### 2. Magnetic properties derived from the Heisenberg Hamiltonian, by using the method of hierarchy of spin algebras.

Another part of our work concerns the study of magnetic materials. For this purpose the Heisenberg Hamiltonian (HeiH) describes in an efficient way most magnetic properties. The usual method applied for the eigenstates of (HeiH) is the effective field theory. However, the solutions so obtained are far from the accurate ones. Using the rotational and translational (permutational) symmetries of these Hamiltonians, we developed a new mathematical methodology, the “Method of the Hirerchy of Spin Algebras” (Physica B 202, 41 (1994), Physica B 202, 47 (1994)). This methodology not only simplifies the calculation of the eigenstates, but also gives their qualitative characteristics from group theoretical considerations. We applied this methodology on many crystals that their magnetic ions involve not only the usual Heisenberg interactions, but also more complex ones. Such cases are: (a) The K2V2O3 crystal which includes a Dzialoshinski-Moriya interaction (Physica A371 (2006) 433). (b) The linear crystal SrNi2V2O8 with spin 1, that has been doped (phys. stat. Sol. (b) 243 1366 (2006)). (c) The crystal CuSb2O6 where we have an energy gap between the ground state and the first excited state because of the weak magnetic interaction between the spin chains (Physica A378 (2007) 273). This energy gap has been calculated theoretically with very good precision. In all the above systems the explanation of the experimental data (mainly the susceptibility and the specific heat) for the magnetic crystals is well mentioned. Using the experimental data we can determine the magnetic interaction parameters j and g. The fact that we have a very good approximation to the experimental results, is due to the large number of eigenstates of the systems we used. The systems with sixteen spins ½ and eight spins 1 have 65536 and 6561 eigenstates respectively and their number is large enough for the statistical determination of thermodynamic quantities.