ISSN 1063-7850, Technical Physics Letters, 2008, Vol. 34, No. 12, pp. 999–1001. © Pleiades Publishing, Ltd., 2008. Original Russian Text © A.E. Dubinov, A.A. Dubinova, 2008, published in Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, 2008, Vol. 34, No. 23, pp. 9–14.
Exact Integral-Free Expressions for the Integral Debye Functions A. E. Dubinova* and A. A. Dubinovab a
Institute of Experimental Physics, Russian Federal Nuclear Center, Sarov, 607188 Russia b Advanced School of General and Applied Physics, Nizhni Novgorod State University, Nizhni Novgorod, 603950 Russia *e-mail:
[email protected] Received March 26, 2008
Abstract—Exact explicit integral-free expressions for the integral Debye functions and an integral-free expression for the heat capacity of the k-dimensional crystal lattice as a function of the temperature are obtained for the first time. The expressions involve polylogarithms and the Riemann zeta function. PACS numbers: 02.30.Gp, 65.40.Ba DOI: 10.1134/S106378500812002X
The integral Debye functions for the energy and heat capacity are defined, respectively, as follows: x
k
k t dt - -------------------- , DE k ( x ) = --------k + 1 exp t – 1 x 0
∫
x
k t exp ( – t ) dt DC k ( x ) = ----k -----------------------------------2 , x 0 [ 1 – exp ( – t ) ]
∫
(1)
k+1
(2)
where multipliers at the integrals can be introduced in different ways. These functions represent mathematical elements of the theory of heat capacity of a k-dimensional crystal [1, 2]. Their importance is illustrated by the fact that, according to the Internet search data, the Debye functions are used in the equations of state of solids in more than 103 articles (see, e.g., [3–5]). Properties of the Debye functions are described practically in all handbooks (see, e.g., [6–8] and monographs (see, e.g., [9–11]) on statistical physics and solid state physics. Function (1) is also introduced in the well known mathematical handbook [12]. In the theory of heat capacity, the Debye functions (1) and (2) appear in integrating of the Planck expressions for the phonon density of states in the crystal lattice. It is commonly believed (and directly stated as, e.g., in [10, 11]) that these functions, even in a particular case of k = 3, cannot be presented in an explicit integral-free form using a finite combination of well-known functions. This circumstance complicates use of the Debye functions in the analytical theory of solids, where expressions containing these functions in the integral form appear as incomplete. For this reason, the Debye functions have been tabulated [12–14]; there are well-
known and new convenient approximations for these functions [14–16]. At the same time, there were attempts at obtaining exact analytical expressions for the Debye functions (1) and (2). It was noticed that integral (1) with x in the upper limit replaced by infinity can be expressed via the Riemann zeta function. Sukheeja [17] used a quite simple procedure (involving expansion in Fourier series) to obtain an exact expression for the limit of function (1) as x ∞ at k = 3. An analogous result was later obtained by Valladares [1] for an arbitrary k. It is also worthy of mentioning the expansion of function (1) in an infinite power series using Bernoulli numbers, which is valid for |x| < 2π and k ≥ 1 (see [12, Section 27.1]): k 1 1 k DE k ( x ) = --- – --- ------------ + -x 2k + 1 x
k
∑ m=1
2m
B 2m x -----------------------------------, ( 2m + k ) ( 2m )!
(3)
and another infinite expansion valid for x > 0 and k ≥ 1 [12]: ∞
k ⎧ exp ( – mx ) DE k ( x ) = --------k + 1⎨ x ⎩m = 1
∑
(4)
k ( k + 1 )x k! ⎫ x kx - + -----------------------------+ … + ----------- . × ---- + -----------2 3 k+1 ⎬ m m m m ⎭ k
k–1
k–2
In his monograph on polylogarithms, Lewin [18, Section 1.12.5] briefly mentioned the Debye function (1) and presented its integral-free expression for k = 2 involving a combination of the Kummer Λ3 function with elementary functions. This gave us grounds to suggest that integral-free expressions for the Debye functions (1) and (2) with arbitrary integer k can be
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DUBINOV, DUBINOVA
obtained on the basis of polylogarithms and/or related functions (including the Riemann zeta function). This task has been solved despite the commonly accepted opinion concerning the impossibility of such representation.
1π 3 3 9 DE 3 ( x ) = – --- ----4- – --- + --- ln ( 1 – exp x ) + -----2 Li 2 ( exp x ) 5x 4 x x (9) 18 18 – -----3- Li 3 ( exp x ) + -----4- Li 4 ( exp x ), x x
Below, we present for the first time such exact explicit integral-free expressions for the Debye functions (1) and (2) in the form of a finite combination of well-known functions.
4π 3x exp ( – x ) DC 3 ( x ) = --- ----3- + ---------------------------------- + 12 ln [ 1 – exp ( – x ) ] 5 x [ exp ( – x ) – 1 ]
The following expressions are valid for any positive integer k: k k - ( – 1 ) k!ζ ( k + 1 ) DE k ( x ) = --------k+1 x
∑ ( –1 )
k–m+1
m=0
∫
k! m k ------ x Li k – m + 1 ( exp x ) – ------------ , m! k+1
k⎧ DC k ( x ) = ----k ⎨ ( k + 1 )!ζ ( k + 1 ) x ⎩ k+1
–
( k + 1 )!
x ∑ -----------------m!
m
m=0
k
∑ ( –1 )
=
m=0
∫
k–m+1
(10)
The writing of Eqs. (9) and (10) is simplified by taking into account that Li 1 ( x ) = – ln ( 1 – x ),
(11)
and that the Riemann zeta function for even argument is defined as
∫ (6)
⎫ Li k – m + 1 [ exp ( – x ) ] ⎬, ⎭
k
t dt -------------------exp t – 1 k+1
k! m t ------ t Li k – m + 1 ( exp t ) – ------------ , m! k+1
(7)
k+1
( k + 1)! m t exp (– t)dt -----------------t Li k – m + 1 [exp (– t)], --------------------------------2 = – m! [1 – exp (– t)] (8) m=0 k+1
72 36 – ------ Li 2 [ exp ( – x ) ] – -----2- Li 3 [ exp ( – x ) ] x x
x Li 0 ( x ) = -----------, 1–x
where ζ(x) is the Riemann zeta function [19], Liv(x) is the polylogarithm [18, 20], and the sums are finite. The validity of Eqs. (5) and (6) can readily be proved. Indeed, the indefinite integrals in expressions (1) and (2) can be written (to within an arbitrary constant) as follows:
∫
4
72 – -----3- Li 4 [ exp ( – x ) ]. x (5)
k
+
4
∑
which can readily be checked by differentiating the original functions and using the rule of differentiation for polylogarithms: (d/dx)Lik(x) = (1/x)Lik – 1(x). Substituting expressions (7) and (8) into Eqs. (1) and (2), using the Newton–Leibnitz formula (with Lik(1) = ζ(k)), and introducing the corresponding multipliers, we eventually arrive at formulas (5) and (6). These functions with k = 3, which are of special importance for the Debye theory of the heat capacity of solids, can be written as follows:
ζ ( 2n ) = ( – 1 )
n – 1 ( 2π )
2n
---------------- B , 2 ( 2n )! 2n
(12)
where B2n are the Bernoulli numbers [12, 21]; in particular, B4 = –1/30 and ζ(4) = π4/90. The negative argument (negative temperature) in both Debye functions have no physical meaning. However, from mathematical standpoint, it is important to know that DEk(x) + k/2(k + 1) is an odd function, while DCk(x) and dDEk(x)/dx are even functions. By direct substitution, one can also readily check the following functional identities: d k ------ [ xDE k ( x ) ] = -------------------------- – kDE k ( x ), dx exp ( x ) – 1
(13)
d DC k ( x ) = xDE k ( x ) – x ------ [ xDE k ( x ) ]. dx
(14)
It should be noted that, according to [9], the integrals calculated above enter into mathematical expressions for virtually all thermodynamic quantities of solids. For example, an exact integral-free dimensionless expression for the heat capacity of the k-dimensional crystal in the Debye theory is as follows: T C V ( T ) = DC k ⎛ -----D-⎞ , ⎝T⎠
(15)
where TD is the Debye temperature. Plots of the heat capacity versus temperature calculated for various k according to this expression are presented in the figure. Note that this figure also shows the temperature dependence of the heat capacity for a 4-dimensional crystal, which is intended (i) to demonstrate the tendency in evolution of the curves with increasing k and (ii) to
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EXACT INTEGRAL-FREE EXPRESSIONS FOR THE INTEGRAL DEBYE FUNCTIONS CV 1.4 1.2 1.0 0.8 0.6
1 2
0.4
4 3
0.2 0
0.2
0.4
0.6
0.8
1.0 TD/T
Plots of the heat capacity CV of a k-dimensional crystal versus reduced inverse temperature TD/T calculated using Eq. (15) for k = 1–4 (curves 1–4, respectively).
recall that, in some theories of solid state (e.g., in the renorm theory of phase transitions [22]), the 4-dimensional crystal occupies a special mathematical position among all k-dimensional crystals. In conclusion, we have obtained for the first time the exact, explicit integral-free expressions for the integral Debye functions. Acknowledgments. One of the authors (A.E.D.) gratefully acknowledges support from the Government of the Nizhni Novgorod Region. REFERENCES 1. A. A. Valladares, Am. J. Phys. 43, 308 (1975). 2. M. N. Magomedov, Fiz. Tverd. Tela (St. Petersburg) 45, 33 (2003) [Phys. Solid State 45, 32 (2003)]. 3. V. Yu. Bodryakov, A. A. Povzner, and O. G. Zelyukova, Fiz. Tverd. Tela (St. Petersburg) 41, 1248 (1999) [Phys. Solid State 41, 1138 (1999)].
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Translated by P. Pozdeev
