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