Energy Levels and Transitions in an AX System

A3B or A3X (depending on magnetic field and hydrogen bonding effect on 8Qh—see Section 4.5) 

Before deriving quantitative expressions for general spin systems, we shall examine qualitatively the energy levels and transitions arising from two spin /2 nuclei that are not coupled or are only weakly coupled (AX system). As usual, we take the static imposed magnetic field to lie along the z axis, and we express the orientation of the ar component of nuclear spin /, as a or fi for Iz = x/2 or —V2, respectively. A system of N nuclei of spin x/2 is described by the 2N possible 

The shorter notation act is often used in place of the notation a,a2 when there is no chance of ambiguity. In such cases it is understood that the nuclei are always given in the same order. 

The parentheses in (4), make it clear that the notation refers to the z spin component of nucleus i, but most literature omits the parentheses, as indicated, for simplicity. We shall usually use the shortened notation in this and similar expressions. The effect of F. on a product function is to generate a + /2 every time an a appears and —l/2 every time a /3 appears, so that the eigenvalue f, represents the total z component of spin of the function. The classification of basis functions according to total spin greatly simplifies the calculation of energy levels, as we shall see. 

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