions

- Lagrangian scheme, expansion of squared length, referred to as the second-power Lagrangian EOS;
- Eulerian scheme, linear expansion of length, referred to as the first-power Eulerian EOS;
- Eulerian scheme, expansion of cubed length, referred to as the third-power Eulerian EOS.

#### 3.1. The Second-Power Lagrangian EOS

#### 3.2. Finite Strains and EOSs from Linear Length

#### 3.3. Finite Strains and EOSs from Cubed Length

## 4. Discussion

#### 4.1. Comparison of Birch–Murnaghan EOSs of Different Orders

^{−1}but rather ${{K}_{{T}_{0}}}^{\u2033}=-0.267$ GPa

^{−1}. Figure 2b,c also show the fourth-order Birch–Murnaghan EOSs of Au and MgO whose second pressure derivatives of the isothermal bulk modulus at zero pressure are set to ${{K}_{{T}_{0}}}^{\u2033}=0$. The pressure at the bottom of the Earth’s mantle is 136 GPa. To obtain such pressure by the third-order Birch–Murnaghan EOSs of Au and MgO, the compression must be $V/{V}_{0}=0.736$ and $V/{V}_{0}=0.678$, respectively. Under these compressions, the fourth-order Birch–Murnaghan EOSs with ${{K}_{{T}_{0}}}^{\u2033}=0$ give 160 and 152 GPa. Thus, approximations that ignore higher-order terms of finite strain and equate higher-order derivatives of the isothermal bulk modulus to zero are not identical in the Birch–Murnaghan EOSs.

_{3}using Au and MgO pressure scales to interpret the phase boundary pressure as the D” layer, their argument contains errors on the order of a few hundred km simply by neglecting ${K}_{{T}_{0}}^{\prime}$ as ${{K}_{{T}_{0}}}^{\u2033}=0$.

#### 4.2. Eulerian Versus Lagrangian Schemes

#### 4.3. Equations of State Obtained from Expansions of Different Powers of Length

#### 4.4. Examination of Equations of State Using Pressure Scale-Free Experimental Data

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A1. Derivation of Parameter ${a}_{2}$

#### Appendix A2. Derivation of Parameter ${\xi}_{1}$

#### Appendix A3. Derivation of Parameter ${\xi}_{2}$

_{2}, we present the third volume derivative of pressure, ${\left({\partial}^{3}P/\partial {V}^{3}\right)}_{T}$. By differentiating Equation (A7) with respect to volume, we have:

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**Figure 1.**Finite uniform compression of a cube. The initial edge length of the cube ${X}_{0}$ is reduced to the final edge length $X={X}_{0}+u$. The volume of the cube accordingly decreases from ${V}_{0}={X}_{0}^{3}$ to $V={X}^{3}$.

**Figure 2.**Comparison of the second-, third- and fourth-order Birch–Murnaghan equations of state (EOSs). (

**a**) NaCl (B1) data are from Matsui et al. [5]; (

**b**) Au data are from Song and Yoneda [6]; (

**c**) MgO data are from Kono et al. [7]. Red and blue colors denote the Eulerian and Lagrangian schemes, respectively. The dashed, solid, and dotted curves are of the second-, third- and fourth-order EOSs, respectively. The green solid curve denotes the pressure-volume relation obtained by integration of the definition of the bulk modulus (Equation (59)).

**Figure 3.**Comparison of the Eulerian and Lagrangian finite strains derived by expansion of the powered length as a function of compression $V/{V}_{0}$. The light red, red, and dark red curves denote the finite strains by expansion of the linear, squared, and cubed lengths in the Eulerian scheme, respectively, and the blue curve denotes those of the squared lengths in the Lagrangian scheme.

**Figure 4.**Pressures given by the second- (light red), third- (red) and fourth- (dark red) power Eulerian and the second-power Lagrangian (blue) third-order EOSs. (

**a**) NaCl; (

**b**) Au; (

**c**) MgO.

**Figure 5.**Hugoniot curves of MgO reproduced by the various EOSs using pressure scale-free data following Tange et al. [11]. The dashed light-red, thin-solid red, dotted dark-red, and solid blue curves are obtained from the first-, second- and fourth- Eulerian and second-power Lagrangian EOSs, respectively. The light-gray square and dark-gray circle are experimental data obtained by Marsh [12] and Duffy and Ahrens [13], respectively.

Material | ${\mathit{K}}_{{\mathit{T}}_{0}}\text{}\left(\mathbf{GPa}\right)$ | ${{\mathit{K}}_{{\mathit{T}}_{0}}}^{\prime}$ | ${{\mathit{K}}_{{\mathit{T}}_{0}}}^{\prime \prime}\text{}\left({\mathbf{GPa}}^{-1}\right)$ | Reference |
---|---|---|---|---|

NaCl | 23.7 | 5.14 | −0.392 | [5] |

Au | 160.44 | 6.56 | 0 | [6] |

MgO | 160.64 | 4.35 | 0 | [7] |

Parameter | 2nd-Power Eulerian EOS (Birch-Murnaghan) | 2nd-Power Lagrangian EOS | 1st-Power Eulerian EOS | 3rd-Power Eulerian EOS |
---|---|---|---|---|

${V}_{0}$ (Å^{3}) | 74.698 (fixed) | 74.698 (fixed) | 74.698 (fixed) | 74.698 (fixed) |

${K}_{{T}_{0}}$ (GPa) | 160.64 | 160.55 | 160.64 | 160.63 |

${K}_{{T}_{0}}^{\prime}$ | 4.221 | 4.909 | 4.293 | 4.347 |

${\theta}_{0}$ (K) | 761 | 761 (fixed) | 761 (fixed) | 761 (fixed) |

${\gamma}_{0}$ | 1.431 | 1.496 | 1.436 | 1.440 |

$a$ | 0.29 | 0 (fixed) | 0.20 | 0.14 |

$b$ | 3.5 | 4.4 | 5.5 |

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**MDPI and ACS Style**

Katsura, T.; Tange, Y. A Simple Derivation of the Birch–Murnaghan Equations of State (EOSs) and Comparison with EOSs Derived from Other Definitions of Finite Strain. *Minerals* **2019**, *9*, 745.
https://doi.org/10.3390/min9120745

**AMA Style**

Katsura T, Tange Y. A Simple Derivation of the Birch–Murnaghan Equations of State (EOSs) and Comparison with EOSs Derived from Other Definitions of Finite Strain. *Minerals*. 2019; 9(12):745.
https://doi.org/10.3390/min9120745

**Chicago/Turabian Style**

Katsura, Tomoo, and Yoshinori Tange. 2019. "A Simple Derivation of the Birch–Murnaghan Equations of State (EOSs) and Comparison with EOSs Derived from Other Definitions of Finite Strain" *Minerals* 9, no. 12: 745.
https://doi.org/10.3390/min9120745