# Spin Classification

One essential parameter for classification of particles is their "spin" or intrinsic angular momentum. Half-integer spin fermions are constrained by the Pauli exclusion principle whereas integer spin bosons are not. The electron is a fermion with electron spin 1/2.

The spin classification of particles determines the nature of the energy distribution in a collection of the particles. Particles of integer spin obey Bose-Einstein statistics, whereas those of half-integer spin behave according to Fermi-Dirac statistics.

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# Fermions

Fermions are particles which have half-integer spin and therefore are constrained by the Pauli exclusion principle. Particles with integer spin are called bosons. Fermions incude electrons, protons, neutrons. The wavefunction which describes a collection of fermions must be antisymmetric with respect to the exchange of identical particles, while the wavefunction for a collection of bosons is symmetric.

The fact that electrons are fermions is foundational to the buildup of the periodic table of the elements since there can be only one electron for each state in an atom (only one electron for each possible set of quantum numbers). The fermion nature of electrons also governs the behavior of electrons in a metal where at low temperatures all the low energy states are filled up to a level called the Fermi energy. This filling of states is described by Fermi-Dirac statistics.

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# Bosons

Bosons are particles which have integer spin and which therefore are not constrained by the Pauli exclusion principle like the half-integer spin fermions. The energy distribution of bosons is described by Bose-Einstein statistics. The wavefunction which describes a collection of bosons must be symmetric with respect to the exchange of identical particles, while the wavefunction for a collection of fermions is antisymmetric.

At low temperatures, bosons can behave very differently than fermions because an unlimited number of them can collect into the same energy state. The collection into a single state is called condensation, or Bose-Einstein condensation. It is responsible for the phenomenon of superfluidity in liquid helium. Coupled particles can also act effectively as bosons. In the BCS Theory of superconductivity, coupled pairs of electrons act like bosons and condense into a state which demonstrates zero electrical resistance.

Bosons include photons and the characterization of photons as particles with frequency-dependent energy given by the Planck relationship allowed Planck to apply Bose-Einstein statistics to explain the thermal radiation from a hot cavity.

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# Bose-Einstein Condensation

In 1924 Einstein pointed out that bosons could "condense" in unlimited numbers into a single ground state since they are governed by Bose-Einstein statistics and not constrained by the Pauli exclusion principle. Little notice was taken of this curious possibility until the anomalous behavior of liquid helium at low temperatures was studied carefully.

When helium is cooled to a critical temperature of 2.17 K, a remarkable discontinuity in heat capacity occurs, the liquid density drops, and a fraction of the liquid becomes a zero viscosity "superfluid". Superfluidity arises from the fraction of helium atoms which has condensed to the lowest possible energy.

A condensation effect is also credited with producing superconductivity. In the BCS Theory, pairs of electrons are coupled by lattice interactions, and the pairs (called Cooper pairs) act like bosons and can condense into a state of zero electrical resistance.

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