Pyramidal HCP · Oblique zig-zag rows
HCP(11-21)
HCP(11-21) combines the rotated basal orientation of (11-20) with a finite c-axis intercept. The result is an anisotropic pyramidal face with staggered ledges.
- Plane
- (11-21)
- Layer character
- Pyramidal · anisotropic
- ASE slab builder
surface
From bulk to facet
How HCP(11-21) is cut
(11-21)
Starting from the (11-20) prism, the l=1 component tilts the plane towards the basal direction. The cut meets a different pair of hexagonal side faces than (10-11).
- Bulk lattice
- Hexagonal close packed
- Plane normal
- reciprocal vector G(11-21)
- What remains
- Oblique zig-zag rows
The plane is drawn through the centre so high-index cuts remain legible. Its orientation is what the indices specify, not its absolute position inside one cell.
Surface geometry
Read the surface from above
L1 is the highest atom-bearing plane, followed by L2 and L3. Support coordinates outside 0–1 are periodic images. The listed Δz describes the supporting atoms, not the marker height. In the interactive model, every numbered site is placed on the same schematic guide plane above the slab; no adsorption distance or energetic ordering is implied.
- ledge ontop1-fold
Above a low-coordinate atom on the ledge.
(u,v) = (0.995, 0.3234)s = s₁ · support shell: mean Δz = +0.000 Å from L1Projection of the L1 atom at (0.995, 0.3234).
Not named in ASE - terrace ontop1-fold
Above an atom within the terrace.
(u,v) = (0.4517, 0.2367)s = s₁ · support shell: mean Δz = -1.534 Å from L1Projection of the L3 atom at (0.4517, 0.2367).
Not named in ASE - lower-row ontop1-fold
Above an exposed atom below the ledge.
(u,v) = (0.89, 0.7801)s = s₁ · support shell: mean Δz = -0.767 Å from L1Projection of the L2 atom at (0.89, 0.7801).
Not named in ASE - ledge bridge2-fold
Between adjacent atoms along the ledge.
(u,v) = (0.495, 0.3234)s = (s₁ + s₂) / 2 · support shell: mean Δz = +0.000 Å from L1Periodic midpoint of L1 (0.995, 0.3234) and L1 (-0.005, 0.3234).
Not named in ASE - terrace bridge2-fold
Between neighbours within the terrace.
(u,v) = (0.39, 0.7801)s = (s₁ + s₂) / 2 · support shell: mean Δz = -0.767 Å from L1Periodic midpoint of L2 (0.89, 0.7801) and L2 (-0.11, 0.7801).
Not named in ASE - cross-ledge bridge2-fold
Between atoms on opposite sides of the ledge.
(u,v) = (0.9425, 0.5518)s = (s₁ + s₂) / 2 · support shell: mean Δz = -0.383 Å from L1Periodic midpoint of L1 (0.995, 0.3234) and L2 (0.89, 0.7801).
Not named in ASE - pyramidal mixed-height pocket3-fold
A pocket characteristic of the pyramidal cut.
(u,v) = (0.1123, 0.1134)s = (Σᵢ sᵢ) / 3 · support shell: mean Δz = -0.767 Å from L1Least-squares centroid of the 3-atom projected shell: L1 (-0.005, 1.3234), L2 (-0.11, 0.7801), L3 (0.4517, 1.2367).
Not named in ASE
Interactive model
Rotate the slab
Sites share a schematic display height · drag to rotate · hover for names
Below the top layer
Why the sites are different
Several projected nearest neighbours occupy different heights because both the basal rotation and pyramidal tilt act at once.
Cell
Geometry at a glance
The constraint \(i=-(h+k)\) gives (11-21), and \(d_{11\bar{2}1}^{-2}=4/a^2+1/c^2\). Its three-index ASE input is (1,1,1).
A site name describes the ideal starting geometry. Relaxation can move an adsorbate away from it.
Practical model
Build it with ASE
The builder creates the slab. Sites marked ASE keyword can be passed directly as a named position; ASE source marks current but inconsistently documented support. Other sites require explicit Cartesian coordinates converted from the fractional construction above.
from ase.build import bulk, surface
mg = bulk("Mg", "hcp", a=3.21, c=5.21)
slab = surface(mg, (1, 1, 1), 10, vacuum=10)
Things that are easy to misread
- Dropping the basal rotation when comparing it with (10-11).
- Counting projected atoms without distinguishing their z levels.