An Apparent Dissociation Transition in Anharmonically Bound 1D Systems

For diatomic molecules and chains bound anharmonically by interactions such a the Lennard Jones and Morse potentials, we obtain analytical expressions for thermodynamic observables including the mean bond length, thermally averaged internal energy, and the coefficient of thermal expansion. These results are valid across the shift from condensed to gas-like phases, a dissociation transition marked by a crossover with no singularities in thermodynamic variables for finite pressures, though singular behavior appears in the low pressure limit. In the regime where the thermal energy $k_{\mathrm{B}} T$ is much smaller than the dissociation energy $D$, the mean interatomic separation scales as $\langle l \rangle = R_{e} + {\mathcal B} (P R_{e}/k_{\mathrm{B}} T)^{-2} e^{-D/k_{\mathrm{B}} T} \left( D/k_{\mathrm{B}} T \right )^{1/2}$ for both the Morse and Lennard Jones potentials where $p$ is a pressure term, $R_{e}$ is the $T = 0$ bond length, and ${\mathcal B}$ is a constant specific to the potential.