Quasi-particle Contribution in Thermal Expansion and Thermal Conductivity in Metals

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1 e-issn:-459 -ISSN:47-6 Quasi-article Contribution in Thermal Exansion and Thermal Conductivity in Metals Edema OG * and Osiele OM ederal Polytechnic, Auchi, Nigeria Delta State University, Abraka, Nigeria Research Article Received date: //7 Acceted date: 9//7 Published date: 9//7 *or Corresondence Edema OG, ederal Polytechnic, Auchi, Nigeria, Tel: edemagregori@gmail.com eywords: Electron gas, Quasi-articles, Electron density arameter, Thermal exansion, Thermal conductivity ASTRACT In this aer the modified Landau theory of ermi quids was used to comute thermal exansion and thermal conductivity of quasi-articles in metals. The result revealed that as temerature increases the thermal exansion of quasi-articles in metals increases in all the metals investigated. It is also observed that as the electron density arameter increases the thermal exansion of quasi-articles increases. This shows that at low density region the thermal exansion of quasi-articles is large. The result obtained for the thermal conductivity of quasi-articles in metals revealed that for all the metals comuted the thermal conductivity of quasi-articles decreases as temerature increases. This seems to suggest that as temerature increases the searation between quasi-articles increases because they are not heavy articles hence, the rate of absorbing heat decreases. The comuted thermal exansion and thermal conductivity of quasi-articles are in better agreement with exerimental values. This suggests that the introduction of the electron density arameter is romising in redicting the contribution of quasi-articles to the bulk roerties of metals. This study revealed the extent to which quasi-articles contribute to the bulk roerties of metals, which assisted their otential alications in materials science and engineering develoment. INTRODUCTION The understanding of the effect of the interactions between electrons on the metallic state which is based on the Landau's ermi-liquid theory rovides the basis for understanding metals in terms of weakly interacting quasi-articles. At zero temerature a ermi liquid has a ermi surface, similar to the non-interacting ermions []. The low lying excitations of the ermi liquid are called quasi-articles (or Landau quasi-articles). Landau quasi-articles consist of electrons, surrounded by a cloud of sin and charge olarization. They share the same quantum numbers as free electrons but their masses can be strongly renormalized by the back flow of the surrounding fluid []. When materials undergo Thermodynamical change, the energy received is in form of heat hence, its temerature rises and thereby changes in dimension []. Heat caacities, thermal exansion, thermal conductivities are few imortant roerties often critical in the ractical and engineering alications of solids. Nodar, and Levan, [4] generalized the Landau s theory of ermi liquids by incororating the de roglie waves diffraction. A newly derived kinetic equation of the ermi articles is used to derive a general disersion relation and the excitation of zero sound is studied. A new mode is found due to the quantum correction. It is shown that the zero sound can exist even in an ideal ermi gas. They also disclose a new branch of frequency sectrum due to the weak interaction. Sykes and rooker [5] derived exact exressions for the transort roerties of a degenerate ermi liquid. The coefficients of shear viscosity, thermal conductivity, diffusion and second viscosity were evaluated giving solution for shear viscosity, and diffusion that agree within 5% with those originally quoted. However, the thermal conductivity is reduced by a factor of about. The coefficient of second viscosity was shown to vary with temerature like T. Gangadharaiah, et al. [6] consider a system of D fermions with a short-range interaction. A straight forward erturbation theory is shown to be ill defined even for an infinitesimally weak interaction, as the erturbative series for the self-energy diverges near the mass shell. They show that the divergences result from the interaction of fermions with the zero-sound collective mode. y re-summing the most divergent diagrams, they obtain a closed form of the self-energy near the mass shell. The sectral function exhibits a threshold feature at the onset of the emission of the zero-sound waves. They also show that the interaction RRJPAP Volume 5 Issue 4 October - December, 7 5

2 e-issn:-459 -ISSN:47-6 with the zero sound does not affect a non-analytic, art of the secific heat. In this work, we modified the ermi liquid theory using the electron density arameter and the modified ermi liquid theory is used to comute the thermal exansion and thermal conductivity of ermi liquid in other to see the redictability of our modified ermi liquid theory. Thermal Exansion of Quasi-Particles THEORY AND CALCULATIONS The delocalized free-electron gas of a metal also contributes to the thermal exansion of the metal, in addition to the an harmonic atomic vibrational contribution, the ressure of the ermi gas is given by [7] UTV (, ) Ρ= V () Where, U( TV, ) is given as, π UT = ε fg( ε) + ( T ) g( ε) () 5 6 π T Or, U ( T ) = Nε + N () 5 4 T Since the volume thermal-exansion coefficient is given by [8], π T Cv = N T Where is the bulk modulus, there is a ositive electronic contribution to the thermal exansion because U (T,V) is an increasing function of T. The thermal-exansion coefficient is given as, π T Cv = N (5) T Eqn. (5) can be written by using, π T Cv = N (6) T and also the ulk modulus equation given by, U N 9V V ε = = (7) E. In atomic units the bulk modulus of metals is given by,.586 = E. 5 rs and, E..85rc =. + (9) r s π T Hence, βt = () Aε (4) (8) A =. E a temerature indeendent constant, r c (a.u) is the Ashcroft core radius and r s is the electron density which varies between and 6 for most metals. Recall that the ermi energy of the metals at the ermi surface is given by [9], E 9π () m 4 rs = Inserting eqn. () into eqn. (), then, the thermal exansion of quasi-articles in terms of the electron density arameter 4 r s is exressed as, π m T 9 π () 4 βt = r 4 s A 4 Thermal Conductivity of Quasi-Particles The oltzmann transort equation is given by [5], n n n +.. =Ι t r r Here n= n(, σ, r) is the distribution function for quasi-articles. The energy ε ε(, σ, r) = of a given quantum level deends uon the distribution of the other articles; therefore ε can vary from lace to lace (even in the absence of an external field) if the liquid is inhomogeneous. () RRJPAP Volume 5 Issue 4 October - December, 7 6

3 e-issn:-459 -ISSN:47-6 The left-hand side of the oltzmann transort equation is exanded to give, ( ε ) n ui uk n T i + δikdiv u + k xk xi k xk T xk n n n ψ n v + k div u + + =Ι t k k xk t The term in the left-hand side of eqn. (4) relevant to the thermal conductivity is given by, n. ( ε ) T + =Ι T n n Where is the ermi energy? If we aroximate to and change ( ε ) to ktt eqn. (5) becomes, n. T ( s + kt ) =Ι (6) where s =. Here v = is a function of energy, and is exanded as a ower series in t. The direction of the olar axis T is taken to be that of T, which means that the left-hand side contains cos Θ as its only sherical harmonic [6] then we will have, dt cos Θ s k Tt kt... J ( t,, φ ) dz = Θ ε (7) Then a solution of the form is alied and is given as, ψ dt dz ( t, Θ, φ) = cos Θq( t) The coefficient of defined to be in q t is made a quantity that is evaluated at the ermi surface, so that all of the variation with energy is q t. Recall that, { } (9) m imφ m m (, Θ, φ) =Β ( cos Θ ) (, ) ψ λψ J t e dx k t x t x s n ns ns ns nm, { } m imφ m m (, Θ, φ) =Β ( cos Θ ) (, ) ψ λψ J t e dx k t x t x a n na na na nm, Then substituting eqn. (8) into eqns. (9) and (), we have, { s s } = Β + Β dx k t, x q t q x s Tt, () { s s } = Β + Β dx k t, x q t q x s Tt, () ε where, λ s = and is defined as, Β= k Β, () 6 (, ) and 8π dω ω η θφ A=Β k = * m kt π cos θ Eqns. () and () are the integral equations that are to be solved in order to find the thermal conductivity. y considering first eqn. () it is shown that qs ( t ) is of the order qa ( t ), so that it is negligible beside qa ( t ). In addition the argument is fairly lengthy. irst we have to evaluate s = T. When a temerature gradient is imosed on the ermi liquid, the hysics of the liquid determines the gradient of, so that it must be ossible to find from the transort equation. The required condition is rovided by the conservation of momentum, given by, dτ Ι= (5) Using eqn. (4) we have, τ dn τ τ π T dt (6) = dt s + kt = s +..., (4) (5) (8) () (4) RRJPAP Volume 5 Issue 4 October - December, 7 7

4 e-issn:-459 -ISSN:47-6 S is the entroy er article and more hysically, = T T rom eqn. (7) it aears that in the resence of a temerature gradient the liquid arranges itself to be at a uniform ressure [5]. Recall that, n dτ ψ = (8) Eqn. (8) can be rewritten as, = τ n dt q t t τ dn τ dn = dt qs ( t) + T dt tq ( t) +..., dt dt Considering eqn. (4), the solution consists of a articular solution and a comlementary function which is zero. The thermal conductivity k is given by, Q= k T () Where Q is the flux energy, given by, n Q= dτε ψ () ε Then from eqn. (8) T = τ dn dt k dt tq t Tv τ dn τ dn = dt tqa( t) + T dt t qs( t) +... dt dt We can see that the contribution of q to the thermal conductivity is two factors of T smaller than that of s function need be considered no further. The odd function is taken into consideration, and using eqn. () and (7) (9) () q a, so the even dn A dt q ( t) t = H ( λ ) () dt λ * m 6 Β Then we have, k = v T Η π λ a where, ( λ ) a ( θφ, ) { cosθ } ( λ ) 8 π dω ωη = *4 m T Η π cos θ λ ( 4n + 5) Η ( λ ) = 4 n+ n+ n+ n+ λ n= { } λ = γ + ln + ψ s + ψ s λ λ and γ =.577 while s and s are given as, s = + 8 λ + and s = 8 λ + (6) Recall that the ermi momentum of the quasi-articles at the ermi level is given as [9], 9π P = k = 4 r s a (4) (5) (7) RRJPAP Volume 5 Issue 4 October - December, 7 8

5 e-issn:-459 -ISSN:47-6 Inserting eqn. (7) into eqn. (4), then, the obtained thermal conductivity of quasi-article in terms of the electron density arameter r s is, 6 8π 9π dω ωη ( θφ, ) k *4 { cos } ( a ) (8) = θ λ m T 4 Η π cos θ rs RESULTS AND DISCUSSION igures and show the relationshi between the comuted thermal exansion of quasi-articles and Landau thermal exansion of quasi-articles with temerature for some metals resectively. The results revealed that the thermal exansion of quasi-articles increases as temerature increases for all the metals investigated. Also, the comuted thermal exansion of quasi-articles in metals is larger than the Landau thermal exansion of quasi-articles in metals. This may be as a result of the electron density arameter used to modify the Landau ermi liquid theory and some aroximations made. In both cases as temerature increases the thermal exansion of quasi-articles is closer to the bulk thermal exansion. ut the comuted thermal exansion of quasi-articles is closer to bulk thermal exansion of metals than the Landau thermal exansion of quasiarticles. This suggests that as temerature increases the amlitude of the lattice vibration of quasi-articles increases, the average interatomic distance becomes greater than the zero-temerature searation and hence thermal exansion is enhanced [7]. In all the metals investigated transition metals have lower thermal exansion than alkali and alkali earth metals. This is due to high concentration of quasi-articles in transition metals. Thermal exansion (x -7 ) At Ag Au Temerature () igure. Variation of Calculated Thermal exansion of quasi-articles with temerature for some metals. Thermal Exansion(x - ) 5 5 Ag Au Temerature () igure. Variation of Landau Thermal exansion of quasi-articles with temerature for some metals. RRJPAP Volume 5 Issue 4 October - December, 7 9

6 e-issn:-459 -ISSN:47-6 igures and 4 show the relationshi between the comuted thermal conductivity of quasi-articles and Landau thermal conductivity of quasi-articles with temerature for some metals. The result revealed that for all the metals investigated the thermal conductivity of quasi-articles decreases as temerature increases. This seems to suggest that as temerature increases the searation between quasi-articles increases because they are not heavy articles hence, the rate of absorbing heat decreases. It is also observed that the comuted thermal conductivity of quasi-articles for monovalent metals is larger in transitions metals (coer (), silver (Ag) and gold (Au)) than most of the alkali metals. This is due to the d-block electrons that have filled electron shell which lies high u in the conduction band in noble metals []. This shows that thermal conductivity of quasi-articles deends largely on the electron concentration of metals. The exerimental thermal conductivity of metals is higher than the comuted thermal conductivity of quasi-articles and the comuted thermal conductivity of quasi-articles is higher than the Landau thermal conductivity of quasi-articles. ut the value of the comuted thermal conductivity of quasi-articles is closer to the value of the exerimental thermal conductivity of metals. This seems to suggest that the modified Landau ermi liquid theory can account and redict very well the contribution of quasi-articles to the thermal conductivity of metals. Thermal Conductivity (W/km) Ag Au Temerature () igure. Variation of Calculated Thermal conductivity of quasi-articles with temerature for some metals. Thermal Conductivity (x -76 ) Temerature () igure 4. Variation of Landau Thermal conductivity of quasiarticles with temerature for some metals. RRJPAP Volume 5 Issue 4 October - December, 7

7 e-issn:-459 -ISSN:47-6 CONCLUSION The thermal exansion of quasi-articles increases as temerature increases for all the metals investigated. In both cases as temerature increases the thermal exansion of quasi-articles is closer to the bulk thermal exansion. ut the comuted thermal exansion of quasi-articles is closer to bulk thermal exansion of metals than the Landau thermal exansion of quasiarticles. Also, it is observed that the exerimental thermal conductivity of metals is higher than the comuted thermal conductivity of quasi-articles and the comuted thermal conductivity of quasi-articles is higher than the Landau thermal conductivity of quasi-articles. ut the value of the comuted thermal conductivity of quasi-articles is closer to the value of the exerimental thermal conductivity of metals. This seems to suggest that the modified Landau ermi liquid theory can account and redict very well the contribution of quasi-articles to the bulk thermal exansion and thermal conductivity of metals. In both roerties the Landau theory of ermi liquid under estimated the contribution of quasi-articles to bulk metals. REERENCES. ittel C. Introduction to Solid State Physics. John and Sons, Inc. 996;4-5.. Animalu AOE. Intermediate Quantum Theory of Crystalline Solids. Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 977; Ashcroft NW and Mermin ND. Solid State Physics. Holt, Rinehart and Winston, London. 976; Nodar LT and Levan NT () Elementary Excitations in Quantum ermi quid. 5. Sykes J and rooker GA. The Transort Coefficients of a ermi quid. Annals of Physics. 97;56: Gangadharaiah S, et al. Interacting ermions in Two Dimensions: eyond the Perturbation Theory. Physical Review Letters. 5;94: Elliot SR. The Physics and Chemistry of Solids. John Willey and Sons, Chichester. 998; Thomas P. Landau's ermi quid Concet to the Extreme: The Physics of Heavy ermions. ; Edema OG, et al. Secific heat and Comressibility of Quasi-articles in Metals. Journal of the Nigerian Association of Mathematical Physics. 6;6: röhlich, et al. A two-dimensional ermi liquid with attractive interactions.. RRJPAP Volume 5 Issue 4 October - December, 7

Week 8 lectures. ρ t +u ρ+ρ u = 0. where µ and λ are viscosity and second viscosity coefficients, respectively and S is the strain tensor:

Week 8 lectures. ρ t +u ρ+ρ u = 0. where µ and λ are viscosity and second viscosity coefficients, respectively and S is the strain tensor: Week 8 lectures. Equations for motion of fluid without incomressible assumtions Recall from week notes, the equations for conservation of mass and momentum, derived generally without any incomressibility