Neutron Scattering TheoryFor Bio-Physicists

Hem MoktanDepartment of Phycis

Oklahoma State University

Particle-wave duality

• de-Broglie wavelength:• Wave number:• Momentum:• Momentum operator:• Kinetic energy:

Schrodinger wave equation

• Time-independent Schrodinger wave equation:

Hψ = EψWhere, H is Hamiltonian operator.H = K.E. + P.E. = T + V

With

Particle in a 1-d box Quantum approach

• Potential:• Solution inside the box:• Boundary conditions: ψ(x=0)=ψ(x=L)=0;• Normalized wave function:• Allowed (Quantized) Energies:• Wave-functions:

Particle waves

• Infinite plane wave: ψ=exp(ikz) = cos kz + i sinkz

• Spherical wave:ψ =

• Scattered wave:

Neutron-Scattering

Model for neutron scattering

Scattering Amplitude

• Wave equation:

• Solution is:• Green’s function satisfies the point source

equation:

• Solution:

The total scattered wave function is an integral equation which can be solved by means of a series of iterative approximations, known as Born Series.

- Zero-order Solution:- First order solution:

And so on…

In real scattering experiment

• Where r is the distance from the target to the detector and r’ is the size of the target.• So we approximate:

• Asymptotic limit of the wave function:

The first Born Approximation

So, the scattering amplitude becomes

And the differential cross section:

Example: Bragg Diffraction

If the potential is spherically symmetric:

So, solving the Schrodinger equation in first-order Born approximation, the differential cross-section is given by above equation for a spherically symmetric potential. The potential is weak enough that the scattered wave is only slightly different from incident plane wave.

For s-wave scattering scattering amplitude = -b scattering length

Question: Use Born approximation for Coulomb potential and derive the classical Rutherford scattering formula.

Scattering Cross Section

Thank you!!

• Reading Materials:• Lectures 1 and 2.• Quantum Mechanics(Text) -Eugen MerzbacherFor SANS:http://www.ncnr.nist.gov/staff/hammouda/

the_SANS_toolbox.pdf