Triclinic (TRI)

Pearson symbol: aP

Triclinic lattice is described by the class TRI.

It is defined by six parameters: \(a\), \(b\), \(c\) and \(\alpha\), \(\beta\), \(\gamma\). with primitive and conventional lattice:

\[ \begin{align}\begin{aligned}\boldsymbol{a}_1 = (a, 0, 0)\\\boldsymbol{a}_2 = (b\cos(\gamma), b\sin(\gamma), 0)\\\boldsymbol{a}_3 = (c\cos(\beta), \frac{c(\cos(\alpha) - \cos(\beta)\cos(\gamma))}{\sin{\gamma}}, \frac{c}{\sin(\gamma)}\sqrt{\sin^2(\gamma) - \cos^2(\alpha) - \cos^2(\beta) + 2\cos(\alpha)\cos(\beta)\cos(\gamma)})\end{aligned}\end{align} \]

Variations

There are four variation of triclinic lattice.

TRI_1a

\(k_{\alpha} > 90^{\circ}, k_{\beta} > 90^{\circ}, k_{\gamma} > 90^{\circ}, k_{\gamma} = \min(k_{\alpha}, k_{\beta}, k_{\gamma})\)

TRI_2a

\(k_{\alpha} > 90^{\circ}, k_{\beta} > 90^{\circ}, k_{\gamma} = 90^{\circ}\)

TRI_1b

\(k_{\alpha} < 90^{\circ}, k_{\beta} < 90^{\circ}, k_{\gamma} < 90^{\circ}, k_{\gamma} = \max(k_{\alpha}, k_{\beta}, k_{\gamma})\)

TRI_2b

\(k_{\alpha} < 90^{\circ}, k_{\beta} < 90^{\circ}, k_{\gamma} = 90^{\circ}\)

In definition of the examples we cheated and defined them through reciprocal lattice parameters.

Example structures

TRI_1a

Default kpath: \(\Gamma-Y-F-L-I\vert I_1-Z-F_1\vert Y-X_1\vert X-\Gamma-N\vert M-\Gamma\).

Brillouin zone and default kpath

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI1a") l.plot("brillouin-kpath") # Save an image: l.savefig( "tri1a_brillouin.png", elev=45, azim=-55, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=45, azim=-55)

Primitive and conventional cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI1a") l.plot("primitive") # Save an image: l.savefig( "tri1a_real.png", elev=31, azim=-20, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=31, azim=-20)

Wigner-Seitz cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI1a") l.plot("wigner-seitz") # Save an image: l.savefig( "tri1a_wigner-seitz.png", elev=9, azim=18, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=9, azim=18)

TRI_2a

Default kpath: \(\Gamma-Y-F-L-I\vert I_1-Z-F_1\vert Y-X_1\vert X-\Gamma-N\vert M-\Gamma\).

Brillouin zone and default kpath

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI2a") l.plot("brillouin-kpath") # Save an image: l.savefig( "tri2a_brillouin.png", elev=10, azim=32, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=10, azim=32)

Primitive and conventional cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI2a") l.plot("primitive") # Save an image: l.savefig( "tri2a_real.png", elev=39, azim=44, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=39, azim=44)

Wigner-Seitz cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI2a") l.plot("wigner-seitz") # Save an image: l.savefig( "tri2a_wigner-seitz.png", elev=30, azim=62, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=30, azim=62)

TRI_1b

Default kpath: \(\Gamma-Y-F-L-I\vert I_1-Z-F_1\vert Y-X_1\vert X-\Gamma-N\vert M-\Gamma\).

Brillouin zone and default kpath

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI1b") l.plot("brillouin-kpath") # Save an image: l.savefig( "tri1b_brillouin.png", elev=30, azim=42, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=30, azim=42)

Primitive and conventional cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI1b") l.plot("primitive") # Save an image: l.savefig( "tri1b_real.png", elev=12, azim=11, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=12, azim=11)

Wigner-Seitz cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI1b") l.plot("wigner-seitz") # Save an image: l.savefig( "tri1b_wigner-seitz.png", elev=21, azim=39, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=21, azim=39)

TRI_2b

Default kpath: \(\Gamma-Y-F-L-I\vert I_1-Z-F_1\vert Y-X_1\vert X-\Gamma-N\vert M-\Gamma\).

Brillouin zone and default kpath

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI2b") l.plot("brillouin-kpath") # Save an image: l.savefig( "tri2b_brillouin.png", elev=40, azim=1, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=40, azim=1)

Primitive and conventional cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI2b") l.plot("primitive") # Save an image: l.savefig( "tri2b_real.png", elev=17, azim=54, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=17, azim=54)

Wigner-Seitz cell

Picture	Code
	import radtools as rad l = rad.lattice_example(f"TRI2b") l.plot("wigner-seitz") # Save an image: l.savefig( "tri2b_wigner-seitz.png", elev=19, azim=13, dpi=300, bbox_inches="tight", ) # Interactive plot: l.show(elev=19, azim=13)

Ordering of parameters

TODO