The product operator model of the evolution of spin states is a rigorous model that predicts precisely what will happen in a given situation for a weakly coupled system. It is also easy to relate it to the vector model pictures ... at least for some types of evolutions. When reading the nmr literature it is almost invariably used to describe the actions of various new pulse sequences. I will introduce it to you and give several examples and then go back to basics and derive it later on.

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For a single spin there are four operators, I_z, I_x, I_y and E/2 (the unity operator .. the "/2" is for normalization purposes). The equilibrium magnetisation is represented by I_z and the transverse magnetisation by I_x and I_y.

If we start with equilibrium magnetisation and issue a b_-x pulse, the vector model says that the magnetisation will be tipped by b degrees.

So, we could write an equation (or more accurately, a transformation):

         M_z ------> M_zcos(b) + M_ysin(b)

If we replace M with I we have the analogous product operator transformation:

         bI_-x
          I_z ------> I_zcosb + I_ysinb

bI_-x represents a pulse which tips the equilibrium magnetisation b degrees about the -x axis, I_zcosb is the residual magnetisation along the z axis and I_ysinb represents the magnetisation along the y axis. So, we can represent a pulse of arbitrary angle and phase and view it as a rotation about the axis specified by the phase, just as in the vector model. The right-hand rule is used, as in the vector model and in vector algebra.

Chemical shift evolution in the transverse plane is represented similarly:

        2pWtI_z
          I_y --------> I_ycos2pWt - I_xsin2pWt

where 2pWt represents the angle of rotation. W is the rotating frame chemical shift frequency and t is the time of rotation.

This is as far as we can go with a single spin. For two spins there are 16 possible product operators:

Again, E/2 represents the unity operator and again, "/2" and "2" are present for normalization purposes. I and S represent the two different spins ... actually, the origin of this representation goes back to the introduction of the INEPT experiment to represent the Insensitive and Sensitive nucleii but over the years the use of these has changed. We can now represent coupling evolution:

        pJt2I_zS_z
          I_y --------------> I_ycospJt - 2I_xS_zsinpJt

where J is the coupling constant and t is the evolution time.

What happens to a two-spin operator as a result of a pulse depends on the pulse spin and the two-spin operator. I-pulses act only on I spins and S-pulses act only on S spins:

           bI_-x
     2I_zS_z ------> 2I_zS_zcosb + 2I_yS_zsinb

Note that in each example the result of a rotation is a 'residual' or 'source' term with a cosine multiplier and an evolved or 'target' term with a sine multiplier. This is generally the case with all product operator evolutions.

Let's look at an example of product operators in action ... specifically, let's look at the spin echo for two spins. The pulse sequence is:

The (p/2)_-x pulse rotates the equilibrium magnetisation, I_z, about the -x axis (using the right-hand-rule remember!) and creates magnetisation along the y axis:

        (p/2)I_-x
          I_z --------> I_y

The cosine term is zero here (cos(p/2)=0) and the sine term is equal to one.

During the delay, t, chemical shift evolution and coupling evolution take place. It doesn't matter what order we do them in because the shift operator, I_z, and the coupling operator, I_zS_z commute with each other. Generally, the order of application of operators is irrelevant if the operators commute. We will do the shift operation first:

        2pwtI_z
          I_y --------> I_ycos(2pwt) - I_xsin(2pwt)

followed by the coupling evolution of each of the shift evolution terms:

                 pJt2I_zS_z
  I_ycos(2pwt) --------------> I_ycos(2pwt)cos(pJt) - 2I_xS_zcos(2pwt)sin(pJt)

                 pJt2I_zS_z
  -I_xsin(2pwt) --------------> -I_xsin(2pwt)cos(pJt) - 2I_yS_zsin(2pwt)sin(pJt)

Equivalently, we could represent the transformations in this manner:

The pI_x pulse will then act on the appropriate operators to give:

Note that no I_x or 2I_xS_z operators are affected by a rotation about the x axis. In this case, only the I_y operator is affected.

Now, the evolution during the second delay time give us:

- Iy*cos(pJt)*cos(2pwt)*cos(pJt)*cos(2pwt) + 2IxSz*sin(pJt)*cos(2pwt)*cos(pJt)*cos(2pwt) + Ix*cos(pJt)*sin(2pwt)*cos(pJt)*cos(2pwt) + 2IySz*sin(pJt)*sin(2pwt)*cos(pJt)*cos(2pwt) - 2IxSz*cos(pJt)*cos(2pwt)*sin(pJt)*cos(2pwt) - Iy*sin(pJt)*cos(2pwt)*sin(pJt)*cos(2pwt) - 2IySz*cos(pJt)*sin(2pwt)*sin(pJt)*cos(2pwt) + Ix*sin(pJt)*sin(2pwt)*sin(pJt)*cos(2pwt) - Ix*cos(pJt)*cos(2pwt)*cos(pJt)*sin(2pwt) - 2IySz*sin(pJt)*cos(2pwt)*cos(pJt)*sin(2pwt) - Iy*cos(pJt)*sin(2pwt)*cos(pJt)*sin(2pwt) + 2IxSz*sin(pJt)*sin(2pwt)*cos(pJt)*sin(2pwt) + 2IySz*cos(pJt)*cos(2pwt)*sin(pJt)*sin(2pwt) - Ix*sin(pJt)*cos(2pwt)*sin(pJt)*sin(2pwt) - 2IxSz*cos(pJt)*sin(2pwt)*sin(pJt)*sin(2pwt) - Iy*sin(pJt)*sin(2pwt)*sin(pJt)*sin(2pwt)

Wow! A pretty complex looking expression, eh? (I just had to sneak some Canadiana in there).Fortunately for us, it can be simplified using some trigonmetric identities:

With these identities the above expression simplifies to:

believe it or not! Useful things, these trig identities. This result is identical to that obtained by using the vector model. Evidently, we can simply leave out the chemical shift evolution in our analysis. If we do this things are much simpler:

- I_ycos(pJt)cos(pJt) + 2I_xS_zsin(pJt)cos(pJt) - 2I_xS_zcos(pJt)sin(pJt) - I_ysin(pJt)*sin(pJt)

Note that for the above analysis, I and S are not the same type of nucleii ...in other words they are heteronucleii, proton and carbon for example. What would be the result if they are the same nucleus? Since, for same nucleii we must apply the p pulse to all nucleii

- I_ycos(pJt)cos(pJt) + 2I_xS_zsin(pJt)cos(pJt) + 2I_xS_zcos(pJt)sin(pJt) + I_ysin(pJt)*sin(pJt)

Using our trigonometric identities this becomes:

If t is set equal to 1/(4J) then the first term vanishes and the second becomes I_xS_z. This, plus a p/2 pulse in I and S at the end of the spin echo is the basis of the INEPT (Insensitive Nucleii Enhancement by Polarisation Transfer):

This is S magnetisation which is antiphase with repect to I. It started out as I magnetisation antiphase with respect to S but through the two p/2 pulses a 'polarisation' transfer from I to S has occurred. If the gyromagnetic ratio of I is higher than that of S, S will gain a sensitivity enhancement.

Another pulse sequence named TBPDF after its (numerous) authors but known more commonly, at least to Bruker users, as DECP90 is:

and the product operator analysis gives:

         p/2I_y
          I_z ------> I_zcos(p/2) + I_xsin(p/2)

Since we are using a p/2 pulse, cos(p/2) equals zero and sin(p/2) equals 1 so we're left with I_x only. Next we wait for a time period, t, which we set equal to 1/(2J). Both chemical shift evolution and coupling evolution will occur:

        2pWtI_z
          I_x ---------> I_xcos(2pWJt) + I_ysin(2pWJt)

                           ptI_zS_z
          I_xcos(2pWJt) ---------> I_xcos(2pWJt)cos(pJt) + 2I_yS_zcos(2pWJt)sin(pJt)

                           ptI_zS_z
          I_ysin(2pWJt) ---------> I_ysin(2pWJt)cos(pJt) - 2I_xS_zsin(2pWJt)sin(pJt)

If, as I said earlier, we set t=1/(2J), the I_x and I_y terms will vanish (cos(pJt)=0) and the antiphase terms will be 2I_yS_zcos(2pWJt) and 2I_xS_zsin(2pWJt) (sin(pJt)=1). Let's look at what will happen to the first of these terms (the same will happent to the second). We now apply the qS_y pulse:

                              qS_y
      2I_yS_zcos(2pWJt) ---------> 2I_yS_zcos(2pWJt)cos(q) + 2I_yS_xcos(2pWJt)sin(q)

The term 2I_yS_x is multiple quantum coherence which is unobservable. If we set q=p/2 then the result of the last expression is simply 2I_yS_xcos(2pWJt) which is unobservable in our spectrometer. A similar fate will befall the 2I_xS_z term. Thus, by setting our delay to 1/(2J) and making changes to the pulse width of the S nucleus we can determine when we reach the p/2 pulse when the spectrum vanishes. This is one of the few times that we don't want to see anything in our nmr spectrum!

A very common two-dimensional experiment is the COSY (COrrelation SpectroscopY. The pulse sequence is deceptively simple:

... two 90 degree pulses separated by a variable delay time. For a two-spin homonuclear system I and S, the spin state after the two pulses and the delay is:

During the acquisition, I_z will not produce any observable magnetisation as is the case with I_xS_y so all we need worry about is the last two terms:

Looking at what happens to them individually is very interesting. First I_x's evolution during the acquistion period (t₂):

+ I_xcos(pJt₂)cos(2PW_It₂)cos(pJt₁)sin(2PW_It₁) + 2I_yS_zsin(pJt₂)cos(2PW_It₂)cos(pJt₁)sin(2PW_It₁) + I_ycos(pJt₂)sin(2PW_It₂)cos(pJt₁)sin(2PW_It₁) - 2I_xS_zsin(pJt₂)sin(2PW_It₂)cos(pJt₂)sin(2PW_It₁)

In this expression we see that in each of the chemical shift evolution terms, the observable magnetisation is being modulated by W_I during both t₁ and t₂. Contrast this to the result of the evolution of 2I_zS_y:

- 2I_zS_ycos(pJt₂)cos(2PW_St₂)sin(pJt₁)*sin(2PW_It₁) + S_xsin(pJt₂)cos(2PW_St₂)*sin(pJt₁)sin(2PW_It₁) + 2I_zS_xcos(pJt₂)sin(2PW_St₂)sin(pJt₁)sin(2PW_It₁) + S_ysin(pJt₂)sin(2PW_St₂)sin(pJt₁)sin(2PW_It₁)

Looking closely at these terms we see that each one is modulated by W_I in t₁ and W_S in t₂

So what, you ask? Well, these results indicate that, if we plot the t₁ modulations the x-axis and the t₂ on the y-axis, the first terms will be along the diagonal and the second set of terms will be 'off diagonal' at chemical shift W_I on the x-axis and W_S on the y-axis.

Even though the product operator method is much easier to use than density matrices it is still a bit tedious and error-prone. To combat this I have produced a product operator calculator ProdOp that will take care of most of the tedious stuff for you.