We present an analytical expression for the dynamical spin-correlation function of the
S=21 linear Heisenberg antiferromagnet based on the fact that the spectrum is dominated by a double continuum of spin-wave excitations. Our expression is not exact, but exact sum rules show that the degree of approximation is small. We predict that in a uniform magnetic field the spectral weight function will display a double-peaked structure. Such a feature is observed in recent neutron-scattering experiments.
The highest and lowest energies as a function of the total spin are computed for the class of "unbound" states in the Bethe formalism for the linear chain of spin-\textonehalf{} atoms with a Heisenberg exchange interaction between nearest neighbors. The lowest energies are used to compute the magnetization curve for the infinite antiferromagnetic chain in the limit of zero temperature. At zero temperature and in zero field, the magnetic susceptibility has the value 0.050661 $\frac{{g}^{2}{\ensuremath{\mu}}^{2}}{J}$, where $\ensuremath{\mu}$ is the Bohr magneton,
g the electron
g factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.
The spin -½ Heisenberg antiferromagnet Hamiltonian with exchange integral and applied external magnetic field is studied with the help of direct exact diagonalization technique. Two site cluster lattices with open boundary conditions are considered and the criteria for the occurrence of a transition from antiferromagnetic (AFM) phase to a ferromagnetic (FM) phase are discussed.However, the results obtained shows that the temperature dependence of the uniform magnetization depends on the strength of the applied magnetic field h. If the applied magnetic field is smaller than the critical field strength, the zero temperature magnetization vanishes, otherwise it is activated.