Zeeman effect - wikipedia Zeeman effect 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 a Fabry - Perot etalon. Zeeman splitting of the 5s level of Rb - 87, including fine structure and hyperfine structure splitting. Here F = J + I, where I is the nuclear spin. ( for Rb - 87, I = 3 / 2 ) Play media 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 polarisation effect is a powerful tool for astronomers to detect and measure stellar magnetic fields. The Zeeman effect ( / ˈzeɪmən / ; Dutch pronunciation : ( ˈzeːmɑn ) ), named after the Dutch physicist Pieter Zeeman, is the effect of splitting a spectral line into several components in the presence of a static 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. 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. 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. It may also be utilized to improve accuracy in atomic absorption spectroscopy. A theory about the magnetic sense of birds assumes that a protein in the retina is changed due to the Zeeman effect. When the spectral lines are absorption lines, the effect is called inverse Zeeman effect. Contents 1 Nomenclature 2 Theoretical presentation 3 Weak field ( Zeeman effect ) 3.1 Example : Lyman alpha transition in hydrogen 4 Strong field ( Paschen -- Back effect ) 5 Intermediate field for j = 1 / 2 6 Applications 6.1 Astrophysics 6.2 Laser cooling 7 See also 8 References 8.1 Historical 8.2 Modern 9 External links 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 an odd half - integer, so that the number of Zeeman sub-levels is even. 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. At higher magnetic fields the effect ceases to be linear. At even higher field strength, when the strength of the external field is comparable to the strength of the atom 's internal field, electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen - Back effect. In the modern scientific literature, these terms are rarely used, with a tendency to use just the `` 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_ ( M ), ) where H 0 ( displaystyle H_ ( 0 ) ) is the unperturbed Hamiltonian of the atom, and V M ( displaystyle V_ ( M ) ) is the perturbation due to the magnetic field : V M = − μ → ⋅ B →, ( displaystyle V_ ( 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 _ ( B ) g ( \vec ( J ) ) ) ( hbar ) ), ) where μ B ( displaystyle mu _ ( 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 _ ( B ) ( g_ ( l ) ( \vec ( L ) ) + g_ ( s ) ( \vec ( S ) ) ) ) ( hbar ) ), ) where g l = 1 ( displaystyle g_ ( l ) = 1 ) and g s ≈ 2.0023192 ( displaystyle g_ ( s ) approx 2.0023192 ) ( the latter is called the anomalous gyromagnetic ratio ; the deviation of the value from 2 is 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 ) ) = left langle sum _ ( i ) ( g_ ( l ) ( \vec ( l_ ( i ) ) ) + g_ ( s ) ( \vec ( s_ ( i ) ) ) ) \right \rangle = left langle ( g_ ( l ) ( \vec ( L ) ) + g_ ( s ) ( \vec ( S ) ) ) \right \rangle, ) where L → ( displaystyle ( \vec ( L ) ) ) and S → ( displaystyle ( \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. If the interaction term V M ( displaystyle V_ ( 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_ ( 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 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, L → ( displaystyle scriptstyle ( \vec ( L ) ) ) and S → ( displaystyle scriptstyle ( \vec ( S ) ) ) are not separately conserved, only the total angular momentum J → = L → + S → ( displaystyle scriptstyle ( \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 scriptstyle ( \vec ( J ) ) ). The ( time - ) `` averaged '' spin vector is then the projection of the spin onto the direction of J → ( displaystyle scriptstyle ( \vec ( J ) ) ) : S → a v g = ( S → ⋅ J → ) J 2 J → ( displaystyle ( \vec ( S ) ) _ ( avg ) = ( \frac ( ( ( \vec ( S ) ) cdot ( \vec ( J ) ) ) ) ( J ^ ( 2 ) ) ) ( \vec ( J ) ) ) and for the ( time - ) `` averaged '' orbital vector : L → a v g = ( L → ⋅ J → ) J 2 J →. ( displaystyle ( \vec ( L ) ) _ ( 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_ ( M ) \rangle = ( \frac ( mu _ ( 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 scriptstyle ( \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 scriptstyle ( \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 scriptstyle 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_ ( M ) & = mu _ ( 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 _ ( B ) Bm_ ( j ) left ( 1 + ( g_ ( S ) - 1 ) ( \frac ( j ( j + 1 ) - l ( l + 1 ) + s ( s + 1 ) ) ( 2j ( j + 1 ) ) ) \right ), & = mu _ ( B ) Bm_ ( j ) g_ ( j ) end ( aligned ) ) ) where the quantity in square brackets is the Landé g - factor g of the atom ( g L = 1 ( displaystyle g_ ( L ) = 1 ) and 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 s = 1 / 2 ( displaystyle s = 1 / 2 ) and j = l ± s ( displaystyle j = l pm s ), the Landé g - factor can be simplified into : 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_ ( 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 e x t m j ( displaystyle ( \begin ( aligned ) E_ ( Z ) ^ ( ( 1 ) ) = langle nljm_ ( j ) H_ ( Z ) ^ ( ' ) nljm_ ( j ) \rangle = langle V_ ( M ) \rangle _ ( Psi ) = mu _ ( B ) g_ ( J ) B_ ( ext ) m_ ( j ) end ( aligned ) ) ) Example : lyman alpha transition in Hydrogen The Lyman alpha transition in hydrogen in the presence of the spin - orbit interaction involves the transitions 2 P 1 / 2 → 1 S 1 / 2 ( displaystyle 2P_ ( 1 / 2 ) \to 1S_ ( 1 / 2 ) ) and 2 P 3 / 2 → 1 S 1 / 2. ( displaystyle 2P_ ( 3 / 2 ) \to 1S_ ( 1 / 2 ). ) In the presence of an external magnetic field, the weak - field Zeeman effect splits the 1S and 2P levels into 2 states each ( m j = 1 / 2, − 1 / 2 ( displaystyle m_ ( j ) = 1 / 2, - 1 / 2 ) ) and the 2P 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 ( displaystyle g_ ( J ) = 2 ) for 1 S 1 / 2 ( displaystyle 1S_ ( 1 / 2 ) ) ( j = 1 / 2, l = 0 ) g J = 2 / 3 ( displaystyle g_ ( J ) = 2 / 3 ) for 2 P 1 / 2 ( displaystyle 2P_ ( 1 / 2 ) ) ( j = 1 / 2, l = 1 ) g J = 4 / 3 ( displaystyle g_ ( J ) = 4 / 3 ) for 2 P 3 / 2 ( displaystyle 2P_ ( 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 values are different. On the left, fine structure splitting is depicted. This 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 _ ( B ) ) ( hbar ) ) ( L_ ( z ) + g_ ( s ) S_ ( z ) ) \right psi \right \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. 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 ( displaystyle Delta E = B mu _ ( B ) Delta m_ ( l ) ) is independent of the unperturbed energies and electronic configurations of the levels being considered. It should be noted that in general ( if s ≠ 0 ( displaystyle s \neq 0 ) ), these three components are actually groups of several transitions each, due to the residual spin - orbit coupling. In general, one must now add spin - orbit coupling and relativistic corrections ( which are of the same order, known as ' fine structure ' ) as a perturbation to these ' unperturbed ' levels. First order perturbation theory with these fine - structure corrections yields the following formula for the Hydrogen atom in the Paschen -- Back limit : E z + f s = E z + α 2 2 n 3 ( 3 4 n − ( l ( l + 1 ) − m l m s l ( l + 1 / 2 ) ( l + 1 ) ) ). ( displaystyle E_ ( z + fs ) = E_ ( z ) + ( \frac ( alpha ^ ( 2 ) ) ( 2n ^ ( 3 ) ) ) left ( ( \frac ( 3 ) ( 4n ) ) - left ( ( \frac ( l ( l + 1 ) - m_ ( l ) m_ ( s ) ) ( l ( l + 1 / 2 ) ( l + 1 ) ) ) \right ) \right ). ) Intermediate field for J = 1 / 2 In the magnetic dipole approximation, the Hamiltonian which includes both the hyperfine and Zeeman interactions is H = h A I → ⋅ J → − μ → ⋅ B → ( displaystyle H = hA ( \vec ( I ) ) cdot ( \vec ( J ) ) - ( \vec ( mu ) ) cdot ( \vec ( B ) ) ) H = h A I → ⋅ J → + μ B ( g J J → + g I I → ) ⋅ B → ( displaystyle H = hA ( \vec ( I ) ) cdot ( \vec ( J ) ) + mu _ ( B ) ( g_ ( J ) ( \vec ( J ) ) + g_ ( I ) ( \vec ( I ) ) ) cdot ( \vec ( B ) ) ) To arrive at the Breit -- Rabi formula, include the hyperfine structure ( interaction between the electron 's spin and the magnetic moment of the nucleus ), which is governed by the quantum number F ≡ F → = J → + I → ( displaystyle F equiv ( \vec ( F ) ) = ( \vec ( J ) ) + ( \vec ( I ) ) ), where I → ( displaystyle ( \vec ( I ) ) ) is the spin angular momentum operator of the nucleus. Alternatively, the derivation could be done with J ( displaystyle J ) only. The constant A ( displaystyle A ) is known as the zero field hyperfine constant and is given in units of hertz. μ B ( displaystyle mu _ ( B ) ) is the Bohr magneton. ħ J → ( displaystyle hbar ( \vec ( J ) ) ) and ħ I → ( displaystyle hbar ( \vec ( I ) ) ) are the electron and nuclear angular momentum operators. g J ( displaystyle g_ ( J ) ) and g F ( displaystyle g_ ( F ) ) can be found via a classical vector coupling model or a more detailed quantum mechanical calculation to be : g 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 ) ( 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 ) ) ) ) g F = g J F ( F + 1 ) + J ( J + 1 ) − I ( I + 1 ) 2 F ( F + 1 ) + g I F ( F + 1 ) − J ( J + 1 ) + I ( I + 1 ) 2 F ( F + 1 ) ( displaystyle g_ ( F ) = g_ ( J ) ( \frac ( F ( F + 1 ) + J ( J + 1 ) - I ( I + 1 ) ) ( 2F ( F + 1 ) ) ) + g_ ( I ) ( \frac ( F ( F + 1 ) - J ( J + 1 ) + I ( I + 1 ) ) ( 2F ( F + 1 ) ) ) ). As discussed, in the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the F, m f ⟩ ( 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, m I, m J ⟩ ( displaystyle I, J, m_ ( I ), m_ ( J ) \rangle ) or just m I, m J ⟩ ( displaystyle m_ ( I ), m_ ( J ) \rangle ) since I ( displaystyle I ) and J ( 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, m F ⟩ ( displaystyle F, m_ ( F ) \rangle ) and m I, m J ⟩ ( displaystyle m_ ( I ), m_ ( J ) \rangle ) basis states. For J = 1 / 2 ( displaystyle J = 1 / 2 ), the Hamiltonian can be solved analytically, resulting in the Breit - Rabi formula. Notably, the electric quadrupole interaction is zero for L = 0 ( displaystyle L = 0 ) ( J = 1 / 2 ( displaystyle J = 1 / 2 ) ), so this formula is fairly accurate. To solve this system, we note that at all times, the total angular momentum projection m F = m J + m I ( displaystyle m_ ( F ) = m_ ( J ) + m_ ( I ) ) will be conserved. Furthermore, since J = 1 / 2 ( displaystyle J = 1 / 2 ) between states m J ( displaystyle m_ ( J ) ) will change between only ± 1 / 2 ( displaystyle pm 1 / 2 ). Therefore, we can define a good basis as : ± ⟩ ≡ m J = ± 1 / 2, m I = m F ∓ 1 / 2 ⟩ ( displaystyle pm \rangle equiv m_ ( J ) = pm 1 / 2, m_ ( I ) = m_ ( F ) mp 1 / 2 \rangle ) We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator L ( displaystyle L ) as L ± ≡ L x ± i L y ( displaystyle L_ ( pm ) equiv L_ ( x ) pm iL_ ( y ) ) These ladder operators have the property L ± L, m L ⟩ = ( L ∓ m L ) ( L ± m L + 1 ) L, m L ± 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 as m L ( displaystyle m_ ( L ) ) lies in the range − L,......, L ( displaystyle ( - L, dots..., L ) ) ( otherwise, they return zero ). Using ladder operators J ± ( displaystyle J_ ( pm ) ) and I ± ( displaystyle I_ ( pm ) ) We can rewrite the Hamiltonian as H = h A I z J z + h A 2 ( J + I − + J − I + ) + μ B B ( g J J z + g I I Z ) ( displaystyle H = hAI_ ( z ) J_ ( z ) + ( \frac ( hA ) ( 2 ) ) ( J_ ( + ) I_ ( - ) + J_ ( - ) I_ ( + ) ) + mu _ ( B ) B ( g_ ( J ) J_ ( z ) + g_ ( I ) I_ ( Z ) ) ) Now we can determine the matrix elements of the Hamiltonian : ⟨ ± H ± ⟩ = − 1 4 h A + μ B B g I m F ± 1 2 ( h A m F + μ B B ( g J − g I ) ) ( displaystyle langle pm H pm \rangle = - ( \frac ( 1 ) ( 4 ) ) hA+ mu _ ( B ) Bg_ ( I ) m_ ( F ) pm ( \frac ( 1 ) ( 2 ) ) ( hAm_ ( F ) + mu _ ( B ) B ( g_ ( J ) - g_ ( I ) ) ) ) ⟨ ± H ∓ ⟩ = 1 2 h A ( I + 1 / 2 ) 2 − m F 2 ( 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, or more easily, with a computer algebra system ) we arrive at the energy shifts : Δ E F = I ± 1 / 2 = − h Δ W 2 ( 2 I + 1 ) + μ B g I m F B ± h Δ W 2 1 + 2 m F x I + 1 / 2 + x 2 ( displaystyle Delta E_ ( F = I pm 1 / 2 ) = - ( \frac ( h Delta W ) ( 2 ( 2I + 1 ) ) ) + mu _ ( B ) g_ ( I ) m_ ( F ) B pm ( \frac ( h Delta W ) ( 2 ) ) ( sqrt ( 1 + ( \frac ( 2m_ ( F ) x ) ( I + 1 / 2 ) ) + x ^ ( 2 ) ) ) ) x ≡ μ B B ( g J − g I ) h Δ W Δ W = A ( I + 1 2 ) ( displaystyle x equiv ( \frac ( mu _ ( B ) B ( g_ ( J ) - g_ ( I ) ) ) ( h Delta W ) ) quad quad Delta W = A left ( I+ ( \frac ( 1 ) ( 2 ) ) \right ) ) where Δ W ( displaystyle Delta W ) is the splitting ( in units of Hz ) between two hyperfine sublevels in the absence of magnetic field B ( displaystyle B ), x ( displaystyle x ) is referred to as the ' field strength parameter ' ( Note : for m = − ( I + 1 / 2 ) ( displaystyle m = - ( I + 1 / 2 ) ) the square root is an exact square, and should be interpreted as + ( 1 − x ) ( displaystyle + ( 1 - x ) ) ). This equation is known as the Breit - Rabi formula and is useful for systems with one valence electron in an s ( displaystyle s ) ( J = 1 / 2 ( displaystyle J = 1 / 2 ) ) level. Note that index F ( displaystyle F ) in Δ E F = I ± 1 / 2 ( displaystyle Delta E_ ( F = I pm 1 / 2 ) ) 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 B = 0 ( displaystyle B = 0 ) otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different F ( displaystyle F ) but equal m F ( displaystyle m_ ( F ) ) ( the only exceptions are F = I + 1 / 2, m F = ± F ⟩ ( displaystyle F = I + 1 / 2, m_ ( F ) = pm F \rangle ) ). Applications Astrophysics 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 0.1 tesla or higher. Today, the Zeeman effect is used to produce magnetograms showing the variation of magnetic field on the sun. Laser cooling The Zeeman effect is utilized in many laser cooling applications such as a magneto - optical trap and the Zeeman slower. See also Magneto - optic Kerr effect Voigt effect Faraday effect Cotton - Mouton effect Polarization spectroscopy Zeeman energy Stark effect Lamb shift Electron configuration says at subshell p ( l = 1 ), there are 3 energy level ml = - 1, 0, 1, but we see only two p1 / 2 and p3 / 2. for subshell s ( l = 0 ), there is only 1 energy level ( ml = 0 ), but here we have 2. l corresponding to fine structure, ml corresponding to hyperfine structure. References Jump up ^ The magnetic compass mechanisms of birds and rodents are based on different physical principles. Journal of the Royal Society Jump up ^ T. Preston, `` Radiation Phenomena in a Strong Magnetic Field `` Transactions Royal Dublin Society, 6 ( 1898 ) 385 - 91 Jump up ^ Paschen, F., Back, E. : Liniengruppen magnetisch vervollständigt. Physica 1, 261 -- 273 ( 1921 ). Jump up ^ Griffiths, David J. ( 2004 ). Introduction to Quantum Mechanics ( 2nd ed. ). Prentice Hall. p. 247. ISBN 0 - 13 - 111892 - 7. OCLC 40251748. Jump up ^ Woodgate, Elementary Atomic Structure, section 9. Jump up ^ first appeared in G. Breit and I. Rabi, Phys. Rev. 38, 2082 ( 1931 ). 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. ( Google Books ) Zeeman, P. ( 1897 ). `` Doubles and triplets in the spectrum produced by external magnetic forces ''. Phil. Mag. 44 ( 266 ) : 55. doi : 10.1080 / 14786449708621028. ( Google Books ) Zeeman, P. ( 11 February 1897 ). `` The Effect of Magnetisation on the Nature of Light Emitted by a Substance ''. Nature. 55 ( 1424 ) : 347. Bibcode : 1897Natur... 55... 347Z. doi : 10.1038 / 055347a0. Modern Feynman, Richard P., Leighton, Robert B., Sands, Matthew ( 1965 ). The Feynman Lectures on Physics. 3. Addison - Wesley. ISBN 0 - 201 - 02115 - 3. CS1 maint : Multiple names : authors list ( link ) Forman, Paul ( 1970 ). `` Alfred Landé and the anomalous Zeeman Effect, 1919 - 1921 ''. Historical Studies in the Physical Sciences. 2 : 153 -- 261. doi : 10.2307 / 27757307. 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. Sobelman, Igor I. ( 2006 ). Theory of Atomic Spectra. Alpha Science. ISBN 1 - 84265 - 203 - 6. Foot, C.J. ( 2005 ). Atomic Physics. ISBN 0 - 19 - 850696 - 1. External links Wikimedia Commons has media related to Zeeman effect. Zeeman Effect Apparatus Manufacturer Retrieved from `` https://en.wikipedia.org/w/index.php?title=Zeeman_effect&oldid=827058958 '' Categories : Spectroscopy Quantum magnetism Foundational quantum physics Hidden categories : CS1 : Julian -- Gregorian uncertainty CS1 maint : Multiple names : authors list Use dmy dates from September 2010 Articles containing video clips Talk Contents About Wikipedia বাংলা Беларуская Български Català Čeština Dansk Deutsch Eesti Español Esperanto فارسی Français Gaeilge Galego 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Italiano עברית Magyar മലയാളം မြန်မာဘာသာ Nederlands 日本 語 Norsk Norsk nynorsk Oʻzbekcha / ўзбекча Polski Português Română Русский Scots Slovenčina Slovenščina Српски / srpski Suomi Svenska தமிழ் Татарча / tatarça ไทย Türkçe Українська Tiếng Việt 中文 Edit links This page was last edited on 22 February 2018, at 14 : 56. 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Question: according to the zeeman effect the splitting of a sunspot's spectral lines is due to
Answer: the presence of a static magnetic field