Preparation and Crystal Structure of PrNb_5O_(14)

ＰｒｅｐａｒａｔｉｏｎａｎｄＣｒｙｓｔａｌＳｔｒｕｃｔｕｒｅｏｆＰｒＮｂ_５Ｏ_（１４）￥ＭａｏＪｉａｎｇ－Ｇａｏ；ＺｈｕａｎｇＨｏｎｇ－Ｈｕｉ；ＨｕａｎｇＪｉｎ－Ｓｈｕｎ（ＳｔａｔｅＫｅｙＬａｂｏｒａｔｏｒｙｏｆＳｔｒｕｃｔｕｒａｌＣｈｅｍｉｓｔｒｙ...     

Synthesis and Crystal Structure of a New Mixed Orthovanadate-Pyrovanadate Series: MBa2V3O11 or MBa2V2PO11 with M = Bi, In, or a Rare Earth

A new series of vanadates with the general formula M Ba₂V₃O₁₁, where M may be Bi, In, or a rare earth, has been synthesized and structurally characterized by single crystal X-ray diffraction and powder X-ray diffraction. The general formula may be rewritten as M Ba₂(VO₄)(V₂O₇) to emphasize that there is one orthovanadate group and one pyrovanadate group in each formula unit. Up to one-third of the vanadium may be replaced by phosphorous, leading to the general formula M Ba₂V₂PO₁₁. However, phosphorous shows no preference between the ortho and pyro groups. Both M Ba₂V₃O₁₁ and M Ba₂V₂PO₁₁ crystallize in the monoclinic system with the space group P2₁/_c and Z = 4. The cell parameters from single crystal X-ray data of BiBa₂V₃O₁₁ are a = 12.332(4) Å, b = 7.750(4) Å, c = 11.279(4) Å, β = 103.22(3)°, V = 1049(1) ų; and for BiBa₂PO₁₁ are a = 12.266(2) Å, b = 7.615(2) Å, c = 11.312(2) Å, β = 103.32(2)°, V = 1028.2(2) ų. The Bi atom coordinates to six oxygen atoms forming a distorted octahedron, and the edge sharing of BiO₆ octahedra results in a BiO₄ chain along the b axis. There are two types of Ba atoms with coordination numbers of 10 and 11. There are three types of tetrahedral (T) atoms in these structures. The nonequivalent T atoms of the pyro group give T-O-T angles of 167 and 171° in BiBa₂V₃O₁₁ and BiBa₂V₂PO₁₁, respectively. Isostructural M Ba₂V₃O₁₁ compounds were prepared in which M is In, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Yb, or Lu.     

The stability of RDDE\ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O

In this paper, the author considered the stability of zero solution of linear RDDE $$\begin{gathered} \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O, \hfill \\ \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) = O \hfill \\ \end{gathered} $$ using Liapunov-Razumikhin functional and transformations and obtained some sufficient conditions for the stability of Eqs.(1) and (2). These results are suitable both for boundedp _i (t),q _i (t) andr(t).i=1,2.     

BRST quantization of a scalar particle in a curved background

The BRST formalism is employed to quantize a scalar particle and interactions with an external scalar field (x ) and vector gauge fieldA (x ) in the background of an arbitrary gravitational field. The second-quantized actions are obtained.