miller index

[Dell]

for the following SC cube, find the miller indexes for the planes:
FAN
KFA
HAN
http://lh6.ggpht.com/_H4Iz7SmBrbk/SvhiU8OeicI/AAAAAAAAB8M/o8QPR2ht2gA/Capture.JPG

what i leared to do was find the points that the plane intercepts the x,y,z axis, i chose axes such that x+ is pointing right and y+ upwards, then i divide to get a ratio, and then ()^-1
i chose the length of a side of the cube to be "a"

FAN -> a,a,a -> 1,1,1 -> 1 1 1
KFA-> a,inf,a -> 1,inf,1 -> 1 0 1

now for the 3rd one i have found the intercept with the z axis which came out as 1.5a
HAN-> a,a,1.5a -> 1,1,1.5 -> 1 1 2/3

my question is if i am meant to do it like that or if i am meant to divide by 1.5a instead of just a
HAN-> a,a,1.5a -> 2/3,2/3,1 -> 1.5 1.5 1

up until this question i have only had whole numbers in the muiller index, so i thought another option could be
HAN-> a,a,1.5a -> 2,2,3 -> 0.5 0.5 1/3

but then i have on;y fractions,

could someone please clear this up for me, which way (if any )) is the correct way?

[Mapes]

It's customary to use all whole numbers.

[Dell]

so then
3 3 2 ????

[Mapes]

Looks good to me.

[Dell]

now my second question is does it matter what i call xyz?? since they arent defined in the question, i chose xyz as i said but a friend of mine chose different and we obviosly got different answers. for example he would get 2 3 3 for the above index.
is there any standard axes system or can i choose any right handed system?

[Mapes]

Your friend's answer is equivalent since, as you suggest, Nature doesn't have a preferred coordinate system.

[Dell]

Mapes, would it not be 223 since it intercepts the z axis at1.5 times the point it intercepts the x,y axes. or is this correct since i apply ()^-1 to the whole index?

[Mapes]

233 looks right to me. If you look at the surface normal, there's one direction whose resolved component is smaller and two whose components are equal and larger. Plus, I agree with your original calculations.

[Dell]

can you explain what you said about the normals a bit more, how do i see that there is one which ils smaller?

[Mapes]

Well, the plane contacts point H instead of point F, so it's not angled towards the z axis as much as it would be if it were a 111 plane. But this is kind of the "casual" way of estimating the indices. The reciprocal approach is the rigorous way, which you showed in your first post.