## 1. Introduction

A positron, the antiparticle of the electron, has the same mass, electric charge (but positive) and spin (1/2) as that of an electron. Like other antiparticles, positrons were produced during the period of baryogenesis when the universe was extremely hot and dense, but now they exist in much lower numbers than its counter part, the electrons. Although not found in normal conditions, they are produced at the galatic center or supernovae events and are found in copious amount in cosmic ray showers and in the ionosphere. Positrons are created naturally in ${\beta}^{+}$ radioactive decays such as from K-40, particle reactions or by pair production from a sufficiently energetic photon interacting with the atomic nuclei in a material. Nevertheless, a small percent of potassium (0.0117%) K-40 is the single most abundant radioisotope in the human body and produces about 4000 natural positrons per day. However, soon after its creation, they annihilate with electrons or form the exotic atom, positronium (Ps), with a very short lifetime, finally decaying to 2 or 3 gamma rays each with energy 511 keV. Ps has a mass of 1.022 MeV/c${}^{2}$ and can form otho-Ps (o-Ps) or a para-Ps (p-Ps) when the electron-positron spins are parallel (spin = 1) or anti-parallel (spin S = 0) respectively. An o-Ps decays to 3 and a p-Ps decays to 2 gamma photons. The energy levels of Ps are similar to those of hydrogen atom. Gamma rays, emitted indirectly by a positron-emitting radionuclide (tracer), are detected in positron emission tomography (PET) scanners used in hospitals. PET scanners create detailed three-dimensional images of metabolic activity within the human body. Positron Annihilation Spectroscopy (PAS) is used in materials research to detect variations in density, defects and displacements within a solid material. It is the detection of 511 keV gamma ray photons that is typically used as the signal for the source of existence or creation of positrons, such as, in the center of our galaxy, Milky Way.

The treatment of electron and positron scattering from atoms are similar and have been studied extensively both theoretically and experimentally. Similar is the case with molecules. The scattering parameters of interest are the scattering cross sections and the spin polarization. The scattered wave function can be obtained by solving the Schrödinger equation using Numerov method (e.g., Reference [

1]), or using other methods mentioned below. The scattering parameters can be determined using the wave function. While extensive set of references are available, this review provides selected references that can lead to details of various approaches and experimental results. Most scattering studies have been carried out for neutral atoms, such as, He, Ne, Mg, Ar, Kr and Xe using various theoretical approaches, such as, polarized orbital method (e.g., References [

2,

3,

4,

5]), modified adiabatic method (e.g., Reference [

6]), variational method (e.g., Reference [

7]) and optical potential method (e.g., References [

1,

8,

9,

10,

11,

12,

13,

14,

15,

16]). There are many experimental studies on positron scattering as well (e.g., References [

17,

18,

19,

20,

21,

22,

23,

24,

25,

26,

27,

28,

29,

30,

31,

32,

33,

34,

35,

36,

37,

38,

39]). Spin polarization for electron scattering was measured by various groups (e.g., References [

40,

41,

42]). Among ions, the study has remained largely on single valence electron ions using Kohn-Feshbach variational method [

43], polarized orbital method [

44,

45] and hybrid method [

46,

47,

48]. Electron/positron scattering from molecules are investigated using spherical complex optical potential (SCOP) as well [

49,

50]. More references can be obtained from the cited articles. Compared to scattering from a positive ion, a neutral target offers consistent scattering features that can help in better understanding the general characteristics. Being less reactive species, noble gases are easy to handle experimentally compared other targets. They are also relatively simple collision systems to approach theoretically. Hence, it will be interesting to review the cross section data on electron/positron scattering from inert gases. The present work will concentrate on the neutral inert gases as well.

Interaction of positron with atoms is dealt mainly with two theoretical approaches; perturbative and non-perturbative. Perturbative methods usually work in the intermediate to high energy region (ionization threshold to about 10 keV), while the non-perturbative theories are capable of accurate calculations at low energies. Among various methods mentioned above, polarized orbital and optical potential methods have been used most widely to calculate scattering parameters for atoms beyond He. The polarized orbital method (e.g., Reference [

51]) ansatzes the distortion in the target wave function. Temkin [

2,

3] first introduced it, where he included long range correlation which has the characteristic behavior of

$-1/{r}^{4}$ of the longest range polarization potential, for the distortion. The method was converted to a hybrid model by Bhatia (e.g., References [

48,

52]), which included the short range correlation and variationally bound energies. His application of the hybrid method to calculate phase shifts, scattering lengths, photo-detachment, photoionization, positron scattering, annihilation and positronium formation produced reasonable results, which showed good agreement with available results. He extended the work to obtain accurate results in the elastic region for S-, P-, and D-wave scattering as well. Bhatia’s investigation using the hybrid method focuses largely on the scattering by single-electron systems (e.g., positron impact excitation of hydrogen [

45]), since the wave function of the target is known exactly. Besides, the possibility of direct annihilation and positronium formation requires a composite wavefunction, which is almost impossible to formulate.

Polarized orbital method for elastic scattering of positrons from noble gases was first developed by McEachran et al. [

4] where they included in principle all multipole moments of the positron-atom interaction. The polarized orbital was calculated by a perturbed Hartree-Fock scheme, which was used to calculate the polarizability of atoms [

53,

54] and the scattered wave was obtained from a potential scattering problem. The method showed good agreement with measured cross sections for for He and Ne atoms [

17,

21]. McEachran et al. [

55] implemented their method successfully for other atoms as well.

One of the most rigorous approach for positron-ion scattering has been the Feshbach projection operator method [

56], where the usual Hartree-Fock and exchange potentials are augmented by an optical potential [

47]. However, the method employs correlation functions that are of Hylleras type and hence do not include long range correlation.

Kohn Variational Principle (KVP) (e.g., Reference [

57]) is usually applied to low-energy positron scattering to obtain elastic and Ps formation cross sections. In this method a two-component trial wave function is chosen, having the correct asymptotic form with enough flexibility to describe all the short-range distortions and correlations of the positron-atom system. This wave function is then used in the Kohn functional, which can be written in terms of the K-matrix elements. From the K-matrix, cross sections can be calculated.

The many-body-theory for a positron-atom interacting system (e.g., References [

58,

59]) is based on the Dyson equation. This is solved by the representation of eigenfunctions of the Hartree-Fock Hamiltonian. The formulated self-energy matrix gives the phase shifts. This approach is used to study low energy positron scattering from atoms.

Schwinger Multichannel Calculations (SMC) is a well-known method to study low energy scattering (e.g., References [

60,

61]). The backbone of the method is the computation of variational expression for the scattering amplitude. SMC describes target polarization through single virtual excitations of the target wave function, explicitly considered in the expansion of the scattering wave function. The Lippmann-Schwinger scattering equations are then solved to obtain the cross sections [

62].

The other method of interest is the close-coupling (CC) approach for the scattered wave and the R-matrix method [

63,

64] to solve the coupled set of integro-differential equations. Jones et al. [

26] used the convergent close-coupling (CCC) approximation, where they solve the equations with a different set of codes than standard R-matrix codes. They use multi-configuration Dirac-Fock program of Grant et al. [

65] to obtain the target wave functions. Their results for positrons scattering from Ne and Ar indicate that, while both polarized orbital method and CCC approximation showed good agreement with experiment in general, the polarized orbital method yielded slightly better cross sections. In the standard close-coupling formalism for molecules (e.g., Reference [

66]), the convergence in the expansion of the three-body wave function is obtained using the exact discrete eigenstates of the atomic target. The technique relies on the expansion of the total wave function in the set of target states of the atom and Ps. The CCC method allows for the examination of the effect of virtual excitation to the continuum as well [

67].

For molecules, the optical potential method (e.g., References [

49,

68]), Born approximations (e.g., References [

69,

70]), and so forth are the most common quantum mechanical perturbative theories used for electron scattering presently. The positron collision studies is an extension of the optical potential method [

71]. In case of non-perturbative theories, the close-coupling or the grid-based method for solving Schrödinger equation are employed. Irrespective of whether the method is perturbative or non-perturbative, the positron-molecule scattering is an extension to the positron-atom interaction technique. One has to consider multi-centre approach to deal with the projectile-target interaction due to the absence of spherical symmetry and due to the complexity of molecules. Here we will discuss few of the most commonly used theoretical methods to deal with positron-atom/molecule interaction.

Theoretical methods to investigate positron scattering and various target molecules studied by these approaches along with references are listed in

Table 1. The list given below is not exhaustive, but gives an overall picture of various studies done so far. Further, this review will elaborate the most common approach, the optical potential method, to study electron and positron scattering from atoms and molecules with reasonable success.

## 2. Scattering Parameters: Cross Sections and Spin Polarizations for Atoms

The characteristic features of the scattering can be observed in the cross section and in spin polarization caused by the projectile. While cross section can be obtained by solving non-relativistic Schrödinger or relativistic Dirac or Dirac-Fock equations, the latter provides accurate treatment for spin polarization parameters. In the present review, we will present relativistic single particle Dirac approach, which has been successful in reproducing the observed scattering phenomena.

The relativistic Dirac equation for a projection of rest mass

${m}_{o}$ and velocity

v traveling in a central field

$V\left(r\right)$ is given by (e.g., Reference [

180,

181]),

where

$\alpha $ and

$\beta $ are the usual 4 × 4 Dirac matrices and

$\psi $ is a four-component (spinor) function,

$\psi =({\psi}_{1},{\psi}_{2},{\psi}_{3},{\psi}_{4})$.

$({\psi}_{1},{\psi}_{2})$ are the large components and

$({\psi}_{3},{\psi}_{4})$ are the small components of

$\psi $. Defining

$\gamma ={(1-{v}^{2}/{c}^{2})}^{-1/2}$, the total energy is

$E=m{c}^{2}={m}_{o}\gamma {c}^{2}={E}^{\prime}+{m}_{o}{c}^{2}$ where

${E}^{\prime}$ is the kinetic energy, and writing the radial function of the large large component as

${G}_{l}=\sqrt{\eta}{g}_{l}\left(r\right)/r$, the equation for the large component can be rewritten as the Dirac equation reduced to the form similar to Schrödinger equation (e.g., Reference [

9]) as

where the effective Dirac potentials due to spin up and spin down respectively are,

and

The prime and double primes represent the first- and the second-order derivatives with respect to

r,

$\eta =(E-V+{m}_{o}{c}^{2})/c\hslash $,

$\delta =(E-V-{m}_{o}{c}^{2})/c\hslash $, and

${K}^{2}=({E}^{2}-{m}_{o}^{2}{c}^{4})/{c}^{2}{\hslash}^{2}$. In atomic unit,

${m}_{o}=e=\hslash =1,\phantom{\rule{3.33333pt}{0ex}}1/c=\alpha $, where

$\alpha $ is the fine structure constant and hence

$\gamma ={(1+{\alpha}^{2}{K}^{2})}^{1/2}$ and

$E=\gamma {c}^{2}=\gamma /{\alpha}^{2}$. The proper solution of the Schrödinger like Dirac equation behaves asymptotically as,

where

${j}_{l}$ and

${n}_{l}$ are spherical Bessel functions of the first and second kind respectively, and

${\delta}_{i}^{\pm}$ are the phase shifts due to collisional interactions. The plus sign corresponds to the incident particles with spin up and the minus sign to those with spin downs.

${\delta}_{i}^{\pm}$ indicates the shifts in the phase of the radial wave function due to the effect of interaction potentials in the scattering. The radial wave function will be “pushed out” if the potential is repulsive and vice-versa with respect to the incoming free radial wave. So from this quantity, we can determine various microscopic quantities like cross section. The values of

${\delta}_{i}^{\pm}$ may be obtained from the values of

${g}_{l}^{\pm}$ at two adjacent points

r and

$r+h$ (

$h<<r$) as

The wave functions

${g}_{l}^{\pm}$ can be obtained by numerical integration of

${g}_{l}^{\pm \u2033}$ using Numerov method and the spherical Bessel functions as described in Nahar and Wadehra [

1]. Schrödinger/Dirac equation, can be solved by various other approaches mentioned above, such as, Kohn-Feshbach variational method [

43], polarized orbitals method [

2,

3,

4,

46,

47,

51], close-coupling approximation [

26,

64] for the scattered wave function from which the phase shift is determined.

The generalized scattering amplitude for the collision process is given by [

9],

where

and

$\widehat{\mathbf{n}}$ is the unit vector perpendicular to the scattering plane. The differential cross section (DCS) for the scattering of the spin-1/2 particles by the spin zero neutral atom is given by

where

${\chi}_{{\nu}^{\prime}}$ represents a spin state and

${\mathbf{P}}_{i}=<{\chi}_{{\nu}^{\prime}}\left|\sigma \right|{\chi}_{\nu}>$ is the incident-beam polarization, which is assumed to be zero. The integrated elastic cross section for the unpolarized incident beam can be obtained as

and the momentum transfer cross section by

The integrated total cross section given by

where

${S}_{l}^{\pm}=exp\left(2i{\delta}_{l}^{\pm}\right)$. The integrated absorption cross section can be obtained from

${\sigma}_{abs}={\sigma}_{tot}-{\sigma}_{el}$Since the spin-orbit interaction is a short-range interaction, the phase shifts of the spin-up and the spin-down particles are equal

$({\delta}_{l}^{+}={\delta}_{l}^{-})$ for the large angular momenta

$l\hslash $. Hence for large

l,

$g\left(\theta \right)$ = 0 and the contribution to the scattering amplitude comes only from

$f\left(\theta \right)$. If Born approximation is used for higher partial wave with

$l>M$,

$f\left(\theta \right)$ can be written as [

9],

where

${f}_{B}(K,\theta )$ is the Born amplitude,

${S}_{Bl}=exp\left(2i{\delta}_{Bl}\right)$ and

${\delta}_{Bl}$ is the Born phase shift. The number of exact phase shifts to be evaluated depends on the impact energy before use of Born approximation. The contribution due to Born approximation should be small. At large distance the interaction potential

$V\left(r\right)$ is dominated by the long range part

${V}_{LR}\left(r\right)=-{\alpha}_{d}/2{r}^{4}$ of the polarization potential and Born phase shift

${\delta}_{Bl}$.

The interaction potential between the spin of the electron or positron and the orbital angular momentum

$\mathbf{L}$, which depends on the velocity and position vector with respect to the target atom, can cause the spin to orient. Hence, even with an unpolarized incident beam the orientation in a preferred direction can give a net spin polarization in the scattered beam. The amount of polarization produced due to the collision in the scattered beam is given by [

182],

The other two spin polarization parameters,

T and

U giving the angle of the component of the polarization vector in the scattering plane are given by [

182],

The three polarization parameters are interrelated through the condition ${P}^{2}+{T}^{2}+{U}^{2}=1$.

## 3. ${\mathit{e}}^{\mathbf{\pm}}$ and Target Atom Interaction Potential

To calculate the scattering parameters, we define the projectile-target interaction and a method to determine the respective wave functions. The scattering can be described in two general categories, elastic (where the total kinetic energy is conserved and the interaction potential is real) and inelastic (where part of the energy is lost due to absorption). For the inelastic processes such as excitation, ionization, positronium formation through electron capture and so forth the absorption potential is developed, which forms the imaginary part of the total complex potential. The total interaction potential between a neutral target (or a single atomic electron) and a projectile electron or positron is assumed to be symmetric or central,

$V\left(r\right)$, which depends on

r only. The general form of

$V\left(r\right)$ is,

where the real part

${V}_{R}\left(r\right)$ represents the elastic scattering and the imaginary part

${V}_{A}\left(r\right)$ represents the absorption of energy through the inelastic channels. When the total kinetic energy is conserved the imaginary part,

${V}_{A}\left(r\right)$, is zero. The absorption potential is negative and typically depends on the local density function.

${V}_{R}\left(r\right)$ has several components: the averaged static potential

${V}_{S}$ (attractive for positrons and repulsive for electrons),

${V}_{P}$ polarization potential (attractive for both electrons and positrons) and an electron-electron exchange potential

${V}_{ex}$ (only for electrons). For a positron there is no exchange probability. The total real potential is represented as,

The static potential,

${V}_{S}$, is obtained by averaging the projectile-target interaction over the target wave function as,

where

${\psi}_{T}$ is the asymmetric Hartree-Fock target wave function,

${\mathsf{\Phi}}_{nlm}\left(\mathbf{r}\right)={\varphi}_{nl}\left(r\right){Y}_{lm}\left(\widehat{\mathbf{r}}\right)$ are the partial atomic orbitals,

${e}_{p}$ is the projectile charge and

${N}_{nlm}$ is the occupancy number of the orbital (

$n,l,m$). The radial part

$\varphi \left(r\right)$ of an orbital can be an analytic expansion, for example, tables of Clementi and Roetti [

183] or in numerical form obtained from configuration interaction atomic structure calculation (e.g., Reference [

65] or a Hylleras type wave function expansion (e.g., Reference [

26]). Use of configuration interaction form is common in close coupling approximation.

The polarization potential usually has a short and a long range part,

where

${r}_{c}$ is the point where the two forms cross each other for the first time. The long range behavior is known to be of the form

${\alpha}_{o}/{r}^{4}$ where

${\alpha}_{o}$ is the polarizability of the target. The short range form can vary. For the electrons scattering from a neutral atom, it could be the parameter free potential, for example, that given by O’Connel and Lane [

184]. They developed the potential on the basis of energy dependent free-electron gas exchange potential and the energy-independent electron-gas correlation potential smoothly joining to the long-range polarization interaction and is given by,

${r}_{s}={[3/\left(4\pi \rho \left(r\right)\right)]}^{1/3}$,

$\rho \left(r\right)$ is the undistorted electronic charge density of the target.

$\rho \left(r\right)$ for the spherically symmetric atom is given by,

where

${N}_{nl}$ is the occupancy number of the orbital (

$nl$).

For the positrons scattering the polarization potential can also be parameter free, such as that by Jain [

185]. It is based on correlation energy of a single positron in a homogeneous electron gas with an asymptotic behavior of the long range polarization potential, and is given by,

The long range form of ${V}_{{p}^{\pm}}\left(r\right)$ is given by ${V}_{LR}\left(r\right)=-{\alpha}_{d}/\left(2{r}^{4}\right)$, where ${\alpha}_{d}$ is the static electric dipole polarizability. In polarized orbital method, the distortion due to polarization is incorporated in the wave function.

The exchange potential,

${V}_{ex}\left(r\right)$ is due to exchange between the projectile electron and the target electrons. One of the common form is given by Riley and Truhlar [

186],

where

${V}_{D}={V}_{S}+{V}_{{P}^{-}}$ is the direct interaction potential and

$\rho \left(r\right)$ is the radial density of the target. Chen et al. [

187] introduced another type of potential which was used for elastic scattering from heavy inert gas, Kr, with reasonable success.

When the impact energy becomes accessible for the inelastic processes (such as excitations of the target, positronium formation, etc.) absorption potential is introduced. The total absorption of energy has been represented by various model potentials with poor to good success for certain atoms (e.g., for Ar [

9]). One major issue was the inclusion of various threshold energies for excitations and electron capture in case of positrons to form positronium. One successful absorption potential model, especially for electron scattering, has been the semi-empirical potential of Staszewska et al. [

188,

189], which are based on qualitative features of an absorption potential at short and long ranges in order to predict accurate differential cross sections. Their later model [

189] has been in use considerably, and is given by,

v is the local velocity of the projectile for

$(E-{V}_{R})\ge 0$,

$\rho $ is the target electron density per unit volume and

$\overline{{\sigma}_{b}}$ is the average quasifree binary cross section for Pauli allowed electron-electron collisions and is obtained non-empirically by using the free-electron gas model for the target as,

where

p is the incident momentum of the projectile and

${k}_{F}$ is the target Fermi momentum. In their third version of

${V}_{A}$, V.3, they define the parameters

$\alpha $ and

$\beta $ as,

where

$\Delta $ is the threshold energy for inelastic scattering and

I is the ionization potential. The factor

$1/2$ in the equation is introduced to account for the exchange between the incident electrons and the atomic electrons of the target. The same absorption potential can be used for the positron scattering with the factor

$1/2$ removed, since there is no exchange effect during the positron scattering. The earlier version of Staszewska et al. [

188] has also shown fair to good representation of absorption potential in reproducing the total cross sections. Various other absorption potential models are also available in literature, but have been only partly successful and hence need improvement.

Figure 1 demonstrate the general features of various components of the real part

${V}_{R}\left(r\right)$ of the total projectile-target interaction potential

$V\left(r\right)$. The components are static potential (repulsive for electron and attractive for positrons), polarization potential (attractive for both

${e}^{\pm}$), exchange potential (only for electrons and attractive) for

${e}^{\pm}$ scattered by the cadmium atoms [

8]. The static potential was obtained using Slater-type orbitals of Roothan-Hartree-Fock wave functions given by Clementi and Roetti [

183]. The same orbital functions were used to obtain the electron density in the absorption potentials. As expected, the static potential dominates near to the nucleus and exchange potential starts away from it, but moves toward it with increasing energy of the projectile.