# Verdet Constant of Magneto-Active Materials Developed for High-Power Faraday Devices
## 2 Characterization of the Verdet Constant as a Function of Wavelength and Temperature
### Model Function for the Verdet Constant Temperature-Wavelength Dependence  
In the next step, after obtaining the Verdet constant data \(V(\lambda_{i},T_{j})\), we need to find a model function for describing the Verdet constant temperature-wavelength dependence \(V(\lambda,T)\). A general model for the Verdet constant may be written as follows [1; 45; 53; 54; 55; 56]  
\[V(\lambda,T)=\sum_{k=1}^{N}\left[\frac{A_{k}\lambda_{0,k}^{3}\lambda^{2}}{( \lambda^{2}-\lambda_{0,k}^{2})^{2}(T-T_{w})}+\frac{B_{k}\lambda_{0,k}^{2}}{ \lambda^{2}-\lambda_{0,k}^{2}}+\frac{C_{k}\lambda_{0,k}^{2}}{(\lambda^{2}- \lambda_{0,k}^{2})(T-T_{w})}\right]+\frac{D}{T-T_{w}}\quad. \tag{10}\]  
The model was derived for solid-state compounds containing rare-earth paramagnetic ions, assuming that: (a) the described spectral range is sufficiently far from any resonance line; and (b) only the contribution of \(N\) dominant transitions at wavelengths \(\lambda_{0,k}(k=1\dots N)\) are considered. It consists of four different contributions (associated with the \(A_{k}\), \(B_{k}\), \(C_{k}\) and \(D\)-terms), arising from various phenomena associated with the Zeeman splitting of the ground and excited states of a material subjected to a magnetic field. A full description of these individual contributions is beyond the scope of this paper, however, a brief explanation of the individual terms is given with the corresponding references to additional literature. The \(A_{k}\)-terms refers to the so-called diamagnetic contributions of the material lattice. This contribution, as well as its mathematical formulation, is discussed in Refs. [1; 53; 55]. The second bracketed \(B_{k}\)-terms stand for the mixing term derived by [PERSON] and [PERSON] based on quantum mechanical considerations [1; 54; 55]. This contribution comes from the overlap of wavefunctions of ground and neighboring excited states which causes the perturbation of the amplitude elements of the electric moment by the magnetic field. The next terms, proportional to \(C_{k}\), are the paramagnetic terms, arising from the different occupation of energy sublevels of the magneto-active ion ground states [1; 55]. The \(D\)-term is called the gyromagnetic contribution, which is the only wavelength-independent term of the model [1; 55]. The origin of this contribution is associated with the magnetic dipole transitions. The influence of this contribution becomes more prominent at longer wavelengths. All contributions except for the mixing terms are temperature-dependent according to the Curie-Weiss law, i.e., \(\propto 1/\left(T-T_{w}\right)\), for the temperatures above the magnetic state phase transition characterized by the Neel temperature [57]. It should be noted that the diamagnetic contributions are often referred as temperature-independent, which is, however, a valid assumption only for higher temperatures (e.g., around the room temperature) or for a narrow range of temperatures. The model in Equation (10) may be easily transformed into a single-variable function of either wavelength (for a fixed temperature \(T_{f}\)) or temperature (at a given wavelength \(\lambda_{f}\)) to get  
\[V(\lambda) =\sum_{k=1}^{N}\left[\frac{E_{k}\left(T_{f}\right)\lambda_{0,k}^{3 }\lambda^{2}}{\left(\lambda^{2}-\lambda_{0,k}^{2}\right)^{2}}+\frac{F_{k}\left( T_{f}\right)\lambda_{0,k}^{2}}{\lambda^{2}-\lambda_{0,k}^{2}}\right]+G\left(T_{f} \right)\quad, \tag{11}\] \[V\left(T\right) =\frac{H\left(\lambda_{f}\right)}{T-T_{w}}+I\left(\lambda_{f} \right)\quad. \tag{12}\]  
When performing a full temperature-wavelength analysis of the Verdet constant, it is often advantageous to fit the data first with the single-variable functions in Equations (11) and (12) to obtain initial estimations of the fitting parameters \((A_{k},B_{k},C_{k},D)\) of the general model in Equation (10). Note that the model in Equation (10) may be considerably simplified.