Fine structure

Short description: Details in the emission spectrum of an atom

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][2][3] The explanation was first given by Niels Bohr in 1914, who suggested that the orbits of electrons in his Bohr model of the atom precessed due to relativistic effects. A successful relativistic formula was given by Arnold Sommerfeld in 1916.[4][5] In the same work, Sommerfeld also introduced the fine-structure constant.[6][7]

## Contents

* 1 Background
* 1.1 Gross structure
* 1.2 Relativistic corrections
* 2 Hydrogen atom
* 2.1 Kinetic energy relativistic correction
* 2.2 Spin–orbit coupling
* 2.3 Darwin term
* 2.4 Total effect
* 2.5 Exact relativistic energies
* 3 See also
* 4 References
* 5 External links

## Background

### Gross structure

The gross structure of line spectra is the structure 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 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.

## 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

The gross structure assumes the kinetic energy term of the Hamiltonian takes the same form as in classical mechanics, which for a single electron means ℋ0=p22me+V where V is the potential energy, p is the momentum, and me 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−mec2=mec2[1+p2me2c2−1] where the first term is the total relativistic energy, and the second term is the rest energy of the electron (c is the speed of light). To further simplify, we note that the term under the square root can be written of the form (1+x)p where the value of x satisfies the conditions to approximate the entire square root term as the first two terms of its Taylor series expansion. Expanding the square root and simplifying yields, T=p22me−p48me3c2+⋯

Although there are an infinite number of terms in this series, the later terms are much smaller than earlier terms, and so we can ignore all but the first two. Since the first term above is already part of the classical Hamiltonian, the first order correction to the Hamiltonian is ℋ′=−p48me3c2

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⟩ where ψ0 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⟩

We can use this result to further calculate the relativistic correction: En(1)=−18me3c2⟨ψ0|p2p2|ψ0⟩=−18me3c2⟨ψ0|(2me)2(En−V)2|ψ0⟩=−12mec2(En2−2En⟨V⟩+⟨V2⟩)

For the hydrogen atom, V(r)=−e24πε0r, ⟨1r⟩=1a0n2, and ⟨1r2⟩=1(ℓ+1/2)n3a02, where e is the elementary charge, ε0 is the vacuum permittivity, a0 is the Bohr radius, n is the principal quantum number, ℓ is the azimuthal quantum number and 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) where we have used: En=−e28πε0a0n2

On final calculation, the relativistic correction to the ground state is −9.056×10−4 eV. In general, En(1)=−1.81×10−4eV×1n4(4nℓ+12−3).

### Spin–orbit coupling

Main article: Physics:Spin–orbit interaction

For a hydrogen-like atom with Z protons (Z=1 for hydrogen), orbital angular momentum 𝐋 and electron spin 𝐒, the spin–orbit term is given by: ℋSO=(Ze24πε0)(gs−12me2c2)𝐋⋅𝐒r3 where gs 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 and μs 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[8] ΔESO=ξ(r)𝐋⋅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) by Kramers–Pasternack recursion relations (named after Hans Kramers and Simon Pasternack) and ⟨𝐋⋅𝐒⟩=ℏ22[j(j+1)−ℓ(ℓ+1)−s(s+1)] the expectation value for the Hamiltonian is: ⟨ℋSO⟩=En2mec2nj(j+1)−ℓ(ℓ+1)−34ℓ(ℓ+12)(ℓ+1)

Thus the order of magnitude for the spin–orbital coupling is: Z4n3(j+12)(j+1)×10−4 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>014π2(Zna0)32 for ℓ=0ℋDarwin=2nmec2En2

The Darwin term affects only the s orbitals. This is because the wave function of an electron with 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 9.057×10−5 eV.

The Darwin term changes potential energy of the electron. 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.[9]

Quantum fluctuations allow for the creation of virtual electron–positron pairs with a lifetime estimated by the uncertainty principle Δt≈ℏ/ΔE≈ℏ/mc2. The distance the particles can move during this time is ξ≈cΔt≈ℏ/mc=λ¯C, their reduced Compton wavelength. The electrons of the atom interact with those pairs. This yields a fluctuating electron position 𝐫+ξ. Using a Taylor expansion, the effect on the potential U can be estimated: U(𝐫+ξ)≈U(𝐫)+ξ⋅∇U(𝐫)+12∑i,jξiξj∂i∂jU(𝐫)

Averaging over the fluctuations ξ ξ‾=0,ξiξj‾=13ξ2‾δij, gives the average potential U(𝐫+ξ)‾=U(𝐫)+16ξ2‾∇2U(𝐫).

Approximating ξ2‾≈λ¯C2, this yields the perturbation of the potential due to fluctuations: δU≈16λ¯C2∇2U=ℏ26me2c2∇2U

To compare with the expression above, plug in the Coulomb potential: ∇2U=−∇2Ze24πε0r=4π(Ze24πε0)δ3(𝐫)⇒δU≈ℏ26me2c24π(Ze24πε0)δ3(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, where ℋCoulomb is the Hamiltonian from the Coulomb interaction.

The total effect, obtained by summing the three components up, is given by the following expression:[10] ΔE=En(Zα)2n(1j+12−34n), where j is the total angular momentum quantum number (j=1/2 if ℓ=0 and j=ℓ±1/2 otherwise). It is worth noting that this expression was first obtained by Sommerfeld based on the old quantum theory; i.e., before modern quantum mechanics was reformulated by Werner Heisenberg and Erwin Schrödinger.

Energy diagram (to scale) of the hydrogen atom for n = 2 corrected by the fine structure and magnetic field. First column shows the non-relativistic case (only kinetic energy and Coulomb potential), the relativistic correction to the kinetic energy is added in the second column, the third column includes all of the fine structure, and the fourth adds the Zeeman effect (magnetic field dependence).

### Exact relativistic energies

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.

The total effect can also be obtained by using the Dirac equation. The exact energies are given by[11] Ejn=−mec2[1−(1+[αn−j−12+(j+12)2−α2]2)−12].

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

1. ↑ A.A. Michelson; E. W. Morley (1887). "On a method of making the wave-length of sodium light the actual practical standard of length". American Journal of Science 34: 427. https://archive.org/details/americanjourna3341887newh/page/427.
2. ↑ A.A. Michelson; E. W. Morley (1887). "On a method of making the wave-length of sodium light the actual practical standard of length". Philosophical Magazine 24: 463. https://archive.org/details/s5philosophicalm24lond/page/463.
3. ↑ Weinberg, Steven (1995-06-30) (in en). The Quantum Theory of Fields. Cambridge University Press. ISBN 978-0-521-55001-7. https://www.google.fr/books/edition/The_Quantum_Theory_of_Fields/doeDB3_WLvwC?hl=en&gbpv=1&dq=weinberg+quantum+field+theory&printsec=frontcover.
4. ↑ "Nobel Prize in Physics 1955" (in en-US). https://www.nobelprize.org/prizes/physics/1955/lamb/lecture/.
5. ↑ (in en) Niels Bohr's Times: In Physics, Philosophy, and Polity. Oxford University Press. 1991-10-17. ISBN 978-0-19-252230-6. https://www.google.fr/books/edition/Niels_Bohr_s_Times/b-VBDgAAQBAJ?hl=en&gbpv=1&dq=michelson+sommerfeld+fine+structure+hydrogen+1916&pg=PA184&printsec=frontcover.
6. ↑ A.Sommerfeld (July 1940). "Zur Feinstruktur der Wasserstofflinien. Geschichte und gegenwärtiger Stand der Theorie" (in german). Naturwissenschaften 28 (27): 417–423. doi:10.1007/BF01490583. Bibcode: 1940NW.....28..417S.
7. ↑ (in en) Niels Bohr's Times: In Physics, Philosophy, and Polity. Oxford University Press. 1991-10-17. ISBN 978-0-19-252230-6. https://www.google.fr/books/edition/Niels_Bohr_s_Times/b-VBDgAAQBAJ?hl=en&gbpv=1&dq=michelson+sommerfeld+fine+structure+hydrogen+1916&pg=PA184&printsec=frontcover.
8. ↑ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. pp. 310–311. ISBN 0-205-12770-3.
9. ↑ Zelevinsky, Vladimir (2011), Quantum Physics Volume 1: From Basics to Symmetries and Perturbations, WILEY-VCH, p. 551, ISBN 978-3-527-40979-2
10. ↑ Berestetskii, V. B.; E. M. Lifshitz; L. P. Pitaevskii (1982). Quantum electrodynamics. Butterworth-Heinemann. ISBN 978-0-7506-3371-0.
11. ↑ Sommerfeld, Arnold (1919). Atombau und Spektrallinien. Braunschweig: Friedrich Vieweg und Sohn. ISBN 3-87144-484-7. German English

* Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. https://archive.org/details/introductiontoel00grif_0.
* 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

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