Lecture 6 Scattering theory Partial Wave Analysisatlas.physics.arizona.edu/~shupe/Indep_Studies_2015/Notes_Goethe... · Lecture 6 Scattering theory Partial Wave Analysis ... we may

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Lecture 6Lecture 6

Scattering theory Scattering theory

Partial Wave AnalysisPartial Wave Analysis

SS2011SS2011: : ‚‚Introduction to Nuclear and Particle Physics, Part 2Introduction to Nuclear and Particle Physics, Part 2‘‘

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Partial wave analysis for elastic scatteringPartial 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 and

after 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 :

(1)

(2)

•We can then examine how each of the partial waves is distorted by V(r ) after the particle

scatters from the potential.

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The most general solution of the Schrödinger equation (3) is

Consider the Schrödinger equation in CM frame:

(3)

(4)

Since V(r ) is central, the system is symmetrical (rotationally invariant) about the z-axis.

The scattered wave function must not then depend on the azimuthal angle ϕϕϕϕ; �� m =0.

Thus, as Yl0(θ,ϕ) ∼ Pl (cosθ), the scattered wave function (4) becomes

(5)

where Rkl (r ) obeys the following radial equation

(6)

Each term in (5), which is known as a partial wave, is a joint eigenfunction of L2 and LZ .

)µE2

khere(2

2

h====

4


A substitution of (2) into

with ϕϕϕϕ=0 (and k=k0 for elastic scattering) gives

(7)

(8)

The scattered wave function is given, on the one hand, by (5) and, on the other hand, by (8).

� Consider the limit

1) 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 jl(kr) for large values of r is given by

∞∞∞∞→→→→r

(9)

�� the asymptotic form of (8) is given by

(10)

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Since

because

one can write (10) as

(11)

(12)

2) To find the asymptotic form of (5), we need first to determine the asymptotic form

of the radial function Rkl (r ). At large values of r, the scattering potential is effectively

zero �� radial equation (6) becomes

(13)

The general solution of this equation is given by a linear combination of the spherical

Bessel and Neumann functions

(14)

where the asymptotic form of the Neumann function is

(15)

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Inserting (9) and (15) into (14), we obtain the asymptotic form of the radial function:

(16)

If V(r)=0 for all r (free particles), the solution of the radial equation (6), rRkl(r ), must

vanish at r=0; thus Rkl(r ) must be finite at the origin (at r=0).

� Since the Neumann function diverges at r=0, the cosine term in (16) does not

represent a physically acceptable solution;

�� one has to introduce the phase shift δδδδl to achieve the regular solution near the origin

by rewriting (14)

(((( )))) (((( )))) (((( )))) (((( ))))[[[[ ]]]]krnsinkrjcosC)r(R lllllkl δδδδδδδδ −−−−====

where we have

in the form (17)

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(18)

With δδδδl =0, the radial function Rkl(r ) of (18) is finite at r =0, since Rkl(r ) in (17)

reduces to jl(kr).

So δδδδl is a real angle which vanishes for all values of l in the absence of the scattering

potential (i.e., V =0); δδδδl is called the phase shift of the l‘th partial wave.

(((( )))) (((( ))))

kr

2

lkrcossin

2

lkrsincos

C)r(Rll

lrkl

−−−−−−−−

−−−−

→→→→∞∞∞∞→→→→

ππππδδδδ

ππππδδδδ

Thus, the asymptotic form of the radial function (16) can be written as

The phase shift δδδδl measures the ‚distortion‘ of Rkl(r ) from the ‚free‘ solution jl(kr)

due to the presence of the potential V(r )

Attractive (repulsive) potentials imply that δδδδl > 0 (δδδδl < 0)

corresponding to the wave being “pulled in” (“pushed out”) by

the scattering center resulting in a phase delay (advance).

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(19)

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 δδδδl.

Since

one can rewrite (19) as

(20)

Compare (20) and (12):

(12)

(21)

We obtain:

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Substituting (21) into (20) and this time equating the coefficient of in the resulting

expression with that of (12), we have

which by combining with

leads to

(22)

(23)

where fl(θ θ θ θ ) is denoted as the partial wave amplitude.

from (20)from (12)

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From (23) we obtain the differential cross sections (24)

and the total cross sections reads:

Using the relation

we obtain from (25):

(25)

(26)

where σσσσl are denoted as the partial cross sections corresponding to the scattering of particles

in various angular momentum states.

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�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=0 everywhere, all the phase shifts δδδδl 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=0, the scattering amplitude (23) becomes

where we have used

(27)

Since f0 does not depend on θθθθ , the differential and total cross sections in the CM

frame are given by the following simple relations:

(28)

)µE2

khere(2

2

h====

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Optical theoremOptical theorem

The total cross section in CM frame can be related to the forward scattering amplitude f (0).

Since for θ=θ=θ=θ=0, eq. (23) leads to

(29)

which - combined with (26) - yields the connection between f (0) and σσσσ :

(30)

This relation is known as the optical theorem.

The physical origin of this theorem is the conservation of particles (or probability):

the beam emerging (after scattering) along the incident direction (θθθθ=0) contains less

particles than the incident beam, since a number of particles have scattered in various other

directions. This decrease in the number of particles is measured by the total cross section σσσσ;

that is, the number of particles removed from the incident beam along the incident

direction is proportional to σσσσ or, equivalently, to the imaginary part of f (0).

Note: although (30) was derived for elastic scattering, the optical theorem (as will be shown

later) is also valid for inelastic scattering.

� 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, Sl (k) is redefined by

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Partial wave analysis for inelastic scatteringPartial wave analysis for inelastic scattering

The scattering amplitude (23) can be rewritten as

(31)

(32)where

(33)with

(34)

(35)

with , then (33) and (31) become

(36)

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Total elastic and inelastic cross sectionsTotal elastic and inelastic cross sections

The total elastic scattering cross section is given by

(37)

(38)

The total inelastic scattering cross section, which describes the loss of flux, is given by

Thus, if ηηηηl (k)= 1 there is no inelastic scattering, but if ηηηηl (k)= 0 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:

(39)

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!

(40)

(41)

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HighHigh--energy scattering from a black diskenergy scattering from a black disk

Consider the example: a black disk is totally absorbing , i.e., ηηηηl (k)= 0.

Assuming the values of angular momentum l do not exceed a maximum value lmax (l < lmax)

and that k is large (high-energy scattering), we have lmax=ka where a is the radius of the disk.

Since ηηηηl = 0, equations (37) and (38) lead to

(42)

�� the total cross section then reads

Classically, the total cross section is a disk equal to ππππa2.

The factor 2 in (43) is due to purely quantum effects, since in the high-energy limit there

are two kinds of scattering: one corresponding to waves that hit the disk, where the cross

section is equal to the classical cross section ππππa2, and the other to waves that are diffracted

- also of size ππππa2.

(43)2

ineleltot a2ππππσσσσσσσσσσσσ ====++++====

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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)

(45)

and also the scattering amplitude:

(46)

In quantum mechanics there is no way

of distinguishing between the particle

that scatters at an angle θ θ θ θ from the one

that scatters at (ππππ-θθθθ ).

Thus, the scattered wave function must be symmetric:

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Scattering of identical bosons

Therefore, the differential cross section is

(47)

interference term - not in the classical case !

For- quantum case

- classical case(48)

If the particles are distinguishable, the differential cross section will be four times

smaller:

(49)

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Scattering of identical fermions

� Consider now the scattering of two identical spin 1/2 particles.

E.g.: electron–electron or proton–proton scattering.

The wave function of a two spin 1/2 particle system is either symmetric or antisymmetric:

• when the spatial wave function is symmetric, that is the two particles are in a spin singlet

state, the differential cross section is given by

(50)

(51)

• when the two particles are in a spin triplet state, the spatial wave function is

antisymmetric, and then

• If the incident particles are unpolarized, the various spin states will be equally likely,

so the triplet state will be three times more likely as the singlet:

(52)

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Scattering of identical fermions

for (53)quantum case

classical case

�� if the incident particles are unpolarized:

(53) � 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:

(23)

We can write(54)

(55)

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Lecture 6 Scattering theory Partial Wave Analysisatlas.physics.arizona.edu/~shupe/Indep_Studies_2015/Notes_Goethe... · Lecture 6 Scattering theory Partial Wave Analysis ... we may