Lecture 20 Scattering theory

Lecture 20

Scattering theory

Scattering theory

Scattering theory is important as it underpins one of the most ubiquitoustools in physics.

Almost everything we know about nuclear and atomic physics hasbeen discovered by scattering experiments,

e.g. Rutherford’s discovery of the nucleus, the discovery ofsub-atomic particles (such as quarks), etc.

In low energy physics, scattering phenomena provide the standardtool to explore solid state systems,

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

As a general topic, it therefore remains central to any advancedcourse on quantum mechanics.

In these two lectures, we will focus on the general methodologyleaving applications to subsequent courses.

Scattering theory: outline

Notations and definitions; lessons from classical scattering

Low energy scattering: method of partial waves

High energy scattering: Born perturbation series expansion

Scattering by identical particles

Bragg scattering.

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 elasticA + B!A + B + C

%inelastic

C absorption

In high energy and nuclear physics, we are usually interested in deepinelastic processes.

To keep our discussion simple, we will focus on elastic processes inwhich both the energy and particle number are conserved –although many of the concepts that we will develop are general.

Scattering phenomena: di!erential cross section

Both classical and quantum mechanical scattering phenomena arecharacterized by the scattering cross section, !.

Consider a collision experiment in which a detector measures thenumber of particles per unit time, N d!, scattered into an elementof solid angle d! in direction (", #).

This number is proportional to the incident flux of particles, jIdefined as the number of particles per unit time crossing a unit areanormal to direction of incidence.

Collisions are characterised by the di!erential cross section definedas the ratio of the number of particles scattered into direction (",#)per unit time per unit solid angle, divided by incident flux,

d!

d!=

N

jI

Scattering phenomena: cross section

From the di"erential, we can obtain the total cross section byintegrating over all solid angles

! =

&d!

d!d! =

& 2!

0d#

& !

0d" sin "

d!

d!

The cross section, which typically depends sensitively on energy ofincoming particles, has dimensions of area and can be separated into!elastic, !inelastic, !abs, and !total.

In the following, we will focus on elastic scattering where internalenergies remain constant and no further particles are created orannihilated,

e.g. low energy scattering of neutrons from protons.

However, before turning to quantum scattering, let us considerclassical scattering theory.

Scattering phenomena: classical theory

In classical mechanics, for a centralpotential, V (r), the angle of scattering isdetermined by impact parameter b(").

The number of particles scattered per unittime between " and " + d" is equal to thenumber incident particles per unit timebetween b and b + db.

Therefore, for incident flux jI, the numberof particles scattered into the solid angled! =2 $ sin " d" per unit time is given by

N d! =2 $ sin " d" N = 2$b db jI

i.e.d!(")

d!# N

jI=

b

sin "

''''db

d"

''''


d!(")

d!=

b

sin "

''''db

d"

''''

For elastic scattering from a hard (impenetrable) sphere,

b(") = R sin% = R sin

($ ! "

2

)= !R cos("/2)

As a result, we find that'' dbd"

'' = R2 sin("/2) and

d!(")

d!=

R2

4

As expected, total scattering cross section is just*

d! d#d! = $R2,

the projected area of the sphere.


For classical Coulomb scattering,

V (r) =&

r

particle follows hyperbolic trajectory.

In this case, a straightforward calculationobtains the Rutherford formula:

d!

d!=

b

sin "

''''db

d"

'''' =&2

16E 2

1

sin4 "/2

At high energy, there is a departure – manifestation of nuclearstructure.

Quantum scattering: basics and notation

Simplest scattering experiment: plane wave impinging on localizedpotential, V (r), e.g. electron striking atom, or % particle a nucleus.

Basic set-up: flux of particles, all at the same energy, scattered fromtarget and collected by detectors which measure angles of deflection.

In principle, if all incoming particles represented by wavepackets, thetask is to solve time-dependent Schrodinger equation,

i! 't#(r, t) =

+! !2

2m$2 + V (r)

,#(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), mayconsider it a plane wave: #(r, t) = ((r)e"iEt/!.

Therefore, seek solutions of time-independent Schrodinger equation,

E((r) =

+! !2

2m$2 + V (r)

,((r)

subject to boundary conditions that incoming component ofwavefunction is a plane wave, e ik·r (cf. 1d scattering problems).

E = (!k)2/2m is energy of incoming particles while flux given by,

j = !i!

2m((!$( ! ($(!) =

!km

Lessons from revision of one-dimension

In one-dimension, interaction of plane wave, e ikx , with localizedtarget results in degree of reflection and transmission.

Both components of outgoing scattered wave are plane waves withwavevector ±k (energy conservation).

Influence of potential encoded in complex amplitude of reflectedand transmitted wave – fixed by time-independent Schrodingerequation subject to boundary conditions (flux conservation).

Scattering in more than one dimension

In higher dimension, phenomenology is similar – consider plane waveincident on localized target:

Outside localized target region, wavefunction involves superpositionof incident plane wave and scattered (spherical wave)

((r) % e ik·r + f (",#)e ikr

r

Scattering amplitude f (",#) records complex amplitude ofcomponent along direction (",#) relative to incident beam.

Scattering phenomena: partial waves

If we define z-axis by k vector, plane wave can be decomposed intosuperposition 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 = ((r) % i

2k

#-

$=0

i$(2) + 1)

+e"i(kr"$!/2)

r! S$(k)

e i(kr"$!/2)

r

,P$(cos ")

where P$(cos ") = ( 4!2$+1 )1/2Y$0(").where |S$(k)| = 1 (i.e.

S$(k) = e2i%!(k) with *$(k) are phase shifts).


((r) % i

2k

#-

$=0

i$(2) + 1)

+e"i(kr"$!/2)

r! S$(k)

e i(kr"$!/2)

r

,P$(cos ")

If we set, ((r) % e ik·r + f (")e ikr

r

f (") =#-

$=0

(2) + 1)f$(k)P$(cos ")

where f$(k) =S$(k)! 1

2ikdefine partial wave scattering amplitudes.

i.e. f$(k) are defined by phase shifts, *$(k), where S$(k) = e2i%!(k).But how are phase shifts related to cross section?

Scattering phenomena: scattering cross section

((r) % e ik·r + f (")e ikr

r

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

j = !i!m

((!$( + ($(!) = !i!m

Re[(!$(]

= !i!m

Re

.+e ik·r + f (")

e ikr

r

,!$

+e ik·r + f (")

e ikr

r

,/

Neglecting rapidly fluctuation contributions (which average to zero)

j =!km

+!k

mer

|f (")|2

r2+ O(1/r3)

Scattering phenomena: scattering cross section

j =!km

+!k

mer

|f (")|2

r2+ O(1/r3)

(Away from direction of incident beam, ek) the flux of particlescrossing area, dA = r2d!, that subtends solid angle d! at theorigin (i.e. the target) given by

Nd! = j · erdA =!k

m

|f (")|2

r2r2d! + O(1/r)

By equating this flux with the incoming flux jI & d!, where jI = !km ,

we obtain the di!erential cross section,

d! =Nd!

jI=

j · er dA

jI= |f (")|2 d!, i.e.

d!

d!= |f (")|2


d!

d!= |f (")|2, f (") =

#-

$=0

(2) + 1)f$(k)P$(cos ")

From the expression for d#d! , we obtain the total scattering

cross-section:

!tot =

&d! =

&|f (")|2d!

With orthogonality relation,

&d! P$(cos ")P$!(cos ") =

4$

2) + 1*$$! ,

!tot =-

$,$!

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

&d!P$(cos ")P$!(cos ")

0 12 34!%!!!/(2$+1)

= 4$-

$

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


!tot = 4$-

$

(2) + 1)|f$(k)|2, f (") =#-

$=0

(2) + 1)f$(k)P$(cos ")

Making use of the relation f$(k) =1

2ik(e2i%!(k) ! 1) =

e i%!(k)

ksin *$,

!tot =4$

k2

#-

$=0

(2) + 1) sin2 *$(k)

Since P$(1) = 1, f (0) =4

$(2) + 1)f$(k) =4

$(2) + 1) ei"!(k)

k sin *$,

Im f (0) =k

4$!tot

One may show that this “sum rule”, known as optical theorem,encapsulates particle conservation.

Method of partial waves: summary

((r) = e ik·r + f (")e ikr

r

The quantum scattering of particles from a localized target is fullycharacterised by the di"erential cross section,

d!

d!= |f (")|2

The scattering amplitude, f ("), which depends on the energyE = Ek , can be separated into a set of partial wave amplitudes,

f (") =#-

$=0

(2) + 1)f$(k)P$(cos ")

where partial amplitudes, f$(k) = ei"!

k sin *$ defined by scatteringphase shifts *$(k). But how are phase shifts determined?

Method of partial waves

For scattering from a central potential,the scattering amplitude, f , must besymmetrical about axis of incidence.

In this case, both scattering wavefunction, ((r), and scatteringamplitudes, f ("), can be expanded in Legendre polynomials,

((r) =#-

$=0

R$(r)P$(cos ")

cf. wavefunction for hydrogen-like atoms with m = 0.

Each term in expansion known as partial wave, and is simultaneouseigenfunction of L2 and Lz having eigenvalue !2)() + 1) and 0, with) = 0, 1, 2, · · · referred to as s, p, d , · · · waves.

From the asymtotic form of ((r) we can determine the phase shifts*$(k) and in turn the partial amplitudes f$(k).


((r) =#-

$=0

R$(r)P$(cos ")

Starting with Schrodinger equation for scattering wavefunction,+

p2

2m+ V (r)

,((r) = E((r), E =

!2k2

2m

separability of ((r) leads to radial equation,+! !2

2m

('2

r +2

r'r !

)() + 1)

r2

)+ V (r)

,R$(r) =

!2k2

2mR$(r)

Rearranging equation, we obtain the radial equation,

+'2

r +2

r'r !

)() + 1)

r2! U(r) + k2

,R$(r) = 0

where U(r) = 2mV (r)/!2 represents e"ective potential.


+'2

r +2

r'r !

)() + 1)

r2! U(r) + k2

,R$(r) = 0

Providing potential su$ciently short-ranged, scattering wavefunctioninvolves superposition of incoming and outgoing spherical waves,

R$(r) %i

2k

#-

$=0

i$(2) + 1)

(e"i(kr"$!/2)

r! e2i%!(k) e

i(kr"$!/2)

r

)

R0(r) %1

kre i%0(k) sin(kr + *0(k))

However, at low energy, kR ' 1, where R is typical range ofpotential, s-wave channel () = 0) dominates.

Here, with u(r) = rR0(r), radial equation becomes,

5'2

r ! U(r) + k26u(r) = 0

with boundary condition u(0) = 0 and, as expected, outside radius

of potential, R, u(r)r%R= A sin(kr + *0).


5'2

r ! U(r) + k26u(r) = 0

Alongside phase shift, *0 it is convenient to introduce scatteringlength, a0, defined by condition that u(a0) = 0 for kR ' 1, i.e.

u(a0) = sin(ka0 + *0) = sin(ka0) cos *0 + cos(ka0) sin *0

= sin *0 [cot *0 sin(ka0) + cos(kr)] % sin *0 [ka0 cot *0 + 1]

leads to scattering length a0 = ! limk&0

1

ktan *0(k).

From this result, we find the scattering cross section

!tot =4$

k2sin2 *0(k)

k&0% 4$

k2

(ka0)2

1 + (ka0)2% 4$a2

0

i.e. a0 characterizes e"ective size of target.

Example I: Scattering by hard-sphere

5'2

r ! U(r) + k26u(r) = 0, a0 = ! lim

k&0

1

ktan *0

Consider hard sphere potential,

U(r) =

7( r < R0 r > R

With the boundary condition u(R) = 0, suitable for an impenetrablesphere, the scattering wavefunction given by

u(r) = A sin(kr + *0), *0 = !kR

i.e. scattering length a0 % R, f0(k) = eikR

k sin(kR), and the totalscattering cross section is given by,

!tot % 4$sin2(kR)

k2% 4$R2

Factor of 4 enhancement over classical value, $R2, due todi"raction processes at sharp potential.

Example II: Scattering by attractive square well

5'2

r ! U(r) + k26u(r) = 0

As a proxy for scattering from a binding potential, let us considerquantum particles incident upon an attractive square well potential,U(r) = !U0"(R ! r), where U0 > 0.

Once again, focussing on low energies, kR ' 1, this translates tothe radial potential,

5'2

r + U0"(R ! r) + k26u(r) = 0

with the boundary condition u(0) = 0.


5'2

r + U0"(R ! r) + k26u(r) = 0

From this radial equation, we obtain the solution,

u(r) =

7C sin(Kr) r < Rsin(kr + *0) r > R

where K 2 = k2 + U0 > k2 and *0 denotes scattering phase shift.

From continuity of wavefunction and derivative at r = R,

C sin(KR) = sin(kR + *0), CK cos(KR) = k cos(kR + *0)

we obtain the self-consistency condition for *0 = *0(k),

K cot(KR) = k cot(kR + *0)


K cot(KR) = k cot(kR + *0)

From this result, we obtain

tan *0(k) =k tan(KR)! K tan(kR)

K + k tan(kR) tan(KR), K 2 = k2 + U0

With kR ' 1, K % U1/20 (1 + O(k2/U0)), find scattering length,

a0 = ! limk&0

1

ktan *0 % !R

(tan(KR)

KR! 1

)

which, for KR < $/2 leads to a negative scattering length.


a0 = ! limk&0

1

ktan *0 % !R

(tan(KR)

KR! 1

)

So, at low energies, the scattering from an attractive squarepotential leads to the ) = 0 phase shift,

*0 % !ka0 % kR

(tan(KR)

KR! 1

)

and total scattering cross-section,

!tot %4$

k2sin2 *0(k) % 4$R2

(tan(KR)

KR! 1

)2

, K % U1/20

But what happens when KR % $/2?


a0 % !R

(tan(KR)

KR! 1

), K % U1/2

0

If KR ' 1, a0 < 0 and wavefunction drawntowards target – hallmark of attractive potential.

As KR " $/2, both scattering length a0 andcross section !tot % 4$a2

0 diverge.

As KR increased, a0 turns positive, wavefunctionpushed away from target (cf. repulsive potential)until KR = $ when !tot = 0 and process repeats.

Why?


In fact, when KR = $/2, the attractive square well just meets thecriterion to host a single s-wave bound state.

At this value, there is a zero energy resonance leading to thedivergence of the scattering length, and with it, the cross section –the influence of the target becomes e"ectively infinite in range.

When KR = 3$/2, the potential becomes capable of hosting asecond bound state, and there is another resonance, and so on.

When KR = n$, the scattering cross section vanishes identically andthe target becomes invisible – the Ramsauer-Townsend e!ect.

Resonances

More generally, the )-th partial cross-section

!$ =4$

k2(2) + 1)

1

1 + cot2 *$(k), !tot =

-

$

!$

takes maximum value if there is an energy at which cot *$ vanishes.

If this occurs as a result of *$(k) increasing rapidly through oddmultiple of $/2, cross-section exhibits a narrow peak as a functionof energy – a resonance.

Near the resonance,

cot *$(k) =ER ! E

%(E )/2

where ER is resonance energy.

Resonances

If %(E ) varies slowly in energy, partial cross-section in vicinity ofresonance given by Breit-Wigner formula,

!$(E ) =4$

k2(2) + 1)

%2(ER)/4

(E ! ER)2 + %2(ER)/4

Physically, at E = ER, the amplitude of the wavefunction within thepotential region is high and the probability of finding the scatteredparticle inside the well is correspondingly high.

The parameter % = !/+ represents typical lifetime, + , of metastablebound state formed by particle in potential.

Application: Feshbach resonance phenomena

Ultracold atomic gases provide arena inwhich resonant scattering phenomenaexploited – far from resonance, neutralalkali atoms interact through short-rangedvan der Waals interaction.

However, e"ective strength of interactioncan be tuned by allowing particles to formvirtual bound state – a resonance.

By adjusting separation between entrancechannel states and bound state throughexternal magnetic field, system can betuned through resonance.

This allows e"ective interaction to betuned from repulsive to attractive simplyby changing external field.

Scattering theory: summary

The quantum scattering of particles from a localized target is fullycharacterised by di"erential cross section,

d!

d!= |f (")|2

where ((r) = e ik·r + f (",#) eikr

r denotes scattering wavefunction.


f (") =#-

$=0

(2) + 1)f$(k)P$(cos ")

where f$(k) = ei"!

k sin *$ defined by scattering phase shifts *$(k).


The partial amplitudes/phase shifts fully characterise scattering,

!tot =4$

k2

#-

$=0

(2) + 1) sin2 *$(k)

The individual scattering phase shifts can then be obtained from thesolutions to the radial scattering equation,

+'2

r +2

r'r !

)() + 1)

r2! U(r) + k2

,R$(r) = 0

Although this methodology is “straightforward”, when the energy ofincident particles is high (or the potential weak), many partial wavescontribute.

In this case, it is convenient to switch to a di"erent formalism, theBorn approximation.

Lecture 21

Scattering theory:Born perturbation series expansion

Recap

Previously, we have seen that the properties of a scattering system,

+p2

2m+ V (r)

,((r) =

!2k2

2m((r)

are encoded in the scattering amplitude, f (",#), where

((r) % e ik·r + f (",#)e ikr

r

For an isotropic scattering potential V (r), the scattering amplitudes,f ("), can be obtained as an expansion in harmonics, P$(cos ").

At low energies, k " 0, this partial wave expansion is dominatedby small ).

At higher energies, when many partial waves contribute, expansionis inconvenient – helpful to develop a di"erent methodology, theBorn series expansion

Lippmann-Schwinger equation

Returning to Schrodinger equation for scattering wavefunction,

8$2 + k2

9((r) = U(r)((r)

with V (r) = !2U(r)2m , general solution can be written as

((r) = #(r) +

&G0(r, r

$)U(r$)((r$)d3r $

where8$2 + k2

9#(r) = 0 and

8$2 + k2

9G0(r, r$) = *d(r ! r$).

Formally, these equations have the solution

#k(r) = e ik·r, G0(r, r$) = ! 1

4$

e ik|r"r!|

|r ! r$|


Together, leads to Lippmann-Schwinger equation:

(k(r) = e ik·r ! 1

4$

&d3r $

e ik|r"r!|

|r ! r$| U(r$)(k(r$)

In far-field, |r ! r$| % r ! er · r$ + · · · ,

e ik|r"r!|

|r ! r$| %e ikr

re"ik!·r!

where k$ # k er .

i.e. (k(r) = e ik·r + f (",#) eikr

r where, with #k = e ik·r,

f (",#) % ! 1

4$

&d3r $ e"ik!·r!U(r$)(k(r

$) # ! 1

4$)#k! |U|(k*


f (",#) = ! 1

4$

&d3r $ e"ik!·r!U(r$)(k(r

$) # ! 1

4$)#k! |U|(k*

The corresponding di"erential cross-section:

d!

d!= |f (",#)|2 =

m2

(2$)2!4|Tk,k! |2

where, in terms of the original scattering potential, V (r) = !2U(r)2m ,

Tk,k! = )#k! |V |(k*

denotes the transition matrix element.

Born approximation

((r) = #(r) +

&G0(r, r

$)U(r$)((r$)d3r $ (+)

At zeroth order in V (r), scattering wavefunction translates to

unperturbed incident plane wave, ((0)k (r) = #k(r) = e ik·r.

In this approximation, (*) leads to expansion first order in U,

((1)k (r) = #k(r) +

&d3r $ G0(r, r

$)U(r$)((0)k (r$)

and then to second order in U,

((2)k (r) = #k(r) +

&d3r $ G0(r, r

$)U(r$)((1)k (r$)

and so on.

i.e. expressed in coordinate-independent basis,

|(k* = |#k*+ G0U|#k*+ G0UG0U|#k*+ · · · =#-

n=0

(G0U)n|#k*

Born approximation

|(k* = |#k*+ G0U|#k*+ G0UG0U|#k*+ · · · =#-

n=0

(G0U)n|#k*

Then, making use of the identity f (",#) = ! 14! )#k! |U|(k*,

scattering amplitude expressed as Born series expansion

f = ! 1

4$)#k! |U + UG0U + UG0UG0U + · · · |#k*

Physically, incoming particle undergoes a sequence of multiplescattering events from the potential.

Born approximation

f = ! 1

4$)#k! |U + UG0U + UG0UG0U + · · · |#k*

Leading term in Born series known as first Born approximation,

fBorn = ! 1

4$)#k! |U|#k*

Setting " = k! k$, where !" denotes momentum transfer, Bornscattering amplitude for a central potential

fBorn(&) = ! 1

4$

&d3r e i"·rU(r) = !

& #

0rdr

sin(&r)

&U(r)

where, noting that |k$| = |k|, & = 2k sin("/2).

Example: Coulomb scattering

Due to long range nature of the Coulomb scattering potential, theboundary condition on the scattering wavefunction does not apply.

We can, however, address the problem by working with the screened

(Yukawa) potential, U(r) = U0e"r/#

r , and taking % "(. For thispotential, one may show that (exercise)

fBorn = !U0

& #

0dr

sin(&r)

&e"r/& = ! U0

%"2 + &2

Therefore, for % "(, we obtain

d!

d!= |f (")|2 =

U20

16k4 sin4 "/2

which is just the Rutherford formula.

From Born approximation to Fermi’s Golden rule

Previously, in the leading approximation, we found that thetransition rate between states and i and f induced by harmonicperturbation Ve i't is given by Fermi’s Golden rule,

%i&f =2$

! |)f|V |i*|2*(!, ! (Ef ! Ei))

In a three-dimensional scattering problem, we should consider theinitial state as a plane wave state of wavevector k and the finalstate as the continuum of states with wavevectors k$ with , = 0.

In this case, the total transition (or scattering) rate into a fixed solidangle, d!, in direction (", #) given by

%k&k! =-

k!'d!

2$

! |)k$|V |k*|2*(Ek ! Ek!) =2$

! |)k$|V |k*|2g(Ek)

where g(Ek) =4

k! *(Ek ! Ek!) = dndE is density of states at energy

Ek = !2k2

2m .

From Born approximation to Fermi’s Golden rule

%k&k! =2$

! |)k$|V |k*|2g(Ek)

With

g(Ek) =dn

dk

dk

dE=

k2d!

(2$/L)31

!2k/m

and incident flux jI = !k/m, the di"erential cross section,

d!

d!=

1

L3

%k&k!

jI=

1

(4$)2|)k$|2mV

!2|k*|2

At first order, Born approximation and Golden rule coincide.


So far, we have assumed that incident particles and target aredistinguishable. When scattering involves identical particles, wehave to consider quantum statistics:

Consider scattering of two identical particles. In centre of massframe, if an outgoing particle is detected at angle " to incoming, itcould have been (a) deflected through ", or (b) through $ ! ".

Classically, we could tell whether (a) or (b) by monitoring particlesduring collision – however, in quantum scattering, we cannot track.


Therefore, in centre of mass frame, we must write scatteringwavefunction in appropriately symmetrized/antisymmetrized form –for bosons,

((r) = e ikz + e"ikz + (f (") + f ($ ! "))e ikr

r

The di"erential cross section is then given by

d!

d!= |f (") + f ($ ! ")|2

as opposed to |f (")|2 + |f ($! ")|2 as it would be for distinguishableparticles.

Scattering by an atomic lattice

As a final application of Born approximation,consider scattering from crystal lattice: At lowenergy, scattering amplitude of particles is againindependent of angle (s-wave).

In this case, the solution of the Schrodinger equation by a singleatom i located at a point Ri has the asymptotic form,

((r) = e ik·(r"Ri ) + fe ik|r"Ri |

|r ! Ri |

Since k|r ! Ri | % kr ! k$ · Ri , with k$ = k er we have

((r) = e"ik·Ri

+e ik·r + fe"i(k!"k)·Ri

e ikr

r

,

From this result, we infer e"ective scattering amplitude,

f (") = f exp [!i" · Ri ] , " = k$ ! k

Scattering by an atomic lattice

If we consider scattering from a crystal lattice, we must sum over allatoms leading to the total di"erential scattering cross-section,

d!

d!= |f (")|2 =

'''''f-

Ri

exp [!i" · Ri ]

'''''

2

For periodic cubic crystal of dimension Ld ,sum translates to Bragg condition,

d!

d!= |f |2 (2$)3

L3*(3)(k$ ! k! 2$n/L)

where integers n known as Miller indices ofBragg planes.


The quantum scattering of particles from a localized target is fullycharacterised by di"erential cross section,

d!

d!= |f (")|2

where ((r) = e ik·r + f (",#) eikr

r denotes scattering wavefunction.


f (") =#-

$=0

(2) + 1)f$(k)P$(cos ")

where f$(k) = ei"!

k sin *$ defined by scattering phase shifts *$(k).


The partial amplitudes/phase shifts fully characterise scattering,

!tot =4$

k2

#-

$=0

(2) + 1) sin2 *$(k)

The individual scattering phase shifts can then be obtained from thesolutions to the radial scattering equation,

+'2

r +2

r'r !

)() + 1)

r2! U(r) + k2

,R$(r) = 0

Although this methodology is “straightforward”, when the energy ofincident particles is high (or the potential weak), many partial wavescontribute.

In this case, it is convenient to switch to a di"erent formalism, theBorn approximation.


Formally, the solution of the scattering wavefunction can bepresented as the integral (Lippmann-Schwinger) equation,

(k(r) = e ik·r ! 1

4$

&d3r $

e ik|r"r!|

|r ! r$| U(r$)(k(r$)

This expression allows the scattering amplitude to be developed as apower series in the interaction, U(r).

In the leading approximation, this leads to the Born approximationfor the scattering amplitude,

fBorn(") = ! 1

4$

&d3re i"·rU(r)

where " = k! k$ and & = 2k sin("/2).

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Lecture 20 Scattering theory