#
Two- and Three-Photon Partial Photoionization Cross Sections of Li^{+}, Ne^{8+} and Ar^{16+} under XUV Radiation

^{*}

## Abstract

**:**

^{+}, Ne

^{8+}and Ar

^{16+}ions, following photoionization from their ground state. The expressions for the cross sections are based on the lowest-order (non-vanishing) perturbation theory for the electric field, while the calculations are made with the use of an ab initio configuration interaction method. The ionization cross section is dominated by pronounced single photon resonances in addition to peaks associated with doubly excited resonances. In the case of two-photon ionization, and in the non-resonant part of the cross section, we find that the

^{1}D ionization channel overwhelms the

^{1}S one. We also observe that, as one moves from the lowest atomic number ion, namely Li

^{+}, to the highest atomic number ion, namely Ar

^{16+}, the cross sections generally decrease.

## 1. Introduction

^{11}Wcm

^{-2}to 5 × 10

^{18}Wcm

^{-2}, photon energies range in (0.02–15) KeV, while average full width at half maximum (FWHM) durations vary from 10 fs to 85 fs. It is also worth noting that three new FELs are under development and set to start user operations soon (E-XFEL, Swiss FEL, PAL) with extra pulse parameter specifications compared to the current ones. Notably, the Swiss FEL will provide pulses as short as 2 fs while E-XFEL as long as 100 fs with a repetition rate between 50–100 times higher than any other existing FEL.

^{2}, and a mean photon energy at the lowest limit of soft X-ray spectrum, say $\omega \sim $ 270 eV, then ${V}_{p}/\omega \simeq 0.025<<1$. Since for current FEL sources the bandwidth, $\Delta \omega $, ranges in (${10}^{-3}$–${10}^{-2})\omega $, LOPT is very well suited to provide reliable quantitative information about various quantities of experimental interest, for example, ion and fluorescence yields.

^{17}W/cm

^{2}and 100 fs, respectively). In this experiment, the ionization of Ne

^{8+}, following the sequential one-photon stripping of the lower charged neon ions, proceeds mainly through a two-photon absorption (the ionization potential of Ne

^{8+}is circa 1362 eV). It is needless to say that the same two-photon ionization channel would have been the dominant ionization channel for any photon in the energy range between 681 eV$\le \omega \le 1362$ eV; thus, the need for two-photon ionization cross sections for a range of photon energies. Of course, similar considerations can be carried over to any atomic system and of any degree of charge in the presence of X-ray radiation.

^{+}with the use of single-channel quantum-defect theory (SQDT) [15], the work of Novikov and Hopersky in neon ions [16], the more recent work by the group of R. Santra [17] (Ne

^{8+}) and the two-photon total cross sections on Ne

^{8+}and Ar

^{16+}, obtained through a Greens-function method calculation [18].

^{+}, Ne

^{8+}and Ar

^{16+}. In addition to this, we have calculated details of their electronic structure. The structure of the text is as follows: in Section 2, we present the theoretical method in sufficient detail for a self-contained formulation of the present study; in Section 3, we present and discuss our results about the calculated energies and the LOPT two- and three-photon ionization partial cross sections. Finally, we conclude with a summary of our findings and a brief discussion of possible further investigations within the present context. In the presentation of the theoretical formulas, we use the atomic-Gaussian unit system ($\hslash ={m}_{e}=e=1/4\pi {\epsilon}_{0}=1$). In the figures, the cross sections and the energies are presented in more traditional units, namely, eV for the energies, and ${\mathrm{cm}}^{4}\mathrm{s}$ and ${\mathrm{cm}}^{6}{\mathrm{s}}^{2}$ for the two- and three-photon cross sections, respectively.

## 2. Theoretical Formulation

^{1}S

_{0}, is spherically symmetric, combined with the selection rules for electric dipole interactions, restricts the states to those of singlet-symmetry with a total magnetic quantum number value of zero. We take advantage of this from the outset in order to simplify the formulation.

#### 2.1. Atomic Structure Calculation

^{+}, 10 for Ne

^{8+}and 18 for Ar

^{16+}.

^{2+}, Ne

^{9+}and Ar

^{17+}. To this end, we adopt a separation of variables approach where the one-electron orbitals are expressed as ${\varphi}_{\u03f5l{m}_{l}}\left(\mathbf{r}\right)=[{P}_{\u03f5}\left(r\right)/r]{Y}_{l{m}_{l}}(\theta ,\varphi )$ where ${Y}_{l{m}_{l}}(\theta ,\varphi )$, are the spherical harmonic functions. Projection of ${\varphi}_{\u03f5l{m}_{l}}\left(\mathbf{r}\right)$ onto the one-electron SE, followed by angular integration leads to the one-dimensional radial differential equation for the unknown radial orbitals, ${P}_{\u03f5}\left(r\right)$:

^{+}, Ne

^{8+}and Ar

^{16+}:

^{1}S ground state, it is concluded that only states with ${M}_{L}=0$, $S=0$ and ${M}_{S}=0$ are involved in the photoionization process. Accordingly, the zero-order states are fully determined if the set of L and $a\equiv ({n}_{1}{l}_{1};{n}_{2}{l}_{2})$ parameters is given. The respective zero-order energy is equal to ${E}_{0}={\u03f5}_{1}+{\u03f5}_{2}$. Therefore, the configuration basis set is comprised of singlet $(S=0)$, spatially antisymmetric, angularly coupled, products of one-electron orbitals with ${m}_{l}=0$:

#### 2.2. Two- and Three-Photon Ionization Cross Section Formulation

## 3. Results and Discussion

^{1}S,

^{1}P,

^{1}D and

^{1}F. The configuration states, ${\mathsf{\Phi}}_{{n}_{1}{l}_{1};{n}_{2}{l}_{2}}^{\left(L\right)}\left({\mathbf{r}}_{1,}{\mathbf{r}}_{2}\right)$, have been constructed according to Equation (5) by one-electron orbitals with angular momenta given in Table 1 and energies sufficiently high to ensure convergence of the results. The order of B-splines was ${k}_{b}=9$ with the total number of B-spline polynomials set at ${n}_{b}=110$ for Li

^{+}and ${n}_{b}=170$ for Ne

^{9+}and Ar

^{16+}. The box radius varied between $R=50-58$ a.u. for Li

^{+}, $R=20-28$ a.u. for Ne

^{8+}and $R=10-15$ a.u. for Ar

^{16+}. The knot sequence of the spatial grid for the B-spline basis was linear. The two-electron wavefunctions, ${\mathsf{\Phi}}_{{n}_{1}{l}_{1};{n}_{2}{l}_{2}}^{\left(L\right)}\left({\mathbf{r}}_{1,}{\mathbf{r}}_{2}\right)$, have been constructed from the zero-order configurations by one-electron orbitals with angular momenta given in the mentioned table and energies determined by the indices ${n}_{1},{n}_{2}$ in the following ranges: $1\le {n}_{1}\le 6$ and $1\le {n}_{2}\le {n}_{b}$. The relationship of the indices ${n}_{1},{n}_{2}$ to the energies of the zero-order wavefunctions depends on the basis size parameters such as the maximum value of the box radius as well as the number of B-spline basis functions used. In summary, the whole basis, for each symmetry, resulted in the inclusion of the following number of functions for each ion: 1650–1940 for Li

^{+}, 1600 for Ne

^{8+}and 2040 for Ar

^{16+}.

^{2+}, Ne

^{9+}and Ar

^{17+}. The degree of the agreement with those reported in the National Institute of Standards and Technology (NIST) atomic spectra database [28] is given in the last row of the table. In all cases, the percentage discrepancies of the ground state energies (between the calculated and those of the NIST database) are of the order 0.1% while the excited states are of similar or even smaller order.

^{+}, Ne

^{8+}and Ar

^{16+}ions with respect to the respective ground state ($1{s}^{2}{\phantom{\rule{0.277778em}{0ex}}}^{1}{S}_{0}$) energy of each ion. In all cases, the percentage discrepancies of the ground state energies (between the calculated and those of the NIST) are of the order 0.1% while the excited states are of similar or even of smaller order. This is not surprising as the role of correlation is more important in the lower energy states, where the electrons are on average closer relative to the higher-energy states.

^{1}S,

^{1}D continua, while in the case of three-photon absorption are the

^{1}P,

^{1}F continua. The cross sections have been evaluated using both the length and the velocity forms of the dipole operator. They generally have excellent agreement throughout all the spectral regions considered, especially for the non-resonant ones. Relative agreement between the length and the velocity forms is important since it provides strong evidence that the dipole matrix elements, contributing in the multiphoton transition amplitude, have been converged.

^{+}, Ne

^{8+}and Ar

^{16+}, respectively, from the ground state, $1{s}^{2}{(}^{1}S)$, to final states of symmetry

^{1}S,

^{1}D. Summing the latter, we obtain the corresponding total two-photon ionization cross sections, ${\sigma}_{2}\left(\omega \right)$. For clarity, we have plotted only the length-form results. Generally, for all three ions, the dominant two-photon ionization channel is the

^{1}D symmetry. The cross sections exhibit strong peak structures, which appear in both the

^{1}S and

^{1}D final symmetries, due to one-photon resonance with the intermediate states $1snp\phantom{\rule{0.166667em}{0ex}}{}^{1}P,n=2,3,...$. Apart from these intermediate-resonance peaks, there are further peaks due to strong configuration mixing of the type $np{n}^{\prime}p,\phantom{\rule{0.166667em}{0ex}}n,{n}^{\prime}=2,3...$, associated with the ${\phantom{\rule{0.166667em}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}S,{\phantom{\rule{0.166667em}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}D$ continua.

^{+}ionization since these would occur at higher photon energy, past about 75 eV, as the lowest post-ionization energy levels are about 150 eV and half that value (two photons) is about 75 eV. When we add the

^{1}S and

^{1}D cross sections to obtain the total cross section, we find a relatively good agreement with that of Ref. [15], which models two-photon ionization of Li

^{+}employing a less elaborate approach, namely, single-channel quantum defect theory; when our shift is accounted for, our peaks occur close to that work (62.2 eV $1s2p{(}^{1}P)$, 69.7 eV $1s3p{(}^{1}P)$, 72.3 eV $1s4p{(}^{1}P)$ and 73.5 eV $1s5p{(}^{1}P)$), and our (partial-wave sum) cross section baseline between 50–55 eV is within the same order of magnitude as in Ref. [15] (both between ${10}^{-53}$–${10}^{-52}$ cm

^{4}·s); however, our shape is slightly different here, being slightly convex (downward) in this region.

^{8+}and Ar

^{16+}, our values are also in good agreement with the Green-function calculations in Ref. [18]. In addition, for Ne

^{8+}, we have also compared our values with the second-order perturbation theory calculation of Novikov and Hopersky [16]; their $1s2p{(}^{1}P)$ and $1s3p{(}^{1}P)$ one-photon resonance peaks (these are their only peaks) are comparable to ours occurring at around 920 eV and 1070 eV, respectively. Their cross section base-line (non-resonant part) circa 600–800 eV is also close to ours, i.e., within the same order of magnitude (both between ${10}^{-56}$–${10}^{-55}$ cm

^{4}·s).

^{+}, Ne

^{8+}and Ar

^{16+}ions from their respective ground state ($1{s}^{2}{\phantom{\rule{0.277778em}{0ex}}}^{1}{S}_{0}$) energy, ${E}_{g}={E}_{1{s}^{2}}$, namely, $\Delta {E}_{P}\equiv {E}_{P}-{E}_{g}$, (i.e., corresponding to peaks (a–d) in Figure 2). Their importance is derived from the fact that these energy differences appear in the denominator of the two-photon cross section expression, Equation (7), i.e., ${E}_{g}+\omega -{E}_{P}^{\prime}=\omega -\Delta {E}_{P}$. It is then immediately evident that the photon energy detuning from these energy differences generates a series of characteristic features in the cross section. A word of caution is necessary at this point: within the current formulation, the height of these peaks becomes infinite in the exact on-resonance case, $\Delta {E}_{P}=\omega $. The first point to note is that we have ignored the inherent spontaneous decay width of the intermediate bound states. This would have served only to remove the unphysical singularities that occur at the resonances positions. However, most importantly, the LOPT cross sections fail to provide the correct ionization yields. In other words, in the resonant case, the LOPT relation for the ionization yield,

^{+}) cannot be safely used in combination with Equation (10). Similar considerations should be assumed for the higher peaks. For completeness, for Ne

^{8+}and Ar

^{16+}, if based again on the appearance of the first peaks in the two-photon cross section (see the first row for the $1s2p$ state in Table 3), the corresponding intervals are scaled upwards to $\pm 9.2$ eV (∼920/100) and $\pm 31$ eV (∼3126/100), respectively. A case that such a discrepancy between the LOPT two-photon ionization cross section [16] and the experimental value [29] is attributed to the bandwidth of the X-ray pulse can be found in Ref. [17]. More specifically, for an X-ray photon energy of 1110 eV (in between (b) and (c) peaks in Figure 2), the reported experimental value was $7\times {10}^{-54}$ cm

^{4}·s [29] while the theoretical cross section based on a Hartree–Fock–Slater (HFS) model was found to be equal to 4.0 × 10

^{−57}cm

^{4}s. Our calculated value for this photon energy is about 4.6 × 10

^{−57}cm

^{4}s for the

^{1}D wave while the

^{1}S value makes a negligible contribution to the total cross section.

^{8+}and Ar

^{16+}, the ’twin’ peaks (see ${A}_{0}$ peaks in Figure 2), exclusive to the

^{1}S symmetry, are due to the strong coupling between the $2s2s$ and $2p2p$ configurations in the expansion Equation (4). Since in the

^{1}D symmetry the $2s2s$ configuration is missing, we observe only one peak (${A}_{2}$ in Figure 2), in between the ${A}_{0}$ ones. To confirm this, we have performed some further tests where, for example, we excluded the ${\mathsf{\Phi}}_{2s2s}^{{}^{1}S}$, ${\mathsf{\Phi}}_{2p2p}^{{}^{1}S}$ zero-order states (separately each time) from the CI wavefunction, Equation (4). By doing this, we obtain a cross section with only one ${A}_{0}$ peak at the same position where the ${A}_{2}$ peak appears. This suggests that the observed (two) ${A}_{0}$ peaks are the result of strong-mixing of the $2s2s$ and $2p2p$ configurations, mainly due to their proximity in energy.

^{1}S symmetry, while the ${B}_{2}$ peak is due to the $3{p}^{2}$ state exclusively. We mention here that these doubly excited (highly correlated) states are also known as autoionizing states as they are associated with a temporal trap of the two excited electrons in the core’s region, eventually leading to the ejection of one of them and the residual (higher-charged ion) to its ground state. In the present (static) context of the CI calculation, these doubly excited states are degenerate with the $1{s}^{2}{\phantom{\rule{0.166667em}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}S$ or $1sd{\phantom{\rule{0.166667em}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}D$ continua, which eventually cause their radiationless (auto)-ionization [30].

^{+}, Ne

^{8+}and Ar

^{16+}, respectively, from the ground state $1{s}^{2}{(}^{1}S)$. The final angular momentum of the ions, following three-photon absorption, are the ${\phantom{\rule{0.166667em}{0ex}}}^{1}P{,}^{1}\phantom{\rule{-0.166667em}{0ex}}F$ continua, all being of singlet symmetry. Similarly, as in the two-photon case, the total three-photon ionization cross section is obtained by the addition of the

^{1}P and

^{1}F partial-wave cross sections. Again, the final state is dominated by configurations with the residual ion in its ground state and the ejected electron with angular momentum $l=1$ for the

^{1}P and $l=3$ for the

^{1}F symmetry. The three-photon cross sections exhibit strong peak structures, which appear in both

^{1}P and

^{1}F final symmetries, due to two-photon resonances with the intermediate states $1s3d,1s4d,...$ (see denominators (${E}_{g}+2\omega -{E}_{L},L=S,D$) in Equations (8) and (9)). In the

^{1}P symmetry, there are additional peaks due to two-photon resonance states of the type $1sns\phantom{\rule{0.166667em}{0ex}}{}^{1}S,n=2,3,...$. Because the $1s3s$ and $1s3d$ states have slightly different energy positions, the intermediate resonance peaks for the

^{1}P and

^{1}F (for example circa 34.4 eV for Li

^{+}) do not generally coincide. Note that the

^{1}P final states are reached by the coherent superposition of two ionization absorption channels: $S\to P\to S\to P$ and $S\to P\to D\to P$. In contrast, the

^{1}F states are reached only via one ionization channel, namely: $S\to P\to D\to F$.

^{+}towards Ar

^{16+}. This observation is rather consistent with the (exact) scaling, $1/{Z}^{4N-2}$, of the N-photon cross section for hydrogenic systems [11]. A second point worth mentioning is that the cross sections ending on the higher symmetry,

^{1}D for two-photon and

^{1}F for three-photon ionization, are proportional to the cross sections for circularly polarized light. To be more specific, due to the dipole selection rules, ionization by circularly polarized light will proceed through intermediate states where ${M}_{L}$ will change either by $+1$ or by $-1$ monotonously. For example, let us assume circularly polarized light which causes a change of the magnetic quantum number by $\Delta {M}_{L}=+1$. This means that if we start from the ground state, where $L=0$ and ${M}_{L}=0$, then the first ionization step will involve only states with ${M}_{L}=+1$. Similarly, the next ionization step will involve states that differ by $+1$ from the previous step, meaning that only states with ${M}_{L}=+2$ will be accessed in this step. In this case, these states will necessarily have $L=2$. Accordingly, if further ionization occurs (three-photon ionization), for the same reason, only states with ${M}_{L}=+3$ will be reached, thus ending necessarily with an $L=3$ total orbital angular momentum. Similar considerations hold had we started by circularly polarized light with opposite helicity, leading to a change of $\Delta {M}_{L}=-1$. In short, for two-photon ionization, only the ionization path $S\to P\to D$ is allowed. Accordingly, for three-photon ionization, only the $S\to P\to D\to F$ ionization channel will occur. Now the crucial observation is that the transition amplitudes by circularly and linearly polarized light for these ionization paths differ only by the total magnetic quantum number, ${M}_{L}$. For linearly polarized light, it is $\Delta {M}_{L}=0,$ while for circularly polarized light, it is $|\Delta {M}_{L}|=1$. Straightforward angular momentum algebra for these transition amplitudes (${T}_{C}\left({T}_{L}\right)$ for circular(linear)-polarized light) of two-electron states shows that they differ only by a proportional factor; namely, $|{T}_{C}\left(D\right)/{T}_{L}\left(D\right)|=\sqrt{3/2}$ for two-photon ionization and $|{T}_{C}\left(F\right)/{T}_{L}\left(F\right)|=\sqrt{5/2}$, for three-photon ionization, where D and F denote the final angular momentum channel for two-photon and three-photon ionization, respectively [31]. Nevertheless, note that circularly polarized light does not guarantee that the total ionization cross section will provide higher rates relative to the ionization by linearly polarized light; the final rate depends on the number of available ionization paths available and the electric field intensity and, in fact, high N-photon ionization by linearly polarized light is more effective.

## 4. Conclusions

^{+}, Ne

^{8+}and Ar

^{16+}ions. These systems are of high experimental and theoretical interest for the study of interaction of strong and ultrashort X-ray radiation at a fundamental level. We have identified that the ionization cross sections are dominated by a series of intermediate (one or two-photon) resonance peaks in addition to peaks due to doubly excited structures. We have noticed the trend that from the lower Z ion, Li

^{+}, to the higher ones, the ionization cross sections generally decrease.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Two-photon partial ionization cross sections of Li

^{+}from its ground state with linearly polarized light. The characteristic peaks in the cross section are associated with the intermediate bound states of the symmetry

^{1}P.

**Figure 2.**Two-photon partial ionization cross sections of Ne

^{8+}from its ground state with linearly polarized light. The small letters $(a,b,c)$ indicate features associated with the intermediate bound states of symmetry

^{1}P while the capital letters $({A}_{i},{B}_{i},i=0,2)$ are associated with final state correlations.

**Figure 3.**Two-photon partial ionization cross sections of Ar

^{16+}from its ground state with linearly polarized light.

**Figure 4.**Three-photon ionization partial cross sections of Li

^{+}from its ground state with linearly polarized light.

**Figure 5.**Three-photon ionization partial cross sections of Ne

^{8+}from its ground state with linearly polarized light.

**Figure 6.**Three-photon ionization partial cross sections of Ar

^{16+}from its ground state with linearly polarized light.

**Table 1.**$({l}_{1},{l}_{2})$ electronic configurations included in the configuration interaction (CI) calculations for the

^{1}L symmetries.

^{1}S | ^{1}P | ^{1}D | ^{1}F |
---|---|---|---|

${s}^{2}$ | $sp$ | $sd$ | $sf$ |

${p}^{2}$ | $pd$ | ${p}^{2}$ | $pd$ |

${d}^{2}$ | $df$ | $pf$ | $pg$ |

${f}^{2}$ | $fg$ | ${d}^{2}$ | $df$ |

${g}^{2}$ | $dg$ | $fg$ | |

${f}^{2}$ |

**Table 2.**Energies of the few lowest states of the one-electron Li

^{2+}, Ne

^{9+}and Ar

^{17+}ions. Energies are given in units of eV. Energies are relative to the single-ionization threshold. $\delta E\left(1s\right)\equiv |E\left(1s\right)-{E}_{g}^{Z+}|/|{E}_{g}^{Z+}|$ is the relative discrepancy (between our calculated ground state values with the energies (${E}_{g}^{Z+}$) listed in the NIST database [28]). The relative discrepancy for excited states is generally less than $\delta E\left(1s\right)$.

State | Li^{2+} | Ne^{9+} | Ar^{17+} |
---|---|---|---|

$1s$ | $-122.451$ | $-1360.57$ | $-4408.2$ |

$2s,2p$ | $-30.613$ | $-340.14$ | $-1102.1$ |

$3p,3d$ | $-13.606$ | $-151.18$ | $-489.8$ |

$4d,4f$ | $-7.653$ | $-85.04$ | $-275.5$ |

$5f$ | $-4.898$ | $-54.42$ | $-176.3$ |

$\delta E\left(1s\right)$ | $2.5\times {10}^{-5}$ | $1.2\times {10}^{-3}$ | $4.1\times {10}^{-3}$ |

**Table 3.**Energy differences of the few lowest states of Li

^{+}, Ne

^{8+}and Ar

^{16+}ions with respect to their respective ground state ($1{s}^{2}{\phantom{\rule{0.277778em}{0ex}}}^{1}{S}_{0}$) energy value. We use boldface for the states’ notation to emphasize that they are listed according the dominant configuration in the CI expansion of Equation (4). In the last row, $\delta E\left(1{s}^{2}\right)\equiv |E\left(1{s}^{2}\right)-{E}_{g}^{Z+}|/|{E}_{g}^{Z+}|$ is the relative discrepancy (between our calculated ground state values and the energies (${E}_{g}^{Z+}$) listed in the NIST database [28]). $E\left(1{s}^{2}\right)$ is given relative to the double ionization threshold.

State | Li^{+} | Ne^{+} | Ar^{16+} |
---|---|---|---|

$1s2s$ | $60.42$ | $914.0$ | $3114.1$ |

$1s2p$ | $61.67$ | $920.4$ | $3126.7$ |

$1s3s$ | $68.73$ | $1070.1$ | $3666.1$ |

$1s3d$ | $69.02$ | $1071.5$ | $3668.7$ |

$1s3p$ | $69.09$ | $1072.0$ | $3669.7$ |

$1s4s$ | $71.54$ | $1124.5$ | $3858.7$ |

$1s4d$ | $71.66$ | $1125.1$ | $3859.9$ |

$1s4p$ | $71.67$ | $1125.3$ | $3860.2$ |

$1s4f$ | $71.67$ | $1125.1$ | $3859.8$ |

$1s5s$ | $72.83$ | $1149.6$ | $3947.7$ |

$1s5d$ | $72.89$ | $1148.9$ | $3948.3$ |

$1s5p$ | $72.90$ | $1150.0$ | $3948.5$ |

$1s5f$ | $72.89$ | $1149.9$ | $3948.3$ |

$E\left(1{s}^{2}\right)$ | $-197.518$ | $-2554.53$ | $-8513.84$ |

$\delta E\left(1{s}^{2}\right)$ | $2.9\times {10}^{-3}$ | $1.36\times {10}^{-3}$ | $3.87\times {10}^{-3}$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hanks, W.; Costello, J.T.; Nikolopoulos, L.A.A. Two- and Three-Photon Partial Photoionization Cross Sections of Li^{+}, Ne^{8+} and Ar^{16+} under XUV Radiation. *Appl. Sci.* **2017**, *7*, 294.
https://doi.org/10.3390/app7030294

**AMA Style**

Hanks W, Costello JT, Nikolopoulos LAA. Two- and Three-Photon Partial Photoionization Cross Sections of Li^{+}, Ne^{8+} and Ar^{16+} under XUV Radiation. *Applied Sciences*. 2017; 7(3):294.
https://doi.org/10.3390/app7030294

**Chicago/Turabian Style**

Hanks, William, John T. Costello, and Lampros A. A. Nikolopoulos. 2017. "Two- and Three-Photon Partial Photoionization Cross Sections of Li^{+}, Ne^{8+} and Ar^{16+} under XUV Radiation" *Applied Sciences* 7, no. 3: 294.
https://doi.org/10.3390/app7030294