We report low temperature specific heat, C, magnetization, M, susceptibility, chi, and electrical resistivity, rho, measurements on high-quality single crystals of the heavy-fermion system YbRh_2(Si_{1-x} Ge_x)_2 (x=0 and 0.05). The undoped compound shows weak antiferromagnetic (AF) order at T_N=70 mK which is suppressed to below 10 mK by a tiny volume expansion in the x=0.05 system. In the latter pronounced deviations from Landau Fermi liquid (LFL) behavior occur, e.g. Delta rho ~ T over three decades in T. Both thermodynamic and magnetic properties show a crossover at about 0.3 K: At 0.3 K <= T <= 10 K we observe C/T ~ log(T_0/T) and a non-Curie behavior chi^{-1} ~ T^alpha with alpha<1 similar to what was found for the prototypical system CeCu_{5.9} Au_{0.1}. Below 0.3 K, chi turns into a Curie-Weiss dependence chi^{-1} ~ (T-Theta) indicating large unscreened Yb^{3+} moments whereas in C(T)/T a pronounced upturn occurs. In the undoped compound the AF order is suppressed continuously by critical fields B_{c0} ~= 0.06 T and 0.7 T applied perpendicular and parallel to the c-axis, respectively. For B>B_{c0} a LFL state with Delta rho = A(B)T^2 and C(T)/T = gamma_0(B) is induced, that fulfills the Kadowaki-Woods scaling A ~ gamma_0^2. Upon reducing the magnetic field to B_{c0} a 1/(B-B_{c0}) dependence of A(B) and gamma_0^2(B) indicates singular scattering at the whole Fermi surface and a divergence of the heavy quasiparticle mass.
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