Blackbody Radiation

Discussion

introduction

Newton's laws of motion and universal gravitation, the laws of conservation of energy and momentum, the laws of thermodynamics, and Maxwell's equations for electricity and magnetism were all more or less nearly complete at the end of the Nineteenth Century. They describe a universe consisting of bodies moving with clockwork predictability on a stage of absolute space and time. They were used to create the machines that launched two waves of industrial revolution — the first one powered by steam and the second one powered by electric current. They can be used to deliver spacecraft to the ends of the solar system with hyper-pinpoint accuracy. They are mathematically consistent in the sense that no one rule would ever violate another. They agree with reality to a high degree of accuracy as tested in experiment after experiment.

At the end of the Nineteenth Century, physics appeared to be at an apex. Several people are reported to have said something like this

There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.

This has been attributed to William Thomson, Lord Kelvin (1824–1907) in an address to the British Association for the Advancement of Science in 1900, but I haven't been able to find the primary source. A similar statement was made twice by the German-American scientist Albert Michelson (1852–1931) as was discussed earlier in this book. It is often reported that Michelson got the idea from Kelvin, but there is little evidence to back this claim up.

At the turn of the century, Kelvin wasn't saying that physics was finished.In fact, I think he was saying quite the opposite. There were two clouds hanging over Nineteenth Century physics.

The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds. I. The first came into existence with the undulatory theory of light, and was dealt with by Fresnel and Dr Thomas Young; it involved the question, How could the earth move through an elastic solid, such as essentially is the luminiferous ether? II. The second is the Maxwell-Boltzmann doctrine regardingthe partition of energy.

Kelvin is describing two problems with the physics of his time. They are highly technical in nature and not something you could easily describe to your grandmother (unless she had some training in physics). The first one refers to the now discredited theory of the luminiferous ether. The second one describes the inability of electromagnetic theory to adequately predict the characteristics of thermal radiation.

In essence, the first argument went like this: light is a wave, waves require a medium, the medium for light was called the luminiferous ether, it must be extremely rigid (since light travels so quickly), it must be extremely tenuous (since we can't detect its drag), rigid and tenuous are adjectives that are incompatible (strong yet soft), Nineteenth Century physics cannot handle this, therefore Nineteenth Century physics is in trouble. The ray of sunshine that dispersed this dark cloud was the theory of relativity devised by Albert Einstein. The major revelations of this theory were that there is no ether, there is no absolute space, there is no absolute time, mass is not conserved, energy is not conserved, and nothing travels faster than light. For awhile, this was the most revolutionary theory in all of physics.

The second dark cloud identified by Kelvin is the subject of the rest of this section and (essentially) most of the rest of this book. I won't breeze through it like I did with the first dark cloud, but I will tell you this, the solution to the problem Kelvin called "the Maxwell-Boltzmann doctrine" lead to the most revolutionary theory in all of physics — quantum mechanics. The major revelations of this theory are that all things are both particles and waves at the same time and that nothing can be predicted or known with absolute certainty.

The arrival of these two revolutionary theories divided physics up into two domains. All theories developed before the arrival of relativity and quantum mechanics and any work derived from them are called classical physics. All theories derived from the basic principles of relativity and quantum mechanics are called modern physics. The word modern was chosen since the foundations of these theories were laid in the first three decades of the Twentieth Century. This the era of modern architecture, modern dance, modern jazz, and modern literature. Modern technologies were starting to appear like electric lights, toasters, refrigerators, sewing machines, radios, telephones, movies, phonograph records, airplanes, automobiles, subways, elevators, skyscrapers, synthetic dyes, nylon, celluloid, machine guns, dynamite, aspirin, and psychology. The early Twentieth Century was filled with revolutionary ideas and inventions. Life now seems unimaginable without them. Modern physics was just one aspect of the modern era.

the failures of classical physics

Frayed edges on the tapestry of classical physics leading to modern physics …

What we know about blackbody radiation

[slide]

John Strutt, Lord Rayleigh (1842–1919) England and James Jeans (1877–1946) England Ultraviolet Catastrophe

A blackbody is an idealized object which absorbs and emits all frequencies. Classical physics can be used to derive an equation which describes the intensity of blackbody radiation as a function of frequency for a fixed temperature — the result is known as the Rayleigh-Jeans law. Although the Rayleigh-Jeans law works for low frequencies, it diverges as ƒ2; this divergence for high frequencies is called the ultraviolet catastrophe.

Wilhelm Wien (1864–1928) Russia–Germany Infrared Catastrophe

In 1896 Wien derived a distribution law of radiation. Planck, who was a colleague of Wien's when he was carrying out this work, later, in 1900, based quantum theory on the fact that Wien's law, while valid at high frequencies, broke down completely at low frequencies.

energy is quantized

Max Planck (1858–1947) Germany. On the Law of Distribution of Energy in the Normal Spectrum. Max Planck. Annalen der Physik 4 (1901): 553.

Proposition …

p = h
λ

And also (from Einstein later, I think) …

p = h
λ

Let's try to derive the blackbody spectrum.

Planck's law is the formula for the spectral radiance of an object at a given temperature as a function of frequency or wavelength. It has dimensions of power per solid angle per area per frequency or power per solid angle per area per wavelength. (Yuck!)

I df =  23   1  

W

c2 ehƒ/kT − 1 sr m2 Hz
 
I dλ =  2hc2   1  

W

λ5 ehckT − 1 sr m2 m

When these functions are multiplied by the total solid angle of a sphere (4π steradian) we get the spectral irradiance. This function describes the power per area per frequency or the power per area per wavelength.

I df =  3   1  

W

c2 ehƒ/kT − 1 m2 Hz
 
I dλ =  hc2   1  

W

λ5 ehckT − 1 m2 m

When either of these functions is integrated over wavelength, the result is the irradiance or the power per area.

L =  5k4  T4
15h3c2

The pile of constants in front of the temperature is known as Stefan's constant.

σ =  5k4  = 5.67040 … × 10−8 W/m2K4
15h3c2

Multiplying the irradiance by the area gives us the essence of the Stefan-Boltzmann law.

L =  P  =  5k4  T4 =  σT4  ⇒  P = σAT4
A 15h3c2

Apply the first derivative test to the wavelength form of Planck's law to determine the peak wavelength as a function of temperature.

λmax =  hc   1
k xT

where x is the solution of the transcendental equation

xex  − 5 = 0
ex − 1

x = 4.96511…

Combine all the constants together and we get the Wien displacement law

λmax = b
T

b = 2.89777 … mmK

Discuss effective temperature. No object emits a mathematically perfect blackbody radiation spectrum. There will always be lumps in the curve. Set the area under intensity-wavelength curve for a real source of radiation equal to the area under the intensity-wavelength curve for an ideal blackbody and solve for temperature. The effective temperature of an object is the temperature of an ideal blackbody that would radiate energy at the same rate as the real body. Different parts of the sun are at different temperatures. When combined, the sun has an effective temperature of 5778 K.

biographical resource

Hierzu ist es notwendig, UN nicht als eine stetige, unbeschränkt teilbare, sondern als eine discrete, aus einer ganzen Zahl von endlichen gleichen Teilen zusammengesetzte Grösse aufzufassen.   Moreover, it is necessary to interpret UN [the total energy of a blackbody radiator] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts.

While in Berlin Planck did his most brilliant work and delivered outstanding lectures. He studied thermodynamics in particular examining the distribution of energy according to wavelength. By combining the formulas of Wien and Rayleigh, Planck announced in 1900 a formula now known as Planck's radiation formula. In a letter written a year later Planck described proposing the formula saying …

the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be.

Within two months Planck made a complete theoretical deduction of his formula renouncing classical physics and introducing the quanta of energy. At first the theory met resistance but due to the successful work of Niels Bohr in 1913, calculating positions of spectral lines using the theory, it became generally accepted. Planck himself in [7] explains how despite having invented quantum theory he did not understand it himself at first:- I tried immediately to weld the elementary quantum of action somehow in the framework of classical theory. But in the face of all such attempts this constant showed itself to be obdurate … My futile attempts to put the elementary quantum of action into the classical theory continued for a number of years and they cost me a great deal of effort. Planck received the Nobel Prize for Physics in 1918.

Nobel Prize in Physics 1918 Presentation Speech by Dr. A. G. Ekstrand, President of the Royal Swedish Academy of Sciences

Ladies and Gentlemen. The Royal Academy of Sciences has decided to award the Nobel Prize for Physics, for the year 1918, to Geheimrat Dr. Max Planck, professor at Berlin University, for his work on the establishment and development of the theory of elementary quanta. From the time that Kirchhoff enunciated the principle "that the intensity of radiation from a black body is dependent only upon the wavelength of the radiation and the temperature of the radiating body, a relationship worth while investigation", the theoretical treatment of the radiation problem has provided a rich, fertile source of fresh discoveries. It is only necessary here to recall the fertile Doppler principle, and further, the transformation of our - concept of the nature of light as seen now in the electromagnetic theory of light formulated by that great man, Maxwell, the deduction of Stefan's Law by Boltzmann, and Wien's Law of Radiation. Since it was clear, however, that this did not correspond exactly with the reality, but was rather, like a radiation law propounded by Lord Rayleigh, only a special case of the general radiation law, Planck sought for, and in 1900 found, a mathematical formula for the latter, which he derived theoretically later on. The formula contained two constants; one, as was demonstrated, gave the number of molecules in a gram molecule of matter. Planck was also the first to succeed in getting, by means of the said relation, a highly accurate value for the number in question, the so-called Avogadro constant. The other constant, the so-called Planck constant, proved, as it turned out, to be of still greater significance, perhaps, than the first. The product hν, where ν is the frequency of vibration of a radiation, is actually the smallest amount of heat which can be radiated at the vibration frequency ν. This theoretical conclusion stands in very sharp opposition to our earlier concept of the radiation phenomenon. Experience had to provide powerful confirmation, therefore, before Planck's radiation theory could be accepted. In the meantime this theory has had unheard-of success….

Source?

Using statistical mechanics, Planck derived an equation similar to the Rayleigh-Jeans equation, but with the adjustable parameter h. Planck found that for 6.63 × 10−34 J s, the experimental data could be reproduced. Nevertheless, Planck could not offer a good justification for his assumption of energy quantization. Physicists did not take this energy quantization idea seriously until Einstein invoked a similar assumption to explain the photoelectric effect.

symbology

There's ℎ and then there's ℏ.

symbol name joules electron-volts
planck constant 6.62606896 × 10−34 J s 4.13566733 × 10−15 eV s
c "h c" 1.986445 × 10−25 J m 1239.842 eV nm
ℏ = h
"h bar", dirac constant, reduced planck constant 1.054571628 × 10−34 J s 6.58211899 × 10−16 eV s
Planck Constant and Variations

planck units

Here we are very near the end of this book and we're talking about the subject that most teachers start a basic physics course with — units. In 1899, at the time when Max Planck first proposed his radical theory of energy quantization, he also proposed building a system of "natural units" (natürliche Maasseinheiten) from a few of the more important constants in physics: the speed of light, the universal gravitational constant, and the two recently identified constants that later came to be known by their discoverers: the Planck and Boltzmann constants. The significance of these quantities is now know to be more than just a way to get the units to work out. The big four fundamental physical constants each tell us something different about the nature of reality.

c = 299,792,458 m/s

The speed of light in a vacuum is a value dictated by nature and thus is a natural unit for speed. It is the universal "speed limit". Nothing may travel faster than the speed of light in a vacuum — not even light itself. Even before we entered the information age, it was recognized that material objects and photons of electromagnetic radiation are, in essence, carriers of information. The speed of light is then a restriction on the speed at which information may travel. More on information theory later.

G = 6.67428 × 10−11 Nm2/kg2

The universal gravitational constant relates mass-energy to space-time curvature. (Although, since general relativity was 15 years away, Planck would not have known this.) It contains in it the natural units for length, mass, and time — the fundamental quantities of mechanics (which, of course, he would have known in 1899). Gravity is obviously an essential characteristic of the universe, which makes the gravitational constant an obvious candidate for one of the fundamental descriptors of reality.

  = 1.054571628 × 10−34 J s

Planck's constant plays two roles. In its traditional form, h is the proportionality constant that relates frequency and energy for electromagnetic radiation. It is sometimes called the quantum of action. In its reduced form, is the quantum of angular momentum. The second form is now considered by many to be the more fundamental of the two, but it did not appear until 1930.Whereas the previous two constants had a long and distinguished history. Planck's constant had never been seen before. His revolutionary paper on blackbody radiation wasn't published until 1901 — two years after he proposed this system of natural units. Can you say "foresight"?

k = 1.3806504 × 10−23 J/K

Boltzmann's constant relates energy and temperature. It has the same unit as entropy and determines the quantum of this quantity. Entropy and information are related. The smallest amount of information is the bit — a choice between one of two things (1 or 0, yes or no, true or false, guns or butter, stay together or break up). The quantum of entropy is thus the entropy of a bit S = k ln 2. Surprisingly, Boltzmann himself never tried to determine the constant that now bears his name. Planck needed the value to complete his model of blackbody radiation and had to determine it himself. (Actually, the constant he used was the ratio h/k, but this fact is not so important.) Adding the last value to the list meant that a natural unit for temperature was now available. Again, the amazing thing about this work is that Planck could see its importance in the first place. Ludwig Boltzmann's work on statistical thermodynamics was based on the assumption that atoms exist. In 1899, this still wasn't widely accepted.

The procedure for generating the Planck units is to combine these four fundamental constants in a way that gives an answer with the right unit. If the unit corresponds to the quantity you desire, you've just made a Planck unit. For example, if it ends in meters it must be the Planck length …

p = √  G  = √  (1.055 × 10−34 J s)(6.673 × 10−11 Nm2/kg2)  = 1.616 × 10−35 m
c3 (2.998 × 108 m/s)3

This is small beyond comprehension. The next biggest material thing is a proton, the diameter of which is on the order of 10−15 m. That's a full 20 orders of magnitude bigger. Think of something that's about 105 m across (100 km). The big island of Hawaii comes to mind. If a proton was blown up to the size of the island of Hawaii, the Planck length would be as big as the original proton.

Next up, the Planck time …

tp = √  G  = √  (1.055 × 10−34 Js)(6.673 × 10−11 N m2/kg2)  = 5.391 × 10−44 s
c5 (2.998 × 108 m/s)5

How long does this last? Think of something very quick — a photon. Think of something very small — a proton. How long does it take a photon to cross the diameter of a proton?

t =  s  =  1 × 10−15 m  = 3 × 10−24 s
c 3 × 108 m/s

We're 20 orders of magnitude short. The universe is 13.8 billion years old. that's about …

t = 13.8 × 109 × 365.25 × 24 × 60 × 60 = 4.35 × 1017 s

Twenty orders of magnitude smaller than that gives you a millisecond. If the time it took a photon to cross the diameter of a proton was slowed to the point where the photon needed the entirety of time itself to complete its task, the Planck time would last a thousandth of a second.

On to the Planck mass …

mp = √  c  = √  (1.055 × 10−34 Js)(2.998 × 108 m/s)  = 2.176 × 10−8 kg
G (6.673 × 10−11 Nm2/kg2)

This one always strikes me as a let down. We're talking 22 µg. That's like a speck of dust. Compare it to an atom of uranium, the heaviest naturally occurring atom …

m = 238 u = 4.0 × 10−25 kg

or the heaviest known subatomic particle, the top quark …

m = 178 GeV/c2 = 3.2 × 10−25 kg

Both of these values are about 17 orders of magnitude smaller than the Planck mass. Whereas the Planck length and Planck time seem to represent some lower limit on how finely space and time can be divided, the Planck mass seems to be an upper limit on how big the small things in nature can be. No elementary particle will ever be more massive than the Planck mass.

With what we've got so far, we can create a whole coherent set of units for mechanics: Planck acceleration, Planck force, Planck pressure, Planck density, and so on. We'll do one more fully described calculation — the Planck temperature — and just summarize the rest in a table.

Tp = √  c5  = √  (1.055 × 10−34 Js)(2.998 × 108 m/s)5  = 8.903 × 1032 K
Gk2 (6.673 × 10−11 Nm2/kg2)(1.381 × 10−23 J/K)

How hot is this? Nothing humans or nature has done recently comes close. The interiors of the hottest stars are close to a billion kelvins (109 K) — 24 orders of magnitude short. The hottest laboratory experiments take place inside very large particle accelerators like the Tevatron at Fermilab near Chicago and the Large Hadron Collider (LHC) at CERN near Geneva. Here we're looking at quadrillions of kelvins (1015 K) and we're still 18 orders of magnitude short. In contrast, the coldest temperatures ever achieved in the lab are a few hundred picokelvins (10−10 K). The entire range of temperatures achieved so far is an astounding 25 orders of magnitude, but we're still short 8 additional zeros. The Planck temperature is so hot as to be meaningless. As we shall soon see, that's the point.

For the next 50 years or so, Planck's notion of a natural unit system — one derived from physical laws, not accidents of human history — was considered an interesting diversion with little or no meaning. The primary reason for this was probably that quantum theory and general relativity were just too new and unfamiliar. (Relativity did not even exist at the time of Planck's publication.) The physics of the Modern era was a strange world that few understood at first.

Max Planck
(1858-1947)
:   Eine neue wissenschaftliche Wahrheit pflegt sich nicht in der Weise durchzusetzen, daß ihre Gegner überzeugt werden und sich als belehrt erklären, sondern vielmehr dadurch, daß ihre Gegner allmählich aussterben und daß die heranwachsende Generation von vornherein mit der Wahrheit vertraut geworden ist.
[A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.]
 
Albert Einstein
(1879-1955)
:   Die Quantenmechanik ist sehr Achtung gebietend. Aber eine innere Stimme sagt mir, dass das noch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, dass der Alte nicht würfelt.
[Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one". I, at any rate, am convinced that He is not playing dice.]
 
Niels Bohr
(1885-1962)
:   Anyone who is not shocked by quantum theory has not understood a single word.
 
Richard Feynman
(1918-1988)
:   Do not keep saying to yourself, if you can possibly avoid it, "But how can it be like that?" because you will get down the drain, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.

The last quote gives you an idea of what eventually happened. People who grew up with the theory applied it in situation after situation and found that it worked. We will end this chapter by addressing the meaning of all of this.

The Planck units have no practical application. No car odometer will be calibrated in Planck lengths, no stopwatch will tick off Planck times, and no thermometer will ever give temperatures as a teeny, tiny fraction of the Planck value. These numbers tell us the limits of physics as we currently know it and maybe even the limit of physics as it could ever be known. That's why it's an important theory.

year quantity interpretation principal scientist
1954 length gravitational limit of quantum theory Oskar Klein
1955 length quantum limit of general relativity John Wheeler
1965 mass upper limit on the mass of elementary particles Moisei Markov
1966 temperature upper limit of temperature ("absolute hot") Andrei Sakharov
1971 mass lower limit on the mass of a black hole Stephen Hawking
1982 density limiting density of matter Moisei Markov
Interpretations of the Planck units

Space and time are generally regarded as smooth and continuous. The number places between any two points is apparently infinite. We pass from one place to another with no sensation of granularity. There is no "screen resolution" to the video game of reality. There is no apparent "frame rate" either. One moment is followed by another with no perceivable jerkiness. Existence does not play itself out like a turn of the century nickelodeon movie. If the universe is some sort of computer program (as some have suggested), it is rendered with an apparently infinite level of detail.

"Apparently" is the key word, however. The Planck length is now generally regarded as the lower limit of space. Distances less than this are meaningless. Likewise, the Planck time is the lower limit of time. No detectable change will occur in a period shorter than this. You cannot cut space and time up into infinitely small parts. Eventually, you will get to the point where the notion of subdividing space and time any further becomes meaningless. Eventually there will be found an "atom" of space. (Recall that atom comes from the Greek ἄ τομος, a tomos, un cuttable.)

That matter is quantized should be evident to everyone with even the tiniest bit of education. Who doesn't know of atoms? It is less likely that the average person would know that energy was quantized, but such knowledge isn't considered exotic. Many people know of photons. Matter and energy are quantized, and as a consequence, so too is the stage on which matter and energy act. Space and time are quantized. This is perhaps the greatest meaning that one could extract from Max Planck's little excursion into units.

quantity symbol value
speed of light1 c 3.00 × 108 m/s
gravitational constant G 6.67 × 10−11 N m2/kg2
reduced planck constant2 1.05 × 10−34 Js
boltzmann constant3 k 1.38 × 10−23 J/K
Fundamental constants
1 c can also be considered the Planck unit of speed.
2 is the quantum of angular momentum.
3 k ln 2 is the quantum of entropy.
quantity expression value
length √(ℏG/c3 ) 1.62 × 10−35 m
mass √(ℏc/G) 2.18 × 10−8 kg
time √(ℏG/c5 ) 5.39 × 10−44 s
temperature √(ℏc5/Gk2 ) 1.42 × 1032 K
The original Planck units
quantity expression value
acceleration √(c7/ℏG) 5.56 × 1051 m/s2
force c4/G 1.21 × 1044 N
momentum √(ℏc3/G) 6.53 kg m/s
energy √(ℏc5/G) 1.96 × 109 J
power c5/G 3.63 × 1052 W
pressure c7/ℏG2  4.63 × 10113 Pa
density c5/ℏG2  5.16 × 1096 kg/m3
angular frequency √(c5/ℏG) 1.86 × 1043 rad/s
Additional Planck units

What about the natural units of electricity and magnetism? Planck never dealt with the subject that I know of. Your natural choice for a natural unit of electric charge might be the elementary charge …

e = 1.602176487 × 10−19 C

but this would not be in keeping with the spirit of Planck's work. After all, the Planck mass isn't related to the mass of an electron, proton, or any other physical thing. The Planck constants are derived from the laws of nature. To that end some have suggested using the coulomb law constant to extend the original system since it's analogous to the universal gravitational constant G.

1  = 8.98755179 × 109 Nm2/C2
4πε0

Including this unwieldy symbol pile gives us the following electromagnetic Planck units. The values for current and voltage look like they could be upper limits. The value for magnetic flux looks like it could be a lower limit. That's nice, I suppose. The value for resistance means … what? Resistance is a bulk property of an object. Subatomic particles or black holes — the kind of things we've been talking about in this section — don't really have a resistance. The value for charge is slightly larger than the elementary charge. Once again, I'm lost. These quantities aren't as easy to interpret as the original Planck units. I don't think anyone is really working on them as a subject of theoretical study.

quantity expression value
electrostatic constant 1/4πε0 8.99 × 109 Nm2/C2
charge √(4πε0c) = √(2ε0hc) 1.88 × 10−18 C
current √(4πε0c6/G) 3.48 × 1025 A
voltage √(c4/4πε0G) 1.04 × 1027 V
resistance 1/4πε0c 30.0 Ω
magnetic flux √(ℏ/4πε0c) = √(h/8π2ε0c) 5.62 × 10−17 Wb
Extended Planck units for electricity and magnetism