Static co 0 and macroscopic q — 0 limits

We will first consider processes which are static on the time scale of electronic transitions (co = 0) but very fast on the time scale of atomic vibrations (co = oo). The macroscopic limit is obtained for infinitely small q values. The main difference between a metal and an insulator lies in the denominator of x(q)- In insulators or semi-conductors, the smallest energy difference E k + q) — E k) between filled and empty levels is equal to the gap width A, while in metals, E k -f q) — E k) may vanish. Using the relationship (Ziman, 1964)  

The plasma frequency cop appears in the numerator of e q = 0, co = 0). It depends on the electron density n in the valence band, in the following way  

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