Quantum electrodynamics of strong fields? - VUwimu/Varying-Constants-Papers/Reinhardt-Greiner... · Quantum electrodynamics of strong fields? JOACHIM REINHARDT and WALTER GREINER

Quantum electrodynamics of strong fields?

JOACHIM REINHARDT and WALTER GREINER Institut fur Theoretische Physik der Johann Wolfgang Goethe Universitiit, 6 Frankfurt am Main 1, Robert Mayer Strasse 10, Germany

Abstract Quantum electrodynamics of strong external fields is investigated in the context

of atomic physics. If the electromagnetic coupling constant Za becomes sufficiently large ( Z > l / a % 137 for point sources or Z> 172 for extended nuclei) bound electron states can join the negative-energy continuum of the Dirac equation. The resulting possibility of spontaneous positron production and the new concept of a charged electron-positron vacuum is discussed under several aspects. The autoionization model and the exact overcritical solutions of the single-particle Dirac equation are contrasted with a quantum-field-theoretical approach. Non-linear field effects are shown to have no influence. Vacuum polarization and the self-screening of extremely strong electric charges are treated explicitly.

Adiabatic collisions of heavy ions leading to the transient formation of quasimolecular electron orbitals are suggested as a means for an experimental test of the special features of strong-field QED, After an introduction to the relativistic two-centre Dirac equation, the x-ray spectroscopy of quasimolecules and the spontaneous and induced positron emission in, for example, U-U collisions are discussed. The importance of several background processes is stressed. Finally, strong-field aspects for nuclear matter and in gravitation are mentioned.

This review was received in October 1976.

+ This work was supported by the Gesellschaft fur Schwerionenforschung (GSI) and the Bundesministerium fur Forschung und Technologie (RMFT).

Rep. Prog. Phys. 1977 40 219-295 16

220 J Reinhardt and W Greiner

Contents

1. Introduction , 2. Theoretical description .

2.1. General discussion of the Dirac equation with a Coulomb potential 2.2. The autoionization model . 2.3. Exact solution of the single-particle Dirac equation 2.4. Quantized description of overcritical states . 2.5. Vacuum polarization in strong external fields . 2.6. Possible influence of non-linear field effects . 2.7. Statistical description of the charged vacuum

3. Experimental tests: QED effects in heavy-ion collisions . 3.1. Intermediate quasimolecules in heavy-ion scattering 3.2. The two-centre Dirac equation (TCD) . 3.3. Quasimolecular x-ray spectroscopy . 3.4. Decay of the neutral vacuum in heavy-ion collisions 3.5. Background effects . References ,

. .

.

.

4. Overcritical phenomena in other fields-outlook .

Page 221 224 224 228 232 241 247 254 257 260 260 263 268 274 280 288 29 1

Quantum electrodynamics of strong $el& 22 1

1. Introduction

Quantum electrodynamics, i.e. the theory of electrons, positrons, and their interaction with the radiation field, has been one of the most successful disciplines of physics, Its formal framework, which essentially dates back to the early works of Dirac, Heisenberg and Weisskopf in the 1930s and was completed in the 1940s by Feynman, Schwinger, Dyson, Tomonaga and others, allows the calculation of atomic properties with virtually arbitrary precision. I n spite of the somewhat unsatisfying divergences in the renormalization scheme, QED may be regarded as a completed theory. There is, however, one phenomenon in QED which only recently has been fully understood and which leads to a qualitatively new concept: the charged vacuum in strong (electrostatic) fields.

The best starting point for a discussion of this idea is the following question: what happens to the atomic electrons if the charge of the nucleus is considerably increased? As we shall see below in detail, relativistic effects will become dominant and will qualitatively alter the level spectrum (leading, for example, to a very large fine- structure splitting). A first attempt to account for them is the famous Somnierfeld fine-structure formula (we mostly set h = c = 1)

with K = i: 1, f 2, . . . and n= 1,2, . . . . It describes the spectrum of electronic bound states in the external Coulomb potential A&) = Za/r of a point charge. I n this case, the appropriate wave equation (the Dirac equation) can be solved analytically, Due to the term L.2- (Zoc)z]l/2 equation (1 . 1) obviously breaks down at Zoc > I K I . For example, the energy levels as a function of Z for the l s 1 / 2 state with Els=m, (1 - 2 2 a 2 ) 1 / 2 and all other states wi th j= 4 cease to exist at Z = l/a- 137, as shown in figure 1. The corresponding wavefunctions become non-normalizable at the origin, This, however, does not imply that the Dirac equation has no solution at high 2, as was first believed. Taking into account the finite extension of the nucleus one can trace any level E,j down to a binding energy of twice the electronic rest mass if the nuclear charge is increased as a parameter. At the corresponding charge number, which we will call critical (Zcr), the state reaches the negative-energy continuum of the Dirac equation (‘Dirac sea’) which, according to the hole-theory hypothesis, is totally occupied by electrons. If the strength of the external field is further increased, the bound state ‘dives’ into the continuum. The overcritical state obtains a width and is spread over the continuum. Still, the electron charge distribution does remain localized.

The related phenomena have been investigated and analysed very carefully. At this point, we will only stress the most important aspect: the overcritical vacuum state is charged. This means the following. As already mentioned, within the hole theory the states of negative energy of the Dirac equation are occupied with electrons. The Fermi surface lies at EF = - mec2 (in QED of weak fields it is convenient to put EF = 0, which is equivalent to EF = - mec2 since there are no (bound) states between E = - m,c2 and E = 0). The negative-energy continuum states occupied with electrons represent the model for the neutral vacuum of QED. The infinite charge of that state is renormal-

222 J Reinhardt and M? Greinev

Figure 1. Lowest bound states of the Dirac equation for nuclei with charge Z . While the Somerfeld fine-structure energies (broken lines) for K = - 1 end at Z N 137 the solutions with extended Coulomb potential (full curves) can be traced down to the negative-energy continuum which is reached at critical charge Zcr. The states entering the continuum obtain a spreading width as indicated by the bars (magnified by a factor of 10). If the state was previously unoccupied, two positrons will be emitted spontaneously leading to a new stable ground state called the charged vacuum (after Muller et al 197213).

ized to zero: in field theory the same result is obtained by symmetrizing, because of charge conjugation symmetry, between the states E < EF (occupied with electrons) and E > EF (occupied with positrons); the vacuum expectation value of the current (charge) operator is then zero: ( O / j p , ) O ) = O . If now an empty atomic states dives into the negative continuum, it will be filled spontaneously with an electron from the Dirac sea with the simultaneous emission of a free positron moving to infinity. The remaining electron cloud of the supercritical atom is necessarily negatively charged. While in ordinary undercritical physics we can define a vacuum state 10) without charges or currents by choosing the Fermi surface (up to which the levels are occupied) below the lowest bound state, this is not possible in the overcritical case ((n I jo l6) # 0). Thus we are led to the concept of the charged vacuum 10) (Muller et al 1972b,c). Note that this is a fundamentally new concept leading to a new understanding of the vacuum as a physical object. Every time when at increasing external field strength an electronic bound state joins the negative continuum, the vacuum undergoes new phase transitions and becomes successively higher charged. In the limit of an overcritical (Za > 1) point charge, infinitely many electronic bound states 'dive'. I n this (academic) case the vacuum is infinitely high charged. I t must now be treated self-consistently, which is already necessary for vacuum charges higher than Z,,, 3 2.

Clearly, the charged vacuum is a new ground state of space and matter. The normal, undercritical, electrically neutral vacuum IO), whose physical existence is reflected by the well known vacuum polarization (displacement of e+ e- charges) in strong fields, is in overcritical fields no more stable: it decays by either positron (for attractive

Quantum electrodynainia of strong fiela3 223

electric fields) or electron (by repulsive electric fields) emission into the new stable but charged vacuum.

It is perhaps useful to draw attention also to the philosophically important aspect of this process. If the vacuum is defined as a part of space free of real particles, this vacuum can be subject to certain conditions, like penetration by fields. If these fields become strong enough, the particle-free vacuum can no longer exist: it must contain real particles (in our special case studied here, these particles are electrons). This also necessitates a more general definition of the concept ‘vacuum’. While the standard definition ‘region of space without real particles’ obviously cannot be true for very strong external fields, the new and better definition ‘energetically deepest and stable configuration of space’ seems to be more appropriate. The stability of the charged vacuum, in our case that of fermions, is assured by the Pauli principle. In the case of a boson vacuum it must come from the interaction between bosons (sometimes effect- ively described by non-linearities in the field equations).

I n the first part of this review we extensively discuss the results of overcritical atomic phenomena which have been obtained since 1968, essentially by two schools in Frankfurt and in Moscow. After some general considerations we introduce a model which treats the decay of the neutral vacuum as an autoionization process of positrons. Section 2.3 quantitatively justifies this picture, giving the exact single-particle solutions of the Dirac equation with an external Coulomb potential for all 2. The important task of quantization of the theory is accomplished in $2.4, still maintaining the concept of overcritical states.

A typical and important feature of the quantized theory is vacuum polarization. Its behaviour in strong fields is shown according to the work of various authors in $2.5. The consequences of certain phenomenologically introduced non-linear exten- sions of the theory are scrutinized in $2.6, namely non-linear electrodynamics of Born-Infeld type and higher order non-linear terms in the spinor field. The results show that diving of electronic levels cannot be prevented by any known causes. Finally, $2.7 gives a short statistical (self-consistent) description of the charged vac- cuum which is interesting in connection with hypothetical very highly charged nuclei.

Though an interesting problem in itself, the physical relevancy of QED of strong fields has yet to be demonstrated ($3). Historically, the first detailed investigations were inspired by the possible existence of superheavy nuclei (Pieper and Greiner 1969). The deformed-shell model of theoretical nuclear physics tells us that nuclei near the magic proton numbers Z= 114 and 164 should have strongly enhanced life- timest. The production of long-lived superheavy elements by fusion of ordinary nuclei would allow the extension of the exact measurements of atomic spectroscopy into a new region of binding energy. The overcritical region (Z> 172) cannot be reached. Luckily, however, it is at least possible to assemble a supercritical charge for a short period of time in the collision of very heavy ions. Such collisions are semi- adiabatic (aionlaelec z 1/10) with respect to the electron motion and therefore the electrons will form molecular orbits during the various steps of such an encounter. This leads us to the concept of intermediate superheavy molecules (Rafelski et al1971). For example, in U-U collisions near the Coulomb barrier the lowest bound state joins

t Recently it has been asserted that superheavy elements of Z E 126 have been found in nature as enclosures in monazite crystals (Gentry et al 1976). If the experimental evidence is correct, it seems that a consistent interpretation of the findings can be given by assuming that these unusual elements are those in the neighbourhood of Z= 164 (Mosel and Greiner 1968, Soff et a1 1976).


the continuum for some 10-19 s. This should be long enough to observe the decay of the neutral vacuum by detecting emitted positrons. In 93.4 we deal with the somewhat delicate dynamical description of this process. It requires the formation of a K hole during the collision which then can autoionize.

Beforehand we have to lay the foundations by discussing the quasimolecular model of atomic collisions (Fano and Lichten 1965, Saris et a1 1972, Mokler et a1 1972) ($3.1) and the two-centre Dirac equation (93.2). The transient formation of superheavy quasimolecules has very much stimulated research during the last five years. Section 3.3 reviews theoretical and experimental work on the spectroscopy of x-rays produced in heavy-ion collisions (Meyerhof et a1 1973, Meyerhof 1974, Green- berg et a1 1973, 1974, Wolfli et a1 1975, Frank et a1 1975, 1976a, b). I t is hoped to extract from such studies information on the behaviour of energy levels in the superheavy region. Section 3.5 is devoted to the discussion of background processes, especially the de-excitation of Coulomb excited nuclei and the shake-off of the vacuum polarization cloud (Soff et a1 1977), which may seriously interfere with the fundamental QED effects we are interested in.

The final section briefly sketches some general aspects of the charged vacuum in QED (which can namely be viewed as an example of spontaneous breaking of charge conjugation symmetry) and shows its relation to other areas of physics. Strongly bound bosonic systems also exhibit overcritical phenomena. Since the Pauli principle is absent in this case, the vacuum becomes unstable against many-particle excitations manifested in, for example, pion and a-meson condensates. Finally, the importance of particle creation in strong gravitational fields will be stressed.

2. Theoretical description

2.1. General discussion of the Dirac equation with a Coulomb p o t e d a l

equation (Dirac 1928)

with the 4 x 4 matrices

The starting point for relativistic quantum theory and especially QED is the Dirac

HD$'(f')E [a.p+pflz] $' (r )=E $'(r) (2.1)

The four-component wavefunction exhibits the properties of a bi-spinor and describes fermions with spin Q. The spectrum of the free Dirac equation consists of a gap between - mc2 and + mcz and two continua of free particles with the dispersion relation

E=hw= zt [(hk)2+(mc2)z]l/z. (2.3) To avoid the decay of electronic states under emission of an infinite amount of energy when turning on the coupling to the radiation fields, Dirac (1930) postulated that all states in the negative continuum are occupied, i.e. he defined a Fermi surface lying in the gap. The vacuum so constructed can be excited by photons, for example, to produce particle-hole pairs at a threshold energy of 2mc2. The hole states then having the properties of a particle with electron mass, positive energy and positive charge are identified with positrons. In the language of second quantization ($2.4) Dirac's prescription takes the form of suitable anticommutation relations between the electron

Quantum electrodynamics of strong fields 225

and positron operators (for an extensive treatment of the formalism of QED refer to Bjorken and Drell (1964, 1965) and Jauch and Rohrlich (1976)).

The electromagnetic field A, is introduced into the Dirac equation by minimal coupling

[ ( Y, p , - e A , 1 1 + m $(r, t ) = O . (2 4)

For stationary states in a static Coulomb potential Ao(r)= V ( r ) this reads

Ea.p+Pm+ V(r)l$(~)=E#(r)- (2.5)

In low-2 atoms it is sufficient to take the potential of a point-like nuclear charge V ( r ) = - Zol/r. V ( r ) may also contain electron-electron interaction terms.

Turning on the binding potential V ( r ) has two effects on the solutions of the Dirac equation. At the edge of the positive continuum bound states emerge, which shift down into the gap with increasing strength of the potential. It is a peculiarity of the long-range l /r potential that an infinite number of bound states are created at once, unlike the situation of, for example, a square well or Yukawa potential, where a few states are truly bound and the remaining ones only show up as resonancest. Also, the continuum states are influenced and deformed by the central Coulomb potential, which attracts electronic and repels positronic wavefunctions. This gives rise to a vacuum displacement charge and is the origin of virtual vacuum polarization.

An explicit derivation of the exact solution of (2.5) for various cases will be given in 52.4. Here we shall look at its qualitative behaviour. T o this end it is advantageous to use the squared form of the Dirac equation, obtained by eliminating one of the components of the bi-spinor. Assuming a spherically symmetric potential the spin- orbit angular momentum operator

K=P (0.L-t-h) ( 2 . 6 ) commutes with the Hamiltonian and the total angular momentum operator, producing a good quantum number K . We separate angular and radial parts by the ansatz

with the spherical spinors xK!

(for detailed treatments of the Dirac equation in central potentials see Rose (1961) and Akhiezer and Berestetzkii (1965)). We are led to the system of first-order differential equations

d K

dr r -u1= - -Ul+(E+m-V)uz

d K

dr r -u2= - (E-m- V ) Ul+- U2.

t Rose (1961) speaks of an additional bound state appearing at Z = 109. This solution has to be ruled out, since the expectation values of kinetic and potential energy do not exist separately.


Eliminating u2 leads to a second-order wave equation

K ( K + ~ ) K dV/dr d2U1 -+ dr2 m + E - V dr dv/dr dui+ ( E - V)Z-m2---- +--) u1=0. (2.10)

1 2 r nz- E+ V

Using the transformation ul(r) = [m + E - V(r)]l/z ~ ( r ) it is further possible to reduce (2.10) to the form of a self-adjoint Schrodinger equation (cf Popov 1971b, 1972c, Zeldovich and Popov 1972):

with

x" + k2(r) x = 0 (2.11)

-_ ______ V' ). (2.12) r m + E - V

The effective potential in this equation is quite a complicated expression. Looking at the asymptotic solution of (2.10) one can easily distinguish between

bound and free states. Assuming that V ( r ) and its derivatives vanish at I --f CO we obtain

u1(r) N (m + E - V)l/2 exp { - r [m2 - ( E - V)2]1/2}. (2.13)

Wavefunctions inside the gap ( - m < E < m) therefore are localized and vanish exponentially:

while continuum solutions (I E I > m) oscillate :

u1(r) w exp [ - (m2- E2)1/2 I ] + 0

ul(r) N exp [ rt ir (E2 - n z 2 ) 1 / 2 ] .

(2 ' 14(aN

(2 ' 14(b)) Particularly interesting is the limit 1 E I = m. The wavefunctions behave like

[ - ir (2m I V(r)1)1/2] for E = +m (2.15(a))

[ - r (2m I V(r)1)1/2] for E = -m. (2.15(b)) ul(r) -

Thus the bound states with negative energy do not display any irregular behaviour and remain localized when reaching the negative continuum.

Now let us turn to the special potential V = - Za/r. The relativistic problem with this potential was first solved by Darwin (1928) and Gordon (1928), leading to the Sommerfeld formula (1.1) already mentioned in $1. The solutions obtained break down at Za > I K I . This can be understood by looking at the behaviour of the wavefunctions.

The form of the wavefunction at small distances Y is deduced from (2.10) or

u1 r[~2-(Z4211/2. (2.16)

When Za> I K J the wavefunction suddenly begins to oscillate near the origin like cos {[(Za)z- ~ 2 1 1 1 2 In r} and thus loses its physical meaning. We can explain the reason for this breakdown of the solution by looking at the effective potential in equation (2.12) which at small r takes the form

(2.11):


For Zol large enough, the l /r2 term becomes attractive and leads to the collapse of the motion. This is a general feature of the relativistic motion in a Coulomb potential. It can already be concluded from the relativistic energy momentum relation

p2+m2-(E- V)2=0 with the effective potential+

(2.18)

(2.19)

which also occurs in classical mechanics. We want to stress, however, that the collapse of the wavefunction is no physically

real effect. Mathematically it means that the Hamiltonian loses its self-adjointness, The solution of the wave equation then has to be specified by an additional parameter as discussed by Case (1950) in a somewhat different context (cf also the work of Alliluev (1972)).

The wavefunction is stabilized in a very natural way by recognizing the finite extension of the potential source. In normal nuclei the potential is practically cut off at the nuclear charge radius R of a few fermis. We can write

for r > R

with

(2.20)

(2 * 21)

corresponding to the model of a spherical shell or a homogeneously charged sphere, respectively. The regular solution of the Dirac equation with potential (2.20) in the interior region r 6 R is easily obtained. Its logarithmic derivative then has to be matched to the exterior solution on the sphere r = R . I n this way a boundary condition is specified for the exterior wavefunction. Therefore we can also construct meaningful solutions at Za =. I K I ,

A simple argument (Zeldovich and Popov 1972) shows that the atomic levels can be traced continuously down to the negative continuum. Let I$(Z)) be any bound- state wavefunction in the gap with energy E(2) and let ZU(r ) be the potential with a strength determined multiplicatively by 2:

(2.22)

Assuming U ( r ) to be an integrable, bounded negative function like (2.20) we can deduce from (2.23) that dE jd2 is finite and negative, so that the energy E ( 2 ) continuously decreases with 2. Note that this reasoning depends on the positive definite- ness of the norm ($[$) in the Dirac theory and would not be valid for Klein-Gordon particles.

Let us now discuss at length the exact energy levels for superheavy nuclei. Strong

t The differencc between (2.17) and (2.18) is the correction of Langer (1937).


binding in a square-well potential was first treated by Schiff et al (1940). The first authors to treat the cutoff potential were Pomeranchuk and Smorodinsky (1945) whose solution, however, was numerically inadequate. Later calculations were performed by Werner and Wheeler (1958) in connection with their speculation on superheavy elements and by Voronkov and Koleznikov (1961) but they had no influence on the modern development and understanding of QED of strong fields.

The first precise calculation of the eigenenergies of the Dirac equation with an extended nuclear potential for all elements up to Z= 170 was performed by Pieper and Greiner (1969). These results are shown in figure 1 and exhibit a number of interesting properties. Substantial deviations from the Sommerfeld energies begin to occur only in the vicinity of Z = l / a - 137. The levels show two kinds of splitting which attain extremely large values. For example, the spin-orbit splitting between the 2pl/a and 2p3p states reaches 800 keV. Also, the exact degeneracy of states which differ only by the sign of K , e.g. 2p1/2 and Z s l p , is broken.

Most interesting, however, is the fact that several of the levels shown do reach the negative-energy continuum. The critical charge numbers for the three lowest levels l s~ /z , 2~112, 3 ~ 1 1 2 are approximately ZcrZ 170, 185,245. The magnitude of the first of these numbers is vital to the idea of testing the decay of the neutral vacuum in heavy-ion collisions, where one can reach 21 + ZZ = 184.

Beyond the critical charge the Dirac equation no longer supports the bound state. It was expressed by Pieper and Greiner (1969) and later by Gershtein and Zeldovich (1969, 1970) that an empty atomic level reaching -mc2 leads to the spontaneous energyless emission of a positron. The physics and mathematics of this ‘diving’ into the continuum and of the overcritical state will be clarified in the next three sections.

T o end our general discussion, we remark that electron screening (Fricke and Greiner 1969) slightly modifies the energies of figure 1. Results of Hartree-Fock- Slater calculations for all elements Z= 100-173 were published by Fricke and Soff (1974, 1977). Although the electron-electron interaction is important for the chemistry and spectroscopy of superheavy elements, it does not influence the inner shells significantly. The most elaborate calculation (Soff et al 1974) for the critical charge leads to ZO= 172 for the lsl/z orbital.

2.2. The autoionization model

When the nuclear charge exceeds the critical value Zc, a bound state vanishes from the gap and becomes degenerate with the negative-energy continuum. T o describe this process we note that it is forinally equivalent to the autoionization of an excited state in atomic or nuclear physics. For the reader familiar with nuclear physics, figure 2 schematically shows a situation where a 2p-2h state is degenerate with a lp-lh continuum state. Both configurations mix because of the residual interaction which leads to the decay of the 2p-2h bound state. One of the excited nucleons jumps back below the Fermi surface while the other one is ejected into the continuum.

Using the mathematical method of Fano (1961) a model was developed (Muller et al 1972a, b, c) describing the overcritical behaviour in terms of the single-particle Dirac equation. The field-theoretical extension will be treated in $2.4.

We are interested in the eigenstates of the overcritical Dirac Hamiltonian

H ( Z ) = H o + V ( Z ) =H(Zcr) + V(Z’) (2.24)

with the free Dirac Hamiltonian HO = iy, ( a/ ax,) + m, the total electromagnetic

Quantum electrodynamics of strong $el& 229

-_ l p - l h __- continuum ____ Energe t i c0 position

of the bound 2 p - 2 h s ide 1 --

Figure 2. Illustration of the autoionization mechanism in nuclear physics. A 2p-2h state can decay into an energetically degenerate lp-1 h continuum state.

potential V ( Z ) and the overcritical extra potential defined by V(Z’)= V ( Z ) - V(Z,,). As the basis for a diagonalization procedure let 14) and I $ E ) denote the bound and

m just reaching the critical charge: scattering states of the at

with <+

Without introducing a serious error, we have restricted our basis to one discrete state since the energetic distance to the next level with the same spherical symmetry (necessary for the interaction) is very large.

For convenience we neglect the weak dependence of the nuclear size on Z and factorize the additional potential

V(Z’)= Z’U(r ) . (2 .27)

Next we need the matrix elements of the overcritical Hamiltonian with our undercritical basis. They can be written

<+ IH(Zcr+Z’)I +>=Eo+Z’ <+lul+> (2 .28) - m f AE

<$EeIH(Zcr+Z’)I +)= <$E*IH(Zcr)/ +?+ <$E~IZ’UI +> = 2’ ($E’/ U I +) 3 VE’ (2.29)

<$’E”IH(Zcr+ z’)l $E’)= <$E”/ H(Zcr) I $E’) f ($E”IZ’U(Y)I $ E # )

E’S(E” -E’) + Z ’ U E N , ~ ~ . (2.30)

230 J Reinhardt and W Gveiner

Using these matrix elements the supercritical state

H(Zcr+Z‘) ITE)=E I T E ) (2.31) can be obtained analytically. Let us expand

I Y d = a ( E ) I +)+ dE’br(E)I $E’) . (2.32)

Inserting this wavefunction and projecting with the basis states (41, < $ E ( I the wave equation (2.3 1) is equivalent to a system of equations for the coefficients

E’< -m

(Eo+ A E ) a ( E ) + j d E ’ b ~ . ( E ) v E ( # = E a ( E ) (2.33(4)

(2*33(b)) V,ya(E)-t-bjy(E) E’+J dE” b p ( E ) Z ’ U E ~ E - = E bn.(E).

The explicit solution of these equations is not possible due to the integral in (2.33(b)). In the following, we neglect this term, which can be justified by two arguments: the matrix element Z’UETE* describes a rearrangement of continuum states due to the extra potential which should not be very significant for 2’ not too large. Furthermore, one can think of the continuum as being pre-diagonalized, in which case our treatment would be exact, implying only a change in the values of V E .

Let us impose upon the wavefunction (2.32) the normalization

(Y,yI Y E ) = S ( E ’ - E ) . (2 * 34) Then the solution of (2.33) neglecting UE*E* takes the form

b ~ f ( E ) = 6 ( E - E ’ ) + ----___ a ( E ) VE‘ E - E‘ - ie

with the principal value integral

(2.36) E’< -9n

and E is a small positive quantity. The convergence of the integral at E + - m will be proven below (equation (2.41)).

Equations (2.35) are the complete solution to our problem of the diving of a level in terms of the reduced basis set I+), / $ E ) . Generalizations of the result to include more than one bound state and more than one continuum are laborious but straightforward.

We are mainly interested in the fate of the previously bound state 14) when the interaction Z’U(r) is turned on. Since the overcritical wavefunction I $ E ) is normalized to unity (equation (2.34)), the square of the coefficient of admixture a ( E ) describes the probability with which 14) is promoted to the energy E < - m under the action of the overcritical additional potential. We obtain

(2.37)

This expression obviously shows resonance behaviour. To further simplify equation (2,37), we neglect the principal value integral F ( E ) . That is justified by the relatively

Quantum electrodynamics of strmg fields 23 1

weak energy dependence of VE and will be supported by the numerical phase-shift analysis in the next section. Writing 2 7 I V~12= I?,

(2.38)

takes the form of a symmetric Breit-Wigner curve centred around E = Eo + AE .e - m with the width I?.

The position of the resonance is determined by

AE= Z’(5, 1 U ( Y ) ( 4 ) - 2 ’ 8 . (2.39)

Therefore the level dives approximately linearly with respect to 2 into the continuum. The width

(2.40)

roughly increases quadratically with the overcritical charge.

calculated (Muller et al 1973a) T o be specific, the following values of the two parameters 8 and y have been

1s1/2, S ~ 2 9 keV, y ~ 0 . 0 4 keV 2~112, 6 z 38 keV, y z 0.08 keV.

T o avoid misapprehensions, however, equation (2.40) has to be interpreted carefully. The matrix element ( $ E [ U(r)I $), although fairly constant at large IE I, vanishes exponentially with E -+ -m. This is clearly caused by the repulsion of the negative- energy continuum wavefunctions by a positively charged nucleus. The amplitude of the positron wavefunction with momentum p near the origin is

1 $E(O) 1 - exp ( - TZamip) (2.41) so that the simple quadratic scaling now does not hold near Zcr. This result also ensures the existence of the integral (2.36) near E N - m where only one branch of the pole lies in the integration region.

Let us now look once more at the physical meaning of the diving process described. At first, we stress the important fact that the charge distribution of a filled K shell remains localized when it enters the continuum. This is illustrated in figure 8 where the 1s1,2 density distributions lvt,h(r)12 for different nuclei are compared. For Z> Zc, the bound state shows up as a strong distortion of the continuum wavefunctions. The collective effects of the continuum states producing a negatively charged cloud still bound to the nucleus has been termed veal vacuum polarization (Muller et al 1972 b,c) in contrast to the normal virtual vacuum polarization where only positive- and negative-charge centroids are displaced in their relative position.

In a normal, undercritical system there are many stationary stable states corresponding to various electron occupations. Especially one can define the electron vacuum state (in the presence of an external field) where all bound states are empty. This is no longer possible in the overcritical situation. Since here the diving state loses its identity and is spread over the continuum (figure 3 shown this schematically in the case of a discretized continuum) it can no longer be defined as empty. If a previously empty bound state becomes overcritical, it will be filled by spontaneous emission of a positron (Pieper and Greiner 1969, Gershtein and Zeldovich 1969, 1970). The resulting new ground state is doubly charged (due to spin degeneracy): the neutral vacuum decays into a charged vacuum (Muller et a1 1972b,c) (of course, charge


Figure 3. Spreading of the bound state (full lines) over the negative-energy continuum. The system is quantized in a finite box. For Z > Z c r the continuum contains one more state (after Muller et a2 197213).

conservation is observed and the obsolete positive charge is carried to infinity by the emitted positron).

The charged vacuum is a basically new physical concept. I t is derived solely from the properties of the relativistic wave equation and applies generally to deeply bound fermionic systems. The overcritical atom is only the most transparent example, where a well known external potential is specified. It is, however, quite interesting to note that in other areas of physics analogous phenomena have been found. For example, Bastard and Noziitres (1976) in the context of solid-state physics investigate the behaviour of a donator level which moves through a band edge into the conduction band? (more precisely, under the influence of increasing external pressure a gap is formed and enlarged while the hole state stays in its position). As in the case of the overcritical atom the bound state becomes quasistationary but a localized charge distribution is retained.

Our treatment of the overcritical phenomena in the autoionization model has been physically very elucidating. I t was, however, not complete in two ways. Most important, the formalism has to be extended and justified by a quantum-field- theoretical treatment ($2.4). Before doing this we will take, in the next section, a closer look at the solutions of the Dirac equation in a central Coulomb potential and verify our statements on energies and wavefunctions.

2.3. Exact solution of the single-particle Brae equation In this section we will investigate the exact solutions of the Dirac equation with a

Coulomb potential generated by a point, or extended nuclear charge distribution. The special case of the wavefunction at E = - m and the determination of the critical radius (or charge, respectively) will be illustrated explicitly. It will turn out in our discussion that the notion of an overcritical point-like Coulomb potential is physically not admissible since the limit R -+ 0 cannot be defined in a unique manner.

We follow the elaborate papers of Pieper and Greiner (1969), Muller et al (1972c, 1973a), Zeldovich and Popov (1972) and the already quoted standard reference books (see also Rein 1969). Throughout we will use the abbreviations

A = (m2 - E2)1/2 (2.42) y = [K2 - (2CY)2]1/2 (2.43) p = ZolE/h. ( 2 . 4 )

t We thank Professor H Thomas, Basel, for drawing our attention to this work.


T o begin with, the bound-state problem for IZal< K has a well known solution due to Gordon (1928) and Darwin (1928). The radial Dirac equation (2.9) with Coulomb potential V ( r ) = - Zoljr can be transformed in the shape of Kummer’s differential equation, leading to confluent hypergeometric functions. In terms of the Whittaker function

Mb, &x) = x*+r exp (- tx) 1F1(& - p + y, 1 + 2y, x) (2.45)

the solution of (2.9) which satisfies the asymptotic boundary condition u ~ ( Y ) ,

u2(r) + 0 for Y -+ CO can be written

(2.46) W ( Y ) = co(m -t- E)1’2 ( ~ X Y ) - ~ / ~ [ Y I ( ~ ) - ~ 2 ( r ) l

u ~ ( T ) = - co(m - E) l / z (2Xr)-ll2 [ q q ( ~ ) +7~,2(1.)]

with

v1=

E q-72=- EK - mp

(1 -2r) r( - p+ r) M,tt, $AY)+ T(1+2y) r( - p- r) Mp+t,-y ( 2 q (2.47)

x q L - h -yW)I

[r (1 -2r) r (1 - p+ r) M ~ - * , ( 2 ~ r ) + r (1 r(i - y )

and CO is a normalization constant determined by the condition

J: ( ~ 1 2 + ~ 2 2 ) dr- 1. (2.48)

The energy E entering the parameters X and p will be determined by thesecond boundary condition, imposed on the wavefunction at small distances. In the case of a l / r potential the solutions (2.46) and (2.47) have to be valid at all r -+ 0. Since the functions Mb*+,-y(2X1.) are irregular at small distances their coefficients in (2.4) have to vanish compared to the regular solution. This condition yields

- p + y = -n (2.49)

which is just the Sommerfeld formula, equation (1 -1). The wavefunctions then become finite polynominals multiplied by (2hr)Y exp (- Xr). T o take into account the finite nuclear dimensions one has to solve the Dirac equation inside the nucleus separately and match the solutions uk and u,(i) at the nuclear radius R(p(r)=O for

(2.50)

Taking the potential of type 11 defined in (2.20) the matching condition (2.50) was solved numerically for the bound-state energies (Pieper and Greiner 1969). As shown in figure 1 each level can be traced down to - m and Za= K has no special significance. Although y becomes imaginary it is immediately shown from the relations

r ( 2 ) = 1-m and M;, p(Z) = M U4 B (x) (2.51)

that the functions (2.47) remain real. For the sake of clarity we will discuss the explicit determination of the critical

charge Zcr. Treating this problem it is sufficient to use the simplified wavefunction (2.46) at the energy E= -m or X -+ 0, p --f - 03. The solution of the accordingly specialized differential equation can be obtained directly.

234 7 Reinhardt and W Greiner

Instead, one can also take the appropriate limit in equation (2.48), e.g. the limit

lim 1Fl(a, b, Z / U ) = d - * b (b ) I b - 1 ( 2 2 / ~ ) (2.52) formula

u - m

can be employed. The Bessel function of the second kind I,(x) may be transformed to the regular solution K,(x). Not bothering about the normalization constant we obtain

q = E & ((8Zar)l/2) (2 .53) and

1 za

u2 =CO - [Q (8Zar)1/2 Ki,' ((8Za1')1/2) + KKir ((8Zar)1/2)]

with v = 2 [(za)z-K2]1/2.

Ki,(x) and Ki,'(x) are the NIacDonald function and its derivative with respect to the argument. Closer inspection of its asymptotic properties proves that the exact electron wavefunction at the diving point is localized (exponential decrease at r + m ) and exhibits an infinite number of oscillations when approaching the origin (r -+ 0). Based on general arguments we had discussed this already in equations (2.14)-(2.16). To obtain physically meaningful solutions the wavefunction inside the nucleus with finite extension R has to be specified.

Taking the square-well potential VI of equation (2.20) for simplicity the solutions at r < R are determined by spherical Bessel functions

where Z 1 = I K I + Q (sgn ( ~ ) - l ) , 12=Z1-sgn ( K ) , pz=(E+Za/R)Z-m2. For the 1s1p level at E = - m the interior solution is

(2 .55 )

Since the nuclear charge radius happens to be small compared to the electron Compton wavelength, R < l /m, we have p N Za/R and the matching condition then reads

cot-=(z) Za 2R Ki,' - ((8Zar)l/2) R Ki, ( ( s z a r ) V j ' (2 .56)

This simple transcendental equation allows the calculation of the l s lp critical charge (at fixed radius R). Equation (2 .56) and several other formulae explicitly treating the level motion and wavefunctions with the approximation A = In l /R% 1 are discussed in the works of Popov (1970, 1971a, b) and Zeldovich and Popov (1972).

Let us now turn to the continuum solutions which are essential for the under-


standing of the overcritical state (Muller et al 1972c, 1973a). I t is useful to transform the radial equations (2.9) by

for energies E > m (2.57) UI = (E+m)llz ($1 +$z) 242 = i (E - m)l/z ($1 - $2)

and ul=(-E-m)l lz ($I+$z) u2= -i (-E+Pz)~/~($I-$z)

for energies E < - m. (2,58)

For a pure - Zm/r potential we obtain the system of coupled differential equations

which may be transformed to the second-order equation

d2 1 d dx2 x dx -$1+- - - -$1 -

(2.59)

(2.60)

We use the abbreviations

(2.61)

The differential equation (2.60), as in the discrete case, has a fundamental system of solutions consisting of Whittaker functions

$I(*) =X*Y exp (-x/2) 1F1 ( I y+ 1 +iy, ~t 2y+ 1, x) (2.62)

ZoiE P

p=(E2-,2)1/2 x = 2ipr y=-,

= x-ll2 M-(ig+*), kY (x) #+) is the regular and $(-) the irregular solution at the origin. Since y= [ K Z - (Z01)2]1/2

becomes imaginary at Zoi > K the continuum functions $(*) again obtain the already encountered infinite oscillations (essential singularity at Y = 0).

At first we treat the case Zol< K , i.e. y real. To construct the most general continuum solution one has to adopt a linear combination

(2.63) $1 = b+M-(i,+*), ,(x) + a-M-(iy+*),-&)I x-1’2

or, defining new constants 7) and N ,

$1 = N [cos 7 exp (ioi+) M-(iv++), ,(x) +sin 7 exp (;a-) M-(i,++), -,,(x)] x-112 (2.64) with

exp ( T h y ) . 1

( K + iym/E) exp ( 2 i 4 = T ( y & iy) (2.65)

$2 is obtained immediately from 41 by the use of (2.59). In (2.64) together with (2.65) the reality condition for u1 and UZ, namely 41=$2*, has been incorporated. The wavefunctions u1, u2 must be real in order to satisfy real boundary conditions on the nuclear surface and to approach a spherical wave at infinity.

If we choose the continuum normalization

J d3x #se+ (x) # ~ , ( x ) = ~ ( E ’ - E ” ) 17

(2.66)

236 J Reinhavdt and W Greiner

the normalization constant N is fixed by comparing the asymptotic behaviour of our solution with

1 $1 --t -__-- exp [i (p + A)]

2 (?Tp)l/Z (2.67)

where A is the Coulomb phase shift.. Finally, the quantity 7 which determines the relative contribution of the regular

(+ y ) and irregular (- y ) Whittaker function is obtained from the matching condition at the nuclear surface. One immediately obtains

(2.68) uz(i) (R) Re [B+(ZipR)] i: [(E-m)/(E+m)]1/2 ul(i) (R) Im [B+(ZipR)] u2(i) (R) Re [B-(ZPR)] & [(E-m)/(E+m)]1/2 ul(i) (R) Im [B-(2ipR)]

___-__ tan q= -~

with the abbreviation

The upper and lower sign apply for E > m and E c - m, respectively.

for the case Za > K . It can be treated in a similar manner. Defining Now we approach the most important part of our analysis: the continuum solution

y = [(,55)2 - $]1/2 = - iy (2.70)

where y is now imaginary, the general solution takes the form

41 = [a+M-(iv+*), i?(x) + a-M-(iy+t), -i~(x)] x - ~ / ~ . (2.71)

Now the reality condition $1 = $2" takes a general form, finally leading to the wavefunction

$1 = N [ exp (iq) M-(i,+i), ip(x> + exp ( - iq - r7) 2 ' ~ M- (iy++), -iT(x)] X-1/2.

(2.72) The matching phase q is again determined by a condition similar to the one obtained above. Inserting

ijj - iy M-(iy++),-iy(x)) (2.73)

.- *

K + iym/E

~M-(ly+t), i&> i: exp (- ..7> + iy m/E

it reads

(2. 74j tan y=-

Since it will be essential for the following discussion let us also give the scattering phase (Muller et aZl973a)

U#) (R) Re [B+(ZipR)] i: [ (E - m)/(E + m)]1/2 ul(i) (R) Im [B+(ZipR)] zkz(i) (R) Im [ B - ~ I ~ [ ( [ ( E - ~ ) / ( E + ~ ) ] ~ / ~ zil(i) (R) Re [B-(Zim*

iy - iy -+exp(-iq+ny) _____- K - iy mlE A = y In 2pr + arg

The A obtained does not vanish in the limit of low energy and has to be renormalized. It turns out that the properly defined physical scattering phase shifts S which satisfy the required limit

lim S = O (2.76) 131-

Quantum electrodynamics of strong fields 23 7

are obtained by subtracting an asymptotic phase, constant for all multipoles (Muller et a1 1973a)

6 = A - 6 1 0 g = A - y (2 * 77)

By the way, it is worthwhile to note that the physical phase shift in the limit of very high energy of the scattering particle does not vanish. Indeed, taking for example a point nucleus with Za < K we have y -t & 20: and the phase shift approaches

lim S = a r g r ( y + l T iZa )~Za( lnZa- l ) -&r [ -yT( l+sgn~) ] -&~. (2.78)

This behaviour is contrary to the results of non-relativistic scattering theory where y = Zajr -t 0 and 6 -+ 0. The discrepancy is easily explained since in the correct relativistic treatment the velocitv remains limited and therefore the particle spends a

E-& m

finite

rg

C UI

N

.-

E l k e V )

Figure 4. The 1~112 bound-state resonance as exhibited in the scattering phase shift S. The embedded state is located at Eo= -926.4 keV 2nd has a width I’=4*8 lreV for the nuclear charge Z= 184 (after Muller et al 1 9 7 2 ~ ) .

Having constructed the exact single-particle solutions of the Dirac equation with a realistic nuclear Coulomb potential we now turn again to the overcritical phenomenon. To this end the phase shift 8 has been calculated numerically for nuclei with very large 2. As anticipated by our earlier results the phase shift undergoes a resonance: 6 ( E r e s ) = QT at a certain energy depending on 2 and R. Figure 4 shows sin2 6 as a function of energy for the nucleus Z=184 yielding a narrow Elreit-Wigner shaped curve with a width of 4.8 keV. In figure 5 the dependence of the location Eres of the resonance on the nuclear charge is illustrated. The broken curve shown for comparison is the ‘linear diving’ result taken from A E = ( + IZ’U(r)[ +) of the autoionization model.

Although this model is logically independent from the phase-shift analysis, both methods can be linked by the relation (Fano 1961, Muller et al 1973a)

P E E - ( E ~ + A E ) - F ~ tan AE= - (2.79)

or

(2.80)


L

\ I-

Figure 5. Diving energies of the lsij2 and 2~112 state in an overcritical extended Coulomb potential. The broken curves indicate the linear diving model while the full curves have been extracted from the exact phase-shift analysis (after Muller et al 1972~).

Employing the parametrization

Er,, N 2’6 - 2’27, F E N z ” y

the following precise numerical results have been obtained for the resonance parameters :

1 S U 2 2p1/e zc r 172 185 S(keV) 29 37.8 ~ ( k e V ) 0.33 0.22 y(keV) 0.04 0.08

T o end this section we take a look at the overcritical point-nucleus problem. Continuum solutions at 20: > K in the potential V= - Zajy cannot be constructed. It can also be shown that the solution with a cutoff potential has no unambiguous limit at R -+ 0. The proof starts from equation (2.74).

The matching phase can be written

tan 7) = -tan [ y In 2pR + E(R) ] (2.81)

where E turns out (for pRQ 1) to be independent of R, leaving the divergent logarithmic term 7 In 2pR. Therefore no point-charge limit can be defined. If we require that the resonance condition is satisfied we find

PresR -+ constant which implies

(constant)2)1:2 -3 constant -__ (2.82) R e Eres+ - (m2+ f22

Hence, all bound states with I K I -= ZE obtain an infinite binding energy when R -+ 0. This, by the way, explains the vertical tangent of, for example, the Is state in the Sommerfeld formula at Za= 1. Figure 6 shows quantitatively the dependence of the critical charge Zcr on the extension Ro of the nucleus, demonstrating that even for a very small radius of about 1 fm Zcr is significantly greater than 137.

Quantum electvodynamics of strong fields 23 9

0 10 20 30 R o l f m l

As Ro+O (point nucleus) Zcr approaches l / a? 137 (after Muller et a2 1972~). Figure 6. The finite extension Pi0 of the nuclear charge determines the critical charge Zcr.

Physically, of course, even in the case of an extremely small overcritical nucleus there will not be infinitely many ‘diving’ states. Screening of the nuclear charge, i.e. electron-electron interaction, will stop this process and produce a certain effective reduced charge of the system (see $2.7). This is very similar to the case of overcritical boson fields, where the energyless pair production is only limited by the mutual particle repulsion. (Contrary to the fermionic situation the Pauli principle is not at work and a single overcritical state is sufficient to produce many particles.)

I n figure 7 two wavefunctions of the negative-energy continuum, computed from the exact solution derived above of the Dirac equation with extended-charge Coulomb potential, are compared. While the ordinary continuum wavefunction ( E = - 1.5 m) decreases at small distances (Coulomb repulsion) the curve for E = - 1.7455 m clearly exhibits a localized bound-state resonance (the dived 1s level). At larger distance the large Is peak joins the oscillating spherical-wave solution. This behaviour can be interpreted in terms of the effective potential (cf equation (2.12)) which has a maximum at intermediate distances. The continuum and bound-state parts of the wavefunction therefore do not join directly but they are connected by a tunnelling process through a potential barrier. For further mathematical details of the effective potential, see the works of Popov and his collaborators quoted in the references.

At this point we take the opportunity to make a remark which is important both from the theoretical and historical point of view. The ideas on the charged vacuum and on spontaneous positron production in strong fields which now seem to be fully understood and which have become liable to experimental examination in fact are based on a number of earlier observations. Soon after the emergence of Dirac’s theory Klein in 1929 investigated the reflection of an electron at a one-dimensional potential barrier of height V. Instead of the usual exponential decay inside the barrier he observed oscillating waves. They emerge in the energy interval m e E < V - m when the potential V is larger than 2m. This phenomenon was first considered as a


r [ f m l Figure 7. Density ui3 + u22 of continuum wavefunctions for an overcritical extended nuclear

Coulomb potential, Z= 180. Note the difference between the curves which belong to energies at (E= - 1.7455 mc2) and off (E= - 1.5 mc2) the resonance. The figure illustrates the tunnelling between the bound state and continuum region of the potential.

puzzling inconsistency of relativistic quantum mechanics and was christened ‘Klein’s paradox’. Later §auter (1931) showed that the transmission into the non-relativistic- ally forbidden region is not caused by the discontinuity of V but happens as well for smoothly rising potential barriers (Nikolsky (1929) came to the same conclusion for an oscillator potential). The penetrability, however, decreases exponentially if the extension of the barrier wing becomes larger than a few Compton wavelengths. Therefore the effect cannot be observed on a laboratory scale with macroscopic electrostatic potentials.

In view of the hole theory Klein’s paradox can be ascribed to the possibility of pair production. The ‘negative-energy’ solution (compared to the position of the gap) describes a positron. To get rid of the unphysical infinitely extended barrier let us

E

look at a potential well. If one draws the potential V ( r ) and the gap with the boun- daries V+ m and V - r ~ . one immediately realizes that the overcritical atomic phenomena are closely connected with Klein’s paradox. A bound state inside the well can aquire an energy value E = - m which belongs to free waves in the negative-energy contin-


uum at large distance r . Both regions are separated by a classically forbidden area where only exponentially decreasing wavefunctions are possible (hence the damping factor of Sauter). Quantum mechanically, tunnelling through this barrier is possible as we have seen in figure 7. This leads to the spontaneous filling of unoccupied levels with E < - m and to the emission of positrons producing a charged-vacuum statc. The arguments presented in the case of the supercritical atom (especially the quantized treatment in 92.4) also apply for more general potentials and are, in fact, a rigorous description of our present understanding of Klein’s ‘paradox’. For earlier references related to this subject, see Heisenberg and Euler (1936), Hund (1954), Beck et a1 (1963) and the recent illuminating paper by Dosch et a1 (1971).

2.4. Quantixed description of overcritical states

A consistent and reliable description of strong-field phenomena certainly must be based on a formally correct quantized theory (Rafelski et a1 1974a). In the present section the decay of the neutral vacuum will be treated field-theoretically. Without loss of generality one can restrict considerations on the problem of one deeply bound state interacting with the negative-energy continuum. Many-particle effects and virtual excitations (for example, vacuum polarization) will be discussed in the next section.

Let us start from the Dirac field operator in a weak external potential. One usually writes

with the anticommutation relations for fermion operators

{b+, bpt)= 8 P P J , (ap+, a,(> = a,pf. (2.84)

All other anticommutators vanish. The wavefunctions qp are eigensolutions of the classical Dirac Hamiltonian

HD=u.p- t -pm+V(r) (2.85)

with energy Ep. The vacuum state is defined by

6, ~0)=&2,0)=0. (2.86)

Since in (2.83) the negative-energy solutions are introduced by a creation operator a,+ this quantization corresponds to the familiar concept of the completely filled Dirac sea. &, and &,+ are annihilation and creation operators for electrons, a, and $+ those for positrons. If the spectrum of H D acquires deeply bound states the expansion (2,83) has to be rethought. The partition of the sum at energy 0 which goes back to the bound-state interaction picture of Furry (1951) is not mandatory when a bound state in the region -m < E, < 0 emerges. Introducing an adjustable Fermi surface (Reinhard 1970, Reinhard et al 1971, Rafelski et a1 1974a) which divides occupied and empty states we may generalize (2.83)

If EF is chosen slightly above the negative-energy continuum at -m, the operator (2,87) describes a bare nucleus with an empty electron shell. Although energy might be gained by filling the deeply bound level this state is stable since charge conserva-


tion prohibits the creation of a single electron. Thus we face a neutral and stable ground state. Generally, the vacuum will be defined as the state with lowest energy which is stable under the given interactions. If the external potential becomes overcritical, i.e. a bound state joins the lower continuum as a resonance, the vacuum defined will be charged.

To proceed with the proof of this statement let us consider two different sets of classical wavefunctions, IQ, Zcr) and 14, Zcr+ Z'), which are eigenstates of the Hamil- tonian H D with near critical (Zcr) or overcritical ( Z o + Z') potential. Both sets are complete since H D is a Hermitian operator (this would not be true for an overcritical point charge, since here the wavefunction is not integrable at the origin). Analogously to the treatment in $2.2 we can expand

] E , Zcr+Z')=a(E)I IS, Zcr)+ J dE'hE<(E)I E' , Zcr) (2.88)

VE' (2.89)

E'<-ni leading to

a(E)= E - E ~ ~ - F ( E ) + in1 vE12

(2.90)

where VE and F ( E ) have been defined in equations (2.29) and (2.36). Coupling to higher bound or continuum states and the rearrangement of the negative-energy continuum have been neglected.

Let us now analyse three different descriptions of the overcritical state.

(i) The atom is initially prepared with a filled K shell. Then the field operator in the undercritical basis is

and lies slightly above El,. The sums are understood to include spin and angular momentum quantum numbers.

The ground state 10) corresponding to (2.91) is stable and normalized, (O)O)= 1. It carries two negative units of charge. This holds as well in the overcritical region. Here, however, the K-hole states &++I 0 j, dlsJ.fl 0) will become unstable.

Adopting the Hamiltonian A ( z ) = :$+HD(z) $: (2.92)

(where normal ordering was introduced to eliminate the infinite energy of the occupied levels below the Fermi surface) the energy of the 1s state is given by

EIS = (0 ldl&(zcr f 2') &+l 0) = {IS, z c r IHD(zcr 4- TI IS, Zcr) = - m - 2'8 (2.93)

with (1s) z c r l u(r)l IS, zcr) .

(ii) If the atom is prepared with an empty K shell, it is appropriate to define the Fermi surface below the 1s level. Again expanding in the undercritical basis the field operator now takes the form


The defined vacuum IO’) (with &l,l 0’) =0, &IO’) =O), though neutral and normalized, is not stable for overcritical external fields. It will mix with the one-electron- one-positron state &I,+ &+I 0’). The resulting eigenstates will be

/#’(E)) =a(E)jO’) + J dE’hp(E) &I,+ &+I 0’). (2.95)

Following Rafelski et al (1974a), we show that the coeficients a ( E ) and h ~ f ( E ) are those of equation (2.35). Let us consider, for convenience, the Hamiltonian

A‘ = : $’+ HI) $‘ : +El , (2.96)

incorporating the energy 231, (this is permissible since a constant term in the Hamil- tonian leads only to an unobservable overall energy shift). By construction the state (2.95) obeys the equation

T o solve (2.95) and (2.97) for the expansion coefficients we need a number of matrix elements of k’ with the vacuum. We employ the explicit form of I?’, namely (dis- carding the states E > E F J

A‘l $’(E))=E I#’(E))* (2 * 97)

P = ( I + his+ hl,) E ~ , + x aEt61,~Et + 2 hl,+ aEt+vEr* E’ < Ep, E’ i E p o

- a E + a E l ~ E t , E . (2.98) E, E‘<Ep,

This leads to the matrix elements (0’1 A’] 0’)=E1, (2.99)

(0’1 ”,+ &+I 0‘) = V E

(0’1 dEf&1&’&l,+ dE.1 O’)=Ea(E-E’)+ U E ’ , B .

(2.100)

(2.101)

These matrix elements are identical with equations (2.28)-(2.30). Therefore the expansion coefficients a(E) , ~ , Y ( E ) of the classical autoionization model given by equations (2.6) and (2.7) also hold in the quantized treatment. The interpretation too remains the same. If the initial state was IO’) it will be spread over the negative- energy continuum. The probability of finding it in the final state #’(E) is

P P I = KO’/ +’P)>l2= la(E)I2* (2.102)

Since #’(E) asymptotically describes a continuum state the bound positron of the undercritical case becomes a free positron moving to infinity.

The state (2.95) was written for only one spin orientation. The spin degeneracy, however, results only in a minor change. We include the two-electron-two-positron states in (2 I 85) and obtain

] # E ’ ) =a(E)lO’) + h ~ t ( E ) &Istf&++l 0’) dE’+ J g E , ( E ) &I,$+ &.++I 0’) dE’ + j f ~ , ~ f / ( E ) &I$ 6lB,++ &t++ &tr+lO’) dE’dE ”. (2.103)

If we consider only the overcritical potential there are no finite matrix elements from the ground state to the two-particle-two-hole state since Z ’ U ( r ) is a one-body operator. The electron-electron interaction on the other hand-which is a two-body force- changes smoothly as a function of 2 and therefore does not lead to a qualitative modification of our results. Thus we can neglect the last term in equation (2.103). Then the solution of (2.103) reduces to that of equation (2.95) up to the trivial factor

244 J Reinhardt and W Greiw

of two. This reflects the fact that both spin states decay independently and two positrons are emitted, yieldingp(E)=2[a(E)[ 2.

(iii) To complete our discussion of the overcritical state we describe the dynamics of the change between the neutral and charged vacua. Consider the occupied 1s state embedded in the negative-energy continuum. The field operator is

$= ~ E * I E’, Z c r + Z ‘ ) + x dp+I E’, Z c r + Z ’ ) . (2,104)

Projecting the overcritical states (which contain the 1s resonance) onto the undercritical basis by means of

14, z c r + z ’ ) = X (n, zcr lq , z c r + z ’ ) l n, z c r ) (2,105)

=E a(n, Q)I n, Zcr)

E‘> -m E ’ < - m

n

we obtain in first order

+ 2 d ~ , + h ~ t ( E ) \ E, Zcr). (2.106) E ” , E < - m

We have employed the approximations

a (Is, E ) z a(E), a (E ’, E ) c hp(E) for E, E’< -m and

+s(E-E‘)zS(E-E’) for E, E’> -m. u(E’, E ) = __ UB‘, E E-E’

Equation (2.106) shows that we can define a generalized (‘collective’) operator for vacancies in the embedded 1s state

&+= S dE‘ d,y+ a(E’). (2.107)

This operator, however, is not linearly independent in the overcritical case (for undercritical fields it just reduces to the 1s-hole operator). If we prepare a bound-state vacancy in this way at a certain time t = 0, the true time-dependent state can be written

I l l i ( t)>=~ ( t ) %,+I 0 ) + J dE‘ WE'(^) ~ E ? + I 0 ) (2.108) 10) is the charged vacuum defined above. The initial state $,+I 0) plays the role of a collective vacuum excitation and corresponds to a 1s hole. ‘The time development of I$(t)) will be described by the Schrodinger equation

E’<-%

Projecting on (01 2lS and (0 coefficients y ( t ) and w ~ ( t ) . The required matrix elements are

we obtain two coupled digerential equations for the

(2,110)


The system of coupled differential equations is therefore

ay(t) 1 awE.(t)aQ(E')dE' (2.111) at wp(t) @*(E') E'dE'=i -+i

at E'< --,In E'<-?ta

(2.112)

Additionally the solution must obey the normalization condition

owl wD=1=IY( t ) l2+ . f IWE+>l2 dE'* (2.113)

After some manipulations the solution of (2.11 1)-(2.113) satisfying the initial conditions y(0) = 1, w ~ ' ( 0 ) = O can be written as

y(t)=exp ( -i€?lst) exp ( -&r l t / ) (2.114)

WE(t)=u(E) [exp (-iEt)-exp (-iE1st) exp (-&rltl)] (2.115)

I' = 27r I VEls 12 is the width at the resonance energy 81, =El, + F(&,). Equations (2.114) and (2.115) evidently describe an exponential decay of the prepared 1s hole. At a given time t the probability to find a localized vacancy is

P(t) = IY @)I2 =exp (- r PI>* (2.116)

The positron spectrum at time t % l/l? approaches

I W E ( 4 l 2 = 14v (2.117)

which, as we know, describes a Breit-Wigner shape. This completes our discussion of the overcritical state. We were able to corrob-

orate the conceptually clear result for the autoionization of positrons. Equation (2.114) proves that the neutral vacuum IO') = &sl+ &++/ 0) becomes unstable under the action of an overcritical potential and decays into the vacuum I 0) bearing two negative units of charge (Muller et al 1972a, Zeldovich and Bopov 1972). The charge distribution in the new vacuum remains localized and closely resembles that of a normal filled electron shell. It may be derived taking the properly understood vacuum expectation value of the operator

rqx) = - 49 [$(x), YO$(X)l (2.118)

(2.119)

where the sum includes all empty bound states. Renormalization problems are avoided if one subtracts the undercritical vacuum charge and integrates over an energy interval in the vicinity of the resonance (Muller et al 197213)

pin&) = &e . f Z Z $E+(x) $E(x) dE- .f," #E+(x) $E(x) dE- &e 2 PBS BS

~ , + A E Pls(x)=e 1 [$E+(x) $ E ( x ) - $-E+ (x) $-E(x) ] dE. (2.120)

%;'Is-AB

Figure 8 shows the charge density for several (collective) supercritical bound-state resonances.

The distribution prz has a shape completely analogous in the undercritical and overcritical cases. With increasing charge the radial extension of the bound states shrinks to about 20 fm (the maximum of pr2) and below. This smoothly continues the 'collapse' of the spatial extension of the wavefunction with increasing Z which is


Figure 8. Electron charge distribution for various overcritical states compared with the K-shell distribution for the still undercritical charge 2= 172. The transition between these cases is smooth. Also shown is the virtual vacuum polarization density Ap multiplied by a factor of 100 (broken curve) (after Muller e t al 1972b).

illustrated in figure 9 for the Is level (§off et al 1974). Figure 8 also demonstrates the drastic difference between the real vacuum charge and the virtual vacuum polarization. The latter (broken curve) is about two orders of magnitude smaller and, of course, has a vanishing space integral (this is not evident from the figure since it shows only the contribution from a finite energy interval).

The discussion of the charged vacuum presented in this section is due to Rafelski et al (1974a). Some of the results have been found independently by Fulcher and

“4

Figure 9. z

Position of the maximum of the 1 ~ 1 1 2 wavefunction showing spatial extension for increasing 2 (after Soff et a2 1974).

a rapid shrinking of the

Quantum electrodynamics of strong jields 247

Klein (1973). Fulcher and Klein (1974) and Klein and Rafelski (1975a,b) present a treatment equivalent to the one described above.

They investigate the reduced Hamiltonian

Qr= - E 2 A,+ b , , z: €,6,,+6,,- U, &,,+6,- x u,*6,+6,, (2.121) U PO PO PO

where the summation runs over energy states - CO < - m - eP < - m and spin orientation U = *.6,+ and 6,,+ are positron creation operators in the bound and continuous region. The energies are measured relative to -m.

Since the Hamiltonian (2,121) is quadratic it can be diagonalized by a linear transformation

AT=AoL--x A, 6,, P

pa, = - A0&+) 60 + c A,&+) 6pg P

to obtain the form a = c W P V + P , 4- I: 4 4 ) P 4 2 P a o .

x = F ( x - 4

4"

This leads to an eigenvalue condition for the bound state

with

(2.122)

(2,123)

(2.124)

The integral equation (2.124) has real solutions only for E > 0 as illustrated in the sketch below,

once again demonstrating the disappearance of the discrete bound state in a supercritical potential. The solutions for the coefficients A in (2.122) are identical with those obtained earlier.

2.5. Vacuum polarization in strong externaljields Quantum electrodynamics of strong fields shows interesting features which are

worthwhile investigating, apart from the overcritical phenomena. For all atoms except hydrogen one has to deal with two different parameters defining the coupling strength, namely 01 and Za. The latter quantity reaches about 0.7 in the heaviest stable elements and can even exceed unity in superheavy systems. Thus the coupling is neither weak (g2< 1) nor strong (gZ$l) and the usual series expansion in (Za)%m becomes questionable. We will therefore describe a method which employs the exact Dirac propagator in the external Coulomb field and includes all orders (201)n. The


coupling to the radiation field characterized by the small constant a will be treated in perturbation theory. We will obtain many-particle equations unifying the Hartree- Fock method of atomic physics with the treatment of vacuum polarization and self- energy using QED (Reinhard et al 1971, Rafelski et al 1974a,b). At the end of this section we will discuss the explicit results for vacuum polarization recently obtained by several authors.

Following Schwinger (1953) we start from the Lagrangian

L = Loe + Loem + Qe {j,, A p ] + ejex ,Ai' (2.126) where Loe and LOem are the standard Lagrangians for the free electron (e) and photon field. A, and j , are the electromagnetic potential and electronic current while jex , describes an external current, e.g. the electrostatic potential due to the nuclear charge distributions. The four-current j , is expressed by the bilinear covariant

I-,= -8($r,+f%J,P)= -m, r,31. (2.127) The equations of motion derived from (2.126) are well known:

(2,128) (2.129)

with the electromagnetic field tensor, F,,= a,A,- a,A,. In the Lorentz gauge a,Ap= 0 (2.129) is a Poisson equation with the formal solution

(2.130) A,(x)=e S Do(x-Y) (j+iex>,(Y) d4y. We adopt the Feynman propagator

(2.131)

1 _- - [(y - x)2 + i E]-1. $79

Insertion of (2.130) in the Dirac equation (2.128) leads to

( Y P - eyAex - m) $ - i e2 { ~ f i $ ( x ) , .f d4y Do(. -Y)~~(Y)} . (2.132)

T o perform actual calculations we expand the field operator in the basis of exact classical solutions of an effective single-particle equation :

and a similar equation for ye(%). The operators b , and d , satisfy fermion anticom- mutator relations. A typical choice of the Fermi surface, EF, is illustrated in figure 10.

The Feynman Dirac propagator connected with equation (2,123) is defined as the time-ordered product of field operators

isFp(% Y) = (0 I T($p(x), $Y(Y))I O > (0 I $$&(4 $U(Y>l 0) for tX > t, (2.134)

- (0 I $ Y W #,(.)I 0) for t , > t X


Figure 10. Single-particle spectrum of the Dirac equation for an adiabatic potential and position of the Fermi surface at EF. The Hartree-Fock, tilde and Uehling sum are defined (after Rafelski et a1 1974a).

and therefore

Obviously the propagator is only uniquely defined by prescribing an additional condition, namely the position of the Fermi surface.

The ground-state expectation value of the currents is (see, for example, Bjorken and Drelll965)

(2.136)

where the propagator at the point x =y is defined by the prescription

< o l j p ( O)=Tr (iSF(% x) Y p )

SF(X, x) = $ lim [SF(X, x + E) + SF(X, x - E)]. (2.137) €-+O

Here €2 > 0 and in a particular Lorentz frame (E’, 0, 0, 0). We thus obtain

(0 Ij,lO> = 3 2 B d X , E,) Y,+& E,) - 8 ‘c B&, E,) Yp+& E,) E q < E g E q > E p

= 5 Bar,+,. (2.138) The ‘tilde sum’ is defined by

HF HP Uh E=+( ’c - 2 ) = E + & ( z - E )ZC+C. (2.139)

It equals the Hartree-Fock sum 2 = over all occupied bound states plus the

‘Uehling sum’. The latter usually accounts for the effects of vacuum polarization since it describes the induced current due to the presence of the external source (Uehling 1935, Serber 1935).

We wish to derive a system of classical self-consistent one-particle equations for the set 4,. Let us consider the matrix elements of the equation of motion between the vacuum and the single-particle (or hole) state bP+lO), d,+/O). Equation (2.132) reduces to

BqCEF Eq>EF 4 Ep<-- ln Eq>--w q q

HF

q -?n<g<EF

( r P - e r & x - m ) +Ax)= (0 IBe2 f d4r Do(x-Y) Li,(Y>’ Y , 9 W b,+10). (2.140)


Using the definition of the current operator (2.127) and the commutation relations of the field operators the equation may be transformed:

( ' Y P - ~ Y A ~ X - ? ~ ) +p(x)=e2y, S d4r W X - Y ) 2 Y"+~w + p ( ~ )

- 4 c 9MY) Y a + P ( 4 +*w P

- e2Ya f d4r (2 0 141) P

The interaction term on the right-hand side consists of a direct and an exchange contribution. Their influence can be understood by calculating the total change in energy which they produce. In first order we have

AEe.x= -e2 J d4x d4y DO(Y s f: Z: $p(y) Ya+p(x) y"+p(y) (2.142) P P

AEd=e2 j d4x d4Y DO(y-x) 4 ii ii $&) Y o + P ( ~ ) $ d x > Y"+a(x). (2.143) P P

The double tilde sum allows a decomposition into three terms -. - U h U h HFUh HFUh HFHF

P ! 7 P P P P P P P q c c=c C+(C c+c D + C c. (2.144)

The first term leads to an energy shift independent of the electron occupation (vacuum correlation energy) and does not lead to observable energy differences between electronic states. The last expression contains the usual Hartree-Fock energy describing the mutual interaction of bound electrons. The mixed double sum in (2.144) leads to electron self-energy in the exchange term and vacuum polarization in the direct term (for a further discussion see Rafelski et al 1974a).

Equation (2,141) may be simplified since we are interested in time-independent (or adiabatic) fields. Separating the time dependence by +p(x)= exp (- iE,t) dq(x) and taking the Fourier transform of the photon propagator (2.13 1)

1 S?mD~(y-~) exp [i(Ep-Eq) (tu-&)] d(t,-t,)=------ 4n Ix-Yl x exp (i I Ep - E , I . I x - y I ) (2.145)

we arrive at the following self-consistent equation:

This equation unifying the treatment of real and virtual electron-electron interactions was obtained by Reinhard (1970) and Reinhard et aZ(l971) who also discuss its derivation from the Schwinger equation for the electron and photon Green function and the proper renormalization procedure.

Equation (2.146) may be represented graphically if one defines the photon propagator -, the exact fermion propagator = and the single-particle electron propagator in the external field -:


This relation follows from the exact Schwinger equations when the exact photon propagator is substituted by the free one. The exact vertex function is approximated in the same way (a similar procedure leads to the Hartree-Fock equations in the non-relativistic case, see Layzer (1963)).

The self-consistent equation (2,146) has not yet been fully exploited numerically. Its extension to two fermionic fields, namely electrons and muons, and the consequences for muonic atoms are discussed by Rafelski et al (1974b). Here we only remark that the electron-electron interactions contained in the self-consistent terms change smoothly as a function of the strength of Vex. The tilde sum is well defined in the undercritical and overcritical case. Therefore it was justified to take eigenfunctions of the single-particle Dirac Hamiltonian when qualitatively discussing the diving process in $2.4.

The remainder of this section will be devoted to the explicit results on vacuum polarization in high-2 atoms recently obtained by several authors. The early result of Serber (1935) and Uehling (1935) applies to a pointlike nucleus. The vacuum polarization charge density and its corresponding potential produced by the nuclear Coulomb field was calculated to first order in Za. The resulting formula is

In case of an extended nuclear charge distribution the Uehling potential can be written :

X- P(Y> (2.149) Ix-Yl'

This result may be obtained from the first-order vacuum-polarization loop correction to the photon propagator. In the momentum representation using the Feynman gauge

D(q2) = - ( q 2 + iE)-l- - (q2+ iE)-l 1; dx( 1 - x)x In [l - q2( 1 - x)x (m2- ie)-1]. (2.150)

The change in interaction energy due to the second term leads to the Uehling potential.

I t is easily seen from (2.148) that VU^ vanishes essentially exponentially for r % l/m. Therefore, vacuum polarization has only an extremely small influence on wavefunctions which have an extension large compared with the electron Compton wavelength. For example, the energy shift due to vacuum polarization in the Lamb shift of hydrogen is only -27 MHz (compared with + 1079 MHz from the lowest order vertex correction). However, it increases strongly if the wavefunction becomes more localized. This is true for muonic atoms where it is the dominant correction and also for atomic states at very high nuclear charge,

The magnitude of the contribution of vacuum polarization in all cases remains small. This can be estimated from the relation of T/'Uh(r) to the inducing Coulomb potential V(r ) . In the small-distance limit of (2.145) we have

2a 7r

18

(2.151)


This quantity remains small for any reasonable value of r. In one of the highest atomic systems accessible for spectroscopy, IooFm, vacuum polarization produces an energy shift of 155 eV out of 142 keV total binding energy for the l s l , ~ state (Fricke et al 1972). At even higher Z the Uehling energy shift was calculated by Pieper and Greiner (1969). It approximately doubles if 2 is increased by 10 charge units and reaches E = -11-83 keV for the ls1p state at 2=171 (Soff et al 1974). Thus it slightly favours the diving process.

The rapid increase of the Uehling energy shift might lead to the apprehension that higher order vacuum polarization becomes dominant. The expansion in Za might break down. In fact, at one time it was believed (based on erroneous arguments) that vacuum polarization increases infinitely as 2 3 Zcr (Panchapakesan 1971).

Explicit calculation has proved that higher order contributions remain exceedingly small even for Za> 1. The primary work on this problem was performed by Wichmann and Kroll (1954, 1956). These authors developed a method to calculate vacuum polarization to all orders 01 (2a)n employing the exact (single-particle) solutions of the Dirac equation in the external field. We will follow the presentation of Gyulassy (1975).

The induced charge density may be expressed by an integral over the trace of the electron Green function. The density pvp is the vacuum expectation value of the current operator (2,127). As already mentioned it can be written

(2.152)

It is useful to consider the electron propagator in an intermediate representation transformed from a time to an energy variable. For time-independent potentials SF depends only on the difference t - t‘:

1 2x1 iSF(x,x’, t-t’)Yo=--: j,dzexp [-i(t-t’)z] G(x,x’,z). (2.153)

The Green function G then satisfies

[a . (p- eA (x)) - x+ eAo(x) + Pnz] G(x, x‘, x) = 63(x-x’). (2.154)

Then (2,148) can be written as e

2x1 pvp= --i J dx T r G(x, x’, z ) /~ , - ,~ . (2.155)

G exhibits branch cuts at z< -m, z>m and simple poles at the various discrete eigenvalues. Otherwise it is a single-valued analytic function of x. As demonstrated in figure 11 the position of the contour C determines which states are occupied. At

I m E I E

Figure 11. Choice of the integration path specifying the Fourier transform of the Feynman propagator. The negative-energy continuum and the states 1 ,2 and 4 are occupied.


Z,, the pole corresponding to the ls1/2 state moves off the physical sheet of the Riemann surface for the Green function G. This again leads to the introduction of a charged vacuum since the contour C is not able to follow the ls1/2 pole and has to remain inside the gap. To evaluate the trace entering (2.155) a spherical-wave decomposition can be used (Mohr 1974):

T r G(x, x', z ) / . . . , + ~ = ~ 2-14 T r G,((xJ, Ix'I, (2.156)

Furthermore, an expansion in Za may be employed for the Green function. Denoting the resolvent in the case of vanishing external potential by G,O we have the Neumann series

G, = Z: (Za)n G,n = Z: (Za)n G,O ( VoG/)n (2.157)

with

4T K

n n

n

.i=l T r G,n(r, r')= J n [dr&Vo(vg>] Tr[G,O(r, rl) . . . G,O(rn, r')]. (2.158)

Deforming the contour C to the imaginary axis Gyulassy (1975) finally obtains the nth-order vacuum-polarization charge density for K = ? I K I

(2.159)

In accordance with Furry's theorem the expression is only valid for n even, otherwise PI, ,n(r ) vanishes. The Feynman diagrams corresponding to p v p are

@--=-U-+* + . . . As it stands the expression for p ( r ) is not meaningful since it is not uniquely defined. For the contributions in (Za), (Za)3 neither the limit x'+x nor the integral over z exists. Therefore the results must be made unambiguous by regularization. One may employ the regulator method of Pauli and Villars, introducing auxiliary masses which make the expressions finite and gauge-invariant. At the end of the calculation the auxiliary masses are taken to be infinite. In the case of an extended nucleus this procedure is not necessary for (Za)n, n 2 5 since the decreasing nuclear charge form factor makes the integral convergent.

The explicit form of the Green function is constructed from the solutions of the Dirac equation with a cutoff Coulomb potential discussed in $2.3, namely Whittaker functions in the exterior and spherical Bessel functions (for model I nuclei) in the interior region.

Gyulassy (1974a, b) investigated the energy shifts due to the term a(Za)3 alone and a(&). for all n 2 3 for the 1 ~ 1 , ~ level near the diving point. He neglected contributions from I K \ # ~ . With a nuclear radius R=lOfm the critical charge obeys Za= 1.274 59. At 2a= 1.274 45 the l s lp energy is just above the negative continuum, namely Elsli2= -0.999. Here the energy shift due to vacuum polarization is AE3= 0.570 keV and AE3+- 1.150 keV. This demonstrates that the Uehling potential contributes most to the strongest energy shift (AE1) and higher orders do in no way qualitatively change the behaviour of the energy level.

The calculations of Gyulassy also verified the smooth transition of the K-shell charge distribution between undercritical and overcritical potentials.


An independent and somewhat different approach to vacuum polarization in strong realistic electric fields was taken by Rinker and Wilets (1973, 1975). They explicitly performed the summation in (2.48) using exact electron wavefunctions. Renormalization was achieved by subtracting the contributions linear in 2. Using a monopole approximation to simulate the U + U quasimolecule Rinker and Wilets (1975) near the diving point found an energy shift of -3.98 keV consisting of AE I= - 4-62 keV from the Uehling potential and AE13+ = + 609 eV, AE$+ = + 34 eV for IKI = 1,2.

Extending the Wichmann-Kroll results a third group treated the higher order vacuum polarization analytically with the approximation of vanishing electron mass. The results of Brown et aZ(l974,1975) on muonic atom energy levels are in agreement with those of the already mentioned work.

Thus we come to the conclusion that vacuum polarization, although rapidly increasing with the nuclear charge, does not behave anomalously near 2 and does not significantly influence the diving process (cf in this connection Migdal (1973a) who considers the Green function G in regions of locally homogeneous field strength, i.e. linearly increasing potentials).

The influence of electronic self-energy is known less reliably. Application of a series expansion of Erickson (1971) gives AE= +2.9 keV (Sof f et aZl974). Thus the self-energy correction is of comparable size to the vacuum polarization shift but of opposite sign. More reliable new results have been based on the work of Brown et aZ(l959). The non-perturbative calculation of Desiderio and Johnson (1971) has recently been applied to superheavy systems. I n this region self-energy increases abnormally. For a pure Coulomb potential it diverges at Za= 1. Taking into account the finite nuclear size Cheng and Johnson (1976) find E- + 12 keV approximately. Thus the energy shift due to the self-energy is approximately cancelling out that due to vacuum polarization.

2.6. Possible injluence of non-linear Jield effects The physics of very strong electrostatic fields permits the study of phenomena not

encountered in ordinary atomic physics. In the following section we will briefly study a few conceivable 'exotic' possibilities of modification of the usual QED,

First we will revisit the limiting-field electrodynamics best known from the theory discussed in the 1930s by Born (1937) and Born and Infeld (1934). This theory started from a Lagrangian density

(2.160)

formed in analogy to the relativistic mechanical Lagrangian which leads to a limiting velocity, c.

Generalizing the model, Rafelski et aZ(1972a) introduced for the electrostatic case a class of Hamiltonian densities (6" = 2 -E.D)

(2.161)

which reduce to the theory of Infeld and Hoffmann (1937) for n=O, of Born and Infeld (n = i) or to Maxwell's theory (n = 1). D = - 86"/aE is the electric displacement satisfying V.D(r) = p ( r ) . While the displacement D also superimposes linearly in


limiting-field electrodynamics this is no longer true for the field strength E. The interaction energy of a two-particle system is then computed according to

The non-linear theories of electrodynamics have the agreeable property that they lead to finite values of the self-energy of a point charge (when n < Q ) . Originally they were proposed in the hope of explaining the rest mass of the electron in the framework of classical electrodynamics. As long as n < & the model (2.161) leads to an upper limit of the electric field strength E determined by the parameter Eo(n).

T o find the effect of the modified electromagnetic interaction the Dirac equation was solved with (2,162) for hig1i-Z atoms using the Thomas-Fermi model. Insertion

5 00

- > h ‘0 - Lu

-500

Figure 12. Energy levels for superheavy atoms obtained from the limiting-field electrodynamics of Born and Infeld (full curves) compared to the result of linear theory. Although Zcr is increased diving will not be avoided (after Rafelski et al 1971).

of the limiting field E,,, derived from the electron rest mass leads to strong energy shifts (Rafelski et a1 1971, 1973). The diving point is shifted up to higher 2 (see figure 12). However, Soff et a1 (1973b) found from a comparison with experimental data of high precision (0.005%) spectroscopic measurements on the upper limit of Emax 2 1 . 7 ~ 1020V cm-1. This value is 140 times larger than the Born result, which is therefore completely ruled out

The realistic value of E,,, may shift the critical charge by, at most, two units. It is interesting to note that the limiting-field theory is a model of maximal vacuum polarization. This is illustrated in figure 13 which shows the effective field as a function of the applied field. I t led Soff et a1 (1973b) to conclude that vacuum polarization cannot alter the diving process in any significant way.

In general, limiting-field theories are not capable to prevent diving because the potential which is responsible for the binding energy of the electrons has no upper


Figure 13. Sketch of the effective field Eefi in dependence of the applied field E. Limiting- field theory is a model of maximum vacuum polarization.

bound. A limiting potential theory could provide this. It may be based on the Lagrange function

9= $ f k l f k l V ( A & 4 V ) -4Tjk#k(AZ Ai) (2.163) with

f k l = 8 k A l - a l A k and #k=J2:(m) dAk[V(A&4")]1/2.

Muller considered the parametrization

V ( A J U ) = (1 f (2.164)

Again, the decisive parameter (TO) has to be chosen so large, TO 3 1000 mz, as to eliminate any significant observable effects in high-2 systems (Soff et al 1974).

Another possible source of deviation from the normal, weak-field QED is the influence of non-linear terms in the spinor field, e.g.

y P (l+-eAp) $+m$+ Add$> $+ ~ ( Y ~ Y S $ Y ~ Y ~ $ ) $=o. (2.165)

The third-order terms can act as a kind of counter potential, most prominent in strongly bound systems where the electron density is highly localized. The values of XI and X Z , however, have to be very small to avoid inconsistency with spectroscopic data.

After discussing non-linearities in the electromagnetic and spinor fields one finally could think of a modification in the coupling. The usual electromagnetic coupling is introduced in the Dirac equation by minimal substitution p , + p , - eA,

[a .p+pm-(E- V ) ] $ = O . (2.166)

The Coulomb potential is the fourth component of a four-vector. Interactions of a different origin might couple with a scalar potential, namely

[a .p+P(m+ V ) - E ] $=O. (2.167)

The Dirac equation with a scalar l / r potential has been solved by Soff et a2 (19934. Its spectrum (figure 14) shows positive and negative energy branches, both electrons and positrons being attracted for V < 0. The levels asymptotically approach E=O. Thus, diving is avoided for a scalar coupling (cf Dosch et al 1971). The argument following equation (2.22) had shown that in the case of an electromagnetic potential


I Is

I

Figure 14. Solutions of the Dirac equation with a purely scalar l / r potential as a function of the scalar coupling constant g, i.e. [ a . p + P(m-g / r ) ] $ = E $ (after Soff et a1 1973 a).

binding energy drops continuously with finite slope as a function of potential strength. For a scalar potential equation (2.23) reads

If the ‘small’ component of the wavefunction becomes comparable to u1 in the potential well, the slope of E(2) will decrease. Of course, figure 14 has nothing to do with observed atomic energies. But it might be possible that a weak scalar interaction is present together with the Coulomb potential. The magnitude of its coupling constant compared to the electromagnetic coupling is limited from experimental knowledge to about 10-8.

Sundaresan and Watson (1972) had proposed scalar coupling caused by a massive scalar boson of about 8 MeV to account for discrepancies in muonic atoms (which have been cleared meanwhile).

At any rate, neither of the proposed phenomenological alterations of QED has been verified by experiment and neither will be able to give rise to essential new contributions even in very strong external fields.

2.7. Statistical description of the charged vacuum

In the previous sections the creation of a charged vacuum via diving of strongly bound electronic levels into the negative-energy continuum has been discussed. We mainly stress the region of field strength which is hoped to be in the scope of ordinary atomic physics. Then only a single (lslp, Zcr= 172) or at most a few (2pli2, 2s1p, Zcl.= 185, 215) levels are involved. It is of considerable principal and theoretical interest to study the opposite limit of a highly charged vacuum state.

Muller and Rafelski (1975) have considered a relativistic Thomas-Fermi statistical model which should be appropriate to describe the real vacuum polarization in very strong external fields.


The charge density of electrons is related to the Fermi momentum K F by

(2.169)

where kF2 has to be a positive quantity. Therefore a unit step function e(EF- V-m) is introduced in the relation between Fermi momentum and Fermi energy EF:

KF~=[(EF- V)2-m2] O(EF-V-m). (2,170)

These equations lead to the following ground-state expectation values of the charge density

(2,171) e (olp(x)l o)= -m [(E-V)2-??i2]3/2 e(EF-v-m).

As usual in the Thomas-Fermi model one can employ the Poisson equation

AV(r) = eprr(r) (2.172) linking the electrostatic potential energy V and the total charge density

P T = P N S p (2.173)

to obtain a non-linear second-order differential equation. This becomes

(2.174)

To describe ordinary neutral atoms one has to take EF=O. Neglecting V compared with m equation (2.174) is the usual Thomas-Fermi equation

2312 3x2

AV=epN-e2m -- V3l2 . (2.175)

In the present context our interest lies in the charge distribution of the autoionized states in the negative-energy continuum. Therefore the Fermi energy is chosen, EF = - m. Then equation (2,174) takes the form

(2.176)

Muller and Rafelski (1975) solved this equation prescribing an external charge distribution p ~ ( r ) of an abnormally large nucleus represented by a homogeneously charged sphere. They took Ro = roA1/3 for the radius with YO = 1.2 fm corresponding to the density of ordinary nuclear matter and assumed symmetry N= 2, i.e. A = 22.

The differential equation (2.176) has to be complemented with suitable boundary conditions. Since the vacuum charge will be locally confined (the autoionized positrons having escaped to infinity) the potential has to behave asymptotically like

e2

3 7r2 AV= e p N - -- (2mV+ V2)3/2 e( - V - 2m).

(2.177)

Here y is the total charge of the system (nucleus + vacuum), and 2- y is the screening charge. The second boundary condition which allows the determination of y is

(2.178)


Figure 15 shows the numerically calculated eigenvalues y as a function of ‘nuclear’ charge 2. The screened charge 7 starts on the curve y= 2 but when diving sets in at 2 values of a few hundred it increases much more slowly than linear. This means that the vacuum charge (curve 2-y) leads to nearly total screening of the pure nuclear Coulomb potential.

The statistical model is only justified if the vacuum charge is large and comparable in size with the external charge 2. It is interesting (and reassuring), however, that the exact single-particle results, namely the values of Zcr for the first (three) diving states denoted by crosses in figure 15, agree quite well with the Thomas-Fermi result.

The effect of the vacuum screening on the electrostatic potential is demonstrated in figure 16. It turns out that V reaches a limiting value if 2 becomes very large. The magnitude of Vlim can be calculated from the condition p ~ ( 0 ) = 0 or

vlim= - n l - (m2fT 37r2 Po) 112 z --== (943 -300 MeV ----. fm (2.179) 10

I I 1 102 10’. i o 5

Z Figure 15. Unscreened charge y and charge of the vacuum 2 - y as a function of the bare

nuclear charge 2. For very high 2 the screening becomes nearly total (after Muller and Rafelski 1975).

Contrary to the earlier conjectures V1im depends on the size and density of the pre- scribed nuclear charge distribution. In ordinary nuclear matter (ro = 1.2 fm) V1im % - 250 MeV. The reason for the potential limit is easily understood. With increasing Z the single-particle binding energy increases only linearly while the electron- electron correlation energy increases with 2 2 .

The charge distribution of p~ essentially obtains the structure of a dipole layer. Except in the vicinity of the nuclear surface the induced charge just cancels the nuclear charge (figure 16). The assumption of symmetric nuclear matter is thereby justified.

The screening of the nuclear Coulomb field leads to a limit of the repulsive energy H / A which has to be overcome when a nucleon is added to a large nucleus. Construct- ing a Hamiltonian from equation (2.176), Muller and Rafelski (1975) obtained the value H / A = 3/8 I Vliml= 112 MeV fm TO-1.

This result is of interest in connection with speculations on abnormal nuclear


Figure

0

I

c1 I E

- -0.L

d

.A-

0

k U

-0 .8

0

1 10 100 1000 r [ f m l

Solutions of the relativistic statistical potential equation (2 . 76) for nuclear charge numbers Z=600 (curve l), 1000 (2), 2000 (3), 5000 (4), IO4 (S), IO6 (€4, 106 (7). (a) The self-consistent potential; (b) the corresponding charge distribution of the vacuum; (c) the total charge densities, scaled with y. Note the large screening of the nuclear charge (after Muller and Rafelski 1975).

states. In the Lce and ’wick model (1974, see also Lee (1975)) the energy functional E[p] for nuclear matter obtains a deep second minimum with 130-500 MeV binding energy per absorbed nucleon. The above result indicates that abnormal nuclei of the Lee-Wick type could not be stopped from growing by the Coulomb repulsion.

3. Experimental tests: QED effects in heavy-ion collisions

3.1. Intermediate quasimolecules in heavy-ion scattering

I n $2 of this review QED modifications of atomic physics in the presence of highly charged nuclei have been discussed. The effect of main interest seems to require a value of 2 of at least ZCr, l8 = 172. Nature does not provide sufficiently large nuclei, if one discards the possible existence of abnormal condensed states. Even stable ‘ordinary’ superheavy nuclei (which cannot be produced artificially by fusion) in the

Quantum electrodynamics of strong jiela’s 26 1

region of the predicted ‘island of stability’ around Z= 164 would not create strong enough fields.

A way out of this dilemma is posed by the idea of using the field of two colliding heavy ions. At high enough bombarding energies the nuclei approach close enough so that the surrounding electron shells experience their combined Coulomb potential. Thus for a short period of time electronic quasimolecules with varying internuclear distance R(t) are formed. In the limit of vanishing R, corresponding to the point of closest approach for collisions at the Coulomb barrier, even the electronic structure of a quasiatom with combined central charge 21 + 2 2 can be reached. Quasimolecular states as a basis set for expanding the time-dependent electronic wavefunction in scattering processes have been applied for a long time (see, for example, the mono- graph of Mott and Massey (1965) and its earlier editions). Heavy-ion physics provides a means of directly testing the formation of quasimolecules. Measurements of the spectra of emitted x-rays, positrons and electrons allow us to draw conclusions on the shape of the molecular correlation diagrams (i.e. the dependence of binding energy on separation) and especially to investigate details of the diving process. These investigations will be impeded somewhat by non-adiabatic and smearing-out effects as a consequence of the finite collision time which is dictated by the Coulomb repulsion of the nuclei.

The idea of superheavy electronic quasimolecules has been proposed in inter- university GSI seminars during 1969 (see, for example, the references in the GSI seminar report by Mokler (1972)) and published by Rafelski et aZ(l971). A discussion of the adiabaticity of the electronic motion was given by Rafelski et aZ(1972b). For the system of U-U the collision time, defined as the time during which the nuclei advance from and recede to a separation of 100 fm, is about 1-8 x 10-19 s. Roughly assuming half the velocity of light for the K-shell electron velocity the relation of collision and orbiting time is of the order of

~ c 0 1 1 / ~ 0 r b 20. (3 .1 ) Therefore the electron configuration should be able to adjust to the nuclear motion and a molecular description of the scattering process is meaningful. The topic of excitation and rearrangement processes occurring in atomic collisions merits a thorough treatment of its own. We refer especially to the reviews of Madison and Merzbacher (19754, Iichten (1975) and of Scheid and Greiner (1977). K-shell excitation in heavy systems is discussed by Meyerhof and Taulbjerg in a forthcoming article. Here we restrict our discussion to a few7 points which are in our area of interest to test atomic physics for very strong fields.

A general description of the collision of complex ions would have to start from a many-body Hamiltonian for the system of nuclei and electrons and to solve the quantum-mechanical scattering problem. For an extensive discussion of the formalism see Smith et al (1975) who stress the analogy with the strong-coupling model of nuclear physics. Adiabaticity of the nuclear motion facilitates the solution, since one can apply the Born-Oppenheimer approximation and separate nuclear and electronic motion. A further simplification is achieved by using the semi-classical approximation. Here the nuclear motion is assumed to follow a definite trajectory, corresponding to a localized wave packet in the wavefunction of relative nuclear position. This treatment is justified if the Sommerfeld parameter (which can be viewed as the relation between distance of closest approach and de Broglie wavelength) 77 = 212aeZ/Av is large compared to unity. In the collision of heavy ions this condition is fulfilled very well


(for instance, 7 > 500 for U-U at the Coulomb barrier). I t is justified to use a Ruther- ford hyperbola for the nuclear trajectory. Even for light systems a classical path may be assumed, but an influence of the electronic state on the motion may be important (e.g. Gaussorgues et al 1975).

With the indicated approximations the problem reduces to the time-dependent Schrodinger equation

W Z , W ) ) I#(h R(t)))=i ; I#(l%, R(t))). (3 .2)

The nuclear separation enters this equation only via the time-dependent parameter R(t), and not as a dynamical variable. In principle one has to use antisymmetrized electron wavefunctions (Slater determinants) and a Hamiltonian including the electron-electron interaction

where ui are the coordinates of the electrons, p1, p~ are those of the nuclei with charge &e, &e. (This Hamiltonian is appropriate for the relativistic problem since the kinetic energy is described by the Dirac operator a.p + /3m. However, it does not contain retardation effects and magnetic energies since it is restricted to the static instantaneous Coulomb interaction.) As a first step when treating the innermost shells of heavy ions the electron-electron interaction may be neglected.

In view of the slow motion of the nuclei it is natural to use the eigenfunctions of the stationary one-electron two-centre Dirac (TCD) problem as a basis for the solution of the time-dependent problem:

[ U . P + Fm + Vdu, P1) + v2 (U, Pa>ll ?,(RI) = E,(R)l %(R))* (3.4) These basis functions Jcp,) form a complete orthonormal set which is parametrically time-dependent. Taking into account the variation of the energy eigenvalue the wavefunction will be expanded like

I # (t>> = c %(t> exp [ - i jt dt’~,(t’)l I ? a m ) . (3.5) U

By projection the time-dependent Schrijdinger equation leads to a coupled system of differential equations for the coefficients

ag= - c a, exp [-iG (-%(t’>--Eg(t’)) dt’l (Vg 1;l 9,). (3.6)

Note that the summation over /3 has to include continuum states. The operator a/at acts on the states I?,) only via their dependence on R(t). Since the will be molecular states oriented along the internuclear axis one has to divide the differentia- tion in a radial and an angular part:

where VR is the radial and o is the angular velocity. Thus in (3,7) transitions are caused by a radial as well as a rotational coupling. The latter originates from the non-inertial character of the used coordinate system. I t is also well known under the name of Coriolis coupling in nuclear physics. Both types of transitions, which


display different radial dependence and different selection rules, are important for electron excitation and ionization.

The coupled-channel problem could be solved numerically to obtain a complete description of the scattering process (in terms of the independent-electron approximation). There is, however, a complication since no simple asymptotic boundary conditions ( t -+ k CO) can be specified. This is due to the fact that the basis functions in (3.5) are eigensolutions of H ( R ( t ) ) in the asymptotic region. The represent stationary molecules while the real ions are moving. Thus the operator ajat has to effect a Galilean transformation and its matrix elements do not vanish as R -+ CO. The first to notice this problem were Bates and McCarroll (1958) who introduced travelling orbitals by multiplying their (atomic) basis functions with a plane wave factor

exp [i ( Pg v . r - $P&zt )] (3.8) where Pt = Zz(Zl+ 22)-1. In this way the spurious transitions at large distances are eliminated. Translational factors have been commonly adopted in atomic collision theory (their use has recently been discussed also for nuclear physics, see Pauli and Wilets (1976)). There remains a certain arbitrariness because factors of the form (3.8) give the electron a boost to the nuclear velocity. This is unrealistic in the interaction region where the electron correlates with both centres. By the introduction of an ad hoc constructed and optimized weight function f ( r , R) several authors (see, for example, Thorson and Levy 1969) tried to remedy this deficiency.

3.2. The two-centre Dirac equation (TCD)

I t is clear from the short discussion of the last section that any quantitative description of adiabatic atomic collisions must be based on the quasimolecular, i.e. two-centre wavefunctions. The solution of this kind of problem is essential for quantum chemistry; therefore it was treated very early in the history of quantum mechanics.

The non-relativistic one-electron Hamiltonian

has been solved by Teller (1930) and Hylleraas (1931) after the first approximate results of Heitler and London (1927). It is essential that the two-centre Schrodinger equation has two constants of motion. Trivially, one of them is the projection L, of angular momentum on the symmetry axis. A second operator commuting with the Hamiltonian has been constructed by Erickson and Hill (1949). In prolate spheroidal coordinates 4 = (rl + rz)/R, 17 = (rl - rz)/R, 9 the Schrodinger equation separates, producing two coupled eigenvalue equations which can be solved for the energy.

The TCD equation is more difficult to handle since there exists no orthogonal coordinate system in which it separates. Correspondingly, no second constant of motion besides the projection of the total angular momentum exists. The solution, then, can only be obtained by numerical methods. Except for a calculation of relativistic corrections in perturbation theory (Luke et al 1969), this problem has been approached only recently. Muller et al (1973b) treated equation (3.4) in prolate spheroidal coordinates and chose an expansion analogous to the non-relativistic case. The basis functions were

$ntsm (t, 17) =exp ( - ~ / 2 ) L m + E a ( x ) W * " 5 ( 1 7 ) xs. (3.10)


Here e5= J 4 for s odd (even) and x is a scaled coordinate, x= (t- l)/a. The variable a serves to satisfy the asymptotic behaviour of the wavefunction glr-texp [-(m2 -E2)1/2r] and therefore is energy-dependent. The matrix elements of the TCD Hamiltonian with the (non-orthogonal) basis (3 . lo) may be obtained analytically. The eigenvalue problem was then diagonalized numerically (for further details and results see Muller and Greiner (1976)).

In an actual calculation the basis, of course, must be truncated and it is difficult to judge the accuracy. In particular, the question of convergence is more critical than in the usual non-relativistic case because the Dirac Hamiltonian is not bounded from below the states of the negative continuum may be admixed.

Recently Rafelski and Muller (1976a) have solved the TCD equation by numerical integration. They have discretized one of the coordinates by of the wavefunction

a multipole expansion

(3.11) K K

Using a multipole expansion of the two-centre potential

V(r, R) = 2 Vi(r, R ) Pi (COS 19) 1

(3.12)

the radial equations become

The coefficients Axzs = ( x K ” 1 Pzl x 5 ~ ‘ ) are determined by angular momentum algebra. After the sum in equation (3.13) had been truncated at a sufficiently large angular momentumjmax the 2(2 jma, + 1) coupled differential equations were integrated. The energy eigenvalue E is determined by an iteration to produce vanishing of the wavefunction at a large distance.

This highly accurate numerical integration method is also convenient since it easily allows the use of modified potentials (3.12) which, for instance, may contain electron screening effects. The influence of finite nuclear radii too is easily calculated (see in this connection Marinov and Popov (197%) who used perturbation theory).

Figure 17 shows a numerically calculated diagram of the energy levels as a function of distance R for the relatively light system Ni-Ni. This type of ‘correlation diagram’ for the two-centre Schrodinger equation has been extensively discussed since the 1920s. The molecular states are classified by the good quantum number j , with eigenvalues Imj I =B, $, $, , . . which are named by U, T, 6, , . . . One often assigns the quantum numbers of the united atomic state ( R +O) to the molecular wavefunction to which it correlates (lslj20, 2p3/p, . . .). For identical nuclei the molecule has a further constant of the motion. Since the parity operator /3p commutes with the Hamiltonian and with j , one can distinguish even (‘gerade’) and odd (‘ungerade’) molecular states.

The principal behaviour of the molecular states can be understood by general symmetry arguments (see Barat and Lichten 1972). Of special interest are those points where the two levels approach each other very closely. If they belong to different symmetry they are allowed to cross. States of equal symmetry, however,


-2

- > 2 -10 I

Lu

- 5 0

R [ f m l

- 3 --- 200 1000 10000

30

3 z p

- 2 s 2p

- I S

Figure 17. Relativistic correlation diagram for the system Ni-Ni showing the dependence of binding energy on internuclear distance for several quasimolecular states (after Muller and Greiner 1976).

may not cross due to the Neumann-Wigner rule (which is not applicable in the non- relativistic case, see Helfrich and Hartmann (1972)). This rule is correctly obeyed by the numerically evaluated molecular levels. As a typical example we examine the 2slp7 and 3d3p.7 states near 8000 fm which are repelled and do not cross (see figure 17).

This interesting behaviour is only of limited practical consequence. A closer analysis of the character of the wavefunctions actually reveals that they cross. In any collision with finite velocity the strong dynamical coupling will cause the electrons not to follow the adiabatic correlation diagram but to follow a ‘diabatic’ molecular orbital. An explicit example of this transition was treated for the asymmetric system N e 0 by Taulbjerg and Briggs (1975).

Figure 18 shows the level diagram of the heavy asymmetric system I-Au with 21 -i- 2 2 = 132. The states do not possess good parity and therefore many pseudo- crossings occur.

The level diagram of U-U, which is of greatest importance to us, is presented in figure 19. Comparison with the Ni-Ni case shows drastic differences due to relativistic corrections. Contrary to the light quasimolecules in U-U all inner-shell levels strongly gain energy when the internuclear distance sinks below a few hundred fermis. This is particularly noticeable for the state which normally ascends in energy to a weakly bound L shell of the united atom. Relativistic binding is also responsible for a level crossing of the 3 ~ 1 ~ 2 0 and 2 ~ 3 ~ 2 0 and a pseudocrossing between 2~3120 and 3 p l p . It is quite clear already from these results that any simple kind of scaling of non- relativistic calculations (e.g. H+-H) will not work for the supercritical systems.

A result vital for the discussion of positron production in $3 is the magnitude of the critical distance, Rcr, where the lslj20 level joins the lower continuum. Here the autoionization process of positrons sets in just as in our previous thought experiment where 2 had been increased beyond ZCr. Numerical solution of the TCD equation leads to Rcr = 37 fm for point-like nuclei and Rcr = 35 fm for extended nuclei. Figure 20 gives a magnified picture of the 1~1120 and 2p1,zu states of three supercritical

266 'J Reinhardt and W Greiner

l 3 2 x

Figure 18. Correlation diagram for the superheavy asymmetric system I-Au (after Muller and Greiner 1976).

molecules U-U, U-Cf and Cf-Cf. For the latter two systems the ls1/2a level dives at 48 fm and at 61 fm, respectively. The 2p1/20 level remains undercritical (for distances below the nuclear Coulomb barrier) for U-U and reaches - m at Rc, = 16 fm for U-Cf and 25 fm at Cf-Cf.

These critical distances are at variance with the values obtained by Popov and co-workers (Popov 1972a, b, 1974a, Marinov et al 1974, Marinov and Popov 1974, 1975a, d), who have employed several approximation methods to determine the critical distance. The TCD problem has been solved analytically in the limit of low super- criticality 8 = (& + Zz - Zcr)Zcr-l+O. Popov further devised a method to obtain the

Figure 19. Correlation diagram for the overcritical system U-U. At about Rcr= 35 fm for extended nuclei and 37 fm for point-like (broken curve) nuclei the l so state reaches E= - m (after Muller and Greiner 1976).


I Coulomb barrier

Figure 20. Energy of the most deeply bound states lso, 2~1120 at small internuclear distances. Broken curves belong to extended nuclei. The dependence of the critical distance Rcr on total nuclear charge is demonstrated. Curve A, U-U (Zi=Zz=92) ; curve B, U-Cf ( 2 1 = 9 2 , 2 ~ = 9 8 ) ; curve C, Cf-Cf (Zi=Zz=98).

two-centre solution at E= -m from a variational principle. He started from basis functions embodying a singularity of the form (for l s l p ~ )

#-([2-~2)-~ with u=l-[l-(~Zcu)2]1/2 (3.14)

which emerges in the relativistic two-centre problem. Variation was performed on a set of one, two or three trial functions. The convergence of this procedure is not quite clear. The latest results are Rcr= 51 fm for U-U and 77-7 fm for Cf-Cf (point nuclei). Since Popov contends that these numbers form a lower limit to the true Rcr, there is a clear disagreement with our result. It seems to us, however, that the good agreement between two independent methods of numerical solution of the TCD equation stands in favour of the above proposed numbers (see also Rafelski and Muller 1976b).

The described relativistic MO are important for many applications in heavy-ion collisions, e.g. excitation and ionization of electrons, quasimolecular x-rays, deviations from Rutherford scattering due to effects of electronic molecular binding (Schafer and Soff 1976) and, finally, autoionization of positrons. For many purposes the one- electron approximation is quite satisfactory. The electron-electron interaction could be incorporated by a relativistic extension of the two-centre Hartree-Fock or Thomas- Fermi method. Larkins (1972) published the first quasimolecular non-relativistic Hartree-Fock correlation diagram. These calculations are very time-consuming on the computer. They are perhaps ‘too accurate’ since they assume total relaxation of the electron shell at every distance R, leading to a discontinuous appearance of the level diagrams.

The quantum-statistical approach for molecules has recently been revived by Eichler and Wille (1965). Gross and Dreizler (1976) obtained a single-particle Thomas-Fermi potential with the object of constructing two-centre orbitals which include electron screening.

19

268 'jf Reinhardt and W Greiner

Especially when dealing with the outer-shell region, however, one has to bear in mind that the adiabaticity condition is not fulfilled in heavy-ion collisions. Since the mean number of electron-electron collisions (see Hofmann et al 1977) is too small, it is not easy to determine to what extent an average potential does build up.

3.3. Quasimolecular x-ray spectroscopy

The spectroscopy of radiative transitions is a tool generally used to extract information on energy levels and wavefunctions. So it may also be applied to quasimolecules transiently formed in heavy-ion collisions. During the last few years this field of research has flourished, both experimentally and theoretically.

Besides gaining valuable insight into the dynamics of atomic excitation and de- excitation processes, the ultimate aim of these studies is to push atomic physics forward into a new region and to investigate the electronic states in superheavy systems. Ideally it might even be possible to follow the change of binding of the l su level with increasing nuclear charge and to identify the diving process by looking at the x-rays produced. It is, however, quite difficult to extract detailed information from the measured spectra due to the finite collision time (Heisenberg broadening). Furthermore, background processes may seriously interfere with the MO effect (see 93.5). We content ourselves with reviewing some of the more relevant work on non- characteristic radiation published recently. The present state of the art is reflected in the conference proceedings at Seattle, 1975 and Freiburg, 1976.

From a theoretical point of view photon emission is easily incorporated in the formalism outlined in $3.1. The electromagnetic potential is introduced into the electronic Hamiltonian via minimal coupling p + p - e/c A . Expanding the time- dependent wavefunction in a way analogous to equation (3 .5 ) one has now also to include photon channels. Since the radiative coupling is a weak perturbation it is sufficient to include one-photon states:

x exp [ - i s" (E,(t')+ 0) dt'] 19,) 0) (3 .15)

where denotes the amplitude for finding the electron configuration I?,) together with an emitted photon with wavevector k and polarization p. By projection one obtains coupled-channel equations:

with the interaction f?r,d=j.A= -ea.A. (3.18)

The differential cross section for photon emission is simply the scattering cross section


multiplied by an incoherent sum of emission probabilities corresponding to different final states and averaged over photon polarization:

(3.19)

Equation (3.17) shows that the emission spectrum is determined by a Fourier transform with variable frequency. To be more specific, let us neglect the small second term on the right-hand side of (3.17) and employ the dipole approximation (however, see below). The vector potential reduces to A = e p and the photon amplitude becomes

a

With the identification (3.20)

(VI1 j 1%) = - ie(rp/?l [Ho, rl I 9J,> = - ieUga(rpj31YI rpa) (3.21) the amplitude C is seen to be a Fourier transform of several time-dependent factors: the dipole strength uPa(rpI1rl rp,), the rotation matrix W( eN(t)) mediating between laboratory and rotating coordinate systems, and the amplitude aJt) describing the time evolvement of the electron configuration.

Even apart from these varying factors the spectrum is influenced by the phase integral s” (UIa(t’) + U) dt‘ which contains the energy difference waa(t) =E,(t) - EI(t’) changing with internuclear separation. The effect of a time-dependent transition frequency has been of interest for a long time in connection with collisional line- broadening (see Weisskopf 1933). I t is obvious that in heavy-ion collisions it will lead to a broad continuum of non-characteristic x-rays located between the corresponding lines of the separated and united atomic systems (figure 21). Since the intensity differences between characteristic and non-characteristic emission are very large (exceeding ten orders of magnitude, for example) the spectral wings of the pure atomic lines have to be treated correctly. A normal Lorentzian decreasing like U-2

could dominate the whole MO spectrum. This is an artificial contribution, however, caused by the assumption that the excitation of a hole takes place suddenly, thus introducing high Fourier frequencies. An improved model calculation of Anholt (1976) yielded U-11 instead of 0-2 for the far wing.

Before proceeding with the theoretical development let us quote a few of the experimental references to make sure that the described effects do exist in nature. After a ‘prehistoric’ result of Coates (1934) which was not pursued further, the first observation was due to Saris et a1 (1972). Mokler et a1 (1972) shortly afterwards observed M-shell radiation in the superheavy system I-Au (Z= 132). Meyerhof et a1 (1973) pioneered the investigation of K radiation by studying Br-Br collisions. He was followed by Greenberg et a1 (1974) with studies on the Ni-Ni system in particular and by Wolfli et aZ(l975, 1976) with experiments on Al, Ca, Fe, Ni, etc.

Studies of the K radiation in superheavy systems were first published by Frank et aZ (1975, 1976a, b) and by Gippner et a1 (1974). This group also observed a sub- division of the non-characteristic x-ray spectrum into two distinct continua which seem to be caused by the separate filling of molecular lso and 2pa holes (Heinig et a1 1976). A convincing proof for the quasimolecular nature of the observed continua was given by Meyerhof et a1 (1975) and, following them, by Folkmann et a1 (1976). Using the Doppler shift they determined the velocity of the source of x-ray emission.

270 J Reinhardt and W Gyeiner

P h o t o n energy ikeV) Figure 21. Quasimolecular K x-ray spectrum for Ni-Ni collisions measured by Greenberg

et al (1974). The photon energies lie above the characteristic K lines of the Ni atoms.

While the characteristic lines are shifted in accordance with the respective ion velocities the MO continuum source moves with the common centre-of-mass velocity.

A necessary prerequisite for the production of MO x-rays is the presence of an inner-shell vacancy (if not otherwise stated we always refer to the K shell and to K x-rays). Two mechanisms have been postulated for its formation. In the one- collision process the hole is produced via excitation or ionization in the same collision in which it decays later on. The two-collision model, on the contrary, assumes the creation of an outer-shell hole in a first collision. By a Fano-Lichten-type electron promotion this hole is transferred to, for example, the K shell of the separated atom and in a second collision becomes an inner-shell (e.g. lsa) hole. The latter process clearly depends on the density of the target material and is only possible in a solid target. Both collision mechanisms have been supported by experiment (see e.g. Laubert et a2 1976, Heitz et aE 1976) and in some region there may be a coexistence between them. For our purpose, however, it is important to notice that a two-collision process is impossible for very heavy systems since the radiative lifetime becomes small compared to the travelling time between neighbouring atoms in the crystal.

Theoretical evaluations of the MO spectra have, up to now, mostly assumed a two- (or many-) collision mechanism. In principle the occupation amplitudes a,(t) should be taken from a solution of the coupled-channel problem and then Fourier- transformed. Less detailed calculations assume a constant or exponentially decreasing amplitude, the latter due to radiative or non-radiative transitions (Smith et al 1975). As a first step the radiation is calculated for a fixed impact parameter. Theory pre- dicts intensity oscillations, the period of which should be non-equidistant and increase with scattering energy (Lichten 1974, Macek and Briggs 1974). The reason is an interference between radiation coherently emitted at different (i.e. ingoing and out-


going) parts of the trajectory. These oscillations so far have not been observed (Tserruya et aZ 1976). A reason for this might be that the electron-electron hole amplitude a,(t) is strongly t-dependent and asymmetric with respect to the point of closest approach (which is usually t = 0), i.e. a ( - t ) # a(t). Since coincidence measurements have been difficult to perform up to now research has mainly concentrated on the singles spectra, integrated over the impact parameter. The oscillations are then smoothed out.

The simplest method to account for the lower energy region of the MO spectrum is the quasistatic or stationary phase approximation. This corresponds to the assumption that at every point of the trajectory only photons of the momentary transition frequency wBa(R(t)) are emitted. The integral in (3.19) is approximated by

j d t e x p [i f l (wg,( t ’ )+o) d t ’ ] f ( t ) -+(;)”’ z f ( t i ) ( 1 dt I t - t i )-I” (3.22) i

where ti is defined by wg,(ti)+w=O. The quasistatic approach is not able to describe the full MO spectrum since it leaves out broadening effects and transitions induced by the collisional dynamics. For example, it leads to an unphysical diver- gence at an energy belonging to the point of closest approach (see (3.22)). An exact treatment of the Fourier integral leads to cross sections uniformly decreasing with w . The high-energy tail (beyond the united-atom limit) has the approximate shape (Muller 1975)

d a -----E w exp (- 42I’) . dw

(3 .23)

The width I? can be simply related to the parameters of the colliding system. Inter- estingly, the width is proportional to the square root of the projectile velocity, contrary to simple A E A t Heisenberg broadening arguments which indicate a linear dependence. Betz et a1 (1975) have obtained a simple analytical formula for the MO tail which quantitatively describes the shape and magnitude of a large number of systems. Since the MO spectra are approaching an exponential shape and have no definite end point only the slope can be taken as a measure for the total nuclear charge (Muller 1975).

Fortunately, there is a simple way to derive more information from the experiments. Measurement of the angular distribution of non-characteristic radiation provides further details of the collision process. Originally Muller et aZ (1974) and Muller and Greiner (1974) suggested the investigation of the photon-energy-dependent anisotropy - -

7 ) s da/dw( 0 = 90 ”) da/dw( e= 0 ”) (3 ,24)

which should exhibit a characteristic behaviour near the united-atom energy due to Coriolis-induced radiative transitions. This work stimulated a great number of experimental and theoretical investigations. However, while it is clear that a rotation of the internuclear axis has to produce an additional current, the originally proposed coupling $2 x v . A does not completely describe this effect. A more detailed analysis was given by Gros et aZ(1976b). At any rate, the induced effect is too weak to account for the large observed anisotropies. The first of these measurements were made by Greenberg et aZ(1974) and for M-shell transitions by Kraft et aZ(l974). Greenberg’s results and those obtained later by a large number of other groups all exhibited a


large peak of positive anisotropy which peaks in the vicinity of the united-atom transition energy (see figure 22). At present there does not exist an a priori theory to explain this behaviour. However, Muller and Greiner (1974) have already noted that the anisotropy strongly depends on the specific admixture of dipole transitions from states with different angular momentum quantum numbers. This alignment is held responsible for the observed angular distribution of MO x-rays (Smith and Greiner 1975). T o obtain good agreement with experiment one of the dynamical couplings

8ok

L O

0

Photon, energy ( k e V J

Figure 22. X-ray energy dependence of the MO emission anisotropy for several collision systems measured by Wolfli (1976). Note the broad bumps above the respective united-atom energy. X , Al-Al (25 MeV); Ca-Ca; A, Fe-Fe; 0, Ni-Ni (40 MeV); +, C1-Ni; 0, Fe-Ni (40 MeV); 0, Kr-Zr (200 MeV).

of equation (3.15) has to be included; the strong rotational coupling between 2pa and 2pn at close distances. This reflects the fact that the electron cannot follow the swift rotation of the internuclear axis at small distances. Parametrizing this 'slippage' effect and assuming a certain alignment Betz et al (1976a) were able to explain the experimental data of Greenberg (Ni-Ni) very well. A recent calculation of Gros et a1 (1976a) using perturbation theory and restricted to slow collisions and light non- relativistic systems verified those results. Assumption of an empty 2p77 subshell produces the characteristic anisotropy peak. For the system H+H a calculation of Briggs (1976) leads to a similar distribution. I n addition to the alignment it is neces-


sary to treat correctly the various interfering paths which lead to the same final configuration.

The angular distribution may leave a door open for the quasimolecular spectroscopy of superheavy two-centre orbitals (Muller et a1 1975). I n a very thorough systematic study Wolfli (1976) has investigated the position of the MO x-ray asym- metry peak (defined by the point of maximal positive slope) and its dependence on the total nuclear charge 2=21+22. For all included systems (with charges between 26 and 94) an accurate scaling with 2 2 is observed so that the peak position always lies slightly above the united-atom energy (see figure 23). However, experimental problems largely increase when entering the superheavy region due to increasing background and small cross sections. It remains to be seen how far the investigations can be followed.

Atomic number

Figure 23. Systematic dependence of the position of the anisotropy bump of MO K x-rays on the total nuclear charge. Note the good agreement with the united-atom transition energies (from Wolfli 1976). Full curve, E (ICaz) ; broken curve, E (Kal).

Perseverance here might be rewarded by the observation of an effect which is probably a unique attribute of the collision of very heavy ions. Since the fast, highly charged nuclei constitute extremely high current densities they are the source of a very strong magnetic dipole field. Ir, a small region the field strength is of the order of 1014 G. Since the inner-shell electrons too are confined to a small volume they will experience a large Zeeman splitting. Rafelski and Muller (1976a) investigated this effect coupling the vector potential

(3.25)

into the Dirac Hamiltonian (this choice of A neglects corrections due to a proper gauge transformation). Matrix elements of the interaction Hamiltonian, leading to


energy shifts of different sign for the two lso electrons with opposite spin orientation, have been calculated with the numerical TCD wavefunctions. For the system U-U the magnetic splitting of lso may exceed 10% of the rest mass while the 2pljzo state is split even stronger. This energy difference of spin-up and spin-down levels may perhaps allow conclusions about the excitation process since it leads to a different ionization and to a polarization of the final atomic characteristic K x-rays. To reduce the background experimental investigation of the magnetic splitting should be done with 2osPb-208Pb.

At the conclusion of this section it is appropriate to mention an effect typical for atomic physics in the superheavy region; the breakdown of the long-wavelength limit together with an unusual behaviour of the radiative transition rates. With increasing 2 the binding energy rapidly increases while the electron cloud shrinks. The photon wavelength eventually will become smaller than the extent of the electron shell so that the dipole approximation exp (ik.u)= 1 will become insufficient. A complete multipole expansion must be applied leading to matrix elements with spherical Bessel functions complicating (especially two-centre) MO calculations. Several different multipoles can contribute to a single transition. While radiation rates increase with 2 4 in the non-relativistic region this scaling does not extend to superheavy atoms. Anholt and Rasmussen (1974) and Soff and Muller (1977) have calculated radiation matrix elements with relativistic Hartree-Fock wavefunctions. They find, for example, that the 2p1p,-ls1jz transition rate has a maximum at about 2=150 (with a lifetime of 0-52 x 10-17 s) and then reduces again roughly to the value of uranium at 2=170. This comes about since the 1 ~ 1 , ~ wavefunction col- lapses and its overlap with outer states is strongly reduced. Other transition rates behave even more irregularly, displaying several extrema and zeros. Normally exotic or forbidden transitions can become dominant due to relativistic binding. One has to bear in mind this behaviour when dealing with x-rays of superheavy atoms or quasimolecules.

3.4. Decay of the neutral vacuum in heavy-ion collisions

The decisive experimental test for the diving of strongly bound electronic states in heavy-ion collisions will be the observation of emitted positrons. Apart from dis- turbances from background processes (§3.5), a complete dynamical calculation of quasimolecular positron formation mechanisms in, for example, U-U collisions is not available at present. All previous investigations assumed a certain probability LO (which was held constant) for the presence of a K hole. Except for systems of two naked nuclei or a naked projectile and an atomic target, or for the possibility of nuclear conversion during the collision, unfortunately no K hole is originally available. It has to be created dynamically via Coulomb excitation. Postponing this problem we at first assume a constant value of LO = 0.01.

In a very slow and truly adiabatic collision positrons could only be produced via the discussed autoionization process. Theory then becomes particularly simple since a static prescription can be applied (Peitz et aZ1973, Popov 1971a, b, 1974a, b, Marinov and Popov 1975a). According to Peitz et a1 (1973), the probability per time interval to emit a positron of energy E at a given distance R is governed by the width of the quasistationary level


in accordance with the results of $2.4. The differential cross section for positron production is obtained from the emission probability by a time integration along the hyperbolic trajectory. We have

with W ( E , 0) = j?tt,,p [R(t, e), E ] dt.

(3.27)

(3 -28)

This result may be improved by considering the filling up of holes during the collision which leads to a small reduction and smearing out of the spectrum. A simple estimation of the width I? compared with collision time reveals that only a small fraction of holes will decay.

Using the monopole approximation for r ( R ) and AE(R) Peitz et a1 (1973) performed the quasistatic calculation and obtained a total positron cross section of

Positive-energy continuum t 8 8

t !

I -m F--

1s;n

----U- - - - - - )$!jO.. Negative-er---,, continuum

Figure 24. Dynamical processes connected with positron production in overcritical heavy-ion collisions, A, B : electron excitation and ionization; C : spontaneous autoionization of positrons; D : induced decay of the vacuum; E: direct pair creation.

2-8 x 10-5 b for U-U and 10-3 b for Cf-Cf. The positron energy distribution exhibits a very sharp peak at the energy which belongs to the point of deepest diving where the radial velocity vanishes. A behaviour of this kind would be very desirable in order to identify the effect experimentally. Popov and co-workers obtained comparable quasistatic results employing two-centre solutions valid in the limit of low super- criticality. Also in their model 1a(E)12 in equation (3.26) is replaced by a delta function.

Further investigations have turned out that the quasistatic results will be grossly modified by dynamical effects. T o illustrate this point figure 24 schematically depicts several types of transitions encountered during the collision process. A and B denote the Coulomb excitation or ionization which produce a hole in the lsa level. This hole may be set free spontaneously (C) during the diving without increasing the energy. Due to the scattering dynamics energy can be drawn from the nuclear motion leading to non-adiabatic filling of the hole even at distances larger than Rcr. This effect may be called an induced transition and will enhance the cross sections, From the shortness of collision time, which is about 2 x 10-21 s, and the corresponding large energy uncertainty the importance of induced transitions is evident. Their influence


may be seen in analogy with the dynamical broadening of MO x-ray spectra (except for the energy threshold for positron creation).

As discussed earlier the description of positron production starts from the time- dependent Schrodinger equation (3.2). In the expansion (3.5) of the wavefunction positron states have to be included, Special care has to be taken of the region of diving, R < Rcr, - tcr < t < tcr. To follow the time development of the diving state we change the representation at R = Rcr and expand the overcritical wavefunction in the stationary basis 1Facr > = Iya(Rcr) > . Time dependence is then contained in the R-dependent Hamiltonian H(R) rather than in matrix elements of the operator, a,/& Coupling to the positron levels is quite weak, so that first-order perturbation theory gives good results. Restricting our discussion to the interaction of 1s level and negative continuum the positron amplitude is determined by the integral

C ( E ) = j?m dtexp [i j!!, (E-E18(t’)) dt’-+ j?-m r(t’) dt’] ME(t) (3.29)

with the matrix element

(3 I 3 0 )

We use the definitions

E(R)= (Ycrl V(R)I ycr), VE= (3LEcrl Av(R)I vcr) (3.31)

where AV(R)=H(R) -H(Rcr) is the overcritical potential. The decay width I’ in the exponential may be calculated from an averaged coupling matrix element I? ( t ) = 2 7 ~ I V(t)lz (see Smith et aZ 1975). Having calculated the amplitude C ( E ) the positron cross section follows immediately from (3.25) with the probability W(E, 0) dE= [ C(E)12 dE for producing a positron of (total) energy between E and E+ dE.

In the work of Smith et aZ(l974) the width of the diving level has been extracted numerically from the one-centre resonance. Figure 25 shows the scaled width y ( E ) defined by y ( E ) =277 I V~12 [ . 2 /~2(R) ] . This quantity is nearly charge-independent and may be conveniently parametrized. Note the influence of electrostatic repulsion at low kinetic energy, E,.

0 O . L 0.8 1.2 EP (MeV1

Figure 25. Dependence of the scaled width y on the diving energy E, obtained from the one- centre phase-shift analysis. y ( E p ) clearly is independent of the amount of overcritical charge. +, 2=184; 0, 2=196; A, Z=208 (after Smith et al 1974).


The resulting energy distribution of positrons obtained by Smith et a1 (1974) for the system U-U is exhibited in figure 26. Here the probability W has been divided into two parts. The integral (3.29) was calculated separately with the limits - a~ < t < - tcr, tcr < t < CO (pre- and after-diving, WPA) and - tcr < t < tcr (during diving, WD). Coherent addition of these contributions yielded the total probability WT. The curves demonstrate that, after all, most of the positrons are emitted in the overcritical region, especially at higher positron energies. The variation of ion energy or scattering angle has no essential influence on the shape of the spectra. Their decrease at high energies is mainly governed by the Fourier transform of the changing two-centre potential, i.e. by the characteristic frequencies of the nuclear motion. The availability of high Fourier frequencies is the reason for strong induced transitions and even brings about direct pair creation between upper and lower continuum (process F in figure 24).

1

Figure 26. Positron emission probability W(Ep , 0) for U-U collisions at 812 MeV CM energy. The total probability WT is divided into contributions from the trajectory regions during diving (R<Rcr ) , WD, and before or after diving (R>Rcr ) , WPA (after Smith et al 1974).

Integration over impact parameter leads to a further broadening of the energy distribution. The cross section du/dEp is given in figure 27(a) at several values of the distance of closest approach (note the logarithmic scale) while figure 27(b) shows the total cross section for positron production. u(Ep) has a kind of threshold at an ion energy (300 MeV) which corresponds to a distance of closest approach of the order of Rcr. Figure 28 demonstrates how the positron production increases with the amount of overcritical charge brought into the collision.

As indicated by the described results dynamical autoionization of positrons in heavy-ion collisions leads to rather broad and not very characteristic spectra. Hence the experimental study of this process will demand careful systematic measurement of the dependence on nuclear charge, icjn energy and positron energy. This is particularly important since strong background processes will be present.

The theory also has to be refined. Up to now a fixed K-vacancy probability, LO, was assumed. Extrapolations of this value ranged from 10-5 (Meyerhof 1974, Foster


Figure 27. (a) Differential cross section for positron emission in U-U collisions du/dEp for several impact energies given by the distances of closest approach Ro(fm) given for each curve. (b) Total cross section as a function of CM ion energy (Lo=O*Ol) (after Smith et a1 1974).

et al 1976) to 10-1 (Burch et aZl974). Apart from the uncertain magnitude a reliable calculation has to consider the variation of the K-hole amplitude with impact parameter and internuclear distance.

Betz et aZ (1976b) recently treated this problem in perturbation theory. They consider the coupling &/aR between lsa and the higher states 2s0, 3sa, . . . . Rotational coupling - iw . j may be neglected since it correlates only with 3 d ~ , . . . and since the matrix elements vanish at small distances. The radial coupling was calculated using exact TCD wavefunctions and the identity

(3.32) (qn 1 a/aR( qm) = (Em - G)-l (w 1 ~J'Tc/~RI p)nt).

0 1000 2000 Ep I k e V i

Figure 28. Positron emission cross section dependence on the amount of overcritical charge (Lo = 0.01). ( U ) Energy spectrum do/dEp at Ro = 15 fm, distance of closest approach. (b) Total cross section plotted against distance of closest approach, Curve A, Cf-Cf; curve B, U-Cf; curve C, U-U (after Smith et uZ 1974).


20 50 100 200 500 1000 2000 50 100 200 5h 10-2 / L 1 _ _ l - L 1 _ _ _ l -

R (fm) Figure 29. (U) Radial coupling matrix elements of the 1su to the 2su, 3so and 4su quasimolecu-

lar levels in U-U as a function of internuclear distance. The broken curves are taken without translation factors, the full curves with translation factors. For comparison the chain curve shows the l s u - 2 ~ ~ matrix element in NI-Ni scaled to U-U. (b) The radial coupling matrix elements between l s u and positive-energy continuum states calculated in monopole approximation (energies are in electron masses) (after Betz et al 1976b).

Figure 29 shows the matrix elements between l s u and 2s0, 3so for U-U. The full curves include the influence of translational factors. Then the coupling vanishes rapidly beyond 2000 fm where the two K shells begin to influence each other. Par- ticularly interesting is a comparison with the broken curve which is an extrapolation from the weakly relativistic system Ni-Ni. Obviously the rapid increase of matrix elements at small distances is a special feature of superheavy systems. Since a point- charge limit does not exist here the wavefunctions are very sensitive to variations of R. This results in a strong Coulomb excitation at small distances. The lsa vacancy probability for a head-on collision of U-U is shown in figure 30 as a function of time. The various curves correspond to different positions of an assumed initial hole. For the coupling to higher states multi-step channels have been added up coherently.

N 0

0.10 1 I

300 100 15 100 1000 3000 R I f m l n i i i i t i

Figure 30. Total probability for Coulomb excitation of a I su electron in U-U at &ab= 1600 MeV into a vacant nso level and into the continuum (&U) at zero impact parameter. One-step and multi-step channels are included (after Betz et aZ1976b).


Also, excitation to the positive continuum has been taken into account. Since these states certainly will be unoccupied the Coulomb ionization result poses a lower bound to the vacancy probability. No relativistic two-centre continuum wavefunctions are available. Therefore the ionization was calculated in monopole approximation. This treatment is justified by a surprisingly good agreement of one-centre and two-centre matrix elements (better than 2% for R e 400 fm) between bound states.

The vacancy probability starts to rise near the distance of closest approach and soon becomes appreciably large. For example, at 1600 MeV and zero impact parameter the probability of having a lsa vacancy in the diving region is predicted to be larger than 0.08. With decreasing ion energies the excitation falls off very steeply until at tandem energies (Elab<200 MeV) they are much less than 10-6. Also, an increase of impact parameter by 20 fm reduces the K excitation by about a factor of two.

These results have to be incorporated in a fully dynamical evaluation of positron production in supercritical heavy-ion collisions. The shapes of the spectra certainly will be altered. However, strong Coulomb excitation can already be seen to guarantee that the cross sections are large enough to be experimentally observable.

3.5. Background effects

Several processes are known to produce x-rays and positrons in heavy-ion collisions. Some of them will be able to severely disturb the observation of quasimolecular phenomena and thus impede the investigation of QED of strong fields. Here we will report on two particularly interesting and important background effects, nuclear bremsstrahlung and conversion of Coulomb-excited nuclear states.

Nucleus-nucleus bremsstrahlung leads to rather small radiative cross sections. Its energy dependence, however, is often weaker than that of electronic processes like MO radiation, radiative electron capture (Kienle et aZ 1973), radiative ionization or electron bremsstrahlung. Therefore there exist combinations of projectile and target for which the high-energy tails of the measured spectra are dominated by nuclear bremsstrahlung.

Intensity and angular distribution of this radiation may be obtained from a straightforward classical calculation (Malkov and Shmushkevich 1961), provided that the Sommerfeld parameter 7 is large. The resulting radiative cross sections for coincidence and singles experiments are presented by Reinhardt et aZ(l976). Employ- ing a multipole expansion up to second order the cross section integrated over impact parameter has the general form

Here M is the nucleon mass, AR is the reduced nucleon number, and Y = W / W O ,

where W O = (2E/(z12ae2)) ( ~ E / M A R ) ~ / ~ is a reciprocial collision time. fA = Z I / A ~ ~ +( - l)A Z2/AzA denotes the electric multipole moment of order A. The g ( v ) are


Kiy(v) is the MacDonald function of order iv. In general the multipole series expansion (3.33) converges rapidly. However, the

first contribution ( A = 1 ) is multiplied by the dipole factor (Z1/A1 - which is small for all heavy systems or even vanishes for identical projectile and target nuclei. Therefore the quadrupole radiation plays an important role. Even if the dipole radiation is much larger than the quadrupole contribution the interference term, which originates from their coherent superposition, influences the angular distribution of x-rays. For example, in the slightly asymmetric system 58Ni-60Ni a fonvard- backward anisotropy of about 5 is introduced (at Elab = 60 MeV, Ex = 30 keV). Recently Trautvetter et aZ(l976) observed this predicted constructive and destructive interference effect for several asymmetric collision systems like 130-58Ni. The results of this experiment are shown in figure 31. This proves that nuclear bremsstrahlung can contribute to the MO x-ray spectra (especially to the anisotropy) and should be subtracted according to (3.33). Figure 32 demonstrates the general dependence of do/dEx on nuclear charge number for symmetric (figure 32(a)) and representative asymmetric (figure 32(b)) systems. The cross section was evaluated at the K, united- atom transition energy, and therefore it decreases with increasing 2, despite the factor of 2 4 .

I n fast collisions of highly charged nuclei a certain fraction of x-rays with energy above 2mc2 is produced. Fourier frequencies larger than the e+e- threshold are well contained in the nuclear motion. Using the bremsstrahlung spectrum and the well known conversion coefficients / ? E ~ ( W ) for electron-positron creation the total pair production cross section via bremsstrahlung can be calculated according to

(3.34)

Using the conversion coefficients in the Born approximation Reinhardt et a1 (1976) obtained a cross section of 3.8 x 10-8 b for 132Xe-238U and 5.8 x 10-8 b for 238U-238u near the Coulomb barrier. This background process can therefore be neglected.

282

Figure

J Reinhardt and W Greiner

o' goo 180°

135' 10-12

110 10 ' " I I I

30 50 70 90 Photon energy (keVJ

High-energy tail of the x-ray spectra produced in several heavy-ion cc ision systems as measured by Greenberg et aZ(1976). Magnitude and angular distribution agree well with the calculated curves (Reinhardt et al 1976) for nuclear bremsstrahlung (full curves). (a) 0-Ni (13 MeV), (b) C-Ni (13.3 MeV), (c) Ni-C (60 MeV).

The described calculation is an approximate treatment of the pair production mechanism in the first of the graphs:

For sake of completeness and clarity we also present in the fourth graph the decay of the vacuum which, of course, should only be understood schematically since a proper Feynman-type description of the vacuum rearrangement is inadequate. The contribution of the second graph often discussed in high-energy physics is expected


100 la I

- .

\ U

' B

0.1 0.1 0 LO 80 0 40 ao

Z ZT

Figure 32. (a) Bremsstrahlung cross section du/dEz in pb keV-1 for various symmetric systems (Z+Z) and its dependence on the charge number Z at A, 1; B, 2 and C, 5 MeV/nucleon projectile energy. The cross sections are calculated at the K x-ray energies of the united-atom limit. (b) The same plot for asymmetric colliding systems (Xe + ZT). The broken curves indicate quadrupole radiation while the full curves give averaged values for the total (dipole plus quadrupole) cross sections (after Reinhardt et al 1976).

to be small in non-relativistic collisions. Substantially more important is the background due to nuclear Coulomb excitation. In close collisions of heavy nuclei, collective degrees of freedom are excited with a probability of nearly one. Within about 10-13 s the populated higher states decay under the emission of a photon. This photon can be converted to emit an atomic electron or create an electron-positron pair.

The differential pair formation cross section is given by the product of the scattering cross section and the positron emission probability of both nuclei:

(3.35)

The probability dW/dEp is a function of the positron kinetic energy Ep and the scattering angle @ion. I t depends on the Coulomb excitation probability PaCb of the initial nuclear level, the branching ratio P z f y for a photon transition into the final state, and the corresponding differential conversion coefficient d/l/dEp

(here the influence of cascading de-excitation has been neglected). Oberacker et al (1976a, b) have performed calculations on Coulomb excitation describing the intrinsic nuclear Hamiltonian by the collective rotation-vibration model (RVM) (Eisenberg and Greiner 1970). For the deformed even-even nucleus 238U two calculations were done. In method 1 all states of the RVM below the fission barrier were taken into account with the restriction to magnetic substates M = 0 which is correct for @ion= 180 '. Method 2 included all magnetic substates but was restricted to the ground-state band and the first f l and y vibrational bands. The excitation probabilities have a magnitude of the order of 10-3. They are obtained from the solution of the time-dependent SchrBdinger equation with a coupling potential including Coulomb and nuclear forces (Oberacker

20

284 3 Reiahardt and W Greiner

et aZ1974). Due to destructive interference, excitation near the barrier may be weaker than at impact energies well below the barrier.

Figure 33 shows the resulting positron cross sections for 238U-238U in dependence of the scattering angle (since the nuclei are not distinguishable du/dQ had to be symmetrized). The broken curves belong to model 1 or 2, as indicated in the caption. These curves must be compared with the full curve which is the cross section for dynamical autoionization of positrons (LO = 0.01). Obviously the ratio between de- sired and background positrons is most favourable at forward angles. Here, however, the nuclei do not, if at all, probe the diving region very deeply. For backward ion angles the cross sections only differ by a factor 2. The total positron cross sections are uvace+e-=5-0 x 10-4 b, uCbe+e-= 1.25 x 10-4 b (method 1) or 2.28 x 10-4 b (method 2).

0 60 120 180 a,,, (degl

Figure 33. The differential pair formation cross section (CM) with respect to ion angle for the symmetric system U-U at 800 MeV. The full curve indicates the positron emission cross section from the spontaneous and induced decay of the vacuum (LO= 0.01 assumed). The broken curves give the background from Coulomb and nuclear excitation calculated with method 1 (lower curve) and method 2 (upper curve) described in the text (after Oberacker et al 1976a).

The incorporation of dynamical K-hole formation should increase the first of these numbers somewhat. On the other hand, the cross section at low scattering angles will be depressed similar to the background.

A clear distinction between the positron spectra from both processes is the behaviour at high kinetic energies of the positron shown in figure 34. In the model considered positrons are produced from transitions between /3, y and the ground- state band with a maximum energy difference of 1-8 MeV. Thus the positron spectrum is cut off above Ep N 800 keV. Inclusion of the rotation-vibration interaction leads to band mixing and allows for higher transition energies but the resulting cross section will decrease rapidly.

The magnitude of the nuclear excitation background precludes the possibility of quasimolecular x-ray spectroscopy in the region of very high Z since the nuclear lines are dominant by several orders of magnitude. The only exception is the very stiff


Figure 34. Pair creation cross sections as a function of the positron kinetic energy Ep. Notation as in figure 33 (after Oberacker et al 1976a).

vibrator nucleus 2o*Pb with a lowest lying 2+ state at 4.086 MeV. On the other hand, the decay of the neutral vacuum can be separated from nuclear conversion pair production provided that the nuclear excitation is systematically studied for various colliding systems both in theory and experiment.

At the end of our discussion we have to draw attention to a further effect producing positrons. In heavy-ion collisions with very large 2 the Dirac continuum states are strongly deformed. Their variation, together with the presence of high Fourier frequencies of the nuclear motion, provides the means for direct continuum- continuum transitions (normal pair production due to the time-dependent Coulomb fields). This process is not hindered by the Pauli principle and its magnitude and behaviour seems to be similar to that for the filling of the lsa state. Soff et a1 (1977) have calculated this process in the framework of the quasimolecular model of 53.1 which leads to another effect of QED, namely a collective type of electron-positron creation, due to the coherent action of the strong, extended time-dependent electric field.

We expand the time-dependent state vector of the system

(3.37)

where IF) denotes the set of occupied states belonging to the initial configuration and ba+, dz+ are creation operators for (quasimolecular) particles and holes, respectively. In first-order perturbation theory, the amplitudes cij are given by

This expression differs from the conventional formulation of pair production in an external electric field, since by using the molecular basis the total zero frequency

286 J Reinhad and W Greiner

part of the electromagnetic potential is diagonalized and the standard methods of quantum field theory in the interaction picture do not apply. In order to obtain a representation of the pair production process which is independent of the specific basis functions yi we introduce the time-dependent density matrix

pi(^, t, R(t); x’, t’, l?(t’) R’) = F~+(X, R)~z(x’, R’) exp [iJf.dT &(R(T))]. (3.39)

One verifies by straightforward calculation

I+ /l),df jm dt‘ d3x d3x‘ I? - pz(x, t, R ; n’, t‘, A’ Ic i j (co)(2=- - m [ A

x A’ -, p j ( ~ , t , R; x’, t‘, R’) . (3.40)

The total number of excited particle-hole pairs N p h after the collision is obtained by summing over i and j , We introduce the density matrices p+ of occupied (positron) and p - of vacant (electron) states by

(3.41)

the charge-symmetrized density matrices p”= +(p+ - p - ) , and p = +(p+ + p - ) . The integrals in equation (3.40) can be interpreted as the trace in the coordinate representation and we can write in matrix formulation:

[ a: 1

p+(x, t , R; x’, t‘, R’)= 1 pj P-= c P.l jsF i $ F

I p(R, R’) +A‘ *(R, R’)] - T r [ ( A aR a p(R, R’)+A’ a p(R, R‘) ,

1 a 1 aR“

(3.42) The last form of equation (3.42) clearly exhibits the nature of the process: the shake-ofl of vacuum polarization and of bound electrons (Soff et a1 1977). Namely p“ can be decomposed into the vacuum polarization charge density matrix and a contribution from the bound electrons:

(3.43)

The term involving p is independent of the initial electron configuration and serves to regularize the divergent expression involving p”. Because (21 + 2 2 ) . > 1 in the collision of very heavy nuclei, infinitely many interactions with the combined electric field of both nuclei must be taken into account. This is contained in the formulation of equation (3.42).

By coherent summation of these diagrams the virtual photon field acquires a collective nature. For 21 + Zz e a--1 this collectivity is small, meaning that the pair production is well described by two-photon exchange. However, €or (21 + 22) . > 1 the collective effect becomes dominant. This is expressed in the 2 dependence of the total cross section: the lowest order diagram increases as 2 4 , whereas in the superheavy region at constant ion velocities the calculation of Soff et aZ(1977) gives


approximately Zn, with n - 18. This immediately sheds light on the average number of photon interactions in the production process.

The total direct electron-positron pair production cross section:

has been calculated in perturbation theory from (3.38), keeping only states with K = F. 1 and employing the monopole approximation. The following table shows the total pair production cross section for symmetric collisions with a total nuclear charge Z = Z l + Z z at ion energies such that the distance of closest approach is always 16 fm (part (a)) while part (b) gives the cross section for Pb+Pb collisions for various ion energies, characterized by the distance of closest approach Rmin.

(4 z (Pb) (b) Rmin (fm) 0 ( ~ b ) 146 5.7 25 3.0 158 33 * 4 20 19.4 164 76-4 16 76.4 168 135 13 193.4

uo-ef is seen to increase rapidly with 2. The energy spectrum of the created positrons and electrons is shown in figure 35(a) for a Pb+Pb collision (2 = 164). The main difference between the electron and the positron distribution, viz the behaviour at small energies, is explained by the Coulomb repulsion of the positron states. The positron distribution peaks at approximately 400 keV kinetic energy with do/dEp - 0.1 pb keV-1. Also more high-energy positrons than electrons are produced.

The impact-parameter dependence of pair creation is plotted in figure 35(b) up to b =40 fm. bP (b) peaks at 3 fm and falls off exponentially for larger impact parameters. The shake-off of the vacuum polarization cloud should be experimentally observable in the collision of very heavy ions. Its study in undercritical systems (Z< 170) is advisable so that it is not disturbed by the decay of the overcritical

E ( k e V ) b fml

Figure 35. (U) The positron (full curve) and electron (broken curve) spectrum of pair production in a 1210 MeV (lab) lead-lead collision. (b) The impact-parameter dependence P(b) of pair production in the same collision (after Soff et al 1977). Curve A, R m i n S 1 3 fm; curve B, R m i n = 1 6 fm; curve C, R m i n = 2 0 fm; curve D, &in= 25 fm.


vacuum. Pb-Pb collisions should be favoured, because positron production through internal conversion of Coulomb-excited nuclei should be minimal in this case. At present it is an open question to what extent the decay of the neutral vacuum can be studied in the collision of ordinary (not fully stripped) heavy ions by qualitative or only by precise quantitative measurements.

4. Overcritical phenomena in other fields-outlook

I n the preceding sections we have discussed the behaviour of the electron (or muon) field under the influence of a strong binding potential. If the binding energy of a state exceeds the threshold for particle creation (2mec2) there results the spontaneous creation of real electron-positron pairs which end as a bound electron and an emitted positron. The charged-vacuum state thus produced is stabilized against further particle creation by the Pauli principle. This is a very fundamental phenomenon, Its investigation and understanding may shed new light on our understanding of field theories and, in fact, may open a new and exciting area for theoretical and experimental investigation.

The concept of overcritical fields and the change of the vacuum state is not restricted to the electron-positron fields and electromagnetic interactions. It may also occur for other fermion fields (nucleons, quarks) and be caused by strong interactions. Overcritical quark fields have recently been discussed (Wolf 1974). They could lead to models of elementary particles, i.e. nucleons, pions, etc, which might turn out as charged quark vacua.

Also for bosons strong binding in a sufficiently deep potential well can lead to the possibility of particle creation. Here, however, the exclusion principle is not in action to stabilize the vacuum. The production mechanism in overcritical boson fields can only be stopped by the mutual interaction of the created particles or by the strong interaction with the source. Assuming a self-interaction of the type ~ 4 ,

Migdal(l972) was the first to treat this effect. Klein and Rafelski (1975b, c) discussed the quantization of overcritical boson fields under the influence of an electromagnetic potential. It turns out that the resulting very strong (mutual) vacuum polarization will, in fact, stabilize the vacuum. Contrary to the fermion case the binding energy as a function of potential depth does level off (if the potential vanishes faster than l/v), not reaching the threshold for real particle production. For a thorough discussion see the review by Rafelski et al (1977).

Migdal (1973b), Sawyer and Scalapino (1973) and other authors investigated the pion spectrum in nuclear matter. At a certain critical density (which plays the role of a critical interaction strength), the normal nuclear matter becomes unstable. A phase transition occurs leading to pion condensation, i.e. a state containing collective particle- hole excitations with the quantum numbers of a pion. The phase transition to pion condensates seems to play an essential role in setting up quasihydrodynamical conditions in relativistic collisions of nuclei on nuclei (Ruck et al 1976, Gyulassy and Greiner 1976), causing shock waves and thus high-density compressions in nuclear matter (Scheid et al 1974, Baumgardt et al 1975, Hofmann and Greiner 1977a). This could lead to the exploration of nuclear matter at high densities, high temperature, and as a highly isobaric gas with density isomerism (Bodmer 1974, Lee and Wick 1974) and limiting temperature (Hagedorn 1971, Hofmann et aE 1976). Detailed calculations will have to treat the time-dependent development of pion condensates


in finite, high-temperature nuclei. Imine (1975) recently discussed pion condensation and some related subjects in this journal.

These remarks indicate that the treatment of vacuum excitation for boson fields requires advanced many-body techniques. The basic phenomena, however, for e.g. pion condensation and the change of the vacuum in QED, are closely related.

As a further example of strong binding let us examine the field of a gravitational source with an event horizon (a black hole). T o understand the nature and development of singularities and horizons, much attention is presently being paid to the interaction of a (classically described) metric with quantized fields of matter, Recently the possibility of particle emission from a black hole for a time-dependent, collapsing (Hawking 1974) or a static (Deruelle and Ruffini 1974, Damour and Ruffini 1976) geometry has been discussed. Christodoulou and Ruffini (1971) treated the pair production semi-classically in the framework of Klein’s paradox.

Guided by the example of the charged vacuum in QED, Soffel et al (1976) have considered the solution of the Dirac (and Klein-Gordon) equation in a Schwarzschild or Reissner-Nordstrom (black hole with charge and angular momentum) background. They found a continuum of solutions for all particle energies, the wavefunctions showing an infinite number of oscillations when approaching the outer coordinate singularity. The appearance of the continuum was studied in the model. of an extended gravitational source. When its radius shrinks to the Schwarzschild radius, all the discrete bound states drop to zero energy. Imbedded in the newly emerged continuum are, as in the QED case, bound-state resonances. Their width results from the possibility of decay into the black hole. The position of these resonances can be understood from the effective potential

where M , Q and L, are the mass, charge and angular momentum projection of the black hole. The effective potential is shown in figure 36. Even in the case of a neutral black hole (figure 36(a)), the gap between the negative and positive continuum (particle and antiparticle states) is narrowed by the attractive gravitational interaction and vanishes for R=rs . I n the case of a charged centre the charge conjugation symmetry is lifted. For a negative central charge (figure 36(b)), the negative continuum is varied in energy. If the Fermi energy exceeds mec2 spontaneous electron emission will occur. The same is true for opposite charges (figure 36(c)). This transition to a charged electron-positron vacuum leads to a limiting stable charge for a black hole, namely

or approximately

Note in (4.2) that only one electron has to be created since the antiparticle is swallowed by the black hole. The limiting charge to mass ratio

(4.4)

reflects the double ratio of the gravitational to the electromagnetic coupling constant.


1

0

-1

- 2

'rs

IC I

Figure 36. Effective potential near a black hole. (a) for the Schwarzschild field with mass M G = ~ (in geometrized units, Mc=(G/cZ) M ) and angular momentum ~ = 5 0 ; (b) for the Reissner-Nordstrom field with charge QG= 1000 (unit, Q G ~ = (G/e4) Q2), M G - 2 and ~ = 4 . Pair creation occurs if h = e Q / r , >meCB. Electrons will be emitted via a tunnelling process; (c ) for QG= - 1000 and K = 10. Positron emission is possible. The horizontal lines indicate the states of the upper continuum (after Soffel et a2 1977).

electron-positron Charged vacuum in' overcritical fields

fi?ld

Figure 37. Schematic drawing of the energy of the system nucleus plus electron cloud against charge, which is taken as being continuous. The quantization of charge at integer values can be imagined to be guaranteed by stabilization potentials. In case (a) the vacuum (ground state) is symmetric under charge conjugation. This symmetry is broken in overcritical fields (case (b)) where the vacuum is charged.


Finally, let us remark that the charged vacuum is an interesting example of spontaneous symmetry-breaking. The Lagrangian of the theory of QED is invariant under charge conjugation. Usually, the ground state exhibits this symmetry too. This is indicated in the symbolic picture drawn in figure 37(a). In the underlying field of a nucleus with a subcritical (positive or negative) charge the state of minimal energy, out of the various different charge states of the ion, has Q = 0. Figure 37(a) is charge- symmetric globally as well as locally in the vacuum state. The latter symmetry is broken if the nuclear field becomes overcritical (figure 37(b)) and the vacuum becomes charged.

The Goldstone boson (Goldstone 1961) associated with this phase transition is an electron-positron pair (positronium) which is, however, extremely unstable in the strong Coulomb field. A hypothetical observer in an overcritical atom would perhaps not even notice that overall C symmetry exists, because his vacuum (ground state) does not show it. This situation is analogous to the magnetization of a ferromagnet at a temperature T > T, (undercritical case) and near zero temperature (overcritical case.)

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