Crystal Structure, Electrical Transport, and Magnetic Properties of Niobium Monophosphide

Large single crystals of NbP have been prepared. A single-crystal X-ray diffraction study shows that it crystallizes in tetragonal symmetry with space group I4(1)md (No. 109) and lattice parameters a = 3.3324(2) ?, c = 11.3705(7) ?, and Z = 4. A full matrix least-squares refinement based on a unique data set of 285 reflections (I> 2sigma(I)) yielded R(F) = 0.017 and R(w)(F(2)) = 0.046 for nine variables. The unit cell consists of one unique Nb and one P, each in trigonal prismatic coordination with the other element. There are two short and four long bond distances of Nb-P. The Nb-Nb bond distances are significantly shorter than R(c) = 4.09 ?, the critical distance required for good Nb-Nb 4d orbital overlap for niobium metal-metal bonds. NbP shows metallic behavior with rho = 4.5 x 10(-)(5) Omega cm at room temperature. Magnetic susceptibility measurements on a collection of randomly oriented single crystals indicate very weak Pauli paramagnetism ( approximately 10(-)(5) emu/mol). A discussion of the structure as well as the physical properties of NbP compared with those of previous results are presented. The band structure of NbP based on the extended Hückel (tight-binding) calculations is presented along with an analysis that reveals that the valence band is built up from three center bonds localized within Nb(3) triangles.     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.