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Appendix Units Appendix Units Throughout this thesis, wherever physical quantities are given, we tried to ensure that they are presented in the most appropriate unit system. Whenever several repre- sentations could prove useful, all of them are shown. This should ensure convenient reading on the one hand and straight-forward comparison of the numbers on the other. Nevertheless, a brief overview of the applied unit systems and of the conversions from one to another is presented in the following. A.1 Atomic Units The International System of Units (abbreviated SI from French: Le Système international d’unités) is not always the most convenient choice. Particularly, for quantities on the molecular or the atomic scale the system of atomic units (au) allows a much better grasp of the occurring quantities. As they are natural units,au are based on universal physical constants as measures. These constants are chosen to represent the dimensions of the electronic motion in an atom. This is underlined by the strong connection between au and the hydrogenic ground state as described by the Bohr model. There are two variants of au1, Rydberg and Hartree units, differing in their choice of the unit of mass and charge. We only make use of the latter, which is gained by setting the electron mass, the elementary charge, the reduced Planck constant and the electrostatic constant to unity: 1 ! me = e = = = 1. (A.1) 4π0 1 The abbreviation au might also lead to confusion with astronomical or arbitrary units. Particularly the latter is thus not used in this work. © Springer International Publishing Switzerland 2015 189 K. Schnorr, XUV Pump-Probe Experiments on Diatomic Molecules, Springer Theses, DOI 10.1007/978-3-319-12139-0 190 Appendix: Units Table A.1 Overview of the numerical values of fundamental atomic units Constant Symbol Dimension SI value −31 Electron rest mass me Mass 9.109 × 10 kg Elementary charge e Charge 1.602 × 10−19 C Reduced Planck constant = h/2π Action 1.054 × 10−34 Js 9 3 2 2 Electrostatic constant 1/4π0 Inverse 0 8.987 × 10 kg m /s C = . × −12 F Here, the vacuum permittivity 0 8 854 10 m is used Table A.2 Overview of the numerical values of some derived atomic units Constant Symbol Dimension SI value Alternative units −11 Bohr radius a0 = /meαc Length 5.292 × 10 m 0.529Å 2 2 −18 Hartree H = meα c Energy 4.360 × 10 J 27.211eV /H Time 2.419 × 10−17s Two convenient alternative representations are given, as discussed in Sect. A.2. Note that the au of time corresponds to the time an electron needs for one revolution on the ground state orbit of hydrogen according to the Bohr model An overview of the numerical values of these constants in SI units is given in TableA.1. From the above definition we can directly infer two important physi- cal constants. The proton mass in SI units can be written m p = 1836.153 · me, thus the au equivalent reads m p ≈ 1836 au. Using the definition of the dimensionless fine-structure constant e2 1 α = = (A.2) 4π0 c 137.036 and the knowledge that dimensionless constants retain their value independent of the unit system, we can give the vacuum speed of light c in atomic units: 1 c = ≈ . α 137 (A.3) From the fundamental atomic units above a full set of units may be derived, a subset of which is given in TableA.2. A.2 Other Commonly Used Units For quantities with the dimension energy, several commonly used representations are available. Besides the SI unit Joule (J) and the atomic unit Hartree (H) (cf. Sect.A.1), the electron volt (eV) is used throughout this work. It is defined as the amount of energy gained by moving an elementary charge e across an electric potential difference of 1V: Appendix: Units 191 − 1eV = 1.602 × 10 19 J. (A.4) It is also common practice to give quantities that are not energies in eV by applying 2 mass-energy equivalence E = mc , with the relativistic mass m = γ m0,therest 2 mass m0, the Lorentz-factor γ = 1/ 1 − β and the velocity in units of the speed of light in vacuum β = v/c. For example, the electron rest mass me is often given 2 as me = 511 keV/c . Additionally, due to the relation between the frequency ν, the wavelength λ, and the energy of a photon, hc E = hν = , λ (A.5) with the Planck constant h = 6.626 × 10−34 J s, the convenient conversion formula 1240 E[eV]≈ (A.6) λ[nm] is widely used. Eventhough, the atomic unit of length, the Bohr radius a0 (cf. Sect. A.1), provides a measure on the atomic scale, another length unit is also commonly used, the Ångström denoted by Å: −10 1Å = 10 m ≈ 2a0. (A.7) Obviously it describes the diameter of the lowest orbit in hydrogen, as a0 is the corresponding radius..
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