###### Abstract

Instability of electron-positron vacuum in strong electric fields is studied.
First, falling to the Coulomb center is discussed at $Z>137/2$ for a spinless
boson and at $Z>137$ for electron. Then, focus is concentrated on description
of deep electron levels and spontaneous positron production in the field of a
finite-size nucleus with the charge $Z>Z_{\rm cr}\simeq 170$. Next, these
effects are studied in application to the low-energy heavy-ion collisions.
Then, we consider phenomenon of “electron condensation” on levels of upper
continuum crossed the boundary of the lower continuum $\epsilon=-m$ in the
field of a supercharged nucleus with $Z\gg Z_{\rm cr}$. Finally, attention is
focused on many-particle problems of polarization of the QED vacuum and
electron condensation at ultra-short distances from a source of charge. We
argue for a principal difference of cases, when the size of the source is
larger than the pole size $r_{\rm pole}$, at which the dielectric permittivity
of the vacuum reaches zero, and smaller $r_{\rm pole}$. Some arguments are
presented in favor of the logical consistency of QED. All problems are
considered within the same relativistic semiclassical approach.

###### keywords:

electron-positron production; supercritical atoms; electron condensation;
polarization of vacuum, zero-charge problem

x 10.3390/—— xx Academic Editor: name Received: date; Accepted: date;
Published: date Electron-positron vacuum instability in strong electric
fields. Relativistic semiclassical approach. D. N. Voskresensky 1,2 D. N.
Voskresensky Correspondence<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
    1. 1.1  A general picture
    2. 1.2  Semiclassical approximation
  2. 2 Relativistic spinless charged particle in static field $A^{\mu}=(A^{0},\vec{0})$
    1. 2.1 Reduction of Klein-Gordon-Fock equation to Schrödinger equation
    2. 2.2 Relativistic spinless charged particle in Coulomb field of point-like center
  3. 3 Dirac equation for particle in static electric field, $A^{\mu}=(A^{0},\vec{0})$
    1. 3.1  Dirac system in case of one-dimensional electric field
    2. 3.2  Dirac system in central-symmetric field
    3. 3.3 Reduction of Dirac system to Schrödinger equation
    4. 3.4 Interpretation of bound states in a weak field
    5. 3.5 Exact solution for electron in Coulomb field of point-like center
    6. 3.6 Avoiding problem of falling to center in realistic treatment. Spherical nucleus of finite size
  4. 4 Semiclassical approach to Dirac equation transformed to second-order differential equation
    1. 4.1 Accuracy of calculation of energy levels in semiclassical approximation
    2. 4.2 Semiclassical approximation to Coulomb field of point-like nucleus
    3. 4.3 Finite nucleus. Semiclassical wave functions and quantization rule.
    4. 4.4 Critical charge of the nucleus
    5. 4.5 Number of levels which crossed boundary of lower continuum
    6. 4.6 Energy of single-particle levels at $\epsilon<-m$
      1. 4.6.1 Energy spectrum for $|\epsilon|-m\ll m$
      2. 4.6.2 Energy spectrum for $|\epsilon|\ll-m$
    7. 4.7 Exponential estimate of probability of spontaneous production of positrons
    8. 4.8 Critical charge of nucleus for muon
  5. 5 Semiclassical approximation to system of linear Dirac equations
    1. 5.1 Semiclassical wave functions
    2. 5.2 Nonrelativistic limit
    3. 5.3 Bohr-Sommerfeld quantization rule
    4. 5.4 Probability of spontaneous production of positrons
    5. 5.5 Semiclassical method for noncentral potentials obeying system of linear Dirac equations
  6. 6 Spontaneous production of positrons in heavy-ion collisions
    1. 6.1 Approach to the problem
    2. 6.2 Electron energy as a function of distance between nuclei
    3. 6.3 Tunneling in the two-center problem. Angular distribution of positrons
    4. 6.4 Screening of K-electron by electron cloud of not fully stripped quasi-molecule
    5. 6.5 Calculation of positron production employing the imaginary-time method
      1. 6.5.1 General description of the method
      2. 6.5.2 Tunneling in slowly time-dependent potential
      3. 6.5.3 Correction on non-adiabaticity to the spontaneous positron production in low-energy heavy-ion collisions
  7. 7 Many-particle semiclassical approximation. Electron condensation in upper continuum
    1. 7.1 Screening of a source of positive charge in presence of external electrons
      1. 7.1.1 Filling of the vacuum shell by electrons
    2. 7.2 A detailed derivation of relativistic Thomas-Fermi equation
    3. 7.3  Weak screening, $1/e^{2}\ll Z\ll 1/e^{3}$
    4. 7.4 Strong screening, $Ze^{3}\gg 1$
    5. 7.5 Falling to the center in relativistic Thomas-Fermi equation
  8. 8 Polarization of vacuum
    1. 8.1 Polarization of vacuum in uniform stationary electric and magnetic fields
    2. 8.2 Noninteracting photon, electron and spin-zero boson propagators
    3. 8.3 Dyson equation for photon propagator
    4. 8.4 Calculation of photon polarization operator
      1. 8.4.1 Case of a weak static electric field. Renormalization of charge
      2. 8.4.2 Case of a strong static electric field
    5. 8.5 Polarization of vacuum and electron condensation
  9. 9 Distribution of charge at super-short distances from the Coulomb center
    1. 9.1 Charge distribution near the charge source of radius $r=r_{0}>r_{\rm L}$
      1. 9.1.1 Electron condensation is not included
      2. 9.1.2  Electron condensation on levels of upper continuum is included
    2. 9.2 Charge source of radius $r=r_{0}<r_{\rm L}$. Polarization of vacuum and electron condensation on levels in lower continuum
    3. 9.3  Distribution of charge of electron
  10. 10 Conclusion

##  1 Introduction

I dedicate this review to the blessed memory of Vladimir Stepanovich Popov who
recently leaved us as the result of a many-year hard illness, which prevented
him to work actively last years. The problem of the electron-positron pair
production when the ground-state electron level dives below energy $-mc^{2}$
($m$ is the electron mass, $c$ is speed of light) was of his interest starting
from the end of 1960-th. Especially he contributed to this problem during
1970-th. For his outstanding contributions to the theory of ionization of
atoms and ions in the field of intense laser radiation and to the theory of
the creation of electron-positron pairs in the presence of superstrong
external fields V. S. Popov was awarded the I. Y. Pomeranchuk Prize in 2019.

We worked together with Vladimir Stepanovich on problems of supercritical
atoms with the charge $Z>Z_{\rm cr}=170-173$ during 1976-1978 when we
developed semiclassical treatment of this problem. These works, cf.
Migdal:1976wx ; Migdal:1977rn ; Eletskii:1977na ; Mur:1978nb ; Mur:1978ke ;
Popov:1978kk ; Popov:1979gq , became a part of my PhD thesis Voskresensky1977
defended in 1977 under the guidance of Arkadi Benediktovich Migdal.

As follows from the Dirac equation in the Coulomb field of a point-like
nucleus with $Z>1/e^{2}$ (in units $\hbar=c=1$, which will be used in this
paper, $e^{2}\simeq 1/137$), the electron occupied the ground-state level
should fall to the center. Following the idea of I. Pomeranchuk and Ya.
Smorodinsky PomeranchukSmorodinsky1945  solution of the problem can be found
taking into account the fact that the real nuclei have a finite radius. The
ground-state electron level in the field of a finite supercritical nucleus
continues to decrease with increasing $Z$ and reaches the energy $-m$. Then
two positrons with the energies $>m$ go off to infinity and electrons with
$\epsilon<-m$ screen the field of the nucleus by the charge $-2e$. Typical
distance characterizing electrons of the vacuum $K$ shell is $\sim 1/(3m)\gg
R_{\rm nucl}$, cf. Popov:1979gq . Reference PomeranchukSmorodinsky1945
essentially overestimated value of the critical charge. The problem got a new
push in the end of 1960-th. Independently W. Pieper and W. Greiner Pieper1969
(in numerical analysis) and V. S. Popov Popov1970ZhETF ; Popov1970YadFiz ;
Popov1971ZhETF32 ; Popov1971ZhETF33 ; ZeldovichPopov1972  (in analytical and
numerical studies) evaluated value of the critical charge to be $Z_{\rm
cr}\simeq 169-173$ depending on assumptions about the charge distribution
inside the nucleus and the ratio $Z/A$. Then there appeared idea to observe
positron production in heavy-ion collisions, where the supercritical atom is
formed for a short time Muller1972 ; PopovJETP1974 . As the reviews of these
problems, I can recommend Popov2001 ; Rafelski1916 ; Voskresensky1990 .

In 1976 with the inauguration of the UNI-LAC accelerator in GSI, Darmstadt, it
became possible to accelerate heavy ions up to uranium below and above the
Coulomb barrier. Instead of a positron line associated with spontaneous decay
of the electron-positron vacuum, mysterious line structures were observed,
which in spite of many attempts did not get a reasonable theoretical
interpretation. The experimental results on the mentioned positron lines
proved to be erroneous. New experiments were done during 1993 – 1995, cf.
Ahmad1995 ; Ganz1996 ; Leinberger1997 . Presence of the line structures was
not observed. Events, which could be interpreted as the effect of the decay of
the QED vacuum with the spontaneous production of the electron-positron pair,
were not selected. In spite of the effect of the spontaneous production of
positrons in the electric field of the supercharged nucleus has been predicted
many decades ago, it was not yet observed experimentally in heavy-ion
collisions.

The spontaneous pair creation could be observed in heavy-ion collisions
provided existence of a nuclear sticking Greiner1985 . However the ideas on
the sticking of nuclei did not find a support. Besides a spontaneous
production of positrons a more intensive induced production of pairs occurs
due to an excitation of nuclear levels. Therefore, the key question is how to
distinguish spontaneous production of positrons originated in the decay of the
electron-positron vacuum from the induced production and other competing phone
processes. New studies of low-energy heavy-ion collisions at the supercritical
regime are anticipated at the upcoming accelerator facilities in Germany,
Russia, and China Gumberidze2009 ; Ter2015 ; Ma2017 . This possibility renewed
theoretical interest to the problem Popov:2020xmd . As one can see from
numerical results reported in Popov:2020xmd , these results support those
obtained in earlier works, although comparison with analytical results derived
in Migdal:1976wx ; Migdal:1977rn ; Eletskii:1977na ; Mur:1978nb ; Mur:1978ke ;
Popov:1978kk ; Popov:1979gq  was not performed. Also let me notice that
recently there appeared statements that spontaneous production of positrons
should not occur in the problem under consideration. I see no serious grounds
for these revisions and thereby will not review these works.

###  1.1  A general picture

States with $|\epsilon|<m$ correspond to the energy
$E=(\epsilon^{2}-m^{2})/2m<0$, see Fig. 1. In terms of the Schródinger
equation these are ordinary bound states. Let the ground state level is empty
and we are able to adiabatically increase the charge of the nucleus $Z$. The
latter means that either the time $\tau_{Z}$ characterizing increase of $Z$ is
much larger compared to $1/|\epsilon_{0}-\epsilon_{njm}|$, where
$\epsilon_{njm}$ are energies of other bound states in the potential well, or
$\tau_{Z}>1/m$ for the case of transitions from the ground-state level,
$\epsilon_{0}$, to the continues spectrum. Empty level with $\epsilon<-m$
becomes quasi-stationary, see Fig. 1. Penetrating the barrier between
continua, see Fig. 2 below, two electrons (with opposite spins) are produced,
which occupy this level, whereas two positrons of opposite energy go off
through the barrier to infinity. In standard interpretation, cf.
ZeldovichPopov1972 , the electron states, $\psi\propto e^{-i\epsilon t}$, with
$\epsilon=\epsilon_{0}+i\Gamma(\epsilon_{0})/2$ for $\epsilon_{0}<-m$,
$\Gamma>0$, cf. Eqs. (3.5), (3.6) in MurPopovFerm , are occupied due to a
redistribution of the charge of the vacuum. The vacuum gets the charge $2e<0$
distributed in the region of the supercritical ion. Two positrons with
$\epsilon_{e^{+}}=-\epsilon_{0}-i\Gamma(\epsilon_{0})/2$ go off to infinity
after passage of a time $\sim\tau_{0}e^{\Gamma t}$, $\tau_{0}\sim R$, where
$R$ is the size of the potential well for
$R\stackrel{{\scriptstyle\scriptstyle>}}{{\phantom{}{}_{\sim}}}1/m$, as it
occurs for any decaying quasi-stationary state, producing a diverging
spherical wave $\psi\propto e^{ikr}$, $k=\sqrt{\epsilon_{e^{+}}^{2}-m^{2}}$
for the positron. For far-distant potentials situation is similar to that for
the charged bosons, cf. Voskresensky1988 . For the case $V=-Ze^{2}/r$ for
$r>R_{\rm nucl}$, one obtains that $\Gamma(-m)=0$.

Figure 1: Typical effective Schrödinger potential of a charged particle in an
electric central-symmetric potential well, $r_{\pm}$ are turning points,
$r_{0}$ corresponds to maximum of $U$. Dashed line describes quasi-stationary
level with $\epsilon<-m$.

For $Z<Z_{\rm cr}$ electrons of the lower continuum (with $\epsilon<-m$) fill
all energy levels according to the Dirac picture of the electron-positron
vacuum. Spatially they are distributed at large distances. For $Z>Z_{\rm cr}$
the process of the tunneling of the electron of the lower continuum to the
empty (a localized) state prepared in the upper continuum with $\epsilon<-m$
can be treated as the tunneling of the virtual positron (electron hole) with
$\epsilon_{e^{+}}=-\epsilon_{0}-i\Gamma/2$ from the region of the well to
infinity, where it already can be observed. If one scatters an external real
positron with a resonance energy $\epsilon_{e^{+}}\simeq-\epsilon_{0}>m$ on
such a potential, this positron for a short time forms a resonance quasi-
stationary state in the effective potential, which after passage of a time
$\sim 1/\Gamma$ is decayed. As the result, positron goes back to infinity.
After that, during a time of the same order of magnitude two positrons, being
produced in a fluctuation together with two electrons, go off to infinity and
those two electrons fill the stationary negative-energy state, as it was
explained.

If the ground state level was initially occupied by two electrons of opposite
spins, then at adiabatic change of the potential (in the sense clarified
above) they remain on this level $\epsilon=\epsilon_{0}$. Production of pairs
does not occur. During a time $\sim 1/\Gamma$ their charge $2e<0$ is
redistributed over the range of energies
$\epsilon_{0}-\Gamma(\epsilon_{0})/2\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}\epsilon\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}\epsilon_{0}+\Gamma(\epsilon_{0})/2$.
This charge is localized at distances ($\sim 1/(3m)$ typical for the ground
state in the Coulomb field Popov:1979gq ). If the experimenter scatters an
external positron with $\epsilon_{e^{+}}\simeq-\epsilon_{0}>m$ on such a
potential, the positron annihilates with one of the two electrons have
occupied the ground-state level. After passage of time $\sim 1/\Gamma$ there
occurs spontaneous production of the one new pair, the electron fills empty
state (after that again two electrons occupy the ground-state level) and the
positron goes to infinity.

###  1.2  Semiclassical approximation

Semiclassical approximation is one of the most important approximate methods
of quantum mechanics Migdal1975 . Classical and semiclassical ideas are widely
used in quantum field theory in problems dealing with spontaneous vacuum
symmetry breaking of bosons, cf. ZuberItzykson1989 ; Voskresensky1990 ;
Voskresensky1988 , in condensed matter physics, cf. LL5 ; LL9 ;
Voskresensky:1993ux , and in physics of nuclear matter Migdal1978 ;
Migdal:1990vm .

As a consequence of the instability of the boson vacuum in a strong external
field, there appears reconstruction of the ground state and there arises
condensate of the classical boson field Migdal1971 ; Migdal1978RevModPhys .
Many-particle repulsion of particles in the condensate provides stability of
the ground state. After that excitations prove to be stable, cf. Migdal1978 ;
Migdal:1990vm . They are also successfully described using semiclassical
methods, e.g. such as the loop expansion ColemanWeiberg ; Voskresensky1988 .

For fermions there exist two possibilities. In the first situation, fermions
heaving attractive interaction, being rather close to each other, may form
Cooper pairs, cf. LL9 . In the second situation, which we focus on here,
electron-positron pairs, being produced in a strong static electric field, are
well separated from each other by the potential barrier. As the result, the
electric potential attracts particles of one sign of the charge and repels
antiparticles. Due to the Pauli principle each unstable single-particle state
is occupied by only one fermion. Therefore it is natural to prolong a single-
particle description in a overcritical region (till there appeared still not
too many dangerous states). Classical approximation does not work for fermions
but semiclassical methods prove to be working. As known, semiclassical
approach is applicable for calculation of the energy of levels with large
quantum numbers and for spatially smooth potentials, when
$d\tilde{\lambda}/dx\ll 1$, where $\tilde{\lambda}=1/p$ is the reduced
electron De Broglie length, $p$ is the momentum, $x$ is coordinate. For the
Coulomb field for the ground-state level a rough estimate yields
$d\tilde{\lambda}/dr\sim 1/(Ze^{2})$ for $r\to 0$. However even for
$d\tilde{\lambda}/dx\sim 1$ semiclassical approximation continues to work not
bad in calculation of the energy levels, with an error not larger than 10% due
to presence of a numerically small parameter $\sim 1/\pi^{2}$, cf. Migdal1975
.

Instability of the vacuum near a nucleus heaving a supercritical charge. It
proves to be that the semiclassical approximation is applicable with an
appropriate accuracy for the description of the electron energy levels in
supercritical field of a nucleus with the supercritical charge $Z>(170-173)$.
Semiclassical approximation allows to find rather simple expressions for the
critical value of the charge, cf. Refs. Krainov1971 ; Marinov1974 ;
Voskresensky1977 , for energies of deep levels as a function of $Z$ and for
probabilities of traversing of the barrier between continua, cf.
Eletskii:1977na ; Mur:1978nb ; Mur:1978ke ; Popov:1978kk ; Popov:1979gq .

Spontaneous positron production in low-energy heavy-ion collisions. Comparison
of theory and experiment should check application of QED in region of strong
fields outside the applicability of the perturbation theory. Description of
spontaneous production of positrons in heavy-ion collisions needs solution of
two-center problem for the Dirac equation. Since variables are not separated
in this case, the problem does not allow for the analytical treatment and
numerical calculations are cumbersome. However usage of semiclassical
approximation results in simple analytical expressions for the energies of the
electron levels, cf. Popov:1978kk ; Popov:1979gq , valid with error less than
few %. Thereby, this is one more example of the efficiency of the
semiclassical approach.

Electron condensation in a field of a supercharged nucleus. In supercritical
fields many energy levels cross the boundary of the lower continuum and the
problem of the finding of the vacuum charge density becomes of purely many-
particle origin. It can be considered within the relativistic Thomas-Fermi
method, cf. Migdal:1977rn . All initially empty states, which crossed the
boundary $\epsilon=-m$, are filled after a while. In this sense one may speak
about “electron condensate”.

Vacuum polarization and electron condensation at super-short distances from
Coulomb center. In spite of the successes in explanation of all purely
electrodynamical phenomena, QED is principally unsatisfactory theory, since
relations between bare mass and charge and observable ones contain diverging
integrals LL4 . As the result, as one thinks, there is no not contradictive
manner to pass from super-short to long distances. In spite of this, as is
well known, it is possible to remove divergencies from all observable
quantities with the help of the renormalization procedure.

One of central problems related to renormalization of the charge is the
problem of so-called “zero charge”. Considering square of the charge of
electron $e^{2}(r)$ as a function of the radius $r$ and assuming finite value
of the bare charge $e^{2}(r_{0})=e_{0}^{2}>0$ for the source-size $r_{0}\to
0$, one derives $e^{2}(r\to\infty)\to 0$ instead of an expected value
$e^{2}(r\to\infty)\to e^{2}=1/137$. The same problem appears, when one
considers the screening of the central source with charge density $n_{\rm
ext}=Z_{0}\delta(r-r_{0})$ for $r_{0}\to 0$, cf. Eletskii:1977na . The problem
of a distribution of the charge near an external source of the charge with
radius $R\ll 1/m$, as well as the problem of the distribution of the charge of
the electron at distances $r\ll 1/m$ are the key principal problems of QED.
Semiclassical approach proves to be very promising in calculation of the
vacuum dielectric permittivity in strong inhomogeneous electric fields
Migdal1972 . The density of the polarized charge is supplemented by the
density from electron condensation Eletskii:1977na ; Migdal1978 . The problem
proves to be specific and depends on weather the radius of the external source
of the charge is larger than a distance $r_{\rm pole}$, where dielectric
permittivity decreases to zero, or smaller $r_{\rm pole}$, cf. V1992 ;
Voskresensky:1993uy . References V1992 ; Voskresensky:1993uy  argued for the
condensation of electron states in the upper continuum at distances larger
than $r_{\rm pole}$ for $r_{0}>r_{\rm pole}$ and for the condensation of
electron states originated in the lower continuum at distances smaller than
$r_{\rm pole}$ (for $r_{0}<r_{\rm pole}$), at which the dielectric
permittivity proves to be negative and $e_{0}^{2}<0$. Semiclassical
consideration of this problem allows to present arguments in favor of a
logical consistency of QED.

Similar effects in semimetals and in stack of graphene layers. Existence of
the Weyl semimetals, i.e. materials with the points in Brillouin zone, where
the completely filled valence and completely empty conduction bands meet with
linear dispersion law, $\epsilon=v_{\rm F}p$, where the Fermi velocity is
$v_{\rm F}\sim 10^{-2}$, has been predicted in AB70 . Systems with
relativistic dispersion law are likely to be realized in some doped silver
chalcogenides, pyrochlore iridates and in topological insulator multilayer
structures. Weyl semimetals are 3-dimensional analogs of graphene Novoselov ,
where the energy of excitations is also approximately presented by the linear
function of the momentum but the electron subsystem is two dimensional one,
whereas the photon subsystem remains three dimensional. Even though the mass
of excitations $m=0$ for ideal graphene and Weyl semimetals without
interactions, a non-zero mass, $m\neq 0$, can be induced in many ways Kotov
resulting in the gapped dispersion relation $\epsilon^{2}=p^{2}v_{\rm
F}^{2}+m^{2}v_{\rm F}^{4}\,,$ In difference with a small value of the fine
structure constant in QED, $e^{2}=1/137$, the effective coupling in Weyl
semimetals and in graphene is $\alpha_{\rm ef}=e^{2}/v_{\rm
F}\varepsilon_{0}$, where $\varepsilon_{0}$ is the dielectric permittivity of
the substance. Coupling constant $\alpha_{\rm ef}$ can be as $\ll 1$ as
$\stackrel{{\scriptstyle\scriptstyle>}}{{\phantom{}{}_{\sim}}}1$, depending on
the substance, and both weak and strong coupling regimes are experimentally
accessible. Thus Weyl semimetals and infinite stack of graphene layers make it
possible to experimentally study various effects have been considered in 3+1
quantum electrodynamics (QED) for weak and effectively strong couplings, cf.
Voskresensky2012 .

Also, electron-positron pair production from the vacuum can be triggered by
the laser electromagnetic fields. However it seems unlikely to realize such a
possibility at least in the nearest future, cf. Palffy:2019scn  and references
therein.

Below, attention is focused on a semiclassical description. I describe
instabilities of the boson and fermion vacua in static potentials, in
particular in the Coulomb field. Then focus is concentrated on the description
of the spontaneous positron production in low-energy heavy-ion collisions.
Next, many-particle semiclassical description of electron condensation is
considered. Finally, modification of the Coulomb field at super-short
distances due to vacuum polarization and electron condensation are studied.

The paper is organized as follows. Sect. 2 starts with a brief discussion of
instability for the charged bosons in static electric fields, in particular in
the Coulomb field of a point-like nucleus with the charge $Z>Z_{\rm
cr}=1/(2e^{2})$. Behavior of deeply bound electrons obeying the Dirac equation
in the strong static electric fields is considered in Sect. 3. First I
consider the case of a one-dimensional field and then of a spherically
symmetric field. The Dirac equation is transformed to equivalent Schrödinger
form in an effective potential and interpretation of the solutions is
discussed. Then in Subsect. 3.5 I demonstrate exact solution of the problem of
bound states in the strong Coulomb field of a point-like center. The focus is
made on the problem of the falling of the electron to the center for a nucleus
with the charge $Z\geq 1/e^{2}$. Subsect. 3.6 describes how the problem is
resolved taking into account that nuclei have a finite size. In Sect. 4 I
introduce a semiclassical approach to the Dirac equation, being transformed to
the second-order differential equation. Electron levels crossed the boundary
of the lower continuum are considered. The mean radius of the K-electron shell
and the critical charge of the nucleus are found for $\epsilon=-m$, as well as
the number of levels crossed the boundary of the lower continuum and their
energies. The critical charge of the nucleus for the muon is also found.
Comparison of semiclassical expressions with much more cumbersome exact
expressions permits to understand merits of the semiclassical approach. In
Sect. 5 semiclassical approximation is developed for the system of linear
Dirac equations. Semiclassical wave functions in classically allowed and
forbidden regions are introduced and the Bohr-Sommerfeld quantization rule is
formulated. Next, probability of the positron production is calculated. Then,
semiclassical approximation is applied to non-central potentials. In Sect. 6
focus is concentrated on problems of spontaneous positron production in low-
energy collisions of heavy ions. Energies of deep levels as a function of the
distance between colliding nuclei and angular distribution of the positron
production are found employing semiclassical approach. Then I consider
screening of the charge at collisions of not fully striped nuclei.
Semiclassical approximation (imaginary time method) is adequate to describe
dynamics of the tunneling of electrons from lower continuum to the upper one.
In such a way a correction on non-adiabaticity to the probability of
production of positrons is found. Electron condensation in the field of a
supercharged nucleus is considered in Sect. 7. Effects associated with
polarization of the electron-positron vacuum in weak and strong fields are
studied in Sect. 8. Then in Sect. 9 I focus on description of the charge
distribution at super-short distances from the charge source. Effects of
polarization of vacuum and electron condensation in upper and lower continua
will be considered. Sect. 10 contains a conclusion.

##  2 Relativistic spinless charged particle in static field
$A^{\mu}=(A^{0},\vec{0})$

###  2.1 Reduction of Klein-Gordon-Fock equation to Schrödinger equation

Consider a spinless negatively charged boson placed in a stationary attractive
potential well $V$. The Klein-Gordon-Fock equation renders

$\Delta\phi+[(\epsilon-V)^{2}-m^{2}]\phi=0\,.$ (1)

With the help of notations

$E=\frac{\epsilon^{2}-m^{2}}{2m}\,,\quad U_{\rm ef}=-\frac{V^{2}-2\epsilon
V}{2m}\,,$ (2)

we may rewrite Eq. (1) in the form of the Schrödinger equation,

$\Delta\phi+2m(E-U_{\rm ef})\phi=0\,.$ (3)

As we see from Eq. (2), for relativistic particles there appears an attractive
term in the effective potential $-V^{2}/(2mc^{2})$, even for a purely
repulsive potential $V$. In the limit case $E\ll m$ and $|V|\ll m$ we have
$\epsilon\simeq m+E$ and $U_{\rm ef}\simeq V$, and we recover Schrödinger
equation for a nonrelativistic particle. For $|\epsilon|<m$ the
“nonrelativistic” energy is $E<0$ that corresponds to bound states in the
interval of energies $-m<\epsilon<m$. For a sufficiently deep potential well
the energy of the ground state level may cross the boundary $\epsilon=-m$. In
a deeper potential other levels cross this boundary. For $\epsilon<-m$, here
$\mbox{Re}E>0$, the levels become quasi-stationary, see Fig. 1.

A comment is in order (D. N. Voskresensky 1974, see comment in MurPopov1976bos
). For a spinless particle under consideration, the ground-state single-
particle level crosses the boundary $\epsilon=-m$ only for far-distant
potentials, when $-V(r\to\infty)>C_{\rm cr}/r^{2}$, for a constant $C_{\rm
cr}>0$. For potentials obeying condition $-V(r\to\infty)<C_{\rm cr}/r^{2}$,
there appears a bound state for antiparticle. In both cases for a broad
potential well of a typical radius $R\gg 1/m$ the vacuum instability occurs at
$|V|\simeq|V|_{\rm cr}\simeq 2m(1\pm O(1/(m^{2}R^{2}))$ either at
$\epsilon_{\rm cr}=-m$ or at $\epsilon_{\rm cr}\simeq-m(1-O(1/(m^{2}R^{2}))$.
In case of a broad potential well solutions of many-particle problems in both
cases are almost the same, cf. Voskresensky1988 . For $-V>-V_{\rm cr}$ there
appears production of pairs. Positively charged antiparticles go off to
infinity and negatively charged particles form a condensate, see Migdal1978 ;
Voskresensky1988 .

Figure 2: Schematic picture of deformation of the upper and lower continua in
a strong external electric field (the boundaries of the continua are shaded).
Electrons belonging to the vacuum shell in upper continuum fill the cross-
hatched region. The states below the curve $\epsilon_{-}/m=V(r)/m-1$ form the
unobservable Dirac sea. The quantity $W$ shows an artificial cutoff energy.

Let us illustrate how deformation of boundaries of upper and lower continua
occurs in a static electric field forming a broad potential well for a
negatively charged particle. To be specific consider spherically symmetric
field. Boundaries of continua, $\epsilon_{\pm}$, are determined by

$\vec{p}^{\,2}(r)=(\epsilon_{\pm}-V)^{2}-m^{2}=0\,.$ (4)

They are shown in Fig. 2. In upper and lower continua $p^{2}(r)>0$, these are
classically allowed regions. In the gap between continua $p^{2}(r)<0$. This is
classically forbidden region. For $V<V_{\rm cr}=-2m-O(1/(m^{2}R^{2}))$ there
arises a region of the overlapping of the continua that means that negatively
charged particle may penetrate from the lower continuum (from exterior of the
potential well) to the upper one (to interior of the well).

With an exponential accuracy the probability of a passage of the one-
dimensional barrier is determined by

$W\sim e^{-2\mbox{Im}S}\sim e^{-2\int_{x_{1}}^{x_{2}}|p|dx}\,,$ (5)

where $x_{1}$ and $x_{2}$ are the turning points at which $p(x)=0$. This
expression is applicable for $W\ll 1$.

As example, consider uniform static electric field $eE=-\nabla V=const$,
$|eE|\ll m^{2}$. Then we have $p\simeq\sqrt{(\epsilon+eEx)^{2}-m^{2}}$ . From
Eq. (5) we immediately obtain

$W\sim e^{-\pi E_{0}/E}\,,\quad E_{0}=m^{2}\,.$ (6)

This expression coincides with the first term of the infinite series solution
Schwinger .

A question arises is it possible to observe a process of production of pairs
already in a weak attractive electric field with the strength $|E|\ll m^{2}$
at $-\delta V>2m$? Critical difference $-\delta V\simeq-2m$ can be easily
reached in the field of the condenser, where $\nabla A_{0}=const$, at increase
of the distance $d$ between plates. Employing $|\nabla A_{0}|=|\vec{E}|\sim
10^{4}$ V$/$cm, the value, which is easily produced in electrical engineering,
we estimate $|\delta V|>2m_{\pi}$ already for
$d\stackrel{{\scriptstyle\scriptstyle>}}{{\phantom{}{}_{\sim}}}10^{3}$ cm.
Here $m_{\pi}\simeq 140$ MeV is the mass of the lightest charged boson, the
pion. However the probability of the production of the pairs $W\sim
e^{-2\mbox{Im}S}$, $\mbox{Im}S=\int_{x_{1}}^{x_{2}}|p|dx$, is negligibly small
at these conditions. Indeed for $V=-eEx$ we get
$\mbox{Im}S=\int_{x_{1}}^{x_{2}}|p|dx=\frac{\pi}{2}\frac{E_{0}}{E}\,.$ for
pions $E_{0}\simeq 3\cdot 10^{20}$ V$/$cm. For electrons $E_{0}\simeq 1.3\cdot
10^{16}{\rm V}/{\rm cm}$.

###  2.2 Relativistic spinless charged particle in Coulomb field of point-like
center

In case of the Coulomb field of a point-like nucleus, $V=-Ze^{2}/r$, with the
help of the replacement $\phi(\vec{r})=R(r)Y_{lm}$ we obtain equation for the
radial wave function $R(r)$ in the form

$\Delta
R_{r}+{2m}\left[E+\frac{(Ze^{2})^{2}}{2mr^{2}}-\frac{l(l+1)}{2mr^{2}}+\frac{\epsilon
Ze^{2}}{mr}\right]R=0\,,\quad\Delta_{r}=\frac{1}{r}\frac{\partial^{2}(rR)}{\partial
r^{2}}\,,$ (7)

where $E=\frac{\epsilon^{2}-m^{2}}{2m}$ is effective nonrelativistic
Schrödinger energy of the particle,

$U_{\rm
ef}(r)=-\frac{(Ze^{2})^{2}}{2mr^{2}}+\frac{l(l+1)}{2mr^{2}}-\frac{\epsilon
Ze^{2}}{mr}$ (8)

is the effective potential, now depending on $l$. Eq. (7) and the ordinary
Schrödinger equation for the radial function in the effective potential
coincide after doing replacements

$l(l+1)-(Ze^{2})^{2}=\lambda(\lambda+1)\,,\quad\epsilon Ze^{2}/m\to
Z^{\prime}e^{2}$ (9)

in the former one. Thus, instead of the energy of the Schrödinger particle in
the Coulomb field we derive

${E}_{n_{r},l}=-\frac{(Z^{\prime}e^{2})^{2}m}{2(n_{r}+\lambda+1)^{2}}\,.$ (10)

Here $n_{r}+\lambda+1=n+\lambda-l$, $n_{r}=0,1,...$ is the radial quantum
number. Solving Eq. (9) and retaining solution with positive-sign square root,
$\lambda=-\frac{1}{2}+\sqrt{(l+\frac{1}{2})^{2}-{(Ze^{2})^{2}}}$, because for
$Z=0$, $l=0$ one should have $\lambda=0$, we find the Sommerfeld formula for a
spinless particle,

$\epsilon^{2}_{n_{r},l}=\frac{m^{2}}{1+\frac{Z^{2}e^{4}}{(n-l-1/2+\sqrt{(l+1/2)^{2}-(Ze^{2})^{2}}\,\,)^{2}}}\,.$
(11)

There are two square-root solutions of this equation. Solution, which yields
$\epsilon\to m$ for $Ze^{2}\ll 1$, $n=1$, $l=0$, describes negatively charged
particle in the attractive Coulomb field ($Z>0)$. Solution, which yields
$\epsilon\to-m$ for $Ze^{2}\ll 1$, $n=1$, $l=0$, $Z>0$, after change of
$\epsilon\to-\epsilon$ describes positively charged particle of the same mass
in the field $Z<0$, since Eq. (1) does not change under simultaneous
replacement $\epsilon\to-\epsilon$ and $Z\to-Z$.

In the limit $Ze^{2}\ll 1$ Eq. (11) for a negatively charged spinless boson,
being in the ground state ($n=1$, $l=0$), produces $\epsilon\to
m-\frac{(Ze^{2})^{2}m}{2n^{2}}$ in accordance with the result for the
Schrödinger particle.

For $Z>Z_{cr}=1/(2e^{2})$ the particle, being in the ground state ($n=1$),
falls down to the center. Let $Ze^{2}=1/2+\delta$ for $0<\delta\ll 1$. Then
choosing positive-sign square root of solution (11) we have for
$Ze^{2}=1/2+\delta$, $\epsilon\simeq\frac{m(1+i\delta)}{\sqrt{2}}$ and the
wave function

$\phi\propto e^{-i\epsilon t}\propto e^{m\delta t/\sqrt{2}}\to\infty\quad{\rm
for}\quad t\to\infty\,,$

being not normalized, reflecting the fact of the falling of the negatively
charged particle to the Coulomb center with $Z>0$ and the falling of the
positively charged particle to the Coulomb center at $Z<0$. We dropped
negative-root solution of Eq. (11) as not physical one, since it appears at
$\epsilon\simeq-m$ already for small $Z>0$. However note that the negative-
root solution of Eq. (11) $-\frac{m(1+i\delta)}{\sqrt{2}}$ for the negatively
charged particle near the Coulomb center for $Z>Z_{\rm cr}=1/(2e^{2})$ yields
$\phi\propto e^{-m\delta t/\sqrt{2}}$, i.e., decreasing at $t\to\infty$. This
implies a possibility of a multi-particle interpretation of the $\epsilon<0$
solution for negatively charged particle in the field $Z>0$. We return to this
question in Subsection 9.2.

The value $Z_{cr}=68.5$. It means that the Mendeleev table would be closed on
element with $Z_{cr}=68$, if nuclei were point-like. As we have mentioned, the
lightest spinless meson is the pion. The radius of the real nucleus with
atomic number $A$ is found from the condition $4\pi\rho_{0}R^{3}/3=A$, where
$\rho_{0}\simeq 0.16$ fm${}^{-3}\simeq 0.48m_{\pi}^{3}$. For a symmetric
nucleus $A\simeq 2Z$ we estimate $R>a_{1\rm B}^{\pi}=1/(m_{\pi}Ze^{2})$
(radius of the ground-state orbit for pion) already for $Z>40$. Then the
lowest pion orbit enters inside the nucleus and approximation of point-like
nucleus becomes invalid.

Note that for $Z=Z_{\rm cr}$, $\epsilon_{\rm part}+\epsilon_{\rm
a.part}=m\sqrt{2}>0$ and thereby pairs are not produced at such conditions.
This peculiarity appears only for the case of the point-like Coulomb field.
For a field, being cut at $R\neq 0$ ($R\ll 1/m_{\pi}$, such that $V=-Ze^{2}/r$
for $r>R$ and $V=-Ze^{2}/R$ (the model I) or for
$V=-\frac{Ze^{2}}{R}(\frac{3}{2}-\frac{r^{2}}{R^{2}})$, the model II at $r<R$,
the ground state particle level continues to decrease with increasing $Z$ and
decreasing $R$ and for $Z=Z_{\rm cr}(R)>Z_{\rm cr}$, it reaches $\epsilon=-m$.
At $Z=Z_{\rm cr}(R)$ the sum $\epsilon_{\rm part}+\epsilon_{\rm a.part}$ is
zero that corresponds to the spontaneous production of the pairs for $Z\geq
Z_{\rm cr}(R)$, at $R<R_{\rm cr}$.

Sommerfeld formula for electron. Electron has spin $1/2$. In absence of the
magnetic field spin and orbit spaces are orthogonal. Thus one may expect that
expression (11) continues to hold also for electron after replacement
$l+1/2\to|\vec{J}|+1/2=|\kappa|$, where $\kappa=-1,0,1...$ is integer number,
since axial vectors of angular momentum and spin are summed up,
$\vec{L}\to\vec{J}=\vec{L}+\vec{s}$. Then we have

$\epsilon^{2}_{n_{r},\kappa}=\frac{m^{2}}{1+\frac{Z^{2}e^{4}}{(n_{r}+\sqrt{\kappa^{2}-(Ze^{2})^{2}}\,\,)^{2}}}\,,$
(12)

where $n_{r}=n-|\kappa|=0,1,...$ is a radial quantum number. Now falling to
the center appears when the ground state level reaches the value $\epsilon=0$.
It occurs for $Z=Z_{\rm cr}=1/e^{2}=137$. For a field cutted at $R\neq 0$,
e.g. for the case $V=-Ze^{2}/r$ for $r>R$ and $V=-Ze^{2}/R$ for $r<R$, the
ground state level continues to decrease with increasing $Z$ and for $Z=Z_{\rm
cr}(R)>Z_{\rm cr}$ it reaches $\epsilon=-m$. After that the sum $\epsilon_{\rm
part}+\epsilon_{\rm a.part}$ reaches zero that corresponds to spontaneous
production of the electron-positron pairs. Two electrons may occupy the
ground-state level and two positrons with $-\epsilon>m$ go off to infinity,

Note that the same expression (12) is derived from the exact solution of the
Dirac equation in the Coulomb field, as we will see in Subsect. 3.5.

##  3 Dirac equation for particle in static electric field,
$A^{\mu}=(A^{0},\vec{0})$

Now we are at the position to focus on the problem of our main interest in
this paper, i.e., to describe behavior of electrons in a strong static
electric field.

Interaction with $4$-vector field $A^{\mu}=(A^{0},\vec{A})$ is constructed
with the help of minimal coupling

$(\hat{p}+e\hat{A}/c-mc)\Psi=0\,,$ (13)

$\hat{A}=\gamma^{\mu}A_{\mu}$, $\gamma_{\mu}$ are ordinary Dirac matrices.

###  3.1  Dirac system in case of one-dimensional electric field

In case of a static one-dimensional electric field ($\vec{A}=0$) using
replacement $\Psi=e^{-i\epsilon t}\tilde{\psi}(x)$ we rewrite Eq. (13) as

$(\epsilon-V+i\gamma^{0}\vec{\gamma}\frac{d}{dx}-\gamma^{0}m+V(x))\tilde{\psi}(x)=0\,.$
(14)

Presenting

$\displaystyle\tilde{\psi}(x)=\left(\begin{array}[]{ccc}G(x)\chi_{\pm}\\\\[8.53581pt]
-iF\sigma_{x}\chi_{\pm}\end{array}\right),\quad\chi_{+}=\left(\begin{array}[]{ccc}1\\\\[8.53581pt]
0\end{array}\right),\quad\chi_{-}=\left(\begin{array}[]{ccc}0\\\\[8.53581pt]
1\end{array}\right)\,,$ (21)

we may rewrite Eq. (14) as

$\displaystyle\psi^{\prime}=\hbar^{-1}\hat{D}\psi\,,\quad\hat{D}=\left(\begin{array}[]{ccc}0&&m+\epsilon-V\\\\[8.53581pt]
m-\epsilon+V&&0\end{array}\right),\quad\psi=\left(\begin{array}[]{ccc}G\\\\[8.53581pt]
F\end{array}\right)\,.$ (26)

For the further convenience we retained here dependence on $\hbar$.

###  3.2  Dirac system in central-symmetric field

Introducing

$\displaystyle{\psi}_{jlm}=\frac{1}{r}\left(\begin{array}[]{ccc}G(r)\Omega_{jlM}(\vec{n})\\\\[8.53581pt]
iF(r)\Omega_{jl^{\prime}M}(\vec{n}^{\,\prime})\end{array}\right),\quad\Omega_{jl^{\prime}M}=-\vec{\sigma}\vec{n}\Omega_{jlM}(\vec{n})\,,$
(29)

where $\Omega_{jlM}$ is the spherical spinor, $j,M$ are full angular momentum
and its projection, $j=l\pm 1/2$, $l$ is orbital angular momentum,
$l+l^{\,\prime}=2j$, $\vec{n}=\vec{r}/r$.

After separation of angular and spin variables the Dirac system becomes

$\displaystyle\psi^{\prime}=\hbar^{-1}\hat{D}\psi\,,\quad\hat{D}=\left(\begin{array}[]{ccc}-\tilde{\kappa}/r&&m+\epsilon-V\\\\[8.53581pt]
m-\epsilon+V&&\tilde{\kappa}/r\end{array}\right),\quad\psi=\left(\begin{array}[]{ccc}G\\\\[8.53581pt]
F\end{array}\right)\,,$ (34)

$\tilde{\kappa}=\hbar\kappa$ , $|\kappa|=j+1/2$. Ground state corresponds to
$\kappa=-1$. One-dimensional result, see (26), follows from (34) provided one
puts $\kappa=0$ and replaces $d/dr\to d/dx$.

###  3.3 Reduction of Dirac system to Schrödinger equation

With the help of replacement

$\phi=(m+\epsilon-V)^{-1/2}G\,,$ (35)

Eq. (34) is reduced to equation of the second-order in $r$-derivative, similar
to the Schrödinger equation,

$\phi^{\,\prime\prime}+p^{2}(r)\phi=0\,,\quad p^{2}=2m(E-U_{\rm ef}(r))\,,$
(36)

where

$E=\frac{\epsilon^{2}-m^{2}}{2m}\,,\quad U_{\rm ef}(r)=\frac{\epsilon
V}{m}-\frac{V^{2}}{2m}+\frac{\kappa(1+\kappa)}{2r^{2}m}+U_{s}\,,$ (37)
$U_{s}=\frac{1}{4m}\left[\frac{V^{\,\prime\prime}}{m+\epsilon-V}+\frac{3}{2}\left(\frac{V^{\,\prime}}{m+\epsilon-V}\right)^{2}-\frac{2\kappa
V^{\,\prime}}{r(m+\epsilon-V)}\right]$ (38)

is the term appeared due to the spin. If $U_{s}$ were zero, after the
replacement $\kappa\to l$ we would recover the Klein-Gordon-Fock equation for
a spinless particle.

At $r\to 0$, for $V=-Ze^{2}/r$ we have $U_{s}\to-\frac{1+4\kappa}{8mr^{2}}$.
For 1s level $\kappa=-1$, $U_{s}\to\frac{3}{8mr^{2}}$. In the latter case

$U_{\rm ef}(r)\to-\frac{(Ze^{2})^{2}}{2mr^{2}}+\frac{3}{8mr^{2}}\,$ (39)

for $r\to 0$. The falling to the center in such a Schrödinger potential occurs
when $U_{\rm ef}(r)<-1/(8mr^{2})$, cf. LL3 , that corresponds to $Ze^{2}>1$.

###  3.4 Interpretation of bound states in a weak field

Dirac equation describes electron and positron simultaneously. Therefore at
appearance of the bound state in a potential well there arises a question does
it relate to the electron or to positron. As example consider case of a weak
external static central-symmetric electric field produced by a static source
of a positive charge distributed in a range $r$. Then $V=-\zeta v(r)<0$ for
the electron, where $\zeta>0$ is a parameter proportional to the depth of the
potential well. As known, for sufficiently small $\zeta$ the Dirac equation,
as the Klein-Gordon-Fock equation, can be transformed to the Schrödinger
equation for a nonrelativistic particle. The bound state for the electron
appears first at a certain value of $\zeta$. At decreasing $\zeta$, this state
is diluted in the continues spectrum with $\epsilon\geq m$.

System of Dirac equations (34) is symmetric in respect to replacements
$\epsilon\to-\epsilon$, $V\to-V$, $\kappa\to-\kappa$, $G\to F$. Equation
describing energy levels does not depend on $G$ and $F$. Thereby it is
symmetric respectively replacements $\epsilon\to-\epsilon$, $V\to-V$,
$\kappa\to-\kappa$. In case of the source of a positive charge the electron
undergoes attraction. In the field of the opposite-sign charge ( $V\to-V$) the
electron undergoes repulsion. Since in the attractive field there appears the
electron energy level going from the upper continuum, in the repulsive field
there appears the electron energy level originating from the lower continuum.
However, since the Dirac equation describes simultaneously electron and
positron, if the electron moves in a repulsive field, then positron moves in
an attractive one. Thereby the electron level going in a repulsive field from
the lower continuum can be interpreted as the positron level
($\epsilon\to-\epsilon$, $\kappa\to-\kappa$) going from the upper continuum
(now in the field of attraction to the positron). It is natural to think that
in a weak repulsive field for the electron for small $\zeta$ a deeply bound
level with $\epsilon\simeq-m$ should not exist. Since such a state
nevertheless exists in the full set of solutions of the Dirac equation, it
after the replacement $\epsilon\to-\epsilon$, $\kappa\to-\kappa$ should be
interpreted as the positron state. This interpretation is confirmed
experimentally. In the field of a proton there are electron bound states lying
near the boundary of the upper continuum but there are no positron states with
$\epsilon\simeq-m$. Vise versa, in the field of an antiproton there exist
positron levels with $\epsilon\simeq m$ but there are no electron levels with
$\epsilon\simeq-m$. This picture is established also by minimization of the
energy in mentioned cases. Namely, in the field of a positive charge presence
of the bound electron is more energetically favorable compared to presence of
the positron.

Statements done above seem obvious except the case, which I shall consider
below in Subsect. 9.2, when polarization of the vacuum may result in a
negative dielectric permittivity and attraction is replaced by repulsion.

###  3.5 Exact solution for electron in Coulomb field of point-like center

Consider discrete spectrum $\epsilon<m$ of the Dirac equation in the potential
$V=-Ze^{2}/r$. We search $G$ and $F$ in Eq. (34) as

$G=\sqrt{m+\epsilon}\,e^{-\tilde{r}/2}\tilde{r}^{g}(Q_{1}+Q_{2})\,,\quad
F=-\sqrt{m-\epsilon}\,e^{-\tilde{r}/2}\tilde{r}^{g}(Q_{1}-Q_{2})\,,$ (40)

where

$\tilde{r}=2r\sqrt{m^{2}-\epsilon^{2}}\,,\quad
g=\sqrt{\kappa^{2}-(Ze^{2})^{2}}\,.$ (41)

This form of the solution follows from asymptotic behavior of $G,F\sim r^{\pm
g}$ at $r\to 0$ and $G,F\sim e^{-\tilde{r}/2}$ at $r\to\infty$. Solutions
$G,F\sim C_{1,2}r^{-g}$ are dropped (i.e. we put $C_{1,2}=0)$ due to
divergence of their contribution to the probability
($\int|\psi|^{2}dr\to\infty$).

Setting (40) in Eq. (34) we obtain system of equations

$\displaystyle\tilde{r}Q_{1}^{\,\prime}+\left(g-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}\right)Q_{1}+\left(\kappa-\frac{Ze^{2}m}{\sqrt{m^{2}-\epsilon^{2}}}\right)Q_{2}=0\,,$
$\displaystyle\tilde{r}Q_{2}^{\,\prime}+\left(g+\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}-\tilde{r}\right)Q_{2}+\left(\kappa+\frac{Ze^{2}m}{\sqrt{m^{2}-\epsilon^{2}}}\right)Q_{1}=0\,.$
(42)

These equations are reduced to

$\displaystyle\tilde{r}Q_{1}^{\,\prime\prime}+\left(2g+1-\tilde{r}\right)Q_{1}^{\,\prime}-\left(g-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}\right)Q_{1}=0\,,$
$\displaystyle\tilde{r}Q_{2}^{\,\prime\prime}+\left(2g+1-\tilde{r}\right)Q_{2}^{\,\prime}-\left(g+1-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}\right)Q_{2}=0\,.$
(43)

As is seen, Eqs. (43) are symmetric under simultaneous replacement
$\epsilon\to-\epsilon$ and $Ze^{2}\to-Ze^{2}$.

Finite solution for $\tilde{r}\to 0$ gets the form

$Q_{1}=AF\left(g-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}\,,2g+1,\tilde{r}\right)\,,\quad
Q_{2}=BF\left(g+1-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}\,,2g+1,\tilde{r}\right)\,,$
(44)

where $F(\alpha,\beta,z)$ is the degenerate hypergeometric function. Setting
$\tilde{r}=0$ in one of Eqs. (43) we find relation

$B=-\frac{g-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}}{\kappa-\frac{Ze^{2}m}{\sqrt{m^{2}-\epsilon^{2}}}}A\,.$
(45)

Both hypergeometrical functions in (44) are reduced to polinomials, otherwise
they would grow as $e^{\tilde{r}}$ for $\tilde{r}\to\infty$, that results in
divergence of the probability. From this requirement follows that $\alpha$ in
$F(\alpha,\beta,z)$ equals to a non-positive integer number, i.e.,

$g-\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}=-n_{r}\,,\quad
n_{r}=1,2,...$ (46)

For $n_{r}=0$ only one of two functions is reduced to a polinomial. Then ,
$g=\frac{Ze^{2}\epsilon}{\sqrt{m^{2}-\epsilon^{2}}}$ and
$\frac{Ze^{2}m}{\sqrt{m^{2}-\epsilon^{2}}}=|\kappa|$. If $\kappa<0$, then
$B=0$ in Eq. (45) and $Q_{2}=0$ and required condition is fulfilled. If
$\kappa>0$, then $B=-A$ and $Q_{2}$ is a divergent function at $n_{r}=0$.
Thereby permitted are $n_{r}=0,1,...$ for $\kappa<0$ and $n_{r}=1,2,...$ for
$\kappa>0$. From (46) also follows solution for negatively charged particle
with $\epsilon<0$ for $Z<0$. In a single particle problem under consideration
one should drop such a solution, since it describes a strongly bound particle
already in a weak field. However such a solution can be appropriately treated
within a many-particle picture with taking into account vacuum polarization
and electron condensation originated in lower continuum, as we argue below in
Subsect. 9.2.

From (46) we obtain Sommerfeld expression

$\epsilon=\pm
m\left[1+\frac{(Ze^{2})^{2}}{(\sqrt{\kappa^{2}-(Ze^{2})^{2}}+n_{r})^{2}}\right]^{-1/2}\,,$
(47)

cf. Eq. (12). Note that for $Z>0$, $Z<1/e^{2}$, only solution $\epsilon>0$
follows from (46), since $n_{r}+g>0$. Thereby “$+$” sign solutions (47)
correspond to particles (electrons) in the field of positively charged Coulomb
center (or to antiparticles – positrons in the field of negatively charged
Coulomb center). The “$-$” sign solutions (47), after replacements
$\epsilon\to-\epsilon$, $\kappa\to-\kappa$ (after that “$-$” sign branch
coincides with “$+$” sign branch) describe antiparticles with $\epsilon>0$ in
the field of negatively charged Coulomb center ($Z<0$).

Ground state 1s-level of electron in the field of positively charged Coulomb
center ($Z>0$) corresponds to $\kappa=-1$, $n_{r}=0$. Its energy is

$\epsilon_{0}=mg_{0}\,,\quad g_{0}=\sqrt{1-(Ze^{2})^{2}}\,.$ (48)

At $Ze^{2}\geq 1$ there occurs falling of the electron to the center. Indeed,
for $r\to 0$ following (40), (43) we get

$G=a_{1}r^{g}+a_{2}r^{-g}\,,\quad F=b_{1}r^{g}+b_{2}r^{-g}\,.$ (49)

For $Ze^{2}=1+\delta>1$ the value $g=i\sqrt{2\delta}$ becomes imaginary and
solutions oscillate as

$C_{1}\cos({|g|}\ln r)+C_{2}\sin({|g|}\ln r)\,,$ (50)

that corresponds to not normalized probability
$\int_{0}^{\infty}|\psi|^{2}dr$. At $Ze^{2}=1+\delta$, $0<\delta\ll 1$,
solution of Eq. (48) yields $\epsilon=+im\sqrt{2\delta}$ and the electron wave
function grows as $\Psi\propto e^{+m\sqrt{2\delta}t}$ indicating falling of
the electron to the center. Solution of opposite sign (see Eq. (47)) rises
from negative continuum at $V\to 0$. In single-particle problem a negative-
energy solution should be dropped. Note that at $Ze^{2}=1+\delta$, it yields
$\epsilon=-im\sqrt{2\delta}$ and $\Psi\to 0$ at $t\to\infty$ that may suggest
an interpretation. However an appropriate interpretation proves to be possible
only beyond the single-particle problem, as will be shown in Subsect. 9.2.

Solutions (47) and (48) hold formally for the positron in the Coulomb
potential of the nucleus with the charge $Z<0$. Within the single-particle
problem under consideration, appropriate interpretation again exists for the
solution, which energy is originated from the upper continuum decreasing with
increasing $-Z$, rather than negative-energy solution, similarly to that
happened for the electron at $Z>0$. Only two (due to Pauli principle)
electrons, if they have occupied the ground state, undergo falling to the
Coulomb center for $Ze^{2}=1$. For levels with quantum number $n_{r}>0$ we
have $\epsilon_{n_{r},\kappa}>0$ for $Z=1/e^{2}$. Now assume that the ground-
state level was empty and we adiabatically increase $Z$. There is no
appropriate solution of the single-particle problem for the point-like nucleus
with $Z>1/e^{2}$ in this case.

Avoiding problem of falling to the center. A reasonable interpretation may
appear, only if one assumes that the nucleus has a size $R\neq 0$ and then we
may decrease $R$. Assume first that $R\ll r_{\Lambda}=1/m$. In the limit
$\Lambda=\ln(r_{\Lambda}/R)\gg 1$ for the ground-state level of the electron
one gets Popov1971ZhETF33 ; ZeldovichPopov1972

$\epsilon_{0}(\zeta<1)=mg_{0}/\mbox{th}(\Lambda g_{0})\,,\quad{\rm
for}\quad\zeta=Ze^{2}<1\,.$ (51)

For $\zeta<1$ the value $\mbox{th}(\Lambda g_{0})\simeq 1+e^{-2\Lambda g_{0}}$
rapidly tends to unity and Eq. (51) coincides with (48). For $R\neq 0$ the
point $\zeta=1$ is already not a singular point for the function
$\epsilon_{0}(\zeta)$. Eq. (51) is analytically continued in the region
$\zeta>1$. For $\zeta$ close to unity we have

$\epsilon_{0}(\zeta>1)=m\tilde{g}_{0}/\mbox{tg}(\Lambda\tilde{g}_{0})\,,\quad{\rm
for}\quad\zeta=Ze^{2}>1\,,$ (52)

where $\tilde{g}_{0}=\sqrt{\zeta^{2}-1}$. One gets
$d\epsilon_{0}(\zeta)/d\zeta\to\infty$ for $R\to 0$. At
$\tilde{g}_{0}=\pi/\Lambda$, Eq. (51) has a spurious pole. At fixed $R$ the
curve $\epsilon_{0}(\zeta>1)$ continues to decrease with increasing $\zeta$
and reaches the boundary of the lower continuum. It occurs at $\zeta_{\rm
cr}=1+\pi^{2}/(2\Lambda^{2})+O(\Lambda^{-3})$.

A comment is in order. Single-particle solution for $R\to 0$ should be
modified. Indeed, for $R$ as small as $R\sim r_{\rm L}\simeq
r_{\Lambda}e^{-3\pi/(2e^{2})}$, the multi-particle effects of the polarization
of the vacuum should be included and the problem goes beyond the single-
particle one, see below consideration in Sect. 8.

###  3.6 Avoiding problem of falling to center in realistic treatment.
Spherical nucleus of finite size

For the Coulomb field with the charge $Z<1/e^{2}$, the electron in the ground
state is typically situated at distances $\sim a_{1\rm B}=1/(Z_{\rm
obs}e^{2}m)>1/m$ and distribution of the charge $Z(r)$ at distances $r\sim
R_{\rm nucl}\ll a_{1\rm B}$ almost does not affect the electron motion. In the
realistic problem the nucleus has a finite size, $R_{\rm nucl}\simeq
r_{N}A^{1/3}\ll a_{1\rm B}$, where $A$ is atomic number, $r_{N}\simeq 1.2$ fm,
and thereby the potential is smoothen at $r<R_{\rm nucl}$. The falling to the
centrum does not occur, as it has been mentioned. Even for $Z\gg 1/e^{2}$ the
electron density remains to be distributed at finite distances.

With taking into account of distribution of the charge inside the nucleus we
have

$V(r)=-\zeta f(r/R_{\rm nucl})/R_{\rm nucl}\quad{\rm for}\quad 0<r<R_{\rm
nucl},\quad V=-\zeta/r\,,\quad{\rm for}\quad r>R_{\rm nucl}\,.$ (53)

Two models have been employed in the literature: model I, when $f(x<1)=1$ that
corresponds to the surface distribution of the charge, and model II, when
$f(x<1)=(3-x^{2})/2$ that describes distribution of protons with the constant
volume density.

The energy shift of the electron level can be found with the help of the
perturbation theory applied to the Dirac system (34). Following
ZeldovichPopov1972 ,

$\beta=\frac{\partial\epsilon}{\partial\zeta}=\int
V(r)(G^{2}+F^{2})dr/\zeta<0\,,$ (54)

i.e., the curve $\epsilon(\zeta)$ decreases monotonically with increasing
$\zeta$ and crosses the boundary of the lower continuum with a finite value
$\beta$. After that $\epsilon(\zeta)$ acquires an exponentially small
imaginary part.

Since exact solution of the Coulomb problem for $r>R$ looks rather cumbersome
and for $r<R$ is impossible for a realistic cut of the potential, it is
natural to use approximate methods. Most economical is semiclassical approach.
Here we should notice that the replacement (35) becomes singular for
$\epsilon<-m$ in the point $V(r_{1})=m+\epsilon<0$. Due to this the effective
potential

$U_{\rm ef}(r,\epsilon)=\frac{3}{8m}(r-r_{1})^{-2}+...\to\infty\,,\quad{\rm
for}\quad r\to r_{1}\,,$ (55)

and semiclassical expressions loose their sense due to divergency of the
integral $\int^{r}2m(E-U_{\rm ef}(r,\epsilon))^{1/2}dr$. However this is only
a formal problem since initial Dirac system (34) has no singularity at $r\to
r_{1}$. To avoid the problem one should bypass the singular point in the
complex plane, as one usually does bypassing turning points, or one may apply
semiclassical consideration straight to the linear Dirac equations. Note that
in the one-dimensional case corresponding to $\kappa=0$, see Eq. (26), the
mentioned singularity occurs in the turning points and one may use standard
semiclassical methods.

Probability of the spontaneous production of positrons is determined by the
width of the corresponding electron level, $\mbox{Im}\,\epsilon$, for
$\mbox{Re}\,\epsilon<-m$. Thus the width is found from the solution of the
Dirac equation. The value $\Gamma$, which determines probability of the
positron production, $W\sim e^{\Gamma t}$, can be expressed directly through
components of the Dirac bispinor ($G$ and $F$). It yields the flux of
particles going to infinity (at normalization on one particle):

$\Gamma=\int\psi^{\dagger}\gamma^{0}\vec{\gamma}\psi
d\vec{f}=2\mbox{Im}(FG^{*})|_{r\to\infty}\,.$ (56)

##  4 Semiclassical approach to Dirac equation transformed to second-order
differential equation

###  4.1 Accuracy of calculation of energy levels in semiclassical
approximation

Substituting $\psi=Ae^{iS/\hbar}$, where $A$ and $S$ are real quantities, in
equation

$\hbar^{2}\Delta\psi+p^{2}(r)\psi=0\,$ (57)

we find two equations

$\hbar^{2}\Delta A+p^{2}A=A(\nabla S)^{2}\,,\quad i\hbar(2\nabla A\nabla
S+A\Delta S)=0\,.$ (58)

For a convenience, dependence on $\hbar$ is here recovered. The Hamilton-
Jacobi equation for the action $(\nabla S)^{2}=p^{2}$ is obtained provided

$\hbar^{2}\frac{A^{\,\prime\prime}}{p^{2}A}\sim\frac{\hbar^{2}}{(pl)^{2}}\sim\left(\frac{d\tilde{\lambda}}{dr}\right)^{2}\ll
1\,,\quad\tilde{\lambda}=\frac{\hbar}{p}\,,$ (59)

where $l$ is typical size of the potential $V$. For the Coulomb potential at
typical distances $r\sim 1/(2m)$ characterising ground-state electron with
$\epsilon\simeq-m$ we have $p\sim\tilde{g}/r$. From estimate (59) we see that
semiclassical approximation for the wave function for such distances is
accurate up to terms $1/\tilde{g}^{2}$,
$\tilde{g}=\sqrt{\zeta^{2}-\kappa^{2}}$ for $\zeta>|\kappa|$.

Using the Bohr-Sommerfeld quantization rule we have

$\frac{\hbar^{2}}{(pl)^{2}}\sim\frac{\hbar^{2}}{(\int_{r_{0}}^{r_{-}}dr)^{2}}\sim\frac{1}{\pi^{2}(n_{r}+\gamma)^{2}}\,,$
(60)

where the phase $\gamma\sim 1$, $n_{r}=0,1,...$, $r_{0}$ and $r_{-}$ are the
turning points separating the classically allowed region. Thus even in
calculation of the energy of the levels with small quantum numbers one may
consider on the error not larger that 10% .

Finally let us notice that the transition from the Dirac equation in the
external field to the corresponding more simple Hamilton-Jacobi equation has
been used in many investigations, cf. Pauli1980 ; Rubinow1963 ; Stachel1977 .
The case of deep levels with binding energy
$\stackrel{{\scriptstyle\scriptstyle>}}{{\phantom{}{}_{\sim}}}m$ was studied
in Eletskii:1977na ; Mur:1978nb ; Mur:1978ke ; Popov:1978kk ; Popov:1979gq .

###  4.2 Semiclassical approximation to Coulomb field of point-like nucleus

In the field $V=-\zeta/r$ for $\zeta<|\kappa|$ semiclassical method results in
exact expression for the energy spectrum. Let us show this. For that we do
replacements

$G=\sqrt{\frac{m+\epsilon}{r}}\,(\chi_{1}+\chi_{2})\,,\quad
F=\sqrt{\frac{m-\epsilon}{r}}\,(\chi_{1}-\chi_{2})\,.$ (61)

Then the system of two Dirac equations (34) reduces to equations

$\chi_{i}^{\,\prime\prime}+p_{i}^{2}(r)\chi_{i}=0\,,\quad i=1,2\,,$ (62)

with

$p_{i}(r)=\left[\epsilon^{2}-m^{2}-\frac{2\epsilon\zeta\pm\sqrt{m^{2}-\epsilon^{2}}}{r}+\frac{\zeta^{2}-\kappa^{2}+1/4}{r^{2}}\right]^{1/2}\,.$
(63)

Adding the Langer correction to the effective potential that results in
replacements $p_{i}\to p_{i}^{*}$ we find

$p_{i}^{*}(r)=\sqrt{-a+2b/r-g^{2}/r^{2}}\,,\quad a=m^{2}-\epsilon^{2}\,,\quad
b=\epsilon\zeta\pm\frac{1}{2}\sqrt{m^{2}-\epsilon^{2}}\,,\quad
g^{2}=\kappa^{2}-\zeta^{2}\,.$ (64)

Then applying the Bohr-Sommerfeld quantization rule we have

$\int_{r_{0}}^{r_{-}}p_{i}^{*}dr=(a^{-1/2}b-g)\pi=(n_{r}+1/2)\pi\,.$ (65)

From here we recover the exact result (47). To get (47) from exact solution of
the Dirac equations we have performed a cumbersome analysis of hypergeometric
functions, whereas the semiclassical approach needs taking only one simple
integral.

After replacements $b\to\epsilon\zeta$, $g^{2}\to(l+1/2)^{2}-\zeta^{2}$, Eq.
(64) is valid also for spinless bosons. Performing integration leads us to the
exact expression (11).

###  4.3 Finite nucleus. Semiclassical wave functions and quantization rule.

Certainly it is possible to apply semiclassical approach also to Eq. (36) with
effective potential in the form (37), (38). In the range, where the parameter
of applicability of semiclassical approximation is $\sim 1$, usage of Dirac
equations presented in different forms leads to slightly different results.
For instance applying (36) to the Coulomb field does not yield the exact
result for the energy of the levels although accuracy of the approximation
proves to be appropriate. For $\epsilon<-m$ replacement (35) leads to the
singularity in the point $r_{1}$ where $V(r_{1})=m+\epsilon<0$ . Near this
point semiclassical expressions become invalid due to divergency of the
contribution to the action $\int[2m(E-U_{\rm ef})]^{1/2}dr$. However, as it
was mentioned, this circumstance is not reflected on the calculation of the
energy levels, since $r_{1}$ is situated under the barrier, where wave
functions prove to be exponentially small.

Electron energy levels can be found with the help of the Bohr-Sommerfeld
quantization rule Eletskii:1977na  applied to the Dirac equation presented in
the form (36) with effective potential in the form (37), (38). We have

$\int_{r_{0}}^{r_{-}}p^{*}dr=(n_{r}+\gamma^{\,\prime})\pi\,.$ (66)

Value $p^{*}$ is obtained from expression (36) after taking into account the
Langer correction, i.e. after doing the replacement
$\kappa(1+\kappa)/r^{2}\to(\kappa+1/2)^{2}/r^{2}$ in the expression for the
effective potential. The value of the phase $\gamma^{\,\prime}$ depends on
whether the turning point is inside the nucleus or outside it. In the latter
case the potential is $V=-\zeta/r$ and $\gamma^{\,\prime}=3/4$ for $\kappa=-1$
and $\gamma^{\,\prime}=1/2$ for $\kappa\neq-1$.

The contribution to the normalization of the semiclassical wave function from
classically forbidden region is usually dropped. In order to understand
accuracy of this approximation consider the probability of the presence of the
electron in sub-barrier region $r_{-}<r<r_{+}$:

$W_{0}=\int_{r_{-}}^{r_{+}}(G^{2}+F^{2})dr\,.$ (67)

To be specific let us put $\epsilon=-m$, and consider $\zeta\gg|\kappa|$. The
wave function in the classically allowed region is Migdal1975 :

$\chi=(c_{0}/\sqrt{p^{*}})\sin(\int_{r_{0}}^{r}p^{*}dr+\pi/4)dr\,.$ (68)

Constant $c_{0}$ is found from the normalization condition Migdal:1977rn ,

$2\int_{r_{0}}^{r_{-}}(\epsilon-V)\chi^{2}dr/m\simeq 1\,.$

Then we expand the effective potential (37) near the turning point. For
$V=-\zeta/r$ we obtain

$U_{\rm
ef}=\zeta/r-\tilde{g}^{2}/(2r^{2}m)=U(r_{-})+4m^{2}(r-r_{-})/\zeta+...,\quad\tilde{g}=\sqrt{\zeta^{2}-\kappa^{2}}\,,$
(69)

for $r-r_{-}\ll r_{-}\sim\zeta$. Solution of Eq. (36) in potential (69) is
expressed through the Airy function

$\chi(r)=(-V)^{-1/2}G=c_{0}Ai(2\zeta^{-1/3}(r-r_{-}))\,.$ (70)

The probability to find the particle in the sub-barrier region is

$W=-\int_{r_{-}}^{\infty}(V/m+1)\chi^{2}dr\simeq
c_{0}^{2}\int_{0}^{\infty}Ai^{2}(2x\zeta^{-1/3})dx=c_{1}\zeta^{-1/3}\,,$ (71)

where $c_{1}=3^{4/3}\Gamma^{2}(2/3)/16\pi\simeq 0.158$.

Thus the probability of a penetration of the electron in classically forbidden
region is numerically small for $\zeta\sim 1$ and falls down with increasing
$\zeta$. This justifies that we neglected the contribution of the region
$r>r_{-}$ at normalization of the wave functions (taking $r_{0}<r<r_{-}$).
Note that quantization rule remains applicable with a larger accuracy,
$1/\zeta^{2}$, since at its derivation it was not used how wave functions are
normalized. Strictly speaking, in case of quasistationary levels the
quantization rule is slightly modified, due to $\mbox{Im}\epsilon\neq 0$, cf.
Sergeev1991 . However changes of the energy levels are exponentially small due
to exponential smallness of the penetrability of the barrier.

With the semiclassical $\chi$ function we obtain expression for the averages
$\overline{r^{\lambda}}$, For $\epsilon=-m$ and $\zeta\gg|\kappa|$ one has
Eletskii:1977na ,

$\overline{r^{\lambda}}=\zeta^{\lambda}\frac{3\pi^{1/2}(\lambda+2)\Gamma(\lambda+1)}{m^{\lambda}2^{\lambda+3}\Gamma(\lambda+5/2)}\left[1-\frac{(\lambda+3/2)\kappa^{2}}{(\lambda+2)\zeta^{2}}+...\right]\,.$
(72)

$\Gamma(x)$ is the Euler $\Gamma$-function. For $\zeta\sim 1$ accuracy of this
expression is not as good but it increases appreciably with increasing
$\zeta$.

The quantity $\overline{r}$ characterizes mean radius of the bound state at
$\epsilon=-m$, values $\overline{r}^{\lambda}$ at $\lambda=1/2,3/2,2$ are met
in the problem of the modification of the value $Z_{\rm cr}$ due to a
screening of the charge by other electrons of the ion (if they are), see below
in Subsect. 6.4. Comparison of the semiclassical expressions with the exact
solutions found numerically shows an appropriate accuracy of the semiclassical
results even for $\zeta\sim|\kappa|\sim 1$. For $\zeta\gg\kappa\gg 1$ the
result (72) coincides with the corresponding asymptotic of the exact solution.

###  4.4 Critical charge of the nucleus

Let us calculate the critical charge of the nucleus (when the electron level
with quantum numbers $n,\kappa$ reaches $\epsilon=-m$). Using the Bohr-
Sommerfeld quantization rule in the form (66) one obtains, cf. Marinov1974 ,

$mR_{\rm nucl}=\tilde{g}^{2}/(2\zeta\mbox{ch}^{2}y)\,,$ (73)

where $y$ is positive root of the equation

$y-\mbox{th}y=\frac{(n_{r}+\gamma_{1})\pi-\tilde{\gamma}}{2\tilde{g}}\,,\quad\tilde{\gamma}=\mbox{arcctg}(\Xi/\tilde{g})\,,$
(74)

$n_{r}=0,1,...$ radial quantum number, $\gamma_{1}=3/4$ for $ns$ levels and
$\gamma_{1}=1/2$ for $\kappa\neq-1$.

In Ref. Marinov1974  quantity $\Xi$ was found from matching of the exact
solution inside the nucleus and semiclassical one outside the nucleus. As was
shown in Voskresensky1977 , usage of the semiclassical solutions both inside
and outside the nucleus does not spoil the accuracy of the result. Therefore
we further follow consideration of Voskresensky1977 .

For the model I, the semiclassical solution inside the nucleus coincides with
the exact one and we find

$\Xi=\beta\mbox{ctg}\beta\,\quad\beta=\sqrt{\zeta(\zeta-2R_{\rm nucl}m)}\,.$
(75)

Note here that a first estimate of $R_{\rm cr}$ in this model was performed in
Krainov1971 , where it was taken $\tilde{\gamma}=\zeta$, that differs from
that follows from (74), (75).

For the model II, analytical expression can be found expanding $p(r<R)$ in the
parameter $\zeta$,

$\tilde{\gamma}=\mbox{arcctg}[((p^{*}(R_{\rm
nucl})/(\tilde{g}m))\mbox{ctg}\int_{0}^{R_{\rm nucl}}p^{*}dr]\,,$ (76)
$\displaystyle\int_{0}^{R_{\rm
nucl}}p^{*}dr=\int_{0}^{1}dx[\zeta^{2}f^{2}(x)-2\zeta R_{\rm
nucl}mf(x)-9/(4f^{2}(x))]^{1/2}$
$\displaystyle=\frac{4}{3}\zeta\left[1-\frac{c_{2}}{\zeta^{2}}-\frac{3}{4}\frac{R_{\rm
nucl}m}{\zeta}+O\left(\frac{1}{\zeta^{4}},\frac{R_{\rm
nucl}^{2}m^{2}}{\zeta^{2}}\right)\right]\,,\quad
c_{2}=\frac{9}{32}\left(1+\frac{1}{\sqrt{3}}\mbox{arth}\frac{1}{\sqrt{3}}\right)\,,\quad\kappa=-1\,,$
(77)

where $f(x)$ follows Eq. (53), here for the model II. Although parameter of
applicability of semiclassical expressions to the Coulomb field is
$\tilde{g}\gg 1$, difference of the above obtained expression with the result
of the exact calculation is less than few percents even at $\zeta=\zeta_{\rm
cr}\simeq 1.24$.

For $\zeta\sim 1$ expanding (74) in $1/y$ and dropping numerically small term
$e^{-2y}$, from Eq. (73) we finally find

$R_{\rm nucl}\simeq\frac{2\tilde{g}_{\rm cr}^{2}}{\zeta_{\rm
cr}m}\left[\mbox{exp}\left(\frac{\pi(n_{r}+\gamma_{1})-\tilde{\gamma}}{\tilde{g}_{\rm
cr}}+2\right)+2\right]^{-1}\,.$ (78)

from where we find $Z_{\rm cr}(R_{\rm nucl})$.

###  4.5 Number of levels which crossed boundary of lower continuum

Now let us find the number of levels $n_{\kappa}$ with fixed quantum number
$\kappa$ and the total number of levels $N$, which have crossed the boundary
$\epsilon=-m$. For this aim Mur:1978ke  we need to use Bohr-Sommerfeld
quantization rule at $\epsilon=-m$. For $\tilde{g}\gg 1$, we have
$d\tilde{\lambda}/dr\ll 1$. For $\zeta\gg 1$ this means that
$\zeta-|\kappa|\gg\zeta^{-1}$, i.e., semiclassical approximation can be
violated only for states with the momenta at which
$\zeta-|\kappa|\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}\zeta^{-1}$.
Accuracy of the semiclassical expressions for the wave function is $\sim
1/\zeta^{2}$, cf. Migdal:1977rn . With taking into account these
approximations employing the Bohr-Sommerfeld quantization rule we obtain

$n_{\kappa}=\frac{1}{\pi}\int(V^{2}+2Vm-\kappa^{2}/r^{2})^{1/2}dr\,.$ (79)

For the potential given by Eq. (53), for $R_{\rm nucl}m\ll 1$ we obtain

$n_{\kappa}=\frac{g}{\pi}[2(\mbox{Arth}\sqrt{1-\eta}-\sqrt{1-\eta})+h(\rho)\,,$
(80)

where $\rho=|\kappa|/\zeta$ , $\eta=R_{\rm nucl}/r_{-}=2R_{\rm
nucl}m/(\zeta(1-\rho^{2}))$, $r_{-}$ is the turning point in the effective
potential, $h(\rho)$ takes into account integral over the interior region of
the nucleus $0<r<R_{\rm nucl}$,

$h(\rho)=(1-\rho^{2})^{-1/2}\int_{x_{0}}^{1}[f^{2}(x)-\rho^{2}x^{-2}]^{1/2}dx\,,$
(81)

where $x_{0}=x_{0}(\rho)$ is the root of equation $xf(x)=\rho$.

For $Ze^{3}\ll 1$ (at this condition distribution of electrons, which fill the
vacuum shell, only slightly modifies the bare potential, as we shall see
below) Eq. (79) correctly determines distribution of electrons with
$\epsilon<-m$ of the supercritical atom over the momenta $j=|\kappa|-1/2$.
Maximum value of $j$ corresponds to $r_{-}=R_{\rm nucl}$, $\eta=1$,

$\kappa_{\rm max}=\zeta-R_{\rm nucl}m+O(R_{\rm nucl}^{2}m^{2}/\zeta)\,.$ (82)

Total number of levels with $\epsilon<-m$,

$N=\sum_{\kappa}n_{\kappa}\,,$ (83)

can be found by replacing the summation by the integration. We should take
into account that in the Dirac equation $|\kappa|\geq 1$. Thereby we sill
should subtract spurious term $\kappa=0$. Thus

$\displaystyle N=\int
dr[\frac{1}{2}(V^{2}+2Vm)r-\frac{1}{\pi}(V^{2}+2Vm)^{1/2}]$
$\displaystyle=A_{1}\zeta^{2}\ln\frac{\zeta}{R_{\rm
nucl}m}+A_{2}\zeta^{2}+A_{3}\zeta\ln\frac{\zeta}{R_{\rm
nucl}m}+A_{3}\zeta+A_{4}+...,$ (84)

where $A_{1}=1/2$, $A_{2}=\int_{0}^{1}f^{2}(x)xdx-\ln 2-1$ , $A_{3}=-1/\pi$,
$A_{4}=-\frac{1}{\pi}(\int_{0}^{1}f(x)dx+\ln 2-2)$.

For the model II, result of this calculation is shown in Fig. 3.

Figure 3: Number of levels with $\epsilon<-m$ for the potential of the model
II. The stepwise broken line represents a numerical solution of the Dirac
equation, while the curve $Q$ was computed according to the semiclassical Eq.
(84).

We again observe excellent accuracy of the semiclassical result, even for
$\zeta\sim 1$.

###  4.6 Energy of single-particle levels at $\epsilon<-m$

####  4.6.1 Energy spectrum for $|\epsilon|-m\ll m$

Expand effective potential in $m+\epsilon$, cf. Eletskii:1977na :

$U_{\rm ef}(r,\epsilon)=\sum_{n=0}^{\infty}(m+\epsilon)^{n}u_{n}(r)\,,$ (85)

where $U_{\rm ef}(r,\epsilon)$ can be taken following Eq. (37). Here
$u_{0}(r)=U_{\rm ef}(r,\epsilon=-m)$. For $n\geq 1$,

$u_{n}=\frac{V}{m^{n}}\delta_{n1}+\frac{1}{mV^{n}}\left[-\frac{V^{\,\prime\prime}}{4V}+\frac{3(n+1)}{8}\left(\frac{V^{\,\prime}}{V}\right)^{2}+\frac{\kappa
V^{\,\prime}}{2rV}\right]\,,$ (86)

where $\delta_{n1}$ is the Kronecker symbol.

Energy of the levels is found from the Bohr-Sommerfeld quantization condition

$\int_{0}^{r_{-}}\sqrt{-2mu_{0}}dr+(m+\epsilon)\int_{0}^{r_{-}}\sqrt{-2u_{1}}dr+O((1+\epsilon/m)^{3/2})=(n_{r}+\gamma^{\prime})\pi\,.$
(87)

As before, $\gamma=3/4$ for levels with $\kappa=-1$ and $\gamma=1/2$ for
$\kappa\neq-1$. With the help of (87) we find

$\epsilon=-m+\beta(\zeta_{\rm cr}-\zeta)+...\,,$ (88)
$\beta=f_{2}/f_{1}\,,\quad
f_{1}=\int_{0}^{r_{-}}\sqrt{-2m^{2}u_{1}}\,dr\,,\quad
f_{2}=\int_{0}^{r_{-}}\frac{\zeta f^{2}/(mR_{\rm
nucl})-f}{\sqrt{-2u_{0}/m}\,R_{\rm nucl}}dr\,.$

Comparison of numerical calculation done following these expressions with that
for the exact Dirac equation again shows a good agreement. Note that the value
$\beta$ determines the threshold behavior of the probability of the production
of positrons.

####  4.6.2 Energy spectrum for $|\epsilon|\ll-m$

This spectrum has been found in Mur:1978ke . For $\zeta\gg\zeta_{\rm cr}$ many
levels have energies $|\epsilon|\ll-m$. In this case, as follows from Eq.
(37), (38), the terms $\propto\kappa$ in centrifugal potential and in spin
term cancel each other. Approximately we have

$p^{*}(r)\simeq[(\epsilon-V)^{2}-\kappa^{2}/r^{2}]^{1/2}\,.$ (89)

For $k=\sqrt{\epsilon^{2}-m^{2}}<\zeta/R_{\rm nucl}$ the turning point $r_{-}$
lies outside the nucleus, $r_{-}>R_{\rm nucl}$. Employing the Bohr-Sommerfeld
quantization condition we get

$\displaystyle k_{n}\simeq|\epsilon_{n}|=c_{0}\zeta R_{\rm
nucl}^{-1}e^{-n\pi/\zeta}=\zeta R_{\rm
nucl}^{-1}e^{-(n-n_{*})\pi/\zeta}\,,\quad n>n_{*}\,,$ (90) $\displaystyle
c_{0}=\mbox{exp}(\int_{0}^{1}f(x)dx-1)\,,\quad
n_{*}=\zeta\pi^{-1}(\int_{0}^{1}f(x)dx-1)\,.$

For deeper levels, $k>\zeta R_{\rm nucl}^{-1}$, classically permitted region
$r_{0}<r<r_{-}$ is completely inside the nucleus. Thereby the spectrum is
entirely determined by the expression for $f(x)$:

$k_{n}=\zeta R_{\rm nucl}^{-1}f(\Xi_{n})\,,\quad 1\ll n\ll n_{*}\,,$ (91)

where $\Xi_{n}$ is the root of equation

$\int_{0}^{\Xi}f(x)dx-\Xi f(\Xi)=n\pi/\zeta\,.$ (92)

For example, for the model II at $1\ll n\ll n_{*}$ we have

$k_{n}=\frac{\zeta}{2R_{\rm nucl}}\left[3-(n/n_{*})^{2/3}\right]\,,\quad
n_{*}=\frac{\zeta}{3\pi}\,.$ (93)

From these expressions it is easy to find expression for the level density
$dn/d\epsilon$. For model II we find

$dn/d\epsilon=Cy^{-1}\,,\quad{\rm for}\quad 0<y<1\,$ (94)

and

$dn/d\epsilon=C(3-2y)^{1/2}\,,\quad{\rm for}\,,\quad 1<y<3/2\,$ (95)

for $y=kR_{\rm nucl}/\zeta$, $C=const$. From here we see the crowding of
levels toward the boundary $\epsilon=-m$ ($k\to 0$).

For levels with arbitrary angular momenta the “Coulomb” part of the spectrum
gets the form

$\epsilon_{n\kappa}=-\zeta R_{\rm
nucl}^{-1}c(\rho)\mbox{exp}(-n\pi/\tilde{g})\,,$ (96)

where $\rho=|\kappa|/\zeta$, $0<\rho<1$. Pre-exponential factor

$c(\rho)=\mbox{exp}\left[\ln(2(e\rho)^{-1}(1-\rho^{2}))-(1-\rho^{2})^{-1/2}\mbox{Arth}(1-\rho^{2})^{1/2}+h(\rho)\right]\,,$
(97)

where $h(\rho)$ given by Eq. (81) depends on the $f(x)$, $e=2.718...$ is the
Euler number. The function $c(\rho)$ monotonically decreases with increase of
$\rho$ from 1 for model I and from $\simeq 1.4$ for model II at $\rho=0$ up to
zero for $\rho=1$ in both models.

Eq. (96) is obtained at the condition that the turning point $r_{-}$ lies
inside the nucleus. Condition of applicability of Eq. (96) is
$\tilde{g}/\pi\ll n<n_{\kappa}$. Since
$n_{\kappa}\simeq(\tilde{g}/\pi)\ln(\zeta/R_{\rm nucl})$, then due to large
values of the logarithm this equation describes most of levels crossed the
boundary $\epsilon<-m$.

The exponential dependence of $\epsilon_{n}$ on $n$ and crowding of levels
near $\epsilon=-m$, as follows from Eqs. (90), (96), are related to the fact
that $U_{\rm ef}\simeq-\tilde{g}^{2}/r^{2}$ for $r\to 0$. If $R$ was zero, the
electrons would collapse to the center. The spectrum of the Schrödinger
equation in such a potential behaves as MorsFeshbach ,

$E_{n}=E_{0}e^{-2\pi n/\tilde{g}}\,,$ (98)

where $E_{0}$ is the energy of the lowest level. In our case
$E\simeq\epsilon^{2}/2m$ and thereby we recover Eq. (96) for $c(\rho)=1$.

###  4.7 Exponential estimate of probability of spontaneous production of
positrons

Since following Dirac the process of production of $e^{-}e^{+}$ pairs can be
treated as the penetration of electrons of the lower continuum into the upper
continuum through classically forbidden region ($p^{2}<0$), the probability of
this process is, as in case of spinless particles, determined by Eq. (5).
Equivalently, one can find the coefficient of transmission of the barrier in
the effective potential or find semiclassical asymptotic of the functions $G$
and $F$ for $r\to\infty$. This single-particle picture is distorted with
deepening of the level and with increase of the number of levels crossed the
boundary $\epsilon=-m$. We may use Eqs. (36), (37), (38) with taking into
account the Langer correction, which improves application of semiclassical
expressions.

In the threshold region of positron energies setting $\epsilon\simeq-m$ in the
expression for the spin term $U_{s}$, we obtain

$p^{*\,2}(r)\simeq(\epsilon-V)^{2}-m^{2}-\kappa^{2}/r^{2}\,,$ (99)

cf. with Eq. (89) we have used for description of the very deep levels. In
case of the Coulomb field $V=-\zeta/r$, replacing (99) in (5) we obtain

$W\sim\mbox{exp}\left[-2\pi\zeta\left(\frac{(m^{2}+k^{2})^{1/2}}{k}-(1-\rho^{2})^{1/2}\right)\right]\,,\quad\rho=\kappa/\zeta\,,\quad
k=\sqrt{\epsilon^{2}-m^{2}}\ll m\,,$ (100)

that coincides with asymptotic of exact solution of the Coulomb problem.

###  4.8 Critical charge of nucleus for muon

For the electron one has $R_{\rm nucl}\ll 1/m$, since $1/m\simeq 386$ fm and
$R_{\rm nucl}\simeq r_{0}A^{1/3}\simeq A^{1/3}/m_{\pi}$, $m_{\pi}\simeq 280m$.
For muon $R_{\rm nucl}\gg 1/m_{\mu}$, $m_{\mu}\simeq 207m_{e}$.

In order to find critical charge for the muon, $\zeta_{\rm cr}^{\mu}$, when
$\mu^{-}$ level reaches $\epsilon=-m_{\mu}$, we continue to apply
semiclassical approximation. For the model I, the turning point lies outside
the nucleus. Let us expand $U_{\rm ef}(r,\epsilon)$ near the turning point.
Using Eq. (70), after the replacement $r_{-}\to r_{0}$, and matching solutions
$G^{\,\prime}/G$ at $r=R_{\rm nucl}$ we find Eletskii:1977na :

$\frac{\mbox{Ai}^{\,\prime}(0)}{\mbox{Ai}(0)}\frac{2}{\zeta^{1/3}}\simeq\frac{\beta\mbox{ctg}\beta}{m_{\mu}R_{\rm
nucl}}\,,\quad\beta=\sqrt{\zeta(\zeta-2R_{\rm nucl}m_{\mu})}$ (101)

for the $ns$ level. From here follows

$\zeta_{\rm cr}^{\mu}\simeq 2Rm_{\mu}+\frac{(n\pi)^{2}}{2R_{\rm
nucl}m_{\mu}}(1+a(R_{\rm nucl}m_{\mu})^{-2/3}+...)\,,\quad
a=-2^{4/3}3^{-5/6}\pi\Gamma^{2}(2/3)\,,$ (102)

that coincides with expression, which follows from direct solution of the
Dirac equation at $\epsilon=-m_{\mu}$. In the model II we obtain $\zeta_{\rm
cr}^{\mu}\simeq 16.7$ that corresponds to $Z_{\rm cr}^{\mu}\simeq 2300$, and
for model I, respectively $Z_{\rm cr}^{\mu}\simeq 3700$.

##  5 Semiclassical approximation to system of linear Dirac equations

###  5.1 Semiclassical wave functions

Let us apply semiclassical expansion to Eq. (34), cf. Popov:1979gq . Parameter
of expansion $\tilde{\lambda}/l$ is $\propto\hbar$, where $l$ is the typical
length for the change of the potential. We present

$\psi=\phi e^{\int ydr}\,,$ (103)
$y(r)=\frac{1}{\hbar}y_{-1}(r)+y_{0}(r)+...\,,\quad\phi=\sum_{n=0}^{\infty}\hbar^{n}\phi^{(n)}\,,$
(104)

and arrive at the chain of equations for $y_{n}$ and $\phi^{(n)}$:

$(\hat{D}-y_{-1})\phi^{(0)}=0\,,\quad(\hat{D}-y_{-1})\phi^{(1)}=\phi^{(0)\,\prime}_{r}+y_{0}\phi^{(0)}\,,...$
(105)

Usually one restricts expansion by consideration of first two terms. Since
semiclassical series is asymptotic one, retaining of too many terms may worsen
convergence of the series to the exact solution.

In order the system of homogeneous equations (105) to have nontrivial
solution, $y_{-1}(r)$ should be an eigenvalue and $\phi^{(0)}\equiv\phi_{i}$,
$i=1,2$, the eigenfunction of one of two-component eigenvectors of the matrix
$\hat{D}(r)$. From the condition $\mbox{det}\hat{D}=0$ we get

$y_{-1}\equiv\lambda_{i}=\pm
i\sqrt{(\epsilon-V)^{2}-m^{2}-\tilde{\kappa}^{2}/r^{2}}\equiv\pm q\,.$ (106)

Replacing $y_{-1}$ back to Eq. (105) we obtain

$\displaystyle\phi_{i}=A\left(\begin{array}[]{ccc}m+\epsilon-V\\\\[8.53581pt]
\lambda_{i}+\kappa/r\end{array}\right)\,=A_{1}\left(\begin{array}[]{ccc}\lambda_{i}-\kappa/r\\\\[8.53581pt]
m-\epsilon+V\end{array}\right)$ (111)

where $A$ and $A_{1}$ are normalization constants.

Since the matrix $\hat{D}$ is not symmetrical, besides the right-hand
eigenvectors $\phi_{i}$ we should introduce the left-hand eigenvectors
$\tilde{\phi}_{i}$:

$\displaystyle(\hat{D}-\lambda_{i})\phi_{i}=\tilde{\phi}_{i}(\hat{D}-\lambda_{i})=0\,,$
(112)
$\displaystyle\tilde{\phi}_{i}=A(m-\epsilon+V,\lambda_{i}+\kappa/r)=A_{1}(\lambda_{i}-\kappa/r,m-\epsilon-V)\,.$

Note that left eigenvectors do not coincide with transposed right eigenvectors
($\tilde{\phi}_{i}\neq\phi_{i}^{T}$) and left-hand and right-hand vectors are
mutually orthogonal,

$(\tilde{\phi}_{i},\phi_{j})=\sum_{\alpha=1}^{2}(\tilde{\phi}_{i})_{\alpha}(\phi_{j})_{\alpha}\sim\delta_{ij}\,.$
(113)

To determine $y_{0}$ let in Eq. (105) put $\phi^{(0)}=\phi_{i}$ and multiply
both sides of equation from the left by $\tilde{\phi}_{i}$. As follows from
the first equation (112), the term with $\phi^{(1)}$ vanishes, and we obtain

$y_{0}=-(\tilde{\phi}_{i},\phi_{i}^{\,\prime})/(\tilde{\phi}_{i},\phi_{i})\,.$
(114)

Further calculations entail no difficulty, cf. Popov:1979gq ;
PopovEletskyMur1976 . The resulting wave functions of the quasistationary
state with energy $\epsilon<-m$ in the region of classically allowed motion
$r_{0}<r<r_{-}$ have the form:

$\displaystyle
G=C_{1}\left[\frac{\epsilon+m-V}{p}\right]^{1/2}\mbox{sin}\theta_{1}\,,\quad
F=\mbox{sgn}\,\kappa\,\cdot C_{1}\left[\frac{\epsilon-
m-V}{p}\right]^{1/2}\mbox{sin}\theta_{2}\,,$ (115) $\displaystyle
p=-iq=\sqrt{(\epsilon-V)^{2}-m^{2}-\frac{\kappa^{2}}{r^{2}}}\,,\quad\theta_{1}=\int_{r}{{}_{0}}^{r}(p+\frac{\kappa
w}{pr})dr+\pi/4\,,$
$\displaystyle\theta_{2}=\int_{r}{{}_{0}}^{r}(p+\frac{\kappa\tilde{w}}{pr})dr+\pi/4\,,\quad
w=\frac{1}{2}\left(\frac{V^{\,\prime}}{m+\epsilon-V}-\frac{1}{r}\right)\,,\quad\tilde{w}=\frac{1}{2}\left(\frac{V^{\,\prime}}{m-\epsilon+V}+\frac{1}{r}\right)\,.$

Here $C_{1}$ is normalization constant. As it was discussed, semiclassical
wave functions can be normalized neglecting penetration of the particle into
the classically forbidden regions $r<r_{0}$ and $r>r_{-}$, i.e.
$\int_{r_{0}}^{r_{-}}(G^{2}+F^{2})dr=1$. Thus we find

$C_{1}=\left[\int_{r_{0}}^{r_{-}}\frac{\epsilon-V}{p}dr\right]^{-1/2}=\left(\frac{2}{T}\right)^{1/2}\,,$
(116)

where $T$ is the period of the particle motion in classically allowed region.

In sub-barrier region $r_{-}<r<r_{+}$, where $p^{2}<0,p=iq$ and $q$, $y_{-1}$
and $y_{0}$ are real, wave functions attenuate exponentially with increasing
$r$. Resulting expressions have different forms in dependence on the sign of
$\kappa$. For $\kappa<0$, i.e. for $\kappa=-1$, we have

$\displaystyle\psi=\left(\begin{array}[]{ccc}G\\\\[8.53581pt]
F\end{array}\right)\,=C_{2-}(Qq)^{-1/2}\,\mbox{exp}\left[-\int_{r_{-}}^{r}\left(q-\frac{V^{\,\prime}m}{2Qq}\right)dr\right]\left(\begin{array}[]{ccc}m+\epsilon-V\\\\[8.53581pt]
-Q\end{array}\right)$ (121)

with $Q=q-\kappa/r$.

For $\kappa>0$, we have

$\displaystyle\psi=C_{2+}(Qq)^{-1/2}\,\mbox{exp}\left[-\int_{r_{-}}^{r}\left(q+\frac{V^{\,\prime}m}{2Qq}\right)dr\right]\left(\begin{array}[]{ccc}-Q\\\\[8.53581pt]
m-\epsilon+V\end{array}\right)$ (124)

with $Q=q+\kappa/r$, $C_{2\pm}$ are normalization constants.

In the region $r>r_{+}$ quasistationary state describes outgoing positron and
represents a diverging wave. For $\kappa<0$:

$\displaystyle\psi=iC_{3-}(Pp)^{-1/2}\,\mbox{exp}\left[-\int_{r_{+}}^{r}\left(ip-\frac{V^{\,\prime}m}{2Pp}\right)dr\right]\left(\begin{array}[]{ccc}m+\epsilon-V\\\\[8.53581pt]
iP\end{array}\right)$ (127)

with $P=p-i\kappa/r$. The flux of particles going off to infinity is then
given by $\Gamma=\mbox{lim}\,\,\mbox{Im}(F^{*}G)$ at $r\to\infty$.

For $\kappa>0$:

$\displaystyle\psi=iC_{3+}(Pp)^{-1/2}\,\mbox{exp}\left[-\int_{r_{+}}^{r}\left(ip+\frac{V^{\,\prime}m}{2Pp}\right)dr\right]\left(\begin{array}[]{ccc}iP\\\\[8.53581pt]
m-\epsilon+V\end{array}\right)$ (130)

with $P=p+i\kappa/r$. $C_{2\pm}$ are normalization constants.

The obtained formulas are valid for all $r$ except regions $\delta r\propto
1/\zeta^{2/3}$ near the turning points. The usual procedure is employed to
match semiclassical solutions. The solution is either expressed in terms of an
Airy function or one may use the Zwaan’s method. As the result, we have

$C_{2\pm}=-iC_{3\pm}=-\frac{\mbox{sgn}\,\kappa}{2}C_{1}\left[\frac{|\kappa|}{mr_{-}+(\kappa^{2}+r_{-}^{2}m^{2})^{1/2}}\right]^{-\mbox{sgn}\,\kappa/2}\mbox{exp}\left[-\int_{r_{-}}^{r_{+}}\left(q+\mbox{sgn}\,\kappa\frac{V^{\,\prime}m}{2Qq}\right)dr\right]\,.$
(131)

Note that the effective potential, which we have used in (36), can be
presented employing function $w$ appeared in (115):

$U_{\rm ef}=-\frac{V^{2}}{2m}+\frac{\epsilon
V}{m}+\frac{\kappa(\kappa+1)}{2r^{2}m}-\frac{\kappa}{rm}w+\frac{1}{2m}(w^{\,\prime}+w^{2}+\frac{w}{r})\,.$
(132)

The terms in Eq. (132), which contain the function $w$, are due to the
electron spin. For $|V|\gg m$ they are small compared to the first three
terms. Then the expression for the effective potential takes the same form as
for a scalar particle. At the turning points $r_{-}$ and $r_{+}$ the effective
potential is not singular.

The action becomes

$S=\int^{r}dr\sqrt{2m(E-U_{\rm ef})}=\int^{r}dr\left[p+\frac{2\kappa
w}{pr}-m(w^{\,\prime}+w^{2}+\frac{w}{r})\right]^{1/2}\,.$ (133)

Expanding $S$ in $1/\zeta\ll 1$ we obtain

$S=\int^{r}dr\left[p+\frac{\kappa w}{pr}+O(1/\zeta^{2})\right]\,,$ (134)

that coincides with Eqs. (115) – (130), which we have derived in this section.

Using Eqs. (115)–(130) for $\epsilon=-m$ we obtain

$\overline{r}=\frac{3(\zeta^{2}-\kappa^{2}+1/4)(\zeta^{2}+2\kappa^{2}/3-5\kappa/3+1)}{10\zeta(\zeta^{2}+(\kappa^{2}-3\kappa/2+1/2)/2)}\,,$
(135)

This expression yields $\overline{r}=0.301/m$ for the ground-state level,
whereas exact result gives $0.303/m$. For $\zeta\gg|\kappa|\gg 1$ result (135)
coincides with that follows from Eq. (135).

###  5.2 Nonrelativistic limit

To be specific consider case $\kappa<0$ and the classically allowed region.
Introducing a nonrelativistic energy $\tilde{\epsilon}=\epsilon-m$ and the
variable $\tilde{q}=(q^{2}+\kappa/r^{2})^{1/2}\simeq q+\frac{\kappa}{2qr^{2}}$
let us transform the factor in exponent (121) as

$\int_{r_{-}}^{r}dr\left(q-\frac{V^{\,\prime}m}{2Qq}\right)=\int_{r_{-}}^{r}dr\left[\tilde{q}-\frac{1}{2q}(V^{\,\prime}/Q+\kappa/r^{2})\right]=\int_{r_{-}}^{r}dr\left[\tilde{q}-\frac{1}{2}(\ln
Q)^{\,\prime}\right]\,,$ (136)

where $Q=q-\kappa/r$. The latter term in the integral cancels with the pre-
exponential factor $Q^{-1/2}$. Now let us take into account that
$\kappa(1+\kappa)=l(1+l)$. a Then we have

$\tilde{q}(r)=\left[2m(-\tilde{\epsilon}+V(r)-l(l+1)/(2mr^{2})\right]^{1/2}\,,\quad
G(r)=\frac{C}{\tilde{q}^{1/2}(r)}\mbox{exp}(-\int_{r_{-}}^{r}dr\tilde{q}(r)\,,$
(137)

where $C=const$ that reproduces the Schrd̈inger wave function in this region.
Note that $\tilde{q}(r)$ enters not $\kappa=\mp(j+1/2)$ but orbital moment
$l$. We formally considered case $\kappa<0$ just to be specific. Case
$\kappa>0$ is considered similarly. Note also that for $\kappa\neq-1$ one
should add to $\tilde{q}(r)$ the Langer correction.

###  5.3 Bohr-Sommerfeld quantization rule

From (121), (124) we derive Popov:1979gq ,

$\int_{r_{0}}^{r_{-}}dr(p+\frac{\kappa w}{pr})=(n+\gamma^{\,\prime})\,.$ (138)

As we have mentioned, value $\gamma^{\,\prime}$ depends on the fact does
$r_{0}$ lie inside the nucleus or outside it. In the latter case
$\gamma^{\,\prime}=1/2$ for $\kappa\neq-1$ and $\gamma^{\,\prime}=3/4$ for
$\kappa=-1$.

Eq. (138) determines real part of the energy $\epsilon_{n\kappa}$. It differs
from ordinary Bohr-Sommerfeld rule used in nonrelativistic quantum mechanics
by expression for relattivistic momentum $p(r)$ and by the term $\propto w$
appeared due to the spin-orbital interaction. Account of the term $\propto w$
is legitimate within semiclassical scheme. Let us show it on example of the
Coulomb field $V=-\zeta/r$. Then
$w(r)=-\frac{m+\epsilon}{2(\zeta+(m+\epsilon)r)}$ and $p(r)$ is determined by
Eq. (115). For $r_{0}<r<r_{-}$ the momentum $p(r)\sim\tilde{g}/r$ and the
ratio $|\frac{\kappa
w}{p^{2}r}|\sim|\kappa\tilde{g}^{-2}rw|\sim|\kappa|/\zeta^{2}$ for deep
levels. Since semiclassical approximation for wave functions is valid up to
$1/\zeta^{2}$, in case of deep levels $|\epsilon|\gg m$ the second term in the
integral (138) should be retained for $|\kappa|\gg 1$ but can be dropped for
$|\kappa|\sim 1$. For $\epsilon=-m$, we have $w=0$.

Note that results of calculations performed using quantization rules (66) and
(138) differ only in correction terms. For instance, from (138) we derive
exactly the same electron energy spectrum as that given by Eqs. (96), (97),
been derived with the help of the quantization rule in the form (66).

###  5.4 Probability of spontaneous production of positrons

Let us calculate probability of spontaneous production of positrons,
$\Gamma=-2\mbox{Im}\epsilon$. Replacing (127), (130) in (56) we find

$\Gamma=\Gamma_{0}e^{-2\int_{r_{-}}^{r_{+}}q(r)dr}\,,\quad\Gamma_{0}={T}^{-1}e^{2\kappa{\rm
Pr}\int_{r_{-}}^{r_{+}}wdr/(qr)}\,.$ (139)

The last integral is understood in the sense of the principal value, denoted
as ${\rm Pr}$, due to singularity at the point where $V(r)=m+\epsilon$.

In the nonrelativistic limit the value $\Gamma_{0}=1/T$ has meaning of number
of impacts per unit time of the particle (localized inside the region
$r_{0}<r<r_{-}$) against the potential barrier at $r=r_{-}$, and the
exponential is the probability of the penetration of the barrier in each
impact. Allowance for the relativistic effects and the spin changes the
expression for the period of the oscillations and adds to (139) a factor
depending on the sign of $\kappa$.

With taking into account that in the region of the barrier $V$ is purely
Coulomb field, for $w=0$ all the integrals are calculated exactly:

$\displaystyle\Gamma=\Gamma_{0}\mbox{exp}\left[-2\pi\zeta\left((m^{2}+k^{2})^{1/2}/k-(1-\rho^{2})^{1/2}\right)\right]\,,$
(140) $\displaystyle
1/\Gamma_{0}=\frac{2\zeta}{k^{2}}\left[(1-\rho^{2})^{1/2}(m^{2}+k^{2})^{1/2}-\frac{m^{2}}{k}\mbox{Arth}\left(k\left(\frac{1-\rho^{2}}{m^{2}+k^{2}}\right)^{1/2}\right)\right]\,.$

For the positron momentum $k=\sqrt{\epsilon^{2}-m^{2}}\to 0$ we have
$\Gamma_{0}=c_{1}=3/[2\zeta(1-\rho^{2})^{1/2}(2+\rho^{2})]$, for $k\to\infty$,
$\Gamma_{0}=c_{2}k=k/[2\zeta(1-\rho^{2})^{1/2}]$. For $k\ll\zeta^{1/2}m$ the
width $\Gamma$ is exponentially small for any $\kappa$. For
$|\kappa|\gg(\zeta/\pi)^{1/2}$ expression simplifies as

$\Gamma\simeq
k[2\zeta(1-\rho^{2})]^{-1/2}\mbox{exp}[-2\pi\zeta(1-(1-\rho^{2})^{1/2})]\,.$
(141)

For
$|\kappa|\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}\kappa_{0}=(\zeta/\pi)^{1/2}$,
the exponential factor in $\Gamma$ becomes of the order of unity, and the
quasiclassical approximation becomes invalid. Note that
$\kappa_{0}/\kappa_{max}=1/\sqrt{\pi\zeta}$. Therefore number of levels
diffused in the continuum, for which $\Gamma$ is not exponentially small, is
tiny for $\zeta\gg 1$.

###  5.5 Semiclassical method for noncentral potentials obeying system of
linear Dirac equations

We described the spectrum of the quasistationary levels in the lower continuum
for a spherical nucleus with charge $Z>Z_{\rm cr}$. The results can be
generalized to the case, when the potential does not obey spherical symmetry
Popov:1979gq . Let us present the Dirac equation as

$-i(\hat{\vec{\alpha}}\nabla)\psi=\hbar^{-1}\hat{D}\psi\,,\quad\hat{D}=\epsilon-m\hat{\beta}-V(\vec{r})\,,$
(142)

where $\hat{\vec{\alpha}}=\gamma^{0}\vec{\gamma}$, $\hat{\beta}=\gamma^{0}$
are Dirac matrices and we recovered dependence on $\hbar$. Let us present
bispinor $\psi$ as $\psi=\phi e^{i\sigma}$ and expand real quantities $\phi$
and $\sigma$ in the parameter proportional to $\hbar$:

$\sigma=\hbar^{-1}\sigma_{-1}+\sigma_{0}+...,\quad\phi=\phi^{(0)}+\hbar\phi^{(1)}+...$
(143)

Replacing these series to Eq. (142) we obtain the chain of equations

$\displaystyle[\hat{D}-(\hat{\vec{\alpha}}\nabla\sigma_{-1})]\phi^{(0)}=0\,,$
(144)
$\displaystyle[\hat{D}-(\hat{\vec{\alpha}}\nabla\sigma_{-1})]\phi^{(1)}=\hat{\vec{\alpha}}\nabla)\phi^{(0)}+(\hat{\vec{\alpha}}\nabla)\sigma_{0})\phi^{(0)}\,,...$

Condition of existence of nontrivial solution $\phi^{(0)}$,

$\mbox{det}[\hat{D}-(\hat{\vec{\alpha}}\nabla\sigma_{-1})]=0\,,$ (145)

results in the Hamilton-Jacobi equation

$(\nabla\sigma_{-1})^{2}=(\epsilon-V)^{2}-m^{2}\,.$ (146)

In difference with spherically-symmetric case, matrix

$\hat{D}-\hat{\vec{\alpha}}\nabla\sigma_{-1}=\epsilon-V(\vec{r})-m\hat{\beta}-\hat{\vec{\alpha}}\nabla
S$ (147)

is Hermitian, therefore its left-hand, $\tilde{\phi}_{i}$, and right-hand,
${\phi}_{i}$, eigenvectors are Hermitian conjugates,
$\tilde{\phi}_{i}=\phi_{i}^{\dagger}$, and

$(\hat{D}-\hat{\vec{\alpha}}\nabla
S)\phi_{i}=\phi_{i}^{\dagger}(\hat{D}-\hat{\vec{\alpha}}\nabla S)=0\,,\quad
i=1,2,3,4\,.$ (148)

With the help of this equation, from (144) we find system of equations for
$\sigma_{0}$,

$\phi_{i}^{\dagger}(\hat{\vec{\alpha}}\nabla\sigma_{0})\phi_{j}=-\phi_{i}^{\dagger}\hat{\vec{\alpha}}\nabla\phi_{j}\,.$
(149)

Bispinors $\phi_{i}$ are found by diagonalizing the matrix
$\hat{D}-\hat{\vec{\alpha}}\nabla S$, so that the right-hand side of (149)
contains known quantities. Determining from this equation $\sigma_{0}$, we
obtain the quasiclassical solution of the Dirac equation

$\psi=\phi_{i}\mbox{exp}(\hbar^{-1}\sigma_{-1}+\sigma_{0})\,.$ (150)

In practice calculation of the functions $\sigma_{-1}$ and $\sigma_{0}$ for
noncentral potentials is a complicated mathematical problem requiring the
solution of first-order differential equations in partial derivatives. In
contrast to the case when $V$ is spherically symmetric, in general case the
result is not expressed in quadratures. If a parameter of “non-sphericity” is
small one may develop a perturbation theory.

##  6 Spontaneous production of positrons in heavy-ion collisions

###  6.1 Approach to the problem

Minimal distance between colliding nuclei with charges $Z_{1}$ and $Z_{2}$ is
as follows LL1 ; PopovJETP1974 ,

$R_{\rm min}=(Z_{1}+Z_{2})^{2}e^{2}/(2E_{\rm
c.m.})+\sqrt{(Z_{1}+Z_{2})^{2}e^{2}/(2E_{\rm c.m.})^{2}+b^{2}}\,,$

where $E_{\rm c.m.}$ is the kinetic energy of colliding nuclei in c.m.
reference frame, $b$ is the impact parameter. In order the energy of the
electron, $\epsilon_{\rm 1s}$, in the quasi-molecule would become $<-m$ the
colliding heavy nuclei should reach distances
$|\vec{r}_{1}-\vec{r}_{2}|=R<R_{\rm cr}$, where $R_{\rm cr}\simeq 33$ fm for
central U$+$U collisions, see below. Thus $R_{\rm cr}$ is approximately twice
larger than $2R_{\rm nucl}$, where $R_{\rm nucl}\simeq 1.2A^{1/3}$ fm is the
radius of the single nucleus $\simeq 7$ fm. On the other hand, $R_{\rm
cr}\ll\overline{r}_{K}\simeq 0.3\zeta_{\rm cr}$, where $\overline{r}$ is
estimated using Eq. (135). For U$+$U collisions $R_{\rm
cr}/\overline{r}_{K}\sim 0.2$. Nuclei move with the velocity
$v_{A}\sim(0.025-0.07)$, cf. PopovJETP1974 , whereas the electron of the
K-shell has typical velocity $v_{e}\simeq 1$. Thereby one may use adiabatic
approximation, i.e., we may use $\epsilon(R(t))$. Since $R_{\rm
cr}/\overline{r}_{K}\sim 0.2\ll 1$, the anisotropy of the potential is not as
large and we may present

$V({r})=-\left(\frac{Z_{1}e^{2}}{r_{1}}+\frac{Z_{2}e^{2}}{r_{2}}\right)=-\frac{Ze^{2}}{r}\left(1+\frac{R^{2}}{(2r)^{2}}P_{2}(\cos\theta)+...\right)\,,$
(151)

where $Z=Z_{1}+Z_{2}$, $\vec{r}_{1,2}=|\vec{r}\pm\vec{R}/2|$, $P_{2}$ is the
second Legendre polinomical, $R(t)$ is the distance between centers of nuclei.
In the second equation and further we for simplicity consider the case
$Z_{1}=Z_{2}$. Otherwise there appear odd-power terms in the expansion. In
inclusive experiments this anisotropy disappears due to the averaging. However
for event-by-event collisions such terms may lead to forward-backward
anisotropy reflecting in some observable effects. In the first approximation
in $(R/2\overline{r}_{K})^{2}$, the problem is reduced to that we have
considered above for the spherical nucleus with the charge $Z=Z_{1}+Z_{2}$.
The effective nucleus radius now is $2R_{\rm nucl}$.

The process of the spontaneous production of positrons also can be described
in adiabatic approximation since as we have argued we may use that
$\epsilon(R(t))$ and since $1/\Gamma(\epsilon(R(t)))\gg\tau_{\rm
col}\stackrel{{\scriptstyle\scriptstyle>}}{{\phantom{}{}_{\sim}}}2R_{\rm
cr}/v_{A}$. The most serious experimental problem is to separate spontaneous
production of positrons from the phone processes. For example, the parameter
$2R_{{\rm nucl}}/R_{{\rm{cr}}}\sim(1/2-1/3)$ is not as small. Therefore a
serious competing time-dependent process is associated with an induced
production of positrons occurring due to excitation of the nuclear levels, cf.
Rafelski1916  and references therein. However difference between
characteristics of the induced and spontaneous production of positrons is
significant. The induced positron production exists both in subcritical and
the supercritical regimes. When the electron level crosses the boundary
$\epsilon=-m$ there appears a narrow energy-line in the positron spectrum
owing to the switching on of the spontaneous positron production occurring in
the tunneling process. Thus there is principal difference between the
subcritical and the supercritical regimes that may help to the experimental
identification of the spontaneous positron production.

Another effect is associated with presence of a magnetic component of the
field. First indication on presence of strong magnetic fields in heavy ion
collisions was performed in VA1980 . For peripheral collisions of heavy ions
at collision energies
$\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}$ GeV$\cdot A$
it yields $h\sim H_{\pi}(Ze^{6})^{1/3}$ for $R\simeq A^{1/3}/m_{\pi}$, $v_{\rm
nucl}\sim 1$, $H_{\pi}=m_{\pi}^{2}/e$. More generally, replacing $1\to
v\gamma$, $\gamma=1/\sqrt{1-v^{2}}$ we have

$eh\sim Ze^{2}v\gamma/R^{2}\,.$ (152)

For collisions with low energies $E\sim(5-10)$MeV$\cdot A$ of our interest
here it follows that $h\sim 10^{15}$G, for $R\sim R_{cr}\simeq(30-50)$ fm, and
$v_{\rm nucl}\sim 10^{-1}$, cf. also RGS1987 .

In presence of a “weak” homogeneous magnetic field, the reduction of $Z_{\rm
cr}$ in case of the supercritical atom has been found by using the
perturbation theory Oraevsky1977 ,

$\zeta_{\rm cr}(h)=\zeta_{\rm cr}(0)-\frac{5\pi^{2}\mu}{6\ln(1/R^{3}_{\rm
nucl})}\frac{h}{H_{0}}\,,$ (153)

$H_{0}=m^{2}/e\simeq 4.4\cdot 10^{13}$G, $\mu\simeq 1/3$ for $\zeta=\zeta_{\rm
cr}$.

For strong fields, numerical evaluations Oraevsky1977 , see also Popov2001 ,
yielded $Z_{\rm cr}=165$ for $h=H_{0}$, and $Z_{\rm cr}=96$ at
$h=10^{2}H_{0}$. For $h=10^{18}$G one gets $Z_{\rm cr}=41$. This effect
appears because of the exact compensation of the diamagnetic and paramagnetic
contributions to the ground state for the electron. Although these estimates
are performed for the case of purely uniform static magnetic field, they show
that a magnetic effect also should be carefully studied for the case of
realistic time-space configuration of the field.

Below I focus only on the description of the spontaneous production of
positrons and simplifying consideration I also ignore mentioned magnetic
effects.

###  6.2 Electron energy as a function of distance between nuclei

Usage of the Bohr-Sommerfeld quantization rule allows to consider the problem
analytically Voskresensky1977 , cf. Popov:1979gq . From (35), (36), (37) with
taking into account the Langer correction resulting in the replacement $p\to
p^{*}$, we have

$p^{*}(r)=F(r,\epsilon)/r\,,\quad
F(r,\epsilon)=\left[(\epsilon^{2}-m^{2})r^{2}+2\epsilon\zeta
r+\frac{\kappa+1}{\tilde{a}}-\frac{3}{4\tilde{a}^{2}}-(\kappa+1/2)^{2}+\zeta^{2}\right]^{1/2}\,,$
(154)

where $\tilde{a}=1+r(m+\epsilon)/\zeta$, $\zeta=Ze^{2}$, $Z=Z_{1}+Z_{2}$.
Applying the quantization rule (66) first for $\epsilon\neq-m$ and then for
$\epsilon=-m$ and subtracting one result from the other we obtain

$\int_{R/2}^{r_{\epsilon}}drF(r,\epsilon)/r=\int_{R_{\rm
cr}/2}^{r_{-m}}drF(r,\epsilon=-m)/r\,.$ (155)

Here $r_{\epsilon}$ is the turning point for the given $\epsilon$ and $r_{-m}$
is the turning point for $\epsilon=-m$. I used that in integration over the
regions $r<R/2$, $r<R_{\rm cr}/2$ dependence on $\epsilon$ can be dropped
since at $|\epsilon|\sim m$ of our interest we have $|V|\gg|\epsilon|$.
Thereby, the specifics of the behavior $V(r)$ in the region $r<R_{\rm cr}/2$
almost does not affect the result. To be specific, we may use $V=const$ for
$r<R_{\rm cr}/2$. Integrals undergo logarithmic diverge at the lower limit.
After their regularization the dependence on $R$ and $R_{\rm cr}$ is separated
in the explicit form:

$\int_{0}^{r_{\epsilon}}dr[F(r,\epsilon)-F(r,\epsilon=-m)]/r+\int_{r_{-m}}^{r_{\epsilon}}drF(r,\epsilon=-m)/r=\tilde{g}\ln\frac{R}{R_{\rm
cr}}\,.$ (156)

Integrals in (156) are calculated numerically. Comparison with exact solution
of two-center Dirac problem shows that the error of the semiclassical result
does not exceed $0.1\%$. We can proceed further using that
$r|m+\epsilon|/\zeta<r_{\epsilon}|m+\epsilon|/\zeta\ll 1$ at least for
$|\epsilon|\sim m$ of our interest. Thereby we expand $\tilde{a}$ in Eq. (154)
in the series of $r$. As the result we find

$\displaystyle F(r,\epsilon)=(\tilde{g}^{2}+br+cr^{2})^{1/2}\,,$ (157)
$\displaystyle b=2\epsilon\zeta-(\kappa-1/2)(m+\epsilon)/\zeta\,,\quad
c=\epsilon^{2}-m^{2}+(\kappa-5/4)(\epsilon+m)^{2}/\zeta^{2}\,.$

From (156) and (157) we obtain

$\frac{R}{R_{\rm
cr}}=-\frac{2\zeta}{b}\left(1+\frac{\tilde{g}^{2}c}{3b^{2}}+O(c^{2})\right)\,.$
(158)

For $|\epsilon+m|\ll m$ we find

$\epsilon=-m-\beta m(R-R_{\rm cr})/R_{\rm
cr}\,,\quad\beta=\left(1-\frac{\kappa-1/2}{2\zeta^{2}}-\frac{\tilde{g}^{2}}{3\zeta^{2}}\right)^{-1}\,.$
(159)

For U$+U$ collisions for the ground-state level we find $\zeta\simeq 1.343$
and $\beta\simeq 0.79$. The slope-parameter $\beta$ determines the probability
of the production of positrons for $|\epsilon+m|\ll m$. The semiclassical
approximation reproduces the $Z$ dependence of $\beta$ correctly, the
difference with exact calculation done within solution of the two-center
problem for the Dirac equation Lisin1980  is about (3-4)%.

Setting $c=0$ in Eq. $(\ref{RRcr})$ we obtain very simple and on the other
hand accurate result Voskresensky1977 ; Popov:1978kk ; Popov:1979gq :

$\epsilon(R)=\epsilon(R/R_{\rm cr})=-m\frac{R_{\rm
cr}/R-(\kappa-1/2)/(2\zeta^{2})}{1-(\kappa-1/2)/(2\zeta^{2})}\,.$ (160)

Difference of this simple expression with exact solution of the two-center
Dirac equation Lisin1980  is less than (1-2)% already for $\zeta\to 1$ when
the parameter of applicability of the semiclassical approximation is $~{}1$.
Such an accuracy is sufficient, therefore I do not present here a more
accurate semiclassical expression Popov:1979gq  obtained without using
expansion in $c^{2}$, which has still higher accuracy. It may be curious to
notice that, when in 1976 I showed the result (160) to Vladimir Stepanovich
Popov, he did not believe in it, saying that one of his collaborators during a
year is trying to solve the Dirac equation for the two-center problem
numerically on ITEP big computer and yet obtained only the result, for
$\zeta=1$. He took slide rule (that time there were no PCs) and confirmed that
the whole curve (160) fully coincides with the result of the exact numerical
calculation. Then, the result (160) was reflected in our publications
Popov:1978kk ; Popov:1979gq . Result (160) is shown in Fig. 4. For
$\zeta=1.343$, $\kappa=-1$ we get $-\epsilon(R/R_{\rm cr})=0.705(R_{\rm
cr}/R)+0.295$.

Figure 4: Solution $\epsilon(R/R_{\rm cr})$ of Eq. (160) for various values of
the parameter $\zeta$.

Expression for the critical distance between nuclei, $R_{\rm cr}$, can be
found from Eq. (78) for a spherical nucleus after replacement of the nucleus
radius $R_{\rm nucl}$ by $R/2$, where now $R$ is the distance between nuclei
and $Z\to Z_{1}+Z_{2}$. As the result, we find

$R_{\rm cr}=\frac{4\tilde{g}^{2}}{\zeta
m}\left[\mbox{exp}\left(\frac{\pi(n+\gamma_{1})-\tilde{\gamma}}{\tilde{g}}+2\right)\right]^{-1}\,.$
(161)

For the case of U$+$U collisions, in the model I we obtain $R_{\rm cr}\simeq
33$ fm, whereas exact solution of the Dirac equation Lisin1980  yields $R_{\rm
cr}\simeq 34.3$ fm.

###  6.3 Tunneling in the two-center problem. Angular distribution of
positrons

The potential of the system of two nuclei (151) contains at $r\gg R$ a
quadrupole correction. In the sub-barrier region the correction is
$\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}(R_{\rm
cr}/(2r_{-}))^{2}\stackrel{{\scriptstyle\scriptstyle<}}{{\phantom{}{}_{\sim}}}10^{-2}$.
Therefore the problem is reduced to the calculation of the penetrability of a
three-dimensional barrier that differs only little from a spherically
symmetrical one. Thus we may use expansion

$V=V_{0}+m^{2}R^{2}V_{1}\,,\quad S=S_{0}+m^{2}R^{2}S_{1}\,.$ (162)

We substitute these expressions to the Hamilton-Jacobi equation and obtain

$\displaystyle(\nabla S_{0})^{2}=2m(E-U_{0})\,,\quad\nabla S_{0}\nabla
S_{1}=-U_{1}\,,$ (163) $\displaystyle
U_{0}(r)=-\left(\frac{\zeta^{2}}{2r^{2}m}+\frac{\zeta\epsilon}{rm}\right)\,,\quad
U_{1}(r)=-\frac{\zeta}{4r^{3}m^{2}}\left(\epsilon+\frac{\zeta}{r}\right)P_{2}(\cos\theta)\,.$

The first equation is easily integrated resulting in

$S_{0}(r,\theta)=\int^{r}pdr+\kappa\theta\,.$ (164)

Taking into account of the first term leads to exponential term in (140).
Second term in (164) is due to anisotropy of the potential.

Equation for $S_{1}$ in the under-barrier region $r_{-}<r<r_{+}$ gets the form

$iq\frac{\partial S_{1}}{\partial r}+\frac{\kappa}{r^{2}}\frac{\partial
S_{1}}{\partial\theta}=-U_{1}(r,\theta)\,,\quad p=iq\,,$ (165)

and it is solved by the method of separation of the variables. Supposing

$r^{2}U_{1}(r,\theta)=u(r)(\frac{3}{4}\cos(2\theta)+\frac{1}{4})$ (166)

and taking into account the boundary condition
$\mbox{Im}S_{1}(r_{-},\theta)=0$, for $r=r_{+}$ we obtain

$\displaystyle\mbox{Im}S_{1}(r_{+},\theta)=aP_{2}(\cos\theta)+a_{1}\,,$ (167)
$\displaystyle
a=\int_{r_{-}}^{r_{+}}dr\frac{u(r)m^{2}}{q(r)}\mbox{ch}\left(2\kappa\int_{r}^{r_{+}}\frac{dr^{\,\prime}}{q(r^{\,\prime})r^{\,\prime\,2}}\right)\,,\quad
a_{1}=-\frac{1}{2}\int_{r_{-}}^{r_{+}}dr\frac{u(r)m^{2}}{q(r)}\mbox{sh}^{2}\left(\kappa\int_{r}^{r_{+}}\frac{dr^{\,\prime}}{q(r^{\,\prime})r^{\,\prime\,2}}\right)\,.$

For the angular asymmetry of the positron production, the constant $a_{1}$ is
immaterial.

A remarkable fact is that the expression for $a$ acquires a hyperbolic cosine
that enhances the angular anisotropy of the emitted particles compared with
the anisotropy of the potential. The cause of this effect is that the sub-
barrier trajectory of a tunneling particle with nonzero angular momentum is
not a straight line due to $\kappa\neq 0$. This constitutes the substantial
difference between the three-dimensional and the one-dimensional problems.

For a Coulomb field integrals (167) can be calculated exactly. However result
looks cumbersome. An estimate shows that
$W(\theta)\simeq\mbox{exp}(-2\mbox{Im}S)=C\mbox{exp}(\alpha
P_{2}(\cos\theta))$, where $C$ is a constant, $\alpha\sim
m^{2}R^{2}\eta^{-1}\mbox{sh}\eta\gg m^{2}R^{2}$, $\eta=2\pi\kappa/g$. For
U$+$U collisions $\alpha\sim 1/3$, and we can expect a noticeable angular
anisotropy. Positrons are predominantly emitted along the axis joining the
nuclei at the instant of their closest approach. This question is worthy for
the experimental study.

Concluding, note that we needed applicability of semiclassical approximation
both for the radial motion and for the angular motion. Strictly speaking, the
latter takes place only for $|\kappa|\gg 1$. However, as it always occurs,
even for $|\kappa|\sim 1$ one may expect a good accuracy of semiclassical
expressions.

###  6.4 Screening of K-electron by electron cloud of not fully stripped
quasi-molecule

If the colliding nuclei are not fully stripped, the quasi-molecule is
surrounded by an electron cloud. Screening weakens the attraction of the
$K$-electron to the nuclei in the quasi-molecule. As a result, the critical
distance $R_{\rm cr}$, at which the K-electron level crosses the boundary
$\epsilon=-m$, is decreased. This effect can be calculated using
nonrelativistic many-particle semiclassical approximation (Thomas-Fermi
method), cf. Voskresensky1977 ; Popov:1979gq . Let us use that

$R_{\rm cr}\ll\overline{r}_{K}\ll a_{\rm
TF}=(9\pi^{2}/128)^{1/3}(Ze^{6})^{-1/3}/m\simeq 30\zeta^{-1/3}/m\,,$ (168)

where $a_{\rm TF}$ is the mean radius of the Thomas-Fermi atom. The shift of
the ground-state electron energy level can be found with the help of the
perturbation theory. We have

$\Delta\epsilon_{0}\simeq\overline{V(\vec{r})-V_{0}(\vec{r})}\,,$ (169)

where $V_{0}(\vec{r})$ is the potential of the two striped nuclei (151) and
$V(\vec{r})$ is the potential of the two not fully striped ions. The typical
size for the change of $\delta V$ is $a_{\rm TF}$. Therefore with accuracy
$\sim(R_{\rm cr}/a_{\rm TF})^{2}\sim 10^{-5}$ the perturbation can be
considered as spherically symmetric. Thus

$V(r)=V(r_{i})-\frac{Ze^{2}\phi(r)}{r}\,,\quad
V(r_{i})=-\frac{Z_{1}e^{2}}{r_{i}}\,,$ (170)

$r_{i}=x_{0}a_{\rm TF}$ is the radius of the ion, $\phi(r)$ is the solution of
the Thomas-Fermi equation LL3 ,

$\phi^{\,\prime\prime}_{x}=x^{-1/2}\phi^{3/2}$ (171)

with boundary conditions $\phi(0)=1$, $\phi(x_{0})=0$, $x=r/a_{\rm TF}$, and
$Z_{1}=-Zx_{0}\phi^{\,\prime}_{x}(x_{0})$ is the observed charge of the two
partially screened nuclei.

Expansion $\phi(x\to 0)$ yields LL3 :

$\phi(x)=1+\phi^{\,\prime}_{x}(0)x+\frac{4}{3}x^{3/2}+...$ (172)

For the case of neutral atoms $\phi^{\,\prime}_{x}(0)=-1.588$.

From (170), (172) for the shift of the ground-state level we obtain

$\displaystyle\Delta\epsilon_{0}=V(r_{i})+\phi^{\,\prime}_{x}(0)\frac{Ze^{2}}{a_{\rm
TF}}=\frac{Ze^{2}}{a_{\rm
TF}}[\phi^{\,\prime}_{x}(0)-\phi^{\,\prime}_{x}(x_{0})]+\frac{4\zeta}{3a_{\rm
TF}^{3/2}}\overline{r^{1/2}}+...$ (173)

Values $\phi^{\,\prime}_{x}(0)$ and $-\phi^{\,\prime}_{x}(x_{0})$ are
tabulated. We estimate $|\Delta R_{\rm cr}/R_{\rm
cr}|\sim|\Delta\epsilon_{0}/\epsilon_{0}|\simeq 10\%$ for ionization parameter
$q=(Z_{1}+Z_{2}-N)/(Z_{1}+Z_{2})\simeq 0.5$, and $\simeq 12\%$ for $q=0$,
where $N$ is the total number of electrons in the quasi-molecule.

###  6.5 Calculation of positron production employing the imaginary-time
method

####  6.5.1 General description of the method

Consider first problem of one-dimensional motion of a relativistic particle in
the potential $V(x,t)$. The Lagrangian is as follows

$L=-m\sqrt{1-\dot{x}^{2}}-V(x,t)+V_{0}\,.$ (174)

The constant is added to recover Lorentz invariance of the action

$S=\int_{t_{1}}^{t_{2}}Ldt\,,$ (175)

since $t$ is not a scalar. At initial time-moment particle was in the point
$x_{1}(t_{1})$ and at final moment, in $x_{2}(t_{2})$.

In the semiclassical approximation the wave function is

$\psi(x)\sim e^{iS(x_{1},x)}=e^{i\mbox{Re}S(x_{1},x)-\mbox{Im}S(x_{1},x)}\,.$
(176)

The action is found from the Hamilton-Jacobi equation.

In the imaginary-time method the sub-barrier motion is formally considered at
imaginary values of the time variable. Performing variable replacement
$\tau=it$ we arrive at the Euclidian action

$S_{\rm
E}=\int_{\tau_{1}}^{\tau_{2}}[m\sqrt{1+(dx/d\tau)^{2}}+V(x,\tau)-V_{0}]d\tau\,.$
(177)

The trajectory $x(\tau)$ in under-barrier motion, where $S_{\rm E}$ is real,
is determined by condition $\delta S=0$. From here one finds equation of
motion, which has a meaning of the Newton equation

$\frac{d\tilde{p}}{d\tau}=\frac{d}{d\tau}\frac{mdx/d\tau}{\sqrt{1+(dx/d\tau)^{2}}}=-\frac{\partial
V_{\rm E}(x,\tau)}{\partial x}\,,\quad V_{\rm E}=-V\,.$ (178)

With exponential accuracy the probability to find the particle in the turning
point of the exit from the barrier, if it initially were in the point of the
entrance in the barrier, is given by

$W(x_{1},x_{2})=e^{-2\mbox{Im}S(x_{1},x_{2})}=e^{-2S_{\rm
E}(x(\tau_{1}),x(\tau_{2}))}\,.$ (179)

This expression can be generalized to take into account pre-exponential
coefficient. However we will restrict ourself by consideration of the
exponential term.

It is essential that the sub-barrier trajectory satisfies the classical
equation of motion but now in the Euclidian time. To find it and to calculate
$S$ and $W$ we may use formally the known equations of the classical physics.

####  6.5.2 Tunneling in slowly time-dependent potential

Case of space-dependent and slowly time-dependent fields was considered in
Popov:1979gq . For simplicity consider a scalar particle in a one-dimensional
field. Let the probability of the tunneling in the static limit is known,

$W=e^{-2\int_{x_{1}}^{x_{2}}|p|dx}\,,$ (180)

where $x_{1}$ and $x_{2}$ are entrance and exit turning points, i.e.
$p(x_{1})=p(x_{2})=0$. Variation of the action due to a weak dependence of the
potential on time $V(x,t)$ yields

$\displaystyle\delta
S=\delta\int_{t_{1}}^{t_{2}}[-m(1-\dot{x}^{2})^{1/2}-V(x,t)]dt$ (181)
$\displaystyle=\int_{t_{1}}^{t_{2}}[p\delta\dot{x}-(\partial V/\partial
x)\delta x-\delta V(t)]dt=-\int_{t_{1}}^{t_{2}}\delta V(x(t))dt\,.$

We used equation of motion and integration by parts. The last integral can be
calculated using imaginary-time method. Thus we obtain

$\delta S_{\rm E}=\int_{\tau_{1}}^{\tau_{2}}\delta V_{\rm E}(x(\tau))d\tau\,.$
(182)

Dependence $x(\tau)$ is determined from (178) as

$\tau(x_{1},x)=\int_{x_{1}}^{x_{2}}dx\frac{\sqrt{m^{2}-\tilde{p}^{2}}}{\tilde{p}}=\int_{x_{1}}^{x_{2}}dx\frac{V-\epsilon}{\sqrt{m^{2}-(\epsilon-V)^{2}}}\,,$
(183)

where we used relation $\tilde{p}^{2}=m^{2}-(\epsilon-V)^{2}$ and that
$\epsilon$ may only adiabatically change with time, i.e. it may depend on
$\tau$ via dependence of one of the parameters.

####  6.5.3 Correction on non-adiabaticity to the spontaneous positron
production in low-energy heavy-ion collisions

As a specific example, consider probability of the spontaneous positron
production in low-energy heavy-ion collisions. Deriving Eqs. (140), (167) we
assumed that during a time of the tunneling
($(r_{+}-r_{-})\sqrt{m^{2}+k^{2}}/k$) the potential $V$ and $\epsilon$ did not
have a time to change. Here please do not mix typical time, for which the
particle passes the barrier, cf. Kolomeitsev:2013du , and time $1/\Gamma$,
being inversed probability to observe the positron. As we see from this simple
estimate, adiabatic approximation does not hold at least for $k\to 0$, i.e. in
the vicinity of the boundary of the continua, $|\epsilon|\simeq m$.

Let us find correction to the penetrability of the Coulomb barrier due to
finite speed of the colliding nuclei Popov:1979gq . Following (151) the $R(t)$
dependent correction to the static Coulomb potential is as follows

$\delta V=-\frac{\zeta}{4r^{3}}P_{2}(\cos\theta)R^{2}(t)\,.$ (184)

Further consider the case when positrons are emitted along the axis that joins
the nuclei, $P_{2}(0)=P_{2}(\pi)=1$. Then, probability of their production is
maximal. Expanding $R(t)$ near the closest approach point we obtain

$R(t)=R_{0}+v^{2}t^{2}/(4R_{0})\,.$ (185)

From (184) and (185) we have

$\delta V=-\frac{\zeta v^{2}}{8r^{3}}t^{2}\,.$ (186)

The imaginary time $\tau=it$ is found from Eq. (183). Thus we obtain

$\tau=\frac{\zeta}{k^{3}}[m^{2}\phi+(m^{2}+k^{2})^{1/2}(m^{2}+\rho^{2}k^{2})^{1/2}\sin\phi]\,,$
(187)

where we introduced variable $\phi=2\mbox{arcsin}[(r_{+}-r)/(r-r_{-})]^{1/2}$,
$0\leq\phi\leq\pi$, $r=r_{+}\cos^{2}(\phi/2)+r_{-}\sin^{2}(\phi/2)$, values
$\tau=0$ and $\phi=0$ correspond to the instant of emergence from under the
barrier. The total imaginary tunneling time is $\tau_{t}=\pi\zeta
m^{2}/k^{3}$, i.e., $\tau_{t}\to\infty$ for the electron energy
$\epsilon\to-m$, whereas for deep electron levels $\tau_{t}$ strongly
diminishes.

Replacement of (187) in (182) yields

$\delta
S_{E}=\delta\mbox{Im}S=-\frac{2Z}{Am_{N}}\frac{\zeta^{2}}{R_{0}v_{p}^{3}}I(\epsilon_{p},\eta)\,,$
(188)

where $\epsilon_{p}=-\epsilon$, $v_{p}=(1-m^{2}/\epsilon_{p}^{2})^{1/2}$ is
the speed of the positron,

$\displaystyle
I(\epsilon_{p},\eta)=-\frac{1}{8}\int_{0}^{\pi}d\phi\left[\frac{\sin\phi+(1-v_{p}^{2})\eta}{\cos\phi+\eta}\right]^{2}\frac{\cos\phi+(1-v_{p}^{2})\eta}{\cos\phi+\eta}\,,$
(189) $\displaystyle\eta=[1-(1-\rho^{2})v_{p}^{2}]^{1/2}\,.$

The ratio

$\delta=\frac{\mbox{Im}\delta S}{\mbox{Im}S_{0}}\,,$ (190)

where $\quad\mbox{Im}S_{0}=\pi\zeta[v_{p}^{-1}-(1-\rho^{2})^{1/2}]\,,$ for
collisions U$+$U ($\zeta=1.343$) is shown in Fig. 5 as a function of the
positron energy $\epsilon_{p}$. It is seen that $\delta<0.1$ for
$\epsilon_{p}>1.65m$. The adiabatic approximation in the problem of
spontaneous production of positrons becomes invalid near $\epsilon_{p}=m$,
where the positron production cross section is any case tiny.

Numerical calculations PopovJETP1974 ; Marinov1974  have shown that $R_{\rm
cr}$ increases rapidly with increasing charge $Z=Z_{1}+Z_{2}$ of colliding
nuclei. The cross section of the spontaneous production of positrons increases
in this case $\propto R_{\rm cr}^{7/2}$, while the correction for the non-
adiabaticity of the tunneling decreases as $1/R_{\rm cr}$ at a fixed
$\epsilon_{p}$. Therefore it would be more convenient to perform experiments
with heavier nuclei, for which $R_{\rm cr}$ is larger.

Figure 5:  Correction on non-adiabaticity of the motion of nuclei, $\delta$,
for collisions U$+$U as a function of the positron energy $\epsilon_{p}$.

##  7 Many-particle semiclassical approximation. Electron condensation in
upper continuum

###  7.1 Screening of a source of positive charge in presence of external
electrons

In a many-particle problem, most of electrons in spherically symmetric
potential well, $V<0$, have angular momenta $l\gg 1$ and thereby spin-effects
can be neglected. Then to find distribution of charge we may deal with a more
simple Klein-Gordon-Fock equation (1). The maximum possible momentum, at which
the last electron still is bound, is given by condition

$p_{max}=\sqrt{(\epsilon_{\rm bound}-V)^{2}-m^{2}}$ (191)

with $\epsilon_{\rm bound}\geq-m$. If there is sufficient amount of external
electrons, the resulting system is charge-neutral. In this case we should put
$\epsilon_{\rm bound}=m$. Then $p_{max}=\sqrt{-2mV+V^{2}}\,,$ and taking into
account that each cell of the phase space can be occupied only by two
electrons of opposite spin we have

$n_{e}=\frac{p_{max}^{3}}{3\pi^{2}}=\frac{(-2mV+V^{2})^{3/2}}{3\pi^{2}}\,.$
(192)

Thus the relativistic Thomas-Fermi equation renders

$\Delta V=4\pi e^{2}\left[n_{\rm
nucl}-\frac{(-2mV+V^{2})^{3/2}}{3\pi^{2}}\right]\,,$ (193)

$n_{\rm nucl}$ is the charged density of the nucleus. It is curious to note
that such equation for neutral atom has been introduced long ago Vallarta1932
but relativistic term was then treated as a small correction in
nonrelativistic limit $|V|\ll m$.

####  7.1.1 Filling of the vacuum shell by electrons

Note that even in absence of external electrons, which may fill the empty
states, in case when the potential well $V<-2mc^{2}$ electrons and positrons
can be created already from the vacuum in absence of any external electrons.
Positrons go off to infinity, whereas electrons screen the initial positive
charge of the source. In this case we should put $\epsilon_{\rm bound}=-m$.
Then the relativistic Thomas-Fermi equation renders, cf. Rafelski1975 ;
Migdal:1976wx ; Migdal:1977rn ,

$\Delta V=4\pi e^{2}\left[n_{\rm
nucl}-\frac{(2mV+V^{2})^{3/2}}{3\pi^{2}}\theta(2mV+V^{2})\right]\,,$ (194)

where $\theta(x)$ is the step-function, with the boundary conditions on the
boarder of the ion

$V(r_{i})=-2m=-Z_{i}e^{2}/r_{i}\,,\quad
V^{\prime}(r_{i})=Z_{i}e^{2}/r_{i}^{2}\,,$ (195)

and with $V(r)=-Z_{i}e^{2}/r$ for $r>r_{i}$. Reference Rafelski1975  presented
numerical solutions. The thorough analytical and numerical study of the
problem of the filling of the vacuum shell by many electrons was performed in
an independent study Migdal:1976wx ; Migdal:1977rn . This phenomenon was
called “electron condensation” demonstrating that all vacuum levels are filled
by electrons of the lower continuum, cf. Migdal1978 .

###  7.2 A detailed derivation of relativistic Thomas-Fermi equation

Electron density can be found by direct summation of squared of the wave
functions Migdal:1977rn :

$n_{e}=-\sum_{n\kappa m}|\psi_{n\kappa m}|^{2}\,,$ (196)

where $\psi_{n\kappa m}$ are semiclassical wave functions presented in Eqs.
(115) – (130). Actually, we need wave functions in classically allowed region
given by (115).

Differentiating quantization rule (138) over $n$ we obtain

$\frac{\partial\epsilon}{\partial
n}\int_{r_{0}}^{r_{-}}\frac{\epsilon-V}{p}dr\simeq\pi\,,$ (197)

where we dropped the term $\frac{\partial}{\partial n}\frac{\kappa w}{pr}$,
which leads only to a small correction $|w|/V^{2}\sim 1/\zeta^{2}$, cf.
Mur:1978ke .

From (197) and (116) we obtain

$C_{1}=\left(\frac{1}{\pi}\frac{\partial\epsilon}{\partial n}\right)^{1/2}\,.$
(198)

Using that $\sum_{m=-j}^{j}|Y_{lm}|^{2}=(2j+1)/(4\pi)$, where $Y_{lm}$ is the
spherical function, from Eq. (196) we have

$n_{e}(r)=-\sum_{n\kappa}\frac{2j+1}{4\pi^{2}}\frac{\partial\epsilon}{\partial
n}\frac{\epsilon-V}{pr^{2}}\,.$ (199)

Here we replaced $\sin^{2}\theta_{1}$ and $\sin^{2}\theta_{2}$ by $1/2$ due to
multiple oscillations. Replacing summation in $n$ by integration we find

$n_{e}=-\frac{1}{4\pi^{2}}\sum_{\kappa}N_{j}=-\frac{1}{4\pi^{2}}\sum_{\kappa}\frac{2(j+1/2)}{r^{2}}\left(\sqrt{(\epsilon-V)^{2}-m^{2}-(j+1/2)^{2}/r^{2}}\right)_{\epsilon_{m}}^{\epsilon_{\rm
bound}}\,,$ (200)

$\epsilon_{m}$ corresponds to the zero of the under-square-root expression.
Doing further integration in $j$ with $\epsilon_{\rm bound}=-m$ we recover
(194).

Now let us estimate number of electrons in the vacuum shell, for which single-
particle approximation fails, i.e. number of levels, for which the width has
no exponential smallness. Integrating (200) over the volume we find number of
levels with momenta $j\leq\kappa-1/2$,

$\delta(\kappa)=\frac{1}{N_{e}}\sum_{j=1/2}^{\kappa-1/2}N_{j}=c\kappa^{2}\,,$
(201)

where

$c=3I_{1}/(2I_{2})\,,\quad I_{1}=\int(V^{2}+2mV)^{1/2}dr\,,\quad
I_{2}=\int(V^{2}+2mV)^{3/2}r^{2}dr\,,$ (202)

$V^{2}\geq-2mV$. In particular for $V=-\zeta/r$ with logarithmic accuracy we
obtain

$I_{1}=\zeta\ln(\zeta/R_{\rm nucl})\,,\quad I_{2}=\zeta^{3}\ln(\zeta/R_{\rm
nucl})\,,\quad c=3/(2\zeta^{2})\,.$ (203)

For $\zeta\gg 1$ and $\kappa_{0}=(\zeta/\pi)^{1/2}$ it follows that

$\delta(\kappa_{0})=3/(2\pi\zeta)\ll 1\,.$ (204)

Getting (201) we counted all states with $|\kappa|<\kappa_{0}$, whereas not
all of them have exponentially suppressed $\Gamma$. Taking into account of a
correction (204) leads to appearance of a numerical factor
$\ln(\kappa_{0}/R_{\rm nucl})/\ln(\zeta/R_{\rm nucl})\simeq 1/4$ for $\zeta\gg
1$ since $R\propto\zeta^{1/3}$. We estimate $\delta\simeq 0.1/\zeta$, i.e.,
$\delta\sim 1\%$ for $Z\sim 1/e^{3}$. Smallness of $\delta$ characterizes
accuracy of Eq (194).

Taking into account of the exchange and correlation corrections in the
relativistic Thomas-Fermi equation is conveniently done by means of a
variational method analogously to that is performed for the nonrelativistic
Thomas-Fermi equation Gambas1949 . We arrive at

$n_{e}\simeq-\frac{1}{3\pi^{2}}[(V^{2}+2mV)^{1/2}-\nu(V+m)]^{3}\theta(V^{2}+2mV)\,,$
(205)

$\nu\simeq e^{2}/\pi$. For $Ze^{3}\ll 1$ this correction can be safely
dropped. For $Ze^{3}\gg 1$ it can be taken into account in Eq (194) by
introducing the renormalized coupling constant $e^{2}\to e^{2}(1+3e^{2}/\pi)$,
cf. Migdal:1977rn .

Also, a correction appears due to that the dielectric permittivity of the
vacuum, $\varepsilon(eE)$, differs from unity, $e\vec{E}=-\nabla V$. Thus one
should replace $\Delta V\to\nabla(\varepsilon(E)\nabla V)$ in Eq. (194).
However this correction, as the correlation correction, is tiny, since
$\varepsilon(eE)=1-(e^{2}/(3\pi))\ln(eE/m^{2})$, and at distances
$r\stackrel{{\scriptstyle\scriptstyle>}}{{\phantom{}{}_{\sim}}}1/(a_{Z})$ of
our interest $\varepsilon(eE)\simeq 1+O(e^{2}/(3\pi))$, cf. LL4  and Eq. (273)
below.

###  7.3  Weak screening, $1/e^{2}\ll Z\ll 1/e^{3}$

Consider screening of the positively charged nucleus of initial proton number
$Z$ and radius $R$ (typically $R_{\rm nucl}\simeq A^{1/3}/m_{\pi}$, $A\sim
2Z$). Assume that inside the nucleus the proton charge density is
$n_{p}^{0}=const$. Introducing $\psi=-V/m-1$ in the region $V<-m$ ($\psi\geq
1$), where electrons of the vacuum shell give some contribution to the
screening of the charge $Z$, from Eq. (194) we obtain

$\Delta(m\psi)=\frac{4e^{2}m^{3}}{3\pi}(\psi^{2}-1)^{3/2}\theta(\psi-1)-4\pi
n_{p}^{0}\theta(R_{\rm nucl}-r)\,,$ (206)

$\theta(x)$ is the step-function. For $r>R_{\rm nucl}$ with the help of the
replacement $x=r/r_{i}$ we obtain

$\psi^{\,\prime\prime}_{x}+2x^{-1}\psi^{\,\prime}_{x}=\mu(\psi^{2}-1)^{3/2}\,,\quad\psi(1)=1\,,\quad\psi^{\,\prime}=-2\,,\quad\mu=\frac{4e^{2}m^{2}r_{i}^{2}}{3\pi}=\frac{(Z_{\rm
obs}e^{3})^{2}}{3\pi}\,.$ (207)

Here $Z_{\rm obs}$ is the charge seen at infinity. Since $\mu\ll 1$, we may
use expansion

$\psi(x,\mu)=\psi_{0}(x)+\mu\psi_{1}(x)+...$ (208)

Then we have equations

$\Delta_{x}\psi_{0}=0\,,\quad\psi_{0}(1)=1\,,\quad\psi_{0}^{\,\prime}(1)=2\,,$
(209)
$\Delta_{x}\psi_{1}=3(\psi_{0}^{2}-1)^{3/2}\psi_{0}\psi_{1}\,,\quad\psi_{1}(1)=0\,,\quad\psi_{1}^{\,\prime}(1)=0\,.$
(210)

Thus we derive

$\psi(x,\mu)=2x^{-1}[1+4\mu(-\ln x+C_{0})+O(x,\mu^{2})]\,,\quad C_{0}=2\ln
2-11/3\,.$ (211)

Inside the nucleus at condition $Ze^{3}\ll 1$ the potential is close to the
bare one. Setting $\psi=\zeta y(\Xi)/(R_{\rm nucl}m)$, $\Xi=r/R$ we obtain

$y_{\Xi}^{\,\prime\prime}+\frac{2y_{\Xi}^{\,\prime}}{\Xi}=\frac{4e^{2}R^{3}}{3\pi\zeta}\left(\frac{\zeta^{2}y^{2}}{R^{2}}-m^{2}\right)^{3/2}-\frac{4\pi
e^{2}}{\zeta}n_{p}^{0}\,,\quad r<R_{\rm nucl}\,.$ (212)

Using that inside the nucleus $|V|\sim\zeta/R_{\rm nucl}\sim Z^{2/3}m\gg m$
and $\frac{4\pi}{3}R_{\rm nucl}^{3}n_{p}^{0}=Z$ we get

$y_{\Xi}^{\,\prime\prime}+\frac{2y_{\Xi}^{\,\prime}}{\Xi}=-3+\nu
y^{3}\,,\quad\nu=\frac{4(Ze^{3})^{2}}{3\pi}\,.$ (213)

Since $\nu\ll 1$ we expand

$y=y_{0}(\Xi)+\nu y_{1}(\Xi)+...$ (214)

and get

$y_{0}(\Xi)=\frac{1}{2}(3-\Xi^{2})\,,\quad
y_{1}(\Xi)=C+\Xi^{-1}\int_{0}^{\Xi}y_{0}^{3}(x)x(\Xi-x)dx\,.$ (215)

Matching of $V$ and $V^{\,\prime}$ at the edge of the nucleus yields

$C=-1-\int_{0}^{1}y_{0}^{3}(x)xdx\,,$ (216)

and

$Z_{\rm
obs}=Z\left[1-\frac{4}{3\pi}(Ze^{3})^{2}(\ln\frac{\zeta}{R}+C_{1})+...\right]\,,$
(217) $C_{1}=\ln 2-\frac{8}{3}+\int_{0}^{1}y_{0}^{3}(x)x^{2}dx\simeq 1.38\,.$
(218)

###  7.4 Strong screening, $Ze^{3}\gg 1$

Continue to consider a nucleus with $Z\sim A/2$ and $R_{\rm nucl}\simeq
Z^{1/3}/m_{\pi}$. Since $R$ grows with $Z$, one may expect that for
sufficiently large $Z$ most of electrons enter the nucleus and the interior