Composite Particles and Statistics
by
Brett Ragozzine
Quantum Mechanics 612
Professor Sergio Ulloa
Monday, May 28, 2007
I. Elementary Particles
An elementary particle is any particle that is not known to have a substructure. Historically, atoms were once elementary particles; so were protons and neutrons. Today’s Standard Model elementary particles are the quarks, leptons, and gauge bosons.
By definition of elementary particles, they are indistinguishable from each other and can combine to form new, composite particles. Each elementary particle is a fermion or a boson; this is known as elementarity or “bare particle”. Fermions have half-integer spin and bosons are integer spin particles. Fermions are usually related to matter and bosons to radiation. In quantum physics, however, they are difficult to separate like that.
The four bosons that are not known to be composite particles are called gauge bosons. Three of these four have been discovered experimentally and they are the photons, W and Z bosons, and gluons. The W and Z bosons mediate the weak interaction while gluons mediate the strong nuclear force.
The fourth elementary boson in the Standard Model, the Higgs boson, is yet undiscovered. It is highly anticipated to be discovered within days or months after the Large Hadron Collider (LHC) begins operation later in 2007 or in 2008. The Higgs boson is hoped to fill in a gap in the Standard Model and shed some light on how elementary particles gain mass.
Figure 1. The Standard Model particles.
II. Composite Particles
A composite particle is made of multiple elementary particles. Composite particles are fermionic if they are made up of an odd number of elementary particles; they are bosons if they have an even number.
Nucleons, protons and neutrons, are fermions because they are made up of three quarks. Helium atoms and alpha particles are also bosonic, but these composite bosons cannot inhabit the same space in the way elementary bosons can. At such a small length scale, comparable to the size of these particles, the nucleon constituents naturally see each other as fermions and cannot violate the Pauli exclusion principle. For this reason, helium and alpha particles are not truly bosons, but have bosonic behavior such as in the superfluid liquid helium. 16O can be thought of as being made up of four alpha particles, but these composite particles also see each other as fermions and thus cannot overlap.
Mesons are particles made up of two elementary particles – a particle with an antiparticle. If a meson is made of a particle-antiparticle pair then it is its own antiparticle. Mesons are made of two elementary particles and are thus bosons.
It is possible to think up forbidden particles. Identical boson composites are forbidden, such as the π-π particle, because no center-of-mass states of definite statistics can be constructed. This also means that there can be no elementary massive bosons in nature; it also means that there are only elementary fermions of half integer spin in nature. Photons are massless bosons. Since photons do not possess the position observable they cannot form composites.
III. Statistics
Two sets of statistics explain behavior of fermions and bosons. Fermions (named after Enrico Fermi, an Italian physicist) follow Fermi-Dirac (FD) statistics.
whereis the number of particles in state i, is the energy of state i,
is the degeneracy of state i (the number of states with energy ), is the chemical potential (sometimes the Fermi energy is used instead, as a low-temperature approximation), is the Boltzmann constant, and is absolute temperature. In the case where μ is the Fermi energy , and , the function is called the Fermi function:
Bosons (named after the Indian physicist Satyendra Nath Bose) follow Bose-Einstein (BE) statistics.
with εi > μ. This reduces to Maxwell-Boltzmann statistics for energies ( εi-μ ) >> kT.
IV. A Look at Lithium
Interestingly, isotopes of an element can form both fermions and bosons, which behave differently from each other. 6Li acts as a fermion since it is formed from nine fermions – three each of protons (composite fermions), neutrons (also composite fermions), and electrons. 7Li acts as a boson because it has ten fermion constituents – one more neutron than 6Li.
It has been shown by Truscott et al. that a gas mixture of 6Li and 7Li in a magnetic trap and lowered to a temperature around 200 nK shows a difference in behavior between the two isotopes.
It can be shown that above a certain critical temperature, bosons fill the finite number of excited states available to it. Below this critical temperature, bosons begin drop into the ground state level, where there is infinite room for them to do so. When the gas temperature reaches T = 0 Kelvin, all of the particles would theoretically be in the ground state.
Figure 2. Catching two gases with one trap. In Truscott et al.'s experiments, 7Li atoms (filled red circles) and 6Li atoms (filled green circles) are simultaneously confined in a magnetic trap (A). Near temperatures of absolute zero, the bosonic 7Li atoms occupy the lowest energy levels. In contrast, only one fermionic 6Li atom can occupy a given quantum state. Fermions must therefore avoid one another, creating an effective "Fermi pressure" that causes the Fermi gas to occupy a much larger volume in the magnetic trap compared with the Bose gas as shown in (B).
V. Parastatistics
Parastatistics is an alternative to FD, BE, and MB statistics. Other alternatives include anyonic statistics and braid statistics, which involve lower spacetime dimensions. Wilczek describes a graphite sheet that behaves very much like a 2D plane, even though a volume of graphite is 3D. This is true because the electrons in graphite can easily move in the plane, but it’s difficult for them to jump between planes, especially at low temperatures. New statistics are needed for new quasiparticles that live in this 2D world.
Fermions and bosons behave in a certain way under rotation and interchange. The phase of these particles under such operations is eiθ. This phase angle θ is what determines if a particle is a fermion or boson; θ=π for fermions and θ=0 for bosons. If the phase angle takes on “any” other value, the particle is called an, “any-on” or anyon. Wilczek coined this term in 1982.
Braid statistics calls these types of particles plektons (which are neither fermions nor bosons). Braid statistics gets it name from the intuitive graphical representation of particle interchange.
Figure 3. The figures above are called geometric braids (braid theory) and represent the exchange of position of two particles. The notation σi3 signifies three exchanges (or looped particles of third order). Both figures show allowed exchanges of particles in different number of positions from each other, but are still indistinguishable.
QCD can be reformulated using parastatistics with quarks as parafermions of order 3 and gluons as parabosons of order 8. But this is different than how we think of quarks obeying anticommutation relations and gluon commutation relations.
VI. Muons
The muon is an elementary particle in the Standard Model. Parastatistics can be used to show that the muon is an elementary particle to high certainty. A parastatistics parameter s is used and its difference from unity is measured. If s = 1 then muon is an elementary particle, a fermion, and not a parafermion. It has been shown through an experiment with muonic helium that the energy difference of 1.574 eV gives a certainty in s of
0 < | 1 – s | < 5 x 10-5
thus it is thought that muon is, indeed, an elementary particle.
References
“Parastatistics as an effective description of complex particles (nucleus-nucleus collisions and muonic atoms)” (J. Phys. G: Nucl. Phys, Vol. 3, No. 8, 1977) by Fink, Muller, and Greiner.
“New Exotic Particle Points to Double Life for Gluons” (Science Magazine) by Gary Taubes.
“Standing Room Only at the Quantum Scale” (Science Magazine) by K. M. O'Hara and J. E. Thomas.
http://en.wikipedia.org/wiki/BE_statistics
http://en.wikipedia.org/wiki/Fermions
http://en.wikipedia.org/wiki/Fermi-Dirac_statistics
http://en.wikipedia.org/wiki/Parastatistics
http://www.iop.org/EJ/article/0295-5075/59/1/094/node2.html
http://adsabs.harvard.edu/abs/1998PhDT.......259M
http://www.sciencewatch.com/interviews/frank_wilczek1.htm