The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. It has units of temperature and is defined as
θ vib = h ν ~ c k B = h ν k B {\displaystyle \theta _{\text{vib}}={\frac {h{\tilde {\nu }}c}{k_{\text{B}}}}={\frac {h\nu }{k_{\text{B}}}}}
where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant, c {\displaystyle c} is the speed of light, ν ~ {\displaystyle {\tilde {\nu }}} is the wavenumber, and ν {\displaystyle \nu } (Greek letter nu) is the characteristic frequency of the oscillator. The vibrational temperature is used commonly when finding the vibrational partition function.
References Statistical thermodynamics University Arizona
See also Rotational temperature Rotational spectroscopy Vibrational spectroscopy Infrared spectroscopy Spectroscopy