Abstract
In this final chapter of Vol. 1 much comes together what has been introduced in the preceding text. In Sects. 10.1–10.2 we give an overview on the classical methods for computing multi-electron wave functions, applying what we have learned in Chap. 3 and Chap. 7. In Sect. 10.3 a short excursion into density functional theory is added. With increasing atomic number Z not only the number of electrons grows – and hence the complexity of the problem. Also the significance of different types of interaction changes: for light atoms exchange interaction was dominant, and LS coupling gave a good description as described in Chap. 6. With increasing spin-orbit interaction, LS coupling is no longer adequate, as we shall illustrate in Sect. 10.4. The energy scale as a whole changes – roughly ∝Z 2. Transition energies for quantum jumps within the outermost electron shells are still in the VIS or UV spectral range; the reader may familiarize him- or herself by way of a few characteristic examples with the ‘zoo’ of energy levels and coupling schemes in complex atoms. Changes in the inner shells are associated with emission or absorption of X-ray radiation, which treated in Sect. 10.5 – complementing Chap. 4. In addition, our understanding of photoionization (Chap. 5) is deepened by some examples of multi-electron systems. Finally, Sect. 10.6 introduces the reader to modern sources for the generation of X-rays.
The last chapter in this volume is dedicated to eigenstates and energies of atoms with many electrons. For these heavy, complex atoms, in principle all electrons have to be treated equally. The repulsion of the electrons among each other is of the same order of magnitude as the Coulomb attraction of the nucleus. So, this problem can no longer be treated by perturbation theory.
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Notes
- 1.
Here and in the following the index μ stands for a set of quantum numbers {n μ ℓ μ m ℓμ } (later also for {n μ ℓ μ m ℓμ m sμ }. If no ambiguities can arise, we abbreviate q j , as well as r j and s j by j – as we also have done in Chap. 7.
- 2.
One may wonder why the sum does not exclude μ′=μ. Note, however, that in this case Coulomb and exchange integral just cancel each other.
- 3.
The quality of the focussing is characterized by the so called emittance. That is the spatial extension of the electron beam multiplied by its divergence angle. As an example, BESSY II is characterized by an emittance of (3 to 6) \(\operatorname{nm} \operatorname{rad}\) in horizontal and \({<}0.1\operatorname{nm} \operatorname{rad}\) in vertical direction.
- 4.
This ‘unit’ may be somewhat confusing. What it implies is simply that B gives the number of photons emitted per second into a solid angle \(\Delta \varOmega=1\operatorname{mrad}^{2}\) from a source area \(\Delta S=1\operatorname{mm}^{2}\) into an energy interval ΔW=10−3 W. Thus, into an arbitrary solid angle and energy interval the source emits
$$\frac{\mathrm{photons}}{\operatorname{s}}=B\times\frac{\Delta S}{\operatorname{mm}^{2}}\times \frac{\Delta \varOmega}{ \operatorname{mrad}^{2}}\times \frac{\Delta W}{10^{-3}W} . $$Expressed in terms of standard spectral radiance (10.56) one would write
$$\frac{\mathrm{photons}}{\operatorname{s}}=\frac{\widetilde{R}_{W}}{\hbar\omega}\times \Delta S\times \Delta \varOmega\times \Delta W , $$so that with (10.56), (10.57) and W=ħω
$$B=10^{-3}W\dfrac{\mathrm {d}^{2} \varPhi}{ \mathrm {d}\varOmega \mathrm {d}W}\operatorname{mrad}^{2}\operatorname{mm}^{2}=10^{-3}\times\widetilde{R}_{W} . $$ - 5.
Of course, the emission angle effective for the experiment also depends crucially on the experimental setup, which includes a certain length of the electron orbit. The angle \(\mathrm{\delta}\theta\) given here thus only holds for the vertical divergence perpendicular to the plane of the ring. In horizontal direction the experimental apertures limit the SR beam.
- 6.
Clearly, this time refers to the light emitted from a single electron and has nothing to do with the actually measurable duration of the SR light pulses from bunches of electrons. Typical pulse duration of modern synchrotrons are several \(\operatorname{ps}\).
- 7.
For readers who want to follow these reformulations: Schwinger uses the esu system, and his formula (II.32) gives the energy which an electron emits along the orbit over χ=2π. For the determination of the brilliance it has to be divided by 2π and by the photon energy. Now, 2πρ/c=t e is the period time for an electron on a circular orbit and, on the other hand, \(\mathcal{N}_{\mathrm{e}}=2\pi\rho I/ ( ec ) \) gives the number of electrons in the ring at a current I. With an emitting area ΔS and the fine structure constant α=e 2/(4πε 0 ħc) one obtains (10.62).
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Author information
Authors and Affiliations
Acronyms and Terminology
- a.u.:
-
‘atomic units’, see Sect. 2.6.2.
- BESSY:
-
‘Berlin Electron Strorage ring for Synchrotron Radiation’, Germany’s third generation synchrotron radiation source in Berlin-Adlerhof.
- CI:
-
‘Configuration interaction’, mixing of states with different electronic configurations in atomic and molecular structure calculations, using linear superposition of Slater determinants (see Sect. 10.2.3).
- DFT:
-
‘Density functional theory’, today one of the standard methods for computing atomic and molecular electron densities and energies (see Sect. 10.3).
- esu:
-
‘electrostatic units’, old system of unities, equivalent to the Gauss system for electric quantities (see Appendix A.3).
- FBA:
-
‘First order Born approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sect. 6.6, Vol. 2 and Sect. 5.5.2, respectively).
- FEL:
-
‘free electron laser’, laser like radiation source using amplification in a spatially oscillating, relativistic electron beam, (see Sect. 10.6.5).
- FLASH:
-
‘Free Electron LASer in Hamburg’.
- FS:
-
‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6).
- FWHM:
-
‘Full width at half maximum’.
- good quantum number:
-
‘Quantum number for eigenvalues of such observables that may be measured simultaneously with the Hamilton operator (see Sect. 2.6.5)’.
- HF:
-
‘Hartree-Fock’, method (approximation) for solving a multi-electron Schrödinger equation, including exchange interaction.
- HHG:
-
‘High harmonic generation’, in intense laser fields.
- LDA:
-
‘Local density approximation’, simplest version of density functional theory.
- LS:
-
‘L’, for orbital and S for spin angular momenta, unfortunately this is used in two opposite contexts: (i) LS interaction characterizes the spin-orbit interaction energy, while (ii) LS coupling denotes an angular momentum coupling scheme for multi-electron systems where all orbital angular momenta and all spin angular momenta of all electrons are first separately coupled together to L and S, respectively, and finally L and S are coupled to J.
- LSDA:
-
‘Local spin density approximation’, similar to LDA but for electrons with one spin orientation only.
- MCHF:
-
‘Multi configuration Hartree-Fock’, method to determine wave functions for multi-electron systems (see Sect. 10.5.4).
- MP2:
-
‘Møller-Plesset correction of 2nd order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials.
- MP3:
-
‘Møller-Plesset correction of 3rd order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials.
- MP4:
-
‘Møller-Plesset correction of 4th order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials.
- NIST:
-
‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html.
- QED:
-
‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction.
- RHF:
-
‘Restricted Hartree-Fock’, assuming all spatial wave functions in a given closed shell to be equal when computing atomic wave functions.
- SASE:
-
‘Self-Amplified Spontaneous Emission’, scheme for FEL operation.
- SCF:
-
‘Self-consistent field’, method for solving coupled integro-differential equations iteratively.
- SI:
-
‘Système international d’unités’, international system of units (m, kg, s, A, K, mol, cd), for details see the website of the Bureau International des Poids et Mésure http://www.bipm.org/en/si/ or NIST http://physics.nist.gov/cuu/Units/index.html.
- SR:
-
‘Synchrotron radiation’, electronmagnetic radiation in a broad range of wavelengths, generated by relativistic electrons on circular orbits.
- TDDFT:
-
‘Time dependent density functional theory’, a modern variety of DFT, allowing also for excited state calculations.
- UHF:
-
‘Unrestricted Hartree-Fock’, allowing different spatial wave functions for each orbital when computing atomic orbitals (specifically in closed shells).
- UV:
-
‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between \(100\operatorname{nm}\) and \(400\operatorname{nm}\) according to ISO 21348 (2007).
- VIS:
-
‘Visible’, spectral range of electromagnetic radiation. Wavelengths between \(380\operatorname{nm}\) and \(760\operatorname{nm}\) according to ISO 21348 (2007).
- VUV:
-
‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between \(10\operatorname{nm}\) and \(200\operatorname{nm}\) according to ISO 21348 (2007).
- XFEL:
-
‘X-ray free electron laser’, as FEL but specifically designed to generate coherent, short pulse X-ray laser like radiation, for details see Sect. 10.6.5.
- XUV:
-
‘Soft X-ray (sometimes also extreme UV)’, spectral wavelength range between \(0.1\operatorname{nm}\) and \(10\operatorname{nm}\) according to ISO 21348 (2007), sometimes up to \(40\operatorname{nm}\).
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Hertel, I.V., Schulz, CP. (2015). Multi-electron Atoms. In: Atoms, Molecules and Optical Physics 1. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54322-7_10
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