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The parallel calculations of fully differential cross section for transfer excitation reactions in fast proton-helium collisions
Ё M.S. Schoffler1, O. Chuluunbaatar2,3, S. Houamer4, A.G. Galstyan5, J. Titze1, T. Jahnke1, L.Ph.H. Schmidt1, Yu.V. Popov6, A. Bulychev2
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Institut fur Kernphysik, University Frankfur t, Frankfur t, Germany Ё 2 Joint Institute for Nuclear Research, Dubna, Russia 3 National University of Mongolia, UlaanBaatar, Mongolia ґ ґ ґ ґ Depar tement de physique, Faculte de Sciences, Universite Ferhat Abbas, Setif, Algeria 5 Physics Faculty, Moscow State University, Moscow, Russia 6 Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia

I N T RO D U C T I O N
From an experimental point of view, single-electron transfer has at least two interesting facets. First, it can be used as a tool for spectroscopy. Energy gain spectroscopy and the related experiments in inverse kinematics which exploit the recoil ion longitudinal momentum for Q -value determination (change in the electron binding energies) give access to the energy levels of highly charged species and to energy levels that do not decay radiatively. These are difficult to access by other spectroscopic techniques. Second, the dynamics of the transfer process itself is of fundamental interest since it combines electron-electron dynamics, correlation, and questions of few-body momentum exchange. In most of the theoretical studies the transfer into an excited state or the transfer combined with a target excitation of a second electron were neglected. Especially at higher impact energies Ep > 100 keV/u, where the final state determination in experiments are difficult, the influence of excitation has not been investigated. Nowadays the modern experimental technique of COLTRIMS (Cold Target Recoil Ion momentum spectroscopy) allows to measure the final electronic states in electron transfer reactions even at high impact energies. We consider here the transfer reaction p+He H+He+ at different high proton energies and present both the experimental single differential cross sections for total excitation of the residual helium ion (n 2), and the calculations within first Born approximation (FBA).

T H E O RY
2 d (1) mp p = d p (2 ) n-1 l FBA |Tnlm |2 n=2 l =0 m=-l n-1 l FBA |Tnlm |2 n=2 l =0 m=-l

R E S U LT S A N D D I S C U S S I O N
(4) We have to work out at least computer code for 9D integration to calculate the SBA or DWBA. Such the FBA calculations of the SDCS2 are presented in [13] for Ep = 300 keV, but the result was about 150 times bigger than the experiment. Our correct results on the base of 9D calculations are presented for the first time in Fig. 2. They quite coincide with both 3D and 6D calculations, and the experiment (n = 2 + 3). See also our discussion in [14, 15].

and d (2) = d p (
2 mp

2 )2

(5)

In theoretical calculations we use four trial ground-state helium wave functions. One is the loosely correlated 1s2 Roothaan et al (RHF) wave function [7] (no angular correlation) with a rather poor groundRHF state energy of EHe = -2.8617 a.u. Second one Silverman et al (SPM) wave function [8] includes angular correlations, but its ground SPM energy is also far from the experiment EHe = -2.8952 a.u. Two another trial functions are highly correlated ones. It is given in Mitroy MMW = -2.9031 a.u. et al (MMW) [9] with a ground-state energy of EHe CPV and that of Chuluunbaatar et al (CPV) [10] with EHe = -2.903721 a.u. Their energies are ver y close to the experimental value. We omit in shor t the mathematical details of description of the symFBA metrized matrix elements |Tnlm |, which are given in [11, 12]. We calculate them with use of 3D and 6D integrals.

Figure 2: SDCS2 calculations versus the scattering angle p at Ep = 300 keV. Colors of cur ves are the same like in Fig. 1.

Next, we calculate 9D integral with the eikonal phase-factor obtained earlier in [12] DWBA 2 Tnlm d 3Re-i R q d 3ei vp 0() d 3r2 (r2)в nlm
[-i /vp f (R ,,r2)] - 1 - e

N U M E R I C A L I N T E G R AT I O N D E TA I L S
The corresponding integrals are calculated using the adaptive subdivision algorithm, and it has a loop that contains four steps: i) determine a new subdivision of the integration region, ii) apply the basic rule to any new subregions, iii) combine new results from step ii) with previous results and iv) check for convergence. For p-processors parallelization of above algorithm we used the Single list algorithm: p-sect region [a1, b1] в [a2, b2] в · · · в [ad , bd ] do parallel apply integration rule to subregions end do parallel do while (error > ) and (number of rule evaluations Nmax) SUBREGION SELECTION do parallel compute new subregion limits apply cubature rule to new subregions end do parallel do parallel remove old subregions from list add new subregions to list update integral approximation and error estimate end do parallel end do Our p-processors parallel calculations are approximately 0.8p times faster than single processor. with



EXPERIMENT
For the swift collisions investigated here, the best resolution is obtained by detecting the recoil ion momentum instead of small change on the large projectile momentum. In the present experiments we have used the COLTRIMS technique to measure both the neutral projectile H0 and the recoil ion He+ in coincidence [1, 2]. The experiment has been performed at the 2.5 MV van de Graaff accelerator at the Institut fur Kernphysik, University of Frankfur t. We used two sets of adjustable slits to collimate the beam to a size of 0.5в0.5 mm2 at the target. Two sets of electrostatic deflectors are placed in front and behind the target. They were used to clean the beam from charge state impurities in front of the target and to analyze the final charge state behind the target. The neutral H0 projectiles were detected on a 40 mm position- and timesensitive multichannel plate (MCP) detector with delay line anode for position read-out [3]. The target is provided by a 2-stage supersonic gas jet. At the interaction point, the gas jet has a diameter of 1.5 mm and areal density of 5в1011 atoms/cm2. The He+ recoil ions produced in the overlap region of gas jet and projectile beam were projected with a weak electrostatic field (4.8 V/cm) onto a 80 mm position- and time-sensitive multi channel plate detector. A three-dimensional time- and space-focusing geometry was applied to maximize the resolution [4]. From the measured data, time of flight (19 µs for He+) and position of impact, we extracted the initial three-dimensional momentum vector. We achieved an overall momentum resolution of 0.1 a.u. which was limited by the target temperature. Our spectrometer geometry and electric fields yielded 4 acceptance angle for all recoil ions with momenta below 9 a.u. In the plane per pendicular to the beam axis, we measure the scattering angle of the projectile and the transverse momentum of the recoiling ion. By momentum conservation they must add to zero. We used this for background suppression. We deduced the scattering angle from the recoil ion transverse momentum, which has a much better momentum resolution in our setup. By gating on the different longitudinal momenta of the recoil ion, we were able to extract the scattering angles for different final electronic states [5]. The small background contribution, mainly from single ionization, has been subtracted.

2 + 0(R - , r2). |R - r2| R

1

(6)

[vp |R - | + vp · (R - )]2 [vp |R - r2| + vp · (R - r2)] f = ln . 2 [v |R - - r | + v · (R - - r )] [vp R + vp · R ] p p 2 2 This phase-factor is a par t of the SBA, and can be considered like modified 4C approximation (in analogy to 3C and 6C). Results for Ep = 300, 630, 1000 and 1200 keV are presented in Fig. 3. We see positive evaluation of the main peak towards the experiment. However, this DWBA can not improve the situation at bigger scattering angles, and full SBA calculations are needed.
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Figure 3: SDCS1 calculations versus the scattering angle p . Solid line: SPM, dashed line: SPM with the 4C phase factor. Ep = 300 keV (left-top), Ep = 630 keV (right-top) Ep = 1000 keV (left-bottom), Ep = 1200 keV (right-bottom).

R E S U LT S A N D D I S C U S S I O N
In Fig. 1 we present experimental and theoretical SDCS1.
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CONCLUSION
In conclusion, we present the SDCS experimental data and FBA theor y for transfer excitation proton-helium collision at 300, 630, 1000 and 1200 keV/u. In calculations both 1s2 and highly correlated trial helium wave functions are used. The 1s2 wave function fails to describe the experiment, while angular correlated functions give practically coincident results and well reproduce the experiment in a vicinity of main peak. 300 keV/u results show the limit for the FBA theory. Also 3D, 6D and 9D calculations give coincident results, close to the experiment. We modified the several standard For tran codes for evaluate a highdimensional integral using the adaptive subdivision method. Now they keep more data in the memor y, can use the complex arithmetics and are adapted for parallel calculations.

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T H E O RY
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Let us denote the projectile proton momentum by pp , the hydrogen momentum by pH , and the recoil-ion momentum by K . We also define the transferred momentum as q = pH - pp . We can deduce its approximate value using the momentum and energy conservation q + K = 0, 2 2 pp pH K2 He + E0 = + + E H + E ion . 2mp 2(mp + 1) 2M (1) (2)

SDCS (a.u.)

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Here k is the ejected electron momentum, the proton mass He = mp = 1836.15, the helium ion mass M 4mp , E0 -2.903724377034 [6]. Now we choose ver y small scattering angles for the outgoing hydrogen (0 p 1.5 mrad). It leads to a practically zero ion velocity K /M in the laboratory frame during the process, and we can consider the ion like immovable. The proton velocity vp = pp /mp varies about a few a.u. for its energy of several hundredths keV. This fact allows one to neglect K 2/2M and q 2/2mp after inser tion of pH = q + pp into Eq. (2). As a result we obtain 12 He (3) vp q = vp + Q ; Q = E0 - E H - E ion , 2 and choose the vector vp as z -axis; there follows the longitudinal component qz = vp /2 + Q /vp . The transverse component of the vector is q = (pH ) mp vp p . We have two types of single differential cross section (SDCS) for TE processes:

Figure 1: Experimental and theoretical data for Ep = 300 keV (left-top panel), Ep = 630 keV (right-top panel) Ep = 1000 keV (left-bottom panel), Ep = 1200 keV (right-bottom panel) p+He collisions. Full squares is the experiment, red line the RHF [7] trial helium wave function, blue line SPM [8], green line MMW [9] and black line CPV [10] practically coincide. n = 2 + 3.

REFERENCES
1 J. Ullrich et al., J. Phys. B 30, 2917 (1997); Rep. Prog. Phys. 66, 1463 (2003). Ё 2 R. Dorner et al., Phys. Rep. 330, 95 (2000). 3 O. Jagutzki et al., Nucl. Instrum. Meth. Phys. Res. A 477, 244, 256 (2002). Ё 4 R. Dorner et al., Nucl. Instrum. Meth. Phys. Res. B 99, 111 (1995). Ё 5 M.S. Schoffler et al., Phys. Rev. A 79, 064701 (2009). 6 O. Chuluunbaatar et al., J. Phys. B 44, L425 (2001). 7 E. Clementi and C. Roetti. Atomic Data and Nuclear Data Tables 14, 177 (1974). 8 J.N. Silverman et al. J. Chem. Phys. 32, 1402 (1960). 9 J. Mitroy et al., J. Phys. B 18, 4149 (1985). 10 O. Chuluunbaatar et al., Phys. Rev. A 74, 014703 (2006). 11 S. Houamer et al., Phys. Rev. A 81, 032703 (2010). 12 H.-K. Kim et al., Phys. Rev. A 85, 022707 (2012). 13 U. Chowdhur y et al., J. Phys. B 45, 035203 (2012). 14 S. Houamer et al., J. Phys. B 46, 028001 (2013). 15 U. Chowdhur y et al., J. Phys. B 46, 028002 (2013).

The shape of the SDCS1 is formed by three FBA terms, one of which (OBK) provides the direct He e+He+ decay mechanism, and two others provide the double decay of the helium in the intermediate state He 2e+He2+. The absolute value of the cross sections is about 1% of that when the residual ion stays in its ground state (charge transfer, CT), and the shape in the case of helium wave functions with angular correlations has the minimum and reminds the case n = 1 [12]. The wave function without angular correlations fails at all. It is interesting to note, that FBA SDCS1 for CT reactions practically does not depend on the trial helium wave function, and even the simplest 1s2 one describes well the main peak. We also see that Ep 500 keV is the boundar y energy when FBA still describes somehow the main peak at ver y small p . The FBA also fails to describe the position of the minimum and the behavior of the cross section beyond it. We need to attract here the SBA or DWBA calculations.