Zeeman effect

The Zeeman effect (pronounced /ˈzeɪmɑːn/) is the splitting of a spectral line into several components in the presence of a static magnetic field. 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 In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field. The Stark effect is the shifting and splitting of Spectral lines of atoms and molecules due to the presence of an external static Electric field. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can 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. Electron paramagnetic resonance (EPR or electron spin resonance (ESR Spectroscopy is a technique for studying Chemical species that have one or more unpaired Mössbauer spectroscopy (Mößbauer is a spectroscopic technique based on the Mössbauer effect.

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. The Netherlands ( Dutch:, ˈnedərlɑnt is the European part of the Kingdom of the Netherlands, which consists of the Netherlands the Netherlands Pieter Zeeman ( Zonnemaire, May 25, 1865 &ndash Amsterdam, October 9, 1943) (ˈzeːmɑn was a Dutch

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. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Atomic physics and Quantum chemistry, electron configuration is the arrangement of Electrons in an Atom, Molecule, or other In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός

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. This article refers to physical states having the same energy The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. 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.

Image:zeeman effect.png

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. Transition rule is a non-standard and rarely used name for Selection rule, as applied to radiative transitions.

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. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J 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:

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

where $\vec{\mu}$ is the magnetic moment of the atom. In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current 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,

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

where $μ B$ is the Bohr magneton, $\vec{J}$ is the total electronic angular momentum, and $g$ is the g-factor. In Atomic physics, the Bohr magneton (symbol \mu_\mathrm{B} is named after the Physicist Niels Bohr. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position For the acceleration-related quantity in mechanics see ''g''-force. 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:

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

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). The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines of a particle or system is the Ratio of its In the case of the LS coupling, one can sum over all electrons in the atom:

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

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. In Quantum mechanics, the procedure of constructing Eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling

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 Atomic physics, the fine structure describes the splitting of the Spectral lines of Atoms due to first order relativistic corrections In the Paschen-Back effect, described below, $V M$ exceeds the LS coupling significantly (but is still small compared to $H 0$ ). The Zeeman effect (ˈzeɪmɑːn is the splitting of a Spectral line into several components in the presence of a static Magnetic field. In Quantum mechanics, the procedure of constructing Eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling 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. Landau quantization in Quantum mechanics is the quantization of the cyclotron orbits of charged particles in magnetic fields 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 $\vec J$ :

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

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 $\vec L = \vec J - \vec S$ 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 $\vec S = \vec J - \vec L$ 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 $J_z = \hbar m_j$ , 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 $g_S \approx 2$ ) and $m j$ is the z-component of the total angular momentum. In Physics, the Landé g-factor is a particular example of a G-factor, namely for an Electron with both spin and Orbital angular 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 Physics, the Lyman series is the series of transitions and resulting Emission lines of the Hydrogen Atom as an Electron goes from Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1

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)

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. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning Louis Karl Heinrich Friedrich Paschen ( January 22, 1865 - February 25, 1947) was a German Physicist, known for his work Ernst Emil Alexander Back ( October 21, 1881 &ndash June 20, 1959) was a German Physicist, born in Freiburg.

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. In Physics and Chemistry, especially in the context of Quantum mechanics, a selection rule is a condition constraining the physical properties of the initial 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. The Stark effect is the shifting and splitting of Spectral lines of atoms and molecules due to the presence of an external static Electric field. Magneto-optic Kerr effect (MOKE is one of the Magneto-optic effects It describes the changes of light reflected from Magnetized media The Voigt Effect, sometimes termed magnetic Birefringence or magnetic double refraction, is a magneto-optical phenomenon whereby the Polarization In Physics, the Faraday effect or Faraday rotation is a Magneto-optical phenomenon or an interaction between Light and a Magnetic The Cotton-Mouton effect refers to the double refraction of light in a liquid in the presence of a constant transverse magnetic field Polarization spectroscopy comprises a set of spectroscopic techniques based on polarization properties of Light (not necessarily visible one UV, X-ray, 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). Events 660 BC - Traditional founding date of Japan by Emperor Jimmu. Year 1897 ( MDCCCXCVII) was a Common year starting on Friday (link will display the full calendar of the Gregorian Calendar (or a Common "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". Alfred Landé ( 13 December, 1888 &ndash 30 October 1976) was a German-American physicist known for his contributions to quantum theory 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). Richard L Liboff is a US physicist who has authored five books and nearly 150 other publications in variety of fields including Plasma physics, 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.

Dictionary

-noun

(physics) the splitting of single spectral lines into three (or more) in the presence of a magnetic field

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