A Method for Simultaneous Determination of Effective Removal Cross-section for Fast Neutrons and Mass Absorption Coefficient for Gamma Rays

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, ∑R (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 ∑R, 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 ∑T (cm2/g) is used. The observed value of the ∑R is roughly 2/3 of ∑T for neutrons having energies in the range of 6-8 MeV [8].


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, µ R (cm 2 /g), effective atomic number (Z eff ), 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 (Z eff ).Z eff 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][3][4][5][6].
The effective removal cross-section, ∑ R (cm 2 /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 ∑ R , 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 ∑ T (cm 2 /g) is used.The observed value of the ∑ R is roughly 2/3 of ∑ T 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][8][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][9][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][12][13][14][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 60 Co radioactive source [16], or from the same source such as 252 Cf [17,18].
It was shown that the ratio of the effective removal crosssection for fast neutrons to the mass absorption coefficient 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 cross-section and the mass absorption coefficient for any material.
The present work aims at developing a simple method based on using the ratio (R) of ∑ R to µ R at 661.6 keV and 1332.5 keV, for elements for determining ∑ R and µ R with the knowledge of Z eff for any material.

Theory
The mass absorption coefficient for gamma rays, µ R and the effective removal cross-section for fast neutrons, ∑ R can be calculated for mixtures, alloys and compounds, with the knowledge of the weight percentages w i , and the values of µ R and ∑ R of the constituting elements.This is achieved by the following simple addition rules [7,19]: ( ) for gamma rays and fast neutrons, respectively.
It was proposed that the following empirical formulas [19]: ( ) ( ) ( ) Can be used to determine ∑ R 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, σ m , per molecule can be written where n i is the number of atoms of the i th constituent element, and σ i is total photon interaction cross section per atom of element i.The total number of atoms in the compound n is given by: Suppose that the cross section per molecule can be written in terms of an effective (average) cross section, σ a , per atom and an effective (average) cross section, σ e , per electron as 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 Z eff electrons.From Eqs. 7 and 9 one obtains 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: Eq. ( 12) is then the basic relation for calculating the effective atomic number of a chemical compound.
A more general expression for Z eff can be obtained by introducing the molar fraction, f i (sometimes expressed in units of atomic percent, at.%).For a chemical compound, one has where Σ i f i =1.Rewriting Eq.12 in terms of f i one has ( ) ( ) 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, σ i , of the i th constituent element is related to the corresponding mass attenuation coefficient, (µ/ρ) i , through the relation where A i is the atomic mass, and N A is the Avogadro's constant.Inserting expression (15) for σ i in the eq.14 gives (16)   Eq. ( 16) can be used for calculating the effective atomic number for both compounds and composites in terms of the mass absorption coefficients.

A Method for Simultaneous Determination of Effective Removal Cross-section for Fast Neutrons and Mass Absorption Coefficient for Gamma Rays
Copyright: © 2014 El Abd and Elkady removal cross-sections Σ R , 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 Z eff were calculated for some compounds, composites and alloys.These  Sn-H, and Zr-H.Also, for some of these materials, the R-values and Z eff 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 Σ R and R-values (calculated relative to µ R at 661.6 keV and 1332.5 keV) coincide with the data for the elements.The calculated values of Z eff 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 Z eff in the mentioned range of energies.Of worth noting, the values of Σ R and µ R (at 661.6 keV and 1332.5 keV) for any material can be determined simultaneously.The determination of Σ R and µ R is based on the knowledge of the corresponding Z eff at either 661.6 keV or 1332.5 keV.Namely, once Z eff is determined for any material, the table containing the values of R including µ R at 661.6 keV and 1332.5 keV and Σ R is searched for the closest value to Z eff .At this value of Z eff , the values of µ R at 661.6 keV and 1332.5 keV and Σ R 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 Σ R and µ R with the knowledge of Z eff 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 Σ R and µ R 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 method.Besides, the determined Σ R , µ R and Z eff for these materials along with those shown in Figure 2 were used to determine Σ R , µ R 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][23][24].The chemical compositions shown in Table 2 were taken from references [22][23][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 Z eff calculated by the traditional method.These values are obtained from values of µ R at 661.6 keV and 1332.5 keV.In such case, the table containing the R-values including µ R at 661.6   Deviations between µ R (at 661.6 keV and 1332.5 keV) and Σ R 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 µ R at 661.6 keV.There are no deviations at 1332.5 keV.For Σ R , deviations do not exceed 17%.

A Method for Simultaneous Determination of Effective Removal Cross-section for Fast Neutrons and Mass Absorption Coefficient for Gamma Rays
It was noticed that when a composite mixture consisting of elements having high atomic numbers (with high weight fraction), and the rest of the constituting elements having low atomic numbers, the resultant value of µ R at 661.6 keV or Σ R 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 Z eff 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.

Figure 1 :
Figure 1:The ratio R at 661.6 keV, 1332.5 keV and Σ R versus Z for elements.

Figure 2 :
Figure 2: The ratio R and Σ R values for some compounds, composites and alloys versus Z eff .Results of Figure. 1 are also included.

Table 1 :
µ R , ∑ R and Z eff calculated by the traditional and the present methods.

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

Method for Simultaneous Determination of Effective Removal Cross-section for Fast Neutrons and Mass Absorption Coefficient for Gamma Rays Copyright: © 2014 El Abd and Elkady keV
and 1332.5 keV and Σ R is searched for the closest value and/ or values to Z eff .For example, for the compound number "12", the calculated values of Z eff are 10.6 and 10.8 and the closest value is 10.7.

Table 3 :
Deviations of µ R at 661.6 and 1332.5 keV, and ∑ R determined in this work from traditional method.