What is the radius of the 1st orbit of He (+)?

Bohr radiusSymbola0 or rBohrNamed afterNiels BohrApproximate values (to three significant digits)SI units5.29×10−11 mnatural units3.27×1024 ℓP

The Bohr radius (a0) 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 atom. Its value is 5.29177210903(80)×10−11 m.[1][2]

Definition and value

The Bohr radius is defined as[3]

a 0 = 4 π ε 0 ℏ 2 e 2 m e = ε 0 h 2 π e 2 m e = ℏ m e c α , {\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{e^{2}m_{\text{e}}}}={\frac {\varepsilon _{0}h^{2}}{\pi e^{2}m_{\text{e}}}}={\frac {\hbar }{m_{\text{e}}c\alpha }},}

where

• ε 0 {\displaystyle \varepsilon _{0}}
  
  is the permittivity of free space,
• ℏ {\displaystyle \hbar }
  
  is the reduced Planck constant,
• m e {\displaystyle m_{\text{e}}}
  
  is the mass of an electron,
• e {\displaystyle e}
  
  is the elementary charge,
• c {\displaystyle c}
  
  is the speed of light in vacuum, and
• α {\displaystyle \alpha }
  
  is the fine-structure constant.

The CODATA value of the Bohr radius (in SI units) is 5.29177210903(80)×10−11 m.[1]

History

In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus 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, 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The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrödinger equation as published in 1926. This is further complicated by spin and quantum vacuum effects to produce fine structure and hyperfine structure. Nevertheless, the Bohr radius formula remains central in atomic physics 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.

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 of the radial distance of the electron, by contrast, is 3/2a0.[4]

Related constants

The Bohr radius is one of a trio of related units of length, the other two being the reduced Compton wavelength of the electron ( λ e / 2 π {\displaystyle \lambda _{\mathrm {e} }/2\pi }

r e {\displaystyle r_{\mathrm {e} }}

r e = α λ e 2 π = α 2 a 0 . {\displaystyle r_{\mathrm {e} }=\alpha {\frac {\lambda _{\mathrm {e} }}{2\pi }}=\alpha ^{2}a_{0}.}

Hydrogen atom and similar systems

The Bohr radius including the effect of reduced mass in the hydrogen atom is given by

  a 0 ∗   = m e μ a 0 , {\displaystyle \ a_{0}^{*}\ ={\frac {m_{\text{e}}}{\mu }}a_{0},}

where μ = m e m p / ( m e + m p ) {\textstyle \mu =m_{\text{e}}m_{\text{p}}/(m_{\text{e}}+m_{\text{p}})}

m p {\displaystyle m_{\text{p}}}

a 0 ∗ ≈ 1.00054 a 0 ≈ 5.2946541 × 10 − 11 {\displaystyle a_{0}^{*}\approx 1.00054\,a_{0}\approx 5.2946541\times 10^{-11}}

This result can be generalized to other systems, such as positronium (an electron orbiting a positron) and muonium (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 a 0 {\displaystyle 2\,a_{0}}

μ e − , e + = m e / 2 {\displaystyle \mu _{{\text{e}}^{-},{\text{e}}^{+}}=m_{\text{e}}/2}

A hydrogen-like atom will have a Bohr radius which primarily scales as r Z = a 0 / Z {\displaystyle r_{Z}=a_{0}/Z}

Z {\displaystyle Z}

μ {\displaystyle \mu }

r Z , μ   = m e μ a 0 Z . {\displaystyle r_{Z,\mu }\ ={\frac {m_{\text{e}}}{\mu }}{\frac {a_{0}}{Z}}.}

A table of approximate relationships is given below.

System	Radius
Hydrogen	1.00054 a 0 {\displaystyle 1.00054\,a_{0}}
Positronium	2 a 0 {\displaystyle 2a_{0}}
Muonium	1.0048 a 0 {\displaystyle 1.0048\,a_{0}}
He+	a 0 / 2 {\displaystyle a_{0}/2}
Li2+	a 0 / 3 {\displaystyle a_{0}/3}

See also

• Bohr magneton
• Rydberg energy

References

1. ^ a b "2018 CODATA Value: Bohr radius". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
2. ^ The number in parenthesis denotes the uncertainty of the last digits.
3. ^ David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 137. ISBN 0-13-124405-1
4. ^ Nave, Rod. "The Most Probable Radius: Hydrogen Ground State". HyperPhysics. Dept. of Physics and Astronomy, Georgia State University. Retrieved 2 October 2021. The Schrodinger equation confirms the first Bohr radius as the most probable radius.

External links

• Length Scales in Physics: the Bohr Radius

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