To understand how the pulsed-gradient spin-echo sequence allows us to measure diffusion, consider the timing diagram for the sequence. 
This sequence is very similar to the spin echo sequence introduced in Chapter 6, except that two gradient pulses have been applied. These two gradient pulses are identical in amplitude, G, and width, 
. The two gradient pulses are separated by a time 
and are placed symmetrically about the 180 degree pulse.
The function of the gradient pulses is to dephase magnetization from spins which have diffused to a new location in the period . These pulses have no effect on stationary spins. For example, a stationary spin exposed to the first gradient pulse, applied along the Z axis, will acquire a phase in radians given by
= 2 
z Gz dt . Consider the following illustration of the effect of the gradient pulses on the phase of stationary and moving spins. The illustration presents the phase of a diffusing spin relative to that of a reference spin and a stationary spin. The reference spin is one which experiences no gradient pulses. The stationary spin is not diffusing during the time illustrated by the sequence. The diffusing spin moves along Z during the sequence. The blue line in the timing diagram represents the time of the 180 degree pulse in the spin echo sequence. When you put the illustration into motion, the stationary spin comes back into phase with the reference one, indicating a positive contribution to the echo. The diffusing spin does not come back into phase with the reference spin so it diminishes the echo height.
The relationship between the signal (S) obtained in the presence of a gradient amplitude Gi in the i direction and the diffusion coefficient in the same direction is given by the following equation where So is the signal at zero gradient. 
)2 Di ( - /3)] The diffusion coefficient is typically calculated from a plot of ln(S/So) versus (G )2 ( - /3). Diffusion in the x, y, or z direction may be measured by applying the gradient respectively in the x, y, or z direction.
There are many pulse sequences which may be used to measure T1 and T2. The inversion recovery, 90-FID, and spin-echo sequences may be used to measure T1. Each technique has its own advantages and disadvantages. The spin-echo sequence may be used to measure T2.
For accurate measurements with each pulse sequence, it is important that the sample experience experience a spatially uniform B1 magnetic field. When inhomogeneous fields are used, inaccurate rotations result and the spins do not follow the general equation for the pulse sequence. One common cause of an inhomogeneous B1 magnetic field in a sample is a sample extending beyond the homogeneous bounds of the RF coil. It is best that the sample be confined to the volume of the RF coil. In practice this means filling an NMR tube with sample to a height no greater than the length of the RF coil, and positioning the tube with sample in the center of the coil. This position will require reshimming the sample if you are concerned about narrow lines widths.
T1 Measurement
Recall the timing diagram for an inversion recovery sequence first presented in Chapter 6.
The signal as a function of TI when the sequence is not repeated is
( 1 - 2eTI/T1 ) .
If the curve is well defined (i.e. if there is a high density of data points recorded at different TI times), the T1 value can be determined from the zero crossing of the curve which is T1 ln2.
Alternatively the relaxation curve as a function of TI may be fit using the equation
This approach is favored when there are fewer data points as a function of TI.
T1 may also be determined from a 90-FID or spin-echo sequence which is repeated at various repetition times (TR). For example, if the 90-FID sequence is repeated many times at TR1 and then many times at TR2, TR3, etc, the plot of signal as a function of TR will be an exponential growth of the form
. This data may be fit to obtain T1.
The difficulty with fitting this data and the inversion recovery data is a lack of knowledge of the value of the equilibrium magnetization or signal So. Other techniques have been proposed which do not require knowledge of the equilibrium magnetization or signal .
T2 Measurement
Measurement of the spin-spin relaxation time requires the use of a spin-echo pulse sequence. The echo amplitude, S, as a function of echo time, TE, is exponentially decaying. Plotting ln(S/So) versus TE yields a straight line, the slope of which is -1/T2. A linear least squares algorithm is often used to find the slope and hence T2 value. This approach can result in lead to large errors in the calculated T2 values when the data has noise. The later points in the decay curve have poorer signal-to-noise ratio than the earlier points, but are given equal weight by the linear least squares algorithm. The solution to this problem is to use a non-linear least squares procedure.
In an asymmetric molecule, the chemical shift is dependent on the orientation. The magnetic field experienced by the nucleus varies as a function of the orientation of the molecule in the magnetic field. The NMR spectrum from a random distribution of fixed orientations, such as in a solid, would look like this. The larger signal at lower field strength is due to the fact that there are more perpendicular orientations. In a nonviscous liquid, the fields at the various orientations average out due to the tumbling of the molecule. The anisotropic chemical shift is one reason why the NMR spectra of solid samples display broad spectral lines. Another reason for broad spectral lines is dipolar broadening. A dipolar interaction is one between two spin 1/2 nuclei. The magnitude of the interaction varies with angle 
and distance r. As a function of , the magnetic field B experienced by the red nucleus is
- 1). A group of dipoles with a random distribution of orientations, as in a solid, gives this spectrum. The higher signal at mid-field strength is due to the larger presence of orientations perpendicular to the direction of the Bo field. This signal is made up of components from the red and blue nuclei in the dipole. In a nonviscous liquid, the interaction averages out due to the presence of rapid tumbling of the molecule.
When the angle in the above equation is 54.7o, 125.3o, 234.7o, or 305.3o, the dipole interaction vanishes. The angle 54.7o is called the magic angle, m.
If all the molecules could be positioned at m, the spectrum would narrow to the fast tumbling limit.
Since this is not possible, the next best thing is to cause the average orientation of the molecules to be m.
Even this is not exactly possible, but the closest approximation is to rapidly spin the entire sample at an angle m relative to Bo. In solid state NMR, samples are placed in a special sample tube and the tube is placed inside a rotor. The rotor, and hence the sample, are oriented at an angle m with respect to the Bo magnetic field. The sample is then spun at a rate of thousands of revolutions per second.
The spinning rate must be comparable to the solid state line width. The centrifigal force created by spinning the sample tube at a rate of several thousands of revolutions per second is enough to destroy a typical glass NMR sample tube. Specially engineered sample tubes and rotors are needed.
m resolution. The reader interested in more information on NMR microscopy is encouraged to read the author's hypertext book on MRI entitled The Basics of MRI located at http://www.cis.rit.edu/htbooks/mri/. If T1 of the two components differ, this may be accomplished by using an inversion recovery sequence, presented in Chapter 6. To eliminate the water signal, choose the TI to be the time when the water signal passes through zero.
In this example, a TI = 1 s would eliminate the water signal.
Another method of eliminating a solvent absorption signal is to saturate it. In this procedure, a saturation pulse similar to that employed in C-13 NMR (See Chapter 9) is used to decouple hydrogen coupling. The frequency of the saturation pulse is set to the solvent resonance. The width of the saturation pulse is very long, so its bandwidth is very small causing it to affect only the solvent resonance.
as a function of magnetic field or Larmor frequency. Therefore, field cycling NMR finds applications in the study of molecular dynamics. The animation window contains an example of results from a field cycling NMR spectrometer. The plot represents the R1 value of the hydrogen nuclei in various concentration aqueous solutions of Mn+2 at 25o as a function of the proton Larmor frequency.
Many different techniques have been used to obtain R1 as a function of magnetic field. Some techniques move the sample rapidly between different magnetic field strengths. One of the more popular techniques keeps the sample at a fixed location and rapidly varies the magnetic field the sample experiences. This technique is referred to as rapid field cycling NMR spectroscopy.
The principle behind a rapid field cycling NMR spectrometer is to polarize the spins in the sample using a high magnetic field, Bp. The magnetic field is rapidly changed to the value at which relaxation occurs, Br. Br is the value at which R1 is to be determined. After a period of time, t, the magnetic field is switched to a value, Bd, at which detection of a signal occurs. Bd is fixed so that the operating frequency of the detection circuitry does not need to be changed. The signal, an FID, is created by the application of a 90o RF pulse. The timing diagram for this sequence can be found in the animation window.
The FT of the FID represents the amount of magnetization present in the sample after relaxing for a period t in Br. A plot of this magnetization as a function of t is an exponentially decaying function, starting from the equilibrium magnetization at Bp and going to the value at Br. When a single type of spin is present, the relaxation is monoexponential with rate constant R1 at Br.
When Br is very large compared to Bd, Bp is often set to zero and the plot of this magnetization as a function of t is an exponentially growing function.
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