bei48482_FM

Nuclear Transformations 445

used in Sec. 5.4,



0 xe

nxdx

Mean free path  (12.21)



0 e

nxdx

Example 12.9

Find the mean free path of thermal neutrons in^113 Cd.

Solution

Since n1.12  104 m^1 here, the mean free path is

8.93 10 ^5 m0.0893 mm

1



1.12 104 m^1

1



n

1



n

Reaction Rate

When we know the cross section for a nuclear reaction caused by a beam of incident

particles, we can find the rate N tat which the reaction occurs in a given sample

of the target material. Let us consider a sample in the form of a slab of area Aand

thickness xthat contains natoms/m^3 , with the particle beam incident normal to one

face of the slab. From Eq. (12.20)

(1enx)

If the slab is thin enough so that none of the nuclear cross sections overlaps any others,

nx 1. Since ey 1 yfor y 1, in this case

nx

N 0



t

N



t

N 0



t

N 0 N



t

N



t

Slow Neutron Cross Sections

A

lthough neutrons interact with nuclei only through short-range nuclear forces, reaction

cross sections for slow neutrons can be much greater than the geometrical cross sections

of the nuclei involved.The geometrical cross section of^113 Cd is 1.06 b, for example but its cross

section for the capture of thermal neutrons is 20,000 b.

When we recall the wave nature of a moving neutron, though, such discrepancies become less

bizarre. The slower a neutron, the greater its de Broglie wavelength and the larger the region of

space through which we must regard it as being spread out. A fast neutron with a wavelength

smaller than the radius Rof a target nucleus behaves more or less like a particle when it interacts

with the nucleus. The cross section is then approximately geometrical, in the neighborhood of R^2.

Less energetic neutrons behave more like wave packets and interact over larger areas. Although

cross sections in the latter case of    ^2 (which is over 10^7 b for a thermal neutron) are rare, cross

sections for nuclear reactions with slow neutrons greatly exceed R^2 , as we have seen.

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