Rules in determining family of planes in Hexagonal

[ralden]

Hi guys, I'm assume that you already know the 7 crystal system, each crystals have unique way of determining the family of planes, for example in cubic, we all know (111) plane is same (-1-1-1), and so on ((-1-11),(-11-1)...) a total of 8, in fact there is pattern to determine how many possible planes does a miller indices (hkl) can have in each crystal, so now my problem is, how to determine all family of planes in hexagonal, based on literature for h is not equal to k, k is not equal to l and h is not equal to l, there are 24 possible planes you could have, but what are those 24? for example in hexagonal you have (123) plane and you could have 24 family of planes, but what are the basis or rules to determine the maximum possible families of planes? please help thanks.

 

[M Quack]

In general, you have to consider the crystallographic point group. By applying all the point group operations to your reciprocal lattice vector (HKL) you
obtain all equivalent vectors, i.e. all other planes in the same family.

For cubic and tetragonal groups you can do that pretty much by inspection. For hexagonal this is a bit more tricky. Sometimes a 4th (redundant) Miller index is used to make lattice planes within the same family look alike:

http://en.wikipedia.org/wiki/Miller_index#Case_of_hexagonal_and_rhombohedral_structures

 

[ralden]

so given a p\lane in hexagonal (134), how could you determine all possible family of plane it have?

 

[read]

This number (24) is also called multiplicity. To calculate it we should calculate the length of your reciprocal lattice vector. In hexagonal case (hkl)^2 = a*^2(h^2+k^2+hk) + c*^2 l. Then we should look at which h,k,l the length remains the same. So the problem is reduced to h^2+k^2+hk=const. This can be done algebraically or geometrically. You can have additionally h'=h-k,k'=-k, etc. Geometrically we have reciprocal a* and b* that make an angle of 60 degrees. This makes 360/60=6 possible combinations, then 6*2=12 because we can exchange h and k, and finally 12*2=24 due to +/-l.
Voila...

 

[ralden]

ahhmm but if you have 110 which have 6 multiplicity, using your equation it only generates 2 multiplicity which is 110, and -1-10.

 

[ralden]

and also 00-4 based on your equation is equal to -100, and -110

 

[read]

24 is a maximal multiplicity when h # k # l # 0. For 110: -110, 0-10,-210,...

00-4 can be equal to -100 only accidentally. Look at the formula. There are also lengths a*, c* which are different.

 

[ralden]

so what are lengths a* and c* ?

 

[read]

reciprocal lattice constants.
Your basis is a*=[b x c]/V, b* =[c x a]/V, c* = [a x b]/V.
V - is cell volume, a,b,c, - basis of the lattice. The hkl that you use are written in the above basis.

 

[ralden]

can you list down all possible families of plane in (123?) I'm still confuse.

 

[read]

1 2 3
-3 1 3
-3 2 3
-2 -1 3
-2 3 3
-1 -2 3
-1 3 3
1 -3 3
2 -3 3
2 1 3
3 -2 3
3 -1 3

and the same as above multiplied by -1.

hope it helps...