Quantization of Physical Properties: Photoelectric and Compton Effects - PDF

1.1 What’s wrong with classical physics? Quantization Particle nature of light Wave nature of matter Quantization of physical properties Why do we need a quantum theory? In the early 1900s, several experimental observations led to the postulation that lig...

1.1 What’s wrong with classical physics? Quantization Particle nature of light Wave nature of matter Quantization of physical properties Why do we need a quantum theory? In the early 1900s, several experimental observations led to the postulation that light comes in quanta. Other results suggested that some other physical properties are quantized... the photo-electric effect the Compton effect discrete line spectra in atomic physics the Franck-Hertz experiment the Stern-Gerlach experiment The photo-electric effect The ejection of electrons from a metal surface by light: Lenard showed that monochromatic light yields electrons of a fixed, definite energy - more light ⇒ more electrons of the same energy! This contradicts expectations based on a wave theory of light. In fact, changing the colour (frequency) of the incident light changes the energy of the ejected electrons. Eel = h(ν − νo) = h̄(ω − ω0) Einstein interpreted this by postulating light quanta: PHOTONS The photon moves with velocity = c, so rest mass = 0. Then E 2 = m2c4 + p2c2 = h̄2ω 2 ⇒ p = h̄ω/c = h̄k Quick question: Which of the following carries the most energy? A. 100 blue-light photons of wavelength 475 nm B. 100 red-light photons of wavelength 650 nm C. an infra-red photon of wavelength 0.3 mm D. an ultra-violet photon of wavelength 10 nm Quick question: The number of electrons emitted from a metal by the photoelectric effect depends on A. the colour of the light B. the brightness of the light C. the length of time for which light is shone on the metal D. the type of metal The Compton effect Compton directed a beam of X rays (photons) at carbon atoms in a graphite block, and measured the scattered beam at various angles. He discovered that the scattered X rays undergo a frequency shift: How does this support the idea of light quanta? Assume the electron is unbound and at rest before the collision. Then: h̄ω = h̄ω 0 + meγc2 − mec2 conservation of energy (1) h̄ω/c = h̄ω 0/c + meγv conservation of momentum (2) Decomposing 2 into parallel and normal components, we get h̄ω/c = h̄ω 0/c cos θ + meγv cos φ h̄ω 0/c sin θ = meγv sin φ Hence we arrive at the Compton scattering formula, 2h̄ 2θ ω − ω0 = ωω 0 sin mc2 2 The Compton scattering formula is normally given in terms of the wavelength λ(= 2πc/ω): 2πh̄ 2 θ λ0 − λ = 2 sin mc 2 The factor 2πh̄/mc = λc is the Compton wavelength of a particle with mass m (in this case, the electron). We can use it as a measure of the size of the particle. The Compton effect not only confirms the quantum nature of light (we wouldn’t get a frequency change in a classical description) - it also demonstrates the conservation of E and p in interactions between light and matter. Quick question: When a photon Compton scatters from an electron, A. the energy given to the electron is independent of the frequency of the incoming light B. the change in the light’s frequency is independent of the fre- quency of the incoming light C. the energy given to the electron is a maximum if the electron scatters at right-angles D. the energy of the scattered photon is a minimum if the electron scatters forwards Modern uses of Compton scattering American Science & Engineering, Inc.’s Z Backscatter system Atomic line spectra In the early part of the 20th century, it was observed that the characteristic spectral lines of particular atoms can be arranged into special series such as the Balmer series in H: Absorption spectrum for hydrogen Emission spectrum for hydrogen Each line is associated with the de-excitation of an excited atomic state: Walter Ritz formulated the Ritz combination principle, which states that new lines can be found through the addition and sub- traction of known lines. Aside: Bohr’s (inaccurate) model of the atom was inspired by Balmer series. He predicted other series of spectral lines which were later observed (the Lyman, Paschen and Brackett series).  h̄ω30 = E3 − E0 = E3 − E1 + E1 − E0 or in terms of ω, ω30 = ω31 + ω10 The existence of discrete line spectra suggests two things: the energies that an atom can have are quantized, and energy can be transferred by light quanta with definite en- ergy (frequency). The Franck-Hertz experiment The Franck-Hertz experiment (1913) provided further evidence for the quantization of energy in atoms: Measure current as a function of voltage... When the energy of the electron reaches 4.9 eV, the current drops. Why? A mercury atom can absorb exactly this much and no less. The Stern-Gerlach experiment Remember that a particle with a magnetic moment is deflected by an inhomogeneous magnetic field. If the neutral Ag atoms are in their ground state, we ex- pect nothing to happen to the beam, but... The magnetic moment has two distinct orientations. Magnetic Resonance Imaging The same physics is behind the medical imaging technique of MRI. In simple terms, you can imagine the proton at the core of a hy- drogen atom as a spinning bar magnet with a magnetic moment which can be “flipped” by a radio frequency transition of the cor- rect energy. Wave aspects of matter If light can be described as particles (pho- tons), can ordinary particles (such as atoms, electrons or nucleons) be described as waves? Louis de Broglie suggested they could, and experimental evidence supports him. In the following, we consider: evidence of wave behaviour the Davisson and Germer experiment Thomson’s experiment de Broglie postulates and the wave nature of matter de Broglie postulated that particles of matter are like particles of light: E p2 E = h̄ω = hν ⇒ ν = = h 2mh h h h p = h̄k = ⇒ λ= =√ λ p 2mE Do we have any evidence for wave-like properties of particles? What would constitute good evidence? c.f. Young’s double slits: Remember: we get constructive interference when d sin θ = mλ Now λ ≈ 5 × 10−7 m, and d is hard to make smaller than 10−4 m. Generally we only see the first few fringes, such that θ is small (≈ 3◦).  Huygens, 1629-1695 Newton, 1643-1727 Destructive interference cannot be explained in any reasonable cor- puscular description of light. So if we can see evidence of interference with beams of particles (other than photons) then we can say that we have evidence of the wave nature of matter. Electron diffraction Two historic experiments showed electron diffraction/interference. 1. Davisson and Germer’s experiment: Laue’s method for X ray diffraction but with electrons: Positions of maxima given by 2d sin φ = 2d sin[(π − θ)/2] = nλ. If the electron is accelerated through V , it has energy eV so: √ nh V sin φ = √ 2d 2me This is confirmed experimentally. So we not only have evidence of wave-like behaviour for electrons, but also that de Broglie’s postu- lates are good. 2. Thomson’s experiment Davisson and Germer used 54 V for acceleration; Thomson and Tartakowski used kV, leading to deeper penetration into the ma- terial and scattering from many atomic planes. Bright rings are given by the Bragg relation, 2d sin θ = nλ ⇒ Dd = 2nLλ which leads to √ 2nLh VD = √ d 2me Quick question: Which will have the closest bright fringes if used in a Davisson-Germer type experiment? A. Electrons accelerated through a 1 V potential B. Electrons accelerated through a 100 V potential C. Protons accelerated through a 100 V potential D. Neutrons accelerated through a 100 V potential X rays diffracted through Electrons diffracted through a thin Al foil the same foil (scaled!!!) What other particles are waves? What about really big stuff? C60 molecules: M. Arnt et al., Nature 401 (1999) 680-682 C60 molecules: M. Arnt et al., Nature 401 (1999) 680-682 But r(C60) ≈ 7 Å!!