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Generalising the Ising model to multiple spin values

  1. Nov 16, 2014 #1
    My tutor asked us today to consider the partition function of the following model as an aside to our topic at the moment.
    I went to work out the maths of it today and I'm quite stuck for how the calculation can proceed.

    It's a 1d closed chain with some number, n, points. Each point has some value of spin associated with it, with the possible values ranging from -s to s in steps of 1.
    With

    H=gμB*(the sum over all of the possible spin configurations)



    where mu is the bohr magneton.
     
  2. jcsd
  3. Nov 18, 2014 #2
    Let's assume "the sum over all of the possible spin configurations" means "the sum of all spins" (I don't see how anything else could make sense as a factor in the Hamiltonian): You can directly write down the partition function. The canonical one is probably the most straightforward. Also, since there is no interaction term, the generic partition function can be simplified greatly (you'll probably see that once you get there - it's an interesting general result). If you don't know how to write down the partition function directly, approach the problem bottom-up. Write it down for 1 particle, then for 2, 3, then hopefully see a pattern. A bit tedious but there's nothing wrong with taking the pedestrian approach.
     
  4. Nov 18, 2014 #3
    Okay thank you! It confuses me that the term means 'the sum of all spins' because in a system with one particle for example, if the spin may equal -s or -s+1 ... etc up to +s surely the sum of the spins is zero?
     
  5. Nov 19, 2014 #4
    The spin may be equal to, say -1 or +1. But in a given state it is either +1 or -1, not both (ignoring QM for a second to keep it simple). I think I see where your problem lies and I thought about ways to explain the issue without giving you too much information about your homework. But I ended up deciding to give you a straight-up reply rather than trying to be cryptic for the sake of not being too clear:
    The partition function for the canonical partition function (for example) is NOT exp(-beta * sum of the energies of all possible states). It is sum_over_all_possible_states[ exp(-beta * energy of the particular state) ]. Hence, your statement that the sum of all possible energies/spins equals zero is correct. But it is irrelevant for your task.
    Sidenote: My assumption above that the sum in your H means the sum of all spins in a given state, and not the sum of all possible total spins of all possible spin combinations, comes from exactly this: Hamiltonians are functions on individual single- or multi-particle states, not on sets of such states.
     
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