AME 60637 Ion Physics The Crystal Lattice
•  The crystal lattice is the organization of atoms and/or molecules in
a solid
simple cubic
body-centered cubic
hexagonal
a
NaCl
Ga4Ni3
tungsten carbide
cst-www.nrl.navy.mil/lattice
•  The lattice constant ‘a’ is the distance between adjacent atoms in
the basic structure (~ 4 Å)
•  The organization of the atoms is due to bonds between the atoms
–  Van der Waals (~0.01 eV), hydrogen (~kBT), covalent (~1-10 eV), ionic
(~1-10 eV), metallic (~1-10 eV)
D.B. Go Slide 1 AME 60637 Ion Physics The Crystal Lattice
•  Each electron in an atom has a particular potential energy
–  electrons inhabit quantized (discrete) energy states called orbitals
–  the potential energy V is related to the quantum state, charge, and
distance from the nucleus
−Z nl e 2
V ( r) =
r
•  As the atoms come together to form a crystal structure, these
potential energies overlap è hybridize forming different, quantized
energy levels è bonds
€
•  This bond is not rigid but more like a spring
potential energy
D.B. Go Slide 2 AME 60637 Ion Physics The Crystal Lattice – Electron View
•  The electrons of a single isolated atom occupy atomic orbitals,
which form a discrete (quantized) set of energy levels
•  Electrons occupy quantized electronic states characterized by four
quantum numbers
–
–
–
–
energy state (principal) è energy levels/orbitals
magnetic state (z-component of orbital angular momentum)
magnitude of orbital angular momentum
spin up or down (spin quantum number
•  Pauli exclusion principle: no 2 electrons can occupy the same exact
energy level (i.e., have same set of quantum numbers)
•  As atomic spacing decreases (hybridization) atoms begin to share
electrons è band overlap
D.B. Go Slide 3 AME 60637 Ion Physics Electrons - Conductors
•  In the atomic structure, valence electrons are in the outer most
shells
–  loosely bonded to the nucleus è free to move!
•  In metals, there are fewer valence electrons occupying the outer
shell è more places within the shell to move
•  When atoms of these types come together (sharing bands as
discussed before) è electrons can move from atom to atom
–  electrons in motion makes electricity! (must supply external force –
voltage, temperature, etc.)
•  In metals the valence electrons are free to move è electrons are
the energy carrier
•  In insulators the valence shells are occupied and there’s nowhere
to move è energy carrier is now the bond (spring) vibrations
(phonons)
D.B. Go Slide 4 AME 60637 Ion Physics Electrons – Free Electron Model
In metals, we treat these electrons as free, independent particles
•  free electron model, electron gas, Fermi gas
•  still governed by quantum mechanics and statistics
free
electron
free electron gas
D.B. Go Slide 5 AME 60637 Ion Physics Electrons – Energy and Momentum
The energy and momentum of a free electron is determined by Schrödinger’s
equation for the electron wave function Ψ
− 2 2 

∇ Ψ( r ) = EΨ( r )
2m
wave function |ψ2| can be thought of as
electron probability (or likelihood of an
electron being there) è Heisenberg
uncertainty principle
We assume a form of the wave function
€

1 ik ⋅ r
Ψ( r ) =
e
∀
k is the wave vector, the velocity of the
electron ‘wave’
From here we determine the electron’s energy and momentum
2 2

k
energy
ε( k ) =
2m


momentum p = k
€
eigenfunction of Shrödinger’s equations
€
D.B. Go Slide 6 AME 60637 Ion Physics Electrons – Energy and Momentum
We can define a relationship between energy (frequency) and momentum
(wave vector) ε = f(k) – we call this the dispersion relation
2k 2
ε( k ) =
2m
dispersion relation for free electron
From
€ the dispersion relation we can determine the density of states, which is
essential to electron emission.
D.B. Go Slide 7 AME 60637 Ion Physics Electrons – Energy and k-space
In the analysis of electrons, the wave function is related to the wave vector via

1 ik ⋅ r
Ψ( r ) =
e
∀
It can be shown, that the wave vector may take only certain discrete states
2πn y
2πn z
2πn x
€
kx =
;k y =
;k z =
⇒ n i = 1,2,3,...
L
L
L
Additionally, for electrons, because of the Pauli exclusion principle, each wave
vector (k state) can only be occupied by 2 electrons (of opposite spin)
€
D.B. Go Slide 8 AME 60637 Ion Physics Electrons – Energy and k-space
We can describe the allowable momentum states in k-space which takes the form of a
circle (2D) or sphere (3D)
D.B. Go Slide 9 AME 60637 Ion Physics Electrons - Density of States
•  The density of states (DOS) of a system describes the number of
states (N) at each energy level that are available to be occupied
–  simple view: think of an auditorium where each tier represents an
energy level
greater available seats (N states) in
this energy level
fewer available seats (N states) in
this energy level
The density of states does not describe if a state is
occupied only if the state exists è occupation is
determined statistically
Simple View: the density of states only describes
the floorplan & number of seats not the number of
tickets sold
D.B. Go http://pcagreatperformances.org/info/merrill_seating_chart/ Slide 10 AME 60637 Ion Physics Electrons – Density of States
Density of States:
D(ε) =
1 dN 1 dN dk
=
∀ dε ∀ dk dε
The number of states is determined by examining k-space
€
(
(
)
4 πk 3
dN
k 3∀
3
=2
= 2
3
dk
3π
2π
L
)
With some manipulation, it can be shown that the 3D density of states for electrons is
€
1 $ 2m '
D(ε) = 2 & 2 )
2π %  (
3
2
ε
€
D.B. Go Slide 11 AME 60637 Ion Physics Electrons – Fermi Levels
•  The number of possible electron states is simply the integral of the
density of states to the maximum possible energy level.
–  at T = 0 K this is the equivalent as determining the number of electrons
per unit volume
–  we put an electron in each state at each energy level and keep filling up
energy states until we run out
εf
n e,0K =
εf
∫ D(ε)dε = ∫
0
0
1 % 2m (
2' 2 *
2π &  )
3
2
ε dε
•  However, the number of electrons in a solid can be determined by
the atomic structure and lattice geometry è known quantity
€•  We call this maximum possible energy level the Fermi energy and
we can similarly define the Fermi momentum, and Fermi
temperature
2
2
2
εF =
2m
(3π ne,0K )
kF = ( 3π n e,0K )
2
D.B. Go TF =
εF
kB
1
3
3
Slide 12 AME 60637 Ion Physics Electrons - Occupation
•  The occupation of energy states for T > 0 K is determined by the
Fermi-Dirac distribution (electrons are fermions)
f (ε) =
1
$ε − µ'
exp&
) +1
% kB T (
electron number density
∞
n elec =
∫ f (ε)D(ε)dε
0
µ ≡ chemical potential ≈ εF
εF = 5 eV
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D.B. Go occupation, f(!)
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electron energy, ! (eV)
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300 K
εF = 5 eV
occupation, f(!)
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€
•  Electrons near the Fermi level can be thermally excited to higher
energy states
1000 K
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electron energy, ! (eV)
)"
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Slide 13 AME 60637 Ion Physics Electrons – Specific Heat
If we know how many electrons (statistics), how much energy for an electron,
how many at each energy level (density of states) è total energy stored by
the electrons! è SPECIFIC HEAT
∞
total electron energy
U=
∫ εf (ε)D(ε)dε
0
∞
∂U
C=
≈ k B2 TD(εF ) ∫
∂T
−ε F
specific heat
€
kB T
z 2e z
π2
T
dz
=
n
k
~T
elec B
2
z
2
TF
(e + 1)
For total specific heat, we combine the
phonon and electron contributions
€
z = εk T
B
€
Ctotal = C phonon + Celectron
Basic relationships
€C
total
Ctotal
D.B. Go = AT 3 + BT → T << θ D (low temperature)
= 3n atomsk B + BT → T >> θ D (high temperature)
Slide 14 AME 60637 Ion Physics Electrons – What We’ve Learned
•  Electrons are particles with quantized energy states
–  store and transport thermal and electrical energy
–  primary energy carriers in metals
–  usually approximate their behavior using the Free Electron Model
•  energy
•  wavelength (wave vector)
•  Electrons have a statistical occupation, quantized (discrete)
energy, and only limited numbers at each energy level (density
of states)
–  we can derive the specific heat!
•  In real materials, the free electron model is limited
because it does not account for interactions with the
lattice
–  energy band is not continuous
–  the filling of energy bands and band gaps determine whether a
material is a conductor, insulator, or semi-conductor
D.B. Go Slide 15