Download An Introduction to Structures and Types of Solids Solids

An Introduction to Structures and Types of Solids
Solids
•
•
Amorphous Solids:
– Considerable disorder in structure
Crystalline Solids:
– Highly regular arrangements of their components (atoms, ions,
molecules) ⇒ Ordered Structures
– Lattice: 3-dim system of points designating positions of
components
– Unit cell: smallest repeating unit of the lattice
Generating a Lattice from a Pattern
In (a) select any point in pattern, in (b) move out through the
pattern, marking additional points and in (c) discard the
original pattern and keep the grid work of points
Crystal Lattice and the Unit Cell
A primitive unit cell (red)
contains 4 x ¼ = 1 lattice point.
The centered unit cell (blue)
contains two lattice points
Checkerboard: 2-dim analogy
There are 7 crystal systems and 14 types of unit cells in
nature
Three Cubic Unit
Cells and the
Corresponding
Lattices
Note that only parts
of spheres on the
corners and faces of
the unit cells reside
inside the unit cell, as
shown by the cutoff
versions.
To define structure we need:
a) type of crystal lattice – we concern only with cubic lattices here
b) number of atoms of each kind in the unit cell – composition of unit cell
c) coordination number for each atom – number of nearest neighbors
Cell Type
Simple cubic (sc)
Body-centered cubic (bcc)
Face-centered cubic (fcc)
Atoms per unit cell
1
2
4
Simple cubic: (8 corners of a cube)(1/8 of each corner atom
within a unit cell)=1 net atom per unit cell.
Body-centered cubic: it has one additional atom within the unit
cell at the cube’s center.
Face-centered cubic: on the faces of the cube, (6 faces of a
cube)(1/2 of an atom within a unit cell) = 3 net face-centered
atoms within a unit cell. Plus one more atom contributed by the
corner atoms.
Face-Centered Cubic
Probing the Structure of Condensed Matter
A well-defined beam of
x-rays beam strike a thin
film of a crystalline solid.
Some of the x-rays are
absorbed, some pass
through unchanged and
others are scattered at
various 2θ angles.
The 2θ scattering angle
is the angle between the
direction of the
diffracted beam and the
direction of the straight
beam.
X rays scattered from two different atoms may reinforce (constructive
interference) or cancel (destructive interference) one another
Incident
Reflected
Bragg Equation
xy + yz =
nλ
Reflection of X rays of wavelength λ from a
pair of atoms in two different layers of a
crystal. The lower wave travels an extra
distance equal to the sum of xy and yz. If this
distance is an integral number of wavelengths
(n = 1, 2, 3, . . .), the waves will reinforce each
other when they exit the crystal.
xy + yz =
2d sin θ
nλ = 2d sin θ
n = integer (1,2,3,…
λ= wavelength of the X rays
d = distance between the atoms
θ = angle of incidence and reflection
Example
In a crystal of Ag, planes of Ag atoms are separated by a distance of 1.4446 Å.
Compute the angles θ for all possible Bragg reflections for these planes if the
wavelength of the incident X-radiation is 0.7093 Å.
Types of Crystalline Solids
•
•
•
Ionic Solids – ions at the points of the lattice that describes the structure of the
solid.
Molecular Solids – discrete covalently bonded molecules at each of its lattice
points.
Atomic Solids – atoms at the lattice points that describe the structure of the
solid.
Structure and Bonding in Metals
Metals
• High thermal conductivity
• High electrical conductivity
• Malleability (can be shaped into something else without
breaking)
• Ductility (can be hammered thin or stretched into wire
without breaking)
A metallic crystal can be pictured as containing spherical metal atoms packed
together and bonded to each other equally in all directions. We can model
such a structure by packing uniform hard spheres in a manner that most
efficiently fills in the space. (closest packing)
The Closest Packing Arrangement of Uniform Spheres
•
•
abab packing – the 2nd layer is like the 1st but it is displaced so that
each sphere in the 2nd layer occupies a dimple in the 1st layer.
The spheres in the 3rd layer occupy dimples in the 2nd layer so
that the spheres in the 3rd layer lie directly over those in the 1st
layer.
The Closest Packing Arrangement of Uniform Spheres
•
•
abca packing – the spheres in the 3rd layer occupy dimples in the
2nd layer so that no spheres in the 3rd layer lie above any in the 1st
layer.
The 4th layer is like the 1st.
Hexagonal Closest Packing
Cubic Closest Packing
fcc
The Indicated Sphere Has 12 Nearest Neighbors
Each sphere in both ccp and
hcp has 12 equivalent
nearest neighbors.
Silver crystallizes in a cubic closest packed structure.
The radius of Ag atom is 1.44 Å (144 pm). Calculate the
density of solid silver.
Body-centered Cubic Packing
Spheres touch along the body
diagonal
unit cell with the center
sphere deleted
One face of the bodycentered cubic unit cell. By
the Pythagorean theorem,
f2 = e2 + e2 =2 e2
The relationship of the body diagonal (b) to
the face diagonal (f) and edge (e)
Bonding Models for Metals
Consistent with main properties of metals (high thermal and electric conductivity,
malleability and ductility) ⇒ strong and non-directional
•
Electron Sea Model
A regular array of cations in a “sea” of mobile valence electrons
21
Bonding Models for Metals
•
Band Model or MO Model
Electrons are assumed to travel around the metal crystal in molecular orbitals
formed from the valence atomic orbitals of the metal atoms.
The molecular orbital energy levels produced
when various numbers of atomic orbitals
interact. Note that for two atomic orbitals two
rather widely spaced energy levels result. As
more atomic orbitals become available to form
MOs, the resulting energy levels become more
closely spaced, finally producing a band of very
closely spaced orbitals.
Mg crystal (hcp)
1s22s22p63s2
Bonding Models for Metals
←Conduction Band
← Valence Band
Thermal energy can excite
electron from the filled MOs
called (valence band) to the
empty MOs (conduction
band)
A representation of the energy levels (bands) in a magnesium crystal. The
electrons in the 1s, 2s, and 2p orbitals are close to the nuclei and thus are
localized on each magnesium atom as shown. However, the 3s and 3p
valence orbitals overlap and mix to form MOs. Electrons in these energy
levels can travel throughout the crystal.
Metal Alloys (contains a mixture of elements and has metallic properties)
•
Substitutional Alloy – some of the host metal atoms are replaced by other
metal atoms of similar size.
~1/3 of Cu replaced by Zn
White gold (75% Au, 25% Ag)
•
Interstitial Alloy – some of the holes in the closest packed metal structure are
occupied by small atoms.
Carbon inclusions provide directional bonding
to the undirectional iron, giving strength
Mild steel: < 0.2% C (ductile and malleable)
Medium steel: 0.2 -0.6% C (structural steel beams)
High-C steel: 0.6 – 1.5% C (springs, cutlery, tools)
Network Atomic Solids
• Contain strong directional covalent bonds
Carbon allotropes (different forms): graphite, diamond, fullerenes (molelular solid)
all electrons are tied
up in sp3 covalent bonds
Diamond
Different Metal
“band gap” diamond = 6eV
no thermal excitation possible at 300 K. insulator
1 eV = 1.6 x10-19 J
Graphite (layers of C atoms)
sp2 hybridized
Leftover unhybridized 2p forms π MOs
benzene
Directional conductivity
Molecular Solids
Fullerenes, C60 , 60 carbon atoms, arranged as 12 pentagons and 20 hexagons
• Intermolecular forces: dipole-dipole, London dispersion and H-bonds.
• Weak intermolecular forces give rise to low melting points.
CO2
I2, P4, S8
Ionic Solids
• Ionic solids are stable, high melting point substances held together by the strong
electrostatic forces that exist between oppositely charged ions.
Three Types of Holes in Closest Packed Structures
Trigonal holes are formed
by three spheres in the same layer.
Tetrahedral holes are formed
when a sphere sits in the dimple of
three spheres in an adjacent layer.
Octahedral holes are formed between two sets
of three spheres in adjoining layers of the
For spheres of a given diameter, the holes
closest packed structures.
increase in size in the order:
trigonal < tetrahedral < octahedral
What is the Radius of the Octahedral Holes
The diagonal of the square, d = R + 2r + R
where r is the radius of the octahedral hole
and R is the radius of the packed spheres.
What is the Radius of the Tetrahedral Holes
The center of the tetrahedral hole (shown in
red) is at the center of the body diagonal b
The four spheres around a tetrahedral hole are (shown in purple).
shown inscribed in a cube. The spheres are
shown much smaller than actual size. They
One packed sphere
actually touch along the face diagonal f.
and its relationship
to the tetrahedral
hole. Note that
(body diagonal)/2 =
R + r.
Guidelines for Filling Tetrahedral and Octahedral Holes
A simple cubic array with X- ions,
with an M+ ion in the center
(in the cubic hole).
The body diagonal b equals R + 2r + R, since Xand M+ touch along this body diagonal.
Structures of Actual Ionic Solids
−
The locations (gray
X) of the octahedral
holes in the facecentered cubic unit
cell
Representation of the unit cell
for solid NaCl. The Cl- ions
(green spheres) have a ccp
arrangement, with Na+ ions
(gray spheres) filling all the
octahedral holes. This
representation shows the
idealized closest packed
structure of NaCl. In the actual
structure, the Cl- ions do not
quite touch.
Empirical formula can be related to the structure by counting the number of cations
and anions contained in one unit cell.
Na+ contained in a unit cell:
1 Na+ in the center of the unit cell) + (1/4 of Na+ in each edge x 12 edges) = net 4
Na+ in a NaCl unit cell. We repeat the same for Cl and the ratio of Na+/ Cl - =1
Example
A unit cell of perovskite is shown below. What is its formula?
Method: Identify the ions present in the unit cell
and their locations within the unit cell. Decide on
the net number of ions of each kind of the cell.
Ca2+ are in the corners of the cube, a Ti 4+ in the
center o the cell and O , a Ti 4+ in the center o the
cell and O in the face centers.
Ca2+ ions: (8 Ca2+ ions at cube corners)(1/8 of
each ion inside unit cell) = 1 net Ca2+ ion
No. of Ti 4+: one ion is in the cube center
No. of O2- ions: (12 O2- in cube edges)(1/4 of each ion inside cell) =
3 net O2- ions in cue edges)(1/4 of each ion inside cell) = 3 net O2-
CaTiO3
Structures of Actual Ionic Solids
The location (red X) of a
tetrahedral hole in the facecentered cubic unit cell
One of the tetrahedral holes.
The unit cell for ZnS, where the S2- ions (yellow) are
closest packed, with the Zn2+ ions (purple) filling
alternate tetrahedral holes
The unit cell for CaF2, where the Ca2+ ions
(purple) form a face-centered cubic
arrangement, with the F- ions (yellow) in all of
the tetrahedral holes.
Lattice Defects
Schottky defects (there are vacant sites)
Frenkel defects (an atom or an ion of either
sign is present at an inappropriate site.
Sometimes involve impurities ⇒ nonstoichiometric compounds
Example
Assume the two-dimensional structure of an ionic compound MxAy is shown
below. What is the empirical formula of this compound?
Assuming the anions A are the larger circles, there are
four anions completely in this repeating square. The
corner cations (smaller circles) are shared by four
different repeating squares. Therefore, there is one
cation in the middle of the square plus 1/4 (4) = 1 net
cation from the corners. Each repeating square has two
cations and four anions. The empirical formula is MA2.
Example
The unit cell for nickel arsenide is shown below. What is the formula of this compound?
The unit cell contains 2 ions of Ni and 2 ions of As, which gives a formula of NiAs.
8 corners × (1/8)/corner + 4 edges ×1/4/edge = 2 Ni ions
2 As ions inside the unit cell
Example
Show that the net composition of each unit corresponds to the correct formula
of the compound