Download CH437 CLASS 7

CH437 CLASS 7
NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY 1
Synopsis. Nuclei in magnetic fields, the NMR experiment, spectral transitions, level populations
and sensitivity, single pulse experiment, rotating frame of reference, free induction decay and
relaxation.
Introduction: Nuclei with Spin
Like electrons, certain atomic nuclei (those with even and odd atomic number Z
and mass number A) possess spin and this circulation of charge generates a
magnetic moment (μ) along the axis of spin. These nuclei are called “magnetic
nuclei” and are characterized by non-zero values of the spin quantum number (I).
Common nuclei and their overall spins are given below.
Nucleus
1H
Spin
quantum 1/2
number
2H
12C
13C
14N
15N
17O
18O
19F
1
0
1/2
1
-1/2
-5/2
0
1/2
Nuclei with even values of A and Z (e.g.
12C
and
16O)
have no spin (I = 0), are
non-magnetic and hence do not participate in NMR experiments. Those with an
odd value of Z and/or A and with I = ±½, behave as spinning spheres (dipoles),
whereas those with I = 1, 3/2, 2 (etc) behave as spinning ellipsoids
(quadrupoles), as shown below. It is these nuclei that are the basis of NMR
spectroscopy, but this course will focus on nuclei with I = ½, especially 1H and
13C.

I = 1/2
No spin
I =0
I = 1, 3/2, 2, etc

I = -1/2
1
Nuclear spins are orientated randomly in the absence of an external magnetic
field (Bo): the spin quantum number (I) and the angular momentum (p) are related
by,
p =
Ih
2
(1)
Each spinning nucleus of spin I  0, has associated with it a magnetic moment
(), given by equation (2)
 = p
(2),
where  is called the magnetogyric ratio. The parameters  and  are determined
by experiment (often by NMR!).
In the presence of an external magnetic field (Bo), nuclear magnetic spins
behave in particular ways that provide the basis for NMR spectroscopy.
Nuclear Spins in an Applied Magnetic Field: the NMR Experiment and
Spectral Transitions (for nuclei with I = ½)
The way in which a nuclear magnetic moment interacts with an applied magnetic
field (Bo) (the Zeeman interaction), can be described using quantum theory:
H
=
=
_ B
o
_  h B .I
o
(3) (by combining equations 1 and 2)
2
H is the Hamiltonian operator.
Here I is a quantum mechanical operator: the solution of this Hamiltonian gives
energy levels in which the interval between adjacent levels is
E = hBo/2 (4).
2
The NMR selection rule allows only transitions only between adjacent levels,
giving one spectral line.
In this equation, m assumes the values -I, -I+1…..I-1 , I. Hence there are 2I + 1
energy levels of a nucleus of spin I, in field Bo. Each of these may be considered
as arising from an orientation of I with respect to Bo, such that the projection of I
on Bo is quantized.
For the very important case when I = ½ (e.g. 1H,
13C, 19F,
etc), there are just two
energy levels (m = +1/2 or –1/2), as shown below, with the ground state (lower
level) corresponding to alignment of nuclear spin with the field.
Note that the gap between lower and upper energy levels increases with
increased applied field strength, which is measured in units of Tesla (T).
The NMR phenomenon arises from transitions between energy levels, just as in
other types of spectroscopy. In NMR transition occurs when nuclei in a strong
external magnetic field Bo are subjected to a radiofrequency field B1 (frequency
), applied in the x direction, with Bo in the z direction.
At 7.04 T, E calculated from equation (4) has values (in the millijoules per mole
region) corresponding to frequencies (since E = h) of only 300 MHz for 1H
(with high magnetogyric ratio), and even lower values of 75.5 MHz and 25.4 MHz
for
13C
and
15N,
respectively. This means, according to Boltzmann’s distribution
law:
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No. of nuclei in lower level
=
exp
No. of nuclei in higher level
E
kT
(5)
.
With such low energy gaps, for all types of nuclei, there is only a very slight
excess of nuclei in the lower energy level (~1 in 500,000) at room temperature.
This fact is the cause of the relative insensitivity of NMR spectroscopy,
compared with other kinds of spectroscopy.
Sensitivity is maximal at high values of applied field B o and radiofrequency field
B1 for nuclei with high magnetogyric ratios, such as 1H and
19F.
Most other nuclei
(e.g. 13C, 15N and 17O) are relatively insensitive.
The Single-Pulse NMR Experiment
It’s nowadays much more common to have the RF field applied as a short pulse,
rather than continuously. This is the basis of Fourier Transform (FT) NMR, as
opposed to continuous wave (CW) NMR spectroscopy. Rather than consider
individual nuclei and their magnetic moments, it is more convenient to consider
the bulk substance under investigation, where it is assumed that all the nuclei are
of one type, with I = ½.
In the presence of a strong magnetic field, the nuclei distribute themselves
between the two energy levels, so that there is a slight excess of nuclei in the
lower level, with spins aligned with the applied field. This results in a bulk
magnetization M, where M is a vector with a fixed z and no x or y components.
If the population levels are disturbed (by an RF pulse), then M becomes a vector
with a fixed z component, but also now with x and y components that vary
sinusoidally, 90o out of phase – that is, executing a circular motion or
precession.
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Bo
Bo
Perturbation
(applicationof
radiofrequency
field B1
z
M
z

M
y
y
x
x
B1
The frequency of this precession (o) is known as the Larmor frequency and is
given by,
o =
 B (radians/sec)
o
=
B
o
(Hz)
(7)
2
After the pulse, the nuclei return to an equilibrium situation, like any other
system.
Larmor frequencies are in the range tens to hundreds of MHz (depending on the
field strength), some typical examples are shown below for 7 tesla field strength.
Effects of pulses
The RF pulse is imposed on the sample along the x-axis, if the magnetic field
direction is along the z-axis. In practice this is done by induction from a current
flowing in a coil wound perpendicular to Bo. A linear RF field oscillation is
generated, which is equivalent to two counter-rotating field vectors in the xyplane:
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If the excitation frequency is exactly equal to the nuclear precession frequency,
the RF field B1 interacts with the magnetization M to produce a torque that moves
the magnetization to the xy-plane (as described previously). Since the precession
frequency (about Bo) is equal to the rotating RF B1 frequency, the magnetization
will remain perpendicular to B1 RF field component in the xy-plane. Hence the
magnetization will precess around the RF field at an angular frequency B1. This
double precession is called nutation.
See the diagram below for illustration. Only one component of B 1 is shown, for
simplicity.
z
Bo

M
y
B1
Mxy
x
The final position of the magnetization will depend on the time duration of the RF
pulse (tp), which is usually a few microseconds. The flip (or tip) angle, through
which the magnetization is flipped from the z-axis is given by
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 = B1tp
Since in practice, the pulse duration that is needed to produce a particular flip
angle varies somewhat from day to day, it is standard practice to refer to this
duration by the effective flip angle rather in time units. Thus a pulse width is
described as 90o (in degrees) or /2 (in radians), rather than in microseconds.
In practice, the relationship between the pulse width and the flip angle is
observed by varying the pulse duration and examining the resultant NMR signal
intensity. Since the magnetization is measured in the xy plane there will be a
sinusoidal variation in the intensity with pulse width, as shown below.
Pulse width
/2

3/2
2
Intensity
Maximum+
0
Maximum -
0
The Rotating Frame of Reference
The nutational motion of magnetization described earlier is rather complex and
hence it is helpful to consider NMR experiments in a rotating reference frame
rather than in a “fixed” laboratory reference frame. The rotating reference frame
is taken to be the precession of the x’ and y’ axes around the z’ axis, described
as the precession of B1 around the z
(Bo) axis, of frequency Bo. The first
precession diagram then reduces to the simpler diagram below:
y
B1
x
Bo
x'
An analogy for the rotating frame is shown below.
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Magnetization lies initially along z’, perpendicular to x’. The precession about B1
is simply a rotation of M in the x’y’ plane. The magnetization will remain
perpendicular to B1 and a /2 (90o) pulse would place M along the y’ axis. If the
Larmor frequency and pulse frequency are identical, then after the pulse, the
magnetization will remain fixed in the x’y’ plane of the rotating frame: it will rotate
at a frequency of Bo with respect to the “fixed” laboratory xy plane.
Rudiments of the NMR Experiment: Detection of Signal and Free Induction
Decay
The coil that supplies the RF field is wrapped around the (laboratory) x-axis and
also acts as a receiver. Hence, the precessing magnetization about the
(laboratory z-axis) will induce an oscillating current in the coil that can be
detected: this is the NMR signal.
A schematic diagram of a typical NMR instrument is shown below.
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However, this signal has a limited lifetime (though long compared with pulse
duration), since it dies as the nuclei reassume their equilibrium distribution
between the two energy levels. This process of signal loss after pulsed excitation
(perturbation) is called Free Induction Decay (FID). The signal resulting from
just one pulse is likely to be very weak and will be hard to differentiate from the
general electronic background (“noise”). Consequently, the pulse, followed by
decay time sequence is repeated many times (until the “signal-to-noise ratio” is
acceptable). The data is accumulated in the computer memory, where noise,
being random, gradually diminishes, whereas signals become more intense.
Fourier transform can then be carried out on the final FID data to produce the
familiar frequency spectrum. The most common representation of an NMR
experiment, in terms of pulse and acquisition times (etc), are as in the figure
below, shown for one of the simplest kinds of NMR experiments. The time axis is
not to scale.
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/2
/2
In practice, this
kind of diagram
is simplified to:
or
AQ
/2
or
AQ
There are variations on these; the pulses often being indicated by shorter, fatter
boxes.
Free Induction Decay and Relaxation
Because a normal organic sample will have nuclei of the same type (e.g. 1H or
13C)
in different magnetic environments (see “Chemical Shifts”, class 8), these
will have slightly different Larmor frequencies. Hence a pulse of RF field at the
nominal precession frequency (say that of 1H or
13C
in TMS, Me4Si) will be
slightly offset from the actual resonant frequencies. Consequently, the FID for
each environmentally identical sets of nuclei will depend on this offset or
frequency difference, as shown below.
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In a real sample, with nuclei in several different magnetic environments, the
individual FIDs combine to give a complex decay pattern, as below.
Fourier transform, a mathematical procedure for separating individual
oscillations from complex combinations, is used to convert the FID (time domain)
into the familiar spectrum (frequency domain). This is done by a computer, using
suitable software (e.g. the Cooley-Tukey algorithm).
Relaxation is the term used to describe the natural tendency of a set of nuclei to
revert to the equilibrium distribution between the energy levels after the excitation
RF pulse: the macroscopic magnetization is restored to its original position along
the z-axis.
From the Bloch equations, which describe the behavior of the macroscopic
magnetization in the presence of an RF field, there are two kinds of relaxation
processes: longitudinal or spin-lattice relaxation and transverse relaxation.
These are characterized by relaxation times, T1 and T2, respectively.
Longitudinal relaxation is characterized by excited nuclei returning to the ground
state by loss of energy caused by non-radiative interactions with their
environment (the “lattice”), thus re-establishing equilibrium. The magnitude of T1
is highly dependent upon the identity of the nucleus, the state of matter of the
sample and the temperature. For liquids, T1 is typically in the range 10-2 – 100 s,
whereas much higher values are typical of solids.
Transverse relaxation results from the loss of the magnetization component in
the xy plane independently of the mechanism for spin-lattice relaxation (i.e.
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without non-radiative exchange of energy with the “lattice”). Hence T2 must be
less than or equal to T1. A major mechanism of transverse relaxation is thought
to involve random exchange of energy between pairs of spins, so that nuclei of
nominally the same kind precess at slightly different rates and the spin ensemble
dephases, with resulting diminution of the xy component of magnetization, as
shown below.
It can be shown from quantum mechanics that the minimum line width for an
NMR signal is proportional to 1/T1. Since T2  T1 and if the if the line shape from
equation (4) is Lorentzian, it can be shown that the observed width of the line at
half maximum (1/2) is given by
1/2 = 1/ T2*,
where T2* is the effective T2 that takes into account magnetic field
inhomogeneity. True T2 values can be determined from “spin echo” experiments.
Spin lattice relaxation times (T1) can be determined from a pulse sequence
experiment, known as the inversion-recovery method, as shown below.
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The first pulse (180o) inverts the magnetization to the –z direction. Partial spin
lattice relaxation occurs during the time delay (), then a 90o pulse rotates the
remaining magnetization to the y’ axis. The system is then allowed to establish
complete equilibrium and the sequence is repeated for another value of , a
relaxation plot may be plotted, from which T1 can be measured. This process
involves purely longitudinal relaxation, because there is no xy component in the
inversion magnetization induced by the first (180o) pulse.
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The plot above is obtained by measurement of peak heights of the signal from a
set of spectra using this inversion-recovery pulse sequence:
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