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**Solid State Physics (1) Phys3710**

Crystal structure 2 Lecture 2 Department of Physics Dr Mazen Alshaaer Second semester 2013/2014 Ref.: Prof. Charles W. Myles, Department of Physics, Texas Tech University

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Crystal structures Crystal: atoms are arranged so that their positions are periodic in all three dimensions Atoms are bound to one another → well defined equilibrium separations; many identical atoms → minimum energy requires every identical atom to be in identical environment → 3D periodicity Ideal crystal: perfect periodicity Real crystals are never perfect: •surface •impurities and defects •thermal motion of atoms (lattice vibrations)

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Basic definitions The periodic array of points is called crystal lattice. For every lattice point there is a group of atoms (or single atom) called basis of the lattice Don't confuse with a1, a2, a3 - basis vectors parallelogram formed by the basis vectors – unit cell if a unit cell contains only one lattice point, it is called a primitive cell (minimum volume) Bravais lattices – all lattice points are equivalent

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Crystal Structure

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Bravais Lattice (1) 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: c b

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Bravais Lattice (2) d c b

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**Bravais Lattice (3) A Bravais lattice has the following property:**

The position vector of all points (or atoms) in the lattice can be written as follows (Translational Lattice Vectors) : Example (1D): Example (2D):

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Bravais Lattice (4)

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Bravais Lattice (5)

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Non-Bravais Lattice Not only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice. Honeycomb The red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right. Red (and blue) sides are equivalent and have the same appearance Red and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º.

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**Five Bravais Lattices in 2D**

General or Oblique Rectangular Centered rectangular Square Hexagonal

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**aren’t necessarily a mutually orthogonal set!**

The Primitive Lattice Vectors a1,a2,a3 aren’t necessarily a mutually orthogonal set! Usually Usually, they are neither mutually perpendicular nor all the same length! For examples, see Fig. 3b (3 dimensions):

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**2-Dimensional Unit Cells**

Unit Cell The smallest component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. S 2D-Crystal a b S S Unit Cell S S S S S S S S S S S S

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**The choice of unit cell is not unique!**

Unit Cell The smallest component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. The choice of unit cell is not unique! 2D-Crystal S S S

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**2-Dimensional Unit Cells**

Artificial Example: “NaCl” Lattice points are points with identical environments.

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**2-Dimensional Unit Cells: “NaCl”**

The choice of origin is arbitrary - lattice points need not be atoms - but the unit cell size must always be the same.

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**2-Dimensional Unit Cells: “NaCl”**

These are also unit cells it doesn’t matter if the origin is at Na or Cl !

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**2-Dimensional Unit Cells: “NaCl”**

These are also unit cells the origin does not have to be on an atom!

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**2-Dimensional Unit Cells: “NaCl”**

These are NOT unit cells - empty space is not allowed!

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**2-Dimensional Unit Cells: “NaCl”**

In 2 dimensions, these are unit cells – in 3 dimensions, they would not be.

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**2-Dimensional Unit Cells Why can't the blue triangle be a unit cell?**

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3-Dimensional Unit Cells

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**3-Dimensional Unit Cells 3 Common Unit Cells with Cubic Symmetry**

Simple Cubic Body Centered Cubic Face Centered Cubic (SC) (BCC) (FCC)

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**Conventional & Primitive Unit Cells**

Simple Cubic (sc) Conventional Cell = Primitive cell Body Centered Cubic (bcc) Conventional Cell ≠ Primitive cell

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Bravais Lattices in 3D There are 14 different Bravais lattices in 3D that are classified into 7 different crystal systems (only the unit cells are shown below)

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**Simple Cubic (SC) Structure**

Simple Cubic Lattice: Unit Cell: It is very cumbersome to draw entire lattices in 3D so some small portion of the lattice, having full symmetry of the lattice, is usually drawn. This small portion when repeated can generate the whole lattice and is called the “unit cell” and it could be larger than the primitive cell

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**Face Centered Cubic (FCC) Structure**

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**Conventional & Primitive Unit Cells Face Centered Cubic Lattice**

(Shaded) Primitive Lattice Vectors a1 = (½)a(0,1,0) a2 = (½)a(1,0,1) a3 = (½)a(1,1,0) Lattice Constant Conventional Unit Cell (Full Cube)

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**Elements That Form Solids with the FCC Structure**

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**Body Centered Cubic (BCC) Structure**

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**Conventional & Primitive Unit Cells Body Centered Cubic Lattice**

Primitive Lattice Vectors a1 = (½)a(1,1,-1) a2 = (½)a(-1,1,1) a3 = (½)a(1,-1,1) Note that the ai’s are NOT mutually orthogonal! Primitive Unit Cell Lattice Constant Conventional Unit Cell (Full Cube)

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**Elements That Form Solids with the BCC Structure**

Note: This was the end of lecture 1

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**Conventional & Primitive Unit Cells Cubic Lattices**

Simple Cubic (SC) Primitive Cell = Conventional Cell Fractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111 Body Centered Cubic (BCC) Primitive Cell Conventional Cell Fractional coordinates of lattice points in conventional cell: ,100, 010, 001, 110,101, , ½ ½ ½

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**Conventional & Primitive Unit Cells Cubic Lattices**

Face Centered Cubic (FCC) Primitive Cell Conventional Cell Fractional coordinates of lattice points in conventional cell: ,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ½1 ½ , 1 ½ ½ , ½ ½ 1

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**Simple Hexagonal Bravais Lattice**

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**Conventional & Primitive Unit Cells Hexagonal Bravais Lattice**

Points of Primitive Cell 120o Hexagonal Bravais Lattice Primitive Cell = Conventional Cell Fractional coordinates of lattice points in conventional cell: , 010, 110, 101, , 000, 001

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**Hexagonal Close Packed (HCP) Structure: (A Simple Hexagonal Bravais Lattice with a 2 Atom Basis)**

The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.

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**General Unit Cell Discussion**

For any lattice, the unit cell &, thus, the entire lattice, is UNIQUELY determined by 6 constants (figure): a, b, c, α, β and γ which depend on lattice geometry. As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only 1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure, each unit cell is associated with (8) (1/8) = 1 lattice point.

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**Primitive Unit Cells & Primitive Lattice Vectors**

In general, a Primitive Unit Cell is determined by the parallelepiped formed by the Primitive Vectors a1 ,a2, & a3 such that there is no cell of smaller volume that can be used as a building block for the crystal structure. As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectors T = n1a1 + n2a2 + n3a3. The Primitive Unit Cell volume can be found by vector manipulation: V = a1(a2 a3) For the cubic unit cell in the figure, V = a3

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**Primitive Unit Cells A 2 Dimensional Example!**

Note that, by definition, the Primitive Unit Cell must contain ONLY ONE lattice point. There can be different choices for the Primitive Lattice Vectors, but the Primitive Cell volume must be independent of that choice. A 2 Dimensional Example! P = Primitive Unit Cell NP = Non-Primitive Unit Cell

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