2.1. Computation Z_eff and Z_eff via the transmission mode

The effective atomic number for any sample is given by

where σ_t-a and σ_t-e are the total effective atomic and total effective electronic cross-section, respectively. One can find the formulation in detail in Ref.[21].

The effective electron density (in electrons·g⁻¹) of the sample can be computed by Eq.(2)

where, where N_A is the Avogadro’s constant; the total number, n, of atoms in the molecule; N_i and A_i are respectively the total number of atoms and the atomic mass of the i^th elemental composition of the material; and ‹A› is the average atomic mass of the sample. Therefore, using the Z_eff value obtained from logarithmic interpolation in Eq. (1), one can calculate the values of N_eff by using Eq. (2). The equations reveal that N_eff and Z_eff can be used interchangeably. The most important difference between them is the value of atomic weights that make up the compounds.

2.2. Computation of Z_eff and N_eff via the elastic mode

If the photon energy and scattering angle are high enough to produce all interatomic interference phenomena negligible, the photon energy is not too high to produce the addition of pair production and nuclear-scattering interactions necessary. This roughly means that sin(θ/2)/λ₀ ≥ 20 nm⁻¹, where λ₀ is the wavelength the primary photons, and E₀ < 1 MeV. In this case elastic scattering can be viewed as a scattering through free atoms whatever configuration of the sample, and the differential cross-section of the Rayleigh scattering is given by [18].

where index a stands for the whole atom, index e stands for a unique electron, index T refers to the Thomson differential cross-sections, which are given as

where r₀ is the classical electron radius, and θ is the scattering angle.

Similarly, the atomic differential cross-sections for Compton process for a bounded electron can be written as [22, 23].

where “K-N” stands for the Klein-Nishina cross-section and S(x, Z) is the inelastic scattering function:

where ω₁ and ω₂⁰ are the angular frequencies of the incident and scattered photons, respectively.

From Eqs. (3) and (5), the elastic-to-inelastic cross-section ratio is then, for one atom, given by

therefore the F²/S factor

The multiplication of (d_eσ/dΩ)_K-N and (d_eσ/dΩ)_T by S(x,Z) and the atomic form factor F(x,Z), respectively allows us to include the effect of electron binding into the accounts. The values of S(x,Z) and F(x,Z) functions are extending from zero to Z. Also, they depend on the momentum transfer parameter x (in Å⁻¹);

where λ₀ and E₀ are the wavelength (in Å⁻¹) and energy (in eV) of the primary photons, respectively. The number of atoms scattered to the detector is given in Eq.(9). It takes into account the experimental set-up, photon flux of the source, and atomic density of the sample η_a=N_Aρ/M_A; where ρ is the target density and M_A is the atomic mass [20].

where d_aσ/dΩ is the atomic differential cross-section, the probability of a photon to be scattered at an angle θ by one atom of the material, within an elementary solid angle dΩ; N₀ is the initial fluency; A is the self-attenuation factor for photons; V is the scattered volume; and ΔΩ is the solid angle.

Using Eqs. (4) and (5) in Eq. (9), the recorded photon numbers for Rayleigh (N_R) and Compton (N_C) by a detector of efficiency ε can be expressed as:

where, A_R=exp[−μ(E₀)L_i+μ(E₀)L_s] and A_C=exp[−μ(E₀)L_i+μ(E_C)L_s] are self-attenuation factors for the Rayleigh and Compton, respectively, with E_C being energy of the photons after a Compton scattering, μ is the linear attenuation coefficient, and L_i and L_s are the path lengths through the sample, along the incident and scattered beams, respectively. The self-attenuation correction factor can be estimated by e^{[μ(EC)−μ(E0)]Ls}

The Rayleigh N_R to Compton photons N_C ratio, R, can be written as the ratio of the net peak areas under the elastic and inelastic peaks, corrected for detector efficiency and pulse losses

where C holds for the ratio C = (d_eσ/dΩ)_T/(d_eσ/dΩ)_K-N.

Thus, for a specific experimental condition characterized by x

For some experimental conditions (E₀,θ), in which E₀ and E_C are roughly equal: this is true for a relatively low scattering angle θ; Eqs. (13) and (14) can be drawn without making self-attenuation corrections [A_R≈A_C]. Also, R_x and Z can be determined by the following relations:

where f _x^D(F²/S) is a discrete function at certain momentum transfer parameter (x), which gives Z as a function of F²/S factor. The points of the discrete function R = f _x^D(Z) for the elements can be calculated by Eq. (15) from the knowledge of F(x,Z) and S(x,Z) using suitable tabulations such as those in Ref. [22].

Figs. 1 shows the discrete points of the functions R_1.5 = f _1.5^D(Z_eff) and Z_1.5 = f _1.5^D(F²/S), obtained by theoretical values of F²/S, for elements of atomic number 1 to 100. Eq. (15) is the basis for the atomic number measurement method by the Rayleigh to Compton scattering ratio, where the R ratio is independent of the attenuation in the sample. It should be stressed that as long as ω₂⁰ describes the average photon energy of Compton scattered photons, Eq. (15) can be successfully applied [18, 24].

Fig.1.Full-screen View

The variation of Rayleigh to Compton ratio R_1.5 as a function of Z_eff (a) and F²/S (b), for x=1.5 Å⁻¹ calculated by ZXCOM.

For compounds and homogenous mixtures containing various elements, we can generalize Eqs. (15) and (16) as follows,

where α_i^at is defined by the mass percentage ω_i and the atomic mass M_i of the i^th element:

Here the function R_.=f_x(Z_eff) is a continuous function, which can be deduced by fitting the discrete f_x^D function. The last relationships enable us to determine theoretical values for R_x and Z_eff from the calculated values of (F²/S)_eff for any material under consideration [19]. Thus, for any element, R_x can be calculated via Eq. (15) at a certain experimental condition. Hence these values can be referred to as “point-wise data”, which can be plotted as continuous functions to determine R_x and (Z_eff)_x for any material with an adequate interpolation method. Under correct experimental conditions, we expect that a good agreement can be achieved between the measured and calculated values.

Practically, details of right experimental compromise for E₀ and θ values were given by many researchers. For example, Duvauchelle et al., showed that E₀ and θ values should be carefully chosen when (i) neglecting the correction factors mentioned in Eq. (12), (ii) obtaining a good counting statics for both Rayleigh and Compton components, and (iii) achieving a good separation between them. They considered that, for example, the choice of photon energy of 59.53 keV and a scattering angle of 35° leads to an acceptable compromise between parameters influencing the measurement [19]. Finally, the effective electron density at certain momentum transfer parameter (x), (N_eff)_x of the sample can be calculated by Eq. (2).

We compared the calculated Z_eff with the Z_eff for some materials measured under different conditions.

2.3. ZXCOM: Windows-based program for Calculating Z_eff

Based on the above theoretical treatment a computer program called ZXCOM has been constructed. It is a MS Windows based program to calculate Z_eff and N_eff using R ratio for any element, compound or mixture, of θ =1°–180° and photon energy of E₀= 1 keV–1 MeV.

2.4 Database

A database file for F and S for 100 elements over the studied energy range, has been compiled from the tabulation in Ref.[23]. The R and Z_eff are calculated from F²/S factors which depend on the momentum transfer parameter, x(E₀,θ), as mentioned in the previous section. The continuous functions of R and Z_eff for any materials are obtained by interpolation of the discrete values of F²/S for the 100 elements.

2.5 Input and output

ZXCOM has a graphical user interface (GUI). The GUI facilitates define and redefine the elements, compounds, and mixtures. In addition, each variable can be redefined within the ongoing run. The main output calculations of ZXCOM are the variation of R_. with incident photon energy E₀ (in keV), at a certain scattering angle θ in degrees, and the variation of R_. with θ at constant E₀. The calculated data are listed in “datagridview object” to a predefined MS Excel template or simple data file. Graphics of the data can be drawn. The execute version of this program can be downloaded via http://photon.yildiz.edu.tr/zxcom.php and http://photon.gelisim.edu.tr/zxcom.php.

3.1. Comparison with values determined by R measurement

Z_eff measured using the elastic to inelastic scattering ratio were collected from Refs. [25]–[29]. Tables 1–3 provide comparisons with experimental results. Tables 1 list the calculated and measured Z_eff for low Z-materials (Z<10) at experimental conditions of (i) E₀=17.44 keV, θ = 90°, x=0.99Å⁻¹ and ΔE=0.58 keV, and (ii) E₀=59.54 keV, θ = 60°, x=2.4Å⁻¹ and ΔE=3.28 keV

Table 1 includes columns of corrected ZXCOM results for self-attenuation corrections of primary and scattered photons. The corrections were calculated with Eq. (12) using attenuation coefficients tabulated in Ref.[32]. The corrections improve the agreement between measured and calculated values, as shown in Table 1. Experimentally, self-attenuation correction factor is dependent on the relative transmission of radiation at various energies as a ratio of photo-peak count rate passing through the reference standard and other materials to that through the air or water of the same geometry [30].

A closer study of Table 1 shows a considerable agreement (differences up to 9.6%) between the calculated and measured Z_eff under experimental conditions of (i), and a good agreement, too, under experimental conditions of (ii).

Generally, the theoretical and experimental results agree with each other better for incident photon energies of >20 keV (see the example of E₀ =17.44 and E₀ =17.4459.54 keV in Table 1). This is a natural result for employing photons in an energy range of scattering dominance. Since the relative interaction probabilities for elastic scattering and photoelectric process are roughly comparable for energies of >20 keV in a matter with atomic numbers of <10 (similar to our studied materials) [17]. Therefore, when determining the Z_eff for low-Z materials by the R ratio measurement, the photon energy should not be too low (>20 keV) in order to depart the photoelectric dominance energy region. If the measurement are performed at lower energies, where the scattering contributions are minor, the R determination becomes a difficult problem, with high uncertainties, especially for low intensity radioactive sources. So the primary photon energy of 59.54 keV is more convenient for scattering measurements, with more reliable results than those of photon energies below 20 keV (Table 1).

Also, we noticed that the deviations between theoretical and experimental results in Tables 1 are always negative, which means that the experimental results are constantly lower than that calculated (as Dev. = (Exp.− Theo.)/ Exp.×100%), this can be attributed to overestimate of Compton scattering contribution as discussed in the following statements.

The origin of differences between theoretical and experimental results comes from the fact that ω₂⁰, in Eq. (6), corresponds to the radiation energy after scattering by a stationary free electron. But the electrons in a real matter are neither stationary nor free. The electron velocities cause the broadening in the energy distribution of scattered photons [so-called “Compton profile”]. As long as ω₂⁰ describes the average energy of the Compton profile, then the general Eq. (7) can be successfully applied for estimating the elastic-to-inelastic cross-section ratio, the (F²/S) factor and Z_eff. In fact, if the binding energy of a particular electron shell, especially K- or L- shells, is of the order of the energy difference of (ω₁−ω₂⁰), ω₂⁰ is no longer the average photon energy after scattering from those electrons [18]. The actual ω₂ (average) is smaller and therefore (d_eσ/dΩ)_K−N overestimates the Compton scattering contribution and, as a result, the obtained values of F²/S and Z_eff are overestimated, as shown in Eqs. (7-b) and (18), respectively. The (d_eσ/dΩ)_K-N overestimate is due to the factor (ω₂⁰/ω₁)² in Eq. (6). As shown in Table 1, the energy difference ω₁−ω₂⁰ (about 0.58 keV) is of the order of the K-shell binding energy of the oxygen (about 0.55 keV), ω₂⁰ is no longer the average photon energy after scattering from oxygen K-shell electrons. So Eq. (18) assigns too much value to the calculated Z_eff. İçelli and Erzeneoğlu showed that the differential cross section ratios decreased with increasing scattering angle [31]. The Z_eff values at 60°are higher than those calculated at 90° (Table 1).

While Z increases, the contributions of K- and L-shell electrons, in Compton scattering, become negligible and this error does not provide any observable impact on the cross-section ratio, hence no overestimate for calculated values of R or Z and no negative deviation between the calculated and measured values (Table 2). The negative deviation value of Lu₂O₃ results from the effect of anomalous scattering. As the energy of the incident radiation (about 60 keV) is close to the to the absorption edge of electron K-shell of Lu (63 keV), the true atomic scattering factor deviates from that employed in ZXCOM program, which is taken from Ref. [32]. Finally, from Table 2, it was noticed that the deviation is significantly lowered by lowering the scattering angle. Also, Table 3 lists the experimental [27] and the ZXCOM results for two alloys. A good agreement is achieved.

Another source of discrepancies arises between the calculation and measurement results because of the electron charge distribution in the molecule due to the effect of binding between different types of atoms. This distortion leads to inaccurate Rayleigh cross-section estimation as a result of limited range of validity of the form factor approximation, which was originally derived to correct the Thomson formula for scattering by a charge distribution rather than a point charge. Specifically, the form factor approximation is not valid for momentum transfers that are higher than the electron binding and therefore the independent atomic model, in which each atom in a material scatters independently of the others, cannot be successfully applied. The effects of interatomic bonding within a molecule, especially for large scattering, should be considered [33].

The deviations may be also originated by experimental errors such the counting statistics errors at the lower counting rates, and the error in designating the scattering angle. The overlap between the signals of coherently and incoherently scattered photons will result in a wrong number of counts, especially at small angles.

3.2. Comparison with experimental values determined by transmission technique

Table 4 shows the experimental Z_eff,T values deduced from transmission technique based on measuring the total attenuation coefficients [34], and ZXCOM results (Z_eff) for low Z-materials. One can see that the Z_eff,T and Z_eff for the most majority of listed materials are in good agreement. However, this agreement is not upon any physical principles, since the Z_eff,T and Z_eff are independent parameters used for characterizing the material uniquely for different photon interaction processes of attenuation and scattering [26]. Moreover, in some cases large deviations were noted, for example, up to 55% for CaO₆C₆H₁₀ and 47% for NaO₂C₂H₃. Del Lama et al. reported a similar large deviation between Z_eff,T and Z_eff values [26]. Therefore these parameters are not directly related. Consequently, the ZXCOM predictions are principally limited to the scattering mode of measurement.