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http://teaching.shu.ac.uk/hwb/chemistry/tutorials/molspec/nmr1.htmIntroduct
ion
Nuclear Magnetic Resonance spectroscopy is a powerful and theoretically complex analytical
tool. On this page, we will cover the basic theory behind the technique. It is important to
remember that, with NMR, we are performing experiments on the nuclei of atoms, not theelectrons. The chemical environment of specific nuclei is deduced from information obtained
about the nuclei.
Nuclear spin and the splitting of energy levels in a magnetic field
Subatomic particles (electrons, protons and neutrons) can be imagined as spinning on their
axes. In many atoms (such as 12C) these spins are paired against each other, such that the
nucleus of the atom has no overall spin. However, in some atoms (such as 1H and 13C) the
nucleus does possess an overall spin. The rules for determining the net spin of a nucleus are asfollows;
1. If the number of neutrons and the number of protons are both even, then the nucleus
has NO spin.
2. If the number of neutrons plus the number of protons is odd, then the nucleus has a
half-integer spin (i.e. 1/2, 3/2, 5/2)
3. If the number of neutrons and the number of protons are both odd, then the nucleus
has an integer spin (i.e. 1, 2, 3)
The overall spin,I, is important. Quantum mechanics tells us that a nucleus of spinIwill have
2I+ 1 possible orientations. A nucleus with spin 1/2 will have 2 possible orientations. In theabsence of an external magnetic field, these orientations are of equal energy. If a magnetic
field is applied, then the energy levels split. Each level is given a magnetic quantum number,m.
When the nucleus is in a magnetic field, the initial populations of the energy levels are
determined by thermodynamics, as described by the Boltzmann distribution. This is very
important, and it means that the lower energy level will contain slightly more nuclei than
the higher level. It is possible to excite these nuclei into the higher level with electromagnetic
radiation. The frequency of radiation needed is determined by the difference in energybetween the energy levels.
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Calculating transition energy
The nucleus has a positive charge and is spinning. This generates a small magnetic field. The
nucleus therefore possesses a magnetic moment, , which is proportional to its spin,I.
The constant, , is called the magnetogyric ratioand is a fundamental nuclear constant whichhas a different value for every nucleus. h is Plancks constant.
The energy of a particular energy level is given by;
WhereB is the strength of the magnetic field at the nucleus.
The difference in energy between levels (the transition energy) can be found from
This means that if the magnetic field,B, is increased, so is E. It also means that if a nucleus
has a relatively large magnetogyric ratio, then Eis correspondingly large.
If you had trouble understanding this section, try reading the next bit (The absorption of
radiation by a nucleus in a magnetic field) and then come back.
The absorption of radiation by a nucleus in a magnetic field
In this discussion, we will be taking a "classical" view of the behaviour of the nucleus - that
is, the behaviour of a charged particle in a magnetic field.
Imagine a nucleus (of spin 1/2) in a magnetic field. This nucleus is in the lower energy level(i.e. its magnetic moment does not oppose the applied field). The nucleus is spinning on its
axis. In the presence of a magnetic field, this axis of rotation will precess around the magnetic
field;
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populations of the higher and lower energy levels will become equal. If this occurs, then there
will be no further absorption of radiation. The spin system issaturated. The possibility of
saturation means that we must be aware of the relaxation processes which return nuclei to the
lower energy state.
Relaxation processes
How do nuclei in the higher energy state return to the lower state? Emission of radiation is
insignificant because the probability of re-emission of photons varies with the cube of the
frequency. At radio frequencies, re-emission is negligible. We must focus on non-radiative
relaxation processes (thermodynamics!).
Ideally, the NMR spectroscopist would like relaxation rates to be fast - but not too fast. If the
relaxation rate is fast, then saturation is reduced. If the relaxation rate is too fast, line-
broadening in the resultant NMR spectrum is observed.
There are two major relaxation processes;
Spin - lattice (longitudinal) relaxation
Spin - spin (transverse) relaxation
Spin - lattice relaxation
Nuclei in an NMR experiment are in a sample. The sample in which the nuclei are held is
called the lattice. Nuclei in the lattice are in vibrational and rotational motion, which creates acomplex magnetic field. The magnetic field caused by motion of nuclei within the lattice is
called the lattice field. This lattice field has many components. Some of these componentswill be equal in frequency and phase to the Larmor frequency of the nuclei of interest. Thesecomponents of the lattice field can interact with nuclei in the higher energy state, and cause
them to lose energy (returning to the lower state). The energy that a nucleus loses increases
the amount of vibration and rotation within the lattice (resulting in a tiny rise in the
temperature of the sample).
The relaxation time, T1 (the average lifetime of nuclei in the higher energy state) is dependanton the magnetogyric ratio of the nucleus and the mobility of the lattice. As mobility increases,
the vibrational and rotational frequencies increase, making it more likely for a component of
the lattice field to be able to interact with excited nuclei. However, at extremely high
mobilities, the probability of a component of the lattice field being able to interact withexcited nuclei decreases.
Spin - spin relaxation
Spin - spin relaxation describes the interaction between neighbouring nuclei with identical
precessional frequencies but differing magnetic quantum states. In this situation, the nuclei
can exchange quantum states; a nucleus in the lower energy level will be excited, while the
excited nucleus relaxes to the lower energy state. There is no net change in the populations of
the energy states, but the average lifetime of a nucleus in the excited state will decrease. This
can result in line-broadening.
Chemical shift
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The magnetic field at the nucleus is not equal to the applied magnetic field; electrons around
the nucleus shield it from the applied field. The difference between the applied magnetic field
and the field at the nucleus is termed the nuclear shielding.
Consider the s-electrons in a molecule. They have spherical symmetry and circulate in the
applied field, producing a magnetic field which opposes the applied field. This means that theapplied field strength must be increased for the nucleus to absorb at its transition frequency.
This upfield shiftis also termed diamagnetic shift.
Electrons in p-orbitals have no spherical symmetry. They produce comparatively large
magnetic fields at the nucleus, which give a low field shift. This "deshielding" is termed
paramagnetic shift.
In proton (1H) NMR, p-orbitals play no part (there aren't any!), which is why only a small
range of chemical shift (10 ppm) is observed. We can easily see the effect of s-electrons on
the chemical shift by looking at substituted methanes, CH3X. As X becomes increasingly
electronegative, so the electron density around the protons decreases, and they resonate at
lower field strengths (increasing H values).
Chemical shiftis defined as nuclear shielding / applied magnetic field. Chemical shift is afunction of the nucleus and its environment. It is measured relative to a reference compound.
For1H NMR, the reference is usually tetramethylsilane, Si (CH3)4.
Spin - spin coupling
Consider the structure of ethanol;
The 1H NMR spectrum of ethanol (below) shows the methyl peak has been split into three
peaks (a triplet) and the methylene peak has been split into four peaks (a quartet). This occursbecause there is a small interaction (coupling) between the two groups of protons. The
spacings between the peaks of the methyl triplet are equal to the spacings between the peaks
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of the methylene quartet. This spacing is measured in Hertz and is called the coupling
constant, J.
To see why the methyl peak is split into a triplet, let's look at the methylene protons. There
are two of them, and each can have one of two possible orientations (aligned with or opposed
against the applied field). This gives a total of four possible states;
In the first possible combination, spins are paired and opposed to the field. This has the effectof reducing the field experienced by the methyl protons; therefore a slightly higher field is
needed to bring them to resonance, resulting in an upfield shift. Neither combination of spins
opposed to each other has an effect on the methyl peak. The spins paired in the direction of
the field produce a downfield shift. Hence, the methyl peak is split into three, with the ratio of
areas 1:2:1.
Similarly, the effect of the methyl protons on the methylene protons is such that there are
eight possible spin combinations for the three methyl protons;
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Out of these eight groups, there are two groups of three magnetically equivalent
combinations. The methylene peak is split into a quartet. The areas of the peaks in the quartet
have the ration 1:3:3:1.
In afirst-orderspectrum (where the chemical shift between interacting groups is much larger
than their coupling constant), interpretation of splitting patterns is quite straightforward;
The multiplicity of a multiplet is given by the number of equivalent protons in
neighbouring atoms plus one, i.e. the n + 1 rule Equivalent nuclei do not interact with each other. The three methyl protons in ethanol
cause splitting of the neighbouring methylene protons; they do not cause splitting
among themselves
The coupling constant is not dependant on the applied field. Multiplets can be easily
distinguished from closely spaced chemical shift peaks.
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2.nmr
http://www.cis.rit.edu/htbooks/nmr/inside.htm
INTRODUCTION
MR
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 learnabout 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-dimensionaltechniques 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 wayto incorporate both static and dynamic figures with hypertext. This book presents a
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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 thesymbol . 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.
SPIN PHYSICS
spin
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What is spin? Spin is a fundamental property of nature like electrical charge or mass. Spin
comes in multiples of 1/2 and can be + or -. Protons, electrons, and neutrons possess spin.
Individual unpaired electrons, protons, and neutrons each possesses a spin of 1/2.
In the deuterium atom ( 2H ), with one unpaired electron, one unpaired proton, and one
unpaired neutron, the total electronic spin = 1/2 and the total nuclear spin = 1.
Two or more particles with spins having opposite signs can pair up to eliminate the
observable manifestations of spin. An example is helium. In nuclear magnetic resonance, it
is unpaired nuclear spins that are of importance.
Properties of Spin
When placed in a magnetic field of strength B, a particle with a net spin can absorb a photon,
of frequency . The frequency depends on the gyromagnetic ratio, of the particle.
= B
For hydrogen, = 42.58 MHz / T.
Nuclei with Spin
The shell model for the nucleus tells us that nucleons, just like electrons, fill orbitals. When
the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, and 126, orbitals are filled.
Because nucleons have spin, just like electrons do, their spin can pair up when the orbitals are
being filled and cancel out. Almost every element in the periodic table has an isotope with a
non zero nuclear spin. NMR can only be performed on isotopes whose natural abundance is
high enough to be detected. Some of the nuclei routinely used in NMR are listed below.Nuclei Unpaired Protons Unpaired Neutrons Net Spin (MHz/T)
1H 1 0 1/2 42.58
2H 1 1 1 6.54
31P 1 0 1/2 17.25
23 Na 1 2 3/2 11.27
14 N 1 1 1 3.08
13C 0 1 1/2 10.71
19F 1 0 1/2 40.08
Energy Levels
To understand how particles with spin behave in a magnetic field, consider a proton. This
proton has the property called spin. Think of the spin of this proton as a magnetic moment
vector, causing the proton to behave like a tiny magnet with a north and south pole.
When the proton is placed in an external magnetic field, the spin vector of the particle aligns
itself with the external field, just like a magnet would. There is a low energy configuration or
state where the poles are aligned N-S-N-S and a high energy state N-N-S-S.
Transitions
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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 ata 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 asa function of the time t after its displacement is:
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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 purepure 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
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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 appearsstationary 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 B 1 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 theequilibrium magnetization down to along the -Z axis.
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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 B o 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 atthe 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 transversemagnetization to dephase. This can be seen by examining the plot of Bo experienced by the
red hydrogens on the following water molecule. The number of molecular motions less than
and equal to the Larmor frequency is inversely proportional to T2.
In general, relaxation times get longer as Bo increases because there are fewer relaxation-
causing frequency components present in the random motions of the molecules.
Spin Exchange
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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 Bnuclei 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.
NMR SPECTROSCOPY
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.
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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 aromaticorbitals creates a magnetic field at the hydrogen nuclei which enhances the B o 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, B o, used toperform 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 threebond lengths, this effect is observable. This effect is called spin-spin coupling or J coupling.
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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 Babsorption 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
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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 5how the FID is converted into a frequency domain spectrum. You willsee inChapter 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 rotatingslower 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.
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2-D TECHNIQUES
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 moleculeA2-C-C-BwhereA andB are spin-1/2 nuclei experiencing resonance. The NMR spectrum from a 90-
FID 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 theA nuclei. There
are two absorptions lines in the spectrum from theA nuclei, one at +J/2 and one at -J/2. Atequilibrium, 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 theB nucleus
which is causing the splitting of theA 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
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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 theA2-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 J-
resolved 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 Type Spectrum
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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 intophase with the reference spin so it diminishes the echo height.
The relationship between the signal (S) obtained in the presence of a gradient amplitude G i in
the i direction and the diffusion coefficient in the same direction is given by the followingequation 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.
For accurate measurements with each pulse sequence, it is important that the sample
experience experience a spatially uniform B1 magnetic field. When inhomogeneous fields are
used, inaccurate rotations result and the spins do not follow the general equation for the pulse
sequence. One common cause of an inhomogeneous B1 magnetic field in a sample is a sample
extending beyond the homogeneous bounds of the RF coil. It is best that the sample be
confined to the volume of the RF coil. In practice this means filling an NMR tube with sample
to a height no greater than the length of the RF coil, and positioning the tube with sample inthe center of the coil. This position will require reshimming the sample if you are concerned
about narrow lines widths.
T1 MeasurementRecall the timing diagram for an inversion recovery sequence first presented inChapter 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 TItimes), the T1 value can be determined from the zero crossing of the curve which is T 1ln2.
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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 proposedwhich do not require knowledge of the equilibrium magnetization or signal .
T2 MeasurementMeasurement 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 non-
linear least squares procedure.
Solid State
We saw inChapter 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 moreperpendicular 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).
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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 m, the spectrum would narrow to the fast tumbling
limit.
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 specialsample 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 MRIlocated 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 = T1ln2
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, asaturation pulse similar to that employed in C-13 NMR (SeeChapter 9) is used to decouple
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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. Bdis 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 forthis 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 exponentiallydecaying 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.
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