Download - Modern Physics Lecture IX
Modern PhysicsLecture IX
The Quantum Hypothesis In this lecture we examine the evidence for In this lecture we examine the evidence for
“light quanta” and the implications of their “light quanta” and the implications of their existenceexistence Waves as ParticlesWaves as Particles
The photoelectric effectThe photoelectric effect Compton scatteringCompton scattering
Particles as WavesParticles as Waves Electron diffractionElectron diffraction
The Double Slit RevisitedThe Double Slit Revisited
Photoelectric effect
When light strikes the When light strikes the cathodecathode, electrons , electrons are emittedare emittedElectrons moving between the two plates Electrons moving between the two plates constitute a currentconstitute a current
Photoelectric Effect
Properties of the photoelectric Properties of the photoelectric effecteffect
Electrons are only emitted above a Electrons are only emitted above a certain “cut-off” frequencycertain “cut-off” frequency
This frequency is different for different This frequency is different for different materialsmaterials
It is called the “work function”It is called the “work function”
Below the “work function” no electrons Below the “work function” no electrons are emitted no matter how intense the light are emitted no matter how intense the light isis
The maximum energy of the ejected The maximum energy of the ejected electron is Kelectron is Kmaxmax=e=eVVss
Photoelectric Effect Properties Properties
No photoelectrons are emitted if the No photoelectrons are emitted if the frequency falls below some cut-off frequency falls below some cut-off frequency frequency ffcc
The maximum energy of the photons is The maximum energy of the photons is independent of the light intensityindependent of the light intensity
The maximum kinetic energy of the The maximum kinetic energy of the photoelectrons increases with increasing photoelectrons increases with increasing frequencyfrequency
Photoelectrons are emitted almost Photoelectrons are emitted almost instantaneously from the surfaceinstantaneously from the surface
Photoelectric Effect ExplanationExplanation
Einstein extended Planck’s explanation for blackbody Einstein extended Planck’s explanation for blackbody radiation to suggest that in fact the quanta of energy used in radiation to suggest that in fact the quanta of energy used in blackbody radiation are in fact localised “particle like” blackbody radiation are in fact localised “particle like” energy packets energy packets
Each having an energy given by Each having an energy given by hf hf Emitted electrons will have an energy given byEmitted electrons will have an energy given by
Where Where is known as the “ is known as the “work function”work function” of the of the materialmaterial
hfKmax
Photoelectric Effect Quantum interpretationQuantum interpretation
If the energy of a photon is less than the work function If the energy of a photon is less than the work function , the , the photon cannot give enough energy to the electron to leave photon cannot give enough energy to the electron to leave the surfacethe surface
KKmaxmax does not depend on light intensity, because doubling the does not depend on light intensity, because doubling the
number of photons would only double the number of number of photons would only double the number of electrons and not double their energyelectrons and not double their energy
KKmaxmax increases with frequency because energy and frequency increases with frequency because energy and frequency
are relatedare related If light is particle-like, then all of the energy will be If light is particle-like, then all of the energy will be
delivered instantaneously thus liberating an electron with no delivered instantaneously thus liberating an electron with no time delay between the light hitting the surface and the time delay between the light hitting the surface and the electron escapingelectron escaping
Inverse photoelectric effect - Production of X-rays
1. Photons are absorbed in whole - electrons can transfer part of their energy.
2. Brehmsstrahlung - electrons decelerate in electromagnetic field of nuclei: Ef = Ei - h , wide distribution
3. Maximal frequency - minimal wavelength (observed empirically first) eV = h max = h c / min
4. Also discrete spectrum (atomic levels)
X-ray photons
Electrons
High voltage
Roentgen Lamp
minmax
2
2v
hc
hfeVm
ACe
Tungsten - wolframTungsten - wolfram
Compton Scattering If light is like a particle does it have momentum?If light is like a particle does it have momentum? In Compton scattering x-rays impart momentum to matter, In Compton scattering x-rays impart momentum to matter,
scattering electrons like billiard balls scattering electrons like billiard balls Thus photons also have momentum. The momentum of a photon Thus photons also have momentum. The momentum of a photon
is given byis given by
IncidentPhoton, 0
ScatteredPhoton, ’
Recoiling electron
)cos1(0 cm
h
e
h
c
hf
c
Ep
Photons and Electromagnetic Waves
How can light be considered a photon (particle) How can light be considered a photon (particle) when we know it is a wavewhen we know it is a wave
Light has a dual nature: it exhibits both wave and Light has a dual nature: it exhibits both wave and particle characteristicsparticle characteristics There is a smooth transition of these properties across the There is a smooth transition of these properties across the
electromagnetic spectrumelectromagnetic spectrum At low frequencies (radio waves) photons have a vanishingly At low frequencies (radio waves) photons have a vanishingly
small energy and the wave properties dominatesmall energy and the wave properties dominate At high frequencies (x-rays, At high frequencies (x-rays, -rays) it is the particle properties that -rays) it is the particle properties that
dominate dominate
But…But…
Louis de Broglie1892 - 1987
Wave Properties of Matter In 1923 Louis de Broglie postulated that perhaps matter In 1923 Louis de Broglie postulated that perhaps matter
exhibits the same “duality” that light exhibitsexhibits the same “duality” that light exhibits Perhaps all matter has both characteristics as wellPerhaps all matter has both characteristics as well Previously we saw that, for photons,Previously we saw that, for photons,
h
c
hf
c
Ep
mv
h
p
h
Which says that the wavelength of light is related to its Which says that the wavelength of light is related to its momentummomentum
Making the same comparison for matter we find…Making the same comparison for matter we find…
de Broglie Wavelength of Electrons We now calculate the wavelength of a charged particle accelerated through potential We now calculate the wavelength of a charged particle accelerated through potential VV Assume that the particles have mass Assume that the particles have mass mm and charge and charge qq Equate kinetic energy of the particles with the electrostatic energyEquate kinetic energy of the particles with the electrostatic energy
KK = = m v m v 22/2 =/2 = q V q Vmomentummomentum pp = = m v m v
We can express kinetic energy in terms of momentumWe can express kinetic energy in terms of momentumKK = = p p 22/(2/(2 m m) =) = q V q V
Reorganise to getReorganise to getpp = (2 = (2 m q V m q V ))1/21/2
de Broglie’s hypothesis gives de Broglie’s hypothesis gives = = h / ph / p
Substitute for Substitute for pp to getto get
1/22 Vqm
hλ
Does Matter Really Have a Wavelength
The wavelength of matter waves is very small. This is The wavelength of matter waves is very small. This is why we do not see them in our every day experiencewhy we do not see them in our every day experience
To see diffraction a grating a very small slit width is To see diffraction a grating a very small slit width is required (eg the space between two atoms in a crystal)required (eg the space between two atoms in a crystal)
This is exactly how electron diffraction was first This is exactly how electron diffraction was first found!found! G. P. Thompson of Scotland and Davisson and Germer from the USA G. P. Thompson of Scotland and Davisson and Germer from the USA
used the close spacing between atoms in a crystal lattice to diffract used the close spacing between atoms in a crystal lattice to diffract electron waves thus proving that matter can also exhibit diffraction and electron waves thus proving that matter can also exhibit diffraction and interference interference
Sir Joseph John (JJ) Thomson
dNi=0.215nmdiffraction
nm165.0sin d
de Broglie
m
peVba 2
2
nm167.02
bameV
h
p
h
C.J.Davisson and L.G.Germer
Example of Measuring the Lattice Spacing Consider an electron accelerated to Consider an electron accelerated to V V == 5050 V scattered through V scattered through
angle angle . Note that 2. Note that 2==180180°, i.e. °, i.e. 9090°-°-/2/2
• The condition for constructive interference is2 d sin = n n integer
Fig
s. f
rom
R. Eis
berg
& R
. R
esn
ick Q
uantu
m P
hysi
cs
Example Electron scattering in nickelElectron scattering in nickel Electrons are accelerated through Electrons are accelerated through VV = = 54V.54V. The maximum of scattering is found to be at The maximum of scattering is found to be at == 6565 ° (° ( == 5050 °)°) Calculate the lattice spacing for nickelCalculate the lattice spacing for nickel
22 dd = = nn sinsin Verify that Verify that d d == 0.0920.092 nmnm
1/22 Vqm
h
Fig
s. f
rom
R. Eis
berg
& R
. R
esn
ick Q
uantu
m P
hysi
cs
a, b, c – computer simulation
d - experiment
Electron interference
Electron Microscope
Electron Waves Electrons with 20ev energy, Electrons with 20ev energy,
have a wavelength of about have a wavelength of about 0.27 nm0.27 nm
This is around the same size as This is around the same size as the average spacing of atoms in the average spacing of atoms in a crystal latticea crystal lattice
These atoms will therefore These atoms will therefore form a diffraction grating for form a diffraction grating for electron “waves”electron “waves”
Several pictures are shown left Several pictures are shown left (see the web links on the course (see the web links on the course home page)home page)
http://www.chem.qmw.ac.uk/surfaces/scc/scat6_2.htm
from this lecture… The solution to the blackbody spectrum leads to the concept of photons, and to a solution The solution to the blackbody spectrum leads to the concept of photons, and to a solution
for the photoelectric effectfor the photoelectric effect The maximum excess energy of a photoelectron isThe maximum excess energy of a photoelectron is
The particle nature of light is also shown by Compton scattering of electrons by photonsThe particle nature of light is also shown by Compton scattering of electrons by photons
Scattering shows that photons have momentum given byScattering shows that photons have momentum given by
This implies that matter also has wavelike properties given by the de Broglie formulaThis implies that matter also has wavelike properties given by the de Broglie formula
The de Broglie wavelength leads to phenomena such as electron diffraction. A common The de Broglie wavelength leads to phenomena such as electron diffraction. A common tool in modern crystallographytool in modern crystallography
hfKmax
)cos1(0 cm
h
e
h
c
hf
c
Ep
mv
h
p
h