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# lec34 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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PARAMAGNETS Spin 1/2 Paramagnet, Microcanonical Ensemble Assume the moment is parallel to the spin. Take n to be the state variable. µ is the magnetic moment of the particle. n H = Hz ˆ µ H µ = µ ˆ ( ) , N = N n, z, = µH N-n - µ H ( ) , N = n, z, = µH µ = µ ˆ 8.044 L34B1
M = µ ( N N ) = µ ( N 2 n ) = M ( n ) N + N = µH ( N 2 n ) = HM ( n ) E N = ↑ ↑ ↓ ↓ H & n N ! S = k ln = S ( n ) n !( N n )! k [ N ln N n ln n ( N n ) ln( N n ) ] 8.044 L34B2

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dS n = k [ 1 ln n +1+ln( N n ) ] = k ln dn N n S = S ( n ) and M = M ( n ) S = S ( M ) 1 ∂S = does not exist! T ∂E M H ∂S dS ∂n n 1 = = = k ln N n 2 µ T ∂M E dn ∂M E 8.044 L34B3
n = e 2 µH/kT 1 n = N N n e 2 µH/kT +1 M = µ ( N 2 n ) = µN tanh( µH/kT ) n N/2 0 µ H/k T 8.044 L34B4

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MAGNETIZATION IN SPIN 1/2 PARAMAGNET 1 2 3 4 5 6 0.2 0.4 0.6
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lec34 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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