International Symposium (NDT-CE 2003)Non-Destructive Testing in Civil Engineering 2003
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Theoretical foundations for reconstruction of the materials density by means of two-energy radiographySergey Naydenov, Institute of Single Crystals of UAS, 60 Lenin avenue, Kharkov, 61001, Ukraine
Vladimir Ryzhikov, Institute of Single Crystals of UAS, Kharkov, Ukraine
In non-destructive testing of building constructions, concrete structures, granular and powder materials, mixtures, alloys, etc. it is often very important to determine their mass and/or surface density. Density decreases or increases with respect to the standard values can substantially lower the quality of functional materials, as well strength of joints and constructions. In this paper, we have shown theoretically that the most complete solution of the inverse problem of quantitative reconstruction of the density profile can be given by means of two-energy radiography. A general expressions has been obtained for the surface density of an arbitrary material. Theoretically, the accuracy of the method is within several percent. An important point is that the approach proposed does not require any preliminary knowledge about physical or chemical composition of the tested object. This allows the use of the developed algorithm for universal testing of various construction objects in building and industry, including situations of prolonged service, when the chemical structure can get substantially changed. Among possible applications of the new method there are control testing in building and building industry, railway and vehicle transport, diagnostics of oil and gas pipes, hydrotechnical constructions (dams), etc.
Control of the density of materials is important for many spheres of civil construction. In many applications, it is related to production and quality check-up of various construction items, ensuring reliability of their functioning and determination of their depreciation degree. Changes in density can reflect the aging of concrete constructions, armature and pipeline corrosion, strength loss of welded seams, etc. The most commonly used methods of density control are ultrasound or X-ray methods of non-destructive testing (see., e.g., [1-3]).
Conventional methods of density control assume the chemical composition of the material to be known. In other words, from general, model or specific practical considerations, the linear attenuation coefficient m(E) of ionizing radiation (with fixed or averaged energy (E) is being assumed for the tested object in one or another form. Thus, in the conventional radiography it is difficult to reconstruct the density profile for objects of variable chemical composition, as well as in other complex objects (e.g., composites, multi-component objects and multi-component welded seams, in constructions with smooth or abrupt density changes, etc. In many cases, nothing at all is known about internal details of the tested object or its atomic and chemical structure. Moreover, even for a known material this structure can be changed during the operation lifetime of the object, e.g., metal corrosion, concrete degradation, destruction of outer layers or coverings by aggressive media, etc. In all these cases, the object is characterized by unknown or variable density, which can be static (variations in space) and/or dynamic (variations in time).
It appears that the most comprehensive way for reconstruction of the materials density, which embraces all the above-described cases, is the method of multi-energy radiography . In this paper, using a simplified physical model, we show that for full reconstruction of the surface density s (bulk density r, multiplied by thickness D of the inspected object in the direction of ray propagation, s = rD), the use of two-energy radiography is sufficient. Appropriate expressions have been obtained for s as function of only radiographic measurements - of the tested object and calibration data.
2. Standard model
Physical foundations of radiography are based on absorption of ionizing radiation (photons) in the substance. Absorption of particles or radiation energy occurs both in the tested object and in the detector. Let us limit ourselves to the monochromatic case, E=const. Effects of non-monochromaticity can be accounted for separately. Let us present the signal V(E) recorded by the detector in the form
where h is the full conversion efficiency of the receiving-detecting circuit; V0(E) - intensity of the source (in the chosen energy units). The linear attenuation coefficient m(E;Z,r) in the object depends upon characteristic radiation energy E and parameters of the physical structure of the material - effective atomic number Zeff and density r. The conversion efficiency h is determined by specific features of the detector design and is also dependent upon energy E. We assume that the radiation energy spectrum is not changed after having passed through the object. This approximation is related to the quantum character of interaction processes of X-ray and gamma-radiation with the matter (substance), meaning spectral elasticity of the transmitted radiation, with multiple scattering neglected. Then h will be practically independent of the properties of the object. This value can be determined in the background mode (i.e., in the absence of an object) or from testing experiments. The loss of spectral elasticity, as well as non-monochromatic character of the radiation used, lower the accuracy of the monitoring.
In digital radiography, a detected signal is finally presented in the digital form. Consequently, it can be transformed to any form and expressed in any (relative) measurement units. It is practically convenient to use the logarithmic scale and to introduce an arbitrary dimensionless signal, i.e., radiography reflex R(E)
It should be noted that in all cases R(E)³0. Let us consider the medium energy range, 0.02MeV£E£0.5MeV (this radiation range is the most widely used in applications). Two independent absorption mechanisms are predominant - photo effect and Compton scattering. Starting from the K-absorption threshold in the given material, the attenuation coefficients are monotonously changed with the radiation energy increase. For these coefficients, their dependences on the effective atomic number are substantially different. The full linear attenuation coefficient can be presented as
where a(E) and b(E) are energy dependences of the absorption cross-section for the photo effect and Compton scattering, respectively.
Combining (1)-(3), we obtain the principal equation
in which the surface density s = rD appears. It is just the value that is actually reconstructed in radiographic density measurements (with known thickness) or, inversely, in measurements of thickness with known density. As distinct from the determination of s or of bulk density r, dimensions (distribution of thickness D) of the object of arbitrary geometry can be fully reconstructed by means of multi-view radiography (tomography). For determination of r in the general case, multi-energy approach is needed.
3. Density reconstruction in conventional radiography
In conventional (non-multi-energy) radiography, they use an experimentally verified fact that the response R is linear with density and thickness, i.e., Rordinary µ s . This is a direct consequence of Equations (1)-(4). Really, for the conventional radiography
so, a coefficient in this direct proportionality is the mass attenuation coefficient mm = m/r. Therefore, the measurement of s is based upon preliminary knowledge or setting a fixed mm for the material of the tested object. Some characteristic examples of the general dependence (5) are shown in Fig. 1. However, if composition or structure of the material that affect the value of mm are unknown, it is not possible to reconstruct the material density in this way.
4. Density reconstruction in 2-energy radiography
For 2-energy radiography, the pair of equations (4) assumes the form
where unknowns x=sZ4 and y=sZ are introduced. Monitoring "constants" ai = a(Ei) and bi = b(Ei) (i=1,2) are singled out - the values that are, in one or another way, known from calibration measurements for 2-radiography. The latter are independent or weakly dependent upon the choice of the object and detectors. Therefore, they can be considered as fixed at a given choice of the radiation ranges, and they can be determined from testing experiments on uniform samples of known composition (Zj and rj) and thickness (Dj)( j =1,2). For the 2-energy case, four such testing measurements would be required - with two objects of different composition measured at two different energies each. Let Cij = R(Ei, Zj,sj) be the results of a testing monitoring. For them, we can also write down an equation system similar to (6):
Solving the systems (6)-(7), after certain transformations, we obtain an expression for the density
It is a function of both radiographic reflexes R1, and R2 their ratio (relative reflex) R = R1/R2. New constants A, B, C, D are related to the old ones C11, C12, C21, C22 by the appropriate relationships, which are not presented here explicitly. These new constants are determined from testing experiments on four uniform samples of unknown composition, but of arbitrary geometry. They are related to determination of the effective atomic number of the material. Specifically,
Denominator N = (C12C21 - C11C22)-1 in (8) plays the role of a normalizing constant, and it can be included into the value of s by changing the system of units, e.g., of length or density. The value of N can also be determined from one more (the fifth) measurement with a sample of known composition, density and geometry. In Fig. 2, dependence (8) is shown for a certain (model) choice of this normalization. It should be noted that this dependence is of universal character for a calibrated 2-energy radiographic system. In other words, using multi-energy radiography, it is possible to determine density of any materials or objects by one and the same introscope, not referring to their physico-chemical composition and its change in time and space. Multi-energeticity allows here substantial broadening of the possibilities of monitoring and non-destructive determination of materials density.
One should note that the reconstructed density, according to (7)-(8), is inversely proportional to Zeff. This means that substances with large atomic number attenuate the radiation more efficiently and require smaller thickness for this. Besides this, going to the normal mode, i.e., E1"E2 and R1"R2, we have C11"C21 and C12"C22. As a result, the normalization becomes non-physical, N (r)¥. Maximum level of the received digital signals is abruptly lowered for the reconstructed physical value (not dependent upon the object geometry) s/N (r)0. This threshold can appear to be lower than the internal noise threshold. Therefore, too much overlapping of the energy ranges, as well as their total coincidence, leads to impossibility of unambiguous reconstruction of the material density.
The expression (8), written down in implicit form as the function of s/R2 dependent on R=R1/R2, is even simpler. The corresponding plot is shown in Fig.3. Using known data for R2 R, such nomograms for calibrated introscopes allow calculations of surface density for arbitrary controlled objects.
Thus, we have shown that the most complete reconstruction of density for different materials and objects , including multi-component objects and objects with non-uniform density distribution or unknown chemical composition, is possible only by means of multi-energy (2-energy) radiography. A theory of 2-energy radiography has been constructed for determination of density distribution in objects of arbitrary composition and geometry. General expressions have been obtained, which could be used for the development of algorithms of density reconstruction in various physical applications. An important feature of the presented theory of multi-energy radiography is that it can be adapted to larger number of external factors that can be encountered with in practice of non-destructive testing. This theory can be generalized for the non-monochromatic case, for the case of choosing different energy ranges of radiation (including high energy range of accelerator radiation in the region of pair formation), account for scattered radiation, etc.
The research described in this publication was made possible in part by Award No. UE2-2484-KH-02 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF).