Magnetic properties of collinear structures of compound La1-xCaxMnO3
  With use of homogeneous model for magnetic sub-lattice of compound La1-x CaxMnO3 allowing for Heisenberg superexchange and double exchange between nearest neighbors the temperature dependence of spontaneous magnetization on a separate site is investigated in molecular field approximation. The difference from the usual magnets with one type of magnetic ions is in the fact that system of equations appears with two unknowns, being the mean spin values of ions Mn3+ and Mn4+, correspondingly. Near ordering temperature this system can be solved analytically and at the same time the expression for the ordering temperature can be obtained. To find the temperature dependence of magnetization in the whole temperature range we used the exact expression for double exchange operator containing spin operators under the square root. The numerical solutions were found in this case. 
B. V. Karpenko, L. D. Falkovskaya, and A. V. Kuznetsov