Nondipole effects in the triply differential cross section for double photoionization of He

Lowest-order nondipole effects are studied systematically in double photoionization (DPI) of the He atom. Ab initio parametrizations of the quadrupole transition amplitude for DPI from the ${}^{1}S_0$ state are presented in terms of the exact two-electron radial matrix elements. Analytic expressions for these matrix elements within lowest-order perturbation theory (LOPT) in the interelectron interaction are also given. The corresponding parametrizations for the dipole-quadrupole triply differential cross section (TDCS) are presented for the case of an elliptically polarized photon. A general analysis of retardation-induced asymmetries of the TDCS including the circular dichroism effect at equal energy sharing is presented. Numerical LOPT estimates of nondipole asymmetries in photoelectron angular distributions for the cases of linear and circular polarization and of the circular dichroism effect at equal energy sharing are presented. We find that experimental observation of nondipole effects at excess energies of the order of tens to hundreds of eV should be feasible in TDCS measurements. Our numerical results exhibit a nondipole forward-backward asymmetry in the TDCS for DPI of He at an excess energy of 450 eV that is in qualitative agreement with existing experimental data.

Figure 1

Geometry suitable for observation of retardation effects in the TDCS for DPI by linearly polarized light. The vectors ${\mathbf{p}}_1$, ${\mathbf{p}}_2$, $\mathbf{ϵ}$, and $\mathbf{k}$ lie in one plane.

Figure 2

Geometries suitable for observation of retardation-induced light polarization effects. (a) Geometry at which the equal energy sharing CD is maximal. Electrons are detected in the xz plane, at angles of $\pm \theta ∕2$ with respect to the $x$ axis. The photon wave vector $\mathbf{k}$ lies in the yz plane and makes an angle 45° with the $z$ axis. (b) Geometry for observation of retardation-induced asymmetries in the TDCS. The first electron is ejected along the $x$ axis and the second one (whose angular distribution exhibits the asymmetry) along $({\theta }_2,{\varphi }_2)$. (Concerning the angles ${\phi }_{xz}$ and ${\phi }_{yz}$, see Fig. 7.)

Figure 3

Schematic diagrams for first-order perturbative contributions to the DPI amplitude. (a) Final-state correlation; (b) ground-state correlation. Two additional contributing diagrams with exchanged ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ are not shown.

Figure 4

Present results for the TDCS for DPI of He at an excess energy of 20 eV for the case of linear polarization and coplanar geometry. Energy sharings and ejection angles are as indicated. Full curves, dipole-quadrupole results; dashed curves, dipole-quadrupole results for the opposite direction of the photon beam; dotted curves, EDA results. (Note that $1.0\mathrm{b}=1.0\times {10}^{-24}{\mathrm{cm}}^2$.)

Figure 5

Present TDCS results for DPI of He at an excess energy of 20 eV for circular polarization and for the geometry shown in Fig. 2b. Full curves, dipole-quadrupole results; dashed curves, dipole-quadrupole results for the opposite direction of the photon beam; dotted curves, corresponding EDA results. (Note that $1.0\mathrm{b}=1.0\times {10}^{-24}{\mathrm{cm}}^2$.)

Figure 6

Present results for the DPI TDCS at an excess energy of 100 eV for linear polarization and coplanar geometry for three energy sharings and ejection angles. Full curves: dipole-quadrupole results; dashed curves: dipole-quadrupole results for the inverted direction of the photon beam; dotted curves: the EDA results. (Note that $1.0\mathrm{b}=1.0\times {10}^{-24}{\mathrm{cm}}^2$.)

Figure 7

Present TDCS results at an excess energy of 100 eV for circular polarization $(\xi =+1)$. Energy sharings and ejection angles are as indicated. Full curves: dipole-quadrupole results; dashed curves: dipole-quadrupole results for the inverted photon beam direction; dotted curves: EDA results. (Note that $1.0\mathrm{b}=1.0\times {10}^{-24}{\mathrm{cm}}^2$.)

Figure 8

Comparison of the present TDCS results for an excess energy of 450 eV for linear polarization and for coplanar geometry with the normalized experimental data of Ref. 12. The direction of the photon wave vector $\widehat{\mathbf{k}}$ and polarization $\widehat{\mathbf{ϵ}}$ is as shown in (a); the electron having momentum $p_1$ (and energy $E_1$) is ejected along $ϵ$. Full curves: dipole-quadrupole results; dashed curves: EDA results. Note that both the experimental data and our nondipole results exhibit a noticeable forward-backward asymmetry as compared to our EDA results: the angular distributions of the fast electron are shifted along the direction of the vector $\mathbf{k}$, especially in the angular ranges $0°<{\theta }_2<90°$ and $270°<{\theta }_2<360°$; cf. Table . (Note that $1.0\mathrm{b}=1.0\times {10}^{-24}{\mathrm{cm}}^2$.)

Figure 9

Present results exhibiting the equal-energy-sharing CD effect, i.e., the sensitivity of the dipole-quadrupole equal-energy-sharing TDCS to the sign of the degree of circular polarization, $\xi$, for three excess energies. The geometry is as shown in Fig. 2a. Full curves: $\xi =+1$; dashed curves: $\xi =-1$. (Note that $1.0\mathrm{b}=1.0\times {10}^{-24}{\mathrm{cm}}^2$.)