In the figure the equilibrium bond distance between two atoms is indicated by the minimum in the potential energy. In a system where the thermal energy is not dominant (i.e., low temperatures) the potential energy can be approximated harmonically, and therefore the displacements, $\Delta r = (r -r_{eq})$, due to the forces will be symmetric in an averaged sense. This is highlighted in the figure inset showing that the harmonic (blue) matches well with the potential curve for small $\Delta r$.
When the thermal energy in the system begins to become significant, the harmonic approximation is no longer appropriate and the displacements are non-symmetric about the equilibrium bond distance. This asymmetry gives rise to thermal expansion of a material. This is further shown in the inset with the anharmonic curve (red) showing better agreement with the potential curve than the harmonic at larger $\Delta r$.
The atomic view of thermal expansion manifest in bulk through the isotropic (i.e. volumetric) thermal expansion of a material which is given by the coefficient of thermal expansion,
$$\alpha = \frac{1}{V}\frac{\partial V}{\partial T}_P $$
where $V$ is the volume, $T$ the temperature, and $P$ indicates the derivative at constant pressure. It is also common to refer to this as the coefficient of thermal expansion (CTE).
The quote for this post is:
Science knows no country, because knowledge has no identity, and therefore exist to illuminate the world.
-Louis Pasteur (modified)
References
When the thermal energy in the system begins to become significant, the harmonic approximation is no longer appropriate and the displacements are non-symmetric about the equilibrium bond distance. This asymmetry gives rise to thermal expansion of a material. This is further shown in the inset with the anharmonic curve (red) showing better agreement with the potential curve than the harmonic at larger $\Delta r$.
The atomic view of thermal expansion manifest in bulk through the isotropic (i.e. volumetric) thermal expansion of a material which is given by the coefficient of thermal expansion,
$$\alpha = \frac{1}{V}\frac{\partial V}{\partial T}_P $$
where $V$ is the volume, $T$ the temperature, and $P$ indicates the derivative at constant pressure. It is also common to refer to this as the coefficient of thermal expansion (CTE).
The quote for this post is:
Science knows no country, because knowledge has no identity, and therefore exist to illuminate the world.
-Louis Pasteur (modified)
References
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