# Pair production as a probe for the dynamics of nuclear fission and $\alpha$
decay

T. Settlemyre  Cyclotron Institute, Texas A&M University, College Station,
Texas 77843, USA  H. Zheng  School of Physics and Information Technology,
Shaanxi Normal University, Xi’an 710119, China  A. Bonasera  Cyclotron
Institute, Texas A&M University, College Station, Texas 77843, USA  Laboratori
Nazionali del Sud, INFN, via Santa Sofia, 62, 95123 Catania, Italy

###### Abstract

Electron-positron pairs can be produced via the Schwinger mechanism in the
presence of strong electric fields. In particular, the fields involved in
$\alpha$ decay and nuclear fission are strong enough to produce them. The
energy of the $e^{+}e^{-}$ pair is related to the relative distance and
velocity of the daughter nuclei. Thus, the energy distribution of the produced
pairs can give information about the dynamics of the fission and $\alpha$
decay processes. A neck model of nuclear fission is used to illustrate how the
pairs can be used as a probe of the dynamics.

The Coulomb force is mediated by the exchange of a virtual photon, which may
result in the creation and annihilation of a virtual $e^{+}e^{-}$ pair. This
higher order effect causes vacuum polarization. The resulting correction to
the Coulomb potential was found by Uehling Uehling:1935uj . This correction is
important in $p$-$p$ scattering prestonbook  and in nuclear scattering systems
more generally Settlemyre:2021tbe , where it is on the order of one percent of
the Coulomb energy. Near the Coulomb barrier, the nuclear and Coulomb energies
roughly cancel, so a small change in the Coulomb energy could significantly
affect the height of the Coulomb barrier. A change in the height of the
Coulomb barrier will affect the cross section of sub-barrier fusion assenbaum
, e.g. Carbon-Carbon fusion in the cores of stars.

When the available energy exceeds twice electron mass, real $e^{+}e^{-}$ pairs
can be created during the dynamics in the presence of strong fields
Schwinger:1951nm ; wongbook ; Blaschke:2019pnj ; carrollbook ;
Voskresensky:2021okp ; popov . This effect may increase the fusion cross
section above the adiabatic limit pairproduction ; Kimura:2004dw ;
Krauss:1987onj ; Engstler:1988tfw ; Shoppa:1993zz ; Musumarra:2001xd ;
Kimura:2005tq . Pair production is also possible in the decay of nuclei. In
this paper, we will derive an approximate result for $dN/dE_{+}$ of the
produced positrons. The dynamics of nuclear fission are not well known bulgac
. We propose that the $e^{+}e^{-}$ pairs produced during fission can shed some
light on the dynamics.

In our previous paper pairproduction , we described a mechanism for pair
production. We briefly recall the main points. In our geometry, two nuclei or
fragments are a distance $R$ apart from each other. The electron is formed in
the middle, and the positron tunnels on an $x$-axis perpendicular to the beam
axis. The Coulomb potential seen by the positron is

$V_{+}(R,x)=\frac{Z_{tot}e^{2}}{\sqrt{\left(\frac{R}{2}\right)^{2}+x^{2}}}-S(x)\frac{e^{2}}{x},$
(1)

where $x$ is the distance between the positron and electron,
$Z_{tot}=Z_{1}+Z_{2}$ is the total charge, $S(x)=1-\exp(-2m_{T}x/\hbar)$ is a
screening factor, and $m_{T}=\sqrt{m^{2}+p_{T}^{2}}$ is the transverse mass of
the positron. Ignoring small spin corrections, the positron satisfies the
Klein-Gordon equation Schwinger:1951nm ; wongbook

$\left[(E_{+}-V_{+}(R,x))^{2}-p_{x}^{2}-m_{T}^{2}\right]\psi=0.$ (2)

This is formally equivalent to the Schrödinger type equation

$\left[\frac{p_{x}^{2}}{2m_{T}}+\frac{m_{T}}{2}-\frac{(E_{+}-V_{+}(R,x))^{2}}{2m_{T}}\right]\psi=0,$
(3)

for a particle of mass $m_{T}$ in an effective potential

$V_{eff}(x)=\frac{m_{T}}{2}-\frac{(E_{+}-V_{+}(R,x))^{2}}{2m_{T}}$ (4)

with effective energy $E_{eff}=0$. The effective potential has a hill that the
positron can tunnel through (Fig. 1). The probability of tunneling is

$\Pi_{t}=[1+\exp(2A/\hbar)]^{-1},$ (5)

where $A$ is the action integrated between the turning points.

Figure 1: $V_{+}$ (top panels) and $V_{eff}$ (bottom panels) for $\alpha$
decay (left side) and symmetric fission (right side) of 252Cf. The relative
distance between the ions is at the moment the $\alpha$ exits the Coulomb
barrier in the case of $\alpha$ decay as dictated by the $Q$ value (6.2 MeV).
For fission, the relative distance is when the two fission fragments are
touching (12.0 fm).

In a small time $dt$, the probability of creating a pair is

$dN=\Pi_{t}\frac{dt}{\Delta\tau},$ (6)

where $\Delta\tau=\frac{\hbar}{2m_{T}}$ is the characteristic tunneling time
from the Heisenberg principle pairproduction . As $E_{+}$ approaches $m_{T}$,
the barrier that the positron must tunnel through becomes infinitely long, so
the tunneling probability goes to zero. From the conservation of energy, the
energy associated with the positron is

$E_{+}=\frac{2Z_{tot}e^{2}}{R}-m_{T}-\Delta E_{k},$ (7)

where $m_{T}$ is the transverse mass of the electron and $\Delta E_{k}$ is the
change in kinetic energy of the ions at the moment of the pair’s creation,
which we showed must be $\geq 2m_{T}$ pairproduction . A simple change of
variables gives

$\frac{dN}{dE_{+}}=\frac{\Pi_{t}R^{2}}{2\Delta\tau|\frac{dR}{dt}|Z_{tot}e^{2}}.$
(8)

Thus the number of pairs produced at a certain energy depends on the distance
between the ions and their relative velocity. Since the ions gain kinetic
energy $\Delta E_{k}$ during the tunneling process, there is some ambiguity
around which value of $dR/dt$ to use: the relative velocity before or after
pair creation (or some average of the two). If we only consider the Coulomb
repulsion between the ions, we can find $dR/dt$ from the Coulomb potential.
Thus, for the Coulomb trajectory,

$\frac{dN}{dE_{+}}=\frac{\Pi_{t}\sqrt{2\mu}Z_{tot}e^{2}}{\Delta\tau(E_{+}+m_{T}+\Delta
E_{k})^{2}}\\\ \times\left(E_{cm}+\beta\Delta
E_{k}-\frac{Z_{1}Z_{2}}{2Z_{tot}}(E_{+}+m_{T}+\Delta E_{k})\right)^{-1/2},$
(9)

where $\mu$ is the reduced mass of the ions, and $0\leq\beta\leq 1$ is a
parameter that determines the relative velocity of the ions based on the
tunneling dynamics. In this way, pair production can give information about
the tunneling process.

Figure 2: $dN/dE_{+}$ for $\alpha$ decay of a selection of nuclei. $\Delta
E_{k}=2m_{e}$ and $\beta=0$ for this case.

We can apply this to the case of $\alpha$ decay. For this work, we assume that
the $\alpha$ particle emerges from the Coulomb barrier at rest, and then it
accelerates away from the daughter nucleus. Thus, Eq. (9) describes the
distribution of positron energies for this case. As an approximation, we will
say that the $\alpha$ particle takes away all the remaining kinetic energy so
that

$E_{\alpha}=Q-E_{+}-m_{T}.$ (10)

Figure 2 shows $dN/dE_{+}$ for the $\alpha$ decay of several heavy nuclei. The
peaks on the left hand side of the graph correspond to positrons that are
created when the $\alpha$ particle is at rest, since $|dR/dt|$ appears in the
denominator of Eq. (8). In reality, the $\alpha$ particle is not likely to be
at rest when it emerges from the Coulomb barrier, so those peaks may not be
reproduced in practice.

Figure 3: $dN/dE_{+}$ for fission and $\alpha$ decay of 252Cf, weighted by the
corresponding branching ratios. Here, $m_{T}=m_{e}$ and $\Delta E_{k}=2m_{e}$.
In the top panel we assume that the extra acceleration ($\Delta E_{k}$) is
given to the ions after the pair creation resulting in the divergence at time
zero, $E_{+}\approx$ 30 MeV, $\beta=0$. The Coulomb divergence disappears if
$\beta=1$, i.e. the ions are accelerated at the beginning of the pair
production process (bottom panel). Similarly for $\alpha$ decay. Figure 4:
$dN/dE_{+}$ for fission of a selection of nuclei. The fission dynamics are
based on the neck model. Here, $m_{T}=m_{e}$, $\beta=0$ and $\Delta
E_{k}=2m_{e}$.

$dN/dE_{+}$ depends sensitively on the distance between the daughter nuclei at
the moment the pair is created. Thus, information about the average dynamics
of fission can be inferred from a careful measurement of the positron
spectrum. A neck model neck  was used to simulate the dynamics for fission of
several heavy nuclei. The neck model gives the distance between the fission
fragments until the neck breaks and Coulomb repulsion takes over. During the
fission process, the total nuclear volume of the fissioning fragments is
assumed constant neckfiss . The fragments separate because of the large
Coulomb repulsion. The separation is slowed down by the surface tension of the
two fragments joined by a neck. Furthermore, nucleon transfer through the neck
results in the famous window dissipation formula randrup . These ingredients
make the dynamics up to the scission point quite different from the simple
Coulomb repulsion of two point particles given by the full (blue) line in Fig.
3. The pair production is strongly affected by the initial kinetic energy of
the fission fragments. If the fragments start at rest, then a peak due to the
Coulomb divergence, should be observed, Eq. (7). The peak disappears if in
such a compact initial configuration some kinetic energy is initially given to
the fragments. This could be due to the action of the electron created between
the two fragments as described above. In Fig. 3, $dN/dE_{+}$ for $\alpha$
decay and asymmetric fission of 252Cf neckfiss  is shown, multiplied by the
corresponding branching ratios. The segment of the plot to the right of the
point labelled “scission" corresponds to pairs created during the fission
process itself. As there are no positrons created at this high energy for
$\alpha$ decay, we can see the effects of the fission dynamics. The blue curve
shows the spectrum obtained if the fission fragments are allowed to accelerate
away from each other without friction and just the Coulomb repulsion. The
friction slows down the fission fragments, increasing $dN/dE_{+}$ for
positrons of the appropriate energy. The positron spectra for symmetric
fission of several heavy nuclei are shown in Fig. 4. Previous experimental
investigations have looked for a coincidence between $e^{+}$ and $e^{-}$
tsunoda . In our approach the coincidence is lost because the electron gets
trapped by the nuclei. We suggest an experiment to look for $e^{+}$ in
coincidence with a fission fragment and not with $e^{-}$.

In conclusion, our model predicts that electron-positron pairs can be created
during $\alpha$ decay or fission. The energy of the created positron is
related through the conservation of energy to the distance between the progeny
nuclei. Careful observation of the energy spectrum of pairs produced during
fission could reveal the dynamics of the fission process, properties of the
vacuum polarization and the tunneling “dynamics.”

## Acknowledgements

This research was funded in part by the United States Department of Energy
under Grant # DE-FG03-93ER40773 and the NNSA Grant No. DENA0003841 (CENTAUR)
and by the National Natural Science Foundation of China (Grant Nos. 11905120
and 11947416).

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