aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b DEBYE MODEL FOR THE SURFACE PHONONS
Yu.M. Poluektov ∗ National Science Center “Kharkov Institute of Physics and Technology”,1, Akademicheskaya St., 61108 Kharkov, Ukraine
A quantum description of the surface waves in an isotropic elastic body without the use of thesemiclassical quantization is proposed. The problem about the surface waves is formulated in theLagrangian and Hamiltonian representations. Within the framework of the generalized Debye model,the contribution of the surface phonons (“rayleighons”) to thermodynamic functions is calculated.It is emphasized that the role of the surface phonons can be significant and even decisive in low-dimensional systems, granular and porous media, and that their contribution to the total heatcapacity increases with decreasing temperature.
Key words : Rayleigh surface waves, phonon, Debye model, entropy, heat capacity
PACS numbers: 63.20.-e, 63.20.Pw, 62.30.+d, 62.65.+k
I. INTRODUCTION
Methods of describing spatially homogeneous condensed media are fairly well developed, a large number of specificproblems have been solved. Although here, too, there remain many unsolved problems associated mainly with thecorrect account for interparticle interactions and the description of phase transitions.In recent years, more and more attention has been paid to both experimental and theoretical study of media whereininterphase boundaries and surfaces play an important role. It is precisely in such essentially inhomogeneous systemsthat there is reason to hope for the discovery of new physical effects and the production of new materials with uniqueproperties on their basis. A simple transfer of the methods of theoretical study of homogeneous systems to the case ofstrongly inhomogeneous media is clearly insufficient, and therefore there is a need for the elaboration and developmentof special approaches. For low temperatures, the quantum-mechanical description of such systems, in particular oftheir electronic and phonon properties, has acquired a fundamental importance.A widespread model of solids, which gives a correct description of thermodynamic properties in the limiting cases oflow and high temperatures, is the well-known model proposed by Debye even before the creation of modern quantumtheory [1]. In the Debye model for an elastic isotropic medium, it is assumed that phonons have a certain averagedvelocity. However, it offers no principal difficulty to generalize the Debye model for an isotropic solid with takinginto account longitudinal and transverse phonons propagating with different velocities [2]. As was shown by Rayleigh[3], in an isotropic elastic body, together with longitudinal and transverse waves, along the body surface there maypropagate surface waves, the amplitude of which decreases with distance from the boundary. The quasiparticlescorresponding to these waves – the surface phonons – will also give a contribution to the thermodynamic functionsof a solid which is proportional to the area of its surface. This contribution can be significant and even decisive forlow-dimensional samples, as well as bodies having a granular or porous structure. The questions of the theory of localand surface vibrations of a crystal lattice, as well as the literature on this issue, are considered in the book [4].The quantization of the surface waves, that is similar to the semiclassical quantization of the photon field by Planck,was carried out by Khalatnikov in the study of heat transfer between a solid and superfluid helium [5]. However, aconsistent quantum theory, as is known, was built only more than a quarter of a century after the introduction of anew world constant by Planck. At the same time, a general prescription for the transition from classical to quantumdescription became clear. It consists in the fact that within the framework of the classical approach there are selectedgeneralized coordinates and canonical momenta, which, upon passing to the quantum description, are considered asoperators acting in the space of wave functions and obeying the commutation relations which generalize the classicalPoisson brackets. This quantization prescription is suitable both for systems of particles interacting by means ofa potential [6,7] and for continuous media (fields) [2,8,9]. For surface waves at the boundary of an isotropic solid,such a quantization method has not yet been implemented, which is the main goal of this work. On the basis ofthe generalized Debye model, taking into account the difference in the velocities of longitudinal and transverse soundwaves [2], we calculate the contribution of the surface phonons to thermodynamic functions at arbitrary temperatures. ∗ Electronic address: [email protected]
II. SURFACE WAVES IN AN ISOTROPIC ELASTIC MEDIUM
Let us first consider the classical surface wave problem. Suppose that an elastic isotropic medium fills the half-space z <
0. Thus, the surface separating the medium and vacuum is z = 0. The deformation vector in an elastic isotropicmedium u ( r , z, t ) satisfies the equation [10] ¨ u = c t ∆ u + (cid:0) c l − c t (cid:1) ∇ div u , (1)where c t = r µρ , c l = s λ + 2 µρ (2)are the velocities of the transverse and longitudinal waves, µ, λ are the Lame coefficients, ρ is the density.Since the system under consideration is spatially homogeneous in the xy -plane and inhomogeneous along the z -axisperpendicular to the surface, we represent the real deformation vector in the form of an expansion in two-dimensionalplane waves u ( r , z, t ) = 1 √ A X k h u ( k , z, t ) e i kr + u ∗ ( k , z, t ) e − i kr i , (3) A is the surface area. Here r and k are the two-dimensional vectors: r ≡ ( x, y ), k ≡ ( k x , k y ). As is known [10], thedeformation vector of a medium can be represented as the sum u = u l + u t , where div u t = 0 and rot u l = 0. Each ofthese vectors can be expressed in terms of scalar and vector potentials u l = ∇ ϕ, u t = rot ψ (4)which satisfy the wave equations ¨ ϕ = c l ∆ ϕ, ¨ ψ = c t ∆ ψ . (5)The potentials can be represented in a form similar to the decomposition (3): ϕ ( r , z, t ) = 1 √ A X k h ϕ ( k , z, t ) e i kr + ϕ ∗ ( k , z, t ) e − i kr i , ψ ( r , z, t ) = 1 √ A X k h ψ ( k , z, t ) e i kr + ψ ∗ ( k , z, t ) e − i kr i . (6)Applying the Fourier transform in time ϕ ( k , z, t ) = Z ∞−∞ ϕ ( k , z, ω ) e − iωt dω, ψ ( k , z, t ) = Z ∞−∞ ψ ( k , z, ω ) e − iωt dω, (7)from the equations (4) we find the solutions bounded at z < ϕ ( k , z, ω ) = A ( k , ω ) exp( γ l z ) , ψ ( k , z, ω ) = B ( k , ω ) exp( γ t z ) , (8)where γ l = s k − ω c l , γ t = s k − ω c t . (9)If the radicands in (9) are positive, then the waves decay exponentially deep into the medium, and if the radicandsare negative, then the solutions (8) have the form of oscillations.The components of the vector of longitudinal deformations, with taking into account the form of the solutions (8)and the formula (4), are determined by the relations: u lx = i √ A X k Z dωk x h A ( k , ω ) e γ l z e i ( kr − ωt ) − A ∗ ( k , ω ) e γ ∗ l z e − i ( kr − ωt ) i ,u ly = i √ A X k Z dωk y h A ( k , ω ) e γ l z e i ( kr − ωt ) − A ∗ ( k , ω ) e γ ∗ l z e − i ( kr − ωt ) i ,u lz = 1 √ A X k Z dω h γ l A ( k , ω ) e γ l z e i ( kr − ωt ) + γ ∗ l A ∗ ( k , ω ) e γ ∗ l z e − i ( kr − ωt ) i . (10)The components of the vector of transverse deformations, with account of (8) and (4), are given by the formulas: u tx = 1 √ A X k Z dω n(cid:2) ik y B z ( k , ω ) − γ t B y ( k , ω ) (cid:3) e γ t z e i ( kr − ωt ) + (cid:2) − ik y B ∗ z ( k , ω ) − γ ∗ t B ∗ y ( k , ω ) (cid:3) e γ ∗ t z e − i ( kr − ωt ) o ,u ty = 1 √ A X k Z dω n(cid:2) − ik x B z ( k , ω ) + γ t B x ( k , ω ) (cid:3) e γ t z e i ( kr − ωt ) + (cid:2) ik x B ∗ z ( k , ω ) + γ ∗ t B ∗ x ( k , ω ) (cid:3) e γ ∗ t z e − i ( kr − ωt ) o ,u tz = i √ A X k Z dω n(cid:2) k x B y ( k , ω ) − k y B x ( k , ω ) (cid:3) e γ t z e i ( kr − ωt ) − (cid:2) k x B ∗ y ( k , ω ) − k y B ∗ x ( k , ω ) (cid:3) e γ ∗ t z e − i ( kr − ωt ) o . (11)The formulas (10), (11) give a general solution for the deformation vector under arbitrary boundary conditions, whichis bounded in the half-space z < f i = σ ik n k [10], where n is the outward normal vector to thesurface, which in the case under consideration is directed along the z -axis, so that n = (0 , , z = 0 the force is equal to zero, we have the boundary conditions σ xz = σ yz = σ zz = 0, which, with taking intoaccount the form of the stress tensor for an isotropic elastic medium, have the form ∂u x ∂z + ∂u z ∂x = 0 , ∂u y ∂z + ∂u z ∂y = 0 , (1 − σ ) ∂u z ∂z + σ (cid:18) ∂u x ∂x + ∂u y ∂y (cid:19) = 0 , (12)where σ is the Poisson coefficient. Using the formulas (10), (11), by means of the boundary conditions (12) we obtainthe algebraic equations for the expansion coefficients in these formulas2 ik x γ l A + k x k y B x − (cid:0) γ t + k x (cid:1) B y + ik y γ t B z = 0 , ik y γ l A + (cid:0) γ t + k y (cid:1) B x − k x k y B y − ik x γ t B z = 0 , (cid:2) (1 − σ ) γ l − σ (cid:0) k x + k y (cid:1)(cid:3) A + i (1 − σ ) γ t (cid:0) k x B y − k y B x (cid:1) = 0 . (13)To shorten the notation we used the designations A ( k , ω ) ≡ A, B ( k , ω ) ≡ B . It is also convenient to introduce thedesignations B + = k − (cid:0) k x B x + k y B y (cid:1) , B − = k − (cid:0) k x B y − k y B x (cid:1) , (14)where k = k x + k y . Then the system of equations (13) takes the form iB z = α t B + , (15)2 iα l A − (cid:0) α t (cid:1) B − = 0 , (2 − y ) A + 2 iα t B − = 0 , (16)wherein y ≡ ω c t k , γ l = α l k, γ t = α t k,α l ≡ p − ξ y, α t ≡ p − y, ξ = (cid:18) c t c l (cid:19) = (1 / − σ )(1 − σ ) . (17)The compatibility condition for a system of linear homogeneous equations gives the equation4 α t α l = (2 − y ) , (18)which, with account of the formulas (17), is reduced to the well-known cubic equation [10] y − y + 8(3 − ξ ) y − − ξ ) = 0 , (19)determining the dispersion law of surface waves. The parameter y is a real number less than one, the value of whichis determined by the ratio of the transverse and longitudinal wave velocities. Thus, the velocity of surface waves c s = √ y c t (20)is less than the velocity of body waves. The radicands in (17) are positive for all values of y and the quantities α l , α t are real, while α l > α t . In the case when there is a relationship ω = ω ( k ) between the frequency and thewavenumber, the amplitudes in the formulas (10), (11) should be taken in the form A ( k , ω ) = ˜ A ( k ) δ (cid:0) ω − ω ( k )) , B ( k , ω ) = ˜ B ( k ) δ (cid:0) ω − ω ( k )) , (21)where ˜ A ( k ) ≡ A (cid:0) k , ω ( k ) (cid:1) , ˜ B ( k ) ≡ B (cid:0) k , ω ( k ) (cid:1) . In our case ω ( k ) = √ yc t k . In the following we will omit the tildesign. The relations (13) – (16) make it possible to express the deformation vector only in terms of one amplitude A ( k ): u x ( r , z, t ) = i √ A X k n k x g ( z, k ) h A ( k ) e i ( kr − ω ( k ) t ) − A ∗ ( k ) e − i ( kr − ω ( k ) t ) io ,u y ( r , z, t ) = i √ A X k n k y g ( z, k ) h A ( k ) e i ( kr − ω ( k ) t ) − A ∗ ( k ) e − i ( kr − ω ( k ) t ) io ,u z ( r , z, t ) = α l √ A X k n k g ( z, k ) h A ( k ) e i ( kr − ω ( k ) t ) + A ∗ ( k ) e − i ( kr − ω ( k ) t ) io . (22)Here, the following functions are defined g ( z, k ) ≡ e α l kz − α l α t α t e α t kz , g ( z, k ) ≡ e α l kz −
21 + α t e α t kz . (23)After defining the time-dependent amplitudes A ( k , t ) = A ( k ) e − iω ( k ) t , A ∗ ( k , t ) = A ∗ ( k ) e iω ( k ) t , (24)and introducing the new amplitudes X ( k , t ) = X ∗ ( − k , t ) ≡ A ( k , t ) + A ∗ ( − k , t ) , (25)the components of the deformation vector can be written as u x ( r , z, t ) = i √ A X k k x g ( z, k ) X ( k , t ) e i kr ,u y ( r , z, t ) = i √ A X k k y g ( z, k ) X ( k , t ) e i kr ,u z ( r , z, t ) = α l √ A X k k g ( z, k ) X ( k , t ) e i kr . (26)Before proceeding to quantizing surface excitations, we present the Lagrangian and the Hamiltonian formulations ofthe problem. III. LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF THE SURFACE WAVES PROBLEM
The density of the Lagrangian function, which leads to the equations for waves in an isotropic elastic medium (1),has the form Λ( r , z, t ) = ρ u − λ (cid:0) div u (cid:1) − µ (cid:0) ∇ j u i ∇ i u j + ∇ j u i ∇ j u i (cid:1) , (27)and to find the complete Lagrangian function one should integrate (27) over the entire volume occupied by the medium L = Z A d r Z −∞ dz Λ( r , z ) . (28)Using the obtained expressions for the components of the deformation vector (26), for which the boundary conditions(12) are fulfilled, and integrating over spatial coordinates, we obtain the Lagrangian function in the form L = ρα l y ) X k h k (cid:12)(cid:12) ˙ X ( k ) (cid:12)(cid:12) − c s k (cid:12)(cid:12) X ( k ) (cid:12)(cid:12) i , (29)where Θ( y ) ≡ (cid:2) − y ) + (2 − y ) (cid:3) (1 − y )(2 − y ) − y )(2 − y ) = y (cid:0) − y + 11 y − y (cid:1) (1 − y )(2 − y ) . (30)Instead of the complex quantities in (29), one should pass to the real quantities X ( k ) = X ′ ( k ) + iX ′′ ( k ) and consider X ′ ( k ) , X ′′ ( k ) as generalized coordinates and ˙ X ′ ( k ) , ˙ X ′′ ( k ) as generalized velocities, on which the Lagrangian functiondepends. However, it should be taken into account that the Lagrangian must be expressed in terms of independentcoordinates and velocities. Meanwhile, due to the condition (25), the coordinates at the oppositely directed wavevectors k and − k are linked by the relations X ′ ( k , t ) = X ′ ( − k , t ) , X ′′ ( k , t ) = − X ′′ ( − k , t ) (31)and therefore are not independent. To pass to independent variables in the Lagrangian (29), we decompose it intotwo terms, each of which contains a summation over all wave vectors except for the oppositely directed ones. Due tothe conditions (31), the contribution to the Lagrangian of each such term will be the same. Therefore, the Lagrangian(29) should be multiplied by two, and the summation should be carried out only over those wave vectors among whichthere are no the oppositely directed ones. For example, we may sum over all wave vectors having k z >
0. As a result,the Lagrangian, expressed in terms of the real independent even and odd (31) coordinates and velocities, takes theform L = ρα l Θ( y )2 ⌢ X k h k ˙ X ′ ( k ) − c s k X ′ ( k ) i + ρα l Θ( y )2 ⌢ X k h k ˙ X ′′ ( k ) − c s k X ′′ ( k ) i . (32)Here, the symbol ⌢ above the sum sign, as explained above, means a summation over vectors among which there areno the oppositely directed ones. The Euler-Lagrange equations give rise to the equations of motion¨ X ′ ( k ) + c s k X ′ ( k ) = 0 , ¨ X ′′ ( k ) + c s k X ′′ ( k ) = 0 . (33)Let us move on to the Hamiltonian description, defining the canonical momentaΠ ′ ( k ) = ∂L∂ ˙ X ′ ( k ) = ρα l Θ( y ) k ˙ X ′ ( k ) , Π ′′ ( k ) = ∂L∂ ˙ X ′′ ( k ) = ρα l Θ( y ) k ˙ X ′′ ( k ) . (34)We introduce the Hamilton function H = ⌢ X k (cid:18) ˙ X ′ ( k ) ∂L∂ ˙ X ′ ( k ) + ˙ X ′′ ( k ) ∂L∂ ˙ X ′′ ( k ) (cid:19) − L = H ′ + H ′′ , (35)where H ′ , H ′′ are the Hamiltonians expressed in terms of the coordinates and momenta with one and two primes: H ′ = ρα l Θ( y )2 ⌢ X k (cid:20) Π ′ ( k ) ρ α l Θ ( y ) + c s k X ′ ( k ) (cid:21) ,H ′′ = ρα l Θ( y )2 ⌢ X k (cid:20) Π ′′ ( k ) ρ α l Θ ( y ) + c s k X ′′ ( k ) (cid:21) . (36)Since these Hamiltonians have the same form, it suffices to consider only the Hamiltonian expressed in terms of thecoordinates and momenta with one prime. From the Hamilton equations˙Π ′ ( k ) = − ∂H ′ ∂X ′ ( k ) , ˙ X ′ ( k ) = ∂H ′ ∂ Π ′ ( k ) (37)there follow the equations of motion˙Π ′ ( k ) = − ρα l Θ( y ) c s k X ′ ( k ) , ˙ X ′ ( k ) = Π ′ ( k ) ρα l Θ( y ) k . (38)These equations with account of the definition of momenta (34), of course, can also be written in the form of thesecond-order equations for an oscillator (33). IV. QUANTIZATION OF SURFACE WAVES
Now we pass from the classical to the consistently quantum description of surface excitations without the use ofthe semiclassical approximation [5]. In this case, the coordinates and momenta should be considered as operatorsobeying the following well-known commutation relations: (cid:2) X ′ ( k ) , Π ′ ( k ′ ) (cid:3) ≡ X ′ ( k )Π ′ ( k ′ ) − Π ′ ( k ′ ) X ′ ( k ) = i ~ ∆( k − k ′ ) , (cid:2) X ′ ( k ) , X ′ ( k ′ ) (cid:3) = (cid:2) Π ′ ( k ) , Π ′ ( k ′ ) (cid:3) = 0 . (39)The same commutation relations hold for the coordinates and momenta with two primes, and all variables withone prime commute with all variables with two primes. It is convenient to pass from the Hermitian operators X ′ ( k ) = X ′ + ( k ) and Π ′ ( k ) = Π ′ + ( k ) to the new non-Hermitian operators a + ( k ) , a ( k ), which, as we will see, have themeaning of the operators of creation and annihilation of surface phonons X ′ ( k ) = C ( k ) a ( k ) + C ∗ ( k ) a + ( k ) , Π ′ ( k ) = B ( k ) a ( k ) + B ∗ ( k ) a + ( k ) , (40)where the coefficients C ( k ) , B ( k ) are c -numbers. The commutation relations (39) will be satisfied if the followingwell-known commutation relations are required to hold (cid:2) a ( k ) , a + ( k ′ ) (cid:3) ≡ a ( k ) a + ( k ′ ) − a + ( k ′ ) a ( k ) = ∆( k − k ′ ) , (cid:2) a ( k ) , a ( k ′ ) (cid:3) = (cid:2) a + ( k ) , a + ( k ′ ) (cid:3) = 0 . (41)In addition, in order for the transformation (40) to be canonical, the following condition should hold C ( k ) B ∗ ( k ) − C ∗ ( k ) B ( k ) = i ~ . (42)In the Hamilton operators (36), which we write in the form (similarly for H ′′ ) H ′ = ⌢ X k h ψ ( k )Π ′ ( k ) + ϕ ( k ) X ′ ( k ) i , (43)where for brevity the designations ψ ( k ) ≡ (cid:14) ρα l Θ( y ) k , ϕ ( k ) ≡ ρα l Θ( y ) c s k (cid:14) a + ( k ) , a ( k ). When substituting the relations (40) into (43) we require that only the operatorsof the form a ( k ) a + ( k ) or a + ( k ) a ( k ) remain in the resulting Hamiltonian, while the operators a ( k ) and a +2 ( k ) dropout. This entails the fulfillment of the condition ψ ( k ) B ( k ) + ϕ ( k ) C ( k ) = 0 . (44)The condition (44) will be satisfied if B ( k ) = 2 (cid:14) ~ ρα l Θ( y ) c s k , C ( k ) = − ~ (cid:14) ρα l Θ( y ) c s k . (45)When extracting the root the signs can be chosen arbitrarily, since this will not affect the final result. Thus, thecanonical transformation (40) takes the form X ′ ( k ) = i ~ k p ~ ρα l y ) c s (cid:2) a ( k ) − a + ( k ) (cid:3) , Π ′ ( k ) = k r ~ ρα l Θ( y ) c s (cid:2) a ( k ) + a + ( k ) (cid:3) . (46)As a result, using the commutation relations (41), we find H ′ = ⌢ X k ~ c s k (cid:20) a + ( k ) a ( k ) + 12 (cid:21) . (47)The diagonalization of the Hamiltonian H ′′ can be carried out in a similar way. When calculating the thermodynamicquantities, the Hamiltonians H ′ and H ′′ make the same contribution, so that one can use the doubled operator (47)and the total Hamiltonian takes the form H = 2 ⌢ X k ~ c s k (cid:20) a + ( k ) a ( k ) + 12 (cid:21) . (48)The Hamiltonian (48) differs from the Hamiltonian of bulk phonons in the value of the velocity and in that thesummation is carried out over two-dimensional wave vectors. The surface two-dimensional phonons, having velocity c s , can naturally be called “rayleighons”. Note that the quanta of vibrations of the surface of liquid helium are calledriplons [11,12]. V. CONTRIBUTION OF “RAYLEIGHONS” TO THERMODYNAMIC FUNCTIONS
The average values of the surface thermodynamic functions are calculated by means of the statistical operator ρ = exp β (cid:0) F − H (cid:1) , (49)where the Hamiltonian is defined by the formula (48), and β = 1 (cid:14) T is the inverse temperature. From the normalizationcondition Sp ρ = 1 we find the surface free energy F = ~ X k c s k + T X k ln (cid:0) − e − β ~ c s k (cid:1) . (50)Here the summation, as noted above, is carried out over the two-dimensional wave vectors k ≡ ( k x , k y ). The firstterm in (50) determines the contribution of zero oscillations. In (50) we pass from summation to integration in whichthe upper limit of integration over the magnitude of the wave vector is determined, as in the bulk case, by the Debyerelation k D = (cid:0) π n (cid:1) / , n = N/V is the particle number density. As a result, we obtain the surface free energy inthe form F = Ak D π (cid:26) Θ s + 32 T h (cid:16) − e − τ − (cid:17) − D (cid:0) τ − (cid:1)i(cid:27) , (51)where Θ s ≡ ~ c s k D is the “surface” Debye energy, τ ≡ T / Θ s , and the Debye functions are defined by the formula D n ( x ) = nx n Z x z n dze z − n ≥ . (52)The surface entropy S = − (cid:0) ∂F/∂T (cid:1) VA has the form S = Ak D π h D (cid:0) τ − (cid:1) − (cid:16) − e − τ − (cid:17)i . (53)From here we find the heat capacity at a constant volume and surface area C VA = T (cid:18) ∂S∂T (cid:19) VA = Ak D πτ (cid:20) τ D (cid:0) τ − (cid:1) − e τ − − (cid:21) . (54)Let us consider the behavior of the surface heat capacity (54) in the limit of low and high temperatures. Since D ( x ) ≈ x − ζ (3) at x ≪
1, we find at τ ≪ C VA ≈ ζ (3) π Ak D (cid:18) T Θ s (cid:19) = 3 ζ (3) π ~ c s AT , (55) ζ ( s ) is the Riemann zeta function. Note that the structure of this formula is similar to the structure of the corre-sponding formula for the bulk case of the usual Debye theory C V ≈ π V k D (cid:18) T Θ D (cid:19) = 2 π ~ c D V T , (56)where c D is the average phonon velocity according to Debye [13,14]. In the two-parameter theory [2], the formula forthe low-temperature bulk heat capacity takes the form C V ≈ π V k D · f ( χ ) (cid:18) T Θ (cid:19) = 2 π ~ (cid:18) c t + c l (cid:19) / V f ( χ ) T , (57)where Θ = 13 (cid:0) t + Θ l (cid:1) , tg χ = √ t Θ l , Θ t = ~ c t k D , Θ l = ~ c l k D , f ( χ ) ≡ / (cid:18) χ + 2 / sin χ (cid:19) . The calculation of the surface contribution to the phonon heat capacity has received a considerable attention. Thequadratic dependence on temperature was obtained in [15], and subsequently the form of the coefficient before T was refined in other works [4]. In [16 – 18], there was obtained a formula for the surface heat capacity in variousapproaches, which in the notation of (17), (19), (20) can be represented in the form [6,16] C A = 3 ζ (3) π ~ c s · y (cid:0) ξ − ξ + 3 (cid:1) (cid:0) − ξ (cid:1) AT . (58)In the formulas (55), (58) the coefficients at T differ in value insignificantly. So, for the Poisson coefficient σ = 1 / ξ = 0 we have C A (cid:14) C VA ≈ . σ = 0 and ξ = 1 / C A (cid:14) C VA ≈ . τ > C VA ≈ Ak D π " − (cid:18) Θ s T (cid:19) . (59)Let there be a sample in the shape of a cube with the edge length L , volume V = L and surface area A ≈ L . Let usestimate the ratio of the surface and bulk heat capacities in such a sample. At high temperatures, when C V = 3 N ,we have C VA C V ≈ Ak D π N = a lL , (60)where l = n − / is the average distance between atoms, a = (cid:0) π (cid:1) / (cid:14) π ≈ .
4. Thus, at high temperatures formacroscopic samples L ≫ l the surface heat capacity is small in comparison with the bulk heat capacity.Let us consider the case of low temperatures, when for the bulk heat capacity the relation C V ≈ π N (cid:18) T Θ (cid:19) holds. In this case C VA C V ≈ ζ (3) π Ak D πN (cid:18) ΘΘ s (cid:19) Θ T . (61)With decreasing temperature this ratio increases, and at a certain temperature T ∗ the surface and bulk heat capacitiesbecome equal. From (61) it follows that T ∗ Θ = 15 ζ (3) π a lL (cid:18) ΘΘ s (cid:19) ≈ . lL, (62)where it is assumed that Θ ≈ Θ s . For samples with sizes of the order of a nanometer L ≈ − cm , and, taking intoaccount that l ≈ − cm , the ratio T ∗ (cid:14) Θ ≈ .
01. Since the Debye energy has the order of magnitude Θ ≈ K ,then the law for the heat capacity C VA ≈ T in samples of such size can be observed at helium temperatures. Thecontribution of the surface heat capacity can also be the main one in porous and granular materials.In the bulk case the low-temperature heat capacity, according to the Debye theory [1], is proportional to the cubeof temperature, and in the two-dimensional case, when the contribution of the surface phonons is taken into account,it is proportional to the square of temperature. This indicates that the temperature exponent in the low-temperaturebehavior of the heat capacity coincides with the dimension of space. In this regard, it is interesting to calculate theentropy and heat capacity of “one-dimensional” phonons. Such a situation with the quasi-one-dimensional phononscould be realized in long cylindrical filaments of small radius. The propagation of the classical surface waves underconditions of cylindrical geometry was studied in many works [19]. For a qualitative study of a one-dimensional systemof phonons, it is sufficient to assume that in the expression for the free energy (50) the summation is performed overwave vectors oriented in one direction. In this case, we find F = Lk D π n Θ + 4 T h ln (cid:16) − e − τ − (cid:17) − D (cid:0) τ − (cid:1)io , (63)where L is the length of a sample, Θ = ~ c k D , c is the velocity of the “one-dimensional” phonons, and as before k D = (cid:0) π n (cid:1) / . From (63) there follow the expressions for the “one-dimensional” entropy and heat capacity: S = Lk D π h D (cid:0) τ − (cid:1) − ln (cid:16) − e − τ − (cid:17)i , (64) C VL = T (cid:18) ∂S∂T (cid:19) VL = Lk D πτ (cid:20) τ D (cid:0) τ − (cid:1) − e τ − − (cid:21) . (65)In the low-temperature limit τ ≪
1, taking into account that D ( x ) ≈ π (cid:14) x , from here we get: S = C VL ≈ πLk D (cid:18) T Θ (cid:19) . (66)As could be expected, the low-temperature entropy and heat capacity of a system of “one-dimensional” phonons areproportional to temperature. VI. CONCLUSION
A method for quantizing surface elastic waves in an isotropic solid without the use of the semiclassical approximationis proposed, and in the Debye approach the contribution of the surface phonons (“rayleighons”) to thermodynamicfunctions is calculated. In agreement with the previous works [15 – 18] it is shown that in the limit of low temperaturesthe contribution of the surface phonons is proportional to the square of temperature, while the value of the propor-tionality coefficient is somewhat different from the earlier results [15 – 18]. It is also shown that in the one-dimensionalcase at low temperatures the dependence of the heat capacity is linear. Thus, the exponent in the temperature de-pendence of the phonon heat capacity in the low-temperature limit is determined by the spatial dimensionality of asystem.There are two extreme points of view on the Debye model. Often this model is given an unduly fundamental meaningand, when processing experimental data, observable quantities are adjusted to the relations of theory assuming thatthe Debye energy depends on temperature. The opposite point of view is that the relations of the Debye theory areconsidered as rough interpolation formulas [13,14]. The Debye model, of course, is an approximate and rather simple(which is its value) model of the solid body, but there is reason to assert that its value is not limited only to thepossibility of constructing a single interpolation formula that would correctly describe the behavior of the solid bodyin the limit of low and high temperatures. This model allows further development and generalization, for example,accounting for the difference in the velocities of longitudinal and transverse phonons [2], the interaction of phonons[8,9], and, as shown in this article, it can be extended to describe surface phenomena in solids.When analyzing thermodynamic and kinetic properties of crystals whose anisotropy is not large and the consideredeffects are not associated with the existence of singled-out directions in crystals, it is possible to use with a goodaccuracy a more simple model of an isotropic medium after choosing its parameters in an optimal way [20]. It wasshown in [2] that the previously proposed method of describing the elastic properties of crystals on the basis of acomparison with an isotropic medium [20] follows from the requirement of the maximal closeness of the free energiesof a crystal and an isotropic medium. The two-parameter Debye model for an isotropic medium with effective elasticmoduli [2] can be a good approximation for describing the properties of crystals. All the general remarks made aboverefer, in particular, to the further development of the considered in this article Debye model for the surface phonons.The author is grateful to A.S. Kovalev for helpful comments.0 [1] P. Debye, Ann. Phys. (4), 789 (1912). doi:10.1002/andp.19123441404[2] Yu.M. Poluektov, East Eur. J. Phys. , №
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