1. 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”
+Ze
electron
Screening electron cloud r
The effective potential energy felt by an electron
Zeff is the effective charge that the electron feels and depends on r. Note that
when r is inside all other electrons
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

2. The Periodic Table
Columns: groups with similar shells, similar properties
Rows: periods with elements with increasingly-full shells
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)
Closed-shell –plus one (alkali) elements: reactive due to loosely-bound outer electron in s-shell

3. The Ionic Bond: Explanation Property
electrostatic force of attraction between positively and negatively charged ions
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.
Brittleness
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.
Hardness
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.
Electrical
conductivity
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.
Melting point
and boiling point
Explanation
Property +
Na+ -
Cl- R
Total energy of ion:
The energy cost to transfer the electron from an alkali to a halogen is
Effective potential
2nd term is repulsion between 2 e- clouds

4. The Covalent Bond
e.g.: F2, HF
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-):
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).

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.

6. 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
1.11, used to model DNA

8. 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;
Rn = n1 a + n2 b
locates an exactly equivalent point, i.e. a point with the same environment as P. This is translational symmetry.
The vectors a, b are known as lattice vectors and (n1, n2) is a pair of integers whose values depend on the lattice point. P
Point D (n1, n2) = (0,2)
Point F (n1, n2) = (0,-1)
Point P (n1, n2) = (3,2)

9. Crystal Structure = Crystal Lattice + Basisα a b C B E D O A y x
a) Situation of atoms at the corners of regular hexagons
b) Crystal lattice obtained by identifying all the atoms in (a)

10. Conventional = Primitive cell Body centered cubic(bcc)
Simple cubic(sc)
Conventional = Primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
Crystal Structure

11. 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

12. Groups: Fill in this Table for CubicsSC
BCC
FCC
Volume of conventional cell a3
Lattice points per cell 1 2 4
Volume, primitive cell
½ a3
¼ a3
# of nearest neighbors 6 8 12
Nearest-neighbor distance a
½ a 3
a/2
# of second neighbors
Second neighbor distance
a2

13. The Cubic Unit Cell looks like
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

14. 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?
1V3

15. 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

16. How many atoms per unit cell?
Perovskites
Perovskites
A-site (Ba)
Oxygen
B-site (Ti)
BaTiO3
Superconductors (YBa2Cu3O7-δ)
Ferroelectrics (BaTiO3)
Colossal Magnetoresistance (LaSrMnO3)
Multiferroics (BiFeO3)
High εr Insulators (SrTiO3)
Low εr Insulators (LaAlO3)
Conductors (Sr2RuO4)
Thermoelectrics (doped SrTiO3)
Ferromagnets (SrRuO3)
Our interface work focuses in the field of complex oxides. In addition to being among the most abundant minerals on earth, complex oxides give some of the most varied and interesting properties. These include their use as dielectric and superconducting materials. Yet, only recently has the research in the field of complex oxides flourished because they were long thought to be …well complex. This complexity comes from the strong coupling with charge, spin, and lattice dynamics, which often results in very full phase diagrams.
Though the coupling may lead to complex behaviors, the structure of these materials can be quite simple, such as the perovskite form shown here, where the A and B sites are typically different cations and X is an anion that bonds to both. This arrangement gives rise to a crystal field potential, hinders the free rotation of the electrons and quenches the orbital angular momentum by introducing the crystal field splitting of the d orbitals. Even among only perovskites, we can see these varied behaviors.
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?

17. Mirror plane indicated by symbol m
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)

18. Rotation Axes Rotation through an angle about a certain axis
Trivial case is 360o rotation
Order of rotation: 2-, 3-, 4-, and 6- correspond 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

19. What rotation axes does a cubic perovskite have?
A-site (Ba)
Oxygen
B-site (Ti)
BaTiO3

20. 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
[2,3,3]
Plane intercepts axes at 2
Reciprocal numbers are:
Indices of the plane (Miller): (2 3 3)
(No commas, commas are for points) 2 y
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

21. Distance between the (111) planes on a cubic lattice

22. 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:
(h, k, l integers)
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
+ cyclic permutations
is volume of unit cell
a’s are not unique, but volume is

23. Constructing the Reciprocal Lattice
Identify the basic planes in the direct space lattice.
Draw normals to these planes from the origin.
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.

24. Reciprocal Lattices to SC, FCC and BCC
Direct lattice Reciprocal lattice Volume of RL SC
BCC
FCC
Direct
Reciprocal
Simple cubic
bcc
fcc

25. 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
( nm)
Wavelength Constant
(600 nm)
Distance d = Constant

26. Scattering Condition We know that G=2/dhkl =2kosin (from the figure)
In a crystal, only significant contributions of this integral arise when G=K.
(Reminder: G is perpendicular to plane.)
Detector
Note: Real space and reciprocal space overlapped
We know that G=2/dhkl
=2kosin (from the figure)
Go through again. We will talk about what  is today.
source ko
Thus, to get diffraction:
2/dhkl =2(2  /λ)sin
or λ=2 dhkl sin 

27. Weigner Seitz Cell: Smallest space enclosed when intersecting the midpoint to the neighboring lattice points.
graphene b1 a1 a2 b2
Real Space
2-atom basis
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
The same perpendicular bisector logic applies in 3D

28. 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
# of nearest neighbors 6 8 12
Nearest-neighbor distance a
½ a 3
a/2
# of second neighbors
Second neighbor distance
a2
FCC Primitive and Conventional Unit Cells

29. Group: Find the structure factor for FCC.
Cubic form:
002
022
220
020
200
202
000
111
Allowed low order reflections are:
111, 200, 220, 311, 222, 400, 331, 310
Forbidden reflections:
100, 110, 210, 211

30. Structure Factor Ni3Al (L12) structure
Simple cubic lattice, with a four atom basis
Again, since simple cubic, intensity at all points.
But each point is ‘chemically sensitive’.

31. 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
Only at 2=0 does f=Z

32. 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

33. 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

34. Monatomic Linear Chain
The force on the nth atom; a a
The force to the right;
The force to the left;
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
Diatomic benzet
Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies

35. Brillouin Zones of the Reciprocal Lattice
Reciprocal Space Lattice:
2p/a
3rd Brillouin Zone
2nd Brillouin Zone
1st Brillouin Zone (BZ=WS)
Each BZ contains identical information about the lattice
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

36. λ=10a
λ=5a

37. Standing wave at the boundaries of the BZ (λ=2a)
Wave velocity
GROUP VELOCITY is velocity of energy transfer
If vphase > vgroup, wave is dispersive
vphase=k/k
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.
Standing wave at the boundaries of the BZ (λ=2a)

38. 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) (n) (n+1) (n+2) K K K K m2 a) m1 m1 m m2 a b)
Un-1 Un
Un+1
Un+2
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.

39. A when the two atoms oscillate in antiphase
As there are two values of ω for each value of k, the dispersion relation is said to have two branches
Optical Branch л / a 2 –л k w A B C
Upper branch is due to the
positive sign of the root.
Acoustical Branch
Negative sign:   k for small k. Dispersion-free 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

40. Neon, FCC Monatomic
NaCl: FCC, Diatomic

41. 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

42. 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
where ij 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
there are only six independent elements.
The vector normal to the corresponding surface
The direction of the force vector acting on that surface

43. We can therefore write:
Similarly, the strain tensor can be written as:
Additional simplification of the stress-strain relationship can be realized through simplifying the matrix notation
for stresses and strains.
We can replace the indices as follows:
Voigt’s notation:

44. For the generalized case, Hooke’s law may be expressed as: where,
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.
But only one-half of the non-diagonal terms are independent constants since Cij = Cji
Independent terms

45. 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:
x=(a-b)/2 or

46. 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!!!

47. Conditions for: elastic scattering
Phonon spectroscopy =
Conditions for: elastic scattering in
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.