Atomic units (au) form a system of units convenient for atomic physics, electromagnetism, and quantum electrodynamics, especially when the focus is on the properties of electrons. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In au, the numerical values of the following six physical constants are all unity by definition:

• Two properties of the electron, its mass and charge;
• Two properties of the hydrogen atom, its Bohr radius and the absolute value of its electric potential energy in the ground state;
• Two constants, Dirac's and that for Coulomb's Law.

Fundamental units

Fundamental Atomic Units
Quantity	Name	Symbol	SI value	Planck unit scale
mass	electron rest mass	$m\_mathrm\{e\}$	9.109 3826(16)×10-31 kg	10-8 kg
length	Bohr radius	$a\_0\; =\; hbar\; /\; (m\_e\; c\; alpha)$	5.291 772 108(18)×10-11 m	10-35 m
charge	elementary charge	$e$	1.602 176 53(14)×10-19 C	10-18 C
angular momentum	Reduced Planck's constant	$hbar\; =\; h/(2\; pi)$	1.054 571 68(18)×10-34 J s	(same)
energy	Hartree energy	$E\_mathrm\{h\}\; =\; m\_mathrm\{e\}\; c^2alpha^2$	4.359 744 17(75)×10-18 J	109 J
electrostatic force constant	Coulomb's constant	$1/(4\; pi\; epsilon\_0)$	8.9875516×109 C-2 N m2	(same)

These six quantities are not independent; to normalize all six quantities to 1, it suffices to normalize any four of them to 1. The normalizations of the Hartree energy and Coulomb's constant, for example, are only an incidental consequence of normalizing the other four quantities.

Some derived units

Derived Atomic Units
Quantity	Expression	SI value	Planck unit scale
time	$hbar\; /\; E\_mathrm\{h\}$	2.418 884 326 505(16)×10-17 s	10-43 s
velocity	$a\_0\; E\_mathrm\{h\}\; /\; hbar$	2.187 691 2633(73)×106 m s-1	108 m s-1
force	$E\_mathrm\{h\}\; /\; a\_0$	8.238 7225(14)×10-8 N	1044 N
current	$e\; E\_mathrm\{h\}\; /\; hbar$	6.623 617 82(57)×10-3 A	1026 A
temperature	$E\_mathrm\{h\}\; /\; k\_mathrm\{B\}$	3.157 7464(55)×105 K	1032 K
pressure	$E\_mathrm\{h\}\; /\; \{a\_0\}^3$	2.942 1912(19)×1013 N m-2	10114 Pa

Comparison with Planck units

Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. To facilitate comparing the two systems of units, the above tables show the order of magnitude, in SI units, of the Planck unit corresponding to each atomic unit. Generally, when an atomic unit is "large" in SI terms, the corresponding Planck unit is "small", and vice versa. It should be kept in mind that au were designed for atomic-scale calculations in the present-day Universe, while Planck units are more suitable for quantum gravity and early-Universe cosmology.

Both au and Planck units normalize the Dirac constant and the Coulomb force constant to 1. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in a vacuum, c. Letting α denote the fine structure constant, the au value of c is α^-1 ≈ 137.036.

Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and a₀, the Bohr radius of the hydrogen atom. Normalizing a₀ to 1 amounts to normalizing the Rydberg constant, R_∞, to 4π/α = 4πc. Given au, the Bohr magneton μ_B=1/2. The corresponding Planck value is e/2m_e. Finally, au normalize a unit of atomic energy to 1, while Planck units normalize to 1 Boltzmann's constant k, which relates energy and temperature.

Quantum mechanics and electrodynamics simplified

The (non-relativistic) Schrödinger equation for an electron in SI units is

$-\; frac\{hbar^2\}\{2m\_e\}\; nabla^2\; psi(mathbf\{r\},\; t)\; +\; V(mathbf\{r\})\; psi(mathbf\{r\},\; t)\; =\; i\; hbar\; frac\{partial\; psi\}\{partial\; t\}\; (mathbf\{r\},\; t)$.

The same equation in au is

$-\; frac\{1\}\{2\}\; nabla^2\; psi(mathbf\{r\},\; t)\; +\; V(mathbf\{r\})\; psi(mathbf\{r\},\; t)\; =\; i\; frac\{partial\; psi\}\{partial\; t\}\; (mathbf\{r\},\; t)$.

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

$hat\; H\; =\; -\; \{\{\{hbar^2\}\; over\; \{2\; m\_e\}\}nabla^2\}\; -\; \{1\; over\; \{4\; pi\; epsilon\_0\}\}\{\{e^2\}\; over\; \{r\}\}$,

while atomic units transform the preceding equation into

$hat\; H\; =\; -\; \{\{\{1\}\; over\; \{2\}\}nabla^2\}\; -\; \{\{1\}\; over\; \{r\}\}$.

Finally, Maxwell's equations take the following elegant form in au:

$nabla\; cdot\; mathbf\{E\}\; =\; 4pirho$

$nabla\; cdot\; mathbf\{B\}\; =\; 0$

$nabla\; times\; mathbf\{E\}\; =\; -alpha\; frac\{partial\; mathbf\{B\}\}\; \{partial\; t\}$

$nabla\; times\; mathbf\{B\}\; =\; alpha\; left(frac\{partial\; mathbf\{E\}\}\; \{partial\; t\}\; +\; 4pi\; mathbf\{J\}\; right)$

(There is actually some ambiguity in defining the atomic unit of magnetic field. The above Maxwell equations use the "Gaussian" convention, in which a plane wave has electric and magnetic fields of equal magnitude. In the "Lorentz force" convention, a factor of α is absorbed into B.)

See also

• Planck units
• Natural units

References

• H. Shull and G. G. Hall, Atomic Units, Nature, volume 184, no. 4698, page 1559 (Nov. 14, 1959)

External links

• CODATA Internationally recommended values of the Fundamental Physical Constants.

The unified atomic mass unit (u), or dalton (Da) or, sometimes, universal mass unit, is a unit of mass used to express atomic and molecular masses. It is the approximate mass of a hydrogen atom, a proton, or a neutron.

The precise definition is that it is one twelfth of the mass of an unbound atom of carbon-12 at rest and in its ground state.

1 u = 1/N_A gram = 1/ (1000 N_A) kg   (where N_A is Avogadro's number)

1 u = =

The atomic mass unit is an older name for the same thing, which differs slightly in definition, and differs in value by one part in 1000.

In biochemistry and molecular biology, when talking about proteins, the term "kilodalton" is used, with the symbol kDa. Because proteins are large molecules, their masses are in kilodaltons, where one kilodalton is 1000 daltons.

The unified atomic mass unit, or dalton, is not an SI unit of mass, but it is accepted for use with SI under either name.

The unit is convenient because one hydrogen atom has a mass of approximately 1 u, and more generally an atom or molecule that contains n protons and neutrons will have a mass approximately equal to n u. (The reason is that a atom contains 6 protons, 6 neutrons and 6 electrons, with the protons and neutrons having about the same mass and the electron mass being negligible in comparison.The mass of the electron is approximately 1/1836 of the mass of the proton.) This is an approximation, since it does not account for the mass contained in the binding energy of an atom's nucleus; this binding energy mass is not a fixed fraction of an atom's total mass. The differences which result from nuclear binding are generally less than , however. Chemical element masses, as expressed in u, would therefore all be close to whole number values (within 2% and usually within 1%) were it not for the fact that atomic weights of chemical elements are averaged values of the various stable isotope masses in the abundances which they naturally occur. For example, chlorine has an atomic weight of because it is composed of 76% and 24% ().

Another reason the unit is used is that it is experimentally much easier and more precise to compare masses of atoms and molecules (determine relative masses) than to measure their absolute masses. Masses are compared with a mass spectrometer (see below).

Avogadro's number (N_A) and the mole are defined so that one mole of a substance with atomic or molecular mass will have a mass of precisely . For example, the molecular mass of a water molecule containing one isotope and two isotopes is , and this means that one mole of this monoisotopic water has a mass of . Water and most molecules consist of a mixture of molecular masses due to naturally occurring isotopes. For this reason these sort of comparisons are more meaningful and practical using molar masses which are generally expressed in g/mol, not u. In other words the one-to-one relationship between daltons and g/mol is true but in order to be used accurately for any practical purpose any calculations must be with isotopically pure substances or involve much more complicated statistical averaging of multiple isotopic compositions.

History

The chemist John Dalton was the first to suggest the mass of one atom of hydrogen as the atomic mass unit. Francis Aston, inventor of the mass spectrometer, later used of the mass of one atom of oxygen-16 as his unit.

Before 1961, the physical atomic mass unit (amu) was defined as of the mass of one atom of oxygen-16, while the chemical atomic mass unit (amu) was defined as of the average mass of an oxygen atom (taking the natural abundance of the different oxygen isotopes into account). Both units are slightly smaller than the unified atomic mass unit, which was adopted by the International Union of Pure and Applied Physics in 1960 and by the International Union of Pure and Applied Chemistry in 1961. Hence, before 1961 physicists as well as chemists used the symbol amu for their respective (and slightly different) atomic mass units. One still sometimes finds this usage in the scientific literature today. However, the accepted standard is now the unified atomic mass unit (symbol u), with: 1 u = 1.000 317 9 amu (physical scale) = 1.000 043 amu (chemical scale). Since 1961, by definition the unified atomic mass unit is equal to one-twelfth of the mass of a carbon-12 atom.

References

See also

• Molecular mass
• Mass-to-charge ratio

External links

• SI website on acceptable non-SI units
• Accepted value of 1u as of 2006
• atomic mass unit