Comparison of 1
There are a number of approaches to study interactions of positrons and electrons with hydrogenic targets. Among the most commonly used are the method of polarized orbital, the close-coupling approximation, and the R-matrix formulation. The last two approaches take into account the short-range and long-range correlations. The method of polarized orbital takes into account only long-range correlations but is not variationally correct. This method has recently been modified to take into account both types of correlations and is variationally correct. It has been applied to calculate phase shifts of scattering from hydrogenic systems like H, He+, and Li2+. The phase shifts obtained using this method have lower bounds to the exact phase shifts and agree with those obtained using other approaches. This approach has also been applied to calculate resonance parameters in two-electron systems obtaining results which agree with those obtained using the Feshbach projection-operator formalism. Furthermore this method has been employed to calculate photodetachment and photoionization of two-electron systems, obtaining very accurate cross sections which agree with the experimental results. Photodetachment cross sections are particularly useful in the study of the opacity of the sun. Recently, excitation of the atomic hydrogen by electron impact and also by positron impact has been studied by this method.
The discovery of an electron by J.J. Thomson in 1897 led to the development of physics beyond the classical physics. Proton was discovered by Rutherford in 1909. Niels Bohr proposed a model of the structure of hydrogen atoms in 1913. Neutron was discovered by Chadwick in 1932. Other important discoveries were of X-rays and radioactivity in 1896. In 1926, Erwin Schrödinger formulated an equation to determine the wave function of quantum mechanical system. According to Max Born, the wave function can be interpreted as a probability of finding a particle at a specific point in space and time. Many processes could be studied due to such developments in physics. For example, an incoming wave behaves like a particle in processes like Compton scattering and photoabsorption. Particularly, Geiger and Bothe, using coincident counters, showed that the time between the arrival of the incident wave and the motion of the electron is of the order of 10−7 second. If the incident wave acted as a wave, the time would have been much longer. Also the experiment of Compton and Simon showed that energy is conserved at every point of the scattering process.
We discuss here scattering of electrons by hydrogenic systems since the wave function of the target is known exactly, and therefore we can test various theories or approximations. When the target consists of more than one electron, a reasonably accurate wave function can only be written using various configurations of the target (called configuration interaction approximation). Among the various approximations for scattering are the exchange approximation , the Kohn variational method , and the method of polarized orbitals  which takes into account the polarization of the target due to the incident electron. The incident electron creates an electric field which results in a change of energy of the target given by , where
2. Scattering function calculations
In the exchange approximation , we write the wave function of incident electron and the target as
In the above equation, the plus sign refers to the singlet state, and the minus sign refers to the triplet state, is the scattering function, and
is the target wave function. The scattering wave function of the incident electron is obtained from
where the polarized target function is given in  and is defined as
is used in this calculation. It can also be of the form
where the exponent and
A comparison of results for singlet and triplet phase shifts obtained using different methods is given in Tables 1 and 2. Results from most methods agree. A comparison of the singlet and triplet phase shifts obtained by the
|EAa||POb||Kohnc||Close-couplingd||R-matrixe||Feshbach methodf||Hybrid theoryg|
|EAa||POb||Kohnc||Close-couplingd||R-matrixe||Feshbach methodf||Hybrid theoryg|
The scattering length
The scattering length is calculated at a distance
Resonance parameters in two-electron systems have been calculated using various approaches. Among them are the stabilization method, the complex-rotation method, the close-coupling method, and the Feshbach projection-operator formalism. In the hybrid theory, phase shifts have been calculated in the resonance region  and are fitted to the Breit-Wigner form
In the above equation,
P-wave phase shifts have been calculated for scattering of electrons from He+ and Li2+ in Ref.  and in Ref. , respectively. Singlet P and triplet P phase shifts are shown in Table 3 for e + He+ and for e + Li2+ scattering. Phase shifts for e + He+ agree well with those obtained by Oza  using the close-coupling approximation. Phase shifts for e + Li2+ agree with those obtained by Gien  using the Harris-Nesbet method.
|e + He+ ||e + Li2+ |
Photodetachment and photoionization are required to calculate radiative-attachment cross sections. The recombination rates are required to calculate the ionization balance in astrophysical plasmas. Cross sections for bound-free transitions of H− are required to account for the absorption in the solar atmosphere . The opacity in the sun is due to photodetachment and free-free absorption of the radiation:
In the first process, after absorption of the radiation by the bound electron, it becomes a free electron in the final state, while in the free-free transition, the electron is in the continuum state in the initial state as well as in the final state. It is possible to have the following reactions which help molecular formation:
The photoabsorption cross section in length form and in units of for the transition from the initial states
In the above equation, is the fine-structure constant,
In Eq. (21),
A comparison of the cross sections of the ground state of He with those obtained in the
Similarly, cross sections have been calculated in  for the (1s2s) 1
Excitation of the 1
In the above equation,
Recombination rate coefficients for a process like that indicated in Eq. (17) have been calculated in Ref.  for the ground states as well as for the metastable states. The attachment cross section is given by
The above relation between the photodetachment and photoionization follows from the principle of detailed balance, where
|0.1||1.3193 (−3)||1.3217 (−3)||5.9268 (−3)||8.5133 (−3)||3.0363 (−3)||8.2703 (−3)|
|0.2||5.0217 (−3)||5.0835 (−3)||6.1299 (−3)||9.0331 (−3)||3.0585 (−3)||8.4642 (−3)|
|0.3||1.0531 (−2)||1.0898 (−2)||6.4446 (−3)||9.8834 (−3)||3.0508 (−3)||8.7011 (−3)|
|0.4||1.7250 (−2)||1.8401 (−2)||6.8511 (−3)||1.1044 (−2)||3.0776 (−3)||9.0700 (−2)|
|0.5||2.4675 (−2)||2.7204 (−2)||7.3028 (−3)||1.2473 (–2)||3.0782 (−3)||9.5041 (−2)|
|0.6||3.2495 (−2)||3.6934 (−2)||7.7904 (−3)||1.4152 (−2)||3.0608 (−3)||1.0009 (−2)|
|0.7||4.0544 (−2)||4.7286 (−2)||8.3087 (−3)||1.6066 (−2)||3.0831 (−3)||1.0622 (−2)|
|0.8||4.8620 (−2)||5.7990 (−2)||8.8420 (−3)||1.8172 (−2)||3.1396 (−3)||1.1380 (−2)|
|0.9||5.6532 (−2)||6.8791 (−2)||9.3860 (−3)||2.0439 (−2)||3.1537 (−3)||1.2151 (−2)|
6. Laser fields
Scattering cross sections have also been calculated in the presence of laser fields . A strong suppression in the laser-assisted cross sections is noted when compared to cross sections in the field-free situation. Further, scattering cross sections have also been calculated in the presence of Debye potential , in addition to the laser field.
7. Positron-hydrogen scattering
Dirac in 1928, combining the ideas of relativity and quantum mechanics, formulated the well-known relativistic wave equation and predicted an antiparticle of the electron of spin ħ/2. At that time only protons and electrons were known. He thought that the antiparticle must be proton. Hermann Weyl showed from symmetry considerations that the antiparticle must have the same mass as an electron. There are many other examples where symmetry played an important role, e.g., Newton’s third law of motion (for every action there is an equal and opposite reaction) and Faraday’s laws of electricity and magnetism (electric currents generate magnetic fields, and magnetic fields generate electric currents). Symmetry laws have some profound implications as shown by Emmy Noether in 1918 that every symmetry in the action is related to a conservation law .
Positrons, produced by cosmic rays in a cloud chamber, were detected by Anderson  in 1932. Positrons can form positronium atoms which annihilate, giving 511 KeV line with a width of 1.6 keV. This line has been observed from the center of the galaxy. Positrons have become very useful to scan the human brain (PET scans). They have been used to probe the Fermi surfaces, and the annihilation of positronium atoms in metals has been used to detect defects in metals.
Calculations of positron-hydrogen scattering should be simpler than the electron-hydrogen scattering because of the absence of the exchange between electrons and positrons. However, the complications arise due to the possibility of positronium atom formation. In electron-hydrogen system, the two electrons are on either side of the proton because of the repulsion between two electrons. However, because of the attraction between a positron and an electron, both the positron and the bound electron tend to be on the same side of the proton. This configuration shows that the correlations are more important in the case of a positron incident on a hydrogen atom. In 1971, we  carried out calculations using the projection-operator formulism of Feshbach  and using generalized Hylleraas-type functions:
Nonlinear parameters are and
|Hybrid theory ||Bhatia et al. ||Schwarz ||Hybrid theory ||Bhatia et al. ||Armstrong |
The incident positron can annihilate the atomic electron with the emission of two gamma rays. The cross section for this process has been given by Ferrell :
9. Positronium formation
Positronium, the bound state of an electron and a positron, was predicted by Mohorovicic  in connection with the spectra of nebulae. Positronium (Ps) formation takes place when the incident positron captures the bound electron of the hydrogen atom:
10. Photodetachment of positronium ion (Ps−)
Photodetachment has been discussed above already. Following the work of Ohmura and Ohmura , Bhatia and Drachman  calculated cross sections (in the length and velocity form) for photodetachment of Ps−. Their result in length and velocity form is
The electron affinity is , where = 0.12651775  and
Lyman-α radiation (2
These cross sections are given in Figure 7 for various photon energies.
11. Binding energies
Resonances formed in the scattering of electrons from atoms are very common. However, they are not that common in positron-target systems. The first successful prediction of
Eigenvalues are complex now. The real part gives position of the resonance, and the imaginary part gives its half width.
A number of Feshbach and shape resonances in Ps− have been calculated by using the complex-rotation method. Parameters of a 1
13. Antihydrogen formation
Antihydrogen can be formed in the collision of Ps with antiproton
In the above equation, represents antiproton and represents antihydrogen. According to the time reversal invariance, the above reaction is related to
From this reaction, the positronium formation cross sections are known from positron-hydrogen scattering. This implies that the cross section for antihydrogen is related to the cross section for Ps formation:
and Ps are momenta of positron and positronium. Humberston et al.
If the antihydrogen is formed in the excited state, then it can decay to the lower states. It would then be possible to verify if the quantum mechanics principles are the same in the antimatter universe. It is thought that gravitational interactions should be the same between matter and antimatter and between antimatter and antimatter. However, there is no experimental confirmation up to now.
14. High-energy cross sections
At high energies, only static potential remains. Therefore, according to the first Born approximation, total cross sections for e—-He and e+-He should be the same. This fact has been verified experimentally by Kauppila et al. , and their results are shown in Table 12. We see that as the incident energy increases, cross sections tend to be equal.
15. Total positron-hydrogen cross sections
Total cross sections for positron scattering from hydrogen atoms have been measured by Zhou et al.  and have been calculated by Walters  using the close-coupling approximation and also by Gien  using the modified Glauber approximation. Their results are given in Table 13. They are fairly close to the experimental results.
16. Threshold laws
When the incident electron or positron has just enough energy to ionize the hydrogen atom, how do the cross sections behave? Wannier , using classical methods and supposing the two electrons emerge opposite to each other, showed that , where
Wigner  has emphasized the importance of long-range forces near the threshold which have been included in these calculations (hybrid theory). At the threshold, the cross section for exciting the 1
In this chapter, we have discussed various interactions of electrons and positrons with atoms, ions, and radiation fields. There are various approximations and theories to calculate scattering functions. Theories which provide variational bounds on the calculated phase shifts are preferable because improved results can be obtained when the number of functions in the closed channels is increased. Such theories are the close-coupling,
The continuum functions obtained using the hybrid theory have been used to calculate photoabsorption cross sections, obtaining results which agree with definitive results obtained using other methods and experiments. Such cross sections are needed to study the opacity in the sun. The resonances play an important role when they are included in the calculations of excitation cross sections, which are important to infer temperatures and densities of solar and astrophysical plasmas.
When Feshbach projection-operator formalism  is used to calculate resonance position, , where , the shift in the resonance position due to its coupling with the continuum, has to be calculated separately . However, in the hybrid theory, the calculated position includes this correction. This is an advantage since the calculation of the shift is nontrivial.
We have indicated that in addition to obtaining accurate phase shifts for positron scattering from a hydrogen atom, we have described calculations of annihilation, positronium, and antihydrogen formation. We have discussed resonances in a positron-hydrogen system. We have discussed photodetachment of a positronium ion and a possibility of observing Lyman-α radiation from a positronium atom when the final state is the 2