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mathematically be described by second-rank tensors, where
the resulting orientation-dependent NMR frequency for each
of the interactions takes the form: 15,18b
δ 2
2
ω ( α , β ) − ω L = ω iso + ( 3 cos β − 1 − η sin β cos 2 α ) . (1)
2
Here, w is the Larmor frequency, w is the isotropic chemical
L iso
shift and the other terms refl ect deviations due to angular-dependent
contributions. The strength of the anisotropic interaction
is specifi ed by d, and h (0≤η≤1) is the asymmetry parameter
that describes the deviation from an axial symmetry. For each
interaction tensor, a coordinate frame of reference (the so-called
principal axes system) can be defi ned, in which the tensor is reduced
to its diagonal elements. Typically, this coordinate system
is directly related to the molecular framework, i.e. its z-axis is
along a bond direction, or normal to an aromatic ring etc. The orientation
of the respective principal axes system with respect to
the external magnetic fi eld is denoted by the polar angles α, β.
The most important spin interaction for achieving chemical site
selectivity is the chemical shift. It results from the shielding of
the magnetic fi eld at the position of the nucleus by surrounding
electrons. Recently, major advances have been achieved in
quantum chemical calculations of this parameter in molecules
and in bulk compounds at different levels of precision and
computational cost, 21,22,23,24 thereby facilitating unambiguous
peak assignments and hence interpretation of experimental
data. The observable NMR frequency is shifted by an isotropic
contribution w iso and by angular-dependent contributions as
described in (Eq. 1). In solution, angular-dependent terms are
often averaged. In rigid solid samples, a so-called powder average
(assuming an equal probability for all directions) yields
a broad, yet characteristic NMR spectrum that refl ects the
chemical-shift anisotropy (cf. Fig 2 left). 15,18 In case of an axially
symmetric chemical-shift tensor, η is zero and the angulardependent
term in Eq. (1) simplifi es to d(3cos 2 β-1)/2. Typically,
the chemical shift in organic solids spans about 20 ppm for 1 H,
200 ppm for 13 C, and 400 ppm for 15 N nuclei.
For abundant nuclei with spin ½ (e.g. 1 H, 19 F) the corresponding
NMR spectrum is often dominated by dipole-dipole couplings
due to interactions among the magnetic moments of two
neighbouring spins (note that there is no isotropic contribution
and η is zero) so that Eq. (1) simplifi es correspondingly. For a
two-spin system, one obtains a Hamiltonian of the form
H
= I I I I
r
⎛
2
μ 0⎞γγ 1 2ℏ⎛3cos
θ −1⎞
��
= ⎜ ⎟ 3 ⎜ ⎟( 3I1
I2 − I1I2) ⎝ 4π⎠
r ⎝ 2 ⎠
⎛
2
μ 0⎞γγ 1 2ℏ⎛3cos
θ −1⎞
��
⎜ ⎟ 3 ⎜ ⎟( 3 1 2 − 1 2)
⎝ 4π⎠
⎝ 2 ⎠
D z z
12
where r is the magnitude of the vector connecting the two
12
spins (e.g. ��
I i
1H-13C), θ is the angle of this vector with respect to
the magnetic fi eld , the are spin operators and the g (i=1,2)
i
are the magnetogyric ratios of the spins. Clearly, the Hamiltonian
can be separated into a spin part (the spin operators) and
a so-called space part that contains angular-dependent terms.
Indeed, this apparently trivial approach constitutes the basis
for rather sophisticated manipulation of the Hamiltonian20 and
is the key to modern solid-state NMR.
The strength of the resulting line-splitting strongly depends
on the distance between the two coupled spins, so that dis-
(2)
UNTERRICHT
tance information can be extracted from such spectra. Homonuclear
interactions among like spins (e.g. 1 H- 1 H, 19 F- 19 F) with
g 1 = g 2 = g and heteronuclear interactions among unlike spins
(e.g. 1 H- 13 C, 1 H- 15 N) with g 1 ≠ g 2 have to be distinguished. In the
latter case, the so-called fl ip-fl op term that is part of the product
in Eq. (2) can be safely neglected. 15 For a powdered
sample, one again has to take all angles b into account yielding
the so-called (dipolar) Pake spectrum, which can span larger
frequency ranges than the chemical shift. Since the dipolar
coupling is a through-space interaction, however, we in principle
have to evaluate the sum over all possible pair interactions
rendering experimental NMR line-shapes for static organic
solids (or rather rigid samples) rather featureless.
2.2. MAGIC-ANGLE SPINNING
The most prominent technique in solid-state NMR for linenarrowing
is magic-angle spinning (MAS). Here, the angular-dependent
part of the interactions is modulated by rapid
mechanical spinning of the sample around an axis inclined at
an angle Q with respect to the magnetic fi eld. If the spinning
axes is chosen along the so-called magic-angle Q m =54.7°, the
relevant scaling factor (3cos 2 Θ -1)/2 becomes zero and the
anisotropic part of the interaction vanishes (Fig. 2). Today, very
fast MAS with rotation frequencies reaching 70 kHz 25 and more
is commercially available, thus providing unprecedented high
spectral resolution, particularly in solid-state 1 H and 19 F NMR.
In addition, MAS largely simplifi es the network of strongly dipolar-coupled
protons 26 that typically precludes the use of the
dipole-dipole couplings among 1 H spins for specifi c structural
investigations in non-spinning (static) solid samples. In order
to effectively reduce or remove line-broadening due to anisotropic
interactions, the rotation frequency w R should be larger
than the magnitude of the respective anisotropic interaction
(e.g. the 1 H- 1 H dipolar coupling w D ). In fact, for most organic
solids, the 1 H- 1 H dipole-dipole couplings, e.g. in CH 2 groups
with a proton-proton distance of 0.18 nm are of the order of
w D » 2p·20.6 kHz. Thus, at fast MAS, the condition w R ≥ w D
Figure 2: Effect of Magic-Angle-Spinning (MAS) on static line-shapes. Left)
The static powder pattern due to chemical shift anisotropy splits into a manifold
of spinning sidebands reflecting inhomogeneous broadening. Clearly,
within the sideband pattern, the signal with highest intensity does not
necessarily represent the isotropic chemical shift. Right) Static line-shape
dominated by homonuclear dipolar couplings e.g. among abundant protons.
Upon increasing MAS, the signal splits into a lesser number of spinning sidebands
where the isotropic center peak is easily identified. This behaviour is
referred to as homogeneous broadening. 18b
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