Bloembergen-Purcell-Pound theory

There are two kinds of popular relaxation times, spin-lattice relaxation time and spin-spin relaxation time T2. Dipolar interactions between nuclear spins in polymers predominantly govern relaxation times. According to the Bloembergen-Purcell-Pound (BPP) theory [5], and T2 vary against the correlation time (t ) of the molecular motion, as indicated in Figure 8.1. Tj has the minimum at = 1/ , in which CO is the resonance angular frequency. If the Tj minimum is observed, the absolute correlation time of the molecular motion can be obtained. On the other hand, T2 decreases with an increment of From these behaviors of the relaxation times, the dynamics of polymer chain can be obtained. 

The rate of relaxation is affected by magnetic field fluctuations in the local environment of excited hydrogen spins. For simple liquids such as water, the diffusion of H2O molecules by Brownian motion allows H spins to encounter and interact with other dipole magnetic moments as described by the Bloembergen-Purcell-Pound, or BPP, theory (Bloembergen et al. 1948). In cementitious materials, there are two main sources of relaxation ... 

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. 

Noting the similarity between NMR relaxation and dielectric relaxation effects, Bloembergen, Purcell, and Pound (35) adapted the Debye theory for the latter to the problems of the former to obtain equation (2). 

Molecules in liquids are in rapid motion with respect to the inverse Larmor frequency. Relaxation in the rapid motion limit was originally treated by Bloembergen, Purcell, and Pound [Biol]. Their theory of relaxation is called the BPP theory [Redl]. 

This implies liquid-like behavior of a dominating water fraction with a liquid-like single correlation time as predicted by the theory of Bloembergen, Purcell, and Pound (BPP) (22). 

This report presents a determination of the fluidity of water adsorbed to saturation in zeolite 13-X. As a measure of the fluidity, the time between molecular jumps or correlation time, r, is employed it has been derived from measurements of nuclear magnetic resonance (NMR) relaxation times via the theory of Bloembergen, Purcell, and Pound (BPP)... 

The first microscopic theory for the phenomenon of nuclear spin relaxation was presented by Bloembergen, Purcell and Pound (BPP) in 1948 [2]. They related the spin-lattice relaxation rate to the transition probabilities between the nuclear spin energy levels. The BPP paper constitutes the foundation on which most of the subsequent theory has been built, but contains some faults which were corrected by Solomon in 1955... 

The relationship between T and and the J s can be derived through time dependent perturbation theory as was done originally by Bloembergen, Purcell, and Pound or by the use of the density matrix which turns out to be equivalent. See the general references in Appendix A, especially Chapter 5 of Slichter. For our case of two spins on a rotating molecule, the relaxation rates are given by... 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

Before discussing some raultiexponential relaxation functions, we will consider some specific aspects of theoretical methods describing the relaxation phenomena in nuclear magnetic resonance. Various methods have been proposed in the literature since the work of Bloembergen, Purcell, and Pound (1). Many of them have some common features like the use of density matrix theory, correlation functions, and their associated spectral density functions. People interested in the foundations of these theories will find some excellent book chapters and review articles in the literature (7-9,21). Slichter s book (Chapter 5) contains a very good introduction to the density matrix which is of crucial importance in all these theories (3). On the other hand, Abragam s book (2) remains a "Bible" in this field. The more recent book of Lenk (5) contains very concise definitions of many terms and concepts widely used in all the relaxation theories. Moreover, this author presents an excellent overview of many recent theories, especially those based on irreversible thermodynamics ... 

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