Blackbody radiation.pdf

BLACKBODY RADIATION

THERMAL RADIATION

Thermal radiation Emitted by a body as a result of its temperature Examples How color of emitted radiation changes with temperature

1000 K

2500 K

5000 K

6500 K

9000 K

ELECTROMAGNETIC SPECTRUM

ELECTROMAGNETIC SPECTRUM

SPECTRAL RADIANCY

Definition Energy emitted by a unit area in unit time as radiation of given frequency (power emitted by a unit area)

BLACKBODY

Blackbody Absorbs all incoming radiation, does not reflect All emitted radiation is produced by the blackbody

A cavity hole is a nice blackbody.

WHY BLACKBODY?

Z

1

RT (⌫)d⌫ = T 4 0 Universal character Every blackbody has the same spectral radiancy The radiancy depends only on wavelength and temperature W = 5.67 ⇥ 10 2 4 m K Kirchhoff’s law of thermal radiation At thermal equilibrium good absorbers are good emitters 8

spectral radiancy of a real body (emitted power)

RTreal ( ) = RT ( ) ↵( ) absorption coefficient (absorbed fraction of incident power)

spectral radiancy of blackbody

OUR SUN IS ALMOST A BLACKBODY

HUMAN EYE EVOLUTION

Adapted to the sun Almost the greatest part of its radiation lies within visible range Most sensitive to wavelengths radiated most intensively

COSMIC MICROWAVE BACKGROUND RADIATION

T = 2.73 K

NIGHT VISION & INFRARED THERMOMETERS

Humans Thermal radiation mainly in infrared

HISTORY OF SCIENCE

Nobel prizes related to blackbody radiation Wilhelm Wien (1911) for his discoveries regarding the laws governing the radiation of heat

Max Planck (1918) in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta Arno Penzias & Robert Wilson (1978) for their discovery of cosmic microwave background radiation

John Mather & George Smoot (2006) for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation

PROPERTIES OF BLACKBODY RADIATION

Stefan-Boltzmann law Power radiated per unit surface area is proportional to T4

PROPERTIES OF BLACKBODY RADIATION

Stefan-Boltzmann law Z 1 Area under the spectral radiancy curve

Z 01

RT (⌫)d⌫ = T 4

RT (⌫)d⌫ = T 4

0

W = 5.67 ⇥ 10 2 4 m K 8

W = 5.67 ⇥ 10 2 4 m K 8

PROPERTIES OF BLACKBODY RADIATION

RTreal ( ) = RT ( ) ↵( )

Wien’s displacement law Peak wavelength is inversely proportional to T

max T

= 2.898 ⇥ 10

3

mK

Wien’s displacement constant temperature peak wavelength of spectral radiancy

RTreal ( ↵( )

)

max T

= RT ( )

= 2.898 ⇥ 10

THEORIST POINT OF VIEW: ENERGY DENSITY RTreal ( ) = RT ( )

↵( )

max T = 2.898 ⇥ 10 Spectral radiancy

3

mK

⇢T ( )

Directly experimentally accessible, emitted power max T

3

=Spectral 2.898 energy ⇥ 10 density mK

⇢T ( ) Energy in a unit volume inside the cavity

RT ( )

4 ⇢T ( ) = R T ( ) c

⇢T ( )

RT ( )

RT ( )

4 ⇢T ( ) = R T ( ) c

mK

⇢T ( )

RT ( )

RAYLEIGH-JEANS THEORY

4 Aim RT ( ) ⇢T ( ) = R T ( ) Calculate, using classical physics, the energy densityc in the cavity Steps 1. Radiation in the cavity forms standing waves 4 8⇡kT 2 2. Find number of standing waves within frequency ⇢T ( ) = R T ( ) ⇢T (⌫)d⌫ = interval ⌫ d⌫ 3 c energy of each standing wave c 3. Find average total 4. Energy density = Point2 x Point3 / volume

8⇡kT 2 ⇢T (⌫)d⌫ = ⌫ d⌫ 3 c

⇢T ( )d =

8⇡kT 4

d

ULTRAVIOLET CATASTROPHE

The catastrophe Classically, energy density diverges for higher frequencies

c3 THE BIRTH OF QUANTUM

8⇡kT ⇢T ( )d = 8⇡kT d 4 ⇢T ( )d = d 4

Planck’s postulate Total energy of an oscillator is quantized, i.e. discrete

E = 0, Eq , 2Eq , 3Eq , . . . E = 0, Eq , 2Eq , 3Eq , . . . Energy quantum is proportional to frequency

Eq = h⌫ Eq = h⌫

T

44

PLANCK THEORY

E = 0, E , 2E , 3E , . . . q q q E = 0, E , 2E , 3E , . . . q

q

q

Aim Calculate the energy density assuming quantization of energy Eq = h⌫ Eq = h⌫ Steps as in the Rayleigh-Jeans theory The average total energy of each standing wave is now different 8⇡⌫ 2 h⌫ ⇢T (⌫)d⌫ = 8⇡⌫3 2 d⌫ h⌫ ⇢T (⌫)d⌫ = c3 exp(h⌫/kT ) 1 d⌫ c exp(h⌫/kT ) 1 ⇢T ( )d =

⇢T ( )d =

8⇡hc 5 8⇡hc

5

1 exp(hc/ 1kT )

exp(hc/ kT )

1

d

1

d

PERFECT AGREEMENT OF PLANCK THEORY WITH EXPERIMENT

Matches the shape of spectral radiancy Stefan-Boltzmann law Integrate over all frequencies Wien’s displacement law Calculate derivate and set it to zero Classical limit Rayleigh-Jeans result for long wavelengths

Eq = h⌫ PLANCK CONSTANT

E = 0, Eq , 2Eq , 3Eq , . . . 8⇡⌫ 2 h⌫ ⇢T (⌫)d⌫ = 3 c exp(h⌫/kT )

Tiny in macroscopic units

Eq = h⌫ 8⇡hc ⇢T ( )d = 5 Planck constant

1

d⌫

1 exp(hc/ kT )

h = 6.626 ⇥ 10

34

Js

1

d

E = 0, Eq , 2Eq , 3Eq , . . . SUMMARY

⇢T ( )d =

8⇡kT 4

d

Eq = h⌫

Total energy of harmonic motion is quantized

E

2 8⇡⌫ h⌫ = 0, E , 2E , 3E , . . . q q q ⇢T (⌫)d⌫ = 3 c exp(h⌫/kT )

1

d⌫

Allowed energies are multiples of the energy quantum ⇢T ( )d =

8⇡hc 5

Eq = h⌫

1 exp(hc/ kT )

h = 6.626 ⇥ 10

34

Js

1

d