Scalar-coupled spin systems can be classified as weakly and strongly coupled, depending on the relative magnitude of the coupling constant, J, and the difference between the chemical shifts of the resonances of the two (or more) coupled spins. The 2-LOMO states of both weakly and strongly coupled spin systems can be prepared independently of the value of J by a frequency-cycling procedure. A pulse sequence that is used to edit LOMO states has been studied previously, which involves frequency-selective inversion of transitions by a shaped (tailored) pulse of appropriate duration and spectral bandwidth, followed by a high power (“hard”) read pulse of fixed phase and flip angle. The phase of the receiver is altered for each transient as the selective inversion frequency is incremented. In the present study, we created 2-LOMO states by executing a two-step frequency cycling. For a system of two spin-1/2 nuclei, each component of the doublet of one spin is selectively inverted in each transient with successive changes in the receiver phase of 90o and −180o. Table 1 in the previous work[4, 5, 21] shows the generation of a 2-LOMO state using this approach and how its effectiveness is completely independent of the value of the coupling constant, J, between the two spins.
The pulse sequence that is used to measure the longitudinal relaxation time of a 2-LOMO state, which has been selected by the frequency-cycling procedure, is shown in Fig. 1. The inversion-recovery pulse sequence was implemented within the frequency-cycling protocol for relaxation analysis. It incorporates a transition-selective inversion of magnetization followed by a 180° hard pulse. The selective inversion of each line of a multiplet in successive transients, combined with a suitable receiver add/subtract cycle, is called frequency cycling. The frequency of the selective inversion pulse is switched to the frequency of an appropriate transition of a multiplet, along with the receiver phase during each transient. Using this procedure, the 2-LOMO state can be distinguished by recording two transients (for which the frequency-selective inversion is performed on two transitions in the AX or AB spin system).
Longitudinal relaxation analysis of a two-spin system
Longitudinal relaxation can be represented by the time-dependent recovery of the diagonal elements of the density matrix to their equilibrium values. This relaxation is characterized by the first block of the “relaxation kite”, and the diagonal elements are in general coupled and their time evolution is given by
where α and β are the eigenstates of the Hamiltonian and σαα and Rααββ are the elements of the density matrix and Redfield relaxation matrix, respectively. Fig. 2 shows the energy-level diagram of the two-spin system together with the corresponding transition probabilities between the levels. Since the diagonal elements of the density matrix represent populations, Eqn (1) can also be written in terms of each population and the relevant transition probabilities. The identical rate equation for a two-spin system that describes the recovery of the population distribution between the various energy levels, relative to their equilibrium values, can be expressed as
Figure 2. Energy level diagram for a two-spin system together with the transition probability for each relaxation connectivity.
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The relationship between the populations denoted by the vector N and various magnetization modes, M, is given by
where V is the transformation matrix connecting the populations to the modes. Thus, the relationship between these modes and the populations for an AX spin system can be written as
and on expanding Eqn (4), we obtain
Hence, the rate of change of the spin modes, expressed in terms of the populations, is
The rate at which each of these populations returns to equilibrium is given as (after rearranging Eqns (2) and (5)):
Substituting Eqn (7) into Eqn (6) gives the relationship between the rate of change of spin modes to the transition probabilities and the populations. Thus, in the resulting equation, replacing the transition probability elements by the relaxation matrix elements and populations by spin modes, we obtain
where ρA and ρX are the direct relaxation rate constant 1/T1A and 1/T1X, respectively, and σAX is the cross-relaxation rate constant.
For an AB spin system, because of strong coupling and no inherent symmetry between the energy levels, a simple set of magnetization modes cannot be defined. Therefore, for such cases one has to retain the matrix of the complete longitudinal multiple-quantum modes.. Hence, for an AB system, Eqn (4) can be written as
Following similar algebraic manipulation to that used to obtain the expression for the longitudinal relaxation term for an AX spin system, we obtain
In this case, the expressions for ρA and ρB cannot be derived because of the absence of pure <Az> and <Bz> magnetization modes.[26-28] However, from the previously mentioned analysis, it is clear that the relaxation rate constants for the 2-LOMO states, ρAX (for 2AzXz) and ρAB (for 2AzBz), depend on the first order-transition probabilities of the two spins and can be prepared independently of the strong coupling parameter using the frequency-cycling procedure.