# Testing Elementary Cycles Formulation of Quantum Mechanics in Carbon
Nanotubes and Superconductivity

Donatello Dolce & Andrea Perali  University of Camerino, Piazza Cavour 19F,
62032 Camerino, Italy.

###### Abstract

Elementary Cycles are intrinsic periodic phenomena, classical in the essence,
whose classical relativistic dynamics reproduce the complete coherence
(perfect recurrences) typically associated to the pure quantum behaviours of
elementary particles. They can be regarded as effective representations of ’t
Hooft Cellular Automata. By means of Elementary Cycles physics we obtain a
consistent, intuitive, novel derivation of the peculiar quantum dynamics of
electrons in Carbon Nanotubes, as well as of Superconductivity fundamental
phenomenology. In particular we derive, from classical arguments, the
essential electronic properties of graphene systems, such as energy bands and
density of states. Similarly, in the second part of the paper, we derive the
Superconductivity fundamental phenomenology in terms of simple geometrical
considerations, directly from the Elementary Cycles dynamics rather than from
empirical aspects and effective quantities connected to the microscopical
characteristics of materials as in the standard approaches to
Superconductivity. With this approach simple geometrical considerations about
the competition between the quantum recurrence and the thermal noise allow for
a novel interpretation of the occurrence of high temperature superconductivity
and the related gauge symmetry breaking mechanism.

##  1 Introduction

As proven by G. ’t Hoof in recent works [1, 2], particular classical systems,
such as Cellular Automata (CA), have much in common with Quantum Mechanics
(QM). This correspondence has been largely tested and extended by the
Elementary Cycles (ECs) theory [3, 4, 5, 6, 7, 8, 9, 10] which has originated
a novel consistent formulation QM in all the its fundamental aspects. For the
scope of this paper an EC can be represented as a (massless) particle moving
very fast on a circle with time period $T$. In order to represent the
electrons in graphene systems, such an EC can also be assumed on a lattice,
similarly to a periodic ’t Hooft CA of $N$ “ontic” sites (“cogwheel mode”).
The relativistic coordinate perametrising this cyclic dynamics is named here
“ontic” time $t$. The EC infinitesimal evolution law in the “ontic” time is in
general $|t\rangle\rightarrow|t+\delta t+\text{mod}~{}T\rangle$.

The periodic dynamics of an EC is described, at a statistical level, by a
wave-function $\Phi(t)$, also named “physical state”, satisfying the Periodic
Boundary Conditions (PBCs) $\Phi(t)=\Phi(t+T)$. An EC can be effectively
regarded as of a continuous periodic CA in which CA evolution law is encoded
in PBCs. The essential idea is that “ _there is a close relationship between a
particle moving on a circle with period $T$ and the Quantum Harmonic
Oscillator (QHO) with the same period_” [1]. This is a fundamental
correspondence as the QHO is the essential ingredient of a second quantised
field, and in turn of the whole Quantum Field Theory (QFT). In this paper we
will use this correspondence to derive applications to condensed matter.

The powerful formalism of ECs theory allows for a direct test, in condensed
matter, of advanced aspects of theoretical physics [3, 4, 5, 8, 7, 9]. To
every EC of period $T$ is associated a fundamental energy
$\hbar\omega=2\pi\hbar/T$, according to the de Broglie phase harmony condition
in the “ontic” time $\omega T=2\pi$. EC is characterised by undulatory
mechanics so that the generic solution constituting the EC physical state has
the form $\phi(t)=e^{-i\omega t}$. The PBCs $\Phi(t)=\Phi(t+T)$ encoding the
EC evolution law in time determine the EC eigenstates
$\phi_{n}(t)=e^{-i\omega_{n}t}/\sqrt{2\pi}$. They form a complete, orthogonal
set with harmonic energy spectrum
$\hbar\omega_{n}=n\hbar\omega=n\frac{2\pi\hbar}{T}$ ($n\in\mathbb{Z}$). Hence
a classical EC naturally defines an Hilbert space of basis $|n\rangle$ such
that $\langle t|n\rangle=\phi_{n}(t)$ with induced inner product $\langle
n|n^{\prime}\rangle=\delta_{n,n^{\prime}}$. Thus an EC turns out to be
represented by the superposition of its eigenstates
$\Phi(t)=\sum_{n\in\mathbb{Z}}\alpha_{n}\phi_{n}(t)$. That is, in the Hilbert
space representation, it is described by a point in the corresponding Hilbert
space $|\Phi\rangle=\sum_{n}\alpha_{n}|n\rangle$. Since
$i\partial_{t}\phi_{n}(t)=\omega_{n}\phi_{n}(t)$ the EC “ontic” time evolution
is given, in this Hilbert space formalism, by the ordinary Schrödinger
equation $i\hbar\partial_{t}|\Phi(t)\rangle=\mathcal{H}|\Phi(t)\rangle$ where
the Hermitian operator $\mathcal{H}$ is defined as
$\mathcal{H}|n\rangle=\hbar\omega_{n}|n\rangle$. Hence, its evolution is given
by the Hilbert operator $\mathcal{U}(dt)=e^{-i\mathcal{H}dt/\hbar}$. We must
bear in mind that the fundamental topology of an EC is that of the circle
$\mathbb{S}^{1}$. This determines, by means of the PBCs, the quantum number
$n$.

Actually, as can be checked in the ECs theory, the temporal dynamics of an EC
correspond to those of the time evolution of a Quantum Harmonic Oscillator
(QHO) of period $T$, except for the fact that the quantum number can assume
negative values, $n\in\mathbb{Z}$ (positive and negative frequencies). This
seems to imply a non positively defined Hamiltonian operator for EC.
Nevertheless, as also argued by ’t Hooft, the negative modes can be
consistently interpreted as the anti-particles of relativistic QM. In
confirmation of this we will in fact find that in graphene physics these
negative modes correspond to the holes in the Fermi sea, i.e. to the negative
modes of the Carbon Nanotubes (CN) energy bands. Indeed, the Elementary Charge
Carriers (ECCs) in a CN actually behave as massless “particles on a circle”,
i.e. they simulate ECs in the essence. In particular, ECCs in a CN with $N$
carbon atoms along the circumference corresponds to ECs of $N$ lattice sites
on a circle. We will use the covariant generalisation of this correspondence
to derive in a simple way, and in agreement with present literature [12, 13],
the essential electronic properties of CNs. CNs can be therefore used to test
ECs theory (and in turn CA models) and thus foundational aspects of QM.

Similarly we will show here that the EC evolution law, i.e. the PBCs of the
physical state, yields the most foundational aspects of SuperConductivity (SC)
such as: quantisation of the magnetic flux in unit of $\varphi_{0}/2=hc/2e$,
the Meissner effect, the Josephson effect, the Little-Park effect, etc.
Indeed, the EC evolution directly implies discretised variations of the
Goldstone field which in turn describe the explicit breaking of the
ElectroMagnetic (EM) gauge invariance typical of SC [3, 14]. In short, the SC
phenomenology will be derived by applying the simple requirement that the wave
function of electrons in the superconductor must have closed space-time
orbits, in analogy with the closed space-time orbits of electrons in the
atomic orbitals (see Bohr atom or Bohr-Sommerfeld quantisation). According to
this description, the electron in the atomic orbitals can be therefore
regarded as being locally in a superconducting regime.

##  2 Testing Elementary Cycles theory by means of Carbon Nanotubes: Compton
clock vibrational modes and controvariant modulations.

As well-known, electrons in a graphene monolayer effectively behave as
massless charged particles (with pseudo-spin), i.e. as relativistic particles
traveling at the Fermi velocity $v_{F}$ (the analogous of the speed of light
for graphene physics). Let us assume that the graphene layer is curled-up in
the same direction of the electron motion. The result will be a massless
“particle moving on a circle” at “light” speed $v_{F}$. Since the CN have a
finite number $N$ of carbon atoms along the circumference $C_{h}$, such a
“circle” is on a lattice of $N$ sites. In short, the electron moving at
velocity $v_{F}$ along a CN circumference characterised by $N$ carbon atoms
can be regarded as an EC with $N$ lattice sites on a circle and time period
$T_{C}=C_{h}/v_{F}$. According to ECs theory this must be identified with the
Compton period of the ECCs implying an effective mass for the electrons.
Indeed CNs are real physical systems simulating ECs (and CA for some aspects).
For the sake of simplicity we start by investigating the continuous lattice
limit. The effective CN electronic properties will be then obtained by passing
to the lattice case. For instance the EC harmonic energy spectrum
$\hbar\omega_{n}=n\hbar\omega=\frac{2\pi\hbar n}{T}$ in the lattice case is
$\hbar\omega_{n}=\frac{N\hbar}{2T}\sin(\frac{\pi n}{N})$.

As a consequence of the compactification of one of the two spatial dimensions
of the graphene layer, the resulting CN effective dynamics are effectively
characterised by a single spatial dimension, i.e. the axial direction
$x_{\parallel}$. In such an effective CN one-dimensional space the electron
moving along the circumference is effectively at rest with respect to the
axial dimension. We denote the momentum of the electron along the axial direct
with $\hbar k_{\parallel}$. The period of the electron moving along the CN
circumference, which is effectively at rest with respect to the axial
direction, is denoted by $T(k_{\parallel}=0)=T_{C}$. According to ECs theory,
this effective rest period corresponds to a rest energy
$\hbar\omega(k_{\parallel}=0)$, such that
$T(k_{\parallel}=0)=2\pi/\omega(k_{\parallel}=0)$. In relativity it is natural
to associate a rest energy to a mass:
$\hbar\omega(k_{\parallel}=0)=m_{*}v_{F}^{2}$. Indeed the rest period $T_{C}$
corresponds to the effective Compton time of the electron in the CN, which in
turn fixes the effective mass scale of the ECCs in CN according to the Compton
relation $m_{*}=\frac{2\pi\hbar}{T_{C}v_{F}^{2}}$ (that is, the Compton time
of a relativistic particle of mass $m$ is $T_{C}=\frac{2\pi\hbar}{mc^{2}}$). A
massless particle generates an effective mass if constrained in a (periodic)
box! [11]. Thus, we find that the effective mass scale of the electrons in the
CNs is determined by the CN circumference according to

$m_{*}=\frac{2\pi\hbar}{C_{h}v_{F}}$ (1)

This effective mass scale is the fundamental rest energy of our ECs of rest
period $T(k_{\parallel}=0)=T_{C}$. The ECCs at rest along the CN axial
direction can be regarded as ECs of rest period $T_{C}$. They simulate the
effective de Broglie internal clocks of the electrons in the CNs. Notice that
the Compton time of an ordinary electron $\sim 10^{-21}$s is effectively
rescaled to about $10^{-15}$s in CNs. Since the latter time scale is
accessible to modern timekeepers, CNs allow for a indirect test of
foundational quantum aspects of elementary particles, as pointed out for
instance in [4].

Furthermore, due to the periodic dynamics, an EC (i.e. a relativistic
continuous CA) is characterised not only by the fundamental energy considered
so far, but also by a whole spectrum of energy levels. In the rest case the EC
rest energy spectrum is
$\hbar\omega_{n}(k_{\parallel}=0)=n\hbar\omega(k_{\parallel}=0)=nm_{*}$. Since
this is a rest energy spectrum, it is natural to associate it to a mass
spectrum $m_{n}=nm_{*}=\frac{2\pi n\hbar}{C_{h}v_{F}}$, with $n\in\mathbb{Z}$.
This yields interesting analogies to the Kaluza-Klein theory as described in
[8]. Similarly to the general case, this harmonic spectrum denotes the
eigenmodes associated to the PBCs in the proper time $\tau$ (i.e. the time
coordinate for the electron at rest with respect to the axial direction) for
the EC describing the ECC in the CN: For the rest EC we have
$\Phi(\tau)=\Phi(\tau+T_{C})$. In short these PBCs encode the evolution law of
the EC evaluated at rest. In the derivation of the lattice limit we have to
consider that, as a consequence of the hexagonal geometry of the graphene
layer, two kind of PBCs are possible for the electron moving along the CN
circumference. Besides the trivial PBCs described above, it is also possible
to have configurations symmetric under lattice rotations of
$\pm\frac{2}{3}\pi$. In this case the PBCs are thus twisted by a corresponding
factor. That is, in general, we can write the PBCs of the ECCs in CNs as

$\Phi(\tau)=e^{i2\pi\alpha}\Phi(\tau+T_{C})\,$ (2)

where $\alpha=0,\pm\frac{1}{3}$. The evolution law of the ECs simulating the
ECCs in CNs from an initial proper time $\tau_{i}$ to a final one $\tau_{f}$
is $|\tau_{i}\rangle\rightarrow
e^{i2\pi\alpha}|\tau_{f}+\text{mod}~{}T_{C}\rangle$. By considering this twist
of the PBCs the resulting mass spectrum of the rest EC turns out to have a
shift $\alpha m_{*}$:

$m_{n}=(n+\alpha)\frac{2\pi\hbar}{T_{C}}=(n+\alpha)m_{*}=(n+\alpha)\frac{2\pi\hbar}{C_{h}v_{F}}~{},~{}~{}(n\in\mathbb{Z})\,.$
(3)

This implies that for $\alpha=0$ the CN is characterised by a massless mode
$m_{0}=m_{*}=0$, whereas all the other modes are massive: $m_{n}=nm_{*}$ with
$n\neq 0$. Indeed this case describes metallic CNs. In the case
$\alpha=\pm\frac{1}{3}$ all the modes are massive. This in fact corresponds to
semiconducting CNs whose characteristic fundamental mode has effective mass

$m_{*}=\frac{2\pi\hbar}{3C_{h}v_{F}}\,.$ (4)

in agreement with present literature [12]. The lattice limit correctly leads
to the effective CN mass spectrum [13]. We find that, for ZigZag (ZZ) and
ArmChair (AC) CNs of $N$ carbon atoms in the perimeter, the resulting mass
spectra turns out to be, respectively,

$\text{ZZ)}~{}~{}~{}m_{n}=m_{*}\frac{N}{\pi}\sin\left(\frac{\pi
n}{N}\right)~{};~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{AC)}~{}~{}~{}m_{n}=m_{*}\frac{N}{3\pi}\left[1+2\cos(\frac{\pi
n}{N})\right]~{}.$ (5)

In order to derive the CN energy bands we must consider the relativistic
transformation of our EC of rest period $T_{C}$. This can be easily done in
analogy with undulatory mechanics. Indeed an EC can be regarded as a “de
Broglie periodic phenomenon”, which at rest reproduces the so called “de
Broglie internal clock”. As originally pointed out by de Broglie, the period
and wavelength of a moving particle, and thus its energy and momentum, are
determined, through Lorentz transformations, by the rest periodicity, i.e. by
the Compton period of a corresponding “periodic phenomenon”. ECs theory shows
that de Broglie idea of internal clocks of particles, at the base of modern
undulatory mechanics and relativistic QM, directly applies to CNs.

The case of ECC with a non zero velocity $v_{\parallel}$ along the CN axial
direction corresponds to an effective Lorentz boost in the effective CN space-
time. That is, the effective period $T(k_{\parallel})$ and wave-length
$\lambda_{\parallel}$ of an electron with non-vanishing momentum $\hbar
k_{\parallel}$ in the axial direction of a CN is obtained from the Compton
time (rest periodicity) $T_{C}$ by means of the Lorentz transformation
$T_{C}=\gamma_{\parallel}T(k_{\parallel})-\beta_{\parallel}\gamma_{\parallel}\lambda_{\parallel}$
in the effective CN space-time, where the Lorentz factor is
$\gamma_{\parallel}=1/\sqrt{1-\beta_{\parallel}}$ with
$\beta_{\parallel}=v_{\parallel}/v_{F}$. In short, the motion along the CN is
described by boosting the rest ECs described above. The effective Compton
period $T_{C}=\frac{C_{h}}{v_{F}}$ implies, by means of Lorentz
transformations, a time period $T(k_{\parallel})$ and a spatial recurrence
$\lambda_{\parallel}$ when the electron moves along the axial direction — as a
consequence, for instance, of an electric potential. These resulting
recurrences in time and space determine the effective energy
$\hbar\omega(k_{\parallel})=\frac{2\pi\hbar}{T(k_{\parallel})}$ and the
momentum $\hbar k_{\parallel}=\frac{2\pi\hbar}{\lambda_{\parallel}}$ of the
electron in the CN, according to undulatory mechanics. Notice that the former
relation is the EC phase harmony relation in time for a moving ECC in the CN
as it relates the time period and fundamental energy for an EC of momentum
$k_{\parallel}$. In analogy to relativistic notations, by introducing the two
tangent two-vector $k_{\mu}=\\{\omega/v_{F},-k_{\parallel}\\}$ (covariant) and
$\lambda^{\mu}=\\{Tv_{F},\lambda_{\parallel}\\}$ (controvariant), the phase
harmony relation can be generalised to the invariant form
$m_{*}v_{F}^{2}T_{C}/\hbar=k_{\mu}\lambda^{\mu}=2\pi$. Thus, the relativistic
constraints for these two vectors are respectively
$m_{*}v_{F}^{2}=\hbar^{2}k_{\mu}k^{\mu}$ and
$\frac{1}{T_{C}^{2}}=v_{F}^{2}\frac{1}{\lambda^{\mu}}\frac{1}{\lambda_{\mu}}$,
denoting the dual relativistic dispersion relations for the fundamental energy
and time period, respectively. The covariant EC evolution law describing an
electron moving in the effective CN space-time $x^{\mu}=\\{t,x_{\parallel}\\}$
is thus given by $|x^{\mu}_{i}\rangle\rightarrow
e^{i2\pi\alpha}|x^{\mu}_{f}+\text{mod}~{}\lambda^{\mu}\rangle$ (the EC
evolution law in time and space is given by the component $\mu=0$ and $\mu=1$,
respectively). In analogy to the relativistic Doppler effect, this implies
that the time period $T(k_{\parallel})=2\pi/\omega(k_{\parallel})$ varies with
$k_{\parallel}$ in such a way that the electron fundamental energy turns out
to satisfy the effective relativistic dispersion relation

$\hbar^{2}\omega^{2}(k_{\parallel})=\frac{(2\pi\hbar)^{2}}{T^{2}(k_{\parallel})}=m^{2}_{*}v_{F}^{4}+\hbar^{2}k_{\parallel}^{2}v_{F}^{2}\,.$
(6)

We have obtained, from simple considerations about the periodicity, that the
motion of the electrons along CNs is the analogous of the relativistic motion,
provided that the mass of the electron is replaced by the effective mass
$m_{*}$ and the speed of light by the Fermi velocity $v_{F}$.

By applying this effective relativistic dispersion relation to every mass
eigenmode $m_{n}$ allowed by the electron effective Compton periodicity
$T_{C}$ we immediately obtain the energy bands structure of CNs in the
continuous limit

$\hbar\omega_{n}(k_{\parallel})=(n+\alpha)\frac{2\pi\hbar}{T(k_{\parallel})}=(n+\alpha)\sqrt{m^{2}_{*}v_{F}^{4}+\hbar^{2}k_{\parallel}^{2}v_{F}^{2}}~{},~{}~{}(n\in\mathbb{Z})$
(7)

The CNs energy bands follow from this result by performing the lattice limit
(and considering that the CN lattice has also a periodicity along the axial
direction characterising Brillouin zones). Thus we have to consider an EC on a
lattice (or the covariant generalisation of a periodic CA). For instance, in
agreement we present literature, in the case ZZ CNs we find the energy bands
structure — or similarly the density of states [4]:

$\text{ZZ)}~{}~{}~{}\hbar^{2}\omega_{n}(k_{\parallel})=m_{*}^{2}v_{F}^{4}\frac{N^{2}}{3\pi^{2}}\left(1+2\sin\frac{\pi
n}{N}\right)^{2}-\frac{2}{3}\hbar^{2}k_{\parallel}^{2}v_{F}^{2}\cos\frac{\pi
n}{N}\,.$ (8)

This description is very interesting to test the validity of ECs theory (and
CA models). Indeed one of the main issues of the CA interpretation of QM is
that the negative modes ($n=-1,-2,\dots$) associated to the CA evolution law
implies a non positively defined Hamiltonian operator. But ECs must be
interpreted as relativistic objects (covariant generalisation of CA). As also
noticed by ’t Hooft, these negative modes can be consistently interpreted as
antiparticles, i.e. holes in the Dirac sea. This is fully confirmed by CNs
physics in which, actually, the negative energy bands corresponding to the
negative ECs vibrational modes, actually describes holes in the Fermi sea.
Thus, in ECs theory the negative modes describe antiparticles.

##  3 Testing Elementary Cycles theory by means of superconductivity: the role
of the temperature

We now introduce the role of the temperature in ECs theory; it can also be
generalised to CA. We must remember that in ECs physics, as in undulatory
mechanics, the time period and the energy are “two faces of the same coin”.
Therefore interactions, i.e. local variations of energy
$\hbar\omega=2\pi\hbar/T$, correspond to local modulations of the period $T$.
Since the temperature implies random collisions among particles (thermal
noise), a quantum system at finite temperature is characterised by chaotic
(Poissonian) breaks of the particle “complete coherences” with characteristic
thermal time $\beta=\hbar/k_{B}\mathcal{T}$, i.e. a dumping
$e^{-\hbar\omega/k_{B}\mathcal{T}}=e^{-2\pi\beta/T}$ of the cyclic behaviours
of periods $T$, being $k_{B}$ the Boltzmann constant and $\mathcal{T}$ the
temperature.

While a free EC at zero temperature, as well as the perfect coherence of pure
quantum systems, are characterised by perfect periodicity $T$ in the
Minkowskian time, thermal quantum systems are characterised by Euclidean time
periodicity of period $\beta=\hbar/k_{B}\mathcal{T}$. These two kinds of
temporal periodicities (which can be related by Wick’s rotation, as for
instance in the Hawking theory of black holes) have fundamental different
physical meanings: they are in competition. In the pure quantum systems at
zero temperature we have a persistent exact recurrence in time, e.g. described
by the phasor $e^{-i\omega_{n}t}$, whereas in the latter case, corresponding
to a non zero temperature, we have a dissipation of the quantum recurrence by
Boltzmann factors $e^{-\omega_{n}/k_{B}\mathcal{T}}=e^{-2\pi n\beta/T}$. That
is a dumping factor of the Minkowskian periodicity. In other words, while the
Minkowskian periodicity $T$ of QM tends to form perfect coherent states
(“periodic phenomena”), the Euclidean periodicity $\beta$ describes a dumping
associated to the thermal noise which tends to destroy the perfect recurrences
of pure quantum systems. Thus, if the $T\ll\beta$ the system can
autocorrelated and give rise to pure quantum phenomena such as SC. In the
opposite limit the thermal noise breaks the the quantum recurrence before it
can give autocorrelation, in this case we have the classical limit, e.g.
ordinary electric resistance.

Another aspect of the temperature which will be used in the following
description is the Boltzmann distribution. In a few words, the dumping factor
$e^{-\omega_{n}/k_{B}\mathcal{T}}=e^{-2\pi n\beta/T}$ can be also regarded as
the probability to populate the nth vibrational mode of the system. Thus, at
very low temperature $T\ll\beta$ only the fundamental mode $n=1$ will be
populated whereas at high temperature many vibrational modes must be
considered (e.g. as in the classical limit in Bohr atom). We will use this
aspect to describe the EM symmetry breaking occurring in superconductivity (in
analogy with the effective gauge symmetry breaking in extra dimensional
extensions of the Standard Model, and bosonic condensation).

These simple arguments, further motivated below, allow us to predict that high
temperature SC happens in quantum systems characterised by very small quantum
recurrences $T$, orders of magnitude smaller than in ordinary SC. This is
confirmed by CNs in which, as we have seen, the quantum recurrence is
explicitly determined by the CN circumference $C_{h}$, so that, in agreement
with [12, 15], the critical temperature is inversely proportional to the
circumference $C_{h}$.

Our description of CNs shows that covariant ECs provide a model describing the
“periodic phenomena” conjectured by de Broglie to describe the quantum
behaviour of elementary particles. Indeed an EC of rest period $T_{C}$
describes the quantum behaviour of a relativistic particle of mass
$m=2\pi\hbar/T_{C}c^{2}$, where we have replaced the Fermi velocity with the
speed of light $c$. In general an EC, similarly to the ideal case of a free
quantum particle at zero temperature, is characterised by a “complete
coherence” in space-time.

Generalising the CN description to the four-dimensional space-time of ordinary
relativity, in a given inertial reference frame the free EC has time period
$T(\vec{k})$ and wave-length $\vec{\lambda}$ which determines its energy
$\hbar\omega(\vec{k})=2\pi\hbar/T(\vec{k})$ and momentum $\hbar
k_{i}=2\pi\hbar/\lambda^{i}$ ($i=1,2,3$). In covariant notation we can
introduce the four vectors $k_{\mu}=\\{\omega/c,-\vec{k}\\}$ and
$\lambda^{\mu}=\\{Tc,\vec{\lambda}\\}$ related by the phase harmony relation
$k_{\mu}\lambda^{\mu}=2\pi\hbar$. This means that the evolution law is
$|x^{\mu}_{i}\rangle\rightarrow|x^{\mu}_{f}+\text{mod}~{}\lambda^{\mu}\rangle$
(notice that in this case a twist fact of $\alpha=1/2$ would reproduce the
vacuum energy $\hbar\omega(\vec{k})/2$). The generic EC solution, in the free
case, is $\Phi(x)=\sum_{n}e^{-i{k_{\mu}}_{n}x^{\mu}}$ and the PBCs, encoding
the evolution law, are $\Phi(x^{\mu})=\Phi(x^{\mu}+\lambda^{\mu})$. These
imply the (normally ordered) energy-momentum spectrum
$\vec{k_{\mu}}_{n}=n\vec{k}_{\mu}$ of a free particle. As can be easily seen
from EC time evolution, $\Phi(t)=\sum_{n}e^{i\omega_{n}t}$ and
$\Phi(t)=\Phi(t+T)$ imply the ordinary (normally ordered) energy spectrum of a
relativistic quantum particle $\omega(\vec{k})=n\hbar\omega(\vec{k})$ (in
analogy with a string vibrating with period $T(\vec{k})$). In short, the EC
evolution law implies that the physical state $\Phi$ describing the wave
function of a free relativistic particle at zero temperature must have closed
orbits along its space-time evolution. That is, we have a correspondence with
the Bohr-Sommerfeld quantisation. Indeed, an interacting EC is characterised
by a locally modulated space-time period so that, in the interacting case, the
generic solution has the form of a locally modulated wave
$\phi(x)=e^{-i\int^{x}dy^{\mu}k^{\mu}(y)}$. The evolution law, i.e. the local
PBCs for this generic solution, in the interacting case yields the
relativistic Bohr-Sommerfeld quantisation condition
$\oint_{\lambda}dy^{\mu}{k_{\mu}}_{n}(y)=2\pi n\hbar$. It is well-known that
in a Coulomb potential this yields the Bohr energy levels of the atomic
orbitals $E_{n}=-13.6~{}\text{eV}/n^{2}$.

Let us assume unitary gauge invariance for an EC (representing electrons in
equilibrium with the EM field in the superconducting material)
$\Phi(\vec{x},t)=U(\vec{x},t)\Phi(\vec{x},t)$, where
$U(\vec{x},t)=e^{-i\frac{e}{\hbar c}\theta(\vec{x},t)}$. The Goldstone
$\theta(\vec{x},t)$ denotes the local invariance of the EM interaction and $e$
is the electric charge. In analogy with the EC description, we now impose PBCs
$\Phi(\vec{x},t)=\Phi(\vec{x},t+T(\vec{k}))$ (for the sake of simplicity we
only consider the time component, furthermore we assume here that the topology
is the orbifold $\mathbb{S}^{1}/\mathcal{Z}_{2}$ associated to the fundamental
topology $\mathbb{S}^{1}$ of the EC, so that the phase invariance is $n\pi$
rather than $2\pi n$). Thus, in this case the PBCs in time of period
$T(\vec{k})$ imply the following evolution law for the Goldstone mode

$\frac{e}{\hbar c}\theta(\vec{x},t)=\frac{e}{\hbar
c}\theta(\vec{x},t+T(\vec{k}))+\text{mod}~{}~{}n\pi.$ (9)

As a consequence of the EC evolution law the Goldstone field can only vary by
finite amounts

$\Delta\theta(\vec{x},t)=\frac{\varphi_{0}}{2}~{},~{}~{}\text{with}~{}~{}\varphi_{0}=\frac{hc}{e}\,.$
(10)

This simple condition yields the characteristic quantum behaviours of a
superconducting regime, as described in [14]. In short, according to our
ansatz, as a consequence of the closed orbits of the wave-function (physical
state) imposed by the EC periodic dynamics (i.e. by the CA evolution law),
${\theta}(\mathbf{x},t)$ can only vary by discrete amounts running along a
conductor contour $\Sigma$ in which the EM field is a pure gauge
$A_{\mu}(\vec{x},t)=\partial_{\mu}\theta(\vec{x},t)$. The Stokes theorem gives
a quantisation of the magnetic flux through the area $\mathcal{S}_{\Sigma}$
limited by $\Sigma$,

$\displaystyle\int_{\mathcal{S}_{\Sigma}}\\!\\!\mathbf{B}(\mathbf{x},t)\cdot
d\mathbf{S}=\oint_{\Sigma_{\mathcal{}}}\\!\\!\mathbf{A}(\mathbf{x},t)\cdot
d\mathbf{x}=\oint_{\Sigma}\\!\\!\mathbf{\nabla}{{\theta}}(\mathbf{x},t)\cdot
d\mathbf{x}=n\frac{\varphi_{0}}{2}\,.$ (11)

Since the magnetic flux is quantised, the electric current cannot smoothly
decay while flowing around the circuit, and there is not electric resistance,
i.e. we have SC, [14] and [5, 6, 7]. The quantisation (10) means that the
Goldstone field $\theta(\mathbf{x},t)$ transforms as the phase of a condensate
of a fermionic pair operator
$\left\langle\epsilon_{\alpha\beta}\Phi^{\alpha}\Phi^{\beta}\right\rangle$ of
charge $-2e$. That is, it plays the role of the usual Cooper pair of the BCS
microscopic theory of SC. From (10) it is also manifest that the quantisation
of the Goldstone field implies a breaking of the EM gauge $U_{EM}(1)$ to
$\mathcal{Z}_{2}$.

Now we consider the EM gauge connection for the EC physical state
$\Phi(x^{\prime})=e^{i\frac{e}{\hbar
c}\int_{x}^{x^{\prime}}A_{\mu}dx^{\mu}}\Phi(x^{\prime})$. Indeed, gauge
interaction has a particular geometrodynamical meaning in ECs theory [7]. This
can be alternatively obtained from the minimal substitution $\hbar
k_{\mu}(x)\rightarrow\hbar k_{\mu}(x)-eA_{\mu}(x)$ in the EC interacting
solution. By imposing the EC evolution law in space-time with orbifold
topology $\mathbb{S}^{1}/\mathcal{Z}_{2}$, i.e. the EC PBCs in space-time,
this yields the Dirac quantisation condition for magnetic monopoles (in
analogy with the Bohr-Sommerfeld quantisation)

$\oint A_{\mu}dx^{\mu}=n\frac{\varphi_{0}}{2}\,.$ (12)

The spatial component yields the quantisation of the magnetic flux described
above whereas the time component yields the Josephson effect. Let us consider
a junction between superconductors with a voltage difference $\Delta V$ (the
BCs in this case is determined by the isolating barrier). Since the Maxwell
equation in this case implies that $\Delta V=-\partial_{t}\theta$, the Stokes’
theorem yields $T_{junct}\Delta V/c=\varphi_{0}/2$ corresponding to the
frequency $f_{junct}=1/T_{junct}$ of the Josephson effect. Alternatively this
can be derived by considering that the Lagrangian at the junction can only
depend on the phase difference of the Goldstone fields
$\mathcal{L}_{junct}=\mathcal{F}(\Delta\theta)$. The Josephson effect follows
from the fact that the evolution law implies that the Goldstone can only vary
by finite steps, see eq.(10), so that
$\mathcal{F}(\Delta\theta)=\mathcal{F}(\Delta\theta+n\varphi_{0}/2)$ [14].

Due to the quantum recurrence, which in our ECs theory originates from the
PBCs, i.e. from the EC evolution law, the EM gauge field $A_{\mu}$, similarly
to the Goldstone field, is constrained to have a periodic behaviour. This
means that it can be expanded in vibrational eigenmodes which, for the sake of
simplicity, we assume here to be harmonic
$A_{\mu}(\vec{x},t)=\sum_{n\in\mathbb{Z}}e^{-in\omega^{\gamma}t}A_{(n)\mu}(\vec{x})$.
This implies that the EM Lagrangian can be expanded in vibrational eigenmodes
as well. In the temporal (unitary) gauge we get

$\mathcal{L}_{YM}=-\frac{1}{4}\int
d^{4}xF_{\mu\nu}F^{\mu\nu}=\sum_{n\in\mathbb{Z}}\int
d^{3}x\left[-\frac{1}{4}F_{(n)\mu\nu}F_{(n)}^{\mu\nu}-\frac{1}{2}\frac{(\omega_{n}^{\gamma})^{2}}{c^{2}}A_{(n)\mu}A_{(n)}^{\mu}\right]$
(13)

However we must consider that at very low temperature $T\ll\beta$ only the
lower mode of the system, that here we will denote by the bar sign, is
populated. This means that the summation of the above Lagrangian must be
truncated to the fundamental mode. As well-known for gauge invariance in extra
dimensional theories (see Higgsless, Composite Higgs or unified gauge-Higgs
models), this implies that the low temperature is no more gauge invariant. So
we find again an effective EM gauge breaking. Roughly speaking, at very low
temperature we can imagine that, similarly to a Bose-Einstein condensation,
all the vibrational modes of energy $\omega_{n}^{\gamma}=n\omega^{\gamma}$
condensate on the fundamental mode.

This symmetry breaking can also be seen from the effective matter Lagrangian
$\mathcal{L}_{S}$ obtained by integrating out from the equations of motion the
contributions of the higher modes as a function of the fundamental one. By
expanding $\mathcal{L}_{S}$ for small perturbations around the fundamental
mode, which now can be effectively written as $\bar{\Phi}=\rho e^{i2e\theta}$,
we get

$\displaystyle\mathcal{L}_{S}$ $\displaystyle\simeq$ $\displaystyle\int
d^{3}x\left[-\frac{1}{2}\bar{\Phi}^{*}|(\vec{\nabla}-i2e\vec{A})|^{2}\bar{\Phi}-\frac{1}{2}\alpha|\bar{\Phi}|^{2}-\frac{1}{4}\beta|\hat{\Phi}|^{4}\right]$
(14) $\displaystyle=$ $\displaystyle\int
d^{3}x\left[-2e^{2}\rho^{2}(\vec{\nabla}-i2e\vec{A})^{2}-\frac{1}{2}(\vec{\nabla}\rho)^{2}-\frac{1}{2}\alpha\rho^{2}-\frac{1}{4}\beta\rho^{4}\right]\,.$

This describes (with appropriate normalisation) the BCS theory of ordinary SC.
The first term describes the fundamental vibrational mode without the
perturbation of the higher modes. Due to gauge transformations it can be
written as
$\bar{\mathcal{L}}_{S}[A_{\mu}-\partial_{\mu}\phi]\sim\frac{V}{\Lambda^{2}}(A_{\mu}-\partial_{\mu}\theta)$,
$V$ is the volume and $\Lambda=\frac{1}{\sqrt{4e^{2}\langle\rho^{2}\rangle}}$
is the penetration length of the Meissner effect. Indeed, deep inside the
superconductor where the EM field is pure gauge, we have
$F_{\mu\nu}(\partial_{\chi}\theta)=0$ and the magnetic field is vanishing
$\vec{B}=0$. $\bar{\mathcal{L}}_{S}$ describes the fact that the energy cost
to expel the magnetic field is small with respect to that of the classical
configuration. By considering that the minimum of $\rho$ is
$\langle\rho^{2}\rangle=-\frac{\alpha}{\beta}$ we get
$\Lambda=\frac{1}{2e}\sqrt{-\frac{\alpha}{\beta}}$, in agreement with ordinary
SC. Finally, by expanding $\rho$ around its minimum
$\rho=\langle\rho\rangle+\rho^{\prime}$ we get
$\vec{\nabla}^{2}\rho^{\prime}=-2\alpha\rho^{\prime}$. It allows us to
introduce the coherence length $\xi=\frac{1}{-2\alpha}$ describing the
stability of the vortices and of the flux-quantised magnetic field. Indeed the
energy density of the superconducting state is lower than the energy density
of the normal state. We have
$\frac{\alpha^{2}}{4\beta}=\frac{1}{32e^{2}\lambda^{2}\xi^{2}}$. Thus, in
agreement with the ordinary description, the case $\xi>\Lambda$ and
$\xi<\Lambda$ describe Type I SC and Type II SC, respectively.

##  4 Conclusions

In this paper we have shown that the interest of the Elementary Cycles theory,
or similarly ’t Hooft Cellular Automata models, is not only limited to our
knowledge of foundational aspects of Quantum Mechanics: it also represents a
powerful tool to derive and describe in a very simple way nontrivial quantum
phenomena of condensed matter, such as the behaviour of the Elementary Charge
Carriers in Carbon Nanotubes and the essential phenomenology of
Superconductivity. Indeed we have derived the Carbon Nanotubes (or similar
graphene systems) electronic properties by simply considering the relativistic
modulations of the vibrational modes characterising the cyclic dynamics of
Elementary Cycles. Similarly we have derived all the fundamental quantum
phenomenology associated to Superconductivity and the role of the temperature
in quantum systems, particularly relevant to interpret the occurrence of High
Temperature Superconductivity. These represent further confirmations of the
overall validity of the Elementary Cycles formulation of Quantum Mechanics.

We acknowledge G. ’t Hooft for useful comments and partial financial support
from the project FAR “ _Control and enhancement of superconductivity by
engineering materials at the nanoscale_ ”.

## References

## References

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  * [4] Dolce D and Perali A 2015 EPJ Plus 103 41 (arXiv:1403.7037)
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  * [6] Dolce D 2011 Found. Phys. 41 178 (arXiv:0903.3680v5)
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  * [8] Dolce D 2012 Annals Phys. 327 2354 (arXiv:1110.0316)
  * [9] Dolce D 2015 Uni ed Description of Relativistic and Quantum Mechanics from Elementary Cycles Theory EJTP  12 IYL15-34 17
  * [10] Dolce D 2016 Introduction to the Quantum Theory of Elementary Cycles Beyond Peaceful Coexistence  (World Scientific) chapter 4 p 93
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  * [14] Weinberg S 1996 The quantum theory of fields. Vol. 2: Modern applications, p 489 (Cambridge Univ. Pr.)
  * [15] White C T and Mintmire J W 1998 Nature 394 29


# Resonant nuclear reaction 23Mg $(p,\gamma)$ 24Al in strongly screening
magnetized neutron star crust

Jing-Jing Liu11affiliation: Corresponding author<EMAIL_ADDRESS>, and
Dong-Mei, Liu11affiliation: Corresponding author<EMAIL_ADDRESS>1
College of Marine Science and Technology, Hainan Tropical Ocean University,
Sanya, Hainan 572022, China

###### Abstract

Based on the relativistic theory of superstrong magnetic field (SMF), by using
three models of Lai (LD), Fushiki (FGP), and ours (LJ), we investigate the
influence of SMFs due to strong electron screening (SES) on the nuclear
reaction 23Mg $(p,\gamma)$ 24Al in magnetars. At relatively low density
environment (e.g., $\rho_{7}<0.01$) and $1<B_{12}<10^{2}$, our screening rates
are in good agreement with those of LD and FGP. However, in relatively high
magnetic fields (e.g., $B_{12}>10^{2}$), our reaction rates can be 1.58 times
and about three orders of magnitude larger than those of FGP and LD,
respectively ($B_{12}$, $\rho_{7}$ are in units of $10^{12}$G,
$10^{7}\rm{g~{}cm^{-3}}$). The significant increase of strongly screening
rates can imply that more 23Mg will escape from the Ne-Na cycle due to SES in
a SMF. As a consequence, the next reaction 24Al $(\beta^{+},\nu)$ 24Mg will
produce more 24Mg to participate in the Mg-Al cycle. Thus, it may lead to
synthesize a large amount of production of $A>20$ nuclides in magnetars,
respectively.

dense matter— nuclear reactions, nucleosynthesis, abundances— stars: magnetic
fields—stars: interiors

††slugcomment: Not to appear in Nonlearned J., 45.

##  1 Introduction

In the dense sites of universe, such as novae, X-ray bursts and supernova,
there are explosive hydrogen burning process in high temperature and high
hydrogen environments. This burning is called the rapid-proton (rp) process
(Wallace et al., 1981). In the stage of hydrogen burning, the proton capture
reactions and $\beta^{+}$-decays (rp-process) will be ignited in the nuclei
whose mass numbers $A>20$. For example, the timescale of the proton capture
reaction of 23Mg in the Ne-Na cycle at sufficient high temperature is shorter
than that of the $\beta^{+}$-decay. Therefore, some 23Mg will kindle and
escape from the Ne-Na cycle by proton capture. The 23Mg leaks from the Ne-Na
cycle into the Mg-Al cycle synthesizing a large amount of heavy nuclei. Thus
the reaction 23Mg $(p,\gamma)$ 24Al in stellar environment is an important
reaction for producing heavy nuclei. Wallace et al. (1981) firstly discussed
the reaction rate of 23Mg $(p,\gamma)$ 24Al. Then, Iliadis et al. (2001) also
investigated this nuclear reaction rate. Kubono et al. (1995) reconsidered the
rate by considering four resonances and the structure of 24Al. Based on some
new experimental information on 24Al excitation energies, Herndl et al.
(1998); Visser et al. (2007), and Lotay et al. (2008) carried out an
estimation of the rate. However, they all seem to have overlooked the
influence of electron screening on nuclear reaction.

In the pre-supernova stellar evolution and nucleosynthesis, the strong
electron screening (SES) is always a challenging and interesting problem. Some
works (Bahcall et al., 2002; Liu, 2013, 2014, 2016; Liu et al., 2017, 2017)
have been done on stellar weak-interaction rates and thermonuclear reaction
rates. In the high-density surrounding, some SES models have been widely
investigated, such as Salpeter model (Salpeter, 1954; Salpeter et al., 1969),
Graboske model (Graboske et al., 1973), and Dewitt model (Dewitt et al.,
1976). Recently these issues were discussed by Liolios et al. (2000, 2001),
Kravchuk et al. (2014), and Liu (2013). However, they neglected the effects of
SES on thermonuclear reaction rate in superstrong magnetic field (SMF).

It is widely known that nuclear reaction rates at low energies play a key role
in energy generation in stars and the stellar nucleosynthesis. The bare
reaction rates are modified in stars by the screening effects of free and
bound electrons. The knowledge of the bare nuclear reaction rates at low
energies is important not only for the understanding of various astrophysical
nuclear problems, but also for assessing the effects of host material in low
energy nuclear fusion reactions in matter.

It is universally accepted that the surface dipole magnetic field strengths of
magnetars are in a range from $10^{13}$ to $10^{15}$G (Peng et al., 2007; Gao
et al., 2011, 2013, 2015, 2017, 2017; Li et al., 2016; Lai, 2001). The
momentum space of the electron gas is modified substantially by so intense
magnetic fields. The electron Fermi energy and nuclear reaction are also
affected greatly by a SMF in magnetars.

Anamalou x-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are
conceived as magnetars, which are a kind of special pulsars powered by their
magnetic energy (Duncan, 1992). The Fermi energy of the electrons will
increase with magnetic field and quantum effects of electron gas will be very
obvious in a SMF. As we all know, the positive energy levels of electrons must
abide by Landau quantization. The distribution of the electron in the momentum
space will be strongly modified by a SMF. Some authors discussed this issue in
detail in strong magnetic fields of magnetars. For instance, Gao et al. (2015,
2017, 2017) investigated not only the spin-down and magnetic field evolutions,
but also the electron Landau level effects on emission properties of
magnetars.

In this paper, according to the relativistic theory in a SMF (Peng et al.,
2007; Gao et al., 2011, 2013, 2015, 2017), we discuss the problem of SES and
then investigate the effect of SES on the thermonuclear reaction within three
different models (i.e., our model (LJ), Lai model (LD)(Lai et al., 1991; Lai,
2001), and Fushiki model (FGP)(Fushiki et al., 1989)) on the surface of
magnetars.

Our work differs from previous work of Liu (2016) about the discussion of
nuclear reaction rates. Firstly, in Liu (2016), though we cited several work
from Gao et al., but it is not familiar to the calculations involved in
electron Fermi energy in a superhigh magnetic field, a non-relativistic
electron cyclotron solution was applied when calculating the rates. Secondly,
our previous work Liu (2016) did not give a comparison among LJ, LD, and FGP
models in the case with a SMF. Finally, we analyze the nuclear reaction rates
in a SMF and also give a comparison for our model with Dewitt model(Dewitt et
al., 1976), and Liolios model(Liolios et al., 2000), in which the SMF were not
taken into consideration. Maybe SES universally occur in pulsars, and the
screening rate calculations in a SMF is of importance to the future studies on
cooling, nucleosynthesis,and emission properties of magnetars.

In this paper, following the works of Peng et al. (2007), and Gao et al.
(2011, 2013, 2015, 2017), we calculate the resonant reaction rates in the case
with SMF and without SMF in several screening models. In the case of the
former, the results from LD and FGP models will be compared with those of our
model, while in the latter case, the results from Dewitti and Liolio models
also will be compared. We derive new results for SES theory and the screening
rates for nuclear reaction in relativistic strong magnetic fields.

The article is organized as follows. In the next Section, we analyse three SES
models in a SMF of magnetars. In Section 3 we discuss the effects of SES on
the proton capture reaction rate of 23Mg, in which the four resonances
contributions will also be considered. The results and discussions will be
shown in Section 4. The article is closed with some conclusions in Section 5.

##  2 The SES in SMF

In astrophysical systems, the SMF may have significant influence on the
quantum processes. In this Section, we will study three models of the electron
screening potential (ESP) in SMF, i.e., LJ model, LD model, and FGP model.

###  2.1 ESP in our model

The rate of nuclear reaction in high density matter is affected by the fact
that the clouds of the electrons surrounding nuclei alter the interactions
among nuclei. The positive energy levels of electrons in SMF are given by
(Landau et al., 1977)

$\frac{\varepsilon_{n}}{m_{\rm{e}}c^{2}}=[(\frac{p_{\rm{z}}}{m_{\rm{e}}c})+1+2(n+\frac{1}{2}+\sigma)b]^{1/2}=(p^{2}_{\rm{z}}+\Theta)^{1/2},$
(1)

where $\Theta=1+2(n+\frac{1}{2}+\sigma)b$, $n=0,1,2,3....$,
$b=\frac{B}{B_{\rm{cr}}}=0.02266B_{12}$, $B_{12}$ is the magnetic fields in
units of $10^{12}$G, i.e., $B_{12}\equiv B/10^{12}$G,
$B_{\rm{cr}}=\frac{m^{2}_{\rm{e}}c^{3}}{e\hbar}=4.414\times 10^{3}$G is the
electron quantum critical magnetic field, and $p_{\rm{z}}$ is the electron
momentum along the field, $\sigma$ is the spin quantum number of an electron,
when $n=0$,$\sigma=1/2$, and when $n\geq 1$, $\sigma=\pm 1/2$.

In an extremely strong magnetic field $(B\gg B_{\rm{cr}})$, the Landau column
becomes a very long and narrow cylinder along the magnetic field. According to
the Pauli exclusion principle, the electron number density should be equal to
its microscopic state density. By introducing the electron Landau level
stability coefficient, the Fermi energy of the electron is given by (Gao et
al., 2013; zhu et al., 2018)

$\displaystyle U_{\rm{F}}$ $\displaystyle=$ $\displaystyle 5.91\times
10^{4}(\frac{B}{B_{{\rm{cr}}}})^{1/6}(\frac{\rho Y_{\rm{e}}}{\rho_{0}\times
0.00564})^{1/3}$ (2) $\displaystyle=$ $\displaystyle 5.91\times
10^{4}(\frac{B}{B_{{\rm{cr}}}})^{1/6}(\frac{n_{\rm{e}}}{0.00564\times\rho_{0}N_{\rm{A}}})^{1/3}\rm{keV},$

where $\rho_{0}=2.8\times 10^{14}{\rm g/cm}^{3}$ is the standard nuclear
density.

In order to evaluate the Thomas-Fermi screening wave-number
$K_{{\rm{TF}}}^{{\rm{LJ}}}$, we defined a parameter
$D^{{\rm{LJ}}}(U_{\rm{e}})$ and according to Eq.(3), we have

$n_{e}=0.00564\rho_{0}N_{A}(\frac{U_{\rm{F}}}{5.91\times 10^{4}b^{1/6}})^{3}$
(3) $\displaystyle D^{{\rm{LJ}}}(U_{\rm{F}})$ $\displaystyle=$
$\displaystyle\frac{\partial n_{\rm{e}}}{\partial
U_{\rm{F}}}=\frac{\partial}{\partial
U_{\rm{F}}}(0.00564\rho_{0}N_{A}(\frac{U_{\rm{F}}}{5.910\times
10^{4}b^{1/6}})^{3})$ (4) $\displaystyle=$ $\displaystyle 4.9913\times
10^{7}n_{\rm{e}}^{2/3}b^{-1/6}~{}~{}~{}\rm{cm^{-3}~{}KeV^{-1}}.$

According to Eq.(5), the Thomas-Fermi screening wave-number
$K_{{\rm{TF}}}^{{\rm{LJ}}}$ is given by (Ashcroft et al., 1976)

$\displaystyle(K_{{\rm{TF}}}^{{\rm{LJ}}})^{2}$ $\displaystyle=$ $\displaystyle
4\pi e^{2}D^{{\rm{LJ}}}(U_{\rm{F}})=4\pi e^{2}\frac{\partial
n_{\rm{e}}}{\partial U_{\rm{F}}}$ (5) $\displaystyle=$ $\displaystyle
6.269\times 10^{7}e^{2}(n_{e})^{2/3}b^{-1/6}~{}~{}~{}\rm{cm^{-3}}.$

By using the uniform electron gas model (Kadomtsev, 1971), the binding energy
of the magnetized condensed matter at zero pressure can be estimated. The
energy per cell can be written as

$E_{\rm{total}}=E_{\rm{k}}+E_{\rm{latt}}=\frac{3\pi^{2}e^{2}z_{j}^{3}}{8b_{1}^{2}r_{i}^{6}}+\frac{9e^{2}z^{\frac{5}{3}}}{10r_{e}}~{}~{}~{}\rm{MeV},$
(6)

where the first term is the kinetic energy and the second term is the lattice
energy. $r_{i}=z^{1/3}r_{\rm{e}}a_{0}$ is the Wigner-Seitz cell radius,
$a_{0}=0.529\times 10^{-8}$cm is the Bohr radius, and $r_{\rm{e}}=(3/4\pi
n_{\rm{e}})^{1/3}$ is the mean electron spacing. $z_{j}$ is the charge number
of the species $j$. $b_{1}=B/B_{0}=425.4B_{12}=1.9773\times 10^{4}b$ and
$B_{0}=m_{\rm{e}}^{2}ce^{3}/\hbar^{3}=2.3505\times 10^{-9}$G is the natural
(atomic) unit for the field strength (Lai, 2001). For the zero-pressure
condensed matter, we require $dE_{\rm{total}}/dr_{i}=0$, so we have

$r_{i}=r_{i0}=0.0371z_{j}^{1/5}b^{-2/5}a_{0}~{}~{}~{}\rm{cm}.$ (7)

By using linear response theory, the energy correction per cell due to non-
uniformity is given by (Lattimer et al., 1985)

$\displaystyle E_{{\rm{TF}}}^{{\rm{LJ}}}(r_{i},z_{j})$ $\displaystyle=$
$\displaystyle-\frac{18}{175}(K_{{\rm{TF}}}^{{\rm{LJ}}}r_{i})^{2}\frac{(z_{j}e)^{2}}{r_{i}}$
(8) $\displaystyle=$ $\displaystyle-\frac{1.30\times
10^{-6}e^{6}(n_{\rm{e}})^{4/3}z_{j}^{9/5}}{b^{11/15}}~{}~{}~{}\rm{MeV}.$

For the relativistic electrons, the influence from exchange free energy were
discussed by Refs.Stolzmann et al. (1996); Yakovlev et al. (1989). Their works
showed that the correlation correction is very small. Therefore, in this paper
we have neglected the correction of Coulomb exchange free energy interaction
in the electron gas model. By taking into consideration of the Coulomb energy
and Thomas-Fermi correction due to non-uniformity of the electron gas, the
energy per cell should be corrected as

$E_{{\rm{s}}}^{{\rm{LJ}}}(r_{i},z_{j})=E_{{\rm{k}}}(r_{i},z_{j})-U_{{\rm{coul}}}(r_{i},z_{j})-E_{{\rm{TF}}}^{{\rm{LJ}}}(r_{i},z_{j}).$
(9)

For two interaction nuclides, the energy required to bring two nuclei with
nuclear charge numbers $z_{1}$ and $z_{2}$ so close together that they
essentially coincide differs from the bare Coulomb energy by an amount which
in the Wigner-Seitz approximation is

$U_{{\rm{sc}}}=E_{{\rm{s}}}(r_{i},z_{12})-E_{{\rm{s}}}(r_{i},z_{1})-E_{\rm{s}}(r_{i},z_{2}),$
(10)

where $z_{12}=z_{1}+z_{2}$. If the electron distribution is rigid, the
contribution to from $E_{\rm{s}}$ the bulk electron energy cancel in
expression (11), and the screening potential is simply given as

$\displaystyle U_{\rm{sc}}$ $\displaystyle=$ $\displaystyle
E_{\rm{coul}}(r_{i},z_{12})-E_{\rm{coul}}(r_{i},z_{1})-E_{\rm{coul}}(r_{i},z_{2})$
(11) $\displaystyle=$ $\displaystyle 6.5984\times
10^{4}b^{2/5}(z_{12}^{9/5}-z_{1}^{9/5}-z_{2}^{9/5})\rm{MeV},$

where we assume the electron density is uniform, and the screening potential
is independent of the magnetic field.

From expression (9), the change of the screening potential due to the
compressibility of the electrons in the zero-pressure magnetized condensed
matter can obtained as

$\displaystyle\delta E_{\rm{TF}}^{\rm{LJ}}$ $\displaystyle=$
$\displaystyle-\frac{18}{175}(K_{\rm{TF}}^{\rm{LJ}}r_{i})^{2}\frac{e^{2}(z_{12}^{2}-z_{1}^{2}-z_{2}^{2})}{r_{i}}$
(12) $\displaystyle=$ $\displaystyle-\frac{1.30\times
10^{-6}e^{6}n_{\rm{e}}^{4/3}(z_{12}^{9/5}-z_{1}^{9/5}-z_{2}^{9/5})}{b^{11/15}}.$

In accordance with the above discussions, the total screening potential is the
sum of the screening potential with a uniformity distribution and a corrected
screening potential with a non-uniformity distribution. The screening
potential in SMF is given by

$U_{\rm{sc}}^{\rm{LJ}}=U_{\rm{sc}}+\delta E_{\rm{TF}}^{\rm{LJ}}.$ (13)

###  2.2 ESP in LD model

Lai (2001) and Lai et al. (1991) discussed the equation of state and the
electron energy in a SMF. In a SMF the electron number density $n_{\rm{e}}$ is
related to the chemical potential $U_{\rm{e}}$ by

$\displaystyle
n_{\rm{e}}=\frac{1}{(2\pi\widehat{\rho})^{2}\hbar}\sum_{0}^{\infty}g_{n0}\int_{-\infty}^{+\infty}fdp_{\rm{z}}$
$\displaystyle=\frac{1}{(2\pi\widehat{\rho})^{2}\hbar}\sum_{0}^{\infty}g_{n0}\int_{-\infty}^{+\infty}[1+\exp(\frac{E-U_{\rm{e}}}{kT})]^{-1}dp_{\rm{z}},$
(14)

where $\widehat{\rho}=(\hbar c/eB)^{1/2}=2.5656\times
10^{-10}B_{12}^{1/2}\rm{cm}$ is the electron cyclotron radius (the
characteristic size of the wave packet), and
$E=[c^{2}p_{z}^{2}+m_{e}c^{4}(1+nb)]^{1/2}$ is the free electron energy,
$g_{n}$ is the spin degeneracy of the Landau level, $g_{00}=1$ and $g_{n0}=2$
for $n\geqslant 1$, and the Fermi-Dirac distribution is given by

$f=[1+\exp(\frac{E-U_{\rm{e}}}{kT})]^{-1}.$ (15)

The electron Fermi energy including the electron rest mass is given by

$n_{\rm{e}}=\frac{1}{2\pi^{3/2}\lambda_{\rm{Te}}\widehat{\rho}^{2}}\sum_{(n=0)}^{\infty}g_{n}I_{-1/2}(\frac{U_{\rm{e}}-n\hbar\omega_{\rm{ce}}}{kT}),$
(16)

where the thermal wavelength of the electron is
$\lambda_{\rm{Te}}=(2\pi\hbar^{2}/m_{\rm{e}}kT)^{2}$, and the Fermi integral
is written as

$I_{n}(y)=\int_{0}^{\infty}\frac{x^{n}}{\exp(x-y)+1}dx.$ (17)

The binding energy of the magnetized condensed matter at zero pressure can be
estimated using the uniform electron gas model. Under the condition of super-
strong magnetic field, the Fermi energy $U_{\rm{F}}$ is less than the
cyclotron energy $\hbar\omega_{\rm{ce}}$, the electrons only occupy the ground
Landau level. According to their viewpoint of (Lai, 2001), the Thomas-Fermi
screening wave-number is given by

$(K_{\rm{TF}}^{\rm{LD}})^{2}=4\pi e^{2}D^{\rm{LD}}(\varepsilon_{\rm{F}})=4\pi
e^{2}\frac{\partial n_{\rm{e}}}{\partial\varepsilon_{\rm{F}}}=4\pi
e^{2}\frac{\partial n_{\rm{e}}}{\partial U_{\rm{F}}},$ (18)

where $\partial n_{\rm{e}}/\partial\varepsilon_{\rm{F}}$ is the density of
states per unit volume at the Fermi surface.
$\varepsilon_{\rm{F}}=P_{\rm{F}}^{2}/2m_{\rm{e}}$. From Eq.(6.16) of Lai
(2001), so we have

$D^{\rm{LD}}=\frac{\partial
n_{\rm{e}}}{\partial\varepsilon_{\rm{F}}}=\frac{3.79\times
10^{6}b^{2}r_{\rm{e}}^{3}}{e^{2}}.$ (19)

The Thomas-Fermi screening wave-number will be given by

$K_{\rm{TF}}^{\rm{LD}}=(\frac{4}{3\pi^{2}})^{1/2}b_{1}r_{\rm{e}}^{3/2}=6.901\times
10^{3}br_{\rm{e}}^{3/2}.$ (20)

Using the linear response theory, the energy correction (in atomic units) per
cell due to non-uniformity can be calculated and gives by (Lai, 2001)

$E_{\rm{TF}}^{\rm{LD}}(r_{i},z_{j})=-\frac{18}{175}(K_{\rm{TF}}^{\rm{LD}}r_{i})^{2}\frac{e^{2}z_{j}^{2}}{r_{i}}=-0.0139b_{1}^{2}r_{i}^{4}z_{j}.$
(21)

The uniform electron gas model can be improved by taking into consideration of
the Coulomb energy and Thomas-Fermi correction due to non-uniformity of the
electron gas. When the electron density is assumed to be uniform, the
screening potential is independent of the magnetic field. The change of the
screening potential due to the compressibility of the electrons for the zero-
pressure magnetized condensed matter can obtained

$\delta E_{\rm{TF}}^{\rm{LD}}=-2.5236\times
10^{-4}b^{2/5}(z_{12}^{9/5}-z_{1}^{9/5}-z_{2}^{9/5}).$ (22)

When we summed of a screening potential with a uniformity distribution and a
corrected screening potential with a non-uniformity distribution, the
screening potential in a SMF is given by

$U_{\rm{s}}^{\rm{LD}}=U_{\rm{sc}}+\delta E_{\rm{TF}}^{\rm{LD}}.$ (23)

###  2.3 ESP in FGP model

The influence of SES in a SMF on nuclear reaction was also discussed in detail
by Fushiki et al. (1989) (hereafter FGP). The electron Coulomb energy by an
amount which in the Wigner-Seitz approximation in a SMF was given by

$U_{\rm{sc}}^{\rm{FGP}}=E_{\rm{atm}}(r_{i},z_{12})-E_{\rm{atm}}(r_{i},z_{1})-E_{\rm{atm}}(r_{i},z_{2}),$
(24)

where $E_{\rm{atm}}(r_{i},z_{j})$ is the total energy of Wigner-Seitz cell. If
the electron distribution is rigid, the contribution to
$E_{\rm{atm}}(r_{i},z_{j})$ from the bulk electron energy cancel, the electron
screening potential at high density can be expressed as

$U_{\rm{sc}}^{\rm{FGP}}=E_{\rm{latt}}(r_{i},z_{12})-E_{\rm{latt}}(r_{i},z_{1})-E_{\rm{latt}}(r_{i},z_{2}),$
(25)

where $E_{\rm{latt}}(r_{i},z_{j})$ is the electrostatic energy of Wigner-Seitz
cell and $E_{atm}(r_{i},z_{j})=-0.9z_{j}^{5/3}e^{2}/r_{\rm{e}}$. Due to the
influence of the compressibility of the electron, the change in the screening
potential is given by (Fushiki et al., 1989)

$\displaystyle\delta
U_{\rm{s}}^{\rm{FGP}}=-\frac{54}{175}(\frac{e^{2}}{r_{\rm{e}}})\frac{1}{n_{\rm{e}}}\frac{\partial
n_{\rm{e}}}{\partial U_{\rm{e}}}[(z_{12})^{7/3}-(z_{1})^{7/3}-(z_{2})^{7/3}]$
$\displaystyle=-\frac{54}{175}(\frac{e^{2}}{r_{\rm{e}}})\frac{1}{n_{\rm{e}}}D^{\rm{FGP}}[(z_{12})^{7/3}-(z_{1})^{7/3}-(z_{2})^{7/3}],$
(26)

where

$D^{\rm{FGP}}=823.1481\frac{r_{\rm{e}}n_{\rm{e}}}{e^{2}}(\overline{\frac{A}{z}})^{4/3}\rho^{-4/3}B_{12}^{2}.$
(27)

The Thomas-Fermi screening wave-number will be given by

$(K_{\rm{TF}}^{\rm{FGP}})^{2}=1.0344\times
10^{4}r_{\rm{e}}n_{\rm{e}}(\overline{\frac{A}{z}})^{4/3}\rho^{-4/3}B_{12}^{2}.$
(28)

Thus, the corresponding result for the changes in the screening potential in a
SMF is

$\displaystyle\delta
U_{\rm{s}}^{\rm{FGP}}=-0.254(\overline{\frac{A}{z}})^{4/3}\rho^{-4/3}B_{12}^{2}[(z_{12})^{7/3}-(z_{1})^{7/3}-(z_{2})^{7/3}]$
$\displaystyle=-494.668(\overline{\frac{A}{z}})^{4/3}\rho^{-4/3}b^{2}[(z_{12})^{7/3}-(z_{1})^{7/3}-(z_{2})^{7/3}]\rm{MeV},$
(29)

where $(\overline{A/z})$ is the average ratio of $A/z$ , which corresponding
to the mean molecular weigh per electron. Thus the electron screening
potential in a SMF of FGP model is given by

$U_{\rm{s}}^{\rm{FGP}}=U_{\rm{sc}}+\delta
E_{\rm{TF}}^{\rm{FGP}}=U_{\rm{sc}}+\delta U_{\rm{s}}^{\rm{FGP}}.$ (30)

##  3 Resonant reaction process and rates

###  3.1 Calculations of resonant reaction rates with and without SES

The reaction rates are summed of contribution from the resonant reaction and
non-resonant reaction. In the case of a narrow resonance, the resonant cross
section $\sigma_{r}$ is approximated by a Breit-Wigner expression (Fowler et
al., 1967)

$\sigma_{\rm{r}}(E)=\frac{\pi\omega}{\kappa^{2}}\frac{\Lambda_{i}(E)\Lambda_{f}(E)}{(E-E_{\rm{r}}^{2})+\frac{\Lambda^{2}_{\rm{total}}(E)}{4}},$
(31)

where $\kappa$ is the wave number, the entrance and exit channel partial
widths are $\Lambda_{i}(E)$ and $\Lambda_{f}(E)$ , respectively.
$\Lambda_{\rm{total}}(E)$ is the total width, and the statistical factor,
$\omega$ is given by

$\omega=(1+\delta_{12})\frac{2J+1}{(2J_{1}+1)(2J_{2}+1)},$ (32)

where the spins of the interacting nuclei and the resonance are $J_{1}$, and
$J_{2}$, respectively, $\delta_{12}$ is the Kronecker symbol.

The partial widths is dependent on the energy, and can be written as(Lane et
al., 1958)

$\Lambda_{i,f}=2\vartheta_{i,f}^{2}\psi_{l}(E,a)=\Lambda_{i,f}\frac{\psi_{l}(E,a)}{\psi_{l}(E_{f},a)}.$
(33)

The penetration factor $\psi_{l}$ is associated with $l$ and $a$, which are
the relative angular momentum and the channel radius, respectively.
$a=1.4(A_{1}^{1/3}+A_{2}^{1/3})$ fm. $\Lambda_{i,f}$ is the partial energy
widths at the resonance process. $E_{\rm{r}}$ and $\vartheta_{i,f}^{2}$ is the
reduced widths, given by

$\vartheta_{i,f}^{2}=0.01\vartheta_{\rm{w}}^{2}=\frac{0.03\hbar^{2}}{2Aa^{2}}.$
(34)

Based on the above analysis, in the phases of explosive stellar burning, the
narrow resonance reaction rates without SES are determined by (Schatz et al.,
1998; Herndl et al., 1998)

$\displaystyle\lambda_{\rm{r}}^{0}=N_{\rm{A}}\langle{\sigma
v}\rangle_{\rm{r}}=1.54\times 10^{11}(AT_{9})^{-3/2}$
$\displaystyle\times\sum_{i}\omega\gamma_{i}\exp(-11.605E_{r_{i}}/T_{9})~{}~{}~{}\rm{cm^{3}mol^{-1}s^{-1}},$
(35)

where $N_{\rm{A}}$ is Avogadro’s constant, $A$ is the reduced mass of the two
collision partners, $E_{r_{i}}$ is the resonance energies and $T_{9}$ is the
temperature in unit of $10^{9}$ K. The $\omega\gamma_{i}$ is the strength of
resonance in units of MeV and given by

$\omega\gamma_{i}=(1+\delta_{12})\frac{2J+1}{(2J_{1}+1)(2J_{2}+1)}\frac{\Lambda_{i}\Lambda_{f}}{\Lambda_{\rm{total}}}.$
(36)

On the other hand, due to SES the reaction rates of narrow resonance is given
by

$\displaystyle\lambda_{\rm{r}}^{s}$ $\displaystyle=$ $\displaystyle
F_{\rm{r}}N_{\rm{A}}\langle{\sigma v}\rangle_{\rm{{r^{\prime}}}}$ (37)
$\displaystyle=$ $\displaystyle 1.54\times
10^{11}(AT_{9})^{-3/2}\sum_{i}\omega\gamma_{i}\exp(-11.605E^{{}^{\prime}}_{r_{i}}/T_{9})$
$\displaystyle=$ $\displaystyle 1.54\times 10^{11}F_{\rm{r}}(AT_{9})^{-3/2}$
$\displaystyle\times\sum_{i}\omega\gamma_{i}\exp(-11.605E_{r_{i}}/T_{9})~{}\rm{cm^{3}mol^{-1}s^{-1}},$

where $F_{\rm{r}}$ is the screening enhancement factor (hereafter SEF). The
values of $E^{{}^{\prime}}_{r_{i}}$ should be measured by experiment, but it
is too hard to provide sufficient data. In general and approximate analysis,
we have $E^{{}^{\prime}}_{r_{i}}=E_{r_{i}}-U_{0}=E_{r_{i}}-U_{\rm{s}}$.

###  3.2 The screening model of resonant reaction rates in the case without
SMF

####  3.2.1 Dewitt model

Dewitt et al. (1976) discussed the problem of thermonuclear ion-electron
screening at some densities. Based on a statistical mechanical theory for the
screening function, the influence of the electron screening on the nuclear
reaction process also was investigated in their paper. The strong electron
screening potential function is given by (Dewitt et al., 1976)

$\displaystyle H_{12}^{\rm{sc}}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{r_{\rm{e}}kT}\\{0.9(\overline{z})^{1/3}(z_{12}^{5/3}-z_{1}^{5/3}-z_{2}^{5/3})$
(38)
$\displaystyle+c_{1}(\overline{z})^{2/3}(z_{12}^{4/3}-z_{1}^{4/3}-z_{2}^{4/3})\\}$
$\displaystyle+[c_{2}(\overline{z})^{-2/3}(z_{12}^{2/3}-z_{1}^{2/3}-z_{2}^{2/3})],$

where $c_{1}=0.2843$ and $c_{2}=0.4600$, and the $\overline{z}$, the average
charge of ionic, is given by

$\overline{z}=\sum_{i}z_{i}f_{i}=\sum_{\rm{i}}z_{i}\frac{n_{i}}{n_{\rm{I}}},$
(39)

where $n_{i}$ and $n_{\rm{I}}$ are the ion densities of nuclear species $i$
and I of the total system, respectively.

The screening enhancement factor (hereafter SEF) in Dewitt model is written as

$F_{r}^{0}(\rm{Dew})=\exp(H_{12}^{\rm{sc}}).$ (40)

####  3.2.2 Liolios model

At astrophysical energies the electron-screening acceleration in laboratory
fusion reactions always play a key role and is an interesting problem for
astrophysics. Based on a mean-field model, Liolios et al. (2000) studied the
screened nuclear reactions at astrophysical energies. The electron screening
potential in Liolios screened Coulomb model is given as (Liolios et al., 2000)

$U_{0}^{\rm{Lios}}=\frac{15}{8}\frac{z_{1}z_{2}e^{2}}{\Xi},$ (41)

where

$\Xi=(\frac{15}{8\pi
z_{i}^{2}})^{1/3}a_{0}=0.8853a_{0}(z_{1}^{2/3}+z_{2}^{2/3})^{1/2},$ (42)

The SEF for the resonant reaction in Liolios model is

$F_{\rm{r}}^{0}(\rm{Lios})=\exp(\frac{11.605U_{0}^{Lios}}{T_{9}}).$ (43)

###  3.3 The screening model of resonant reaction rates in SMFs

In this Subsection, we will discuss the screening potential in the strong
screening limit. The dimensionless parameter $(\Gamma)$, which determines
whether or not correlations between two species of nuclei $(z_{1},z_{2})$ are
important, is given by

$\Gamma=\frac{z_{1}z_{2}e^{2}}{(z_{1}^{1/3}+z_{2}^{1/3})r_{\rm{e}}kT},$ (44)

Under the conditions of $\Gamma\gg 1$, the nuclear reaction rates will be
influenced appreciably by SES. According to the above three SES models (LD,
FGP, LJ) in SMFs, the three enhancement factors for resonant reaction process
in SMFs can be expressed as follows

$F_{\rm{r}}^{\rm{B}}(\rm{LD})=\exp(\frac{11.605U_{\rm{s}}^{\rm{LD}}}{T_{9}}),$
(45)
$F_{\rm{r}}^{\rm{B}}(\rm{FGP})=\exp(\frac{11.605U_{\rm{s}}^{\rm{FGP}}}{T_{9}}),$
(46)
$F_{\rm{r}}^{\rm{B}}(\rm{LJ})=\exp(\frac{11.605U_{\rm{s}}^{\rm{LJ}}}{T_{9}}).$
(47)

##  4 Numerical results

###  4.1 Analysis of the results on a SEF

The strong magnetic fields modify significantly the properties of the matter
and always play a critical role in astronomical conditions. Figure 1 presents
the variations of ESP as a function of $B_{12}$ for our SES model. The SMF has
only a slight influence on ESP when $B_{12}>3\times 10^{3}$ and $\rho_{7}<1$.
But the ESP increases greatly when $B_{12}<1.4\times 10^{3}$ and $\rho_{7}<1$
($B_{12}$, $\rho_{7}$ are in units of $10^{12}$G, $10^{7}\rm{g~{}cm^{-3}}$,
respectively). Numerical results in our model show that the maximum value of
ESP reaches to $0.1$ MeV. Figure 2 (a) presents the ESP in LD model as a
function of $B_{12}$. The ESP increases rapidly and reaches the maximum value
of $0.008442$ MeV at $B_{12}=80$, then decreases with increasing of a SMF.

Based on the Thomas-Fermi and Thomas-Fermi-Dirac approximations, Fushiki et
al. (1989) analyzed the electron Fermi energy, electron Landau level, and SES
problem in a SMF. The results show that, as a consequence of the field
dependence of the screening potential, magnetic fields can significantly
increase nuclear reaction rates (Fushiki et al., 1989). According to electron
screening model of Ref.(Fushiki et al., 1989) (hereafter FGP model ) in a SMF,
Figure 2 (b) shows the ESP as a function of $B_{12}$ under some typical
astrophysical conditions. The ESP increases greatly when $B_{12}<10^{3}$ and
gets to the maximum value of $0.0188$ MeV at $B_{12}=580.7$ and
$\rho_{7}=0.1$. Then the ESP decreases around two orders of magnitude when
$10^{3}<B_{12}<2\times 10^{3}$ at $\rho_{7}=0.1$.

The influence of SES in a SMFs on nuclear reaction is mainly reflected by the
SEF. We discuss the influence of SES on SEF by three models (LD, FGP, LJ) from
Figure 3 to Figure 4. One finds that the SEF of LD model is a sensitive
parameter for a SMF and temperature. The maximum value of a SEF is about
$1.632$ for $B_{12}=78.17$ and $T_{9}=0.2$, as shown in Figure 3, where
$T_{9}$ is the temperature in units of $10^{9}$K. But for $B_{12}>219.3$ the
SEF is less than $1.001$. Figure 4 presents the SEF as function of $B_{12}$ of
FGP and LJ models. From sub-figures 4(a) and 4 (b), one find that the shifty
trend of SEF in FGP model is in good agreement with those of LD at low density
(e.g. $\rho_{7}=0.01$). The maximum value of SEF of FGP model is about $1.66$
for $B_{12}=84.18$ and $\rho_{7}=0.01$. On the contrary, the SEF increases
with increasing of $B_{12}$ at relatively high density (e.g. $\rho_{7}=1$),
then gets to the maximum value of $5.166$ at $T_{9}=0.2$. Sub-figures 4(c) and
4(d) show that in LJ model show that the SEF increases with increasing of
$B_{12}$, and the maximum value will reach up to $5.056$ for $B_{12}=1000$,
$T_{9}=0.2$ and $\rho_{7}=0.01$.

In Figure 5, some comparisons of the resonant SEF are shown among the models
of LJ, LD, and FGP for typical astronomical conditions in a SMF. The results
of LD model are well agreement with those of FGP for relatively low density
(e.g., $\rho_{7}\leqslant 0.01$). Nevertheless, the SEF of our model decreases
placidly with the increasing of $B_{12}$ and $T_{9}$ to compare with those of
LD and FGP.

The SES problem always plays important roles in stellar evolution process.
Based on a statistical mechanical theory for the screening function, Dewitt et
al. (1976) investigated the influence of the electron screening on nuclear
reaction. Based on a mean-field model, Liolios et al. (2000) also studied the
effect about screened nuclear reactions. However, they neglected the influence
of SMFs on SES. We compare the SEF of the two models (Dewitt, and Liolios
model) with those of LD, FGP, and LJ. One can conclude that the SEF of Dewitt
model is larger than those of other three SES models for $B_{12}<140$,
$\rho_{7}=0.01$ and $T_{9}<0.17$, shown as in Figure 6. However, when
$T_{9}<0.18$, $\rho_{7}=0.01$, the results of our model are larger than those
of Dewitt and Liolios. At a relatively high density (e.g., $\rho_{7}=0.1$),
the SEFs of LD, FGP and LJ models decrease due to SMFs and is lower than those
of Dewitt model. The results obtained by Dewitt et al. (1976) amount to an
overestimation of the screening effect because of their neglect of spatial
dependence of the screening function.

Table 1 shows some information of SEF for the five typical models at some
astronomical conditions. The results of LD, FGP, and LJ are always lower than
those of Liolios and Dewitt due to a SMF. The SEF of our model decreases very
greatly with increasing of density and temperature when $B_{12}=10^{3}$. It is
because that the ESP increases very rapidly as SMF increases. The higher the
ESP, the larger the influence on SES becomes. On the contrary, the SEF of LD
decreases with increasing of magnetic fields because ESP is reduced. The SEF
of FGP model gets to the maximum of $1.929$ when
$B_{12}=10^{3},\rho_{7}=1,T_{9}=0.5$ and then decreases slowly as the density
and temperature increase.

The Thomas-Fermi screening wave-number $K_{\rm{TF}}$ is a very key parameter,
which strongly depends on the electron number density and ESP. In consequence
the electron number density and ESP will play important roles in a SMF. Lai et
al. (1991), analyzed in detail the electron Fermi energy and electron number
density in a SMF based on the works of Canuto et al. (1968, 1971); Kubo.
(1965), and Pathria (2003). By using the uniform electron gas model and linear
response theory, Lai (2001)discussed the electron energy (per cell)
corrections due to non-uniformity in a SMF. According to their theory, we
study the ESP and the SES model (i.e., LD model). The results show that the
ESP decreases as the magnetic fields increase due to the diminution of
electron chemical potential. The LD model is valid only in the condition of
$K_{\rm{TF}}r_{i}\ll 1$ at lower densities, because they investigated the non-
uniformity effect only through detailed electronic (band) structure
calculations.

The electron chemical potential is a pivotal parameter, which is closely
related to the electron number density and exchange energy. Based on Thomas-
Fermi-Dirac approximation, it is given as (Fushiki et al., 1989)

$U_{\rm{F}}=U_{\rm{e}}=\frac{\partial w_{\rm{ex}}}{\partial
n_{\rm{e}}}=\frac{r_{\rm{cyc}}}{\pi a_{0}}\hbar\omega_{0}nI(n),$ (48)

where $w_{\rm{ex}}$ is the exchange energy and $I(n)$ can be found in Ref.
(Fushiki et al., 1989). By using the linear response theory, Fushiki et al.
(1989) discussed the exchange energy and electron chemical potential in the
lowest Landau level for non-uniformity electron gas in a SMF. They analyzed
the SES problem in a SMF and their results shown that a SMF only the lowest
Landau level is occupied by electrons on the condition of
$r_{\rm{e}}>(3\pi/8)^{1/3}r_{\rm{cyc}}$ or equivalently $\rho<7.04\times
10^{3}B_{12}^{3/2}$(A/z)$\rm{g/cm^{3}}$. The cyclotron radius in the lowest
Landau level orbital is give by $r_{\rm{cyc}}=(2\hbar c/eB)^{1/2}\simeq
3.36\times 10^{-10}B_{12}^{-1/2}$. FGP used the expression of
$n_{\rm{e}}\partial n_{\rm{e}}/\partial U_{\rm{F}}=(3/2)n_{\rm{e}}/U_{\rm{F}}$
in dealing with $\partial n_{\rm{e}}/\partial U_{\rm{F}}$. In FGP model, they
thought at high density the exchange correction is very small, thus they
neglected the exchange correction to $\partial n_{\rm{e}}/\partial U_{\rm{F}}$
and had $n_{\rm{e}}\partial n_{\rm{e}}/\partial
U_{\rm{F}}=(1/2)n_{\rm{e}}/U_{\rm{F}}$ in a SMF. Due to different ways of
dealing with exchange correction under this condition, the SEF of FGP model
has some difference compared with other SES models.

According to statistical physics the microscopic state number
$dxdydzdp_{x}dp_{y}dp_{z}$ can be given by $dxdydzdp_{x}dp_{y}dp_{z}/h^{3}$ in
a 6-dimension phase-space. The number of states occupied by completely
degenerate relativistic electrons per volume is calculated by (Canuto et al.,
1968, 1971)

$\displaystyle
N_{\rm{phase}}=\sum_{p_{x}}\sum_{p_{y}}\sum_{p_{z}}=\frac{1}{h^{3}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dp_{x}dp_{y}dp_{z}$
$\displaystyle=\frac{1}{h^{3}}\int_{0}^{p_{\rm
F}}dp_{z}\int_{0}^{\infty}p_{\bot}dp_{\bot}\int_{0}^{2\pi}d\theta=\frac{\pi
p_{\rm F}}{h^{3}}\int_{0}^{\infty}dp_{\bot}^{2},$ (49)

where $\theta~{}=~{}tan^{-1}p_{y}/p_{x}$, $p_{\bot}^{2}\rightarrow
m^{2}c^{4}\frac{B}{B_{\rm cr}}2n$, So,
$\int_{0}^{\infty}dp_{\bot}^{2}\rightarrow\sum_{n=0}^{\infty}\omega_{n}$, and
the $\omega_{n}$ is the degeneracy of the $n$-th electron Landau level in
relativistic magnetic field, and can be calculated by (Canuto et al., 1971;
Kubo., 1965; Pathria, 2003)

$\displaystyle\omega_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{h^{2}}\int_{0}^{2\pi}d\phi\int_{k_{1}<p_{\bot}^{2}<k_{2}}p_{\bot}dp_{\bot}=\frac{2\pi}{h^{2}}\frac{(k_{2}-k_{1})}{2}$
(50) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}(\frac{\hbar}{m_{e}c})^{-2}\frac{B}{B_{\rm
cr}}=\frac{b}{2\pi}(\frac{\hbar}{m_{e}c})^{-2}~{}~{},$

where $k_{1}=2nm_{e}^{2}c^{2}\frac{B}{B_{\rm cr}}=2nbm_{e}^{2}c^{2}$, and
$k_{2}=2(n+1)bm_{e}^{2}c^{2}$.

Based on the works of Peng et al. (2007); Gao et al. (2013), which introduced
the Dirac $\delta$-function and considered Pauli exclusion principle, we
discuss the SES problem in a SMF. Our results show that the stronger the
magnetic field, the higher Fermi energy of electrons becomes. The ESP
increases with SMF and the maximum value of ESP is 0.1 MeV in a SMF. The SEF
also increases greatly and its maximum approaches to 5.0 MeV (e.g.
$\rho_{7}=0.01,T_{9}=0.2,B_{12}=10^{3}$G).

###  4.2 Investigation of the nuclear reaction rates

In the explosive hydrogen burning stellar environments, the nuclear reaction
23Mg$(p,\gamma)$24Al plays a key role because of breaking out the Ne-Na cycle
to heavy nuclear species (i.e., Mg-Al cycle). Therefore, it is very important
to accurate determinate the rates for the reaction 23Mg$(p,\gamma)$24Al.
However, the resonance energy has a large uncertainty due to the inconsistent
24Mg(3He,t)24Al measurements mentioned. So it may lead to a factor of 5
variation in the reaction rate at $T_{9}=0.25$ because of its exponential
dependence on $E_{\rm{r}}$ (Visser et al., 2007). Some authors discussed the
contributions from several important resonance states, such as (Wallace et
al., 1981; Wiescher et al., 1986; Kubono et al., 1995; Visser et al., 2007).
In order to reduce the uncertainty of the reaction rates in this paper, we
reference some information about this reaction and the values of the
$E_{\rm{{r_{i}}}},E_{\rm{x}}$ and corresponding to $\omega\gamma_{i}$ and some
average values of $\omega\gamma_{i}$ are adopted and listed in Table 2.
According to these information, we analysis the total rates for these five SES
models.

Table 1: The comparisons of the resonant SEFs for Dewitt, Liolios, LD, FGP and LJ models in several typical astronnomical conditions. The former two models are in the case without SES and SMFs, while the latter three models are in the case with SES and SMFs. |  |  |  | $B_{12}=10$ |  | $B_{12}=10^{3}$
---|---|---|---|---|---|---
$\rho_{7}$ | $T_{9}$ |  $F_{\rm{r}}^{0}$(Lios) |  $F_{\rm{r}}^{0}$(Dew) |  $F_{\rm{r}}^{\rm{B}}$(LD) |  $F_{\rm{r}}^{\rm{B}}$(FGP) |  $F_{\rm{r}}^{\rm{B}}$(LJ) |  |  $F_{\rm{r}}^{\rm{B}}$(LD) |  $F_{\rm{r}}^{\rm{B}}$(FGP) |  $F_{\rm{r}}^{\rm{B}}$(LJ)
0.01 | 0.1 | 1.7475 | 3.8973 | 1.6956 | 1.6964 | 0.1749 |  | 1.0725e-15 | 1.3472e-13 | 25.5680
0.05 | 0.1 | 1.7475 | 10.9451 | 1.6956 | 1.7013 | 8.4513e-4 |  | 1.0725e-15 | 0.6045 | 19.5717
0.1 | 0.2 | 1.3219 | 4.3605 | 1.3021 | 1.3045 | 0.0022 |  | 3.2750e-8 | 2.4894 | 3.8848
0.1 | 0.3 | 1.2051 | 2.5873 | 1.1922 | 1.1941 | 0.0174 |  | 1.0221e-5 | 1.8371 | 2.4713
0.2 | 0.3 | 1.2045 | 3.3934 | 1.1924 | 1.1939 | 9.1604e-4 |  | 1.0236e-5 | 2.4990 | 2.1361
0.3 | 0.4 | 1.1497 | 2.8170 | 1.1411 | 1.1422 | 7.7822e-4 |  | 1.8097e-4 | 2.1178 | 1.6055
1.0 | 0.5 | 1.1181 | 3.5124 | 1.1113 | 1.1122 | 6.2630e-7 |  | 0.0011 | 1.9290 | 0.9512
10 | 0.7 | 1.0830 | 7.2692 | 1.0780 | 1.0791 | 6.2012e-9 |  | 0.0072 | 1.6142 | 0.0894
Table 2: Resonance parameters for the reaction 23Mg $(p,\gamma)$ 24 Al. $E_{\rm{x}}$ (MeV) 111is adopted from Ref. (Endt, 1998) |  $E_{\rm{x}}$ (MeV) 222from Ref.(Visser et al., 2007) | $J^{\pi}$ |  $E_{\rm{{r_{i}}}}$ (MeV) 333 from Ref.(Audi et al., 1995) | $\Gamma_{\rm{p}}$ | $\Gamma_{\gamma}$ |  $\omega\gamma_{i}$(meV) 444 from Ref.(Herndl et al., 1998) |  $\omega\gamma_{i}$(meV) 555from Ref.(Wiescher et al., 1986) |  $\omega\gamma_{i}$(meV) 666is adopted in this paper
---|---|---|---|---|---|---|---|---
2.349$\pm$0.020 |  2.346$\pm$0.000 | $3^{+}$ | 0.478 | 185 | 33 | 25 | 27 | 26
2.534$\pm$0.013 |  2.524$\pm$0.002 | $4^{+}$ | 0.663 | 2.5e3 | 53 | 58 | 130 | 94
2.810$\pm$0.020 |  2.792$\pm$0.004 | $2^{+}$ | 0.939 | 9.5e5 | 83 | 52 | 11 | 31.5
2.900$\pm$0.020 |  2.874$\pm$0.002 | $3^{+}$ | 1.029 | 3.4e4 | 14 | 12 | 16 | 14
Table 3: Comparisons of the rates of $\lambda^{0}_{r}$, which are in the case without SES with those of the LD ($\lambda^{\rm{scB}}_{\rm{r}}(\rm{LD})$), FGP ($\lambda^{\rm{scB}}_{\rm{r}}(\rm{FGP})$) and our calculations $\lambda^{\rm{scB}}_{\rm{r}}(\rm{LJ})$ in the case with SES for some typical astronomical conditions at $B_{12}=10$, respectively. $S_{i}=\lambda_{\rm{r}{i}}^{\rm{scB}}/\lambda_{\rm{r}}^{0}$, $i=1,2,3$ denote the rates of LD, FGP, and LJ model, respectively. $B_{12}=10$ |  |  |  |
---|---|---|---|---
$\rho_{7}$ | $T_{9}$ | $\lambda^{0}_{\rm{r}}$ | $\lambda^{\rm{scB}}_{\rm{r}}(\rm{LD})$ | $\lambda^{\rm{scB}}_{\rm{r}}(\rm{FGP})$ | $\lambda^{\rm{scB}}_{\rm{r}}(\rm{LJ})$ | $S_{1}$ | $S_{2}$ | $S_{3}$
0.01 | 0.1 | 1.0942e-19 | 1.8552e-19 | 1.8561e-19 | 1.9138e-20 | 1.6956 | 1.6964 | 0.1749
0.02 | 0.1 | 1.0942e-19 | 1.8552e-19 | 1.8598e-19 | 4.0569e-21 | 1.6956 | 1.6998 | 0.0371
0.03 | 0.1 | 1.0942e-19 | 1.8552e-19 | 1.8608e-19 | 1.0413e-21 | 1.6956 | 1.7007 | 0.0095
0.03 | 0.2 | 4.2967e-8 | 5.5949e-8 | 5.6034e-8 | 4.1916e-9 | 1.3021 | 1.3041 | 0.0976
0.04 | 0.2 | 4.2967e-8 | 5.5949e-8 | 5.6041e-8 | 2.2448e-9 | 1.3021 | 1.3043 | 0.0522
0.05 | 0.2 | 4.2967e-8 | 5.5949e-8 | 5.6044e-8 | 1.2491e-9 | 1.3021 | 1.3043 | 0.0291
0.1 | 0.2 | 4.2967e-8 | 5.5949e-8 | 5.6051e-8 | 9.3383e-11 | 1.3021 | 1.3045 | 0.0022
0.2 | 0.4 | 0.0163 | 0.0186 | 0.0186 | 8.5713e-5 | 1.1411 | 1.1422 | 0.0053
0.3 | 0.5 | 0.1925 | 0.2140 | 0.2141 | 6.2713e-4 | 1.1114 | 1.1122 | 0.0033
0.5 | 0.6 | 0.9764 | 1.0663 | 1.0669 | 8.5404e-4 | 1.0920 | 1.0927 | 8.7465e-4
0.7 | 0.8 | 7.2550 | 7.7500 | 7.7537 | 0.0079 | 1.0682 | 1.0687 | 0.0011
1.0 | 0.9 | 14.0604 | 14.9100 | 14.9162 | 0.0050 | 1.0604 | 1.0609 | 3.5785e-4
Table 4: Comparisons of the rates of $\lambda^{0}_{\rm{r}}$, which are in the case without SES and SMFs with those of the LD ($\lambda^{\rm{scB}}_{\rm{r}}(\rm{LD})$), FGP ($\lambda^{\rm{scB}}_{\rm{r}}(\rm{FGP})$) and our calculations $\lambda^{\rm{scB}}_{\rm{r}}(\rm{LJ})$ in the case with SES for some typical astronomical conditions at $B_{12}=10^{3}$, respectively. The $S_{i}$ is the same as in Table 3. $B_{12}=10^{3}$ |  |  |  |
---|---|---|---|---
$\rho_{7}$ | $T_{9}$ | $\lambda^{0}_{\rm{r}}$ | $\lambda^{\rm{scB}}_{\rm{r}}(\rm{LD})$ | $\lambda^{\rm{scB}}_{\rm{r}}(\rm{FGP})$ | $\lambda^{\rm{scB}}_{\rm{r}}(\rm{LJ})$ | $S_{1}$ | $S_{2}$ | $S_{3}$
0.01 | 0.1 | 1.0942e-19 | 1.1735e-34 | 1.4740e-32 | 2.7975e-18 | 1.0725e-15 | 1.3472e-13 | 25.5680
0.02 | 0.1 | 1.0942e-19 | 1.1735e-34 | 6.4612e-24 | 2.5883e-18 | 1.0725e-15 | 5.9052e-5 | 23.6555
0.03 | 0.1 | 1.0942e-19 | 1.1735e-34 | 1.5306e-21 | 2.4177e-18 | 1.0725e-15 | 0.0140 | 22.0969
0.03 | 0.2 | 4.2967e-8 | 1.4072e-15 | 5.0819e-9 | 2.0198e-7 | 3.2750e-8 | 0.1183 | 4.7007
0.04 | 0.2 | 4.2967e-8 | 1.4072e-15 | 1.7120e-8 | 1.9575e-7 | 3.2750e-8 | 0.3984 | 4.5559
0.05 | 0.2 | 4.2967e-8 | 1.4072e-15 | 3.3408e-8 | 1.9009e-7 | 3.2750e-8 | 0.7775 | 4.4240
0.1 | 0.2 | 4.2967e-8 | 1.4072e-15 | 1.0696e-7 | 1.6692e-7 | 3.2750e-8 | 2.4894 | 3.8848
0.2 | 0.4 | 0.0163 | 2.9459e-6 | 0.0324 | 0.0288 | 1.8097e-4 | 1.9876 | 1.7669
0.3 | 0.5 | 0.1925 | 1.9523e-4 | 0.3509 | 0.2812 | 0.0010 | 1.8227 | 1.4604
0.5 | 0.6 | 0.9764 | 0.0031 | 1.6579 | 1.1949 | 0.0032 | 1.6979 | 1.2237
0.7 | 0.8 | 7.2550 | 0.0976 | 10.8789 | 7.8159 | 0.0135 | 1.4995 | 1.0773
1.0 | 0.9 | 14.0604 | 0.3053 | 20.2526 | 13.6742 | 0.0217 | 1.4404 | 0.9725

Tables 3 and 4 give a brief description of the factor $S_{i}$ $(i=1,2,3)$ for
LD, FGP, and LJ models when $B_{12}=10,10^{3}$, respectively. As the density
and temperature increase, the results of LD model are in good agreement with
those of FGP, but disagreement with our results at $B_{12}=10$. This is
because that the electron Fermi energy of our model is lower than those of LD
and FGP in relatively low magnetic fields. As the magnetic fields increase
from $B_{12}=10$ to $10^{3}$, the factor $S_{3}$ increases about $2\sim 3$
orders magnitude (i.e., from 0.1749 to 25.5680 and from 0.0022 to 3.8848) when
$\rho_{7}=0.01,T_{9}=0.1$ and $\rho_{7}=0.1,T_{9}=0.2$, respectively. When
$B_{12}=10^{3}$ the factor $S_{3}$ is about 39.74, 5.69, 1.56 times larger
than $S_{2}$ (FGP model) at $\rho_{7}=0.03,T_{9}=0.2$,
$\rho_{7}=0.05,T_{9}=0.2$ and $\rho_{7}=0.1,T_{9}=0.2$, respectively. From
what has been discussed above, the LD model maybe only adapts to the
relatively low magnetic field and low density surroundings. The FGP and LD
models are both unadapted to relatively low density, and high magnetic field
surroundings (e.g. $\rho_{7}<0.1,B_{12}>10^{2}$). However, our model can be
well adapted to relatively high magnetic field and low density surroundings
(e.g. $B_{12}>10^{2},\rho_{7}<0.05$).

Summing up the above discussions, our calculations show that this SES effect
in a SMF can increase nuclear reaction rates of 23Mg $(p,\gamma)$24Al by
several orders magnitude. A more precise thermonuclear rates of 23Mg
$(p,\gamma)$24Al will help us to constrain the determination of nuclear flow
out of the Ne-Na cycle, and production of $A\geq 20$ nuclides, in explosive
hydrogen burning over a temperature range of $0.2\leq T\leq 1.0$ GK.

##  5 Conclusions

In this paper, based on the relativistic theory in a SMF, we investigate the
problem of SES, and the SES influence on the nuclear reaction of 23Mg
$(p,\gamma)$24Al by LD, FGP, and LJ strong screening models in a SMF. The
results show that the SES thermonuclear reaction rates have a remarkable
increase in a SMF. The rates can increase by around three orders of magnitude.
For example, when $B_{12}$ increases from 10 to $10^{3}$, the rates increase
from 0.1749 to 25.5680 at $\rho_{7}=0.01,T_{9}=0.1$, and from 0.0022 to 3.8848
at $\rho_{7}=0.1,T_{9}=0.2$. The considerable increase in the reaction rates
for 23Mg $(p,\gamma)$ 24Al implies that more 23Mg will escape the Ne-Na cycle
due to SES in a SMF. Then it will make the next reaction convert more 24Al
$(\beta^{+},\nu)$ 24Mg to participate in the Mg-Al cycle. It may lead to
synthesizing a large amount of heavy elements at the crust of magnetars. These
heavy elements, which are produced from the nucleosynthesis process, may be
thrown out due to the compact binary mergers of double neutron star (NS-NS) or
black hole and neutron star (BH and NS) systems. On the other hand, our model
for the rates is in good agreement with those of LD and FGP models at
relatively low density (e.g., $\rho_{7}<0.01$) and $B_{12}<10^{2}$. In
relatively low magnetic fields (e.g., $B_{12}<1$), the SES of LD and FGP
models have strong influence on the rates compare to our model. However, the
rates in our model can be about 1.58 times and three orders magnitude higher
than those of FGP and LD in relatively high magnetic fields and low density
surroundings (e.g., $B_{12}\geq 10^{2}$, $\rho_{7}<0.05$), respectively. The
results we derived, may have very important implications in some astrophysical
applications for the nuclear reaction, the thermal evolution, and numerical
simulation of magnetars.

We would like to thank the anonymous referee for carefully reading the
manuscript and providing some constructive suggestions which are very helpful
to improve this manuscript. This work was supported in part by the National
Natural Science Foundation of China under grants 11565020, and the Counterpart
Foundation of Sanya under grant 2016PT43, the Special Foundation of Science
and Technology Cooperation for Advanced Academy and Regional of Sanya under
grant 2016YD28, the Scientific Research Starting Foundation for 515 Talented
Project of Hainan Tropical Ocean University under grant RHDRC201701, and the
Natural Science Foundation of Hainan Province under grant 114012.

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Figure 1: The electron screening potential as a function of $B_{12}$ of LJ
model for some typical astronomical condition.

Figure 2: The electron screening potential as a function of $B_{12}$ in LD,
and FGP models for some typical astronomical condition. Figure 3: The resonant
SEF for LD model as a function of $B_{12}$ in the case with SES and SMF.

Figure 4: The resonant SEF for FGP and LJ models as a function of $B_{12}$ in
the case with SES and SMF.

Figure 5: The comparisons are plotted for some typical astronomical condition
of the resonant SEF among the three models of LJ, LD, and FGP in the case with
SES and SMF.

Figure 6: The comparisons of the resonant SEF for the model of Liolios, Dewitt
with those of models of LD, FGP, and LJ.