Relativistic calculations for spin-polarization of elastic electron—mercury scattering

The Dirac equation for motion of the projectile electron with a velocity v in a central field V(r) is given in [37] as

$$\begin{eqnarray}&&[c{\boldsymbol{\alpha }}.p+\beta {m}_{o}{c}^{2}+V(r)]\psi ({\bf{r}})=E\psi ({\bf{r}})\end{eqnarray} \tag{ 1 }$$

where, $E={m}_{o}\gamma {c}^{2}={E}_{i}+{m}_{o}{c}^{2}$ is the total energy, m_o is the rest mass of the projectile, $\gamma ={(1-{v}^{2}/{c}^{2})}^{-1/2}$, c is the velocity of light in vacuum, E_i is the kinetic energy of the incident particle, α and β are the usual 4 × 4 Dirac matrices. The relativistic wave function $\psi ({\bf{r}})$, a four-component spinor with quantum numbers κm, describing the motion of the scattered electron, is given by

$$\begin{eqnarray}&&{\psi }_{E\kappa m(\vec{r})}=\displaystyle \frac{1}{r}\left(\begin{array}{c}{P}_{E\kappa }(r){{\rm{\Omega }}}_{\kappa ,m}(\hat{r})\\ {{\rm{i}}{Q}}_{E\kappa }(r){{\rm{\Omega }}}_{-\kappa ,m}(\hat{r})\end{array}\right).\end{eqnarray} \tag{ 2 }$$

Here, ${P}_{E\kappa }(r)$ and ${Q}_{E\kappa }(r)$ represent, respectively the radial parts of the large and small components of the scattering wave function and ${{\rm{\Omega }}}_{\kappa ,m}(\hat{r})$ are the spherical spinors. The relativistic quantum number κ is defined as $\kappa =(l-j)(2j+1)$, where j and l are the total and orbital angular momentum quantum numbers that are both determined by the value of κ as $j=| \kappa | -1/2$, $l=j+\kappa /(2| \kappa | )$.

The radial functions ${P}_{E\kappa }(r)$ and ${Q}_{E\kappa }(r)$ satisfy separately the following set of coupled differential equations [38]:

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{P}}_{E\kappa }}{{\rm{d}}{r}}=-\displaystyle \frac{\kappa }{r}{P}_{E\kappa }(r)+\displaystyle \frac{{E}_{{i}}-V+2{m}_{o}{c}^{2}}{c\hslash }{Q}_{E\kappa }(r),\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{Q}}_{E\kappa }}{{\rm{d}}{r}}=-\displaystyle \frac{{E}_{{i}}-V}{c\hslash }{P}_{E\kappa }(r)+\displaystyle \frac{\kappa }{r}{Q}_{E\kappa }(r).\end{eqnarray} \tag{ 4 }$$

The scattering information is determined from the asymptotic form of the large component ${P}_{E\kappa }(r)$ of the scattering wave function which can be expressed in terms of the complex phase-shift ${\delta }_{\kappa }$ as

$$\begin{eqnarray}&&{P}_{E\kappa }(r)\simeq \sin \left({kr}-l\displaystyle \frac{\pi }{2}+{\delta }_{\kappa }\right).\end{eqnarray} \tag{ 5 }$$

The equations (3) and (4) satisfying the asymptotic condition (5) are solved numerically using the subroutine package RADIAL [39]. In the relativistic partial wave analysis, the phase shift ${\delta }_{\kappa }$ may be calculated for angular momentum value up to the maximum l_max = 25 000. The value of l_max, however, depends on the incident energy and rate of convergence. For energy above 5 MeV, where the convergence of partial wave series is slow, approximate factorization methods are used for the calculation of ${e}^{-}$–atom scattering.

The relativistic treatment of electron collisions enables us to calculate the polarization, which describes the measured spin-up and spin-down asymmetries in the number of scattered electrons. The expressions for the spin-polarization in terms of the scattering amplitudes is given by [3, 28]

$$\begin{eqnarray}&&S(\theta )={\rm{i}}\displaystyle \frac{f(\theta ){g}^{* }(\theta )-{f}^{* }(\theta )g(\theta )}{| f(\theta ){| }^{2}+| g(\theta ){| }^{2}}.\end{eqnarray} \tag{ 6 }$$

Here, $f(\theta )$ and $g(\theta )$, the direct and spin-flip scattering amplitudes [40], are respectively given as

$$\begin{eqnarray}&&f(\theta )=\displaystyle \frac{1}{2{\rm{i}}{k}}\displaystyle \sum _{l=0}^{\infty }\{(l+1)[\exp (2{\rm{i}}{\delta }_{\kappa =-l-1})-1]+l[\exp (2{\rm{i}}{\delta }_{\kappa =l})-1]\}{P}_{l}(\cos \theta ),\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&g(\theta )=\displaystyle \frac{1}{2{\rm{i}}{k}}\displaystyle \sum _{l=0}^{\infty }[\exp (2{\rm{i}}{\delta }_{\kappa =l})-\exp (2{\rm{i}}{\delta }_{\kappa =-l-1})]{P}_{l}^{1}(\cos \theta ).\end{eqnarray} \tag{ 8 }$$

Here, ${P}_{l}(\cos \theta )$ and ${P}_{l}^{1}(\cos \theta )$ denote respectively the Legendre polynomials and associated Legendre functions, k is the relativistic wave number of the projectile that is related to the momentum p and the kinetic energy E_i by

$$\begin{eqnarray}&&\hslash k=p,\,\,\,\,\,\,{(c\hslash k)}^{2}={E}_{i}({E}_{i}+2{m}_{o}{c}^{2}).\end{eqnarray} \tag{ 9 }$$

Equation (6) suggests that the behavior of the scattering amplitudes f (θ) and g(θ) respectively in equations (7) and (8) determines the total polarization points S = ±1 near the critical minima of DCS given by $\sigma (\theta )=| f(\theta ){| }^{2}+| g(\theta ){| }^{2}$.

The other spin-polarization parameters $U(\theta )$ and $T(\theta )$, treated in the present work, are given [3, 28] as

$$\begin{eqnarray}&&U(\theta )=\displaystyle \frac{f(\theta ){g}^{* }(\theta )+{f}^{* }(\theta )g(\theta )}{| f(\theta ){| }^{2}+| g(\theta ){| }^{2}},\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}&&T(\theta )=\displaystyle \frac{| f(\theta ){| }^{2}-| g(\theta ){| }^{2}}{| f(\theta ){| }^{2}+| g(\theta ){| }^{2}}.\end{eqnarray} \tag{ 11 }$$

$U(\theta )$ and $T(\theta )$ describe the relation of polarization vector during the scattering process [2]. The three spin functions obey the following conservation relation [19, 20]

$$\begin{eqnarray}&&{S}^{2}(\theta )+{T}^{2}(\theta )+{U}^{2}(\theta )=1.\end{eqnarray} \tag{ 12 }$$

The values of $U(\theta )$ and $T(\theta )$ are dependent on $S(\theta )$ and hence are useful indicators of total polarization, $S(\theta )$ = ±1.

The complex optical model potential V_opt(r), for the ${e}^{-}$–atom effective interaction, is given by [41]

$$\begin{eqnarray}&&{V}_{{\rm{opt}}}={V}_{{\rm{st}}}(r)+{V}_{{\rm{ex}}}(r)+{V}_{{\rm{cp}}}(r)-{{\rm{i}}{W}}_{{\rm{abs}}}(r).\end{eqnarray} \tag{ 13 }$$

The real components V_st(r), V_ex(r), V_cp(r) in (13) are the static, exchange and correlation-polarization potentials respectively. As stated earlier, the component W_abs(r) represents absorptive part which accounts for the loss of particles into various inelastic channels during the collisions. The static potential V_st(r) has been generated from the procedure adopted in Salvat et al [42], where the proton and electron densities are respectively taken as Fermi nuclear charge distribution [43] and Hartree–Fock (HF) analytical density function [44]. For e⁻–Hg scattering, we have used the semi-classical exchange potential of Furness and McCarthy [45] derived directly from the formal expression of the non-local exchange interaction by using a WKB like approximation for the wave functions. This is expressed as

$$\begin{eqnarray}&&{V}_{{\rm{ex}}}(r)=\displaystyle \frac{1}{2}[{E}_{i}-{V}_{{\rm{st}}}(r)]-\displaystyle \frac{1}{2}\{{[{E}_{i}-{V}_{{\rm{st}}}(r)]}^{2}+4\pi {a}_{o}{e}^{4}\rho (r)\}{}^{1/2}.\end{eqnarray} \tag{ 14 }$$

Here E_i is the incident energy of electron, a_o is the Bohr radius.

The electron density function $\rho (r)$, represented by an analytical function F(r) by [44], was calculated from the numerical HF wave functions, subject to some important constraints. Thus

$$\begin{eqnarray}&&\rho (r)\cong F(r)={f}_{o}(r)+\displaystyle \sum _{i=1}^{{N}_{f}}{c}_{i}{f}_{i}(r),\end{eqnarray} \tag{ 15 }$$

where $\{{c}_{i}\}$ are mixing coefficients,

$$\begin{eqnarray}&&{f}_{0}(r)=\rho (0)\exp (-2{Zr}),\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray}&&{f}_{i}(r)={r}^{{n}_{i}}\exp (-{\zeta }_{i}r),\,\,\,\,\,\,\,(i=1,\,\ldots ,\,{N}_{f}).\end{eqnarray} \tag{ 17 }$$

In equation (17), ${n}_{i}\gt 0$ and ${\zeta }_{i}\gt 0$ are used.

The approximation (15) for electron density consists of ${N}_{f}+1$ terms and includes linear $\{{c}_{i}\}$ and nonlinear $\{{n}_{i},{\zeta }_{i}\}$ parameters taken from [44]. The electron density function $\rho (r)$ satisfies the following normalized relation

$$\begin{eqnarray}&&\displaystyle \int \rho (r)4\pi {r}^{2}{\rm{d}}{r}=Z,\end{eqnarray} \tag{ 18 }$$

with Z being the atomic number of the target atom.

In the present work, we choose a global polarization potential V_cp [41], a combination of the parameter free long range polarization potential V_cps and LDA correlation potential V_co. Accordingly,

$$\begin{eqnarray}{V}_{{\rm{cp}}}(r)\equiv \left\{\begin{array}{lll}\max \{{V}_{{\rm{co}}}(r),{V}_{{\rm{cps}}}(r)\} & {\rm{if}} & r\lt {r}_{c}\\ \ \qquad \qquad {V}_{{\rm{cps}}}(r) & {\rm{if}} & r\geqslant {r}_{c},\end{array}\right.\end{eqnarray} \tag{ 19 }$$

where r_c is the outer radius at which short-range V_co(r) and long-range V_cps(r) intersect first.

The long range part V_cps(r) i.e. when the projectile is far from the atom, is approximated as [36]

$$\begin{eqnarray}&&{V}_{{\rm{cps}}}(r)=-\displaystyle \frac{\alpha }{2{({r}^{2}+{d}^{2})}^{2}},\end{eqnarray} \tag{ 20 }$$

where α = 5.1 × 10⁻²⁴ cm³ is the atomic static polarizability for Hg atom and for e⁻-scattering the constant d can be determined by

$$\begin{eqnarray}&&{V}_{{\rm{cps}}}(0)=-\alpha /2{d}^{4}\approx {V}_{{\rm{co}}}(0),\end{eqnarray} \tag{ 21 }$$

then

$$\begin{eqnarray}&&d={(-\alpha /2{V}_{{\rm{co}}}(0))}^{1/4}.\end{eqnarray} \tag{ 22 }$$

In LDA, the correlation energy of the projectile at r is the same as if it were moving within a free electron gas (FEG) of density $\rho (r)$ equal to the local atomic electron density. Following Padial and Norcross [32], the correlation potential is calculated as the functional derivative of the FEG correlation energy with respect to $\rho (r)$. It is convenient to introduce the density parameter

$$\begin{eqnarray}&&{r}_{s}\equiv \displaystyle \frac{1}{{a}_{o}}{\left[\displaystyle \frac{3}{4\pi \rho (r)}\right]}^{\displaystyle \frac{1}{3}}.\end{eqnarray} \tag{ 23 }$$

The parameterization of the correlation potential, for the electron scattering, given in Perdew and Zunger [46] is adopted as

$$\begin{eqnarray}&&{V}_{{\rm{co}}}(r)=-\displaystyle \frac{{e}^{2}}{{a}_{o}}(0.0311\mathrm{ln}({r}_{s})-0.0584+0.00133{r}_{s}\mathrm{ln}({r}_{s})-0.0084{r}_{s}),\end{eqnarray} \tag{ 24 }$$

for ${r}_{s}\lt 1$ and for ${r}_{s}\geqslant 1$,

$$\begin{eqnarray}&&{V}_{{\rm{co}}}(r)=-\displaystyle \frac{{e}^{2}}{{a}_{o}}{\beta }_{o}\displaystyle \frac{1+(7/6){\beta }_{1}{r}_{s}^{\tfrac{1}{2}}+(4/3){\beta }_{2}{r}_{s}}{{\left(1+{\beta }_{1}{r}_{s}^{\tfrac{1}{2}}+{\beta }_{2}{r}_{s}\right)}^{2}},\end{eqnarray} \tag{ 25 }$$

where ${\beta }_{0}=0.1423$, ${\beta }_{1}=1.0529$ and ${\beta }_{2}=0.3334$.

For electron scattering, W_abs(r), originally proposed by Salvat [41] by means of LDA, is given by

$$\begin{eqnarray}&&{W}_{{\rm{abs}}}(r)=\sqrt{\displaystyle \frac{2{({E}_{L}+{m}_{o}{c}^{2})}^{2}}{{m}_{o}{c}^{2}({E}_{L}+2{m}_{o}{c}^{2})}}\times {A}_{{\rm{abs}}}\displaystyle \frac{\hslash }{2}[{v}_{L}\rho (r){\sigma }_{{\rm{bc}}}({E}_{L},\rho ,{\rm{\Delta }})].\end{eqnarray} \tag{ 26 }$$

Here, v_L is the velocity with which the projectile interacts as if it were moving within a homogeneous gas of density ρ and is given by ${v}_{L}={(2{E}_{L}/{m}_{o})}^{1/2}$ corresponding to the local kinetic energy ${E}_{L}(r)={E}_{i}-{V}_{{\rm{st}}}(r)-{V}_{{\rm{ex}}}(r)$. ${\sigma }_{{\rm{bc}}}({E}_{L},\rho ,{\rm{\Delta }})$ is the cross section for binary collision of electron with the degenerate FEG [45] involving energy transfers greater than a certain energy gap Δ.

In the present calculation, the value of the empirical parameter A_abs is chosen to be 1/2. The value of the energy gap Δ in this calculation is adopted as the first excitation energy of Hg atom, ${\epsilon }_{1}=4.67\,\mathrm{eV}$, taken from NIST Physics Reference Data [47]. The factor 1/2 in the absorption potential for electron arises due to the exchange effect in the binary collision of the incident electron with the electron gas.

The present DRPWA results of the spin-polarization $S(\theta )$ calculated within the framework of complex OPM are illustrated in figures 1–8. Here, the spin-asymmetry parameter is calculated over a wide range of energies and compared with the experiments of Dümmler et al [17] at 1.0, 1.5 and 1.9 eV; Kaussen et al [18] at 6.0, 9.0, 11.0, 12.2, 14.0, 17.0, 24.0 and 180.0 eV; Berger and Kessler [19] at 25.0, 35.0, 50.0 and 150.0 eV; Hanne et al [21] at 8.0, 10.0, 12.0, 15.0 and 18.0 eV; Duweke et al [48] at 1.4, 2.4 and 3.9 eV; Hanne et al [49] at 30.0 and 125.0 eV; Eitel et al [50] at 100.0, 150.0, 170.0, 190.0, 230.0, 400.0, 425.0, 450.0, 475.0 and 500.0 eV; Deichsel et al [51] at 3.5 eV; Jost and Kessler [52] at 180.0, 260.0, 280.0, 300.0, 320.0, 340.0, 900.0 and 1700.0 eV.

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The DRPWA results are also compared with the theoretical calculations due to the RCCC approach of Bostock et al [22] at 1.0, 1.5, 1.9, 2.4, 3.9, 6.0, 9.0, 11.0, 12.2 and 14.0 eV; the relativistic complex OPM calculations of Kelemen and Remeta [5] (denoted by RSEPA) at 150.0, 900.0 and 1700.0 eV; the GKS theory of Fritsche et al [25] at 1.0, 1.5 and 1.9 eV and Haberland and Fritsche [26] at 6.0, 9.0, 11.0, 12.2, 14.0, 17.0, 24.0 and 180.0 eV; RDW of McEachran and Stauffer [27] at 25.0, 35.0, 50.0, 150.0 and 180.0 eV; and the relativistic Hartree–Fock (RHF) method of Walker [53] at 3.5, 8.0, 100.0, 150.0, 200.0, 260.0, 300.0, 340.0 and 500.0 eV. As far as we know, at 75.0, 700.0, 1000.0, 1500.0 and 2000.0 eV, there are neither any measured data nor any other calculated values for $S(\theta )$ available in literature to compare with. It is anticipated that this investigation might motivate future experimental and theoretical studies shortly.

The comparison shows an excellent agreement between DRPWA results and experiments at 12.0, 12.2, 14.0 and 17.0 eV in figure 3; 30.0, 35.0, and 50.0 eV in figure 4; 100.0, 150.0, 170.0, 180.0 and 190.0 eV in figure 5; 230.0, 260.0, 280.0, 300.0 and 320.0 eV in figure 6; 340.0, 400.0, 425.0, 475.0 and 500.0 eV in figure 7; and 900.0 and 1700.0 eV in figure 8. These results also present an overall good agreement between the DRPWA results and experimental data for the energies of 8.0, 10.0, 11.0, 15.0, 18.0, 24.0, 25.0 and 125.0 eV at all angles except the regions of the absolute maxima in the positive and negative excursions. At these energies, the DRPWA results along with other theoretical calculations and experiments exhibit oscillations at about the same scattering angles with differences in magnitude. These differences may be due to the different choice of the interaction potential and different procedures among the theoretical studies. It is worth-mentioning that the DRPWA results at 9.0 eV are in well agreement with experiment despite the fact that 5d⁹ core excited states are listed in the NIST tables in this region at 8.79 and 9.54 eV [47]. At 200.0 eV, both the DRPWA and RHF [53] results are in excellent agreement with each other (figure 6(a)).

At lower energy region (≤6.0 eV) the DRPWA results produce less satisfactory agreement with the experiment as seen in figures 1 and 2. Before the onset of the inelastic threshold, there will be a substantial contribution of the resonance elastic scattering corresponding to the isolated levels of the composite system, Hg⁻ $\equiv \,({\rm{H}}{\rm{g}}+{e}^{-})$. This invasion of the indirect process cannot be modeled accurately in the OPM approach. Moreover, at the onset of the inelastic threshold, there may be an interplay between the real and imaginary parts of OP due to the dispersion effect leading to a deviation from the smooth variation in the OP parameters with energy. This trait is not also considered in the present study. Nonetheless, the present DRPWA calculations show reasonably good agreement with the data at lower scattering angles up to $\theta =90^\circ$ for this energy region and describe correctly the oscillatory features of the spin-polarization at E_i = 1.0 eV (figure 1(a)), E_i = 1.4 eV (figure 1(b)), E_i = 1.5 eV (figure 1(c)) and E_i = 1.9 eV (figure 1(d)). Figure 1(a) depicts the overestimations of the RCCC results due to Bostock et al [22] in describing the oscillation of the experimental data. However, the present S(θ) results also show good agreement with those due to RHF [53] at E_i = 3.5 eV (figure 1(f)) and 8.0 eV (figure 2(c)). Our DRPWA findings agree reasonably with those due to GKS of [26] at E_i = 6.0 and 9.0 eV presented respectively in figures 2(b) and (d).

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Regarding the polarization dynamics in the scattering process our study supports the work of Kelemen and Remeta [5]. The angular positions of the critical minimum (CM) in the DCS distributions of [5] at 10 energy-points in the range 9 eV $\leqslant {E}_{i}\leqslant 2$ keV are linked to the corresponding total spin-polarizations $S=\pm 1$. The aforesaid connection is effectively employed to examine the conjecture [2, 5] that the total $S=\pm 1$ values occur in the vicinity of CM. This behavior is also evident for incident energies approaching 1 keV and beyond (see figures 8(b) and (e)). However, at lower energies, a more appropriate statement regarding the angular positions of CM and the corresponding extremum S-values may be that S(θ) swings through the S = 0 value with its maximum values in the positive and negative excursions with their angular positions determined by the derivative of the DCS values. For E_i = 9.0, 12.2, 35.0, 50.0, 100.0 and 150.0 eV with CM at $\theta \approx 100^\circ$, 75°, 83°, 80°, 130° and 118° respectively in figures 2(d), 3(b), 4(d), 4(e), 5(a) and 5(c), maximum value, near to S = 1, in the positive excursion is obtained with the corresponding extremum value in the negative excursion much lesser than S = −1 excepting the case of E_i = 50.0 eV where $S\approx 0.33$, the maximum value, even in the positive excursion. In all the above-mentioned 6 energy-points, the quantum spin-entaglement [6–8] along with other quantum effects is likely to dilute the maximum value ofS(θ) in the negative excursion.

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Our DRPWA results of the angular variation of parameters U (θ) and T(θ) at few selected energies (25.0, 35.0, 50.0, 150.0, 260.0 and 500.0 eV) are depicted respectively in figures 9 and 10. The angular and energy dependence of these parameters are defined by the real and imaginary parts of the direct and spin-flip scattering amplitudes. The present results at E_i = 25.0 eV (figures 9(a), 10(a)), E_i = 35.0 eV (figures 9(b), 10(b)), E_i = 50.0 eV (figures 9(c), 10(c)) and E_i = 150.0 eV (figures 9(d), 10(d)) are compared with the only available experimental data of Berger and Kessler [19] and the theoretical RDW results of McEachran and Stauffer [27]. As no experimental data and other theoretical studies of U and T parameters are available in literature at E_i = 260.0 and 500.0 eV, we have displayed only our present DRPWA results providing further impetus for experimental studies.

Figure 9 depicts the parameter of U similar to the spin-polarization S curves (figures 1–8). Our comparison shows that our DRPWA results produce excellent agreement with both the experiment and the RDW findings. But at the minimum position near $\theta =110^\circ$ for the impact energy of 150.0 eV, the present results significantly disagree with them. Apart from slight shift in the scattering angle of the minimum position at 25.0 eV, in figure 10, the DRPWA results of T parameter agree quite well with those of both the experiment [19] and the RDW calculations of [27].

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S(θ), T(θ) and U (θ) arise from the interference effect of the direct f (θ) and spin-flip g(θ) amplitudes in equations (7) and (8) respectively and hence they are sensitive to both the spin-dependent and correlation interactions. In our DRPWA approach, S(θ), T(θ) and U (θ), although connected through equation (12), are calculated using separate expressions in (6), (10) and (11). The derived results not only agree satisfactorily with the experimental data but also the sum of their squares for a particular scattering angle at a collision energy is found to be close to unity. The sum for all the scattering angles at 500.0 eV is typically illustrated in figure 10(f), which justifies the validity of our methodology and the scattering potentials used.

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