physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, natural units are

physical units A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...

of measurement in which only universal

physical constants A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...

are used as defining constants, such that each of these constants acts as a

coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deri ...

unit of a quantity. For example, the elementary charge may be used as a natural unit of

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...

, and the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

may be used as a natural unit of

speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...

. A purely natural

system of units A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement i ...

has all of its units defined such that each of these can be expressed as a product of powers of defining physical constants. Through

nondimensionalization Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...

, physical quantities may then redefined so that the defining constants can be omitted from mathematical expressions of physical laws, and while this has the apparent advantage of simplicity, it may entail a loss of clarity due to the loss of information for

dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...

. It precludes the interpretation of an expression in terms of constants, such as and , unless it is ''known'' which units (in dimensionful units) the expression is supposed to have. In this case, the reinsertion of the correct powers of , , etc., can be uniquely determined.

Systems of natural units

Planck units

The Planck unit system uses the following defining constants: :, , , , where is the

, is the

reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...

, is the gravitational constant, and is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...

. Planck units form a system of natural units that is not defined in terms of properties of any prototype, physical object, or even

elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, a ...

. They only refer to the basic structure of the laws of physics: and are part of the structure of

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, and is at the foundation of

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

. This makes Planck units particularly convenient and common in theories of quantum gravity, including string theory. Planck considered only the units based on the universal constants , , , and _B to arrive at natural units for length,

time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

, and

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...

, but no electromagnetic units. The Planck system of units is now understood to use the reduced Planck constant, , in place of the Planck constant, .Tomilin, K. A., 1999,
Natural Systems of Units: To the Centenary Anniversary of the Planck System
", 287–296.

Stoney units

The Stoney unit system uses the following defining constants: :, , , , where is the

, is the gravitational constant, is the Coulomb constant, and is the elementary charge. George Johnstone Stoney's unit system preceded that of Planck. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the

British Association The British Science Association (BSA) is a charity and learned society founded in 1831 to aid in the promotion and development of science. Until 2009 it was known as the British Association for the Advancement of Science (BA). The current Chie ...

in 1874. Stoney units did not consider the

Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...

, which was discovered only after Stoney's proposal. Stoney units are rarely used in modern physics for calculations, but they are of historical interest.

Atomic units

The Hartree atomic unit system uses the following defining constants: :, , , . The Coulomb constant, , is generally expressed as when working with this system. These units are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom, and are widely used in these fields. The Hartree units were first proposed by

Douglas Hartree Douglas Rayner Hartree (27 March 1897 – 12 February 1958) was an English mathematician and physicist most famous for the development of numerical analysis and its application to the Hartree–Fock equations of atomic physics and the ...

. The units are designed especially to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, in Hartree atomic units, in the

Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...

of the hydrogen atom an electron in the ground state has orbital radius (the

Bohr radius The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...

) ₀ = 1 , orbital velocity = 1 ⋅, angular momentum = 1 ⋅⋅, ionization energy = ⋅⋅, etc. The unit of

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...

is called the Hartree energy in the Hartree system. The

is relatively large in Hartree atomic units ( = ⋅ ≈ 137 ⋅) since an electron in hydrogen tends to move much more slowly than the speed of light. The gravitational constant is extremely small in atomic units ( ≈ 10⁻⁴⁵ ⋅⋅), which is due to the gravitational force between two electrons being far weaker than the Coulomb force between them. A less commonly used closely related system is the system of Rydberg atomic units, in which are used as the defining constants, with resulting units = = , = , = 2, = .

Natural units (particle and atomic physics)

This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants: :, , , , where is the

, _e is the

electron mass The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of ...

, is the

, and ₀ is the vacuum permittivity. The vacuum permittivity ₀ is implicitly used as a

constant, as is evident from the physicists' expression for the

fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...

, written , which may be compared to the same expression in SI: .

Quantum chromodynamics units

Defining constants: :, , , . Here, is the proton rest mass. ''Strong units'', also called quantum chromodynamics (QCD) units, are "convenient for work in QCD and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".

Geometrized units

Defining constants: :, . The geometrized unit system, used in

, is an incompletely defined system. In this system, the base physical units are chosen so that the

and the gravitational constant are coherent units and often used for nondimensionalization. Other units may be treated however desired. Planck units and Stoney units are examples of geometrized unit systems.

Summary table

where: * is the

( ≈ 0.007297) * ≈ * ≈ *A dash (–) indicates where the system is not sufficient to express the quantity.

Notes and references

External links

The NIST website
(

National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...

) is a convenient source of data on the commonly recognized constants.
K.A. Tomilin: ''NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System''
A comparative overview/tutorial of various systems of natural units having historical use.
Pedagogic Aides to Quantum Field Theory
Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units.
Natural System Of Units In General Relativity (PDF)
by Alan L. Myers (University of Pennsylvania). Equations for conversions from natural to SI units. {{DEFAULTSORT:Natural Units Metrology bs:Prirodne jedinice sr:Природне јединице sh:Prirodne jedinice