Fine structure explained

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]} laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.^[3]

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$$\backslash mathcal^0\; =\; \backslash frac\; +\; V$$where is the potential energy,

p is the momentum, and 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,$$T\; =\; \backslash sqrt\; -\; m\_e\; c^2\; =\; m\_e\; c^2\; \backslash left[\backslash sqrt\{1\; +\; \backslash frac\{p^\{2\}\}\{m\_e^2\; c^2\}\}-1\backslash right]$$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). Expanding the square root for large values of c , we find$$T\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$$

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$$\backslash mathcal\text{'}\; =\; -\backslash frac$$

Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.$$E\_n^\; =\; \backslash left\backslash langle\backslash psi^0\backslash right\backslash vert\; \backslash mathcal\text{'}\; \backslash left\backslash vert\backslash psi^0\backslash right\backslash rangle\; =\; -\backslash frac\; \backslash left\backslash langle\backslash psi^0\backslash right\backslash vert\; p^4\; \backslash left\backslash vert\backslash psi^0\backslash right\backslash rangle\; =\; -\backslash frac\; \backslash left\backslash langle\backslash psi^0\backslash right\backslash vert\; p^2\; p^2\; \backslash left\backslash vert\backslash psi^0\backslash right\backslash rangle$$where

\psi ^{0} is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see

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

For the hydrogen atom,$$V(r)\; =\; \backslash frac,$$ $$\backslash left\backslash langle\; \backslash frac\; \backslash right\backslash rangle\; =\; \backslash frac,$$ and $$\backslash left\backslash langle\; \backslash frac\; \backslash right\backslash rangle\; =\; \backslash frac,$$where

e is the elementary charge, \varepsilon _{0} is the vacuum permittivity, a _{0} is the Bohr radius, n is the principal quantum number, \ell 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 iswhere we have used:$$E\_n\; =\; -\; \backslash frac$$

On final calculation, the order of magnitude for the relativistic correction to the ground state is

-9 . 056 x 10 ^{-4} eV .

Spin–orbit coupling

See main article: 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:$$\backslash mathcal\_\backslash mathrm\; =\; \backslash left(\backslash frac\backslash right)\; \backslash left(\backslash frac\backslash right)\backslash frac$$where 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 and \boldsymbol\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$$\backslash Delta\; E\_\; =\; \backslash xi\; (r)\; \backslash mathbf\; L\; \backslash cdot\; \backslash mathbf\; S$$

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$$\backslash left\backslash langle\; \backslash frac\; \backslash right\backslash rangle\; =\; \backslash frac\; \backslash frac$$by Kramers–Pasternack relations and$$\backslash left\backslash langle\; \backslash mathbf\; L\; \backslash cdot\; \backslash mathbf\; S\; \backslash right\backslash rangle\; =\; \backslash frac\; \backslash left[j(j\; +\; 1)\; -\; \backslash ell(\backslash ell\; +\; 1)\; -\; s(s\; +\; 1)\backslash right]$$the expectation value for the Hamiltonian is:$$\backslash left\backslash langle\; \backslash mathcal\_\; \backslash right\backslash rangle\; =\; \backslash frac\; ~n~\; \backslash frac$$

Thus the order of magnitude for the spin–orbital coupling is:$$\backslash frac\; 10^\backslash text$$

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:

The Darwin term affects only the s orbitals. This is because the wave function of an electron with

\ell > 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 .

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.

\Delta t ≈ \hbar/\Delta E ≈ \hbar/mc ^{2} . The distance the particles can move during this time is \xi ≈ c\Delta t ≈ \hbar/mc = λ _{c} , the Compton wavelength. The electrons of the atom interact with those pairs. This yields a fluctuating electron position r + \boldsymbol \xi . Using a Taylor expansion, the effect on the potential U can be estimated:$$U(\backslash mathbf\; r\; +\; \backslash boldsymbol\backslash xi)\; \backslash approx\; U(\backslash mathbf\; r)\; +\; \backslash xi\backslash cdot\backslash nabla\; U(\backslash mathbf\; r)\; +\; \backslash frac\; 1\; 2\; \backslash sum\_\; \backslash xi\_i\; \backslash xi\_j\; \backslash partial\_i\; \backslash partial\_j\; U(\backslash mathbf\; r)$$

Averaging over the fluctuations

\boldsymbol \xi $$\backslash overline\backslash xi\; =\; 0,\; \backslash quad\; \backslash overline\; =\; \backslash frac\; 1\; 3\; \backslash overline\; \backslash delta\_,$$gives the average potential$$\backslash overline\; =\; U\; +\; \backslash frac\; \backslash overline\; \backslash nabla^2\; U\backslash left(\backslash mathbf\; r\backslash right).$$

Approximating

\overline{\boldsymbol\xi ^{2}} ≈ λ ^{2} _{c} , this yields the perturbation of the potential due to fluctuations:$$\backslash delta\; U\; \backslash approx\; \backslash frac16\; \backslash lambda\_c^2\; \backslash nabla^2\; U\; =\; \backslash frac\backslash nabla^2\; U$$

To compare with the expression above, plug in the Coulomb potential:$$\backslash nabla^2\; U\; =\; -\backslash nabla^2\; \backslash frac\; =\; 4\backslash pi\; \backslash left(\backslash frac\backslash right)\; \backslash delta^3(\backslash mathbf\; r)\; \backslash quad\backslash Rightarrow\backslash quad\; \backslash delta\; U\; \backslash approx\; \backslash frac\; 4\backslash pi\; \backslash left(\backslash frac\backslash right)\; \backslash delta^3(\backslash mathbf\; r)$$

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$$\backslash mathcal=\backslash mathcal\_\backslash text\; +\; \backslash mathcal\_+\backslash mathcal\_+\backslash mathcal\_,$$where

l{H} _{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] $$\backslash Delta\; E\; =\; \backslash frac\backslash left(\backslash frac\; -\; \backslash frac\; \backslash right)\backslash ,,$$where

j is the total angular momentum quantum number (j = 1 / 2 if \ell = 0 and j = \ell \pm 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.

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] $$E\_\; =\; -m\_\backslash textc^2\backslash left[1\; -\; \backslash left(1\; +\; \backslash left[\backslash frac\{\backslash alpha\}\{n\; -\; j\; -\; \backslash frac\{1\}\{2\}\; +\; \backslash sqrt\{\backslash left(j\; +\; \backslash frac\{1\}\{2\}\backslash right)^2\; -\; \backslash alpha^2\}\}\backslash right]^2\backslash right)^\backslash right].$$

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

• Book: Griffiths, David J. . Introduction to Quantum Mechanics (2nd ed.) . Prentice Hall . 2004 . 0-13-805326-X . registration .
• Book: Liboff, Richard L. . Liboff, Richard L. . Introductory Quantum Mechanics . Addison-Wesley . 2002 . 0-8053-8714-5.

External links

• Hyperphysics: Fine Structure
• University of Texas: The fine structure of hydrogen

Notes and References

1. On a method of making the wave-length of sodium light the actual practical standard of length. A.A. Michelson. Albert A. Michelson. E. W. Morley. Edward W. Morley. American Journal of Science. 34. 427. 1887.
2. On a method of making the wave-length of sodium light the actual practical standard of length. A.A. Michelson. Albert A. Michelson. E. W. Morley. Edward W. Morley. Philosophical Magazine. 24. 463. 1887.
3. A.Sommerfeld. Arnold Sommerfeld. Zur Feinstruktur der Wasserstofflinien. Geschichte und gegenwärtiger Stand der Theorie. Naturwissenschaften. July 1940. 28. 27. 417–423. german. 10.1007/BF01490583. 45670149 .
4. Book: Berestetskii , V. B. . Butterworth-Heinemann . 978-0-7506-3371-0 . E. M. Lifshitz . L. P. Pitaevskii . Quantum electrodynamics . 1982.
5. Book: Sommerfeld, Arnold . Atombau und Spektrallinien'. Friedrich Vieweg und Sohn . Braunschweig . 1919 . 3-87144-484-7. German English