تقدير فقدان الطاقة الكلي لبوزترونات في عنصري النحاس والنيكل

صباح محمود أمان الله 1 ايناس محمد سلمان2 رافع عبدالله عباد 1
1 قسم الفيزياء – كلية العلوم –جامعة تكريت- صلاح الدين – العراق
2 قسم الفيزياء – كلية العلوم للبنات –جامعة بابل - بابل – العراق

الخلاصة :
قدمنا في هذا البحث قيم قدرة الإيقاف الإشعاعية ,التصادمية و القدرة الكلية , طول المسار الإشعاعي وزمن التوقف لبوزترونات ?+ الساقطة على مادتي النحاس و النيكل باستخدام معادلة بيث –بلوخ النسبية ولمدى طاقة يبدأ من0.1 ميكا إلكترون فولت 10 ميكاالكترون فولت وقد بينت النتائج التي حصلنا عليها الهيمنة الكبيرة لقدرة الإيقاف الإشعاعية أكثر من قدرة الإيقاف التصادمية لقيم قدرة الإيقاف الكلية وبالمقارنة النتائج مع الكود العالمي Estar فقد وجدت انها في تطابق جيد مع البرنامج العالمي أي ستار .
الكلمات المفتاحية: معادلة بيث-بلوخ , قدرة الايقاف ,الاشعاعية ,التصادمية ,معدل طاقة التأيين , النيكل, النحاس ,برنامج اي ستار .
Estimation of the total energy loss of positrons in Copper and Nickel
1Sabah Mahmoud Aman Allah .
2Enas Mohamad Salaman
1Rafea Abduala Abad
1Physics Department - College of Science –Tikrit University –Salah din –Iraq.
2Physics Department - College of Science for Women –Babylon University –Babylon –Iraq.
Abstract:
In present paper , the values of radiative Srad and collisional Scoll stopping powers and the total stopping power positrons ?+ 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: Bethe-Bloch ,stopping power ,stopping time , radiative ,collision, mean excitation energy, Nickel , copper , Estar code.
1-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[1]. 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 ?+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 [2-6]:
……………….(1)
Where the signs (+) and (-) refers to positron and electron respectively. An extensive study [7-14] 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 , and the electron density [15].
……………(2)
r0 is the classical radius , c the speed of the light , and r the position vector .The correlation function describes the increase 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 [16] ……………(3)
Where for positron

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 , ? = v/c is related to the kinetic energy T by for [11]:

………………………….(5)
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[17]:
n= NA Z ?/ A Mu………… (6)
Where NA is Avogadro s number , ? 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 [12].
………………..(7)
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/?dx 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 ?+ 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 [9].
………………….(8)
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 [ 7]:
………… (9)

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 [19]:
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.
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 [20]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 materials . However the 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.

Fig(1): Comparison of the present work radiative Srad and Estar stopping power values for copper in units MeV .g-1.cm2.


Fig(2): Comparison of the present work collisional Scoll and Estar stopping power values for nickel in units MeV .g-1.cm2.


Fig(3): Comparison of the present work collisional Scoll and Estar stopping power values for copper in units MeV .g-1.cm2.

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


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 non-relativistic 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 ?/ A Mu--------------------(4)
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 large-angle 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[22 ]:
-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
?= 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 [19]
1/X_0 =4?r_e^2 N_A/A {Z^2 [L_rad-f(Z)+Z?L^ ?_rad ] } ……….(11)
The function f(Z) is called the Coulomb correction function given by:
f(Z)= a^2 [1/((1+a^(2 ) ) )+0.20206-0.0369a^(2 )+0.0083a^(4 )+0.02a^(6 ) ]…………...(12)
Where a is a=? Z, and ?=?(1/37) is the fine structure ,its values given by the table(1)
Table(1): parameters values of Coulomb correction functions of Tsai method.
Elements L_rad ?L^ ?_rad
H1 5.31 6.144
He2 4.79 5.621
Li3 4.74 5.805
Be4 4.71 5.924
Others ln(184.15Z-1/3) ln(1194Z-2/3)


Dahl s relation ,which requires less computations than equation (12) is given by[20]:


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.
?
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