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Screening effects in electron–positron pair production with capture in ultrarelativistic collisions
15 May 2000
Physics Letters A 269 Ž2000. 325–332 www.elsevier.nlrlocaterpla
Screening effects in electron–positron pair production with capture in ultrarelativistic collisions A.B. Voitkiv 1, N. Grun ¨ ) , W. Scheid Institut fur ¨ Theoretische Physik der UniÕersitat ¨ Giessen, Heinrich-Buff-Ring 16, Giessen, Germany Received 23 February 2000; received in revised form 12 April 2000; accepted 12 April 2000 Communicated by B. Fricke
Abstract We study the influence of the shielding of the atomic nucleus by atomic electrons on positron–electron pair production with capture in ultrarelativistic nucleus-atom collisions. The importance of the shielding is shown to increase with the collision energy and with the atomic number of the target atom. We report calculations of cross sections for the pair production with capture in collisions of 160 GeVrnucleon Pb 82q projectiles with different atomic targets ranging from Be to Au. Depending on the atomic number of the target the shielding is shown to reduce the cross sections by 2.5–14 percent at this collision energy. q 2000 Elsevier Science B.V. All rights reserved. PACS: 34.10.q x; 34.50.-s; 34.50.Fa
An investigation of the process of pair production in collisions between charged particles is of interest, both from the point of view of checking of quantum electrodynamics at small distances and for the determination of the interaction between fast projectiles and matter. Perturbative calculations of cross sections for the production of free electron–positron pairs in relativistic heavy ion collisions have a long history w1–6x. The development of heavy particle accelera-
)
Corresponding author. Fax: q641-99-33339. E-mail address: [email protected] ŽN. Grun ¨ .. 1 Permanent address: Arifov Institute of Electronics, 700143 Tashkent, Uzbekistan.
tors in the recent twenty years has revived the interest to the topic and numerous calculations for pair production in relativistic nucleus-nucleus collisions were done during the past two decades using both perturbative and non-perturbative methods Žfor the review and references therein, see Refs. w7–9x.. In almost all these papers both colliding nuclei were considered as bare ones, without any electrons. However, in some experimental situations one of the colliding partners can be a neutral atom. Then the screening effect should, in principle, be taken into account. Recently the screening effects in free electron–positron pair production were studied in Refs. w7;9x. The conclusions made by these authors are quite opposite. Bertulani and Baur w7x claimed that for collisions with neutral target atoms the screening
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 7 9 - 6
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A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
effect is very important for lower energies of the projectile nuclei and decreases in importance for higher energies. They had also found that when the screening is present the cross section for free electron–positron pair production will always be smaller by at least a factor of 1.5–2 also for very high projectile energies Žg F 10 5, g is the Lorentz factor for the projectile.. In contrast, Wu et al. w9x have concluded that the screening effect becomes more important when the projectile energy increases. They have found that the screening effect becomes of considerable importance only at extremely high collision energies. For example, it follows from their calculations that in Au79qq Au0 collisions the screening reduces the cross section for free pair production by modest 4.5 percent at a collision energy of E s 200 GeVrnucleon and by 31.4 percent at a collision energy of E s 200 TeVrnucleon. In the present Letter we consider the screening effects in the bound–free pair production when the electron created in the collision is captured in a bound state of the projectile. The goal of the present Letter is to estimate the influence of the screening on the bound–free pair production rather than to give very precise calculations for this process. While some non-perturbative approaches to treat the process are available at present Žsee w8;10x., here we focus on the screening effect and describe the influence of the target atom on the pair production process in the first order of the perturbation theory. The field of the projectile will be treated exactly. It will be seen below that the screening effect becomes important for very energetic collisions where g is very large. For such collisions one may expect that the difference between results of the first order treatment and of more refined theories is already not very large w10x. In addition, the screening effect arises mainly from relatively large impact parameters where the perturbative and non-perturbative approaches should give close results. The most complete set of perturbative results for the process of bound–free pair production was presented, to our knowledge, in the paper of Anholt and Becker w11x where the cross sections for bound–free pair production were given for a variety of projectile-target pairs for collisions with energies corresponding to g F 1000. However, because of large momentum transfers needed to create a positron–
electron pair, the screening effects were assumed in w11x to give negligible changes to the cross sections in this region of g . Here, we first consider collisions with an atom which does not change its internal state in the collision. The effect of electron transitions in the atom known as the antiscreening effect will be included in an approximate way by multiplying the resulting w11x. It is convecross section by a factor 1 q Zy1 2 nient to consider the collision in the rest frame of the ion. The nucleus of the ion with charge Z1 is at rest and is taken as the origin. The relative motion of the neutral atom is treated classically. The nucleus of the incident neutral atom with atomic number Z2 moves on a straight-line trajectory R Ž t . s b q zt, where b is the impact parameter and z s Ž0,0,Õ . the collision velocity. This nucleus is ‘dressed’ by the electrons, which are ‘frozen’ with respect to the nucleus. The coordinates of the electrons with respect to the origin are r j s R Ž t . q j j , where j j are the coordinates of the electrons with respect to the moving nucleus of the atom Žas seen in the rest frame of the ion.. The atom creates the field, described by scalar and vector potentials which obey the Maxwell equations. We first consider the atom as a beam consisting of the nucleus and point-like electrons which move with the same velocity Õ. The quantum nature of the electrons, ‘frozen’ in the ground state of the incident atom, will finally be taken into account in a standard way by averaging the obtained potentials over the quantum electron density in the ground state of the incident atom. In this Letter we use the Coulomb gauge in which the Maxwell equations read
DF Ž r ,t . s y4pr Ž r ,t . , D A Ž r ,t . y 4p sy c
1 E 2A Ž r ,t . c2
E t2
J Ž r ,t . q
1 c
EF =
Et
.
Ž 1.
The charge and the current densities of the incident atom are given by
r Ž r ,t . s Z2 d Ž r y R Ž t . . y Ý d Ž r y R Ž t . y j j . , j
J Ž r ,t . s r Ž r ,t . z ,
Ž 2.
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
where the sum runs over all atomic electrons. In order to solve Ž1. it is convenient to use the four-dimensional Fourier transformation:
F Ž r ,t . s
1
Similarly one finds for the Fourier transform of the vector potential G Ž k, v . 2 d Ž v y kz . exp Ž yi kb .
q` 2
4p
d3 k
H Hy`
327
d v FŽ k,v .
GŽ k,v . s k2y
v
1 4p
ž
=exp Ž i Ž k P r y v t . . .
Ž 3.
Using the representation of the d-function by a Fourier integral one gets
r Ž r ,t . s
1
yz
/
j
3
ž
= Z2 y Ý exp Ž yi k P j j . j
/
1
3
2p 2
Hd k exp Ž i k P Ž r y b y zt . . kv
zy 1
=
q` 3
Hd k Hy` d vd Ž v y kz .
8p 3
2
k y
k2 v2 c
ž
ž
/
= Z2 y Ý exp Ž yi k P j j . . j
j
2
Ž 8. Ž 4.
Inserting the first equations of Ž3. and Ž4. into the first equation of Ž1., we find the Fourier transform for the scalar potential F Ž k, v . 2 d Ž v y kz . exp Ž yi kb . k2
ž
= Z2 y Ý exp Ž yi k P j j . j
/
Ž 5.
and obtain the following integral representation for the scalar potential: 1 2p
ž
3
2
Hd k
2
Z2 y Ý exp Ž yi k P j j . j
™Z
2eff
Ž k. .
Ž 9.
The quantity Z2eff Ž k . is the effective charge of the atom and is proportional to the elastic form-factor of the atom
Ž 10 .
where the elastic form-factor reads Žsee e.g. w12x.
/
= Z2 y Ý exp Ž yi k P j j . . j
In order to transform the electromagnetic potentials, created by the incident beam of the equivelocity point-like particles, to the potentials, created by the moving atom being in the ground state, the potentials Ž6. and Ž8. should be averaged over the atomic electron charge density Žas seen in the rest frame of the ion.. The resulting electromagnetic potentials are given by the expressions Ž6. and Ž8. with the replacement
Z2eff Ž k . s Z2 fel Ž k . ,
exp Ž i k P Ž r y b y zt . . k
/
Z2 y Ý exp Ž yi k P j j . .
=exp Ž i Ž k P Ž r y b . y v t . .
F Ž r ,t . s
Ž 7.
Hd kexp Ž i k P Ž r y b y zt . .
8p 3
FŽ k,v . s
/
2
It fo llo w s fro m Ž 7 . th a t k P G ; k P Ž k vrk 2 y z . d Ž v y kz . ' 0 as it should be in the ‘transverse’ gauge where = P A s 0. For the vector potential we have the following integral representation
A Ž r ,t . s
s
k2
= Z2 y Ý exp Ž yi k P j j . .
q` 3
Hd k Hy` d v G Ž k , v .
2
kv
c2
=exp Ž i Ž k P r y v t . . , A Ž r ,t . s
ž
Ž 6.
fel Ž k . s 1 y
1 Z2
Hd j r
el
Ž j . exp Ž yi k P j . .
Ž 11 .
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
328
Here rel Ž j . is the charge density of the electrons in the incident atom. It is important to note that rel Ž j . in Eq. Ž11. is the density as seen in the rest frame of the ion. Within the first order of the perturbation theory in the interaction with the atom, the amplitude for the production of an electron in the ground state c 0 of the ion and a positron in a continuum state cp of the ion reads
field. In Eq. Ž13. the vector Q, 0 F Q - `, is a two-dimensional vector which is perpendicular to the collision velocity, Q P z s 0. Assuming that the positron states are normalized on the ‘energy’ scale, the cross section for the bound–free pair production, with electron capture into the ground state, is given by
Ý Hd 2 bHd EpHd V p N a p ,0 N 2 ,
s Ž 1s . s
Ž 14 .
sp , se
a p ,0 s
i 2p
q`
2
=
1
where the sums run over the spin quantum numbers s e and s p of the electron and positron, respectively. The two-dimensional integration over the impact parameter in Ž14. is performed with the help of the relation
Hy` d texp Ži v 0 , p t . Hd3 k
² c 0 Ž r . Nexp Ž i kPr . N c p Ž r . : k2
ž
² c 0 Ž r . Nexp Ž i kPr . a P zy
q c
k2 y
kP Ž kPz . k
2
N cp Ž r . :
Ž kPz . 2 c
2
=Z2eff Ž k . exp Žyi kP Ž bq zt .. .
0
iZ2
pÕ
=
1
Hd2 Q ² c 0 Ž r . Nexp Ž i kPr . N c p Ž r . : k2
ž
² c 0 Ž r . Nexp Ž i kPr . a P zy
q c
=fel Ž k . exp Žyi kPb . .
k2 y
v 02 , p c2
Hd bexp Ž i Ž Q y Q . P b . s Ž 2p .
k v0, p k2
/
N cp Ž r . :
0 Ž 13 .
Now k s Ž Q,k I . where k I s v 0, prÕ is the minimum momentum transfer to the electron–positron
2
d Ž Q y QX . .
Ž 15 . Then one obtains Ž fel Ž k . is assumed as real.
Ž 12 .
Here, v 0, p s E0 q Ep is the transition frequency with E0 being the electron energy in the ground state c 0 and with Ep the positron energy in the continuum state cp . In Eq. Ž12. and below the indices 0 and p denote all quantum numbers of the corresponding states, including spin. Integration over time in Ž12. results in the factor 2pd Ž kz y v 0, p . which permits to integrate over the component, k I s kzrÕ, of the momentum transfer k. We obtain a p ,0 s
X
2
/
s Ž 1s . s
4Z22 Õ
2
`
Ý Hd 2 QHd V pHmc d Ep fel2 Ž k . 2
sp , s e
=
² c 0 Ž r . Nexp Ž i kPr . N c p Ž r . : k2
ž
² 1 c 0 Ž r . Nexp Ž i kPr . a P zy q c
k2 y
v 02 , p
k v0, p k2
2
/
N cp Ž r . : .
c2
Ž 16 . Since we are not interested in the angular spectra of positrons and in the polarization of the electron state c 0 , we are allowed to take the vector k as the quantization z-axis in evaluating Ž16.. The operator expŽi k P r . is even with respect to reflection on a plane parallel to k and the operator a P Ž z y k v 0 , p rk 2 . exp Ži k P r . is odd since k P Ž z y k v 0, prk 2 . s 0. Therefore, taking k as the quantization axis, one sees that these operators connect the state c 0 with different positron states, and the corresponding matrix elements in the integrand on the right hand side of Ž16. can be squared separately Žsee also the discussion in w13–15x.. Further, if one assumes that the collision velocity z lies in
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
the xz-plane, then one has a P Ž z y k v 0, prk 2 . s 2 2 Ž . Õa x k 2 y v 0, p rÕ rk and the cross section 16 is rewritten as
(
s Ž 1s . s
4Z22 Õ
`
Ý Hd 2 QHd V pH
2
d Ep fel2 Ž k .
mc 2
sp , se
° N ²c =~
0
Ž r . N exp Ž i k P r . N cp Ž r . : N 2
given in a paper of Salvat et al. w16x. In that paper analytical Dirac–Hartree–Fock–Slater screening functions were given for neutral atoms with atomic numbers Z2 s 1 y 92. Using the results of the paper of Salvat et al. and taking into account that the atom moves with the relativistic velocity z in the rest frame of the ion, the form-factor Ž11. can be written as fel Ž k .
k4
¢
Õ q
2
c k
Q 2
ž
ž
k2y
2
2 v 0, p 2
c
A i k i2
3
s 1y 2
329
Ý
2
k i2 q k 2 y Ž kz . rc 2
is1
3 2
s Ž k 2 y Ž kz . rc 2 .
2
/ ¶ •
= N ² c 0 Ž r . N exp Ž i k P r . a x N cp Ž r . : N 2 .
ß
Ž 17 . Eq. Ž17. represents the cross section for the bound– free pair production as a sum of the so called ‘longitudinal’ Žthe first term in Ž17.. and ‘transversal’ Žthe second term in Ž17.. contributions w15x. The ‘longitudinal’ contribution is due to the action of the unretarded non-relativistic Coulomb potential. The ‘transversal’ contribution is caused by the vector potential and can be regarded as occurring due to the exchange of virtual photons which have transversal polarization. The pair production process occurs mainly through the ‘transversal’ contribution2 . In order to evaluate the cross section Ž17. we need to know the elastic atomic form-factors fel Ž k .. Although there are several analytical representations for the form-factors, the very convenient one is that
Ai
Ý
2 2 is1 k i q k y
.
2 Ž kz . rc 2 Ž 18 .
Here A i and k i are real constants for a given atom which are tabulated for all atomic elements in w16x. For obtaining the second line on the right hand side of Ž18. the condition Ý i A i s 1 Žsee w16x. was used. The ‘retardation term’ yŽ kz . 2rc 2 appears in Eq. Ž18. because of the Lorentz transformation of the atomic electron density from the rest frame of the atom to the rest frame of the ion. Inserting Ž18. into Ž17. we get:
s Ž 1s . s
4Z22 Õ
`
Ý Ý A i A jHd V pH
2
=
2
Q q
½ž
2
2 v 0, p 2 2
Õg
Q q
q k i2
2 v 0, p
g 2 Õ2
/ž
2
Q q
2 v 0, p
Õ 2g 2
q k j2
/
2
/
N ² c 0 Ž r . N exp Ž i k P r . N cp Ž r . : N 2 k4 Õ2 Q2
q
H
1
ž =
d Ep d 2 Q
mc 2
se , s p i , j
=
2
We note that sometimes the ‘longitudinal’ and ‘transversal’ contributions are regarded as the ‘electric’ and ‘magnetic’ ones. However, this is not quite correct because in relativistic collisions an important part of the electric field of the projectile is given by the time derivative of the vector potential. Some additional confusion appears when the ‘longitudinal’ and ‘transversal’ parts of the transition amplitude, obtained in the Coulomb gauge, are regarded as parts arising from the scalar and vector potentials, respectively, in the Lorentz gauge.
/
c2 k 2
N ² c 0 Ž r . N exp Ž i k P r . a x N cp Ž r . : N 2
5
Ž 19 . with g s Ž1 y Õ 2rc 2 .y1 r2 .
330
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
If we set all k i equal to zero in Ž19., we recover the cross section for the bound–free pair production in collisions between two bare nuclei w17x. In collisions between two bare nuclei at asymptotically high values of g , the main contribution to the cross section is given by the range of relatively small Q, 0 F Q Q v 0, prÕg . This results in the logarithmic growth of the cross section with g : s Ž1s. ; lng Žsee w8;11x.. In collisions with neutral atoms, where the screening constants k i are different from zero, both the ‘longitudinal’ and ‘transversal’ parts of the cross section are reduced in magnitude compared to collisions with bare atomic nuclei. This reduction is relatively more important for the ‘transversal’ part, which contributes most to the cross section for pair production. From Eq. Ž19. it is easy to see that the effect of the screening depends on the ratio between the screening constants k i and v 0, prŽg Õ .. Although the minimum momentum transfer v 0, prÕ is very large on the atomic energy scale, the quantity v 0, prŽg Õ . can be smaller than k i at high enough g . In such a case the screening will be very important and will considerably reduce the cross section for bound–free pair production. For very high g this can be true even for collisions with light atoms, where all screening constants are of the order of a few atomic units. In an experiment of Belkacem et al. w18x cross section for bound–free pair production in collisions of 10 GeVrnucleon Au79q projectiles with Al, Cu, Ag and Au atomic targets were measured. Our calculations for the free-bound pair production show that the screening effect is still very weak at this collision energy Žonly up to 2 percent for the heaviest target.. Fig. 1 shows a comparison between the experimental data of Krause et al. w19x and theoretical results. In this experiment cross sections were measured for total electron capture by 160 GeVrnucleon Pb 82q projectiles colliding with different solid state targets ranging from Be to Au. The bound–free pair production cross sections were extracted from the measured data. The theoretical results include the perturbative calculations of the present Letter and those of Anholt and Becker w11x, where the screening effects were not taken into account. We give two sets of our results for the cross section which were obtained with and without including the screening.
Fig. 1. Cross sections for bound–free pair production in collisions of 160-GeVrnucleon Pb 82q with solid state targets as a function of the target atomic number. Open squares: experimental data from w19x, open and solid diamonds: nonperturbative results cited in w19x, dotted curve: calculations of Anholt and Becker w11x, dashed curve: our calculations without the screening, solid curve: our calculations with the screening included. See also comments in the text.
Both results of Anholt and Becker w11x and ours include a multiplication of the cross section for the capture to the ground state by a factor of 1.2 which takes into account the possibility for the electron to be captured into excited states of the ion and, also, a multiplication by a factor 1 q Zy1 which accounts 2 for the bound–free pair production due to the antiscreening effect of the atomic electrons w11x. We also show in this figure two non-perturbative results for the capture to the ground state of a lead ion Ž s Ž1s. s 46 and 50 b. in collisions between bare Pb and Au nuclei. These results were cited in the paper of Krause et al. w19x as private communications. In order to avoid partial wave expansion for the exact Dirac continuum states, which is slowly convergent at high g values, we use the semi-relativistic
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
Darwin wavefunction for the ground state of the electron and Sommerfeld–Maue wavefunctions for positron continuum states in our calculations. Our unscreened numerical results are close to those of Anholt and Becker w11x. With the screening included the cross section is reduced by 2.5 percent for collisions with Be and C targets. The screening effect increases with the atomic number of the target and reduces the cross section already by 14 percent for collisions with the Au target. Thus, at this collision energy the screening can already be of noticeable importance if the atomic number of the target is not much less than that of the projectile. In Fig. 2 we present energy spectra of positrons created in the process of bound–free pair production in collisions of 160 GeVrnucleon Pb 82q ions with Au79q and Au0 . As it can be expected the screening effect influences mainly the lower energy part of the positron spectrum.
It follows from Fig. 1 that the inclusion of the screening effects does not lead to closer agreement between the perturbative calculations and the experimental results. It was claimed in w19x that the overall uncertainty in the experimental data is about "10 percent. In collisions with light targets, like Be and C, where the perturbative theory is expected to be valid, the perturbative calculations yield cross sections which are smaller than the experimental ones by about of 20 percent. In collisions with heavy targets, like Sn and Au, where noticeable deviations from the perturbative treatment could be expected, the nonscreened perturbative results seem to be, at the first glance, in agreement with the experiment. However, as it was pointed out in w19x, in collisions in solid targets the contributions to measured capture cross sections from excited ion states are small for the heaviest targets. Correspondingly, these authors estimated that, in order to account for the possibility for the electron to be captured into excited states of Pb 81q in collisions with a solid state gold target, the cross section s Ž1s. should be multiplied by a factor of f 1.06 instead of 1.2 3. If one takes this into account, then one sees that the perturbative results underestimate the experimental ones by more than 10 percent. On the other hand, if one assumes that, in order to compare the nonperturbative results s Ž1s. s 46 and 50 b with the experiment w19x, these nonperturbative results should also be multiplied by 1.06, then one sees that the nonperturbative results overestimate the experimental ones by more than 10 percent. Since the screening effects reduce the cross sections one can argue that including the screening effect into nonperturbative calculations could shift the non-perturbative results to closer agreement with the experiment. We have also estimated cross sections for bound–free pair production in collisions of 1000 GeVrnucleon Pb 82q projectiles with Be and Au targets. For this collision energy, neglecting the screening effect, we found s Ž1s. s 0.18 and 56b for Be and Au targets, respectively. Including the
3
Fig. 2. Spectra of positrons created in the process of bound–free pair production in collisions of 160-GeVrnucleon Pb 82q with Au79q Ždashed line. and Au0 Žsolid line. targets.
331
Surprisingly, despite their own statement the authors of w19x still compared their experimental data with results of Anholt and Becker for 1.2= s Ž1s. for all, including the heaviest targets Žsee Fig. 3a of w19x..
332
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
screening effect these cross sections are reduced to s Ž1s. s 0.15 and 37b, respectively. Thus, at this collision energy the screening reduces the cross section by 16% for Be targets and by 33% for Au targets. Our calculations show considerably larger screening effects in the bound–free pair production than it was found in w9x for free pair production. For example, Wu et al. w9x obtained in their perturbative calculations that in collisions of 200 GeVrnucleon Au79q projectiles with Au target the screening effect reduces the cross section for free pair production just by 4.5 percent. Our calculations for the bound–free pair production in the same collision system show that the screening reduces the bound–free pair production by about 16 percent. The reason for such a considerable increase of the screening effect for the bound–free pair production compared to the case of the free pair production can be attributed to the fact that the process of free pair production involves larger momentum transfers because both positron and electron are in continuum states. In conclusion, we have estimated the screening effects in the process of bound–free pair production in collisions between a point-like ion and a neutral atom. We have shown that the screening becomes more important for collisions with heavy atoms and when the collision energy increases. We have found that, in contrast to previous statements w11x, the screening effects can be of importance for collisions with g ; 100–1000. Comparing the results of our calculations with those for the free pair production w9x one may conclude that the screening is more important for the bound–free pair production.
Acknowledgements A.B.V. acknowledges with thanks the support from the Alexander von Humboldt Stiftung.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x
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Physics Letters A 269 Ž2000. 325–332 www.elsevier.nlrlocaterpla
Screening effects in electron–positron pair production with capture in ultrarelativistic collisions A.B. Voitkiv 1, N. Grun ¨ ) , W. Scheid Institut fur ¨ Theoretische Physik der UniÕersitat ¨ Giessen, Heinrich-Buff-Ring 16, Giessen, Germany Received 23 February 2000; received in revised form 12 April 2000; accepted 12 April 2000 Communicated by B. Fricke
Abstract We study the influence of the shielding of the atomic nucleus by atomic electrons on positron–electron pair production with capture in ultrarelativistic nucleus-atom collisions. The importance of the shielding is shown to increase with the collision energy and with the atomic number of the target atom. We report calculations of cross sections for the pair production with capture in collisions of 160 GeVrnucleon Pb 82q projectiles with different atomic targets ranging from Be to Au. Depending on the atomic number of the target the shielding is shown to reduce the cross sections by 2.5–14 percent at this collision energy. q 2000 Elsevier Science B.V. All rights reserved. PACS: 34.10.q x; 34.50.-s; 34.50.Fa
An investigation of the process of pair production in collisions between charged particles is of interest, both from the point of view of checking of quantum electrodynamics at small distances and for the determination of the interaction between fast projectiles and matter. Perturbative calculations of cross sections for the production of free electron–positron pairs in relativistic heavy ion collisions have a long history w1–6x. The development of heavy particle accelera-
)
Corresponding author. Fax: q641-99-33339. E-mail address: [email protected] ŽN. Grun ¨ .. 1 Permanent address: Arifov Institute of Electronics, 700143 Tashkent, Uzbekistan.
tors in the recent twenty years has revived the interest to the topic and numerous calculations for pair production in relativistic nucleus-nucleus collisions were done during the past two decades using both perturbative and non-perturbative methods Žfor the review and references therein, see Refs. w7–9x.. In almost all these papers both colliding nuclei were considered as bare ones, without any electrons. However, in some experimental situations one of the colliding partners can be a neutral atom. Then the screening effect should, in principle, be taken into account. Recently the screening effects in free electron–positron pair production were studied in Refs. w7;9x. The conclusions made by these authors are quite opposite. Bertulani and Baur w7x claimed that for collisions with neutral target atoms the screening
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 7 9 - 6
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A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
effect is very important for lower energies of the projectile nuclei and decreases in importance for higher energies. They had also found that when the screening is present the cross section for free electron–positron pair production will always be smaller by at least a factor of 1.5–2 also for very high projectile energies Žg F 10 5, g is the Lorentz factor for the projectile.. In contrast, Wu et al. w9x have concluded that the screening effect becomes more important when the projectile energy increases. They have found that the screening effect becomes of considerable importance only at extremely high collision energies. For example, it follows from their calculations that in Au79qq Au0 collisions the screening reduces the cross section for free pair production by modest 4.5 percent at a collision energy of E s 200 GeVrnucleon and by 31.4 percent at a collision energy of E s 200 TeVrnucleon. In the present Letter we consider the screening effects in the bound–free pair production when the electron created in the collision is captured in a bound state of the projectile. The goal of the present Letter is to estimate the influence of the screening on the bound–free pair production rather than to give very precise calculations for this process. While some non-perturbative approaches to treat the process are available at present Žsee w8;10x., here we focus on the screening effect and describe the influence of the target atom on the pair production process in the first order of the perturbation theory. The field of the projectile will be treated exactly. It will be seen below that the screening effect becomes important for very energetic collisions where g is very large. For such collisions one may expect that the difference between results of the first order treatment and of more refined theories is already not very large w10x. In addition, the screening effect arises mainly from relatively large impact parameters where the perturbative and non-perturbative approaches should give close results. The most complete set of perturbative results for the process of bound–free pair production was presented, to our knowledge, in the paper of Anholt and Becker w11x where the cross sections for bound–free pair production were given for a variety of projectile-target pairs for collisions with energies corresponding to g F 1000. However, because of large momentum transfers needed to create a positron–
electron pair, the screening effects were assumed in w11x to give negligible changes to the cross sections in this region of g . Here, we first consider collisions with an atom which does not change its internal state in the collision. The effect of electron transitions in the atom known as the antiscreening effect will be included in an approximate way by multiplying the resulting w11x. It is convecross section by a factor 1 q Zy1 2 nient to consider the collision in the rest frame of the ion. The nucleus of the ion with charge Z1 is at rest and is taken as the origin. The relative motion of the neutral atom is treated classically. The nucleus of the incident neutral atom with atomic number Z2 moves on a straight-line trajectory R Ž t . s b q zt, where b is the impact parameter and z s Ž0,0,Õ . the collision velocity. This nucleus is ‘dressed’ by the electrons, which are ‘frozen’ with respect to the nucleus. The coordinates of the electrons with respect to the origin are r j s R Ž t . q j j , where j j are the coordinates of the electrons with respect to the moving nucleus of the atom Žas seen in the rest frame of the ion.. The atom creates the field, described by scalar and vector potentials which obey the Maxwell equations. We first consider the atom as a beam consisting of the nucleus and point-like electrons which move with the same velocity Õ. The quantum nature of the electrons, ‘frozen’ in the ground state of the incident atom, will finally be taken into account in a standard way by averaging the obtained potentials over the quantum electron density in the ground state of the incident atom. In this Letter we use the Coulomb gauge in which the Maxwell equations read
DF Ž r ,t . s y4pr Ž r ,t . , D A Ž r ,t . y 4p sy c
1 E 2A Ž r ,t . c2
E t2
J Ž r ,t . q
1 c
EF =
Et
.
Ž 1.
The charge and the current densities of the incident atom are given by
r Ž r ,t . s Z2 d Ž r y R Ž t . . y Ý d Ž r y R Ž t . y j j . , j
J Ž r ,t . s r Ž r ,t . z ,
Ž 2.
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
where the sum runs over all atomic electrons. In order to solve Ž1. it is convenient to use the four-dimensional Fourier transformation:
F Ž r ,t . s
1
Similarly one finds for the Fourier transform of the vector potential G Ž k, v . 2 d Ž v y kz . exp Ž yi kb .
q` 2
4p
d3 k
H Hy`
327
d v FŽ k,v .
GŽ k,v . s k2y
v
1 4p
ž
=exp Ž i Ž k P r y v t . . .
Ž 3.
Using the representation of the d-function by a Fourier integral one gets
r Ž r ,t . s
1
yz
/
j
3
ž
= Z2 y Ý exp Ž yi k P j j . j
/
1
3
2p 2
Hd k exp Ž i k P Ž r y b y zt . . kv
zy 1
=
q` 3
Hd k Hy` d vd Ž v y kz .
8p 3
2
k y
k2 v2 c
ž
ž
/
= Z2 y Ý exp Ž yi k P j j . . j
j
2
Ž 8. Ž 4.
Inserting the first equations of Ž3. and Ž4. into the first equation of Ž1., we find the Fourier transform for the scalar potential F Ž k, v . 2 d Ž v y kz . exp Ž yi kb . k2
ž
= Z2 y Ý exp Ž yi k P j j . j
/
Ž 5.
and obtain the following integral representation for the scalar potential: 1 2p
ž
3
2
Hd k
2
Z2 y Ý exp Ž yi k P j j . j
™Z
2eff
Ž k. .
Ž 9.
The quantity Z2eff Ž k . is the effective charge of the atom and is proportional to the elastic form-factor of the atom
Ž 10 .
where the elastic form-factor reads Žsee e.g. w12x.
/
= Z2 y Ý exp Ž yi k P j j . . j
In order to transform the electromagnetic potentials, created by the incident beam of the equivelocity point-like particles, to the potentials, created by the moving atom being in the ground state, the potentials Ž6. and Ž8. should be averaged over the atomic electron charge density Žas seen in the rest frame of the ion.. The resulting electromagnetic potentials are given by the expressions Ž6. and Ž8. with the replacement
Z2eff Ž k . s Z2 fel Ž k . ,
exp Ž i k P Ž r y b y zt . . k
/
Z2 y Ý exp Ž yi k P j j . .
=exp Ž i Ž k P Ž r y b . y v t . .
F Ž r ,t . s
Ž 7.
Hd kexp Ž i k P Ž r y b y zt . .
8p 3
FŽ k,v . s
/
2
It fo llo w s fro m Ž 7 . th a t k P G ; k P Ž k vrk 2 y z . d Ž v y kz . ' 0 as it should be in the ‘transverse’ gauge where = P A s 0. For the vector potential we have the following integral representation
A Ž r ,t . s
s
k2
= Z2 y Ý exp Ž yi k P j j . .
q` 3
Hd k Hy` d v G Ž k , v .
2
kv
c2
=exp Ž i Ž k P r y v t . . , A Ž r ,t . s
ž
Ž 6.
fel Ž k . s 1 y
1 Z2
Hd j r
el
Ž j . exp Ž yi k P j . .
Ž 11 .
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
328
Here rel Ž j . is the charge density of the electrons in the incident atom. It is important to note that rel Ž j . in Eq. Ž11. is the density as seen in the rest frame of the ion. Within the first order of the perturbation theory in the interaction with the atom, the amplitude for the production of an electron in the ground state c 0 of the ion and a positron in a continuum state cp of the ion reads
field. In Eq. Ž13. the vector Q, 0 F Q - `, is a two-dimensional vector which is perpendicular to the collision velocity, Q P z s 0. Assuming that the positron states are normalized on the ‘energy’ scale, the cross section for the bound–free pair production, with electron capture into the ground state, is given by
Ý Hd 2 bHd EpHd V p N a p ,0 N 2 ,
s Ž 1s . s
Ž 14 .
sp , se
a p ,0 s
i 2p
q`
2
=
1
where the sums run over the spin quantum numbers s e and s p of the electron and positron, respectively. The two-dimensional integration over the impact parameter in Ž14. is performed with the help of the relation
Hy` d texp Ži v 0 , p t . Hd3 k
² c 0 Ž r . Nexp Ž i kPr . N c p Ž r . : k2
ž
² c 0 Ž r . Nexp Ž i kPr . a P zy
q c
k2 y
kP Ž kPz . k
2
N cp Ž r . :
Ž kPz . 2 c
2
=Z2eff Ž k . exp Žyi kP Ž bq zt .. .
0
iZ2
pÕ
=
1
Hd2 Q ² c 0 Ž r . Nexp Ž i kPr . N c p Ž r . : k2
ž
² c 0 Ž r . Nexp Ž i kPr . a P zy
q c
=fel Ž k . exp Žyi kPb . .
k2 y
v 02 , p c2
Hd bexp Ž i Ž Q y Q . P b . s Ž 2p .
k v0, p k2
/
N cp Ž r . :
0 Ž 13 .
Now k s Ž Q,k I . where k I s v 0, prÕ is the minimum momentum transfer to the electron–positron
2
d Ž Q y QX . .
Ž 15 . Then one obtains Ž fel Ž k . is assumed as real.
Ž 12 .
Here, v 0, p s E0 q Ep is the transition frequency with E0 being the electron energy in the ground state c 0 and with Ep the positron energy in the continuum state cp . In Eq. Ž12. and below the indices 0 and p denote all quantum numbers of the corresponding states, including spin. Integration over time in Ž12. results in the factor 2pd Ž kz y v 0, p . which permits to integrate over the component, k I s kzrÕ, of the momentum transfer k. We obtain a p ,0 s
X
2
/
s Ž 1s . s
4Z22 Õ
2
`
Ý Hd 2 QHd V pHmc d Ep fel2 Ž k . 2
sp , s e
=
² c 0 Ž r . Nexp Ž i kPr . N c p Ž r . : k2
ž
² 1 c 0 Ž r . Nexp Ž i kPr . a P zy q c
k2 y
v 02 , p
k v0, p k2
2
/
N cp Ž r . : .
c2
Ž 16 . Since we are not interested in the angular spectra of positrons and in the polarization of the electron state c 0 , we are allowed to take the vector k as the quantization z-axis in evaluating Ž16.. The operator expŽi k P r . is even with respect to reflection on a plane parallel to k and the operator a P Ž z y k v 0 , p rk 2 . exp Ži k P r . is odd since k P Ž z y k v 0, prk 2 . s 0. Therefore, taking k as the quantization axis, one sees that these operators connect the state c 0 with different positron states, and the corresponding matrix elements in the integrand on the right hand side of Ž16. can be squared separately Žsee also the discussion in w13–15x.. Further, if one assumes that the collision velocity z lies in
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
the xz-plane, then one has a P Ž z y k v 0, prk 2 . s 2 2 Ž . Õa x k 2 y v 0, p rÕ rk and the cross section 16 is rewritten as
(
s Ž 1s . s
4Z22 Õ
`
Ý Hd 2 QHd V pH
2
d Ep fel2 Ž k .
mc 2
sp , se
° N ²c =~
0
Ž r . N exp Ž i k P r . N cp Ž r . : N 2
given in a paper of Salvat et al. w16x. In that paper analytical Dirac–Hartree–Fock–Slater screening functions were given for neutral atoms with atomic numbers Z2 s 1 y 92. Using the results of the paper of Salvat et al. and taking into account that the atom moves with the relativistic velocity z in the rest frame of the ion, the form-factor Ž11. can be written as fel Ž k .
k4
¢
Õ q
2
c k
Q 2
ž
ž
k2y
2
2 v 0, p 2
c
A i k i2
3
s 1y 2
329
Ý
2
k i2 q k 2 y Ž kz . rc 2
is1
3 2
s Ž k 2 y Ž kz . rc 2 .
2
/ ¶ •
= N ² c 0 Ž r . N exp Ž i k P r . a x N cp Ž r . : N 2 .
ß
Ž 17 . Eq. Ž17. represents the cross section for the bound– free pair production as a sum of the so called ‘longitudinal’ Žthe first term in Ž17.. and ‘transversal’ Žthe second term in Ž17.. contributions w15x. The ‘longitudinal’ contribution is due to the action of the unretarded non-relativistic Coulomb potential. The ‘transversal’ contribution is caused by the vector potential and can be regarded as occurring due to the exchange of virtual photons which have transversal polarization. The pair production process occurs mainly through the ‘transversal’ contribution2 . In order to evaluate the cross section Ž17. we need to know the elastic atomic form-factors fel Ž k .. Although there are several analytical representations for the form-factors, the very convenient one is that
Ai
Ý
2 2 is1 k i q k y
.
2 Ž kz . rc 2 Ž 18 .
Here A i and k i are real constants for a given atom which are tabulated for all atomic elements in w16x. For obtaining the second line on the right hand side of Ž18. the condition Ý i A i s 1 Žsee w16x. was used. The ‘retardation term’ yŽ kz . 2rc 2 appears in Eq. Ž18. because of the Lorentz transformation of the atomic electron density from the rest frame of the atom to the rest frame of the ion. Inserting Ž18. into Ž17. we get:
s Ž 1s . s
4Z22 Õ
`
Ý Ý A i A jHd V pH
2
=
2
Q q
½ž
2
2 v 0, p 2 2
Õg
Q q
q k i2
2 v 0, p
g 2 Õ2
/ž
2
Q q
2 v 0, p
Õ 2g 2
q k j2
/
2
/
N ² c 0 Ž r . N exp Ž i k P r . N cp Ž r . : N 2 k4 Õ2 Q2
q
H
1
ž =
d Ep d 2 Q
mc 2
se , s p i , j
=
2
We note that sometimes the ‘longitudinal’ and ‘transversal’ contributions are regarded as the ‘electric’ and ‘magnetic’ ones. However, this is not quite correct because in relativistic collisions an important part of the electric field of the projectile is given by the time derivative of the vector potential. Some additional confusion appears when the ‘longitudinal’ and ‘transversal’ parts of the transition amplitude, obtained in the Coulomb gauge, are regarded as parts arising from the scalar and vector potentials, respectively, in the Lorentz gauge.
/
c2 k 2
N ² c 0 Ž r . N exp Ž i k P r . a x N cp Ž r . : N 2
5
Ž 19 . with g s Ž1 y Õ 2rc 2 .y1 r2 .
330
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
If we set all k i equal to zero in Ž19., we recover the cross section for the bound–free pair production in collisions between two bare nuclei w17x. In collisions between two bare nuclei at asymptotically high values of g , the main contribution to the cross section is given by the range of relatively small Q, 0 F Q Q v 0, prÕg . This results in the logarithmic growth of the cross section with g : s Ž1s. ; lng Žsee w8;11x.. In collisions with neutral atoms, where the screening constants k i are different from zero, both the ‘longitudinal’ and ‘transversal’ parts of the cross section are reduced in magnitude compared to collisions with bare atomic nuclei. This reduction is relatively more important for the ‘transversal’ part, which contributes most to the cross section for pair production. From Eq. Ž19. it is easy to see that the effect of the screening depends on the ratio between the screening constants k i and v 0, prŽg Õ .. Although the minimum momentum transfer v 0, prÕ is very large on the atomic energy scale, the quantity v 0, prŽg Õ . can be smaller than k i at high enough g . In such a case the screening will be very important and will considerably reduce the cross section for bound–free pair production. For very high g this can be true even for collisions with light atoms, where all screening constants are of the order of a few atomic units. In an experiment of Belkacem et al. w18x cross section for bound–free pair production in collisions of 10 GeVrnucleon Au79q projectiles with Al, Cu, Ag and Au atomic targets were measured. Our calculations for the free-bound pair production show that the screening effect is still very weak at this collision energy Žonly up to 2 percent for the heaviest target.. Fig. 1 shows a comparison between the experimental data of Krause et al. w19x and theoretical results. In this experiment cross sections were measured for total electron capture by 160 GeVrnucleon Pb 82q projectiles colliding with different solid state targets ranging from Be to Au. The bound–free pair production cross sections were extracted from the measured data. The theoretical results include the perturbative calculations of the present Letter and those of Anholt and Becker w11x, where the screening effects were not taken into account. We give two sets of our results for the cross section which were obtained with and without including the screening.
Fig. 1. Cross sections for bound–free pair production in collisions of 160-GeVrnucleon Pb 82q with solid state targets as a function of the target atomic number. Open squares: experimental data from w19x, open and solid diamonds: nonperturbative results cited in w19x, dotted curve: calculations of Anholt and Becker w11x, dashed curve: our calculations without the screening, solid curve: our calculations with the screening included. See also comments in the text.
Both results of Anholt and Becker w11x and ours include a multiplication of the cross section for the capture to the ground state by a factor of 1.2 which takes into account the possibility for the electron to be captured into excited states of the ion and, also, a multiplication by a factor 1 q Zy1 which accounts 2 for the bound–free pair production due to the antiscreening effect of the atomic electrons w11x. We also show in this figure two non-perturbative results for the capture to the ground state of a lead ion Ž s Ž1s. s 46 and 50 b. in collisions between bare Pb and Au nuclei. These results were cited in the paper of Krause et al. w19x as private communications. In order to avoid partial wave expansion for the exact Dirac continuum states, which is slowly convergent at high g values, we use the semi-relativistic
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
Darwin wavefunction for the ground state of the electron and Sommerfeld–Maue wavefunctions for positron continuum states in our calculations. Our unscreened numerical results are close to those of Anholt and Becker w11x. With the screening included the cross section is reduced by 2.5 percent for collisions with Be and C targets. The screening effect increases with the atomic number of the target and reduces the cross section already by 14 percent for collisions with the Au target. Thus, at this collision energy the screening can already be of noticeable importance if the atomic number of the target is not much less than that of the projectile. In Fig. 2 we present energy spectra of positrons created in the process of bound–free pair production in collisions of 160 GeVrnucleon Pb 82q ions with Au79q and Au0 . As it can be expected the screening effect influences mainly the lower energy part of the positron spectrum.
It follows from Fig. 1 that the inclusion of the screening effects does not lead to closer agreement between the perturbative calculations and the experimental results. It was claimed in w19x that the overall uncertainty in the experimental data is about "10 percent. In collisions with light targets, like Be and C, where the perturbative theory is expected to be valid, the perturbative calculations yield cross sections which are smaller than the experimental ones by about of 20 percent. In collisions with heavy targets, like Sn and Au, where noticeable deviations from the perturbative treatment could be expected, the nonscreened perturbative results seem to be, at the first glance, in agreement with the experiment. However, as it was pointed out in w19x, in collisions in solid targets the contributions to measured capture cross sections from excited ion states are small for the heaviest targets. Correspondingly, these authors estimated that, in order to account for the possibility for the electron to be captured into excited states of Pb 81q in collisions with a solid state gold target, the cross section s Ž1s. should be multiplied by a factor of f 1.06 instead of 1.2 3. If one takes this into account, then one sees that the perturbative results underestimate the experimental ones by more than 10 percent. On the other hand, if one assumes that, in order to compare the nonperturbative results s Ž1s. s 46 and 50 b with the experiment w19x, these nonperturbative results should also be multiplied by 1.06, then one sees that the nonperturbative results overestimate the experimental ones by more than 10 percent. Since the screening effects reduce the cross sections one can argue that including the screening effect into nonperturbative calculations could shift the non-perturbative results to closer agreement with the experiment. We have also estimated cross sections for bound–free pair production in collisions of 1000 GeVrnucleon Pb 82q projectiles with Be and Au targets. For this collision energy, neglecting the screening effect, we found s Ž1s. s 0.18 and 56b for Be and Au targets, respectively. Including the
3
Fig. 2. Spectra of positrons created in the process of bound–free pair production in collisions of 160-GeVrnucleon Pb 82q with Au79q Ždashed line. and Au0 Žsolid line. targets.
331
Surprisingly, despite their own statement the authors of w19x still compared their experimental data with results of Anholt and Becker for 1.2= s Ž1s. for all, including the heaviest targets Žsee Fig. 3a of w19x..
332
A.B. VoitkiÕ et al.r Physics Letters A 269 (2000) 325–332
screening effect these cross sections are reduced to s Ž1s. s 0.15 and 37b, respectively. Thus, at this collision energy the screening reduces the cross section by 16% for Be targets and by 33% for Au targets. Our calculations show considerably larger screening effects in the bound–free pair production than it was found in w9x for free pair production. For example, Wu et al. w9x obtained in their perturbative calculations that in collisions of 200 GeVrnucleon Au79q projectiles with Au target the screening effect reduces the cross section for free pair production just by 4.5 percent. Our calculations for the bound–free pair production in the same collision system show that the screening reduces the bound–free pair production by about 16 percent. The reason for such a considerable increase of the screening effect for the bound–free pair production compared to the case of the free pair production can be attributed to the fact that the process of free pair production involves larger momentum transfers because both positron and electron are in continuum states. In conclusion, we have estimated the screening effects in the process of bound–free pair production in collisions between a point-like ion and a neutral atom. We have shown that the screening becomes more important for collisions with heavy atoms and when the collision energy increases. We have found that, in contrast to previous statements w11x, the screening effects can be of importance for collisions with g ; 100–1000. Comparing the results of our calculations with those for the free pair production w9x one may conclude that the screening is more important for the bound–free pair production.
Acknowledgements A.B.V. acknowledges with thanks the support from the Alexander von Humboldt Stiftung.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x
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