Talk:Neutron scattering lengths

Origin of the scattering lengths

The following description is adapted from [www.ncnr.nist.gov/programs/sans/pdf/polymer_tut.pdf Boualem Hammouda's (NCNR) SANS tutorial].

Consider a neutron of energy $E_{i}$ interacting with a nucleus, which exhibits an attractive square well of depth $-V_{0}$ and width $2R$; where $V_{0}\gg E_{i}$. The Schrödinger equation is:

$\left[-{\frac {h^{2}}{8\pi ^{2}m}}\nabla ^{2}+V(r)\right]\psi =E\psi$

Outside of the square-well ($r>R$), $V(r)=0$, and so the equation is solved as simply:

$\psi _{s,\mathrm {out} }={\frac {\sin(kr)}{kr}}-b{\frac {e^{ikr}}{r}}$

where <math>k=\sqrt{2mE_i} 2 \pi/h.