Cubic Lattice

Author

Eric W. Weisstein
July 26, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Lattices/CubicLattice.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CubicLattice.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Michael Trott

Graphic

g=With[{n=4},Show[Graphics3D[{PointSize[.02],
Table[Point[{x,y,z}],{x,-4,4},{y,-4,4},{z,-4,4}]}],AspectRatio->Automatic,Axes->True]
]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["cublat.m",g]

Lattice

LatticeData["*Cubic*"]

Select[LatticeData[],StringMatchQ[#,"*Cubic"]&]

LatticeData[{"IntegerLattice",3},"Image"]

LatticeData[{"IntegerLattice",2},"Image"]

LatticeData[{"IntegerLattice",2},"Notation"]

LatticeData[{"IntegerLattice",3},"Notation"]

Brillouin Zones

Michael Trott
Mon Oct 24 11:16:31 2005

3D cubic lattices are infinite arrangements of points that are invariant under translations and rotations. 3D cubic lattices are Bravais lattices.

<<Graphics`Shapes`

There are three kinds of cubic lattices.

Simple cubic lattices (SC):
It is spanned by the three basis vector
(Subscript[a, 1])=(Subscript[e, x])
(Subscript[a, 2])=(Subscript[e, y])
(Subscript[a, 3])=(Subscript[e, z])

Show[Graphics3D[{EdgeForm[],  SurfaceColor[Hue[0.06], Hue[0.32], 2.1],
Table[TranslateShape[Sphere[1/5, 12, 12], 
               i ({1, 0, 0} ) +
               j ({0, 1, 0} ) +
               k ({0, 0, 1} )], {i, -1, 4}, {j, -1, 4}, {k, -1, 4}]}
               ], PlotRange -> {{0, 2}, {0, 2}, {0, 2}}]

Here are the first four Brillouin zones of this lattice.

Po forms this lattice.

The packing density (using nonintersecting sphres of maximal size) is π /6≈52%.

(π/6.)

Face-centered cubic lattices (FCC):
It is spanned by the three basis vector
(Subscript[a, 1])=((Subscript[e, y])+(Subscript[e, z]))/2
(Subscript[a, 2])=((Subscript[e, x])+(Subscript[e, z]))/2
(Subscript[a, 3])=((Subscript[e, x])+(Subscript[e, y]))/2

Show[Graphics3D[{EdgeForm[], SurfaceColor[Hue[0.45], Hue[0.87], 2.2],
Table[TranslateShape[Sphere[1/5, 12, 12], 
               i/2 ({0, 1, 0} + {0, 0, 1}) +
               j/2 ({1, 0, 0} + {0, 0, 1}) +
               k/2 ({1, 0, 0} + {0, 1, 0})], {i, -1, 4}, {j, -1, 4}, {k, -1, 4}]}
               ], PlotRange -> {{0, 2}, {0, 2}, {0, 2}}]

Here are the first four Brillouin zones of this lattice.

Al, Cu form this lattice. And diamonds.

The packing density (using nonintersecting sphres of maximal size) is π /(2(Sqrt[3]))≈74%.

((π )/(3.(Sqrt[2])))

Body-centered cubic lattices (BCC):

It is spanned by the three basis vector
(Subscript[a, 1])=(-(Subscript[e, x])+(Subscript[e, y])+(Subscript[e, z]))/2
(Subscript[a, 2])=((Subscript[e, x])-(Subscript[e, y])+(Subscript[e, z]))/2
(Subscript[a, 3])=((Subscript[e, x])+(Subscript[e, y])-(Subscript[e, z]))/2

Show[Graphics3D[{EdgeForm[], SurfaceColor[Hue[0.32], Hue[0.76], 1.9],
Table[TranslateShape[Sphere[1/5, 12, 12], 
               i/2 (-{0, 1, 0} + {0, 1, 0} + {0, 0, 1}) +
               j/2 ( {1, 0, 0} - {0, 1, 0} + {0, 0, 1}) +
               k/2 ( {1, 0, 0} + {0, 1, 0} - {0, 0, 1})], {i, -3, 8}, {j, -3, 8}, {k, -3, 8}]}
               ], PlotRange -> {{0, 2}, {0, 2}, {0, 2}}]

Here are the first four Brillouin zones of this lattice.

Na and Fe form this lattice.

The packing density (using nonintersecting sphres of maximal size) is π (Sqrt[3])/8≈68%.

((π (Sqrt[3]))/8.)