Single-quantum coherence transitions

In contrast, in a two-spin system the two nuclei coupled with each other by the coupling constant, J, will have four energy levels available for transitions (Fig. 5.56). Such a system not only has single-quantum coher-... 

Coherence A condition in which nuclei precess with a given phase relationship and can exchange spin states via transitions between two eigenstates. Coherence may be zero-quantum, single-quantum, double-quantum, etc., depending on the AM of the transition corresponding to the coherence. Only single-quantum coherence can be detected directly. 

Equation (33) means also that spectroscopic terms (energy levels) can be assigned to the product functions and the transitions defined between them have well determined energies (Figure 6). These transitions define the precession frequency of the corresponding single quantum coherences. 

The average and the individual density matrices are different from each other very much in the case of non-mutual exchanges as the former one is built on the product functions of the spin system while the latter one is defined on the basis functions of the spin set yielding in a significant difference in dimensions of the matrices. In this case, the two matrices can be compared in the space spanned by the intramolecular transitions of the spin system. This space is the direct sum of the space spanned by the single quantum coherences of the spin set with itself (rn times in case of m conformers). This space is spanned by the single quantum coherences of the conformers. 

Note that each of these coherences corresponds to a transition between two energy levels (e.g., c ci corresponds to the /3h c to q hQ c transition) in the two-spin energy diagram. The four SQCs correspond to the four peaks in the 13 C and XH spectra (two doublets), and the MQCs are not observable. Later, we will see that all of these numbers (the four populations, the four single-quantum coherences, and the ZQC and DQC) can be fit into a 4 x 4 matrix that provides a succinct summary of everything we could ever want to know about the spin state of this ensemble of N spin pairs. This matrix is called the density matrix. 

In-phase single-quantum coherence (SQC) is represented by nonzero values for the matrix elements that correspond to the single-quantum transitions. For example, I spin (1H) SQC corresponds to a superposition of the cqa s and /3 (xs states (row 1 and column 3), and the a Ps and A/ s states (row 2 and column 4). Real numbers are used for magnetization on the xf axis, and imaginary numbers are used for magnetization on the y axis. Notice that the downward transition A s — cqa s has a matrix element that is the complex conjugate of the upward transition oqas —> A s-... 

From Eqs. 6.29 and 6.34 we know that the frequencies of the single quantum transitions include both the chemical shift difference and the coupling constant, and we saw in Eq. 11.54 that the single quantum coherence terms evolve at those frequencies. From Eq. 6.29 we can see that the expression for the double quantum frequency E4 — E, would not depend on J, and the difference 3 — E2 likewise does not depend on J for weakly coupled spins. Thus zero quantum and double quantum coherences evolve as though there were no spin coupling. 

Figure 5.27. Net transverse magnetisation is produced from the bunching of individual magnetic moments which gives rise to an observable NMR signal. These spins are said to posses phase coherence, and because only single-quantum spin transitions (a <-> ) ate involved in generating this state, it is termed single-quantum coherence.

The first QST procedure dedicated to quadrupolar systems was developed by Kampermann and Veeman for spin 3/2 nuclei [8]. It is a direct adaptation from the method used for spin 1/2, but uses transition selective pulses instead of spin selective pulses. Similar to the spin 1 /2 method, the transition selective pulses are used to bring an specific set of populations and coherences to the reading position of the deviation density matrix (single quantum coherences) and after that the NMR spectrum is acquired. The transformations (transition selective pulses) that are applied to the deviation density matrix Ap in order to bring the unknown elements to the reading positions are. 

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