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40
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84927321976
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This form for the rotating frame spin spin coupling terms is valid for dipolar coupling and for indirect coupling of the form AIxIx' + BIyIy' +CIzIz' in the laboratory frame. In the latter case, the resulting secular terms in the rotating frame have the form 1 / 2(A+B) vecI cdot vecI' + [C-1 / 2(A+B)] IzIz'.
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46
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84927321975
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Ref. 1, Chap. XII.
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47
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84927321974
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The like spin second moment for spin echo decay is <Δω2>A = 1 / 8fA sumjαij2, where fA is the isotopic abundance of A spins, and 1 / 8 is I(I+1)/3 for a fictitious spin of one half times 1 / 2 to include only spins which are in the pm1 / 2 state.
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49
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84927321973
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In order of increasing decay rate, the eigenvectors for Eq. (9) are [1,1,1,1], [3,1,-1,-3], [1,-1,-1,1] and [1,-3,3,-1].
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50
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84927321972
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We adopt a simplified notation for the m states, [3 / 2, 1 / 2, -1 / 2, -3 / 2] → [1,2,3,4], and defining f2 = exp(-2W0t), f3 = exp(-6W0t), f4 = exp(-12W0t), we have, for the Pm',m's, P1,1 = P4,4 = (5+9f2+f3+5f4)/20, P2,2 = P3,3 = (5+f2+9f3+5f4)/20, P1,2 = P2,1 = P3,4 = P4,3 =(5+3f2-3f3-5f4)/20, P1,3 = P3,1 = P2,4 = P4,2 = (5-3f2+3f3-5f4)/20, P1,4 = P4,1 = (5-9f2-f3+5f4)/20, P2,3 = P3,2 = (5-f2-9f3+5f4)/20.
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51
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84927321971
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One can use here, for example, the motional narrowing formula from Ref. 1, Chap. X: T2-1 = <Δω2> int0sup ∞ g(τ )dτ, where g(τ ) is the correlation function of the interaction which contributes the second moment < Δω2>.
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52
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84927321970
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In generating these decay curves it became quite evident that the long time decay function is quite sensitive to fine details about the distribution of interaction strengths employed. It was necessary to average results from several sets of random couplings to obtain smooth curves.
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57
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84927321969
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We make estimates of the Cu hyperfine constants Aab, Ac, and B using data from the literature. The coincidence of T1 curves for doped and undoped LSCO at high temperatures (Ref. onlineciteimaihiT) suggests similar transverse hyperfine parameters for these systems. Thus, we assume the value Aab-4B = -139gab = -286 kG/spin from the undoped case (Ref. onlineciteimaihiT), where we take gab = 2.06 (Ref. 12). For Ac we compare shift (Ref. onlineciteimaiTh) and susceptibility (Refs. 10,12) results to find dKcsup s/d chicsup s = 1.0 (emu/mol)-1. This, in turn, gives Ac+4B approx13 kG/spin. Finally, estimating Kabs = 0.35 and taking chiabs = 146×10-6 (emu/mol) (Ref. onlineciterewbss), we obtain Aab+4B approx 294 kG/spin. This number may vary if portions of Kabs and chiabs are allocated to a second band in accord with the findings of this paper and Ref. 12. Using Aab+4B = 294 kG/spin, one finds Aab = 4, Ac = -277, and B = 72.5, all in kG/spin.
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58
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84927321968
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Orientation of the LSCO crystal with respect to the vecq is unspecified in Ref. onlineciteaep2. We shall interpret data given for chi''0(ω ,T) as a powder average. This is expected to be correct to better than 10 chiα(rij) [Eq. (14)] is taken to be chiα(rij) propto gαsup 2, where gab,c = (2.06, 2.27) (see Ref. 12).
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60
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84927321967
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T. Imai, Ph.D. thesis, The University of Tokyo, 1991.
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61
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84927321966
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At large exchange frequencies, the effect of the m = pm1 / 2 fluctuations on the echo decay weakens, and these levels merge into an effectively m = 0 level. However, the T1 fluctuations between the pm3 / 2 levels and the effective m = 0 level now take on a larger amplitude, compensating for the loss of the pm1 / 2 level contribution. The contrast between 63Cu and 65Cu effects in Fig. ,7(b) apparently depends on subtle differences between like spin and unlike spin relaxation terms.
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