buket NMR spectroskopy

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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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