Page 343 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN010C-493 July 19, 2001 20:30

708 Nuclear Magnetic Resonance (NMR)

the absence of shielding, and the term −γB 0 σ xx is the fre- sociated with the fact that there are other magnetic nuclei
quency shift due to the anisotropic shielding along the x present. These nuclei act as little magnets, or magnetic
axis of the shielding ellipsoid. The observed resonance dipoles, and provide an additional local field to the nu-
frequencies with the external field parallel to the y and z cleus in question. This field represents a classical “through
axes of the shielding ellipsoid would be correspondingly space” interaction, and has no relation to the electronic
different if the values of σ yy , and σ zz differ from σ xx . With chargecloudpresentinmolecules.Theshapeofthis“dipo-
the convention σ xx >σ yy >σ zz , and B 0 in some direction lar field” due to a nuclear magnet is of the form of the
other than parallel to one of these axes, the observed reso- pattern that iron filings take when spread around a bar
nance frequency for a given nucleus will lie between that magnet. Recall that this pattern varies both with direction,
with the field parallel to the x, and that with the field par- and distance from the magnet. Thus the effect of this field
allel to the z axes of the shielding ellipsoid. The spectral upon a neighboring nucleus will depend on where this
width (i.e., the range of resonance frequencies associated neighbor is located within the dipolar field. The shift in
with the anisotropic shielding interaction) for a sample in NMR frequency due to this dipolar field will similarly de-
which a nucleus is described by a shielding ellipsoid σ, pend on where the nucleus feeling the field is with respect
and all orientations of σ are present for fixed B 0 , will be to the nucleus producing the field. The observed shift in
angular resonance frequency ω dip of a nucleus in the
ω = γB 0 (σ zz − σ xx ).
presence of another nuclear magnetic dipole has spatial
Thus, a powdered sample of solid benzene, C 6 H 6 (solid), dependence
in which all protons are chemically identical in that they 2 3
ω dip = const · (1 − 3 cos θ)/r ,
are all aromatic protons on a single benzene ring, will have
a proton NMR spectrum associated with the shielding in- where θ is the angle between the line connecting the inter-
teraction that is a powder average of the individual lines acting nuclei and the external magnetic field, and r is the
associated with the specific orientations for each benzene internuclear distance. The constant is proportional to the
molecule. This spectrum has been found to be roughly 6 magnitudesofthemagneticmomentsoftheinteractingnu-
ppm wide and looks roughly as shown in Fig. 4e, Sec- clei. We thus see that the resonance frequency shift due to
tion IV. What happens now when the solid sample is con- the dipolar interaction, if it could be measured without the
verted to a liquid? The benzene molecules are free to rotate interference of other interactions, provides a measure of
isotropically in solution. This means that the shielding el- nuclear geometries. It is notable that an interacting pair of
lipsoids characterizing the NMR lines associated with the dipolar nuclei have their resonance frequencies shifted by
shielding interaction are rotating isotropically. When the the inverse cube of their internuclear distance, so this shift
rotation frequency is faster than the spectral width char- is very sensitive to distance. Note that the above spatial
acterized by the difference (σ zz − σ xx ), then the observed dependence of the dipolar interaction between two nuclei
shielding frequency is characterized by the isotropic av- does not contain the azimuthical angle φ. This statement
erage σ iso = (σ xx + σ yy + σ zz )/3, and the observed NMR translates into the fact that the ellipsoid describing the
spectrum associated with the shielding interaction is a sin- anisotropy of the dipolar interaction between two mag-
gle line at angular frequency ω iso = γ B 0 (1 − σ iso ). netic nuclei has two axes that are equal (i.e., there will
In exactly the same manner that the anisotropic shield- be a plane in which all resonance frequencies due to the
ing is represented by an ellipsoid with three unequal axes, pairwise dipolar interaction will be the same). This result
the anisotropy of the other three interactions may be so enforces a particular symmetry upon the NMR spectrum
represented. A complete specification of an anisotropic associated with a powdered sample of interacting dipolar
interaction would include the lengths of the three axes pairs, which is illustrated in Section IV, Fig. 3a. It is a fact
of the interaction ellipsoid, and the three angles that ori- that the isotropic value of the dipolar interaction, which
ent this ellipsoid with respect to some coordinate sys- is D iso = (D xx + D yy + D zz )/3 is zero. This fact will be
tem, such as the molecular framework in which the nu- subsequently important when the effects of motion on the
cleus resides, or a fixed coordinate system within the dipolar interaction are considered.
laboratory. Therefore, six independent pieces of infor- A large number of interacting dipoles in a powdered
mation completely specify an anisotropic interaction in sample would yield many resonance frequencies that
general. would reflect the powder average of the angular distri-
2
bution, (1 − 3 cos θ), and the sum of all pairs i, j, with
internuclear distance r ij , so the spectrum of nuclei in such
B. Dipolar Interaction
a sample due to dipolar interactions alone would be a
In addition to the shielding field, mentioned previously, a broad, featureless spread just reflecting average geome-
nucleus in a molecule will experience a magnetic field as- tries and distances, as shown in Fig. 4c.