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Physics 3 (PHYF144)
Chap 11: Modern Physics
-1–
11.1 The photoelectric effect
The emission of electrons when light of enough energy strikes a surface. The emitted electrons
are called photoelectrons. These liberated electrons absorbed energy from the incident radiation
and are thus able to overcome the attraction of positive charges.
Light
E
C
e
i
A
i
The figure shows an apparatus in which the
photoelectric effect can occur. When the phototube
is kept in the dark, the ammeter reads zero,
indicating no current in the circuit.
However, when monochromatic light of appropriate
wavelength shines on the plate E, a current is
detected by ammeter, indicating a flow of charges
across the gap between E and C. Electrons are
emitted from the surface.
V
Variable power supply
Einstein postulated that a beam of light consists of small packages of energy called photons or
hc
quanta, and the energy of a photon is
E hf
A photon arriving at the surface is absorbed by an electron, and this energy transfer is an all-ornothing process; the electron gets all the photon’s energy or none at all.
The minimum energy needed to remove an electron from the surface of the target material
is called the work function, , of the material.
Thus, the maximum kinetic energy for an emitted electron is the energy hf gained from a
photon minus the work function :
1 2
Kmax hf
(1)
mvmax
2
If the frequency of the light is below certain cut-off or threshold frequency fc, no
photoelectrons are emitted. Thus, the work function can be written as
hf c
(2)
The greater intensity of incident light at particular frequency means greater number of photons
per second, thus a proportionally greater number of electrons emitted per second and greater
the photoelectric current.
Current
Constant f. E= hf
2I
I
Vs
0
Applied voltage, V
The figure shows photoelectric current versus
applied voltage V for two light intensities.
When V is sufficiently large and positive, the
current reaches a maximum value, showing that
all the emitted electrons are being collected. The
current levels off at a higher value as the light
intensity increases.
When the V is negative, the current drops to a very low value because most of the emitted
electrons are repelled by the negative plate C. Only those electrons with kinetic energy
greater than eV will reach C; eV is the potential difference set up by the applied voltage.
We can determine the maximum kinetic energy of the emitted electrons by making V just
negative enough so that the current stops. The magnitude of this voltage is called the stopping
potential, Vs. Thus,
(3)
K max eVs
Trimester 1, 2010/2011
Physics 3 (PHYF144)
Chap 11: Modern Physics
eVs
(3) in (1),
h
f
e
Vs
Vs
hf
hf
h
e
f
0
K max
Kmax
e
slope,
-2–
f
0
fc
fc
Note: The work function and thus the cut-off frequency fc are characteristic of the material.
The stopping potential Vs depends on the frequency of the radiation; Greater f, greater Vs.
Example 1: A sodium surface is illuminated with light of wavelength 300 nm. The work
function for sodium metal is 2.16 eV. Find (a) the kinetic energy of the ejected photoelectrons
and (b) the cut-off wavelength for sodium.
(a) The energy of each photon in the illuminating light beam is
E
hc
hf
(6.626 10 34 Js )(3.00 108 m/s)
300 10
K max
hf
9
6.63 10 19 J
m
1 eV
4.14 eV
1.60 10 19 J
4.14 eV 2.46 eV 1.68 eV
(b) The work function is
hf c
hc
. Thus the cut-off wavelength is
c
hc
(6.626 10 34 Js )(3.00 108 m/s)
c
2.46 eV
1.60 10
1 eV
19
5.05 10 7 m
505 nm
J
Exercise: What is the maximum speed of a photoelectron emitted from a surface whose work
function is 5.0 eV when illuminated by a light whose wavelength is 200 nm? Ans: 6.50 105 m/s
11.2The Compton effect
In 1919, Einstein extended the photon idea. The photon not only carries energy E= hf but also
carries a momentum p = E/c. When light interacts with matter, not only energy but also linear
momentum is transferred via photons
1923, Compton performed an experiment as shown in figure, which supported the idea. When
the x-rays of single wavelength (= 71.1 pm) strike a target made of carbon, the radiation is
scattered from electron in various directions, just as visible light falling on a rough surface
undergoes diffuse reflection.
X ray
Electron
Detector
Before
Scattered
x rays
’
At rest, v = 0
’
X ray
Incident
x-rays
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Carbon
Target
After
Electron
v
Physics 3 (PHYF144)
Chap 11: Modern Physics
-3–
Compton and his co-workers discovered that the scattered radiations have smaller frequencies
(longer wavelengths) than the incident radiation. The scattered x-rays contain a range of
wavelengths with two prominent intensity peaks.
One peak is centred about the incident wavelength , the other about a wavelength ’ that is
longer than by amount of , which called the Compton shift. The change in wavelength
depends on the scattering angle :
'
h
(1 cos )
mc
This expression is known as the Compton shift equation, where m is the mass of the electron;
C = h/mc is called the Compton wavelength C of the electron. If the radiation is scattered by
proton, then m is the mass of the proton.
Example 2: X-rays of wavelength o = 0.200 nm are scattered from a block of material. The
scattered x-rays are observed at angle of 45.0 to the incident beam. Calculate the wavelength
of the scattered x-rays.
The shift in wavelength of the scattered x-rays is
h
(1 cos )
mc
7.10 10 13 m
6.626 10 34 Js
(9.11 10 31 kg)(3.00 108 m/s)
0.000710nm
Hence, the wavelength of the scattered x-ray at this angle is
'
0.200 710 nm
o
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(1 cos 45.0  )
Physics 3 (PHYF144)
Chap 11: Modern Physics
-4–
Exercise: Find the fraction of energy lost by the photon in this collision.
Ans: 0.003 54
11.3 Bohr’s quantum model of the atom
At the beginning of the 20th century, scientists were perplexed by the failure of classical
physics in explaining the characteristics of atomic spectra: Why did atoms of a given element
emit only certain spectral lines, and absorb only those wavelengths they emitted? In 1913,
Niels Bohr proposed a model of hydrogen atom (the simplest atom) of what he thought must be
the atom’s structure. The basic ideas of the Bohr theory:
1. Electron moves in circular orbit around the proton under influence of the Coulomb force
of attraction, which is the centripetal force.
2. Only certain electron orbits are stable and possible.
3. Radiation (photon) is emitted by the atom when the electron makes a transition (‘jump’)
from more energetic orbit to a lower orbit. The energy of the emitted photon is
hc
hf
Ei
Ef
(1)
where Ei = energy of initial state, Ef = energy of final state, and Ei
Ef
4. The orbital angular momentum of electron about the nucleus is quantized, and given by
(2)
mvr n
n 1,2,3,... ; 
h/2
Each value of n corresponds to a permitted value of the orbit radius.
Using these four assumptions, we can determine
 The size of the allowed electron orbits,
 The allowed energy levels, and thus
 The emission wavelengths
Derivation:
The total energy (kinetic + potential) of hydrogen atom:
E
K U
1 2
mv
2
e2
ke
r
(3)
The Coulomb attraction force is the centripetal force that allows the electron to move in
circular orbit:
ke
e2
r2
mv 2
r
or
ke
e2
r
mv2
(4)
(4) in (3),
E
ke e 2
2r
ke
e2
r
ke e 2
2r
(5)
The total energy is negative, indicating that the electron-proton system is stably bound.
Divide (4) to (2), and solve for v, we get
Form (1), v 2
n
mr
Therefore, we get
Trimester 1, 2010/2011
vn
2
, and
from (4), v 2
rn
ke e 2
n
ke e 2
.
mr
n2 2
mke e 2
: n
1,2,3,...
Physics 3 (PHYF144)
Chap 11: Modern Physics
-5–
These equations show that the orbit radii and speeds in Bohr’s model have discrete values, or
are quantized.
The smallest orbit corresponds to n = 1. We denote this minimum radius, called the Bohr
radius, as ao:
2
Alternatively, rn n 2 ao
r1 ao
5.29 10 11 m .
2
mke e
n
E (eV)
0.00
4
3
16ao
Paschen
series
2
0.85
1.51
3.40
Balmer
series
9ao
4ao
ao
+e
Lyman
series
1
The circular orbits for hydrogen atom
13.6
An energy level diagram for hydrogen atom
From (5), the allowed energy levels for hydrogen atom are
En
kee 2 1
2a o n 2
13.6
eV
n2
13.6
eV
n2
En
or
n = 1: The lowest energy level is called the ground state with E1 = 13.6 eV
n = 2: The next energy level is called the 1st excited state with E2 = E1 / 22 = 3.4 eV
.
.
n = : The uppermost level is the state of zero total energy for which the electron is no longer
bound to the atom.
The minimum energy required to remove an electron from the atom is called the ionisation
energy. Then, the energy required to ionise hydrogen atom when it is in the ground state is
13.6 eV.
When electron jumps from an outer orbit to an inner orbit, a photon is emitted with frequency:
Ei E f
kee 2 1
1
f
,
2
h
2a o h n f ni2
and the corresponding wavelength
1
f
c
Trimester 1, 2010/2011
kee 2
1
2a o hc n 2f
1
ni2
is
RH
1
n 2f
1
ni2
or
1
RH
1
n 2f
1
ni2
Physics 3 (PHYF144)
Chap 11: Modern Physics
-6–
The remarkable fact is that the theoretical constant ( k e e 2 2ao hc ) is found to be identical to
the experimentally determined Rydberg constant, RH = 1.0974 107 m-1.
Example 3: The Balmer series for the hyhorgen atom corresponds to electronic transitions that
terminate in the state of quantum number n = 2, as shown in Figure. Find (a) the longestwavelength photon emitted in this series and determine its energy.
n
5
E (eV)
0.00
-0.54
4
-0.85
3
-1.51
The longest-wavelength photon in the Balmer series
results from the transition from n = 3 to n = 2.
Thus,
1
RH
max
2
-3.40
max
Balmer series
The energy of this photon is
E
(6.626 10 34 Js )(3.00 108 m/s)
hc
hf
36
5RH
656.3 10
max
9
1
1
2
2
2
3
5
RH
36
36
5(1.097 107 m 1 )
656.3 nm
3.03 10 19 J 1.89 eV
m
(b) Find the shortest-wavelength photon in the Balmer series.
The shortest-wavelength photon in the Balmer series is emitted when the electron makes a
transition from n = to n = 2.
1
Thus,
RH
min
min
4
RH
1
2
2
RH
4
1
2
4
1.097 10 7 m
1
364.6 nm
Exercise 1: For a hydrogen atom in its ground state, use the Bohr model to compute (a) the
orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the electric potential
energy of the atom.
Ans: 2.19 106 m/s; 13.6 eV; -27.2 eV
Exercise 2: Energy of 13.6 eV is needed to ionise an electron from the ground state of a
hydrogen atom. What is the wavelength of a photon accomplishes this task?
Ans: 91 nm
Bohr extended his model for hydrogen atom to other elements with a single electron orbiting a
fixed nucleus of charge +Ze, where Z is the atomic number of the element (the number proton
in the nucleus).
a
rn n 2 o
Z
kee 2 Z 2
Z2
En
13
.
6
eV
: n 1,2,3,...
2a o n 2
n2
Trimester 1, 2010/2011