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Neutron Scattering Theory

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:

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Neutron Scattering Theory

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  1. Neutron Scattering Theory For Bio-Physicists Hem Moktan Department of Phycis Oklahoma State University

  2. Particle-wave duality • de-Broglie wavelength: • Wave number: • Momentum: • Momentum operator: • Kinetic energy:

  3. Schrodinger wave equation • Time-independent Schrodinger wave equation: Hψ = Eψ Where, H is Hamiltonian operator. H = K.E. + P.E. = T + V With

  4. 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:

  5. Particle waves • Infinite plane wave: ψ=exp(ikz) = coskz + isinkz • Spherical wave:ψ = • Scattered wave:

  6. Neutron-Scattering

  7. Model for neutron scattering

  8. Scattering Amplitude • Wave equation: • Solution is: • Green’s function satisfies the point source equation: • Solution:

  9. 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…

  10. In real scattering experiment • So we approximate: • Where r is the distance from the target to the detector and r’ is the size of the target.

  11. Asymptotic limit of the wave function:

  12. The first Born Approximation • So, the scattering amplitude becomes • And the differential cross section:

  13. Example: Bragg Diffraction

  14. 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

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

  16. Scattering Cross Section

  17. Thank you!! • Reading Materials: • Lectures 1 and 2. • Quantum Mechanics(Text) -EugenMerzbacher For SANS: http://www.ncnr.nist.gov/staff/hammouda/the_SANS_toolbox.pdf

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