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By D. R. Uhlmann
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204) is positive at t > t2 a n d has a maximum. By substituting Eq. (207) for S(t) denned in Eqs. (203) a n d (204), by rearranging a n d using Eq. -<'-'i>A2 = b > 0. Assume 1/τχ > 1 / τ 2 ; then (i2 - tl)(\/rl - 1 / T 2 ) > 0, and e ^ - ' i X i A i - i A z ) = c > 1. Using the introduced notations S(t) 4- b - ac - b/c) = (wxw2/c)(c = wxw2(a - l)(b - ac), (211) f where wxw2 > 0 a n d (c - l)/c > 0. T h e third term of the product, b —a c = -U-n)/n e = _ e(-('-'l)Al + ('2-'l)Al-(i2-'l)A2) -('-a)A2(l _ ^-C-'2)Al+('-i2)A2) e = ^ - ( ' - a ) A 2 ( l _ it-t2)(l/r e 2-\/ri)y (212) f Since 1 / τ 2 - 1/τχ < 0 a n d ( / - ί 2 ) ( 1 / τ 2 - 1/τχ) < 0, e < ' - ' 2 K i A 2 - i A i > < 1?
81) ;= 1 T h e coefficients wi and τ,· were determined using the described procedure a n d the spectrum Η was found from Eq. (80). The result of the calculation is compared in Fig. 19 with the Η calculated from Eq. (68), the exact solution for the relaxation function given by Eq. 00 ' 10 R * V ι 11111 10 1 2 Ι 1 1 1 111! 10 3 ι ι ι M I 10 1 4 1 1 1 llll I0 ι 5 ι ι ι IVR» ^ • " • J 6 10 1 1 1 1 III 10 relaxation time (sec) F I G . 19. The exact spectrum given by Eq. (68) (curve 1), the approximate one given by Eq.
19 with the Η calculated from Eq. (68), the exact solution for the relaxation function given by Eq. 00 ' 10 R * V ι 11111 10 1 2 Ι 1 1 1 111! 10 3 ι ι ι M I 10 1 4 1 1 1 llll I0 ι 5 ι ι ι IVR» ^ • " • J 6 10 1 1 1 1 III 10 relaxation time (sec) F I G . 19. The exact spectrum given by Eq. (68) (curve 1), the approximate one given by Eq. (73) (curve 2), and the spectrum obtained by a proposed technique, Eqs. (76), (78), and (80) (circles). ] β 2S (α) Ο Ο (b) οοο boo (c) oobo v (d) ie ιβ* ι·* ιβ' ιβ* «· β relaxation time (sec) F I G .