Crystal Structures Crystal is constructed by the continuous repetition in space of an identical...
-
Upload
franklin-johns -
Category
Documents
-
view
217 -
download
0
Transcript of Crystal Structures Crystal is constructed by the continuous repetition in space of an identical...
Crystal Structures
Crystal is constructed by the continuous repetition in space of an identical structural unit.
Lattice: a periodic array of mathematical points
Basis: periodically repeated arrangement of a set of atoms, ions, or molecules
lattice basis
lattice + basis = unit cell
repeated by translations to cover the whole crystal
cubic tetragonal orthorhombic hexagonal
rhombohedral monoclinic triclinic
The Bravais Lattices
14 different types (translational symmetry) 7 crystal systems (type of conventional unit cell)
Bravais lattice: arrangement of structureless points so thateach point bears the same geometrical relation to all other points.
cubic (simple)rhombohedral
tetragonal (simple)hexagonal
orthorhombicmonoclinic
7 crystals systems and 14 Bravais lattices
triclinic
body centered
face centered
base centered
base centered
b-c
b-c f-c
a
aa
aa
a
a
aa
simple cubic body-centered Cubic face-centered cubic
Number of Bravais lattices Conditions
simplebody-centered cubic face-centered cubic
a1 = a2 = a3
= = = 90
Cubic
Number of Bravais lattices Conditions
simplebody-centered
a1 = a2 ≠ a3
= = = 90
c
aa
c
aa
simple tetragonal body-centered tetragonal
Tetragonal
Number of Bravais lattices Conditions
simplebody-centeredface-centeredbase-centered
a1 ≠ a2 ≠ a3
= = = 90
c
ba
c
ba
simple body-centered face-centered
c
ba
c
ba
base-centered
Orthorhombic:
Number of Bravais lattices Conditions
simple = = 90
= 120
simple hexagonal
Hexagonal
a1 = a2 ≠ a3
simple rhombohedral
Number of Bravais lattices Conditions
simple a1 = a2 = a3
120 90
Rhombohedral
Number of Bravais lattices Conditions
simplebase-centered
a1 ≠ a2 ≠ a3
90
simple monoclinic base-centered monoclinic
Monoclinic
4 rectangular faces and 2 parallelogram faces
Number of Bravais lattices Conditions
simple a1 ≠ a2 ≠ a3
simple triclinic
Triclinic
Primitive Vectors: vector between any two lattice points
R ma nb lc
m, n, l : integers
ex) a simple cubic lattice
a
aa
simple cubic
ˆ ˆ ˆ, , a ax b ay c az
Assignment of primitive vectors is not unique.
face-centered cubic
a
aa
rotated by 90ºprimitive cell
a
a
aa
face-centered cubic rotated by 90ºprimitive Cell
a
a
a
a
body-centered cubicunit cell
primitive cell
Primitive Unit Cell
Parallelepipes formed by the three primitive vectors
ucV a b c
Volume of the unit cell remains the same no matter how the primitive vectors are chosen.
Wigner-Seitz Unit Cell
bcc fcc diamond
Indexing procedure
1
z
x
y2
3
( 632 )
Miller Index
1) Divide each intercept value by the unit cell length along the respective coordinate axis 1, 2, 3
2) Invert the intercept values 1/1, 1/2, 1/3
3) Using an appropriate multiplier, convert the (1/intercept) set to the smallest possible set of whole numbers4) Enclose the whole-number set in curvilinear brackets (632)
1/ 1 6, 1 / 2 6, 1 / 3 6
ha kb lc
h, k, l integers
orientation of crystal planedirection perpendicular to crystal plane
z
x
y
(110)
a
a
a
Crystal planes of body-centered cubic
z
x
y
(100)
a
a
a
Basis Made of Two or More Atoms
basis
Arranging Na+ and Cl- ions alternatively at the lattice points of a simple cubic latticeEach ion surrounded by six nearest neighbors of the opposite charge
FCC with basisNa+ at (1/2,1/2,1/2) , Cl- at (0, 0, 0)
Simple Cubic with basisCs+ at (0,0,0) , Cl- at (1/2, 1/2, 1/2)
Cesium Chloride (CsCl) Sodium Chloride (NaCl)
Diamond structure (Zincblende)
FCC with basisone at (0,0,0) the other at (1/4, 1/4, 1/4)
YBa2Cu3O7 (Yttrium-barium-copper oxide)high temperature superconductor:
superconducting at temperature below 91 K
simple orthorhombic lattice with basis containing 13 atoms
top view
http://www.als.lbl.gov/pics/154graphene01.png
Graphite (C)
band structureeffective mass of an electron
, very high conductivity (theory).
2 2
1
/E k
*e 0m
http://www.rsc.org/images/FEATURE-graphene-390_tcm18-116226.jpg
Fullerene (Cx, usually C60), CNT (Carbon Nano Tube), and graphite
SWNT (Single Wall Nano Tube)
covalent bond between atoms loose bond between planes
graphite
chiral vector h 1 2 C ma na
CNTaxis: perpendicular to chiral vector
magnitude of : perimeter of tube
hC
tube diameter
2 2C-Cht 3
aCd m mn n
C-C C-C 0.142 nm, 3 0.246 nma a a
Cross-sectional area of SWCNT 2 2 2
c C-C 3 3 A Da a m mn n
graphene layer