Precise Values for Critical Fields in Quantum Electrodynamics * Gerhard Soff and Berndt Müller Institut für Theoretische Physik der Universität Frankfurt am Main. Germany and Johann Rafelski ** Department of Physics, University of Pennsylvania, Philadelphia, Pa 19174 (Z. Natur forsch. 29 a, 1267-1275 [1974] ; received May 30, 1974) A careful investigation of different corrections to binding energies of electrons in almost critical fields is performed. We investigate quantitatively the influence of the nuclear charge parameters, nuclear mass, degree of ionization on the value of the critical charge of the nucleus. Rather quali- tative arguments are given to establish the contribution of the quantumelectrodynamic corrections, which are found to be small. Some phenomenological modifications of QED are quantitatively in- vestigated and found to be of negligible influence on the value of the critical field. For heavy ion collisions with ZlJrZ2^> Zc r the critical separations between ions are given as results of precise solutions of the relativistic two coulomb center problem. Corrections due to electron-electron inter- action are considered. We find (with present theoretical accuracy) Zcr = 173 + 2, in the heavy ion collisions RCT(U-U) = 34.7 ±2 fm and RCT (U-C f) = 47.7 ± 2 fm. We shortly consider the pos- sibility of spontaneous muon production in muonic supercritical fields. 1. Introduction where r0 = 1.2fm. The atomic mass A has been approximated for superheavy elements by If the binding energy of a state is greater than 2 me c2, while this state is vacant spontaneous free A = 0.00733 Z2 + 1.3 Z + 63.6. (1.2) positron creation occurs and the vacuum becomes charged1-3. The external electromagnetic potential The Is i/o state was found for values of Z 169. for which the spontaneous positron creation may At this point the binding energy of the lsi/2 state occur is called critical. In order to investigate the reached 2 me c2. For the 2pi/o electron the critical positron creation in a heavy ion collision it is value was found to be Z = 182. If the critical value necessary to know the precise least nuclear charge of Z could be increased by more than 15 units due which produces the corresponding critical potential. to nuclear properties, the electron - electron inter- The solution of the Dirac equation in the case of action and quantumelectrodynamical effects the as- a pure Coulomb potential leads to the Sommerfeld sociated experiments1'2 may become impossible. finestructure formula which allows for lsj/o electrons The exact theoretical and experimental determina- only solutions for Z < a-1 = h c/e2 = 137.03602. tion of the critical charge seems also to be a valuable Beyond this value of Z no lsi/9 state is found in test for relativistic quantum mechanics and quan- the discrete spectrum. tum electrodynamics of strong fields. By a precise For extended nuclei the Dirac equation was first measurement of electronic transition energies in solved by Pomeranchuk and Smorodinsky 4 (Their superheavy atoms or quasiatoms formed during the work suffered from various numerical errors) and collision of heavy ions with Zx + Z.2~ Zcr, it may by Pieper and Greiner 5. The nuclear charge distri- be possible to decide if selfenergy- and vacuum- bution was given by a homogeneous charge sphere polarization corrections are described well within with a radius the frame of the generalized Hartree-Fock equation R (1.1) given by Reinhard 6 and also Rafelski and Müller 7 (r I*p(z)yK&p(z) o(J d3zyK 8 71 \x-z 3 K F(\x-z\, ;(p)) (1.3) fd z!y <Pp(x)<Pp(z)yK ®q{z) v x-z\ Reprint requests to Dr. G. Soff, Institut für Theoretische * Work supported by Deutsche Forschungsgemeinschaft, Physik der Universität Frankfurt/M., D-6000 Frankfurt/ by Bundesministerium für Bildung und Wissenschaft, Main 1, Robert-Mayer-Straße 8—10. and by Gesellschaft für Schwerionenforschung (GSI). ** Supported by US-Atomic Energy Comission. 1268 G. Soff et al. • Precise Values for Critical Fields in Quantum Electrodynamics with sin(<jr' |X — Z |) F(\x-z\, ep-eq, sg(p)) = f d?< si' (1.4) + sg(p) {q'-ia) + 1 if P>F, 1.5 units of the critical value for positron creation ;(P) = -1 if p<F, when including the electron-electron interaction in the Thomas-Fermi approximation. Numerically it is F means the Fermi energy. possible to find bound state wavefunctions up to This paper is organized as follows. In the next the region Z = Zer — 0.01. In Fig. 1 one can see the section we discuss the one center problem in the frame of the Dirac equation. We present several numerical results concerning the influence of dif- Z = 172 .U, E = -1021.966 keV ferent parameters on the critical potential. In Sect. 3 we solve the two center problem and approximate the electron-electron interaction and obtain nu- IWI2-102 merical results for critical radii. In Sect. 4 we then consider the quantum electrodynamical corrections for one and two center potentials. In the following section we discuss the possibility of muonic super- critical states and show, that they cannot be created in heavy ion collisions. We close our discussion of the possible effects which could change the value of the critical nuclear charge discussing several phe- nomenological modifications of classical electro- dynamics and QED in Section 6. We show that no essential change of the critical quantities is to be Fig. 1. The radial density | r |2 of the lsj/j electron and the nuclear charge distribution is shown for an assumed expected from such modifications if the values of nucleus with Z = 172.44. The binding energy differs only by the phenomenological parameters are restricted by 44 eV from the critical value —2 m c2 . present experiments. In the last section we sum- marize our results and give a short outlook on radial density of the lsj/9 electron of Z = 172.44 future calculations. (c == 9.9 fm, t = 2.2 fm, £=-1021.964 keV) in comparison to the nuclear charge density assumed. A very remarkable fact is the relatively localized 2. Zcr within the Framework of the Dirac appearance of the wave function. For the diving Equation velocity defined as = AE/dZ\z=Zor we obtain 30.1 keV/unit charge. In our calculations we use a realistic nuclear charge distribution of a Fermi type with the surface In Fig. 2 the maximum of the radial density 2 thickness parameter t fixed usually to 2.2 fm and ' xpr of the lsj/o electron as function of the nuclear the half density parameter c which was derived charge is shown. For Z ZCT it reaches a value from the equivalent radius of about 25 fm. In a high energy collision of heavy naked nuclei = r0 Äh (2.1) with Z = Z1 + Z2> 137 one could assume because of with r0 = 1.2fm. The nuclear mass A was chosen compression of nuclear matter that it is possible to according to Eq. (1.2) of Pieper and Greiner4. In observe positron creation for example in a Pb + Pb order to take into account the electronic interaction (Zx + Z2 = 164 <Zcr) collision or with even lighter we first employed a Thomas-Fermi-approximation8. projectiles. In Fig. 3 we show the binding energy We found numerical solutions for the lsi/o state for of the lsj/2 electron as a function of Z where we Z 172 (c = 9.234 fm), the binding energy for have assumed as charge distribution two concentric Z = 172 is -1018.139 keV. That means a shift by superposed homogeneously charged spheres with in- 1269 G. Soff et al. • Precise Values for Critical Fields in Quantum Electrodynamics max If r| [fm] cal equivalent radius of a fermi charge distribution with t = 2.2 fm as function of Z is shown. We note 1s-electron that a change in the nuclear radius of 10% corre- sponds to a shift in the diving point by one unit. As the equivalent radius is determined from the atomic mass, we can also plot the A dependence of the energy of the lsj/2 state — see Figure 4 b. In 160 Z Fig. 2. The maximum of | ip r ]2 of the lsx/2 electron as function of Z. Fig. 4. a) The critical equivalent radius of a fermi charge distribution with t = 2.2 fm as function of Z. b) and c) Energies of the lsj/2 state as function of the atomic mass number A and of the surface thickness t of the nucleus for Z = 171. Fig. 4 c one can further see the energy as function of the surface thickness parameter t, where c was fixed to 9.218 fm and Z=171. With the value of the diving velocity = 30.1 keV/charge unit it is easy to recover the corresponding effect on the shift in the value of Zcr. The determination of Zcr seems to be uncertain by one unit due to uncertainties in Fig. 3. The energy eigenvalue of the lsj/o electron as func- the nuclear charge distribution. As an example of a tion of Z in a high energy heavy ion collision. The nuclear charge distribution consists of superposed homogeneous numerical solution to an overdimensioned nucleus charge spheres of the collision partners with increasing we show the radial density \iyr 2 of the lsj/g elec- charge in the smaller nucleus. Diving occurs at Z=167. tron for Z = 184 (92U + 92U) in Fig. 5 with a as- creasing charge in the smaller particle. Diving then sumed nuclear radius corresponding to c = 20.3 fm.