Question

Math

Posted 3 months ago

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证明以下立方晶格中单位体积交换能增量的公式
$\Delta E_{\text {交 }}=\frac{A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Given the formula for the exchange energy increment per unit volume in a cubic lattice:
$\Delta E_{\text{交}} = \frac{A s^{2}}{a} \left[ \left( \nabla_{\alpha_{x}} \right)^{2} + \left( \nabla_{\alpha_{y}} \right)^{2} + \left( \nabla_{\alpha_{z}} \right)^{2} \right]$

step 2

Identify the terms in the formula:
- $A$ is a constant
- $s$ is the spin
- $a$ is the lattice constant
- $\nabla_{\alpha_{x}}, \nabla_{\alpha_{y}}, \nabla_{\alpha_{z}}$ are the gradients in the x, y, and z directions respectively

step 3

The formula represents the sum of the squares of the gradients of the spin in the x, y, and z directions, scaled by the factor $\frac{A s^{2}}{a}$

step 4

To prove this formula, we need to show that the energy increment per unit volume is proportional to the sum of the squares of the gradients of the spin in each direction. This involves understanding the physical meaning of each term and how they contribute to the total energy

step 5

The term $\frac{A s^{2}}{a}$ acts as a scaling factor that adjusts the contribution of the gradients to the total energy. The gradients $\nabla_{\alpha_{x}}, \nabla_{\alpha_{y}}, \nabla_{\alpha_{z}}$ represent the rate of change of the spin in each direction, and their squares indicate the energy contribution from these changes

step 6

Therefore, the formula $\Delta E_{\text{交}} = \frac{A s^{2}}{a} \left[ \left( \nabla_{\alpha_{x}} \right)^{2} + \left( \nabla_{\alpha_{y}} \right)^{2} + \left( \nabla_{\alpha_{z}} \right)^{2} \right]$ correctly represents the exchange energy increment per unit volume in a cubic lattice

Answer

The formula for the exchange energy increment per unit volume in a cubic lattice is correctly given by $\Delta E_{\text{交}} = \frac{A s^{2}}{a} \left[ \left( \nabla_{\alpha_{x}} \right)^{2} + \left( \nabla_{\alpha_{y}} \right)^{2} + \left( \nabla_{\alpha_{z}} \right)^{2} \right]$.

Key Concept

Exchange energy increment per unit volume

Explanation

The formula represents the sum of the squares of the gradients of the spin in the x, y, and z directions, scaled by a factor that includes the spin, lattice constant, and a constant. This shows how the energy changes with the spin configuration in a cubic lattice.

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