Theory of Scattering
Lecture 3
Free Particle:
Energy
, In Cartesian and spherical
Coordinates.
Wave function:
(plane waves in Cartesian system)
(spherical waves in spherical system)
Plane waves can be expressed in terms of spherical
wave states
When wave vector k is along z-axis and m = 0,
Partial Wave Analysis: Consider that the incident plane
wave is travelling along z-axis, we may express it as
----------(1)
We can write the above plane wave as superposition of
angular momentum Eigen states each with definite
angular momentum number ℓ, so we write
Rayleigh’s Formula
---------(2)
So we expressed in above Eq. incident plane wave as sum of
partial waves which will be distorted by scattering potential.
We know write the solution to the Schrodinger Eq
-----------(3)
As we know the solution of above Eq. can be written using
separable variable technique. So we write
----------(4)
Foe central potential the system is symmetrical about z-axis
and hence rotational invariant and the scattered wave will
not depend upon azimuthal angle φ and thus m = 0.
Also
-----------(5)
The radial wave function in Eq. (4) obeys following Eq.
--------(6)
Also as we have discussed the total wave function after
Scattering is written by superposition of incident and
Scattered wave i.e. we write
-----(7)
Using (2) in (7), we get
-----------(8)
Note that in Eq. (8) we have no dependence on φ.
Note that Eq. (4) and (8) are two ways of writing the
total scattered wave.
In scattering experiments the distance of detector from
the scattering center is large as compared to size of target.
We now utilize this fact to find the scattering amplitude
and differential cross-section.
For large value of r, the Bessel function can be approximated
as,
---------(9)
Using (9) in (8), we get the asymptotic form
--------------(10)
For large value of r, the radial wave equation (6) can be
written as (Radiation Zone)
-----------(11)
The general solution of above equation consist of linear
combination of spherical Bessel and Neumann functions
------------(12)
Bessel’s function of first kind
Bessel’s function of 2nd kind
Known as Neumann function
In asymptotic limit the Neumann function is written as
-----(13)
Using Eq. (9) and (13) in Eq. (12), we get
-----(14)
If V(r) = 0 for all r, solution
It means at r = 0, Rkl should vanish. Now Neumann functions
have property that they diverge when argument is zero i.e.
at r = 0 they will diverge. Hence in this case we cannot
consider contribution of Neumann function to solution.
We introduce the phase shift to achieve the regular solution
near the origin by rewriting
where
Asymptotic form of radial function is written as
-----(15)
Note that when
in Eq. (15) the Rkl reduces to Bessel
function which is finite at r = 0. The
becomes zero in
absence of scattering potential. It is called phase shift.
The phase shift measure the degree to which
is
different from
. Remember that
was the
Radial wave function in absence of scattering.
We now use Eq (15) in Eq. (4) and write the asymtotic form
Of solution as,
-------------(17)
The wave function defined in above Eq. represent the
distorted plane wave because it contain the phase shift
which comes into picture because of scattering.
Using Equation
----------(17)
We write Eq. (16) as
-------(18)
Above Eq. Gives us the asymptotic form of Eq. (4).
Also we have obtained earlier the Eq. (10) as asymptotic
form of Eq. (7), which is
-------(19)
Comparing coefficients of
Eq. (18) and (19), we get
Or
----------(20)
Using Eq. (20) in Eq. (18), we get
-------------(21)
Comparing coefficients of
we get,
in Eq. (21) with Eq. (10),
Using
---------(22)
above Eq. becomes
-------------(23)
is known as partial wave amplitude.
Using (23), the Differential cross-section is written as
---------(24)
Total cross-section is written as
------------(25)
Using
We get
----------(26)
gives us the partial cross-section corresponding to scattering
of particles in various angular momentum states.
For the case of scattering between particles which are in
S-state, ℓ = 0 (low energy scattering). The scattering
amplitude using (23) can be written as,
-----------(27)
Differential and total cross-section, using (27) are
-----------(28)
For forward scattering
Forward scattering amplitude is written as,
-----(29)
Using Eq. (26) and (29), we get
-------(30)
Above Eq. is known as optical theorem.
The physical origin of this theorem is the conservation of
particles (or probability):
the beam emerging (after scattering) along the incident
direction (θ=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 σ;
that is, the number of particles removed from the incident
beam along the incident
direction is proportional to σ or, equivalently, to the
imaginary part of f (0).