Zeeman effect

Dictionary: Zeeman effect

The splitting of single spectral lines of an emission spectrum into three or more polarized components when the radiation source is in a magnetic field.

[After Pieter ZEEMAN.]

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Sci-Tech Encyclopedia: Zeeman effect

A splitting of spectral lines when the light source being studied is placed in a magnetic field. Discovered by P. Zeeman in 1896, the effect furnishes information of prime importance in the analysis of spectra. Each kind of spectral term has its characteristic mode of splitting, and the types of terms are most definitely identified by this property, Furthermore, the effect allows an evaluation of the ratio of charge to mass of the electron and an evaluation of its precise magnetic moment.

The normal Zeeman effect is a splitting into two or three lines, depending on the direction of observation, as shown in the illustration. The light of these components is polarized in ways indicated in the illustration. The normal effect is observed for all lines belonging to singlet systems, those for which the spin quantum number S = 0. The change of frequency of the shifted components can be evaluated on classical electromagnetic principles.

Triplet observed in normal Zeeman effect. ν₀ = unshifted frequency; Δν_n = frequency shift.

The anomalous Zeeman effect is a more complicated type of line splitting, so named because it did not agree with the predictions of classical theory. It occurs for any spectral line arising from a combination of terms of multiplicity greater than one. Since multiplicity in spectral lines is caused by the presence of a resultant spin vector S of the electrons, the anomalous effect must be attributed to a nonclassical magnetic behavior of the electron spin.

The quadratic Zeeman effect, which depends on the square of the field strength, is of two kinds. The first results from second-order terms, and the second from the diamagnetic reaction of the electron when revolving in large orbits.

The inverse Zeeman effect is the Zeeman effect of absorption lines. It is closely related to the Faraday effect, the rotation of plane-polarized light by matter situated in a magnetic field. See also Faraday effect.

The Zeeman effect in molecules is, in general, so small as to be unobservable, even for molecules which have a permanent magnetic moment. An exception occurs for some light molecules where the magnetic moment is coupled so lightly to the frame of the molecule that it can orient itself freely in the magnetic field just as for atoms.

A clear Zeeman effect also can be observed in many crystals with sharp spectrum lines in absorption or fluorescence. Such crystals are found particularly among rare-earth salts.

The magnetic moment of the nucleus causes a Zeeman splitting in atomic spectra which is of an order of magnitude a thousand times smaller than the ordinary Zeeman effect. This Zeeman effect of the hyperfine structure usually is modified by a nuclear Paschen-Back effect. See also Paschen-Back effect.

Britannica Concise Encyclopedia: Zeeman effect

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Splitting of a spectral line (see spectrum) into two or more lines of different frequencies. The effect occurs when the light source is placed in a magnetic field. It has helped identify the energy levels in atoms; it also provides a means of studying atomic nuclei and electron paramagnetic resonance (see magnetic resonance) and is used in measuring the magnetic field of the Sun and other stars. It was discovered in 1896 by Pieter Zeeman (1865 – 1943); he shared the second Nobel Prize for Physics (1902) with Hendrik Antoon Lorentz, who had hypothesized that a magnetic field would affect the frequency of the light emitted.

For more information on Zeeman effect, visit Britannica.com.

Columbia Encyclopedia: Zeeman effect

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Zeeman effect, splitting of a single spectral line (see spectrum) into a group of closely spaced lines when the substance producing the single line is subjected to a uniform magnetic field. The effect was discovered in 1896 by the Dutch physicist Pieter Zeeman. In the so-called normal Zeeman effect, the spectral line corresponding to the original frequency of the light (in the absence of the magnetic field) appears with two other lines arranged symmetrically on either side of the original line. In the anomalous Zeeman effect (which is actually more common than the normal effect), several lines appear, forming a complex pattern. The normal Zeeman effect was successfully explained by H. A. Lorentz using the laws of classical physics (Zeeman and Lorentz shared the 1902 Nobel Prize in Physics). The anomalous Zeeman effect could not be explained using classical physics; the development of the quantum theory and the discovery of the electron's intrinsic spin led to a satisfactory explanation. According to the quantum theory all spectral lines arise from transitions of electrons between different allowed energy levels within the atom, the frequency of the spectral line being proportional to the energy difference between the initial and final levels. Because of its intrinsic spin, the electron has a magnetic field associated with it. When an external magnetic field is applied, the electron's magnetic field may assume only certain alignments. Slight differences in energy are associated with these different orientations, so that what was once a single energy level becomes three or more. Practical applications based on the Zeeman effect include spectral analysis and measurement of magnetic field strength. Since the separation of the components of the spectral line is proportional to the field strength, the Zeeman effect is particularly useful where the magnetic field cannot be measured by more direct methods.

Wikipedia: Zeeman effect

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The Zeeman effect (pronounced /ˈzeɪmən/) 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. It may also be utilized to improve accuracy in Atomic absorption 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.

1 Introduction
2 Theoretical presentation
3 Weak field (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

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. Therefore, a line produced by a transition from a, b or c to d, e or f will now be split into several components between different combinations of a, b, c and d, e, f. However, not all transitions will be possible (in the dipole approximation), as governed by the selection 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 + V_M,\$

where $H 0$ is the unperturbed Hamiltonian of the atom, and $V M$ is perturbation due to the magnetic field:

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

where $\vec{\mu}$ 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,

$\vec{\mu} = -\mu_B g \vec{J}/\hbar,$

where $μ B$ is the Bohr magneton, $\vec{J}$ 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:

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

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:

$g \vec{J} = \left\langle\sum_i (g_l \vec{l_i} + g_s \vec{s_i})\right\rangle = \left\langle (g_l\vec{L} + g_s \vec{S})\right\rangle,$

where $\vec{L}$ and $\vec{S}$ 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 $H 0$ ). 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, $\scriptstyle \vec L$ and $\scriptstyle \vec S$ are not separately conserved, only the total angular momentum $\scriptstyle \vec J = \vec L + \vec S$ is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector $\scriptstyle \vec J$ . The (time-)"averaged" spin vector is then the projection of the spin onto the direction of $\scriptstyle \vec J$ :

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

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

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

Thus,

$\langle V_M \rangle = \frac{\mu_B}{\hbar} \vec J(g_L\frac{\vec L \cdot \vec J}{J^2} + g_S\frac{\vec S \cdot \vec J}{J^2}) \cdot \vec B.$

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

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

and: using $\scriptstyle \vec S = \vec J - \vec L$ and squaring both sides, we get

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

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

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

where the quantity in square brackets is the Lande g-factor g_J of the atom ( $g L = 1$ and $\scriptstyle g_S \approx 2$ ) 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

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

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 ( $m j = 1 / 2, - 1 / 2$ ) and the 2P_3/2 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

1 S 1 / 2

(j=1/2, l=0)

g J = 2 / 3

for

2 P 1 / 2

(j=1/2, l=1)

g J = 4 / 3

for

2 P 3 / 2

(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. 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 $[H 0, S] = 0$ . This allows the expectation values of $L z$ and $S z$ to be easily evaluated for a state $|A\rangle$ :

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

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 $Δ E = B μ B Δ m l$ 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. (11 February 1897). "The Effect of Magnetisation on the Nature of Light Emitted by a Substance" (^{[dead link]}). Nature 55: 347. doi:10.1038/055347a0. http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Zeeman-effect.html.

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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