Random Distribution of Two Types of Objects with Different Coordination Numbers
eal...@virginia.edu
answer should be easy but I can't figure it out. This may be because
it is actually difficult, or because I'm just making it harder that it
really is. It would be very much appreciated if someone with more
math knowledge than I could help me out.
Consider a random distribution of two different types of spheres, or
atoms in my world, A and B. There is no defined lattice. The
constraints are that A atoms must be surrounded by Za nearest
neighbors and B atom be surrounded by Zb NNs. I am trying to look at
the probabilities PAi and PBi, of finding a ceratin number i B atoms
around A and B atoms.
I know that for Za=Zb=Z the probabilities are simply the probability
of finding i B atoms and Z-i A atoms around a given atom, multiplied
by the degeneracy for this value of i. If the fraction of A and B
atoms in my system are Xa and Xb, respectively, the probabilities
associate with a random arrangment of atoms are
PAi=PBi={Z!/[(Z-i)!*i!]}*Xa^(Z-i)*Xb^i
The sum of the probabilities from 0 to Z must equal one, which is true
for the above equation.
However, if Za is not equal to Zb, the above probability is not
correct. This is because the number of B atoms around A atoms must be
equal to the number of A atoms around B atoms or
Xa*Sum(i*PAi, i=0..Za)=Xb*Sum[(Zb-i)*PBi, i=0..Zb]
I think that the total number of A-B pairs should be
Xa*Xb*(Za+Zb)/2
but I have yet to figure out how determine the new PAi and PBi using
this constraint and at the same time keeping the sum of the
probabilities equal to 1.
If I haven't stated the problem clearly, please let me know and I will
try to clarify. Any help would be very much appreciated.
Thanks so much,
Eric
UVa
francois
why do you think the total number of A-B pairs should be Xa*Xb*(Za+Zb)/
2 ?
by the way, if N is the total number of atoms, do you mean N*Xa*Xb*(Za
+Zb)/2, or N*N*Xa*Xb*(Za+Zb)/2 ?
(since Xa and Xb are just fractions)
François