Efficiency of Control Rods Which Absorb Only Thermal Neutrons

Efficiency of Control Rods Which Absorb Only Thermal Neutrons 

Abstract. Simplified pile equations are derived under the assumption that the fast neutrons have at every collision with the moderator the same probability of reaching thermal energy. The approximation involved is much less accurate than Fermi s whose pile equations assume that the slowing down occurs in a definite number of collisions. However, the resulting equations are much more easily solved than Fermi s since they are ordinary instead of partial differential equations. 

Solutions are given for the case of a control rod at the center of a cylindrical pile and the result compared with Murray s calculation, carried out on the basis of Fermi s equations. Second, the efficiency of a regular array of control rods is calculated and compared with older work and with an experimental value. The agreement is good in each case. 

If one replaces the lattice arrangement by a homogeneous mixture one can describe the pile by the equations 

In these Ut is the density of thermal neutrons multiplied by their velocity, q t) is proportional to the slowing down density t is the age of neutrons, to the age of thermal neutrons, k the multiplication constant, Kf the reciprocal diffusion length of thermal neutrons in the mixture. The above equations were derived by Fermi on the assumption that the number of collisions which a fission neutron suffers before it becomes thermal is the same for all neutrons and that the distance between two collisions is also a well-defined quantity depending only on the energy. 

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E. P. Wigner, A. M. Weinberg, and R. R. Williamson, Efficiency of Control Rods Which Absorb Only Thermal Neutrons , CP-1461, February 24,1944. 

For the very small radii of the control rod, the ratio of the efficiency of the rod which absorbs only thermal neutrons is only + times as great... 

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