Download Scattering theory

Lecture:
Scattering theory
30.05.2012
SS2012: ‚Introduction to Nuclear and Particle Physics, Part 2‘
1
Part I:
Scattering theory:
Classical trajectories and
cross-sections
Quantum Scattering
2
I. Scattering experiments
Scattering experiment:
A beam of incident scatterers with a given flux or intensity (number of particles per unit
area dA per unit time dt ) impinges on the target (described by a scattering potential);
the flux can be written as
The number of particles per unit time which are detected in a small region of
the solid angle, dΩ, located at a given angular deflection specified by (θ, φ), can
be counted as
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Scattering cross section
The differential cross-section for scattering is defined as the number of particles scattered
into an element of solid angle dΩ in the direction (θ,φ) per unit time :
d σ ( ϑ ,φ )
1 dN sc
=
dΩ
J inc d Ω
[dimensions of an area]
(1.1)
Jinc - incident flux
The total cross-section corresponds to scatterings through any scattering angle:
(1.2)
Most scattering experiments are carried out in the laboratory (Lab) frame in which the
target is initially at rest while the projectiles are moving. Calculations of the cross sections
are generally easier to perform within the center-of-mass (CM) frame in which the center
of mass of the projectiles–target system is at rest (before and after collision)
one has to know how to transform the cross sections from one frame into the other.
Note: the total cross section σ is the same in both frames, since the total number of
collisions that take place does not depend on the frame in which the observation is carried
out. However, the differential cross sections dσ/dΩ are not the same in both frames, since
the scattering angles (θ,φ) are frame dependent.
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Connecting the angles in the Lab and CM frames
Elastic scattering of two structureless particles in the Lab and CM frames:
Note:
consider for simplicity
NON-relativistic
kinematics!
r
r
V CM || V 1 L
To find the connection between the Lab and CM cross sections, we need first to find
how the scattering angles in one frame are related to their counterparts in the other.
If
denote the position of m1 in the Lab and CM frames, respectively,
and if
denotes the position of the center of mass with respect to the Lab frame,
we have
A time derivative of this relation leads to
(1.3)
where
are the velocities of m1 in the Lab and CM frames before collision
and
is the velocity of the CM with respect to the Lab frame. Similarly, the
velocity of m1 after collision is
(1.4)
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Connecting the angles in the Lab and CM frames
r
r
Since V CM || V 1 L
(1.5)
(1.6)
Dividing (1.6) by (1.5), we end up with
(1.7)
Use that CM momenta
Since
r
r
r
(1.8)
p CM = p 1 L + p 2 L
r
r
r
( m 1 + m 2 )V CM = m 1V 1 L + m 2V 2 L
from (1.8) (1.9)
or from (1.9) (1.10)
(1.11)
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Connecting the angles in the Lab and CM frames
Since the center of mass is at rest in the CM frame, the total momenta before and after
r
r
r
r
r
v
collisions are separately zero: p = p + p
= p′ = p′ + p′ = 0
C
1C
2C
C
1C
2C
(1.12)
(1.13)
Since the kinetic energy is conserved:
(1.14)
(1.15)
In the case of elastic collisions, the speeds of the particles in the CM frame are the
same before and after the collision;
From (1.11) (1.16)
Dividing (1.9) by (1.16) (1.17)
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Connecting the angles in the Lab and CM frames
Finally, a substitution of (1.17) into (1.7) yields
(1.17)
and using
we obtain
(1.18)
Note: In a similar way we can establish a connection between θ2 and θ.
From (1.4) we have
+ using
the x and y components of this relation are
(1.19)
(1.20)
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Connecting the Lab and CM Cross Sections
The connection between the differential cross sections in the Lab and CM frames can be
obtained from the fact that the number of scattered particles passing through an
infinitesimal cross section is the same in both frames:
(1.21)
What differs is the solid angle dΩ :
in the Lab frame:
in the CM frame:
(1.22)
Since there is a cylindrical symmetry around the direction of the incident
beam
(1.23)
From (1.18) (1.24)
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Connecting the Lab and CM Cross Sections
Thus :
(1.25)
From (1.23) and (1.20) (1.26)
Limiting cases:
1)
the Lab and CM results are the same,
since (1.17) leads to θ1=θ 2)
(1.27)
then from (1.20) from (1.25) (1.28)
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From classical to quantum scattering theory
1.
Classical theory:
- scattering of hard ‚spheres‘
- individual well-defined trajectories
2.
Quantum theory:
- scattering of wave pakeges  wave–particle duality
- probabilistic origin of scattering process
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II. Classical trajectories and cross-sections
Scattering trajectories, corresponding to
different impact parameters, b give different
scattering angles θ.
All of the particles in the beam in the hatched
region of area dσ = 2π bdb are scattered into the
angular region (θ, θ + dθ)
The equations of motion for incident particle:
(2.1)
(Newton’s law)
+ initial conditions:
defines the trajectory (2.2)
(2.3)
since initial kinetic energy
For a particle obeying classical mechanics:
the trajectory for the unbound motion, corresponding to a scattering event, is
deterministically predictable, given by the interaction potential and the initial conditions;
the path of any scatterer in the incident beam can be followed,
is determined as precisely as required
and its angular deflection
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Classical trajectories and cross-sections
The number of particles scattered per unit time into the angular region (θ, θ+dθ) with
any value of φ can be written as
(2.4)
(2.5)
The knowledge of b(θ), obtained directly from Newton’s laws or other methods, is then
sufficient to calculate the scattering cross-section.
For the scattering from nontrivial central forces, the trajectory can be obtained from the
equations of motion by using energy- and angular momentum- conservation methods
E.g., one can rewrite
in the form
L - angular momentum
(2.6)
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Classical trajectories and cross-sections
The angular momentum can be written via the angular velocity
from (2.6)
(2.7)
the angle through which the particle moves between two radial distances r1 and r2 :
(2.8)
• Scattering trajectories in a central potential :
rmin - the distance of closest approach,
Θ – deflection angle:
(2.9)
Use that the initial angular momentum
and
from (2.8), (2.9)
(2.10)
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Classical Coulomb scattering
The potential corresponding to the scattering of charged particles via Coulomb’s law can be
written in the general form
(2.11)
where the constant
point-like charges Z1e and Z2e
corresponds to the electrostatic interaction of
from (2.10)
(2.12)
Note that for large energies (E→∞) or vanishing charges (A → 0), one has rmin → b
Use that
(2.13)
from (2.12)
(2.14)
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Classical Coulomb scattering
Use that
(2.15)
Rutherford formula - characteristic for Coulomb scattering from a point-like charge:
(2.16)
Note: If one attempts to evaluate the total cross-section using Eq. (2.16), one obtains an
infinite result. This divergence is due to the infinite range of the 1/r potential, and the
“infinity” in the integral comes from scatterings as θ → 0 which corresponds to arbitrarily
large impact parameters where the Coulomb potential causes a scatter through an
arbitrarily small angle, which still contributes to the total cross-section.
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III. Quantum Scattering
The scattering amplitude of spinless, nonrelativistic particles:
Consider the elastic scattering between two spinless, nonrelativistic particles of masses
m1 and m2. During the scattering process, the particles interact with each other. If the
interaction is time independent, we can describe the two-particle system with stationary
states
(3.1)
where ET is the total energy and
Schrödinger equation:
is a solution of the time-independent
(3.2)
is the potential representing the interaction between the two particles.
In the case where the interaction between m1 and m2 depends only on their relative
distance
one can reduce the eigenvalue problem
(3.2) to two decoupled eigenvalue problems:
one for the center of mass (CM), which moves like a free particle of mass M= m1+m2
(which is of no concern to us here)
and another for a fictitious particle with a reduced mass
which
moves in the potential
(3.3)
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Scattering of spinless particles
In quantum mechanics the incident particle is described by means of a wave packet
that interacts with the target.
After scattering, the wave function consists of an unscattered part propagating in the
forward direction and a scattered part that propagates along some direction (θ,ϕ)
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Scattering of spinless particles
One can consider (3.3) as representing the scattering of a particle of mass µ from a fixed
scattering center described by V(r), where r is the distance from the particle µ to the center
of V(r):
(3.3)
assume that V(r) has a finite range a:
 ≠ 0 , if r ≤ a
V̂ ( r ) = 
 = 0 , if r > a
Outside the range, r > a, the potential vanishes, V(r)= 0; the eigenvalue problem (3.3)
then becomes
(3.4)
In this case µ behaves as a free particle before the collision and can be described
by a plane wave
(3.5)
where
is the wave vector associated with the incident particle;
A is a normalization factor.
Thus, prior to the interaction with the target, the particles of the incident beam
are independent of each other; they move like free particles, each with a
momentum
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Scattering of spinless particles
When the incident wave (3.5) collides or interacts with the target, an outgoing wave
is scattered out.
In the case of an isotropic scattering, the scattered wave is spherically symmetric, having the
form
In general, however, the scattered wave is not spherically symmetric; its amplitude depends
on the direction (θ,ϕ) along which it is detected and hence
(3.6)
where f (θ,ϕ) is called the scattering amplitude,
is the wave vector associated with the scattered particle,
and θ is the angle between
After the scattering has taken place, the total wave consists of a superposition of the incident
plane wave (3.5) and the scattered wave (3.6):
(3.7)
where A is a normalization factor; since A has no effect on the cross section, as will
be shown later, we will take it equal to one.
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Scattering amplitude and differential cross section
Introduce the incident and scattered flux densities:
(3.8)
(3.9)
Inserting (3.5) into (3.8) and (3.6) into (3.9) and taking the magnitudes of the expressions
thus obtained, we end up with
(3.10)
We recall that the number dN(θ,ϕ) of particles scattered into an element of solid angle
dΩ in the direction (θ,ϕ) and passing through a surface element dA= r2dΩ per unit time
is given as follows:
(3.11)
Using (3.10) (3.12)
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Scattering amplitude and differential cross section
By inserting (3.12) and
in (1.1), we obtain:
(3.13)
Since the normalization factor A does not contribute to the differential cross section, it
will be taken to be equal to one.
For elastic scattering k0 is equal to k; so (3.13) reduces to
(3.14)
The problem of determining the differential cross section dσ/dΩ therefore reduces
to that of obtaining the scattering amplitude f (θ,ϕ).
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Part II:
Scattering theory:
Born Approximation
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Scattering amplitude
We are going to show here that we can obtain the differential cross section in the CM frame
from an asymptotic form of the solution of the Schrödinger equation:
(1.1)
Let us first focus on the determination of the scattering amplitude f (θ, φ), it can be
obtained from the solutions of (1.1), which in turn can be rewritten as
where k 2 =
2 µE
h2
(1.2)
The general solution of the equation (1.2) consists of a sum of two components:
1) a general solution to the homogeneous equation:
(1.3)
In (1.3)
is the incident plane wave
2) and a particular solution of (1.2) with the interaction potential
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General solution of Schrödinger eq. in terms of Green‘s function
The general solution of (1.2) can be expressed in terms of Green’s function.
(1.4)
is the Green‘s function corresponding to the
operator on the left side of eq.(1.3)
The Green‘s function is obtained by solving the point source equation:
(1.5)
(1.6)
(1.7)
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Green‘s function
A substitution of (1.6) and (1.7) into (1.5) leads to
(1.8)
The expression for
can be obtained by inserting (1.8) into (1.6)
(1.9)
(1.10)
To integrate over angle in (1.10) we need to make the variable change x=cosθ
(1.11)
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Method of residues
Thus, (1.9) becomes
(1.12)
(1.13)
The integral in (1.13) can be evaluated by the method of residues by closing the contour
in the upper half of the q-plane:
The integral is equal to 2π i times the residue of the integrand at the poles.
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Green‘s functions
Since there are two poles, q =+k, the integral has two possible
values:
the value corresponding to the pole at q =k, which lies
inside the contour of integration in Figure 1a, is given by
(1.14)
the value corresponding to the pole at q =-k,
Figure 1b, is
(1.15)
Green’s function
the function
represents an outgoing spherical wave emitted from r‘ and
corresponds to an incoming wave that converges onto r .
Since the scattered waves are outgoing waves, only
is of interest to us.
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Born series
Inserting (1.14) into (1.4) we obtain for the total scattered wave function:
(1.16)
This is an integral equation.
All we have done is to rewrite the Schrödinger (differential) equation (1.1) in an integral
form (1.16), which is more suitable for scattering theory.
Note that (1.16) can be solved approximately by means of a series of successive or iterative
approximations, known as the Born series.
the zero-order solution is given by
the first-order solution
is obtained by inserting
into the integral
of (1.16):
(1.17)
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Born series
the second order solution is obtained by inserting
into (1.16):
(1.18)
continuing in this way, we can obtain
to any desired order;
the n order approximation for the wave function is a series which can be obtained
th
by analogy to (1.18) .
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Asymptotic limit of the wave function
In a scattering experiment, since the detector is located at distances (away from the target)
that are much larger than the size of the target, we have r>>r‘, where r represents the
distance from the target to the detector and r’ the size of the target.
If r >> r‘ we may approximate:
(1.19)
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Asymptotic limit of the wave function
Substitute (1.19) to (1.16):
(1.16)
for r>>r‘ =
r
φ inc ( r ) −
µ
2π h
∫
e
ikr
e
r
rr
− ik r '
r
r
V ( r ′ )ψ ( r ′ )d 3 r ′
From the previous two approximations (1.19), we may write the asymptotic form of (1.16)
as follows:
(1.20)
(1.21)
where
ia a plane wave and k is the wave vector of scattered wave;
the integration variable r‘ extends over the spacial degrees of freedom of the target.
The differential cross section is given by
(1.22)
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The first Born approximation
If the potential V(r) is weak enough, it will distort only slightly the incident plane wave.
The first Born approximation consists then of approximating the scattered wave function
Ψ(r ) by a plane wave.
This approximation corresponds to the first iteration in the Born series of (1.16):
(1.16)
that is, Ψ(r ) is given by (1.17):
(1.23)
Thus, using (1.21) we can write the scattering amplitude in the first Born approximation
as follows:
(1.24)
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The first Born approximation
Using (1.23), we can write the differential cross section in the first Born approximation
as follows:
(1.25)
where
is the momentum transfer;
are the linear momenta of the incident and scattered particles, respectively.
•In elastic scattering, the magnitudes of
are equal (Figure 1); hence
(1.26)
Figure 1:
Momentum transfer for elastic scattering:
34
The first Born approximation
If the potential
is spherically symmetric,
along q (Figure 1), then
and therefore
and choosing the z-axis
(1.27)
Inserting (1.27) into (1.24) and (1.25) we obtain
(1.28)
(1.29)
In summary, we have shown that by solving the Schrödinger equation (1.1) in first-order Born
approximation (where the potential V(r ) is weak enough that the scattered wave function is
only slightly different from the incident plane wave), the differential cross section is given by
equation (1.29) for a spherically symmetric potential.
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Validity of the firts Born approximation
The first Born approximation is valid whenever the wave function Ψ(r ) is only slightly
different from the incident plane wave;
that is, whenever the second term in (1.23) is very small compared to the first:
(1.23)
(1.30)
Since
we have
(1.31)
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Validity of the firts Born approximation
In elastic scattering k0= k and assuming that the scattering potential is largest near r=0,
we have
(1.32)
(1.33)
Since the energy of the incident particle is proportional to k (it is purely kinetic,
we infer from (1.33) that
)
the Born approximation is valid for large incident energies and weak scattering potentials.
That is, when the average interaction energy between the incident particle and the scattering
potential is much smaller than the particle’s incident kinetic energy, the scattered wave can be
considered to be a plane wave.
37
Born approximation for Coulomb potential
Let‘s calculate the differential cross section in the first Born approximation for a
Coulomb potential
(1.34)
where Z1e and Z1e are the charges of the projectile and target particles, respectively.
In a case of Coulomb potential, eq.(1.29):
(1.35)
becomes
(1.36)
(1.37)
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Born approximation for Coulomb potential
Now, since
an insertion of (1.37) into (1.36) leads to
(1.38)
where
is the kinetic energy of the incident particle.
Eq. (1.38) is Rutherford formula
Note: (1.38) is identical to the purely classical case! – cf. Part I, eq.(2.16)
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