# Investigating the muon catalyzed fusion in muonic lithium hydride

Qian Wu  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou
730000, China  Xurong Chen<EMAIL_ADDRESS>Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science
and Technology, University of Chinese Academy of Sciences, Beijing 100049,
China  Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China

###### Abstract

In this work, we consider the muonic LiH as a new stage of the $\mu$ catalyzed
fusion. We calculate the bound states of the muonic LiH and LiH+ and their
wave functions based on the three and four body approximation. In order to
solve the Schr$\ddot{\rm o}$dinger equation, we apply the Gaussian expansion
method which gives us both the eigen energies and the wave functions. The
existences of the bound states in muonic LiH and LiH+ are confirmed. We find
the dynamical behaviors between the 7Li and proton in muonic LiH and LiH+ are
similar as the ones in dt$\mu_{2}$ and dt$\mu$, respectively. The $\alpha$
sticking probability in the muonic LiH+ after the nuclear fusion is studied.
The calculated $\alpha$ sticking probability of being captured to the ground
state of He$\mu$ is 5 times smaller than the one in dt$\mu$.

$\mu$ catalyzed fusion, LiH, $\alpha$ stick

##  I Introduction

The nuclear fusion, known as the nuclear reaction between the light nuclei,
gives out a few MeVs energy due to the mass difference between the initial and
final nuclei. Thus, it is considered to be an important way to obtain nuclear
power from nature. However, in order to make the nuclear fusion happen within
the nuclear distance (a few fms), the coulomb barrier between two nuclei has
to be overcome which requires a high temperature or pressure. And by far, no
design of nuclear fusion has produced more fusion energy than the input
electrical power.

The $\mu$ catalyzed fusion ($\mu$CF) has been studied as a reliable low energy
nuclear fusion for a long history Jackson1957 ; Zelddovich1960 ; Vesman ;
kamimura1988 ; kamimura1993 ; Kami2022 ; Kami1989 . Among all of the $\mu$CF
studies, the dt$\mu$ system is considered to be a possible resource of fusion
which happens in a deuterium and tritium (D-T) mixture. The idea of $\mu$CF
originates from the fact that the $\mu$ mass (105.66 MeV) is almost 207 times
of the electron mass (0.55 MeV). Then, the radius of the $\mu^{-}$ atom is
nearly 1/207 times smaller than the normal atom. Considering a molecular
consisting of the deuteron, triton and $\mu$, the distance between two nuclei
should be much smaller than a normal D2 or T2 molecular.

The first key process of dt$\mu$ $\mu$CF is the formation of the t$\mu$ atom.
When the $\mu$s go into the D-T mixture, they will immediately ($\sim
10^{-11}\mathrm{~{}s}$) replace the electrons inside the deuterium and tritium
to formate the d$\mu$ and t$\mu$ due to a much larger binding energy with the
deuteron and triton. More importantly, since the mass of the triton is heavier
than the deuteron, the binding energy of t$\mu$ is larger than the d$\mu$.
Thus, the muon inside d$\mu$ will be taken by the triton to form t$\mu$, which
is known as the so-called muon transfer reaction Hiyama2003GEM .

Then, the t$\mu$ atom, which is electrically neutral, enters into a D2
molecule and is captured by a deuteron to form a dt$\mu$ molecule. It should
be noted that the formation of the dt$\mu$ molecular happens via the Vesman
mechanism Vesman  in ${\rm t}\mu(2s)-{\rm D}_{2}$ scattering,
$\mathrm{t}\mu(2\mathrm{~{}s})+\mathrm{D}_{2}\rightarrow[(\mathrm{dt}\mu)\mathrm{dee}]$.
If the energy of the dt$\mu$ molecular with respect to the t$\mu$-d threshold
is less than the dissociation energy of ${\rm D}_{2}\simeq 4.56$ eV, the
released energy during the formation of the dt$\mu$ can be absorbed by the
electrons in the D2 which make it transfer into the excited states without
breaking the whole D2 molecular. In Refs. Vesman ; dtmform1982 , they gave the
muon molecular’s formation rate of $10^{8}$ s-1 which is almost 200 times
larger than the muon decay rate.

Finally, in the dt$\mu$ molecule, fusion reaction ${\rm d}+{\rm
t}\rightarrow\alpha+{\rm n}+17.6$ MeV takes place immediately $\left(\sim
10^{-12}\mathrm{~{}s}\right)$ due to the small distance between deuteron and
triton Kami2022 ; Kami1989 . After that, the $\mu$ becomes free again and
continues to catalyze the fusion reaction until its life time is exhausted.
Unfortunately, with a small probability $(\sim 1\%)$ alphastick1986 ;
Zelddovich1960 , $\mu$ is captured by the generated $\alpha$ particle from the
fusion reaction and exhausts its lifetime in the atom, although there is a
probability of that the muon could be recaptured by the D2 or T2 during the
time sticking with the $\alpha$ particle. This sticking behavior of the $\mu$
causes a major reduction of the fusion times for each muon.

In order to produce one $\mu$, the input energy is estimated to be $\sim 5$
GeV 1980Petrov . If $N_{f}$ is the number of fusions catalyzed by one muon,
since one d-t fusion generates 17.6 MeV, we easily judge that $N_{f}\sim 280$
is necessary to reach the scientific break-even. Experiments performed so far
show that $N_{f}$ increases almost linearly with the density of the
$\mathrm{D}_{2}/\mathrm{T}_{2}$ mixture, reaching $N_{f}\sim 150$ at the
density of liquid hydrogen. However, the fusion number comes to a limitation
if we can’t solve the alpha sticking problem.

In terms of the alpha sticking issue, we consider a muonic lithium hydride
(LiH) molecular as the fusion reaction stage instead of the dt$\mu$. When a
muonic LiH or LiH+ is formed, the fusion reaction ${}^{7}{\rm Li}+{\rm
p}\rightarrow 2\alpha+17.35$ MeV could happen. According to Ref. Jackson1957 ,
the alpha sticking probability strongly depends on the velocity of the
outgoing alpha particle. In the nuclear fusion reaction ${}^{7}{\rm Li}+{\rm
p}\rightarrow 2\alpha$, the outgoing alpha particle carries more momentum than
the one in the d+t$\rightarrow\alpha$+n. In this case, it may be more
difficult for the alpha particle to capture the muon than in the D-T mixture.
Besides, since the 7Li3+ has three plus electric charge and the muon may have
a large probability to be recaptured by the 7Li3+ with the muon transfer
reaction after the fast moving He$-\mu$ is slowed down.

In this work, we investigate the alpha sticking probability after the fusion
reaction happens in the muonic LiH and LiH+. First, we study the bound system
of the Lip$\mu_{4}$ and Lip$\mu_{3}^{+}$. In order to solve the few body
Schr$\ddot{\rm o}$dinger equation, we apply the Gaussian expansion method
(GEM), which is a reliable variational method and is used in nuclear and
atomic physics Hiyama2003GEM ; 35hiyama1997ptp . The strict five body or six
body Li+p+$\mu$s systems are difficult to solve with GEM, however, due to the
extremely large number of Gaussian basis which goes beyond our temporary
supercomputer’s power. Then, we take an approximation. As we know, the first
ionization energy of the Li is 5.39 eV while the second and third ionization
energy are 75.64 eV and 122.45 eV, respectively. Thus, we give a vague picture
of the bound Lip$\mu_{4}$ and Lip$\mu_{3}$ molecular by approximately treat
the Li$\mu_{2}$ as a whole part. The bound system of muonic LiH and LiH+ are
discussed in the next section. Second, in section III, we calculate the alpha
sticking probability in muonic LiH+ after the fusion reaction. After that, we
give a summary in the final section.

##  II Bound state of muonic LiH and LiH+

As we mentioned in the first section, we treat the muonic LiH as a four body
Li$\mu_{2}$-p-$\mu$-$\mu$ system and the muonic LiH+ as a three body system.
Then, the four body hamiltonian is written as:

$H=T+\frac{e^{2}}{r_{1}}+\frac{e^{2}}{r_{2}}-\frac{e^{2}}{r_{3}}-\frac{e^{2}}{r_{5}},$
(1)

where $T$ is the kinetic energy and $r_{1}~{}r_{5}$ are the relative
coordinates in Fig. 1. Then, in order to solve this equation, we apply the
Gaussian expansion method and the trivial wave function is constructed with a
sum of 8 Jacobian coordinates channels, shown in Fig. 1 and the whole
Li$\mu_{2}$ part is named as the Li in short. The total wave function of the
munic LiH with angular momentum J and its z-component is written as follows:

$\displaystyle\Phi_{JM}({\rm LiH})$ $\displaystyle=\sum_{c=1}^{8}\sum_{n\ell
NLv\lambda\Lambda IS}C^{c}_{n\ell NLv\lambda\Lambda
IS}\mathcal{A}[[[\psi_{n\ell}(\mathbf{r_{c}})$ (2)
$\displaystyle\otimes\phi_{NL}(\mathbf{R_{c}})]_{\Lambda}\otimes\varphi_{\nu\lambda}(\rho_{c})]_{I}[\chi^{\mu_{1}}_{1/2}\chi^{\mu_{2}}_{1/2}]_{S}]_{JM}$

where the $C^{c}_{n\ell NLv\lambda\Lambda IS}$ are the parameters and
$\mathcal{A}$ is the antisymmetric operator between the two muons.
$\chi^{\mu}_{1/2}$ is the spin wave function of the muon. In this work, we
omit the spin of the all nuclei we mention. The basis functions have the
Gaussian radial shape multiplied by spherical harmonics.

$\psi_{i\ell}(\mathbf{r})=r^{\ell}e^{-\alpha_{i}r^{2}}Y_{\ell
m}(\hat{\mathbf{r}}),$ (3)

and similarly for $\phi(\mathbf{R})$ and $\chi(\rho)$. The Gaussian range
parameters are taken to lie in a geometrical progression. The parameters C are
then obtained with applying Eq. 2 into the Rayleigh-Ritz variational method.
As for the three body Li$\mu_{2}$-p-$\mu$ system, its wave function can be
written as:

$\displaystyle\Phi_{JM}({\rm LiH^{+}})=\sum_{c=1,3}\sum_{n\ell NL}C^{c}_{n\ell
NL}[\psi_{n\ell}(\mathbf{r_{c}})\otimes\phi_{NL}(\mathbf{R_{c}})]_{JM}$ (4)

where c=1 and 3 are the similar ones in Fig. 1 with taking $\rho_{1}$ and
$\rho_{3}$ out. Here we omit the spin of the single $\mu$.

Figure 1:  Jacobian coordinates of four body Li$\mu_{2}$-p-$\mu$-$\mu$ system.
Here Li stands for the Li$\mu_{2}$ in short.

It should be noted that we calculate the binding energy of the muonic LiH with
respect to the Li$\mu$-p$\mu$ threshold, which is $-2763.1$ eV for
Li$\mu_{2}$-$\mu$ and $-2528.9$ eV for p-$\mu$. And the binding energy for
muonic LiH+ is calculated with respect to the Li$\mu_{2}$-$\mu$. Here, the
binding energy between the Li$\mu_{2}$ and $\mu$ shows one major short come of
our approximation. As shown in Table.1, the whole four body calculation of
Li$\mu_{3}$ gives the first ionization energy with 861.4 eV while it gives
2763.1 eV when we treat the Li$\mu_{2}$ as a whole. It is reasonable since the
electrons inside Li has a 1S22S1 structure.

For comparison, we calculate the ionization energy of Lithium atom and muonic
Lithium atom, which are shown in Table. 1. The first, second and third
ionization energy of Lithium atom are close to the experimental values. As for
the muonic Lithium atom, the 2763.1 eV, which is calculated within two body
Li+-$\mu$ system, certainly gives the energy in the 1S orbit, which is not far
from the 4 times of the first ionization energy of Li$\mu_{3}$, 861.5 eV. The
way we treat the four body Li$\mu\mu\mu$ calculation is given in the next
section.

Table 1: The ionization energy of Lithium atom and muon Lithium atom. The experimental values of Lithium atom are taken from Ref. ionization . The units are in eV. |  Li($e_{3}$) |  Li($\mu_{3}$) |  Exp. ionization
---|---|---|---
First |  5.36 |  861.50 |  5.392
Second |  75.64 |  15369.00 |  75.640
Third |  122.44 |  24916.50 |  122.454
Total |  203.44 |  41146.99 |  203.486

Once we treat the Li$\mu_{2}$ as a whole part, the structures of muonic LiH
and LiH+ may be similar as the dt$\mu_{2}$ and dt$\mu$, respectively. Since
they are all composed of two heavy and positive charged particles and two
(one) light and negative charged particles. This also reminds of the D2 and H2
molecular where the two hydrogen atoms are bound with the covalent bond and
the two electrons are shared by the two protons.

In Table. 2, we show the binding energy of the Lip$\mu_{4}$, Lip$\mu_{3}$,
dt$\mu_{2}$ and st$\mu$. And we only show the ground states’ energy (J=0). The
calculated binding energy of muonic LiH is 478.5 eV with respect to the
Li$\mu_{3}$-p$\mu$ threshold and the binding energy of muonic LiH+ is 207.1
eV. Together with this, the binding energies of the ground state of
dt$\mu_{2}$ and dt$\mu$ are also shown and all three states have the same
angular momentum as J=0. The binding energy for the dt$\mu$ is 319.1 eV with
respect to the t$\mu$-d threshold. And the binding energy for the doubly $\mu$
dt$\mu_{2}$ molecular is 536.8 eV, with respect to the t$\mu$-d$\mu$
threshold. This also agrees with the one calculated in Ref. kamimura2001 .

Table 2: Binding energies of the ground state of the Lip$\mu_{4}$, Lip$\mu_{3}$, dt$|mu_{2}$ and dt$\mu$. Here d, t and Li represent the deuteron (2H), triton (3H) and 7Li nucleus, respectively. $\Psi(0)$ is the wave function with the relative distance between two nuclei being equal to 0. The $a_{\mu}$ represents the $\mu$ atomic unit which is equal to $\hbar^{2}/e^{2}m_{\mu}$. Molecular |  $E_{00}$ (eV) |  $\Psi(0)$ ($a_{\mu}^{-3/2}$)
---|---|---
Lip$\mu_{4}$ |  478.5 |  0.016
Lip$\mu_{3}$ |  207.1 |  0.003
dt$\mu_{2}$ |  536.8 |  0.021
dt$\mu$ |  319.1 |  0.003

It is reasonable to assume that the wave functions between the two nuclei in
muonic LiH and dt$\mu_{2}$ are more compact than ones in the dt$\mu$ and
muonic LiH+. In the third column of table 2, we show $\Psi(0)$ of these four
systems which $\Psi(0)$ is defined as follows:

$\Psi(0)=\int dR_{1}d\rho_{1}\Psi(r_{1}=0,R_{1},\rho_{1}).$ (5)

The $\Psi(0)$ in Lip$\mu_{4}$ and dt$\mu_{2}$ is larger than the ones in
dt$\mu$ and muonic LiH+. Moreover, we calculate the density distribution
$\rho(r)$ of these four systems which is defined as:

$\rho(r)=\int d\hat{r}dRd\rho\Psi(r,R,\rho).$ (6)

In Fig. 2, the density distributions as a relative distance between
Li$\mu_{2}$ and the proton in muonic LiH and LiH+ are shown in the red solid
and dash-dotted line. And the ones between deuteron and triton in dt$\mu_{2}$
and dt$\mu$ are shown in the black dashed and dash-dotted line. The density
distribution and the $\Psi(0)$ of muonic LiH are similar as the ones in
dt$\mu_{2}$. And similar behavior exists between the muonic LiH+ and dt$\mu$.

It is obvious that the major part of the wave function is located around 1 to
3 $a_{\mu}$ which is approximately around 100 to 500 fm. This allows a much
more significant wave function around the 1 to 10 fm than the normal
molecular. In terms of the exact nuclear fusion rate, we plan to discuss it in
our future work.

Figure 2:  Density distributions $\rho(r)$ as a function of the relative
distance between Li and proton (black) and between deuteron and triton (red).
The black solid and dash-dotted line represent the density distribution of the
Lip$\mu_{3}$ and Lip$\mu_{4}$, respectively. The red solid and dash-dotted
line represent the density distribution of the dt$\mu\mu$ and dt$\mu$,
respectively. $a_{\mu}$ represents the $\mu$ atomic unit which is equal to
$\hbar^{2}/e^{2}m_{\mu}$.

##  III $\alpha$ sticking probability

In this section, we calculate the $\alpha$ sticking probability after the
nuclear fusion reaction happens in the muonic LiH+. As we mentioned in the
first section, our computer resource can’t afford a full five body
A$\mu\mu\mu\mu$ calculations. In this case, we only calculate the sticking
probability in the muonic LiH+ and solve a four body A$\mu\mu\mu$ system.

According to Ref. Zelddovich1960 , when the nuclear fusion happens, the
nucleus 7Li and proton becomes close enough. Then, the whole system can be
regard as a core nucleus with the mass equal to Li+p and the positive charge
4. At this moment, the wave function of one single $\mu$ is given with solving
the four body A$\mu_{4}$ system, where A4+ is a combination of the 7Li and the
proton.

When the fusion reaction:

${}^{7}\rm{Li}+\rm{p}\rightarrow 2\alpha,\;Q=17.35\;\rm{MeV}$ (7)

happens, the two $\alpha$ particles receive a recoil energy E and spread with
a velocity v. Thus, when the $\mu$ is captured by the fast-moving $\alpha$
particle to its ground state, its final state is

$\phi_{f}(\bf{r})=e^{ip\cdot\bf{r}/\hbar}(\frac{8}{\pi
a_{\mu}^{3}})^{1/2}e^{2r/a_{\mu}}.$ (8)

Here $p=m_{\mu}v$ and $r$ is the relative distance between the $\mu$ and the
$\alpha$ particle. Since the function $e^{ip\cdot\bf{r}/\hbar}$ oscillates
strongly, the major contribution comes from the short-range part of $r$
($r\leq\hbar/p$). And the wave function at $r=0$ is propagational to
$1/n^{3}$. One the other hand, the item $r^{l}$ in the excited states ($l\neq
0$) should strongly reduce the integration than the $r^{0}$. Thus, the
probability of being captured to the excited state of He-$\mu$ is much smaller
than being captured to the ground state. Therefore, in this work, we only
study the alpha sticking probability of $\mu$ being captured to the ground
state of He$-\mu$.

The sticking probability S is given as:

$S=\left|\int d\bf{r}^{3}\phi_{f}(\bf{r})\varphi_{\mu}(r)\right|^{2}$ (9)

where $\varphi_{\mu}(\bf{r})$ is the wave function of the ground state of
$\mu$ particle in the A$\mu_{4}$. In order to solve this equation, we write
the oscillation part $e^{ip\cdot\bf{r}/\hbar}$ as:

$e^{i\rm{p}\cdot\bf{r}/\hbar}=4\pi\sum_{\lambda=0}^{\infty}\sum_{\beta=-\lambda}^{\lambda}(-i)^{\lambda}j_{\lambda}(-|r||p|/\hbar)Y^{*}_{\lambda\beta}(\hat{\bf{p}})Y_{\lambda\beta}(\hat{\bf{r}})$
(10)

where j${}_{\lambda}(x)$ is the spherical Bessel function. Thus, the
integration over the $\bf{r}$ leads to a nonzero value only when
$\lambda$=$\beta$=0. Then, Eq. 9 turns to

$S=4\pi\int_{0}^{\infty}\varphi_{\mu}(r)(\frac{8}{\pi
a_{\mu}^{3}})^{1/2}e^{2r/a_{\mu}}\frac{sin(pr/\hbar)}{p\hbar}rdr.$ (11)

Besides, the relation between the p and the recoil energy E is

$p=\sqrt{2EM_{\mu}^{2}/M_{\alpha}}.$ (12)

where Mμ and Mα are the mass of the $\mu$ and $\alpha$ particle, respectively.

In order to obtain the wave function of the $\mu$, we need to solve the four
body A$\mu_{4}$ system. Here we use the Gaussian expansion method and the wave
function is written as

$\displaystyle\Phi_{JM}$ $\displaystyle=\sum_{c=1}^{4}\sum_{n\ell
NLv\lambda\Lambda IsS}C^{c}_{n\ell NLv\lambda\Lambda
IsS}\mathcal{A}[[[\psi_{n\ell}(\mathbf{r_{c}})$ (13)
$\displaystyle\otimes\phi_{NL}(\mathbf{R_{c}})]_{\Lambda}\otimes\chi_{\nu\lambda}(\rho_{c})]_{I}[[\chi^{\mu_{1}}_{1/2}\chi^{\mu_{2}}_{1/2}]_{s}\chi^{\mu_{3}}_{1/2}]_{S}]_{JM}.$

where c$=1\sim 4$ is the channels of the Jacobian coordinates in Fig.3 and
$\mathcal{A}$ is the antisymmetric operator between the three $\mu$s. In Fig.
3, the A represents the heavy nuclei which is 7Li$+$p.

Figure 3:  Jacobian coordinates of four body A-$\mu$-$\mu$-$\mu$ system.

Accordingly, the wave function of one single $\mu$ is defined as

$\varphi_{\mu}(r)=\left|\int
d\hat{r_{1}}dR_{1}d\rho_{1}\Phi(r,R,\rho)\right|^{1/2}$ (14)

And as mentioned in the former section, the same mode space and method are
used in calculating the ionization energies of Lithium atom and muonic Lithium
atom with simply replacing A with Li3+.

We then calculate the alpha stick probability of the following two nuclear
reactions:

$\displaystyle{}^{7}\rm{Li}+\rm{p}\rightarrow$ $\displaystyle
2^{4}\rm{He},\quad E=8.67\;\rm{MeV}.$ (15) $\displaystyle d+t\rightarrow$
$\displaystyle{}^{4}\rm{He}+n,\quad E=3.5\;\rm{MeV}.$

where E is the recoil energy of the $\alpha$ particle. It should be noted that
in the dt$\mu$, the wave function of the $\mu$ when the dt fusion happens can
be regarded as the wave function of the Helium atom Zelddovich1960 .

Shown in Table. 3, the $\alpha$ sticking probabilities of these two reactions
are calculated to be 0.0098 and 0.0021, respectively for dt$\mu$ and
Lip$\mu_{3}$. As we can see, the sticking probability is almost five times
smaller than the one in the dt$\mu$. As we mentioned in the introduction
section, the sticking probability have a strong dependence of fourth power of
the velocity of the outgoing $\alpha$ particle Zelddovich1960 . Thus, it is
reasonable to have a much smaller sticking probability in LiH since the recoil
energy in the fusion reaction is much larger than the one in dt$\mu$.

Table 3: Alpha sticking probability of dt$\mu$ and LiH$\mu_{4}$. system |  probability
---|---
dt$\mu$ |  0.0098
Lip$\mu_{3}$ |  0.0021

##  IV Summary

In order to give a possible solution to the $\alpha$ sticking issue in the
normal dt$\mu$ $\mu$CF, we put forward the muonic LiH and LiH+ as the stage of
the $\mu$CF. The goal fusion reaction is ${}^{7}{\rm Li}+{\rm p}\rightarrow
2\alpha+17.35$ MeV. We calculate the ground state energies and the wave
functions of the muonic LiH and LiH+ based on a four body Li$\mu_{2}+{\rm
p}+\mu+\mu$ and three body Li$\mu_{2}+{\rm p}+\mu$ model. To make the
calculation practicable, we treat the Li$\mu_{2}$ as one whole part due to the
large binding energy of the two $\mu$s and the Li3+. We find that the
dynamical behaviors between 7Li and proton in muonic LiH and LiH+ are similar
as the ones in dt$\mu_{2}$ and dt$\mu$, respectively. Thus, the nuclear
reaction in the muonic LiH and LiH+ might happen immediately when the muonic
LiH or LiH+ is formed. It should be noted that another nuclear reaction
${}^{7}{\rm Li}+{\rm p}\rightarrow^{7}{\rm Be}+n$ could happen in muonic LiH.
And the detailed investigation concerning the fusion channels and rates in
muonic LiH will be our next work.

Then, we calculate the $\alpha$ sticking probability after the fusion
reaction. We find that the sticking probability of the $\mu$ being captured to
the ground state of He$\mu$ in the muonic LiH+ is almost five times smaller
than the one in the dt$\mu$. However, it may not ease the current circumstance
of the $\mu$CF concerning the $\alpha$ sticking issue as we explained in the
first section. We give further discussions as follows. On the one hand, in the
muonic LiH or LiH+, it requires four or three $\mu$s and two $\alpha$
particles are generated after the fusion. Therefore, the total improvement of
to the whole $\mu$CF cycle in muonic LiH or LiH+ could be marginal. On the
other hand, if the He$-\mu$ is slowed down in the LiH system, the $\mu$
particle could be recaptured by the Li3+ with the $\mu$ transfer reaction. The
similar reaction also happens in d$\mu$-t$\mu$ transfer reaction Hiyama2003GEM
. This calculation of the $\mu$ transfer reaction between the He$\mu$ and
Li$\mu$ is our next work.

We have to admit that one major impediment of our conjecture is that the
formation of the muonic LiH or LiH+. Since LiH4+ has four positive charge, it
has to capture at least three $\mu$s which we think is very difficult in the
actual design. But we hope modern experimental technology can help solve this
problem.

###### Acknowledgements.

This work is supported by the Strategic Priority Research Program of Chinese
Academy of Sciences under the Grant NO. XDB34030301, the National Natural
Science Foundation of China under the Grant NOs. 12005266, 12075288, 11735003,
11961141012 and Guangdong Major Project of Basic and Applied Basic Research
No. 2020B0301030008. It is also supported by the Youth Innovation Promotion
Association CAS.

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