GY 302: Crystallography & Mineralogy GY 302: Crystallography & Mineralogy
ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic...
Transcript of ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic...
![Page 1: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/1.jpg)
•A basic knowledge of crystallography is essentialfor solid state physicists;
–- to specify any crystal structure and-–- to classify the solids into different types according
to the symmetries they possess.
•Symmetry of a crystal can have a profoundinfluence on its properties.
•We will concern in this course with solids withsimple structures.
ELEMENTARYCRYSTALLOGRAPHY
13
![Page 2: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/2.jpg)
CRYSTAL LATTICE
What is crystal (space) lattice?In crystallography, only the geometrical properties of thecrystal are of interest, therefore one replaces each atom bya geometrical point located at the equilibrium position ofthat atom.
Platinum Platinum surface Crystal lattice andstructure of Platinum(scanning tunneling microscope)
14
![Page 3: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/3.jpg)
•- An infinite array ofpoints in space,
•- Each point hasidenticalsurroundings to allothers.
•- Arrays arearranged exactly ina periodic manner.
Crystal Lattice
α
a
bCB ED
O A
y
x
15
![Page 4: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/4.jpg)
Crystal Structure•Crystal structure can be obtained by attaching atoms,
groups of atoms or molecules which are called basis(motif) to the lattice sides of the lattice point.
Crystal Structure = Crystal Lattice + Basis
16
![Page 5: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/5.jpg)
A two-dimensional Bravais lattice withdifferent choices for the basis
17
![Page 6: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/6.jpg)
E
HO A
CB
Fb G
D
x
y
a
α
a
bCB ED
O A
y
x
b) Crystal lattice obtained byidentifying all the atoms in (a)
a) Situation of atoms at thecorners of regular hexagons
Basis A group of atoms which describe crystal structureA group of atoms which describe crystal structure
18
![Page 7: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/7.jpg)
Crystal structure
•- Don't mix up atoms withlattice points
•- Lattice points areinfinitesimal points in space
•- Lattice points do notnecessarily lie at thecentre of atoms
Crystal Structure = Crystal Lattice + Basis
19
![Page 8: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/8.jpg)
Crystal Structure 8
Crystal Lattice
Bravais Lattice (BL) Non-Bravais Lattice (non-BL)
All atoms are of the same kind All lattice points are equivalent
Atoms can be of different kind Some lattice points are not
equivalentA combination of two or more BL
20
![Page 9: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/9.jpg)
Types Of Crystal Lattices
is an infinite array of discrete points withBravais lattice1)an arrangement and orientation that appears exactlythe same, from whichever of the points the array isviewed. Lattice is invariant under a translation
Nb film
21
![Page 10: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/10.jpg)
Types Of Crystal Lattices
•- The red side has a neighbour to itsimmediate left, the blue one insteadhas a neighbour to its right.
•- Red (and blue) sides are equivalentand have the same appearance
•-Red and blue sides are not equivalent.Same appearance can be obtainedrotating blue side 180º.
2)2) NonNon--Bravais LatticeBravais LatticeNot only the arrangement but also the orientation mustappear exactly the same from every point in a bravais lattice.
Honeycomb
22
![Page 11: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/11.jpg)
Translational Lattice Vectors – 2D
A space lattice is a set of points suchthat a translation from any point inthe lattice by a vector;
Rn = n1 a + n2 blocates an exactly equivalent point,i.e. a point with the same environmentas P . This is translational symmetry.The vectors a, b are known as latticevectors and (n1, n2) is a pair ofintegers whose valuesdepend on the lattice point.
P
Point D(n1, n2) = (0,2)Point F (n1, n2) = (0,-1)
23
![Page 12: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/12.jpg)
•bandtwo vectors aThe-form a set of lattice
vectors for the lattice.•- The choice of lattice
vectors is not unique. - Thusone could equally well takethe vectors a and b’ as a latticevectors.
Lattice Vectors – 2D
24
![Page 13: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/13.jpg)
Lattice Vectors – 3DAn ideal three dimensional crystal is described by 3fundamental translation vectors a, b and c. If there isa lattice point represented by the position vector r,there is then also a lattice point represented by theposition vector where u, v and w are arbitraryintegers.
r’ = r + u a + v b + w c (1)
25
![Page 14: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/14.jpg)
Five Bravais Lattices in 2D
26
![Page 15: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/15.jpg)
Unit Cell in 2D
•The smallest component of the crystal (group of atoms, ions ormolecules), which when stacked together with pure
translational repetition reproduces the whole crystal.
Sa
b
S
S
S
S
S
S
S
S
S
S
S
S
S
S
27
![Page 16: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/16.jpg)
Unit Cell in 2D
•The smallest component of the crystal (group ofwhen stackedatoms, ions or molecules), which
with pure translational repetitiontogetherreproduces the whole crystal.
S
S
The choice ofunit cell
is not unique.
a
Sb
S
28
![Page 17: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/17.jpg)
2D Unit Cell example -(NaCl)
We define lattice points ; these are points with identicalenvironments
29
![Page 18: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/18.jpg)
Choice of origin is arbitrary - lattice pointsunit cell size shouldbut-need not be atoms
.always be the same
30
![Page 19: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/19.jpg)
This is also a unit cell -it doesn’t matter if you start from Na or Cl
31
![Page 20: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/20.jpg)
- or if you don’t start from an atom
32
![Page 21: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/21.jpg)
even though they are all theNOT a unit cellThis issame - empty space is not allowed!
33
![Page 22: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/22.jpg)
In 2D, this IS a unit cellIn 3D, it is NOT
34
![Page 23: ELEMENTARY CRYSTALLOGRAPHY - University of Babylon › eprints › paper_11_24458_28.pdf · A basic knowledge of crystallography is essential "for solid state physicists; - to specify](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f1058637e708231d448a67d/html5/thumbnails/23.jpg)
Q : Why can't the blue triangle be a unit cell?
35