Modern Physics Lecture IX

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Modern PhysicsLecture IX
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The Quantum Hypothesis

• In this lecture we examine the evidence for
  light quanta and the implications of their
  existence
• Waves as Particles
• The photoelectric effect
• Compton scattering
• Particles as Waves
• Electron diffraction
• The Double Slit Revisited

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Photoelectric effect

• When light strikes the cathode, electrons are
  emitted
• Electrons moving between the two plates
  constitute a current

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Photoelectric Effect

• Properties of the photoelectric effect
• Electrons are only emitted above a certain
  cut-off frequency
• This frequency is different for different
  materials
• It is called the work function
• Below the work function no electrons are
  emitted no matter how intense the light is
• The maximum energy of the ejected electron is
  KmaxeDVs

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Photoelectric Effect

• Properties
• No photoelectrons are emitted if the frequency
  falls below some cut-off frequency fc
• The maximum energy of the photons is independent
  of the light intensity
• The maximum kinetic energy of the photoelectrons
  increases with increasing frequency
• Photoelectrons are emitted almost instantaneously
  from the surface

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Photoelectric Effect

• Explanation
• Einstein extended Plancks explanation for
  blackbody radiation to suggest that in fact the
  quanta of energy used in blackbody radiation are
  in fact localised particle like energy packets
• Each having an energy given by hf
• Emitted electrons will have an energy given by
• Where f is known as the work function of the
  material

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Photoelectric Effect

• Quantum interpretation
• If the energy of a photon is less than the work
  function f, the photon cannot give enough energy
  to the electron to leave the surface
• Kmax does not depend on light intensity, because
  doubling the number of photons would only double
  the number of electrons and not double their
  energy
• Kmax increases with frequency because energy and
  frequency are related
• If light is particle-like, then all of the energy
  will be delivered instantaneously thus liberating
  an electron with no time delay between the light
  hitting the surface and the electron escaping

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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)

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Roentgen Lamp
Tungsten - wolfram
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Compton Scattering

• If light is like a particle does it have
  momentum?
• In Compton scattering x-rays impart momentum to
  matter, scattering electrons like billiard balls
• Thus photons also have momentum. The momentum of
  a photon is given by

Recoiling electron
f
q
Incident Photon, l0
Scattered Photon, l
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Photons and Electromagnetic Waves

• How can light be considered a photon (particle)
  when we know it is a wave
• Light has a dual nature it exhibits both wave
  and particle characteristics
• There is a smooth transition of these properties
  across the electromagnetic spectrum
• At low frequencies (radio waves) photons have a
  vanishingly small energy and the wave properties
  dominate
• At high frequencies (x-rays, g-rays) it is the
  particle properties that dominate
• But

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Louis de Broglie1892 - 1987
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Wave Properties of Matter

• In 1923 Louis de Broglie postulated that perhaps
  matter exhibits the same duality that light
  exhibits
• Perhaps all matter has both characteristics as
  well
• Previously we saw that, for photons,

• Which says that the wavelength of light is
  related to its momentum
• Making the same comparison for matter we find

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de Broglie Wavelength of Electrons

• We now calculate the wavelength of a charged
  particle accelerated through potential V
• Assume that the particles have mass m and charge
  q
• Equate kinetic energy of the particles with the
  electrostatic energy
• K m v 2/2 q V
• momentum p m v
• We can express kinetic energy in terms of
  momentum
• K p 2/(2 m) q V
• Reorganise to get
• p (2 m q V )1/2
• de Broglies hypothesis gives
• l h / p
• Substitute for p to get

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Does Matter Really Have a Wavelength

• The wavelength of matter waves is very small.
  This is why we do not see them in our every day
  experience
• To see diffraction a grating a very small slit
  width is required (eg the space between two atoms
  in a crystal)
• This is exactly how electron diffraction was
  first found!
• G. P. Thompson of Scotland and Davisson and
  Germer from the USA used the close spacing
  between atoms in a crystal lattice to diffract
  electron waves thus proving that matter can also
  exhibit diffraction and interference

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Sir Joseph John (JJ) Thomson
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C.J.Davisson and L.G.Germer
dNi0.215nm
diffraction
de Broglie
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Example of Measuring the Lattice Spacing

• Consider an electron accelerated to V 50 V
  scattered through angle q. Note that 2 f q
  180, i.e. f 90-q /2

Figs. from R. Eisberg R. Resnick Quantum Physics

• The condition for constructive interference is
• 2 d sinf n l n integer

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Example

• Electron scattering in nickel
• Electrons are accelerated through V 54V.
• The maximum of scattering is found to be at f
  65 (q 50 )
• Calculate the lattice spacing for nickel
• 2 d n l / sinf
• Verify that d 0.092 nm

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Electron interference
a, b, c computer simulation d - experiment
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Electron Microscope
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Electron Waves

• Electrons with 20ev energy, have a wavelength of
  about 0.27 nm
• This is around the same size as the average
  spacing of atoms in a crystal lattice
• These atoms will therefore form a diffraction
  grating for electron waves
• Several pictures are shown left (see the web
  links on the course home page)

http//www.chem.qmw.ac.uk/surfaces/scc/scat6_2.htm
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from this lecture

• The solution to the blackbody spectrum leads to
  the concept of photons, and to a solution for the
  photoelectric effect
• The maximum excess energy of a photoelectron is
• The particle nature of light is also shown by
  Compton scattering of electrons by photons
• Scattering shows that photons have momentum given
  by
• This 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 tool in modern
  crystallography