PART V. NUCLEAR SPIN LEVEL SPECTROSCOPY
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where v0 is the frequency (in units of radian per second) of the electromagnetic
radiation, corresponding to the difference between the spin energy levels,
known as Larmor frequency. Equation (2) also predicts linear dependence of
the energy level difference on the external magnetic field B0. For the currently
available magnetic fields, the energy difference falls into the radio frequency
(rf) portion of the electromagnetic radiation. When the system is at thermal
equilibrium, the population of the given energy level is given by the Boltzmann
distribution
ÀEi
Ni ¼ exp
kB T
(3)
where Ni is the number of nuclei occupying energy level Ei, kB is the Boltzmann
constant, and T is the absolute temperature. In contrast to the electronic transitions, at temperatures close to the room temperature, the product of kBT is
several orders of magnitude larger than the difference between the Zeeman
levels DE. Consequently, even random thermal fluctuations can very efficiently
induce the nuclear spin transitions, nearly equalizing the number of nuclei occupying each Zeeman energy level. Being directly proportional to the population
difference, the NMR signal strength is significantly weaker compared with other
atomic or molecular spectroscopies involving electron levels transitions. Out of
200,000 1H spins placed into the external magnetic field of 11.7 T (500 MHz 1H
frequency), approximately 100,001 spins will occupy the lower energy level
and 99,999 the higher one. Effectively, only 1/200,000 of the sample gives
rise to the observable NMR signal. When compared with other spectroscopies,
NMR is a relatively insensitive technique.
Another property of nuclear transitions stands apart from spectroscopies
involving electronic transitions. It is the relative isolation of the nuclear system
from its surroundings. This simplifies the NMR response of the system under
the study and, typically, one can understand and simulate the outcome of any
NMR experiment based on first principles considerations. When out of thermal
equilibrium, nuclear spins tend to spontaneously return to their equilibrium
states. This relaxation process is driven by random fluctuations of variables
affecting the nuclear transitions. In many cases, encountered especially in
solid-state NMR (SSNMR), the relaxation might be prohibitively long. It is not
uncommon for the return to equilibrium to take in the order of hours or longer,
prohibiting the practical observation of the signal. On the other hand, unlike
the electronic spectroscopies, the nuclear spins can form coherent states for
long periods of seconds or longer. Modern NMR multidimensional experiments
take excellent advantage of this phenomenon. The slow decay of coherent states
allows one to design rather complex pulse sequences manipulating the coherent
states of nuclear spins to obtain the targeted information about the system under
the study.
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Nuclear Spin Interactions
To understand the applications of the NMR spectroscopy in both the solution and
solid phases, it is useful first to review the various interactions affecting the
nuclear spins. Nuclear spins can interact with each other or with internal or external electric and magnetic fields (1). The internal electric and magnetic fields arise
due to the effect of electrons surrounding the nuclei of interest in atoms or molecules. The possibility to selectively exploit the nuclear interactions makes NMR
an extremely powerful tool, finding its applications across many scientific
disciplines.
In quantum mechanics, all the interactions of a system of interest are represented by corresponding Hamiltonians. The state of the system is given by its
wave function and all the measurable quantities are represented by their operators. The most basic NMR interaction, the Zeeman interaction, was already
introduced in the previous section. Zeeman interaction causes the nuclear spins
to interact with the external magnetic field. This interaction is responsible for
observing the NMR phenomenon. Based on the Correspondence Principle, the
classical description from Equation (1) can be rewritten using the quantum mechanical Zeeman Hamiltonian HZ
HZ ¼ ÀgI^Z B0
(4)
where I^Z is the spin angular momentum operator. The energy levels of the nuclei
are directly proportional to the applied external field B0. It turns out that the external magnetic field has to be extremely homogeneous to resolve the fine features
of NMR spectra. A great deal of attention is paid to this issue. Every modern
NMR spectrometer is equipped with a set of shimming coils correcting the
local imperfections of the field.
NMR would not be a very chemically useful technique if only Zeeman
interaction existed, because all spins from any compound would resonate at
the same frequency. The useful NMR properties arise from other nuclear interactions. All the interactions exist simultaneously and can be described by the
total Hamiltonian (2 5)
HT ẳ HZ ỵ HCS ỵ HJ ỵ HD ỵ HQ ỵ Hrf
(5)
where the subscripts CS, J, D, Q, and rf denote the chemical shift interaction,
indirect spin-spin (J-coupling), dipolar coupling, quadrupolar coupling, and rf
irradiation, respectively. It should be noted that these nuclear interactions do
not represent all the interactions that nature provides. They are, however, sufficient to facilitate the majority of discussions of NMR applications in this
review. In terms of magnitude, the Zeeman interaction is typically, but not
always, the strongest, followed by quadrupolar, dipolar, chemical shift, and
J-coupling. This is just a general order, which may change strongly depending
on the type and shape of molecules. For example, in many asymmetric molecules
with quadrupolar nuclei, the quadrupolar coupling can be larger than the Zeeman
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interaction, especially when working in weaker magnetic fields. Such nuclei
may be difficult to observe by NMR but are a great subject to nuclear quadrupolar
resonance (NQR) studies, a spectroscopic technique, which does not require a
presence of an external magnetic field, because it deals only with the quadrupolar states. On the other hand, highly symmetrical molecules containing
quadrupolar nuclei can show vanishing quadrupolar contribution.
In general, in the laboratory frame, the Hamiltonian representing a particular interaction Hl can be characterized as a product of two spin operators with a
Cartesian tensor Alof rank two, described by the (3 Â 3) matrix:
0
Alxx
Alxy
B
Hl ¼ Cl (I^x , I^y , I^z ) Á @ Alyx
Alyy
Alzx
Alzy
1
S^ lx
B C
Alyz C
A Á @ S^ l A
Alxz
1 0
y
Alzz
(6)
S^ lz
where Cl is a constant dependent on the interaction l, Iˆ ¼ (Iˆx, ˆIy, ˆIz) is the
nuclear spin angular momentum operator of spin I, and Sˆ ¼ (Sˆx, Sˆy, Sˆz) can be
either the nuclear spin operator of spin S or B0 in the special case of l ¼ CS
(the chemical shift interaction). From this notation, it is evident that the
nuclear interactions depend on properties that exhibit a dependence on a
particular orientation in the physical space and have to be described by
tensors rather than scalars (numbers). For example, the Alzx element of the
tensor A, describes the reaction of the system along the z direction when a
“force” represented by the operator Sˆlx is applied in the x direction. In a
special frame of reference called the principal axis system (PAS), the symmetrical part of the second rank tensor takes a diagonal form, that is, the only nonzero
components lie along the diagonal:
0
Al11
0
B
Al (PAS) ¼ B
@ 0
0
0
Al0 (PAS)
¼
Al22
Al33
1
0 0
C
1 0C
A
1
B
0
dl11
B
Al2 (PAS) ẳ B
@ 0
0
dlii ẳ Alii AlISO
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C
l
l
0 C
A ¼ A0 (PAS) þ A2 (PAS)
0
AlISO B
@0
0
1
0
0
0
dl22
0
1
0
1
C
0 C
A
dl33
(7)
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The orientation of the principal axis system of a tensor Al depends on the
local molecular symmetry and even for the same nucleus can, in general,
change depending on the type of interaction l. The PAS orientation for given
coupling l is fixed with respect to the molecular frame of the compound.
Because Tr(Al2 ) ¼ 0, only two parameters are necessary for a complete
description of Al2 :
dl ¼ dl33
hl ¼
dl22 À dl11
dl
(8)
where hl is usually referred to as the asymmetry parameter with values limited
to 0 hl 1. An arbitrary second rank tensor is then defined in its respective
PAS by its isotropic part Aliso, by its anisotropic contribution dl, and the asymmetry parameter hl.
The chemical shift interaction is a coupling of the nuclear spins I with the
external magnetic field B0 mediated through the electronic environment. Diamagnetic currents in the electron orbitals and partial unquenching of their paramagnetism generate local field Bind proportional to B0, Bind ¼ 2ACSB0. The chemical
shift Hamiltonian is defined as
^ ind ¼ gIA
^ CS B0
HCS ¼ ÀgIB
(9)
By comparison of Equation (9) and Equation (6), the proportionality constant CCS
is simply equal to g. The values of the components of chemical shift tensor ACS
depend, in general, on the orientation and local symmetry of the molecule. The
remarkable property of the chemical shift tensor is that the sum of its diagonal
elements (known as its trace) does not vanish, thus giving rise to the isotropic
shift
Á
1 À
diso ¼ Tr ACS = 0
3
(10)
This is why even the fast isotropic tumbling of molecules in solution does not
average out the chemical shift to zero. Different nuclei in the same molecule
experience different local fields. Based on the chemical shifts alone, many molecules can be uniquely identified.
At the atomic level, a given nucleus affects, and is also affected by, the
local dipolar fields of its neighbors. The indirect spin –spin J-coupling can be
described by Hamiltonian in Equation (11).
HJ ¼ I^ Á AJ Á S^
(11)
It is called indirect, because it acts through the electronic environment, rather
than directly though space. Similar to the chemical shift, the trace of the
J-coupling tensor is not zero. Therefore we can observe and measure finite
J-coupling values even in solution. Moreover, for majority of practical
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applications, the J-coupling tensor AJ can be approximated as being symmetrical with vanishing anisotropy and as such can be regarded as scalar (a number)
rather than tensor.
Nuclear spins can also interact directly through space. Such an interaction
is referred to as dipolar coupling (or dipole – dipole coupling) and can be
described by the dipolar Hamiltonian:
"
(I^1 r)(I^2 r)
HD ¼ D I^1 I^2 À 3
r2
#
(12)
where r is the internuclear vector and D is the dipolar coupling constant in frequency units, defined as
D¼
m0 g1 g2 hÀ
4p 2p r 3
(13)
m0 is the permeability of vacuum constant. The trace of the dipolar tensor
vanishes, which is why dipolar coupling has no direct effect on isotropic solution
NMR spectra apart from relaxation-induced processes.
Spin I ¼ 1/2 nuclei possess only magnetic dipole moments, because all the
higher magnetic and electric multipole moments vanish. Therefore, all their
^ However, apart from the magnetic couplings
Hamiltonians are linear in I.
encountered in spin I ¼ 1/2 isotopes, nuclei with I . 1/2 are distinguished by
a nonzero nuclear quadrupole moment Q, which gives origin to the interaction
with the surrounding electric field gradients (EFG). The quadrupolar coupling
between the quadrupole moment and the electric field gradient is described by
the quadrupolar Hamiltonian:
HQ ¼
eQ
^ Q I^
IA
2I(2I À 1)hÀ
(14)
The symbol e designates the elementary charge. Similar to the dipolar coupling,
the trace of EFG tensor vanishes having no effect on solution NMR signals except
through the relaxation processes. The majority (72%) of all the magnetically
active nuclides in the Periodic Table possess half-integer spin higher than 1
(Fig. 1) (6). This clearly underlines the importance as to why we should study
these spins (7 – 9).
The chemical shift interaction, J, dipolar, and quadrupolar couplings are
called internal interactions, because they represent couplings between the
various fields present internally within a molecule. On the other hand, Zeeman
interaction and the rf irradiation are external interactions, because they represent
the interactions of the nuclear spins with the external magnetic fields.
In order to disturb the spin system from its thermodynamic equilibrium,
an oscillating rf field B1 has to be applied in the direction perpendicular to that
of B0. If the frequency of this oscillating field is v and its phase in the plane
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Figure 1 Magnetically active isotopes in the Periodic Table. The majority (72%) of the magnetically active nuclides are half-integer quadrupolar nuclei. If one plans to use solid state NMR
for studying ceramics, semi-conductors, catalysts, glasses, superconductors or metal binding
processes it is necessary to deal with the spectroscopy of these spins. For elements with
several different nuclides, only those with the highest natural abundance are considered.
perpendicular to B0 is w, then the B1 vector can be fully described by:
0
1
cos w cos (vt)
B1 (t) ¼ 2B1 @ sin w cos (vt) A
0
(15)
The irradiation Hamiltonian representing the coupling between the oscillating rf
field and the particular components of the magnetic dipole is in the laboratory
frame given by:
Hrf ¼ v1 ẵI^x cos(vt ỵ w) ỵ I^y sin(vt ỵ w)
(16)
where v1 ¼ gB1 is the nutational frequency directly proportional to the amplitude
of the irradiation field. This value, typically given in kHz, is used frequently in the
NMR literature to describe the strength of irradiation. The irradiation field is typically calibrated based on the experimentally observed duration of 908 pulse (t90),
which is related to v1 by v1 ¼ 1/(4t90). In practice, its value depends on the available power delivered from the rf amplifier (v1 is directly proportional to the rf
voltage amplitude) and on the internal characteristics of the probe, such as coil
geometry. Of a great interest is the homogeneity of the B1 field throughout the
coil, strongly influencing performance of many pulse sequences.
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SSNMR OF POWDER SOLIDS
Introduction
Most NMR applications have traditionally involved solution samples. In solution,
all the valuable anisotropic information is lost by the fast tumbling of the liquid.
Unlike in solution, the full extent of the anisotropic interactions is observed in the
solid state. In essence, there is more NMR information available on the system
when in the solid state. However, SSNMR possess much greater technical challenge to mine this information, which typically requires selective “turning on”
and “off” the various strong nuclear interactions discussed in the preceding
chapter. The great advantage of SSNMR is the “concentration” of the samples.
One does not have to rely on dissolution properties of compounds to introduce
them into the NMR coil.
The NMR spectra of stationary (not spinning) polycrystalline or disordered
solids are known as powder patterns. It is possible to predict their line shapes by
numerical simulations based on the a priori known nuclear coupling parameters.
Figure 2 and Figure 3 show the typical line shapes of chemical shift and
quadrupolar interactions. In simulating these spectra, only a single spin contribution was considered. Because more than one magnetically inequivalent
nucleus is typically present in a realistic system, the static spectra can quickly
become overcrowded and difficult or impossible to interpret. The available
Figure 2 Examples of static powder patterns for chemical shielding as a function of
asymmetry parameter hCS. The magnitude of chemical shielding dCS was arbitrarily set
to 143 kHz, and the isotropic shift diso ẳ 0.
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Figure 3 Examples of static powder patterns for quadrupolar coupling to the first- and
second-order as a function of asymmetry parameter hQ for I ¼ 7/2. The quadrupolar coupling constant e 2qQ/h was arbitrarily set to 22 MHz, the Larmor frequency (important for
second-order spectra) used in these simulations was v0 ¼ 47.7 MHz (59Co at 4.7 T), and
no chemical shift was considered.
signal is also spread over broad range of frequencies, which diminishes the
achieved sensitivity in terms of the signal-to-noise ratio (S/N ). Therefore, a technique improving the resolution of the SSNMR spectra of crystalline solids has
long been sought. The two key factors precluding widespread use of SSNMR
in the past were sensitivity and resolution. A boost in solid-state applications
appeared after a combination of the line-narrowing technique called magic
angle spinning (MAS) (10 – 12) and sensitivity-enhancing technique called
cross-polarization (CP) (13) was introduced. Cross-polarization magic angle
spinning (CPMAS) (14) brought about the necessary sensitivity enhancement
to make SSNMR spectroscopy of natural abundance samples practicable. Spinning around the “magic” angle, which is the root of second-order Legendre polynomial, x ¼ 54.748 (Fig. 4), removes completely the second rank anisotropies if
these are much smaller than the rotational frequency or may otherwise lead to a
spinning sideband manifold. Example of this behavior is shown in Figure 5 on
carbon spectra DL -alanine. At spinning speed of 1.0 kHz, all three alanine
carbons show significant spinning sidebands. At 2.5 kHz, no appreciable
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Figure 4 The most conventional example of spinning about a single axis, the magic
angle spinning. The time average of kP2(cos b)l over integer multiples of rotor period
vanishes.
Figure 5 The 11.7 T carbon CPMAS spectra of DL -alanine at variable spinning speeds of
1.0, 2.5, 4.0, 5.5, and 7.0 kHz. The spinning sidebands are spaced in integer multiples of
the spinning speed (highlighted by the arrowhead lines). Abbreviation: CPMAS, crosspolarization magic angle spinning.
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spinning sideband intensities are observed for the methyl group. The methine
carbon still shows signs of spinning sidebands, albeit with greatly reduced intensities. Even at the maximum spinning speed of 7.0 kHz (the maximum specified
spinning speed of standard wall 7.0 mm Bruker-Biospin rotors), the carboxylic
carbon signal is still spread between spinning sidebands. The spinning sidebands
in this example arise due to the modulation of the carbon chemical shift anisotropies by the MAS spinning. Clearly, the anisotropic contribution of methyl group
is smaller than that of methine, which is still smaller than that of carboxylic
carbon. In general, the magnitude of the chemical shift anisotropy for different
carbons approximately follows the rank order of CH3,CH2,CH,aromaticcarbonyl, which happens to coincide with the typical rank order of their isotropic
shifts. The contribution of methyl groups to spinning sidebands is further reduced
due the fact that they exhibit fast wheel-like rotation around the C –C axis, even
in the solid state at temperatures close to room temperature.
Spins higher than 12 show additional quadrupolar coupling contribution to
their spectral line shapes. Two cases should be distinguished. For half-integer
spins, the central transition (21/2 $ ỵ1/2) is void of quadrupolar coupling
to the first order. For compounds with high molecular symmetry (small nuclear
quadrupole moment), MAS is likely to remove most of the quadrupolar
anisotropies. To get high resolution for compounds with moderately strong quadrupolar coupling, such techniques as double rotation (15,16), dynamic angle
spinning (17), or multiple quantum MAS (18,19) have to be applied. There is
no central transition for nuclei with integer spins. The full effect of the quadrupolar coupling shows in their spectra. There are at least two pharmaceutically
relevant spin 1 nuclei: 2H and 14N. Deuterium quadrupole moment tends to be
small and spinning around magic angle is typically sufficient to average out
the anisotropies, leaving only the extensive spinning sideband manifold. Due
to the low 2H natural abundance, isotope enrichment is required for any deuterium studies. The easiest way to introduce 2H labels is through deuterium
exchange of the exchangeable protons. Unlike 2H, 14N is a nitrogen isotope
with almost 100% natural abundance. However, its widespread applications
have been largely hindered by the presence of rather sizeable nuclear quadrupole
moment. This is why NQR technique has been successfully applied to detect
nitrogen-containing explosives. Several NMR approaches to get better 14N
resolution have been tried, but no simple method has been found yet (20 – 27).
Proton SSNMR
In contrast to 1H spectroscopy in solutions, proton NMR in the solid state has
faced significant technical challenges, mainly due to the extensive line broadening originating from strong, through-space dipole –dipole coupling. This interaction is, to a very good approximation, completely averaged out in isotropic
solution as molecules tumble fast and completely randomly on the NMR timescale. Recently, it was shown that the dipolar coupling may not get completely
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averaged out over long distances, if the interacting spins do not share the same
diffusion sphere (28 –31). This contribution is, however, very small and for
most applications of practical interest can be disregarded. The only remaining
effect that the dipolar coupling has on isotropic solutions is through the incoherent relaxation. The distance measurements in solution rely on dipolar relaxationinduced nuclear Overhauser effect (NOE) (32). NOE-derived structural
constraints are the basis for protein structural determination by NMR. Due to
the type and magnitude of motions involved, NOE effect plays much smaller
role in the solid state.
In the crystalline solids, molecules are locked in their crystal lattices.
Equation (12) predicts the dipolar coupling to be directly proportional to the
product of the gyromagnetic ratios of the two involved nuclei (two protons in
this case) and inversely proportional to the cube of the distance between the
nuclei (r 23). The gyromagnetic ratio of protons is among the biggest of all
nuclei. Moreover, the typical organic solids contain many protons in the close
vicinity of each other. Both of these factors translate into very strong proton
dipolar coupling. In the absence of extensive molecular motions, the dipole–
dipole interaction is usually significantly larger than the chemical shielding
effect. When out of the equilibrium state, the through-space dipole –dipole
interaction induces fast exchange of magnetization between the protons, effective
over many molecules. This magnetization exchange is called a spin diffusion
(2,4). The term of diffusion is very appropriate as it refers to the diffusion
of the NMR signal over the net of nuclei, albeit without any physical
movement of the molecules. The signal transfer due to spin diffusion can be
described by conventional equations for diffusive motion [Equation (17)],
where D is the diffusion coefficient, t is the time during which the diffusion is
effective, and kLl is the average distance traveled by the NMR signal.
kLl ¼
pﬃﬃﬃﬃﬃﬃﬃﬃ
6Dt
(17)
Thus, domain sizes of phase-separated solids may be estimated from the
rates of spin diffusion. Unless the dipolar coupling is averaged out by significant
molecular motions or by very fast MAS rotation, the spin diffusion is fast on the
NMR timescale. A typical proton static line shape of organic molecules is a single
peak, up to 50 kHz wide, precluding resolution of the chemical shifts (2 – 4)
(Fig. 6; bottom trace). As a result of the spin diffusion, the anisotropic proton
lineshape broadening is referred to as being homogeneous. In contrast to the
inhomogeneous broadening arising from, for example, the chemical shift or
quadrupolar coupling (Figs. 2 and 3), selective excitation of arbitrarily narrow
part of the proton spectrum leads to saturation (or disappearance) of the whole
spectrum. The spin diffusion also counteracts the MAS averaging effect (Fig. 6;
second trace from bottom). Because MAS narrows down the proton line shape
only marginally, other line-narrowing techniques have been sought. Combined
rotation and multiple-pulse spectroscopy (CRAMPS) (33 –39) offers a partial
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