Yeah, the "problem" with this system is that there are multiple ways that you can write the same direction, as you noted. In order to get around this redundancy, this system only uses the combination such that h+k+i=0 (where you have [hkil]). So [0010] is never a valid index, since 0+0+1 ≠ 0. But [11-20] does work, because 1+1-2=0.
According to Ashcroft & Mermin, Miller indices are used for the reciprocal lattice only, so (101) and {101} are Miller indices. A&M call [101] and just directions in the direct lattice, and that's what you are doing. So I am a little bit confused by the title of your video. Otherwise very good, thanks a lot.
@@TaylorSparks My question is, are the numbers in directions like [101] in the direct lattice also called Miller indices? A&M say [101] refers to the direct lattice, (101) refers to the reciprocal lattice, and only the latter numbers are called Miller indices.
I start by showing that uvw 3 indix notation will label it [110] but I point out the problem with this too (it's not permutation invariant meaning it should be grouped in family but it would not be). Instead, I show why uvtw is a better system as it creates permutation invariant representations of crystal directions, planes etc. See the 5:00 min mark and see how I took [100] in red ink and [110] in blue ink directions and converted them both into permutation invariant 4 index versions
@@TaylorSparks nah I get that, this was before you converted them. In the very beginning you label the 3 indice in a direction [110], then right before you do the conversion you label the same direction with a 3 indice of [100]. Idk how and why the same direction can be both simultaneously, or if they even should
@@coreyfarrell7347 can you give me a time stamp? I'm not seeing this. At 1:27 I show you how the diagonal red line would be labeled in a 3 indices system [110] but I don't show it as [100]. Instead, I say it should be a permutation of such as [010], [001] etc but that's not possible in 3 indices system which is why we use 4 indices system in hexagonal unit cells.
![How to draw directions from given miller indices in HCP [1120], [1100] hexagonal directions](/img/n.gif)
Very straightforward and understandable 🙌🙌🙃my go-to guy for mse concepts
thanks for the excellent explanation! One question though - couldn't you also say [0010] instead of [1,1,-2,0]? How is this redundancy handled?
Yeah, the "problem" with this system is that there are multiple ways that you can write the same direction, as you noted. In order to get around this redundancy, this system only uses the combination such that h+k+i=0 (where you have [hkil]). So [0010] is never a valid index, since 0+0+1 ≠ 0. But [11-20] does work, because 1+1-2=0.
@@MatSciStudent that's cool, thanks for clarifying!
This Explained it perfectly! thank you!
glad to help!
Well done Taylor
And on the other hand, how would it be?PToPas To change the IM from hexagonal to cubic, I can't understand that.
nice video
According to Ashcroft & Mermin, Miller indices are used for the reciprocal lattice only, so (101) and {101} are Miller indices. A&M call [101] and just directions in the direct lattice, and that's what you are doing. So I am a little bit confused by the title of your video. Otherwise very good, thanks a lot.
I'm not quite sure I follow your question.
@@TaylorSparks My question is, are the numbers in directions like [101] in the direct lattice also called Miller indices? A&M say [101] refers to the direct lattice, (101) refers to the reciprocal lattice, and only the latter numbers are called Miller indices.
@@herbmuell okay, I see. They refer to direct space as well.
@@TaylorSparks Okay, thanks.
I used this method to do the opposite direction calculation. [012] -> [-12-16], it's not correct!!
In the beginning of the video you said the 3 índice was [110], then later in the same direction, you label it [100]. I’m confused
I start by showing that uvw 3 indix notation will label it [110] but I point out the problem with this too (it's not permutation invariant meaning it should be grouped in family but it would not be). Instead, I show why uvtw is a better system as it creates permutation invariant representations of crystal directions, planes etc. See the 5:00 min mark and see how I took [100] in red ink and [110] in blue ink directions and converted them both into permutation invariant 4 index versions
@@TaylorSparks nah I get that, this was before you converted them. In the very beginning you label the 3 indice in a direction [110], then right before you do the conversion you label the same direction with a 3 indice of [100]. Idk how and why the same direction can be both simultaneously, or if they even should
@@coreyfarrell7347 can you give me a time stamp? I'm not seeing this. At 1:27 I show you how the diagonal red line would be labeled in a 3 indices system [110] but I don't show it as [100]. Instead, I say it should be a permutation of such as [010], [001] etc but that's not possible in 3 indices system which is why we use 4 indices system in hexagonal unit cells.
very well explainded sorrrrrrr
اطرش بالزفة 🥲💔
Very straightforward and understandable 🙌🙌🙃my go-to guy for mse concepts