There are various kinds of approximation methods in band theory. In my opinion, Bloch theorem implies the existence of energy band. From nearly-free electron approximation or tight binding method, we can calculate the energy band. They can tell us the information of band gap and band width. However, I am quite confused with k.p method and envelop function method. Starting from some particular point in the band, we can calculate the effective mass from k.p method. But why do we need to do so? Also, it seems hard to calculate energy band from k.p method. Lost in all these approximation methods, i want to ask the following questions: What is the major task in energy band theory? To calculate band structure and band gap? What is the main differences in application (not in theory, I know how to derive these methods. ) between these methods: neatly-free, OPW, PP, TB, k.p, envelop function...?