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: H ψ = E ψ 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: x.pdf