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The magnetization of three Pb 1Ϫx Eu x Se samples, with xϭ1.3,3.0, and 4.1 %, was measured at 30 and 50 mK in magnetic fields H up to 50 kOe, and at 0.6 K in fields up to 180 kOe. For xϭ1.3% and with Hʈ͓100͔, a magnetization step ͑MST͒ due to an energy-level crossing for isolated Eu 2ϩ ions was observed at 30 and 50 mK. The magnetic field at this MST, 1.76Ϯ0.2 kOe, was close to the predicted value H c ϭ1.98 kOe. At the same low temperatures ͑30 and 50 mK͒ but at higher fields, a magnetization ''ramp'' due to pairs was observed in all samples. For xϭ3.0 and 4.1 % this ramp consisted of well-resolved MST's arising from pairs. A ramp due to open triplets was also observed in these two samples. The MST's due to pairs were used to obtain the value J/k B ϭϪ0.24Ϯ0.03 K for the dominant antiferromagnetic exchange constant. Comparisons between the measured magnetization curves at 30 or 50 mK and theoretical simulations indicates that this J is the nearest-neighbor ͑NN͒ exchange constant J 1 . At 0.62 K the magnetization of each of the three samples rose rapidly with H in fields below several kOe. At higher fields a rounded ramp due to pairs and triplets was present. This ramp ended near 40 kOe, and complete saturation was achieved near 50 kOe. A model which includes only the NN exchange constant J 1 gave a reasonably good account for all the data at 0.62 K. Calculated magnetization curves for pairs, and for open and closed triplets, at various values of k B T/͉J͉ are presented. The effects of the single-ion and dipole-dipole anisotropies on the MST's due to pairs are also discussed.

Several exchange constants J i between Mn 21 ions which are not nearest neighbors were determined in Zn 12x Mn x X (X S, Se, Te) from magnetization steps at 20 mK. When the J i 's are listed in order of decreasing size, ratios between successive J i 's are material dependent, and differ from all predictions. The measured J i 's were identified by comparing the magnetization curves with simulations which assumed a random Mn distribution. Contrary to existing theories the second-largest exchange constant is not J 2 between next-nearest neighbors. The most likely alternative is J 4 , between fourth neighbors. [S0031-9007 (98)06413-8] PACS numbers: 75.30.Et, 75.50.Ee, 75.50.Pp, 75.60.EjThe distance dependence of the d-d exchange constants J i in dilute magnetic semiconductors (DMS's) has been discussed for more than a decade [1][2][3][4][5][6][7][8][9]. The focus has been on Mn-based II-VI DMS's with the zinc-blende structure. It has been established that the largest J i is the nearest-neighbor (NN) exchange constant J 1 . This J 1 is antiferromagnetic (AF), and is of order 210 K [5,6]. It is generally accepted that the second-neighbor (next-nearestneighbor) exchange constant J 2 , third-neighbor constant J 3 , etc., are all AF. What is at issue are the ratios J 1 : J 2 : J 3 : J 4 , etc.All existing theories, conjectures, and reported data as interpreted by their authors maintain that J 2 is the secondlargest exchange constant, after J 1 . The theory of Larson et al. [1] predicts that J 2 : J 1 , J 3 : J 2 , and J 4 : J 3 are all about 0.08. In the modified version by Rusin [9], J 2 : J 1 Х 0.08, and both J 3 and J 4 are less than 0.1J 2 . According to Bruno and Lascaray (BL), J 3 : J 2 J 4 : J 3 1͞2 (no prediction for J 2 : J 1 ) [4]. A power law dependence of J i on distance, J i~r

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