SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Classical electron radius</title>

</head>

Classical electron radius

</h1>

</header>

The classical electron radius is a combination of fundamental <a href="Physical_quantity" title="Physical quantity">physical quantities</a> that define a <a href="Length_scale" title="Length scale">length scale</a> for problems involving an electron interacting with <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>. It links the classical electrostatic self-interaction energy of a homogeneous charge distribution to the electron's <a href="Rest_mass_energy" class="mw-redirect" title="Rest mass energy">rest mass energy</a>. According to modern understanding, the electron has no internal structure, and hence no size attributable to it. Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The <a href="CODATA" class="mw-redirect" title="CODATA">CODATA</a> value for the classical electron radius is<a href="#cite_note-physconst-re-1">[1]</a>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=}</annotation>

</semantics>

</math><img src="./780fc2a6c28fd23ab3bad8663ca1d0c480a5bb20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.931ex; height:6.343ex;" alt="{\displaystyle r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=}" loading="lazy"> 2.8179403205(13)×10−15 m</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> is the <a href="Elementary_charge" title="Elementary charge">elementary charge</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{e}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{\text{e}}}</annotation>

</semantics>

</math><img src="./c45a5eb082ea4dafd3cb43ca39e033989e4a52eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.009ex;" alt="{\displaystyle m_{\text{e}}}" loading="lazy"> is the <a href="Electron_mass" title="Electron mass">electron mass</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is the <a href="Speed_of_light" title="Speed of light">speed of light</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

</semantics>

</math><img src="./acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"> is the <a href="Vacuum_permittivity" title="Vacuum permittivity">permittivity of free space</a>.<a href="#cite_note-2">[2]</a> This is about three times larger than the <a href="Proton_radius" class="mw-redirect" title="Proton radius">charge radius of the proton</a>.

The classical electron radius is sometimes known as the <a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a> radius or the <a href="Thomson_scattering" title="Thomson scattering">Thomson scattering</a> length. It is one of a trio of related scales of length, the other two being the <a href="Bohr_radius" title="Bohr radius">Bohr radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation>

</semantics>

</math><img src="./693ad9f934775838bd72406b41ada4a59785d7ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}" loading="lazy"> and the <a href="Reduced_Compton_wavelength" class="mw-redirect" title="Reduced Compton wavelength">reduced Compton wavelength</a> of the electron ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \!\!\!{\bar {}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \!\!\!{\bar {}}}</annotation>

</semantics>

</math><img src="./55eccfa96819e2faedb93ba0b79c6c16b6beb1c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.176ex;" alt="{\displaystyle \lambda \!\!\!{\bar {}}}" loading="lazy">⁠. Any one of these three length scales can be written in terms of any other using the <a href="Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">:

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{e}}=\lambda \!\!\!{\bar {}}\alpha =a_{0}\alpha ^{2}.}</annotation>

</semantics>

</math><img src="./4cbe89441fbaed0942c2be5cc5d7f2149e36fb6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.525ex; height:3.009ex;" alt="{\displaystyle r_{\text{e}}=\lambda \!\!\!{\bar {}}\alpha =a_{0}\alpha ^{2}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2></div>

The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> into a sphere of a given radius ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">⁠.<a href="#cite_note-3">[3]</a> The electrostatic potential at a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> from a charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r}}.}</annotation>

</semantics>

</math><img src="./d026b2835c2fbfc8f5f4f771fd607cfcd3bb8820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.764ex; height:5.676ex;" alt="{\displaystyle V(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r}}.}" loading="lazy"></dd></dl>

To bring an additional amount of charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dq}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dq}</annotation>

</semantics>

</math><img src="./212a4169ef66be3955464b7b64e185e2b78ef365.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle dq}" loading="lazy"> from infinity adds energy ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dU}</annotation>

</semantics>

</math><img src="./fad9083e6f626fcc9e4217e8fd3d751e1126da9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.998ex; height:2.176ex;" alt="{\displaystyle dU}" loading="lazy">⁠ to the system:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dU=V(r)dq.}</annotation>

</semantics>

</math><img src="./0821873336402048f886c1cced8639f6972a5335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.674ex; height:2.843ex;" alt="{\displaystyle dU=V(r)dq.}" loading="lazy"></dd></dl>

If the sphere is assumed to have constant <a href="Charge_density" title="Charge density">charge density</a>, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">⁠, then

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=\rho {\frac {4}{3}}\pi r^{3}}</annotation>

</semantics>

</math><img src="./d07a00e7e2aa7f4f72899a3e51d145d14c14fa1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.803ex; height:5.176ex;" alt="{\displaystyle q=\rho {\frac {4}{3}}\pi r^{3}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dq=\rho 4\pi r^{2}dr.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dq=\rho 4\pi r^{2}dr.}</annotation>

</semantics>

</math><img src="./bc162247c84219e30ed6373aefb895ce73d469d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:3.176ex;" alt="{\displaystyle dq=\rho 4\pi r^{2}dr.}" loading="lazy"></dd></dl>

Integrating ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dU}</annotation>

</semantics>

</math><img src="./fad9083e6f626fcc9e4217e8fd3d751e1126da9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.998ex; height:2.176ex;" alt="{\displaystyle dU}" loading="lazy">⁠ for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> from zero to a final radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r'}</annotation>

</semantics>

</math><img src="./20f4d06078c3550f2cd0812005ba6301d12cc4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}" loading="lazy"> yields the expression for the total energy ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">⁠, necessary to assemble the total charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q'}</annotation>

</semantics>

</math><img src="./912405e5d416048908ea7978929975843d2ee4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle q'}" loading="lazy"> uniformly into a sphere of radius ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r'}</annotation>

</semantics>

</math><img src="./20f4d06078c3550f2cd0812005ba6301d12cc4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}" loading="lazy">⁠:

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

<mrow>

</mrow>

</msup>

<msup>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U={\frac {3}{5}}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {q'^{2}}{r'}}.}</annotation>

</semantics>

</math><img src="./3b0416e53b6871b823cf122e7d9e83679a79f85b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.417ex; height:6.176ex;" alt="{\displaystyle U={\frac {3}{5}}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {q'^{2}}{r'}}.}" loading="lazy"></dd></dl>

This is called the electrostatic <a href="Self-energy" title="Self-energy">self-energy</a> of the object. Interpreting the charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q'}</annotation>

</semantics>

</math><img src="./912405e5d416048908ea7978929975843d2ee4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle q'}" loading="lazy"> as the electron charge, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -e}</annotation>

</semantics>

</math><img src="./07ba5105b33e9ac88f605cc50663a9e2c16fcc63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.892ex; height:2.176ex;" alt="{\displaystyle -e}" loading="lazy">⁠, and equating the total energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> with the energy-equivalent of the electron's rest mass, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{e}}c^{2}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{\text{e}}c^{2}}</annotation>

</semantics>

</math><img src="./1b24ce4c2bf079105ab2aade239cafc1dd26f3db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.064ex; height:3.009ex;" alt="{\displaystyle m_{\text{e}}c^{2}}" loading="lazy">⁠, and solving for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r'}</annotation>

</semantics>

</math><img src="./20f4d06078c3550f2cd0812005ba6301d12cc4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}" loading="lazy">⁠:

<msup>

</msup>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r'={\frac {3}{5}}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}.}</annotation>

</semantics>

</math><img src="./a457ea65b3a7ce6af225dc7da112aa411fcd1903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.845ex; height:6.343ex;" alt="{\displaystyle r'={\frac {3}{5}}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}.}" loading="lazy"></dd></dl>

The numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density (e.g., for a charged spherical surface, this factor is 1/2). The resulting radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r'}</annotation>

</semantics>

</math><img src="./20f4d06078c3550f2cd0812005ba6301d12cc4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}" loading="lazy"> adjusted to ignore this factor is then defined to be the classical electron radius, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{e}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{e}}}</annotation>

</semantics>

</math><img src="./7b5a847a05d815b4a89c83e1bb4a12a39d9029c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.009ex;" alt="{\displaystyle r_{\text{e}}}" loading="lazy">⁠, and one arrives at the expression given above.

Note that this derivation does not say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{e}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{e}}}</annotation>

</semantics>

</math><img src="./7b5a847a05d815b4a89c83e1bb4a12a39d9029c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.009ex;" alt="{\displaystyle r_{\text{e}}}" loading="lazy"> is an indication of the actual radius of an electron. It only establishes a link between electrostatic self-energy and the energy-equivalent of the rest mass of the electron, and neglects the energy in the magnetic dipole field of an electron, which if considered, leads to a substantially larger calculated radius.

<div class="mw-heading mw-heading2"><h2 id="Discussion">Discussion</h2></div>

The classical electron radius appears in the <a href="Classical_limit" title="Classical limit">classical limit</a> of modern theories as well, including non-relativistic <a href="Thomson_scattering" title="Thomson scattering">Thomson scattering</a> and the relativistic <a href="Klein%E2%80%93Nishina_formula" title="Klein–Nishina formula">Klein–Nishina formula</a>. Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{e}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{e}}}</annotation>

</semantics>

</math><img src="./7b5a847a05d815b4a89c83e1bb4a12a39d9029c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.009ex;" alt="{\displaystyle r_{\text{e}}}" loading="lazy"> is roughly the length scale at which <a href="Renormalization" title="Renormalization">renormalization</a> becomes important in <a href="Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. That is, at short-enough distances, quantum fluctuations within the vacuum of space surrounding an electron begin to have calculable effects that have measurable consequences in atomic and <a href="Particle_physics" title="Particle physics">particle physics</a>.

Based on the assumption of a simple mechanical model, attempts to model the electron as a non-point particle have been described by some as ill-conceived and counter-pedagogic.<a href="#cite_note-curtis74-4">[4]</a>

<ul><li><a href="Electromagnetic_mass" title="Electromagnetic mass">Electromagnetic mass</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?re">"2022 CODATA Value: classical electron radius"</a>. The NIST Reference on Constants, Units, and Uncertainty. <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a>. May 2024. Retrieved 2024-05-18.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFD._J._Griffiths1995" class="citation cs2"><a href="David_J._Griffiths" title="David J. Griffiths">D. J. Griffiths</a> (1995), Introduction to Quantum Mechanics, Prentice-Hall, p. 155, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-124405-1</bdi></cite>

</li>

<cite id="CITEREFYoung2004" class="citation book cs1">Young, Hugh (2004). University Physics, 11th Ed. Addison Wesley. p. 873. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-8684-X</bdi>.</cite>

</li>

<cite id="CITEREFCurtis2003" class="citation book cs1">Curtis, L.J. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KmwCsuvxClAC&pg=PA74">Atomic Structure and Lifetimes: A Conceptual Approach</a>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 74. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-53635-9</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFArthur_N._Cox1999" class="citation cs2">Arthur N. Cox, ed. (1999), Allen's Astrophysical Quantities (4th ed.), Springer</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/lengths.html#classical_electron_radius">Length Scales in Physics: the Classical Electron Radius</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-06" href="https://en.wikipedia.org/wiki/?title=Classical_electron_radius&oldid=1294168735">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>