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This can be ascribed to a spinor-type property of the wave function under a rotation in the wave vector space . , its spin is always quantized into the direction of its wave vector, and therefore each scattering corresponds to a spin rotation. , UA{t) = UB{T) and U'^{T) — U'Q{T) = 0, a matrix element for scattering is separable into a product of that of the impiurity potential and a spin rotation. A back-scattering corresponds to a spin rotation by +(2n4-l)7r with n being an appropriate integer and its time reversal process corresponds to a spin rotation by -(2n+l)7r.

In the limit r —+ oo absorption spectra except for intraband Drude terms is proportional to T> r ^=0/ M e^ r27/€,^n)i2 22 T. 0 --^ ' • -r . 0 Energy (units of 47ry/3L) Energy (units of 4iiy3L) Fig. 19: Calculated optical absorption spectra for the parallel polarization in a metallic (left) and semiconducting (right) CN. M\]go{\nw\/2f, (64) where d{x) is the step function defined by 9{x) = 1 for a; > 0 and 0{x) = 0 for x < 0. Figure 19 shows the calculated results of Re cr^^°{(^) of a metalhc CN for

P in the expression of 5(p+i)" because of Eq. (97) since 0{-kj) always appears in a pair. By making the replacement ^(k)—>^(—k)—TT and 0(—k)-^^(k)+7r, we can immediately obtain g{p+i)f' ^ _5(p+i)/ =, ^(p+i)*. (107) To see the above explicitly, we have {s\R[ei-k)-27r]R-^[ei-ki)]\si) = {si\R[0{ki)]R-'[eik)]\sr, {sp\R[e{-kp)]R-'[e{k)]\s) = {s\R[e{-k)]R-'[e{kp)]\sp)*. : (m which is either real or pure imaginary. Because the number of matrix elements for different bands is always even in the expression of 5^^"^^^, we see immediately that 5(p+i) is given by a real number.