Difference between revisions of "Lattice:BCC"

Latest revision as of 13:50, 15 October 2014

BCC or body-centered cubic is a cubic lattice where the symmetry involves an atom/particle sitting in the center of the conceptual unit cell.

BCC unit cell.

Canonical BCC

Symmetry

Crystal Family: Cubic
Crystal System: Cubic
Bravais Lattice: I (bcc)
Crystal class: Hexoctahedral
Point Group: m3m
Space Group: Im3m
Particles per unit cell: $n=2$
Volume of unit cell: $V_{d}=a^{3}$
Dimensionality: $d=3$
Projected d-dimensional volume: $v_{d}=a^{3}$
Solid angle: $\Omega _{d}=4\pi$
Nearest-neighbor distance: $d_{nn}={\sqrt {3}}a/2$
Assuming spherical particles of radius R:
- Particle volume fraction: $\phi =8\pi R^{3}/\left(3a^{3}\right)$
- Maximum volume fraction: $\phi _{max}=\pi {\sqrt {3}}/8\approx 0.680$ when $R=a{\sqrt {3}}/4$

Structure

Body-centered cubic (BCC) is a cubic lattice where a cube-shaped unit cell has particles at each corner of the cube, plus a particle in the center of this cube. The minimal unit cell has only two particles.

The extended lattice can be thought of in terms of two inter-penetrating simple cubic lattices:

One can also consider the BCC to be a distorted FCC:

Neighbors

Each particle is equivalent (there is nothing special about the 'center' particle in a homogeneous BCC). For a lattice of edge-length a, each particle has 8 "diagonal" nearest-neighbors at a distance of:

d_{\mathrm {nn} }={\sqrt {3}}/2\times a\approx 0.866a

and another 6 "direct" next-nearest-neighbors at a distance of $a=(2/{\sqrt {3}})d_{nn}$ .

Reciprocal-space Peaks

Allowed reflections:
$f_{hkl}=2\,\,\,\mathrm {for} \,\,\,h+k+l=\mathrm {even}$
Peak multiplicities:
$m_{h00}=6$

$m_{hh0}=12$

$m_{hhh}=8$

$m_{hk0}=24$

$m_{hhk}=24$

$m_{hkl}=48$
Peak positions:
$q_{hkl}={\frac {2\pi }{a}}{\sqrt {h^{2}+k^{2}+l^{2}}}$

For a = 1.0:

peak    q value         h,k,l   m       f       intensity
1:      8.885765876317  1,1,0   12      2       48
2:      12.566370614359 2,0,0   6       2       24
3:      15.390597961942 2,1,1   24      2       96
4:      17.771531752633 2,2,0   12      2       48
5:      19.869176531592 3,1,0   24      2       96
6:      21.765592370811 2,2,2   8       2       32
7:      23.509526717078 3,2,1   48      2       192
8:      25.132741228718 4,0,0   6       2       24
9:      26.657297628950 4,1,1   36      2       144
10:     28.099258924163 4,2,0   24      2       96
11:     29.470751386686 3,3,2   24      2       96
12:     30.781195923885 4,2,2   24      2       96
13:     32.038084488828 5,1,0   72      2       288
14:     34.414423257273 5,2,1   48      2       192
15:     35.543063505267 4,4,0   12      2       48
16:     36.636951272563 5,3,0   48      2       192
17:     37.699111843078 6,0,0   30      2       120
18:     38.732155490827 6,1,1   72      2       288
19:     39.738353063184 6,2,0   24      2       96
20:     40.719694735877 5,4,1   48      2       192
21:     41.677936304377 6,2,2   24      2       96
22:     42.614636098416 6,3,1   48      2       192
23:     43.531184741621 4,4,4   8       2       32
24:     44.428829381584 5,5,0   60      2       240
25:     45.308693596556 6,4,0   24      2       96
26:     46.171793885827 6,3,3   48      2       192
27:     47.019053434156 6,4,2   48      2       192
28:     49.473850582607 6,5,1   48      2       192
29:     51.044838738971 5,5,4   24      2       96
30:     51.812473373661 6,4,4   24      2       96
31:     52.568899858234 6,5,3   48      2       192
32:     53.314595257900 6,6,0   12      2       48
33:     54.775539595071 6,6,2   24      2       96
34:     58.267863475287 6,5,5   24      2       96
35:     58.941502773372 6,6,4   24      2       96
36:     65.296777112432 6,6,6   8       2       32

Realspace Spacings

Note that in general:

{\begin{alignedat}{2}\left({\frac {1}{d_{hkl}}}\right)^{2}=\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}\end{alignedat}}

Since $q=2\pi /d$ , and since $a=b=c$ , the realspace spacing of planes is:

{\begin{alignedat}{2}d_{hkl}&={\frac {a}{\sqrt {h^{2}+k^{2}+l^{2}}}}\end{alignedat}}

Examples

Elemental

3. Lithium (Li) (a = 3.509 Å)

11. Sodium (Na) (a = 4.291 Å)

19. Potassium( K) (a = 5.247 Å) (at 78 K)

23. Vanadium (V) (a = 3.024 Å)

24. Chromium (Cr) (a = 2.884 Å)

26. Iron (Fe) (a = 2.866 Å)

37. Rubidium (Rb) (a = 5.605 Å) (at 78 K)

41. Niobium (Nb) (a = 3.300 Å)

42. Molybdenum (Mo) (a = 3.147 Å)

55. Caesium (Cs) (a = 6.067 Å) (at 78 K)

56. Barium (Ba) (a = 5.025 Å)

63. Europium (Eu) (a = 4.606 Å)

73. Tantalum (Ta) (a = 3.306 Å)

74. Tungsten (W) (a = 3.165 Å)

Atomic

TBD

Nano

TBD

Body-centered Two-particle

A lattice where the unit cell has two distinct atoms/particles, arranged with one at the center of the simple cubic lattice defined by the other, has simple cubic symmetry. This lattice is often called CsCl.

Symmetry

Crystal Family: Cubic
Crystal System: Cubic
Bravais Lattice: P (pcc)
Crystal class: Hexoctahedral
Point Group: m3m
Space Group: Pm3m
Particles per unit cell: $n=2$ (distinct)
Volume of unit cell: $V_{d}=a^{3}$
Dimensionality: $d=3$
Projected d-dimensional volume: $v_{d}=a^{3}$
Solid angle: $\Omega _{d}=4\pi$
Nearest-neighbor distance: $d_{nn}={\sqrt {3}}a/2$

Examples

Atomic

Cesium chloride (CsCl) (a = 4.11 Å)

Body-centered with Edges and Faces

The BCC lattice, where a second particle type occupies positions along edges and faces. The symmetry is the same as the canonical BCC. (The particles at the face position are effective 'edge' particles with respect to the center particle...)

Particle Positions

There are 27 positions, with 8 particles in the unit cell

Particle A

These are the usual BCC positions. There are 9 positions. In total there are 2 particles in the unit cell:

$8\,\mathrm {corners} \,\times \,{\frac {1}{8}}=1$ $8\,\mathrm {corners} \,\times \,{\frac {1}{8}}=1$
- $\left(0,0,0\right),(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,0,1),(1,1,0),(1,1,1)$
$1\,\mathrm {center} \,\times \,1=1$ $1\,\mathrm {center} \,\times \,1=1$
- $\left({\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}}\right)$

Particle B

There are 18 positions, with 6 particles in the unit cell.

$6\,\mathrm {faces} \,\times \,{\frac {1}{2}}=3$ $6\,\mathrm {faces} \,\times \,{\frac {1}{2}}=3$
- $\left(0,{\frac {1}{2}},{\frac {1}{2}}\right),\left({\frac {1}{2}},0,{\frac {1}{2}}\right),\left({\frac {1}{2}},{\frac {1}{2}},0\right),\left(1,{\frac {1}{2}},{\frac {1}{2}}\right),\left({\frac {1}{2}},1,{\frac {1}{2}}\right),\left({\frac {1}{2}},{\frac {1}{2}},1\right)$
$12\,\mathrm {edges} \,\times \,{\frac {1}{4}}=3$ $12\,\mathrm {edges} \,\times \,{\frac {1}{4}}=3$
- $\left({\frac {1}{2}},0,0\right),\left(0,{\frac {1}{2}},0\right),\left(0,0,{\frac {1}{2}}\right)$
- $\left(1,0,{\frac {1}{2}}\right),\left(1,1,{\frac {1}{2}}\right),\left(0,1,{\frac {1}{2}}\right),\left({\frac {1}{2}},1,0\right),\left(1,{\frac {1}{2}},0\right),\left({\frac {1}{2}},0,1\right),\left(0,{\frac {1}{2}},1\right),\left(1,{\frac {1}{2}},1\right),\left({\frac {1}{2}},1,1\right)$

@@ Line 1: / Line 1: @@
 '''BCC''' or '''body-centered cubic''' is a cubic [[lattice]] where the symmetry involves an atom/particle sitting in the center of the conceptual [[unit cell]].
+[[Image:Bcc02-unit cell.png|thumb|right|300px|BCC unit cell.]]
 ==Canonical BCC==
 ===Symmetry===
@@ Line 93: / Line 96: @@
 </pre>
 [[Image:Lattice_peaks-BCC.png|450px]]
+===[[Realspace]] Spacings===
+Note that in general:
+:<math>
+\begin{alignat}{2}
+\left( \frac{1}{d_{hkl}} \right )^2 = \left(\frac{h}{a}\right)^2 + \left(\frac{k}{b}\right)^2 + \left(\frac{l}{c}\right)^2
+\end{alignat}
+</math>
+[[Q value|Since]] <math>q=2 \pi / d</math>, and since <math>a=b=c</math>, the realspace spacing of planes is:
+:<math>
+\begin{alignat}{2}
+d_{hkl}
+  & = \frac{a}{\sqrt{ h^2 + k^2 + l^2 }}
+\end{alignat}
+</math>
 ===Examples===
@@ Line 136: / Line 155: @@
 ===Examples===
-====Atomics====
+====Atomic====
 * [http://en.wikipedia.org/wiki/Cesium_chloride Cesium chloride (CsCl)] (''a'' = 4.11 Å)

Difference between revisions of "Lattice:BCC"

Latest revision as of 13:50, 15 October 2014

Contents

Canonical BCC

Symmetry

Structure

Neighbors

Reciprocal-space Peaks

Realspace Spacings

Examples

Elemental

Atomic

Nano

Body-centered Two-particle

Symmetry

Examples

Atomic

Body-centered with Edges and Faces

Particle Positions

Particle A

Particle B

See Also

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