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

16.4 Phonons 457Definingand√ ( ma k = ω k X k + i )2¯hω k m P −ka ∗ k = a∗ −k ,the Hamiltonian assumes the formwhereH = ∑ kω k =−ω −k(¯hω k N k + 1 )2N k = a ∗ k a kA general state of the system is constructed from a superposition of the eigenstates:ψ = ∑ ∑C kn ψ knk nwhere |C kn | 2 represents the probability that the system is in the state ψ kn . Fromthe customary quantum mechanical relation Eψ = Hψ, the expectation value ofthe total energy of the system is〈ψ |E| ψ〉 = ∑ ∑|C nk | 2¯hω (k n k + 1 )k n2The expectation value of the square of the momentum operator is found from〈ψ ∣ ∣P 2∣ ∣ ψ〉 =m ∑ k〈ψ |X k |ψω 2 k = ∑∑ |C nk | 2¯hω k n k =〈ψ |E| ψ〉−E 0where E 0 is the zero-point energy.From the expectation state of the position operator, a state ψ nk can be coupledonly to the state ψ (n+1)k or ψ (n−1)k . Consequently the time dependence will behaveas 2 cos (ω k t). The term phonon can be defined in the following manner: a phonon isemitted or absorbed when a system of harmonically coupled quantum mechanicalparticles executes a transition from a state ψ nk to a state ψ (n–1)k (which results inemission) or ψ (n+1)k (which results in absorption).The absorption of ultrasonic waves in solids are attributable to a number ofdifferent causes, each one of which is characteristic of the physical propertiesof the material concerned. They can be classified as (a) losses characteristic ofpolycrystalline solids, (b) absorption due to lattice imperfections, (c) absorption inferromagnetic and ferroelectric materials, (d) absorption due to electron–phononinteractions, (e) absorption due to phonon–phonon interactions, and (f) absorptiondue to other possible causes. It is also interesting to note that a rapid decrease in attenuationoccurs at the critical temperature for superconductivity. This variation ofattenuation with temperature has been explained in a Noble-prize winning paperby Bardeen, Cooper, and Schrieffer through the B.C.S. theory (so-called after the