
333Collisions and Scatterings
© 2010 Taylor & Francis Group, LLC
The quantity cos(sin
–1
b/R) needs some special attention. Let y = sin
–1
(b′/R); then sin(y) = b′/R, and
cos(y) = (1 – sin
2
y)
1/2
= (1 – b′
2
/R
2
)
1/2
.
That is,
cossin (/
/
−
′
=−
′
2
12
1bR bR
and Equation 11.68 becomes
b
R
b
R
b
R
=−
′
+
′
1
2
2
12/
si
os
from which we obtain
bb bb R
22
2
+
′
−
′
=co
in
. (10.50)
Eliminating b′ from this equation and from Equation 10.50, we obtain the impact parameter b as
a function of the scattering angle θ:
b
Rp p
pp pp
=
′
+
′
−
′
(/)sin(/)
(/)(/)cos(
θ
θ
2
12 2
2
.
The range of θ varies from θ = 0 to θ = θ
max
, where θ
max
is determined by setting b = R in Equation
10.50. ...