fulltext

Protonium formation in laser-assisted

antiproton–hydrogen-atom collision

M. BHATTACHARYA

Department of Theoretical Physics, Indian Association for the Cultivation of Science Jadavpur, Calcutta 700 032, India

(ricevuto il 23 Novembre 1995; approvato il 22 Ottobre 1996)

Summary. — A detailed analysis has been given for the study of protonium formation in laser-assisted antiproton–hydrogen-atom collision within the framework of first Born approximation. The dressing effects of the hydrogen and the protonium atoms have been considered to first order in the A Q p gauge for field direction parallel to the incident momentum. The differential cross-sections for protonium formation have been presented for no-photon and one-photon exchange (l 40, 61) at 50, 100 and 200 keV energies with laser frequency v 40.043 a.u. for both the field strengths e04 0 .002 a.u. and e04 0 .02 a.u., respectively, together with the corresponding field-free results. It has been observed that in the case of absorption (l 421) as the incident energy decreases the cross-section values corresponding to e04 0 .002 a.u. are greater than the corresponding field-free results. It has also been noted that the contributions from almost the entire angular range 0 –180 7 are important for the calculation of the cross-sections, a characteristic usually seen in light particle scattering.

PACS 34.80.Qb – Laser-modified scattering.

PACS 34.50.Rk – Laser-modified scattering and reactions. PACS 34.70 – Charge transfer.

PACS 25.43 – Antiproton-induced reactions.

1. – Introduction

In recent times, the formation of protonium—a bound state of proton and antiproton—in antiproton–hydrogen-atom collision has been a subject of particular interest because of its importance in many branches of physics. Bracci et al. [1] have investigated the rearrangement collisions of antiproton with proton, neutral and negatively ionised hydrogen atom as targets in the low-energy region using a purely classical method. The classical trajectory Monte Carlo (CTMC) approach has been employed by Cohen [2] for the calculation of protonium formation cross-sections in antiproton–hydrogen-atom collision at very low energy. Only recently Deb and Roy [3] have given, using Born approximation, the quantum-mechanical analysis for the rearrangement collision in which protonium is formed in the ground state in antiproton–hydrogen-atom collisions in the intermediate energy range (50–250 keV).

mechanically. The laser-modified wave functions of the antiproton and the free protonium atom have been given by the Volkov solutions. The field-dressed bound-state wave functions have been obtained by solving the coupled Schrödinger equation in the A Q p gauge through first-order time-dependent perturbation theory assuming the field strength e0 to be much less than the characteristic intra-atomic

electric field.

Making use of these laser-modified wave functions and considering the field-free interaction (post form) to be the perturbation responsible for the collision, we have obtained closed-form expressions for the protonium formation cross-sections. Atomic units have been used throughout the calculation.

2. – Theory

The rearrangement collision of interest is the following: p 1H(1S) K ( pp)(1S)1e2

(1)

which has taken place in the presence of a laser field treated classically as a spatially homogeneous, single mode and linearly polarised electric field given by

e_{(t) 4}e0sin (vt) .

(2)

The corresponding vector potential in the A Q p gauge is

A(t) 4A0cos (vt)

(3)

with A04 ce0O v.

In the first Born approximation, the S-matrix element for the above rearrangement process

(

eq. (1)

)

may be written as

S_if( pp)_{4 2i}



2Q 1Q dt aCfNVN Cib , (4) where (cf. fig. 1) Ci4 xki(rp, t) c H 0(r1, t) , (5a) Cf4 xkf(r , t) c ( pp) 0 (R , t) , (5b) V 4 1 r2 2 1 r1 . (5c)

Fig. 1. – Coordinate representation.

In the above equations, xki and xkf are the Volkov state wave functions corresponding

to the incoming antiproton and the protonium atom in the presence of the laser field and are given by

xki(rp, t) 4 (2p) 23 O 2_exp

k

_i

m

_k iQ rp2 kiQa0sin vt 2 k2 i 2 mi t

nl

(6) with a04e0O miv2, xkf(r , t) 4 (2p) 23 O 2_exp

k

_i

m

_k fQ r 2 kf2 2 mf t

nl

. (7)

miand mfin eqs. (6) and (7) are the reduced masses in the initial and the final channel,

respectively. It is to be noted that in eq. (7) the effect of the laser field on the motion of the protonium atom is absent because of its neutral character. In writing the Volkov solutions (eqs. (6) and (7)), the terms involving A2 _{have been omitted since we have}

restricted ourselves to first order in e0.

Also cH

0 and c( pp)0 in eq. (5), the laser-dressed ground-state wave functions of the

target hydrogen atom and the protonium atom embeded in the laser field, are obtained in the AQp gauge using the time-dependent first-order perturbation theory [4]

cH

0(r1, t) 4exp [2iW0Ht] ]fH0(r1) 2cos vt fAH0(r1)( ,

(8)

where WH

0 is the eigenenergy of the ground-state hydrogen atom,

fH0(r1) 4 (p)21 O 2exp [2l1r1] (9) with l14 1 and fAH 0(r1) 4 il1e0Q r×1 vHv fH 0(r1) , (10)

where vHis the average excitation energy of the dressed hydrogen atom and r×1is the

unit vector of r1. In a similar way, the wave function c( pp)0 in eq. (5b) has been

constructed where l24 M O 2, M being the proton (antiproton) mass.

Making use of eqs. (5)-(8), the time integration in eq. (4) has been performed following our earlier work [5] with the help of addition theorem of Bessel functions

Q

k

Iwd1 l

m

kiQa0

miv2

n

(I_dH_{1 Id}( pp))

l

. Here Iwd refers to the term without dressing and IdH, Id( pp)represent the terms due to

dressing of hydrogen and protonium atoms, respectively. In eqs. (11) and (12), l is the number of photon exchanged and Jl is the Bessel function of order l.

We choose the direction of the laser field e0 as our polar axis throughout the

calculations. Now, in order to evaluate the integrals Iwd, IdH and Id( pp) in eq. (12),

consider the following mother integrals:

I04



d3r1d3r2exp [2ix1Q r1] exp [2ix2Q r2] exp [2l 1r1] r1 exp [2l2r12] r12 , (13)

I14



d3r1d3r2exp [2ix1Q r1] exp [2ix2Q r2] exp [2l 1r1] r1 exp [2l2r12] r12 1 r1 , (14)

I24



d3r1d3r2exp [2ix1Q r1] exp [2ix2Q r2]

exp [2l1r1] r1 exp [2l2r12] r12 1 r2 , (15) with x14 kfO 2 2 [M O (M 1 1 ) ] ki and x24 kfO 2 1 ki.

The space integrations in the above equations have been evaluated using the Fourier transform techniques and the properties of Dirac delta function and we obtain I04 16 p2 [Nx11x2N21 l21][x221 l22] , (16) I14 16 p2 Nx11x2N [x221 l22] tan21

m

Nx11x2N l1

n

, (17) I24 8 p2(b22 ag)21 O 2ln

y

b 1 (b2 2 ag)1 O2 b 2 (b22 ag)1 O2

z

, (18) where b 4l2[Nx11x2N21 l21] 1l1[x221 l22] , (19) ag 4 [Nx11x2N21 l21][x211 (l11 l2)2][x221 l22] . (20)

parametric differentiations of I0, I1 and I2. Thus we have Iwd4 ˇ ˇl1 ˇ ˇl2 I21 ˇ ˇl2 I0, (21) IdH4 l1e0 vHv ˇ ˇx1 z ˇ ˇl2 (I22 I1) , (22) I_d( pp)_{4 2} l2e0 v( pp)v

k

ˇ ˇx1 z 2 ˇ ˇx2 z

l

g

ˇ ˇl1 I21 I0

h

. (23)

Now, in view of eqs. (12) and (21)-(23), the amplitude for protonium formation with the transfer of l photons between the laser field and the scattering system may be written as (24) A_l( pp)₄ mf 2 p2

g

M 2

h

3 O2 (21)l_J l

g

kiQa0 miv2

hk

Iwd1 l

g

kiQa0 miv2

h

21 ]IdH1 I ( pp) d (

l

.

The differential cross-section is then given by dsl dV 4 kf ki mi mf NAlN 2 . (25)

3. – Results and discussions

We have studied the laser-assisted protonium formation differential cross-sections for field direction parallel to the incident momentum, i.e. e0V ki considering the

dressing effects of the hydrogen and the protonium atoms to first order. Results have been computed for no-photon and one-photon exchange (l 40, 61) with laser frequency v 40.043 a.u. for both the field strengths e04 0 .002 a.u. and e04 0 .02 a.u. at

50, 100 and 200 keV incident energies.

Figure 2 displays the differential cross-sections for protonium formation in laser-assisted antiproton–hydrogen-atom collisions as functions of the scattering angle (u) for 50 keV incident energy with no-photon exchange (l 40), one-photon emission (l 411) and one-photon absorption (l421) for both the field strengths e04

0 .002 a.u.

(

fig. 2(a)

)

and e040 .02 a.u.

(

fig. 2(b)

)

together with the corresponding

field-free results. Figures 3 and 4 are the same as fig. 2 but for 100 and 200 keV energies, respectively. The qualitative behaviour of each of the four curves in any figure is more or less the same. Each of the curves falls smothly from a maximum followed by a sharp fall and a dip minimum at u C907. The remaining portion of each curve (uC90–1807) is almost the mirror image of the first part. The interference of the two interactions is possibly responsible for this kind of nature of the curves. It has been observed that the cross-section values in the extreme backward region are slightly greater than the corresponding results in the extreme forward region. It is interesting to note that in fig. 2(a) the curve corresponding to l 421 lies above the field-free curve throughout the entire angular region considered. This may be attributed to the effect of dressing which is, in this particular case, strong enough to enhance the undressed cross-sections quite significantly. Thereby the sum total of the differential cross-sections

Fig. 2. – (a) The first Born differential cross-sections (a2

0sr21) for protonium formation in the ground state in the laser-assisted p 1H collisions with e0Vki for no-photon exchange (l 40) (– P P – P P –), one-photon absorption (l421) (——) and one-photon emission (l411) (– P – P – P –) at 50 keV incident energy with e04 0 .002 a.u. and v 4 0 .043 a.u. along with the corresponding field-free cross-sections (- - - - -). (b) Same as (a) but for e04 0 .02 a.u.

corresponding to different l values is greater than the corresponding field-free cross-section results and thus the so-called sum rule does not hold good in this case. The same feature is noted in fig. 3(a) for e04 0 . 002 a.u. and 100 keV incident energy

Fig. 3. – (a) Same as fig. 2(a) but for 100 keV incident energy. (b) Same as (a) but for e04 0 .02 a.u.

Fig. 4. – (a) Same as fig. 2(a) but for 200 keV incident energy. (b) Same as (a) but for e04 0 .02 a.u.

though the relative separation between the curve corresponding to l 421 and the field-free curve reduces. But in fig. 4(a) which is for e04 0 .002 a.u. and 200 keV energy

the curves corresponding to l 40, 61 are all below the corresponding field-free curve and this pattern in also observed in fig. 2(b), 3(b) and 4(b) which are for the field strength, e04 0 .02 a.u.

In fig. 5, the variation of the protonium formation cross-sections corresponding to one-photon absorption (l 421) with incident energy has been shown for e04

0 .002 a.u., v 40.043 a.u. and u4107. It has been noted that up to 159 keV incident energy the laser-modified cross-sections are greater than the corresponding field-free cross-section results, while for the energy range 160–250 keV the situation is reversed.

Thus, in case of one-photon absorption (l 421) it may be concluded from the presented results that for relatively weaker field strength as the incident energy decreases the probability of formation of protonium in the presence of a laser field becomes higher than the corresponding field-free probability.

Moreover, a comparative study of the curves in fig. 2, 3 and 4 shows that the absorption and the emission curves corresponding to l 421 and l411, respectively, are always above the no-photon exchange (l 40) curve irrespective of the field strength and incident energy. It demonstrates the effects of dressing since there is no dressing in case of l 40. The relative separation between the curves for l40, 61 becomes larger with increasing field strength and incident energy. One interesting characteristic has been noticed in comparing the differential cross-section curves in fig.

Fig. 5. – Protonium formation differential cross-sections (a2

0sr21) as functions of the incident energy (keV) for l 421, u4107, e04 0 .002 a.u. and v 4 0 .043 a.u.

2, 3 and 4. It is well known that in heavy-particle collision, the major contribution usually comes from below the angular region of one mrad. But here, contributions from almost the entire angular region ( 0 7–1807) except the region near 907 are important for the calculation of cross-section. This kind of behaviour is normally seen in light particle scattering.

Since our calculation for protonium formation in laser-assisted antiproton– hydrogen-atom collision is the first attempt, no direct comparison is possible. However, as a check, we have reproduced the field-free protonium formation differential cross-sections of Deb and Roy [3].

4. – Conclusions

In the first Born approximation, we have studied the laser-assisted rearrangement process in which protonium is formed in the ground state in antiproton–hydrogen-atom collision. Two interesting features are noted in the protonium formation differential cross-sections. Firstly, the cross-sections corresponding to absorption of one-photon for lower incident energy and weaker field strength are greater than the corresponding field-free cross-section values thereby disobeying the sum rule. This demonstrates the importance of the dressing effects. Secondly, contributions from almost the entire angular range ( 0 –180 7) except near u4907 are important for the calculation of the cross-sections which is a familiar characteristic in light particle scattering.

Finally, it may be mentioned that the FBA is a rough approximation and the cross-section values obtained for the present investigation using this approximation are rather small. But the qualitative understanding of the main features of the process in which we are interested is a primary need. As a first attempt we have tried to achieve that. It seems that extension of the investigation for lower incident energies and for

higher excited states of the protonium will be of interest and in what follows will be done in future work.

* * *

The author gratefully acknowledges many fruitful discussions with Prof. S. SARKAR. He is also thankful to Prof. N. C. SILfor careful reading of the manuscript. The author

would like to thank the Council of Scientific and Industrial Research, India, for providing partial financial support.

R E F E R E N C E S

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[3] DEB N. C. and ROY P., Ind. J. Phys. B, 68 (1994) 245; 69 (1995) 443.

[4] SHU-MINLI, ZI-FANGZHOU, JIAN-GEZHOUand YAO-YANGLIU, Phys. Rev. A, 47 (1993) 4960. [5] BHATTACHARYA M., SINHA C. and SILN. C., Phys. Rev. A, 40 (1989) 567.