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The 16 operators in the shift basis for a system of two spin-1/2 nuclei are ðÞ 1 = 2 E Iþ I I 0 Sþ S S 0 IþS 0 IS 0 I 0 S 0 I ...
T ABLE 2.3 Product Operators in the Shift Basis for a Two-Spin System 1 2 E ¼ 1 2 1000010000100001 2 664 3 775 I^0 ¼ 1 2 10 0 00 ...
The preceding examples illustrate the concept ofcoherence, which is one of the most fundamental aspects of NMR spectroscopy. As ...
h jj i¼h jj i¼0, and there isno coherence between the two states. Therefore, as has been stated previously, the equilibrium de ...
the spin state of one nucleus changing while the spin state of the other nucleus remains constant. The spectrum can be convenien ...
transitions 1–2 and 2–4). The values ofmifor the stationary states are m 1 ¼þ3/2,m 2 ¼þ1/2, andm 4 ¼1/2. The states represented ...
eigenstates increases unless implemented numerically on a computer. Unfortunately as well, the density matrix formalism provides ...
Hilbert vector space and vector algebra (2, 12). The orthogonality condition is Tr ByjBk no ¼BjjBk ¼jkBkjBk : ½ 2 : 202 Unnor ...
Section 2.3. These solutions can be applied as a recipe by using a simple set of rules, which are presented in Section 2.7.3. Th ...
where is the resonance offset,is the phase of an applied rf pulse, and ! 1 ¼ B 1 , and B 1 is the rf field strength ( 13 ). Th ...
eigenstates,ji , (^) , (^) , and (^) , where the first symbol in each represents the state of theI spin and the second symbol re ...
The operators have physical interpretations; for example, E¼I^ S^ þI^ S^ þI^ S^ þI^ S^ ½ 2 : 219 denotes equal populations of ...
2.7.3 EVOLUTION IN THEPRODUCTOPERATORFORMALISM The goal of the product operator formalism is to derive the evolution of a spin s ...
Iy ) 2 JISIzSzt Iycos 2ðÞJISt 2 IxSzsin 2ðÞJISt, ½ 2 : 227 Iz ) 2 JISIzSzt Iz: ½ 2 : 228 Equations [2.226] and [2.227] ...
expression describing the pulses can be written asHt¼ Ixor Iy, for an x-pulse ory-pulse, respectively, and is the flip angle of ...
Ix Iy Ix Iy Iz Ix Iy Iz Ix 2IzSz 2IySz 2IxSz 2IzSz Iy a b c d e Iz Ωt a a 2 pJIS t 2 pJIS t FIGURE 2.6 Transformations of produc ...
Because I and S operators commute, a nonselective pulse can be represented by a pulse onIfirst, followed by a pulse onS(or by a ...
The trace of this result with the observation operator, ignoring for convenience the constants of proportionality, yields Trfexp ...
observable and generate in-phase resonance signals. Operators of the formI 1 z...Iði 1 ÞzIixIðiþ 1 Þz...IkzandI 1 z...Iði 1 Þz ...
coherence is represented by suitable combinations of bilinear product operators, 1 2 ðI þSþþISÞ¼ 1 2 ð^2 IxSx^2 IySyÞ¼DQx, ½^ ...
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