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IN A DIATOMIC MOLECULEbecause a photon has one unit of
angular momentum,the rotational quantum number
must change(vib is not angular motion)
r = 0 not allowed
** + Vibration + Rotation
Ocean Optics: Nitrogen N2
~ 0.3 eV
~ 0.4 eV
Some Molecular ConstantsMolecule Equilibrium
Distance
Ro (Å)
Dissociation NRG
Do (eV)
Vibrational freq
v a (cm-1)
Moment of Inertia
Bb (cm-1)
H2+ 1.06 2.65 2297 29.8
H2 0.742 4.48 4395 60.8
O2 1.21 5.08 1580 1.45
N2 1.09 9.75 2360 2.01
CO 1.13 9.60 2170 1.93
NO 1.15 5.3 1904 1.70
HCl 1.28 4.43 2990 10.6
NaCl 2.36 4.22 365 0.190
Notes: a) vibrational frequency in table is given as f / c b) moment of inertia in table is given as hbar2/(2I) / hc
Types of Solids
Harris 9.4
TYPES OF SOLIDS (ER 13.2)
CRYSTALINE BINDING
• molecular
• ionic
• covalent
• metallic
Molecular Solids
• orderly collection of molecules held together by v. d. Waals• gases solidify only at low Temps• easy to deform & compress• poor conductors
most organicsinert gases
O2 N2 H2
Ionic Solids• individ atoms act like closed-shell, spherical, therefore binding not so directional• arrangement so that minimize nrg for size of atoms
• tight packed arrangement poor thermal conductors• no free electrons poor electrical conductors• strong forces hard & high melting points• lattice vibrations absorb in far IR• to excite electrons requires UV, so ~transparent visible
NaClNaIKCl
Covalent Solids
• 3D collection of atoms bound by shared valence electrons
• difficult to deform because bonds are directional• high melting points (b/c diff to deform)• no free electrons poor electrical conductors• most solids adsorb photons in visible opaque
Ge Si
diamond
Metallic Solids
• (weaker version of covalent bonding)• constructed of atoms which have very weakly
bound outer electron• large number of vacancies in orbital (not enough
nrg available to form covalent bonds)• electrons roam around (electron gas )• excellent conductors of heat & electricity• absorb IR, Vis, UV opaque
Fe Ni Co
config dhalf full
Free-Electron Model and
Band Formation
Harris 9.5
‘Free-Electron’ Models
• Free Electron Model • Nearly-Free Electron Model
– Version 1 – SP221– Version 2 – SP324a– Version 3 – SP324b
• graphite is conductor, diamond is insulator• variation in colors of x-A elements• temperature dependance of resistivity• resistivity can depend on orientation of crystal & current I direction• frequency dependance of conductivity• variations in Hall effect parameters• resistance of wires effected by applied B-fields
• .• .• .
Nearly-Free Electron Modelversion 1 – SP221
Nearly-Free Electron Modelversion 1 – SP221
2/2
2/k
a
2/k
2/2
2/k
a
Nearly-Free Electron Modelversion 1 – SP221
2/2
2/k
a
2/k
2/2
2/k
a
Nearly-Free Electron Model version 2 – SP324a
Nearly-Free Electron Model version 2 – SP324a
• Bloch Theorem
• Special Phase Conditions, k = +/- m /a
• the Special Phase Condition k = +/- /a
This treatment assumes that when a reflection occurs, it is 100%.
(x) ~ u e i(kx-t)
(x) ~ u(x) e i(kx-t)
~~~~~~~~~~
amplitude
In reality, lower energy waves are sensitive to the lattice:
Amplitude varies with location
u(x) = u(x+a) = u(x+2a) = ….
Bloch’sTheorem
u(x+a) = u(x)
(x+a) e -i(kx+ka-t) (x) e -i(kx-t)
(x) ~ u(x) e i(kx-t)
(x+a) e ika (x)
Something special happens with the phase when
e ika = 1
ka = +/ m m = 0 not a surprise m = 1, 2, 3, …
...,2,aa
k
What it is ?
ak
Consider a set of waves with +/ k-pairs, e.g.
k = + /a moves k = /a moves
This defines a pair of waves moving right & left
Two trivial ways to superpose these waves are:
+ ~ e ikx + e ikx ~ e ikx e ikx
+ ~ 2 cos kx ~ 2i sin kx
+ ~ 2 cos kx ~ 2i sin kx
Kittel
+|2 ~ 4 cos2 kx |2 ~ 4 sin2 kx
Free-electron Nearly Free-electron
Kittel
Discontinuities occur because the lattice is impacting the movement of electrons.
Effective Mass m*
A method to force the free electron model to work in the situations where
there are complications
*2
22
m
k
free electron KE functional form
Effective Mass m* -- describing the balance between applied ext-E and lattice site reflections
2
2
2
1
*
1
km
m* a = Fext
q Eext
No distinction between m & m*, m = m*, “free electron”, lattice structure does not apply additional restrictions on motion.
m = m*
greater curvature, 1/m* > 1/m > 0, m* < m net effect of ext-E and lattice interaction provides additional acceleration of electrons
greater |curvature| but negative,net effect of ext-E and lattice interaction de-accelerates electrons #1
At inflection pt
1)
2)
*
2222
22 m
k
m
k latticefromonperturbatiapply
Another way to look at the discontinuities
Shift up implies effective mass has decreased, m* < m, allowing electrons to increase their speed and join faster electrons in the band.The enhanced e-lattice interaction speeds up the electron.
Shift down implies effective mass has increased, m* > m, prohibiting electrons from increasing their speed and makingthem become similar to other electrons in the band.The enhanced e-lattice interaction slows down the electron
From earlier: Even when above barrier, reflection and transmission coefficients can increase and decrease depending upon the energy.
change in motiondue to reflections is more significant
than change in motiondue to applied field
change in motion
due to applied field enhanced by change in reflection coefficients
Nearly-Free Electron Model version 3
From earlier:Even when above barrier, reflection and transmission coefficients can increase and decrease depending upon the energy.
Nearly-Free Electron Model version 3
Nearly-Free Electron Model version 3
à la Ashcroft & Mermin, Solid State Physics
This treatment recognizes that the reflections of electron waves off lattice sites can be more complicated.
A reminder:
Waves from the left behave like:
iKxiKx
leftthefrom ere
iKx
leftthefrom et
m
K
2
22
Waves from the right behave like:
iKxiKx
rightthefrom ere
iKx
rightthefrom et
m
K
2
22
rightleftsum BA
Bloch’s Theorem defines periodicity of the wavefunctions:
xeax sumika
sum
xeax sumika
sum
unknown weights
Related toLattice spacing
xeax sumika
sum xeax sumika
sum
Applying the matching conditions at x a/2
A + B
left right
A + B
left right
A + B
left right
A + B
left right
iKaiKa et
et
rtka
2
1
2cos
22
m
K
2
22
And eliminating the unknown constants A & B leaves:
For convenience (or tradition) set:
221 rt iett ierir
ka
t
Kacos
cos
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7 8
a
valu
e
COS(electron)
COS(lattice)
ka
t
Kacos
cos
Related topossible
Lattice spacings
Related toEnergy
m
K
2
22
allowed solution regions
ka
t
Kacos
cos
Related topossible
Lattice spacings
Related toEnergy
m
K
2
22
allowed solution regions
allo
wed
sol
utio
n re
gion
s
allo
wed
sol
utio
n re
gion
s
Conductors, Insulators, Semiconductors
Harris 9.6-9.9
The transistor is the result of "reverse engineering" of the electronic remains of the UFO that landed in Roswell in 1947?
32 km track550,000 km since 1984Design speed 550 km/h
NOTE(061204): I’m not so sure this track is superconducting. The MagLev planned for the Munich area will be. France is also thinking about a sc maglev.
Maglev Frog
A live frog levitates inside a 32 mm diameter vertical bore of a Bitter solenoid in a magnetic field of about 16 Tesla at the Nijmegen High Field Magnet Laboratory.
MAGSAFE will be able to locate targets without flying close to the surface.Image courtesy Department of Defence.
http://www.csiro.au/science/magsafe.html
Finding 'objects of interest' at sea with MAGSAFE
MAGSAFE is a new system for locating and identifying submarines.
Operators of MAGSAFE should be able to tell the range, depth and bearing of a target, as well as where it’s heading, how fast it’s going and if it’s diving.
Building on our extensive experience using highly sensitive magnetic sensors known as Superconducting QUantum Interference Devices (SQUIDs) for minerals exploration, MAGSAFE harnesses the power of three SQUIDs to measure slight variations in the local magnetic field.
MAGSAFE has higher sensitivity and greater immunity to external noise than conventional Magnetic Anomaly Detector (MAD) systems. This is especially relevant to operation over shallow seawater where the background noise may 100 times greater than the noise floor of a MAD