Oklahoma StateUniversity Lecture 3: Bonding, molecular and lattice vibrations:http://physics.okstate.edu/jpw519/phys5110/index.htm
Revisit 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 nearMolecular orbital.
Crystalline SolidsPeriodicity of crystal leads to the following properties of thewave function: 1-dim. ψ(x+L)= ψ(x); ψ‘(x+L)= ψ‘(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.) Ground state: T=0 Particles and Holes: T>0 K-space mg 2d ( EF ) = π 2
Distribution functions for T>0•Particle-hole excitations are increased as T increases•Particles are promoted from within k T of E to an unoccupied B F single particle state with E>EF•Particles are not promoted from deep within Fermi SeaProbability of finding a single-particle (orbital) state of particularspin with energy E is given by Fermi-Dirac distribution 1 f ( E , µ ,T ) = E −µ k T e B +1 µ-chemical potential
Fermi-Dirac (FD) DistributionAs T 0, FD distribution approaches a step functionFermi gas described by a FD distribution that’s almoststep like is termed degenerate T=0
X-RAYS TO CONFIRM CRYSTAL STRUCTURE• Incoming X-rays diffract from crystal planes. de te c ” to “1 in ra r ys co ys reflections must X- ra m be in phase to X- ” “2 in detect signal “1 g g in λ Adapted from Fig. ” extra o g “2 distance θ ut θ ” 3.2W, Callister 6e. travelled o by wave “2” spacing d between planes• Measurement of: x-ray Critical angles, θc, intensity d=nλ/2sinθc (from for X-rays provide detector) atomic spacing, d. θ θc 20
X-Ray Diffraction nλ = S Q + Q T = 2d hkl sin θ ad hkl = 2 h +k +l 2 2
THE PERIODIC TABLE• Columns: Similar Valence Structure inert gases give up 1e give up 2e accept 2e accept 1e give up 3e Metal Nonmetal H He Li Be Intermediate Ne O F Na Mg Adapted S Cl Ar from Fig. 2.6, K Ca Sc Se Br Kr Callister 6e. Rb Sr Y Te I Xe Cs Ba Po At Rn Fr Ra Electropositive elements: Electronegative elements: Readily give up electrons Readily acquire electrons to become + ions. to become - ions. 6
IONIC BONDING• Occurs between + and - ions.• Requires electron transfer.• Large difference in electronegativity required.• Example: NaCl Na (metal) Cl (nonmetal) unstable unstable electron Na (cation) + - Cl (anion) stable Coulombic stable Attraction 8
COVALENT BONDING• Requires shared electrons• Example: CH4 shared electrons H C: has 4 valence e, from carbon atom CH4 needs 4 more H: has 1 valence e, H C H needs 1 more shared electrons Electronegativities H from hydrogen are comparable. atoms Adapted from Fig. 2.10, Callister 6e.
METALLIC BONDING• Arises from a sea of donated valence electrons (1, 2, or 3 from each atom). + + + Electrons are “delocalized” + + + •Electrical and thermal conductor •Ductile + + +• Primary bond for metals and their alloys 12
SECONDARY BONDINGArises from interaction between dipoles• Fluctuating dipoles asymmetric electron ex: liquid H2 clouds H2 H2 + - secondary + - H H H H secondary bonding Adapted from Fig. 2.13, Callister 6e. bonding• Permanent dipoles-molecule induced Adapted from Fig. 2.14, secondary -general case: + - + - Callister 6e. bonding secondary Adapted from Fig. 2.14, -ex: liquid HCl H Cl bonding H Cl Callister 6e. secon -ex: polymer dary bond ing 13
Secondary bonding or physical bonds Van der Waals, Hydrogen bonding, Hyrophobic bonding• Self assembly – how biology builds…• DNA hybridization• Molecular recognition (immuno- processes, drug delivery etc. )
SUMMARY: PRIMARY BONDSCeramics Large bond energy(Ionic & covalent bonding): large Tm large EMetals Variable bond energy(Metallic bonding): moderate Tm moderate EPolymers Directional Properties(Covalent & Secondary): Secondary bonding dominates small T secon dary bond small E ing 18
Oklahoma State Energy bands in crystalsUniversity More on this next lecture!! 2 2 jk ⋅r − 2m ∇ + V (r )φ k ( r ) = Eφ k ( r ) φ k (r ) = e U n (k , r ) (Bloch function) Ref: S.M. Sze: Semiconductor Devices Ref: M. Fukuda, Optical Semiconductor Devices
Interatomic ForcesNet Forces Fr = − dE / dr E = ∫ Fdr Potential Energy: E
ENERGY AND PACKING• Non dense, random packing Energy typical neighbor bond length typical neighbor r bond energy• Dense, regular packing Energy typical neighbor bond length typical neighbor r bond energy Dense, regular-packed structures tend to have lower energy. 2
PROPERTIES FROM BONDING: TM• Bond length, r • Melting Temperature, TmF F Energy (r) r• Bond energy, Eo ro r Energy (r) smaller Tm unstretched length ro larger Tm r Eo= Tm is larger if Eo is larger. “bond energy” 15
PROPERTIES FROM BONDING: C• Elastic modulus, C cross sectional length, Lo Elastic modulus area Ao undeformed F ∆L ∆L =C Ao Lo deformed FEnergy• C ~ curvature at ro E is larger if Eo is larger. unstretched length ro r smaller Elastic Modulus larger Elastic Modulus 16
Vibrational frequencies of moleculesFor small vibrations, can use the Harmonic approximation: ∂2E E (r ) = Eo (ro ) + 2 ( r − ro ) 2 ∂r r o where ( r − ro ) Represents small oscillations from ro Oscillation frequency of two k masses connected by spring m11 m2 ∂2E ω=(k/ µ)1/2 where k= 2 ∂r ro µ=m1m2/(m1+m2)-reduced mass
Quantized total energy (kinetic + potential): n + 1 ω where n = 0,1, 2,... 2 Vibrational energies of molecules ω[1013 Hz] µ[10-27 kg] k [N/m] C2H2 C~~H 8.64 1.53 450 C C H H C2D2 C~~D 6.42 2.85 463 C16O 12 C~~O 5.7 11.4 1460 C O C18O 13 C~~O 5.41 12.5 1444
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) 1 M PE = ∑ k ( un+1 − un ) 2 KE = ∑ un 2 2n 2 n d 2u n M 2 = k ( un+1 + un−1 − 2un ) dt un-1 un un+1 k k k k k k k