Generalized Off-Axis Distributions from Disk Sources of Radiation

Abstract

In the field of shielding it is frequently necessary to calculate biological radiation doses at any point in space due to disk sources of radiation. Such a source is the leakage radiation from an end of a cylindrical reactor. If G(R) is the dose at distance R from a unit point source (e.g., G∼e^−σR/R²) then the dose due to a disk source of radius ``a'' with source strength S(ρ) is given by $$\mathrm{D(z,a,\epsilon )=}{\displaystyle {\int }_0^a}{\displaystyle {\int }_0^{\text{2π}}}{\mathrm{G[(z}}^2{\mathrm{+\rho }}^2{\mathrm{+\epsilon }}^2\text{−2ερ\hspace{0.17em}}\cos {\mathrm{\theta )}}^{\frac{1}{2}}\mathrm{]S(\rho )\rho d\rho d\theta }$$ , where (z,ρ,θ) are the usual cylindrical coordinates, z=0 is the source plane, and (z,ε,0) locates the dose point. This integral is quite general, arising in many physical problems. In counter measurements, use of G=z/(4πR³), S=1 yields the solid angle subtended by the counter. In neutron physics the slowing down density of neturons from a circular source of finite radius can be calculated in the age approximation with G∼exp(−R²/τ). Since for most G's the integral cannot be evaluated in closed form, there are derived three alternative series for D(z,a,ε) each of which has a direct physical interpretation and a different region of convergence. The results are obtained for a class of analytic functions G(R), the possible singularities of which occur only at R=0 for finite R; and for S(ρ) of the form Σ_pS_pρ^2p. The approach in each case is to expand either the double integral or one of the two repeated integrals in a power series in cosθ, ε, or ρ prior to performing the detailed integrations. Numerical results are given for the case of a typical shielding function.