Miller indices for hexagonal crystal systems

In summary: Einfache_HerleitungIn summary, The conversation discusses the use of Miller indices in crystals and the derivation of the formulas for finding the direction of a given Miller index. It is mentioned that the 4th index can be dropped as it corresponds to the reciprocal space "d*" axis. The physical reason for using 4 indices is explained and a German Wikipedia link is provided as a resource for a simple derivation of the formula. The use of the vector-component method is also mentioned but it does not work for finding the results.
  • #1
Mind----Blown
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Hi everyone, to find the draw the direction for a given miller index say, [1234] we first convert this miller index consisting of 4 indices into one containing 3 indices.
To do so, we have a set of formulae prescribed in almost every book. Sadly I haven't been able to come across a single book the gives the derivation of those formulae!
I thought that i could use vector-component method to get the results but that gives totally weird formulae not even close to the ones i see in my textbooks. (have a look at the attached image)

So, can anyone suggest me a textbook, a link or anything that can help me understand the derivation? I'm not finding the enthusiasm for rote-memorising the formulae if i don't know where they come from...

Thanks!
 

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  • #2
The 4 Miller indices often used with hexagonal crystals are just a convenience. The 3rd index can be dropped. 4 indices are used to make planes with the same symmetry also "look alike" in the 4-notation.
The physical reason is that the a- and b- axes form a 120 degree angle. There is a third axis within the basal plane that has the exact same symmetry. You get this axis, let's call it "d", by rotating the b-axis by another 120 degrees. So instead of using a and be as basis vectors, you could just as well use b and d, or d and a. Absolutely nothing would change, as the 120 degree rotation is the defining feature of a hexagonal crystal.
The redundant 3rd Miller index corresponds to the reciprocal space "d*" axis. The 4th index is along the c-axis.
 
  • #3
M Quack said:
The 4 Miller indices often used with hexagonal crystals are just a convenience. The 3rd index can be dropped. 4 indices are used to make planes with the same symmetry also "look alike" in the 4-notation.
The physical reason is that the a- and b- axes form a 120 degree angle. There is a third axis within the basal plane that has the exact same symmetry. You get this axis, let's call it "d", by rotating the b-axis by another 120 degrees. So instead of using a and be as basis vectors, you could just as well use b and d, or d and a. Absolutely nothing would change, as the 120 degree rotation is the defining feature of a hexagonal crystal.
The redundant 3rd Miller index corresponds to the reciprocal space "d*" axis. The 4th index is along the c-axis.
yes, i get that but still, how do we arrive at those formulae?
i tried using vectors and their components method but it doesn't work
 

1. What are Miller indices for hexagonal crystal systems?

Miller indices for hexagonal crystal systems are a way to describe the orientation and spacing of crystal planes in a hexagonal lattice. They are represented by three numbers (hkl), where h, k, and l are integers that indicate the intercepts of the plane with the crystallographic axes.

2. How do you determine Miller indices for a specific plane in a hexagonal crystal?

To determine Miller indices for a specific plane in a hexagonal crystal, you must first identify the intercepts of the plane with the crystallographic axes. Then, take the reciprocals of these intercepts and reduce the resulting fractions to the smallest possible integers. The resulting numbers will be the Miller indices for that plane.

3. What is the significance of Miller indices in hexagonal crystal systems?

Miller indices provide a way to describe the orientation and spacing of crystal planes in a hexagonal lattice. This information is important for understanding the physical and chemical properties of materials, as well as for predicting their behavior in various environments.

4. Can Miller indices be negative in hexagonal crystal systems?

No, Miller indices cannot be negative in hexagonal crystal systems. Since the intercepts of a plane with the crystallographic axes are always positive, the reciprocals used to determine Miller indices will also be positive integers.

5. How are Miller indices for hexagonal crystal systems different from those in other crystal systems?

The main difference between Miller indices for hexagonal crystal systems and those in other crystal systems is the presence of a third axis in the hexagonal lattice. This means that hexagonal crystals have three Miller indices instead of the typical four for cubic or orthorhombic crystals.

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