Research Article Open Access
A Method for Simultaneous Determination of Effective Removal Cross-section for Fast Neutrons and Mass Absorption Coefficient for Gamma Rays
Abdelghany A. El AbdM1 and Ashraf S. Elkady1,2*
1Egyptian Atomic Energy Authority (EAEA), NRC, Cairo, Egypt
2Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
*Corresponding author: Ashraf S. Elkady, Department of Physics, Faculty of Science for girls, King Fahd road, P.O. 9470, Jeddah 21413, Saudi Arabia, Tel: +966-543547574; E-mail: aelkady@kau.edu.sa
Received: August 19, 2014; Accepted: September 30, 2014; Published: October 16, 2014
Citation: El Abd AA, Elkady AS (2014) A Method for Simultaneous Determination of Effective Removal Cross-section for Fast Neutrons and Mass Absorption Coefficient for Gamma Rays . SOJ Mater Sci Eng 2(2): 1-6.
AbstractTop
A method for determining the effective removal cross section (ΣR) of fast neutrons and mass absorption coefficient (μR) for gamma rays at energies 661.6 keV and 1332.5 keV is proposed. It is based on relating the ratio (R) of ΣR to μR at 661.6 keV and 1332.5 keV to effective atomic number (Zeff) for elements, compounds, composite materials and alloys. It is shown that the R-values versus Zeff for the most studied materials lie on two master curves. The method was tested by determining ΣR and μR for several materials. There is a good agreement between results obtained from the present developed method and the traditional method. Some deviations from the values obtained from the traditional method are discussed as well.

Keywords: Effective removal cross-section; Mass absorption coefficient; Fast neutrons; Gamma rays; Composite materials; Alloys
Introduction
The study of photon and neutron interactions with matter is an important issue for several applications, e.g. in industry, medical radiation dosimetry, security inspections, radiation shielding and nuclear engineering materials. In this aspect, an interdisciplinary science between nuclear physics and materials science emerges, which can much help in engineering novel materials for the required application. Mass attenuation coefficient, μR (cm2/g), effective atomic number (Zeff), effective electron density, and photon mean free path are the most important quantities for determining the penetration of X-ray and gamma rays in matter. Theoretical values of the mass attenuation coefficients for elements, compounds and mixtures (composites) from 1 keV to 100 GeV can be obtained using WinXCom software [1]. Besides, the scattering and absorption of X-ray and gamma radiation in matter are related to the densities and atomic numbers of its elemental constituents. However, when such scattering and absorption take place in composite materials, it is then related to the density and the effective atomic number (Zeff). Zeff is introduced to describe the properties of composite materials in terms of equivalent elements. Values of the effective atomic number for many composite materials and alloys have been reported [2-6].

The effective removal cross-section, ΣR (cm2/g) is the probability that a fast or fission energy neutron undergoes a first collision, which removes it from the group of penetrating, uncollided neutrons. It is considered to be approximately constant for neutron energies between 2 and 12 MeV [7]. To use the concept of ΣR, the shielding material under investigation should contain some scattering atoms. However, when there are no scattering atoms, another quantity i.e. the total mass neutron cross-section ΣT (cm2/g) is used. The observed value of the ΣR is roughly 2/3 of ΣT for neutrons having energies in the range of 6-8 MeV [8].

Non-Destructive Testing (NDT) of materials is a well-known technique applied in several fields such as inspection of luggage and containers. NDT methods are mainly based on gamma or X-ray scanners, which produce high resolution images. In addition, photons inspection provides materials recognition when traditional transmission measurements at fixed energy are implemented with special technologies as in the case of the so-called "dual energy radiography", "backscattering imaging" or "computed tomography" [7-9]. Materials recognition in such applications is based on the atomic number Z dependence of the relevant photon absorption coefficients: it is a well-established method at low photon energy where the photoelectric effect dominates, while it becomes critical for increasing photon energy, as it is required in order to increase the penetration of radiation to inspect thick objects [8-10]. A drawback in utilizing photon-based inspection approaches is that when the photon penetration is not sufficient, the object’s image appears black. Thus, developing new approaches to overcome the latter problem is a concern in security inspections.

Therefore, in most cases when photon irradiation is unable to disentangle the problem of inspections, the use of neutrons as probing radiation has been often proposed. To this end, sophisticated techniques have been developed in order to enhance materials recognition, especially for low-atomic-number materials, in an effort to optimize the detection of explosives and drugs in customs operation. Examples of such developments

are represented by the "combined fast-neutron and gammaradiography" and the "fast neutron resonance radiography" [11-15], or by detection of neutron induced gamma rays [15]. Combined fast-neutron and gamma radiography systems [9] perform materials recognition by transmission measurements of fast neutrons and gamma rays [16,17]. Neutrons and gamma rays are obtained from either separate sources such as 14 MeV neutrons (produced by a D+T generator) and gamma rays from an intense 60Co radioactive source [16], or from the same source such as 252Cf [17,18].

It was shown that the ratio of the effective removal crosssection for fast neutrons to the mass absorption of gamma rays, R can be utilized for the purpose of non-destructive testing for materials recognition [11,12,15,17,18]. However, this ratio was not applied before for determining the effective removal crosssection and the mass absorption coefficient for any material.

The present work aims at developing a simple method based on using the ratio (R) of ΣR to μR at 661.6 keV and 1332.5 keV, for elements for determining ΣR and μR with the knowledge of Zeff for any material.
Theory
The mass absorption coefficient for gamma rays, μR and the effective removal cross-section for fast neutrons, ΣR can be calculated for mixtures, alloys and compounds, with the knowledge of the weight percentages wi, and the values of μR and ΣR of the constituting elements. This is achieved by the following simple addition rules [7,19]:
μ R =  i w i ( μ R ) i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH8oqBpaWaaSbaaSqaa8qacaWGsbaapaqabaGcpeGaeyypa0Ja aeiiaiabggHiL=aadaWgaaWcbaWdbiaadMgaa8aabeaakiaadEhada WgaaWcbaWdbiaadMgaa8aabeaakiaacIcapeGaeqiVd02damaaBaaa leaapeGaamOuaaWdaeqaaOGaaiykamaaBaaaleaapeGaamyAaaWdae qaaaaa@4588@
R = i w i ( R ) i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHris5paWaaSbaaSqaa8qacaWGsbaapaqabaGcpeGaeyypa0Ja eyyeIu+damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaadEhapaWaaS baaSqaa8qacaWGPbaapaqabaGcdaqadaqaa8qacqGHris5paWaaSba aSqaa8qacaWGsbaapaqabaaakiaawIcacaGLPaaadaWgaaWcbaWdbi aadMgaa8aabeaaaaa@4510@
for gamma rays and fast neutrons, respectively.
It was proposed that the following empirical formulas [19]:
R = 0.21 A 0.56 c m 2 g 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHris5paWaaSbaaSqaa8qacaWGsbaapaqabaGcpeGaeyypa0Ja aeiiaiaaicdacaGGUaGaaGOmaiaaigdacaWGbbWdamaaCaaaleqaba WdbiabgkHiTiaaicdacaGGUaGaaGynaiaaiAdaaaGccaWGJbGaamyB a8aadaahaaWcbeqaa8qacaaIYaaaaOGaam4za8aadaahaaWcbeqaa8 qacqGHsislcaaIXaaaaaaa@4851@
R = 0.00662 A 1/3 +0.33  A 2/3 0.211  A 1 c m 2 g 1 ( A>12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHris5paWaaSbaaSqaa8qacaWGsbaapaqabaGcpeGaeyypa0Ja aeiiaiaaicdacaGGUaGaaGimaiaaicdacaaI2aGaaGOnaiaaikdaca WGbbWdamaaCaaaleqabaWdbiabgkHiTiaaigdacaGGVaGaaG4maaaa kiabgUcaRiaaicdacaGGUaGaaG4maiaaiodacaqGGaGaamyqa8aada ahaaWcbeqaa8qacqGHsislcaaIYaGaai4laiaaiodaaaGccqGHsisl caaIWaGaaiOlaiaaikdacaaIXaGaaGymaiaabccacaWGbbWdamaaCa aaleqabaWdbiabgkHiTiaaigdaaaGccaWGJbGaamyBa8aadaahaaWc beqaa8qacaaIYaaaaOGaam4za8aadaahaaWcbeqaa8qacqGHsislca aIXaaaaOWdamaabmaabaWdbiaadgeacqGH+aGpcaaIXaGaaGOmaaWd aiaawIcacaGLPaaaaaa@5F62@
R = 0.190  Z 0.743   c m 2 g 1 ( Z  8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHris5paWaaSbaaSqaa8qacaWGsbaapaqabaGcpeGaeyypa0Ja aeiiaiaaicdacaGGUaGaaGymaiaaiMdacaaIWaGaaeiiaiaadQfapa WaaWbaaSqabeaapeGaeyOeI0IaaGimaiaac6cacaaI3aGaaGinaiaa iodaaaGccaGGGcGaaiiOaiaadogacaWGTbWdamaaCaaaleqabaWdbi aaikdaaaGccaWGNbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGc paWaaeWaaeaapeGaamOwaiaabccacqGHKjYOcaqGGaGaaGioaaWdai aawIcacaGLPaaaaaa@5330@
R = 0.125  Z 0.565  c m 2 g 1  ,  ( Z>8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHris5paWaaSbaaSqaa8qacaWGsbaapaqabaGcpeGaeyypa0Ja aeiiaiaaicdacaGGUaGaaGymaiaaikdacaaI1aGaaeiiaiaadQfapa WaaWbaaSqabeaapeGaeyOeI0IaaGimaiaac6cacaaI1aGaaGOnaiaa iwdaaaGccaGGGcGaam4yaiaad2gapaWaaWbaaSqabeaapeGaaGOmaa aakiaadEgapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaakiaaccka caGGSaGaaiiOaiaacckapaWaaeWaaeaapeGaamOwaiabg6da+iaaiI daa8aacaGLOaGaayzkaaaaaa@5435@
Can be used to determine ΣR as a function of the atomic weight A and the atomic number, Z.
The effective atomic number for composites, compounds and alloys for gamma rays can be determined as follows [20]: The total photon interaction cross section, σm, per molecule can be written
σ m = i σ i n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaGcpeGaeyypa0Zd amaaqahabaWaaSbaaSqaaiaadMgaaeqaaaqaaaqaaaqdcqGHris5aO Wdbiabeo8aZ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaWGUbWd amaaBaaaleaapeGaamyAaaWdaeqaaaaa@431F@
where ni is the number of atoms of the ith constituent element, and σi is total photon interaction cross section per atom of element i.
The total number of atoms in the compound n is given by:
n = i n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGUbGaaeiiaiabg2da98aadaaeabqaamaaBaaaleaacaWGPbaa beaaaeqabeqdcqGHris5aOWdbiaad6gapaWaaSbaaSqaa8qacaWGPb aapaqabaaaaa@3E39@
Suppose that the cross section per molecule can be written in terms of an effective (average) cross section, σa, per atom and an effective (average) cross section, σe, per electron as
  σ m = n σ a = n  Z eff σ e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWdbiab g2da9iaabccacaWGUbGaeq4Wdm3damaaBaaaleaapeGaamyyaaWdae qaaOWdbiabg2da9iaabccacaWGUbGaaeiiaiaadQfapaWaaSbaaSqa a8qacaWGLbGaamOzaiaadAgaa8aabeaak8qacqaHdpWCpaWaaSbaaS qaa8qacaWGLbaapaqabaaaaa@4A75@
Eq. (9) can be regarded as the definition of the effective atomic number. Essentially, it assumes that the actual atoms of a given molecule can be replaced by an equal number of identical (average) atoms, each of which having Zeff electrons. From Eqs. 7 and 9 one obtains
σ a =( i n i σ i ) /n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeGaeyypa0Za aeWaaeaapaWaaabqaeaadaWgaaWcbaGaamyAaaqabaaabeqab0Gaey yeIuoak8qacaWGUbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiab eo8aZ9aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPa aacaqGGaGaai4laiaad6gaaaa@46D1@ σ e =( i n i σ i / Z i ) /n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGLbaapaqabaGcpeGaeyypa0Za aeWaaeaadaaeabqaamaaBaaaleaacaWGPbaabeaaaeqabeqdcqGHri s5aOGaamOBa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqaHdpWC paWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaai4laiaadQfapaWaaS baaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaaeiiaiaa c+cacaWGUbaaaa@49AA@
It follows from the last equality of Eq. (9) that the effective atomic number can be written as the ratio between the atomic and electronic cross sections:
Z eff = σ a / σ e =( i n i σ i ) /( i n i σ i / Z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGAbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqabaGc peGaeyypa0Jaeq4Wdm3damaaBaaaleaapeGaamyyaaWdaeqaaOWdbi aac+cacqaHdpWCpaWaaSbaaSqaa8qacaWGLbaapaqabaGcpeGaeyyp a0ZaaeWaaeaadaaeabqaamaaBaaaleaacaWGPbaabeaaaeqabeqdcq GHris5aOGaamOBa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqaH dpWCpaWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaa Gaaeiiaiaac+cadaqadaqaamaaqaeabaWaaSbaaSqaaiaadMgaaeqa aaqabeqaniabggHiLdGccaWGUbWdamaaBaaaleaapeGaamyAaaWdae qaaOWdbiabeo8aZ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaGG VaGaamOwa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawIcaca GLPaaaaaa@5BD0@
Eq. (12) is then the basic relation for calculating the effective atomic number of a chemical compound.
A more general expression for Zeff can be obtained by introducing the molar fraction, fi (sometimes expressed in units of atomic percent, at.%). For a chemical compound, one has
f i =  n i / i n i =  n i /n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGMbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da9iaa bccacaWGUbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaac+cada aeabqaamaaBaaaleaacaWGPbaabeaaaeqabeqdcqGHris5aOGaamOB a8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpcaqGGaGaam OBa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaGGVaGaamOBaaaa @483A@
where Σifi =1. Rewriting Eq.12 in terms of fi one has
Z eff =( i f i σ i ) /( i f i σ i / Z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGAbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqabaGc peGaeyypa0ZaaeWaaeaapaWaaabqaeaadaWgaaWcbaWdbiaadMgaa8 aabeaaaeqabeqdcqGHris5aOWdbiaadAgapaWaaSbaaSqaa8qacaWG PbaapaqabaGccqaHdpWCdaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbi aawIcacaGLPaaacaqGGaGaai4lamaabmaabaWdamaaqaeabaWaaSba aSqaa8qacaWGPbaapaqabaaabeqab0GaeyyeIuoak8qacaWGMbWdam aaBaaaleaapeGaamyAaaWdaeqaaOGaeq4Wdm3aaSbaaSqaa8qacaWG PbaapaqabaGcpeGaai4laiaadQfapaWaaSbaaSqaa8qacaWGPbaapa qabaaak8qacaGLOaGaayzkaaaaaa@5407@
Eq. (14) is then the basic relation for calculating the effective atomic number for all types of materials, compounds as well as composites.
The atomic cross section, σi, of the ith constituent element is related to the corresponding mass attenuation coefficient, (μ/ρ) i, through the relation
σ i =  (µ/ρ) i A i / N A   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaabaaaaaaaaapeGaamyAaaWdaeqaaOWdbiabg2da9iaabcca paGaaiika8qacaWG1cGaai4laiabeg8aY9aacaGGPaWaaSbaaSqaa8 qacaWGPbaapaqabaGcpeGaamyqa8aadaWgaaWcbaWdbiaadMgaa8aa beaak8qacaGGVaGaamOta8aadaWgaaWcbaWdbiaadgeaa8aabeaak8 qacaGGGcaaaa@4758@
where Ai is the atomic mass, and NA is the Avogadro’s constant. Inserting expression (15)

for σi in the eq. 14 gives
Z eff =( i f i A i (μ/ρ) i ) /( i f i (μ/ρ) i A i / Z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGAbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqabaGc peGaeyypa0ZaaeWaaeaapaWaaabqaeaadaWgaaWcbaGaamyAaaqaba aabeqab0GaeyyeIuoak8qacaWGMbWdamaaBaaaleaapeGaamyAaaWd aeqaaOWdbiaadgeapaWaaSbaaSqaa8qacaWGPbaapaqabaGccaGGOa GaeqiVd02dbiaac+cacqaHbpGCpaGaaiykamaaBaaaleaapeGaamyA aaWdaeqaaaGcpeGaayjkaiaawMcaaiaabccacaGGVaWaaeWaaeaapa WaaabqaeaadaWgaaWcbaGaamyAaaqabaaabeqab0GaeyyeIuoak8qa caWGMbWdamaaBaaaleaapeGaamyAaaWdaeqaaOGaaiikaiabeY7aT9 qacaGGVaGaeqyWdi3daiaacMcadaWgaaWcbaWdbiaadMgaa8aabeaa k8qacaWGbbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaac+caca WGAbWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcpeGaayjkaiaawMca aaaa@5FD5@
Eq. (16) can be used for calculating the effective atomic number for both compounds and composites in terms of the mass absorption coefficients.
Results and Discussions
The physical data of most elements, atomic masses, mass removal cross-sections ΣR, and mass absorption coefficients for gamma rays at 661.6 keV and 1332 keV for elements needed for the calculations [1,7,8,19], were prepared and stored in an Excel spread sheets. The equations written above were implemented in the sheets. They were used to calculate mass absorption coefficients, effective removal cross-sections and effective atomic numbers for the chosen materials.

The estimated ratios of effective removal cross section to mass absorption coefficients at the gamma ray energies 661.6 keV and 1332 keV for most elements (R), as well as effective removal cross section versus atomic number for elements (Z=1- 92) are shown in Figure 1. As one can see, the R-values at 1332.5 keV are higher than those at 661.6 keV.

The R-values and Zeff were calculated for some compounds, composites and alloys. These are : H2O, B4C, CO, CO2, MgO, MgCO, SO3, MgCO3, NaCl, SO2, NaO,Na2O, SiO2, P2O5, Al2O3, CaO, CaO3, TiO2, MnO2, K2O, FeO, Fe2O3, NiO, CuO, ZnO, RbO, SrO, H-Li , H2Li, H3-Li, H4Li, HB, H2B, HBe, HC, HO, CH2 HNa, Zn-Cu-Ni, Mn- Zn-As, Zr-Mo-Cd, Ag-Sn-Nd, As-Ga, Mn-Sc, Mn-Zn-Rb, Mn-Zn-Sr, Dy-H, Dy-Lu, Lu-W-Au, Hf-Pb, H-Pb, Sn-Pb, Sn-Pb, Zr-Pb, Zn-Pb, Sn-H, and Zr-H. Also, for some of these materials, the R-values and Zeff were calculated for different concentrations of their constituents. Figure 2 shows these results along with those for the corresponding elements. As one can see, the ΣR and R-values (calculated relative to μR at 661.6 keV and 1332.5 keV) coincide with the data for the elements. The calculated values of Zeff at 661.6 keV and 1332.5 keV for any of the above mentioned materials were found roughly the same, which implies an energy independence of Zeff in the mentioned range of energies. Of worth noting, the values of ΣR and μR (at 661.6 keV and 1332.5 keV) for any material can be determined simultaneously. The determination of ΣR and μR is based on the knowledge of the corresponding Zeff at either 661.6 keV or 1332.5 keV. Namely, once Zeff is determined for any material, the table containing the values of R including μR at 661.6 keV and 1332.5 keV and ΣR is
Figure 1: The ratio R at 661.6 keV, 1332.5 keV and ΣR versus Z for elements.
Figure 2: The ratio R and ΣR values for some compounds, composites and alloys versus Zeff. Results of Figure. 1 are also included.
searched for the closest value to Zeff. At this value of Zeff, the values of μR at 661.6 keV and 1332.5 keV and ΣR are the required ones for such material.
To check the proposed method, the results (for compounds, alloys and mixtures stated above), were used to determine ΣR and μR with the knowledge of Zeff for the following materials [21]: 1%, 5%, 5.45%, and 30% borated polyethylene, 7.5% Lithium Polyethylene, 78.5% and 90% bismuth-loaded polyethylene, borated silicone, flexi-boron shielding, borated Hydrogen- Loaded castable dry mix, borated hydrogenated mix, boratedlead polyethylene, K-resin, resin 250WD, SUS304, krafton-HB, and premadex. The numbers from 1 to 17 in Table 1 refer to them respectively. The chemical composition of these composites was taken from reference [21]. The obtained results of ΣR and μR in comparison with the corresponding values calculated by the traditional method (Eqs.1-16) are listed in Table 1. In most cases, good agreement can be noticed between the determined values by the proposed method and the traditional methods. Besides, the determined ΣR, μR and Zeff for these materials along with those shown in Figure 2 were used to determine ΣR, μR for natural Fiber–Plastic (FP), Fiber–Plastic–Lead (FPPb), Cement–Fiber (CF) and Cement–Fiber–Magnetite (CFM) composites, along with the shielding materials of dolomite-sand, barite-barite, magnetite-limonite, ilmenite–ilmenite [22-24]. The chemical compositions shown in Table 2 were taken from references [22- 24]. The results obtained by the present method along with those obtained by the traditional method are listed in Table 1. One can note good agreements between the values obtained by using the two methods. It can also be noticed in Table 1 for 78.5% bismuthloaded polyethylene (number "6"), compound number "12", FPPb, Barite-barite, and Magnetite-limonite that, there are two values of Zeff calculated by the traditional method. These values are obtained from values of μR at 661.6 keV and 1332.5 keV. In such case, the table containing the R-values including μR at 661.6 keV and 1332.5 keV and ΣR is searched for the closest value and/
Table 1: μR, ΣR and Zeff calculated by the traditional and the present methods.

Compound/ mixture/alloy number or name

mR, ∑R  in (cm2/g) , and Zeff calculated by traditional method

mR  and R  in (cm2/g) , calculated in this work

661.6

(keV)

1332.5

(keV)

R

Zeff

661.6 (keV)

1332.5

(keV)

R

Zeff

1

0.0809

0.0576

0.0718

4.60

0.0818

0.0583

0.0735

4.50

2

0.0857

0.0611

0.1119

3.00

0.0784

0.0559

0.104

3.00

3

0.0811

0.0578

0.0728

4.53

0.0818

0.0583

0.0735

4.50

4

0.0817

0.0582

0.1001

3.22

0.0886

0.0612

0.1029

3.43

5

0.0818

0.0583

0.0927

3.50

0.0831

0.0592

0.0926

3.50

6

0.1080

0.0581

0.0358

8.40

8.66

0.0772

0.0769

0.0549

0.0548

0.0398

0.0394

8.40

8.66

7

0.1110

0.0575

0.0222

15.85

0.0741

0.0526

0.0259

15.6

8

0.0803

0.0572

0.0641

5.20

0.0727

0.0519

0.0562

5.20

9

0.0778

0.0554

0.0591

5.45

0.0727

0.0519

0.0562

5.20

10

0.0850

0.0606

0.1010

3.43

0.0831

0.0858

0.0592

0.0612

0.0926

0.103

3.40

3.34

11

0.0799

0.0564105

0.05507794

6.1

0.0774

0.0552

0.0578

6.00

12

0.1060

0.0587

0.0302

10.6/10.8

0.0774

0.0551

0.0341

10.7

13

0.0934

0.0666

0.129

2.96

0.0881

0.0628

0.1289

2.67

14

0.0825

0.0587

0.0844

3.89

0.0831

0.0592

0.0926

3.50

15

0.0740

0.0522

0.0214

25

0.0718

0.0507

0.0204

24.5

16

0.0850

0.0605

0.107

3.11

0.0784

0.0559

0.104

3.00

17

0.0859

0.0612

0.109

3.12

0.0784

0.0559

0.104

3

FP

0.0836

0.0595

0.0939

3.51

0.0831

0.0592

0.0962

5.5

FPPb

0.0912

0.0571

0.0598

5.03/4.92

0.0727

0.0519

0.0562

5.2

Dolomite -

sand

0.0776

0.0552

0.0403

8.6

0.0769

0.0548

0.0394

 

5.66

Barite-barite

0.0779

0.0526

0.0287

12.5/12.4

0.0741

0.0526

 

0.0285

12.7

Magnetite-limonite

0.0766

0.0543

0.0365

10/10.1

0.0767

0.0544

0.0354

10.00

Ilmenite -ilmenite

0.0755

0.0536

0.0327

11.2

0.0776

0.0552

0.0331

10.02

CF

0.0799

0.0568

0.0509

6.74

0.0782

0.0558

0.0499

6.50

CFM

0.0772

0.0548

0.0448

7.63

0.0795

0.0522

0.0429

7.74

Table 2: Weight fractions of some materials used in neutron and γ-ray shielding

Element

FP

FPPb

Dolomite -

sand

Barite-barite

Magnetite-limonite

Ilmenite –ilmenite

 CF

CFM

H

0.0860

0.043500

0.0082538

0.006

0.011460

0.0064622

0.0284

0.01780

B

-

0.149100

 

 

 

 

 

0.09290

C

0.5774

0.323700

0.0839755

0.0029

0.000076

 

0.1091

 0.09457

O

0.3333

0.128800

0.5098378

0.33128

0.392600

0.384150

0.4323

0.32820

Zn

0.0010

0.000800

 

 

 

 

 

 

Pb

 

0.353700

 

 

 

 

 

 

Ca

-

-

0.2611081

0.061000

0.090200

0.053300

0.3162

0.126800

Mg

-

-

0.069146

0.004200

0.003860

0.001721

0.00602

0.005000

Na

 

 

0.002732

0.003100

0.006800

0.006944

 

 

K

 

 

0.0002803

 

0.000446

0.002232

 

 

Fe

 

 

0.004129

0.00300

0.421100

0.280000

0.0294

0.274200

P

 

 

0.0000164

0.000005

0

0.000785

 

 

Si

 

 

0.05412

0.022441

0.0694738

0.015600

0.0644

0.025800

S

 

 

0.0004093

0.106300

0.0002412

0.000854

0.0043

0.001700

Al

 

 

0.00064078

0.011200

0.0043540

0.003720

0.0151

0.006100

Ba

 

 

 

0.443965

 

 

 

 

Cl

 

 

 

0.004757

 

 

 

 

Ti

 

 

 

 

 

0.243800

 

0.028700

Mn

 

 

 

 

 

0.001550

 

 

Ni

 

 

 

 

 

0.000366

 

 

Cr

 

 

 

 

 

 

 

0.001640

or values to Zeff. For example, for the compound number "12", the calculated values of Zeff are 10.6 and 10.8 and the closest value is 10.7.

Deviations between μR (at 661.6 keV and 1332.5 keV) and ΣR determined by the present approach and the traditional method are noticed for some mixtures (Table 3). These can be noticed for 78.5% bismuth-loaded polyethylene (0.785 Bi, 0.184 C, 0.0309 H), 90% bismuth-loaded Polyethylene (0.9 Bi, 0.0866 C, 0.0144 H), borated lead polyethylene (0.8 Pb, 0.0122 Ca, 0.0047 Si, 0.042 O, 0.1071 C, 0.061 B, 0.0179 H) and Fiber–Plastic–Lead (FPPb) in Table2. These deviations are only for μR at 661.6 keV. There are no deviations at 1332.5 keV. For ΣR, deviations do not exceed 17%.
It was noticed that when a composite mixture consisting of elements having high atomic numbers (with high weight
Table 3: Deviations of μR at 661.6 and 1332.5 keV, and ΣR determined in this work from traditional method.

Compound/ mixture/alloy number or name

mR( this work)/ mR(traditional)

at 661.6 keV

mR( this work)/mR(traditional)

at 1332.5 keV

R( this work)/ R (traditional)

1

1.01

1.01

1.02

2

0.92

0.92

0.93

3

1.01

1.01

1.01

4

1.08

1.05

1.02

5

1.02

1.02

1.00

6

0.72

0.95

1.11

7

0.67

0.92

1.17

8

0.91

0.91

0.88

9

0.93

0.94

0.95

10

0.98

0.98

0.92

11

0.97

0.98

1.05

12

0.73

0.94

1.13

13

0.94

0.94

1.00

14

1.01

1.01

1.10

15

0.97

0.97

0.95

16

0.92

0.92

0.97

17

0.91

0.91

0.95

FP

0.99

1.00

1.02

FPPb

0.78

0.91

0.94

Dolomite -

sand

0.99

0.99

0.98

Barite-barite

0.95

1

0.990

Magnetite-limonite

1.00

1.00

0.97

Ilmenite -ilmenite

1.03

1.03

1.01

CF

0.98

0.98

0.98

CFM

1.03

0.95

0.96

fraction), and the rest of the constituting elements having low atomic numbers, the resultant value of μR at 661.6 keV or ΣR for the mixture deviates from the traditional method. This is noticed for 78.5% and 90% bismuth loaded polyethylene, borated lead polyethylene and Fiber-Plastic-Lead (FPPb) composites. Also, it is noticed that the values of Zeff determined for these mixtures at 661.6 keV and 1332.5 keV are slightly different. The deviations noticed in this work can be minimized, e.g. by preparing separate tables containing data for mixtures consisting of elements having high and low atomic numbers. Namely, for every mixture, compound and/or alloy of interest, a separate table should be prepared. These concerns along with electron density calculations, for composites containing light and heavy elements, will be considered in a forthcoming work.

Importantly, the new developed method would be beneficial not only in the area of nuclear physics but in materials science and engineering as well, as it would help a lot in carrying out the necessary calculations for the design of new materials used in radiation shielding and detection. Actually, the recent advances in materials science and nanotechnology allowed for the creation of new materials with superior and enhanced characteristics that qualify them to be used in radiation detection and shielding [25,26]. However, in order to understand the behavior of such advanced nanomaterials under the influence of radiation, it is necessary to know preliminary information about their characteristic parameters influencing, e.g. their radiation detection efficiency. Among these important parameters are the nanocomposites mass attenuation coefficients and effective removal cross-sections. Whence, the current developed method for estimating such parameters is important in understanding their radiation detection, or shielding characteristics measured at different radiation doses.
Conclusions
In this work, a method was developed for determining the effective removal cross-section for fast neutrons and the mass absorption coefficients for gamma rays at 661.6 keV and 1332.5 keV for any compound, alloy and/or composite material. The effective atomic number should be known at one of these energies. In most cases, good agreement is obtained between the values determined by the proposed approach and those obtained by the traditional method.
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