Find Miller Indexes of SC Cube Planes: FAN, KFA, HAN

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Discussion Overview

The discussion revolves around finding the Miller indices for specific planes (FAN, KFA, HAN) in a simple cubic (SC) lattice. Participants explore the methodology for determining these indices, including the choice of axes and the treatment of intercepts with the coordinate axes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes their method for finding Miller indices by determining the intercepts of the planes with the x, y, and z axes, using a side length of the cube as "a".
  • For the FAN plane, the participant calculates the indices as (1, 1, 1).
  • For the KFA plane, the indices are proposed as (1, ∞, 1), which is interpreted as (1, 0, 1) in Miller index notation.
  • The participant expresses uncertainty about the calculation for the HAN plane, initially proposing (1, 1, 1.5) and later considering (2/3, 2/3, 1) and (2, 2, 3) as alternative representations.
  • Another participant suggests that using whole numbers for Miller indices is customary.
  • There is a discussion about the choice of coordinate axes, with one participant questioning whether different choices could lead to different indices, while another participant asserts that the choice of axes does not affect the equivalence of the indices.
  • A participant raises a question about the interpretation of the surface normal and how it relates to the calculated indices.
  • Another participant emphasizes the difference in angles of the plane relative to the axes, suggesting that the rigorous method of calculating indices should be considered.

Areas of Agreement / Disagreement

Participants generally agree on the importance of using whole numbers for Miller indices, but there is no consensus on the correct representation of the indices for the HAN plane. The discussion about the choice of axes also reveals differing opinions, indicating that multiple views remain on this topic.

Contextual Notes

Participants mention the potential for different interpretations based on axis choices and the implications for the resulting Miller indices. The discussion includes various calculations and representations, but no definitive resolution is reached regarding the correct approach for the HAN plane.

Dell
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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?
 
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It's customary to use all whole numbers.
 
so then
3 3 2 ?
 
Looks good to me.
 
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?
 
Your friend's answer is equivalent since, as you suggest, Nature doesn't have a preferred coordinate system.
 
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?
 
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.
 
can you explain what you said about the normals a bit more, how do i see that there is one which ils smaller?
 
  • #10
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.
 

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