NMR Spectroscopy

The Inversion Recovery 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 nucleus 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 spins in the xy plane randomizes and both M_x and M_y decay to zero.

How does one experimentally measure T₁ ?

Detection in NMR spectroscopy employs coils along the x and y axes. Therefore one only detects magnetization in the xy plane. That is, one can only measure M_x and M_y. Because T₁ is the time constant for changes in M_z and there is no direct measurement of M_z, one must perform the measurement indirectly.

T₁ values are measured using the Inversion Recovery experiment. The pulse sequence of this experiment is

180°_x - τ - 90°_x - FID

In between the two pulses, the system is allowed to evolve for a time τ . The Inversion Recovery experiment is an example of 2D NMR spectroscopy, because the data set contains two dimensions: t and τ. For each application of this pulse sequence, one obtains an FID that is a function of time t. Each application of this pulse sequence uses a different delay time τ . A Fourier transform is performed on each FID, resulting in a different spectrum for each value of τ .

During the delay time τ, M_z changes and the bulk magnetization evolves towards its equilibrium value, M_eq. Recall that, in the absence of a pulse, M_z follows first-order kinetics as it changes to reach the equilibrium state. The time constant is T₁.

The 180°_x pulse has the effect of inverting M_z from the initial equilibrium value of M_eq to -M_eq. Following the 180°_x pulse, the first-order rate law for how M_z changes over the period of time τ is

M_z = M_eq ( 1 - 2 exp( - τ / T₁ ))

Because M_z can be positive or negative just before the 90°_x - FID portion of the sequence, the peak in the spectrum can be positive or negative. In practice, one measures the peak height associated with various values for τ. The data set is then curve-fit to an exponential decay to obtain the time constant T₁.

A short cut is to perform the experiment for various value of τ until one identifies the value for which the spectrum is completely flat (no peak, positive or negative). The flat spectrum results from M_z = 0 just before the 90°_x pulse. At this value of τ

1 = 2 exp( - τ / T₁ )

T₁ =

τ ln 2

Experiment

The animation below simulates the Inversion Recovery experiment. There is only one type of nuclei, which precesses at 300. Hz in the rotating frame. The simulation executes one step at a time, so you can visualize what is happening in each step. Only the FID portion of the pulse sequence is graphed. After completing the pulse sequence, click on the Plot Spectrum button to perform the FFT on the FID and graph the resulting spectrum.

Carefully observe the behavior of M (represented by the green arrow in the animation).
How is this pulse sequence able to measure T₁ ?
How is it possible to measure relaxation of M_z using only measurements of M_x and M_y ?

Perform the simulation several times using different values of τ between 0 and 10. sec and observe the effect of τ on the spectrum.

In practice, the 90°_x pulse would only require about 10 μsec and the 180°_x pulse about 20 μsec. The animation is 100,000 times slower to allow you to clearly see the behavior of M.

Although the FID is acquired over a realistic period of time, the precession around the z axis occurs 100 time slower than is represented in the spectrum.

Answer the following questions.

What is the effect of the 180°_x pulse?
What happens during the delay time τ?
What is the reason for varying the value of τ?
What is the effect of the 90°_x pulse?
How does the value of τ affect the appearance of the spectrum?
Note that the spectrum is correctly phased before being graphed.
What is the value of T₁ ?