(a) consider a 1-dimensional monatomic lattice of atomic spacing, a. by consideration of a simple harmonic (hooke's law) approach between nearest neighbours only, show that the relationship between the angular frequency, o, of a lattice disturbance and its wave number, k, (the dispersion relation) can be written in the form b2. o-n sin(ka/2) (the identity cos(ka)- 1-2 sin°(ka/2) may be useful.) [10 marks] (b) show that m can be written as om= 2(w/m)^, with m the atomic mass and u the force constant between neighbours. describe, with the aid of a sketch of o versusk, the physical significance of o indicate the first brillouin zone on your sketch [4 marks] (e) for solid argon the maximum phonon energy is 6 mev. given that argon has a mass number of 40 and that the mass of the proton (and neutron) is 1.67 x 10 kg, calculate the force constant in units of kgs2. (e=1.60 x 10*19; a 1.05 x 104 js) 3 marks (d) with the dispersion relation given in part (a), show that the group and phase velocities are equal in the long wavelength limit.