Number of atoms - Crystal Struture

[Mathew Murdock]

Homework Statement

Determine the relative amount of low-coordinated atoms (atoms at surfaces, edges, corners) in a cuboid of n atoms edge length. Assume a simple cubic crystal structure.

Relevant Equations

Crystal Structure

Do I do it right guys?
The relative amount of low-coordinated atoms is
The number of atoms at 8 corners: n_corner=8 1/8=1.
The number of atoms at 12 edges: n_edge=12 (n-2)/4=3n-6.
The number of atoms at 6 surfaces: n_surface=6 (n^2-4n)/2=3n^2-12n.
The relative amount of low-coordinated atoms is
n_total=n_corner+n_edge+n_surface=3n^2-9n-5.

 

[BvU]

Any double counting ?
Check your expression for a few small n. Like n = 3 $\Rightarrow$ n_total = -5 ?

 

[Mathew Murdock]

Any double counting ?
Check your expression for a few small n. Like n = 3 $\Rightarrow$ n_total = -5 ?

I got it wrong, the n surface must be (6 (n-2)^2)/2. Thank you for helping me

 

[BvU]

You might want to consider another approach: how many non- low-coordinated atoms are hiding underneath the outer layer of an n $\times$ n $\times$ n cube ? How about (n-2)³ ?

 

[BvU]

n³ - (n-2)³ is not (6 (n-2)^2)/2

Alternatively: (6 (n-2)^2)/2 does not yield 1 for n = 3 !

What does (6 (n-2)^2)/2 represent according to you ?

 

[mjc123]

This is not a unit cell, where you assume atoms on the borders are shared with other unit cells. It is a distinct lump of matter, all of whose atoms belong fully to it (that is the point of "low-coordinated" - they are not shared with another lump). So there are 8 corner atoms, 12(n-2) edge atoms etc.

@BvU's post#5 gives you perhaps an easier approach.

Also note that you are asked for the relative amount of low-coordinated atoms - not the absolute number.

 

[Mathew Murdock]

n³ - (n-2)³ is not (6 (n-2)^2)/2

Alternatively: (6 (n-2)^2)/2 does not yield 1 for n = 3 !

What does (6 (n-2)^2)/2 represent according to you ?

I mean my final answer is 3n^2-9n+7

 

[BvU]

I would thoroughly check that if I were you. You have lost me completely.
Also, bear in mind what @mjc123 pointed out!

 

[Mathew Murdock]

This is not a unit cell, where you assume atoms on the borders are shared with other unit cells. It is a distinct lump of matter, all of whose atoms belong fully to it (that is the point of "low-coordinated" - they are not shared with another lump). So there are 8 corner atoms, 12(n-2) edge atoms etc.

@BvU's post#5 gives you perhaps an easier approach.

Also note that you are asked for the relative amount of low-coordinated atoms - not the absolute number.

Thank you for helping me. I just thought that if it is as you have pointed out for me, then the problem seems to be easier than I thought, and that makes me a little bit nervous

 

[Mathew Murdock]

I would thoroughly check that if I were you. You have lost me completely.
Also, bear in mind what @mjc123 pointed out!

ok I got what you mean by now, just find the whole atoms, minus the non low coordinated atoms and we will get the answer.
So as you said, I have made difficulties myself? :)
I just thought that if it is just about calculating the "area" of this, the teacher does not need to give me this homework in my first solid state physics class.
hope to get your reply

 

[BvU]

hope to get your reply

Reply to what question, precisely ?

 

[Mathew Murdock]

Reply to what question, precisely ?

I mean the low-coordinated atoms made me confused and therefor I thought that I had to calculate the atoms that are entirely inside the unit cells, so there were terms 1/8, 1/4, 1/2 in my answer. You guys then explained that we just basically find the number of atoms on the outer layer, and foget about the unit cells, and the problem is even more simpler :)

 

[Mathew Murdock]

Reply to what question, precisely ?

So do I get your point of view now?

 

[mjc123]

ok I got what you mean by now, just find the whole atoms, minus the non low coordinated atoms and we will get the answer.

The answer you require is the RELATIVE number. You have not yet shown that you understand that.
(You might like to think: how does the relative number scale with n? What are the implications for very small lumps? Perhaps that's why your teacher gave you this question.)

 

[BvU]

So do I get your point of view now?

Point of view ? This is a mathematical exercise with a straightforward answer. You haven't convinced me yet that you have solved it correctly, but I don't see what I can add to help you further. Advice: post your step-by step solution and solicit comments.

 

[Mathew Murdock]

Point of view ? This is a mathematical exercise with a straightforward answer. You haven't convinced me yet that you have solved it correctly, but I don't see what I can add to help you further. Advice: post your step-by step solution and solicit comments.

Ok let me start again. The total atoms is n^3, the atoms lie inside the cube is (n-2)^3, then the atoms on the outer layer is n^3 - (n-2)^3 = 6n^2-12n+8, for which yield 8 for n=2, 26 for n=3 etc... Whats wrong with my solution?
And mjc123 says about relative number, can you help me figure it out what it means? To me, I think that I have to put the approximately equal sign in my answer? My english is not good so that is what I can do now. Sometimes I get stuck with what the question is asking, that is why I need all your contribute comments.

 

[BvU]

Ah ! That explains a lot !
The relative number is the number you got (correctly), divided by the total, so divided by n³.
That's all !

 

[Mathew Murdock]

The answer you require is the RELATIVE number. You have not yet shown that you understand that.
(You might like to think: how does the relative number scale with n? What are the implications for very small lumps? Perhaps that's why your teacher gave you this question.)

I still cannot figure it out by now, please have a look at my new solution and I am looking forward to hearing from you

 

[Mathew Murdock]

Ah ! That explains a lot !
The relative number is the number you got (correctly), divided by the total, so divided by n³.
That's all !

Why do I have to divide it by n^3, could you explain it? is that the relative's meaning? I thought that it was done. Thank you

 

[Mathew Murdock]

Ah ! That explains a lot !
The relative number is the number you got (correctly), divided by the total, so divided by n³.
That's all !

Ah I get it now, thank you :D

 

[Mathew Murdock]

Ah ! That explains a lot !
The relative number is the number you got (correctly), divided by the total, so divided by n³.
That's all !

If the question is "the percentage of", I won't ask you guys ))

 

[BvU]

Determine the relative amount of low-coordinated atoms (atoms at surfaces, edges, corners) in a cuboid of n atoms edge length. Assume a simple cubic crystal structure.

Never been any good at telepathy