Electron impact excitation of Sn$^{2+}$

Abstract

The relativistic convergent close-coupling method is applied to calculate integrated cross sections for the electron-impact excitation of Sn$^{2+}$. Cross sections have been calculated for excitations to all states in the 5s5p, $5p^2$, 5s6s and 5s5d manifolds from the ground state for projectile electron energies ranging from 24 eV to 500 eV. The total discrete inelastic scattering cross section is also presented.

Graphical abstract

1 Introduction

Cross sections for electron collisions with low-charge states of tin are currently sparse in literature, while the demand for such data is increasing. Research on the design and performance of tokamak fusion devices such as ITER and DEMO is presently very active, and tin has been noted as an important material in the manufacturing of plasma-facing components. The erosion of plasma-facing components due to bombardment from the fusion plasma is a significant concern for tokamak fusion devices. The divertor region of the ITER fusion reactor, which is currently constructed using a tungsten monoblock design, is particularly prone to sustaining damage from erosion [1]. Tin may be included within a sub-surface layer of the plasma-facing components, from where it can be used as an indicator for the erosion as it enters the fusion plasma after some material has eroded [2]. Tin is also considered for use in liquid-metal divertors which are currently being studied and may be implemented in the European DEMO fusion reactor [3, 4]. Current designs propose a 3D-printed porous tungsten armour layer filled with liquid tin, which will have a longer lifetime compared to the presently adopted tungsten monoblock implementation of the divertor. This is due to its ability to automatically replenish the plasma-facing surface, and also because it is less sensitive to neutron damage. Tin is currently the best choice for use as the liquid metal as its tritium retention is sufficiently low, it is safe to use with water-cooling and provides a large operational range of temperatures over which evaporation is low. When surface material is vaporised it can form a dense secondary plasma above the plasma-facing components which protects them from further damage [5]. This phenomenon is called vapour shielding, and is a particularity favourable outcome of the use of liquid metal divertors which will lead to further protection from erosion. For a greater understanding of the effectiveness of tin as a diagnostic tool for monitoring damage and also as a protective layer against the erosion of plasma-facing components, accurate plasma simulations must be developed which will require collision datasets for tin and its ions. Collision data for ions of tin is also of interest to applications in the nano-lithography industry, particularity in the development of extreme ultra-violet (EUV) light sources [6, 7]. A dense tin plasma is used to produce the EUV light required for use in EUV lithography, which can be better studied with accurate collision data for tin and its ions. In particular, the meta-stable triplet states of the Sn$^{2+}$ ion have recently been noted for their important role in EUV light sources [8].

Table 1 First 16 excitation energies (eV) for the Sn$^{2+}$ spectrum compared with the recommended values from NIST [14]

Previous studies on electron collisions with Sn$^{2+}$ have been primarily focused on ionisation while there have been no studies on excitations. Here, we present integrated cross sections (ICS) for the electron-impact excitation of doubly charged tin to states in the 5s5p, $5p^2$, 5s6s and 5s5d manifolds. The total discrete inelastic scattering cross section is also produced. Calculations are performed using the Relativistic Convergent Close-Coupling (RCCC) method for the $5s^2$ $^1S_0$ ground state over a projectile energy range of 24 eV to 500 eV. First, details of the target structure and scattering calculations are discussed. Results are then presented alongside the discussion.

2 RCCC method

2.1 Structure model

Sn$^{2+}$ is modelled as a quasi two-electron system and its electron configuration is [Kr]$4d^{10}5s^2$. Due to its large atomic number, we adopt a fully relativistic approach to the structure calculations for Sn$^{2+}$. First, the core-electron orbitals ([Kr]$4d^{10}$) are obtained using the GRASP [9] package. To build the two-electron configurations which model the Sn$^{2+}$ valence spectrum, one-electron orbitals must first be obtained. For these, the Sn$^{3+}$ Dirac-Fock Hamiltonian is diagonalised in an L-spinor basis [10] which generates sets of one-electron functions for all symmetries from $s_{1/2}$ to $g_{9/2}$. To further improve the quality of these orbitals, one-electron polarisation potentials as described in [11] are included. This model for the polarisation of the target requires a value for the static dipole-polarisability of the core (which in this case is Sn$^{4+}$). This has been calculated to be 2.37 $a_0^3$ using the methods described in Sect. 2.1.1. The j-dependent fall-off radii for this polarisation potential have then chosen to best model the energy levels of Sn$^{3+}$, which leads to the present set of one-electron orbitals.

From each symmetry, one-electron orbitals with principal quantum numbers up-to $n=20$ are used in standard two-electron CI calculations. There are two types of configurations included; frozen core and correlation. Frozen core configurations are included to best model highly excited and continuum pseudo-states of Sn$^{2+}$. They are of the form $5s_{1/2}nl_j$ where $nl_{j}$ are the one-electron states generated from diagonalisation and included in the CI calculations. The correlation configurations are included to better model the low-lying spectrum and generate the doubly excited $5p^2$ states. These configurations have the form $nl_jn'l'_{j'}$ where $nl_j$ and $n'l'_{j'}$ are restricted to the $5s_{1/2}$, $6s_{1/2}$ and $5p_{1/2,3/2}$ orbitals. A total of 329 states are produced, which range in total electronic angular momentum from $J=0$ to $J=5$ for both negative and positive parities. This includes bound states and also continuum pseudo-states with energies of up-to 242 eV. To further improve the structure model, a phenomenological two-electron polarisation potential is included [12]. The fall-off radius for this potential is chosen such that the presently calculated optical oscillator strength (OOS) for the resonance transition $5s^2$ $^1S_0$ $\rightarrow$ 5s5p $^1P_1$ is fitted to an accurate value of 1.40 [13]. With a fall-off radius of $r^{diel}_c=1.21$, the RCCC OOS value for this transition is calculated to be 1.397. The presently calculated energies for the first 16 states in the Sn$^{2+}$ spectrum are presented in Table 1. These states are named based on the dominant configurations arising for each one in the LSJ coupling scheme.

The RCCC structure calculations produce a value of 11.35 $a_0^3$ for the static dipole polarisability of Sn$^{2+}$ when all 329 states are included. A few other values exist in literature, the first being 17.75 $a_0^3$ as calculated by [15] using a Thomas-Fermi model. Another value of 15.526 $a_0^3$ was calculated by [16] using a relativistic many-body singly and doubly approximated coupled-cluster methodology (among others). The RCCC value for this static dipole polarisability is smaller than other calculations from literature as the present structure model does not include excitations of the inner-core electrons.

2.1.1 Static dipole-polarisability of the Sn$^{4+}$ core

Limited literature is available for values of static dipole polarisability of Sn$^{4+}$, with the most recent values being calculated by [17] as 3.27 $a_0^3$ using a Thomas-Fermi model and [18] as 2.168 $a_0^3$ using a coupled Hartree-Fock approximation. We calculated the static dipole polarisability of Sn$^{4+}$ using a non-relativistic, fully quantum mechanical approach. A configuration-interaction (CI) representation of the wave-functions is utilised, for which radial functions are obtained from a mixture of Multi-Configuration Hartree-Fock [19] calculations and Laguerre basis functions [20]. In our model, one-electron and two-electron excitations are allowed out of the 4d orbital. The configurations included for the one-electron excitations are $4d^{9}nl$, where l is p, f and n is taken up-to 35. The included configurations for the two-electron excitations are $4d^{8}5snl$ and $4d^{8}5pn'l'$ where l is p, f and $l'$ is s, d, and both n and $n'$ span up-to a value of 35. The CI coefficients are produced by diagonalising the Sn$^{4+}$ Hamiltonian in a basis of multi-electron configurations generated using the previously obtained radial functions. Because only electric-dipole transitions contribute to the calculation of the static dipole polarisability, only electron configurations with odd parity are included. Optical oscillator strengths were then calculated for all produced electric-dipole transitions, using which the static dipole polarisability of Sn$^{4+}$ is obtained [21]. Our calculations yield a value of 2.37 $a_0^3$ which is substantially lower than 3.27 $a_0^3$ ([17]), but in good agreement with 2.168 $a_0^3$ ([18]).

2.2 Scattering calculation details

Scattering calculations are performed using the RCCC method which has been thoroughly documented in literature [22,23,24,25]. The key-feature of this method is the expansion of the total scattering wave-function in a set of target pseudo-states, leading to a set of coupled relativistic Lippmann-Schwinger equations which are then solved to obtain the T-matrix. This expansion is done in a partial-wave form for each total angular momentum J and parity $\Pi$ of the total scatting system (electron and Sn$^{2+}$ ion). The obtained T-matrix is then used to produce integrated cross sections.

Convergence in the calculated cross sections is established by increasing the number of target-states included in the expansion of the total scattering wave-function and in the number of partial waves the T-matrix is obtained for. Typically, calculations are performed up-to $J=20.5$ for both positive and negative parities and an analytical Born-extrapolation technique is utilised to speed up the convergence with increasing J [26]. Various scattering models are produced to test for convergence of the relativistic close-coupling calculations in the number of included target states. These models include 70, 113, 139, 199 and 253 states and are denoted as RCCC(70), RCCC(113), RCCC(139), RCCC(199) and RCCC(253), respectively. The RCCC(70) model includes only bound states while the RCCC(113), RCCC(139), RCCC(199), and RCCC(253) models include positive-energy pseudo-states with energies up-to 10, 18, 50, and 90 eV, respectively. Cross sections are only presented at projectile energies above 24 eV. This is due to the large number of resonances present in the low projectile-energy regime which will be studied elsewhere. Here we concentrate on the intermediate to high projectile energy range for this system where coupling with the ionisation continuum is important.

There is no data in literature to compare the RCCC cross sections with for all transitions studied here. Therefore, we present relativistic first-order coulomb-wave Born approximation (FCBA) calculations alongside the RCCC cross sections where possible to illustrate the importance of inter-channel coupling effects (absent in the first-order method) and determine the projectile energy range where the first-order technique becomes accurate. In a non-relativistic formalism, FCBA cross sections for transitions from singlet to triplet states are zero due to the lack of exchange. In the present approach, the Sn$^{2+}$ target states are modelled within a fully relativistic framework which leads to effective mixing between the singlet and triplet manifolds. Therefore, the relativistic FCBA cross section for a transition from the singlet ground state to a triplet excited state does not necessarily have to be zero but often is small.

3 Results

3.1 Excitations to the 5s5p manifold

The 5s5p manifold includes four states. There are three states originating from the triplet 5s5p $^3P_{0,1,2}^\circ$ configurations which have distinct energies due to relativistic effects, and one state originating from the singlet 5s5p $^1P_1^\circ$ configuration. The $^1P_1^\circ$ and $^3P_1^\circ$ states allow an electric-dipole transition from the $5s^2$ $^1S_0$ ground state, while transitions to the $^3P_0^\circ$ and $^3P_2^\circ$ states are dipole forbidden. This is reflected in the integrated cross sections for these transitions shown in Fig.  1. The cross sections for transitions to the $^3P_0^\circ$ and $^3P_2^\circ$ states are much smaller and decrease quickly compared to the cross sections for transitions to the $^1P_1^\circ$ and $^3P_1^\circ$ states. The cross section for excitation to the $^1P_1^\circ$ state is much larger in magnitude than the one for excitation to the $^3P_1^\circ$ state, however they have similar behaviour at high energies. In Fig. 1, the FCBA results presented for excitations to the $^3P_1^\circ$ and $^1P_1^\circ$ states are larger than the RCCC cross sections at projectile energies less than 300 eV, and at higher energies the two results are the same. This shows that the RCCC cross sections converge to the correct high energy result, while at lower energies the inter-channel coupling effects are important.

Fig. 1

RCCC ICS for excitations to the 5s5p $^3P_0^\circ$, $^3P_1^\circ$, $^3P_2^\circ$ and $^1P_1^\circ$ states from the $5s^2$ $^1S_0$ ground state. Relativistic first-order coulomb-wave Born (FCBA) results are also presented for comparison

Fig. 2

Convergence of the RCCC ICS for excitation to the 5s5p $^1P_1^\circ$ state from the $5s^2$ $^1S_0$ ground state

Fig. 3

Convergence of the RCCC ICS for excitation to the 5s5p $^3P_2^\circ$ state from the $5s^2$ $^1S_0$ ground state

Figures 2 and 3 illustrate the convergence with an increasing number of states of the ICS for excitations to the $^1P_1^\circ$ and $^3P_2^\circ$ states, respectively. The cross section for the dipole-allowed transition to the $^1P_1^\circ$ state converges rather quickly as demonstrated by a small difference between the RCCC(139) and RCCC(199) models (shown in Fig. 2). This indicates that this transition is not strongly affected by coupling with high-lying continuum states. The differences between the cross sections of each of the RCCC models is more pronounced for the excitation to the $^3P_2^\circ$ state (Fig. 3). This figure also illustrates the importance of the inclusion of the continuum as the cross sections for the larger models are quite different compared to the RCCC(70) model in the projectile energy range of about 40 eV to 100 eV. These two figures show that the cross sections for the dipole allowed and exchange transitions are well converged with the RCCC(253) model. A smooth and converged cross section is produced by combining the various models at appropriate projectile energies. More specifically, in the 24 eV to 32 eV projectile energy range, the RCCC(70) and RCCC(113) models are utilised as they are more stable compared to the larger models and still sufficiently convergent. At higher projectile energies, the RCCC(199) and RCCC(253) models are adopted. This methodology is used for all other excitation cross sections presented in this paper, as the convergence for all of them is similar.

3.2 Excitations to the $5p^2$ manifold

This manifold includes the $5p^2$ $^3P_0$, $^3P_1$, $^3P_2$, $^1D_2$ and $^1S_0$ states. Transitions to all of these levels from the ground state are dipole forbidden due to no change in parity. The calculated ICS for excitations to these levels are presented in Fig. 4. These cross sections are relatively small, with the largest one being for the excitation to the $^1D_2$ level. FCBA cross sections are presented for transitions to the $^3P_0$, $^3P_2$, $^1D_2$ and $^1S_0$ states. For excitations to the $^3P_2$ and $^1D_2$ states, the FCBA cross sections are larger than RCCC at projectile energies less than about 300 eV, and at higher energies they converge (the same as what was described for the 5s5p manifold). However, this is not the case for excitations to the $^3P_0$ and $^1S_0$ states. For the $^3P_0$ state, the FCBA cross section is significantly smaller than RCCC over the entire projectile energy range, which is typical for the exchange dominated transitions. For the transition to the $^1S_0$ state the FCBA cross section is smaller than RCCC for the same reasons.

Fig. 4

RCCC ICS for excitations to the $5p^2$ $^3P_0$, $^3P_1$, $^3P_2$, $^1D_2$ and $^1S_0$ states from the $5s^2$ $^1S_0$ ground state. Relativistic first-order coulomb-wave Born (FCBA) results are also presented for comparison

3.3 Excitations to the 5s6s manifold

The 5s6s manifold is comprised of the $^3S_1$ and $^1S_0$ states, and transitions to both these states are dipole forbidden due to no change in parity. The ICS for excitations to these two states from the ground state is presented in Fig. 5. The FCBA cross section for excitation to the $^1S_0$ state compares very similarly with RCCC as described previously for the 5s5p manifold. It is larger than the RCCC cross section in the intermediate projectile energy range and at higher energies they converge.

Fig. 5

RCCC ICS for excitations to the 5s6s $^3S_1$ and $^1S_0$ states from the $5s^2$ $^1S_0$ ground state. Relativistic first-order coulomb-wave Born (FCBA) results are also presented for comparison

3.4 Excitations to the 5s5d manifold

The 5s5d $^3D_1$, $^3D_2$, $^3D_3$ and $^1D_2$ states make up this manifold. Excitations from the $5s^2$ $^1S_0$ ground state to these 5s5d states are all dipole forbidden and the corresponding cross sections are small as shown in Fig. 6. The cross section for excitation to the $^1D_2$ state is the largest and falls off slower with increasing projectile energy compared to the ICS for excitation to the triplet states. The ICS for the $^3D_1$, $^3D_2$ and $^3D_3$ states decreases quickly with increasing projectile energy. The FCBA cross section for the $^1D_2$ state overestimates RCCC at lower energies and they converge at about 70 eV.

Fig. 6

RCCC ICS for excitations to the 5s5d $^3D_1$, $^3D_2$, $^3D_3$ and $^1D_2$ states from the $5s^2$ $^1S_0$ ground state. Relativistic first-order coulomb-wave Born (FCBA) results are also presented for comparison

3.5 Total discrete inelastic scattering

The total discrete inelastic scattering cross section for the ground state of Sn$^{2+}$ is presented in Fig. 7. It includes contributions from all excitations to bound levels included in the scattering calculations. The largest contribution to this cross section comes from the electric-dipole transition from the ground state to the 5s5p $^1P_1^\circ$ state.

Fig. 7

RCCC total discrete inelastic scattering cross section for the ground state of Sn$^{2+}$. The RCCC cross section for excitation to the 5s5p $^1P_1^\circ$ state is also included

4 Conclusions

A detailed set of electron impact excitation cross sections for Sn$^{2+}$ has been calculated using the RCCC method. This includes integrated cross sections for excitations from the ground state to all states in the 5s5p, $5p^2$, 5s6s and 5s5d manifolds over a projectile energy range of 24 eV to 500 eV. Such cross section data for excitations Sn$^{2+}$ have not previously been studied, and so no comparison with previous literature could be made. FCBA results were presented alongside RCCC to show the importance of inter-channel coupling, and determine the incident energies at which the high energy approximation becomes valid. The total discrete inelastic scattering cross section is also presented, which includes contributions from excitations to all bound states. We hope that this data will be useful in plasma modelling applications, and it will be available in electronic form upon request.