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==Brillouin Function==
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


The '''Brillouin function'''<ref name=Kittel>C. Kittel, ''Introduction to Solid State Physics'' (8th ed.), pages 303-4 ISBN 978-0-471-41526-8</ref><ref>{{Cite journal
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
  | last = Darby
* Only registered users will be able to execute this rendering mode.
  | first = M.I.  
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  | author-link =
  | title = Tables of the Brillouin function and of the related function for the spontaneous magnetization
  | journal = Brit. J. Appl. Phys.
  | volume = 18
  | issue = 10
  | pages = 1415–1417
  | year = 1967
  | doi =10.1088/0508-3443/18/10/307
  | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->  |bibcode = 1967BJAP...18.1415D }}</ref> is a [[special function]] defined by the following equation:
<blockquote style="border: 1px solid black; padding:10px;">
:<math>B_J(x) = \frac{2J + 1}{2J} \coth \left ( \frac{2J + 1}{2J} x \right )
                - \frac{1}{2J} \coth \left ( \frac{1}{2J} x \right )</math>
</blockquote>
The function is usually applied (see below) in the context where ''x'' is a real variable and ''J'' is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as <math>x \to +\infty</math> and -1 as <math>x \to -\infty</math>.


The function is best known for arising in the calculation of the [[magnetization]] of an ideal [[paramagnet]]. In particular, it describes the dependency of the magnetization <math>M</math> on the applied [[magnetic field]] <math>B</math> and the [[total angular momentum quantum number]] J of the microscopic [[magnetic moment]]s of the material. The magnetization is given by:<ref name=Kittel/>
Registered users will be able to choose between the following three rendering modes:  
:<math>M = N g \mu_B J \cdot B_J(x)</math>


where
'''MathML'''
*<math>N</math> is the number of atoms per unit volume,
:<math forcemathmode="mathml">E=mc^2</math>
*<math>g</math> the [[g-factor (physics)|g-factor]],
*<math>\mu_B</math> the [[Bohr magneton]],
*<math>x</math> is the ratio of the [[Zeeman effect|Zeeman]] energy of the magnetic moment in the external field  to the thermal energy <math>k_B T</math>:
::<math>x = \frac{g \mu_B J B}{k_B T}</math>
*<math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> the temperature.


Note that in the SI system of units <math>B</math> given in Tesla stands for [[magnetic induction]], <math>B=\mu_0 H</math>, where <math>H</math> is the applied [[magnetic field]] given in A/m and <math>\mu_0</math> is the [[permeability of vacuum]].
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
'''source'''
!Click "show" to see a derivation of this law:
:<math forcemathmode="source">E=mc^2</math> -->
|-
|A derivation of this law describing the magnetization of an ideal paramagnet is as follows.<ref name=Kittel/> Let '''z''' be the direction of the magnetic field. The z-component of the angular momentum of each magnetic moment (a.k.a. the [[azimuthal quantum number]]) can take on one of the 2J+1 possible values -J,-J+1,...,+J. Each of these has a different energy, due to the external field '''B''': The energy associated with quantum number ''m'' is
:<math>E_m = -mg \mu_B B = -k_BTxm/J</math>
(where ''g'' is the [[g-factor (physics)|g-factor]], μ<sub>B</sub> is the [[Bohr magneton]], and ''x'' is as defined in the text above). The relative probability of each of these is given by the [[Boltzmann factor]]:
:<math>P(m)=e^{-E_m/(k_BT)}/Z=e^{xm/J}/Z</math>
where ''Z'' (the [[Partition function (statistical mechanics)|partition function]]) is a normalization constant such that the probabilities sum to unity. Calculating ''Z'', the result is:
:<math>P(m) = e^{xm/J}/\left(\sum_{m'=-J}^J e^{xm'/J}\right)</math>.
All told, the [[expectation value]] of the azimuthal quantum number ''m'' is
:<math>\langle m \rangle = (-J)\times P(-J) + \cdots + J\times P(J) = \left(\sum_{m=-J}^J m e^{xm/J}\right)/ \left(\sum_{m=-J}^J e^{xm/J}\right)</math>.
The denominator is a [[geometric series]] and the numerator is a type of [http://planetmath.org/encyclopedia/ArithmeticGeometricSeries.html arithmetic-geometric series], so the series can be explicitly summed. After some algebra, the result turns out to be
:<math>\langle m \rangle = J B_J(x)</math>
With ''N'' magnetic moments per unit volume, the magnetization density is
:<math>M = Ng\mu_B\langle m \rangle = NgJ\mu_B B_J(x)</math>.
|}


==Langevin Function==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


[[File:Langevin function.png|300px|thumb|right|Langevin function (red line), compared with
==Demos==
<math>\tanh(x/3)</math> (blue line).]]


In the classical limit, the moments can be continuously aligned in the field and <math>J</math> can assume all values (<math>J \to \infty</math>). The Brillouin function is then simplified into the '''Langevin function''', named after [[Paul Langevin]]:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


<blockquote style="border: 1px solid black; padding:10px; width:230px">
:<math>L(x) = \coth(x) - \frac{1}{x}</math>
</blockquote>
For small values of {{math|''x''}}, the Langevin function can be approximated by a truncation of its [[Taylor series]]:
:<math>
  L(x) = \tfrac{1}{3} x - \tfrac{1}{45} x^3 + \tfrac{2}{945} x^5 - \tfrac{1}{4725} x^7 + \dots
</math>
An alternative better behaved approximation can be derived from the
[[Gauss's continued fraction#The series 0F1 2|Lambert's continued fraction]] expansion of {{math|tanh(''x'')}}:
:<math>
L(x) = \frac{x}{3+\tfrac{x^2}{5+\tfrac{x^2}{7+\tfrac{x^2}{9+\ldots}}}}
</math>
For small enough {{math|''x''}}, both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from [[Loss of significance]].


The inverse Langevin function can be approximated to within 5% accuracy
* accessibility:
by the formula<ref name="Cohen">{{cite journal |title=A Padé approximant to the inverse Langevin function |last=Cohen |first=A. |journal=[[Rheologica Acta]] |volume=30 |issue=3 |pages=270–273 |year=1991 |doi=10.1007/BF00366640 }}</ref>
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
:<math>
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
  L^{-1}(x) \approx x \frac{3-x^2}{1-x^2},
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
</math>
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
valid on the whole interval (-1, 1).
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
For small values of x, better approximations are the [[Padé approximant]]
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
:<math>
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
  L^{-1}(x) = 3x \frac{35-12x^2}{35-33x^2} + O(x^7)
</math>
and the
[[Taylor series]]<ref name="Johal">{{cite journal |title=Energy functions for rubber from microscopic potentials |last=Johal |first=A. S. |last2=Dunstan |first2=D. J. |journal=[[Journal of Applied Physics]] |volume=101 |issue=8 |page=084917 |year=2007 |doi=10.1063/1.2723870 |bibcode = 2007JAP...101h4917J }}</ref>
:<math>
  L^{-1}(x) = 3 x + \tfrac{9}{5} x^3 + \tfrac{297}{175} x^5 + \tfrac{1539}{875} x^7 + \dots
</math>


==High Temperature Limit==
==Test pages ==


When <math>x \ll 1</math> i.e. when <math>\mu_B B / k_B T</math> is small, the expression of the magnetization can be approximated by the [[Curie's law]]:
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
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*[[Help:Formula]]


:<math>M = C \cdot \frac{B}{T}</math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
where <math>C = \frac{N g^2 J(J+1) \mu_B^2}{3k_B}</math> is a constant. One can note that <math>g\sqrt{J(J+1)}</math> is the effective number of Bohr magnetons.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
==High Field Limit==
 
When <math>x\to\infty</math>, the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field:
 
:<math>M = N g \mu_B J</math>
 
== References ==
<references/>
 
[[Category:Articles with inconsistent citation formats]]
[[Category:Magnetism]]
 
[[de:Langevin-Funktion]]
[[fr:Fonction de Langevin]]
[[ja:ブリルアン関数とランジュバン関数]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


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Bug reporting

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