Lattice:BCC - Revision history

KevinYager: /* Examples */

2014-10-15T18:50:32Z

‎Examples

KevinYager at 18:46, 14 October 2014

2014-10-14T18:46:39Z

KevinYager: /* Realspace Spacings */

2014-09-02T18:23:25Z

‎Realspace Spacings

KevinYager: /* Reciprocal-space Peaks */

2014-09-02T18:20:47Z

‎Reciprocal-space Peaks

130.199.3.165 at 14:50, 20 June 2014

2014-06-20T14:50:55Z

KevinYager: /* Canonical BCC */

2014-06-18T14:55:06Z

‎Canonical BCC

KevinYager: Created page with "'''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. ==Canonical B..."

2014-06-18T14:51:06Z

Created page with "'''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. ==Canonical B..."

New page

'''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]].
==Canonical BCC==
===Symmetry===
* Crystal Family: Cubic
* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: [[Lattice:Cubic|Cubic]]
* [http://en.wikipedia.org/wiki/Bravais_lattice#Bravais_lattices_in_3_dimensions Bravais Lattice]: I (bcc)
* Crystal class: Hexoctahedral
* [http://en.wikipedia.org/wiki/Crystallographic_point_group Point Group]: m3m
* Space Group: Im3m
* Particles per unit cell: <math>n=2</math>
* Volume of unit cell: <math>V_d=a^3</math>
* Dimensionality: <math>d=3</math>
* Projected ''d''-dimensional volume: <math>v_d=a^3</math>
* Solid angle: <math>\Omega_d=4\pi</math>
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/2</math>
* Assuming spherical particles of radius ''R'':
** Particle volume fraction: <math>\phi=8 \pi R^3/\left(3a^3\right)</math>
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/8\approx0.680</math> when <math>R=a\sqrt{3}/4</math>

===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.

[[Image:Bcc01.png|300px]] [[Image:Bcc02-unit cell.png|300px]]

The extended lattice can be thought of in terms of two inter-penetrating [[Lattice:SC|simple cubic]] lattices:

[[Image:Bcc03-lattice.png|400px|left]] [[Image:Bcc04-two lattices.png|400px]]

One can also consider the BCC to be a [[FCC_to_BCC|distorted FCC]]:

[[Image:Fcc to bcc075.png|400px]]

===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:
::<math>d_{\mathrm{nn}} = \sqrt{3}/2 \times a \approx 0.866 a</math>
and another 6 "direct" next-nearest-neighbors at a distance of <math>a=(2/ \sqrt{3}) d_{nn}</math>.

[[Image:Bcc05-mini cell.png|300px]] [[Image:Bcc05-mini cell2.png|300px]] [[Image:Bcc05-mini cell3.png|300px]]

===[[Reciprocal-space]] Peaks===
* Allowed reflections:
*: <math>f_{hkl} = 2 \,\,\, \mathrm{for} \,\,\, h+k+l=\mathrm{even}</math>
* Peak multiplicities:
*:: <math>m_{h00}=6</math>
*:: <math>m_{hh0}=12</math>
*:: <math>m_{hhh}=8</math>
*:: <math>m_{hk0}=24</math>
*:: <math>m_{hhk}=24</math>
*:: <math>m_{hkl}=48</math>
* Peak positions:
*: <math>q_{hkl}=\frac{2\pi}{a}\sqrt{h^2+k^2+l^2}</math>
*: For ''a'' = 1.0:
<pre>
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
</pre>
[[Image:Lattice_peaks-BCC.png|450px]]

===Examples===
====Elemental====
: 3. [http://en.wikipedia.org/wiki/Lithium Lithium (Li)] (''a'' = 3.509 Å)
: 11. [http://en.wikipedia.org/wiki/Sodium Sodium (Na)] (''a'' = 4.291 Å)
: 19. [http://en.wikipedia.org/wiki/Potassium Potassium( K)] (''a'' = 5.247 Å) (at 78 K)
: 23. [http://en.wikipedia.org/wiki/Vanadium Vanadium (V)] (''a'' = 3.024 Å)
: 24. [http://en.wikipedia.org/wiki/Chromium Chromium (Cr)] (''a'' = 2.884 Å)
: 26. [http://en.wikipedia.org/wiki/Iron Iron (Fe)] (''a'' = 2.866 Å)
: 37. [http://en.wikipedia.org/wiki/Rubidium Rubidium (Rb)] (''a'' = 5.605 Å) (at 78 K)
: 41. [http://en.wikipedia.org/wiki/Niobium Niobium (Nb)] (''a'' = 3.300 Å)
: 42. [http://en.wikipedia.org/wiki/Molybdenum Molybdenum (Mo)] (''a'' = 3.147 Å)
: 55. [http://en.wikipedia.org/wiki/Caesium Caesium (Cs)] (''a'' = 6.067 Å) (at 78 K)
: 56. [http://en.wikipedia.org/wiki/Barium Barium (Ba)] (''a'' = 5.025 Å)
: 63. [http://en.wikipedia.org/wiki/Europium Europium (Eu)] (''a'' = 4.606 Å)
: 73. [http://en.wikipedia.org/wiki/Tantalum Tantalum (Ta)] (''a'' = 3.306 Å)
: 74. [http://en.wikipedia.org/wiki/Tungsten 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 [[Lattice:Simple Cubic|simple cubic]] symmetry. This lattice is often called '''CsCl'''.

[[Image:Bcc-two particle.png|300px]]

===Symmetry===
* Crystal Family: Cubic
* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: [[Lattice:Cubic|Cubic]]
* [http://en.wikipedia.org/wiki/Bravais_lattice#Bravais_lattices_in_3_dimensions Bravais Lattice]: P (pcc)
* Crystal class: Hexoctahedral
* [http://en.wikipedia.org/wiki/Crystallographic_point_group Point Group]: m3m
* Space Group: Pm3m
* Particles per unit cell: <math>n=2</math> (distinct)
* Volume of unit cell: <math>V_d=a^3</math>
* Dimensionality: <math>d=3</math>
* Projected ''d''-dimensional volume: <math>v_d=a^3</math>
* Solid angle: <math>\Omega_d=4\pi</math>
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/2</math>

===Examples===
====Atomics====
* [http://en.wikipedia.org/wiki/Cesium_chloride 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...)

[[Image:Bcc-faces and edges.png|400px]]

===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:
* <math> 8 \, \mathrm{corners} \, \times \, \frac{1}{8} = 1</math>
** <math>\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)</math>
* <math> 1 \, \mathrm{center} \, \times \, 1 = 1</math>
** <math>\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)</math>

====Particle B====
There are 18 positions, with 6 particles in the unit cell.
* <math> 6 \, \mathrm{faces} \, \times \, \frac{1}{2} = 3</math>
** <math>\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)</math>
* <math> 12 \, \mathrm{edges} \, \times \, \frac{1}{4} = 3</math>
** <math>\left(\frac{1}{2},0,0\right) , \left(0,\frac{1}{2},0\right) , \left(0,0,\frac{1}{2}\right) </math>
** <math>\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) </math>

==See Also==
* [http://en.wikipedia.org/wiki/Cubic_crystal_system Wikipedia: Cubic crystal system]

@@ Line 95: / Line 95: @@
 ===[[Realspace]] Spacings===
-[[Q value|Since]] <math>q=2 \pi / d</math>, the realspace spacing of planes is:
+Note that in general:
 :<math>
 \begin{alignat}{2}
-\mathbf{q}_{hkl}
+\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
-  & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\
+\end{alignat}
-  & = (2 \pi h)\begin{bmatrix} \frac{1}{a} \\ \frac{1}{\sqrt{3}a} \\ 0 \end{bmatrix}
+</math>
-    + (2 \pi k)\begin{bmatrix} 0 \\ \frac{2}{\sqrt{3}b} \\ 0 \end{bmatrix}
-    + (2 \pi l)\begin{bmatrix} 0 \\ 0 \\ \frac{1}{c} \end{bmatrix} \\
+[[Q value|Since]] <math>q=2 \pi / d</math>, and since <math>a=b=c</math>, the realspace spacing of planes is:
-  & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi h}{\sqrt{3}a} + \frac{4 \pi k}{\sqrt{3}b} \\ \frac{2 \pi l}{c} \end{bmatrix}  \\
+:<math>
-   & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi (h + 2 k)}{\sqrt{3}a} \\ \frac{2 \pi l}{c} \end{bmatrix}
+\begin{alignat}{2}
 \end{alignat}
 </math>

@@ Line 155: / Line 155: @@
 ===Examples===
-====Atomics====
+====Atomic====
 * [http://en.wikipedia.org/wiki/Cesium_chloride Cesium chloride (CsCl)] (''a'' = 4.11 Å)

@@ 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]].
 ==Canonical BCC==
 ===Symmetry===

@@ Line 93: / Line 93: @@
 </pre>
 [[Image:Lattice_peaks-BCC.png|450px]]
 ===Examples===

@@ Line 123: / Line 123: @@
 ===Symmetry===
 * Crystal Family: Cubic
-* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: [[Lattice:Cubic|Cubic]]
+* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: Cubic
 * [http://en.wikipedia.org/wiki/Bravais_lattice#Bravais_lattices_in_3_dimensions Bravais Lattice]: P (pcc)
 * Crystal class: Hexoctahedral