Finding out and exploration of two new equations for calculating the
dead-time of neutron detectors and the energy of slow-downed neutrons
SEYED ALIREZA MOUSAVI SHIRAZI
Department of Physics
South Tehran Branch, Islamic Azad University
Shahid Deh-Haghi AVE, Fifth Bridge, Abouzar Blvd, Pirouzi AVE, Tehran, Iran. Postal Code:
1777613651.
IRAN
Abstract: - One of the most important issues in nuclear science and technology is neutron detection and optimized usage of
neutron detectors. The significance of this issue is to the extent that accurate neutron detection is the most desirable issue
in nuclear energy engineering including in the area of nuclear reactors. To better design a neutron detector, many items
should be taken into account. One of the items is neutron detector dead-time and its calculation. Nowadays, the dead-time
of nuclear radiation detectors is among less-discussed objects and it may usually be neglected. In this research, a new
equation for calculating the dead-time of neutron detectors has been found out in a way that applying this equation, the
dead-time, which is a very significant issue in radiation detection, is calculated as accurately as possible. In addition, in
this paper, the equation associated with the energy of a slow-downed incident neutron is specified. By this equation, the
energy of an incident neutron that moves across a path undergoes slowing down and deposits its energy is obtained.
Key-Words: - Equation; Dead-time; Detector; Neutron; Slowing down.
Received: August 12, 2021. Revised: March 21, 2022. Accepted: April 23, 2022. Published: June 3, 2022.
1 Introduction
- The neutron is a subatomic particle and has a behavior
like a proton. It is disintegrated into proton and neutrino
within 12 minutes according to this reaction:
pn
(1) [1].
A few numbers of radioisotopes generate neutrons. There
is a heavy element like 252Cf that emits neutrons because
of spontaneous fission [2].
The half-life of californium is 2.5 years. The neutron
sources are required to calibrate the energy in neutron
spectrometry and also calibration of dosimeters, flux
meters, etc. 226Ra, 210Po, 239Pu, and 241Am along with
beryllium are applied as alpha emitters [3]. The reaction
between α and Be is as follows:
)75.5(
1
0
12
6
9
4
4
2MevQnCBeHe
(2)
In this situation, the high-energy neutrons within the
energy range 10-13 MeV are generated. The rate of the
fast neutrons emitted from the neutron source like Po-Be
is
sec
103.2 6n
for one Curi of decay of Po meaning
that of 16000 Po atom decay, and consequently
generation of the alpha particle, only one interaction
between alpha and Be is created, and one neutron is
generated [4]. For the neutron source Ra-Be, the rate of
emitted neutrons is
sec
107.1 7n
for one Cu of atom
disintegration.
In the reaction (
, n) associated with the sources, the
photons having energy above 2.2MeV reacts in the
following reaction in collision with Be [5].
Be
9
4
Hen4
2
1
02
(3)
More gamma photon energy is lost in collision with the
beryllium nucleus, thus, the generated low energy
neutrons have energy within the energy range of a few
keV. For instance, the Sb-Be neutron source generates
neutrons having 24keV but the neutrons are categorized
to various energies based on scattering in collision with
Be [6, 7]. The nuclear reactions cause to generate
neutrons, which can be applied through charged particles
produced by accelerators. The research associated with
neutron detectors in the nuclear research association has
so far continued in a way that regardless of efforts related
to a modern reactor design on how to apply the reactor,
this research continues [8].
The neutron sources, which can be used, are 252Cf, D-T,
and D-Be. In the neutron therapy practice, a mono-energy
neutron source had better be used, thus, a source D-T,
which generates the neutrons with energy of about
14MeV, is usually used. Therefore, a neutron source like
Am-Be that is having an energy peak is not appropriate
for an experiment [9].
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Of course, it is better at first, the number of neutrons
emitted from a laboratory neutron source is calibrated
with another neutron counter. ICRP
1
has specified 5rem
for maximum acceptable yearly radiation. On the other
hand, 100mrem/week for 2.5mrem/hour per week is
considered. The equivalent dose (H) is also used as
quality-specifying [10].
2 Materials and Methods
2.1. Neutron and its interactions with materials
The neutrons are neutral and lack electric charge while
they have mass. But, they can not directly make
ionization in a detector, thus, can not directly be detected.
This means that the neutron detectors must rely on a
conversion stage in a way that a collision between a
neutron and a nucleus causes a secondary electric charge
to be generated. Then, these secondary electric charges
are directly detected, and through inferring them, the
existence of the neutrons is concluded. An incident
neutron may not react with a hydrogen material but rather
it may react with the constituent elements of that material
according to the following reactions [11,12].
1- elastically (n,n), 2- inelastically (n,n’), 3- capturing the
next emit of a photon (like gamma) and charged particles
like proton based on the reaction (n,γ). For the neutrons
having energy less than 14MeV, elastic scattering with
hydrogen results in maximum storage of energy in
hydrogen and maximum neutron moderation in tissue and
other hydrogen materials such as polyethylene that are
applied in detection system research. The energy and
neutron emission angle after the collision is very
significant in these studies [13].
To obtain the information relating to neutron penetration,
the absorbed energy by tissue, and the angle of scattered
nucleus considering neuron crossing from phantom
layers, precise information of collision cross-section and
angular distribution of scattered neutrons are required.
The angular distribution can give the relative
probabilities of scattered neutrons in various directions.
The neutron absorption cross-section depends on some
parameters like target material and neutron energy. Of
course, there may be multiple reactions because of the
collision of emitted neutrons with compositions and
existing elements in a tissue [14]. For instance, due to a
collision between fast neutron and nitrogen existing in a
tissue, the reaction 14N(n,p)14C occurs. Besides, due to
colliding thermal neutron with nitrogen, the reaction (n,γ)
may happen. The occurrence of every reaction is required
to have conditions and the amount of binding energy,
kinetic energy, and ΔE (the excited energy level of the
compound nucleus). For example, in the resonance
region, if the exciting energy of the neutron absorber
nucleus (BE+KE) equals one of the exciting levels (ΔE),
then, an absorption occurs, and the absorbed neutron (in
the nucleus) may cause the capturing and (n,γ) reaction.
Also, after the formation of the compound nucleus, the
nucleus may release a neutron such that its energy is very
lower than the initial neutron energy, and the remainder
nucleus returns to a stable level through emitting
radiation that is an indicator of an inelastic reaction [15].
When ΔE>BE+KE, no absorption occurs, and the neutron
gives a small amount of its energy to the target nucleus,
and it will be scattered with a new angle based on the
reaction (n,n’). Of course, the emitted neutron can result
in the recoil of the proton of the target nucleus and is
sometimes able to cause the target nucleus to be moved.
Therefore, some reactions such as (n,), (n,p), (n,α), (n,n),
and (n,n’) may occur. Of course, for the energies higher
than 10MeV, the (n,α) has a more fraction [16].
There are three main types of collisions between neutron
with carbon and hydrogen. These collisions are elastic,
inelastic, and radioactivity capturing, respectively. The
neutron collision with the nuclei of carbon and hydrogen
results in the transfer of energy from the neutron to the
target nucleus. The recoiled nucleus passes a short
distance within the material length and deposits its energy
within the path. The main problem is the calculation of
recoiled nuclei energies, and it necessitates the diffusion
equation [17].
High LET is because of the protons, which have been
generated as a result of capturing thermal neutrons and
nitrogen atoms [14N(n,p)14C], and also the protons
resulting from the reactions between fast neutrons and
hydrogen atoms. Additionally, the high LET of protons is
also because of fast neutron scattering. But, the low LET
of gamma radiations is a result of capturing the thermal
neutrons and tissue and also hydrogen atoms [1H(n,γ)2H]
[18]. By detecting each proton or an alpha particle, there
can infer the existence of a neutron. An electric field can
be made by both positive and negative electrodes to
collect protons (having the positive charge) via cathode
(negative electrode). In this stage, the detection of
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protons, which have been absorbed by the cathode, is
acted. In that case, by changing the intensity of the
electric field and changing the voltage between cathode
and anode, the efficiency of ions collection can increase
[19]. When the voltage is zero, the collected electric
charge is zero because after ion-pairs getting formed, they
can easily be collected gain with each other. This process
is named ion recombination. Even if a low-voltage is
applied, some electric chargers may yet be collected. The
efficiency of ions collection is a fraction of collected
charges to charges released by initial ionization [20]. The
more the voltages of two electrodes increase, the fewer
and fewer charges are recombined with each other by the
time almost all of the freed ions are collected, and the
efficiency reaches 100%. When the voltage between
electrodes increases, the number of collected electric
charges increases by raising the voltage. This event is a
result of ionization increment or on the other hand
secondary ionization. Therefore, the voltage of two
electrodes should be so regulated that it can be reached an
efficiency of 100% [21, 22].
One of the items that must be considered in a neutron
detector is the calculation of detector dead-time. The
following equation can be written [23].
t s d
L t t
(4)
Where:
Lt: neutron lifetime (when a neutron is born by the time it
annihilates because of escape or absorption)
ts: neutron moderation time
td: neutron diffusion time
: mean-free-path-length in every collision or on the
other hand the mean distance in which a neutron moves
during successive collisions until moderation.
3 Results and Discussion
The equations, which are applied to find out and
exploration of two new mentioned equations for
calculating the dead-time of neutron detectors and the
energy of slow-downed neutrons, the following equations
are used.
For calculation ts, the number of neutron collisions within
a time range dt:
dt
s
vdt

(5)
0
ln ln (ln )E E E
(6)
ln
s
dE vdt
EE

(7)
E
dE
v
dt
s
1
(8)
Where:
v: the velocity of neutron
s: mean-free-path-length of neutrons
ξ: logarithmic energy decrement of neutron after a
collision
2
2
1mvE
(9)
m
E
v2
(10)
thth E
E
s
E
Es
t
vE
dE
E
dE
v
dtt
00
11
0
(11)
th
E
E
s
sE
m
t
0
2
1
2
1
2
2
(12)
0
211
n
ssth
m
tEE





(13)
E
EE
E
E
nth
))(ln(
ln 0
(14)
3
2
2
A
(15)
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mn: the mass of neutron
Σs: macroscopic scattering cross-section
E0: initial energy of neutron
Eth: final energy of neutron
To obtain td that is related to the reactor core and fuel
region, and is not associated with an external neutron
source.
0
0
() 11
() ( ) (0.025 )
( ) ( )
ad
aa
E
t E t l
v E v ev
v E E

(16)
2
1
0
2
)025.0()( 0
T
T
Ea
a
(17)
00 KTE
(18)
KTE
(19)
2
1
00
T
T
v
vT
(20)
Since
)/(2200
0smv
, thus:
a
T
dv
t
2
(21)
M
a
F
aa
(22)
T
M
a
F
a
dv
lt
)(2
1
(23)
As
dst ttL
and
sd tt 
, so
dt tL
. Thus:
() (1 )
t tM
l l f

(24)
If the initial energy of an incident neutron (En) and
deposited energy (ER) are considered respectively, the
following equation can be written. In this equation, n is
the number of neutron collisions.
n
EE
EE Rn
RnewR
)(
(25)
A schematic view of track length and slowing down
of an incident neutron in the successive collisions is
shown in Fig.1.
Fig 1. The collision of an incident neutron from an
external source
To analyze the energy deposited within a path, the
ER is obtained as follows.
n
nR eEE
(26)
2
(1 )
3
as
tr
sl sl
A
n


(27)
Through merging the above-mentioned equations, the
final equation is extracted.
2
(1 ) 2
3
2
3
as
sl
A
A
Rn
E E e

(28)
4 Conclusion
Therefore, concerning an external source and according
to Eq.13, the span of ts can be taken into consideration as
detector dead-time, and it can be considered as a delay
time in the clock-pulse of a shift-register circuit to which
the outputs of analog to digital (A/D) is connected. After
emitting the neutrons from an external source and more
absorption, they enter the polyethylene region and then
are slowed. The period of their slowing-down is
equivalent to the detector dead-time, and it can be
obtained from the above equations. As well, based on
Eq.28, the energy of an incident neutron that moves
across a path and undergoes slowing down, and deposits
its energy is specified.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Seyed Alireza Mousavi Shirazi has carried out
all of the scientific works belonging to this
research consisting of idea, finding out the
equations, and extraction of the results. In
addition, he has authored and organized the
paper.
Sources of funding for research
presented in a scientific article or
scientific article itself
There are no potential sources of funding for this
research.
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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