Multiplication and Addition to get an integer

[antonni]

Hi, to understand finally the Laue equation for diffraction I am missing something :

h*p+k*q+l*r = integer. Given that p,q,r are integers how come h,k,l MUST BE INTEGERS as well?

Say p=q=r=2, than h=k=l=1/2 works just fine. I understand that there is something about a common multiplier, but what I really want to see is a SOLID MATHEMATICAL proof to that statement. I am not a mathematician but an engineer and have no clue there to start to look for such a proof . . .

help?Thank you,

Anton

 

[scottdave]

I did a little reading about this subject. I need to do more to understand all of what is going on, but maybe it has to do with what these numbers represent (something in the crystal structure) rather than what is mathematically possible.

 

[antonni]

Well, these are very important numbers (h,k,l), true. But we start without restrictions, only then we get the condition for constructive interference:

h*p+k*q+l*r = integer (or more precisely h*p+k*q+l*r = integer*2*pi) knowing that p,q,r are integers. From here it is pure mathematics (THE PART I AM MISSING)
to show that h,k,l MUST BE INTEGERS to fulfill this condition and once they are, only THEN they will become IMPORTANT and gain more meaning.

 

[antonni]

Well, this question was moved from the General Math forum here. Reason: "Physics question, mathematically they could be fractions".

I keep insisting that NOT, this is a MATHEMATICS question, simple as that. If mathematicaly they could be fractions, the there is
something very wrong with the physics :)

AGAIN: Proof that for the statement h*p+k*q+l*r = integer then p,q,r are integers to be true, we MUST HAVE h,k,l =INTEGERS.

MATHEMATICS!

 

[f95toli]

This all has to do with physics; not maths (mathematically they could be anything, and this should be fairly obvious).

The equation gives you a condition(relation for when the phases of the incoming and outgoing waves coincide. Hence, we get this result because we are working with (periodic) waves; there is no way to derive this equation without keeping in mind what it represents.

Have you looked at the wiki for the Laue equation? Is there something in the derivation you do not understand?

 

[nasu]

Can you post a link for that derivation? Or maybe to the source?
In the derivations I know, this condition does not come up at all.

 

[antonni]

Hi watch this

for example. It comes up always. Start from 16:30 and pay special attention to 18:55.

I WILL SAY AGAIN: We find the vector (S-S0)/lambda, this has units of 1/length and ONLY THEN WE EXPRESS IT as a vector in the reciprocal space.
Then we WRITE THE CONDITION FOR CONSTRUCTIVE interference and ONLY THEN WE CONCLUDE THAT h,kl are integers FOR THAT CONDITION TO BE TRUE.

I understand the derivation perfectly but this MATHEMATICAL statement I CANNOT PROOVE.