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Partition Function of a Single Magnetic Particle

  1. Oct 20, 2008 #1

    M@B

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    1. The problem statement, all variables and given/known data

    For a magnetic particle with an angular momentum "quantum number", j, the allowed values of the z component of a particles magnetic moment are:

    µ = -jδ, (-j + 1)δ, ..., (j-1)δ, jδ

    δ is a constant, and j is a multiple of 1/2

    Show that the partition function of a single magnetic particle is

    Z = sinh[βδB(j+1/2)] / sinh[(βδB)/2]


    2. Relevant equations

    in general, Z = Σ exp(β·E(s))

    and for a magnetic particle: E(s) = -µB

    1 + x + x2 + .... +xn = 1 - xn+1 / 1 - x


    3. The attempt at a solution

    If i did things correctly, I can get to an equation:

    Z = [1 - exp(-βδB(j+1/2))] / [1 - exp(-βδB/2)]

    I got this just by x = exp(-βδB/2) and noticing that the n in the finite sum is 2j. (if you add j to all µ to get a sequence from 0 to 2j instead of -j to j). Then I plugged into the mathematical identity I have above. The problem is converting this in to the sinh term that the question asks for. Unless of course, it is completely wrong, in which case I'm rather lost on the subject.

    Thanks for teh help in advance,
    M@
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 20, 2008 #2
    I see two problems. First, if you have a sum of powers from -j to +j, e.g.
    [tex]x^{-j} + x^{-j+1} + ... + x^j[/tex]
    when you add j to each index, you're really multiplying each term by [tex]x^j[/tex]. In order to shift the indices, you need to divide the thing by [tex]x^j[/tex]. Second, for some reason you have a negative sign in your exponentials rather than a positive sign.
     
  4. Oct 20, 2008 #3

    M@B

    User Avatar

    Thank you very much for your insight. I had completely overlooked the fact that adding j to the index was actually a multiplication. I've managed to make it work out properly by taking that into consideration.

    Thanks again,
    M@
     
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