For longitudinal phonons in a string of atoms the dispersion relation of the kinetic energy in a 1-dimensional k -space, as shown in Figure 2, is given by E = 2 ℏ ℏ ω ω 0 | sin ⁡ ⁡ ( k a 2 ) | where ω ω 0 = k F / m is the oscillator frequency, m the mass of the atoms, k F the inter-atomic force constant and a inter-atomic spacing. For small values of k ≪ ≪ π π / a the dispersion relation is linear: E = ℏ ℏ ω ω 0 k a