$\alpha.$ A gauge boson carries colors $\alpha\beta^{\ast}.$ There are the
following sorts of the $s-SU(5)$ color-single states.

2-fermion states: $\alpha$$\alpha$∗ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$,
$\alpha\neq\beta.$ When the spin of $\alpha$$\alpha$∗ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
is zero, we denote $\underline{\alpha}\underline{\alpha}^{\ast}$ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
by $s-\pi$. When the spin of $\alpha$$\alpha$∗ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
is 1, we denote $\underline{\alpha}\underline{\alpha}^{\ast}$ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
by $s-\rho$. Analogously to $QCD,$ we have
$m\left(s-\pi\right)<m\left(s-\rho\right)$ (because of color-magnetic
interaction) and can suppose $m(s-\pi)\sim m(\pi)$ to be minimum in all
$s-SU(5)$ color-single states. In contrast with a given $\pi-meson,$ $s-\pi$
must be stable, because there is no the electroweak interaction in the case
$\varpi_{s}=0$.

3-fermion states: $\left(\underline{AB}\right)(\underline{CD})$$E$ or
$\left(\underline{AB}\right)\underline{A}^{\ast}\underline{B}^{\ast}.$ When
the spin of $3-fermion$ states is $1/2$, we denote them by $s-N$. $s-N$ is
stable, and we can suppose $m(s-N)\sim m(N).$ $N$ denotes a given nucleon.

4-fermion states: $\left(\underline{AB}\right)\underline{C}\underline{D}$$E$.

5-fermion states: $A$$B$$\underline{C}\underline{D}$$E$ or
$\left(\underline{AB}\right)(\underline{BC})\left(\underline{CD}\right)(\underline{DE})\left(\underline{EA}\right).$

Gauge boson single-states:
$\left(\underline{\alpha\beta^{\ast}}\right)\left(\underline{\alpha^{\ast}\beta}\right)$
or
$\left(\underline{\alpha\beta^{\ast}}\right)\left(\underline{\beta\gamma^{\ast}}\right)\left(\underline{\gamma\alpha^{\ast}}\right)$
etc., $\alpha\neq\beta\neq\gamma.$

Fermion-gauge boson single-states.
$\alpha^{\ast}$$\left(\underline{\alpha\beta^{\ast}}\right)$$\beta$,
$\alpha^{\ast}$$\left(\underline{\alpha\gamma^{\ast})(\gamma\beta^{\ast}}\right)$$\beta$
etc..

Of course, there are the $s-antiparticles$ corresponding to the $s-colour$
single states above as well.

###  IX.2 The characters of the s-SU(5) color single states in V-breaking

We can only qualitatively discuss the $SU(5)$-color single states by comparing
the $SU(5)$-color single states with the $SU(3)$-color single states, because
there is no mature theory of strict $SU(5)$ symmetry up till now. The
following inferences are obtained on the basis of the sameness of the $SU(5)$
and $SU(3)$ groups, hence they are reliable.

The characters of the $s-SU(5)$ color single states in the $V-breaking$ are as
follows.

1\. If the masses of the $s-SU(5)$ fermions only originate from the coupling
of the $s-SU(5)$ fermions with the $s-SU(5)$ Higgs fields, they all must be
zero because $\langle\omega\rangle_{s0}=0$ in the $V-breaking$. If such mass
terms $m\overline{\psi}\psi+MTr\overline{\Psi}\Psi$ are added to $\mathcal{L}$
($\mathcal{L}_{V}$ and $\mathcal{L}_{S}$), the masses of the $s-SU(5)$
fermions are non-zero. Here $\psi$ and $\overline{\Psi}$ are respectively the
representations $\underline{5}$ and $10$. The masses of all $SU(3)$ color
single states (include gluon balls ) are non-zero. Consequently, we can
determine that all $SU(5)$-color single states can get their suitable masses.

2\. In contrast with $SU(3)$ color single states among which there are the
electromagnetic and weak interactions due to the gauge group is $SU(3)\times
U(1)$, there is no electroweak interaction among the $s-SU(5)$ color single
states because $s-SU(5)$ is a simple group. Hence the $s-SU(5)$ color single
states with the minimum masses must be stable.

3\. The interaction radius of the $s-SU(5)$ color single states must be finite
and much smaller ($\sim 1\times 10^{-15}m$) than the radius of hydrogen atoms
($\sim 1\times 10^{-10}m$). This is because the interactions can come into
being only by exchanging the $s-SU(5)$ color single states, and the masses of
all $s-SU(5)$ color single states are non-zero.

Thus we can approximately regard the $s-SU(5)$ color single states as ideal
gas without collision.

In a word, the $s-SU(5)$ color single states are analogous to $v-neutrinos$,
but in contrast with the $v-neutrinos$, there is the repulsion between the
$s-SU(5)$ color single states and $v-matter.$

From the characters of the $s-SU(5)$ color single states mentioned above, we
will see that the $s-SU(5)$ color single states in the $V-breaking$ cannot
form clusters and must loosely distribute in space, but it is possible they
form $s-superclusterings$ as the neutrinos.

##  X Dynamics of v-structure formation and the distributive form of the
s-SU(5) color single states

Generalizing equations governing nonrelativistic fluid motion[8] to present
model, we have

$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v}\mathbf{\cdot\nabla}\right)\mathbf{v}_{v}=-\frac{\nabla
p_{v}}{\rho_{v}}-\nabla\Phi,$ (128) $\frac{\partial}{\partial
t}\rho_{v}+\nabla\cdot\left(\rho_{v}\mathbf{v}_{v}\right)=0$ (129)
$\nabla^{2}\Phi=4\pi G\left(\rho_{v}-\rho_{s}\right),$ (130)
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s}\mathbf{\cdot\nabla}\right)\mathbf{v}_{s}=-\frac{\nabla
p_{s}}{\rho_{s}}+\nabla\Phi,$ (131) $\frac{\partial}{\partial
t}\rho_{s}+\nabla\cdot\left(\rho_{s}\mathbf{v}_{s}\right)=0,$ (132)

in the $V-breaking,$ where $\partial/\partial
t+\mathbf{v}_{v}\mathbf{\cdot\nabla}$ is call the convective derivative. We
can produce the linearized equations of motion by collecting terms of first
order in perturbations about a homogeneous background
$\rho_{v}=\rho_{v0}+\delta\rho_{v}$ etc. Let

$\displaystyle\mathbf{v}_{v}$
$\displaystyle=\mathbf{v}_{v0}+\delta\mathbf{v}_{v},\quad\ \ \ \
\mathbf{v}_{s}=\mathbf{v}_{s0}+\delta\mathbf{v}_{s}^{\prime}$ (133a)
$\displaystyle\rho_{v}$ $\displaystyle=\rho_{v0}+\delta\rho_{v},\quad\ \ \ \
\rho_{s}=\rho_{s0}+\delta\rho_{s},$ (133b) $\displaystyle\delta_{v}$
$\displaystyle=\frac{\delta\rho_{v}}{\rho_{v0}},\quad\ \ \ \
\delta_{s}=\frac{\delta\rho_{s}}{\rho_{s0}},$ (133c)

considering
$\mathbf{v}_{v0}=H\mathbf{x=}\left(\overset{\cdot}{a}/a\right)\mathbf{x,}$ we
get

$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v0}\mathbf{\cdot\nabla}\right)\delta\mathbf{v}_{v}=-\frac{\nabla\delta
p_{v}}{\rho_{v0}}-\nabla\delta\Phi-\left(\delta\mathbf{v}_{v}\cdot\nabla\right)\mathbf{v}_{v0},$
(134) $\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v0}\cdot\nabla\right)\delta_{v}=-\nabla\delta\mathbf{v}_{v},$
(135) $\nabla^{2}\Phi=4\pi
G\left(\rho_{v0}\delta_{v}-\rho_{s0}\delta_{s}\right),$ (136)
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s0}\mathbf{\cdot\nabla}\right)\delta\mathbf{v}_{s}=-\frac{\nabla\delta
p_{s}}{\rho_{s0}}+\nabla\delta\Phi-\left(\delta\mathbf{v}_{s}\cdot\nabla\right)\mathbf{v}_{s0},$
(137) $\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s0}\cdot\nabla\right)\delta_{s}=-\nabla\delta\mathbf{v}_{s},$
(138)

Defining the comoving spatial coordinates

$\mathbf{x}(t)=a\left(t\right)\mathbf{r}\left(t\right),\quad\ \ \ \
\delta\mathbf{v}\left(t\right)=a\left(t\right)\mathbf{u}\left(t\right),$ (139)

we have $\nabla_{x}=\nabla_{r}/a.$ Let

$\displaystyle\delta_{vk}$
$\displaystyle=\delta_{vk}\left(t\right)\exp\left(-i\mathbf{k}_{v}\cdot\mathbf{r}\right),\quad\
\ \ \ \
\delta_{sk}=\delta_{sk}\left(t\right)\exp\left(-i\mathbf{k}_{s}\cdot\mathbf{r}\right),$
(140a) $\displaystyle c_{v}^{2}$ $\displaystyle=\frac{\partial
p_{v}}{\partial\rho_{v}},\quad\ \ \ \ \ \ c_{s}^{2}=\frac{\partial
p_{s}}{\partial\rho_{s}},$ (140b)

from $\left(134\right)-\left(140\right)$ can get

$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v0}\mathbf{\cdot\nabla}\right)^{2}\delta_{vk}+2\frac{\overset{\cdot}{a}}{a}\overset{\cdot}{\delta}_{vk}=4\pi
G\left(\rho_{v0}\delta_{vk}-\rho_{s0}\delta_{sk}\right)-\frac{c_{v}^{2}k_{v}^{2}}{a^{2}}\delta_{vk}.$
(141) $\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s0}\mathbf{\cdot\nabla}\right)^{2}\delta_{sk}+2\frac{\overset{\cdot}{a}}{a}\overset{\cdot}{\delta}_{sk}=4\pi
G\left(\rho_{s0}\delta_{sk}-\rho_{v0}\delta_{vk}\right)-\frac{c_{s}^{2}k_{s}^{2}}{a^{2}}\delta_{sk}.$
(142)

It is necessary that $\delta_{sk}<0$ when $\delta_{vk}>0$ because there is
only repulsion between $s-matter$ and $v-matter.$ Consequently,

$\displaystyle\rho_{v0}\delta_{vk}-\rho_{s0}\delta_{sk}$
$\displaystyle=\rho_{v0}\delta_{vk}+\rho_{s0}\mid\delta_{sk}\mid,$ (143a)
$\displaystyle\rho_{s0}\delta_{sk}-\rho_{v0}\delta_{vk}$
$\displaystyle=\rho_{s0}\delta_{sk}+\rho_{v0}\mid\delta_{vk}\mid.$ (143b)

According to the present model, there must be
$\overset{\cdot}{a}/a=\left(-K/a^{2}+\eta\rho_{g}\right)^{1/2}\sim 0,$ because
$\rho_{v}=\rho_{vm}+\rho_{v\gamma},$ $\rho_{vm}\varpropto R^{-3},$
$\rho_{v\gamma}\varpropto R^{-4},$ $\rho_{s}=\rho_{sm}>\rho_{vm},$ and
$\rho_{sm}\varpropto R^{-3}$ so that $\rho_{g}=\rho_{v}-\rho_{s}=0$ and $K=0.$
There possibly is $\delta_{vk}\left(t\right)>\delta_{sk}\left(t\right)$ when
$\rho_{v}=\rho_{s}$, because $s-SU(5)$ color single states in the $V-breaking$
can be regarded as ideal gas without collision. The ideal gas has the effect
of free flux damping for clustering. Ignoring $\delta_{sk}$ in
$\left(141\right),$ for $\overset{\cdot}{a}/a\sim 0$ and $\left(4\pi
G\rho_{v0}-c_{v}^{2}k_{v}^{\prime 2}\right)>0,$ from $\left(141\right)$ we get

$\overset{\cdot\cdot}{\delta}_{vk}\left(t\right)=\left(4\pi
G\rho_{v0}-c_{v}^{2}k_{v}^{\prime 2}\right)\delta_{vk},\quad\ \ \
\delta_{vk}\left(t\right)=\exp\left(t/\tau\right)\ ,$ (144)

where $k_{v}^{\prime}=k_{v}/a,$ $\tau=1/\sqrt{4\pi
G\rho_{v0}-c_{v}^{2}k_{v}^{\prime 2}}.$ We see that
$\delta_{vk}\left(t\right)$ will exponentially grow for long-wavelength. We
cannot get the result as $\left(144\right)$ for $\delta_{sk}$ from
$\left(142\right)$, because the velocity $v_{s0}$ of a $s-SU(5)$ color single
state is invariant because there is no collision and is very big. Let the
duration in which $\overset{\cdot}{R}\sim 0$ be $\triangle t,$ the distance
$l$ to be damped out is $l=v_{s0}\triangle t.$ The perturbation whose size is
less than $l$ cannot form. Thus $s-SU(5)$ color single states can only form
superclusterings. When $\rho_{s0}>\rho_{v0},$
$\rho_{s0}\delta_{sk}-\rho_{v0}\delta_{vk}>0$ is possible. Consequently, the
$s-SU(5)$ color single states possibly form superclusterings, because when
$\left(\overset{\cdot}{a}/a\right)$ is large, the perturbation will slowly
grow in power rules.

From the above mentioned, we see that the $s-SU(5)$ color single states must
loosely distribute in space or form $s-superclustering$, i.e., huge $v-voids$,
in which $\rho_{s}\gg\rho_{v}.$

To sum up, because of the following two reasons, the perturbation
$\delta_{vk}$ will grow faster or earlier than that determined by the
conventional theory.

1\. There is such a stage in which $\rho_{s}-\rho_{v}\sim 0$ and
$K\left(\rho\right)\sim 0$ so that
$\overset{\cdot}{R}/R=\overset{\cdot}{a}\sim 0,$ because the gravitational
masses of $s-matter$ and $v-matter$ are opposite and $K\left(\rho\right)$ is
changeable. Consequently, from $\left(141\right)$ and $\left(144\right)$ we
see that $\delta_{vk}\left(t\right)$ will exponentially grow for long-
wavelength in the stage.

2\. From $\left(141\right),$ $\left(143\right)$ and $\left(144\right)$ we see
that $\delta_{vk}\left(t\right)$ can grow faster than that determined by
$\left(144\right)$ because
$\rho_{v0}\delta_{vk}+\rho_{s0}\mid\delta_{sk}\mid>\rho_{v0}\delta_{vk}.$ The
origin is the repulsion between $s-matter$ and $v-matter$.

##  XI Some guesses, new predictions and an inference

###  XI.1 Some guesses

####  XI.1.1 The universe is composed of infinite s-cosmic islands and
v-cosmic islands

If the whole universe is the world in the $S-breaking$ or the the world the
-$V-breaking$, the analysis above is correct. But according to the present
model, as mentioned before, there are the two sorts of symmetry breaking which
are different in essence. Thus it is possible that there are different
breaking forms in different regions of the universe.

As mentioned above, $v-matter$ and $s-matter$ are symmetric and mutually
repulsive, $v-matter$ in the $V-breaking$ can form $v-galaxies$ and $s-matter$
in the $S-breaking$ can form $s-galaxies$.

From this we present a new cosmic model as follows.

The universe is composed of infinite $s-cosmic$ islands and and $v-cosmic$
islands. $\langle\omega_{v}\rangle=0$ and
$\langle\omega_{s}\rangle=\langle\omega_{s}\rangle_{0}$ in $s-cosmic$ islands,
and $\langle\omega_{s}\rangle=0$ and
$\langle\omega_{v}\rangle=\langle\omega_{v}\rangle_{0}$ in the the $v-cosmic$
islands.

A $s-cosmic$ island or a $v-cosmic$ island must be finite. There must be a
transitional region ($T-region$) between a $s-cosmic$ island and a $v-cosmic$
island. In the $T-region,$ it is necessary $\langle\omega_{s}\rangle$ and
$\langle\omega_{v}\rangle$ change from
$\langle\omega_{s}\rangle=\langle\omega_{s}\rangle_{0}$ and
$\langle\omega_{v}\rangle=0$ into $\langle\omega_{s}\rangle=0$ and
$\langle\omega_{v}\rangle=\langle\omega_{v}\rangle_{0},$ respectively.
Consequently, the expectation values $\langle\omega_{s}\rangle_{T}$ and
$\langle\omega_{v}\rangle_{T}$ inside the $T-region$ must satisfy

$0<\mid\langle\omega_{s}\rangle_{T}\mid<\mid\langle\omega_{s}\rangle_{0}\mid,\text{
\ \
}0<\mid\langle\omega_{v}\rangle_{T}\mid<\mid\langle\omega_{v}\rangle_{0}\mid.$
(145)

There must be only $v-cosmic$ islands neighboring a $s-cosmic$ island. This is
because that if two $s-cosmic$ islands are neighboring, they must form one new
larger $s-cosmic$ island.

It is obvious that if there is only one sort of breaking, it is impossible
that such cosmic islands exist.

Based on the following reasons, the probability is very little that a
$v-observer$ accepts messages from a $s-cosmic$ island.

1. The probability must be very small that a $s-particle$ (a quark, a lepton or a photon) in the $s-cosmic$ island comes into the $v-cosmic$ island, because a $s-particle$ in a $s-cosmic$ island is $s-SU(5)$ non-color single state. If a $s-particle$ comes into the $v-cosmic$ island, it would still be non-color single state and would get very big mass. This is impossible due to color confinement. But a bound state of the $s-particles,$ e.g. $\left(u\overline{u}\mp d\overline{d}\right)/\sqrt{2}$ which is a color single state in both $V-breaking$ and $S-breaking,$ possibly comes into the $v-cosmic$ island. It is hardly funded by a $v-observer$ because the boson $\left(u\overline{u}\mp d\overline{d}\right)/\sqrt{2}$ is a $s-colour$ single state in the $v-cosmic$ island which is a particle of dark energy. .

2\. The probability must be very small that a $v-particle$ in the $s-cosmic$
island come into the $v-cosmic$ island as well. The $v-particle$ (a fermion or
a gauge boson) in the $s-cosmic$ island must be massless. If the massless
$v-particle$ comes into the $v-cosmic$ island, it would get its mass. Thus its
static mass will change from $m_{0}=0$ to $m_{0}>0$ so that it must suffer a
strong-repulsive interaction, hence it hardly comes into the $v-cosmic$
island.

3\. Higgs particles in the $s-cosmic$ island must decay fast into fermions or
gauge bosons, hence they cannot come to the $v-cosmic$ island.

4\. $T-regions$ is so big that the probability through which a particle passes
is very small..

The probability must be very small that particles leave the $v-cosmic$ island
because of the same reasons.

A $v-cosmic$ islands and a $s-cosmic$ island can influence on each other by
the Higgs potential in the $T-region$ between both.

As a consequence a $v-observer$ in the $v-cosmic$ island can regard the
$v-cosmic$ island as the whole cosmos. It is possible that Some cosmic islands
are forming or expanding, and the other cosmic islands are contracting.

Thus, according to the present model the cosmos as a whole is infinite and its
properties are always unchanging, and there is no starting point or end of
time.

####  XI.1.2 Mass redshifts

Hydrogen spectrum is

$\omega_{nk}=(E_{n}-E_{k})/\hbar=-\frac{\mu
e^{4}}{2\hbar^{3}}(\frac{1}{n^{2}}-\frac{1}{k^{2}}),\text{ \ \
}\mu=\frac{mM}{m+M},$ (146)

where $m$ is the mass of an electron, and $M$ is the mass of a proton.
According the unified model, $m\propto\upsilon_{e}$, the mass of a quark
$m_{q}\propto\upsilon_{q},$ where $\upsilon_{e}$ and $\upsilon_{q}$ are the
expectation values of the Higgs fields coupling with the electron and the
quark, respectively. $M\propto m_{q}.$

If there are some galaxies inside a $T-region,$ from $\left(145\right)$ we see
that the mass $m_{T}$ of an electron and the mass $M_{T}$ of a proton inside
the $T-region$ must be

$m_{T}<m,\text{ \ \ }M_{T}<M.$

Thus we have

$\mu_{T}<\mu,\text{ \ \
}\bigtriangleup\omega_{nk}=\omega_{nk}-\omega_{nkT}=-\frac{(\mu-\mu_{T})e^{4}}{2\hbar^{3}}(\frac{1}{n^{2}}-\frac{1}{k^{2}})<0.$
(147)

The sort of red-shifts is called $mass$ $redshift$. The mass redshift is
essentially different from the cosmological red-shift mentioned before. Thus,
the photons coming from the star in a $T-region$ must have larger red-shift
than that determined by the Hubble formula at the same distance. Thereby we
guess that some quasars are just the galaxies in the $T-region$ of our cosmos
island and they have the mass redshifts. The fine-structure constant is
considered to change based on the redshifts of some quasars according to the
conventional theory. In contrast with the conventional, we consider that the
mass of electrons possibly changes, but is not the fine-structure constant to
change.

An ordinary $s-galaxy$ and a $s-quasar$ can be neighboring, because a
$T-region$ must be neighboring to an ordinary region.

###  XI.2 New predictions

####  XI.2.1 It is possible that huge voids are equivalent to huge concave
lenses. The densities of hydrogen and helium in the huge voids predicted by
the present model must be more less than that predicted by the conventional
theory.

Hot dark matter, e.g. the neutrinos, can form huge voids, but cannot explain
evolution of structure with middle and small scales. Cold dark matter can
explain evolution of structure with the middle and the small scales, but
cannot explain the huge voids. The problem of huge voids remains unsettled.

According the present model, huge $v-voids$ in the $V-breaking$ are, in fact,
superclusterings of $s-particles$. The huge $v-voids$ not empty, and in which
there is $s-matter$ with a bigger density,
$\rho_{s}^{\prime}\gg\rho_{v}^{\prime},$ $\rho_{s}^{\prime}>\rho_{s}$ and
$\rho_{v}^{\prime}<\rho_{v},$ here $\rho_{s}^{\prime}$ and $\rho_{v}^{\prime}$
denote the densities in the huge $v-voids$. Because there is the repulsion
between $s-matter$ and $v-matter$ and there is the gravity among
$s-particles,$ a huge $v-void$ can form. The forming process is analogous to
that of the neutrino-superclusterings. The characters of such a huge $v-void$
are as follows.

$A$. A $v-void$ must be huge, because there is no other interaction among the
$s-SU(5)$ color single states except the gravity and the masses of the
$s-SU(5)$ color single states are very small.

$B$. When $v-photons$ pass through such a huge $v-void$, the $v-photons$ must
suffer repulsion from $\rho_{s}^{\prime}$ and are scattered by
$\rho_{s}^{\prime}$ as they pass through a huge concave lens. Consequently,
the galaxies behind the huge $v-void$ seem to be darker and more remote.

$C$. Both density of matter and density of dark matter in huge voids must be
more lower than those predicted by the conventional theory. Specifically, the
density of hydrogen in the huge voids must be more less than that predicted by
the conventional theory. Right or mistake of the predict can be confirmed by
the observation of distribution of hydrogen.

It is seen that the present model can well explain the characters of some huge
voids. This is a decisive prediction which distinguishes the present model
from other models.

####  XI.2.2 The gravitation between two galaxies distant enough will be less
than that predicted by the conventional theory.

There must be $s-matter$ between two $v-galaxies$ distant enough, hence the
gravitation between the two $v-galaxies$ must be less than that predicted by
the conventional theory due to the repulsion between $s-matter$ and
$v-matter$. When the distance between the two $v-galaxies$ is small, the
gravitation is not influenced by $s-matter,$ because $\rho_{s}$ must be small
when $\rho_{v}$ is big.

####  XI.2.3 A black hole with its mass and density big enough will transform
into a white hole

Let there be a $v-black$ hole with its mass and density to be big enough so
that the critical temperature $T_{cr}$ can be reached in the $V-breaking.$ If
its mass is so big that its temperature $T_{v}\gtrsim 2\mu/\sqrt{\lambda}$
since the black hole contracts by its self-gravitation, then the expectation
values of the Higgs fields inside the $v-black$ hole will change from
$\varpi_{v}=\varpi_{v0}$ and $\varpi_{s}=0$ into $\varpi_{v}=\varpi_{s}=0$.
Consequently, inflation must occur. After inflation, the highest symmetry will
transit into the $V-breaking$ or the $S-breaking$. No matter which breaking
appears, the energy of the black hole must transform into both $v-energy$ and
$s-energy$. Thus, a $v-observer$ will find that the black hole disappears and
a white hole appears.

In the process, part of $v-energy$ transforms into $s-energy.$ A $v-observer$
will consider the energy not to be conservational because he cannot detect
$s-matter$ except by repulsion. The transformation of black holes is different
from the Hawking radiation. This is the transformation of the vacuum
expectation values of the Higgs fields. There is no contradiction between the
transformation and the Hawking radiation or another quantum effect, because
both describe different processes and based on different conditions (the
density and mass of the black hole must be large enough so that its
temperature $T_{v}\gtrsim 2\mu/\sqrt{\lambda}$). According to the present
model, there still are the Hawking radiation or other quantum effects of black
holes. In fact, the universe is just a huge black hold. The universe can
transform from the $S-breaking$ into the $V-breaking$ because of its
contraction or expansion. This transformation is not quantum effects.

Let there be a $v-cosmic$ island neighboring on a $s-cosmic$ island. It is
possible that the $v-cosmic$ island transforms into a $s-cosmic$ island after
it contracted. In this case, a $s-observer$ in the $s-cosmic$ island must
observe a very huge white hole.

####  XI.2.4 The transformation of the cosmic ultimate

As mentioned before, in the $V-breaking$, when $\rho_{s}>\rho_{v}$, space will
expand with an acceleration. It seems that the universe will always expand
with an acceleration. This is impossible. In the expanding process of the
universe, galaxies can be continue to exist and $v-matter$ will gather so that
a huge $v-black$ hole with its mass and density big enough can be formed after
long enough time, because the repulsion between s-matter and v-matter and the
gravitation among $v-particles$. After the temperature of the black hole
reaches the critical temperature $T_{cr}$, the expectation values of all Higgs
fields will tend to zero and the local space will expand. In the case, both
$V-breaking$ and $s-breaking$ are possibly realized. If the density of
$v-matter$ around the huge $v-black$ hole is little enough, the $S-breaking$
will is realized. Consequently the $V-breaking$ transforms into the
$S-breaking$ and the $v-world$ transforms the $s-world$. If the density of
$v-matter$ around the huge $v-black$ hole is very large, the $V-breaking$ will
is realized. This is because the transformation of symmetry breaking must
cause the transformation of existing form of matter. $V-matter$ whose
temperature is low around the huge $v-black$ hole will prevent the
transformation of the $V-breaking$ into the $S-breaking.$ After the average
density of v-matter is little because space expands and a huge $v-black$ hole
with its mass and density big enough formed, the transformation of the
$V-breaking$ into the $S-breaking$ can occur.

It is seen that in the both cases that the universe is little enough because
space contracts or is large enough because space expands, it will occur that
one sort of breaking transforms into the other and the world transforms the
other world.

###  XI.3 An inference :$\lambda_{eff}=\lambda=0,$ although $\rho_{0}\neq 0$

The effective cosmological constant $\lambda_{eff}=\lambda+\rho_{g0}.$ The
conventional theory can explain evolution with a small $\lambda_{eff}.$ Such a
small $\lambda_{eff}$ cannot be derived from a elementary theory. According to
the conventional quantum field theory, $\lambda_{eff}=\lambda+\rho_{0},$
$\rho_{0}\ggg\lambda_{eff},$ In fact, $\rho_{0}$ is divergence. According to
the conventional gravitational theory, $\rho=\rho_{g}=\rho$ ($c=1$).
Consequently, the issue of the cosmological constant appears.

$\rho_{0}=0$ can be obtained by some supersymmetric model, but it is not a
necessary result. On the other hand, the particles predicted by the
supersymmetric theory have not been found, although their masses are not
large.

$\rho_{0}=0$ is a necessary result of our quantum field theory without
divergence${}^{[6]}.$ In this theory, $\rho_{0}=0$ is naturally obtained
without normal order of operators, there is no divergence of loop corrections,
and dumpling dark matter is predicted${}^{[7]}.$

According to the present model, $\lambda_{eff}=\lambda=0,$ although $\rho_{0}$
is still very big according to the conventional quantum field theory.

Proof: $\lambda_{eff}=\lambda=0,$ although $\rho_{0}\neq 0.$

Applying the conventional quantum field theory to the present model, we have
$\rho_{0}=\rho_{s0}+\rho_{v0}.$ Both $\rho_{s0}$ and $\rho_{v0}$ must be two
constants. According to the conjecture 1, $s-particles$ and $v-particles$ are
strictly symmetric in essence. Hence

$\rho_{s0}=\rho_{v0}=\rho_{0}/2.$

According to the conjecture 1, the gravitational mass of $s-matter$ is
opposite to that of $v-matter$. i.e., $\rho_{gs}=-\rho_{gv}$ when
$\rho_{s}=\rho_{v}$. Hence we have

$\displaystyle\rho_{0}$ $\displaystyle=\rho_{s0}+\rho_{v0}=2\rho_{s0}\neq 0,$
(148a) $\displaystyle\rho_{g0}$ $\displaystyle=\rho_{sg0}+\rho_{vg0}=0.$
(148b)

Thus, there is no the fine tuning problem, even if $\lambda_{eff}\neq 0$.

$\lambda_{eff}=0$ is a necessary inference, because evolution of the cosmos
can be explained by the present model without $\lambda_{eff}$. Consequently,
although $\rho_{0}\neq 0,$ we have still

$\lambda_{eff}=0=\lambda+\rho_{g0}=\lambda.$ (149)

This is an direct inference of the present model, and independent of a quantum
field theory. Thus, the cosmological constant issue has been solved.

For the vacuum state in the $S-breaking$ or the $V-breaking$, the Einstein
field equation is reduced to

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-8\pi
G\left(T_{s0\mu\nu}-T_{v0\mu\nu}\right)=-8\pi
G\left(T_{v0\mu\nu}-T_{s0\mu\nu}\right)=0.$ (150)

This is a reasonable result.

##  XII Primordial nucleosynthesis

###  XII.1 F-W dark matter model

The $F-W$ dark matter model[7] is a necessary inference of the quantum field
theory without divergence. The $F-W$ dark matter model is similar with the
mirror dark matter model.

According to the mirror dark matter model, it is impossible that the density
of matter is equal to that of dark matter in order to explain the primordial
nucleosynthesis and $CMBR$. This is too difficultly understood, because matter
and mirror matter are symmetric and both can transform from one into another
when temperature is high enough.

In contrast with the mirror dark matter model, according to the $F-W$ matter
model, $F-matter$ (ordinary matter) and $W-matter$ (dark matter) are not only
symmetric, but also $\rho_{vF}=\rho_{vW}.$ If the total density of matter and
dark energy is $\rho_{t}$ and the ratio of the density of dark energy
$\rho_{de}$ to $\rho_{t\text{ }}$ is $0.73$,
$\left(\rho_{vF}+\rho_{vW}\right)/\rho_{t}=2\rho_{vF}/\rho_{t}=0.27.$
Considering only $F-baryon$ matter is visible and the radio of the density
$\rho_{vFB}$ to $\rho_{t}$ is $\rho_{vFB}/\rho_{t}=0.04,$ dark matter can be
classified into the following three sorts: invisible $F-matter$ for a time
whose density is $\rho_{vFu}=\left(0.27/2-0.4\right)\rho_{t}=0.095\rho_{t},$
invisible $W-baryon$ matter whose density is $\rho_{vWB}=0.04\rho_{t},$ and
invisible $W-non-baryon$ matter whose density is $\rho_{vWu}=0.095\rho_{t}.$
$\rho_{vWB}$ can form dark galaxies and can be observed, $\rho_{vWu}$ and
$\rho_{vFu}$ cannot form any dumpling. The $vFu-particles$ is possibly
observed in future. $\rho_{vF}$ and $\rho_{vW}$ can transform from one into
another when temperature is high enough.

According to the present cosmological model, there are $s-matter$ and
$v-matter$ which are symmetric in principle. After symmetry spontaneously
breaking, $s-matter$ and $v-matter$ are no longer symmetric. In the
$V-breaking$, $v-matter$ corresponds visible matter and dark matter,
$s-matter$ corresponds to so-called dark energy and must exist in the form of
the $s-SU(5)$ color single states. The masses of all color single states are
non-zero. But the masses are different from each other. Some masses of color
single states (their density are denoted by $\rho_{sl}$) are possibly less
than $1Mev$ so that they may be ignored when temperature $T\gtrsim 1Mev,$ and
the others (their density are denoted by $\rho_{sm}$) are larger than $1Mev$.
$\rho_{sm}$ and $\rho_{sl}$ can be determined based on observation.

When the $F-W$ dark matter model and the present cosmological model are
simultaneously considered, the primordial nucleosynthesis and CMBR can be
explained.

###  XII.2 Primordial nucleosynthesis

According to the $F-W$ matter model[7] which is similar with the mirror dark
matter model, the mechanism of primordial nucleosynthesis is the same as the
conventional theory. But the mechanism of space expansion of the present model
is different from that of the conventional theory. For short, we consider only
influence of space expansion on the primordial nucleosynthesis and $CMBR.$

The primordial helium abundance $Y_{4}$ is determined by $n_{n}/n_{p}^{[8]},$

$\displaystyle Y_{4}$
$\displaystyle=2/\left[1+\left(n_{n}/n_{p}\right)^{-1}\right],$ (151a)
$\displaystyle n_{n}/n_{p}$ $\displaystyle=exp\left(-\bigtriangleup
m/kT_{1}\right),\text{ }\bigtriangleup m=m_{n}-m_{p},$ (151b)

where $n_{n}/n_{p}$ is the neutron-proton ratio in the unit comoving volume at
the freeze-out temperature $T_{1}.$ $T_{1}$ is determined by
$\Gamma_{1}=\Gamma\left(T_{1}\right)$ and
$H_{1}=\eta\rho_{g}\left(T_{1}\right),$ here .$\Gamma$ is the interaction rate
experienced by a particle.

As mentioned above,

$\rho_{vFm}=\rho_{vWm},\text{ \ }\rho_{vF\gamma}=\rho_{vW\gamma},\text{\
}\rho_{v}=2\rho_{vFm}+2\rho_{vF\gamma},\text{ \
}\rho_{s}=\rho_{sm}=\rho_{sm}^{\prime}+\rho_{sl},$

and in the $V-breaking,$ $T_{1}=T_{v1}$ and in general $T_{v1}\neq T_{s1}.$
Since the masses of all $s-color$ single states are non-zero, it is possible
there are some $s-color$ single states with their masses $m_{sm}\gtrsim 1Mev$
and the others with their masses $m_{sl}$ and $1Mev>m_{sl}>1ev.$ Thus, when
$T_{s}\gtrsim 1Mev,$ $m_{sl}$ may be ignored so that $\rho_{sl}\varpropto
T_{s}^{4},$ and when $T_{s}\lesssim 1ev$,
$\rho_{s}=\rho_{sm}^{\prime}+\rho_{sl}\varpropto T_{s}^{3}.$

Considering $m_{p}\sim m_{n}\sim 1Gev,$ $m_{e}=0.511Mev,$ $m_{\gamma}=0$ and
$m_{\nu i},$ $i=e,$ $\mu$ and $\tau$, are regarded as zero, $g_{e}=7/2,$
$g_{\gamma}=2$ and $g_{\nu i}=7/4$ so that $g^{\ast}=10.75.$ When $T\sim
T_{v1}\sim T_{s1}\sim 1Mev,$ $m_{e}$ may be ignored and the universe is
dominated by $\rho_{sl}$ and $\rho_{s\gamma},$ we have

$\displaystyle H_{1}^{2}$
$\displaystyle=\frac{\overset{\cdot}{R}^{2}\left(T_{1}\right)}{R^{2}\left(T_{1}\right)}=\eta\rho_{g}\left(T_{1}\right)=\eta\left[\rho_{v}\left(T_{v1}\right)-\rho_{s}\left(T_{s1}\right)\right]$
$\displaystyle=\eta\left[\left(\rho_{vFm}+\rho_{vWm}+\rho_{vF\gamma}+\rho_{vW\gamma}\left(T_{v1}\right)\right)-\left(\rho_{sm}^{\prime}+\rho_{sl}\left(T_{s1}\right)\right)\right]$
$\displaystyle\simeq\eta\left[\rho_{vF\gamma}+\left(\rho_{vW\gamma}\left(T_{v1}\right)-\rho_{sl}\left(T_{s1}\right)\right)\right]$
$\displaystyle=\eta\left[\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}+\left(\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}-\rho_{sl}\left(T_{s1}\right)\right)\right].$
(152) $\rho_{sl}\left(T_{s}\right)/\rho_{sl}\left(T_{s1}\right)\varpropto
T_{s}^{4}/T_{s1}^{4}=\left(R_{1}/R\right)^{4}=T_{v}^{4}/T_{v1}^{4},\text{ \
when }T_{v}\gg T_{vdec},$ (153)

where $T_{vdec}$ is the $v-photon$ decoupling temperature (see the following).
$\rho_{sl}$ and $\rho_{sm}^{\prime}$ are two parameters which should be
determined by observations. $\rho_{sl}\left(T_{s1}\right)$ can be so chosen
that

$\displaystyle\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}-\rho_{sl}\left(T_{s1}\right)$
$\displaystyle\sim 0,$ (154a) $\displaystyle H_{1}^{2}$
$\displaystyle\sim\eta\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}.$ (154b)

For the freeze-out temperature $T_{v1}$ is determined by $\Gamma_{1}$ and
$H_{1}$. $\Gamma_{1}$ in the present model is the same as that in the
conventional theory. $\left(154b\right)$ is the same as that in the
conventional theory as well. Hence the present model can explain the
primordial nucleosynthesis and $Y_{4}$ as the conventional theory. For
example, taking the rough approximation $\Gamma_{1}=H_{1},$ considering
$\left(154b\right),$ we get the equation to determine $T_{v1}$

$\Gamma_{1}\left(T_{v1}\right)\sim
G_{F}^{2}T_{v1}^{5}=H_{1}=\left[\eta\left(\rho_{v}\left(T_{v1}\right)-\rho_{s}\left(T_{s1}\right)\right)\right]^{1/2}\simeq\left(\eta\frac{\pi^{2}}{30}g_{v}^{\ast}T_{v1}^{4}\right)^{1/2}.$
(155)

This result and $n_{n}/n_{p}$ and $Y_{4}$ corresponding to this are the same
as those of the conventional theory[19].

It is seen that although ordinary matter and dark matter are completely
symmetric so that $\rho_{vF}=\rho_{vW}$, we can still obtained the result of
conventional theory, provided the F-W dark matter model and the present model
are simultaneously considered. This is different from the mirror dark matter
model.

##  XIII Cosmic microwave background radiation ($CMBR)$

###  XIII.1 The recombination temperature $T_{rec}$

It is the same as the conventional theory that there are the inflation and big
bang processes in the present model. Hence there must be the cosmic microwave
background radiation $(CMBR).$

The recombination temperature of the present model is the same as that of the
conventional theory, because it is independent of s-matter in the V-breaking.
From the following formulas[19] we can determine the recombination temperature
$T_{rec},$

$\frac{1-\chi}{\chi^{2}}=1.1\times 10^{-8}\xi
T_{v}^{3/2}\exp\left(B/T_{v}\right),\text{ \ }A\equiv T_{v}/Tv0.$

where $\chi=n_{e}/n=n_{p}/n,$ $n=n_{p}+n_{H}$ to be number density and
$B=13.6ev$ is the ionization potential of hydrogen. Taking $\xi\sim 5\times
10^{-10}$ and $\chi=0.1,$ considering $n_{e}=n_{p}$,
$n_{\gamma}=\left(\zeta\left(3\right)/\pi^{2}\right)g_{\gamma}T^{3}=\left(2.4/\pi^{2}\right)T^{3},$
$n=\xi n_{\gamma},$ $T_{v}=Tv0\left(T_{v}/Tv0\right)$ and $Tv0=2.35\times
10^{-4}ev,$ we have[19]

$\displaystyle T_{rec}$ $\displaystyle=3423.5K=0.295ev.$ (156a)
$\displaystyle\left(1+z_{rec}\right)$ $\displaystyle=T_{rec}/Tv0=1255.$ (156b)

###  XIII.2 The temperature $T_{eq}$ of matter-radiation equality

In contrast with the conventional theory, according to the $F-W$ dark matter
model, $F-matter$ and $W-matter$ are completely symmetric so that not only
there are $F-photon$ (ordinary photons), but also $W-photons$ (dark-matter
photons). $\rho_{v\gamma}=\rho_{vF\gamma}+\rho_{vW\gamma}=2\rho_{vF\gamma},$
$\rho_{vm}=\rho_{vFm}+\rho_{vWm}=2\rho_{vFm}.$ From this we can estimate the
temperature $T_{eq}$ of matter-radiation equality as follows.

When only the photons and the three species of neutrinos are considered (here
the three species of neutrinos are regarded as massless)., we have[18]

$\rho_{vF\gamma}=\rho_{vW\gamma}=\frac{\pi^{2}}{30}g_{\gamma}^{\ast}\left(\frac{kT_{v}}{\hbar
c}\right)^{4}.$ (157a) $g_{\gamma}^{\ast}=2+\frac{7}{8}\times
6\times\left(\frac{4}{11}\right)^{4/3}=3.36.$ (157b)

Thus, considering $T_{v0}=T_{0}=2.728K=2.35\times 10^{-4}ev,$ we get

$\displaystyle\rho_{vr0}$
$\displaystyle=2\rho_{vFr0}=2\times\left[\frac{3.36}{2}\times\frac{\pi^{2}}{30}\times
2\times\left(2.35\times 10^{-13}Gev\right)^{4}\right]$
$\displaystyle=6.7425\times 10^{-51}Gev^{4}.$ (158)

Observation shows that the total density of matter and dark matter is
$\rho_{0}=\Omega_{0}\rho_{c}=0.27\rho_{c}.$ According to the $F-W$ model, this
implies

$\displaystyle\rho_{v0}$
$\displaystyle=2\rho_{vFm0}+2\rho_{vFr0}\simeq\rho_{vm0}=\Omega_{0}\rho_{c}$
$\displaystyle=1.8789\times 10^{-26}h^{2}\Omega_{0}\cdot kg\cdot
m^{-3}=9.238\times 10^{-48}Gev^{4},\text{ \ when }h=0.65.$ (159)

where $h=0.5-0.8,$ $\rho_{vm0}=\rho_{vFm0}+\rho_{vWm0},$ and$\
\rho_{vr0}=\rho_{vF\gamma 0}+\rho_{vWr0}=2\rho_{vF\gamma 0}=2\rho_{r0}.$

From $\left(158\right)-\left(159\right)$ $T_{veq}$ can be determined as
follows

$\displaystyle\rho_{vmeq}\left(T_{veq}\right)$
$\displaystyle=\rho_{vm0}\left[R_{0}/R_{eq}\left(T_{veq}\right)\right]^{3}=\rho_{v\gamma
0}\left[R_{0}/R_{eq}\left(T_{veq}\right)\right]^{4}=\rho_{v\gamma
eq}\left(T_{veq}\right),\text{ }$ (160a) $\displaystyle\frac{R_{0}}{R_{eq}}$
$\displaystyle=\frac{T_{veq}}{T_{v0}}=\frac{\rho_{vm0}}{\rho_{v\gamma
0}}=\frac{\Omega_{0}\rho_{c}}{2\rho_{vFr0}}=\left(1+z_{eq}\right)=1370,\text{
}$ $\displaystyle T_{veq}$ $\displaystyle=0.32ev,\text{ when }h=0.65,$ (160b)
$\displaystyle\frac{R_{0}}{R_{eq}}$
$\displaystyle=\frac{T_{veq}}{T_{v0}}=\frac{\rho_{vm0}}{\rho_{v\gamma
0}}=1272\text{, }$ $\displaystyle T_{veq}$
$\displaystyle=0.295ev=T_{rec},\text{ when }h=0.624.$ (160c)

According to the present model, $\left(160\right)$ is a crude approximation,
$\rho_{v}-\rho_{s}$ is changeable not only because of $\rho_{vm}\varpropto
R^{-3},$ $\rho_{v\gamma}\varpropto R^{-4}$ and $\rho_{sm}\varpropto R^{-3},$
but also because of $\left(65c\right).$

According to the conventional theory, $\rho_{\gamma 0}=\rho_{vr0}/2$,
$\rho_{0}=\rho_{v0},$ hence if $h=0.65$ and $\Omega_{0}=0.27$,

$T_{eq}^{\prime}=T_{0}R_{0}/R_{eq}^{\prime}=\left(1+z_{eq}^{\prime}\right)T_{0}=\left(\rho_{m0}/\rho_{r0}\right)T_{0}=2\left(\rho_{m0}/\rho_{vr0}\right)T_{0}=0.64ev=2T_{veq}.$
(161)

It is seen that $T_{eq}^{\prime}$ is remarkably different from $T_{veq}.$ When
$T_{v}\sim T_{veq},$ the universe is not matter-dominated.

###  XIII.3 Decoupling temperature

Let when $T_{v}=T_{veq},$ $T_{s}=T_{sq},$
$\rho_{sm}\left(T_{sq}\right)=t_{mq}\rho_{vm}\left(T_{veq}\right)$ and the
masses of all s-color single states cannot be ignored, i.e.
$\rho_{sm}\left(T_{sq}\right)=\rho_{s}\left(T_{sq}\right).$ Considering

$\frac{T_{sq}}{T_{s}}=\frac{R}{R_{eq}}=\frac{T_{veq}}{T_{v}},$ (162)

ignoring the transformation $\rho_{v}$ into $\rho_{s},$ we can rewrite
$\rho_{vm},$ $\rho_{v\gamma}$ and $\rho_{sm}$ as follows.

$\displaystyle\rho_{vm}$
$\displaystyle=\frac{\rho_{vm}}{\rho_{vm0}}\frac{\rho_{vm0}}{\rho_{c}}\rho_{c}=\frac{R_{0}^{3}}{R^{3}}\Omega_{m0}\rho_{c}=A^{3}\Omega_{m0}\rho_{c},\text{
}A\equiv\frac{Tv}{Tv0}=1+z,$ (163a) $\displaystyle\rho_{v\gamma}$
$\displaystyle=\frac{\rho_{v\gamma}}{\rho_{v\gamma eq}}\frac{\rho_{v\gamma
eq}}{\rho_{vmeq}}\frac{\rho_{vmeq}}{\rho_{vm0}}\frac{\rho_{vm0}}{\rho_{c}}\rho_{c}=\left(\frac{T_{v0}}{Tveq}\right)A^{4}\Omega_{m0}\rho_{c},$
(163b) $\displaystyle\rho_{sm}$
$\displaystyle=\frac{\rho_{sm}}{\rho_{sm}q}\frac{\rho_{sm}q}{\rho_{vme}q}\frac{\rho_{vmeq}}{\rho_{vm0}}\frac{\rho_{vm0}}{\rho_{c}}\rho_{c}=t_{mq}A^{3}\Omega_{m0}\rho_{c},$
(163c)

Ignoring $K,$ considering $H_{0}=\sqrt{\eta\rho_{c}}=65km\cdot\left(s\cdot
Mpc\right)^{-1}=1.4\times 10^{-42}Gev,$ $\Omega_{vm0}=0.27,$ and
$T_{v0}/Tveq=\left(2.35\times 10^{-4}\right)/0.32=0.734\times 10^{-3},$we get

$\displaystyle H^{2}$
$\displaystyle=\eta\rho_{g}=\eta\left(\rho_{vm}+\rho_{v\gamma}-\rho_{sm}\right)$
$\displaystyle=0.27\times 1.4^{2}\times
10^{-82}A^{3}\left(1-t_{mq}+0.734\times 10^{-3}A\right)Gev.$ (164)

This is the only deference between the present model and the conventional
theory in order to determine the decoupling temperature.

We have the same interaction rate experienced by one photon and Saha equation
as the conventional theory[19]

$\Gamma=n_{vFe}\sigma_{th}=\chi\xi
n_{vF\gamma}\sigma_{th}=\frac{2.4}{\pi^{2}}\xi T_{v0}^{3}\sigma_{th}\chi
A^{3}=5.4\times 10^{-36}\xi\chi A^{3}Gev,$ (165)
$\frac{1-\chi}{\chi^{2}}=1.1\times 10^{-8}\xi
T_{v}^{3/2}\exp\frac{13.6}{T_{v}}=3.96\times 10^{-14}\xi
A^{3/2}\exp\frac{57872}{A},$ (166)

where $T_{v}=T_{v0}\left(T_{v}/T_{v0}\right)=2.35\times 10^{-4}A$ is
considered. Only for comparison of the present model with the conventional
theory, we use the same equation $\left(166\right)$ and the same crude
approximation $\Gamma=H$ to evaluate the decoupling temperature. Taking
$\Gamma=H$, and $t_{mq}=1.5,$ from $\left(164\right)-\left(165\right)$ we have

$\displaystyle\chi A^{3/2}$ $\displaystyle=\xi^{-1}1.347\times
10^{-7}\left(1-t_{mq}+0.734\times 10^{-3}A\right)^{1/2}$
$\displaystyle=269\left(-0.5+0.734\times 10^{-3}A\right)^{1/2}.,\text{ \ when
\ }\xi=5\times 10^{-10},$ (167)

Substituting $\left(167\right)$ into $\left(166\right)$, we get

$\displaystyle A^{3/2}$ $\displaystyle=\xi^{-1}1.347\times
10^{-7}\left(1-t_{mq}+0.734\times 10^{-3}A\right)^{1/2}$
$\displaystyle+\xi^{-1}7.185\times 10^{-28}\times\left(1-t_{mq}+0.734\times
10^{-3}A\right)\times\exp\frac{57872}{A}$ (168)

Taking $t_{mq}=1.5$ and $\xi=5\times 10^{-10}$, we get

$A=1+z_{dec}=1097,\text{ \ \ }T_{vdec}=0.258ev,\text{ \ }\chi=0.004.$ (169)

Taking $t_{mq}=1.5$ and $\xi=3\times 10^{-10}$, we get

$A=1+z_{dec}=1108,\text{ \ }T_{vdec}=0.260ev,\text{ \ }\chi=0.0068,$ (170)

Taking place of $\left(164\right)$ by the equation in the conventional theory

$H^{2}=\eta\rho_{g}=\eta\rho_{vm}=0.27\times 1.4^{2}\times 10^{-82}A^{3}Gev,$
(171)

we have

$\displaystyle A$ $\displaystyle=1+z_{dec}=1121,\text{ \
}T_{vdec}=0.263ev,\text{ \ }\chi=0.007,$ $\displaystyle\text{when }t_{mq}$
$\displaystyle=1.5\text{, \ }\xi=5\times 10^{-10},$ (172) $\displaystyle A$
$\displaystyle=1+z_{dec}=1132,\text{ \ }T_{vdec}=0.266ev,\text{ \
}\chi=0.012,$ $\displaystyle\text{when }t_{mq}$ $\displaystyle=1.5\text{, \
}\xi=3\times 10^{-10}.$ (173)

$z_{dec}$ is not susceptible for change of $t_{mq}$ in the scope $1.1-1.7.$

Considering

$\frac{\rho_{vmdec}}{\rho_{vmeq}}=\left(\frac{T_{vdec}}{T_{veq}}\right)^{3},\text{
\ \ }\frac{\rho_{v\gamma dec}}{\rho_{v\gamma
eq}}=\left(\frac{T_{vdec}}{T_{veq}}\right)^{4},\text{ \ \
}\rho_{vmeq}=\rho_{v\gamma eq},$ (174)

we have

$\frac{\rho_{v\gamma
dec}}{\rho_{vmdec}}=\frac{T_{vdec}}{T_{veq}}=\frac{0.258}{0.32}=0.81.$ (175)

It is seen from $\left(175\right)$ that in the decoupling stage,
$\rho_{vmdec}\sim\rho_{v\gamma dec}$ and the universe is not matter-dominated.
This is different from the conventional theory.

###  XIII.4 Space-time is open, i. e. $K<0$.

The first peak of the $CBMR$ power spectra is the evidence of existence of the
elementary wave. The elementary wave began at reheating ($T=T_{reh}$) and
ended at recombination after $3.8\times 10^{5}$ years ($T=T_{rec}$) according
to the conventional theory. Let the temperature of reheating is $T_{reh}$. In
the period $T_{reh}$ descends into $T_{rec},$ baryons exist in plasma. The
sound speed of plasma is $c_{s}=\partial p/\partial\rho=\sqrt{5T_{b}/3m_{p}}.$
Let the period in which $T_{reh}$ descends into $T_{rec}$ is $\bigtriangleup
t_{hc}$ according to the present model and that according to the conventional
theory is $\bigtriangleup t_{hc}^{\prime}=3.8\times 10^{5}a,$ there must be

$\bigtriangleup t_{hc}>\bigtriangleup t_{hc}^{\prime}.$ (176)

The reasons are as follows.

That $\left(152\right)$ holds implies

$\displaystyle\rho_{g}\left(T_{v1}\right)$
$\displaystyle=\rho_{vFm}\left(T_{v1}\right)+\rho_{vWm}\left(T_{v1}\right)+\rho_{vFr}\left(T_{v1}\right)-\rho_{sm}\left(T_{s1}\right)+\left(\rho_{vWr}\left(T_{v1}\right)-\rho_{sl}\left(T_{s1}\right)\right)$
$\displaystyle\sim\rho_{vFm}\left(T_{v1}\right)+\rho_{vWm}\left(T_{v1}\right)+\rho_{vFr}\left(T_{v1}\right)-\rho_{sm}\left(T_{s1}\right)\sim\rho_{vF\gamma}\left(T_{1}\right)$
$\displaystyle\sim\rho_{m}\left(T_{1}\right)+\rho_{\gamma}\left(T_{1}\right)\sim\rho_{\gamma}\left(T_{1}\right)=\rho_{g}\left(T_{1}\right),$
(177)

where $\rho_{m}\left(T\right)$ and $\rho_{\gamma}\left(T\right)$ are the mass
density in the conventional theory $(T=T_{v}),$ and
$\rho_{m}\left(T_{1}\right)\ll\rho_{\gamma}\left(T_{1}\right)$ etc are
considered. It is necessary that when $T_{v}\geq T_{v1},$ $\left(152\right)$
or $\left(177\right)$ still holds. This is because
$\rho_{sl}\left(T_{s}\right)\varpropto R^{-4},$
$\rho_{vF\gamma}\left(T_{v}\right)=\rho_{vW\gamma}\left(T_{v}\right)\varpropto
R^{-4}$ and $\rho_{v\gamma}\left(T_{v}\right)\gg\rho_{vm}\left(T_{v}\right)$
and $\rho_{sl}\left(T_{s}\right)\gg\rho_{sm}\left(T_{s}\right)$ when
$T_{v}\geq T_{v1}.$ When $T_{v}<T_{v1},$ it is necessary that
$\rho_{vW\gamma}\left(T_{v}\right)<\rho_{sl}\left(T_{s}\right)$. This is
because
$\rho_{vF\gamma}\left(T_{v}\right)=\rho_{vW\gamma}\left(T_{v}\right)\varpropto
R^{-4}$ still holds, but $\rho_{sl}\left(T_{s}\right)\varpropto R^{-3}$ (due
to $1Mev>m_{sl}\gtrsim 1ev$) when $T_{v}<T_{v1}.$ Hence when $T_{v}<T_{v1},$

$H^{2}\left(T_{v}\right)=\eta\rho_{g}\left(T_{v}\right)<\eta\left(\rho_{m}\left(T\right)+\rho_{\gamma}\left(T\right)\right)=\eta\rho_{g}^{\prime}\left(T\right)=H^{\prime
2}\left(T\right).$ (178)

This implies $\left(176\right)$ to hold. On the other hand, the sound speed
$c_{vs}=\partial p_{v}/\partial\rho_{v}$ is determined by only $p_{v}$ and
$\rho_{v}$ or $T_{v}$ and $m_{p},$ and is independent of $\rho_{s}.$ Hence
$c_{vs}\sim c_{s}^{\prime}$ when $T_{reh}$ descends into $T_{rec}$, here
$c_{s}^{\prime}$ is the sound speed in the conventional theory when
$T_{reh}\geq T\geq T_{rec}$. Thus when temperature descends from $T_{reh}$
descends into $T_{rec},$ the propagating distance of the elementary sound wave
must be longer according to the present model than that according to the
conventional theory.

Based on $\bigtriangleup t_{hc}^{\prime}=3.8\times 10^{5}a,$ space-time is
flat or $K=0$ as the conventional theory, but based on $\bigtriangleup
t_{hc}>\bigtriangleup t_{hc}^{\prime}$ and $c_{s}=c_{s}^{\prime}=\partial
p/\partial\rho,$ space-time is open or $K<0$ as the the present model. This is
consistent with the present model according which $K<0$ when
$\rho_{g}=\rho_{v}-\rho_{s}<0$ in present stage. We will discuss the issue in
detail in the following paper.

##  XIV Conclusions

The new conjectures are proposed that there are $s-matter$ and $v-matter$
which are symmetric, whose gravitational mass densities are opposite to each
other and whose energies are all positive. Both can transform from one to
another when temperature $T\gtrsim T_{cr}$. Consequently there is no
singularity in the model and the cosmological constant
$\lambda=\lambda_{eff}=0$ is determined although the energy density of the
vacuum state is still large and there is no the fine tuning problem, even if
$\lambda_{eff}\neq 0$.

There are two sorts of breaking modes, i.e. the $S-breaking$ and the
$V-breaking$. In the $V-breaking$ $v-SU(5)$ symmetry is broken into
$v-SU(3)\times U\left(1\right)$ for $v-particles$ and $s-SU(5)$ symmetry is
still strictly kept for $s-particles.$ Consequently $v-particles$ get their
masses determined by the $SU(5)$ $GUT$ and form the $v-galaxies$ etc., while
$s-particles$ are massless and form $s-SU(5)$ color-single states which
loosely distribute in space and can cause space to expand with an
acceleration. In contrast with the dark energy, the gravitational masses of
$s-matter$ is negative in the $V-breaking$.

The conjectures are not in contradiction with all given experiments and
astronomical observations up to now, although the conjecture 1 violates the
equivalence principle.

The curvature factor $K$ in the RW metric is regarded as a function of the
gravitational mass density in the comoving coordinates.

Based on the present model, the space evolving process is as follows. Firstly,
in the $S-breaking$, $\rho_{Sg}=\rho_{s}-\rho_{v}>0$ and $K>0,$ hence space
contracts and $T_{s}$ rises. When $T_{v}\sim T_{s}=T_{cr}$,
$\langle\omega_{s}\rangle=\langle\omega_{v}\rangle=0,$ both $v-SU(5)$ and
$s-SU(5)$ symmetries is strictly kept (this state have the highest symmetry),
and the masses of all particles originating from the couplings with the Higgs
fields are zero so that $\rho_{s}$ and $\rho_{v}$ can transform from one into
another. As a consequence $\rho_{s}=\rho_{v},$ $T_{v}=T_{s}$ and
$\widetilde{\rho}_{Sg}=\rho_{s}-\rho_{v}+V_{0}=V_{0}$ so that inflation must
occur. After the inflation, the phase transition of the vacuum, i.e. the
reheating process, occurs. After the reheating process, this state with the
highest symmetry transits to the state with the $V-breaking$. Space in the
$V-breaking$ have three evolving stages. Space firstly expands with a
deceleration because $\rho_{Vg}=\rho_{v}-\rho_{s}>0$ and $K>0$; then comes to
static because $\rho_{Vg}=0$ and $K=0$; and finally expands with an
acceleration up to now because $\rho_{Vg}<0$ and $K<0$. The results above is
still valid when $V\rightleftarrows S$ and $v\rightleftarrows s.$ It is seen
that the world in the $S-breaking$ and the world in the $V-breaking$ can
transform from one into another. The evolving process of the cosmos is
different from that determined by the conventional theory. $w=p/\rho$ changes
from $w>0$ into $w<-1$ is obtained.

There are the critical temperature $T_{cr}$, the highest temperature
$T_{\max}$, the least scale $R_{\min}$ and the largest energy density
$\rho_{\max}$ in the universe. $R_{\min}$ and $T_{cr}$ are two new important
constants, $T_{\max}$ and $\rho_{\max}$ are determined by
$R\left(T_{cr}\right)$.

A formula is derived which well describes the relation between a luminosity
distance and its redshift.

Generalizing equations governing nonrelativistic fluid motion to the present
model, the equations of $v-structure$ formation have been derived. According
to the equations, galaxies formed earlier and more easily than the
conventional theory.

Two guesses have been presented. The universe is composed of infinite
$s-cosmic$ islands and $v-cosmic$ islands. Some huge redshifts (e.g. the big
redshifts of quasi-stellar objects) are explained as the mass redshifts which
is caused by $m_{eT}<m_{e},$ here $m_{eT}$ is the mass of an electron in a
transiting region and $m_{e}$ is the given mass of an electron in the normal
region.

Three new predicts have been given. Huge $v-voids$ in the $V-breaking$ are not
empty, but are superclusterings of $s-particles$, there must be $s-matter$
with its density $\rho_{s}\gg\rho_{v}$ in the $v-voids$. It is possible that
huge voids is equivalent to a huge concave lens. The density of hydrogen and
the density of helium in the huge voids predicted by the present model must be
more less than that predicted by the conventional theory. It is possible that
a $v-black$ hole with its big enough mass and density can transform into a
huge white hole by its self-gravitation.

The primordial nucleosynthesis and $CMBR$ are explained based on the $F-W$
dark matter model (or the mirror dark matter model) and this cosmological
model.

The first peak of the $CBMR$ power spectra is the evidence of existence of the
elementary wave. The elementary wave began at reheating and ended at
recombination after $\bigtriangleup t_{hc}^{\prime}\equiv 3.8\times 10^{5}$
years according to the conventional theory. But according to the present
model, it is necessary that $\bigtriangleup t_{hc}>\bigtriangleup
t_{hc}^{\prime},$ because the sound speed $c_{s}\sim c_{s}^{\prime}$ and
$H=\eta\rho_{g}<H^{\prime}=\eta\rho_{g}^{\prime}=\eta\rho^{\prime}.$
Consequently, space-time is open or $K<0$ according to the present model

###### Acknowledgement 5

I am very grateful to professor Zhao Zhan-yue and professor Wu Zhao-yan for
their helpful discussions and best support.

## References

  * (1) S. W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambidge University Press, (1999) p7, p98, p101, p137, p256-298.
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  * (4) P.J.E. Peebles and B. Ratra, Astrophys. J.325 (1988) L17; B Ratra and P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406; P.J.E., Peebles and B. Ratra, Rev. Mod. Phys. 75 (2003) 559.
  * (5) L.J, Hall, Y. Nomura, and S.J. Oliver, Phys. Rev. Lett. 95 (2005) 141302.
  * (6) S-H. Chen, ‘Quantum Field Theory Without Divergence A’, (2002a) hep-th/0203220; S-H, Chen, ‘Significance of Negative Energy State in Quantum Field Theory A’ (2002b) hep-th/0203230; S-H, Chen, ‘Quantum Field Theory :New Research’, O. Kovras Editor, Nova Science Publishers, Inc. (2005a) p103-170.
  * (7) S-H, Chen, ‘A Possible Candidate for Dark Matter’, (2001) hep-th/0103234; S-H, Chen, ‘Progress in Dark Matter Research’ Editor: J. Val Blain, (2005b) pp.65-72. Nova Science Publishers, Inc. arXiv: 1001.4221.
  * (8) J. A, Peacock, Cosmological Physics, Cambridge University Press, (1999) p579, p458, (3.44) in p78, (3.81) in p90, (3.78) in p89, p460-464, p664, 296 (9.81).
  * (9) S-H Chen, ‘Discussion of a Possible Universal Model without Singularity’, (2009) arXiv. 0908.1495; S-H, Chen, ‘A Possible Universal Model without Singularity and its Explanation for Evolution of the Universe’, (2006) hep-th/0611283.
  * (10) G. W. Gibbons, and S. W. Hawking, Phys. Rev D, 15, (1977) 2752\.
  * (11) M Chaichian and N.F.Nelipa, ‘Introduction to Gauge Field Theories’ Springer-Verlag Berlin Heidelberg, (1984) p269; G. G. Ross, ‘Grand Unified Theories’, The Benjamin/Cummings Publishing Company, INC, (1984) p177-183.
  * (12) S. Weinberg, Gravitation and Cosmology, New York, Wiley (1972) Chanter 12 section 3.
  * (13) M. Carmeli, ‘Classical Fields: General Relativity and Gauge Theory’, World Scientific Publishing, (1982) p89-92, (3.2.1), (3.2.7), (3.2.8), (3.2.9), (3.2.1), (3.2.15), (3.2.16)-(3.2.18).
  * (14) S. Coleman and E.J.Weiberg Phys. Rev.D7. (1973) 1888; R.H. Brandenberg, Rev. of Mod. Phys. 57. (1985)1.
  * (15) Liu Liao, Jiang Yuanfang and Qian Zhenhua, Progress in Physics, V.9, No. 2, (1989) p159.
  * (16) H.C. Ohanian, and R. Ruffini, Gravitation and Spacetime (2nd ed.), W.W. Norton and Company, Inc. (1994) Section 9.9.
  * (17) A. H. Guth, Phys. Rev. D 23, (1981) 347.
  * (18) A.R. Liddle and D.H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press 2000, p20 (2.27).
  * (19) Yu Yunqiang, Lectures in Cosmological Physics, Peking University Press, 2002, p151, (6.21), p170-172..


[]

Conceptualization of this study, Methodology, Software

[type=editor, auid=000,bioid=1, ]

[1] 1]organization=Saha Institute of Nuclear Physics,addressline=1/AF, Bidhan
Nagar, city=Kolkata, postcode=700064, country=India

2]organization=Homi Bhabha National Institute,addressline=Anushaktinagar,
city=Mumbai, postcode=400094, country=India

3]organization=Variable Energy Cyclotron Centre,addressline=1/AF, Bidhan
Nagar, city=Kolkata, postcode=700064, country=India

4]organization=Analytical Chemistry Division, Bhabha Atomic Research Centre,
Variable Energy Cyclotron Centre,addressline=1/AF, Bidhan Nagar, city=Kolkata,
postcode=700064, country=India

[cor1]Corresponding author

# Calculation of true coincidence summing correction factor for a Broad Energy
Germanium (BEGe) detector using standard and fabricated sources

Ashish Gupta  M. Shareef  Munmun Twisha  Saikat Bhattacharjee  Gopal Mukherjee
Satya Samiran Nayak  Sansaptak Basu  S. Dasgupta  J. Datta  S. Bhattacharyya
A. Mukherjee  [  [  [  [

###### Abstract

The true coincidence summing (TCS) correction factor for a Broad Energy
Germanium (BEGe) detector has been calculated at far and close geometry
measurement using multi-energetic radioactive $\gamma$-ray sources 60Co, 133Ba
and 152Eu. The correction factors were calculated using experimental method
and analytical method. Photopeak efficiency and total efficiency required to
calculate the correction factor were obtained using Geant4 Monte Carlo
simulation code. A few standard as well as fabricated mono-energetic sources
were also included in the $\gamma$-ray efficiency measurements. The simulated
efficiencies of mono-energetic $\gamma$-ray sources were matched to
experimental $\gamma$-ray efficiencies by optimizing the detector parameters.
The same parameters were used to obtain the photopeak and total efficiency for
$\gamma$-ray of our interest and coincident $\gamma$-ray. Analytical
correction factors and experimental correction factors were found in good
agreement with each other.

###### keywords:

$\gamma$-ray detection efficiency Close geometry measurement Coincidence
summing Mono-energetic source Muti-energetic source Geant4 simulation

Close geometry measurement for $\gamma$-ray detection efficiency

Coincidence summing correction for BEGe (Broad Energy Germanium) detector

##  1 Introduction

In nuclear astrophysics arena, the nuclear reactions of stellar evolution
occurs within the Gamow window which lies much below the coulomb barrier. The
cross-sections of these reactions are very less $\sim nb-pb$ [1, 2], thereby
making the measurements very difficult and therefore requires extra care in
performing the experiments. In such measurements, the resulting low yields of
the $\gamma$-rays originating from the decay of the excited states to ground
states of the product nuclei, lead to insufficient statistics and hence
increase the uncertainty in the measurement of the $\gamma$-ray cross-
sections. The $\gamma$-ray cross-sections are obtained from the detected yield
of the $\gamma$-ray, using the relation:

$\sigma=\frac{Y}{\epsilon N_{b}N_{t}}$ (1)

where, $Y$, $\epsilon$, $N_{b}$ and $N_{t}$ are the yield, detector
efficiency, beam current and number of target nuclei per unit area,
respectively. A better statistics in the measurement may be achieved by
increasing the yield, which is proportional to $N_{b}$, $N_{t}$ and
$\epsilon$. Among these factors, $N_{b}$ and $N_{t}$ can be increased, but
there are practical limitations. It is also possible to enhance the efficiency
of the detection system. One way to increase the efficiency is to place the
source close to the face of the detector. However, such close geometry
measurements may introduce coincidence summing effect in $\gamma$-ray
efficiency measurements.

It is well known fact that many unstable nuclei de-excite to the ground states
via the emission of multiple $\gamma$-rays from its higher energy states. When
two or more $\gamma$-rays emitted in cascade from the excited nucleus are
detected within the resolving time of the detector then this phenomenon is
referred as the true coincidence summing (TCS). This TCS is not a random
summing, the latter effect is related to pulse pileup where photons of
different cascades sum their energies randomly because of relatively high
pulse rates. Where as TCS is independent of pulse rate but depends on the
source to detector distance and the decay scheme [3] of the source. The origin
of TCS observed is not only due to the simultaneous detection of two or more
cascading $\gamma$-rays but may also be due to the simultaneous detection of a
$\gamma$-ray and a X-ray [4]. This X-ray may be emitted from internal
conversion of any $\gamma$-ray from decay scheme or electron capture
transitions.

Even though various modes for an excited nucleus are possible to decay to its
ground state, three types of phenomenon may results in TCS. As an example, in
the case of coincidence of two $\gamma$-rays, if the first $\gamma$-ray
deposits all of its energy and the coincident $\gamma$-ray deposits only part
of its energy in the detector, then there is a loss in the count from first
$\gamma$-ray photopeak. This is referred as "summing-out". If both
$\gamma$-rays deposit all of their energies in the detector, a sum peak
appears in the spectrum, at an energy corresponding to the sum of the energies
of the two $\gamma$-ray and this is referred as "summing-in". If a nucleus
independently emits a $\gamma$-ray with an energy equal to the energy of the
sum peak then we observe extra counts in the area under the curve of that
photopeak energy. The simultaneous detection of two $\gamma$-rays which
deposits full of its energy is less probable. Last case is when both the
coincident $\gamma$-rays deposit only parts of their energies in the detector,
it contributes only to the background and neither influence the peak intensity
nor the efficiency.

Figure 1: Schematic diagram of BEGe detector geometry with dimensions as
provided by the manufacturer

In close geometry, the coincidence summing effect increases as the distance
between the source and the detector decreases. This coincidence summing is an
unwanted effect which contributes to the miscalculation of photopeak area and
hence the photopeak efficiency. As cross-section in the astrophysical energy
domain are very small, close geometry measurements become essential. Hence in
such cases, coincidence summing correction for the detection system become
necessary to have an accurate knowledge of detector efficiency. The method for
calculating the coincidence summing correction factor was first described in
Ref. [5] and later improved in Ref. [6, 7, 8, 9]. The detailed procedure for
calculating the correction factor with this method is described in Ref. [10,
11].

Figure 2: Experimental setup of activation method for mono-energetic source
preparation at VECC, Kolkata

The method described here is an analytical approach to calculate the TCS
correction factors. This method requires the photopeak efficiency, total
efficiency and decay parameters, such as the mode of the parent nuclei decay,
energies of the $\gamma$-transitions, $\gamma$-ray emission probability,
K-capture probability (in $e^{-}$ capture decay), mean energies of the X-rays,
fluorescence yield, total and K conversion coefficients. Information of all
these factors are required to calculate the probability of simultaneous
emission of two or more $\gamma$-rays. The correction factor can be calculated
to desired accuracy and to a first order approximation the factor is given by
[12],

$k_{TCS}=\frac{1}{1-\Sigma_{i=1}^{i=n}p_{i}\epsilon_{ti}}$ (2)

where $n$ is number of $\gamma$-rays in coincidence with the $\gamma$-ray of
interest, $p_{i}$ is the probability of simultaneous emission of
$i^{th}~{}\gamma$ and $\gamma$-ray of interest and $\epsilon_{ti}$ is the
total efficiency of the $i^{th}~{}\gamma$-ray. The $\gamma$-ray photopeak
efficiencies can be obtained over a wide energy region using multi-energetic
$\gamma$-ray sources but the measurement of total efficiency is very
complicated with multi-energetic radioactive sources since the spectrum cannot
be decomposed into well defined components belonging to $\gamma$-rays with
distinct energies. So we can only use the mono-energetic radioactive sources
which emits single photopeak energy ($E_{\gamma}$). However, the availability
of mono-energetic $\gamma$-ray sources are very limited and their half-lives
($t_{1/2}$) are often very small, around days, so we need to replace them
periodically.

Detector parameters |  Manufacturer dimensions (mm) |  Optimized dimensions (mm)
---|---|---
Crystal radius |  30.35 |  29.45
Crystal length |  30 |  29.5
Deal layer |  0.045 |  1.8
Al end cap thickness |  1.2 |  1.2
Al end cap to crystal distance ($d_{alc}$) |  12 |  14.5
Table 1: Detector dimension parameters provided by the manufacturer and
optimized by Geant4 Monte Carlo simulation

The other method is, to simulate the $\gamma$-ray spectrum at each energy of
interest for a given detector geometry and calculate the photopeak and total
efficiency from the simulated $\gamma$-ray spectrum. This simulation is
independent of decay-scheme of radioactive source, hence its results are free
from coincidence summing effect. The exact and precise knowledge of detector
dimensions like crystal diameter, crystal length, dead layer thickness,
aluminium end cap thickness and aluminium end cap to crystal distance are
required to simulate the $\gamma$-ray spectrum. These parameters provided by
the manufacturer may not be accurate or may change with the course of time. In
a standard procedure, these detector parameters are optimized using measured
efficiencies with the mono-energetic $\gamma$-ray sources. After optimizing
the detector parameters, one can estimate the efficiencies corresponding to
the unexplored energy region as well in the simulation.

Source |  Reaction |  $A_{t}$ ($\mu$Ci) |  $E_{b}$ (MeV) |  $E_{\gamma}$ (keV) |  $T_{t}$ ($mg/cm^{2}$)
---|---|---|---|---|---
51Cr |  51V(p,n)51Cr |  1.445 |  10.5 |  320.08 |  2.4
65Zn |  65Cu(p,n)65Zn |  1.724 |  10.5 |  1115.3 |  10.375
109Cd |  109Ag(p,n)109Cd |  0.556 |  9.8 |  88.033 |  4.7
Table 2: Fabricated monoenergetic sources, corresponding reactions, activity
($A_{t}$), Beam energy ($E_{b}$), $\gamma$-ray energy ($E_{\gamma}$),
thickness of the target ($T_{t}$)
---
Figure 3: Experimental efficiency as a function of $\gamma$-ray energy at
source to detector distances, d = (a) 25 cm, (b) 10 cm, (c) 8 cm, (d) 5 cm,
(e) 3 cm and (f) 1 cm. Fitting curve of fifth order log-log polynomial is
depicted in red line colour.

The aim of this work is to obtain TCS correction factor ($k_{TCS}$) for 152Eu,
133Ba, 60Co radioactive sources using the analytical method as well as the
experimental method. Probabilities for the coincidence of two or more
$\gamma$-rays were calculated from the available decay schemes. The photopeak
and total efficiencies were determined using Geant4 (G4) Monte Carlo
simulation. The monoenergetic $\gamma$-ray sources were fabricated at VECC,
Kolkata using proton beam from the K-130 cyclotron accelerator. These sources
were used for experimental validation of TCS correction factor ($k_{TCS}$)
obtained analytically.

Figure 4: Efficiencies of monoenergetic sources at source to detector
distances, d = 25 cm, 10 cm, 8 cm, 5 cm, 3 cm and 1 cm. Experimental
efficiency shown in black colour data point, red colour data point shows the
Geant4 (G4) simulated efficiency with the detector parameter provided by
manufacturer (MP) and green color data point shows the Geant4 simulated
efficiency with the optimized detector parameter (O).

##  2 Absolute efficiency measurement using standard $\gamma$-ray sources

The absolute efficiency curve for an electrically cooled Falcon 5000 BEGe [13,
14] detector has been determined in this work. This detector is coupled with a
pre-amplifier, a buit-in MCA, Genie-2000 software to collect the spectrum and
tablet connected though a LAN cable is used to control the detector functions
as well as display the spectrum. The Falcon 5000 BEGe detector is a complete
self reliant system with its own in-built High Voltage Power Supply (HVPS)
unit, Digital Signal Processing (DSP) unit and analysis software Genie-2000. A
-3500 voltage of reverse bias was applied to the detector which was supplied
by the HVPS unit. The DSP unit is responsible for input pulse processing and
amplification. Genie-2000 [15] software acquires and analyses the data from
Multichannel Analyzers (MCAs). Genie 2000 has adequate set of capabilities to
handle multiple functions like MCA control, spectral display, basic spectrum
analysis and detector operations. A schematic diagram of detector geometry is
shown in Fig. 1. The $\gamma$-ray spectrum were recorded using Genie 2000
software and it was utilized for the preliminary analysis of recorded data.
The final offline analysis was performed using ROOT data analysis framework
[16] software.

The data was taken initially by placing the standard $\gamma$-ray sources at
25 and 10 cm away from the detector face [17]. The calibrated standard
radioactive $\gamma$-ray point sources 152Eu, 133Ba, 60Co, 241Am and 137Cs
were used. The large distances of 25 cm and 10 cm between source and detector
were chosen to avoid any influence of coincidence summing effect. The close
geometry measurement for source to detector distance d = 8, 5, 3 and 1 cm were
also performed. The experimental efficiency for a photopeak is given by,

$\epsilon=\frac{C/sec}{I_{\gamma}A_{t}}$ (3)

where $\epsilon,C,I_{\gamma}$ and $A_{t}$ are the efficiency of detector, area
under a $\gamma$-ray photopeak, intensity of a $\gamma$-ray photopeak and
activity of source, respectively [18]. The $\gamma$-ray intensities were taken
from the available decay scheme in the literature [19].

As discussed in previous section, close geometry measurements introduces
summing effect in efficiency measurements. One way to avoid is to use mono-
energetic $\gamma$-ray sources for efficiency measurements so that there is no
cascading $\gamma$-rays to contribute in coincidence summing. However, the
availability of standard mono-energetic $\gamma$-ray sources having
sufficiently long half-lives is very limited and also not sufficient for broad
energy region efficiency measurement. The other way to deal with the summing
effect is do a TCS correction for detector efficiency measurement. This TCS
correction factor can be calculated by Eq. 2. To calculate the correction
factor using this method we need the probability of the coincidence of
$\gamma$-rays, photopeak and total efficiency of corresponding $\gamma$-ray.
Total efficiency for each $\gamma$-ray photopeak was obtained by simulating
the detector response at different source to detector distances using Geant4
simulation toolkit.

##  3 Geant4 Monte Carlo simulation

A Monte Carlo simulation code Geant4 [20] was used to study the response of
the Falcon 5000 BEGe detector. Geant4 is a toolkit for simulating the passage
and interaction of different particles through different matter [21]. It has a
vast range of functionality including tracking, geometry, physics model and
particle generation. The various physical models like electromagnetic, decay,
optical, transportation are offered in Geant4 code. A wide variety of
particles, materials, elements, geometry and energy range from 250 eV to
$\sim$ TeV has been accommodated in Geant4. Until now Geant4 has been found to
be very useful to perform reliable Monte Carlo simulations from low energy
physics to high energy physics.

To simulate the BEGe detector response at different source to detector
distances several in-built classes in Geant4 were used. The geometry of the
detector was constructed using G4VUserDetectorConstruction class, material and
elements were incorporated using the in-built NIST library G4NistManager
class. The volume of the Ge crystal in the detector was chosen as the only
scoring volume. In principle, this scoring volume is responsible for energy
deposition by the interacting $\gamma$-ray in the detector crystal and further
store the deposited energy in the output data file. The structural dimensions
of the detector were optimized using a well-checked procedure in order to
obtain accurately the full energy peak efficiency for any energy and
measurement geometry. The detector dimensions parameters provided by the
manufacturer and optimized detector parameters have been tabulated in Table 1.
A schematic diagram of the Falcon 5000 BEGe detector geometry has been shown
in  Fig. 1. For particle generation G4VUserPrimaryGeneratorAction class was
used in which G4ParticleGun class was the function responsible to generate the
particles. The $\gamma$-rays were generated randomly and isotropically in all
directions. In each run, 108 particles were sampled to reduce any statistical
uncertainties in the simulations. Our primary particles were the $\gamma$-rays
which interact with matter through various processes, such as photo-electric
effect, Compton scattering and pair production. These interactions were
included in the simulation using the G4EmStandardPhysics class. The particle
was tracked throughout the world volume and the energy deposited in the
detector is collected in every step and later added at the end of the event.
This deposited energy is stored in a root file at the end of each event and
further analysis was done using ROOT data analysis framework [16] software.
The resolution of the detector was measured experimentally at different
energies using photopeaks of 152Eu, 133Ba radioactive sources. The photopeak
energies were fitted with Gaussian function and the corresponding standard
deviations ($\sigma_{G}$) were obtained [22]. The variation of standard
deviation ($\sigma_{G}$) as function of energy was fitted with a function
given by,

$\sigma_{G}=a+b*E^{1/2}$ (4)

where E is energy of $\gamma$-ray in keV, a and b are the constants. The
fitted value of a = 0.19259 and b = 0.0177498 were used in the simulation of
detector resolution. The detector resolution was incorporated in the simulated
results by redistributing the simulated spectrum with a random variable biased
to a Gaussian distribution of standard deviation obtained from Eq. 4.

---
Figure 5: Experimental efficiency and Geant4 (G4) simulated efficiency (in red
colour line) as a function of $\gamma$-ray energy at source to detector
distances, d = (a) 25 cm, (b) 10 cm, (c) 8 cm, (d) 5 cm, (e) 3 cm and (f) 1
cm.

##  4 Fabrication of mono-energetic $\gamma$-ray sources

Geant4 simulation needs an experimental verification such that the simulated
results are reliable in close geometry measurement. In close geometry
measurement, we can not rely on standard multi-energetic sources because of
the coincidence summing effect so it is required to fabricate some mono-
energetic sources that are free from coincidence summing.

Eγ (keV) |  Exp eff |  G4 eff (MP) |  G4/Exp (MP) |  G4 eff (O) |  G4/Exp (O)
---|---|---|---|---|---
d = 25 cm |  |  |  |  |
59.54 |  0.0025 |  0.0030 |  1.21 |  0.0025 |  1.01
88.03 |  0.0025 |  0.0030 |  1.17 |  0.0026 |  1.01
320.08 |  0.0011 |  0.0013 |  1.15 |  0.0011 |  1.02
661.66 |  0.0005 |  0.0006 |  1.18 |  0.0005 |  1.06
1115.5 |  0.0003 |  0.0004 |  1.11 |  0.0003 |  1.01
Avg. ratio |  |  |  1.16 |  |  1.02
d = 10 cm |  |  |  |  |
59.54 |  0.0125 |  0.0154 |  1.22 |  0.0125 |  0.99
88.03 |  0.0125 |  0.0151 |  1.21 |  0.0127 |  1.01
320.08 |  0.0052 |  0.0060 |  1.15 |  0.0052 |  1.00
661.66 |  0.0022 |  0.0027 |  1.22 |  0.0024 |  1.07
1115.5 |  0.0015 |  0.0016 |  1.12 |  0.0015 |  0.99
Avg. ratio |  |  |  1.18 |  |  1.01
d = 8 cm |  |  |  |  |
59.54 |  0.0180 |  0.0221 |  1.22 |  0.0178 |  0.99
88.03 |  0.0179 |  0.0216 |  1.21 |  0.0180 |  1.01
320.08 |  0.0073 |  0.0084 |  1.16 |  0.0072 |  1.00
661.66 |  0.0032 |  0.0038 |  1.18 |  0.0033 |  1.03
1115.5 |  0.0020 |  0.0023 |  1.14 |  0.0020 |  1.00
Avg. ratio |  |  |  1.18 |  |  1.00
d = 5 cm |  |  |  |  |
59.54 |  0.0347 |  0.0440 |  1.27 |  0.0348 |  1.01
88.03 |  0.0355 |  0.0425 |  1.20 |  0.0349 |  0.98
320.08 |  0.0134 |  0.0160 |  1.19 |  0.0135 |  1.01
661.66 |  0.0063 |  0.0072 |  1.15 |  0.0062 |  0.99
1115.5 |  0.0039 |  0.0044 |  1.11 |  0.0038 |  0.96
Avg. ratio |  |  |  1.18 |  |  0.99
d = 3 cm |  |  |  |  |
59.54 |  0.0628 |  0.0803 |  1.28 |  0.0627 |  1.00
88.03 |  0.0633 |  0.0774 |  1.22 |  0.0626 |  0.99
320.08 |  0.0235 |  0.0281 |  1.20 |  0.0235 |  1.00
661.66 |  0.0107 |  0.0127 |  1.19 |  0.0107 |  1.01
1115.5 |  0.0071 |  0.0077 |  1.08 |  0.0065 |  0.92
Avg. ratio |  |  |  1.19 |  |  0.98
d = 1 cm |  |  |  |  |
59.54 |  0.1253 |  0.1716 |  1.37 |  0.1313 |  1.05
88.03 |  0.1315 |  0.1661 |  1.26 |  0.1319 |  1.00
320.08 |  0.0495 |  0.0599 |  1.21 |  0.0488 |  0.99
661.66 |  0.0207 |  0.0272 |  1.32 |  0.0224 |  1.08
1115.5 |  0.0138 |  0.0164 |  1.19 |  0.0136 |  0.99
Avg. ratio |  |  |  1.27 |  |  1.02
Table 3: The comparison of experimental efficiency (Exp eff), Geant4
efficiency (G4 eff) with manufacturer provided dimension (MP) and optimized
parameters (O) at source to detector distance, d = 25, 10, 8, 5, 3 and 1 cm

###  4.1 Experiment

A few mono-energetic $\gamma$-ray sources were fabricated via irradiation
method, using proton beams obtained from the K-130 cyclotron at VECC, Kolkata
[23]. The accelerated p beams were used to bombard the natural 51V, 65Cu and
109Ag targets to fabricate the mono-energetic $\gamma$-ray sources 51Cr, 65Zn
and 109Cd. The targets were mounted perpendicular to the beam direction and
placed in beam line with aluminum foil and an aluminium (Al) plate of 1 mm
thickness, which prevents the end flange getting activated on prolonged use.
This end flange was being cooled by low conductivity water (LCW) to avoid any
heating of target material during long irradiation runs. The duration of
irradiation of each target varied from 8 hrs to 60 hrs depending on the
reaction specifications such as target thickness, reaction cross-section and
desired $\gamma$-ray yield. The vacuum inside the irradiation setup was
between $10^{-5}-10^{-6}$ mbar through out the experiment [24]. A schematic
diagram of the experimental setup is shown in Fig. 2. The beam current was
around 400-500 nA during the course of irradiation. Beam was dumped on Al
plate and beam current was measured directly from the end flange using a
current integrator. The details of the monoenergetic radioactive sources,
corresponding reactions, beam energy ($E_{b}$), thickness of the target
material ($T_{t}$), observed the $\gamma$-ray ($E_{\gamma}$) after irradiation
of target and calculated activity ($A_{t}$) has been tabulated in the  Table
2.

###  4.2 Activity measurement of the fabricated mono-energetic $\gamma$-ray
sources

The measured efficiency curve obtained at distances of 25 cm and 10 cm using
standard 152Eu, 133Ba, 60Co and 137Cs point radioactive sources were used to
standardize the fabricated mono-energetic $\gamma$-ray sources. The
efficiencies of point sources at these large distance were fitted with log-log
polynomial function of fifth order. Activities (disintegration rates) of
fabricated source were calculated by interpolating the efficiencies from these
curves and average of the activities at two distances were chosen as the
actual activity of the fabricated sources. The fabricated standard mono-
energetic sources thus obtained were also used for the efficiency measurements
at close geometries where the source to detector distances were d = 8, 5, 3, 1
cm, respectively. These measurements were used in validation of Geant4
simulation.

Source |  Eγ (keV) |  $k^{Analytical}_{TCS}$ |  $k^{Exp}_{TCS}$ |  $k^{Analytical}_{TCS}$ |  $k^{Exp}_{TCS}$ |  $k^{Analytical}_{TCS}$ |  $k^{Exp}_{TCS}$ |  $k^{Analytical}_{TCS}$ |  $k^{Exp}_{TCS}$ |  $k^{Analytical}_{TCS}$ |  $k^{Exp}_{TCS}$ |  $k^{Analytical}_{TCS}$ |  $k^{Exp}_{TCS}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  d = 1 cm |  |  d = 3 cm |  |  d = 5 cm |  |  d = 8 cm |  |  d = 10 cm |  |  d = 25 cm |
60Co |  1173.2 |  1.074 |  1.029 |  1.035 |  1.009 |  1.003 |  0.981 |  1.01 |  0.971 |  1.007 |  0.96 |  1.001 |  0.949
|  1332.5 |  1.077 |  1.042 |  1.036 |  1.023 |  1.003 |  0.988 |  1.011 |  0.981 |  1.008 |  0.97 |  1.001 |  0.969
133Ba |  53.16 |  1.208 |  1.239 |  1.091 |  1.078 |  1.05 |  1.061 |  1.026 |  1.034 |  1.018 |  1.041 |  1.001 |  1.008
|  80.99 |  1.13 |  1.166 |  1.059 |  1.03 |  1.033 |  1.001 |  1.017 |  0.98 |  1.012 |  1.011 |  1 |  0.968
|  276.39 |  1.277 |  1.301 |  1.118 |  1.094 |  1.063 |  1.058 |  1.032 |  1.013 |  1.022 |  1.053 |  1.003 |  0.947
|  302.85 |  1.176 |  1.23 |  1.079 |  1.086 |  1.043 |  1.051 |  1.022 |  1.045 |  1.015 |  1.047 |  1.002 |  1.031
|  356.01 |  1.157 |  1.186 |  1.071 |  1.06 |  1.039 |  1.034 |  1.02 |  1.022 |  1.014 |  1.047 |  1.001 |  1.022
|  383.85 |  0.938 |  0.977 |  0.969 |  0.976 |  0.982 |  0.979 |  0.991 |  1.007 |  0.993 |  1.028 |  0.998 |  1.011
152Eu |  121.78 |  1.08 |  1.124 |  1.037 |  1.09 |  1.021 |  0.999 |  1.011 |  0.98 |  1.008 |  0.998 |  1.002 |  0.95
|  244.69 |  1.17 |  1.225 |  1.075 |  1.134 |  1.04 |  1.059 |  1.021 |  1.041 |  1.015 |  1.054 |  1.003 |  1.023
|  344.27 |  1.066 |  1.054 |  1.031 |  1.058 |  1.017 |  0.999 |  1.009 |  0.995 |  1.007 |  1.022 |  1.001 |  0.978
|  411.11 |  1.176 |  1.159 |  1.078 |  1.087 |  1.044 |  1.024 |  1.023 |  1.009 |  1.016 |  0.998 |  1.003 |  1.034
|  443.96 |  1.29 |  1.262 |  1.122 |  1.158 |  1.065 |  1.039 |  1.033 |  1.006 |  1.023 |  1.047 |  1.004 |  1.011
|  778.9 |  1.103 |  1.104 |  1.047 |  1.109 |  1.027 |  1.028 |  1.014 |  1.025 |  1.01 |  1.043 |  1.002 |  1.036
|  867.37 |  1.255 |  1.301 |  1.108 |  1.124 |  1.058 |  1.098 |  1.029 |  1.065 |  1.02 |  1.054 |  1.004 |  1.017
|  964.07 |  1.227 |  1.205 |  1.098 |  1.132 |  1.052 |  1.047 |  1.026 |  1.008 |  1.018 |  1.038 |  1.003 |  0.985
|  1085.86 |  1.008 |  0.993 |  1.002 |  1.06 |  1 |  1 |  1 |  1.024 |  1 |  1.044 |  1 |  1.044
|  1089.73 |  1.101 |  1.041 |  1.046 |  1.067 |  1.026 |  0.988 |  1.014 |  0.987 |  1.01 |  1.027 |  1.002 |  1.044
|  1112.07 |  1.203 |  1.152 |  1.087 |  1.123 |  1.046 |  1.039 |  1.023 |  1.011 |  1.016 |  1.067 |  1.003 |  0.972
|  1212.94 |  1.418 |  1.377 |  1.165 |  1.197 |  1.087 |  1.13 |  1.043 |  1.029 |  1.03 |  1.048 |  1.006 |  1.015
|  1299.14 |  1.104 |  1.122 |  1.047 |  1.079 |  1.029 |  1.046 |  1.014 |  1.027 |  1.01 |  1.055 |  1.002 |  0.989
|  1408 |  1.22 |  1.203 |  1.094 |  1.162 |  1.051 |  1.07 |  1.025 |  1.047 |  1.018 |  1.065 |  1.003 |  0.984
Table 4: Coincidence summing correction factors using analytical
($k^{Analytical}_{TCS}$) and experimental ($k^{Exp}_{TCS}$) method at source
to detector distances, d = 1, 3, 5, 8, 10 and 25 cm

##  5 Results and discussion

The measured efficiencies for the source to detector distances d = 25, 10, 8,
5, 3 and 1 cm are shown in Figs. 3(a-f) respectively. The experimental
efficiencies were determined using both the standard and fabricated mono-
energetic sources like 241Am, 109Cd, 51Cr, 137Cs and 65Zn. The standard
radioactive mono-energetic sources 241Am and 137Cs were available in the
laboratory, whereas 109Cd, 51Cr, 65Zn mono-energetic radioactive sources were
prepared at K-130 cyclotron facility at VECC, Kolkata. Experimental
efficiencies were also determined using the multi-energetic sources 60Co,
133Ba and 152Eu. In the close geometry measurements, the experimental
efficiencies determined from mono-energetic sources are free from TCS whereas
the experimental efficiencies from multi-energetic sources will not be free
from the TCS. The sources with complex decay scheme may have larger summing
effect than the sources with simple decay scheme.

These experimental efficiency curves were fitted using fifth order log-log
polynomial at each source to detector distances, which is shown using solid
red line in Figs. 3 (a-f). It is observed from the experimental efficiency
curve that the efficiency of BEGe detector and fitting curve matches well with
each other for the source to detector distances from d = 25 up to 8 cm.
However the measured efficiencies are found to be deviating from the fit curve
as the source to detector distance decreases further. Fig. 3(f) shows the
experimental efficiencies corresponding to the source to detector distance of
1 cm and its deviation from the fit curve which clearly depicts the presence
of coincidence summing effect at close geometry measurements.

---
Figure 6: TCS corrected experimental efficiency and Geant4 (G4) simulated
efficiency (in red line colour) as function of $\gamma$-ray energy at source
to detector distances, d = (a) 25 cm, (b) 10 cm, (c) 8 cm, (d) 5 cm, (e) 3 cm
and (f) 1 cm.

Efficiency of the detector at close geometry without the effect of coincidence
summing may be determined using mono-energetic radioactive sources having
desired $\gamma$-ray energy, but there are limitations with these sources as
mentioned earlier. On the other hand, we can use multi-energetic $\gamma$-ray
sources and calculate the coincidence summing correction factor for each
$\gamma$-ray energy and determine the sum corrected efficiency. As discussed
in Section 1, to calculate the coincidence summing correction factor for any
$\gamma$-ray energy we need to perform the simulation to obtain the BEGe
detector response for each $\gamma$-ray energy at each source to detector
distance (d). Initially, the simulations were performed for all five
monoenergetic sources at source to detector distances d = 25, 10, 8, 5, 3 and
1 cm, using the dimension of detector as provided by manufacturer. Fig. 4
shows the experimental efficiency, Geant4 simulated efficiency with detector
dimension provided by manufacturer (MP) and optimized detector dimension (O)
by Geant4 Monte Carlo simulation for monoenergetic radioactive sources. It is
evident from the Fig. 4 that the simulated efficiencies (with manufacturer
provided detector dimension) are higher than the experimental efficiency. The
similar trend in efficiency were also reported in previous measurements [25,
26, 27, 28]. The $\gamma$-ray energies emitted by each mono-energetic source
used in this study lie in the range between 59-1115 keV. The average ratio of
Geant4 (MP) to experimental efficiency for d = 25 is 1.164 and it increases up
to 1.27 at d = 1 cm. The average ratio for $\gamma$-rays of mono-energetic
sources at each source to detector distance has been tabulated in Table 3.
This over-prediction in efficiency by Geant4 than the experimental efficiency
may be due to the inaccurate dimensions of internal structure of the detector
used in the simulations. The structural dimensions of the detector involves
crystal radius, crystal length, dead layer thickness, Al end cap thickness and
Al end cap to crystal distance ($d_{alc}$). As detector dimensions provided by
the manufacturer may not be accurate or it might be possible to change with
the period of time, one needs to optimize these parameters using the measured
efficiency data for all the distances. It is clear that the crystal radius has
uniform effect on each $\gamma$-ray photopeak where as the crystal length
affects the photopeaks due to high energy $\gamma$-rays [25]. It is discussed
in Ref. [29] that a long period of operation may increase the dead layer
thickness of the detector multiple times than of its original thickness.
Optimizing the dead layer thickness of the detector is important as it is
found to largely affect the low energy $\gamma$-ray photopeaks. Proper
knowledge of Al end cap to crystal distance (dalc) is necessary to reduce the
uncertainty in the source to detector distance (d). A small variation in this
parameter may cause a significant variation in efficiencies in close geometry
measurements. The detector parameters were tuned systematically and optimized
within the limits of uncertainty using measured efficiencies.

Figure 7: Experimental and analytical correction factors at d = 1, 3, 5, 8 cm

The efficiencies calculated from the Geant4 simulations using the optimized
parameters are shown in Fig. 4, which well reproduces the experimental
efficiencies at each source to detector distances within the error bars. The
same detector parameters were used to simulate the $\gamma$-ray photopeak
efficiency corresponding to multi-energetic source 60Co, 133Ba and 152Eu also.
Since the $\gamma$-ray efficiencies predicted by Geant4 are independent of
decay scheme of the source, it is free from the coincidence summing effect.
Hence the ratio of the efficiency obtained form the Geant4 simulation to that
from the measurement is the experimental correction factor ($k^{Exp}_{TCS}$)
for coincidence summing effect, which is given by,

$k^{Exp}_{TCS}=\frac{Geant4~{}efficiency}{Experimental~{}efficiency}$ (5)

The experimental efficiency and Geant4 efficiency for $\gamma$-ray photopeak
energy at each source to detector distances are shown in Figs. 5(a-f). From
this figure, it is clearly seen that the Geant4 simulated efficiency using the
optimized detector parameters for all $\gamma$-ray photopeak energy is in good
agreement with the measured efficiencies at large source to detector distances
for d = 8 to 25 cm. However, a significant deviation of Geant4 efficiency can
be observed from the measured values when the source to detector distance
approaches smaller values from d = 8 cm and the magnitude of deviation is
found to be larger at d = 1 cm, which again confirm the coincidence summing at
close geometry measurements. It may be noticed from the Figs. 5(a-f) that
Geant4 simulated efficiency match well to the experimental efficiency obtained
using monoenergetic sources at all the measured source to detector distances.

The experimental values of TCS correction factor for the $\gamma$-ray
photopeaks of 60Co, 133Ba and 152Eu radioactive sources are given in Table 4.
The coincidence correction factor for these radioactive sources were also
calculated using analytical method at each source to detector distance as
discussed in Section 1. The probability ($p_{i}$) of coincidence of two
cascading $\gamma$-rays was calculated using appropriate parameters obtained
from the decay schemes of the radioactive sources available in the
literature[30]. The internal conversion coefficients, fluorescence yield and
k-capture probability were taken from ref. [19]. The photopeak and total
efficiency for each $\gamma$-ray of interest and coincident $\gamma$-ray were
computed within the framework of Geant4 simulation. Coincidence summing
corrected experimental efficiency is shown in Figs. 6(a-f) for each distance
as function of $\gamma$-ray energy along with the Geant4 simulated values. The
correction factors obtained from the analytical method for $\gamma$-ray
photopeak of 60Co, 133Ba and 152Eu radioactive sources have been given in
Table 4.

The correction factors estimated using the experimental and analytical method
are shown in Fig. 7 for all the source to detector distances chosen in this
work. It is observed that the correction factors obtained from both the
methods lies very close to unity for d = 8 cm. As the source to detector
distance decreases further, we notice that the deviation of correction factor
from unity is significant and it is predominant at d = 1. It is evident from
Fig. 7 that the correction factor increases with decrease in the source to
detector distance. It is also found that, the BEGe detector may be suitable to
use in $\gamma$-ray spectroscopy experiments without considering the effect of
coincidence summing if the detector is at 8 cm or above from the gamma
emitting source.

##  6 Conclusion

The FEP efficiency measurement of an electrically cooled BEGe detector has
been carried out using standard mono-energetic and multi-energetic point
sources as well as fabricated mono-energetic sources for different source to
detector distances. The detector response was simulated and FEP efficiency was
also obtained using Geant4 Monte Carlo code. The internal structural
dimensions of the detector were optimized using measured efficiencies obtained
from the mono-energetic sources as the detector specification provided by the
manufacturers are insufficient. It is found that for the source to detector
distance from 8 cm and above, there were no effect of coincidence summing in
the spectra. However as the source to detector distance decreases below 8 cm,
the presence of coincidence summing is evident. The true coincidence summing
correction factors for radioactive sources 60Co, 133Ba and 152Eu have been
determined using experimental and analytical method for BEGe detector at
different source to detector distances. The coincidence summing is a very
important effect and must be taken into account when performing close geometry
measurements for source emitting gamma rays in cascade. The highest correction
factor was observed for 1212 keV $\gamma$-ray photopeak energy of 152Eu
radioactive source and it was estimated to be 1.418 using analytical method
and 1.377 using experimental method. No summing correction is required for the
BEGe detector if the detector is at 8 cm or above from the $\gamma$ emitting
sources.

##  7 Acknowledgment

The authors gratefully acknowledge the cyclotron crew of VECC, Kolkata for
providing excellent quality beam throughout the irradiation experiment. One of
the authors (A.G.) acknowledge the useful discussions with Prof. Supratik
Mukhopadhyay and Subhendu Das regarding simulations.

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