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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A Phonon is a Quantized Lattice Vibration<br />
• Consider linear chain of particles of mass M coupled by<br />
springs. Force on n’th particle is<br />
Fn = α 0 un<br />
+ α1(<br />
un−1<br />
+ un+<br />
1)<br />
+ α 2(<br />
un−2<br />
+ un+<br />
2)<br />
+ ...<br />
• Equation of motion is<br />
• Solution is:<br />
a<br />
First neighbor force constant displacements<br />
2π<br />
4π<br />
N 2π<br />
q =<br />
0 , ± , ± ,...... ±<br />
L L 2 L<br />
F = Mu&<br />
&<br />
i ( qna−ωt<br />
)<br />
2<br />
un ( t)<br />
= Aqe<br />
with ωq<br />
=<br />
Phonon Dispersion Relation:<br />
Measurable by inelastic neutron scattering<br />
n<br />
u<br />
n<br />
4<br />
M<br />
∑<br />
ν<br />
α<br />
ν<br />
sin<br />
2<br />
1<br />
( νqa)<br />
2<br />
1.4<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
w<br />
-1 -0.5 0.5 1<br />
qa/2π
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