1000 Solved Problems in Modern Physics

(Tina Meador) #1
7.3 Solutions 401

Fig. 7.14Collision of a
heavy charged particle with
an electron

From the velocity triangle which is isosceles as in Fig. 7.14,
v 2 = 2 vccosφ= 2 m 1 vcosφ/(m 1 +m 2 )≈ 2 vcosφ
(Becausem 2 =m<<m 1 )
The energy acquired by the electron
W=(m/2) (2vcosφ)^2 = 2 mv^2 cos^2 φ (1)
If the recoil angle of the electron isφ∗in the CMS thenφ=φ∗/2 and
φ∗=π–θ∗, whereθ∗is the scattering angle of the incident particle in the
CMS. And so
cos^2 φ=sin^2 (θ∗/2)/2(2)
W= 2 mv^2 sin^2 (θ∗/2) (3)
dW=mv^2 sinθ∗dθ∗ (4)
But Rutherford’s formula for scattering in the CMS is
σ(θ∗)=dσ/dΩ∗=z^2 e^4 / 4 μ^2 v^2 sin^4 θ∗/2(5)
where we have put Z=−1 for electron. Since the electron mass is negligible
compared to that of the incident particle,μ≈m. Further, the element of solid
angle dΩ∗=^2 πsinθ∗dθ∗.
Formula (5) becomes
dσ=(2πsinθ∗dθ∗z^2 e^4 )/(4m^2 v^4 sin^4 (θ∗/2)) (6)
Using (3) and (4) in (6)
dσ/dW= 2 z^2 e^4 /mv^2 W^2 (differential energy spectrum)
This gives us the cross-section for finding the delta rays(emitted electrons)
of energyWper unit energy interval.

7.23 When the charged particle just grazes the nucleus


rmin=R=^1 / 2 R 0 [1+(1+ 4 b^2 /R 02 )^1 /^2 ](1)
Solving forb, we obtain
b^2 =R^2 – RR 0 (2)
Denoting the cross-section byσ=πb^2 ,
σ/σg=πb^2 /πR^2 =1–R 0 /R

7.24 The closest distance of approachris given by


rmin=(R 0 /2)(1+cosec (θ/2))=(R 0 /2)(1+2)= 3 R 0 / 2
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