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25.2: Fermi-Dirac Statistics and the Fermi-Dirac Distribution Function

  • Page ID
    151986
  • Let us consider the total probability sum for a system of particles that follows Fermi-Dirac statistics. As before, we let \({\epsilon }_1\), \({\epsilon }_2\),, \({\epsilon }_i\),. be the energies of the successive energy levels. We let \(g_1\), \(g_2\),, \(g_i\),. be the degeneracies of these levels. We let \(N_1\), \(N_2\),, \(N_i\),. be the number of particles in all of the degenerate quantum states of a given energy level. The probability of finding a particle in a quantum state depends on the number of particles in the system; we have \(\rho \left(N_i,{\epsilon }_i\right)\) rather than \(\rho \left({\epsilon }_i\right)\). Consequently, we cannot generate the total probability sum by expanding an equation like

    \[1={\left(P_1+P_2+\dots +P_i+\dots \right)}^N.\]

    However, we continue to assume:

    1. A finite subset of the population sets available to the system accounts for nearly all of the probability when the system is held in a constant-temperature environment.
    2. Essentially the same finite subset of population sets accounts for nearly all of the probability when the system is isolated.
    3. All of the microstates that have a given energy have the same probability. We let this probability be \({\rho }^{FD}_{MS,N,E}\).

    As before, the total probability sum will be of the form \[1=\sum_{\{N_i\}}{W^{FD}\left(N_i,{\epsilon }_i\right)}{\rho }^{FD}_{MS,N,E}\]

    Each such term reflects the fact that there are \(W^{FD}\left(N_i,{\epsilon }_i\right)\) ways to put \(N_1\) particles in the \(g_1\) quantum states of energy level \({\epsilon }_1\), and \(N_2\) particles in the \(g_2\) quantum states of energy level \({\epsilon }_2\), and, in general, \(N_i\) particles in the \(g_i\) quantum states of energy level \({\epsilon }_i\). Unlike Boltzmann statistics, however, the probabilities are different for successive particles, so the coefficient \(W^{FD}\) is different from the polynomial coefficient, or thermodynamic probability, \(W\). Instead, we must discover the number of ways to put \(N_i\) indistinguishable particles into the \(g_i\)-fold degenerate quantum states of energy \({\epsilon }_i\) when a given quantum state can contain at most one particle.

    These conditions can be satisfied only if \(g_i\ge N_i\). If we put \(N_i\) of the particles into quantum states of energy \({\epsilon }_i\), there are

    1. \(g_i\) ways to place the first particle, but only
    2. \(g_i-1\) ways to place the second, and
    3. \(g_i-2\) ways to place the third, and
    4. \(g_i-\left(N_i-1\right)\) ways to place the last one of the \(N_i\) particles.

    This means that there are

    \[\left(g_i\right)\left(g_i-1\right)\left(g_i-2\right)\dots \left(g_i-\left(N_i+1\right)\right)=\]

    \[=\frac{\left(g_i\right)\left(g_i-1\right)\left(g_i-2\right)\dots \left(g_i-\left(N_i+1\right)\right)\left(g_i-N_i\right)\dots \left(1\right)}{\left(g_i-N_i\right)!}=\frac{g_i!}{\left(g_i-N_i\right)!}\]

    ways to place the \(N_i\) particles. Because the particles cannot be distinguished from one another, we must exclude assignments which differ only by the way that the \(N_i\) particles are permuted. To do so, we must divide by \(N_i!\). The number of ways to put \(N_i\) indistinguishable particles into \(g_i\) quantum states with no more than one particle in a quantum state is \[\frac{g_i!}{\left(g_i-N_i\right)!N_i!}\]

    The number of ways to put indistinguishable Fermi-Dirac particles of the population set \(\{N_1\mathrm{,\ }N_2\mathrm{,\dots ,\ }N_i\mathrm{,\dots }\}\) into the available energy states is

    \[W^{FD}\left(N_i,g_i\right)=\left[\frac{g_1!}{\left(g_1-N_1\right)!N_1!}\right]\times \left[\frac{g_2!}{\left(g_2-N_2\right)!N_2!}\right]\times \dots \times \left[\frac{g_i!}{\left(g_i-N_i\right)!N_i!}\right]\times \dots =\prod^{\infty }_{i=1}{\left[\frac{g_i!}{\left(g_i-N_i\right)!N_i!}\right]}\]

    so that the total probability sum for a Fermi-Dirac system becomes

    \[1=\sum_{\{N_j\}}{\prod^{\infty }_{i=1}{\left[\frac{g_i!}{\left(g_i-N_i\right)!N_i!}\right]}{\left[{\rho }^{FD}\left({\epsilon }_i\right)\right]}^{N_i}}\]

    To find the Fermi-Dirac distribution function, we seek the population set \(\{N_1\mathrm{,\ }N_2\mathrm{,\dots ,\ }N_i\mathrm{,\dots }\}\) for which \(W^{FD}\) is a maximum, subject to the constraints

    \[N=\sum^{\infty }_{i=1}{N_i}\] and \[E=\sum^{\infty }_{i=1}{N_i}{\epsilon }_i\]

    The mnemonic function becomes

    \[F^{FD}_{mn}=\sum^{\infty }_{i=1}{ \ln g_i!\ } -\sum^{\infty }_{i=1}{\left[\left(g_i-N_i\right){ \ln \left(g_i-N_i\right)\ }-\left(g_i-N_i\right)\right]}-\sum^{\infty }_{i=1}{\left[N_i{ \ln N_i-N_i\ }\right]+\alpha \left[N-\sum^{\infty }_{i=1}{N_i}\right]} +\ \beta \left[E-\sum^{\infty }_{i=1}{N_i}{\epsilon }_i\right]\]

    We seek the \(N^{\textrm{⦁}}_i\) for which \(F^{FD}_{mn}\) is an extremum; that is, the \(N^{\textrm{⦁}}_i\) satisfying

    \[ \begin{align*} 0&=\frac{\partial F^{FD}_{mn}}{\partial N_i}=\frac{g_i-N^{\textrm{⦁}}_i}{g_i-N^{\textrm{⦁}}_i}+{ \ln \left(g_i-N^{\textrm{⦁}}_i\right)\ }-1-\frac{N^{\textrm{⦁}}_i}{N^{\textrm{⦁}}_i}-{ \ln N^{\textrm{⦁}}_i\ }+1-\alpha -\beta {\epsilon }_i \\[4pt] &={ \ln \left(g_i-N^{\textrm{⦁}}_i\right)\ }-{ \ln N^{\textrm{⦁}}_i\ }-\alpha -\beta {\epsilon }_i \end{align*}\]

    Solving for \(N^{\textrm{⦁}}_i\), we find

    \[N^{\textrm{⦁}}_i=\frac{g_ie^{-\alpha }e^{-\beta {\epsilon }_i}}{1+e^{-\alpha }e^{-\beta {\epsilon }_i}}\]

    or, equivalently,

    \[\frac{N^{\textrm{⦁}}_i}{g_i}=\frac{1}{1+e^{\alpha }e^{\beta {\epsilon }_i}}\]

    If \(1\gg e^{-\alpha }e^{-\beta {\epsilon }_i}\) (or \(1\ll e^{\alpha }e^{\beta {\epsilon }_i}\)), the Fermi-Dirac distribution function reduces to the Boltzmann distribution function. It is easy to see that this is the case. From

    \[N^{\textrm{⦁}}_i=\frac{g_ie^{-\alpha }e^{-\beta {\epsilon }_i}}{1+e^{-\alpha }e^{-\beta {\epsilon }_i}}\approx g_ie^{-\alpha }e^{-\beta {\epsilon }_i}\]

    and \(N=\sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i}\), we have

    \[N=e^{-\alpha }\sum^{\infty }_{i=1}{g_i}e^{-\beta {\epsilon }_i}=e^{-\alpha }z\]

    It follows that \(e^{\alpha }={z}/{N}\). With \(\beta ={1}/{kT}\), we recognize that \({N^{\textrm{⦁}}_i}/{N}\) is the Boltzmann distribution. For occupied energy levels, \(e^{-\beta {\epsilon }_i}=e^{-\epsilon_i}/{kT}\approx 1\); otherwise, \(e^{-\beta \epsilon_i}=e^{-\epsilon_i/kT}<1\). This means that the Fermi-Dirac distribution simplifies to the Boltzmann distribution whenever \(1\gg e^{-\alpha }\). We can illustrate that this is typically the case by considering the partition function for an ideal gas.

    Using the translational partition function for one mole of a monatomic ideal gas from Section 24.3, we have

    \[\begin{align*} e^{\alpha } &=\frac{z_t}{N}=\left[\frac{2\pi mkT}{h^2}\right]^{3/2} \frac{\overline{V}}{\overline{N}} \\[4pt] &=\left[\frac{2\pi mkT}{h^2}\right]^{3/2} \frac{kT}{P^0} \end{align*}\]

    For an ideal gas of molecular weight \(40\) at \(300\) K and \(1\) bar, we find \(e^{\alpha }=1.02\times {10}^7\) and \(e^{-\alpha }=9.77\times {10}^{-8}\). Clearly, the condition we assume in demonstrating that the Fermi-Dirac distribution simplifies to the Boltzmann distribution is satisfied by molecular gases at ordinary temperatures. The value of \(e^{\alpha }\) decreases as the temperature and the molecular weight decrease. To find \(e^{\alpha }\approx 1\) for a molecular gas, it is necessary to consider very low temperatures.

    Nevertheless, the Fermi-Dirac distribution has important applications. The behavior of electrons in a conductor can be modeled on the assumption that the electrons behave as a Fermi-Dirac gas whose energy levels are described by a particle-in-a-box model.