The Basics of NMR

The Basics of NMR

Chapter 1

INTRODUCTION

NMR

Spectroscopy

Units Review

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:

1. can be thought of as a small magnetic field, and
2. 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.

The Basics of NMR

Chapter 2

THE MATHEMATICS OF NMR

Exponential Functions

Trigonometric Functions

Differentials and Integrals

Vectors

Matrices

Coordinate Transformations

Convolutions

Imaginary Numbers

The Fourier Transform

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.

The Basics of NMR

Chapter 3

SPIN PHYSICS

Spin

Properties of Spin

Nuclei with Spin

Energy Levels

Transitions

Energy Level Diagrams

Continuous Wave NMR Experiment

Boltzmann Statistics

Spin Packets

T1 Processes

Precession

T2 Processes

Rotating Frame of Reference

Pulsed Magnetic Fields

Spin Relaxation

Spin Exchange

Bloch Equations

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.