1
Lecture 6
Lecture 6
Scattering theory
Scattering theory
Partial Wave Analysis
Partial Wave Analysis
SS2011
SS2011:
: ‚
‚Introduction to Nuclear and Particle Physics, Part 2
Introduction to Nuclear and Particle Physics, Part 2‘
‘
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1
Lecture 6
Lecture 6
Scattering theory
Scattering theory
Partial Wave Analysis
Partial Wave Analysis
SS
SS
Introduction to Nuclear and Particle Physics, Part 2Introduction to Nuclear and Particle Physics, Part 2
2
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
The
Born approximation
for the differential cross section is
valid
if the interaction
between the projectile particle and the scattering potential
V
r )
is considered to be small
compared with the energy of the incident particle (cf. Lecture 5).
Let‘s obtain the
cross section without imposing any limitation on the strength of
V(r )
We assume here the
potential
to be
spherically symmetric
The
angular momentum
of the incident particle will therefore be
conserved
, a particle
scattering from a central potential will have the same angular momentum before andafter the collision.Assuming that the
incident plane wave
is in the
z
-direction and hence
we may express it in terms of a
superposition of angular momentum eigenstates
, each
with a definite angular momentum number
l
We can then
examine
how each of the partial waves is
distorted by
V(r )
after the particle
scatters from the potential
4
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
A substitution of (2) intowith
ϕϕϕϕ
=0 (and
k=k
0
for elastic scattering) gives
The scattered wave function is given, on the one hand, by (5) and, on the other hand, by (8).
Consider the
limit
Since in almost all scattering experiments detectors are located at distances from the
target that are much larger than the size of the target itself.The
limit of the Bessel function
j
l
(kr
) for large values of
r
is given by
∞∞∞∞
→→→→
r
the
asymptotic form
of (8) is given by
5
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
Sincebecauseone can write (10) as
To find the asymptotic form of (5), we need first
to determine the asymptotic form
of the radial function
R
kl
r
At large values of
r
, the scattering potential is effectively
zero
radial equation (6) becomes
The general solution of this equation is given by a linear combination of the
spherical
Bessel and Neumann
functions
where the asymptotic form of the Neumann function is
7
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
With
δδδδ
l
=0, the radial function
R
kl
(r )
of (18) is
finite at
r
, since
R
kl
(r )
in (17)
reduces to
j
l
(kr)
So
δ δ
δ δ
l
is a real angle
which vanishes for all values of
l
in the absence of the scattering
potential (i.e.,
V
δδδδ
l
is called the
phase shift of the l‘th partial wave
((((
))))
( (
( (
))))
kr
l
kr
cos
sin
l
kr
sin
cos
C
r
R
l
l
l
r
kl
∞∞∞∞
→→→→
π π
π π
δδδδ
ππππ
δδδδ
Thus, the
asymptotic form of the radial function
(16) can be written as
The phase shift
δδδδ
l
measures the
‚distortion‘
of
R
kl
(r )
from the ‚free‘ solution
j
l
(kr)
due to the presence of the potential
V(r )
Attractive (repulsive) potentials
imply that
δδδδ
l
δ δ
δ δ
l
corresponding to the wave being “pulled in” (“pushed out”) bythe scattering center resulting in a
phase delay (advance).
8
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
Using (17) we can write the
asymptotic limit of
the
scattered wave function
(5) as
This wave function (19) is known as a
distorted plane wave
, which differs from a plane
wave by the phase shifts
δδδδ
l
Sinceone can rewrite (19) as
Compare (20) and (12):
We obtain:
10
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
From (23) we obtain the
differential cross sections
and the
total cross sections reads:
Using the relationwe obtain from (25):
where
σσσσ
l
are denoted as the
partial cross sections
corresponding to the scattering of particles
in various angular momentum states.
11
Partial wave analysis for elastic scattering Partial wave analysis for elastic scattering
The
differential cross section
(24) consists of a superposition of terms with different
angular momenta; this gives rise to
interference
patterns between different partial waves
corresponding to different values of
l
The
interference
terms
go away in the total cross
section when the integral over
θθθθ
is
carried out.
Note that
when
V=
everywhere, all the phase shifts
δδδδ
l
vanish, and hence the partial and
total cross sections, (24) and (26), are
zero.
In the case of low energy
scattering
between particles, that are in their respective
s states
, i.e.
l=
, the scattering amplitude (23) becomes
where we have used
Since
f
0
does not depend on
θθθθ
, the
differential and total cross sections in the CM
frame
are given by the following simple relations:
μ
E
k
here
2
2
h
In the case where there is
no flux loss
, we must have
However, this requirement is not valid whenever there is
absorption
of the incident
beam. In this case of
flux loss
S
l
k
) is redefined by
13
Partial wave analysis for inelastic scattering Partial wave analysis for inelastic scattering
The scattering amplitude (23) can be rewritten as
where
with
with
, then (33) and (31) become
14
Total elastic and inelastic cross sections Total elastic and inelastic cross sections
The
total elastic
scattering cross section
is given by
The
total inelastic
scattering cross section
, which describes the loss of flux, is given by
Thus, if
ηηηη
l
(k)=
1 there is
no inelastic scattering
, but if
ηηηη
l
(k)=
we have
total absorption
although there is still elastic scattering in this partial wave.The sum of (37) and (38) gives the
total cross section
Using (31) and (35) we get:
A comparison of (40) and (39) gives the
optical theorem relation
Note that the
optical theorem is also valid for inelastic scattering
16
Scattering of identical bosons
Let‘s consider the
scattering of two identical
bosons
in their center of mass frame.
Classically
the cross section for the scattering of two identical particles whose interaction
potential is central is given by (44)
and also the scattering amplitude:
In quantum mechanics
there is no way
of distinguishing between the particlethat scatters at an angle
θθθθ
from the one
that scatters at (
ππππ
θ θ
θ θ
Thus, the scattered wave function must be symmetric:
17
Scattering of identical bosons
Therefore, the differential cross section is
interference term -
not in the classical case!
For
- quantum case- classical case
If the particles are distinguishable
, the differential cross section will be four times
smaller:
19
Scattering of identical fermions
for
quantum caseclassical case
if the incident particles are
unpolarized:
this
quantum
differential cross section for the scattering of identical fermions is
half the classical expression
, and four times smaller than the quantum differential cross
section for the scattering of two identical bosons (48) -
Note that,
in the case of partial wave analysis for elastic scattering
, using the relations
and inserting them into (23) leads to:
We can write