The Zeeman effect () is the splitting of a spectral line into several components in the presence of a static magnetic field. It is caused by the interaction of the magnetic field with the magnetic moment of the atomic electron associated with its Angular momentum and spin; this interaction shifts some orbital energies more than others, resulting in the split spectrum. The effect is named after the Netherlands physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel Prize in Physics for this discovery. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field. Also, similar to the Stark effect, transitions between different components have, in general, different intensities, with some being entirely forbidden (in the dipole approximation), as governed by the .
Since the distance between the Zeeman sub-levels is a function of magnetic field strength, this effect can be used to measure magnetic field strength, e.g. that of the Sun and other or in laboratory plasmas.
When illuminated by a slit-shaped source, the grating produces a long array of slit images corresponding to different wavelengths. Zeeman placed a piece of asbestos soaked in salt water into a Bunsen burner flame at the source of the grating: he could easily see two lines for sodium light emission. Energizing a 10-kilogauss magnet around the flame, he observed a slight broadening of the sodium images.
When Zeeman switched to cadmium as the source, he observed the images split when the magnet was energized. These splittings could be analyzed with Hendrik Lorentz's then-new Lorentz force. In retrospect, we now know that the magnetic effects on sodium require quantum-mechanical treatment.
Zeeman and Lorentz were awarded the 1902 Nobel Prize; in his acceptance speech Zeeman explained his apparatus and showed slides of the spectrographic images.
At higher magnetic field strength the effect ceases to be linear. At even higher field strengths, comparable to the strength of the atom's internal field, the electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen–Back effect.
In modern scientific literature, these terms are rarely used, with a tendency to use just the "Zeeman effect". Another rarely used obscure term is inverse Zeeman effect,
A similar effect, splitting of the nuclear energy levels in the presence of a magnetic field, is referred to as the nuclear Zeeman effect.
H = H_0 + V_\text{M},where is the unperturbed Hamiltonian of the atom, and is the perturbation due to the magnetic field:
V_\text{M} = -\vec{\mu} \cdot \vec{B},where 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 here. Therefore,
\vec{\mu} \approx -\frac{\mu_\text{B} g \vec{J}}{\hbar},where is the Bohr magneton, is the total electronic angular momentum, and is the Landé g-factor.
A more accurate approach is to take into account that the operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum and the spin angular momentum , with each multiplied by the appropriate gyromagnetic ratio:
\vec{\mu} = -\frac{\mu_\text{B} (g_l \vec{L} + g_s \vec{S})}{\hbar},where , and (the anomalous gyromagnetic ratio, deviating from 2 due to the effects of quantum electrodynamics). In the case of the LS coupling, one can sum over all electrons in the atom:
g \vec{J} = \Big\langle\sum_i (g_l \vec{l}_i + g_s \vec{s}_i)\Big\rangle = \big\langle(g_l \vec{L} + g_s \vec{S})\big\rangle,where and are the total spin 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 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, exceeds the LS coupling significantly (but is still small compared to ). In ultra-strong magnetic fields, the magnetic-field interaction may exceed , in which case the atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are intermediate cases that are more complex than these limit cases.
\vec S_\text{avg} = \frac{(\vec S \cdot \vec J)}{J^2} \vec J,and for the (time-)"averaged" orbital vector:
\vec L_\text{avg} = \frac{(\vec L \cdot \vec J)}{J^2} \vec J.
Thus
\langle V_\text{M} \rangle = \frac{\mu_\text{B}}{\hbar} \vec J \left(g_L\frac{\vec L \cdot \vec J}{J^2} + g_S\frac{\vec S \cdot \vec J}{J^2}\right) \cdot \vec B.Using 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 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 , we obtain the magnetic potential energy of the atom in the applied external magnetic field:
V_\text{M} &= \mu_\text{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] \\ &= \mu_\text{B} B m_j \left[1 + (g_S - 1) \frac{j(j + 1) - l(l + 1) + s(s + 1)}{2j(j + 1)}\right] \\ &= \mu_\text{B} B m_j g_J,\end{align} where the quantity in square brackets is the Landé g-factor of the atom ( ), and is the z component of the total angular momentum.
For a single electron above filled shells, with and , the Landé g-factor can be simplified to
g_J = 1 \pm \frac{g_S - 1}{2l + 1}.
Taking to be the perturbation, the Zeeman correction to the energy is
E_\text{Z}^{(1)} = \langle nljm_j | H_\text{Z}^' | nljm_j \rangle = \langle V_\text{M} \rangle_\Psi = \mu_\text{B} g_J B_\text{ext} m_j.
In the presence of an external magnetic field, the weak-field Zeeman effect splits the and levels into 2 states each () and the level into 4 states (). The Landé g-factors for the three levels are
g_J &= 2 & &\text{for}\ 1\,^2\!S_{1/2}\ (j = 1/2, l = 0), \\ g_J &= 2/3 & &\text{for}\ 2\,^2\!P_{1/2}\ (j = 1/2, l = 1), \\ g_J &= 4/3 & &\text{for}\ 2\,^2\!P_{3/2}\ (j = 3/2, l = 1).\end{align}
Note in particular that the size of the energy splitting is different for the different orbitals because the gJ values are different. Fine-structure 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.
+ Dipole-allowed Lyman-alpha transitions in the weak-field regime
! Initial state | j, m_j\rangle
! Final state | j, m_j\rangle
! Energy perturbation |
\left>\frac{1}{2}, \pm\frac{1}{2} \right\rangle | \left>\frac{1}{2}, \pm\frac{1}{2} \right\rangle | |
\left>\frac{1}{2}, \pm\frac{1}{2} \right\rangle | \left>\frac{1}{2}, \mp\frac{1}{2} \right\rangle | |
\left>\frac{3}{2}, \pm\frac{3}{2} \right\rangle | \left>\frac{1}{2}, \pm\frac{1}{2} \right\rangle | |
\left>\frac{3}{2}, \pm\frac{1}{2} \right\rangle | \left>\frac{1}{2}, \pm\frac{1}{2} \right\rangle | |
\left>\frac{3}{2}, \pm\frac{1}{2} \right\rangle | \left>\frac{1}{2}, \mp\frac{1}{2} \right\rangle |
When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume . This allows the expectation values of and to be easily evaluated for a state . The energies are simply
The above may be read as implying that the LS-coupling is completely broken by the external field. However, and are still "good" quantum numbers. Together with the for an electric dipole transition, i.e., this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the selection rule. The splitting is independent of the unperturbed energies and electronic configurations of the levels being considered.
More precisely, if , each of these three components is actually a group of several transitions due to the residual spin–orbit coupling and relativistic corrections (which are of the same order, known as 'fine structure'). The first-order perturbation theory with these corrections yields the following formula for the hydrogen atom in the Paschen–Back limit:
+Dipole-allowed Lyman-alpha transitions in the strong-field regime
!Initial state
()
!Initial energy perturbation !Final state () !Final energy perturbation | |||
\left>1, \frac{1}{2}\right\rangle | \left>0, \frac{1}{2}\right\rangle | ||
\left>0, \frac{1}{2}\right\rangle | \left>0, \frac{1}{2}\right\rangle | ||
\left>1, -\frac{1}{2}\right\rangle | \left>0, -\frac{1}{2}\right\rangle | ||
\left>-1, \frac{1}{2}\right\rangle | \left>0, \frac{1}{2}\right\rangle | ||
\left>0, -\frac{1}{2}\right\rangle | \left>0, -\frac{1}{2}\right\rangle | ||
\left>-1, -\frac{1}{2}\right\rangle | \left>0, -\frac{1}{2}\right\rangle |
In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the basis. In the high field regime, the magnetic field becomes so strong that the Zeeman effect will dominate, and one must use a more complete basis of or just since and will be constant within a given level.
To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the and basis states. For , the Hamiltonian can be solved analytically, resulting in the Breit–Rabi formula (named after Gregory Breit and Isidor Isaac Rabi). Notably, the electric quadrupole interaction is zero for (), so this formula is fairly accurate.
We now utilize quantum mechanical , which are defined for a general angular momentum operator as
These ladder operators have the property
as long as lies in the range (otherwise, they return zero). Using ladder operators and We can rewrite the Hamiltonian as
We can now see that at all times, the total angular momentum projection will be conserved. This is because both and leave states with definite and unchanged, while and either increase and decrease or vice versa, so the sum is always unaffected. Furthermore, since there are only two possible values of which are . Therefore, for every value of there are only two possible states, and we can define them as the basis:
This pair of states is a two-level quantum mechanical system. Now we can determine the matrix elements of the Hamiltonian:
Solving for the eigenvalues of this matrix – as can be done by hand (see two-level quantum mechanical system), or more easily, with a computer algebra system – we arrive at the energy shifts:
where is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field , is referred to as the 'field strength parameter' (Note: for the expression under the square root is an exact square, and so the last term should be replaced by ). This equation is known as the Breit–Rabi formula and is useful for systems with one valence electron in an () level.First appeared in:
Note that index in should be considered not as total angular momentum of the atom but as asymptotic total angular momentum. It is equal to total angular momentum only if otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different but equal (the only exceptions are ).
The Zeeman effect may also be utilized to improve accuracy in atomic absorption spectroscopy.
The sodium vapor can be created by sealing sodium metal in an evacuated glass tube and heating it while the tube is in the magnet.
Alternatively, salt (sodium chloride) on a ceramic stick can be placed in the flame of Bunsen burner as the sodium vapor source. When the magnetic field is energized, the lamp image will be brighter. However, the magnetic field also affects the flame, making the observation depend upon more than just the Zeeman effect. These issues also plagued Zeeman's original work; he devoted considerable effort to ensure his observations were truly an effect of magnetism on light emission.
When salt is added to the Bunsen burner, it dissociates to give sodium and chloride. The sodium atoms get excited due to Photon from the sodium vapour lamp, with electrons excited from 3s to 3p states, absorbing light in the process. The sodium vapour lamp emits light at 589nm, which has precisely the energy to excite an electron of a sodium atom. If it was an atom of another element, like chlorine, shadow will not be formed. When a magnetic field is applied, due to the Zeeman effect the spectral line of sodium gets split into several components. This means the energy difference between the 3s and 3p Atomic orbital will change. As the sodium vapour lamp don't precisely deliver the right frequency anymore, light doesn't get absorbed and passes through, resulting in the shadow dimming. As the magnetic field strength is increased, the shift in the spectral lines increases and lamp light is transmitted.
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