Order 1238142: Condensed matter

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Close-packed structures

Greengrocers all over the

world know that simple cubic

is not a good idea – close packed

stacking good idea

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Close packed structures: HCP and FCC

Hexagonal close-packed

Face-centred-cubic:

Both structures have

hexagonal planes for

views along certain

crystallographic directions

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The fcc structure

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Close-packed structures: fcc and hcp

hcp

ABABAB...

fcc

ABCABCABC...

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Close-packed structures: fcc and hcp

hcp

ABABAB...

fcc

ABCABCABC...

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Close-packed structures: fcc and hcp

hcp

ABABAB...

fcc

ABCABCABC...

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Close-packed structures: fcc and hcp

hcp

ABABAB...

fcc

ABCABCABC...

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Close-packed structures: fcc and hcp

hcp

ABABAB...

fcc

ABCABCABC...

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Close-packed structures: fcc and hcp

hcp

ABABAB...

fcc

ABCABCABC...

• The hexagonal close-packed (hcp) and face-centred cubic (fcc) and structure have the same packing fraction

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Close-packed structures

• Close-packed structures are found for inert solids and for metals.

• For metals, the conduction electrons are smeared out and directional bonding is not important. Close-packed

structures have a big overlap of the wave functions.

• Most elements crystallize as hcp (36) or fcc (24).

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Non close-packed structures

• covalent materials (bond direction more important than packing)

diamond (only

34 % packing) graphite graphene

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The reciprocal lattice

This is now formal and may be difficult to understand what it

means but it is central to the whole solid state physics.

Ch5 Ashcroft and Mermin

Important physics of waves in solids (vibrational and electron)

is best described in reciprocal space.

Consider a set of points R constituting a Bravais lattice and a

Plane wave:

For a general k such a wave will not have the periodicity of the

Bravais lattice, but for certain special choices of wave vector it will.

The set of all wave vectors G that yield plane waves with the

periodicity of a given Bravais lattice is known as its reciprocal

lattice.

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The reciprocal lattice

Analytically, G belongs to the reciprocal lattice of a Bravias lattice of points

That is, the reciprocal lattice is defined as the set of vectors G

for which

or

The reciprocal lattice is also a Bravais lattice

provided for any r and all R in the

Bravais lattice

(l is an integer)

*Note: instead of G, sometimes K is used.

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The reciprocal lattice

construction of the reciprocal lattice

with this it is easy to see why

a useful relation is

Note: the Bravais lattice that determines the given reciprocal

lattice is often referred to as the direct lattice

Kronecker ij =1 for i=j, 0 otherwise

Volume of unit cell in

real space

The reciprocal of the reciprocal lattice

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Since the reciprocal lattice is itself a Bravais lattice,

we can construct its reciprocal lattice.

Its reciprocal lattice is just the original direct lattice

If the v is the volume of the primitive cell in direct space, then

the primitive cell of the reciprocal lattice has a volume (2)3/v

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The reciprocal lattice

example 1: in two dimensions

|a1|=a

|a2|=b

|b2|=2π/b

|b1|=2π/a

The first Brillouin zone

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The Wigner-Seitz primitive cell of the reciprocal lattice is known

as the first Brillouin zone (BZ).

In practice only “first Brillouin zone” is only

applied to k-space (reciprocal space).

Because the reciprocal of the

bcc lattice is fcc, the first BZ of

the bcc lattice is just the fcc Wigner-Seitz cell.

And since the reciprocal of the

fcc lattice is bcc, the first BZ of

the fcc lattice is the bcc Wigner-Seitz cell.

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The reciprocal lattice in 3D

example 2: in three dimensions bcc and fcc lattice

The fcc lattice is the reciprocal of the bcc lattice and

vice versa.

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Find reciprocal lattice of the bcc direct lattice

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The reciprocal lattice in 3D

The reciprocal to a simple hexagonal Bravais lattice with lattice constants

a and c is another simple hexagonal lattice with lattice constants 4/(3 a) and 2/c rotated through 30o about the c-axis with respect to the direct lattice.

Homework for Advanced – show this is the case

Lattice Planes

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A lattice plane (or crystal plane) is a plane containing at least three

noncolinear, and therefore infinite number of points of a lattice

A family of lattice planes is an infinite set of equally separated lattice planes

which taken together contain all points in the lattice

Advanced:

see

Ashcroft &

Mermin Ch5,

p90,91 for

proof

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Lattice Planes

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For any family of lattice planes separated by a

distance d, there are reciprocal lattice vectors

perpendicular to the planes, the shortest of which

has a length of G=2/d.

Conversely, for any reciprocal lattice vector G, there

is a family of lattice planes normal to G and separated

by a distance d, where 2/d is the length of the shortest reciprocal lattice vector parallel to G

Advanced: see Ashcroft &

Mermin Ch5, p90,91 for proof

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Labelling crystal planes (Miller indices)

1. determine the intercepts

with the axes in units of the

lattice vectors

2. take the reciprocal of

each number

3. reduce the numbers to

the smallest set of integers

having the same ratio.

These are then called the

Miller indices.

step 1: (2,1,2)

step 2: ((1/2),1,(1/2))

step 3: (1,2,1)

Describes the orientation of a plane by giving a vector normal to the plane

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Examples

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More examples

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Example

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Another example

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; including(100), (010), (001) - - -

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- - -

Homework – list the directions belonging to <111>