The Zeeman effect (Dutch: [ˈzeːmɑn]) 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 moments of the atomic electrons associated with their orbital motion and spin; this interaction shifts some orbital energies more than others, resulting in the split spectrum. The effect is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received the Nobel Prize in Physics for it in 1902. 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 selection rules. 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 stars or in laboratory plasmas.
Discovery In 1896, Zeeman learned that his laboratory had one of Henry Augustus Rowland's highest resolving diffraction gratings. Zeeman had read James Clerk Maxwell's article in Encyclopædia Britannica describing Michael Faraday's failed attempts to influence light with magnetism. Zeeman wondered if the new spectrographic techniques could succeed where early efforts had not. 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 in the sodium light emission spectrum. Switching on a strong (about one tesla) electromagnet around the flame, he observed a slight broadening of the lines. When Zeeman switched to cadmium as the source, he observed the spectral lines split when the magnetic field was applied. These splittings could be analyzed with Hendrik Lorentz's then-new electron theory. It is now known 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.
Nomenclature Historically, one distinguishes between the normal and an anomalous Zeeman effect (discovered by Thomas Preston in Dublin, Ireland). The anomalous effect appears on transitions where the net spin of the electrons is non-zero. 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. Wolfgang Pauli recalled that when asked by a colleague as to why he looked unhappy, he replied: "How can one look happy when he is thinking about the anomalous Zeeman effect?" 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, referring to the Zeeman effect in an absorption spectral line. A similar effect, splitting of the nuclear energy levels in the presence of a magnetic field, is referred to as the nuclear Zeeman effect.
Theoretical presentation The total Hamiltonian of an atom in a magnetic field is
H = H 0 + V M , {\displaystyle H=H_{0}+V_{\text{M}},}
where H 0 {\displaystyle H_{0}} is the unperturbed Hamiltonian of the atom, and V M {\displaystyle V_{\text{M}}} is the perturbation due to the magnetic field:
V M = − μ → ⋅ B → , {\displaystyle V_{\text{M}}=-{\vec {\mu }}\cdot {\vec {B}},}
where μ → {\displaystyle {\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 here. Therefore,
μ → ≈ − μ B g J → ℏ , {\displaystyle {\vec {\mu }}\approx -{\frac {\mu _{\text{B}}g{\vec {J}}}{\hbar }},}
where μ B {\displaystyle \mu _{\text{B}}} is the Bohr magneton, J → {\displaystyle {\vec {J}}} is the total electronic angular momentum, and g {\displaystyle g} 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 L → {\displaystyle {\vec {L}}} and the spin angular momentum S → {\displaystyle {\vec {S}}} , with each multiplied by the appropriate gyromagnetic ratio:
μ → = − μ B ( g l L → + g s S → ) ℏ , {\displaystyle {\vec {\mu }}=-{\frac {\mu _{\text{B}}(g_{l}{\vec {L}}+g_{s}{\vec {S}})}{\hbar }},}
where g l = 1 {\displaystyle g_{l}=1} , and g s ≈ 2.0023193 {\displaystyle g_{s}\approx 2.0023193} (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 J → = ⟨ ∑ i ( g l l → i + g s s → i ) ⟩ = ⟨ ( g l L → + g s S → ) ⟩ , {\displaystyle 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 L → {\displaystyle {\vec {L}}} and S → {\displaystyle {\vec {S}}} 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 V M {\displaystyle V_{\text{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 {\displaystyle V_{\text{M}}} exceeds the LS coupling significantly (but is still small compared to H 0 {\displaystyle H_{0}} ). In ultra-strong magnetic fields, the magnetic-field interaction may exceed H 0 {\displaystyle H_{0}} , 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.
Weak field (Zeeman effect) If the spin–orbit interaction dominates over the effect of the external magnetic field, L → {\displaystyle {\vec {L}}} and S → {\displaystyle {\vec {S}}} are not separately conserved, only the total angular momentum J → = L → + S → {\displaystyle {\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 J → {\displaystyle {\vec {J}}} . The (time-)"averaged" spin vector is then the projection of the spin onto the direction of J → {\displaystyle {\vec {J}}} :
S → avg = ( S → ⋅ J → ) J 2 J → , {\displaystyle {\vec {S}}_{\text{avg}}={\frac {({\vec {S}}\cdot {\vec {J}})}{J^{2}}}{\vec {J}},}
and for the (time-)"averaged" orbital vector:
L → avg = ( L → ⋅ J → ) J 2 J → . {\displaystyle {\vec {L}}_{\text{avg}}={\frac {({\vec {L}}\cdot {\vec {J}})}{J^{2}}}{\vec {J}}.}
Thus
⟨ V M ⟩ = μ B ℏ J → ( g L L → ⋅ J → J 2 + g S S → ⋅ J → J 2 ) ⋅ B → . {\displaystyle \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 L → = J → − S → {\displaystyle {\vec {L}}={\vec {J}}-{\vec {S}}} and squaring both sides, we get
S → ⋅ J → = 1 2 ( J 2 + S 2 − L 2 ) = ℏ 2 2 [ j ( j + 1 ) − l ( l + 1 ) + s ( s + 1 ) ] , {\displaystyle {\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 S → = J → − L → {\displaystyle {\vec {S}}={\vec {J}}-{\vec {L}}} and squaring both sides, we get
L → ⋅ J → = 1 2 ( J 2 − S 2 + L 2 ) = ℏ 2 2 [ j ( j + 1 ) + l ( l + 1 ) − s ( s + 1 ) ] . {\displaystyle {\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 = ℏ m j {\displaystyle J_{z}=\hbar m_{j}} , we obtain the magnetic potential energy of the atom in the applied external magnetic field:
V M = μ B B m j [ g L j ( j + 1 ) + l ( l + 1 ) − s ( s + 1 ) 2 j ( j + 1 ) + g S j ( j + 1 ) − l ( l + 1 ) + s ( s + 1 ) 2 j ( j + 1 ) ] = μ B B m j [ 1 + ( g S − 1 ) j ( j + 1 ) − l ( l + 1 ) + s ( s + 1 ) 2 j ( j + 1 ) ] = μ B B m j g J , {\displaystyle {\begin{aligned}V_{\text{M}}&=\mu _{\text{B}}Bm_{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}}Bm_{j}\left[1+(g_{S}-1){\frac {j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}}\right]\\&=\mu _{\text{B}}Bm_{j}g_{J},\end{aligned}}}
where the quantity in square brackets is the Landé g-factor g J {\displaystyle g_{J}} of the atom ( g L = 1 , {\displaystyle g_{L}=1,} g S ≈ 2 {\displaystyle g_{S}\approx 2} ), and m j {\displaystyle m_{j}} is the z component of the total angular momentum. For a single electron above filled shells, with s = 1 / 2 {\displaystyle s=1/2} and j = l ± s {\displaystyle j=l\pm s} , the Landé g-factor can be simplified to
g J = 1 ± g S − 1 2 l + 1 . {\displaystyle g_{J}=1\pm {\frac {g_{S}-1}{2l+1}}.}
Taking V M {\displaystyle V_{\text{M}}} to be the perturbation, the Zeeman correction to the energy is
E Z ( 1 ) = ⟨ n l j m j | H Z ′ | n l j m j ⟩ = ⟨ V M ⟩ Ψ = μ B g J B ext m j . {\displaystyle 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}.}
Example: Lyman-alpha transition in hydrogen The Lyman-alpha transition in hydrogen in the presence of the spin–orbit interaction involves the transitions 2 2 P 1 / 2 → 1 2 S 1 / 2 {\displaystyle 2\,^{2}\!P_{1/2}\to 1\,^{2}\!S_{1/2}} and 2 2 P 3 / 2 → 1 2 S 1 / 2 . {\displaystyle 2\,^{2}\!P_{3/2}\to 1\,^{2}\!S_{1/2}.}
In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1 2 S 1 / 2 {\displaystyle 1\,^{2}\!S_{1/2}} and 2 2 P 1 / 2 {\displaystyle 2\,^{2}\!P_{1/2}} levels into 2 states each ( m j = + 1 / 2 , − 1 / 2 {\displaystyle m_{j}=+1/2,-1/2} ) and the 2 2 P 3 / 2 {\displaystyle 2\,^{2}\!P_{3/2}} level into 4 states ( m j = + 3 / 2 , + 1 / 2 , − 1 / 2 , − 3 / 2 {\displaystyle m_{j}=+3/2,+1/2,-1/2,-3/2} ). The Landé g-factors for the three levels are
g J = 2 for 1 2 S 1 / 2 ( j = 1 / 2 , l = 0 ) , g J = 2 / 3 for 2 2 P 1 / 2 ( j = 1 / 2 , l = 1 ) , g J = 4 / 3 for 2 2 P 3 / 2 ( j = 3 / 2 , l = 1 ) . {\displaystyle {\begin{aligned}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{aligned}}}
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.
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 strong to disrupt the coupling between orbital ( L → {\displaystyle {\vec {L}}} ) and spin ( S → {\displaystyle {\vec {S}}} ) angular momenta. This effect is the strong-field limit of the Zeeman effect. When s = 0 {\displaystyle s=0} , the two effects are equivalent. The effect was named after 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 {\displaystyle [H_{0},S]=0} . This allows the expectation values of L z {\displaystyle L_{z}} and S z {\displaystyle S_{z}} to be easily evaluated for a state | ψ ⟩ {\displaystyle |\psi \rangle } . The energies are simply
E z = ⟨ ψ | H 0 + B z μ B ℏ ( L z + g s S z ) | ψ ⟩ = E 0 + B z μ B ( m l + g s m s ) . {\displaystyle E_{z}=\left\langle \psi \left|H_{0}+{\frac {B_{z}\mu _{\rm {B}}}{\hbar }}(L_{z}+g_{s}S_{z})\right|\psi \right\rangle =E_{0}+B_{z}\mu _{\rm {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. However, m l {\displaystyle m_{l}} and m s {\displaystyle m_{s}} are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., Δ s = 0 , Δ m s = 0 , Δ l = ± 1 , Δ m l = 0 , ± 1 {\displaystyle \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 Δ m l = 0 , ± 1 {\displaystyle \Delta m_{l}=0,\pm 1} selection rule. The splitting Δ E = B μ B Δ m l {\dis