On Resonance Enhancement of E1-E2 Nondipole Photoelectron Asymmetries in Low-Energy Ne 2p Photoionization

Abstract

Earlier, a significant enhancement of the nondipole parameters ${\gamma }_{2p}$, ${\delta }_{2p}$, and ${\zeta }_{2p}={\gamma }_{2p}+3{\delta }_{2p}$ in the photoelectron angular distribution for Ne $2p$ photoionization was predicted, owing to resonance interference between dipole ($E1$) and quadrupole ($E2$) transitions. This enhancement manifests as narrow resonance spikes in the parameters due to the low-energy $2s\to 3p$ and $2s\to 4p$ dipole, as well as the $2s\to 3d$ quadrupole autoionizing resonances. Given the unique nature of this predicted enhancement, it requires further validation, specifically regarding whether these narrow spikes in ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ will or will not retain their values for experimental observation if one accounts for a typical finite frequency spread in the ionizing radiation. To address this, we revisit the previous study, now incorporating the effect of frequency spread in the ionizing radiation, assuming a spread as large as 5 meV at the half-maximum of the radiation’s intensity. In the present paper we demonstrate that while the frequency spread does affect the resonance enhancement of ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$, these parameters still retain quantitatively significant values to be observed experimentally. The corresponding calculations were performed using the random phase approximation with exchange, which accounts for interchannel coupling in both dipole and quadrupole photoionization amplitudes.

1. Introduction

Understanding both the qualitative and quantitative aspects of the interference between electric dipole ($E1$) and electric quadrupole ($E2$) transitions in the photoionization process is important not only for the interpretation of angle-resolved photoelectron spectroscopy but also from a fundamental point of view, as it offers a unique tool for developing, testing and refining many-body theories of atomic physics that account for nondipole excitation channels in low-photon-energy ionization processes. The $E1$-$E2$ interference arises from the first-order correction term $i\mathbf{k}·\mathbf{r}$ to the dipole approximation for the photoionization matrix element between initial and final states: $M_{if}=\langle f|(1+i\mathbf{k}·\mathbf{r})\mathbf{e}·\widehat{\mathbf{p}}|\mathbf{i}\rangle$, where $\mathbf{k}$ and $\mathbf{e}$ are the photon momentum and polarization vector, and $\mathbf{r}$ and $\widehat{\mathbf{p}}$ are the electron position vector and momentum operator, respectively [1,2,3,4] (and references therein). Such studies have particularly gained momentum after the breakthrough in experimental techniques that have made these interference effects well observable [5,6,7,8] (and references therein). Their unexpectedly strong significance, both predicted and measured, in angle-resolved spectra of photoelectrons at photon energies of not only a few keV but also at energies of tens of eV has led to the discovery of the breakdown of the dipole approximation, which was considered the only significant “player” at such energies. Even nondipole effects from higher-order electric-octupole and pure-electric-quadrupole excitation channels have been predicted and experimentally confirmed to be significant at photon energies ranging from only 100 eV to 1500 keV, in some cases [9]. The reader is referred, e.g., to a comprehensive review paper in [10] as well as a recent topical review in [11], where more details and references are presented. Here, we only note that, initially focused on photoelectron angular asymmetries in the one-electron single-photon photoionization of atoms, research on $E1$-$E2$ interference has expanded in recent years to cover its impact on a variety of phenomena, including harmonic generation [11], double-electron photoionization [12], photoionization time delays [13,14], ionization by twisted radiation [15], sequential two-photon atomic double ionization [16,17] and strong-field atomic ionization [18] (and references therein).

Low-energy angle-resolved photoemission has thus been under continuing scrutiny by theorists and experimentalists over a period of several decades, and it continues to be of interest.

In the present paper, we re-evaluate the previously predicted [19] $2s\to 3p$ and $2s\to 4p$ dipole, as well as the $2s\to 3d$ quadrupole, autoionizing resonances in the nondipole photoelectron angular asymmetry parameters (referred to as the ${\gamma }_{n\ell }$, ${\delta }_{n\ell }$ and ${\zeta }_{n\ell }$ parameters, as defined in [3]) for Ne $2p$ photoionization. While the corresponding resonance spikes in ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ were predicted to be significant, they were also narrow, and previous calculations did not account for the frequency spread in the ionizing radiation. The question of how such frequency spread might affect these resonance spikes—e.g., “could it potentially wipe the spikes out of the spectrum, thus obliterating nondipole effects in the angular asymmetry of $2p$ photoelectrons?”—has remained unexplored until now.

It is the aim of the present paper to address this gap in knowledge by performing a long-overdue refined calculation of the nondipole ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ angular asymmetry parameters in Ne $2p$ photoionization through the region of the $2s\to 3p$, $2s\to 4p$ and $2s\to 3d$ resonances, now incorporating the effects of frequency spread in the photon beam. We demonstrate that the frequency spread in the ionizing radiation does quantitatively affect the resonance spikes in ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$. Nevertheless, the spikes remain sufficiently strong to be experimentally detected.

We note that there are examples of calculated ${\gamma }_{n\ell }$, ${\delta }_{n\ell }$ and ${\zeta }_{n\ell }$ in the region of dipole and quadrupole resonances where the resonance spikes in these parameters are significantly wider and quantitatively larger than those observed in Ne. Notable examples include the gigantic resonance enhancements of these parameters near the $3p\to 3d$ dipole giant autoionizing resonance in Cr and Mn, as well as the $4p\to 4d$ resonance in Mo and Tc [20], or, especially, near the $3s\to 3d$ giant quadrupole resonance in the $3p$ photoionization of Ca [21]. In the latter case, the nondipole parameters were found to increase drastically, reaching as much as $65\%$ of the dipole counterpart. However, these remarkable examples relate to atoms in the metallic group of the periodic table, and as we understand it, experiments with metallic atom vapors are difficult to perform. In contrast, Ne is a noble gas, for which it is easier to conduct experiments. For this reason, we focus on the photoionization of Ne in the present work.

Corresponding calculations were performed in the framework of the Random Phase Approximation with Exchange (RPAE) [22], accounting for interchannel coupling in both dipole and quadrupole photoionization amplitudes. We have chosen RPAE because this ab initio approximation is well-established in atomic studies in general, and performs well for both dipole autoionizing resonant and continuum photoionization spectra in Ne in particular, as supported by detailed analyses and data in [23]. Additionally, good agreement has been demonstrated between experimentally measured data [7] on the one hand and RPAE- and relativistic RPA-calculated data [24] on the other, of the $E1$-$E2$ nondipole ${\gamma }_{2p}$ and ${\zeta }_{2p}$ photoelectron angular asymmetries in Ne $2p$ photoionization at photon energies up to 1000 eV. Consequently, as in prior works, RPAE is well-suited for our study. However, if, for example, studying $3s\to np$ autoionizing resonances in Ar, additional multielectron contributions beyond pure RPAE would be necessary, as noted long ago in [22,23,25,26] and relatively recently demonstrated in a study of a dipole attosecond time delay in Ar in [27] as well.

Atomic units (a.u.) ($\left|e\right|=m_{\mathrm{e}}=ħ=1$), where e and $m_{\mathrm{e}}$ are the electron’s charge and mass, respectively, are used throughout the paper unless specified otherwise.

2. Theory

To calculate the Ne ${\sigma }_{n\ell }$ photoionization cross section and the dipole ${\beta }_{n\ell }$ as well as the nondipole ${\gamma }_{n\ell }$ and ${\delta }_{n\ell }$ angular-asymmetry parameters, we use well known formulas. In a one-electron approximation, for $100\%$ linearly polarized light, the angle-differential photoionization cross section, ${\displaystyle \frac{d{\sigma }_{n\ell }}{d\Omega }}$, is given by [3]

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\sigma }_{n\ell }}{d\Omega }}={\displaystyle \frac{{\sigma }_{n\ell }}{4\pi }}\left[1+{\displaystyle \frac{{\beta }_{n\ell }}{2}}(3{cos}^2\theta -1)\right]+\Delta E_{12}.\end{array}$$

(1)

Here, $d\Omega$ is a solid angle, ${\sigma }_{n\ell }$ is the dipole photoionization cross section of the $n\ell$-subshell, and $\Delta E_{12}$ is the $E1$-$E2$ interference correction term:

$$\begin{array}{c}\hfill \Delta E_{12}={\displaystyle \frac{{\sigma }_{n\ell }}{4\pi }}({\delta }_{n\ell }+{\gamma }_{n\ell }{cos}^2\theta )sin\theta cos\varphi .\end{array}$$

(2)

Here, the spherical angles $\theta$ and $\varphi$ are defined in relation to directions of the photon momentum k, photoelectron momentum p, and photon polarization vector e.

The ${\sigma }_{n\ell }$, ${\beta }_{n\ell }$, ${\gamma }_{n\ell }$ and ${\delta }_{n\ell }$ are, in turn, given by [3]

$$\begin{array}{c}{\sigma }_{n\ell }={\displaystyle \frac{4{\pi }^2\alpha }{3(2\ell +1)}}{N}_{n\ell }\omega [\ell d_{\ell -1}^2+(\ell +1)d^2\ell +1],\hfill \end{array}$$

(3)

$$\begin{array}{c}{\beta }_{n\ell }={\displaystyle \frac{\ell (\ell -1)d_{\ell -1}^2+(\ell +1)(\ell +2)d_{\ell +1}^2}{(2\ell +1)[l{d}_{\ell -1}^2+(\ell +1)d_{\ell +1}^2]}}-{\displaystyle \frac{6\ell (\ell +1)d_{\ell -1}{d}_{\ell +1}cos({\eta }_{\ell +1}-{\eta }_{\ell -1})}{(2\ell +1)[\ell d_{\ell -1}^2+(\ell +1)d_{\ell +1}^2]}},\end{array}$$

(4)

$$\begin{array}{c}{\gamma }_{n\ell }={\displaystyle \frac{3k}{2[\ell d_{\ell -1}^2+(\ell +1)d_{\ell +1}^2]}}\sum _{{\ell }^{\prime },{\ell }^{″}}{A}_{{\ell }^{\prime },{\ell }^{″}}{d}_{{\ell }^{\prime }}{q}_{{\ell }^{″}}cos({\eta }_{{\ell }^{″}}-{\eta }_{{\ell }^{\prime }}),\hfill \end{array}$$

(5)

$$\begin{array}{c}{\delta }_{n\ell }={\displaystyle \frac{3k}{2[\ell d_{\ell -1}^2+(\ell +1)d_{\ell +1}^2]}}\sum _{{\ell }^{\prime },{\ell }^{″}}{B}_{{\ell }^{\prime },{\ell }^{″}}{d}_{{\ell }^{\prime }}{q}_{{\ell }^{″}}cos({\eta }_{{\ell }^{″}}-{\eta }_{{\ell }^{\prime }}).\hfill \end{array}$$

(6)

Here, $\alpha$ is the fine structure constant, $N_{n\ell }$ is the number of electrons initially in the ionized $n\ell$ subshell of the atom, $\omega$ and k are the photon energy and momentum, respectively, and $d_{{\ell }^{\prime }}$ and $q_{{\ell }^{″}}$ are the radial dipole and quadrupole photoionization amplitudes, respectively:

$$\begin{array}{c}\hfill d_{{\ell }^{\prime }}={\int }_0^{\infty }{P}_{ϵ{\ell }^{\prime }}\left(r\right)r{P}_{n\ell }\left(r\right)dr,\end{array}$$

(7)

$$\begin{array}{c}\hfill q_{{\ell }^{″}}={\int }_0^{\infty }{P}_{ϵ{\ell }^{″}}\left(r\right)r^2{P}_{n\ell }\left(r\right)dr.\end{array}$$

(8)

Here, ${\ell }^{\prime }=\ell \pm 1$, ${\ell }^{″}=\ell ,\ell \pm 2$, $P_{n\ell }\left(r\right)/r$ and $P_{ϵ\lambda }\left(r\right)/r$ are the radial parts of the electron wave functions in the bound $n\ell$ state and in the continuous $ϵ\lambda$ spectrum, respectively, ${\eta }_{\lambda }$ are the phase shifts of the wave functions of photoelectrons in the field of the positive ionic core, and the coefficients $A_{{\ell }^{\prime },{\ell }^{″}}$ and $B_{{\ell }^{\prime },{\ell }^{″}}$ depend on the combination of only orbital quantum numbers ${\ell }^{\prime }$ and ${\ell }^{″}$ and are tabulated in [3].

In the present work, we account for electron correlation in the form of various initial-state and final-state interchannel couplings between the dipole $2s\to ϵ\left(n\right)p$ and $2p\to ϵ\left(n\right)d$ and $2p\to ϵ\left(n\right)s$ on the one hand and, on the other hand, between the quadrupole $2s\to ϵ\left(n\right)d$ and $2p\to ϵ\left(n\right)f$ and $2p\to ϵ\left(n\right)p$ transitions into discrete and continuum spectra of Ne. To meet the goal, we employ a non-relativistic RPAE [22]. We choose RPAE because it is a well established method that has been used in atomic studies with a great success for decades.

In RPAE, the dipole and quadrupole matrix elements become complex. To account for this, the following substitution must be made in the above equations for ${\sigma }_{n\ell }$, ${\beta }_{n\ell }$, ${\gamma }_{n\ell }$ and ${\delta }_{n\ell }$ [28]:

$$\begin{array}{c}\hfill d_{\lambda }^2\to {\left|D_{{\lambda }^{\prime }}\right|}^2,\\ \hfill d_{\lambda }{w}_{{\lambda }^{\prime }}cos\Delta \eta \to & & (D_{{\lambda }^{\prime }}^{\prime }{W}_{{\lambda }^{\prime }}^{\prime }+D_{{\lambda }^{\prime }}^{″}{W}_{{\lambda }^{\prime }}^{″})cos\Delta \eta +(D_{{\lambda }^{\prime }}^{″}{W}_{{\lambda }^{\prime }}^{\prime }-D_{l^{\prime }}^{\prime }{W}_{{\lambda }^{\prime }}^{″})sin\Delta \eta ,\hfill \\ \hfill \Delta \eta ={\eta }_{{\lambda }^{\prime }}-{\eta }_{\lambda }.\end{array}$$

Here, w (W) stands either for a quadrupole matrix element q and Q, or a dipole matrix element d and D, calculated in a Hartree–Fock (HF) or RPAE approximation, respectively; $D^{\prime }$ and $Q^{\prime }$ are real parts of corresponding matrix elements, whereas $D^{\prime ″}$ and $Q^{\prime ″}$ are their imaginary parts.

The key points of RPAE are as follows (the reader is referred to [22] for details): RPAE uses the HF basis as the vacuum state. With the aid of the Feynman diagrammatic technique, the RPAE photoionization amplitude $\langle k\left|W\right|i\rangle \equiv W_{ik}$ (whether dipole or quadrupole) of the ith subshell of an atom is shown in Figure 1.

Diagrams (c) and (d) represent RPAE corrections to the HF photoionization amplitude $\langle k|\widehat{w}|i\rangle \equiv w_{ik}$ [diagram (b)]. The corresponding RPAE equation for $\left(k\right|\widehat{W}\left|i\right)$ is [22]

$$\begin{array}{c}\hfill \langle k|\widehat{W}|i\rangle =\langle k|\widehat{w}|i\rangle +\sum _{j,k^{\prime }}\langle j|\widehat{W}|k^{\prime }\rangle {\chi }_{j{k}^{\prime }}\langle k^{\prime }i|U|jk\rangle .\end{array}$$

(9)

Here,

$$\begin{array}{c}\hfill {\chi }_{j{k}^{\prime }}={\displaystyle \frac{1}{ħ\omega -{\omega }_{j{k}^{\prime }}+i\zeta }}-{\displaystyle \frac{1}{ħ\omega +{\omega }_{j{k}^{\prime }}-i\zeta }},\end{array}$$

(10)

$$\begin{array}{c}\hfill \langle k^{\prime }i\left|U\right|jk\rangle =\langle k^{\prime }i\left|V\right|jk\rangle -\langle k^{\prime }i\left|V\right|kj\rangle {\delta }_{{\mu }_j{\mu }_i}.\end{array}$$

(11)

In the above equations, ${\omega }_{j{k}^{\prime }}$ is the excitation energy of the $j\to k^{\prime }$ transition, V is the Coulomb interaction operator, ${\delta }_{{\mu }_j{\mu }_i}$ is Kronecker’s delta, ${\mu }_j$ (${\mu }_i$) is the electron’s z-spin-projection in a state j (i), the summation is performed over all possible core intermediate states j and excited virtual states $k^{\prime }$ (including integration over the energy of continuous states), and $i\zeta$ ($\zeta \to +0$) indicates the path of integration around the pole in Equation (9).

In these calculations, to maintain the equality of length and velocity gauges in the application of the RPAE, we use the HF values of the ionization potentials, $I_{n\ell }$, of Ne: $I_{2s}\approx 52.53$ eV and $I_{2p}\approx 23.14$ eV. Therefore, ${\sigma }_{2p}$, ${\beta }_{2p}$, ${\gamma }_{2p}$ and ${\delta }_{2p}$, calculated near the $2s\to n\ell$ autoionizing resonances should be shifted by a corresponding amount to match the resonance positions determined experimentally, if needed.

Finally, to calculate ${\beta }_{n\ell }$, ${\gamma }_{n\ell }$, ${\delta }_{n\ell }$ and ${\zeta }_{n\ell }={\gamma }_{n\ell }+3{\delta }_{n\ell }$, taking into account the frequency spread in the ionizing radiation, we use the Gaussian function, $G(\omega -{\omega }^{\prime })$, for a normal probability distribution with dispersion ${\sigma }^2$ [29]:

$$\begin{array}{c}\hfill G(\omega -{\omega }^{\prime })={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}exp\left[-{\displaystyle \frac{{(\omega -{\omega }^{\prime })}^2}{2{\sigma }^2}}\right].\end{array}$$

(12)

The full width at half maximum for this Gaussian is determined by finding the half-maximum points ${\omega }_0$. This leads to

$$\begin{array}{c}\hfill \mathrm{FWHM}=2\sqrt{2ln2\sigma }\approx 2.3548\sigma .\end{array}$$

(13)

Correspondingly, assuming, in our work, that FWHM = 5 meV, we find that $\sigma \approx 2.1233$ meV.

To apply the frequency distribution to convolute ${\beta }_{n\ell }$, ${\gamma }_{n\ell }$, ${\delta }_{n\ell }$ and ${\zeta }_{n\ell }$, we first need to convolute the angle-differential photoionization cross section, ${\displaystyle \frac{d{\sigma }_{n\ell }}{d\Omega }}$, determined by Equations (1) and (2):

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\sigma }_{n\ell }^{*}}{d\Omega }}=\int G(\omega -{\omega }^{\prime }){\displaystyle \frac{d{\sigma }_{n\ell }\left({\omega }^{\prime }\right)}{d\Omega }}d{\omega }^{\prime }.\end{array}$$

(14)

After simple and obvious mathematical derivations, we obtain, for example, for convoluted ${\gamma }_{n\ell }$ to be labeled as ${\gamma }_{n\ell }^{*}$,

$$\begin{array}{c}\hfill {\gamma }_{n\ell }^{*}\left(\omega \right)={\displaystyle \frac{1}{{\sigma }_{n\ell }^{*}\left(\omega \right)}}\int {\sigma }_{n\ell }\left({\omega }^{\prime }\right){\gamma }_{n\ell }\left({\omega }^{\prime }\right)G(\omega -{\omega }^{\prime })d{\omega }^{\prime }.\end{array}$$

(15)

Here, ${\sigma }_{n\ell }^{*}$ is a convoluted photoionization cross section:

$$\begin{array}{c}\hfill {\sigma }_{n\ell }^{*}=\int {\sigma }_{n\ell }\left({\omega }^{\prime }\right)G(\omega -{\omega }^{\prime })d{\omega }^{\prime }.\end{array}$$

(16)

Expressions for convoluted ${\beta }_{n\ell }^{*}$, ${\delta }_{n\ell }^{*}$ and ${\zeta }_{n\ell }^{*}$ are similar to Equation (15) with obvious substitutions: ${\gamma }_{n\ell }\to {\beta }_{n\ell }$, ${\delta }_{n\ell }$ and ${\zeta }_{n\ell }$, respectively.

3. Results and Conclusions

Calculated ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$, with and without accounting for FWHM = 5 meV frequency spread in the ionizing radiation, are depicted in Figure 2 in the vicinity of the $2s\to 3d$ quadrupole autoionizing resonance.

One can see that frequency spread has a significant impact on ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$. For instance, its account reduces the maximum value of ${\gamma }_{2p}\approx 0.12$ to ${\gamma }_{2p}^{*}\approx 0.06$, which corresponds to approximately $6\%$ of ${\beta }_{2p}$. Similarly, ${\zeta }_{2p}\approx 0.22$ decreases to ${\zeta }_{2p}^{*}\approx 0.12$, making it about $12\%$ of ${\beta }_{2p}$ at its maximum. Despite these reductions, the nondipole angular asymmetries in Ne $2p$ photoelectrons remain within experimental detection limits. Indeed, for example, in [8], even much weaker interference effects—accounting for only around $0.5\%$ of the dipole contribution—were successfully observed experimentally.

Figure 3 illustrates the calculated values of ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$, with and without accounting for frequency spread in the ionizing radiation, through the region of the $2s\to 3p$ and $2s\to 4p$ dipole resonances.

It is evident that the frequency spread in the ionizing radiation has a minimal impact on the ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ parameters in the region of the $2s\to 3p$ dipole resonance, except at the very tips of the spikes in these parameters. This is because the overall energy widths of the spikes are significantly broader than the 5 meV frequency spread of the radiation. However, even at the resonance tips, the frequency spread only slightly affects ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$.

Next, the effect of frequency spread on ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ becomes more pronounced in the region of the higher-lying, narrower $2s\to 4p$ resonance, particularly impacting ${\gamma }_{2p}$. Nevertheless, the resonance enhancement of ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ remains noticeable, even after accounting for the frequency spread.

In conclusion, this paper demonstrates that the $2s\to 3d$ quadrupole resonance, as well as the $2s\to 3p$ and $2s\to 4p$ dipole resonances, in the angular asymmetry parameters ${\gamma }_{2p}$, ${\delta }_{2p}$ and ${\zeta }_{2p}$ for Ne $2p$ photoionization, remain sufficiently large to be experimentally observable, even with a frequency spread of up to 5 meV in the incident radiation. We encourage experimentalists to conduct such an experiment.