Editing Classical electron radius

{{Use American English|date = March 2019}}
{{Short description|Physical constant providing length scale to interatomic interactions}}
The '''classical electron radius''' is a combination of fundamental [[Physical quantity|physical quantities]] that define a [[length scale]] for problems involving an electron interacting with [[electromagnetic radiation]]. It links the classical electrostatic self-interaction energy of a homogeneous charge distribution to the electron's relativistic mass-energy.  According to modern understanding, the electron is a [[point particle]] with a [[Point particle#Point charge|point charge]] and no spatial extent.  Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems.  The classical electron radius is given as
: <math>r_\text{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\text{e}} c^2} = 2.817 940 3227(19) \times 10^{-15} \text{ m} = 2.817 940 3227(19) \text{ fm} ,</math>
where <math>e</math> is the [[elementary charge]], <math>m_{\text{e}}</math> is the [[electron mass]], <math>c</math> is the [[speed of light]], and <math>\varepsilon_0</math> is the [[vacuum permittivity|permittivity of free space]].<ref>[[David J. Griffiths]], ''Introduction to Quantum Mechanics'', Prentice-Hall, 1995, p. 155. {{ISBN|0-13-124405-1}}</ref>  This numerical value is several times larger than the [[proton radius|radius of the proton]].

The classical electron radius is sometimes known as the [[Hendrik Lorentz|Lorentz]] radius or the [[Thomson scattering]] length.  It is one of a trio of related scales of length, the other two being the [[Bohr radius]] <math>a_0</math> and the [[reduced Compton wavelength]] of the electron {{math|''ƛ''<sub>e</sub>}}.   Any one of these three length scales can be written in terms of any other using the [[fine-structure constant]] <math>\alpha</math>:
: <math>r_\text{e} = </math> {{math|''ƛ''<sub>e</sub>}} <math>\alpha = a_0 \alpha^2.</math>

== Derivation ==
The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge <math>q</math> into a sphere of a given radius <math>r</math>.<ref>
{{cite book
 | title=University Physics, 11th Ed.
 | last=Young | first=Hugh
 | publisher=Addison Wesley
 | year=2004
 | isbn=0-8053-8684-X
 | location=
 | pages=873
 | quote=
}}</ref>  The electrostatic potential at a distance <math>r</math> from a charge <math>q</math> is
: <math>V(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r} .</math>

To bring an additional amount of charge <math>dq</math> from infinity necessitates putting energy into the system, {{tmath| dU }},  by an amount 
: <math>dU = V(r) dq .</math>   

If the sphere is ''assumed'' to have constant [[charge density]], <math>\rho</math>, then 
: <math>q = \rho \frac{4}{3} \pi r^3</math> and <math>dq = \rho 4 \pi r^2 dr .</math> 

Integrating for <math>r</math> from zero to the final radius <math>r</math> yields the expression for the total energy {{tmath| U }}, necessary to assemble the total charge <math>q</math> into a uniform sphere of radius {{tmath| r }}:
: <math>U = \frac{1}{4\pi\varepsilon_0} \frac{3}{5} \frac{q^2}{r} .</math>  

This is called the electrostatic [[self-energy]] of the object. The charge <math>q</math> is now interpreted as the electron charge, {{tmath| e }}, and the energy <math>U</math> is set equal to the energy-equivalent of the electron's rest mass, {{tmath| m_\text{e} c^2 }}, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density. The radius <math>r</math> is then ''defined'' to be the classical electron radius, {{tmath| r_\text{e} }}, and one arrives at the expression given above.

Note that this derivation does not say that <math>r_\text{e}</math> is the actual radius of an electron.   It only establishes a link between electrostatic self-energy and the energy due to the rest mass of the electron.

== Discussion ==
The classical electron radius appears in the [[classical limit]] of modern theories as well, including non-relativistic [[Thomson scattering]] and the relativistic [[Klein–Nishina formula]].  Also, <math>r_\text{e}</math> is roughly the length scale at which [[renormalization]] becomes important in [[quantum electrodynamics]].  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 [[particle physics]].

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.<ref name="curtis74">  
{{cite book
 | last = Curtis | first = L.J.
 | year = 2003
 | title = Atomic Structure and Lifetimes: A Conceptual Approach
 | url = https://books.google.com/books?id=KmwCsuvxClAC&pg=PA74
 | page = 74
 | publisher = [[Cambridge University Press]]
 | isbn = 0-521-53635-9
}}</ref>
 
== See also ==
* [[Electromagnetic mass]]

== References ==
{{reflist}}

== Further reading==
* 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]

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