bei48482_FM

Our first step is to note that all alpha particles approaching a target nucleus with

an impact parameter from 0 to bwill be scattered through an angle of or more, where

is given in terms of bby Eq. (4.29). This means that an alpha particle that is initially

directed anywhere within the area b^2 around a nucleus will be scattered through 

or more (Fig. 4.32). The area b^2 is accordingly called the cross sectionfor the

interaction. The general symbol for cross section is , and so here

Cross section b^2 (4.30)

Of course, the incident alpha particle is actually scattered before it reaches the imme-

diate vicinity of the nucleus and hence does not necessarily pass within a distance b

of it.

Now we consider a foil of thickness tthat contains natoms per unit volume. The

number of target nuclei per unit area is nt, and an alpha-particle beam incident upon

an area Atherefore encounters ntAnuclei. The aggregate cross section for scatterings

of or more is the number of target nuclei ntAmultiplied by the cross section for

such scattering per nucleus, or ntA. Hence the fraction fof incident alpha particles

scattered by or more is the ratio between the aggregate cross section ntAfor such

scattering and the total target area A. That is,

f



ntb^2

Substituting for bfrom Eq. (4.30),

fnt

2

cot^2 (4.31)

In this calculation it was assumed that the foil is sufficiently thin so that the cross sec-

tions of adjacent nuclei do not overlap and that a scattered alpha particle receives its

entire deflection from an encounter with a single nucleus.





2

Ze^2



4  0 KE

ntA



A

aggregate cross section



target area

alpha particles scattered by or more



incident alpha particles

Rutherford Scattering 155

Figure 4.32The scattering angle decreases with increasing impact parameter.

θ

Target nucleus

Area = πb^2

b

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