The Basics of NMRChapter 1 INTRODUCTIONNMR Spectroscopy Units Review
NMRNuclear 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.
SpectroscopySpectroscopy 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 ReviewBefore 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 NMRChapter 2 THE MATHEMATICS OF NMRExponential Functions Trigonometric Functions Differentials and Integrals Vectors Matrices Coordinate Transformations Convolutions Imaginary Numbers The Fourier Transform
Exponential FunctionsThe 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 FunctionsThe 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 IntegralsA 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.
VectorsA 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
MatricesA 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 TransformationsA coordinate transformation is used to convert the coordinates of a vector in one coordinate system (XY) to that in another coordinate system (X"Y").
ConvolutionThe 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 NumbersImaginary 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 TransformsThe 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 NMRChapter 3 SPIN PHYSICSSpin 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
SpinWhat 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 SpinWhen 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 SpinThe 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