Blackbody radiation

How does a solid contain thermal energy?

Can a vacuum be “hot”, have a temperature?

Why does solid glow when it’s hot?

22energy 1 1

volume 2 2oo

Bu E

Yes its fields (photons) can be in equilibrium with objects at T

Hot vacuum oven with a cold object inside

Definitions

( ) ( )

( ) ( , ) ( , ) ( )

I T u T

u T T d E T g d

3

3

3

/ 2 (Hz)

( ) energy density (J/m )

( , ) spectral energy density ( J/m /Hz)

( , ) energy per mode(J/mode)

( ) frequency mode density (modes/Hz/m )

u T

T

E T

g

E/M wave modes in a vacuum or laser

1-D waves (used for lasers!)

Derive modes per hertz per length

1

modes 2( )Dg Hz m c

If we double L, number of modes/Hz doubles, but modes/Hz/L is constant

large L

small L

Derive mode spacing =Dn fundamental frequency

Laser cavities have this 1-D g(n)

1. Gain medium2. Laser pumping energy3. High reflector (R14. Output coupler (R <1)5. Laser beam

Depends on L

of atoms in gas, crystal

Most laser emits many lines (modes).

1/decay time: Sharper for higher Routput

How could we get just one cavity mode to lase?

E/M 3D wave modes in a vacuum

Derive 3-D density of modes2

3 3

modes 8( )g d d

Hz m c

Independent of box size used for derivation

Build and from these pieces

Classical thermo theory and the ultraviolet catastrophe

Equipartition theorem (classical) gives infinite u

2

3

8( )g d d

c

Every “degree of freedom” in u (E2 and B2) is filled with energy of kT/2!

E kT ( )

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volume 2 2oo

Bu E

( , )class T ( , )classu T

Quantum resolution

Planck:

2

3

( , ) ( , ) ( )

8( , )

T E T g

h n Tc

1

1h

kT

n

e

( )

E n h ( ) ( )1. A mode accepts energy only in precise quanta of ____

2. The avg no of quanta in a mode goes down fast if _____

3. Equipartition (classical) works only for modes where _____

When is a body not a “blackbody”?

2

3

8( )g d d

c

Sun’s corona is 106 K. Why don’t we die of x-rays?

Emissivity e(n) comes from complex index n(n),k(n) and roughness. Where absorption is high, emission is high.

4

4

( , )

( ) ( , )

theory

practical eff

I AT T d

I e AT e T d

blackbody I made

In a vacuum, which frequency region has the most g (modes per Hz per m3)? (Find relative factors)

a) same b) 0.5 eV c) 2 eV d) 4 eV

Compare three frequency regions near different photon energies: hn= 0.5 eV (IR), hn = 2 eV (red light) and hn = 4 eV (soft UV). They are in equilibrium with the surface of the sun at kT≈ 1 eV

1

1h

kT

n

e

( )

2

3

( , ) ( , ) ( )

8( , )

T E T g

h n Tc

In a vacuum, which frequency region has the most n (photons per mode)? (Find the n’s…take Euler’s e ≈ 3)

a) same b) 0.5 eV c) 2 eV d) 4 eV

1

1h

kT

n

e

( )

2

3

( , ) ( , ) ( )

8( , )

T E T g

h n Tc

Compare the three frequency regions in equilibrium with the surface of the sun at kT≈ 1 eV

In a vacuum, which frequency region has the most spectral density r (energy per Hz per m3)?

a) same b) 0.5 eV c) 2 eV d) 4 eV

Compare the three frequency regions in equilibrium with the surface of the sun at kT≈ 1 eV

1

1h

kT

n

e

( )

2

3

( , ) ( , ) ( )

8( , )

T E T g

h n Tc

When is a body not a “blackbody”?

If you throw a piece of transparent glass into a glowing furnace, and it stays transparent glass. As it comes to equilibrium with the furnace, you will see the glass glow ___ the furnace wallsa) less than b) more than c) the same as