Z. PhysikA 285, 49- 52 (1978)
Zeitschrift
for P h y s i k A
9 by Springer-Verlag 1978
Spontaneous Vacuum Decay of Supercritical Nuclear Composites*
Johann Rafelski
GSI, Darmstadt, Germany
Berndt Mtiller and Walter Greiner
Institut fiir Theoretische Physik der Johann Wolfgang Goethe-Universit/it
Frankfurt am Main, Germany
Received October 12, 1977
We show that in deep inelastic heavy ion collisions with Z1 +Z2 > 173 the spontaneous
decay of the neutral vacuum by emission of positrons may be isolated from other
competing positron producing processes. From the details of the spontaneous positron
spectrum information about the nature (lifetime, shape and angular momentum) of the
composite nuclear system may be derived.
It has been known theoretically for some time now
that when the binding energy of an electron in an
external (supercritical) field exceeds twice the electron
rest mass me, the neutral vacuum will decay to the
charged vacuum state by emission of a positron [1].
The observation of such a spontaneous positron production in supercritical electron fields is the only
unique experimental way to test the correctness of
the theory of strong fields [1]. Supercritical potentials will for example be created when two heavy
nuclei are brought close to each other with the united
charge Z u exceeding 173 and the nuclear separation
being smaller than the critical distance Ror(Z,); the
value of Rcr for two Uranium nuclei is Rcr~35 fm
[1].
Experiments proposed previously [1] to test this
prediction of quantum electrodynamics suggested
elastic scattering of nuclei at energies below
1600 MeV for U on U collision. In such collisions the
time for which the supercritical fields are created is
very short as compared with the natural lifetime of
the resonant positron emitting state. Thus only a
small fraction of positrons emitted is spontaneous.
Furthermore the spontaneous and induced amplitudes are hardly distinguishable from other competing, positron producing mechanisms; in particular
from the direct pair production (shake off of vacuum
polarisation) by the time dependent Coulomb field of
the moving nuclei [1]. The shape of the positron
* Supported by the Bundesministeriumftir Forschungund Technologie (BMFT)
spectrum is determined mainly by the dynamics of
the collision.
In collisions of heavy nuclei the formation of a composite nuclear system is another possible and well
investigated mode of nuclear collisions [2, 3]. We
show that the emission of positrons from the spontaneous decay of the vacuum in inelastic collisions
may be distinguished from other competing processes. This is possible since the sticking time of the
nuclei in the inelastic part of the collision is several
times longer than the elastic collision time within the
critical radius. We note that extrapolations based on
various models and presently available data may lead
to even more pronounced extension of the nuclear
reaction time, which would be to our advantage. In
order to discriminate against all other non-nuclear
channels for positron production we propose to select
only the inelastic collisions, that is to measure the
coincident heavy ion cross section d 2 ainel/d~2~o,~dEp
where Ep is the kinetic energy of the positrons. From
experiments with lighter ions it is known that the
deep-inelastic cross section is concentrated in a kinematically rather well defined region of large energy
loss [2]. For the given scattering system U on U the
relevant positron kinetic energies are in the interval
200 < Ep < 500 keV. This situation is shown schematically in Figure 1, where the energy of the l s a quasimoleeular orbital in an inelastic collision is
depicted as a function of time. For t l < t < t 2 the
nuclei stick together; the energy of the electronic
states does not change and the emission of positrons
occurs at a fixed energy. The splitting in the energy of
0340-2193/78/0285/0049/$01.00
50
J. Rafelski et al.: Spontaneous Vacuum Decay of Supercritical Nuclear Composites
E(~0) is the total kinetic energy of the heavy ion
system and E(0)-Ecou~(R): = E r ~ 300 MeV is a typical value for the energy available for nuclear reactions. If most of the rotational energy is used up in
internal nuclear excitations, the simplest ~0-dependence of E((p) is given by
E
me
0
E(~0) - Ecou~ - ~ r ( 1 - ~0/~ o).
(2)
Upon integration of (1) we find for the lifetime of the
nuclear composite:
-
_
_
iiI
F~-1 - z r = (2 J / E r ) 1/2 ~oo.
Fig. 1. Schematic representation of the correlation diagram for the
inner electronic shells in inelastic U - U-collision as function of
time. The resonances in the positron continuum are indicated
together with the relevant widths. The insert representsthe associated positron width
the quasimolecular state is due to magnetic interaction of the nuclear currents with the electronic spin.
In the insert to the figure a typical positron spectrum
that we expect in the proposed experiment is shown
qualitatively. An estimate of the positron background
expected from the pair conversion of nuclear ~-rays
based on preliminary experimental 7-multiplicities
[4] shows that the signal to noise ratio should be
better than seven to one. If the existence of the
spontaneous positron emission process will be determined by the observation of such a positron spectrum, incidental information about the shape, angular
momentum, and lifetime of the composite nuclear
system may be obtained from the location , separation
and the width of the positron peaks. This will lead to
a better understanding of possible fusion or supertransfer mechanisms which may be an important step
in the search for superheavy nuclei [5].
We first turn to the discussion of the collision time in
deep inelastic heavy ion U on U collision. We follow
here closely the general ideas of [6] which are consistent with the experimental findings in lighter colliding systems [3]. Assuming that the scattering system consists of rigidly rotating nuclei separated by
the distance R with the reduced mass #, the moment
of inertia is given by J = # R 2. Henceforth we consider as an example two Uranium nuclei sticking to
each other with R = 14 fro. Assuming angular motion
only, the lifetime of such a rotating system is [6]
q~0
0
9 ~ = S &o(r
o) - Ecou~(R))- ~/~
(1)
(3)
Theoretical models [6] of the inelastic collisions lead
to a typical value ~Oo~rC. Thus we find that the
reaction width is F ~ 1 6 0 k e V . This has to be compared with the typical value of the positron dynamical width in elastic collisions of 600 keV [1]. Thus the
nuclear friction slows down the collision by about a
factor four in the situation described above. It is
quite possible that additional effects, not considered
here (e.g. considerably increased mass in the overlap
region [5]) may allow for even smaller F~. We note
that the natural positron line width Fp is known [1]
rather accurately. It has a typical value of 3 keV in
the example of two sticking Uranium nuclei discussed above.
For given nuclear charges Z1, Z 2 the average location/~v of the resonance in the positron spectrum is
determined by the separation R of the nuclei. To
obtain a precise prediction we have solved the relativistic two center problem for finite nuclei, including for the first time the average screening arising
from the electron-electron interaction. We have taken
only the inner-electron shells (30electrons) that can
follow the nuclear motion and have sufficiently short
relaxation times, so that even the electron-electron
interactions can be treated adiabatically. At R = 14 fm
we find for the average kinetic energy /~p of the
positron resonance in U on U deep inelastic collision
/~p= 350keV. The value of/~p depends sensitively on
R(AF, v / A R ~ 3 O k e V / f m ) and is depicted in Figure2
for several values of Z,. The dashed lines give the
electromagnetic (Rein) and grazing (Rg~az) "touching"
distances between the nuclei, respectively. The uncertainty in the numerical results for /~p is of the
order of 20 keV, which could be reduced substantially
by greater numerical efforts. Thus measurement of/~p
will determine R to better than I fro.
A splitting of the positron lines AE M originates from
the coupling of the electron spin to the nuclear
rotation that generates a strong magnetic field. We
now show that despite the reduction in the rotational
velocity of the sticking nuclei A E ~ remains constant
as long as R does not change. AE M is given by the
J. Rafelski et al.: Spontaneous Vacuum Decay of Supercritical Nuclear Composites
14
{
9
16
L
18
L
L
Rgr~
100
20 R[fm]
L
~
~Zu=180
20Q
300
4OO
/
/
/
190
500
600
[p, [keV]
Fig. 2. Average kinetic energy of positrons spontaneously emitted
fiom nuclear composite systems of united charge Z,=180, 184,
190. R is the distance between the centers of mass of spherical
nuclei. Rein, Rgraz(Zu) are the electromagnetic, resp. nuclear "touching" distances
expression:
AE M = - 2 ~ d 3 x d 3 x' L(x) Ix - x'l- 1jN(X')
(4)
where Je, N a r e the electron (nuclear) currents respectively. The main contribution in (4) comes from
the magnetic dipole interaction. Thus as long as the
nuclear currents are proportional to the mass density,
AEM is time-independent as a consequence of the
conservation of the total angular momentum.
The formalism of [7] may now be used to obtain the
splitting of the electronic states associated with the
composite system:
AE M =
2 IJ~ol [2(E(0)
-
Ecoul(a))/l,f]
1/2.
(5)
Taking from [7] [~g/l~[=l.8 me, (me is the electron
mass) we find AEM=130keV. Since the dynamical
width of the nuclear system is found to have the same
value (Eq.(3)), it is possible that the magnetic splitting
of the two spin states can be resolved experimentally,
giving an independent measure of the nuclear angular
momentum in the collision.
In the above discussion we have made the most likely
assumptions about the nature of the nuclear composites. We shall now discuss several conceivable
experimental situations characterized by the relative
size of the reaction width F~ and the positron escape
width Fp. In all cases discussed below the spontaneous positrons stemming from the vacuum decay
can be identified.
In the last case more incidental information about
the nuclear composites could be obtained, since the
lifetime of the composite becomes comparable to the
lifetime of the supercritical neutral vacuum in this
particular example.
51
1) F p ~ F ~ A E M : In this most likely case the width of
the overlapping double positron peak will be dominantly determined by F~. This will allow for a
precise measurement of the composite nucleus lifetime. From the average positron energy E v the
value of R can be deduced. Information about the
angular momentum may be obtained from the splitting of the positron spectrum. Only a fraction of
available K-holes will decay by positron emission.
Assuming F ~ 5 0 F p the positron cross section in the
interesting energy interval due to positrons from
spontaneous decay of the vacuum in the field of the
composite nucleus will be at least an order of magnitude larger than the contributions from induced and
spontaneous decay expected for the usual Coulomb
scattering [1,8] in the above considered example of
two sticking Uranium nuclei. Since the number of
vacancies in the two l so- states in the entrance
channel is about 0.02 per collision, we expect 4
x 10 -~ positrons per inelastic collision in the energy
interval 200 < E v < 500 keV. The abundance of positrons produced by conversion of nuclear 7-rays can be
estimated as follows: Given a v-multiplicity N ~ 3 0
and an exponential decay of the V spectrum [4] we
write:
n(co) = N exp ( - co/coo).
(6)
o) 0
In order to obtain a value for the fall-off constant coo
we argue that most of the approx. 300MeV excitation energy will be carried away by particle emission. Some 8 MeV in each fragment will be emitted
in form of nuclear y-rays. Thus the total v-ray energy
yield Scodcon(co) must be 2 x 8 = 1 6 M e V leading to
coo ~ 500 keV. With this value of N, coo and the shape
(6) of the spectrum we find a positron yield of 0.8
x 10 -4 per inelastic collision in the energy interval
O<Ep<5OOkeV. We have obtained this number
using the E2 conversion coefficients and Coulomb
correction factors given in [10]. Thus the background
from t h e conversion of nuclear 7-rays does not alter
the
structure
of the
spontaneous
positron
spectrum.
2) Fp~F~<AEM: This case corresponds to a true
nuclear molecule lasting for several rotations. In this
case the spontaneous positron distribution will separate into at least two distinct peaks (other peaks
may be generated by other reaction c h a n n e l s - t h a t is
by other average values of L, R.) From the precise
structure of such spectra one may deduce important
information about the nuclear molecular potential.
The nuclear background is completely negligible.
3) I;>F~: The case of a compound nucleus would
probably give rise to a broader distribution than
52
J. Rafelski et at.: Spontaneous Vacuum Decay of Supercriticat Nuclear Composites
expected (~Fv) from the folding of the widths. This
b r o a d e n i n g c o r r e s p o n d s to the p o p u l a t i o n of several
a n g u l a r m o m e n t u m states at fixed n u c l e a r charge
- p e r unit of lost a n g u l a r m o m e n t u m an a d d i t i o n a l
u n r e s o l v e d line s e p a r a t e d b y 0.1 k e V w o u l d a p p e a r .
H o w e v e r , different c o m p o u n d nuclei c o r r e s p o n d i n g
to different nuclear charges c o u l d be resolved. Per
unit of charge the s e p a r a t i o n in the p o s i t r o n p e a k s is
40keV. T h u s from the intensity of those lines inform a t i o n a b o u t f o r m a t i o n rates of different c o m p o u n d
nuclei could be deduced.
W e n o t e t h a t in the u n d e r c r i t i c a l collisions (Z,~ < 173)
the c r e a t i o n of c o m p o s i t e systems with lifetimes longer t h a n 1 0 - 1 8 s could be d e t e c t e d by m e a s u r e m e n t
of the q u a s i - m o l e c u l a r K - X - r a y s [11].
W e have shown t h a t the o b s e r v a t i o n of s p o n t a n e o u s
p o s i t r o n s is p o s s i b l e in h e a v y ion c o i n c i d e n t scattering m e a s u r e m e n t s . A successful e x p e r i m e n t w o u l d
u n a m b i g u o u s l y establish the p r e d i c t i o n s o f the t h e o r y
of strong fields. In a d d i t i o n the p o s i t r o n s p e c t r u m
w o u l d be a source of i m p o r t a n t i n f o r m a t i o n a b o u t the
s u p e r h e a v y c o m p o s i t e nuclei.
References
1. For details see the following recent review artices:
Miiller, B.: Ann. Rev. Nucl. Sci. 26, 351 (1976)
Reinhardt, J., Greiner, W.: Rep. Prog. Phys. 40, 219 (1977)
Rafelski, J., Fulcher, L., Klein, A.: to appear in Phys. Rep.
(1977)
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8. In sub-Coulomb barrier heavy ion collisions positron production in fair agreement with the theory has been found at
UNILAC, Darmstadt, by the joint GSI-Darmstadt-Miinchen
experiments
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Handschug, L., Hegberger, F., Kankeleit, E., Kienle, P., Kozhuharov, Ch., Nakayama, Y., Richter, L., Stettmeier, H., Vincent, P., Weik, F., Willwater, R., GSI-preprint 1977
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Johann Rafelski
Gesellschaft fiir Schwerionenforschung mbH
Postfach 541
D-6100 Darmstadt 1
Federal Republic of Germany
Berndt Miiller
Walter Greiner
Institut f6r Theoretische Physik
der Johann Wolfgang Goethe-Universitiit
Robert-Mayer-StraBe 8 - 10
D-6000 Frankfurt/Main 1
Federal Republic of Germany