solid state physics lecture 1
TRANSCRIPT
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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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2-Dimensional Unit Cells
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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2-Dimensional Unit Cells: NaCl
These are also unit cells -
it doesnt matter if the origin is atNa orCl !
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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 areNOTunit cells - empty space is not allowed!
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2-Dimensional Unit Cells: NaCl
In 2 dimensions, these areunit cellsin 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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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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Conventional & Primitive Unit Cells
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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Conventional & Primitive Unit Cells
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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Conventional & Primitive Unit Cells
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