Solid State Physics Lecture 1

8/13/2019 Solid State Physics Lecture 1

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Crystal Structure Continued!

NOTE!!Much of the discussion & many figures in what follows was

constructed from lectures posted on the web by Prof. Be!ireGNLin Turkey. She has done an excellent job of covering manydetails of crystallography & she illustrates her topics with

many very

nice pictures of lattice structures. Her lectures on this are posted Here:http://www1.gantep.edu.tr/~bgonul/dersnotlari/.

Her homepage is Here:http://www1.gantep.edu.tr/~bgonul/.


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Crystal Lattices

Bravais Lattices(BL)

Non-Bravais Lattices(non-BL)

All atoms are the same kind

All lattice points are equivalent

Atoms are of different kinds. Somelattice points are not equivalent.

Atoms are of different kinds.Some lattice points arent equivalent.

A combination of 2 or more BL

2 d examples


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In General

Mathematically, a lattice is defined by 3 vectors called

Primitive Lattice Vectors

a1, a2, a3are 3d vectors which depend on the geometry.

Once a1, a2, a3are specified, thePrimitive Lattice Structure

is known.

The infinite lattice is generated by translating through a

Direct Lattice Vector: T = n1a1 + n2a2 + n3a3

n1,n2,n3are integers. T generates the lattice points. Each

lattice point corresponds to a set of integers (n1,n2,n3).

Lattice Translation Vectors


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Consider a 2-dimensional lattice(figure).Let the

2 Dimensional Translation Vectorbe Rn= n1a+ n2b

(Sorry for the notation change!!)

a&bare 2 d Primitive Lattice Vectors, n1, n2are integers.

2 Dimensional Lattice Translation Vectors

Once a & bare specified by thelattice geometry & an origin ischosen, all symmetrically equivalent

points in the lattice are determined bythe translation vector Rn. That is, the

lattice has translational symmetry.Note that the choice of Primitive

Lattice vectors is not unique!So,one could equally well take vectors a& b'as primitive lattice vectors.

Point D(n1, n2) = (0,2)

Point F(n1, n2) = (0,-1)


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The Basis

(or basis set)

!The set of atoms which, when placed at each

lattice point, generates the Crystal Structure.

Crystal Structure!

Primitive Lattice + BasisTranslate the basis through all possible lattice vectorsT = n1a1 + n2a2 + n3a3

to get theCrystal Structure

or the

DIRECT LATTICE


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The periodic lattice symmetry is such that theatomic arrangement looks the same from an arbitrary

vector positionras when viewed from the point

r' = r + T (1)

where Tis the translation vector for the lattice:

T = n1a1 + n2a2 + n3a3

Mathematically, the lattice & the vectorsa1,a2,a3are

Primitive if any 2 points r& r'always satisfy (1)with a

suitable choice of integers n1,n2,n3.


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In 3 dimensions, no 2 of the 3 primitive lattice vectorsa1,a2,a3can be along the same line. But,

Dont think of a1,a2,a3as a mutually orthogonal set!

Usually, they are neither mutually perpendicular nor all

the same length!

For examples, see Fig. 3a (2 dimensions):


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The Primitive Lattice Vectorsa1,a2,a3

arent necessarily a mutually orthogonal set!Usually

Usually, they are neither mutually perpendicular nor all

the same length!


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Bravais Lattice !

An infinite array of discrete points with an arrangement& orientation that appears exactly the same, from

whichever of the points the array is viewed.

A Bravais Latticeis invariant under a translation

T = n1a1 + n2a2 + n3a3

Crystal Lattice Types

Nb film


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Non-Bravais LatticesIn a Bravais Lattice,not only the atomic arrangement

but also theorientationsmust appear exactly the samefrom every lattice point.

2 Dimensional Honeycomb Lattice The red dots each have a neighbor to

the immediate left. The blue dot has aneighbor to its right.

The red (& blue) sidesareequivalent& have the sameappearance. But,the red &blue dotsare not equivalent.

If the blue side is rotated through 180the lattice is invariant.

"The Honeycomb Lattice is NOT

a Bravais Lattice!! HoneycombLattice


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It can be shown that, in 2 Dimensions, there are Five(5) & ONLY FiveBravais Lattices!


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2-Dimensional Unit Cells

Unit Cell!

The smallest componentof the crystal (group of

atoms, ions or molecules),which, when stacked together withpure translational repetition, reproduces the whole crystal.

2D-CrystalS

ab

S

S

S

S

S

S

S

S

S

S

S

S

S

S


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Unit Cell !The smallest componentof 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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Artificial Example: NaCl

Lattice pointsare 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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These are also unit cells -

it doesnt matter if the origin is atNa orCl !


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These are also unit cells -

the origin does not have to be on an atom!


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These areNOTunit cells - empty space is not allowed!


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In 2 dimensions, these areunit cellsin 3 dimensions, they would not be.


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Why can't the blue triangle be a unit cell?


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Example: 2 Dimensional, Periodic Art!

A Painting by Dutch Artist Maurits Cornelis Escher (1898-1972)

Escher was famous for his so

called impossible

structures,such as

Ascending & Descending,

Relativity, ..

Can you find the Unit Cell in this painting?


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3-Dimensional

Unit Cells


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3-Dimensional Unit Cells3 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

Unt Cell Types

Primitive

A single lattice point per cellThe smallest area in 2 dimensions, orThe smallest volume in 3 dimensions

Simple Cubic (sc)Conventional Cell= Primitive cell

More than one lattice point per cellVolume (area) = integer multiple of

that for primitive cell

Conventional(Non-primitive)

Body Centered Cubic (bcc)Conventional Cell"Primitive cell


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Face Centered Cubic (FCC) Structure


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Conventional Unit Cells

A Conventional Unit Celljust fills space when

translated through a subset of Bravais lattice vectors.

The conventional unit cell is 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.

The full cubeis the

Conventional Unit Cellfor the FCC Lattice

FCC Bravais Lattice


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Face Centered Cubic Lattice

Primitive Lattice Vectors

a1 = (!)a(0,1,0)

a2 = (!)a(1,0,1)

a3 = (!)a(1,1,0)

Note that theais are

NOT Mutually

Orthogonal!

Conventional Unit Cell(Full Cube)

Primitive Unit Cell(Shaded)

LatticeConstant


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Elements That Form Solidswith the FCC Structure


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Body Centered Cubic (BCC) Structure


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Body Centered Cubic Lattice

Primitive Lattice Vectorsa1 = (!)a(1,1,-1)

a2 = (!)a(-1,1,1)

a3 = (!)a(1,-1,1)Note that theais are

NOT mutually

orthogonal!

Primitive Unit Cell

Lattice

Constant

Conventional Unit Cell(Full Cube)


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Elements That Form Solidswith the BCC Structure


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Conventional & Primitive Unit CellsCubic Lattices

Simple Cubic (SC)Primitive Cell =Conventional CellFractional 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 inconventional cell:

000,100, 010, 001, 110,101, 011111, ###

Primitive Cell =Rombohedron


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Conventional & Primitive Unit CellsCubic Lattices

Face Centered Cubic (FCC)Primitive Cell #Conventional Cell

Fractional coordinates of lattice points in conventional cell:000,100, 010, 001, 110,101, 011,111, ##0, #0 #, 0 ##

#1 #, 1 ##, ##1


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Simple Hexagonal Bravais Lattice


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Hexagonal Bravais LatticePrimitive Cell =Conventional Cell

Fractional coordinates of lattice

points in conventional cell:100, 010, 110, 101, 011111, 000, 001

Points of Primitive Cell

12

0o


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

The HCP latticeis not a Bravais

lattice,because the orientation ofthe 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, isUNIQUELY

determined by6 constants(figure):a, b, c, $, %and &

which depend on lattice geometry.

As well see, we sometimes want tocalculate the number of atoms in aunit cell. To do this, imagine stackinghard spheres centered at each lattice point& just touching each neighboring sphere.Then, for the cubic lattices, only1/8 of

each lattice point in a unit cell assignedto that cell. In the cubic lattice in thefigure, each unit cell is associatedwith (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 theparallelepiped formed by thePrimitive Vectorsa1,a2, & a3

such that there is no cell ofsmaller volumethat can be used

as a building block for thecrystal structure.

As weve discussed, a Primitive

Unit Cellcan be repeated to fill

space by periodic repetition of it

through the translation vectors

T = n1a1 + n2a2 + n3a3.

The Primitive Unit

Cellvolume 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 Note that,by definition,the Primitive Unit Cell must

contain ONLY ONE lattice point. There can be different choices for thePrimitive Lattice

Vectors,but the Primitive Cell volumemust be independent

of that choice.

A 2 DimensionalExample!

P = Primitive Unit CellNP = Non-Primitive Unit Cell

Solid State Physics Lecture 1

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