Photoionization of atoms confined in giant single-walled and multiwalled fullerenes

Results of a theoretical study of photoionization cross sections and photoelectron angular distribution asymmetry parameters of an atom $A$ confined in spherical multiwalled fullerenes are presented and compared with those obtained for the free atom $A$ as well as for $A$ confined in giant spherical single-walled fullerenes ${\mathrm{C}}_{n⩾240}$. Each individual nested fullerene cage is represented by a spherical, short-range, attractive potential characterized by the inner radius, width, and potential depth. Trends in the strength and location of confinement resonances in the photoionization spectra, as a function of the number of nested fullerenes, are delineated. As examples, calculations of the $1s$ photoionization of Ar confined in ${\mathrm{C}}_{60}$, ${\mathrm{C}}_{240}$, ${\mathrm{C}}_{540}$, ${\mathrm{C}}_{60}@{\mathrm{C}}_{240}$, and ${\mathrm{C}}_{60}@{\mathrm{C}}_{240}@{\mathrm{C}}_{540}$ are presented. The calculations are performed within the framework of a Hartree-Fock approximation which is appropriate for a deep inner-shell photoionization. The trends seen should be a general occurrence for an inner-shell photoionization of virtually any atom confined in spherical fullerenes not only Ar.

Figure 1

Angles $\theta$ and $\varphi$ in Eqs. (4, 6) as relative to the photoelectron momentum $\mathbf{p}$, photon momentum $\mathbf{k}$, and photon polarization vector $\mathbf{e}$.

Figure 2

Calculated HF results for the $1s$ photoionization cross section ${\sigma }_{1s}$ of free Ar (dashed line), as well as of Ar from $\mathrm{Ar}@{\mathrm{C}}_{60}$, $\mathrm{Ar}@{\mathrm{C}}_{240}$, $\mathrm{Ar}@{\mathrm{C}}_{540}$, $\mathrm{Ar}@{\mathrm{C}}_{60}@{\mathrm{C}}_{240}$, and $\mathrm{Ar}@{\mathrm{C}}_{60}@{\mathrm{C}}_{240}@{\mathrm{C}}_{540}$ formations, as indicated.

Figure 3

Calculated results for a nondipole photoelectron angular-asymmetry parameter ${\gamma }_{1s}\left(\omega \right)$ for the $1s$ photoionization of free Ar (dashed line), as well as $\mathrm{Ar}@{\mathrm{C}}_{60}$, $\mathrm{Ar}@{\mathrm{C}}_{60}@{\mathrm{C}}_{240}$, and $\mathrm{Ar}@{\mathrm{C}}_{60}@{\mathrm{C}}_{240}@{\mathrm{C}}_{540}$, as indicated.

Figure 4

Calculated results for the dipole $D_{1s\to ϵp}$ and quadrupole $Q_{1s\to ϵd}$ photoionization amplitudes (in atomic units), the ratio $\rho =\mid D_{1s\to ϵp}\mid ∕\mid Q_{1s\to ϵd}\mid$, as well as $\cos \Delta$ ($\Delta$ being the phase difference between the dipole and quadrupole photoionization amplitudes) for confined Ar, as indicated. A dashed line, when present, relates to the calculated data for free Ar.