Half-integer spins

To treat systems of half-integer spin we go back to the discussion of the possible eigenvalues for a symmetry operator. If the operator is a rotation operator corresponding to a rotation angle of 2 jt/n, then an n-fold repetition of the rotation operation corresponds to a total rotation of 2it. For normal functions a rotation through 2 jt is equal to no operation, that is the identity E, giving the possible eigenvalues /. as  

For halfinteger spin systems, however, rotation through 2jc does not equal the identity. Rather, as is well known, half-integer spin states change sign under a 2ji rotation and only go into themselves under a 4ji rotation. This means, that for halfinteger spin, instead of the condition  

This holds for rotation operators. The inversion operation is still a simple operator with eigenvalues 1, while reflection, which can be taken as the product of a twofold rotation with inversion, will have the extra eigenvalues e 1Il/2 = i. 

Apart from this change in possible eigenvalues (and some extra complications related to time-reversal symmetry and treated in the following section), the treatment of halfinteger spin is exactly similar to the way integer spin was treated. We shall illustrate this with a few examples. 

We start again with the pointgroup D4. The eigenvalues under the C4 operation are now e m/4 and e l3n/4. The commutation relations are as for integer spin and operation with Q transforms an eigenstate X4) into another eigenstate X4 ). Again there is freedom in the choice of phases. Since now, however, we have  

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Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

Only integer and half-interger values ean range from j to -j in steps of unity. Speeies with integer spin are known as Bosons and those with half-integer spin are ealled Fermions. 

There are two possible cases for the wavefunction of a system of identical fundamental particles such as electrons and photons. These are the symmetric and the antisymmetric cases. Experimental evidence shows that for fermions such as electrons and other particles of half integer spin the wavefunction must be anti-symmetric with respect to the interchange of particle labels. This... 

The presence of low-lying excited levels can greatly increase the efficiency of the relaxation processes, especially in the case of paramagnetic centers with half-integer spins. 

FIGURE 5.9 Effective g-values for half-integer spin systems in axial symmetry. The scheme gives the values for all transitions within the Kramer s doublets of S = nl2 systems assuming gmal = 2.00 and S S S B. 

However, the effect of introducing a hnite rhombicity is very different from that for half-integer spins in Figure 5.12 the non-Kramer s doublets are actually split by the rhombic zero-held interaction, and this results in a change in gNeff in the direction of higher values, that is, in the direction of lower helds. For pronounced rhombicities all h1- " disappear in inhnity (zero held) in other words, the rhombicity-induced... 

Equation 12.3 is readily rewritten for any other half-integer spin. For integer spins we have one singulet (the ms = 0 level) and S doublets, but we do not bother writing down the modified equation, because spin counting for integer-spin systems in the weak-field limit is near-to-impossible (Hagen 2006). 

Note the exchange of E and E° compared to Equation 13.12.) There are also quite a number of systems that can occur in three subsequent oxidation states, usually with the intermediate state being half-integer spin, such as Coin/Con/Co, or Movi/Mov/Moiv. In a reductive titration, the EPR intensity initially increases, and subsequently decreases again according to... 

Despite the benefits of high resolution, measurements of wideline spectra of quadrupolar nuclei under static or MAS conditions are still commonly used in a variety of applications. For both integer and half-integer spins, simulations of quadrupolar lineshapes can yield full sets of NMR parameters associated with quadrupolar and chemical shift tensors and can be used for studying molecular dynamics. 

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