Download Atoms, ions and molecules

PHYSICS of Materials (PHYS132)
Revision Session
N.B. Full course of lectures should be used in preparation for the exam
7 topics (13 lectures)
•
Introduction: definitions, structure (2)
•
inter-atomic forces (2)
•
Thermal properties: States of matter, latent heat, thermal expansion (3)
•
Mechanical properties: elasticity (1)
•
Magnetic properties (2)
•
Electrical properties: band theory, semi-conductors (2)
•
Optical properties: colour (1)
Books (recommended, not required):
Fundamentals of Physics (Haliday, Resnick, Walker):
Electrical and magnetic properties of matter
Properties of Matter (Flowers and Mendoza):
Structure, interatomic potentials, thermal properties, mechanical properties
Gases, Liquids and Solids (Tabor):
Structure, interatomic potentials, thermal properties
Properties of Materials (White):
Optical properties
1
Atoms, ions and molecules
Macroscopic matter is made up of assemblies of atoms,
ions and molecules
ATOMS
The smallest particle of an ELEMENT consists of NUCLEUS (Z
Protons + N Neutrons) surrounded by Z Electrons. Electrically
neutral. e.g. Cu, H, Ne.
MOLECULES
The smallest particle of a compound consists of a combination of
atoms with some rearrangement of the electrons. Electrically neutral
e.g. CuCl2, H2 etc.
IONS
Atoms (or combinations of atoms) which have lost or gained one or
more electrons. Charged e.g. Cu++, H+, OH–
1. Introduction to Materials > 1.1 Definitions
2
1
A mole is the amount of a substance that contains as many elementary
entities (atoms, molecules, etc.) as there are atoms in 0.012kg of
carbon-12 (12C)
Or equivalently: A mole is the amount of a substance that contains
6.022x1023 atoms/molecules.
=> AVOGADRO CONSTANT NA=6.022x1023 mol-1
1. Introduction to Materials > 1.1 Definitions
3
Molar mass: the mass of a mole of a substance. Symbol: M, unit: kg/mol
Relative atomic (molecular) mass:
The ratio of the mass of one atom (molecule) of an element (compound) to 1/12
of the mass of one atom of 12C. Symbol: Ar (or Mr for molecules).
Unit ≡ number of atomic mass units (a.m.u. or u)
Absolute mass
of 1 unit:
1u =
1
0.012
0.001
×
=
= 1.66 ×10-27 kg
23
23
12 6.023×10
6.023×10
Molar mass versus relative
atomic (molecular) mass
Molar volume:
V0 =
(
)
M = Ar ×10−3 kg/mol or Mr ×10−3 kg/mol
M
where ρ is the density of a substance
ρ
Volume occupied by one molecule = Vo /NA
1. Introduction to Materials > 1.1 Definitions
4
2
Close-packed structure
Consider atoms (as spheres) arranged in a plane.
Most closely “packed” if they form a 2D hexagonal lattice.
Atoms in the second layer locate themselves in the hollow sites
between 3 adjacent atoms.
1. Introduction to materials > 1.2 Structure: packing and order
5
Stacking sequence cont...
The second layer can be located in two ways (B or C)
If the second layer was placed with all the atoms at hollow sites B,
the third layer can be placed at sites A (ABABAB stacking
sequence, hexagonal close-packed structure)
or sites C (ABCABC stacking sequence, face-centred cubic structure)
12 neighbours are touching any particular sphere in these
close-packed structures.
1. Introduction to materials > 1.2 Structure: packing and order
6
3
Density of a crystalline solid
•
•
•
ρ
a0 = lattice constant
volume of cell (a0)3
Density
= (mass of unit cell) / (volume of unit cell)
= (effective number of atoms in unit cell) x (mass of atom) / (volume of unit cell)
ρ=
N eff ⋅ M N A
a 30
M = molar mass
Neff = effective number of atoms in unit cell
NA= 6.022x1023 mol-1 –Avogadro's number
1. Introduction to materials > 1.2 Structure: packing and order
7
Packing fraction
≡ Volume of (touching) atoms accommodated by a unit cell
divided by the volume of the cell itself is the packing fraction.
Example: fcc
Conventional unit cell seems to contain 14 atoms:
However atoms are shared with neighbouring cells!
• 8 “corner”-atoms, shared between 8 cells
• 6 “face”-atoms shared between 2 cells
Take: Volume cell = a3
Atoms touch along the side-diagonal of the cell, hence
Radii of atomic spheres = √2 a/4
a
Total Volume of spheres = [8x(1/8)+6x(1/2)] × [(4/3) π (√2 a/4 )3]
= (√2 π a3)/6
effective number of atoms in unit cell
Packing fraction = {(√2 π a3 )/6} / a3 = √2 π /6 = 0.74
1. Introduction to materials > 1.2 Structure: packing and order
8
4
Attractive Interatomic Forces
• 4 main types: ionic, covalent, metallic and van der Waals
IONIC (e.g. Na+Cl–)
In ionic materials there is a complete transfer of electrons between atoms
=> +ve and –ve ions
=> Coulomb forces (F ~ 1/r2): attractive or repulsive
∴ attractive forces dominate
COVALENT (e.g. diamond)
In covalent materials atoms share electrons.
+ve nuclei are attracted to the electrons between them and hence to each other
=> attractive forces between atoms
METALLIC (e.g. Na)
Metals consist of a lattice of +ve ions in a gas of free electrons.
attraction between ions and electrons (not unlike covalent bonds but
electron are shared between many atoms)
=>attractive force
2. Interatomic Force Models > 2.1 Origins and potentials
9
Attractive force cont.
VAN DER WAALS (e.g. Ar, N2, HCl )
VdW forces arise from an attraction between the dipole moment (either
permanent or instantaneous) of one atom (molecule) and that (either
permanent or induced) of another
theory =>
F ∝
1
r7
occur between all types of atoms or molecules, including neutral*
atoms or molecules.
(*when, although it is a weak force it is important because other
attractive forces are absent)
2. Interatomic Force Models > 2.1 Origins and potentials
10
5
Repulsive Force
• Has both ELECTROSTATIC and QUANTUM-MECHANICAL origins.
As atoms get close together their electron clouds overlap.
(i) +ve nuclear charges no longer screened => Coulomb repulsion
(ii) Electrons near to each-other cannot have the same quantum number
(Pauli Exclusion Principle)
∴ cannot have the same energy and position
∴ some electrons have to change (increase) their energy
=> repulsion
i) and ii) combined =>
F∝e
or more conveniently
F∝
−ra
1
n
r
where n = 10 – 13
2. Interatomic Force Models > 2.1 Origins and potentials
11
Interatomic forces: Representation
• Consider force F between an atom (or molecule) at the origin and another a
distance r away.
– require F → 0 as r → ∞ , ∴ F ∝ 1/rp
• Sign convention: repulsive force + ve ; attractive force – ve
Total force F(r)=FR(r)+FA (r)
m (typically 13) > n
n: 2 for ionic, > for VdW
FR dominates at short distance (r<a)
FA dominates at longer distances (r>a)
At r = a, F(a)=0, I.e.when | FR |=| FA |
a ≡ equilibrium separation
2. Interatomic Force Models > 2.1 Origins and potentials
repulsion
m
const
a
FR = + m = A  m
repulsive force
r FR=+const=+A
 r( ar)
F
rm
a
0
(a)n
attractive
force F
A=+const.Õ=-A
attraction
r
rn
const
a
FA = − n = B 
r
r
n
12
6
Interatomic potential energies
• Instead of F(r) it is often more convenient to use V(r), the potential
energy of the two atoms (molecules).
• V(r) is the work done (on the system) in bringing one atom from
∞ to r (where V(∞)≡0),
r
∫
∴ V(r) = − F(r)dr
F(r) = −
or
∞
dV(r )
dr
• N.B. System changes to minimize V.
• If
a
F (r ) = const  
r
m
then
a
V (r ) = const '  
r
m −1
2. Interatomic Force Models > 2.1 Origins and potentials
13
The Mie potential
• In 1907 Mie proposed a very simple form of the inter-atomic
potential
V(r) =
• A,B,m,n are constants.
first term is repulsive
second attractive.
m>n
• At equilibrium,
dV
=0
dr
∴ F =0
2. Interatomic Force Models > 2.1 Origins and potentials
A B
m − n
r
r
V(r)
very steep
equilibrium
separation
less steep
equilibrium potential energy
14
7
The Lennard-Jones 6-12 Potential (van
der Waals solid)
• A good approximation to the potential energy.
 a0 12  a0  6 
V (r ) = ε   − 2  
 r  
 r 
(equivalent to Mie potential with m=12, n=6, A=εa012, B=2εa06)
V(r)
• At r = ∞ V(r) = 0
At r = a0 , (equilibrium separation):
V(r) = -ε,
F = -dV/dr = 0
• The L-J 6-12 potential describes the
interaction between two isolated atoms
(molecules)
2. Interatomic Force Models > 2.1 Origins and potentials
15
L-J potential: in solids or liquids
• The forces described by L-J 6-12 are very short range.
e.g. V(a0) = -ε,
V(2a0) ~ -ε/30
∴ to a good approximation, only NEAREST NEIGHBOUR
interactions need to be considered.
=> Concept of Coordination number n,
the number of NEAREST NEIGHBOURS of any particular molecule
• SOLIDS
n has a definite value (dependent on structure)
e.g. for close-packed structure (hcp or fcc) n = 12,
for bcc n = 8 etc
• LIQUIDS
n has an average value, typically 1 – 2 smaller than in solids
2. Interatomic Force Models > 2.1 Origins and potentials
16
8
Interatomic potential for ionic crystals
∴ Electrostatic potential energy per pair of ions in an ionic crystal
equals p.e. of an isolated pair of neighbouring ions multiplied by the
Madelung constant α.
• Including the p.e. due to the short range repulsive forces the p.e. per
pair of ions in the crystal as a function of nearest neighbour separation
r
A
αe 2
V (r ) = + m −
r
4πε 0 r
α is the MADELUNG CONSTANT which depends on the particular
arrangement of ions. (typicaly 1-2)
for 1D line α = 1.38; for 3D NaCl structure α = 1.75
2. Interatomic Force Models > 2.2 Interatomic potential for ionic crystal
17
Topic 3.1 Thermodynamic aspects of stability
• Solid: - definite volume and shape
• Gas: - volume and shape dependent on the container
• Liquid:- definite volume, shape dependent on the container
This behaviour relates to:
COMPRESSIBILITY (C) : response to an attempt to change the volume
VISCOSITY (V) and RIGIDITY (R): response* to an attempt to change
the shape
(*N.B. the measurement time-scale is important)
• Solid: - low C, infinite V, high R
• Gas: - high C, low V, zero R
• Liquid:- low C, intermediate V, low R
These properties relate to the PACKING and ORDER of the atoms:• Solid: close packing, long range order
• Gas: dilute packing, no order
• Liquid: close packing, short range order
3. Thermal Properties > 3.1 Thermodynamics: states and phases
18
9
States of Matter
• The State of a substance depends on the values of P, V and T
(pressure, volume and thermodynamic (i.e. Kelvin) temperature).
• For a particular amount (e.g. 1 mole) of substance, in a
particular state, these quantities are linked by an EQUATION
of STATE (e.g. PV = RT for ideal gas)
• Possible values of P,V and T form a surface in PVT space.
Different regions correspond to different STATES
• Boundaries between these regions
correspond to transitions between
STATES
• The links between PVT and STATE
are usually displayed on P-T or P-V
PHASE DIAGRAMS.
3. Thermal Properties > 3.1 Thermodynamics: states and phases
19
P-T Phase Diagram
•
•
•
•
The range of variables where the solid,
liquid and gas phases exist are shown as
areas.
Lines are boundaries and represent
conditions where phase transitions take
place
TP: Triple Point : Solid, liquid, gas coexist
CP: Critical Point : highest temperature at
which liquid can exist
Changes at constant P: isobars
Changes at constant T: isotherms
Examples: 3 isotherm line α, β and γ
α
GAS -> SOLID (below triple point
temperature, sublimation)
β
GAS -> LIQUID -> SOLID
γ
GAS -> SOLID (above critical
temperature, no liquid phase)
3. Thermal Properties > 3.1 Thermodynamics: states and phases
α
γ
β
P
P
SOLID
SOLID
CP
CP
TP
TP
LIQUID
LIQUID
GAS
GAS
T
Brown: sublimation curve
Blue: melting/fusion curve
Red: vaporization/condensation curve
(also vapour pressure)
20
10
P-V Phase Diagram
T information can be included by ISOTHERMS
GAS
S+L
S+G
Critical point curve (fixed T)
α
L
L+G
Triple point line
(fixed T)
SOLID
ISOTHERMS:
α:
compression of gas (ideal gas: P ∝ 1/V)
⇒ solidification (g+s)
⇒ compression of solid
β:
compression of gas
⇒ liquefaction/condensation (g+l)
⇒ compression of liquid
⇒ solidification/fusion (l+s)
⇒ compression of solid
γ:
compression of gas
⇒ solidification/sublimation (s+g)
⇒ compression of solid
β
SOLID+GAS
γ
Notice in all (phase transition) regions of mixed phase, P constant along isotherm.
3. Thermal Properties > 3.1 Thermodynamics: states and phases
21
What determines the state of a system?
(Microscopically)
The balance between the interatomic (intermolecular) potential energy*
and the kinetic energy+ (thermal energy) of the atoms (molecules)
*depends on separation (i.e. P or V)
+ depends on temperature (T)
• interatomic p.e. dominant (low T or high P)
• thermal energy dominant (high T or low P)
• both important (intermediate T & P)
3. Thermal Properties > 3.1 Thermodynamics: states and phases
=> SOLID
=> GAS
=> LIQUID
22
11
Critical temperature
(For the L-J 6-12 van der Waals potential)
 a0 12  a0  6 
V (r ) = ε   − 2  
 r  
 r 
•
-ε is the potential energy between two molecules at their equilibrium separation
∴ +ε is the BINDING ENERGY of the pair
≡ energy needed to separate them (to ∞)
• THERMAL ENERGY ≡ kBT
(Boltzmann’s constant: kB = 1.38×10–23 J/deg)
when THERMAL ENERGY > BINDING ENERGY molecules do not stay
together
∴ substance does not liquefy irrespective of pressure applied
∴ at CRITICAL TEMPERATURE
k BTC ≅ ε
∴ TC ≅
3 Thermal Properties > 3.2 Critical temperature and latent heat
ε
kB
23
Latent Heat (van der Waals solid/liquid)
• The MOLAR LATENT HEAT OF SUBLIMATION (or
VAPORISATION) is the energy required to change one mole (NA
molecules) of a substance from solid (or liquid) to a gas.
• In 1 mole there are 1/2 n NA pairs.
– factor of 1/2 ensures pairs are not counted twice,
– n is coordination number (lecture 2.1).
For sublimation use n for solid ; For vaporisation use n for liquid
MOLAR LATENT HEAT
L0 =
1
nN Aε
2
∴ using assumptions that kinetic energy is small compared to potential energy
(i.e. low temperatures) and only nearest neighbours are considered
For ionic crystal:
L0 = N Aε
3 Thermal Properties > 3.2 Critical temperature and latent heat
24
12
Thermal expansion in solids
MACROSCOPICALLY (HRW)
Coefficient of linear expansion
Coefficient of volume expansion
∆L
= α ∆T
L
β = 3α
∆V
= β ∆T
V
MICROSCOPICALLY (QUALITATIVE)
Thermal expansion arises from
increasing atomic vibration through
increasing temperature
in combination with
the fact that the V(r) curve is not
symmetric
∴ increasing temperature
⇒
increasing mean separation
3. Thermal properties > 3.3 Thermal expansion
+ε
V(r)
0
a0
r
Increasing
T
-ε
T=0
∴ anharmonicity ⇒ thermal expansion
25
4: Mechanical Properties:
Topic 4.1: Elasticity
• Elasticity is a macroscopic property related to how the potential energy of
(≡ forces between) the atoms changes as a result of changes in separation from
equilibrium. Elasticity is the change in shape or volume of a body in response to
external forces.
• For small changes (~0.1→ 1 %) the behaviour is REVERSIBLE (ELASTIC) and
the original shape or volume is restored when the forces are removed.
• For very small changes:
Change in shape or volume ∝ force producing it
Elastic modulus = Stress / Strain
Elastic modulus is a constant
Strain = fractional change in relevant dimension
Stress = (change in) pressure producing it
4 Mechanical Properties > 4.1 Elasticity
26
13
There are 3 main elastic moduli corresponding to different types of
deformation:
Shear Modulus (G)
Young Modulus (E):
F
A
F/A
E=
∆L / L
∆x
L
G=
F/A
∆x / L
Bulk Modulus (K)
V
dV
 ∆P 
K = −V 

 ∆V 
Connection with
interatomic energy:
4 Mechanical Properties > 4.1 Elasticity
∴ K r = a0
d 2E
d 2 E  dr 
=V
≅V


dV 2
dr 2  dV 
N.B. assumptions:
1) low temperatures
2) small changes from equilibrium
2
27
• External field B0 causes magnetisation of material
r r
r
B = B0 + µ0 M
• Resulting magnetic field in the material
• There are 3 main types of magnetic behaviour of materials
– Diamagnetism (a very small effect exhibited by all materials)
• the ORBITAL motion of all electrons changes in a way that gives
• Î very small magnetic moment opposite to B0,
• Î B very slightly less than B0
– Paramagnetism (in atoms/ions of certain elements with unpaired electrons)
• The atom magnetic moments are aligned by external magnetic field B0
• Î small magnetic moment parallel to B0
• Î B slightly greater than B0
– Ferromagnetism (a few elements and their alloys which have non-zero atomic
magnetic moments and exchange coupling)
• Magnetic domains formed due to exchange coupling are aligned by external
magnetic field B0
• Î large magnetic moment parallel to B0
• Î B greater than B0
– (effects can remain when B0 is taken back to 0 !)
5. Magnetic properties > 5.2: Magnetism in matter
28
14
Magnetization M = (magnetic moment of sample)/(volume of sample)
(Units of M: (Am2)/m3 = Am-1)
Curie law: M = C
B0
T
(C = Curie
constant )
Complete alignment (“saturation”) ⇒ magnetic moment of sample: µmax = Nµatom
where N is the number of atoms in sample
1
M/M max
0.5
Magnetization curve for paramagnetic materials
Curie
law
saturation
deviation from
Curie law
Curie law holds
1
3
2
B0 / T (T/K)
4
5. Magnetic properties > 5.2: Magnetism in matter
29
Hysteresis
• In FERROMAGNETIC materials the magnetization curve does not
retrace itself for B0 increasing then B0 decreasing.
• This phenomenon results from the irreversibility of:
– moving domain boundaries
– changing domain
magnetization direction
BM
x→
• x =BM without B0
I.e. PERMANENT
MAGNETISM or
MAGNETIC MEMORY
B0
• The above curve is called a HYSTERESIS LOOP
– Broad loop means high reminiscent magnetisation (used for permananent magnets)
– Narrow loop means low reminiscent magnetisation (used in transformers)
5. Magnetic properties > 5.2 Magnetism in matter
30
15
Section 6: Electronic properties
Topic 6.1.1 The Band Theory of Conduction
Conduction of Electricity in Solids
• Conduction is measured by resistance R where for a block of area A,
length L, the resistance (R = V/I) is given by
R=ρ
L 1 L
=
A σ A
• METAL
INSULATOR
SEMICONDUCTOR
(ρ = resistivity, σ = conductivity )
ρ ≈ 2 x 10-8 Ωm
ρ ≈ 1 x 1016 Ωm
ρ ≈ 3 x 103 Ωm
e.g. Copper
e.g. Diamond
e.g. Silicon
• Also the variation with temperature is very distinctive:
α=
1 dρ
ρ dT
Copper : α ≈ 4 x10 −3 K −1 , Silicon : α ≈ − 7 x10 −2 K −1
• Finally the density of charge carriers is very different
Copper: n ≈ 9x1028 m-3, Silicon: n ≈ 1x1016 m-3
6. Electronic Properties > 6.1 Band theory
31
Energy Levels in a Crystal
•
In a solid (typically N ~ 1028 atoms/m3) we have many atoms and hence electron wavefunctions brought close together, modifying the energy levels:
• The most significant effect is at the least tightly bound energy levels where the
overlap will be greatest.
• Due to the Pauli Principle each electron must still occupy its own unique state,
however these states have almost infinitesimally closely packed energy levels.
•
The quasi-continuous
distribution of energy
levels (arising from a
single atomic energy
level) is called an
energy band.
•
Energy
Example: electron levels in Na
solid
3s: 1 electron
2p: 6 electrons
It is the occupation of
the outermost band
with electrons that
determines the
electrical properties.
6. Electronic Properties > 6.1 Band theory
isolated atom
2s: 2 electrons
1s: 2 electrons
a0
Separation
32
16
Occupation Probability
The probability of occupation of the states in a band is given by a
function p(E).
This is called the
Fermi function.
( E − E F ) / kT
p( E ) =
1
+1
e
T = 0K
P(E) 1
T > 0K
P(E) 1
0.5
0
EF
0
E
The highest occupied energy at T = 0K
is called the Fermi level EF.
EF
E
At T > 0K the Fermi function ‘smears’ out.
(At the Fermi level p(E) = ½)
6. Electronic Properties > 6.1 Band theory
33
Metals
Energy
Metal
EF
partially
full
full
• In metals, the highest occupied energy levels lie in the middle of an energy band
which is called the conduction band.
• There are therefore plenty of adjacent empty states almost infinitesimally higher
energies for an electron to be raised to by the application of an electrical potential
(or by heat) so that the Pauli principle poses no obstacle to conduction.
6. Electronic Properties > 6.1 Band theory
34
17
Insulators
• When the levels in one band are exactly filled and the lowest energy level in
the next band is at a considerably higher energy, a material is an insulator.
• This is because any externally applied electrical field would have to be very
high to excite any electrons into the next available energy levels.
• Also, as the band-gap, Egap is much greater than the typical thermal
excitation energy there will be no electrons thermally excited to the (empty)
conduction band. (e.g. diamond Egap = 5.5eV)
Insulator
empty
Energy
conduction band
EF
valence band
Egap >> kT
Example diamond: Egap = 5.5eV
⇒ p(EC ) =
1
( EC − EF ) / kT
e
+1
≈ e−( EC −EF )/ kT
=e
full
− Egap / 2kT
≈ 6 ×10−47 at T = 300K
6. Electronic Properties > 6.1 Band theory
35
Semiconductors
• When Egap is near kT, we have a semiconductor.
• Although p(EC) may be small at room temperature, we can still have a
significant number of electrons in the conduction band through thermal
excitation.
• These electrons (and the holes they leave in the top of the valence band!)
explain the conduction properties of semiconductors.
• Because there is an exponential dependence of p(E) on temperature, the
value of α is large and negative for semiconductors, i.e. better conductivity
at higher T.
Example silicon: Egap = 1.1eV
Energy
semi conductor
empty
conduction band
EF
–
+
– +
– +
– +
valence band
6. Electronic Properties > 6.1 Band theory
Egap ~ eV
full
⇒ p(EC ) =
1
( EC − EF ) / kT
e
+1
−( EC − EF ) / kT
≈e
=e
− Egap / 2kT
≈ 6 ×10−10 at T = 300
36
18
Difference in colour between metals
• The colours of silver and gold are quite distinct from one another.
• This is due to differences in the number of states above the Fermi edge.
• For silver there are plenty of states
available to reflect photons of all
wavelengths of visible light with high
efficiency.
• Gold does not reflect much high-energy
visible light (blue and violet) because of
an absence of energy levels in this
region. Since gold predominantly
reflects at the low-energy (longwavelength) end of the visible range, it
appears yellow.
silver
Reflection
100%
300
gold
λ (nm)
700
7. Optical Properties > 7.1 Colour
37
Colours of pure semiconductors
• For pure semiconductors the colour depends on the energy gap Eg.
• If Egap< the lowest energy of visible light (λ = 700nm,
red, E = 1.7eV) then any wavelength will be absorbed
and the colour will be black or metallic (depending on
the efficiency of re-radiation):
– Si (Egap= 1.1eV) ⇒ grey metallic
– GaAs (Egap= 1.4eV) ⇒ black
• If Egap > the highest energy of visible light (λ = 400nm,
violet, E = 3eV), then no visible light is absorbed and the
material is colorless/transparent (e.g. diamond, Egap=
5.4eV).
• If Egap falls in the range of visible light, the colour is the
complementary colour to the absorbed wavelengths.
– HgS (Egap= 2.1eV): λTRANSMITTED> 590nm ⇒ red
HgS is a red pigment “Vermilion”.
7. Optical Properties > 7.1 Colour
38
19