The Bohr radius (''a''_{0}) is a

Length Scales in Physics: the Bohr Radius

{{Authority control Atomic physics Physical constants Units of length Niels Bohr Atomic radius

physical constant
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, ...

, approximately equal to the most probable distance between the nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucle ...

and the electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...

in a hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...

in its ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...

. It is named after Niels Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...

, due to its role 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 Syste ...

of an atom. Its value is The number in parenthesis denotes the uncertainty
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable or ...

of the last digits.
Definition and value

The Bohr radius is defined as $$a\_0\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; ,$$ where *$\backslash varepsilon\_0$ is thepermittivity of free space
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...

,
*$\backslash hbar$ 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 equivale ...

,
*$m\_$ is the mass of an electron
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 ab ...

,
*$e$ is the elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...

,
*$c$ is 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 ...

in vacuum, and
*$\backslash alpha$ is 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 ele ...

.
The CODATA value of the Bohr radius (in SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...

) is
History

In theBohr 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 Syste ...

for atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...

ic structure, put forward by Niels Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...

in 1913, electrons
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...

orbit a central nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucle ...

under electrostatic attraction. The original derivation posited that electrons have orbital angular momentum in integer multiples of the reduced Planck constant, which successfully matched the observation of discrete energy levels in emission spectra, along with predicting a fixed radius for each of these levels. In the simplest atom, hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...

, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. (It is not ''exactly'' the Bohr radius due to the reduced mass effect. They differ by about 0.05%.)
The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...

as published in 1926. This is further complicated by spin and quantum vacuum effects to produce fine structure
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...

and hyperfine structure
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucl ...

. Nevertheless, the Bohr radius formula remains central in atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...

calculations, due to its simple relationship with fundamental constants (this is why it is defined using the true electron mass rather than the reduced mass, as mentioned above). As such, it became the unit of length in atomic units
The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry calculations. They are named after the physicist Douglas Hartree. By definition, the following f ...

.
In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. The expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

of the radial distance of the electron, by contrast, .
Related constants

The Bohr radius is one of a trio of related units of length, the other two being thereduced Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...

of the electron ($\backslash lambda\_\; /\; 2\backslash pi$) and the classical electron radius
The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...

($r\_$). Any one of these constants can be written in terms of any of the others using the fine-structure constant $\backslash alpha$:
:$r\_\; =\; \backslash alpha\; \backslash frac\; =\; \backslash alpha^2\; a\_0.$
Hydrogen atom and similar systems

The Bohr radius including the effect ofreduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...

in the hydrogen atom is given by
:$\backslash \; a\_0^*\; \backslash \; =\; \backslash fraca\_0\; ,$
where $\backslash mu\; =\; m\_\backslash text\; m\_\backslash text\; /\; (m\_\backslash text\; +\; m\_\backslash text)$ is the reduced mass of the electron–proton system (with $m\_\backslash text$ being the mass of proton). The use of reduced mass is a generalization of the classical two-body problem
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...

when we are outside the approximation that the mass of the orbiting body is negligible compared to the mass of the body being orbited. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly ''larger'' than the Bohr radius ($a^*\_0\; \backslash approx\; 1.00054\backslash ,\; a\_0\; \backslash approx\; 5.2946541\; \backslash times\; 10^$ meters).
This result can be generalized to other systems, such as positronium
Positronium (Ps) is a system consisting of an electron and its antimatter, anti-particle, a positron, bound together into an exotic atom, specifically an onium. Unlike hydrogen, the system has no protons. The system is unstable: the two parti ...

(an electron orbiting a positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...

) and muonium
Muonium is an exotic atom made up of an antimuon and an electron,
which was discovered in 1960 by Vernon W. Hughes
and is given the chemical symbol Mu. During the muon's lifetime, muonium can undergo chemical reactions. Due to the mass diffe ...

(an electron orbiting an anti-muon) by using the reduced mass of the system and considering the possible change in charge. Typically, Bohr model relations (radius, energy, etc.) can be easily modified for these exotic systems (up to lowest order) by simply replacing the electron mass with the reduced mass for the system (as well as adjusting the charge when appropriate). For example, the radius of positronium is approximately $2\backslash ,a\_0$, since the reduced mass of the positronium system is half the electron mass ($\backslash mu\_\; =\; m\_\backslash text/2$).
A hydrogen-like atom
A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such as ...

will have a Bohr radius which primarily scales as $r\_=a\_0/Z$, with $Z$ the number of protons in the nucleus. Meanwhile, the reduced mass ($\backslash mu$) only becomes better approximated by $m\_\backslash text$ in the limit of increasing nuclear mass. These results are summarized in the equation
:$r\_\; \backslash \; =\; \backslash frac\backslash frac\; .$
A table of approximate relationships is given below.
See also

*Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mathrm ...

* Rydberg energy
In spectroscopy, the Rydberg constant, symbol R_\infty for
heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...

References

External links

Length Scales in Physics: the Bohr Radius

{{Authority control Atomic physics Physical constants Units of length Niels Bohr Atomic radius