Spin 1/2 particles in magnetic Field B can take energies E = ± μ B Ba) For N non interacting spin -1/2 particles at temperature T, derive the partition function Z and Helmholtz free energy F.b) Calculate the magnetic moment of this system, given by m = − ( ∂ F ∂ B ) T

Question

Spin 1/2 particles in magnetic Field B can take energies   E  =  ±      μ    B    Ba) For N non interacting spin -1/2 particles at temperature T, derive the partition function Z and Helmholtz free energy F.b) Calculate the magnetic moment of this system, given by   m  =  −  (            ∂      F              ∂      B            )    T

Gallichi5mtwt · Accepted Answer

Part (a):The partition function of single particle is given by,      Z    1    =  ∑      e          −              E                  k          T                            Z    1    =      e          −                                    μ            B                    B                          k          T                      +      e                                        μ            B                    B                          k          T                            Z    1    =  2  cosh  ⁡  (                    μ        B            B              k      T        )Now for N particles the partition function is,      Z    N    =  (      Z    1        )    N          Z    N    =  (      e          −                                    μ            B                    B                          k          T                      +      e                                        μ            B                    B                          k          T                          )    N          Z    N    =  (  2  cosh  ⁡  (                    μ        B            B              k      T        )      )    N    F  =  −  k  T  ln  ⁡  (  Z  )    F  =  −  k  T  ln  ⁡  [  (  2  cosh  ⁡  (                    μ        B            B              k      T        )      )    N    ]    F  =  −  N  k  T  ln  ⁡  [  2  cosh  ⁡  (                    μ        B            B              k      T        )  ]

Eve Dunn · Accepted Answer

Part (b)The magnetic moment is,  m  =  −      ∂          ∂      B        F  =      ∂          ∂      B        [  −  k  T  ln  ⁡  (  Z  )  ]    m  =      ∂          ∂      B        k  T  ln  ⁡  [  (  2  cosh  ⁡  (                    μ        B            B              k      T        )      )    N    ]    F  =  (  N  k  T  )      ∂          ∂      B        ln  ⁡  [  2  cosh  ⁡  (                    μ        B            B              k      T        )  ]    F  =  (  N  k  T  )  [            μ      B              k      T        (            sinh      ⁡      (                                    μ            B                    B                          k          T                    )              cosh      ⁡      (                                    μ            B                    B                          k          T                    )        )  ]    F  =  N      μ    B    (  tanh  ⁡  (                    μ        B            B              k      T        )  )

Spin 1/2 particles in magnetic Field B can take energies E = ± &#x03B

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