# Atomistic Modeling of the Negative Thermal Expansion in δ- Plutonium Based on the Two-State Description

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## Abstract

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## 1. Introduction

## 2. Simulation Methods

#### 2.1. Two-State Model Description of Many-Body Interacting Systems

**Figure 1.**Binding energy curves ${E}_{1}\left(V\right)$ and ${E}_{2}\left(V\right)$ for the two competing electronic states in an effective two-state description of δ-Pu. $\Delta V$ is the difference between the equilibrium volumes ${V}_{1}$ and ${V}_{2}$, while $\Delta E$ (=$E\left({V}_{2}\right)-E\left({V}_{1}\right)$) is the energy separation at equilibrium for the two states.

#### 2.2. Computational Details

## 3. Simulation Results

**Figure 2.**Calculated temperature dependence of the relative occupation of state 2 for a varying value of the mixing energy Δ (in eV). A similar curve obtained from the simple two-level statistics (with $\Delta E/{k}_{\mathrm{B}}\sim $700 K and ${g}_{1}/{g}_{2}=1$) is also included for reference (labeled as “Schottky”).

**Figure 3.**Calculated temperature dependence of the heat capacity per atom, scaled with the Boltzmann constant, for a varying value of the mixing energy Δ (in eV). The experimental data [48] in the stability range of δ-Pu are also included for comparison.

**Figure 4.**Calculated temperature dependence of the linear coefficient of thermal expansion for a varying value of the mixing energy Δ (in eV). The experimental data [48] in the stability range of δ-Pu are also included for comparison.

**Figure 5.**Schematic illustration of how volume contraction occurs in the Weiss two-state picture. The overall volume of the system is reduced as more atoms are excited to the small-volume state at higher temperature. This effect is in competition with the usual volume expansion due to the anharmonic effect of lattice vibrations. The fraction of small-volume atoms is exaggerated for visual effect in this representation.

**Figure 6.**Calculated temperature dependence of the atomic volume for a varying value of the mixing energy Δ (in eV). The experimental data [48] in the stability range of δ-Pu are also included for comparison.

## 4. Conclusions

## Acknowledgements

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Lee, T.; Baskes, M.I.; Lawson, A.C.; Chen, S.P.; Valone, S.M. Atomistic Modeling of the Negative Thermal Expansion in δ- Plutonium Based on the Two-State Description. *Materials* **2012**, *5*, 1040-1054.
https://doi.org/10.3390/ma5061040

**AMA Style**

Lee T, Baskes MI, Lawson AC, Chen SP, Valone SM. Atomistic Modeling of the Negative Thermal Expansion in δ- Plutonium Based on the Two-State Description. *Materials*. 2012; 5(6):1040-1054.
https://doi.org/10.3390/ma5061040

**Chicago/Turabian Style**

Lee, Tongsik, Michael I. Baskes, A. C. Lawson, Shao Ping Chen, and Steven M. Valone. 2012. "Atomistic Modeling of the Negative Thermal Expansion in δ- Plutonium Based on the Two-State Description" *Materials* 5, no. 6: 1040-1054.
https://doi.org/10.3390/ma5061040