Limiting charge for point nuclei

Volume 95B, number 2

PHYSICS LETTERS

22 September 1980

LIMITING CHARGE F O R POINT NUCLEI ~ P. GJ~RTNER and B. MULLER Institut fur Theoretische Physik der Johann Wolfgang Goethe-Universitat, 6 Frankfurt am Main, West Germany and J. REINHARDT and W. G R E I N E R 1 A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06511, USA Received 25 March 1980

We investigated the behavior of the vacuum charge around a supercritical nucleus with Z > 137 in dependence of the nuclear radius. The screening effects are considered in an effective potential approximation. We show that in the point nucleus limit the nuclear charge is screened by the vacuum charge up to Z = 137. This means that the coupling strength of a point charge in QED cannot be larger than 1. The influence of heavier leptons is also discussed briefly.

The behavior o f an electron in the electrostatic field o f a nucleus is described by the Dirac equation, which leads to the Sommerfeld fine-structure formula for the spectrum o f electron bound states in the external Coulomb potential Ao(r ) = -Ztx/r of a point charge (we set/[ = c = 1):

En/= m e {1 + [Za/n - I K I + [K2 - ( Z a ) 2] 1/2] 2 ) - 1 / 2 , with K = +1, +2, ... and n = 1,2, .... Due to the term [K2 - (Za) 2 ] 1/2 this expression becomes imaginary for Zct > IKI. For example, all states with / = 1/2 cease to exist at Z = 1/a ~ 137. The corresponding wavefunctions become non-normalizable at the origin. Mathematically, this means that the hamiltonian loses its self-adjointness. In this letter we • investigate if point nuclei with Z > 137 can exist if one takes the effects due to the decay of the neutral vacuum into a charged vacuum into account. The wavefunction is stabilized in a very natural way by rec-

This work has been supported by the Bundesministerium fur Forschung und Technologic and by the Gesellschaft flir Schwerionenforschung (GSI). t Permanent address: Institut f/it Theoretische Physik der J.W. Goethe-Universit~it, Frankfurt a. M., West Germany.

ognizing the finite extension of the nucleus. In normal nuclei the potential is practically cut off at the nuclear charge radius R. In this case one can trace any level En] down to a binding energy of twice the electronic rest mass if the nuclear charge is increased as a parameter. At the critical charge Zer the Is-state reaches the negative energy continuum o f the Dirac equation. If the strength of the external field is further increased, the bound state "dives" into the continuum and appears as a resonance in the scattering phase shifts. This reflects the fact that the normal, neutral vacuum becomes unstable and decays into a charged vacuum state The latter is stable due to the Pauli principle. The electron charge distribution remains localized. The charged vacuum corresponds to the occupation o f the dived bound states (see the recent review articles [ 1 - 3 ] and references therein). If the nuclear charge radius R is decreased, an infinite number o f states can reach the negative energy continuum, while Zcr approaches 1/ct ~. 137 [4]. We have investigated the solutions of the Dirac equation in the case o f an extended nucleus with Za > 1 using the nucleus charge radius as a parameter. With the transformation

181

Volume 95B, number 2

PHYSICS LETTERS

22 September 1980 E • [MeV]

u I = ( - E - me)l/2q51 + ( - E - me)l/2qs2,

:

osT

u 2 = - i ( - E + me)l/2q~ 1 + i ( - E + me)l/2~2, the negative energy continuum solutions o f the Dirac equation in the case Za > 1 are given by [5] :

d~1 = Nx-1/2 IeinM_y_ +

e-i(n-~r3")

1

-g~r~/EM_ y_ l/2, - 3 ' j .

(1)

X 1/2±3".

With this restriction, formula (1) becomes. ~1=

3"-y

N[eirlx3"+e-i(n-~r3")x-3"K +ym----~/E]"

This leads to the following electronic vacuum charge distribution in the vicinity of the nucleus: r2p = 4N2e -~r3" [A - B sin(2 3' In(r/R)

+ a)].

(2)

Here A and B are constants depending on the resonance energy, while the phase a m addition depends on the shape o f the nuclear charge distribution. Thus, the vacuum charge distribution has an oscillating structure in the variable In(r/R) with the frequency 23' = 2 [K2 - (Za) 2] 1/2. This means that the vacuum charge shrinks together with the nuclear charge when R goes to zero. The number of oscillations is given by K = n - 1. Formula (2) does not depend on the sign of K, so that Sl/2- and Pl/2-states can be treated as equivalent. Explicit numerical calculation of solutions of the Dirac equation confirms the vahdity of the approximations which lead to formula (2). These calculations were performed with a cutoff Coulomb potential that corresponds to the model of a homogeneously charged sphere of radius R. The nuclear charge radius R m determined by the condition R = R 0 (2.5 X Z)I/3 where R 0 = 1.2 fm corresponds to the density of ordinary nuclear matter. We obtained the result that the resonance 182

iUl 5

--

_

3s,,2 1S1/2

-10 f -25 -50

We have used the abbreviations: x = 2ipr, y = i ZaE/p, 3' = [K2 - (Ze) 2 ] 1/2. M(x) denotes the Whittaker function, and r~ is the matching phase, which is determined by the boundary condition on the charge surface. We are interested only in the region in which the vacuum charge is localized. Therefore, we consider only the limit pr "~ 1, so that we can set: M _ y _ 1/2,± 3"(X) =

, --------=

-2 5, -5#

1/2,3"(x)

3' - Y

K

,

,

0 4~

Fig. 1. Single-particle energies of some electronic states in the field of a nucleus with charge Z = 150 as a function of the nuclear radius parameter R e. With decreasing Ro a growing number of nsl/2- and nPl/2-states joins the lower continuum. energy of a supercritical state is proportional to R0-1 (fig. 1), and the vacuum charge distribution shows the shell structure of formula (2) (fig. 2) which shrinks together with the nucleus according to the scaling variable In(fiR). Due to the logarithmic stretching of the radial scale in eq. (2), we can estimate that nearly all of the electronic charge of a state is localized at the outermost maximum of the charge distribution. Thus we can describe the vacuum charge distribution of a supercritical nucleus by a series of concentric shells each containing the charge ~ e (due to the equivalence o f states with the same IKr), taking into account that for Z < 274 only electrons with angular momentum / = 1/2 can reach the negative energy continuum. Now the screening of the Coulomb potential can be considered in the

]05~r?P~

ls

(1-2)

,o,//// v \ 1

,

,, ;,

10-8

,

,

R~ 10-6

,

10-4

k

10-2

1

r[}~ 102

Fig. 2. Single-particle electronic charge distribution of a shrunk nucleus with Z = 150 and R o = 10-s fm for various states.

Volume 95B, number 2

PHYSICS LETTERS

following way: we replace in formula (2) the nuclear charge Z by an effective Z e f f ( r ) = Z - Qvac(r), where Qvac(r) is the vacuum charge contained in the sphere with radius r. Because Z appears in the argument o f the sine, this replacement leads to a broadening of the outer maxima in the charge distribution. The replacement of Z by Zef f is an approximation which can be checked by numerical methods. Calculations performed with a Dirac-Hartree program agree with the results obtained by calculating the charge distribution with Z e f f. A theoretical p r o o f for the validity of this approximation in the frame of a T h o m a s - F e r m i model is under investigation. As a consequence o f the discussed screening, a state which the outermost maximum screened by more than Z = 137 electrons does not become supercritical. Using formula (2), the position o f the maxima and the diving point R~ rit , where a level reaches the negative energy continuum, can be calculated. Fig. 3 shows that for Z < 137 + 4(n - 1) the nsl/2 and the (n + 1)Pl/2 states never become supercritical, even if the nucleus shrinks to a point. The innermost Z - 137 electrons in the vacuum shell of a shrinking nucleus follow the sin(ln(r/R))-law and reach an arbitrarily large binding energy. The outer electrons never reach the negative energy continuum; the

22 September 1980

inner maxima o f their charge distribution also follow the sin(ln(r/R))-law, while their outer maxtma which contain nearly the whole charge behave like a normal wavefunction o f an undercritical atom with Z <~ 137. This indicates that no point nucleus with Z > 137 can exist because the screening vacuum charge reaches the limit Z - 137 and is degenerate to a point too. Beside electrons, the heavier leptons must be considered. The replacement o f m e by another mass m x in the Dirac equation is equivalent to a scale transformation: The Dirac equation for electrons On units/~ = c = m e = 1) has the same solutions as for other particles (in units h = c = m x = 1) with the nuclear radius R replaced b y R(mx/me). This means that the diving point R8 tit is shifted b y a factor me/m x. The recoil of the nucleus, which should be considered when the lepton mass is comparable to the nuclear mass, is neglected. For R ~ 0 the wavefunction o f heavier particles has the same position and the same binding energy (in MeV) as the electronic one. This is seen from the Dirac equation which is Independent o f the mass in the limit IEI -* ~ . With mz = 105.6595 MeV, m r ~ 1500 MeV and Z = 150, we get the diving points R ~ r ~ ) = 2 X 10 - 4 fm, R~r(r) = 1.2 X 10 - 5 fm for the lSl/2-single-particle state. Adopting the formalism for electrons, we can

RCr~ o ~7 B0 121--I

F

150 F

II','4 1

160 t / i~ /

11,,,i, 1oi 1 10;°

//

/

170 .~f-';

180 ..k...-'• /

//

L'lJlj/ i/JJ/

137140

150

160

170

180

190

201-I II /I I ~o-I ILI #:11145~I,/~Z9

__

z"

Fig. 3. (a) Position of the maxima of the vacuum charge distribution, solid lines, including screening; dashed lines" no screening. (b) Value of the critical radius parameter R ~r for the four lowest s1/2-states as a function of Z. 183

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conclude that each vacuum-shell contains the charge of 4 electrons, 4 muons, etc., until the shrinking nucleus is screened to a charge o f 137. These considerations neglect the effects o f the normal vacuum polarization and the weak interaction. The problem o f the point nucleus including those effects is under investigation, but we do not expect a change of our conclusion reached here: Point charges with charges Z > 137 cannot exist, or, expressed differently: The coupling constant Za for point charges cannot be larger than 1.

184

22 September 1980

References [1] J. Reinhardt and W. Greiner, Rep. Prog. Phys. 40 (1977) 279. [2] J. Rafelski, L.P. Fulcher and A. Klein, Phys. Rep. 38C (1978) 227. [3] S. Brodsky and P.J. Molar, in: Structure and collisions of ions and atoms, ed. I.A. Sellin (Springer, Berlin, 1978) p. 3. [4] B. Mtilier, J. Rafelski and W. Greiner, Z. Phys. 257 (1972) 183. [5] B. Muller, J. Rafelski and W. Greiner, Nuovo Cimento 18A (1979) 551.