Atomic units
The atomic units, abbreviated a.u. is a set of units used in atomic calculations.^{[1]}^{[2]} In the a.u. system any four of the five quantities charge e, mass m_{e}, action ℏ, length a_{0}, and energy E_{h} may be taken as base quantities, and other quantities are derived.^{[3]} A listing of numerical values in terms of SI units can be found on the NIST website.^{[4]}
Example units
The expressions for a few example atomic units in terms of formulas in SI units are discussed next.
Mass
The atomic unit of mass is the electron mass, m_{e}.^{[5]} However, confusingly, the terminology "atomic unit of mass" often is taken instead to refer to the unified atomic mass unit, the Dalton, symbol u. The Dalton is defined as 1/12 the mass of a free carbon 12 atom,^{[6]} with a value of 1.660538921(73) × 10^{-27} kg.,^{[7]} or 931.494 061(21) MeV.^{[8]} A proton has a mass of approximately 1.007 276 466 812 u.^{[9]} The electron mass is about 5.485 799 0946 × 10^{-4} u.^{[10]}
Length
In SI units the a.u. unit of length, the Bohr radius a_{0} or bohr, is:^{[11]}
where R_{∞} is the Rydberg constant, e is the elementary charge, ε_{0} is the electric constant, ℏ is the reduced Planck's constant h/(2π), m_{e} is the electron mass, and where the (dimensionless) fine structure constant α is given by (in SI units):
and has the value:^{[12]}
The Bohr radius was the distance of an electron from the nucleus of a hydrogen atom predicted by the Bohr theory of the atom, which required an integer number of wavelengths around the electron orbit.^{[13]} In modern quantum mechanics the Bohr radius is the distance of maximum likelihood for finding the electron in the hydrogen atom in its ground state. ^{[14]}
Energy
The unit of energy, the hartree, is the energy of two a.u. charges separated by one bohr in a medium of permittivity given by 1 a.u. of permittivity, 4πε_{0}:^{[15]}
Time
Somewhat unusually, time is a derived quantity, ℏ/E_{h}, with the interpretation as the period of an electron circling in the first Bohr orbit divided by 2π.^{[16]}^{[17]}
Velocity
Using the unit of time, and the expression for the hartree, the a.u. unit of velocity is one bohr per a.u. unit of time:
Here c_{0} is the SI units defined speed of light in classical vacuum and α is the fine structure constant. Its value is:^{[18]}
The a.u. unit of velocity changes size with refinement in measurement of the fine structure constant. However, the defined value of c_{0} = v_{B}/α is unaffected by such refinements. See the articles speed of light and metre for more detail about the defined value c_{0} = 299 792 458 m/s (exactly).
Tabulation
Basic atomic units ^{[4]} | |||
---|---|---|---|
Name | Symbol | Quantity | Value in SI units |
elementary charge | e | charge | 1.602 176 565(35) × 10^{−19} C |
Bohr radius (bohr) | a_{0} | length | 0.529 177 210 92(17) × 10^{−10} m |
electron mass | m_{e} | mass | 9.109 382 91(40) × 10^{−31} kg |
reduced Planck constant | ℏ | action | 1.054 571 726(47) × 10^{−34} Js |
Hartree energy (hartree) | E_{h} | energy | 4.359 744 34(19) × 10^{−18} J |
Evidently in atomic units, if we choose e, ℏ, m_{e}, and a_{0} as the four basic units, then these entities all become 1 in algebraic expressions, and because a_{0} = (4πε_{0})(ℏ/e)^{2}/m_{e}, it follows that 4πε_{0} = 1 as well, defining the a.u. unit of permittivity. The hartree, E_{h} = (4πε_{0})^{−1} e^{2}/a_{0} also is automatically 1.
Derived atomic units ^{[4]}^{[1]}^{[19]} | |||
---|---|---|---|
Name | Formula | Quantity | Value in SI units |
a.u. velocity | v_{B} = αc_{0} = a_{0}E_{h}/ℏ | velocity | 2.187 691 263 79(71) × 10^{6} m/s |
a.u. time | ℏ/E_{h} | time | 2.418 884 326 502(12) × 10^{−17} s |
a.u. current | eE_{h}/ℏ | current | 6.623 617 95(15) × 10^{−3} A |
a.u. electric potential | E_{h}/e | electric potential | 27.211 385 05(60) V |
a.u. magnetic flux density | ℏ/ea_{0}^{2} | magnetic flux density | 2.350 517 464(52) × 10^{5} T |
a.u. magnetic dipole moment | ℏe/m_{e} = 2μ_{B} | magnetic dipole moment | 1.854 801 936(41) × 10^{−23} J T^{−1} |
a.u. permittivity | e^{2}/a_{0}E_{h} = 10^{7}/c_{0}^{2}=4πε_{0} | permittivity | 1.112 650 056... × 10^{−10} F m^{−1} (exact) |
Here, c_{0} = SI units defined value for the speed of light in classical vacuum, ε_{0} is the electric constant, α = fine structure constant and μ_{B} is the Bohr magneton.^{[20]}
Notes
- ↑ ^{1.0} ^{1.1} For an introduction, see Gordon W. F. Drake (2006). “§1.2 Atomic units”, Springer handbook of atomic, molecular, and optical physics, Volume 1, 2nd ed. Springer, p. 6. ISBN 038720802X.
- ↑ Tabulated values from (2008) Barry N. Taylor, Ambler Thompson: International System of Units (SI), NIST special publication 330 • 2008 ed. DIANE Publishing, Table 7, p.34. ISBN 1437915582.
- ↑ According to Taylor, as cited above, page 33: "any four of the five quantities charge, mass, action, length and energy are taken as base quantities."
- ↑ ^{4.0} ^{4.1} ^{4.2} Fundamental physical constants: atomic units. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Fundamental physical constants: atomic unit of mass, m_{e}. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-08.
- ↑ Ambler Thompson (2009). “Table 7: Non-SI units accepted for use with the SI”, Guide for the Use of the International System of Units (SI) (rev. ): The Metric System. DIANE Publishing, p. 9. ISBN 1437915590.
- ↑ Fundamental physical constants: atomic mass unit-kilogram relationship 1 u. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-08.
- ↑ Fundamental physical constants: atomic mass unit-electron volt relationship 1 u (c_{0}^{2}). The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Fundamental physical constants: proton mass in u: m_{p}. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Fundamental physical constants: electron mass in u: m_{e}. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Fundamental physical constants: atomic unit of length a_{0}. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Fundamental physical constants: fine-structure constant α. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ A discussion can be found in W. Demtröder (2006). “Chapter 3: Development of quantum physics”, Atoms, molecules and photons: an introduction to atomic-, molecular-, and quantum-physics. Birkhäuser, pp. 112ff. ISBN 3540206310.
- ↑ See previously cited work W. Demtröder (2006). “§5.1.4 Spatial distributions and expectation values of the electron in different quantum states”, Atoms, molecules and photons: an introduction to atomic-, molecular-, and quantum-physics. Birkhäuser, p. 164.
- ↑ Fundamental physical constants: atomic unit of energy E_{h}. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Volker Schmidt (1997). “§6.1 Atomic units”, Electron spectrometry of atoms using synchrotron radiation. Cambridge University Press, pp. 273 ff. ISBN 052155053X.
- ↑ Fundamental physical constants: atomic unit of time ℏ/E_{h}. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ Fundamental physical constants: atomic unit of time a_{0}E_{h}/ℏ. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
- ↑ PJ Mohr, BN Taylor, and DB Newell (2008). "CODATA recommended values of the fundamental physical constants: 2006; Table LIII". Rev. Mod. Phys. vol. 80 (No. 2): p. 717.
- ↑ An overview of the importance and determination of the fine structure constant is found in G. Gabrielse (2010). “Determining the fine structure constant”, B. Lee Roberts, William J. Marciano, eds: Lepton dipole moments. World Scientific, pp. 195 ff. ISBN 9814271837.