lec20-21_compressed scatter.pdf

8/10/2019 lec20-21_compressed SCATTER.pdf

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Lecture 20

Scattering theory


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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. Rutherfords discovery of the nucleus, the discovery of sub-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.


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Scattering theory: outline

Notations and denitions; lessons from classical scattering

Low energy scattering: method of partial waves

High energy scattering: Born perturbation series expansionScattering by identical particles

Bragg scattering.


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Scattering phenomena: background

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


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Scattering phenomena: di ff 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 ux of particles, j Idened as the number of particles per unit time crossing a unit areanormal to direction of incidence.

Collisions are characterised by the diff erential cross section denedas the ratio of the number of particles scattered into direction ( , )per unit time per unit solid angle, divided by incident ux,

d d

= N j I


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Scattering phenomena: cross section

From the diff erential, we can obtain the total cross section by

integrating over all solid angles

= d d d = 2

0d

0d sin

d d

The cross section, which typically depends sensitively on energy of incoming 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.


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Scattering phenomena: classical theory

In classical mechanics, for a central

potential, 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 ux j I , 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 d b j I

i.e. d ()

d

N j I

= b sin

db d


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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 nd that db d = R

2 sin(/ 2) and

d ()d =

R 2

4

As expected, total scattering cross section is just d d d = R 2,the projected area of the sphere.


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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 21

sin4 / 2


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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: ux of particles, all at the same energy, scattered fromtarget and collected by detectors which measure angles of deection.

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

i t (r , t ) = 2

2m2 + V (r) (r, t )

and nd probability amplitudes for outgoing waves.


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Quantum scattering: basics and notation

However, if beam is switched on for times long as compared withencounter-time, steady-state conditions apply.

If wavepacket has well-dened 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

2m2 + V (r) (r)

subject to boundary conditions that incoming component of wavefunction is a plane wave, e i kr (cf. 1d scattering problems).

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

j = i

2m ( ) =

km


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Lessons from revision of one-dimension

In one-dimension, interaction of plane wave, e ikx , with localized

target results in degree of reection and transmission.

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

Inuence of potential encoded in complex amplitude of reectedand transmitted wave xed by time-independent Schr odingerequation subject to boundary conditions (ux conservation).


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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)

ikr


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Scattering phenomena: partial waves

If we dene 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,

i kr = i

e i (kr / 2)

e i (kr / 2)


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(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 i kr + f ()e ikr

r

f () =

=0

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

where f (k ) = S (k ) 1

2ik dene partial wave scattering amplitudes .

i.e. f (k ) are dened by phase shifts, (k ), where S (k ) = e 2i (k ) .But how are phase shifts related to cross section?


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Scattering phenomena: scattering cross section

(r) e i kr + f () e ikr

r

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

j = i

m ( + ) = i

m Re [ ]

= i

mRe e i kr + f ()

e ikr

r e i kr + f ()

e ikr

r

Neglecting rapidly uctuation contributions (which average to zero)

j = km

+ k m

e r |f ()|2

r 2 + O (1/ r 3)


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d d

= |f ()|2, f () =

=0

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

From the expression for d d , we obtain the total scatteringcross-section:

tot =

d =

|f ()|2d

With orthogonality relation, d P (cos )P (cos ) = 42 + 1 , tot =

,

(2 + 1)(2 + 1) f (k )f (k )

d P (cos )P (cos )

4 / (2 +1)= 4 (2 + 1) |f (k )|2


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tot = 4 (2 + 1) |f (k )|2, f () =

=0

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

Making use of the relation f (k ) = 12ik

(e 2i (k ) 1) = e i (k )

k sin ,

tot = 4k 2

=0

(2 + 1) sin 2 (k )

Since P (1) = 1, f (0) = (2 + 1) f (k ) = (2 + 1) e i (k )

k sin ,

Im f (0) = k 4 tot

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


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Method of partial waves: summary

(r) = e i kr

+ f ()e ikr

r

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

d d = |f ()|2

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

f (

) =

=0 (2 + 1)f

(k

)P

(cos)

where partial amplitudes, f (k ) = e i

k sin dened by scatteringphase shifts (k ). But how are phase shifts determined?


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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 simultaneous

eigenfunction of L

2

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 ).


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(r) =

=0

R (r )P (cos )

Starting with Schr odinger equation for scattering wavefunction,

p2

2m + V (r ) (r) = E (r), E =

2k 2

2m

separability of (r) leads to radial equation,

2

2m 2r +

2r r

( + 1)r 2

+ V (r ) R (r ) = 2k 2

2m R (r )

Rearranging equation, we obtain the radial equation,

2r + 2r r

( + 1)r 2

U (r ) + k 2 R (r ) = 0

where U (r ) = 2 mV (r )/ 2 represents eff ective potential.


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2r + 2r r

( + 1)r 2

U (r ) + k 2 R (r ) = 0

Providing potential suffi 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 e 2i (k )

e i (kr / 2)r

R 0(r ) 1kr

e i 0(k ) sin(kr + 0(k ))

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

Here, with u (r ) = rR 0(r ), radial equation becomes,

2r U (r ) + k 2 u (r ) = 0

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


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2r U (r ) + k 2 u (r ) = 0

Alongside phase shift, 0 it is convenient to introduce scatteringlength , a0, dened 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(ka

0) + cos( kr )] sin

0 [ka

0 cot

0 + 1]

leads to scattering length a0 = limk 0

1k

tan 0(k ).

From this result, we nd the scattering cross section

tot = 4k 2

sin2 0(k ) k 0 4k 2(ka0)

2

1 + ( ka0)2 4 a20

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


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Example I: Scattering by hard-sphere

2r U (r ) + k 2 u (r ) = 0 , a0 = limk 0

1k

tan 0

Consider hard sphere potential,

U (r ) = r < R 0 r > R

With the boundary condition u (R ) = 0, suitable for an impenetrable

sphere, the scattering wavefunction given byu (r ) = A sin(kr + 0), 0 = kR

i.e. scattering length a0 R , f 0(k ) = e ikR

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

tot 4 sin2(kR )

k 2 4 R 2

Factor of 4 enhancement over classical value, R 2, due todiff raction processes at sharp potential.


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Example II: Scattering by attractive square well

2r U (r ) + k 2 u (r ) = 0

As a proxy for scattering from a binding potential, let us consider

quantum particles incident upon an attractive square well potential,U (r ) = U 0(R r ), where U 0 > 0.

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

2r + U 0(R r ) + k

2

u (r ) = 0with the boundary condition u (0) = 0.


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2r + U 0(R r ) + k

2u (r ) = 0

From this radial equation, we obtain the solution,

u (r ) = C sin(Kr ) r < R

sin(kr + 0) r > R where K 2 = k 2 + U 0 > k 2 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)


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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 = k 2 + U 0

With kR 1, K U 1/ 20 (1 + O (k 2/ U 0)), nd scattering length,

a0 = limk 01k tan 0 R

tan( KR )KR 1

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


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a0 = limk 01k tan 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 4k 2

sin2 0(k ) 4 R 2 tan( KR )KR 1

2

, K U 1/ 20

But what happens when KR / 2?


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a0 R tan( KR )

KR 1 , K U 1/ 20

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

As KR / 2, both scattering length a0 andcross section tot 4 a20 diverge.

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


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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 inuence of the target becomes eff ectively innite 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 ff ect .


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Resonances

More generally, the -th partial cross-section

= 4k 2

(2 + 1) 1

1 + cot 2 (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 ) = E R E

(E )/ 2where E R is resonance energy.


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Resonances

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

(E ) = 4k 2

(2 + 1) 2(E R )/ 4

(E E R )2 + 2(E R )/ 4

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

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


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Application: Feshbach resonance phenomena

Ultracold atomic gases provide arena in

which resonant scattering phenomenaexploited far from resonance, neutralalkali atoms interact through short-rangedvan der Waals interaction.

However, eff 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 eld, system can betuned through resonance.

This allows eff ective interaction to betuned from repulsive to attractive simplyby changing external eld.


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Scattering theory: summary

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

d d = |f ()|

2

where (r) = e i kr + f (, ) e ikr

r denotes scattering wavefunction.


f () =

=0

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

where f (k ) = e i

k sin dened by scattering phase shifts (k ).


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The partial amplitudes/phase shifts fully characterise scattering,

tot = 4k 2

=0

(2 + 1) sin 2 (k )

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

2r + 2r r

( + 1)r 2

U (r ) + k 2 R (r ) = 0

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

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


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Lecture 21

Scattering theory:Born perturbation series expansion

R


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Recap

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

p2

2m + V (r ) (r) = 2

k 2

2m (r)

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

(r) e i kr + 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 diff erent methodology, theBorn series expansion

Li S h i i


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Lippmann-Schwinger equation

Returning to Schrodinger equation for scattering wavefunction,

2 + k 2 (r) = U (r) (r)

with V (r) = 2U (r)

2m , general solution can be written as

(r) = (r) +

G 0(r, r )U (r ) (r )d 3r

where 2 + k 2 (r) = 0 and 2 + k 2 G 0(r, r ) = d (r r ).

Formally, these equations have the solution

k(r) = e i kr , G 0(r, r ) =

14

e ik | r r |

|r r |

Li S h i i


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Together, leads to Lippmann-Schwinger equation:

k(r) = e i kr 1

4 d 3r e

ik | r

r |

|r r | U (r )k(r )

In far-eld, |r r | r e r r + ,

e ik | r r ||r r |

e ikr

r e i k r

where k k e r .

i.e. k(r) = e i kr + f (, )e ikr

r where, with k = e i kr ,

f (, ) 14 d 3r e i k r U (r )k(r ) 14 k |U |k

Li S h i i


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f (, ) = 14 d

3r e

i k rU (r )k(r )

14 k |U |k

The corresponding diff erential cross-section:

d

d = |f (, )|2 =

m2

(2 )2 4 |T k,k |2

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

2m ,

T k,k = k |V |k

denotes the transition matrix element.


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B i ti


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Born approximation

f = 14

k |U + UG 0U + UG 0UG 0U + |k

Leading term in Born series known as rst Born approximation ,

f Born = 1

4k |U |k

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

f Born ( ) = 14 d

3r e i

r U (r) =

0rdr

sin( r ) U (r )

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

E ample: Co lomb scattering


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Example: Coulomb scattering

Due to long range nature of the Coulomb scattering potential, the

boundary condition on the scattering wavefunction does not apply.We can, however, address the problem by working with the screened(Yukawa) potential, U (r ) = U 0

e r /

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

f Born = U 0

0 dr sin( r )

e r /

= U 0

2 + 2

Therefore, for , we obtain

d

d

= |f ()|2 = U 20

16k 4

sin4

/ 2which is just the Rutherford formula .

From Born approximation to Fermis Golden rule


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Previously, in the leading approximation, we found that thetransition rate between states and i and f induced by harmonic

perturbation Ve i t

is given by Fermis Golden rule,

i f = 2

| f |V | i |2 ( (E f E i ))

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

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

k k =k d

2 | k |V |k |2 (E k E k ) =

2 | k |V |k |2g (E k )

where g (E k ) = k (E k E k ) = dndE is density of states at energy

E k = 2 k 2

2m .



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From Born approximation to Fermi s Golden rule

k k = 2 | k |V |k |2g (E k )

With

g (E k ) = dndk

dk dE

= k 2d

(2 / L)31

2k / m

and incident ux j I = k / m , the diff erential cross section,

d d

= 1L3

k k

j I=

1(4 )2

| k |2mV

2 |k |2

At rst order, Born approximation and Golden rule coincide.

Scattering by identical particles


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So far, we have assumed that incident particles and target are

distinguishable. 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) deected through , or (b) through .

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



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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 diff 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


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Scattering by an atomic lattice

As a nal 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 i k(r R i ) + f e ik | r R i |

|r R i |

Since k |r R i | kr k R i , with k = k e r we have

(r) = e i kR i e i kr + fe i (k k)R i e ikr

r From this result, we infer eff ective scattering amplitude,

f () = f exp[ i R i ] , = k k


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

d

d = |f ()|2

where (r) = e i kr + f (, ) e ikr

r denotes scattering wavefunction.


f () =

=0

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

where f (k ) = e i

k sin dened by scattering phase shifts (k ).



53/54


The partial amplitudes/phase shifts fully characterise scattering,

tot = 4k 2

=0

(2 + 1) sin 2 (k )

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

2r + 2r r

( + 1)r 2

U (r ) + k 2 R (r ) = 0

Although this methodology is straightforward, when the energy of incident particles is high (or the potential weak), many partial wavescontribute.In this case, it is convenient to switch to a diff erent formalism, theBorn approximation.



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Formally, the solution of the scattering wavefunction can be

presented as the integral (Lippmann-Schwinger) equation,

k(r) = e i kr

14 d 3r 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,

f Born ( ) = 1

4 d 3re i r U (r)

where = k k and = 2 k sin(/ 2).

lec20-21_compressed scatter.pdf

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