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2012-10-30T22:51:46Z

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			{{Refimprove\|date=November 2008}}
			{{More footnotes\|date=November 2008}}
			}}

			The '''classical electron radius''', also known as the [[Hendrik Lorentz\|Lorentz]] radius or the [[Thomson scattering]] length, is based on a classical (i.e., non-[[quantum mechanics\|quantum]]) [[special relativity\|relativistic]] model of the [[electron]]. According to modern research, the electron is assumed to be a [[point particle]] with a [[Point particle#Point charge\|point charge]] and no spatial extent.<ref name="curtis74">
			{{cite book
			\| last = Curtis \| first = L.J.
			\| year = 2003
			\| title = Atomic Structure and Lifetimes: A Conceptual Approach
			\| url = http://books.google.com/?id=KmwCsuvxClAC&pg=PA74
			\| page = 74
			\| publisher = [[Cambridge University Press]]
			\| isbn = 0-521-53635-9
			}}</ref> However, the classical electron radius is calculated as

			:<math>r_\mathrm{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\mathrm{e}} c^2} = 2.817 940 3267(27) \times 10^{-15} \mathrm{m}</math>

			where <math>e</math> and <math>m_{\mathrm{e}}</math> are the [[electric charge]] and the [[mass]] of the electron, <math>c</math> is the [[speed of light]], and <math>\varepsilon_0</math> is the [[permittivity]] of free space.<ref>[[David J. Griffiths]], ''Introduction to Quantum Mechanics'', Prentice-Hall, 1995, p. 155. ISBN 0-13-124405-1</ref>

			In [[Gaussian units\|cgs units]], this becomes more simply

			:<math>r_\mathrm{e} = \frac{e^2}{m_e c^2} = 2.817 940 3267(27)\times 10^{-13} \mathrm{cm}</math>

			with (to three significant digits)

			:<math>e = 4.80 \times 10^{-10} \mathrm{esu}, m_e = 9.11 \times 10^{-28} \mathrm{g}, c =3.00 \times 10^{10} \mathrm{cm/s} \, </math>.

			Using classical [[electrostatics]], the energy required to assemble a sphere of constant [[charge density]], of radius <math>r_e</math> and charge <math>e</math> is

			:<math>E=\frac{3}{5}\,\,\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r_\mathrm{e}}</math>.

			If the charge is on the surface the energy is

			:<math>E=\frac{1}{2}\,\,\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r_\mathrm{e}}</math>.

			Ignoring the factors 3/5 or 1/2, if this is equated to the relativistic energy of the electron (<math>E=mc^2</math>) and solved for <math>r_e</math>, the above result is obtained.

			In simple terms, the classical electron radius is roughly the size the electron would need to have for its mass to be completely due to its electrostatic potential energy - not taking quantum mechanics into account. We now know that quantum mechanics, indeed quantum field theory, is needed to understand the behavior of electrons at such short distance scales, thus the classical electron radius is no longer regarded as the actual size of an electron. Still, the classical electron radius is used in modern classical-limit theories involving the electron, such as non-relativistic [[Thomson scattering]] and the relativistic [[Klein-Nishina formula]]. Also, the classical electron radius is roughly the length scale at which [[renormalization]] becomes important in [[quantum electrodynamics]].

			The classical electron radius is one of a trio of related units of length, the other two being the [[Bohr radius]] <math>a_0</math> and the [[Compton wavelength]] of the electron <math>\lambda_e</math>. The classical electron radius is built from the [[electron mass]] <math>m_e</math>, the [[speed of light]] <math>c</math> and the [[electron charge]] <math>e</math>. The Bohr radius is built from <math>m_e</math>,
			<math>e</math> and [[Planck's constant]] <math>h</math>. The [[Compton wavelength]] is built from <math>m_e</math>, <math>h</math> and <math>c</math>. Any one of these three lengths can be written in terms of any other using the [[fine structure constant]] <math>\alpha</math>:

			:<math>r_e = {\alpha \lambda_e \over 2\pi} = \alpha^2 a_0</math>

			Extrapolating from the initial equation, any mass <math>m_0</math> can be imagined to have an 'electromagnetic radius' similar to the electron's classical radius.

			:<math>r=\frac{k_{C}e^2}{m_0 c^2}=\frac{\alpha\hbar}{m_0 c}</math>

			where <math>k_C</math> is [[Coulomb's constant]], <math>\alpha</math> is the [[fine structure constant]] and <math>\hbar</math> is the reduced [[Planck's constant]].

			== References ==
			<references/>
			* [[CODATA]] value for the [http://physics.nist.gov/cgi-bin/cuu/Value?re\|search_for=classical+electron+radius classical electron radius] at [[NIST]].
			* Arthur N. Cox, Ed. "Allen's Astrophysical Quantities", 4th Ed, Springer, 1999.

			== External links ==
			* [http://math.ucr.edu/home/baez/lengths.html#classical_electron_radius Length Scales in Physics: the Classical Electron Radius]
			* [http://www.alternativephysics.org/book/ElectronStructure.htm Structure and radius of electron], an intuitive explanation

			[[Category:Physical constants]]
			[[Category:Atomic physics]]
			[[Category:Electron]]

en>TakuyaMurata: /* Commutative rings */ format

2012-06-09T12:27:58Z

Commutative rings: format

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en>TakuyaMurata: /* Commutative rings */ format