Parity nonconservation effect in the resonance elastic electron scattering on heavy He-like ions

Investigations of the parity violation in the domain of atomic physics originate from consideration of the parity nonconservation (PNC) effects in neutral systems (see [1–3] and references therein). The most accurate up to date measurement of the PNC was achieved for ¹³³Cs atom [4, 5]. These experimental data when coupled with the corresponding theoretical calculations of the same accuracy level (see [6–8] and references therein) provided the best verification of the electroweak sector of the Standard Model at the low-energy regime. However, the precise calculations of the PNC effects in neutral systems are very difficult to achieve. For this reason, the investigations of the PNC effects in heavy few-electron systems where the interelectronic interaction can be calculated accurately by means of the perturbation theory in the parameter $1/Z$ (Z is the nuclear charge number) seem very promising.

Gorshkov and Labzowsky [9] were first who considered highly-charged ions as a proper tool for measuring the PNC effect. To date, various theoretical scenarios were proposed to study the P-odd asymmetry in highly-charged ions. The PNC effect in the process of Auger decay of the He-like uranium was studied by Pindzola [10]. Gribakin et al [11] discussed the parity violation in the process of dielectronic recombination of polarized electrons with H-like ions. A similar process for the case of He-like ions was investigated in [12]. The PNC effect in the process of radiative recombination of electrons with H-like ions was studied in several works [13–15]. The parity violation on the laser-induced transition was considered for heavy He-like ions in [16, 17] and for heavy Be-like ions in [18].

Though the PNC effect in highly-charged ions was extensively studied, the influence of the weak interaction on the process of electron scattering by a heavy ion has not yet been investigated. In the present work we study the PNC effect in the elastic scattering of polarized electrons by heavy He-like ions, being initially in the ground state. In order to enhance the parity violation we assume that the energy of the incident electron is tuned in resonance with close-lying opposite-parity ${\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}$ and ${\left[{\left(1s2{p}_{1/2}\right)}_{0}n\kappa \right]}_{1/2}$ states of the corresponding Li-like ions [19].

The relativistic units (${m}_{{\rm{e}}}={\hslash }=c=1$) and the Heaviside charge unit ($\alpha ={e}^{2}/(4\pi )$) are used in the paper.

We consider the resonance elastic scattering of an electron with asymptotic four-momentum $\left(\varepsilon ,{{\bf{p}}}_{i}\right)$ and polarization ${\mu }_{i}$ by a heavy He-like ion being initially in the ground ${(1s)}^{2}$ state. It is assumed that the electron energy is tuned in resonance with doubly-excited quasidegenerate opposite-parity d₁ or d₂ states. The scattered electron is characterized by four-momentum $(\varepsilon ,{{\bf{p}}}_{f})$ and polarization ${\mu }_{f}$.

Let us start with the consideration of the parity conserving part of the process amplitude. We construct this amplitude by means of the $1/Z$ perturbation theory up to the second order:

$$\begin{eqnarray}&&{\tau }_{{\mu }_{f}{\mu }_{i}}^{\mathrm{PC}}={\tau }_{{\mu }_{f}{\mu }_{i}}^{(0)}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(1,\mathrm{dir})}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(1,\mathrm{exc})}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(2)},\end{eqnarray} \tag{ 1 }$$

where the first order contribution is separated into two terms which correspond to the direct and exchange parts of the interelectronic interaction. The sum of the zero-order and direct first-order terms can be written as follows [20]:

$$\begin{eqnarray}&&{\tau }_{{\mu }_{f}{\mu }_{i}}^{(0)}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(1,\mathrm{dir})}={\chi }_{1/2{\mu }_{f}}^{\dagger }({\bf{n}})(A+2B{\boldsymbol{\eta }}\cdot {\bf{S}}){\chi }_{1/2{\mu }_{i}}({\boldsymbol{\nu }}),\end{eqnarray} \tag{ 2 }$$

where ${\bf{S}}$ is the spin operator, ${\boldsymbol{\nu }}$ and ${\bf{n}}$ are the unit vectors in the ${{\bf{p}}}_{i}$ and ${{\bf{p}}}_{f}$ directions (see figure 1), respectively, and ${\boldsymbol{\eta }}=[{\boldsymbol{\nu }}\times {\bf{n}}]/| [{\boldsymbol{\nu }}\times {\bf{n}}]|$. The two-component ${\chi }_{1/2{\mu }_{i}}({\boldsymbol{\nu }})$ function is an eigenfunction of the ${\bf{S}}\cdot {\boldsymbol{\nu }}$ operator with an eigenvalue ${\mu }_{i}$ and ${\chi }_{1/2{\mu }_{f}}({\bf{n}})$ satisfies $({\bf{S}}\cdot {\bf{n}}){\chi }_{1/2{\mu }_{f}}({\bf{n}})={\mu }_{f}{\chi }_{1/2{\mu }_{f}}({\bf{n}})$. The scattering amplitudes A and B are defined as [21]:

$$\begin{eqnarray}A & = & \displaystyle \frac{1}{2{\rm{i}}{p}_{f}}\underset{l=0}{\overset{\infty }{\displaystyle \sum }}\left\{(l+1)\left[\mathrm{exp}(2{\rm{i}}{\delta }_{l+1/2,l})-1\right]\right.\\ & & +\left.l\left[\mathrm{exp}(2{\rm{i}}{\delta }_{l-1/2,l})-1\right]\right\}{P}_{l}(\mathrm{cos}\theta ),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&B=\displaystyle \frac{1}{2{p}_{f}}\underset{l=1}{\overset{\infty }{\displaystyle \sum }}\left[\mathrm{exp}(2{\rm{i}}{\delta }_{l+1/2,l})-\mathrm{exp}(2{\rm{i}}{\delta }_{l-1/2,l})\right]{P}_{l}^{1}(\mathrm{cos}\theta ).\end{eqnarray} \tag{ 4 }$$

Here p_f is the momentum of the scattering electron, P_l and P_l¹ are the Legendre polynomials and associate Legendre functions, respectively, and θ is the scattering angle. The phase shifts ${\delta }_{j,l}$ for the total angular j and the orbital l momenta are determined from the asymptotic behavior of the Dirac equation solutions in the scattering potential $V(r)={V}_{\mathrm{nuc}}(r)+{V}_{\mathrm{scr}}(r)$. Here ${V}_{\mathrm{nuc}}$ is the electrostatic potential of the extended nucleus and ${V}_{\mathrm{scr}}$ is the screening potential of the ${(1s)}^{2}$ shell:

$$\begin{eqnarray}&&{V}_{\mathrm{scr}}(r)=2\alpha {\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{dr}\prime }{{r}_{\gt }}\left[{G}_{1s}^{2}(r\prime )+{F}_{1s}^{2}(r\prime )\right],\end{eqnarray} \tag{ 5 }$$

where ${r}_{\gt }$ is the greater of r and r', ${G}_{1s}(r)$ and ${F}_{1s}(r)$ are the upper and lower components of the radial wave function of one-electron $1s$ state, respectively. Since $V(r)\sim (Z-2)/r$ for large r, the scattering amplitudes defining by equations (3) and (4) are divergent as they stand. Nevertheless, one can obtain the convergent expression for A and B utilizing the regularization procedure [22–25] which deals with the pure Coulomb potential. The deviation of the scattering potential from the Coulomb one is accounted for using the method described in [26].

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The exchange first-order amplitude ${\tau }_{{\mu }_{f}{\mu }_{i}}^{(1,\mathrm{exc})}$ is constructed by subtraction of the terms corresponding to the direct part of the interelectronic interaction from ${\tau }_{{\mu }_{f}{\mu }_{i}}^{(1)}={(2\pi )}^{2}\varepsilon \langle {\Psi }_{f}| I| {\Psi }_{i}\rangle$ (see [27, 28] for details). Here I is the operator of the interelectronic interaction, $\left|{\Psi }_{i}\right.\rangle$ and $\left|{\Psi }_{f}\right.\rangle$ are the wave functions of the initial and final states of the system, respectively. Due to the fact that for heavy highly-charged ions the electron–electron interaction is suppressed by a factor $1/Z$ compared to the electron–nucleus Coulomb interaction, we can utilize the one-electron approximation. In this approach the wave functions of the initial and final states are given by

$$\begin{eqnarray}{\Psi }_{{\bf{p}}\mu ,{JM}}\left({{\bf{x}}}_{1},{{\bf{x}}}_{2},{{\bf{x}}}_{3}\right) & = & {A}_{N}\underset{{\mathcal{P}}}{\displaystyle \sum }{(-1)}^{{\mathcal{P}}}{\mathcal{P}}\underset{{m}_{1}{m}_{2}}{\displaystyle \sum }{C}_{{j}_{1}{m}_{1},{j}_{2}{m}_{2}}^{{JM}}{\psi }_{{n}_{1}{\kappa }_{1}{m}_{1}}\\ & & \times \left({{\bf{x}}}_{1}\right){\psi }_{{n}_{2}{\kappa }_{2}{m}_{2}}\left({{\bf{x}}}_{2}\right){\psi }_{{\bf{p}}\mu }\left({{\bf{x}}}_{3}\right).\end{eqnarray} \tag{ 6 }$$

Here ${\psi }_{n\kappa m}$ is the one-electron bound-state Dirac wave function and ${\psi }_{{\bf{p}}\mu }$ is the continuum Dirac state wave function with asymptotic momentum ${\bf{p}}$ and helicity μ (spin projection onto the momentum direction). The normalization factor ${A}_{N}=1/\sqrt{2\cdot 3!}$ for equivalent bound electrons and ${A}_{N}=1/\sqrt{3!}$ otherwise, ${C}_{{j}_{1}{m}_{1},{j}_{2}{m}_{2}}^{{JM}}$ is the Clebsch–Gordan coefficient, ${(-1)}^{{\mathcal{P}}}$ is the permutation parity and ${\mathcal{P}}$ is the permutation operator. The explicit expression for the continuum Dirac wave function can be written as [29, 30]

$$\begin{eqnarray}{\psi }_{{\bf{p}}\mu }^{(\pm )}({\bf{r}}) & = & \displaystyle \frac{1}{\sqrt{4\pi }}\cdot \displaystyle \frac{1}{\sqrt{\varepsilon p}}\underset{\kappa {m}_{j}}{\displaystyle \sum }{C}_{l0,1/2\mu }^{j\mu }{{\rm{i}}}^{l}\\ & & \times \sqrt{2\;l+1}{e}^{\pm {\rm{i}}{\delta }_{j,l}}{D}_{{m}_{j}\mu }^{j}({\bf{z}}\to {\bf{p}}){\psi }_{\varepsilon \kappa {m}_{j}}({\bf{r}}),\end{eqnarray} \tag{ 7 }$$

where the upper (lower) sign corresponds to the incoming (outgoing) electron and $\kappa ={(-1)}^{j+l+1/2}(j+1/2)$ is the Dirac quantum number. The Wigner matrix ${D}_{{{MM}}^{\prime }}^{J}({\bf{z}}\to {\bf{p}})$ (see [29, 31] for details) rotates the ${\bf{z}}$ axis into the ${\bf{p}}$ direction.

The second-order amplitude, corresponding to the dielectronic recombination into one of doubly excited d₁ or d₂ states with subsequent Auger decay, is given by [27, 28]

$$\begin{eqnarray}&&{\tau }_{{\mu }_{f}{\mu }_{i}}^{(2)}={(2\pi )}^{2}\varepsilon \underset{k=1,2}{\displaystyle \sum }\underset{{M}_{{d}_{k}}}{\displaystyle \sum }\displaystyle \frac{\left\langle{\Psi }_{f}\left|I\right|{\Psi }_{{d}_{k}}\right\rangle\left\langle{\Psi }_{{d}_{k}}\left|I\right|{\Psi }_{i}\right\rangle}{{E}_{i}-{E}_{{d}_{k}}+{\rm{i}}{\Gamma }_{{d}_{k}}/2},\end{eqnarray} \tag{ 8 }$$

where ${E}_{{d}_{k}}$ is the energy of the d_k state, ${E}_{i}={E}_{{(1s)}^{2}}+\varepsilon$ is the energy of the initial state, ${\Gamma }_{{d}_{k}}$ is the total width and ${M}_{{d}_{k}}$ is the momentum projection of the d_k state. The wave functions of the d₁ and d₂ states in the one-electron approximation are given by

$$\begin{eqnarray}{\Psi }_{J(J\prime )M}\left({{\bf{x}}}_{1},{{\bf{x}}}_{2},{{\bf{x}}}_{3}\right) & = & {B}_{N}\underset{{\mathcal{P}}}{\displaystyle \sum }{(-1)}^{{\mathcal{P}}}{\mathcal{P}}\underset{M\prime {m}_{3}}{\displaystyle \sum }\\ & & \times \underset{{m}_{1}{m}_{2}}{\displaystyle \sum }{C}_{J\prime M\prime ,{j}_{3}{m}_{3}}^{{JM}}{C}_{{j}_{1}{m}_{1},{j}_{2}{m}_{2}}^{J\prime M\prime }\\ & & \times {\psi }_{{n}_{1}{\kappa }_{1}{m}_{1}}\left({{\bf{x}}}_{1}\right){\psi }_{{n}_{2}{\kappa }_{2}{m}_{2}}\left({{\bf{x}}}_{2}\right){\psi }_{{n}_{3}{\kappa }_{3}{m}_{3}}\left({{\bf{x}}}_{3}\right),\end{eqnarray} \tag{ 9 }$$

where B_N is the normalization factor.

Having constructed all the relevant parity conserving amplitudes, we now turn to the evaluation of the parity violation in the resonance elastic electron scattering. The dominant contribution to the PNC effect in the process of interest is provided by the nuclear spin-independent part of the weak interaction, which can be described by the following effective Hamiltonian [1]

$$\begin{eqnarray}&&{H}_{{\rm{W}}}=-\left({G}_{{\rm{F}}}/\sqrt{8}\right){Q}_{{\rm{W}}}{\rho }_{{\rm{N}}}(r){\gamma }_{5}.\end{eqnarray} \tag{ 10 }$$

Here ${Q}_{{\rm{W}}}\approx -N+Z\left(1-4{\mathrm{sin}}^{2}{\theta }_{{\rm{W}}}\right)$ is the weak charge of the nucleus, ${\rho }_{{\rm{N}}}$ is the nuclear weak-charge density (normalized to unity), ${G}_{{\rm{F}}}$ is the Fermi constant, and ${\gamma }_{5}$ is the Dirac matrix. To account for the weak interaction we have to modify the wave functions:

$$\begin{eqnarray}&&\left|{\Psi }_{{d}_{1}}\right.\rangle\to \left|{\Psi }_{{d}_{1}}\right.\rangle+\displaystyle \frac{\left\langle{\Psi }_{{d}_{2}}\left|{H}_{{\rm{W}}}\right|{\Psi }_{{d}_{1}}\right\rangle}{{E}_{{d}_{1}}-{E}_{{d}_{2}}}\left|{\Psi }_{{d}_{2}}\right.\rangle,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\left|{\Psi }_{{d}_{2}}\right.\rangle\to \left|{\Psi }_{{d}_{2}}\right.\rangle+\displaystyle \frac{\left\langle{\Psi }_{{d}_{1}}\left|{H}_{{\rm{W}}}\right|{\Psi }_{{d}_{2}}\right\rangle}{{E}_{{d}_{2}}-{E}_{{d}_{1}}}\left|{\Psi }_{{d}_{1}}\right.\rangle.\end{eqnarray} \tag{ 12 }$$

To simplify the notations we define the admixing parameter ${\rm{i}}\xi =\left\langle{\Psi }_{{d}_{1}}\left|{H}_{{\rm{W}}}\right|{\Psi }_{{d}_{2}}\right\rangle/\left({E}_{{d}_{2}}-{E}_{{d}_{1}}\right)$. Substituting equations (11) and (12) into equation (8) and keeping only the linear terms in ξ one obtains the parity violating amplitude

$$\begin{eqnarray}{\tau }_{{\mu }_{f}{\mu }_{i}}^{\mathrm{PNC}} & = & {\rm{i}}{(2\pi )}^{2}\varepsilon \xi \underset{{M}_{d}}{\displaystyle \sum }\left(\langle {\Psi }_{f}| I| {\Psi }_{{d}_{2}}\rangle \langle {\Psi }_{{d}_{1}}| I| {\Psi }_{i}\rangle \right.\\ & & -\left.\langle {\Psi }_{f}| I| {\Psi }_{{d}_{1}}\rangle \langle {\Psi }_{{d}_{2}}| I| {\Psi }_{i}\rangle \right)\\ & & \times \left(\displaystyle \frac{1}{{E}_{i}-{E}_{{d}_{1}}+{\rm{i}}{\Gamma }_{{d}_{1}}/2}-\displaystyle \frac{1}{{E}_{i}-{E}_{{d}_{2}}+{\rm{i}}{\Gamma }_{{d}_{2}}/2}\right).\end{eqnarray} \tag{ 13 }$$

Here we have utilized the fact that the weak interaction conserves the total momentum projection and, as a result, M_d stands for ${M}_{{d}_{1}}={M}_{{d}_{2}}$.

One should point out that the nuclear spin-independent part of the weak interaction provides one more contribution to the PNC effect of the process studied. This contribution is related to the scattering by the direct electron-nucleus weak interaction and can be expressed by the amplitude ${(2\pi )}^{2}\varepsilon \left\langle{\Psi }_{f}\left|{H}_{{\rm{W}}}\right|{\Psi }_{i}\right\rangle$. However, we omit this term since it is negligibly small in the framework of the approximations considered. Thus, the amplitude of the resonance elastic electron scattering can be written in the following form

$$\begin{eqnarray}&&{\tau }_{{\mu }_{f}{\mu }_{i}}={\tau }_{{\mu }_{f}{\mu }_{i}}^{\mathrm{PC}}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{\mathrm{PNC}}\end{eqnarray} \tag{ 14 }$$

with ${\tau }_{{\mu }_{f}{\mu }_{i}}^{\mathrm{PC}}={\tau }_{{\mu }_{f}{\mu }_{i}}^{(0)}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(1,\mathrm{dir})}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(1,\mathrm{exc})}+{\tau }_{{\mu }_{f}{\mu }_{i}}^{(2)}$ being the parity conserving contribution. Examining the introduced amplitudes with respect to the spatial symmetry leads to the following rules

$$\begin{eqnarray}&&{\tau }_{\mu \mu }^{\mathrm{PC}}={\tau }_{-\mu -\mu }^{\mathrm{PC}},\quad {\tau }_{\mu -\mu }^{\mathrm{PC}}=-{\tau }_{-\mu \mu }^{\mathrm{PC}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\tau }_{\mu \mu }^{\mathrm{PNC}}=-{\tau }_{-\mu -\mu }^{\mathrm{PNC}},\quad {\tau }_{\mu -\mu }^{\mathrm{PNC}}={\tau }_{-\mu \mu }^{\mathrm{PNC}}=0.\end{eqnarray} \tag{ 16 }$$

In order to enhance the PNC effect in the elastic scattering of polarized electrons by He-like ions, being in the ground state, we assume that the energy of the incident electron is tuned in resonance with doubly-excited opposite-parity ${d}_{1}\equiv {\left[{\left(1s2{p}_{1/2}\right)}_{0}n\kappa \right]}_{1/2}$ and ${d}_{2}\equiv {\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}$ states of the corresponding Li-like ions. The quasidegeneracy of these states was found for several $n,\kappa$ and Z in [19].

We study the influence of the parity violation on the differential cross section (DCS) ${\sigma }_{{\mu }_{f}{\mu }_{i}}\equiv d{\sigma }_{{\mu }_{f}{\mu }_{i}}/d\Omega ={| {\tau }_{{\mu }_{f}{\mu }_{i}}| }^{2}$ of the scattering process. Let us introduce the non-spin-flip ${\sigma }_{\mathrm{nsf}}=\frac{1}{2}\left({\sigma }_{1/2\ 1/2}\right.+\left.{\sigma }_{-1/2\ -1/2}\right)$ and the spin-flip ${\sigma }_{\mathrm{sf}}=\frac{1}{2}\left({\sigma }_{1/2\ -1/2}+{\sigma }_{-1/2\ 1/2}\right)$ cross sections. Then, the total DCS is ${\sigma }_{0}={\sigma }_{\mathrm{nsf}}+{\sigma }_{\mathrm{sf}}$. According to the rules (16), the weak interaction modifies the cross section only in the case when the helicities of the incident and outgoing electrons coincide (${\mu }_{i}={\mu }_{f}$). As a result, the presence of the PNC effect manifests in deviation of the P-odd contribution ${\sigma }_{\mathrm{PNC}}=\frac{1}{2}\left({\sigma }_{1/2\ 1/2}-{\sigma }_{-1/2\ -1/2}\right)$ to the cross section from zero. In the present work we consider two scenarios. In the first scenario, the polarization of the outgoing electron is assumed to be detected and only the non-spin-flip contribution to the cross section is considered. In the second scenario the polarization remains unobserved and both ${\sigma }_{\mathrm{nsf}}$ and ${\sigma }_{\mathrm{sf}}$ are taken into account. The luminosity of the first (I) and second (II) scenarios can be expressed as follows [11, 13]

$$\begin{eqnarray}&&{L}_{I,{II}}=\displaystyle \frac{{\sigma }_{I,{II}}+{\sigma }_{I,{II}}^{({\rm{B}})}}{2{\sigma }_{\mathrm{PNC}}^{2}{\eta }^{2}T}.\end{eqnarray} \tag{ 17 }$$

Here ${\sigma }_{I}={\sigma }_{\mathrm{nsf}}$ while ${\sigma }_{{II}}={\sigma }_{0}$, ${\sigma }_{I,{II}}^{({\rm{B}})}$ corresponds to the background signal, T is the data collection time, and η is the desired relative uncertainty of the PNC effect measurement. In the present analysis we set ${\sigma }_{I,{II}}^{({\rm{B}})}=0$, T equals to two weeks, and $\eta =1\%$.

In figure 2, the PNC asymmetry coefficients ${{\mathcal{A}}}_{I}={\sigma }_{\mathrm{PNC}}/{\sigma }_{\mathrm{nsf}}$ and ${{\mathcal{A}}}_{{II}}={\sigma }_{\mathrm{PNC}}/{\sigma }_{0}$ for the elastic electron scattering on He-like samarium (Z = 62) are displayed as functions of the scattering angle θ in the case of resonance with the ${\left[{(1s2s)}_{0}7s\right]}_{1/2}$ and ${\left[{\left(1s2{p}_{1/2}\right)}_{0}7s\right]}_{1/2}$ states. Since these coefficients are directly related to the magnitude of the PNC effect, one can conclude that for the first scenario the parity violation is expected to become most significant at large scattering angles. In the case when the polarization of the scattered electron is not detected (second scenario) the most promising situation occurs for $\theta \sim 60^\circ$, while at larger scattering angles a strong suppression of the P-odd asymmetry is observed. This is due to the fact that at large scattering angles the dominant contribution to the DCS is provided by the P-even spin-flip amplitude, which does not interfere with the PNC amplitude according to equations (16).

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In figure 3, the parity violating asymmetry of the resonance electron scattering on He-like samarium (Z = 62) is depicted as a function of the incident electron energy for three different scattering angles (60, 110 and 175 degrees). From this figure one can see that the peak magnitude of the P-odd asymmetry is expected for the energy of the scattering electron close to resonance which is provided by the ${\left[{(1s2s)}_{0}7s\right]}_{1/2}$ state. Here it is worth mentioning that the parameters, being related to the maximum magnitude of the parity violating asymmetry, may not provide the best value of the luminosity, and vice versa. In order to find the optimal relation we propose the following procedure. First, one should pick out scattering angles at which the P-odd asymmetry has the same order of magnitude as the maximal one. Among them the optimal relation is provided by such an angle which corresponds to the minimum of the luminosity. As an example, let us consider the scenario where the polarization of the outgoing electron is detected (first scenario). For the case of the samarium ion (see figure 3) the maximal value of ${{\mathcal{A}}}_{I}$ is expected for the scattering angle $175^\circ$ and equals $-2.2\times {10}^{-7}$, while L_I for these parameters is equal to $7.1\times {10}^{33}$ cm⁻² s⁻¹. The optimal relation between ${{\mathcal{A}}}_{I}$ and L_I is expected for $\theta \sim 108^\circ$ where they take the values $-1.2\times {10}^{-7}$ and $6.5\times {10}^{31}$ cm⁻² s⁻¹, respectively.

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In tables 1 and 2 we present the numerical results for the parameters n, κ and Z which seem to be most promising for measuring the PNC effect in the process of resonance elastic electron scattering on He-like ions. It is assumed that the energy of the incident electron is tuned in vicinity of the resonance, being related to the ${\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}$ state, to provide the peak value of the P-odd asymmetry. In table 1 we present the results for the case when the polarization of the scattered electron is measured (first scenario). The results for the second scenario, where the polarization of the outgoing electron is not detected, are represented in table 2.

Table 1.  Cross section of the resonance elastic electron scattering on He-like ions for parameters n, κ and Z which seem to be most promising for measuring the PNC effect. The energy of the incident electron is tuned in vicinity of resonance corresponding to the ${\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}$ state. It is assumed that the polarization of the scattered electron is detected. The energy difference $\Delta E={E}_{{\left[{\left(1s2{p}_{1/2}\right)}_{0}n\kappa \right]}_{1/2}}-{E}_{{\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}}$ is taken from [19]. The scattering angle θ provides the optimal relation between the P-odd asymmetry ${{\mathcal{A}}}_{I}={\sigma }_{\mathrm{PNC}}/{\sigma }_{\mathrm{nsf}}$ and the luminosity L_I, which is defined by equation (17). ${\sigma }_{0}$ and ${\sigma }_{\mathrm{nsf}}$ are the total and non-spin-flip cross sections, respectively, and ${\sigma }_{\mathrm{PNC}}$ stands for the parity violating contribution to the cross section.

Z	$n\kappa$	$\Delta E$ (eV)	${\varepsilon }_{i}$ (keV)	θ (deg)	${{\mathcal{A}}}_{I}$	LI (cm−2 s−1)	${\sigma }_{0}$ (b)	${\sigma }_{\mathrm{nsf}}$ (b)	${\sigma }_{\mathrm{PNC}}$ (b)
60	$6s$	−0.222(56)	36.40	163	$-1.0\times {10}^{-7}$	$2.7\times {10}^{33}$	$4.8\times {10}^{3}$	$1.5\times {10}^{2}$	$-1.5\times {10}^{-5}$
62	$7s$	−0.103(64)	39.56	108	$-1.2\times {10}^{-7}$	$6.5\times {10}^{31}$	$1.0\times {10}^{4}$	$4.7\times {10}^{3}$	$-5.5\times {10}^{-4}$
90	$6s$	2.51(47)	88.36	64	$-3.3\times {10}^{-8}$	$1.7\times {10}^{32}$	$2.5\times {10}^{4}$	$2.2\times {10}^{4}$	$-7.2\times {10}^{-4}$
	$7s$	1.75(47)	89.22	57	$3.8\times {10}^{-8}$	$9.3\times {10}^{31}$	$3.4\times {10}^{4}$	$3.0\times {10}^{4}$	$1.2\times {10}^{-3}$
92	$5s$	2.97(28)	91.43	66	$-3.8\times {10}^{-8}$	$1.4\times {10}^{32}$	$2.3\times {10}^{4}$	$2.0\times {10}^{4}$	$-7.6\times {10}^{-4}$
	$6s$	−1.07(28)	92.95	74	$-1.0\times {10}^{-7}$	$3.1\times {10}^{31}$	$1.6\times {10}^{4}$	$1.3\times {10}^{4}$	$-1.3\times {10}^{-3}$

Table 2.  Cross section of the resonance elastic electron scattering on He-like ions for parameters n, κ and Z which seem to be most promising for measuring the PNC effect. The energy of the incident electron is tuned in vicinity of resonance corresponding to the ${\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}$ state. It is assumed that the polarization of the scattered electron is not detected. The energy difference $\Delta E={E}_{{\left[{\left(1s2{p}_{1/2}\right)}_{0}n\kappa \right]}_{1/2}}-{E}_{{\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}}$ is taken from [19]. The scattering angle θ provides the optimal relation between the P-odd asymmetry ${{\mathcal{A}}}_{{II}}={\sigma }_{\mathrm{PNC}}/{\sigma }_{0}$ and the luminosity L_II, which is defined by equation (17). ${\sigma }_{0}$ and ${\sigma }_{\mathrm{nsf}}$ are the total and non-spin-flip cross sections, respectively, and ${\sigma }_{\mathrm{PNC}}$ stands for the parity violating contribution to the cross section.

Z	$n\kappa$	$\Delta E$ (eV)	${\varepsilon }_{i}$ (keV)	θ (deg)	${{\mathcal{A}}}_{{II}}$	LII (cm−2 s−1)	${\sigma }_{0}$ (b)	${\sigma }_{\mathrm{nsf}}$ (b)	${\sigma }_{\mathrm{PNC}}$ (b)
60	$6s$	−0.222(56)	36.40	43	$2.2\times {10}^{-8}$	$4.0\times {10}^{31}$	$2.1\times {10}^{5}$	$1.9\times {10}^{5}$	$4.6\times {10}^{-3}$
62	$7s$	−0.103(64)	39.56	45	$4.9\times {10}^{-8}$	$1.0\times {10}^{31}$	$1.6\times {10}^{5}$	$1.5\times {10}^{5}$	$8.0\times {10}^{-3}$
90	$6s$	2.51(47)	88.36	59	$-2.7\times {10}^{-8}$	$1.8\times {10}^{32}$	$3.2\times {10}^{4}$	$2.8\times {10}^{4}$	$-8.4\times {10}^{-4}$
	$7s$	1.75(47)	89.22	58	$3.5\times {10}^{-8}$	$1.0\times {10}^{32}$	$3.2\times {10}^{4}$	$2.9\times {10}^{4}$	$1.1\times {10}^{-3}$
92	$5s$	2.97(28)	91.43	62	$-3.2\times {10}^{-8}$	$1.5\times {10}^{32}$	$2.7\times {10}^{4}$	$2.4\times {10}^{4}$	$-8.5\times {10}^{-4}$
	$5{p}_{1/2}$	−0.511(27)	91.44	46	$2.2\times {10}^{-8}$	$1.2\times {10}^{32}$	$6.8\times {10}^{4}$	$6.3\times {10}^{4}$	$1.5\times {10}^{-3}$

From table 1 one can see that for the first scenario the PNC effect is expected to be most pronounced for scattering on the samarium (Z = 62) ion at the energy of the incident electron tuned in vicinity of resonance corresponding to the ${\left[{(1s2s)}_{0}7s\right]}_{1/2}$ state. In this case, the optimal values of the asymmetry and the luminosity equal to $-1.2\times {10}^{-7}$ and $6.5\times {10}^{31}$ cm⁻² s⁻¹, respectively, and are achieved for the scattering angle $\sim 108^\circ$. This system seems also to be most preferable for the second scenario (table 2), where the polarization of the scattered electron is not detected. In this scenario the optimal values ${{\mathcal{A}}}_{{II}}=4.9\times {10}^{-8}$ and ${L}_{{II}}=1.0\times {10}^{31}$ cm⁻² s⁻¹ are obtained at $\theta \sim 45^\circ$. From these tables one can conclude that the observation of the outgoing electron polarization does not allow to increase significantly the PNC effect.

Polarized beams of electrons are produced nowadays by photo-ionization from surfaces of semiconductors, such as GaAsP, with circularly polarized laser light with a help of the Fano effect [32]. Another approach uses Penning ionization from polarized excited helium atoms [33]. In both cases the beam intensity is limited typically to 1 mA. The degree of the electron beam polarization may reach 80–90%. Polarization of such beams can be determined by Mott [34, 35] and bremsstrahlung [36–39] polarimetry techniques, albeit with low efficiencies. Therefore, in the following we consider experimental possibilities of the second approach only. In this approach the spin state of the scattered initially polarized electron is not determined.

Ion beams of any charge state are nowadays routinely available in the storage ring ESR at GSI in Darmstadt and in the near future they will be available in the storage rings HESR and CRYRING at FAIR. A target of polarized electrons is being developed for these storage rings. Its areal density will be ${\rho }_{e}={10}^{8}$ cm⁻². In the following we estimate the number of electrons scattered in the interval of the scattering angles $40^\circ \lt \theta \lt 50^\circ$:

$$\begin{eqnarray}&&{R}_{\mathrm{scat}}={\sigma }_{0}{\rho }_{e}{N}_{i}R\Omega ,\end{eqnarray} \tag{ 18 }$$

where ${N}_{i}={10}^{8}$ is the number of He-like samarium ions circulating in the ring, R = 1 MHz is their revolution frequency and $\Omega =1$ is a solid angle covered by the detectors of the scattered electrons. With such parameters we estimate the rate of the scattered electrons ${R}_{\mathrm{scat}}=100$ Hz, which is not sufficient for this experiment.

In the following we consider another scheme of a laboratory experiment which uses a modified electron beam ion source (EBIS) [40–42]. In a typical EBIS the ions are produced in collisions with a mono-energetic electron beam. They are trapped radially by a space charge of the beam and axially by electrostatic potentials. The electron beam is compressed by a magnetic field produced by a superconducting solenoid to a typical diameter of ${D}_{i}=50\;\mu$m. Such setups are capable of producing large quantities of ions in nearly any charge state. For measuring the electron scattering cross sections we propose to modify this setup. Two electron beams should be injected into the EBIS—a strong unpolarized beam, detuned from the DR resonance, to produce and trap the ions and an overlapping polarized beam, tuned into the resonance, to probe the elastic scattering cross section. Furthermore, in addition to the uniform magnetic field in the ion trapping section we propose to use two additional coils to increase the field at the edges of this section, see figure 4. The stronger fields will act as magnetic mirrors reflecting electrons scattered in the central section of the trap. Electrons scattered at an angle θ inside the trap will be reflected at the trap magnetic mirrors 1 and 2 with corresponding magnetic fields B₁ and B₂ if

$$\begin{eqnarray}&&\mathrm{sin}{\theta }_{\mathrm{1,2}}\gt \sqrt{{B}_{0}/{B}_{\mathrm{1,2}}},\end{eqnarray} \tag{ 19 }$$

where B₀ is the field inside the trap. Thus, if the magnetic fields of the mirrors are tuned to reflect electrons with ${\theta }_{1}\gt 50^\circ$ and ${\theta }_{2}\gt 40^\circ$, this setup will reflect the electrons scattered within the trap in the intervals of angles $40^\circ \gt \theta \gt 50^\circ$ and $\theta \gt 130^\circ$, see inset in figure 4. The primary electron beam will pass both mirrors. The reflected electrons can be separated from the incoming electron beam with a help of a Wien filter and detected with an efficiency close to 100%. The rate of detected scattering events in this experiment can be estimated as

$$\begin{eqnarray}&&{R}_{\mathrm{scat}}=\displaystyle \frac{{\sigma }_{0}}{{D}_{e}^{2}}\cdot \displaystyle \frac{{I}_{e}}{e}{N}_{i}\Omega ,\end{eqnarray} \tag{ 20 }$$

where I_e = 1 mA is the current of the polarized electron beam, $e=1.6\times {10}^{-19}$ C is the electron charge, ${N}_{i}={10}^{7}$ is the number of the trapped He-like samarium ions and $\Omega =1$ is the solid angle covered by the interval $40^\circ \gt \theta \gt 50^\circ$. The rate of the detected scattering events is ${R}_{\mathrm{scat}}=3\times {10}^{7}$ Hz. With this rate identification of the parity-violating contribution to the electron scattering is possible within 6 months of continuous operation. However, a number of experimental challenges associated with a construction of a dedicated modified EBIS with a more complex magnetic field configuration and two overlapping electron beams should be addressed beforehand. Moreover, the high rate of the scattering events will pose their detection difficulties.

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In the present work the PNC effect has been studied in the elastic scattering of polarized electrons by heavy He-like ions, being initially in the ground state. In order to enhance the parity violation effect, the energy of the incident electron has been chosen to provide a resonance with one of the quasidegenerate doubly-excited ${\left[{(1s2s)}_{0}n\kappa \right]}_{1/2}$ and ${\left[{\left(1s2{p}_{1/2}\right)}_{0}n\kappa \right]}_{1/2}$ states of the corresponding Li-like ion. We have considered two different scenarios. In the first scenario we assume that the polarization of the scattered electron was measured. In the second one the polarization was supposed to be unobservable. It has been found that for both variants the PNC effect occurs to be most pronounced for scattering on samarium ion at the energy of the incident electron tuned in vicinity of resonance, which is related to the ${\left[{(1s2s)}_{0}7s\right]}_{1/2}$ state. In the case of the first scenario the peak value of the PNC asymmetry equals to $-1.2\times {10}^{-7}$ at $\theta \sim 108^\circ$, while in the second scenario the P-odd asymmetry is $4.9\times {10}^{-8}$ for the scattering angle $\theta \sim 45^\circ$. We have also proposed and discussed the scheme of a laboratory experiment for the modified EBIS which does in principle allow the measurement of these values.

It is worth noting that some difficulties of the process under investigation can be avoided by studying inelastic electron scattering, where one could get rid of the dominant zero-order (in $1/Z$) contribution to the PC amplitude, thus reducing the suppression of the PNC effect. Additionally, one may also think, that the corresponding investigations with other heavy few-electron ions can lead to a bigger effect. We expect that the calculations performed in the present paper can serve as a proper basis for further study in these directions.

Fruitful discussions with D A Telnov are gratefully acknowledged. This work was supported by the grant of the President of the Russian Federation (Grant No. MK-1676.2014.2), by RFBR (Grant No. 13–02-00630), by SPbSU (Grants No. 11.38.269.2014 and No. 11.38.237.2015), and by SAEC Rosatom. Additional support was received from DFG, GSI, and DAAD. The work of V A Z was supported by the German-Russian Interdisciplinary Science Center (G-RISC). V A Z and A V M acknowledge financial support by the 'Dynasty' foundation. S T acknowledges support by the German Research Foundation (DFG) within the Emmy Noether program under Contract No. TA 740 1-1.