Rate Constants and Kinetic Energy Releases in Unimolecular Processes, Detailed Balance Results Klavs Hansen Göteborg University and Chalmers University of Technology Igls, march 2003

Realistic theories: RRKM, treated elsewhere Detailed Balance V. Weisskopf, Phys. Rev. 52, (1937) Same physics Different formulae Same numbers? Yes (if you do it right)

Physical assumptions for application of detailed balance to statistical processes 1) Time reversal, 2) Statistical mixing, compound cluster/molecule: all memory of creation is forgotten at decay General theory, requires input: Reaction cross section, Thermal properties of product and precursor

Detailed balance equation Number of states (parent) Evaporation rate constant Number of states (product) Formation rate constant Density of state of parent, product

Detailed balance (continued) (single atom evaporation) Important point: Sustains thermal equilibrium, Extra benefit: Works for all types of emitted particles. D = dissociation energy = energy needed to remove fragment, OBS, does not include reverse activation barrier. Can be incorporated (see remark on cross section later, read Weisskopf)

Ingredients 1) Cross section 2) Level densities of parent 3) Level density of product cluster 4) Level density of evaporated atom Observable Known Angular momentum not considered here.

Microcanonical temperature Total rates require integration over kinetic energy releases Define OBS: T m is daughter temperature

Total rate constants, example Geometrical cross section:

Numerical examples Evaporated atom Au = geometric cross section = 10Å 2 Evaporated atom C = geometric cross section = 10Å 2 (Monomer evaporation) g = 2 g = 1

Dimer evaporation Replace the free atom density of states with the dimer density of states (and cross section) Integrations over vibrational and rotational degrees of freedom of dimer give rot and vib partition function:

Kinetic energy release Given excitation energy, what is the distribution of the kinetic energies released in the decay? Depends crucially on the capture cross section for the inverse process, Stating the cross section in detailed balance theory is equivalent to specifying the transition state in RRKM  Measure or guess

Kinetic energy release General (spherical symmetry): Geometric cross section: Langevin cross section: Capture in Coulomb potential: Simple examples:

Kinetic energy release Special cases: Motion in spherical symetric external potentials. Capture on contact.

If no reverse activation barrier, values between 1 and 2 k B T m : Geometric cross section: 2 k B T m Langevin cross section: 3/2 k B T m Capture in Coulomb potential: 1 k B T m OBS: The finite size of the cluster will often change cross sections and introduce different dependences. Average kinetic energy releases

No reverse activation barrier Reverse activation barrier Barriers and cross sections Reaction coordinate = 0 for < E B EBEB

Level densities Vibrational degrees of freedom dominates Calculated as collection of harmonic oscillators. Typically quantum energy << evaporative activation energy At high E/N: (E 0 = sum of zero point energies) More precise use Beyer-Swinehart algorithm, but frequencies normally unknown

Warning: clusters may not consist of harmonic oscillators Level densities Examples of bulk heat capacities:

Level Densities Heat capacity of bulk water

What did we forget? Oh yes, the electronic degrees of freedom. Not as important as the vibrational d.o.f.s but occasionally still relevant for precise numbers or special cases (electronic shells, supershells) Easily included by convolution with vib. d.o.f.s (if levels known), or with microcanonical temperature