Temporal Evolution of Cluster Ensembles, with Focus on Power Laws Olof Echt, University of New Hampshire March 17, 2003 Igls, ECCN We would like to have very basic talks on various statistical models and their applications...

Consider decay of an ensemble of clusters: A n-1 + Aevaporation A n + + eelectron emission A p + A q fission A n (E)  Similarly for parent ions A n - or A n z+ Abundance of parents Yield of products Exponential decay if k(E) constant in time

Example 1: State selected clusters Once isolated, C-14 decays strictly exponentially (k = const)

Example 2: Microcanonical ensemble Energy distribution k = const

Example 3: Canonical ensemble k = const

Example 1b: State selected clusters

k  const !!

Example 2b: Microcanonical ensemble Energy distribution k  const !!

Example 3b: Canonical ensemble k  const once ensemble is isolated

Example 4: Evaporative ensemble EE Note: This would not necessarily apply to C 60 * or C 60 + formed from C 60

5. The EEE ( everything-else-ensemble ) Power law with exponent p = - 1 Assumptions: g o (E) flat over range where k(E)  1/t Energy equilibrated Only one decay channel "Evaporative" cooling Distribution at time = 0

Some early examples for delayed electron emission (time-of-flight mass spectra) No fit Leisner et al., JCP 1993 Fit: Two rate constants Campbell et al., PRL 1992 C 60  C 60 + W n  W n +

Hansen et al., PRL 2001 Ag 5 -  Ag 5 Some systems, like Ag 5 - and C n -, exhibit power law with p  -1 over several orders of magnitude in time

t -1 Deviation from t -1 due to radiative cooling Andersen et al., PRL 1996 C 60 -  C 60 Competing channels: electron emission and radiation data Loss in electron emission rate may be used to quantify rate of radiative cooling Electron emission only: Electron emission k e plus radiation k r:

Hansen & Echt, PRL 1997 Electron TOF reflects time of formation, not KER Wurz & Lykke, JPC 1992 Spectra reconstructed from C 60  C e Competing channels: electron emission and dissociation E(k q+1 ) – E(k q ) = h ??

Deng & Echt, unpublished data C 60  e

Electron emission only Electron emission (k e ) plus C 2 evaporation (k a ) Assumptions to proceed: Arrhenius relation for k e (E) and k a (E), Observed channel (electrons) parasitic to dark channel (C 2 emission), i.e. k e « k a No other dark channels (radiation) g o (E) flat over range where k(E)  1/t Energy equilibrated Model for competition between two activated processes

General Result: Electron yield follows power law with p  -E e /E a (to first order) Advantage: Find E a if E e is known. No knowledge of Arrhenius prefactors required Specific Results: Experiment: p = 0.64 ± 0.1. Ionization energy of C 60 is E e = 7.6 eV  E a = 11.9 ± 1.9 eV for neutral C 60 (11.4 eV for ion) Internal consistency, and consistency with other data, is not trivial: k e « k a (assumption), but probability that highly excited C 60 emits electron rather than C 2, is 2.6 % (measurement), k e « k a (assumption), but E e much smaller than E a (result), hence prefactors A a » A e

82  82 + Rohmund et al., JCP 2001Lassesson et al., JCP 2002 C 60  C 60 +

Two issues: 1. Radiative cooling

2. Structured energy distribution g o (E) Overall width Graininess 1 photon 2 photons ???

Delayed electron emission from C 60 Summary of experimental observations C 60 exhibits power law but p significantly different from –1 Competition between electron emission and C 2 loss some dependence of p on method of excitation Initial Energy distribution not sufficiently flat? strong deviation from power law for t » 10  s i) Geometric discrimination against ions that have moved? ii) Radiative cooling? iii) Nonstatistical effects (long-lived triplet state, bottleneck)? Suggested reading: Andersen, Bonderup, Hansen, J. Phys. B (2001)

All done But don't forget, there are other methods of thermometry: For example, kinetic energy release: Delayed decay of C 3 H 8 + Matt et al., IntJMS, 2003 Temperature of transition state