handout4 - Handout 4 Lattices in 1D, 2D, and 3D In this...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 4 Lattices in 1D, 2D, and 3D In this lecture you will learn: • Bravais lattices • Primitive lattice vectors • Unit cells and primitive cells • Lattices with basis and basis vectors August Bravais (1811-1863) ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice A fundamental concept in the description of crystalline solids is that of a “Bravais lattice”. A Bravais lattice is an infinite arrangement of points (or atoms) in space that has the following property: The lattice looks exactly the same when viewed from any lattice point A 1D Bravais lattice: b A 2D Bravais lattice: b c
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice A 2D Bravais lattice: A 3D Bravais lattice: b d c ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice A Bravais lattice has the following property: The position vector of all points (or atoms) in the lattice can be written as follows: 2 1 a m a n R r r r + = 3 2 1 a p a m a n R r r r r + + = 1 a n R r r = 1D 2D 3D Where n , m , p = 0, ±1, ±2, ±3, ……. And the vectors, are called the “primitive lattice vectors” and are said to span the lattice. These vectors are not parallel. 3 2 1 and , , a a a r r r Example (1D): x b a ˆ 1 = r Example (2D): b c x b a ˆ 1 = r y c a ˆ 2 = r x y b
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice b d c Example (3D): x b a ˆ 1 = r y c a ˆ 2 = r z d a ˆ 3 = r The choice of primitive vectors is NOT unique: b c x b a ˆ 1 = r y c a ˆ 2 = r x b a ˆ 1 = r y c x b a ˆ ˆ 2 + = r All sets of primitive vectors shown will work for the 2D lattice ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice All lattices are not Bravais lattices: Example (2D): The honeycomb lattice
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4 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Wigner-Seitz Primitive Cell • The Wigner-Seitz primitive cell of a Bravais lattice is a region in space around a lattice point that consists of all points in space that are closer to this lattice
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handout4 - Handout 4 Lattices in 1D, 2D, and 3D In this...

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