bei48482_FM

152 Appendix to Chapter 4

Appendix to Chapter 4

Rutherford Scattering

R

utherford’s model of the atom was accepted because he was able to arrive at a

formula to describe the scattering of alpha particles by thin foils on the basis of

this model that agreed with the experimental results. He began by assuming that

the alpha particle and the nucleus it interacts with are both small enough to be consid-

ered as point masses and charges; that the repulsive electric force between alpha particle

and nucleus (which are both positively charged) is the only one acting; and that the nu-

cleus is so massive compared with the alpha particle that it does not move during their

interaction. Let us see how these assumptions lead to Eq. (4.1).

Scattering Angle

Owing to the variation of the electric force with 1r^2 , where ris the instantaneous sep-

aration between alpha particle and nucleus, the alpha particle’s path is a hyperbola with

the nucleus at the outer focus (Fig. 4.30). The impact parameterbis the minimum

distance to which the alpha particle would approach the nucleus if there were no force

between them, and the scattering angleis the angle between the asymptotic direc-

tion of approach of the alpha particle and the asymptotic direction in which it recedes.

Our first task is to find a relationship between band .

As a result of the impulse Fdtgiven it by the nucleus, the momentum of the

alpha particle changes by pfrom the initial value p 1 to the final value p 2. That is,

pp 2  p 1  F dt (4.24)

Because the nucleus remains stationary during the passage of the alpha particle, by hy-

pothesis, the alpha-particle kinetic energy is the same before and after the scattering.

Hence the magnitudeof its momentum is also the same before and after, and

p 1 p 2 m

Figure 4.30Rutherford scattering.

Target nucleus

Alpha particle

θ

b

θ = scattering angle

b = impact parameter

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