In its simplest form, thegeometric modelof crystal growth is a third‐order, nonlinear, ordinary differential equation forθ(s,ε):Aneedle crystalis a solution that satisfies boundary conditionsThe geometric model admits a needle‐crystal solution forε= 0; for smallε, it admits an asymptotic expansion that is valid to all orders for such a solution. Even so, we prove that the geometric model in this form admits no needle crystal for any small, nonzeroε, a fact that lies beyond all orders of the asymptotic expansion. A more complicated version of the geometric model iswhereαrepresentscrystalline anisitropy. We show that for 0<α<1, the geometric model admits needle crystals for a discrete set of values ofα. The number of such values ofαincreases l
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机译：在最简单的形式中，晶体生长的几何模型是θ（s，ε）的三阶非线性常微分方程：A针状晶体是满足边界条件的解几何模型允许ε=0的针状晶体解;对于 smallε，它允许对此类解的所有阶数都有效的渐近展开。即便如此，我们还是证明了这种形式的几何模型不允许任何小的、非零ε的针状晶体，这一事实超出了渐近展开的所有阶数。几何模型的一个更复杂的版本是其中α表示晶体各向异性。我们表明，对于 0<α<1，几何模型允许针晶体用于一组离散的 α 值。α 的此类值的数量增加 l
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