Bulk Structures of Crystals
Prof. Dr. Ulrich Jonas
Macromolecular Chemistry
Department Chemistry - Biology
University of Siegen
7 crystal systems
can be further subdivided into 32 crystal classes...
see Simon Garrett, "Introduction to Surface Analysis CEM924": http://www.cem.msu.edu/~cem924sg/LectureNotes.html
Bulk Structures of Crystals 2
lattice features of ideal crystals:
"asymmetric unit": elementary building blocks of a crystal, the "basis"
(atoms, molecules, proteins, colloids...)
"space lattice": 3D infinite array of points (locations of asymmetric unit assemblies),
surrounded in an identical way by neighbors
"crystal structure": blue print of the exact positions of asymmetric units in the
space lattice (location, orientation)
"unit cell": fundamental unit (arrangement of one ore more asymmetric units)
from which the entire crystal can be generated by translational
symmetry operations (like the bricks in a regular wall)
2D lattices:
Bulk Structures of Crystals 3
C2
C1
C4
C3
(unit cells)
C5
C6
C7
dense packing only possible with Cn : n = 1,2,3,4,6
14 Bravais lattices:
Important Basic Crystal Structures
The three most common basic crystal structures are fcc, hcp, and bcc.
first layer: hexagonally close packed
fcc: face centred cubic,
coordination number CN = 12
=
second layer: sits in hollow sites
occupied volume = 74%
hcp: hexagonal
close packed, CN = 12
bcc: body centred
cubic, CN = 8
third layer: two possible locations
hcp
fcc
..ABAB..
..ABCA..
"on top"
"on hole"
occupied volume = 74%
occupied volume = 68%
see Roger Nix, "An Introduction to Surface Chemistry": http://www.chem.qmw.ac.uk/surfaces/scc/
Crystal Planes
Miller Indices
The planes of ideal crystals (a cut through the lattice) are closely related to ideal crystal
surfaces. To specifiy a particular plane most commonly Miller indices are used.
2)
1)
note: hexagonal and trigonal
lattices use 4 Miller indices
by convention (redundant)!
3)
1 / (1,
,
) = (1 0 0)
Procedure:
1) Identify intercepts of plane on the x , y , and z axes.
1 x a, x b, x c
(example: cubic lattice a=b=c, = = =90°)
2) Specify intercepts in fractional coordinates of unit cell parameters a, b, c.
(1 x a) / a, ( x b) / b, ( x c) / c
1, ,
3) Take reciprocal of fractional intercepts, clear fractions.
1 / (1, , ) = (1 0 0)
bar for negative values h̄ k̄ l̄
R = (120)
for cubic lattice:
(100), (010), (001)
are identical
Simple 2D Lattices
translation vectors, unit cell vectors: a, b
example: Au (100) surface
conventional
bulk unit cell
primitive
surface unit
cell
primitive cell obeys translation symmetry
see also "Wallpaper Groups: lattices": http://aleph0.clarku.edu/~djoyce/wallpaper/lattices.html
low index surfaces:
fcc
Ideal Crystal Surfaces
high index surfaces:
bcc
(100)
(755)
(100)
(111)
Pt(755) or Pt S [7(111)x(100)]
(110)
stepped surface
(100) step 1 atom high
(111) terrace 7 atoms wide
(111)
(10 8 7)
(310)
(111)
compound surface:
NaCl(100)
Pt(10 8 7) or Pt S [7(111)x(310)]
(steps with kinks)
Real Crystal Surfaces
Relaxation:
Reconstruction and Superlattices
Reconstruction:
Superlattices:
non relaxed
unit surface cell different from the
bulk projected substrate
Substantial rearrangement
relaxed
distance change < 10%
bulk atom
satisfying dangling bond
Si(100) (2x1)
driving force:
unbalanced forces compared to bulk
surface atom
( 2x 2),R45°
Si(111) (7x7)
Surface Defects and Structures
monoatomic step
terrace
adatom (mobile)
step adatom
step vacancy (mobile)
kink
surface defects are important for crystal growth:
traps for newly adsorbed atoms / molecules
restructuring of surfaces happens at defects first
kink , step , and terrace atoms have large equilibrium concentrations on real surfaces
(hard to get perfect surface...)
isolated adatoms and vacancies are important for atomistic transport (restructuring), but
equilibrium concentration is low (< 1% of monolayer even at Tmelt)
Persistent Defects: Dislocations
"dislocations": stacking faults that disrupt regularity of crystal
kinetically: adsorbing species does not have enough time to find
thermodynamic equilibrium (correct position)
composition: adsorbing impurity atom disrupts packing
point defect, missing atom, impurity:
screw dislocation:
propagates through crystal
step dislocation, line defect:
dislocations contribute to mechanical properties (ductility, brittleness) and influence
crystallization speed (trapping sites)
Other Defect Structures
mosaic: composition of "real" (imperfect) crystal of many small "ideal" (perfect) crystallites
(diameter about 10 7 m) with slight misalignment of crystal axes
(like rectangular bricks in an irregular wall)
domain borders relate to step dislocations
Frenkel Schottky defect: in ionic crystals, explains high electric conductivity in such materials
Frenkel: dislocation of a single ion
+
+
to an interlattice site
+
+
+
+
+
+
+
+
+
Schottky: dislocation of single ion to crystal
surface or stacking defect
+
color center: alkali halogenide crystals > excess of metal atoms with free electrons located
in the lattice
> optical absorption (specific for lattice, not for metal type: K or Na)
Surface Sites and Adsorbate Surface Structures
Adsorption Sites:
Adsorbate Surface Structures:
bridge
four fold
hollow
three fold
hollow
fcc (100) c(2x2)
or fcc(100) ( 2x 2)R45
on top
four fold
hollow
on top
bridge
fcc (100) p(2x2)
or fcc(100) (2x2)
short bridge
2 fold hollow
fcc(111)
( 3x 3)R30
long bridge
hcp hollow (ABA...)
fcc hollow (ABC...)
fcc (100) (2x1)
Wood’s Notation and Matrix Notation
Description of the ordered adlayer (adsorbed atoms and molecules) in terms of the relationship to the
underlying ideal crystal plane.
examples:
fcc(100)
alternatively:
for p( 2x 2)R45
Electronic and Vibronic Features of Surfaces
molecule
1D solid
bulk
potential energy
1D solid with surface
atom spring
model
vibronic band
structure
electronic band
structure
probability
density
phonon wave vector: kvib = 2
electron wave vector: k = 2
vib
with
with
vib
: phonon wave length, a : lattice constant
: electron wave length, a : lattice constant
Epot : potential energy, Evib : vibonic energy, Eel : electronic energy , Ess : energy of the surface state
surface
atom
Electronic and Vibronic Features of Surfaces 2
STM image of Fe atoms on Cu(111):
electronic surface states
(particel in a box)
SPR (surface plasmon resonance) setup:
optical coupling in electronic surface states
(fluorescence)
SAW (surface acoustic wave) sensor:
horizontally propagating acoustic surface waves