Eng 45 - Chapter 1 - Structure(28)

Eng 45 - Chapter 1 - Structure(28) - Materials Scien ce...

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Materials Science Fall, 2007 Page 27 where ¬ is the wavelength of the electron state, h is Planck's constant, Ó = h/2 π , and k = 2 π /¬ is the wavenumber . It follows from the Heisenberg Uncertainty Principle that the allowed values of k are separated from one another by a difference of the order Îk « 2 π a 2.7 where a is the interatomic separation. They vary from zero to the maximum value needed to accommodate two electrons per atom; since each state can hold electrons of either spin, the number of discrete values of k that are needed to describe the states that arise from a given atomic state is equal to the number of atoms in the solid. The momentum difference between successive states is of the order Îp « 10 -24 kg^m/sec, which is a small number. As the momentum, p, increases, the kinetic energy, T, increases as well, where T = p 2 2m* = Ó 2 k 2 2m* 2.8 This has the consequence that the discrete energy levels of the isolated atoms broaden into bands of allowed energy states, as illustrated in Fig. 2.8. Each band contains
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