Theory of vibrational cooling in molecular crystals: Application to crystalline naphthalene

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Abstract
The process of vibrational cooling (VC) is theoretically investigated in the molecular crystal naphthalene. Specificially we consider the process where a highly excited vibration cools by emitting lower energy vibrations (vibrational relaxation, or VR) and phonons. We also consider the subsequent cooling of emitted optic phonons by emission of acoustic phonons. Using previously determined vibrational lifetimes [J. R. Hill et al., J. Chem. Phys. 8 8, 949 (1988)], a consistent transition rate matrix is obtained which describes VR of all vibrations and optic phonons at all temperatures. Then a Master equation is solved numerically to obtain the time dependent vibrational populations of all states following impulse excitation of a high frequency vibration. These results are compared to a previously derived analytic model for VC in molecular crystals [J. R. Hill and D. D. Dlott, J. Chem. Phys. 8 9, 830 (1988)]. In that theory, which is shown to be in good agreement with the naphthalene calculation, the excess vibrational excitation moves to lower energy states and broadens as time increases. The motion toward lower energy states is described by a temperature independent ‘‘vibrational velocity’’ (emitted energy per unit time). In naphthalene, the vibrational velocity is V0 ≊9 cm−1 /ps. The VC process occurs on a time scale as much as an order of magnitude longer than an individual VR step. Although VR is highly temperature dependent, VC is not. The VC calculations are used to predict the decay from the initial state, the time dependent populations of transient vibrational excitations, and the return to the vibrationless ground state. All these quantities are directly related to experimental observables such as incoherent anti‐Stokes Raman scattering and hot luminescence.