Intermetallic crystal structures: what do L1₂, D0₂₂ and B2 mean?

Where the positions come from

Every site is a Wyckoff position (a named set of equivalent sites in the space group) from the AFLOW Library of Crystallographic Prototypes (Mehl et al. 2017) and the older NRL Crystal Lattice Structures pages. The full cell is filled by adding the centring shifts: (0,½,½), (½,0,½), (½,½,0) for the F cells (D0₃, L2₁) and (½,½,½) for the I cells (D0₂₂, D0₂₃).

L1₂ (Cu₃Au, cP4, Pm3̄m): Au 1a (0,0,0); Cu 3c (0,½,½), (½,0,½), (½,½,0). “All of the atoms are located on the sites of a face centered cubic lattice” (NRL).

D0₂₂ (Al₃Ti, tI8, I4/mmm): Al 2a (0,0,0); Ti 2b (0,0,½); Al 4d (0,½,¼). “When c = 2a the atoms are on the sites of a face-centered cubic lattice” (AFLOW).

D0₂₃ (Al₃Zr, tI16, I4/mmm): Al 4c, 4d, Al 4e (0,0,z₃), Zr 4e (0,0,z₄). With c = 4a, z₃ = 3/8 and z₄ = 1/8 the atoms sit on fcc sites (AFLOW). The NRL page uses these ideal values because it had not seen measured ones for Al₃Zr; this page does the same.

L1₀ (CuAu, tP2, P4/mmm): Au 1a (0,0,0), Cu 1d (½,½,½). AFLOW notes that earlier workers doubled this cell and gave it the Pearson symbol tP4. NRL: “a tetragonal distortion of the fcc structure”, on fcc sites when c = a. The viewer draws the same atoms in the larger face-centred cell (axes turned 45°), with c = a.

B2 (CsCl, cP2, Pm3̄m): 1a (0,0,0) and 1b (½,½,½), on bcc sites. D0₃ (AlFe₃ in NRL, BiF₃ in AFLOW, cF16, Fm3̄m): Al 4a (0,0,0); Fe 4b (½,½,½) and 8c (¼,¼,¼), (¾,¾,¾). L2₁ (Heusler, AlCu₂Mn, cF16): Al 4a, Mn 4b, Cu 8c. Both sit on bcc sites, and L2₁ turns into D0₃ if Mn is replaced by Cu (NRL).

D0₁₉ (Ni₃Sn, hP8, P6₃/mmc): Sn 2c (⅓,⅔,¼); Ni 6h (x,2x,¼). When x = 5/6 the atoms sit on hcp sites with a_hcp = a/2; in Ni₃Sn and Ti₃Al x is about 5/6 (NRL). The viewer uses x = 5/6; its cell height is drawn for shape only (schematic).

The names

NRL: “A structures are supposed to be monatomic, B's are diatomic with equal numbers of atoms of each type, C's have a 2-1 abundance ratio, D0's are 3-1”, with D for A_mB_n compounds, E to K for more complex ones and L for alloys. The Pearson symbol gives the crystal family, the cell type and the number of atoms in the cell (IUCr): cP4 is cubic, primitive, 4 atoms. Other names in the same system include the Laves phases C15 (Cu₂Mg, cF24, Fd3̄m) and C14 (MgZn₂, hP12, P6₃/mmc) (AFLOW).

Building D0₂₂ and D0₂₃

“The D0₂₂ unit cell consists of two L1₂ cubes stacked along the z direction, with a [½ ½ 0] antiphase shift between the cubes”, and “the D0₂₃ structure consists of a stacking of four L1₂ cubes with the same antiphase shifting every two cubes” (Jia et al. 2016). Leineweber et al. (2020) put it as antiphase boundaries “on one set of {001} fcc planes”. Measured cells (Knipling et al. 2006, Table 1): D0₂₂ Al₃Ti a = 3.848 Å, c = 8.596 Å, so c/a = 2.234; D0₂₃ Al₃Zr a = 4.014 Å, c = 17.321 Å, so c/a = 4.315. Their cubic L1₂ forms have a = 3.967 Å (Al₃Ti) and 4.08 Å (Al₃Zr), both metastable.

Concentration waves and special points

The static concentration wave method, proposed by Khachaturyan, writes the chance n(r) that site r holds atom A as n(r) = c + ½ Σ_j [Q(k_j) exp(ik_j·r) + Q*(k_j) exp(−ik_j·r)], with c the atom fraction and k_j wave vectors in the first Brillouin zone (Paudyal et al. 2002). The wave amplitude η “serves as a long-range order parameter”: it is zero in a solid solution (Fultz 2020). The page uses the real form n = c + Σ a_j η cos(2π k_j·r), k in units of 2π/a and r in units of a, with amplitudes it chose so that n is exactly 0 or 1 at η = 1: L1₀ ½ (X); L1₂ ¼ for each of the three X waves; L1₁ ½ (L); D0₂₂ ¼ (X) and −½ (W); B2 ½ (H); D0₃ ¼ (H) and ½ (P). It then checks every site against the AFLOW cells of the 3D viewer (L1₁ is not in the viewer, so it is not checked).

Special points “are defined as the points for which any k-space function with the symmetry of the reciprocal lattice must have extrema, simply due to symmetry reasons”; for fcc they are (000), (100), (1½0) and ½(111) (Wolverton and Zunger 1995). The Lifshitz condition says the free energy may hold no terms of the form η(∂η/∂x), and “second-order transitions to commensurate phases must be driven by an order parameter belonging to an IR associated with a k point of symmetry”; incommensurate phases need only a weaker condition (Stokes, Hatch and Nelson 1993). A second-order transition also needs the Landau condition (no third-degree terms) (same source), so the page does not claim any of these transitions is second order. Structures with ordering vectors only at high-symmetry points are called Lifshitz structures (Chepulskii et al. 2012). Paudyal et al. assign X to L1₂ and L1₀, and L to L1₁; Woodgate et al. (2024) write that (0,0,1) and equivalent waves describe L1₂ and that (0,0,1) together with (0,½,1), an X and a W wave, gives D0₂₂; Woodgate et al. (2025) give k = (0, 0, 2π/a), the H point, for B2. That D0₃ needs an H plus a P wave is the page's own result from its check. The page scales all waves of a pattern with one η (its own simplification). Point coordinates and zone shapes: Hadley (TU Graz). Short-range order peaks need not sit on a special point: for Cu-Pd alloys near Cu₃Pd, measured and calculated peaks are “split off of the special-point (100) along the (1ξ0) line” (Wolverton and Zunger 1995); see the short-range order page.

The structure factor

F_hkl = Σ f_j exp[2πi(hx_j + ky_j + lz_j)], summed over the atoms of the cell, with f_j the scattering factor of atom j and (x_j, y_j, z_j) its position as fractions of the cell; the intensity is |F|² (Wikipedia, Structure factor, after Warren 1969). For CsCl it gives f_Cs + f_Cl when h + k + l is even and f_Cs − f_Cl when odd. The page applies the same sum to the positions above: for L1₂ it gives f_A + 3f_B when h, k, l are all even or all odd (fundamental) and f_A − f_B when they are mixed (superlattice); for the 16-atom D0₃ cell, 4(f_A + 3f_B) and 4(f_A − f_B). The f sliders are in relative units chosen by the page; real f values depend on the element, the angle and the radiation.