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To investigate the magnetic properties of SmFe$_{12}$, we construct aneffective spin model, where magnetic moments, crystal field (CF) parameters,and exchange fields at 0 K are determined by first principles. Finitetemperature magnetic properties are investigated by using this model. Wefurther develop an analytical method with strong mixing of states withdifferent quantum number of angular momentum $J$ ($J$-mixing), which is causedby strong exchange field acting on spin component of 4$f$ electrons. Comparingour analytical results with those calculated by Boltzmann statistics, weclarify that the previous analytical studies for Sm transition metal compoundsover-estimate the $J$-mixing effects. The present method enables us to makequantitative analysis of temperature dependence of magnetic anisotropy (MA)with high-reliability. The analytical method with model approximations revealsthat the $J$-mixing caused by exchange field increases spin angular momentum,which enhances the absolute value of orbital angular momentum and MA constantsvia spin-orbit interaction. It is also clarified that these J-mixing effectsremain even above room temperature. Magnetization of SmFe$_{12}$ shows peculiarfield dependence known as first-order magnetization process (FOMP), where themagnetization shows an abrupt change at certain magnetic field. The result ofthe analysis shows that the origin of FOMP is attributed to competitive MAconstants between positive $K_1$ and negative $K_2$. The sign of $K_{1(2)}$appears due to an increase in CF potential denoted by the parameter$A_2^0\langle r^2\rangle$ ($A_4^0\langle r^4\rangle$) caused by hybridizationbetween 3$d$-electrons of Fe on 8$i$ (8$j$) site and 5$d$ and 6$p$ valenceelectrons on Sm site. It is verified that the requirement for the appearance ofFOMP is given as $-K_2<K_1<-6K_2$.

Statistical and Analytical Approaches to Finite Temperature Magnetic Properties of SmFe$_{12}$