Also at ]Department of Physics and NTIS – European Centre of Excellence,
University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic. Also
at ]Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Břehová 7, Prague 11519, Czech Republic. Also at
]National Institutes for Quantum Science and Technology (QST), Kansai Photon
Science Institute, 8-1-7 Umemidai, Kizugawa, Kyoto 619-0215, Japan.

# Electron-positron pairs and radioactive nuclei production by irradiation of
high-Z target with -photon flash generated by an ultra-intense laser in the
$\lambda^{3}$ regime

David Kolenatý  [<EMAIL_ADDRESS>Prokopis Hadjisolomou  Roberto Versaci
Tae Moon Jeong  Petr Valenta  [  Veronika Olšovcová  Sergei Vladimirovich
Bulanov  [  ELI Beamlines Centre, Institute of Physics, Czech Academy of
Sciences, Za Radnicí 835, 25241 Dolní Břežany, Czech Republic

###### Abstract

This paper studies the interaction of laser-driven -photons and high energy
charged particles with high-Z targets through Monte-Carlo simulations. The
interacting particles are taken from particle-in-cell simulations of the
interaction of a tightly-focused ultraintense laser pulse with a titanium
target. Lead is chosen as the secondary high-Z target owing to its high cross
section of the giant dipole resonance and electron-positron pair production.
The results reveal an ultra-short ultra-relativistic collimated positron
population and their energy spectra, angular distribution, and temporal
profile are found. We investigate the target thickness dependence of the
resulting total numbers and total kinetic energies of various particle species
emitted from the lead target irradiated with laser-generated -photons and
charged particles separately. We plot the charts of residual high-Z nuclides
generated by irradiation of the lead target. Owing to the short pulse
duration, the -photon, electron-positron, and neutron sources can find
applications in material science, nuclear physics, laboratory astrophysics,
and as injectors in laser-based accelerators of charged particles.

††preprint: APS/123-QED

##  I Introduction

The invention and development of the Chirped Pulse Amplification (CPA)
technique [1] coupled with the Kerr lens mode-locking technique [2] led to the
rapid growth of laser power during the last few decades [3]. The record power
of a 10 PW laser has been recently announced by ELI-NP [4] and another 10 PW
laser is being finalized at ELI-Beamlines. Current worldwide activities on PW
laser systems and plans to construct lasers beyond 100 PW are outlined in this
review article [3]. Intensities as high as $10^{23}\text{ W/cm}^{2}$ have been
recently reported [5]. The highest intensity achievable by a laser of certain
energy corresponds to a tightly-focused single-cycle laser pulse as proposed
by [6], referred to as the $\lambda^{3}$ regime, where $\lambda$ is the laser
wavelength.

The laser-matter interactions at such ultrahigh intensities result not only in
acceleration of electrons [7] and ions [8] but also in the generation of high-
energy -photons and electron-positron pairs [9]. As theoretically foreseen
[10, 11] and recently calculated by three-dimensional (3D) Particle-In-Cell
(PIC) simulations [12, 13, 14], the laser-matter interactions at ultrahigh
intensities result in high conversion efficiency of the laser energy to
-photon energy, up to 50% in a time comparable to the laser pulse duration.
The generation of such -photon flash is one of the main goals of multi-PW
laser facilities [15].

All aforementioned laser-generated high-energy particles undergo many
interactions with the surrounding matter [16]. They generate further electrons
through ionization [17] but also electron-positron pairs through pair
production in the Coulomb field of nuclei [18] and atomic electrons [19].
Additionally, -photons are produced by Bremsstrahlung [20] of electrons,
positrons, and ions or come from nuclear interactions of heavy ions [21]. All
-photons undergo Compton scattering [22] on electrons. Furthermore, -photons,
neutrons, protons, ions, and radioactive nuclides are produced through
photonuclear [23, 24, 25] and electronuclear [26] reactions, and through
nuclear interactions of heavy ions [21]. All these interactions and properties
of generated particles can be simulated by Monte Carlo (MC) particle transport
codes.

The pulsed sources of particles with such high energies can be used in the
field of fundamental science as a positron injector for plasma-based wakefield
accelerators [27], for electron-positron pair plasma studies of astrophysical
phenomena [28, 29, 30, 31], but also can find applications in medicine and
material science as positron annihilation lifetime spectroscopy [32, 33, 34],
neutron resonance spectroscopy [35, 36], neutron diffraction [37, 38], and
nuclear waste management [39, 40]. Activation of residual nuclides can be used
for the study of nuclear physics and astrophysics [41], and for direct
applications in nuclear medicine [42].

In this work we couple a 3D PIC simulation with MC simulations to examine the
interaction of the -photon flash and high energy charged particles generated
by an ultra-intense laser in the $\lambda^{3}$ regime with a high-Z target,
specifically lead. We consider the possible interactions of these laser-driven
high energy particles with a lead cylindrical target of varying thickness in
order to estimate the activation of produced residual nuclei and the
properties of particles emitted from the target, with the focus on positrons
in view of the aforementioned applications.

This paper is organized as follows. In Section II, we describe the simulation
setup of the PIC and the MC simulations and the result of the PIC simulation
of laser-matter interaction. In Section III, we show and discuss the results
of the MC simulations in terms of generated nuclides and emitted particles
dependence on the thickness of the lead target, the energy and angular
distributions of positrons emitted from the target with various thicknesses,
the temporal profile of collimated emission of ultra-relativistic positrons,
and the distribution of generated residual nuclides. In Section IV, we
conclude our findings.

##  II Simulation Setup

The output from the PIC simulation of ultra-intense laser pulse interaction
with a titanium target is imported into MC simulations as the primary
particles (-photons, electrons, positrons, and ions) to compute their
interactions with a lead target and the resulting products. The geometrical
setup of the MC simulations in respect to the PIC coordinates is designed to
irradiate the lead target with almost all of the particles moving in the
forward direction, i.e., with a positive momentum component in the laser
propagation direction, see Fig. 1. The titanium target for the PIC simulation
is chosen since its electron number density is optimum for high conversion
efficiency of the laser energy to -photon energy [13]. Natural lead ($Z=82$,
$A=207.19$) is chosen as target material for the MC simulations owing to high
cross sections for electron-positron pair production, proportional to $Z^{2}$,
and for the GDR being the dominant photonuclear interaction for the energies
considered. The detailed descriptions of the simulations are given in the
following subsections.

Figure 1:  The drawing illustrates the geometrical setup and the scales of PIC
and MC simulation regions. The laser-matter interaction resulting in the
generation of high energy particles is computed with the PIC code using a
titanium target. The simulation employs the $\lambda^{3}$ regime, where a
radially polarized near-single-cycle pulse is focused to a spherical-like
intensity of $\sim 0.5\lambda$ FWHM spatial profile and $\sim 3.4\text{ fs}$
pulse duration. An $\sim 80\text{ PW}$ power laser under the tight focusing
scheme used reached an intensity of $10^{25}\text{ Wcm}^{-2}$. The
interactions of laser-driven high energy charged particles and -photons with
high-Z matter and the resulting products were computed with a MC code using a
lead target. The deposited energy to the lead target is represented by color
mapping.

###  II.1 Particle in Cell Simulation

This subsection provides a short description of the PIC simulation presented
in Ref. [13]. The simulation is performed by the 3D EPOCH [43] PIC code. The
simulation employs the $\lambda^{3}$ regime [6], where a radially polarized
near-single-cycle pulse is focused by an $f$-number $1/3$ parabola [44, 45] to
a spherical-like intensity of $\sim 0.5\lambda$ FWHM spatial profile and $\sim
3.4\text{ fs}$ pulse duration. An $\sim 80\text{ PW}$ power laser under the
tight focusing scheme used reached an intensity of $10^{25}\text{ Wcm}^{-2}$.

The radius of the target, $r=2.4\kern 4.30554pt\mathrm{\mu m}$, is large
enough for the target destruction by the laser not to reach the target edges.
The highest conversion efficiency of the laser energy to -photon energy
corresponds to a relatively thick target ($2\kern 4.30554pt\mathrm{\mu m}$)
and an electron number density of $\sim 1.25\times 10^{24}\text{ cm}^{-3}$,
similar to that of titanium. The cubic simulation box is split into $512^{3}$
cubic cells with an edge size of 20 nm containing 8 macroparticles each. The
size of the simulation box of $10.24\kern 4.30554pt\mathrm{\mu m}$ and the 16
fs duration of the simulation are both large enough for the conversion of
laser energy to the energy of all species to saturate before the fields or
particles exit the simulation box.

As we see from Fig. 2, the PIC simulation of the ultra-intense laser-matter
interaction under the aforementioned conditions results in the generation of
high-energy -photons, electrons, positrons, and titanium ions, which are
assumed to be fully ionized. The distributions of all the particle species
exhibit a cylindrical symmetry with respect to the laser propagation direction
owing to the radial polarization of the laser pulse. Approximately 57% and 26%
of the laser energy was transferred to the kinetic energy of the particles
moving in the forward direction and in the backward direction (i.e., with a
negative momentum component in the laser propagation direction), respectively.
For MC simulations we employ only the particles moving in the forward
direction consisting of -photons, electrons, positrons, and titanium ions; the
total numbers of these particle species are approximately $3.2\times 10^{13}$,
$4.4\times 10^{13}$, $4.5\times 10^{11}$, $1.3\times 10^{12}$, respectively,
and the total kinetic energies of these particle species are approximately 93,
27, 10, and 30 Joules, respectively. It is worth mentioning the temperatures
of the particles with the energies $>500$ MeV: $\sim 140$ MeV for the
-photons, $\sim 160$ MeV for electrons, and $\sim 420$ MeV for positrons.
Moreover, all the particles apart from positrons are strongly collimated along
with the laser propagation axis, see Fig. 2(a).

Figure 2:  (a) The energy fluence per solid angle and (b) energy spectra of
PIC simulation output particles accelerated in the forward direction and
imported to the MC simulation as primary particles. -photons, electrons,
positrons, and fully ionized titanium ions (energy and particle number are per
nucleon) are denoted by the violet, blue, red, and green the color,
respectively. The distributions of all the particle species exhibit a
cylindrical symmetry with respect to the laser propagation direction owing to
the radial polarization of the laser pulse.

###  II.2 Monte Carlo Simulation

MC simulations are performed using the FLUKA code [46, 47] and FLAIR graphical
interface [48]. The input file of the FLUKA simulation consists of a variable
number of commands, each consisting of one or more lines, also called cards.
All the options and possibilities of the FLUKA code are described in the FLUKA
manual [49]. A user-written source routine is used to import all the PIC
output particles (species, position, momentum, and weight) as MC primary
particles. -photons, electrons, positrons, and ions are simulated using $\sim
1.4\times 10^{8}$, $\sim 1.9\times 10^{8}$, $\sim 1.9\times 10^{6}$, and $\sim
2.4\times 10^{6}$ primary particles, respectively. FLUKA provides simulations
results per primary particle, then the results are normalized to the real
number of primaries given in Subsection II.1, corresponding to a single laser
pulse. The cylindrical lead target with a diameter of 10 cm and thickness
varying from 1 mm to 10 cm is positioned at a distance of 1 mm from the focal
spot coordinates, as shown in Fig. 1. The given geometrical setup with a large
acceptance angle of $\sim 89^{\circ}$ is designed to irradiate the lead target
with almost all PIC output particles moving in the forward direction.

The DEFAULTS card is set to PRECISIOn with the particle transport threshold
set at 100 keV (particles below the threshold are not transported and their
energy is deposited on the spot), except neutrons ($10^{-5}\text{ eV}$).
Further models of nuclear interactions explicitly added on top of the defaults
include the activation of gamma and electron interactions with nuclei, the
Relativistic Quantum Molecular Dynamics (RQMD) [50] and a Dual Parton Model to
simulate heavy ion interactions (DPMJET) [51], the evaporation and coalescence
of nuclei during nuclear interactions, and the treatment of deuterons as two
separate nucleons.

To detect the energy and angular distributions of particles passing a chosen
plane, we use the USRBDX card. Employing the USRYIELD card, we obtain the
temporal profile of the particles passing the given plane. Furthermore, we
apply the RESNUCLE card to compute the generated residual nuclei fully de-
excited down to the ground or isomeric state. For definition and description
of cards used here, see the FLUKA manual in [49].

##  III Results and Discussion

In this section, we show and discuss the results of the MC simulations of
laser-driven high energy particles interacting with the lead target. The
results are divided into the following subsections: the target thickness
dependence of the generated nuclides and emitted particles, the energy and
angular distribution of the emitted positrons, the temporal profile of
collimated pulse of ultra-relativistic positrons, and the distribution of
generated residual nuclides. Laser-driven particles accelerated in the forward
direction resulting from the PIC simulation are referred to as primary
particles in the whole section.

###  III.1 Target thickness dependence of generated nuclides and emitted
particles

The kinetic energy of the laser-accelerated particles interacting with the
solid matter can be transformed to electron-positron pair production and
transferred to other particles. The particles with kinetic energy lower than
the transport threshold are assumed to deposit their energy at the position of
their production. As we see from panels (a) and (b) in Fig. 3, the deposited
energy from all primary particle species saturates at the thickness of the
lead target of 10 cm.

Figure 3: The total energy and the number of particles moving in the forward
direction beyond the target as function of the thickness of the lead target.
Irradiation by -photon primary particles only are plotted in panels (a) and
(c), respectively, and irradiation by the remaining primary particles (i.e.,
electrons, positrons, and $\text{Ti}^{+}$ ions) are plotted in panels (b) and
(d), respectively. Panels (a) and (c) show also energy deposited to the
target; panels (c) and (d) show the total number of residual nuclides in the
target. The percentage of the initial energy of primary particles and the
yield per initial primary particle are displayed on the right-hand side
vertical axes. The primary particles irradiating the target are denoted with
the black dashed lines; -photons, electrons, positrons, neutrons, energy
deposited to the target, and generated nuclides are denoted with the violet,
blue, red, green, orange, and the vine solid lines, respectively. Note the
multiplying factors for the neutrons and generated nuclides.

The total number of electrons and positrons emitted in the forward direction
from the target irradiated with -photons plotted in Fig. 3 (c) implies that
the majority of the generated electrons originate from the pair production.
The ionization process generates predominantly low energy electrons, since the
cross section increases with the decreasing -photon energy. For -photons of
the energies below 1 MeV, the ionization becomes the dominant interaction.
Although many electrons are excited to the state of a free electron by
ionization, their kinetic energy is usually not high enough to escape the
target. As a consequence, the ratio between electrons and positrons total
kinetic energies is close to unity for all the simulated thicknesses, see Fig.
3 (a). The target thickness of 5 mm is optimal for the conversion of primary
-photons to positrons escaping the target in the forward direction, where both
the conversion efficiency of total kinetic energy and the yield per primary
-particle reach $10\%$. Taking into account the charged primary particle
species as well, plotted in panels (b) and (d) in Fig. 3, the values of the
total number and the total kinetic energy are higher for electrons than for
positrons as a result of a considerable high number of primary electrons
penetrating the target. However, these values decrease with the increasing
target thickness because of its shielding effect.

The primary -photons are most efficient in the generation of neutrons escaping
the target and residual nuclides staying in the target for all target
thicknesses, see the data plotted in panels (c) and (d) in Fig. 3. The chart
of generated nuclides is given in Subsection III.4. While neutrons and
residual nuclides can be produced in ions interactions, their vast majority
are produced via photonuclear interactions, either directly by primary
-photons or indirectly by secondary -photons. The secondary -photons play an
important role for photonuclear interactions since the cross section peak for
the total GDR photoabsorption is narrow with FWHM of $\sim 4\text{ MeV}$ and
peaks at the -particle energy of $\sim 13\text{ MeV}$ for lead. Although the
integral cross section for the total GDR photoabsorption is proportional to
$NZ/(N+Z)$ [23], where N is the number of neutrons, the narrow energy range
stays in the same order of magnitude for all elements. Since the
Bremsstrahlung from high-energy electrons and positrons is the major process
of secondary -photons generation, the primary -photons require a relatively
high target thickness of 1 cm for the maximum number of emitted secondary
-photons, while the primary charged particles reach the maximum number of
emitted secondary -photons already at 5 mm.

The primary -photons irradiating the target with the thickness of 5 cm optimal
for the neutron emission produced 71% of all the neutrons emitted from the
target and 19% of the total kinetic energy of all the neutrons emitted from
the target, see Fig. 3. The latter is due to the aforementioned narrow cross
section of the GDR process at relatively low -photon energies. The remaining
neutrons, especially the high energy ones, are generated mostly by titanium
ions. High-flux neutron pulse sources find potential applications in nuclear
waste management [39, 40] and material science as neutron resonance
spectroscopy [35, 36] and neutron diffraction [38]. Photonuclear interaction
of laser-driven -rays for neutron source generation is simulated using PIC and
MC in [52].

###  III.2 Energy and angular distribution of emitted positrons

Fig. 4 panel (a) shows the energy and angular distribution of positrons moving
in the forward direction from the PIC output. These high energy primary
positrons are mostly deviated by $60^{\circ}-90^{\circ}$ from the laser
propagation direction with a total number of particles of $1.6\times 10^{11}$
and the total kinetic energy of 10.3 J. The major fraction of primary
positrons is already shielded by a target thickness of 1 mm. On the other
hand, irradiation of the target by any primary particle species can result in
the generation of positrons escaping the target in the forward direction.

Figure 4: The energy and angular distributions of positrons moving in the
forward direction for the lead target thickness of (a) 0 mm, (b) 1 mm, (c) 2
mm, and (d) 5mm are denoted by color mapping. The energy fluences of positrons
moving in the forward direction per solid angle are overplotted with gray
lines. The cylindrical symmetry of the primary particles with respect to the
laser propagation direction is transferred to all emitted positrons from the
target.

The total kinetic energy of positrons emitted in the forward direction reaches
the highest value of 12.5 J for the target thickness of 2 mm; however, a
significant fraction of $\sim 29\%$ of this energy comes from non-collimated
primary positrons. The target thickness of 5 mm leads to the maximum number of
positrons emitted in the forward direction with the value of $2.49\times
10^{12}$, which is by a factor of $\sim 16$ higher than the number of primary
positrons. The electromagnetic cascade showers do not occur since this target
thickness is comparable to the radiation length of lead of $\sim 5\text{ mm}$.
Moreover, the kinetic energy fluence per solid angle along the laser
propagation axis is the highest for a target thickness of 5 mm as well; the
total kinetic energy of 12 J is still higher than the total kinetic energy of
primary positrons by a factor of $\sim 1.2$. The spectrum of all positrons
emitted from the 5 mm thick target in the forward direction exhibits two
temperatures. The population with an energy below 500 MeV, mostly secondaries
produced by primary -photons and electrons, has a temperature of $\sim
80\text{ MeV}$. The population with an energy above 500 MeV mostly formed by
primary positrons has a temperature of $\sim 410\text{ MeV}$.

It is shown in Fig. 3 that the major fraction of positrons emitted from the
target in the forward direction and the major fraction of total kinetic energy
of positrons emitted from the target in the forward direction are caused by
target irradiation with primary -photons for all simulated target thicknesses.
The charged particles are less efficient since they have to first produce a
Bremsstrahlung -photon to generate electron-positron pair. In particular, for
the target thickness of 5 mm optimal for positron emission, the primary
-photons, electrons, and positrons are responsible for $\sim 80\%$, $\sim
10\%$, and $\sim 10\%$ of positrons escaping the target, respectively;
titanium ions produce an insignificant number of positrons. According to the
doubly differential Bethe-Heitler cross section for the electron-positron pair
production [53], the most probable scattering angle between incident -photon
and produced positron decreases with the increasing -photon energy and the
positron fraction of total pair kinetic energy. For -photon energies
$<10\text{ MeV}$, the most probable scattering angles of produced positrons
are $>5^{\circ}$. Nevertheless, the peak of the angular distribution of the
scattering probability is asymmetric with a slower decrease towards the
smaller angles than the bigger angles.

Owing to the aforementioned properties of electron-positron pair production
cross section and strongly collimated energy fluence of primary -photons, see
Fig. 2, the highest fluences per solid angle of positrons escaping the target
and of their total kinetic energy are computed in the laser propagation
direction for all target thicknesses plotted in panels (b), (c), and (d) in
Fig. 4. The highest population of resulting spectra along with the laser
propagation are in the energy range of $10-100\text{ MeV}$.

###  III.3 Temporal profile of collimated pulse of ultra-relativistic
positrons

Fig. 5 shows the pulse profile of positrons with the energy $>10\text{ MeV}$
escaping a 5 mm thick target and entering the disc detector with a maximum
divergence of 50 mrad from the target normal; the detector with a diameter of
1 mm is aligned concentrically and in parallel to the target at a distance of
1 mm. These thresholds and target-to-detector geometry are designed for the
detection of ultra-relativistic positrons collimated with the laser
propagation axis that can be further focused or moderated or both for the
requirements of a particular application. The detected positron pulse exhibits
a short duration of 35 fs (FWHM) and predominantly originates from primary
-photons, as expected from the previous discussion. The positrons decay
exponentially with the time constant of $\sim 43\text{ fs}$ and $\sim 34\text{
fs}$ for positrons originating from primary -photons and electrons,
respectively. Calculated brightness of positron beam is $\sim 1.1\kern
1.99997pt\mathrm{A/mm^{2}/mrad^{2}}$.

Figure 5: The flux of positrons per solid angle escaping from a 5 mm thick
target in the forward direction and entering the disc detector with a diameter
of 1 mm aligned concentrically and in parallel to the target at a distance of
1 mm. The minimum energy threshold is 10 MeV and the maximum threshold of
divergence angle from the target normal is 50 mrad. The data for positrons
originating from primary -photons, electrons, and positrons are denoted with
violet, blue, and red color, respectively; the bands represent errors of
estimated values. The inset figure displays the results with a linear scale.
The zero point of the time axes is placed at the first detection of positrons.
Note the positron data are below the logarithmic range of the vertical axis in
the main panel and overlap the horizontal axis in the inset.

The data for electrons and positrons in Fig. 3 implies that the ratio between
high energy electrons and positrons emitted from the target converges to unity
with increasing target thickness owing to the higher shielding effect for
primary electrons and positrons. For all simulated target thicknesses, see
Subsection III.1, the dominant process of electron generation is electron-
positron pair production which is a symmetric interaction in electron and
positron angular and energy distributions. By neglecting the scattering
effect, the most probable mean opening angle in radians is
$\theta=1.6/k_{\text{MeV}}$, where $k_{\text{MeV}}$ is the energy of pair
generating -photon in MeV [54]. As an example, the average electron and
positron both with energy of 10 MeV are emitted with an opening angle of $\sim
80\text{ mrad}$, i.e. $\sim 4.6\text{ deg}$. Owing to the strong collimation
of primary high energy -photons [13], the emission of high-energy electrons
and positrons with low mean opening angles is predominantly directed along the
laser propagation axis. The pulse of electrons and positrons emitted from the
target with the presented properties can be characterized as high-flux
relativistic pair jets [30] and used for the electron-positron pair plasmas
studies, where apart from other criteria the high and equal densities of
electrons and positrons in the volume of their co-existence are required to
achieve quasi-neutrality and collective behavior [28, 29, 31]. Studies on
high-flux relativistic pair jets and electron-positron pair plasmas open new
areas of research, particularly in laboratory astrophysics.

###  III.4 Distribution of generated residual nuclides

The results of the MC simulation show that all species of primary particles
irradiating the lead target produce not only stable but also radioactive
residual nuclides. Fig. 6 displays the numbers of residual high-Z nuclides
created in the target with the highest simulated thickness of 10 cm irradiated
by primary -photons, electrons/positrons, and titanium ions, individually. The
natural lead used as a target is comprised of three stable isotopes denoted by
squares with black edges in the row $Z=82$ with a standard atomic weight of
$A=207.19$. The majority of residual nuclides are produced through
photonuclear processes, predominantly through GDR, either directly by primary
-photons or indirectly by secondary Bremsstrahlung -photons from fast
electrons and positrons, see Subsection III.1. The remaining minor processes
generating residual nuclides are various nucleus-nucleus interactions with
heavy ions and electonuclear interactions.

Figure 6: Chart of residual high-Z nuclides generated in the lead target with
a thickness of 10 cm irradiated by primary a) -photons, b) electrons and
positrons, and c) titanium ions from the single laser pulse. Stable nuclides
are indicated with squares with black edges.

Among all generated nuclides in the lead target, the most abundant ones are
lead isotopes, protons, and alpha particles. One of the most numerous lead
radioactive isotopes generated is ${}^{203}{\rm Pb}$ where $\sim 4\times
10^{9}$ nuclides are produced by all primary particles from a single laser
pulse. The direct decay of ${}^{203}{\rm Pb}$ to stable ${}^{203}_{81}{\rm
Tl}$ through electron capture with half-life of $\sim 52$ hours, emitted
-photon energy of $\sim 279.2\text{ keV}$, and the absence of hadron emission
makes it particularly suitable for medical imaging [55]. High number of
thallium isotopes ($Z=81$) is also generated in the target with a total count
of $\sim 10^{9}$ nuclei per single laser pulse. Among them, the most useful
radioisotope being historically used extensively for nuclear medicine is
${}^{201}{\rm Tl}$ where $\sim 1.5\times 10^{8}$ nuclides are produced. The
long half-life of $\sim 73$ hours and the direct decay to stable
${}^{201}_{80}{\rm Hg}$ by electron capture provide convenient shelf storage
and successful imaging over a period of hours [56]. Other interesting
radioactive isotopes of thallium produced for medical applications are
${}^{206}{\rm Tl}$, ${}^{200}{\rm Tl}$, ${}^{199}{\rm Tl}$, ${}^{198}{\rm
Tl}$, and ${}^{197}{\rm Tl}$ with half-lives of 4.2 min, 26.1 hours, 7.4 h,
5.3 hours, and 2.8 hours, respectively.

##  IV Conclusions

We have examined the interaction of the -photon flash and high energy charged
particles generated by an ultra-intense laser in the $\lambda^{3}$ regime with
a high-Z target by using a 3D PIC (EPOCH) simulation coupled with MC (FLUKA)
simulations. We have found that the interaction of laser-driven high energy
particles with the cylindrical lead target can produce an ultra-short pulse of
ultra-relativistic positrons emitted from the target collimated to the laser
propagation axis and a broad source of radioactive nuclides. The highest
population of resulting spectra along with the laser propagation axis are in
the energy range of $10-100\text{ MeV}$. The computed pulse of ultra-
relativistic positrons with energies $>10\text{ MeV}$ and the divergence angle
from the target normal $<50\text{ mrad}$ originate predominantly from primary
-photons and exhibit a very short duration of 35 fs (FWHM).

Such an ultra-short and ultra-relativistic positron beam can be further
modified and utilized as a probe for high-resolution and volumetric scanning
of materials using positron annihilation lifetime spectroscopy [32, 33, 34].
The ultra-short duration of the pulsed positron sources is necessary for fine
temporal resolution in annihilation lifetime measurement since the pulse
duration should be significantly smaller than a timescale of interest. The
energy range of tens and hundreds of MeV can extend this technique to the
study of millimeter-centimeter thick materials. Further modified properties of
the collimated source of high energy positrons can find a potential
application as the injector in laser-based accelerators of charged particles.
Laser-driven positron beams can potentially become injectors of reduced cost
and size for future high energy accelerators and lepton colliders [27]. The
ultra-relativistic positrons are accompanied by electrons in the same
direction as the opening angles for ultra-relativistic pairs are very low. The
pulse of electrons and positrons emitted from the target with presented
characteristics can be used for the electron-positron pair plasmas studies
[28, 29, 31] opening new areas of research, particularly in laboratory
astrophysics research. The properties such as high density of pairs due to the
short pulse duration, high energy of pairs reaching hundreds of MeV, and low
opening angle emission, characterize the electron-positron pulses, referred to
as high-flux relativistic pair jets [30], well suited for the study of scaled
high energy astrophysical phenomena including relativistic shocks formation in
laboratory experiments.

We have also investigated the target thickness dependence of the number and
total kinetic energy of various particles emitted from the lead target
irradiated by laser-generated -photons and charged particles separately. Such
high-intensity neutron pulse sources can find potential applications in
nuclear waste management [39, 40] and material science as neutron resonance
spectroscopy [35, 36] for probing the interior ion temperature of opaque
materials and neutron diffraction [38] for analysis of the atomic and magnetic
structure of materials. The short pulse duration and the ability to dope only
a given section of the material enable high temporal and spatial resolution of
the analysis. Owing to the short pulse duration the continuous spectrum can be
used with the time-of-flight detection method.

The majority of residual nuclides are produced through photonuclear processes
dominantly by the giant dipole resonance. Activation of residual nuclides can
be used for the study of nuclear physics and astrophysics, and for direct
applications in nuclear medicine. For nuclear medicine, the important
radioactive isotopes produced are ${}^{203}{\rm Pb}$ and ${}^{201}{\rm Tl}$
with a total count of $\sim 4\times 10^{9}$ and $\sim 1.5\times 10^{8}$ nuclei
produced in a 10 cm thick lead target by all primary particles per single
laser pulse. These nuclides undergo direct decay to stable nuclides through
electron capture with a half-life of tens of hours. They emit -photons of
suitable energies for medical imaging.

###### Acknowledgements.

This work is supported by the projects High Field Initiative
(CZ.02.1.01/0.0/0.0/15_003/0000449) from the European Regional Development
Fund and “e-INFRA CZ” (ID:90140) from the Ministry of Education, Youth and
Sports of the Czech Republic. The EPOCH code is in part funded by the UK EPSRC
grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/M022463/1.

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