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| − | ==Origin of the scattering lengths==
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| − | The following description is adapted from [http://www.ncnr.nist.gov/programs/sans/pdf/polymer_tut.pdf Boualem Hammouda's (NCNR) SANS tutorial].
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| − | Consider first the energies of neutrons used in scattering experiments. A thermal neutron
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| − | , the energy for even a thermal neutron (1.8 Å wavelength) is
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| − | Consider a neutron of energy <math>E_i</math> interacting with a nucleus, which exhibits an attractive square well of depth <math>-V_0</math> and width <math>2R</math>; where <math>V_0 \gg E_i</math>. The [http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation] is:
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| − | :<math>
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| − | \left[ - \frac{h^2}{8 \pi^2 m}\nabla^2 + V(r) \right] \psi(r) = E \psi(r)
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| − | </math>
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| − | Outside of the square-well (<math>|r|>R</math>), <math>V(r)=0</math>, and so the equation is solved as simply:
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| − | :<math>
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| − | \psi_{s,\mathrm{out}} = \frac{\sin(kr)}{kr} - b \frac{e^{ikr}}{r}
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| − | </math>
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| − | where <math>k=\sqrt{2mE_i} 2 \pi/h</math>. Inside the square-well (<math>|r|<R</math>), the potential is <math>V(r)=-V_0</math>, and the solution becomes:
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| − | :<math>
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| − | \psi_{s,\mathrm{in}} = A \frac{\sin(qr)}{qr}
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| − | </math>
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| − | where <math>q=\sqrt{2m(E_i+V_0)} 2 \pi/h</math>. The two solutions are subject to a continuity boundary condition at <math>|r|=R</math>:
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| − | :<math>\begin{alignat}{2}
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| − | \psi_{s,\mathrm{out}} (r=R) & = \psi_{s,\mathrm{in}} (r=R) \\
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| − | \frac{\mathrm{d}}{\mathrm{d}r} \psi_{s,\mathrm{out}} (r=R) & = \frac{\mathrm{d}}{\mathrm{d}r} \psi_{s,\mathrm{in}} (r=R)
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| − | \end{alignat}
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| − | </math>
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| − | Note that the mass of a neutron is ~10<sup>−27</sup> kg
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| − | Note that <math>kR = \sqrt{2 m E_i} R 2\pi/h \ll 1</math> for , and
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