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Chapter 3
Nuclear Magnetic
Resonance Spectroscopy
3.1
Nuclear Spins
Nuclei may possess a spin angular momentum and the observation of nuclear
spins in Stern-Gerlach type experiments played a large role in the development
and acceptance of particle physics. The original Stern Gerlach experiment in
1922 probed the electron spin in silver atoms evaporated in an oven and flying
through a magnetic field (Fig. 3.1).
N
Ag, (700º C)
S
Figure 3.1: Stern and Gerlach observed the quantized electron spin νz = ± 21 · ge νB
in a beam of neutral silver atoms.
Silver atoms have 47 electrons, (electronic shell: 1s2 2s2 2p6 3s2 3p6 4s2 4p6 3d10 5s1 ).
The single outer electron was expected to have zero angular momentum (l=0)
1
2 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
Nucleus
1
H
Abundance
Spin
Spin projections
γN (107 T1·s )
99.98%
I= 12
mI = ± 21
26.752
± 21
6.727
13
C
1.11%
I= 12
14
N
99.64%
I=1
I= 12
I= 12
19
F
100%
31
P
100%
mI =
mI = 0, ±1
mI =
mI =
± 21
± 21
1.933
25.177
10.840
Table 3.1: Nuclear spins regularly used for NMR spectroscopy.
and no interaction with the magnetic field was expected. The experiment, however, showed two discreet trajectories indicating a quantized magnetic property
of the atoms. Both, the presence of an inherent electron spin, and its quantized
property in an external magnetic field came as a surprise and affected the development and wide acceptance of quantum physics. Nuclear spins are about 3
orders of magnitude weaker and were later observed in similar experiments. The
observation of nuclear spins gave first indications that nuclei have structure.
Nuclei are composed of protons and neutrons, both of which have inherent
spins of I = 12 . Hence, nuclei with an odd number of protons or neutrons must
have a spin I6=0; if the sum of protons+neutrons is odd, then the spin must be
a half-integer number I = n + 12 (n=0,1,...). The nuclear spin quantum number I for nuclei commonly encountered in nuclear magnetic resonance (NMR)
spectroscopy is shown in table 3.1.
EmI Spin state energy
γN gyromagnetic moment
B0 Magnetic field
EmI
= −γN ~B0 mI
(3.1)
mI spin projection
In absence of a magnetic field, the nuclear spin is not an observable property.
In presence of a magnetic field B0 , the nuclear spin adopts quantized values
mI = -I, -I+1 ... -I, which explains the discreet trajectories observed in the
Stern-Gerlach experiments. There exists no classical interpretation for the spin
quantization. The energy of the spin states in a magnetic field can be calculated
using the nuclear gyromagnetic moment γN in table 3.1 using the equation 3.1.
Nuclear Spin Signals
The energy difference ∆E = γN ~B0 between proton spin states in a strong
magnetic field of 1 T can be directly estimated as ∆E = 2.675 · 108 ~s−1 corresponding to a frequency of 42.6 MHz. This so called ”Larmor frequency” is
directly proportional the the magnetic field strength. Irradiation of a sample
3.2. NMR SPECTRA
3
with the Larmor frequency of its nuclei will lead to spin excitation and absorption of the radiation.
1
H gyro. moment
Magnetic field
Planck constant
Boltzmann const.
Temperature
γN = 2.7 · 108
1
Ts
B0 = 1.0 T
~ = 1.05 · 10−34 Js
J
k = 1.4 · 10−23
K
T = 300 K
Nα
Nβ
Nα
Nβ
γN ~B0
= exp −
kT
≈ 1 − 6.7 · 10−6
(3.2)
MHz and GHz frequencies are comfortably in the range of radiowave emitters
and detectors, so spectroscopic characterization is straightforward. Original experiments used a fixed frequency radiowave oscillator and observed absorption
while scanning the magnetic field strength induced by electromagnetic coils.
The size of the observed signal is proportional to the population difference between the lower (α) and upper (β) spin state, and to the energy per absorbed
photon (the latter is directly proportional to B0 ). We can use the Boltzmann
distribution as shown in equation 3.2 to estimate population differences. With
the approximation that e−x ≈ 1 − x for small x, the population difference is
also proportional to B0 , and we find that the signal depends quadratically on
the field strength. This explains the the great efforts to obtain ever stronger
magnetic fields for NMR and the early adoption of superconductor materials in
the corresponding magnetic coils. Modern NMR spectrometers operate at field
strength up to 20 T.
3.2
NMR Spectra
Chemical Shift
When an external magnetic field is applied to a sample, not all nuclei feel the
same field strength. The electron spins with their much higher magnetic moment
get spin polarized and induce a local magnetic field which can shield the nucleus
or increase the field at the nucleus (Fig. 3.2). Both, the nuclear polarization
and the electron polarization are proportional to the magnetic field, therefore
the shielding effect leads to a frequency shift which is a constant fraction of the
resonance frequency. It is therefore advantageous to plot NMR spectral shifts
6
0
as parts per million (ppm) on a proportional scale δ = ν−ν
ν0 · 10 , invariant to
the field strength of a particular experiment.
The discovery of the chemical shift moved the investigation of nuclear spins
into the domain of chemistry, whereas originally such investigation were of interest only to nuclear physicists who wanted to get insight into the structure
of atomic nuclei. In 1952, only 6 years after the publication of the first NMR
spectra, Bloch and Purcell were awarded the Nobel Prize in Physics for their
4 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
B0
Bloc=B0-σB0
ν=
γN
(1 − σ )B0
2π
Bel=-σB0
Figure 3.2: The applied magnetic field B0 interacts with the electron angular momentum and induces a magnetization Bel . This magnetization affects the local magnetic
field acting onto the nucleus and shifts the corresponding resonance frequency ν.
work. Through the direct measurement of local magnetic fields, the nuclear
spins are perfect probes for local electronic structure. Nuclear spins are always
reported relative to a chemical standard, typically tetramethyl-silane (TMS) to
remove the need for an absolute calibration. Chemical shifts for some chemical
groups are shown in figure 3.3.
H
R
O
R
H
N
(TMS)
H
O
R
HC
CH3
H3C3
Si
R
O
R
H
O
H
CH3
R
H
H
H
R
H
R
H
11 10 9
8
higher ν
7
6
δ
5
H
4
3
2
R
1
H
H
0
Figure 3.3: Chemical shifts for a number of chemical groups. Electron withdrawing groups ”deshield” the protons and lead to smaller shifts and higher excitation
frequencies.
Spin-Spin Coupling
Beyond the interaction with the electron angular momentum, nuclear spins also
interact with other nuclear spins leading to the so called ”fine structure” or J-J
splitting. This interaction can increase, or reduce the energy of a spin state by
3.2. NMR SPECTRA
5
1
4 J.
If we consider a system of two coupled spins A and X, then the combined
system can be in an αA αX , αA βX , βA αX , or βA βX state. The excitation energy
of either spin from αA αX in the example in Fig. 3.4 is reduced by 2 · 14 J, but
the excitation energy from αA βX or βA αX to βA βX is increased by 2 · 14 J. The
excitation energies for the nuclear systems A and X are therefore now split into
two lines and split by exactly the same energy ± 21 J = J.
uncoupled spins
+¼ J
coupled spins
ββ
βX
uncoupled spins
+¼ J
βA
hνA +½ J
h νX
hνX +½ J
αX β A
hνA
αA
-¼ J
+¼ J
βα
αβ
hνA - ½ J
h νA
-¼ J
αA
hνX
hνX - ½ J
αα
βX
+¼ J
αX
Figure 3.4: Coupled versus uncoupled spin system: Uncoupled spins A and X can
be excited independently, e.g. first A, then X (left) or first X, then A (right). In the
coupled system, the spins can stabilize (αβ, βα) or destabilize (αα, ββ) each other by
1
J, leading to a splitting of each transition line.
4
Nuclear spins are magnetic dipoles and spin-spin interaction therefore falls of
very rapidly, proportional to r16 . Through-space interaction is therefore limited
to very short ranges (< 5Å), and the spins couple most strongly through the
electronic system. The coupling through bonding electrons is illustrated in Fig.
3.5: The Fermi coupling between nuclear and electronic spin in a single atom
leads to a preferentially antiparallel alignment of nuclear and electronic spin.
According to Pauli, the two electrons in the bonding orbital to a neighboring
atom must be of opposite spin, carrying the spin information to the next atom.
The Hund rule indicates that electrons in different orbitals preferentially adopt
the same spin state, carrying the spin correlation into further bonds. However,
through-bond spin coupling is is quickly decreasing with the distance, i.e. the
number of atoms between the nuclei.
The JJ splitting is independent of the magnetic field (6= chemical splitting)
and is usually given in Hertz (Hz). Typical coupling constants for a number of
vicinal (neighboring) and geminal (non-neighboring) JJ-interactions are given
in Figure 3.6. JJ Coupling across more than 3 (for JHH ) or 4 (for JCH ) bonds
is rarely observed.
When a spin is coupled to two identical spins (e.g. in a system R-C(H)=CH2 ),
then two identical splittings lead to the observation of a triplet with intensity
ration of 1:2:1 with identical spacing of J. If a spin is coupled to two different
6 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
hydrogen
H
e- e-
ethylene
Hund rule
H
H
Fermi
coupling
Pauli
rule
C
Fermi
coupling
C
H
Figure 3.5: Coupling of nuclear spins occurs predominantly via the electronic system.
The Fermi coupling, Pauli’s rule and Hund’s rule determine whether parallel spins
(e.g. hydrogen) or antiparallel spins (e.g. ethylene) are lower in energy.
spins x and y, then the interaction will lead to a splitting by Jx and another by
Jy and therefore result in 4 lines of equal intensity for the single spin. The fine
structure can become very convoluted if multiple couplings are present (e.g. 18
lines in the spectrum of CH3 -CH2 -CH2 -NO2 ), complicating the analysis.
Spin-spin coupling also occurs through space, but the magnetic dipole-dipole
interaction falls off with the 6t h power of the distance. Therefore only interactions across small distances are observed and no discreet splittings can be
assigned. However, the through space coupling of spins A and B reduces the
intensity of both spin signals. The intensity of spin A can be recovered by saturating the transition for spin B. This effect is called the ”Nuclear Overhauser”
effect and allows to characterize through-space coupling.
Spin decoupling
To interpret convoluted spectra, it would be desirable to observe only selected
JJ couplings which can be unambiguously assigned. This can be done by irradiating the sample with a continuous, strong RF pulse at the Larmor frequency
of a chosen spin x. The selected spin will undergo multiple spin flips due to
absorption and stimulated emission. If the spin flips occur mich faster than the
coalescence time, then the neighboring spins will interact with a time-averaged
spin 12 α + 12 β = 0, hence the splitting with spin x is removed. All other spins
still couple, so the missing couplings can be assigned to interaction with spin x.
3.3
Fourier Transform NMR
Instead of sweeping the magnetic field B0 for a constant RF frequency ωf , or
sweeping ωf for constant B0 , it is possible to excited multiple spins coherently
and to observe the temporal evolution of the induced magnetization. To excite
multiple transitions, the excitation pulse must contain all the corresponding
frequencies. A short pulse always contains many frequencies (remember the
uncertainty principle: ∆E∆t ≥ ~) and can be considered to be the superposition
3.4. NMR IMAGING
7
R
H
R
R'
H
H
JHH = 12..15 Hz
H
RR
R R
H
R'
H
R
1J
CH
6..8 Hz
H
H
= 125..200 Hz
R
6..9 Hz
H
R
R
H
R
R
H
H
H
R
H
R3C
2J
7..12 Hz
R
R
160 Hz
250 Hz
R
R
JHH = 0-3 Hz
H
R
H
CH
R
R3C
CH
RC
R2C
H
H
= -2 ..-5 Hz
13..18 Hz
3J
H
R
H
-2.5 Hz
R3C
H
R
H
R
CH3
40-66 Hz
R3C
= 4 ..8 Hz
7-13 Hz
H
3 Hz
Figure 3.6: Selection of typical homonuclear JHH and heteronuclear JCH coupling
constants.
of discrete frequencies. The creation of such RF pulses is no major challenge
and is illustrated in Fig. 3.8.
If a large ensemble of spins interacts with the electromagnetic field, then
the superposition of particle waves will resemble a classical particle and instead
of quantized properties and a probabilistic description of transitions, we can
consider the motion of a quasi-classical particle. We therefore discuss the excitation and observation of a ”spin wavepacket” in the ground (α) and excited (β)
state and talk about the total magnetization as if it were a true property of the
→
−
−
→
−
system. In the x-y plane, the external field leads to a torque T = →
µ × B0 on
the magnetization vector. If the field is oriented along the z-axis, this results in
a spinning of the magnetization (superposition of the magnetic moments) in the
x-y plane with the difference frequency between α and β, the Larmor frequency.
P→
−
→
−
For easy interpretation, the total magnetization M =
µi is considered and
i
plotted as shown in Fig. 3.9. The rotating magnetization vector induces currents
into a detector coil, i.e. emits electromagnetic radiation which can be detected.
3.4
NMR Imaging
Imaging in one Dimension
The resonance (Larmor) frequency for a spin transition is proportional to the
γ
magnetic field strength: µL = ( 2π
· B0 . If a field gradient B = B0 + x · Bx is
applied along one axis, then the resonance frequency is a linear function of the
spin positions within the sample. A NMR spectrum will then directly measure
the spin density along the selected spatial coordinate and offer a one-dimensional
8 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
image of the sample. Please note, that possible frequency shifts due to chemical
shift and fine-structure are very small and are ignored. Measuring multiple
one-dimensional projections along different axes allows to reconstruct a two- or
three-dimensional image as shown in Fig. 3.10.
All biological tissue contains water and other proton sources, hence the contrast of 1 H NMR for biological imaging may be unsatisfactory. But a second
observable is directly available: the T2 spin relaxation time which is a sensitive
probe to diffusion properties, pH, proton mobility, and other factors. T2 is preferentially measured in the time domain using a π − mixing − time − π/2 pulse
sequence as shown in Fig. 3.11. Because signals measured with the π − π/2
pulse sequence may show different phases for spins with long/short τ1 lifetimes,
selection of a suitable mixing time can give excellent contrast in imaging experiment and allows the easy distinction of different types of biological tissue as
seen in the T2 enhanced brain image in figure 3.14, center.
Imaging in Multiple Dimensions
If imaging is performed in the time domain, multiple pulsed field gradients can
be used to mark spins in more than one dimension. In a first approach, we can
excite only a selected slice of the sample along axis x with a field gradient Bx and
a frequency selective excitation pulse. After free induction decay is complete, a
π pulse leads to a spin echo, which is detected while applying a field gradient
By . The detected frequencies thus encode the y position within a slice x, as
shown in Fig. 3.12.
There is, however, a more elegant way for multidimensional imaging: a
detected coherent spin signal is due to the larmor precession of the corresponding
spin magnetization vector sin(ω·t+δ0 ) with the larmor frequency ω = γN ·B0 . If
a field gradient B0 (x) = B0 + x · ∆Bx is applied, the Larmor frequency will vary
as a function of x. As illustrated in Fig. 3.13, the precessing spins now dephase
with time t by ∆ω · tx = γN · B0 (x) · tx . After the field is switched off, we can
observe the unperturbed frequencies, but with a phase shift of δ0 = δ0 + ∆ω · tx .
A measurement of the phase shift is therefore a direct measurement of the
position along the axis x. In a sequence of spin-echo measurements, multiple
phase shifts can be induced along different axes using field gradients Bx , By and
Bz to obtain 3-dimensional spatial information.
Instead of discrete field gradient pulses along x,y and z, is is preferable to
use sinusoidally varying B-fields Bx,y,z ∼ sin((ax + by + cz) · t). The corresponding frequency components a, b and c are superimposed on the actual spin
frequencies and can be recovered by the fourier transform. This allows the measurement of three-dimensional images in a single experiment, but requires that
the experiment can resolve the applied frequency components from linewidth
effects.).
3.4. NMR IMAGING
9
Medical Imaging
The potential use of NMR tomography in medicine was a large motivator from
the early days of NMR imaging. NMR tomography offers a non-invasive and
non-destructive imaging method for biological tissue which easily penetrates a
human body. As compared to X-ray tomography, NMR offers unique capabilities
beyond the simple imaging of tissue density. First and foremost, we should
consider the spin relaxation time τ1 , which is easily measured with a scheme
as shown in 3.11. The spin relaxation time is a function of rotational mobility
(e.g. high in water, low in fat), pH (≥ 7 in normal cells, often lower in cancer
cells), and diffusion. The contrast can therefore be optimized for a particular
diagnostic problem. With spin lifetimes of seconds, it is even possible to follow
the flow of fluids. Figure 3.14 shows the image of a brain after a stroke, using
different imaging methods to diagnose the blood flow or lack thereof in the
affected brain regions (1).
The ability to observe blood flow and metabolic activity resulted in a novel
medical field, the so called ”functional imaging” or fNMR, where local brain
activity is monitored and correlated with mental activity. Fig. 3.15 shows
two examples, the brain activity associated with moving a finger and the more
abstract brain activity of imagining pain (1, 2). These methods offer unprecedented insight into neurological processes.
Direct spectroscopic measurements can characterize metabolites and properties such as pH (via analysis of phosphate signals). In some cases, metabolic
pathways can be followed, such as the chemical pathway of drugs.
10 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
δ 1.02
7.2 Hz
triplet
triplet
7.2 Hz
quadruplet
δ 4.35
7.5 Hz
triplet
Figure 3.7: Example for spin decoupling in the molecule 1-nitropropane: The convoluted band at δ2 appears to be a sextet. Decoupling at the frequency of the δ1.02 band
(bottom) reveals a triplet due to coupling with the CH2 NO2 group. Decoupling of the
band δ4.35 leads to a quadruplet due to coupling with the CH3 moiety (second from
top). The apparent sextet is therefore due to 12 lines, which cannot be experimentally
resolved.
©1999 William Reusch, All rights reserved,
http://www.cem.msu.edu/~reusch/VirtualText/Spectrpy/spectro.htm#contnt
δ 2.04
7.5 Hz
3.4. NMR IMAGING
11
Figure 3.8: 5 Sinusoidal pulses with frequency differences of approximately π/10 (top)
and their sum (bottom). The sum is an apparently shorter pulse, and indeed we expect
a frequency uncertainty according to the uncertainty principle (h∆ν · ∆t ≥ h).
B0
B0
z
z
Μ
μi
π/2zy pulse
μi
Μ
y
y
x
x
Figure 3.9: Nuclear spins µi and total magnetization M =
P
µi in a magnetic field
B0 (left), and after excitation of a coherent spin wavepacket with a π2 zy pulse.
12 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
Bo
ωL
B
o
Bo
ωL
ωL
Figure 3.10: NMR Zeugmatogram: If a linear magnetic field gradient B0 is applied
along an axis, then the spin frequencies ωL in the sample are directly proportional
to the position along the axis. The NMR spectrum then measures spin density as
function of position. Measuring multiple projections along different axes allows the
reconstruction of an image (Lauterbur 1973, Nobelprice for medicine together with
Mansfied in 2003).
B0 z
π
Μ
t1
y
x
z
z
z
π/2
τ1 short
y
y
x
τ1 short
x
τ1 long
y
x
τ1 long
Figure 3.11: Measurement of τ1 : after spin inversion with a π pulse, waiting time
t1 allows for spin relaxation. With a π/2 pulse, spins are turned into the x-y plane
for detection of the free induction decay (FID) signal. Signals which decayed by more
than 50% have opposite sign of the magnetization vector before the π/2 pulse and
have opposite phase in the FID.
3.4. NMR IMAGING
13
frequency / space
selective excitation
ωX
gh
hi
frequency / space
encoded detection
w
lo
q.
fr e
BY
q.
fre
BX
π (ωX)
π/2
BX
BY
Figure 3.12: Selective excitation of a yz plane is achieved by applying a field gradient
x and selecting a suitable excitation frequency ωx . The spin echo is detected after
a π/2 pulse while applying a field gradient along the y axis. Y-positions within the
selected yz-plane are now encoded in the frequencies of the free-induction decay.
x=0: ν = γN⋅B0/2π
x≠0: ν = γN⋅B(t)/2π
B(t) = B0
π/2
B(t) = B0 + ΔB⋅x
π
B(t) = B0
π
ΔB⋅x
Figure 3.13: Phase encoding of the spatial position: Varying the magnetic field
strength B along one spatial coordinate x will lead to gradual dephasing of the magnetic
vectors along this axis. The phase shift is directly proportional to the field and the
encoded phases can be analyzed by detecting a spin echo signal after a π pulse.
14 CHAPTER 3. NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY
Prichard, J. W et al. BMJ 1999;319:1302
Copyright ©1999 BMJ Publishing Group Ltd.
Figure 3.14: Magnetic resonance images after clinical stroke (blocked blood vessels
P.L. Jackson et al., Neuropsychologia 44, 752–761 (2006).
Copyright ©1999 BMJ Publishing Group Ltd.
Prichard, J. W et al. BMJ 1999;319:1302
in the brain). Left: ”Diffusion weighted imaging” shows damaged tissue. Center: conventional T2 weighted image. Right: Magnetic resonance angiography (measurement
of blood flow) shows flow void in blood vessels.
Figure 3.15: (A) Functional NMR image of the brain during finger movement shows
increased blood flow to particular brain regions. (B) fNMR imaging of the brain during
the abstract thought about pain.
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