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Lecture 20

Scattering theory

Scattering theory

Scattering theory is important as it underpins one of the most ubiquitous

tools in physics.

Almost everything we know about nuclear and atomic physics has

been discovered by scattering experiments,

e.g. Rutherford’s discovery of the nucleus, the discovery of

sub-atomic particles (such as quarks), etc.

In low energy physics, scattering phenomena provide the standard

tool to explore solid state systems,

e.g. neutron, electron, x-ray scattering, etc.

As a general topic, it therefore remains central to any advanced

course on quantum mechanics.

In these two lectures, we will focus on the general methodology

leaving applications to subsequent courses.

Scattering phenomena: background

In an idealized scattering experiment, a sharp beam of particles (A)

of definite momentum k are scattered from a localized target (B).

As a result of collision, several outcomes are possible:

A + B −→

A + B elastic

A + B

∗

A + B + C

inelastic

C absorption

In high energy and nuclear physics, we are usually interested in deep

inelastic processes.

To keep our discussion simple, we will focus on elastic processes in

which both the energy and particle number are conserved –

although many of the concepts that we will develop are general.

Scattering phenomena: differential cross section

Both classical and quantum mechanical scattering phenomena are

characterized by the scattering cross section, σ.

Consider a collision experiment in which a detector measures the

number of particles per unit time, N dΩ, scattered into an element

of solid angle dΩ in direction (θ, φ).

This number is proportional to the incident flux of particles, j I

defined as the number of particles per unit time crossing a unit area

normal to direction of incidence.

Collisions are characterised by the differential cross section defined

as the ratio of the number of particles scattered into direction (θ, φ)

per unit time per unit solid angle, divided by incident flux,

dσ

dΩ

N

j I

Scattering phenomena: classical theory

In classical mechanics, for a central

potential, V (r ), the angle of scattering is

determined by impact parameter b(θ).

The number of particles scattered per unit

time between θ and θ + dθ is equal to the

number incident particles per unit time

between b and b + db.

Therefore, for incident flux j I , the number

of particles scattered into the solid angle

dΩ =2 π sin θ dθ per unit time is given by

N dΩ =2 π sin θ dθ N = 2πb db j I

i.e.

dσ(θ)

dΩ

N

j I

b

sin θ

db

dθ

Scattering phenomena: classical theory

dσ(θ)

dΩ

b

sin θ

db

dθ

For elastic scattering from a hard (impenetrable) sphere,

b(θ) = R sin α = R sin

π − θ

= −R cos(θ/2)

As a result, we find that

db

dθ

R

2

sin(θ/2) and

dσ(θ)

dΩ

R

2

As expected, total scattering cross section is just

dΩ

dσ

dΩ

= πR

2 ,

the projected area of the sphere.

Quantum scattering: basics and notation

Simplest scattering experiment: plane wave impinging on localized

potential, V (r), e.g. electron striking atom, or α particle a nucleus.

Basic set-up: flux of particles, all at the same energy, scattered from

target and collected by detectors which measure angles of deflection.

In principle, if all incoming particles represented by wavepackets, the

task is to solve time-dependent Schr¨odinger equation,

iℏ ∂ t Ψ(r, t) =

[

2

2 m

2

• V (r)

]

Ψ(r, t)

and find probability amplitudes for outgoing waves.

Quantum scattering: basics and notation

However, if beam is “switched on” for times long as compared with

“encounter-time”, steady-state conditions apply.

If wavepacket has well-defined energy (and hence momentum), may

consider it a plane wave: Ψ(r, t) = ψ(r)e

−iEt/ℏ .

Therefore, seek solutions of time-independent Schr¨odinger equation,

E ψ(r) =

[

2

2 m

2

• V (r)

]

ψ(r)

subject to boundary conditions that incoming component of

wavefunction is a plane wave, e

ik·r (cf. 1d scattering problems).

E = (ℏk)

2 / 2 m is energy of incoming particles while flux given by,

j = −i

2 m

(ψ

∗ ∇ψ − ψ∇ψ

∗ ) =

ℏk

m

Scattering in more than one dimension

In higher dimension, phenomenology is similar – consider plane wave

incident on localized target:

Outside localized target region, wavefunction involves superposition

of incident plane wave and scattered (spherical wave)

ik·r

e

ikr

Scattering phenomena: partial waves

If we define z-axis by k vector, plane wave can be decomposed into

superposition of incoming and outgoing spherical wave:

If V (r ) isotropic, short-ranged (faster than 1/r ), and elastic

(particle/energy conserving), scattering wavefunction given by,

e

ik·r = ψ(r) %

i

∞ ∑

i

$ (2) + 1)

[

e

−i(kr −$π/2)

− S

$ (k)

e

i(kr −$π/2)

]

P

$ (cos θ)

Scattering phenomena: scattering cross section

ψ(r) % e

ik·r

• f (θ)

e

ikr

r

Particle flux associated with ψ(r) obtained from current operator,

j = −i

m

(ψ

∗ ∇ψ + ψ∇ψ

∗ ) = −i

m

Re[ψ

∗ ∇ψ]

= −i

m

Re

[

e

ik·r

• f (θ)

e

ikr

r

]

∗

[

e

ik·r

• f (θ)

e

ikr

r

]

Neglecting rapidly fluctuation contributions (which average to zero)

j =

ℏk

m

ℏk

m

ˆe r

|f (θ)|

2

r

2

• O(1/r

3 )

Scattering phenomena: scattering cross section

j =

ℏk

m

ℏk

m

ˆe r

|f (θ)|

2

r

2

• O(1/r

3 )

(Away from direction of incident beam, ˆe k

) the flux of particles

crossing area, dA = r

2 dΩ, that subtends solid angle dΩ at the

origin (i.e. the target) given by

NdΩ = j · ˆe r dA =

ℏk

m

|f (θ)|

2

r

2

r

2 dΩ + O(1/r )

By equating this flux with the incoming flux j I × dσ, where j I

ℏk

m

we obtain the differential cross section,

dσ =

NdΩ

j I

j · ˆe r dA

j I

= |f (θ)|

2 dΩ, i.e.

dσ

dΩ

= |f (θ)|

2

Scattering phenomena: partial waves

σ tot = 4π

$

(2) + 1)|f $ (k)|

2 , f (θ) =

∞ ∑

$=

(2) + 1)f $ (k)P $ (cos θ)

Making use of the relation f $ (k) =

2 ik

(e

2 iδ!(k) − 1) =

e

iδ!(k)

k

sin δ $

σ tot

4 π

k

2

∞ ∑

$=

(2) + 1) sin

2 δ $ (k)

Since P $ (1) = 1, f (0) =

$

(2) + 1)f $ (k) =

$

e

iδ !

(k)

k

sin δ $

Im f (0) =

k

4 π

σ tot

One may show that this “sum rule”, known as optical theorem,

encapsulates particle conservation.

Method of partial waves: summary

ψ(r) = e

ik·r

• f (θ)

e

ikr

r

The quantum scattering of particles from a localized target is fully

characterised by the differential cross section,

dσ

dΩ

= |f (θ)|

2

The scattering amplitude, f (θ), which depends on the energy

E = E

k , can be separated into a set of partial wave amplitudes,

f (θ) =

∞ ∑

$=

(2) + 1)f $ (k)P $ (cos θ)

where partial amplitudes, f $ (k) =

e

iδ !

k

sin δ $ defined by scattering

phase shifts δ $ (k). But how are phase shifts determined?