Magnetic Dipolar Broadening of Rigid Lattices
A number of physical phenomena may contribute to the width of a resonance line. The most prosaic is the lack of homogeneity of the applied static magnetic field. By dint of hard work and clever techniques, this source can be reduced to a few milligauss ou
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3.1 Introduction A number of physical phenomena may contribute to the width of a resonance line. The most prosaic is the lack of homogeneity of the applied static magnetic field. By dint of hard work and clever techniques, this source can be reduced to a few milligauss out of 104 Gauss, although more typically magnet homogeneities are a few tenths of a Gauss. The homogeneity depends on sample size. Typical samples have a volume between 0.1 cc to several cubic centimeters. Of course fields of ultrahigh homogeneity place severe requirements on the frequency stability of the oscillator used to generate the alternating fields. Although these matters are of great technical importance, we shall not discuss them here. If a nucleus possesses a nonvanishing electric quadrupole moment, the degeneracy of the resonance frequencies between different m-values may be lifted, giving rise to either resolved or unresolved splittings. The latter effectively broaden the resonance. The fact that Tl processes produce an equilibrium population by balancing rates of transitions puts a limit on the lifetime of the Zeeman states, which effectively broadens the resonance lines by an energy of the order of lilTl' In this chapter, however, we shall ignore all these effects and concentrate on the contribution of the magnetic dipole coupling between the various nuclei to the width of the Zeeman transition. This approximation is often excellent, particularly when the nuclei have spin! (thus a vanishing quadrupole moment) and a rather long spin-lattice relaxation time. A rough estimate of the effect of the dipolar coupling is easily made. If typical neighboring nuclei are a distance r apart and have magnetic moment Jl, they produce a magnetic field Hl oc of the order (3.1)
By using r =2 A and Jl = 10- 23 erg/Gauss (10- 3 of a Bohr magneton), we find Hloc 9t 1 Gauss. Since this field may either aid or oppose the static field Ho, a spread in the resonance condition results, with significant absorption occurring 1 Gauss. The resonance width on this argument is indepenover a range of H dent of Ho, but for typical laboratory fields of 104 Gauss, we see there is indeed a sharp resonant line. Since the width is substantially greater than the magnet inhomogeneity, it is possible to study the shape in detail without instrumental limitations.
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65 C. P. Slichter, Principles of Magnetic Resonance © Springer-Verlag Berlin Heidelberg 1990
3.2 Basic Interaction The classical interaction energy E between two magnetic moments J.Ll and J.L2 is (3.2)
where r is the radius vector from J.Ll to J.L2. (fhe expression is unchanged if r is taken as the vector from J.L2 to J.Ll.) For the quantum mechanical Hamiltonian we simply take (3.2), treating J.Ll and J.L2 as operators as usual: J.Ll = "fI1iIl
J.L2 = "f21iI2
,
,
(3.3)
where we have assumed that both the gyromagnetic ratios and spins may be different. The general dipolar contribution to the Hamiltonian for N spins then becomes Hd=!
EE[J.Lj~J.L1e
2 j=1k=l
3(J.Lj.rjle~(J.LIe.rjle)]
_
rjle
rjle
(3.4)
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