Lecture 15 - 2 Scattered Bodies in a Ditch
•Hyperbolic motion e > 1
•Scattering Cross Sections (Section 3.10)
•Quick look at the Lab point of view (Section 3.11)
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Hyperbolic Orbits and Scattering Angles in Physics - Prof. M. W. Bromley, Study notes of Physics
The concept of hyperbolic orbits and scattering angles in physics. It covers the hyperbolic motion equation, the solid angle expression, and the hyperbolic orbit equation. The document also discusses rutherford scattering and the infinite total cross section in the context of coulomb scattering.
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Pre 2010
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Lecture 15 - 2 Scattered Bodies in a Ditch
Hyperbolic motion
e >
Scattering Cross Sections (Section 3.10)
Quick look at the Lab point of view (Section 3.11)
Solid Angle Expression
Want to find an expression for
s ) , take
so we go back to the equation of orbital motion
θ ( r ) =
θ 0
r
r 0
dr
r 2 √
2 ℓ mE 2
2 ℓ mV 2
r 2 1
with
r 0 =
, θ
0
π
and for
θ ( r min
θ min
π
−
π
−
θ min
∞
r min
dr
r 2 √
2 ℓ mE 2
2 ℓ mV 2
r 2 1
rewrite using
s √
2 mE
AND
π
= Ψ + Ψ + Θ
s ) =
π
−
(^2)
∫
u min
0
s du
V (^) ( u )
E
(^) s 2 u 2
hyperbolic orbit angles
with hyperbolic orbit
u
=
r 1
=
mZZ
′ e
2
ǫ (^) cos
(^) θ
−
we find
from when
r → ∞
ie.
ǫ (^) cos
(^) θ ∞
−
ǫ^1
= cos
(^) θ ∞
= cos (
π
−
(^) Ψ) = cos (Ψ) = cos
π
−
(^) Θ
= sin
Note
θ ∞
π/
π
when large
ǫ ie. large
E, s
Scattering Angle
So
π
−
and
cos Ψ =
ǫ^1 = sin Θ
and since
ǫ 2 = 1
(^) sin
2 = csc
2 = cot
2 +^
we write
cot
) = √ ǫ 2 −
Es
ZZ
e′ 2
thus
s
ZZ
′ e 2
E
cot
Backwards scattering (ie.
> π/
) occurs for small
s, E
Infinite Total Cross Section
Defn: Total scattering cross section
σ T = ∫ 4 π σ ( Ω ) d Ω
note: surface area of a sphere is
πr
2 →
π
steradians
which is infinite for pure Coulomb scattering:
σ T
= 2
π
∫
π
0
σ (Θ) sin Θ
d Θ
π
0
csc
4 (
2 Θ )
sin Θ
d Θ =
[
csc
2 (
)]
0 π
Coulomb field is a long-range force!
σ T
only non-zero when potential cuts off at some
r max
Units of
σ
t
are area, ’size’ of reaction
• Example II: Potential scattering....
fun problem
V
r ≤
a ) =
− V 0 , V (
r > a
contd.
contd.
• Example III: homework problem....
tough, but fun, problem
V
r ) =
r k
−
a k
, V (^) ( r > a
Flashback to Lab Frame
Define
ϕ
as the angle measured in the lab
ie. can convert between
and
ϕ
:
(sinceIn collision, the scattering particle slows down
m
2 gains kinetic energy
T
elastic scattering is where total kinetic energy is constant
eg. the best fast neutron moderators use light elements.
(eg. excitation of scattering particle and or target)Inelastic scattering is where energy is transferred