Angular Momentum and Magnetic Moments

• In our discussion of hydrogen-like atoms the only interaction we have considered is the Coulomb force between the nucleus and the electron. Each object is treated as a classical point charge.

• There are more subtle aspects to the electromagnetic interaction between the nucleus and the electrons in atoms. Some have to do with the presence of more than one electron. These will be treated later. Here we will discuss the magnetic moments associated with a single electron.

• Thinking classically, a charged particle in a circular orbit is like a current loop, where the current is the charge times the frequency of the orbit: I = qf. In classical physics such a loop creates a magnetic dipole moment whose magnitude is the current times the area of the loop: m = IA = pr²I, where r is the radius of the orbit. Thus, using w = 2pf,

• The angular momentum of the orbit is L = mrv = mr²w, so we have

• The ratio m/L is the gyromagnetic ratio, which we see in this case is q/2m. Any system of charges executing orbital motion will create magnetic dipole moments. The value of the gyromagnetic ratio will vary according to the details.

• Since the electron has negative charge, we expect the direction of m to be opposite to that of L. For an electron in a quantum state the angular momentum is quantized, of course. We write for the magnitude of m:

while the z-component of L is



(Here we use m_l for the quantum number, since m represents the mass.)
 

• The quantity

is called the Bohr magneton, and provides the unit of magnetic moment for the electron. In SI units it is about 10^-23.

Effects of an External B-Field

• The properties of the electron's magnetic moment are easiest to observe by means of an external magnetic field. The classical interaction between such a field and a pointlike magnetic moment is given by the alignment potential energy

• If we take the direction of the B-field to define the z axis, then this becomes

• If the B-field is uniform over the region occupied by the magnetic moment then there is no net force, but there is a torque, the familiar t = m ¥ B.

• A different effect arises if the magnetic moment is in a non-uniform B-field. Then there is a net force given by

• We will take B to be in the z direction, and varying with z. Then the force is also in the z direction:

• For positive (negative) values of m_l this is in the positive (negative) z direction. Thus a single-electron atom whose electron state has a positive (negative) value of m_l will be deflected in the positive (negative) z direction by the field.

• If a beam of such atoms is passed through the field, we expect it to be split into 2l + 1 parts, corresponding to the available values of m_l. This might provide a way to measure the value of l for the state of the atom.

• As we will see, most atoms in their ground state have l = 0, so one would expect this splitting of the beam not to occur in most cases.

The "Normal" Zeeman Effect

• As we have seen, the magnetic moment associated with the electron's "orbital" state interacts with an external uniform B-field, adding an interaction energy proportional to m_l to the energy of the electron's state.

• This splits the original energy level into 2l + 1 levels, which introduces complications into the observed emission spectrum of the atom.

• If the upper level has quantum number l₂ and the lower level l₁, one might expect the original single spectral line to split into (2l₂ + 1)(2l₁ + 1) lines. But it is not that complicated, because there are selection rules that forbid some of the transitions, ond some of the others have the same energy difference and thus give the same emitted frequency.

• The selection rules that apply here (given without proof, but which follow from conservation laws) are

• These restrictions are sufficient to ensure that the original single spectral line splits into only three when the atom is in the external B-field.

• This splitting into three lines was observed by Zeeman before 1900, and is called the "normal" Zeeman effect, because it can be given a classical explanation.

• Later it was discovered that for many atoms the splitting is into an even number of lines. This is the "anomalous" Zeeman effect.

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Electron Spin; the Stern-Gerlach Experiment

• In 1922 (before the quantum rules for angular momentum were known) Stern and Gerlach did the experiment just described, using a beam of silver atoms.

• They observed a splitting of the beam, but into two parts, one above and one below the original beam.

• If the atoms are in a state with l = 0 there should be no splitting. For any value of l there should be an odd number of split beams. Thus the result of the experiment was a puzzle.

• The solution of the puzzle is that the electron has an intrinsic magnetic moment, associated with an intrinsic spin angular momentum. This magnetic moment is responsible for the splitting of the atomic beam, even when the "orbital" angular momentum is zero.

• In place of the quantum numbers l  and m_l we introduce the spin quantum numbers s and m_s. But we assign to s the single value 1/2, so that m_s can take only the values 1/2 and -1/2. This accounts for the splitting into only two beams.

• The observable consequences of electron spin fall into two categories:

• those having to do with the magnetic moment associated with the spin, including the Stern-Gerlach experiment, fine stucture of atomic spectra, and the anomalous Zeeman effect;

• those having to do with the doubling of the number of possible states for an electron, including the structure of the periodic table.

• A "two-valuedness" of the electron state was proposed in 1925 by Pauli in formulating his Exclusion Principle. In the same year the first published suggestion that there is an intrinsic angular momentum was made by Uhlenbeck and Goudsmit.

The Electron's Intrinsic Magnetic Moment

• Later versions of the Stern-Gerlach experiment with hydrogen atoms showed that the extent of the beam splitting was anomalously large. From the discussion above, the force causing the splitting should be (for l = 0 ground state atoms)

• Expeiment showed that the force was twice this large. Thus the electron's gyromagnetic ratio seemed to be twice the classical value e/2m.

• Support for this "anomalous" magnetic moment came from atomic spectra (fine structure, anomalous Zeeman effrect). These will be discussed below.

• Much later it was found that the anomaly in the gyromagnetic ratio is not exactly a factor of two, and the deviation from that value has provided one of the best tests of our current quantum theory of radiation, called quantum electrodynamics.

Total Angular Momentum and Magnetic Moment

• An isolated atom is free from external torques, so its total angular momentum must be conserved. Neglecting any angular momentum of the nucleus, this means that L + S is conserved. This sum is usually denoted by J.

• If there are more electrons, then J is the total angular momentum of them all, and it is conserved, i.e., [H,J] = 0.

• Like the other angular momenta, J obeys the quantum rules. Its eigenvalues are

Here j can be either a non-negative integer or half-integer, and m_j runs in steps of one from -j to +j.

• For given L and S there are restrictions on J, of course. In terms of j these are:

• For a single electron (s = 1/2) we have only j = l + 1/2 or j = l - 1/2. If l = 0 then of course we have only j = 1/2.

• The total magnetic moment of an electron is the sum of its orbital and spin parts:

• Because this is not proportional to J it generally does not commute with H. This causes considerable complication of the details in atomic spectra and in the Zeeman effect.

The Spin-Orbit Interaction

• There is an interaction in the atom that depends on the relative directions of L and S. It arises from the motion of the electron's magnetic moment through the electric field of the nucleus and the other electrons.

• This is easy to understand qualitatively in terms of a classical electron orbit, using a reference frame fixed to the electron. (This is a noninertial frame, which changes the details of the argument but not its overall validity.) The electron "sees" the nucleus orbit around it; the current thus circulating produces a B-field at the electron's location.

• The orientation of this B-field is the same as that of the (apparent) angular momentum of the nuclear "orbit". This direction is also that of the electron's angular momentum in the frame of the nucleus, so it is the direction of L. We take this to define the z direction.

• The z component of the electron's intrinsic magnetic moment can take only two values: one positive and one negative The orientation energy (-m*B) is thus proportional to L·S.

• This is called the spin-orbit interaction. A detailed analysis shows that it is given by

Here U_c is the electrostatic potential energy experienced by the electron. (For a single electron atom it is just the Coulomb potential energy of interaction with the nucleus; for multi-electron atoms it includes the average effect of the other electrons as well.)
 
• This interaction gives the "doublet" structure of the spectrum of alkali elements. In Na, for example, there is a strong yellow spectral line (which give sodium light its dominant color) arising from the transition of the valence electron from a state with n = 3, l = 1 to the ground state with n = 3, l = 0. Because of U_s-o, the upper state is split into two levels, while the lower state is unsplit (because L = 0). The resulting spectrum has two closely spaced lines. (This also occurs in hydrogen, but the splitting is very small.)

• The general term for such splittings is fine structure. Note that U_s-o vanishes if c becomes infinite, so fine structure is fundamentally a relativisitic "correction".