Question 22.1: Suppose that the components of the angular neutron current a......
Suppose that the components of the angular neutron current along the x-axis, the y-axis, and the z-axis in a Cartesian coordinate system are given by J_x = 0.25 × 10^{14} neutrons/cm²/s, J_y = 0.433 × 10^{14} neutrons/cm²/s, and J_z = 0.866 × 10^{14} neutrons/cm²/s. What is the neutron density n(r, E, Ω, t) at r = 0 if the average velocity of the neutrons is 2200 m/s?
The magnitude of the neutron density n(r, E, Ω, t) and the magnitude of the neutron current are related by J(r, E, Ω, t) = v ⋅ n(r, E, Ω, t). The magnitude of the neutron density is therefore given by n(r, E, Ω, t) = J(r, E, Ω, t)/|v|, where J(r, E, Ω, t) = \sqrt{(J^{2}_{x} + J^{2}_{y} + J^{2}_{z})} . The neutron density is then (r, E, Ω, t) = \sqrt{(J^{2}_{x} + J^{2}_{y} + J^{2}_{z})}|\text{v}| = 1 × 10^{ 14} neutrons/cm²/s ÷ 2.2 × 10^{16} cm/s = 4.55 × 10^7 neutrons/cm³.