## Magnetization Transfer

Magnetization transfer is the central process in all advanced NMR experiments, both 1D and 2D. By an appropriate combination of pulses and/or waiting periods ("delays"), we can make net magnetization "jump" from one nucleus in a molecule to another. I will not attempt to explain the details of how this happens, I will just "narrate" the process in general terms and you will have to take it mostly on faith. Just as there are two kinds of net magnetization, z magnetization and coherence, there are two ways to transfer magnetization: NOE (transfer of z magnetization) and INEPT (coherence transfer). The NOE transfer occurs directly through space from one proton in a molecule to a nearby proton. The distance between them must be less than 5 A and the efficiency of transfer is proportional to the inverse 6th power of the distance between them (1/r6). In this way we can measure distances within a molecule and make conclusions about stereochemistry and conformation. The INEPT (insensitive nuclei enhanced by polarization transfer) transfer occurs via J couplings, which means it is a through-bond effect between atoms that are two or three (occasionally more) bonds apart in the covalent bonding network of a molecule. As J coupling values depend on the dihedral angle for vicinal (three-bond) relationships (Karplus relation), we can learn about conformation as well as covalent connectivity. These are the structural relationships we can discover using NMR, and the key to connecting one spin to another via these relationships is magnetization transfer.

The NOE works like this: if you perturb the z magnetization of one proton in a molecule so that it is no longer at equilibrium (i.e., no longer +Mo), this perturbation will propagate over time (0.2-1 s for small molecules) to other protons in the molecule, creating perturbations of their z magnetization away from equilibrium. For small molecules (<1000 Da) the effect is "negative": reducing the z magnetization of one proton will lead to the buildup of an increase in z magnetization of nearby spins. There are several experiments that can be designed to perturb one proton's z magnetization and to convert the transferred z magnetization into a measurable signal (enhanced peak height in the spectrum) and thereby determine the distances between protons. The initial perturbation can be created by a long, low power radio frequency signal at the exact resonant frequency of one proton in the spectrum ("saturation") or by a 180° pulse which rotates the net magnetization of one proton

in the molecule to the negative z axis. The NOE process can be viewed as magnetization transfer as the z magnetization difference (perturbation from equilibrium) of one proton is transferred to another during the "mixing time" of the experiment, leading to a z magnetization difference in the second proton. Two-dimensional experiments that use NOE as a means of magnetization transfer include the NOESY and ROESY experiments.

INEPT coherence transfer works differently. We create coherence on one nucleus with a 90o pulse and then wait a period of time (equal to 1/(27), where J is the coupling constant). At this point, the two components of the doublet signal are opposite in phase: an FID acquired at this point would give a spectrum of a doublet with one component pointing up and one pointing down (Fig. 7.3, left). This "antiphase" state has a very special property: if we subject it to 90o pulses simultaneously affecting both nuclei in the J-coupled pair, the coherence will "jump" from one nucleus to the other. We will now have antiphase coherence on the J-coupled nucleus and no coherence on the starting nucleus (Fig. 7.3, right). This can be applied to any pair of nuclei that are J coupled: two protons on adjacent carbons (vicinal relationship), a proton and its directly bonded carbon (1-bond heteronuclear coupling), a proton and the carbon next to its own carbon (2-bond heteronuclear J coupling), and so on. The INEPT transfer is used in advanced 1D experiments such as DEPT, as well as in a number of 2D experiments (COSY, DQF-COSY, HETCOR, HSQC, HMBC).

7.3 THE PRODUCT OPERATOR FORMALISM: INTRODUCTION

It is very difficult to describe coherence transfer using the vector model. To understand it we will need to expand our theoretical picture to include product operators. Product operators are a shorthand notation that describes the spin state of a population of spins by dividing it into symbolic components called operators. You might wonder why you would trade in a nice pictorial system for a bunch of equations and symbols. The best reason I can give is that the vector model is useless for describing most of the interesting NMR experiments, and product operators offer a bridge between the familiar vectors and the more formal and mathematical matrix representation. Later on you will see that product operators are just shorthand for the elements of the density matrix.

Using the vector model, when we want to describe the spin state of a particular nucleus, we can draw a vector in three-dimensional space, or we can describe the projection of that vector onto the three axes (the "components" of the vector). For example, a vector of length Mo on the —X axis could be described as

Note that this description requires that we make three statements about the components of the vector. In the product operator formalism, we simply say that the spin state is —Ix. We do not have to say anything about the y and z components because if we do not see Iy or Iz in the spin state, we just assume that those components are zero. If we wish to talk about two different spins at the same time in the vector model, we need two different sets of coordinate axes; for example, one for XH and one for 13C, to keep it straight which one we are talking about. With product operators, we usually use I for 1H and S for 13 C, so we could describe a spin state as —Iz + Sx, meaning that the 1H net magnetization is on the —z axis and the 13C net magnetization is on the +x axis. If we want to talk about two protons, we can use either I and S or Ia and Ib to describe Ha and Hb. As you can see, we are going to move away from the pictorial representation on a coordinate system (vectors) to a symbolic representation using the product operators. These symbols can be easily manipulated using simple math and rules about how they behave with pulses (rotations about the B1 vector) and delays (rotation about the z axis, also known as "evolution"). The rules all refer back to the vectors, so if we understand how the vectors behave we can manipulate these symbols very easily.

A more complicated case is that of the 13C doublet of a methine (13C-1H) group in the absence of 1H decoupling. In the vector model we draw two vectors: one for the net magnetization of all 13 C spins whose 1H coupling partner is in the a state, and the other for the net magnetization of all 13 C spins whose 1H coupling partner is in the j state. This corresponds to the two lines in the 13C spectrum (without 1H decoupling) for the CH doublet: the 13C spins with a 1H partner in the a state give rise to the left-hand peak of the doublet, and the 13C spins with a 1H partner in the j state give rise to the right-hand peak of the doublet. If both the "a" vector and the " j" vector are on the — x axis, we would call this state "in-phase magnetization on the — x axis." In the product operator notation, we would call this —Sx, meaning that the 13C magnetization, regardless of the spin state (a or j) of its 1H coupling partner, is on the — x axis (Fig. 7.4). If the "a" vector is on the y axis and the " j" vector is on the — y axis, we call this state "antiphase." In the product operator notation, we represent the spin state with the symbol 2SyIz, which we read as "13C magnetization on the y axis, antiphase with respect to its coupling partner 1H." Everyone who sees this for the first time is completely mystified. First of all, the 2 is just a normalization constant, so you can ignore it. It is necessary any time you have two operators multiplied together (hence the name "product" operators). The Sy means that the 13C magnetization from carbons attached to 1H in the a state is on the +y axis. The multiplication by Iz is the hard part. This Iz says nothing about the net magnetization of 1H, so do not make the mistake of thinking that the 1H net magnetization vector is on the +z axis. The Iz multiplier represents the microscopic spin state of each individual proton: half are in the a state (Iz = +1/2, nuclear magnet

pointing "up," along the +z axis) and half are in the 5 state (Iz = —1/2, nuclear magnet pointing "down," along the — z axis), and we are multiplying 2Sy by this number. So as we add up the individual nuclear magnetic dipoles (vectors) to yield the net magnetization, we are saying that the individual 13C vectors are aligned with the +y axis if the attached 1H is in the a state (multiply the 2Sy by +1/2), and the individual 13 C vectors are aligned with the —y axis if the attached 1H is in the 5 state (multiply the 2Sy by —1/2). We end up with the two net magnetization vectors, one on the +y axis for those 13C nuclei whose attached protons are in the a state, and one on the —y axis for those 13C nuclei whose attached protons are in the 5 state. Later, when we consider the density matrix representation, we will see that multiplication by Iz is really multiplication by a 4 x 4 matrix, and all the math works out perfectly to generate the vector model picture of two opposed vectors. For now, you can think of the Iz multiplier as a +1/2 (for I = a) or a —1/2 for (I = ¡). Iz by itself still represents the 1H net magnetization on the +z axis (Mz (1H) = +Mo).

7.4 SINGLE SPIN PRODUCT OPERATORS: CHEMICAL SHIFT EVOLUTION

If there is only one NMR line in the spectrum (a population of identical nuclei) there are only three product operators, and they correspond to the three components of the net magnetization vector. A complete description of a population of spins can be given by the spin state a:

a = Ix + Cy Iy + Cz Iz where cx, cy, and cz are coefficients equal to Mx/Mo, My /Mo, and Mz/Mo. This is the product operator representation of the spin state of a population of spins. Compare this to the vector representation of the net magnetization (sum of individual spin vectors):

M = Mx i + My j + Mz k where i, j, and k are the unit vectors along the x, y, and, z axes. The product operators Ix, Iy, and Iz can be viewed as pure spin states. Iz is the equilibrium state, Iy is the spin state immediately following a 90o pulse on the — X axis, and Ix is the spin state immediately following a 90o pulse on the y' axis. In the literature you will find that pulses are regarded as counterclockwise rotations for product operators, so that a 90o pulse on the X axis rotates

the Iz state into the —Iy state. This is opposite to the convention sometimes used for the vector model. For consistency, in this book all pulse rotations are counterclockwise.

These simple product operators precess in the X-y' plane of the rotating frame at a frequency corresponding to the chemical shift in hertz relative to the center of the spectral window (the resonance offset Av = vo — vr). The chemical shift frequency Av can also be represented as the angular velocity Q in units of rad/s (Q = 2nAv). Using Q allows us to skip all the 2n terms.

-^ Iz (no precession ror z magnetization)

The first two changes represent the circular motion described by the components Mx and My of the net magnetization vector in the X-y' plane (Fig. 7.5), a process called evolution. The projection of the net magnetization vector on the x axis, relative to Mo, gives the factor in front of Ix and the projection on the y axis, relative to Mo, gives the factor in front of Iy. Note that in every evolution period, the spin state you start with is multiplied by a cosine term (cos© = 1 for © = 0; cos© = 0 for © = 90o) and the spin state you are moving toward (by 90o counterclockwise rotation in the X-y' plane) is multiplied by a sine term (sin © = 0 for © = 0, sin © = 1 for © = 90o). If the NMR line is upfield of the center of the spectral window (vo < vr, negative value of Q), the sine terms will start negative and the cosine terms will start out positive, reflecting the clockwise rotation. So we always think of the motion as counterclockwise and let the sign of Q correct for any clockwise rotations:

{starting state}—t delay, Q evolution ^ {starting state} cosQt + {Next ccw stop} sinQT

Any complicated representation of a spin system in terms of Ix, Iy, and Iz can be described after a delay t by substituting the corresponding expression on the right side every time one of the terms on the left side (Ix, Iy, or Iz) occurs in the representation. For example, if we start with the spin state

Ix — Iy + Iz at time t = 0, we will have at time t = t as a result of chemical shift evolution

{lx cos^T + ly sin^T} + {—ly cos^T + lx sin^T} + lz = lx (cos^T + sin^T) + ly (sin^T — cos^T) + lz l y

Each operator in the original spin state is replaced by that operator ("where we are starting"), multiplied by the cosine term, plus the next operator we encounter moving counterclockwise in the x-y plane ("where we are headed"), multiplied by the sine term. Because z magnetization is stationary during delays, Iz does not undergo evolution.

The beauty of the product operators is that we never have to deal with any more than one operator at a time. If we know how Ix, Iy, and Iz behave for a pulse or delay, we know how any combination of these operators behaves as we just replace each operator with the result of the pulse or delay on that operator alone. Instead of the net magnetization vector, which can point anywhere in 3D space, we have reduced the problem to understanding how pulses and delays affect the three simple components: Ix, Iy , Iz.

So far we have not included the relaxation processes (Ti and T2), and for many pulse sequences we can leave out this aspect to make the math simpler. We know that relaxation is going on, but in many cases this is merely a technicality and is not essential in understanding the pulse sequence. In general, pulses are on the timescale of microseconds (^s), delays for evolution are on the order of milliseconds (ms), and delays for buildup of NOE can be hundreds of ms. For organic-sized molecules, we can safely ignore relaxation for delays in the ¡is or ms range. Of course, for some experiments such as NOE, the relaxation process is central to the experiment so we cannot ignore it.

The chemical shift evolution (precession of spins as a result of chemical shift) can be represented as a circle with the rotation rate (in radians per second) written in the center (Fig. 7.6). This is the same as the motion of the net magnetization vector, viewed from the +z axis. Homonuclear product operators (Ha represented by Ia and Hb represented by Ib) undergo chemical shift evolution in the same way:

Note that the rate (and direction) of precession in the rotating frame is controlled by the chemical shift Q term, which is specific to each different proton in the molecule (each different resonance or peak in the spectrum).

Qt radians s-1

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