World Library  
Flag as Inappropriate
Email this Article

Bohr radius

Article Id: WHEBN0000174396
Reproduction Date:

Title: Bohr radius  
Author: World Heritage Encyclopedia
Language: English
Subject: Conversion of units, Niels Bohr, Natural units, Bohr model, Unit of length/BigsmalllP
Collection: Atomic Physics, Physical Constants, Units of Length
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Bohr radius

Bohr radius
Symbol: a0
Named after: Niels Bohr
Value in meters: ≈ 5.29×1011m
Value in picometers: ≈ 52.9 pm
Value in angstroms: ≈ 0.529 Å
Value in natural units: ≈ 2.68×104 1/eV

The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the proton and 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 5.2917721092(17)×10−11 m[1][note 1]

Contents

  • Definition & value 1
  • Use 2
  • Related units 3
  • Reduced Bohr radius 4
  • See also 5
  • Notes 6
  • References 7
  • External links 8

Definition & value

In SI units the Bohr radius is:[2]

a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_{\mathrm{e}} e^2} = \frac{\hbar}{m_{\mathrm{e}}\,c\,\alpha}

where:

a_0 is the Bohr radius,
\varepsilon_0 \ is the permittivity of free space,
\hbar \ is the reduced Planck's constant,
m_{\mathrm{e}} \ is the electron rest mass,
e \ is the elementary charge,
c \ is the speed of light in vacuum, and
\alpha \ is the fine structure constant.

In Gaussian units the Bohr radius is simply

a_0=\frac{\hbar^2}{m_e e^2}

According to 2010 CODATA the Bohr radius has a value of 5.2917721092(17)×10−11 m (i.e., approximately 53 pm or 0.53 angstroms).[1][note 1]

Use

In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy. In the simplest atom, hydrogen, a single electron orbits the nucleus and its smallest possible orbit, with 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 0.1%.)

Although the Bohr model is no longer in use, the Bohr radius remains very useful in atomic physics calculations, due in part to its simple relationship with other fundamental constants. (This is why it is defined using the true electron mass rather than the reduced mass, as mentioned above.) For example, it is the unit of length in atomic units.

An important distinction is that the Bohr radius describes the most probable radial distance of the electron, not its expected radial distance. The expected radial distance is actually 1.5 times the Bohr radius, as a result of the long tail of the radial wave function.

Related units

The Bohr radius of the electron is one of a trio of related units of length, the other two being the Compton wavelength of the electron \lambda_{\mathrm{e}} \ and the classical electron radius r_{\mathrm{e}} \ . The Bohr radius is built from the electron mass m_{\mathrm{e}}, Planck's constant \hbar \ and the electron charge e \ . The Compton wavelength is built from m_{\mathrm{e}} \ , \hbar \ and the speed of light c \ . The classical electron radius is built from m_{\mathrm{e}} \ , c \ and e \ . Any one of these three lengths can be written in terms of any other using the fine structure constant \alpha \ :

r_{\mathrm{e}} = \frac{\alpha \lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0.

The Compton wavelength is about 20 times smaller than the Bohr radius, and the classical electron radius is about 1000 times smaller than the Compton wavelength.

Reduced Bohr radius

The Bohr radius including the effect of reduced mass in the hydrogen atom can be given by the following equation:

\ a_0^* \ = \frac{\lambda_{\mathrm{p}} + \lambda_{\mathrm{e}}}{2\pi\alpha},

where:

\lambda_{\mathrm{p}} \ is the Compton wavelength of the proton.
\lambda_{\mathrm{e}} \ is the Compton wavelength of the electron.
\alpha \ is the fine structure constant.

In the above equation, the effect of the reduced mass is achieved by using the increased Compton wavelength, which is just the Compton wavelengths of the electron and the proton added together.

See also

Notes

  1. ^ a b The number in parenthesis (17) denotes the uncertainty of the last digits.

References

  1. ^ a b "CODATA Value: Bohr radius". Fundamental Physical Constants.  
  2. ^ David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 137. ISBN 0-13-124405-1

External links

  • Length Scales in Physics: the Bohr Radius
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.