Point Lattices: Bravais Lattices - MIT

6.730 Physics for Solid State Applications

Lecture 6: Periodic Structures

Tuesday February 17, 2004

Outline

• Point Lattices
•Crystal Structure= Lattice + Basis
•Fourier Transform Review
•1D Periodic Crystal Structures: Mathematics

6.730 Spring Term 2004
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Point Lattices: Bravais Lattices

Bravais lattices are point lattices that are classified topologically
according to the symmetry properties under rotation and reflection,
without regard to the absolute length of the unit vectors.

A more intuitive definition:
At every point in a Bravais lattice the “world” looks the same.

1D: Only one Bravais Lattice

-2a -a 0 a 2a 3a

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2D Bravais Lattices

hexagonal

rectangular

square

oblique

6.730 centered rectangular
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3D: 14 Bravais Lattices

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Lattice and Primitive Lattice Vectors

A Lattice is a regular array of points {Rl } in space which must
satisfy (in three dimensions)

The vectors ai are know as the primitive lattice vectors.

A two dimensional lattice with different possible choices of primitive lattice vectors.

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Cubic

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PSSA http://cst-www.nrl.navy.mil/lattice/struk.picts/a_h.s.png

Body Centered Cubic

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PSSA http://cst-www.nrl.navy.mil/lattice/struk.picts/a2.s.png

FACE CENTERED CUBIC

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PSSA http://cst-www.nrl.navy.mil/lattice/struk.picts/a1.s.png

Wigner-Sietz Cell

1. Choose one point as the origin and draw lines from the origin to each of
the other lattice points.

2. Bisect each of the drawn lines with planes normal to the line.

3. Mark the space about the origin bounded by the first set of planes
that are encountered. The bounded space is the Wigner-Sietz unit cell.

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Wigner-Sietz Cell

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Wigner-Sietz Cell

FCC BCC

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Crystal Structure = Lattice + Basis

6.730 A two dimensional lattice with a basis of three atoms
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Cubic Structure with atomic
basis

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Diamond Crystal Structure

6.730 http://cst-www.nrl.navy.mil/lattice
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Zincblende Crystal Structure

6.730 http://cst-www.nrl.navy.mil/lattice
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Diamond Crystal Structure: Silicon

Bond angle = 109.5º

6.730 • 2 atom basis to a FCC Lattice
PSSA • Tetrahedral bond arrangement
• Each atom has 4 nearest neighbors and

12 next nearest neighbors

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Perovskite and Related
Structures

6.730 Spring Term 2004 YBa2Cu3O7-x
PSSA High-Tc Structure

Fourier Transform Review

EE-convention for Fourier Transforms

S(t) S(f)

(1) (1) (1) (1) (1)

t 2 1 0 1 f
T T T
-2T -T 0 T 2T 2
T

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Physics Convention for Fourier Transforms

In general,

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1D Periodic Crystal Structures

Sa(x) Sa(q)

(1) (1) (1) (1) (1)

x 4π 2π 0 q
a a
-2a -a 0 a 2a 2π 4π
aa

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1D Periodic Crystal Structures

M(x) Sa(x) Ma(x)

(1) (1) (1) (1) (1)

=x x x

-2a -a 0 a 2a -2a -a 0 a 2a -a 0 a 2a

M(q) Sa(q) Ma(q)

4π 2π 0 2π = 4π 2π 0 2π 4πq 4π 2π 0 2π
a a a a a a a a
a a

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Reciprocal Lattice Vectors

1. The Fourier transform in q-space is also a lattice
2. This lattice is called the reciprocal lattice
3. The lattice constant is 2 π / a
4. The Reciprocal Lattice Vectors are

K-2 K-1 K0 K1 q
K2

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Periodic Function as a Fourier Series

Recall that a periodic function and its transform are

Define then the above is a Fourier Series:

and the equivalent Fourier transform is

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1D Periodic Crystal Structures with a basis

M(x) Sa(x) Ma(x)

(1) (1) (1) (1) (1) d1 d2

=x x x

-2a -a 0 a 2a -2a -a 0 a 2a -a 0 a 2a

M(q) Sa(q) Ma(q)

4π 2π 0 2π = 4π 2π 0 2π 4πq 4π 2π 0 2π
a a a a a a a a
a a

Delta functions still only occur where the reciprocal lattice is, independent of

6.7t3h0e basis. But the basis determines the weights.

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Atomic Form Factors &
Geometrical Structure Factors

Ma(x) Different Atoms (1) and (2)

d1 d2

x

-a 0 a 2a

Ma(x) Same atoms at d1 and d2

d1 d2 Spring Term 2004
x

-a 0 a 2a

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