NMR Relaxation and Molecular Dynamics: Theory

General Concepts

The fact that a system of nuclear spins interacts with magnetic fields and is able to absorb and dissipate energy is a fundamental aspect of nuclear magnetic resonance (NMR). The energy exchange within the nuclear spin system and between the system and the surrounding environment is commonly referred to as relaxation and is one very important problem in NMR.

The concept of relaxation is closely associated with the concept of evolution of populations in the different energy levels. When in the steady state the population of the spin system energy levels follows a Boltzmann distribution. In the case of spins I = 1/2 the ratio between the populations of the high and low energy levels, p+ and p_, respectively, with p+ + p- = 1, is given by p+/p_ = exp(hy B0/(kT)), where T is the temperature of the sample. The interaction of the spin systems with electromagnetic waves with frequency ш = ш0 = hy B0 (i.e., equal to the Larmor frequency) disturbs the equilibrium state, and a new equilibrium results from the competition between the radio-frequency (RF) irradiation which

NMR ofLiquid Crystal Dendrimers

Carlos R. Cruz, Joao L. Figueirinhas, and PedroJ. Sebastiao Copyright © 2017 Pan Stanford Publishing Pte. Ltd.

ISBN 978-981-4745-72-7 (Hardcover), 978-981-4745-73-4 (eBook) www.panstanford.com

Energy levels of an I = 1/2 spin system and a lattice reservoir, both with the same energy gap

Figure 6.1 Energy levels of an I = 1/2 spin system and a lattice reservoir, both with the same energy gap.

promotes the increase of p+/p_ and the relaxation process which tends to decrease this ratio toward the initial value. A new state is reached with p+/p_ = exp(hy B0/(kTS)) when the RF irradiation stops and the system relaxes toward the original equilibrium. TS in the Boltzmann factor represents the new "temperature” of the spin system at the end of the RF pulse and it is clear that TS > T. This definition of "spin temperature” can be extended also to spin systems with I > 1/2. This concept is not strictly valid when the RF is being applied, since there are transverse components of the magnetization which are not compatible with the description of the statistical behavior of spin systems by the population of its energy levels. One important aspect of the "spin temperature” concept is related with the saturation state when p+ = p_, which corresponds to an "infinite temperature” case. Surprising is the case observed when p+ < p_, which leads to the concept of a "negative.” This situation is observed when a 180° pulse is applied and the nuclear magnetization is inverted. Obviously, a "negative” TS corresponds to a "hotter” state in comparison with the "infinite temperature” state (Abragam, 1961).

The balance between the populations in the two energy levels of I = 1/2 spin systems can be used also to introduce the concept of relaxation time. In the relaxation process the rate of change of one energy level is obviously coupled with the rate of change of the other energy level. For the I = +1/2 we have an energy E + = _(1/2)hy B0p+ and for I = _1/2 we have E_ = +(1/2)hy B0p_. For an isolated spin system the rate of change of p+ can be written as

and clearly the transition probabilities per unit of second for the transitions (—) ^ (+) and (+) ^ (—), respectively, W; and W^, must be equal, as it can be demonstrated by the time-dependent perturbation theory in quantum mechanics (Slichter, 1992).

In the case a nuclear spin system coupled with the latticed reservoir only the transitions (—) ^ (+) with (a) ^ (b), with probability per second W^ and (+) ^ (—) with (b) ^ (a), with probability per second W^ are allowed according to basic quantum mechanical principles. The total number of transitions per second in a steady state gives p pbW^. = p+ paW.^ and since W^ = W^ it happens that p /p+ = pa/pb, where pa and pb are the populations of the (a) and (b) energy levels of the lattice reservoir.

At equilibrium the total number of transitions per second can also be written in the form p W; = p+ W^ with W; = pbW^. and W| = p„W^. Since W^ = W^ the upward and downward transition probabilities per second for the spin systems coupled with the lattice are different.

The rate of change of the difference between the population in the two spin states n = p+ — p can be written as

with n0 = (W; — Wt)/(W; + Wt) and T1 = W; + Wt (Slichter,

is

1992). T1 is the spin-lattice relaxation time and its inverse, Tj-1, is spin-lattice relaxation rate.

The quantification of the spin-lattice relaxation rate, as well as the calculation of other relaxation rates that will be introduced below are only possible within the quantum mechanics formalism, in particular within the time-dependent perturbation theory. One often used theoretical approach is that of the Bloch-Wangsness-Redfield (BWR) density operator perturbation for weak collisions (Abragam, 1961). The spin Hamiltonian is written as

I

where H0 is the static part of the Hamiltonian and H1(t) is the perturbation Hamiltonian usually associated with the spin interactions.

According to this theory the density operator of the spin system a can be used to estimate the expectation value of any observable Q

of the system according with the usual quantum theory expression

For the calculus of the density operator is necessary to solve the differential equation

This very formal expression is more often used in a more practical form in the interaction representation of the operators. Using the unitary quantum operator U = exp(-(//h)H0t), and the relations H0 = UH'0U-1 = H'0, H1 = UH[U-1, and a' = UaU-1 it is possible to write

valid for t ^ tc where tc is the characteristic time of the spin interactions (Slichter, 1992) and for a stochastically fluctuation perturbation H'(t) = 0.

The perturbation Hamiltonian corresponding to the spin interactions most often considered in NMR relaxation, can be written the very compact form

where KA is the interaction strength. T2,m represents the spin part of the Hamiltonian terms, and F2, m(r(t)) the spatial part of the time fluctuating Hamiltonian.

Since the hydrogen nucleus has spin 1/2 and hydrogen is the most abundant chemical element in organic materials, the following theoretical treatment will focus on the dipolar interaction between spins. For 1H spins the remaining spin interactions are less relevant for NMR relaxation studies.

 
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