Formation of superconducting pair correlations
in spherical even-even nuclei

V. A. Kuz’mina) and T. V. Teterevab)

a) Bogoliubov Laboratory of Theoretical Physics, Joint Institute
for Nuclear Research, Dubna, Moscow region, 141980, Russia
b) Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow
State University, Moscow, 119992, Russia

###### Abstract

The appearance of like nucleon pair correlations in the ground state of
spherical even-even nuclei is considered within a special Bogoliubov
transformation. It is confirmed that in closed subshell nuclei superconducting
pair correlations start to form if the coupling constant $G$ exceeds a certain
threshold value. Rough upper and lower estimates are obtained for the
threshold value. It is shown, that superconducting correlations exist in open
subshell nuclei at any positive $G$. In this case, nucleon pairs are
distributed over all subshells participating in the pairing interaction.

## Introduction

It was stressed in one of the very first papers [1] on the application of the
idea of superconducting correlations to the atomic nuclei spectroscopy that a
pair correlations in nuclei can appear only if the coupling constant exceeds a
certain value. It was also mentioned that the existence of such a threshold
value distinguishes atomic nuclei from the infinite systems considered in the
theory of superconductivity. This remark was later repeated several times [2,
3].

The necessity of a threshold was revealed by the following reasoning [1, 2].
In the simplest case when all the matrix elements of the two-body pairing
interactions are replaced by the same positive constant $G$, the equations for
the correlation function $C$ and the chemical potential $\lambda$ can be
written as

$\displaystyle\frac{G}{2}\sum_{s}\frac{1}{\sqrt{(E_{s}-\lambda)^{2}+C^{2}}}=1;$
(1) $\displaystyle 2\sum_{s}v_{s}^{2}=\mathcal{N},\qquad
v_{s}^{2}=\frac{1}{2}\left[1-\frac{E_{s}-\lambda}{\sqrt{(E_{s}-\lambda)^{2}+C^{2}}}\right].$
(2)

Here $\mathcal{N}$ is the number of particles in the system. The summation is
carried out over pairs of doubly-degenerate single particle states in the
nuclear mean field. These states are related to each other by the operation of
time reversal [2]. The inequality

$\frac{G}{2}\sum_{s}\frac{1}{\left|E_{s}-\lambda\right|}>1,$ (3)

follows from Eq. (1) for non-zero $C$. Inequality (3) is often considered as a
relation that determines the minimal value of $G$, starting from which
superconducting pair correlations can exist in the system. The reasoning
contains a tacit assumption that $\lambda$ can not approach one of the
$E_{s}$’s close enough to increase considerably the sum
$\displaystyle\sum_{s}1/\left|E_{s}-\lambda\right|$ as $G$ decreases. The
correctness of the assumption is not evident because both $\lambda$ and $C$
are calculated by solving the nonlinear system of Eqs. (1) and (2). Therefore,
inequality (3) is a useful hint rather than a proof.

In this paper, we study the conditions of the formation of superconducting
pair correlations between like nucleons in the ground state of even-even
spherical nuclei. Since Eqs. (1) and (2) for many-level systems can only be
solved numerically, we first consider the solution of equations with one and
two subshells and then use the obtained results in the discussion of the many-
subshell problem.

The paper contains the introduction, three sections and the conclusion. The
equations for calculating normal and abnormal single-particle densities are
derived in the first section. The contribution of pairing interaction to the
single-particle energy is taken into account explicitly. The appearance of
pair correlations in single subshell and two-subshell systems is discussed in
the second section. The threshold value of $G$ is calculated for the system
with closed subshell, too. Many-subshell nuclei are discussed in the third
section. The main results of the paper are summed up in the conclusion.

##  1 Model Hamiltonian and main equations

The Hamiltonian of the neutrons (or protons) in the nucleus with spherical
symmetry is chosen [2] as

$H=\sum_{k}\sum_{m_{k}=-j_{k}}^{j_{k}}E_{k}a^{{\dagger}}_{k,m_{k}}a_{k,m_{k}}-\\\
-\frac{G}{4}\sum_{k,l}\sum_{m_{k}=-j_{k}}^{j_{k}}(-1)^{j_{k}-m_{k}}a^{{\dagger}}_{k,m_{k}}a^{{\dagger}}_{k,-m_{k}}\sum_{m_{l}=-j_{l}}^{j_{l}}(-1)^{j_{l}-m_{l}}a_{l,-m_{l}}a_{l,m_{l}}.$
(4)

Here the indices $k$ label single-particle states with energy $E_{k}$ and
angular momentum $j_{k}$; $m_{k}$ is the third projection of angular momentum,
$m_{k}=-j_{k},-j_{k}+1,\dots j_{k}-1,j_{k}$; the operators
$a^{{\dagger}}_{k,m_{k}}$ and $a_{k,m_{k}}$ are fermion operators of the
creation and annihilation of a particle in the state $(k,m_{k})$. We follow
the tradition and call the subshell a set of $2j_{k}+1$ single-particle states
with the same $E_{k}$ and $j_{k}$. We consider the attractive interaction,
i.e. $G>0$.

The quasiparticle operators are introduced by the special Bogoliubov
transformation with the real coefficients [2]

$a_{k,m_{k}}=u_{k}\alpha_{k,m_{k}}+(-1)^{j_{k}-m_{k}}v_{k}\alpha^{{\dagger}}_{k,-m_{k}}.$
(5)

If the operators $\alpha_{k,m_{k}}$ and $\alpha^{{\dagger}}_{k,m_{k}}$ are the
fermion annihilation and creation operators, the transformation coefficients
satisfy the conditions

$u^{2}_{k}+v^{2}_{k}=1.$ (6)

The quasiparticle vacuum $|\rangle$ is determined by the equations
$\alpha_{k,m_{k}}|\rangle=0$ for any $k$ and $m_{k}$. If the quasiparticle
operators are introduced by Eq. (5), the angular momentum of $|\rangle$ is
zero and its parity is positive. This is easy to prove by applying the
operators of the total angular momentum of the system of like nucleons and the
parity operator on $|\rangle$. As a result, the product of neutron and proton
vacuum states has zero angular momentum and positive parity, the values
coincide with the angular momentum and parity of ground states of all even-
even spherical nuclei. For this reason, we approximate the wave function of
the ground state of a system containing an even number $\mathcal{N}$ of
identical nucleons (either neutrons or protons) by an appropriate
quasiparticle vacuum $|\rangle$.

The quasiparticle vacuum is connected with the state without nucleons [2]:

$|\rangle=\prod_{k,m_{k}>0}\left(u_{k}+(-1)^{j_{k}-m_{k}}v_{k}a^{{\dagger}}_{k,m_{k}}a^{{\dagger}}_{k,-m_{k}}\right)|0\rangle,$
(7)

where $|0\rangle$ is the state without nucleons, $a_{k,m_{k}}|0\rangle=0$ for
any $k$ and $m_{k}$. If for a certain $k$ the coefficients $u_{k}$ and $v_{k}$
differ simultaneously from zero ($u_{k}v_{k}\neq 0$), the function (7)
contains components with a different number of nucleon pairs.

The coefficients of transformation (5) can be determined from the condition of
minimum of the system energy $\langle|H|\rangle$. In searching for an energy
extremum, one should satisfy the constraint that the average of the nucleon
number operator over quasiparticle vacuum is equal to the number of particles
in the system,

$\langle|N|\rangle=\langle|\sum_{k,m_{k}}a^{{\dagger}}_{k,m_{k}}a_{k,m_{k}}|\rangle=\mathcal{N}.$

Therefore, the functional to be minimized is

$\langle|H|\rangle-\lambda\left[\langle|\sum_{k,m_{k}}a^{{\dagger}}_{k,m_{k}}a_{k,m_{k}}|\rangle-\mathcal{N}\right].$

Here $\lambda$ is the Lagrange multiplier usually called the chemical
potential. One can see that the inclusion of a supplementary condition
modifies the operator $H$ into

$H^{\prime}=H-\lambda\sum_{k,m_{k}}a^{{\dagger}}_{k,m_{k}}a_{k,m_{k}}.$

The average of $H^{\prime}$ over the quasiparticle vacuum is equal to

$\langle|H^{\prime}|\rangle=\sum_{k}(2j_{k}+1)(E_{k}-\lambda)v^{2}_{k}-\frac{G}{2}\sum_{k}(2j_{k}+1)v^{4}_{k}-\frac{G}{4}\left[\sum_{k}(2j_{k}+1)u_{k}v_{k}\right]^{2}\\!\\!.$
(8)

The transformation coefficients $u_{k}$ and $v_{k}$ enter into the expression
as the products $v^{2}_{k}$ and $u_{k}v_{k}$, only. We use the products as new
variables

$w_{k}=v^{2}_{k}\quad\mathrm{and}\quad t_{k}=u_{k}v_{k}\,.$ (9)

Sometimes they are called the normal density and abnormal density because

$w_{k}=\langle|a^{{\dagger}}_{k,m_{k}}a_{k,m_{k}}|\rangle\quad\mathrm{and}\quad
t_{k}=(-1)^{j_{k}-m_{k}}\langle|a_{k,-m_{k}}a_{k,m_{k}}|\rangle.$

The presence of non-zero $\langle|a_{k,-m_{k}}a_{k,m_{k}}|\rangle$ is a sign
of a superconducting state. It follows from Eq. (6) that the densities obey to
the inequalities

$0\leqslant w_{k}\leqslant 1\quad\mathrm{and}\quad 0\leqslant
t_{k}\leqslant\frac{1}{2}$

and are connected by the relations

$w^{2}_{k}+t^{2}_{k}=w_{k}\,.$ (10)

The matrix element (8) is expressed via $w_{k}$ and $t_{k}$ as

$\langle|H^{\prime}|\rangle=\sum_{k}(2j_{k}+1)(E_{k}-\lambda)w_{k}-\frac{G}{2}\sum_{k}(2j_{k}+1)w^{2}_{k}-\frac{G}{4}\left[\sum_{k}(2j_{k}+1)t_{k}\right]^{2}.$
(11)

The supplementary condition for the particle number is

$\sum_{k}(2j_{k}+1)w_{k}=\mathcal{N}.$ (12)

For convenience of further calculations, we write $w_{k}$ as a sum

$w_{k}=\xi_{k}+\frac{1}{2},\qquad\left(-1/2\leqslant\xi_{k}\leqslant
1/2\right).$ (13)

Now relations (10) are simplified

$\xi^{2}_{k}+t^{2}_{k}=\frac{1}{4}$ (14)

and can be satisfied by the trigonometric functions

$\xi_{k}=\frac{1}{2}\cos\varphi_{k}\,,\quad
t_{k}=\frac{1}{2}\sin\varphi_{k}\,,\qquad\mathrm{with}\ \
0\leqslant\varphi_{k}\leqslant\pi.$

Please note that $t_{k}>0$ with $0<\varphi_{k}<\pi$. The matrix element (8)
depends on the unknown $\varphi_{k}$ only,

$\langle|H^{\prime}|\rangle=\sum_{k}\bigl{(}j_{k}+\frac{1}{2}\bigr{)}\bigl{(}E_{k}-\lambda-\frac{G}{4}\bigr{)}+\sum_{k}\bigl{(}j_{k}+\frac{1}{2}\bigr{)}\bigl{(}E_{k}-\frac{G}{2}-\lambda\bigr{)}\cos\varphi_{k}-\\\
-\frac{G}{4}\sum_{k}\bigl{(}j_{k}+\frac{1}{2}\bigr{)}\cos^{2}\varphi_{k}-\frac{G}{4}\left[\sum_{k}\bigl{(}j_{k}+\frac{1}{2}\bigr{)}\sin\varphi_{k}\right]^{2}.$
(15)

For any $\varphi_{k}$ from the interval $0<\varphi_{k}<\pi$ the condition of
extremum $\langle|H^{\prime}|\rangle$ is

$\frac{\partial\langle|H^{\prime}|\rangle}{\partial\varphi_{k}}=0\,.$ (16)

When $\varphi_{k}$ approaches $0$ or $\pi$, the usual derivatives should be
replaced by the one-sided ones. The simplest way to arrange the limiting
processes is to map the entire number axis to the segment
$0\leqslant\varphi_{k}\leqslant\pi$ by
$\varphi_{k}=\mathop{\mathrm{arccot}}\nolimits x_{k}$ with
$\infty>x_{k}>-\infty$. Afterwards, Eq. (16) transforms into

$\frac{\partial\langle|H^{\prime}|\rangle}{\partial
x_{k}}=\frac{\partial\langle|H^{\prime}|\rangle}{\partial\varphi_{k}}\frac{d\varphi_{k}}{dx_{k}}=\frac{\partial\langle|H^{\prime}|\rangle}{\partial\varphi_{k}}\left(-\frac{1}{1+x^{2}_{k}}\right)=0\,.$
(17)

Three stationary points are possible for each $k$:

$\displaystyle x_{k}=\infty,\quad\varphi_{k}=0,\quad\xi_{k}=\frac{1}{2},\
w_{k}=1,\ t_{k}=0;$ (18) $\displaystyle
x_{k}=-\infty,\quad\varphi_{k}=\pi,\quad\xi_{k}=-\frac{1}{2},\ w_{k}=0,\
t_{k}=0;$ (19) $\displaystyle x_{k}\
\mathrm{finite},\quad\frac{\partial\langle|H^{\prime}|\rangle}{\partial\varphi_{k}}=0,\quad
0<\varphi_{k}<\pi,\quad t_{k}>0,\quad 0<w_{k}<1.$ (20)

Stationary points (18) and (19) describe the states with normal density at
subshell $k$ strictly equal to either $1$ or $0$ and abnormal density equal to
zero (normal solutions). Equation (20) describes the states with $t_{k}>0$ and
$0<w_{k}<1$ (superconducting solutions). Equation (20) can be written as

$\bigl{(}E_{k}-\frac{G}{2}-\lambda\bigr{)}\sin\varphi_{k}-\frac{G}{2}\cos\varphi_{k}\sin\varphi_{k}+\frac{G}{2}\left[\sum_{l}\bigl{(}j_{l}+\frac{1}{2}\bigr{)}\sin\varphi_{l}\right]\cos\varphi_{k}=0,$

or

$\bigl{(}E_{k}-\frac{G}{2}-\lambda\bigr{)}t_{k}-G\xi_{k}t_{k}+GD\xi_{k}=0.$
(21)

Here the notation

$D=\sum_{l}S_{l}\,t_{l}\,,\quad S_{l}=j_{l}+\frac{1}{2}\,.$

is used. For each subshell $l$ the quantities $S_{l}$ are equal to the number
of different particle pairs that form the state (7).

Remark 1. Usually in Eq. (8) the terms proportional to $v^{4}_{k}$ are
discarded (or absorbed into $E_{k}$ by the modification of single-particle
energies [2]). In this case, Eq. (21) is transformed into

$\bigl{(}E_{k}-\lambda\bigr{)}t_{k}=-GD\xi_{k}.$

We square both parts of the equation, use Eq. (14) and obtain equations
analogues to Eqs. (1) and (2) with the correlation function $C=GD$.

Remark 2. Extremum conditions (18) and (19) at $\varphi_{k}=0$ and
$\varphi_{k}=\pi$ correspond to the normal solutions with $t_{k}=0$. These
solutions allow one to describe the system having $t_{l}>0$ for a certain
subshell and $t_{m}=0$ for others subshells. If $t_{k^{\prime}}=0$ for a
certain subshell $k^{\prime}$ then from Eqs. (21) alone it would follow that
$D=0$ and the rest of the abnormal densities would be zero.

Remark 3. The contribution of components with a different particle number to
$|\rangle$ is estimated by the particles number variance

$\mathcal{V}=\langle|N^{2}|\rangle-\langle|N|\rangle^{2}=2\sum_{k}(2j_{k}+1)\,t^{2}_{k}=4\sum_{k}S_{k}t^{2}_{k}.$

The particle number variance is determined by non-zero $t_{k}$ only.

##  2 One and two subshells

Let us start with the simplest cases.

###  2.1 Single subshell

If the system contains a single subshell with the energy $E_{0}$ and angular
momentum $j_{0}$, Eqs. (21) and (12) are reduced to

$\displaystyle\bigl{(}E_{0}-\frac{G}{2}-\lambda\bigr{)}t_{0}-G\xi_{0}t_{0}+GS_{0}t_{0}\xi_{0}=0,$
(22) $\displaystyle 2S_{0}\left(\xi_{0}+\frac{1}{2}\right)=\mathcal{N}.$ (23)

If $\mathcal{N}=2S_{0}$ (all single-particle states are occupied or the
subshell is closed), the particle number equation (23) has the solution
$\xi_{0}=1/2$, and therefore $t_{0}=0$. Only the normal solution is possible
for any $G$.

If $\mathcal{N}=2S_{0}-2P_{0}$, here $P_{0}$ is the number of particle pairs
removed from the closed subshell. The particle number equation (23) has the
solution

$\xi_{0}=1/2-p_{0},\quad\mathrm{with}\
p_{0}=P_{0}/S_{0},\quad\mathrm{and}\quad w_{0}=1-p_{0},\quad w_{0}<1,$

and only superconducting solution exists. The abnormal density and correlation
function are equal to

$\displaystyle t_{0}=\sqrt{p_{0}(1-p_{0})},$ $\displaystyle
C=GD=GS_{0}t_{0}=GS_{0}\sqrt{p_{0}(1-p_{0})}=\frac{G}{2}\sqrt{\mathcal{N}(2S_{0}-\mathcal{N})}.$

In the case of single open subshell, the correlation function depends linearly
on $G$, because both $\xi_{0}$ and $t_{0}$ are independent of $G$. The
particle number variance is also constant

$\mathcal{V}=4S_{0}t^{2}_{0}=4S_{0}p_{0}(1-p_{0})=\frac{\mathcal{N}(2S_{0}-\mathcal{N})}{S_{0}}.$

The chemical potential is a linear function of $G$,

$\lambda=E_{0}+G\left[\left(\frac{1}{2}-p_{0}\right)S_{0}-\left(1-p_{0}\right)\right].$

###  2.2 Two subshells

For the system having two subshells, the set of equations is

$\displaystyle\begin{split}\bigl{(}E_{1}-\frac{G}{2}-\lambda\bigr{)}t_{1}-Gt_{1}\xi_{1}+G\left(S_{1}t_{1}+S_{2}t_{2}\right)\xi_{1}&=0,\\\
\bigl{(}E_{2}-\frac{G}{2}-\lambda\bigr{)}t_{2}-Gt_{2}\xi_{2}+G\left(S_{1}t_{1}+S_{2}t_{2}\right)\xi_{2}&=0,\end{split}$
(24) $\displaystyle
2\left(S_{1}\xi_{1}+S_{2}\xi_{2}\right)=\mathcal{N}-S_{1}-S_{2}.$ (25)

We label subshells by the indices ”1” and ”2” so that $E_{2}>E_{1}$. If
$E_{2}=E_{1}$ we face, due to definition (5), the single subshell system with
$S_{0}=S_{1}+S_{2}$. It is convenient to introduce the dimensionless chemical
potential and coupling constant

$\mu=\frac{\lambda-E_{1}+G/2}{E_{2}-E_{1}}\quad\mathrm{and}\quad
g=\frac{G}{E_{2}-E_{1}}.$

And equations (24) can be written as

$\displaystyle-\mu
t_{1}-gt_{1}\xi_{1}+g\left(S_{1}t_{1}+S_{2}t_{2}\right)\xi_{1}$
$\displaystyle=0,$ (26)
$\displaystyle\bigl{(}1-\mu\bigr{)}t_{2}-gt_{2}\xi_{2}+g\left(S_{1}t_{1}+S_{2}t_{2}\right)\xi_{2}$
$\displaystyle=0.$

####  2.2.1  $S_{1}=S_{2}$ and $\mathcal{N}=2S_{1}$

Consider the system with $S_{1}=S_{2}$. The particle number equation (25) is
simplified

$2S_{1}\left(\xi_{1}+\xi_{2}\right)=\mathcal{N}-2S_{1}.$

When $\mathcal{N}=2S_{1}$, it gives $\xi_{2}=-\xi_{1}$ and $t_{2}=t_{1}$.
Equations (26) are transformed into

$\displaystyle-\mu t_{1}-gt_{1}\xi_{1}+2gS_{1}t_{1}\xi_{1}$ $\displaystyle=0,$
(27)
$\displaystyle\bigl{(}1-\mu\bigr{)}t_{1}+gt_{1}\xi_{1}-2gS_{1}t_{1}\xi_{1}$
$\displaystyle=0.$

We sum them and get the equation

$(1-2\mu)\,t_{1}=0,$

having two solutions: $\mu=1/2$ and $t_{1}=0$.

The $t_{1}=0$ solution corresponds to the normal state with $\xi_{1}=\pm 1/2$
and $\xi_{2}=\mp 1/2$. The energy minimum is reached at $\xi_{1}=1/2$
($w_{1}=1$) and $\xi_{2}=-1/2$ ($w_{2}=0$). The solution exists for any
positive $G$. The chemical potential $\mu$ is arbitrary.

Let us consider the $\mu=1/2$ solution with $t_{1}\neq 0$. The chemical
potential is

$\lambda=\frac{1}{2}\left(E_{1}+E_{2}-G\right).$

It follows from the first of Eqs. (27) for $t_{1}\neq 0$ that

$\xi_{1}=\frac{1}{2}\,\frac{1}{g\left(2S_{1}-1\right)}.$

Due to Eq. (14) the real non-zero $t_{1}$ exists for $|\xi_{1}|<1/2$;
therefore, the superconducting pair correlations may appear if

$g>\frac{1}{2S_{1}-1}.$

It is convenient to introduce special notation for the critical values of the
interaction constant

$\mathop{g_{\mathrm{cr}}}\nolimits=\frac{1}{2S_{1}-1}\quad\mathrm{and}\quad\mathop{G_{\mathrm{cr}}}\nolimits=\frac{E_{2}-E_{1}}{2S_{1}-1}.$

The solutions are

$\displaystyle\xi_{1}=\frac{1}{2}\frac{\mathop{g_{\mathrm{cr}}}\nolimits}{g}=\frac{1}{2}\frac{\mathop{G_{\mathrm{cr}}}\nolimits}{G},$
$\displaystyle
t_{1}=\sqrt{\frac{1}{4}-\xi^{2}_{1}}=\frac{1}{2}\sqrt{\frac{(g-\mathop{g_{\mathrm{cr}}}\nolimits)(g+\mathop{g_{\mathrm{cr}}}\nolimits)}{g^{2}}}.$

For small $g$,
$0<g-\mathop{g_{\mathrm{cr}}}\nolimits\ll\mathop{g_{\mathrm{cr}}}\nolimits$,
we have (the approximate equality $g+\mathop{g_{\mathrm{cr}}}\nolimits\approx
2g$ is taken into account) $\displaystyle
t_{1}\approx\sqrt{\frac{g-\mathop{g_{\mathrm{cr}}}\nolimits}{2g}}=\sqrt{\frac{G-\mathop{G_{\mathrm{cr}}}\nolimits}{2G}}$,
and $C=2GS_{1}t_{1}\approx
S_{1}\sqrt{2G(G-\mathop{G_{\mathrm{cr}}}\nolimits)}$. Here one can not expand
the correlation function $C$ in the Taylor series in $G$ near
$\mathop{G_{\mathrm{cr}}}\nolimits$. As in statistical mechanics, this example
demonstrates the non-analytical dependence of correlation function on coupling
constant. The $\mathop{G_{\mathrm{cr}}}\nolimits$ is the breaking point of
$C(G)$

$C(G)=\begin{cases}0,&\mathrm{if}\
G\leqslant\mathop{G_{\mathrm{cr}}}\nolimits;\\\
GS_{1}\sqrt{1-(\mathop{G_{\mathrm{cr}}}\nolimits/G)^{2}},&\mathrm{if}\
G>\mathop{G_{\mathrm{cr}}}\nolimits.\end{cases}$

For very large $G$, $G\gg(E_{2}-E_{1})/(2S_{1}-1)$,

$C=GS_{1}\sqrt{1-\frac{(E_{2}-E_{1})^{2}}{(2S_{1}-1)^{2}\,G^{2}}}\approx
GS_{1}\left(1-\frac{1}{2}\frac{(E_{2}-E_{1})^{2}}{(2S_{1}-1)^{2}\,G^{2}}+\ldots\right).$

The first term of the expansion in powers of
$\mathop{G_{\mathrm{cr}}}\nolimits/G$ does not depend on the difference
$(E_{2}-E_{1})$ and coincides with the correlation function of one subshell
system having $S_{0}=2S_{1}$ and $p_{0}=1/2$.

The calculated critical value of the interaction constant turned out to be
proportional to the ratio $1/(2S_{1}-1)$. It is not clear whether
$\mathop{g_{\mathrm{cr}}}\nolimits$ depends on the particle number or on the
number of vacancies in the system.

Remark 4. Both normal ($w_{1}=1$, $w_{2}=0$, $t_{2}=t_{1}=0$) and
superconducting ($w_{1}=1/2+\mathop{G_{\mathrm{cr}}}\nolimits/(2G)$,
$w_{2}=1/2-\mathop{G_{\mathrm{cr}}}\nolimits/(2G)$, $t_{2}=t_{1}>0$) solutions
are possible for $G>\mathop{G_{\mathrm{cr}}}\nolimits$ in the considered
example. The difference of their energies is easily calculated by Eq. (11)
with $\lambda=0$,

$\langle|H|\rangle_{w_{2}=0}-\langle|H|\rangle_{w_{2}>0}=S_{1}\left(S_{1}-\frac{1}{2}\right)G\left(1-\frac{\mathop{G_{\mathrm{cr}}}\nolimits}{G}\right)^{2}.$

The subscripts $w_{2}=0$ and $w_{2}>0$ indicate that the Hamiltonian is
averaged over the wave function of either the normal state or the
superconducting one, respectively. One can see that for
$G>\mathop{G_{\mathrm{cr}}}\nolimits$ the energy of the normal state exceeds
the energy of the superconducting one.

####  2.2.2 $S_{1}\neq S_{2}$ and $\mathcal{N}=2S_{1}$

Let us figure out how the value of the critical constant depends on the number
of particles in the closed subshell $E_{1}$ and on the number of empty single
particle states in the subshell $E_{2}$.

For $\mathcal{N}=2S_{1}$ equation (25) is

$2(S_{1}\xi_{1}+S_{2}\xi_{2})=S_{1}-S_{2}\quad\mathrm{or}\quad
S_{1}(\xi_{1}-\frac{1}{2})+S_{2}(\xi_{2}+\frac{1}{2})=0.$ (28)

To simplify the calculations, we introduce the new variables $\delta_{1}$ and
$\delta_{2}$ such as

$\xi_{1}=\frac{1}{2}-\delta_{1}\quad\mathrm{and}\quad\xi_{2}=-\frac{1}{2}+\delta_{2}.$

It follows from Eq. (28) that the unknown $\delta_{1}$ and $\delta_{2}$ are
connected by the equation

$S_{1}\delta_{1}=S_{2}\delta_{2},$

which can be taken into account by substitution

$\delta_{1}=S_{2}\delta,\quad\delta_{2}=S_{1}\delta.$

As $-1/2\leqslant\xi_{1,2}\leqslant 1/2$, the $\delta$ should be inside the
interval

$0\leqslant\delta\leqslant\min\left(\frac{1}{S_{1}},\frac{1}{S_{2}}\right)\,.$

It follows from Eq. (14) that

$\displaystyle t_{1}$
$\displaystyle=\sqrt{\delta_{1}(1-\delta_{1})}=\sqrt{S_{2}\delta(1-S_{2}\delta)}\,,\quad\xi_{1}=\frac{1}{2}-S_{2}\delta\,,$
$\displaystyle t_{2}$
$\displaystyle=\sqrt{\delta_{2}(1-\delta_{2})}=\sqrt{S_{1}\delta(1-S_{1}\delta)}\,,\quad\xi_{2}=-\frac{1}{2}+S_{1}\delta\,.$

The unknown $\xi_{1,2}$ and $t_{1,2}$ expressed in terms of $\delta$ can be
substituted into Eq. (26), the chemical potential can be excluded and the
algebraic equation of sixth degrees can be obtained. The analysis of solutions
of these equations is prohibitively difficult, and we narrow the problem by
looking for values of the interaction constant $G$ at which the
superconducting correlations will start to form. In other words, the unknown
$\delta$ will be an infinitely small positive number.

The abnormal densities $t_{1,2}\approx\sqrt{S_{2,1}\delta}$ for small
$\delta$. Therefore, the first of Eqs. (26) changes into

$-\mu\sqrt{S_{2}\delta}+g(S_{1}-1)\sqrt{S_{2}\delta}\xi_{1}+gS_{2}\sqrt{S_{1}\delta}\xi_{1}=0.$

Since we consider $\delta>0$ ($\delta=0$ corresponds to the normal solution),
we can divide both sides of the equation by $\sqrt{S_{2}\delta}$. We transform
the second equation of Eqs. (26) in a similar way and obtain the simplified
equations

$\displaystyle-\mu+g\left(S_{1}-1+\sqrt{S_{1}S_{2}}\right)\xi_{1}$
$\displaystyle=0,$ $\displaystyle
1-\mu+g\left(\sqrt{S_{1}S_{2}}+S_{2}-1\right)\xi_{2}$ $\displaystyle=0.$

The linear equation for $\delta$ follows:

$\left[\left(S_{1}-1+\sqrt{S_{1}S_{2}}\right)S_{2}+\left(S_{2}-1+\sqrt{S_{1}S_{2}}\right)S_{1}\right]\delta\\\
\hbox{}\qquad=\frac{1}{2}\left(\sqrt{S_{1}}+\sqrt{S_{2}}\right)^{2}-1-\frac{1}{g}\,.$
(29)

The coefficient for $\delta$ in the left-hand side of the equation is
independent of $g$ and is positive because $S_{1,2}\geqslant 1$. Therefore,
$\delta$ is positive if the right-hand side of the equation is positive. We
rewrite the right hand side as

$\frac{1}{2}\left(\sqrt{S_{1}}+\sqrt{S_{2}}\right)^{2}-1-\frac{1}{g}=\frac{1}{\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}-\frac{1}{g}=\frac{g-\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}{\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})\,g}\,,$

here

$\displaystyle\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})=\frac{1}{\frac{1}{2}\left(\sqrt{S_{1}}+\sqrt{S_{2}}\right)^{2}-1}.$
(30)

Thus, $\delta$ will be an infinitesimal positive number if

$0<g-\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})\ll
g\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2}).$

The expression for $\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})$ shows that
the particle number (the size of the completely filled subshell $2S_{1}$) and
the vacancy number of the completely empty subshell ($2S_{2}$) equally affect
the critical value of the interaction constant. Such a somehow unexpected
result can be explained by the observation that the Hamiltonian (4) has equal
matrix elements for the processes of particle pair creation and destruction
inside the $E_{1}$ subshell and the pair creation in the $E_{2}$ subshell.

Now we calculate the correlation energy, chemical potential and particle
number variance for infinitely small $\delta$, in other words, for $G$
satisfying the inequalities:

$0<G-\mathop{G_{\mathrm{cr}}}\nolimits(S_{1},S_{2})\ll\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})G,\quad\mathrm{where}\quad\mathop{G_{\mathrm{cr}}}\nolimits(S_{1},S_{2})=\left(E_{2}-E_{1}\right)\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2}).$

The $\delta$ is calculated from Eq. (29), afterwards $\delta_{1}$ and
$\delta_{2}$ are determined

$\displaystyle\delta_{1}$
$\displaystyle=\frac{S_{2}}{R(S_{1},S_{2})}\left(\frac{1}{\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}-\frac{1}{g}\right)=\frac{S_{2}}{R(S_{1},S_{2})}\frac{g-\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}{g\,\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})},$
$\displaystyle\delta_{2}$
$\displaystyle=\frac{S_{1}}{R(S_{1},S_{2})}\left(\frac{1}{\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}-\frac{1}{g}\right)=\frac{S_{1}}{R(S_{1},S_{2})}\frac{g-\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}{g\,\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})},$
$\displaystyle\mathrm{where}\quad
R(S_{1},S_{2})=\left(S_{1}-1+\sqrt{S_{1}S_{2}}\right)S_{2}+\left(S_{2}-1+\sqrt{S_{1}S_{2}}\right)S_{1}\,.$

The correlation energy is

$\displaystyle C$ $\displaystyle=G\left(S_{1}t_{1}+S_{2}t_{2}\right)\approx
G\left(S_{1}\sqrt{\delta_{1}}+S_{2}\sqrt{\delta_{2}}\right)=$
$\displaystyle=G\left(\sqrt{S_{1}}+\sqrt{S_{2}}\right)\sqrt{\frac{S_{1}S_{2}\left(g-\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})\right)}{R(S_{1},S_{2})\,g\,\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}}.$

Evidently, $C$ can not be expanded in a Taylor series around
$\mathop{g_{\mathrm{cr}}}\nolimits$. The chemical potential is

$\lambda=E_{1}+\frac{1}{2}\left(S_{1}+\sqrt{S_{1}S_{2}}-2\right)G\\\
+\frac{\left(S_{1}+\sqrt{S_{1}S_{2}}-1\right)S_{2}}{R(S_{1},S_{2})}\left(1-\frac{g}{\mathop{g_{\mathrm{cr}}}\nolimits(S_{1},S_{2})}\right)(E_{2}-E_{1}).$

The particle number variance is

$\mathcal{V}=4\left(S_{1}t_{1}^{2}+S_{2}t_{2}^{2}\right)=8\frac{S_{1}S_{2}}{R(S_{1},S_{2})}\left(\frac{1}{\mathop{g_{\mathrm{cr}}}\nolimits}-\frac{1}{g}\right),$

therefore, $\mathcal{V}$ is proportional to the difference
$\left(g-\mathop{g_{\mathrm{cr}}}\nolimits\right)$, if $g$ slightly exceeds
$\mathop{g_{\mathrm{cr}}}\nolimits$.

####  2.2.3  $\mathcal{N}<2S_{1}$

Let the number of particles be $\mathcal{N}=2S_{1}-2P_{1}$, that is, the
$P_{1}$ pair of particles is removed from the low closed subshell. We have
shown in subsection 2.1 that the abnormal density $t_{1}$ is positive for any
$G$ in this case. One needs to find out at what $G$ the abnormal density
$t_{2}$ will be non-zero. Two variants are possible. Either $t_{2}>0$ at any
positive coupling constant or a certain critical value $\tilde{G}$ exists such
as $t_{1}>0$ and $t_{2}=0$ if $0<G<\tilde{G}$.

With $\mathcal{N}=2S_{1}-2P_{1}$ particle number equation (25) is

$2S_{1}\xi_{1}+2S_{2}\xi_{2}=S_{1}-2P_{1}-S_{2}\quad\mathrm{or}\quad
S_{1}\left(\xi_{1}-\frac{1}{2}+p_{1}\right)+S_{2}\left(\xi_{2}+\frac{1}{2}\right)=0.$

Here $p_{1}=P_{1}/S_{1}$. We put $\xi_{1}=1/2-p_{1}-\delta_{1}$ and
$\xi_{2}=-1/2+\delta_{2}$ and obtain

$t_{1}=\sqrt{\left(p_{1}+\delta_{1}\right)\left(1-p_{1}-\delta_{1}\right)}\quad\mathrm{and}\quad
t_{2}=\sqrt{\delta_{2}\left(1-\delta_{2}\right)}.$

New variables $\delta_{1}$ and $\delta_{2}$ are related to each other by
$S_{1}\delta_{1}=S_{2}\delta_{2}$, which can be satisfied if
$\delta_{1}=S_{2}\delta$ and $\delta_{2}=S_{1}\delta$. The density $t_{2}>0$
if $\delta>0$. The boundaries (13) of $\xi_{1,2}$ lead to the inequalities

$0\leqslant\delta\leqslant\min\left(\frac{1-p_{1}}{S_{2}},\frac{1}{S_{1}}\right).$

As in the previous example, Eqs. (26) can be transformed into an algebraic
equation of the sixth degree with respect to $\delta$. We obtain simplified
equations for infinitesimal positive $\delta$. For small $\delta$

$\displaystyle t_{1}$
$\displaystyle=\sqrt{p_{1}(1-p_{1})}+\frac{1}{2}\frac{1-2p_{1}}{\sqrt{p_{1}(1-p_{1})}}S_{2}\delta+o(\delta),$
$\displaystyle t_{2}$
$\displaystyle=\sqrt{S_{1}\delta}\left(1-\frac{1}{2}S_{1}\delta+o(\delta)\right).$

The symbol $o(x)$ stands for the functions of $x$ that
$\displaystyle\frac{o(x)}{x}\to 0$ when $x\to 0$.

For $0<\delta<1$ the inequalities $0<\delta<\sqrt{\delta}<1$ are satisfied;
therefore, the expansion of $t_{1,2}$ should be carried for $\sqrt{\delta}$
not for $\delta$ itself. Keeping the terms with zero and first powers of
$\sqrt{\delta}$, we obtain the approximate

$\displaystyle t_{1}$ $\displaystyle=\sqrt{p_{1}(1-p_{1})},\quad$
$\displaystyle\xi_{1}$ $\displaystyle=1/2-p_{1},$ $\displaystyle t_{2}$
$\displaystyle=\sqrt{S_{1}\delta},$ $\displaystyle\xi_{2}$
$\displaystyle=-1/2.$

The approximate equations follow from the exact ones (26)

$\displaystyle-\mu-g\left(\frac{1}{2}-p_{1}\right)+g\left(S_{1}+S_{2}\sqrt{\frac{S_{1}\delta}{p_{1}(1-p_{1})}}\,\right)\left(\frac{1}{2}-p_{1}\right)=0,$
$\displaystyle\left(1-\mu\right)\sqrt{S_{1}\delta}+\frac{g}{2}\sqrt{S_{1}\delta}-\frac{g}{2}\left(S_{1}\sqrt{p_{1}(1-p_{1})}+S_{2}\sqrt{S_{1}\delta}\right)=0.$

The solutions of the system of approximate equations are

$\displaystyle\mu$
$\displaystyle=g\left(\frac{1}{2}-p_{1}\right)\left(S_{1}-1+S_{2}\sqrt{\frac{S_{1}\delta}{p_{1}(1-p_{1})}}\,\right),$
$\displaystyle t_{2}$
$\displaystyle=\sqrt{S_{1}\delta}=\frac{g}{2}\,\frac{S_{1}\sqrt{p_{1}(1-p_{1})}}{1+g\left(1+p_{1}(S_{1}-1)-\left(S_{1}+S_{2}\right)/2\right)}.$

When calculating $t_{2}$, we ignored the term proportional to $S_{1}\delta$
produced in the second equation from the product $\mu\sqrt{S_{1}\delta}$.

The obtained solutions show that the abnormal density $t_{2}$ is the
infinitely small positive number for any positive infinitely small $g$. The
correlation function is approximately equal to

$C\approx
GS_{1}\sqrt{p_{1}(1-p_{1})}\left(1+\frac{G}{2\left(E_{2}-E_{1}\right)}S_{2}\right).$

The linear on $g$ part of $t_{2}$ is taken into account here. The chemical
potential calculated with the same accuracy is

$\lambda\approx
E_{1}+G\left[\left(\frac{1}{2}-p_{1}\right)S_{1}+p_{1}-1+\frac{G}{2\left(E_{2}-E_{1}\right)}\left(\frac{1}{2}-p_{1}\right)S_{1}S_{2}\right].$

It is easy to see that the chemical potential calculated to the first order in
$G$ is equal to the chemical potential for a system consisting of a single
open subshell.

####  2.2.4  $\mathcal{N}=2S_{1}+2Q,\ 0<Q<S_{2}$

Let the number of particles be sufficient to fill the subshell $E_{1}$
completely and the subshell $E_{2}$ – partially. Equation (25) gives

$\displaystyle 2\left(S_{1}\xi_{1}+S_{2}\xi_{2}\right)=S_{1}-S_{2}+2Q,$
$\displaystyle
S_{1}\left(\xi_{1}-\frac{1}{2}\right)+S_{2}\left(\xi_{2}+\frac{1}{2}-q\right)=0,\quad
q=\frac{Q}{S_{2}}.$

We introduce $\varepsilon_{1,2}$ such that $\xi_{1}=1/2-\varepsilon_{1}$ and
$\xi_{2}=-1/2+q+\varepsilon_{2}$. The new variables $\varepsilon_{1}$ and
$\varepsilon_{2}$ are connected by the equation
$S_{1}\varepsilon_{1}=S_{2}\varepsilon_{2}$, which can be solved by
substitutions $\varepsilon_{1}=S_{2}\varepsilon$ and
$\varepsilon_{2}=S_{1}\varepsilon$.

As in the previous cases, in order to study the appearance of superconducting
pair correlations, we keep the terms proportional to the zero and first
degrees of infinitesimal $\sqrt{\varepsilon}$ and obtain

$\displaystyle\xi_{1}$
$\displaystyle=\frac{1}{2}-S_{2}\varepsilon\approx\frac{1}{2},$
$\displaystyle\quad t_{1}$
$\displaystyle=\sqrt{S_{2}\varepsilon\left(1-S_{2}\varepsilon\right)}\approx\sqrt{S_{2}\varepsilon},$
$\displaystyle\xi_{2}$
$\displaystyle=-\frac{1}{2}+q+S_{1}\varepsilon\approx-\frac{1}{2}+q,$
$\displaystyle\quad t_{2}$
$\displaystyle=\sqrt{\left(q+S_{1}\varepsilon\right)\left(1-q-S_{1}\varepsilon\right)}\approx\sqrt{q\left(1-q\right)}.$

We substitute these approximate values into Eqs. (26), divide both sides of
the second equation by non-zero $t_{2}$, and determine $\mu$ and $t_{1}$ from
the coupled linear equations

$\displaystyle\mu\approx
1+g\left(q-\frac{1}{2}\right)\left[S_{2}-1+S_{1}\frac{t_{1}}{t_{2}}\right],$
$\displaystyle
t_{1}\approx\sqrt{S_{2}\varepsilon}=\frac{g}{2}\,\frac{S_{2}\sqrt{q\left(1-q\right)}}{1+g\left[1+\left(S_{2}-1\right)q-\left(S_{1}+S_{2}\right)/2\right]}\approx\frac{g}{2}S_{2}\sqrt{q(1-q)}.$

The obtained $t_{1}$ and $t_{2}$ are used for calculation of the correlation
function and chemical potential

$\displaystyle C$ $\displaystyle\approx
GS_{2}\sqrt{q\left(1-q\right)}\left(1+\frac{G}{2(E_{2}-E_{1})}S_{1}\right),$
$\displaystyle\lambda$ $\displaystyle\approx
E_{2}+G\left[\left(q-\frac{1}{2}\right)S_{2}-q+\frac{G}{2(E_{2}-E_{1})}\left(q-\frac{1}{2}\right)S_{1}S_{2}\right].$

Please note that the correlation function and chemical potential for small $g$
reproduce the exact solutions for the single open subshell.

The last two examples dealing with two-subshell systems having one open
subshell show that both abnormal densities will be non-zero (the
superconducting solutions exist for both subshells) at any small positive
coupling constant. The chemical potential is found to be near the energy of
the single particle states forming the open subshell.

##  3 Several subshells

Let the system have $\mathcal{M}$ subshells, $\mathcal{M}>2$. We number the
subshells so that $E_{k}\leqslant E_{l}$ if $k<l$. In the model of independent
particles, as the number of particles grows the subshells are gradually
filled: from the subshells with lower single-particle energy to the subshells
with larger ones. We denote by $F$ the number of the largest energy subshell
which still has particles in it. Therefore, the total number of particles in
the system is

$\mathcal{N}=\sum_{k=1}^{F}2S_{k}-2P,$

here $P$ is the number of particle pairs required to fill the subshell $F$
completely. If $P=0$, we have the closed subshell system. If $0<P<S_{F}$, the
system is the open subshell system.

The average number of particles (12) can be written as

$\sum_{k=1}^{F-1}S_{k}\left(\xi_{k}-\frac{1}{2}\right)+S_{F}\left(\xi_{F}-\frac{1}{2}+p_{F}\right)+\sum_{k=F+1}^{\mathcal{M}}S_{k}\left(\xi_{k}+\frac{1}{2}\right)=0.$

Here $p_{F}=P/S_{F}$. Instead of variables $\xi_{k}$ we introduce new unknowns
$\delta_{k}$

$\xi_{k}=\begin{cases}\frac{1}{2}-\delta_{k},&1\leqslant k<F;\\\
\frac{1}{2}-p_{F}-\delta_{k},&k=F;\\\
-\frac{1}{2}+\delta_{k},&F<k\leqslant\mathcal{M}.\end{cases}$

They are connected by

$\sum_{k=1}^{F}S_{k}\delta_{k}=\sum_{l=F+1}^{\mathcal{M}}S_{l}\delta_{l}.$
(31)

The conditions $-1/2<\xi_{k}<1/2$, valid for any $k$, dictate the inequalities
for $\delta_{k}$:

$0<\delta_{k}<1\ \mathrm{for}\ k\neq F,\ \mathrm{and}\
-p_{F}<\delta_{F}<1-p_{F}.$

Please note that now $\delta_{F}$ can be not only a positive but also a
negative number. It follows from Eq. (14) that the abnormal densities are

$t_{k}=\begin{cases}\sqrt{\delta_{k}(1-\delta_{k})},&k\neq F;\\\ \vskip 3.0pt
plus 1.0pt minus
1.0pt\cr\sqrt{(p_{F}+\delta_{F})(1-p_{F}-\delta_{F})},&k=F.\end{cases}$

Equations (21) can be written as the system

$\displaystyle\left[E_{k}-G\left(1-\delta_{k}\right)-\lambda\right]t_{k}+GD\left(\frac{1}{2}-\delta_{k}\right)$
$\displaystyle=0\,,\quad$ $\displaystyle 1\leqslant k<F,$ (32)
$\displaystyle\left[E_{F}-G\left(1-p_{F}-\delta_{F}\right)-\lambda\right]t_{F}+GD\left(\frac{1}{2}-p_{F}-\delta_{F}\right)$
$\displaystyle=0\,,\quad$
$\displaystyle\left[E_{l}-G\delta_{l}-\lambda\right]t_{l}-GD\left(\frac{1}{2}-\delta_{l}\right)$
$\displaystyle=0\,,\quad$ $\displaystyle F<l\leqslant\mathcal{M}\,.$

that connects $\lambda$, $\delta_{k}$ and $t_{k}$, $k=1,\ldots,\mathcal{M}$.

###  3.1 $p_{F}>0$

For the open subshell nuclei $p_{F}>0$. We are looking for the conditions
under which all $\delta_{k}$ with $k\neq F$ will be positive infinitely small
numbers. If follows from Eq. (31) that $\delta_{F}$ will also be an
infinitesimal number of the same order as $\delta_{k}$. The abnormal densities
are

$\displaystyle t_{F}$
$\displaystyle=\sqrt{p_{F}(1-p_{F})}+\frac{1}{2}\frac{1-2p_{F}}{\sqrt{p_{F}(1-p_{F})}}\delta_{F}+o(\delta_{F}),$
$\displaystyle t_{k}$
$\displaystyle=\sqrt{\delta_{k}}\left(1-\frac{1}{2}\delta_{k}+o(\delta_{k})\right),\quad
k\neq F.$

For $\delta_{k}$ with $k\neq F$ the inequalities
$0<\delta_{k}<\sqrt{\delta_{k}}$ are satisfied; therefore, we consider
$\sqrt{\delta_{k}}$ as having the first order of smallness and will keep
$\sqrt{\delta_{k}}$ in the zeroth and first degrees. As a result, we have

$\displaystyle t_{k}\approx\sqrt{\delta_{k}},\quad k\neq F,$ $\displaystyle
t_{F}\approx\sqrt{p_{F}\left(1-p_{F}\right)},$ $\displaystyle D\approx
S_{F}\sqrt{p_{F}\left(1-p_{F}\right)}+\sum_{l\neq
F}S_{l}\sqrt{\delta_{l}}\approx S_{F}\sqrt{p_{F}\left(1-p_{F}\right)}.$

The terms proportional to $S_{l}\sqrt{\delta_{l}}$ are neglected in the last
expression because they are infinitely small in comparison with the finite
term $S_{F}\sqrt{p_{F}\left(1-p_{F}\right)}$. We substitute the approximate
expressions into Eq. (32) and obtain

$\displaystyle\lambda=E_{F}+\left[\left(\frac{1}{2}-p_{F}\right)S_{F}-(1-p_{F})\right]G,$
$\displaystyle
t_{k}=\frac{1}{2}S_{F}\sqrt{p_{F}(1-p_{F})}\frac{G}{E_{F}-E_{k}}+o\left(\frac{G}{E_{F}-E_{k}}\right),\quad
1\leqslant k<F,$ $\displaystyle
t_{l}=\frac{1}{2}S_{F}\sqrt{p_{F}(1-p_{F})}\frac{G}{E_{l}-E_{F}}+o\left(\frac{G}{E_{l}-E_{F}}\right),\quad
F<l\leqslant\mathcal{M}.$

The correlation function and the particle number variance are

$\displaystyle C\approx
G\,S_{F}\sqrt{p_{F}(1-p_{F})}\left(1+\frac{1}{2}\sum_{k=1}^{F-1}S_{k}\frac{G}{E_{F}-E_{k}}+\frac{1}{2}\sum_{l=F+1}^{\mathcal{M}}S_{l}\frac{G}{E_{l}-E_{F}}\right),$
$\displaystyle\mathcal{V}\approx
S_{F}\,p_{F}(1-p_{F})\left[4+\sum_{k=1}^{F-1}S_{F}S_{k}\left(\frac{G}{E_{F}-E_{k}}\right)^{2}+\sum_{l=F+1}^{\mathcal{M}}S_{F}S_{l}\left(\frac{G}{E_{l}-E_{F}}\right)^{2}\right].$

For a small coupling constant both the correlation function and the particle
number variance are mostly determined by the abnormal density of the open
subshell. The following inequalities are valid:

$C>G\,S_{F}\sqrt{p_{F}(1-p_{F})}\quad\mathrm{and}\quad\mathcal{V}>4S_{F}\,p_{F}(1-p_{F}).$

We would like to note that the particle number variance $\mathcal{V}$ is
bounded below by the positive number when $G\to 0$.

One can see that the interaction spreads the influence of non-zero $t_{F}$
over all subshells in the system. The abnormal densities differ from zero at
all subshells enveloped by the interaction. In the system of like nucleons
with the open subshell the superconducting solution exists at any arbitrarily
small constant of attractive interaction.

###  3.2 $p_{F}=0$

Now we consider the closed subshell system with $p_{F}=0$. Let us assume for a
moment that all $E_{k}$ with $1\leqslant k<F$ are equal to $E_{F}$ and all
$E_{l}$ with $F<l\leqslant\mathcal{M}$ are equal to $E_{F+1}$. By this
assumption we return to the two-subshell problem with

$\widetilde{S}_{F}=\sum_{k=1}^{F}S_{k}\quad\mathrm{and}\quad\widetilde{S}_{F+1}=\sum_{l=F+1}^{\mathcal{M}}S_{l}.$

We have shown in subsection 2.2.2 that in the present case the superconducting
pair correlations start to appear if the coupling constant $G$ exceeds the
threshold value (30) which is here

$\mathop{G_{\mathrm{cr}}^{\prime}}\nolimits=\frac{E_{F+1}-E_{F}}{\frac{1}{2}\left(\sqrt{\widetilde{S}_{F}}+\sqrt{\widetilde{S}_{F+1}}\right)^{2}-1}.$

Please note that $2\widetilde{S}_{F}=\mathcal{N}$. During the development of
the superconducting correlations, pairs of particles begin to jump from fully
occupied subshells into free subshells. The difference
$\left(E_{F+1}-E_{F}\right)$ is the lowest energy of such transitions.
Therefore, in the initial $\mathcal{M}$-subshell system the constant
$\mathop{G_{\mathrm{cr}}^{\prime}}\nolimits$ gives the lower bound for the
actual critical value of the interaction constant
$\mathop{G_{\mathrm{cr}}}\nolimits$.

On the other hand,

$\mathop{G_{\mathrm{cr}}^{\prime\prime}}\nolimits=\frac{E_{F+1}-E_{F}}{\frac{1}{2}\left(\sqrt{S_{F}}+\sqrt{S_{F+1}}\right)^{2}-1},$

is the critical constant for the system with only two interacting subshells
taken into account, and the contributions of other subshells are ignored.
Therefore, $\mathop{G_{\mathrm{cr}}^{\prime\prime}}\nolimits$ gives the upper
bound for $\mathop{G_{\mathrm{cr}}}\nolimits$.

This consideration shows that in the closed subshell system only the normal
solutions are possible if the interaction constant satisfies
$0<G<\mathop{G_{\mathrm{cr}}^{\prime}}\nolimits$. All abnormal densities are
equal to zero in this case. If $G$ exceeds
$\mathop{G_{\mathrm{cr}}^{\prime\prime}}\nolimits$, the superconducting
solution is possible and all abnormal densities become positive numbers. The
intermediate case with
$\mathop{G_{\mathrm{cr}}^{\prime}}\nolimits<G<\mathop{G_{\mathrm{cr}}^{\prime\prime}}\nolimits$
requires additional study.

Remark 5. We have used the simplest model Hamiltonian (4) with the constant
attractive interaction. The realistic Hamiltonian can be written [4] as

$H=\sum_{k}\sum_{m_{k}=-j_{k}}^{j_{k}}E_{k}a^{{\dagger}}_{k,m_{k}}a_{k,m_{k}}-\\\
-\frac{1}{4}\sum_{k,l}G_{k,l}\sum_{m_{k}=-j_{k}}^{j_{k}}\sum_{m_{l}=-j_{l}}^{j_{l}}(-1)^{j_{k}-m_{k}}a^{{\dagger}}_{k,m_{k}}a^{{\dagger}}_{k,-m_{k}}\,(-1)^{j_{l}-m_{l}}a_{l,-m_{l}}a_{l,m_{l}}.$

If all matrix elements $G_{k,l}$ are positive numbers, our qualitative
conclusions about the system with open subshell will survive. The expressions
for $\lambda$ and $C$ will be more complicated of course.

Remark 6. We have considered the spherical nuclei. To obtain formulae for
deformed nuclei, one should put $j_{k}=1/2$ and $S_{k}=1$ for all subshells.
The formulae obtained for the closed subshell spherical nuclei are suitable
for deformed nuclei. The deformed nucleus can be open subshell nucleus only if
the energies $E_{F}$ and $E_{F+1}$ coincide.

## Conclusion

We have considered the appearance of superconducting pair correlations in
spherical even-even nucleus using the simplest model Hamiltonian. The
influence of the monopole pairing interaction on the energy of single-particle
states was taken into account.

It is shown that the emergence of pair correlations depends on the particle
number and shell structure.

In the open subshell system non-zero abnormal densities appear for any small
positive coupling constant. The new result obtained in the present paper is
that at infinitely small positive $G$ the abnormal densities differ from zero
at each subshell participating in the pairing interaction.

In the closed subshell system, superconducting pair correlations begin to form
when the coupling constant exceeds a certain critical value
$\mathop{G_{\mathrm{cr}}}\nolimits$. If the coupling constant is less than
$\mathop{G_{\mathrm{cr}}}\nolimits$, the normal solution is the only solution.
All abnormal densities and correlation function equal to zero. The rough lower
and upper bounds of $\mathop{G_{\mathrm{cr}}}\nolimits$ are obtained. More
accurate estimations of $\mathop{G_{\mathrm{cr}}}\nolimits$ are required. Both
normal and superconducting solutions are possible for $G$ larger than
$\mathop{G_{\mathrm{cr}}}\nolimits$.

## References

  * [1] Belyaev S. T. Effect of pairing correlations on nuclear properties // Mat. Fys. Medd. Dan. Vid. Selsk. 1959. 31. No. 11.
  * [2] Soloviev V. G. Teoriya sloghnyh yader. // 1971. Nauka. GRFML. M. [English translation: Theory of Complex Nuclei. 1976. Pergamon Press.]
  * [3] Fifty Years of Nuclear BCS: Pairing in Finite Systems / Ed.: Broglia R. A. and Zelevinsky V.: World Scientific Publishing Co. Pte. Ltd. 2013.
  * [4] Volya A., Brown B. A., Zelevinsky V. Exact solution of the nuclear pairing problem // Phys. Lett. B. 2001. V. 509. P. 37.


# Muon Hunter: a Zooniverse project

R. Bird1 M. K. Daniel2 H. Dickinson3 Q. Feng4,7 L. Fortson3 A. Furniss5 J.
Jarvis6 R. Mukherjee7 R. Ong1 I. Sadeh8 and D. Williams9 1University of
California, Los Angeles, Department of Physics and Astronomy, Physics and
Astronomy Building, 430 Portola Plaza, Box 951547, Los Angeles, CA 90095.USA.
2Whipple Observatory, 670 Mt Hopkins Road, Amado, AZ 85645. USA.  3University
of Minnesota, School of Physics and Astronomy, 116 Church St. SE, Minneapolis,
MN 55455. USA.  4Physics Department, McGill University, 3600 University
Street, Montreal, QC H3A 2T8. Canada.  5California State University East Bay,
25800 Carlos Bee Blvd, Hayward, CA 94542. USA.  6ASTERICS, DECS Workpackage,
School of Physical Sciences, Robert Hooke Building, Open University, Walton
Hall, Milton Keynes, Buckinghamshire, MK7 6AA. UK.  7Columbia University,
Barnard College, Deparment of Physics and Astronomy, 3009 Broadway, New York,
NY 10027. USA.  8Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6,
15738 Zeuthen, Germany.  9SCIPP, Room 337 Natural Sciences 2, University of
California, 1156 High Street, Santa Cruz, CA 95064. USA.
<EMAIL_ADDRESS>
###### Abstract

The large datasets and often low signal-to-noise inherent to the raw data of
modern astroparticle experiments calls out for increasingly sophisticated
event classification techniques. Machine learning algorithms, such as neural
networks, have the potential to outperform traditional analysis methods, but
come with the major challenge of identifying reliably classified training
samples from real data. Citizen science represents an effective approach to
sort through the large datasets efficiently and meet this challenge. Muon
Hunter is a project hosted on the Zooniverse platform, wherein volunteers sort
through pictures of data from the VERITAS cameras to identify muon ring
images. Each image is classified multiple times to produce a “clean” dataset
used to train and validate a convolutional neural network model both able to
reject background events and identify suitable calibration data to monitor the
telescope performance as a function of time.

##  1 Introduction

The Very Energetic Radiation Imaging Telescope Array System (VERITAS) is
composed of four 12 m diameter imaging atmospheric Cherenkov telescopes
(IACTs) sited at the Fred Lawrence Whipple Observatory(FLWO) in southern
Arizona (USA) [1]. Each dish is a tessellated reflector made up of 350
individual mirror facets in a Davis-Cotton configuration focused onto a
499-pixel photomultiplier tube camera for a total field of view of
$3.5^{\circ}$. The primary field of research of VERITAS is in the ground based
detection of very high energy (VHE, E$\gtrsim$100 GeV) gamma-rays under a
seemingly overwhelming 1000:1 background of cosmic-rays. In both cases the
primary very energetic particle will cause a particle cascade in the
atmosphere known as an extensive air shower. As the air shower particles move
through the atmosphere they in turn generate Cherenkov light which can be
picked up and imaged by the telescopes on the ground. The cosmic ray
background is reduced to approximately 1:1 through the stereo imaging
technique, where cosmic rays will tend to have larger, blotchier, more
variable light distribution patterns to the smooth, elliptical images expected
in the camera for gamma rays. The cosmic ray showers will also result in a
large number of muon particles being generated, muons that pass close by the
optical axis of the telescope will cause ring like images in the camera. As
the muon path moves further from the centre of the camera, or as the direction
of travel moves off from the optical axis, the images will only be partial
rings, which can readily mimic the elliptical gamma-ray primary images,
causing false positive classification. Muon rings, however, can also be
useful: the amount of light expected in the ring is a well known quantity and
so they can be used to provide an absolute calibration of the camera
digitisation. It is for these two reasons – background event rejection and
calibration event identification – that we are interested in being able to
pick out the muon ring images from the non-muon cosmic ray and gamma ray ones.

Classification is a common task in experimental physics and with inherently
large datasets it is easy to see why the use of machine learning algorithms
has become increasingly popular. With a trigger rate of O(400 Hz) and over
10,000 hours of observational data for the VERITAS telescopes it is not
possible for any one person to sift through all of the data for events of
interest. One powerful machine learning algorithm, convolutional neural
networks (CNN), was used on a small batch of VHE gamma-ray data to detect and
characterize muon events [2]. For developing supervised machine learning
algorithms like the CNN, it is essential to assign correct labels to a large
training dataset. In this work, we describe Muon Hunter111www.muonhunter.org,
a citizen-science project where volunteers label and parameterize muon and
non-muon images in VHE gamma-ray data to provide high purity machine learning
training samples.

##  2 Muon Hunter

The Muon Hunter experiment was developed in collaboration with the ASTERICS
Horizon2020 project222www.asterics2020.eu, and the classification interface is
hosted by the Zooniverse333www.zooniverse.org platform, where researchers in
many disciplines can easily design and deploy a citizen-science project
through the Zooniverse Project Builder444www.zooniverse.org/labs. The
Zooniverse is the world’s largest and most powerful platform for people-
powered research. In the ten years since the launch of the first Zooniverse
project, Galaxy Zoo [3], the platform has expanded to over 100 projects in a
wide variety of research areas from the physical to the medical and social
sciences, and liberal arts. Via the Zooniverse anyone can be a researcher; you
do not need any specialised background, training, or expertise to participate
(though training in the project task is given on the website). One further
noteworthy aspect of the Zooniverse is that it is not simply an outreach tool,
volunteers and professionals make real discoveries together – many of which
result in published research papers and open source data sets. The Zooniverse
projects are constructed with the specific aim of converting volunteers’
efforts into measurable results and scientifically significant discoveries [3,
4, 5, 6].

The key components of a Zooniverse project are data and workflows. The
workflows are a sequence of specific tasks tailored by the researchers to gain
insights on the data. The workflows are streamlined and optimised to balance
quality of experience for the volunteers with the need to obtain accurate
results. The workflow for identifying the presence of rings in an image is
illustrated in Figure 1. To help new users, a short tutorial, a mini course,
and a detailed ‘About’ page were available. Images were retired once a total
of 15 volunteers had examined it and finished the workflow. A total of 137,515
VERITAS single-telescope images were served on the Muon Hunter website, with
examples given in Figure 2. Most images are preprocessed using the standard
VERITAS two-level cleaning [7] based on the signal-to-noise ratio of each
pixel, but a subset of uncleaned images were also uploaded to explore the
effect of image cleaning on the results of classification. From 5,734
volunteers we received a total of $\sim$2.1 million classifications, half
within the first week after the official launch of the project.

Figure 1: The workflow for classifying images as muons or non-muons.

It was possible to draw multiple rings on one image, as sometimes more than
one muon was recorded in an image. But this also allowed room for human error,
especially when a user was unfamiliar with the workflow. After a user had
completed the workflow of a subject image there was an option to discuss it in
the Talk board. The Talk board enables interactions between experts and
volunteers regarding specific images. Collections of interesting images are
also established by users, including double muon rings and composite images
with a muon ring and a background air shower, or particularly noteworthy,
unexpected or unusual images that may be worthy of further investigation
later.

Figure 2: Example muon ring images. Standard analysis results are shown in
magenta and Muon Hunter volunteer input in yellow.

Figure 3 summarizes the input we received from the volunteers of Muon Hunter.
The distribution of the number of classifications each user made roughly
follows a log-normal distribution (shown as the red dashed curve), with a
median of 30 images per user. Assuming the number of classifications from one
user is roughly proportional to the time spent, this log-normal distribution
is of similar nature to the dwell time of internet users on social media
articles [8]. There are 16 volunteers who classified more than 10,000 images,
while there are 724 volunteers who only classified one image. Of 134,000
images 12% had unanimous votes that a ring was present, 73% were unanimous a
ring was not present, and the remainder were split. The 15 votes for each
image allowed us to estimate the confidence of the users’ classifications. The
distribution of the fraction of votes for the presence of a ring in each image
is shown in the right plot of Figure 3.

Figure 3: Left: the distribution of the number of classifications each user
made. Right: distribution of votes for muon presence in the images.

The volunteers are also engaged through an active social media presence
including an online blog555muonhunterblog.wordpress.com (to date 1301 reads
from 55 countries) giving details about the experiment from the researchers
involved; Facebook666www.facebook.com/muonhunters/ (to date 103 likes and 104
follows) and Twitter777twitter.com/ZooniMuonHunter (to date 25 followers and 7
likes) feeds; and material like postcards for dispersal at public outreach
events, e.g. school fairs, and the FLWO visitor center. The Muon Hunter Launch
event was also advertised on Facebook for a week, the $15 ad reached 9,299
people in 50 countries and received 80 event responses. The tracking aspects
of these social media sites can also help us to get some insight into the
demographics of the people we are reaching and engaging: for example as a
function of gender and age group for our Facebook page, see Table 1, we
reached an audience composed close to parity of 47% of women and 52% men, but
see that we engaged (with the Facebook page) only 35% women and 64% men. These
insights can help us understand where we need to focus in the future to even
out or encourage participation.

Table 1: Muon Hunter Facebook page reach and engagement as a function of age group and gender. Age Group | People Reached | People Engaged
---|---|---
| Female | Male | Female | Male
13-17 | 20% | 19% | 23% | 32%
18-24 | 13% | 17% | 4% | 14%
25-34 | 7% | 10% | 5% | 8%
35-44 | 3% | 3% | 2% | 5%
45-54 | 2% | 1% | 0.629% | 2%
55-64 | 0.91% | 0.836% | 0.629% | 2%
65+ | 0.495% | 0.579% | 0% | 0.629%
|  |  |  |

##  3 CNN: training & evaluation

One purpose of this project is to train a reliable CNN model to classify muon
rings. Two sources of labels, one provided by the VERITAS analysis [9] and the
other by the Muon Hunter user input, can be used for the training, validation,
and testing of the models. We randomly selected 16 observation runs, each 30
minutes in duration, and analysed them using one of the standard VERITAS data
analysis packages, with the muon identification proceedure outlined in [8]. A
detailed description of the CNN model can be found in [2]. Treating all images
with 10 or more of the 15 votes for muons as muon events and the rest as non-
muon events, we were able to train a CNN model from the Muon Hunter volunteers
with a test accuracy of $\sim$97%; comparing favourably to the VERITAS
standard analysis labels test accuracy of $\sim$95%.

##  4 Summary and future work

Citizen science is a great resource for both outreach and practical science.
Having a simple, clearly defined project task helped a lot with the success of
the Muon Hunter project, for which we received a phenomenal response from our
volunteers. The input from the volunteers not only helped us train a more
efficient machine learning model and gain insight into where the standard
analysis may be lacking, it also identified interesting new avenues to
research on interesting and/or unexpected images that would be missed in a
purely machine processed context which could, in turn, become the focus of
future citizen science projects.

The authors gratefully acknowledge all the Muon Hunter volunteers who
contributed to this effort without whom this work would not be possible. Muon
Hunters was developed with the help of the ASTERICS Horizon2020 project.
ASTERICS is a project supported by the European Commission Framework Programme
Horizon 2020 Research and Innovation action under grant agreement №653477.
VERITAS research is supported by grants from the U.S. Department of Energy
Office of Science, the U.S. National Science Foundation and the Smithsonian
Institution, and by NSERC in Canada. We acknowledge the excellent work of the
technical support staff at the Fred Lawrence Whipple Observatory and at the
collaborating institutions in the construction and operation of the
instrument. The VERITAS Collaboration is grateful to Trevor Weekes for his
seminal contributions and leadership in the field of VHE gamma-ray
astrophysics, which made this study possible.

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