Tesseral fields

The fields with m = 0 are independent of the angle zonal fields. The others are called tesseral fields. Some shim coils for production of tesseral and zonal fields are sketched in Fig. 2.3.2 [Krel, Mori], Room temperature shims may provide correction fields up to = 5, while cryoshims frequently provide corrections only up to n = 1. In this way homogeneities of 2 ppm can be reached over 30 cm and 5 ppm over 50 cm diameter volumes. 

We will illustrate the development of the even part of the crystal-field potential for the symmetries C4 and D3, First of all we determine which tesseral harmonics have the same symmetry elements as the point group C4. In C4, only a fourfold rotation axis is present. This axis can generate the symmetry operations C, C (= C2), C and the identity. The tesseral harmonics Z q, and Zgf, have cylindrical symmetry and are present in the even part of every crystal-field potential. The C4 axis is a symmetry element for the harmonics Z44, Z44, Zg4 and Zg4. We have thus the following coefficients in the even part of the crystal-field potential for C4 Bl, Bq, B, B l, 5 and B. The potential can be written as... 

We want to give here also some comments on the definitions of the crystal-field parameters by Wybourne (1965). His crystal-field formalism is sometimes a little bit confusing. First, he introduces the parameters which have the same transformation properties as spherieal harmonies. These parameters are in general complex numbers. Both fij and oeeur. Moreover, his parameters transform as tesseral harmonics, like our parameters do. It is therefore easy to confuse and S. Prather (1961) always uses tesseral harmonies for his parameters and also for his operators. 

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