Physics

Problem

Consider a solid sphere of pure

^{235}

U having radius

R

and density

\rho = 1.9\times10^{4}\,\text{kg m}^{-3}

. Each fission liberates on average

\nu = 2.5

neutrons. Assume that the probability that a neutron produced anywhere inside the sphere will escape without causing a subsequent fission is

e^{-R/L}

, where

L = 10\,\text{cm}

is the effective diffusion length of neutrons in the material. a. (i) Establish the condition on

R

for which the sphere becomes critical (exactly self-sustaining). b. (ii) Hence derive an expression for the critical radius

R_c

in terms of

L

and

\nu

. c. (iii) Calculate

R_c

for the data given and hence estimate the corresponding critical mass of

^{235}

U. (Use the volume of a sphere

=\frac{4}{3}\pi R^{3}

.) d. (iv) Explain qualitatively how surrounding the sphere with a thick neutron reflector would affect the critical mass obtained above.