Neutron Scattering
Theory
For Bio-Physicists
Hem Moktan
Department 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 = Eψ
Where, 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 Merzbacher
For SANS:
http://www.ncnr.nist.gov/staff/hammouda/the_SANS_toolbo
x.pdf