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Applying the pulsed ion chamber methodology to full range reactor ...

Applying the pulsed ion chamber methodology to full range reactor ...

It should be noted that

It should be noted that these coupled equations, when used, generally constitute a nonlinear pair of equations with no possible analytical solution. However, for the densities of interest in this study, the equations may be markedly simplified. The two ion density regions o^ interest are those controlled by the free diffusion regime and the ambi- polar-volume recombination regime. Second order loss mechanisms can be neglected at ion densities where fre.s diffusion dominates and the asymtotic form of equation 2-2 reduces to; M n. + S = (2-3) When this equation is applied to cylindrical geometry and is subject to the boundary conditions, + + where a and b are the outside boundary of the inside electrode and the inside boundary of the outer electrode, respectively, the exact solution n + (r) - 4 D~TnTb7a7 E(^-a 2 )^ " ln ( -'^^ r ' + ^ lr:b ~ b lna]. (2-4) At ion densities greater than ^10 cm" ambipolar diffusion is con- trolling the mechanism for spatial distribution of charged particles and volume recombination becomes the major loss mechanism. Because of ambi- polar diffusion, the ion-electron diffusional losses become approximately

equal, thus validating the approximation n, = n_ at such densities. This approximation leads to, DVn. - »„nj + S 0, (2-5) which is the decoupled ion density equation for n>jQ . This equation, when written in cylindrical coordinates, is the Emden differential equa- tion which cars be solved using only numerical methods The PIC voltage signal amplitude v(t ), at the time t„, when all of 3 -1 r c c the ions are collected, for a large RC cathode circuit time constant (large with respect to t ), and for coaxial detector electrode geometry, is analytically and experimentally related to the steady state ionization density, n + , in the chamber gas. by, r b v(tj=^/ rn.(r)dr, (2-6) where; 1 = length of the chamber, C = cathode circuit capacitance, e = the unit of electron charge. The final link in relating the measured vol Lace peak signal v(t ) to the neutron and gamma flux, is provided by the equations; ana EaN & -_*- , (2-7) 5 « qly, (2-8) . g