Blackbody radiation and Plank’s law“blackbody” problem: calculating the intensity of radiation at a given wavelength emitted by a body at a specific temperaturecalculating the intensity of radiation at a given wavelength emitted by a body at a specific temperature

Max Planck, 1900quantization of energy of radiation-emitting oscillators:only certain energies for the radiation-emitting oscillators in the cavity wall are allowed

Albert Einstein, 1905extended quantization of energy

Niels Bohr, 1913quantum model of the atomextended quantization of energy

of radiation-emitting oscillators to quantization of light → photons

quantum model of the atom

explained the photoelectric effect

Blackbody radiation and Plank’s law

blackbody is an object that absorbs all electromagnetic radiation falling on it an consequently appears black

the opening to the cavity is a good approximation of a blackbody:after many reflections all of the incident energy is absorbed

radiation is in thermal equilibrium with the cavity (e.g. oven cavity) because radiation has exchanged energy with the walls many times

similar to the thermal equilibrium

spectral energy densityis energy per unit volume per unit frequency of the radiation within the blackbody cavity

( , )u f Tof a fluid with a container

depends only on temperature and light frequencyand not on the physical and chemical makeup of the blackbody

( , )u f T

all the objects in the oven, regardless of their chemical nature, size, or shape, emit light of the same color

blackbody is a perfect absorber and ideal radiatorblackbody is a perfect absorber and ideal radiator

Max Planck, 1900spectral energy density of blackbody radiation

towards “ultraviolet” catastrophe

classical

3

3

8 1( , )1Bhf k T

hfu f Tc eπ

=−

Rayleigh-Jeans law

d in

tens

ity

quantumPlanck law1c e

34

23

6.626 10 J s is Planck's constant1.380 10 J/K is Boltzmann's constantB

hk

−

−

= × ⋅

= ×f

radi

ated

limiting behaviors

f

at high frequencies 1hf k T >> at low frequencies 1hf k T <<

3 3

at high frequencies 11

18 1 8

B

B

B

hf k Thf k T

hf k T

ee

hf hfπ π

−

>>

≈−

3 3 2

at low frequencies 11 1

1 1 ... 1

8 1 8 8

B

B

Bhf k T

B

hf k Tk T

e hf k T hf

hf hf k T f

<<

= ≈− + + −

3 3

8 1 8( , )1

B

B

hf k Thf k T

hf hfu f T ec e cπ π −= ≈

−

3 3 2

3 3 3

8 1 8 8( , )1B

BBhf k T

hf hf k T fu f T k Tc e c hf cπ π π

= ≈ =−

Wien’s exponential law, 1893 classical Rayleigh-Jeans law

Rayleigh-Jeans law is the classical limit obtained when h→0

Max Planck:blackbody radiation is produced by vibrating submicroscopic electric charges,which he called resonatorswhich he called resonators

the walls of a cavity are composed of resonators vibrating at different frequency

Classical Maxwell theory:Classical Maxwell theory:An oscillator of frequency f could have any value of energy and could change its amplitude continuously by radiating any fraction of its energy

Planck: the total energy of a resonator with frequency f could only be an integer multiple of hf.(During emission or absorption of light) resonator can change its energy only by the quantum of energy ∆E=hf

where 1,2,3,...E nhf nE hf= =

∆ =

Ε

3hf4hf

h t b ti

Planck: allowed energy levels of a resonator

f

hf

2hf

0

photon absorption

photon emission

0

all systems vibrating with frequency f are quantized and lose or gain energy in discrete packets or quanta ∆E=hf

Consider a pendulum m=0.1 kg, l=1 m, displaced by θ=100

( ) ( ) ( ) ( )2 0 2(1 cos ) 0.1 kg 9.8 m/s 1 m 1 cos10 1.5 10 JE mgl θ −= − = − = ×( ) ( ) ( ) ( )

( )( )

2

34 1 34

( ) g

1 1 9.8 m/s 0.5 Hz2 2 1 m

g

gflπ π

= = =

( )( )34 1 34

3432

2

6.63 10 J s 0.5 s 3.3 10 J

3.3 10 J 2.2 101.5 10 J

E hf

EE

− − −

−−

−

∆ = = × ⋅ = ×

∆ ×= = ×

× classical physics: ti ti i b bl d

quantum of energy

quantization is unobservable and energy loss or gain looks continuum

∆E is large ← f is largeE i ll i ll quantum physics E is small ← m is small qua tu p ys cs

energy change of atomic oscillator sending out green light

( )( )34 8196.63 10 J s 3 10 m/s

3 68 10 2 3hchf−× ⋅ ×

∆( )( ) 19

9 3.68 10 J 2.3 eV540 10 m

hcE hfλ

−−∆ = = = = × =

×191 eV 1.602 10 J−= ×a more appropriate unit of energy for describing atomic processes

Max Planck:towards “ultraviolet” catastrophe

classical( ) ( )f df f dfnumber of oscillators with

average energy emitted per oscillator

classicalRayleigh-Jeans law

inte

nsity

quantumPlanck law

2

3

( , ) ( )8( )

u f T df EN f dffN f df dfπ

=

=

frequency between f and f+df

radi

ated

a c aw3

11Bhf k T

c

E hfe

=−

Plankdistribution function

f2

3

8( , )1Bhf k T

f hfu f T df dfc eπ

=−

in classical Rayleigh-Jeans theory BE k T=28( ) fu f T df k Tdfπ

=

5

8 1( , )1Bhc k T

hcu T d de λ

πλ λ λλ

=−

3( , ) Bu f T df k Tdfc

=

for f→∞ classical theory predicts unlimited energy emission in the ultraviolet region“ultraviolet catastrophe”

cfλ

= ultraviolet catastrophe

in quantum theory “ultraviolet catastrophe” is avoided:tents to zero at high f because the first allowed energy level E=hf is so large for large f

d t th th l il bl k TE

fλ

compared to the average thermal energy available kBTthat occupation of the first excited state is negligibly small

spectral energy densityis energy per unit volume per unit frequency of the radiation within the blackbody cavity

power densityis power emitted per unit area per unit frequency

( , )fe J f T=( , ) ( , )

4cJ f T u f T=

( , )u f T

Wien’s displacement law, 1893: the wavelength marking the maximum power emission of a blackbody, λmax, shifts towards

5

8 1( , )1Bhc k T

hcu Te λ

πλλ

=−

emission of a blackbody, λmax, shifts towards shorter wavelengths with increasing temperature

1max

3

~

2.898 10 m K

T

T

λ

λ

−

−= × ⋅

Stefan-Boltzmann law, 1879:The total power per unit area emitted at all frequencies by a blackbody, etotal,

max 2.898 10 m KTλ

p p q y y, total,is proportional to the forth power of its temperature

4

08 2 46 10

total fe e df Tσ∝

= =∫h f l8 2 45.67 10 W m Kσ − − −= × ⋅ ⋅ ← The Stefan-Boltzmann constant

( )2 4 4 3 5 4

4 42 3 2 350 0 0

2 2 2( , )4 1 151B

B Btotal xhc k T

c hc k T x ke u T d d dx T Tc h e c he λ

π π πλ λ λ σλ

∝ ∝ ∝= = = = =

−∫ ∫ ∫( )0 0 04 1 151B c h e c heλ −−∫ ∫ ∫

Bx hc k Tλ= 4 15π=

for perpendicular radiated energy

radiated power

( , ) ( , )4cJ f T u f T=derivation of

1 volumebecause half the power will be going in the –x direction

radiated power

δx=cδt area A

[ ]1 volumeradiated power = energy density2 time

1 1( , ) ( , ) ( , )2 2 2

A x A x cu f T u f T u f T At x cδ δδ δ

× =

⋅ ⋅= = =

2 2 2t x cδ δ

θ [ ]1 volumeperpendicular radiated power = energy density cos2 time

θ× × =for any angle

θ

21

2 time1 1( , ) cos ( , ) cos ( , ) cos2 2 cos 2

A x A x cu f T u f T u f T At x cδ δθ θ θδ δ θ−

⋅ ⋅= = =

⋅

averaging over θand dividing by

2( , ) cosradiated power 2( , ) ( , )A 4

c u f T A cJ f T u f TA

θ= = =

A to get power density

2cos 1 2θ =

Estimate the surface temperature of the Sun and find λmax for the Sun emission.The Sun radius RS=7.0x108 mThe Earth-to-Sun distance R=1.5x1011 mThe total power from the Sun at the Earth etotal=1400 W/m2

4

2 2

( )

( ) 4 ( ) 4total S

t t l S S t t l

e R T

e R R e R R

σ

π π

=

⋅ = ⋅

← Stefan-Boltzmann law

← conservation of energy1 41 4 2

2

( ) 4 ( ) 4

1 1 ( )( )

total S S total

totaltotal S

S

e R R e R R

e R RT e RR

π π

σ σ⎛ ⎞⋅⎛ ⎞= ⋅ = ⋅⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

conservation of energy

( ) ( )( )

1 422 11

28 2 4 8

1400 W/m 1.5 10 m1 5800 K5.67 10 W/m K 7.0 10 m

T −

⎛ ⎞⋅ ×⎜ ⎟= ⋅ =

× ⋅⎜ ⎟×⎝ ⎠3 3

max2.898 10 m K 2.898 10 m K

Tλ

− −× ⋅ × ⋅= = 9500 10 m 500 nm

5800 K−= × =

eye’s sensitivity peak

Wien’s displacement law

Planck, 1900: oscillators in the walls of the blackbody are quantized

Einstein, 1905: light itself is composed of quanta of energy

E 1 photon

photon – a quantum of electromagnetic radiation (a quantum of light)

E

E 1 photonE=hf

2 photons

xx

B

E=2hf

3 photons3 photonsE=3hf

photonE hf=

the photoelectric effect

Einstein, 1906:in addition to carrying energy E=hfa photon carries a momentum p=E/c=hf/c directed along its line of motion

Peter Debye, Arthur Holly Compton, 1923:scattering of x-rays photons from electrons could be explained by treating photons as particles with energy hf and momentum hf/c and particles with energy hf and momentum hf/c and by conserving energy and momentum of the photon-electron pair in a collision

particle properties of lightp p p g

x-rays – are electromagnetic waves with short wavelengthswaves with short wavelengths

x-rays were discovered byWilhelm Roentgen in 1895: X t d h

Frequency (f)X-rays are generated whenhigh-speed electrons strike a metal target and give up some of their energy when they interact with the

bi l l fh t 12 k Vhc hchf orbital electrons of an atom10photon energy 12 keV10 m

hfλ −= ≈ ≈