Comparison of 1S phase shifts obtained in different methods.
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 α is the polarizabilty of the target and is equal to 4.5 in the case of a hydrogen target. The polarization is possible only when the incident electron is outside the target, according to the method of polarized orbitals . However, this method includes only the long-range −1/r4 potential and not the short-range correlations, and the method is not variationally correct. It is possible to use the Feshbach formalism  to take into account the short-range correlations via an optical potential. This method gives rigorous lower bonds on the phase shifts, i.e., they are lower than the exact phase shifts. However, it is difficult to include the long-range correlations at the same time. The close-coupling approach takes into account both the long-range and short-range correlations  and is variationally correct. However, a large number of target states must be included to obtain converged results and the correct polarizability. A Feshbach resonance below the n = 2 threshold of hydrogen atom was first discovered in a close-coupling calculation. A method which has been applied extensively to atomic, molecular, and nuclear physics is the R-matrix method introduced by Wigner and Eisenbud in 1947. In this method, configuration space is divided in such a way that all correlations are described within a radius r = a, and outside this radius simple continuum functions can be used while matching the inside and outside functions at the boundary r = a. This method was introduced in atomic physics by Burke. A method called the hybrid theory  has been introduced in which the short-range and long-range correlations are taken into account at the same time and the polarization takes place whether the incident electron is outside or inside the target. This method is variationally correct. The equations for the scattering function are very detailed and are given in .
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 H is the Hamiltonian and E is the energy of e-target system. In Rydberg units,
where k2 is the kinetic energy of the incident electron and Z is the nuclear charge of the target, assumed fixed. The scattering equation is
Phase shift η (radians) of a partial wave l is calculated from the asymptotic value of the scattering function.
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 β, another parameter which is a function of k, can be used to optimize the phase shifts. Both cutoffs (11) and (12) are used in calculations on scattering. is a short-range correlation function  which can be written using the Euler angle decomposition  for all partial waves. This formulism allows the separation of the angular parts and the radial parts consisting of radial coordinates , , and . This facilitates writing the equation for radial functions . The function in Eq. (10) is given by
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 R-matrix and the hybrid theory is given in Figure 1. The curves for the two methods cannot be distinguished, showing that accurate result can be obtained using the hybrid theory .
|k||EAa||POb||Kohnc||Close-couplingd||R-matrixe||Feshbach methodf||Hybrid theoryg|
|k||EAa||POb||Kohnc||Close-couplingd||R-matrixe||Feshbach methodf||Hybrid theoryg|
The scattering length a is defined by.
The scattering length is calculated at a distance R, and there is a correction for the long-range polarization .
where a is the true scattering length and α is the polarizability of the hydrogen atom. At R = 117.3088, 1S scattering length a(R) is 5.96554, obtained using . The corrected 1S scattering length is 5.96611. At R = 349.0831, the 3S scattering length is 1.781542, obtained using , and it is 1.76815, after correction for the long-range polarization given by . The 1S and 3S scattering lengths are therefore 5.96611 and 1.76815, which agree with those calculated by Schwartz , using the Kohn variational method. In this method, scattering lengths have upper bounds, while phase shifts do not have any bounds in this method (Kohn variational method). Similar calculations for the phase shifts for e-He+ and e-Li2+ have been carried out in Ref. .
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, E = k2 is the incident energy, are the calculated phase shifts, and , A, , and are the fitting parameters. and represent the resonance position and resonance width, respectively. We find that in the hybrid theory, the He singlet resonance is at = 57.8481 eV with respect to the ground state of He and = 0.1233 eV. They agree well with = 57.8435 eV and = 0.125 eV obtained using the Feshbach projection-operator formalism . A similar calculation  has been carried out for Li2+ resonance parameters. The resonance parameters agree with those obtained in .
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 I to the final state f is given by
In the above equation, is the fine-structure constant, k is the momentum of the outgoing electron, = is the energy of the incident photon, and I is the ionization potential of the system absorbing the photon. Photoabsorption cross sections for H−, He, and Li+ are given in  and in Table 4. The correlated wave functions for H−, He, and Li+ are of Hylleraas form having terms 364, 220, and 165, respectively. A Hyllerraas wave function is given by
In Eq. (21), C’s are the linear parameters, while and are nonlinear parameters, and . Ohmura and Ohmura  have calculated the photodetachment cross section of H−, using the effective range theory and the loosely bound structure of hydrogen ion. These cross sections are shown in Figure 2. It should be noticed that with the outgoing energy of the electron going to zero, the cross section goes to zero when the remaining system is neutral and is finite when the remaining system is an ion .
A comparison of the cross sections of the ground state of He with those obtained in the R-matrix  is shown in Figure 3. The agreement is very good except at k = 0.6 where the R-matrix result is slightly lower. It seems that the cross section has not been calculated accurately at this k. The precision measurements of the cross sections of photoionization of He by Samson et al.  are also shown in Figure 3, showing that the agreement with the cross sections obtained using the hybrid theory is very good.
Similarly, cross sections have been calculated in  for the (1s2s) 1S and (1s2s) 3S metastable states of He and Li+. The number of Hylleraas terms used are 455 and 364 for He singlet and triplet states, respectively. For Li+, 120 and 220 terms are used for the singlet and triplet states, respectively. These cross sections are comparable to those obtained for the ground state. A comparison of photoionization cross sections of the ground state, metastable states singlet S, and triplet S is shown in Figure 4. The cross section for the singlet (1s2s) state is highest at k = 0.1, and then the cross sections decrease rapidly.
Excitation of the 1S state of atomic hydrogen to the 2S state has been calculated using the hybrid theory in the distorted-wave approximation . The total excitation cross section is written in the form
In the above equation, ki and kf are the incident and final momenta, and Tfi is the matrix element for the excitation of the initial state i to the final state f and is given by
Z is the nuclear charge, and r1 and r2 indicate the position of the incident and target electron, respectively. The excitation cross sections at k(Ry) = 0.8–2 are shown in Figure 5. There is a maximum at k = 0.907 and another at k = 1.50 (Table 5).
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 g(i) and g(f) are the weight factors for the initial and final states, and pe = k is the electron momentum. The attachment cross sections are in general smaller than the photoabsorption cross sections. Recombination rate coefficients, averaged over the Maxwellian velocity distribution, are given in Table 6 for H−, He, and Li+ for a temperature range from 1000 to 40,000 K.
Elastic P-wave scattering from e- Be3+, C5+, and O7+ and photoionization in two-electron systems have been carried out. Phase shifts and photoionization cross sections are given in  for the ground as well as metastable states. In order to calculate photoionization cross sections from P-state of an atom, both S-wave and D-wave continuum functions are needed. We give a few phase shifts for 1D and 3D in Table 7 for scattering from H, He+, and Li2+.
|k||1D, e-H||3D, e-H||1D, e-He+||3D, e-He+||1D, e-Li2+||3D, e-Li2+|
|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 MeV 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 C’s are the linear coefficients. The results obtained agree with those obtained by Schwartz . However, the long-range correlations could not be taken into account in  at the same time and had to be added separately, with the result that the final phase shifts ceased to have any bounds. In the hybrid theory, as indicated earlier, both short-range and long-range correlations can be taken into account at the same time. It should be noted that in the case of positrons, the sign before the second term in Eq. (10) is plus instead of minus, as in the case of electrons. The phase shifts obtained in two approaches are given in Table 8. We used 84 terms in the Hylleraas wave function previously, and improved results are obtained now with shorter expansions  as indicated in the table. A comparison of the results obtained with different approaches is shown in Figure 6. Using a fewer number of terms, higher phase shifts have been obtained in the hybrid theory . P-wave shifts have been calculated in the hybrid theory  and are compared with those using the Feshbach projection-operator formalism  and with those obtained by Armstrong .
|k||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 :
where α is the fine-structure constant and
|k||Zeff (l = 0)||Zeff (l = 1)||Zeff (l > 1)||Total|
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 k is the momentum of the outgoing electron. These cross sections are much larger than that for the photodetachment of the negative hydrogen ion.
Lyman-α radiation (2P 1S) at 1216 Å has been seen from astrophysical sources and the sun. It has been observed from Voyager measurements . Similarly, Lyman-α radiation (2P 1S) at 2416 Å is expected when in the photodetachment of Ps−, the remaining positronium is left in the 2P state. Following Ohmura and Ohmura , photodetachment cross sections have been calculated when the remaining atom is in 2p, 3p, 4p, 5p, 6p, and 7p states .
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 S-wave Feshbach resonance in positron-hydrogen system is by Doolen et al. , who, using the complex-rotation method, obtained the position −0.2573741 and for width 0.0000677 Ry. In this method
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 1P shape resonance above n = 2 have been calculated by Bhatia and Ho . They obtained −0.06217 and 0.000225 Ry for the position and width of the resonance. These results have been confirmed experimentally by Michishio et al. .
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 E is the excess energy. As indicated earlier in the positron-hydrogen scattering, positron and electron tend to be on the same side of the nucleus. This shows the cross section at threshold cannot be the same for positrons and electrons. The threshold behavior of the positron impact on hydrogen has been analyzed by Klar . He finds that at threshold, .
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 1S state of hydrogen atom to the 2S state is proportional to .
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, R-matrix, and hybrid theory. It should be possible to formulate the hybrid theory for more complicated systems.
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 2p state.