Positron emission in PbPb and PbU collisions

Volume 78B, number 2,3 PHYSICS LETTERS 25 September 1978

POSITRON EMISSION IN Pb-Pb AND P b - U COLLISIONS ~

J. REINHARDT, V. OBERACKER, B. MklLLER, W. GREINER Institut fiir Theoretische Physik, Johann Wolfgang Goethe Univers#dt, Frankfurt am Main, Germany

and

G. SOFF Gesellschaft fiir Schwerionenforschung, Darmstadt, Germany

Received 4 May 1978

We present quantitative calculations of positron emission dynamically induced in the collision of Pb--Pb and Pb-U. The results are compared with recent measurements, yielding evidence for new processes characteristic for quantum electrodynamics of strong fields.

During the last decade quantum electrodynamics in the presence of strong external fields has found in- creasing attention. If the nuclear charge of an atom is made to exceed the critical value Zcr = 173, the deepest bound state joins the antiparticle continuum of the Dirac equation. A previously present K-hole will then be emitted as a free positron [1,2]. This is called the spontaneous decay of the neutral vacuum [1 -3 ] . The required high concentration of nuclear charge can only be assembled temporarily in the collision of very heavy ions. Therefore only recently the theory of QED of strong fields has been subject to experimental test, mainly at the UNILAC accelerator at GSI [4,5[.

During the close approach of two colliding heavy ions electronic quasimolecules are formed. If the combined charge reaches the domain Z 1 +Z 2 ~> 1/a = 137 strong field effects will contract the quasimolecular orbitals leading to a large increase of binding and of coupling matrix elements at small internuclear distance. Even before spontaneous positron production is possible, this behaviour results in several observable effects:

(a) A large ionization rate for inner shell states [6-8] which in turn favors quasimolecular X-ray

Work supported by tile Bundesministerium ftir Forschung und Technologie (BMFT) and by the Gesellschaft ffir Schwerionenforschung (GSI).

emission [8]. Measurements of the characteristic X-rays support the prediction of high vacancy production probability at small impact parameters [9].

(b) An enhanced production of electron-positron pairs [10] by direct excitation which has been explained as the shake-off o f the vacuum polarization charge distribution.

(c) The induced emission o f positrons by filling of inner shell vacancies. Henceforth we will abbreviate processes (b) and (c) by "shake-off ' and "induced" positron production, respectively.

In this letter we present fully dynamical calculations treating vacancy formation and induced positron production coherently, in contrast to former works which assumed constant (in time)hole probability [11 ]. The results are compared with improved calculations of the background [121 from pair conversion following Coulomb excitation of nuclei and with recent experi- ments [4,5]. We mainly concentrate on the experimentally most favourable system Pb-Pb; the spontaneous positron production in systems with overcriti- cal charge, Z 1 +Z 2 >Zcr = 173, requires a special treat- ment and will be subject of a future publication.

Since the effects of the strong electrostatic field in the vicinity of two colliding heavy nuclei cannot be described in perturbation theory, we treat the electronic excitation process in the quasimolecular model.

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The wave function of the electron-positron field is ex- panded in the basis of eigenstates of the stationary two-center Dirac equation. The time-dependent Dirac equation then yields a set of coupled differential equa- tions for the expansion coefficients of the form

ai/ - a o / exp

+k~Faiklkl~t]lexp[i_f dt'(Ek-E] )} (1)

t

-k~Fak/lil~tk)exp{iLdt'(Ei-Ek) }

where aij is the amplitude for a particle in state i and a hole in state ]. We restrict ourselves to l p - l h states, E i are the single particle energies, a 0 denotes the amplitude for retaining the original ground state and the summation goes over originally occupied (set F, first sum), and empty (second sum) levels. The negative energy continuum is contained in the set F. The first term in (1) describes the creation of particle hole pairs from the ground state while the sums represent "re- scattering" of excRed holes or electrons, respectively. To investigate the production of positrons we employ time-dependent perturbation theory and keep only the most important terms. Thus we obtain the electron-positron pair amplitudes

o¢1

a (1)Ee,E p - f dt e ~-[ exp {i(E e - Ep)t } (2)

and

a(2E:,Ep = + dtaEe ' n (t) ~-

{: } X exp i dT(En-Ep)

(3)

with

aEe, n=- L dt e ~'[ exp 1 r (E e - E n ) . (4)

~Ee, ~Ep, ~0n are wave functions of free electrons, free positrons and bound electrons, respectively. Eq. (2) represents the direct creation of pairs [ 11 ]. Since, however, the inner shell bound states are strongly de- formed by the combined Coulomb field of the nuclei, their influence will become important. Inner shell

states are usually filled initially, so the lowest order non-vanishing contribution comes from two-step processes where a hole is first created, e.g. by direct ionization (4) and later transferred to the negative energy continuum where it emerges as an emitted positron (3). Both processes, leading to the same final configuration, cannot be distinguished and must be added in phase. The total cross section for positron emission is obtained by integrating the squared total pair amplitude over electron and positron energy and over impact parameter b and by summing over angular momentum K :

_ m ¸

o=27r~(2Ji+l ) f bdb f dEp K

0 - - ~

(5) e ¢

m n'~'F Ee 'Ep "

In the numerical calculations the K sum was restrict- ed to angular momentum ] = 1/2(K = + 1), since these states are most severely affected by the strong field. Furthermore, we have taken into account radial coupling (R3/OR) only since the corresponding matrix elements become large at small internuclear distance, Rotational coupling (11 .]) is not expected to be ef- fective in coupling levels which are energetically far apart as must be achieved to produce pairs. It could play a role in the formation of inner-shell vacancies, especially in the nPl/2o-states. Still then its influence at very small impact parameters must be small (~2 vb/R2), and only these contribute significantly to pair creation. A more detailed investigation of the possible influence of rotational coupling must be deferred to future coupled channel calculations.

The wave functions and matrix elements have been calculated in the monopole approximation, i.e. only the spherically symmetric term in the multipole expansion of the two-center potential has been retained. The effect of finite nuclear extension was found to reduce coupling matrix elements at small internuclear distance by up to 30 percent. For bound states the monopole approximation to the two-center potential

184


is known [13] to introduce an error of less than 5 percent in the range covered by the present calculation. For continuum states there is no reason to expect larger errors.

Using these approximations we have integrated eqs. (2 ) - (5 ) numerically prescribing Rutherford trajectories for the nuclear motion. Several typical results are dis- played in the figures. Theoretical and experimental results concerning the rate of inner shell ionization in collisions of very heavy ions are presented in refs. [6 -91 and [14]. The overall agreement on the ionization rates lends confidence to the use of vacancy amplitudes calculated according to eq. (4) as intermediate step for pair creation. (However, see the note at the end.)

The positron background from pair conversion following nuclear Coulomb excitation can be dealt with by folding the measured 7-ray-spectrum with the conversion coefficient [15], if the multipolarities of the nuclear transitions are known [4,5,16]. We have calculated Coulomb excitation of the nuclei 2°8pb and 238U with a coupled channel code. Due to the very low level density of the double magic nucleus 2°8pb the collision system 208pb-2°Spb is optimal both in view of low background cross section and of theoretical tractability. Only two levels in 208pb are of importance ( 3 - at 2.615 MeV and 2 + at 4.086 MeV).

The calculated excitation is in excellent agreement with measurements [4].

The nucleus 238U has a complicated band structure which we have described in terms of the collective rotation-vibration model (RVM). The energy levels are found to be in accordance with experimental data even at high spins. In calculating the positron emission the additional population of nuclear levels by 7 - and e - - cascades has been considered. The calculations on nuclear Coulomb excitation and following internal e+, e--pair conversion can be tested independently [12, 16] in lower-Z system like 136Xe+238U or 136Xe + 232Th, where the QED processes give no significant contributions to the total positron production.

Fig. 1 a shows the impact parameter dependence of the positron emission probability for 208pb-208pb collisions with distance of closest approach 2a = 20.9 fro. The dashed line contains the coherent sum of shake-off and induced pair creation in the channels K = -+ 1. Included as intermediate bound states are the six lowest K = -+ 1 levels, namely lso, 2so, 3se and 2Pl/2O , 3p1/2a , 4pl/2O. The curve decreases mono- tonically with impact parameter, similar to that obtained for the ionization probability [14]. The slope, however, is even steeper, so that 90 percent of the cross section comes from collisions with b < 30 fm.

10-" Pe'(b) ZOepb _ ZoBpb

20 = 2&g lm

1[~1 - ~ [~e% ZoBpb_ ZoBpb 2a=ZOgfm

, . - . . ZoSpb_ z38 u / \\ Zo : 2ZOfm / , \ ~.A.',.4 \ , V " • L - 1 r "-I \

10 -7

10- 8

102 L-I \ Z38U-[ .~ \

' \ "L~ [ _X___. " "~ i \ i . . . . -_-Z25-~- \ I

2+(&O86M \, _: . - - - - - flED positrons& I / - '~-Pb(3 ) ] \ I / ' ~ ' . ', 10" ! (shake off * X I • /" ' -" ' - . ",#-(z.61s.~ \v~\ ' ~nd~t6~tat~s]l \L / / L! .... ~"\

" \ \ ' \ ",, . . . . . . . nuclear background ~ /! - - - - - - (shakeQEO positrOnSoff * II [ \ \ \ sum ' '

" " /.t-2"~/~O86MeV] . . . . . . . . . . . . . ~-'%'--I \ x induced[gstatesDll ] - '\ \ \ ' \ - 10 4 i . . . . . . . nuclear backgro~i'" j

. . . . . . . . . '\ \ ! / " I I .... Dockground ~ ~ " t i - - - - - - {IED positrons \ \ r ' I , / ',

j . I --sum ~, ~ , , ~kin,. ,,, I

Impact Parameter b[fm] ,I/ Lp [KeVl l 10"" . . . . " ' ' ' ~ . . . . . ' ' ' 1'0 2'0 " 3'0 " 40 0 500 1000 1500 2000 0

i

kin ! L---- I Ep keV] I [

. . . . i . . . . . . . . i r , ~

l[Jg 500 1000 1500 2000

Fig. 1. (a) Probability for positron emission as function of impact parameter in 2°8pb-2°8pb collisions with E = 4.45 MeV/u, Dashed line: coherent sum of positrons from the shake-off of the vacuum polarization and of induced positrons from the quasimolecular orbitals 1 sa up to 4pl/2 o. Dotted fines: background positrons from the conversion of ,,/-rays following nuclear Coulomb excitation. Full curve: total sum. (b) Energy spectrum of positrons from the same collision as in (a). (c) Energy distribution dee+/ dEp of positrons produced in E = 4.45 MeV/u 2°8pb-23Su collisions. Same notation.

185


Since the radial coupling matrix elements show a pro- nounced maximum at the distance of closest approach most of the ionization and positron production takes place in a closely localized region. For example, in head-on collisions the induced positron emission from the 1 sa state peaks at about R = 40 fm on the outgo- ing branch of the trajectory and remains nearly constant for R > 100 fm. At larger impact parameters the oscillations reach further out and only a small fraction of the intermediate virtual excitation survives asymp- totically. The dotted curves in fig. 1 a show the nuclear background which must be added incoherently giv- ing the solid curve. The appearance of a bump at about b = 7 fm is caused by the reorientation of the 3 - (2.615 MeV) level. In the computations we used the quadrupole moment Q(3 - ) = -0 .42 e. b given in ref. [17].

Figs. lb, c show representative spectra dcre+/dEp of emitted positrons with respect to kinetic positron energy E kin for the systems 208pb-2°8pb and 2°8pb -238U a~E = 4.45 MeV/u. The spectrum of QED positrons has a maximum at about E kin = 400 keV and falls off exponentially at higher energies. The nuclear background in 208pb-208pb originates essentially from pair conversion of two 3,-lines and therefore looks considerably different. Setting a window at positron energies below 1 MeV will greatly suppress the background. In the 208pb-238U system, on the other hand, many 3,-transitions occur between 1 and 3 MeV showing a serrate structure with an overall shape similar to that for QED positrons. Fig. 2 gives our results for the most comprehensive data, the total positron cross sections o e+ as a function of projectile energy. The calculated curve is compared with total cross sections measured at GSI [5]. In both collision systems the experimental values lie above theory by up to a factor of two. The dependence on bombarding energy, however, agrees very well. Most convincing in this respect are the Pb -Pb data where the slope of the experimental curve can be explained only by the sum of QED and nuclear'positrons. This is also true for the differential cross sections dae+/df2ion at scattering angle Ola b = 45 ° measured by the same group. Here the absolute agreement is better leaving a discrepancy of only about 30 percent.

With a different experimental set up Kienle et al. [51 measured the differential probability for the erhis- sion of 500 keV positrons at (lab) scattering angles

cre

10 3

10 2

[pb]

Pb-Pb

-% ,N

11'5 . . . . 2'0' ' ' '> . . . . !art 25 30 6'.0 5.5 510 4'5 4.0 3.5 E~AtM~V~1

Total Positron Cross Section

+ Pb-O

\ ~ + + \

"N \NN~x + ":\, \ \

Exger~ment(GSI) ~N

te/at posHron yield "N ~ 1 QEalshoke off • " 2a[f . . . . indu~ed[SslntesO

. . . . . . nucleor background ] 15 20 25 30

5:o55 51o ~5 ~o 3 . ~ v / ~ Fig. 2. Total positron cross sections for (a) 2°8pb-2°8pb and (b) 2°8pb-238U collisions in dependence of bombarding energy or, equivalently, of distance of closest approach 2a. The experimental points are taken from ref. [4].

between 30 ° and 70 °. Their results for Pb -Pb and P b - U are shown in fig. 3 together with our theoretical values. In this case it is possible to subtract (within an uncertainty discussed in ref. [5]) the positrons from nuclear transitions. Therefore the theoretical lines con- tain only QED processes.

Let us now discuss in more detail the various con-

I gLll

U-Pb

r / l / ~ 0E0p0s, t~00~ P / # l ,T,oor,,,

4 , ,I , , , , I0 300 6~ o 900

Fig. 3. Probability for positron emission as a function of projectile cm scattering angle for E = 5.9 MeV/u 2°spb-2°8pb and 238U-2°spb collisions in the energy window Ee+ = 440 -550 keV. Theory is compared with measurements from ref. [5] where nuclear background has been subtracted. Both curves have been symmetrized with respect to projectile and target nucleus.

186


Table 1 Total positron cross section in gb for several collision systems with total charge Z1 + Z2 = 154,164,174. The distance of closest approach 2a = 20.9 fm is kept fixed. For the channels K = -1 and K = +1 the direct (labeled by (1)) and two-step (labeled by (2)) contributions and the total coherent sum are listed separately. While the sum includes all I KI= 1 states up to 4pl/2 a the columns a(2) display only the lsa and 2Pl/2a contribution.

Z I+Z2 K = - I ~ =+I

a (1) a (2) ~ a (1) q (2) a

154 1.53 0.33 3.76 1.55 0.29 3.3 164 5.6 2.8 14.8 4.5 2.6 13.9 174 25.4 20.3 56 12.8 18 50

tributions from QED to the positron spectrum. Our results presented so far contained shake-off positrons and those induced from several inner shell levels up to 4Pl/2. Table 1 lists the separate contributions of the two lowest states to the total cross section for the systems with total charge Z 1 + Z 2 = 154, 164 and 174 at energies corresponding to a distance of closest approach 2a = 20.9 fro. We observe that (a) the interference between shake-off and induced positrons becomes de- structive, when the considered intermediate state is deeply bound, (b) the two-step contr ibution increases much faster with nuclear charge than the direct process. This clearly demonstrates the non-perturbative character of positron creation in the strong fields en- countered in collisions of very heavy ions. Let us note that it seems to be difficult to isolate the various contributions to the cross section, experimentally. Their dependence on bombarding energy and impact parameter is similar, since in each case the same amount of energy has to be transferred to the created pair in about the same interaction region.

Table 2 lists the induced positron product ion from various levels separately for the chosen parameters Z 1 + Z 2 = 164, 2a = 20.9 fm. Although the probabil i ty for vacancy product ion in higher orbitals becomes very large, the largest individual contributions to the positron cross section originate from the deepest bound states. For comparison the right column of table 2 lists the induced positron cross sections calculated with a constant hole amplitude (4) set equal to unity. The contr ibution from the l so level is very large and by far dominant, justifying the name "induced decay of the neutral vacuum".

The results for induced positron emission presented so far have been based on hole amplitudes resulting

Table 2 TOtal cross section for induced positron production from various bound states for the system Pb-Pb at 2a = 20.9 fro. Left column: Two-step process (no vacancy is present initially). Right column: Calculation assuming totally empty shells.

e+ e+ step) State Crmd (2 step) amd (1 [gb] [#b]

lsa 2.8 1790 K = -1 2so 1.26 27

3sa 0.28 3.5 dkect. 5.6 5.6

2pl/2b 2.6 121 t~ = +1 3pl/2 ° 0.32 4.7

4pl/2 a 0.10 1.24 direct 4.5 4.5

from ionization and calculated in time dependent perturbation theory. To treat the influence of higher bound states and their mutual coupling more accurately, coupled channel calculations admitting various intermediate and final configurations should be performed. In particular, vacancy formation will be enhanced by radial coupling with outer shell bound states which may be initially empty, depending on the charge state of the projectile atom. To get an idea of the magnitude of this effect we have employed vacancy amplitudes calculated by the coupled channel method assuming initial holes in the 4so or 5so level, resp. The resulting additional positron yield varies somewhat slower with bombarding energy and shows an oscillating structure in dependence of impact parameter b. For the collision treated in table 2 the total cross section (~ ; - 1 ) increases by about 30 percent (4sa) or 8 percent (5so) respectively. A similar result is expected for the K = +1 channel. For an assessment of the validity of these

187


results one should compare the calculated vacancy formation rates with experiment.

Summarizing we conclude that the study of positron emission in,almost critical heavy ion collisions

provides a sensitive test for the theory of QED of strong fields. We have shown that induced emission via bound states contributes a sizable fraction to the total positron production. Experimantal evidence (par- ticularly from P b - P b collisions) establishes the exist- ence of the QED effects and confirms in particular the importance of induced positrons.

To obtain more detailed information one should

carefully study the subtraction procedure for nuclear background e.g. by comparing several collision systems

with the same total charge Z 1 + Z2, where QED positron emission is the same but the background depends

on the details of nuclear structure. Furthermore one

should investigate the positron excitation function for various positron energies Ee+ and one should go to systems with even higher charge (e.g. up to A m - C f collisions) or to higher impact energies with prolonged lifetimes of the nuclear compound system [18]. Ulti- mately, the availability of very heavy nuclei totally stripped from electrons would allow an unambiguous identification of the production mechanisms for QED positrons.

We are grateful to P. Schltiter for making available to us his calculations on pair conversion coefficients.

We thank H. Backe, H. Bokemeyer, J.S. Greenberg, E. Kankeleit, P. Kienle and J. Rafelski for many help-

ful discussions.

References

[ 1 ] W. Pieper and W. Greiner, Z. Physik 218 (1969) 327; B. Miiller et al., Phys. Rev. Lett. 28 (1972) 1235; B. Miiller, J. Rafelski and W. Greiner, Z. Physik 257 (1972) 62 and 183.

[2] Ya.B. Zeldovich and V.S. Popov, Sov. Phys.-Usp. 14 (1972) 673.

[3] The recent status is the subject of the following review articles: B. Miiller, Ann. Rev. Nucl. Sci. 26 (1976) 351; J. Reinhardt and W. Greiner, Rep. Prog. Phys. 40 (1977) 219; J. Rafelski, L. Fulcher and A. Klein, Phys. Rep. 38C (1978) 228.

[4] H. Backe et al., Phys. Rev. Lett. 40 (1978) 1443; and private communication.

[5] Ch. Kozhuharov, P. Kienle, E. Berdermann, H. Bokemeyer, J.S. Greenberg, Y. Nakayama, P. Vincent, L. Handschug, E. Kankeleit and H. Backe, Positrons from 1.4 GeV Uranium- Atom-Collisions, preprint; and private communication.

[6] W. Betz et al., Phys. Rev. Lett. 37 (1976) 1046. [7] G. Soft, B. Mtiller and W. Greiner, Phys. Rev. Lett. 40

(1978) 54O. [8] J. Kirsch et al., Phys. Lett. 72B (1978) 298; see also: W.

Meyerhof, Contribution to the Int. Syrup. on Superheavy Elements; Lubbock, Texas, March 1978.

[9] J.S. Greenberg et al., Phys. Rev. Lett. 39 (1977) 1404; J.R. Macdonald et al., Z. Physik A284 (1978) 57.

[10] G. Soft et al., Phys. Rev. Lett. 38 (1977) 597. [11] K. Smith et al., Phys. Rev. Lett. 32 (1974) 554;

D.H. Jakubassa and M. Kleber, Z. Physik A277 (1976) 41. [12] V. Oberacker, G. Soft and W. Greiner, Phys. Rev. Lett.

36 (1976) 1024; and Nucl. Phys. A259 (1976) 324. [13] G. Soft et al., Physica Scripta 17 (1978) 417. [14] G. Soft et al., Phys. Lett. 65A (1978) 19. [ 15 ] P. Schliiter, G. Soft and W. Greiner, Z. Physik A286

(1978) 149. [16] W.E. Meyerhof et al., Phys. Lett. 69B (1977) 41. [17] A.M.R. Joye et al., Phys. Rev. Lett. 38 (1977) 807. [18] J. Rafelski, B. Mtiller and W. Greiner, Z. Physik A285

(1978) 49.

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Positron emission in PbPb and PbU collisions