Atkins & de Paula:
Atkins’ Physical Chemistry 9e
Chapter 19: Materials 2: Solids
Chapter 19: Materials 2: Solids
Crystallography
19.1 Lattices and unit cells
 space lattice, the pattern formed by points representing the locations of structural
motifs (atoms, molecules, or groups of atoms, molecules, or ions).
 unit cell, an imaginary parellelepiped that contains one unit of a translationally
repeating pattern.
unit cell
Chapter 19: Materials 2: Solids
 crystal system, a classification based on the rotational symmetry elements of a unit
cell.
 essential symmetry, the elements a unit cell must possess to belong to a particular
crystal system.
cubic system
monoclinic system
triclinic system
Trigonal
Chapter 19: Materials 2: Solids
 Bravais lattice, the 14 distinct space lattices in three dimensions.
 primitive unit cell (P), formed by joining neighbouring lattice points by straight lines.
 body-centred unit cell (I), with lattice points at the corners and at the centre.
 face-centred unit cell (F), with lattice points at the corners and on each face.
 side-centred unit cell (A,B,C), with lattice points at the corners and on two opposite faces.
Chapter 19: Materials 2: Solids
19.2 The identification of lattice planes
 Miller indices (hkl), indices that distinguish planes in a lattice.
How to define Miller Indices
5. Negative directions are denoted with a bar on top of the number, e.g. 100
Chapter 19: Materials 2: Solids
(110)
(110)
(230)
(111)
(110)
(010)
Chapter 19: Materials 2: Solids
Some Examples
Chapter 19: Materials 2: Solids
Common crystallographic terms
 (hkl); parenthesis designate a crystal face or a family of planes throughout a crystal lattice.
 [uvw]; square brackets designate a direction in the lattice from the origin to a point. Used
to collectively include all the faces of a crystals whose intersects (i.e., edges) parallel each
other. These are referred to as crystallographic zones and they represent a direction in the
crystal lattice.
 {hkl}; "squiggly" brackets or braces designate a set of faces that are equivalent by the
symmetry of the crystal.
[111]
{111}
Chapter 19: Materials 2: Solids
 separation of planes (d-spacing)
1
d hk2 0
h2  k 2

a2
or d hk 0 
1
h2  k 2  l 2

2
d hkl
a2
1
h2 k 2 l 2
 2 2 2
2
d hkl a
b
c
a
(h 2  k 2 )1/ 2
or d hkl 
a
(h 2  k 2  l 2 )1/ 2
Chapter 19: Materials 2: Solids
19.3 The investigation of structure
19.3(a) X-ray diffraction
 diffraction, interference caused by an object in the path of waves.
 diffraction pattern, the pattern of varying intensity that results from diffraction.
 Bremsstrahlung, X–radiation generated by the deceleration of electrons.
 K–radiation, X–radiation emitted when an electron falls into a K shell.
Chapter 19: Materials 2: Solids
 four-circle diffractometer, a device used in X–ray crystallography.
Chapter 19: Materials 2: Solids
19.3(b) Bragg’s law
 reflection, an intense beam arising from constructive interference.
 glancing angle, θ, the angle of incidence of a beam of radiation.
 Bragg’s law, λ = 2d sin θ.
AB+BC = 2d sin θ
nλ = 2d sin θ
n = 1; first-order reflection
19.3(c) Scattering factors, f, a measure of the ability of an atom to diffract radiation
sin kr 2
4
r dr, k 
sin 
0
kr

 (r ) : electrondensitydistribution

 f  4   (r )
 f; equal to the total # of e in the atom at θ=0
(Justification 19.1)
Chapter 19: Materials 2: Solids
19.3(d) The electron density
 Structure factor, overall amplitude of a wave diffracted by the {hkl} planes.
A
ax
a/h
B
A
a
2d sin   2 sin     2ax sin   hx; differencein path
h
hx
phasedifference( )  2 
 2hx

A  f Aeit  f B ei (t  )
I  A  ( f Ae it  f B e i (t  ) )( f Aeit  f B ei (t  ) )  f A2  f B2  2 f A f B cos
2
 A  f A  f B e i
Fhkl   f j eihkl ( j )
where
hkl ( j )  2 (hxj  ky j  lz j )
j
Example 19.2
Fhkl  4( f   f  )
for even h, k , l
Fhkl  4( f   f  )
for odd h, k , l
Fhkl  0
for other h, k , l
Chapter 19: Materials 2: Solids
 systematic absences, reflections with values of h + k + l that are absent from the
powder diffraction pattern.
*
I hkl  Fhkl
Fhkl  ( f A  f B e ihkl )( f A  f B eihkl )
I hkl  f A2  f B2  f A f B (eihkl  e ihkl )  f A2  f B2  2 f A f B coshkl
cubicI ; ( x, y, z )  ( 12 , 12 , 12 )  hkl  (h  k  l )
no reflection from odd (h  k  l ) of cubicP
Au Nanooctahedron
Chapter 19: Materials 2: Solids
 Fourier synthesis, the construction of the electron density distribution from structure
factors .
1
 (r )   Fhkl e 2i ( hx  kylz)
V hkl
 phase problem, the ambiguity in phase
of structure factors obtained from
intensities.
 structure refinement, the adjustment of
structural parameters to give the best fit
between the observed intensities and
those calculated from the model of the
structure deduced from the diffraction
pattern.
 Neutron and electron diffraction
Chapter 19: Materials 2: Solids
19.5 Metallic solids
19.5(a) Close packing
 close-packed, a layer of spheres with maximum utilization of space.
 polytype, structures that are identical in two dimensions but differ in the third dimension.
 hexagonally close-packed (hcp), the sequence of layers ABABAB....
 cubic close-packed (ccp), the sequence of layers ABCABC....
hcp
ccp
Chapter 19: Materials 2: Solids
 coordination number, the number of nearest neighbours.
 packing fraction, the fraction of space occupied by hard spheres.
19.5(b) Less closely packed structures; bcc (cubic I) & cubic P
Chapter 19: Materials 2: Solids
CCP
Primitive Cubic
Coordination Number 6
52% Packing Fraction
Close-Packed (CCP or HCP)
Coordination Number 12
74% Packing Fraction
Body-Centered Cubic
Coordination Number 8
68% Packing Fraction
HCP
Chapter 19: Materials 2: Solids
19.6 Ionic solids
19.6(a) Structure
 (n+,n–)–coordination, the number of nearest neighbours of opposite charge; n+ is the
coordination number of the cation and n– that of the anion. .
 caesium-chloride structure, an ion of one charge occupies the centre of a cubic unit cell
with eight counter ions at its corners: (8,8)–coordination
 rock-salt structure, of two interpenetrating slightly expanded fcc arrays, one of cations
and the other of anions: (6,6)–coordination
 radius ratio, γ = rsmaller/rlarger.
 radius-ratio rule, a rule suggesting which type of structure is likely based on the radius
ratio: γ < 0.414 (zinc blende); 0.414 < γ < 0.732 (rock salt); γ > 0.732 (caesium chloride).
Chapter 19: Materials 2: Solids
19.6(b) Energies
 lattice energy, the difference in potential energy of ions packed together in a solid and
widely separated as a gas.
 lattice enthalpy, ΔHL, the change in molar enthalpy for MX(s)  Mz+(g) + Xz–(g).

1  z 2e 2 z 2e 2 z 2e 2 z 2e 2
E p (cation) 








4 0  d
2d
3d
4d

z 2e 2  1 1 1
z 2e 2


1       
 ln 2
4 0 d  2 3 4
4 0 d

z 2e 2
E p (cation)  2 ln 2 
4 0 d
1
z 2 N Ae 2
E p  N A E p (cation)  E p (anion)  2 ln 2 
2
4 0 d
For 3D array; E p   A 
z A z B N Ae 2
4 0 d
( A : Made lu n gcon stan)t
Chapter 19: Materials 2: Solids
 Born–Mayer equation, for the total potential energy of an ionic crystal.
 Born–Haber cycle, a closed path of transformations starting and ending at the same point,
one step of which is the formation of the solid compound from a gas of widely separated
ions.
*
d / d
repulsivecontribution E p  N AC e
Minim untotal E , E p ,min  
*
z A z B N Ae 2
4 0 d
d*
(1  ) A
d
Chapter 19: Materials 2: Solids
19.7 Molecular solids and covalent networks
 covalent network solid, a solid in which covalent bonds in a definite spatial orientation
link the atoms in a network extending through the crystal.
 molecular solid, a solid consisting of discrete molecules held together by van der Waals
interactions.
Graphite
Diamond
Ice
(molecular solid)
CNT
Chapter 19: Materials 2: Solids
Impact on Nanotechnology; CNTs
 Strong & light: 100 times stronger than steel but 1/6 as heavy.
 High electrical & thermal conductivities: far better than Cu.
[CVD growth]
[MWCNTs]
[SWCNTs]
metallic
semiconducting
[Nanotube field-effect transistor]
[Electrical properties of CNTs]
[Mechanical properties of CNTs]
Chapter 19: Materials 2: Solids
I19.1 X-ray crystallography of biological macromolecules
DNA
Proteins
TLR3
Chapter 19: Materials 2: Solids
THE PROPERTIES OF SOLIDS
19.8 Mechanical properties
 stress, the applied force divided by the area to which
it is applied.
 strain, the distortion of a sample resulting from an
applied stress.
 rheology, the study of the relation between stress and
strain.
 uniaxial stress, stress applied in one direction.
 hydrostatic stress, a stress applied simultaneously in
all directions.
 pure shear, a stress that tends to push opposite faces
of the sample in opposite directions.
uniaxial stress
shear stress
hydrostatic stress
Chapter 19: Materials 2: Solids
 elastic deformation, a deformation that disappears when the stress is removed.
 plastic strain, a strain from which recovery does not occur when the stress is removed.
 Young’s modulus, E = (normal stress)/(normal strain).
 bulk modulus, K = pressure/(fractional change in volume).
 shear modulus, G = (shear stress)/(shear strain).
 Poisson’s ratio, vP = (transverse strain)/(normal strain).
Chapter 19: Materials 2: Solids
19.9 Electrical properties
 metallic conductor, a substance with an
electrical conductivity that decreases as the
temperature is raised.
 semiconductor, a substance with an electrical
conductivity that increases as the temperature
is raised.
 insulator, a semiconductor with a very low
electrical conductivity.
 superconductor, a solid that conducts
electricity without resistance.
Chapter 19: Materials 2: Solids
19.9(a) The formation of bands
 nearly-free-electron approximation, a model of a metal in which the valence electrons are
assumed to be trapped in a box with a periodic potential.
 tight-binding approximation, a model of a metal in which the valence electrons are
assumed to occupy molecular orbitals delocalized throughout the solid.
 s- and p-bands, a band formed from overlap of s- and p-orbitals, respectively.
 band gap, a range of energies to which no orbital corresponds.
From Hückel secular determinant
k
Ek    2 cos
k  1,2,..., N
N 1
E N  E1  4 as N  
Chapter 19: Materials 2: Solids
19.9(b) The occupation of orbitals
 Fermi level, the highest occupied molecular orbital in a solid at T = 0.
 Fermi–Dirac distribution, P = 1/(e(E – μ)/kT + 1); a version of Boltzmann distribution that
takes into account the effect of the Pauli principle.
Chapter 19: Materials 2: Solids
19.9(c) Insulators and semiconductors
 valence band, a filled band in a solid.
 conduction band, an empty band in a solid.
 intrinsic semiconductor, where semiconduction is a property of the pure material.
 compound semiconductor, an intrinsic semiconductor being a compound of different elements.
 extrinsic semiconductor, becomes semiconducting when it is doped with other atoms.
 dopant, introduced atoms.
 p- and n-type semiconductivity, conduction by holes and particles, respectively.
 p–n junction, a junction between p- and n-type semiconductors.
p-type
n-type
Reverse bias
Forward bias
Chapter 19: Materials 2: Solids
19.10(a) Light absorption by excitions in molecular solids
 exciton, an electron–hole pair.
 Frenkel exciton, the electron and hole jump together from molecule to molecule.
 Wannier exciton, the electron and hole are on different but nearby molecules.
 exciton bands, the structure of an absorption spectrum due to exciton formation: there are N
exciton bands when there are N molecules in each unit cell
 Davydov splitting, the splitting between exciton bands.
Chapter 19: Materials 2: Solids
19.10(b) Light absorption by metals and semiconductors
19.10(c) Nonlinear optical phenomena
 frequency doubling (or second harmonic generation), the process in which an intense
laser beam is converted to radiation with twice its initial frequency as it passes though a
suitable material.
 optical Kerr effect, the change in refractive index of a well chosen medium (Kerr medium)
when it is exposed to intense electric fields .
 Kerr lens, the self-focussing of the laser beam by using the Kerr effect.
frequency doubling
1
2
  E  E 2    : hyperpolarizability
1
2
E 2  E 0 2 cos2 t  E 2 (1  cos 2t )
Chapter 19: Materials 2: Solids
19.11 Magnetic properties
19.11(a) Magnetic susceptibility
 magnetization, the magnetic dipole moment density, M = χH.
 volume magnetic susceptibility, the proportionality constant χ.
 molar magnetic susceptibility, χm = χVm.
 magnetic flux density, B = μ0(H + M) = μ0(1 + χ) H.
 paramagnetic, a material for which χ is positive.
 diamagnetic, a material for which χ is negative.
 magnetizability, ξ, a measure of the extent to which a
magnetic dipole moment may be induced in a molecule.
 Curie law, χm = A + C/T, A = NAμ0ξ and C = NAμ0m2/3k.
 Gouy balance, a device for determining the magnetic
susceptibility of a sample.
 superconducting quantum interference device (SQUID), a
superconducting device for determining the magnetic
susceptibility of a sample.
Gouy balance
Chapter 19: Materials 2: Solids
19.11(b) The permanent magnetic moment
 ferromagnetism, strong, persistent magnetization arising from the cooperative alignment of
spins.
 antiferromagnetic phase, a phase in which spins are locked into a low–magnetization
arrangement.
 Curie temperature, the temperature of a ferromagnetic transition.
 Néel temperature, the temperature of an antiferromagnetic transition.
 Temperature–independent paramagnetism (TIP), orbital paramagnetism.
paramagnetic
ferromagnetic
antiferromagnetic
Chapter 19: Materials 2: Solids
19.12 Superconductors
 superconductor, a substance that conducts electricity without resistance.
 high-temperature superconductor (HTSC), a substance that is superconducting at
relatively high temperatures.
 Type I superconductor, a superconductor that shows an abrupt loss of superconductivity
when exposed to a magnetic field above a critical value; completely diamagnetic below Hc
 Meissner effect, the exclusion of a magnetic field from a superconductor.
 Type II superconductor, a superconductor that shows a gradual loss of superconductivity
when exposed to a magnetic field.
 Cooper pair, a pair of electrons that exists as a result of interactions with the lattice.
YBa2Cu3O7
Cooper pair