## Materials Science and Engineering: An Introduction

$c_{v}$ rises with increasing temperature at temperature near 0 K because at this temperature, the allowed vibrational energy levels of the lattice waves are far apart relative to the thermal energy. As temperatures increases, more of the lattice waves are excited by the available thermal energy, and the ability of the material to absorb energy increases.
$c_{v}$ rises with increasing temperature at temperature near 0 K because at this temperature, the allowed vibrational energy levels of the lattice waves are far apart relative to the thermal energy. As temperatures increases, more of the lattice waves are excited by the available thermal energy, and the ability of the material to absorb energy increases.