Chapter 2 Particle properties of waves 1
2.1 Electromagnetic waves Coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behavior 2
James Clerk Maxwell 1831 1879 Developed the electromagnetic theory of light Developed the kinetic theory of gases Explained the nature of color vision Explained the nature of Saturn s rings Died of cancer 3
Electromagnetic Waves In empty space, q = 0 and I = 0 Maxwell predicted the existence of electromagnetic waves The electromagnetic waves consist of oscillating electric and magnetic fields The changing fields induce each other which maintains the propagation of the wave A changing electric field induces a magnetic field A changing magnetic field induces an electric field 4
Maxwell showed that electromagnetic waves propagated with the speed 1 8 c 2.98810 m / 0 0 This is exactly the speed of light, and Maxwell concluded light consists of electromagnetic waves. Note that light waves are only a small part of the electromagnetic spectrum. s 5
Plane em Waves We will assume that the vectors for the electric and magnetic fields in an em wave have a specific space-time behavior that is consistent with Maxwell s equations Assume an em wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction 6
Plane em Waves, cont The x-direction is the direction of propagation Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only 7
em Wave Representation This is a pictorial representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction E and B vary sinusoidally with x 8
GNUPLOT: equation plotting program E y Asin( x / t) Vertically (y axis) polarized wave having an amplitude A, a wavelength of and an angular velocity (frequency * 2) of, propagating along the x axis. 9
Plane-polarized light Vertical E y Asin( x / t) Horizontal E z Asin( x / t) 10
Circularly polarized light Right circular E y Asin( x / t 90) E z Asin( x / t) E y Left circular Asin( x / t 90) E z Asin( x / t) 11
Heinrich Rudolf Hertz 1857 1894 Greatest discovery was radio waves 1887 Showed the radio waves obeyed wave phenomena Died of blood poisoning 12
The Electromagnetic Spectrum 13
Figure 2.2 The spectrum of electromagnetic radiation 14
Electromagnetic spectrum The transition wavelengths are a bit arbitrary. Visible light used to be 400 to 700 nm, but now it s officially 380 to 780 nm. 15
The electromagnetic spectrum 16
60-Hz radiation from power lines Yes, this very-low-frequency current emits 60-Hz electromagnetic waves. No, it is not harmful. A very flawed epidemiological study in 1979 claimed otherwise, but no other study has ever found such results. Also, electrical power generation has increased exponentially since 1900; cancer incidence has remained essentially constant. Also, the 60-Hz electrical fields reaching the body are small; they re greatly reduced inside the body because it s conducting; and the body s own electrical fields (nerve impulses) are much greater. 60-Hz magnetic fields inside the body are < 0.002 Gauss; the earth s magnetic field is ~ 0.4 G. 17
Microwave ovens Microwave ovens operate at 2.45 GHz, where water absorbs very well. Percy LeBaron Spencer, Inventor of the microwave oven 18
Visible light Wavelengths and frequencies of visible light 19
The sun s spectrum 20
The eye s response to light and color The eye s cones have three receptors, one for red, another for green, and a third for blue. 21
The Ultraviolet The UV is usually broken up into three regions, UVA (320-380 nm), UVB (290-320 nm), and UVC (220-290 nm). UVC is almost completely absorbed by the atmosphere. You can get skin cancer even from UVA. 22
Very short-wavelength regions Vacuum-ultraviolet (VUV) 180 nm > l > 50 nm Absorbed by <<1 mm of air Ionizing to many materials Soft x-rays 5 nm > l > 0.5 nm Strongly interacts with core electrons in materials Extreme-ultraviolet (XUV or EUV) 50 nm > l > 5 nm Ionizing radiation to all materials 23
Fast electrons impacting a metal generate x-rays. High voltage accelerates electrons to high velocity, which then impact a metal. Electrons displace electrons in the metal, which then emit x-rays. The faster the electrons, the higher the x-ray frequency. 24
X-rays penetrate tissue and do not scatter much. Roentgen s x-ray image of his wife s hand (and wedding ring) 25
Fig. 2.3 (a) In constructive interference, superposed waves in phase reinforce each other. (b) In destructive interference, waves out of phase partially or completely cancel each other. 26
Fig. 2.4 Origin of the interference pattern in Young s experiment. Constructive interference occurs where the difference in path lengths from the slits to the screen is, 2, Destructive interference occurs where the path difference is /2, 3/2, 5/2 27
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/2 29
Each of these wavefronts will cancel each other, ie destructively interfere at some distance point 30
2.2 Blackbody Radiation Only the quantum theory of light can explain its origin 31
Fig. 2.5 A hole in the wall of a hollow object is an excellent approximation of a blackbody 32
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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. 34
500 K or less -- essentially all of the radiation is at frequencies less than those of visible light. (Infrared ROY G. BiV ) The blackbody is black. 2000 K -- visible light of appreciable intensity, but mostly at the red (low frequency) end of the spectrum. 3000 K -- red predominates, but significantly more blue. This is about the temperature of an incandescent lamp filament. 6500 K -- distribution is more nearly uniform, therefore "white hot." Example -- the surface of the sun. 10000 K and above -- blue light predominates. "Blue hot." Example -- some of the hotter stars. 35
Trouble in Physics The Blackbody spectrum in late 1890s Boltzmann 36
Trouble in Physics The Blackbody spectrum classical theory Max Planck 1900 Quantisation E vib = nhf 37
The intensity also depends on the temperature of the blackbody. Why does the blue curve correspond to the hotter blackbody? A range of frequencies are emitted. The intensity depends on the frequency. 38
Rayleigh-Jeans Formula experiment well The Ultraviolet Catastrophe! 39
Wien s Displacement Law The spectral intensity I(, T) 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. 40
Stefan-Boltzmann Law The total power radiated increases with the temperature: This is known as the Stefan-Boltzmann law, with the constant σ experimentally measured to be 5.6705 10 8 W / (m 2 K 4 ). 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. 41
Wilhelm Wien s law u W ( f, T ) Af 3 e f T Energy per volume per frequency u: energy per unit frequency 42
Rayleigh-Jeans formula U-V catastrophe u ( f ) 8 c f RJ 3 2 k B T 43
Planck s Radiation Law In 1900, Max Planck derived his now famous relationship that explains blackbody radiation 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. u P 8hf ( f ) 3 c 3 e hf k 1 B T 1 Planck s radiation law 44
Planck made two modifications to the classical theory: The oscillators (of electromagnetic origin) can only have certain discrete energies, E n = nh, where n is an integer, 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 = h 45
2.3 : 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. 46
Photo-electric Effect Experimental Setup 47
Discovery: Heinrich Hertz and Phillip Lenard Back in 1887 Hertz clarified Maxwell s electromagnetic theory of light: Proved that electricity can be transmitted in electromagnetic waves. Established that light was a form of electromagnetic radiation. First person to broadcast and receive these waves. 48
The Spark Gap Generator First observed the effect while working with a spark-gap generator ~ accidentally, of course Illuminated his device with ultraviolet light: This changed the voltage at which sparks appeared between his electrodes! 49
Hertz s Spark Gap Generator: 50
Hertz s Experiment An induction coil is connected to a transmitter The transmitter consists of two spherical electrodes separated by a narrow gap 51
Hertz s Experiment, cont The coil provides short voltage surges to the electrodes As the air in the gap is ionized, it becomes a better conductor The discharge between the electrodes exhibits an oscillatory behavior at a very high frequency From a circuit viewpoint, this is equivalent to an LC circuit 52
Hertz s Experiment, final Sparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties Interference, diffraction, reflection, refraction and polarization He also measured the speed of the radiation 53
Lenard Goes Further His assistant, Phillip Lenard, explored the effect further. He built his own apparatus called a phototube to determine the nature of the effect: 54
Lenard s Photoelectric Apparatus 55
The Experiment: By varying the voltage on a negatively charged grid between the ejecting surface and the collector plate, Lenard was able to: Determine that the particles had a negative charge. Determine the kinetic energy of the ejected particles. 56
Lenard s Findings: Thus he theorized that this voltage must be equal to the maximum kinetic energy of the ejected particles, or: KE max = ev? ev stopping Perplexing Observations: The intensity of light had no effect on energy There was a threshold frequency for ejection Classical physics failed to explain this, Lenard won the Nobel Prize in Physics in 1905. 57
An example of the kind of apparatus used to study the photoelectric effect (but not Hertz s apparatus) is described. Start with a glass tube. Insert a couple of metal plates, with vacuum feedthroughs carrying wires to the outside. Pump out all * the air and other gases. *You never get all the gases out, but you can get most out. 58
+ - V x Seal off the tube (or keep pumping to maintain a vacuum). Connect the metal plates to a variable high voltage power supply. The positive plate is the anode and the negative plate is the cathode.* Connect an ammeter to measure the current between the plates. A *Hint for remembering which is which. Cathode ray tubes have electron guns. Electrons come from the cathode. Electrons are negative. Therefore the cathode is negative. The other one must be positive. 59
The Photoelectric Effect V x A 60
Light strikes the anode V x A 61
Light striking the anode causes electrons to be emitted from the anode. + - V A x (Large current) The electrons are emitted with kinetic energy, and some of them reach the cathode (in spite of its negative voltage). You can measure the current with the ammeter. Wait a minute aren t electrons attracted to + and repelled from -? They are, but if their initial KE is large enough, they can reach the cathode anyway. 62
+ - V A x (small current) You can increase the retarding potential (make the cathode voltage more negative relative to the anode) and see how that affects the current. 63
+ - further increase the retarding potential V A x (zero current) We can make the cathode voltage negative enough so that no photoelectrons reach the cathode. 64
Our Text book Fig. 2.10 Photoelectron current is proportional to light intensity I for all retarding voltages. The stopping potential V 0, which corresponds to the maximum photoelectron energy, is the same for all intensities of light of the same frequency. 65
Our Text book Fig. 2.11 The stopping potential V 0, and hence the maximum photoelectron energy, depends on the frequency of the light. When the retarding potential is V=0, the photoelectron current is the same for light of a given intensity regardless of its frequency. 66
Our Text book 67
Parameters: intensity (S) and frequency (f) of light, applied voltage (V), measured photocurrent (I) stopping voltage I I V V 0 V V 0 (f 2 ) V 0 (f 1 ) retarding intensity of light increases, f =const f 2 > f 1 intensity = const, f increases Three experimental findings cannot be explained by the em theory of light. Observations: For a given material of the cathode, the stopping voltage does not depend on the light intensity. 1. The saturation current is proportional to the intensity of light at f =const. 2. Material-specific red boundary f 0 exists: no photocurrent at f < f 0. 3. Practically instantaneous response no delay between the light pulse and 68 the photocurrent pulse (many applications are based on this property). stopping voltage cesium calcium red boundary f 0 f
Photo-electric Effect Classical Theory The kinetic energy of the photoelectrons should increase with the light intensity and not depend on the light frequency. Classical theory also predicted that the electrons absorb energy from the beam at a fixed rate. So, for extremely low light intensities, a long time would elapse before any one electron could obtain sufficient energy to escape. Initial observations by Heinrich Hertz 1887 69
Einstein s s Relations Einstein predicted that a graph of the maximum kinetic energy versus frequency would be a straight line, given by the linear relation: KE = hv - Φ Therefore light energy comes in multiples of hv 70
Einstein s Interpretation A new theory of light: Electromagnetic waves carry discrete energy packets The energy per packet depends on wavelength, explaining Lenard s threshold frequency. More intense light corresponds to more photons, not higher energy photons. This was published in his famous 1905 paper: On a Heuristic Point of View About the Creation and Conversion of Light 71
Photons Einstein suggested that the electro-magnetic radiation field is quantized into particles called photons. Each photon has the energy quantum: E h where is the frequency of the light and h is Planck s constant. Alternatively, E where: h /2 72
Einstein s Theory Conservation of energy yields: Electron kinetic energy h 1 2 2 mv where 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 v max (not v): h m 1 2 2 v max 73
Graph of KE max vs. frequency 74
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Maximum photoelectron energy versus frequency of incident light: straight line; y = mx + b 76
77 work function: h 0 6.4 Platinum 4.7 Silver 4.7 Copper 3.2 Calcium 2.5 Lithium 2.3 Sodium 2.2 Potassium 1.9 Cesium Pt Ag Cu Ca Li Na K Cs function Work Symbol Metal ) ( max max 0 0 h KE h KE h
2.5 X-RAYS It is found by Roentgen in 1895. 78
Sources of light Accelerating charges emit light! Linearly accelerating charge Synchrotron radiation light emitted by charged particles deflected by a magnetic field U Bremsstrahlung (Braking radiation) light emitted when charged particles collide with other charged particles 79
But the vast majority of light in the universe comes from molecular vibrations emitting light. Electrons vibrate in their motion around nuclei. High frequency: ~10 14-10 17 cycles per second. Nuclei in molecules vibrate with respect to each other. Intermediate frequency: ~10 11-10 13 cycles per second. Nuclei in molecules rotate. Low frequency: ~10 9-10 10 cycles per second. 80
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: E E h f i E f E i h 81
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 82
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. 83
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. brems braking, i.e., decelerating strahlung radiation 84
Figure 2.16 and 2.17 show the x-ray spectra that result when tungsten and molybdenum targets are bombarded by electrons at several different accelerating potentials. The curves exhibit two features electromagnetic theory cannot explain: The same Fig. 2.17. X-ray spectra of tungsten and molybdenum at 35 kv accelerating potential. 1. In the case of molybdenum, intensity peaks occur that indicate the enhanced production of x-rays at certain wavelengths. The peaks occur at specific wavelengths for each target material and originate in rearrangements of the electron structures of the target atoms after having been disturbed by the bombarding electrons. 85
Fig. 2.16 X-ray spectra of tungsten at various accelerating potentials. 2. (a) The x-rays produced at a given accelerating potential V vary in wavelength, but none has a wavelength shorter that a certain value min. (b) Increasing V decreases min. (c) At a particular V, min is the same for both the tungsten and molybdenum targets. (see Fig. 2.17) 86
87 m V V ev hc function work neglecting by hc h ev Theory m V V 6 min min max 6 min 10 1.24 ) ( 10 1.24 Hz v nm m kv V If 19 max 11 min 10 1.21 0.0248 10 2.48, 50 equation derived experimentally by Hunt Ex.2.3
Inverse Photoelectric Effect (production of X-rays) X-rays: nm f Hz E kev 16 19 0.01 10 3 10 3 10 ph 0.1 100 -rays X-rays UV 0.01 nm 10 nm Production: c 1 2 hfmax h ev Ke mec min The upper cut-off of the spectrum corresponds to the full conversion of the electron kinetic energy into the photon energy. 34 8 6.610 310 / min 19 5 hc J s m s ev 1.610 C 110 V 11 1.2 10 m 0.12A 12.4nm o min VkV 88
Inverse Photoelectric Effect Conservation of energy requires that the electron kinetic energy equal the maximum photon energy (neglect the work function because it s small compared to the electron potential energy). This yields the Duane-Hunt limit, first found experimentally. The photon wavelength depends only on the accelerating voltage and is the same for all targets. ev h 0 max hc min 89
Continuous x-ray spectrum bremsstrahlung X-ray line spectrum? Measured x-ray spectra have extra sharp lines. The total x-ray spectrum looks like this. 90
Schematic picture of an atom K + L 91
Photons also have momentum! Use our expression for the relativistic energy to find the momentum of a photon, which has no mass: E ( mc ) p c 2 2 2 2 2 E h h p c c Alternatively: p h 2 k 2 When radiation pressure is important: Comet tails (other forces are small) Viking spacecraft (would've missed Mars by 15,000 km) Stellar interiors (resists gravity) 92
2.6 X-ray Diffraction X-ray interacts with electrons in crystal scattered on crystal plane (Bragg plane) Fig. 2.18 The scattering of electromagnetic radiation by a group of atoms. Incident plane wave are reemitted as spherical waves. 93
Fig. 2.19 Two sets of Bragg planes in a NaCl crystal. 94
Incident x-ray Bragg plane 2d sin n : n 1,2,3, constructive interference Fig. 2-20 X-ray scattering from a cubic crystal. 95
Fig. 2.21 X-ray spectrometer. 96
2.7 Compton Effect 1922 Further confirmation of the photon model (Nobel Prize in 1927) According to the quantum theory of light, photon behave like particles except for their lack of rest mass. How far can this analogy be carried? For instance, can we consider a collision between a photon and an electron as if both were billiard balls? 97
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Compton Scattering 99
Wave Picture f in f out E F = ee in = out f in = f out 100
Photon Picture E out E in E electron 101
Compton scattering Classical picture: oscillating electromagnetic field causes oscillations in positions of charged particles, which re-radiate in all directions at same frequency and wavelength as incident radiation. (Thomson scattering) Change in wavelength of scattered light is completely unexpected classically Incident light wave Oscillating electron Emitted light wave Compton s explanation: billiard ball collisions between particles of light (X-ray photons) and electrons in the material Before Incoming photon p Electron After p e θ p scattered photon scattered electron 102
Compton Effect Photons have energy and momentum: E p, p p E hc / p h / 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. E e, p e E p, p p This yields the change in wavelength of the scattered photon, known as the Compton effect: 103
Compton Effect (1922) X-ray tube X-ray detector target (light atoms, e.g. graphite) e - 104
interaction between photon and electron conservation of the relativistic energy and momentum!! Figure 2.22 shows such a collision: an x-ray photon strikes an electron (assumed to be initially at rest in the laboratory coordinate system) and is scattered away from its original direction of motion while the electron receives an impulse and begins to move. It was assumed that the scattering particle is able to move freely. Figure 2.22 105
Recall relativity But isn t the (rest) mass of a photon = 0? E 2 = (pc) 2 + (mc 2 ) 2 So for a photon E 2 = (pc) 2 E= pc (photon momentum) pc = hf p = h/ But doesn t E = hf also.. 106
photon model wave model E = hf p = h 107
Loss in photon energy = gain in electron energy h -h = KE (2.14) From Chap 1, we recall that the momentum of a massless particle is related to its energy by the formula E = pc (1.25) Since the energy of a photon is h, its momentum is Photon momentum p E h (2.15) c c 108
(1) Conservation of momentum h c 0 ' h c cos p cos (2.16) 0 ' h c sin psin (2.17) (2) Conservation of total relativistic energy target electron Scattered electron h 2 ' 2 4 2 2 mc h m c p c (2.18) incident photon Scattered photon 109
Compton effect h : ' ( 1 cos) m c 0 (2.22) c : Compton wavelength Equation (2.21) was derived by Arthur H. Compton in the early 1920s. c 2.426 10 12 m 0.02427 for an electron ' c (1 cos ) (2.23) From Eq. (2.23) we note that the greatest wavelength change possible corresponds to = 180 O, when the wavelength change will be twice the Compton wave-length c. 110
Figure 2.23 Experimental demonstration of the Compton effect. 111
atom recoils? Figure 2.24 Experimental confirmation of Compton scattering. The greater the scattering angle, the greater the wavelength change, in accord with Eq. (2.21). 112
The spectrum of scattered X-rays contains not only the original line, but also a shifted line. The wavelength shift (Compton shift) depends on but does not depend on the target material and the initial X-ray wavelength. 113
The results, shown in Fig. 2.24, exhibit the wavelength shift predicted by Eq. (2.21), but at each angle the scattered x-rays also include many that have the initial wavelength. This is not hard to understood. In deriving Eq. (2.21) it was assumed that the scattering particle is able to move freely, which is reasonable since many of the electrons in matter are only loosely bounded to higher parent atoms. 114
Unshifted peak Note that, at all angles there is also an unshifted peak. This comes from a collision between the X-ray photon and the nucleus of the atom ' h mn c (1 cos ) 0 since m n >>m e 115
How about recoil of atoms?? ' very small( undetectable) Other electrons, however, are very tightly bound and when struck by a photon, the entire atom recoils instead of the single electron. In this event the value of m to use in Eq. (2.21) is that of the entire atom, which is tens of thousands of times greater that that of an electron, and the resulting Compton shift is accordingly so small as to be undetectable. 116
2.8 Pair Production Energy into matter 117
Pair Production In 1932, C. D. Anderson observed a positively charged electron (e + ) in cosmic radiation. This particle, called a positron, had been predicted to exist several years earlier by P. A. M. Dirac. A photon s energy can be converted entirely into an electron and a positron in a process called pair production: Paul Dirac (1902-1984) 118
Pair Production Fig. 2.25 In the process of pair production, a photon of sufficient energy materialized into an electron and a positron. 119
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E&M energymatter. No violation of energy and momentum conservation! Need the help of nucleus. Why not in free space? 121
Pair Production in Matter In the presence of matter, the nucleus absorbs some energy and momentum. The photon energy required for pair production in the presence of matter is: h E E K. E.( nucleus) h m c MeV 2 2 e 1.022? 122
The rest energy m o c 2 of an electron or positron is 0.51 MeV, hence pair production requires a photon energy of at least 1.02 MeV. m o c 2 8.1910 (9.109510 14 J 31 kg)(310 8 m s ) 2 8.1910 14 19 1.6 10 0.51 MeV 123
2 2 m c m c 0.51 MeV 0.51 MeV 1. 02 0 0 MeV Minimum energy for pair production 12400( ev A) 12 1.2 10 m 1.2 6 1.02 10 ( ev ) pm 1.2 pm Gamma rays 124
Pair Production in Empty Space Conservation of energy for pair production in empty space is: h E E h E E + The total energy for a particle is: So: E pc This yields a lower limit on the photon energy: h p c p c Momentum conservation yields: h p ccos( ) p ccos( )? This yields an upper limit on the photon energy: h p c p c A contradiction! And hence the conversion of energy and momentum for pair production in empty space is impossible! 125
Pair annihilation e e (No nucleus, no other particle is needed.) 126
Pair Annihilation A positron passing through matter will likely annihilate with an electron. The electron and positron can form an atom-like configuration first, called positronium. Pair annihilation in empty space produces two photons to conserve momentum. Annihilation near a nucleus can result in a single photon. hv hv 127
Gamma rays result from matter-antimatter annihilation. An electron and positron self-annihilate, creating two gamma rays whose energy is equal to the electron mass energy. e - e + Direction?? h = 511 kev More massive particles create even more energetic gamma rays. Gamma rays are also created in nuclear decay, nuclear reactions and explosions, pulsars, black holes, and supernova explosions. 128
Pair Annihilation Conservation of energy: 2 2mc e hv1hv2 Conservation of momentum: hv c hv c 1 2 0 So the two photons will have the same frequency: v v v 1 2 The two photons from positronium annihilation will move in opposite directions with an energy: hv m c e 2 0.511 MeV hv hv 129
Positron emmision tomography - PET e+ e- 130
Ultraviolet Photon Absorption Photoelectric effect hv Atom e x-ray Compton scattering hv e hv' Gamma ray pair production hv e e Fig. 2.27 X- and gamma rays interact with matter chiefly through the photoelectric effect, Compton scattering, and pair production. Pair production requires a photon energy of at least 1.02 MeV. 131
The relative probabilities of the photoelectric effect, Compton scattering, and pair production as functions of energy in carbon (a light element) and lead (a heavy element) Carbon Lead 132
The intensity I of an x- or gamma-ray beam is equal to the rate at which it transports energy per unit cross-sectional area of the beam. The fractional energy di/i lost by the beam in passing through a thickness dx of a certain absorber is found to be proportional to dx: di ds I The proportionality constant is called the linear attenuation coefficient and its value depends on the energy of the photons and on the nature of the absorbing material. Radiation intensity I I 0 e x Absorber thickness x I ln( 0 I ) 133
Figure 2.29 Linear attentuation coefficients for photons in lead. 134
2.9 Photons and Gravity Although they lack rest mass, photons behave as though they have graviational mass. Another way to approach the gravitational behavior of light follows from the observation that, although a photon has no rest mass, it nevertheless interacts with electrons as though it has the inertial mass Photon mass m p v h 2 c (We recall that, for a photon, p=h/c and =c.) According to the principle of equivalence, gravitational mass is always equal to inertial mass, so a photon of frequency ought to act gravitationally like a particle of mass h/c 2. 135
stone photon Final photon energy = initial photon energy + increase in energy h ' h mgh and so ' h h h gh 2 c Photon energy after falling through height H 1 2 mv 2 mgh v 2gH ' h h 1 gh 2 c Fig. 2.30 A photon that falls in a gravitational field gains energy, just as a stone does. This gain in energy is manifested as an increase in frequency from to. 136
Gravitational red shift If the frequency associated with a photon moving toward the earth increases, then the frequency of a photon moving away from it should decrease. f (red shift) Fig. 2.31 The frequency of a photon emitted from the surface of a star decreases as it moves away from the star. Gravitational red shift ' ' 1 GM c 2 R The photon has a lower frequency at the earth, corresponding to its loss in energy as it leaves the field of the star. 137
Gravitational Red-shift Another test of general relativity is the predicted frequency change of light near a massive object. A light pulse traveling vertically upward from the surface of a massive body will gain potential energy and lose kinetic energy, as with a rock thrown straight up. Since a light pulse s energy depends on its frequency through E = h, as the light pulse travels vertically, its frequency decreases. This phenomenon is called gravitational red-shift. 138