WDS'08 Proceedings of Contributed Papers, Part III, 168–172, 2008. ISBN 978-80-7378-067-8 © MATFYZPRESS

Protonium Formation in – Collisions J. Eli´aˇsek Charles University in Prague, Faculty of Mathematics and Physics, Czech Republic.

Abstract. We present a computation of the cross section of protonium formation in antiproton-hydrogen collisions using the nonlocal resonance model. It is based on restoration of the Born-Oppenheimer approximation, when the adiabatic electronic basis is replaced with the so called diabatic one. This allows us to separate the and parts of our problem at the cost of a nonlocal potential in hadron dynamics.

Introduction Many groups used different methods to describe collisions of a hydrogen and negatively charged heavy . The first article about the formation of an atom which has a heavy instead of an electron was written by Fermi and Teller. In their work (studying the capture of heavy negative particles in ) [Fermi and Teller, 1947] they also deduced the critical distance between the and the heavy particle now called the Fermi-Teller radius. If the distance is smaller then this radius then the potential is not strong enough to hold the electron bound. This means that we can not use the adiabatic approximation, because all electronic bound states disappear in continuum (see figure 1). More sophisticated methods must be used. Older works are summarized in [Cohen, 2004]. Advanced adiabatic methods were used by Ovchinnikov and Macek [2005] and by Zˇd´ansk´aetˇ al. [2004]. Hesse et al. [2004] employed the hyperspherical close coupling method. The wave packet evolution and semiclassical approach were used by Sakimoto [2002a, 2001, 2002b]. Yamanaka and Ichimura [2006] solved Faddeev equations for this three body system and calculated probabilities of protonium formation. Some of the approaches scale down the mass of the antiproton to be able to use their method and get some qualitative results, because the antiproton is captured in states with n ≈ 40 and l remains approximately constant in the collision. The reaction ¯p+ H → p¯p+ e− is also a prototype of the associative detachment reaction studied in atomic physics [Bieniek and Dalgrano, 1979; C´ıˇzekˇ et al., 1998]. As a one electron system, free of electron correlation, it is vital for the understanding of errors that are caused by some approximations used in such calculations. Next, we introduce the theoretical background of our calculation of hadron dynamics during the collision and show some results for a preliminary fit to complex level–shift which we took from our previous calculation for electron motion with fixed .

Theory The nonlocal resonance model for collisions of negative particles with an atom was developed else- where (see [Domcke, 1991; C´ıˇzekˇ , 1999]) and here we summarize it. The Hamiltonian of our problem is (we use atomic units unless stated otherwise)

H = Tp¯p + Hel, = P 2 Tp¯p 2µ , (1) p2 1 1 1 Hel = − + − , 2 rp−e rp¯−e R where p and P denote the momenta of the electron and hadrons in the center of mass system respectively. We introduce an expansion of the wave function

Ψ = ψi(R)φi(r; R), (2) Xi where R is the p-¯pdistance, r is the position of the electron and ψi and φi are the nucleonic and electronic parts of the wave function respectively. The electronic part depends parametrically on R. An expansion similar to this is used in the adiabatic approximation: there, φi are eigenfunctions of the electronic part

168 ELIA´SEK:ˇ PROTONIUM FORMATION IN ANTIPROTON-HYDROGEN COLLISIONS

0

-0.2

-0.4

-0.6 Potential energy [au]

-0.8 m=0 m=1 m=2 Coulombic attraction Polarization -1 1 10 Proton-antiproton distance [au]

Figure 1. The solid line indicates the Coulombic interaction between the proton and the antiproton (continuum threshold). Bound states of the electron in the proton-antiproton potential with the projec- tion of angular momentum on p-¯paxis m between 0 and 2, i.e. adiabatic potential curves, are shown. Polarization has the same asymptotic behaviour as the ground state of the electron in the field of the proton and the antiproton. We can clearly see that bound states are converging to electronic continuum with the decrease of proton-antiproton distance.

of the Hamiltonian Hel for each fixed R. In our approach we build this basis differently. We fix one state φd called the discrete state, in our case the ground state of the hydrogen atom, and the rest is assumed to form a continuum φk orthogonal to φd. By doing this we separate our Hilbert space into two parts — a part corresponding to the discrete state (we will call it Q) and a part corresponding to the rest (P ). Let us define projection operators to these subspaces

Q = |φdihφd| ⊗ 1p¯p,P = |φkihφk| ⊗ 1p¯p = 1 − Q. (3)

This procedure ensures, with properly chosen discrete state, that φi will change slowly with R and that the main idea of the Born-Oppenheimer approximation may be used, namely that Q commutes with the kinetic energy operator Tp¯p. Now let us focus on the dynamics. We write the Hamiltonian in the form

H = H0 + HI ,H0 = PHP + QHQ, HI = PHQ + QHP, (4) so that the Lippmann-Schwinger equation reads

(+) −1 (+) |Ψ i = |Ψii + [E − HP − HQ + iǫ] [PHQ + QHP ]|Ψ i, (5) where HP and HQ stand for PHP and QHQ respectively. If we multiply this equation by the projection operators P , Q and eliminate P |Ψ(+)i, we get an inhomogeneous Schr¨odinger equation with the effective Hamiltonian Heff = HQ + HQP GP (E)HPQ. (6) In the Q-space we can write the effective Hamiltonian in the form

Heff = Tpp¯ + Vd + F (E), (7) where Vd = hφd|Hel|φdi is the energy of the system in the discrete state and

′ −1 ∗ ′ hR|F (E)|R i = dedΩVde(R)(E − Tp¯p − V0 − e + iǫ) Vde(R ) (8) Z

169 ELIA´SEK:ˇ PROTONIUM FORMATION IN ANTIPROTON-HYDROGEN COLLISIONS

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0.8

0.6

0.4

0.2

0

∆ , - Γ /2 [au] -0.2

-0.4

-0.6 R=0.5 R=1.0 R=2.0 -0.8 R=0.5 R=1.0 R=2.0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 E[au]

Figure 2. A fit of the data from [Eli´aˇsek, 2007] used in tuning calculations. In the upper part there is Γ(E,R) and in lower part there is ∆( ) for a few values of the parameter R. The input data are − 2 E,R displayed by dots, the fit by lines. We used three terms in the series (11).

is the R-representation of the nonlocal potential with V0 being the interaction potential between the hadrons, i.e. V0 = 1/R and Vde is the coupling between the discrete state and the continuum state φk with the electron energy e = k2/2. The nonlocal potential is often split into Hermitian and antihermitian parts ( ′) = ∆( ′) i Γ( ′) (9) F E,R,R E − Tp¯p − V0,R,R − 2 E − Tpp¯ − V0,R,R .

The electronic part of the Hamiltonian is fully parametrized by the functions V0(R), Vd(R), Γ and ∆. To calculate the nonlocal potential (9) we used the energy shift ∆(E,R) and the decay width Γ(E,R) from previous work [Eli´aˇsek, 2007]. There we calculated F (E,R) — complex level shift in fixed nuclei approach, using the formula

¯ F (E,R) = hφd|HelP G(E)PHel|φdi, (10) where G¯(E) is Green’s function of the electron in a fixed nuclei potential calculated in P part of the full Hilbert space. This problem is separable in spheroidal coordinates and was solved in [Eli´aˇsek, 2007]. For the numerical solution of nucleon dynamics we assume that ∆ and Γ can be written as series of separable terms 2 ′ 2 Γ(E,R) = fi(E)gi (R), ∆(E,R) = fi (E)gi (R). (11) Xi Xi The equation (9) is then rewritten as

( ′) = ( )[ ′( ) i ( )] ( ′) ′ (12) F E,R,R hR|νigi R fi E − Eν − 2fi E − Eν gi R hν|R i Xν where Eν are the eigenvalues and |νi are the eigenvectors of Tp¯p + V0(R). In our problem there is an infinite number of bound states. Our approach to this problem was to enclose the protonium in a box. This limits the number of bound states to a finite value. We can easily add more states by changing the size of the box and check the convergence. Practically, we are searching for zero points in solutions of a differential equation for Couloubic problem at the border of the box. Once we set up the nonlocal potential F , we can write the final Lippmann-Schwinger equation

(+) (+) (+) |ψ i = |ψi + GQ (E)F (E)|ψ i, (13)

170 ELIA´SEK:ˇ PROTONIUM FORMATION IN ANTIPROTON-HYDROGEN COLLISIONS

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-0.2

-0.4

-0.6 RFT -0.8

-1 Potential energy[au]

-1.2 V0 Vd V -1.4 loc Polarization 0 2 4 6 8 10 R[au]

Figure 3. The potential curves of our problem. V0 is the potential of the proton-antiproton interaction, Vd is the energy of the discrete state and Vloc is the adiabatic potential curve of the ground state within our model.

(+) where GQ is the Green’s function for Hamiltonian HQ, and calculate cross section = 2(2 )5 2 ∗ (+) 2 (14) σ π µ |hν|Vdef |ψi | Model We fit the data from [Eli´aˇsek, 2007] for the level shift function F (E,R) by the series (11) shown in figure 2. For the testing of the numerical parameters of the calculation we used three terms in this series (lines in figure 2). The accuracy is about 10% for small R and decreases for larger R. The most important region for nucleon dynamics is around the Fermi-Teller radius RFT = 0.639 au, where the lowest vanishes.

As stated previously, Vd,V0 and F contain full information about the electronic problem. Specially, it is possible to find the energy Vloc(R) of the bound state for R > RFT , i.e. the adiabatic potential. As shown by Domcke [1991], Vloc = Eres + V0, where Eres is the solution of the equation

Eres(R) = Vd(R) − V0 + ∆(Eres,R,R). (15) This potential curve is shown in figure 3. The function should be the same as the lowest potential curve in figure 1. We see that it does not have the same asymptotic behaviour (as polarization) and it joins the continuum at R greater than the Fermi-Teller radius, which is due to inaccuracy of our fit for F (E,R). Results and future prospects We calculated the integral cross section of the formation of protonium (fig.4) from this model using equations (13), (14). Values of the total cross section are of the same order as results of calculations of Tong et al. [2007]. We can clearly see the threshold at ED = 0.5 au that corresponds to the opening of an ionization channel. The structures at lower energies correspond to the possibility of creation of protonium with different principal quantum numbers (of this hydrogen-like atom), i.e. they are excitation thresholds. Below the lowest excitation threshold (roughly 0.5 eV) the cross section exhibits a minor deviation from monotonic behaviour. We believe that this has no physical significance and is probably caused by an inaccurate fit for F (E,R). Now we are working on a better fit including more terms in (11) to get better agreement for Γ and ∆ and to repair correct polarization asymptote and the point of disappearance of the bound state. Next, we would like to include final electronic states with non-zero angular momentum (now we include only an outgoing electronic s-wave). This leads to a system of coupled Lippmann-Schwinger equations. And finally we would like to use more discrete states in this problem. This will introduce a multicomponent wave function and enable the possibility of studying antiproton collisions with an excited hydrogen atom.

Acknowledgments. This work is supported by GAUK through Grant No 116-10/143107. I would like to thank M. C´ıˇzekˇ for substantial support during the preparation of this paper.

171 ELIA´SEK:ˇ PROTONIUM FORMATION IN ANTIPROTON-HYDROGEN COLLISIONS

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7

6 E

] D

2 5

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cross section [Å 3

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0 0 2 4 6 8 10 12 14 16 E[eV]

Figure 4. Dependence of the total cross section of the formation of protonium in antiproton-hydrogen collision on H-¯pkinetic energy. Dissociative threshold is indicated by the arrow at 13.6 eV. Thresholds corresponding to the possibility of protonium creation in different internal states (different principal quantum number and therefore energy - protonium is hydrogen-like atom) can be recognized at lower energies. For reason of lucidity, we don’t depict cross section value greater than 8 A˚2. The energy dependence in this region exhibits almost monotonic behaviour with limiting a value of 16 A˚2, although small oscillations can be observed below the lowest excitation threshold (roughly 0.5 eV). We believe that these structures are of no physical importance being probably caused by an inaccurate fit for complex level shift.

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