THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

16.4 Phonons 455Debye described the role of phonons in his explanation of thermal conductivity,and in that same year (1912) Frederick Landemond correlated lattice vibrationsto thermal expansion and melting of solids, both of which are attributable to thenonlinearities of forces between atoms in solids.Consider a one-dimensional array of masses m j ( j = 0,...,N + 1) interconnectedwith ideal springs of stiffness s j ( j = 0,...,N). A spring s j interconnectsmass m j and mass m j+1 . Let ξ j denote the displacement of mass m j . The displacementsξ 0 and ξ N+1 provide the boundary conditions at each end of the system.Within the boundaries, the motion of each mass is assumed to follow Hooke’s lawof linear elasticity and Newton’s law:d 2 ξ jm jdt =−s j−1(ξ 2 j − ξ j−1 ) + s j (ξ j+1 − ξ j ) (16.15)The motion can be assumed amenable to Fourier analysis, so that it assumes atime dependence e iωt . The left term of Equation (16.15) then becomes –m j ω 2 ξ j .The solution to this equation, through the application of the Bloch theorem forthe normal modes of a coupled system, is a linear combination of the normalmodes:ξ j = ∑ (eik( ja) X k + e −ik( ja) )X −k eiω k tkwherek = eigenvalues of the system determined by the boundary conditionsa = periodic spacing, or the “stretch” of the “springs” connecting the massesX k = ξ 1 − ξ 0 e −ika2i sin(ka)The values of ξ 0 , ξ 1 , X k and the restrictions on k are established by boundary andinitial conditions. For example, consider a system clamped at the ends, i.e., x 0 =0, x N+1 = 0. Then( )ξ1ξ j = sin[k( ja)]sin(ka)The eigenvalue k is quantitized withk =n πwith n = 1, 2,...,NN + 1 aIn an infinite system or in a system with periodic boundary conditions, it isreadily established that X k = 0orX −k = 0. If we apply a coordinate system withthe mass m 0 located at its origin, then the location of the jth mass is x = ja, andwe haveξ k (x) = e ikx X k (16.16)Equation (16.16) represents the customary Bloch wave result. The subscript k wasadded in Equation (16.16) to serve as a label for the normal mode. The solution is