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      The above equation can be simplified to:

       (2.10)

      2.3.4 Mode Frequency

      To be used as a coil for MR imaging, the exploited mode of the resonator must be adjusted to the working frequency. In this framework estimating the mode frequency with a simple method is very useful for predesigning the ceramic probe. Methods with low computational costs provide an approximated value of the TE_01δ mode frequency for disk resonators. For example, a commonly used expression, derived from simulation data, is the following:

(2.11)

The expression in Equation 2.11 is valid with an accuracy around 2% or below in the following domain: [1]. A diversity of alternative simplified methods can be found in the literature, some extrapolated from experimental data [25,26], some using rough approximations about the boundary conditions at the dielectric interfaces [27] or implying the numerical solving of a transcendental equation [17], and others combining the above-mentioned approaches [21].

These estimation methods are valid for disk resonators, while the dielectric probe is described as a high-permittivity (ϵ_d = ϵ_rϵ₀) ring filled with a lower permittivity (ϵ_d = ϵ_r,sampϵ₀) sample. In the specific case of the TE_01δ mode, equating the field distribution of the probe and the disk resonators is a reasonable approximation. Indeed, between the disk and the probe, the TE_01δ mode features are affected in proportion to the electric and magnetic energies stored in the sample volume V_samp [28]. The mode frequency varies from the value f_disk for the disk to f_probe for the probe according to:

       (2.12)

As the region where the sample is located corresponds to the lowest values of the E-field distribution, the frequency shift due to the sample is limited. For the TE_01δ mode, the ceramic probe is expected to be robust with respect to the sample loading while the permittivity contrast and the sample volume (or sample over disk radii ratio) have reasonable values. This is demonstrated in Figure 2.4.