# 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, *T _{1}* , characterizes the rate at which

*M*changes to reach the equilibrium value of

_{z}*M*. Changes in magnetization along the

_{eq}*z*axis require changes in the energy of the spin system, thus

*T*characterizes the rate at which energy is transferred from the precessing nucleus to the surroundings.

_{1}The spin-spin relaxation time, *T _{2}* , characterizes the rate at which a collection of precessing nuclei scramble their spin components in the

*xy*plane. Changing

*M*and/or

_{x}*M*does not involve energy transfer with the surroundings. When there is a component of

_{y}**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*and

_{x}*M*decay to zero.

_{y}How does one experimentally measure *T _{1}* ?

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*. Because

_{y}*T*is the time constant for changes in

_{1}*M*and there is no direct measurement of

_{z}*M*, one must perform the measurement indirectly.

_{z}*T _{1}* 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*. Recall that, in the absence of a pulse,

_{eq}*M*follows first-order kinetics as it changes to reach the equilibrium state. The time constant is

_{z}*T*.

_{1}The 180°_{x} pulse has the effect of inverting *M _{z}* from the initial equilibrium value of

*M*to -

_{eq}*M*. Following the 180°

_{eq}_{x}pulse, the first-order rate law for how

*M*changes over the period of time τ is

_{z}*M _{z}* =

*M*( 1 - 2 exp( - τ /

_{eq}*T*))

_{1}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

*T*.

_{1}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 / *T _{1}* )

*T*=

_{1}## 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 *T _{1}* ?

How is it possible to measure relaxation of

*M*using only measurements of

_{z}*M*and

_{x}*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
*T*?_{1}

*InversionRecovery.html version 2.0*

*© Copyright 2013, 2014, 2023 David N. Blauch*