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OklahomaOklahomaState State
UniversityUniversity
Lecture 3:
Bonding, molecular and lattice vibrations:
http://physics.okstate.edu/jpw519/phys5110/index.htm
Revisit 1-dim. caseRevisit 1-dim. caseLook at a 30 nm segment 0f a single walledcarbon nanotube (SWNT)Use STM noting that tunneling current is proportional toLocal density of states (higher conductance when near Molecular orbital.
Crystalline SolidsCrystalline Solids
Periodicity of crystal leads to the following properties of the wave function: 1-dim. (x+L)= (x); ‘(x+L)= ‘(x)
In 2-dim.
Periodic Boundary Conditions in a solid leads to traveling waves instead of standing waves
Excitations in Ideal Fermi Gas (2-dim.)
K-space
22 m
Eg Fd )(
Ground state: T=0 Particles and Holes: T>0
Distribution functions for T>0Distribution functions for T>0
•Particle-hole excitations are increased as T increases
•Particles are promoted from within kBT of EF to an unoccupied single particle state with E>EF
•Particles are not promoted from deep within Fermi Sea
Probability of finding a single-particle (orbital) state of particularspin with energy E is given by Fermi-Dirac distribution
-chemical potential
1
1
Tk
E
Be
TEf
),,(
Fermi-Dirac (FD) DistributionFermi-Dirac (FD) DistributionAs T 0, FD distribution approaches a step functionFermi gas described by a FD distribution that’s almost step like is termed degenerate
T=0
Crystal SystemsCubica=b=c°
Hexagonala=b≠c°°
Tetragonala=b≠c°
Rhombohedrala=b=c=≠90°
Orthorhombica≠b≠ca=b=g=90°
Monoclinica≠b≠c°≠
Triclinica≠b≠c≠≠≠
11
• ABCABC... Stacking Sequence• 2D Projection
A sites
B sites
C sitesB B
B
BB
B BC C
CA
A
• FCC Unit CellA
BC
FCC STACKING SEQUENCE
Point Coordinates
Crystallographic Directions[u,v,w] (integers)
d=n/2sinc
x-ray intensity (from detector)
c20
• Incoming X-rays diffract from crystal planes.
• Measurement of: Critical angles, c, for X-rays provide atomic spacing, d.
Adapted from Fig. 3.2W, Callister 6e.
X-RAYS TO CONFIRM CRYSTAL STRUCTURE
reflections must be in phase to detect signal
spacing between planes
d
incoming
X-rays
outg
oing
X-ra
ys
detector
extra distance travelled by wave “2”
“1”
“2”
“1”
“2”
X-Ray Diffraction sinhkldTQQSn 2
222 lkh
adhkl
6
• Columns: Similar Valence Structure
Electropositive elements:Readily give up electronsto become + ions.
Electronegative elements:Readily acquire electronsto become - ions.
He
Ne
Ar
Kr
Xe
Rn
iner
t ga
ses
ac
cept
1e
ac
cept
2e
give
up
1e
give
up
2e
give
up
3e
F Li Be
Metal
Nonmetal
Intermediate
H
Na Cl
Br
I
At
O
S Mg
Ca
Sr
Ba
Ra
K
Rb
Cs
Fr
Sc
Y
Se
Te
Po
Adapted from Fig. 2.6, Callister 6e.
THE PERIODIC TABLE
Na (metal) unstable
Cl (nonmetal) unstable
electron
+ - Coulombic Attraction
Na (cation) stable
Cl (anion) stable
8
• Occurs between + and - ions.• Requires electron transfer.• Large difference in electronegativity required.• Example: NaCl
IONIC BONDING
• Requires shared electrons• Example: CH4
C: has 4 valence e, needs 4 more
H: has 1 valence e, needs 1 more
Electronegativities are comparable.
shared electrons from carbon atom
shared electrons from hydrogen atoms
H
H
H
H
C
CH4
Adapted from Fig. 2.10, Callister 6e.
COVALENT BONDING
12
• Arises from a sea of donated valence electrons (1, 2, or 3 from each atom).
• Primary bond for metals and their alloys
+ + +
+ + +
+ + +
METALLIC BONDING
Electrons are “delocalized”
•Electrical and thermal conductor
•Ductile
13
Arises from interaction between dipoles
• Permanent dipoles-molecule induced
• Fluctuating dipoles
+ - secondary bonding + -
H Cl H Clsecondary bonding
secondary bonding
HH HH
H2 H2
secondary bonding
ex: liquid H2asymmetric electron clouds
+ - + -secondary bonding
-general case:
-ex: liquid HCl
-ex: polymer
Adapted from Fig. 2.13, Callister 6e.
Adapted from Fig. 2.14, Callister 6e.
Adapted from Fig. 2.14, Callister 6e.
SECONDARY BONDING
Secondary bonding or physical bondsVan der Waals, Hydrogen bonding,
Hyrophobic bonding
• Self assembly – how biology builds…
• DNA hybridization
• Molecular recognition (immuno- processes, drug delivery etc. )
18
Ceramics(Ionic & covalent bonding):
Metals(Metallic bonding):
Polymers(Covalent & Secondary):
secondary bonding
Large bond energylarge Tm
large E
Variable bond energymoderate Tm
moderate E
Directional PropertiesSecondary bonding dominates
small Tsmall E
SUMMARY: PRIMARY BONDS
14
Type
Ionic
Covalent
Metallic
Secondary
Bond Energy
Large!3-5 eV/atom
Variablelarge-Diamondsmall-Bismuth1-7 ev/atom
Variablelarge-Tungstensmall-Mercury0.7-9 eV/atom
Smallest.05-0.5 ev/atom
Comments
Nondirectional (ceramics,NaCl, CsCl)
Directionalsemiconductors, ceramicsDiamond, polymer chains)
Nondirectional (metals)
Directionalinter-chain (polymer)
inter-molecular
SUMMARY: BONDING
Energy bands in crystalsMore on this next lecture!!
(Bloch function)
Ref: S.M. Sze: Semiconductor Devices Ref: M. Fukuda, Optical Semiconductor Devices
)()()(2
22
rErrVm kk
),()( rkUer n
rkjk
OklahomaOklahomaState State
UniversityUniversity
Interatomic Forces
Net Forces
Potential Energy: E
FdrE
drdEFr /
Potential Energy CurveE(r)
2
• Non dense, random packing
• Dense, regular packing
Dense, regular-packed structures tend to have lower energy.
Energy
r
typical neighbor bond length
typical neighbor bond energy
Energy
r
typical neighbor bond length
typical neighbor bond energy
ENERGY AND PACKING
15
• Bond length, r
• Bond energy, Eo
F F
r
• Melting Temperature, Tm
Eo=
“bond energy”
Energy (r)
ro r
unstretched length
r
larger Tm
smaller Tm
Energy (r)
ro
Tm is larger if Eo is larger.
PROPERTIES FROM BONDING: TM
16
• Elastic modulus, C
• C ~ curvature at ro
cross sectional area Ao
L
length, Lo
F
undeformed
deformed
L F Ao
= C Lo
Elastic modulus
r
larger Elastic Modulus
smaller Elastic Modulus
Energy
ro unstretched length
E is larger if Eo is larger.
PROPERTIES FROM BONDING: C
Vibrational frequencies of moleculesFor small vibrations, can use the Harmonic approximation:
where orr Represents small oscillations from ro
=(k/ )1/2 where k=
Oscillation frequency of two masses connected by spring
=m1m2/(m1+m2)-reduced mass
m11 m2
k
orrE
2
2
22
2
ooo rrrE
rErEor
)()(
Quantized total energy (kinetic + potential):
,...,, 21021
nwheren
C2H2 C~~H 8.64 1.53 450
C2D2 C~~D 6.42 2.85 463
12C16O C~~O 5.7 11.4 1460
13C18O C~~O 5.41 12.5 1444
C O
CH C
[1013 Hz] [10-27 kg] k [N/m]
H
Vibrational energies of molecules
k k k kk k k
un-1 un un+1
Lattice vibrations in Crystals•Equilibrium positions of atoms on lattice points (monatomic basis)•Small displacements from equilibrium positions•Harmonic Approximation•Vibrations of atoms slow compared to motion of electrons- Adiabatic Approximation•Waves of vibration in direction of high symmetry of crystal – q•Nearest neighbor interactions (Hooke’s Law)
n
nn uukPE 2
121
nnnn uuuk
dtud
M 2112
2
n
nuM
KE 2
2