- #1

opaka

- 16

- 0

## Homework Statement

Consider a network of N = 1006 non-interacting spin 1/2 particles fixed to the sites of a 1D lattice. The network is placed in an external uniform magnetic field so that its total (fixed) energy is given by E = -(N up- N down)ε = -100ε where ε is a positive constant describing how the magnetic moment of each particle couples to the external magnetic field. Now divide the network into a very large component ("the reservoir") and a very small one ("the system"), with the system containing 6 spins and the reservoir containing the remaining 1000.

a. compute the probability of finding the system in each of its allowed 64 microstates

b. compute the average energy of the system

c. use the fact that the probability of finding the system in each of its allowed microstates is given by P

_{a}= exp (-βE

_{a})/Z where Z is a normalization constant to compute the temperature of the reservoir in units of ε. The simplest way of doing this is to take the ratio of the probabilities of two microstates (i.e. the one having all spins up and the one having all spins down).

## Homework Equations

the probability of finding a microstate for a given energy P

_{a}= g

_{a}exp(-βE

_{a})/Z

the partition function Z

_{a}= Ʃ

_{a}g

_{a}exp(-βE

_{a})

average energy U= partial derivative with respect to β of ln Z

_{a}

## The Attempt at a Solution

No problem with part a, I just used N!/(N-n)!n! to find the degeneracies of each microstate - for all spin up and all spin down that would be 1, and E = ±6ε

And while messy, part b was straightforward, since Z

_{a}would be summing up the 7 individual microstates Z and then taking the partial derivative.

c is where I get stuck: I use P(6 up)/P(6 down) = exp(-β6ε)/exp(β6ε). Unfortunately, too much cancels out, and I end up with neither ε or temperature. Any ideas where I'm approaching this incorrectly.