Introduction

Because ionsing radiation is widely used whether this is in a hospital, laboratory, nuclear power plant, or nuclear waste treatment location, it is very important to measure and quantify how much radiation is produced by any radioactive sources and how much radiation is absorbed by materials. This requires methods that allow determination of relevant radiation interaction parameters. For example, absorbed dose in matter can be obtained by solving radiation transmission equations. For gamma ray radiation, narrow beam geometry can be employed including the Beer–Lambert law (Swinehart 1962). In fact, narrow beam geometry is considered an essential setup to study the gamma ray interaction with matter by determining linear attenuation coefficients or mass attenuation coefficients. The latter coefficients are used to describe the probability of photon–matter interactions per unit mass and unit area (McNair 1981; Chantler 2000). The \(\mu _{\text {m}}\) values are essential to derive atomic cross sections, electronic cross sections, and effective atomic numbers for a compound or mixture. The notion of effective atomic number was introduced first by Hine who noticed that the effective atomic number of a mixture or compound is not constant but variable with photon energy (Hine 1951). Nowadays, it is well known that \(\mu _{\text {m}}\) and \(Z_{\text {eff}}\) are the most important parameters for selection of suitable materials for radiation dosimetry and radiation detection.

Many attempts have been made to identify materials to be used as radiation dosimeters. Reddy studied the radiation response of several liquid and solid materials (Kumar and Reddy 1997), using elemental data from Hubbell and Seltzer (1995), Chantler (1995). Shivaramu calculated the effective atomic numbers of some thermoluminescent dosimetric (TLD) materials such as LiF, \(\hbox {CaCO}_3\), and \(\hbox {CaSO}_4\) (Ramprasath et al. 2000). He concluded that effective atomic numbers vary linearly with photon energy. The mass attenuation coefficients of some TLD materials were also experimentally determined at different photon energies in a narrow beam geometry using a hyper pure germanium detector (Gowda et al. 2004). The radiological features of some gel materials were compared against water with respect to their \(Z_{\text {eff}}\) values (Taylor et al. 2008). In addition, gamma-ray buildup factors for TLD materials were reported and compared with ICRU standard dosimeters (Manohara et al. 2010; Kucuk et al. 2013; Singh and Badiger 2016). Recently, \(\mu _{\text {m}}\) values for some polymers were determined using Monte Carlo simulation at different photon energies (Singh et al. 2015). In addition, there are numerous similar studies dealing with various materials using different methods including measurements, theoretical considerations, or simulations (Sidhu et al. 2012; Özdemir and Kurudirek 2009; Kurudirek 2014; Bootjomchai and Laopaiboon 2014; Singh et al. 2014).

Nonlinear optical NLO materials were found to be useful for many applications: for smart filtering (Dini et al. 2016), for photonics (Lin et al. 2014), and for biomedicine and radiation detection (Liu et al. 2017). NLO materials might possibly be used in biological dosimetry because of the following properties: their strength, lightness, low cost, and ease of fabrication. Very recently, Awasarmol studied some organic NLO materials in terms of \(\mu _{\text {m}}\) and \(Z_{\text {eff}}\) (Awasarmol et al. 2017). He found that the NLO materials can support radiation measurement due to their low \(Z_{\text {eff}}\). At present, fundamental gamma interaction parameters for selected NLO materials have not yet been published.

This prompted the present study in which dosimetric parameters of various NLO materials such as such as potassium dihydrogen phosphate, ammonium dihydrogen phosphate, zinc tris-thiourea sulphate, and zinc thiourea chloride were investigated. Both measurements and Monte Carlo simulations were performed in narrow beam geometry in an effort to determine mass attenuation coefficients. These mass attenuation coefficients were then utilized to compute the effective atomic numbers and effective electron densities of the selected materials. The present investigation is considered useful for selection of NLO materials to be used for biological dosimetry and other applications.

Experimental and computational methods

Experimental detials

Narrow beam transmission geometry was arranged, as shown in Fig. 1, to measure \(\mu _{\text {m}}\) for the NLO materials shown in Table 1. Six radioactive sources were used, namely, \(^{22}\hbox {Na}\), \(^{54}\hbox {Mn}\), \(^{57}\hbox {Co}\), \(^{60}\hbox {Co}\), \(^{133}\hbox {Ba}\), and \(^{137}\hbox {Cs}\). Gamma ray spectrometry was performed by means of a scintillation detector (\(2\times 2\) inch) to measure \(\mu _{\text {m}}\). The system showed an energy resolution of \(8.2\%\) at 662 keV and included a 16K multichannel analyzer (MCA). All samples were shaped as pellets with different thicknesses of 0.5–1.5 cm, and then irradiated by gamma ray energies of 122, 356, 511, 662, 840, 1170, 1275, and 1330 keV. The radioactive sources were within a lead cylinder with a 0.52 cm orifice to collimate and produce a pencil beam. The detector was also housed in a lead shield to minimize any background radiation. The spectrometer was calibrated using the 356.1 keV line emitted by \(^{133}\hbox {Ba}\), the 122 keV line emitted by \(^{57}\hbox {Co}\), the 511 and 1275 keV lines emitted by \(^{22}\hbox {Na}\), the 661.9 keV line emitted by \(^{137}\hbox {Cs}\), the 840 keV line emitted by \(^{54}\hbox {Mn}\), and the 1173.2 and 1332.5 keV lines emitted by \(^{60}\hbox {Co}\). The laboratory temperature was kept constant at \(25^{\circ }\hbox {C}\) (\(77^{\circ }\hbox {F}\)). The distance between source and detector was kept 35 cm and the sample was placed in-between. Archimedes principle was utilized to obtain the density of the studied materials, where measurements were performed at room temperature with ethylene as the reference immersion liquid.

Table 1 Chemical composition of the investigated NLO materials
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Fig. 1
figure 1

Experimental setup for narrow beam geometry

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GEANT4 simulation

The GEANT4 code can be applied for simulation of a narrow beam geometry with various photon energies (Allison et al. 2016). Determination of mass attenuation coefficients for any compound or mixture requires knowledge on the chemical or elemental composition and density of the material. The GEANT4 simulation developed for the present study included a mono-energetic photon beam imposing on a sample, as shown in Fig. 2. The mass attenuation coefficients for the material under investigation were determined using the transmission method following Lambert’s Beer law (\({I}={I}_0\hbox {e}^{-\mu _{\text {m}}x}\)), where \({I}_0\) and I are the incident and attenuated photon intensity, respectively, \(\mu _{\text {m}}\) (in \(\hbox {cm}^2 \hbox {g}^{-1}\)) is the mass attenuation coefficient and x is the thickness of the studied material in \(\hbox {g}/\hbox {cm}^2\).

Fig. 2
figure 2

GEANT4 setup for narrow beam geometry

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For the simulation of mass attenuation coefficients, an input file has been prepared including beam properties, irradiation parameters, sample compositions, and physical settings. A beam of \(10^6\) photons was simulated to attenuate in the NLO samples. The code was run to obtain the transmission values, \({I}/{I}_0\), for each of the investigated NLO samples.

Effective atomic number

Effective atomic numbers refer to the attenuation of photons, which occurs due to photon interactions with matter, is is given by the following ratio (Singh et al. 2014; Manohara et al. 2009):

$$\begin{aligned} Z_{\text {eff}} = \frac{\sigma _{\text {a}}}{\sigma _{\text {e}}}, \end{aligned}$$
(1)

where \(\sigma _{\text {a}}\) is the total atomic cross section and \(\sigma _{\text {e}}\) is the total electronic cross section. The total atomic cross section can be obtained from the mass attenuation coefficient as follows:

$$\begin{aligned} \sigma _{\text {a}} = \frac{\mu _{\text {m}}}{N_{\text {A}} \sum _{i}{w_i /A_i}}, \end{aligned}$$
(2)

where \(\hbox {w}_i\) is the weight fraction of the ith element in the sample. The total electronic cross section is given by the following equation:

$$\begin{aligned} \sigma _{\text {e}} = \frac{1}{N_{\text {A}}}\sum _{i}\frac{f_i A_i}{Z_i}(\mu _{\text {m}})_{i}, \end{aligned}$$
(3)

where \(N_{\text {A}}\) is Avogadro‘s number, \(f_i\) is the fractional abundance of the each constituent element given that \(\sum _{i}{f_i}=1\), \(Z_i\) is the atomic number, and \(A_i\) is the atomic weight.

Results and discussion

Mass attenuation coefficients for the selected NLO materials are shown in Fig. 3, while corresponding effective atomic numbers are shown in Fig. 4 . Effective atomic numbers as a function of effective electron density are shown in Fig. 5, and measured and simulated mass attenuation coefficients are compared in Table 2. In Table 3, effective atomic numbers of the investigated NLO materials, TLD materials, polymers, and various human tissues are compared.

Mass attenuation coefficients

Figure 3 shows the mass attenuation coefficients (\(\mu _{\text {m}}\)) versus photon energy for the investigated NLO materials as obtained by measurement and simulation, with photon energies ranging from 1 to 1400 keV. It is to be noted that the \(\mu _{\text {m}}\) values for NLO materials decrease with increasing photon energy down to about \(0.06\ \hbox {cm}^2/\hbox {g}\) at a photon energy of 1400 keV. It was found that the \(\mu _{\text {m}}\) values are lowest for ADP and highest for ZTC, in the photon energy region of interest. The variation of the mass attenuation coefficients with photon energy can be explained using the photoelectric effect, Compton scattering and pair production (Chantler 2000; Singh et al. 2014; Tonguc et al. 2018; Özdemir and Kurudirek 2009).

Fig. 3
figure 3

Experimental and simulated values of mass attenuation coefficients as a function of photon enrgy for the investigated NOL materials

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The observed behaviour of the \(\mu _{\text {m}}\) values with photon energy can be divided into three energy regions, i.e., low-energy (\({E}\le 100\ \hbox {keV}\)), intermediate-energy (\(0.1\le {E}<1022\ \hbox {keV}\)), and high-energy (\({E} \ge 1022\ \hbox {keV}\)). It is observed that in the low-energy region, the \(\mu _{\text {m}}\) values sharply decrease with increasing photon energy for all the selected NLO materials. In this low-energy region, the predominant interaction is described by the photoelectric effect, where photon interaction depends upon atomic number and photon energy. In the intermediate-energy region, the \(\mu _{\text {m}}\) values only slightly decrease with increasing photon energy because of Compton scattering. In fact, Compton scattering is found to be roughly independent upon photon energy. In contrast, in the high-energy region, the \(\mu _{\text {m}}\) values slightly increase and become constant due to pair-production being the dominant process. The highest \(\mu _{\text {m}}\) values among the selected NLO materials were found for ZTC, which obviously provides superior shielding properties for the investigated photon energies.

Table 2 compares measured and simulated mass attenuation coefficients. The table shows that both the measured and simulated coefficients are very close to each other within \(< 15\%\) of standard deviation. Therefore, it is concluded that GEANT4 simulations allow for an investigation of photon interaction processes in compound or complex materials. It is also concluded that GEANT4 simulations can replace experimental investigations of photon interaction with materials.

Effective atomic number and electron density

Figures 4 and  5 show effective atomic numbers (\(Z_{\text {eff}}\)) as a function of photon energy and effective electron density (\(N_{\text {eff}}\)), respectively. It is observed that the \(Z_{\text {eff}}\) values for ZTC are the highest, while those for ADP are the lowest. The observed dependence of \(Z_{\text {eff}}\) on photon energy is due to the dependence of \(Z_{\text {eff}}\) on the atomic number of constituent elements and their compositions (see Eqs. 2 and  3). As compared to ADP, ZCT contains higher atomic number elements with higher chemical abundance, and therefore, \(Z_{\text {eff}}\) values are also found to be higher. It is to be noted that the relation between \(Z_{\text {eff}}\) and \(N_{\text {eff}}\) is linear and \(Z_{\text {eff}}\) values increase with increasing \(N_{\text {eff}}\). Figure 4 shows a comparison of \(Z_{\text {eff}}\) values for the investigated NLO materials with those of water. It is found that the \(Z_{\text {eff}}\) values of ADP are comparable those of water. Therefore, in terms of dosimetry ADP can be considered as tissue equivalent.

In Table 3, \(Z_{\text {eff}}\) values of the invvestigated NLO materials are compared with those of TLD materials, polymers, and human tissues. It is observed that the \(Z_{\text {eff}}\) values of TLD materials, polymers, and human tissues are found comparable to those of the investiged NLO materials. Therefore, these NLO materials might be suitable for biological dosimetery and other radiation applications.

Fig. 4
figure 4

Comparison of effective atomic number (\(Z_{\text {eff}}\)) values of NLO materials with those of water

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Fig. 5
figure 5

Variation of effective atomic number (\(Z_{\text {eff}}\)) with effective electron density (\(N_{\text {eff}}\))

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Table 2 Measured (Exp.) and simulated (GEANT4) mass attenuation coefficient (\(\mu _{\text {m}}\)) values, for the investigated NLO materials
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Table 3 Effective atomic number (\(Z_{\text {eff}}\)) values of the investigated NLO materials as compared to those of TLD materials, polymers, and human tissues
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Conclusion

Mass attenuation coefficients for some NLO materials (potassium dihydrogen phosphate, ammonium dihydrogen phosphate, zinc tris-thiourea sulphate, and zinc thiourea chloride) were measured by using a NaI(Tl) detector for selected photon energies. GEANT 4 simulations were carried out to investigate the same parameters. As a result, good agreement between the experimental and simulated results was observed. Effective atomic numbers of the NLO materials were compared with those of TLD materials, polymers and human tissues, and found to be comparable . This suggests that the investigated NLO materials could be used for dosimetric applications. In addition, it is concluded that GEANT4 simulations offer an alternative for experimental investigations of photon interaction with matter.