In a three dimensional semiconductor crystal with a perfect lattice. Assume the distance between atoms is 'b' center meter. The reciprocal lattice is what we call k-space. It's Fourier transform

of original lattice. We solve the Schrödinger equation of electron wave in original periodical lattice with one unit cube of side length of b and obtain the electron wave mode. We can represent all solutions in the k-space with each point of the k-value as one solution i.e. one mode. This problem is asking you to derive the mode density in this crystal. a). Assume the electrical potential distribution is infinite near the atoms. This results in a periodic solution with periodic unit (T/b, T/b, T/b) in kx, ky and kz directions of k-space. Write done the volume of the unit cell in the k-space. (5 points) c). How many cells in the spherical shell in total? (5 points) Due to k can only be positive, how many cells in the first quadrant? (5 points) Each k point can correspond to two modes for electron spin up and spin down. So how many modes in the first quadrant in k-space. (5 points) d). if we related the results to the density of modes, we can equal it to p(k)*dk, where the p(k) is the density of the modes. Please find the p(k) based on c) result. (5 points) e). Based on d) result, please normalized it to unit volume mode density. (5 points) b). Imagine in k-space, there is a spherical shell, the radius of the inner sphere is k, the thickness of the shell is the differential dk, write down the volume of the spherical shell. (5 points) f). Based on e), convert unit volume normalized p(k) to p(v) by using p(k)dk=p(v)dv relationship.It's more convenient to use frequency v instead of k in the real practice. (5 points)