NMR Spectroscopy

Spin Echo Experiment

The rate at which the bulk magnetization changes to reach the equilibrium state is characterized by two relaxation times.

The spin-lattice relaxation time, T₁ , characterizes the rate at which M_z changes to reach the equilibrium value of M_eq. Changes in magnetization along the z axis require changes in the energy of the spin system, thus T₁ characterizes the rate at which energy is transferred from the precessing nuclei to the surroundings.

The spin-spin relaxation time, T₂ , characterizes the rate at which a collection of precessing nuclei scramble their spin components in the xy plane. Changing M_x and/or M_y does not involve energy transfer with the surroundings. When there is a component of M in the xy plane, there is a slight excess of nuclei with their spins oriented in a particular direction in the xy plane. Over time, the direction of individual nuclear spins in the xy plane randomizes and both M_x and M_y decay to zero.

How does one experimentally measure T₂ ?

One might think measuring T₂ to be an easy task. The FID is a product of an oscillating signal and an exponential decay. Unfortunately, the exponential decay has the time constant T₂* not T₂. That is, the rate of decay of M_x and M_y, and thus the width of the NMR peak, depend on both the intrinsic rate of randomization of the individual μ_x and μ_y (characterized by T₂) and instrumental contributions to decay (or peak broadening), such as inhomogenieties in the magnetic field B.

The actual experimental time constant for decay of the FID is given T₂*, known as the effective spin-spin relaxation constant. T₂* is related to T₂ and the NMR spectral peak width w by

The challenge is separate the fundamental spin-spin relaxation characterized by T₂ from instrumental effects that speed the dephasing of the individual nuclear magnetic moments in the xy plane.

The Spin Echo experiment is a pulse sequence designed to reject instrumental contributions to peak broadening. There are several variations in the Spin Echo pulse sequence. The form used in this simulation is

90°_x - τ - 180°_y - τ - FID

The sequence begins with the 90°_x pulse that is used in the single-pulse experiment. Beginning with the system at equilibrium, the effect of this pulse is to rotate M from the z axis onto the y axis.

During the first delay of duration τ seconds, M precesses around the z axis and M_x, M_y and M_z all relaxation toward their equilibrium values.

What is the purpose of the 180°_y pulse? And how does this pulse and the subsequent delay time eliminate instrumental effects?

The simulation below lets you step through each component of the Spin Echo sequence and observe the effect on the bulk magnetization. Perform the simulation several times and carefully observe the effect of the 180°_y pulse and the second delay time.

Experiment

The animation below simulates the Spin Echo experiment. The simulation lets you step through each component of the sequence and observe the effect on the bulk magnetization.

The context of the simulation is that there is a single type of nucleus with Larmor frequency f in the rotating frame. However, owing to instrumental effects such as variations in the magnitude of B at various regions of the sample, not all nuclei precess at frequency f. Some precess faster. Some precess slower. The animation depicts three arrows representing the bulk magnetization attributable to three groups of nuclei. Each group has the same Larmor frequency, but variations in B cause each to precess at a different frequency.

Initially, the system is at equilibrium and all three bulk magnetization components are oriented along the positive z axis. The 90°_x pulse rotates all three arrows onto the positive y axis. At this point, these three components are all in phase. That is, they all point the same direction. However, this coherence is lost, to some extent, as they precess during the first delay period, because each group of nuclei precesses at a different frequency.

At the end of the first delay period, the 180°_y pulse flips the position of each arrow along the x axis. A large negative M_x before the pulse becomes a large positive M_x after the pulse. M_y is unaffected by the pulse, because the y axis is the axis of rotation. This 180° pulse occurs in about 20 μsec, which is over 100 times faster than the time required to precess one cycle, so there is negligible precession about the z axis during the 180°_y pulse.

After a second delay period of time τ, two things have happened to the arrows.

First, they have all reconverged and are once more in phase. Why did this occur? How did the pulse sequence eliminate the loss of coherence resulting from different precession frequencies?

Second, the arrows are all shorter. This effect is straightforward. It is now a time 2 τ after the magnetization was rotated into the xy plane and during that time the magnetization has relaxed toward the equilibrium value of zero. Here is the key to this pulse sequence: This relaxation of the xy component of the bulk magnetization is characterized by T₂ and is unaffected by the instrumental effects that produced the differing rates of precession.

Finally, the FID is acquired. The FFT is performed and the spectrum plotted. The nuclei are all the same, so there is only one peak. However, the peak is fairly broad, owing to the instrumental broadening. The height (or area) of the peak, however, depends upon how far M_x and M_y have relaxed toward equilibrium and that depends only upon T₂. Let S be the height of the peak in the NMR spectrum. S_o is the height when τ = 0. The decrease in S with increasing τ follows first-order kinetics.

S   =   S_o exp( - 2 τ / T₂ )

Perform the simulation using values of τ between 0 and 1 sec and observe the effect of τ and the 180°_y pulse on the behavior of the arrows and on the resulting spectrum.

• The graph shows the x detector signal for both delay periods and the FID. The vertical blue lines mark the end of each delay period.
• The 90°_x pulse takes 10 μsec and the 180°_y pulse takes 20 μsec. On the graph, this time is less than the thickness of the vertical blue lines. In the simulation, however, these pulses take one or two seconds to allow you to clearly visualize the effect of the pulse.
• The Larmor frequency is 30.0 Hz. The animation, however, precesses 10 times slower so that the movement of the arrows can be seen.
• T₁ has been set to an unrealistically long time to keep the magnetization in the xy plane where the effects of the delay times and the 180°_y pulse can be more clearly observed. You may wish to view the animation along the z axis during the delay times to obtain the best view of the dephasing and rephasing.
• When the spectrum is displayed, clicking the left mouse button shows the position of the cursor. You may use this feature to accurately measure the height of the peak. Click the Peak Width button to have the computer show the FWHM. Because the baseline is not zero, the half maximum of the peak is halfway between the baseline and the top of the peak.

• How does the Spin Echo sequence allow one to measure T₂ rather than T₂*?
• Determine the value of T₂.
• What is the natural line width?
  The natural line width is the peak width expected in the absence of instrumental effects.
• Measure the peak width and determine T₂*.
• How large is the instrumental broadening, w_inst?

In order to obtain T₂, one can measure values of S for several values of τ and then fit the data to the exponential decay described above. An alternate approach is to determine half-life for decay of M_x and M_y. First, perform the simulation with τ = 0 and measure the peak height (height above the baseline). Then perform the simulation with various values of τ until you locate the value of τ for which S is exactly one-half the height for τ = 0. This half-life is τ_½ , which is related to T₂ by