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PART V. NUCLEAR SPIN LEVEL SPECTROSCOPY

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





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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