, Solutions to problems from chapter 1- Collective Behavior, From Particles to Fields
1. The binary alloy: A binary alloy (as in β brass) consists of NA atoms of type A, and
NB atoms of type B. The atoms form a simple cubic lattice, each interacting only with its
six nearest neighbors. Assume an attractive energy of −J (J > 0) between like neighbors
A − A and B − B, but a repulsive energy of +J for an A − B pair.
(a) What is the minimum energy configuration, or the state of the system at zero temper-
ature?
• The minimum energy configuration has as little A-B bonds as possible. Thus, at zero
temperature atoms A and B phase separate, e.g. as indicated below.
A B
(b) Estimate the total interaction energy assuming that the atoms are randomly distributed
among the N sites; i.e. each site is occupied independently with probabilities pA = NA /N
and pB = NB /N .
• In a mixed state, the average energy is obtained from
E = (number of bonds) × (average bond energy)
= 3N · −Jp2A − Jp2B + 2JpA pB
2
NA − NB
= −3JN .
N
(c) Estimate the mixing entropy of the alloy with the same approximation. Assume
NA , NB ≫ 1.
• From the number of ways of randomly mixing NA and NB particles, we obtain the
mixing entropy of
N!
S = kB ln .
NA !NB !
Using Stirling’s approximation for large N (ln N ! ≈ N ln N − N ), the above expression can
be written as
S ≈ kB (N ln N − NA ln NA − NB ln NB ) = −N kB (pA ln pA + pB ln pB ) .
1
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller COURSEHERO2. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $11.39. You're not tied to anything after your purchase.
