2.4 PHYSICAL CONSTANTS USED AS ATOMIC UNITS
Sometimes fundamental physical constants, or other well defined physical quantities, are used as though they were units in certain specialized fields of science. For example, in astronomy it may be more convenient to express the mass of a star in terms of the mass of the sun. In atomic and molecular physics it is similarly more convenient to express masses in terms of the electron mass, $m_{\rm{e}}$, or in terms of the unified atomic mass unit, $1\ \rm{u}$, and to express charges in terms of the elementary charge $e$, and energies in terms of the electronvolt, $\rm{eV}$.
The electronvolt is the kinetic energy acquired by an electron in passing through a potential difference of $1\ \rm{V}$ in vacuum, $1\ \rm{e}V = 1.602\ 176\ 634 \times 10^{−19}\ \rm{J}$. The numerical value of a quantity expressed in this unit may be converted into its value when expressed in the SI by multiplication with the value of the physical constant in the SI.
The dalton and the unified atomic mass unit are alternative names for the same unit, therefore $1\ \rm{u} = 1 \rm{Da} \approx 1.6605 \times 10^{−27}\ \rm{kg}$. The dalton may be combined with the SI prefixes to express the masses of large molecules in kilodalton ($\rm{kDa}$) or megadalton ($\rm{MDa}$).
| Physical quantity | Physical constant | Symbol for unit | Value in SI units | Notes |
|---|---|---|---|---|
| mass | electron mass | $m_{\rm{e}}$ | $\approx 9.1094 \times 10^{-31}\ \rm{kg}$ | |
| charge | elementary charge | $e$ | $= 1.602\ 176\ 634 \times 10^{-19}\ \rm{C}$ | |
| action, (angular momentum) | Planck constant divided by $2\pi$ | $\hbar$ | $= 1.054\ 571\ 817 ... \times 10^{-34}\ \rm{J s}$ | 1 |
| length | bohr | $a_{0}$ | $\approx 5.2918 \times 10^{-11}\ \rm{m}$ | 1 |
| energy | hartree | $E_{\rm{h}}$ | $\approx 4.3597 \times 10^{-18}\ \rm{J}$ | 1 |
| time | $\hbar/E_{\rm{h}}$ | $\approx 2.1489 \times 10^{-17}\ \rm{s}$ | ||
| speed | $a_{0}E_{\rm{h}}/\hbar$ | $\approx 2.1877 \times 10^{6}\ \rm{m\ s}^{-1}$ | 2 | |
| electric field strength | $E_{\rm{h}}/ea_{0}$ | $\approx 5.1422 \times 10^{11}\ \rm{V\ m}^{-1}$ | ||
| electric dipole moment | $ea_{0}$ | $\approx 8.4784 \times 10^{-30}\ \rm{C\ m}$ | ||
| electric quadrupole moment | $ea_{0}^2$ | $\approx 4.4866 \times 10^{-40}\ \rm{C\ m}^2$ | ||
| electric polarizability | $ea_{0}^2/E_{\rm{h}}$ | $\approx 1.6488 \times 10^{-41}\ \rm{C}^2\ \rm{m}^2\ \rm{J}^{-1}$ | ||
| magnetic flux density | $\hbar/ea_{0}^2$ | $\approx 2.3505 \times 10^{5}\ \rm{T}$ | ||
| magnetic dipole moment | $e\hbar/m_{\rm{e}}$ | $\approx 1.8548 \times 10^{-23}\ \rm{J\ T}^{-1}$ | 3 |
(1) $\hbar = h/2\pi; a_{0} = 4\pi \varepsilon_{0} \hbar^2/m_{e}e^2; E_{\rm{h}} = \hbar^2/m_{e}a_{0}^2$.
(2) The numerical value of the speed of light, when expressed in atomic units, is equal to the reciprocal of
the fine-structure constant $\alpha$; $c/(a_{0}E_{\rm{h}}/\hbar) = c\hbar/a_{0}E_{\rm{h}} = \alpha^{-1} = 137.035\;999\;084(21)$
(3) The atomic unit of magnetic dipole moment is twice the Bohr magneton, $\mu+{\rm{B}}$.