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# lec35 - 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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Simple Quantum Paramagnet, Canonical Ensemble Origin of magnetic moments: Electron spin and orbital angular momentum S + h/ 2 m e c L J µ = g J µ B J µ B e ¯ Nuclear angular momentum h/ 2 m p c I µ = g I µ N I µ N e ¯ 8.044 L35B1
m = B H m 2J + 1 m = J, J 1 , · · · − J g µ B H Z 1 ( T, H ) = J e m /kT = J η ) m = sinh[( J + 1 2 ) η ] sinh[ 1 2 η ] ( e m = J m = J η B H = level spacing kT kT Note Z 1 = Z 1 ( η ) Z = Z 1 ( η ) N = Z ( η ) at fixed N 8.044 L35B2

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e m /kT /Z 1 = p ( m ) = e η m /Z 1 < µ > = m ( B m ) e η m Z 1 = B 1 ∂Z 1 Z 1 ∂η B JB J ( η ) M = N < µ > = B NJB J ( η ) 1 B J ( η ) = J 1 ∂Z 1 Z 1 ∂η 1 1 = 1 ( J + 1 ) coth[( J + ) η ] coth[ 1 η ] J 2 2 2 2 This is called the “Brillouin Function”.
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