Photons, Electrons, and
Atoms
Montwood High School
Physics
R. Casao
Wave – Particle Duality
OR
Emission & Absorption of Light
• The internal energy of atoms is quantized; it is
emitted and absorbed in particle-like packages of
definite energy called quanta.
• For a given kind of individual atom the energy can’t
just have any value; only discrete values called
energy levels are possible.
• Unique sets of wavelengths are emitted and
absorbed by gaseous elements.
• If a light source is a hot solid (like a light bulb
filament) or liquid, the electromagnetic (EM)
spectrum emitted is continuous; light of all
wavelengths are present.
• Continuous Spectrum
• If the light source is a gas carrying an electric
discharge (think neon sign) or a salt heated in a
flame (think flame test in chemistry), only a few
colors appear in the form of isolated sharp parallel
lines.
• Each line corresponds to a definite wavelength and
frequency.
• Each element in its gaseous state has a unique set
of wavelengths in its line spectrum.
• Scientists find the line spectrum to be a valuable
tool to identify elements and compounds.
Line Spectrum
Line Emission Spectrum of Hydrogen Atoms
Photons: The Quanta of Light
• All electromagnetic radiation is quantized and occurs in finite
"bundles" of energy which we call photons.
• The quantum of energy for a photon is the product of
Planck’s constant h and the frequency;
– E photon = h·f = 6.626 x 10-34 J·s ·f
– Planck’s constant also equals 4.136 x 10-15 eV·s
• The quantization implies that a photon of blue light of given
frequency or wavelength will always have the same size
quantum of energy.
– For example, a photon of blue light of wavelength 450 nm will always
have 2.76 eV of energy.
– It occurs in quantized chunks of 2.76 eV, and you can't have half a
photon of blue light - it always occurs in precisely the same sized
energy chunks.
Electron-Volt
• The electron-volt, eV, is a unit of energy used in atomic and
nuclear processes.
• The electron-volt is the energy given to an electron by
accelerating it through 1 V of electric potential difference.
The work done on the charge is given by the charge times
the voltage difference (W = q·V), which in this case is:
• The work done on the charge increases the kinetic energy of
the charge;
1
q V 
2
 m v 2
The Photoelectric Effect
• Photoelectric effect: emission of electrons by a
substance (generally a metal), when illuminated by
electromagnetic radiation (EM radiation).
– Electrons are ejected when illuminated by the EM
radiation.
– Ejected electrons are called photoelectrons.
• Subsequent investigations have shown that all
substances exhibit photoemission of electrons.
• Energy contained within the incident light is
absorbed by electrons within the metal, giving the
electrons sufficient energy to overcome the
attraction of the positive ions in the metal and
escape into the surrounding space.
First Law of Photoelectric Emission
• First Law of Photoelectric Emission: the rate of
emission of photoelectrons is directly proportional to
the intensity of the incident light.
– Einstein proposed that the incident light consisted of
individual quanta, called photons, that interacted with the
electrons in the metal like discrete particles, rather than as
continuous waves. For a given frequency, or 'color,' of
the incident radiation, each photon carried the energy
E = h·f, where h is Planck's constant and f is the
frequency.
– Increasing the intensity of the light increased the number
of incident photons per unit time (flux), while the energy
of each photon remained the same (as long as the
frequency of the radiation was held constant).
First Law of Photoelectric Emission
– increasing the intensity of the
incident radiation would cause
greater numbers of electrons
to be ejected, but each
electron would carry the same
average energy because each
incident photon carried the
same energy.
– This assumes that individual
photons are being absorbed
by the material and result in
the ejection of a single
electron.
Second Law of Photoelectric Emission
• Second Law of Photoelectric Emission: the kinetic
energy of photoelectrons is independent of the
intensity of the incident light.
– increasing the frequency f, rather than the intensity, of
the incident radiation increases the average kinetic energy
of the emitted electrons.
– The maximum kinetic energy of photoelectrons increases
with the frequency of the light illuminating the emitter.
– For each kind of surface, there is a cut-off or threshold
frequency fo below which the photoelectric emission of
electrons ceases. In other words, no electrons are
emitted if the frequency falls below fo.
Work Function
• Work function, : the minimum energy
required to remove an electron from the
surface of a material.
– The work required to free the most weakly bound
electron is the work function.
– If an electron acquires less energy than the work
function of the metal, the electron cannot be
ejected.
– If an electron acquires more energy than the
work function, the electron is emitted from the
surface of the metal and the excess energy
appears as kinetic energy of the photoelectron.
Work Function
– The greater the kinetic energy of the photoelectron, the
greater the speed of the photoelectron.
– Photoelectrons ejected from atom layers below the
surface will lose energy thru collisions in reaching the
surface and then must give up energy equal to the work
function of the metal in escaping thru the surface.
– Photoelectrons ejected from the surface layer of metals
only lose the energy necessary to overcome the surface
attractions.
– In any photoelectric phenomenon, photoelectrons are
emitted at various velocities ranging up to a maximum
value possessed by electrons having their origin in the
surface layer of atoms.
Work Function Equations
•
•
•
•
h = 6.626 x 10-34 J·s = 4.136 x 10-15 eV·s
fo = cut-off or threshold frequency
o = cut-off or threshold wavelength
c = 3 x 108 m/s
hc
photon energy E  h  f 
λ
hc
work function   h  fo 
λo
Cut-Off Wavelength
• Cut-off or threshold wavelength – the wavelength
for which the energy of the photon is exactly equal
to the work function .
– The cut-off (threshold) wavelength of the incident light is
related to the cut-off (threshold) frequency by c = ·fo.
– The electron is freed but has zero kinetic energy.
hc
  h  fo 
λo
– Nanometer (nm) to meter (m) conversion:
1 nm = 1 x 10-9 m
• Example: 535 nm = 535 x 10-9 m = 5.35 x10-7 m
Cut-Off (Stopping) Potential
• Cut-off or Stopping Potential, Vo: a negative
potential on the collector of a photoelectric cell that
reduces the photoelectric current to zero.
– A negative potential on the collector plate repels the
photoelectrons, tending to turn them back to the emitter
plate.
– Only those electrons having enough kinetic energy and
velocity to overcome the repulsion reach the collector.
– As the cut-off potential is approached, only those
photoelectrons with the highest velocity reach the
collector. These electrons have the maximum kinetic
energy and are emitted from the surface layer.
– At the cut-off potential, even these electrons are repelled.
Cut-Off Potential Vo
– The cut-off potential Vo measures the kinetic
energy of the fastest photoelectrons.
– Work done by the cut-off potential = maximum
kinetic energy of electrons:
K  q  Vo
• q = charge on electron; q = 1.602 x 10-19 C
• Mass of electron = 9.11 x 10-31 kg
• Kinetic energy:
2
K  0.5  m  v
q  V  0.5  m  v
2
Third Law of Photoelectric Emission
• Third Law of Photoelectric Emission: the
maximum kinetic energy of photoelectrons
varies directly with the difference between
the frequency of the incident light and the
cut-off frequency.
– For photoelectric emission from any surface, the
incident light radiation must contain frequencies
higher than the cut-off frequency characteristic of
the surface.
– The maximum kinetic energy of photoelectrons
emitted from any surface can be increased only
by raising the frequency of the illuminating light.
Third Law of Photoelectric Emission
• Equations:
K  photon energy - work function
K  h  f  h  fo
h c h c
K

λ
λo
h  f  energy excited state - energy ground state
q  Vo  h  f  h  f o
h c h c
q  Vo 

λ
λo
• A photon arriving at the surface is absorbed by an
electron.
• The energy transfer is an all-or-nothing process; the
electron gets all of the photon’s energy or none at
all.
• If the photon energy is greater than the work
function , the electron may escape from the
surface.
• Greater intensity at the same frequency means a
proportionally greater number of photons per
second absorbed and a greater number of electrons
emitted per second and a greater current.
Experimental Data Supporting Photoelectric Effect
• Photoelectron Kinetic Energy vs. Frequency
Experimental Data Supporting Photoelectric Effect
• Threshold frequency fo is the point where the line
crosses the x-axis.
• Work function  is the point where the
line crosses the y-axis.
Experimental Data Supporting Photoelectric Effect
• Photoelectron Kinetic Energy vs. Frequency
Frequency and Photoelectrons
• For potassium, the work function  = 2 eV.
– Photon energies less than 2 eV will not produce photoelectrons.
– Photon energies greater than or equal to 2 eV produce photoelectrons;
energy above 2 eV increases the kinetic energy of the photoelectrons.
Work Functions for Photoelectric Effect
Element
Aluminum
Work Function
(eV)
4.08
Element
Work Function
(eV)
Element
Work Function
(eV)
Copper
4.7
Niobium
4.3
Beryllium
5
Gold
5.1
Potassium
2.3
Cadmium
4.07
Iron
4.5
Platinum
6.35
Calcium
2.9
Lead
4.14
Selenium
5.11
Carbon
4.81
Magnesium
3.68
Silver
4.73
Cesium
2.1
Mercury
4.5
Sodium
2.28
Cobalt
5
Nickel
5.01
Zinc
4.3
Application
• Photoelectric cell acts like a switch in an electric circuit in that
it produces a current in an external circuit when light of
sufficiently high frequency falls on the cell, but it does not
allow a current in the dark.
• Burglar alarms: beams of UV light pass from a source to a
photosensitive surface; the current produced is amplified and
used to energize an electromagnet that attracts a metal rod.
When the light beam is broken, the electromagnet switches
off and the spring pulls the iron rod to the right. In this
position, a completed electric circuit allows current to pass
and activate the alarm system. See figure on next slide.
• Some garage door systems and fire alarm systems also
employ photoelectric sensors as part of their operating
system.
Burglar Alarm System
Application
• Photoelectric cell acts like a switch in an electric circuit in that
it produces a current in an external circuit when light of
sufficiently high frequency falls on the cell, but it does not
allow a current in the dark.
• Burglar alarms: beams of UV light pass from a source to a
photosensitive surface; the current produced is amplified and
used to energize an electromagnet that attracts a metal rod.
When the light beam is broken, the electromagnet switches
off and the spring pulls the iron rod to the right. In this
position, a completed electric circuit allows current to pass
and activate the alarm system. See figure on next slide.
• Some garage door systems and fire alarm systems also
employ photoelectric sensors as part of their operating
system.
Photoelectric Effect Example
•
A sodium surface is illuminated with light of
wavelength 300 nm. The work function for
sodium is 2.46 eV.
a. Determine the kinetic energy of the ejected
photoelectrons.
hc
K  hf  

λ
K
4.136 x 10 15 eV  s  3 x 108 m
300 x 10 9 m
s  2.46 eV
K  4.136 eV  2.46 eV  1.676 eV
Photoelectric Effect Example
b. Determine the cut-off wavelength for sodium.
K  h  f 
h  f 

0  h  f 
h c
4.136 x 10


15
h c


eV  s  3 x 10 m / s
2.46 eV
  5.0439 x 10 m
7
8
Photon Momentum
• A photon of any EM radiation with frequency f
and wavelength  has energy:
E h  f 
h c

• Every particle that has energy must also have
a momentum, even if it has no rest mass;
photons have zero rest mass.
• A photon with energy E has momentum p
given by E = p∙c.
E hf h


• Photon momentum: p 
c
c

• The direction of the photon’s momentum is the
direction in which the EM wave is moving.
De Broglie Waves
• Nature loves symmetry.
• Light is dualistic in nature, behaving in some
situations like a wave and in others like a particle.
• If nature is symmetric, the dual nature should also
hold for particles.
• Electrons, which we usually think of as a particle,
may in some situations behave like a wave.
• If a particle acts like a wave, it should have a
wavelength and a frequency.
• De Broglie proposed that a particle with mass m,
moving with speed v, should have a wavelength 
related to its momentum p = m∙v just like a photon:
h

p
• The de Broglie wavelength of a particle is:
h
h
 
p m v
• The electrons in orbit around the nucleus do not radiate
energy.
• The electrons can be thought of as a standing wave fitted
around a circle in one of the orbits.
• For the wave to come out even and join onto itself smoothly,
the circumference of the circle must include some whole
number of wavelengths.
Compton Scattering
• When a photon strikes matter, some of
the radiation is scattered, just like light
falling on a rough surface undergoes
diffuse reflection.
• Some of the scattered radiation has
smaller frequency (longer wavelength)
than the incident radiation and that the
change in wavelength depends on the
angle through which the radiation is
scattered.
• If the scattered radiation emerges at an angle θ
with respect to the incident direction, if  is the
wavelength of the incident radiation, and  is the
wavelength of the scattered radiation:
h
 -
 1- cos  
me  c
'
• Imagine the scattering process as a collision of two
particles, the incident photon and an electron that is
initially at rest.
• The incident photon disappears, giving part of
its energy and momentum to the electron,
which recoils as a result of the impact.
• The remaining energy and momentum goes
to a new scattered photon that has less
energy, smaller frequency, and longer
wavelength than the incident photon.
• The Compton effect is a quantum
phenomenon that cannot be explained by
classical physics.
• Compton effect depends on the assumption
that while a photon does not possess mass, it
does have momentum that can be transferred
during a collision.
• The change in kinetic energy for the electron
equals the difference in energy for the
incident photon, h∙fi, and the scattered
photon,h∙ff.
K  h  f f  h  f i 
h c
f

h c
i
• The change in wavelength between the incident and
scattered photon, the Compton shift, depends on
the scattering angle θ (the angle between the
incident and scattered photon).
• If the scattering angle θ = 90°, the Compton shift
has a definite, numerical value called the Compton
wavelength.
h
12

 2.43 x 10 m
me  c
• Momentum and energy of the incident
photon:
p
h

and
E  p c
• Momentum and energy of the scattered
electron before collision:
p 0
and
E  m c
2
 is θ in the other images
Compton Scattering video link Youtube:
http://www.youtube.com/watch?feature=player_embedd
ed&v=zYZNTzviBxs
Derivation of Compton Scattering Equation
• Momentum and energy of the scattered photon:
p 
'
h

'
and
E  p c
'
• Momentum and energy of the scattered electron
after collision:
momentum  pe
and

E  m c
2
2
  p
2
e
c 
– the energy equation is derived from Einstein’s
theory of relativity for particles that have a
velocity that approaches the speed of light.
2
• Conservation of energy:
p  c  m  c  p ' c  E
2
E  p  c  m  c  p ' c
2
• Square both sides to get E2:

E  p  c  m  c  p ' c
2
2

2
• Substitute E2 = (m·c2)2+(pe·c)2:

(m ·c )  ( pe ·c )  p  c  m  c  p ' c
2 2
2

2
m ·c  pe ·c  p  c  m  c  p ' c
2
4
2
2
2


2
2
Conservation of momentum:
0  p  p ' pe
• Apply conservation of vector momentum and the
law of cosines:
 is θ in the other images
pe2 = p2 + p´2 – 2·p·p´·cos 
• Substitute pe2 into the conservation of energy
equation:

m ·c  pe ·c  p  c  m  c  p ' c
2
4
2

2
2


2

m ·c  p  p´ – 2· p· p´·cos  ·c  p  c  m  c  p ' c
2
4
2
2
2
2


2
m ·c  p ·c  p´ ·c – 2· p· p´·c ·cos   p  c  m  c  p ' c
2
4
2
2
2
2
2
2
• Squaring the (p·c + m·c2 – p´·c)2 term:



p  c  m  c 2  p ' c  p  c  m  c 2  p ' c 
p c  p  p c  p m c  p  p c  p c
2
2
'
2
3
'
2
 p'  m  c 3  p  m  c 3  p'  m  c 3  m 2  c 4
'2
2

2
• Combining like terms:
p c  2 p  p c  2 p m c  p c
2
2
'
2
2  p  m  c  m  c
'
3
2
3
'2
2
4
• Combining both sides of the equation:
m 2 ·c 4  p 2 ·c 2  p´2 ·c 2 – 2· p· p´·c 2 ·cos   p 2  c 2
2  p  p '  c 2  2  p  m  c 3  p '2  c 2  2  p '  m  c 3  m 2  c 4
• The m2·c4 terms cancel out:
p 2 ·c 2  p´2 ·c 2 – 2· p· p´·c 2 ·cos   p 2  c 2
2  p  p  c  2  p  m  c  p  c  2  p  m  c
'
2
3
'2
2
'
3
• Divide both sides of the equation by c2:
p 2 ·c 2  p´2 ·c 2 – 2· p· p´·c 2 ·cos 

2
c
p 2  c 2  2  p  p '  c 2  2  p  m  c 3  p '2  c 2  2  p '  m  c 3
c2
gives us
p 2  p´2 – 2· p· p´·cos   p 2  2  p  p '
2  p  m  c  p '2  2  p '  m  c
• The p2 and p´2 terms cancel out:
– 2· p· p´·cos   2  p  p '  2  p  m  c  2  p '  m  c
• The 2 cancels out:
– p· p´·cos    p  p  p  m  c  p  m  c
'
'
• Combine like terms (p·p´ and m·c) by adding p·p´
to both sides:
p  p – p· p´·cos   p  m  c  p  m  c
'
'
• Factor out p·p´ and m·c:

p  p  1  cos    m  c  p  p
'
'

• Replace the p’s with the h/λ terms, combine like
terms, and find a common denominator:
h h
h h 
 '  1  cos    m  c    ' 
 
  
h  h  
h
 1  cos    m  c    '  '  
' 
 
   
2
'
 h  h 
h
 1  cos    m  c  

' 
'
 
   
2
'
h
from both sides and cancel the
'
 
• Factor out
term:
h
h
'
 h  1  cos    m  c 
  
'
'
 
 


h  1  cos    m  c    
'

• Solve for the change in wavelength, λ´ – λ:
h
  
 1  cos  
m c
'

Summary of Photons
Photons can be thought of as
“packets of light” which behave as
a particle.

To describe interactions of light with matter, one generally
has to appeal to the particle (quantum) description of light.


A single photon has an energy given by
E = h·c/, where:
h = Planck’s constant = 6.6x10-34 [J·s]
c = speed of light = 3x108 [m/s]
 = wavelength of the light (in m)
Photons also carry momentum. The momentum is related to
the energy by: p = E/c = h/

One might ask:
Matter Waves ?
“If light waves can behave like a particle, might particles act like waves”?
The short answer is YES. The explanation lies in the realm of
quantum mechanics, and is beyond the scope of this course.
However, you already have been introduced to the answer.
Particles also have a wavelength given by:
 = h/p = h / mv
 That is, the wavelength of a particle depends on its momentum,
just like a photon!
 The main difference is that matter particles have mass, and
photons don’t !
Matter Waves
Compute the wavelength of a 1 [kg] block moving at 1000 [m/s].
 = h/mv = 6.6x10-34 [J s]/(1 [kg])(1000 [m/s]) = 6.6x10-37 m.
 This is immeasureably small. So, on a large scale, we cannot
observe the wave behavior of matter
Compute the wavelength of an electron (m=9.1x10-31 [kg])
moving at 1x107 [m/s].
 = h/mv = 6.6x10-34 [J s]/(9.1x10-31 [kg])(1x107 [m/s])
= 7.3x10-11 m.
This is near the wavelength of X-rays