Post on 24-Feb-2021
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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 −= ≈ ≈