2.1 Basic Concepts and Formulae
51
Relations of Quantities in the Lab System (LS) and Centre of Mass System
(CMS)
Quantities that are unprimed refer to LS and primed refer to CMS.
Lab system
CM system
m 1 : u 1 ;
u
∗
1 =
m 2 u 1
m 1 + m 2
(2.19)
m 2 : u 2 = 0; u
∗
2 =
m 1 u 1
m 1 + m 2
(2.20)
Scattering Angle
Because of elastic scattering
v
∗
1 = u
∗
1 ; v
∗
2 = u
∗
2
The centre of mass velocity
v c = −
m 1 u 1
m 1 + m 2
= −u
∗
2
(2.21)
tan θ =
sin θ ∗
cos θ ∗ + m 1 /m 2
(2.22)
tan θ
∗
=
sin θ
cos θ − m 1 /m 2
(2.23)
If m 1 < m 2 , all scattering angles for m 1 in LS are possible.
If m 1 = m 2 , scattering only in the forward hemisphere (θ ≤ 90 ◦ ) is possible.
If m 1 > m 2 , the maximum scattering angle is possible, θ max , being given by
θ max = sin
−1
(m 2 /m 1 )
(2.24)
Recoil Angle
tan ϕ =
sin ϕ ∗
cos ϕ ∗ + 1
= tan
ϕ ∗
2
(2.25)
Or φ =
1
2
ϕ
∗
(2.26)
Recoiling angle is limited to ϕ ≤ 90 ◦ .