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Unformatted text preview: Ξ» V Writing β β= i i i i M m M L ) 3 / 2 ( Ξ» Ξ³ Differentiating with respect to m j β β β ββ β = = β β j i i j j i i j m M m M m M m L / ) 3 / 2 ( ) 3 / 2 ( / / From solid geometry, we also have (the not particularly obvious relationship) that β β β + = = β β j i i j j j m M m M M m V / ) 3 / 1 ( ) 3 / 1 ( / Or β β β = j i i j m M m M / ) 2 / 1 ( Then, combing this we get β β β β β= β ββ β = = β β j i i i j i i j i i j m M m m M m m M m L / } { / / / Since the derivative is in general not zero, this has a solution = i i m / The shape then only scales up or down as we vary Ξ», i.e. the shape is the same no matter what the size is. One caveat about this proof. Technically it only proves that the Wulff construction is a stationary point, not that it is a true minimum. Mathematicians worry about this, we donβt need to....
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 Winter '10
 MatSci
 Crystallography, Surface tension, Max von Laue, wulff construction, Wulffsche Satz fΓΌr, mi βM

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