This post is intended to be the first in a series explaining the theory and application of nuclear spin relaxation in nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI). This part is intended to introduce relaxation and get as far as a general equation for evolution of the density operator including relaxation. The theory introduced here is known as Redfield theory, or Bloch-Wangsness-Redfield theory after its originators.

## What is relaxation?

Consider a system of nuclear spins in an applied magnetic field, left unperturbed and at at equilibrium. There will be a polarisation of nuclear spin states with an excess population in low-energy eigenstates of the Zeeman Hamiltonian, given by the Boltzmann distribution. This gives rise to a bulk magnetic moment parallel or antiparallel (depending on the sign of the gyromagnetic ratio) to the direction of the applied magnetic field. With nothing to create any coherence of spin phase, all phases (directions of nuclear spin magnetic moments in the transverse plane) are equally probable, so there is no transverse magnetisation. Now suppose a 90 degree radiofrequency (RF) pulse is applied along +y axis. The polarisation of spin states is removed - states are equally probable. A coherence is created along the x-axis, which precesses about the z-axis. But such a situation does not persist for long. The polarisation is quickly re-established, and, even quicker, the coherence in the transverse plane is lost. The re-establishment of the Boltzmann distribution of spin states, and therefore equilibrium ensemble z-magnetisation, is longitudinal relaxation. The loss of phase coherence in the transverse plane (re-randomnisation) is transverse relaxation.

## Why does relaxation happen?

Fundamentally, no system likes being displaced from equilibrium. It will always try to achieve thermal equilibrium with its surroundings. A system heated above the temperature of its surroundings will soon cool down. Longitudinal (T1) relaxation is an example of this enthalpic process. Systems also do not like having order imposed upon them and more readily lose order than gain it. Transverse (T2) relaxation is an example of this entropic process.

## How does relaxation happen?

Spin relaxation occurs due to random (stochastic) modulation of the local magnetic environment, so that each spin experiences a randomly changing magnetic field. The random changes are small compared to the size of the static applied field. It is useful to make the distinction between the spin interactions which ultimately determine the local magnetic field, and the molecular processes that determine the instantaneous spin interactions. The same system, with the same Hamiltonian, would relax differently if the molecular motions or processes changed (for example due to a change in temperature or phase).

The magnetic environment is a function of molecular orientation and configuration through spin interactions, namely chemical shift anisotropy, dipolar coupling and quadrupolar coupling. Any change in molecular orientation relative to the applied field, or change in molecular configuration due to internal bond motions, will change the instantaneous field experienced by a particular spin. In the case of water, for example, relaxation is dominated by the intra-molecular dipolar coupling between its two protons. The dipolar coupling means that the two protons mutually affect the magnetic field experienced by one another. As the dipolar coupling is a rank-2 interaction, it is spatially dependent so causes relaxation as random re-orientations of the molecule occur. In a solution-state system this "random re-orientation" is dominated by rotational diffusion but internal molecular dynamics are also important.

Random changes in local field on timescales at or near spin-state transition frequencies cause T1 relaxation by inducing changes in spin quantum number with a preference towards high-energy to low-energy transitions. The Boltzmann distribution is thus restored. The same stochastically varying fields cause irreversible dephasing amongst the ensemble of spins. This is T2 relaxation. Returning to the example of water, as each molecule tumbles randomly in solution, the changing dipolar coupling between the two protons means that a random distribution of local fields is sampled in a given unit of time by each molecule. It is the nature of this distribution of molecular orientations and local fields which ultimately determines the relaxation properties.

## Formal theory of relaxation: the interaction frame

Relaxation theory in NMR and MRI is semi-classical, meaning that the spin interactions are treated quantum mechanically whereas the surroundings are treated as a classical "lattice" or heat sink. It is based on the separation of timescales between molecular processes modulating spin interactions and relaxation itself. The assumption invoked several times is that molecular processes are on a much faster (shorter) timescale than relaxation. This is usually a safe assumption. For example, a protein in solution may have timescale of rotational diffusion of a few nanoseconds, but relaxation times in the millisecond to second range, comfortably seprating timescales by several orders of magnitude. The separation is often greater for smaller molecules in solution. Even in a paramagnetic spin system, where spins near a paramagnetic ion may have relaxation times reduced to microseconds, there is still a comfortable separation of timescales. In solids, "internal" motions such as rotations about certain bond vectors may be the relevant molecular processes. These still have very short timescales compared to millisecond to second timescales of relaxation. $T_1$ relaxation is often slower in solids (minutes), which can extend the theory's domain of validity.

The link between quantum mechanical spin dynamics and macroscopic observables for a pure ensemble is made by the Liouville-von Neumann equation

with H the (nuclear) spin Hamiltonian and σ the density operator. Square brackets indicate the commutator. To account for relaxation, we use time-dependent perturbation theory. This involves separating the Hamiltonian into a sum of two terms, the first H0 containing static or periodic components, the second H1(t) containing the stochastic parts relevant to relaxation, such that

We can then define an "interaction frame" Hamiltonian which evolves according to

It is often the case that the interaction frame is the same as the rotating frame (rotating at the chemically shifted Larmor frequency) we are used to seeing. In the interaction frame the Liouville-von Neumann equation is then

To solve this equation, and thus relate the molecular dynamics of the system to its spin relaxation parameters, we introduce several simplifying assumptions. Firstly, we assume that the stochastic perturbations are small. Secondly, we assume that the stochastic Hamiltonian varies more rapidly than the density operator. Thirdly, we assume that the correlation time, which will be discussed shortly and in later posts, is also short compared to the timescale of change in the density operator (equivalently, that we apply the theory only to longer timescales than the correlation time, or that relaxation is much slower than molecular motion). The above can then be expanded and integrated using a __Dyson series__, which to second order is:

Differentiating and setting

Each spin has a different random Hamiltonian, for each spin follows a different random trajectory, even if its statistics are the same. Therefore we need to average this equation over the ensemble, which we denote with an overbar:

The interaction-frame stochastic Hamiltonian is defined so that

Since we will only be interested in

we will drop the overbar and take it as implicit. We then introduce some simplifying assumptions: We assume that any terms higher than second-order in the Dyson series expansion can be neglected. We assume that the interaction frame Hamiltonian is not correlated with the interaction frame density operator, with the consequence that we can replace $\sigma^I(0)$ with $\sigma^I(t)$ and average inside the integral when averaging the commutators over the ensemble. We assume that we can extend the integral to infinity. Then we have:

This is almost our equation of motion for the interaction frame stochastic density operator. However, it is possible to show that this predicts that the density operator (in the interaction frame) decays to zero. This deficit is because the surrounding "lattice" has been treated classically. In reality, it relaxes only to thermal equilibrium. A correction (that can be proved by a complete quantum theoretical treatment, e.g. in __Abgragam__) is to subtract the equilibrium density operator from the interaction frame density operator wherever it appears, which we will take as implicit from here on. All equations are really for the quantity

## Justification of assumptions used

The simplifications are justified by the condition that we apply the theory to significantly longer timescales than the correlation time (i.e. that dynamics are on a faster timescale than relaxation). Although precise meaning will be given to the correlation time later in the context of specific models, for now it is enough to say that it is the characteristic timescale on which a system loses correlation with its initial configuration due to its stochastic motions, such as rotational diffusion, bond vector vibrations etc. It is therefore the timescale of molecular/sub-molecular processes driving the stochastic changes in the stochastic Hamiltonian. We will see that it is often in the picosecond to nanosecond range, whereas relaxation is on the millisecond to second range. The justification of the assumption of no correlation between the Hamiltonian and density operator and that we have:

is then made as follows: we can expect there to be correlation up to $\tau_c$ but not beyond. However, we require the correlation time to be much shorter than the integration limit so extension to infinity will be acceptable. Any correlation will already have gone. The same condition, that there is correlation between the density operator and Hamiltonian only at very short times, much shorter than those of interest, is also why it is permissible to replace he initial density operator with its value at time t and carry out the ensemble average inside the integral. The theory is dealing with the interaction frame Hamiltonian losing correlation with itself, but not with the density operator.

## Expansion in a basis of operators

Similarly to the "conventional" theory of NMR, we can represent the random Hamiltonian (before transformation into the interaction frame) as an expansion of products of irreducible spherical tensor operators with spatial functions:

Here, the terms

are the spin tensor operators of rank k, whilst the terms

are stochastic spatial functions depending upon the particular type of spin interaction under consideration. The spin tensor operators are themselves represented in terms of a set of operators as

and obey the commutation property

The following properties also exist:

With the definition of the interaction frame:

Note that we have incorporated a second index in the definition of the basis operators and frequencies with which operators evolve to take account of degenerate terms. This is relevant in the treatment of homonuclear dipolar couplings (amongst others), of which the two protons of a water molecule constitute a relevant example. In cases where this index can be ignored (e.g. CSA, heteronuclear dipolar coupling), we have:

The phase convention is also not consistent across treatments of relaxation. Whether the $(-1)^q$ phase factor is included explicitly depends on the definitions of the operator basis, into which it can be absorbed. If we absorb it into the operator basis we can write:

Now we substitute this into our master equation:

The quantity on the final line is the correlation function for the stochastic spatial parts of the Hamiltonian. We will dedicate some time to its meaning and calculation for particular models of molecular motion later.

For now, let us accept an abbreviation for it:

Since its Fourier transform appears, it is useful to include another definition, the spectral density function J, representing the density of motion occurring at a given frequency:

The second term here is the dynamical shift and is usually ignored, as it is small. The master equation is then:

This summation over 4 indices takes into account cross-correlation between operators of different rank. A simplification is possible by making use fo the secular approximation, which neglects terms that oscillate quickly and therefore average to zero. We can impose such a condition by requiring that

Given that

we therefore require p=p' and q=-q' and we have:

By imposing the secular approximation, we have retained only auto-correlation. To re-cap, the index q refers to basis operators in the expansion of the stochastic Hamiltonian (whilst p indexes expanison of basis operators to separate them by frequency). We have therefore neglected cross-correlation among the irreducible basis operators of the stochastic Hamiltonian. This defines the Liouville (relaxation) superoperator:

## In the laboratory frame

With the relaxation superoperator, and being explicit about the fact that we are calculating for the displacement of the density operator from equilibrium, in the laboratory frame we now have:

This is the equation of motion for the density operator, in the laboratory frame, including the effects of relaxation. The first term represents oscillations at various frequencies due to spin interactions (as we are used to seeing), whilst the second term, under the relaxation superoperator, represents the re-equilibration due to relaxation.

## Applying the theory

In order to apply the Redfield theory of relaxation, we need to know several things: we must know the Hamiltonian for the spin interaction(s) for which we wish to study the behaviour of relaxation, and represent such a Hamiltonian in the expansion of operators with associated frequencies described. We will also need to know the initial state of the density operator from which the system will relax. Finally, we will need a model of the system's stochastic dynamics, from which we must derive the spectral density function. Therefore, in applying Redfield theory, we typically do so for a particular spin interaction (e.g. dipolar or CSA interaction), for a particular operator (e.g. the raising operator), and for a particular dynamical process (e.g. isotropic rotational diffusion).

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