# A high-efficiency proton-boron fusion scheme taking into account the effects
of quantum degeneracy

S. J. Liu  Institute for Fusion Theory and Simulation, School of Physics,
Zhejiang University, Hangzhou, 310058, China  D. Wu<EMAIL_ADDRESS>Key
Laboratory for Laser Plasmas and School of Physics and Astronomy, and
Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai, 200240, China  T. X. Hu  Institute for Fusion Theory and
Simulation, School of Physics, Zhejiang University, Hangzhou, 310058, China
T. Y. Liang  Institute for Fusion Theory and Simulation, School of Physics,
Zhejiang University, Hangzhou, 310058, China  X. C. Ning  Institute for Fusion
Theory and Simulation, School of Physics, Zhejiang University, Hangzhou,
310058, China  J. H. Liang  Key Laboratory for Laser Plasmas and School of
Physics and Astronomy, and Collaborative Innovation Center of IFSA (CICIFSA),
Shanghai Jiao Tong University, Shanghai, 200240, China  Y. C. Liu  Institute
for Fusion Theory and Simulation, School of Physics, Zhejiang University,
Hangzhou, 310058, China  P. Liu  Institute for Fusion Theory and Simulation,
School of Physics, Zhejiang University, Hangzhou, 310058, China  X. Liu  Key
Laboratory for Laser Plasmas and School of Physics and Astronomy, and
Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai, 200240, China  Z. M. Sheng<EMAIL_ADDRESS>Institute
for Fusion Theory and Simulation, School of Physics, Zhejiang University,
Hangzhou, 310058, China  Y. T. Zhao  MOE Key Laboratory for Nonequilibrium
Synthesis and Modulation of Condensed Matter, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  D. H. H. Hoffmann  MOE Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  X. T. He
Institute for Fusion Theory and Simulation, School of Physics, Zhejiang
University, Hangzhou, 310058, China  J. Zhang  Key Laboratory for Laser
Plasmas and School of Physics and Astronomy, and Collaborative Innovation
Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai, 200240,
China  Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

The proton-boron (p-11B) reaction is regarded as the holy grail of advanced
fusion fuels, since the primary reaction produces three $\alpha$ particles
with few neutrons and induced radio-activities from second order reactions.
Compared to the Deuterium-Tritium reaction a much higher reaction temperature
is required. Moreover, bremsstrahlung energy losses due to the high nuclear
charge of boron deem it seemingly apparent than a fusion reactor based on
Deuterium-Tritium plasma in equilibrium is to say the least very difficult. It
is becoming more appealing to collide intense laser beams or accelerated
proton beams with a boron target to produce p-11B reactions. The fusion yield
of p-11B reactions is closely related to proton beam parameters and boron
target conditions such as density, temperature, and ingredients. Quantum
degeneracy will increase fusion yields by reducing the stopping power of
injected protons. In this work, we suggest a high-efficiency scheme for beam-
target p-11B fusions via injecting a MeV proton beam into a highly compressed
quantum degenerated boron target. Such a boron target can be achieved via
quasi-isentropic compression of solid boron by using precisely shaped laser
pulses. Our results indicate that for densities ranging from $10^{3}$ to
$10^{4}\rho_{s}$, where $\rho_{s}$ is the density of solid boron,
contributions of bound and free electrons to the stopping of protons can be
completely disregarded and dramatically reduced respectively. The result is an
increase in fusion yield by orders of magnitude. Furthermore, in order to
achieve multiplication factor $F$ greater than one, with $F$ defined as the
ratio of output fusion energy to the energy of injected protons, it is found
there exits a minimum possible density of boron target, which is $2.15\times
10^{4}\rho_{s}$ when the kinetic energy of injected protons is $0.8$ MeV.

##  I Introduction

Apart from the advances in fusion research that made the headlines in public
non-scientific journals, such as the record in confinement time at high
temperature of the Hefei Experimental Advanced Superconducting Tokamak (Xinhua
Updated: 2021-12-31 17:14 May 28, 2021), the 59 Megajoules of fusion power
reported from the Joint European Torus in a Deuterium Tritium fusion
experiment (JET, 2021 FAZ 2022-02-09), and the burning plasma in an inertial
fusion experiment from the National Ignition Facility of the Lawrence
Livermore National Laboratory ( LLNL, 2021; New York Times 2021-08-17), there
was a great number of more quiet achievements that demonstrate an accelerated
pace towards the final goal of fusion energy Tikhonchuk  _et al._ (2021); Gus’
kov  _et al._ (2021); Liu  _et al._ (2021); Weber  _et al._ (2021). While the
mainstream of research and technological development is directed towards the
Deuterium-Tritium reaction for fusion energy, the 11B(p,$\alpha$)2$\alpha$
process got renewed attention Hora  _et al._ (2021); Ribeyre  _et al._ (2022);
Margarone  _et al._ (2022), since non-equilibrium conditions, as they are
available in laser generated plasma, may turn out to be favorable to enhance
the fusion yield Labaune  _et al._ (2013). The cross-section of the
11B(p,$\alpha$)2$\alpha$ together with reactions like 12C(e,e’p)11B provides a
direct probe for nuclear structure properties of 12C Hoffmann  _et al._
(1981); Calarco  _et al._ (1984); Allas  _et al._ (1964); Segel  _et al._
(1965). Moreover, the abundance of 11B in the universe is still an unresolved
problem. Therefore, the investigation of these reactions will also contribute
to solve the mystery of the low astrophysical abundances of the light elements
Li, Be, and B in young main-sequence F and G stars Lamia  _et al._ (2011);
Hoffmann  _et al._ (1983). Besides, the 11B(p,$\alpha$)2$\alpha$ reaction
provides a new method of cancer treatment Cirrone  _et al._ (2018).

Despite the recent progress, there is still a long way to go until fusion
energy will finally be the solution the global energy problem. All routes to
fusion energy, as there are magnetic confinement fusion and inertial fusion
carry different inherent problems and at the moment it is not clear where the
chances of success are highest. Therefore, all possible routes should
presently be investigated thoroughly. However, there are not only technical
problems or unsolved physics details, there is also the problem of supply of
fusion fuel, especially Tritium in case of the DT-fusion reaction. The start
of the ITER reactor in the 2030s will use the world’s stockpile of tritium and
it will take a while until tritium may be supplied by using fusion neutrons
and lithium within the reactor. Therefore, we deem it important to investigate
alternative routes using the 11B(p,$\alpha$)2$\alpha$ reaction. This reaction
is a good candidate since the boron is readily available and neither in the
entrance channel nor in the exit channel of the reaction radioactivity is
involved Liu  _et al._ (2021). There are of course reactions of second order,
where neutrons are present and induced radioactivity may occur, but on a low
and acceptable level Weber  _et al._ (2021). Therefore, at government level in
Europe a discussion started to license fusion power plants outside of the
restrictions that are valid for fission power plants because it releases few
radioactive products (Markus Roth at Fusion Day at GSI Darmstadt Nov.
10${}^{\text{th}}$, 2022). But there is a price to pay. The physical
conditions to be met for fusion power based on proton boron fusion are much
more demanding than those of DT. On the one hand, p11B requires much higher
ignition temperature as compared to the D-T reaction, and on the other hand
radiation losses due to bremsstrahlung are overwhelmingly high and thus
prevent a burning plasma in equilibrium conditions and equimolar fuel
constituents, resulting eventually in no or very little net energy output
Moreau (1977); Putvinski  _et al._ (2019). However, with advances in high-
intensity laser technology, the laser-induced proton-boron beam target nuclear
fusion is gradually becoming more attractive.

Figure 1: (a)-(b) Diagram of a high efficiency p-11B fusion scheme, (c) p-11B
fusion cross section as a function of center-of-mass energy, where the data of
black square is extracted from the work of Nevins and Swain Nevins and Swain
(2000), (d) the ionization degree of boron target as a function of temperature
and density of boron target.

Based on this idea, a number of groups Belyaev  _et al._ (2005); Labaune  _et
al._ (2016); Kimura  _et al._ (2009); Giuffrida  _et al._ (2020); Labaune  _et
al._ (2013) have performed a series of experiments on p-11B fusion reactions
and measured the yields of $\alpha$ particles. The yields have been increased
from about $10^{5}\ \mathrm{sr}^{-1}$ in 2005 Belyaev  _et al._ (2005); Kimura
_et al._ (2009) to about $10^{10}\ \mathrm{sr}^{-1}$ in 2020 Giuffrida  _et
al._ (2020). Giuffrida et al. Giuffrida  _et al._ (2020) have investigated the
p-11B beam-target fusion reactions and calculated the fusion yields. However,
the stopping process of protons in detail has not been analyzed and there,
thus still leaving open ambiguities involving the interaction between the
intense proton beams and the boron target. This interaction depends largely on
the intensity of proton beams and the conditions of the boron target such as
temperature, density, composition and others. Beam intensity is influencing
the stopping process as has recently been demonstrated Ren  _et al._ (2020).
The same is obviously true for the target and plasma parameters. It is
important to uncover the relationship between these factors and their
respective influence on the reaction probability. Among these factors, the
degree of degeneracy is defined as $\Theta={T_{F}}/{T_{e}}$, with $T_{F}$ and
$T_{e}$ representing the Fermi energy and thermal temperature, respectively.
Electrons have to obey Fermi-Dirac statistics. For $\Theta\ll 1$ Boltzmann
statistics may be applied Zwicknagel  _et al._ (1999).

In this paper, the quantitative relationship between the yields of p-11B beam-
target nuclear reactions and the density of boron targets is derived. We find
that the effect of quantum degeneracy will increase the fusion yields mediated
by the effect of reduced the stopping power of the beam protons Ning  _et al._
(2022). Based on that, we suggest a high efficiency scheme for beam-target
p-11B fusions, where a MeV proton beam is injected into a highly compressed
quantum degenerated boron target. An outline of the suggested scheme is
displayed in Fig. 1. A highly compressed quantum degenerated boron target can
be achieved via quasi-isentropic compression of a solid boron by using
precisely shaped laser pulses. Our results indicate that for boron target of
densities ranging from $10^{3}$ to $10^{4}\rho_{s}$, contributions of bound
and free electrons to the stopping of protons can be completely disregarded
and dramatically reduced respectively, which therefore results in orders of
magnitudes increment of fusion yields. Furthermore, in order to achieve
multiplication factor $F$ greater than one, we find that there exits a minimum
required boron target density of $2.15\times 10^{4}\rho_{s}$ when the kinetic
energy of injected protons is $0.8$ MeV.

The structure of this paper is organized as follows. In section II, a
quantitative relationship between the reaction yields of p-11B beam-target
nuclear reactions as a function of proton stopping power per unit density is
derived. In section III, contributions from free electrons, bound electrons
and ions to the stopping power of protons are analyzed and compared with PIC
simulations. In section IV, in order to achieve multiplication with $F>1$, the
relation between the minimum possible density of compressed boron and the
kinetic energy of injected protons is analyzed. Finally, conclusions and
discussions are displayed in section V.

##  II Beam-target fusion yields

In general, to calculate the reaction yields of the proton-boron nuclear
fusion, we first need to integrate the relative velocity distribution
according to the cross section of the proton-boron nuclear fusion under the
center of mass system to get the average reaction rate Atzeni and Meyer-ter
Vehn (2004)

$\langle\sigma v\rangle=\int_{0}^{\infty}\sigma(v)vf(v)dv,$ (1)

where $f(v)$ is the distribution function of the relative velocities of
protons to boron nuclei, and $\sigma(v)$ is the corresponding cross section
with $v$, which is the relative velocity of protons to boron nuclei. The
number of reactions per unit time per unit volume, namely the volumetric
reaction rate, is then calculated as $R=n_{p}n\langle\sigma{v}\rangle$, where
$n_{p}$ is the number density of protons, and $n$ is the number density of
boron nuclei. It shows that the volumetric reaction rate is proportional to
the density of protons and boron nuclei Atzeni and Meyer-ter Vehn (2004).
Finally, the total reaction number under a certain volume in a certain energy
confinement time is obtained by multiplying $R$ with the total volume and
confinement time.

However, this method is strictly applicable only when the proton boron plasma
is in thermal equilibrium. For non-equilibrium states, such as the process of
projectile-target interaction, the relative velocity distribution changes
rapidly. Therefore, the average reaction rate also changes with time. Here we
propose a simple model to calculate the nuclear yields of beam-target
reactions. In this model, we just consider the process of projectile-target
interaction in p-11B beam-target fusion while neglecting thermonuclear
contributions. The previous work of Giuffrida Giuffrida  _et al._ (2020) also
proved that thermonuclear contributions to the fusion yields are negligible in
the beam-target process. The density of incident protons is considered to be
low, therefore the action of the proton beam on the boron target can be
regarded as small perturbation. The temperature of the proton beam and boron
target are kept at low levels, meaning that the relative velocities of protons
and boron nuclei equal the injected velocities of the proton beam. Due to the
interaction of the proton beam and boron target, when the protons move inside
the boron target, their velocities will be decelerated gradually.

We assume a small cloud of protons to be injected into the boron target, and
the proton number in this cloud is $N_{p}$. According to the proton energy,
the deceleration process of this cloud inside of the boron target can be
divided into many segments with infinitesimal volume, and the energy of
protons is considered constant within a respective segment. In the $i$-th
segment, the density of protons $n_{pi}$ and the volume $V_{i}$ will satisfy
$N_{p}=n_{pi}V_{i}$. We only need to calculate the reaction number of every
segment and sum them up. Then we will get the total reaction number during the
deceleration process of protons.

During the deceleration process, the energy of protons has slowed down from
$E_{pi}$ to $E_{pi}-\delta E_{p}$ in laboratory coordinates in the $i$-th
segment. The average reaction rate in the $i$-th segment becomes
$\sigma(v_{i})v_{i}$, and then we can express the volumetric reaction rate in
the $i$-th segment as $R_{i}=n_{pi}n\sigma(v_{i})v_{i}$. The reaction number
in the $i$-th segment during the time internal $\delta t_{i}$ is $R_{i}\delta
t_{i}V_{i}=nN_{p}\sigma({v_{i}})v_{i}\delta t_{i}=nN_{p}\sigma(E_{pi})\delta
z_{i}$, where $\delta t_{i}$ and $\delta z_{i}$ are the deceleration time and
the deceleration distance of the proton cloud in the $i$-th segment
respectively, satisfying $\delta z_{i}=v_{i}\delta t_{i}$. We have

$\delta z_{i}=\int_{E_{pi}}^{E_{pi}-\delta E_{p}}\frac{1}{dE/dx}dE,$ (2)

where $dE/dx=-S$, and $S$ is the stopping power of protons, which is also
called the stopping force.

By integrating all segments, we can obtain the total reaction number as

$R_{T}=nN_{p}\int_{0}^{E_{p}}\frac{\sigma(E)}{S}dE,$ (3)

where $\sigma(E)$ is the p-11B fusion cross section as a function of center-
of-mass energy, as is shown in Fig. 1(c). The data of black square is
extracted from the work of Nevins and Swain Nevins and Swain (2000). Eq. (3)
elucidates the relationship between the number of reactions and the energy
loss of protons in a boron target, which can be written into the form,

$R_{T}=N_{p}P=N_{p}\int_{0}^{E_{p}}\frac{\sigma(E)}{S/n}dE,$ (4)

where $P$ is the rate of a single proton triggering p-11B nuclear reactions
during the whole deceleration process, and $E_{p}$ is the initial energy of
injected protons. A similar formula as Eq. (4) had also been derived by
Giuffrida Giuffrida  _et al._ (2020).

It is worth noticing that since the cross section of the p-11B fusion can not
be changed, $P$ only depends on the stopping power per unit density
$S_{a}=S/n$. Obviously, $S_{a}$ is fully determined by the thermodynamic state
(density and temperature) of the boron target Skupsky (1977); Fano (1963);
Clauser and Arista (2018); Hayes  _et al._ (2020).

##  III The Stopping power of protons

Contributions to the stopping power of injected protons can be divided into
three parts, i.e., stopping by collisions with free electrons, bound electrons
and nuclei. Generally speaking, the contribution from nuclear stopping is
small unless at low incident energy. However, at highly compressed and quantum
degenerate plasmas, the nuclear contribution can not be ignored easily before
a detailed analysis of the specific situation. In order to comprise all the
three contributions, we here take the stopping power of protons as,

$S=S_{f}+S_{b}+S_{n},$ (5)

where $S_{f}$ is from free electrons, $S_{b}$ is from bound electrons, and
$S_{n}$ is from nuclei. As free and bound electrons contribute separately, to
distinguish them, the ionization degree of boron target with given density and
temperature needs to be determined. According to the shielded hydrogen model
and the single electron counting model Heltemes and Moses (2012), the
ionization degree as a function of density and temperature is displayed in
Fig. 1 (d). For highly compressed and quantum degenerated (low temperature)
plasmas, the degree of ionization depends only on the density. In our
following analysis, the initial ionization state of boron target is chosen
according to Fig. 1 (d).

###  III.1 The free electronic stopping power

The stopping contribution from free electrons is closely related to quantum
degeneracy. To indicate the effect of degeneracy, here both theoretical
analysis and computer simulations are displayed and compared to each other. We
divide the part of theoretical analysis into semi-classical and quantum parts
respectively. Simulations are also divided into two parts: the classical part,
and the quantum part considering both Fermi-Dirac statistics and the Pauli
exclusion principle.

As for the stopping contribution from free electrons, it was intensively
analyzed with both the semi-classical partial wave scattering (SPWS) method
Sigmund (1982); Arista and Sigmund (2007); Clauser and Arista (2013, 2018) and
the dielectric function method Fernández-Varea and Arista (2014); Lindhard and
Winther (1964); Maynard and Deutsch (1982, 1985); Skupsky (1977). Within the
dielectric formalism, the free electronic stopping power of a bare ion of mass
$m_{b}\gg m_{e}$ and charge $Z_{b}e$ ($m_{e}$ and $e$ are the electron mass
and the elementary charge, respectively) moving with velocity $v_{p}$ is given
by Maynard and Deutsch (1985); Fernández-Varea and Arista (2014); Deutsch and
Maynard (2016),

$S_{f}=\frac{2}{\pi}\left(\frac{Z_{b}e}{v_{p}}\right)^{2}\int_{0}^{\infty}\frac{dk}{k}\int_{0}^{kv_{p}}d\omega\omega\operatorname{Im}\left(\frac{-1}{\varepsilon(\omega,k)}\right),$
(6)

where $\varepsilon(k,\omega)$ is the complex dielectric function of the medium
which depends on the wave number $k$ and angular frequency $\omega$ of the
electromagnetic disturbance caused by the bypassing projectile,

$\varepsilon(\omega,\boldsymbol{k})=1+\frac{4\pi}{k^{2}}\Pi_{0}(\omega,\boldsymbol{k})$
(7)

where $\Pi_{0}(\omega,\boldsymbol{k})$ is the free-electron density response
function. The proportionality of $S_{f}$ with $Z_{b}^{2}$ is a signature of
linear-response theory.

Figure 2: The stopping power per unit density from free electrons as a
function of densities and injected velocities of protons, where analytical
results and values extracted from simulation by LAPINS code are compared with
each other, in which the analytical curves including ‘Wigner-Poisson’ and the
semi-classical ‘Vlasov-Poisson’ methods, and the numerical curves including
values extracted from ‘Quantum PIC’ and ‘Classical PIC’ simulations.

Semi-classically, the stopping power of free electrons that dismisses quantum
wave effect can be derived from the Vlasov-Poisson equations,

$(\partial_{t}+\frac{\boldsymbol{p}}{m_{e}}\cdot\partial_{r})f(\boldsymbol{r},\boldsymbol{p},t)+q\nabla\phi\cdot\partial_{\boldsymbol{p}}f(\boldsymbol{r},\boldsymbol{p},t)=0,$
(8)

and

$\nabla^{2}\phi=-4\pi
e\left[Z_{b}\delta(\boldsymbol{r}-\boldsymbol{v}_{p}t)-\frac{1}{m_{e}}\int
d\boldsymbol{p}f+Zn\right].$ (9)

The $\delta$ function stands for the projectile ion moving with velocity
$\boldsymbol{v}_{p}$. The last term in the Poisson equation represents the
static plasma ion background. The $f$ involved in the Vlasov-Poisson equations
is Fermi-Dirac distribution. The semi-classical response function is

$\Pi_{0}(\omega,\boldsymbol{k})_{s.c}=-\int
d\boldsymbol{p}\frac{\boldsymbol{k}\cdot\partial_{\boldsymbol{p}}f(\boldsymbol{p})}{\omega-\boldsymbol{k}\cdot\boldsymbol{v}}.$
(10)

In quantum analysis, the stopping contribution from free electrons can be
derived from the quantum-mechanical dielectric function method, which is also
called the random-phase-approximation (RPA) method. With quantum wave effects,
Wigner-Poisson equations Hu  _et al._ (2022); Bertsch and Gupta (1988) are
usually used. It is convenient to replace the Vlasov equation with the Wigner
equation,

$\displaystyle(\partial_{t}+\frac{\boldsymbol{p}}{m}\cdot\partial_{r})f(\boldsymbol{r},\boldsymbol{p},t)$
$\displaystyle=$ $\displaystyle\frac{e}{i\hbar}\int
d\boldsymbol{\xi}\int\frac{d\boldsymbol{p^{\prime}}}{(2\pi\hbar)^{3}}e^{i(\boldsymbol{p^{\prime}}-\boldsymbol{p})\cdot\boldsymbol{\xi}/\hbar}$
$\displaystyle\times$
$\displaystyle[\phi(\boldsymbol{r}+\frac{\boldsymbol{\xi}}{2})-\phi(\boldsymbol{r}-\frac{\boldsymbol{\xi}}{2})]f(\boldsymbol{r},\boldsymbol{p^{\prime}},t).$

The $f$ involved in the Wigner equation is Fermi-Dirac distribution. In this
case the quantum response function is

$\Pi_{0}(\omega,\boldsymbol{k})_{q}=-\int
d\boldsymbol{p}\frac{f(\boldsymbol{p})-f(\boldsymbol{p}+\hbar\boldsymbol{k})}{\hbar\omega-\boldsymbol{k}\cdot\hbar\boldsymbol{v}-\hbar^{2}k^{2}/2m}.$
(12)

Taking the response function into $\varepsilon(\omega,k)$ and solving Eq. 6
numerically, we will obtain the stopping contribution from free electrons.
Especially, in the limit of low projectile velocities, $v_{p}\ll
v_{\text{ave}}$, with $v_{\text{ave}}$ the average electrons velocity,
$v_{\text{ave}}=(v_{th}^{2}+v_{F}^{2})^{1/2}$, where $v_{th}$ is the thermal
velocity of electrons, and $v_{F}$ is the Fermi velocity, the free electronic
stopping power is written as

$S_{f}=\frac{4\pi
Z_{b}^{2}e^{4}n_{f}}{m_{e}v_{F}^{3}}v_{p}\mathrm{ln}(\Lambda_{f}),$ (13)

where $n_{f}$ is the number density of free electrons $n_{f}=Zn$, and
$\mathrm{ln}(\Lambda_{f})$ is the Coulomb logarithm, which changes little with
the slowing down of protons. In the limit of high projectile velocities,
$v_{p}\gg v_{\text{ave}}$, the stopping power of free electrons simplifies to
Fernández-Varea and Arista (2014)

$S_{f}=\frac{4\pi
Z_{b}^{2}e^{4}}{m_{e}v_{p}^{2}}n_{f}\mathrm{ln}(\Lambda_{f}),$ (14)

which already provides $1\%$ accuracy for $v_{p}>2v_{\text{ave}}$. See
Appendix A for detailed information.

The statistic model used in the LAPINS code Wu  _et al._ (2019a, 2017, 2021,
b) to deal with the case that disregards quantum degeneracy is based on the
classical Boltzmann equation, where the average energy of electrons is only
determined by the thermal temperature $T_{e}$. The model used in the LAPINS
code to deal with quantum degeneracy is based on the first principle
Boltzmann-Uhling-Uhlenbeck (BUU) equation Wu  _et al._ (2020). BUU collisions
can ensure that the evolution of degenerate particles is enforced by the Pauli
exclusion principle. This principle prevents degenerate particles being
scattered into an energy state that is already occupied. By using this code,
we have simulated the free electronic stopping power in different densities of
boron target. See Appendix B for relevant information.

In Fig. 2, the stopping power per unit density of free electrons $S_{e}/n$, as
a function of proton velocity is shown. The range of proton velocity
considered is $0.001-0.05c$, with $c$ the speed of light. The densities of
boron target in Fig. 2 (a-e) are respectively $\rho_{s}$, $10\rho_{s}$,
$100\rho_{s}$, $10^{3}\rho_{s}$, $10^{4}\rho_{s}$.

When we compare the cases disregarding and considering quantum degeneracy in
Fig. 2, i.e., “Classical PIC” and “Quantum PIC”, we can find the effect of
degeneracy does decrease the stopping power per unit density of free electrons
$S_{e}/n$. The reason is due to the following two reasons. Firstly, as the
most probable distribution of electrons is the Fermi-Dirac distribution, the
average energy of electrons is much higher than the thermal temperature, which
is

$T_{\mathrm{eff}}=T_{e}\left[\frac{2}{\sqrt{\pi}}F_{1/2}(\eta)(1+e^{-\eta})\right]^{2/3},$
(15)

where $F_{1/2}(\eta)=\int_{0}^{\infty}x^{1/2}(e^{x-\eta}+1)^{-1}dx$,
$\eta=\mu/T_{e}$, $\mu$ is chemical potential. Secondly, the Pauli-exclusion
principle ensure that only electrons on the boundary of the Fermi surface
contribute to the stopping power, as those electrons deep inside the Fermi
surface are frozen. Therefore, when the density of boron plasma target
increases, the stopping power per unit density of free electrons is decreased
under the influence of quantum degeneracy.

In the limit of low projectile velocities, the average electron stopping power
$S_{e}/n$ predicted by “Wigner-RPA” is proportional to the velocity of
protons. In the limit of high projectile velocities, it is inversely
proportional to the square of the velocity. It is shown that the results of
“Wigner-RPA” and “Quantum PIC” agree with each other quite well. Due to the
limitations of semi-classic stopping power theory, results of “Vlasov-RPA”
departs from “Wigner-RPA” and “Quantum PIC”, but still shows a similar trend.
Especially, the result of “Vlasov-RPA” is well consistent with that of
“Wigner-RPA” when the density of boron target is lower than 100$\rho_{s}$.

###  III.2 The bound electronic stopping power

The model used in both analysis and LAPINS code to calculate the bound
electron stopping power is based on the work of Fano Fano (1963), Trujillo
Cabrera-Trujillo  _et al._ (1997) and Gil Gil  _et al._ (2020). Generally, we
can write the bound electronic stopping power as

$S_{b}=\frac{4\pi
Z_{b}^{2}e^{4}}{m_{e}v_{p}^{2}}(A-Z)n\mathrm{ln}(\Lambda_{b}),$ (16)

where

$\mathrm{ln}(\Lambda_{b})\equiv\mathrm{ln}\left[\frac{2{\gamma}^{2}m_{e}v_{p}^{2}}{\bar{I}(Z,A)}\right]-{\beta}^{2}-\frac{C_{K}}{A}-\frac{\delta}{2},$
(17)

in which $A$ is the atomic number of stopping medium, $Z$ is the ionization
degree of the background plasma, $\gamma$ is the relativistic factor of the
injected ions, and $\bar{I}(Z,A)$ is the average ionization potential
considering the degeneracy effect Rosmej  _et al._ (2021); Hu (2017)

$\bar{I}(Z,A)=U-\bigtriangleup U+T_{F},$ (18)

with $U$ the isolated ionization potential Mehlhorn (1981), and
$\bigtriangleup U$ the ionization potential depression (IPD) Ciricosta  _et
al._ (2012). As the density of the boron target is increased, the Fermi energy
is also increased, and ionizing the bound electrons needs to overcome an extra
energy of $T_{F}$. In Eq. (17), the latter two terms are related to shell
corrections and density effect corrections, respectively. These two terms are
based on Fano’s original work Fano (1963), to which the definitions of
$C_{K}/A$ and $\delta/2$ can be referred.

Fig. 3 (a) shows the stopping power per unit density of bound electrons
$S_{b}/n$ as a function of proton velocity during the deceleration process in
different densities of the boron target. It is clear that $S_{b}/n$ is
decreased when the density of the boron target is increased. Especially, it
can be completely disregarded when the density of the boron target exceeds
$100\rho_{s}$. Moreover, as the density of the boron target is increased, the
average ionization potential $\bar{I}(Z,A)$ is increased as well, which is
also reflected by the increasing peak positions of the bound electronic
stopping power per unit density when the density of boron target is increased.

###  III.3 The nuclear stopping power

As for the nuclear stopping power, the typical binary collision method Sigmund
(2014) is usually used,

$S_{n}=\frac{4\pi
Z_{b}^{2}Z_{t}^{2}e^{4}}{m_{t}v_{p}^{2}}n\mathrm{ln}(\Lambda_{n}),$ (19)

where $Z_{t}$ is the nuclear charge of the target particle, $m_{t}$ is the
mass of the target particle, and $\mathrm{ln}(\Lambda_{n})$ is the Coulomb
logarithm Starrett (2018),

$\mathrm{ln}(\Lambda_{n})=\frac{1}{2}\left[\mathrm{ln}(1+\frac{b_{\text{max}}^{2}}{b_{\text{min}}^{2}})-\frac{b_{\text{max}}^{2}/b_{\text{min}}^{2}}{1+b_{\text{max}}^{2}/b_{\text{min}}^{2}}\right],$
(20)

with

$b_{\text{min}}=\frac{2Z_{b}Z_{t}e^{2}}{M_{0}v_{r}^{2}},$ (21)

in which $M_{0}=m_{b}m_{t}/(m_{b}+m_{t})$ is the reduced mass,
$b_{\text{min}}$ represents the closest distance that two charged particles of
the same sign, $Z_{b}$ and $Z_{t}$, with relative velocity $v_{r}$, can reach,
and $b_{\text{max}}$ is an effective maximum impact parameter for nuclear
collisions, with $b_{\text{max}}=max(\lambda_{\text{D}},r_{\text{WS}})$. Due
to the effects of high density and low temperature, the Debye length
$\lambda_{\text{D}}$ is usually smaller than the Wigner-Seitz radius
$r_{\text{WS}}=({4\pi}/{3n})^{-{1}/{3}}$.

Fig. 3 (b-e) show the contributions of all the three components (free
electrons, bound electrons, nuclei) to the stopping power per unit density of
injected protons in the boron target. As can be seen in Fig. 3, at the low
density region, $S_{n}/n$ is far smaller than the electronic stopping power
per unit density and can be ignored. With the increasing of boron target
density, the contribution of nuclei becomes important. Meanwhile, the peak of
$S_{b}/n$ moves to the right and the contribution of bound electrons dwindles.
When the target density is $10^{3}$-$10^{4}\rho_{s}$, the contributions of
free electrons and nuclei determine the stopping power per unit density,
especially $S_{e}/n$ dominates at high proton velocity while $S_{n}/n$
dominates at low proton velocity.

Figure 3: (a) The bound electronic stopping power per unit density as a
function of densities and velocities of injected protons. (b)-(e) The stopping
power per unit density as a function of densities and velocities of injected
protons, with contributions from the total, free electrons, bound electrons,
and nuclei plotted, respectively. The label is shown in Fig. (b).

##  IV Multiplication factor of beam-fusion reactions

With above analysis, we can easily obtain the total reaction number of beam-
target nuclear fusion

$R_{T}=N_{p}P=N_{p}\int_{0}^{E_{p}}\frac{\sigma(E)}{(S_{f}+S_{b}+S_{n})/n}dE.$
(22)

For the p-11B beam-target reaction, the total number of reactions is closely
related to the stopping power per unit density of protons in the target. As
can be seen in Fig. 3, the total stopping power per unit density $S/n$
decreases when the density of boron target is increased. Especially when the
density of boron target reaches $10^{3}\rho_{s}$, $S_{b}/n$ almost disappears,
the total stopping power per unit density is now determined by

$\frac{S}{n}=\frac{4\pi
Z_{b}^{2}Z_{t}^{2}e^{4}}{m_{t}v_{p}^{2}}\mathrm{ln}(\Lambda_{n})+\frac{4\pi
Z_{b}^{2}e^{4}Z}{m_{e}v_{F}^{3}}v_{p}\mathrm{ln}(\Lambda_{f}).$ (23)

The reason for the decrease is due to the following two facts. Firstly, as
$b_{\text{max}}$ is inversely proportional to $n^{1/3}$,
$\mathrm{ln}(\Lambda_{n})$ decreases when the density of boron target is
increased. Secondly, as $v_{F}^{3}$ is proportional to $n_{f}=Zn$, the second
term of Eq. (23) $\propto\mathrm{ln}(\Lambda_{f})/n$ also decreases when the
density of boron target is increased.

Figure 4: (a) The reaction probability of the p-11B nuclear fusion as a
function of boron target density. The initial energy of proton beam is 1 MeV.
(b) The multiplication factor $F$ of p-11B beam-target fusion as a function of
boron target density and initial proton kinetic energy. $F$ is defined as the
ratio of the fusion energy produced during the deceleration of proton beam to
the overall energy of injected protons.

Numerically, we have run simulations with different initial densities by using
the LAPINS code, with density of $\rho_{s}$, $10\rho_{s}$, $100\rho_{s}$,
$10^{3}\rho_{s}$, and $10^{4}\rho_{s}$, respectively. The initial kinetic
energy of the proton beam is fixed at $1$ MeV. Fig. 4 (a) shows the
probability of the p-11B beam-target fusion as a function of the boron target
density. The comparisons between theoretical analysis and numerical
simulations coincide with each other quite well. Due to high degree of quantum
degeneracy, the effect of temperature is quite small on the reaction
probability.

Quantitatively, it is critical to consider the energy multiplication factor
$F$ in order to achieve the net energy gain. Here $F$ is a fundamental
quantity in beam-target fusion,

$F=\frac{P(E_{p})Q}{E_{p}},$ (24)

where $E_{p}$ is the initial proton energy, $Q$ is the fusion $Q$-value. For
the p-11B fusion reaction, $Q$ equals 8.7 MeV. Theoretical analysis leads to
the conclusion that for injected protons of energy of 1 MeV, the threshold
density of boron target is $2.35\times 10^{4}\rho_{s}$, beyond which the
energy multiplication factor $F$ would be greater than one.

Moreover, the cross section of nuclear fusion depends on the center of mass
energy, which is a function of the injected proton energy. Therefore,
different injected energy of proton beams may lead to different reaction
probabilities. We have calculated the fusion probability with varying injected
proton kinetic energies. In order to make the $F$ factor more than one, we
find when the energy of injected protons is around 800 keV, there exists a
minimum possible compressed density, which is $2.15\times 10^{4}\rho_{s}$, as
shown in Fig. 4 (b).

##  V Conclusion

In conclusion, we suggest a high-efficiency scheme for beam-target p-11B
fusions via injecting a MeV proton beam into a highly compressed quantum
degenerated boron target. The degeneracy effect is found to have an effect on
the number of fusion reactions by decreasing the stopping power per unit
density of protons in the boron target. At low boron target densities, free
electrons and bound electrons dominate the stopping power per unit density.
Especially when $E_{p}=1$ MeV and $\rho=10^{3}-10^{4}\rho_{s}$, $S_{b}/n$ and
$S_{e}/n$ can be completely disregarded and dramatically reduced,
respectively, which therefore results in orders of magnitude increments in
fusion yields. When the injected proton beam has an energy of around 800 keV,
there exists a minimum possible compressed density, which is $2.15\times
10^{4}\rho_{s}$ to make the $F$ factor greater than one.

##  VI Acknowledgements

This work was supported by the Strategic Priority Research Program of Chinese
Academy of Sciences (Grant No. XDA250050500), the National Natural Science
Foundation of China (Grant No. 12075204, No. 11875235, and No. 61627901), and
Shanghai Municipal Science and Technology Key Project (No. 22JC1401500). Dong
Wu thanks the sponsorship from Yangyang Development Fund.

##  Appendix A The calculation of the free electron stopping power from
Wigner-RPA equations

Consider a homogeneous free electrons gas (FEG) with density $n_{f}$ and Fermi
wave number $k_{F}=(3\pi^{2}n_{f})^{1/3}$, the FEG can be characterized by the
Lindhard parameter $\chi^{2}$; these dimensionless quantities are linked to
$k_{F}$ through Maynard and Deutsch (1985); Fernández-Varea and Arista (2014)

$\chi^{2}=(\pi k_{F}a_{0})^{-1},$ (A-1)

where $a_{0}=\hbar^{2}/m_{e}e^{2}$ is the Bohr radius ($\hbar$ is the reduced
Plank constant) and $\alpha=(4/9\pi)^{1/3}$. It is convenient to replace $k$
and $\omega$ by the dimensionless variables

$z=\frac{k}{k_{F}}\ \text{and}\ u=\frac{\omega}{kv_{F}},$ (A-2)

where $v_{F}=\hbar k_{F}/m_{e}$ stands for the Fermi velocity. In this case
the dielectric function can be written as Maynard and Deutsch (1985)

$\varepsilon(z,u)=1+\frac{\chi^{2}}{z^{2}}[f_{1}(z,u)+if_{2}(z,u)],$ (A-3)

with

$f_{1}(u,z)=-\frac{\pi}{8z\Theta}[F(u+z)-F(u-z)].$ (A-4)

When $\Theta\gg 1$,

$F(p)=2p\left[\frac{1}{2}+\frac{1-p^{2}}{4p}\log\frac{p+1}{p-1}\right],$ (A-5)

and

$f_{2}(u,z)=-\frac{\pi}{8\Theta
z}\log\frac{1+\exp\left({\mu}/{T}-\Theta(u+z)^{2}\right)}{1+\exp\left({\mu}/{T}-\Theta(u-z)^{2}\right)}.$
(A-6)

The free electron stopping power can be written in the form

$S_{f}=\frac{4\pi Z_{b}^{2}e^{4}}{m_{e}v_{p}^{2}}n_{f}L_{e},$ (A-7)

with

$\displaystyle
L_{e}=\frac{6}{\pi\chi^{2}}\int_{0}^{v_{p}/v_{F}}udu\int_{0}^{\infty}zdz\operatorname{Im}\frac{1}{\varepsilon(z,u)},$
(A-8)

which depends on the number density and temperature of target nuclei as well
as the velocity of protons, satisfying

$L_{e}=\left(\frac{v_{p}}{v_{r}}\right)^{3}\mathrm{ln}(\Lambda_{f}).$ (A-9)

Here $v_{r}$ is the relative velocity between protons and free electrons, and
$v_{r}=(v_{\text{ave}}^{2}+v_{p}^{2})^{1/2}$. In the limit of low projectile
velocity, $v_{p}\ll v_{\text{ave}}$, the Eq. (A-9) becomes

$L_{e}=\left(\frac{v_{p}}{v_{r}}\right)^{3}\mathrm{ln}(\Lambda_{f})\simeq\left(\frac{v_{p}}{v_{F}}\right)^{3}\mathrm{ln}(\Lambda_{f}),$
(A-10)

and the stopping power of free electrons simplifies to Fernández-Varea and
Arista (2014)

$S_{f}=\frac{4\pi
Z_{b}^{2}e^{4}n_{f}}{m_{e}v_{F}^{3}}v_{p}\mathrm{ln}(\Lambda_{f}).$ (A-11)

In the limit of high projectile velocity, $v_{p}\gg v_{\text{ave}}$, $L_{e}$
becomes

$\displaystyle
L_{e}=\left(\frac{v_{p}}{v_{r}}\right)^{3}\mathrm{ln}(\Lambda_{f})\simeq\mathrm{ln}(\Lambda_{f}),$
(A-12)

and the stopping power of free electrons simplifies to Fernández-Varea and
Arista (2014)

$S_{f}=\frac{4\pi
Z_{b}^{2}e^{4}}{m_{e}v_{p}^{2}}n_{f}\mathrm{ln}(\Lambda_{f}),$ (A-13)

Note, Eq. (A-12) already gives a one percent accuracy for
$v_{p}>2v_{\text{ave}}$.

##  Appendix B Simulation of free electron stopping power with the LAPINS code

The statistic model used in the LAPINS code Wu  _et al._ (2019a, 2017, 2021,
b) to deal with the case that disregards quantum degeneracy is based on the
classical Boltzmann equation,

$\frac{\partial f}{\partial t}+\boldsymbol{v}_{k}\cdot{\frac{\partial
f}{\partial\boldsymbol{r}}}+q_{k}(\boldsymbol{E}+\boldsymbol{v}_{k}\times\boldsymbol{B})\cdot{\frac{\partial
f}{\partial\boldsymbol{p}_{t}}}=\frac{\partial f}{\partial t}|_{\text{coll}},$
(B-1)

where the subscript $k$ indicates the species of particles,
$f=f(\boldsymbol{r},\boldsymbol{p},t)$ is the distribution function,

Figure B-1: (a) The proton energy as a function of time and incident distance,
simulated by the LAPINS code considering Pauli exclusion principle, with boron
density of $10^{3}\rho_{s}$ and temperature of $10$ eV. The duration of the
beam is set long enough to ensure that the beam is still injected when the
protons in the front are stopped. We therefore can get stable data describing
the proton energy as a function of the incident distance during the
deceleration process. (b) The proton energy as a function of the incident
distance during the deceleration process.

$\boldsymbol{r}$ is the position, $\boldsymbol{p}$ is the momentum, $t$ is the
time, $\boldsymbol{v}$ is the velocity, $\boldsymbol{E}$ is the electric
field, $\boldsymbol{B}$ is the magnetic field and the collision term
${\partial f}/{\partial t}|_{\text{coll}}$ is Boltzmann collision integral. On
this occasion, the average energy of electrons is only determined by thermal
temperature $T_{e}$, and the quantum effect of electrons is completely not
considered.

The model used in the LAPINS code to deal with quantum degeneracy is based on
the first principle Boltzmann-Uhling-Uhlenbeck (BUU) equation Wu  _et al._
(2020); Domps  _et al._ (1997),

$\frac{\partial f}{\partial t}+\boldsymbol{v}_{k}\cdot{\frac{\partial
f}{\partial\boldsymbol{r}}}+q_{k}(\boldsymbol{E}+\boldsymbol{v}_{k}\times\boldsymbol{B})\cdot{\frac{\partial
f}{\partial\boldsymbol{p}_{t}}}=\frac{\partial f}{\partial
t}|_{\text{coll}}^{\text{BUU}},$ (B-2)

where the BUU collisions can ensure that evolution of degenerate particles is
enforced by the Pauli exclusion principle. This principle prevents degenerate
particles being scattered into an energy state that is already occupied.

In order to give the visual stopping power per unit density of free electrons,
we do a series of simulations by the LAPINS code. The initial energy of proton
beams is set to 1 MeV. The densities of boron target are respectively
$\rho_{s}$, $10\rho_{s}$, $100\rho_{s}$, $10^{3}\rho_{s}$, $10^{4}\rho_{s}$,
where $\rho_{s}$ is the density of solid boron target equal to $2.34$
$\mathrm{g/cm^{3}}$. The temperatures of the former three cases are set to be
1 eV, while the temperatures of the later two are set to be 10 eV and 100 eV,
respectively. The reason why a higher temperature is set for the last two
cases is to avoid numerical errors in the integral due to large degeneracy
parameter $\Theta$. In fact, when the density of boron target is over 100
$\rho_{s}$, such changes of temperature will not influence the result a lot.

For each simulation, the density of proton beam is set low enough to make sure
the influence can be regarded as perturbations. Duration of the beam is set
long enough to ensure that the beam is still being injected when the velocity
of the protons in the front plane is slowed down to zero. As shown in Fig. B-1
(a), we can get the proton energy at any position in the target at any time.
On this occasion, as is shown in Fig. B-1 (b), we can obtain the data of
proton average energy $E_{pi}$ as a function of distance in the deceleration
process. We will then get the free electronic stopping power of to protons by
$\delta E_{pi}/\delta z_{i}$. Dividing it by $n$, we can obtain the free
electronic stopping power per unit density $S_{e}/n$.

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