  The anisotropic s=1/2 anti-ferromagnetic Heisenberg chain in the presence of
an external magnetic field is studied by using the standard quantum
renormalization group. We obtain the critical line of the transition from
partially magnetized (PM) phase to the saturated ferromagnetic (SFM) phase. The
crossover exponent between the PM phase and anti-ferromagnetic Ising (AFI)
phase is evaluated. Our results show that the anisotropy($\d$) term is relevant
and causes crossover. These results indicate that the standard RG approach
yields fairly good values for the critical points and their exponents. The
magnetization curve, correlation functions and the ground state energy per site
are obtained and compared with the known exact results.
