Chapter 25
Waves and ParticlesMidterm 4 UTC 1.132
Wave Phenomena
• Interference• Diffraction• Reflection
l – 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: 2f: (rad/s)
Tv l
Tf 1
fv l
Wave Description
E E0 cos t E0 cos2T
t
Wave: Variation in Time
xExEE
l
l 2cos2cos 00
Wave: Variation in Space
xEE
l2cos0
t
TEE 2cos0
xt
TEE
l 22cos0
‘-’ sign: the point on wave moves to the right
Wave: Variation in Time and Space
xt
TEE
l 22cos0
But E @ t=0 and x =0, may not equal E0
l xt
TEE 22cos0
phase shift, =0…2
Two waves are ‘out of phase’
Wave: Phase Shift
tEt
TEE cos2cos 00
(Shown for x=0)
In many cases we are interested only in E at certain location:can ignore dependence on x:
tEt
TEE cos2cos 00
Using angular frequency makes equation more compact
Wave: Angular Frequency
t
y(x,t) Asin(kx t)
2T
k 2l
tEE cos0
E0 is a parameter called amplitude (positive). Time dependenceis in cosine function
Often we detect ‘intensity’, or energy flux ~ E2. For example: Vision – we don’t see individual oscillations
Intensity I (W/m2):20EI
Works also for other waves,such as sound or water waves.
Wave: Amplitude and Intensity
Superposition principle: The net electric field at any location isvector 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:
E1
E2
Fields in crossing point
tEE
tEE
coscos
02
01
Superposition: tEEEE cos2 021
Amplitude increases twice: constructive interference
Interference: Constructive
Two emitters:
E1
E2
tEEEE cos2 021
What about the intensity (energy flux)?
Energy flux increases 4 times while two emitters produce onlytwice more energy
There must be an area in space where intensity is smaller than thatproduced by one emitter
Interference: Energy
E1
E2
ttEEEE coscos021
tEEtEE
coscos
02
01
tcos
0
Two waves are 1800 out of phase: destructive interference
Interference: Destructive
Superposition principle: The net electric field at any location isthe vector sum of the electric fields contributed by all sources.
Interference
tEE
tEE
coscos
02
01
tEEEE cos2 021
Amplitude increases twice
Constructive: Energy flux increases 4 times while two emitters produce only twice more energy
ttEEEE coscos021
tEEtEE
coscos
02
01
Two waves are 1800 out of phase
Constructive: Destructive:
Intensity at each location depends on phase shift between twowaves, 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, l, 2 l…)
Minima with zero amplitude occur when phase shift between two waves is , 3, 5 …(Or path difference is 0, l/2, 3l/2…)
Can we observe complete destructive interference if 1 2 ?
Interference
Predicting Pattern For Two SourcesPoint 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: )sin(dl
If l = 0, l, 2l, 3l, 4l … - maximumIf l = l/2, 3l/2, 5l/2 … - minimum
Predicting Pattern For Two SourcesC
normal
Path difference: )sin(dl
If l = 0, l, 2l, 3l, 4l … - maximumIf l = l/2, 3l/2, 5l/2 … - minimum
What if d < l ?
complete constructive interferenceonly at =00, 1800
What if d < l/2 ?
no complete destructive interference anywhere
Note: largest l for =/2
d = 4.5 l
Why is intensity maximum at =0 and 1800 ?
Why is intensity zero at =90 and -900 ?
What is the phase difference at Max3?
Intensity versus AnglePath difference: )sin(dl
If l = 0, l, 2l, 3l, 4l … - maximumIf l = l/2, 3l/2, 5l/2 … - minimum
Path difference: )sin(dl
If l = 0, l, 2l, 3l, 4l … - maximumIf l = l/2, 3l/2, 5l/2 … - minimum
d = l/3.5
Two sources are l/3.5 apart. What will be the intensity pattern?
Intensity versus Angle
Path difference:
If l = 0, l, 2l, 3l, 4l … - maximumIf l = l/2, 3l/2, 5l/2 … - minimum
)sin(dl
L=2 m, d=0.5 mm, x=2.4 mmWhat is the wavelength of this laser?
)sin(l ddl )sin(
Lx
)tan(
Small angle limit: sin() tan()
Lx
d
lnm m 600106 7
Lxdl
Two-Slit Interference
Using interference effect we can measure distances with submicronprecision
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
Acceleratedelectrons
Copper
X-rays
Electrons knock out innerelectrons in Cu. When theseelectrons fall back X-rayis 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 anglephase shift = 0
X-Ray: Constructive Interference
Reflection from the second layer will not necessarily be in phase
Path difference:
sin2dl
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 l nd sin2
X-Ray: Constructive Interference
crystalturn crystal
x-ray diffracted
l nd sin2
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
l nd sin2
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