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 elsewhereDetailed BalanceV. Weisskopf, Phys. Rev. 52, 295-303 (1937) Same physicsDifferent 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) 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) (single atom evaporation) Important point: Sustains thermal equilibrium, Extra benefit: Works for all types of emitted particles.
Ingredients Observable Observable Observable Known 1) Cross section 2) Level densities of parent 3) Level density of product cluster 4) Level density of evaporated atom Angular momentum not considered here.
Microcanonical temperature Total rates require integration over kinetic energy releases Define OBS: Tm is daughter temperature
Total rate constants, example Geometrical cross section:
Numerical examples (Monomer evaporation) Evaporated atom Au = geometric cross section = 10Å2 Evaporated atom C = geometric cross section = 10Å2 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, s(e) Measure or guess Stating the cross section in detailed balance theory is equivalent to specifying the transition state in RRKM
Kinetic energy release Simple examples: General (spherical symmetry): Geometric cross section: Langevin cross section: Capture in Coulomb potential:
Kinetic energy release Special cases: Motion in spherical symetric external potentials. Capture on contact.
Average kinetic energy releases If no reverse activation barrier, values between 1 and 2 kBTm: Geometric cross section: 2 kBTm Langevin cross section: 3/2 kBTm Capture in Coulomb potential: 1 kBTm OBS: The finite size of the cluster will often change cross sections and introduce different dependences.
Barriers and cross sections No reverse activation barrier Reverse activation barrier EB Reaction coordinate Reaction coordinate = 0 for < EB
Level densities Vibrational degrees of freedom dominates Calculated as collection of harmonic oscillators. Typically quantum energy << evaporative activation energy At high E/N: (E0 = sum of zero point energies) More precise use Beyer-Swinehart algorithm, but frequencies normally unknown
Level densities Warning: clusters may not consist of harmonic oscillators 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