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[Statistical Physics] Spin-1 atoms in uniform magnetic field

  1. Jan 16, 2012 #1
    1. The problem statement, all variables and given/known data

    A crystal contains [itex]N[/itex] atoms which posses spin 1 and magnetic moment [itex]\mu[/itex]. Placed in a uniform magnetic field [itex]B[/itex] the atoms can orient themselves in three directions: parallel, perpendicular, and antiparallel to the field. If the crystal is in thermal equilibrium at temperature [itex]T[/itex] find an expression for its mean magnetic moment [itex]M[/itex], assuming that only the interactions of the dipoles with the field [itex]B[/itex] need be considered. [This is a literal transcription of exercise 3.1 from the second edition of Statistical Physics by F. Mandl. ]

    3. The attempt at a solution

    First, I wrote down the partition function, henceforth denoted [itex]Z[/itex]. There are two perpendicular states, which have no interaction energy. There are also two parallel states, separated by a minus sign. Therefore, if I take [itex]x = \beta \mu B[/itex] I get

    [itex] Z = e^0 + e^0 + e^{-\beta \mu B} + e^{\beta \mu B} = 2 + \cosh(x). [/itex]

    Now I get confused. How can I calculate the mean magnetic moment if the book gives me the magnetic moment for each atom? Surely, [itex] M = N \mu [/itex] is a little too simple. Besides, the answer is given in the back of the book as

    [itex] M = N \mu \frac{2sinh(x)}{1+2cosh(x)} [/itex].
     
  2. jcsd
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