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Hard spheres are widely used as model particles in the statistical mechanical theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimick the extremely strong repulsion that atoms and spherical molecules feel at very close distances.
Formal definition
Hard spheres of diameter <math>\sigma<math> are particles with the following pair-wise interaction potential
- <math>V(\mathbf{r}_1,\mathbf{r}_2)=\left\{ \begin{matrix}0 & \mbox{if}\quad |\mathbf{r}_1-\mathbf{r}_2| \geq \sigma \\ \infty & \mbox{if}\quad|\mathbf{r}_1-\mathbf{r}_2| < \sigma \end{matrix} \right. <math>
where <math>\mathbf{r}_1<math> and <math>\mathbf{r}_2<math> are the positions of the two particles.
Virial coefficients
The first 3 virial coefficients for hard spheres can be determined analyically
| <math>\frac{B_2}{v_0}<math> | = | <math>4{\frac{}{}}<math>
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| <math>\frac{B_3}{{v_0}^2}<math> | = | <math>10{\frac{}{}}<math>
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| <math>\frac{B_4}{{v_0}^3}<math> | = | <math>-\frac{712}{35}+\frac{219 \sqrt{2}}{35 \pi}+\frac{4131}{35 \pi} \arccos{\frac{1}{\sqrt{3}}}\approx 18.365<math>
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Higher order ones can be determined numerically using monte carlo integration. We list
| <math>\frac{B_5}{{v_0}^4}<math> | = | <math>28.24 \pm 0.08<math>
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| <math>\frac{B_6}{{v_0}^5}<math> | = | <math>39.5 \pm 0.4<math>
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| <math>\frac{B_7}{{v_0}^6}<math> | = | <math>56.5 \pm 1.6<math>
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Literature
J. P. Hansen and I. R. McDonald Theory of Simple Liquids Academic Press, London (1986)
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