Difference between revisions of "Talk:Neutron scattering lengths"
KevinYager (talk | contribs) (Created page with "==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]. ...") |
KevinYager (talk | contribs) (→Origin of the scattering lengths) |
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==Origin of the 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]. | + | The following description is adapted from [http://www.ncnr.nist.gov/programs/sans/pdf/polymer_tut.pdf Boualem Hammouda's (NCNR) SANS tutorial]. |
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: | 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> | </math> | ||
| − | Outside of the square-well (<math>r>R</math>), <math>V(r)=0</math>, and so the equation is solved as simply: | + | Outside of the square-well (<math>|r|>R</math>), <math>V(r)=0</math>, and so the equation is solved as simply: |
:<math> | :<math> | ||
\psi_{s,\mathrm{out}} = \frac{\sin(kr)}{kr} - b \frac{e^{ikr}}{r} | \psi_{s,\mathrm{out}} = \frac{\sin(kr)}{kr} - b \frac{e^{ikr}}{r} | ||
</math> | </math> | ||
| − | where <math>k=\sqrt{2mE_i} 2 \pi/h. | + | 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: |
| + | :<math> | ||
| + | \psi_{s,\mathrm{in}} = A \frac{\sin(qr)}{qr} | ||
| + | </math> | ||
| + | where <math>q=\sqrt{2m(E_i+V_0} 2 \pi/h</math>. | ||
Revision as of 01:18, 6 June 2014
Origin of the scattering lengths
The following description is adapted from Boualem Hammouda's (NCNR) SANS tutorial.
Consider a neutron of energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_i} interacting with a nucleus, which exhibits an attractive square well of depth Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -V_0} and width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2R} ; where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0 \gg E_i} . The Schrödinger equation is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ - \frac{h^2}{8 \pi^2 m}\nabla^2 + V(r) \right] \psi = E \psi }
Outside of the square-well (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|>R} ), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)=0} , and so the equation is solved as simply:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{s,\mathrm{out}} = \frac{\sin(kr)}{kr} - b \frac{e^{ikr}}{r} }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\sqrt{2mE_i} 2 \pi/h} . Inside the square-well (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|<R} ), the potential is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)=-V_0} , and the solution becomes:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{s,\mathrm{in}} = A \frac{\sin(qr)}{qr} }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=\sqrt{2m(E_i+V_0} 2 \pi/h} .
