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

Abdelghany A. El AbdM; Ashraf S. Elkady

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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. Elkady^1,2*

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. DOI: http://dx.doi.org/10.15226/sojmse.2014.00113

AbstractTop

A method for determining the effective removal cross section (Σ_R) of fast neutrons and mass absorption coefficient (μ_R) for gamma rays at energies 661.6 keV and 1332.5 keV is proposed. It is based on relating the ratio (R) of Σ_R to μ_R at 661.6 keV and 1332.5 keV to effective atomic number (Z_eff) for elements, compounds, composite materials and alloys. It is shown that the R-values versus Z_eff for the most studied materials lie on two master curves. The method was tested by determining Σ_R and μ_R 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, μ^R (cm²/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 [2-6].

The effective removal cross-section, Σ_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 Σ_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 Σ_T (cm²/g) is used. The observed value of the Σ_R is roughly 2/3 of Σ_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" [7-9]. 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 252_Cf [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 Σ_R to μ_R at 661.6 keV and 1332.5 keV, for elements for determining Σ_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, Σ_R can be calculated for mixtures, alloys and compounds, with the knowledge of the weight percentages wi, and the values of μ_R and Σ_R of the constituting elements. This is achieved by the following simple addition rules [7,19]:

μ_{R} = \sum_{i} w_{i} {(μ_{R})}_{i}

\sum_{R} = \sum_{i} w_{i} {(\sum_{R})}_{i}

for gamma rays and fast neutrons, respectively.
It was proposed that the following empirical formulas [19]:

\sum_{R} = 0.21 A^{- 0.56} c m^{2} g^{- 1}

\sum_{R} = 0.00662 A^{- 1 / 3} + 0.33 A^{- 2 / 3} - 0.211 A^{- 1} c m^{2} g^{- 1} (A > 12)

\sum_{R} = 0.190 Z^{- 0.743} c m^{2} g^{- 1} (Z \leq 8)

\sum_{R} = 0.125 Z^{- 0.565} c m^{2} g^{- 1}, (Z > 8)

Can be used to determine Σ_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

σ_{m} = \sum_{}^{}_{i} σ_{i} n_{i}

where ni 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:

n = \sum_{i} n_{i}

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

σ_{m} = n σ_{a} = n Z_{e f f} σ_{e}

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

σ_{a} = (\sum_{i} n_{i} σ_{i}) / n

σ_{e} = (\sum_{i} n_{i} σ_{i} / Z_{i}) / n

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_{e f f} = σ_{a} / σ_{e} = (\sum_{i} n_{i} σ_{i}) / (\sum_{i} n_{i} σ_{i} / Z_{i})

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, fi (sometimes expressed in units of atomic percent, at.%). For a chemical compound, one has

f_{i} = n_{i} / \sum_{i} n_{i} = n_{i} / n

where Σ_ifi =1. Rewriting Eq.12 in terms of f_i one has

Z_{e f f} = (\sum_{i} f_{i} σ_{i}) / (\sum_{i} f_{i} σ_{i} / Z_{i})

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 i^th 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}

where A_i is the atomic mass, and N_A is the Avogadro’s constant. Inserting expression (15)

for σi in the eq. 14 gives

Z_{e f f} = (\sum_{i} f_{i} A_{i} {(μ / ρ)}_{i}) / (\sum_{i} f_{i} {(μ / ρ)}_{i} A_{i} / Z_{i})

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 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 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 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

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

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.

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 methods. 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 [22-24]. 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 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 keV and 1332.5 keV and Σ_R is searched for the closest value and/

Table 1: μR, Σ_R and Z_eff calculated by the traditional and the present methods.

Compound/ mixture/alloy number or name	m_R, ∑_R in (cm²/g) , and Z_eff calculated by traditional method				m_R and ∑_R in (cm²/g) , calculated in this work
Compound/ mixture/alloy number or name	661.6 (keV)	1332.5 (keV)	∑_R	Z_eff	661.6 (keV)	1332.5 (keV)	∑_R	Z_eff
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 μ_R at 661.6 and 1332.5 keV, and Σ_R determined in this work from traditional method.

Compound/ mixture/alloy number or name	m_R( this work)/ m_R(traditional) at 661.6 keV	m_R( this work)/m_R(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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