Bose-Einstein statistics and Fermi-Dirac statistics

Introduction:

The quantum statistics was first formulated in 1924 by Satyendra Nath Bose in the deduction of Planck's law of radiation by purely statistical reasoning on the basis of fundamental assu mptions that were radically different from those of classical statistics. Einstein in the same year used the same principles in evolving kinetic theory of gasses, as a substitute for classical Boltzmann Statistics. Thus a new quantum statistics, known as Bose-Einstein statistics, came to be accepted. In 1926, Fermi and Dirac independently modified Bose-Einstein statistics in certain cases, on the basis of additional principle, the Pauli's Exclusion Principle. In statistical mechanics, this exclusion principle takes the form that two or more phase points cannot possibly occupy the same phase cell. This led to the recognition of the second kind of quantum statistics, called the Fermi-Dirac statistics.

Thus Quantum statistics can be further divided into two sub-classes:

(a) Bose-Einstein statistics:

This is applicable to the identical, indistinguishable particles of zero or integral spin. These particles are called Bosons. The examples of Bosons are helium atoms at low temperature and the photons.

Since Bosons are identical and non-distinguishable, we must focus on cells rather than individual particles.Each cell may have any number of particles.

(b) Fermi-Dirac statistics:

This is applicable to the identical, indistinguishable particles of half spin. These particles obey Pauli's exclusion principle and are called Fermions. The examples of Fermions are electrons, protons, neutrons, etc.

Fermions are indistinguishable, so we must focus on cells rather than individual particles. Since fermions also obey Pauli's exclusion principle, according to which it is not possible to have more than one representative point in any one cell, each cell will contain only one particle.

Bose-Einstein Statistics

Conditions:

(i) The particles are indistinguishable from one another so that there is no distinction between the different ways in which \(n_i\) particles are can be chosen.

(ii) Each eigenstate of the i^th quantum state may contain 0,1,2...upto \(n_i\) identical particles.

(iii) The sum of energies of all the particles in different quantum groups taken together constitutes the total energy of the system.

Let us consider N particles in the system. Let particles are in the 1^st, 2^nd,...i^th cell with average energies \(\varepsilon_1,\varepsilon_2,...,\varepsilon_i\) for each particle. Let g is the degeneracy factor.

Suppose that \(n_i\) particles are arranged in a row and distributed among \(g_i\) quantum states with \(g_i-1\) partitions in between. The total number of possible arrangements of particles and partitions is equal to the total number of permutations of (\(n_i+g_i-1\)) objects in a row. Therefore, the total possible way o arranging \(n_i\) particles with \(g_i-1\) partitions\(=(n_i+g_i-1)!\)

As the particles are identical and indistinguishable the possible number of distinct arrangement\(=\frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}\)

The total number of different and distinguishable ways of arranging N particles in all the available energy states is given by,

\begin{align*}\Omega=\frac{(n_1+g_1-1)!}{n_1!(g_1-1)!}\times \frac{(n_2+g_2-1)!}{n_2!(g_2-1)!}\times ... \end{align*}\begin{align*}\therefore \Omega=\prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}\rightarrow 1\end{align*}Also we know, logx!=xlogx-x, for large x\begin{align*} From\space 1, log\Omega=\sum_i[log(n_i+g_i-1)!-lohn_i!-log(g_i-1)!]\end{align*}\begin{align*}\therefore log\Omega =\sum_i[(n_i+g_i-1)log(n_i+g_i-1)-n_ilogn_i-(g_i-1)log(g_i-1)]\rightarrow 2 \end{align*}\begin{align*}\therefore \delta log\Omega=\sum_i[log(n_i+g_i-1)\delta n_i-logn_i\delta _i]=0(for\space maxima) \end{align*}\begin{align*}\therefore \sum_i[-log(n_i+g_i-1)+logn_i]\delta n_i=0\rightarrow 3 \end{align*}\begin{align*}Also,\sum_i\delta n_i=0\rightarrow 4 \end{align*}\begin{align*}and,\sum_i\varepsilon_i\delta n_i=0\rightarrow 5 \end{align*}Multiplying 4 by \(\alpha\) and 5 by \(\beta\) and adding to 3, we get\begin{align*} \sum_i[-log(n_i+g_i-1)+logn_i+\alpha +\beta\varepsilon_i]\delta n_i=0\end{align*}\begin{align*}or,\space -log(n_i+g_i-1)+logn_i+\alpha +\beta\varepsilon_i =0\end{align*}\begin{align*}\therefore -log(n_i+g_i)+logn_i+\alpha +\beta\varepsilon_i=0(1<<<n_i+g_i) \end{align*}\begin{align*}\therefore log\left ( \frac{n_i}{n_i+g_i}\right )=-\alpha-\beta\varepsilon_i \end{align*}\begin{align*}or,\space\frac{n_i+g_i}{n_i}=e^{\alpha +\beta\varepsilon_i} \end{align*}\begin{align*}or,\space 1+\frac{g_i}{n_i}= e^{\alpha +\beta\varepsilon_i}\end{align*}\begin{align*}or,\space\frac{g_i}{n_i}= e^{\alpha +\beta\varepsilon_i}-1 \end{align*}\begin{align*}\therefore n_i=\frac{g_i}{e^{\alpha +\beta\varepsilon_i}-1} \end{align*}This is Bose-Einstein distribution law.

Fermi-Dirac Statistics

Conditions:

(i) The particles are indistinguishable from each other so that there is no distinction between the different ways in which \(n_i\) particles are chosen.

(ii)The particles obey Pauli's exclusion principle according to which each sub-level or cell may contain 0 to 1 particle. Then \(g_i\geq n_i\).

(iii)The sum of energies of all the particles in the different quantum groups taken together constitutes the total energy of the system.

Let \(n_i\) be the number of particles for energy level \(\varepsilon_i\). The first particle can be placed in anyone of the available \(g_i\) states. Thus the first particle can be distributed in \(g_i\) different ways and process continues.

Thus the total number of different ways of arranging \(n_i\) particles among the \(g_i\) states with energy level \(\varepsilon_i\) is\(=g_i(g_i-1)...[g_i-(n_i-1)]=\frac{g_i!}{(g_i-n_i)!}\)

Since, the particles are indistinguishable, there is no possibility to detect any difference when \(n_i\) particles are reshuffled into different states occupied by them in the energy level \(\varepsilon_i\).

\(\therefore\) The total number of different ad distinguishable ways is,\(\frac{g_i!}{n_i!(g_i-n_i)!}\)

\(\therefore\) The total number of different and distinguishable ways of getting the distribution \(n_1,n_2,n_3,...,n_i\) among the various energy levels \(\varepsilon!,\varepsilon_2,...,\varepsilon_i\) etc can be calcuated by multiplying the various factors.\begin{align*}\therefore \Omega=\frac{g_1!}{n_1!(g_1-n_1)!}.\frac{g_2!}{n_2!(g_2-n_2)!}....=\prod_i \frac{g_i!}{n_i!(g_i-n_i)!}\end{align*}\begin{align*}\therefore log\Omega=\sum_i [logg_i!-logn_i!-log(g_i-n_i)!] \end{align*}\begin{align*}\therefore log\Omega=\sum_i[(g_ilogg_i-g_i)-(n_ilogn_i-n_i)-((g_i-n_i)log(g_i-n_i)-(g_i-n_i))] \end{align*}\begin{align*}\therefore \delta log\Omega=\sum_i[logn_i-log(g_i-n_i)]\delta n_i \end{align*}For maximum \(\delta (log\Omega )=0\)\begin{align*}\implies \sum_i[logn_i-log(g_i-n_i)]\delta n_i =0\rightarrow 1 \end{align*}\begin{align*}But\space \sum_i\delta n_i=0\rightarrow 2 \end{align*}\begin{align*}and\space \sum_i \varepsilon_i\delta n_i=0\rightarrow 3 \end{align*}Multiplying 2 by \(\alpha\) and 3 by \(\beta\) and adding to 1,\begin{align*} \sum_i[logn_i-log(g_i-n_i)+\alpha+\beta\varepsilon_i]\delta n_i =0\end{align*}\begin{align*}or,\space logn_i-log(g_i-n_i)+\alpha+\beta\varepsilon_i=0 \end{align*}\begin{align*}or,\space log\frac{n_i}{g_i-n_i}=-(\alpha+\beta\varepsilon_i) \end{align*}\begin{align*}or,\space \frac{g_i}{n_i}=1+e^{\alpha+\beta\varepsilon_i} \end{align*}\begin{align*}\therefore n_i=\frac{g_i}{1+e^{\alpha+\beta\varepsilon_i} } \end{align*}This is Fermi-Dirac Distribution law.

Results of three Statistics

The expressions for the most probable distributions in the three statistics are:

(1) Maxwell-Boltzmann statistics:

\(\frac{g_i}{n_i}=e^{\alpha+\beta\varepsilon_i}\rightarrow 1\)

(2) Bose-Einstein statistics:

\(\frac{g_i}{n_i}+1=e^{\alpha+\beta\varepsilon_i}\rightarrow 2\)

(3) Fermi-Dirac statistics:

\(\frac{g_i}{n_i}-1=e^{\alpha+\beta\varepsilon_i}\rightarrow 3\)

For large\(\frac{g_i}{n_i}\)(i.e.\(\frac{g_i}{n_i}\)>>>1),

\(\frac{g_i}{n_i}=\frac{g_i}{n_i}+1=\frac{g_i}{n_i}-1\)

i.e. for large value of\(\frac{g_i}{n_i}\), Bose-Einstein and Fermi-Dirac Distributions approach the Maxwell-Boltzmann distibution. This is the case for normal existence of gases when the temperature is not too low and pressure is not too high.

Electron gas

Electrons are fermions as they obey Pauli's exclusion principle. Energy levels of electrons in metals are grouped in bands. Practically at al temperature, the lower level energy bands are filled with electrons and the upper-level energy bands are filled with electrons. The distribution of electrons is considered only in upper bands called conduction bands.The zero energy level is taken at the lowest level of the conduction band. It is assumed that the electrons have free movements within the conductor, provided the energy associated with the electrons is of the order of upper energy bands.

From Fermi-Dirac distribution law,

\( n_i=\frac{g_i}{1+e^{\alpha+\beta\varepsilon_i}}\)

where n_i is the number of particles in i^th cell and g_i is the degeneracy factor for i^_th cell.

As the energy of electron in the conduction band is continuous the degeneracy factor g_i is replaced by g(E)dE and n_i by n(E)dE.

\begin{align*}i.e.,\space n(E)dE= \frac{g(E)dE}{1+e^{\alpha}e^{\frac{E}{KT}}}\end{align*}

Here n(E) refers to number of electrons and g(E) refers to number of phase cells. But g(E)dE in terms of momentum is,

$$g(P)dP=\frac{8\pi v}{h^3}P^2dP$$
\begin{align*}=\frac{8\pi v}{h^3}P.PdP \end{align*}\begin{align*}But\space E=\frac12mv^2\space and\space P=mv \end{align*}\begin{align*}P=(2mE)^{1/2} \end{align*}\begin{align*}\therefore P^2=2mE\implies 2PdP=2mdE \end{align*}\begin{align*}\implies PdP=mdE \end{align*}\begin{align*}\therefore g(E)dE= \frac{8\pi v}{h^3}(2mE)^{1/2} mdE\end{align*}\begin{align*}=\frac{8\sqrt 2\pi v}{h^3} m^{3/2}E^{1/2}dE\end{align*}\begin{align*}\therefore n(E)dE=\frac{8\sqrt 2\pi v}{h^3}\frac{m^{3/2}E^{1/2}dE}{1+e^{\alpha}e^{\frac{E}{KT}}}\end{align*}

$$g(P)dP=\frac{8\pi v}{h^3}P^2dP \$$

\begin{align*}=\frac{8\pi v}{h^3}P.PdP \end{align*}\begin{align*}But\space E=\frac12mv^2\space and\space P=mv \end{align*}\begin{align*}P=(2mE)^{1/2} \end{align*}\begin{align*}\therefore P^2=2mE\implies 2PdP=2mdE \end{align*}\begin{align*}\implies PdP=mdE \end{align*}\begin{align*}\therefore g(E)dE= \frac{8\pi v}{h^3}(2mE)^{1/2} mdE\end{align*}\begin{align*}=\frac{8\sqrt 2\pi v}{h^3} m^{3/2}E^{1/2}dE\end{align*}\begin{align*}\therefore n(E)dE=\frac{8\sqrt 2\pi v}{h^3}\frac{m^{3/2}E^{1/2}dE}{1+e^{\alpha}e^{\frac{E}{KT}}}\end{align*}

But for electron gas, \(\alpha =-\frac{E_F}{KT}\), \(E_F\) is the Fermi energy.

\begin{align*}\therefore n(E)dE=\frac{8\sqrt 2\pi v}{h^3}\frac{m^{3/2}E^{1/2}dE}{1+e^{\frac{E-E_F}{KT}}}\end{align*}

This is the exact form of Fermi-Dirac law of energy distribution electrons. At the temperature T=0K, the number of electrons is equal to total number of energy states occupied by electrons from 0 to \(E_F\).\begin{align*}\therefore n=\int_{0}^{E_F}g(E)dE \end{align*}\begin{align*}=\frac{8\sqrt 2\pi v}{h^3} m^{3/2}\int_{0}^{E_F} E^{1/2}dE \end{align*}\begin{align*}=\frac{8\sqrt 2\pi vm^{3/2}}{h^3}\left [ \frac23 E^{2/3} \right ]_0^{E_F}= \frac{16\sqrt 2\pi vm^{3/2}}{3h^3}E_F^{3/2}\end{align*}\begin{align*}or,\space E_F^{3/2}=\frac{3nh^3}{16\sqrt 2\pi vm^{3/2}} \end{align*}\begin{align*} \therefore E_F=\frac{h^2}{2m}\left ( \frac{3n}{8\pi v}\right )^{2/3}\end{align*}

Fermi Level and Fermi Energy

The topmost filled energy level at absolute zero temperature is called Fermi level and corresponding energy at topmost level(Fermi level) is called Fermi energy i.e. the energy values upto which all the states are full at 0K and above which all the energy states are empty is known as Fermi Energy.

\begin{align*} \end{align*}\begin{align*} \end{align*}\begin{align*} \end{align*}\begin{align*} \end{align*}\begin{align*} \end{align*}