Download Estimation Of the Total Energy Loss of Positrons in Copper and Nickel

Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
Estimation Of the Total Energy Loss
of Positrons in Copper and Nickel
Sabah Mahmoud Aman Allah
Physics Department - College of Science –Tikrit University –Salah din –Iraq.
[email protected]
Enas Mohamad Salaman
Physics Department-College of Science for Women–Babylon University Babylon
[email protected]
Rafea Abduala Abad
Physics Department - College of Science –Tikrit University –Salah din –Iraq.
[email protected]
Abstract:
In present paper , the values of radiative Srad and collisional Scoll stopping powers and the total
stopping power positrons β+ for copper and nickel by employing Bethe-Bloch relativistic formula in
the energy range of 0.1MeV-10MeV. The results showing a that the radiative stopping power dominate
more than the collisional stopping power in the of the total stopping power which are in good
agreement with Estar universal code results .
Keywords: Theoretical Physics, Bethe-Bloch ,stopping power ,stopping time , radiative ,collision,
mean excitation energy, Nickel , copper , Estar code.
: ‫اﻟﺨﻼﺻﺔ‬
‫ ﻁﻭل ﺍﻟﻤﺴﺎﺭ ﺍﻹﺸﻌﺎﻋﻲ ﻭﺯﻤﻥ ﺍﻟﺘﻭﻗﻑ‬, ‫ﺍﻟﺘﺼﺎﺩﻤﻴﺔ ﻭ ﺍﻟﻘﺩﺭﺓ ﺍﻟﻜﻠﻴﺔ‬, ‫ﻗﺩﻤﻨﺎ ﻓﻲ ﻫﺫﺍ ﺍﻟﺒﺤﺙ ﻗﻴﻡ ﻗﺩﺭﺓ ﺍﻹﻴﻘﺎﻑ ﺍﻹﺸﻌﺎﻋﻴﺔ‬
‫ ﻤﻴﻜـﺎ‬٠.١‫ ﺍﻟﺴﺎﻗﻁﺔ ﻋﻠﻰ ﻤﺎﺩﺘﻲ ﺍﻟﻨﺤﺎﺱ ﻭ ﺍﻟﻨﻴﻜل ﺒﺎﺴﺘﺨﺩﺍﻡ ﻤﻌﺎﺩﻟﺔ ﺒﻴﺙ –ﺒﻠﻭﺥ ﺍﻟﻨﺴﺒﻴﺔ ﻭﻟﻤﺩﻯ ﻁﺎﻗﺔ ﻴﺒﺩﺃ ﻤﻥ‬β+ ‫ﻟﺒﻭﺯﺘﺭﻭﻨﺎﺕ‬
‫ ﻤﻴﻜﺎﺍﻟﻜﺘﺭﻭﻥ ﻓﻭﻟﺕ ﻭﻗﺩ ﺒﻴﻨﺕ ﺍﻟﻨﺘﺎﺌﺞ ﺍﻟﺘﻲ ﺤﺼﻠﻨﺎ ﻋﻠﻴﻬﺎ ﺍﻟﻬﻴﻤﻨﺔ ﺍﻟﻜﺒﻴﺭﺓ ﻟﻘﺩﺭﺓ ﺍﻹﻴﻘﺎﻑ ﺍﻹﺸﻌﺎﻋﻴﺔ ﺃﻜﺜﺭ ﻤﻥ ﻗﺩﺭﺓ‬١٠ ‫ﺇﻟﻜﺘﺭﻭﻥ ﻓﻭﻟﺕ‬
‫ ﻓﻘﺩ ﻭﺠﺩﺕ ﺍﻨﻬﺎ ﻓﻲ ﺘﻁﺎﺒﻕ ﺠﻴﺩ ﻤﻊ ﺍﻟﺒﺭﻨﺎﻤﺞ‬Estar ‫ﺍﻹﻴﻘﺎﻑ ﺍﻟﺘﺼﺎﺩﻤﻴﺔ ﻟﻘﻴﻡ ﻗﺩﺭﺓ ﺍﻹﻴﻘﺎﻑ ﺍﻟﻜﻠﻴﺔ ﻭﺒﺎﻟﻤﻘﺎﺭﻨﺔ ﺍﻟﻨﺘﺎﺌﺞ ﻤﻊ ﺍﻟﻜﻭﺩ ﺍﻟﻌﺎﻟﻤﻲ‬
. ‫ﺍﻟﻌﺎﻟﻤﻲ ﺃﻱ ﺴﺘﺎﺭ‬
. ‫ﺑﺮﻧﺎﻣﺞ اي ﺳﺘﺎر‬, ‫ اﻟﻨﺤﺎس‬,‫ اﻟﻨﯿﻜﻞ‬, ‫ﻣﻌﺪل ﻃﺎﻗﺔ اﻟﺘﺄﯾﯿﻦ‬, ‫اﻟﺘﺼﺎدﻣﯿﺔ‬, ‫اﻻﺷﻌﺎﻋﯿﺔ‬, ‫ ﻗﺪرة اﻻﯾﻘﺎف‬, ‫ﺑﻠﻮخ‬-‫ ﻣﻌﺎدﻟﺔ ﺑﯿﺚ‬:‫اﻟﻜﻠﻤﺎت اﻟﻤﻔﺘﺎﺣﯿﺔ‬
Introduction:
Information on stopping power(s.p) is essential in many fields involving
radiation. Their accuracy may critically affect calculations, measurements and
interpretation of experiments. Research concerning the stopping power has taken the
position of the basic theme in the fields of ion-matter interactions for a long time.
Despite the long history of stopping power research, the current knowledge, both
experiment and theoretical, is far from being complete, and is often inadequate for the
determination of stopping power values of a variety of materials and for a wide range
of particle energies( Sing 2013) . The study of s.p. to positron and electron through
matter is an effective tool for exploring the structure of matter s.p. in materials are of
interest in many research fields, such as in nuclear physics, atomic physics, solid-state
physics, radiation dosimetry and nuclear technique applications. During the last two
decades, it has attracted a great deal of attention. The s.p. calculations for β+are
studied in two different ways : the first is to consider the interactions of incoming of
the positron with target electron , which is called collisional s.p. which may end with
annihilation radiation if the conditions are met, while the second is considered the
fact that decelerated charged particles as approach nuclear field ,which is called
radiative s.p. Or Bremsstrahlung Loss which will discussed in the next section in
details. The total stopping power is given as follows (Pal 1986 , Bethe 1932) :
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014



Stotal
(E)  Scoll
(E) Srad
(E) ……………….(1)
Where the signs (+) and (-) refers to positron and electron respectively. An
extensive study ( Moler 1932 , ICRU) exists in the literatures. The aim of this work is
to determine the positron s.p. of copper and nickel which they have a huge
importance in a variety of applications such as radiation physics, Chemistry, Biology
and Medicine.
2-Methodology :
The interactions of positron with matter depends upon its life time  which is
a function of the electron density at the annihilation site .The annihilation rate  ,
which is the reciprocal of the positron life time  is given by the overlap of the
positron density n  ( r )  

(r )
2
, and the electron density n  ( r )
(Nieminen 1979).
 
2
1
  r02 c    ( r ) n  ( r )  dr ……………(2)

r0 is the classical radius , c the speed of the light , and r the position vector .
The correlation function    n  ( r )   1 
 n
describes the increase
n
 n  in the electron density due to the Coulomb attraction between a positron and
electron .The interactions of positron with matter can be summarized as the following
:
2-1The Collisional s. p. scoll
The collisional s.p. of both types of positron particles can be written as the
following( Tsai 1974):


4k 2 e4  mc2   2
 dE 
  
ln
 F  (  ) ……………(3)
2 2 
I 2
 dx coll mc  

Where for positron
F (  )  ln2
2 
14
10
4 


..........
.(4)
23
..........
2
24   2   2   23 
Equations (3) is a dimensionless functions depending on the kinetic energy T of
the incident electron and the atomic number Z of the stopping medium and
Kinetic energy meaning rest mass ,
(Btra 1970):
=
is related to the kinetic energy T by for
 1

T  m0c2
1
 1 2 


and
………………………….(5)
1/ 2
2
 

  1 
  1

T
  m c2  
  0 
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
The symbols in equation (2) are defined as following:
v = velocity of particle ,c = speed of light in vacuum ,E= energy of the incident
particle
x= distance traveled by the particle in material ,I= mean excitation potential of target
material, k = Coulomb constant, n= electron density of material which can be
calculated by the equation(Tufan 2006):
n= NA Z ρ/ A Mu………… (6)
Where NA is Avogadro's number , ρ density of target material and Mu is the molar
mass constant . In the equation (2) after substitution the above values , we obtain a
more simplified formula in units of MeV/ cm ( Jablonski 2006).

 dE
 
 dx 

5.081031n  3.61105   2

ln

F
(

)
coll 

 ………………..(7)
2
I


Where(–dE) is the energy lost in the infinitesimal material thickness of dx ,thus
higher s.p. means shorter range in material that the particle can penetrate . The
stopping power is proportional inversely with the incident particle velocity and
ionization energy .This means that the mass s.p. of a material is obtained by dividing
the stopping power by density .Common units for mass stopping power –dE/ρdx are
MeV .g-1.cm2 .The mass stopping power is a useful quantity because it expresses the
rate of energy loss of charged particle per g.cm-2 of the medium traversed.
2-2The Radiative stopping power srad
The deceleration of positron β + near a nuclear field (due to columbic repulsion) is
known as beam braking or Bremsstrahlung. Bethe and Heitler obtained an
approximate relation between the collisional Scoll and radiative Srad stopping power by
the relation (Fohrlch 1953).


S rad
 S coll
(
TZ
) ………………….(8)
800
Where Z is the atomic number of the target atom and T is the energy of the
incident positron or electron in MeV. By combining the equations (1) and (10) we get
(Moler 1932):
TZ


Stotal
 Scoll
(1 
) ………… (9)
800
2-3The mean excitation energies I
The mean excitation energies I has been calculated from the quantum mechanics
definition that obtained in derivation of Bethe formula, the following approximates
empirical formula can be used to estimate I values in eV for element with atomic
number Z ( Turner 2007):
I= 52.8+ 8.71 Z
Z > 13………………(10)
Here , we had employed the mean ionization energy because the value of I is
different for different electronic shells, so we take into consideration the mean value
given in equation (10). The average ionization potential INi  2967 eV and ICu  3054
eV respectively.
3-Results and discussion
The positron life time very short and its behavior during penetration through
matter is the same as electron behavior in regard to loss of energy , but the absorption
of positrons in a medium is important for checking the effect of annihilation which
causes the difference between the stopping powers for both electron and positron.
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
The present results indicates that the Srad (figures1and2) increases with increasing
particle incident energies, this behavior can be explained with the increasing the
incident energy will approaches the nuclear field of target atom due to the columbic
repulsion between the incident and target particles .While in contrast for Scoll up to
1.75MeV(figures3 and 4) for Cu and Ni respectively. This behavior can be
interpreted that a slower projectile(less energetic) spends more time in the proximity
of the target(electron field), hence has a higher probability of interaction, while a swift
particle(more energetic) can sweep through the target or its potential field without
being affected much, due to this behavior produces the ionization and excitation of
medium atoms.
The comparison of the total stopping powers for Ni and Cu(figures 5 and 6)
reveals their values decreases as the atomic number Z of the absorber increases. This
occurs because substances of high Z have fewer electrons per gram(equation 6), and
these are more tightly bound. Consequently ,the range tends to decrease as Z
increases. But as Z increases, the multiple scattering of the positrons decreases. The
effect of multiple scattering is to reducing the actual path of the positron in a
substance .This tends to decrease the range which is the linear distance through the
medium. These two effects act to balance each other, so that the density of a substance
gives one a good idea of its relative ability to stop positrons.
All the obtained results are compared with Estar-code(ESTAR 2010)which run on
PC. This code was prepared by combining previously data bases for Scoll, Srad and
STotal .It uses chemical structure of element or atomic number as input for matee
deviation is not too large which may because of ionization energies(INi  2967 eV
and ICu  3054 eV) adopted by present work and Estar-code .
4-Conclusions
The cross section for annihilation of fast positrons is quite small , but increases
with increasing energy .Therefore , a positron tends to lose all its energy by slowing
down (bremsstrahlung)before being annihilated .A positron can also be absorbed ,
hence annihilated by a bound electron in a atom as the velocity of positron increase .
The energy loss components depend sensitively on the charge number Z and the
average ionization potential of the absorber materials, the number density N, the
relativistic velocity of the electrons(  = v/c ) with the rest mass m0.
7.00E-01
Estar
6.00E-01
work
Srad(MeV-cm2/g
5.00E-01
4.00E-01
3.00E-01
2.00E-01
1.00E-01
0.00E+00
1.00E-01
1.00E+00
Energy(MeV)
1.00E+01
Fig(1): Comparison of the present work radiative Srad and Estar stopping
power values for copper in units MeV .g-1.cm2.
1976
Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
Fig(2): Comparison of the present work collisional Scoll and Estar stopping
power values for nickel in units MeV .g -1.cm2.
3.00E+00
Estar
Scoll(MeV-cm2/g
2.50E+00
2.00E+00
1.50E+00
1.00E+00
5.00E-01
0.00E+00
1.00E-01
1.00E+00
Energy(MeV)
1.00E+01
Fig(3): Comparison of the present work collisional Scoll and Estar stopping
power values for copper in units MeV .g-1.cm2.
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
Fig(4): Comparison of the present work collisional Scoll and Estar stopping
power values for nickel in units MeV .g-1.cm2.
Fig(5): Comparison of the present work and Estar total stopping power values
Stotal for copper in units MeV .g-1.cm2
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
Fig(6): Comparison of the present work Stotal and Estar total stopping power
values for nickel in units MeV .g-1.cm2.
When a charged particle penetrates in matter, it will interact with the electrons and
nuclei present in the material through the electromagnetic force. If the charged
particle is a proton, an alpha particle or any other charged hadron
(discussed in Chap. 1),it can also undergo a nuclear interaction and this will be
discussed in Sect. 2.5. In the present section we ignore this possibility. If the particle
has 1 MeV or more as energy, as is typical in nuclear phenomena, the energy is large
compared to the binding energy of the electrons in the atom. To a first approximation,
matter can beseen as a mixture of free electrons and nuclei at rest. The charged
particle will feel the electromagnetic fields of the electrons and the nuclei and in this
way undergo elastic collisions with these objects. The interactions with the electrons
and with the nuclei present in matter will give rise to very different effects. Let us
assume for the sake of definiteness that the charged particle is a proton. If the proton
collides with a nucleus, it will transfer some of its energy to the nucleus and its
direction will be changed. The proton is much lighter than most nuclei and the
collision with a nucleus will cause little energy loss. It is easy to show, using nonrelativistic kinematics and energy– momentum conservation, that the maximum
energy transfer in the elastic collision of a proton of mass ‘m’ with nucleus of mass
‘M’ is given by The microscopic interactions undergone by electrons or any charged
particle vary somewhat randomly, resulting in a statistical distribution of energy loss
and number of collisions along its path. The stopping power for electrons decreases as
the atomic number Z of the absorber increases. This occurs because substances of
high Z have fewer electrons per gram
n= NA Z ρ/ A Mu--------------------(4)
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
and these are more tightly bound. Consequently ,the range tends to increase as Z
increases. But as Z increases, the multiple scattering of the electrons increases. The
effect of multiple scattering is to increase the actual path of the electron in a
substance .This tends to decrease the range which is the linear distance through the
medium. These two effects act to balance each other, so that the density of a substance
gives one a good idea of its relative ability to stop electrons.
- Electrons are light mass particles, electrons are therefore scattered easily in all
directions due to their interactions with the atomic electrons of the absorber material.
This results into more energy loss per scattering event.electrons often undergo largeangle deflections along their paths due to their small mass. This leads to the
phenomenon of backscattering, in which an electron entering an absorber may
undergo sufficient deflection such that it re-emerges from the surface through which it
entered. These backscattered electrons do not deposit all of their energy in the
absorber, and therefore the backscattering process can have a significant impact on
absorbed dose. Electrons with high incident energy and absorbers with low atomic
number have the lowest probability for backscattering. Therefore, backscattering
typically occurs when low-energy electrons enters a region of high atomic number or
high mass density (Knoll 2000). Electrons backscatter by nuclear elastic scattering,
which is the glancing of an electron off an atomic nucleus. Nuclear elastic scattering
takes place when the relative size of the atomic nucleus is large and the relative
electron charge density of the atom (Z/A) is low. Lower values of Z/A generally occur
for large atomic mass numbers (A). The microscopic interactions undergone by
electrons or any charged particle vary somewhat randomly, resulting in a statistical
distribution of energy loss and number of collisions along its path.
The ratio between these two components depends on the energy of the electron
beam E and the charge Z of the absorber material.
4-stopping time:
The Stopping time is the time interval required to stop a charge particle in an
absorber .This time can be expressed in terms of the stopping power by using the
chain of differentiation( Ahad 2007) :
-dE/dt = -(dE/dx)/(dt/dx) = v(-dE/dx)……………………..(14)
Where v = dx/dt is the velocity of the particle .A rough estimate can be made of
the time it takes a heavy charged particle to stop in matter, if one assumes that the
slowing-down rate is constant. For a particle with kinetic energy T, this time is
approximately
τ= T/(-dE/ dx)t = T/v ( dE / dx )sec ……………………. (15)
-4Radiation length:
Tsai Method ,which is employed for estimation the radiative which depends on
radiation length (9)
……….(11)
The function f(Z) is called the Coulomb correction function given by:
1980
Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
=
…...(12)
Where a is
, and
=
is the fine structure ,its values given by the
table(1)
Table(1): parameters values of Coulomb correction functions of Tsai method.
Elements
6.144
5.31
H1
5.621
4.79
He2
5.805
4.74
Li3
5.924
4.71
Be4
ln(1194Z- ln(184.15Z-1/3)
Others
2/3
)
'
Dahl s relation ,which requires less computations than equation (12) is given
by(ESTRA 2010):
716 . 4 A
.......( 13 )
Z (1  Z ) ln( 287 / Z )
Here, X0 is in g/cm2 this relation gives reasonable results for elements with low
and moderate atomic numbers . In the equations 12and 13 , X0 is a quantity that
characterizes how charged particles interact in a material. It depends on the density
and the charge of the nucleus
-Bremsstrahlung.Any charged particle undergoing acceleration will emit
electromagnetic radiation. If a high-energy charged particle deviates from its
trajectory due to a collision with a nucleus, this collision is necessarily accompanied
by electromagnetic radiation. The emission is strongly peaked in the direction of flight
of the charged particles.
-The microscopic interactions undergone by positrons or any charged particle vary
somewhat randomly, resulting in a statistical distribution of energy loss and number
of collisions along its path.
-. If we analyzing the radiative stopping power figure (3) and table 2 illustrates the
values of Srad increasing this two situation can be explained that the nuclear braking
begin due to the incident electrons approaches the nuclear field of the target atoms
and by observing the total stopping power in figure (4) which take the same behavior
of the SColl in figure (2) which we can conclude that for STotal values , the Srad has
little contribution due to that we had used low and moderate electron energies.
X
0

1981
Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
5-References:
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Zhenyu Tan ,Yueyuan Xia , Mingwmino . Zhao and Xian gdong Liu (2008)."
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