Physics of Solids Flashcards
principle quantum number
n
from 1 to infinity
angular momentum quantum number (azimuthal)
l
between 0 and n-1
magnetic quantum number
m
between -l and l
spin quantum number
s
-1/2 or +1/2
greater distance so appear
point like
two atoms get close
repel each other
what forces can exist between two neutral atoms?
mutual, non uniform repulsion of electrons, creates charge distribution
this attraction is the van der waals force
force for point charges
F prop. to 1/r^2
force for dipoles
F prop. to 1/r^7
ie much much smaller
cooling atoms so that their kinetic energy is low enough allows..
van der waals forces to bind them together as a liquid or a van der waals solid
van der waals solids unstable because
forces very weak
when atoms get really close
wave functions overlap
symmetric eigenstate
1/root2(1+2)
energy of the joint state is lowered
this is a bonding state
anti-symmetric eigenstate
1/root2 (1-2)
energy of the joint state is riased
this is an anti-bonding state
most elements bond
metallically
when there is no longer an energetic advantage of bonding covalently
metallic bonding - each orbital overlaps several other orbitials. Collections of states overlap to form
bands
metallic bonding - provided some of the bands are not full…
an infinitesimal change in energy allows the electron to change state
metallic - treat system as
continuum of electron states surrounding a regular grid or lattice of positive ions
take into account Pauli - different quantum states
why beryllium is stronger than lithium
Be gives 2 electrons to the lattice leaving a 2+ ion
why aren’t elements that form large scale covalent structures metallic
either have band gaps which come from their lattice structure or completed outer orbitals which prevent delocalised electron cloud from forming
why aren’t elements like Nitrogen and Oxygen metallic
covalently bond into stable molecules with full bonding states
why aren’t noble gases metallic
full outer electron shells
can still show van der waals bonding and can be made solid if cold enough
electronegativity describes
how much atoms attract electrons
two atoms with different electronegativity can ‘take’ electrons from each other
ionisation potential
energy it costs to remove electron
electron affinity
energy gained by gaining electron
binding energy per bond
energy released when a positive and negative ion combine
work out from electromagnetism due to the electric field between them
all outer shells are full so how does this form a solid?
because it is very polarised
in carbon, the four spatially oriented covalent bonds allow it to
act like a scaffold for all sorts of structures
what covalent bonding depends on
unfilled anti-bonding states
I angstrom
10^-10m
crystalline solids
atoms have regular periodic arrangement
amorphous solids
atoms are disordered though can be ordered on a short range
DVD-RAMs
laser heats phase change material allowing it to go from crystalline to amorphous phase and vice versa
pretty much all elements will form crystals if
they are allowed to cool slowly enough and to a low enough temp
most pure substances will form
crystals
if you cool things too quickly, even pure substances
you get an amorphous solid
there is a timescale required for the ordered structure to form
unit cell
pattern that repeats without transformation
ie no flips
what is a unit cell made up of
a basis (eg an atoms or ions)
a lattice
lattice
array of points periodically repeated in space
each lattice point can have one or more atoms associated with it
most efficient packing
hexagonal
eg honeycomb
hexagonal close packing
hexagonal sheets will overlap in alternating layers
ABAB layer structure
has hexagonal symmetry
Face centred cubic
ABC layer structure
next later goes in the space that hasn’t been used
has cubic symmetry
body centred cubic
alternating layers of cubic atoms not truly close packed, no hexagonal symmetry
next layer above spaces in previous. ABAB
produces a cubic lattice with one extra atom in the middle
simple cubic
directly above previous layer
aka primitive cubic
looks like regular cube
lattice constant, a
gives size of unit cell
crystal structure is a convolution of
a lattice and a basis
how many non-degenerate lattice symmetries are there in nature
14
these are the bravais lattices
how many atoms in the unit cell
work out how many cells the corners and faces are shared with
8 x corner share + 6 x face share
coordination number
nearest neighbours
packing fraction
fraction of the structure occupied by atoms
=volume of atoms / volume of cell
use lattice parameter =a and atomic radius = r0
characteristics of simple cubic
coordination =6
atoms per unit cell =1
packing fraction=52%
characteristics of body centred cubic
coordination=8
atoms per unit cell=2
packing fraction=68%
characteristics of face-centred cubic
coordination=12
atoms per unit cell=4
packing fraction=74%
characteristics of hexagonal close packed
coordination=12
atoms per unit cell=6
packing fraction=74%
‘face’ atom is shared between
two unit cells
‘corner’ atoms is shared among
eight unit cells
coordination number
number of nearest neighbours
lattice can be described by three vectors
these vectors are chosen such that
the lattice looks identical for an integer translation along these vectors
r’=r+sa+tb
a and b ‘primitive lattice vectors’
if the cell is defined by primitive vectors, we can find any point inside the cell as
a fraction of the primitive vectors
drawing unit cells
collapse 3D cell on to xy-plane
mark each atom/ion with its z coordinates
don’t draw top layer if it is a repeat of bottom layer
miller indices are used to
identify atomic planes
miller indices steps
pick a cell
identify the intercepts (if never intercepts, intercept at infinity)
write down intercepts
take reciprocals
multiply through to get integers
reduce to lowest common fraction
replace -ve signs with bars
normal distance between planes from the miller index
d=a/ sqrt(h^2+l^2+k^2)
crystal directions are denoted
[hkl] rather than (hkl)
we can specify direction in crystals by
T=Ua1+Va2+Wa3
where [UVW] is defined such that [100] is x-axis, [010] is y-axis
for cubic crystals only, [hkl] direction is
perpendicular to face of (hkl) plane
equivalent lattice directions
directions that are identical under symmetry operations
there can also be equivalent lattice planes
notations
miller indices for planes (hkl)
notation for directions [hkl]
equivalent planes {hkl}
equivalent directions <hkl></hkl>
the lennard-jones potential
atoms have an equilibrium spacing
as atoms get closer, pauli exclusion principle forces electrons to fill higher energy states
attractive potential energy from van der waals force
real crystals - if system cools quickly, likely that
sometimes crystals will get stuck in local minima
don’t have energy to overcome the potential barrier
when atoms end up in the ‘wrong’ place
they create defects which propagate through the structure as grain boundaries
this can also be caused by impurities
the size of the crystal grain tends to correlate to
how fast they cool
quick cooling = lots of defects = lots of small grains
glasses (like obsidian) are
amorphous solids
they have no crystalline order
molten material –> flash freezing –> snapshot of liquid state
water expands as it freezes due to
frustration and residual entropy in its crystal structure
polar so +ve must be next to -ve
no way to do this across crystal
millions of ways to almost do it - raise the entropy of the system
normally increased pressure
holds solids together
for water, increased pressure breaks the solid apart
quasi crystals
crystals that have order, but no repetition
based on odd-number symmetry (pentagons, heptagons etc)
bulk modulus of elasticity quantifies
how much the solid deforms
(kapa has dimensions of pressure)
real tensile strength values are mostly
around 100x lower than prediction
only break this way in perfect crystals
before solids break, they
bend
due to elastic distortion
to be ductile, a solid needs to to able to
change the position of atoms without much change in energy
(metals tend to be ductile)
in general, alloys and compounds are less ductile than
pure metals but not always - structure dependent
covalent and ionically bonded materials tend to be very
brittle since bonding is directionally dependent
the difference between yield stress and ultimate tensile strength is a measure of
ductility, ie how much a material can be plastically deformed
x-ray diffraction - the bragg law
atoms act as 3D diffraction grating
pattern of diffracted radiation allows for determination of internal structure of solid
consider two planes separated by distance, d
for reflected radiation to be in phase with incident, the anlge of incidence must
equal the angle of reflection
for diffraction, need
lambda < or = 2d
a collimated beam has
parallel wavefronts
if they reflect off different planes we can calculate the path difference
general equation for a plane
hx+ky+lz=a
miller indices tell us where planes cut the axes
the distance between miller planes is
the length of the vector normal to the plane that goes from the origin to the plane
separation of miller planes, d=
a / sqrt(h^2+k^2+l^2)
bright x-ray scattering when
n lambda = 2asin theta / sqrt(h^2+k^2+l^2)
x-ray generated by projecting
accelerated electrons at a metal target
electrons ejected from lower energy states
x-rays are produced by e in higher state dropping
spectrum of x-rays has sharp lines due to
electron transitions
rotation photography
crystal alligned in particular direction and rotated with x-ray detector
uses monochromatic x-rays
laue photography
continuum of x-rays directed at thin sample
bragg condition always met at perpendicular angles, creating spots on screen
powder photography
uses monochromatic x-rays on sample of fine powder
some grains will be at correct angle to satisfy bragg
x-rays emitted in series of cones
destructive interference can arise from
equivalent planes in more complex crystal structures
if we take the square of both sides of the Bragg law
not all values of {hkl} are visible
for FCC and BCC latices, destructive interference within unit cell
cuts out certain reflections
general rules for destructive interference cutting out certain reflections
FCC: h,k,l must all be odd or all even (0 counts as even)
BCC: h+k+l must be even
making a monochromatic x-ray beam
at certain angle, only one wavelength will be diffracted
this separates a single wavelength from a spread
relativistic momentum, E=
p^2c^2 + m^2c^4
electron microscopy
charged so interact strongly with ions in the crystal
comparing wavelength to diameter of an atom, small diffraction angles
neutron diffraction
no electric charge
magnetic dipole moment
need large samples as have a weak interaction with matter
can be used to probe magnetic atoms
conduction electrons
in metals, free electrons can carry energy around the metal and also electric current
free electron model
ignore the positive ions
fermi energy
corresponds to the highest occupied energy level
intrinsic property of the metal (more electrons = higher Ef)
electrons near fermi energy dominate electric current
density of states
the number of states per unit energy
when electric field applied to electron sea in metals
each e accelerates
acquires a drift velocity in addition to fermi velocity
resistivity
intrinsic property, independent of shape
conductivity
1/resistivity
current density units
coulombs per square metre per second
current density
current across a unit area
drude model
e accelerated by E field
in addition to normal motion
time between collisions small so gained v «_space;normal v
as well as collisions with the lattice, electrons will scatter from
defects in the crystal
important at low temperatures
residual resistivity that depends on
material purity and quality
hall effect
electron path between collisions is curved
charge builds up on one side of the conductor
this creates E field that opposes further transverse drift
steady electric potential maintained across material as long as current flowing
momentum or k-space is known as
reciprocal space
nearly free electron model
taking into accounts effects of lattice
Brillouin zones
map all possible k-vectors into a range between -pi/a and pi/a (first B zone)
“folding back to the middle zone”
the square of the wavefunction gives
probability of finding electron there
symmetric state
electrons sit mainly at atom sites
lower energy
anti-symmetric state
electrons sit away from atom sites
higher energy
band gap
no e in this energy range - no solutions with this energy
as wavelength gets closer to a, electron converges on two states:
- between atom sites
or
- bound to atom sites
at long wavelengths its an average of the two so
they cancel and the electron behaves as if completely free
band gaps open up
band gaps occur in all materials
the effect depends on
the filling of the bands
an intrinsic semiconductor is typically considered conducting if
kbT is approx a tenth of the bandgap
then approx 0.1% of electrons are in conduction band which passes threshold for conductivity to start
valence band
where electrons are involved in covalent bonds, but do not contribute to electric current
n type doping
add atoms that have extra valence electron
donor atoms donate electron
provides excess of electrons to help with bonding
p type doping
add atoms that have one less valence electron
results in a positive hole which can carry electric current
dopant atoms add
extra ‘dopant bands’ which are closer to the conduction/valence band
smaller band gap
conduct better at lower temps
for metals there is an overlap in conduction and valence bands so
conduction band partially filled by electrons
for semiconductors there is
an energy gap between bands
at absolute zero, the valence band of semiconductors
is full of electrons and conduction band empty
principle energy gap
Eg=Ec-Ev
shining long wavelength at a semiconductor
valence electrons cannot absorb photons and scatter to high energy states so photons will pass right through
for wavelengths where the photon energy is greater than Eg, each photon will
raise a valence electron into conduction band
E=hv=hc/lambda > Eg
indirect bandgaps
in some semiconductors there is an offset between the top of the valence band and the bottom of the conduction band (in momentum space)…
the change in p is too large for photon to take up so phonon also needed
photon carries almost all energy
phonon almost all momentum
paramagnetic sample suspended in a magnetic field
due to electron spin interacting with magnetic field
stronger effect comes from interaction with unpaired electrons
diamagnetic sample suspended in a magnetic field
weak effect due to the fact that electrons are paired with partners of opposite spin
magnetic materials M
magnetisation of a material
A/m
magnetic materials H
magnetic field strength
A/m
magnetic materials B
magnetic induction or magnetic flux density
T
paramagnetism
unpaired electrons
diamagnetism
paired electrons in the same orbital
ferromagnetism
d shells incompletely filled
atoms have permanent magnetic moments
for ferro magnets, all atomic magnetic moments are
aligned in the same direction
Curie temperature
critical temp above which the alignment of magnetic moments is destroyed and materials are no longer ferromagnetic
examples of ferromagnetic materials
Fe, Co and Ni
antiferromagnet materials
adjacent magnetic moments are aligned in opposite directions
antiferromagnetic materials have no net
magnetic moment in the bulk, but neutron diffraction confirms that the materials are
magnetically ordered, but with a period of twice the inter-atomic spacing.
The physics of magnetic ordering depends on
the coupling energy between neighbouring magnetic moments
E=-Ju1.u2 where J is the exchange coupling constant
if J>0
parallel u1 and u2 give lowest possible energy which corresponds to ferromagnetic order
if J<0
antiparallel moments give the lowest energy which leads to antiferromagnetism
Neutron diffraction studies indicate
how atomic magnetic moments are distributed within
magnetic crystals.
