Thermal Physics · Problem 7

Propagation — Charles Kittel & Herbert Kroemer

Physics — Statistical Mechanics · critical-size-of-nuclear-reactor

Problem

Critical size of nuclear reactor. Extend the considerations of the preceding problem to particle diffusion, and assume that there is a net particle generation rate

g_n

that is proportional to the local particle concentration,

g_n = n/t_0

, where

t_0

is a characteristic time constant. Such behavior describes the neutron generation in a nuclear reactor. The value of

t_0

depends on the concentration of

^{235}

U nuclei; if no surface losses took place, the neutron concentration would grow as

\exp(t/t_0)

. Consider a reactor in the shape of a cube of volume

L^3

and assume that surface losses pin the neutron surface concentration at zero. Show that Eq. (3), if augmented by a generation term

g_n = n/t_0

, has solutions of the form

n(x,y,z,t) \propto \exp(t/t_1)\cos(k_x x)\cos(k_y y)\cos(k_z z)\,, \tag{67}

where

k_x L

k_y L

and

k_z L

are integer multiples of

\pi

. Give the functional dependence of the net time constant

t_1

k_x

k_y

k_z

and

t_0

, and show that for at least one of the solutions of the form (67) the neutron concentration grows with time if

L

exceeds a critical value

L_{\text{crit}}

. Express

L_{\text{crit}}

as a function of

D_n

and

t_0

. In actual nuclear reactors this increase is ultimately halted because the neutron generation rate

g_n

decreases with increasing temperature.

Related theory

Nuclear Size And Shape

Browse the problem library