The Nucleus in a Magnetic Field

Finally, remember that different isotopes of the same element can have different nuclear spins, some of which are detectable by NMR, others of which are not. 

Just when we begin to understand that a nuclear spin 

Each of these 2/ + 1 states has its own spin quantum number m in the range m = -/,-/+ 1./- 1, / (listed in the order of decreasing energy and increasing stability). Thus, for nuclei with I = the multiplicity is 2, and these two states are m = and m = 

As we have said, a nucleus with nuclear spin I adopts 2/ + 1 nondegenerate spin orientations in a magnetic field. The states separate in energy, with the largest positive m value corresponding to the lowest energy (most stable) state. It is this separation of states in a magnetic field that is the essence of the nuclear Zeeman effect. 

The energy of the ith spin state ( ,) is directly proportional to the value of m, and the magnetic field strength Bq (that is, energy is quantized in units of ykR /ln) 

And now you realize why values of 4 were picked for m (and s too, for that matter). It is so that the difference in energy between two neighboring spin states will always be an integer multiple of y h B0/27t. 

In Table 2.1 are listed many of the common isotopes examined by NMR techniques, together with their nuclear constants. Notice that a bare proton has the largest y value of any nuclear particle, while heavier nuclei, surrounded by many subvalence electrons, tend toward lower values. This 

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According to classical electrodynamics, the energy of the nucleus in a magnetic field can be written in the form... 

Larmor precession The motion describing the rotation of the axis of the spin of a nucleus in a magnetic field. 

Spin quantum number Number indicating the number of allowed orientations of a particular nucleus in a magnetic field. For example, H has an I value of lk, allowing for two possible orientations, whereas 14N has an I of 1, allowing three possible orientations. 

A simple kind of symmetry which applies to a large number of crystals is axial symmetry that is, two directions are equivalent, such as x and y. When the quadrupole interaction is small compared to the energy separation between the 27 - - 1 energy levels of a nucleus in a magnetic field Ho, the effect of the quadrupole interaction is to destroy the equal spacing. For axial symmetry the effect may be described in terms of a single parameter (5, 90) eq, as follows ... 

In brief, a magnetic atomic nucleus in a magnetic field may be either lined up with or opposed to the external field, and the transition from one state to the other corresponds to an amount of energy which can be provided by electromagnetic radiation in the radiofrequency range. 

NMR spectra result from the transition of an atomic nucleus in a magnetic field from a low-energy to a high-energy state [235], There are two aspects to the quantum-mechanical calculation of NMR spectra [301] calculation of shielding (chemical shifts) and calculation of splitting (coupling constants). Most of the... 

THE TWO SPIN STATES OF A HYDROGEN NUCLEUS IN A MAGNETIC FIELD. 

Figure 5.1 The effect of quadrupole interactions on an / = 3/2 nucleus in a magnetic field. The Zeeman interaction splits the levels by an equal amount, cot, (the Larmor frequency in frequency units). The central +1 /2 to -1 /2 transition is unaffected by first order coupling co j however, second order coupling, co , affects all transitions.

For a nucleus of spin /, the energy of a nucleus in a magnetic field is split into 2/ +1 Zeeman levels. A proton, and other nuclei with spin, have just two possible levels ... 

The relative populations of the separate energy levels for a nucleus in a magnetic field and the relaxation processes from a higher energy level to a lower one are also of fundamental interest. On the basis of Boltzmann s law and equation (5.3.2) the ratio of the number of nuclei in the higher energy level to that in the lower level for a proton (7=1 /2) is... 

Larmor frequency (Vj ) - The precession frequency of a magnetic dipole in an applied magnetic field. In particular, a nucleus in a magnetic field of strength B has a Larmor frequency of yBUn, where y is the magnetogyric ratio of the nucleus. 

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