Physics of Collinear Inelastic Collision

1. Energy Transfer in Inelastic and Reactive Collisions using Algebraic methods Tim Wendler, Manuel Berrondo, Jean-Francois Van Huele Physics of Collinear Inelastic Collision Results of Inelastic Collision Collinear Reactive Collision • Dynamics for diatomic molecule reduced to single coordinate with reduced mass • Natural coordinates s and x smoothly connect reactants and product reduced mass schemes • Hamiltonian transformed to natural coordinates and their conjugate momenta • Curvature function k (s) in kinetic energy • Diatomic state-to-state transition probability over t • Initial: ground state • Final: L.C. of time dependent states • Remove center of mass and rescale energy units • Treat translationcoordinate x classically • Treat vibrational coordinate yquantumly • Probability landscape for transitions single initial state Classical Trajectories Initial Superposition of States • Classical reaction dynamics simple to analyze in natural coordinates • Anharmonicity in quantum coordinate more accurately models the dynamics curvature Hamiltonian • Harmonic oscillator for the quantum variable • Landau-Teller model applied to classical variable • Classical coordinate x coupled to expectation value exhibits asymmetry (see also vib. phase space) Ensemble of Colliding Oscillators Inelastic Collision Internal Energy Lost Lie Algebra • Algebraic approach  both phase space dynamics and transitions • Both Hamiltonian and time evolution operator constructed using four Lie Algebra basis elements: f i Internal Energy Gained Complete Dissociation • Equation of motion for U(t) follows from Hamiltonian: • Initial Boltzmann distribution of diatomic SHO’s before and after colliding with atoms at 40K • a non-equilibrium redistribution of states: temperature undefined Quantum Effects • Quantum harmonic oscillator in x • Resonances appear with certain initial velocities • Quantum transition rates allude to quanta being transferred to new bond Equations of Motion for a’s • Time evolution dynamics at initial 40K towards non-equilibrium • Algebraic approach produces coupled ODEs: solved numerically • Same at 300K