Page 182 - Principles and Applications of NanoMEMS Physics

170 Chapter 4

where i labels the nuclei, and ı denotes the so-called chemical shifts,
i
which characterize the fact that distinctly different precession frequencies
are exhibited by identical atomic species within a given molecule, when the
shielding environment produced by their surrounding electrons results in a
different magnetic field, B . They also point out [194] that typical chemical
0
shifts range in the order of a few kilohertz, compared to the precession
frequencies, which range in the MHz.
In addition to isolated spin nuclei, liquid-state NMR includes the presence
of coupled spins. These are characterized by either a direct or an indirect
coupling mechanism. The direct coupling is of the magnetic dipole-dipole
nature, similar to the interaction between two adjacent bar magnets and, for
nuclei i and j , separated by a distance r , is given by [194],
ij
ª º
µ Ȗ Ȗ = G G 3 G G G G
»
H = ¦ 0 i j « I i ⋅ I j − (I i ⋅r )(I j ⋅r ) , (15)
D G 3 « G 2 ij ij »
i < j 4ʌ r r
ij ¬ ij ¼
G
i
where µ is the free space magnetic permeability, and I is the magnetic
0
moment vector of spin i . Under certain conditions, Eq. (15) may be
simplified. For instance, for large precessing frequencies it reduces to [194],
µ 0 Ȗ i Ȗ = G G G G
j
H D = ¦ G 3 ( − cos31 2 ș ij )[ I3 i ⋅ I j − (I i ⋅ I j )], (16)
i < j 8ʌ r
ij
G
where ș is the angle between B and r , whereas if ω i − ω is much
j
ij 0 ij 0 0
greater than the coupling strength it reduces to [191],
µ 0 Ȗ i Ȗ =
j
H D = ¦ G 3 ( − cos31 2 ș ij ) II i z z j . (17)
i < j 4ʌ r
ij
The indirect coupling is characterized by a strength J, which captures the
overlap of electronic wavefunctions between two atomic nuclei, and has
values ranging from several Hz, for three- to four-bond couplings, to several
KHz for one-bond coupling. The indirect coupling Hamiltonian takes the
form [194],
G G
H = = ¦ 2ʌʌ I i I j = = ¦ J ( II i j + I i I j + I i I j ), (18)
J ij ij x x y y z z
i < j i < j
where J characterizes the coupling between spins i and j. Simplification of
ij
this expression is also possible in certain circumstances, in particular, when