Transcript Multielectron Atoms – The Independent Particle Approximation

The Independent Particle Approximation
• We approximate the strong electrostatic forces between e-s by treating the force on
each electron independently, which includes force from nucleus and other electrons
• Inner electrons can shield the nuclear charge, leading to “screening”
Screening
electron
cloud
The effective potential energy felt by an electron
+Ze
r
electron
ke2
U (r )   Z eff (r )
r
Zeff is the effective charge that the electron feels and depends on r. Note that
Z eff  Z
when r is inside all other electrons
Z eff  1
when r is outside all other electrons
Unlike in hydrogen, in multielectron atoms the
dependence of the potential energy on r due to
screening lifts the degeneracy between the n states
The Periodic Table
Columns: groups with similar
shells, similar properties
Rows: periods with elements
with increasingly-full shells
Closed-shell –plus one
(alkali) elements: reactive
due to loosely-bound outer
electron in s-shell
Closed-shell–minus-one elements (halogens):
elements with high electron affinity A (energy
gained when an additional electron is added to a
neutral atom); will easily form negative ions (take
additional electron) in remaining p-shell state due
to large nuclear charge; these elements are very
reactive (e.g., F- with e.a.=3.4 eV)
The Ionic Bond:
Property
Melting point
and boiling point
Electrical
conductivity
Hardness
Brittleness
electrostatic force of attraction between
positively and negatively charged ions
Explanation
The melting and boiling points of ionic compounds
are high because a large amount of thermal
energy is required to separate the ions which are
bound by strong electrical forces.
Solid ionic compounds do not conduct electricity
when a potential is applied because there are no
mobile charged particles.
No free electrons causes the ions to be firmly
bound and cannot carry charge by moving.
Most ionic compounds are hard; the surfaces of
their crystals are not easily scratched. This is
because the ions are bound strongly to the lattice
and aren't easily displaced.
Most ionic compounds are brittle; a crystal will
shatter if we try to distort it. This happens because
distortion cause ions of like charges to come close
together
then sharply repel.
Cl
R
+
Na+
Total energy of ion:
The energy cost to transfer the electron from an alkali to a halogen is
2
E  Ionization Energy (alkali)  Electron Affinity (halogen)
Effective potential    ke + B
n
r
r
ke2
BE  E ( R0 ) 
 E
2nd term is repulsion between 2 e- clouds
R
e.g.: F2, HF
•
•
•
•
The Covalent Bond
The covalent bond is formed by sharing of outer shell electrons between
atoms rather than by electron transfer.
This lowers the energy of the system since electrons are attracted to both
nuclei (stronger effective Coulomb potential)
As an example, consider the H2+ molecular ion (two protons, one e-):
y 1 (r )  Ae r / a y 2 (r )  Ae r / a
1
B
2
B
•
As the distance between the atoms is decreased, significant interference
between the wave functions occur
• In the bonding (symmetric) y+ state electron has a larger probability of being
attracted by both protons – this state is the one responsible for the molecule
formation. Therefore, the bonding state has a lower energy than the
antibonding (antisymmetric).
+
y +y
+
y -y
2.5
+
6
2.0
5
1.5
4
2
1.0
|y(r)|
y(r)
0.5
0.0
3
2
-0.5
1
-1.0
-1.5
0
-2.0
-5
-5
-4
-3
-2
-1
0
r (aB)
1
2
3
4
5
 2
|y +y |
+
 2
|y -y |


-4
-3
-2
-1
0
r(aB)
1
2
3
4
5
Comparison of Ionic and Covalent Bonding
The type of bonding in a solid is determined mainly by the degree of
overlap between the electronic wavefunctions of the atoms involved.
van der Waals
From charge fluctuations in atoms due to zero-point motion (from
Heisenberg uncertainty principle); creates attractive dipole moments
Always present, but significant only when other bonding not possible
Typical strength ~1% of other bonds, short range, varying as r -6
To model the van der Waals interaction,
considered two harmonic oscillators. Each
dipole consists of a pair of opposite charges
with a restoring force acting between each
pair of charges.
We wrote down the Hamiltonian for the
oscillators. Transforming to normal
coordinates decoupled the energy into a
symmetric and antisymmetric
contributions.
Calculated the frequencies and bond energy
C2: Translational Lattice Vectors – 2D
A lattice is a set of points such that a
translation from any point in the lattice by
a vector;
P
Rn = n1 a + n2 b
locates an exactly equivalent point, i.e. a
point with the same environment as P.
This is translational symmetry.
Point D (n1, n2) = (0,2)
Point F (n1, n2) = (0,-1)
Point P (n1, n2) = (3,2)
The vectors a, b are known as lattice
vectors and (n1, n2) is a pair of integers
whose values depend on the lattice point.
Crystal Structure = Crystal Lattice
+ Basis
y
B
C
α
b
O
a) Situation of atoms at the
corners of regular hexagons
a
D
A
E
x
b) Crystal lattice obtained by
identifying all the atoms in (a)
UNIT CELL
Primitive
§ Single lattice point per cell
§ Smallest area in 2D, or
§Smallest volume in 3D
Simple cubic(sc)
Conventional = Primitive cell
Conventional & Non-primitive
§ More than one lattice point per cell
§ Integral multibles of the area of
primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
Crystal Structure
10
Face-centered Cubic (FCC)
• Close-packed planes are
perpendicular to cube
diagonal
• Stacking (ABCAB…)
reduces symmetry to
three-fold
• Four 3-fold rotation axes
+ mirror plane, therefore
Oh (octahedral symmetry)
• Examples: Cu, Ag, Au, Ni,
Pd, Pt, Al
Groups: Fill in this Table for Cubics
SC
BCC
FCC
Volume of conventional cell
a3
a3
a3
Lattice points per cell
1
2
4
Volume, primitive cell
a3
½ a3
¼ a3
# of nearest neighbors
6
8
12
Nearest-neighbor distance
a
½ a 3
a/2
# of second neighbors
12
6
6
Second neighbor distance
a2
a
a
Many common semiconductors have
Diamond or Zincblende crystal structures
Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn).
Basis set: 2 atoms. Lattice  face centered cubic (fcc).
Diamond or Zincblende  2 atoms per fcc lattice point.
Diamond: The 2 atoms are the same. Zincblende: The 2 atoms are different.
The Cubic Unit Cell looks like
For ABCABC… stacking it is called zinc blende
Group: CsCl
• The figure shows the crystal structure of CsCl.
Take the lattice constant as a, all the bonds
shown have the same length. The grey atoms are
Cs and the green ones are Cl.
• What are the primitive Bravais lattice and the
associated basis for this crystal (including the
locations of these atoms in terms of lattice
parameter a)?
• What is the distance to the nearest neighbors of
Cs?
If CsCl is Simple Cubic, what is NaCl?
• CsCl: similar to bcc but atom at
center of cube is different
• NaCl: interpenetrating fcc
structures
– One atom at (0,0,0)
– Second atom displaced by (1/2,0,0)
• Majority of ionic crystals prefer NaCl
structure despite lower
coordination (what is coordination?)
– Radius of cations much smaller than
anions
– For very small cations, anions can not
get too close in CsCl structure
– This favors NaCl structure where
anion contact does not limit structure
as much
CsCl
NaCl

Perovskites
A-site (Ba)
Perovskites
Oxygen
• Superconductors (YBa2Cu3O7-δ)
BaTiO3
• Ferroelectrics (BaTiO3)
• Colossal Magnetoresistance (LaSrMnO3)
• Multiferroics (BiFeO3)
B-site (Ti)
• High εr Insulators (SrTiO3)
• Low εr Insulators (LaAlO3)
• Conductors (Sr2RuO4)
• Thermoelectrics (doped SrTiO3)
• Ferromagnets (SrRuO3)
 Formula unit – ABO3
A atoms (bigger) at the corners
 O atoms at the face centers
 B atoms (smaller) at the body-center
How many atoms
per unit cell?
Reflection Plane
• A plane in a cell such that, when a mirror
reflection in this plane is performed (e.g., x’=x, y’=y, z’=z), the cell remains invariant.
• Mirror plane indicated by symbol m
• Example: water molecule has 2 mirror planes
sv (xz)
sv (yz)
Rotation Axes
• Rotation through an angle about a
certain axis
• Trivial case is 360o rotation
• Order of rotation: 2-, 3-, 4-, and 6correspond to 180o, 120o, 90o, and 60o.
– These are only symmetry rotations allowed
in crystals with long-range order;
incompatible with translational symmetry
– Small aggregates (short-range order) or
molecules can also have 5-, 7-, etc. fold
rotational symmetry
What rotation axes does a cubic
perovskite have?
A-site (Ba)
BaTiO3
B-site (Ti)
Oxygen
The density of lattice points on each plane
of a set is the same and all lattice points
are contained on each set of planes.
z
3a , 2b , 2c
1 1 1
Reciprocal numbers are:
, ,
3 2 2
[2,3,3]
Plane intercepts axes at
2
2
y
Indices of the plane (Miller): (2 3 3)
(No commas, commas are for points)
Indices of the direction: [2,3,3]
x
3
The vector perpendicular to the plane shares
the same coordinates.
Miller indices still apply for a non-cubic system
Distance between the (111)
planes on a cubic lattice
Review: Reciprocal Lattice
The reciprocal lattice is composed of all points lying at positions
from the origin, so that there is one point in the reciprocal lattice for
each set of planes (hkl) in the real-space lattice.
Suppose G can be decomposed into basis vectors: G  hg1 + kg 2 + lg3 (h, k, l integers)
G rn  2m
d hkl
2

Ghkl
gi a j  2ij
Note: a has dimensions of length, g
has dimensions of length-1
Ghkl is perpendicular to (hkl) plane
The basis vectors gi define a reciprocal lattice:
1. for every real lattice there’s a reciprocal lattice
2. reciprocal lattice vector g1 is perpendicular to plane
defined by a2 and a3
g1  2
a  a  a
a 2  a3
a1  a 2  a3 
 is volume of unit cell
+ cyclic permutations
a’s are not unique, but volume is
Constructing the Reciprocal Lattice
1. Identify the basic planes in the direct space lattice.
2. Draw normals to these planes from the origin.
3. Note that distances from the origin along these
normals is proportional to the inverse of the
distance from the origin to the direct space planes.
Reciprocal Lattices to SC, FCC and BCC
Direct lattice
SC
 a1  ax

a 2  ay
a  az
 3
Reciprocal lattice
 b1  2 / a x

b 2  2 / a y
 b  2 / a z
 3
Volume of RL
2 / a 3
Direct
Reciprocal
Simple cubic
Simple cubic
bcc
fcc
fcc
bcc
a1  1 ax + y  z 
2
BCC 
1
a 2  2 a x + y + z 
a  1 ax  y + z 
 3 2
 b1  2 y + z 
a

2
b 2  a x + z 
b  2 x + y 
 3 a
 a1  1 ax + y 
2

FCC a  1 ay + z 
 2 2
 a  1 a z + x 
 3 2
b1  2  x + y  z 
a

2
b 2  a x  y + z 
b  2 x + y  z 
 3 a
42 / a 3
22 / a 3
DIFFRACTION
• Diffraction is a wave phenomenon in which
the apparent bending and spreading of
waves when they meet an obstruction is
measured.
• Light, radio, sound and water waves.
• Diffraction is optimally sensitive to the
periodic nature of the solid’s atomic
structure.
Width Variable
(500-1500 nm)
Wavelength Constant
(600 nm)
Distance d = Constant
Scattering Condition
In a crystal, only significant contributions of this integral arise when G=K.
(Reminder: G is perpendicular to plane.)
I (K)   e
ir( GK )
2
dr
K  k  k0
Note: Real space and
reciprocal space overlapped
We know that G=2/dhkl
=2kosin (from the figure)
ko
source
Thus, to get diffraction:
2/dhkl =2(2  /λ)sin
or λ=2 d sin 
Weigner Seitz Cell: Smallest space enclosed when
intersecting the midpoint to the neighboring lattice points.
graphene
b1
a1
a2
Real Space
2-atom basis
The same perpendicular
bisector logic applies in 3D
b2
k Space
Wigner-Seitz Unit Cell of Reciprocal Lattice
= First Brillouin zone, whose construction
exhibits all the wavevectors k which can be
Bragg-reflected by the crystal
First Brillouin Zone of the FCC Lattice
The BZ
reflects
lattice
symmetry
Note: fcc lattice in reciprocal space is a bcc lattice
SC
BCC
FCC
6
8
12
Nearest-neighbor distance
a
½ a 3
a/2
# of second neighbors
12
6
6
Second neighbor distance
a2
a
a
FCC Primitive and
# of nearest Unit
neighbors
Conventional
Cells
Group: Find the structure factor for FCC.
S hkl
Cubic form:
1 1 
1
 1 1
1
Four atom basis: r  0,0,0 , r   , ,0 , r   ,0,  & r   0, , 
2
2 2 
2
 2 2


SFhkl f 1+ exp  i h + k  + exp  i k + l + exp  i h + l
So:
022
002
S hklF=4f if h,k,l all even or odd
S hklF=0 if h,k,l are mixed even or odd
202
000
Allowed low order reflections are:
111
020
111, 200, 220, 311, 222, 400, 331, 310
Forbidden reflections:
100, 110, 210, 211
200
220
Structure Factor
Ni3Al (L12) structure
Simple cubic lattice, with a four atom basis
1 1 
1
 1 1
1
rAl  0,0,0 , rNi   , ,0 , rNi   ,0,  & rNi   0, , 
2
2 2 
2
 2 2




 
 
F  fal + fNi exp  i h + k  + exp  i k + l  + exp  i h + l 
So :
F=fAl +3fNi
F=fAl -fNi
if h,k,l all even or odd
if h,k,l are mixed even or odd
Again, since simple cubic, intensity at all points.
But each point is ‘chemically sensitive’.
Atomic Scattering Factor f
(aka Structure or Form Factor)
Atoms are of a comparable size to the wavelength of the x-rays and so
the scattering is not point like. There is a small path difference between
waves scattered at either side of the electron cloud. Increases with 
• For x-rays, scattering strength depends on electron density
• Core electrons localized around nucleus, so density profile ~spherical
atom
 iGr  cos 2

r  dr d cos  d
  (r )e
atom
40
Mean Atomic Scattering Factors
 iG r 

f    (r )e
dr   
Zn
30
20
Zr
10
0
Ca
0
0.5
1.0
-1
[sin()]/ (Å )
Only at 2=0 does f=Z
1.5
Diffraction Methods
• Any particle will scatter and create a
diffraction pattern
• Beams are selected by experimentalists
depending on sensitivity
–X-rays not sensitive to low Z elements, but
neutrons are
–Electrons sensitive to surface structure if
energy is low
– Atoms (e.g., helium) sensitive to surface only
Electron
X-Ray
Neutron
λ = 1A°
λ = 1A°
λ = 2A°
E ~ 104 eV
E ~ 0.08 eV
E ~ 150 eV
interact with electron
Penetrating
interact with nuclei
Highly Penetrating
interact with electron
Less Penetrating
Lattice Vibrations
When a wave propagates along one direction, 1D problem.
Use harmonic oscillator approx., meaning amplitude vibration small.
Atoms are tied via bonds, so they can't vibrate independently. The
vibrations take the form of collective modes which propagate.
Phonons are quanta of lattice vibrations.
Longitudinal Waves
Transverse Waves
Monatomic Linear Chain
The force on the nth atom;
a
a
•The force to the right;
K (u n+1  u n )
•The force to the left;
K (u n  u n1 )
Un-1
Un
Un+1
The total force = Force to the right – Force to the left
Thus, Newton’s equation for the nth atom is
mun  K un  un1   K un  un+1 
..
m u n  K (un+1  2un + un1 )
Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies
un  A exp i  kx  t  
0
n
..
u n   2un
Brillouin Zones of the Reciprocal Lattice
Reciprocal
Space Lattice:
2/a

4K
M
k

4
a

3
a

2
a


0
a

a
2
a
3
a
4
a
1st Brillouin Zone (BZ=WS)
Each BZ contains
identical
information
about the lattice
q  20 sinka / 2
2nd Brillouin Zone
3rd Brillouin Zone
0 
K
m
There is no point in saying that 2
adjacent atoms are out of
phase by more than  (e.g., 1.2
=-0.8 )
Modes outside first Brillouin zone
can be mapped to first BZ
m
m
m
m
m  1
λ=10a
m  2
λ=5a
Wave velocity
• GROUP VELOCITY is velocity of energy transfer
• If vphase > vgroup, wave is dispersive
• vphase=k/k
vgroup k   dk / dk  o a coska / 2
• The slope of the dispersion curve gives the group
velocity.
• Near the origin k = 0 the phase and group velocity
must be the same (dispersionless)
• The edges of the FBZ correspond to neighboring
atoms moving in opposite directions. The energy
cannot propagate along the crystal.
vk kmax   0 Standing wave at the boundaries of the BZ (λ=2a)
Diatomic Chain(2 atoms in primitive basis)
2 different types of atoms of masses m1 and m2 are connected by identical springs
(n-2)
(n-1)
K
(n)
K
m1
m2
(n+1)
K
K
m1
(n+2)
m
m2
a)
a
b)
Un-2
Un-1
Un
Un+1
Un+2
Since a is the repeat distance, the nearest neighbors separations is a/2
Two equations of motion must be written;
One for mass m1, and One for mass m2.
m1un1  K 2un1  un2  un1,2 
m2un2  K 2un2  un1  un+1,1 

   +    + 
2
2
1
2
2
2
1

2 2
2

 4  sin qa / 2
2
1
2
2
2
1/ 2
•
As there are two values of ω for each value of k, the dispersion
relation is said to have two branches

A
Optical Branch
Upper branch is due to the
positive sign of the root.
B
C
Acoustical Branch
–л/a
0
л/a
2л/a
k
Negative sign:   k for small k. Dispersionfree propagation of sound waves
• At C, M oscillates and m is at rest.
• At B, m oscillates and M is at rest.
• This result remains valid for a chain containing an arbitrary number of
atoms per unit cell.
A when the two atoms
oscillate in antiphase
Neon, FCC Monatomic
NaCl: FCC, Diatomic
3D Dispersion curves
• Every crystal has 3 acoustic branches, 1
longitudinal and 2 transverse
• Every additional atom in the primitive basis
contributes 3 further optical branches (again 2
transverse and 1 longitudinal)
• P atoms/primitive unit cell means 3 acoustic
branches and 3(p-1) optical branches=3p
branches
• One for each degree of freedom
Stress Tensor
Forces divided by an area are called stresses.
The stresses/tractions tk (or k) along axis k are
... in components we can write this as
 tk1 
 
t k   tk 2 
t 
 k3 
ti  s ij n j
where sij is the stress tensor and nj is a surface normal.
The stress tensor describes the forces acting on planes within a body.
Due to the symmetry condition
s ij  s ji
there are only six independent elements.
s ij
xx

 yx
 zx

xy
yy
zy
xz 

yz 
z z 
The vector normal to the corresponding surface
The direction of the force vector acting on that surface
 s11 s12 s13   s11

 
s 21 s 22 s 23   s12
s s s  s
 i  Sijs j
 31 32 33   13
Similarly, the strain tensor can be written as:
We can therefore write:
s i  Cij  j
s12 s13 

s 22 s 23 
s 23 s 33 
 11 12 13   11 12 13 

 

  21  22  23    12  22  23 
       
 31 32 33   13 23 33 
Voigt’s notation:
Additional simplification of the stress-strain
11  1 23  4
relationship can be realized through
simplifying the matrix notation
22  2 13  5
for stresses and strains.
33  3 12  6
We can replace the indices as follows:
• For the generalized case, Hooke’s law may be expressed
as: s  C  where,
i
ij j
C  Stiffness (or Elastic cons tan t )
 i  Sijs j
S  Compliance
• Both Sijkl and Cijkl are fourth-rank tensor quantities.
• The consequence of the symmetry in the stress and strain
tensors is that only 36 components of the compliance
and stiffness tensors are independent and distinct terms.
 s 1   C11
  
 s 2   C21
s   C
 3    31
 s 4   C41
s   C
 5   51
 s 6   C61
C12
C22
C13 C14
C23 C24
C32
C42
C52
C62
C33
C43
C53
C63
C34
C44
C54
C64
C15 C16  1  But only one-half of the
 
C25 C26   2  non-diagonal terms are
C35 C36   3  independent constants
 
since Cij = Cji
C45 C46   4 
30



C55 C56  5
+ 6  21
 
2
C65 C66   6 
Independent terms
abc
      90o
The cubic axes are equivalent, so the diagonal
components for normal and shear distortions must
be equal.
And cubic is not elastically isotropic because a
deformation along a cubic axis differs from the stress
arising from a deformation along the diagonal.
e.g., [100] vs. [111]
Zener Anisotropy Ratio:
2C44
C44
A

C11  C12
x
x=(a-b)/2 or
C44 
C11  C12
2
• These quantized normal modes of vibration are called
PHONONS
• PHONONS are massless quantum mechanical particles which
have no classical analogue.
– They behave like particles in momentum space or k space.
• Phonons are one example of many like this in many different areas of
physics. Such quantum mechanical particles are often called
“Quasiparticles”
Examples of other Quasiparticles:
Photons: Quantized Normal Modes of electromagnetic waves.
Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids
Excitons: Quantized Normal Modes of electron-hole pairs
Polaritons: Quantized Normal Modes of electric polarization excitations in solids
+ Many Others!!!
Phonon spectroscopy =
Conditions for: inelastic scattering
Constraints:
Conservation laws of
Momentum
Energy
In all interactions involving phonons, energy must be conserved and crystal
momentum must be conserved to within a reciprocal lattice vector.