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
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