11. Superheavy Quasimolecules

Lacking the existence of stable nuclei with charge Z > 173 there is no immediate way to test the ideas about supercritical fields in QED. However, in a collision of two very heavy ions for a short period of time the electrons will experience the combined potential of both nuclei. If the nuclear motion is not too fast a super­heavy quasi molecule may be formed. As a prerequisite for dynamical calcula­tions, in this chapter the electron states of a hypothetical stationary molecule are discussed by solving the two center Dirac equation. Special attention is devoted to the critical internuclear distance Ref> where the bound levels reach the lower continuum.

11.1 Heavy-Ion Collisions: General Remarks

The discussion of the binding energy of the K-shell electrons showed that in­stability of the neutral vacuum state occurs for nuclei with more than 170 protons. As this is far beyond the charge of stable nuclei and even the trans­actinides, a new kind of atoms, the so-called super heavy elements, would be required before laboratory tests of the theory of strong fields could be under­taken. For a considerable time there was much hope that certain (lighter) super­heavy nuclei around Z = 114 could be stable and be produced in the laboratory [Ni 69a, Mo 68]. Mosel and Greiner even suggested that a second island of truly superheavy ("giant") nuclei around Z = 164 might be quasistable [Mo 69]. However, the recent unsuccessful attempts all over the world to synthesize super­heavy elements (i.e. elements around Z = 114) through nuclear reactions have made such hopes slim.

Nevertheless, supercritical fields can be created temporarily in a collision of very heavy atoms, such as uranium and uranium. When the two nuclei have ap­proached each other to a distance of, say 20 fm, in a head-on or nearly head-on collision, the electric field surrounding them very much resembles that which would surround a giant nucleus with 92 + 92 = 184 protons. In fact, the multipole expansion of the Coulomb potential (in the Coulomb gauge) for two equal point­like nuclei with charge Z separated by a distance R is (Fig. 11.1) [Ra 76b, So 79a]

W. Greiner et al., Quantum Electrodynamics of Strong Fields© Springer-Verlag Berlin Heidelberg 1985

11.1 Heavy-Ion Collisions: General Remarks 301

Fig. 11.1. Definition of the coordinates in the multi­pole expansion of the potential

z

{

2Ze2 [(2r)2 ] - R12 1 + Ii P2(COSO) + ... , V(r, cos 0) = 2 2

_ 2~e [1+(~)P2(COSO)+ ... J' r::5R12} .

r~R12

(11.1)

Clearly, at distances r > R the potential differs little (to be precise, by less than one third) from the Coulomb potential of a nucleus with charge (2Z). It is also seen that the potential difference comes from the quadrupole and higher multi­pole terms and falls off rapidly with increasing distance from the nuclei.

That the strong electric fields temporarily created in heavy-ion collisions could be utilized for laboratory tests of the theoretical predictions of QED of supercritical fields was proposed about 15 years ago independently by the two groups at Frankfurt [Gr69b, Ra 71, Mil 72a-c] and Moscow [Ge70, Ze72]. They also argued why these transient fields should be able to produce supercritical binding, reviewed in the rest of this chapter. Chapters 12 and 13 then deal with the refined treatment of electronic phenomena in heavy-ion collisions and with the present status of experimental efforts in this field.

Three basic questions have to be addressed before concluding that the transient strong electric fields in a heavy-ion collision actually provide super­critical binding to the innermost electrons.

1) Is the collision slow enough so electrons can adjust to the increasing strength of the potential as the two nuclei approach? A first answer to this question compares the velocity of the scattering nuclei with the "orbiting" velocity of the electrons. The K-shell electrons in an uranium atom are bound by more than 100 keV. Due to the virial theorem, the binding energy in a Coulomb potential equals the average kinetic energy of the particle. Hence the ratio of the K-electron velocity Vel to the speed of light c in an uranium atom is estimated as vellc - (2IEl/mec2)112 > 0.6. For the nuclei to move at this relative speed would require a laboratory bombarding energy of approximately 200 MeV Inucleon. This is much more than one likes to have, because at projectile energies above 6 MeV Inucleon violent nuclear reactions set in that could provide a formidable background, e.g. in positron spectra. At energies up to 6 MeV Inucleon the Coulomb barrier slows down the approaching nuclei, thereby considerably reducing the relative velocity at close internuclear distances. We conclude that the nuclear motion is slower by a factor of 5 - 10 than the electronic motion in relevant collisions.

302 11. Super heavy Quasimolecules

103

E = 3: -;:: ~ ~

/

102 /

/ /

/

/'R TC

/

/ /

/ /

500 1000

Fig. 11.2. Change of the average radii (r> of selected inner-shell wave functions in the Pb + Pb quasi molecule with varying inter­nuclear distance. Observe that the 1 sll2 elec­tron moves effectively around both nuclei up to RTC ;;::: 500 fm

2) Are the electrons moving in the region of very strong electric fields? Stated otherwise: under what circumstances will the nuclei approach so closely that the innermost electrons effectively orbit around both nuclei, seeing one giant quasi­nucleus? As we are primarily interested in the change of the vacuum state, we estimate the average radius of the electronic charge distribution when the binding energy reaches the threshold E = 2me' Again, we use of the virial theorem: lEI must be equal to the average kinetic energy (T) of the electron, which, in turn, is related to the average squared momentum (p2) by (T) = (m; + (p2» 112 - me' From the uncertainty relation, for the ground state one has (p2) <r2) - 1.

Putting everything together

(11.2)

The validity of this estimate is shown in Fig. 11.2, where the average distance < Ifflr I Iff) of the strongly bound electrons from the centre of the Pb + Pb quasi­molecule is shown as function of the nuclear separation. When the two Pb nuclei touch, the average extension of the K shell is about 150 fm.

Thus, when the nuclei are much closer together than this distance, the elec­trons will feel essentially the monopole part of the two-centre potential (11.1). When this is the case, the system comprising the two nuclei and the electrons attached to them is usually referred to as a quasiatom. A more general term is quasimolecule, which applies to a much longer phase of the collision, as long as the electrons have a substantial chance of being found at both nuclei. The quasi­molecular states may be strongly deformed, whereas quasi-atomic states should have approximately good angular momentum, indicating that they exist in an almost spherical potential.

3) Even when the electron orbit becomes supercritical, is there enough time for the neutral vacuum state to decay? Because the dived electronic bound state is rather weakly coupled to the positron scattering states, the resonance width r is relatively small, typically a few keY, Chap. 2. This corresponds to a vacuum decay time of Tvac - r- 1 - 10 -19 s, following Chaps. 6, 10. The normal collision time is much shorter, a reasonable estimate being Tcoll = Ref/Vion, where Ref is the internuclear distance at which the bound state becomes supercritical, and Vion the

11.2 The Two-Centre Dirac Equation 303

asymptotic ion velocity estimated before. From (11.2) follows that Ref must be considerably smaller than 135 fm (for U + U the true value is about 30 fm, see below), hence the collision time 'eoll is less than 10- 21 s. This result indicates that a positron would be spontaneously emitted in not more than one out of a hundred normal collisions of two uranium atoms. In addition, the positron dis­tribution must be expected to be quite broad, typically ,;11 -1 MeV, according to Sect. 10.2, making the identification of the positrons from the spontaneous vacuum decay difficult.

This may change radically when the two nuclei touch and stick together for a certain period of time under the action of the attractive nuclear force, forming a nuclear molecule. Such molecular configurations are well known from the scat­tering of light nuclei, e.g. C + C, 0+0, and evidence is mounting for their presence even in collisions of the heaviest nuclei [AI 60, Sche 70, Pa 81]. If a com­posite nuclear system forms the nuclear sticking time T must be added to the collision time for Coulomb scattering, 'coli. When T becomes comparable with the characteristic time for spontaneous positron emission 'vae, the superheavy nuclear molecule has for our purposes essentially the same properties as a stable supercritical nucleus - vacancies decay by spontaneous emission of positrons of well-defined energy. How the shape of the positron line approaches the static limit with increasing T was discussed in Sects. 10.2, 3.

The qualitative picture is thus as follows. During the heavy-ion collision the deeply bound electrons adjust almost adiabatically to the momentary position of the nuclei on their scattering trajectory. When two uranium nuclei approach closer than ca. 50 fm, their electric field reaches the critical threshold. If a vacan­cy is brought into a temporarily supercritical bound state, it has a large chance of being emitted as a positron only if the nuclei are kept together by nuclear forces for about 10 -19 s. These crude estimates are made quantitative in the following sections dealing with the collision dynamics. For the moment, we accept the adiabaticity assumption as a working hypothesis, but wish to emphasize one very important point: the K-shell vacancies that are absolutely necessary for the experimental observation of the vacuum decay can originate only when the colli­sion process deviates from the strict adiabatic model. Solely because the nuclei move can we expect that sometimes an inner-shell electron is excited out of its quasi-molecular state. Therefore, there must be a delicate balance between adiabaticity and dynamics of the heavy-ion collision. That we find ourselves precisely in the favourable region must be considered a very fortunate circum­stance.

11.2 The Two-Centre Dirac Equation

In the following two sections we leave aside all dynamical processes occurring during a heavy-ion - atom collision. The question asked is: if some "deus ex machina" kept two nuclei with charges ZI and Z2, respectively, fixed at a dis-

304 11. Superheavy Quasimolecules

tance R apart, what would the motion of the electrons in the electrostatic field of these two nuclei be? Since the potential would be static we could search for stationary states that are solutions of the two-centre Dirac equation [~1ti 73a, 76b]

EIf/(r) = HTco(r) If/ = {ca· p + PmecZ+ Vt (Ir+ 17R I) + Vz[ Ir-(1-17)R I1}If/(r)

= [ca· p + PmecZ+ Vt (r1) + Vz(rz)] If/(r) , (11.3)

where Vt and Vz are the potentials due to the two nuclei. Outside the nuclear radius (}i (i = 1,2) there are pure Coulomb potentials

Z·ez V;( Ix I) = - -'-( Ixl > (}i)'

Ix I (11.4)

In (11.3) the origin of the coordinate system is conveniently chosen at the centre of mass of the two nuclei, which gives the value M1 /(M1 + M z) for the parameter 17 if R is taken to point from nucleus 1 to nucleus 2. We have neglected the mutual interaction between electrons in (11.3). The effect of this interaction - mainly screening the Coulomb potential - is discussed below.

Before discussing the properties of the solutions of the two-centre Dirac equation (11.3), it is useful to elaborate on the analogous non-relativistic problem, the two-centre Schrodinger equation

Elf/ = (:~ + Vt + Vz) If/. (11.5)

This problem was solved by Teller [Te 30] and Hylleraas [Hy 31] after the first approximate results from Heitler and London [He 27]. The two-centre Schrodin­ger equation must have two constants of motion. Trivially, one of them is the projection L z of angular momentum on the symmetry axis. A second operator commuting with the Hamiltonian was constructed by Erickson and Hill [Er 49] for point-like nuclei:

where Oi are the angles between ri and the internuclear axis, whereas L 1 and L z are the angular momentum operators with respect to the two centres. In prolate spheroidal coordinates e = (r1 + rz)/ R, 17 = (r1 - rz)/ R, qJ, the Schrodinger equation separates, producing two coupled eigenvalue equations which can be solved for the energy. For nuclear molecules the two-centre shell model has been developed at Frankfurt (1969-1973) to describe similar phenomena in nuclear fission and nuclear reactions at low energy. For a review of this active field see [Ci 81] and for the theoretical aspects, [pa 85].

The two-centre Dirac equation is more difficult to handle since no second constant of motion exists besides the projection of the total angular momentum [Co 67]. A complete proof for the non-existence has recently been given by

11.2 The Two-Centre Dirac Equation 305

Schluter et al. [Wi 83] and the statement is supported by the numerical results (discussed below) that the relativistic two-centre states do not cross as a function of R. This would not happen if a second constant of motion existed, according to the famous non-crossing theorem of von Neumann and Wigner [Ne 29].

Let us next discuss the symmetries and common nomenclature of the rela­tivistiv two-centre problem. Whereas the total angular momentum of the electron orbital is not conserved, its projection Jz along the axis that joins the nuclei com­mutes with the Dirac operator H TCD since the azimuthal symmetry around this axis has been maintained. Therefore, letting the z coordinate pass through the nuclei as shown in Fig. 11.1,

J L az n 'n a az n z= z+- = -1 -+- , 2 oqJ 2

(11.6)

an operator that commutes with H TCD ' Every electronic orbital is characterized therefore by a unique eigenvalue f..l

(11. 7)

Further, f..l can assume the values

f..l = ± 112, ±312, ±5/2, ....

The solutions with ± If..ll are degenerate in energy since there is no difference in rotation around the internuclear axis. The value of f..l is commonly denoted by a, 1C, 0, ... for Jz = ± 112, ± 312, ± 512, ....

When the two-centre potential V is, in addition, parity invariant, i.e. if both nuclei are identical

V(e, z) = V(e, - z) , (11.8)

(e is the cylindrical radial coordinate as shown in Fig. 11.1) then the eigensolu­tions have good parity and are referred to either as "gerade" (even parity) or "ungerade" (odd parity) states, abbreviated by "g" and "u", respectively.

To specify a complete set of solutions of (11.3), the two-centre states must be characterized by an additional label, usually corresponding to the quantum numbers of the asymptotic atomic states continuously reached in the united atom limit, e.g. a full description of the bound electron states in a heavy-ion collision would be 1 S112 a g, 2P3/31Cg, etc. This procedure is unique for the discrete spectrum; i.e. for the bound states, but not for the molecular continuum states, because there an infinite number of degenerate states exists for every energy.

We should, however, point out that the united atom label (e.g. ls1/2) is not very useful from a physical point of view, except in a few cases. This is illustrated in Fig. 11.3 where a selected region of the two-centre diagram (binding energies versus internuclear distance) is shown for the system Pb + em. Due to the forbid­den crossing of states with the same symmetry, the properties of the wave func­tion often change abruptly along a continuous line. For example, at the avoided

306 11. Superheavy Quasimolecules

a E !keV]

111:

~ % 6l: - State Pb -em <f~> x··l /·;.i .......

: \ I ' i

50

o 1000 2000 3000 R(fm]

Fig. 1l.3a, b. A selected region of the Pb + em molecular correlation diagram. At the avoided crossings seen in (a), the nature of the adiabatic wave functions changes abruptly, shown in (b) for the 62: state as a function of R

crossing at 1300 fm the continuous state labelled 6L: changes from an approxi­mate d312 (x = 2) configuration to an approximate PII2 - P312 (x = + 1, - 2) symmetry (Fig. 11.3). The opposite is true for the other state, labelled 5 E, that participates in the avoided crossing. One therefore labels the states often only by consecutive numbers, 1 L:(g), 2L:(g), 3 E(g), etc. The lowest states of each exact symmetry (of Jz and, possibly, parity) do not participate in avoided crossings, so that for these states the united atom nomenclature is commonly used and meaningful (ls1/2O', 2P3127C, etc.).

The two-centre Dirac equation (11.3) was first solved by Muller et al. [Mil 73a, 76b, Fr 76b] by diagonalization in a basis of functions of the prolate spheroidal coordinates e, YJ and {fJ:

If/n/!1(e, YJ, (fJ) = (e- n1-(I.uI±+)exp (e- l ) Llfl±+( e-l )p/(YJ) ei{Jl±+)q.>. 2a a (11.9)

Due to the presence of the negative energy continuum states, however, this approach, through correctly predicting the critical distance Ref' was not success­ful for good convergence of the wave functions [Ra 76c]. Similar attempts by Marinov et al. encountered the same difficulties and even gave much too large values for Ref [Ma 74, 75a, b].

Satisfactory results were obtained later by Rale/ski and Muller [Ra 76b] with a numerical integration technique, described in the following. The dependence of the wave function on the angular variables is discretized by passing to the conjugate variable, i.e. angular momentum. This is achieved by expanding the wave function If/(r, e, (fJ) into an infinite series of multi pole components

11.2 The Two-Centre Dirac Equation 307

( 0 ) ~ ~ ( gAr) x~(O, (O) ) (11.10) ifill r, , (O = '-' IfIxll= '-' 'f () Il (ll ) .

X X= ± 1 1 x r X - x u, (O

Here X~ are the spinor spherical harmonics, Sect. 3.3. We now also introduce a multi pole expansion of the two-centre potential

00

J!i(r1)+ J!2(r2) = 1: Vj(r,R)P1(cosO) 1=0

(11.11)

and substitute (11.10, 11) into the two-centre Dirac equation (11.3). Because the spinor spherical harmonics X~ form a complete basis of two-component angular functions, we derive an equivalent set of coupled differential equations for the radial functions?, Ar), fAr) by projecting with the X~:

(11.12)

d x+l -gx(r) = (E+ me)f,/r) - --gAr) - 1:h(r) Vj(r,R)A -xl-A' dr r AI

The coefficients AxIA=JdQX~(Q)tl1(cosf})xf(Q) can be evaluated by angular momentum algebra. After the sum in (11.11) has been truncated at a suf­ficiently large angular momentum jrnax, the 2 (2jrnax+ 1) coupled differential equations are integrated. The energy eigenvalue E is determined by an iteration to make the wave function vanish at a large distance from the two Coulomb centres.

This highly accurate numerical integration method is also convenient since it easily allows the use of modified potentials (11.11) which, for instance, may contain electron screening effects. The influence of finite nuclear radii is readily calculated.

Calculations using this method have been systematically carried out on a broad scale during the last few years by Betz et al. [Be 76a, So 79a, Be 80b]. As a prototype we discuss the diagram for the uranium - uranium (U + U) system in some detail (Fig. 11.4). As is obvious from the diagram, the various states can be classified in two types, the first made up by those states with symmetry S112 or P112

(x = ± 1) in the united atom limit. These states continuously gain binding energy when the internuclear separation decreases below several hundred fermis. This is typical for all superheavy quasimolecules (Z1 + Z2 > 137) and is caused by the "collapse to the centre" for I x I < Z a, making the wave function very sensitive to the extension of the nuclear charge distribution.

The states with I x I > Z a approach a constant value of the binding energy at rather large internuclear separation (300 fm). This behaviour is known as the "runway" effect. Throughout this region the wave functions very closely resemble those of the united atom (Z1 + Z2)' The difference between the solid and the dashed-dotted lines in Fig. 11.4 is caused by the finite extension of the in­dividual nuclei. As may be expected, this effect considerably lowers the binding

308 11. Superheavy Quasimolecules

-'0 r----'r-5 ----.,3O~50'----'loor-_-"':mTI -"'~::r-, _'~qro~~3OOl~1~_ R(fm]

-30 3sIr

-50

-300 -5(lj

-1000 -1500

E [keY]

Fig. 11.4. The U + U correlation diagram (Zt = Z2 = 92). (_. -) include the effect of the finite nuclear radius, (-) for point nuclei [So 79a, Be 80b]

energy for distances below several times the nuclear radii (R ::S 50 fm). It is felt only for the S112 and P1I2 states.

It is possible to show [Mil 76b] that the angular symmetry of the molecular states is profoundly changed at a typical nuclear separation of 500 fm. For larger distances the quadrupole part of the two-centre potential ~ (r) becomes stronger than the spin-orbit force, changing the angular momentum coupling, rearranging the states of approximately good total angular momentum J into states of approximately good orbital angular momentum L and spin parallel or antiparal­leI to the molecular axis. For very large nuclear separations another recoupling occurs, resulting in (atomic) states with good angular momentum JI> J2 with respect to the individual nuclei.

We finally discuss an asymmetric system, Pb + Cm (Z1 = 82, Z2 = 96). A superficial look at Fig. 11.5 shows that the details are more complex due to the loss of parity symmetry. The behaviour of the strongest bound states, however, is practically unaffected by this; it is dominated by the continuous gain in binding energy upon approach of the nuclei. It is quite illuminating to follow the density distribution of some of the wave functions as a function of variable internuclear separation. The 1s112 0' wave function (Fig. 11.6), e.g., starts out as the K shell of the isolated Cm atom and becomes strongly polarized as the Pb nucleus ap­proaches. For distances below ca. 500 fm, however, the wave function very closely resembles the K shell of the united atom, Z = 178. This would also happen for a lighter system (Z1 + Z2 < 137), but then the united atom wave func­tion would not behave characteristically for a superheavy system, namely the radial extension of the density distribution shrinks further with decreasing distance between the nuclei (the "fall to the centre", compare the configurations at R = 18 fm and R = 100 fm). Similar behaviour is found for the 2P1120' and 2s112 0' states, with a rather complicated region of intermediate distances, and a simple shape of the density distribution for distances below R = 100 fm.

A new method for solving the two-centre Dirac equation has recently been developed by Wietschorke et al. [Wi 83]. This method is similar in spirit to that employed by Rafelski and Maller [Ra 76b] , i.e. the dependence of the wave

-20

-50

-100

-200

-500

-1000

E [keY]

Dz 40 2.0

16 30 50

Or-~~~~~~~-

t A IOOlm

l~~~. ~Jri\~: ~J ~ 0.8 0.4

51JOfm

OL-~~~~~~~~~

0_6 01, 02

11lXl 1m

O~~ , /\:,:, 2~1lXl"m O~Lt~_~~~~~~~~_ 04

Pb

11.2 The Two-Centre Dirac Equation 309

3roJ R[fm] Fig. 11.5. The Pb + Cm correla­tion diagram (Z\ = 82, Z2 = 96)

Pb+ em

Fig. 11.6. Evolution of the density distribution of the 1 Sll2 and 2P1l2 wave functions in the Pb + Cm system as a function of nuclear separation R

function on the angular variable is discretized by expansion on a set of angular functions, and the remaining coupled radial equations are solved by numerical integration. However, Wietschorke et al. use the finite element technique with spline functions for the angular variable, which allows them so treat the Dirac equation also in other orthogonal coordinate systems (prolate spheroidal coor­dinates, Cassini coordinates, etc.). This method is therefore much more flexible, especially in the limit of large nuclear separation. Their method has also recently

310 11. Superheavy Quasimolecules

given numerical solutions for the two-centre Dirac continuum states for the first time [Wi 84].

11.3 The Critical Distance R cr

The internuclear distance at which the binding energy of the lowest quasi­molecular state (the 1 sa state) reaches 2mec2 is of greatest immediate concern to us. Only when this critical distance Rer exists and when the collision is sufficiently energetic to let the nuclei approach closer than Rer can spontaneous pair creation occur: for R < Rer the quasi-molecular 1 sa state is supercritical. (For very large values of Z1 + Z2 the 2P112a state may also become supercritical.) This explains the importance of precisely determining Rer as a function of the nuclear charges Z1 and Z2.

Figure 11.4 shows that the critical distance in the U + U system is of the order of 30 fm, agreeing with our estimate in Sect. 11.1 (R er ~ 135fm). The precise value, however, depends on a number of assumptions. For example, for two point-like uranium nuclei Rer = 37 ± 1 fm [Mil 73a, Li 77] 1. Inclusion of finite nuclear size reduces the critical distance to 35 fm, meaning that for such small internuclear separation the quadrupole and higher multi pole contributions to the two-centre potential are unimportant (they affect Rer by only 2 fm). This makes it possible to include also electron screening effects in the approximation by taking only the spherically symmetric part of the potential (the so-called monopole approximation) [Be 76b]. Assuming identical nuclei with radius R N , the mono­pole part of the two-centre potential is (Z1 = Z2 = Z/2)

2 (r>~R+RN) -Ze Ir 2

--- -- -R-RN -R+3RN -- -R+RN 2Ze2 [ 1 (1 )3 (1 ) 1 (1 )2

R~ 16r 2 2 4 2

Vo(r) = (1 ) 3 (1 2 2) X TR-2RN +8r "4R -RN

--Rr +-r -R+RN>r>-R-RN 1 2 1 3J (1 1) 8 16 2 2

2Ze2 (1 ) ~ r<TR - RN .

(11.13)

1 Popov [Po 73a) estimated Rer .. 45 fm for the double uranium system. Later, Marinov et al. [Ma 74, 75a, b) claimed that improved calculations gave a critical distance of about 51 fm, in con­trast to the results of Maller et al. [Mu 73a, Ra 76c) , who predicted Rer = 36.8 fm. Lisin et al. [Li 77) finally obtained Rer .. 38.5 fm, which is in reasonable agreement considering numerical uncertainties.

60

170

d Z-2

Z-10

150 100 10

200

11.3 The Critical Distance Rer 311

Fig. 11.7. Dependence of the critical distance Rer as func­tion of combined nuclear charge (Zt + Z2) in the mono­pole approximation. The different lines correspond to various degrees of ionization of the quasiatom [Wi 79]

Modifications of this potential due to screening have been considered within two models: the Thomas-Fermi and the Hartree-Fock methods.

The Thomas-Fermi method (density functional method) for molecular con­figurations has been investigated in great detail by Gross and Dreizler, who also attempted to include relativistic effects [Gr 79b, 83d]. Taking into account the influence of 30 electrons, the Thomas-Fermi method gives a critical distance of 30 fm [deR 81]. However, for such strong potentials the (non-relativistic) Thomas-Fermi method is known to underestimate the influence of the innermost electrons considerably. This is corroborated by Hartree-Fock-Slater calculations by Wietschorke et al. [Wi 79], who find a critical distance of 26 fm when all 184 electrons are present. However, in a violent heavy-ion collision a large number of electrons from the inner shells are also ionized, considerably reducing the screen­ing effect. Taking this into account by allowing for various degrees of ionization, the results of Wietschorke and So!! predict a possible range of 26 - 31 fm for the critical distance in the U + U system. About 2 fm should be added to these values to account for the influence of the higher muitipole terms of the two-centre potential. Allowing for further corrections due to vacuum polarization, self­energy, etc., Rcr(U + U) = 30 ± 2 fm seems reasonable.

The variation of Rer with combined nuclear charge Z = Zl + Z2 is shown in Fig. 11.7. The curves are calculated within the monopole approximation, but with variable degrees of ionization, by Wietschorke and Soff. The values should be taken with an uncertainty of 2 - 3 fm due to the corrections discussed above. One may conclude that the system Pb + em (Zl + Z2 = 178) is just barely super­critical (Ref -18 fm), and one has to go to heavier systems to penetrate into the region where spontaneous positron emission is possible during a heavy collision.

312 11. Superheavy Quasimolecules

Bibliographical Notes

The formation of super heavy quasimolecules in heavy-ion collisions and their applications in the physics of strong fields was discussed by MUller et al. [Mil 72b, c]. The solution of the two-centre Dirac equation and its properties are contained in [Mil 76b].

The validity of the use of an adiabatic basis was investigated in detail by [Th 79, 81].