Download The Basics of NMR

The Basics of NMR
Chapter 1
INTRODUCTION
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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
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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
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Spin
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
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)
H
1
0
1/2
42.58
H
1
1
1
6.54
P
1
0
1/2
17.25
Na
1
2
3/2
11.27
N
1
1
1
3.08
C
0
1
1/2
10.71
F
1
0
1/2
40.08
1
2
31
23
14
13
19
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 relaxationcausing 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.
The Basics of NMR
Chapter 4
NMR SPECTROSCOPY
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Chemical Shift
Spin-Spin Coupling
The Time Domain NMR Signal
The +/- Frequency Convention
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-)
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, .
 = ( - REF) x106 / REF
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 (+). Vectors rotating
slower than the rotating frame are said to be rotating at a negative frequency relative to the
rotating frame (-).
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.
The Basics of NMR
Chapter 5
FOURIER TRANSFORMS
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Introduction
The + and - Frequency Problem
The Fourier Transform
Phase Correction
Fourier Pairs
The Convolution Theorem
The Digital FT
Sampling Error
The Two-Dimensional FT
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-i2
t
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(2
t), Imaginary: -sin(2
t)
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.
The Basics of NMR
Chapter 6
PULSE SEQUENCES
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
Introduction
The 90-FID Sequence
The Spin-Echo Sequence
The Inversion Recovery Sequence
Introduction
You have seen in Chapter 5 how a time domain signal can be converted into a frequency
domain signal. In this chapter you will learn a few of the ways that a time domain signal can
be created. Three methods are presented here, but there are an infinite number of
possibilities. These methods are called pulse sequences. A pulse sequence is a set of RF
pulses applied to a sample to produce a specific form of NMR signal.
The 90-FID Sequence
In the 90-FID pulse sequence, net magnetization is rotated down into the X'Y' plane with a
90o pulse. The net magnetization vector begins to precess about the +Z axis. The
magnitude of the vector also decays with time.
A timing diagram is a multiple axis plot of some aspect of a pulse sequence versus time. A
timing diagram for a 90-FID pulse sequence has a plot of RF energy versus time and another
for signal versus time.
When this sequence is repeated, for example when signal-to-noise improvement is needed,
the amplitude of the signal after being Fourier transformed (S) will depend on T1 and the time
between repetitions, called the repetition time (TR), of the sequence. In the signal equation
below, k is a proportionality constant and is the density of spins in the sample.
S = k ( 1 - e-TR/T1 )
The Spin-Echo Sequence
Another commonly used pulse sequence is the spin-echo pulse sequence.
Here a 90o pulse
is first applied to the spin system. The 90o degree pulse rotates the magnetization down into
the X'Y' plane. The transverse magnetization begins to dephase. At some point in time after
the 90o pulse, a 180o pulse is applied. This pulse rotates the magnetization by 180o about the
X' axis. The 180o pulse causes the magnetization to at least partially rephase and to produce
a signal called an echo.
A timing diagram shows the relative positions of the two radio frequency pulses and the
signal.
The signal equation for a repeated spin echo sequence as a function of the repetition time,
TR, and the echo time (TE) defined as the time between the 90o pulse and the maximum
amplitude in the echo is
S = k ( 1 - e-TR/T1 ) e-TE/T2
The Inversion Recovery Sequence
An inversion recovery pulse sequence can also be used to record an NMR spectrum. In this
sequence, a 180o pulse is first applied. This rotates the net magnetization down to the -Z axis.
The magnetization undergoes spin-lattice relaxation and returns toward its equilibrium
position along the +Z axis. Before it reaches equilibrium, a 90o pulse is applied which
rotates the longitudinal magnetization into the XY plane. In this example, the 90o pulse is
applied shortly after the 180o pulse. Once magnetization is present in the XY plane it rotates
about the Z axis and dephases giving a FID.
Once again, the timing diagram shows the relative positions of the two radio frequency
pulses and the signal.
The signal as a function of TI when the sequence is not repeated is
S = k ( 1 - 2e-TI/T1 )
It should be noted at this time that the zero crossing of this function occurs for TI = T1 ln2.
The Basics of NMR
Chapter 7
NMR HARDWARE
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Hardware Overview
Magnet
Field Lock
Shim Coils
Sample Probe
RF Coils
Gradient Coils
Quadrature Detector
Digital Filtering

Safety
Hardware Overview
The graphics window displays a schematic representation of the major systems of a nuclear
magnetic resonance spectrometer and a few of the major interconnections. This overview
briefly states the function of each component. Some will be described in detail later in this
chapter.
At the top of the schematic representation, you will find the superconducting magnet of the
NMR spectrometer. The magnet produces the Bo field necessary for the NMR experiments.
Immediately within the bore of the magnet are the shim coils for homogenizing the Bo field.
Within the shim coils is the probe. The probe contains the RF coils for producing the B1
magnetic field necessary to rotate the spins by 90o or 180o. The RF coil also detects the signal
from the spins within the sample. The sample is positioned within the RF coil of the probe.
Some probes also contain a set of gradient coils. These coils produce a gradient in Bo along
the X, Y, or Z axis. Gradient coils are used for for gradient enhanced spectroscopy (See
Chapter 11.), diffusion (See Chapter 11.), and NMR microscopy (See Chapter 11.)
experiments.
The heart of the spectrometer is the computer. It controls all of the components of the
spectrometer. The RF components under control of the computer are the RF frequency source
and pulse programmer. The source produces a sine wave of the desired frequency. The pulse
programmer sets the width, and in some cases the shape, of the RF pulses. The RF amplifier
increases the pulses power from milli Watts to tens or hundreds of Watts. The computer also
controls the gradient pulse programmer which sets the shape and amplitude of gradient fields.
The gradient amplifier increases the power of the gradient pulses to a level sufficient to drive
the gradient coils.
The operator of the spectrometer gives input to the computer through a console terminal with
a mouse and keyboard. Some spectrometers also have a separate small interface for carrying
out some of the more routine procedures on the spectrometer. A pulse sequence is selected
and customized from the console terminal. The operator can see spectra on a video display
located on the console and can make hard copies of spectra using a printer.
The next sections of this chapter go into more detail concerning the magnet, lock, shim coils,
gradient coils, RF coils, and RF detector of nuclear magnetic resonance spectrometer.
Magnet
The NMR magnet is one of the most expensive components of the nuclear magnetic
resonance spectrometer system. Most magnets are of the superconducting type. This is a
picture of a 7.0 Tesla superconducting magnet from an NMR spectrometer. A
superconducting magnet has an electromagnet made of superconducting wire.
Superconducting wire has a resistance approximately equal to zero when it is cooled to a
temperature close to absolute zero (-273.15o C or 0 K) by emersing it in liquid helium. Once
current is caused to flow in the coil it will continue to flow for as long as the coil is kept at
liquid helium temperatures. (Some losses do occur over time due to the infinitesimally
small resistance of the coil. These losses are on the order of a ppm of the main magnetic field
per year.)
The length of superconducting wire in the magnet is typically several miles. This wire is
wound into a multi-turn solenoid or coil. The coil of wire is kept at a temperature of 4.2K by
immersing it in liquid helium. The coil and liquid helium are kept in a large dewar. This
dewar is typically surrounded by a liquid nitrogen (77.4K) dewar, which acts as a thermal
buffer between the room temperature air (293K) and the liquid helium. A cross sectional
view of the superconducting magnet, depicting the concentric dewars, can be found in the
animation window.
The following image is an actual cut-away view of a superconducting magnet. The magnet
is supported by three legs, and the concentric nitrogen and helium dewars are supported by
stacks coming out of the top of the magnet. A room temperature bore hole extends through
the center of the assembly. The sample probe and shim coils are located within this bore hole.
Also depicted in this picture is the liquid nitrogen level sensor, an electronic assembly for
monitoring the liquid nitrogen level.
Going from the outside of the magnet to the inside, we see a vacuum region followed by a
liquid nitrogen reservoir. The vacuum region is filled with several layers of a reflective mylar
film. The function of the mylar is to reflect thermal photons, and thus diminish heat from
entering the magnet. Within the inside wall of the liquid nitrogen reservoir, we see another
vacuum filled with some reflective mylar. The liquid helium reservoir comes next. This
reservoir houses the superconducting solenoid or coil of wire.
Taking a closer look at the solenoid it is clear to see the coil and the bore tube extending
through the magnet.
Field Lock
In order to produce a high resolution NMR spectrum of a sample, especially one which
requires signal averaging or phase cycling, you need to have a temporally constant and
spatially homogeneous magnetic field. Consistency of the Bo field over time will be
discussed here; homogeneity will be discussed in the next section of this chapter. The field
strength might vary over time due to aging of the magnet, movement of metal objects near
the magnet, and temperature fluctuations. Here is an example of a one line NMR spectrum of
cyclohexane recorded while the Bo magnetic field was drifting a very significant amount.
The field lock can compensate for these variations.
The field lock is a separate NMR spectrometer within your spectrometer. This spectrometer
is typically tuned to the deuterium NMR resonance frequency. It constantly monitors the
resonance frequency of the deuterium signal and makes minor changes in the Bo magnetic
field to keep the resonance frequency constant. The deuterium signal comes from the
deuterium solvent used to prepare the sample. The animation window contains plots of the
deuterium resonance lock frequency, the small additional magnetic field used to correct the
lock frequency, and the resultant Bo field as a function of time while the magnetic field is
drifting. The lock frequency plot displays the frequency without correction. In reality, this
frequency would be kept constant by the application of the lock field which offsets the drift.
On most NMR spectrometers the deuterium lock serves a second function. It provides the
=0 reference. The resonance frequency of the deuterium signal in many lock solvents is well
known. Therefore the difference in resonance frequency of the lock solvent and TMS is also
known. As a consequence, TMS does not need to be added to the sample to set =0; the
spectrometer can use the lock frequency to calculate =0.
Shim Coils
The purpose of shim coils on a spectrometer is to correct minor spatial inhomogeneities in
the Bo magnetic field. These inhomogeneities could be caused by the magnet design,
materials in the probe, variations in the thickness of the sample tube, sample permeability,
and ferromagnetic materials around the magnet. A shim coil is designed to create a small
magnetic field which will oppose and cancel out an inhomogeneity in the Bo magnetic field.
Because these variations may exist in a variety of functional forms (linear, parabolic, etc.),
shim coils are needed which can create a variety of opposing fields. Some of the functional
forms are listed in the table below.
Shim Coil Functional Forms
Shim Function
Z0
Z
Z2
Z3
Z4
Z5
X
XZ
XZ2
X2Y2
XY
Y
YZ
YZ2
XZ3
X2Y2Z
YZ3
XYZ
X3
Y3
By passing the appropriate amount of current through each coil a homogeneous Bo magnetic
field can be achieved. The optimum shim current settings are found by either minimizing the
linewidth, maximizing the size of the FID, or maximizing the signal from the field lock. On
most spectrometers, the shim coils are controllable by the computer. A computer algorithm
has the task of finding the best shim value by maximizing the lock signal.
Sample Probe
The sample probe is the name given to that part of the spectrometer which accepts the
sample, sends RF energy into the sample, and detects the signal emanating from the sample.
It contains the RF coil, sample spinner, temperature controlling circuitry, and gradient coils.
The RF coil and gradient coils will be described in the next two sections. The sample spinner
and temperature controlling circuitry will be described here.
The purpose of the sample spinner is to rotate the NMR sample tube about its axis. In doing
so, each spin in the sample located at a given position along the Z axis and radius from the Z
axis, will experience the average magnetic field in the circle defined by this Z and radius.
The net effect is a narrower spectral linewidth. To appreciate this phenomenon, consider the
following examples.
Picture an axial cross section of a cylindrical tube containing sample. In a very homogeneous
Bo magnetic field this sample will yield a narrow spectrum. In a more inhomogeneous field
the sample will yield a broader spectrum due to the presence of lines from the parts of the
sample experiencing different Bo magnetic fields. When the sample is spun about its z-axis,
inhomogeneities in the X and Y directions are averaged out and the NMR line width
becomes narrower.
Many scientists need to examine properties of their samples as a function of temperature. As
a result many instruments have the ability to maintain the temperature of the sample above
and below room temperature. Air or nitrogen which has been warmed or cooled is passed
over the sample to heat or cool the sample. The temperature at the sample is monitored with
the aid of a thermocouple and electronic circuitry maintains the temperature by increasing or
decreasing the temperature of the gas passing over the sample. More information on this
topic will be presented in Chapter 8.
RF Coils
RF coils create the B1 field which rotates the net magnetization in a pulse sequence. They
also detect the transverse magnetization as it precesses in the XY plane. Most RF coils on
NMR spectrometers are of the saddle coil design and act as the transmitter of the B1 field
and receiver of RF energy from the sample. You may find one or more RF coils in a probe.
Each of these RF coils must resonate, that is they must efficiently store energy, at the Larmor
frequency of the nucleus being examined with the NMR spectrometer. All NMR coils are
composed of an inductor, or inductive elements, and a set of capacitive elements. The
resonant frequency, , of an RF coil is determined by the inductance (L) and capacitance (C)
of the inductor capacitor circuit.
RF coils used in NMR spectrometers need to be tuned for the specific sample being studied.
An RF coil has a bandwidth or specific range of frequencies at which it resonates. When you
place a sample in an RF coil, the conductivity and dielectric constant of the sample affect the
resonance frequency. If this frequency is different from the resonance frequency of the
nucleus you are studying, the coil will not efficiently set up the B1 field nor efficiently detect
the signal from the sample. You will be rotating the net magnetization by an angle less than
90 degrees when you think you are rotating by 90 degrees. This will produce less transverse
magnetization and less signal. Furthermore, because the coil will not be efficiently detecting
the signal, your signal-to-noise ratio will be poor.
The B1 field of an RF coil must be perpendicular to the Bo magnetic field. Another
requirement of an RF coil in an NMR spectrometer is that the B1 field needs to be
homogeneous over the volume of your sample. If it is not, you will be rotating spins by a
distribution of rotation angles and you will obtain strange spectra.
Gradient Coils
The gradient coils produce the gradients in the Bo magnetic field needed for performing
gradient enhanced spectroscopy, diffusion measurements, and NMR microscopy. The
gradient coils are located inside the RF probe. Not all probes have gradient coils, and not all
NMR spectrometers have the hardware necessary to drive these coils.
The gradient coils are room temperature coils (i.e. do not require cooling with cryogens to
operate) which, because of their configuration, create the desired gradient. Since the vertical
bore superconducting magnet is most common, the gradient coil system will be described for
this magnet.
Assuming the standard magnetic resonance coordinate system, a gradient in Bo in the Z
direction is achieved with an antihelmholtz type of coil. Current in the two coils flow in
opposite directions creating a magnetic field gradient between the two coils. The B field at
the center of one coil adds to the Bo field, while the B field at the center of the other coil
subtracts from the Bo field.
The X and Y gradients in the Bo field are created by a pair of figure-8 coils. The X axis
figure-8 coils create a gradient in Bo in the X direction due to the direction of the current
through the coils. The Y axis figure-8 coils provides a similar gradient in Bo along the Y
axis.
Quadrature Detector
The quadrature detector is a device which separates out the Mx' and My' signals from the
signal from the RF coil. For this reason it can be thought of as a laboratory to rotating frame
of reference converter. The heart of a quadrature detector is a device called a doubly
balanced mixer. The doubly balanced mixer has two inputs and one output. If the input
signals are Cos(A) and Cos(B), the output will be 1/2 Cos(A+B) and 1/2 Cos(A-B). For this
reason the device is often called a product detector since the product of Cos(A) and Cos(B) is
the output.
The quadrature detector typically contains two doubly balanced mixers, two filters, two
amplifiers, and a 90o phase shifter. There are two inputs and two outputs on the device.
Frequency and o are put in and the MX' and MY' components of the transverse
magnetization come out. There are some potential problems which can occur with this device
which will cause artifacts in the spectrum. One is called a DC offset artifact and the other is
called a quadrature artifact.
Digital Filtering
Many newer spectrometers employ a combination of oversampling, digital filtering, and
decimation to eliminate the wrap around artifact. Oversampling creates a larger spectral or
sweep width, but generates too much data to be conveniently stored. Digital filtering
eliminates the high frequency components from the data, and decimation reduces the size of
the data set. The following flowchart summarizes the effects of the three steps by showing
the result of performing an FT after each step.
Let's examine oversampling, digital filtering, and decimation in more detail to see how this
combination of steps can be used to reduce the wrap around problem.
Oversampling is the digitization of a time domain signal at a frequency much greater than
necessary to record the desired spectral width. For example, if the sampling frequency, fs, is
increased by a factor of 10, the sweep width will be 10 times greater, thus eliminating
wraparound. Unfortunately digitizing at 10 times the speed also increases the amount of raw
data by a factor of 10, thus increasing storage requirements and processing time.
Filtering is the removal of a select band of frequencies from a signal. For an example of
filtering, consider the following frequency domain signal. Frequencies above fo could be
removed from this frequency domain signal by multipling the signal by this rectangular
function. In NMR, this step would be equivalent to taking a large sweep width spectrum
and setting to zero intensity those spectral frequencies which are farther than some distance
from the center of the spectrum.
Digital filtering is the removal of these frequencies using the time domain signal. Recall from
Chapter 5 that if two functions are multiplied in one domain (i.e. frequency), we must
convolve the FT of the two functions together in the other domain (i.e. time). To filter out
frequencies above fo from the time domain signal, the signal must be convolved with the
Fourier transform of the rectangular function, a sinc function. (See Chapter 5.) This process
eliminates frequencies greater than fo from the time domain signal. Fourier transforming the
resultant time domain signal yields a frequency domain signal without the higher
frequencies. In NMR, this step will remove spectral components with frequencies greater
than +fo and less than -fo.
Decimation is the elimination of data points from a data set. A decimation ratio of 4/5 means
that 4 out of every 5 data points are deleted, or every fifth data point is saved. Decimating the
digitally filtered data above, followed by a Fourier transform, will reduce the data set by a
factor of five.
High speed digitizers, capable of digitizing at 2 MHz, and dedicated high speed integrated
circuits, capable of performing the convolution on the time domain data as it is being
recorded, are used to realize this procedure.
Safety
There are some important safety considerations which one should be familiar with before
using an NMR spectrometer. These concern the use of strong magnetic fields and cryogenic
liquids.
Magnetic fields from high field magnets can literally pick up and pull ferromagnetic items
into the bore of the magnet. Caution must be taken to keep all ferromagnetic items away
from the magnet because they can seriously damage the magnet, shim coils, and probe. The
force exerted on the concentric cryogenic dewars within a magnet by a large metal object
stuck to the magnet can break dewars and magnet supports. The kinetic energy of an object
being sucked into a magnet can smash a dewar or an electrical connector on a probe. Small
ferromagnetic objects are just as much a concern as larger ones. A small metal sliver can get
sucked into the bore of the magnet and destroy the homogeneity of the magnet achieved with
a set of shim settings.
There are additional concerns regarding the effect of magnetic fields on electronic circuitry,
specifically pacemakers. An individual with a pacemaker walking through a strong magnetic
field can induce currents in the pacemaker circuitry which will cause it to fail and possibly
cause death. A person with a pacemaker must not be able to inadvertently stray into a
magnetic field of five or more Gauss. Although not as important as a pacemaker, mechanical
watches and some digital watches will also be affected by magnetic fields. Magnetic fields of
approximately 50 Gauss will erase credit cards and magnetic storage media.
The liquid nitrogen and liquid helium used in NMR spectrometers are at a temperature of
77.4 K and 4.2 K respectively. These liquids can cause frostbite, which is not a concern
unless you are filling the magnet. If you are filling the magnet or if you are operating the
spectrometer, suffocation is another concern you need to be aware of. If the magnet
quenches, or suddenly stops being a superconductor, it will rapidly boil off all its cryogens,
and the nitrogen and helium gasses in a confined space can cause suffocation.
The Basics of NMR
Chapter 8
PRACTICAL CONSIDERATIONS
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Introduction
Sample Preparation
Sample Probe Tuning
Determining a 90 Degree Pulse
Field Shimming
Phase Cycling
1-D Hydrogen Spectra
Integration
SNR Improvement
Variable Temperature
Troubleshooting
Cryogen Fills
Unix Primer
Introduction
In previous chapters, you have learned the basic theory of nuclear magnetic resonance. This
chapter emphasizes some of the spectroscopic techniques. While some of these may be easy
for you to understand based on the simple theory you have learned in previous chapters, there
may be specific points discussed which are less obvious because they are based on theories
not presented in this hypertext book.
When comparing two NMR spectra, always keep in mind the subtle differences in the way
the spectra were recorded. One obvious example is the effect of field strength. As the Bo field
increases in magnitude (i.e. 1.5T, 4.7T, 7T) the signal-to-noise ratio generally increases. The
shape of the spectrum may also change. For example, consider the hydrogen NMR spectrum
from three coupled nuclei A, B, and C with the following chemical shifts and J coupling
constants.
Nuclei (ppm)
A
1.89
B
2.00
C
2.08
Interaction J (Hz)
AB
4
BC
8
Compare the 100 MHz and 400 MHz NMR spectra. The spectral lines from the B type
spins are colored red. You can see how easy it would be to make the wrong choice as to the
structure of the molecule based on the 100 MHz spectrum, although the chance of error
might be reduced if you had further information, eg. the relative areas under the peaks. This
topic is described in a later section of this chapter.
Sample Preparation
NMR samples are prepared by dissolving an analyte in a deuterium lock solvent. Several
deuterium lock solvents are available . Some of these solvents will readily absorb moisture
from the atmosphere and give water signal in your spectrum. It is therefore advisable to keep
bottles of these solvents tightly capped when not in use.
Most routine high resolution NMR samples are prepared and run in 5 mm glass NMR tubes.
Always fill your NMR tubes to the same height with lock solvent. This will minimize the
amount of magnetic field shimming required. The animation window depicts a sample tube
filled with solvent such that it fills the RF coil.
The concentration of your sample should be great enough to give a good signal-to-noise ratio
in your spectrum, yet minimize exchange effects found at high concentrations. The exact
concentration of your sample in the lock solvent will depend on the sensitivity of the
spectrometer. If you have no guidelines for a specific spectrometer, use one drop of analyte
for liquids and one or two crystals for solid samples.
The position of spectral absorption lines can be solvent dependent. Therefore, if you are
comparing spectra or trying to identify an unknown sample by comparison to reference
spectra, use the same solvent. The hydrogen NMR spectrum of ethanol is a good example of
this solvent dependence. Compare the positions of the CH3, CH2, and OH absorption lines in
a hydrogen NMR spectrum of ethanol in the lock solvents CDCl3 and D2O . Notice also
that the relative peak heights are not the same in the two spectra. This is because the
linewidths are not equal. The area under a peak, not the height of a peak, is proportional to
the number of hydrogens in a sample. This point will be emphasized later in this chapter.
Variations in the polarity and dielectric constant of the lock solvent will also effect the tuning
of the probe. The correction of these effects are covered in the next section of this chapter on
sample probe tuning.
Sample Probe Tuning
Variations in the polarity and dielectric constant of the lock solvent will affect the probe
tuning. For this reason the probe should be tuned whenever the lock solvent is changed.
Tuning the probe entails adjusting two capacitors on the RF probe. One capacitor is called
the matching capacitor and the other the tuning capacitor. The matching capacitor matches
the impedance of the loaded probe to that of the 50 Ohm cable coming from the
spectrometer. The tuning capacitor changes the resonance frequency of the RF coil.
Most spectrometers have a probe tuning mode of operation. This mode of operation presents
a display of reflected power vs. frequency on the screen. The goal is to adjust the display so
that the reflected power from the probe is zero at the resonance frequency of the nucleus you
are examining.
As the polarity and dielectric constant of the lock solvent changes, so does the bandwidth of
the RF probe. This is significant because it affects the amount of RF power needed to
produce a 90 degree pulse. The larger the bandwidth, the more power is needed to produce
the 90 degree rotation.
Determinining a 90o Pulse
As pointed out in the previous section of this chapter, changes in the polarity and dielectric
constant of the lock solvent affect the bandwidth of the RF probe which in turn affects the
amount of RF power needed to produce a 90 degree rotation. Most NMR spectrometers will
not allow you to change the RF power, but they will permit you to change the pulse length.
Therefore, if the bandwidth of the RF probe increases, you will need to increase the RF pulse
width to produce a 90 degree pulse.
To determine the pulse width needed to produce a 90 degree pulse, you should perform the
following experiment using a sample which has a single absorption line and a relatively short
T1. Record a series of spectra with incrementally longer RF pulse widths. Fourier transform
the time domain signals and plot these lines as a function of pulse width. The peak height
should vary sinusoidally with increasing pulse width. The 90 degree pulse width will be the
first maximum. The 180 degree pulse width will be the first zero crossing. Many
spectrometers have routines which will automatically record the data necessary to produce
these plots.
You should also be aware of the effect of varying the width of the RF pulse on the
distribution of frequencies being delivered to your sample. Recall from the discussion of the
convolution theorem in Chapter 5 that the Fourier pair of a sine wave which is turned on and
off is a sinc function centered at the frequency of the sine wave. When you apply an RF
pulse of width t in the time domain, you apply a distribution of frequencies to your sample.
Not all of these frequencies will have sufficient B1 magnitude to produce a 90 degree
rotation. The range of frequencies from the center of the distribution to the first zeros in the
distribution is +/- 1/t. As your pulse width increases, the width of the distribution of
frequencies in your pulse decreases. If the distribution is too narrow, you may not be
applying the desired rotation to the entire sample.
Field Shimming
The purpose of shimming a magnet is to make the magnetic field more homogeneous and to
obtain better spectral resolution. Shimming can be performed manually or by computer
control. It is not the intent of this section to teach you a step-by-step procedure for shimming,
but to present you with the basic theory so that you can, with the aid of your NMR
instruction manual, shim your magnet. The reader is encouraged to write down or save the
current shim settings before making changes to any of the current shims coil settings.
Broad lines, asymmetric lines, and a loss of resolution are indications that a magnet needs to
be shimmed. The shape of an NMR line is a good indication of which shim is misadjusted.
Consider a single narrow NMR line. If we zoom in on this line we might see the following
shape. . The following series of spectra depict the appearance of this spectral line in the
presence of various inhomogeneities.
Shim
Spectrum
2
Z
Z3
Z4
X, Y, ZX, or ZY
XY or X2-Y2
In general, asymmetric lineshapes result from mis-adjusted even-powered Z shims. This can
be seen by looking at the shape of a Z2 shim field. As you go further away from the center
of the sample in the +Z or -Z direction, the field increases, giving more components of the
spectral line at higher fields. The higher the power of the Z inhomogeneity, the further away
the asymmetry is from the center of the line.
Symmetrically broadened lines are from mis-adjusted odd-powered Z shims. Consider the
shape of the Z3 shim field. The top of the sample (+Z) is at a higher field, resulting in
higher field spectral components, while the bottom (-Z) is at a lower field, giving more lower
field spectral components. Transverse shims (X,Y) will cause large first order or second
order spinning sidebands when the sample is spun. The shape of these inhomogeneities cause
the sample, when it is spun, to experience a periodic variation in the magnetic field. Those
shims (XY or X2-Y2) causing a spinning sample to experience two variations per cycle will
create second order spinning sidebands.
Phase Cycling
There are a few artifacts of the detection circuitry which may appear in your spectrum if you
record a single FID and Fourier transform it. Phase cycling is the technique used to eliminate
these artifacts. The artifact will be introduced first, followed by the technique used to
eliminate it.
Electronic amplifiers often have small offsets in their output when no signal is being put in.
This is referred to as the DC offset of the amplifier. A DC offset in the time domain is
equivalent to a peak at zero frequency in the frequency domain. If there is an FID on top of
a DC offset, its Fourier transform will have an additional peak at zero frequency in the
spectrum. This picture has been simplified by presenting only the real part of the signal.
The DC offset could be eliminated by spending thousands of dollars on better quality
amplifiers. Alternatively, the artifact can be removed by taking an FID recorded with a 90
degree pulse applied along +X' , an FID recorded with a 90 degree pulse applied along -X'
(note the phase change in the FID) , multiplying the FID recorded with a 90 degree pulse
along -X' by -1 , adding the two FIDs, and Fourier transforming. This process only costs a
little extra time and a few extra lines of computer code.
Another type of artifact is caused by having unequal gains on the real and imaginary outputs
of the quadrature detector. For a Fourier transform to produce a proper spectrum, it requires
true real and imaginary inputs. When the inputs are equal in amplitude, there are no
negative frequency artifacts in the spectrum. If the two inputs are different, the negative
frequency components of a signal do not cancel. You can tell a negative frequency artifact
because it appears to be the mirror image (but smaller) of a peak from the opposite sign end
of the spectrum.
Negative frequency artifacts can be removed by recording an FID with Mx or the real signal
(My or the imaginary signal) from channel 1 (2) of the quadrature detector. Another FID is
recorded with Mx or the real signal (My or the imaginary signal) from channel 2 (1) of the
quadrature detector. The two FIDs are then averaged. As a result, the amplitude of the real
and imaginary inputs to the FT are equal, so when the FIDs are Fourier transformed, there are
no negative frequency artifacts.
The averaging described above can be achieved by applying a 90 degree pulse about +X and
a 90 degree pulse about +Y, and adding the two resulting FIDs together. To eliminate all
possible errors from different combinations of these types of pulses, phase cycling is applied.
Phase cycling adds together eight FIDs recorded with the following phases to eliminate all
the possible quadrature artifacts.
1-D Hydrogen Spectra
There are several parameters, in addition to the ones already discussed in this chapter, which
must be set before a spectrum can be recorded. These include the width of the spectrum,
number of data points in the spectrum, and the receiver gain. Some of these are automatically
set to default values on some spectrometers. You are encouraged to refer to Chapter 5 for a
deeper appreciation of the significance of these parameters.
Once an FID is recorded and Fourier transformed, the resultant spectrum must be phased so
that all the absorption lines are positive. You are encouraged to review Chapter 5 for an
explanation of the need to phase correcting a spectrum. There are various automatic and
manual phase correction algorithms on most NMR spectrometers.
Here are a few examples of simple hydrogen NMR spectra to demonstrate the capabilities of
NMR spectroscopy. As you become more knowledgeable about NMR, you will learn the
relationship between peak locations, peak splitting, and molecular structure in NMR spectra.
Molecule
cyclohexane
Formula
C6H12
Solvent Spectrum
CDCl3
benzene
C6H6
CDCl3
toluene
C6H5CH3
CDCl3
ethyl benzene
C6H5CH2CH3
CDCl3
acetone
CH3(C=O)CH3
CDCl3
methyl ethyl ketone CH3(C=O)CH2CH3
CDCl3
water
H2O
D2O
ethanol
CH3CH2OH
CDCl3
ethanol
CH3CH2OH
D2O
1-propanol
CH3CH2CH2OH
CDCl3
2-propanol
(CH3)2CHOH
CDCl3
t-butanol
(CH3)3COH
CDCl3
2-butanol
CH3CH2CH(OH)CH3 CDCl3
pyridine
C5H5N
CDCl3
Integration
In addition to chemical shift and spin-spin coupling information, there is one additional piece
of information which the chemist can use in determining the structure of a molecule from an
NMR spectrum. This information is the relative area of absorption peaks in the spectrum.
Here an absorption peak is defined as the family of peaks centered at a particular chemical
shift. For example, if there is a triplet of peaks at a specific chemical shift, the number is the
sum of the area of the three. The rule is that peak area is proportional to the number of a
given type of spins in the molecule and in the sample. An example should help you
understand this relationship.
Consider the methyl ethyl ketone (CH3CH2(C=O)CH3) molecule and its hydrogen NMR
spectrum. When the -CH2- ( = 2.25 ppm), -CH3 ( = 2.0 ppm), and CH3- ( = 0.9 ppm)
peaks are integrated we get the following spectrum. The areas under the three types of
peaks on this spectrometer are 26:39:39. Dividing each number by 13, we obtain a 2:3:3 ratio
which is proportional to the number of -CH2- to -CH3 to CH3- hydrogens.
There are a few assumptions which were made in presenting this rule.




The T1 and T2 values of all the spins are equal.
There is no spin decoupling being performed.
The signal-to-noise ratio is good.
There is no sloping baseline in the spectrum.
Spin decoupling will be discussed in Chapter 9.
You may correct for a sloping baseline by performing a baseline correction to the spectrum.
A poor signal-to-noise ratio may be improved by performing signal averaging, discussed
next.
SNR Improvement
The signal-to-noise ratio (SNR) of a spectral peak is the ratio of the average height of the
peak to the standard deviation of the noise height in the baseline. Often spectroscopists
approximate this quantity as the average peak height divided by the amplitude of the noise in
the baseline. The signal to noise ratio may be improved by performing signal averaging.
Signal averaging is the collection and averaging together of several spectra. The signals are
present in each of the averaged spectra so their contribution to the resultant spectrum add.
Noise is random so it does not add, but begins to cancel as the number of spectra averaged
increases. The signal-to-noise improvement from signal averaging is proportional to the
square root of the number of spectra (N) averaged.
SNR N1/2
Because of the need to perform phase cycling, you will need to have the number of averages
equal to a multiple of the minimum number of phase cycling steps. Compare the results of
averaging together the following number of spectra of a very dilute solution of methyl ethyl
ketone.
N
N1/2 Spectrum
1
1.00
8
2.83
16 4.00
80 8.94
800 28.28
Variable Temperature
Many NMR spectrometers have the ability to control the temperature of the sample in the
probe. A schematic representation of the variable temperature hardware on an NMR
spectrometer is depicted in the animation window. All of these spectrometers permit you to
set the temperature to values above room temperature by just entering the desired
temperature. You should be careful not to exceed the maximum temperature allowable for
your probe because doing so will melt adhesives and components in the probe. Controlling
the temperature below room temperature requires the use of hardware to cool the gas flowing
over the sample. If this gas is air, it must be dry air to avoid condensation of water on the
sample. Once the sample and probe have been cooled or heated, you should slowly return the
probe to room temperature. Do not expose a cold probe to the moist atmosphere;
condensation will result.
Troubleshooting
By now you may realize that an NMR spectrometer is a complex piece of instrumentation
with many sub systems which must be functioning properly in order to record a useable
NRM spectrum. The intent of this section is to provide you with a systematic method of
identifying a problem with the spectrometer. Once a problem is identified, you are not
necessarily expected to be able to solve it, but you will at least be able to describe the steps
you took to diagnose the problem when speaking to a system administrator or a service
representative from the manufacturer of your spectrometer. Click on this icon to start the
diagnosis process in the animation window.
Cryogen Fills
Superconducting magnets require liquid nitrogen (N2) and liquid Helium (He). Because it is
difficult to make a perfect dewar to hold these cryogens, they need to be periodically
replenished. Liquid nitrogen is typically filled every 7 to 10 days and liquid helium every 200
to 300 days. Cryogen fills must be performed correctly to avoid injury to you and the magnet.
The injuries to you from cryogenic liquids were described in Chapter 7. Injury to a magnet
could include breaking a seal on a dewar or quenching a magnet. Both forms of magnet
injuries are repairable, but at the least entail recharging the magnet; at the most, they can
entail replacing the magnet.
When filling the magnet with liquid nitrogen, you must be sure not to exceed the
recommended fill pressure and rate for your magnet. If your magnet has two liquid nitrogen
ports, one should be used for filling and the other for venting the boil-off gaseous nitrogen
and overfill liquid nitrogen. A piece of tubing is typically placed on the vent port to direct the
overfill liquid nitrogen away from the magnet seals, probe, and electronics. It is highly
recommended that your liquid nitrogen tanks be made of non-magnetic stainless steel.
Liquid helium fills are typically a two-person operation. Because they are done so
infrequently, it is good to review the process before each fill. The fill requires a supply dewar
of liquid helium, a special liquid helium transfer line, and a tank of pure compressed helium
gas. Liquid helium is transferred from the liquid helium supply dewar up through the
transfer line, into the helium dewar of the magnet.
The transfer line goes into the top of the liquid helium supply dewar, but should never rest on
the bottom of the dewar. The bottom of the dewar may contain frozen water, oxygen, and
nitrogen which will be forced into your magnet if the transfer line touches the bottom during
the transfer process. The compressed helium gas, mentioned earlier, is for pressurizing the
liquid helium supply dewar with about 2 to 4 psi of pressure. Gauges on helium supply
dewars can be very inaccurate, so do not count on them to give you an accurate reading. A
helium pressure above the liquid forces the Helium into the magnet dewar.
The transfer line is usually inserted into the magnet until it contacts a transfer flange in the
bottom of the magnet. The nitrogen ports on the magnet should be plugged with a check
valve during filling of the helium dewar of the magnet. This step prevents cryopumping, a
process whereby nitrogen, water, and oxygen are condensed out of the atmosphere into the
nitrogen dewar due to the magnet stacks being cooled by the helium. Many labs loosely plug
the helium vents with tissue during the fill. This cuts down on cryopumping should the flow
of the venting He drop.
The best way to determine if the magnet is full is to look for a change in the gas cloud
coming out of the magnet vents. When the magnet is full the cloud becomes very thick with a
deep white center plume with a slight blue tint. The helium vents on the magnet should be
closed promptly after the magnet is full.
Unix Primer
Most NMR spectrometers are controlled by a computer workstation. The NMR program
which gives your spectrometer the look and feel you are used to is running on this computer.
This computer is most likely running a UNIX operating system. The operating system is
equivalent to DOS on a Microsoft system or OS-5 on a Macintosh system. Although you may
be able to perform all the file transfer and manipulation commands from your NMR program,
you may find it useful to know a few UNIX commands. This chapter is intended to give you
enough information about UNIX to perform simple tasks in the UNIX operating system.
The UNIX file system is divided into directories, which are equivalent to folders in some
operating systems. Because UNIX is a multi-user system, there must be a way to keep your
directories separate (and safe) from someone else's. To achieve this, there are accounts with
passwords and ownership of directories. For example, you have an account which has a
password. Logging on under your account gives you access to your directories and to other
directories for which you have access (permission).
The most useful, but least used command in UNIX is man. This is short for manual and gives
you on-line help on every UNIX command. The more you use it, the easier it is to use. The
animation window contains a table of a few simple UNIX commands. Entries in italics are
examples and can be any string of characters or numbers.
The Basics of NMR
Chapter 9
CARBON-13 NMR





Introduction
Decoupling
NOE
Population Inversion
1-D C-13 Spectra
Introduction
Many of the molecules studied by NMR contain carbon. Unfortunately, the carbon-12
nucleus does not have a nuclear spin, but the carbon-13 (C-13) nucleus does due to the
presence of an unpaired neutron. Carbon-13 nuclei make up approximately one percent of the
carbon nuclei on earth. Therefore, carbon-13 NMR spectroscopy will be less sensitive (have
a poorer SNR) than hydrogen NMR spectroscopy. With the appropriate concentration, field
strength, and pulse sequences, however, carbon-13 NMR spectroscopy can be used to
supplement the previously described hydrogen NMR information. Advances in
superconducting magnet design and RF sample coil efficiency have helped make carbon-13
spectroscopy routine on most NMR spectrometers.
The sensitivity of an NMR spectrometer is a measure of the minimum number of spins
detectable by the spectrometer. Since the NMR signal increases as the population difference
between the energy levels increases, the sensitivity improves as the field strength increases.
The sensitivity of carbon-13 spectroscopy can be increased by any technique which increases
the population difference between the lower and upper energy levels, or increases the density
of spins in the sample. The population difference can be increased by decreasing the sample
temperature or by increasing the field strength. Several techniques for increasing the carbon13 signal have been reported in the NMR literature.
Unfortunately, or fortunately, depending on your perspective, the presence of spin-spin
coupling between a carbon-13 nucleus and the nuclei of the hydrogen atoms bonded to the
carbon-13, splits the carbon-13 peaks and causes an even poorer signal-to-noise ratio. This
problem can be addressed by the use of a technique known as decoupling, addressed in the
next section.
Decoupling
The signal-to-noise ratio in an NMR spectrometer is related to the population difference
between the lower and upper spin state. The larger this difference the larger the signal. We
know from chapter 3 that this difference is proportional to the strength of the Bo magnetic
field.
To understand decoupling, consider the familiar hydrogen NMR spectrum of HC-(CH2CH3)3.
The HC hydrogen peaks are difficult to see in the spectrum due to the splitting from the 6 CH2- hydrogens. If the effect of the 6 -CH2- hydrogens could be removed, we would lose the
1:6:15:20:15:6:1 splitting for the HC hydrogen and get one peak. We would also lose the
1:3:1 splitting for the CH3 hydrogens and get one peak. The process of removing the spinspin splitting between spins is called decoupling. Decoupling is achieved with the aid of a
saturation pulse. If the affect of the HC hydrogen is removed, we see the following spectrum.
Similarly, if the affect of the -CH3 hydrogens is removed, we see this spectrum.
A saturation pulse is a relatively low power B1 field left on long enough for all magnetization
to disappear. A saturation pulse applied along X' rotates magnetization clockwise about X'
several times. As the magnetization is rotating, T2 processes cause the magnetization to
dephase. At the end of the pulse there is no net Z, X, or Y magnetization. It is easier to see
this behavior with the use of plots of MZ, MX', and MY' as a function of time. Since the B1
pulse is long, its frequency content is small. It therefore can be set to coincide with the
location of the -CH2- quartet and saturate the -CH2- spin system. By saturating the -CH2spins, the -CH2- peaks and the splittings disappear, causing the height of the now unsplit HCand -CH3 peaks to be enhanced.
Now that the concept of decoupling has been introduced, consider the carbon-13 spectrum
from CH3I. The NMR spectrum from the carbon-13 nucleus will yield one absorption peak in
the spectrum. Adding the nuclear spin from one hydrogen will split the carbon-13 peak into
two peaks. Adding one more hydrogen will split each of the two carbon-13 peaks into two,
giving a 1:2:1 ratio. The final hydrogen will split each of the previous peaks, giving a
1:3:3:1 ratio. If the hydrogen spin system is saturated, the four lines collapse into a single
line having an intensity which is eight times greater than the outer peak in the 1:3:3:1 quartet
since 1+3+3+1=8 . In reality, we see a single line with a relative intensity of 24. Where
did the extra factor of three come from?
NOE
The answer to the question raised in the previous paragraph is the nuclear Overhauser effect
(NOE). To understand the NOE, consider a set of coupled hydrogen and carbon-13 nuclei.
Assume that the red-green nuclei are carbon-13 and the blue-pink nuclei are hydrogen.
T1CC is T1 relaxation due to interactions between carbon-13 nuclei. T1HH is T1 relaxation due
to interactions between hydrogen nuclei. T1CH is T1 relaxation due to interactions between
carbon-13 and hydrogen nuclei.
MZ(C) is the magnetization due to carbon-13 nuclei. Mo(C) is the equilibrium magnetization
of carbon-13. MZ(H) is the magnetization due to hydrogen nuclei. Mo(H) is the equilibrium
magnetization of hydrogen.
The equations governing the change in the Z magnetization with time are:
If we saturate the hydrogen spins, MZ(H) = 0.
Letting the system equilibrate, d MZ(C) /dt = 0 .
Rearranging the previous equation, we obtain an equation for MZ(C) .
.
Note that MZ(C) has increased by Mo(H) T1CC / T1CH which is approximately 2 Mo(C), giving
a total increase of a factor of 3 relative to the total area of the undecoupled peaks. This
explains the extra factor of three (for a total intensity increase of 24) for the carbon-13 peak
when hydrogen decoupling is used in the carbon-13 spectrum of CH3I.
The following spin-echo sequence has been modified to decouple the hydrogen spins from
the carbon-13 spins. The signal is recorded as the second half of the echo.
Population Inversion
Another method of improving the NMR signal in systems with spin-spin coupling is
population inversion. To understand the concept of a population inversion, recall from
Chapter 3 that Boltzmann statistics tell us that there are more spins in the lower spin state
than the upper one of a two spin state system. Population inversion is the interchange of the
populations of these two spin states so that there are more spins in the upper state then the
lower one.
To understand how a population inversion improves the signal-to-noise ratio in a spectrum,
consider the CHI3 molecule. CHI3 will have four energy levels (L1, L2, L3, and L4) due to
C-H spin-spin coupling. There are two carbon-13 absorption frequencies f1 and f2 and two
hydrogen absorption frequencies f3 and f4. The population distribution between the four
levels is such that the lowest state has the greatest population and the highest the lowest
population. The two intermediate states will have populations between the outer two as
indicated by the thickness of the levels in the accompanying diagram. The four lines in the
spectrum will have intensities related to the population difference between the two levels
spanned by the frequency. The two carbon-13 absorption lines (f1 and f2) will have a lower
intensity than the hydrogen lines (f3 and f4) due to the smaller population difference between
the two states joined by f1 and f2.
If the populations of L3 and L1 are inverted or interchanged with a frequency selective 180
degree pulse at f3, the signal at f2 will be enhanced because of the greater population
difference between the states joined by f2. It should be noted that the signal at f1 will be
inverted because the upper state of the two joined by f1 has a greater population than the
lower one. An example of a population inverting pulse sequence designed to enhance the
carbon-13 spectral lines is depicted in the animation window. The 180 degree pulse at f3
has a narrow band of frequencies centered on f3 that selectively rotates only the
magnetization at f3 by 180 degrees.
1-D C-13 Spectra
The following table of compounds contains links to their corresponding one-dimensional
carbon-13 NMR spectra. The spectra were recorded on a 300 MHz NMR spectrometer with a
delay time between successive scans of two seconds. This relatively short delay time may
cause differences in the peak heights due to variations in T1 values. Other differences may be
caused by variations in the nuclear Overhauser effect. In spectra recorded with deuterated
chloroform (CDCl3) as the lock solvent, the three peaks at = 75 are due to splitting of the
CDCl3 carbon-13 peak by the nuclear spin = 1 deuterium nucleus.
Molecule
Formula
Solvent Spectrum
cyclohexane
C6H12
CDCl3
benzene
C6H6
CDCl3
toluene
C6H5CH3
CDCl3
ethyl benzene
C6H5CH2CH3
CDCl3
acetone
CH3(C=O)CH3
CDCl3
methyl ethyl ketone CH3(C=O)CH2CH3
CDCl3
ethanol
CH3CH2OH
CDCl3
ethanol
CH3CH2OH
D2O
1-propanol
CH3CH2CH2OH
CDCl3
2-propanol
(CH3)2CHOH
CDCl3
t-butnol
(CH3)3COH
CDCl3
2-butanol
CH3CH2CH(OH)CH3 CDCl3
pyridine
C5H5N
CDCl3
The Basics of NMR
Chapter 10
2-D TECHNIQUES




Introduction
J-resolved
COSY
Examples
Introduction
In Chapter 6 we saw the mechanics of the spin echo sequence. Recall that a 90 degree pulse
rotates magnetization from a single type of spin into the XY plane. The magnetization
dephases, and then a 180 degree pulse is applied which refocusses the magnetization.
When a molecule with J coupling (spin-spin coupling) is subjected to a spin-echo sequence,
something unique but predictable occurs. Look at what happens to the molecule A2-C-C-B
where A and B are spin-1/2 nuclei experiencing resonance. The NMR spectrum from a 90FID sequence looks like this.
With a spin-echo sequence this same molecule gives a rather peculiar spectrum once the echo
is Fourier transformed. Here is a series of spectra recorded at different TE times. The
amplitude of the peaks have been standardized to be all positive when TE=0 ms.
To understand what is happening, consider the magnetization vectors from the A nuclei.
There are two absorptions lines in the spectrum from the A nuclei, one at +J/2 and one at J/2. At equilibrium, the magnetization vectors from the +J/2 and -J/2 lines in the spectrum
are both along +Z.
A 90 degree pulse rotates both magnetization vectors into the XY plane. Assuming a
rotating frame of reference at o = , the vectors precess according to their Larmor frequency
and dephase due to T2*. When the 180 degree pulse is applied, it rotates the magnetization
vectors by 180 degrees about the X' axis. In addition the +J/2 and -J/2 magnetization
vectors change places because the 180 degree pulse also flips the spin state of the B nucleus
which is causing the splitting of the A spectral lines.
The two groups of vectors will refocus as they evolve at their own Larmor frequency. In
this example the precession in the XY plane has been stopped when the vectors have
refocussed. You will notice that the two groups of vecotrs do not refocus on the -Y axis. The
phase of the two vectors on refocussing varies as a function of TE. This phase varies as a
function of TE at a rate equal to the size of the spin-spin coupling frequency. Therefore,
measuring this rate of change of phase will give us the size of the spin-spin coupling
constant. This is the basis of one type of two-dimensional (2-D) NMR spectroscopy.
J-resolved
In a 2-D J-resolved NMR experiment, time domain data is recorded as a function of TE and
time. These two time dimensions will referred to as t1 and t2. For the A2-C-C-B molecule, the
complete time domain signals look like this.
This data is Fourier transformed first in the t2 direction to give an f2 dimension, and then in
the t1 direction to give an f1 dimension.
Displaying the data as shaded contours, we have the following two-dimensional data set.
Rotating the data by 45 degrees makes the presentation clearer. The f1 dimension gives us J
coupling information while the f2 dimension gives chemical shift information. This type of
experiment is called homonuclear J-Resolved 2-D NMR. There is also heteronuclear Jresolved 2-D NMR which uses a spin echo sequence and techniques similar to those
described in Chapter 9.
COSY
The application of two 90 degree pulses to a spin system will give a signal which varies with
time t1 where t1 is the time between the two pulses. The Fourier transform of both the t1 and
t2 dimensions gives us chemical shift information. The 2-D hydrogen correlated chemical
shift spectrum of ethanol will look like this. There is a wealth of information found in a
COSY spectrum. A normal (chemical shift) 1-D NMR spectrum can be found along the top
and left sides of the 2-D spectrum. Cross peaks exist in the 2-D COSY spectrum where there
is spin-spin coupling between hydrogens. There are cross peaks between OH and CH2
hydrogens , and also between CH3 and CH2 hydrogens hydrogens. There are no cross
peaks between the CH3 and OH hydrogens because there is no coupling between the CH3 and
OH hydrogens.
Heteronuclear correlated 2-D NMR is also possible and useful.
Examples
The following table presents some of the hundreds of possible 2-D NMR experiments and the
data represented by the two dimensions. The interested reader is directed to the NMR
literture for more information.
2-D Experiment (Acronym)
Information
f1
f2
Homonuclear J resolved
J
Heteronuclear J resolved
JAX
X
Homoculclear correlated spectroscopy (COSY)
A
A
Heteronuclear correlated spectroscopy (HETCOR)
A
X
Nuclear Overhauser Effect (2D-NOE)
H, JHH
H, JHH
2D-INADEQUATE
A
+
X
X
The following table of molecules contains links to their corresponding two-dimensional
NMR spectra. The spectra were recorded on a 300 MHz NMR spectrometer with CDCl3 as
the lock solvent.
Molecule
Formula
methyl ethyl ketone CH3(C=O)CH2CH3
Type Spectrum
COSY
ethanol
CH3CH2OH
COSY
1-propanol
CH3CH2CH2OH
COSY
2-propanol
(CH3)2CHOH
COSY
2-butanol
CH3CH2CH(OH)CH3 COSY
ethyl benzene
C6H5CH2CH3
COSY
pyridine
C5H5N
COSY
The Basics of NMR
Chapter 11
ADVANCED SPECTROSCOPIC TECHNIQUES
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Introduction
Diffusion
Spin Relaxation Time
Solid State
Microscopy
Solvent Suppression
Field Cycling NMR
Introduction
Nuclear magnetic resonance spectroscopy is one of the richest spectroscopies available. In
previous chapters you have seen how it can be used to elucidate chemical structure. In this
chapter you will see some of the other more advanced applications of NMR. Two of these
techniques, solid state NMR and gradient enhanced spectroscopy, will assist us further in the
determination of molecular structure. Two additional techniques will assist us in studying
molecular dynamics, or the rotational and translational motions of molecules. The last
technique, NMR microscopy, will enable us to determine the spatial distribution of nuclear
spins in a sample.
Diffusion
Diffusion is the motion of particles due to Brownian motion. The diffusion coefficient, D, is
a measure of the diffusion. The pulsed-gradient spin-echo sequence permits us to measure the
diffusion coefficient. The sequence is in theory capable of measuring both the rotational and
translational diffusion coefficients, but is used primarily for studying translational diffusion.
To understand how the pulsed-gradient spin-echo sequence allows us to measure diffusion,
consider the timing diagram for the sequence. This sequence is very similar to the spin echo
sequence introduced in Chapter 6, except that two gradient pulses have been applied. These
two gradient pulses are identical in amplitude, G, and width, . The two gradient pulses are
separated by a time and are placed symmetrically about the 180 degree pulse.
The function of the gradient pulses is to dephase magnetization from spins which have
diffused to a new location in the period . These pulses have no effect on stationary spins.
For example, a stationary spin exposed to the first gradient pulse, applied along the Z axis,
will acquire a phase in radians given by
=2
z Gz dt .
The spin will acquire an equal but opposite phase from the second pulse since the pulses are
on different sides of the 180 degree RF pulse. Thus, their effects cancel each other out.
Consider the following illustration of the effect of the gradient pulses on the phase of
stationary and moving spins. The illustration presents the phase of a diffusing spin relative
to that of a reference spin and a stationary spin. The reference spin is one which experiences
no gradient pulses. The stationary spin is not diffusing during the time illustrated by the
sequence. The diffusing spin moves along Z during the sequence. The blue line in the timing
diagram represents the time of the 180 degree pulse in the spin echo sequence. When you put
the illustration into motion, the stationary spin comes back into phase with the reference one,
indicating a positive contribution to the echo. The diffusing spin does not come back into
phase with the reference spin so it diminishes the echo height.
The relationship between the signal (S) obtained in the presence of a gradient amplitude Gi in
the i direction and the diffusion coefficient in the same direction is given by the following
equation where So is the signal at zero gradient.
S/So = exp[-(Gi
)2 Di (
- /3)]
The diffusion coefficient is typically calculated from a plot of ln(S/So) versus (G )2 (
/3). Diffusion in the x, y, or z direction may be measured by applying the gradient
respectively in the x, y, or z direction.
-
Spin Relaxation Time
The spin-lattice and spin-spin relaxation times, T1 and T2 respectively, of the components of
a solution are valuable tools for studying molecular dynamics. You saw in Chapter 3 that T1-1
is proportional to the number of molecular motions at the Larmor frequency, while T2-1 is
proportional to the number of molecular motions at frequencies less than or equal to the
Larmor frequency. When we are dealing with solutions these motions are predominantly
rotational motions.
There are many pulse sequences which may be used to measure T1 and T2. The inversion
recovery, 90-FID, and spin-echo sequences may be used to measure T1. Each technique has
its own advantages and disadvantages. The spin-echo sequence may be used to measure T2.
T1 Measurement
Recall the timing diagram for an inversion recovery sequence first presented in Chapter 6.
The signal as a function of TI when the sequence is not repeated is
S = k ( 1 - 2eTI/T1 ) .
If the curve is well defined (i.e. if there is a high density of data points recorded at different
TI times), the T1 value can be determined from the zero crossing of the curve which is T1 ln2.
Alternatively the relaxation curve as a function of TI may be fit using the equation
S = So (1 - 2e-TI/T1).
This approach is favored when there are fewer data points as a function of TI.
T1 may also be determined from a 90-FID or spin-echo sequence which is repeated at
various repetition times (TR). For example, if the 90-FID sequence is repeated many times at
TR1 and then many times at TR2, TR3, etc, the plot of signal as a function of TR will be an
exponential growth of the form
S = k ( 1 - eTR/T1 ) .
This data may be fit to obtain T1.
The difficulty with fitting this data and the inversion recovery data is a lack of knowledge of
the value of the equilibrium magnetization or signal So. Other techniques have been proposed
which do not require knowledge of the equilibrium magnetization or signal .
T2 Measurement
Measurement of the spin-spin relaxation time requires the use of a spin-echo pulse sequence.
The echo amplitude, S, as a function of echo time, TE, is exponentially decaying. Plotting
ln(S/So) versus TE yields a straight line, the slope of which is -1/T2. A linear least squares
algorithm is often used to find the slope and hence T2 value. This approach can result in lead
to large errors in the calculated T2 values when the data has noise. The later points in the
decay curve have poorer signal-to-noise ratio than the earlier points, but are given equal
weight by the linear least squares algorithm. The solution to this problem is to use a nonlinear least squares procedure.
Solid State
We saw in Chapter 4 that the magnitude of the chemical shift is related to the extent to which
the electron can shield the nucleus from the applied magnetic field. In a spherically
symmetric molecule, the chemical shift is independent of molecular orientation. In an
asymmetric molecule, the chemical shift is dependent on the orientation. The magnetic field
experienced by the nucleus varies as a function of the orientation of the molecule in the
magnetic field.
The NMR spectrum from a random distribution of fixed orientations, such
as in a solid, would look like this. The larger signal at lower field strength is due to the fact
that there are more perpendicular orientations. In a nonviscous liquid, the fields at the various
orientations average out due to the tumbling of the molecule.
The anisotropic chemical shift is one reason why the NMR spectra of solid samples display
broad spectral lines. Another reason for broad spectral lines is dipolar broadening. A dipolar
interaction is one between two spin 1/2 nuclei. The magnitude of the interaction varies with
angle and distance r.
As a function of , the magnetic field B experienced by the red
nucleus is
(3cos2 - 1).
A group of dipoles with a random distribution of orientations, as in a solid, gives this
spectrum. The higher signal at mid-field strength is due to the larger presence of
orientations perpendicular to the direction of the Bo field. This signal is made up of
components from the red and blue nuclei in the dipole. In a nonviscous liquid, the interaction
averages out due to the presence of rapid tumbling of the molecule.
When the angle in the above equation is 54.7o, 125.3o, 234.7o, or 305.3o, the dipole
interaction vanishes. The angle 54.7o is called the magic angle, m.
If all the molecules could be positioned at
tumbling limit.
m,
the spectrum would narrow to the fast
Since this is not possible, the next best thing is to cause the average orientation of the
molecules to be m.
Even this is not exactly possible, but the closest approximation is to rapidly spin the entire
sample at an angle m relative to Bo. In solid state NMR, samples are placed in a special
sample tube and the tube is placed inside a rotor. The rotor, and hence the sample, are
oriented at an angle m with respect to the Bo magnetic field. The sample is then spun at a
rate of thousands of revolutions per second.
The spinning rate must be comparable to the solid state line width. The centrifigal force
created by spinning the sample tube at a rate of several thousands of revolutions per second is
enough to destroy a typical glass NMR sample tube. Specially engineered sample tubes and
rotors are needed.
Microscopy
NMR microscopy is the application of magnetic resonance imaging (MRI) principles to the
study of small objects. Objects which are studied are typically less than 5 mm in diameter.
NMR microscopy requires special hardware not found on conventional NMR spectrometers.
This includes gradient coils to produce a gradient in the magnetic field along the X, Y, and Z
axes; gradient coil drivers; RF pulse shaping software; and image processing software.
Resultant images can have 20 to 50 m resolution. The reader interested in more information
on NMR microscopy is encouraged to read the author's hypertext book on MRI entitled The
Basics of MRI located at http://www.cis.rit.edu/htbooks/mri/.
Solvent Suppression
Occasionally, it becomes necessary to eliminate the signal from one constituent of a sample.
An example is an unwanted water signal which overwhelms the signal from the desired
constituent. If T1 of the two components differ, this may be accomplished by using an
inversion recovery sequence, presented in Chapter 6. To eliminate the water signal, choose
the TI to be the time when the water signal passes through zero.
TI = T1 ln2
In this example, a TI = 1 s would eliminate the water signal.
Another method of eliminating a solvent absorption signal is to saturate it. In this procedure,
a saturation pulse similar to that employed in C-13 NMR (See Chapter 9) is used to decouple
hydrogen coupling. The frequency of the saturation pulse is set to the solvent resonance. The
width of the saturation pulse is very long, so its bandwidth is very small causing it to affect
only the solvent resonance.
Field Cycling NMR
Field cycling NMR spectroscopy is used to obtain spin-lattice relaxation rates, R1, where
R1 = 1/T1 ,
as a function of magnetic field or Larmor frequency. Therefore, field cycling NMR finds
applications in the study of molecular dynamics. The animation window contains an example
of results from a field cycling NMR spectrometer. The plot represents the R1 value of the
hydrogen nuclei in various concentration aqueous solutions of Mn+2 at 25o as a function of the
proton Larmor frequency.
Many different techniques have been used to obtain R1 as a function of magnetic field.
Some techniques move the sample rapidly between different magnetic field strengths. One of
the more popular techniques keeps the sample at a fixed location and rapidly varies the
magnetic field the sample experiences. This technique is referred to as rapid field cycling
NMR spectroscopy.
The principle behind a rapid field cycling NMR spectrometer is to polarize the spins in the
sample using a high magnetic field, Bp. The magnetic field is rapidly changed to the value at
which relaxation occurs, Br. Br is the value at which R1 is to be determined. After a period of
time, , the magnetic field is switched to a value, Bd, at which detection of a signal occurs. Bd
is fixed so that the operating frequency of the detection circuitry does not need to be
changed. The signal, an FID, is created by the application of a 90o RF pulse. The timing
diagram for this sequence can be found in the animation window.
The FT of the FID represents the amount of magnetization present in the sample after
relaxing for a period  in Br. A plot of this magnetization as a function of  is an
exponentially decaying function, starting from the equilibrium magnetization at Bp and going
to the value at Br. When a single type of spin is present, the relaxation is monoexponential
with rate constant R1 at Br.
When Br is very large compared to Bd, Bp is often set to zero and the plot of this
magnetization as a function of  is an exponentially growing function.
Glossary
Artifact
A feature which appears in an NMR spectrum of a molecule which should not be
present based on the chemical structure and pulse sequence used. [Chapter 7]
Chemical Screening
The screening of an applied magnetic field experienced by a nucleus due to the
electron cloud around an atom or molecule. [Chapter 4]
Chemical Shift
A variation in the resonance frequency of a nuclear spin due to the chemical
environment around the nucleus. Chemical shift is reported in ppm. [Chapter 4]
Coil
One or more loops of a conductor used to create a magnetic field. In NMR, the term
generally refers to the radiofrequency coil. [Chapter 7]
Convolution
A mathematical operation between two functions. [Chapter 2]
Complex Data
Numerical data with a real and an imaginary component. [Chapter 2]
Continuous Wave (CW)
A form of spectroscopy in which a constant amplitude electromagnetic wave is
applied. [Chapter 3]
Coordinate Transformation
A change in the axes used to represent some spatial quantity. [Chapter 2]
Cryopumping
The condensation of air onto a surface cooled by a cryogenic liquid . [Chapter 8]
Dephasing Gradient
A magnetic field gradient used to dephase transverse magnetization. [Chapter 11]
Digital Filtering
A feature found on may newer spectrometers which eliminates wraparound artifacts
by filtering out the higher frequency components in the time domain spectrum.
[Chapter 7]
Doubly balanced mixer
An electrical device, often referred to as a product detector, which is used in NMR to
convert signals from the laboratory frame of reference to the rotating frame of
reference. [Chapter 7]
Echo
A form of magnetic resonance signal from the refocusing of transverse magnetization.
[Chapter 6]
Echo Time ( TE )
The time between the 90 degree pulse and the maximum in the echo in a spin-echo
sequence. [Chapter 6]
Exchange, Chemical
The interchange of chemically equivalent components on a molecule. [Chapter 3]
Exchange, Spin
The interchange of spin state between two nuclei. [Chapter 3]
Figure-8 Coil
A magnetic field gradient coil shaped like the number eight. [Chapter 7]
Free induction decay ( FID )
A form of magnetic resonance signal from the decay of transverse magnetization.
[Chapter 4]
Fourier transform ( FT )
A mathematical technique capable of converting a time domain signal to a frequency
domain signal and vice versa. [Chapter 5]
Gradient ( G )
A variation in some quantity with respect to another. In the context of NMR, a
magnetic field gradient is a variation in the magnetic field with respect to distance.
[Chapter 7]
Gyromagnetic Ratio
The ratio of the resonance frequency to the magnetic field strength for a given
nucleus. [Chapter 3]
Imaginary Component
The component of a signal perpendicular to the real signal. [Chapter 5]
Imaging Sequence
A specific set of RF pulses and magnetic field gradients used to produce an image.
[Chapter 11]
Inversion Recovery Sequence
A pulse sequence producing signals which represent the longitudinal magnetization
present after the application of a 180o inversion RF pulse. [Chapter 6
Inversion Time (TI)
The time between the inversion pulse and the sampling pulse(s) in an inversion
recovery sequence. [Chapter 6]
K-Space
That image space represented by the time and phase raw data. The Fourier transform
of k-space is the magnetic resonance image. [Chapter 5]
Larmor frequency
The resonance frequency of a spin in a magnetic field. The rate of precession of a
spin packet in a magnetic field. The frequency which will cause a transition between
the two spin energy levels of a nucleus. [Chapter 3]
Longitudinal Magnetization
The Z component of magnetization. [Chapter 3]
Lorentzian Lineshape
A function obtained from the Fourier transform of an exponential function. [Chapter
5]
Magnitude
The length of a magnetization vector. In NMR, the square root of the sum of the
squares of the Mx and My components, i.e. the magnitude of the transverse
magnetization. [Chapter 2]
Magnetic Resonance Imaging (MRI)
An imaging technique based on the principles of NMR. [Chapter 11]
Negative Frequency Artifact
The appearance of smaller in amplitude peaks in one half of the spectrum which are
the mirror image of ones in the opposite half. [Chapter 8]
Net Magnetization Vector
A vector representing the sum of the magnetization from a spin system. [Chapter 3]
Nuclear Magnetic Resonance (NMR)
A spectroscopic technique used by scientists to elucidate chemical structure and
molecular dynamics. [Chapter 1]
Pixel
Picture element. [Chapter 1]
Precess
A rotational motion of a vector about the axis of a coordinate system where the polar
angle is fixed and the azmuthal angle changes steadily. [Chapter 3]
Proportionality Constant
A constant used to convert one set of units to another. [Chapter 8]
Pulse Sequence
A series of RF pulses and/or magnetic field gradients applied to a spin system to
produce a signal whose behavior gives information about some property of the spin
system. [Chapter 4]
Quadrature Detection
Detection of Mx and My simultaneously as a function of time. [Chapter 9]
Radio Frequency
A frequency band in the electromagnetic spectrum with frequencies in the millons of
cycles per second. [Chapter 3]
Raw data
The Mx and My data as a function of time and/or other parameters in an NMR pulse
sequence. This is also called k-space data. [Chapter 10]
Real
The component of a signal perpendicular to the imaginary signal. [Chapter 2]
Repetition Time
The time between repetitions of the basic sequence in a pulse sequence. [Chapter 6]
Resonance
An exchange of energy between two systems at a specific frequency. [Chapter 3]
RF Coil
An inductor-capacitor resonant circuit used to set up B1 magnetic fields in the sample
and to detect the signal from the sample. [Chapter 7]
RF Pulse
A short burst of RF energy which has a specific shape.
Rotation Matrix
A matrix used to describe the rotation of a vector. [Chapter 3]
Sample Probe
That portion of the NMR spectrometer containing the RF coils and into which the
sample is placed. [Chapter 7]
Saddle Coil
A coil geometry which has two loops of a conductor wrapped around opposite sides
of a cylinder. [RF: Chapter 7]
Spin
A fundamental property of matter responsible for NMR and MRI. [Chapter 3]
Spin Density
The concentration of spins. [Chapter 6]
Spin-Echo
An NMR sequence whose signal is an echo resulting from the refocusing of
magnetization after the application of 90o and 180o RF pulses. [Chapter 6]
Spin-Lattice Relaxation
The return of the longitudinal magnitization to its equilibrium value along the +Z
axis. [Chapter 3]
Spin-Lattice Relaxation Time ( T1 )
The time to reduce the difference between the longitudinal magnitization and its
equilibrium value by a factor of e. [Chapter 3]
Spin Packet
A group of spins experiencing the same magnetic field. [Chapter 3]
Spin-Spin Relaxation
The return of the transverse magnitization to its equilibrium value (zero). [Chapter 3]
Spin-Spin Relaxation Time
The time to reduce the transverse magnetization by a factor of e. [Chapter 3]
Sinc Pulse
An RF pulse shaped like Sin(x)/x. [Chapter 5]
Superconduct
To have no resistance. A perfect superconductor can carry an electrical current
without losses. [Chapter 7]
T2*
Pronounced T-2-star. The spin-spin relaxation time composed of contributions from
molecular interactions and inhomogeneities in the magnetic field. [Chapter 3]
Timing Diagram
A multiaxis plot of some aspects of a pulse sequence as a function of time. [Chapter
6]
Transverse magnetization
The XY component of the net magnetization. [Chapter 3]
List of Symbols
Symbol
Definition
Å
Angstrom (10 meters)
Bo
Static magnetic field
B1
The radio frequency magnetic field
C
Contrast
C
Electrical eapacitance
COSY
2-D correlated spectroscopy
CW
Continuous wave
D
Deuterium
D
Diffusion coefficient
-10
Chemical shift
Magnetic field gradient pulse width
Magnetic field gradient pulse separation
Chemical shift
E
Energy
f
Frequency
FID
Free induction decay
FT
Fourier transform
Gi
Magnetic field gradient in the i direction
Gyromagnetic ratio
h
Planck's constant
H
Hydrogen
IFT
Inverse Fourier transform
IM
Imaginary part of a complex number
J
Joule
J
Spin-spin coupling constant
k
Boltzmann constant
k
kilo (103)
k
Proportionality constant
K
Kelvin temperature
L
Electrical Inductance
m
milli (10-3)
M
mega (106)
micro (10-6)
Mo
Equilibrium magnetization
MX
X component of magnetization
MX'
X' component of magnetization
MY
Y component of magnetization
MY'
Y' component of magnetization
MZ
Z component of magnetization
MXY
Transverse component of magnetization
MRI
Magnetic resonance imaging
Resonance frequency in Hertz
N
Number of averages
N+
Spin population in low energy state
N-
Spin population in high energy state
NMR
Nuclear magnetic resonance
Resonance frequency in radians per second
Ohm, impedance
3.14159...
Phase angle
ppm
Parts per million
R1
Spin-lattice relaxation rate
RE
Real part of a complex number
RF
Radio frequency
s
Second
Chemical shielding constant
SAR
Specific absorption rate
Sinc
Sin(x)/x
SNR
Signal-to-noise ratio
T
Temperature
T
Tesla
T1
Spin-lattice relaxation time
T2
Spin-spin relaxation time
T2*
T2 star
T2inhomo
Inhomogeneous T2
Rotation angle
TE
Echo Time
TI
Inversion Time
TR
Repetition Time
X
X axis in laboratory coordinate system
X'
Rotating frame X axis
Y
Y axis in laboratory coordinate system
Y'
Rotating frame Y axis
Z
Z axis in laboratory coordinate system
1-D
One-dimensional
2-D
Two-dimensional
References
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E. Fukushima and S.B.W. Roeder, Experimental Pulse NMR, Addison-Wesley,
Reading, MA 1981.
T.C. Farrar, An Introduction To Pulse NMR Spectroscopy, Farragut Press, Chicago,
1987.
R.C. Jennison, Fourier Transforms and Convolutions, Pergamon Press, NY 1961.
E.O.Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ
1974.
A. Carrington and A.D. McLachlan, Introduction To Magnetic Resonance, Chapman
and Hall, London 1967.
B. Noble and J.W. Daniel, Applied Linear Algebra, Prentice-Hall Englewood Cliffs,
NJ.
G.B. Thomas, Calculus, Addison-Wesley, Reading, MA 1969.
H. Gunther, "Modern pulse methods in high-resolution NMR spectroscopy." Angew.
Chem.. Int. Ed. Engl.22:350-380 (1983).
R.K. Harris, Nuclear Magnetic Resonance Spectroscopy, Pitman, London 1983.
T.C. Farrar, An Introduction to Pulse NMR Spectroscopy, Farragut Press, Chicago
1987.
T.C. Farrar and E.D. Becker, Pulse And Fourier Transform NMR, Academic Press,
1971.
D. Shaw, Fourier Transform NMR Spectroscopy, Elsevier, NY 1976.
J.W.Akitt, NMR and Chemistry, An Introduction to the Fourier Transform Multinuclear Era. Chapman and Hall, London, 1983.
R. Freeman, A Handbook of Nuclear Magnetic Resonance, Longman Scientific &
Technical, Essex, England, 1988.
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Woodrow W. Conover, "Magnet Shimming," Technical Report, Nicolet Magnetics
Corporation
C.E. Hayes, W.A. Edelstein, J.F. Schenck, "Radio Frequency Resonators." in
Magnetic Resonance Imaging, ed. by C.L. Partain, R.R. Price, J.A. Patton, M.V.
Kulkarni, A.E. James, Saunders, Philadelphia, 1988.
G.M. Bydder, "Clinical Applications of Gadolinium-DTPA." in in Magnetic
Resonance Imaging, ed. by D.D. Stark and W.G. Bradley, C.V. Mosby Co., St. Louis,
MO 1988.
J. Granot, J. Magn. Reson. 70:488-492 (1986).
S.R. Thomas, L.J. Busse, J.F. Schenck, "Gradient Coil Technology." in Magnetic
Resonance Imaging, ed. by C.L. Partain, R.R. Price, J.A. Patton, M.V. Kulkarni, A.E.
James, Saunders, Philadelphia, 1988.
J. Gong and J.P. Hornak, "A Fast T1 Algorithm." Magn. Reson. Imaging 10:623-626
(1992).
X. Li and J.P. Hornak, "T2 Calculations in MRI: Linear versus Nonlinear Methods." J.
Imag. Sci. & Technol. 38:154-157 (1994).
Peter Stilbs, "Fourier transform pulsed-gradient spin-echo studies of molecular
diffusion." Progress in NMR Spectroscopy 19:1-45 (1987).
J.R. Dyer, Applications of Absorption Spectroscopy by Organic Compounds,
Prentice-Hall, Englewood Cliffs, NJ 1965.
D.A. Skoog, F.J. Holler, T.A. Nieman, Principles of Instrumental Analysis, 5th ed.,
Saunders College Publishing, New York, 1992.
D.E. Leyden, R.H. Cox, Analytical Applications of NMR, Wiley, NY 1977.
A.E. Derome, NMR Techniques for Chemistry Research, Pergamon Press, NY 1987.
Atta-ur-Rahman, Nuclear Magnetic Resonance, Basic Principles, Springer-Verlag,
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F. Noack, "NMR field-cycling spectroscopy: principles and applications." Progress in
NMR Spectroscopy, 18171-276 (1986).
Field Cycling NMR Relaxometry: Review of Technical Issues and Applications,
STELAR s.r.l., via E. Fermi 4, 27035 Mede (PV), Italy. © 2001,
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E. Anoardo, G. Galli, G. Ferrante, "Fast Field Cycling NMR: Applications and
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FAQ
Although the questions below are of a general nature, the instruments mentioned
are the Varian spectrometers currently used in the NMR Centre of the Australian
National University).
Please email your ideas for other questions to be included in this list to Chris Blake.
There is a much more comprehensive Basics of NMR document available at the
Center for Imaging Science, Rochester Institute of Technology.




If you haven't used any NMR spectrometers at the NMR Centre before.
Come down to room G48 or G42 in the Chemistry Faculties building, and ask
Chris or Peta to arrange a training session for you. Even if you have used
Varian spectrometers before, our setup here is likely to be slightly different, so
in the long run you will save yourself time by having a short lesson. More
information is available in the introduction to the NMR Centre. It is NMR
Centre policy that you will not be given a license to operate a spectrometer
until we assess your competence with Varian spectrometers.
How much solvent volume should I use?
To get good resolution you need at least 0.7 ml of solvent in a 5 mm NMR
tube. If you have a limited amount of sample you can increase its effective
concentration by reducing the solvent volume to 0.4 - 0.5 ml however you will
need to spend more time shimming. If you have a sample that will give a
proton spectrum in 15 minutes when disolved in 0.7 ml, there is little point
reducing sample volume if it means you need to spend an extra 10 minutes
shimming! On the other hand if you then want to run a carbon spectrum of
that sample it would certainly be worth reducing the sample volume. For 13C
HMQC or HMBC runs with very small amounts of material, a 4mm tube and
rotor is available.
What does the "ADC overflow" error message mean?
The signal recieved from the NMR sample is first amplified by the reciever
and then digitised by the analog to digital converter (ADC). If the signal is too
strong for them to handle, either the receiver or ADC will "overflow", causing a
RECEIVER OVERFLOW or ADC OVERFLOW message to be displayed. The
acquired FID is likely to be clipped, resulting in a distorted spectrum. The
solution is to use autogain (type gain='n' or on a Gemini GAIN=N) or to type in
a lower value for the receiver gain. If overflow still occurs when the gain is set
to zero, reduce the observe pulsewidth (PW) to half its present value. If
overflow still occurs dilute your sample, or if the solvent signal is causing the
ADC overflow use a solvent suppression technique.
How do I shim / tune the spectrometer?
First of all, let's get our terminology straight. Shimming is adjusting the
resolution of the signal by optimizing the homogeneity of the magnetic field.
Tuning is adjusting the impedance of the probe. A poorly tuned probe reflects
a lot of the power of the pulses, so that what should be a 90 degree pulse is
in reality only (say) a 50 degree pulse. Probe tuning does not affect the
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resolution, however the signal to noise of a standard spectrum will be worse.
Also, experiments such as DEPT or COSY that rely on accurate 90 degree
pulses may produce artefacts or not work at all. (Note however that it is
possible to adjust the pulse width to give a 90 degree pulse on a poorly tuned
probe). Poor shimming on the other hand, results in broad NMR resonances.
People often talk about "tuning the resolution" which is where some confusion
between shimming and probe tuning arises. Shimming is adjusting the
homogeneity of the magnetic field, so that every part of the sample in the
NMR tube experiences exactly the same field strength.
OK, so how do I tune the probe?
If you're using the broadband Gemini or Mercury spectrometers, you never
need to tune the probe. The probes of these intruments are tuned at the
factory, and further tuning is a specialised operation. For best results, you
should tune all other spectrometers before acquiring a spectrum. Frequency,
solvent and sample height all affect probe tuning. If you were running a set of
similar samples in the same solvent, you might only bother to tune the probe
before running the first spectrum. If however, half your samples were disolved
in chloroform and half in D2O, you might run all of the chloroform samples and
then quickly adjust the tuning after inserting the first D2O sample. Tuning
involves setting up for the nucleus of interest and minimizing the reflected
power shown on the meter on the magnet leg. Some recabling is required. Do
not attempt to do this unless an NMR staff member has given you a lesson.
This doesn't mean that Geminis have some great "automatic tuning"
technology. It just means they are left in a state of tune that is good enough
for the run-of-the-mill experiments they were designed for. On other
spectrometers, tuning is necessary because
o You can get the best possible tuning for your sample,
o You may not know what nucleus the previous user left the probe tuned
to, or whether he/she completely messed up the tuning,
o More sophisticated experiments such as HMQC, HMBC etc. work best
when the probe is tuned and short 90 degree pulses are required.
How do I tune for carbon or phosphorus?
As mentioned above, if you're using a Gemini spectrometer, you don't need to
tune the probe. First, check whether the probe you are using requires a tuning
stick to be inserted. Tuning sticks are kept separate from the probe, and have
a small capacitor on the end to change the tuning range of the probe. If a
tuning stick is required, select the stick for the observe frequency and screw it
gently all the way into the probe. You can find the observe frequency by
setting up for the nucleus of interest and reading the value of sfrq from the dg
display. Then make the cable connections for tuning, and adjust both the
tuning and matching rods. These two tuning rods affect each other, so it is
usually necessary to go back and forth between them to get a good minimum.
There is a bit of a knack to it, so persevere! (Hint: make the tuning worse with
one rod, then better with the other. Each dual operation should result in better
tuning than before).
Also note that if you are decoupling protons while observing carbon or
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phosphorus, it is a good idea to check the proton tuning. If the probe is poorly
tuned to protons, some decoupler power may be reflected, resulting in an
improperly decoupled spectrum. On the Inova spectrometers you can tune
the observe and decoupler channels at the same time. On older
spectrometers you need to set up for and tune protons, then set up for and
tune carbon or phosphorus. (Hint: always tune the highest frequency first and
the lowest frequency last).
Which spectrometer should I use for carbon?
When measuring carbon spectra, the main concern is usually signal to noise.
You would expect higher field spectrometers to have a decisive advantage for example a 500 Mhz spectrometer when compared to a 300 MHz
spectrometer should have an advantage of (5/3) squared, or 2.8 times the
signal to noise. However there are other considerations, including for example
the type of probe. An indirect detection probe has the proton observe coil on
the inside (that is, closer to the sample than the coil used for carbon). This
improves the proton signal to noise, however if you use an indirect detection
probe for directly observing carbon, the signal to noise will of course be worse
than a standard probe which has the carbon coil on the inside. Regardless of
the probe design, carbon and protons use different coils, and since the
electronic circuit for the two nuclei is different it makes no sense to compare
proton signal to noise on two instruments and extrapolate the results to
carbon.
Also, signal to noise tests are usually performed by collecting a single scan
on a concentrated sample, however this does not give the best indication of
the results obtainable on "real" samples where the sample is scanned for
several hours. When a sample is repeatedly pulsed, the relaxation times of
the various carbons must be taken into consideration. Nuclei take longer to
relax at higher fields, so the gain in signal to noise is less than expected. Also
note that carbons that do not have directly bonded protons (i.e. carbonyls and
quaternaries) have much longer relaxation times than protonated carbons.
In order to see how some of the spectrometers compare under "real life"
conditions, a dilute sample was run for 256 scans on the Inova 500 (PFG
indirect detection probe), broadband Gemini, and VXR300. A D1 delay of 1
second with a 45 degree pulse was used, and 16 dummy pulses were given
to bring the system to a steady state before starting acquisition. The signal to
noise ratios of three resonances were then measured.
CDCl3 CH3 quarternary
VXR300
21.3
16.5 2.2
Gemini BB 16.9
15.8 4.3
Inova 500
22.4 5.9
29.9
It can be seen that there is not a large difference in the signal to noise you
can expect to see on these instruments. Also remember that
if there is not much sample available, you should reduce the amount of
solvent. (See How much solvent volume should I use?) A 4mm tube
and rotor is available. This will allow you to use even less solvent than
is necessary in a 5mm tube.
o if you are interested in quarternary carbons, a longer D1 delay of 3
seconds or more is advisable.
o If the signal to noise of your carbon spectrum is too low, try running a
short and/or long range proton-carbon 2D correlation experiment. It
has been known for a long time that this can give dramatic
improvements in S/N. See J. Am. Chem. Soc. 101, 4481 - 4484
(1979).
Why are some of the peaks in my APT missing?
The APT experiment relies as much on the size of the 1JCH coupling as the
number of attached protons to generate the spectral pattern. This is because
the delays in the experiment are matched to the inverse of the size of 1JCH. If
1J
1
CH is much larger than the default JCH of the experiment (usually set to 140
Hz which is the average of 1JCH for sp3 and sp2 carbons) then peaks will either
disappear or appear with incorrect phase. Carbons that may show this
behaviour are terminal ethynyl groups (1JCH = 250 Hz approx.), epoxide
carbons (1JCH = 175 Hz), furan, pyrone and isoflavone carbons (1JCH = 200
Hz), 2-unsubstituted pyridine and pyrolle carbons (1JCH = 180 Hz) and 2unsubstituted imidazole and pyrimidine carbons (1JCH > 200 Hz).
I can't lock on.
o You are using a deuterated solvent aren't you?
o Can you see a lock signal? If not, make sure the lock is turned off, turn
the lock power and lock gain to their maximum values, and look for a
sine wave by adjusting Z0. If you find a sine wave, adjust Z0 until its
frequency becomes zero. Then reduce the lock power (to avoid
saturating the lock) and try to lock on.
o If it loses lock as soon as you try to lock on, turn the lock off and adjust
the lock phase as shown in the manual.
o Is your tube spinning? It might not be spinning because you inserted
the tube too quickly, causing it to break. Take the tube out and check
that it is in one piece. While you have it out, use a depth gauge to
check that the sample is centred in the probe.
It won't shim.
o Check the linewidth of the narrowest line in your spectrum. If there are
some broad lines and some narrow lines, the broad lines are probably
broad because they are undergoing chemical exchange, not because
the resolution is poor. Broad lines may also be caused by quadrupolar
broadening if your compound has a transition metal.
o If you have not already done so, load the standard shims. You don't
know what sort of state the previous user left the shims in! All
o
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spectrometers in the NMR Centre have a macro rtss which loads the
standard shims. This macro is equivalent to typing rts('stdshm') su. On
the Inova 300 and 500, check that the probe parameter is set to the
probe you are using, since the rtss macro uses this value to determine
which shims to load.
o If it still won't shim, take the tube out and inspect your sample. Is the
tube scratched? Is there anything floating in the sample? Is the sample
centred in the coil? If you are using a small amount of solvent to
improve the concentration, you may need to add some more solvent to
make it easier to shim.
o Do you have paramagnetic ions in your sample?
o Have other people been getting poor resolution? If so, report it to a
member of the NMR staff. If not, change NMR tubes, filter your
sample, and try again. If changing tubes solves the problem, throw the
old tube away.
o Have you placed your NMR tube in an oven to dry? If so, throw the
tube away as it has distorted. (Remember, glass is a liquid. It flows at
high temperature). The correct way to dry a tube is via a stream of dry
nitrogen through a glass wool filter.
My tube broke when I inserted it into the magnet.
During the day, phone an NMR staff member. Tell them the solvent and any
hazards posed by your compound. After normal working hours tell the
watchmen who will call in someone. Leave a note to warn others not to use
the spectrometer. Remember - the more quickly you lower a tube into the
magnet, the more likely it is to break! If necessary, use two hands on the
sample eject button to make it easier to lower the tube slowly.
I can't phase correct my spectrum.
The aph (automatic phase correction) command usually does a good job of
correcting the phase, and should be the first thing you try. Sometimes (for
example in noisy spectra) the aph command is unable to correct the phase,
and in these situations it often leaves lp at a high value (say one or two
thousand). In these situations you will have to correct the phase manually.
First a couple of obvious things: if you ran a DEPT or APT experiment or
something similar, there will be some positive and some negative peaks, so
don't try and phase them all positive! Similarly in a 1:1 binomial solvent
suppression sequence, half the spectrum will be positive and half negative.
Having established that you are not running an exotic pulse sequence that
produces strange phases, the next thing to consider is foldback. Are you sure
that you used a large enough spectral width when acquiring the spectrum? If
one or more resonances occurred outside the observe region, the method
used to digitise the signal results in these resonances appearing within the
observed spectral width, but with a phase error. If in doubt, double or triple
the spectral width, run the spectrum again, and see if the resonance that
could not be phase corrected remains at the same chemical shift as before.
To perform manual phase correction, proceed as follows:
Type lp=0 rp=0. This sets the left phase and right phase to zero. On Varian
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spectrometers, "right phase" and "left phase" equate very roughly to the zeroorder and first-order phase adjustments respectively. The zero-order phase
affects the entire spectrum equally, while the first-order phase is frequency
dependent. The zero-order phase should always be in the range -360° to
+360° and the first order phase should also usually be in this range. If you
have a first order phase correction of more than a thousand degrees, not only
is it probably incorrect, but you will also probably be generating baseline roll.
On the Gemini spectrometer, type QP to get into quick phase mode. On Sun
based spectrometers or data stations, click the "phase" button with the
mouse. Perform a zero-order phase correction on the largest peak as
described in the manual for the spectrometer you are using. Now choose
another peak some distance from the largest peak, and adjust the first-order
phase. On Sun based systems you only get one shot at adjusting the zeroorder phase - all subsequent corrections are made to the first-order phase, so
there is no point clicking on the largest peak again. If you want to readjust the
zero-order phase, get out of the phase-correction routine (by for example
typing ds) then click on the phase button again. If for some reason a large
first-order phase correction is required, it may be easier to choose a peak for
the first-order adjustment that is close to the peak you used for the zero-order
adjustment. On Sun based systems, set the phasing parameter to 100. This
causes the effect of the phase values to be shown for the entire spectrum as
you are making the adjustments, thus making it easier to see what you are
doing.
I need to run my spectrum at a higher field to get better resolution.
No you don't! The resolution of a high field spectrometer may even be worse
than a low field spectrometer. What a high field instrument has more of is
dispersion. This means that resonances with different chemical shifts are
further apart. Multiplets due to coupling will not show any improvement unless
the higher field instrument separates overlapping multiplets with different
chemical shifts, or the multiplet showed strong coupling effects at lower field.
Some nuclei such as 31P may have worse resolution because of a property
called chemical shift anisotropy which increases with field strength.
There are no parameters for the solvent I want to use.
If you're running a proton spectrum, set up for 1H / CDCl3, double the spectral
width, run a quick spectrum, and put the two cursors around the spectrum.
Then do a movesw and acquire the final spectrum. If you're running a carbon
spectrum, set up for 13C / CDCl3, increase the spectral width by 20 percent,
and run as normal. If your solvent has carbon nuclei which show up quickly,
reference the solvent and check that the observed spectral range is correct.
If you are running a phosphorus spectrum, set up for 31P / CDCl3, increase
the spectral width by 20 percent, and run as normal. Then supply an NMR
tube containing the solvent to the NMR staff so that they can set up H 3PO4
referencing parameters for you.
How can I suppress a strong solvent resonance in a proton spectrum?
If the solvent signal is less than two or three times the size of the largest
signal from your compound, it may not be worth bothering. On the Gemini, the
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
usual method is to presaturate the solvent signal using the decoupler
(instructions for doing this are in the folders near the spectrometer). Although
it is simple, this method has the disadvantage that NH or OH protons that are
exchanging with water also have their signals reduced or eliminated. Another
method is the 1:1 binomial pulse sequence. The signals on one side of the
solvent resonance are of opposite phase to the other side when this method
is used. On the Inova spectrometers, the method of choice if a pulsed field
gradient probe is in use, is watergate solvent suppression. Simply set the
observe transmitter on the solvent position, type wgate and acquire a
spectrum. The watergate sequence set up by the wgate macro uses hard
pulses and therefore does not require pulse phases etc. to be optimised. The
other watergate technique available uses shaped pulses. Simply type
autowatergate, and wait while it automatically optimises the parameters and
runs a final spectrum. Watergate is only available on the Inovas because it
uses pulsed field gradients. If chemical exchange is very rapid, watergate
may not be suitable, in which case a binomial pulse sequence is the best
choice.
What is nuclear spin?
All nuclei carry a charge. In some nuclei this charge "spins", causing the
nucleus to behave like a tiny bar magnet. This is why it aligns with or against
the magnetic field of an NMR spectrometer. However unlike a bar magnet,
the low energy state is aligned with the field and the high energy state is
aligned against the field. Up to now we have been talking about nuclei with a
uniform spherical charge distribution. These nuclei are said to have a spin of
½. Protons, 13C and 31P are all spin half nuclei. Note that the most common
isotope of carbon, 12C, has no spin and can therefore not be observed using
NMR. Nuclei with a non-spherical charge distribution have a spin number I of
1, 3/2 or higher (in steps of ½ ), and are referred to as quadrupolar nuclei.
Spin ½ nuclei have two orientations (with or against the field). Spin 1 nuclei
have three orientations, spin 3/2 nuclei have 4 orientations, etc. Deuterium is
an example of a spin 1 nucleus. Although deuterium is chemically the same
as hydrogen, for the purposes of NMR it is completely different. For example
a carbon spectrum of CDCl3 is a 1:1:1 triplet regardless of whether you turn
on the proton decoupler. This is because the deuterium attached to the
carbon can have three orientations, and occurs at a different frequency to
protons.
What is a double quantum coherence?
When you put your sample in the magnet, all the spin half nuclei align either
with or against the magnetic field. The population difference between these
two orientations (known as the Boltzman distribution) is field dependent, and
is determined by their energy difference. An NMR signal is observed when
nuclei flip from one orientation to the other. This is a single quantum
coherence. When two nuclei are coupled, they can flip together as though
they were a single unit. If they flip in opposite directions, the flips "cancel each
other out" (sort of) resulting in a zero quantum coherence. If they both flip the
same way, you get a double quantum coherence. The frequency of a zero

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
quantum coherence is between zero and a few kilohertz, so it is not directly
observed. Similarly the frequency of a double quantum coherence is roughly
twice the normal observe frequency, so that is not observed directly either.
You can also have triple quantum coherences from groups of three coupled
nuclei. The effect of double and triple quantum coherences can only be
observed by inserting pulses or delays into a pulse sequence to convert them
to single quantum coherences before acquisition of the NMR signal. Do not
confuse double quantum coherences with coupling in a normal spectrum. A
doublet for example, arises when there are two coupled spins, but only one of
these spins flips.
What are pulsed field gradients?
Imagine if you could really mess up the Z1 resolution for a few milliseconds
then restore it to its proper value during the course of the pulse sequence.
This is an oversimplification, since pulsed field gradients do not use the
normal shim circuits. A special PFG probe, and a PFG amplifier are
necessary. By applying a gradient to the magnetic field, the top of the sample
experiences a slightly different magnetic field to the bottom of the sample.
Since magnetisation precesses at different rates in different fields, it is
possible after a 90 degree pulse and a PFG of a few milliseconds to have the
magnetisation vectors along the length of the tube pointing in all directions
instead of nicely aligned along one axis of the rotating frame. Obviously if the
magnetisation vectors are pointing in all directions, there is no net signal. The
vectors are said to be dephased. If you now apply a PFG of opposite sign for
the same time, you will rephase the magnetisation, and get your signal back.
You could achieve the same thing by giving the dephased vectors a 180
degree pulse, then applying a PFG of the same sign. The other thing to be
aware of is that double quantum coherences dephase at twice the rate of
normal single quantum coherences, so by adjusting the strength or duration
of pulsed field gradients, you can select single, double or triple quantum
coherences. The "old fashioned" way of selecting certain types of coherences
is to use elaborate phase cycles which cause the unwanted signals to cancel
out on successive scans. The PFG method acquires only the desired signal
on each scan, resulting in fewer artifacts and allowing fewer scans. The old
method can be thought of as "cancellation of unwanted signals over time"
whereas the PFG method can be thought of as "cancellation of unwanted
signals over space" where "time" refers to successive scans, and "space"
refers to the physical length of the sample in an NMR tube.
What is the Nuclear Overhauser Effect?
Glad you asked. Have a look at our NOE guide.
How do I run a quantitative spectrum?
A quantitative spectrum is simply a spectrum where you can trust the integral
ratios. In other words, if the integral of resonance A is twice the height of the
integral of resonance B, you can say with certainty that resonance A is due to
twice the number of nuclei as resonance B. Why do we use integrals?
Because it is the area of the resonances that is proportional to the number
nuclei. The height of a broad line may be less than that of a sharp line, but its
area may be greater. How do we get accurate integrals? By ensuring that all
resonances are equally excited, well digitised, and properly relaxed.
o Equally excited : if the pulse power is not high enough, some
resonances far from the observe frequency may experience a reduced
flip angle, resulting in a smaller observed signal.
o Well digitised : if the number of data points in the spectrum is too low,
there will not be enough points to accurately define each resonance,
resulting in inaccurate integrals (and peak heights).
o Properly relaxed : resonances that are not fully relaxed give a weaker
signal than fully relaxed resonances. The nuclei in your compound will
not all relax at the same rate, so if you pulse too rapidly the quickly
relaxing resonances will appear stronger than the slowly relaxing ones.
To be sure of obtaining accurate integrals, you need to measure the
relaxation times of your compound, and set a delay equal to 5 times
the longest relaxation time. Fortunately it is easy to run an inversion recovery experiment to measure relaxation times.
It is harder to obtain quantitative carbon spectra, because carbon relaxes
more slowly than protons, is less intense, and steps have to be taken to
eliminate the Nuclear Overhauser Effect which builds up when protons are
decoupled.

What is digital resolution?
Digital resolution is simply the separation in hertz between each data point in
your spectrum. It has nothing to do with shimming! Say, for example, you set
the number of points np to 32,768 and acquire a normal 1 dimensional FID.
The number of points in the spectrum you see will be 16384, since half the
data points are imaginary. Now if the spectral width (sw) is 6000, the digital
resolution will be 6000/16384, or 0.366 hz per point. (Before you grab your
calculator to measure your own digital resolution, note that the number of
points in the spectrum is not always simply np/2. See the section below on
the Fourier number). The Vnmr command to display the digital resolution is
dres. If you place the cursor on a peak and type dres, two values will be
displayed:
o the linewidth which is the width of the peak at half-height, and depends
on shimming, weighting functions and the natural width of the line. Also
the
o digital resolution, which is what this section is all about.
The dres command may give a different linewidth value for every peak you
put the cursor on, but the digital resolution value will always be the same,
unless you change the Fourier number fn and do another Fourier transform. If
the natural linewidth of a resonance is comparable to the digital resolution,
the resonance may only be defined by one or two data points. If you expand a
line like this, it will look more like a spike than a proper Lorentzian line.
Consequently the height of the line may appear less then it really is, the
integral will be inaccurate, and even the chemical shift value will be less
accurate than it should be. Also, if the separation between two resonances is
comparable to the digital resolution, they may appear as a single resonance
in the spectrum, because no data point falls in the space between the tops of
the two peaks.
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What is the fourier number?
Mathematicians can do a Fourier transform of any number of points. NMR
spectrometers speed things up by using the Cooley-Tukey fast fourier
transform algorithm. As implemented on NMR spectrometers, this requires
the number of points to be a power of two. So what happens if the number of
points np is not a power of two? On Varian spectrometers this can be
controlled by the Fourier number (fn) parameter. If it is used, fn can only be
set to powers of 2, and the value of fn is the number of points that are actually
used in the Fourier transform. If fn is less than np, some points on the end of
the FID are not used in the Fourier transform. If fn is greater than np, the end
of the FID is padded with zeros to increase the number of points. This is
referred to as zero filling. Zero filling does not write extra zeros on to the end
of the FID file on the disk where the FID is stored, it merely adds the zeros in
memory just before doing the transform. It is also possible to set the Fourier
number to n (not used). In this case, the spectrometer uses the first power of
2 which is higher than np when doing the Fourier transform. So for example if
np was 16385 (that is, 214 + 1) it would use 32768 (i.e. 215) points for the
Fourier transform.
What is the relaxation time?
It would be an oversimplification to say that the relaxation time is the time
taken for a nucleus to relax to equilibrium. After a pulse, a nucleus relaxes
toward its equilibrium value at an exponential rate. The value quoted as the
relaxation time is actually the time constant of this exponential curve. It takes
five time constants for the magnetisation to relax to 95% of its equilibrium
value. There are two basic types of relaxation, T 1 and T2. In the T1 process,
the magnetization remaining along the z-axis relaxes back to its equilibrium
value. This is also known as spin-lattice relaxation because relaxation occurs
by the loss of energy from the excited nuclear spins to the surrounding
molecular lattice. In the T2 process, the magnetization in the x-y plane fans
out out until the net magnetization is zero. This is also known as spin-spin
relaxation because it is due to the excited spins exchanging energy with each
other.
What NMR Simulation Programs are Available?
o To simulate a normal (non-exchanging) spin system, you can perform
the simulation using the same Vnmr program that you use for data
processing. There are instructions in the folders. The first step is to
decide what sort of spin system you have - AB, A2X, ABCXY etc. The
letters are not important to Vnmr, so it doesn't matter whether you tell it
that you have an ABC or an AMX system. Vnmr only needs to know
the values of the chemical shifts and coupling constants.
o
To simulate a dynamic (exchanging) spin system, the program you use
depends on the type of experiment you ran. If you ran a series of
normal spectra at different temperatures, then you need to simulate the
lineshape. This is done using the DNMR5 program. There is another
program, dnmr5input, to help you create the input file for DNMR5.
Instructions are in the folders by the Sun computers.
If you ran a series of
selective inversion pulse -- delay -- hard pulse -- acquire
experiments to use magnetization transfer information to determine
rate constants, we have a program provided by Prof. Brian Mann of
Sheffield University that you can use to analyse your data.