3. Determination of α in the E�(1+α) epithermal neutron spectrum

In a nuclear research reactor, the distribution of epithermal neutrons per unit energy interval is considered inversely proportional to the neutron energy. However, this assumption is only valid if the following conditions are satisfied [11].


In practical situations, these conditions are almost not satisfied in a nuclear reactor. Accordingly, the deviations from the 1/E distribution of epithermal neutron can occur in irradiation channels, and errors could be induced in the resonance integral defined as

$$I = \int\_{E\_{\rm Gd}}^{\bullet} \frac{\sigma(E)}{E} dE \tag{5}$$

Qð Þ¼ α

1, 2

can be determined by the following equation [11].

<sup>f</sup> <sup>¼</sup> <sup>ϕ</sup>th ϕepi !

capture reaction is defined as follows:

integral is defined as:

where

Ið Þ α σ0

<sup>¼</sup> <sup>k</sup>1ε<sup>1</sup> k2ε<sup>2</sup>

equality between the quantities f1, 2 and f1, 3 leads to the following equation:

a ¼ Asp, <sup>1</sup>k2ε<sup>2</sup>

b ¼ Asp,1k3ε<sup>3</sup>

¼ ð Þ Q � 0:426 Er

By using the co-irradiation of two suitable standard foils, denoted as 1 and 2, the flux ratio f

When three resonance detectors (foils), denoted as 1, 2 and 3, are irradiated under the same experimental conditions, Eq. (12) can be rewritten for detector couples 1–2 and 1–3. Making

� �= Asp, <sup>2</sup>k1ε<sup>1</sup>

� �= Asp,3k1ε<sup>1</sup>

In an ideal 1/E epithermal neutron spectrum, the resonance integral cross section for a neutron

σð Þ E

The coefficient α would be experimentally determined by solving the Eqs. (12) and (14).

I<sup>0</sup> ¼ ð ∞

I0ð Þ¼ α

The relation between I0 and I0(α) is defined as the following expression:

<sup>I</sup>0ð Þ¼ <sup>α</sup> ð Þ <sup>1</sup>eV <sup>α</sup> <sup>I</sup><sup>0</sup> � <sup>0</sup>:426σ<sup>0</sup>

ECd

ð ∞

ECd

Eα r � �

where σ (E) is the neutron capture cross section as a function of neutron energy E, and ECd is the cadmium cut-off energy to be 0.55 eV when Cd shield thickness is 1 mm. In a non-ideal epithermal neutron spectrum which can be approximated by 1/E1+<sup>α</sup> distribution, the resonance

<sup>σ</sup>ð Þ <sup>E</sup> ð Þ <sup>1</sup>eV <sup>α</sup>

þ

0:426σ<sup>0</sup>

Asp, <sup>1</sup> Asp, <sup>2</sup>

� � Asp, <sup>1</sup>

Q1ð Þ� α

�<sup>α</sup> <sup>þ</sup>

Q2ð Þ α

Fð Þ¼ α ð Þ a � b Q1ð Þ� α ð Þ a þ 1 Q2ð Þþ α ð Þ b þ 1 Q3ð Þ¼ α 0 (14)

0:426 2α þ 1

Asp, <sup>2</sup>

� � � <sup>1</sup> � ��<sup>1</sup> (15)

� � � <sup>1</sup> � ��<sup>1</sup> (16)

<sup>E</sup> dE, (17)

<sup>E</sup><sup>1</sup>þ<sup>α</sup> dE, (18)

ð Þ <sup>2</sup><sup>α</sup> <sup>þ</sup> <sup>1</sup> ð Þ ECd <sup>α</sup> : (19)

E�<sup>α</sup>

Monte Carlo Simulation of Correction Factors for Neutron Activation Foils

Q ¼ I=σ<sup>0</sup> (11)

� <sup>k</sup>1ε<sup>1</sup> k2ε<sup>2</sup> � ��<sup>1</sup>

k ¼ γσ0θ=M: (13)

Cd (10)

59

http://dx.doi.org/10.5772/intechopen.76984

, (12)

where: ECd = 0.55 eV is the Cadmium cut off energy.

In order to take into account, the correction for deviation effect in the expression of the resonance integral, the 1/E1+<sup>α</sup> distribution has been introduced [11] where α is an energyindependent coefficient but its values dependent on the neutron source configuration. Therefore, the effective resonance integral is rewritten as follows:

$$I(\alpha) = \int\_{E}^{\bullet} \frac{\sigma(E)}{E^{1+\alpha}} dE \tag{6}$$

Experimental determination of the coefficient α for a specific neutron irradiation channel or facility is required for exact estimation of the corresponding effective resonance integral. The relationship between resonance integral I and effective resonance integral I(α) is expressed as the following equation [11]

$$I(\alpha) = (I - 0.426\sigma\_0) \left(\overline{E\_r}\right)^{-\alpha} + \frac{0.426\sigma\_0}{2\alpha + 1} E\_{Cd'}^{-\alpha} \tag{7}$$

where Er is the effective resonance energy, and σ<sup>0</sup> is the 2200 m/s neutron capture cross section. The method for experimental instantaneous determination of α value, based on co-irradiation of three suitable resonance monitors, has been introduced by F. De CORTE [11], which represent as follows.

The specific count rate for an interesting γ-peak, emission from an irradiated sample, is defined as

$$A\_{sp} = \frac{1}{m} \frac{\mathbb{C}\lambda}{(1 - e^{-\lambda t\_1})(e^{-\lambda t\_2})(1 - e^{-\lambda t\_3})} \,\text{}\tag{8}$$

where t1, t2, t3 are the irradiation time, decay time, measurement time, respectively, C the number of counts under γ-peak, and m the weight of irradiated sample. The specific count rate, Asp, can be also calculated from the following expression:

$$A\_{sp} = [f + Q(\alpha)] \phi\_{\rm epi} \sigma\_0 \varepsilon \gamma \theta \mathcal{C} / M \tag{9}$$

where M, θ, γ, ε are atomic weight, isotope abundance, γ-ray absolute intensity, and the efficiency of the detector used in the gamma-ray spectrum measurement, respectively. Q(α) is the ratio of the resonance integral in the 1/E(1+α) epithermal neutron spectrum to the (n,γ) reaction cross section σ0; f is the ratio of thermal to epithermal neutron flux [11].

Monte Carlo Simulation of Correction Factors for Neutron Activation Foils http://dx.doi.org/10.5772/intechopen.76984 59

$$Q(a) = \frac{I(a)}{\sigma\_0} = (Q - 0.426)\overline{E\_r}^{-a} + \frac{0.426}{2a + 1}E\_{\text{Cd}}^{-a} \tag{10}$$

$$\mathbf{Q} = \mathbf{I}/\sigma\_0 \tag{11}$$

By using the co-irradiation of two suitable standard foils, denoted as 1 and 2, the flux ratio f can be determined by the following equation [11].

$$f = \left(\frac{\phi\_{th}}{\phi\_{spi}}\right)\_{1,2} = \left[\frac{k\_1\varepsilon\_1}{k\_2\varepsilon\_2}Q\_1(\alpha) - \frac{A\_{sp,1}}{A\_{sp,2}}Q\_2(\alpha)\right] \left[\frac{A\_{sp,1}}{A\_{sp,2}} - \frac{k\_1\varepsilon\_1}{k\_2\varepsilon\_2}\right]^{-1},\tag{12}$$

$$
\mathbf{k} = \gamma \sigma\_0 \Theta / \mathbf{M}.\tag{13}
$$

When three resonance detectors (foils), denoted as 1, 2 and 3, are irradiated under the same experimental conditions, Eq. (12) can be rewritten for detector couples 1–2 and 1–3. Making equality between the quantities f1, 2 and f1, 3 leads to the following equation:

$$F(a) = (a-b)Q\_1(a) - (a+1)Q\_2(a) + (b+1)Q\_3(a) = 0\tag{14}$$

where

In practical situations, these conditions are almost not satisfied in a nuclear reactor. Accordingly, the deviations from the 1/E distribution of epithermal neutron can occur in irradiation

σð Þ E

<sup>E</sup> dE (5)

<sup>E</sup><sup>1</sup>þ<sup>α</sup> dE (6)

Cd , (7)

, (8)

channels, and errors could be induced in the resonance integral defined as

where: ECd = 0.55 eV is the Cadmium cut off energy.

58 Advanced Technologies and Applications of Neutron Activation Analysis

the following equation [11]

sent as follows.

defined as

fore, the effective resonance integral is rewritten as follows:

I ¼ ð ∞

Ið Þ¼ α

Ið Þ¼ α ð Þ I � 0:426σ<sup>0</sup> Er

Asp <sup>¼</sup> <sup>1</sup> m

rate, Asp, can be also calculated from the following expression:

ECd

In order to take into account, the correction for deviation effect in the expression of the resonance integral, the 1/E1+<sup>α</sup> distribution has been introduced [11] where α is an energyindependent coefficient but its values dependent on the neutron source configuration. There-

> ð ∞

σð Þ E

� ��<sup>α</sup> <sup>þ</sup>

0:426σ<sup>0</sup> <sup>2</sup><sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>E</sup>�<sup>α</sup>

Asp ¼ ½ � f þ Qð Þ α ϕepiσ0εγθC=M (9)

ECd

Experimental determination of the coefficient α for a specific neutron irradiation channel or facility is required for exact estimation of the corresponding effective resonance integral. The relationship between resonance integral I and effective resonance integral I(α) is expressed as

where Er is the effective resonance energy, and σ<sup>0</sup> is the 2200 m/s neutron capture cross section. The method for experimental instantaneous determination of α value, based on co-irradiation of three suitable resonance monitors, has been introduced by F. De CORTE [11], which repre-

The specific count rate for an interesting γ-peak, emission from an irradiated sample, is

where t1, t2, t3 are the irradiation time, decay time, measurement time, respectively, C the number of counts under γ-peak, and m the weight of irradiated sample. The specific count

where M, θ, γ, ε are atomic weight, isotope abundance, γ-ray absolute intensity, and the efficiency of the detector used in the gamma-ray spectrum measurement, respectively. Q(α) is the ratio of the resonance integral in the 1/E(1+α) epithermal neutron spectrum to the (n,γ)

reaction cross section σ0; f is the ratio of thermal to epithermal neutron flux [11].

Cλ 1 � e ð Þ �λt<sup>1</sup> eð Þ �λt<sup>2</sup> 1 � e ð Þ �λt<sup>3</sup>

$$a = \left[ \left( A\_{sp,1} k\_2 \varepsilon\_2 \right) / \left( A\_{sp,2} k\_1 \varepsilon\_1 \right) - 1 \right]^{-1} \tag{15}$$

$$b = \left[ \left( A\_{\circ \flat, 1} k\_3 \varepsilon\_3 \right) / \left( A\_{\circ \flat, 3} k\_1 \varepsilon\_1 \right) - 1 \right]^{-1} \tag{16}$$

The coefficient α would be experimentally determined by solving the Eqs. (12) and (14).

In an ideal 1/E epithermal neutron spectrum, the resonance integral cross section for a neutron capture reaction is defined as follows:

$$I\_0 = \int\_{E\_{\rm Cd}}^{\infty} \frac{\sigma(E)}{E} dE\_{\prime} \tag{17}$$

where σ (E) is the neutron capture cross section as a function of neutron energy E, and ECd is the cadmium cut-off energy to be 0.55 eV when Cd shield thickness is 1 mm. In a non-ideal epithermal neutron spectrum which can be approximated by 1/E1+<sup>α</sup> distribution, the resonance integral is defined as:

$$Io(\alpha) = \int\_{E\_{\rm cl}}^{\infty} \frac{\sigma(E)(1eV)^{\alpha}}{E^{1+\alpha}} dE\_{\prime} \tag{18}$$

The relation between I0 and I0(α) is defined as the following expression:

$$I\_0(a) = (1eV)^a \left[ \frac{I\_0 - 0.426\sigma\_0}{E\_r^a} \right] + \frac{0.426\sigma\_0}{(2\alpha + 1)(E\_{Cd})^a}.\tag{19}$$

The experimental values of resonance integral I0(α) for target x can be determined relative to that of 197Au(n,γ) 198Au as a standard reaction by the following relation [11]:

$$I\_0(\boldsymbol{\alpha})\_x = I\_0(\boldsymbol{\alpha})\_{A\boldsymbol{u}} \frac{(\boldsymbol{g}\sigma\_0)\_x}{(\boldsymbol{g}\sigma\_0)\_{A\boldsymbol{u}}} \frac{(\boldsymbol{R}\_{\mathbb{C}\boldsymbol{d}} - \boldsymbol{F}\_{\mathbb{C}\boldsymbol{d}})\_{A\boldsymbol{u}}}{(\boldsymbol{R}\_{\mathbb{C}\boldsymbol{d}} - \boldsymbol{F}\_{\mathbb{C}\boldsymbol{d}})\_{\boldsymbol{x}}} \frac{\boldsymbol{G}\_{\text{epi},Au}}{\boldsymbol{G}\_{\text{epi},x}} \frac{\boldsymbol{G}\_{\text{th},x}}{\boldsymbol{G}\_{\text{th},Au}},\tag{20}$$

The deviation of the epithermal neutron spectrum from the 1/E shape parameter can be experimentally determined by the 'Cd-ratio for multi-monitor' method, using the monitors of 197Au, 59Co, 186W and 55Mn. When a set of n monitors are irradiated with and without Cd-cover, the αparameter can be obtained as the slope (�α) of the straight-line logTi versus log(Er,i) [11].

$$T\_i = \frac{E\_{r,i}^{-a}}{(F\_{\mathbb{C}d,i}R\_{\mathbb{C}d,i} - 1)Q\_{0,i}(\alpha)G\_{\mathbb{C},i}G\_{\mathbb{H},i}},\tag{21}$$

The neutron capture reaction rate R(E), at neutron energy E per one atom of the irradiating

where σ<sup>γ</sup> (E) is capture cross section. During the irradiation processes, strong resonance reactions deplete the neutron spectrum at the resonance energy due to absorption and scattering. For the energy regions just lower than the resonance energies, the neutron distribution is raised due to multiple scatterings from the resonances. Therefore, the capture reaction rate is affected because of the perturbation in the neutron spectrum. It is assumed that, for a sample or monitor with finite thickness but infinitely diluted, the neutron distribution inside this sample or monitor is a non-perturbed spectrum. The correction factor a real sample with

R Eð Þ¼ ϕð Þ E σγð Þ E , (22)

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61

φ0ð Þ E σγð Þ E dE, (23)

sample in a neutron flux distribution ϕ(E), is defined as the following expression:

Figure 1. A typical block diagram for Monte-Carlo simulation of neutron transport process.

thickness 't' can be calculated as the following ratio.

ENDF/B-VII nuclear data library.

G tðÞ¼

E ð2

E1

φð Þ E σγð Þ E dE=

E ð2

E1

where ϕ0(E) is the original or non-perturbed neutron spectrum; ϕ(E) represents the perturbed neutron spectrum inside the real irradiating sample; E1 and E2 are, respectively, the lower and the upper limits of the neutron spectrum; for Gth: E1 = 1e�<sup>5</sup> eV and E2 = 0.5 eV; for Gepi: E1 = 0.5 eV and E2 = 2e-1 MeV. The non-perturbed and perturbed neutron spectrum inside a real sample can be calculated by Monte-Carlo simulation using the MCNP5 code based on the

where i denotes the ith isotope, FCd the cadmium transmission factor, RCd the Cd-ratio, Er the effective resonance energy in eV, Gth and Gepi the self-shielding factor for thermal and epithermal neutrons, and Q0(α) is the ratio of the resonance integral in 1/E1+<sup>α</sup> epithermal neutron spectrum to the capture cross section σ<sup>0</sup> for 2200 m/s neutrons. The parameter Ti is a function of the α parameter, which can be determined by an iterative least square fit to the regression line.

## 4. Monte Carlo simulation method for neutron self-shielding calculations

Monte Carlo (MC) simulation is known as an essential numerical method for performing the statistical process of radiation interaction with material. The principle MC simulation in this subject is random selection of particle properties and its interaction behaviours from their probability distribution functions. By tracking the history of each particle during the interaction process, the information of particle fluxes, energy spectra and energy deposition in a specific cell of the simulating model can be obtained. Accordingly, the radiation dose rate at any position in the environment of the experiment can be estimated with statistical uncertainty. A typical block diagram for process of neutron particle transport are shown in the Figure 1 [5].

#### 5. Neutron self-shielding correction factors

In the neutron activation analysis experiments, the thickness of the samples and monitors may not be thin enough for ignoring the variance of neutron flux distribution and should be considered for correction. Therefore, the correction factors for thermal (Gth), epithermal (Gepi) and resonance (Gres) neutron self-shielding effects should be determined exactly. In this work, the Monte-Carlo code MCNP5 was used for calculations with specified case of irradiation foils with different thickness from 1�10�<sup>5</sup> to 2 mm.

Figure 1. A typical block diagram for Monte-Carlo simulation of neutron transport process.

The experimental values of resonance integral I0(α) for target x can be determined relative to

198Au as a standard reaction by the following relation [11]:

The deviation of the epithermal neutron spectrum from the 1/E shape parameter can be experimentally determined by the 'Cd-ratio for multi-monitor' method, using the monitors of 197Au, 59Co, 186W and 55Mn. When a set of n monitors are irradiated with and without Cd-cover, the α-

> r,i ð Þ FCd,iRCd,i � 1 Q0,ið Þ α Ge,iGth,i

where i denotes the ith isotope, FCd the cadmium transmission factor, RCd the Cd-ratio, Er the effective resonance energy in eV, Gth and Gepi the self-shielding factor for thermal and epithermal neutrons, and Q0(α) is the ratio of the resonance integral in 1/E1+<sup>α</sup> epithermal neutron spectrum to the capture cross section σ<sup>0</sup> for 2200 m/s neutrons. The parameter Ti is a function of the α parameter, which can be determined by an iterative least square fit to the

4. Monte Carlo simulation method for neutron self-shielding calculations

Monte Carlo (MC) simulation is known as an essential numerical method for performing the statistical process of radiation interaction with material. The principle MC simulation in this subject is random selection of particle properties and its interaction behaviours from their probability distribution functions. By tracking the history of each particle during the interaction process, the information of particle fluxes, energy spectra and energy deposition in a specific cell of the simulating model can be obtained. Accordingly, the radiation dose rate at any position in the environment of the experiment can be estimated with statistical uncertainty. A typical block diagram for process of neutron particle transport are shown in the

In the neutron activation analysis experiments, the thickness of the samples and monitors may not be thin enough for ignoring the variance of neutron flux distribution and should be considered for correction. Therefore, the correction factors for thermal (Gth), epithermal (Gepi) and resonance (Gres) neutron self-shielding effects should be determined exactly. In this work, the Monte-Carlo code MCNP5 was used for calculations with specified case of irradiation foils

parameter can be obtained as the slope (�α) of the straight-line logTi versus log(Er,i) [11].

Ti <sup>¼</sup> <sup>E</sup>�<sup>α</sup>

ð Þ RCd � FCd Au ð Þ RCd � FCd <sup>x</sup>

Gepi,Au Gepi,x

Gth,x Gth,Au

, (20)

, (21)

ð Þ gσ<sup>0</sup> <sup>x</sup> ð Þ gσ<sup>0</sup> Au

I0ð Þ α <sup>x</sup> ¼ I0ð Þ α Au

60 Advanced Technologies and Applications of Neutron Activation Analysis

5. Neutron self-shielding correction factors

with different thickness from 1�10�<sup>5</sup> to 2 mm.

that of 197Au(n,γ)

regression line.

Figure 1 [5].

The neutron capture reaction rate R(E), at neutron energy E per one atom of the irradiating sample in a neutron flux distribution ϕ(E), is defined as the following expression:

$$\mathbf{R}(\mathbf{E}) = \phi(\mathbf{E})\sigma\_{\gamma}(\mathbf{E}),\tag{22}$$

where σ<sup>γ</sup> (E) is capture cross section. During the irradiation processes, strong resonance reactions deplete the neutron spectrum at the resonance energy due to absorption and scattering. For the energy regions just lower than the resonance energies, the neutron distribution is raised due to multiple scatterings from the resonances. Therefore, the capture reaction rate is affected because of the perturbation in the neutron spectrum. It is assumed that, for a sample or monitor with finite thickness but infinitely diluted, the neutron distribution inside this sample or monitor is a non-perturbed spectrum. The correction factor a real sample with thickness 't' can be calculated as the following ratio.

$$G(t) = \int\_{E\_1}^{E\_2} \varphi(E)\sigma\_\gamma(E)dE / \int\_{E\_1}^{E\_2} \varphi\_0(E)\sigma\_\gamma(E)dE,\tag{23}$$

where ϕ0(E) is the original or non-perturbed neutron spectrum; ϕ(E) represents the perturbed neutron spectrum inside the real irradiating sample; E1 and E2 are, respectively, the lower and the upper limits of the neutron spectrum; for Gth: E1 = 1e�<sup>5</sup> eV and E2 = 0.5 eV; for Gepi: E1 = 0.5 eV and E2 = 2e-1 MeV. The non-perturbed and perturbed neutron spectrum inside a real sample can be calculated by Monte-Carlo simulation using the MCNP5 code based on the ENDF/B-VII nuclear data library.

The results of MCNP5 simulations for perturbed and none-perturbed neutron spectra, ϕ(E) and ϕ0(E), inside the real irradiating samples of Au-197 and Co-60 in different thickness is represented in Figures 2 and 3.

In order to make validation for the simulation procedure, the comparison between the results Gres factors with the literature data for foil materials of Au-197 and Co-59 have been carried out with different sample thicknesses as presented in Figures 2 and 3. Up to present, from the literature overview, the available data of neutron self-shielding factors are almost for the study of resonance neutron self-shielding correction Gres which is considered only in the energy region of primary resonance peak of reaction cross-section, but in NAA the thermal and epithermal neutron (or effective) self-shielding correction factor should be taken into account. As shown in Figures 4 and 5, the result of present simulated Gres factors for Gold and Cobalt foils has reasonable agreement with the experimental and calculated data by Gongalves [9],

Monte Carlo Simulation of Correction Factors for Neutron Activation Foils

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63

In order to provide effective information of neutron self-shielding correction factors in NAA, the validated simulation procedure has been applied for obtaining Gth, Gepi and Gepi factors for the activation nuclides of Au-97, Co-59, Mn-55 and W-186. In this work, the foil samples with 1.3 cm in diameter and thickness varying from 10�<sup>5</sup> mm to 2 mm were used for simulations. The experimental configurations were simulated for the real condition of a reactor-based neutron activation experiment in which the irradiation channel can be described as a cylindrical isotropic neutron sources with sample flat to the channel axis. The dimensions of the irradiating channel are 30 cm length and 2.4 cm in diameter. These simulations were conducted for three case studies of neutron energy spectrum: (i) the Maxwellian distribution with average energy of 0.025 eV was

Figure 4. The calculated resonance neutron self-shielding correction factors for Gold foils of different thickness, in

comparison with published data by Gongalves [9] and Brose [3].

Brose [12] and Eastwood [2].

Figure 2. MCNP5 simulated results for perturbed and none-perturbed epithermal neutron spectrum in Gold foils of different thickness.

Figure 3. MCNP5 simulated results for perturbed and none-perturbed epithermal neutron spectrum in Cobalt foils of different thickness.

In order to make validation for the simulation procedure, the comparison between the results Gres factors with the literature data for foil materials of Au-197 and Co-59 have been carried out with different sample thicknesses as presented in Figures 2 and 3. Up to present, from the literature overview, the available data of neutron self-shielding factors are almost for the study of resonance neutron self-shielding correction Gres which is considered only in the energy region of primary resonance peak of reaction cross-section, but in NAA the thermal and epithermal neutron (or effective) self-shielding correction factor should be taken into account. As shown in Figures 4 and 5, the result of present simulated Gres factors for Gold and Cobalt foils has reasonable agreement with the experimental and calculated data by Gongalves [9], Brose [12] and Eastwood [2].

The results of MCNP5 simulations for perturbed and none-perturbed neutron spectra, ϕ(E) and ϕ0(E), inside the real irradiating samples of Au-197 and Co-60 in different thickness is

Figure 2. MCNP5 simulated results for perturbed and none-perturbed epithermal neutron spectrum in Gold foils of

Figure 3. MCNP5 simulated results for perturbed and none-perturbed epithermal neutron spectrum in Cobalt foils of

represented in Figures 2 and 3.

62 Advanced Technologies and Applications of Neutron Activation Analysis

different thickness.

different thickness.

In order to provide effective information of neutron self-shielding correction factors in NAA, the validated simulation procedure has been applied for obtaining Gth, Gepi and Gepi factors for the activation nuclides of Au-97, Co-59, Mn-55 and W-186. In this work, the foil samples with 1.3 cm in diameter and thickness varying from 10�<sup>5</sup> mm to 2 mm were used for simulations. The experimental configurations were simulated for the real condition of a reactor-based neutron activation experiment in which the irradiation channel can be described as a cylindrical isotropic neutron sources with sample flat to the channel axis. The dimensions of the irradiating channel are 30 cm length and 2.4 cm in diameter. These simulations were conducted for three case studies of neutron energy spectrum: (i) the Maxwellian distribution with average energy of 0.025 eV was

Figure 4. The calculated resonance neutron self-shielding correction factors for Gold foils of different thickness, in comparison with published data by Gongalves [9] and Brose [3].

Thickness (mm) Neutron self-shielding correction factors for Au-197 foils

Thickness (mm) Neutron self-shielding correction factors for Mn-55 foils

Thickness (mm) Neutron self-shielding correction factors for Co-60 foils

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Monte Carlo Simulation of Correction Factors for Neutron Activation Foils

Thickness (mm) Neutron self-shielding correction factors for W-186 foils

Gth Gepi Gres Gth Gepi Gres

Gth Gepi Gres Gth Gepi Gres

0.00001 1.000 1.000 1.000 0.00001 1.00 0.995 1.000 0.00002 1.000 1.000 1.000 0.00002 1.00 0.996 1.000 0.00004 1.000 1.000 1.000 0.00004 1.00 0.995 1.000 0.00006 1.000 1.000 1.000 0.00006 1.00 0.996 1.000 0.00008 1.000 1.000 1.000 0.00008 1.00 0.999 1.000 0.0001 1.000 1.000 1.000 0.0001 1.00 0.999 0.990 0.0002 1.000 1.000 1.000 0.0002 1.00 1.006 1.000 0.0004 1.000 1.000 1.000 0.0004 1.00 1.001 0.992 0.0006 1.000 1.000 1.000 0.0006 1.00 0.994 0.982 0.0008 1.000 1.000 1.000 0.0008 1.00 0.990 0.976 0.001 0.999 1.000 1.000 0.001 1.00 0.993 0.989 0.002 0.998 1.000 1.000 0.002 1.00 0.973 0.951 0.004 0.997 1.000 1.000 0.004 1.00 0.947 0.911 0.006 0.996 1.000 1.000 0.006 1.00 0.916 0.865 0.008 0.995 1.000 1.000 0.008 0.99 0.891 0.827

0.006 0.984 0.696 0.671 0.006 0.989 0.902 0.008 0.981 0.646 0.618 0.008 0.987 0.880 0.01 0.977 0.607 0.575 0.01 0.985 0.992 0.858 0.02 0.961 0.486 0.445 0.02 0.974 0.988 0.762 0.04 0.934 0.384 0.334 0.04 0.957 0.981 0.642 0.06 0.912 0.335 0.282 0.06 0.941 0.977 0.562 0.08 0.893 0.306 0.251 0.08 0.928 0.974 0.506 0.1 0.876 0.286 0.229 0.1 0.916 0.971 0.465 0.2 0.805 0.234 0.175 0.2 0.865 0.961 0.355 0.4 0.707 0.194 0.135 0.4 0.792 0.946 0.259 0.6 0.636 0.175 0.116 0.6 0.736 0.935 0.214 0.8 0.58 0.162 0.104 0.8 0.690 0.925 0.186 1 0.535 0.152 0.096 1 0.651 0.915 0.164 2 0.388 0.124 0.073 2 0.516 0.873 0.108

Table 1. The results of neutron self-shielding correction factors foils of Au-197 and Co-60.

Figure 5. The calculated resonance neutron self-shielding correction factors for Cobalt foils of different thickness, in comparison with published data by Gongalves [9] and Eastwood [2].

applied for thermal neutrons, (ii) the pure 1/E distribution in energy range from 0.55 eV to 0.2 MeV was applied for epithermal neutrons, and (iii) the primary resonance energy peaks for resonance neutron spectrum. The results of thermal and epithermal neutron self-shielding correction factors for the activation foils of Au-197, W-186, Co-60 and Mn-55 in different thickness are presented in Tables 1 and 2.



Table 1. The results of neutron self-shielding correction factors foils of Au-197 and Co-60.

applied for thermal neutrons, (ii) the pure 1/E distribution in energy range from 0.55 eV to 0.2 MeV was applied for epithermal neutrons, and (iii) the primary resonance energy peaks for resonance neutron spectrum. The results of thermal and epithermal neutron self-shielding correction factors for the activation foils of Au-197, W-186, Co-60 and Mn-55 in different thickness

Gth Gepi Gres Gth Gepi Gres

0.00001 1 1 1 0.00001 1.000 1.001 1.000 0.00002 1 0.991 0.99 0.00002 1.000 1.000 0.00004 0.999 0.989 0.988 0.00004 1.000 1.000 0.00006 0.999 0.987 0.986 0.00006 1.000 1.000 0.00008 0.999 0.985 0.984 0.00008 0.999 1.000 0.0001 0.999 0.983 0.982 0.0001 0.999 1.001 1.000 0.0002 0.999 0.973 0.971 0.0002 0.999 1.000 0.0004 0.998 0.953 0.949 0.0004 0.998 1.000 0.0006 0.997 0.935 0.93 0.0006 0.997 1.000 0.0008 0.996 0.92 0.913 0.0008 0.997 0.997 0.001 0.996 0.906 0.898 0.001 0.996 0.998 0.989 0.002 0.993 0.846 0.834 0.002 0.995 0.996 0.958 0.004 0.988 0.76 0.74 0.004 0.992 0.930

Thickness (mm) Neutron self-shielding correction factors for Co-60 foils

Figure 5. The calculated resonance neutron self-shielding correction factors for Cobalt foils of different thickness, in

are presented in Tables 1 and 2.

Thickness (mm) Neutron self-shielding correction factors for Au-197 foils

comparison with published data by Gongalves [9] and Eastwood [2].

64 Advanced Technologies and Applications of Neutron Activation Analysis



Author details

Pham Ngoc Son1

References

\* and Bach Nhu Nguyen2

2 Representative Office of MIC in Da Nang City, Danang, Vietnam

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Monte Carlo Simulation of Correction Factors for Neutron Activation Foils

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[10] Pomme S, Hardeman F, Robouch P, Etxebarria N. Neutron Activation Analysis with K0- Standardization: General Formalism and Procedure. SCK-CEN report. 1997; BLG-700 [11] Benjelloun M, Paulus JM. α-Determination in the E�(1+α) epithermal neutron spectrum of the CNR Reactor (Strasbourg). Journal of Radioanalytical and Nuclear Chemistry. 1987;

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\*Address all correspondence to: pnson.nri@gmail.com

1 Nuclear Research Institute, Dalat, Vietnam

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Table 2. The results of neutron self-shielding correction factors foils of Mn-55 and W-186.
