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 quarks are also fermions with spin 1/2. The photon is a boson with spin 1, which is a typical boson spin. Exceptions are the graviton with spin 2 and the Higgs boson with spin 0.

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.

Carroll describes fermions and bosons as follows: "Particles come in two types: the particles that make up matter, known as 'fermions', and the particles that carry forces, known as 'bosons'. The difference between the two is that fermions take up space, while bosons can pile on top of one another. You can't just take a pile of identical fermions and put them all at the same place; the laws of quantum mechanics won't allow it." The fact that two identical fermions can't occupy the same state is further described by the Pauli exclusion principle, and the fact that bosons can is further described as Bose-Einstein condensation.

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 include 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.

Another aspect of the nature of fermions is discussed by Carroll: ordinary matter including the elements of the periodic table is made up of just three types of fermions, the electron and the up and down quarks. They are responsible for the great difference in scale between the nucleus and the atom. Quantum mechanically, an electron can be considered to be a wave-packet that obtains its relatively small mass by interacting with the Higgs field. The small mass energy translates to a relatively long characteristic wavelength, so the packet is spread out in space. This gives a relatively large size to the atom as a whole. The up and down quarks which make up protons and neutrons in the nucleus have a relatively much larger mass from a stronger interaction with the Higgs field. The characteristic wavelengths of their quantum mechanical wave packets are much smaller. They make up the nucleus as an entity much smaller than the atom. This approach to the scale of the nucleus and atom should be compared to the discussion of particle confinement and the uncertainty principle.

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.

The elementary particles which carry forces in the Standard Model of Particle Physics are bosons, like the photon which carries the electromagnetic force. These bosons do not take up space or constitute solid objects since they can condense into the same location in space.

• Vector bosons with spin =1 are the gauge bosons for the Standard Model.
  • γ The photon for the electromagnetic force
  • g The eight types of gluons which participate in the strong interaction
  • Z The neutral boson in the weak interaction
  • W⁺ and W^- The charged bosons of the weak interaction
• The scalar boson H⁰ , the Higgs boson with spin = 0.
• The proposed tensor boson G, the graviton with spin = 2 for gravity.

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.

The conditions for achieving a Bose-Einstein condensate are quite extreme. The participating particles must be considered to be identical, and this is a condition that is difficult to achieve for whole atoms. The condition of indistinguishability requires that the deBroglie wavelengths of the particles overlap significantly. This requires extremely low temperatures so that the deBroglie wavelengths will be long, but also requires a fairly high particle density to narrow the gap between the particles.

Since the 1990s there has been a surge of research into Bose-Einstein condensation since it was discovered that Bose-Einstein condensates could be formed with ultra-cold atoms. The use of laser cooling and the trapping of ultra-cold atoms with magnetic traps has produced temperatures in the nanokelvin range. Cornell and Wieman along with Ketterle of MIT received the 2001 Nobel Prize in Physics "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates". Cornell and Wieman led an active group at the University of Colorado, Boulder which has produced Bose-Einstein condensates with rubidium atoms. Other groups at MIT, Harvard and Rice have been very active in this rapidly advancing field.