bei48482_FM

Here is the alpha-particle velocity far from the nucleus.

From Fig. 4.31 we see that according to the law of sines,



Since sin ()cos

and sin2 sin cos

we have for the magnitude of the momentum change

p 2 msin (4.25)

Because the impulse  Fdtis in the same direction as the momentum change p,

its magnitude is

 F dt Fcos dt (4.26)

where is the instantaneous angle between Fand palong the path of the alpha

particle. Inserting Eqs. (4.25) and (4.26) in Eq. (4.24),

2 msin 





Fcos dt

To change the variable on the right-hand side from tto , we note that the limits of

integration will change to 2 ^1 () and^12 (), corresponding to at t

and trespectively, and so

2 msin 

() 2

() 2

Fcos d (4.27)

dt



d





2





2





2





2





2





2

1



2

m



sin 



2





p



sin

Rutherford Scattering 153

Figure 4.31Geometrical relationships in Rutherford scattering.

p 2

p 1

∆p

θ

1

2 (π – θ)

∆p

1

2 (π – θ)

b

Path of alpha particle

Target nucleus

φ

F

1

2 (π – θ)

θ

Alpha

particle

b

bei48482_ch04.qxd 1/14/02 12:21 AM Page 153