The magnetic phase diagram of Ce͑Ru 1−x Rh x ͒ 2 Si 2 contains a spin-density-wave ͑SDW͒ magnetic ordering temperature approaching T = 0 at both x Ϸ 0.03 and 0.35-0.4 ͑i.e., a dome-shaped phase boundary͒ and long range, local moment antiferromagnetism, T N = 36 K, in pure CeRh 2 Si 2 suppressed with Ru doping to T =0 at x Ϸ 0.6-0.65. This latter possible second quantum critical point ͑QCP͒, and the possible interplay between the fluctuations caused at each of the two QCP's, are investigated here using specific heat, resistivity, and dcmagnetic susceptibility data over a broad range of composition, from x = 0.04 to 0.8. One principal result is that the specific heat divided by temperature, C / T, for x = 0.6 ͑0.65͒ near T local moment → 0 is proportional to log T over 2 1 / 2 decades of temperature down to the lowest temperature of measurement, 0.04 ͑0.3͒ K, indicative of strong fluctuations near a QCP. For the region in the phase diagram x = 0.4-0.5, i.e., near the reported T SDW → 0 composition, C / T measured down to 0.08 K in the present work, as well as literature data, show a distinctly different behavior with temperature, a saturation in C / T which can be fit by ␥ − a ͱ T. Such a temperature dependence is consistent with a nearby QCP with weakly interacting spin fluctuations as proposed, e.g., in the theory of Moriya. In the region between the two QCP's, at x = 0.55, specific heat data down to 0.1 K are not well fit by C / T = ␥ − a ͱ T and are consistent with C / T ϳ log T only down to 0.6 K, i.e., this composition displays intermediate behavior. The residual resistivity, 0 , vs x shows two strong peaks, at x = 0.4 and 0.65, consistent with the existence of two quantum critical points. The exponent ␣ in = 0 + AT ␣ indicates non-Fermi liquid behavior, with ␣ varying monotonically-in contrast to 0-from 1.5 to ϳ 0.9 between x = 0.3 and 0.65. The Fermi liquid exponent, ␣ = 2, is recovered at x = 0.8. These results taken together indicate two distinct quantum critical points in the phase diagram of Ce͑Ru 1−x Rh x ͒ 2 Si 2 , with different fluctuation strengths at x ͑T SDW → 0͒ = 0.4 and x͑T local moment → 0͒Ϸ0.6-0.65.
