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”

The effective potential energy felt by an electron

rkerZrU eff

2

)()(

Zeff is the effective charge that the electron feels and depends on r. Note that

ZZeff

1effZ

when r is inside all other electrons

when r is outside all other electrons

+Ze

electron

Screening electron

cloud

r

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 propertiesRows: 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)

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.

Electricalconductivity

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

ExplanationProperty

The Ionic Bond:

Effective potential nrB

rke

+2

(halogen)Affinity Electron (alkali)Energy Ionization E

ERkeREBE

0

2

0 )(2nd term is repulsion between 2 e- clouds

The energy cost to transfer the electron from an alkali to a halogen isTotal energy of ion:

+Na+

-Cl-

R

electrostatic force of attraction between positively and negatively charged ions

• 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).

The Covalent Bond

BarAe /1

1)( ry BarAe /2

2)( ry

-5 -4 -3 -2 -1 0 1 2 3 4 5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

y(r)

r (aB)

y++y

y+-y

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

1

2

3

4

5

6

|y++y|2

|y+-y|2

|y (r

)|2

r(aB)

e.g.: F2, HF

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 WaalsFrom charge fluctuations in atoms due to zero-point motion (from Heisenberg uncertainty principle); creates attractive dipole momentsAlways present, but significant only when other bonding not possibleTypical 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;

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)

α

a

bCB ED

O A

y

x

b) Crystal lattice obtained by identifying all the atoms in (a)

a) Situation of atoms at the corners of regular hexagons

Crystal Structure = Crystal Lattice + Basis

Crystal Structure 10

Body centered cubic(bcc)Conventional ≠ Primitive cell

Simple cubic(sc)Conventional = Primitive cell

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 CubicsSC 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 a2 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

NaCl

CsCl

PerovskitesPerovskites

• 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)

A-site (Ba) Oxygen

B-site (Ti)

BaTiO3

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 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

What rotation axes does a cubic perovskite have?

A-site (Ba) Oxygen

B-site (Ti)

BaTiO3

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.

Reciprocal numbers are: 21 ,

21 ,

31

Plane intercepts axes at cba 2 ,2 ,3

Indices of the plane (Miller): (2 3 3)(No commas, commas are for points)

Indices of the direction: [2,3,3]3

2

2[2,3,3]

The vector perpendicular to the plane shares the same coordinates.

Miller indices still apply for a non-cubic system

x

y

z

Distance between the (111) planes on a cubic lattice

Review: Reciprocal Lattice

Suppose G can be decomposed into basis vectors: 321 gggG lkh ++ (h, k, l integers)

mn 2rGijji 2ag

The basis vectors gi define a reciprocal lattice: 1. for every real lattice there’s a reciprocal lattice2. reciprocal lattice vector g1 is perpendicular to plane defined by a2 and a3

Note: a has dimensions of length, g has dimensions of length-1

321

321 2

aaaaag

+ cyclic permutations

321 aaa is volume of unit cell a’s are not unique, but volume is

Ghkl is perpendicular to (hkl) planehkl

hkl Gd 2

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.

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 BCCDirect lattice Reciprocal lattice

Volume of RL

SC

BCC

FCC

zayaxa

aaa

3

2

1

+

+

+

xzazyayxa

a

a

a

21

3

21

2

21

1

+

++

+

zyxazyxa

zyxa

a

a

a

21

3

21

2

21

1

zbybxb

aaa

/2/2/2

3

2

1

+

+

+

yxb

zxb

zyb

a

a

a

23

22

21

+

+

+

zyxb

zyxb

zyxb

a

a

a

23

22

21

3/2 a

3/22 a

3/24 a

Direct Reciprocal

Simple cubic Simple cubic

bcc fcc

fcc bcc

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

Detector

2)()( rK r deI i KG

0kkK

source

In a crystal, only significant contributions of this integral arise when G=K.(Reminder: G is perpendicular to plane.)

ko

Note: Real space and reciprocal space overlapped

We know that G=2/dhkl =2kosin (from the figure)

Thus, to get diffraction: 2/dhkl =2(2 /λ)sin

or λ=2 dhkl sin

graphene

Real Space

2-atom basis

a2

a1

b2

b1

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

Weigner Seitz Cell: Smallest space enclosed when intersecting the midpoint to the neighboring lattice

points.

k Space

First Brillouin Zone of the FCC Lattice

FCC Primitive and Conventional Unit Cells

SC BCC FCC

# of nearest neighbors 6 8 12

Nearest-neighbor distance a ½ a 3 a/2

# of second neighbors 12 6 6

Second neighbor distance a2 a a

Note: fcc lattice in reciprocal space is a bcc lattice

The BZ reflects lattice

symmetry

Four atom basis: r 0,0,0 , r 12,12,0

, r 1

2,0,12

& r 0,12,12

F f 1+exp i h+k

+exp i k+ l

+exp i h+ l

So: F=4f if h,k,l all even or odd F=0 if h,k,l are mixed even or odd

Group: Find the structure factor for FCC.

hklS

hklShklS

002 022

220

020

200

202

000 111Allowed low order reflections are:111, 200, 220, 311, 222, 400, 331, 310

Forbidden reflections:100, 110, 210, 211

Cubic form: hklS

Structure FactorNi3Al (L12) structure

rAl 0,0,0 , rNi 12

,12

,0

, rNi

12

,0,12

& rNi 0,1

2,12

F fal + fNi exp i h+k

+exp i k+ l

+exp i h+ l

So: F=fAl+3fNi if h,k,l all even or odd F=fAl-fNi if h,k,l are mixed even or odd

Simple cubic lattice, with a four atom basis

Again, since simple cubic, intensity at all points. But each point is ‘chemically sensitive’.

Atomic Scattering Factor f(aka Structure or Form Factor)

Only at 2=0 does f=Z

0

10

20

30

40

0 0.5 1.0 1.5

Zr

Zn

Ca

[sin()]/ (Å-1)

Mea

n A

tom

ic S

catte

ring

Fact

ors

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

)( rr rG def i

atom

2cos cos)( ddrdre riGr

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

Lattice Vibrations

Longitudinal Waves

Transverse Waves

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.

The force on the nth atom;

)( 1 nn uuK +

• The force to the right;

• The force to the left;

)( 1 nn uuKThe total force = Force to the right – Force to the left

0)2( 11

..+ + nnnn uuuCum

a a

Un-1 Un Un+1

Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies

Monatomic Linear Chain

)2( 11

..

+ + nnnn uuuKum

Thus, Newton’s equation for the nth atom is

11 + nnnnn uuKuuKum

0expn nu A i kx t ..

2n nu u

Brillouin Zones of the Reciprocal Lattice

1st Brillouin Zone (BZ=WS)

k

a

a2

a

a2

0

MK4

a3

a4

a3

a4

2nd Brillouin Zone

3rd Brillouin Zone

Each BZ contains identical

information about the lattice

2/a

Reciprocal Space 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 2/sin2 0 kaq m

K0

m

m

m

m 1mλ=10a

2m λ=5a

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.

2/cos/ kaadkdkv okgroup

0max kvk 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

Un-2Un-1 Un Un+1 Un+2

K K K Km1 m1m2 m2m a)

b)

(n-2) (n-1) (n) (n+1) (n+2)

a

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.

2,12111 2 nnnn uuuKum 1,11222 2 + nnnn uuuKum

•As there are two values of ω for each value of k, the dispersion relation is said to have two branches

Upper branch is due to thepositive sign of the root.

Negative sign: k for small k. Dispersion-free propagation of sound waves

Optical Branch

Acoustical Branch

• This result remains valid for a chain containing an arbitrary number of atoms per unit cell.

0 л/a 2л/a–л/a k

A

BC

2/1 22

221

222

21

22

21

2 2/sin4 qa ++

A when the two atoms oscillate in antiphase

• At C, M oscillates and m is at rest.• At B, m oscillates and M is at rest.

NaCl: FCC, Diatomic

Neon, FCC Monatomic

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 TensorForces divided by an area are called stresses.

The stresses/tractions tk (or k) along axis k are

3

2

1

k

k

k

k

ttt

t

... in components we can write this as

jiji nt swhere 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

jiij ss

there are only six independent elements.

ijs The vector normal to the corresponding surface

The direction of the force vector acting on that surface

zzyy

xx

zyzxyzyxxzxy

We can therefore write:

332313

232212

131211

333231

232221

131211

sssssssss

sssssssss

Similarly, the strain tensor can be written as:

332313

232212

131211

333231

232221

131211

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:

333222

111

612513423

jiji

jiji

S

C

s

s

Voigt’s notation:

• 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.

jiji

jiji

S

C

s

s

ComplianceStconsElasticorStiffnessC

)tan(

6

5

4

3

2

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

6

5

4

3

2

1

ssssss

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC But only one-half of the

non-diagonal terms are independent constants

since Cij = Cji

Independent terms

2162

30 +

o

cba

90

x=(a-b)/2 or

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]

2CCC 1211

44x

CCC

CA 44

1211

442

Zener Anisotropy Ratio:

• 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 solidsExcitons: Quantized Normal Modes of electron-hole pairs

Polaritons: Quantized Normal Modes of electric polarization excitations in solids

+ Many Others!!!

Phonon spectroscopy =

Constraints:Conservation laws of

Momentum Energy

Conditions for: elastic scattering in

In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector.