Study of Doppler broadening Compton scattering and cross section determination for the elements Fe, Zn, Ag, Au and Hg

Abstract


Introduction
Compton effect is a scattering of gamma or x-ray waves by a charged particle in which a portion of its energy is given to the charged particle in an elastic collision. The major physical effects characterizing the backwardscattered spectra are besides attenuation, Compton shift and material-specific Doppler broadening of the photon spectrum in the sample.
The shape of a Compton-broadened peak can reveal information about the electron momentum of an elementary or, even to some extent, of a chemical compound. Essentially, two types of methods were considered for describing the broadening of the characteristic lines. The first calculation procedure evaluated scattered spectra based on tabulated material specific Compton profiles.
The second model was a phenomenological approach; it fitted a function consisting of Gaussian curves above a linearly approximated background to the curvature of the 2nd derivative of the Compton spectrum. These models were experimentally validated on Compton profiles of a variety of sample materials containing period 2 and period 3 elements [1]. It was proven that, in principle, a comparing of measured with calculated spectra provides high material differentiation capabilities, but for most molecules tabulated Compton profiles are not available and the independent atom approximation causes deviations. The phenomenological method was employed to extract Gaussian curve fitparameters to distinguish measured materials quantitatively. Most of the samples could be distinguished from each other based on their profile structure. Compton energy absorption cross sections are calculated using the formulas based on a relativistic impulse approximation to assess the contribution of Doppler broadening and to examine the Compton profile literature and explore what, if any, effect our knowledge of this line broadening has on the Compton component in terms of mass-energy absorption coefficient. The electron momentum prior to the scattering event should cause a Doppler broadening of the Compton line [1]. Measurement of differential scattering cross sections for x-rays is useful in the studies of radiation attenuation, transport and energy deposition and plays an important role in medical physics, reactor shielding, industrial radiography in addition to x-ray crystallography. Coherent (Rayleigh) scattering accounts for only a small fraction of the total cross section, contributing to the most of 10% in heavy elements, just below the K-edge energy.
Incoherent (Compton) scattering accounts for the rest of the total cross section. For low Z materials, this process dominates over most part of the energy range. The Compton profile-provides detailed information about the electron momentum distribution in the scatterer. The technique is particularly sensitive to the behavior of the slower moving outer electrons involved in bonding in condensed matter and can be used to test their quantum-mechanical description. Mendelsohn, et al., (1974) [2], had made calculations of relativistic Hartree-Fock(HF) Compton profile J (q) for the rare gases and Pb, for values of q between 0 and 100 and compared with the non-relativistic calculations.
Comparison with experimental profile data was made for Ar and Kr. For q between 0.0 and 0.4 in Kr, much closer agreement with the experiment was obtained when the relativistic HF wave function was used to perform the profile calculation, when the non-relativistic HF. R Benesch (1976) [3], calculated Compton profile by HF wave functions for the neutral atoms As (Z = 33) to Yb (Z = 70) were the comparison of the results with relativistic HF wave functions indicates that the overall effect of using the relativistic functions is to produce the total J (q), which are flatter at the center than those computed from non-relativistic HF wave functions. Rao et al., (2002) [4], had been calculated the Compton energy absorption cross sections using the formulas based on a relativistic impulse approximation to assess the contribution of Doppler broadening and examine the Compton profile. Using these cross sections, the Compton component of the massenergy absorption coefficient is derived in the energy region from 1 keV to 1 MeV for all elements with Z=51 -100. Also the Compton broadening was estimated using the non-relativistic formula in the angular region 1°-180°, for 17.44, 22.1, 58.83, and 60 keV photons for few elements H, C, N, O, P, S, K, and Ca. Rao et al., (2004) [5], had been used the relativistic and non-relativistic Compton profile cross sections for H, C, N, O, P, and Ca for a few important biological materials such as water, polyethylene, lucite, polystyrene, nylon, polycarbonate, bakelite, fat, bone and calcium hydroxyapatite that estimated for a number of, K, x-ray energies for 59.54 keV ( 241 Am) photons. These values are estimated around the centroid of the Compton profile with an energy interval of 0.1 and 1.0 keV for 59.54 keV photons. Prem Singh (2011) [6], had been studied the Compton scattering differential cross-sections for the 19.648 keV photons in a few elements with (6 ≤ Z ≤ 50).The measured Compton scattering cross-sections were compared with the theoretical Klein-Nishina cross-sections corrected for the non-relativistic HF incoherent scattering function S(X, Z). Hossain, et al., (2012) [7], had been studied Compton scattering using NaI (TI) scintillator detector and a collimated 137 Cs source producing gamma rays with an energy of 662 keV scattered incoherently by Al and Cu materials through the angles from (0 to 120°). The Compton scattering effect was investigated and found that the energy of the scattered gamma ray decreases as the scattering angle increases. The differential scattering cross-section as a function of scattering angles was also measured. The experimental results of differential scattering cross-sections for Al and Cu materials were compared with a function of a scattering angle and found to coincide at the higher angle region, although scattering cross-sections for Cu are larger than Al scatter at the lower angles region. In the present work, a proposal for calculating Compton profile required for the calculations of Doppler broadening and Double differential cross section is presented and tested. Different elements and materials are used and tested in this work together with the NaI(Tl) measurement and analysing system. The Compton profile can be determined from measurements of the partial differential cross section by performing a constant energy scan through all possible scattering angle at the scattering angle (180°).

Theory
The differential cross section for the scattering of gamma photons with free electrons was first derived in 1928 by Oskar Klein and Yoshio Nishina [8] using the quantum electrodynamics approach. The angular distribution of the scattered photon is known as the Klein-Nishina cross-section formula and can be given by [9]: and ° is the classical electron radius and 180°. However, what has been stated earlier is valid assuming that the gamma interacts with an electron at rest. But this is an approximation. If the electron momentum is taken into account, neither the angle-energy relations nor the Klein-Nishina formula, Eq. (1) are completely valid. The electron pre-collision momentum creates a broadening in the energy spectrum of the scattered photon which is known as the Doppler broadening effect [10]. The distance travelled by the electron ejected from the atom is shorter than the spatial resolution of today's solid state detectors, which make it impossible to measure the energy and momentum of the electron in order to correct this effect. This effectively introduces an error on the calculated Compton scattering angle .

The angular
Klein-Nishina is integrated to give the total cross section for a given energy which is given as [11]: The Doppler broadening effect imposes an inherent limitation on the angular resolution. In general the interaction takes place with a bound, moving electron. The momentum of the electron results in a broadening of the gamma spectrum lines and this leads to an error on the computation of the Compton scattering angle. The equation that accounts for the electron movement is [12] ′ ′ ′ ′ where is the electron momentum, is the Compton energy calculated from (through Eq.3) before the interaction, projected upon the gamma momentum transfer vector. This equation reduces exactly to Eq. (3) taking = 0. The angular distribution is also affected by the electron motion and the Klein-Nishina cross-section must be modified giving the following expression [13]: where J( ), known as Compton profile which is the electron momentum distribution on the material. The values of Compton profiles are important as they give insight of the electron movement in the atom. They have been calculated using the HF method [13,14]. DuMond (1933) [15] likened the Dopplerbroadening process to the reflection of light from a moving mirror (the electron) with the addition of a wavelength shift fixed by the scattering angle. J( ) and J( )mag. are properties of the scattering electrons. These are the Compton and magnetic Compton profiles. They are defined as the one-dimensional projections of the electron and spin momentum density distributions, respectively. By including the normalizing pre factors 1/N and 1/µ, where N and µ represent the total number of electrons and the spin-unpaired electrons only, we choose the integrated profile areas to be equal to the total charge and spin of the system. Thus, ∬ ∞ ∞ In an isotropic system the expression is commonly rewritten in terms of the radial momentum distribution I (p) = 4πp 2 n(p) and a scalar momentum variable, q, as where n(p), n↑ (p) and n↓ (p) are the three-dimensional electron momentum distributions for all, majority-spin (↑)      wide range of element samples, one can observe found the necessity of extending these profiles to include lower energy photon scattering, which is a significant source of energy uncertainty. This limits the choice of scatter detectors to low-Z materials, where the Doppler broadening is less and the relative probability of Compton scattering is higher. Also, We have shown the importance of the energy resolution of the absorption detector. 2-The Compton profile data are useful to assess the contribution of Doppler broadening and to calculate the Compton component of the mass energy absorption coefficient. 3-The effect of Compton broadening is significant at energies below 100 keV and must be considered, since it spreads the counts in the neighboring isogonic regions.