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Formation of negative hydrogen ion in Positronium - Hydrogen collisions

The importance of the excited states of Positronium (Ps) in the formation cross sections(both differential and total) of the negative hydrogen ion (H^-) are investigated theoretically for the charge transfer reaction, Ps (n = 1, 2) + H -->e+ + H^- for a wide range of incident energies (e. g., threshold - 500 eV) . The calculations are performed in the frame work of a qualitative model, the post collisional Coulomb Modified Eikonal Approximation (CMEA) . A comparative study is also made between the capture from ground and excited states of the Ps. The present CMEA model takes account of higher order effects which is essential for a rearrangement process where the First Born type Approximation (Coulomb Born for the ionic case) is not supposed to be adequate. At low incident energies, the excited states of Ps (2s, 2p) are found to play a dominant role in the H^- formation cross sections . Significant deviations are noted between the present CMEA and the Coulomb Born (CBA) results even at very high incident energies (e.g., Ei = 500 eV), indicating the importance of higher order effects. At high incident energies the present CMEA differential cross section (DCS) exhibits a double peak structure which is totally absent in the CBA and could again be attributed to higher order effects.

g., E i = 500 eV ) , indicating the importance of higher order effects. At high incident energies the present CMEA differential cross section ( DCS ) exhibits a double peak structure which is totally absent in the CBA and could again be attributed to higher order effects.  [ 1 ] . Further , it was suggested [ 2 ] that the main source of opacity in the atmosphere of the Sun at red and infrared wavelengths was absorption by the − H ion. Negative ions also play a major role in a number of areas of Physics & Chemistry involving weakly ionized gases and plasmas [ 1 -4 ]. Particularly the − H ion finds important biological applications as it can be used as an efficient antioxidant in human body.

PACS
The present study addresses the charge transfer reaction in the simplest Ps -atom system e. g., formation cross sections ( both differential and total ) between the ground and the excited states , particularly in respect of the incident energy and the angular momentum quantum number ( l ).
Further, the present theoretical estimates, in absence of any other results , could provide some guidelines to the future experiments involving individual excited Ps states , the latter being already feasible for the Ps formation process [ 5 ] . The excited states of the H atom on the other hand, are expected to be less important than those of the Ps in the − H ion formation. The importance of the Ps excited states was also noted [ 6 ] in the production of low energy antihydrogen ( H ) atoms by collision with antiprotons ( ) where the H production was found to increase rapidly with the excitation of Ps. Although to our knowledge, there exists no experimental work in the literature as yet for collisions involving the excited Ps as target , measurements on the production of excited states Ps were reported much earlier [ 7,8 ] .
To obtain reliable results at very low incident energies, it is essential to take into account the effect of all possible channels and a coupled channel approach is therefore needed. Therefore, the importance of the inclusion of the above rearrangement channel in the scattering process of the PS + H system in the frame work of a coupled channel approach can not be over emphasized .
Further, since the negative hydrogen ion does not possess any discrete excited state, this particular charge transfer process has an added advantage both in respect of experimental and theoretical aspects.
However, despite such paramount importance, theoretical study of this process is quite limited [ 9,10 ] probably because of the complexity lying with the four body problem. As such , in the absence of any experimental data or any sophisticated theoretical calculations , a comparatively simple but a consistent model could provide a reasonable estimate for this important reaction and provide some guidelines for the future experiments as well as for the sophisticated ( e. g. , CC , R Matrix ) theories.
Theoretically, Ps -atom scattering is one of the most difficult problems since both the Biswas [ 9 ] . The present CMEA model takes account of the higher order distortion effects in the asymptotic region as well as in the collisional region and as such unlike the plain eikonal , it can be pushed even up to quite low incident energies . In fact, inclusion of higher order effects is essential for a rearrangement process where the First Born type approximation ( Coulomb Born for the ionic case) is not supposed to be adequate. Further, the present Coulomb Born ( CB ) results , extracted from the present CMEA code could also serve the purpose of reliable inputs to the more sophisticated Close Coupling ( CC ) approximation . However, it should be mentioned here, that at extreme low energies ( near threshold ) the present model might not yield quite reliable results and as already mentioned , a more sophisticated calculation ( e.g., CC or R Matrix ) is needed.
Since the − H ion can exist only in its ground singlet state ( S = 0 ) , the above charge transfer channel occurs in the electronic spin singlet only. The present work gives special emphasis on electron capture from the excited states of Ps ( 2s, 2p ) since few results for the ground state capture were reported earlier [ 10 ] . To the best of our knowledge, this work is the first theoretical attempt for the negative hydrogen ion formation from the excited states of Ps.

Theory:
The prior and the post forms of the transition amplitude for the above process are given as The asymptotic initial channel wave function The initial channel perturbation i V in equation (1a) is given by where pT V refers to the positron -target interaction while . The corresponding final state wave function in the Coulomb Born approximation ( CBA ) is given as : function of Chandrasekhar [ 4 ] and is given by : with N = 0.3948, α = 1.03925 , β = 0.28309 . The ground state energy of the − H ion for this wave function [ 4 ] is E = -0.513 a. u. Although the present work uses the simpler wave function ( − H ion ) of Chandrasekhar [ 4 ] that was widely used successfully [ 9, 12 -15 ] , more sophisticated wave functions are available in the literature [ 12,16 ] involving explicit correlation ( e -e ) terms and producing more accurate binding energy of the − H ion. However, the correlated wave function is expected to be more suitable and in fact unavoidable for the two electron transition processes that are mainly governed by the e -e correlation [ 17,18 ] . Since the present work refers to single electron capture and the − H ion is formed in the final state, to our belief , the use of this simpler representation of the − H ion [ 4 ] which partly takes account of the e -e correlation [ 13,14 ] is quite justified and is supposed to give reasonable description of the physics of the target for this particular process where the electron is transferred from the Ps atom to the one electron target to form the − H ion in the final state. In fact , the choice of this simpler wave function is mainly dictated by the feasibility of the present calculation which is already quite involved for a four body problem. Even in the simple first Born approximation ( FBA ) , the mathematical expression for the scattering amplitude becomes quite complex with the correlated wave function [12].
Using equations ( 2 ) -( 6 ) in equation 1(a) and then after much analytical reduction [ 19 ] , the transition amplitude is finally reduced to a two -dimensional integral to be evaluated numerically [ 19,20 ]. The differential cross section for the process studied is given by where i v and Since the threshold energy for this particular process is 6.45 eV for the ground ( n = 1 ) Ps state and 1.35 eV for the excited ( n = 2 ) Ps states, the DCS results at a much lower incident Ps energy ( e.g., 2 eV or 5 eV ) are exhibited in figure 1 for the excited states ( 2s, 2p ) of Ps only.       The appearance of the double peak structure in the 1s & 2s DCS at higher E i could be attributed to the higher order effect that is taken into account through the post collisional eikonal approximation. For the 2p state on the other hand , this prominent secondary peak is again somewhat suppressed resulting in some hump like structures ( fig. 6 ) , due to the same reason as stated before for the first minimum . It should be pointed out here that no such secondary structure occurs in the CBA results ( vide the inset of fig. 6 ) even at very high incident energy ( e. g., 300 eV ).     are always found to be higher than the CMEA ones except at the threshold region. Further, the significant discrepancy between the present CMEA ( both in the partial and the integrated TCS ) and the CB results even at a very high incident energy ( e. g. , i E = 500 eV ) indicates the importance of higher order effects for a rearrangement process. Table 1 also reveals that at lower incident energies, the discrepancy between the CMEA and the CB results is much higher for the 1s and 2s states than for the 2p one, while at higher i E , the reverse is true, i. e. , the deviation is most for the 2p state.
To our knowledge, the only theoretical results available for this particular charge transfer reaction ( for the ground state Ps only ) is due to Biswas [ 9 ] in the framework of coupled two channel approximation using their [ 9 ] model exchange method as well as the abinitio exchange method. As stated before , the large deviations ( vide Table 1 ) noted between the two theories ( present and that of Biswas [ 9 ] ) could probably be attributed to the Author's [ 9 ] neglect of the long range Coulomb attraction between the + e and the − H in the final channel. Further, the neglect of this interaction leads to an unphysical behaviour , e. g. , the TCS of Biswas [ 9 ] converge to the FBA results while , the present CMEA TCS tend to converge to the Coulomb Born results at very high incident energies, as is expected physically for the present process leading to an ion formation in the final channel . It should be emphasized that this long range Coulomb interaction between the charged particles should in no way be neglected to obtain reliable results.

Conclusions :
At low and intermediate incident energies, the excited states of the Ps play a dominant role in the − H ion formation cross sections while at higher energies, the capture from the ground state Ps dominates. Particularly the Ps 2p state is found to be the dominant process among the three states ( 1s, 2s, 2p ) at very low incident energies.
The distinct double peak structures occurring at higher incident energies ( E i ) in the 1s, 2s DCS ( which are absent in the CBA ) could be the manifestation of higher order effects. The signature of the double peak becomes more and more prominent with increasing i E indicating the increasing importance of the higher order effects with incident energy. For the 2p state however , the double peak structure is not so prominent as in the case of 1s , 2s because of the mdegenerate states.
The discrepancy between the present CMEA and the CBA TCS arising due to higher order effects retains even at a very high incident energy as is expected for a rearrangement process.
For a more reliable results at very low incident energies ( near threshold ) , the polarization of the Ps atom should be taken into account explicitly and a more sophisticated calculation with proper inclusion of the final channel long range coulomb interaction is highly needed.