Chapter 19: Materials 2: Solids
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Transcript Chapter 19: Materials 2: Solids
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
2hx
A f Aeit f B ei (t )
I A ( f Ae it f B e i (t ) )( f Aeit 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 eihkl ( 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 ihkl )( f A f B eihkl )
I hkl f A2 f B2 f A f B (eihkl e ihkl ) f A2 f B2 2 f A f B coshkl
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 2i ( hx kylz)
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 2t )
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