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Fine structure

From Wikipedia, the free encyclopedia
Details in the emission spectrum of an atom
Interference fringes, showing fine structure (splitting) of a cooleddeuterium source, viewed through aFabry–Pérot interferometer.

Inatomic physics, thefine structure describes the splitting of thespectral lines ofatoms due toelectron spin andrelativistic corrections to the non-relativisticSchrödinger equation. It was first measured precisely for thehydrogen atom byAlbert A. Michelson andEdward W. Morley in 1887,[1][2] laying the basis for the theoretical treatment byArnold Sommerfeld, introducing thefine-structure constant.[3]

Background

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Gross structure

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Thegross structure of line spectra is the structure predicted by the quantum mechanics of non-relativistic electrons with no spin. For ahydrogenic atom, the gross structure energy levels only depend on theprincipal quantum numbern. However, a more accurate model takes into account relativistic and spin effects, which break thedegeneracy 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, whereZ is theatomic number andα is thefine-structure constant, adimensionless number equal to approximately 1/137.

Relativistic corrections

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The fine structure energy corrections can be obtained by usingperturbation theory. To perform this calculation one must add three corrective terms to theHamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to thespin–orbit coupling, and the Darwin term coming from the quantum fluctuating motion orzitterbewegung of the electron.

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

Hydrogen atom

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

Kinetic energy relativistic correction

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The gross structure assumes the kinetic energy term of theHamiltonian takes the same formas in classical mechanics, which for a single electron meansH0=p22me+V{\displaystyle {\mathcal {H}}^{0}={\frac {p^{2}}{2m_{e}}}+V}whereV is thepotential energy,p{\displaystyle p} is the momentum, andme{\displaystyle m_{e}} is theelectron rest mass.

However, when considering a more accurate theory of nature viaspecial relativity, we must use a relativistic form of the kinetic energy,T=p2c2+me2c4−mec2=mec2[1+p2me2c2−1]{\displaystyle 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]}where the first term is the total relativistic energy, and the second term is therest energy of the electron (c{\displaystyle c} is thespeed of light). Expanding the square root for large values ofc{\displaystyle c}, we findT=p22me−p48me3c2+⋯{\displaystyle T={\frac {p^{2}}{2m_{e}}}-{\frac {p^{4}}{8m_{e}^{3}c^{2}}}+\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 ordercorrection to the Hamiltonian isH′=−p48me3c2{\displaystyle {\mathcal {H}}'=-{\frac {p^{4}}{8m_{e}^{3}c^{2}}}}

Using this as aperturbation, 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)}=\left\langle \psi ^{0}\right\vert {\mathcal {H}}'\left\vert \psi ^{0}\right\rangle =-{\frac {1}{8m_{e}^{3}c^{2}}}\left\langle \psi ^{0}\right\vert p^{4}\left\vert \psi ^{0}\right\rangle =-{\frac {1}{8m_{e}^{3}c^{2}}}\left\langle \psi ^{0}\right\vert p^{2}p^{2}\left\vert \psi ^{0}\right\rangle }whereψ0{\displaystyle \psi ^{0}} is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we seeH0|ψ0⟩=En|ψ0⟩(p22me+V)|ψ0⟩=En|ψ0⟩p2|ψ0⟩=2me(En−V)|ψ0⟩{\displaystyle {\begin{aligned}{\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{aligned}}}

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⟩){\displaystyle {\begin{aligned}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_{e}c^{2}}}\left(E_{n}^{2}-2E_{n}\langle V\rangle +\left\langle V^{2}\right\rangle \right)\end{aligned}}}

For the hydrogen atom,V(r)=−e24πε0r,{\displaystyle V(r)={\frac {-e^{2}}{4\pi \varepsilon _{0}r}},}⟨1r⟩=1a0n2,{\displaystyle \left\langle {\frac {1}{r}}\right\rangle ={\frac {1}{a_{0}n^{2}}},} and⟨1r2⟩=1(ℓ+1/2)n3a02,{\displaystyle \left\langle {\frac {1}{r^{2}}}\right\rangle ={\frac {1}{(\ell +1/2)n^{3}a_{0}^{2}}},}wheree{\displaystyle e} is theelementary charge,ε0{\displaystyle \varepsilon _{0}} is thevacuum permittivity,a0{\displaystyle a_{0}} is theBohr radius,n{\displaystyle n} is theprincipal quantum number,ℓ{\displaystyle \ell } is theazimuthal quantum number andr{\displaystyle r} is the distance of the electron from the nucleus. Therefore, the first order relativistic correction for the hydrogen atom isEn(1)=−12mec2(En2+2Ene24πε01a0n2+116π2ε02e4(ℓ+12)n3a02)=−En22mec2(4nℓ+12−3){\displaystyle {\begin{aligned}E_{n}^{(1)}&=-{\frac {1}{2m_{e}c^{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_{e}c^{2}}}\left({\frac {4n}{\ell +{\frac {1}{2}}}}-3\right)\end{aligned}}}where we have used:En=−e28πε0a0n2{\displaystyle E_{n}=-{\frac {e^{2}}{8\pi \varepsilon _{0}a_{0}n^{2}}}}

On final calculation, the order of magnitude for the relativistic correction to the ground state is−9.056×10−4 eV{\displaystyle -9.056\times 10^{-4}\ {\text{eV}}}.

Spin–orbit coupling

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Main article:Spin–orbit interaction

For ahydrogen-like atom withZ{\displaystyle Z} protons (Z=1{\displaystyle Z=1} for hydrogen), orbital angular momentumL{\displaystyle \mathbf {L} } and electron spinS{\displaystyle \mathbf {S} }, the spin–orbit term is given by:HSO=(Ze24πε0)(gs−12me2c2)L⋅Sr3{\displaystyle {\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}}}}wheregs{\displaystyle g_{s}} is the sping-factor.

Thespin–orbit correction can be understood by shifting from the standardframe of reference (where theelectron orbits thenucleus) 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 itsintrinsic angular momentum. The two magnetic vectors,B{\displaystyle \mathbf {B} } andμs{\displaystyle {\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 formΔESO=ξ(r)L⋅S{\displaystyle \Delta E_{\mathrm {SO} }=\xi (r)\mathbf {L} \cdot \mathbf {S} }

Notice that an important factor of 2 has to be added to the calculation, called theThomas 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){\displaystyle \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)}}}by Kramers–Pasternack relations and⟨L⋅S⟩=ℏ22[j(j+1)−ℓ(ℓ+1)−s(s+1)]{\displaystyle \left\langle \mathbf {L} \cdot \mathbf {S} \right\rangle ={\frac {\hbar ^{2}}{2}}\left[j(j+1)-\ell (\ell +1)-s(s+1)\right]}the expectation value for the Hamiltonian is:⟨HSO⟩=En2mec2 n j(j+1)−ℓ(ℓ+1)−34ℓ(ℓ+12)(ℓ+1){\displaystyle \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)}}}

Thus the order of magnitude for the spin–orbital coupling is:Z4n3(j+12)(j+1)10−4 eV{\displaystyle {\frac {Z^{4}}{n^{3}\left(j+{\frac {1}{2}}\right)\left(j+1\right)}}10^{-4}{\text{ eV}}}

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

Darwin term

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There is one last term in the non-relativistic expansion of theDirac equation. It is referred to as the Darwin term, as it was first derived byCharles 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{\displaystyle {\begin{aligned}{\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}{na_{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{aligned}}}

The Darwin term affects only the s orbitals. This is because the wave function of an electron withℓ>0{\displaystyle \ell >0} vanishes at the origin, hence thedelta function has no effect. For example, it gives the 2s orbital the same energy as the 2p orbital by raising the 2s state by9.057×10−5 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 tozitterbewegung, or rapid quantum oscillations, of the electron. This can be demonstrated by a short calculation.[4]

Quantum fluctuations allow for the creation ofvirtual electron-positron pairs with a lifetime estimated by theuncertainty principleΔt≈ℏ/ΔE≈ℏ/mc2{\displaystyle \Delta t\approx \hbar /\Delta E\approx \hbar /mc^{2}}. The distance the particles can move during this time isξ≈cΔt≈ℏ/mc=λc{\displaystyle \xi \approx c\Delta t\approx \hbar /mc=\lambda _{c}}, theCompton wavelength. The electrons of the atom interact with those pairs. This yields a fluctuating electron positionr+ξ{\displaystyle \mathbf {r} +{\boldsymbol {\xi }}}. Using aTaylor expansion, the effect on the potentialU{\displaystyle U} can be estimated:U(r+ξ)≈U(r)+ξ⋅∇U(r)+12∑i,jξiξj∂i∂jU(r){\displaystyle 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} )}

Averaging over the fluctuationsξ{\displaystyle {\boldsymbol {\xi }}}ξ¯=0,ξiξj¯=13ξ2¯δij,{\displaystyle {\overline {\xi }}=0,\quad {\overline {\xi _{i}\xi _{j}}}={\frac {1}{3}}{\overline {{\boldsymbol {\xi }}^{2}}}\delta _{ij},}gives the average potentialU(r+ξ)¯=U(r)+16ξ2¯∇2U(r).{\displaystyle {\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).}

Approximatingξ2¯≈λc2{\displaystyle {\overline {{\boldsymbol {\xi }}^{2}}}\approx \lambda _{c}^{2}}, this yields the perturbation of the potential due to fluctuations:δU≈16λc2∇2U=ℏ26me2c2∇2U{\displaystyle \delta U\approx {\frac {1}{6}}\lambda _{c}^{2}\nabla ^{2}U={\frac {\hbar ^{2}}{6m_{e}^{2}c^{2}}}\nabla ^{2}U}

To compare with the expression above, plug in theCoulomb potential:∇2U=−∇2Ze24πε0r=4π(Ze24πε0)δ3(r)⇒δU≈ℏ26me2c24π(Ze24πε0)δ3(r){\displaystyle \nabla ^{2}U=-\nabla ^{2}{\frac {Ze^{2}}{4\pi \varepsilon _{0}r}}=4\pi \left({\frac {Ze^{2}}{4\pi \varepsilon _{0}}}\right)\delta ^{3}(\mathbf {r} )\quad \Rightarrow \quad \delta U\approx {\frac {\hbar ^{2}}{6m_{e}^{2}c^{2}}}4\pi \left({\frac {Ze^{2}}{4\pi \varepsilon _{0}}}\right)\delta ^{3}(\mathbf {r} )}

This is only slightly different.

Another mechanism that affects only the s-state is theLamb shift, a further, smaller correction that arises inquantum 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

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The full Hamiltonian is given byH=HCoulomb+Hkinetic+HSO+HDarwin,{\displaystyle {\mathcal {H}}={\mathcal {H}}_{\text{Coulomb}}+{\mathcal {H}}_{\text{kinetic}}+{\mathcal {H}}_{\mathrm {SO} }+{\mathcal {H}}_{\text{Darwin}},}whereHCoulomb{\displaystyle {\mathcal {H}}_{\text{Coulomb}}} is the Hamiltonian from theCoulomb interaction.

The total effect, obtained by summing the three components up, is given by the following expression:[5]ΔE=En(Zα)2n(1j+12−34n),{\displaystyle \Delta E={\frac {E_{n}(Z\alpha )^{2}}{n}}\left({\frac {1}{j+{\frac {1}{2}}}}-{\frac {3}{4n}}\right)\,,}wherej{\displaystyle j} is thetotal angular momentum quantum number (j=1/2{\displaystyle j=1/2} ifℓ=0{\displaystyle \ell =0} andj=ℓ±1/2{\displaystyle j=\ell \pm 1/2} otherwise). It is worth noting that this expression was first obtained by Sommerfeld based on theold Bohr theory; i.e., before the modernquantum mechanics was formulated.

Energy diagram (to scale) of the hydrogen atom forn=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 theZeeman effect (magnetic field dependence).

Exact relativistic energies

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

The total effect can also be obtained by using the Dirac equation. The exact energies are given by[6]Ejn=−mec2[1−(1+[αn−j−12+(j+12)2−α2]2)−12].{\displaystyle 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].}

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 thehyperfine structure corrections, which are due to interactions with the nuclear spin. Other corrections fromquantum field theory such as theLamb shift and theanomalous magnetic dipole moment of the electron are not included.

See also

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References

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  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.
  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.
  3. ^A.Sommerfeld (July 1940). "Zur Feinstruktur der Wasserstofflinien. Geschichte und gegenwärtiger Stand der Theorie".Naturwissenschaften (in German).28 (27):417–423.doi:10.1007/BF01490583.S2CID 45670149.
  4. ^Zelevinsky, Vladimir (2011),Quantum Physics Volume 1: From Basics to Symmetries and Perturbations, WILEY-VCH, p. 551,ISBN 978-3-527-40979-2
  5. ^Berestetskii, V. B.; E. M. Lifshitz; L. P. Pitaevskii (1982).Quantum electrodynamics. Butterworth-Heinemann.ISBN 978-0-7506-3371-0.
  6. ^Sommerfeld, Arnold (1919).Atombau und Spektrallinien'. Braunschweig: Friedrich Vieweg und Sohn.ISBN 3-87144-484-7.{{cite book}}:ISBN / Date incompatibility (help)GermanEnglish

External links

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