Lecture 12 Debye Theory

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1 Lecture 12 Debye Theory 12.1 Background As an improvement over the Einstein model, we now account for interactions between particles they are really coupled together by springs. Consider the 3N normal modes of vibration of the crystal. These mechanical vibrations are called sound waves. What are some of the normal vibrational modes that we can get? There will be high frequency modes whre atoms are moving opposite to their neighbour. There will also be low frequency elastic waves whose wavelength λ is very large compared to the interatomic spacing a of the crystal, i.e. such that large groups of atoms are moving together. λ a (12.1) If we then focus on low-frequency modes, we can ignore the discrete-like atomic structure of the solid and treat as a homogeneous medium. Elastic waves are then propogated with a constant frequency-independent velocity, the velocity of sound, which is determined by the elastic stiffness constants of the medium. We are assuming that the crystal is isotropic, that is there is no preferred direction in space. Hence, the problem reduces to finding the normal modes for low frequencies Integrals over normal modes In direct analogy to what we did in the BBR method, we shall convert a summation over the number of elastic modes to an integral this is the essence of Debye Theory:

2 to consider the possible free vibrations for a continuous solid = 1 4πn 2 dn (12.2) 8 Note the following important points: n 1. For the BBR case, where we are dealing with electromagnetic radiation, we only had 2 transverse modes (or two polarizations). For the elastic solid, we have 2 transverse and one longitudinal mode giving rise to three possible polarizations that our sound waves can be polarized. 2. For the BBR case, there was no upper limit on the integral over modes it was infinite light waves can have very short wavelengths. For the elastic solid, the integral has an upper limit because the total number of modes is bounded, since the sound waves in the solid can not have wavelengths shorter than twice the atomic spacing. 3. Each mode n of oscillation has a particular wave pattern characterized by a number of bumps (n = number of bumps). So by the point above, each bump must contain at least one atom. If we are dealing with a 3D cube, with N atoms along each length, then this means that the maximum number of modes is n = 3 N. For a sphere, we find: = 3 nd 4πn 2 dn = 3N (12.3) 8 n where n D refers to the maximum number of elastic modes (D representing Debye). Performing the above integral, we can therefore solve for n D, n D = ( ) 1/3 6N (12.4) π A couple of points: 1. As noted above, the upper limit or maximum number of modes also corresponds to the fact that there is a maximum frequency corresponding to the minimal possible wavelength for which waves may be propagated in a solid of lattice spacing a, i.e. λ min = 2a. 2. Related to this point, we have made a big assumption that we can extend our treatment beyond the low frequency (large wavelength) to the high frequency range. We will come back to this point later.

3 12.3 Thermal energies and Heat Capacity 1. Energy Following along exactly as we did for the BBR treatment, we may calculate the total thermal energy by summing over all possible modes, U = n ɛ n = n s n ω n (12.5) where, recall that, 1 s n = e β~ω n 1 was the Planck Distribution which gives the average number of phonons (or photons) in a given mode n. Converting to integral form gives us, U = 3 nd 8 4π n 2 ω n dn e β~ωn 1 (12.6) For the BBR treatment with EM waves (standing waves) in a cavity the frequencies were given as, ω n = nπc L Now that we are dealing with sound waves, we replace c by the velocity of sound v, ω n = nπv (12.7) L As we did before, let x = β ω n and we find, Thus, the energy becomes: n 2 = x2 L 2 (π vβ) 2, dn = Ldx π vnβ U = 3π 2 = 3π 2β nd ( L π vβ where x D is defined in terms of n D as, x 2 L 2 (π vβ) 2 ) 3 Ldx π V nβ x β e x 1 x D = β ω nd = π vn Dβ L ( ) 6Nπ 2 1/3 = vβ V (12.8) where V = L 3 is the volume.

4 In analogy with the Einstein temperature, Θ E, we define the Debye temperature, Θ D as, x D = Θ D T (12.9) Now, rewriting our energy in terms of x D we have, ( ) ( ) 3 3V 1 1 U = 2 β v π 2 β = 3 6Nk B T xd 2 x 3 D e x 1 Therefore, the energy is given as, e x 1 U = 9Nk BT x 3 D e x 1 (12.1) Show that if you include the zero point energy term that you obtain an additional factor: 9 8 Nk BΘ D (12.11) 2. Heat Capacity Since we may perform differentiation underneath the integral sign, the heat capacity is found as, [ 3 xd C v = 3Nk B x 3 D ] x 4 e x dx (e x 1) 2 (12.12) 12.4 Out of the Box: High and Low Temp limits 1. High Temperature Limit T Θ D, x D 1: We now proceed to calculate the high temperature behaviour of the heat capacity: e x x 4 (e x 1) 2 = x 4 (e x 1)(e x 1)e x = = = x 4 (e x 1)(1 e x ) x 4 2(cosh x 1) x 4 2( x2 2! + x4 4! + )

5 In this range of integration, < x < x D, x D 1; therefore, we can keep the leading order term in the denominator to give us: e x x 4 (e x 1) 2 x2 So we find that the heat capacity becomes: C v 3Nk B [ 3 x 3 D ] x 2 dx = 3Nk B (12.13) This is known as Dulong and Petit s Law which is applicable in the high temperature, classical limit. Note once again that we obtain the result anticipated by the Equipartition theorem. 2. Low Temperature Limit, Θ D T, x D 1 (a) Energy Since x D is so large, we encounter minimal error by replacing the upper limit, x D of our energy integral by (since these extra modes that we are adding in don t contribute because of the large exponential in the denominator). Thus, the integral becomes, as we have encountered before. Thus, the energy is given as, x 3 dx e x 1 e x 1 = π4 15 U 9Nk BT x 3 D ( ) π Nk B T 4 π 4 5 Θ 3 D (12.14) and we see that the energy in the low temperature limit is proportional to T 4 (b) Heat Capacity Taking the derivative of the energy above, we obtain the following form for the heat capacity, C v 12 5 Nk Bπ 4 ( T Θ D ) 3 (12.15) which is in agreement with experiment: (d) Why does it work so well? At both low and high temperature limits, the Debye approximation is exact. Debye theory is really an interpolation between these limits.

6 Table 12.1: Comparison of the heat capacity between experiment and theory Element Θ D (K) Theory Θ D (K) Experiment Al (c) Cu Pb At low temperature, it is reasonable that it should work well since only the low frequency modes of oscillation contribute to the heat capacity and for these modes it is reasonable to replace the characteristic modes of the crystal with the elastic waves (or sound waves) in a continous medium. The fact that it works so well beyond low temperature is amazing, since the continuum picture should break down for higher frequency modes where the wavelength becomes comparable to the atomic spacing and, hence, the approximation of a continuous elastic medium is no longer valid. The key point is that at high temperatures all that matters is that we account for the total number of modes or degrees of freedom, in agreement with the equipartition theorem. Note: In a metal, the conduction electrons contribute to the heat capacity and, hence, in addition to the T 3 dependence there is also a linear dependence on T : Metals: C v at + bt 3 (12.16)

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