A better way of proving the Weiss Zone Law?

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In summary, the Bravais lattice in 3D is described by 3 basis vectors and a discrete point in the lattice can be generated with these basis vectors and integers. A direction is specified by a triplet and a plane is specified by a triplet. The question is to show that if a direction lies in a plane, then their coefficients satisfy a certain equation. One method is to use the definition of the plane and the direction and solve for the coefficients. Another method is to use the reciprocal basis and the dot product to show that the coefficients of the direction and the plane must satisfy the same equation.
  • #1
etotheipi
Homework Statement
Prove the Weiss Zone Law
Relevant Equations
N/A
For some context, a Bravais lattice in 3D is described by 3 basis vectors ##\vec{a}_1##, ##\vec{a}_2##, ##\vec{a}_3## s.t. a discrete point in the lattice can be generated with ##\vec{R} = \sum R_i \vec{a}_i##, with ##R_1##, ##R_2##, ##R_3## being integers. A direction is specified by a triplet ##[u_1 \, u_2 \, u_3]## (i.e. a vector along this direction is ##\vec{u} = \sum u_i \vec{a}_i##) and a plane is specified by a triplet ##(h_1 \, h_2 \, h_3)## s.t. the intersections of the plane with our coordinate system are at ##x_i = \phi a_i / h_i##, where ##a_i = |\vec{a}_i|## and ##\phi## is just some multiplicative constant (specifically, the lowest common denominator of all the ##a_i/x_i##) by which all of the ##a_i/x_i## are multiplied, so that the tuple ##(h_1 \, h_2 \, h_3)## contains integers).

The question is to show that if a direction ##[u_1 \, u_2 \, u_3]## lies in the plane ##(h_1, h_2, h_3)##, then ##u_1 h_1 + u_2 h_2 + u_3 h_3 = 0##. I thought my way was a bit clunky. The plane can be defined by two vectors i.e. ##\vec{\Pi}(\alpha, \beta) = \vec{\Pi}_0 + \alpha \vec{v}_1 + \beta \vec{v}_2## which are$$\vec{v}_1 = \frac{\vec{a}_3}{h_3} - \frac{\vec{a}_1}{h_1}, \quad \vec{v}_2 = \frac{\vec{a}_3}{h_3} - \frac{\vec{a}_2}{h_2}$$A normal to the plane is$$\vec{n} = \vec{v}_1 \times \vec{v}_2 = \frac{\vec{a}_1 \times \vec{a}_2}{h_1 h_2} - \frac{\vec{a}_1 \times \vec{a}_3}{h_1 h_3} - \frac{\vec{a}_3 \times \vec{a}_2}{h_3 h_2}$$If ##\vec{u} = u_1 \vec{a}_1 + u_2 \vec{a}_2 + u_3 \vec{a}_3## lies within ##(h_1, h_2, h_3)## then ##\vec{u} \cdot \vec{n} = 0##, i.e.$$\frac{u_3}{h_1 h_2} (\vec{a}_1 \times \vec{a}_2) \cdot \vec{a}_3 - \frac{u_2}{h_1 h_3} (\vec{a}_1 \times \vec{a}_3) \cdot \vec{a}_2 - \frac{u_1}{h_3 h_2} (\vec{a}_3 \times \vec{a}_2) \cdot \vec{a}_1 = 0 $$ $$\begin{align*}\left[ \frac{u_3}{h_1 h_2} + \frac{u_2}{h_1 h_3} + \frac{u_1}{h_2 h_3} \right] (\vec{a}_1 \times \vec{a}_2) \cdot \vec{a}_3 = 0 &\implies\frac{u_3}{h_1 h_2} + \frac{u_2}{h_1 h_3} + \frac{u_1}{h_2 h_3} = 0\\ \\
&\implies u_3 h_3 + u_2 h_2 + u_1 h_1 = 0
\end{align*}$$It works, but it seems long winded. Does anyone have a better way?
 
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  • #2
I would just use the definition of the plane and the definition of being a direction in the plane directly. If u is a direction in the plane then it has to be the difference of two points in the plane. You know exactly one property those points need to satisfy. Can you do anything with it?
 
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  • #3
I think the vector ##\vec{v}=h_1 \vec{a}_1 + h_2 \vec{a}_2 + h_3 \vec{a}_3 ## is normal to the plane(s) regardless of whether the coordinates are orthogonal or not. If you write the equation for the plane passing through the origin that is parallel to the set of planes, the zone law follows immediately.
Edit: Upon further consideration, I think the first sentence may be in error, but the methodology in post 4 should still work.
 
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  • #4
To add to post 3, note one parallel plane will have the form ## h_ 1 u_1+h_2 u_2 +h_3 u_3=1 ##, where I'm using the ## u's ## as coordinates. Note when ##u_1= 1/h_ 1 ##, then ##u_2=0 ##, and ##u_3=0 ## for the axis intercept. Similarly for ## u_2 ## and ## u_3 ## intercepts. I think you only need to look at the plane that is parallel to this one that passes through the origin for the result you need.
 
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  • #5
Office_Shredder said:
I would just use the definition of the plane and the definition of being a direction in the plane directly. If u is a direction in the plane then it has to be the difference of two points in the plane. You know exactly one property those points need to satisfy. Can you do anything with it?

Thanks, that works! I interpreted that to mean, for some ##\alpha, \beta##,$$u_1 \vec{a}_1 + u_2 \vec{a}_2 + u_3 \vec{a}_3 = \alpha \vec{v}_1 + \beta \vec{v}_2 = \alpha \left( \frac{\vec{a}_3}{h_3} - \frac{\vec{a}_1}{h_1} \right) + \beta \left( \frac{\vec{a}_3}{h_3} - \frac{\vec{a}_2}{h_2} \right)$$if the set ##\{ \vec{a}_1, \vec{a}_2, \vec{a}_3 \}## is linearly independent then we can deduce$$u_1 = -\frac{\alpha}{h_1}, \quad u_2 = -\frac{\beta}{h_2}, \quad u_3 = \frac{1}{h_3}(\alpha + \beta)$$from which it immediately follows that
$$u_1 h_1 + u_2 h_2 + u_3 h_3 = -\alpha - \beta + (\alpha + \beta) = 0$$
Charles Link said:
I think the vector ##\vec{v}=h_1 \vec{a}_1 + h_2 \vec{a}_2 + h_3 \vec{a}_3 ## is normal to the plane(s) regardless of whether the coordinates are orthogonal or not.

I think this only holds in a cubic lattice, for the direct lattice vectors. On the other hand for the reciprocal lattice vectors ##\{\vec{b}_1, \vec{b}_2, \vec{b}_3 \}##, it is always true that the plane ##(h_1 \, h_2 \, h_3)## is orthogonal to ##h_1 \vec{b}_1 + h_2 \vec{b}_2 + h_3 \vec{b}_3##. There is some justification here: https://en.wikipedia.org/wiki/Miller_index
 
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  • #6
Yes, I think my statement of the vector always being normal is incorrect. (I will mark it with an "Edit" in post 3). In any case, I think my post 4 still applies, where parallel planes can be used to come up with the Weiss zone law formula. The direction vector's coefficients become the coordinates when the plane passes through the origin.
 
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  • #7
In case anyone was interested, I realized earlier that this law is actually pretty trivial if we use the reciprocal basis. The plane with Miller indices ##(h_1 h_2 h_3)## is orthogonal to ##h_1 \vec{a}_1^* + h_2\vec{a}_2^* + h_3 \vec{a}_3^*##, where ##\vec{a}_i^* = \frac{\vec{a}_j \times \vec{a}_k}{[\vec{a}_i, \vec{a}_j, \vec{a}_k]}## are the reciprocal basis. The zone law follows from constraining$$(u_1\vec{a}_1 + u_2\vec{a}_2 + u_3 \vec{a}_3) \cdot (h_1\vec{a}_1^* + h_2\vec{a}_2^* + h_3\vec{a}_3^*) = 0$$and we can simply use that ##\vec{a}_i \cdot \vec{a}_j^* = \delta_{ij}## to end up with$$u_1 h_1 + u_2 h_2 + u_3 h_3 = 0$$
 
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Related to A better way of proving the Weiss Zone Law?

1. What is the Weiss Zone Law and why is it important?

The Weiss Zone Law is a physical law that describes the relationship between the magnetic properties of a material and its atomic structure. It is important because it helps us understand and predict the behavior of magnetic materials, which have numerous technological applications.

2. How is the Weiss Zone Law currently proven?

The Weiss Zone Law is currently proven through experimental measurements of the magnetic properties of various materials and their corresponding atomic structures. This data is then compared to the predictions of the law to confirm its validity.

3. What is the current limitation of proving the Weiss Zone Law?

The current limitation of proving the Weiss Zone Law is that it relies heavily on experimental data, which can be time-consuming and expensive to obtain. This makes it difficult to study a large number of materials and their magnetic properties.

4. Is there a better way to prove the Weiss Zone Law?

Yes, there are ongoing efforts to develop computational models and simulations that can accurately predict the magnetic properties of materials based on their atomic structures. This would provide a faster and more cost-effective way of proving the Weiss Zone Law.

5. What impact would a better way of proving the Weiss Zone Law have?

A better way of proving the Weiss Zone Law would have a significant impact on the field of materials science and engineering. It would allow for a deeper understanding of magnetic materials and their properties, leading to the development of more efficient and advanced technologies such as magnetic storage devices and magnetic sensors.

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