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 verticle 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].

 

[charng]

Hello, thank you for reaching out. The Miller-Bravais scheme is a method used to describe the orientation of crystal planes and directions in hexagonal crystals. It involves a 4-index system, where the first three indices represent the direction of the vector and the fourth index represents the plane normal to the vector.

In the case of hexagonal crystals, the a1 and a2 axes are not perpendicular to each other, which can make it confusing to determine the direction of a vector. The steps you have listed are correct, but it is important to remember that the a1 and a2 axes are not independent of each other. This means that the ratio between them will affect the final 4-index representation of the vector.

In step 3, when you reduce the ratio of a1:a2 to the lowest integer, you are essentially finding the direction of the vector with respect to the a1 and a2 axes. This will affect the final 4-index representation, as shown in step 5.

In your example, the vector [11 (-2) 1] is correct. This means that the vector is parallel to the a1 axis and perpendicular to the a2 axis, with a plane normal of 1. The ratio between a1 and a2 is 1:2, which is why the fourth index is 1. This is not the same as the z axis, which is independent of the ratio between a1 and a2.

I hope this helps to clarify any confusion. If you have any further questions, please feel free to ask. Thank you.