neutron scattering theory
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
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