Chapter 25 Waves and Particles

1621: Refraction, W. Snell 1664: Interference: color in thin films, R. Hooke 1665: Diffraction, F. Grimaldi 1677: Wave theory, C. Huygens 1704: Particles, I. Newton 17th Century Optics takes off:

Christian Huygens (1629 – 1695) Isaac Newton (1643 – 1727) Particles or Waves?

18th Century Corpuscular theory rules…

19th Century 1801: Interference, Thomas Young –Famous double-slit experiment –Color in thin films –Diffraction of light –Diffraction grating 1814: Fresnel ‘rediscovers’ interference and diffraction 1865: Maxwell equations, light is electromagnetic wave 1886: Hertz, discovery of radio waves But: observed photoelectric effect! Augustin-Jean Fresnel ( )

Classical electromagnetic theory of light cannot explain the observed spectrum of black body! Birthday of quantum theory 1900: Meeting of German Physical Society. Max Karl Ernst Ludwig Planck reads his paper “On the theory of the energy distribution law of the normal spectrum” Nobel prize in 1918 Black Body Radiation Max Planck ( )

Wave Phenomena Interference Diffraction Reflection

– wavelength: distance between crests (meters) T – period: the time between crests passing fixed location (seconds) v – speed: the distance one crest moves in a second (m/s) f – frequency: the number of crests passing fixed location in one second (1/s or Hz)  – angular frequency: 2  f: (rad/s) Wave Description

The national public radio station (NPR) broadcasts at a frequency of 920 kHz in Lafayette. What is the wavelength of this radiation? FM radio station broadcasts at MHz. What is the wavelength? The wavelength of green light is about 530 nm. What is its frequency and period? Exercises

Wave: Variation in Time

Wave: Variation in Space

‘-’ sign: the point on wave moves to the right Wave: Variation in Time and Space

But t=0 and x =0, may not equal E 0 phase shift,  =0…2  Two waves are ‘out of phase’ Wave: Phase Shift (Shown for x=0)

In many cases we are interested only in E at certain location: can ignore dependence on x: Using angular frequency makes equation more compact Wave: Angular Frequency tt

E 0 is a parameter called amplitude (positive). Time dependence is in cosine function Often we detect ‘intensity’, or energy flux ~ E 2. For example: Vision – we don’t see individual oscillations Intensity I (W/m 2 ): Works also for other waves, such as sound or water waves. Wave: Amplitude and Intensity

Superposition principle: The net electric field at any location is vector sum of the electric fields contributed by all sources. Can particle model explain the pattern? Laser: source of radiation which has the same frequency (monochromatic) and phase (coherent) across the beam. Two slits are sources of two waves with the same phase and frequency. Interference

Two emitters: E1E1 E2E2 Fields in crossing point Superposition: Amplitude increases twice: constructive interference Interference: Constructive

Two emitters: E1E1 E2E2 What about the intensity (energy flux)? Energy flux increases 4 times while two emitters produce only twice more energy There must be an area in space where intensity is smaller than that produced by one emitter Interference: Energy

E1E1 E2E2 Two waves are out of phase: destructive interference Interference: Destructive

Superposition principle: The net electric field at any location is the vector sum of the electric fields contributed by all sources. Interference Amplitude increases twice Constructive: Energy flux increases 4 times while two emitters produce only twice more energy Two waves are out of phase Constructive:Destructive:

Intensity at each location depends on phase shift between two waves, energy flux is redistributed. Maxima with twice the amplitude occur when phase shift between two waves is 0, 2 , 4 , 6  … (Or path difference is 0,, 2 …) Minima with zero amplitude occur when phase shift between two waves is , 3 , 5  … (Or path difference is 0, /2, 3 /2…) Can we observe complete destructive interference if  1   2 ? Interference

Predicting Pattern For Two Sources Point C on screen is very far from sourcesC normal Need to know phase difference Very far: angle ACB is very small Path AC and BC are equal Path difference: If  l = 0,, 2, 3, 4 … - maximum If  l = /2, 3 /2, 5 /2 … - minimum

Predicting Pattern For Two Sources C normal Path difference: If  l = 0,, 2, 3, 4 … - maximum If  l = /2, 3 /2, 5 /2 … - minimum What if d < ? complete constructive interference only at  =0 0, What if d < /2 ? no complete destructive interference anywhere Note: largest  l for  = 

d = 4.5 Why is intensity maximum at  =0 and ? Why is intensity zero at  =90 and ? What is the phase difference at Max 3 ? Intensity versus Angle Path difference: If  l = 0,, 2, 3, 4 … - maximum If  l = /2, 3 /2, 5 /2 … - minimum

Path difference: If  l = 0,, 2, 3, 4 … - maximum If  l = /2, 3 /2, 5 /2 … - minimum d = /3.5 Two sources are /3.5 apart. What will be the intensity pattern? Intensity versus Angle

Path difference: If  l = 0,, 2, 3, 4 … - maximum If  l = /2, 3 /2, 5 /2 … - minimum L=2 m, d=0.5 mm, x=2.4 mm What is the wavelength of this laser? Small angle limit: sin(  )  tan(  )  Two-Slit Interference

Using interference effect we can measure distances with submicron precision laser Detector Application: Interferometry

Coherent beam of X-rays can be used to reveal the structure of a crystal. Why X-rays? - they can penetrate deep into matter - the wavelength is comparable to interatomic distance Diffraction = multi-source interference Multi-Source Interference: X-ray Diffraction

Diffraction = multi-source interference lattice X-ray Electrons in atoms will oscillate causing secondary radiation. Secondary radiation from atoms will interfere. Picture is complex: we have 3-D grid of sources We will consider only simple cases Multi-Source Interference

Accelerated electrons Copper X-rays Electrons knock out inner electrons in Cu. When these electrons fall back X-ray is emitted. (Medical equipment) Synchrotron radiation: Electrons circle around accelerator. Constant acceleration leads to radiation Generating X-Rays

Simple crystal: 3D cubic grid first layer Simple case: ‘reflection’ incident angle = reflected angle phase shift = 0 X-Ray: Constructive Interference

Reflection from the second layer will not necessarily be in phase Path difference: Each layer re-radiates. The total intensity of reflected beam depends on phase difference between waves ‘reflected’ from different layers Condition for intense X-ray reflection: where n is an integer X-Ray: Constructive Interference

crystal turn crystal x-ray diffracted May need to observe several maxima to find n and deduce d Simple X-Ray Experiment

X-ray of Tungsten

Suppose you have a source of X-rays which has a continuum spectrum of wavelengths. How can one make it monochromatic? crystal incident broadband X-ray reflected single-wavelength X-ray Using Crystal as Monochromator

Powder contains crystals in all possible orientations polycrystalline LiF Note: Incident angle doesn't have to be equal to scattering angle. Crystal may have more than one kind of atoms. Crystal may have many ‘lattices’ with different d X-Ray of Powdered Crystals

(Myoglobin) 1960, Perutz & Kendrew X-Ray of Complex Crystals

The spacing between neighboring layers in a particular crystal is 2 Å. A monochromatic X-ray beam of wavelength 0.96 Å strikes the crystal. At what angle might one expect to find a diffraction maximum?  = 13.9 o, 28.7 o, 46.1 o, Exercise

Why do we see reflection of light from any smooth surface? Condition for intense X-ray reflection: where n is an integer Visible light: ~ 6000 Å >> interatomic spacing Reflection from many layers is almost in-phase Reflection of Visible Light

Constructive interference: The only possible difference in path length is zero. There will be maxima only when incident angle is equal to scattering angle. Reflection of Visible Light

Mobile electrons – easy to accelerate No spring – no resonance Field ~ -qa  Electron accelerates in the direction opposite to incident E and re-radiates E in opposite direction to incident E – in the forward direction the net field will be decreasing rapidly due to superposition principle. Constructive interference – in reflected beam. Reflection on Metal Surfaces

Most sources produce sinusoidal waves which have ‘short’ total length L: 2 6 Phase correlation for long path distances is lost and there will be no interference. Coherence length: the length of the wave along which the wave is coherent, i.e. knowing phase at one location we can predict phase at the other location Coherence Length

Thin films such as soap bubbles are often colored: interference Consider thin /2-thick film There are ~3000 atomic layers Layer 1 and (N/2+1): destructive interference For each layer i=1…N/2 there is a layer i+N/2 which re-radiates with phase shift resulting in zero intensity – there will be no reflection of light for this particular wavelength Thin-Film Interference

Destructive interference: for film thickness n /2. Constructive interference: for film thickness /4, 3 /4, 5 /4… Why are soap bubbles so colorful? Why after a while soap bubbles lose their color? Why there is no such effect for thick glass plates? Other examples or thin-film interference: oil or gasoline on water butterfly wings (in some cases) bird feathers (in some cases) Thin-Film Interference

Wavelength of light in dense materials is shorter than in vacuum. Atoms get polarized due to the E induced by EM wave and due to the field created by other polarized atoms. The crest-to-crest distance in the net electric field is reduced v= f since is reduced, the speed v is slower Index of refraction: n=c/v, or v=c/n Index of Refraction

Index of refraction: n=c/v, or v=c/n Water: n=1.33 Glass: n~1.5 Frequency of light: not affected v= fWavelength: ’ = /n 1 = /n 1 2 = /n 2 1 n 1 = 2 n 2 X-rays: very high frequency, barely polarize atoms, speed almost not affected Index of Refraction