Negative thermal expansion , coordination

In most solids vibrations parallel to bond directions decrease in frequency as the volume increases and the entropy (eq. (11.14)) increases with volume (dS/dV)T and the thermal expansion are positive. Negative thermal expansion is usually associated with more open structures where coordination numbers are low and vibrations perpendicular to bond directions can dominate the change in entropy with volume and thus the derivative (dS/dV)T. 

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