We obtained the analytic expressions of the specific heat at constant volume of metals which are dependent on the value of q-deformed parameters. Results calculated for the specific heat at constant volume for some kinds of alkali and transition metals show good agreement with those obtained by other theories and in experiments as well.
JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol 58, No 7, pp 117-124 This paper is available online at http://stdb.hnue.edu.vn INVESTIGATION OF THE SPECIFIC HEAT AT CONSTANT VOLUME OF FREE ELECTRONS IN METALS USING q-DEFORMED FERMI- DIRAC STATISTICS Vu Van Hung1 , Duong Dai Phuong2 and Luu Thi Kim Thanh3 Viet Nam Education Publishing House, Hanoi Tank Armour Officers Training School, Tam Duong, Vinh Phuc Faculty of Physics, Hanoi University of Education No 2, Xuan Hoa, Vinh Phuc Abstract The contribution of free electrons to the specific heat at constant volume of metals in low temperature was investigated using q-deformed Fermi-Dirac statistics We obtained the analytic expressions of the specific heat at constant volume of metals which are dependent on the value of q-deformed parameters Results calculated for the specific heat at constant volume for some kinds of alkali and transition metals show good agreement with those obtained by other theories and in experiments as well Keywords: Specific heat at constant volume, q-deformed fermi-dirac statistics, alkali metal, transition metal Introduction In metals, many electrons can move freely throughout the crystal which oftentimes makes the metal a high electrical conductivity candidate with an electrical conductivity of around 106 to 108 Ω−1 m−1 For instance, if each atom in a material contains only one free electron, there would be about 1022 conduction electrons per cm3 Depending on which distribution function is used to consider the free-electron gas, different theories could be established: (i) If free electrons are considered to be a simple classical gas settling on the same energy, Drude’s theory can be used to analyze issues arising related to the metal; (ii) When using the Maxwell-Boltzmann distribution function for a classical gas, the metal can be described in the framework of Lorentz’s theory; (iii) In the quantum feature with the Fermi-Dirac distribution function being used, Sommerfeld’s theory is proposed instead In light of these theories, the specific heat at constant volume of free electrons in the metals had been studied in detail [4-6] Received August 12, 2013 Accepted October 1, 2013 Contact Duong Dai Phuong, e-mail address: vanha318@yahoo.com 117 Vu Van Hung, Duong Dai Phuong and Luu Thi Kim Thanh Obtained results for those theories [7, 10] show that at low temperature the specific heat at constant volume of free electrons in metals is linearly dependent on temperature T The specific heat at constant volume in metals is mainly due to the total number of the free electrons In this work, we propose another way to apply the statistical distribution of Fermi-Dirac -q deformation to investigate the specific heat at constant volume of free electron gas in metals at low temperature We will point out the analytical expressions of the specific heat at constant volume of free electrons in metals as well as the q-deformed parameters Present theoretical calculations of the specific heat at constant volume for some kinds of alkali and transition metals have shown good agreement with those observed in experimental as well as from the other theories [5, 7, 8, 10] Content 2.1 Theory At low temperature, free electron gas in metals via the Fermi-Dirac statistics and the specific heat at constant volume is ratio linear with absolute temperature T [4, 5] ( ) ∂E π2 N k2T e = CV = ⇒ CVe = γT, (2.1) ∂T V µ0 where µ0 is the chemical potential at T = 0K, β = , k is a Boltzmann constant, T is kT the absolute temperature, N is the total number of free electrons, E is the total energy of free electron gas and γ is a constant In the q-deformed Fermions, oscillator operators satisfy the commutative relations, [1, 2, 9] ˆbˆb+ + qˆb+ˆb = q −Nˆ { } { } ˆb+ˆb = N ˆ , ˆbˆb+ = N ˆ +1 , (2.2) q q ˆ is an oscillator number operator and q is a deformation parameter where N With q-deformed Fermions, we obtained the following equation q −n − (−1)n q n {n}q = q + q −1 (2.3) In statistical physics the thermal average expression of the operator Fˆ is given as [1, 2] ( { } ) ˆ N) ˆ Fˆ ⟨ ⟩ T r exp −β(H−µ ( { }) , Fˆ = (2.4) ˆ N) ˆ T r exp −β(H−µ ˆ is the Hamiltonian of the system where µ is the chemical potential and H 118 Investigation of the specific heat at constant volume of free electrons From equations (2.4), the average number of particles has been calculated following ( { } ) ˆ N) ˆ N ˆ ⟨ ⟩ T r exp −β(H−µ ˆ ( { }) N = (2.5) ˆ N) ˆ T r exp −β(H−µ Using expression (2.3), the calculation yield has been written ( ∞ { } { }) ∑ { } ˆ ˆ − µN ˆ) N ˆ ˆ |n⟩ ∗T r exp −β(H = ⟨n|e−β(ε−µ)N N q = ∞ ∑ ⟨n|e−β(ε−µ)n {n}q |n⟩ = n=0 ∞ ∑ e−β(ε−µ)n {n}q n=0 = ∞ ∑ e−β(ε−µ)n n=0 = q n=0 q −n − (−1)n q n q + q −1 e−β(ε−µ) + (q − q −1 )e−β(ε−µ) − e−2β(ε−µ) (2.6) On the other hand, we have: ∞ ( { }) ∑ ˆ ˆ − µN ˆ) ∗T r exp −β(H = ⟨n|e−β(ε−µ)N |n⟩ n=0 = ∞ ∑ ∞ ∑ ⟨n|e−β(ε−µ)n |n⟩ = e−β(ε−µ)n = n=0 n=0 1− e−β(ε−µ) (2.7) Substituting equations (2.6) and (2.7) into equation (2.5), we obtain the Fermi-Dirac distribution function q-deformed Fermi-Dirac as follows: ⟨ ⟩ ˆ = n ¯ (ε) = N eβ(ε−µ) − e2β(ε−µ) + (q − q −1 )eβ(ε−µ) − (2.8) The total number of free electrons and the total energy of free electron gas at temperature T respectively are [4] ∫∞ N= ρ(ε).¯ n(ε)dε (2.9) ε.ρ(ε).¯ n(ε)dε, (2.10) ∫∞ E= 119 Vu Van Hung, Duong Dai Phuong and Luu Thi Kim Thanh where ρ(ε) is the density of states defined as ρ(ε) = g(ε).V (2m)3/2 ε1/2 , 4π (2.11) and n(ε) is the average number of particles with energies ε and g(ε) is the multiple degeneracy of each energy level ε V.(2m)3/2 Using equations (2.8), (2.9), (2.10) and we can rewrite α = 2π ε−µ ∫∞ e kT − E = α ε3/2 ε − µ dε (2.12) ε−µ e kT + (q − q −1 )e kT − ε−µ e kT − ∫∞ N =α ε1/2 dε ε−µ ε−µ e kT + (q − q −1 )e kT − From equation (2.12), (2.13), at temperature T = 0K, we obtain: (2.13) 3/2 (2.14) N = α.µ0 , 3 5/2 E0 = α.µ0 = µ0 N (2.15) 5 These results show good agreement between our works and the results presented in [4] Where µ0 is the chemical potential of the Fermi energy when T → 0K and given by ( 3π N µ0 = 2m V )2 ( = 3N 2α )2 (2.16) At very low temperature T ̸= 0K with integrals in equations (2.12), (2.13), we can evaluate approximately and, using the expression (2.14), (2.15), (2.16) to perform transformations, the contribution of the q-deformed is also taken into account and, when < q < 1, the total energy of free electron gas at temperature T have been determined to be [ ] F (q)(kT )2 E = E0 + (2.17) µ20 Substituting equation (2.15) into equation (2.17), we obtain the following equation for total energy of electron [ ] F (q)(kT )2 (2.18) E = N µ0 + 5 µ20 120 Investigation of the specific heat at constant volume of free electrons where F (q) is the function depending on the q-deformed parameter, that could be expressed as the following [ ] ∞ ∞ ∞ ∞ k k k k ∑ ∑ ∑ ∑ −1 (q) (−q) (q) (−q) F (q) = q(q − 1) + (1 + q) −q + 2 q +1 k k k k3 k=1 k=1 k=1 k=1 (2.19) From (2.18) we obtain the specific heat at constant volume of free electrons gas in metals and, when taking into account the contribution of the deformation-q parameter, it can be determined as follows ( ) N F (q)k T ∂E e CV = = = γ bd T (2.20) ∂T V µ0 So, at very low temperature, the specific heat at constant volume of free-electron gas in metals, when using the q-deformed Fermi-Dirac statistics, is also ratio linearly with absolute temperature T The result obtained in equation (2.20) shows agreement with the one released in [4, 7, 10] From equation (2.20), we obtain the expression of F(q) F (q) = µ0 γ bd , 6N k (2.21) Replacing results of the experimental data γ = γ bd , µ0 , N, k taken from [4, 5] into the right-hand side of the equation (2.21), F (q) for each metal can be determined Then using the Maple program for equation (2.19), we evaluate the values of the q-deformed parameter presented in Table From equation (2.21), we infer the expression of the free electrons thermal constants in metals depended on the q-deformed parameter as follows: γ bd = N.k F (q) µ0 (2.22) 2.2 Numerical results and discussions Data for the Fermi energy and electron thermal constants taken from [5] are shown in Table Our results suggest that for alkali metals with the same number of outer electrons layer, the values of parameter q and function F (q) are larger than those of the transition metal From equation (2.20), we also showed that the contribution to heat capacity of free electrons is larger for transition metals in which the outer electron layer of layers d, f, with the values of deformation parameter q and functionF (q) that are smaller than those of the alkali metals, leading to the free electron contribution into the specific heat at the constant volume is smaller 121 Vu Van Hung, Duong Dai Phuong and Luu Thi Kim Thanh Table The experimental values of the Fermi energy and electron thermal constants of the metals Metal Cs K Na Ba Sr Ca µ0 (eV) 1.58 2.12 3.23 3.65 3.95 4.68 γ(mJ.mol−1 K −2 ) 3.20 2.08 1.38 2.7 3.6 2.9 Li 4.72 1.63 Ag 5.48 0.646 Metal µ0 (eV) γ(mJ.mol−1 K −2 ) Be 11.1 0.17 Mg 17.1 1.3 Au 5.51 0.72 Cu 7.0 0.59 Cd 7.46 0.68 Zn 9.39 0.64 Ga 10.35 0.596 Al 11.03 1.35 Table Experimental and theoretical values of parameters and deformation parameters of the electrons in metals Metal γ T N (mJ.mol−1 K −2 ) γ theory (mJ.mol−1 K −2 ) q F(q) Na 1.38 1.379 0.642 1.036662 K 2.08 2.079 0.627 1.025545 Rb 2.41 2.409 0.642 1.036954 Cs 3.20 3.199 0.835 1.175845 Be 0.17 0.215 0.279 0.559054 Cu 0.595 0.594 0.563 0.968659 Ag 0.646 0.645 0.442 0.823320 Au 0.729 0.728 0.531 0.934189 Cd 0.688 0.686 0.570 0.975432 Figure Temperature dependence of the specific heat at constant volume of free electrons for potassium 122 Investigation of the specific heat at constant volume of free electrons Figure Temperature dependence of the specific heat at constant volume of free electrons for gold Figure Temperature dependence of the specific heat at constant volume of free electrons for sodium We obtained different values for the q-deformation parameter with various metal groups Thus, the contribution of electrons to specific heat at constant volume depends on the outermost electron Table shows that the values of −q is the same, equaling 0.642, for the alkali metals, and the values of q are the same equaling 0.564 for the transition metals The dependence of the specific heat at constant volume on temperature for Potassium, Gold and Sodium metals has been shown in Figures 1, and Our calculated results, when compared to available experimental date and with those of other theories [5, 7, 10], showing good agreement 123 Vu Van Hung, Duong Dai Phuong and Luu Thi Kim Thanh Conclusion In this paper, we used q-deformed the Fermi - Dirac statistics to study specific heat at constant volume of free electrons in metals We showed the temperature dependence of the specific heat at constant volume of free electrons at low temperature which is a linear ratio which parallels that of absolute temperature T Our results show good agreement when compared to experimental results and also compared with other theoretical studies Acknowledgements This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2011.16 REFERENCES [1] Duc D V., 1994 Generalized q-deformed oscillators and their statisti Preprint ENSLAPP - A - 494/94, Annecy France [2] Duc D V., Thanh L T K., 1997 On the q- deformed multimode oscillators Comm In Phys No 1.2, pp 10-14 [3] Hunklinger S and A K Raycharhuri, 1998 In progress in low temperature physics, p 145 [4] Hung V V., 2006 Statistical Physics Hanoi National University of Education Publishing House, pp 69-77 (in Vietnamese) [5] Kittel C., 1996 Introduction to Solid State Physics Eighth Edition, pp 131-157 [6] Kittel C., 1999 Einfuhrung in die Festkorper Physics Abb.6.3 Abb.6.4; Abb.6.8; Abb.6.9 [7] Nimtz G and J P Stadler, 1985 Heat capacity of free electrons at the 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Temperature dependence of the specific heat at constant volume of free electrons for potassium 122 Investigation of the specific heat at constant volume of free electrons Figure Temperature... dependence of the specific heat at constant volume of free electrons for gold Figure Temperature dependence of the specific heat at constant volume of free electrons for sodium We obtained different