1. Modern PhysicsLecture IX

2. 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

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

4. 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 Kmax=eDVs

5. 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

6. Photoelectric Effect • 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

7. 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

8. Inverse photoelectric effect - Production of X-rays • 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)

9. Roentgen Lamp Tungsten - wolfram

10. 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’

11. 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…

12. Louis de Broglie1892 - 1987

13. 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…

14. 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 Broglie’s hypothesis gives l= h / p Substitute for pto get

15. 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

16. Sir Joseph John (JJ) Thomson

17. C.J.Davisson and L.G.Germer dNi=0.215nm diffraction de Broglie

18. Example of Measuring the Lattice Spacing • Consider an electron accelerated to V =50V scattered through angle q. Note that 2f+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 ln integer

19. Figs. from R. Eisberg & R. Resnick Quantum Physics 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 2d =nl/ sinf • Verify that d =0.092nm

20. Electron interference a, b, c – computer simulation d - experiment

21. Electron Microscope

22. 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

23. 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