## 1 Introduction
### Correlative Theory  
The technique of X-ray transmission is widely used in such domains as medicine, security inspection, industry, etc. When an ideal narrow beam, single energy and monochromatic X-ray penetrates through a material with thickness \(t\), its energy intensity is attenuated exponentially in accordance with the Beer\(-\)Lambert Law [19] as follows:  
\[I=I_{0}e^{-\mu t}=I_{0}e^{-\mu_{m}t} \tag{1}\]  
where \(I_{0}\) (keV) and \(I\) (keV) are, respectively, the measured values of X-ray energy intensity before and after penetration, \(\mu\) (cm\({}^{-1}\)) is the linear attenuation coefficient, \(t\) (cm) is the material thickness, \(\mu_{m}\) (cm\({}^{2}\)/g) is the mass attenuation coefficient, \(\rho\) (g/cm\({}^{3}\)) is the material density, and \(\rho t\) (g/cm\({}^{2}\)) is also called mass thickness. From Equation (1), it can be concluded that:  
\[\ln(I_{0}/I)=\mu t=\sigma(Z,E)\frac{N_{A}}{A}\rho t \tag{2}\]  
where \(\sigma\) (cm\({}^{2}\)/atom) is cross-section, \(N_{A}\) (atom/mole) is Avogadro constant, and \(A\) (g/mole) is relative atomic mass. The cross-section \(\sigma\) is a function of atomic number \(Z\) and energy \(E\)[27]. For dual-energy monochromatic X-ray transmission, it can be obtained from Equation (2) that:  
\[\left\{ \begin{array}{ll}\ln(I_{10}/I_{1})=\mu_{1}t=\sigma(Z,E_{1})\frac{N_{ A}}{A}\rho t\\ \ln(I_{20}/I_{2})=\mu_{2}t=\sigma(Z,E_{2})\frac{N_{A}}{A}\rho t\end{array} \right. \tag{3}\]  
The R_value of dual-energy monochromatic XRT is defined as follows:  
\[R=\frac{\ln(I_{1}/I_{01})}{\ln(I_{2}/I_{02})}=\frac{\ln(I_{01}/I_{1})}{\ln(I_ {02}/I_{2})}=\frac{\mu_{1}}{\mu_{2}}=\frac{\sigma(Z,E_{1})}{\sigma(Z,E_{2})} \tag{4}\]  
where \(I_{01}\) and \(I_{1}\) are, respectively, the measured values of the energy intensity of monochromatic X-ray before and after attenuation of energy \(E_{1}\), \(I_{02}\) and \(I_{2}\) are, respectively, the measured values of the energy intensity of monochromatic X-ray before and after attenuation of energy \(E_{2}\). The two ratios of \(I_{1}/I_{01}\) and \(I_{2}/I_{02}\) can be obtained by measurement. \(E_{1}\) and \(E_{2}\) are the energy of known constant monochromatic X-ray. The R_value is the ratio of the attenuation cross-section of two kinds of energy, which changes monotonously with \(Z\). From Equation (4), it can be seen that the R_value has nothing to do with the mass density and thickness of the material. Therefore, the atomic number \(Z\) of materials can be identified by R_value, so as to identify the material types. For continuous spectrum X-rays that pass through a homogeneous material, using an energy-integrating detector, the value of the transmitted signal after passing through the object is:  
\[T=\int_{0}^{E_{in}}N(E)P_{d}(E)Ee^{\mu(E)t}dE \tag{5}\]  
where \(E\) is the energy of X-ray photons, \(E_{in}\) is the highest energy of input continuous spectrum,  
Similarly, the R_value of continuous spectrum DE-XRT is defined as:  
\[R=\frac{\ln(T_{L}/T_{L0})}{\ln(T_{H}/T_{H0})} \tag{7}\]  
where \(T_{L0}\) and \(T_{H0}\) are, respectively, the measured values of low-energy and high-energy integral of transmission signal before attenuation. \(T_{L}/T_{L0}\) and \(T_{H}/T_{H0}\) are defined as low-energy and high-energy transparency, respectively. The following R_values are defined by Equation (7). Because of the influence of integration, both the coefficient of integration term and the linear attenuation coefficient in exponential term in Equation (6) are functions of energy \(E\) of integration variable, so the calculation of the R_value in Equation (7) cannot remove the influence of thickness \(t\), that is, the R_value is not only related to the atomic number \(Z\), relative atomic mass \(A\) and density \(\rho\), but also related to the thickness \(t\) of the material. Therefore, DE-XRT method cannot directly identify the atomic number and the material according to the R_value calculated by Equation (7), and the influence of thickness needs to be removed.