Wulff_Construction_Math

# Wulff_Construction_Math - Ξ V Writing β β-= i i i i M m...

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Wulff Construction Based on an analysis by Max von Laue, (1943): ‘Der Wulffsche Satz für die Gleichgewichtsform von Kristallen', Z. Kristallogr. 105, pp. 124-133 (the same person who received a nobel for discovering x-ray diffraction); this paper was published at a time when he was one of the few scientists left in Germany who resisted the Nazi regime. Suppose we consider our crystal as (with respect to some center) being composed of a series of facets “i” each of which has an area of M i and is a distance m i from the origin of the Wulff construction as shown below. For each facet we have a surface free energy of γ i . The total surface energy of the system can be written as S = Σ γ i M i and the total volume as: V = (1/3) Σ m i M i Since we can break the body down into a series of pyramids, each of which has a volume of 1/3 Base x Height. We solve this in a standard way by finding the minimum of S at constant volume using a Lagrangian method, L = S – 2

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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 β--β 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 β-β 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 donβt need to....
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Wulff_Construction_Math - Ξ V Writing β β-= i i i i M m...

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