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I'm currently in an introductory thermal physics course, and today we

learned about the Boltzmann factor and how to use it to calculate

various quantities with the corresponding probability distribution. All was going well until we got to the subject of paramagnetism.

We supposed a system of independent magnetic spins, pointing either

parallel or anti-parallel to the Magnetic field given by [itex]\vec{B}[/itex]. In other

words, there are N spins each with magnetic moment [itex]\vec{μ}[/itex] in contact with a

thermal reservoir of temperature T.

I understand that each spin has a potential energy given by

E[itex]_{n}[/itex]=[itex]\pm[/itex][itex]\vec{μ}[/itex][itex]\vec{B}[/itex] (A basic E&M result). However, I DON'T really understand how we can relate this

quantity to the Boltzmann distribution: P[itex]_{n}[/itex]= e^([itex]\frac{-E_{n}}{kT}[/itex])/Z where Z is the partition function which normalizes the distribution.

How can the thermal energy of the reservoir (kT) can be related to the

purely magnetic state of each magnetic spin? I'm clearly missing

something since this makes no sense to me. Is energy also going into

the thermal energy of these spins and we are just not considering it?

I'm quite confused by this concept and any clarification would be most

appreciated.

learned about the Boltzmann factor and how to use it to calculate

various quantities with the corresponding probability distribution. All was going well until we got to the subject of paramagnetism.

We supposed a system of independent magnetic spins, pointing either

parallel or anti-parallel to the Magnetic field given by [itex]\vec{B}[/itex]. In other

words, there are N spins each with magnetic moment [itex]\vec{μ}[/itex] in contact with a

thermal reservoir of temperature T.

I understand that each spin has a potential energy given by

E[itex]_{n}[/itex]=[itex]\pm[/itex][itex]\vec{μ}[/itex][itex]\vec{B}[/itex] (A basic E&M result). However, I DON'T really understand how we can relate this

quantity to the Boltzmann distribution: P[itex]_{n}[/itex]= e^([itex]\frac{-E_{n}}{kT}[/itex])/Z where Z is the partition function which normalizes the distribution.

How can the thermal energy of the reservoir (kT) can be related to the

purely magnetic state of each magnetic spin? I'm clearly missing

something since this makes no sense to me. Is energy also going into

the thermal energy of these spins and we are just not considering it?

I'm quite confused by this concept and any clarification would be most

appreciated.

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