Numerical Analysis of the Relationship between the Photoelectric Effect and Energy of the X-Ray Photons in CT
Purpose: Attenuation of x-rays in any object is a combined effect of scattering and absorption of x-ray photons by the atoms in the material. While the scattering part dependents linearly upon the material’s electron density (ρe) and is weakly dependent on the energy (E) of the photon, the
photoelectric part varies as [ρe Zx eff /Ey] where Zeff is the effective atomic number of the material. We aim to determine the exponents (x,y) which are crucial for many radiological studies.
Methods: In order to obtain exponent ‘y’, we find an equation in which the exponent x does not appear and the dependence on ‘y’ appears only linearly. Having thus reduced the problem to that of ‘one parameter fit’ in ‘y’, we generate the ‘data’ numerically from the numerical values of physical constants given in the NIST (The National Institute of Standards and Technology, or NIST) tables and determine ‘y’ from linear regression. The effect of the source spectrum on the effective value of ‘y’, denoted by ‘ym’ is studied for different low Zeff substances in different energy ranges.
Results: It is seen that with x-ray sources operating at 80, 100,120,140 kVp, the effective exponent ‘y’ progressively decreases, as the x-ray source spectrum is pushed to higher energy side. However, for most practical purposes y=2.99 may be used for a wide variety of low Zeff substances. For practical cases with different source spectra, the effective energy of the source and the effective photoelectric exponent are seen to increase as the thickness of the aluminum filter increases.
Conclusion: It thus follows that for applications such as DECT inversion, the appropriate values of ‘ym’ ought to be used that takes into account the appropriate source spectrum.
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