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Interference fringes, showing fine structure (splitting) of a cooled deuterium source, viewed through a Fabry–Pérot interferometer.

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887^[1] laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.^[2]

Background

Gross structure

The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum number n. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the energy levels and split the spectral lines. The scale of the fine structure splitting relative to the gross structure energies is on the order of (Zα)², where Z is the atomic number and α is the fine-structure constant, a dimensionless number equal to approximately 1/137.

Relativistic corrections

The fine structure energy corrections can be obtained by using perturbation theory. To perform this calculation one must add the three corrective terms to the Hamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to the spin-orbit coupling, and the Darwin term coming from the quantum fluctuating motion or zitterbewegung of the electron.

These corrections can also be obtained from the non-relativistic limit of the Dirac equation, since Dirac's theory naturally incorporates relativity and spin interactions.

The hydrogen atom

This section discusses the analytical solutions for the hydrogen atom as the problem is analytically solvable and is the base model for energy level calculations in more complex atoms.

Kinetic energy relativistic correction

Classically, the kinetic energy term of the Hamiltonian is

T=p22medisplaystyle T=frac p^22m_e

where pdisplaystyle p is the momentum and medisplaystyle m_e is the electron rest mass.

However, when considering a more accurate theory of nature via special relativity, we must use a relativistic form of the kinetic energy,

T=p2c2+me2c4−mec2displaystyle T=sqrt p^2c^2+m_e^2c^4-m_ec^2

where the first term is the total relativistic energy, and the second term is the rest energy of the electron (cdisplaystyle c is the speed of light). Expanding this in a Taylor series ( specifically a binomial series ), we find

T=p22me−p48me3c2+⋯displaystyle T=frac p^22m_e-frac p^48m_e^3c^2+cdots

Then, the first order correction to the Hamiltonian is

Hkinetic=−p48me3c2displaystyle H_mathrm kinetic =-frac p^48m_e^3c^2

Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.

En(1)=⟨ψ0|H′|ψ0⟩=−18me3c2⟨ψ0|p4|ψ0⟩=−18me3c2⟨ψ0|p2p2|ψ0⟩displaystyle E_n^(1)=leftlangle psi ^0rightvert H'leftvert psi ^0rightrangle =-frac 18m_e^3c^2leftlangle psi ^0rightvert p^4leftvert psi ^0rightrangle =-frac 18m_e^3c^2leftlangle psi ^0rightvert p^2p^2leftvert psi ^0rightrangle

where ψ0displaystyle psi ^0 is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see

H0|ψ0⟩=En|ψ0⟩(p22me+V)|ψ0⟩=En|ψ0⟩p2|ψ0⟩=2me(En−V)|ψ0⟩displaystyle beginalignedH^0leftvert psi ^0rightrangle &=E_nleftvert psi ^0rightrangle \left(frac p^22m_e+Vright)leftvert psi ^0rightrangle &=E_nleftvert psi ^0rightrangle \p^2leftvert psi ^0rightrangle &=2m_e(E_n-V)leftvert psi ^0rightrangle endaligned

We can use this result to further calculate the relativistic correction:

En(1)=−18me3c2⟨ψ0|p2p2|ψ0⟩En(1)=−18me3c2⟨ψ0|(2me)2(En−V)2|ψ0⟩En(1)=−12mec2(En2−2En⟨V⟩+⟨V2⟩)displaystyle beginalignedE_n^(1)&=-frac 18m_e^3c^2leftlangle psi ^0rightvert p^2p^2leftvert psi ^0rightrangle \E_n^(1)&=-frac 18m_e^3c^2leftlangle psi ^0rightvert (2m_e)^2(E_n-V)^2leftvert psi ^0rightrangle \E_n^(1)&=-frac 12m_ec^2left(E_n^2-2E_nlangle Vrangle +leftlangle V^2rightrangle right)endaligned

For the hydrogen atom,

V(r)=−e24πε0rdisplaystyle V(r)=frac -e^24pi varepsilon _0r, ⟨1r⟩=1a0n2displaystyle leftlangle frac 1rrightrangle =frac 1a_0n^2, and ⟨1r2⟩=1(l+1/2)n3a02displaystyle leftlangle frac 1r^2rightrangle =frac 1(l+1/2)n^3a_0^2 ,

where edisplaystyle e is the elementary charge , ε0displaystyle varepsilon _0 is the vacuum permittivity, a0displaystyle a_0 is the Bohr Radius, ndisplaystyle n is the principal quantum number, ldisplaystyle l is the azimuthal quantum number and rdisplaystyle r is the distance of the electron from the nucleus. Therefore, the first order relativistic correction for the hydrogen atom is

En(1)=−12mec2(En2+2Ene24πε01a0n2+116π2ε02e4(l+12)n3a02)=−En22mec2(4nl+12−3)displaystyle beginalignedE_n^(1)&=-frac 12m_ec^2left(E_n^2+2E_nfrac e^24pi varepsilon _0frac 1a_0n^2+frac 116pi ^2varepsilon _0^2frac e^4(l+frac 12)n^3a_0^2right)\&=-frac E_n^22m_ec^2left(frac 4nl+frac 12-3right)endaligned

where we have used:

En=−e28πε0a0n2displaystyle E_n=-frac e^28pi varepsilon _0a_0n^2

On final calculation, the order of magnitude for the relativistic correction to the ground state is −9.056×10−4 eVdisplaystyle -9.056times 10^-4 texteV.

Spin-orbit coupling

For a hydrogen-like atom with Zdisplaystyle Z protons (Z=1displaystyle Z=1for hydrogen), orbital angular momentum L→displaystyle vec L and electron spin S→displaystyle vec S, the spin-orbit term is given by:

HSO=12(Ze24πε0)(gs2me2c2)L→⋅S→r3displaystyle H_mathrm SO =frac 12left(frac Ze^24pi varepsilon _0right)left(frac g_s2m_e^2c^2right)frac vec Lcdot vec Sr^3

where gsdisplaystyle g_s is the spin g-factor.

The spin-orbit correction can be understood by shifting from the standard frame of reference (where the electron orbits the nucleus) into one where the electron is stationary and the nucleus instead orbits it. In this case the orbiting nucleus functions as an effective current loop, which in turn will generate a magnetic field. However, the electron itself has a magnetic moment due to its intrinsic angular momentum. The two magnetic vectors, B→displaystyle vec B and μ→sdisplaystyle vec mu _s couple together so that there is a certain energy cost depending on their relative orientation. This gives rise to the energy correction of the form

ΔESO=ξ(r)L→⋅S→displaystyle Delta E_mathrm SO =xi (r)vec Lcdot vec S

Notice that an important factor of 2 has to be added to the calculation, called the Thomas precession, which comes from the relativistic calculation that changes back to the electron's frame from the nucleus frame.

Since

⟨1r3⟩=Z3n3a031l(l+12)(l+1)⟨L→⋅S→⟩=ℏ22[j(j+1)−l(l+1)−s(s+1)]displaystyle beginalignedleftlangle frac 1r^3rightrangle &=frac Z^3n^3a_0^3frac 1lleft(l+frac 12right)(l+1)\leftlangle vec Lcdot vec Srightrangle &=frac hbar ^22[j(j+1)-l(l+1)-s(s+1)]endaligned

the expectation value for the Hamiltonian is:

⟨HSO⟩=En2mec2 n j(j+1)−l(l+1)−34l(l+12)(l+1)displaystyle leftlangle H_mathrm SO rightrangle =frac E_n^2m_ec^2~n~frac j(j+1)-l(l+1)-frac 34lleft(l+frac 12right)(l+1)

Thus the order of magnitude for the spin-orbital coupling is Z4n3(j+1/2)10−5 eVdisplaystyle frac Z^4n^3(j+1/2)10^-5text eV.

When weak external magnetic fields are applied, the spin-orbit coupling contributes to the Zeeman effect.

Darwin term

There is one last term in the non-relativistic expansion of the Dirac equation. It is referred to as the Darwin term, as it was first derived by Charles Galton Darwin, and is given by:

HDarwin=ℏ28me2c24π(Ze24πε0)δ3(r→)⟨HDarwin⟩=ℏ28me2c24π(Ze24πε0)|ψ(0)|2ψ(0)=0 for l>0ψ(0)=14π2(Zna0)32 for l=0HDarwin=2nmec2En2displaystyle beginalignedH_mathrm Darwin &=frac hbar ^28m_e^2c^2,4pi left(frac Ze^24pi varepsilon _0right)delta ^3left(vec rright)\langle H_mathrm Darwin rangle &=frac hbar ^28m_e^2c^2,4pi left(frac Ze^24pi varepsilon _0right)

The Darwin term affects only the s orbitals. This is because the wave function of an electron with l>0displaystyle l>0 vanishes at the origin, hence the delta function has no effect. For example, it gives the 2s orbital the same energy as the 2p orbital by raising the 2s state by 6977145109124427590♠9.057×10⁻⁵ eV.

The Darwin term changes the effective potential at the nucleus. It can be interpreted as a smearing out of the electrostatic interaction between the electron and nucleus due to zitterbewegung, or rapid quantum oscillations, of the electron. This can be demonstrated by a short calculation.^[3]

Quantum fluctuations allow for the creation of virtual electron-positron pairs with a lifetime estimated by the uncertainty principle Δt≈ℏ/ΔE≈ℏ/mc2displaystyle Delta tapprox hbar /Delta Eapprox hbar /mc^2. The distance the particles can move during this time is ξ≈cΔt≈ℏ/mc=λcdisplaystyle xi approx cDelta tapprox hbar /mc=lambda _c, the Compton wavelength. The electrons of the atom interact with those pairs. This yields a fluctuating electron position r→+ξ→displaystyle vec r+vec xi . Using a Taylor expansion, the effect on the potential Udisplaystyle U can be estimated:

U(r→+ξ→)≈U(r→)+ξ⋅∇U(r→)+12∑ijξiξj∂i∂jU(r→)displaystyle U(vec r+vec xi )approx U(vec r)+xi cdot nabla U(vec r)+frac 12sum _ijxi _ixi _jpartial _ipartial _jU(vec r)

Averaging over the fluctuations ξ→displaystyle vec xi

ξ¯=0,ξiξj¯=13ξ→2¯δij,displaystyle overline xi =0,quad overline xi _ixi _j=frac 13overline vec xi ^2delta _ij,

gives the average potential

U(r→+ξ→)¯=U(r→)+16ξ→2¯∇2U(r→).displaystyle overline Uleft(vec r+vec xi right)=Uleft(vec rright)+frac 16overline vec xi ^2nabla ^2Uleft(vec rright).

Approximating ξ→2¯≈λc2displaystyle overline vec xi ^2approx lambda _c^2, this yields the perturbation of the potential due to fluctuations:

δU≈16λc2∇2U=ℏ26me2c2∇2Udisplaystyle delta Uapprox frac 16lambda _c^2nabla ^2U=frac hbar ^26m_e^2c^2nabla ^2U

To compare with the expression above, plug in the Coulomb potential:

∇2U=−∇2Ze24πε0r=4π(Ze24πε0)δ(r→)⇒δU≈ℏ26me2c24π(Ze24πε0)δ(r→)displaystyle nabla ^2U=-nabla ^2frac Ze^24pi varepsilon _0r=4pi left(frac Ze^24pi varepsilon _0right)delta (vec r)quad Rightarrow quad delta Uapprox frac hbar ^26m_e^2c^24pi left(frac Ze^24pi varepsilon _0right)delta (vec r)

This is only slightly different.

Another mechanism that affects only the s-state is the Lamb shift, a further, smaller correction that arises in quantum electrodynamics that should not be confused with the Darwin term. The Darwin term gives the s-state and p-state the same energy, but the Lamb shift makes the s-state higher in energy than the p-state.

Total effect

The full Hamiltonian is given by

H=HCoulomb+Hkinetic+HSO+HDarwin,displaystyle H=H_rm Coulomb+H_mathrm kinetic +H_mathrm SO +H_mathrm Darwin ,!

where HCoulombdisplaystyle H_rm Coulomb is the Hamiltonian from the Coulomb interaction.

The total effect, obtained by summing the three components up, is given by the following expression:^[4]

ΔE=En(Zα)2n(1j+12−34n),displaystyle Delta E=frac E_n(Zalpha )^2nleft(frac 1j+frac 12-frac 34nright),,

where jdisplaystyle j is the total angular momentum (j=1/2displaystyle j=1/2 if l=0displaystyle l=0 and j=l±1/2displaystyle j=lpm 1/2 otherwise). It is worth noting that this expression was first obtained by Sommerfeld based on the old Bohr theory; i.e., before the modern quantum mechanics was formulated.

Relativistic corrections (Dirac) to the energy levels of a hydrogen atom from Bohr's model. The fine structure correction predicts that the Lyman-alpha line (emitted in a transition from n=2 to n=1) must split into a doublet.

Exact relativistic energies

The total effect can also be obtained by using the Dirac equation. In this case, the electron is treated as non-relativistic. The exact energies are given by^[5]

Ejn=−mec2[1−(1+[αn−j−12+(j+12)2−α2]2)−1/2].displaystyle E_j,n=-m_textec^2left[1-left(1+left[dfrac alpha n-j-frac 12+sqrt left(j+frac 12right)^2-alpha ^2right]^2right)^-1/2right].

This expression, which contains all higher order terms that were left out in the other calculations, expands to first order to give the energy corrections derived from perturbation theory. However, this equation does not contain the hyperfine structure corrections, which are due to interactions with the nuclear spin. Other corrections from quantum field theory such as the Lamb shift and the anomalous magnetic dipole moment of the electron are not included.

See also

• Angular momentum coupling


• Fine electronic structure

References

• 
  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.
  

External links

• Hyperphysics: Fine Structure


• University of Texas: The fine structure of hydrogen