Physical Chemistry Third Edition

24.5 Fourier Transform NMR Spectroscopy 1029

Example 24.17). In order to describe the COSY experiment we need to envision

a Cartesian coordinate system that is rotating counterclockwise around thezaxis at the

reference frequency. The magnetic moment of an unshielded proton would be station-

ary in this rotating coordinate system. The vector sum of the proton magnetic moments

of all protons is denoted byM, as before. Before the first pulseMis parallel to thez

axis. The first 90◦pulse rotatesMonto the positiveyaxis, just as in the simple NMR

experiment and the spin–echo experiment.

z

y

x MB

MA

Figure 24.11 (a) The Magnetization

Contributions MAand MBin the Rotat-

ingCoordinateSystem.

z

y

x

MB

MA

MBx

MAx

MBy MAy

Figure 24.11 (b) The Magnetization

Contributions MA and MB in the

Rotating Coordinate System Repre-

sented by TheirxandyComponents.

We first consider a substance that has two uncoupled protons, A and B, that have

different chemical shifts (different shielding constants). We letMAbe the contribution

toMof the A protons in all of the molecules of the sample and letMBbe the contribution

toMof all of the B protons. Because of their different shielding constants,MAandMB

precess at slightly slower rates than an unshielded proton, and lag behind the rotation of

the coordinate system. They therefore move clockwise in thex−yplane of the rotating

coordinate system. We assume that proton B has a larger shielding constant than proton

A, so it lags farther behind the rotating coordinate system than does proton A and moves

through a greater angle in the rotating coordinate system, as shown in Figure 24.11a.

We can expressMAandMBin terms of theirxandycomponents since they now lie

in thexyplane.

MAiMAx+jMAy (24.5-3)

and

MBiMBx+jMBy (24.5-4)

whereiis the unit vector in thexdirection andjis the unit vector in theydirection.

The components are also depicted in Figure 24.11b. At timet 1 a second 90◦pulse is

applied. We can find the effect of this pulse by considering thexandycomponents of

the magnetization separately. The pulse has no effect on thexcomponents, and rotates

theycomponents clockwise onto the negativezaxis, where they are not sensed by the

detecting coil, which senses only magnetization in thex−yplane. Thexcomponents

MAxandMAyat the time of the second pulse (tt 1 ) are:

MAx(t 1 )MA(0)sin(2πνAt 1 ) (24.5-5)

and

MBx(t 1 )MB(0)sin(2πνBt 1 ) (24.5-6)

whereMA(0) is the magnitude ofMAat the end of the first pulse. The precession

frequency of proton A is denoted byνA, and that for proton B is denoted byνB. These

xcomponents are the only magnetization vectors that remain in thex−yplane after

the second pulse. They continue to precess after the second pulse and produce the

FID signal that is recorded as a function oft 2 , the time that elapses after the second

pulse.The FID is crudely depicted in Figure 24.10 along with the pulses.

The strength of the signal is determined byMAx(t 1 ) andMBx(t 1 ), the value of thex

components at the time of the second pulse. If the value oft 1 is such that sin(2πνBt 1 )

is equal to zero or has a small value, proton B will make little or no contribution to the

FID signal that is detected after the second pulse. Similarly, if the value oft 1 is such

that sin(2πνAt 1 ) is equal to zero or has a small value, there will be little or no signal

detected from proton A. The signals from proton A and proton B have effectively been

separated from each other, occurring at different values oft 1. This dependence ont 1 is

the reason that the method is calledcorrelation spectroscopy.