Miller bravais indices

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?

DrDu
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.

member 392791
The redundancy in coordinate implies the relationship

nasu
Gold Member
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?

Yes they are are all with a 60 degree angle relative to each other.

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.

nasu
Gold Member
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.

yes exactly, 3 vectors in the hexagonal plane, one in the c-direction

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).