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Section 5: Composite Bosons and Fermions

Whether atoms and molecules can condense into the same quantum state, as Einstein predicted in 1924, depends on whether they are bosons or fermions. We therefore have to extend Pauli's definition of bosons and fermions to composite particles before we can even talk about things as complex as atoms being either of these. As a start, let's consider a composite object made from two spin-1/2 fundamental particles. The fundamental particles are, of course, fermions; but when we combine them, the total spin is either or . This correctly suggests that two spin-1/2 fermions may well combine to form a total spin of 1 or 0. In either case, the integer spin implies that the composite object is a boson.

Why the caveat "may well"? As in many parts of physics, the answer to this question depends on what energies we are considering and what specific processes might occur. We can regard a star as a structureless point particle well described by its mass alone if we consider the collective rotation of billions of such point masses in a rotating galaxy. However, if two stars collide, we need to consider the details of the internal structure of both. In the laboratory, we can think of the proton (which is a composite spin-1/2 fermion) as a single, massive, but very small and inert particle with a positive unit charge, a fixed mass, and a spin of 1/2 at low energies. But the picture changes at the high energies created in the particle accelerators we first met in Unit 1. When protons moving in opposite directions at almost the speed of light collide, it is essential to consider their internal structures and the new particles that may be created by the conversion of enormous energies into mass. Similarly two electrons might act as a single boson if the relevant energies are low enough to allow them to do so, and if they additionally have some way to actually form that composite boson; this requires the presence of "other particles," such as an atomic nucleus, to hold them together. This also usually implies low temperatures. We will discuss electrons paired together as bosons below and in Unit 8. A hydrogen atom, meanwhile, is a perfectly good boson so long as the energies are low compared to those needed to disassemble the atom. This also suggests low temperatures. But, in fact, Einstein's conditions for Bose condensation require extraordinarily low temperatures.

Protons in LHC collisions (left) and electrons in a superconductor (right) are examples of composite fermions and bosons.

Figure 18: Protons in LHC collisions (left) and electrons in a superconductor (right) are examples of composite fermions and bosons.

Source: © CERN. More info

Helium as boson or fermion

When is helium a boson? This is a more complex issue, as the helium nucleus comes in two isotopes. Both have Z = 2, and thus two protons and two electrons. However, now we need to add the neutrons. The most abundant and stable form of helium has a nucleus with two protons and two neutrons. All four of these nucleons are spin-1/2 fermions, and the two protons pair up, as do the two neutrons. Thus, pairing is a key concept in the structure of atomic nuclei, as well as in the organization of electrons in the atom's outer reaches. So, in helium with mass number A = 4, the net nuclear spin is 0. Thus, the 4He nucleus is a boson. Add the two paired electrons and the total atomic spin remains 0. So, both the nucleus and an atom of helium are bosons.

The two isotopes of helium: a fermion and a boson.

Figure 19: The two isotopes of helium: a fermion and a boson.

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The chemical symbol He tells us that Z = 2. But we can add the additional information that by writing the symbol for bosonic helium as 4He, where the leading superscript 4 designates the atomic number. This extra notation is crucial, as helium has a second isotope. Because different isotopes of the same element differ only in the number of neutrons in the nucleus, they have different values of A. The second stable isotope of helium is 3He, with only one neutron. The isotope has two paired protons and two paired electrons but one necessarily unpaired neutron. 3He thus has spin-1/2 overall and is a composite fermion. Not surprisingly, 3He and 4He have very different properties at low temperatures, as we will discuss in the context of superfluids below.

Atomic bosons and fermions

More generally, as the number of spin-1/2 protons equals the number of spin-1/2 electrons in a neutral atom, any atom's identity as a composite fermion or boson depends entirely on whether it has an odd or even number of neutrons, giving a Fermi atom and a Bose atom, respectively. We obtain that number by subtracting the atomic number, Z, from the mass number, A. Note that A is not entirely under our control. Experimentalists must work with the atomic isotopes that nature provides, or they must create novel ones, which are typically unstable; which isotopes these are depends on the rules for understanding the stability of nuclei. A difficult task as neutrons and protons interact in very complex ways, which befits their composite nature, and are not well represented by simple ideas like the electrical Coulomb's Law attraction of an electron for a proton.

As this chart shows, not every imaginable nucleus is stable.

Figure 20: As this chart shows, not every imaginable nucleus is stable.

Source: © Courtesy of Brookhaven National Laboratory. More info

We are now in a position to understand why one would need bosonic atoms to make a BEC: Fermions cannot occupy the same macroscopic quantum state, but bosons can. And, we know how to recognize which atoms will actually be bosons. The first atomic gaseous Bose-Einstein condensates were made in Boulder, Colorado, using rubidium (Rb) atoms; in Cambridge, Massachusetts, using atomic sodium (Na); and in Houston, Texas, using atomic lithium (Li). These three elements are members of the alkali metal family, and share the property of having a single (and thus unpaired) electron outside a fully closed shell of electrons. These unpaired outer shell electrons with their magnetic moments allow them to be caught in a magnetic trap, as seen in Unit 5. If we want gases of these atoms to Bose condense, we must think counterintuitively: We need isotopes with odd values of A, so that the total number of spin-1/2 fermions—protons, electrons, and neutrons—is even. These alkali metals all have an odd number of protons, and a matching odd number of electrons; therefore, we need an even number of neutrons. This all leads to A being an odd number and the alkali atom to being a boson. Thus the isotopes 7Li, 23Na, and 87Rb are appropriate candidates, as they are composite bosons.