Definitions

# Zeeman effect

The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field. The Zeeman effect is very important in applications such as nuclear magnetic resonance spectroscopy, electron spin resonance spectroscopy, magnetic resonance imaging (MRI) and Mössbauer spectroscopy.

When the spectral lines are absorption lines, the effect is called Inverse Zeeman effect.

The Zeeman effect is named after the Dutch physicist Pieter Zeeman.

## Introduction

In most atoms, there exist several electronic configurations that have the same energy, so that transitions between different pairs of configurations correspond to a single spectral line.

The presence of a magnetic field breaks the degeneracy, since it interacts in a different way with electrons with different quantum numbers, slightly modifying their energies. The result is that, where there were several configurations with the same energy, now there are different energies, which give rise to several very close spectral lines.

Without a magnetic field, configurations a, b and c have the same energy, as do d, e and f. The presence of a magnetic field splits the energy levels. A line produced by a transition from a, b or c to d, e or f now will be several lines between different combinations of a, b, c and d, e, f. Not all transitions will be possible, as regulated by the transition rules.

Since the distance between the Zeeman sub-levels is proportional to the magnetic field, this effect is used by astronomers to measure the magnetic field of the Sun and other stars.

There is also an anomalous Zeeman effect that appears on transitions where the net spin of the electrons is not 0, the number of Zeeman sub-levels being even instead of odd if there's an uneven number of electrons involved. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it at the time that Zeeman observed the effect.

If the magnetic field strength is too high, the effect is no longer linear; at even higher field strength, electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen-Back effect.

## Theoretical presentation

The total Hamiltonian of an atom in a magnetic field is

$H = H_0 + H_M$,

where $H_0$ is the unperturbed Hamiltonian of the atom, and $H_M$ is perturbation due to the magnetic field:

$V_M = -vec\left\{mu\right\} cdot vec\left\{B\right\}$,

where $vec\left\{mu\right\}$ is the magnetic moment of the atom. The magnetic moment consists of the electronic and nuclear parts, however, the latter is many orders of magnitude smaller and will be neglected further on. Therefore,

$vec\left\{mu\right\} = -mu_B g vec\left\{J\right\}$,

where $mu_B$ is the Bohr magneton, $vec\left\{J\right\}$ is the total electronic angular momentum, and $g$ is the g-factor. The operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum $vec l$ and the spin angular momentum $vec s$, with each multiplied by the appropriate gyromagnetic ratio:

$vec\left\{mu\right\} = -mu_B \left(g_l vec\left\{l\right\} + g_s vec\left\{s\right\}\right)$,

where $g_l = 1$ or $g_s approx 2.0023192$ (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the relativistic effects). In the case of the LS coupling, one can sum over all electrons in the atom:

$g vec\left\{J\right\} = leftlanglesum_i \left(g_l vec\left\{l_i\right\} + g_s vec\left\{s_i\right\}\right)rightrangle = leftlanglevec\left\{L\right\} + g_s vec\left\{S\right\}rightrangle$,

where $vec\left\{L\right\}$ and $vec\left\{S\right\}$ are the total orbital momentum and spin of the atom, and averaging is done over a state with a given value of the total angular momentum.

If the interaction term $V_M$ is small (less than the fine structure), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen-Back effect, described below, $V_M$ exceeds the LS coupling significantly (but is still small compared to $H_\left\{0\right\}$). In ultrastrong magnetic fields, the magnetic-field interaction may exceed $H_0$, in which case the atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are, of course, intermediate cases which are more complex than these limit cases.

## Weak field (Zeeman effect)

If the spin-orbit interaction dominates over the effect of the external magnetic field, $vec L$ and $vec S$ are not separately conserved, only the total angular momentum $vec J = vec L + vec S$ is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector $vec J$. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of $vec J$:

$vec S_\left\{avg\right\} = frac\left\{\left(vec S cdot vec J\right)\right\}\left\{J^2\right\} vec J$.

and for the (time-)"averaged" orbital vector:

$vec L_\left\{avg\right\} = frac\left\{\left(vec L cdot vec J\right)\right\}\left\{J^2\right\} vec J$.

Thus,

$langle V_M rangle = frac\left\{mu_B\right\}\left\{hbar\right\} vec J\left(g_Lfrac\left\{vec L cdot vec J\right\}\left\{J^2\right\} + g_Sfrac\left\{vec S cdot vec J\right\}\left\{J^2\right\}\right) cdot vec B$.

Using $vec L = vec J - vec S$ and squaring both sides, we get

$vec S cdot vec J = frac\left\{1\right\}\left\{2\right\}\left(J^2 + S^2 - L^2\right) = frac\left\{hbar^2\right\}\left\{2\right\}\left[j\left(j+1\right) - l\left(l+1\right) + s\left(s+1\right)\right]$,
and: using $vec S = vec J - vec L$ and squaring both sides, we get

$vec L cdot vec J = frac\left\{1\right\}\left\{2\right\}\left(J^2 - S^2 + L^2\right) = frac\left\{hbar^2\right\}\left\{2\right\}\left[j\left(j+1\right) + l\left(l+1\right) - s\left(s+1\right)\right]$

Combining everything and taking $J_z = hbar m_j$, we obtain the magnetic potential energy of the atom in the applied external magnetic field,

$V_M = mu_B B m_j left\left[g_Lfrac\left\{j\left(j+1\right) + l\left(l+1\right) - s\left(s+1\right)\right\}\left\{2j\left(j+1\right)\right\} + g_Sfrac\left\{j\left(j+1\right) - l\left(l+1\right) + s\left(s+1\right)\right\}\left\{2j\left(j+1\right)\right\} right\right]$,

where the quantity in square brackets is the Lande g-factor gJ of the atom ($g_L = 1$ and $g_S approx 2$) and $m_j$ is the z-component of the total angular momentum. For a single electron above filled shells $s = 1/2$.

### Example: Lyman alpha transition in hydrogen

The Lyman alpha transition in hydrogen in the presence of the spin-orbit interaction involves the transitions

$2P_\left\{1/2\right\} to 1S_\left\{1/2\right\}$ and $2P_\left\{3/2\right\} to 1S_\left\{1/2\right\}$.

In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S1/2 and 2P1/2 states into 2 levels each ($m_j = 1/2, -1/2$) and the 2P3/2 state into 4 levels ($m_j = 3/2, 1/2, -1/2, -3/2$). The Lande g-factors for the three levels are:

$g_J = 2$ for $1S_\left\{1/2\right\}$ (j=1/2, l=0)

$g_J = 2/3$ for $2P_\left\{1/2\right\}$ (j=1/2, l=1)

$g_J = 4/3$ for $2P_\left\{3/2\right\}$ (j=3/2, l=1)

Note in particular that the size of the energy splitting is different for the different orbitals, because the gJ values are different. On the left, fine structure splitting is depicted. This splitting occurs even in the absence of a magnetic field, as it is due to spin-orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.

## Strong field (Paschen-Back effect)

The Paschen-Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. This occurs when an external magnetic field is sufficiently large to disrupt the coupling between orbital and spin angular momenta. This effect is the strong field generalization of the Zeeman effect. The effect was named for the German physicists Friedrich Paschen and Ernst E. A. Back.

When the magnetic-field perturbation significantly exceeds the spin-orbit interaction, one can safely assume $\left[H_\left\{0\right\}, S\right] = 0$. This allows the expectation values of $L_\left\{z\right\}$ and $S_\left\{z\right\}$ to be easily evaluated for a state $|Arangle$:

$langle A| left\left(H_\left\{0\right\} + frac\left\{B_\left\{z\right\}mu_B\right\}\left\{hbar\right\}\left(L_\left\{z\right\}+g_\left\{s\right\}S_z\right) right\right) |A rangle = E_\left\{0\right\} + B_zmu_B \left(m_l + g_\left\{s\right\}m_s\right)$.

The above may be read as implying that the LS-coupling is completely broken by the external field. The $m_l$ and $m_s$ are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., $Delta S = 0, Delta m_s = 0, Delta L = pm 1, Delta m_l = 0, pm 1$ this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the $Delta m_l = 0, pm 1$ selection rule. The splitting $Delta E = B mu_B Delta m_l$ is independent of the unperturbed energies and electronic configurations of the levels being considered.

## References

### Historical

• Condon, E. U.; G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4. (Chapter 16 provides a comprehensive treatment, as of 1935.)
• Zeeman, P. (1897). "On the influence of Magnetism on the Nature of the Light emitted by a Substance". Phil. Mag. 43 226.
• Zeeman, P. (1897). "Doubles and triplets in the spectrum produced by external magnetic forces". Phil. Mag. 44 55.
• Zeeman, P. "The Effect of Magnetisation on the Nature of Light Emitted by a Substance". Nature 55 347.

### Modern

• Forman, Paul (1970). "Alfred Landé and the anomalous Zeeman Effect, 1919-1921". Historical Studies in the Physical Sciences 2 153–261.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
• Sobelman, Igor I. (2006). Theory of Atomic Spectra. Alpha Science. ISBN 1-84265-203-6.

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