Editing Fine structure

{{Short description|Details in the emission spectrum of an atom}}
[[File:Fabry Perot Etalon Rings Fringes.png|right|thumb|200px|[[Interference (wave propagation)|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 line]]s of [[atoms]] due to [[electron spin]] and [[special relativity|relativistic correction]]s 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,<ref>{{cite journal|title=On a method of making the wave-length of sodium light the actual practical standard of length|author1=A.A. Michelson|author-link1=Albert A. Michelson|author2=E. W. Morley|author-link2=Edward W. Morley|journal=American Journal of Science|volume=34|page=427|year=1887|url=https://archive.org/details/americanjourna3341887newh/page/427}}</ref><ref>{{cite journal|title=On a method of making the wave-length of sodium light the actual practical standard of length|author1=A.A. Michelson|author-link1=Albert A. Michelson|author2=E. W. Morley|author-link2=Edward W. Morley|journal=Philosophical Magazine|volume=24|page=463|year=1887|url=https://archive.org/details/s5philosophicalm24lond/page/463}}</ref> laying the basis for the theoretical treatment by [[Arnold Sommerfeld]], introducing the [[fine-structure constant]].<ref>{{cite journal|author1=A.Sommerfeld|author-link1=Arnold Sommerfeld|title=Zur Feinstruktur der Wasserstofflinien. Geschichte und gegenwärtiger Stand der Theorie|journal=Naturwissenschaften|date=July 1940|volume=28|issue=27|pages=417–423|language=german|doi=10.1007/BF01490583|s2cid=45670149 }}</ref>

== 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 [[Degenerate energy level|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'')<sup>2</sup>, where ''Z'' is the [[atomic number]] and ''α'' is the [[fine-structure constant]], a [[dimensionless quantity|dimensionless number]] equal to approximately 1/137.

=== Relativistic corrections ===
The fine structure energy corrections can be obtained by using [[Perturbation theory (quantum mechanics)|perturbation theory]]. To perform this calculation one must add three corrective terms to the [[Hamiltonian (quantum mechanics)|Hamiltonian]]: the leading order relativistic correction to the kinetic energy, the correction due to the [[Spin–orbit interaction|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 (physics)|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 Mechanics|Hamiltonian]] takes the same form [[Kinetic energy#Kinetic energy of rigid bodies|as in classical mechanics]], which for a single electron means
<math display="block">\mathcal{H}^0 = \frac{p^{2}}{2m_e} + V</math>
where {{mvar|V}} is the [[potential energy]], <math>p</math> is the momentum, and <math>m_e</math> is the [[Electron mass|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,
<math display="block">T = \sqrt{p^2 c^2 + m_e^2 c^4} - m_e c^2 = m_e c^2 \left[ \sqrt{1 + \frac{p^{2}}{m_e^2 c^2}}-1\right]</math>
where the first term is the total relativistic energy, and the second term is the [[rest energy]] of the electron (<math>c</math> is the [[speed of light]]). Expanding the square root for large values of <math>c</math>, we find
<math display="block">T = \frac{p^2}{2m_e} - \frac{p^4}{8m_e^3 c^2} + \cdots</math>

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
<math display="block">\mathcal{H}' = -\frac{p^{4}}{8m_e^{3}c^{2}}</math>

Using this as a [[Perturbation theory (quantum mechanics)|perturbation]], we can calculate the first order energy corrections due to relativistic effects.
<math display="block">E_n^{(1)} = \left\langle\psi^0\right\vert \mathcal{H}' \left\vert\psi^0\right\rangle = -\frac{1}{8m_e^3c^2} \left\langle\psi^0\right\vert p^4 \left\vert\psi^0\right\rangle = -\frac{1}{8m_e^3c^2} \left\langle\psi^0\right\vert p^2 p^2 \left\vert\psi^0\right\rangle</math>
where <math>\psi^{0}</math> is the unperturbed wave function.  Recalling the unperturbed Hamiltonian, we see
<math display="block">\begin{align}
                     \mathcal{H}^0 \left\vert\psi^0\right\rangle &=          E_n \left\vert\psi^0\right\rangle \\
  \left(\frac{p^2}{2m_e} + V\right)\left\vert\psi^0\right\rangle &=          E_n \left\vert\psi^0\right\rangle \\
                               p^2 \left\vert\psi^0\right\rangle &= 2m_e(E_n - V)\left\vert\psi^0\right\rangle
\end{align}</math>

We can use this result to further calculate the relativistic correction:
<math display="block">\begin{align}
  E_n^{(1)} &= -\frac{1}{8m_e^3 c^2}\left\langle\psi^0\right\vert p^2 p^2 \left\vert\psi^{0}\right\rangle \\[1ex]
            &= -\frac{1}{8m_e^3 c^2}\left\langle\psi^0\right\vert (2m_e)^2 (E_n - V)^2\left\vert\psi^0\right\rangle \\[1ex]
            &= -\frac{1}{2m_ec^2}\left(E_n^2 - 2E_n\langle V\rangle + \left\langle V^2\right\rangle \right)
\end{align}</math>

For the hydrogen atom,
<math display="block">V(r) = \frac{-e^2}{4\pi \varepsilon_0 r},</math> <math display="block">\left\langle \frac{1}{r} \right\rangle = \frac{1}{a_0 n^2},</math> and <math display="block">\left\langle \frac{1}{r^2} \right\rangle = \frac{1}{(\ell + 1/2) n^3 a_0^2},</math>
where <math>e</math> is the [[elementary charge]], <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>a_0</math> is the [[Bohr radius]], <math>n</math> is the [[principal quantum number]], <math>\ell</math> is the [[azimuthal quantum number]] and <math>r</math> is the distance of the electron from the nucleus.  Therefore, the first order relativistic correction for the hydrogen atom is
<math display="block">\begin{align}
  E_{n}^{(1)} &= -\frac{1}{2m_ec^2}\left(E_n^2 + 2E_n\frac{e^2}{4\pi \varepsilon_0}\frac{1}{a_0 n^2} + \frac{1}{16\pi^2 \varepsilon_0^2}\frac{e^4}{\left(\ell + \frac{1}{2}\right) n^3 a_0^2}\right) \\
              &= -\frac{E_n^2}{2m_ec^2}\left(\frac{4n}{\ell + \frac{1}{2}} - 3\right)
\end{align}</math>
where we have used:
<math display="block"> E_n = - \frac{e^2}{8 \pi \varepsilon_0 a_0 n^2} </math>

On final calculation, the order of magnitude for the relativistic correction to the ground state is  <math> -9.056 \times 10^{-4}\ \text{eV}</math>.

=== Spin–orbit coupling ===
{{Main|Spin–orbit interaction}}

For a [[hydrogen-like atom]] with <math>Z</math> protons (<math>Z = 1</math> for hydrogen), orbital angular momentum <math>\mathbf L</math> and electron spin <math>\mathbf S</math>, the spin–orbit term is given by:
<math display="block">\mathcal{H}_\mathrm{SO} = \left(\frac{Ze^2}{4\pi\varepsilon_0}\right) \left(\frac{g_s - 1}{2m_e^2 c^2}\right)\frac{\mathbf{L} \cdot \mathbf{S}}{r^3}</math>
where <math>g_s</math> is the spin [[g-factor (physics)|g-factor]].

The [[Spin (physics)|spin]]–orbit correction can be understood by shifting from the standard [[frame of reference]] (where the [[electron]] orbits the [[Atomic nucleus|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, <math>\mathbf{B}</math> and <math>\boldsymbol\mu_s</math> 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<ref>{{cite book |last1=Levine |first1=Ira N. |title=Quantum Chemistry |date=1991 |publisher=Prentice-Hall |isbn=0-205-12770-3 |pages=310-311 |edition=4th}}</ref>
<math display="block"> \Delta E_{\mathrm{SO}} = \xi (r) \mathbf L \cdot \mathbf S</math>

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
<math display="block">
\left\langle \frac{1}{r^3} \right\rangle = \frac{Z^3}{n^3 a_0^3} \frac{1}{\ell \left(\ell + \frac{1}{2}\right) (\ell + 1)}
</math>
by Kramers–Pasternack relations and
<math display="block">
\left\langle \mathbf L \cdot \mathbf S \right\rangle = \frac{\hbar^2}{2} \left[j(j + 1) - \ell(\ell + 1) - s(s + 1)\right]
</math>
the expectation value for the Hamiltonian is:
<math display="block">\left\langle \mathcal{H}_{\mathrm{SO}} \right\rangle = \frac{E_n{}^2}{m_e c^2} ~n~ \frac{j(j + 1) - \ell(\ell + 1) - \frac{3}{4}}{\ell \left( \ell + \frac{1}{2}\right) (\ell + 1) }</math>

Thus the order of magnitude for the spin–orbital coupling is:
<math display="block">\frac{Z^4}{n^3 \left(j + \frac{1}{2}\right)\left(j + 1\right)} 10^{-4}\text{ eV}</math>

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:
<math display="block">\begin{align}
                   \mathcal{H}_{\text{Darwin}} &= \frac{\hbar^2}{8m_e^2 c^2}\,4\pi\left(\frac{Ze^2}{4\pi \varepsilon_0}\right) \delta^3{\left(\mathbf r\right)} \\
   \langle \mathcal{H}_{\text{Darwin}} \rangle &= \frac{\hbar^2}{8m_e^2 c^2}\,4\pi\left(\frac{Ze^2}{4\pi \varepsilon_0}\right)| \psi(0)|^2 \\[3pt]
                                 \psi (0) &= \begin{cases}
                                                 0 & \text{ for } \ell > 0 \\
                                                 \frac{1}{\sqrt{4\pi}} \, 2 \left( \frac {Z}{n a_0} \right)^\frac{3}{2} & \text{ for } \ell = 0
                                             \end{cases}\\[2pt]
                   \mathcal{H}_{\text{Darwin}} &= \frac{2n}{m_e c^2}\,E_n^2
\end{align}</math>

The Darwin term affects only the s orbitals. This is because the wave function of an electron with <math>\ell > 0</math> vanishes at the origin, hence the [[Dirac delta function|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 {{val|9.057|e=-5|u=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.<ref>{{citation| last = Zelevinsky | first = Vladimir | title = Quantum Physics Volume 1: From Basics to Symmetries and Perturbations | publisher = WILEY-VCH | year = 2011 | isbn = 978-3-527-40979-2 | p = 551}}</ref>

[[Quantum fluctuation]]s allow for the creation of [[Virtual particle|virtual]] electron-positron pairs with a lifetime estimated by the [[uncertainty principle]] <math>\Delta t \approx \hbar/\Delta E \approx \hbar/mc^2</math>. The distance the particles can move during this time is <math>\xi \approx c\Delta t \approx \hbar/mc = \lambda_c</math>, the [[Compton wavelength]]. The electrons of the atom interact with those pairs. This yields a fluctuating electron position <math>\mathbf r + \boldsymbol \xi</math>. Using a [[Taylor expansion]], the effect on the potential <math>U</math> can be estimated:
<math display="block"> U(\mathbf r + \boldsymbol\xi) \approx U(\mathbf r) + \xi\cdot\nabla U(\mathbf r) + \frac 1 2 \sum_{i,j} \xi_i \xi_j \partial_i \partial_j U(\mathbf r) </math>

Averaging over the fluctuations <math>\boldsymbol \xi</math>
<math display="block"> \overline\xi = 0, \quad \overline{\xi_i\xi_j} = \frac 1 3 \overline{\boldsymbol\xi^2} \delta_{ij}, </math>
gives the average potential
<math display="block"> \overline{U\left(\mathbf r + \boldsymbol\xi\right)} = U{\left(\mathbf r\right)} + \frac{1}{6} \overline{\boldsymbol\xi^2} \nabla^2 U\left(\mathbf r\right). </math>

Approximating <math>\overline{\boldsymbol\xi^2} \approx \lambda_c^2</math>, this yields the perturbation of the potential due to fluctuations:
<math display="block"> \delta U \approx \frac16 \lambda_c^2 \nabla^2 U = \frac{\hbar^2}{6m_e^2c^2}\nabla^2 U </math>

To compare with the expression above, plug in the [[Coulomb potential]]:
<math display="block">
  \nabla^2 U = -\nabla^2 \frac{Z e^2}{4\pi\varepsilon_0 r} = 4\pi \left(\frac{Z e^2}{4\pi\varepsilon_0}\right) \delta^3(\mathbf r)
               \quad\Rightarrow\quad \delta U \approx \frac{\hbar^2}{6m_e^2c^2} 4\pi \left(\frac{Z e^2}{4\pi\varepsilon_0}\right) \delta^3(\mathbf r)
</math>

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
<math display="block">\mathcal{H}=\mathcal{H}_\text{Coulomb} + \mathcal{H}_{\text{kinetic}}+\mathcal{H}_{\mathrm{SO}}+\mathcal{H}_{\text{Darwin}},</math>
where <math>\mathcal{H}_\text{Coulomb}</math> is the Hamiltonian from the [[Coulomb's law|Coulomb interaction]].

The total effect, obtained by summing the three components up, is given by the following expression:<ref>
{{cite book
 | publisher = Butterworth-Heinemann
 | isbn = 978-0-7506-3371-0
 | last = Berestetskii
 | first = V. B.
 | author2 = E. M. Lifshitz |author3=L. P. Pitaevskii
 | title = Quantum electrodynamics
 | year = 1982
}}</ref>
<math display="block">\Delta E = \frac{E_{n}(Z\alpha)^{2}}{n}\left( \frac{1}{j + \frac{1}{2}} - \frac{3}{4n} \right)\,,</math>
where <math>j</math> is the [[total angular momentum quantum number]] (<math>j = 1/2</math> if <math>\ell = 0</math> and <math>j = \ell \pm 1/2</math> otherwise). It is worth noting that this expression was first obtained by Sommerfeld based on the [[Old quantum theory|old Bohr theory]]; i.e., before the modern [[quantum mechanics]] was formulated.
[[File:Hydrogen fine structure energy 2.svg|center|thumb|800x800px|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 ===
[[File:Hydrogen fine structure.svg|thumb|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 {{nowrap|1=''n'' = 2}} to {{nowrap|1=''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<ref name="Sommerfeld">{{cite book|first=Arnold |last= Sommerfeld|title=Atombau und Spektrallinien'|  publisher=Friedrich Vieweg und Sohn | location=Braunschweig | year=1919 | isbn=3-87144-484-7}} [https://archive.org/stream/atombauundspekt00sommgoog German] [https://archive.org/embed/0244-pdf-atomic-structure-and-spectral-lines-arnold-sommerfeld-2 English]</ref>
<math display="block">E_{j\,n} = -m_\text{e}c^2\left[1 - \left(1 + \left[\frac{\alpha}{n - j - \frac{1}{2} + \sqrt{\left(j + \frac{1}{2}\right)^2 - \alpha^2}}\right]^2\right)^{-\frac{1}{2}}\right].</math>

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 ==
{{reflist}}
* {{cite book |author=Griffiths, David J. |title=Introduction to Quantum Mechanics (2nd ed.) |publisher=Prentice Hall |year=2004 |isbn=0-13-805326-X |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0 }}
* {{cite book | author=Liboff, Richard L. | author-link=Liboff, Richard L. | title=Introductory Quantum Mechanics | publisher=Addison-Wesley | year=2002 | isbn=0-8053-8714-5}}

== External links ==
* [http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/hydfin.html#c1 Hyperphysics: Fine Structure]
* [http://farside.ph.utexas.edu/teaching/qmech/lectures/node107.html University of Texas: The fine structure of hydrogen]

[[Category:Atomic physics]]