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Physics 334 Modern Physics Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were based on the textbook “Modern Physics” by Thornton and Rex. I have replaced some images from the adopted text “Modern Physics” by Tipler and Llewellyn. Others images are from a variety of sources (PowerPoint clip art, Wikipedia encyclopedia etc) and were part of original lectures. Contributions are noted wherever possible in the PowerPoint file. The PDF handouts are intended for my Modern Physics class. 1 CHAPTER 3 Quantization of Charge, Light, and Energy 3-1 Quantization of Electric Charge 3-2 Blackbody Radiation 3-2.1 The Photon 3-3 The Photoelectric Effect 3-4 X Rays and the Compton Effect 3-4.1 Pair Production and Pair Annihilation Max Karl Ernst Ludwig Planck (1858-1947) We have no right to assume that any physical laws exist, or if they have existed up until now, or that they will continue to exist in a similar manner in the future. An important scientific innovation rarely makes its way by gradually winning over and converting its opponents. What does happen is that the opponents gradually die out. - Max Planck Photo: http://www.spaceandmotion.com/quantum-theory-max-planck-quotes.htm 2 3.1: Discovery of the X-Ray and the Electron In the 1890s scientists and engineers were familiar with “cathode rays.” These rays were generated from one of the metal plates in an evacuated tube with a large electric potential across it. Wilhelm Röntgen (1845-1923) J. J. Thomson (1856-1940) It was surmised that cathode rays had something to do with atoms. It was known that cathode rays could penetrate matter and were deflected by magnetic and electric fields. Rontgen: www.aip.org/history/ electron/jjsound.htm Thomson: www.haverford.edu/.../ songs/ionsmine.htm 3 Observation of X Rays Wilhelm Röntgen studied the effects of cathode rays passing through various materials. He noticed that a phosphorescent screen near the tube glowed during some of these experiments. These new rays were unaffected by magnetic fields and penetrated materials more than cathode rays. He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube. Wilhelm Röntgen www.peoples.ru/science/ physics/rontgen/ 4 Röntgen’s X-Ray Tube Röntgen constructed an x-ray tube by allowing cathode rays to impact the glass wall of the tube and produced x rays. He used x rays to make a shadowgram the bones of a hand on a phosphorescent screen. X-ray photo: http://pt.wikipedia.org/wiki/Imagem:Roentgen-x-ray-von-kollikershand.jpg 5 3-1 Quantization of Electric Charge Thomson used an evacuated cathode-ray tube to show that the cathode rays were negatively charged particles (electrons) by deflecting them in electric and magnetic fields. Thomson’s method of measuring the ratio of the electron’s charge to mass was to send electrons through a region containing a magnetic field perpendicular to an electric field. 6 Calculation of e/m Exercise 1: An electron is moving under the influence of electric and magnetic fields. Show that the charge to mass ratio is given by; q E = m RB 2 The B filed can be computed by measuring the current using an ammeter, the electric field can be computed by measuring the voltage and the radius R can be measured experimentally by a rod 11 q / m = 0.7 × 10 C / kg This is independent of the nature of gas or metal for the Cathode. Lorentz called the charge electron e as one unit of negative charge 7 Determination of Electron Charge Millikan’s oil-drop experiment Robert Andrews Millikan (1868 – 1953) Millikan was able to show that electrons had a particular charge. http://imglib.lbl.gov/ImgLib/COLLECTIONS/BERKELEYLAB/images/pg08_Millikan.lowres.jpeg 8 Calculation of the oil drop charge Exercise 2: For Millikan’s experiment derive an expression for the charge of an electron as a function of mdrop, acceleration due to gravity g, plate separation d and voltage V. Then using Stoke’s law to determine the terminal velocity and thus the mass of the drop, calculate the charge of an electron. e = 1.602 x 10-19 C 9 3-2 Blackbody Radiation When matter is heated, it emits radiation. A blackbody is a cavity with a material that only emits thermal radiation. Incoming radiation is absorbed in the cavity. Blackbody radiation is theoretically interesting because the radiation properties of the blackbody are independent of the particular material. Physicists can study the properties of intensity versus wavelength at fixed temperatures. 10 Wien’s Displacement Law The spectral intensity R(λ,Τ ) is the total power radiated per unit area per unit wavelength at a given temperature. Wien’s displacement law: The maximum of the spectrum shifts to smaller wavelengths as the temperature is increased. λmT = constant=2.898 ×10-3 m.K 11 Stefan-Boltzmann Law The total power radiated increases with the temperature: R = εσ T 4 This is known as the Stefan-Boltzmann law, with the constant σ experimentally measured to be 5.6705 × 10−8 W / (m2 · K4). The emissivity є (є = 1 for an idealized blackbody) is simply the ratio of the emissive power of an object to that of an ideal blackbody and is always less than 1. 12 Example – Wein’s Displacement Law How Hot is a Star? Measurement of the wavelength at which spectral distribution R(λ) from the Sun is maximum is found to be at 500nm, how hot is the surface of the Sun? 13 Example – Stefan Boltzmann Law How Big is a Star? Measurement of the wavelength at which spectral distribution R(λ) from a certain star is maximum indicates that the star’s surface temperature is 3000K. If the star is also found to radiate 100 times the power Psun radiated by the Sun, how big is the star? Take the Sun’s surface temperature as 5800 K. 14 Rayleigh-Jeans Formula Lord Rayleigh used the classical theories of electromagnetism and thermodynamics to show that the blackbody spectral distribution should be: u (λ ) = 8π kT λ −4 u (λ ) = 8π kT λ −4 Integrating u (λ ) from 0 → ∞, we find ∞ ∫ u(λ )d λ → 0 when λ → 0 0 It approaches the data at longer wavelengths, but it deviates badly at short wavelengths. This problem for small wavelengths became known as the ultraviolet catastrophe and was one of the outstanding exceptions that classical physics could not explain. 15 Planck’s Radiation Law Planck assumed that the radiation in the cavity was emitted (and absorbed) by some sort of “oscillators.” He used Boltzman’s statistical methods to arrive at the following formula that fit the blackbody radiation data. 8π hcλ −5 u (λ ) = hc λ kT e −1 Planck’s radiation law Planck made two modifications to the classical theory: The oscillators (of electromagnetic origin) can only have certain discrete energies, En = nhf, where n is an integer, ν is the frequency, and h is called Planck’s constant: h = 6.6261 × 10−34 J·s. The oscillators can absorb or emit energy in discrete multiples of the fundamental quantum of energy given by: ∆E = hf 16 Planck’s Radiation Law Exercise 3 Derive Planck’s radiation law. Exercise 4 Show that Stefan’s Boltzmann law, Wein’s displacement law and Rayleigh-Jean’s law can be derived from Planck’s law Application: The Big Bang theory predicts black body radiation. This radiation was discovered in 1965 by A. Penzias & R. Wilson. Cosmic Background Explorer (COBE) and Wilkinson Microwave Anisotropy Probe (WMAP) detected this radiation field at 2.725± 0.001 °K This data supports the Big Bang Theory 17 Example Planck’s Radiation Law Exercise 5 What is the average energy of an oscillator that has a frequency given by hf=kT according to Planck’s calculations? 18 What is a Photon? Planck introduced the idea of a photon or quanta. A cavity emits radiation by way of quanta. How does the radiation travel in space? We think that radiation is a wave phenomenon however the energy content is delivered to atoms in concentrated groups of waves (quanta). 19 Properties of Photon If a photon is to be considered as a particle we must be able to describe its mass, momentum, energy, statistics etc. Energy of a photon = × -34 = × -15 E hf ; h=6.626 10 J .s 4.136 10 eV .s E = Energy, f = frequency limiting value 1/2hf otherwise integral multiple of hf Interaction of photons with matter Complete absorption or partial absorption with the photon adjusting its frequency to remain as particle Intensity of photon Intensity ∝ number of photons intensity has nothing to do with the energy of photons 20 Properties of Photons Constant h of photon h defines the smallest quantum angular momentum of a particle ∴ E = hf = mc 2 ⇒ m = hf hf hf and p c = = c2 c2 c Exercise 6 Show that h has units of angular momentum Mass and momentum of photon photons move with velocity v=c E = hf = mc 2 = m0 2 1− v c 2 c 2 ⇒ m0 = ( ) 1 − v2 c2 • hf =0 c2 Photons have no rest mass m0 Photon is not a material particle since rest mass is zero, it is a wave structure that behaves like a particle 21 Properties of Photons Charge of a photon photons do not carry charge, however they can eject charge particles from matter when they impinge on atoms Photon Statistics Consider the radiation as a gas of photon. Photons move randomly like molecules in a gas and have wide range of energies but same velocity Statistics of photons is described by Bose-Einstein. We can talk about intensity and temperature in the same way as density and temperature. 22 3-3 Photoelectric Effect Methods of electron emission: Thermionic emission: Applying heat allows electrons to gain enough energy to escape. Secondary emission: The electron gains enough energy by transfer from another high-speed particle that strikes the material from outside. Field emission: A strong external electric field pulls the electron out of the material. Photoelectric effect: Incident light (electromagnetic radiation) shining on the material transfers energy to the electrons, allowing them to escape. We call the ejected electrons photoelectrons. Image adapted from http://sol.sci.uop.edu/~jfalward/particlesandwaves/particlesandwaves.html 23 Photo-electric Effect Experimental Setup Image adapted from http://www.chem1.com/acad/webtut/atomic/qprimer/#Q8 24 Photo-electric effect observations Electron kinetic energy The kinetic energy of the photoelectrons is independent of the light intensity. The kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light. Classically, the kinetic energy of the photoelectrons should increase with the light intensity and not depend on the frequency. Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png 25 Photoelectric effect observations Electron kinetic energy There was a threshold frequency of the light, below which no photoelectrons were ejected (related to the work function φ of the emitter material). The existence of a threshold frequency is completely inexplicable in classical theory. Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png 26 Photoelectric effect observations (number of electrons) When photoelectrons are produced, their number is proportional to the intensity of light. Also, the photoelectrons are emitted almost instantly following illumination of the photocathode, independent of the intensity of the light. Classical theory predicted that, for extremely low light intensities, a long time would elapse before any one electron could obtain sufficient energy to escape. We observe, however, that the photoelectrons are ejected almost immediately. Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png 27 Einstein’s Theory: Photons Einstein suggested that the electro-magnetic radiation field is quantized into particles called photons. Each photon has the energy quantum: E = hf where f is the frequency of the light and h is Planck’s constant. Alternatively, E = hω where: h = h 2π 28 Einstein’s Theory Conservation of energy yields: 1 hf = φ + mv 2 2 where φ is the work function of the metal (potential energy to be overcome before an electron could escape). In reality, the data were a bit more complex. Because the electron’s energy can be reduced by the emitter material, consider fmax (not f): 1 eV0 = mv 2 = hf − φ 2 max 29 Example – Photoelectric Effect An experiment shows that when electromagnetic radiation of wavelength 270 nm falls on an aluminum surface, photoelectrons are emitted. The most energetic of these are stopped by a potential difference of 0.46 volts. Use this information to calculate the work function of aluminum in electron volts. 30 Example – Photoelectric Effect The threshold wavelength of potassium is 558 nm. What is the work function for potassium? What is the stopping potential when light of 400 nm is incident on potassium? 31 Example – Photoelectric Effect Light of wavelength 400 nm and intensity 10-2 W/m2 is incident on potassium. Estimate the time lag expected classically. 32 3-4 X-Ray Production: Theory An energetic electron passing through matter will radiate photons and lose kinetic energy, called bremsstrahlung. Since momentum is conserved, the nucleus absorbs very little energy, and it can be ignored. The final energy of the electron is determined from the conservation of energy to be: Ei Ef hν E f = Ei − hf 33 3-4 X-Ray Production: Theory If photons can transfer energy to electrons, can part or all of the kinetic energy of electron be converted into photons? Ei “The Inverse photoelectric effect” Ef This was discovered before the work of Planck and Einstein hν 34 Observation of X Rays 1895: Wilhelm Röntgen studied the effects of cathode rays passing through various materials. He noticed that a phosphorescent screen near the tube glowed during some of these experiments. These new rays were unaffected by magnetic fields and penetrated materials more than cathode rays. He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube. Wilhelm Röntgen www.peoples.ru/science/ physics/rontgen/ 35 X-Ray Production: Experiment Current passing through a filament produces copious numbers of electrons by thermionic emission. If one focuses these electrons by a cathode structure into a beam and accelerates them by potential differences of thousands of volts until they impinge on a metal anode surface, they produce x rays by bremsstrahlung as they stop in the anode material. 36 Electromagnetic theory Predicts X-Ray Accelerated charges produce electromagnetic waves, when fast moving electrons are brought to rest, they are certainly accelerated 1906: Barkla found that X-Ray show Polarization, this establishing that X-Rays are waves. X-Rays have wavelength range 0.1 nm – 100 nm Even though classical theory predicts x-ray’s, the experimental data is not explainable. 37 Two distinctive features 1. Some targets have enhanced peaks. For example Molybdenum shows two peaks at specific wavelengths. a) This is due to rearrangement of electrons of the target material after bombardment. b) X-ray ‘s have a continuous spectrum 2. No matter what the target , the threshold wavelength depends on the accelerating potential 38 Inverse Photoelectric Effect Since the work function of the target is of the order of few eV, whereas the accelerating potential is thousand of eV 1 eV0 = mv 2 = hf − φ 2 max eVo = hf max = From photoelectric effect hc λmin Duane-Hunt rule 39 X-Ray Scattering Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed. Crystals act as three-dimensional gratings, scattering the waves and producing observable interference effects. 40 Bragg’s Law William Lawrence Bragg interpreted the x-ray scattering as the reflection of the incident x-ray beam from a unique set of planes of atoms within the crystal. There are two conditions for constructive interference of the scattered x rays: 1) The angle of incidence must equal the angle of reflection of the outgoing wave. 2) The difference in path lengths must be an integral number of wavelengths. Bragg’s Law: nλ = 2d sin θ (n = integer) 41 The Bragg Spectrometer A Bragg spectrometer scatters x rays from crystals. The intensity of the diffracted beam is determined as a function of scattering angle by rotating the crystal and the detector. When a beam of x rays passes through a powdered crystal, the dots become a series of rings. 42 Examples Exercise 1: What is the shortest wavelength present in a radiation if the electrons are accelerated to 50,000 volts? Exercise 2:The spacing of one set of crystals planes in common salt is d=0.282nm. A monochromatic beam of X-rays produces a Bragg maximum when its glancing angle is 7 degrees. Assuming that this is the first order maximum (n=1), find the wavelength of the X-rays, what is the minimum possible accelerating voltage Vo that produced the Xrays? 43 Compton Effect If photons behave like particles than collisions between a photon and a particle can be studied just like billiard ball collisions. hc / λ Classical theory predicts that the scattered frequency of the photon should be the same as the incident photon. When a photon enters matter, it can interact with one of the electrons. The laws of conservation of energy and momentum apply, as in any elastic collision between two particles. The momentum of a particle moving at the speed of light is: p= Ee hc / λ ′ E hν h = = c c λ The electron energy is: This yields the change in wavelength of the scattered photon, known as the Compton effect: ∆λ = λ2 − λ1 = h (1 − cos θ ) mc 44 Compton Effect Intensity versus wavelength for Compton scattering at several angles. The first peak results from photons of the original wavelength that scattered by tightly bound electrons, which have an effective mass equal t that of the atom. The separation in wavelength of the peaks is given by the Compton scattering equation 45