The decay of the vacuum in supercritical fields of giant nuclear systems

The Decay of the Vacuum in Supercritical Fields of Giant Nuclear Systems J. FINK)J. A. MARUHN)B. MOLLER, U. MOLLER, L. NEISE, J. REINHARDT)T. DE REUS, A. SCHAFER, P. SCHLI~TER, W. SCHMIDT, S. SCHRAMM, G. SOFF, D. VASAK AND W. GREINER* lnstitut fiir Theoretiache Physik, Johann Wolfgang Goethe Universit?it, 6000 Frankfurt am Main, F.R.G.

ABSTRACT

The structure of the vacuum has become of general importance in modern physics. In this context the vacuum of quantum electrodynamlcs plays a special role because it changes drastically in strong external electric fields and these changes can be followed in a controllable way in the laboratory. It is reviewed how in supercritical electric fields the electron-positron-vacuum, i.e., the ground state of QED is changing from a neutral to a charged vacuum. The theoretical concepts, developed during the last 15 years) can be tested in collisions of very heavy ions, like U+U. U+Cm) etc. The /mportant theoretical and experimental stages are described; in particular it is shown that the vacuum decay might have recently been experimentally verified: Sharp line structures in the positron spectra with pronounced threshold effects are the signals of this fundamental phase transition. Furthermore, the observed ~-ray and 6-electron spectra, the non-appearance of a massive line structure in undercrltical systems like U+Ta and the quantitative understanding of the innershell ionization phenomena in giant quasimolecules (their agreement with theoretical predictions) give confidence into this interpretation.

At the same time the positron line structures possibly indicate the existence of rather long living giant nuclear system with Z - 180-190. It is shown that most likely giant nuclear molecules have been formed as doorway states, but that other new not yet completely understood nuclear structure of giant complexes is indicated. The structure of the giant complexes can eventually be investigated spectroscopically by the positrons from the vacuum: The positron spectrum should not only show the vacuum decay llne, but also Raman and Stokes satellites revealing the nuclear collective modes of the giant system formed. Various minima in the collective potential of giant systems should reveal the vacuum decay and their own local nuclear structure through corresponding classes of positron lines. The seemingly constancy with (ZT + Zp) of the principal positron line structures observed so far could indicate the formation of always the same giant object. Other possibilities to explain this effect (vacuum vibrations) excitation of the same nuclear mode in various giant systems and its supercrltical conversion) are also discussed. In particular, the possibility of the creation and decay of a yet unknown particle causing constant positron line structure is shown to be in contradiction with atomic precision experiments. Finally'the overcritical phenomenon in other areas of physics (e.g., overcritical gluo-electrlc fields, overcritical gravitational fields) is shortly discussed.

*)

Invited Speaker

2

J. Fink et al. KEYWORDS Structure of the vacuum; heavy ion atomic physics; supercritical fields; atomic clock; spontaneous positron emission; giant nuclear systems; fission of quark bags; strong gravitational fields.

I.

INTRODUCTION

During the last decade the understanding of the structure of the vacuum in field theories has become of general importance. The conjecture that the vacuum is not just an empty piece of space, and its philosophical implications have a very long history, dating back to the Greek philosophers of the Eleatic school and to Aristotle in particular. Only with the advent of a quantum field theory it became clear that the vacuum is a physical object, which can be subject of physical experiments. In modern gauge theories the vacuum necessarily has to have certain structures; most of them not yet completely understood. Let us remind you only on some of them: i) In the ~4 theories the vacuum is degenerate, because the V(~) potential looks as in figure i, and degenerate groundstates are obviously possible. li) In some gauge theories the vacuum has to be assumed to contain Higgs fields in order t o produce masses for the particles in a gauge-invariant way. Such Higgs fields have not been observed to date. Hence one does not know whether they really exist, but we do not k n o w a n y better way to construct invariant, renormallzable theories allowing partlcles to obtain masses. iii) Particularly in quantum chromodynamics one distinguishes a true vacuum, which expels gluo-electric field lines in a similar way as a super-conductor repels magnetic field lines. Then, an assembly of real quarks, which are the sources of the gluo-electric field lines, would have to form a bubble (bag) around the true vacuum. In that bubble the true vacuum cannot exist because of the presence of gluo-electric fields and the bubble will be filled with a simple vacuum (sometimes also called perturbative vacuum). We illustrate these ideas in figure 2. We shall come back to the structure of the simple vacuum in case of supercritical gluo-fields appearing during the deformation and fission of a bag at the end of the talk. The problem with these vacuum ideas described so far is that these vacuum models are a priori given. We w o r k with them, but we have so far no tool to change in a relatively easy and controlfable way one vacuum structure into another one. Such 'experimenting' with the vacuum is necessary to substantiate the underlying ideas, or, to say it differently, to convert the models and ideas into true physics. This is precisely what is achieved with the vacuum of the best field theory we have up to n ~ , i.e.,

iV(~)

F~&ure 1. The g r o u n d s t a t e (vacuum) i n a t ~ t h e o r i e

is degenerate.

Figure Z. The true vacuum of QCD is liquid-like, with gluon and quark balls of a certain size forming and transforming. Quarks can only exist within the simple vacuum inside; they are caught inside the bubbles.

Giant Nuclear Systems II.

3

THE V A C U U H O F QUANTUHELECTRODYNAHICS

It is the electron-positron-photon groundstate, which is best described by lookin E at the single particle spectrum (flgure 3) of electrons in an external field. The spectrum dlvides into a positive and a negative energy spectrum divided by the energy gap of 2m,c 2, within which one finds the bound states for attractive external potentials. Let us call the negative energy states @n' the positive energy states #p. Then the field operator at time t = 0 is defined as

~(x,t=o)

=

Zp%p ~p(X) + Zn A+n @n (x) "

(1)

b p are the annihilation operators for electrons and a+ n the creation operators for positrons. This definition guarantees that the energy H D of the electron-positron field is positive definite, i.e.,

The vacuum

lO>, characterized by the Fermi energy EF, i.e., by In> with E n < E F filled with

electrons, is best illustrated in hole theory (figure 4). A hole in this 'Dirac sea' is then interpreted as a positron. Note, the position Of the Fermi surface, which divides the single particle states into those which are to be counted as electronic and those which are positronlc states, is up to nature and can be deduced experimentally by observing, e.g., the threshold for + e e -pair production. It cannot be chosen at will. This so defined vacuum I0> is neutral: If one calculates the vacuum expectation value of the charge operator

=

½ [~+(x,O), ~(x,0)].

,

(3)

which i s t h e zero-component o f the c u r r e n t - f o u r v e c t o r

J~ = ½ 1~, ~ l

,

(4)

one finds <01 ~ I0>

~

Pvac pol

=

~e (Zn ~n@n + + - Zp ~p~p) .

(5)

This vanishes for the field-free case because of s ~ m e t r y (equal number and structure of n and p states), but gives the vacuum polarization charge: Pvac pol (x) in case an external potential is present. The latter gives rise to a part of the Lamb shift and is well established. We can say that in weak fields the vacuum is a polarizable medium, characterized under weak external fields by the displacement charge Pvac pol (x)' for which J d'x Pvac pol ( x )

=

0

(6)

~£ positive continuum

unbound (free)st0tes

me c 2

÷mec~' bound states

-Ze -moc z

Fermi surface negative .continuum

- me C2 !

c~ ~ .-.-

unbound (f~e)st0tes

Figure 3. The slmple particle spectrum of the Dirac-equatlon in an attractive external field. Figure 4. The neutral vacuum of QED can be viewed as the states of negative energy filled with electrons. The infinite charge is renormallzed to zero.

J. Fink et al.

T

a)

Figure 5. The vacuum polarization charge around a central nucleus (a) and around two colliding heavy nuclei (h). In the former case the static s electrons are shifted in energy somewhat due to the modification of the Coulomb potential by Pvac pol (this is part of the Lamhshift); in the latter case the vacuum polarizatlon the ions.

charge is partially stripped off because of the motion of

This vacuum polarization displacement charge can be illustrated as in figure 5. Of particular interest to us here is the stripping-off of the wacuumpolarization charge in case of the moving ions. This leads to the ejection of the e+e'-palrs, which goes like (ZT + Zp)~"-power as

a

function of the colliding charges (Soff, 1977; Relnhardt, 1977; for a review cf. also Reinhardt, 1985). It has been observed experimentally by Backe, Kankeleit, and coworkers (Backe, 1978) and by Kienle, Greenberg, and associates (Kozhuharov, 1979).

III.

THE CHARGED VACUUM IN SUPERCRITICAL FIELDS

If the charge of the central nucleus is increased, the spectrum looks for point and extended nuclei as in figure 6. For point nuclei the we11-known fine-structure formula results, which has no solutions beyond Z = 137. This puzzle is postponed to the so-called critical charge Zcr = 173 for extended nuclel, where the Is½ state 'dives' into the negative continuum. For even higher central charge, at Z'cr = 183 the 2p~ state dives, etc. The significance of this is the following: In the overcritical case the dived bound state becomes degenerate with the (oc+ cupied) negative electron states. Hence spontaneous e e -pair creation becomes possible, where an electron from the Dirac sea occupies the bound state, leaving a hole in a continuum state~ which escapes as a positron. This is a fundamentally new process, which can also be expressed in the following way: The neutral vacuum of QED becomes unstable in overcritical electric fields. It decays in about 10 -2" sec into a charged vacuum. The charged vacuum is stable due to the Pauli principle. It is two times charged, because of the spin degeneracy two electrons (~,~) can occupy the dived shell. After the 2p~ shell dived beyond Z'cr = 183, the vacuum is four times charged, etc. This cha~ge of the vacuum structure is absolutely fundamental; it is not a perturbative effect, as all other known QED effects (vacuum polarization, self energy, etc.), but a massive phase change. The vacuum polarization charge contains now a real component, not only a displacement charge, and we now have < charged vac. I ~ I charged vac. >

=

2e, 4e, etc.

(7)

The significance of this vacuum charge is made more transparent in a different way in figure 7. The space in the box is pumped empty by an elementary partlcle pump. There is only the central nucleus as a source for the electric field left as a spectator. The empty space represents the neutral vacuum. In the undercrltical case, it is stable. In the overcritical case, however, two positrons are emitted into free states, travelling around in the box and simultaneously two electrons are created in a strongly bound state around the nucleus. The positrons, being in free states, are easily pumped away. The electron cloud represents the charged vacuum. It is the stable groundstate in the overcritical field case.

Giant Nuclear Systems Figure 6. Lowest bound states of the Dirac equation for nuclei with charge Z. While the Sommerfeld finestructure energies (broken lines) for K = -I end at Z = 137, the solutions for extended Coulomb potentials (full curves) can be traced down to the negative energy continuum at the critical charge Z . The bound states entering cr the continuum obtain a spreading width.

[k,V] I~ positiveenergycontinuum 50C

0

-SOC

~ -

, -

e, , . ~

itehtectrons - - \ : \ - -

-1000

During the course of the last decade the following questions have been answered: 1. Can vacuum polarization hinder the vacuum decay, i.e., prevent diving of the bound levels? The answer g l v e n b y Soff (1973) and Gyulassy (1975) is no. The energy shlft due to vacuum polarization is of the order of a few keV at the diving point. 2. Can self energy prevent diving? The answer is, again, no. It was given by Soff and colleagues (1982) only 4 years ago. 3. Can non-linear effects in the electromagnetic fie]d (e.g., of the Born-lnfeld-type, limiting electric fields in nature) or in the electron-positron-field (e.g., of the Heisenberg-type) become eventually so large in supercritical fields that the vacuum decay is prevented or, at least, substantially shifted toward higher central charges ? The answer has been given many years ago by Soff, M~ller, and Greiner (cf. Soff, 1973) and is again no, if the excellent agreement between QED of weak fields and high precision experiments (Lambshift, g-atoms, etc.) is not to be destroyed (el. also Reinhardt, 1977, 1985). 4. What happens in the point limit, i.e., in the case when the radius of an overcritical charge R ~ O. This very interesting problem was solved by G~rtner, Heinz, M~ller, and Greiner (1981) years ago and the following answer was obtained: As R ~ 0, more and more bound levels with K = il dive, in an onion-type fashion (figure 8). The vacuum becomes higher and higher charged; the charge distribution of the vacuum, onion-type structured, shrinks to zero, so that in the limit H ~ 0 the effective charge Zef f of the system, consisting of the central charge Z minus the vacuum charge Zva c approaches 137: Zef f

=

Z - Zva c

~

137

for

R ~ 0

(8)

This is a most interesting result, telling us, that QED does not allow point charges with charge larger than 132. In other words, the coupling constant of QED (for point particles) can never become larger than i.

o) undercriticot

pump

b) overcriticoL

pump

Figure 7. The vacuum in the box is penetrated by the electric field of a central nucleus with charge Z. The nucleus is solely a spectator, furnishing the source of the electric field.

J. Fink et al.

///////////// //

.........

//

Z- 153 /

//.

,..%

//

"+~'""

r~rn~ining

//Z's~o~uum ~

R

[nr

Fi&ure 8. As the radius R of the central charge shrinks to zero, the vacuum becomes onlon-shell-type higher and higher charged, shielding the central charge. In this way point charges higher than 137 are prevented in QED. Figure 9. The formation and change'of quasimolecules in the course of a heavy ion collision. The various innershell processes are illustrated (from left): ionization, quasimolecular X rays, characteristic X rays. Not shown are positron creation processes (for those see figure

ll). IV.

SUPERHEAVY~UASIMOLECULES

Now we come down to earth and ask ourselves whether this can all be experimentally tested; can one make physics out of the theoretical ideas? A crucial suggestion was made in 1969 by Grelner (and also by Zeldovlch and Gershtein) (cf. Relnhardt, 1977, 1985) that during the course of the collision of two heavy ions, superheavy quasimolecules are formed (see figure 9). This is, because of the adiabaticity (Vion/Vel = 1/20) of the fast electrons orbiting the colliding slow nuclei. The proper quantum-mechanical formulation of the transiently formed superhesvy molecules and the various processes therein required the solution of the Ywo-Centre-Dirac-equat~on, which was first achieved by M~ller and Greiner (1976). A typical TCD-level diagram is depicted in figure I0 for the U+Cm case. The critical distance, at which the Is½ level is diving, is about 35 fm. Notice the s~rou E ]es s p ] i ~ n E of, e.g., the p½ and p ~ order of 600

-

800

keV !

18 30 50 100 I ,

" ~?,--

-5~ -/~

-I~00

-2~ E

--

states in the

i

300 500 1000 3000 R[frnL ,~pus t a

~_--

~ .

l

-

• ~~------h .

Figure I0. The Two-Centre-Dirac-level diagram for U+Cm. The K shell is diving at a critical distance of about 35 fm. Notice the rather steep diving (double logarithmic scaler) and the enormous fine structure splitting in the giant system (Z = ZT+Z P = 188).

Giant Nuclear Systems

7

PosHiveEnergyC~inuum

fl

I

-mec z

t

--U ~e

Ene~C~,~uum

Figure Ii. Illustration of the various processes in the time dependent TCD-Ievel diagram. The vertical arrows indicate dynamical processes connected with the change of the molecular wavefunction durlng the course of the collision. The horizontal arrow represents the vacuum decay. It always occurs in an overcrltical system.

A time-dependent view of such a level diagram during the course of a Rutherford scattering is taken in figure ii; t = 0 corresponds to the time of closest approach. Due to the time dependence and the thus introduced Fourier frequencies, a number of processes happen. They are indicated by vertical arrows. We draw attention particularly to the K ionization (processes a and h) and to the dynamiuslposltron creation (processes d, e, and f) which represents the formerly mentioned shake-off of the vacuum polarization cloud (see figure 5b). The vacuum decay process, indicated by the horizontal arrow c, would also go on if all the dynamical processes came t o a stand-still, i.e., if a giant nucleus is fused, slttin E there forever. The studies of quasimolecular X rays, in particular the work of W. Meyerhof and coworkers (1975) and of J.S. Greenber E snd P. Vincent (cf. Vincent, 1983), have uniquely proven the formation of these intermediate systems in heavy ion collisions. To give some impression of the agreements between theory and experiments in this field we show in figure 12 the impact parameter dependence of the K-hole production probability for the Pb+Cm system (measurements by F. Bosch and coworkers; cf. Liesen, 1980; Bosch, 1982), in figure 13 the same for the system Xe+Pb (Anholt, 1982), and in figure 14 a delta-electron spectrum for the J+Pb system (measurements by W. Koeni E and colleagues; cf. G~ttner, 1982; Kozhuharov, 1982). The excellent agreement with the (parameter-free) theory of Soff, Reinhardt, de Reus, and colleagues (Soff, 1980; de Reus, 1984) is astoundin E. However, there remain small discrepancies concernin E the bomhardin E energy dependence of K-hole production (figure 13). They seem to stem from the neglect of the rotational couplin E and some particular level crossin E effects, which are currently under investigation. The details of the calculations and the measurements can be found in the literature (Soff, 1980; de Reus, 1984; Anholt, 1982; G~ttner, 1982; Kozhuharov, 1982). We remark

t h a t many details of the experiments (e.g., ZT+Zp-dependence) are reproduced by the

theory, so that one can justly state that these processes are quantitatively understood.

V.

POSITRONS FROM HEAVY ION COLLISIONS

The semiclassical treatment of electron-positron excitation processes is based on the time-dependent two-centre Dirac equation

i~ a / a t ~i ( R C t ) ) where __ ~CD

=

%cD(RCt)) ~i(RCt))

,

(9)

is the relativistic two-centre Hamiltonian dependin E on the time-dependent clas-

sical internuclear separation R(t). At non-relativistlc hombardin E energies it is useful to expand the wavefunctions ~i into Born-Oppenheimer states Cj given by the instantaneous molecular eigenstates of the Hamiltonian:

J. Fink et al.

I P(b)

10-

I K - holes

E'b"5"&HeV/n ~

I

Pb -Cm ELob=5.9 MeVI u F = 3sG

l

-

-

¢,P(b)

,o-'

....

,'o'''

',o'o'

'

i

10

i

10

ZO

31]

I

40

i

50

~

b[~)

FiEure 12. Probability for K-hole creation per Pb+Cm collision at 5.9 HeV/u laboratory energy as a function of impact parameter b. The experimental data are taken from Liesen and coworkers (1980; cf. also Bosch, 1982), the theory is from Th. de Reus, U. H~ller, J. Reinhardt, and colleaEues (de Reus, 1984). The calculation is based on Two-Centre-Dirac-Hartree-Fock states conraining the electron-electron interaction. No parameters are adjusted. The insert shows the agreement for small impact parameters, which has only recently experimentally been clarified.

Fi&ure 13. Probability for 1so-hole formation in the system Xe+Pb versus impact parameter b, F = 3. The dependence of Plsa(b) on different bombardln E energies ELa h = 4.6, 5.9, 7.2, and 8.5 MeV/u is illustrated. The experimental data (+) were measured by Anholt and coworkers (1982). The data for 4.6 HeV/u, e.E., have to be multiplied by 8.

d£tedEd2°

106

r~bL;r.ksVt. . . . . . . J - Pb

105 "

~

ELob=500MeV

104 ~ . . . . % . . . .

)-

103 .~..:~, _ 102

"'",,%,,,

K- shell

101 ,

i

,

i

,

I

i

i

,

~

i

i

,

,

•

J

100 150 200 250 3 0 350 t,00 450 Es. [ keVl

Fi&ure 14. Total 8-electron distribution

•

(triangles) and 6 electrons in coincidence with K vacancies for the J+Ph system versus kinetic 8-electron energy are results of Koeni E and coworkers (G~ittnsr, 1982; Kozhuharov, 1982). The latter data are compared with absolute values of a calculation by de Reus, Soff, and coworkers (de Reus, 1984). No parameters are adjusted.

Giant Nuclear Systems )i(g(t))

9

Ej aij(t) Sj(R(t)) exp {-ixj(t)} .

=

(10)

lhe s u m ! - c ] u d A a a n integration over continuum states of positive and neEative frequencies. The phase factors X i are conveniently chosen as x .Ct)3

=

~6 I t d t '

<¢jCgCt'))l%CD(gCt°))lSj(R(t°))>

.

(11)

Inserting the expansion of eq. (10) into eq. (9) and projecting with stationary eigenfunctions we obtain a set of coupled d~fferential equations for the amplitudes aij(t ) aij(t)

=

- Zk# j

aik (t) <~j 1 % / a t

I Ck >

exp (i(Xj-Xk)} ,

(12)

with the initial condition a..(--) = 8... :J :J After splitting t h e t i m e - d e r i v a t i v e o p e r a t o r i n terms o f a r a d i a l and a r o t a t i o n a l c o u p l i n g and n e g l e c t i n g t h e l a t t e r one, t h e coupled e q u a t i o n s (12) are s o l v e d by n u m e r i c a l i n t e g r a t i o n . I n the independent-particle approximation excitations of the many-electron system are described by incoherent summation over one-electron transition probabilities. After the collision the number of particles occupying a state above the Fermi level F, up to which the quasimolecular levels are initially filled, is Np

=

2

Zk
(p < F) ,

(13a)

while the number of holes in a state below the Fermi level is

Np

=

2 Zk> F

)akp(+-)l'

(p < F) .

(13b)

An adequate description of positron production in supercritical collision systems, where (Zp + ZT) exceeds Zcr ~ 173, requires a slight modification of the formalism set forth here. In a supercritical system the is state is represented as a resonance in the positron s-wave continuum and not by a single eigenstate of the Hamiltonian % C D " A formalism that avoids those difficulties and moreover has heuristic value for the interpretation of the positron creation process was developped by Reinhardt and colleagues (1981a) and later also discussed by Tomoda and Weidenm~ller (1982). The method is based on the observation that the continuum wavefunctlon of the supercritical system at resonance energy E = E R is quite similar to the discrete Is state in the subcritical case except for an oscillating tail, small in amplitude but reaching: out to infinity. This structure reflects the occurrence of a tunnelling process through the barrier separating the particle and antiparticle solutions of the Dirac equation in a semlclassical picture. Apart from the asymptotic behaviour the is wavefunction retains many of its properties, e.g., the strong localization and the radial matrix elements which may be continued smoothly to the supercritical region if the tail of the wavefunction is neglected. This idea can be used to develop a general method to treat resonance scattering. In this context Wang and Shakln (1970) introduced a projection formalism for resonances in the nuclear continuum shell model: After having defined a normalizable quasibound wavefunction CR at resonance energy ER, a new negative energy continuum Ce + is constructed which spans a subspace orthogonal to CR and replaces the old continuum Ce +. The modified continuum states satisfy the original Dirac equation supplemented by an inhomogeneous term that ensures orthogonality with respect to the resonance wavefunction #R:

(f%CD" Ee+)

l~e+>

=

<~Rl~CDl;e +> l~R> '

(:4)

If the states #R and #e + are used as part of the basis in eq. (I0) the Is state SR couples to the new positron continuum by two separate coupling operators

<¢Ria/aRl;e+>

+

i/S

<¢R]~LrCD]~e+> .

(15)

The second matrix element arises since ¢R and #e + are no longer elgenstates of the two-centre Hamiltonlan % C D "

It is independent on the nuclear motion and leads, in the static limit R(t) =

const = R ° < Rcr , to an exponential decay of a hole prepared in CR" The decay width

r(Ro)

=

2~ I<¢%RI % c D ( R o )

ISR>I 2

(16)

10

J. Fink et aL is identical

to the width of the resonance

in the unmodified positron

continuum.

The f o r m a l i s m t h u s n a t u r a l l y l e a d s t o t h e e m e r g e n c e o f 'induced' and 'spontaneous" positron creation, the latter resultlnE from the presence of an unstable state ~R in the expansion

basis. In practice, however, this does not result in a marked threshold behaviour at the border of the supercrltical region for two strongly related reasons. Firstly, both couplings enter via their Fourier transforms depending on the time development of the heavy ion colllslon. Their contributions have to be added coherently so that in a given colllsion there is no physical way to distinguish between them. Secondly, in collisions below the Coulomb barrier the rapid variation of the quasimolecular potential, especially in the supercritical region, causes significant contributions from the dynamical coupling, whereas the period of time for which the internuclear distance R(t) is less than Rcr is usually very short (~ i0 -2z sec) as compared with the decay time of the ls resonance (- i0 -~I sac). Therefore, the predicted production rates and energy spectra of positrons continue smoothly from the subcrltlcal to the supercritical region. Qualitative deviations of the posltron production rate in supercrltical collision systems are expected only under favourable conditions: Since the 'spontaneous' and 'dynamical' couplings exhibit a different functional dependence on the nuclear motion, an increase in collision time can be expected to provide a clear slgnature for supercrltlcal colllslons. Therefore Rafelski, M~11er, end Grelner (1978) suggested the study of positron emission in heavy ion reactions at bombarding energies above the Coulomb barrier, where the formation of a di-nuclear system or of a compound nucleus would eventually lead to a time delay within the bounds of the critical distance Rcr. During this sticking time T the spontaneous decay of the 1so resonance, by filling dynamically created K-shell holes under emission of positrons, might be strongly enhanced. This idea is i11ustrated in figure 15. Several generations of experiments concerning positron creation have been performed at the Gesellschaft Fdr Schwerlonenforschung (GSI) in Darmstadt. During the past six years three different collaborations, the groups at the Orange- (headed by P. Kienle), at the EPOS- (headed by H. Bokemeyer, 3.S. Greenberg, and D. Schwalm), and at the TORI-spectrometers (headed by E. Kankelelt), concentrated their efforts to investigate positron emission in heavy ion collisions for restrictive kinematlcal conditions. Here we wish to concentrate on theoretical results end comparisons with selected experiments concerning non-Coulombic collisions. First of all, however, some general remarks. We have integrated the modified system of differential equations (12), (15) in the framework of the monopole approximation including up to 8 bound states end ~ 20 states in each, the upper and the lower continuum, separately for the two angular momentum channels m = -1 and ~ = +i (i.e., s½ end p½ states, respectively), which are dominant for the production of positrons. The integration was performed with a standard Hamming predlctor-corrector routine takinE about 1500 steps in time. Positron emission rates (Reinhardt, 1981a) increase very fast with total nuclear charge, flattenln E somewhat for the highest Zu values. If parametrized by a power law (ZT + Zp) n, the expo-

\\. -m

,

/I---

°

Fisure 15. Due to the formation of a giant nuclear system with a certain 1lie time T the spontaneous emission of positrons is enhanced. For large T a positron llne develops surpassing the contribution of dynamically induced positrons by several orders of magnitude.

Giant Nuclear Systems

11

nent takes values of 20 down to 13, if an initial Fermi level above 3so, 4p~0 is assumed, or even

= 29

for bare

nuclei

(F = 0). This highly nonlinear behaviour clearly expresses the

non-perCurbative nature o f the mechanism of positron production in such giant systems. Mainly r e s p o n s i b l e f o r t h e enhancement f o r f u l l y s t r i p p e d n u c l e i i s t h e c o n t r i b u t i o n o f t h e l s s t a t e which i n normal c o l l i s i o n s (F > 0) i s suppressed by t h e s m a l l K-vacancy p r o b a b i l i t y . If the K s h e l l i s empty from t h e v e r y b e g i n n i n g i t becomes t h e dominant f i n a l s t a t e f o r p a i r p r o d u c t i o n due t o t h e s t r o n g c o u p l i n g between t h e l s s t a t e and the a n t i p a r t i c l e continuum which i t approaches and even e n t e r s i n t h e s u p e r c r i t i c a l r e g i o n . In sub-Coulomb b a r r i e r c o l l i s i o n s s½ and p~o waves c o n t r i b u t e

about e q u a l l y t o t h e t o t a l

result.

g e n e r a t i o n o f e x p e r i m e n t s (Backs, 1978; Kozhuharov, 1979) e s t a b l i s h e d t h e dependence

The f i r s t

of positron excitation rates on the combined charge Zu as we]] as on the kinematic conditions.

The Zu-dependent

increase, which spans an order of a magnitude while AZu/Z u is only 12%, is

well described by theory. Also the theoretical, total positron production probabilities, measured as a function of the ion scattering angle, are in overall good agreemant with the experimental data. In the Pb+Pb system and, for smaller distances of closest approach, in Pb+U collisions the data agree also in absolute values. In the highest accessible system U+Cm (Zu = 188) and for larger distances Rmin, theory has had a tendency to overestimate the measured data by up to 25%. At this point we must address the major problem in analysing the experimental data. In the course of the collision, even for bombarding energies well below the Coulomb barrier E C (E/E C ~ .8), the nuclei can be excited by Coulomb and nuclear forces. In the final state virtually emitted photons with energy above 1022 keY can undergo internal pair conversion. Thus one has to measure the ~ spectrum simultaneously and to fold it with the conversion coefficients. Here one has to know - or to assume - the ~-ray multipolarlty. MonopoJe conversion cannot be handled by this method. Up to now, all conclusions on positron production in heavy ion collisions had to rely on the described procedure for background subtraction. The EPOS collaboration (Schweppe, 1983; Cowan, 1985) measured ties integrated over kinetic energies from 100 keV to 1 H e V for ure 16, part a), 5.9 HeV/u U+Pb (b), 5.9 MeV/u U+U (c), and 6.05 For the asymmetric systems the data are displayed versus the approach Rmin, for the heavier, more symmetric systems versus

positron production probabili5.9 MeV/u U+Sm collisions (figMeV/u U+Cm collisions (part d). Rutherford distance of closest the ion scattering angle 8La b.

Here a rather fair agreement even in absolute height for the heaviest collision systems has been achieved.

Po~tron Produc%4on:100 keY < E.. < I MeV ,

,

(c)

,

,

,

.

J.

,

.

,

.

,

"v • '~

asymmetric systems U+Sm and U+Pb in parts a and b, the distance of closest approach Rmin, for the more symmetric systems U+U and

",,~ j0 ~

(o)

I •

I

"~"

m u .

U+Cm (parts angle 8La b.

6

I

[(d)

~

t,...,..~.. ~"'~.i 104

'~z"

o ah,.~e"

[b)

;0

;+

Ro [ fm ]

16. Positron production probabillties integrated over an energy range of 100 keV ~ Ee+ ~ 1 HeV as a function of, for the Figure

;0

1

0

.,.:... ~ • '~c~

c and d) the ion scattering Dots: measured total positron

yield; open symbols: measured positron production after subtraction of nuclear background deduced from the ~-ray spectra (Schweppe, 1983; Schwalm, 1984; Cowan, 1985). The dotted line represents the theoretical results calculated within the coupled channel formalism, the solid lines display the sum of calculated results (MUller, U., 1983) plus nuclear background (dashed lines).

12

J. Fink et al. From these data, however, no qualitative signature for the 'diving' of the 1so state in U+U and U+Cm collisions could he extracted, in agreement with theoretical predictions. More sensitive information can he obtained by the m e a s u r e m e n ~ o f e n e r g y s p e c t r a o f p o s i t r o n s detected in coincidence with the scattered ions. Thelrknowledge is most useful if one wants to find deviations hinting to the positron creation mechanism. Figure 17 shows the earliest published positron spectra of Backe, Kankeleit, and coworkers (Backe, 1983a) for three collision systems, U+Pd, U+Pb, and U+U, at 5.9 HeV/u hombardln 8 energy; the ions are detected in an angular window 8La b = 45 e ± I0 e. For U+Pd (Zu = 138) no atomic positrons are expected, the data are fully accounted for by nuclear conversion (dashed llne). Extrapolating this procedure to the U+Ph system (dashed curve), the sum of background and calculated QED positron rates (full curve) is in excellant agreement with the observed emission spectra. Figure 18 shows a similar comparison of theoretical predictions (M~ller, U., 1983) with measured data, now from the EPOS collaboration (Schweppe, 1983; Schwalm, 1984; Cowan, 1985). In addition to the systems U+Ph and U+U, the supercritical system U+Cm was investigated at a homharding energy of ELa h = 6.05 HeV/u. The scattered particles have been detected in an ion lab angular window of 2 5 1 K

8La h ~ 65 I. The displayed,

even in absolute numbers fair agreement,

however, conld not be achieved for all experimental positron spectra, where discrepancies up to ± 30~ might occur (Clemente, 1984).

3[e:::'.,,..-',o.' ® J 11 [.l~"~,

" /

. 5.9~v,,., I

I#'lh

5.gMevlu/

Tots[ Positron Production: 25 ° < 6LAe < 65 °

J

> 1.~/ r I~,

,.oI ILN,,

®J

(o) [.~.

""

U°U 5.9MeVlu

3

1.0' T~@

~'~

0

•

•

.

2~U ÷ " S i n 5.9 IdeV/imu

(b)

"u +

(d}

mO + 2°'Pb 5.9 IdeV/llmu

1

T >G) .3£

0 3

(c)

/

2zqj ~ 2{eCru

:

0.5 0.1 1.0 0.5

1

0.1

0

0.5

1.0 1.5 Ee, I MeV

2.0

0 200 ~00 500 800 1000

200

LO0 600

800

1000 1200

[ keY ]

17. Earliest spectra of emitted positrons in 5.9 HeV/u collisions measured by Backe, Kankelelt, and associates (Backe, 1983a) in coincidence with ions scattered in the angular window 8La b = 45" ± i0". The spectrum in the lightest system, U+Pd, is explained by nuclear pair F~gure

conversion alone (dashed llne). In the U+Pb and U+U systems the sum (full lines) of nuclear and calculated atomic positron production rates is displayed. Figure

Total positron production probabilities within an ion scattering lab angular window 8La b S 65' for the four collision systems already displayed in figure 16. Again the

18.

of 2 5 1 K

experimental data are represented by dots, whereas the full curves denote the sum of nuclear background (dashed lines) plus theoretical results..

Giant Nuclear Systems

13

A possible source which could cause qualitative deviations in the shape of the positron spectra from the results presented so far will he discussed in the following. To obtain theoretical predictions for positron production it is essential to include the 'spontaneous' coupling for collisions where the iso state joins the lower continuum, since at the same time, when the spontaneous coupling becomes important, the 'induced' radial coupling is reduced drastically. If one of both is left out of the calculations the resulting positron spectra would be strongly altered. Both contributions add up coherently and cannot be observed separately. A promising strategy, however, to get a clear qualitative signature for the diving process is to try to modify the time structure of the nuclear scattering process and to select heavy ion collisions with prolonged nuclear contact time. Such nuclear reactions are expected to occur at energies close to or above the Coulomb barrier. The nuclear delay time T should provide a handle to distlnguish supercritical from subcritical systems. Using a schematic model for the trajectory, U. MUller, J. Reinhardt, T. de Reus, and coworkers (Reinhardt, 1981b; MUller, U., 1983) have performed coupled channel calculations for the four heavy ion collision systems Pb+Pb, Pb+U, U+U, and U+Cm, corresponding to Z u = 164, 174, 184, and 188, respectively. Independently on assumptions on the incoming and outgoing path, of dissipation of nuclear kinetic energy or of angular momentum during the reaction, and of the position of the initial Fermi level, all positron spectra exhibit the following features: In subcritical

collision

systems (Z T + Zp ~ 173) a n u c l e a r d e l a y t i m e T d u r i n g the course o f a

Rutherford scatterin& causes modulations in the positron spectrum with a width AE : 2~t~/T. In figure 19a positron spectra are displayed for a Ph+Pb collision (ELa h = 6.2 MeV/u, b = 0, F =

3so, 4p½0) with delay times T = 0 (pure Rutherford scattering), 3x, 6x, and 10 x i0 -t* sac. The modulations are due to interference effects in much the same way as predicted for the 8-electron spectra in deep-inelastic heavy ion collisions (Soff, 1979, and figure 20). Due to the time delay of the two sticking nuclei the 8-electron and subcritical positron spectra show oscillations, because the incoming and outgoing amplitudes for 8-electron and positron creation, respectively, do now interfere with a phase proportional to the sticking time. The oscillation energy &E thus can be directly related to the delay time T. A

Pb+Pb o4

U+U

b -.....

ELab- 6.20MeV/u b-O F-3

~5

i? .~ \

.

.

.

.

.

.

T.6[*.o

.

3 .i, / I'~

\.

,01

T=3 "~. - ...... T~6 .I0 ...... T=IOJ

; r%,,

//

,,..,.,,.. ,'

"

\

,

,

I

500

,

,

J

,

I

~,,,,

,..., ,

,

L

,

1000 Ee.(keV)

500

1000 E~(keV)

Fi&ure 19. Spectra of positrons created in subcrltical (part a) and supercritical heavy ion collisions (part b) assuming almost grazln E Coulomb trajectories (full lines) and nuclear reactions leading to delay times 3x, 6x, and 10 x 10 -2* sec, respectively, using a schematic model for the trajectory (cf. Relnhardt, 1981b). Whereas for the lighter collislon systems modulations in the positron spectra are present, a distinct peak at the resonance energy Elso(Rmln) builds up for systems with Zu > 173.

14

J. Fink et al.

I0 -a

XI*Pb

I0-,~

%"

\\\_ 10": ,.,

o..,

\iv

£

Figure 20. lso-coincldent 8-electron spectra without (straight line) and with time delay in a grazing Xe+Pb collislon at ELa b = 7 HeV/u (from Soff, 1979). The

A?*-

1

"v,'v"

/

t; :i/ 10-7.

,

,

,

t.O0

800

dash-dotted lined 6-electron spectrum 10 -~° sec.

,

however, represents a total for F = 3so and delay time T =

','t....

1200

1600

In addition to the interference patterns an enhancement of positron production in tlme-delayed supercrltlcaJ collisions is observed, where the binding energy of the lowest bound states exceeds the value 2mec z. For long delay times a diatinctpeak in the positron spectrum is found at the location of the supercrltlcal bound state resonance (binding energy minus 2mmc 2) due to the spontaneous pair creation process. A detailed analysis of the spectra reveals that this peak emerges gradually as Z u exceeds Zcr. Positron spectra for the supercritlcal system U+U (ELab = 6.2 HeV/u, b = 0, F = 3) are shown in figure 19b. With increasing delay time the position of the maximum drifts slowly from the kinematic maximum to the 'resonance energy', which depends on the combined nuclear charge, the separation of the two nuclei, the nuclear charge distribution, and on the degree of electronic shielding effects. However, for any chosen set of experimental parameters, the nuclear reaction time T w111 not be sharp, but distributed over a certain range with a time distribution function f(T). As the most 'pessimistic' assumption we took an exponentlal decay in time fCT)

=

The r e s u l t i n g

( F c / 5 ) exp ( - F c T / ~ ) positron

.

(17)

s p e c t r a f o r n u c l e a r decay w i d t h s Fc = 50, 25, and 12.5 keV are d i s p l a y e d

in figure 21 for a head-on U+Cm collision at ELa b = 6.2 HeV/u. Part a shows the emission of plo positrons. Since these states are subcritical the time delay does not result in an enhancement of the probability but only in interferences which are completely smeared out by an exponential distribution f(T). In part b the spectra of positrons emitted in the angular momentum channel s½ are displayed for the exponential (thick lines) and a sharp time distribution f(T) = 8(T-~/rc) (thin lines). Again the oscillatory patterns are completely damped out in case of a folding with a time distribution function. The resulting positron line, originating from the spontaneous part of the positron production mechanisms, however, seems to be invariant: It is even narrower and more dominant than for a sharp time distribution. Thus we can conclude that the appearance of several oscillations in the spectrum can be expected only for sufficiently sharp nuclear reaction times. Considering the spontaneous contribution only, it is not immediately clear how much information on the distribution function f(T) can be gained by the experiment. Additional broadening and distortion of the line will result from experlmental effects (Doppler broadening). Also, the position of the line is very sensitive to the nuclear configuration during the reaction. Therefore a realistic reaction model probably will produce a superposition of lines centered at various (neighbouring) energies making the analysls more complex.

Giant Nuclear Systems

1°t

i!

'

U+Cm '~ E~-E20MeV/~ /~

b.0

~

~

~

~_

Ira-

15

A

~

'

U.Cm ' ~, E~.6.Z0MeV/u ,'

b-0

IL~I

,i!

j.I,ii, II! :i

oi,l,

~,

,,\ ,°, /

Figure 21. Spectra of positrons

~o-7

I q.i ',~ It.l,~ III

emitted in the subcritica± p½o channel

(part a) and in the

supercritical s½ channel (part h) in head-on U+Cm collisions at 6.2 MeV/u impact energy, calculated under the assumption that a nuclear reaction takes place. Predictions for an exponential (full lines) and a sharp (thin curves) distribution of reaction times, i.e., f(T) = (Fc/~) exp (-FcT/~) and f(T) = 6(T-~/Fc) , respectively, are compared. Nuclear widths r c = 50, 25, and 12.5 keV (from Reinhardt, 1983).

A similar united atom effect is expected in the continuum spectrum of quasimolecular X rays. Calculations for the Pb+Pb system (figure 22) show that a line emerges at the united atom K transition energF if the delay time became sufficiently long (Kitsch, 1983). The results for the U+U system are quite similar, but there is a formidable background from nuclear ~ rays which probably makes an experimental observation very difficult.

dP [ i I

Pb.,-Pb

I1

,.,I

,tl,

.~81

- - ~o-~L

i; i,,"-:L:~/

,uF

-

-

"

I

l

ii#I

"- 0

l

3.1o-2°

l

I

500

I

I

I

I

I

1000

b~!~..i.

,

,

,

~'

,

I

1500-

Figure 22. Quasimolecular X rays of the Pb+Pb system for various sticking times (from Kirsch, 1983).

16

J. Fink

et al.

The results described so far were obtained within a schematic description for the nuclear motion, which facilitates a systematic study of the time delay effect and allows for an investigation of the conceptionally interesting limit of large reaction times. For the description of a given experiment, however, it is more convincing to adapt trajectories calculated from a nuclear model which is consistent with the elastic and inelastic heavy ion scattering data. For a detailed comparison of theoretical (Reinhardt, 1981b; H~ller, U., 1984) and experimental data (Backe, 1983b; Krieg, 1985; Stoller, 1984) concerning the investigation of atoslc excitation processes during deep-inelastic heavy ion collisions of U+U at bombarding energies high above the Coulomb barrier, we refer to the contribution of T. de Reus (1985). Here we want to summarize only a few theoretically predicted effects: K-hole production is influenced very sensitively by nuclear delay times in the order of a few 10 -2* sec. The spectra of 6 electrons emitted during deep-inelastic collisions decrease considerably in their high-energy part compared with spectra calculated for Rutherford scattering and their fall-off is steeper. Positron spectra show a gradual enhancement at their maxima as well as a drift towards lower kinetic positron energies. The comparison between theoretical K-vacancy probabilities and 6 electron spectra and the corresponding experimental data (cf. de Reus, 1985) shows qualitative agreement supporting the applicability of trajectories from nuclear friction models within the semiclassical approximation. For completion in figure 23 the experimental positron data (Krieg, 1985) for heavy ion collisions of U+U at 8.4 HeV/u are displayed together with theoretical results. For the upper part, elastic scattering events have been measured, thus we have to compare with theoretical results for positrons emitted along a Rutherford scattering-path. For the lower part, also nuclear friction model trajectories (Schmidt, 1978; cf., e.g., H~ller, U., 1984; de Reus, 1985) have been assumed. Here, the theoretical values have been scaled by a factor of ~ .8. The shape of the experimental positron data only shows agreement with theoretical results when taking into account small delay times in the order of some ~ i0 -21 sec as achieved in friction model calculations (Schmidt, 1978; MUller, U., 1984). From another experiment in the lower united charge domain W. Koenig (Heidelberg) reported that he found even oscillations in the 6-electron spectra in the deeply inelastic reaction of l+Au. He measured coincidences of the 6 electrons with projectile and target nucleus in a certain Q-value window. Furthermore the ~ spectrum has been measured and the electrons stemming from conversion were subtracted. The thus resulting 6-electron spectrum showed weak oscillations indicating a delay time of the order of T ~ i0 -2° sec. These reported results are preliminary. They could be the beginning of an interesting experimental endeavour, bringing quantitative (absolute) time scales into nuclear reactions. Actually, the 'atomic clock' should work in the range between i0 -*~ sec and 10 -2* sec.

m U ÷ ZUU II./, MeV/u (pr~imirmry) -

10-°

i'

2 ~IO-T 2 5

e

,

Figure

~IO-T

6OO

IOO0

PSO0

Enorgy/keV

2000

23. Experimental spectra (Backe, 1983b; Krieg, 1985) of positrons emitted in U+U collisions at 8.4 HeV/u in coincidence with elastic scattering (upper part) and with nuclear fission (lower part). The data are compared with theoretical results based on Rutherford scattering (dashed lines) and on a nuclear friction model (Schmidt, 1978; full lines). The theoretical spectra in part b have been scaled by a factor of 0.8.

Giant Nuclear Systems

17

As another interesting theoretical problem one might ask about the presence of nuclear coll~sions with very long reaction times. What would positron spectra look like if, at a given scattering angle, a superposition of Rutherford scattering and long-lasting nuclear reaction is assumed. If, for the sake of simplicity, nuclear scatterlng'with definite delay time T is consideredj a ratio q ~ 1 can be used as a measure for the relative cross section of reactions leading to long contact times, as compared with the Rutherford cross section. As positron production is very strongly enhanced for reaction times larger than 10 -2" sec, a peak superimposed on the smooth spektrum of positrons emitted in the much more frequent 'distant' Coulomb collisions could emerge. A rough estimate shows that for long delay times a peak may be prominent even if the differential reaction cross section is less than i~ of the Rutherford cross section. Position, width, and strength of the spontaneous positron line are determined by the nuclear reaction process: The kinetic energy of spontaneously created positrons by the shape of the giant nuclear molecule

and by the effective charge Z elf = Zu - Z . , if the nuclear u screenlng reaction time is long enough; the line width by the nuclear reaction time T for not too short nuclear reactions - and for asymptotically long reaction times T ~ ~/F by F itself; and the relative strength by the life time of the nuclear composite system and by the ratio q defining the number of nuclear molecule residues per elastically scattered ion in the angular window considered. The shape of the giant nuclear molecules with total charges Z u is parametrized by the two-centre distance R. Investigating the heavy ion collision systems Th+Th, U+Th, U+U, Th+Cm, and U+Cm, we take as equilibrium distance R the distance of half density overlap for the most faro ourable elongated orientation ~Seiwert, 1985a), i.e., R ~ 16.7 fm (Th+Th), ~ 16.7 fm (U+Th), o - 16.8 fm (U+U), ~ 16.3 fm (Th+Cm), and ~ 16.3 fm (U+Cm). Thus we have assumed that the same type of nuclear reaction occurs in all systems discussed. Electron screening effects are taken into account within the Hartree-Fock-Slater (HFS) formalism, assuming a 50-fold ionisation of the

quasimolecules

molecular

(de

Reus,

1984).

This determines

the binding energies E~s~(Ro)

of the

iso states to be IE~so(Ro)l ~ 1080 keV for Th+Th, - 1135 keV for U+Th, ~ 1195 keV

for U+U, ~ 1270 keV for Th+Cm, and ~ 1340 keV for U+Cm. The corresponding energetic positions of the spontaneous lines are E + ~ 60 keV, ~ 115 keV, ~ 170 keV, ~ 250 keV, and ~ 315 keV, e respectively. For figure 24a, the life time of the T = 10 -19 sec. Doppler broadening account. Thus the width of all lines q = 1.3 x 10 -3 for the U+Cm system.

giant nuclear molecules is assumed to have the sharp value as well as other broadening effects are not taken into (FWHM) approximately is ~ 40 keV. For the ratio q we take This corresponds to a nuclear reac'ion cross section for

the long-lasting events of o N ~ 22 mb assuming a symmetric break-up of the nuclear composite system. For the other collision systems q is renormalized slightly to yield the same nuclear reaction cross section. As explained above, the position of the lines is determined by the shape of the nuclear molecule for given degree of ionisatlon and for life times not too short. While for the chosen parameters, in the spectrum of U+Cm the spontaneous positron line at E + ~ 315 keV exceeds the e ordinary positron spectrum by a factor of nearly 3, the relative strength of the peak gets reduced step by step for decreasing charge Z u . Finally, for Th+Th the remaining structure at Ee+ ~ 60 keV is strongly suppressed due to the small decay width for spontaneous positron creation. Assuming equal conditions for the nuclear reaction process and a delay time of T = I0 -Is sec, the model of spontaneous positron production from a giant nuclear molecule thus predicts a shift of the positron line and a distinct reduction in intensity if Z u is decreased. Now we investigate as an alternative assumption that the life time of the giant nuclear molecule greatly exceeds the spontaneous decay time of a hole in the resonant Iso state, i.e., T ~ • /F. Then all the holes, dynamically created in the iso state during the Rutherford scattering before the nuclear reaction process takes place, will decay into positrons completely. Due to the similar kinematical conditions for all scattering systems considered, the probabilities to create a i s o hole prior to the nuclear reaction, Pls0(t=0), are nearly the same, in between

18

J. Fink et al. - 6.0% (for Th+Th) and ~ 5.4% (for U+Cm). For asymptotically long nuclear reaction times, T ~ ~/r, these values define the emission probabilities for spontaneous positrons. Thus, the cross section in the spontaneous positron lines is nearly Zu independent, as figure 24b displays. To

emphasize the contrast to part a, in part b the nuclear cross sections o N are reduced in such a manner that the cross section in the spontaneous positron lines for U+Cm is identical. For the other scattering systems the ratio q has been adjusted correspondingly. For asymptotically long reaction times the widths of the spontaneous positron lines are determined by their natural decay width r = However, since up to now it is experimentally not possible to measure

r(Zu,Ro).

dP/dE(e') .

,

.

dP/dE(e')

[lO-7/keV]

,

.

,

.

,

.

,

.

,

g.O

U+

[lO'7/keV]

.

g.O

U+

Cm 6.0

6.0

Cm

EL,~ " 6.05 N e V / u

4.0

2.0

O0

.

,

.

,

•

,

.

,

,

,

.

,

.

8.0

Th + C m

Th + C m

5.0

E ~ = 6.0 MeV/u

40 3.0 20

20 10

O0

O0 2.5

A

2.0

]

u.u

U+U ELib = 5.9 IvleV/u

1.5

tO

2°I

05 0.0

•

,

.

,

.

,

.

,

.

t5

,

.

,

0.0

.

S.O

Th + U

I

Th+U

6.o

1.0

EL.,= = 5.8 MeV/u

05

0.0

.

,

.

,

.

,

.

,

.

,

Th +

t.2

.

,

0.0

.

Th

Th

+

Th

tO Eu~ = 5.75 MeV/u O@ 0.6 04 0.2

,° L

0.0

O0 200

400

600

$00

E(e °) [keY]

Figure

1000

1200

1400

0

20o

400

6o0 coo moo E(e') [keV]

raoo ~oo

24. Superposition of positrons emitted during ordinary Rutherford scattering events (broad smooth lines) and spontaneous positrons from long-lived giant nuclear molecules. In part a) a reaction time T = 10 -zl sec is assumed, for part h) T ~ / F , F = F(Zu,Ro) (cf. description in the text).

GiantNuclearSystems

19

widths of a few keV, we have broadened the lines for all systems from r ~ 3 ... 4 keV to ~ 40 keV as in figure 24a, thus simulating Doppler broadening as well as other similar effects. As expected under these conditions for the nuclear reaction, the spontaneous positron lines now stand out from the dynamically created positron 'background' for all systems investigated. To obtain the full shape of the positron spectrum, an assumption about the angular distribution doN/d8 of the nuclear reaction component is needed. We shall discuss here the isotropic breakup of the compound system. Focussing the reaction fragments into a narrow angular window would require a quantum mechanical treatment of the nuclear motion together with a partial wave analysis of the scattering cross section (see later). If the reaction products were emitted isotropically, a line should, in principle, be observable in the positron spectrum at all ion scattering angles. However, it would be most pronounced at 8 = 90 °, being suppressed at other angles relative to the elastic scattering cross section. cm Figure 25 shows spectra of positrons, reduced by a factor ~ 2/3, for 5.79 MeV/u U+U collisions for several fixed scattering angles. A long-lived nuclear reaction component (dashed lines) with T = 4 x I0 -2° sec and admixture q = 2.4 x 10 -3 (at 8 = 90 °) has been added under the cm assumption of an isotropic distribution in the reaction plane. Two experimental groups, one headed by P. Kienle (cf., e.g., Clemente, 1984), the other one by Backe, Bethge, Bokemeyer, Greenberg, and Schwalm (cf. Schweppe, 1983; Schwalm, 1984; Cowan, 1985), during the last years have performed experiments with Th+Th, U+Th, U+U, Th+Cm, and U+Cm at energies close to the Coulomb barrier. Contrary to the early results of Backe, Kankeleit, and associates (Backe, 1983a), their positron spectra show remarkable structures. Meanwhile, also the Kankeleit group has reported on the positron line structure in U+U (Krieg, 1985). Figure 26 shows positron spectra in U+U collisions (Clemente, 1984) at ELa b = 5.7, 5.9, and 6.2 MeV/u, each for two different angular windows determining the detection range of the scattered ions. The spectrum for ELa b = 5.7 MeV/u, 26 o < 8La b < 38 o , seems to he broad and mostly structureless (Clemente, 1984), whereas the positron spectrum at 38 o < 8La b < 520 for the same bombarding energy shows a peak at ~ 310 keV with a width of about 60 keV. Looking at the other spectra in figure 26, it seems that position and width of the peak depend on the scattering angle and beam energy. If the observed structure is of quasimolecular origin, it must be produced in very long-lasting nuclear reactions because of the small width. Comparing the experimental width with spectra from coupled channel calculations based on the schematic reaction model, a minimum value T ~ 7 x I0 -2° sec is required. Due to additional broadening effects (such as Doppler broadening) the reaction time T would have to be even longer. The probability for positron production in a delayed (T = 7 x 10 -2° sec) collision must be compared with the observed probability. Thus a fraction of q = Pe+(exp)/Pe+(theor) ~ I.i x I0-' delayed collisions per elastically scattered ion is sufficient to produce the observed effect in figure 26a. This number should serve only for a general orientation, since it depends on the details of the model.

I

d± -I l dee.

I

I

'"

"I0"7

O'U ,

E,~b- 5.79 HeV/u

IL

2.0

OL~b"

~~ 't

\450

1.0 ~ 500

1000

Ee.[keV]

1500

Figure 25. Spectra of positrons emitted in 5.79 MeV/u U+U collisions in coincidence with a scattered nucleus for three selected lab ion angles. The fully drawn curves are calculated assuming Rutherford scattering only. The dashed lines show the effect of an additional nuclear reaction with a life time T = 4 x 10 -2° sec. A relative fraction of q = 2.4 x 10-* reactions per elastically scattered ion (at 8La b = 45 °) has been assumed.

20

J. Fink et al. dP/d£(e') 5n

•

,

.

[10 "7 k e V "1] ,

'~

.

,

.

,

.

,

T

,~

.

,

.

U+U

~

d P / d E ( e +) [10 -7 k e Y - l ]

5~

,

5.7 MeV/u

.,~

,.5~

li

,~ ~

dP/dE(e') • , . ,

U + U

38

°

<

e

<

52

3~

•

~

40"5° < e < 49 °

35

.

,

.

,

U+U 6.2 MeV/u

5.9 MeV/u i

~

[10 -7 k e V "1] . , . , . ,

7n

6.0

405*

< 0 < 490

5~

3~ 4.0

3.0 t5

A T [ I - F : s " 1.18 / 0.5

0~

]

T - 7 * 10-ms

i

%o

q

•

,

.

,

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.

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.

,

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,

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.

,

.

.

.

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800

0~

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.

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.

.

.

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.

,

.

12

*

.

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.

,

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,

400 500 600 E(e') [keV]

i

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,

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.

,

.

300

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,

.

,

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.

.

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800

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.

,

.

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.

U+ U

3s

MeV/u

.

400 500 600 E(e') [keY]

dP/dE(e')

U+U ~

~

200

.

6.2 MeV/u

3~

25.50 < 0 < 3 8 0

260 < e < 380

-

ATDHF: s

lO

10 "s

,

U+U 5Y M e V / u

21~

q

'

100

[10 -7 k e V "I] .

~-5.10-~.

0.5 I - / .

2.0

,T~: ,-_?.01

~/

10 "~

400 500 300 E(e') [keV]

dP/dE(e') 3.0

1.1

=l- /

~ ±

e

25.5 ° <

< 32"

2~ k

,

T!.T o

/i

/

03

05

I-/--

T

/ 0.0

, i , i . , , , 2oo

3oo

.

.

.

4oo ~oo 6oo E(e') [keV]

.

.

700

ATDHF: " 6

q

-

s *

=

/

200

300

••

•

400 500 800 £(e'} [keV]

T - 5.5 * 10-ms

p

3 * 10 "4

q - 3.6 * 10-' 00

I]0

°

.88

lO-2Os

.

go~

~

700

800

,

100

,

200

.

,

300

,

,

,

,

.

.

.

t,o0 500 600 E(e °) [ k e V ]

.

.

i 700

800

Figure 26. Positron spectra of U+U collisions measured by H. Clemente, C. Kozhuharov, P. Kienle, and coworkers (Clemente, 1984) at three different bombarding energies and, in each case, at two different angular windows. The experimental data still include a smooth nuclear background. The broad-shaped spectra represent positrons calculated for Rutherford scattering only, scaled by a factor s due to a least-square fit to reproduce best the high-energy part of the experimental data. The peaked lines show in addition the contribution of atomic positrons emitted from a giant nuclear complex with life time T. The ratio of nuclear cross section o N and Rutherford cross section o R within the corresponding angular windows is given by q. Similar effects have been observed in 5.9 HeV/u U+Th collision~ (figure 27b,c) ca£ried out by the same group (Clemente, 1984~. Thes% s;ectra have to be comT~ared with the Pb+Th spectra in part a (figure 27). While the former systems are supercritical, the latter is undercriticaland should, accordingly not show line structures (except for Raman lines - s e e later) This, indeed, seems to be the case. Striking structures were detected in the experiment of T. Cowan, K.E. Stieb[ng, J. Schweppe, H. Bokemeyer, J.S. Greenberg, D. Schwalm, and coworkers (Schweppe, ]983; Schwalm ]984; Cowan 1985) in U+U and U+Cm collisions. A comparison of the theory with the U+Cm experiment is shown in figure 28. Again two'parameters the width r c and th. area (ccoss section) under the line, were fitted. They correspond to the mean life time of the giant system and to the range of impact parameters (reaction cross section) leading to fusion of the giant system, respectively.

Giant Nuclear Systems

21

In figure 29 the experimental positron data from the EPOS collaboration (Schweppe, 1983; Cowan, 1985) for heavy ion collisions in a range of united charge Zu from Zu = 180 to Zu = 188 are collected. These data have to be compared with the theoretical results displayed in figure 24. While in the latter the position of the peak structure depends sensitively on the collision system considered due to the assumption of giant nuclear systems with similar nuclear charge distributions and states of ionisation (vertical dashed lines in figure 29), the experimental data exhibit nearly a degeneracy in peak energies, E + ~ 330 keV (Cowan, 1985), perhaps with e the exception of Th+Cm. So the naive reaction model assuming equal conditions for all giant nuclear systems fails to describe even qualitatively the measured positron spectra in figure 29. Including the data from the Orange spectrometer, figure 26 (Clemente, 1984), there exists, however, a contrasting spreading in peak energy even for similar bombarding energy and angular detection window. One should keep in mind that the EPOS group performs by hand certain cuts through their data selecting special angular windows to make the positron llne optimal. It looks to us likely that other positron lines at different energies may appear if different cuts are choosen. Final conclusions can at this stage certainly not be drawn. 6

i

l

l

=

l

v

,

,

v

=

Pb.

I

'

I

I

I

I

I

I

3 30

I

~o

: : : : : : : 1

U+

q2.

"~ .

~qM~V/u

20

C

2.0 >

~

1.2

E

0.~ ?

0.~

10

~0.5~01o~-~ ~

U•

~

'

"%

~qM~V

/u

-~ n

O 20

Q,

0.6 10

O~

22,<®,o~< 32, 0.2

I

,

i

,

,

I

n n

% n v

20O ~00 6O0 800 1000 Ee.[c m ] /key

0 0

200

~00

600

800

1000

1200

E(e +) [keV]

Pigure 27. Positron spectra of the Kienle group (Clemente, 1984) for Pb+Th and U+Th in various angular windows. Curves b indicate the dynamical spectrum according to theory, whereas curves a display theoretical spectra with superimposed spontaneous emission. Pb+Th, being nndercritical, should not show spontaneous positron lines (except for Raman satellites).

F~gure 28. The measurements of T. Cowan, K.E. Stiebing, J. Schweppe, H. Bokemeyer, J.S. Greenberg, D. Schwalm~ and colleagues (Schweppe, 1983; Schwalm~ 1984; Cowan, 1985) for U+Cm positron spectra at 6.05 HeV/u show a remarkable structure when selecting backward scattered ions (upper part). Theoretical results (H~ller, U., 1983) (at ELa b = 5.8 MeV/u, folded with the detector efficiency) are compared with the experimental positron yield. explained by a nuclear reaction component with a delay time T = ratio q = 1.3 x i0 -~ . The Doppler broadening is taken into account. more forward scattered ions (lower part) is in good agreement with ering Rutherford scattering only.

The line structure can be i0 -xl sac and an admixture - The positron spectrum for theoretical results consid-

22

J. Fink et al.

50~.i

I 6.07MeV/uZ. - 188.

Th+Th (180) at bombarding energies indicated taken from the EPOS group at GSI (Cowan, 1985). Dynamically emitted positrons from Rutherford scattering have relatively been reduced by the choice of we11-selected kinematic cuts. A positron line at E + ~ 330 keV (~ 370 keV for Th+Cm), i.e., e nearly independent of Zu, is clearly visible. The use of Cm,

3O 2O ~0 0 60 -

40

mi MeV/ 6.Th02+- C186 "

,o

I

0

i .q,) >-

5.8 MeV/u

Zu 194 "

3o 20

0

lO

(3_

o ~.0

50

563 PleV/u

Z~" 182

30 20 10 o

Th + T h -

I00

U, and Th as collision partners avoids the possibility that one of those was responsible for a trivial source of the line structure. Possible shapes of long-lived giant nuclear systems held responsible to yield the experimentally observed positron lines are indicated schematically. The assumption of nuclear charge distributions and degree of ionisation as in figure 24, on the other hand, would lead to kinetic energies for spontaneous positron emission indicated by the vertical dashed lines.

U+U

,

zo

C 0 s-

Figure 29. Positron energy spectra for the collision systems U+Cm (Zu = 188), Th+Cm (186), U+U (184), Th+U (182), and

5Y5 IVeV/u-

80 " / 0 ~ " 6O

Also, the experimental equipments seem to be not sensitive enough for detecting with high efficiency and resolution positrons emitted in the low energy region, as expected for spontaneous positron emission for 'small' united charges Zu . Thus, to resolve this open question also a high-efficiency detection of positrons with small kinetic energies is required yielding positron data of high resolution and with trustable statistics at all positron and various beam energies. Recently, H. Tsertos, P. Kienle, and collaborateurs (Kienle, 1984; Tsertos, 1985) very precisely measured the dependence of positron line energy, width, and cross section on the heavy ion scattering angle, performing 5.9 MeV/u U+U and U+Th collisions. Line energy and width seem - within statistical uncertainties indicated in figure 30a and b - to be independ = ent on 8. (Energetic position of the positron line in the rest frame and its intrinsic width are also given in the figure.) The differential line cross section seems to be well reproduced by a Vsine law as expected for an isotropic break-up of the compound system in the reaction plane (part

c).

4O 20 0

~'00 41111 600

E,+

800

1000

Before we deal with the exiting physics of up to now unexpected giant nuclear systems we want to collect and discuss some additional experimental hints zoncerning the nature of the measured positron lines.

[keV]

VI.

COULD THE LINE STRUCTURE BE OF TRIVIAL ORIGIN ?

One might suppose that those peaks are of trivial origin, e.g., they might be caused by nuclear background processes. In this Section we thus will discuss the theoretical framework of conversion processes in sin81e atoms and in supercritical compound systems and the corresponding experimental data. In collisions of very heavy ions with ELa b > 3 MeV/u both nuclei are Coulomb excited. Transfer reactions or even deep-inelastic nuclear reactions can take place which lead to additional excitations of the nuclei. This internal excitation energy may be carried away by a photon or may be transferred to a bound electron or to an electron of the negative energy continuum, which leads to ionization and electron-positron pair creation, respectively. The latter process requires nuclear transition energies ~ larger than twice the electron rest mass. Nuclear E0

Giant Nuclear Systems 360

E / A : 5.9NeVlu ~I I

i

I

I

I

,

I

I

U+U

• >

23

28O

I

,

l

I

~, ,(, '{'

,

+

U-Th

+

~- 28t

,

i i .i (280 -*8)[c.m]lkeV '~ ~ +

÷/ +

(277±Sl[cmllkeV

2o0 :

:

;

:

:

;

I

I

I

I

I

I

U+U 100 l

60

.3

1~.o

I

'\'

'

i

[80± 8) key

'

'

U*Th

100

+

+ + ,,s:8~,.v

.....

,

aD =c

.

;

:

~

,

,

,

,

,

,

I=U*U I=U*Th - - Fit. s-~nlep

2& -

2.

d

@

8 0

i

30

i

/.0

I

I

50

I

I

60

I

I

70

8p{c m ]ldegree

I

I

80

I

90

Figure 30. Energy, width, and differential cross section of the positron line observed in 5.9 HeV/u U+U and U+Th collisions by Tsertos, Kienle, and collaborateurs (Kienle, 1984; Tsertos, 1985) as a function of the cm scattering angle. While the kinetic energy as well as the width of the lines seem to be independent on 8, the cross section in the positron line seems to follow a ~sin8 law.

transitions are characterized by the absence of single photon emissions, because a photon m u s t carry at least one unit of angular momentum. Such processes form the main source of non-atomic positrons, and they have to be well understood (cf. Soff, 1981; Schl~ter, 1983), if one wants to draw firm conclusions about the presence or absence of spontaneous pair creation. The basic processes under investigation are depicted schematically in figure 31. The nucleus which makes an EO transition is labelled by its initial and final state angular momenta Ji' Jf = Ji' and eigenenergies Ei, Ef = E i - w (~ = i). Process a describes the

electron-posltron pair

creation. An electron of the negative energy continuum (s = -E < -m,c 2) with Dirac quantum number K is lifted to the positive energy continuum. The final state energy obviously amounts to E' = ¢ + w, whereas the angular momentum quantum number remains unchanged. Since neither the initial state energy nor the final state energy are fixed one expects a continuous energy distribution for the emitted positrons. Process b indicates the conversion of a K-shell electron (n = I, £ = 0, j = ½, K = -i) with energy eigenvalue Els ½. Thus bound states with definite energies are involved. Energy conservation then simply causes monoenergetic lepton emission for a fixed nuclear transition energy w. Process c symbolizes monoenergetic positron production. ][ere an electron of the negative energy continuum is excited to a bound state, e.g.~ to the is} state. This represents a rather rare process, which can be neglected. Thus we focus our attention on: i) The pair conversion coefficient 8, defined as the ratio of the pair production probability (process a) compared wi~h that of photon emission for a specific nuclear transition with energy w. Since the energy of the electron and the positron takes continuous values we may express also as integral of the differential pair conversion coefficient d~/dE. The lower bound of the integral is determined by the rest mass of the electron, which corresponds to vanishing kinetic energy, while the upper bound is given by the nuclear transition energy w minus m0c s.

ppNY-B

24

J. 1<% - 1

,

J' - - - - ~

~,

El%

E

- ~

Fink et al.

=

•

1<='1<

E'=Ef*w

Figure 31. Schematic representation of electron conversion processes accompanying nuclear EO transition from a state {Ei,Ji} to a state (Ef = Ei-w ,

,me2 lsl~ :>

--

lSl/z h --- 0

Jf =

- - -me2 Jr=J, f

E~,

o)

b)

Ji}:

a)

Electron-positron

pair

production leading to a continuous energy distribution of positrons and electrons; b) Conversion of K-shell electrons - a monoenergetic electronproduction mechanism; and c) Monoenergetic positron production - a negligible process (after Schl~ter, 1983).

- - - - £=£1s-W V2 1<=-1 C)

ii) The conversion coefficient a, defined as the ratio of the probabilities of inner-shell vacancy formation (process b) and photon emission. In particular, this mechanism is important for low energy nuclear transitions. iii) The ratio , of the two conversion probabilities for electron-positron pair creation and for the ionization of hound state electrons. This ratio is completely determined by the density of the electron wavefunctions at the nuclear origin, thus being independent of the nuclear wavefunction. We computed the differential conversion coefficient d~/dE for nuclear E1 and E2 transitions. For the nucleus ~2U the energy distribution of emitted positrons is shown in figures 32a,b. Nuclear transition energies of 1323 keV, 1423 keV, 1523 keV, and 1623 keV are considered. Figure 32c shows the equivalent differential conversion coefficient d~/dE for nuclear E0 transitions. As bound state only the atomic K shell has been taken into account. The conversion probability of higher bound states is at least one order of magnitude smaller.

10 9

20

dE LkeVJ

8

Z= 92 E1

7

/

Ij

"

i

6

.

.

S

)

,

•

I

L'~'~'j '

3

I

:

Z=92 E2

J /

lt

2 I I

t I

1

~,

I I I

I I I I

I

I

L

I

eVl i

IL

100 20O 300 ~0

50O 6O0

~',

,

Fisure 32. a,b) Differential conversion coefficient d ~ / d E w i t h respect to the kinetic positron energy E for nuclear E1 (part a) and E2 (part b) transitions in szU. Nuclear transition energies w = 1323 keV, 1423 keV, 1523 keV, and 1623 keV are considered, corresponding to maximum kinetic positron energies of E = 300 keV, 400 keV, 500 keV, and 600 keV, respectively. - c) max Differential conversion probability ratio d~/dE with respect to the kinetic positron energy E for nuclear E0 transitions in U. The same transition energies as in parts a and b.

Giant Nuclear Systems

25

Now we can discuss the possibility whether the observed structures in positron spectra may originate from nuclear E0 transitions. One convincing argument against this interpretation is + related to the shape of the e -energy distribution. According to figure 32, the halfwidth of the spectra should be at least 150 keV. However, the observed structure is much narrower, cf. figure 33. The second argument is connected with the energy distribution of emitted ~ rays and 6 electrons. Nuclear transitions of, e.g., multipolarity E1 or E2 should also be observable in the emitted photon spectra provided that proper Doppler shift corrections are performed. This is shown for the U+U system in figure 34a and for the U+Cm system in figure 34b. In both cases the expected ~ line is indicated if the positron structure would be due to conventional conversion. Obvlously, this can be ruled out. If the observed structures are caused by nuclear EO transitions, one should also observe a distinct peak in the 6-electron distribution. Such a peak does not exist, as measurements of the Kienle-Kozhuharov and of the Bokemeyer-Bethge-Greenberg-Schwalm group reveal - figure 35. Since both, ~-ray and 6-electron spectra do not show any substantial structure which could eventually be connected with the positron line, we can conclude that the sharp positron line structure does not stem from a trivial conversion in a single collision partner.

These c o n c l u s i o n s are s u b s t a n t i a t e d by two o t h e r experimental f a c t s : The Orange group (cf. Clemente, 1984) measured the emission probability in the positron line as a function of the azimuthal angle relative to the scattering plane. If positrons were emitted from a system moving with the velocity of the centre of mass, the emission probability for s given positron energy does not depend on the azimuthal angle between the scattering plane and the positron detection. Thus, the ratio N(in-plane) / N(off-plane) should be constant. For

238U + 248Cm 5.8 MeV/omu

-

'" "ll ....

•

J)l I1 I]

-

~

Eo: ; .,

" 'U"'

in U-like Nucleus X2/f,2.41, Px, 0.018

" '

E ,,0ns.,392 (20) keV

~

-

U-Like

11~

NuclIm~

-

XZ/f'"34. Px"0-24

-

0

° '°

/ tll,

T

_

I

'11

30

~(

Zs

llr~

Jl~

Llnle Ernlslsion Fro~ Cm- Like Nucleus X 2 =~'33, Ps" 0.030 I E0"293(6)

_

[~

Line Erniss6on From

-~

r~'l

" "

~ ~l

Ouasimolecule

" -

X~'/f "0~81, Px'O. 56 " , Eo 316(5), r<4OkeV.

0 ¢J

0

200

400

600

800

I000

Ee~ (keY]

0

ZOO

400

600

800

I000

1200

Es+ (JmV ]

Figure 33. Various fits to the positron llne structure according to the possible various interpretations indicated in figure 31. If the positron line stems from the giant nuclear system (vacuum decay), by far the highest confidence (Px = 0.56) is reached - see lower right corner (according to the Greenberg-Schwalm group).

26

J. Finket al.

Figure 34. Part a: The ~-ray spectrum for U+U according to Kienle and coworkers (Clemente, 1984). The expected ~ lines are shown, if the positron line structure would be due to conversion. - Part b: ~-ray spectrum (Bokemeyer, 1984) from ~ 6.05 MeV/u U+Cm collisions measured in 1981. The data shown are coincident with 10-s scattered particles with the same kinematic condi! ' ' ' iJ~41_#l~Ll'SJu'Jpti:on!E~ ,~ = ,,, , : ' ' ' tions as yield the peak in the positron energy U*U spectrum. The expected ~-ray peak is plotted for t|l~ Assumption:E 1 10 "6 the assumption that the positron structure is due to monoenergetic pair conversion of an El or E2 transition in the uranium nucleus. Two K holes at E/A I0 -? the moment of conversion have been assumed. The calculated ~ peak includes Doppler shift, detector 40°< 8p < 50* ".}.lra' ~ efficiency and resolution. - Part c: Same as in £ tO-' part b, now for the assumption that the positron structure is due to nuclear pair conversion of an E1 or E2 transition in either one of the nuclei, '','' I , , I J ,t,I'lll~9 I~i'~ I0 -~ uranlum or curium (from Bokemeyer, 1984). 0 t000 2000 3000 I.O00 "~" - Energy [keVJ

=5.9M e V / ~

l - R a y Energy Spectrum

T-Ray Energy Spectrum ;

. . . .

,,'I" I ~ ' ' Pa; C~v;mon' ~

103 ,

1@

,"" I. \}/ " '~ :,',',

"

z3°U end 2~3rn

~

....

E2

f: "6

: 102 ? 5

~,o' t 6 05 MeV/amu

~

,

6.05 MeV/amu

'

'

3 2 8o0

=

1000

,

,

1200 EI

i

,

,

,

1400 1600 (keY)

•

,

1000

•

i

2000

8oo

looo

i

1200

i

i

i

1400 1600 E, [keV]

i

WOO

2000

positron emission from the separated collision partners, however, the ratio would obey the course of the curve displayed in figure 36 (Kienle, 1984). As an important preliminary result one could determine 'the source velocity in direction of the beam to be somewhat smaller than the cm velocity.' The EPOS group (Schweppe, 1983; Schwalm, 1984; Cowan, 1985) on the other hand used the widths of the peaks to determine the emitter velocity. Due to the similarity in the line shape for all angular detection windows ~aken, the widths appear to be independent of the scattering angle for all collision systems investigated. Moreover, it was found that the observed widths correspond to the minimum broadening expected from Doppler broadening itself (figure 37). T. Cowan and colleagues (1985) thus concluded that the source has to move with the centre of momentum velocity. Since the experimental facts contradict an emission of the positron line from the separated atoms, we should speculate on electron-positron pair conversion in supercrltlcal compound systems. A supercrltical nucleus Z > Zcr which undergoes a transition with w > 2m0c 2 during the nuclear reaction period T may transfer this excitation energy to one of the electrons in the negative energy continuum. The remaining hole represents a positron. But also the K-shell electron can be lifted to the upper continuum. If T is longer than the spontaneous decay width of the K-shell resonance the K vacancy will be filled again leading to spontaneous positron emission. This is a sort of 'positron-gun' firing several shots, the energy of which is supported by the excitation energy of the giant nuclear system. For the nuclear charge distribution a homogeneously charged sphere with a radius R N = 10.88 fm

Giant Nuclear Systems 2]

•

27 i

•

,

•

,

.

i

•

,

.

I

In3 ,v h ~ ' ~ '"n~

U + U •r

lO'a'

"

'

5.9 MeV/u "

'

"

'

EO Conversion

(Op ~, 49 °) '

"

"

'

Transformed (B"-y'~d ~-~

J

<

2 10"5

>.

for

(0

mmu,.mcD

"~ 1025

ee

"0 •~ 10 4

~

61 MeV/amu I0i

i

10-7

,x i

,

1000

800

1200

1400

H~oo

1800

2000

soo

0-Ray Kinetic Energy ltab) /keY

'i

~' =

i

s~unn*Jd:

5

2

~o

"'

I

8oo

i

X~o I

~j~l

I I

Iooo

i

I

I

12oo E,_ [keV]

l

14oo

I

i

il

I 'hll l "

lsoo

moo

Fisure 35. Part a: 6-electron spectrum for U+U according to Kienle and colleagues (Clemente, 1984). The full curve shows the theory of de Reus and coworkers. If the observed line structure in the positron spectrum would be due to E0 conversion, the indicated bump in the 8-electron spectrum should he seen. Obviously the 6-electron spectrum is smooth. Consequently the positron structure must come from the giant nuclear system. - Part b: Electron spectrum from 6.13 HeV/u U+Cm collisions measured June, 1983. The data are coincident with scattered particles in the kinematic region which yields a 316 keV peak in the positron energy spectrum (cf. figure 28). The expected internal conversion electron peak is plotted over the data assuming that the positron structure is due to pair conversion (PC) or monoenergetic pair conversion (HPC) of an EO transition in the uranium nucleus. (Note: The positron line width argues against the PC process.) The plotted peaks are calculated for one K hole at the moment of E0 conversion. In order to explain both the positron peak intensity and the structureless electron spectrum, there must be at least 1.85 K holes at the moment of E0 conversion (95% efficiency and detector resolution) (from Bokemeyer, 1984; Cowan, 1985). has been assumed. In figure 38 we show the differential pair conversion coefficient dS/dE as function of the positron energy E. Nuclear transitions with w = 4m0 c2 and of multipolarity E1 are considered. The appearance of the pronounced peak at E = E R is striking. The dashed curves represent the various contributions of electron states with K = -I and ~ # -i. As expected, the resonance shows up only in the ~ = -i channel. Similar ratios d~/dE are obtained for other multipolarlties. Further details can be found in the articles of Soff (1981) and Schl~ter (1983). Since a giant nuclear compound system very likely is not a static object, its internal dynamics may influence the spectrum of emitted positrons. Clearly, a rigorous treatment of such effects cannot be based on the semiclassical approximation, but requires a fully quantum mechanical reaction theory for the nuclear scattering. Therefore we postpone a qualitative discussion of electron-positron pair conversion in supercritical compound systems to Section VIII.

VII.

ON THE EXISTENCE OF GIANT NUCLEI AND GIANT NUCLEARNOLECULES

The question for us nuclear physicists is how it can come about that two such very heavy nuclei can stick together for a time longer then 5 x 10 -2" sec, probably even longer than 10 -*~ sec. These are the typical times deduced from the sharp line structure of the positron spectra. At first this seems so unlikely that one would like to dismiss such a proposal right away. We would like to present some ideas we have worked out with H. Seiwert, P.O. Hess, and J.A. Haruhn (Frankfurt) and V. Oberacker (Vanderbilt University) (cf., e.g., Seiwert, 1985a; Hess, 1984a, 1984b). The problem is to calculate the nucleus-nucleus interaction potential. We first followed the path known from literature and first described many years ago by Scheid and Greiner, namely to calculate the folding potential of two approaching deformed nuclei. Using the H3Y interaction V(I,2) whose parameters were fitted to medium heavy elastic ion-ion scattering by Satchler, one calculates V(R)

=

$ dxxdx2 ?(rz,R) V(1,2) p(r2,R)

•

(18)

28

J. Fink et al. l

1.4

_ N~in

I

I

I

..

ptone)

i

i

U+Cm

18

Ne~OffPLone) 4 - ~ ,

:~ T h + C m

I

1.6

1.2

ProjectiLe-Like

"?><

:£ 1l,

1.0 IJJ

0.8

. . ...'"

RecoiL-Like "\

1.2 o,,.

..'""

Center

of

Mass

1.0

0.6

0.8

e" Vl)

I

I,

300

280

320

"'"

0.6

I EL°bIke

25

I

I

l

I

I

30

35

~0

~5

50

,%Lab

340

-

proj

[deg]

Figure 36. Ratio of positrons emitted in the scattering plans (0 m K ¢ S 450 ) to those emitted off-plane (&5" ~ ¢ ~ 90°), as a function of positron energy. For emission from the combined system moving into the beam direction this ratio should be constant (= i.), whereas for an emission from the separated collision partners the ratio should follow the full line. From the Kienle group (1984),

Figure 37. Doppler broadening of the positron line expected from projectile, target, and centre-of-momentum emission in binary scattering events from 6.05 geV/u U+Cm and 6.02 H e V / u T h + C m collisions plotted as a function of the laboratory projectile scattering angle. Broadening is expressed as a fraction of the calculated width for cm emission. Obviously, the line width is independent of the scattering angle ruling out positron line emission from the separated collision partners, but affirming emission from a system travelling with cm velocity. From Cowan (1985).

dP E1

dE

Z- ~1.

10(

I

R. • W, H h R

10

6= ~3

F~gure 38. Differential pair conversion coefficient dR/dE as function of the positron energy E. Nuclear transitions with w = 4m,c 2

/;-, 1.0

1.$

,

,

Z.O

2.S

e / ).0

and of multipolarity El in a supercritical nucleus Z = 184 with a radius ~ = 10.88 fm are considered. The dashed curves denote the various contributions of the electron angular momentum states.

Giant Nuclear Systems

29

ORNL OWG 8~.16124

I

I --T--1

-I-

~ 238 g2 U

/'/'~'\.~I = ¢t2 : -/

90"

•

1 .... [ + 238 92 U

IO I'A t

/F

/

i

\\~0

-

lO0

I/ I I

Ci 3 650

\

\

15.1

I I I I

-

6,0 MeV/A

=

~ •

,,\~, \

Figure 39.

Folding potentials of two deformed Uranium nuclei for various orientations. The arrows indicate half density overlap.

/

t

I

I ~ / ~

i

16

14

I

18

I

__L___

20

)2

rum]

The densities p(r,R,c,t) are deformed Fermi densities; c is the radius parameter c = c0(l + Ep =2U Y2B (e'¢))' and t is the surface thickness. A typical result is shown in figure 39 for U+U and various orientations of the two Uranium nuclei. Clearly, one can believe such potentials only up to half density overlap, indicated by arrows in figure 39; the extremely negative binding at small distances is unrealistic. The question now arises how to take the wrong parts out of the potential. To proceed we recall two facts: First, note that for a homogeneous spherical charge distribution ~(rl) the integral I d~,d~2 ~(r,,R) V(r,,r2) ~(r2,R)

=

av (4~/3)R ~ + as 4~R 2 ,

=

-3.884 [MeV fm-*]

(19)

where av

=

2 1 vi~ i 2 . ½V, pa 2 2~p02 Zi=

as

=

2 1 vi~ i ~ -~p02 ~i=

=

1.399 [MeV fm -~] ,

, ASth

-10[

I

. . . . . 0

Figure 4 0 .

, 0.5

,,,

J . . . . 1.0

I, 1.5

J

1,5,,

A

i

,

I

,_ 20 2.0-13

.

.

.

.

.

r[fm]~

Surface thickness correction for U+U as a function of distance.

J. Fink et al.

30

e5o ,V[~] \\ ~,

/

~pb.~

\ ,

8501V[MeV]

nZlh.Z~U

VlMw ]

It'~," m

O0

00---

~.p,.,~s,,

•

O0

£-%"

/ 0 0 - - ~,.p,.4so ',.~T,,,o,,-'{ ",,, O0 .... ~-~,-~

800";

r~

k~),

',

nSU-m U

793 •,,

\<""

-

700

',x

\

,\

7~ -----T

~

"

O~e

5~

V[MeV] I~ 850

900IVlMeV] \ \ X /\I \

Z3Z~mt~Cm

\'\, \ \ \

OO--

Ibl~,"0°

O0 . . . . ~.~,.~s,,

800

O0

00.--

',, ,,~ >---,

80C-

\\

p.p,-IF

~.~,-~s0 ,

,

',

\--\

750 700

850[ \ \

Z38U.z~.6Cm

k k 5.91~/~

\,~ ',',

~

" 7OC ~X\

65~

i

i

Figure 41. Nucleus-nucleus potentials with surface thickness corrections for various orientations of the colliding nuclei. Several combinations leading to the giant nucleus domain are presented. The calculations were carried out by M. Seiwert (1985a).

,

\'

6~

1}

.....

',~ ',3",

,=Z"7 ,T, lZ, 15

18 rlfm]

"

i.e., one gets a volume and a surface term which are functions of the force parameters. Second, remember that we have examples already in physics how to get rid of wrong results: We renorma]ize. This is done so in quantum electrodynamics and, perhaps more familiar to us, in the calculation of the shell corrections. In the latter case we renormalize by subtracting from the single particle sum a smoothed sum, i.e., Asc(R) = Zi zi(R ) - Z i ~i(R,~) .

(20)

Here ~ is a more or less phenomenological smoothing parameter. The shell correction Asc(R) is then added to the 'average potential' calculated in the empirical liquid drop model. In this way we ensure that the binding properties (Q values) are properly contained in the potentlal. We do now the same with the folding potential by calculating Asth(R)

=

I dxzdzz p(rz,R) V(1,2) p(r2,R) -

a $ dzzdz2 ~(rl,R) V{l,2) ~(rz,R)

.

(21)

We call Asth(R) surface thickness correction. Note that we have taken the volume and surface terms out of the folding potential and simply keep the effect of the surface thickness. The factor a is determined such that Asth(R )

~

0

for

R ~ 0

(22)

If the surface thickness vanishes, Asth(R) ~ 0. The typical result for Asth{R) is shown in figure 40.

Giant Nuclear Systcms

31

o} Figure 42. (a) Stretched U4.U bended against each other. In the latter case the two nuclei are coming closer to each other. The Coulomb energy in case (b) is certainly higher than in case (a). Because of the quadrupole and hexadecupole deformation of the U nuclei there are special orientations where the nuclear touching is especially intense. Adding this to the liquid drop potential, as in the well-known shell correction method, the p o t a n t / a l s for U+U of figure 41 result. Again, various orientations are shown. Obviously bir~dJn8 pockets of molecular type appear. They have the following interesting properties: i) For the head-on configuration the potential pocket is about 20 HeV deep (this depends, of course, on the strength and range of the interaction, which is taken from Satchler's fits, as stated above). It lles precisely at the energy where the positron experiments are carried out, i.e., in the vicinity of 5.8 HeV/u. - ii) For other orientations the pocket appears at higher energy, is deeper, and the outer barriez is narrower. The rise in energy is essentially a Coulomb energy effect: In the non-aligned orientations of figure 41 the nuclei approach closer and hence the Coulomb energy rises (see figure 42). At those orientations where the nuclear touch is especially intensive (this is because of the quadrupole and hexadecupole deformation of the Uranium nuclei), the potential pockets are considerably deeper and the outer barrier becomes narrower (see figure 43, where this effect is once more stressed). Clearly this effect can be c a l l e d nuclear cohesion. We are now led to the picture that two nuclei form a nuclear molecule of the type illustrated in figures 42 - 44. Butterfly and belly-dancer modes appear, also relative vibrations of the two nuclei against each other (B vibrations of the giant system). In the latter case the distance R is oscillatlng. The former case has similarity with ~ vibrations of the giant system. They represent the remnants of the free rotations which the nuclei would have at large distance. At small distance only those 'hindered' rotations survive. There are also the modes of the indivldaal B and ~ vibrations of the individual two nuclei. Hence we see that the spectrum of collective modes of the giant molecules is extremely rich. At the barrier, where the positron experiments are carried out, we have a highly dense collective spectrum with probably overlapping levels, thus giving rise to the large time delay observed (see figure 45). Because the moment of inertia is so large, the rotational bands are quenched, i.e., the first 2+ state has an energy level of only a few keV. Similarly the B vibrations have low energy of the order of 100 - 800 keV. F.O. Hess and colleagues (1984a, 1984b) investigated these spectra. Counting for all ~ and ~ vibrations onlp in the giant nuclear molecule U+U, there exist about 520 band heads with much more than 30,000 rotational states having decay widths in between 1 keV and 1 H e V , i.e., an enormous richness of level structure.

o} Figure 43. Schematic explanation of nuclear cohesion. In orientation (b) the nuclear cohev

R

pPWP-Bt

sion force is much more active than in configuration (a).

32

J. Fink et al.

b) Figure 44, Butterfly (a) and belly-dancer (b) modes of a giant molecule consisting of two deformed nuclei. Their dynamical properties have been theoretically investigated by P. Hess and colleagues (1984a, 1984b).

We would like to mention again that a number of these levels can possibly be investlgated experimentally in the future by positron spectroscopy. The positron line due to spontaneous vacuum decay may acquire satellites which result from the de-excitation of the glantmolacula (Ramanand Stokes lines), as will be dlscussadintha next Section. It is remarkable how close the calculated pocket harrier for the aligned position of the U+U or U+Cm system is compared with the enarglas of 5.7 - 6.2 MeV/u for the U beams, at which the positron resonances have been observed. Too far under the barrier the nuclei undergo only Coulomb deflections. At the Coulomb barrier only tha 'nose-to-nose' orientations can overcome the barrier and the positron llna should be most pronounced. Too high above tha barrier the life time of the giant system becomes smaller, and also many more oriantatlons with rather different distances of closest approach lead to a smearing out of the positron line structure. (See also the more quantitative discussion in connectlon with figures 49 and 50.) This is, indeed, what has first been observed by Kienle and coworkers for the U+U system and by Graenberg, Schwalm, and their associates for the U+Cm system (sae figure 46). This threshold effect gives further evidence for the existence of a nuclear pocket. The calculated U+Cm pocket gives exactly the posltion where the dominant positron line is observed (~ 320 keV). This is not so for the U+U and the other colllslon systems, where the prasant calculatlons indicate that the spontaneous positron lines should be located wall below E + ~ 320 keV (see figure 24). This and other open quese tlons will be discussed in Section IX.

c-J

7 U+Cm I ~5-

i

\

R

Figure 45. Schematic drawing of the spectrum revesllng the high density of collectlve states of the giant molecule near the harrier.

5.9

6.0

6.1

6.2

Ebmm[MeV/u]

6.3

Figure 46. The threshold effect of the appearance of the positron line structure as a function of the projectile energy (Schweppe, 1983; Schwalm, 1984; Bokemeyer, 1984; Cowan, 1985) may indicate a nuclear pocket.

Giant Nuclear Systems

33

VIII. EXTENSIONS OF THESEHI-CLASSICAL APPROACH THE INTERPLAY BETNEENREACTION DYNAMICS AND POSITRON SPECTROSCOPY I t i s c o n c e i v a b l e t h a t t h e modes o f t h e g i a n t n u c l e a r s y s t e m a r e c o u p l e d t o t h e p o s i t r o n d e c a y . Thus a f u l l y quantum m e c h a n i c a l r e a c t i o n t h e o r y f o r t h e n u c l e a r s c a t t e r i n g i s r e q u i r e d . U. H e i n z , J . R e i n h a r d t , and c o w o r k e r s ( H e i n z , 1983a, 1984a) e x t e n d e d t h e t h e o r y t o w a r d s a quantum m e c h a n i c a l t r e a t m e n t o f t h e n u c l e a r r e l a t i v e m o t i o n , w i t h t h e aim t o i n c o r p o r a t e f i r s t i n a quantum m e c h a n i c a l way t h e n u c l e a r p h y s i c s r e s p o n s i b l e f o r t h e s t i c k i n g o f t h e two n u c l e i , g a i n i n g i n s i g h t i n t o t h e quantum m e c h a n i c a l o r i g i n o f t h e c l a s s i c a l n u c l e a r t i m e d e l a y . A v e r y g e n e r a l framework, s u i t a b l e f o r t h e d e s c r i p t i o n o f any n u c l e a r and a t o m i c i n t e r f e r e n c e phenomen a , was d e r i v e d ( H e i n z , 1983a). W i t h i n t h a t framework i t was shown t h a t unde r t h e a s s u m p t i o n t h a t a t o m i c and n u c l e a r e x c i t a t i o n s can be l o c a l i z e d i n d i f f e r e n t s p a t i a l r e g i m e s , t h e a t o m i c e x c i t a t i o n amplitude from the two-centre elgenstate i to state f in collisions with nuclear sticking can be written as (Heinz, 1983b, 1984a, 1984b)

£ ai~f(-,)

"£ +£ "~ -~• , - Zn ai+n(-,Rm) exp (2i6~} _ an~f~m,

=

(23)

if the scattering occurs wlth angular momentum £. Here ai~n(Rm,- ) is the atomic excitation amplitude on the way from R = - to R = H sions),

m

(where R

is a matching radius of nuclear dimen-

m

and an, +£ f~,m, r, ,~• is the atomic amplitude on the way out from Rm to -. The a±£ can be n

determined semiclasslcally with sufficient accuracy.

6£ is a nuclear phase shift, describing

the nuclear scattering at distances H < Rm; its dependence on the intermediate electronic channel n is via the available energy in that channel at Rm:

6£n ~

6£(E.En(Rm))

.

(24)

These can, i n p r i n c i p l e , be d e t e r m i n e d from t h e i n t e r n u c l e a r p o t e n t i a l . We c o n s i d e r t h e c a s e o f a p o c k e t i n t h a t p o t e n t i a l (as i n f i g u r e 41) which can s u p p o r t many v i b r a t i o n a l and r o t a t i o n a l s t a t e s . Then we can p a r a m e t r i z e 6£ as a s u p e r p o s i t i n n of B r e i t - W i g n e r s ha pe d r e s o n a n c e s in

a

set

1,...,N),

of,

say,

N rotational

bands on t o p o f v i b r a t i o n a l

states

whose e n e r g i e s a r e En£ ffi En + g£(£+1)~ 2 (g i s t h e r o t a t i o n a l

for the U+U quaslmolecule), S£(E)

E

exp {216£(E)}

and whose widths we call =

w i t h e n e r g i e s En (n ffi c o n s t a n t , a bout 0.7 keV

rn£:

~Nn=l [(E'Ent'½1rn£)

/ (E'En£+½irnt)]

"

(25)

These resonances will be very dense, and in experiment one will always average over many resonances (e.g., the GSI beam has an anergyspread of the order of 10 HeV). The thus averaged excitatlon cross sectlon with (23) takes the form dd°(i~f) ~

-t' an'*f +£' ai+n -£* an*f +t* [Zt£' ffi ~(4Kz) ~nn' al~l' x (2£+1)(2£'+1)

P£(cosg) P~,(cosg) exp {21(o~,-o£))]

.

(26)

0£ is the Coulomb phase shift, which is slowly varying with the beam energy and can be taken out of the energy average. Also the a

±t

of the ~ sum and energy average.

are weakly dependent on E and £ and have thus been taken out * defines a nuclear autocorreJaclon function; its

Fourier transform with respect to the difference in the energy arguments can be interpreted as a distribution of nuclear delay times (Relnhardt, 1983). It can be analytically calculated from the model (Heinz, 1983b, 1984a, 1984b), using

m

(bE/I)

J" dE f(¢) / [(t-E)Z+AE 2] ,

with AE ffi I0 HeV, as the prescription for the energy average. cross section can be written as (Heinz, 1983b, 1984a, 1984b) do(i'f)

(6)

ffi

d°(delaved~(8,T ) . ,Y" dT l a i ~ f , T Iffi d2 N

(27) The result for the excitation

(28)

I t s e p a r a t e s ~ n c o h e r e n t l F i n t o a d i r e c t p a r t , due t o p u r e Coulomb s c a t t e r i n g w i t h o u t t i m e delay, and a delayed part due to resonance scattering. In the latter contribution lai~f,T [~ is

34

J. Fink et al. t h e s e m i c l a s s i c a l e x c i t a t i o n p r o b a b i l i t y c o r r e s p o n d i n g t o a s h a r p c l a s s i c a l d e l a y time T. I t i s weighted w i t h a d i f f e r e n t delayed n u c l e a r c r o s s s e c t i o n f o r every T (Heinz, 1983b, 1984a, 1984b): d°(delayed)(e,T)

=

~(4K z)

IZ t

( 2 t + l ) exp ( 2 i o t} P t ( c o s 0 )

o~(T)I a

,

(29)

where s~(T) r e s u l t s from t h e model eq. (25) as aT(T)

ffi

zN n=l

[~2"AE"rnt /

{E-Ent+l(AE+½rnt)} ] exp {-i(Ent-½1rnt)T)

x ~N mfn [(Ent'Em£'½i(rnt+rmt))

/ (Ent'Emt'½i(rnt'rmt)}]

"

(30)

In figures 47 - 50 we show the results from a sample calculation for U+Cm collisions at 6.2 MeV/u beam energy (Ecru = 750 MeV) with the assumption that the maximum of the potential barrier is at Vma x = 725 MeV and that the pocket supports one rotational band with band head energy 8 MeV below the barrier. The widths have been computed by the Hill-Wheeler formula for a parabolic potential barrier. In figure 47 we show the time distribution for the delayed nuclear cross section eq. (29) at 8 = 45 e . The regular peaks can be interpreted as a lighthouse effect generated by a nuclear molecule rotating with a rather well defined mean angnlar momentum t = 226 ~ and decaying after a different number of revolutions under the chosen scattering angle. The width of the state with ~ = 226 ~ is 24 keY, explaining the possibility of many revolutions. The position of the peaks can be explained by observing that pure Coulomb scattering (T = 0) corresponds to 8C(£=226~ ) = 151 °. In figure 48 we show the angular distribution of the nuclear delayed cross section for fixed times. One sees very pronounced dependence oa the scat ° taring angle. The position of the peaks, being reflected by the angular dependent appearance of positron lines (which is sharp for small T, but smeared out for ion E T) is consistent with the llghthouse interpretation. We will right away demonstrate that the occurence of llne structures due to spontaneous posltron emission is strongly dependent on beam energy, with the line appearing only in a rather narrow energy window around the Coulomb barrier. Let us assume that the potential pocket contains l0 rotational bands with band energies 1, 2, ..., 10 MeV below the Cop of the barrier Vma x. We varied the relevant parameter E - Vma x by shifting the whole internuclear potential instead of the beam energy. Since the energy dependence of the delayed atomic excitation amplitudes is comparatively weak, this procedure is a good approximation for the computation of an 'excitation function I . In figure 49 we show three positron spectra for beam energies (i) far above the potential barrier, (li) barely below the barriers and (lil) far below the barrier, as indicated in the insert. The llne structure at E + ~ 300 keV, attributed to spontaneous positron emission, e clearly shows a threshold behaviour. This is also demonstrated in figure 50 which shows the 'excitation function' for spontaneous positrons. The posltron line is strongest if the cmenergy is ~ 20 MeV above the top of the (t=0) barrier; since typical angular momenta are of the order of 200 ~, then E is barely below the effective potential barrier. CAn

~. 10° ~_~

U.Cm

B.2 MeV/n 8.45 o

'°-' I o

..8

10.6 _

10.-8-

l

Figure 47.

L

3

T [lO-2°s]

i

5

t

i

6

Delayed n u c l e a r c r o s s s e c t i o n as a f u n c t i o n o f s t i c k i n g time a t 8 = 45 e f o r one band o f r o t a t i o n a l s t a t e s . The U+Cm system a t 6.2 MeV/u i s c o n s i d e r e d . A t y p ical l i g h t h o u s e e f f e c t (damped) can be r e c o g n i z e d .

Giant Nuclear Systems

l-

',

/

\

l',,z.~-=,\

,-

/ ,,

,:

.10-~°, /~

.,L......, -1==

\

',,

;..:/

35

48 (left). Angular distribution of delayed nuclear cross section for different sticking times of the U+Cm system at 6.2 HeV/u.

Figure

Figure 49 (bottom left). Positron spectra for

=

)..'i "

".;

..~

L

"-.

I-.'" .,/......"f.~,

~-6

/

L "¢"

.... "*;

',

5ilO''s

,-' \"

i

\\

.,-, !

1 --jf~

I/

/_~ iI I1 ~,,10- s J |

"",

three values of E - Vma x, at O = 60". Only the contribution from scattering events with delay times 0 < T ~ 6 x 10 -*t sec, normalized to the Rutherford cross section, is shown. The background from Rutherford events is not added.

Figure

_st

10[

u+Cm

~\ /".'\\\

I ,

, , I ,\,

&ZMeVln

[" ....

0

50

'~}/

\

i

60 ° .

, , i , ," ,]

100 8(deg)

,

50 (bottom right). The height of the maximum in the positron spectra as in figure 49 is shown as a function of cm energy for 8 =

150

,

i

_ _

i

"/

/

X

i

Vm=[MeV] 730

?50

770

',,._L

~4-

710

\°-°°°17

/

\

1

U+Cm, 0-60 °

~3lu W

k

_ 0

690

...........

v__. 690MeV 73o M~v 760MeV

t

100

300

500 700 Ee+[keV]

900

1100

O/J

-20

r

I

0

t

I

I

I

20 40 E-V,~x[MeV]

t

l/

60

The beam energy dependence of the spectra has its origin in the delayed nuclear cross section and is determined by the fact that only those resonances Eat, which lie within an energy interval f3E around the mean incident energy E, contribute to the delayed cross section. Therefore, at energies far belOW the potential barrier, resonances are too narrow to be excited, and the cross section for delayed scattering events (and thus the chance to see spontaneous positron emission) is negligible. Far above the barrier, the beam can only hit the high-lying, short-llved resonances, again leading to a very low chance for spontaneous positron emission. Only of the beam energy is barely below the barrier, there is a reasonable probability to be captured by • resonance with sufficiently long life time to lead to enhanced production of spontaneous positrons.

36

J. Fink et al.

T h i s model so f a r o n l y c o n s i d e r s e l a s t i c n u c l e a r s c a t t e r i n g and n e g l e c t s i n t e r n a l n u c l e a r e x c i t a t i o n s . S. Schramm, J . R e i n h a r d t , and c o w o r k e r s (1985) e x t e n d e d t h e framework t r e a t i u E two d i f f e r e n t models quantum m e c h a n i c a l l y . I n t h e f i r s t c a s e one p o t e n t i a l p o c k e t i s assumed and the giant system may undergo a transition from one state in t h e p o t e n t i a l p o c k e t to another one (figure 51a). Obviously two positron lines result: one corresponding to the positrons from vacuum decay, the other one to the same process with the emltted positron carryinE in addltion the de-excitatlon energy of the giant nuclear complex. This is illustrated in the central part of figure 52. Also in the case of a subcrltlcal charge, Z < Zcr , positron lines could be produced by monoenergetlc conversion of excited nuclear states filling a hole in, e.g., the is, 2p}, at6. inner life time T state. For right part released by amount E N .

shell levels. The width of the llne again would be inversely proportional to the of the nuclear system, provided this is shorter than the decay time of the excited supercrltlcal Z also a new physlcal phenomenon is possible, as indicated in the of figure 52. In an inverse conversion process the nucleus may take up energy the filling of a Is hole. The emitted positron has an enerEyreduced by the absorbed

In ordinary stable nuclel these processes are rather slow, having transition times in the order of > 10 -I= sec (corresponding to partlal decay widths F < 10 -3 eV). The contraction of cony bound and continuum w a v e f u n c t i o n s i n s u p e r h e a v y atoms s t r o n g l y e nha nc e s t h i s p r o c e s s . T h i s i s most p r o m i n e n t i n e l e c t r i c monopole (E0) c o n v e r s i o n , where t h e w i d t h t o a good a p p r o x i m a t i o n i s g i v e n by t h e s i m p l e e x p r e s s i o n

Fconv(i~f,E0 )

=

2~ IV6 e = @f(0) @i(0) R=p[ = ,

(31)

with the nuclear EO-transitlon matrix element

U-U dPe./dE[lO-6/keV] ELo b-5.6MeV/u~7°F\

(R)[MeV]

20

Figure

51. P a r t a: S c a t t e r i n g p o t e n t i a l o f a U+U c o l l i s i o n s y s tem and r e s u l t i n g p o s i t r o n s p e c trum showing two l i n e s t r u c t u r e s which o r i g i n a t e from d i f f e r e n t processes. The f i r s t pe a k a t E + = 200 keV o r i g i n a t e s from t h e e

spontaneous decay of a hole in the overcrltical is state. The 15 20 Rlfm] second peak at E + = 1200 kaV e emerges due to conversion - an electron fro~ the Dirac sea is getting excited into the ls state [ I I i i I 4" I ~,,I creating a positron. The speco0 E[mcZ] -trum shown results after averaging over a beam energy interval AE = 200 keV to include the whole 61, dP~/dE influence of the chosen resonance [MeV] (from Schramm, 1985). - Part b: Assumed scattering potential i n c l u d i n g two p o c k e t s and r e s u l t i n g positron spectrum of a U+U collision, showing two line ~ \ structures a t E + = 190 keV and e 10 15 R[fm] 280 keV. Both peaks originate from the spontaneous emission of positrons due to the diving of the ls state into the lower contlnuum. Since the Is blndlnE energy varies with R, each pocket generates a peak at a different ' ~ i J i positron energy. The high ampliE[ke~ tude at E + = 280 keV arises from e the enhanced spontaneous emission a t s m a l l e r i n t e r n u c l e a r d i s ~ a n c e s . On t h e o t h e r hand, r e s o n a n c e s o f t h e l e f t p o c k e t l y i n g more d e e p l y w i l l p a r t l y be s c r e e n e d by t h e o u t e r one (from Schramm, 1985),

500

[1041keY]

1000

690~R)

U-U

4 E~b'5"~M~V~j~ 6501~-/

100

2OO

300

400

Giant Nuclear Systems

Z
37

Z•Zcr

"

EIcE N. . . . . .

--

U',.- I 1

. . . . . .

10) --*

I0}

-.~

ENt~,d~ "10)

E1s-Monoenergetic pair conversion filling e hole in the Is state, induced by a nuclear transition with energy EN, in a suhcrltical (left) and supercritlcal (centre) system. In the

Figure 52.

latter case also the inverse process is possible~ where the nucleus becomes excited while a positron with reduced energy is emitted (right).

p

=

(32)

Zp ! dV Sf (rp/R) z $i "

r is proportional to the electron (positron) densities in the initial and final state at cony the origin. A simple inspection of these densities assuming #i to be e state in the positron continuum and tf the Is state leads to the observation that rconv(E0) increases by more than five orders of magnitude when going from a single U nucleus to the combined Z=184 system, assuming constant #. Depending on the strength of the nuclear matrix element this means that the width of the conversion process can approach the spontaneous decay width within one order of magnitude. Let us now assume that there exist two potential pockets 3 as schematically shown in figure 53. Then (Schramm, 1985) two principal positron lines emerge (figure 51h). For each one the Stokes satellltes of figure 51a might appear. In a provocative picture the following scenario might be expected: families of llnes corresponding to various shape isomeric giant nuclear structure might be observable in the future, e spectroscopy of the collective states of the giant nuclear system could, indeed, be within reach.

AV[MeV] 9 0 0 ~

238U246Cm

800~isomeric X j .. co~tigurotion ~giont molecule

giant nucleus

X

600

O

501 30[~

molecutor distance determined by the positronline I

100

,,,, o.5 .... 1.o , .... 1,5 I , , I ,210 • ~= p

20

L

i

i

30

i

40

O i r[fm],_ 50--

Figure 53. Potential with giant molecular and giant nucleus pocket for U+U. The dashed line shows speculations on more spherical exotic giant structures.

38

J. Fink et al. IX.

OPENPROBLEMS

Nuclear PhysicsAspects The most puzzling problem is that in all supercrltical systems studied so far the main posltron llne seems to have the wrong scaling with (Zp + ZT). Taking the data from the EPOS group alone (figure 29) it seems - perhaps with the exception of Th+Cm - to appear around 310 - 320 keV. A small side peak having the correct scaling expected for a giant molecule (cf. figure 24) is statistically irrelevant st present time. Figure 54 summarizes the position of observed positron peaks from the Orange and EPOS groups. Obviously both deviate in energy by about 50 - 70 keV. The question arises : Are these a11 different lines seen under different conditions or are - due to experimental shortcomings - all lines indeed the same? Suppose the model leading to the theoretical positron lines in figure 24 is correct. What ere then the observed structures at E + ~ 320 keV in the spectra of lower united charge systems? e Well, one could imagine that those lines are Reman lines, reflecting the excitations of the giant molecular system. Their strong intensity speaks, at the present level of our understanding, against such an interpretation. However, the results of Schramm, Reinhardt, and coworkers (see figure 51) indicate that also such satellites can be rather pronounced. It could also be possible that there exist other, more spherical quasibound configurations (isomers) of the giant system, which then would emit spontaneously positrons at higher energies. The standard, homogeneously charged giant compound nuclei, stabilized due to shell corrections are conventionally expected at an energy of 900 NeV, which is about 180 MeV too h l g h t o be considered as a possible candidate for those positron emitters (see figure 53). However, due to the strong Coulomb repulsion it is not unllkely that rather hollow or torus-like structures could exist as giant nuclear systems. Such speculations are indicated by dashed lines /n figure 53. L. Neise~ 3. Fink, and J. Maruhn investigated these possibllities. They all were unsuccessful in the sense that the lowerlng of the potential at the spherical point from 900 HeY to approximately 720 MeV (see figure 53) could not be obtained. A new fit of the Bass formula including the binding energy of giant nuclel as a principal input, however, was possible, with a good least mean square deviation per degree of freedom. Haruhn, Friedrich, and colleagues (Seiwert, 1985b) obtained this result, if the surface contribution to the nuclear compression constant is reversed in sign. This means that giant nuclei should be easier compressible than ordinary ones, approaching the nuclear matter limit from above as the nucleon number A ~ -. The microscopic interpretation of such a behaviour seems to be in a phase transition of giant clusters of nuclear Batter as a function of A, leading to a loosening of quark-gluon confinement in nucleons, similar to energy band structure for electrons in crystals (metals), and finally to a quark-gluon plasma. Derreth, Zhang, Sch~fer, and coworkers (Derreth, 1985) are, already for more than a year, investigating this important nuclear structure problem. Energy bands for quarks and gluons emerge if the nuclear cubic crystal is large enough. One can then speak of quark and gluon (or colour) conductivity. Also the binding energy per nucleon seems to increase, It could very well turn out that an increase in Z and A in giant nuclei leads to an unusual blow-up in the nuclear radius of such systems, thus resulting in a nearly constant binding energy for the ls electrons. Such a mechanism could therefore result in a positron resonance energy nearly independent of Z in the giant nuclei domain.

Th,Th

U*Th

400

LAU //"

Th*Cm //

U÷Crn

Else[keY] 300

200

100

180

m 182

J 184

i 186

I Ztot 188

Figure 54. Summary of positions of observed positron lines in the experiments on the EPOS (o) (Schweppe, 1983; Schwalm, 1984; Cowan, 1985) and Orange-spectrometers (~) (Clemente~ 1984) as a function of united charge Z u . The full line shows the

theoretical dependence of the Tesonance energy assuming two sticking nuclei at half density overlap for the most elongated configuration (figure 42a), the long-dashed llne stands for the belly-tobelly configuration (figure 42b). The dashed line represents calculations for the most amalgamated configuration of a spherlcal nucleus with combined nuclear charge Z u. Screening effects within the HFS formalism included.

(de

Reus,

1984)

are

Giant Nuclear Systems

39

Figure 55.

Two different possibilities of producing approximately the same nuclear excitation in giant systems: a) vibration; b) The same cluster is formed (e.g., a polarized = particle) which has, of course, always the same excitation energy.

It is crucial to know whether there occurs positron line structure also in undercritical s y s t e n s . E a r l y measurements o f some such s y s t e m s ( s e e f i g u r e 27a) d i d n o t i n d i c a t e any l i n e . Kienle and colleagues measured recently the undercritical U+Ta system with high statistics. According to Kienle (1985) there is no llne structure seen. A small fluctuation around 260 keV, if at all relevant, has only a cross section which is by a factor 40 smaller than the cross section of the principal line structure in the U+U system. In this case the ~ spectrum seems to be smooth everywhere. Positron resonances with such small cross sections can appear easily in undercritical systems due to conversion in the giant system. On the other hand the EPOS group (Bokemeyer, 1985) claims to observe a sharp line structure for the Th+Ta system. In this case, however, also structure at the corresponding energy of the ~ spectrum is observed. This indicates a conversion process and only a very careful analysis (not available at present) can reveal whether there is something of the llne structure left or not. It would further be interesting to know the magnitude of the cross section. If one can safely say that in undercritical systems no principal lines appear, then a rather constant principal line in overcritical systems could indicate that the same giant system is formed in all cases. The nuclear structure leading to a rather stable, spherical giant Th+Th system is presently unknown. According to Martin Greiner and W. Scheid (1985), the dynamically induced and quasistatic a decay of giant systems can be expected to be fast enough so that within iO -*~ sec neighbouring giant systems can be formed. A rather constant principal line structure could also be obtained if the various giant systems have (approximately) the same collective state, e.g., a 0+ state like a ~ vibration at energy E~ - 1.3 HeV (cf.

figure 55a).

Such states can indeed be expected to be nearly independent of (Z,A}, because nuclear shell structure should not play a role for the vibrational spectra. A nearly classical behaviour can be expected. P. Schl~ter and colleagues (1983, 1985) have studied the subcritical and the supercritical conversion of such a state and obtained the positron spectra shown in figure 56. Obviously various lines are then present. That line, however, which stems from the pair conversion process depositing the electrons in the 3s, 4s, ..., etc. levels, is nearly independent of the system. All other conversion lines should be strongly dependent on the system studied (figure 57). For overcritical systems always the spontaneous decay line should also be present; it has been omitted, however, in part b. Also in the electron spectrum a number of conversion lines from the giant system should be seen, particularly in coincidence with positrons from the llne structure. Obviously it is most important that an experimental search for other positron resonances is started and that it is reliably decided whether other line structures are there or not. Backe (private communications) had speculated whether not always the same small cluster has been formed in a certain excited state of 1300 keV (cf. figure 55a) giving rise to a constant conversion line. This, however, is ruled out by the results of Schl~ter and coworkers (1983; cf. also Soff, 1981) who demonstrated that pair conversion spectra in light nuclei are broad and could never lead to a narrow line structure.

New Elementary P a r t i c l e s Seen i n the P o s i t r o n Spectra ? Let us now discuss, whether it is possible that the positron lines are caused by the decay of a new, previously unknown particle produced in these heavy ion collisions. Three different scenarios, investigated by A. Sch~fer, J. Reinhardt, and colleagues (1984, 1985) can be conceived in this context to explain the positron line. (a) A neutral particle X is formed which decays into an electron-positron pair X * e+ + e'. The mass of X is determined by m X = 2(me+E), where E is the kinetic energy of the positron. Assuming

J. Fink et al.

40 E = 330 keV we h a r e m X = 1680 keY.

(b) The unknown particle carries positive charge and decays into a posltronend a neutrino, X ~ + + v. Its mass is m X = m e + E + [E(E+2me) ]} = 1510 keV. + (c) A heavy unknownparticle decays into a lighter one, emitting a positron: X * X' + e .

e

The last alternative has been mentioned for completeness only. We will not consider it further because it calls for a whole family of unknown particles. We also do not wish to discuss the posslbility (h) because it is hard to imagine that a charged light particle could hitherto have escaped detection. However, there m a y b e s related mechanism which completely avoids the introduction of a new particle (Sch~fer, 1984). This is based on the fact that the Glashow-SalamWeinberg (GSW) model of electroweak interaction includes a 'hidden' charged particle, namely the Goldstone hoson. As discussed by Sch~fer (1984), in sufficiently strong external electromagnetic potentials the Higgs vacuum of the GSW model is expected to become unstable against the condensation of negatively charged Higgs particles and formation of positively charged Higgs vacuum excitations that finally decay into a positron and electron neutrino (figure 58). This phenomenon is expected to be similar to the decay of the neutral vacuum in QED, since the nuclear charge has to surpass a critical value Z' for which, unfortunately, no quantitative cr predictions exist yet. As the massive Higgs vacuum excitations can only appear if the Higgs vacuum changes locally, no such particles should be observed in other (low-Z) experiments.

I

~ ' PbC -m

=o ~

Y

'A 2s

!

u-

Cm

3s

I l I

f

[

I

I I I $

I

I 3:~'sl-~

6

2

3 E[mc2]4

Figure 56. Conversion spectra in giant nuclear systems for given nuclear transition energy of w = 2.5 [mot2 ] . Here, the positron llne at E = 1.63 [mot2 ] is caused by the conversion of electrons from the Dirac sea to a bundle of bound states F ~ 3s0 (cf. figure 57). Vacancy rates obtained within coupled channel calculations are assumed for the bound states at the moment of conversion. Clearly, various conversion lines are then present.

Fi&ure 57. Various monoenergetic conversion processes in a giant, for reasons of lucidity here subcrltical system (after P. SchlUter and coworkers, 1985). Full lines denote processes leading to (monoenergetlc) positron creation by conversion into bound electron states, whereas the dashed lines show conversion of bound electrons into the upper continuum. Electron-positron pair creation is denoted b y t h e dash-dotted line.

Giant Nuclear Systems Zl
e~ . . . v

a ~ " v -~'e-÷v --~

/

// Z2< Zcr

41

','~ p*"v-~ a

/< - - o :

e*

Figure 58. A hypothetical experiment to observe the decay of the Higgs vacuum, where two nuclei with Z, < Zcr and Z2 < Zcr collide and form a compound system with Zu > Zcr. Here, Zcr depends on the unknown self-coupling parameter X and is thus related to the Higgs mass. We now discuss the scenario involving the production of a new neutral particle. The new p a r t l cle may be produced by two different sources. It may couple to the atomic electrons (including the virtual electron-posltron pairs excited during the collision) or, alternatively, the source may be the nuclei, i.e., the quarks. While one can hardly disprove the existence of anunknownpartlcle, it is possible to put quite stringent limits on the production cross section. This is required in order to avoid contradictions to the existing body of well established experimental facts. Experimental searches for anomalous decays of elementary particles and excited nuclei constitute an important source of information. However, in all such experiments searching for the production of pseudoscalar particles a thick shielding was placed between the source and the detectors. Therefore particles with a life time longer than about % = 10 -II sec would not have been seen. Since the mass of the proposed particle is very light, it contributes to virtual processes. This enables us to deduce upper limits for the coupling strength from the high-precision experiments of QED. We will discuss two such arguments. (A) The anoma]ous ma&netic moment of the electron (or muon) is known and understood to a high degree of accuracy. The presence of a new neutral particle coupling to the electron (muon) will necessarily provide an additional contribution to the g factor. Let us assume an interaction of the form Le,X

=

gi#eriSe¢X ,

(33)

where the index i enumerates the Lorentz-invariant couplings of scalar, pseudoscalar, vector, axial vector, and tensor type, i.e., rS = i, rp = z,, rv = zu' rA = ~u~" rT = a.v" The addi-

tional contribution to the electron anomaly, which enters the g factor as g = 211 + =QED + ax]' is easily calculated from the vertex correction to the electron-photon interaction (figure 59a). The result depends on the ratio between the mass of the fermion and that of the virtual boson and therefore is different for electrons and muons. In Table 1 (cf. also figure 60) we give the numbers K i defined by a X = (al/2~) Ki, where a i = gi2/4~. Assuming a mass m X = 1.68

.,.

X / l e,p

get

Y pin

\ \

t ger e,p.

gqr

X

/r __x_~~.~,° b

q (a)

e,p

(b)

I¢I

Figure 59. Feynman g r a p h s f o r (a) t h e e l e c t r o n o r muon anomalous m a g n e t i c moment, (b) t h e interaction of electrons assumed new boson X.

(muons) w i t h q u a r k s , (c) t h e de c a y X ~ 2~. The broke n l i n e s d e n o t e t h e

J. F i n k et al.

42

Teldle I. The numerical constants K~ determine the contribution to the fermion anomalous magnetic moment assuming the presence of a new light particle of type L They are given for the electron, the muon, and in the limiting case of infinite fermion mass m I~ rex. The last two columns give an upper limit for the coupling strength deduced from the present uncertainty of the measured & factors. Kr Coupling (0

electron

Scalar(S) Pseudmcalar (P) Vector(V) Axialvector (A)

MeV, t h e c o e f f i c i e n t s

-0.16 0.12 0.045 -0.25

al m ~, m x

muon - 1.4 0.50 0.95 -24.4

-3/2 1/2 1 -oo

Ki a r e g i v e n f o r t h e e l e c t r o n ,

electron

muon

< 1.1 x 10 - s < 1.6 x 10-9 < 4 x 10 - 9 < 8 x 10 - I °

< 4 x 10 - s < 1 x 10 -~ < 6 x 10 - s

< 2 x 10 - s

t h e muon, and i n t h e l i m i t o f i n f i n i t e

fer-

mion m a s s . Due t o t h e mass r a t i o t h e a n o m a l o u s m a g n e t i c moment o f t h e muon i s more s e n s i t i v e t o the radiative correction. This advantage, however, is outweighed by the higher experimental p r e c i s i o n o f t h e e l e c t r o n d a t a . The l a s t two c o l u m n s o f t h e T a b l e g i v e t h e u p p e r l i m i t s f o r t h e c o u p l i n g c o n s t a n t s a i w h i c h r e s u l t from t h e maximum a l l o w a b l e d i s c r e p a n c y b e t w e e n t h e o r y a n d e x p e r i m e n t (A~ < 3 x 10 -11 f o r t h e e l e c t r o n and Aa < 9 x 10 - s f o r t h e muon, c f . I t i s o b v i o u s from t h e T a b l e t h a t t h e c o u p l i n g s a i a r e s u p p r e s s e d b y a t l e a s t magnitude compared with the electromagnetic e r e n c e we n o t e t h a t stant through ¢

=

the life

fine-structure

constant

Picasso, 1983). seven orders of

~ = V137. F o r f u r t h e r

ref-

t i m e o f t h e d e c a y X ~ e + + e" i s d e t e r m i n e d b y t h e c o u p l i n g c o n -

5/[½BX¢ i F(mx/me) ]

=

7.8 x 10 - ' z

/

[a i FCmx/me) ]

[sec]

,

(34)

where t h e f u n c t i o n F(mx/me) i s q u i t e independent o f t h e t y p e o f c o u p l i n E ( i = S , P , V , A , T ) , t a k i n g on v a l u e s b e t w e e n 0 . 5 and 0.9 a t t h e 8 i v a n p a r t i c l e

mass.

I f we assume t h a t t h e source o f t h e p a r t i c l e X i s t h e e l e c t r o n s h e l l , then t h e emission i n t e n s i t y can be d e t e r m i n e d by a d a p t i o n o f t h e f o r m a l i s m f o r e l e c t r o n i c b r e m s s t r a h l u n 8 o r m o l e c u l a r o r b i t a l X - r a y r a d i a t i o n (MOX). To o b t a i n an o r d e r o f magnitude e s t i m a t e one may s i m p l y m u l t i p l y t h e cross s e c t i o n f o r MOX r a d i a t i o n a t photon e n e r g i e s E~ > mX by t h e r a t i o o f c o u p l i n g constunts at/e. this

For 5 . 8 MeV/u Pb+Pb c o l l i s i o n s

(cf.

Kirsch,

1980) assuming p s e u d o s c a l a r c o u p l i n g

estimate g i v e s

=

Ox

(ai/a)

o~(E~>l.6MeV)

<

2 . 2 x 10 -v x 400 pb

=

8 x 10 - I x b .

(35)

This number must be compared with the production cross section of the positron llne of several ~b. In vlew of the large discrepancy between these two numbers, the hypothesis that an unknown neutral particle coupling to the electrons only is responsible for the positron llne can safely be ruled out. (However, note that this argument could be circumvented by introducing several new particles with masses and coupllng constants fine-tuned in such a manner that their contributions to g cancel each other. Although we consider this concept to be unattractlve we cannot exclude it a prlor~.) (B) If we assume that the particle X is created from the nuclear current during the heavy lon colllslon, some conclusions can be drawn from hlgh-precislon QED measurements of atomic blndln 8 energies. In addition to the electron-X coupling we also have the quark-X interaction

Lq ' x where f effective "

=

Ef g [ ~fri*f# X ,

(36)

c o u n t s t h e q u a r k f l a v o u r s . T a k i n g i n t o a c c o u n t two f l a v o u r s one may c a l c u l a t e i n t e r a c t i o n e n e r g y f o r t h e o n e - b o s o n e x c h a n g e p r o c e s s d e p i c t e d i n f i g u r e 59b: :

"$ d*x d ' y g : ~ e ( X ) r i * e ( X )

D ( x - y ) { ~ i Zp ~ p ( y ) r i * p ( y )

n + g i En ~ n ( Y ) r i * n ( Y ) }

the

' (37)

OiantNuclcar Systems

43

where t h e c o u p l i n g c o n s t a n t s f o r p r o t o n and n e u t r o n , gP and g n have been i n t r o d u c e d . D(x-y) i s t h e p r o p a g a t o r f o r a boson o f mass mX. In t h e n o n r e l a t i v i s t i c limit for the nucleon current, eq. (37) l e a d s t o an a d d i t i o n a l

¥ukawa-type i n t e r a c t i o n

potential

o f r a n g e VmX. I t c o u p l e s t h e

atomic electron to the scalar nuclear density (for i = S,V) or to the nuclear spin density (i = A,T). For pseudoscalar coupllng (i = P) the interaction vanishes in the non-relatlvlstlc limit. Ne have investigated the consequences of such anaddltional potential in three cases. (1) The experimental value of the Lamb shift in hydrogen AE = 1057.845 ± 0.009 Yflizagrees with theoretical predictions to better than 0.03 F~Iz. From this we deduce a limit for the combined coupling constant gegp < 2 x I0 -8. (il) The transition energies between high-lying states in muonic atoms (e.g., 599/2 ~ 4f7/2 in pPb) are understood with an accuracy better than bE = I0 eV. This leads to a combined coupling constant gegp/n < 1 0 _ 7 where gp/n represents an average of g P a n d gn. (ill) The electronic K a transition e n e r g y i n heavy atoms, e.g., ZeaFm, is not affected significantly if gegp/n < IO.G. At energies below the nuclear Coulomb barrier, we expect that the emission of light particles is mainly caused by the collective deceleration of the collidlng nuclei, i.e., by a mecbanlsm of bremsstrahlung tFpe (Vasak, 1980). To o b t a L n a n e s t l m a t e of the emission rate a calculatlon based on the semlclassical approximation for the nuclear motion (cf. Reinhardt, 1976) has been performed. Let us assume that the nuclei are localized (pointlike) sources of the X-particle field with a strength proportional to the nuclear number, moving on Rutherford hyperbola. This leads to the energy spectra of emitted X particles shown in figure 61. The total production cross sections for scalar and pseudoscalar particles are

0 xS ffi 6 . 4 x 1 0 - ' ( d / n / 4 ~ ) 2

[b]

(38) P oX

ffi 5.3 x 10 -7 (g1~Jnl4~)2 x (spinlnucleon) 2

[b]

E

d•[Qrb.mitsi Twobodydecw

IT" V

~ ' ~

X - e'÷e-

Exponentiolspectrum

e

I

J

._.E~.~-

m

10-1

-I

-2

10-z 500

F~gure 60. Coefficients K i as function of the ratio ~ / m

1000

1500

(cf. Table 1).

F~gure 61. Spectra of positrons, emitted in the rest frame of the decaying particle X at a sharp energy, for various valu"..~ o.~. the llfe time of X.

44

J. Fink et al. Let us investigate, whether the creation of s new particle through the discussed bremsstrahlung type mechanism can explaln the measured positron lines. In the rest frame of the decaying particle the positron is emitted at the sharp e n e r E y E +. The spectrum measured in the laboratory e will be broadend and, however, depends to some degree on the experimental set-up. It will have a maximum at E = E + a n d fall off at higher energies, depending on the velocity distribution of e the X particles. If the positron detector is sensitive to a limited spatial region AR around the target only, the spectrum will be narrowed as fast particles will leave the detection region before they decay. This effect is demonstrated in figure 61 for various values of the llfe time assumin E an exponentially fallln E energy spectrum of the X particles. First we note that the spectra do/dE X displayed in figure 61 without further assumptions will certainly not lead to a narrow positron llne.

Due to the suppression of do/dE X at small energies, the posi-

tron energy distribution deduced from the bremsstrahlung model will be broader than that displayed in the figure, where a purely exponentially spectrum was assumed. Only in the case of long llfe time, i.e., when most of the fast partlcles decay outside of the detector, a narrow structure can emerge. Taking into account the llfe time argument, to explain the positron cross section we have to identify o~y

~

o x f1 - e x p ( - T D / , ~ ]

.

(39)

The e x p e r i m e n t a l Tsertos, ically

cross s e c t i o n (Schweppe, 1983; Schwalm, 1984; Cowan, 1985; Clemente, 1984; exp 1985) i s o f t h e o r d e r Oe+ ~ 50 ~b. The ' e f f e c t i v e d e t e c t o r escape t i m e ' TD i s t y p -

o f t h e o r d e r TD = 10-*" sac. I f ,

as r e q u i r e d from t h e l i n e w i d t h argument, ~ ~ 10-*" sec,

exp we have t o r e q u i r e Oe+ ~ oX TD/¢. Let aS ~

us

first

= I0-*

discuss

the

case

of scalar particles.

Using z = 10-2~/a e sec, we a r r i v e

at

This conclusion is independent of the assumption of long life time. In the

opposite limiting case we would have found separately a~ > 10 -*l (from ~ < 10-*" sec) and N exp a S ~ 10-' (from Oe+ = O ) which gives the same result for =~ u~. For this product of coupllng constants, however, an upper limit of =~ a~ found from t h e a t o m i c p h y s i c s d a t a . C l e a r l y ,

these c o n s t r a i n t s

10.xs has been

are n o t c o m p a t i b l e .

I f we t u r n t o t h e case o f pseudoscalar p a r t i c l e s , no upper l i m i t i s a v a i l a b l e f o r t h e p r o d u c t o f Ne P ap ep. The v e r y s m a l l v a l u e o f OX, however, i s s u f f i c i e n t t o r u l e out t h i s p o s s i b i l i t y . Using oe+eXp = o~,_ i . e . , o m i t t i n g

the additional

r e d u c t i o n f a c t o r due t o t h e long l i f e

times,

we have a~ ~ I0 4 if the average spin per nucleon is (quite arbitrarily) taken as 0.i. Clearly such a large coupling constant is impossible. In the previous estimate we have made the assumption that the e+e" decay is dominant compared with the possible two-neutrlno and the two-photon decays. This becomes a real problem for the pseudoscalar case, which we have not excluded. Here the ~u decay is forbidden (if lepton number is conserved), hut the 2~ decay must be considered. Assuming that it proceeds mainly via a vlrtual nucleon-antinucleon pair (cf. figure 59c), the decay width is F ( X ~ ) eV, corresponding to a llfe time ¢ ( X ~ ) and e

= 3.6xlO-X"/m pin sec.

= 1.8 x I0 -6 x =p/n =p/n

Hence for the values of

discussed here the 2~ decay is slower than the decay into e + e " pairs.

Finally it is also worth mentioning that the couplin E constants =q to heavy quarks are bounded by the decay data of the J/~ and the ¥ meson which have led to the upper limits =c < 4xI0.7 and a b < 2x10 -s. Again, this conclusion applles if the llfe time is larger than = I0-** sec.

Giant Nuclear Systems X. Overcritical

45

SUPERCRITICAL PHENOMENA IN OTHER FIELDS OF PHYSICS

Gluo-Electric

FieJds

As a first example of supercritical fields in other areas of physics we shall discuss the fission of heavy quarkonla, which has been worked out by D. Vasak and B. MUller (Vasak, 1983). In the MIT-bag model (Chodos) 1974) a hadron is viewed as a bubble in the 'true' QCD vacuum, within which the quarks and gluons are confined. On its boundary their wavefunctions must fulfill a special (the so-called 'linear HIT') boundary condition, and additionally balance the external vacuum pressure B ~. A heavy quarkonium consists of a quark and an antiquark (c- or b-flavoured) with large mass (we take m c = 1.35 GeV and mb = 4.75 GeV), thus moving non-relatlvistically. Their densities are approximated by 6 functions. Their interaction with the boundary is mediated by the colour fields. In a reasonable approximation these fields can be taken to be abelian in this case. The electric potential produced by the heavy quark Q and antiquark Q = -Q) placed at the foci of an ellipsoidal bag (focal length d) is shown in figure 62 for different shapes specified by the ratio x = a/b of the semi-axes (a ~ b). The shape parameters of the hag are adjusted to minimize the sum of the volume energy BV and the field energy I dV E 2 for fixed QQ separation R = 2d. By increasing R also this minimum energy U(R) increases. In the adiabatic (Born-Oppenheimer) approach, U(R) is the potential energy of the slow relative motion of the heavy QQ pair (cf. figure 63). The strong coupling constant a s = 0.289 and the vacuum pressure B ~ = 235 HeV are chosen to reproduce the upsilonium and charmonium masses. For R ~ 0.5 fm we observe the linear behaviour U(R) - R , which reflects the confinement built into the theory by hand through the boundary conditions: the quarks Q and ~ cannot be pulled apart without spendinE an infinite amount of energy in the limit R ~ ~. On the other hand, only a finite amount of energy is needed to produce a light quark-antiquark pair (~q) within the baE. If we solve the Dirac equation for a light quark (mass 0, charge ±q = i~) within this ellipsoidal cavity, containing the confined potential produced by the heavy QQ pair, the energy eigenvalue depends on R. At R = R = 2.2 fm the lowest eigenenergy approaches zero: The i0 state goes over into the cr HIT value 2.04 for the free 1S½ state at R = 0, where the charges O and "O cancel each other (figure 64). Note, however, that the curves for negative and positive energies do not cross, but repel each other. At this point the wavefunction for the negatively charged I0 solution with negative energy 'migrates' from the external point charge -Q to the charge +~ (cf. figure 65): by deforming the bag the lines of force are squeezed within the cavity until the colour-

o)

°

2

.

-0

~

.

d

7

I

~

d

~-~

Q

/

Z

Figure 62. The equipotentlsl lines of the potential produced by two charges, Q and -Q, and modified by the confining boundary, is plotted here (in unlts Q/4~d) as well as the lines of force of the colour-electrlc field (lines with arrows) for two deformations: (a) x = 1.2, (b) x = 1.9. d is the focal length. The arrows indicate the direction of the colour-electric field, their density is, however, not related to the field strength.

46

J. Fink et al.

• , ~' ' , , • , , , . , , , , 3

•

> qD

m :D _J

<_

.

,

•

.

,

.

,

•

2

,

I-Z

i lel (3

,

3

U~ t--

-1 .

L

.

i

,

1

i

.

i

.

,

.

2

,

.

,

3

i

0 4

1.0

1.2

1,~

1.8

lJ

2.0

BAG DEFORMATION x

QI~-SEPARATION R (fm) Figure 63.

The relation x(R) between the deformation x and the interquark distance R = 2d was obtained by minimizing the total hadron energy (i.e., the volume and the field energy in our approximation)

in x for fixed d. The usual NIT-bag pressure B ~ = 145 MeV and, alternatively,

B ~ = 235 MeV was used. Through this relation also the potential energy of the relative QQ motion is given.

Figure 64.

The scale-independent energy eigenvalue bE nO. Due to the axial symmetry of the bag the magnetic quantum number is conserved (Jz = ½ is indicated by 0); n is the principal quantum

number. In the spherical case the point charges cancel each other and the curves end in the NIT values 2.04 (ISI/2 state for lo) and 3.20 (2P3/2 state for 20), respectively. To gnide the eye the calculated points are connected by straight lines. Here again u

s

= 0.289.

electric potential becomes strong enough to 'put1' the wavefunction from one location to the other. In the language of field theory this can be interpreted as screening of the external charges Q In the ground state all negative energy states are (by definition) occupied. This renders the vacuum R-dependent. BeloW Rcr we have

and -Q by spontaneous pair creation:

<0,R+6R i 0,R>

ffi I

for

R, R+gR < Rcr ,

(40a)

because the rearrangement of the solution is not large. For R = Rcr the situation is different; by rearrangement of the lowest (1o) wavefunction we have instead <0,Rcr-½6R I 0,Rcr+½6R>

~

1 ,

(40b) ffi 1 because the subcritical state, llo,~'o,Rcr-½6R>,

is obtained from the corresponding vacuum by

annihilation of the negatively charged lo solution with negative energy (localized at -Q) and creation of the corresponding positive energy solution (localized at Q); this state is similar to the supercrltical ground state, 10,Rcr+½6R>, where the wavefunction has migrated from one side to the other. The charge distribution within the bag also changes. Above the critical separation the heavy quarks are screened by the newly produced light quarks. There is no (attractive) colour-electric force now between Q and Q for R > Rcr, and the potential energy U(R) must be modified there, becoming a constant for R ~ Rcr which is fixed by the mass value of two atom-like mesons Q~ + ~q (cf. figure 66). It is interesting to observe that no ~b and ~c resonances have been found yet above the barrier U(Rcr ) = 1.8 GeV, although the precise shape of U(R) is not known yet, because the ~q interaction has been neglected and the bag shapes restricted to prolate ellipsoids (no necks allowedl). This is presently remedied by Vasak and Paulus.

Giant Nuclear Systems

47

2 ,E

I

------/~oo.

-8~(~0ZoF

/Y(g.46)

)

~-I

i

[

I

,

=

,

I )R~,

'

I

'

I

2

1

3

-1

4

I

J

I

,

0 4,/

~

0

I

'

°'

0u(3.74)

I

I

IR~. I

I

I

1 2 3 QO-Sep~0ti0n R [fro]

,

4

F i g u r e 65.

I n t h e s p h e r i c a l case ~he p o t e n t i a l i s z e r o and t h e s o l u t i o n s w i t h d i f f e r e n t es d e g e n e r a t e ( 1 ) . I n t h e vacuum s t a t e a l l n e g a t i v e e n e r g y s t a t e s are o c c u p i e d ( s o l i d When t h e source charges are p u l l e d a p a r t , t h e w a v e f u n c t i o n s b e g i n t o l o c a l i z e (2 and 3 ) . the critical d e f o r m a t i o n t h e c o l o u r f i e l d i s s t r o n g enough t o ' p u l l ' t h e w a v e f u n c t i o n s opposite sides (4).

F i g u r e 66.

Using the potential

e n e r g y U(R)

(solid

curve) with ¢

S

chargdots). Beyond to the

= 0.289 and B~ = 235 MeV i n

t h e S c h r ~ d i n g e r e q u a t i o n d e s c r i b i n g t h e r e l a t i v e m o t i o n o f two h e a v y quarks Q and Q, t h e b o t t o monium (a) and t h e charmonlum (b) masses can be c a l c u l a t e d , p r o v i d e d t h e c o n s t i t u e n t masses o f t h e c o r r e s p o n d i n g quarks are p r o p e r l y a d j u s t e d ( h e r e mc = 1.35 GeV and mb = 4 . 7 5 GeV) and added

to the eigenenergies. The results agree well with experimental data which are indicated by thick horizontal lines. The potential will be modified at R = Rcr, since beyond this point the QQ system will prefer to exist in the form of two atom-like mesons Q~ + Qq. The masses of these mesons (B-mesons b5 + bu and D-mesons cO + ~u) have to coincide with the asymptotic value of U(R) for R ~ Rcr (these values are given by 2(mB-mb) or 2(mD-mc) , respectively). Note that resonances with energies above the barrier U(Rcr) = 1.8 GeV have not yet been discovered.

Strong Gravitational F i e l d s We first consider (cf. Sorrel, 1982) a very simple gravitating system: an arbitrarily thin mass shell of total mass M and radius r 0. The metrical (i.e., gravitational) field of the mass shell is given in spherical coordinates by I g~v

diag {[l-(2M/r)],

-[l-(2M/r)] "I, -r=, -r'sin=~}

r ~ r0

=

(41) diag ([l-(2~/r0)],

-i, -r a, -r=sin2~)

r < r=

where M = GM/c 2. The behaviour of spin-½ particles in a gravitational background field is governed by the covariant Dirac equation i~ ~ ~ ; p

-

(mc/~)

¢

=

0

.

(42)

The colon ";" denotes the covariant derivative of the bi-spinor ¢, ~P are the generalized Dirac matrices defined via the Clifford algebra [~B,$~] = 2gBg. In a time-independent gBv field stationary states of the Dirac equation may be defined via the usual time separation ~ = exp (-i/~ wit ) X(r,8,#). Figure 67 shows the lowest eigenfrequencies of the Dirac equation

J. F i n k et al.

48

(42) in the fleld of the mass shell. The radius r, of the mass shell is varied (r0 > 2~) and we can see that all elgenfrequencies tend to zero as ~a approaches the event horizon r

= 2M of the s gravitating system. Due to the charge conjugation invariance of the gravitational interaction, the anti-partlcle states lie symmetric with respect to the ~ = 0 axis. For r, ~ r the system s becomes dynamical (it is forced to collapse by the overwhelming gravitational force) and eigenfrequencies can no longer be defined. It is mainly the gravitational red-shift which is responsible for the behaviour of the eigenfrequencies since they represent eigenenergles for an asymptotic observer (sitting at r ~ -). This can be confirmed by solving the point-nucleus problem in an external Schwarzschild field neglecting all derivatives of gp v and the interaction between the gpv and the Coulomb field of the protons. If Ra denotes the position of the nucleus in the Schwarzschild field (R, > rs) one obtains

~i for

=

g~,CR.) w~

(43) H w i ; wi a r e j u s t

the Dirac eigenvalues

the ordinary

hydrogen atom frequencies.

Clearly,

eq.

( 4 3 ) c a n b e u n d e r s t o o d t o show t h e r e d - s h i f t of frequencies in the external Schwarzschild field. It must be emphasized, however, that this is not the effect of the choice of the coordinate system but expresses the effect of gravitational binding. We s e e t h a t t h e s t a t i c overcritical situation does not occur for the This result can be generalized: since particles and antiparticles are a mass distribution in Einstein's theory of gravity, the groundstate not decay in a globally static gravitational field. Pair creation g r a v i t y is always connected with the dynamics of the g~v fie]d. The

simple mass shell model. equally well attracted by of a quantum field will in Einstein's theory of behaviour of particles in

the pure Schwarzschild or Relssner-NordstrSm geometry (charged black hole) can be understood by means of effective potentials. The effective potential, which governs the behaviour of classlcal test partlcles, reads for the charged black hole

a)

-%

/3~/2

~

e

s

.-~2si/2 'r2 --~2Pl lsl/2

.4~,,,,,,, "-~ ~..m ~ , . - i11

~

i12

113

i

-%

II/.,

des

115 ~ rolrs

. I ~~ /2pI,

3si/2

Figure 67. The lowest boundstates of the Dirac equation in the field of a gravitating mass shell, as function of the radius of the shell r,. ~

Figure 68. Part (a) shows the effective potential for the Schwarzschild field. The effective potential for a negatively or positively charged black hole is demonstrated in (b) and (c), respectively. In (a), L = 50, M G = 2; in (b) L = 4, M G = 2, Q = -I000; and in (c) L = 10, M G = 2, Q = 1000.

2O

3O

~

r

Giant Nuclear Systems ~eff(r,L)

=

49

(eQlr) ± [l-(2Mlr)] ½ [mlc4+(L21r2)] ½

(44)

L is the angular momentum of the test particle, Q the total charge of the black hole. The effective potential (44) is shown in figure 68 for the uncharged and charged black hole. From these simple pictures one may learn the following: (i) In the pure Schwarzschild or Reissner-Nordstr~m field only resonant states can exist because particles can be absorbed by the black hole with a certain probability. (ii) Pair creation occurs in the case of a highly charged black hole due to the decay of the vacuum in analogy to the processes encountered in QED. Let r+ denote the event horizon for the charged black hole, then pair creation occurs if Qe/r+ > mc =.

We conclude that the decay of

the vacuum state leads to a limiting charge to mass ratio of a black hole of 8 x I0 -*5 Cg-1.

New strong field aspects arise in the ECSK theory of gravity with torsion. In this theory the spin density of matter generates torsion of space-time by a contact interaction. Strong external torsion fields are able to create pairs of particles with nonvanishing spin. We can understand this mechanism again using a simple picture. We prescribe a background spin density distribution pEG(X) of negligible mass (we set g~v = nN9, the Minkowski tensor) and solve the Dirac equation [ i ~ P ~ + 3~LpP~G(X)~ ~ s - mc/~] T

=

0

(45)

in the resulting torsion field (Lp = (fi~/c*)½ = I 0 - "

cm is the Planck length)

If the back-

ground spin distribution is polarized with, say, spin in the negative z-direction (spin 4), then all particles with spin # are attracted while those with spin + are repelled by the torsion field. For simplicity we discuss eq. (45) in one spatial dimension with the potential I 3~LpP~G

=

[0,0,0,V(z)]

;

V(z)

-V,

z S za

0

z > za

=

(46)

The first boundstates in this square well potential are shown in figure 69. Boundstates with spin # dive into the negative energy continuum which is filled with particles of the same spin. If the original boundstate was empty it is filled in the overcritical case while the resulting hole escapes from the system as an antiparticle with spin 4. In four space-time dimensions complications arise due to the lack of symmetry. Still, qualitative results can be obtained by statistical methods, which support the results just discussed. However, the overcritical situation occurs for extremely high spin densities (104' g c m - * for polarized electrons) and therefore this decay mechanism of the vacuum could only be interesting in the earliest phase of the big bang or in the latest phase of gravitational collapse.

/ 2/

2/

,-,

Figure 69. The first four boundstates of the two-dimensional Dirac equation in a square well

torsion background with flat geometry are shown.

50

J. Fink et al. Strong Time-Dependent G r a v i t a t i o n a l F i e l d s C a u s a l i t y f o r b i d s a s t e l l a r o b j e c t w i t h t o t a l mass g r e a t e r than 1.4M t o he s t a b i l i z e d when the o nucle ar f u e l i s burnt o u t . The s t a r collapses through i t s event h o rizo n and e v e n t u a l l y forms a p o i n t s i n g u l a r i t y in apace-time. The r e s u l t i n g o b j e c t is c a l l e d a black hole. Hawking (1974) d i s c o v e r e d t h a t a b l a c k h o l e e v a p o r a t e s mass due t o p a r t i c l e p r o d u c t i o n and b e h a v e s as i f i t had an e f f e c t i v e t e m p e r a t u r e T = hc*/8~GM a f t e r t h e c o l l a p s e ha s t a k e n p l a c e . (M i s t h e g r a v i t a t i n g mass o f t h e b l a c k h o l e and T t h e t e m p e r a t u r e measured by an a s y m p t o t i c o b s e r v e r . ) B l a c k h o l e s o r i g i n a t e d by g r a v i t a t i o n a l c o l l a p s e be ha ve as h o t b o d i e s and i t was s u g g e s t e d t h a t t h e i r t h e r m a l e m i s s i o n l e a d s t o a d e c r e a s e i n mass of t h e b l a c k h o l e i f i t s mass is sufficiently s m a l l . For a b l a c k h o l e o f s o l a r mass t h e Hawking t e m p e r a t u r e i s much lower t h a n t h e 3 K t e m p e r a t u r e of t h e cosmic b a c k g r o u n d r a d i a t i o n and t h u s a b l a c k h o l e o f t h i s s i z e i s e x p e c t e d t o a b s o r b , r a t h e r t h a n e m i t , r a d i a t i o n , t h e r e b y i n c r e a s i n g i t s mass. These r e s u l t s

have been c o n f i r m e d by means of a t w o - d i m e n s i o n a l model of a c o l l a p s i n g mass

s h e l l . D e f i n i n g a s u i t a b l e vacuum s t a t e [ i n , v a c > t h e components o f < i n , v a c [ T~v I i n , v a c > , r e g u l a r i z e d by t h e c o v a r i a n t p o i n t s p l i t t i n g method, a r e ( D a v i e s , 1976): (i) before the onset of the collapse:

=

(24w)'* [(TM2/r 4) - (4M/r*)]

,

<~ 1>

=

0 ,



=

- ( 2 4 ~ ) "~ [ 1 - ( 2 H / r ) ] TM (M2/rd) ;

(47a)

(ii) long after the collapse: =

( 2 4 , ) "* [ ( 7 M = / r ' )



=

]/(768 H=) ,



=

-(24~)'*

j

Thus

- (4M/r*) + ]/(32 ~2)]

,

1

the

collapse

(47b)

[ 1 - ( 2 M / r ) ] TM [(M=/r 4) - ]/(32 ~ 2 ) ]

through

the

event horizon produces an outgoing energy flux of strength

(768~M=) "l. The energy flux in thls two-dimensional model can he expressed as (~=c=G=K=I) (768M=) "I

=

(2~) "z J "

w dw {exp (8~Mw) - 1) "I ,

(48)

showing Hawking's temperature directly. The temperature does not depend on the details of the acceleration of the collapsing mass distribution, if only an event horizon is formed. Neither does it depend on the coupling strength between the quantum field and the collapsing matter. The Hawking effect can be compared with the direct particle production process (transitions between the negative and positive frequence continua) in the QED of heavy ion collisions. We can therefore expect that the inclusion of bound states in a dynamical picture of gravitational collapse may be of importance as well. We introduce a slmple model and choose an irrotational, incoherent matter distribution to generate an asymptotically flat background geometry. The distribution is assumed to be spherlcally symmetric and initially at rest. The gravitational field is time-dependent in the interior and the static Schwarzschild field in the exterior of the collapsing dust cloud. In an asymptotic coordinate system where g~v is independent of t, in the exterior region an asymptotic observer must wait infinitely long until the edge of the dust cloud reaches the event horizon at r = r s. (In a co-moving coordinate system the dust cloud falls through its event horizon to a point singularity in finite time.) Now we are looking for solutions of the Klein-Gordon equation using the above metric as background geometry. The reason why we choose a boson field is that the expected particle creation in the strong gravitational field will give rise to many-body phenomena which are more easily described for bosons. The Klein-Gordon equation m a y b e linearized in the spirit of Feshhach and Villars in(a/~t) t

=

~,

~ .

(49)

Giant Nuclear Systems +m

(~

positivefrequencyr~tinuum _~_

51 20. Various excitation mechanisms for particle-antiparticle pairs in the gravitational field of a spherically symmetric collapsing matter distribution are shown.

Yiaure

~ _ _

) t /

-m

~

negutivefrequencycontinuum

In adiabatic approximation (we fix the time coordinate and solve for the elgenstates O f ~ v ) , we find a spectrum of boundstates besides the continuum (figure 70). We note that these results are not equivalent to the overcritical situation in QED, but nevertheless 8ive rise to induced transitions in the spectrum of bound states. Due to the many-body aspect of our problem a formalism of second quantization must be applied. However, in constructing a QFT in curved space-time one is confronted with two basic difficulties (de dirt, 1975): (i) The division of field operators into positive and negative frequency parts depends on the coordinate system and therefore leads to the problem of defining a physically meaningful Fock

space of states. (ii) The divergent expectation values of operators containing ~ in bi-linear form have to he regularized. Since the usual normal ordering procedure is coordinate dependent, two ways of thought are possible: first, one may construct a regularization prescription by means of symmetry principles (gauge invariance, etc.). Such a regularization scheme was used to derive for the mass shell model of the two-dimensional collapse. The second way is to give up reg general invariance by selecting a special coordinate system and using normal ordering for regularization. We choose the second way and consider the problem of collapse only in asymptotic coordinates. It is clear that in a realistic collapse model it is difficult to solve the problem of pair production exactly. We therefore reduce the degrees of freedom of our quantum system in a first attempt and consider only coherent states of the boson field. The calculation shows that the amplitude A(t) of the coherent state increases drastically for late times, indicating massive pair production IX(t)l

=

X, exp of ~ dr'

V(t')

.

(50)

Here V(t) is a transition matrix element between boundstates which is large for late times. Calculations which take into account the back reaction of the particle pairs onto the gravitational field are in progress. For details see Soffel (1982).

XI.

SUMMARY ANDOUTLOOK

We shall summarize our presentation with t h e following statements: i) The vacuum structure is most fundamental for the understanding of the physical world. 2) In overcritical external Saute fields the vacuum undergoes massive changes; the neutral vacuum decays into a charged vacuum by emission of antiparticles. The particle creation process will continue until either the potential difference in the gauge field is reduced or the Pauli principle prevents further particle creation.

52

J. Fink et al. 3) If the Dirac field occupies a new ground state, then we speak of spontaneously broken symmetry. The new ground state is called supercritical vacuum (charged vacuum). In QED, the neutral electron-positron vacuum decays, for Z > Z = 173, into the charged vacuum. cr 4) Symmetry breaking in this sense does no occur in strong gravitational fields (without torsion) because gravity does not distinguish between different kinds of particles nor between particles and antiparticles. Therefore, globally static gravitational fields can never lead to spontaneous pair creation. This very property of the gravitational interaction is also responsible for the inevitable breakdown of global time-independence when the strength of gravitational fields exceeds a certain limit, so that an event horizon is formed. The Schwarzschild radius separates then a region of static field from a region where space-time is intrinsically time-dependent. This dependence leads to pair creation, but this process is of basically different (dynamic) nature compared to the case of supercrltical electric fields. 5) Due to the presence of an event horizon it may be understood as being of topological origin as one has discussed for the rather trivial case of Rindler space. 6) Supercritica] gluo-electric fields in deformed bags lead to spontaneous ~q creation connected with the fission of the bag. This helps to understand how confinement works. 7) Quantum electrodynamics of strong, supercritica] fields plays a distinguished role as the one example that is amenable to tests in the laboratory. 8) Important for the test in the laboratory is that in collisions of very heavy ions (ZT + Zp > Zcr ) superheavy (giant) electronic

quasimolecules are formed. The Two-Centre-Dirac-Equation

and the corresponding correlation diagram constitute the theoretical basis of these quasimolecules. 9) The quasimolecules have been tested experimentally through the study of quasimolecular X rays, inner shell (particularly K-) vacancy formation and 8-electron production. The agreement between theory and many experiments is even quantitatively exciting. 10) The positron production spectrum in a heavy ion collision consists of a dynamical and a spontaneous component. For Rutherford trajectories the spontaneous component, though important, can hardly be distinguished from the dynamical spectrum. The latter ('shake-off' of vacuumpolarization) increases with a very high power of the total charge, namely (ZT+Zp)=°. II) All the theoretically predicted features of the dynamical positron spectrum have been experimentally confirmed. 12) In order to obtain signals for the vacuum decay (spontaneous positrons) a new idea had to be invoked, namely the formation of giant nuclear systems (sticking of nuclei, time delay, formation of giant molecules or other exotic structures like hollow nuclei or toroidal nuclei,

etc.). 13) Line s t r u c t u r e s in t h e p o s i t r o n spectrum have indeed been e x p e r i m e n t a l l y v e r i f i e d by K i e n l e , Xozhuharov, and a s s o c i a t e s , and by Schweppe, Cowan, Bokemeyer, Bethge, S t i e b i n g , Greenberg, Schwalm, and coworkers. 14) The positron line structures are experimentally shown to stem from a system travelling with a velocity smaller than or equal to the centre-of-momentum velocity. One can consistently assume a g i a n t n u c l e a r m o l e c u l e as the e m i t t e r . The arguments a r e : a) t h e main s t r u c t u r e s seem t o appear o n l y i n o v e r c r i t i c a l systems, b) t h e 6 - e l e c t r o n spectrum i s smooth, c) t h e X - r a y spectrum i s smooth, d) a t y p i c a l t h r e s h o l d e f f e c t as a f u n c t i o n o f t h e i o n energy seems t o be ' o b s e r v e d , and e) t h e w i d t h i s v e r y s m a l l (~ 80 k e Y ) . 15) N u c l e a r p o c k e t s i n t h e i n t e r a c t i o n p o t e n t i a l o f v e r y heavy i o n s seem indeed t o be t h e o r e tically possible. For deformed nuclei they are orientation dependent. The predicted energy for the spontaneous emission line agrees with experiment for U+Cm, but not for the other systems investigated experimentally. 16) Additional Raman-type structures due to supercritical conversion are theoretically predicted, as are classes of vacuum decay lines reflecting the nuclear structure of shape isomeric

Giant Nuclear Systems

53

minima if such pockets exist. This opens up the possibility of a spectroscopy of the giant systams, which may appear in various isomeric forms. 17) The formation of giant nuclear molecules with old (~ and ~ vibrations, rotations) and new (butterfly, belly-dancer) type modes of excitations are certainly the one species of giant systems, which immediately explains the observed break-up of that system close to the entrance configuration. Such giant molecules would always be the doorway states. 18) A first quantum mechanical theory reproduces (in a yet schematic approach) all essential experimental features and helps to understand the observations. In particular, it is predicted that those states of the giant nuclear system are most dominant, which are at (or slightly under) the outer barrier. 19) It could very well be that the not yet understood (Zp+ZT)-dependence of the positron line reflects up to now not completely understood structures of the giant nuclear system: a) quarkgluon-gas nuclei with unusual nuclear radii depending on Z and A, different from the ordinary R = r A v3 scaling; b) always the same giant system after emission of = particles and other o clusters; c) nearly the same collective excited state of giant nuclei and its conversion; etc. 20) We also have studied the hypothesis that the positron lines in heavy ion collisions were caused by the decay of a neutral massive boson of mass m x = 1.68 HeV. The properties of this particle would not he compatible with the theoretically postulated standard axion. Avoiding the need to postulate the existence of a whole family of new particles with adjusted masses and coupling constants, the relatively large production cross section required to explain the positron line was found to lead to conflict with well-established high-precision data of atomic physics. 21) We conclude, that it is most likely that the wacuum decay in supercritical electric fields of giant nuclei has been observed, but that the structure of giant systems is far from being understood. 1"his conclusion can be drawn from the presently existing experimental facts. The spontaneous positrons will in future serve as a tool for a spectroscopy of the giant systems, which may, indeed, have life times T ~ I0 -*s sec. Thus a newly discovered fundamental process of field theory may help to make an equally basic discovery in nuclear physics. This opens the possibility for a most exciting future consisting in the identification of the wealth of structure, nature hides in giant nuclear systems. Information coming from clusters of nuclear matter practically twice as large as available in the present periodic system can shed light on our understanding of nuclear matter, but can he quite valuable for astrophysics as well.

ACKNOWLEDGEMENTS The last 18 years during which this research was going on, starting first with vague theoretical ideas around 1967, becoming more and more concrete and quantitative and finally physics by performing experiments and comparing them in very many details to the theory, were most rewarding for us. We would first like to thank our friends and collaborators, above all P. G~rtner, U. Heinz, P.O. Hess, W. Pieper, J. Rafelski, M. Seiwert, and K.H. Wietschorke, as well as T. Pinkston (Vanderbilt University) and several others, who contributed so importantly to the theory. Without them, their energy, enthusiasm and reliable work, the complex theory of these involved processes would not he as developed as it is. We would equally like to thank a number of experimental friends and colleagues, in particular H. Backe, K. Bethge, H. Bokemeyer, F. Bosch, H. Clemente, T. Cowan, J.S. Greenherg, E. Kankeleit, P. Kienle, W. Koenig, Ch. Kozhuharov, D. Liesen, W.E. Heyerhof, D. Schwalm, J. Schweppe, P. Senger, K.E. Stiebing, P. Vincent, and their many associates for their enormous work and dedication. They followed a courageous path against many stumbling blocks, setbacks, and other discouragements. This path was paved with a number of ingenious inventions. The close contact with them was of mutual benefit and for us personally most gratifying.

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J. Fink et al. REFERENCES

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P~-C

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