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Debye function
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<h2 id="mathematical-properties">Mathematical properties</h2> <h3>Relation to other functions</h3> <p>The Debye functions are closely related to the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a>. </p> <h3>Series expansion</h3> <p>They have the series expansion<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> D n ( x ) = 1 − n 2 ( n + 1 ) x + n ∑ k = 1 ∞ B 2 k ( 2 k + n ) ( 2 k ) ! x 2 k , | x | < 2 π , n ≥ 1 , {\displaystyle D_{n}(x)=1-{\frac {n}{2(n+1)}}x+n\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k+n)(2k)!}}x^{2k},\quad |x|<2\pi ,\ n\geq 1,} where B n {\displaystyle B_{n}} is the n-th <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli number</a>. </p> <h3>Limiting values</h3> <p> lim x → 0 D n ( x ) = 1. {\displaystyle \lim _{x\to 0}D_{n}(x)=1.} If Γ {\displaystyle \Gamma } is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> and ζ {\displaystyle \zeta } is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>, then, for x ≫ 0 {\displaystyle x\gg 0} ,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> D n ( x ) = n x n ∫ 0 x t n d t e t − 1 ∼ n x n Γ ( n + 1 ) ζ ( n + 1 ) , Re n > 0 , {\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}\,dt}{e^{t}-1}}\sim {\frac {n}{x^{n}}}\Gamma (n+1)\zeta (n+1),\qquad \operatorname {Re} n>0,} </p> <h3>Derivative</h3> <p>The derivative obeys the relation x D n ′ ( x ) = n ( B ( x ) − D n ( x ) ) , {\displaystyle xD_{n}^{\prime }(x)=n\left(B(x)-D_{n}(x)\right),} where B ( x ) = x / ( e x − 1 ) {\displaystyle B(x)=x/(e^{x}-1)} is the Bernoulli function. </p> <h2 id="applications-in-solid-state-physics">Applications in solid-state physics</h2> <h3>The Debye model</h3> <p>The <a href="/facts/Debye_model/Hak5Spp8">Debye model</a> has a <a href="/facts/Density_of_states/bwVTyhTf">density of vibrational states</a> g D ( ω ) = 9 ω 2 ω D 3 , 0 ≤ ω ≤ ω D {\displaystyle g_{\text{D}}(\omega )={\frac {9\omega ^{2}}{\omega _{\text{D}}^{3}}}\,,\qquad 0\leq \omega \leq \omega _{\text{D}}} with the Debye frequency <i>ω</i>D. </p> <h3>Internal energy and heat capacity</h3> <p>Inserting <i>g</i> into the internal energy U = ∫ 0 ∞ d ω g ( ω ) ℏ ω n ( ω ) {\displaystyle U=\int _{0}^{\infty }d\omega \,g(\omega )\,\hbar \omega \,n(\omega )} with the <a href="/facts/Bose%E2%80%93Einstein_distribution/Utb2jN3n">Bose–Einstein distribution</a> n ( ω ) = 1 exp ( ℏ ω / k B T ) − 1 . {\displaystyle n(\omega )={\frac {1}{\exp(\hbar \omega /k_{\text{B}}T)-1}}.} one obtains U = 3 k B T D 3 ( ℏ ω D / k B T ) . {\displaystyle U=3k_{\text{B}}T\,D_{3}(\hbar \omega _{\text{D}}/k_{\text{B}}T).} The heat capacity is the derivative thereof. </p> <h3>Mean squared displacement</h3> <p>The intensity of <a href="/facts/X-ray_diffraction/WtI0F5DN">X-ray diffraction</a> or <a href="/facts/Neutron_diffraction/fYXAunDW">neutron diffraction</a> at wavenumber <i>q</i> is given by the <a href="/facts/Debye-Waller_factor/t7NCltVL">Debye-Waller factor</a> or the <a href="/facts/Lamb-M%C3%B6ssbauer_factor/ghxHIGmw">Lamb-Mössbauer factor</a>. For isotropic systems it takes the form exp ( − 2 W ( q ) ) = exp ( − q 2 ⟨ u x 2 ⟩ ) . {\displaystyle \exp(-2W(q))=\exp \left(-q^{2}\langle u_{x}^{2}\rangle \right).} In this expression, the <a href="/facts/Mean_squared_displacement/85lYfXRp">mean squared displacement</a> refers to just once Cartesian component <i>ux</i> of the vector u that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> one obtains 2 W ( q ) = ℏ 2 q 2 6 M k B T ∫ 0 ∞ d ω k B T ℏ ω g ( ω ) coth ℏ ω 2 k B T = ℏ 2 q 2 6 M k B T ∫ 0 ∞ d ω k B T ℏ ω g ( ω ) [ 2 exp ( ℏ ω / k B T ) − 1 + 1 ] . {\displaystyle 2W(q)={\frac {\hbar ^{2}q^{2}}{6Mk_{\text{B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\text{B}}T}{\hbar \omega }}g(\omega )\coth {\frac {\hbar \omega }{2k_{\text{B}}T}}={\frac {\hbar ^{2}q^{2}}{6Mk_{\text{B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\text{B}}T}{\hbar \omega }}g(\omega )\left[{\frac {2}{\exp(\hbar \omega /k_{\text{B}}T)-1}}+1\right].} Inserting the density of states from the Debye model, one obtains 2 W ( q ) = 3 2 ℏ 2 q 2 M ℏ ω D [ 2 ( k B T ℏ ω D ) D 1 ( ℏ ω D k B T ) + 1 2 ] . {\displaystyle 2W(q)={\frac {3}{2}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\text{D}}}}\left[2\left({\frac {k_{\text{B}}T}{\hbar \omega _{\text{D}}}}\right)D_{1}{\left({\frac {\hbar \omega _{\text{D}}}{k_{\text{B}}T}}\right)}+{\frac {1}{2}}\right].} From the above <a href="/facts/Power_series/vYh6sTps">power series</a> expansion of D 1 {\displaystyle D_{1}} follows that the mean square displacement at high temperatures is linear in temperature 2 W ( q ) = 3 k B T q 2 M ω D 2 . {\displaystyle 2W(q)={\frac {3k_{\text{B}}Tq^{2}}{M\omega _{\text{D}}^{2}}}.} The absence of ℏ {\displaystyle \hbar } indicates that this is a <a href="/facts/Classical_Physics/gGnINVPE">classical</a> result. Because D 1 ( x ) {\displaystyle D_{1}(x)} goes to zero for x → ∞ {\displaystyle x\to \infty } it follows that for T = 0 {\displaystyle T=0} 2 W ( q ) = 3 4 ℏ 2 q 2 M ℏ ω D {\displaystyle 2W(q)={\frac {3}{4}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\text{D}}}}} (<a href="/facts/Zero-point_energy/Kuusl57a">zero-point motion</a>). </p> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Milton_Abramowitz/6yRyez7P">Abramowitz, Milton</a>; <a href="/facts/Irene_Stegun/FjSS1LDR">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a href="http://www.math.ubc.ca/~cbm/aands/page_998.htm">"Chapter 27"</a>. <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 998. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61272-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/65012253">65-12253</a>.</li> <li><a href="http://mathworld.wolfram.com/DebyeFunctions.html">"Debye function" entry in MathWorld</a>, defines the Debye functions without prefactor <i>n</i>/<i>xn</i></li></ul> <h2 id="implementations">Implementations</h2> <ul><li>Ng, E. W.; Devine, C. J. (1970). <a href="https://doi.org/10.1090%2FS0025-5718-1970-0272160-6">"On the computation of Debye functions of integer orders"</a>. <i>Math. Comp</i>. 24 (110): 405–407. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0025-5718-1970-0272160-6">10.1090/S0025-5718-1970-0272160-6</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0272160">0272160</a>.</li> <li>Engeln, I.; Wobig, D. (1983). "Computation of the generalized Debye functions delta(x,y) and D(x,y)". <i>Colloid & Polymer Science</i>. 261: 736–743. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01410947">10.1007/BF01410947</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:98476561">98476561</a>.</li> <li>MacLeod, Allan J. (1996). <a href="https://doi.org/10.1145%2F232826.232846">"Algorithm 757: MISCFUN, a software package to compute uncommon special functions"</a>. <i>ACM Trans. Math. Software</i>. 22 (3): 288–301. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F232826.232846">10.1145/232826.232846</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:37814348">37814348</a>. <a href="http://www.netlib.org/toms/757">Fortran 77 code</a></li> <li><a href="http://www.csit.fsu.edu/~burkardt/f_src/toms757/toms757.html">Fortran 90 version</a></li> <li>Maximon, Leonard C. (2003). "The dilogarithm function for complex argument". <i>Proc. R. Soc. A</i>. 459 (2039): 2807–2819. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2003RSPSA.459.2807M">2003RSPSA.459.2807M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frspa.2003.1156">10.1098/rspa.2003.1156</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122271244">122271244</a>.</li> <li>Guseinov, I. I.; Mamedov, B. A. (2007). "Calculation of Integer and noninteger n-Dimensional Debye Functions using Binomial Coefficients and Incomplete Gamma Functions". <i>Int. J. Thermophys</i>. 28 (4): 1420–1426. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2007IJT....28.1420G">2007IJT....28.1420G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs10765-007-0256-1">10.1007/s10765-007-0256-1</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120284032">120284032</a>.</li> <li><a href="https://www.gnu.org/software/gsl/doc/html/specfunc.html#debye-functions">C version</a> of the <a href="/facts/GNU_Scientific_Library/WKX9L554">GNU Scientific Library</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 27". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 998. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. <a href="978-0-486-61272-0" target="_blank">978-0-486-61272-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "3.411.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. pp. 355ff. ISBN 978-0-12-384933-5. LCCN 2014010276. <a href="978-0-12-384933-5" target="_blank">978-0-12-384933-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Ashcroft & Mermin 1976, App. L, <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>