# The Basics of nmr chapter 1 introduction

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## PULSE SEQUENCES

### Introduction

You have seen in Chapter 5 how a time domain signal can be converted into a frequency domain signal. In this chapter you will learn a few of the ways that a time domain signal can be created. Three methods are presented here, but there are an infinite number of possibilities. These methods are called pulse sequences. A pulse sequence is a set of RF pulses applied to a sample to produce a specific form of NMR signal.

### The 90-FID Sequence

In the 90-FID pulse sequence, net magnetization is rotated down into the X'Y' plane with a 90o pulse. The net magnetization vector begins to precess about the +Z axis. The magnitude of the vector also decays with time.

A timing diagram is a multiple axis plot of some aspect of a pulse sequence versus time. A timing diagram for a 90-FID pulse sequence has a plot of RF energy versus time and another for signal versus time.

When this sequence is repeated, for example when signal-to-noise improvement is needed, the amplitude of the signal after being Fourier transformed (S) will depend on T1 and the time between repetitions, called the repetition time (TR), of the sequence. In the signal equation below, k is a proportionality constant and is the density of spins in the sample.

S = k ( 1 - e-TR/T1 )

### The Spin-Echo Sequence

Another commonly used pulse sequence is the spin-echo pulse sequence. Here a 90o pulse is first applied to the spin system. The 90o degree pulse rotates the magnetization down into the X'Y' plane. The transverse magnetization begins to dephase. At some point in time after the 90o pulse, a 180o pulse is applied. This pulse rotates the magnetization by 180o about the X' axis. The 180o pulse causes the magnetization to at least partially rephase and to produce a signal called an echo.

A timing diagram shows the relative positions of the two radio frequency pulses and the signal.

The signal equation for a repeated spin echo sequence as a function of the repetition time, TR, and the echo time (TE) defined as the time between the 90o pulse and the maximum amplitude in the echo is

S = k ( 1 - e-TR/T1 ) e-TE/T2

### The Inversion Recovery Sequence

An inversion recovery pulse sequence can also be used to record an NMR spectrum. In this sequence, a 180o pulse is first applied. This rotates the net magnetization down to the -Z axis. The magnetization undergoes spin-lattice relaxation and returns toward its equilibrium position along the +Z axis. Before it reaches equilibrium, a 90o pulse is applied which rotates the longitudinal magnetization into the XY plane. In this example, the 90o pulse is applied shortly after the 180o pulse. Once magnetization is present in the XY plane it rotates about the Z axis and dephases giving a FID.

Once again, the timing diagram shows the relative positions of the two radio frequency pulses and the signal.

The signal as a function of TI when the sequence is not repeated is

S = k ( 1 - 2e-TI/T1 )

It should be noted at this time that the zero crossing of this function occurs for TI = T1 ln2.

## NMR HARDWARE

### Hardware Overview

The graphics window displays a schematic representation of the major systems of a nuclear magnetic resonance spectrometer and a few of the major interconnections. This overview briefly states the function of each component. Some will be described in detail later in this chapter.

At the top of the schematic representation, you will find the superconducting magnet of the NMR spectrometer. The magnet produces the Bo field necessary for the NMR experiments. Immediately within the bore of the magnet are the shim coils for homogenizing the Bo field. Within the shim coils is the probe. The probe contains the RF coils for producing the B1 magnetic field necessary to rotate the spins by 90o or 180o. The RF coil also detects the signal from the spins within the sample. The sample is positioned within the RF coil of the probe. Some probes also contain a set of gradient coils. These coils produce a gradient in Bo along the X, Y, or Z axis. Gradient coils are used for for gradient enhanced spectroscopy (See Chapter 11.), diffusion (See Chapter 11.), and NMR microscopy (See Chapter 11.) experiments.

The heart of the spectrometer is the computer. It controls all of the components of the spectrometer. The RF components under control of the computer are the RF frequency source and pulse programmer. The source produces a sine wave of the desired frequency. The pulse programmer sets the width, and in some cases the shape, of the RF pulses. The RF amplifier increases the pulses power from milli Watts to tens or hundreds of Watts. The computer also controls the gradient pulse programmer which sets the shape and amplitude of gradient fields. The gradient amplifier increases the power of the gradient pulses to a level sufficient to drive the gradient coils.

The operator of the spectrometer gives input to the computer through a console terminal with a mouse and keyboard. Some spectrometers also have a separate small interface for carrying out some of the more routine procedures on the spectrometer. A pulse sequence is selected and customized from the console terminal. The operator can see spectra on a video display located on the console and can make hard copies of spectra using a printer.

The next sections of this chapter go into more detail concerning the magnet, lock, shim coils, gradient coils, RF coils, and RF detector of nuclear magnetic resonance spectrometer.

### Magnet

The NMR magnet is one of the most expensive components of the nuclear magnetic resonance spectrometer system. Most magnets are of the superconducting type. This is a picture of a 7.0 Tesla superconducting magnet from an NMR spectrometer. A superconducting magnet has an electromagnet made of superconducting wire. Superconducting wire has a resistance approximately equal to zero when it is cooled to a temperature close to absolute zero (-273.15o C or 0 K) by emersing it in liquid helium. Once current is caused to flow in the coil it will continue to flow for as long as the coil is kept at liquid helium temperatures. (Some losses do occur over time due to the infinitesimally small resistance of the coil. These losses are on the order of a ppm of the main magnetic field per year.)

The length of superconducting wire in the magnet is typically several miles. This wire is wound into a multi-turn solenoid or coil. The coil of wire is kept at a temperature of 4.2K by immersing it in liquid helium. The coil and liquid helium are kept in a large dewar. This dewar is typically surrounded by a liquid nitrogen (77.4K) dewar, which acts as a thermal buffer between the room temperature air (293K) and the liquid helium. A cross sectional view of the superconducting magnet, depicting the concentric dewars, can be found in the animation window.

The following image is an actual cut-away view of a superconducting magnet. The magnet is supported by three legs, and the concentric nitrogen and helium dewars are supported by stacks coming out of the top of the magnet. A room temperature bore hole extends through the center of the assembly. The sample probe and shim coils are located within this bore hole. Also depicted in this picture is the liquid nitrogen level sensor, an electronic assembly for monitoring the liquid nitrogen level.

Going from the outside of the magnet to the inside, we see a vacuum region followed by a liquid nitrogen reservoir. The vacuum region is filled with several layers of a reflective mylar film. The function of the mylar is to reflect thermal photons, and thus diminish heat from entering the magnet. Within the inside wall of the liquid nitrogen reservoir, we see another vacuum filled with some reflective mylar. The liquid helium reservoir comes next. This reservoir houses the superconducting solenoid or coil of wire.

Taking a closer look at the solenoid it is clear to see the coil and the bore tube extending through the magnet.

### Field Lock

In order to produce a high resolution NMR spectrum of a sample, especially one which requires signal averaging or phase cycling, you need to have a temporally constant and spatially homogeneous magnetic field. Consistency of the Bo field over time will be discussed here; homogeneity will be discussed in the next section of this chapter. The field strength might vary over time due to aging of the magnet, movement of metal objects near the magnet, and temperature fluctuations. Here is an example of a one line NMR spectrum of cyclohexane recorded while the Bo magnetic field was drifting a very significant amount. The field lock can compensate for these variations.

The field lock is a separate NMR spectrometer within your spectrometer. This spectrometer is typically tuned to the deuterium NMR resonance frequency. It constantly monitors the resonance frequency of the deuterium signal and makes minor changes in the Bo magnetic field to keep the resonance frequency constant. The deuterium signal comes from the deuterium solvent used to prepare the sample. The animation window contains plots of the deuterium resonance lock frequency, the small additional magnetic field used to correct the lock frequency, and the resultant Bo field as a function of time while the magnetic field is drifting. The lock frequency plot displays the frequency without correction. In reality, this frequency would be kept constant by the application of the lock field which offsets the drift.

On most NMR spectrometers the deuterium lock serves a second function. It provides the =0 reference. The resonance frequency of the deuterium signal in many lock solvents is well known. Therefore the difference in resonance frequency of the lock solvent and TMS is also known. As a consequence, TMS does not need to be added to the sample to set =0; the spectrometer can use the lock frequency to calculate =0.

### Shim Coils

The purpose of shim coils on a spectrometer is to correct minor spatial inhomogeneities in the Bo magnetic field. These inhomogeneities could be caused by the magnet design, materials in the probe, variations in the thickness of the sample tube, sample permeability, and ferromagnetic materials around the magnet. A shim coil is designed to create a small magnetic field which will oppose and cancel out an inhomogeneity in the Bo magnetic field. Because these variations may exist in a variety of functional forms (linear, parabolic, etc.), shim coils are needed which can create a variety of opposing fields. Some of the functional forms are listed in the table below.

Shim Coil Functional Forms

 Shim Function Z0 Z Z2 Z3 Z4 Z5 X XZ XZ2 X2Y2 XY Y YZ YZ2 XZ3 X2Y2Z YZ3 XYZ X3 Y3

By passing the appropriate amount of current through each coil a homogeneous Bo magnetic field can be achieved. The optimum shim current settings are found by either minimizing the linewidth, maximizing the size of the FID, or maximizing the signal from the field lock. On most spectrometers, the shim coils are controllable by the computer. A computer algorithm has the task of finding the best shim value by maximizing the lock signal.

### Sample Probe

The sample probe is the name given to that part of the spectrometer which accepts the sample, sends RF energy into the sample, and detects the signal emanating from the sample. It contains the RF coil, sample spinner, temperature controlling circuitry, and gradient coils. The RF coil and gradient coils will be described in the next two sections. The sample spinner and temperature controlling circuitry will be described here.

The purpose of the sample spinner is to rotate the NMR sample tube about its axis. In doing so, each spin in the sample located at a given position along the Z axis and radius from the Z axis, will experience the average magnetic field in the circle defined by this Z and radius. The net effect is a narrower spectral linewidth. To appreciate this phenomenon, consider the following examples.

Picture an axial cross section of a cylindrical tube containing sample. In a very homogeneous Bo magnetic field this sample will yield a narrow spectrum. In a more inhomogeneous field the sample will yield a broader spectrum due to the presence of lines from the parts of the sample experiencing different Bo magnetic fields. When the sample is spun about its z-axis, inhomogeneities in the X and Y directions are averaged out and the NMR line width becomes narrower.

Many scientists need to examine properties of their samples as a function of temperature. As a result many instruments have the ability to maintain the temperature of the sample above and below room temperature. Air or nitrogen which has been warmed or cooled is passed over the sample to heat or cool the sample. The temperature at the sample is monitored with the aid of a thermocouple and electronic circuitry maintains the temperature by increasing or decreasing the temperature of the gas passing over the sample. More information on this topic will be presented in Chapter 8.

### RF Coils

RF coils create the B1 field which rotates the net magnetization in a pulse sequence. They also detect the transverse magnetization as it precesses in the XY plane. Most RF coils on NMR spectrometers are of the saddle coil design and act as the transmitter of the B1 field and receiver of RF energy from the sample. You may find one or more RF coils in a probe.

Each of these RF coils must resonate, that is they must efficiently store energy, at the Larmor frequency of the nucleus being examined with the NMR spectrometer. All NMR coils are composed of an inductor, or inductive elements, and a set of capacitive elements. The resonant frequency, , of an RF coil is determined by the inductance (L) and capacitance (C) of the inductor capacitor circuit.

RF coils used in NMR spectrometers need to be tuned for the specific sample being studied. An RF coil has a bandwidth or specific range of frequencies at which it resonates. When you place a sample in an RF coil, the conductivity and dielectric constant of the sample affect the resonance frequency. If this frequency is different from the resonance frequency of the nucleus you are studying, the coil will not efficiently set up the B1 field nor efficiently detect the signal from the sample. You will be rotating the net magnetization by an angle less than 90 degrees when you think you are rotating by 90 degrees. This will produce less transverse magnetization and less signal. Furthermore, because the coil will not be efficiently detecting the signal, your signal-to-noise ratio will be poor.

The B1 field of an RF coil must be perpendicular to the Bo magnetic field. Another requirement of an RF coil in an NMR spectrometer is that the B1 field needs to be homogeneous over the volume of your sample. If it is not, you will be rotating spins by a distribution of rotation angles and you will obtain strange spectra.

The gradient coils produce the gradients in the Bo magnetic field needed for performing gradient enhanced spectroscopy, diffusion measurements, and NMR microscopy. The gradient coils are located inside the RF probe. Not all probes have gradient coils, and not all NMR spectrometers have the hardware necessary to drive these coils.

The gradient coils are room temperature coils (i.e. do not require cooling with cryogens to operate) which, because of their configuration, create the desired gradient. Since the vertical bore superconducting magnet is most common, the gradient coil system will be described for this magnet.

Assuming the standard magnetic resonance coordinate system, a gradient in Bo in the Z direction is achieved with an antihelmholtz type of coil. Current in the two coils flow in opposite directions creating a magnetic field gradient between the two coils. The B field at the center of one coil adds to the Bo field, while the B field at the center of the other coil subtracts from the Bo field.

The X and Y gradients in the Bo field are created by a pair of figure-8 coils. The X axis figure-8 coils create a gradient in Bo in the X direction due to the direction of the current through the coils. The Y axis figure-8 coils provides a similar gradient in Bo along the Y axis.

The quadrature detector is a device which separates out the Mx' and My' signals from the signal from the RF coil. For this reason it can be thought of as a laboratory to rotating frame of reference converter. The heart of a quadrature detector is a device called a doubly balanced mixer. The doubly balanced mixer has two inputs and one output. If the input signals are Cos(A) and Cos(B), the output will be 1/2 Cos(A+B) and 1/2 Cos(A-B). For this reason the device is often called a product detector since the product of Cos(A) and Cos(B) is the output.

The quadrature detector typically contains two doubly balanced mixers, two filters, two amplifiers, and a 90o phase shifter. There are two inputs and two outputs on the device. Frequency and o are put in and the MX' and MY' components of the transverse magnetization come out. There are some potential problems which can occur with this device which will cause artifacts in the spectrum. One is called a DC offset artifact and the other is called a quadrature artifact.

### Digital Filtering

Many newer spectrometers employ a combination of oversampling, digital filtering, and decimation to eliminate the wrap around artifact. Oversampling creates a larger spectral or sweep width, but generates too much data to be conveniently stored. Digital filtering eliminates the high frequency components from the data, and decimation reduces the size of the data set. The following flowchart summarizes the effects of the three steps by showing the result of performing an FT after each step.

Let's examine oversampling, digital filtering, and decimation in more detail to see how this combination of steps can be used to reduce the wrap around problem.

Oversampling is the digitization of a time domain signal at a frequency much greater than necessary to record the desired spectral width. For example, if the sampling frequency, fs, is increased by a factor of 10, the sweep width will be 10 times greater, thus eliminating wraparound. Unfortunately digitizing at 10 times the speed also increases the amount of raw data by a factor of 10, thus increasing storage requirements and processing time.

Filtering is the removal of a select band of frequencies from a signal. For an example of filtering, consider the following frequency domain signal. Frequencies above fo could be removed from this frequency domain signal by multipling the signal by this rectangular function. In NMR, this step would be equivalent to taking a large sweep width spectrum and setting to zero intensity those spectral frequencies which are farther than some distance from the center of the spectrum.

Digital filtering is the removal of these frequencies using the time domain signal. Recall from Chapter 5 that if two functions are multiplied in one domain (i.e. frequency), we must convolve the FT of the two functions together in the other domain (i.e. time). To filter out frequencies above fo from the time domain signal, the signal must be convolved with the Fourier transform of the rectangular function, a sinc function. (See Chapter 5.) This process eliminates frequencies greater than fo from the time domain signal. Fourier transforming the resultant time domain signal yields a frequency domain signal without the higher frequencies. In NMR, this step will remove spectral components with frequencies greater than +fo and less than -fo.

Decimation is the elimination of data points from a data set. A decimation ratio of 4/5 means that 4 out of every 5 data points are deleted, or every fifth data point is saved. Decimating the digitally filtered data above, followed by a Fourier transform, will reduce the data set by a factor of five.

High speed digitizers, capable of digitizing at 2 MHz, and dedicated high speed integrated circuits, capable of performing the convolution on the time domain data as it is being recorded, are used to realize this procedure.

### Safety

There are some important safety considerations which one should be familiar with before using an NMR spectrometer. These concern the use of strong magnetic fields and cryogenic liquids.

Magnetic fields from high field magnets can literally pick up and pull ferromagnetic items into the bore of the magnet. Caution must be taken to keep all ferromagnetic items away from the magnet because they can seriously damage the magnet, shim coils, and probe. The force exerted on the concentric cryogenic dewars within a magnet by a large metal object stuck to the magnet can break dewars and magnet supports. The kinetic energy of an object being sucked into a magnet can smash a dewar or an electrical connector on a probe. Small ferromagnetic objects are just as much a concern as larger ones. A small metal sliver can get sucked into the bore of the magnet and destroy the homogeneity of the magnet achieved with a set of shim settings.

There are additional concerns regarding the effect of magnetic fields on electronic circuitry, specifically pacemakers. An individual with a pacemaker walking through a strong magnetic field can induce currents in the pacemaker circuitry which will cause it to fail and possibly cause death. A person with a pacemaker must not be able to inadvertently stray into a magnetic field of five or more Gauss. Although not as important as a pacemaker, mechanical watches and some digital watches will also be affected by magnetic fields. Magnetic fields of approximately 50 Gauss will erase credit cards and magnetic storage media.

The liquid nitrogen and liquid helium used in NMR spectrometers are at a temperature of 77.4 K and 4.2 K respectively. These liquids can cause frostbite, which is not a concern unless you are filling the magnet. If you are filling the magnet or if you are operating the spectrometer, suffocation is another concern you need to be aware of. If the magnet quenches, or suddenly stops being a superconductor, it will rapidly boil off all its cryogens, and the nitrogen and helium gasses in a confined space can cause suffocation.