Geometric theory of defects in solids
Geometric theory of defects in solids
Geometric theory of defects in solids
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Frank vector<br />
ij<br />
ω ( x)<br />
- is not cont<strong>in</strong>uous !<br />
ω<br />
ij<br />
μ<br />
( x)<br />
ij<br />
⎧∂ ⎪ μω<br />
= ⎨<br />
ij<br />
⎪⎩ lim∂μω<br />
- outside the cut<br />
- on the cut<br />
- SO(3)-connection<br />
(cont<strong>in</strong>uous on the cut)<br />
ij μ ij μ ν ij ij<br />
∫<br />
μ ∫∫<br />
μ ν ν μ<br />
Ω = dx ω = dx ∧ dx ( ∂ ω −∂ ω )<br />
ij ij ik j<br />
R μν μ ν μ ν k<br />
=∂ ω −ω ω −( μ ↔ν)<br />
- the Frank vector<br />
-curvature<br />
ij<br />
Ω =<br />
∫∫<br />
dx<br />
μ<br />
∧<br />
ν<br />
dx R<br />
ij<br />
μν<br />
- def<strong>in</strong>ition <strong>of</strong> the Frank vector<br />
<strong>in</strong> the geometric <strong>theory</strong><br />
Back to the sp<strong>in</strong> structure: if<br />
2<br />
n∈ then SO(3) → SO(2)<br />
7
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