Question

Math

Posted about 2 months ago

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证明在以下两种立方晶格中单位体积交换能增量的公式:
筒立方: $\Delta E_{\text {文 }}=\frac{A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$
面心立方: $\Delta E_{\text {文 }}=\frac{4 A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$
```

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Answer from Sia

Posted about 2 months ago

Solution by Steps

step 1

Identify the given equations for the two types of cubic lattices

step 2

For the simple cubic lattice, the formula is given by:
$\Delta E_{\text{文}} = \frac{A s^{2}}{a} \left[ \left( \nabla_{\alpha_{x}} \right)^{2} + \left( \nabla_{\alpha_{y}} \right)^{2} + \left( \nabla_{\alpha_{z}} \right)^{2} \right]$

step 3

For the face-centered cubic lattice, the formula is given by:
$\Delta E_{\text{文}} = \frac{4 A s^{2}}{a} \left[ \left( \nabla_{\alpha_{x}} \right)^{2} + \left( \nabla_{\alpha_{y}} \right)^{2} + \left( \nabla_{\alpha_{z}} \right)^{2} \right]$

step 4

Note the difference in the coefficients between the two formulas. For the simple cubic lattice, the coefficient is $\frac{A s^{2}}{a}$, while for the face-centered cubic lattice, the coefficient is $\frac{4 A s^{2}}{a}$

Answer

The formulas for the unit volume exchange energy increment in simple cubic and face-centered cubic lattices are correctly identified and differentiated by their coefficients.

Key Concept

Unit Volume Exchange Energy Increment

Explanation

The unit volume exchange energy increment formulas for simple cubic and face-centered cubic lattices differ by a factor of 4 in their coefficients, reflecting the different structural properties of the lattices.

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