Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 159–167, 2002

159

Nuclear Shells in the Superheavy Region within Meson Field Theory
W. Greiner∗
Institut für Theoretische Physik, J.W. Goethe-Universität, D-60054 Frankfurt, Germany
Received: January 16, 2002; In Final Form: January 16, 2002
The extension of the periodic system into various new areas is investigated. Experiments for the synthesis of superheavy elements and the predictions of magic numbers with modern meson field theories are reviewed. Furtheron,different channels of nuclear decay are discussed including cluster radioactivity, cold fission, and cold multifragmentation. A perspective for future research is given.

1. Introduction
There are fundamental questions in science, like e.g. “how did
life emerge” or “how does our brain work” and others. However, the most fundamental of those questions is “how did theworld originate?”. The material world has to exist before life
and thinking can develop. Of particular importance are the substances themselves, i.e. the particles the elements are made of(baryons, mesons, quarks, gluons), i.e. elementary matter. The
vacuum and its structure is closely related to that. On this I want
to report today. I begin with the discussion of modern issues innuclear physics.
The elements existing in nature are ordered according to their
atomic (chemical) properties in the periodic system which wasdeveloped by Mendeleev and Lothar Meyer. The heaviest element of natural origin is Uranium. Its nucleus is composed of
Z = 92 protons and a certain number of neutrons (N = 128–150).
They are called the different Uranium isotopes. The transuranium elements reach from Neptunium (Z = 93) via Californium(Z = 98) and Fermium (Z = 100) up to Lawrencium (Z = 103).
The heavier the elements are, the larger are their radii and their
number of protons. Thus, the Coulomb repulsion in their interior increases, and they undergo fission. In other words: thetransuranium elements become more instable as they get bigger.
In the late sixties the dream of the superheavy elements arose.
Theoretical nuclear physicists around S. G. Nilsson (Lund)1 andfrom the Frankfurt school2–4 predicted that so-called closed proton and neutron shells should counteract the repelling Coulomb
forces. Atomic nuclei with these special “magic” proton andneutron numbers and their neighbours could again be rather
stable. These magic proton (Z) and neutron (N) numbers were
thought to be Z = 114 and N = 184 or 196. Typical predictions of
their lifetimes varied between seconds and many thousand years.
Figure 1 summarizes the expectations at the time. One can see
the islands of superheavy elements around Z = 114, N = 184 and
196, respectively, and the one around Z = 164, N = 318. The
important question was how to produce these superheavy nuclei. There were many attempts, but only little progress was
made. It was not until the middle of the seventies that the Frankfurt school of theoretical physics together with foreign guests(R. K. Gupta (India), A. Sandulescu (Romania))6 theoreticallyunderstood and substantiated the concept of bombarding of double magic lead nuclei with suitable projectiles, which had been
proposed intuitively by the Russian nuclear physicist Y. Oganessian.7 The two-center shell model, which is essential for thedescription of fission, fusion, and nuclear molecules, was developed in 1969–1972 together with my then students U. Mosel
and J. Maruhn.8 It showed that the shell structure of the two final
fragments was visible far beyond the barrier into the fusioning
nucleus. The collective potential energy surfaces of heavy nuclei, as they were calculated in the framework of the two-center
shell model, exhibit pronounced valleys, such that these valleys∗E-mail: greiner@th.physik.uni-frankfurt.de.

provide promising doorways to the fusion of superheavy nuclei
for certain projectile-target combinations (Figure 2). If projectile and target approach each other through those “cold” valleys,
they get only minimally excited and the barrier which has to be
overcome (fusion barrier) is lowest (as compared to neighbouring projectile-target combinations).
2. Cold Valleys in the Potential
In this way the correct projectile and target combinations
for fusion were predicted. Indeed, Gottfried Münzenberg and
Sigurd Hofmann and their group at GSI9 have followed this approach. With the help of the SHIP mass-separator and the position sensitive detectors, which were especially developed bythem, they produced the pre-superheavy elements Z = 106, 107,
. . . , 112, each of them with the theoretically predicted projectiletarget combinations, and only with these. Everything else failed.
This is an impressive success, which crowned the laborious
construction work of many years. The before last example ofthis success, the discovery of element 112 and its long α-decay
chain, is shown in Figure 3. Very recently the Dubna-Livermore
group produced two isotopes of Z = 114 element by bombarding 244 Pu with 48 Ca and also Z = 116 by 48 Ca + 248 Cm (Figure 4).
Also these are cold-valley reactions (in this case due to the combination of a spherical and a deformed nucleus), as predicted
by Gupta, Sandulescu, and Greiner10 in 1977. There exist also
cold valleys for which both fragments are deformed,11 but thesehave yet not been verified experimentally. The Z = 118 isotope
claimed to be fused with the cold valley reaction13 86 Kr + 208 Pb
by Ninov et al.14 could not be reproduced in later experiments.
3. Shell Structure in the Superheavy Region
Studies of the shell structure of superheavy elements in theframework of the meson field theory and the Skyrme-Hartree-

Figure 1. The periodic system of elements as conceived by the Frankfurt school in the late sixties. The islands of superheavy elements(Z = 114, N = 184, 196 and Z = 164, N = 318) are shown as dark hatchedareas.

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

160 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Greiner

Figure 2. The collective potential energy surface of 184 114, calculated within the two center shell model by J. Maruhn et al., shows clearly the cold2
denotes the mass asymmetry: η = 0
valleys which reach up to the barrier and beyond. Here R is the distance between the fragments and η = AA11−A+A2
corresponds to a symmetric, η = ±1 to an extremely asymmetric division of the nucleus into projectile and target. If projectile and target approachthrough a cold valley, they do not “constantly slide off” as it would be the case if they approach along the slopes at the sides of the valley. Constantsliding causes heating, so that the compound nucleus heats up and gets unstable. In the cold valley, on the other hand, the created heat is minimized.
The colleagues from Freiburg should be familiar with that: they approach Titisee (in the Black Forest) most elegantly through the Höllental and notby climbing its slopes along the sides.

Figure 3. The fusion of element 112 with 70 Zn as projectile and 208 Pb as target nucleus has been accomplished for the first time in 1995/96 by
S. Hofmann, G. Münzenberg, and their collaborators. The colliding nuclei determine an entrance to a “cold valley” as predicted as early as 1976 by
Gupta, Sandulescu, and Greiner. The fused nucleus 112 decays successively via α emission until finally the quasi-stable nucleus 253 Fm is reached.
The α particles as well as the final nucleus have been observed. Combined, this renders the definite proof of the existence of a Z = 112 nucleus.

(T1/2, f)

1nsf 1msf 1msf 1sf 1ksf

1sf 1ksf 1Msf

118

293
175118

Proton Number

289

285

277
165112
272

110

269

268

265

264

Hs

110 CN271

110 CN

273

Hs

277

110

269

273

Hs
a (E=9.78 MeV,
t=1.203 s)

107
265

269

Sg

265

Hs

a (E=10.29 MeV,
t=1.23 s)
283

112

112

284

~
sf (TKE~195
MeV,
t=9.33 min)
280

110

a (E=10.18 MeV,
t=5.738 ms)

109

267

281

CN
a (E=2.41 MeV,
t=1.201 ms)a

111 CN287
288
289
114 175114
173114114

a (E=11.31 MeV,
t=0.708 ms)

112

104

292

116

116 CNa (E=11.63 MeV,
t=1243 ms)

114

106

CN

a (E=12.37 MeV,
t=281 ms)

116

108

(T1/2,a)
1msa
1msa

110

281

112

285

CN

CN

a (E=9.71 MeV,
t=30 s)112

a (E=8.67 MeV,
t=15 min)

1sa
1ksa
b-

stable110a (E=8.83 MeV,
t=1.6 min)
277

Hs

~
sf (TKE~205
MeV)

Sg

Rf

150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180

Neutron Number
Figure 4. The Z = 106–112 isotopes were fused by the Hofmann-Münzenberg (GSI) group. The two Z = 114 isotopes and the Z = 116 isotope were
produced by the Dubna-Livermore group. It is claimed that three neutrons are evaporated. Obviously the lifetimes of the various decay products
are rather long (because they are closer to the stable valley), in crude agreement with early predictions3, 4 and in excellent agreement with the recent
calculations of the Sobiczewski group.12 The Z = 118 isotope claimed to be fused by V. Ninov et al. at Berkeley could not be reproduced in later
experiments.

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161

Figure 5. Grey scale plots of proton gaps (left column) and neutron gaps (right column) in the N–Z plane for spherical calculations with the forces
as indicated. The assignment of scales differs for protons and neutrons, see the uppermost boxes where the scales are indicated in units of MeV.
Nuclei that are stable with respect to β decay and the two-proton dripline are emphasized. The forces with parameter sets SkI4 and NL-Z reproduce
the binding energy of 264
156 108 (Hassium) best, i.e. |δE/E | < 0.0024. Thus one might assume that these parameter sets could give the best predictions
for the superheavies. Nevertheless, it is noticed that NL-Z predicts only Z = 120 as a magic number while SkI4 predicts both Z = 114 and Z = 120 as
magic numbers. The magicity depends — sometimes quite strongly — on the neutron number. These studies are due to Bender, Rutz, Bürvenich,Maruhn, P.G. Reinhard et al.15
304
120
magic numbers Z=120 N=184

0.16

Z=120, N=172

NL-Z2

0.14
0.12

-3

[fm ]

0.1

Edef(MeV)

40

0.08
0.06

20

0.04

0

0.02
0.0

-20

-10

-5

0
r [fm]

5

10

-40
-60

Fock approach have recently shown that the magic shells in the
superheavy region are very isotope dependent5, 15 (see Figure 5).
According to these investigations Z = 120 being a magic proton number seems to be as probable as Z = 114. Additionally,

recent investigations in a chirally symmetric mean-field theory
result also in the prediction of these two magic numbers,42, 44 seealso below. The corresponding magic neutron numbers are predicted to be N = 172 and — as it seems to a lesser extend —0

Figure 6. Potential energy surface as a function of reduced elongation(R − Ri )/(Rt − Ri ) and mass asymmetry η for the double magic nucleus
304
120184 .

Figure 7. Typical structure of the fullerene 60 C. The double bindings
are illustrated by double lines. In the nuclear case the Carbon atoms are
replaced by α particles and the double bindings by the additional neutrons. Such a structure would immediately explain the semi-hollownessof that superheavy nucleus, which is revealed in the mean-field calculations within meson-field theories. The radial density of the nucleus
with 120 protons and 172 neutrons, as emerging from a meson-fieldcalculation with the force NL-Z2 is shown on the right side. Note thatthe semi-bubble structure is mostly pronounced for this nucleus. When
going to higher neutron numbers, this structures becomes less and less.

6
1.
2
1. R i)80. (R t/4
0. -R )i
(R

0.8

-0.8

-0.4

0
η

0.4

304

304

120

120

208
172

Pb + 96Sr

Yb + 132Sn

162 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Greiner

N = 184. Thus, this region provides an open field of research.
R. A. Gherghescu et al. have calculated the potential energy
surface of the Z = 120 nucleus. It utilizes interesting isomericand valley structures (Figure 6). The charge distribution of theZ = 120, N = 184 nucleus, calculated with mean-field models, indicates a hollow inside. This leads us to suggest that it might beessentially a fullerene consisting of 60 α particles and one additional binding neutron per alpha. This is illustrated in Figure 7.
The protons and neutrons of such a superheavy nucleus are distributed over 60 α particles and 60 neutrons (forgetting the last
4 neutrons). The potential energy surfaces of superheavy elements, as they emerge from selfconsistent calculations within
mean-field models in axial symmetry, exhibit some interesting
features.24
Nuclei in the vicinity of the nucleus with Z = 108 protons and
Z = 162 have prolate ground-states and barriers. Going upward
in proton and neutron numbers, one encounters transitional systems with two shallow minima, one on the oblate, one on the
prolate side. Nuclei with proton numbers Z = 120 and neutron numbers N = 178. . . 184 exhibit no pronounced deformation. Mean-field forces predict either a clear spherical shape ora rather soft potential energy surface around zero deformation
with small wiggles. For these nuclei, however, triaxial degrees
of freedom might become important and change the picture considerably.
The barriers correspond to a simple-humped structure foralmost all forces. Isomeric states appear in the reflectionsymmetric solutions but disappear when allowing for shapes including odd multipole moments. Globally, barriers calculatedwith Skyrme-forces appear to be up to twice as high as the ones
emerging from RMF calculations. This effect has already been
seen in former studies.23 It indicates the need for a deeper understanding of these selfconsistent approaches. One might further
ask how collective motions of these spherical superheavy elements might look like. We will take a first look at these aspects
in the following section.
4. Vibrational Modes in Spherical Superheavy Nuclei
We consider vibrational collective properties of the putative
double magic SH nucleus 292 120 as predicted by the RMF axialsymmetric model and compare them with those of the wellknown double magic heavy nucleus 208 Pb (Ref. 25). As one cansee in Figure 8, the nucleus 208 Pb has a pronounced harmonic behaviour, at least for the three vibrational states, i.e. 0+ , 2+ , andthe triplet 0+ , 2+ , 4+ . In contrast to that the SHE 292 120, computed also with the force NL-Z2, exhibits a clear prolate-oblate
asymmetry and consequently the sequence of states follows a
4.0

12

+

3.5

4

10
3.0
+

4

8

E[MeV]E[MeV]

2.5
6
+

2

2

2.0

+

1.5
4
1.0
+

2

0
-0.2

0

+

0

RMF+NL-Z2
H.A.
-0.15

-0.1

-0.05

0.5

208

Pb

0.0
2

0.05

0.1

0.15

0.2

0.0
-0.2

RMF+NL-Z2
H.A.
-0.15

-0.1

-0.05

292

120

0.0

0.05

0.1

0.15

0.2

2

Figure 8. Potential well and first three vibrational states of the potential, calculated in the frame of the RMF model with NL-Z2 force(RMF + NL-Z2) and in the Harmonic approximation (HA) for two nuclei. The wave functions of the states are also shown. The left panel
represents the case of 208 Pb where the harmonic approximation works
quite well. The right panel shows the putative double-magic nucleus
292
120 for which the anharmonic distortions in the potential are inducing
a sensitive departure of the collective level spacing from the equidistantharmonic behaviour.

non-equidistant behaviour. This result was expected because
the SHE are less stable (calculations give barriers up to 5 times
smaller when the first symmetric barrier of 292 120 is comparedwith that of 208 Pb). Therefore the departure of the deformationenergy curve from the harmonic oscillator well will be larger.
It is important to stress that in view of the width and height ofthe potential well in the β2 coordinate, no more than two phononstates exist. Clearly, the future observation of such β-vibrational
states will yield further useful information about the structure ofthese nuclei. Also the sensitivity of this structure to the underlying effective forces is interesting.
The determination of the chemistry of superheavy elements,
i.e. the calculation of the atomic structure — which is in thecase of element 112 the shell structure of 112 electrons due tothe Coulomb interaction of the electrons and in particular the
calculation of the orbitals of the outer (valence) electrons — hasbeen carried out as early as 1970 by B. Fricke and W. Greiner.16
Hartree-Fock-Dirac calculations yield rather precise results.
5. Asymmetric and Superasymmetric Fission — Cluster
Radioactivity
The potential energy surfaces, which are shown prototypically for Z = 114 in Figure 2, contain even more remarkable
information that I want to mention cursorily: if a given nucleus, e.g. Uranium, undergoes fission, it moves in its potential
mountains from the interior to the outside. Of course, this happens quantum mechanically. The wave function of such a nucleus, which decays by tunneling through the barrier, has maxima where the potential is minimal and minima where it has
maxima.
The probability for finding a certain mass asymmetry η =A1 −A2of the fission is proportional to ψ∗ (η)ψ(η)dη. Generally,
A1 +A2
this is complemented by a coordinate dependent scale factor forthe volume element in this (curved) space, which I omit for thesake of clarity. Now it becomes clear how the so-called asymmetric and superasymmetric fission processes come into being. They result from the enhancement of the collective wave
function in the cold valleys. And that is indeed, what one observes. For large mass asymmetry (η ≈ 0.8, 0.9) there exist very
narrow valleys. They are not as clearly visible in Figure 2, butthey have interesting consequences. Through these narrow valleys nuclei can emit spontaneously not only α particles (Helium
nuclei) but also 14 C, 20 O, 24 Ne, 28 Mg, and other nuclei. Thus,
we are led to the cluster radioactivity (Poenaru, Sandulescu,
Greiner17 ).
By now this process has been verified experimentally by research groups in Oxford, Moscow, Berkeley, Milan, and otherplaces. Accordingly, one has to revise what is learned inschool: there are not only 3 types of radioactivity (α-, β-, γradioactivity), but many more. Atomic nuclei can also decay
through spontaneous cluster emission (that is the “spitting out”
of smaller nuclei like carbon, oxygen, . . . ). Figure 9 depicts
some examples of these processes.
The knowledge of the collective potential energy surface andthe collective masses Bi j (R, η), all calculated within the TwoCenter-Shell-Model (TCSM), allowed H. Klein, D. Schnabel,and J. A. Maruhn to calculate lifetimes against fission in an “abinitio” way.18 The discussion of much more very interesting
new physics cannot be pursued here. We refer to the literature.19–22, 26–28
The “cold valleys” in the collective potential energy surface
are basic for understanding this exciting area of nuclear physics!
It is a master example for understanding the structure of elementary matter, which is so important for other fields, especially astrophysics, but even more so for enriching our “Weltbild”, i.e. the status of our understanding of the world aroundus.

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163

Figure 9. Cluster radioactivity of actinide nuclei. By emission of 14 C, 20 O, . . . , “big leaps” in the periodic system can occur, just contrary to the
known α, β, γ radioactivities, which are also partly shown in the figure.

6. Extension of the Periodic System into the Sections ofHyper- and Antimatter

The chances for hyperons and antibaryons to meet gets rapidly
worse with increasing number. In order to produce multi-Λ nuclei and antimatter nuclei, one has to look for a different source.

Nuclei that are found in nature consist of nucleons (protons
and neutrons) which themselves are made of u (up) and d (down)quarks. However, there also exist s (strange) quarks and evenheavier flavors, called charm, bottom, top. The latter has justrecently been discovered. Let us stick to the s quarks. They
are found in the ‘strange’ relatives of the nucleons, the so-called
hyperons (Λ, Σ, Ξ, Ω). The Λ particle, e.g., consists of one u, d,and s quark, the Ξ particle even of an u and two s quarks, whilethe Ω (sss) contains strange quarks only.
If such a hyperon is taken up by a nucleus, a hyper-nucleus
is created. Hyper-nuclei with one hyperon have been known for
20 years now, and were extensively studied by B. Povh (Heidelberg).31 Several years ago, Carsten Greiner, Jürgen Schaffner,and Horst Stöcker32 theoretically investigated nuclei with many
hyperons, hypermatter, and found that the binding energy perbaryon of strange matter is in many cases even higher than thatof ordinary matter (composed only of u and d quarks). This
leads to the idea of extending the periodic system of elements inthe direction of strangeness.
One can also ask for the possibility of building atomic nuclei
out of antimatter, that means searching e.g. for anti-helium,
anti-carbon, anti-oxygen. Figure 10 depicts this idea. Due to thecharge conjugation symmetry antinuclei should have the same
magic numbers and the same spectra as ordinary nuclei. However, as soon as they get in touch with ordinary matter, theyannihilate with it and the system explodes.
Now the important question arises how these strange matterand antimatter clusters can be produced. First, one thinks of collisions of heavy nuclei, e.g. lead on lead, at high energies (energy per nucleon ≥ 200 GeV). Calculations with the URQMDmodel of the Frankfurt school show that through nuclear shockwaves33–35 nuclear matter gets compressed to 5–10 times of its
usual value, ρ0 ≈ 0.17 fm3 , and heated up to temperatures ofkT ≈ 200 MeV. As a consequence about 10000 pions, 100 Λ’s,
40 Σ’s and Ξ’s, and about as many antiprotons and many otherparticles are created in a single collision. It seems conceivablethat it is possible in such a scenario for some Λ’s to get captured
by a nuclear cluster. This happens indeed rather frequently for
one or two Λ particles; however, more of them get built into nuclei with rapidly decreasing probability only. This is due to thelow probability for finding the right conditions for such a capture in the phase space of the particles: the numerous particles
travel with every possible momenta (velocities) in all directions.

In the framework of meson field theory within the mean-fieldapproximation the energy spectrum of baryons in a nucleus has apeculiar structure, depicted in Figure 11. It consists of an upper
and a lower continuum, as it is known from the electrons (seee.g. Reference 30). The upper well represents the nuclear shellmodel potential. It describes the overall structure throughout the

Figure 10. The extension of the periodic system into the sectors of
strangeness (S, S̄) and antimatter (Z̄, N̄). The stable valley winds outof the known proton (Z) and neutron (N) plane into the S and S̄ sector,
respectively. The same can be observed for the antimatter sector. In
the upper part of the figure only the stable valley in the usual proton(Z) and neutron (N) plane is plotted, however, extended into the sectorof antiprotons and antineutrons. In the second part of the figure it has
been indicated, how the stable valley winds out of the Z–N plane into
the strangeness sector. This is due to an additional term proportional to( AA − SA0 )2 in the mass formula.

164 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002
nuclear table very well.
Of special interest in the case of the baryon spectrum is thepotential well, built of the scalar and the vector potential, whichrises from the lower continuum. It is known since P. A. M.
Dirac (1930) that the negative energy states of the lower continuum have to be occupied by particles (electrons or, in ourcase, baryons). Otherwise our world would be unstable, because
the “ordinary” particles are found in the upper states which candecay through the emission of photons into lower-lying states.
However, if the “underworld” is occupied, the Pauli-principle
will prevent this decay. Holes in the occupied “underworld”(Dirac sea) are antiparticles. This has been extensively discussed in the context of QED of strong fields (overcritical fields,
decay of the vacuum from a neutral one into a charged one30 ).
The occupied states of this underworld including up to 40000
occupied bound states of the lower potential well represent thevacuum. The peculiarity of this strongly correlated vacuum
structure in the region of atomic nuclei is that — depending on
the size of the nucleus — more than 20000 up to 40000 (occupied) bound nucleon states contribute to this polarization effect. Obviously, we are dealing here with a highly correlated
vacuum. A pronounced shell structure can be recognized.36–38
Holes in these states have to be interpreted as bound antinucleons (antiprotons, antineutrons). If the primary nuclear density rises due to compression, the lower well increases while
the upper decreases and soon is converted into a repulsive barrier (Figure 12). This compression of nuclear matter can only
be carried out in relativistic nucleus-nucleus collision with thehelp of shock waves, which have been proposed by the Frankfurt school33, 34 and which have since then been confirmed extensively (for references see e.g. Reference 39). These nuclear
shock waves are accompanied by heating of the compressed nuclear matter. Indeed, density and temperature are correlated in

Figure 11. Baryon spectrum in a nucleus. Below the positive energy
continuum exists the potential well of real nucleons. It has a depth of
50–60 MeV and shows the correct shell structure. The shell model of
nuclei is realized here. However, from the negative continuum anotherpotential well arises, in which about 40000 bound particles are found,
belonging to the vacuum. A part of the shell structure of the upper welland the lower (vacuum) well is depicted in the lower figures.

Greinerterms of the hydrodynamic Rankine-Hugoniot equations. Heating as well as the violent dynamics cause the creation of manyholes in the very deep (measured from −MB c2 ) vacuum well andan equal number of particles (baryons) in the upper continuum.
This is analogous to the dynamical e+ e− pair creation in heavy
ion collisions.39
These numerous bound holes resemble antimatter clusters
which are bound in the medium; their wave functions have large
overlap with antimatter clusters. When the primary matter density decreases during the expansion stage of the heavy ion collision, the potential wells, in particular the lower one, disappear.
The bound antinucleons are then pulled down into the (lower)
continuum. In this way antimatter clusters may be set free. Of
course, a large part of the antimatter will annihilate on ordinary
matter present in the course of the expansion. However, it isimportant that this mechanism for the production of antimatterclusters out of the highly correlated vacuum does not proceed
via the phase space. The required coalescence of many particles
in phase space suppresses the production of clusters, while it isfavoured by the direct production out of the highly correlated
vacuum. In a certain sense, the highly correlated vacuum is akind of cluster vacuum (vacuum with cluster structure). The
shell structure of the vacuum levels (see Figure 11) supports thislatter suggestion. Figure 13 illustrates this idea.
The mechanism is similar for the production of multi-hyper
nuclei (Λ, Σ, Ξ, Ω). Meson field theory predicts also for theΛ energy spectrum at finite primary nucleon density the existence of upper and lower wells. The lower well belongs to thevacuum and is fully occupied by Λ’s. Dynamics and temperature then induce transitions (e.g. ΛΛ̄ creation) and deposit many
Λ’s in the upper well. These numerous bound Λ’s (and similarly other hyperons) are sitting close to the primary baryons:in a certain sense a giant multi-Λ hypernucleus has been created. When the system disintegrates (expansion stage) the Λ’s
distribute over the nucleon clusters (which are most abundant inperipheral collisions). In this way multi-Λ hypernuclei can beformed. Also clusters of hyperons alone (Λ, Σ, ...) seem possible and quasistable5, 32 and the Bethe-Weizsäcker mass formularequires at least one additional term proportional to ( fS − fS0 )2 ,where fS /A is the strangeness content in a hypernucleus.
Of course this vision has to be worked out and probably refined in many respects. This means much more and thoroughinvestigation in the future. It is particularly important to gain

Figure 12. The lower well rises strongly with increasing primary nucleon density, and even gets supercritical (spontaneous nucleon emission and creation of bound antinucleons). Supercriticality denotes thesituation, when the lower well enters the upper continuum.

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165

1
4

LBM = ∑B ψB iγµ ∂µ − gωB γµ ωµ − gφB γµ φµ
1− gρB γµ τB ρµ − eγµ (1 + τB )Aµ − m∗B ψB ,2
1
1
1
Lvec = m2ω ωµ ωµ + m2ρ ρµ ρµ + m2φ φµ φµ ,
2
2
2
1
1
1
1
L0 = − m2s s2 − m2z z2 − bs3 − cs4 .
22
34Chiral Lagrangian:− Fµν F µν ,1111∂µ σ∂µ σ + ∂µ ζ∂µ ζ + ∂µ χ∂µ χ − Bµν Bµν
2
2
2
4
1
1
µν
µν
− Gµν G − Fµν F ,44LBM = ∑B ψB iγµ ∂µ − gωB γµ ωµ − gφB γµ φµ
1− gρB γµ τB ρµ − eγµ (1 + τB )Aµ − m∗B ψB ,2
1 2 χ21 2 χ21 χ2µLvec = mω 2 ωµ ω + mρ 2 ρµ ρµ + m2φ 2 φµ φµ
2 χ0
2 χ0
2 χ0
Lkin =

Figure 13. Due to the high temperature and the violent dynamics,
many bound holes (antinucleon clusters) are created in the highly correlated vacuum, which can be set free during the expansion stage into
the lower continuum. In this way, antimatter clusters can be produced
directly from the vacuum. The horizontal arrow in the lower part of
the figure denotes the spontaneous creation of baryon-antibaryon pairs,
while the antibaryons occupy bound states in the lower potential well.
Such a situation, where the lower potential well reaches into the upper
continuum, is called supercritical. Four of the bound holes states (boundantinucleons) are encircled to illustrate a “quasi-antihelium” formed. It
may be set free (driven into the lower continuum) by the violent nuclear
dynamics.

more experimental information on the properties of the lower
well by (e, e p) or (e, e pp ) and also ( p̄c pb , pc p̄b ) reactions athigh energy ( p̄c denotes an incident antiproton from the continuum, pb is a proton in a bound state; for the reaction products
the situation is just the opposite).40 Also the reaction (p, p d),(p, p 3 He), (p, p 4 He) and others of similar type need to be investigated in this context. The systematic scattering of antiprotons on nuclei can contribute to clarify these questions: Timelike momentum transfer is required here! The Nambu-JonaLasigno (NJL) model seems to give much smaller lower wells,
but does not describe the shell model potentials. Studies of
I. Mishustin, L. Satarov et al. to improve the NJL model for applications to nuclear and baryon-meson sectors are on the way.
Problems of the meson field theory (e.g. Landau poles) canthen be reconsidered. An effective meson field theory has to beconstructed. Various effective theories, e.g. of Walecka-type on
the one side and theories with chiral invariance on the other side,
seem to give different strengths of the potential wells and also
different dependence on the baryon density.41 The Lagrangiansof the Dürr-Teller-Walecka-type and of the chirally symmetricmean field theories look quantitatively quite differently. We
exhibit them — without further discussion — in the following
equations:
L = Lkin + LBM + Lvec + L0 + LSB .

Non-chiral Lagrangian:
Lkin =

11
1
1∂µ s∂µ s + ∂µ z∂µ z − Bµν Bµν − Gµν Gµν
2
2
44+ g44 (ω4 + 6ω2 ρ2 + ρ4 ) ,σ41
L0 = − k0 χ2 (σ2 + ζ2 ) + k1 (σ2 + ζ2 )2 + k2 ( + ζ4 )22
1 4 χ4 δ σ2 ζ2
4
+ k3 χσ ζ − k4 χ + χ ln 4 + ln 2 2 ,4
χ0 3 σ0 ζ0 2  √1χ
LSB = −
2m2K fK − √ m2π fπ ζ .
m2π fπ σ +χ02The non-chiral model contains the scalar-isoscalar field s andits strange counterpart z, the vector-isoscalar fields ωµ and φµ ,and the ρ-meson ρµ as well as the photon Aµ . For more details see Reference 41. In contrast to the non-chiral model, theSU(3)L × SU(3)R Lagrangian contains the dilaton field χ introduced to mimic the trace anomaly of QCD in an effective Lagrangian at tree level (for an explanation of the chiral model see
References 41, 42).
The connection of the chiral Lagrangian with the Waleckatype one can be established by the substitution σ = σ0 − s (andsimilarly for the strange condensate ζ). Then, e.g. the differencein the definition of the effective nucleon mass in both models(non-chiral: m∗N = mN − gs s, chiral: m∗N = gs σ) can be removed,
yielding:m∗N = gs σ0 − gs s ≡ mN − gs s
for the nucleon mass in the chiral model. Nevertheless, if theparameters in both cases (e.g. gs , gω , gρ , ms , b, c in the nonchiral case) are adjusted such that ordinary nuclei (binding energies, radii, shell structure, . . . ) and properties of infinite nuclear matter (equilibrium density, compression constant K, binding energy) are well reproduced, the prediction of both effective
Lagrangians for the dependence of the properties of the correlated vacuum on density and temperature is remarkably similar
as long as the same mesons are considered. The question of
the nucleonic substructure (form factors, quarks, gluons) and its
influence on the highly correlated vacuum structure has to bestudied. The nucleons are possibly strongly polarized in the correlated vacuum: the ∆ resonance correlations in the vacuum areprobably important. Is this highly correlated vacuum state, especially during the compression, a preliminary stage to the quarkgluon cluster plasma? To which extent is it similar or perhaps
even identical with it? It is well known for more than 10 years
that meson field theories predict a phase transition qualitatively
and quantitatively similar to that of the quark-gluon plasma43 —see Figure 14.
7. Concluding Remarks — Outlook
The extension of the periodic system into the sectors hypermatter (strangeness) and antimatter is of general and astrophys-166 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Greiner

1.0

X
0.8
0.6
0.4
0.2

0

4.

5

3.

0

3.

5

2.

T[

0

M

2.

eV

5

]

1.

0

1.
x

5

2

10

0.

1.0

2.0

3.0

4.0

ρ[fm-3]

5.0

6.0

10-1

x

Figure 14.
The strong phase transition inherent in Dürr-TellerWalecka-type meson field theories, as predicted by J. Theis et al.43 Notethat there is a first order transition along the ρ-axis (i.e. with density),
but a simple transition along the temperature T -axis. Note also that
this is very similar to the phase transition obtained recently from the
Nambu-Jona-Lasinio approximation of QCD.45

ical importance. Indeed, microseconds after the big bang the
new dimensions of the periodic system we have touched upon,certainly have been populated in the course of the baryo- andnucleo-genesis. Of course, for the creation of the universe, evenhigher dimensional extensions (charm, bottom, top) come into
play, which we did not pursue here. It is an open question, how
the depopulation (the decay) of these sectors influences the distribution of elements of our world today. Our conception of theworld will certainly gain a lot through the clarification of thesequestions.
For the Gesellschaft für Schwerionenforschung (GSI), which
I helped initiating in the sixties, the questions raised here could
point to the way ahead. Working groups have been instructed by
the board of directors of GSI, to think about the future of the laboratory. On that occasion, very concrete (almost too concrete)suggestions are discussed — as far as it has been presented tothe public. What is necessary, as it seems, is a vision on a longterm basis. The ideas proposed here, the verification of which
will need the commitment for 2–4 decades of research, could
be such a vision with considerable attraction for the best young
physicists. The new dimensions of the periodic system madeof hyper- and antimatter cannot be examined in the “stand-by”
mode at CERN (Geneva); a dedicated facility is necessary forthis field of research, which can in future serve as a home for theuniversities. The GSI — which has unfortunately become much
too self-sufficient — could be such a home for new generationsof physicists, who are interested in the structure of elementary
matter. GSI would then not develop just into a detector laboratory for CERN, and as such become obsolete. I can already see
the enthusiasm in the eyes of young scientists, when I unfold
these ideas to them — similarly as it was 30 years ago, when the
nuclear physicists in the state of Hessen initiated the construction of GSI.
I am grateful to Dipl.-Phys. Thomas Bürvenich for helping
me in the technical production of these proceedings.
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