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Modern Physics Lecture III

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Modern PhysicsLecture III

- 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

- Waves as Particles

Photoelectric effect

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

- 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 Kmax=eDVs

- 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

- Explanation
- Einstein extended Planck’s 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

- 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

- Photons are absorbed in whole - electrons can transfer part of their energy.
- Brehmsstrahlung - electrons decelerate in electromagnetic field of nuclei: Ef = Ei - h , wide distribution
- Maximal frequency - minimal wavelength (observed empirically first) eV = h max = h c / min
- Also discrete spectrum (atomic levels)

Tungsten - wolfram

- 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’

- 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…

- 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…

- 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 Broglie’s hypothesis gives

l= h / p

Substitute for pto get

- 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

dNi=0.215nm

diffraction

de Broglie

a, b, c – computer simulation

d - experiment

- 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

Wave Properties

Reflection, Refraction

A property of both particles and waves

Interference and Diffraction

Young’s double slits

Waves Only

Polarisation

Waves Only

Particle properties

Consists of discreet particles

(atoms or molecules)

Momentum

A well defined trajectory

Does not diffract or interfere

1 particle + 1 particle = 2 particles

Conclusion: Our initial hypothesis is incorrect. We need to form a new hypothesis for the differences between matter and light. One could be that matter has a rest mass!

- 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