The Calculation of Self Attenuation Factors for Simple Bodies in the Far Field Approximation

Microsoft Excel spreadsheet functions have been developed for the calculation of the self attenuation of gamma-rays in simple bodies viewed from afar. The cases are a uniform rod viewed along its axis, a sphere viewed along a diameter and a cylinder (or disk) viewed along a mid-plane diameter. The results for the former two cases can be expressed in closed form while the algorithm used for the third is described. We also develop useful expressions for small and large lump cases. The self attenuation functions have been used along with other numerical methods to generate and validate test data for exercising a proposed new lump correction algorithm based on exploiting the differential attenuation of different energy γ-rays emitted by an item. Introduction The measurement of special nuclear materials such as U and Pu, in radioactive waste by the application of high resolution gamma-ray spectrometry is a widely used technique [1]. If the nuclides are present the form of “lumps” (such as shavings, chips, pellets, foundry spills, crevice accumulations and the like) rather than as dilutely distributed activity in and on the bulk waste matrix, then self attenuation may occur. Self attenuation is not accounted for by the usual transmission source and weight based matrix correction factor methods. This is because dense lumps of significant size, sufficient to affect the assay result, are still physically small in relation to the overall size of most waste container types [2]. Therefore, traditional gross matrix correction methods are not sensitive to the presence of lumps. If the presence of lumps goes unrecognized then an assay system calibrated using dilute (minimally absorbing) standards will underreport when put into operation. This is because the number of γ-rays emerging from the lump per unit mass of nuclide present will be less than assumed. In order to examine the importance of self attenuation by numerical simulation it is desirable to have a simple way to calculate the attenuation factor for lumps in a variety of shapes. In essence, a shape defines a particular distribution of emergent path lengths. By using a given lump shape or combination of shapes various measurement scenarios can be played out to examine the impact of self attenuation as a function of photon energy and nuclide mass given assumptions on the nature (chemical composition, density, enrichment) of the lumps. This is important to do as part of the assessment of a reasonable and justifiable total measurement uncertainty and also when bounding or limiting assay results are to be reported. Self Attenuation Factors Expressions for the self attenuation factors (SAF’s) for several common shapes have been given by Dixon [3]. The SAF is the ratio of the emergent photon rate to the corresponding rate if the specimen were perfectly transparent. In particular Dixon considered: 1) a rod (or linear) source viewed end on, 2) a right cylindrical (elliptical cylinder or disk) source viewed along a diameter, and 3) a spherical source viewed along a diameter. In all cases the radioactivity was assumed to be uniformly distributed within the uniform and homogenous material of the solid body. The source to detector separation was assumed to be large in relation to either the size of the detector or the size of the source so that only rays (for all practical purposes) parallel to the direction of viewing are detected. In addition to this far field assumption the photon spectrometer is assumed to confer the conditions of “good geometry” (perfect energy selection) onto the measurement so that only full energy photons are of interest. Under the specified conditions the SAF, f, may be written as: ( ) ( ) ( ) ( ) ( ) ( ) [ ] x L x I x p x f x f p p p − ⋅ ⋅ + Γ = = = 2 1 1 0 where Г is the gamma function [4], Ip is the modified Bessel function of order p [5] and Lp is the Modified Struve function of order p [5]. The value of p takes on the value 1⁄2 for rod sources, 1 for cylindrical (elliptical) sources and /2 for spherical sources. The parameter x is the “thickness” of the source. Denoting the linear attenuation coefficient by, μ, then x is given by (μ·L) for a rod source of length L; by (2·μ·a) for an (elliptical) cylinder of principal axis 2·a (diameter in the case of a right circular cylinder); and by (2·μ·a) for a sphere of diameter (2·a). The spherical case is of particular interest because the SAF for a sphere is the lowest of any uniform solid. This is a result of the high volume to surface area quotient. Because of this spherical lumps represent a worst case assumption and can be used to derive bounding assay values. The cylindrical case is of practical interest because many calibration samples are in the form of small cylindrical containers filled with powder (e.g. PuO2, U3O8 etc...). These are often arranged perpendicular to the detector axis in a geometry which closely approximates far field conditions. We note in passing that the evaluation of escape probabilities is closely related to the problem of calculating the collision probabilities for external irradiation. Results for certain other shapes are therefore also available [see 8 and references therein] for the case of isotropic incoming irradiation. In general the use of such results for SAF evaluation would therefore correspond to viewing an ensemble of randomly orientated bodies in the far field approximation in order for an integrated over angle (IOA) signal to be obtained.