Introduction to Nuclear Spin Cross-relaxation and Cross-correlation Phenomena in Liquids

DANIEL CANET

Université de Lorraine, France Email: daniel.canet@univ-lorraine.fr

Nuclear magnetic resonance (NMR) cross-relaxation and cross-correlation phenomena are part of the general nuclear spin relaxation processes. What is meant by relaxation is very common in physics, chemistry, biochemistry, etc. and is related to the recovery of a system that has been subjected to different constraints. As far as NMR is concerned, the nuclear spin system tends to recover toward its equilibrium configuration, which consists of the so-called macroscopic magnetization, collinear with the polarizing static magnetic field B0. This magnetization originates from the magnetic momentum associated with each spin momentum. A nuclear spin system can be moved from its equilibrium configuration by changing, non-adiabatically, the B0 field value or, more commonly, by applying pulses of an oscillating magnetic field called also the radio-frequency field. The latter is generally denoted B1 and, with a frequency close to the nuclear resonances, can induce the NMR signal. As a matter of fact, the consideration of nuclear spin relaxation was a key issue for the first NMR experiments. Moreover, although the prime interest of NMR for chemists was its ability to reveal the molecular structure via chemical shifts and coupling constants, it was rapidly realized that the relaxation parameters could also provide not only dynamical but structural information.

Very early (in 1948, whereas the first NMR experiments were performed in 1946), Bloembergen, Purcell and Pound were able to interpret the two major relaxation parameters: T1, the spin–lattice relaxation time (longitudinal relaxation time), related to the nuclear magnetization component along the B0 field; and T2, the spin–spin relaxation time (transverse relaxation time), related to the nuclear magnetization components perpendicular to the B0 field. These two relaxation times are involved in the famous Bloch equations that, in a phenomenological way, accounts for the evolution of the three components of the nuclear magnetization (or polarization, magnetization being the polarization times the gyromagnetic ratio). These equations are perfectly valid if the system encompasses a single spin species. However, as soon as one is dealing with a multi-spin system, it is mandatory to consider a polarization for each spin species and, possibly, further quantities describing different spin states. Although T1 and T2 remain active for each individual polarization (they will be referred to as auto-relaxation parameters), it turns out that all spin states, including the polarization for each species, may be coupled by various spin relaxation pathways. The corresponding parameters include the so-called cross-relaxation and cross-correlation relaxation rates, which are the subject of this book. They arise from different relaxation mechanisms (considered in Section 1.1) and are active through the so-called spectral densities, a concept also developed in Section 1.1. How these parameters may be involved in dedicated experimental procedures is not a simple matter. It requires some knowledge of spin quantum mechanics (Section 1.2), which will be used for a detailed approach of cross-relaxation (Section 1.3) and cross-correlation (Section 1.4). Finally, the type of molecular information that can be gained from cross-relaxation and cross-correlation parameters will be surveyed in Section 1.6.

1.1A Survey of Nuclear Spin Relaxation Mechanisms and the Concept of Spectral Densities

One way to perturb a spin system from its equilibrium configuration is to induce transitions among its energy levels. The same process can be envisioned for restoring this equilibrium configuration. Inducing transitions can be achieved by the application of a radio-frequency field or, without having recourse to an external constraint, through local fluctuating magnetic fields. It is the latter that gives rise to relaxation phenomena. Consider, for simplicity, a single spin 1/2 system, that is a system involving only two energy levels. In order to induce a transition between these two states, the experimenter can apply a radio-frequency field at the Larmor frequency v0 (or close to this frequency) which is such that: hv0 = ?E = ?hB0/2p, where ?E is the energy difference between these two energy levels, ? the gyromagnetic ratio of the considered nucleus and h the Planck constant (the shielding coefficient, responsible for the chemical shift effect, has been omitted). The constants appearing in the latter expression arise from the relationship between the magnetic moment and the spin operator I: µ=?hI (h stands for the Planck divided by 2p, with these notations and for a spin 1/2, the length of the vector I is [square root of 3/4] and its projection on a given axis can take the value +½ or -½). However, within a sample, an elementary nuclear magnetic moment µ is subjected to a local magnetic field b(t) originating from the various interactions to which this magnetic moment is subjected. Due to molecular motions, this local field is time dependent and consequently may be able to induce transitions, thus constituting a relaxation mechanism. It must however mimic the action of a radio-frequency field and therefore fulfil the following conditions: (i) present some degree of coherence, and (ii) be active at the frequency of the considered transition.

A first global treatment, considering simple randomly fluctuating magnetic fields (without specifying their characteristics), will be presented first. Depending on the origin of b(t), specific mechanisms can be considered and the ones relevant to this book will be detailed thereafter. It can be borne in mind that each of them will possibly contribute to relaxation rates in an additive way. The limited relaxation mechanisms treated here are those which concern spin 1/2 nuclei. For the sake of simplicity, nuclei of spin greater than 1/2 (also called quadrupolar nuclei) will not be considered, although, in some instances, they could be involved in cross-relaxation and cross-correlation relaxation rates.

1.1.1 Interaction with Local Randomly Fluctuating Magnetic Fields

b(t) is, for instance, the magnetic field created by other spins (nuclear spins or spins of unpaired electrons). However, we shall disregard here the origin of b(t) and just rely on the fact that, due to molecular motions, it randomly fluctuates (we shall further assume an isotropic medium throughout):

(i) the three components of a local magnetic...