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A Model System

Some of the more peculiar properties of temperature can be illustrated by a simple example:

Certain particles such as electrons have ``spin ${1 \over 2}$'' which (it turns out) prevents their spins from having any orientation in a magnetic field  $\Vec{B}$  other than parallel to the field (``spin up'') or antiparallel to it (``spin down''). Because each electron has a magnetic moment  $\vec{\mu}$  (sort of like a tiny compass needle) lined up along its spin direction, there is an energy   $\varepsilon = - \vec{\mu} \cdot \Vec{B}$  associated with its orientation in the field.<sup>15.21</sup> For a ``spin up'' electron the energy is   $\varepsilon_{\uparrow} = + \mu B$  and for a ``spin down'' electron the energy is   $\varepsilon_{\downarrow} = - \mu B$.

Consider a system consisting of  N  electrons in a magnetic field and neglect all other interactions, so that the total energy  U  of the system is given by

$$U = (N_{\uparrow} - N_{\downarrow}) \; \mu B$$

where   $N_{\uparrow}$  is the number of electrons with spin up and   $N_{\downarrow}$  is the number of electrons with spin down. Since   $N_{\downarrow} = N - N_{\uparrow}$,  this means

$$U = (2 N_{\uparrow} - N) \; \mu B \quad \hbox{\rm or}$$

$$N_{\uparrow} = {N \over 2} \, + \, {U \over 2 \mu B}$$ (15.15)

-- that is,   $N_{\uparrow}$  and  U  are basically the same thing except for a couple of simple constants. As   $N_{\uparrow}$  goes from  0  to  N,  U  goes from  $-N \mu B$  to  $+N \mu B$.

This system is another example of the binomial distribution whose multiplicity function was given by Eq. (1), with   $N_{\uparrow}$  in place of  n.  This can be easily converted to  $\Omega(U)$.  The entropy  $\sigma(U)$  is then just the logarithm of  $\Omega(U)$,  as usual. The result is plotted in the top frame of Fig. 15.2 as a function of energy. Note that the entropy has a maximum value for equal numbers of spins up and down - i.e. for zero energy. There must be some such peak in  $\sigma(U)$  whenever the energy is bounded above - i.e. whenever there is a maximum possible energy that can be stored in the system. Such situations do occur [this is a ``real'' example!] but they are rare; usually the system will hold as much energy as you want.

  

Figure: Entropy, inverse temperature and temperature of a system consisting of  N=32  spin-$\onehalf$ particles (with magnetic moments  $\mu$) in a magnetic field  B.