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The kinetic theory to problem of attenuation transversal sound waves in metal

Earlier electron influence on sound absorption in metal on basis of the assumption of deformation of Fermi-surfaces under the influence of a sound wave was considered. In the present work other approach to this problem will be considered. Our approach based on the account of dynamic (kinetic) interaction of electronic gas with lattice fluctuations. The analysis of influence of electric field on process of attenuation of sound fluctuations is carried out. It is shown that in the long-wave limit this influence is essential.


Introduction
Electron influence on sound absorption in metal was considered on basis of the assumption of deformation of Fermi-surfaces under the influence of a sound wave [1]. However process of change of Fermi-surfaces is caused by interaction of electronic gas with lattice and inevitably depends on characteristics of this interaction. This process cannot be strictly speaking to be considered in static approach. Dynamical and kinetical processes should be considered at the analysis of formation of Fermi-surfaces at propagation of a sound wave in metal. In the present work the approach to this problem will be considered, based on the account of dynamic (kinetic) interaction of electronic gas with lattice fluctuations.
Let us consider propagation of the transversal sound in the isotropic metal. As well as in the previous works, we will assume, that interaction of a sound wave with electron conductivity not renders appreciable influence on speed of a sound wave. Our purpose there will be a question consideration how electrons conductivity of metal influence on process of attenuation of the sound wave.
The problem of attenuation of a sound wave in metal was considered in works [2]-[10].

Coefficient of attenuation of a sound wave
The transversal sound wave creates a field of velocities in metal u = u 0 e i(kr−ωt) , ku = 0, ω = s tr k.
Here s tr is the velocity of transversal sound oscillations, k is the wave vector.
Let us choose an axis x along a direction of distribution of a sound wave k = k(1, 0, 0), and an axis y along a direction of velocityd u. Then u = u y (0, 1, 0), where u y = u y (x, t) = u 0 e i(kx−ωt) .
The kinetic Vlasov Boltzmann equation with integral of collisions of relaxation type for electrons will have the following form

1)
where f eq is the equilibrium Fermi-distribution in degenerate plasmas. Equilibrium distribution corresponds to a condition of electronic gas which has zero velocity comparatively of lattice. We will consider a case of isotropic metal with undisturbed spherical Fermi-surface. The transversal sound wave leads to distortion of lattice, that in turn can lead to distortion equilibrium Fermi-surface of metal. In the given work we will not to be consider this effect. Thus, in the considered case we have Here E(v − u) is the electron energy, Θ(x) is the unit step of Heaviside, quantities e and m are charge and mass of electron сorrespondingly. At such statement of this problem we consider, that electrons at scattering to pass in equilibrium condition with the lattice which possesses the local speed u = u(x, t).
At such approach to this problem there is no necessity to enter "fictitious forces" [5], responsible for interaction electronic gas with lattice fluctuations.
At considered statement of the problem electric field E has only one nonzero to a component E = E y (0, 1, 0), where Because of an electroneutrality the volume density of a charge of ions (lattice) equals −enu, where n = const is the electron concentration, the current density is equal −en. Then the equation on the field looks like Here j y (x, t) is the electron current density along axis y, In linear approach the distribution electron function in the field sound wave we will search in the form Here f 0 is the absolute Fermian, The energy flux (stream), a transferable by sound wave is equal where ρ is the density of metal.
Because of attenuation of a sound wave where Γ is the coefficient of attenuation. The coefficient of attenuation Γ e is defined by following expression Here Q e is the density of energy dissipation of a sound wave. The dissipation is caused by nonlinearity of fluctuations of a lattice and interaction of sound fluctuations of electronic component and field generated by a electromagnetic wave. The quantity Q e is calculated as follows Q e = −F y u y . (1.5) Here force F y is caused by two different factors. First, on the charged ions operate the electric force induced by electric currents σE. Here σ is the volume density of a charge of a lattice. Because of a metal electroneutrality it is had σ = −ne.
On the other hand on a lattice operates force of a friction from the party of electronic gas F e . At each scattering of electrons, having velocity v, on the average the momentum m(u − v) is transferred to a lattice. Considering, that force F e has only one the nonzero component, directed along the axis lengthways y, we have Here it has been considered, that the numerical density (concentration) of electrons is equal also the average quantity of y-components of electron velosity along an axis y is entered,v

Thus, force of the friction is equal
Therefore the density of energy dissipation of a sound wave is the following It is obvious, that the velocity of of electron current and average quantity y-component of electrov velocity are connected among themselves by equalities j y = env y andv y = (1/en)j y .
Let us consider further, that . Let us make averaging on time (see, for example, [2]) equalities (1.5). Passing to the real variables, we receive Here the top index "asterisk" means complex conjugation. According to (1.4) and (1.6) coefficient of attenuation of a sound wave it is equal 2. The solution of system of the equations and electric current density After linearization of locally equilibrium distribution function on quantity of speed u we receive By means of (2.1) and (2.2) the equation (1.1) it will be transformed to the following equation In the right part of the equation (1.2) there is a density of current, which taking into account equality (2.1) it is equal From the equation (2.3) we find function Let us substitute expression (2.5) in the previous expression (2.4) for electric current density. Then for current density it is received following expression The density of longitudinal current now equals Let us consider that fact, that velocity of the sound is much less than electron velocity on Fermi's surfaces: s tr ≪ v F . Then the quantity Ω is small parameter where ε 1 is so small parameter, Taking into account this fact the expression for ϕ(q, Ω, y) becomes simpler and looks like Now expression for density of an electric current (2.7) becomes simpler, and its amplitude it is equal
Let us take advantage of equality (3.4) and we will allocate the real and imaginary part at fraction D .
Therefore the quantity E 0 equals .

(3.6)
From here we receive the real part of this quantity .

Coefficient of attenuation of a sound wave
We note that according to (3.6) we have Hence, according to (2.8) for amplitude of density of an electric current it is received From here for average quantity of electron velocity it is had Let us return to the formula (1.7) for coefficient of attenuation and we will present it in the form .
Let us pass to a finding Q 2 . According to (4.1) it is found Let us transform the previous expression to the form The coefficient of attenuation of a sound wave according to (4.2) is equal Let us transform the formula (4.3) to the form where K(q, y) is the dimensionless coefficient of attenuation, Let us present this coefficient in an explicit form From expression (4.5) it is visible, that the dimensionless coefficient of attenuation consists of two components K(q, y) = K 1 (q, y) + K 2 (q, y).
Here K 1 (q, y) is the contribution to coefficient of attenuation which brings electric field, is the contribution to coefficient of attenuation which brings electronic friction with a crystal lattice, .
Investigating behavior of dimensionless factor K(q, y), it is possible to find out, that with the big accuracy coefficient K(q, y) it is possible to replace by coefficient K 0 (q, y) = 1 + 3y 2q 3 ϕ 0 (q, y).
Let us underline, that all plots in the present work are constructed for gold.
From Figs. 1 and 2 it is visible, that the coefficient K 0 (q, y) effectively approximates coefficient K(q, y) at all values of wave numbers. Comparison of plots on Fig. 1 and 2 shows, that quantity of the deviation of coefficient K 0 (q, y) from K(q, y) decreases at transition of dimensionless frequency of collisions from 10 −4 to 10 −3 . More below we investigate is more thin structure of dimensionless coefficient, representing the sum of coefficients K 1 (q, y) and K 2 (q, y).
Let us enter critical value of dimensionless wave number q 0 , such, that This critical wave number breaks an interval of values of the dimensionless wave number on two interval: the first interval I 1 = {0 < q < q 0 } and the second interval I 2 = {q > q 0 }. Let us estimate typical values of critical wave number for gold. We take two values of dimensionless frequency of collisions y = 10 −3 and y = 10 −4 . Corresponding values of critical wave number are equal q 0 = 1.25 · 10 −7 , and q 0 = 1.25 · 10 −6 .
From Fig. 6 it is visible, that the critical wave number is equal in accuracy q = q 0 = 1.25 · 10 −6 .
On Fig. 6 dependence of coefficients K 1 (q, y) and K 2 (q, y) from wave number at the fixed value of frequency of electron collisions is presented.
From Fig. 6 it is visible, that graphics of coefficients K 1 (q, y) and K 2 (q, y) are crossed in the point q 0 . The point crossings q 0 of graphics is the critical value of wave number. At change wave number from zero to q 0 the attenuation coefficient is defined by the electric field (the coefficient K 1 (q, y)), and at q > q 0 the attenuation coefficient is defined by friction of electrons on the crystal lattice (the coefficient K 2 (q, y)).
Let us begin with the first interval. We will find asymptotics of attenuation coefficient in the case 0 < q ≪ q 0 . We will take advantage of the 5.0x10  Fig. 6. The contribution to attenuation of the transversal sound wave. Curves of 1 and 2 according correspond to electric field and friction of electrons on crystal lattice, y = 10 −4 . At q < q 0 the contribution to attenuation is defined by electric field, and at q > q 0 the contribution to attenuation is defined by an electronic friction.
previous decomposition. At small q → 0 for fraction D 1 we have From this equality we obtain that We note that y 2 /ε 2 = 1/q 2 0 , q 0 ≪ y, therefore in the considered interval 22 Therefore finally we have From the formula (4.1) for average electron velocity at small values of wave number we receive From here we find that Further we receive that From here it is visible, that on absolute value in the limit at q → 0 average electron velocity tends to velocity of the lattice Now we will find the coefficient K(q, y). According to (4.4) it is had So, at small values of wave number the attenuation coefficient is equal Γ e (q, y) = νnm ρs tr q 2 5y 2 − 3q 4 35y 4 + · · · = = νnmq 2 5ρs tr y 2 1 − 3q 2 7y 2 + · · · , or, with dimensional parameters, From here in square-law approach (on k) Γ e (k) = Γ 0 k 2 (k → 0), where Let us estimate one after another this quantity for gold. The mass of electron is equal m = 9 · 10 −28 gr, the gold density is equal ρ = 19.32gr/cm 3 , we consider, that ν ∼ 10 13 1/sec, other data we take from resulted above the table. We receive, that Γ 0 = 9 · 10 −28 (1.4 · 10 8 ) 2 (8.2 · 10 7 ) 3 15π 2 (19.3)10 13 (1.7 · 10 5 ) ∼ 2 · 10 −9 1 cm .
On Fig. 7 the behaviour of factors K 1 (q, y) and K 2 (q, y) is presented. We will notice, that in the range of wave numbers, biger critical, the coefficient K 1 (q, y) has the unique maximum, and then quickly decreases to zero. From Fig. 8 it is visible, that the maximum quantity decreases with growth of frequency of electron collisions.
We obtain that Thus, at small values of wave number the contribution to attenuation coefficient it is defined by presence of electric field.
Let us consider the second interval of values of dimensionless wave number q > q 0 . This interval we will break into three intervals: q 0 ≪ q ≪ y, y ≪ q ≪ 1 and q ∼ 1.
Let us observe the third case small q: y ≪ q ≪ 1. In this case Hence, the dimensionless attenuation coefficient is equal The dimensional attenuation coefficient is equal Now we will observe the case, when q ∼ 1. In this case The dimensionless attenuation coefficient now is equal The dimensional attenuation coefficient is computed under the same formula, as in the previous interval.
On Fig. 9 the behaviour of function K(q, y)/q 2 depending on the wave number at various values of dimensionless electron collision frequencies is presented.
Graphics show dependence of coefficient Γ 0 from magnitude of a collision frequency of electrons. For comparison on Fig. 10 dependence of coefficient K 1 (q, y) is shown at q → 0 at various values of collision frequency of electrons. Let us note, that magnitudes K(q, y)/q 2 and K 1 (q, y) decrease with growth of collision frequency.

Comparision with Pippard' results
Let us result of damping coefficient discovered by Pippard [5] α T = nm ρs tr τ 2 3 Here Let us this result to our denotations We investigate the case of small values of the wave number. We have Therefore from here we obtain α T = νnmq 2 ρs tr y 2 1 + This effect is comparable with our effect Γ e (q, y) = νnmq 2 5ρs tr y 2 1 − 3q 2 7y 2 + · · · , Both coefficients diminish at q → 0 as q 2 , but with various coefficients (Fig. 11) Fig. 11. Comparison of dimensionless coefficients of attenuation at small values dimensionless wave number. The curve 1 corresponds to coefficient K(q, y), obtained in the present operation, the curve 2 corresponds to coefficient, discovered by Pippard' (see [5]), y = 0.001.

Conclusion
In summary we will observe the case when the electric field misses. Then Last expression will easily be converted to the form Γ e = νnm ρs tr K 0 (q, y), где K 0 (q, y) = 1 + 3y 2q 3 ϕ 0 (q, y). As it was already specified, coefficient K 0 (q, y), obtained without electrical field, the attenuation dimensionless coefficient K(q, y) effectively approximates. However, the analysis carried out above shows, that for the thin analysis of a damping coefficient its representation in the form of the total K(q, y) = K 0 (q, y) + K 1 (q, y) is required.
The present work represents the kinetic approach to damping coefficient investigation sound wave into degenerate plasma, somewhat alternative to the approach Pippard' [5]. Investigation of coefficient is carried out of attenuations of a soundc wave on the basis of the kinetic (dynamic) approach of interraction of degenerate electronic gas with oscillations of lattice in given work. [9] Becker K., Fulde P., Keller J. Thalmeier P. Theory of ultrasonic attenuation in metallic rare earth systems// Journal de Physique Colloques, 1979, 40 (C5), pp. C5-35-C5-37.
[10] Lüthi B. Physical Acoustics in the Solid State. Springer, Berlin Heidelberg New York. Springer Series in solid-state sciences. 420 pp.