Download Lecture. Photoelectric Effect

Lecture 4. Particle properties of waves
v/c
Relativistic
mechanics,
El.-Mag.
(1905)
Relativistic
quantum
mechanics
(1927-)
Classical
physics
Quantum
mechanics
(1920’s-)
h/s
S – the action=momentumdistance,
units kgm2/s
Outline:
Light: waves vs. particles
Photons
Photoelectric effect
Historical Development
Newton (Opticks, 1704): light as a stream of (classical) particles (corpuscles).
Descartes (1637), Huygens, Young, Fresnel (1821), Maxwell: by mid-19th
century, the wave nature of light was firmly established (interference and
diffraction, transverse nature of e.-m. waves).
Physics of the 19th century - mostly investigation of light waves; physics of the
20th century – interaction of light with matter.
One of the challenges – understanding the “blackbody spectrum” of thermal radiation (to
be considered later in the course).
Planck (1900) suggested a solution based a revolutionary new idea: emission and
absorption of electromagnetic radiation by matter has quantum nature. The energy of a
quantum of e.-m. radiation emitted or absorbed by a harmonic oscillator with the frequency f
is given by the famous Planck’s formula:
E h f
h is the Planck’s constant
h  6.63 1034 J  s
- at odds with the “classical” tradition, where energy was always associated with amplitude,
not frequency
In terms of the
angular frequency:
  2 f
E 
where

h
2
 1.05 1034 J  s
Historical Development (cont’d)
This progress leads to the concept of photons as quanta of the
electromagnetic field. However, Planck thought that the quantum nature of light
reveals itself only in the processes of interaction with matter. Otherwise, he
thought, “classical” Maxwell’s equations adequately describe all e.-m. processes.
Einstein (1905) put forward even more radical idea: light itself consists of a collection of
particles with wave-like properties.
He is purported to have said, “I know that beer comes in pint bottles” (in referring to the
quantum features of blackbody radiation). “What I want to know is whether all beer comes in
pint bottles” (that is, whether the “quantum character” of the electromagnetic field has its
origins in the atoms or as an intrinsic property of the field).
H.J. Kimble, Strong Interactions of Single Atoms and Photons in Cavity QED, Physica Scripta T76, 127
(1998) (posted on our Webpage).
Even when light propagates in free space, it can be thought of as beam of quantum particles
(not “old” classical particles, as Newton thought).
“Light” – a shorthand notation for any e.-m. radiation ( from 0 to ).
Waves
Wave equation in one dimension
for any quantity :
2
 2
2  
v
2
t
x 2
Solution: a plane wave traveling in the
negative (positive) direction x with velocity v:
phase  x  vt
constant phase  x  vt  const
t=0
  f  x  vt 
Harmonic plane wave traveling in the positive direction x:
x
Electromagnetic waves:
(transverse in free space)
2 E
1

t 2  0 0
2 B
1

t 2  0 0
2
2 E
2  E
c
x 2
x 2
2
2 B
2  B
c
x 2
x 2
wave k  2
number

v

k
x

A0
t 
  A sin  x  vt   A sin  2      A sin  kx  t 
   T 
2
angular

 2 f
frequency
T
vt0
 v  x/t
v – the phase velocity

t = t0
x
-A0

A0

t
E  x, t 
B  x, t 
-A0
T
E  x, t   cB  x, t 
Photons
According to the quantum theory of radiation, photons are massless
particles of spin 1 (in units ħ). They move with the speed of light :
E ph  h f
 E ph    cp ph    mphc2   0
E ph  cp ph
2
2
2
Quantum character of this equation is illustrated by the fact that the energy is
associated with the frequency of oscillations rather than their amplitude.
The phase t  kr is a Lorentz-invariant quantity,
the (scalar) product of two 4-vectors:
Particle properties of light
i
E ph
c
, p ph
 ict, r 
  
i ,k 
 c 
t  kr  invariant
Wave properties of light
i

c
,k
- both the time-like and space-like
components of these 4-vectors should
transform under L.Tr. in a similar way
Thus, if Planck’s idea E=ħ is correct, than we must conclude that
p ph  k 
h

p ph  k
Some numbers
Visible light:  = 0.4 ↔ 0.8 micrometers
violet red
c 3 108 m
14
f  violet   

7.5

10
Hz
7
 4 10 m
3 108 m
f  red  
 4.3 1014 Hz
7
7 10 m
c
3 108 m / s
5 1019 J
34
E ph  violet   h  6.62 10 J  s

 3.1eV
7
19

4 10 m 1.6 10 eV / J
Eph  red   1.6eV
hf 6.6 1034 J  s  7.5 1014 Hz
27 kg  m
p ph  violet   

1.65

10
c
3 108 m / s
s
for comparison, the momentum of a (non-relativistic) electron with K =3.1eV:
pe  K  3.1eV   2me K  2  9.11031 kg  3.1eV 1.6 1019 eV / J  9.5 1025
kg  m
s
more than two orders of magnitude greater than for a photon with this energy!
(this difference, however, is diminished when one considers an ultra-relativistic
electron and equally energetic photon).
Photoelectric Effect
Historical Note: The photoelectric effect was accidentally discovered by
Heinrich Hertz in 1887 during the course of the experiment that discovered
radio waves. Hertz died (at age 36) before the first Nobel Prize was
awarded.
Observation: when a negatively charged body was illuminated with
UV light, its charge was diminished.
J.J. Thomson (Nobel 1906) and P. Lenard (Nobel 1905) determined the ration e/m for
the particles emitted by the body under illumination – the same as for electrons.
The effect remained unexplained until 1905 when Albert Einstein postulated the
existence of quanta of light -- photons -- which, when absorbed by an electron near the
surface of a material, could give the electron enough energy to escape from the material.
Robert Milliken carried out a careful set of experiments, extending over ten
years, that verified the predictions of Einstein’s photon theory of light.
Einstein was awarded the 1921 Nobel Prize in physics: "For his services to
Theoretical Physics, and especially for his discovery of the law of the
photoelectric effect." Milliken received the Prize in 1923 for his work on the
elementary charge of electricity (the oil drop experiment) and on the
photoelectric effect.
Photoelectric Effect (cont’d)
Parameters: intensity (S) and frequency (f) of light,
applied voltage (V), measured photocurrent (I)
eObservations:
I
+ V
_
I
stopping
voltage
V0
V
1. For a given material of the cathode, the “stopping”
voltage does not depend on the light intensity.
2. The saturation current is proportional to the intensity
of light at f =const.
3. Material-specific “red boundary” f0 exists: no
photocurrent at f < f0.
4. Practically instantaneous response – no delay
between the light pulse and the photocurrent pulse
(many applications are based on this property).
I
V
V0(f2)
V0(f1)
f2 > f1
intensity = const,
f increases
stopping voltage
intensity of light
increases, f =const
retarding
“red
boundary” f0
f
Attempts to explain Ph.E. using the wave approach
(and the classical model of matter) fail !
1. Classical equation of motion of
an electron in the light wave
me x  eE cos  2 ft   eE cos t 
2
me v 
eE

sin t 
2
me v 2
1  eE 
S
E
2
K

sin

t




 
 
2
2me   
2
 
Observation: Kmax (and, thus, the stopping potential) does not depend on the
intensity S, and is proportional to .
2.
The photocurrent for sodium can be observed at S as low as 10-6 W/m2. Number of
electrons in the volume 1m1m  1 monolayer ~ 1019. Thus, for a single electron,
dK
~ 1025 J / s ~ 106 eV / s
dt
t  K ~ 1eV   106 s ~ 2weeks
Observation: almost instantaneous response
Photon - based model of Ph. E.
Absorption of a photon by an electron
in metal (inelastic collision between
these particles)
me c 2  K
me c 2
hf
after
before
However, we’ve concluded that a free electron cannot absorb a photon!
hf
me c 2  K
me c 2
the rest RF of an electron
after the collision
after
before
 mec2  Eph  mec2
 1
E ph  0
What’s wrong? The electron is not “free”, it is embedded in metal, and the chunk of metal is
the third “body” that participates in the collision:
Eph  mec2  M met c2  mec2  Ke  M met c2  Kmet energy conservation
p ph  pe  pmet
p ph  pe ~ pmet
momentum conservation
M met  me
vmet
(see Slide 6)
Thus, while the electron is
still inside metal
m
~ ve e
M met
2
K met ~
E ph  K e
M met  me 
me
v

K e  K e
 e

2  M met  M met
energy conservation
p ph  pe  pmet momentum conservation
The photon energy is absorbed by an electron (the energy absorbed by metal is negligibly small),
but the momentum exchange between electron and metal is crucial for momentum conservation.
Photon - based model of Ph. E. (cont’d)
In the experiment, the electron is observed outside the metal. It takes some energy to escape:
(consider an attraction between an electron and the positive
“image” charge induced on the metal surface)
The “escape” energy: the work function W (material-specific)
q+
q-
metal
Thus, for the electron
outside metal
K e  E ph  W
Ke  f   hf  W
“red” boundary of Ph. E.
Ke  0  hf 0  W
Ke  f   h  f  f 0 
Planck’s constant
measurements:
K e  f  eV0  f 
h

f  f0
f  f0
W
f0
Millikan with Michelson, Kinsley, and Gale,
circa 1910
Photon - based model of Ph. E. (cont’d)
K e  E ph  W
E
metal
For a given material of the cathode, the “stopping”
voltage does not depend on the light intensity
– the energy of photons is determined by the
light frequency, not intensity
vacuum
Ke  f 
E ph  hf
W
Observations:
energy of
a free
electron in
vacuum
with Ke =0

The saturation current is proportional to the intensity of
light at f =const
– the saturation current is proportional to the
number of photons, thus to the light intensity


Material-specific “red boundary” f0 exists: no
photocurrent at f < f0
– at f < f0 (hf < W) the photon energy is
insufficient to extract an electron from metal
Practically instantaneous response – no delay between
the light pulse and the photocurrent pulse
– single act of e-ph collision

Well, not so simple...
1921: Einstein received the Nobel prize for his interpretation of Ph.E.
However, even much later in his life, he considered the concept of the quanta
as heuristic at best: incomplete and not fully compatible with his own physical
picture of the world. The title of his paper: “On a heuristic point of view
concerning the production and transformation of light”.
1969: Lamb and Scully in “The photoelectric effect without photons” (in
Polarisation, Matiere et Rayonnment, Presses University de France, 1969)
showed that one can account for the Ph.E without the concept of photons,
and, thus, the Ph.E. does not provide proof of the existence of photons. The
theory of Lamb and Scully treated atoms quantum-mechanically, but
regarded light as being a purely classical electromagnetic wave (see the link
on our Webpage).
So what would give us a proof? The study of statistical properties of photons.
“Although surely the correct description of the electromagnetic field is a quantum one, just
as surely the vast majority of optical phenomena are equally well described by a
semiclassical theory, with atoms quantized but with a classical field. ... The first
experimental example of a manifestly quantum or nonclassical field was provided in 1977
with observation of photon anti-bunching for the fluorescent light from a single atom (PRL
39, 691 (1977))”. H.J. Kimble, Physica Scripta T76, 127 (1998).
Excellent reading on this subject: G. Greenstein and A.G. Zajonc, “The Quantum Challenge”
(Jones and Bartlett Publishers, 2005).
In hindsight ...
We should not forget, however, that when Einstein suggested his model of photoelectric
effect, quantum mechanics had not been invented yet. Even a solid foundation of the
atomistic theory was brand-new at that time.
Einstein (1900), after reading the work of Ludwig Boltzmann on the molecular theory of
gases: “The Boltzmann is absolutely magnificent. I am firmly convinced of the correctness
of the principles of his theory, i.e., I am convinced that in the case of gases we are really
dealing with discrete particles of definite finite size which move accordingly to certain
conditions”.
Planck (1882): “Despite the great success that the atomic theory has so far enjoyed,
ultimately it will have to be abandoned in favor of the assumption of continuous matter””.
After Einstein published his theoretical explanation of Brownian motion in
terms of atoms (1905), Jean Perrin did the experimental work to test and
verify Einstein's predictions, thereby settling the century-long dispute
about the physical reality of molecules.
Perrin received the Nobel Prize in Physics in 1926 for his work on “the
discontinuous structure of matter” (including the experimental finding
of the Avogadro’s number).
http://nobelprize.org/nobel_prizes/physics/laureates/1926/perrin-lecture.html
Problems
1. The work function of tungsten surface is 5.4eV. When the surface is illuminated by light
of wavelength 175nm, the maximum photoelectron energy is 1.7eV. Find Planck’s constant
from these data.
K e  hf  W  h
c

W
K e  W   1.7eV  5.4eV  1.75 107 m

h

 4.11015 eV  s
c
3 108 m / s
 4.11015 eV  s 1.6 1019 J / eV  6.6 1034 J  s
2. The threshold wavelength for emission of electrons from a given metal surface is 380nm.
(a) what will be the max kinetic energy of ejected electrons when  is changed to 240nm?
(b) what is the maximum electron speed?
(c) the loss of electrons due to the photoelectric effect will cause an isolated sphere of this
metal to acquire a positive charge. Find the largest electric potential (in Volts) that could
be achieved by this sphere for  = 240nm.
(a)
(b)
Ke  me v / 2
2
hf  W  eV
h
c
0
W
1 1
Ke  h  W  h  h  hc     1.9eV
1
1
0
 1 0 
c
c
c
2 Ke
v
 8.2 105 m / s (c) metal
me
hf  W K e
 V

 1.9V
e
e
hf
E
vacuum
Ke  f
W

electron
energy in the
(repulsive)
electric field
Inverse Photoelectric Effect (production of X-rays)
  0.01 10 nm
X-rays:
-rays
X-rays
0.01nm
UV
f  31016  31019 Hz
E ph  0.1 100 keV

10nm
Production:
hf max  h
min
c
min
 eV  K e  me c 2    1
The upper “cut-off” of the spectrum corresponds
to the full conversion of the electron kinetic
energy into the photon energy.
hc 6.6 1034 J  s  3 108 m / s
11
o



1.2

10
m

0.12A
eV
1.6 1019 C 1105V
min 
12.4  nm 
V  kV 
Problem (Midterm 1, 2008)
(5) Light of wavelength =410-7 m is incident perpendicularly on a flat metal surface at the
rate of 10-6 W/m2. How many photons are incident per square meter per second?
(10) A metal surface illuminated by light of wavelength 3.510-7m emits electrons whose
maximum kinetic energy is 0.60 eV. The same surface illuminated by light of wavelength
2.510-7 m emits electrons whose maximum kinetic energy is 2.02eV. From these data find
Planck’s constant and the work function of the metal in eV
(a)
Intensity  N ph hf
where Nph is the number of photons per square meter per second
c 3 108 m / s
15 1
f  

0.75

10
s
7
 4 10 m
(b) hf  K  W
1
max1
hf 2  Kmax 2  W
106 J /  s  m2 
Intensity
N ph 

 2 1012 photons /  s  m2 
34
15 1
hf
6.6 10 J  s  0.75 10 s
h  f1  f2   Kmax1  Kmax2
h
K max1  K max 2 K max1  K max 2

f1  f 2
1 1 
c  
 1  2 
2.02eV  0.6eV
h
 4.14 1015 eV  s
1
1


3 108 m / s 


8
8
 2.5 10 m 3.5 10 m 
c
4.14 1015 eV  s  3 108 m / s
W  h  K max1 
 0.6eV  2.95eV
7
1
3.5 10 m