Effect of Electric and Magnetic Fields On Quantum Systems

Type of physical science: Atomic physics

Field of study: Nonrelativistic quantum mechanics

A quantum system, such as an atom or a molecule, has its behavior modified by the presence of an external field in ways that are not always intuitive. The concept of the spin of the electron is a great aid in describing these phenomena.

Overview

A quantum system, such as an atom or a molecule, ordinarily resides in a quantum state defined by the values of a set of quantum numbers. If some other state exists that has lower energy than the original state of the system, there may be a transition from the initial state into the final state, with the excess energy being emitted in the form of a photon, whose frequency is proportional to the energy difference between the initial and the final states. Every time that the system makes the same transition, the same energy (and therefore frequency and wavelength) of light is emitted, leading to the observation of a spectral line. If the system is immersed in an externally applied magnetic field, it often happens that the spectral line is split into several components; this is called the Zeeman effect (named for Pieter Zeeman). If the spectral line is split because the system was immersed in an electric field, it is called the Stark effect (named for Johannes Stark).

The easiest form of the Zeeman effect to understand is the case where the spectral line splits into three components: One is at the same wavelength as the original transition; the other two are shorter and longer, respectively, by the same amount. The amount of the shift in the energy of the system is proportional to the strength of the applied magnetic field; the constant of proportionality is called the Bohr magneton (named for Niels Bohr), and it is a constant of nature that is independent of material. When this description applies to a system, one can say that it is the normal Zeeman effect. Anything else that happens to spectral lines in a reasonably weak magnetic field is called the anomalous Zeeman effect. The use of the word "anomalous" in this context does not mean "rare"; rather, it means "difficult to explain." The anomalous Zeeman effect is more common than the normal variety.

The normal Zeeman effect occurs only for chemical elements that have an even number of valence electrons. Some typical examples with two valence electrons include helium, calcium, magnesium, and mercury. Even for these elements, however, not every spectral line exhibits the normal Zeeman effect. The key to understanding the phenomenon lies in the recognition that the electron has spin in addition to the angular momentum that it has by virtue of its orbital motion.

For elements with an odd number of electrons, the spins can never combine to cancel, so the Zeeman effect includes both spin and orbital angular momentum interacting with the magnetic field, and the effect is anomalous. For an element with an even number of valence electrons, the spins may or may not cancel each other. If they do, then the orbital motion alone couples with the magnetic field to produce the normal Zeeman effect.

In quantum mechanics, one can calculate the rate of transition of the system from one state to another. Sometimes, the symmetry of the system causes the rate to vanish for certain pairs of states; then it is said that the transition is forbidden. If the rate is not zero, then the transition is allowed. For the normal Zeeman effect, the orbital angular momentum quantum number must change by one unit, and its projection along the direction of the magnetic field must stay the same or change by one unit. This statement expresses the "dipole selection rules" for the normal Zeeman effect.

For the anomalous Zeeman effect, the selection rules are more complicated. The orbital angular momentum quantum number must once again change by one unit. The total angular momentum (orbital plus spin) must change by zero or one unit. The projection of the total angular momentum onto the field direction must change by zero or one unit. Transitions in which there is a change in the spin are not exactly forbidden but are less likely than those in which the spin remains the same. In the anomalous Zeeman effect, the proportionality between the energy shift and the magnetic field is no longer expressed simply by the Bohr magneton. An additional factor called the gyromagnetic ratio must be inserted; its value can be calculated from quantum mechanics.

For a typical transition with no magnetic field, the light emitted is unpolarized.

Nevertheless, when the magnetic field is turned on, the various Zeeman components show polarization, dependent on whether the projection of angular momentum onto the field direction (magnetic quantum number) changes. If one looks perpendicular to the magnetic field, the transitions with no change in magnetic quantum number will have their light polarized parallel to the magnetic field. The other transitions will have their light polarized perpendicular both to the field and to the line of sight. If one looks parallel to the field (perhaps through a small hole in the pole piece of the magnet), no transitions will be visible for which the magnetic quantum number stays the same. The transitions that correspond to a change in the magnetic quantum number will give rise to circularly polarized light.

It sometimes happens that an atom has two levels that are close together in energy, leading to "fine structure" in the spectrum. If a magnetic field is then applied to the system, the Zeeman splitting can cause the upper levels of the lower state to try to cross the lower levels of the upper state. Quantum mechanics predicts that such a crossing cannot occur. What happens instead is that the levels begin to mix. If the magnetic field is made even stronger, one comes into a regime where a different set of quantum numbers must be used, leading to different selection rules. The result is called the Paschen-Back effect.

The Stark effect is the name for the splitting of spectral lines caused by an electric field. As in the Zeeman case, quantum mechanics can be used to perform the calculation of the energy shifts that result from the interaction with the field. These calculations are, in general, more difficult than those for the Zeeman effect; they give results for selection rules that are similar but not identical to those of the Zeeman effect.

The electric and magnetic fields considered thus far have been static, that is, constant in time. If the fields are time-varying, then electric and magnetic fields will both be present.

Quantum mechanics enables the calculation for this case as well, and the result is that a quantum system can absorb a photon from the field if the energy of the photon is equal to the energy difference between the initial state and a final state. If the final state has higher energy than the initial state, one says that resonant absorption has occurred. If the final state has lower energy, then in the final state there will be two photons with the same energy, and one says that stimulated emission has occurred.

Applications

The Zeeman effect has very direct application to astronomy, since it enables the measurement of magnetic fields in stars, including the sun. For reference, the magnetic field of the earth is about 0.5 gauss. By studying the spectrum of light from the sun and observing Zeeman splitting in known lines, one finds that the average magnetic field near the surface of the sun is 1 or 2 gauss. It is much higher than that in sunspots, where fields of several thousand gauss are common. Sunspots are essentially magnetic storms. White dwarf stars can have magnetic fields on the order of a million gauss, as measured by Zeeman splitting.

A logical extension of the fine structure of the atom and the anomalous Zeeman effect is the topic of hyperfine structure. The electron is not the only particle that has spin. Protons and neutrons also have spin and, therefore, many nuclei have spin. The magnetic properties of nuclei are weak in comparison with those of electrons because the nuclear magneton replaces the Bohr magneton as the natural unit of coupling to an external field; the nuclear magneton is smaller than the Bohr magneton by a factor of 1,836, the ratio of the proton mass to the electron mass.

When a spectral line from an atomic transition is examined with very good resolution, the hyperfine splitting is revealed as very closely spaced structure in the line. By simply counting the multiplicities of the splits, one can determine the spin of a nucleus. If an external magnetic field is applied, the lines are split even finer. The details of the calculations for this structure proceed in a way completely analogous to that used for the anomalous Zeeman effect. Because there were already multiple levels that were very close together, the Zeeman effect in hyperfine structure is limited to rather weak fields; the hyperfine Paschen-Back region sets in very soon as the magnetic field rises.

A very fertile application of electromagnetic fields acting on quantum systems is provided by magnetic resonance phenomena. For example, a nucleus that has spin will interact with an externally applied magnetic field to produce two (or perhaps more) states that are close together in energy. If the nucleus is also exposed to radio-frequency electromagnetic waves at the frequency corresponding to the energy difference between the two states, transitions will be induced and nuclei will absorb photons as they go from one state to another. This effect is called nuclear magnetic resonance. It has many uses as a diagnostic tool in medicine and in other areas.

It is one way to measure the strength of a magnetic field.

The selection rule concerning spin not changing in anomalous Zeeman transitions is more nearly valid the lighter the element under discussion. Helium is the lightest element with two different spin states (zero and one), and transitions between the two are not observed. At one time, it was seriously suggested that helium was really two different chemical elements: one with spin zero (normal Zeeman effect) called parhelium and the other with spin one (anomalous Zeeman effect) called orthohelium. Eventually, it was realized that there was really only one element with two different spin states and that transitions between the two are forbidden.

The Stark effect has an interesting application in that it can explain why the dipole selection rules are so good for atomic physics. One could conceive of more complicated selection rules that would govern transitions that are forbidden as dipole transitions but that could proceed at a slower rate. In practice, such transitions really occur for nuclear γ-decay transitions; in atomic physics, however, they are not important. The reason is that the atoms are in motion, and they often collide either with other atoms in the sample or with the walls of the container.

Whenever such collisions occur, the atom is subjected to strong localized electric fields from the atom with which it collided. The Stark effect serves to mix other quantum states with the initial state, and some of the blended states can couple with the final state by dipole rules. Therefore, the transition can go rapidly, often without emission of a photon, since the other atom can carry off energy and momentum. This effect is called collisional deexcitation.

Context

The Zeeman effect was discovered experimentally by the Dutch physicist Pieter Zeeman in 1896. Soon afterward, the normal Zeeman effect was explained by the Dutch theorist Hendrik Antoon Lorentz using a thoroughly classical model of an electric charge oscillating in a magnetic field. This model is unable to explain the anomalous Zeeman effect. Zeeman and Lorentz shared the 1902 Nobel Prize in Physics based on this work.

Niels Bohr's theory of the atom with quantized electron orbits (awarded the 1922 Nobel Prize in Physics) was able to describe the normal Zeeman effect with great success, using the planetary orbits instead of the classical harmonic oscillator model of Lorentz. The Bohr theory was also capable of a description of the Stark effect (discovered in 1913 by the German physicist Johannes Stark, who received the 1919 Nobel Prize in Physics), which had resisted attempts at understanding within a classical framework. The anomalous Zeeman effect remained unexplained.

In 1916, Albert Einstein constructed the first theory to contain both resonant absorption and stimulated emission. It was a general theory of the interaction of an oscillating electromagnetic field with a quantum system; at the time, the stimulated emission was considered a theoretical curiosity. Eventually, however, the invention of the maser and the laser were to show the genuineness and the importance of stimulated emission.

By the early 1920's, the anomalous Zeeman effect was becoming one of the major puzzles of atomic physics. The concept of spin was introduced to explain it, and that was a step in the right direction. The invention of quantum mechanics in 1925 was another big advance; it was soon realized that with this new theory, one could explain the anomalous Zeeman effect, apart from an extra factor of two needed for the gyromagnetic ratio of the electron.

In 1928, Paul Adrien Maurice Dirac published his discovery of the Dirac equation, which merges relativity with quantum mechanics. This equation has the additional properties that it gives spin to the electron, and it also gets the correct gyromagnetic ratio for the electron.

Principal terms

ANOMALOUS ZEEMAN EFFECT: Zeeman splitting into a number of components other than three because of spin fine structure; splitting of a spectral line without an external field because of electron spin

HYPERFINE STRUCTURE: splitting of a spectral line because of nuclear spin

NORMAL ZEEMAN EFFECT: splitting into three components, understandable without spin

PASCHEN-BACK EFFECT: splitting of close-spaced spectral lines in a strong magnetic field

PHOTON: a package containing the smallest amount of energy that light of that frequency can possess

QUANTUM NUMBER: a label for one dimension of a quantum state; a set of quantum numbers defines the state

SELECTION RULE: a statement of how the quantum numbers may change as a quantum state makes a transition to a different quantum state

SPECTRAL LINE: light emitted when a quantum system makes a transition from one state to another

STARK EFFECT: splitting of a spectral line in an electric field

ZEEMAN EFFECT: splitting of a spectral line in a weak magnetic field

Bibliography

Born, Max. ATOMIC PHYSICS. New York: Hafner, 1959. This work was written by a Nobel laureate, one of the leaders in the development of quantum mechanics. Uses a minimum of mathematics, relegating the advanced details to appendices.

Finkelnburg, Wolfgang. STRUCTURE OF MATTER. Berlin: Springer-Verlag, 1964. This book provides an overview of atomic physics, including the interaction of fields with atomic systems. Contains a minimum of mathematics.

Herzberg, Gerhard. ATOMIC SPECTRA AND ATOMIC STRUCTURE. New York: Dover, 1944. Shows original results on the Zeeman and Stark effects. Some mathematics, but it is highly localized. One can skip the mathematics and read the rest of the text with understanding.

Krane, Kenneth S. MODERN PHYSICS. New York: John Wiley & Sons, 1983. This is a standard text for undergraduates. The sections on electric and magnetic fields acting on quantum systems are less mathematical than some other parts of the book.

Pais, Abraham. INWARD BOUND: OF MATTER AND FORCES IN THE PHYSICAL WORLD. Oxford, England: Oxford University Press, 1986. This book is a discursive, authoritative, philosophical, and historical treatise on the origins of atomic physics. Very interesting reading.

The Measurement of Magnetic Fields

Sunspots and Stellar Structure