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Section 4: Spin, Bosons, and Fermions

In spite of having no experimentally resolvable size, a single electron has an essential property in addition to its fixed charge and mass: a magnetic moment. It may line up, just like a dime store magnet, north-south or south-north in relation to a magnetic field in which it finds itself. These two possible orientations correspond to energy levels in a magnetic field determined by the magnet's orientation. This orientation is quantized in the magnetic field. It turns out, experimentally, that the electron has only these two possible orientations and energy levels in a magnetic field. The electron's magnetic moment is an internal and intrinsic property. For historical reasons physicists called it spin, in what turned out to be a bad analogy to the fact that in classical physics a rotating spherical charge distribution, or a current loop as in a superconducting magnet, gives rise to a magnetic moment. The fact that only two orientations of such a spin exist in a magnetic field implies that the quantum numbers that designate spin are not integers like the quantum numbers we are used to, but come in half-integral amounts, which in this case are 1/2. This was a huge surprise when it was first discovered.

The Stern-Gerlach experiment demonstrated that spin orientation is quantized, and that "up" and "down" are the only possibilities.

Figure 14: The Stern-Gerlach experiment demonstrated that spin orientation is quantized, and that "up" and "down" are the only possibilities.

Source: © Wikimedia Commons, GNU Free Documentation License, Version 1.2. Author: Peng, 1 July 2005. More info

What does this mean? A value of 1/2 for the spin of the electron pops out of British physicist Paul Dirac's relativistic theory of the electron; but even then, there is no simple physical picture of what the spin corresponds to. Ask a physicist what "space" spin lives in, and the answer will be simple: Spins are mathematical objects in "spin space." These spins, if unpaired, form little magnets that can be used to trap and manipulate the atoms, as we have seen in Unit 5, and will see again below. But spin itself has much far-reaching implications. The idea of "spin space" extends itself into "color space," "flavor space," "strangeness space," and other abstract (but physical, in the sense that they are absolutely necessary to describe what is observed as we probe more and more deeply into the nature of matter) dimensions needed to describe the nature of fundamental particles that we encountered in Units 1 and 2.

Spin probes of unusual environments

Is spin really important? Or might we just ignore it, as the magnetic moments of the proton and electron are small and greatly overwhelmed by their purely electrical interactions? In fact, spin has both straightforward implications and more subtle ones that give rise to the exclusion principle and determine whether composite particles are bosons or fermions.

The more direct implications of the magnetic moments associated with spin include two examples in which we can use spins to probe otherwise hard-to-reach places: inside our bodies and outer space. It turns out that the neutron and proton also have spin-1/2 and associated magnetic moments. As these magnetic moments may be oriented in only two ways in a magnetic field, they are usually denoted by the ideograms and for the spin projections +1/2 (spin up) and -1/2 (spin down). In a magnetic field, the protons in states and have different energies. In a strong external magnetic field, the spectroscopy of the transitions between these two levels gives rise to Nuclear Magnetic Resonance (NMR), a.k.a. MRI in medical practice (see Figure 3). Living matter contains lots of molecules with hydrogen atoms, whose nuclei can be flipped from spin up to spin down and vice versa via interaction with very low energy electromagnetic radiation, usually in the radiowave regime. The images of these atoms and their interactions with nearby hydrogen atom nuclei provide crucial probes for medical diagnosis. They also support fundamental studies of chemical and biochemical structures and dynamics, in studies of the folding and unfolding of proteins, for example.

Another example of the direct role of the spins of the proton and electron arises in astrophysics. In a hydrogen atom, the spin of the proton and electron can be parallel or anti-parallel. And just as with real magnets, the configuration ( has lower energy than . This small energy difference is due to the hyperfine structure of the spectrum of the hydrogen atom reviewed in Unit 5. The photon absorbed in the transition or emitted in the transition has a wavelength of 21 centimeters, in the microwave region of the electromagnetic spectrum. Astronomers have used this 21 centimeter radiation to map the density of hydrogen atoms in our home galaxy, the Milky Way, and many other galaxies.

Our galaxy, imaged at many different wavelengths.

Figure 15: Our galaxy, imaged at many different wavelengths.

Source: © NASA. More info

Electrons are fermions

However, there is more: That a particle has spin 1/2 means more than that it has only two possible orientations in a magnetic field. Fundamental particles with intrinsic spin of 1/2 (or any other half-integer spin, such as 3/2 or 5/2 or more whose numerators are odd numbers) share a specific characteristic: They are all fermions; thus electrons are fermions. In contrast, fundamental particles with intrinsic spin of 0, 1, 2, or any integral number are bosons; so far, the only boson we have met in this unit is the photon.

Is this a big deal? Yes, it is. Applying a combination of relativity and quantum theory, Wolfgang Pauli showed that identical fermions or bosons in groups have very different symmetry properties. No two identical fermions can be in the same quantum state in the same physical system, while as many identical bosons as one could wish can all be in exactly the same quantum state in a single quantum system. Because electrons are fermions, we now know the correct arrangement of electrons in the ground state of lithium. As electrons have only two spin orientations, or , it is impossible to place all three electrons in the lowest quantum energy level; because at least two of the three electrons would have the same spin, the state is forbidden. Thus, the third electron in a lithium atom must occupy a higher energy level. In recognition of the importance of spin, the Lewis representation of the elements in the first column of the periodic table might well be H, Li, Na, K, and Rb, rather than his original H, Li, Na, K, and Rb. The Pauli symmetry, which leads to the exclusion principle, gives rise to the necessity of the periodic table's shell structure. It is also responsible for the importance of Lewis's two electron chemical pair bond, as illustrated in Figure 16.

Spin pairing in the molecules H2 and Li2.

Figure 16: Spin pairing in the molecules H2 and Li2.

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The Pauli rules apply not only to electrons in atoms or molecules, but also to bosons and fermions in the magnetic traps of Unit 5. These may be fundamental or composite particles, and the traps may be macroscopic and created in the laboratory, rather than electrons attracted by atomic nuclei. The same rules apply.

Conversely, the fact that bosons such as photons have integral spins means that they can all occupy the same quantum state. That gives rise to the possibility of the laser, in which as many photons as we wish can bounce back and forth between two mirrors in precisely the same standing wave quantum state. We can think of this as a macroscopic quantum state. This is certainly another example of particles in an artificial trap created in the laboratory, and an example of a macroscopic quantum state. Lasers produce light with very special and unusual properties. Other bosons will, in fact, be responsible for all the known macroscopic quantum systems that are the real subject of this and several subsequent units.

Photons are bosons

When Max Planck introduced the new physical constant h that we now call Planck's constant, he used it as a proportionality constant to fit the data known about blackbody radiation, as we saw in Unit 5. It was Albert Einstein who noted that the counting number n that Planck used to derive his temperature dependent emission profiles was actually counting the number of light quanta, or photons, at frequency , and thus that the energy of one quantum of light was . If one photon has energy , then n photons would have energy n. What Planck had unwittingly done was to quantize electromagnetic radiation into energy packets. Einstein and Planck both won Nobel prizes for this work on quantization of the radiation field.

What neither Planck nor Einstein realized at the time, but which started to become clear with the work of the Indian physicist Satyendra Bose in 1923, was that Planck and Einstein had discovered that photons were a type of particle we call "bosons," named for Bose. That is, if we think of the frequency as describing a possible mode or quantum state of the Planck radiation, then what Planck's n really stated was that any number, n = 0, 1, 2, 3..., of photons, each with its independent energy could occupy the same quantum state.

The spectrum of blackbody radiation at different temperatures.

Figure 17: The spectrum of blackbody radiation at different temperatures.

Source: © Wikimedia Commons, GNU Free Documentation license, Version 1.2. Author: 4C, 04 August 2006. More info

In the following year, Einstein also suggested in a second paper following the work of Bose, that atoms or molecules might be able to behave in a similar manner: Under certain conditions, it might be possible to create a gas of particles all in the same quantum state. Such a quantum gas of massive particles came to be known as a Bose-Einstein condensate (BEC) well before the phenomenon was observed. Physicists observed the first gaseous BEC 70 years later, after decades of failed attempts. But, in 1924, even Einstein didn't understand that this would not happen for just any atom, but only for those atoms that we now refer to as bosons. Fermionic atoms, on the other hand, would obey their own exclusion principle with respect to their occupation of the energy levels of motion in the trap itself.