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Contributions of Debye Functions to Bosons and Its Applications on Some Nd Metals, Part Ii

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Contributions of Debye Functions to Bosons and Its Applications on Some Nd Metals, Part Ii International Journal of Pure and Applied Mathematics Volume 102 No. 3 2015, 429-445 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v102i3.3 ijpam.eu CONTRIBUTIONS OF DEBYE FUNCTIONS TO BOSONS AND ITS APPLICATIONS ON SOME ND METALS, PART II Muhammad A. Al-Jalali Department of Physics Faculty of Science Taif University Taif, AL-Haweiah, P.O. Box 888, Zip Code 21974, KINGDOM OF SAUDI ARABIA Abstract: Internal thermal energy in solids contributes to vibrations (phonons) energy; spin waves (magnons) energy if solid has magnetism and fermions en- ergy across very complicated mechanisms. Debye functions, mathematically, was estimated because they are considered a main term which controls in all equations of those contributions. Semi-empirical equation has been obtained to nd (n=3,4,5) transition met- als specific heat to calculate some important physical constants. Numerical analyses gives an agreement with experimental results on nd transition metals. Comparison between theoretical and experimental was in- vestigated. AMS Subject Classification: 62H10, 62P35, 74F05, 74H45, 81V19, 82C10, 82D35, 82D40 Key Words: Debye functions, Bose–Einstein distribution, Phonons, magnons, fermions c 2015 Academic Publications, Ltd. Received: January 6, 2015 url: www.acadpubl.eu 430 M.A. Al-Jalali 1. Introduction In solid-state physics and statistical mechanics, bosons mean phonons, magnons, and photons, which subject to Bose–Einstein statistics [1]. There are two kinds of Debye functions family, the first belong to bosons energy, and the second is bosons specific heat. In mathematics, the family of Debye functions defined as [2]: x n tn E(n, x) = dt, xn+1 et − 1 Z0 x (1) n tn+1et C(n, x) = dt, xn (et − 1)2 Z0 Here Peter Debye in 1912, analytically computed the heat capacity, with n = 3 (of what is now called the Debye model in solid state). Most components of relationship (1) come from, Bose–Einstein distribution at low temperatures, or Maxwell-Boltzmann distribution at high temperatures [3, 4, 5, 6, 7], and according to the type of particles or quasi-particles, whether they were phonons or magnons. The temperature-dependent thermal energy or specific heat in most kinds of metals and their alloys must have in its relationship part of equation (1). The aim of this paper is to mathematically estimate equations (1), and compare this with experimental results. In addition, to get a semi-empirical equation to nd transition metals specific heat to calculate some of the physical constants. 2. Quantum Statistical Mechanics Background Quantum Debye model of phonons thermal energy in solids for all possible frequencies up to the maximum frequency given by [8, 9]: ωmax V ω2 ~ω U = d. (2) phonons 2 3 ~ω/K T 2π υs e B − 1 Z0 If υs is identical for all three polarizations, and suppose that: ~ω ~ωmax ~ωD Θ x ≡ ⇒ xmax = ⇒ xD = ≡ , KBT KBT KBT T CONTRIBUTIONS OF DEBYE FUNCTIONS TO BOSONS AND... 431 where xD ~ 3 ωD T 3 x Θ = ⇒ Uphonons = 9NKBT ( ) x dx = 3NKE(3, xD). (3) KB Θ e − 1 Z0 Derivative phonons energy in equation (2) will give phonons specific heat as follows [10]: xD T 3 x4ex Cphonons = 9NkB dx = 3NkC(3, xD). (4) Θ (ex − 1)2 Z0 For anharmonic phonons energy and specific heat, general expressions may be written as: xD xn U =CT n+1 dx, phonons ex − 1 Z 0 (5) xD xn+1ex C =CT n dx n > 3. v(phonons) (ex − 1)2 Z0 In addition, for magnons (spin waves) thermal energy in magnetic solids [11] at temperature T is given by: xm 3/2 3/2 KBT KBT x Umagnons = 2 x dx, 4π Dstiff e − 1 (6) Z0 2 where x =[∆ + Dstiff k ]/KB T. Here the dispersion relation for magnon in a spin system is [12, 13, 14]: 2 ω(k) = ∆ + Dstiff k , (7) Dstiff (Alternatively, D) is the spin-stiffness constant, which is a linear combi- nation of the exchange integrals, ∆is being the anisotropy spin wave gap. The magnons specific heat is usually obtained by thermal energy derivative for long-wavelength of spin waves, which are the dominant excitations at low temperatures, as: xm 3/2 5/2 x dUmagnons KB KBT x e Cmagnons = = 2 2 dx. (8) dT 4π Dstiff (ex − 1) Z0 432 M.A. Al-Jalali For anharmonic magnons (spin waves) thermal energy and specific heat, general expressions may be written as: xD n n+1 x Umagnons =CT dx, ex − 1 Z 0 (9) xD xn+1ex 3 C =CT n dx n > . v(magnons) (ex − 1)2 2 Z0 Whereas Planck obtained Photons thermal energy law in solids black body radiation’s, or may be sometimes named the Stefan-Boltzmann law as follows: ∞ 8πK4 T 4 x3 U = B dx (10) (hc)3 ex − 1 Z0 where x = hυ = β. KBT In addition, the thermal conductivity is given by [15, 16, 17]: xD 4 x kB kB 3 x e Kphonons = T τ(x) dx, (11) 2π2υ ~ (ex − 1)2 Z0 υ is the velocity of phonons in solids,τ(x)is the relaxation time. Thermal con- ductivity entirely connected with specific heat as follows: 1 K = C υℓ, (12) phonons 3 phonons ℓis the mean free path. Finally, the electronic (fermions) specific heat, which subject to Fermi-Dirac distribution given be this equation: Similarly, with bosons specific heat, fermions specific heat may be written as follows: ∞ df(ε) C = dε(ε − ε )D(ε) , fermions F dT Z0 1 (13) f(ε) = , e(ε−µ)/KT + 1 3 N(ε) D(ε) = . 2 εF CONTRIBUTIONS OF DEBYE FUNCTIONS TO BOSONS AND... 433 As was doing in relation (3), electronic (fermions) specific heat given as follows: ∞ x2ex C =K2TD(ε ) dx, fermions F (ex + 1)2 −εFZ/KT ∞ x2ex π (14) dx = for KT ≪ ε , (ex + 1)2 3 F −∞Z π C = K2TD(ε ), fermions 3 F where general formula of fermions specific heat given by: ∞ xn+1 C = AT n dx. (15) v(fermions) (ex + 1)2 Z0 3. Mathematical Integrals Results of Debye Functions In all equations log(x) is the natural logarithm and Lin (x) = polylog (n, x) is the polylogarithm function given by: ∞ xn x2 x3 n! Li (x) = = x + + + .... , (16) s ns 2s 3s r!(n − r)! n=1 X where x Li (x) = , Li (x) = − log(1 − x). 0 1 − x 1 In addition, Riemann zeta function describes by [18]: ∞ ∞ s−1 − 1 x ζ(s) = lim n s or ζ(s) = dx, (17) n→∞ Γ(s) ex − 1 nX=1 Z0 where Lis(1) = ζ(s), Γ(s) is Gamma function. Many attempt was made to, mathematically, simplify Debye functions [2], the integral formula for Debye energy (1) was given as follows: π4 3 3 9 18 18 E(3, x) = − − + (ln(1 − ex)+ Li (ex)− Li (ex)+ Li (ex). (18) 5x5 4 x x2 2 x3 3 x4 4 434 M.A. Al-Jalali Moreover, for Debye specific heat [2] was written as follows: 4 −x 4π 3xe − 36 − C(3, x) = + + 12 ln(1 − e x) − Li (e x) 5x5 (e−x − 1) x 2 72 − 72 − − Li (e x) − Li (e x). (19) x2 3 x3 4 However, for more simplicity, Mathematical programs may help to give solutions for all complicated functions. For n = 1, 2, 3, 4, 5, phonon first kind integrals of Debye functions may be given from (1) by the following expressions: x n tn E(n, x) = dt, n = 1, 2, 3, 4, 5, ..., xn et − 1 Z0 1 x2 ζ(2)E (1, x) = − + xLog [1 − ex] + PolyLog [2, ex] − , x 2 π2 Li (1) =ζ(2) = ≃ 1.645, (20) 2 6 2 x3 ζ(3)E (2, x) = [− + x2Log [1 − ex] + 2xPolyLog [2, ex] x2 3 − 2PolyLog [3, ex] − 2], Li3(1) =ζ(3) = 1.202, (21) 3 x4 ζ(4)E(3, x) = [− + x3Log [1 − ex] + 3x2PolyLog [2, ex] x3 4 − 6xPolyLog [3, ex] + 6PolyLog [4, ex] −6], π4 Li (1) =ζ(4) = ≃ 1.082, (22) 4 90 4 x5 ζ(5)E (4, x) = − + x4Log [1 − ex] + 4x3PolyLog [2, ex] x4 5 − 12x2PolyLog [3, ex] + 24xPolyLog [4, ex] −24PolyLog [5, ex] − 24] , Li5(1) =ζ(5) = 1.037, (23) 5 x6 ζ(6)E (5, x) = − + x5Log [1 − ex] + 5x4PolyLog [2, ex] x5 6 − 20x3PolyLog [3, ex] + 60x2PolyLog [4, ex] − 120xPolyLog [5, ex] +120PolyLog [6, ex] − 120] , CONTRIBUTIONS OF DEBYE FUNCTIONS TO BOSONS AND... 435 π6 Li (1) =ζ(6) = = 1.017. (24) 6 945 For n = 1, 2, 3, 4, 5, phonon second kind integrals of Debye functions may be given from (1) by the following expressions where C′(n, x) represents integral part of C(n, x): x n+1 t n t e n ′ C(n, x) = dt = C (n, x), xn (et − 1)2 xn Z0 ′ x C (1, x) = − x − + Log [1 − ex] + Constant, −1 + ex Li1(1) =ζ(1) = ∞, (25) x 2 ′ e x π C (2, x) =x + 2Log [1 − ex] + 2PolyLog [2, ex] − , 1 − ex 3 π2 Li (1) =ζ(2) = ≃ 1.645, (26) 2 6 x ′ e x C (3, x) =x2 + 3Log [1 − ex] + 6xPolyLog [2, ex] 1 − ex − 6PolyLog [3, ex] − 7.212, Li3(1) =ζ(3) = 1.202, (27) 4 ′ x C (4, x) = − x4 − + 4x3Log [1 − ex] + 12x2PolyLog [2, ex] −1 + ex 24 − 24xPolyLog [3, ex] + 24PolyLog [4, ex] − π4, 90 π4 Li (1) =ζ(4) = ≃ 1.082, (28) 4 90 5 ′ x C (5, x) = − x5 − + 5x4Log [1 − ex] + 20x3PolyLog [2, ex] −1 + ex − 60x2PolyLog [3, ex] + 120xPolyLog [4, ex] − 120PolyLog [5, ex] − 124.44, Li5(1) =ζ(5) = 1.037.
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