en>I dream of horses: /* top */ Fixed link to disambiguation page

2014-09-09T07:25:04Z

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en>Jamesmcmahon0: /* Relation to the Polygamma function */Typo fixing, typo(s) fixed: etc, → etc., using AWB

2014-02-13T17:41:48Z

Relation to the Polygamma function: Typo fixing, typo(s) fixed: etc, → etc., using AWB

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	~~In [[mathematics]],~~ the ~~family~~ of '~~''Debye functions''' is defined by~~		Historical past of the of the writer was Adrianne Quesada. Her husband doesn't like it then the way she truly but what she honestly likes doing is that will bake but she's pondering on starting something new. After being out of their job for years the person became an order clerk. Vermont is almost certainly where her house has always been. Her husband and her always maintain a [http://Search.Huffingtonpost.com/search?q=website&s_it=header_form_v1 website]. You might prefer to check it out: http://[http://search.About.com/?q=prometeu.net prometeu.net]<br><br>My site - [http://prometeu.net clash of clans hack android]

	~~:<math>D_n(x) = \frac{n}{x^n} \int_0^x \frac{~~t~~^n}{e^t - 1}\,dt.</math>~~

	~~The functions are named in honor of [[Peter Debye]], who came across this function (with ''n'' = 3) in 1912 when he analytically computed~~ the ~~[[heat capacity]] of~~ what is ~~now called the [[Debye model]]~~.

	~~== Mathematical properties ==~~

	~~=== Relation to other functions ===~~

	~~The Debye functions are closely related to~~ the ~~[[Polylogarithm]]~~.

	~~=== Limiting values ===~~

	~~For <math>x \rightarrow 0</math> :~~
	~~:<math>D_n(0)=1.</math>~~

	~~For <math>x >> 1</math> : <math>D_n</math>~~ is ~~given by the [[Gamma function]] and the [[Riemann zeta function]]:~~
	~~:<math>D_n(x)\propto\int_0^\infty{\rm d}t\frac{t^{n}}{\exp(t)-1} = \Gamma(n + 1) \zeta(n + 1)~~. ~~\quad [\Re \, n > 0]</math> <ref>Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series,~~ and ~~Products (Academic, New York, 1980), (3.411).</ref>~~

	~~== Applications in solid-state physics ==~~

	~~=== The Debye model ===~~

	~~The [[Debye model]] has~~ a [~~[density of vibrational states]]~~
	:~~<math>g_{\rm D}(\omega)=\frac{9\omega^2}{\omega_{\rm D}^3}</math> for <math>0\le\omega\le\omega_{\rm D}<~~/~~math>~~
	~~with the ''Debye frequency'' ''ω''<sub>D<~~/~~sub>~~.

	~~=== Internal energy and heat capacity ===~~

	~~Inserting ''g'' into the internal energy~~
	~~:<math>U=\int_0^\infty{\rm d}\omega\,g(\omega)\,\hbar\omega\,n(\omega)</math>~~
	~~with the [[Bose-Einstein distribution]]~~
	~~:<math>n(\omega)=\frac{1}{\exp(\hbar\omega/k_{\rm B}T)-1}</math>~~.
	~~one obtains~~
	~~:<math>U=3 k_{\rm B}T\, D_3(\hbar\omega_{\rm D}~~/~~k_{\rm B}T)</math>.~~
	~~The heat capacity is the derivative thereof.~~

	=== ~~Mean squared displacement ===~~

	~~The intensity of [[X-ray diffraction]] or [[neutron diffraction]] at wavenumber ''q'' is given by~~
	~~the [[Debye-Waller factor]] or the [[Lamb-Mössbauer factor]~~].
	~~For isotropic systems~~ it ~~takes the form~~
	:~~<math>\exp(-2W(q))=\exp(-q^2\langle u_x^2\rangle</math>).~~
	~~In this expression, the [[mean squared displacement]] refers to just once Cartesian component~~
	~~''u<sub>x</sub>'' of the vector '''u''' that describes the displacement of atoms from their equilibrium positions.~~
	~~Assuming harmonicity and developing into normal modes,<ref>Ashcroft & Mermin 1976, App. L,</ref>~~
	~~one obtains~~
	:<math>2W(q)=\frac{\hbar^2 q^2}{6M k_{\rm B}T}\int_0^\infty{\rm d}\omega\frac{k_{\rm B}T}{\hbar\omega}g(\omega)\coth\frac{\hbar\omega}{2k_{\rm B}T}=\frac{\hbar^2 q^2}{6M k_{\rm B}T}\int_0^\infty{\rm d}\omega\frac{k_{\rm B}T}{\hbar\omega}g(\omega)\left[\frac{2}{\exp(\hbar\omega/~~k_{\rm B}T)-1}+1\right].<~~/~~math>~~
	~~Inserting the density of states from the Debye model, one obtains~~
	:<math>2W(q)=\frac{3}{2}\frac{\hbar^2 q^2}{M\hbar\omega_{\rm D}}\left[2\left(\frac{k_{\rm B}T}{\hbar\omega_{\rm D}}\right)D_1\left(\frac{\hbar\omega_{\rm D}}{k_{\rm B}T}\right)+\frac{1}{2}\right]</math>.

	~~==References==~~
	~~{{Reflist}}~~

	~~==Further reading==~~

	* {{AS ref\|27\|998}}
	* [http://~~mathworld~~.~~wolfram~~.com/~~DebyeFunctions~~.~~html "Debye function" entry in MathWorld~~]~~, defines the Debye functions without prefactor ''n''/''x~~<~~sup~~>n<~~/sup~~>''

	~~== Implementations ==~~
	* [http://www.netlib.org/toms/757 Fortran 77 code] by [[Allan MacLeod]] from [[Transactions on Mathematical Software]]
	* [http://www.csit.fsu.edu/~burkardt/f_src/toms757/toms757.html Fortran 90 version]
	* [http://~~www.gnu.org/software/gsl/manual/html_node/Debye-Functions~~.~~html C version]~~ of ~~the [[GNU Scientific Library]]~~

	~~[[Category:Special functions]~~]

en>Newyorkadam: clean up, replaced: the the → the using AWB

2014-01-29T21:31:04Z

clean up, replaced: the the → the using AWB

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en>AvocatoBot: r2.7.1) (Robot: Adding hu:Clausen-függvény

2012-04-23T13:31:29Z

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Clausen function - Revision history

en>I dream of horses: /* top */ Fixed link to disambiguation page

en>Jamesmcmahon0: /* Relation to the Polygamma function */Typo fixing, typo(s) fixed: etc, → etc., using AWB

en>Newyorkadam: clean up, replaced: the the → the using AWB

en>AvocatoBot: r2.7.1) (Robot: Adding hu:Clausen-függvény