Chapter 25 Waves and Particles Midterm 4 UTC 1.132

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

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