arXiv:1402.3370v4 [physics.hist-ph] 28 Feb 2015 Fermi-Dirac Statistics Shyamal Biswas * School of Physics, University of Hyderabad, CR Rao Road, Gachibowli, Hyderabad-500046, India (Dated: March 3, 2015) Here we have discussed on Fermi-Dirac statistics, in particular, on its brief historical progress, derivation, consequences, applications, etc. Importance of Fermi-Dirac statistics has been discussed even in connection with the current progresses in science. This article is aimed mainly for under- graduate and graduate students. PACS numbers: 01.65.+g, 03.75.Ss, 05.30.-d, 01.55.+b I. INTRODUCTION AND BRIEF HISTORICAL PROGRESS Fermi-Dirac statistics describes energy distribution of a non (or weakly) interacting gas of identical particles (now called fermions, eg. neutrino, electron, quark, proton, neutron, 6 Li, 40 K, etc ) that obey the Pauli exclusion principle [1]. It is named after Enrico Fermi who derived it in 1926 and Paul Dirac who derived it independently a little later in the same year [2, 3]. It was a very important time for the development of quantum mechanics as well as of modern physics particularly for the revolutionary transition from the old quantum theory to the new quantum theory. While the old quantum mechanics mostly was commanded by Bohr-Sommerfeld quantization [4, 5], new quantum mechanics is commanded by Schrodinger’s wave mechanics [6] and Heisenberg-Born-Jordan matrix mechanics [7]. This time spin (1/2) of an electron although was proposed, yet not well understood [8]. Nonetheless, the Pauli exclusion principle worked- it clearly explained the structure of the periodic table [9], the atomic spectra (fine structure [10], anomalous Zeeman effect [11] and Paschen-Back effect [12]), etc. While for a single-particle system, an operator (eg. Hamiltonian) is quantized using the (first) commutation relation between position and momentum, a successful attempt of quantizing the (energy) distribution of particles, for a many- particle system, was made by Fermi using Pauli exclusion principle as another (second) quantization rule for identical particles not even well knowing about the new quantum mechanics. Dirac, of course, was well aware of new quantum mechanics when he obtained a similar distribution formula. Fermi and Dirac essentially established a connection between (classical) Maxwell-Boltzmann distribution [13] and (quantum) Pauli exclusion principle [1]. They perhaps were motivated to do such a brilliant work from the qualitative prediction of Nernst on the ‘degeneracy’ of gases at low temperature [14]. ‘Degeneracy’ of the non (or weakly) interacting (Fermi) gas, of course, is a natural outcome of their brilliant connection. It should be mentioned in this regard, that, a similar brilliant successful attempt (towards ‘degeneracy’ of a gas at low temperature) for another type of gas of identical particles (now called bosons eg. photon, gluon, π 0 , 1 H, 4 He, 7 Li, 14 N, 16 O, 23 Na, H 2 O, 52 Cr, W ± bosons, 87 Rb, Z-boson, Higgs boson, etc ) was also made by Bose and Einstein even earlier than Fermi did [15, 16]. Spin of particles, other than electron, was of course, not known at that time. Spin of photon was observed to be 1 in 1931 by Raman and Bhagavantam [17]. Connection between spin of identical particles and (Fermi-Dirac or Bose-Einstein) statistics was known from the spin statistics theorem proposed in around 1940 by Fierz and Pauli [18]. Categorization of atoms to either bosons or fermions started around this time [19, 20]. Later on, quantum field theory came to us as a complete! description of a many particle system interpreting interparticle interaction among fermions as exchange of bosons eg. exchange of photon leads to Coulomb interaction between two electrons [21], exchange of Z-boson leads to weak interaction between electron and neutrino [22], exchange of pi-meson (π 0 ) or gluons leads to strong interaction between proton and neutron [23] or more strongly among quarks [24], etc. That, atoms of integral spin obey Bose-Einstein statistics, was confirmedly verified much later in 1995 [25], and that, atoms of half integral spin obey Fermi-Dirac statistics, was verified even later in 1999 [26]. However, historical development of Fermi-Dirac statistics as well as Pauli exclusion principle was mainly associated with that of atomic spectral lines. Let us now discuss on some previous history in this regard. Existence of spectral lines was known to us since the beginning of the nineteenth century when Wollaston and Fraunhofer first observed the dark absorption lines in the spectrum of the Sun [27]. It took almost eighty years, * Electronic address: sbsp [at] uohyd.ac.in