Decay-rate power-law exponent as a link between dissociation energy and temperature

Indium-cluster anions ${\mathrm{In}}_n^{-}$ are probed for delayed dissociation by photoexcitation in a multi-reflection time-of-flight device. In addition to prompt dissociation with below-microsecond decay constants, we observe reactions on timescales of several tens to hundreds of microseconds. These time-resolved decay-rate measurements reveal a power-law behavior in time which can be traced back to the clusters' energy distribution due to their production by laser ablation in high vacuum. Modeling energy distributions from such a production allows us to connect the cluster-specific dissociation energy with the ensemble temperature through experimentally determined power-law exponents.

Figure 1

Experimental setup including laser-ablation source and MR-ToF analyzer. The ion flight path is indicated in red, the source- and excitation-laser beams in green.

Figure 2

Reference (a) and photodissociation spectrum of ${\mathrm{In}}_{12}^{-}$ (b) and ${\mathrm{In}}_{15}^{-}$ (c) after 400 revolutions. For details, see text.

Figure 3

(a) Relative abundances of fragment sizes $(n-1)$ through $(n-4)$ after photoexcitation of ${\mathrm{In}}_n^{-}$ clusters at $\lambda =532\mathrm{nm}$. (b) Ratio of electron detachment vs dissociation after photoexcitation.

Figure 4

(a) ToF spectrum of delayed ${\mathrm{In}}_{12}^{-}$ signals from photoexcitation of ${\mathrm{In}}_{13}^{-}$ after 96.84-ms flight time. The precursor is stored for four post-excitation laps (two lift transitions per lap) before ejection. (b) Ion counts of delayed ${\mathrm{In}}_{12}^{-}$ signals and power-law fit to the data. (c) Power-law exponent from measurements after different ion flight times prior to photoexcitation. Red lines indicate weighted mean and its uncertainty.

Figure 5

(a) Theoretical distribution of total vibrational energy of an ensemble of ${\mathrm{In}}_{13}$ clusters ($f=3n-6=33$ degrees of freedom). The two examples show ensembles with temperatures $T=300\mathrm{K}$ (blue) and $1000\mathrm{K}$ (orange). (b) Cluster decay time for $D=2.64\mathrm{eV}$ [47] and ${\nu }_D=129\mathrm{K}{k}_B/\hslash$ [50] calculated from the above energy distributions for single-photon absorption. The experimental window for observation of delayed fragmentation is shown in gray.

Figure 6

(a) Calculated power-law exponent for ${\mathrm{In}}_{13}^{-}$ cluster ensembles as a function of temperature for $D=2.64\mathrm{eV}$ (black line); experimental $p$ value in red. (b) Dissociation energies as a function of ensemble temperature for experimental power-law exponent $p=-1.06\left(3\right)$.

Figure 7

Probability density of vibrational energy for cluster sizes $n=3$, 4, 6, 10, 18 ($f=3n-6$) resulting from (A16) for $T=1000\mathrm{K}$. Gaussian functions following (A17) are added for each distribution (dashed red lines).

Figure 8

(a) Distributions of vibrational energy for cluster size $n=13$ for $T=800$, 1000, $1200\mathrm{K}$. (b) Distribution of lifetimes with respect to loss of a neutral monomer ($D=2.64\mathrm{eV}$ [47], ${\nu }_D=129\mathrm{K}{k}_B/\hslash$ [50]) resulting from the energy distributions of (a) and additional photoexcitation with photon energy $2.33\mathrm{eV}$.

Figure 9

Individual decay rates calculated from the lifetime distributions shown in Fig. 8 (red lines) and total sum of all rates (blue, orange, and yellow lines). A power-law curve is fitted to the sum in the experimental observation window (black lines).