2.5 Internal Interactions
2.5.2 The Dipolar Interaction
This ability to measure accurately the subtle changes in frequencies is fundamental to NMR, and the major contributions to the chemical shift come from the low lying excited electronic states. Hence, it can be seen that the chemical shift range for different nuclei can vary significantly. It has been established that heavier atoms tend to have a larger chemical shift ranges (on the order of thousands of ppm), whereas lighter atoms might have much smaller shift ranges in the tens of ppm. However, there can also be a contribution from directly bonded electronegative atoms, and the chemical shift can be further influenced by the next nearest neighbours.
2.5.2
The Dipolar Interaction
The Dipolar interaction arises from direct dipole-dipole contact between nuclei. As each nuclear spin possess a magnetic moment this has an associated magnetic field with it that propagates through space and interacts with the magnetic fields of the surrounding nuclei. This is a mutual interaction as the magnetic field of a spin interacts with the field of a second spin, the field of the second spin will also interact with the field generated by the first spin. This interaction is generally broken into two different categories; like spins that interact (homonuclear) and unlike spins that interact (heteronuclear). The magnitude of the heteronulcear and homonuclear dipolar couplings are both dependant on the inverse cube of the distance between the nuclei and the orientation of the resultant vector due to the nuclei involved. Both of these cases give slightly different outcomes and will be discussed later but now the interaction will be dealt with in a general sense.
The dipole Hamiltonian can be expressed in the form of equation as
HD =−2I·D·S (2.59)
has principle values -d/2, -d/2 and d, where d is defined as
d=~µ0 4π
1
r3γIγs (2.60)
The tensorD describes the interaction between the spinsIand Swith respect to each other and the distance between them. However the full solution to this is rather lengthy so a slightly different approach will be invoked here. Fortunately it can be approached from a more classical point of view.
Classically, the energy of two interacting magnetic dipoles is given by
Ed= µ0 4π µ1.µ2 r3 −3 (µ1.r)(µ2.r) r5 (2.61)
By substituting in the quantum mechanical operator (see Equation 2.1) for the nuclear magnetic moment into Equation 2.61 an expression for the Hamiltonian due to dipole- dipole interactions is obtained
HD = µ0 4π γIγS r3 ~ II.IS (II.r)(IS.r) r2 (2.62)
As the interaction between the spins depends on the internuclear vector between the spins, it is common to express the dipole Hamiltonian in spherical polar coordinates
HD =
µ0
4π γIγS
r3 ~(A+B +C+D+E +F) (2.63)
This leads to a complicated expression for the interaction, and terms A-F (often referred to as the dipolar alphabet) are
A=IzSz(3cos2θ−1) (2.64) B =−1 4[I+S−+I−S+](3cos 2θ−1) (2.65) C = 3 2[IzS++I+Sz]sinθcosθexp (−iφ) (2.66) D= 3 2[IzS−+I−Sz]sinθcosθexp (+iφ) (2.67)
E = 3 4[I+S+]sin 2 θexp(−2iφ) (2.68) F = 3 4[I−S−]sin 2 θexp(+2iφ) (2.69)
With I+ and I− being the raising and lowering operators I+ = Ix + iIy and I− = Ix
– iIy likewise for S+ and S−. It is now convenient to consider the Hamiltonian for the
dipole-dipole interactions separately for homonuclear and heteronuclear coupling, as there are slight differences in the physical effects.
In the case of homonuclear coupling it is necessary to transfer the Hamiltonian to the rotating frame of reference with respect to the observed spin. As both spins are of the same kind this has to act on both. By using the rotation operator defined in Equation 2.50 that describes a rotation about z for the spin system, then
HD =−
µ0
4π γIγS
r3 ~[(A+B) +Rz(−φ)(C+D+E+F)Rz(φ)] (2.70)
When transferred to the rotating frame A and B remain unaffected, however the ex- pressions C-F obtain a time dependence at frequencies ω and 2ω and can for these purposes be discarded. The raising and lowering operators are redefined in terms of the Cartesian spin operators. Therefore only the time independent or secular part remains and the Hamiltonian for homonuclear dipole coupling becomes
HD =− µ0 4π γIγS r3 ~ IzSz− 1 2(IxSx+IySy) (3cos2θ−1) (2.71)
However if the same process is applied in the case of hetronuclear coupling, when transformed to the rotating frame, as the spins are different, it only acts on the observed spin I and the Hamiltonian becomes
HD =−
µ0
4π γIγS
r3 ~[A+Rz(−φ)(B +C+D+E+F)Rz(φ)] (2.72)
It can now be seen that the B term has also become time dependant, and when the time dependencies are removed the heteronuclear dipole Hamiltonian reduces to
HD =− µ0 4π γIγS r3 ~IzSz(3cos 2θ−1) (2.73)
The implications from the two different types of coupling manifest themselves in their dependence on the B term, while the A term remains integral in both the homonuclear and heteronuclear cases. The A term represents an energy shift of the spin levels while the B term is often termed a flip-flop term. This flip-flop process is a continuous exchange of energy between the different spin states of the coupled spins, implying that the different spins can spontaneously change from−1
2 to + 1
2 in a two spin system.
However this only happens when energy in the system is conserved; if the nuclear species are different then γ is different and energy is no longer conserved, hence the removal of this effect in the heteronuclear case. This effect does not change the longitudinal magnetisation as this process is continuous, however it does affect the observed signal, as what is observed during this process is a large range of transition frequencies which translates as a broadened spectrum tending to a Gaussian line shape (see Figure 2.7a). This depends critically on the strength of the dipole-dipole interaction and this process is called homogeneous broadening.
In the case of heteronuclear coupled spins the flip-flop term is not present and there is no mixing of the Zeeman states, and therefore the cause of homogeneous broadening is not present. However there is a shift in the energy levels due to the coupling which has an angular dependence of ±(3cos2θ −1) which consequently exhibits the same
features that are found in a CSA pattern, though there will also be a mirror image (see Figure 2.7b) with one feature arising from the + transitions and the other from the – transitions.
In addition, it is possible for a hetronuclear coupled spin system to contribute to homogeneous broadening if it is strongly bonded to another spin which is part of a homonuclear network. This will effect one or more of the spins in the homonuclear network and will influence the flip-flop interaction, and therefore will contribute to the homogeneous broadening of the final spectrum.
As was the case with the chemical shift anisotropy, the dipolar interaction is motion- ally averaged out in liquids. In polycrystalline materials the effects of dipolar coupling can cause significant resolution problems. They can be removed or reduced by magic angle spinning (see Chapter 3.2), however this is dependant on the magnitude of the interaction and it is often necessary to use special pulse sequences simultaneously to
Figure 2.7: (a) Hypothetical powder pattern showing the influence of homonuclear dipole coupling on the line shape, with the bottom line shape exhibiting no dipole coupling, middle slight broadening due to coupling and top tending to a Gaussian shape due to significant homonuclear dipole-dipole coupling. (b) Simulated line shape due to hetronuclear dipole coupling, resembles a Pake doublet, essentially a CSA line shape with a mirror image due to the ±(3cos2θ − 1) contribution to the transition
levels, the splitting of the horns is equal to the dipole coupling constant
suppress the interaction or irradiate with rf during signal acquisition the spins that are not being observed to uncouple them from the spin system under investigation. In many cases it is actually desirable to probe dipole couplings through the use of spe- cial pulse experiments, as the dipole-dipole interaction is a function of distance; this can give valuable structural information as to what chemical positions are spaciously proximate to others.