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Zeeman effect

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Spectral line splitting in magnetic field

The spectral lines of mercury vapor lamp at wavelength 546.1 nm, showing anomalous Zeeman effect. (A) Without magnetic field. (B) With magnetic field, spectral lines split as transverse Zeeman effect. (C) With magnetic field, split as longitudinal Zeeman effect. The spectral lines were obtained using aFabry–Pérot interferometer.
Zeeman splitting of the 5s level of87Rb, including fine structure and hyperfine structure splitting. HereF = J + I, whereI is the nuclear spin (for87Rb,I = 3⁄2).
This animation shows what happens as a sunspot (or starspot) forms and the magnetic field increases in strength. The light emerging from the spot starts to demonstrate the Zeeman effect. The dark spectra lines in the spectrum of the emitted light split into three components and the strength of the circular polarisation in parts of the spectrum increases significantly. This polarization effect is a powerful tool for astronomers to detect and measure stellar magnetic fields.

TheZeeman effect (Dutch:[ˈzeːmɑn]) is the splitting of aspectral line into several components in the presence of a staticmagnetic field. It is caused by the interaction of the magnetic field with themagnetic moment of the atomicelectron associated with itsorbital motion andspin; this interaction shifts some orbital energies more than others, resulting in the split spectrum. The effect is named after theDutch physicistPieter Zeeman, who discovered it in 1896 and received aNobel Prize in Physics for this discovery. It is analogous to theStark effect, the splitting of a spectral line into several components in the presence of anelectric field. Also, similar to the Stark effect, transitions between different components have, in general, different intensities, with some being entirely forbidden (in thedipole approximation), as governed by theselection 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 theSun and otherstars or in laboratoryplasmas.

Discovery

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In 1896 Zeeman learned that his laboratory had one ofHenry Augustus Rowland's highest resolvingdiffraction gratings. Zeeman had readJames Clerk Maxwell's article inEncyclopædia Britannica describingMichael Faraday's failed attempts to influence light with magnetism. Zeeman wondered if the new spectrographic techniques could succeed where early efforts had not.[1]: 75 

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 aBunsen burner flame at the source of the grating: he could easily see two lines forsodium light emission. Energizing a 10-kilogauss magnet around the flame, he observed a slight broadening of the sodium images.[1]: 76 

When Zeeman switched tocadmium as the source, he observed the images split when the magnet was energized. These splittings could be analyzed withHendrik Lorentz's then-newelectron theory. In retrospect, we now know that the magnetic effects on sodium require quantum-mechanical treatment.[1]: 77  Zeeman and Lorentz were awarded the 1902 Nobel Prize; in his acceptance speech Zeeman explained his apparatus and showed slides of the spectrographic images.[2]

Nomenclature

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Historically, one distinguishes between thenormal and ananomalous Zeeman effect (discovered byThomas Preston in Dublin, Ireland[3]). The anomalous effect appears on transitions where the netspin of theelectrons 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?"[4]

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 thePaschen–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 isinverse Zeeman effect,[5] referring to the Zeeman effect in an absorption spectral line.

A similar effect, splitting of thenuclear energy levels in the presence of a magnetic field, is referred to as thenuclear Zeeman effect.[6]

Theoretical presentation

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The totalHamiltonian of an atom in a magnetic field isH=H0+VM,{\displaystyle H=H_{0}+V_{\text{M}},}whereH0{\displaystyle H_{0}} is the unperturbed Hamiltonian of the atom, andVM{\displaystyle V_{\text{M}}} is theperturbation due to the magnetic field:VM=−μ→⋅B→,{\displaystyle V_{\text{M}}=-{\vec {\mu }}\cdot {\vec {B}},}whereμ→{\displaystyle {\vec {\mu }}} is themagnetic 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→≈−μBgJ→ℏ,{\displaystyle {\vec {\mu }}\approx -{\frac {\mu _{\text{B}}g{\vec {J}}}{\hbar }},}whereμB{\displaystyle \mu _{\text{B}}} is theBohr magneton,J→{\displaystyle {\vec {J}}} is the total electronicangular momentum, andg{\displaystyle g} is theLandé 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 theorbital angular momentumL→{\displaystyle {\vec {L}}} and thespin angular momentumS→{\displaystyle {\vec {S}}}, with each multiplied by the appropriategyromagnetic ratio:μ→=−μB(glL→+gsS→)ℏ,{\displaystyle {\vec {\mu }}=-{\frac {\mu _{\text{B}}(g_{l}{\vec {L}}+g_{s}{\vec {S}})}{\hbar }},}wheregl=1{\displaystyle g_{l}=1}, andgs≈2.0023193{\displaystyle g_{s}\approx 2.0023193} (theanomalous gyromagnetic ratio, deviating from 2 due to the effects ofquantum electrodynamics). In the case of theLS coupling, one can sum over all electrons in the atom:gJ→=⟨∑i(gll→i+gss→i)⟩=⟨(glL→+gsS→)⟩,{\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 },}whereL→{\displaystyle {\vec {L}}} andS→{\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 termVM{\displaystyle V_{\text{M}}} is small (less than thefine structure), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen–Back effect, described below,VM{\displaystyle V_{\text{M}}} exceeds theLS coupling significantly (but is still small compared toH0{\displaystyle H_{0}}). In ultra-strong magnetic fields, the magnetic-field interaction may exceedH0{\displaystyle H_{0}}, in which case the atom can no longer exist in its normal meaning, and one talks aboutLandau levels instead. There are intermediate cases that are more complex than these limit cases.

Weak field (Zeeman effect)

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If thespin–orbit interaction dominates over the effect of the external magnetic field,L→{\displaystyle {\vec {L}}} andS→{\displaystyle {\vec {S}}} are not separately conserved, only the total angular momentumJ→=L→+S→{\displaystyle {\vec {J}}={\vec {L}}+{\vec {S}}} is. The spin and orbital angular momentum vectors can be thought of asprecessing about the (fixed) total angular momentum vectorJ→{\displaystyle {\vec {J}}}. The (time-)"averaged" spin vector is then the projection of the spin onto the direction ofJ→{\displaystyle {\vec {J}}}:S→avg=(S→⋅J→)J2J→,{\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→)J2J→.{\displaystyle {\vec {L}}_{\text{avg}}={\frac {({\vec {L}}\cdot {\vec {J}})}{J^{2}}}{\vec {J}}.}

Thus⟨VM⟩=μBℏJ→(gLL→⋅J→J2+gSS→⋅J→J2)⋅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}}.}UsingL→=J→−S→{\displaystyle {\vec {L}}={\vec {J}}-{\vec {S}}} and squaring both sides, we getS→⋅J→=12(J2+S2−L2)=ℏ22[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 usingS→=J→−L→{\displaystyle {\vec {S}}={\vec {J}}-{\vec {L}}} and squaring both sides, we getL→⋅J→=12(J2−S2+L2)=ℏ22[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 takingJz=ℏmj{\displaystyle J_{z}=\hbar m_{j}}, we obtain the magnetic potential energy of the atom in the applied external magnetic field:VM=μBBmj[gLj(j+1)+l(l+1)−s(s+1)2j(j+1)+gSj(j+1)−l(l+1)+s(s+1)2j(j+1)]=μBBmj[1+(gS−1)j(j+1)−l(l+1)+s(s+1)2j(j+1)]=μBBmjgJ,{\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 theLandé g-factorgJ{\displaystyle g_{J}} of the atom (gL=1,{\displaystyle g_{L}=1,}gS≈2{\displaystyle g_{S}\approx 2}), andmj{\displaystyle m_{j}} is thez component of the total angular momentum.

For a single electron above filled shells, withs=1/2{\displaystyle s=1/2} andj=l±s{\displaystyle j=l\pm s}, the Landé g-factor can be simplified togJ=1±gS−12l+1.{\displaystyle g_{J}=1\pm {\frac {g_{S}-1}{2l+1}}.}

TakingVM{\displaystyle V_{\text{M}}} to be the perturbation, the Zeeman correction to the energy isEZ(1)=⟨nljmj|HZ′|nljmj⟩=⟨VM⟩Ψ=μBgJBextmj.{\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

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TheLyman-alpha transition inhydrogen in the presence of thespin–orbit interaction involves the transitions22P1/2→12S1/2{\displaystyle 2\,^{2}\!P_{1/2}\to 1\,^{2}\!S_{1/2}} and22P3/2→12S1/2.{\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 the12S1/2{\displaystyle 1\,^{2}\!S_{1/2}} and22P1/2{\displaystyle 2\,^{2}\!P_{1/2}} levels into 2 states each (mj=+1/2,−1/2{\displaystyle m_{j}=+1/2,-1/2}) and the22P3/2{\displaystyle 2\,^{2}\!P_{3/2}} level into 4 states (mj=+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 aregJ=2for 12S1/2 (j=1/2,l=0),gJ=2/3for 22P1/2 (j=1/2,l=1),gJ=4/3for 22P3/2 (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 thegJ 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
(n=2,l=1){\displaystyle (n=2,l=1)}
|j,mj⟩{\displaystyle |j,m_{j}\rangle }		Final state
(n=1,l=0){\displaystyle (n=1,l=0)}
|j,mj⟩{\displaystyle |j,m_{j}\rangle }		Energy
perturbation
|12,±12⟩{\displaystyle \left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }|12,±12⟩{\displaystyle \left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }∓23μBB{\displaystyle \mp {\frac {2}{3}}\mu _{\text{B}}B}
|12,±12⟩{\displaystyle \left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }|12,∓12⟩{\displaystyle \left|{\frac {1}{2}},\mp {\frac {1}{2}}\right\rangle }±43μBB{\displaystyle \pm {\frac {4}{3}}\mu _{\text{B}}B}
|32,±32⟩{\displaystyle \left|{\frac {3}{2}},\pm {\frac {3}{2}}\right\rangle }|12,±12⟩{\displaystyle \left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }±μBB{\displaystyle \pm \mu _{\rm {B}}B}
|32,±12⟩{\displaystyle \left|{\frac {3}{2}},\pm {\frac {1}{2}}\right\rangle }|12,±12⟩{\displaystyle \left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }∓13μBB{\displaystyle \mp {\frac {1}{3}}\mu _{\text{B}}B}
|32,±12⟩{\displaystyle \left|{\frac {3}{2}},\pm {\frac {1}{2}}\right\rangle }|12,∓12⟩{\displaystyle \left|{\frac {1}{2}},\mp {\frac {1}{2}}\right\rangle }±53μBB{\displaystyle \pm {\frac {5}{3}}\mu _{\text{B}}B}

Strong field (Paschen–Back effect)

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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. Whens=0{\displaystyle s=0}, the two effects are equivalent. The effect was named after theGermanphysicistsFriedrich Paschen andErnst E. A. Back.[7]

When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume[H0,S]=0{\displaystyle [H_{0},S]=0}. This allows the expectation values ofLz{\displaystyle L_{z}} andSz{\displaystyle S_{z}} to be easily evaluated for a state|ψ⟩{\displaystyle |\psi \rangle }. The energies are simply

Ez=⟨ψ|H0+BzμBℏ(Lz+gsSz)|ψ⟩=E0+BzμB(ml+gsms).{\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,ml{\displaystyle m_{l}} andms{\displaystyle m_{s}} are still "good" quantum numbers. Together with theselection rules for anelectric dipole transition, i.e.,Δs=0,Δms=0,Δl=±1,Δml=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 theΔml=0,±1{\displaystyle \Delta m_{l}=0,\pm 1} selection rule. The splittingΔE=BμBΔml{\displaystyle \Delta E=B\mu _{\rm {B}}\Delta m_{l}} isindependent of the unperturbed energies and electronic configurations of the levels being considered.

More precisely, ifs≠0{\displaystyle s\neq 0}, 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:[8]

Ez+fs=Ez+mec2α42n3{34n−[l(l+1)−mlmsl(l+1/2)(l+1)]}.{\displaystyle E_{z+fs}=E_{z}+{\frac {m_{e}c^{2}\alpha ^{4}}{2n^{3}}}\left\{{\frac {3}{4n}}-\left[{\frac {l(l+1)-m_{l}m_{s}}{l(l+1/2)(l+1)}}\right]\right\}.}

Example: Lyman-alpha transition in hydrogen

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In this example, the fine-structure corrections are ignored.

Dipole-allowed Lyman-alpha transitions in the strong-field regime
Initial state

(n=2,l=1{\displaystyle n=2,l=1})

∣ml,ms⟩{\displaystyle \mid m_{l},m_{s}\rangle }

Initial energy perturbationFinal state

(n=1,l=0{\displaystyle n=1,l=0})

∣ml,ms⟩{\displaystyle \mid m_{l},m_{s}\rangle }

Final energy perturbation
|1,12⟩{\displaystyle \left|1,{\frac {1}{2}}\right\rangle }+2μBBz{\displaystyle +2\mu _{\rm {B}}B_{z}}|0,12⟩{\displaystyle \left|0,{\frac {1}{2}}\right\rangle }+μBBz{\displaystyle +\mu _{\rm {B}}B_{z}}
|0,12⟩{\displaystyle \left|0,{\frac {1}{2}}\right\rangle }+μBBz{\displaystyle +\mu _{\rm {B}}B_{z}}|0,12⟩{\displaystyle \left|0,{\frac {1}{2}}\right\rangle }+μBBz{\displaystyle +\mu _{\rm {B}}B_{z}}
|1,−12⟩{\displaystyle \left|1,-{\frac {1}{2}}\right\rangle }0{\displaystyle 0}|0,−12⟩{\displaystyle \left|0,-{\frac {1}{2}}\right\rangle }−μBBz{\displaystyle -\mu _{\rm {B}}B_{z}}
|−1,12⟩{\displaystyle \left|-1,{\frac {1}{2}}\right\rangle }0{\displaystyle 0}|0,12⟩{\displaystyle \left|0,{\frac {1}{2}}\right\rangle }+μBBz{\displaystyle +\mu _{\rm {B}}B_{z}}
|0,−12⟩{\displaystyle \left|0,-{\frac {1}{2}}\right\rangle }−μBBz{\displaystyle -\mu _{\rm {B}}B_{z}}|0,−12⟩{\displaystyle \left|0,-{\frac {1}{2}}\right\rangle }−μBBz{\displaystyle -\mu _{\rm {B}}B_{z}}
|−1,−12⟩{\displaystyle \left|-1,-{\frac {1}{2}}\right\rangle }−2μBBz{\displaystyle -2\mu _{\rm {B}}B_{z}}|0,−12⟩{\displaystyle \left|0,-{\frac {1}{2}}\right\rangle }−μBBz{\displaystyle -\mu _{\rm {B}}B_{z}}

Intermediate field for j = 1/2

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In the magnetic dipole approximation, the Hamiltonian which includes both thehyperfine and Zeeman interactions is[citation needed]

H=hAI→⋅J→−μ→⋅B→{\displaystyle H=hA{\vec {I}}\cdot {\vec {J}}-{\vec {\mu }}\cdot {\vec {B}}}
H=hAI→⋅J→+(μBgJJ→+μNgII→)⋅B→{\displaystyle H=hA{\vec {I}}\cdot {\vec {J}}+(\mu _{\rm {B}}g_{J}{\vec {J}}+\mu _{\rm {N}}g_{I}{\vec {I}})\cdot {\vec {\rm {B}}}}

whereA{\displaystyle A} is the hyperfine splitting at zero applied magnetic field,μB{\displaystyle \mu _{\rm {B}}} andμN{\displaystyle \mu _{\rm {N}}} are theBohr magneton andnuclear magneton, respectively (note that the last term in the expression above describes thenuclear Zeeman effect),J→{\displaystyle {\vec {J}}} andI→{\displaystyle {\vec {I}}} are the electron and nuclear angular momentum operators andgJ{\displaystyle g_{J}} is theLandé g-factor:gJ=gLJ(J+1)+L(L+1)−S(S+1)2J(J+1)+gSJ(J+1)−L(L+1)+S(S+1)2J(J+1).{\displaystyle g_{J}=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)}}.}

In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the|F,mf⟩{\displaystyle |F,m_{f}\rangle } 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|I,J,mI,mJ⟩{\displaystyle |I,J,m_{I},m_{J}\rangle } or just|mI,mJ⟩{\displaystyle |m_{I},m_{J}\rangle } sinceI{\displaystyle I} andJ{\displaystyle J} 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|F,mF⟩{\displaystyle |F,m_{F}\rangle } and|mI,mJ⟩{\displaystyle |m_{I},m_{J}\rangle } basis states. ForJ=1/2{\displaystyle J=1/2}, the Hamiltonian can be solved analytically, resulting in theBreit–Rabi formula (named afterGregory Breit andIsidor Isaac Rabi). Notably, the electric quadrupole interaction is zero forL=0{\displaystyle L=0} (J=1/2{\displaystyle J=1/2}), so this formula is fairly accurate.

We now utilize quantum mechanicalladder operators, which are defined for a general angular momentum operatorL{\displaystyle L} as

L±≡Lx±iLy{\displaystyle L_{\pm }\equiv L_{x}\pm iL_{y}}

These ladder operators have the property

L±|L,mL⟩=(L∓mL)(L±mL+1)|L,mL±1⟩{\displaystyle L_{\pm }|L_{,}m_{L}\rangle ={\sqrt {(L\mp m_{L})(L\pm m_{L}+1)}}|L,m_{L}\pm 1\rangle }

as long asmL{\displaystyle m_{L}} lies in the range−L,…...,L{\displaystyle {-L,\dots ...,L}} (otherwise, they return zero). Using ladder operatorsJ±{\displaystyle J_{\pm }} andI±{\displaystyle I_{\pm }} We can rewrite the Hamiltonian as

H=hAIzJz+hA2(J+I−+J−I+)+μBBgJJz+μNBgIIz{\displaystyle H=hAI_{z}J_{z}+{\frac {hA}{2}}(J_{+}I_{-}+J_{-}I_{+})+\mu _{\rm {B}}Bg_{J}J_{z}+\mu _{\rm {N}}Bg_{I}I_{z}}

We can now see that at all times, the total angular momentum projectionmF=mJ+mI{\displaystyle m_{F}=m_{J}+m_{I}} will be conserved. This is because bothJz{\displaystyle J_{z}} andIz{\displaystyle I_{z}} leave states with definitemJ{\displaystyle m_{J}} andmI{\displaystyle m_{I}} unchanged, whileJ+I−{\displaystyle J_{+}I_{-}} andJ−I+{\displaystyle J_{-}I_{+}} either increasemJ{\displaystyle m_{J}} and decreasemI{\displaystyle m_{I}} or vice versa, so the sum is always unaffected. Furthermore, sinceJ=1/2{\displaystyle J=1/2} there are only two possible values ofmJ{\displaystyle m_{J}} which are±1/2{\displaystyle \pm 1/2}. Therefore, for every value ofmF{\displaystyle m_{F}} there are only two possible states, and we can define them as the basis:

|±⟩≡|mJ=±1/2,mI=mF∓1/2⟩{\displaystyle |\pm \rangle \equiv |m_{J}=\pm 1/2,m_{I}=m_{F}\mp 1/2\rangle }

This pair of states is atwo-level quantum mechanical system. Now we can determine the matrix elements of the Hamiltonian:

⟨±|H|±⟩=−14hA+μNBgImF±12(hAmF+μBBgJ−μNBgI)){\displaystyle \langle \pm |H|\pm \rangle =-{\frac {1}{4}}hA+\mu _{\rm {N}}Bg_{I}m_{F}\pm {\frac {1}{2}}(hAm_{F}+\mu _{\rm {B}}Bg_{J}-\mu _{\rm {N}}Bg_{I}))}
⟨±|H|∓⟩=12hA(I+1/2)2−mF2{\displaystyle \langle \pm |H|\mp \rangle ={\frac {1}{2}}hA{\sqrt {(I+1/2)^{2}-m_{F}^{2}}}}

Solving for the eigenvalues of this matrix – as can be done by hand (seetwo-level quantum mechanical system), or more easily, with a computer algebra system – we arrive at the energy shifts:

ΔEF=I±1/2=−hΔW2(2I+1)+μNgImFB±hΔW21+2mFxI+1/2+x2{\displaystyle \Delta E_{F=I\pm 1/2}=-{\frac {h\Delta W}{2(2I+1)}}+\mu _{\rm {N}}g_{I}m_{F}B\pm {\frac {h\Delta W}{2}}{\sqrt {1+{\frac {2m_{F}x}{I+1/2}}+x^{2}}}}
x≡B(μBgJ−μNgI)hΔWΔW=A(I+12){\displaystyle x\equiv {\frac {B(\mu _{\rm {B}}g_{J}-\mu _{\rm {N}}g_{I})}{h\Delta W}}\quad \quad \Delta W=A\left(I+{\frac {1}{2}}\right)}

whereΔW{\displaystyle \Delta W} is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic fieldB{\displaystyle B},x{\displaystyle x} is referred to as the 'field strength parameter' (Note: formF=±(I+1/2){\displaystyle m_{F}=\pm (I+1/2)} the expression under the square root is an exact square, and so the last term should be replaced by+hΔW2(1±x){\displaystyle +{\frac {h\Delta W}{2}}(1\pm x)}). This equation is known as theBreit–Rabi formula and is useful for systems with one valence electron in ans{\displaystyle s} (J=1/2{\displaystyle J=1/2}) level.[9][10]

Note that indexF{\displaystyle F} inΔEF=I±1/2{\displaystyle \Delta E_{F=I\pm 1/2}} should be considered not as total angular momentum of the atom but asasymptotic total angular momentum. It is equal to total angular momentum only ifB=0{\displaystyle B=0}otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with differentF{\displaystyle F} but equalmF{\displaystyle m_{F}} (the only exceptions are|F=I+1/2,mF=±F⟩{\displaystyle |F=I+1/2,m_{F}=\pm F\rangle }).

Applications

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Astrophysics

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Zeeman effect on a sunspot spectral line

George Ellery Hale was the first to notice the Zeeman effect in the solar spectra, indicating the existence of strong magnetic fields in sunspots. Such fields can be quite high, on the order of 0.1tesla or higher. Today, the Zeeman effect is used to producemagnetograms showing the variation of magnetic field on the Sun,[citation needed] and to analyze the magnetic field geometries in other stars.[11]

Laser cooling

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The Zeeman effect is utilized in manylaser cooling applications such as amagneto-optical trap and theZeeman slower.[12]

Spintronics

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Zeeman-energy mediated coupling of spin and orbital motionsis used inspintronics for controlling electron spins in quantum dots throughelectric dipole spin resonance.[13]

Metrology

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Old high-precision frequency standards, i.e. hyperfine structure transition-based atomic clocks, may require periodic fine-tuning due to exposure to magnetic fields. This is carried out by measuring the Zeeman effect on specific hyperfine structure transition levels of the source element (cesium) and applying a uniformly precise, low-strength magnetic field to said source, in a process known asdegaussing.[14]

The Zeeman effect may also be utilized to improve accuracy inatomic absorption spectroscopy.[citation needed]

Biology

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A theory about themagnetic sense of birds assumes that a protein in the retina is changed due to the Zeeman effect.[15]

Nuclear spectroscopy

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The nuclear Zeeman effect is important in such applications asnuclear magnetic resonance spectroscopy,magnetic resonance imaging (MRI), andMössbauer spectroscopy.[citation needed]

Other

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Theelectron spin resonance spectroscopy is based on the Zeeman effect.[citation needed]

Demonstrations

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Diagram of a Zeeman effect demonstration

The Zeeman effect can be demonstrated by placing a sodium vapor source in a powerful electromagnet and viewing a sodium vapor lamp through the magnet opening (see diagram). With magnet off, the sodium vapor source will block the lamp light; when the magnet is turned on the lamp light will be visible through the vapor.

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.[16]

Alternatively, salt (sodium chloride) on a ceramic stick can be placed in the flame ofBunsen burner as the sodium vapor source. When the magnetic field is energized, the lamp image will be brighter.[17] However, the magnetic field also affects the flame, making the observation depend upon more than just the Zeeman effect.[16] 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.[18]

When salt is added to the Bunsen burner, itdissociates to givesodium andchloride. The sodium atoms get excited due tophotons 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.[19][failed verification] When a magnetic field is applied, due to the Zeeman effect thespectral line of sodium gets split into several components. This means the energy difference between the 3s and 3patomic orbitals 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.[citation needed]

See also

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Wikimedia Commons has media related toZeeman effect.

References

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  1. ^abcPais, Abraham (2002).Inward bound: of matter and forces in the physical world (Reprint ed.). Oxford: Clarendon Press [u.a.]ISBN 978-0-19-851997-3.
  2. ^Pieter, Zeeman (1902)."Pieter Zeeman Nobel Lecture".The Nobel Prize.Archived from the original on 15 November 2018. Retrieved1 March 2024.
  3. ^Preston, Thomas (1898)."Radiation phenomena in a strong magnetic field".The Scientific Transactions of the Royal Dublin Society. 2nd series.6:385–391.
  4. ^"Niels Bohr's Times: In Physics, Philosophy, and Polity" By Abraham Pais, page 201.
  5. ^Jenkins, Francis; White, Harvey (3 December 2001).Fundamentals of Optics (4 ed.). McGraw-Hill Education.ISBN 978-0-07-256191-3.
  6. ^Dunlap, Richard A. (2023). "Hyperfine interactions—part III: the magnetic dipole interaction and the nuclear Zeeman effect".The Mössbauer Effect (2 ed.). IOP Publishing. pp. 7-1 –7-9.doi:10.1088/978-0-7503-6039-5ch7.ISBN 978-0-7503-6039-5. Retrieved4 March 2024.
  7. ^Paschen, F.; Back, E. (1921). "Liniengruppen magnetisch vervollständigt" [Line groups magnetically completed [i.e., completely resolved]].Physica (in German).1:261–273. Available at:Leiden University (Netherlands)
  8. ^Griffiths, David J. (2004).Introduction to Quantum Mechanics (2nd ed.).Prentice Hall. p. 280.ISBN 0-13-111892-7.OCLC 40251748.
  9. ^Woodgate, Gordon Kemble (1980).Elementary Atomic Structure (2nd ed.). Oxford, England: Oxford University Press. pp. 193–194.
  10. ^First appeared in:Breit, G.; Rabi, I.I. (1931). "Measurement of nuclear spin".Physical Review.38 (11):2082–2083.Bibcode:1931PhRv...38.2082B.doi:10.1103/PhysRev.38.2082.2.
  11. ^Kochukhov, Oleg (December 2021). "Magnetic fields of M dwarfs".The Astronomy and Astrophysics Review.29 (1): 1.arXiv:2011.01781.doi:10.1007/s00159-020-00130-3.ISSN 0935-4956.
  12. ^Bowden, William; Gunton, Will; Semczuk, Mariusz; Dare, Kahan; Madison, Kirk W. (18 April 2016)."An adaptable dual species effusive source and Zeeman slower design demonstrated with Rb and Li".Review of Scientific Instruments.87 (4): 043111.arXiv:1509.07460.doi:10.1063/1.4945567.ISSN 0034-6748.
  13. ^Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Coherent single electron spin control in a slanting Zeeman field, Phys. Rev. Lett.96, 047202 (2006)
  14. ^Verdiell, Marc (CuriousMarc) (31 October 2022).How an Atomic Clock Really Works, Round 2: Zeeman Alignment (YouTube video). Retrieved11 March 2023.
  15. ^Thalau, Peter; Ritz, Thorsten; Burda, Hynek; Wegner, Regina E.; Wiltschko, Roswitha (18 April 2006)."The magnetic compass mechanisms of birds and rodents are based on different physical principles".Journal of the Royal Society Interface.3 (9):583–587.doi:10.1098/rsif.2006.0130.PMC 1664646.PMID 16849254.
  16. ^abCandle flame is repelled by magnets (and Zeeman follow-up), retrieved27 February 2024
  17. ^Candle flame is repelled by magnets (and Zeeman follow-up), retrieved27 February 2024
  18. ^Kox, A J (1 May 1997)."The discovery of the electron: II. The Zeeman effect".European Journal of Physics.18 (3):139–144.Bibcode:1997EJPh...18..139K.doi:10.1088/0143-0807/18/3/003.ISSN 0143-0807.S2CID 53414643.
  19. ^Suzuki, Masatsugu Sei; Suzuki, Itsuko S. (2011)."Lecture Note on Senior Laboratory Zeeman effect in Na, Cd, and Hg".ResearchGate.

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