Download Interaction of Light and Matter

Stellar
Astrophysics:
The Interaction of
Light and Matter
The Photoelectric Effect
Methods of electron emission
•  Thermionic emission: Application of heat allows electrons to gain
enough energy to escape
•  Secondary emission: The electron gains enough energy by transfer
from another high-speed particle that strikes the material from outside
•  Field emission: A strong external electric field pulls the electron out of
the material
•  Photoelectric effect: Incident light (electromagnetic radiation) shining
on the material transfers energy to the electrons, allowing them to
escape
The Photoelectric Effect
•  Electromagnetic radiation interacts with electrons within metals
and gives the electrons increased kinetic energy
•  Light can give electrons enough extra kinetic energy to allow
them to escape
•  The ejected electrons are called photoelectrons
•  The minimum extra kinetic energy which allows escape is
called the work function φ of the material
Experimental Setup
Minimum voltage for which I = 0
is called stopping potential V0
Experimental Results
•  The kinetic energies of the photoelectrons are independent of the
light intensity
•  The maximum kinetic energy of the photoelectrons, for a given
emitting material, depends only on the frequency of the light
•  The smaller the work function φ of the emitter material, the smaller
is the threshold frequency of the light that can eject photoelectrons
Experimental Results
•  When the photoelectrons are produced, however, their number is
proportional to the intensity of light
•  The photoelectrons are emitted almost instantly following
illumination of the photocathode, independent of the intensity of
the light
Einstein s Theory
• 
• 
Einstein suggested that the electromagnetic
radiation field is quantized into particles called
photons
Each photon has the energy quantum
Ephoton = h ν =
• 
Albert Einstein (1879 - 1955)
hc
λ
where ν is the photon frequency and h is Planck s constant
The photon travels at the speed of light in a vacuum, and its
wavelength is given by
λ =
c
ν
Einstein s Theory
• 
Conservation of energy yields
h ν = φ + Kelectron
• 
Explicitly, the energy of the most energetic electrons is
Kmax = h ν – φ
• 
The retarding potentials measured in the photoelectric effect are the
opposing potentials needed to stop the most energetic electrons
e V0 = Kmax
Quantum Interpretation
• 
The kinetic energy of the electron does not depend on the light
intensity at all, but only on the light frequency and the work
function of the material
Kmax = e V0 = h ν – φ
• 
Einstein in 1905 predicted that
the stopping potential was
linearly proportional to the light
frequency, with a slope h, the
same constant found by Planck
The Compton Effect
• 
• 
• 
A photon impinging on a material will scatter
from an atomic electron
For high photon energies, we can neglect
the binding energy and treat the collision as
an elastic collision between the photon and
electron
For the photon energy we can write
Ephoton = h ν =
hc
=pc
λ
Arthur Compton (1892 - 1962)
The Compton Effect
Scattering of photons off loosely bound atomic electron
Conservation laws
h νi + me c 2 = h νf + ( pe2 c 2 + (me c 2 ) 2 ) 1/2
h / λi = h / λf ⋅ cos θ + pe⋅ cos φ
h / λf ⋅ sin θ = pe⋅ sin φ
The Compton Effect
• 
The scattering off the electron yields a change in wavelength of
the scattered photon which is known as the Compton Effect
h
Δ λ = λf – λI =
me c
( 1 – cos θ )
= λC ( 1 – cos θ )
with Compton wavelength
• 
λC = 0.00243 nm
The radiation pressure discussed previously is caused by the
Compton Effect
Line Spectra
• 
• 
Chemical elements were observed to produce unique
wavelengths of light when burned or excited in an electrical
discharge
Emitted light is passed through a diffraction grating with
thousands of lines per cm and diffracted according to its
wavelength λ by the equation
d sinθ = n λ with n = 0, 1, 2, …
where d is the distance
between grating lines
Balmer Series
• 
In 1885, Johann Balmer found an empirical
formula for wavelengths of the visible hydrogen
line spectrum
1
λ
1
1 ⎞
⎛
= RH
– 2
⎝ 4
n ⎠
Johann Balmer (1825 - 1898)
where n = 3,4,5… and RH = 1.097 · 107 m-1 Rydberg constant
De Broglie Waves
• 
• 
Prince Louis V. de Broglie suggested that massive particles
should have wave properties similar to electromagnetic radiation
He was guided by the relations for photons
hν=pc=pλν
• 
Similarly for photons, the wavelength and momentum are related
λ=h/p
• 
The wavelength of the particle, deduced from its
momentum, is called the de Broglie wavelength
Louis de Broglie (1892 - 1987)
The Classical Atomic Model
Let s consider atoms as a planetary model
•  The force of attraction on the electron by the nucleus and Newton s
second law give
1
4 π ε0
e2
= –µ
2
r
υ2
r
where υ is the tangential velocity of the electron
and the reduced mass
µ =
• 
me mp
me + mp
Can estimate υ, taking r = 5 × 10-11 m
⇒ υ ≈ 2 × 106 m/s < 0.01 c
The Classical Atomic Model
• 
The total energy of the atom can be written as the sum of kinetic and
potential energy
1
E =K+U =
8 π ε0
1
=–
8 π ε0
e2
1
–
r
4 π ε0
e2
r
e2
r
Problems with the Planetary Model
• 
• 
From classical E&M, an accelerated electric charge radiates energy
(electromagnetic radiation)
So the total energy must decrease ⇒ radius r must decrease
Electron crashes into the nucleus in ~ 10-9 s !?
The Bohr Model of
the Hydrogen Atom
Bohr s assumptions
•  Stationary states in which orbiting electrons do not
radiate energy exist in atoms
•  Emission/absorption of energy occurs along with
atomic transition between two stationary states
ΔE
Niels Bohr (1885 - 1962)
= E1 - E2 = h ν
•  Classical laws of physics do not apply to transitions
between stationary states
•  The angular momentum of the atom in a stationary
state is a multiple of h/2π
L
=
n
h
≡
2π
n ћ
n =
principle quantum number
Bohr s Quantization Condition
• 
• 
The electron is a standing wave in an orbit around the proton
This standing wave will have nodes and be an integral number of
wavelengths
h
2πr =nλ =n
p
• 
The angular momentum becomes
nh
L=rp = 2π = nћ
Bohr Radius
From
• 
h
L
=
n
≡
n
ћ
2π
2
and
e
µυ2
K=
= 2
8π
ε
0
r
The radius of the hydrogen atom for stationary states is
4 π ε0 n 2 ћ 2
rn=
= n 2 a0
µe2
quantized orbit (position)
2
where the Bohr radius is
• 
• 
4
πε
0
ћ
-
10
x
a0
=
0
.
5
3
10
m
2
=
µ e
The smallest radius of the hydrogen atom occurs when n = 1
This radius gives its lowest energy state (called the ground state of
the atom)
The Hydrogen Atom
The energies of the stationary states (with E0 = 13.6 eV)
e2
e2
E0
En = –
=–
=–
8 π ε0 rn
8 π ε0 a0 n2
n2
Emission of light occurs when the atom in an
excited state decays to a lower energy state
(nu → nl)
h ν = Eh – El
where ν is the frequency of the photon
1
λ
=
ν
c
=
Eh – El
hc
and RH the Rydberg constant
1 ⎞
⎛ 1
= RH
⎝ nl 2 – nh 2 ⎠
Transitions in the Hydrogen Atom
Lyman series (invisible)
The atom exists in the excited state
for a short time before making a
transition to a lower energy state with
emission of a photon. At ordinary
temperatures, almost all hydrogen
atoms exist in n = 1 state. Absorption
therefore gives the Lyman series.
Balmer series (visible)
When sunlight passes through the
atmosphere, hydrogen atoms in water
vapor absorb wavelengths of the
Balmer series giving dark lines in the
absorption spectrum.
Limitations of the Bohr Model
The Bohr model was a great step for the new quantum theory,
but it had its limitations.
• 
• 
Works only for single-electron atoms
Could not account for the intensities or the fine structure of the
spectral lines
• 
Could not explain the binding of atoms into molecules