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Ey

Bz

x

Chapter 2: Particle Properties of Waves

Electromagnetic Waves

coupled Electric and Magnetic Oscillations

harmonic waves a.k.a. sine waves

E ˆ y Emax sin(kx t)B ˆ z Bmax sin(kx t)

k 2

2f

f k

v 1

Emax vBmax

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electromagnetic spectrum

= c/f

100 105 1010 1015 1020 1025

105 100 10-5 10-10 10-15

Rad

io

mic

row

ave

mil

lim

eter

infr

ared

visi

ble

ultr

avio

let

x-ra

y

Gam

ma

Visible Light

f (Hz)

(m)

4.3 x1014 - 7.5 x1014 (Hz)

700 nm - 400 nm

ROYGBIV

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Principle of Superposition: add instantaneous amplitudes

Constructive interference Destructive interference

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Diffraction of light waves

S1

S2

r1

r2

d

r

r d sinconstructive: n d sinn 0,1,2,

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Ideal thermal radiator: Blackbody Radiator

I

I

ICobserved contiunuous spectrum vs. classical theory:

Ultraviolet Catastrophe

Observed:

I P

A T 4

max T

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Theoretical black body: standing wave modes in a (3-d) cavity

Density of States:G( ) 8 2

c3d

Statistical Mechanics: P( ) e kT

: microscopic energy

k : Boltzmann's constant1.281x10 23 J/K

kT : average energy, classically via E.T.

Equipartion Thm :1

2kT per D.O.F.

Planck: Quantization of energy

nh , n 0,1,2

heh kT 1

h 6.63x10 34 J s

4.14x10 15 eV s

u( )d 8h

c3

3deh kT 1

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Example 2.1 A certain 660 Hz tuning fork can be considered as a harmonic oscillator with a vibrational energy of 0.04J. Compare its energy quantum of energy for the tuning fork with its vibrational energy. Compare the fork’s quantum of energy with those of an atomic oscillator which emits a frequency of 5.00x14 Hz.

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Photoelectric effect

Classical problems:

no delay in emission of electrons

KE of electrons indepentdent of intensity

KE of electrons depends upon frequency of light

effect occurs only above threshold frequency 0

Einstein: quantize light (photons)

1 photon absorbed => 1 photoelectron released

Conservation of Energy

A

KEmax eVs h hc

h 0 work function~ a few eV

hc 1240eV nm

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Example 2.2 Ultraviolet light of wavelength 350 nm and an intensity of 1.00 W/m2 is directed at a potassium surface ( = 2.2eV). (a) Find the maximum KE of the photoelectrons. (b) If 0.50 percent of the incident photons produce photoelectrons, how many are emitted per second if the potassium surface area is 1.0 cm2?

Thermionic Emission: kT ~ f

=> random motion kicks electrons loose

Wave-particle “duality”

interference and diffraction: wave phenomena

photoelectric effect, etc.: particle phenomena

=> intensity ~ probability for individual photons

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X-Ray production: bremsstrahlung (braking radiation)

“inverse” photoelectric effect

1 electron (KE) in => photon (E = hf) out

V

maximum energy photon get all of electron’s KE

electron KE from accelerating potential

KE eV (qV PE)

eV h max hc

min

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Example 2.3 Find the shortest wavelength present in the radiation from an X-ray machine whose accelerating potential is 50 kV.

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wavelength

rela

tive

inte

nsit

y

wavelength

rela

tive

inte

nsit

y

Typical continuous x-ray spectrum

Some target materials produce sharp maximum in the x-ray spectrum

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X-ray diffraction: how to measure the wavelength of x-rays

i r

Constructive Interference when i = r (0th order)

i

d

Path difference = 2d sin i Constructive Interference n = 2d sin i

n = 1, 2, 3 ...note: path deflected by 2

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Compton effect: an elastic collision between a phton and a charged particle initially at rest

e e

e

’

particle nature of light (photons)

+ (Relativistic) conservation of energy and momentum

E tot h m0c2

ptot hc

E2 p2c2 m0

2c4 E KE m0c

2

Etot h ' m0c2 KE

ptot x h '

ccos pcos

ptot y h '

csin psin

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p2c2 E 2 m0

2c4 KE m0c2 2

m0

2c4 KE2 2KE m0 c2

p2c2 h h ' 2 2 h h ' m0c2

h h ' m0c2 h h ' 1 cos '

h

m0c1 cos C 1 cos

h h 'KE

hc

h '

ccos pcos pccos h h 'cos

0 h '

csin psin pcsin h sin

p2c2 h 2 2 h h ' cos h ' 2

C = 2.426 pm for electrons

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Example 2.4 X-rays of wavelength 10.0 pm are scattered from a target. (a) Find the wavelength of the x-rays scattered through an angle of 45 degrees. (b) Find the maximum wavelength of the scattered x-rays. (c) Find the maximum KE of the recoil electrons.

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pair production:

conservation of energy, momentum + other conservation laws

E = mc2

Ne

p

creation of particle and antiparticle (antimatter)

antiparticle has same mass, opposite charge etc.

particle/antiparticle pair can anihilate to create a pair of photons: e + e > +

e

e

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Example 2.5: Show that pair production cannot occur in empty space. (Hint: look at relativistic conservation of energy and momentum)

Example 2.6: An electron and a positron are moving side by side in the +x direction at 0.500c when they annihilate each other. Two photons are produced that move along the x -axis. (a) Do both photons move in the +x direction? (b) What is the energy of each photon?

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Photon Absorption

Three chief mechanisms for x-ray and gamma ray photons to interact with matter

photoelectric effect (photon absorbed)

Compton scattering (photon energy decreased)

pair production (photon converted to pair)

The dominant mechanism depends upon material and photon energy

A slab of material will reduce intensity:

dI

I dx linear attenuation coeefficient

I = I0e x or x ln I 0 I

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Example 2.7: The linear attenuation coefficient for 2.0 MeV gamma rays in water is 4.9 m-1. (a) Find the relative intensity of the gamma rays after it has passed through 10 cm of water. (b) How far must the beam travel in water before being reduced to 1 percent of its original value?

Problems: 2,5,6,8,11,12,15,17,19,20,21,22,23,26,27,29,32,39,43,45,46,47

skip 2.9 or read at your liesure