Dynamical aspects: Coherent production of positrons in heavy ion collisions

Dynamical Aspects: Coherent Production of Positrons in Heavy Ion Collisions I J. REINHARDT, G. SOFF, B. MOLLER and W. GREINER 2 Institut fflr Theoretisehe Physik der Johann Wolfgang Goethe-Universit~t, Frankfurt am Main, West Germany

ABSTRACT Collisions of very heavy ions are discussed as a means to investigate quantum electrodynamics in the presence of strong external fields. The action of the combined Coulomb field of the closely approaching nuclei leads to strong binding of the inner electron shells and to large induced transitions. The resulting mechanisms for the production of positrons are discussed in detail and compared with recent experiments. The importance of multi-step excitations is stressed and first results of coupled channel calculations are presented.

KEYWORDS

Spontaneous and induced positron production, quasimolecular model, single-particle amplitudes, multi-step excitation.

INTRODUCTION In this lecture we want to investigate the behaviour of electrons in "atomic" systems with a very high charge (Z > i/~=137) and to study Quantum Electrodynamics of strong fields experimentally. Nuclei, of course, with such charge number are not stable. However, the typical time-scale for atomic processes to evolve in such an atom is of the order 10 -20 - 10-18 sec. It is therefore sufficient to form a "quasimolecule" for a very short instant of time. 3 This is possible in heavy ion collisions, where two nuclei can be brought together as close as 20 fm for a typical time of ca. 10 -21 sec. It has been found that the most suitable way to describe the various stages of the scattering process is the adiabatic picture: (For

ISupported by Bundesministerium f~r Forschung und Technologie (BMFT), and by the Gesellschaft f~r Schwerionenforschung (GSI). Invited Speaker at the Erice School on Heavy Ion Interactions at High Energies, Erice (Italy) 26 March - 6 April, 1979. The idea of forming superheavy quasimolecules in heavy ion collision has been suggested by W. Greiner in the GSI-seminars during 1969/1970. It has been published by Rafelski, Fulcher and Greiner, (1971) and independently by Zeldovich and Gershteln (1969). 547

548

J. Reinhardt

et al.

the physics of s u p e r c r i t i c a l fields, see the first lecture and the following review articles: M Q l l e r (1976), R e i n h a r d and G r e i n e r (1977), Rafelski, Fulcher and Klein (1978)) the electrons are e n v i s a g e d as m o v i n g very fast (at velocity v~c) around two slowly m o v i n g nuclei (at v~c/iO). Then one takes a snapshot of this "quasimolecule" at every i n t e r n u c l e a r separation R. The time d e v e l o p m e n t is d e s c r i b e d by the scattering t r a j e c t o r y R(t). W h e n one solves the (relativistic) Dirac equation for an electron in the electric field of two nuclei (MQller, Rafelski and G r e i n e r 1973), one obtains the quasim o l e c u l a r c o r r e l a t i o n diagrams of which two typical examples are shown in Fig. i:

-5

~ 5 0 Ill lllS~lllll . . . . . . Pb.Pb

~ R~

-I~

-3C

-1~

E1

Fig.

i.

A d i a b a t i c c o r r e l a t i o n d i a g r a m of the q u a s i - m o l e c u l a r system U - U (b) and Pb-Pb (a). R is the distance between the two nuclei, E is the elctronic binding energy. At R~35 fm in U-U the isO state becomes supercritical.

(a) for a Pb-Pb c o l l i s i o n and (b) for a U-U collision. (Betz, 1976). The m o s t interesting states are the lowest ones, in p a r t i c u l a r the one d e n o t e d IsO. At small d i s t a n c e s R~20 fm the q u a s i - m o l e c u l a r states approach those of an atom of charge ZI+Z 2 (i.e. 164 and 184 in our examples). From the b e h a v i o u r of the wave functions one finds that in the region R < 2 0 0 fm the system resembles much more a "quasi-atom" w i t h a big b l o w n - u p nucleus of radius R/2 than a molecule. We conclude that in a heavy ion scattering process q u a s i - a t o m i c systems of nuclear charge up to 184-190 become a v a i l a b l e and it is p o s s i b l e to study them. The m a j o r p r o c e s s e s allowing for an experimental i n v e s t i g a t i o n are the following: (a~ Pair creation and p o s i t r o n emission, when the b i n d i n g e n e r g y of the isO state approaches i M e V (will be d i s c u s s e d in detail). (b) Ionization of high energy (delta-electrons) out of the low-lying states. By a careful analysis of the n u m e r i c a l c a l c u l a t i o n s and by c o m p a r i s o n with analytical models, one finds (Soff et al. 1978, Betz et al. 1976, M Q l l e r et al. 1978) that the ionization p r o b a b i l ity P of the iso orbital versus impact p a r a m e t e r b can be d i r e c t l y related to the

Coherent Production of Positrons

549

binding energy E~S(R ) at closest approach. A systematic measurement of P(b) as function of bombarding energy E~, and quasi-atomic charge Z=ZI+Z o could therefore be utilized to determine the b i ~ i n g energy E~S(R,Z) right up in{o the region of supercritical binding (E_> i MeV). B These ideas have been successfully tested during the last year, notably in experiments performed at GSI. The first are due to Greenberg, Bokemeyer and Schwalm (1977) who used a clever Doppler shift technique to determine the ion scattering angle. Predicted rates in the percent region have been observed. More detailed results can be obtained by ion-X-ray coicidence experiments performed by the Behncke, Liesen, Mokler, Armbruster group at GSI (1978). Just recently a systemmtlc measurement has been used to extract binding energies in the quasi-molecular diagram (Behnke et al. 1978). The results are in rather good agreement, indicating that a spectroscopy of quasi-atoms up to Z~I80 seems to be feasible. Another way to i n v e s t i l ~ e the quasi-atomic wave functions is delta-electron spectroscopy. The high-energy part of the spectrum of ionized electrons can be related to the high-momentum components of the quasi-atomic wave functions (Soff et al. 1978). It is only because the inner wave functions in atoms beyond Z~140 show structures of the order 20-30 fm, that very high energy electrons (E>I MeV) are produced. (c) X-ray transitions between quasi-molecular states when inner-shell vacancies have been formed. Since an inner-shell (e.g. IsO) vacancy must be created before radiative decay can occur, ionization is a necessary prerequisite. Unfortunately, the transition energy between two levels varies rapidly with time (see Fig. i) and the overall time-dependence introduces an additional broadening of the X-ray energy. Therefore, the quasi-molecular X-ray spectra usually have an exponentially falling shape. Very little information can be obtained in this way. However, it was predicted (M~ller and Greiner 1974) and later experimentally confirmed (Greenberg et al. 1974), that the X-ray spectrum exhibits a pronounced directional anisotropy around the transition energy at the distance of closest approach. This phenomenon has to do with the swift rotation of the internuclear axis in the moment of closest approach of the two nuclei. The systematics of the anisotropy peak have been investigated by W61fli's group at Z6rich for light and medium light systems (Stoller et al. 1977), and up to Z=184 for outer transitions. (W61fli et al. 1978). If plotted versus Z=ZI+Z 2 the peak allows for a spectroscopy of the quasi-atomic states in heavy and superheavy atoms. There is hope, that it will be possible to continue it right through Z=I70. Calculations for the systems Pb-Pb are in progress. The framework for the description of these phenomena is the same, viz. time-dependent perturbation theory on the adiabatic quasi-molecular basis. (J. Kitsch et al. 1978). In the following we will discuss the theory of positron production in detail, keeping in mind that the treatment may be taken as a guideline for the description of ionization and X-ray emission, too.

THE QUESTION OF COHERENCE The first attempts to calculate the processes occurring during a heavy ion collision were based on the quasi-static approximation. A typical example is the calculation of positron production of Peitz et al. (1973). Here one assumed that a constant IsO vacancy probability was available during the whole collision. Furthermore, the positron emission was calculated as if the system had infinite time to settle down at every internuclear distance. Thus energy conservation was built in from the beginning and only spontaneous processes were investigated. The stationary phase approximation (see e.g. Macek and Briggs (1974) for X-ray emission) was an improvement only insofar as the coherence between approaching and receding part of the trajectory could be accounted for. The improved understanding of the collision dynamics showed, however, that the lack of full adiabaticity leads to the occurrence of induced process, i.e.

550

J. Reinhardt et al.

phenomena where energy is transferred from the nuclear motion into other channels. These effects can be treated by the evaluation of the full Fourier integral for, e.g., the positron amplitudes (Smith et al. 1974, Soff et al. 1977). In general, this procedure led to a broadening of the calculated spectra, which was observed in the Mo X-ray experiments and attributed to the time-energy uncertainty relation. Often also a considerable increase in the total cross-section was found. In the following it became clear that also the last approximation, namely that of constant vacancy amplitudes, had to be dismissed. Especially in heavy systems or at higher bombarding energy, the vacancies are created during the collision and therefore are rapidly oscillating functions of time. Such calculations, where ionization and the subsequent process are treated coherently, could be performed only after the theoretical (and experimental) understanding of the direct excitation process had been achieved. Examples are the calculations of Kirsch et al. (1978) for X-ray emission and of M011er et al. (1978), Reinhardt et al. (1978) for positron creation. In general, the effect of coherence is to reduce to probability. This is exemplified in Fig. 2, where we have chosen the process of induced positron emission in Pb-Pb collisions. The solid line accounts for the fully coherent calculation, the expression + 2 pe = 2 dE I (2) [ is p aEe,Ep;iS

/dEe/

of eq. (22) below. The dashed line represents the calculation, where ionization Pls and positron emission pe are calculated separately with full intrinsic coherence, but the product is taken incoherently: + + ~els = Pls

.pe

= 2f dEe[ a Ne,ls[2.fdEp]aEp 'IsI2

P. would be observable as the asymptotic isO ionization probability. The incoIs . seen to overestlmate , pe1+ by a factor 3 to 4, but to glve . herent product is a reasonable impact parameter dependence. ~e conclude that full coherence is necessary to obtain reliable values for processes like induced or spontaneous positron creation.

Fig. 2.

Induced positron emission probability P e+ in Pb-Pb collisions v ~ s u s impact parameter. Ionization (first step) and pair creation (second step) are treated coherently (solid line) and incoherently (dashed line).

t

i

Pb - Pb I~I- ~

2o ~ 20.9 fm TM

\\ \\ \\\\

10-~ coherent

i

1

1

10

20

30

b[fm] \

40

Coherent Production of Positrons

551

FORMALISM To describe the production of inner-shell holes and the emission of positrons by the various processes qualitatively discussed in the last section it is essential to account for the strong Coulomb force experienced by the electron-positron field at small distances of the colliding nuclei. Since the adiabacity criterion is satisfied for the motion of inner-shell electrons, it is most natural to adopt the quasi-molecular picture. Thus we start from the complete set of solutions of the stationary Two Centre Dirac equation (M~ller et el. 1973, M~ller and Greiner 1976)

~TC~ ~

where

i"~

+ ~t4~ ~

V(~,~)

and q runs over the+set of bound states and the upper and lower continuum. The basis functions ~ (R(t))depend parametrically on time via the changing internuclear distance ~(t). q The nuclear motion, therefore, will induce transitions between the adiabatic states 9 . To treat the dynamics ~f the collision and calculate the various possible excitations, in principle one has to solve a problem with infinitely many particles, since according to Dirac's hole picture the negative continuum is occupied with electrons. In addition various bound states may be occupied depending on the charge state of the colliding ions. If one neglects the electron-electron interaction it turns out, however, that it is sufficient to solve the one-electron problem and to include the effect of the Pauli-principle only afterwards. Let us therefore look at the fate of a single electron which initially occupies the level ~. before the collision. The time development is determined by the l time-dependent Dirac equation

with

the

initial

condition

:

TO obtain the functions ~i(t) one expands in the adiabatic basis ~j(~(t)) :

with the phase

~

[~) ~

~'

£~ ( ~ ) )

Inserting in (2) and using the orthonormality of the basis functions, one obtains a set of coupled differential equations for the expansion parameters a.. : x3

The initial condition is a.k(-~) = ~. . Eq.(4) can be solved numerically after the infinite set of basis s~ates ~. ~ been suitably truncated. The squared amplitudes la.~(~) I2 then give the3probability that the electron is excited from l state i to sta~e f dur£ng the collision. To discuss the many-particle problem we note, that the functions $.(t) form a l complete orthonormal set at every instant t. This follows from the hermitecity of the Hamiltonian HTCD, i.e. from the unitarity of the time development operator:

J. Reinhardt et ~ .

552 or equivalently

~"

,x,c;.

O-~;.

"-"

. O't,~ ~ ; . ~

=

~,c~

•

(6)

,,,7,

Therefore the set ~. (t), already containing the dynamical excitations, can be used 1 as a basis for solvlng the many-particle problem. To do this, one can expand the total wave function in a basis of many-electron configurations which may be represented by Slater determinants of the single particle basis functions ~ . The . . . . 1 amplitude for exciting a final configuration starting from a glven znltlal configuration, turns out to be just the determinant of the corresponding single-particle amplitudes a.. (t). l More formally one c~n use the language of quantum f~eld theory to construct a state vector I~- > in the Heisenberg picture, i.e. ~ I~H > = O. (7) The (time-dependent) field operator may be expande~Cas ~

~

~

~ )

and

~

~

~

~

~)

(8)

where the creation and destruction operators satisfy the anticommutation relation A

~ 4

Now we define a set of initially occupied states (e.g. the negative energy continuum and several bound states) denoting them by q < F. The complementary set of empty levels is described by q > F. As usual operators for holes can be introduced by the canonical transformation ^+ q

=b

q for q < F

~+ = q leading

to

(10)

q

following

representation

~<~

of

the

field

operator

~>~

(11) J

The so defined operators _q , dq both annihilate the state vector

bql~H > q

= 0

for

q > F

[~H > = O

for

q < F.

I~H >

(12)

Note that the number of "particles" and "holes" defined by the basis set ~q always remains zero: ^+^

^+~

< ~HEbqbql~.> = < ~H1dq ql~. > = O. The "physical" particles and holes are described by the wave functions.~ from Eq. (I) . For the corresponding operators we will write ~ , ~+ , ~ , ~ + q so that the field operator reads in this representation: q q q q

~

~m~

(13)

Inserting (3) into (11) we obtain the transformation between both sets of creation and destruction operators:

Coherent

Production

of Positrons

553

To calculate the number of particles (or holes) excited in a p a r t i c u l a r level one + • ~ + has to take the e x p e c t a t i o n value of the number operator n = ~ ~ or n = ~ ~ , respectively. This is immediately found using (12) and th~ ant~co~nutat~on r~la~ tions.

N u m b e r of particles:

Np =

N u m b e r of holes:

Nq

~Hlb p

=<~I

F

, p > F

~H> = s>F ~ lasq I~ , ~ <

F

(is)

Thus, if one is interested only in the number of created p a r t i c l e s (holes) it is sufficient to calculate the single particle t r a n s i t i o n p r o b a b i l i t i e s for the various initially o c c u p i e d (empty) states and to sum them incoherently. If one is interested in the number of correlated p a r t i c l e - h o l e pairs one has to calculate the e x p e c t a t i o n value of ~ .~ . This leads to ^ P P

w

&>w

This formula should be applied to analyze experiments like those p r e s e n t l y performed for c o i n c i d e n c e s between electrons and characteristic X-rays. Eq. (16) holds also for p a r t i c l e - p a r t i c l e or hole-hQle correlations if the sign of the second term is inverted. If the calculation involves channels w h i c h are treated incoherently, e.g. spin up and spin down states w h e n neglecting rotational coupling and magnetic effects, eq. (12) has to be supplemented by incoherent terms to account for random coincidences. The m e n t i o n e d case of spin d e g e n e r a c y leads to on enhancement by m o r e than a factor of two, namely N = 2N;~ + 2N; °. N. ° ONO o where N~ , ,N~.] are calculated from (iO) (12) for one spin state and N is summed over all combinations. To isolate the various contributions to p o s i t r o n p r o d u c t i o n d i s c u s s e d in the last section, let us w o r k out (15) and (16) in p e r t u r b a t i o n theory. The single particle amplitude for producing a pair consisting of a hole in level q and a particle in level p reads to first order =

~

(17a)

and to second order

In particular, taking p and q as electron- and p o s i t r o n - s t a t e s in the continuum, Eq. (17a) d e s c r i b e s the direct p a i r creation due to the rapidly varying C o u l o m b field (process f in Fig. 3). In Eq.(17b) deeply b o u n d inner shell states, w h i c h are initially occupied, will be the m o s t important intermediate states

554

J. Reinhardt et

E

aZ.

l

Positive~rgy Continuum

*m

-m

° l

NegotiveEnergyContinuum

Fig. 3.

Schematic representation of pair-production processes in heavy ion collisions. ~,b: ionization, ~: spontaneous and d,£: induced vacuum decay, ~: vacuum polarization shake-off.

r = Is , 2Pi/2 , ... (process b and d in Fig. 3). It follows naturally that the one-step process and two-step contributions with various intermediate states r cannot be distinguished and must be added in phase, i.e. a = a(1)+ a(2)+ . . . . In perturbation theory the number of particle-hole pairs i~qjust pq Pq

Np,q

=

lag P 12

=

)ap

q

1, 2.

(18)

which follows from the second term of Eq. (16) using (~r$4< 4

if

~

~ , ~t£ ~ 4

if

0-= 5 .

The number of holes is Nq

p F

N

(19)

This is only valid in perturbation theory, where many particle-many hole excitations are negligible. The total cross section for the emission of electron-positron pairs is obtained by integrating the squared pair amplitude over electron and positron energy and over impact parameter b and by summing over angular momentum

o

where 2j

-~

~

(20)

+ I denotes the multiplicity of the initial state.

NUMERICAL RESULTS The actual numerical calculations of positron emission in heavy ion collisions performed up to now have employed several approximations. (a) The amplitudes a (t) have been calculated in perturbation theory, i.e. the P~

Coherent Production of Positrons

555

integrals (17a,b) have been evaluated numerically prescribing Rutherford trajectories for the nuclear motion (Reinhardt et al. 1978). The full solution of the coupled channel equations (4) lead to the important discovery that multistep processes between bound states and also continuum states (continuum rearrangemenO are most important (see section 5). The "rescattering" of continuum particles may deform the shape of the emission spectra of ~-electrons and positrons. (b) The sum over ~£ was restricted to angular momentum j = i / 2 ( ~ = ± i), since these states are most severely affected by the strong field. Furthermore, we have taken into account radial coupling (R a/aR) only since the corresponding m@t~ix elements become large at small internuclear distance. Rotational coupling (~.j) is not expected to be effective 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/2~-states (Heiligenthal et al. 1978) and at large impact parameters. 2Still t ~ n its influence at very small impact parameters must be small (Q~vb/R), and only these contribute significantly to pair creation. Vacancy formation on the outgoing branch of the nuclear trajectory, on the other hand, is not relevant for positron production. (c) All wave functions and matrix elements have been calculated in the monopole approximation (Soff et al. 1978), which was proven to be remarkably good for the very heavy systems of interest for positron production. With the availability of full two centre-continuum wave functions, however, a considerable improvement in the description of special features (such as angular distributions, etc.) could be achieved. The effect of the finite nuclear size is included and was found to reduce coupling matrix elements by up to 30%. (d) The influence of the transverse electromagnetic field is not included in the instantaneous Coulomb potential of Eq. (I), i.e. magnetic and retardation effects are neglected. Furthermore, the energy shifts due to QED radiative corrections (vacuum polarization, self energy) are expected (Gyulassy 1974, Rinker and Wilets 1975, Cheng and Johnson 1976) to be much too small to influence the positron production rates significantly. (e) As already implied in the formalism of the last section we neglect the electron-electron interaction. Its main importance lies in the lowering of binding energies which could be included e.g. by using an effective Thomas-Fermi screening function to modify V(r). The nondiagonal couplings due to electron-electron interaction may be neglected for inner-shell excitation since they lead to relaxation times large compared to the collision time. (Rihan et al. 1978). Before discussing the resulting positron emission probabilities, let us first take a look at the b e h a v i o u ~ of the coupling matrix elements. Fig. 4 shows the radial matrix element < E I~-J~In > (responsible for the induced emission of positrons with various energi~s'°a'~) from the inner shell bound states n = Is, 2s, for Pb-Pb (Z=ZI+Z 2 = 164) andPPb-U (Z=174) collisions. The calculation made use of the Hellman-Feynman identity (E l ~ E2) :

where ~1 , ~p are eigenstates of the same Hamiltonian H. Most st~ikin~ is the strong increase of the matrix elements at very small internuclear distances R, clearly demonstrating the contraction of the electron and positron wave functions when the external nuclear charge aistribution begins to approach the point charge limit. The slope of the curves ~ even larger than that for the corresponding ionization matrix elements < nI~-~=IE > • it is steep~R e ' est for the lowest values of positron kinetic energy, reflecting the smaller energy d e n o m i ~ t o r in (21). The absolute value of the induced positron matrix element < E l ~ I n > lies about one order of magnitude below the ionization matrix element < n~d-~IE > due to the Coulomb repulsion of the positron wave function. It deoreasesD~it~ the principal quantum number of the bound state. Fig. 5 presents the matrix elements for direct pair creation, < E I~_IE > , where no intermediate bound states are involved, for various p o s i ~ r ~ e~ergies

556

J. Reinhardt

e~; a~.

E . The full lines b e l o n g t o ~ = -i (sl/o) ~ n t i n u u m waves, the dashed lines to ~ +2 (p.,~) waves. The m a g n i t u d e of ~/~ i ~ r E > grows with charge Z=Z +Z : Going f r ~ Z z = 264 (Pb-Pb) to Z = 174 (Pb~U~, w~ich corresponds to a 6% ~ha~ge increase, leads to a strong enhancement of the m a t r i x elements (cf. Fig. 5b). This, of course, will be reflected in a high Z - d e p e n d e n c e of the p o s i t r o n emission cross section. In Fig. 5 the electron energy is kept fixed at E - m = 10.22 keY. e V a r i a t i o n of E would shift the c u r ~ s but it will not introduce additlonal structure since it ~urns out that E I~--J E > is nearly independent of E . On the ~R e e other hand, the d e p e n d e n c e on p o s i t r o n energy is a result of the C o u l o m b r e p u l s i o n factor (see Fig. 6). Rod~olMotrtxELement(£pl~lE.)

Z-164 ls Fig.

4.

Z-164 2Pl/z

M a t r i x elements for induced emission from the is, 2s, 2p, 3p states versus two-center distance R.

:?,

?!! -14

IJ

Z.174 2Pl/z

zC ¢ -35 -2G -19

q2 100

50

Dis/Qnce[fm ]

RodiolMolrixElement

10-1

E,=1.02

5

50 Fig.

5.

100

150 0 .... R[frn]

5'0 ....

100 ....

M a t r i x elements for direct p a i r creation t w o - c e n t e r d i s t a n c e R.

150' R[fm]

versus

50

Coherent Production of Positrons

0"08I naturalunits

557

Radiol MQtrix Element (Epl~lE e) I U~

Z=164

w

Ee-'1.02

0.06

OJ04

//

_

__--

/I

~ - - ~ -

// 0.02

///

. . . . . ----

-

2 Fig. 6.

, -

-

-

3

-

-

-

%---1 "N.=+I _

4

Same as fig. 6, but as a function of positron energy E . P

In perturbation theory (Eq.(17)) the transition amplitudes for positron emission are Fourier transforms of the coupling matrix elements where the frequency E - E is governed by the energy matching of initial and final state. We have numerical ~ ly solved the integrals (17) using the matrix elements and energies of the monopole approximation, prescribing Rutherford trajectories for the nuclear motion. (Reinhardt et al. 1978). The radial matrix elements were fitted by a power law at distances larger than 200 fm to account for the limited range of validity of the monopole approximation. In addition, they were cut off with a Gaussian factor at R > iOOO fm to ensure convergence of the integral. This may be thought to simulate the influence of translational factors, which arise from the transformation between standing and "travelling" molecular orbitals but cannot be determined in ~ ? monopole approximation. In the calculation of the shake-off amplitudes aE ~ " an integration by parts was employed to make the integral convergent (SSf~ et al. 1977). Before turning to the calculated emission probabilities, let us inspect the timedependence of the transition amplitudes. Their general behaviour is quite similar to the amplitudes for vancy production (Fig. 7). Due to a steeper increase of the matrix elements, however, the radial scale is contracted and the maximal value of lal 2 is reached at distances R less than 1OO fm. The path of the time dependent positron amplitude in the complex plane is characteristic for the one step process. With the phase convention ~p(t=O) = 0 the final amplitude (t -~ ~) is purely imaginary since the.integrand is an odd function of time. This is valid both for the direct process a (I) and for the induced positron emission from E ,E e p (I) bound orbitals brought empty into the collision, a For the two-step process on the other hand:

n,Ep"

(t) h

(22)

558

J. Reinhardt

et

~Z.

015 Fig.

7.

Path d e s c r i b e d by the IsO ionization amplitude in the complex plane.

Im OE

0.12

0.08 t:O<

(104

-0.0~

Re OE i

-0.04 the i n t e g r a n d i s w e i g h t e d w i t h the t i m e - d e p e n d e n t i o n i z a t i o n

I

0

i

I ~ _

0.04 (1)

a m p l i t u d e a E ,n ( t ) ,

which starts from zero and builds up during the collision. T h e r e f o r e theeshape of the figure of aE(2~_ (t) is d i s t o r t e d and its symmetry is lost. Fig. 8 shows ,~ In a (2) (t~ a~ the energies E = I 02, E = -i 7 for the c o l l i s i o n Pb-Pb at E ,E ;Is e " p " impact ~ne~gy Ela h = 5.85 MeV/ (distance of closest approach 2a = 15.8 fm) and impact p a r a m e t e H - 5 = O. The final amplitude (t + ~) has a complex value defining a phas 9 angle ~ i relative to the i m a g i n a r y axis. It is important to k n o w the value of this angle since the t w o - s t e p amplitudes m u s t be added c o h e r e n t l y to each other and to the o n e - s t e p amplitude a (I) The latter is positive imaginary, E ,E . . e . . . . therefore the s h a k e - o f f p r o c e s s and the induce~ p o s l t r o n emlsslon via the is-state interfere somewhat desptructively. F u r t h e r investigations reveal that the phase angle A~ increases (i) with increasing e l e c t r o n energy, (2) with d e c r e a s i n g (kin . netic) p o s i t r o n energy, (3) with increasing b i n d i n g energy of the intermediate b o u n d state. The last point is d e m o n s t r a t e d in Fig. 9 where the complex amplitudes a E n (~) for induced emission of an e l e c t r o n - p o s i t r o n pair via the bound state n ~ndPthe d i r e c t amplitude a~l) E" • ~ " f o r ~ = + i are shown. The p a r a m e t e r s are the same as e in Fig. 9 except for t~e slightly h i g h e r e l e c t r o n e n e r g y E = 1.2. Fig. IO e x p l i c i t l y d e m o n s t r a t e s the time d e p e n d e n c e of t h e e d i f f e r e n t i a l probability of induced p o s i t r o n emission from the is-level integrated over electron en-

for various energies E . As we have a l r e a d y stressed the e x c i t a t i o n happens at very small i n t e r n u c l e a ~ d i s t a n c e s r e f l e c t i n g the strong d e f o r m a t i o n of the electronic wave functions. We note that, although there is no way of m e a s u r i n g the e x c i t a t i o n d u r i n g the course of the collision, the curves of Figs 8 and 10 have some p h y s i c a l meaning. In a g e d a n k e n e x p e r i m e n t one could imagine a m e c h a n i s m which stops the relative nuclear m o t i o n at an a r b i t r a r y distance R . Then the radial coupling vanishes and the e x c i t a t i o n amplitudes are frozen O at their

559

Coherent Production of Positrons

PositronQmp[~ude OE~l,EctsTWOstepprocess

~|mo

Pb- Pb , 20 15.8fm , E, =102 , Ep--1.7

L .

Fig. 8.

v

Reo

-2

Path described in the complex plane by the amplitude for induced positron emission out of the IsG state.

Positron omp[itudes

lira o

Im 0

Ep--l.5, E,- 1.2

10-3 '14. = -1

direct 2s

3s

2P~Jz

3PV2

t~-~ep

Fig. 9.

Phase and amplitude of the various second-order processes for s-wave (~=-I) and p-wave ( ~ = + i ) .

momentary values. E.g. such a mechanism would be provided by the nuclear interactions if the nuclei underwent fusion. The "high Fourier frequencies" of the collision then would show up particularly in a slower decrease of the high energy part of the emitted particle (electron or positron) spectrum since here the virtual excitations are large (cf. Fig. 10). In deep inelastic nuclear reactions the nuclei stay together for a prolonged time interval At during which the radial velocity R is small. The effect of such a time delay on the electron-positron field is to change the interference between

560

J. Reinhardt

#t ~ [ .

excitations taking place at the incoming and outgoing branches of the trajectory. This is readily d e m o n s t r a t e d looking again at the t i m e - d e p e n d e n t path of the amplitude in the complex plane. Fig. 11 shows how the amplitude a~ (2) m l~(t) of Fig. 8 changes if various delay times At are introduced b e t w e e ~ the branches of an otherwise unchanged R u t h e r f o r d hyperbola. O b v i o u s l y the spiral path b e l o n g i n g to the outgoing nuclear m o t i o n is rotated around the point a(t=O). K n o w l e d g e of the

wlO -6' JOE., %, ls (t)I z --imch2 4 Fig.

10.

Pb - Pb

Time evolution of the p r o b a b i l i t y for positron emission. Most oscillations vanish after i n t e g r a t i o n over p o s i t r o n energy.

Zo = 15.8 fm b=O Ee = 102

Ep, - 20

# -20

II

O0

10

20

3.0

40

50

60

70 80 2 f ,r,e t [f~/mc ]

Oistonce R [fm]

158

OE.&#{t)

Positron omplitude

f~ veri~Js detoy times At

1(j3! -Iron

Fig.

Ii.

The influence of a time delay at closest approach of the nuclei on the induced positron amplitude. The p a t h of a

/ /

/

\

\

/~ - , , I ~ ~

(~) is shown as solid line, the amplitude a(t) in two cases as d a s h e d curve.

/

/

~t ~?.o2 / \

",b~"i i k~

I\

\\

-

/ /iI

6"~i"

_~3

usual amplitudes a(t=O) and a(t + ~) for ~t=O allows a simple analytic determination of the influence of a delay time. The full curve shows the m o t i o n of the

Coherent Production of Positrons

56!

final values of the amplitude (t ÷ ~) in the complex plane as a function of delay time At(noted in multiples of IO-21sec). The epioycloid path depends on the two, phases (E -E )At and (E~ -E )At. For the first order amplitude we have only one phase angle ~nd the fina~ a~plitude moves on a simple circle around a(O). For a fixed time interval At the discussed effect leads to marked oscillations in the excitation spectra. Summation over various channels (e.g. intermediate states n or electron energies E , if the electron is not detected) will partially smear out thls znterference pattern. The investigation of this behaviour is interesting in connection with the suggestion to use deep inelastic collisions as a means to enhance the rate of spontaneous positron production in supercritical collisions (Rafelski et el. 1978). The amplitude of the latter process grows (to first order) linearly with At while the induced contribution gives rise to oscillations. Therefore both processes may be distinguished provided that collisions with At large enough do take place. In the following figures we will show results on positron production in undercritical systems in first and second order perturbation theory. Later we shall discuss the importance of multistep processes. We assume that all inner shell states are occupied initially, and have to be emptied during the collision by direct ionization. The positron emission probability is obtained from the squared coherent sum of one-step and two-step amplitudes la~1) E + ~ aE(2) calculated in e" p n e'Ep ;n •

.

e

12

perturbation theory. The results for the s- and p-channels are then added and multiplied by the degenercy factor 2. As intermediate states we take n=is, 2s, 3s (%6= -i) and n = 2PH~{~ert '3pl/2' 4Pl 2 ()6 = + i). This basis set should embody most of the relevant ~pace, only slightly underestimate the cross sections. Another source of possible errors is the neglect of mutual coupling between bound states. This effect can only be treated by a coupled channel calculation. Fi~+ 12 shows the differential positron spectra with respect to kinetic energy do for Pb-Pb collisions at various bombarding energies ranging from 3.68 to dE P 5.8 MeV/u. The spectra peak between 300 keV and 400 keV kinetic positron energy and fall off almost exponentially at±higher energies. The same behaviour is found for the positron energy spectrum dpeT (b) at various impact parameters (Fig. 13). dE The slope of the exponential is determined mainly by the Fourier decomposition of the nuclear motion R(t). Therefore it becomes steeper for distant collisions, i.e. at large impact parameters or at low incident energies. • Figs. 14 a,b show representative QED posltron spectra dO e - /dE for the system Pb-Pb and Pb-U at E = 4.45 MeV/u in comparison with the backgroundPfrom nuclear Coulomb excitation. The nuclear background in 208pb-208pb originates essentially from pair conversion of only two y-lines and therefore looks quite different. Setting a window at positron energies below i MeV will greatly suppress this background. In the 208Pb-238U system, on the other hand, many y-transitions occur between 1 and 3 MeV showing a serrate structure with an overall shape similar to that for QED positrons. + An important information is contained in the impact parameter dependence pe (b) shown in Figs. 15 a,b. The curves decrease monotonically with impact parameter, similar to those obtained for the ionization probability. The slope, however, is even steeper, so that 90 percent of the cross section comes from collisions with b < 30 fm. This is the consequence of the pronounced maximum of the coupling matrix elements at small internuclear distances. The nuclear background (Fig. 16) has a different impact parameter dependence and falls off even more steeply with b (note that we are comparing a process on the "atomic" scale with a nuclear process). The appearance of a bump at about b = 7 fm i~_~ig. 16a is caused by the reorientation effect of the 3- (2.615 MeV) level of -U~Pb. Two experiments measuring positron in collisions of very heavy ions recently have been performed at GSI. Backe, Bokemeyer, Greenberg et al. (1978) employed a

{ence

J. Reinhardt e # aZ.

562

10-1 ~. e. 2o

:

15fro

104 10e 2a

:o'b ~. 4| 1~

0 Fig.

12.

10+ I/ F . , .\\\ \ \\\\" " ° + °'°

\ \\'"" OEO positrons shak+ ott, induced 16sWtes)

500

\

1000

\

" 22J,f~ X

-

+0.~t

1500

r~l llJ~i d,~"" dE~ [keVJ

'%-~Pb

F

13.

Fig.

Differential positron cross-section in Pb-Pb for various bombarding energies.

ZosPb-~o8U A e \\

/'----- 0E8 positrons\ ~ucleor

II/ --so,

I

20 =22.0fro ~ \

2~+,.Ii .o,.\\

HI

I

I r

/ /_

~ckground % ~

/'"

I

'\\\J

nudeer bQckgreund

!

li/,+" Fig. 14.

\

L.4___\___~ Pb(3) [ \ L.--1

. . . . . . .

100

"

, -- -- QEDpositrons ] / (sheke off • inducedl6stolesDLJ

.......... E--

Ii /

"

~/i

ii

\

10100

Same as fig. 13, but for various impact parameters.

t_

2 o =20.9fm

HI Lt/i ..

, Ep (keY 1,%0 2000

i

500

bi+ mevl 500

1000

1500

/

l=kin!keV]

i ,

2000 0

500

1000

1500

2000

Energy distribution doe of positrons produced in dE 2OSpb and 208pb - 238uPcollisions. Dashed line: positrons from all quasi-molecular (QED) processes. Dotted li nes: background positrons from the conversion of y-rays following nuclear Coulomb excitation. Full curve: total sum.

Coherent Production of Positrons

P"{b) ,

i04

TO"4

,,o . . , _ . .

,,,,o,

J

563

,

[

,

'"o" o,,.=,

i0-s

10-6

107

lgOl

,o Fig.

2o

3o

~o

oLrmj I

,'o

o

z'o

3'o

~o

Impact parameter dependence of QED positrons production in Pb-Pb and Pb-U collisions for various b o m b a r d i n g energies.

15.

P"{b)_

Z°ePb - Z°SPb

ID' ~ b )

~pb_Z~U

2o = 20.9lm

2o•22frn

"~"~ -"-" \\

10°

/

nudeor

~ 10! ,....~

.... beckground

---

positrons - - OEO sum

"~"-~

",,

',,3"{2.615MeV)~

"

'

"

l[j B

-

r 10~71~-

'\

'-

',,

"3"'",

\

X '~

L

,

Fig.

16.

,

10

,

ml

Impnct'Porometer \ \ b[f ,

20

,,

~

30

,', i

40

b[fm] - --- % Impoc~Porometer "

_of

1~

u

'

.L

IU

'

'

20

"'

'

30

'

40

Impact parameter dependence of p a i r - p r o d u c t i o n in Pb-Pb and Pb-U collisions. N u c l e a r transitions (dotted lines), the QED p r e d i c t i o n (dashed line) and the total curve (solid line) are shown.

solenoidal spectrometer to collect the positrons. They m e a s u r e d total cross sectlons and dlfferential cross sections with respect to the scatt d i in two angular windows. The nuclear b a c k g r o u n d is well u n d e r s t o o d in Pbb collisions and can be subtracted. Fig. 17 shows the result at 2a = 19.2 fm (4.85 MeV/u)

564

J. Reinhardt

et a~.

103 a¢' [~]

dF~labLsrj

Fig.

17.

Differential positron c r o s s - s e c t i o n in Pb-Pb collisions as function of (averaged) scattering angle. T h e experimental points are taken from ref. 32, and are corrected for the nuclear background.

Pb- Pb gel] pasitrans shake aft*induced (6states Zo : 192 fm

10z

4-10

ela~

10o ' 2'0o ' 4'00

600

8'00

compared with the theoretical prediction. The two bumps of the c a l c u l a t e d curve originate from s y m m e t r i z a t i o n w i t h respect to p r o j e c t i l e and target, the left one being due to forward scattering of the p r o j e c t i l e nucleus. The m e a s u r e d values are some 30% above theory while the angular d e p e n d e n c e agrees very well (the nuclear b a c k g r o u n d looks much different). W i t h a d i f f e r e n t e x p e r i m e n t a l set up, Kienle, Kozuharov, G r e e n b e r g et al. (1979) p e r f o r m e d c o i n c i d e n c e m e a s u r e m e n t s for several systems at 5.85 M e V / u p r o j e c t i l e energy. Using a solenoid s p e c t r o m e t e r they singled out an energy w i n d o w of IOO keV width near the expected peak of the p o s i t r o n spectrum. We refer to the excellent lectures by P r o f e s s o r Kienle for all the d i f f i c u l t i e s and e x p e r i m e n t a l skill n e c e s s a r y for p e r f o r m i n g these experiments. Fig. 18 shows the p r o b a b i l i t y dP e+ as a function of c.m. scattering angle. The nuclear background, which has dE be~n estimated from the m e a s u r e d y - s p e c t r u m using an effective c o n v e r s i o n coefficient has been subtracted. The theoretical curves represent the ratio of the symmetrized positron cross section to the scattering cross section. T h e o r y now lies above e x p e r i m e n t but, again, the a n g u l a r d i s t r i b u t i o n agrees ~uite well. Fig. 19 gives our results for the total p o s i t r o n cross sections o e as a function of p r o j e c t i l e energy. The c a l c u l a t e d curve is compared with total cross sections m e a s u r e d at GSI (Backe et al. 1978). In both c o l l i s i o n systems the experimental values lie above theory by up to a factor of two. The d e p e n d e n c e on b o m b a r d i n g energy, however, agrees very well. Most c o n v i n c i n g 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 n u c l e a r positrons, but not by the individual contributions alone. This is also true for the d i f f e r e n t i a l cross sections d d e + / d ~ i o n at scattering angle 01a b = 45 ° m e a s u r e d by the same group. The absolute agreement is better, leaving a d i s c r e p a n c y of only about 30 percent. Let us now discuss in more detail the various contributions to the p o s i t r o n spectrum. The c o n t r i b u t i o n s of the various channels are d i s p l a y e d in Fig. 20 over an energy scale. In the left part of this figure the cross section for induced positron p r o d u c t i o n via the level n s a (circles) or n Pl/2 O (crosses) have been

Coherent Production of Positrons

565

~P"l-LM-~J 17 Probability for positron emission as a function of projectile cm scattering angle for (5.9 MeV/u) 208pb-208pb and

Fig. 18.

Er- 0.44....0.55 HeM

~

U-Pb

238U-208pb c o l l i s i o n s in the energy window E + = 440-550 keV. T h e o ~ is compared with measurements from ref. 33 where nuclear background has been subtracted. Both curves are symmetrized with respect to projectile and target nucleus.

/

•

k

~ 50o

Pb-Pb

I~

hl I , 300

,b]

L

L Ocm~ 90°

lotol Positron Cross Section

103

+

Pb-Pb

Pb-U

\ "'~ ++

I0z "',.\\ \ ~ +

",\,,"~

10

Experiment (GSI)"., lotol positron y i e l d " - - - IlEO (shoke oft • induced [6stotes]] . . . . . . . nudeor bQckground

+

--

% i

,

i

L

~

J

20 [tml!

2Q Jim L

,

,

i

,

,

,

,

i

115 20 25 25 30 15 2O 610.55 5[0 415 ~.0 3'.5 E~AtMeW~] 610 ~5 S'.O 4'.5 4'0 Fig.

19.

,

,

,

,

i

30

33 e~,c.eWo~j

Total positron cross sections for 208pb-208pb and 208pb" 2 3 ~ collisions in dependence of bombarding ~ energy or, equivalently, of distance of closest approach 2a. The experimental points are taken from ref. 32.

marked at the corresponding binding energies in the united atom limit. In addition the right half of the figure shows the differential cross section for direct pair creation ( M = + I ) which is seen to decrease exponentially with electron energy. The lower set of points has been calculated assuming initially occupied

566

J. Reinhardt

e~ al.

states higher

(two step process). A l t h o u g h the p r o b a b i l i t y for vacancy p r o d u c t i o n in orbitals b e c o m e s very large; the largest individual contributions to the positron cross section o r i g i n a t e from the deepest b o u n d states. This is even more p r o n o u n c e d in the collision of naked nuclei (upper set of points). Here the contribution from the isO level is very large and by far dominant, justifying the name "induced decay of the neutral vacuum" (Smith et al. 1974).

10 3 ~"[pb] -

"~

Positron Cross Section

\

\

\

10 z

20 = Z0.9 fm

Pb - Pb \

d~'" [..~bl dEp [keVJ 1

\

"\2P"2

T

\

Empty shells

\ \.Zs \\

Occupied shells

1

\

.10-~

\%,3p1/~ " "~m2S % \

E-

'!:';',

F i0 -I !_

, ~.~2 103

Fl

. . . .

-1000

Fig.

20.

4~)blt;

J

-500

%:%: .I|

", ,

,

",

,

Electron Energy

500

V a r i o u s contributions to the QED p o s i t r o n production: d i r e c t p a i r c r e a t i o n (right side) and induced vacuum decay (left side). The contributions from the bound states are shown over the maximal b i n d i n g energy. The d a s h e d lines b e t w e e n the dots are there to guide the eye.

The contributions to p o s i t r o n p r o d u c t i o n from the various d i s c u s s e d channels cannot be identified in a single e x p e r i m e n t since their b e h a v i o u r is quite similar. It is, however, interesting to study the v a r i a t i o n of their m a g n i t u d e and relative p r o p o r t i o n with total n u c l e a r charge Z + Z 2 Since in the region Z~ > i the 1 b i n d i n g of inner shells increases strongly and the i n n e r m o s t levels b e g i n to app r o a c h the lower c o n t i n u u m one expects a steep rise of the induced p o s i t r o n emission. This is d e m o n s t r a t e d in Fig. 21 where the p o s i t r o n p r o b a b i l i t y P e+ at imp a c t p a r a m e t e r b = 0 is shown for symmetric (or n e a r l y symmetric) collisions with the distance of closest approach 2a = 20.9 fm. The induced c o n t r i b u t i o n s from IsO and 2p - o grow very fast with charge Z.I + Z_. The curves for the charge de2 p e n d e n c e o~/~he total cross section look very similar, their slope being somewhat smaller due to the c o n t r i b u t i o n of h i g h e r impact parameters. At 2a = 20.9 we roughly find the scaling b e h a v i o u r Z n with n ~ 1 9 . If the incident v e l o c i t y is kept fixed the exponent is smaller, e.g. n ~ 15 at E/A = 5.9 MeV/u.

THE IMPORTANCE

OF MULTI

All the results we have d i s c u s s e d t h e o r y (in powers of ~/~). While

STEP PROCESSES EXCITATIONS

IN HEAVY

ION INNER SHELL

up to now have b e e n o b t a i n e d using p e r t u r b a t i o n the p r e d i c t e d c h a r a c t e r i s t i c s of the e x c i t a t i o n

Coherent Production of Positrons

Fig. 21.

Charge-dependence of the various quasi-molecular mechanisms of pair production. The distance of closest approach is kept fixed.

16&

567

}"(b = Oi Zo = ZO.9fm

10.5

10-s

'o'5-

10-7

I

10"8150

160

1~'0

180

ZI,.Z z ] 190 200

process have been found in excellent agreement with recent experiments on K-shell excitation, there remained an unexplained discrepancy in absolute magnitude: experimental date lay above theory by a factor of 3-5. In this section we point out that it is important to go beyond perturbation theory to obtain quantitative agreement. We demonstrate the importance of multi-step processes, where an electron is excited via various intermediate states, by full coupled channel calculations. We base our discussion on the formalism developed in section 3, eqs (2)...(16). Having established the relevance of the single particle amplitudes we must turn to the solution of the coupled differential eqs.(4). For any practical use the basis has to be restricted to a finite and not too large set. In particular the continuum must be discreti ed into finite segments. This is most simply achieved by replacing the integral ~ E ~(E) by a finite sum.~l-AE;~ ~ Since the matrix elements vary smoothly with energy the allowed spacing ~ & is limited by the oscillations of the phase ~ (t) : ~ = ~'~% {~ <
~l!

z

= <~(a~4~=l~s>

.

We found that it is satisfied sufficiently if states up to 2MeV kinetic energy are included. About half of the coupling strength goes into the continuum. For coupling between two continuum states the radial matrix elements

exhibit a pole at E=E . energy integral in (4).

In this case, one has to take the principal value of the

J. R e i n h a r d t

568

~

~.

S o l v i n g the d i s c r e t i z e d s i n g l e - p a r t i c l e c o u p l e d channel e q u a t i o n s n u m e r i c a l l y we have p e r f o r m e d c a l c u l a t i o n s of i n n e r - s h e l l hole production, 6 - e l e c t r o n e m i s s i o n and p o s i t r o n creation. While a d e t a i l e d a c c o u n t will be p u b l i s h e d s e p a r a t e l y we show the m a i n d i f f e r e n c e s c o m p a r e d to p r e v i o u s p e r t u r b a t i v e c a l c u l a t i o n s (Betz et al. 1976, M ~ l l e r et al. 1978, S o f f et al. 1978). The coupling m a t r i x elements were o b t a i n e d in m o n o p o l e a p p r o x i m a t i o n (Soff et al. 1978) w h i c h was p r o v e n to be r e m a r k a b l y good in h e a v y c o l l i s i o n systems for ~ and ~ L ~ states (angular m o m e n t u m q u a n t u m n u m b e r ~ = ~ 4 ) . All radial m a t r i x elements were e x t r a p o l a t e d to d i s t a n c e s R>2oo fm by a p o w e r law and cut off s m o o t h l y with a G a u s s i a n factor at R > 1 5 o o fm to simulate the effect of t r a n s l a t i o n factors. The i n t e g r a t i o n was p e r f o r m e d along R u t h e r f o o d t r a j e c t o r i e s up to R = 30oo fm using a s t a n d a r d p r e d i c t o r - c o r r e c t o r H a m m i n g routine. The m o s t r e m a r k a b l e new result of the c o u p l e d channel t r e a t m e n t is an increase of v a c a n c y f o r m a t i o n rates by up to a factor of 5 due to m u l t i - s t e p excitations, the effect b e i n g l a r g e s t for very h e a v y systems. T a b l e I d e m o n s t r a t e s the d e p e n d e n c e of the i $ ~ hole p r o b a b i l i t y (normalized to 2 due to spin degeneracy) on the number N of ~ $ ~ b o u n d state channels i n c l u d e d for a Pb Pb c o l l i s i o n at b=O and 2a = 20 fm (Elab/A = 4.65 MeV/u).

N(3S)

N(is)

N1s

i$

is

Is

2.96 %

-

-

2s

4.80 %

-

19.92 %

-

21.4o %

N

(N)

3s

6.84 %

4s

8.08 %

9.24 %

6s

9.o6 %

II.oo %

22.56 %

8s

9.42

11.34 %

22.68

%

22.12

%

%

The left column includes e x c i t a t i o n s to the c o n t i n u u m o n l y w h i l e in the m e d i a n (right) column the Fermi surface was taken above the 3 f O (IsO) state. All calculations include 16 c o n t i n u u m states. A n y further increase of this n u m b e r results only in m i n u t e changes of the e x c i t a t i o n rates. The d e p e n d e n c e of K - h o l e formation on the initial p r e p a r a t i o n of the s y s t e m can be read off the various columns in table I. To f a c i l i t a t e a c l o s e r i n s p e c t i o n of the e f f e c t of c o u p l i n g b e t w e e n c o n t i n u u m states Fig. 22 d i s p l a y s the (single particle) ~ - e l e c t r o n spectrum I~6¢,S~¢l~ The first order result (dashed line) shows a smooth, r o u g h l y e x p o n e n t i a l decline. I n c l u s i o n of the c o n t i n u u m - c o n t i n u u m c o u p l i n g shifts the s p e c t r u m to the right, i.e. the e m i t t e d e l e c t r o n s are r e - s c a t t e r e d to h i g h e r energy. In a simple static p i c t u r e such a b e h a v i o u r is expected: since the e l e c t r o n is c r e a t e d in the combined C o u l o m b field of the c l o s e l y a p p r o a c h i n g nuclei p a r t of its p o t e n t i a l energy may be l i b e r a t e d d u r i n g the s e p a r a t i o n of the ions. The o p p o s i t e effect, h o w e v e r m u c h weaker, should show up in the s p e c t r u m of positrons. The i n c l u s i o n of interm e d i a t e b o u n d states (solid line) is seen to increase the i o n i z a t i o n by s u p p l y i n g m a i n l y low energy e l e c t r o n s so that the net e f f e c t of m u l t i p l e e x c i t a t i o n s is a rise of the e l e c t r o n s p e c t r u m w i t h o u t change of shape. Fig. 23 shows the impact p a r a m e t e r d e p e n d e n c e of K - v a c a n c y p r o d u c t i o n for Xe-Pb c o l l i s i o n s at E / = 4.6 M e V / u c a l c u l a t e d in p e r t u r b a t i o n t h e o r y (dashed line) and in the c o u p l e d channel a p p r o a c h (full lines) for the (last occupied) Fermi level F = 3sO. The m u l t i - s t e p e x c i t a t i o n is seen to lead to on o v e r - a l l increase of P(b). A l s o shown are recent e x p e r i m e n t a l results o b t a i n e d by two g r o u p s in Darmstadt and S t a n f o r d (Greenberg et al. 1977, A n h o l t et al. 1979). T h e o r y is in rem a r k a b l e a g r e e m e n t w i t h these data and also w i t h m e a s u r e m e n t s in other systems like Xe-Au. This also holds true for the d e p e n d e n c e on b o m b a r d i n g e n e r g y w h i c h is shown in Fig. 23. Our theory, however, does not show a peak at small impact

Coherent Production of Positrons

569

parameter. This special feature was observed in one of the experiments and was tentatively explained in the Briggs model (Amundsen 1978) using perturbation theory in the united atom basis. It is ascribed to the influence of a dipole term which is not included in our present calculations. Q

~

I

t

Fig. 22.

Squared amplitude for 6-electron emission from the iso orbital calculated in perturbation theory (dashed line) and with coupled channels taking into account only the is~ state (chain line) or a set of 8 bound nsO states (full line).

Fig. 23. a) Probability for isO excitation as a function of impact parameter b for Xe-Pb collisions at 4.6 MeV/u lab energy. Full line: coupled channel result assuming a Fermi level above F=3sO, dashed line: perturbation theory. Also shown are experimental results of Greenberg et al. (1977) (triangles). b) Energy dependence of _ Plso (b) at varlous fixed impact parameters.

..... 1 r0 ,. ,~

Perturbationtheory 1so' • Cont. 1so'... 8sdr • Cont.

Ed~]

10"~ ''''

-%Jb)

'

~O''''

1000'"

- P~s'o-(b) '

54Xe%zPb

F. 3s~

/

// /*

iiiII ' 4'o

I

b[fm]

6b'

o

0

/ ,

I

I

/

2

4

6

Fig. 24 gives the prediced Is~ vacancy probability for the very heavy system Pb-Cm (zl+z2=178). For various positions of the Fermi surface. Note, in particular, the strong impact parameter dependence and the steep increase with bombarding energy. We have demonstrated the importance of multi-step processes in the quasimolecular description of heavy ion inner-shell excitations. The nonadiabacity of the electronic motion is the origin of violent couplings between neighbouring states which have the effect of increasing the excitation rates. (This contrasts with calculations in the atomic basis where the neglect of increased binding energy leads to

P.P.N.P, VOL 4 - - T

570

J. Reinhardt

~

al.

cross sections which are g e n e r a l l y too large.) The c h a r a c t e r i s t i c s of the excitation like the electron spectrum, impact p a r a m e t e r - and energy-dependence, remain almost unchanged. The n u m b e r of vacancies depends m o n o t i o n i c a l l y on the number of i n i t i a l l y occupied states. To o b t a i n accurate results for vacancy formation (and thus for p o s i t r o n production) the initial d i s t r i b u t i o n of electrons (i.e. the p o s i t i o n of the Fermi surface) m u s t be known. The u n d e r s t a n d i n g of the dynamics of electronic excitations in h e a v y ion collision~ is important for an e n v i s a g e d s p e c t r o s c o p y of s u p e r h e a v y quasimolecules, m o l e c u l a r orbital X-rays, ~-electron s p e c t r o s c o p y and p o s i t r o n creation by strong and supercritical fields. D e t a i l e d studies of the impact parameter, energy and charge dep e n d e n c e of these p r o c e s s e s as a function of the b i n d i n g energies and coupling strengths that go into the coupled channel c a l c u l a t i o n s will h e l p to come closer to this goal.

j,-..//..

/Uo Fig.

-50 -24.

DYNAMICAL

100

150

2

4

6

Same as Fig. 2 for Pb-Cm collisions at 4.7 MeV/u lab energy. The various curves in a) b e l o n g to d i f f e r e n t p o s i t i o n s of the assumed Fermi level F as indicated in the figure.

TREATMENT

OF S U P E R C R I T I C A L

COLLISIONS

The formulation of our theory of p o s i t r o n p r o d u c t i o n in heavy sented before, is applicable only in u n d e r c r i t i c a l collisions states have joined the lower continuum. We have e x p a n d e d the tion (Eq.(3)) in the adiabatic m o l e c u l a r basis c o m p r i s i n g the I~ E > , various bound states I~ n > and the p o s i t r o n c o n t i n u u m to ~oupled d i f f e r e n t i a l treated in p e r t u r b a t i o n of the bound states has function is smeared out

ion collisions prewhere no bound electron wave funcelectron continuum I~ E >. This leads

equations for the expansion coefficients w~ich have been theory. In the supercritical situation one, or several, d i s a p p e a r e d from the discrete spectrum I~ n >. Its wave in the c o n t i n u u m I~E > as a B r e i t - W i g n e r resonance with

a certain width F, indicating the p o s s i b i l i t ~ of spontaneous decay. L e t us discuss three d i f f e r e n t m e t h o d s of solving the p r o b l e m of dynamical transitions

Coherent Production of Positrons

571

involving such a resonance. I. Since the continuum wave functions in the vicinity of a resonance differ drastically from their normal shape, the use of the states I ~ > in the expansion (3) leads to a pathological and nearly singular bound state - ~ o n t i n u u m and continuum - continuum coupling which is highly impractical for numerical calculations. Nevertheless, one might artificially discretize the spectrum of the two centre Dirac operator, e.g. by imposing a boundary condition on the wave functions, and solve the coupled channel equations. The motion of the resonance state through the lower continuum leads to a great number of avoided crossings with transition probabilities very close to unity (see Fig. 25). Since we are interested in the (very small) fraction of holes which finally remain in the lower continuum, it will be difficult to obtain reliable results. The choice of this basis is obl~iously not well suited to solve the dynamics. II. For an alternative treatment of the problem, one has to change the basis. One simple way to do this is to freeze the basis near the critical internuclear distance Rcr, where the bound state joins the continuum. The so defined undercritical wave functions I~O> are, of course, no longer eigenstates to the Hamiltonian HTcD(R) if R < Rcr. Therefore the coupled system (4) is modified to

k 4

(23) where AV(R) = H(R) - H(Rcr) = V(R) - V(R ). This equation in principle is cr exact. It turns out, however, that its treatment In first order is not sufficient since the strong distortion of the adiabatic wave functions under the action of AV is important. While the true binding energy E of the diving state resonance (as obtained from a phase shift analysis of I~E >) res.increases strongly for closely approaching nuclei, the first order approximation E(1)n - EnO = < ~ I A V I ~

>

mc 2

-mc 2/

Fig. 25.

--t

Schematic view of the quasi-bound overcritical is-state penetrating the discretized negative frequency continuum. Each avoided crossing contributes to spontaneous positron creation.

572

J. Reinhardt

et al.

remains small. It turns out that this m e t h o d cannot be applied even in the context of a coupled channel approach. The convergence of the matrix elements of the C o u l o m b p o t e n t i a l AV is simply too slow to be useful for p r a c t i c a l calculations. III. We want to find a d y n a m i c a l d e s c r i p t i o n of the excitations in the presence of a "diving" level which is a smooth e x t e n s i o n of the treatment of u n d e r c r i t i c a l collisions. To achieve this, we observe that the resonance w a v e function, i.e. the c o n t i n u u m solution [~E >, closely resembles a normal bound state except for an oscillatres ing tail of small amplitude indicating the o c c u r r e n c e of tunneling through the particle-antiparticle g a p b e t w e e n E - V(r) + m and E - V(r) - m. Therefore it m a y be p o s s i b l e to c o n s t r u c t a "reasonable" b o u n d state I~ > by cutting off [~ E > at a c a r e f u l l y chosen distance r = r 0 and normaliziRg the t r u n c a t e d wave r S func~lon. The p r o b l e m then is to c o n s t r u c t a new continuum I~ E > which is orthogonal to the r e s o n a n c e

state

[¢R >

< ~.

I~R >

P

= o

(24)

P

and complete. Such a p r o b l e m has b e e n treated in the context of n u c l e a r p h y s i c s using a projection o p e r a t o r technique (Wang and S h a k i n 1970, see also M i c k l i n g h o f f 1977). In this language we o r i g i n a l l y have an operator p r o j e c t i n g onto the set of free positron states Po = I~E > < ~E I (25a) P P and the c o m p l e m e n t

where

l~e> includes (H - Ep)

all bound

states

[~n > and the upper

continuum

[~E >" e

We have

19 E > = O P

(H - E ) I~ > and

p2 P , 2 o = o Qo = Qo

(26) = 0 ' Po Qo = O

,

Po + Qo = i .

o u r o b j e c t i v e is to remove the resonance c o n t a i n e d from P-space and to transfer it to Q-space: Q = P =

l~e > < ~e 1 + i -Q

[~ E

> p

J~R > < ~R I

(28a)

.

Here and in the following d e r i v a t i o n s I~R > is o r t h o g o n a l to all states I ~ < ~I~R

in

(27)

> = O

(28b) we make >

the important

assumption

that

(29)

so that P and Q are also p r o j e c t i o n operators. E x c e p t for this p r o p e r t y the choice of I~R > is a r b i t r a r y and m u s t be m a d e on physical, not mathematical, grounds. We come back to this in a moment. We now define the m o d i f i e d c o n t i n u u m I~E > by the eigen-value equation of the p r o j e c t e d H a m i l t o n i a n PHP: p

Coherent Production of Positrons

573

(PHP- Ep) ] Ep > = o Using

(28b),

(28a),

(3o)

(26) and (29) this equation can be transformed to

(~-~p)t+Pp>

= <,~,~t~t~r> t ~ >

(31)

Thus the modified continuum satisfies the original Dirac equation with an additional inhomogeneous term proportional to I~R > . Eq. (31) may be solved by help of the Green's function G = (H-E) -I The formal solution

=

~,+,~I+, i,+,,>-) t e~,, >

(32)

may be employed to derive the orthogonality of the modified continuum states :

For numerical purposes a different approach is more useful. Due to its special form (it has a degenerate kernel) the integro-differential equation (31) can be solved by integration. We make the ansatz

t +"+-,, > = o_ t,..++,, > + ~ I ~ > where

I+E

(34)

> is the solution of the homogeneous equation (26a) and

+ E ( i ) > solves

the inhom~geneous equation with an arbitrary constant ~ # O (e.g. 7 = ~) :

The constants a and b are then uniquely defined by the orthogonality requirement (24)

I e"+,+> --- It'.+,,,.>_ and by the asymptotic normalization condition of the wave function I+E

>

o

The non-diagonal matrix elements of H, i.e. those coupling the P and ~ s p a c e ,

<+,,.I~ l~p'> -- -,t 2i,,.~,.,4"~ "'+'. . The new basis set

t+E

>,

I+n >,

t+R>,

I+E

are

(3~)

> can be used to expand the time-

dependent wave functio e (3). We obtain thePusual set of differential equations with one additional coupling between the resonance state and the modified continuum: For example Eq. (17a) is modified to

,:,.,,,,,+, t+>

+ <+,,.,,, +,> ].

The second term can be understood to describe a spontaneous transition with the decay width (see first lecture, eq. (41))

l" (+~,,") "+

Z~[ <-~ ,P+,. tt..

Iz

(39)

Apart from the spontaneous decay also induced transitions are present and the theory allows, in principle, a separate investigation of these processes. The main problem in this formalism is the proper choice of the resonating bound state wave function I~R >. As indicated above, this may be done by truncating a continuum wave function in the resonance region. Numerical investigations have shown, that the modified continuum wave functions l~Ep > are quite sensitive to

574

J. R e i n h a r d t

et al.

the p r e c i s e nature of the cut-off. While the decay width F(E ) comes out correctly in various procedures, a sharp cut-off at some value rre~eads to an irregular R - d e p e n d e n c e of the m a t r i x elements. Furthermore, if r ° is kept fixed, the wave function I~R > does not go over into ; I~is > at the critical o internuclear distance, so that there is no smooth transltlon b e t w e e n the undercritical and o v e r critical regime. A g e n t l e r way to truncate the wave function, was found to be the use of a m o d i f i e d p o t e n t i a l like

V(r)

with

E

res

=

I

- V(r+)

V(r)

r < r+ for

V(r+)__

+ m

r > r+

= 0

°

The value of r + is d e f i n e d as the outer edge of the c l a s s i c a l l y forbidden tunneling region b e t w e e n electron and p o s i t r o n states. At larger d i s t a n c e s the potential is kept fixed so that the wave function goes to zero. W i t h this choice discontinuities in I~R > are avoided. C o u p l e d channel calculations with the matrix elements (eq. 38) Have recently been carried out by J. Reinhardt. We will dem o n s t r a t e the m a i n results in the following figures. The m a i n effect resulting from the use of amplitudes which are solutions of the coupled eqs. (4) is an increase of e x c i t a t i o n probability. It is less pronounced, however, than in the case of K-hole production, never e x c e e d i n g a factor of .. Figs. 26 and 27 show the resulting p o s i t r o n p r o b a b i l i t i e s for four d i f f e r e n t systems with total charge Z I + Z_ = 164, 174, 184, 190 at b o m b a r d i n g energy 5.9 M e V / ~ 2 Here and in the followin~ figuresFis taken to lie above the 3sO state. The last two systems are supercritical. One observes again a very steep increase with nuclear charge but there are no striking q u a l i t a t i v e d i f f e r e n c e s b e t w e e n the shapes of the p o s i t r o n spectra (Fig. 26) or the impact p a r a m e t e r d e p e n d e n c e (Fig. 27), drawn as a function of m i n i m u m i n t e r n u c l e a r distance Rmin = a(l+(i+(bTa) 2"). To obtain these results, however, it has b e e n essential to include the "spontaneous" coupling for collisions where the isO state joins the lower continuum. If it is left out of the calculation the resulting p o s i t r o n spectrum (Fig. 28) or impact p a r a m e t e r d e p e n d e n c e (Fig. 29) d e n o t e d by the d a s h e d lines is strongly altered. The "induced" radial coupling (first term in eq.(38)) is changed at the same time as the spontaneous coupling (second term) becomes important. Both contributions add up c o h e r e n t l y and can not be observed separately. P r o b a b l y the only way to get a clear q u a l i t a t i v e signature for spontaneous p o s i t r o n p r o d u c t i o n will be to go above the Coulomb b a r r i e r and select collisions w i t h p r o l o n g e d nuclear contact time. (Rafelski et al. 1978a). As already m e n t i o n e d in section 4 the nuclear delay time At provides a handle to d i s t i n g u i s h both types of coupling. For At in the order of the decay time of the resonance a sharp and intense (several p e r c e n t probability) p o s i t r o n line would be generated. The onset of this effect m a y be o b s e r v e d for contact times of the order 10 -20 sec as d e m o n s t r a t e d in Fig. 30. (Note that only the isO c o n t r i b u t i o n " laE ,E ;isoI 2 calculated in p e r { u r b a t i o n theory is shown. Full coupled channel calculations will m o d i f y the curves but not the general trend.). Whereas the o b s e r v a t i o n of a sharp line would require the formation of a l o n g - l i v e d s u p e r h e a v y compound nucleus times of the order Io-2Osec m a y be r e a l i s t i c a l l y expected in deep inelastic collisions.

SUMMARY We have d i s c u s s e d the theory of p o s i t r o n p r o d u c t i o n in very heavy ion collisions as an example to show the importance of coherence for q u a s i - m o l e c u l a r processes. The theory agrees well with p r e s e n t experimental data from GSI. It is note-worthy

Coherent Production of Positrons

575

~-z~ p'"

E/A = 5.9 MeV/u b=O

~~~~

~4 |uu,,u 164 ~5 U- £t .IJ

I(] s

lOe

R,,,llm]

156 Fig. 26.

,

,

t

.

500

,

,

.

i

,

1000

,

.

.

,

,

.

-

1500E,,~eV] Fig. 27.

Coupled channel calculations of energy spectra of positrons emitted in 5.9 MeV/u collisions of various systems.

Coupled channel calculations of the positron emission probability as a function of distance of closest approach for 5.9 MeV/u collisions.

i

I

2g = 17fm b=0

b =0

10-4 I

10-~

i

£,[mc2]

Ep[mC2]

L

-2 Fig. 28.

-'3

-4

Energy spectra of emitted positrons in supercritical U - U and U-Cf collisions only the ~=-i contributions have been drawn. Black lines: Full coupled channel calculations. Dashed curves: The spontaneous coupling has been omitted.

576

J. Reinhardt

el

c:l,

[1so-Positrons U-U ZailZ,r], b:Stm, Ro=16im --T

P'" (gcm) ST/zcontributiono~/./~

~t = (3

I

f,t J.11] z,~c

/ \u-c~ L a.~.lO-Z~sec

61

Al's'°"~I

/

0 Fig.

29.

40

80

o -'J

120 160

Positron emission p r o b a b i l i t y for supercritical collisions as a function of scattering angle. Meaning of the curves as in Fig. 30.

-~

Fig.

30.

-2 E~.imdl-3 -1

\,,

-2 Ee[~z]-3

Spectrum of emitted p o s i t r o n s for UU collisions at E=8.5 MeV/u and b=8 fm for various p r o l o n g e d nuclear contact times At. A schematic nuclear trajectory has been assumed consisting of two hyperbolic and a circular orbit. D a s h e d curves: spontaneous coupling only, chain curves: induced coupling only, full curves: sum of both. Only the c o n t r i b u t i o n of the 1sO state c a l c u l a t e d in p e r t u r b a t i o n theory is shown.

that the agreement could not be achieved w i t h o u t treating the full coherence between ionization and pair-creation. This insight has equal b e a r i n g for related phenomena, e.g. m u l t i p l e excitation p r o c e s s e s in inner shells or m o l e c u l a r X-ray transition. We also note that the agreement b e t w e e n the observed spectra and theoretical predictions constitute the first evidence for the novel aspects of quantum electrodynamics of strong fields: the n o n - p e r t u r b a t i v e character of direct pair production (the "shake-off of the vacuum p o l a r i z a t i o n cloud") and the onset of the change of the vacuum state from neutral to being charged. The identification of the spontaneous vacuum decay, however, m u s t await further calculations and future m e a s u r e m e n t s w i t h the h e a v i e s t collision systems, such as U-Cf and perhaps more refined experiments with deep inelastic collisions.

REFERENCES

Amundsen, P.A. (1978). J. Phys. B II, L737. Anholt, R., W.E. Meyerhof, and Ch. Stoller (1979).

Preprint.

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577

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