Miller-Bravais scheme for Hexagonal crystals

[murasame]

Hello, sorry to bother again but I've been giving this much thought as well and am very confused..

Please refer to the following page:

http://x5.freeshare.us/119fs641427.jpg

If I am supposed to calculate the direction vector for the hexagonal crystal, I was told to:

1) Calculate the line of projection of the vector (from origin to X) onto the base plane

2) Calculate the new line of projection of this projected line with respect to a1 and a2 axis.

3) Reduce the ratio of a1: a2 to the lowest integer

4) Calculate the line of projection of the vector onto the vertical z axis

5) Use the 3-index system to 4-index system formula to convert it and eventually get [11 (-2) 1] for the mentioned vector.

However, this would imply that the z vector is independent of the ratio in a1 and a2 since using this method we would have gotten 0.5 unit length for a1 and a2 and 1 unit length for z. If not, shouldn't it be [11 (-2) 2] instead?

I'm sorry if I sound confusing, because I'm very confused myself as well. But if anyone knows about this, please help, thanks!

 

[Astronuc]

the direction vector for the hexagonal crystal

I am not sure what one means by the direction vector. There is a unique planar orientation, the basal pole (normal to the basal plane), which is in the c-direction of the hexagonal (hcp) crystal.

Then there are the three a-directions. Besides that, there are prismatic and pyramidal planes and associated directions.

IIRC, there is [11$\bar{2}$0].