Sexta-feira, 21 de Setembro de 2007

NMR

Nuclear magnetic resonance, or NMR as it is abbreviated by scientists, is a phenomenon which occurs when the nuclei of certain atoms are immersed in a static magnetic field and exposed to a second oscillating magnetic field. Some nuclei experience this phenomenon, and others do not, dependent upon whether they possess a property called spin. You will learn about spin and about the role of the magnetic fields in Chapter 2, but first let's review where the nucleus is.

Most of the matter you can examine with NMR is composed of molecules. Molecules are composed of atoms. Here are a few water molecules. Each water molecule has one oxygen and two hydrogen atoms. If we zoom into one of the hydrogens past the electron cloud we see a nucleus composed of a single proton. The proton possesses a property called spin which:

can be thought of as a small magnetic field, and

will cause the nucleus to produce an NMR signal.

Not all nuclei possess the property called spin. A list of these nuclei will be presented in Chapter 3 on spin physics.

Spectroscopy

Spectroscopy is the study of the interaction of electromagnetic radiation with matter. Nuclear magnetic resonance spectroscopy is the use of the NMR phenomenon to study physical, chemical, and biological properties of matter. As a consequence, NMR spectroscopy finds applications in several areas of science. NMR spectroscopy is routinely used by chemists to study chemical structure using simple one-dimensional techniques. Two-dimensional techniques are used to determine the structure of more complicated molecules. These techniques are replacing x-ray crystallography for the determination of protein structure. Time domain NMR spectroscopic techniques are used to probe molecular dynamics in solutions. Solid state NMR spectroscopy is used to determine the molecular structure of solids. Other scientists have developed NMR methods of measuring diffusion coefficients.

The versatility of NMR makes it pervasive in the sciences. Scientists and students are discovering that knowledge of the science and technology of NMR is essential for applying, as well as developing, new applications for it. Unfortunately many of the dynamic concepts of NMR spectroscopy are difficult for the novice to understand when static diagrams in hard copy texts are used. The chapters in this hypertext book on NMR are designed in such a way to incorporate both static and dynamic figures with hypertext. This book presents a comprehensive picture of the basic principles necessary to begin using NMR spectroscopy, and it will provide you with an understanding of the principles of NMR from the microscopic, macroscopic, and system perspectives.

Units Review

Before you can begin learning about NMR spectroscopy, you must be versed in the language of NMR. NMR scientists use a set of units when describing temperature, energy, frequency, etc. Please review these units before advancing to subsequent chapters in this text.

Units of time are seconds (s).

Angles are reported in degrees (o) and in radians (rad). There are 2 radians in 360o.

The absolute temperature scale in Kelvin (K) is used in NMR. The Kelvin temperature scale is equal to the Celsius scale reading plus 273.15. 0 K is characterized by the absence of molecular motion. There are no degrees in the Kelvin temperature unit.

Magnetic field strength (B) is measured in Tesla (T). The earth's magnetic field in Rochester, New York is approximately 5x10-5 T.

The unit of energy (E) is the Joule (J). In NMR one often depicts the relative energy of a particle using an energy level diagram.

The frequency of electromagnetic radiation may be reported in cycles per second or radians per second. Frequency in cycles per second (Hz) have units of inverse seconds (s-1) and are given the symbols or f. Frequencies represented in radians per second (rad/s) are given the symbol . Radians tend to be used more to describe periodic circular motions. The conversion between Hz and rad/s is easy to remember. There are 2 radians in a circle or cycle, therefore

2 rad/s = 1 Hz = 1 s-1.

Power is the energy consumed per time and has units of Watts (W).

Finally, it is common in science to use prefixes before units to indicate a power of ten. For example, 0.005 seconds can be written as 5x10-3 s or as 5 ms. The m implies 10-3. The animation window contains a table of prefixes for powers of ten.

In the next chapter you will be introduced to the mathematical beckground necessary to begin your study of NMR.

Exponential Functions

The number 2.71828183 occurs so often in calculations that it is given the symbol e. When e is raised to the power x, it is often written exp(x).

ex = exp(x) = 2.71828183x

Logarithms based on powers of e are called natural logarithms. If

x = ey

then

ln(x) = y,

Many of the dynamic NMR processes are exponential in nature. For example, signals decay exponentially as a function of time. It is therefore essential to understand the nature of exponential curves. Three common exponential functions are

y = e-x/t

y = (1 - e-x/t)

y = (1 - 2e-x/t)

where t is a constant.

Trigonometric Functions

The basic trigonometric functions sine and cosine describe sinusoidal functions which are 90o out of phase.

The trigonometric identities are used in geometric calculations.

Sin() = Opposite / Hypotenuse

Cos() = Adjacent / Hypotenuse

Tan() = Opposite / Adjacent

The function sin(x) / x occurs often and is called sinc(x).

Differentials and Integrals

A differential can be thought of as the slope of a function at any point. For the function

the differential of y with respect to x is

An integral is the area under a function between the limits of the integral.

An integral can also be considered a sumation; in fact most integration is performed by computers by adding up values of the function between the integral limits.

Vectors

A vector is a quantity having both a magnitude and a direction. The magnetization from nuclear spins is represented as a vector emanating from the origin of the coordinate system. Here it is along the +Z axis.

In this picture the vector is in the XY plane between the +X and +Y axes. The vector has X and Y components and a magnitude equal to

( X2 + Y2 )1/2

Matrices

A matrix is a set of numbers arranged in a rectangular array. This matrix has 3 rows and 4 columns and is said to be a 3 by 4 matrix.

To multiply matrices the number of columns in the first must equal the number of rows in the second. Click sequentially on the next start buttons to see the individual steps associated with the multiplication.

Coordinate Transformations

A coordinate transformation is used to convert the coordinates of a vector in one coordinate system (XY) to that in another coordinate system (X"Y").

Convolution

The convolution of two functions is the overlap of the two functions as one function is passed over the second. The convolution symbol is . The convolution of h(t) and g(t) is defined mathematically as

The above equation is depicted for rectangular shaped h(t) and g(t) functions in this animation.

Imaginary Numbers

Imaginary numbers are those which result from calculations involving the square root of -1. Imaginary numbers are symbolized by i.

A complex number is one which has a real (RE) and an imaginary (IM) part. The real and imaginary parts of a complex number are orthogonal.

Two useful relations between complex numbers and exponentials are

e+ix = cos(x) +isin(x)

and

e-ix = cos(x) -isin(x).

Fourier Transforms

The Fourier transform (FT) is a mathematical technique for converting time domain data to frequency domain data, and vice versa.

The Fourier transform will be explained in detail in Chapter 5.

Spin

What is spin? Spin is a fundamental property of nature like electrical charge or mass. Spin comes in multiples of 1/2 and can be + or -. Protons, electrons, and neutrons possess spin. Individual unpaired electrons, protons, and neutrons each possesses a spin of 1/2.

In the deuterium atom ( 2H ), with one unpaired electron, one unpaired proton, and one unpaired neutron, the total electronic spin = 1/2 and the total nuclear spin = 1.

Two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin. An example is helium. In nuclear magnetic resonance, it is unpaired nuclear spins that are of importance.

Properties of Spin

When placed in a magnetic field of strength B, a particle with a net spin can absorb a photon, of frequency . The frequency depends on the gyromagnetic ratio, of the particle.

= B

For hydrogen, = 42.58 MHz / T.

Nuclei with Spin

The shell model for the nucleus tells us that nucleons, just like electrons, fill orbitals. When the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, and 126, orbitals are filled. Because nucleons have spin, just like electrons do, their spin can pair up when the orbitals are being filled and cancel out. Almost every element in the periodic table has an isotope with a non zero nuclear spin. NMR can only be performed on isotopes whose natural abundance is high enough to be detected. Some of the nuclei routinely used in NMR are listed below.

Nuclei Unpaired Protons Unpaired Neutrons Net Spin (MHz/T)

1H 1 0 1/2 42.58

2H 1 1 1 6.54

31P 1 0 1/2 17.25

23Na 1 2 3/2 11.27

14N 1 1 1 3.08

13C 0 1 1/2 10.71

19F 1 0 1/2 40.08

Energy Levels

To understand how particles with spin behave in a magnetic field, consider a proton. This proton has the property called spin. Think of the spin of this proton as a magnetic moment vector, causing the proton to behave like a tiny magnet with a north and south pole.

When the proton is placed in an external magnetic field, the spin vector of the particle aligns itself with the external field, just like a magnet would. There is a low energy configuration or state where the poles are aligned N-S-N-S and a high energy state N-N-S-S.

Transitions

This particle can undergo a transition between the two energy states by the absorption of a photon. A particle in the lower energy state absorbs a photon and ends up in the upper energy state. The energy of this photon must exactly match the energy difference between the two states. The energy, E, of a photon is related to its frequency, , by Planck's constant (h = 6.626x10-34 J s).

E = h

In NMR and MRI, the quantity is called the resonance frequency and the Larmor frequency.

Energy Level Diagrams

The energy of the two spin states can be represented by an energy level diagram. We have seen that = B and E = h , therefore the energy of the photon needed to cause a transition between the two spin states is

E = h B

When the energy of the photon matches the energy difference between the two spin states an absorption of energy occurs.

In the NMR experiment, the frequency of the photon is in the radio frequency (RF) range. In NMR spectroscopy, is between 60 and 800 MHz for hydrogen nuclei. In clinical MRI, is typically between 15 and 80 MHz for hydrogen imaging.

CW NMR Experiment

The simplest NMR experiment is the continuous wave (CW) experiment. There are two ways of performing this experiment. In the first, a constant frequency, which is continuously on, probes the energy levels while the magnetic field is varied. The energy of this frequency is represented by the blue line in the energy level diagram.

The CW experiment can also be performed with a constant magnetic field and a frequency which is varied. The magnitude of the constant magnetic field is represented by the position of the vertical blue line in the energy level diagram.

Boltzmann Statistics

When a group of spins is placed in a magnetic field, each spin aligns in one of the two possible orientations.

At room temperature, the number of spins in the lower energy level, N+, slightly outnumbers the number in the upper level, N-. Boltzmann statistics tells us that

N-/N+ = e-E/kT.

E is the energy difference between the spin states; k is Boltzmann's constant, 1.3805x10-23 J/Kelvin; and T is the temperature in Kelvin.

As the temperature decreases, so does the ratio N- /N+. As the temperature increases, the ratio approaches one.

The signal in NMR spectroscopy results from the difference between the energy absorbed by the spins which make a transition from the lower energy state to the higher energy state, and the energy emitted by the spins which simultaneously make a transition from the higher energy state to the lower energy state. The signal is thus proportional to the population difference between the states. NMR is a rather sensitive spectroscopy since it is capable of detecting these very small population differences. It is the resonance, or exchange of energy at a specific frequency between the spins and the spectrometer, which gives NMR its sensitivity.

Spin Packets

It is cumbersome to describe NMR on a microscopic scale. A macroscopic picture is more convenient. The first step in developing the macroscopic picture is to define the spin packet. A spin packet is a group of spins experiencing the same magnetic field strength. In this example, the spins within each grid section represent a spin packet.

At any instant in time, the magnetic field due to the spins in each spin packet can be represented by a magnetization vector.

The size of each vector is proportional to (N+ - N-).

The vector sum of the magnetization vectors from all of the spin packets is the net magnetization. In order to describe pulsed NMR is necessary from here on to talk in terms of the net magnetization.

Adapting the conventional NMR coordinate system, the external magnetic field and the net magnetization vector at equilibrium are both along the Z axis.

T1 Processes

At equilibrium, the net magnetization vector lies along the direction of the applied magnetic field Bo and is called the equilibrium magnetization Mo. In this configuration, the Z component of magnetization MZ equals Mo. MZ is referred to as the longitudinal magnetization. There is no transverse (MX or MY) magnetization here.

It is possible to change the net magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states. If enough energy is put into the system, it is possible to saturate the spin system and make MZ=0.

The time constant which describes how MZ returns to its equilibrium value is called the spin lattice relaxation time (T1). The equation governing this behavior as a function of the time t after its displacement is:

Mz = Mo ( 1 - e-t/T1 )

T1 is therefore defined as the time required to change the Z component of magnetization by a factor of e.

If the net magnetization is placed along the -Z axis, it will gradually return to its equilibrium position along the +Z axis at a rate governed by T1. The equation governing this behavior as a function of the time t after its displacement is:

Mz = Mo ( 1 - 2e-t/T1 )

The spin-lattice relaxation time (T1) is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e.

Precession

If the net magnetization is placed in the XY plane it will rotate about the Z axis at a frequency equal to the frequency of the photon which would cause a transition between the two energy levels of the spin. This frequency is called the Larmor frequency.

T2 Processes

In addition to the rotation, the net magnetization starts to dephase because each of the spin packets making it up is experiencing a slightly different magnetic field and rotates at its own Larmor frequency. The longer the elapsed time, the greater the phase difference. Here the net magnetization vector is initially along +Y. For this and all dephasing examples think of this vector as the overlap of several thinner vectors from the individual spin packets.

The time constant which describes the return to equilibrium of the transverse magnetization, MXY, is called the spin-spin relaxation time, T2.

MXY =MXYo e-t/T2

T2 is always less than or equal to T1. The net magnetization in the XY plane goes to zero and then the longitudinal magnetization grows in until we have Mo along Z.

Any transverse magnetization behaves the same way. The transverse component rotates about the direction of applied magnetization and dephases. T1 governs the rate of recovery of the longitudinal magnetization.

In summary, the spin-spin relaxation time, T2, is the time to reduce the transverse magnetization by a factor of e. In the previous sequence, T2 and T1 processes are shown separately for clarity. That is, the magnetization vectors are shown filling the XY plane completely before growing back up along the Z axis. Actually, both processes occur simultaneously with the only restriction being that T2 is less than or equal to T1.

Two factors contribute to the decay of transverse magnetization.

1) molecular interactions (said to lead to a pure pure T2 molecular effect)

2) variations in Bo (said to lead to an inhomogeneous T2 effect

The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T2 star and is given the symbol T2*. The relationship between the T2 from molecular processes and that from inhomogeneities in the magnetic field is as follows.

1/T2* = 1/T2 + 1/T2inhomo.

Rotating Frame of Reference

We have just looked at the behavior of spins in the laboratory frame of reference. It is convenient to define a rotating frame of reference which rotates about the Z axis at the Larmor frequency. We distinguish this rotating coordinate system from the laboratory system by primes on the X and Y axes, X'Y'.

A magnetization vector rotating at the Larmor frequency in the laboratory frame appears stationary in a frame of reference rotating about the Z axis. In the rotating frame, relaxation of MZ magnetization to its equilibrium value looks the same as it did in the laboratory frame.

A transverse magnetization vector rotating about the Z axis at the same velocity as the rotating frame will appear stationary in the rotating frame. A magnetization vector traveling faster than the rotating frame rotates clockwise about the Z axis. A magnetization vector traveling slower than the rotating frame rotates counter-clockwise about the Z axis .

In a sample there are spin packets traveling faster and slower than the rotating frame. As a consequence, when the mean frequency of the sample is equal to the rotating frame, the dephasing of MX'Y' looks like this.

Pulsed Magnetic Fields

A coil of wire placed around the X axis will provide a magnetic field along the X axis when a direct current is passed through the coil. An alternating current will produce a magnetic field which alternates in direction.

In a frame of reference rotating about the Z axis at a frequency equal to that of the alternating current, the magnetic field along the X' axis will be constant, just as in the direct current case in the laboratory frame.

This is the same as moving the coil about the rotating frame coordinate system at the Larmor Frequency. In magnetic resonance, the magnetic field created by the coil passing an alternating current at the Larmor frequency is called the B1 magnetic field. When the alternating current through the coil is turned on and off, it creates a pulsed B1 magnetic field along the X' axis.

The spins respond to this pulse in such a way as to cause the net magnetization vector to rotate about the direction of the applied B1 field. The rotation angle depends on the length of time the field is on, , and its magnitude B1.

= 2 B1.

In our examples, will be assumed to be much smaller than T1 and T2.

A 90o pulse is one which rotates the magnetization vector clockwise by 90 degrees about the X' axis. A 90o pulse rotates the equilibrium magnetization down to the Y' axis. In the laboratory frame the equilibrium magnetization spirals down around the Z axis to the XY plane. You can see why the rotating frame of reference is helpful in describing the behavior of magnetization in response to a pulsed magnetic field.

A 180o pulse will rotate the magnetization vector by 180 degrees. A 180o pulse rotates the equilibrium magnetization down to along the -Z axis.

The net magnetization at any orientation will behave according to the rotation equation. For example, a net magnetization vector along the Y' axis will end up along the -Y' axis when acted upon by a 180o pulse of B1 along the X' axis.

A net magnetization vector between X' and Y' will end up between X' and Y' after the application of a 180o pulse of B1 applied along the X' axis.

A rotation matrix (described as a coordinate transformation in #2.6 Chapter 2) can also be used to predict the result of a rotation. Here is the rotation angle about the X' axis, [X', Y', Z] is the initial location of the vector, and [X", Y", Z"] the location of the vector after the rotation.

Spin Relaxation

Motions in solution which result in time varying magnetic fields cause spin relaxation.

Time varying fields at the Larmor frequency cause transitions between the spin states and hence a change in MZ. This screen depicts the field at the green hydrogen on the water molecule as it rotates about the external field Bo and a magnetic field from the blue hydrogen. Note that the field experienced at the green hydrogen is sinusoidal.

There is a distribution of rotation frequencies in a sample of molecules. Only frequencies at the Larmor frequency affect T1. Since the Larmor frequency is proportional to Bo, T1 will therefore vary as a function of magnetic field strength. In general, T1 is inversely proportional to the density of molecular motions at the Larmor frequency.

The rotation frequency distribution depends on the temperature and viscosity of the solution. Therefore T1 will vary as a function of temperature. At the Larmor frequency indicated by o, T1 (280 K ) < T1 (340 K). The temperature of the human body does not vary by enough to cause a significant influence on T1. The viscosity does however vary significantly from tissue to tissue and influences T1 as is seen in the following molecular motion plot.

Fluctuating fields which perturb the energy levels of the spin states cause the transverse magnetization to dephase. This can be seen by examining the plot of Bo experienced by the red hydrogens on the following water molecule. The number of molecular motions less than and equal to the Larmor frequency is inversely proportional to T2.

In general, relaxation times get longer as Bo increases because there are fewer relaxation-causing frequency components present in the random motions of the molecules.

Spin Exchange

Spin exchange is the exchange of spin state between two spins. For example, if we have two spins, A and B, and A is spin up and B is spin down, spin exchange between A and B can be represented with the following equation.

A() + B() A() + B()

The bidirectional arrow indicates that the exchange reaction is reversible.

The energy difference between the upper and lower energy states of A and of B must be the same for spin exchange to occur. On a microscopic scale, the spin in the upper energy state (B) is emitting a photon which is being absorbed by the spin in the lower energy state (A). Therefore, B ends up in the lower energy state and A in the upper state.

Spin exchange will not affect T1 but will affect T2. T1 is not effected because the distribution of spins between the upper and lower states is not changed. T2 will be affected because phase coherence of the transverse magnetization is lost during exchange.

Another form of exchange is called chemical exchange. In chemical exchange, the A and B nuclei are from different molecules. Consider the chemical exchange between water and ethanol.

CH3CH2OHA + HOHB CH3CH2OHB + HOHA

Here the B hydrogen of water ends up on ethanol, and the A hydrogen on ethanol ends up on water in the forward reaction. There are four senarios for the nuclear spin, represented by the four equations.

Chemical exchange will affect both T1 and T2. T1 is now affected because energy is transferred from one nucleus to another. For example, if there are more nuclei in the upper state of A, and a normal Boltzmann distribution in B, exchange will force the excess energy from A into B. The effect will make T1 appear smaller. T2 is effected because phase coherence of the transverse magnetization is not preserved during chemical exchange.

Bloch Equations

The Bloch equations are a set of coupled differential equations which can be used to describe the behavior of a magnetizatiion vector under any conditions. When properly integrated, the Bloch equations will yield the X', Y', and Z components of magnetization as a function of time.

Chemical Shift

Chemical Shift

When an atom is placed in a magnetic field, its electrons circulate about the direction of the applied magnetic field. This circulation causes a small magnetic field at the nucleus which opposes the externally applied field.

The magnetic field at the nucleus (the effective field) is therefore generally less than the applied field by a fraction .

B = Bo (1-s)

In some cases, such as the benzene molecule, the circulation of the electrons in the aromatic orbitals creates a magnetic field at the hydrogen nuclei which enhances the Bo field. This phenomenon is called deshielding. In this example, the Bo field is applied perpendicular to the plane of the molecule. The ring current is traveling clockwise if you look down at the plane.

The electron density around each nucleus in a molecule varies according to the types of nuclei and bonds in the molecule. The opposing field and therefore the effective field at each nucleus will vary. This is called the chemical shift phenomenon.

Consider the methanol molecule. The resonance frequency of two types of nuclei in this example differ. This difference will depend on the strength of the magnetic field, Bo, used to perform the NMR spectroscopy. The greater the value of Bo, the greater the frequency difference. This relationship could make it difficult to compare NMR spectra taken on spectrometers operating at different field strengths. The term chemical shift was developed to avoid this problem.

The chemical shift of a nucleus is the difference between the resonance frequency of the nucleus and a standard, relative to the standard. This quantity is reported in ppm and given the symbol delta, .

d = (n - nREF) x106 / nREF

In NMR spectroscopy, this standard is often tetramethylsilane, Si(CH3)4, abbreviated TMS. The chemical shift is a very precise metric of the chemical environment around a nucleus. For example, the hydrogen chemical shift of a CH2 hydrogen next to a Cl will be different than that of a CH3 next to the same Cl. It is therefore difficult to give a detailed list of chemical shifts in a limited space. The animation window displays a chart of selected hydrogen chemical shifts of pure liquids and some gasses.

The magnitude of the screening depends on the atom. For example, carbon-13 chemical shifts are much greater than hydrogen-1 chemical shifts. The following tables present a few selected chemical shifts of fluorine-19 containing compounds, carbon-13 containing compounds, nitrogen-14 containing compounds, and phosphorous-31 containing compounds. These shifts are all relative to the bare nucleus. The reader is directed to a more comprehensive list of chemical shifts for use in spectral interpretation.

Spin-Spin Coupling

Nuclei experiencing the same chemical environment or chemical shift are called equivalent. Those nuclei experiencing different environment or having different chemical shifts are nonequivalent. Nuclei which are close to one another exert an influence on each other's effective magnetic field. This effect shows up in the NMR spectrum when the nuclei are nonequivalent. If the distance between non-equivalent nuclei is less than or equal to three bond lengths, this effect is observable. This effect is called spin-spin coupling or J coupling.

Consider the following example. There are two nuclei, A and B, three bonds away from one another in a molecule. The spin of each nucleus can be either aligned with the external field such that the fields are N-S-N-S, called spin up , or opposed to the external field such that the fields are N-N-S-S, called spin down . The magnetic field at nucleus A will be either greater than Bo or less than Bo by a constant amount due to the influence of nucleus B.

There are a total of four possible configurations for the two nuclei in a magnetic field. Arranging these configurations in order of increasing energy gives the following arrangement. The vertical lines in this diagram represent the allowed transitions between energy levels. In NMR, an allowed transition is one where the spin of one nucleus changes from spin up to spin down , or spin down to spin up . Absorptions of energy where two or more nuclei change spin at the same time are not allowed. There are two absorption frequencies for the A nucleus and two for the B nucleus represented by the vertical lines between the energy levels in this diagram.

The NMR spectrum for nuclei A and B reflects the splittings observed in the energy level diagram. The A absorption line is split into 2 absorption lines centered on A, and the B absorption line is split into 2 lines centered on B. The distance between two split absorption lines is called the J coupling constant or the spin-spin splitting constant and is a measure of the magnetic interaction between two nuclei.

For the next example, consider a molecule with three spin 1/2 nuclei, one type A and two type B. The type B nuclei are both three bonds away from the type A nucleus. The magnetic field at the A nucleus has three possible values due to four possible spin configurations of the two B nuclei. The magnetic field at a B nucleus has two possible values.

The energy level diagram for this molecule has six states or levels because there are two sets of levels with the same energy. Energy levels with the same energy are said to be degenerate. The vertical lines represent the allowed transitions or absorptions of energy. Note that there are two lines drawn between some levels because of the degeneracy of those levels.

The resultant NMR spectrum is depicted in the animation window. Note that the center absorption line of those centered at A is twice as high as the either of the outer two. This is because there were twice as many transitions in the energy level diagram for this transition. The peaks at B are taller because there are twice as many B type spins than A type spins.

The complexity of the splitting pattern in a spectrum increases as the number of B nuclei increases. The following table contains a few examples.

Configuration Peak Ratios

A 1

AB 1:1

AB2 1:2:1

AB3 1:3:3:1

AB4 1:4:6:4:1

AB5 1:5:10:10:5:1

AB6 1:6:15:20:15:6:1

This series is called Pascal's triangle and can be calculated from the coefficients of the expansion of the equation

(x+1)n

where n is the number of B nuclei in the above table.

When there are two different types of nuclei three bonds away there will be two values of J, one for each pair of nuclei. By now you get the idea of the number of possible configurations and the energy level diagram for these configurations, so we can skip to the spectrum. In the following example JAB is greater JBC.

The Time Domain NMR Signal

An NMR sample may contain many different magnetization components, each with its own Larmor frequency. These magnetization components are associated with the nuclear spin configurations joined by an allowed transition line in the energy level diagram. Based on the number of allowed absorptions due to chemical shifts and spin-spin couplings of the different nuclei in a molecule, an NMR spectrum may contain many different frequency lines.

In pulsed NMR spectroscopy, signal is detected after these magnetization vectors are rotated into the XY plane. Once a magnetization vector is in the XY plane it rotates about the direction of the Bo field, the +Z axis. As transverse magnetization rotates about the Z axis, it will induce a current in a coil of wire located around the X axis. Plotting current as a function of time gives a sine wave. This wave will, of course, decay with time constant T2* due to dephasing of the spin packets. This signal is called a free induction decay (FID). We will see in Chapter 5 how the FID is converted into a frequency domain spectrum. You will see in Chapter 6 what sequence of events will produce a time domain signal.

The +/- Frequency Convention

Transverse magnetization vectors rotating faster than the rotating frame of reference are said to be rotating at a positive frequency relatve to the rotating frame (+n). Vectors rotating slower than the rotating frame are said to be rotating at a negative frequency relative to the rotating frame (-n).

It is worthwhile noting here that in most NMR spectra, the resonance frequency of a nucleus, as well as the magnetic field experienced by the nucleus and the chemical shift of a nucleus, increase from right to left. The frequency plots used in this hypertext book to describe Fourier transforms will use the more conventional mathematical axis of frequency increasing from left to right.

Introduction

A detailed description of the Fourier transform ( FT ) has waited until now, when you have a better appreciation of why it is needed. A Fourier transform is an operation which converts functions from time to frequency domains. An inverse Fourier transform ( IFT ) converts from the frequency domain to the time domain.

Recall from Chapter 2 that the Fourier transform is a mathematical technique for converting time domain data to frequency domain data, and vice versa.

The + and - Frequency Problem

To begin our detailed description of the FT consider the following. A magnetization vector, starting at +x, is rotating about the Z axis in a clockwise direction. The plot of Mx as a function of time is a cosine wave. Fourier transforming this gives peaks at both + and - because the FT can not distinguish between a + and a - rotation of the vector from the data supplied.

A plot of My as a function of time is a -sine function. Fourier transforming this gives peaks at + and - because the FT can not distinguish between a positive vector rotating at + and a negative vector rotating at - from the data supplied.

The solution is to input both the Mx and My into the FT. The FT is designed to handle two orthogonal input functions called the real and imaginary components.

Detecting just the Mx or My component for input into the FT is called linear detection. This was the detection scheme on many older NMR spectrometers and some magnetic resonance imagers. It required the computer to discard half of the frequency domain data.

Detection of both Mx and My is called quadrature detection and is the method of detection on modern spectrometers and imagers. It is the method of choice since now the FT can distinguish between + and -, and all of the frequency domain data be used.

The Fourier Transform

An FT is defined by the integral

Think of f() as the overlap of f(t) with a wave of frequency .

This is easy to picture by looking at the real part of f() only.

Consider the function of time, f( t ) = cos( 4t ) + cos( 9t ).

To understand the FT, examine the product of f(t) with cos(t) for values between 1 and 10, and then the summation of the values of this product between 1 and 10 seconds. The summation will only be examined for time values between 0 and 10 seconds.

=1

=2

=3

=4

=5

=6

=7

=8

=9

=10

f()

The inverse Fourier transform (IFT) is best depicted as an summation of the time domain spectra of frequencies in f().

Phase Correction

The actual FT will make use of an input consisting of a REAL and an IMAGINARY part. You can think of Mx as the REAL input, and My as the IMAGINARY input. The resultant output of the FT will therefore have a REAL and an IMAGINARY component, too.

Consider the following function:

f(t) = e-at e-i2t

In FT NMR spectroscopy, the real output of the FT is taken as the frequency domain spectrum. To see an esthetically pleasing (absorption) frequency domain spectrum, we want to input a cosine function into the real part and a sine function into the imaginary parts of the FT. This is what happens if the cosine part is input as the imaginary and the sine as the real.

To obtain an absorption spectrum as the real output of the FT, a phase correction must be applied to either the time or frequency domain spectra. This process is equivalent to the coordinate transformation described in Chapter 2

If the above mentioned FID is recorded such that there is a 45o phase shift in the real and imaginary FIDs, the coordinate transformation matrix can be used with = - 45o. The corrected FIDs look like a cosine function in the real and a sine in the imaginary.

Fourier transforming the phase corrected FIDs gives an absorption spectrum for the real output of the FT. This correction can be done in the frequency domain as well as in the time domain.

NMR spectra require both constant and linear corrections to the phasing of the Fourier transformed signal.

= m + b

Constant phase corrections, b, arise from the inability of the spectrometer to detect the exact Mx and My. Linear phase corrections, m, arise from the inability of the spectrometer to detect transverse magnetization starting immediately after the RF pulse.

In magnetic resonance, the Mx or My signals are displayed. A magnitude signal might occasionally be used in some applications. The magnitude signal is equal to the square root of the sum of the squares of Mx and My.

Fourier Pairs

To better understand FT NMR functions, you need to know some common Fourier pairs. A Fourier pair is two functions, the frequency domain form and the corresponding time domain form. Here are a few Fourier pairs which are useful in NMR. The amplitude of the Fourier pairs has been neglected since it is not relevant in NMR.

Constant value at all time

Real: cos(2t), Imaginary: -sin(2t)

Comb Function (A series of delta functions separated by T.)

Exponential Decay: e-at for t > 0.

A square pulse starting at 0 that is T seconds long.

Gaussian: exp(-at2)

Convolution Theorem

To the magnetic resonance scientist, the most important theorem concerning Fourier transforms is the convolution theorem. The convolution theorem says that the FT of a convolution of two functions is proportional to the products of the individual Fourier transforms, and vice versa.

If f() = FT( f(t) ) and g() = FT( g(t) )

then f() g() = FT( g(t) f(t) ) and f() g() = FT( g(t) f(t) )

It will be easier to see this with pictures. In the animation window we are trying to find the FT of a sine wave which is turned on and off. The convolution theorem tells us that this is a sinc function at the frequency of the sine wave.

Another application of the convolution theorem is in noise reduction. With the convolution theorem it can be seen that the convolution of an NMR spectrum with a Lorentzian function is the same as the Fourier Transform of multiplying the time domain signal by an exponentially decaying function.

The Digital FT

In a nuclear magnetic resonance spectrometer, the computer does not see a continuous FID, but rather an FID which is sampled at a constant interval. Each data point making up the FID will have discrete amplitude and time values. Therefore, the computer needs to take the FT of a series of delta functions which vary in intensity.

Sampling Error

The wrap around problem or artifact in a nuclear magnetic resonance spectrum is the appearance of one side of the spectrum on the opposite side. In terms of a one dimensional frequency domain spectrum, wrap around is the occurrence of a low frequency peak which occurs on the high frequency side of the spectrum.

The convolution theorem can explain why this problem results from sampling the transverse magnetization at too slow a rate. First, observe what the FT of a correctly sampled FID looks like. With quadrature detection, the spectral width is equal to the inverse of the sampling frequency, or the width of the green box in the animation window.

When the sampling frequency is less than the spectral width, wrap around occurs.

The Two-Dimensional FT

The two-dimensional Fourier transform (2-DFT) is an FT performed on a two dimensional array of data.

Consider the two-dimensional array of data depicted in the animation window. This data has a t' and a t" dimension. A FT is first performed on the data in one dimension and then in the second. The first set of Fourier transforms are performed in the t' dimension to yield an f' by t" set of data. The second set of Fourier transforms is performed in the t" dimension to yield an f' by f" set of data.

The 2-DFT is required to perform state-of-the-art MRI. In MRI, data is collected in the equivalent of the t' and t" dimensions, called k-space. This raw data is Fourier transformed to yield the image which is the equivalent of the f' by f" data described above.

For more topics see: http://www.cis.rit.edu/htbooks/nmr/inside.htm

Nuclear magnetic resonance, or NMR as it is abbreviated by scientists, is a phenomenon which occurs when the nuclei of certain atoms are immersed in a static magnetic field and exposed to a second oscillating magnetic field. Some nuclei experience this phenomenon, and others do not, dependent upon whether they possess a property called spin. You will learn about spin and about the role of the magnetic fields in Chapter 2, but first let's review where the nucleus is.

Most of the matter you can examine with NMR is composed of molecules. Molecules are composed of atoms. Here are a few water molecules. Each water molecule has one oxygen and two hydrogen atoms. If we zoom into one of the hydrogens past the electron cloud we see a nucleus composed of a single proton. The proton possesses a property called spin which:

can be thought of as a small magnetic field, and

will cause the nucleus to produce an NMR signal.

Not all nuclei possess the property called spin. A list of these nuclei will be presented in Chapter 3 on spin physics.

Spectroscopy

Spectroscopy is the study of the interaction of electromagnetic radiation with matter. Nuclear magnetic resonance spectroscopy is the use of the NMR phenomenon to study physical, chemical, and biological properties of matter. As a consequence, NMR spectroscopy finds applications in several areas of science. NMR spectroscopy is routinely used by chemists to study chemical structure using simple one-dimensional techniques. Two-dimensional techniques are used to determine the structure of more complicated molecules. These techniques are replacing x-ray crystallography for the determination of protein structure. Time domain NMR spectroscopic techniques are used to probe molecular dynamics in solutions. Solid state NMR spectroscopy is used to determine the molecular structure of solids. Other scientists have developed NMR methods of measuring diffusion coefficients.

The versatility of NMR makes it pervasive in the sciences. Scientists and students are discovering that knowledge of the science and technology of NMR is essential for applying, as well as developing, new applications for it. Unfortunately many of the dynamic concepts of NMR spectroscopy are difficult for the novice to understand when static diagrams in hard copy texts are used. The chapters in this hypertext book on NMR are designed in such a way to incorporate both static and dynamic figures with hypertext. This book presents a comprehensive picture of the basic principles necessary to begin using NMR spectroscopy, and it will provide you with an understanding of the principles of NMR from the microscopic, macroscopic, and system perspectives.

Units Review

Before you can begin learning about NMR spectroscopy, you must be versed in the language of NMR. NMR scientists use a set of units when describing temperature, energy, frequency, etc. Please review these units before advancing to subsequent chapters in this text.

Units of time are seconds (s).

Angles are reported in degrees (o) and in radians (rad). There are 2 radians in 360o.

The absolute temperature scale in Kelvin (K) is used in NMR. The Kelvin temperature scale is equal to the Celsius scale reading plus 273.15. 0 K is characterized by the absence of molecular motion. There are no degrees in the Kelvin temperature unit.

Magnetic field strength (B) is measured in Tesla (T). The earth's magnetic field in Rochester, New York is approximately 5x10-5 T.

The unit of energy (E) is the Joule (J). In NMR one often depicts the relative energy of a particle using an energy level diagram.

The frequency of electromagnetic radiation may be reported in cycles per second or radians per second. Frequency in cycles per second (Hz) have units of inverse seconds (s-1) and are given the symbols or f. Frequencies represented in radians per second (rad/s) are given the symbol . Radians tend to be used more to describe periodic circular motions. The conversion between Hz and rad/s is easy to remember. There are 2 radians in a circle or cycle, therefore

2 rad/s = 1 Hz = 1 s-1.

Power is the energy consumed per time and has units of Watts (W).

Finally, it is common in science to use prefixes before units to indicate a power of ten. For example, 0.005 seconds can be written as 5x10-3 s or as 5 ms. The m implies 10-3. The animation window contains a table of prefixes for powers of ten.

In the next chapter you will be introduced to the mathematical beckground necessary to begin your study of NMR.

Exponential Functions

The number 2.71828183 occurs so often in calculations that it is given the symbol e. When e is raised to the power x, it is often written exp(x).

ex = exp(x) = 2.71828183x

Logarithms based on powers of e are called natural logarithms. If

x = ey

then

ln(x) = y,

Many of the dynamic NMR processes are exponential in nature. For example, signals decay exponentially as a function of time. It is therefore essential to understand the nature of exponential curves. Three common exponential functions are

y = e-x/t

y = (1 - e-x/t)

y = (1 - 2e-x/t)

where t is a constant.

Trigonometric Functions

The basic trigonometric functions sine and cosine describe sinusoidal functions which are 90o out of phase.

The trigonometric identities are used in geometric calculations.

Sin() = Opposite / Hypotenuse

Cos() = Adjacent / Hypotenuse

Tan() = Opposite / Adjacent

The function sin(x) / x occurs often and is called sinc(x).

Differentials and Integrals

A differential can be thought of as the slope of a function at any point. For the function

the differential of y with respect to x is

An integral is the area under a function between the limits of the integral.

An integral can also be considered a sumation; in fact most integration is performed by computers by adding up values of the function between the integral limits.

Vectors

A vector is a quantity having both a magnitude and a direction. The magnetization from nuclear spins is represented as a vector emanating from the origin of the coordinate system. Here it is along the +Z axis.

In this picture the vector is in the XY plane between the +X and +Y axes. The vector has X and Y components and a magnitude equal to

( X2 + Y2 )1/2

Matrices

A matrix is a set of numbers arranged in a rectangular array. This matrix has 3 rows and 4 columns and is said to be a 3 by 4 matrix.

To multiply matrices the number of columns in the first must equal the number of rows in the second. Click sequentially on the next start buttons to see the individual steps associated with the multiplication.

Coordinate Transformations

A coordinate transformation is used to convert the coordinates of a vector in one coordinate system (XY) to that in another coordinate system (X"Y").

Convolution

The convolution of two functions is the overlap of the two functions as one function is passed over the second. The convolution symbol is . The convolution of h(t) and g(t) is defined mathematically as

The above equation is depicted for rectangular shaped h(t) and g(t) functions in this animation.

Imaginary Numbers

Imaginary numbers are those which result from calculations involving the square root of -1. Imaginary numbers are symbolized by i.

A complex number is one which has a real (RE) and an imaginary (IM) part. The real and imaginary parts of a complex number are orthogonal.

Two useful relations between complex numbers and exponentials are

e+ix = cos(x) +isin(x)

and

e-ix = cos(x) -isin(x).

Fourier Transforms

The Fourier transform (FT) is a mathematical technique for converting time domain data to frequency domain data, and vice versa.

The Fourier transform will be explained in detail in Chapter 5.

Spin

What is spin? Spin is a fundamental property of nature like electrical charge or mass. Spin comes in multiples of 1/2 and can be + or -. Protons, electrons, and neutrons possess spin. Individual unpaired electrons, protons, and neutrons each possesses a spin of 1/2.

In the deuterium atom ( 2H ), with one unpaired electron, one unpaired proton, and one unpaired neutron, the total electronic spin = 1/2 and the total nuclear spin = 1.

Two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin. An example is helium. In nuclear magnetic resonance, it is unpaired nuclear spins that are of importance.

Properties of Spin

When placed in a magnetic field of strength B, a particle with a net spin can absorb a photon, of frequency . The frequency depends on the gyromagnetic ratio, of the particle.

= B

For hydrogen, = 42.58 MHz / T.

Nuclei with Spin

The shell model for the nucleus tells us that nucleons, just like electrons, fill orbitals. When the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, and 126, orbitals are filled. Because nucleons have spin, just like electrons do, their spin can pair up when the orbitals are being filled and cancel out. Almost every element in the periodic table has an isotope with a non zero nuclear spin. NMR can only be performed on isotopes whose natural abundance is high enough to be detected. Some of the nuclei routinely used in NMR are listed below.

Nuclei Unpaired Protons Unpaired Neutrons Net Spin (MHz/T)

1H 1 0 1/2 42.58

2H 1 1 1 6.54

31P 1 0 1/2 17.25

23Na 1 2 3/2 11.27

14N 1 1 1 3.08

13C 0 1 1/2 10.71

19F 1 0 1/2 40.08

Energy Levels

To understand how particles with spin behave in a magnetic field, consider a proton. This proton has the property called spin. Think of the spin of this proton as a magnetic moment vector, causing the proton to behave like a tiny magnet with a north and south pole.

When the proton is placed in an external magnetic field, the spin vector of the particle aligns itself with the external field, just like a magnet would. There is a low energy configuration or state where the poles are aligned N-S-N-S and a high energy state N-N-S-S.

Transitions

This particle can undergo a transition between the two energy states by the absorption of a photon. A particle in the lower energy state absorbs a photon and ends up in the upper energy state. The energy of this photon must exactly match the energy difference between the two states. The energy, E, of a photon is related to its frequency, , by Planck's constant (h = 6.626x10-34 J s).

E = h

In NMR and MRI, the quantity is called the resonance frequency and the Larmor frequency.

Energy Level Diagrams

The energy of the two spin states can be represented by an energy level diagram. We have seen that = B and E = h , therefore the energy of the photon needed to cause a transition between the two spin states is

E = h B

When the energy of the photon matches the energy difference between the two spin states an absorption of energy occurs.

In the NMR experiment, the frequency of the photon is in the radio frequency (RF) range. In NMR spectroscopy, is between 60 and 800 MHz for hydrogen nuclei. In clinical MRI, is typically between 15 and 80 MHz for hydrogen imaging.

CW NMR Experiment

The simplest NMR experiment is the continuous wave (CW) experiment. There are two ways of performing this experiment. In the first, a constant frequency, which is continuously on, probes the energy levels while the magnetic field is varied. The energy of this frequency is represented by the blue line in the energy level diagram.

The CW experiment can also be performed with a constant magnetic field and a frequency which is varied. The magnitude of the constant magnetic field is represented by the position of the vertical blue line in the energy level diagram.

Boltzmann Statistics

When a group of spins is placed in a magnetic field, each spin aligns in one of the two possible orientations.

At room temperature, the number of spins in the lower energy level, N+, slightly outnumbers the number in the upper level, N-. Boltzmann statistics tells us that

N-/N+ = e-E/kT.

E is the energy difference between the spin states; k is Boltzmann's constant, 1.3805x10-23 J/Kelvin; and T is the temperature in Kelvin.

As the temperature decreases, so does the ratio N- /N+. As the temperature increases, the ratio approaches one.

The signal in NMR spectroscopy results from the difference between the energy absorbed by the spins which make a transition from the lower energy state to the higher energy state, and the energy emitted by the spins which simultaneously make a transition from the higher energy state to the lower energy state. The signal is thus proportional to the population difference between the states. NMR is a rather sensitive spectroscopy since it is capable of detecting these very small population differences. It is the resonance, or exchange of energy at a specific frequency between the spins and the spectrometer, which gives NMR its sensitivity.

Spin Packets

It is cumbersome to describe NMR on a microscopic scale. A macroscopic picture is more convenient. The first step in developing the macroscopic picture is to define the spin packet. A spin packet is a group of spins experiencing the same magnetic field strength. In this example, the spins within each grid section represent a spin packet.

At any instant in time, the magnetic field due to the spins in each spin packet can be represented by a magnetization vector.

The size of each vector is proportional to (N+ - N-).

The vector sum of the magnetization vectors from all of the spin packets is the net magnetization. In order to describe pulsed NMR is necessary from here on to talk in terms of the net magnetization.

Adapting the conventional NMR coordinate system, the external magnetic field and the net magnetization vector at equilibrium are both along the Z axis.

T1 Processes

At equilibrium, the net magnetization vector lies along the direction of the applied magnetic field Bo and is called the equilibrium magnetization Mo. In this configuration, the Z component of magnetization MZ equals Mo. MZ is referred to as the longitudinal magnetization. There is no transverse (MX or MY) magnetization here.

It is possible to change the net magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states. If enough energy is put into the system, it is possible to saturate the spin system and make MZ=0.

The time constant which describes how MZ returns to its equilibrium value is called the spin lattice relaxation time (T1). The equation governing this behavior as a function of the time t after its displacement is:

Mz = Mo ( 1 - e-t/T1 )

T1 is therefore defined as the time required to change the Z component of magnetization by a factor of e.

If the net magnetization is placed along the -Z axis, it will gradually return to its equilibrium position along the +Z axis at a rate governed by T1. The equation governing this behavior as a function of the time t after its displacement is:

Mz = Mo ( 1 - 2e-t/T1 )

The spin-lattice relaxation time (T1) is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e.

Precession

If the net magnetization is placed in the XY plane it will rotate about the Z axis at a frequency equal to the frequency of the photon which would cause a transition between the two energy levels of the spin. This frequency is called the Larmor frequency.

T2 Processes

In addition to the rotation, the net magnetization starts to dephase because each of the spin packets making it up is experiencing a slightly different magnetic field and rotates at its own Larmor frequency. The longer the elapsed time, the greater the phase difference. Here the net magnetization vector is initially along +Y. For this and all dephasing examples think of this vector as the overlap of several thinner vectors from the individual spin packets.

The time constant which describes the return to equilibrium of the transverse magnetization, MXY, is called the spin-spin relaxation time, T2.

MXY =MXYo e-t/T2

T2 is always less than or equal to T1. The net magnetization in the XY plane goes to zero and then the longitudinal magnetization grows in until we have Mo along Z.

Any transverse magnetization behaves the same way. The transverse component rotates about the direction of applied magnetization and dephases. T1 governs the rate of recovery of the longitudinal magnetization.

In summary, the spin-spin relaxation time, T2, is the time to reduce the transverse magnetization by a factor of e. In the previous sequence, T2 and T1 processes are shown separately for clarity. That is, the magnetization vectors are shown filling the XY plane completely before growing back up along the Z axis. Actually, both processes occur simultaneously with the only restriction being that T2 is less than or equal to T1.

Two factors contribute to the decay of transverse magnetization.

1) molecular interactions (said to lead to a pure pure T2 molecular effect)

2) variations in Bo (said to lead to an inhomogeneous T2 effect

The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T2 star and is given the symbol T2*. The relationship between the T2 from molecular processes and that from inhomogeneities in the magnetic field is as follows.

1/T2* = 1/T2 + 1/T2inhomo.

Rotating Frame of Reference

We have just looked at the behavior of spins in the laboratory frame of reference. It is convenient to define a rotating frame of reference which rotates about the Z axis at the Larmor frequency. We distinguish this rotating coordinate system from the laboratory system by primes on the X and Y axes, X'Y'.

A magnetization vector rotating at the Larmor frequency in the laboratory frame appears stationary in a frame of reference rotating about the Z axis. In the rotating frame, relaxation of MZ magnetization to its equilibrium value looks the same as it did in the laboratory frame.

A transverse magnetization vector rotating about the Z axis at the same velocity as the rotating frame will appear stationary in the rotating frame. A magnetization vector traveling faster than the rotating frame rotates clockwise about the Z axis. A magnetization vector traveling slower than the rotating frame rotates counter-clockwise about the Z axis .

In a sample there are spin packets traveling faster and slower than the rotating frame. As a consequence, when the mean frequency of the sample is equal to the rotating frame, the dephasing of MX'Y' looks like this.

Pulsed Magnetic Fields

A coil of wire placed around the X axis will provide a magnetic field along the X axis when a direct current is passed through the coil. An alternating current will produce a magnetic field which alternates in direction.

In a frame of reference rotating about the Z axis at a frequency equal to that of the alternating current, the magnetic field along the X' axis will be constant, just as in the direct current case in the laboratory frame.

This is the same as moving the coil about the rotating frame coordinate system at the Larmor Frequency. In magnetic resonance, the magnetic field created by the coil passing an alternating current at the Larmor frequency is called the B1 magnetic field. When the alternating current through the coil is turned on and off, it creates a pulsed B1 magnetic field along the X' axis.

The spins respond to this pulse in such a way as to cause the net magnetization vector to rotate about the direction of the applied B1 field. The rotation angle depends on the length of time the field is on, , and its magnitude B1.

= 2 B1.

In our examples, will be assumed to be much smaller than T1 and T2.

A 90o pulse is one which rotates the magnetization vector clockwise by 90 degrees about the X' axis. A 90o pulse rotates the equilibrium magnetization down to the Y' axis. In the laboratory frame the equilibrium magnetization spirals down around the Z axis to the XY plane. You can see why the rotating frame of reference is helpful in describing the behavior of magnetization in response to a pulsed magnetic field.

A 180o pulse will rotate the magnetization vector by 180 degrees. A 180o pulse rotates the equilibrium magnetization down to along the -Z axis.

The net magnetization at any orientation will behave according to the rotation equation. For example, a net magnetization vector along the Y' axis will end up along the -Y' axis when acted upon by a 180o pulse of B1 along the X' axis.

A net magnetization vector between X' and Y' will end up between X' and Y' after the application of a 180o pulse of B1 applied along the X' axis.

A rotation matrix (described as a coordinate transformation in #2.6 Chapter 2) can also be used to predict the result of a rotation. Here is the rotation angle about the X' axis, [X', Y', Z] is the initial location of the vector, and [X", Y", Z"] the location of the vector after the rotation.

Spin Relaxation

Motions in solution which result in time varying magnetic fields cause spin relaxation.

Time varying fields at the Larmor frequency cause transitions between the spin states and hence a change in MZ. This screen depicts the field at the green hydrogen on the water molecule as it rotates about the external field Bo and a magnetic field from the blue hydrogen. Note that the field experienced at the green hydrogen is sinusoidal.

There is a distribution of rotation frequencies in a sample of molecules. Only frequencies at the Larmor frequency affect T1. Since the Larmor frequency is proportional to Bo, T1 will therefore vary as a function of magnetic field strength. In general, T1 is inversely proportional to the density of molecular motions at the Larmor frequency.

The rotation frequency distribution depends on the temperature and viscosity of the solution. Therefore T1 will vary as a function of temperature. At the Larmor frequency indicated by o, T1 (280 K ) < T1 (340 K). The temperature of the human body does not vary by enough to cause a significant influence on T1. The viscosity does however vary significantly from tissue to tissue and influences T1 as is seen in the following molecular motion plot.

Fluctuating fields which perturb the energy levels of the spin states cause the transverse magnetization to dephase. This can be seen by examining the plot of Bo experienced by the red hydrogens on the following water molecule. The number of molecular motions less than and equal to the Larmor frequency is inversely proportional to T2.

In general, relaxation times get longer as Bo increases because there are fewer relaxation-causing frequency components present in the random motions of the molecules.

Spin Exchange

Spin exchange is the exchange of spin state between two spins. For example, if we have two spins, A and B, and A is spin up and B is spin down, spin exchange between A and B can be represented with the following equation.

A() + B() A() + B()

The bidirectional arrow indicates that the exchange reaction is reversible.

The energy difference between the upper and lower energy states of A and of B must be the same for spin exchange to occur. On a microscopic scale, the spin in the upper energy state (B) is emitting a photon which is being absorbed by the spin in the lower energy state (A). Therefore, B ends up in the lower energy state and A in the upper state.

Spin exchange will not affect T1 but will affect T2. T1 is not effected because the distribution of spins between the upper and lower states is not changed. T2 will be affected because phase coherence of the transverse magnetization is lost during exchange.

Another form of exchange is called chemical exchange. In chemical exchange, the A and B nuclei are from different molecules. Consider the chemical exchange between water and ethanol.

CH3CH2OHA + HOHB CH3CH2OHB + HOHA

Here the B hydrogen of water ends up on ethanol, and the A hydrogen on ethanol ends up on water in the forward reaction. There are four senarios for the nuclear spin, represented by the four equations.

Chemical exchange will affect both T1 and T2. T1 is now affected because energy is transferred from one nucleus to another. For example, if there are more nuclei in the upper state of A, and a normal Boltzmann distribution in B, exchange will force the excess energy from A into B. The effect will make T1 appear smaller. T2 is effected because phase coherence of the transverse magnetization is not preserved during chemical exchange.

Bloch Equations

The Bloch equations are a set of coupled differential equations which can be used to describe the behavior of a magnetizatiion vector under any conditions. When properly integrated, the Bloch equations will yield the X', Y', and Z components of magnetization as a function of time.

Chemical Shift

Chemical Shift

When an atom is placed in a magnetic field, its electrons circulate about the direction of the applied magnetic field. This circulation causes a small magnetic field at the nucleus which opposes the externally applied field.

The magnetic field at the nucleus (the effective field) is therefore generally less than the applied field by a fraction .

B = Bo (1-s)

In some cases, such as the benzene molecule, the circulation of the electrons in the aromatic orbitals creates a magnetic field at the hydrogen nuclei which enhances the Bo field. This phenomenon is called deshielding. In this example, the Bo field is applied perpendicular to the plane of the molecule. The ring current is traveling clockwise if you look down at the plane.

The electron density around each nucleus in a molecule varies according to the types of nuclei and bonds in the molecule. The opposing field and therefore the effective field at each nucleus will vary. This is called the chemical shift phenomenon.

Consider the methanol molecule. The resonance frequency of two types of nuclei in this example differ. This difference will depend on the strength of the magnetic field, Bo, used to perform the NMR spectroscopy. The greater the value of Bo, the greater the frequency difference. This relationship could make it difficult to compare NMR spectra taken on spectrometers operating at different field strengths. The term chemical shift was developed to avoid this problem.

The chemical shift of a nucleus is the difference between the resonance frequency of the nucleus and a standard, relative to the standard. This quantity is reported in ppm and given the symbol delta, .

d = (n - nREF) x106 / nREF

In NMR spectroscopy, this standard is often tetramethylsilane, Si(CH3)4, abbreviated TMS. The chemical shift is a very precise metric of the chemical environment around a nucleus. For example, the hydrogen chemical shift of a CH2 hydrogen next to a Cl will be different than that of a CH3 next to the same Cl. It is therefore difficult to give a detailed list of chemical shifts in a limited space. The animation window displays a chart of selected hydrogen chemical shifts of pure liquids and some gasses.

The magnitude of the screening depends on the atom. For example, carbon-13 chemical shifts are much greater than hydrogen-1 chemical shifts. The following tables present a few selected chemical shifts of fluorine-19 containing compounds, carbon-13 containing compounds, nitrogen-14 containing compounds, and phosphorous-31 containing compounds. These shifts are all relative to the bare nucleus. The reader is directed to a more comprehensive list of chemical shifts for use in spectral interpretation.

Spin-Spin Coupling

Nuclei experiencing the same chemical environment or chemical shift are called equivalent. Those nuclei experiencing different environment or having different chemical shifts are nonequivalent. Nuclei which are close to one another exert an influence on each other's effective magnetic field. This effect shows up in the NMR spectrum when the nuclei are nonequivalent. If the distance between non-equivalent nuclei is less than or equal to three bond lengths, this effect is observable. This effect is called spin-spin coupling or J coupling.

Consider the following example. There are two nuclei, A and B, three bonds away from one another in a molecule. The spin of each nucleus can be either aligned with the external field such that the fields are N-S-N-S, called spin up , or opposed to the external field such that the fields are N-N-S-S, called spin down . The magnetic field at nucleus A will be either greater than Bo or less than Bo by a constant amount due to the influence of nucleus B.

There are a total of four possible configurations for the two nuclei in a magnetic field. Arranging these configurations in order of increasing energy gives the following arrangement. The vertical lines in this diagram represent the allowed transitions between energy levels. In NMR, an allowed transition is one where the spin of one nucleus changes from spin up to spin down , or spin down to spin up . Absorptions of energy where two or more nuclei change spin at the same time are not allowed. There are two absorption frequencies for the A nucleus and two for the B nucleus represented by the vertical lines between the energy levels in this diagram.

The NMR spectrum for nuclei A and B reflects the splittings observed in the energy level diagram. The A absorption line is split into 2 absorption lines centered on A, and the B absorption line is split into 2 lines centered on B. The distance between two split absorption lines is called the J coupling constant or the spin-spin splitting constant and is a measure of the magnetic interaction between two nuclei.

For the next example, consider a molecule with three spin 1/2 nuclei, one type A and two type B. The type B nuclei are both three bonds away from the type A nucleus. The magnetic field at the A nucleus has three possible values due to four possible spin configurations of the two B nuclei. The magnetic field at a B nucleus has two possible values.

The energy level diagram for this molecule has six states or levels because there are two sets of levels with the same energy. Energy levels with the same energy are said to be degenerate. The vertical lines represent the allowed transitions or absorptions of energy. Note that there are two lines drawn between some levels because of the degeneracy of those levels.

The resultant NMR spectrum is depicted in the animation window. Note that the center absorption line of those centered at A is twice as high as the either of the outer two. This is because there were twice as many transitions in the energy level diagram for this transition. The peaks at B are taller because there are twice as many B type spins than A type spins.

The complexity of the splitting pattern in a spectrum increases as the number of B nuclei increases. The following table contains a few examples.

Configuration Peak Ratios

A 1

AB 1:1

AB2 1:2:1

AB3 1:3:3:1

AB4 1:4:6:4:1

AB5 1:5:10:10:5:1

AB6 1:6:15:20:15:6:1

This series is called Pascal's triangle and can be calculated from the coefficients of the expansion of the equation

(x+1)n

where n is the number of B nuclei in the above table.

When there are two different types of nuclei three bonds away there will be two values of J, one for each pair of nuclei. By now you get the idea of the number of possible configurations and the energy level diagram for these configurations, so we can skip to the spectrum. In the following example JAB is greater JBC.

The Time Domain NMR Signal

An NMR sample may contain many different magnetization components, each with its own Larmor frequency. These magnetization components are associated with the nuclear spin configurations joined by an allowed transition line in the energy level diagram. Based on the number of allowed absorptions due to chemical shifts and spin-spin couplings of the different nuclei in a molecule, an NMR spectrum may contain many different frequency lines.

In pulsed NMR spectroscopy, signal is detected after these magnetization vectors are rotated into the XY plane. Once a magnetization vector is in the XY plane it rotates about the direction of the Bo field, the +Z axis. As transverse magnetization rotates about the Z axis, it will induce a current in a coil of wire located around the X axis. Plotting current as a function of time gives a sine wave. This wave will, of course, decay with time constant T2* due to dephasing of the spin packets. This signal is called a free induction decay (FID). We will see in Chapter 5 how the FID is converted into a frequency domain spectrum. You will see in Chapter 6 what sequence of events will produce a time domain signal.

The +/- Frequency Convention

Transverse magnetization vectors rotating faster than the rotating frame of reference are said to be rotating at a positive frequency relatve to the rotating frame (+n). Vectors rotating slower than the rotating frame are said to be rotating at a negative frequency relative to the rotating frame (-n).

It is worthwhile noting here that in most NMR spectra, the resonance frequency of a nucleus, as well as the magnetic field experienced by the nucleus and the chemical shift of a nucleus, increase from right to left. The frequency plots used in this hypertext book to describe Fourier transforms will use the more conventional mathematical axis of frequency increasing from left to right.

Introduction

A detailed description of the Fourier transform ( FT ) has waited until now, when you have a better appreciation of why it is needed. A Fourier transform is an operation which converts functions from time to frequency domains. An inverse Fourier transform ( IFT ) converts from the frequency domain to the time domain.

Recall from Chapter 2 that the Fourier transform is a mathematical technique for converting time domain data to frequency domain data, and vice versa.

The + and - Frequency Problem

To begin our detailed description of the FT consider the following. A magnetization vector, starting at +x, is rotating about the Z axis in a clockwise direction. The plot of Mx as a function of time is a cosine wave. Fourier transforming this gives peaks at both + and - because the FT can not distinguish between a + and a - rotation of the vector from the data supplied.

A plot of My as a function of time is a -sine function. Fourier transforming this gives peaks at + and - because the FT can not distinguish between a positive vector rotating at + and a negative vector rotating at - from the data supplied.

The solution is to input both the Mx and My into the FT. The FT is designed to handle two orthogonal input functions called the real and imaginary components.

Detecting just the Mx or My component for input into the FT is called linear detection. This was the detection scheme on many older NMR spectrometers and some magnetic resonance imagers. It required the computer to discard half of the frequency domain data.

Detection of both Mx and My is called quadrature detection and is the method of detection on modern spectrometers and imagers. It is the method of choice since now the FT can distinguish between + and -, and all of the frequency domain data be used.

The Fourier Transform

An FT is defined by the integral

Think of f() as the overlap of f(t) with a wave of frequency .

This is easy to picture by looking at the real part of f() only.

Consider the function of time, f( t ) = cos( 4t ) + cos( 9t ).

To understand the FT, examine the product of f(t) with cos(t) for values between 1 and 10, and then the summation of the values of this product between 1 and 10 seconds. The summation will only be examined for time values between 0 and 10 seconds.

=1

=2

=3

=4

=5

=6

=7

=8

=9

=10

f()

The inverse Fourier transform (IFT) is best depicted as an summation of the time domain spectra of frequencies in f().

Phase Correction

The actual FT will make use of an input consisting of a REAL and an IMAGINARY part. You can think of Mx as the REAL input, and My as the IMAGINARY input. The resultant output of the FT will therefore have a REAL and an IMAGINARY component, too.

Consider the following function:

f(t) = e-at e-i2t

In FT NMR spectroscopy, the real output of the FT is taken as the frequency domain spectrum. To see an esthetically pleasing (absorption) frequency domain spectrum, we want to input a cosine function into the real part and a sine function into the imaginary parts of the FT. This is what happens if the cosine part is input as the imaginary and the sine as the real.

To obtain an absorption spectrum as the real output of the FT, a phase correction must be applied to either the time or frequency domain spectra. This process is equivalent to the coordinate transformation described in Chapter 2

If the above mentioned FID is recorded such that there is a 45o phase shift in the real and imaginary FIDs, the coordinate transformation matrix can be used with = - 45o. The corrected FIDs look like a cosine function in the real and a sine in the imaginary.

Fourier transforming the phase corrected FIDs gives an absorption spectrum for the real output of the FT. This correction can be done in the frequency domain as well as in the time domain.

NMR spectra require both constant and linear corrections to the phasing of the Fourier transformed signal.

= m + b

Constant phase corrections, b, arise from the inability of the spectrometer to detect the exact Mx and My. Linear phase corrections, m, arise from the inability of the spectrometer to detect transverse magnetization starting immediately after the RF pulse.

In magnetic resonance, the Mx or My signals are displayed. A magnitude signal might occasionally be used in some applications. The magnitude signal is equal to the square root of the sum of the squares of Mx and My.

Fourier Pairs

To better understand FT NMR functions, you need to know some common Fourier pairs. A Fourier pair is two functions, the frequency domain form and the corresponding time domain form. Here are a few Fourier pairs which are useful in NMR. The amplitude of the Fourier pairs has been neglected since it is not relevant in NMR.

Constant value at all time

Real: cos(2t), Imaginary: -sin(2t)

Comb Function (A series of delta functions separated by T.)

Exponential Decay: e-at for t > 0.

A square pulse starting at 0 that is T seconds long.

Gaussian: exp(-at2)

Convolution Theorem

To the magnetic resonance scientist, the most important theorem concerning Fourier transforms is the convolution theorem. The convolution theorem says that the FT of a convolution of two functions is proportional to the products of the individual Fourier transforms, and vice versa.

If f() = FT( f(t) ) and g() = FT( g(t) )

then f() g() = FT( g(t) f(t) ) and f() g() = FT( g(t) f(t) )

It will be easier to see this with pictures. In the animation window we are trying to find the FT of a sine wave which is turned on and off. The convolution theorem tells us that this is a sinc function at the frequency of the sine wave.

Another application of the convolution theorem is in noise reduction. With the convolution theorem it can be seen that the convolution of an NMR spectrum with a Lorentzian function is the same as the Fourier Transform of multiplying the time domain signal by an exponentially decaying function.

The Digital FT

In a nuclear magnetic resonance spectrometer, the computer does not see a continuous FID, but rather an FID which is sampled at a constant interval. Each data point making up the FID will have discrete amplitude and time values. Therefore, the computer needs to take the FT of a series of delta functions which vary in intensity.

Sampling Error

The wrap around problem or artifact in a nuclear magnetic resonance spectrum is the appearance of one side of the spectrum on the opposite side. In terms of a one dimensional frequency domain spectrum, wrap around is the occurrence of a low frequency peak which occurs on the high frequency side of the spectrum.

The convolution theorem can explain why this problem results from sampling the transverse magnetization at too slow a rate. First, observe what the FT of a correctly sampled FID looks like. With quadrature detection, the spectral width is equal to the inverse of the sampling frequency, or the width of the green box in the animation window.

When the sampling frequency is less than the spectral width, wrap around occurs.

The Two-Dimensional FT

The two-dimensional Fourier transform (2-DFT) is an FT performed on a two dimensional array of data.

Consider the two-dimensional array of data depicted in the animation window. This data has a t' and a t" dimension. A FT is first performed on the data in one dimension and then in the second. The first set of Fourier transforms are performed in the t' dimension to yield an f' by t" set of data. The second set of Fourier transforms is performed in the t" dimension to yield an f' by f" set of data.

The 2-DFT is required to perform state-of-the-art MRI. In MRI, data is collected in the equivalent of the t' and t" dimensions, called k-space. This raw data is Fourier transformed to yield the image which is the equivalent of the f' by f" data described above.

For more topics see: http://www.cis.rit.edu/htbooks/nmr/inside.htm

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