Photoionization profiles of metal clusters and the Fowler formula

Metal-cluster ionization potentials are important characteristics of these “artificial atoms,” but extracting these quantities from cluster photoabsorption spectra, especially in the presence of thermal smearing, remains a big challenge. Here we demonstrate that the classic Fowler theory of surface photoemission does an excellent job of fitting the photoabsorption profile shapes of neutral In${}_n$${}_{=3-34}$ clusters [Wucher et al., New J. Phys. 10, 103007 (2008)]. The deduced ionization potentials extrapolate precisely to the bulk work function, and the internal cluster temperatures are in close agreement with values expected for an ensemble of freely evaporating clusters. Supplementing an earlier application to potassium clusters, these results suggest that the Fowler formalism, which is straightforward and physical, may be of significant utility in metal-cluster spectroscopy. It is hoped also that the results will encourage a comprehensive theoretical analysis of the applicability of bulk-derived models to cluster photoionization behavior, and of the transition from atomic and molecular-type to surface-type photoemission.

Figure 1

Appearance energies of K${}_{30\le n\le 101}$ clusters determined by a Fowler fit [11] to photoionization profiles [12]. With $R=r_{\mathrm{s}}{n}^{1/3}$ ($r_s=4.86a_0$), linear regression gives $E_I=2.30$ eV $+$ 0.34$e^2/R$ for all points, or 2.29 eV + 0.38$e^2/R$ for the spherical closed-shell clusters. The potassium metal work function is 2.3 eV. Inset: examples of threshold data [12] and finite-temperature Fowler functions (solid lines) [11].

Figure 2

Examples of Fowler fits for indium clusters (top: In${}_{33}$, bottom: In${}_{14}$). On the right are Fowler plots following Eq. (1) (lines: finite-temperature Fowler functions, dots: experimental data [18]). On the left are the corresponding ionization threshold profiles (line: Fowler functions, dots: experimental data, arrows: deduced appearance energies $E_I$).

Figure 3

Blue solid circles: appearance energies of In${}_{3\le n\le 34}$ clusters determined by a Fowler fit to photoionization profiles [18], as described in the text and illustrated in Fig. 2. With $R=r_{\mathrm{s}}$(3$n$)${}^{1/3}$ ($r_s=2.41a_0$), linear regression for $n>$12 (heavy blue line) gives $E_I$$=$ 4.04 eV + 0.46$e^2/R$. The $y$ intercept is in excellent agreement with the polycrystalline work function of indium, 4.09 eV. The grey diamonds are $E_I$ values deduced from the same data via a thermal-oscillator fit in Ref. [18]. The triangles are $E_I$ values from Ref. [23] estimated from measurements using a cold laser-vaporization cluster source. For reference, the atomic ionization potential is marked on the right-hand axis.