Zeeman effect

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The Zeeman effect (IPA [zeɪmɑn]) is the splitting of a spectral line into several components in the presence of a 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 electron spin resonance spectroscopy, magnetic resonance imaging (MRI) and Mössbauer Spectroscopy.

1 Introduction
2 Theoretical presentation
3 Weak field (anomalous Zeeman effect)
- 3.1 Example: Lyman alpha transition in hydrogen
4 Strong Field (Paschen-Back effect)
5 See also
6 References
- 6.1 Historical
- 6.2 Modern

[edit] 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 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 — see 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.

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

[edit] Theoretical presentation

The total Hamiltonian of an atom in a magnetic field is:

<math>H = H_0 + H_1

= H_0 + \sum_\alpha \xi(\vec{r_\alpha}) \vec{L} \cdot \vec{S} -\sum_\alpha \vec{\mu_{\alpha}} \cdot \vec{B}</math>

where <math>H_0</math> is the unperturbed Hamiltonian of the atom, and the sums over α are sums over the electrons in the atom. The term

<math>\xi(\vec{r_\alpha}) \vec{L} \cdot \vec{S}</math>

is the spin-orbit interaction, or LS-coupling, for each electron (indexed by α) in the atom. If there is only one electron, the sum contains just a single term. The magnetic potential energy

<math>V_M = -\vec{\mu_{\alpha}} \cdot \vec{B} = \frac{\mu_{B}}{\hbar}(g_L \vec{L} + g_S \vec{S}) \cdot \vec{B}</math>

is the energy due to the magnetic moment μ of the α-th electron. It can be written as a sum of the contributions of the orbital angular momentum <math>\vec L</math> and the spin angular momentum <math>\vec S</math>, with each multiplied by the appropriate gyroscopic or Landé g-factor, g_L or g_S. By projecting the vector quantities onto the z-axis, the Hamiltonian may be written as

<math>H = H_0 + \xi(r) \vec{L}\cdot \vec{S} + \mu_B (g_L L_z+ g_s S_z) B_z

\approx H_{at} + \frac{\mu_B}{\hbar}(J_z + S_z) B_z</math>

where the approximation results from taking the g-factors to be <math>g_L=1</math> and <math>g_S \approx 2</math>. The summation over the electrons was omitted for readability. Here, <math>J_z=L_z+S_z</math> is the total angular momentum, and the spin-orbit (LS) coupling term has been combined with <math>H_0</math> and written as <math>H_{at}</math>.

The size of the interaction term H ' is not always small, and can induce large effects on the system. In the Paschen-Back effect, described below, H ' cannot be treated as a perturbation, as its magnitude is comparable to or larger than the unperturbed system <math>H_{at}</math>. The H ' term does not commute with <math>H_{at}</math>. In particular, <math>S_z</math> doesn't commute with the spin-orbit interaction in <math>H_{at}</math>.

[edit] Weak field (anomalous Zeeman effect)

If the spin-orbit interaction dominates over the effect of the external magnetic field, <math>\vec L</math> and <math>\vec S</math> are not separately conserved; instead only the total angular momentum vector <math>\vec J = \vec L + \vec S</math> is conserved. We can write the "averaged" magnetic energy for a single electron spin as

<math>\langle V_M \rangle = \frac{\mu_B}{\hbar}(\vec J + \vec S_{avg}) \cdot \vec B</math>

The spin and orbital angular momentum vectors can be thought of as precessing rapidly about the (fixed) total angular momentum vector <math>\vec J</math>. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of <math>\vec J</math>:

Using <math>\vec L = \vec J - \vec S</math> and squaring both sides, we get

for spin-1/2. Combining everything and taking <math>J_z = \hbar m_j</math>, we obtain the magnetic potential energy of the atom in the applied external magnetic field,

<math>V_M = \mu_B B m_j \left[ 1 + \frac{j(j+1) - l(l+1) + 3/4}{2j(j+1)} \right]</math>,

where the quantity in square brackets is the Lande g-factor g_J of the atom. Note that g_J depends on the quantum numbers j and l.

[edit] Example: Lyman alpha transition in hydrogen

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

<math>2P_{1/2} \to 1S_{1/2}</math> and <math>2P_{3/2} \to 1S_{1/2}</math>.

In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S_1/2 and 2P_1/2 states into 2 levels each (<math>m_j = 1/2, -1/2</math>) and the 2P_3/2 state into 4 levels (<math>m_j = 3/2, 1/2, -1/2, -3/2</math>). The Lande g-factors for the three levels are:

<math>g_J = 2</math> for <math>1S_{1/2}</math> (j=1/2, l=0)

<math>g_J = 2/3</math> for <math>2P_{1/2}</math> (j=1/2, l=1)

<math>g_J = 4/3</math> for <math>2P_{3/2}</math> (j=3/2, l=1)

Note in particular that the size of the energy splitting is different for the different orbitals, because the g_J values are different.

[edit] Strong Field (Paschen-Back effect)

To simplify the solution, it is useful to assume that <math>[H_{at},S_{z}]=0</math>, so that <math>L_{z}</math> and <math>S_{z}</math> have a set of common eigenfunctions with respect to <math>H_{at}</math>. This allows the expectation values of <math>L_{z}</math> and <math>S_{z}</math> to be easily evaluated on a general state <math>|A\rangle </math>:

<math> \left( H_{at} + \frac{B_{z}\mu_B}{\hbar}(L_{z}+2S_z) \right) |A \rangle = (E_{at} + B_z\mu_B (m_l + 2m_s)|A\rangle </math>

The above may be read as implying that the LS-coupling is completely broken by the external field. The system re-arranges substantially according to the <math>B_z</math> field. The <math>m_l</math> and <math>m_s</math> are still "good" quantum numbers. This implies that the selection rules obtained from <math>\Delta S = 0, \Delta L = \pm 1</math> are still very likely for the system. In particular, apart from the line splittings one might normally expect, only three spectral lines will be visible, corresponding to the <math>\Delta m = \pm 1</math> transition rule. The splitting depends upon the l level being considered. The spectral lines depend on the transition frequencies, that is, on the difference of energy.

[edit] See also

Stark effect

[edit] References

[edit] Historical

Condon, E. U., G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 521-09209-4. (Chapter 16 provides a comprehensive treatment, as of 1935.)
Zeeman, P. (1897). "(Title unknown)". Phil.Mag. 43: 226.
Zeeman, P. (11 February 1897). "The Effect of Magnetisation on the Nature of Light Emitted by a Substance". Nature 55: 347.

[edit] 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.ca:Efecte Zeeman