Wurtzite 2H

Crystal system: Hexagonal

Space Group: P6₃mc

Stacking Sequence: AB

A: ¹⁄₃ ²⁄₃ 0

B: ¹⁄₃ ²⁄₃ ³⁄₈

Example: GaN

Wurtzite GaN unit cell containing four atoms. The tetrahedral dark box is the asymmetric unit cell.

Hexagonal 4H

Crystal structure:¹⁄₂ Hexagonal – ¹⁄₂ Cubic

Space Group: P6₃mc

Stacking Sequence: ABCB

A₁: 0 0 0

A₂: ¹⁄₃ ²⁄₃ ⁴⁄₁₆

B₁: 0 0 ³⁄₁₆

B₂: ¹⁄₃ ²⁄₃ ⁷⁄₁₆

Example: ZnS

Note:

Clearly the fractions of the z component can be simplified further but that would not show the coherent symmetry of the model. One can see clearly in this form that the component ascends by four steps and alternates between even and odd numbers for the numerator but with equal denominator.

If one is even more precise in their observation, by correlating the 2H-4H-6H denominators, one can see how the pattern just repeats with the denominator being respectively half and three times less the size from 2H, something that makes sense since 4H, 6H are two and three times the size of 2H. This is truly a wonderful depiction of the spatial symmetry that these structures have.

The same applies for 3C, being a diamond structure and needing only 2 components and 9R which exhibits the same translational and spatial symmetry by alternating its x and y components and following the principles of the common denominator, representing a part of a whole.

Hexagonal 6H

Crystal structure: ²⁄₃ Hexagonal – ¹⁄₃ Cubic

Space Group: P6₃mc

Stacking Sequence: ABCACB

A₁: 0 0 0

A₂: ¹⁄₃ ²⁄₃ ⁴⁄₂₄

A₃: ²⁄₃ ¹⁄₃ ⁸⁄₂₄

B₁: 0 0 ³⁄₂₄

B₂: ¹⁄₃ ²⁄₃ ⁷⁄₂₄

B₃: ²⁄₃ ¹⁄₃ ¹¹⁄₂₄

Example: SiC

Note:

Clearly the fractions of the z component can be simplified further but that would not show the coherent symmetry of the model. One can see clearly in this form that the component ascends by four steps and alternates between even and odd numbers for the numerator but with equal denominator.

If one is even more precise in their observation, by correlating the 2H-4H-6H denominators, one can see how the pattern just repeats with the denominator being respectively half and three times less the size from 2H, something that makes sense since 4H, 6H are two and three times the size of 2H. This is truly a wonderful depiction of the spatial symmetry that these structures have.

The same applies for 3C, being a diamond structure and needing only 2 components and 9R which exhibits the same translational and spatial symmetry by alternating its x and y components and following the principles of the common denominator, representing a part of a whole.

Zincblende 3C

Crystal system: Cubic

Space group: F43m

Stacking Sequence: ABC

A: 0 0 0

B: ¹⁄₄ ¹⁄₄ ¹⁄₄

Example: GaN

Rhombohedral 9R

Crystal structure: ²⁄₃ Cubic – ¹⁄₃ Hexagonal

Space Group: R3m(Hexagonal axes)

Stacking Sequence: CBABACACB

A₁: 0 0 0

A₂: ²⁄₃ ¹⁄₃ ⁴⁄₃₆

A₃: ¹⁄₃ ²⁄₃ ⁸⁄₃₆

B₁: 0 0 ³⁄₆₀

B₂: ²⁄₃ ¹⁄₃ ⁷⁄₃₆

B₃: ¹⁄₃ ²⁄₃ ¹¹⁄₃₆

Example: SiC

Note:

Clearly the fractions of the z component can be simplified further but that would not show the coherent symmetry of the model. One can see clearly in this form that the component ascends by four steps and alternates between even and odd numbers for the numerator but with equal denominator.

If one is even more precise in their observation, by correlating the 2H-4H-6H denominators, one can see how the pattern just repeats with the denominator being respectively half and three times less the size from 2H, something that makes sense since 4H, 6H are two and three times the size of 2H. This is truly a wonderful depiction of the spatial symmetry that these structures have.

The same applies for 3C, being a diamond structure and needing only 2 components and 9R which exhibits the same translational and spatial symmetry by alternating its x and y components and following the principles of the common denominator, representing a part of a whole.