Determination of effective atomic number, electron density and Kerma of some ferroelectric materials using mass attenuation coefficients in the energy range 1 keV -100 GeV

The effective atomic number Zeff and the corresponding effective electron density Nel of ferroelectric materials (BaCoF4, LiNbO3, KNbO3, BaTiO3, PbZrO3, PbTiO3, KNbO3, Bi4Ti3O12, KHPO4, PbBi2Nb2O9, SrBi2Ta2O9, Ba0.73Sr0.27TiO3 and Mg3B7O13Cl) has been calculated in the extended energy region from 1 keV – 100 GeV using WinXCom program. The total photon interaction with coherent and incoherent one can distinguish three energy regions are approximately E<1 keV, 10 keV10 GeV. The main photon interaction processes in these regions are photoelectric absorption, incoherent (Compton) scattering and pair production. Zeff and Nel values also have been obtained for different processes such as coherent and incoherent scattering pair production in nuclear and electric field, photoelectric absorption and the variations are shown graphically. The variations are due to the presence of different elements in a compound. The behaviour of the energy dependence of electron density is similar to the energy dependence of effective atomic number. The photoelectric absorption is dominant for total coherent process for Kinetic energy released per unit mass (Kerma). Direct method is one of the method helpful to calculate effective atomic number for all types of materials, for all energies greater than 1 keV.


Introduction
Ferroelectric materials are utilized widely in a variety of applications due to their spontaneous electrical polarization [1].Numerous beneficial features are available in ferroelectric materials.Ferroelectric hysteresis (used in nonvolatile memories), high permittivities (in capacitors), high piezoelectric effects (in sensors, actuators, and resonant wave devices like radio-frequency filters), high pyroelectric coefficients (in infrared detectors), strong electro-optic effects (in optical switches), and anomalous temperature coefficients of resistivity (in electric-motor overload-protection circuits) are a few of these.Ferroelectrics may also be produced in a wide range of materials, such as ceramics, single crystals, polymers, and thin films, which expands their use.The fundamental theories underlying the ferroelectric effect and the major kinds of ferroelectric materials are covered in this work, along with how their properties relate to their composition and various manufacturing processes.[2].
The total mass attenuation coefficient, which is the sum of absorption and scattering coefficients, is the most commonly used parameter in photon interaction with matter.The total mass attenuation coefficient is a measure of the probability of incident photons interacting with a matter of unit mass per unit area.The atomic number of the sample, the sample density, the incident photon energy and the chemical structure of the sample significantly affect this parameter.The total mass attenuation coefficient is an essential tool to obtain many other parameters like atomic, molecular and electronic cross sections, effective atomic number, electron density, kinetic energy released per unit 1300 (2024) 012013 IOP Publishing doi:10.1088/1757-899X/1300/1/012013 2 mass (Kerma) [3] and molar extinction coefficient [4].The measurement or calculation of these parameters is a pioneer attempt for many applied fields.These parameters are widely used in medical, radiation, nuclear, plasma and space physics, biological and agricultural industries, so accurate data of these parameters have a great importance [4,5].X-rays, Gamma rays and particle radiation such as electrons, protons or heavy ions are used in a variety of applications, which include radioisotope monitoring, cross-section studies of electromagnetic radiation absorption, scattering and attenuation, and testing multi-component, heterogeneous, and composite materials [6].A thorough knowledge of the interaction of photons with ferroelectric materials is desirable.In radiation biology, as well as medical diagnosis and therapy, photons in the keV range are important [7].Radiography and medical imaging require photons in the MeV range, while astrophysics and cosmology are interested in photons in the GeV range.
The incoming photon energy and the atomic number of the materials utilized determine how photons interact with matter.G. J. Hine [8] has pointed out that, unlike pure elements, a single number cannot uniquely represent the atomic number across the whole energy range in composite materials for photon interactions.The "effective atomic number" (Zeff) for composite materials is a variable value that changes with energy.If certain constants are known, the energy absorption in a given medium can be computed.As a result, Zeff is a parameter that varies with photon energy based on the interaction mechanism involved, rather than a genuine constant for a specific material.The electron density, which is defined as the number of electrons per unit mass of the absorber, is closely related to the effective atomic number [9].
Attenuation, absorption coefficients and photon interaction cross sections in materials, as well as effective atomic number and electron density, have been studied extensively in the field of radiation.X-ray and Gamma ray attenuation coefficients and interaction cross sections of various materials are typically calculated using the well-known XCOM program.Berger and Hubbell [10] published the first version of this program for estimating photon interaction cross sections and attenuation coefficients for each element or compound in the energy range 1 keV-100 GeV.Gerward et al [11] have released a new version of this program called WinXCOM, which runs on the Windows operating system.For incoherent and coherent scattering, photoelectric absorption, and pair production, the program offers total cross-sections and attenuation coefficients as well as partial cross sections.XmuDat, a similar program, calculates mass attenuation coefficients for elements, compounds and mixtures in the energy range of 1 to 50 MeV photons [12].WinXCOM is another program that determines the removal and attenuation coefficients of transmitted fast neutrons, gamma rays, via mixtures and compounds [13].The direct or interpolation approaches are commonly used to obtain the effective atomic number [9].Z eff is determined by the ratio of atomic or electronic cross sections in the direct method.Many writers have lately programmed this method to make computing Zeff for any compound/material in a desired energy range.Taylor et al, on the other hand, used the interpolation method to report Auto-Zeff soft quick determination of effective atomic numbers for any materials or compounds [14,15].
In this study, the mass attenuation coefficients, effective atomic numbers and electron densities of all the selected ferroelectric materials have been selected in the energy range of 1 keV to 100 GeV were evaluated.In the energy range of 1 keV to 20 MeV, another crucial photon interaction feature known as Kerma values has also been investigated.We also explain the significance of the WinXCom program's single values for effective atomic numbers and electron densities for all photon interactions [coherent, incoherent, photoelectric, pair generation, total photon interaction (with coherent and without coherent)].The acquired Zeff, and Nel values are useful in medicine, research and dosimetry.

Theoretical background
The methods for calculating the Ferroelectric materials effective atomic number, electron density and Kerma (BaCoF4, LiNbO3, KNbO3, BaTiO3, PbZrO3, PbTiO3, KNbO3, Bi4Ti3O12, KHPO4, PbBi2Nb2O9, SrBi2Ta2O9, Ba0.73Sr0.27TiO3and Mg3B7O13Cl) [16,17] are described in the following subsections.Using WinXCom [11,18] we computed mass attenuation coefficients and photon interaction cross sections in the energy range of 1 keV to 100 GeV.This programme uses the same underlying crosssectional database as Hubbell and Seltzer's well-known tabulation [19].WinXCom allows you to export cross-sectional data to a predetermined MS Excel template, which makes future graphical and numerical data analysis, much easier.

Mass attenuation coefficients
When a beam of γ -photons experiences attenuation via scattering (coherent or incoherent scattering) and absorption (photoelectric effect, pair and triplet production) when passing through a material of given thickness [20].The decrease in the intensity of photons is given by Beer-Lambert's law, I and I are the incident and attenuated photon intensities respectively, t is the thickness of the absorber (cm) and µ is the linear attenuation coefficient (cm -1 ).The total photon mass attenuation coefficient comp ) / ( ρ µ has been estimated by the following 'mixture rule' with WinXCOM program for a chemical compound or mixture [21], [22].
where i ω is the weight fraction of the i th constituent element present in the given compound and i ) / ( ρ µ is the photon mass attenuation.For a material composed of multi elements the fraction by weight is given by where i A is the atomic weight of the i th element and i n is the number of formula units of the i th element.

Total atomic cross section
The total cross section (σ ) and differential partial cross-sections are related by the relation where coh σ is the coherent and incoh σ is the incoherent scattering cross-sections, respectively.τ is the atomic photoelectric cross-section, κ is the positron electron pair production cross-section and n Ph, σ is the photonuclear cross-section [23].

Molecular cross section
The effective molecular cross section ( m σ ) is estimated using the values of mass attenuation coefficients comp ) / ( ρ µ by the following relation: where N is the Avogadro's number, i n and i A are the total number of atoms and atomic weight of the i th element in a molecule respectively [24].

Atomic and electronic cross-section
The effective atomic cross-section (σa) and effective molecular cross-section ( m σ ) are related by the following equation [25,26]: Similarly, electronic cross-section ( e σ ) is given by the following equation [27]: where and are the atomic number and fractional abundance of the constituent element

Effective atomic number
Effective atomic number (Zeff) is the ratio of the atomic and electronic cross-sections and it is given by The definition of the effective atomic number can be found in equation (8).It is assumed that the molecule's actual atoms can be substituted by an equal number of identical (average) atoms, each with Zeff electrons [28].Effective atomic numbers Zeff can also be calculated from the direct method of the low-Z materials for total photon interaction [29].The formula is given below: Where i f is molar fraction in the mixture/compound with linear attenuation coefficient µ and mass attenuation coefficient ( ρ µ / ), A is atomic weight, Z is atomic number.The ratio (A/Z), of the atomic mass and the atomic number is approximately constant.

Electron density
Electron density (Nel) is the number of electrons per unit mass and is closely related to the effective atomic number and it is given by the following equation [30] , ( )

Kerma relative to air
Kerma is the initial kinetic energy of all secondary charged particles liberated per unit mass by uncharged radiation at a site of interest (ICRU180, Attix 1986).It applies to photons (x-rays, gamma rays, bremsstrahlung, and neutrons) and uses the same measure as the absorbed dosage, J/kg = Gy.Photon fluence and the likelihood of an interacting medium are both closely related to Kerma [31] .To understand the relationship between mass and energy absorption coefficient ( Therefore, the kerma is the product of the mass energy absorption coefficient and the energy fluence.The kerma of ferroelectric compounds with respect to air can be calculated as ) Now the computation of Kerma is calculation of mass energy-absorption coefficient.

Results and Discussions
The variation of mass attenuation coefficient against optical energy of the compounds obtained by cross sectional data which were exported to MS Excel from.The molecular formula of crystalline materials is presented in Table 1.

Effective atomic number and electron density
The calculation of Zeff and Nel of ferroelectric materials were carried out using above mass attenuation coefficients in the energy region 1 keV-100 GeV.These data shows that both the parameters depend upon chemical composition of the given molecule or compound.The variations of Nel with photon energy in all the ferroelectric materials is same for partial and total interaction processes are similar to that of Zeff shown in Figures and can be explained on a similar manner as that of Zeff.The average values of Zeff and Nel are given in Table 2 and 3 for the selected ferroelectric materials.From Fig. 1 it can be easily seen that in the continuous energy range (1 keV-100 GeV) the Zeff and Nel are mainly dominated by different partial photon interaction processes [33][34][35][36].Energy below 0.3 MeV, Zeff has maximum values, where photoelectric interaction is dominant.Zeff drops to a lower value typically for Compton scattering, in the energy region 0.3 MeV-2 MeV.Compton scattering or incoherent scattering is the main interaction process between 2-40 MeV energy region in this intermediate energy range and Zeff is about constant and equal to mean (average) atomic number [37,38].In the transition region from 40 to 500 MeV, Zeff increases with increasing energy as pair production gradually becomes dominant.At high energies above 500 MeV, Zeff assumes an almost constant value determined by pair production.
The Zeff varies from a higher value at lower energies to a lower value at higher energies, with a peak due to photoelectric effect near the K-edge of the high Z element present in the ferroelectric materials, then becoming constant with a minimum value at intermediate energies; further, there is an increasing trend in Zeff values due to the relative dominance of photon interaction processes in various energy regions.The Z dependency of total atomic cross-sections explains all changes, leading to effective atomic numbers such as Z 4-5 for photoelectric absorption, Z for Compton scattering, and Z 2 for pair creation.
As a result of the above, the photoelectric absorption cross-section, which is proportional to Z 4-5 , gives larger weight to the high Z elements than the other processes.The greatest values of Zeff in the low energy zone are readily explained by this state.In contrast to photoelectric absorption and pair production processes, the Compton scattering cross-section is proportional to Z, giving high Z elements less weight [39].
Electron density is closely related to the effective atomic number, shown in Fig. 2. As a result, Nel qualitative energy dependence is similar to Zeff.Furthermore, within the dominance region of any of the three major photon interaction processes, the effective atomic number values remain constant, but the values of effective atomic number vary in the regions where the dominance region for one process shifts to the other.The presence of absorption edges of the constituent parts creates an exception in the lower energy zone [20].The Nel values where maximum upto 0.1 MeV, where photoelectric absorption is dominant.The values decrease from 0.1 MeV-1.5 MeV, where Compton scattering starts increasing attains minimum value between 1.5 MeV-18 MeV.Nel values gains minimum values between 18 MeV-500 MeV and becomes independent of energy above 500 MeV.In this energy region Pair production is the dominant interaction process.

Total photon interaction (with incoherent)
From Fig. 3, Zeff values have maximum between 0.03 MeV-1 MeV, where photoelectric absorption is dominant.From 1 MeV-14 MeV it gradually decreases attains minimum values in the range 14 MeV-6 GeV, where Compton scattering is dominant interaction process.Above that pair production is dominant interaction process.Similarly, in Fig. 4 the Nel values have maximum values between 0.01 MeV-1.5 MeV and after that gradually decreases and attains low values of Zeff in the range 18 MeV-6 GeV.Above these values Zeff values become independent of energy with small variation in energy.

Coherent scattering
The Fig. 5 shows the variation of Zeff with photon energy for coherent scattering.It is apparent from the figure that Zeff increases as energy increases from 1 keV to 0.005 MeV.After that it is independent of energy.There is drastic variation in Zeff values for BaCOF4, PbTiO3 and PbBi2Nb2O9 [40].For the same energy range these compounds show maxima and minima values and above 600 MeV they attain constant values.Similary, the variation of Nel with photon energy for coherent scattering is shown in Fig 6 which indicates that Nel increases as energy increases from 1 keV-0.003MeV and becomes independent of energy after that.

Incoherent scattering
The Fig. 7 shows the variation of Zeff with photon energy for coherent scattering.It is apparent from the figure that Zeff increases as energy increases from 1 keV to 100 MeV.After that it is independent of energy.There is drastic variation in Zeff values for ferroelectric materials from 1 keV-100 MeV due to elemental composition in them [40].For the same energy range these compounds show maxima and minima values and below 600 MeV they attain constant values.Similary, the variation of Nel with photon energy for incoherent scattering is shown in Fig 8 which indicates that Nel increases as energy increases from 1 keV-100 MeV and becomes independent of energy.

Photoelectric absorption
A plot of Zeff vs photon energy, especially for medium and high-Z materials, shows the distinctive absorption edges as the binding energy of each electron subshell is reached and a new pathway for photoexcitation is energetically enabled [17].Fig 9 demonstrate the fluctuation of Zeff with photon energy for photoelectric absorption, indicating that Zeff is nearly independent of photon energy.There is drastic variation in Zeff values for BaCOF4, PbTiO3 and PbBi2Nb2O9.This is because; photoelectric process is predominant at low energies (1 MeV) and for materials of higher atomic numbers [42].The Nel graph for photoelectric absorption process in Fig. 10 shows drastic variation for all the ferroelectric materials.

Pair production in nuclear field
Pair production in the nuclear field, the change of Zeff with photon energy is shown in Fig. 11, which demonstrates that Zeff increases significantly with increasing photon energy from 0.1 MeV to 10 MeV before becoming almost energy independent [41].It could be because pair production in the nuclear field is Z 2 dependent.Due to the inclusion of elements with a wide range of atomic numbers, the

Pair production in electric field
The Zeff and Nel values increases with increasing photon energies up and then it is independent of energy thereafter for all ferroelectric materials.Similary, the variation of Zeff and Nel with photon energy for electric field is shown in Fig. 13 and 14.

Kerma relative to air
The variation of Kerma (Ka) values of the molecules for photon energy 1 keV to 20 MeV are shown in Fig. 15.The average of Kerma values are given in Table 4.It is to be noted that the Ka values show sharp peak due to photoelectric absorption of the high-Z constituent of the compound.From Fig. 15 one can observe that there is gradual increase in Kerma with photon energy, attains maximum value at 0.05 MeV.After that gradually decreases upto 20 MeV.The relative significance of the partial photon interaction processes.

Conclusions
Study has been undertaken to get information of the mass attenuation coefficients, effective atomic number Zeff and electron density Nel of the selected ferroelectric materials in the energy range 1 keV to 100 GeV using WinXCom program and its underlying database of atomic photon interaction cross sections.Variation of Zeff and Nel can be distinguished in three energy regions as low, intermediate and high energies.The maximum value Zeff and Nel are found at low-energy region where photoelectric absorption is the main interaction process and minimum values at intermediate energies than higherenergy region.At low-enrgy region (E<0.
energy fluence of mono-energetic photon passing through area, A of an absorber is ] [ 2 − Jm ψ and is used to derive Kerma.Where Adx en ) (ψµ is the photon energy transmitted from uncharged ionizing gamma radiation to charge particles in a volume over a short distance dx behind the area, where en µ is mass energy-absorption coefficient.Because the mass in a volume with density ρ is equals Adx ρ , kerma equals

Figure 1 .
Figure 1.Plots of effective atomic number as a function of energy of ferroelectric materials (with coherent).

Figure 2 .
Figure 2. Plots of Electron density a function of energy of ferroelectric materials (with coherent).

Figure 3 .Figure 4 .
Figure 3. Plots of effective atomic number as a function of energy of ferroelectric materials (without coherent) Figure 4. Plots of Electron density a function of energy of ferroelectric materials (without coherent).

Figure 5 .
Figure 5. Plots of effective atomic number as a function of energy of ferroelectric materials (coherent).

Figure 6 .
Figure 6.Plots of Electron density a function of energy of ferroelectric materials (coherent).

Figure 7 .Figure 8 .
Figure 7. Plots of effective atomic number as a function of energy of ferroelectric materials (incoherent) Figure 8. Plots of Electron density a function of energy of ferroelectric materials (incoherent).

Figure 9 .
Figure 9. Plots of effective atomic number as a function of energy of ferroelectric materials (photoelectric absorption).

Figure 10 .
Figure 10.Plots of Electron density a function of energy of ferroelectric materials (photoelectric absorption)..
.1088/1757-899X/1300/1/012013 10 increase in Zeff with energy is more significant.Similarly, Nel values increases from 1 keV-5 MeV, after that they are independent of energy as shown in Fig.12.

Figure 11 .Figure 12 .
Figure 11.Plots of effective atomic number as a function of energy of ferroelectric materials (pair nuclear) Figure 12.Plots of Electron density a function of energy of ferroelectric materials (pair nuclear).

Figure 13 .
Figure 13.Plots of effective atomic number as a function of energy of ferroelectric materials (pair electron).

Figure 14 .
Figure 14.Plots of Electron density a function of energy of ferroelectric materials (pair electron).

Figure 15 .
Figure 15.Plots of Kerma with energy for ferroelectric materials in the energy range 1 keV-20 MeV.

3
MeV), photoelectric absorption is dominant process.At intermediate energies (2 MeV<E> 40 MeV), the Compton scattering is dominant process.At highenergy region (E<500Mev), pair production is dominant process.We have also calculated Zeff and Nel for individual photon interactions such as pair production in the nuclear and electric field, photoelectric absorption, coherent and incoherent scattering.Obtained value of Zeff and Nel depends on the chemical composition of the sample we use.Therefore the calculated values of effective atomic number and electron densities of three molecules are the same.Set of formulas which are used for calculating the Nel and Zeff in the present work can be used for any other materials.The kerma values shows peak due to photoelectric absorption at at 0.05 MeV.Calculated values of Nel and Zeff will of ferroelectric materials hopefully helpful in the field of medical imaging, thin film industry, supercapacitors, memories, sensors, and actuators.

Tabe 1
Name and molecular formula of ferroelectric materials.

Table 2 .
Average values of effective atomic number of ferroelectric materials.

Table 3 .
Average values of electron density x1024(electrons/g) of ferroelectric materials.

Table 4
Kerma values of ferroelectric materials for the energy range 1 keV-20 MeV.