1. Orbital Ordering and Exchange Interactions in RMnO 3 Perovskites J. B. Goodenough, U.T. Austin DMR-0555663

2. Crystallographic structure of RMnO 3 LaPrNdSmEuGdTb Dy Y HoErTmYbLu Synthesized under ambient pressure Synthesized under high pressure Synthesized under ambient pressure O3O3 O4O4 O1O1 O4O4 R2R2 O2O2 O4O4 O4O4 O3O3 R2R2 R2R2 R1R1 Mn 3+ ion MnO 5 Bipyramid O3O3 O4O4 O4O4 a 3z 2 - r (x 2 -y 2 ) ± ixy yz ± ixz 2 c xy, yz, zx 3x 2 - r 2 y 2 - z 2 MnO 6 Octahedra 1 cc Pbnm P6 3 cm 

3. Magnetic phase diagram of RMnO 3 Perovskites Although the same orbital ordering takes place over entire family of RMnO 3 perovskites, there are three magnetic phases: Type-A spin ordering, No classic spin ordering, Type-E spin ordering.

4. The evolution of structural distortions versus ionic size of the A-site Rare Earth The structure refinement from data of neutron and x-ray diffraction.

5. The relationship between structural distortions and magnetic couplings in perovskite RMnO 3 J ab = J  ab -J  ab J c = -J  c -J  c J  is sensitive to  J  is much less sensitive to    = (  1 + 2  2 )/3  180  -  J  ab ~ cos 4 (  /2)cos 2 (  /2)|  3x 1 2 - r 1 2 |H’|x 2 2 -z 2 2  | 2 /  

6. Check some numbers from Structure, T N and T JT (For Type-A spin ordering phase) dlnT N /d(IR)  9 Ǻ -1 dln(cos 2 (  /2))/d(IR)  0.4 Ǻ -1 dln(cos 4 (  /2))/d(IR), wrong sign dln(   -1 ))/d(IR)  7 Ǻ -1 (   ~ kT JT, mean-field theory) Conclusions * Type-A spin ordering phase for R=La-Gd J  ab is weakened primarily by  , not by the angle . * No classic spin ordering phase for R= Tb, Dy A |J  ab | ~ |J  ab | leads to the exchange interaction frustration. * Type-E spin ordering phase for R= Y-Lu Segregation into J  ab dominant and J  ab dominant Mn-O-Mn bonds occurs.