Unit Cell in 3D - University of Babylon · Wigner-Seitz Method A simply way to find the primitive...
Transcript of Unit Cell in 3D - University of Babylon · Wigner-Seitz Method A simply way to find the primitive...
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Unit Cell in 3D
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Unit Cell in 3D
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Three common Unit Cell in 3D
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UNIT CELL
Primitive Conventional & Non-primitive
Single lattice point per cell
Smallest area in 2D, or
Smallest volume in 3D
More than one lattice point per cell
Integral multibles of the area of
primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
Simple cubic(sc)
Conventional = Primitive cell
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The Conventional Unit Cell
•- A unit cell just fills space when
translated through a subset of
Bravais lattice vectors.
•-The conventional unit cell is
chosen to be larger than the
primitive cell, but with the full
symmetry of the Bravais lattice.
•- The size of the conventional
cell is given by the lattice
constant a.
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Primitive and conventional cells of FCC
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1
2
3
1ˆ ˆ ˆ( )
2
1ˆ ˆ ˆ( )
2
1ˆ ˆ ˆ( )
2
a x y z
a x y z
a x y z
Primitive and conventional cells of BCC
Primitive Translation Vectors:
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a
b c
Simple cubic (sc):
primitive cell=conventional cell
Fractional coordinates of lattice points:
000, 100, 010, 001, 110,101, 011, 111
Primitive and conventional cells
Body centered cubic (bcc):
conventional primitive cell
a
b cFractional coordinates of lattice points in
conventional cell:
000,100, 010, 001, 110,101, 011, 111, ½ ½ ½
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Body centered cubic (bcc):
primitive (rombohedron) conventional cell
a
b
c
Fractional coordinates: 000, 100, 101, 110, 110,101, 011, 211, 200
Face centered cubic (fcc):
conventional primitive cell
a
bc
Fractional coordinates: 000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½
0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1
Primitive and conventional cells
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Hexagonal close packed cell (hcp):
conventional primitive cell
Fractional coordinates:
100, 010, 110, 101,011, 111,000, 001
points of primitive cell
a
b
c
Primitive and conventional cells-hcp
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•The unit cell and, consequently, the entire lattice, is uniquely determined by the six lattice constants: a, b, c, α, β and γ.
•Only 1/8 of each lattice point in a unit cell can actually be assigned to that cell.
•Each unit cell in the figure can be associated with 8 x 1/8 = 1 lattice point.
Unit Cell
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•- A primitive unit cell is made of primitive translation vectors a1 ,a2, and a3 such
smaller volume there is no cell ofthat that can be used as a building block for crystal structures.
•- A primitive unit cell will fill space by repetition of suitable crystal translation vectors. This defined by the parallelpiped a1, a2 and a3. The volume of a primitive unit cell can be found by
•V = a1.(a2 x a3) (vector products)
Cubic cell volume = a3
Primitive Unit Cell and vectors
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The primitive unit cell must have only one lattice point.
There can be different choices for lattice vectors , but the
volumes of these primitive cells are all the same.
P = Primitive Unit Cell
NP = Non-Primitive Unit Cell
Primitive Unit Cell
1a
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Wigner-Seitz Method
A simply way to find the primitive
cell which is called Wigner-Seitz
cell can be done as follows;
1- Choose a lattice point.
2- Draw lines to connect these
lattice point to its neighbours.
3-At the mid-point and normal to
these lines draw new lines.
The volume enclosed is called as a
Wigner-Seitz cell. 49
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Wigner-Seitz Cell - 3D
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( wigner - seitz) زاٌتس -خلٌة وٌغنر
:ويتم تحديدها وفقا لما يليوهً خلٌة حجمها ٌساوي إلى حجم الخلٌة البدائٌة
نختار عقدة ما من عقد الشبكة البلورٌة ثم نصل بٌن هذه العقدة والقد المجاورة لهل مباشرة بخطوط 1.
.مستقٌمة
.نرسم من منتصفات هذه الخطوط مستوٌات عمودٌة علٌها 2.
(انظر الشكل.)زاٌتس – وٌغنرٌكون اصغر حجم بٌن هذه المستوٌات المنصفة هو خلٌة 3.