Zero quantum transition

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

Figure 1.44 Transitions between various energy levels of an AX spin system. A, and Aj represent the single-quantum relaxations of nucleus A, while Xi and Xj represent the single-quantum relaxations of nucleus X. W2 and are double- and zero-quantum transitions, respectively.

In order to define and pi(j) it is convenient to refer to Fig. 7.1 and to define w[ and w( as the transition probabilities between two states involving a single quantum transition either of spin 7 or J wo is the zero quantum transition and corresponds to the —I— + — transition and vice versa wj corresponds to... 

Figure 2 The four-level diagram for a system of two interacting spins, in this case an electron (S) and nucleus with a positive gyromagnetic ratio (/). The intrinsic electron and nuclear spin relaxation are given by p and w°, respectively, and the dipolar and/or scalar interactions between the electron and nuclear spin are represented by p, w0, w, and w2. The transition w0 is known as the zero-quantum transition, while w, is the singlequantum transition and w2 is the double-quantum transition. Nuclear and electronic relaxation through mechanisms other than scalar or dipolar coupling are denoted with w° — 1/Tio and p — 1/Tie, where Ti0 and T1e are the longitudinal relaxation times of the nucleus and electron in the absence of any coupling between them. Since much stronger relaxation mechanisms are available to the electron spin, the assumption p>p can be safely made. Adapted with permission from Ref. [24].

Now look at the energy differences corresponding to the double-quantum and zero-quantum transitions ... 

The four single-quantum transitions relax at a rate Wi, or more specifically for the Ha transitions and W for the Hb transitions. The double-quantum transition (aa /3/3) relaxes with rate W2 and the zero-quantum transition (otft fiot) relaxes at a rate W0 (Fig. 10.2). For... 

Note that the equilibrium population difference for the zero-quantum transition is zero because the two states have (essentially) the same energy. Now we can substitute the (indirectly) measurable quantities Ma and M for the population differences. For the disequilibrium of we have... 

Suppose that we are talking about a double-quantum transition in which both the proton and carbon change from the a state to the p state. This transition is thus from the aH c state to the PuPc state ol l lc two-spin, four-state system. This transition corresponds to DQC. Likewise, if the proton flips from ft to a while the carbon simultaneously flips from a to P, we have a zero-quantum transition (P ac to a Pc) because the total number of spins in the excited (ft) state has not changed. This transition corresponds to ZQC. What can we say about these mysterious coherences In Section 7.10, we encountered ZQC and DQC as intermediate states in coherence transfer, created with pulses from antiphase SQC ... 

Note that as is just the Larmor frequency and, because real numbers are associated with the x axis and imaginary numbers with the y axis, time evolution is simply rotation in the x-y plane at the offset frequency. For double-quantum transitions, > = a> + >s, and for zero-quantum transitions co = coi — cos. For example, a 90° pulse on the y axis followed by a delay r would give... 

Equation 6.33 is completely general. For the two-spin system, it results in the transitions we identified in Fig. 6.2, while the double quantum transition between and 4, and the zero quantum transition between 02 and 03 are forbidden. Note that this statement is true for this treatment, which employs stationary state wave functions and time-dependent perturbations, but as we shall see in Chapter 11, it is easy with suitable pulse sequences to elicit information on zero quantum and quantum double processes. For our present purposes in the remainder of this chapter we accept the validity of Eq. 6.33. 

In Eq. 11.48 we saw that the basis functions for our density matrix are divided into three groups with fz = 1,0, and —1, respectively. As we saw in Chapter 6, transitions between energy levels El - E2, E3 E4, Ex E3, and E2 -> E4 each result in Afz = 1 and are called single quantum transitions, while transitions Ex E4 and E2 E3 are termed double quantum and zero quantum transitions, respectively. The usual selection rules from time-dependent perturbation theory show that only single quantum transitions are permitted in such simple experiments as excitation by a 90° pulse. Moreover, for weakly coupled nuclei, the single quantum transitions each involve only a single type of nucleus, I or 5, as indicated in Fig. 6.2. 

In Table 11.1 we sketch the form of the density matrix for the two-spin system to show the significance of the elements. Px —P4 refer to the populations of the four states, I and S represent single quantum I and S transitions, and Z refers to zero quantum transitions and D to double quantum transitions. We saw in Eq. 11.9 that an off-diagonal element pm is nonzero only if there is a phase coherence between states m and n, and in Eq. 10.19 we saw that pmn, evolves with a frequency determined by the difference in energies Em — En. Thus, these off-diagonal elements represent not only transitions, but single quantum, double quantum, and zero quantum coherences, which evolve in free precession at approximate frequencies of v, vs, vt + vs, and vt — vs. In Eq. 11.53 we see that p(r) has... 

The second form of relaxation is called spin-spin relaxation. This form involves any change in the quantum state of the spin. Thus, any of the transitions shown in Figure 3.1 can cause spin-spin relaxation. In particular, the exchange of magnetization between spins via a zero quantum transition is a very effective mechanism for spin-spin relaxation. Thus, spin-spin relaxation is analogous to fluorescence energy transfer. Because spin-spin relaxation limits the lifetime of the excited state, it affects the line width of the observed resonance lines due to the uncertainty principle shortlived states have ill-defined frequencies. The actual relationship between the spin-spin relaxation rate and the line width (Av) is given by R2, the rate of spin-spin relaxation T2 is the time constant for spin-spin relaxation,... 

The key point is the effect of molecular weight on the spectral density function. As the molecular size increases, the intensity of fluctuations with a frequency close to the zero quantum transitions also increases. Hence, the spin-spin relaxation rate increases as the molecular weight increases. This has two very important consequences. First, the spectral line width will increase as the molecular size increases. Consequently, the NMR spectra of larger proteins show increased degeneracy because of the increased number of resonances and the increased line width. The second consequence of the shortened lifetime of the excited state is a reduction in the efficiency by which magnetization can be passed from one nucleus to another... 

Zero quantum transition—change in the population levels that does not result in the net change of the quantum state. 

In the case of transverse relaxation, the transitions involved include zero quantum transitions, i.e. M — 0, and their respective frequencies then appear in the relevant spectral density hence transverse relaxation proceeds even for very low-frequency motions. 

Fig. 2.8 In a two-spin system there is one double quantum transition (1-4) and one zero-quantum transition (2-3) the frequency of neither of these transitions are affected by the size of the coupling between the two spins.

There are also six zero-quantum transitions in which M does not change. Like the double quantum transitions these group in three pairs, but this time centred around the difference in the Larmor frequencies of two of the spins. These zero-quantum doublets are split by the difference of the couplings to the spin which does not flip in the transitions. There are thus many similarities between the double- and zero-quantum spectra. 

It turns out that in such a system it is possible to have relaxation induced transitions between all possible pairs of energy levels, even those transitions which are forbidden in normal spectroscopy why this is so will be seen in detail below. The rate constants for the two allowed I spin transitions will be denoted W/11 and W/2), and likewise for the spin S transitions. The rate constant for the transition between the aa and /J/J states is denoted W2, the "2" indicating that it is a double quantum transition. Finally, the rate constant for the transition between the a/3 and j8a states is denoted W0, the "0" indicating that it is a zero quantum transition. 

When the zero-quantum transition is greater than double-quantum Wi, the nOe enhancements will be negative. Similarly, when is greater than W(), the resultant nOe will have a positive sign. The predominance of Wi and over one another depends on the molecular motion. It is known that the Wo transition is maximal when the molecule tumbles at a rate of about 1 KHz, while the VW transition is fastest at a tumbling rate of about 800 MHz. On this basis, a rough idea of the sign of nOe can be obtained. For example, small molecules in nonviscous... 

Flip-flop transition. Syn. zero quantum transition, Wg transition, zero quantum spin flip. When two spins undergo simultaneous spin flips such that the sum of their spin quantum numbers is the same before and after the transition takes place. For example, if spins A and B undergo a flip-flop transition, then if spin A goes from the a to the p spin state, then spin B must simultaneously goes from the a to the p spin state. 

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