NationMaster - Encyclopedia: Zeeman effect

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 nuclear magnetic resonance spectroscopy, electron spin resonance spectroscopy, magnetic resonance imaging (MRI) and Mössbauer spectroscopy. Articles with similar titles include the NATO phonetic alphabet, which has also informally been called the â€œInternational Phonetic Alphabetâ€. For information on how to read IPA transcriptions of English words, see IPA chart for English. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... In physics, a magnetic field is a force field that surrounds electric current circuits. ... The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ... It has been suggested that optical field be merged into this article or section. ... Pacific Northwest National Laboratorys high magnetic field (800 MHz, 18. ... Electron Paramagnetic Resonance (EPR) or Electron Spin Resonance (ESR) is a spectroscopic technique which detects species that have unpaired electrons, generally meaning that the molecule in question is a free radical if it is an organic molecule, or that it has transition metal ions if it is an inorganic complex. ... Magnetic Resonance Image showing a median sagittal cross section through a human head. ... MÃ¶ÃŸbauer spectroscopy is a spectroscopic technique based on the MÃ¶ssbauer effect. ...

The Zeeman effect is named after the Dutch physicist Pieter Zeeman. Pieter Zeeman (May 25, 1865 â€“ October 9, 1943) (pronounced zÄmÃ¤n) was a physicist who shared the 1902 Nobel Prize in Physics with Hendrik Lorentz for his discovery of the Zeeman effect. ...

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

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. Properties In chemistry and physics, an atom (Greek á¼„Ï„Î¿Î¼Î¿Ï‚ or Ã¡tomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... Electron configuration is the arrangement of electrons in an atom, molecule or other body. ...

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. The energy levels of two or more physical states are said to be degenerate when they have the same value. ... e- redirects here. ... A quantum number describes the energies of electrons in atoms. ...

Created by me; released under GFDL. File links The following pages link to this file: Zeeman effect User:Bogdangiusca/Foto Categories: GFDL images | NowCommons ...

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. The transition rules (or selection rules) describe possible state transitions of a quantum mechanical system, expressed by changes of the quantum numbers. ...

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. In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... e- redirects here. ... e- redirects here. ...

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 Paschen-Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. ...

Theoretical presentation

The total Hamiltonian of an atom in a magnetic field is: In physics, Hamiltonian has distinct but closely related meanings. ...

$H = H_0 + H_1 = H_0 + sum_alpha &# 0;vec{r_alpha}) vec{L} cdot vec{S} -sum_alpha vec{mu_{alpha}} cdot vec{B}$

where $H 0$ is the unperturbed Hamiltonian of the atom, and the sums over α are sums over the electrons in the atom. The term

$&# 0;vec{r_alpha}) vec{L} cdot vec{S}$

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 ...

$V_M = -vec{mu_{alpha}} cdot vec{B} = frac{mu_{B}}{hbar}(g_L vec{L} + g_S vec{S}) cdot vec{B}$

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 $vec L$ and the spin angular momentum $vec S$ , 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 The Azimuthal quantum number (or orbital angular momentum quantum number) l is a quantum number for an atomic orbital which determines its orbital angular momentum. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... In physics, the LandÃ© g-factor, , relates the magnetic dipole moment to the angular momentum of a quantum state. ...

$H = H_0 + &# 0;r) vec{L}cdot vec{S} + frac{mu_B}{hbar} (g_L L_z+ g_s S_z) B_z approx H_{at} + frac{mu_B}{hbar}(J_z + S_z) B_z$

where the approximation results from taking the g-factors to be $g L = 1$ and $g_S approx 2.0023192$ (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the relativistic effects). The summation over the electrons was omitted for readability. Here, $J z = L z + S z$ is the total angular momentum, and the spin-orbit (LS) coupling term has been combined with $H 0$ and written as $H a t$ .

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 $H a t$ . The H ' term does not commute with $H a t$ . In particular, $S z$ doesn't commute with the spin-orbit interaction in $H a t$ . The Paschen-Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. ... Spin-orbit interaction, in quantum mechanics, is a shift in energy levels due to the potential energy of the spin magnetic moment of the electron in the magnetic field it feels as it moves through the electric field of the nucleus. ...

Weak field (anomalous 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; instead only the total angular momentum vector $vec J = vec L + vec S$ is conserved. We can write the "averaged" magnetic energy for a single electron spin as

$langle V_M rangle = frac{mu_B}{hbar}(vec J + vec S_{avg}) cdot vec B$

The spin and orbital angular momentum vectors can be thought of as precessing rapidly 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 $vec J$ :

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

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

$vec S cdot vec J = frac{1}{2}(J^2 + S^2 - L^2) = frac{hbar^2}{2}[j(j+1) - l(l+1) + 3/4]$

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

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

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. The Landé g-factor, , is a multiplicative term in the lifting of the energy degeneracy in for an atom in a weak uniform external magnetic field. ...

Example: Lyman alpha transition in hydrogen

The Lyman alpha transition in hydrogen in the presence of the spin-orbit interaction involves the transitions In physics, the Lyman series is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n â‰¥ 2 to n = 1 (where n is the principal quantum number referring to the energy level of the electron). ... General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ...

$2P_{1/2} to 1S_{1/2}$ 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.

Strong Field (Paschen-Back effect)

To simplify the solution, it is useful to assume that $[H a t, S z] = 0$ , so that $L z$ and $S z$ have a set of common eigenfunctions with respect to $H a t$ . This allows the expectation values of $L z$ and $S z$ to be easily evaluated on a general state $|Arangle$ :

$left( H_{at} + frac{B_{z}mu_B}{hbar}(L_{z}+2S_z) right) |A rangle = (E_{at} + B_zmu_B (m_l + 2m_s)|Arangle$

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 $B z$ field. The $m l$ and $m s$ are still "good" quantum numbers. This implies that the selection rules obtained from $Delta S = 0, Delta L = pm 1$ 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 $Delta m = pm 1$ 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.

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". Nature 55: 347.

February 11 is the 42nd day of the year in the Gregorian calendar. ... 1897 (MDCCCXCVII) was a common year starting on Friday (see link for calendar). ...

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.

Categories: Atomic physics | Magnetism | Foundational quantum physics | Physical phenomena Alfred LandÃ© was a German physicist (1888-1976) known for his contributions to Quantum Theory. ... Richard L. Liboff is a U.S. physicist who has authored five books and nearly 150 other publications in variety of fields, including plasma physics, planetary physics, cosmology, quantum chaos, and quantum billiards. ...

Results from FactBites:

Pieter Zeeman (973 words)
	Pieter Zeeman (Zonnemaire (Schouwen-Duiveland), Netherlands, May 25, 1865 – Amsterdam, October 9, 1943) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Hendrik Lorentz for his discovery of the Zeeman effect.
	Pieter Zeeman was born in Zonnemaire, a small town on the island of Schouwen-Duiveland, Netherlands to Catharinus Forandinus Zeeman, a minister of the Dutch Reformed Church, and Willemina Worst.
	Zeeman, then a student of the highschool in Zierikzee, made a drawing and description of the phenomenon and submitted that to Nature, where it was published.

Zeeman Effect (757 words)
	First observed by Pieter Zeeman, this splitting is attributed to the interaction between the magnetic field and the magnetic dipole moment associated with the orbital angular momentum.
	The Zeeman effect for the hydrogen atom offered experimental support for the quantization of angular momentum which arose from the solution of the Schrodinger equation.
	While the Zeeman effect in some atoms (e.g., hydrogen) showed the expected equally-spaced triplet, in other atoms the magnetic field split the lines into four, six, or even more lines and some triplets showed wider spacings than expected.

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