Introduction

Soils have chemical compositions characterized by the presence of major compounds, such as SiO2 (40–90 %), Al2O3 (5–20 %), Fe2O3 (2–20 %) and minor compounds such as CaO, K2O, MnO, TiO2, and so on. The major compounds in soils present concentrations that exceeds 100 mg kg−1 [1, 2]. The most important quantity characterizing the penetration and diffusion of gamma radiation through soil is the mass attenuation coefficient (μ/ρ). It is the fundamental parameter to derive many other parameters of dosimetric interest such as molecular, atomic (σ a) and electronic (σ e) cross sections, effective atomic number (Z eff) and electron density (N el). These quantities can be evaluated theoretically and experimentally. In literature, a variety of works relevant to μ/ρ and Z eff estimations for different compound materials has been published by several authors in different categories. Recently, a significant number of papers about experimental and theoretical determinations of these parameters in various elements, compounds and mixtures were published [3–11]. Attenuation of photons has been reported as the most accurate and convenient technique for non destructive measurements of soil parameters such as: water retention curve, water content, bulk density and porosity [12–20].

In heterogeneous media such as soil the atomic number cannot be represented by a single number, as in the case of elements. This number in such materials is called effective atomic number. Z eff is considered the most basic and important quantity among the parameters determining the constitutive structure of composite materials. It gives basic information about the characteristics of multi elemental materials. Attenuation (due to scattering or absorption) of photons in composite materials depends basically on two factors, Z eff and N el. Thus, Z eff has proved to be a convenient parameter for interpreting the γ-ray attenuation by soil. Moreover, the study of Z eff provides important information on the composition of the soil. For example, higher values of Z eff, soils tend to contain more inorganic compounds and metals than the smaller values [11].

The objective of this study deals with the theoretical and experimental determination of mass attenuation coefficients of soils with different textures. Other parameters such as molecular, atomic and electronic cross sections, effective atomic number and electron density were also calculated for photon energies in the range 1 keV–100 GeV. Theoretical results were compared with the measurements obtained with photon energies of 59.5 keV (241Am) and 661.6 keV (137Cs). The soils under consideration were collected from Southeast and South of Brazil.

Theoretical basis and simulations

Mathematical basis

For materials composed of several elements, it is assumed that the contribution of each element to total photon attenuation is additive. In such cases, μ/ρ of any material with density ρ is related to (μ/ρ)i values of its constituents by the mixture rule, \( \mathop \sum \limits_{i} {c}_{i} \left( {\frac{\mu}{\rho}} \right)_{i} \), where c i is the proportion by weight of the i-th constituent element. The total molecular cross-section (σ m) can be calculated from the knowledge of μ/ρ by using the following relation [9]:

$$ \sigma_{\text{m}} = \left( {\frac{\mu }{\rho }} \right)\frac{M}{{N_{A} }}, $$
(1)

where \( {\text{M}} = \mathop \sum \limits_{i} n_{i} A_{i} \), is the molecular weight of the compound, N A is the Avogadro number, A i is the atomic weight of the i-th element and n i is the number of formula units in the molecule. The average atomic cross-section can be obtained by dividing σ m by the total number of formula units as follows:

$$ \sigma_{\text{a}} = \sigma_{\text{m}} \frac{1}{{\mathop \sum \nolimits_{i} n_{i} }}. $$
(2)

Similarly, the average electronic cross-section is given by

$$ \sigma_{\text{e}} = \frac{1}{N}_{\text{A}} \sum\nolimits_{i} {\frac{{f_{i} A_{i} }}{{Z_{i} }}\left( {\frac{\mu }{\rho }} \right)_{i} } , $$
(3)

where \( f_{i} = {\text{n}}_{i} /\sum\nolimits_{j} {n_{j} } \) and Z i are fractional abundance and atomic number of the constituent element, n j is the number of atoms of the constituent element and \( \sum\nolimits_{j} {n_{j} = n} \) is the total number of atoms present in the molecular formula. Therefore, the effective atomic number can be now defined as:

$$ Z_{\text{eff}} = \frac{{\sigma_{\text{a}} }}{{\sigma_{\text{e}} }}. $$
(4)

Finally, the effective electron number or electron density (number of electrons per unit mass) of the material can be derived from:

$$ N_{\text{el}} = \frac{{\left( {\frac{\mu }{\rho }} \right)}}{{\sigma_{\text{e}} }} = \left( {\frac{{Z_{\text{eff}} }}{M}} \right)N_{\text{A}} \sum\nolimits_{i} {n_{i} } . $$
(5)

XCOM software

Calculations of μ/ρ of samples were carried out by the XCOM program [21]. The software can generate cross-sections and attenuation coefficients for elements, compounds or mixtures in the energy range 1 keV–100 GeV. The calculated data can be given in the form of total cross-sections and attenuation coefficients as well as partial cross-sections of the following processes: incoherent and coherent scatterings, photoelectric absorption and pair production in the field of the atomic nucleus and electrons. The program possesses a comprehensive database for all elements over a wide range of energies, constructed through the combination of photoelectric absorption, incoherent and coherent scatterings and pair production (nuclear and electric field) cross-sections.

GEANT4 simulation code

The GEANT4 code is based on object-oriented programming and allows user to derive classes to describe the detector geometry, primary particle generator and physics processes models along electromagnetic, hadronic and decay physics. These simulations are based on theory, materials and elements, experimental data or parameterizations [22].

The proposed simulation model depends on narrow beam geometry with the various photon energies. The model consists of a mono-energetic photon beam impinging on a slab of one of the selected materials. Soil mass attenuation of investigated materials are determined by the transmission method according to Beer Lambert’s law (\( {I} = I_{0} e^{ - \mu x} \)), where I 0 and I are the incident and attenuated photon intensity, respectively, μ (cm−1) is the linear attenuation coefficient and x is the thickness of the slab. The thickness x is optimized according to I 0, in order to prevent the complete photon absorption by the slab or its transmission without interaction. The primary photons emerging unperturbed from the slab are counted. The energy range of incident photons varies from 1 keV to 100 GeV. Attenuation of photons is calculated by simulating all relevant physical processes and interactions before and after inserting the investigated sample [23–25].

Materials and methods

Samples collection and preparation

In the investigated study, soil samples were collected from regions located in Southeast (soils 1–3) and South (soils 4–8) of Brazil (tropical and sub-tropical climate). All samples were taken from the surface layer (0–10 cm) of every region. The physical properties of the investigated soil samples are shown in Table 1. Soil samples (1–3) were collected from an area covered by grasses and weeds cultivated with orange trees (Soil 1), coffee field (Soil 2) and mixed forest cropped with sugarcane (Soil 3). Soils (4–8) were collected from an area used for pasture (Soil 4), cultivated by tobacco field (Soil 5), maize field (Soil 6), soybean field (7) and cultivated with maize and soybean (Soil 8). Collected soil samples were dried in oven at 105 °C for 2 days and sieved through a 1 mm mesh sieve to obtain a homogeneous sample. Samples were kept in containers with silica-gel throughout the experimental analyses.

Table 1 Some physical properties of the soils studied
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Elemental analysis of samples

Elemental analysis of soil samples was carried out using energy dispersive X-ray fluorescence (EDXRF) spectrometer (Shimadzu, EDX-720) equipped with Rh tube (5–50 kV; 1–1,000 μA). Samples were ground in mortar and put into proper containers supplied by the equipment manufacturer and sealed with mylar (6 μm thick). The spectra was obtained for a time of 100 s in the energy bands of Na–Sc (15 kV) and Ti–U (50 kV). All measurements were carried out with pressure under 30 Pa. EDXRF analyses of the investigated samples are presented in Table 2.

Table 2 EDXRF analyses results of the dry soil samples
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Experimental setup

A NaI(Tl) solid scintillation detector of plain type (7.62 × 7.62 cm) was used to detect gamma photons emitted from γ-ray sources of 137Cs (11.1 GBq) and 241Am (7.4 GBq). Circular lead collimators (2–4 and 4.5 mm), mounted between source and detector, were utilized to collimate the radiation beam. The source and detector distance were kept constant (23 cm) and the acrylic container used to assembly the sample is 4.0 cm thick. The schematic arrangement of the experimental setup in the present investigation is shown in Fig. 1. Incident and transmitted photons for each sample were measured for sufficiently large fixed preset time to reduce statistical uncertainty. The counting time adopted in μ experimental measurements was 600 s for both of 137Cs and 241Am sources. The detected spectra can be translated to ASCII and processed with custom-made programs based on ROOT.

Fig. 1
figure 1

Schematic diagram of the set up used to the experimental measurements of the soil mass attenuation coefficient. 1 241Am γ-ray source; 2 137Cs γ-ray source; 3 Pb shields; 4 Pb circular collimators; 5 Soil sample inside an acrylic box; 6 NaI(Tl) detector and 7 photomultiplier tube. z and x represent the distance between the radioactive source and detector and the soil sample thickness

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Results and discussion

The obtained results of μ/ρ for different types of soil were compared with XCOM and GEANT4 calculations as tabulated in Table 3. There is a notable deviation between the experimental and theoretical values of XCOM for most soil samples at low photon energy (59.5 keV). It is clear that the representative GEANT4 Monte Carlo generated data for μ/ρ of all samples are in agreement with experimental values. Discrepancy of the calculated and the experimental μ/ρ values could be due to deviations from narrow beam geometry in the source-detector arrangements [23].

Table 3 Comparison among GEANT4, XCOM and experimental mass attenuation coefficients of the investigated soil samples at 59.5 and 661.6 keV
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It is notable from Fig. 2 the correlations among experimental and theoretical (XCOM and GEANT4) methods. The best correlations were obtained for μ/ρ simulated through GEANT4 in comparison to the experimental values (Fig. 2a, b). A strong correlation was also found for μ/ρ simulated through XCOM for the 241Am γ-ray source (Fig. 2c), however, only a weak correlation was observed for the 137Cs γ-ray source (Fig. 2d). This is due to the uncertainty results of XCOM in low energy (less than 50 keV) is smaller. These results show the importance of Monte Carlo simulations to predict γ-ray attenuation coefficients of soils [26].

Fig. 2
figure 2

Analysis of correlation among the experimental mass attenuation coefficient (μ/ρ) and theoretical μ/ρ values simulated through XCOM and GEANT4 codes for the 241Am (a–c) and 137Cs (b–d) γ-ray sources

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The result of total μ/ρ of the studied soils as function of photon energy is presented in Fig. 3. Figure 3 shows the strong dependence of this parameter with photon energy. It can be easily seen that there are three energy ranges, where photoelectric absorption, Compton scattering and pair production, respectively, are the dominating attenuation processes. Experimental as well as theoretical values of other attenuation parameters (σ a, σ e, Z eff and N e) are presented in Table 4. Good agreement has been achieved between measured and calculated values.

Fig. 3
figure 3

Variation of photon mass attenuation coefficient of the investigated soil samples with photon energy for total photon interaction calculated by XCOM in the energy range 1 keV–100 GeV

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Table 4 The experimental and theoretical values of atomic cross-section (σ a), electronic cross-section (σ e), effective atomic number (Z eff) and effective electron number (N e) for the investigated soil samples
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The strongest energy dependence was observed in low photon energy region (1–100 keV). In this range, μ/ρ have the highest values, where the photoelectric absorption is significant and its cross-section is proportional to Z 4,5. In the intermediate energy region, the Compton scattering is significant, the linear Z-dependence of incoherent scattering is demonstrated and μ/ρ is found to be constant. In the high energy region, μ/ρ increases again, where the pair production is significant and μ/ρ is proportional to Z 2. The present theoretical results were in similar to the observations of Medhat [9] and Manohara and Hanagodimath [11].

For total photon interaction process, the variations of Z eff and N el with photon energy are shown in Figs. 4 and 5. From Fig. 4, it is clear that Z eff values of the soils change with photon energy and soil chemical structures. In low energy range (1–10 keV), soils 2, 5, 6, 7 and 8 have almost constant values of Z eff, while the other soils are slightly lower in their Z eff values. The almost constant values for soils 2, 5, 6, 7 and 8 can be associated with the similarities between silt and sand contents among them.

Fig. 4
figure 4

Variation of Z eff with photon energy of the selected soils for total photon interaction (with coherent) calculated by XCOM in the energy range 1 keV–100 GeV

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Fig. 5
figure 5

Variation of N el with photon energy of the selected soil for total photon interaction (with coherent) calculated by XCOM in the energy range 1 keV–100 GeV

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In the energy range (10–200 keV), Z eff values are decreased reaching their minimum values between 2 and 4 keV among soils. From the obtained Z eff values, it is possible to observe that the clayey soils presented the same range of variation. Regions of Z eff stabilization can be observed for all the soils studied.

It is interesting to observe the differences of Z eff values among soils which can be associated sometimes with their textures. However, in the case of the loamy soils (1–4) it is hard to establish some kind of correlation due to the large variety of combinations of silt, sand and clay contents to form these types of soils. However, soils (4–5) presented similarity in Z eff behavior with energy and these loamy soils have in common almost the very same values of Al2O3, Fe2O3, K2O and SiO2.

The variations of N el with photon energy for total interaction processes (Fig. 5) are similar to that of Z eff and can be explained on the similar manner. The N el values show photon–energy dependence similar to that observed for Z eff. This result is confirmed due to the positive correlations between Z eff and N el values obtained from the theoretical calculation carried out by Kucuk and collaborators [3]. These authors found small differences in Z eff and N el among soils independent from the texture. In the present work the discrepancies among soils for these two physical parameters can be associated with differences in the nature of clays and management systems of the Brazilian soils [27].

It is important to have in mind that different proportions of sand, silt and clay will result in different soil textures. Soil texture is an important physical property related to several dynamic phenomena that take place in this porous media. As the soil is characterized by a broad distribution of particle sizes, soils with different textures can also attenuate the radiation in a different manner. For example, the loamy soils (1, 4 and 5) presented the smallest values of μ/ρ and clayey soils were characterized by the largest ones. These results are mainly related to the Fe2O3 content.

Considering the clay content and the soil major components it is possible to observe medium to strong positive correlations between experimental μ/ρ values, clay and Fe2O3 contents (Fig. 6a, c). This strong positive correlation between μ/ρ and Fe2O3 is important because the Brazilian soils are rich in Fe. On the other hand, a medium negative correlation between μ/ρ and SiO2 was observed (Fig. 6b). Similar results were measured by AkarTarim and collaborators [26] for the same γ-ray energy (241Am). Only a weak positive correlation was found between μ/ρ and Al2O3 (Fig. 6d). For the case of the 137Cs γ-ray source, only weak positive and negative correlations were observed between μ/ρ, clay and the major soil compounds. The results were also similar to those observed by Akar Tarim et al. [26].

Fig. 6
figure 6

Analysis of correlation between the experimental mass attenuation coefficient (μ/ρ), clay (a) and the major soil compounds (b–d) for the 241Am γ-ray source

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Conclusion

The present study has been undertaken to get some information about mass attenuation coefficients and related parameters, effective atomic numbers and electron density for different types of soils used in agriculture purposes. The electron density and effective atomic number are closely related and they are qualitative energy dependent. A good agreement was observed between the theoretical calculations and experimental results of μ/ρ. The low incident energy region represents the one with the most remarkable differences in μ/ρ among soils. This fact is related to the Z dependence for the photoelectric absorption mechanism that governs the photon interaction in this region.