5.2.2 Samples and standards induced gamma-activities

Different disk shaped red beet samples and standards were irradiated with 14 MeV neutrons. Standards were prepared by mixing pure graphite and high purity chemical compounds powders (NaCl, Na2HPO4 and K2CO3) [14]. The induced gamma activities on the sodium, potassium, chlorine and phosphorus elements have been experimentally measured by means of hyper-pure germanium spectrometer.

The analyzed beet samples and standards have a 23 mm diameter and a 6 mm thickness.

The different parameters of the nuclear reactions used (cross section, isotopic abundance, etc.) are summarized in Table 3.

After the irradiation (irradiation time = tirr) and cooling times (td), the number of produced radionuclides is given by:

$$N(t\_{irr} + t\_d) = \frac{TN\_0 \sigma \phi}{\ln 2} \theta I\_\chi \left(1 - e^{-\left(\frac{\ln 2}{T}\right)t\_{irr}}\right) . e^{-\left(\frac{\ln 2}{T}\right)t\_d} \tag{16}$$

where σ is the nuclear reaction cross section, N0 is the number of target nuclei, T is the half-life of the produced radionuclide, θ is the isotopic abundance of the studied element, Iγ, is the emitted gamma rays intensity and ϕ the neutron flux. To take into account the activity measuring time, relation Eq. (16) should be multiplied by the term 1 � e � ln<sup>2</sup> ð Þ <sup>T</sup> tm .


Table 3.

The produced nuclear reactions by irradiating the samples (standards) with 14 MeV neutrons.

#### 5.2.3 Calculation method

The transmission probability of a gamma photon generated from point P1 (Figure 7) after crossing a path length l1 = P1S1 in a homogeneous disk shaped sample (standard) of radius Rt in cm, depth D in cm, and density ρ<sup>1</sup> in g.cm�<sup>3</sup> is given by:

$$P(l\_1) = e^{-\mu\_1 l\_1}$$

where μ<sup>1</sup> is the total attenuation coefficient of the gamma rays in the irradiated sample.

The interaction probability of a gamma photon after crossing a path l2 = Q2S2 (Figure 7) in a disk shaped detector of radius R2 in cm, depth e in cm and density ρ<sup>2</sup> in g.cm�<sup>3</sup> is given by:

$$P(I\_2) = \left(1 - e^{-\mu\_2 l\_2}\right) \tag{17}$$

By combining relations Eq. (16) and Eq. (19) we get the number of the detected

The total attenuation coefficients are calculated by using respectively the Klein-Nishina theory formula [12], Allen Brodsky's approximation [15], and Max Born's approximation [16], for the Compton, photoelectric and pair production effects. The calculation of the paths lengths l1 and 12 consists firstly on generating random numbers by using a programme based on a congruentia 1 method.

� ln<sup>2</sup> ð Þ <sup>T</sup> td : <sup>1</sup> � <sup>e</sup>

� ln<sup>2</sup> ð Þ <sup>T</sup> tm � � (20)

� ln<sup>2</sup> ð Þ <sup>T</sup> tirr � �: <sup>e</sup>

Gamma Rays: Applications in Environmental Gamma Dosimetry and Determination Samples…

gamma rays.

where:

given by:

where

125

O2Q<sup>2</sup> 2 ¼ r 2

> O0 <sup>2</sup>Q<sup>3</sup> 2 ¼ r 2 <sup>2</sup> þ e 2 : tan <sup>2</sup>

distance from the axis of the detector and:

Nc <sup>¼</sup> TN0σϕ

DOI: http://dx.doi.org/10.5772/intechopen.85503

The path length l1 is given by [17]:

O0 <sup>1</sup>O<sup>1</sup> 2 <sup>¼</sup> <sup>r</sup><sup>2</sup> <sup>1</sup> þ t

with four uniform random numbers [17]:

ln 2 <sup>θ</sup>Iγω <sup>1</sup> � <sup>e</sup>

l<sup>1</sup> ¼

And : l rð Þ¼ ; θ; ψ

t

8 ><

>:

cos<sup>θ</sup> if R<sup>1</sup> <sup>≥</sup> <sup>O</sup><sup>0</sup>

R2 <sup>1</sup> � r<sup>2</sup>

r<sup>1</sup> ¼ R<sup>1</sup>

t ¼ Dξ<sup>2</sup> cosð Þ¼ θ ξ<sup>3</sup> ψ ¼ 2πξ<sup>4</sup> ð Þ With 0≤ ξ<sup>i</sup> ≤ 1

We have developed the following theory to calculate the path length 12 which is

cos<sup>θ</sup> if R<sup>2</sup> . <sup>O</sup>2Q<sup>2</sup> and R<sup>2</sup> <sup>≥</sup> <sup>O</sup><sup>0</sup>

<sup>2</sup>Q<sup>3</sup>

0 if R<sup>2</sup> ≤ O2Q<sup>2</sup>

X<sup>2</sup> if O2Q<sup>2</sup> , R<sup>2</sup> , O<sup>0</sup>

: tan <sup>2</sup>

h is the distance between the sample and the detector (Figure 7). r2 is the

e

<sup>1</sup> <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>h</sup> <sup>2</sup>

8 >>>><

>>>>:

q

l rð Þ ; θ; ψ if R<sup>1</sup> , O<sup>0</sup>

<sup>1</sup>O<sup>1</sup>

<sup>2</sup>: tan <sup>2</sup>ð Þþ <sup>θ</sup> <sup>2</sup>r1:t:tangð Þ<sup>θ</sup> :cosð Þ <sup>ψ</sup>

<sup>1</sup> sin <sup>2</sup>ð Þ ψ

sin ð Þθ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffi ξ1 p

The uniform random sampling of the emission point P1 and emission direction is achieved by computing the distance from the center r1, the depth t, cos(θ) and ψ,

<sup>1</sup>O<sup>1</sup>

� r<sup>1</sup> cosð Þ ψ

<sup>2</sup>Q<sup>3</sup>

ð Þþ θ 2r1:ð Þ t þ h :tangð Þθ :cosð Þ ψ

ð Þþ θ 2r2:e:tangð Þθ :cosð Þ ψ

where μ<sup>2</sup> is the total attenuation coefficient of the gamma photon in the detector. The absorption probability of a gamma photon in the detector is given by:

$$P = e^{-\mu\_1 l\_1} \left(\mathbf{1} - e^{-\mu\_2 l\_2}\right) \tag{18}$$

Consequently the detection rate for N photons is:

$$\overline{\boldsymbol{\omega}} = \frac{\mathbf{1}}{N} \sum\_{i=1}^{N} e^{-\mu\_1 l\_{1i}} \left( \mathbf{1} - e^{-\mu\_2 l\_{2i}} \right) \tag{19}$$

Gamma Rays: Applications in Environmental Gamma Dosimetry and Determination Samples… DOI: http://dx.doi.org/10.5772/intechopen.85503

By combining relations Eq. (16) and Eq. (19) we get the number of the detected gamma rays.

$$N\_c = \frac{TN\_0 \sigma \phi}{\ln 2} \,\,\theta I\_\gamma \overline{o} \left(\mathbf{1} - e^{-\left(\frac{\ln 2}{\overline{T}}\right) t\_{\overline{tr}}}\right) \,\,e^{-\left(\frac{\ln 2}{\overline{T}}\right) t\_d} \cdot \left(\mathbf{1} - e^{-\left(\frac{\ln 2}{\overline{T}}\right) t\_m}\right) \tag{20}$$

The total attenuation coefficients are calculated by using respectively the Klein-Nishina theory formula [12], Allen Brodsky's approximation [15], and Max Born's approximation [16], for the Compton, photoelectric and pair production effects.

The calculation of the paths lengths l1 and 12 consists firstly on generating random numbers by using a programme based on a congruentia 1 method.

The path length l1 is given by [17]:

$$l\_1 = \begin{cases} \frac{t}{\cos \theta} \text{ } \text{if } R\_1 \ge \overline{O\_1'O\_1} \\\\ l(r, \theta, \psi) \text{ } \text{if } R\_1 < \overline{O\_1'O\_1} \end{cases}$$

where:

5.2.3 Calculation method

in g.cm�<sup>3</sup> is given by:

given by:

sample.

Figure 7.

124

The transmission probability of a gamma photon generated from point P1 (Figure 7) after crossing a path length l1 = P1S1 in a homogeneous disk shaped sample (standard) of radius Rt in cm, depth D in cm, and density ρ<sup>1</sup> in g.cm�<sup>3</sup> is

P lð Þ¼ <sup>1</sup> e

where μ<sup>1</sup> is the total attenuation coefficient of the gamma rays in the irradiated

The interaction probability of a gamma photon after crossing a path l2 = Q2S2 (Figure 7) in a disk shaped detector of radius R2 in cm, depth e in cm and density ρ<sup>2</sup>

where μ<sup>2</sup> is the total attenuation coefficient of the gamma photon in the detector. The absorption probability of a gamma photon in the detector is given by:

�μ1l<sup>1</sup> <sup>1</sup> � <sup>e</sup>

�μ1l1<sup>i</sup> <sup>1</sup> � <sup>e</sup>

P Ið Þ¼ <sup>2</sup> 1 � e

P ¼ e

The irradiated sample (standard) to γ-detector arrangement used in the activities calculation.

<sup>ω</sup> <sup>¼</sup> <sup>1</sup> <sup>N</sup> <sup>∑</sup> N i¼1 e

Consequently the detection rate for N photons is:

Use of Gamma Radiation Techniques in Peaceful Applications

�μ1l<sup>1</sup>

�μ2l<sup>2</sup>

�μ2l<sup>2</sup>

�μ2l2<sup>i</sup>

(17)

(18)

(19)

$$\overline{O\_1'O\_1}^2 = r\_1^2 + t^2. \tan^2(\theta) + 2r\_1. t. \text{t.} \text{t.} \text{arg}(\theta). \cos\left(\psi\right).$$

$$\text{And}: l(r, \theta, \psi) = \frac{\sqrt{R\_1^2 - r\_1^2 \sin^2(\psi)} - r\_1 \cos\left(\psi\right)}{\sin\left(\theta\right)}$$

The uniform random sampling of the emission point P1 and emission direction is achieved by computing the distance from the center r1, the depth t, cos(θ) and ψ, with four uniform random numbers [17]:

$$r\_1 = R\_1 \sqrt{\xi\_1}$$

$$t = D \xi\_2$$

$$\cos\left(\theta\right) = \xi\_3$$

$$\boldsymbol{\nu} = 2\pi \xi\_4$$

$$(\mathbf{With}) \ 0 \le \xi\_i \le 1$$

We have developed the following theory to calculate the path length 12 which is given by:

$$\begin{cases} \mathbf{0} \text{ } \text{if } R\_2 \le \overline{O\_2 Q\_2} \\\\ \frac{e}{\cos \theta} \text{ } \text{if } R\_2 > \overline{O\_2 Q\_2} \text{ and } \ R\_2 \ge \overline{O\_2' Q\_3} \\\\ X\_2 \text{ } \text{if } \overline{O\_2 Q\_2} < R\_2 < \overline{O\_2' Q\_3} \end{cases}$$

where

$$\overline{O\_2Q\_2}^2 = r\_1^2 + (t+h)^2.\tan^2(\theta) + 2r\_1.(t+h).\text{tag}\,(\theta).\cos\left(\psi\right)$$

$$\overline{O\_2'Q\_3}^2 = r\_2^2 + e^2.\tan^2(\theta) + 2r\_2.e.\text{tag}\,(\theta).\cos\left(\psi\right)$$

h is the distance between the sample and the detector (Figure 7). r2 is the distance from the axis of the detector and:

Use of Gamma Radiation Techniques in Peaceful Applications

We notice that the results obtained by the two methods (experimental and calculation) are in good agreement with each other. The calculation method has the advantage of being accurate (error is smaller than 3%) and rapid (the calculation

Gamma Rays: Applications in Environmental Gamma Dosimetry and Determination Samples…

In the first part of the chapter, a careful study of the correcting factors linked to

In the second part, the calculation method was developed. It is very accurate, rapid, adapted to the experimental conditions, it does not necessitate the use of a very expensive detection chain, and can be used to determine the trace element concentrations in materials. This technique is a good test for neutron activation analysis experiments. It allows these experiments to be calibrated in cases where it

the environmental and experimental conditions is performed.

time is of about 2 min).

DOI: http://dx.doi.org/10.5772/intechopen.85503

is difficult to achieve standards.

6. Conclusion

Author details

Hassane Erramli<sup>1</sup>

127

Mohammed V, Rabat, Morocco

\* and Jaouad El Asri<sup>2</sup>

\*Address all correspondence to: hassane@uca.ac.ma

provided the original work is properly cited.

1 Faculty of Sciences Semlalia, University Cadi Ayyad, Marrakech, Morocco

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Nuclear Reactor and Nuclear Security, Faculty of Sciences, University

Figure 8. Scheme shows changes in gamma-ray direction in the case of multiple interactions.

$$\begin{aligned} X\_2 &= \begin{cases} 0 \text{ if } \tan\theta \ge \tan\theta\_{\text{max}}\\ \sqrt{R\_2^2 - r\_2^2 \sin^2\Phi} - r\_2 \cos\theta \end{cases} \\ \text{(With) } \tan\theta\_{\text{max}} &= \frac{R\_2 - R\_1}{t + h} \end{aligned}$$

To complete this study, we have developed another program based on the EGC method [18]. This program results in determining the energy loss predominant phenomenon that occurs when gamma rays interact with the absorber.

For this, we compare the range x to the relaxation length (λ) of the γ-rays in the material (Figure 8).

### 5.3 Results and discussions

The measured Nm and calculated Nc activities of some different irradiated standards containing Na, K, Cl, and P are shown in Table 4.


#### Table 4.

Data obtained for different irradiated standards with a 14 MeV neutron flux by experimental Nm and calculation Nc methods.

Gamma Rays: Applications in Environmental Gamma Dosimetry and Determination Samples… DOI: http://dx.doi.org/10.5772/intechopen.85503

We notice that the results obtained by the two methods (experimental and calculation) are in good agreement with each other. The calculation method has the advantage of being accurate (error is smaller than 3%) and rapid (the calculation time is of about 2 min).
