Miller Indices (hklj): Hexagonal Lattice Explained

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In summary, the conversation discusses the relationship between the miller indices and the basis vectors in a hexagonal lattice. The third miller index is redundant and can be expressed as a linear combination of the other two indices. Understanding the basis vectors and constructing the reciprocal space basis vectors is essential in determining the relationship between the miller indices.
  • #1
aaaa202
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There are four miller indices (hklj) for the hexagonal lattice, the third being redudant:

l=-(h+k) (1)

Given the basis vectors a1,a2,a3 I can certainly see that:

a3=-(a1+a2)

But how does this immidiatly lead me to the relation (1) between the miller indices?
 
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  • #2
Spontaneously, I don't know the answer, but I think it is relatively easy to work out. Have you tried?
If you really encounter problems, we all are willing to help you.
 
  • #3
The redundancy in coordinate implies the relationship
 
  • #4
aaaa202 said:
Given the basis vectors a1,a2,a3 I can certainly see that:

a3=-(a1+a2)

But how does this immidiatly lead me to the relation (1) between the miller indices?

How do you define these basis vectors? Is a3 in the same plane as a1 and a2?
 
  • #5
Yes they are are all with a 60 degree angle relative to each other.
 
  • #6
The problem is how exactly to relate the miller indices given (-a1+a2)=a3. After all miller indices for a plane are obtained as inverses of the coordinates for the intersection of the lattice vectors with the plabe.
 
  • #7
Oh, so these are the a1,a2,a3 in the system with four indices. You have and a4 as well. Right?
Sorry, I was confused.
 
  • #8
yes exactly, 3 vectors in the hexagonal plane, one in the c-direction
 
  • #9
Hint: Miller indices refer to a point in reciprocal space, hence you have to construct the reciprocal space basis vectors.

G = H a* + K b* + L c*(for the "normal" 3 Miller indices).
 

What are Miller Indices?

Miller indices, also known as Miller indices notation, is a system for describing the orientation of planes or directions in a crystal lattice. They are used to describe the arrangement of atoms or molecules in a crystal structure.

What is the significance of Miller Indices in a hexagonal lattice?

In a hexagonal lattice, Miller indices are used to describe the orientation of the hexagonal planes. This notation is important because it helps scientists understand the crystal structure and properties of hexagonal materials.

How do you determine the Miller Indices for a plane in a hexagonal lattice?

The Miller Indices for a plane in a hexagonal lattice are determined by finding the intercepts of the plane with the three axes of the lattice. These intercepts are then converted to fractions and enclosed in parentheses, with no commas, to form the Miller indices.

What does the Miller Index (0001) represent in a hexagonal lattice?

The Miller Index (0001) represents the basal plane in a hexagonal lattice. This plane is perpendicular to the c-axis and has the highest atomic density in the lattice.

How are Miller Indices used in crystallography?

In crystallography, Miller indices are used to identify and describe different crystal structures. They are also used to determine the symmetry and properties of crystals, and to predict the behavior of materials in various applications.

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