Transcript BAND THEORY OF SOLIDS

Wave / Particle Duality
PART I
• Electrons as discrete Particles.
– Measurement of e (oil-drop expt.) and e/m (e-beam expt.).
• Photons as discrete Particles.
– Blackbody Radiation: Temp. Relations & Spectral Distribution.
– Photoelectric Effect: Photon “kicks out” Electron.
– Compton Effect: Photon “scatters” off Electron.
PART II
• Wave Behavior: Diffraction and Interference.
• Photons as Waves: l = hc / E
– X-ray Diffraction (Bragg’s Law)
• Electrons as Waves: l = h / p = hc / pc
– Low-Energy Electron Diffraction (LEED)
Page 1
Electrons: Quantized Charged Particles
• In the late 1800’s, scientists discovered that electricity was
composed of discrete or quantized particles (electrons) that had a
measurable charge.
• Found defined amounts of charge in electrolysis experiments, where
F (or Farad) = NA e.
– One Farad (96,500 C) always decomposes one mole (NA) of
monovalent ions.
• Found charge e using Millikan oil-drop experiment.
• Found charge to mass ratio e/m using electron beams in cathode ray
tubes.
Page 2
Electrons: Millikan’s Oil-drop Expt.
• Millikan measured quantized charge values for oil droplets, proving
that charge consisted of quantized electrons.
– Formula for charge q used terminal velocity of droplet’s fall
between uncharged plates (v1) and during rise (v2) between
charged plates.
Charged oil droplets

mg  v2
q
  1
E  v1

Charged Plates
Scope to measure droplet terminal velocity.
Page 3
Electron Beam e/m : Motion in E and B Fields
Circular Motion of electron in B field:
mv 2
(or Fcentrip )  evB (or FB )
r
mv
m
r

eB
e
FE  eE
FB  ev  B
 Larger e/m gives smaller r, or larger deflection.
Electron (left hand)
Proton (right hand)
FB
FB
v
v
B
B
Page 4
Electron Beam e/m: Cathode Ray Tube (CRT)
• Tube used to produce an electron beam, deflect it with
electric/magnetic fields, and then measure e/m ratio.
• Found in TV, computer monitor, oscilloscope, etc.
J.J. Thomson
Charged Plates
(deflect e-beam)
Deflection  e/m
(+) charge
Cathode
(–) charge
Slits
(hot filament
produces electrons) (collimate beam)
Fluorescent Screen
(view e-beam)
Page 5
Ionized Beam q/m: Mass Spectrometer
• Mass spectrometer measures q/m for unknown elements.
1.
1
2qV
mv 2  qV  v 2 
2
m
2.
mv
R
qB
Ions accelerated by E field.
Ion path curved by B field.
2 2
2
m
v
m
 2qV 
2
R  2 2  2 2
q B
q B  m 
q
2V

 2 2
m
BR
2.
1.
Page 6
Photons: Quantized Energy Particle
• Light comes in discrete energy “packets” called photons.
Energy of
Single Photon
hc 1240 eV nm
E  hf  
l
l (nm)
From Relativity:
 
E   pc   mc
For a Photon (m = 0):
Momentum of
Single Photon
2
2
2 2
Rest mass
E   pc   0  E  pc
2
2
E hc h
p 

c lc l
Page 7
Photons: Electromagnetic Spectrum
400 nm
Gamma Rays
Ultraviolet
Visible
Infrared
Microwave
Short Radio Waves
Wavelength
Frequency
X-Rays
Visible
Spectrum
TV and FM Radio
AM Radio
Long Radio Waves
700 nm
Page 8
Photoelectric Effect: “Particle Behavior” of Photon
PHOTON IN  ELECTRON OUT
• Photoelectric effect experiment shows quantum nature of light, or
existence of energy packets called photons.
– Theory by Einstein and experiments by Millikan.
• A single photon can eject a single electron from a material only if it has
the minimum energy necessary (or work function f.
– For example, if 1 eV is necessary to remove an electron from a metal
surface, then only a 1 eV (or higher energy) photon can eject the
electron.
Page 9
Photoelectric Effect: “Particle Behavior” of Photon
PHOTON IN  ELECTRON OUT
• Electron ejection occurs instantaneously, indicating that photons cannot
be “added up.”
– If 1 eV is necessary to remove an electron from a metal surface, then
two 0.5 eV photons cannot add together to eject the electron.
• Extra energy from the photon is converted to kinetic energy of the
outgoing electron.
– For example above, a 2 eV photon would eject an electron having
1 eV kinetic energy.
Page 10
Photoelectric Effect: Apparatus
• Photons hit metal cathode and eject electrons with work function f.
• Electrons travel from cathode to anode against retarding voltage VR
(measures kinetic energy Ke of electrons).
• Electrons collected as
“photoelectric” current at anode.
• Photocurrent becomes zero when
retarding voltage VR equals
stopping voltage Vstop,
i.e. eVstop = Ke
Cathode
Anode
Light
Page 11
Photoelectric Effect: Equations
• Total photon energy =
e– ejection energy + e– kinetic energy.
2
hc
mv
f 
 f  eVstop
l
2
– where hc/l = photon energy, f = work
function, and eVstop = stopping energy.
• Special Case: No kinetic energy (Vo = 0).
– Minimum energy to eject electron.
Emin 
hc
lmin
f
Page 12
Photoelectric Effect: IV Curve Dependence
Intensity I
dependence
Vstop= Constant
f1 > f2 > f3
Frequency f
dependence
f1
f2
f3
Vstop f
Page 13
Photoelectric Effect: Vstop vs. Frequency
eVstop  hf  f
Vstop  0 
hfmin
hf min  f
Slope = h = Planck’s constant
f
Page 14
Photoelectric Effect: Threshold Energy Problem
If the work function for a metal is f = 2.0 eV, then find the threshold energy Et and
wavelength lt for the photoelectric effect. Also, find the stopping potential Vo if the
wavelength of the incident light equals 2lt and lt /2.
At threshold, Ek = eVo = 0 and the photoelectric equation reduces to:
Et  f  2 eV
and
hc 1240eVnm
lt 

 620 nm
Et
2 eV
For 2lt, the incoming light has twice the threshold wavelength (or half the threshold
energy) and therefore does not have sufficient energy to eject an electron. Therefore, the
stopping potential Vo is meaningless because there are no photoelectrons to stop!
For lt/2, the incoming light has half the threshold wavelength (or twice the threshold
energy) and can therefore eject an electron with the following stopping potential:
For l 
lt
2
(or E  2Et ),
eVo 
hc
 f  2Et  f  2(2)  2 eV  2 eV
l
Page 15
Compton Scattering: “Particle-like” Behavior of Photon
• An incoming photon (E1) can inelastically scatter from an electron and lose
energy, resulting in an outgoing photon (E2) with lower energy (E2 < E1.
• The resulting energy loss (or change in wavelength Dl) can be calculated
from the scattering angle q.
Incoming X-ray
Scattered X-ray
Scattering
Crystal
Angle
measured
Page 16
Compton Scattering: Schematic
PHOTON IN  PHOTON OUT (inelastic)
hc
EE2  hc
2
ll22
E1 
hc
l1
Page 17