Two Coupled Spins

We shall defer a discussion of symmetry until Section 6.11. The other operator is Fz, which was defined in Eq. 6.2. Because a and /3 are eigenfunctions of Iz, the product functions f are eigenfunctions of F,. By using the well-established commutation rules for angular momentum, it can be shown that Fz and X commute, so Eq. 6.20 is applicable. For the two-spin case, Eq. 6.1 shows that the four basis functions are classified according to Fz = 1, 0, or —1. Only (f)2 and / 3, which have the same value of Fz, can mix. Thus only X23 and X-i2 might be nonzero all 10 other off-diagonal elements of the secular equation are clearly zero and need not be computed. 

We are now in position to complete our calculation of the AB system in general, with no restrictions whatsoever regarding the magnitude of the coupling constant Jab- By virtue of the factoring due to Fz, the secular equation is 

The secular equation is always symmetric about the principal diagonal hence X23 = X32. We thus have five matrix elements to evaluate. Because X — 2f(0) + X(, we need evaluate only the portion of the matrix elements arising from XJ and then add the portion evaluated in Section 6.5 from X.  

Similar computations for the other diagonal elements give 

The lone nonzero off-diagonal element may be evaluated in a similar manner  

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Schematic COSY spectrum of a two coupled spins, denoted A and X. For convenience, the normal one-dimensional spectrum is plotted alongside the F and F2 axes and the diagonal (F t = F2) is indicated by a dashed line. This spectrum shows two types of multiplets those centred at the same F t and F2 frequencies, called diagonal-peak multiplets, and those centred at different frequencies in the two dimensions, called cross-peak multiplets. Each multiplet has four component peaks. The appearance of a cross-peaked multiplet centred at I = A, F2 = 8x indicates that the proton with shift A is coupled to the proton with shift A. This observation is all that is required to interpret a COSY spectrum.

In this review, we focus our discussion on organic molecules with more than two coupled spins (S > 3/2). They are arbitrarily called high-spin molecules (Itoh, 1978 Weltner, 1983). Molecular solids in which more than two unpaired electrons are aligned in parallel among neighbouring molecules will also be discussed. 

For a spin-1/2 nucleus, such as carbon-13, the relaxation is often dominated by the dipole-dipole interaction with directly bonded proton(s). As mentioned in the theory section, the longitudinal relaxation in such a system deviates in general from the simple description based on Bloch equations. The complication - the transfer of magnetization from one spin to another - is usually referred to as cross-relaxation. The cross-relaxation process is conveniently described within the framework of the extended Solomon equations. If cross-correlation effects can be neglected or suitably eliminated, the longitudinal dipole-dipole relaxation of two coupled spins, such... 

Replacement of Jn by — J,2 gives four lines of the same frequencies as (8.69) the observed spectrum is independent of the sign of J n. The spectrum of two coupled spin- nuclei consists of four lines symmetrically placed about the frequency p0(1 — ia2) as shown in Fig. 8.5. The... 

The Hamiltonian for a system of two coupled spin 1/2 species I and S in a magnetic field is given by4... 

Product operators Spin states for two coupled spins... 

In the limit of infinitely strong coupling Hartmann-Hahn limit), only the terms and I yl2x hxhy) are created from o-(O) = (see Fig. 1C). The most remarkable property of this limit is that the transfer of polarization between the two coupled spins iscomplete. In the simulation of Fig. 1C a coupling constant 7,2 = 10 Hz was assumed. In this case, the initial polarization of the first spin is completely transformed into polarization Ilx of the second spin after 50 ms, which is equal to l/fZ/jj). 

The effective coupling tensor between two coupled spins in the toggling frame is only a good approximation of the effective coupling tensor in the (doubly) rotating frame if the higher order contributions in the Baker-Campbell-Hausdorff expansion [see Eq. (119)] can be neglected. This is the case if the term... 

A number of theoretical transfer functions have been reported for specific experiments. However, analytical expressions were derived only for the simplest Hartmann-Hahn experiments. For heteronuclear Hartmann-Hahn transfer based on two CW spin-lock fields on resonance, Maudsley et al. (1977) derived magnetization-transfer functions for two coupled spins 1/2 for matched and mismatched rf fields [see Eq. (30)]. In IS, I2S, and I S systems, all coherence transfer functions were derived for on-resonance irradiation including mismatched rf fields. More general magnetization-transfer functions for off-resonance irradiation and Hartmann-Hahn mismatch were derived for Ij S systems with N < 6 (Muller and Ernst, 1979 Chingas et al., 1981 Levitt et al., 1986). Analytical expressions of heteronuclear Hartmann-Hahn transfer functions under the average Hamiltonian, created by the WALTZ-16, DIPSI-2, and MLEV-16 sequences (see Section XI), have been presented by Ernst et al. (1991) for on-resonant irradiation with matched rf fields. Numerical simulations of heteronuclear polarization-transfer functions for the WALTZ-16 and WALTZ-17 sequence have also been reported for various frequency offsets (Ernst et al., 1991). 

Homonuclear Hartmann-Hahn transfer functions for off-resonant CW irradiation have been derived for two coupled spins 1 /2 (Bazzo and Boyd, 1987 Bothner-By and Shukla, 1988 Elbayed and Canet, 1990) and for the AX 2 spin system (Chandrakumar et al., 1990). In the multitilted frame, Hartmann-Hahn transfer functions under mismatched effective fields are related to polarization- and coherence-transfer functions in strongly coupled spin systems (Kay and McClung, 1988 McClung and Nakashima, 1988 Nakai and McDowell, 1993). Numerical simulations of homonuclear... 

In a system consisting of two coupled spins 1/2, Hartmann-Hahn transfer can be conveniently analyzed based on the one-to-one correspondence between the evolution of the density operator in the zero-quantum space that is spanned by the operators (ZQ),, (ZQ)y, and (ZQ) [Eq. (14)] and the magnetization trajectory of a single, uncoupled spin in the usual rotating frame (Muller and Ernst, 1979 Chingas et al., 1981 Kadkhokaei et al., 1991 see Fig. 2). This equivalence can be used for the construction of zero-quantum analogs of well-known composite pulses. Effective phase shifts of the zero-quantum field can be implemented by short periods of precession about the z axis of the zero-quantum frame... 

The most direct assessment of Hartmann-Hahn mixing sequences is based on the efficiency of polarization or coherence transfer. Fortunately, the transfer efficiency between two coupled spins is usually sufficient to characterize the transfer properties in extended coupling networks. Several offset-dependent quality factors have been proposed that are based on magnetization-transfer functions T r) (see Section VI) between two spins i and j with offsets v, and Vj. 

In contrast to MLEV-16, WALTZ-16 is relatively insensitive to phase errors. In the absence of amplitude imbalances, coherence transfer is isotropic. Although coherence transfer is possible over a relatively large bandwidth, the transfer efficiency decreases rapidly if the offset difference li j — Vj of two coupled spins i and j is larger than 0.6 i / (see Fig. 23A). In analogy to MLEV-17, a nonisotropic WALTZ-17 sequence was con-... 

Fig. 2.2.12 Spectrum (a) and energy level diagram (b) of two coupled spins j, A and X. The arrows indicate the orientation of the magnetic moments in the magnetic field. The continuous lines connecting the energy levels correspond to observable single-quantum transitions. The broken lines indicate forbidden multi-quantum transitions. The splitting of the resonance lines gives the strength 7ax of the interaction.

Magnetic resonance also has an effect on photoexcitation dynamics as the absorption of microwaves will equalize the populations of the two coupled spin sublevels. Consider two triplet sublevels X and Y with sublevel populations generation rates and recombination rates 7 ,= l/r,. Under steady state con-... 

The stimulated transitions between the four energy levels of two coupled spins result in two distinct types of relaxation. The first is called spin-lattice relaxation. This form of relaxation involves the net loss of energy from the excited state and is analogous... 

Figure 2.16. The influence of spin-echoe.s on scalar coupling as illustrated for two coupled spins A and X. (a) A homonuclear spin-echo (in which both spins experience a 180° pulse) allows the coupling to evolve throughout the sequence, (b) A heteronuclear spin-echo (in which only one spin experiences a 180° pulse) causes the coupling to refocus. If both heteronuclear spins experience 180° pulses, the heteronuclear coupling evolves as in (a) (see text).

The generation of double-quantum coherence requires an antiphase disposition of coupling vectors, which here develop during a period A = l/2Jcc- This is provided in the form of a homonuclear spin-echo to make the excitation independent of chemical shifts. The ti period represents a genuine double-quantum evolution period in which these coherences evolve at the sums of the rotating-ffame frequencies of the two coupled spins, that is, at the sums of their offsets... 

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