$\beta$–V2N ($L'3_{2}$) Structure: AB2_hP9_162_ad_k

Picture of Structure; Click for Big Picture
Prototype : $\beta$–V2N
AFLOW prototype label : AB2_hP9_162_ad_k
Strukturbericht designation : $L'3_{2}$
Pearson symbol : hP9
Space group number : 162
Space group symbol : $\mbox{P}\bar{3}\mbox{1m}$
AFLOW prototype command : aflow --proto=AB2_hP9_162_ad_k
--params=
$a$,$c/a$,$x_{3}$,$z_{3}$


Other compounds with this structure

  • $\epsilon$-Fe2N, Cr2N, $\beta$-Nb2N, and $\beta$-Ta2N

  • (Parthé, 1993) is the only reference we have seen with a $L'3_{2}$ Strukturbericht designation. (Christensen, 1979) states that $\epsilon$-Fe2N is the prototype for this structure. We will instead follow (Villars, 1991), which uses $\beta$-V2N as the prototype. The ternary version of this structure is rosiaite, PbSb2O6.

Trigonal Hexagonal primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{x}} - \frac{\sqrt{3}}{2} \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2} \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}}\\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \mathbf{\hat{x}} + 0 \mathbf{\hat{y}} + 0 \mathbf{\hat{z}} & \left(1a\right) & \mbox{N I} \\ \mathbf{B}_{2} & =&\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(2d\right) & \mbox{N II} \\ \mathbf{B}_{3} & =&\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(2d\right) & \mbox{N II} \\ \mathbf{B}_{4} & =&x_{3} \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{3}& =&\frac12 \, x_{3} \, a \, \mathbf{\hat{x}}- \frac{\sqrt3}2 \, x_{3} \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(6k\right) & \mbox{V} \\ \mathbf{B}_{5} & =&x_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&\frac12 \, x_{3} \, a \, \mathbf{\hat{x}}+ \frac{\sqrt3}2 \, x_{3} \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(6k\right) & \mbox{V} \\ \mathbf{B}_{6} & =&- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(6k\right) & \mbox{V} \\ \mathbf{B}_{7} & =&- x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&- \frac12 \, x_{3} \, a \, \mathbf{\hat{x}}- \frac{\sqrt3}2 \, x_{3} \, a \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(6k\right) & \mbox{V} \\ \mathbf{B}_{8} & =&- x_{3} \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{3}& =&- \frac12 \, x_{3} \, a \, \mathbf{\hat{x}}+ \frac{\sqrt3}2 \, x_{3} \, a \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(6k\right) & \mbox{V} \\ \mathbf{B}_{9} & =&x_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(6k\right) & \mbox{V} \\ \end{array} \]

References

  • A. Norlund Christensen and B. Lebech, The structure of beta–Vanadium Nitride, Acta Crystallogr. Sect. B Struct. Sci. 35, 2677–2678 (1979), doi:10.1107/S0567740879010141.
  • E. Parthé, L. Gelato, B. Chabot, M. Penso, K. Cenzula, and R. Gladyshevskii, Gmelin Handbook of Inorganic and Organometallic Chemistry: Standardized Data and Crystal Chemical Characterization of Inorganic Structure Types (Springer-Verlag, Berlin, Heidelberg, 1993), 8th edn., doi:10.1007/978-3-662-02909-1_3.
  • P. Villars and L. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases (ASM International, Materials Park, OH, 1991), 2nd edn.

Found in

  • P. Villars and L. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases (ASM International, Materials Park, OH, 1991), 2nd edn., pp. 4503.

Geometry files


Prototype Generator

aflow --proto=AB2_hP9_162_ad_k --params=

Species:

Running:

Output: