Product operator (NMR)

In the various fields of nuclear magnetic resonance, the product operator mathematical formalism is often used to simplify both the design and the interpretation of often very complex sequences of radio frequency electromagnetic pulses applied to samples under study. Basically, it is a short hand mathematical construct, a set of equations, that is used in place of more complex, although equivalent, matrix multiplication. The formalism uses a rotating frame of reference, in which the central irradiation frequency, say 800 MHz, is fixed on the X- or Y-axis, and the magnetic field, by convention, points towards the postive Z-axis. By convention, I and S indicate magnetic vectors associated with protons or heteroatom, respectively. Subscripts are used to indicate the axial orientation of the magnetic vector. At equilibrium, the net proton magnetic vector is thus I_z. Although even 2 pulse experiments on only 2 protons can lead to equations with 32 parts in the final equation due to chemical shifts, pulses and various forms of coupling, the use and knowledge of phase-cycling techniques or gradients to wipe out most of the terms leads to simplified final equations once accounted for.

Single Pulses (rotations)

Arbitrary pulses (rotations)

$I_{x}\;{\stackrel {\theta _{x}}{\longrightarrow }}\;I_{x}$

$I_{x}\;{\stackrel {\theta _{y}}{\longrightarrow }}\;I_{x}cos(\theta )-I_{z}sin(\theta )$

$I_{x}\;{\stackrel {\theta _{z}}{\longrightarrow }}\;I_{x}cos(\theta )+I_{y}sin(\theta )$

$I_{y}\;{\stackrel {\theta _{x}}{\longrightarrow }}\;I_{y}cos(\theta )+I_{z}sin(\theta )$

$I_{y}\;{\stackrel {\theta _{y}}{\longrightarrow }}\;I_{y}$

$I_{y}\;{\stackrel {\theta _{z}}{\longrightarrow }}\;I_{y}cos(\theta )-I_{x}sin(\theta )$

$I_{z}\;{\stackrel {\theta _{x}}{\longrightarrow }}\;I_{z}cos(\theta )-I_{y}sin(\theta )$

$I_{z}\;{\stackrel {\theta _{y}}{\longrightarrow }}\;I_{z}cos(\theta )+I_{x}sin(\theta )$

$I_{z}\;{\stackrel {\theta _{z}}{\longrightarrow }}\;I_{z}cos(\theta )$

90 degree pulses

So called 90 degree ($\pi$/2) pulses, in which magnetization is rotated from one axis to another, are the most widely used single pulses in NMR spectroscopy and the above equations simplify to the following for such pulses.

$I_{x}\;{\stackrel {90_{x}}{\longrightarrow }}\;I_{x}$

$I_{x}\;{\stackrel {90_{y}}{\longrightarrow }}\;-I_{z}$

$I_{x}\;{\stackrel {90_{z}}{\longrightarrow }}\;I_{y}$

$I_{y}\;{\stackrel {90_{x}}{\longrightarrow }}\;I_{z}$

$I_{y}\;{\stackrel {90_{y}}{\longrightarrow }}\;I_{y}$

$I_{y}\;{\stackrel {90_{z}}{\longrightarrow }}\;-I_{x}$

$I_{z}\;{\stackrel {90_{x}}{\longrightarrow }}\;-I_{y}$

$I_{z}\;{\stackrel {90_{y}}{\longrightarrow }}\;I_{x}$

$I_{z}\;{\stackrel {90_{z}}{\longrightarrow }}\;I_{z}$

Chemical Shift Operators

Nuclei rotate around the XY plane at different frequencies. For example, assuming an 800 MHz central proton frequency, some protons will rotate 800 Hertz, or 1 part-per-million (ppm), faster, while others will rotate about the field more slowly. This difference from the central frequency, expressed in ppm, is called a chemical shift, which is sybolized as $\delta$. The actual frequency, is sybolized as $\omega$ in radians/second or $\nu$ if expressed in Hertz. $\omega$= 2$\pi \nu$. Remembering that the central frequency is fixed on the X-axis, the chemical shifts of each proton will cause them to rotate away from the X-axis towards the Y-axis for faster frequencies and towards the minus Y-axis for slower frequencies. The total angle of the rotation is time dependent, so that during time delay $\tau$, the angle extended is

Note that chemical shifts only evolve in the XY plane.

$I_{x}\;{\stackrel {\delta (\tau )}{\longrightarrow }}\;I_{x}cos(\omega \tau )+I_{y}sin(\omega \tau )$

$I_{y}\;{\stackrel {\delta (\tau )}{\longrightarrow }}\;I_{y}cos(\omega \tau )-I_{x}sin(\omega \tau )$

$I_{z}\;{\stackrel {\delta (\tau )}{\longrightarrow }}\;I_{z}$

The one pulse experiment

Ignoring relaxation, coupling and other effects then, for a simple single proton starting at equilibrium, excited with a 90_y pulse, the time-dependent signal observed is:

$I_{z}\;{\stackrel {90_{y}}{\longrightarrow }}\;{\stackrel {\delta \tau }{\longrightarrow }}\;I_{x}cos(\omega \tau )+I_{y}sin(\omega \tau )$

Relaxation Operators

Although the previous equations imply that the NMR signal would ring out indefinitely, a number of relaxation processes cause the excited NMR states to relax back to equilibrium. Although these effects can be separated into rates R₁ and R₂, longitudinal (Z-axis) and transverse (XY-plane) effects, respectively, for simplicity one can consider the effect relaxation rate, R, and its characteristic relaxation time T = 1/R. Relaxation effects cause an exponential decay of the observable signal. The effects of relaxation then are expressed as:

The one pulse experiment with relaxation

$I_{x}\;{\stackrel {\delta (\tau )}{\longrightarrow }}\;{\stackrel {R(\tau )}{\longrightarrow }}\;I_{x}cos(\omega \tau )e^{-\tau /T_{R}}+I_{y}sin(\omega \tau )e^{-\tau /T_{R}}$

J-coupling Operators

Many types of NMR spectra exhibit line-splitting due to coupling of energy states mediated by through-bond electron-electron interactions. This coupling is referred to as J-coupling. Both heteronuclear and homonuclear coupling occur. The splitting due to J-coupling can be thought of in a manner similar to chemical shift. That is, the split resonances can stay on resonance, rotate faster or rotate slower than the central chemical shift. The relative intensities of signals due to J-coupling approximately follow the binomial expansion, 1, 1:1, 1:2:1, 1:2:2:1, 1:2:3:2:1, and so on.

Heteronuclear J-coupling

Consider a single proton, denoted I, bonded to a heteroatom, S. A proton-carbon pair of benzene is a useful example. No coupling occurs when the magnetic vectors of both nuclei are aligned with the Z-axis (that is, state I_zS_z), but the moment one of them is excited onto the XY-plane, heteronuclear J-coupling becomes important.

Heteronuclear J-coupling: A Doublet On Resonance

Assuming the proton of interest is exactly on resonance, thus ignoring chemical shift effects, and also ignoring relation and other effects, J-coupling to a single nuclei causes a single resonance (I) to split into typically two, three or four lines, although each of these may then also be split again by addition J-coupling to other nuclei into very complex patterns. Heteronuclear J-coupling operators are designated like I_xS_z. The heteronuclear J-coupling operators for doublets have the following effects:

$I_{x}\;{\stackrel {I_{x}S_{z}(\tau )}{\longrightarrow }}\;1/2{\Big (}I_{x}cos(J\pi \tau )+I_{y}sin(J\pi \tau ){\Big )}+1/2{\Big (}I_{x}cos(-J\pi \tau )+I_{y}sin(-J\pi \tau ){\Big )}$

It is important to notice that the first two terms correspond to one resonance frequency (J/2) while the last two terms refer to the second resonance frequency (-J/2) of the doublet NMR signal, separated by frequence J. Because rotations are additive, one can apply either the chemical shift operator or the J-coupling operators first, and then perform the other operation second. In addition, the signal of the heteroaton will be split in a similar manner. One only need calculate the resultant final signal for the actual nuclei type being observed.

Homonuclear J-coupling

Most splitting of NMR resonance signals is due to homonuclear J-coupling due to neighboring proton nuclei. For illustrating homonuclear coupling, indices for proton species must be included, giving rise to resonance states such as I_1xI_2z, where the 1 indicates one nuclei and the 2 indicates a second nuclei, and the two nuclei are J-coupled.