Investigation of Gamma-Ray’s Transmission Geometries for the Measurement of Attenuation Coefficients

1 Introduction

The precise measurement of gamma (γ)-ray mass attenuation coefficients (μ_m) of absorber needs narrow-beam transmission geometry [1]. Both liner attenuation coefficient (μ) and mass attenuation coefficient (μ_m, cm² g⁻¹) are very useful parameters for non-destructive analysis of materials and in medical diagnostics such as the computed tomography scan [1]. The γ-ray shielding parameters (GSP) such as effective electron density (N_el,eff), Klein-Nishina cross section (σ_c) and scatter acceptance angle (θ_SC) are explicit functions of the attenuation coefficients. Thus, the accuracy in the μ_m measurement is desirable for analysis of the γ-ray shielding behaviour (GSB) of any material. A good-geometry (narrow-beam) condition is recommended for the analysis of GSB of absorbers [1], [2], [3], [4], [5]. In practice, the γ-ray transmission geometry deviates from the good-geometry conditions. Multiple factors cause such deviations, i.e. the size of collimator aperture, thickness of the sample, various distances [source to absorber (SA), absorber to detector (AD) and source to detector distance (STD)], buildup factors (B) and scatter acceptance angle (θ_SC) [6]. These factors are termed as geometrical parameters (GPs). In other words, it is impossible to design a collimator which allows an γ-ray beam with unit photon flux to pass, and at the same time it is very difficult to make the absorber’s thickness on an atomic scale. Practically during γ-ray measurements for the absorber, many photons pass through the finest collimator and thereby interact randomly with a large number of atoms during their traverse through the absorber. Thus, probability plays a role in the attenuation of γ-rays. The aim of the present work was to provide a parameter which provides quantitative information about the nature of geometry. The deviation of transmission geometry from a perfect narrow-beam condition results in the intermixing of scattered photons with the transmitted beam by the absorber. It causes fluctuations in the transmitted γ-ray intensity reaching the detector. This is the major cause of errors in μ_m measurements [5], [6]. In order to parameterise the magnitude of contribution of the scattered photons, White (1950) introduced buildup factor, which is the ratio of measured to computed intensities of γ-rays reaching the detector [7]. In the present study, the combined influence of GPs and B on the μ_m measurement has been investigated using scattered-to-transmitted ratio (STR).

2 Objective

The objective of this study was to develop a parameter which quantitatively describes the nature of the experimental setup (transmission geometry) used for the μ_m measurement of material.

3 Materials and Methods

3.4 Computer Program

For theoretical computations, a computer program (GRIC3 toolkit) has been developed by modifying the GRIC2 toolkit [5] (see Supplementary Material, Fig. 1). The GRIC3 toolkit (see Supplementary Material, Fig. 2) has the ability to compute the θ_SC, buildup factor (B) and STR for a given experiential setup, i.e. for different values of the collimator aperture, SA and SD.

Figure 1:

Schematic representation of transmission geometry used in the present investigations.

Figure 2:

Description of the variations in STR (%) and scatter acceptance angle with the absorber’s (CW) thickness (linear in cm and optical in mfp).

4 Methodology

According to the Midgley condition for narrow-beam geometry, θ_SC ≤ 3° [8], the nature of the transmission geometry has been decided. Mann [5], [6] used this condition for deciding the optimum thickness [0.5 mean free path (mfp)] of the sample for its precise μ measurements. The following section provides the step-by-step information and methodology used in the investigation of the STR.

5 Results and Discussion

Figure 2 describes the variation of STR and θ_SC with absorber thickness (CW) by keeping constant values for the size of the collimator apertures, SA and SD. The value of STR for which the Midgley condition [8] was satisfied is termed as the critical value of STR (CSTR). It is indicated from Figure 2 that for the CW sample, CSTR is 0.0075 and the corresponding absorber thickness, i.e. optimum thickness = 24 mm (2.18 mfp). Thus, for accurate μ measurement for the energy range (661.66–1332.5 keV), the experimental setup should be such that its STR remains below the CSTR. Thus, the STR includes various parameters such as absorber’s thickness, collimator aperture sizes, buildup factor and γ-ray energy, which otherwise are required to quantify the nature of geometry for an experimental setup.

Figures 3 and 4 describe the simultaneous variations in STR and θ_SC for the absorber (CW) with different aperture values of C2 and C3, respectively, for γ-ray energy 661.66 keV. Other GPs have been kept constant during these variations. The inverse relationship has been found between the size of the aperture and the optimum thickness value of the absorber in light of the Midgley condition [8].

Figure 3:

For monoenergetic γ-rays (661.66 keV), the description of variations in scatter acceptance angle with CW sample thickness and collimator C3’s aperture.

Figure 4:

For monoenergetic γ-rays (661.66 keV), the description of variations in scatter acceptance angle with CW sample thickness and collimator C2’s aperture.

Figure 5 provides the simultaneous variations of STR and θ_SC with absorber thickness for three γ-ray energies. The similar trend of variation in STR with absorber thickness at three γ-ray energies for the same geometrical setup and GPs has been observed.

Figure 5:

For fixed geometry, description of the combined variations in STR (%) and θ_SC with the absorber’s (CW) thickness at three γ-ray energies.

6 Conclusions

It has been concluded that the STR provides a quantitative description of the nature of γ-ray transmission geometry used for the measurements of attenuation coefficients of materials. It summarises multiple parameters related to the geometry such as thickness of sample, size of collimator apertures, SA, STD, AD, θ_SC and B. STR provides better information of scattered γ-rays by shielding material than buildup factor (B). In addition, the CSTR can serve as the borderline between the narrow-beam and broad-beam transmission geometries. For a given energy range (661.66–1332.50 keV), it has been found that for the chosen low-Z samples, CSTR is 0.075.

This study can be extended to the wider energy range and for high-Z materials after the reestablishment of American Nuclear Standards [10].