Method for fitting crystal field parameters and the energy level fitting for Yb³⁺ in crystal Sc₂O₃

A method to compute the numerical derivative of eigenvalues of parameterized crystal field Hamiltonian matrix is given, based on the numerical derivatives the general iteration methods such as Levenberg–Marquardt, Newton method, and so on, can be used to solve crystal field parameters by fitting to experimental energy levels. With the numerical eigenvalue derivative, a detailed iteration algorithm to compute crystal field parameters by fitting experimental energy levels has also been described. This method is used to compute the crystal parameters of Yb³⁺ in Sc₂O₃ crystal, which is prepared by a co-precipitation method and whose structure was refined by Rietveld method. By fitting on the parameters of a simple overlap model of crystal field, the results show that the new method can fit the crystal field energy splitting with fast convergence and good stability.