An Introduction to Neutron Scattering - Spallation Neutron Source
An Introduction to Neutron Scattering - Spallation Neutron Source
An Introduction to Neutron Scattering - Spallation Neutron Source
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<strong>Scattering</strong> From Fractal Systems<br />
• Fractals are systems that are “self-similar” under a change of scale I.e. R -> CR<br />
• For a mass fractal the number of particles within a sphere of radius R is<br />
proportional <strong>to</strong> R D where D is the fractal dimension<br />
Thus<br />
4pR<br />
2<br />
dR.<br />
G(<br />
R)<br />
= number of particles between distance R and R + dR = cR<br />
D−3<br />
∴ G(<br />
R)<br />
= ( c / 4π<br />
) R<br />
r<br />
r r r<br />
iQ.<br />
R 2π<br />
and S(<br />
Q)<br />
= ∫ dR.<br />
e G(<br />
R)<br />
= dR.<br />
R.<br />
sin QR.(<br />
c / 4 ) R<br />
Q ∫ π<br />
c 1<br />
= D<br />
2 Q<br />
dx.<br />
x<br />
const<br />
For a surface fractal, onecan<br />
prove that S(<br />
Q)<br />
∝ which reduces <strong>to</strong> the Porod<br />
6−<br />
Ds<br />
Q<br />
form for smooth surfaces<br />
∫<br />
D−2<br />
const<br />
. sin x = D<br />
Q<br />
of dimension 2.<br />
D−3<br />
D−1<br />
dR
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