Decoherence-Induced Universality in Simple Metal Cluster Photoelectron Angular Distributions

Measured angular distributions of photoelectrons from size-selected copper and sodium cluster anions are demonstrated to exhibit a universal behavior independent of the initial electron state, cluster size, or material, which can be traced back to momentum conservation upon photoemission. Quantum simulations reproduce the universality under the assumption that multielectron dynamics localizes the emission on the cluster surface and renders the cluster opaque to photoelectrons, thereby quenching interference effects that would otherwise obscure this almost classical behavior.

Figure 1

(a) Anisotropy parameter of the angular distributions for the 1g-derived states of ${\mathrm{Na}}_{55}^{-}$ and ${\mathrm{Cu}}_{55}^{-}$ as a function of the final photoelectron momentum $k$ (see Fig. S1 for the photoelectron spectra). (b) Same data as a function of the reduced momentum $kR$ ($R$: cluster radius). (c) Anisotropy parameter as a function of $kR$ for different states of different sodium cluster anions. (d) Same data as a function of $kR/L$, where $L$ is the approximate angular momentum of the initial state of the electron. Gray dashed line: classical model. (e) Emission probability as a function of the angle $\theta$ between the emission direction and the electric field vector of the laser light for three values of the anisotropy parameter $\beta$. (f) Classical model: the electron is emitted from the cluster surface with an angle $\alpha$ to the surface normal, as determined by its momentum $k$ and the surface projection $k_{||}=L/R$. The absorption probability depends on the angle $\vartheta$, the emission angle $\theta$ on $\alpha$, $\vartheta$, and the azimuthal emission angle $\zeta$ (see text for details).

Figure 2

(a) Anisotropy parameter as a function of the scaled final electron momentum $kR/L$ for different initial angular momentum states of the electron, assuming vanishing photoelectron-cluster interaction. (b) Same as (a) but taking photoelectron-cluster interaction into account, by assuming a spherical box potential of 6 eV depth, for two initial angular momentum states and two cluster radii (of ${\mathrm{Na}}_{33}^{-}$ and ${\mathrm{Na}}_{55}^{-}$).

Figure 3

(a) Comparison of three model calculations of angular distributions (electron-cluster interaction neglected): classical calculation, assuming outward electron emission with conservation of tangential momentum (dashed red line); WKB-like treatment of the photoemission, assuming complete opaqueness of the cluster for the photoelectrons (dashed red and blue line), giving results identical to the classical ones in the direct emission regime; quantum calculation for different initial angular momentum states assuming free electron continuum waves subject to an absorptive boundary condition at the cluster surface (green, dark red, and dark gray line). The results converge to the classical and semiclassical ones for large $L$. (b) Similar calculation, additionally taking the exchange-correlation interaction between the emitted electron and the cluster into account [12], for ${\mathrm{Na}}_{55}^{-}$ and four initial angular momenta in comparison to the data from Fig. 1 and copper cluster results (see Fig. S5 for further examples).