A Study of the Sensitivity of Radiation Flaw Detection in Inhomogeneous Multicomponent Materials

V. A. VOROB'EV

(Presented to the scientific seminar of the Research Institute of Electronic Introscopy)

The introduction into industry of flaw detectors based on the use of penetrating radiation may proceed by creating unique specialized installations for high-speed inspection of products or materials of a single type. In this case the main requirements imposed on the installation will be high sensitivity, high productivity, maximum automation of inspection and process-control operations on the basis of current data obtained in the course of inspection, and high reliability of the entire system as a whole. Another path is directed toward the development and creation of universal flaw detectors intended for the inspection of unique products or parts manufactured at a single enterprise. The main requirements for such types of flaw detectors will be reliability, sensitivity, and universality.

By universality is meant the possibility of rapidly readjusting the flaw detector to another operating mode, to the inspection of another type of product or material. The main conditions that must be fulfilled when readjusting to a new mode are reproducibility of the operating characteristics of the installation in the new mode and their rapid and reliable determination. Determining the sensitivity of a flaw detector when changing the operating mode and the parameters of the inspected object, when these are determined experimentally, requires much time and must be carried out by experienced specialists. The use of the equipment of a flaw-detection laboratory for experimental determinations of operating characteristics and for selecting the optimum operating mode of an installation under new conditions requires additional sets of measuring apparatus, additional premises, and entails considerable expense. All this is eliminated if a theoretical calculation of the necessary parameters is applied, proceeding from experimental data obtained when inspecting another product, or from specially performed experiments on specimens made of specially selected materials that are taken as standards. To carry out such calculations it was necessary to develop a theory and to find an expression relating the sensitivity—which can be expressed as the minimum size of a detectable defect—to the parameters of the object under study and to the permissible limits of their variation, under the condition that the specified value of sensitivity is ensured.

Such an expression, based on the attenuation equation for the intensity of a narrow beam of penetrating radiation in a multicomponent material containing an inclusion with parameters differing from those of the object itself, can be obtained from the condition:

\[ I = I_0 e^{-\mu_{\mathrm{e}}(x_{\mathrm{e}}-\delta)-\mu_{\delta}\delta}. \tag{1} \]

The equivalent linear absorption coefficient of radiation by a multicomponent inhomogeneous medium may be written as

\[ \mu_{\mathrm{e}} = \frac{ \displaystyle \sum_{i=1}^{n} \alpha_i m_i }{ \displaystyle \sum_{i=1}^{n} \frac{\alpha_i}{\rho_i} }, \tag{2} \]

where \(\alpha\) is the weight content of each component per unit weight of the material under investigation, \(m_i\) and \(\rho_i\) are, respectively, the mass absorption coefficient of radiation by the substance of the component and the density of the \(i\)-th component.

In inhomogeneous multicomponent materials there is a large number of individual simple substances. This necessitates extensive calculations. The calculations are simplified if one introduces a grouping of components that are close in their properties or that enter into the composition of the material in a bound form. An example of such grouped components may be, for instance, gravel or crushed stone in concrete, which consists of many elements of the periodic table, but in calculations is considered as a single grouped component.

Writing the law of passage of radiation through an absorber for two objects and taking their ratio, we obtain

\[ \frac{I_1}{I_2} = \frac{I_{0_1}}{I_{0_2}} e^{ M_2\rho_2(x_2\varphi_2-\delta_2) - M_1\rho_1(x_1\varphi_1-\delta_1) - m_{\Delta_2}\rho_{\Delta_2}\delta_2 - m_{\Delta_1}\rho_{\Delta_1}\delta_1 }. \tag{3} \]

Expressing the increments of the quantities relative to their corresponding quantities, from [3] we obtain an expression relating the parameters of two multicomponent media, the radiation parameters, the sensitivity of the instruments, and the magnitude of the smallest detectable defects in two materials under inspection.

For the smallest detectable defect size,

\[ \frac{\Delta \delta_1}{\delta_1} > \eta \frac{ \displaystyle \frac{ \Delta \frac{I_1}{I_2} }{ \frac{I_1}{I_2} } + \frac{ \Delta \frac{I_{0_1}}{I_{0_2}} }{ \frac{I_{0_1}}{I_{0_2}} } + x_1\varphi_1\rho_1 M_1 \left[ \left( 1-\frac{\delta_1}{x_1\varphi_1} \right) \left( \frac{\Delta \rho_1(\rho)}{\rho_1} + \frac{\Delta \varphi_1(\alpha)}{\varphi_1} + \frac{\Delta M_1(m)}{M_1} + \frac{\Delta M_1(\alpha)}{M_1} \right) + \frac{\Delta x_1}{x_1} + \frac{\Delta \varphi_1}{\varphi_1} \right] }{ \displaystyle M_1\rho_1\delta_1 + m_{\Delta_1}\rho_{\Delta_1}\delta_1 } \]

\[ + \frac{ \displaystyle x_2\varphi_2 M_2\rho_2 \left[ \left( 1+\frac{\delta_2}{x_2\varphi_2} \right) \left( \frac{\Delta \rho_2(\rho)}{\rho_2} + \frac{\Delta \varphi_2(\alpha)}{\varphi_2} + \frac{\Delta M_2(m)}{M_2} + \frac{\Delta M_2(\alpha)}{M_2} \right) + \frac{\Delta x_2}{x_2} + \frac{\Delta \varphi_2}{\varphi_2} + \frac{\Delta \delta_2}{x_2\varphi_2} \right] }{ \displaystyle M_1\rho_1\delta_1 + m_{\Delta_1}\rho_{\Delta_1}\delta_1 } \]

\[ + \frac{ \displaystyle m_{\Delta_1}\rho_{\Delta_1}\delta_1 \left( \frac{\Delta m_{\Delta_1}}{m_{\Delta_1}} + \frac{\Delta \rho_{\Delta_1}}{\rho_{\Delta_1}} \right) + m_{\Delta_2}\rho_{\Delta_2}\delta_2 \left( \frac{\Delta m_{\Delta_2}}{m_{\Delta_2}} + \frac{\Delta \rho_{\Delta_2}}{\rho_{\Delta_2}} + \frac{\Delta \delta_2}{\delta_2} \right) }{ \displaystyle M_1\rho_1\delta_1 + m_{\Delta_1}\rho_{\Delta_1}\delta_1 }. \tag{4} \]

where

\[ \Delta \frac{\dfrac{I_1}{I_2}}{\dfrac{I_1}{I_2}} \]

— is the relative error introduced into the measurements by the inaccuracy in determining the ratio of the intensities of the radiation fluxes and by the change of this ratio during the inspection process as a result of a change in the intensity of one of the fluxes; otherwise, it may be characterized as a change in the ratio of the relative sensitivities of the recording instruments.

\[ \Delta \frac{\dfrac{I_{0_1}}{I_{0_2}}}{\dfrac{I_{0_1}}{I_{0_2}}} \]

— by carrying out analogous reasoning, is defined as the relative instability of the source intensity during the inspection of the object under study;

\(x_1\) and \(x_2\) — are the thicknesses of the first and second objects, respectively;

\[ \frac{\Delta x_1}{x_1} \quad \text{and} \quad \frac{\Delta x_2}{x_2} \]

— are the accuracy of measuring the thickness and the accuracy of maintaining the constancy of the radiographed thickness over the entire inspection region for the first and second objects, respectively;

\(\varphi_1\) and \(\varphi_2\) — are the filling coefficients of the first and second materials, respectively;

\[ \frac{\Delta \varphi_1}{\varphi_1} \quad \text{and} \quad \frac{\Delta \varphi_2}{\varphi_2} \]

— are the accuracy of measuring the filling coefficients and the accuracy of maintaining the constancy of the calculated porosity value over the entire inspected region of the first and second materials;

\(\alpha_1\) and \(\alpha_2\) — are the weight concentrations of the group components entering into the material of the first and second objects, respectively;

\[ \frac{\Delta \alpha_1}{\alpha_1} \quad \text{and} \quad \frac{\Delta \alpha_2}{\alpha_2} \]

— are the accuracy of measuring the concentrations of the individual group components in the material and the accuracy of maintaining the constancy of the concentrations throughout the entire inspected region of the first and second objects, respectively;

\[ \mu_1 = M_1 \rho_1 \]

and

\[ \mu_2 = M_2 \rho_2 \]

— are the equivalent linear attenuation coefficients of radiation by the material of the first and second objects, respectively;

\(M_1\) and \(M_2\) — are the equivalent mass attenuation coefficients of radiation by the material of the first and second objects, respectively;

\(\rho_1\) and \(\rho_2\) — are the equivalent densities of the materials of the first and second objects, respectively;

\(\Delta \rho_1(\alpha)\) and \(\Delta \rho_2(\alpha)\) — are the changes in the equivalent density of the materials of the first and second objects, respectively, due to changes in the concentrations of the individual group components in the material;

\[ \frac{\Delta \rho_1(\alpha)}{\rho_1} \quad \text{and} \quad \frac{\Delta \rho_2(\alpha)}{\rho_2}, \]

where

\[ \frac{\Delta \rho(\alpha)}{\rho} = \frac{ \displaystyle \sum_{i=1}^{n} \frac{\alpha_i}{\rho_i}\cdot \frac{\Delta \alpha_i}{\alpha_i} }{ \displaystyle \sum_{i=1}^{n} \frac{\alpha_i}{\rho_i} } \]

— the accuracy of determining and the accuracy of maintaining constant the value of the equivalent density of the material throughout the entire inspected volume of the first and second objects, respectively, taking into account only the change in the concentrations of individual group components;

\[ \Delta \rho_1(\rho) \quad \text{and} \quad \Delta \rho_2(\rho) \]

— the change in the equivalent density of individual group components;

\[ \frac{\Delta \rho_1(\rho)}{\rho_1} \quad \text{and} \quad \frac{\Delta \rho_2(\rho)}{\rho_2}, \]

\[ \frac{\Delta \rho(\rho)}{\rho} = \frac{ \displaystyle \sum_{i=1}^{n} \frac{\alpha_i}{\rho_i}\cdot \frac{\Delta \rho_i}{\rho_i} }{ \displaystyle \sum_{i=1}^{n} \frac{\alpha_i}{\rho_i} } \]

— the accuracy of determining and the accuracy of maintaining constant the value of the equivalent density of the materials of the first and second objects, respectively, when taking into account exclusively the change in equivalent density due to the change in the density of individual group components;

\[ \Delta M_1(\alpha) \quad \text{and} \quad \Delta M_2(\alpha), \]

— the change in the equivalent mass absorption coefficient of radiation by the material of the first and second objects, respectively, due to changes in the concentrations of individual group components in the materials;

\[ \frac{\Delta M_1(\alpha)}{M_1} \quad \text{and} \quad \frac{\Delta M_2(\alpha)}{M_2}, \]

where

\[ \frac{\Delta M(\alpha)}{M} = \frac{ \displaystyle \sum_{i=1}^{n} \alpha_i m_i \frac{\Delta \alpha_i}{\alpha_i} }{ \displaystyle \sum_{i=1}^{n} \alpha_i m_i } \]

— the accuracy of determining and the accuracy of maintaining constant the value of the equivalent mass absorption coefficient of radiation by the materials of the first and second objects, respectively, when taking into account exclusively the change in the magnitude of the absorption coefficients due to changes in the concentrations of individual group components in the materials under study;

\[ \Delta M_1(m) \quad \text{and} \quad \Delta M_2(m) \]

— the change in the equivalent mass absorption coefficient of radiation by the materials of the first and second objects, respectively, due to changes in the mass coefficients of radiation absorption by the individual group components, the materials;

\[ \frac{\Delta M_1(m)}{M_1} \quad \text{and} \quad \frac{\Delta M_2(m)}{M_2}, \]

where

\[ \frac{\Delta M(m)}{M} = \frac{ \displaystyle \sum_{i=1}^{n} \alpha_i m_i \frac{\Delta m_i}{m_i} }{ \displaystyle \sum_{i=1}^{n} \alpha_i m_i } \]

— the accuracy of determining and the accuracy of maintaining constant the value of the equivalent mass absorption coefficient of radiation by the material of the first and second objects, respectively, when taking into account exclusively the change in the magnitude of the equivalent absorption coefficient due to changes in the mass coefficients of radiation absorption of the individual group components of the materials under consideration;

\(m_{\Delta_1}\) and \(m_{\Delta_2}\) — mass attenuation coefficients of radiation by the substance filling the cavities of defects in the first and second objects, respectively;

\(\dfrac{\Delta m_{\Delta_1}}{m_{\Delta_1}}\) and \(\dfrac{\Delta m_{\Delta_2}}{m_{\Delta_2}}\) — accuracy of measurement or accuracy of the theoretical calculation of the values of the mass attenuation coefficient of radiation by the material filling the cavities of defects in the first and second objects, respectively;

\(\rho_{\Delta_1}\) and \(\rho_{\Delta_2}\) — densities of the substance filling the cavities of defects in the first and second objects, respectively;

\(\dfrac{\Delta \rho_{\Delta_1}}{\rho_{\Delta_1}}\) and \(\dfrac{\Delta \rho_{\Delta_2}}{\rho_{\Delta_2}}\) — accuracy of measurement or theoretical calculation of the density of the substance filling the cavities of defects in the first and second objects, respectively;

\(\delta_1\) and \(\delta_2\) — linear dimensions of defects in the direction of radiation propagation in the first and second objects, respectively (reference defect dimensions);

\(\Delta\delta_1\) and \(\Delta\delta_2\) — change in the linear dimension of the defect in the direction of radiation propagation, recorded by the detecting instrument (the smallest detectable defect size when calculating defect detectability);

\(\dfrac{\Delta\delta_1}{\delta_1}\) and \(\dfrac{\Delta\delta_2}{\delta_2}\) — ratio of the smallest detectable defect size to the reference size when calculating defect detectability. The accuracy of determining the defect values in the first and second objects, respectively, when studying the conditions for modeling one object by another;

\(x_1\varphi_1\) and \(x_2\varphi_2\) — thickness of the material of the first and second objects, respectively, i.e., the thickness of the object in the direction of radiation propagation, excluding the total thickness of air in the pores in a multicomponent inhomogeneous material;

\(\dfrac{\delta_1}{x_1\varphi_1}\) and \(\dfrac{\delta_2}{x_2\varphi_2}\) — ratio of the linear dimension of the reference defect in the direction of radiation propagation to the thickness of the absorber material for the first and second objects, respectively;

\(\dfrac{\Delta\delta_1}{x_1\varphi_1}\) and \(\dfrac{\Delta\delta_2}{x_2\varphi_2}\) — ratio of the linear dimension of the smallest detectable defect size in the direction of propagation to the thickness of the absorber material.

Note: The quantity \(\Delta\delta\) characterizes the size of the smallest detectable defect size for the given values of the parameters entering into the equation. The condition \(\Delta\delta > \delta\) means that the specified reference defect size \(\delta\), under the given conditions and given parameters, is not recorded by the detecting instrument.

From (4), any of the terms may be determined, provided that the others are specified or determined theoretically or experimentally.

When electromagnetic radiation passes through the object being inspected, as a result of the processes of interaction of the radiation with the substance, along with absorption and scattering of the primary radiation, secondary radiation accumulates. This increase in the secondary radiation fluxes with the thickness of the absorber, being superposed on the flux of primary radiation, which is the carrier of information about the defect, reduces sensitivity. The deterioration of detectability due to the accumulation of secondary radiation can be taken into account through the buildup factor and will be proportional to the increase in the buildup factor in passing from the reference object to the object under study, i.e.,

\[ \Delta \delta \sim \frac{B_1}{B_2}. \]

Taking into account that \(B_2 = B_1 + \Delta B\), we can write

\[ \Delta \delta \sim 1 + \frac{\Delta B}{B}. \]

In analyzing the performance characteristics of flaw detectors, when inspecting materials with different thicknesses and effective atomic numbers, the influence of the buildup factor on sensitivity can be taken into account by means of the expression

\[ \frac{\Delta \delta}{\delta} = \eta \frac{\Delta \delta_1}{\delta_1} \left( 1 + \frac{\Delta B}{B} \right), \tag{5} \]

where \(\dfrac{\Delta \delta_1}{\delta}\) is calculated from equation (4),

\(\eta\) is the modeling coefficient,

\(\dfrac{\Delta \delta_2}{\delta}\) is the change in the magnitude of the buildup factor in passing from the reference material to the material under study.

The above-considered method of theoretical investigation of the sensitivity of radiation flaw detection makes it possible to take into account the influence of the parameters entering into (4) and (5) on detectability and to calculate the expected sensitivity when radiographing objects with known parameters.