Talk:Planck's law/Archive 1

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Derivation (Statistical Mechanics)

Hey PAR, you didn't define $f$ at the end of your derivation.

Hey, my friend, its our derivation, mostly yours. I fixed the f, it was just the factor of 2 for the photon spin. PAR 18:34, 19 May 2005 (UTC)

Spin of photons is 1

I remember that it shall be 1, NOT 2.--GyBlop 08:52, 24 February 2006 (UTC)

And in another cases,3/2...1/2....5/2 something etc. like those,are for Fermion.--GyBlop 11:27, 24 February 2006 (UTC)

expression of U and R

In Beiser's and Blatt's books,it is said of U(frenquency)=something,and seems to be NOT get a R(something)=stuff... making others confused.

Actually I believe that U(frenquency)=something proposed in a place/page only will be better than proposed both U and R in a one. At least,it will be better in someplaces of math proceses.--GyBlop 08:58, 24 February 2006 (UTC)

EXAMPLE:

for$u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

and we can calculate an expression of certain R which contains of those above.

$R={\mathcal {\,}}{\frac {c}{4}}u\,$

Thus

$R(\nu ,T)={\frac {cu}{4}}{\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}={\frac {2\pi h\nu ^{3}}{c^{2}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

Where this formula is not good for teaching when talking about itself and Energy of EM-w in a cavity at the same time. Or easily gets confused. --GyBlop 09:28, 24 February 2006 (UTC)

It doesn't have to be confusing if the article is very clear in defining symbols and showing the relationships among them. -- Metacomet 12:42, 24 February 2006 (UTC)

---

The wavelength is related to the frequency by

$\lambda ={c \over \nu }.$

The law is sometimes written in terms of the spectral energy density

$u(\nu ,T)={4\pi \over c}I(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz).

The spectral energy density can also be expressed as a function of wavelength:

$u(\lambda ,T)={8\pi hc \over \lambda ^{5}}{1 \over e^{hc/\lambda kT}-1}$

as shown in the derivation below.

• For those editing by the original author.

I am not sure what you did was for Stefan's law ${\mathcal {\,}}R={\sigma }T^{4}={\frac {c}{4}}\rho _{T}\,$ or not. If it was,then some formula like :${\mathcal {\,}}R({\nu },T)\,$ should be going to represent Planck's $U={\frac {8{\pi }V\nu ^{2}}{c^{3}}}{\frac {h\nu }{e^{h{\nu }/kT}-1}}$ (Of-course it is after some process of math)

It is because that I is for a radiance of a volume,but R is for a radiance of a surface.--GyBlop 17:34, 26 February 2006 (UTC)

A Little Suggestion

My information was from books of Modern Physics of Blatt's and Beiser's. I suggest the author could refer to one of their books for rewritting some parts of the article.

At least,for myself,a little confussion on me while my first and second times getting in to see some descriptions of yours.--GyBlop 17:45, 26 February 2006 (UTC)

Boxes-- new guideline?

Whats with the boxes outlining some of the equations/ Is this a new WP guideline?--Light current 21:10, 8 August 2006 (UTC)

I hope not! :) When I revised the derivation, inserted more details etc., I left the boxes with the main results intact. Count Iblis 21:24, 8 August 2006 (UTC)

You mean you yourself did not add the boxes?--Light current 21:49, 8 August 2006 (UTC)

NO, I didn't do it. Perhaps user:PAR might have done it, not sure though... Count Iblis 22:26, 8 August 2006 (UTC)

NO, I didn't do it, and I wouldn't mind seeing them gone. PAR 22:30, 8 August 2006 (UTC)

OK that sounds like a consensus. Now who is going to do the dirty deed?--Light current 22:44, 8 August 2006 (UTC)

I'm too busy editing a real article that was provisionally accepted (the referee wanted some more explanations which means changing large chunks of the text) :( So, why not have a go if you have time and we'll take a look later... Count Iblis 22:50, 8 August 2006 (UTC)

OK done it!--Light current 23:01, 8 August 2006 (UTC)

GOOD PAR 23:40, 8 August 2006 (UTC)

Table

Do we really need a table to define the variables? i think it tends to dominate the page. 8-(--Light current 21:55, 8 August 2006 (UTC)

My opinion: Get rid of it, but make sure you explain everything in the text. The explanation of units/dimensions is a waste of editing space. That's almost primary school level information. And it's irrelevant, because the whole point of units is that you can choose your own favorite units for the relevant quantities. Count Iblis 22:37, 8 August 2006 (UTC)

Not quite primary school level, but they could just be listed normally 8-|--Light current 22:49, 8 August 2006 (UTC)

The use of Stirling's formula in the theory of black body radiation

The Bench, the Round Table, and the Outset

0) This point refers to the basics of statistics as used in the original derivation of the black body formula. It has to be added because otherwise nobody will ever understand Planck . Step one was that he put his energy elements into boxes. The same can be done 'when the lady of the house has 13 guests for dinner and is worried about the arrangement of seats' - Willoughby calculates the result as 13! (read 13 factorial). This is correct if each guest takes one seat.

In another example Willoughby gets a formula similar to the one used by Planck for his resonators and energy elements. On p.557 of the original edition of his work in Ann. d. Phys., to calculate the thermodynamic probability W used in his Equation 3, Planck used the equation

$W_{P1}\,=\,\ {(n+p-1)! \over (n-1)!\,p\,!}$ ,

where index P1 refers to the first expression to describe the example system used by Planck. The actual symbol used to mean "the number of complexions" in Planck's paper was different. It was a Gothic letter which I can not find in the available fonts. In the system, Planck assumed that p=100 elements were randomly placed in n=10 boxes. Willoughby considers what happens "if you wish to arrange $n=2$ boys and $p=3$ girls along a bench and are interested only in the relative positions of boys and girls" and finds out that the number is

$W\,=\,{(n+p)! \over n!\,p!}={5! \over 2!\,3!}=10$

This is correct again if we assume that each person takes a separate seat on the bench. Although the expression is similar to the one used by Planck, it is not the same. Let us see below why it is so.

What is important in this case is that each problem of this kind has its solution given as a factorial or the result of multiplication or division of factorials. If "the lady of the house" has a round table, there are 'only' 12! arrangements instead of 13!. This is because certain arrangements are equivalent. A simpler example will clarify this result . Three persons a, b and c are seated along a bench. There are $W=3!=6$ ways or arranging them: abc, bca, cab, acb, cba and bac. However, at a round table there are only $W=(3-1)!=2$ ways, because the first three are the same order and the next three are also equivalent.

We will now see what happens if 2 boys and 3 girls seated on Willoughby's bench are arranged at a round table.

Bench: b,b,g,g,g; b,g,b,g,g; b,g,g,b,g; b,g,g,g,b; g,b,b,g,g; g,b,g,b,g; g,b,g,g,b; g,g,b,b,g; g,g,b,g,b; g,g,g,b,b

Round table: b,b,g,g,g; b,g,b,g,g

The first arrangement for the round table is the same order as the first, fourth, fifth, eighth and tenth for the bench. The other bench arrangements are the second round table arrangement.

What does Planck's formula predict?

$W=(n+p-1)!/(n-1)!p!=(2+3-1)!/(2-1)!3!=24/6=4$

Four does not equal two, so the round table arranged with boys and girls who have no identity is not what Planck started with. If we remember that the result for the round table arrangement for 13 guests was 12! instead of 13 !, we can calculate W for the case when all the persons around the table can be identified (e.g. have names). The result will be

$W=(n+p-1)!$

which is again not Planck's starting point. What will happen if we:

(1) know the two boys' names while the girls' names are unknown, (2) know the three girls' names while the boys' names are not known?--C. Trifle 18:38, 29 November 2006 (UTC) The answer to the first question is that the result is as predicted by Planck's original equation. As Wikipedia is not intended as a primary source of information, I don't think that one should include the calculation (which is easy) unless the editors agree.

It is a good idea to refer to the example with donuts used in combinatorics . The equation used by Planck is demonstrated there using an example of buying confectionery items. "If you have ten types of donuts to choose from and you want three donuts there are (10 + 3 − 1)! / 3!(10 − 1)! = 220 ways to choose". The explanation is conditionally correct. Interestingly, as trifles are also confectionery items, a trifle (meaning detail) has usually been omitted in books on statistics which use this equation. To learn more about the consequences of certain trifles a serious researcher should study Othello, where the word was used by Shakespeare. Anyway, failing to remember the original purpose of the black body in physics combined with failing to notice the condition on the validity of the first statistical equation used by Planck has led to a century long misunderstanding about the real nature of quanta. More details to come in the section on "Hungry Photons". --C. Trifle 09:11, 8 December 2006 (UTC)

Tentative Conclusions

The conclusion which I think should remain on this page is below:

S1) If one starts with separate elements of any kind put in boxes of any kind, even in multidimensional space, but arranged in such a way that an element is completely enclosed in a box, then the number of 'states of the system' can be calculated by using a formula appropriate for the case. A deflection of the number measured by experiment from the one predicted by theory is a measure of the lack of separation (so - interaction) between the elements. --C. Trifle 23:52, 28 November 2006 (UTC) S2) The study of approximations in the work of Planck, Einstein, Debey, Bose, De Broglie and other physicists proves that neither the black body radiation formula nor the photoelectric equation were derived from the idea of separate elements of radiation. This does not mean that the concept of elements was refuted. It means that those elements interact and are not separated.--C. Trifle 21:49, 29 November 2006 (UTC) And if we introduce Bose-Einstein statistics (which was later) to Planck's law (which was earlier) then why not compare the very basic starting points in statistics of the writers concerned?--C. Trifle 23:14, 3 December 2006 (UTC) Perhaps it could be explained by someone doing research in the field where Planck's law is frequently used ? [4] [5] --C. Trifle 16:57, 11 December 2006 (UTC)

01) So from the point of view of elements completely enclosed in boxes it did not matter what kind of space it was in which the problem was solved (Planck , Debye , Bose , and de Broglie , and others (see Phys.R. Lett.). Neither did it matter what kind of elements was used as long as the element was completely inside the box and completely filling it in. It could be a cube or a narrow pipe from minus to plus infinity as well. All these problems have factorials as solutions. But what was necessary was powering. --C. Trifle 00:43, 29 November 2006 (UTC)(the bibliographical data added today (Bose still missing))--C. Trifle 10:24, 1 December 2006 (UTC)

An Equation by Bose Reduced to a Multi-Bench Problem in the Frequency Domain

(Bose (the paper) rediscovered last night in the attic. I must apologize for misquoting Bose. The explanation will follow below.) --C. Trifle 18:57, 3 December 2006 (UTC)

Bose's reference No 1 was Debye , but his purpose and approach were different. He wrote his formula for thermodynamic probability W on p. 386.

From the point of view of basic statistics, the important difference between the problem of the "bench" and Bose's equation is that instead of one bench with "n boys and p girls" one can see great many benches, and they all affect the "thermodynamic probability". Π in the domain of frequency ν is the product of probability expressions for individual "benches", i.e. factorials related to numbers of "boys and girls" located on the "benches", each placed in a small frequency area between a certain frequency ν and ν + dν . We skip Bose's physics, however fascinating it is to specialists, as we are only interested in the basic statistics of the frequency "benches". For each of them we can introduce the respective factorials of "boy and girl numbers" n_νk and p_νk, whatever they might physically mean. For the first two components one can then rewrite Bose's equation very simply as

$W_{B}\,=\,\ {(n_{1}+p_{1})! \over n_{1}\,!\,p_{1}\,!}\,\ {(n_{2}+p_{2})! \over n_{2}\,!\,p_{2}\,!}$

etc. for more "benches".

If you multiply the components of each factorial by components of the other factorials, for example for the first three frequencies, you will get $3!*3!*3!=(1*1*1)*(2*2*2)*(3*3*3)$ , which is one cubed times two cubed times three cubed and so on. So you can look at it as a product of the various components each raised to a certain power (as high as many frequencies you take), however, if you look at each individual frequency, the factorial is still there. How to integrate the expression over the frequency domain and how to choose some interesting frequencies in this concept is another story. What is mainly interesting from the point of view of girls and boys' statistics is that instead of one single bench you have a whole park with a lot of benches, each of them affecting the others a bit. --C. Trifle 21:27, 3 December 2006 (UTC)

Bose's idea of 1924 can be linked to the wavepacket , "photon clumps" (see below) and, as frequency in Bose's equation is basically not limited, to the idea of the Fourier series . --C. Trifle 19:27, 10 December 2006 (UTC)

Hungry Light Quanta and The Photon Clump Model

02) For the 'light quanta' we also come to the conclusion that they were not separated, even though the situation is different. Einstein did not put light quanta into boxes a priori. However, if one considers the equation for W as a function of volume and the number of quanta n, there is no factorial there. We can unexpectedly see powering. Einstein explains the idea as 'the case when all n 'moving points' (quanta) which were in the (bigger) volume v ₀ come independently of each other to the small volume v . The situation in which this is supposed to happen is described as a dilute solution of a gas (light quanta were a gas of moving points) in another gas. Einstein continued this approach in his 1906 work and later.

The model is different than that for the black body, but it is difficult to interpret it as a separate light quanta model. The reason is that if one divides volume v ₀ into , let us say, P parts, then we either have the same problem as in the black body, which results in factorials, or we have a different probability situation. In the first case, assuming that the photoelectric equation has indeed been found in agreement with experiment, if the experimentally tested equation is obtained by using powering in the probability expression, the quanta must not be separate. On the other hand, we can think of a probability model that gives a quantity W which contains powering for elements which are separate. What kind of real situation does it represent? Why should n quanta meet in that small part of the volume?

This expression might apply to a situation in which one repeats an experiment always with the same probability. For example 'the lady of the house' would have to excuse all the guests from the room and calculate the probability that, when the door is opened, they all jump together onto one (the same) seat not disturbing each other. The problem is why this exercise should be more justified as a model for the photoelectric effect, whereas for the black body the guests were originally postulated to behave in a civilized way - which proved not to lead to the correct formula though. --C. Trifle 02:27, 29 November 2006 (UTC) (But sources do indicate that more than one photon are necessary to eject an electron from the surface, so this might be a link to "photon clumps" )--C. Trifle 22:01, 29 November 2006 (UTC) "Photon clumps" were connected using a certain force resulting from the uncertainty principle. --C. Trifle 17:13, 10 December 2006 (UTC)

One can now compare the example where an act of buying donuts was used to explain the equation used by Planck. Let us see what will happen if a customer who wishes to buy three donuts of different kinds comes to a small shop where donuts are available according to the example given by Planck . The supply of donuts is shown below: --C. Trifle 10:12, 8 December 2006 (UTC)

Box (kind of donut) 1 2 3 4 5 6 7 8 9 10

Number of donuts 7 38 11 0 9 2 20 4 4 5

If the customer wishes to buy a donut from box 4, he will not get it (the number available for box 4 is 0). The equation is only valid for selling donuts if there is no limit on supply. The black body in physics is both an emitter and an absorber. If we imagine a circular (or spherical) cavity with a small opening in the front wall, and the number of n "boxes" or "shelves" where p elements that fall inside are stored, the equation used by Planck is properly chosen.

If we then imagine light quanta as molecules of a gas (Einstein's "moving points") rising from the shelves in the cavity and flying around, the use of the ideal gas equation seems justified. If the black body is kept at a constant temperature, n quanta are to leave the cavity with the same probablity per unit time, so the use of powering instead of factorial seems justified. (A few quanta would have to meet in the small area near the 'cavity door' around a certain time ). Certainly, all these models are not real things.

Even though they are only models, they are used as a basis for equations that are recognized as fundamental laws. Understanding the differences between the meaning of simple statistical models that were used at the beginning of the development of the quantum theory could help appreciate certain modern "entangled photon" concepts ‹The template Talkfact is being considered for merging.›  which otherwise could be misunderstood. --C. Trifle 11:16, 8 December 2006 (UTC)

Growth phenomena of different kind could be described using the original Planck's concept. And these here look like the real thing [6] . They could be explained using one of several possible interpretations,i.e. how particles of this type "jump together on one seat" and begin to form a drop that grows. A crystal could begin to grow spontaneously in a similar way, or maybe a soliton. A tornado too. --C. Trifle 22:40, 12 December 2006 (UTC) This interpretation is possible if we assume that what Planck called "first order Stirling approximation" did actually represent a transition from one state state of the system to another.--C. Trifle 18:58, 13 December 2006 (UTC)

Many-world interpretation [7] is not excluded either, if you look at it from a different point of view. --C. Trifle 22:40, 12 December 2006 (UTC) This interpretation is possible if we assume that the equation postulated by Planck ($n!=n^{n}$ ) is a steady state equation. Then our system elements which we assume to be solely represented 'here' would in fact appear also in other systems. That would occur at a different place and/or time. A 'hybrid' approach is also possible - a number of correlated systems could become unstable from time to time, which would result in the growth of phenomena like droplets, solitons etc. on different world trajectories in the various coupled systems. --C. Trifle 18:58, 13 December 2006 (UTC)

Certainly, the 'confectionery' example is not the final solution to the problem. It was only shown that different numbers W for 'energy elements' and 'light quanta' depend on the model used. Different models could be justified, so dramatic differences are not a mystery. However, this indicates that some changes in gas entropy should be observed. --C. Trifle 01:22, 9 December 2006 (UTC)

There has been some misunderstanding about the nature of 'quanta' in popular books published since the beginning of the twentieth century. The descriptions mainly concentrated on the geometrical features of 'particles' which were usually models synthesised from several authentic models by different writers. One could read about "streams of little bundles of energy" whose energy was equal to hν , which were neither used by Planck nor by Einstein in their breakthrough articles. At the same time the statistical aspect (which mainly did the job) was skipped. This occurred even in otherwise very good textbooks ‹The template Talkfact is being considered for merging.› . --C. Trifle 10:40, 11 December 2006 (UTC)

Einstein's Footnote - The Ideal Gas Equation vs Black Body

The footnote on p.142 in Einstein's paper is interesting too. It links entropy in the blackbody, the photoelectric equation and the ideal gas equation, so all those models must be valid in the three basic areas they concern. How about the modern "entangled photons" and "qubits"? Do they match this criterion? I am afraid not. --C. Trifle 23:14, 3 December 2006 (UTC)

Boltzmann's Approximation

Planck's success in formulating the experimentally confirmed radiation law is usually linked to Boltzmann's statistical interpretation of the second law of thermodynamics . However, it is rarely reported ‹The template Talkfact is being considered for merging.›  that Boltzmann made an approximation which dramatically changed the meaning of his work. The approximation on p. 103 requires Bolzmann's gas particles to have some unexpected properties in order to comply with the first order equation derived therein.--C. Trifle 18:11, 7 December 2006 (UTC)

In the past, Boltzmann's writing seemed to have had an almost hypnotic effect on most readers. Even those who, like Ernst Zermelo , noticed that something was defintely incomplete in Boltzmann's explanation of the second law of thermodynamics acted as if they were unable to find certain approximataions in the work they had read. Boltzmann's writing keeps the reader entangled. I observed that several years ago when the only person who proved to remember the approximation discussed in class was a female student who had been absent when the rest of the group studied the first part of Boltzmann's statistical Universe. On the other hand, the approximation is only important in the context of the complete model described by Ludwig Boltzmann in his work. It should also be remembered that Boltzmann himself indicated that his work was not complete. --C. Trifle 13:18, 8 December 2006 (UTC)

Neglecting certain components of Taylor series is by no means an unusual operation in mathematics. However, trying to maintain the general validity of the truncated equation usually leads to the need for reexamination of the initial conditions. In fact, if an analysis of shape and motion distortion of gas molecules was performed, it could trace the effect of rejection of some higher order components ‹The template Talkfact is being considered for merging.› . Both Zermelo and Boltzmann were in a way right. Boltzmann function remained and is still in use but nature itself gave the answer and the rejected components had to be mathematically transferred into molecules of Boltzmann gas. In the theories being closer to the real world, moving points or collections of points became sophisticated mechanisms with a kind of built-in PID system, some elementary memory and intuition, and tentacles of orbitals spreading both into the future and the past. This was only discovered after Boltzmann's death.--C. Trifle 01:00, 9 December 2006 (UTC)

The higher terms of Stirling formula in electromagnetic radiation

Assuming that the approximation used in black body radiation theory was indeed based on the "first order Stirling approximation", what effect do the higher terms of the series named after Stirling have on electromagnetic radiation ? The Fourier series has a clear interpretation. It would be interesting to link certain observable phenomena ‹The template Talkfact is being considered for merging.›  for the Stirling series as well. --C. Trifle 19:26, 13 December 2006 (UTC)

For the radio frequency range, the electric and magnetic fields in the immediate vicinity of an antenna are greater in magnitude, and differ in phase from the radiation field in the distance. The induction fields must be added to the radiation field in order to give the fields actually present there . As quanta are now usually assumed to be present in the whole frequency domain, there might have been some attempts to match the radiation near the antenna to the quantum expressions and perhaps to the higher terms of the Stirling series ‹The template Talkfact is being considered for merging.›  . The numbers of quanta would be enormous there.--C. Trifle 22:02, 13 December 2006 (UTC)

Black body theory vs radio antenna does not mean that one has to cool the antenna to single Kelvin degrees. The idea refers to a standing wave at room temperature. Technical literature related to near field measurement theory uses the wave concept which provides a solid experimental basis. This is because one can hardly imagine practical field measurements of single UHF quanta even assuming that they exist. On the other hand, if one assumes that a node might be interpreted as a low entropy state and an antinode as a high entropy state, then perhaps some correspondence between the various black body entropy expressions and the antenna field distribution could be found ‹The template Talkfact is being considered for merging.›  . Then, if higher order Stirling series components do indeed affect entropy, one should see it where there are very many quanta, i.e. in the near field .--C. Trifle 20:48, 14 December 2006 (UTC)

Otherwise, if the higher terms of the Stirling series do not produce any measurable effect on radiation, one could ask why the first term was so important. Maybe Planck and the other 20th century physicts really got it wrong and one should rather do the bench, the round table, and the donut shop exercises once again carefully. But if it indeed were so, one would have to accept once again that the so-called fundamental laws are often useful, but often become fundamental by mistake. Maybe someone heard about some research into Stirling vs near field ‹The template Talkfact is being considered for merging.›  ? --C. Trifle 21:58, 14 December 2006 (UTC)

Lewis photon model vs ring counter

A photon model proposed by Lewis, if the "mystery variable" used there to remember photon absorption history is given in bits and the structure is extended to a parallel-in, serial-out shift register connected as a ring counter, could account for several different entropy expressions given by Planck and other writers quoted herein. The model could accumulate heat in a way similar to that described in the Boltzmann model, and release a quantum for a given frequency when the counter is full. --C. Trifle 18:03, 27 December 2006 (UTC)

By saying the above, it is not assumed that the idea of a certain quantity, expressed in the 1926 article by Gilbert N. Lewis, could explain on its own the different values of entropy calculated by the historic writers quoted on this page. In 1926, Lewis's idea had no (or very little) practical value if we remember that the word bit was only used 21 years later as an acronym to mean the binary digit, and it took years to develop more advanced logic circuits. Neither it is suggested herein that a real photon is indeed equipped with a built-in ring counter.

However, a circuit of this kind, or its software equivalent, could be helpful to demonstrate a physical phenomenon of the parallel absorption of portions of an input quantity (such as heat in units or fractions of kT) through the "surface" (or the "input points") of the model, storing that quantity, processing it and finally producing an output pulse of hν when all the inputs have been reached. This is provided that the 'receptors',or the inputs of the model, were blocked (as Lewis wanted, although perhaps not indefinitely) after each individual absorption or emission act.

Indeed, if various statistical models are currently used to explain Planck's and Boltzmann's work which were developed tens of years after (or even over a hundred years after) the original papers by Max Planck and Ludwig Boltzmann, why not use a software implementation of a digital circuit, or the circuit itself, to demonstrate the operation of the "light particle"? This idea is submitted as a possible research topic, or perhaps as a graduate project, especially in the area of Computer Graphics, and Physics. --C. Trifle 11:27, 28 December 2006 (UTC)

Artists

Displaying the translated article by Max Planck on this page was an excellent idea. It would also be a good idea to describe the work of the people who made the main contributions to the theory as work of artists not idols. Their theories were not correct and complete descriptions of reality and it is not necessary to show them as such. To understand their achievement one must see both the authentic idea and the approximations. They had different visions of the world. They were like writers or painters. They had visions of each other's work too.

A good example for the latter: see "Self-Portrait with Portrait of Gauguin" by Emile Bernard , and two paintings by Gauguin "Self-Portrait with Portrait of Bernard" and " Van Gogh Painting Sunflowers" [8] .

--C. Trifle 16:37, 10 December 2006 (UTC)

Other Topics and Comments

I decided to divide the section into subsections because it was getting difficult to manage. --C. Trifle 17:33, 7 December 2006 (UTC)

Topics proposed on this page were used as input for students' essays in Scientific English written during a university course in Poland. Students were given material copied from authentic research papers and compared it with some textbook versions in their essays on (generally) Advantages and Disadvantages of Scientific Theories. Over 300 essays were written. --C. Trifle 10:40, 11 December 2006 (UTC)

Interestingly, in their JAT essays most students treated the approximations in the historic papers as "errors", even though I insisted that the approximations must have described a physical quantity (one or more) of some kind. Although a number of students seemed really interested in the topic, there were also some who were disappointed and in some cases refused to write on the approximations, as if they were either not interested or afraid that their writing might offend somebody (not quite clear who though). In my opinion, the problem resulted partly from the fact that most undergraduate courses use various versions of natural laws "named after" their authors and do not include the authentic material at all. --C. Trifle 12:45, 30 December 2006 (UTC)

As the JAT course was finished in January 2006, this text has been written as part of the teaching materials to be used within the "consultation hours" in English for postgraduate students in the autumn semester. It has been submitted to Instytut Informatyki, WEiTI, Politechnika Warszawska, with a special request to promote further research and help those interested in pursuing the topic (especially students interested in simulating physical phenomena in the Computer Graphics Lab). By permission of the Institute, it is also offered under the GNU license to anybody interested in this research, individually or in co-operation, if they find the information given here of any value. --C. Trifle 12:02, 28 December 2006 (UTC)

This is where talk Stirling's approximation originally started

I added the text above later, so this line is a kind of deconfuser --C. Trifle 08:22, 4 December 2006 (UTC)

1) How about adding a link to what Planck called the "first order" Stirling's approximation? I added two paragraphs under "Stirling's formula" yesterday. That Stirling's thing is visible not only in Planck's original paper which is now Ref.2 here, but it was also used by Debye, Bose and Louis de Broglie. It worked, so there must have been a reason for $n!=n^{n}$ , which is what I tried to teach for some time. A possible reason follows qualitatively in point 2 with a proposed correction in 3 below.

2) If you change the number W in Planck's equation 3, you change the entropy of the whole system. Imagine that you keep the temperature constant and manage to change the entropy just by changing the number W (you change the experimental set up, an element is absorbed, etc). Then the system energy has to change by very little (Boltzmann constant is small). This is because of the definition of entropy: delta S equals delta Q over T. The change of energy due to the change of the number W does not seem to be related to a single element, but rather goes to the whole system. Consequently, the way in which Planck, Debye and Bose used Stirling's formula led to a situation where they added some extra variable(-s) (in addition to energy)to the system. That worked as the black-body equation matched the results. One can speculate that the three writers thus managed to include some interactions that were unknown at the time of their writing (e.g. spin, the uncertainty principle, Pauli's exclusion principle, etc?) and possibly something that is not known so far. All that or anything of that could have been hidden in the "first order Stirling's" disguise (which at that time served the developing theory very well). As a result, the various (then unknown) quantities were eliminated from the black-body equation. Not completely, however, as they are still there, in the approximations, so that, in conclusion, the quanta in the black-body equation are not completely separated from each other.

3) Therefore I propose the following corrections to the history section. It must not be too long here. So how about:

a) A small change to the words "a theoretical derivation" made by Bose. Perhaps "an alternative" or "a new" instead of "a" would be better, because how to call Planck's work then - only "practical"?

b) A sentence between the two paragraphs could refer the reader to the Stirling's formula. For example:

Interestingly, Planck and other early writers on the black-body radiation used the "first order" Stirling's formula which provided a greater number of interactions among the elements of the system than the number calculated from the idea of completely separated system elements. The approximation was strongly nonlinear for a small number of elements. It could have been equivalent to a joint effect of some variables or phenomena discovered later in quantum mechanics, such as spin or the uncertainty principle .

Any comments please? (That was me but the signature and time is sadly missing. So let me add now -- C. Trifle 21:41, 5 October 2006

I decided to change the above "interactions among ... the ... completely separated elements" to "complexions", as Planck wrote, because otherwise, if the elements were completely separated, the number of interactions should be zero, which would surely make all that mathematics useless. C. Trifle, 5 October 2006 partitioning the history section here--C. Trifle 22:45, 28 November 2006 (UTC)

references

Are there different standards for citing sources on mathematical derivations? I know the steps should be self-evident, and the math is in the public domain, but the way it looks now it appears to have come off the top of the authors' heads. I don't want to be pedantic, and if nobody cares, I can remove the little templates. Good writing, btw. Ojcit 15:55, 3 October 2006 (UTC)

I gave a self-contained derivation suitable for wikipedia. You won't find this derivation anywhere in the literature, because textbooks usually assume that the student knows the basic stuff and they'll proceed from there. Giving the standard derivation that you can find in textbooks serves no purpose, because the people who can follow that are precisely the ones who already know it. What I did here was to imagine that the reader knows nothing except what he/she can find on wikipedia. He/she must, of course, have some basic mathematical skills.

So, perhaps a sentence needs to be included at the start of the derivation saying that we give a self contained derivation and that other derivations can be found in books such as.... Count Iblis 16:29, 3 October 2006 (UTC)

Alternative derivation of energy density (with little mathematics)

The energy due to equilibrium radiation in a little volume $d^{3}x\,$ has three contributing factors:

• Energy of a light quantum, the so called photon:

$h\nu \,$

• Possible states:

$2{\frac {dp_{x}dx\,dp_{y}dy\,dp_{z}dz}{h^{3}}}=2{\frac {d({\frac {h\nu _{x}}{c}})dx\,d({\frac {h\nu _{y}}{c}})dy\,d({\frac {h\nu _{z}}{c}})dz}{h^{3}}}=2{\frac {d^{3}\nu \,d^{3}x}{c^{3}}}$

The factor 2 is due to the possible polarisations, the rest due to the fact that phase space $dpdx\,$ could be occupied only in multiples of $h\,$. In thermodynamic equilibrium no particular direction will be favored. Therefore we can integrate over all frequency values lying on a sphere

$2{\frac {4\pi }{c^{3}}}\nu ^{2}\mathrm {d} \nu \,d^{3}x$

• Expected occupation number for a state:

In thermodynamic equilibrium a photon with energy $h\nu \,$ is emitted and absorbed by an electron jumping between two energy states separated by an energy gap of $h\nu \,$. The higher of the energy states will be less populated by electrons than the lower - at least in a thermodynamic equilibrium where the upper energy level has a lower occupation by electrons given by the factor

$e^{-{\frac {h\nu }{kT}}}$

The expected occupation number $b\,$ of the photon's quantum state determines how probable it is that a photon kicks an electron from the lower energy state up to the higher. Without external influence an electron will drop back to the lower. However, photons could additionally stimulate the electron to drop down. Assuming that the occupation of a photon state has the same influence in kicking electrons up as kicking them down the energy gap the equilibrium condition is

${b=e}^{-{\frac {h\nu }{kT}}}(1+b)$

determining the occupation number of a photon state in equilibrium:

$b={\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

With these three factors we get the energy in a little volume $d^{3}x\,$ and the frequency interval $d\nu \,$ :

$(h\nu )\,(2{\frac {4\pi }{c^{3}}}\nu ^{2}\mathrm {d} \nu d^{3}x)\,({\frac {1}{e^{\frac {h\nu }{kT}}-1}})={\frac {8\pi h}{c^{3}}}{\frac {\nu ^{3}\mathrm {d} \nu }{e^{\frac {h\nu }{kT}}-1}}\,d^{3}x$

Matthias Unverzagt 21:46, 11 November 2006 (UTC)

Warnings posted 11 Jan 2007

I posted two warnings in The use of Stirling's formula in the theory of black body radiation.

The first regards the language of the section, which is too informal for a Wikipedia article. The language should be cleaned up.

The second warning deals specifically with the Lewis photon model and the logical variable section. The relevance of this section to the rest of the article is unclear, as it discusses the history of the concept of the photon instead of discussing Planck's law directly. The relevance of the extended discussion to Planck's law should be stated clearly or the discussion should be moved to another page. If the purpose is to give background information on the concept of the photon, then that should be done in another article. Dr. Submillimeter 08:40, 11 January 2007 (UTC)

Warnings posted 13 Jan 2007

Aside from the other problems with the The use of Stirling's formula in the theory of black body radiation section, the section is also off-topic. Stirling's approximation is more commonly applied in derivations of the Boltzmann distribution. While the Boltzmann distibution is used in derivations of Planck's law, descriptions of the derivation of the Boltzmann distribution really belong in the Boltzmann distribution article.

I will remove the section on 20 Jan 2007 unless I receive strong objections. Dr. Submillimeter 22:34, 13 January 2007 (UTC)

Off-topic, inappropriately-written discussion removed

Following a lengthy discussion at Wikipedia talk:WikiProject Physics, the majority of people wanted to see The use of Stirling's formula in the theory of black body radiation removed from this article. I have cut the section and placed it here. Dr. Submillimeter 07:34, 20 January 2007 (UTC)

==The use of Stirling's formula in the theory of black body radiation==

{{off-topic}} {{story}}

===Controversy over the meaning of the "first order Stirling approximation" in black body radiation===

Interestingly, Planck, followed by some other early twentieth-century writers on the black-body radiation such as Debye and de Broglie, used the "first order" Stirling's approximation in a rather unusual way. Instead of using the expression $n!\approx (n/e)^{n}$ the writers, who all were excellent physicists, preferred either to write it as $n!=n^{n}$ or leave some components out immediately below. Two other eminent physicists, Einstein and Bose each took a different approach in their dealings with the problem and used other approximations. Although the left side of the equation $n!=n^{n}$ is obviously not equal to its right side, which would lead to a gross error in simple calculation (for n=1, 1=1; for n=2, 2=4; for n=3, 6=27 etc., which does not seem reasonable), the substitution $n!=n^{n}$ should be considered in terms of the mathematical operations that followed.

For Planck, they were: (1) n=n+1, (2) substitution to the fraction, (3) neglecting certain terms, (4) differentiation, and (5) integration. Planck's approximation resulted in a kind of "approximation of shape". The relationship between the idea of "energy elements" and the curve that was verified by measurement was demonstrated, but the elements were not separated. In fact the total of the approximations provided a far greater number of "complexions", as Planck called the system states, than the number calculated from the idea of completely separated system elements. This implied some extra microstates or interactions between the elements. The approximation was strongly nonlinear for a small number of elements. The same applies to the derivation of the photoelectric equation by Albert Einstein (the 'light quanta' were not separated).

Actually, none of the papers in question was devoted to the radiation of microsystems and that was perhaps why the writers chose not to give the reason for the approximation. One can only speculate that it could have been equivalent either to a kind of distant interaction being a joint effect of some variables or phenomena not yet known at the time of their writing and discovered later in quantum mechanics, such as spin or the uncertainty principle, or to some still unknown hidden variables . However, a relationship between the lack of separability implied by the early 20th-century thermodynamical analyses of the black body and the contemporary quantum entanglement theories is still missing.

===Simulating heat radiation by using a monster===

====Lewis photon model and the logical variable====

{{off-topic}}

In his 1926 article in Nature Gilbert N. Lewis hypothesized about the properties of the "atom of light" for which he coined the name "photon". While the name was generally accepted, an idea of the logical variable in the model was not. In fact, at that time Lewis's idea was not clear and was thought as one of no (or very little) practical value if we remember that the word bit was only used 21 years later as an acronym to mean the binary digit. It then took years to develop advanced logic circuits and programming.

It was not noticed that Lewis's "mystery variable" had actually been present in the photon since 1901 when it was born in its previous incarnation as the "energy element" in Max Planck's work on black body radiation. The use of the "first order Stirling approximation" by Planck could in fact be treated as certain steps of an algorithm where the comment and some other steps were missing. The model described below uses a "software monster " which could accumulate heat using an algorithm based on the approximations in Planck's work and certain ideas described in Boltzmann gas model, and then release a quantum for a given frequency when the evolving "counter" is full. The monster ‹The template Talkfact is being considered for merging.›  is a semi-educated rebel who has been taught a lesson of how to live with the second law. All his efforts to beat the second law have failed. However, he is semi-educated in the sense that he only checks the value of his entropy with his mouth closed. With his mouth open, he always swallows the same number of bits. His entropy is defined according to the function used by Planck.

Lewis proposed photon as "a new type of atom, an identifiable entity, uncreatable and indestructible, which acts as the carrier of radiant energy and, after absorption, persists as an essential constituent of the absorbing atom until it is later sent out again bearing a new amount of energy." The structure proposed by Lewis is not energy. It "spends only a minute fraction of its existence as a carrier of radiant energy, while the rest of the time it remains as an important structural element within the atom." The minimum requirement for a carrier able to perform such a variety of tasks is to have an algorithm and an ability of a certain kind to store and implement it, though perhaps not memory in the meaning used in psychology.

By saying the above, it is not assumed that the idea of a certain quantity, expressed by Lewis, could on its own explain the different values of entropy calculated by the historic writers on black body. Neither it is suggested that a real photon is indeed equipped with a built-in ring counter or shift register.

====Einstein's footnote====

The idea expressed in 1905 in the footnote on p.142 in Einstein's paper is used herein. The footnote links the second law of thermodynamics with entropy in the blackbody, the photoelectric equation and the ideal gas equation. It is assumed that the growing object can connect small particles of heat both in the cavity and in a gas.

====Monster algorithm====

An assumption that photons are generated in the cavity in a kind of "clumps" leads to an idea of computer simulating a similar process. Planck obtained the equation for thermodynamic probability from the following formula

$W_{1}=(n+p-1)!/(n-1)!\,p!$

where n was a number equal to the number of resonators and p a number equal to the number of energy elements. Then he postulated n=n+1 and n!=nⁿ (which was called "first order Stirling's approximation"), and obtained

$W_{2}=(n+p)^{(n+p)}/n^{n}\,p^{p}$

The only reason given for the two operations was that they were "approximations". The logarithm of the second expression was then calculated and multiplied by k to mean entropy S.

Trying to interpret somehow these operations one could imagine that they together describe a growth process in which the "mouth" of a growing object would "swallow" portions of n's and p's. The object would be similar to a jumping spiral. W₁ could be associated with the entropy of elements to be incorporated by the "growing object". The object could absorb energy from the walls, in units or fractions of kT , attaching bits 0 (for "tails") and bits 1 (for "heads"). A unit of kT is awarded at bit 1, but it can only be digested by the object (which is to become a monster or just a droplet) when a corresponding bit 0 is attached. The operation n = n+1 opens the arrangement in which the entry bits are stored. The monster swallows them, shuts its mouth and checks whether the total of n equals the total of p. If it does, it is not hungry, because the expression W₂ reaches a maximum and the requirements of the second law of thermodynamics are met. (The entropy of an isolated system - "mouth shut"- has reached a maximum.)

The numbers (n + p -1) and (n + p) are the values of a quantity which represents the number of performed operations (the state of a counter). The outcome of of each operation is either "heads" or "tails". Another logical variable checks whether n = p (looks for the maximum).

The monster does not know the binary code. For example, it reads 1001001 and 1110000 as one state of "3 heads and 4 tails", at which it is still hungry. Attaching a "1" anywhere in this example would lead to many different binary numbers. All of them are just one state for the monster - "4 heads and 4 tails", at which n = p and it is not hungry. Like many famous monsters it is interested in food rather than in any digits defined by the computer user. If it happens to operate in an integrated circuit, the user will normally not detect it. It only takes a kT from a bit of the program, so its appetite does not usually interfere with the circuit and its program. If n in the first portion does not equal p, the growth process continues until the totals of n and p are equal.

When this happens, we toss a coin once again (or the decision is taken as a result of a certain process). The outcome lets the object either stay and keep on growing at the same or a new location or collect all its p units of kT and leave the cavity as a photon hν = pkT. For a monster that has operated in a semiconductor wafer, the energy of hν might be sometimes sufficient to charge a deep level trap inside a device or a trap in the oxide coating. Elsewhere this simple algorithm could produce effects roughly similar to the growth of various objects very different in size, such as drops, solitons, lightning , tornadoes or solar flares and sunspots in the macroscopic world. The numbers n and p could represent the numbers of states provided for elements of any two chosen physical quantities which interact in an isolated or open system, such as kinetic and potential energy of a mechanical oscillator or electric and magnetic field, etc., which have been connected to the system.

A similar algorithm could be used for any two quantities that do not interact ("a sleeping monster") and only be activated when they start to interact ("take notice of each other"). The two entropy expressions used in the historic derivations could be associated with absorption through parallel inputs (factorials) and interactions inside the growing object (powering).

====Tossing the coin====

In a coin-tossing version of the "experiment" performed on the table, with one coin, assuming the value of kT = 0.026 eV as calculated for T = 300 K, the following values have been obtained for the first ten thus generated "photons": 0.116 eV (total 8 bits i.e. 4 ones and 4 zeroes, three locations), 0.026 eV (2 bits, 1 location), 0.926 eV (36 ones and 36 zeroes, total 72 bits, two locations), 0.052 eV (6 bits, three locations), 0.039 eV (4 bits, one location), 0.013 eV (2 bits, one location), 0.906 eV (6 locations), 0.039 eV (1 location), 0.440 eV (two locations) and 0.699 eV (two locations). The first "cookie" eaten was assumed to have energy equal to kT, the second 1/2 kT, third 3/2 kT, the fourth and all the following one kT each. The different values for the second and third bit p could be associated with switching the process on, but a more realistic approach could be proposed.

If we look at these numbers more closely, we can see that the results of coin tossing are not satisfactory. To have a better correlation with the real 300K black body spectrum one must not have so many "photons" whose energies exceed 200 meV. The results can easily be improved if we remember that, according to Einstein's postulate, the light quantum (when imagined as a particle of a gas) must only be emitted or absorbed as a whole. So one should not have given the "spiral jumping monster" the second chance to grow. Once the maximum of W₂ is reached for n = p , the "photon" is formed. If one includes this assumption, there are 22 "photons" instead of 10, and the correlation improves, although some still have energies above 200 meV, one as high as 600 meV.

The geometry of the cavity is not included. The monster can also act outside the cavity and collect units of kT from the colliding gas molecules. The reason why these results are similar to the real black body spectrum was that Planck used equations that closely remind the equation of the spiral. Equations of this type are not uncommon in nature. The assumption that units of kT should be convertible into hν does not seem unreasonable, if that is associated with collisions of gas particles which have an internal electromagnetic structure. The way in which the second law is included seems logical too.

One could ask, firstly, why should a quantity such as entropy lead to the growth of anything productive? Entropy is usually associated with the dissipation of useful energy, growing disorder, and losses of activity. The secret seems to be that, at the time when the concept of entropy was created, heat was considered as the loss of the useful energy of the moving machine parts. Entropy described that process well. However, the "loss of activity" did actually increase the energy of small particles. This meant the production of quanta of higher energy than the quanta of the electromagnetic field that powered, for example, the electric motor. Secondly, it could seem rather puzzling why the origin of so many apparently different objects as, let us list the examples again, drops, solitons, lightning , tornadoes , solar flares and sunspots could be associated with very similar functions. The answer is that the function postulated by Planck has its maximum when, for a given sum of n and p, n = p. This fact is not connected with the lack of symmetry, and it is the latter that is often regarded as disorder by human observers.

Indeed, many people might regard a state in which a long line of elements "p" is followed by a line of "n's" of the same length as "perfectly ordered", as well as a line arranged as "npnpnp...", for which n = p, whereas many other arrangements for which n = p but there is no evident symmetry will be treated as disorder. The function W₂ has its maximum at n = p irrespective of symmetry. The lack of symmetry may lead to various shapes, which on the one hand increases the range of objects where the function could be useful, but on the other hand the same may lead to unpredictable shapes in modelling (or to different shapes of natural objects).

It is expected that using a more accurate bar-graph version of Maxwell-Boltzmann distribution and appropriate software could show more accurately the correlation between that distribution and Planck's law. Gas molecules had discrete values of energy in Boltzmann's 1872 work , but the difference between those values was definitely smaller than kT. It would be interesting to see the correlation resulting from the second law of thermodynamics, and especially to point out some components still missing in the correlation. If the simulation produced results where the values calculated from Maxwell-Boltzmann distribution could be converted into Planck's law, one might conclude that Stirling approximation has no physical meaning in the black body radiation, at least as far as the density function is concerned.

Requested move

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

Planck's law of black body radiation → Planck's law — First, most textbooks do not use "Planck's law of blackbody radiation" to identify the law. Several shorter variants ("Planck's law", "Planck's radiation formula", etc.) are generally used. "Planck's law" itself is used by the Rybicki and Lightman reference in this article. Second, this is the only law (or at least the most well-known law) named after Max Planck. Planck's law itself is currently a redirect to this article. This article simply does not need such a lengthy name. Dr. Submillimeter 09:30, 2 April 2007 (UTC)

Survey

Add  # '''Support'''  or  # '''Oppose'''  on a new line in the appropriate section followed by a brief explanation, then sign your opinion using ~~~~. Please remember that this survey is not a vote, and please provide an explanation for your recommendation.

Survey - in support of the move

• Support - See my reasoning above. Dr. Submillimeter 09:32, 2 April 2007 (UTC)
• Support simpler names for titles. If the simpler name in common use already redirects then it should be the title. --Polaron | Talk 16:22, 2 April 2007 (UTC)
• Support per Polaron. I don't forsee any new laws coming from Dr. Planck so there is no worry about disambiguation. Planck's law is by far the most common name and is quite suitable as an article title. 205.157.110.11 06:38, 3 April 2007 (UTC)
• Support - it's generally referred to as Planck's law, in my experience. Also, the explanation belongs in the article rather than the title. Mike Peel 21:50, 4 April 2007 (UTC)

Survey - in opposition to the move

• Oppose. Nothing wrong with longer more explanatory titles. Leave as is, including the redirect. --Bduke 12:06, 2 April 2007 (UTC)

Discussion

Add any additional comments:

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

This article has been renamed from Planck's law of black body radiation to Planck's law as the result of a move request. --Stemonitis 09:05, 8 April 2007 (UTC)

Citation style

User:Paul August changed the citation style of the article to list footnotes in a separate section from the references. The problem that I have with this style is that it leaves the reader guessing as to whether the reference is used for the whole article or just for specific passages. I would prefer to revert to the original reference style, where the references were explicitly connected to specific passages through the footnotes, thus leaving no ambiguity as to how the references were used. Would anyone object if I revert to the original citation style? Dr. Submillimeter 07:40, 15 May 2007 (UTC)

History

Any one think history should be near the top of the page?--Light current 23:47, 8 August 2006 (UTC)

Yes, I think that would be better. Count Iblis 00:34, 9 August 2006 (UTC)

It would be if it were a real history. Currently all we have is a short litany of errors in the usual telling, which would look rather silly at the beginning. I'd suggest a real history except that there's already a quite serviceable one at Black body. --Vaughan Pratt (talk) 01:06, 16 August 2008 (UTC)

Missing $\pi$ ???

The equation for spectral intensity that is given in many sources contains a $\pi$ that is missing in the equation given here. The spectral energy density is fine, so the question is wether

$I=u{\frac {c}{4}}$,

or

$I=u{\frac {c}{4\pi }}$.

The difference is presumably wether the spectral intensity is per unit solid angle or not. However, the solid angle subtended by a detector with an unrestricted field of view is $2\pi$, so then there is factor of two missing. Would someone please add an explanation for how to correctly account for solid angle? Cfn137 02:06, 8 March 2007 (UTC)

Update:

I will call the spectral intensity that is not per steradian $R$, since that notation was used in the discussion above. After doing some geometry,

$R=u\,c\left({\frac {\alpha }{4\pi }}-{\frac {\alpha ^{2}}{16\pi ^{2}}}\right)$,

where $\alpha$ is the solid angle. This gives $R=u{\frac {c}{4}}$ for $\alpha =2\pi$, and $I=u{\frac {c}{4\pi }}$ in the limit of small solid angle. Surely there is a good reference that can verify this answer.

It is easier to write these relations in terms of the detector's field of view (FOV):

$R=u{\frac {c}{4}}\sin ^{2}{\frac {FOV}{2}}$.

The solid angle is calculated from the field of view using:

$\alpha =4\pi \sin ^{2}{\frac {FOV}{4}}$.

Cfn137 03:15, 8 March 2007 (UTC)

I think its the idea of the detector that is the problem. A detector with an unrestricted field of view will have the entire 4π availiable to it. The detector you are thinking of is an area element which only collects from one half that.

I restrict the detector to 2π because the flux of photons from the other half is negative, and thus there is no net flux. Cfn137 19:37, 8 March 2007 (UTC)

You really shouldn't think of it in terms of a detector. The intensity is defined independently of any detector. The intensity (radiance) is the infinitesimal power per unit area directed into an infinitesimal solid angle, and if its per unit frequency, then its spectral intensity (radiance). Here's a derivation - consider a small volume element which is a cylinder of height dr and cross sectional area dA. It contains a total energy of U dr dA and of this energy, a fraction dΩ/4π is directed along the axis of the cylinder into solid angle dΩ It moves at velocity c so the time it takes to travel thru the face of the cylinder is dt=dr/c. The intensity is this energy divided by dt, dA, and dΩ

$I=(U\,dr\,dA)\left({\frac {d\Omega }{4\pi }}\right)\left({\frac {c}{dr}}\right)\left({\frac {1}{d\Omega }}\right)\left({\frac {1}{dA}}\right)={\frac {Uc}{4\pi }}$

The intensity, as a concept, is independent of any detector. In order to determine what a detector sees, you must integrate the intensity over its entire field of view, taking into account the fact that the face of the detector may not be perpendicular to all of the impinging radiation. PAR 05:24, 8 March 2007 (UTC)

I don't have any problem with an intensity that is defined as directed into an infinitesimal solid angle. The problem I have is that many authors define the intensity as the total flux through an area with no restriction to an infinitesimal solid angle. The radiometry article gives a table listing different radiometry quantities and SI units. The discussion here is basically wether spectral intensity refers to spectral radiance (I) or spectral irradiance (R). As PAR points out, you integrate I to calculate R, but you must take into account the angle the face of the detector makes with the radiation. That is exactly how I got the formulas above.

Plank's law really refers to spectral radiant energy density. (See his 1901 paper) I think the equation for u should be presented at the top of the page instead of the equation for I. I also think that spectral radiance and spectral irradiance should be given equal treatment, since a given reference is likely to refer to only one or the other and either one may be considered spectral intensity. Finally, shouldn't the variables in this article be consitent with the radiometry article? Cfn137 19:37, 8 March 2007 (UTC)

The equations given for Planck's law at the top of the page come straight from Rybicky & Lightman, a reliable radiative transfer textbook commonly used throughout professional astronomy. Even if it is not the form of the equation that Planck originally developed, it is the form that is most commonly used my astronomers. (I have a lot of professional experience with using Planck's law myself.) If this is true in other branches of physics, then I suggest leaving the equation in its current form. Changing the equation to a historical but uncommonly used form will only confuse people, including my students. Dr. Submillimeter 23:10, 8 March 2007 (UTC)

I've always seen the radiance expressed first, then less commonly, the energy density. Most practical problems need the radiance, especially when using a detector. The question of what does "intensity" mean is a much argued subject, and I think that the terms radiance and irradiance should be used instead, unless one specifically links intensity to one or the other. Also, I agree, the variables here and in the radiometry article should be consistent. PAR 06:19, 9 March 2007 (UTC)

I think that spectral radiance and spectral irradiance should both be in the article. I was quite confused when the formulas presented disagreed with all of my source texts with no apparent reason. Possibly some of this discussion should be

included as well. naturalnumber 06:22, 6 October 2007 (UTC)

I agree with that, going to change this now. Han-Kwang (t) 06:34, 30 August 2008 (UTC)

Just a little note

This is just a minor thing, but:

The article mistaknely states $I(\nu )d\nu =I(\lambda )d\lambda ,$

In actuality it is $I(\nu )d\nu =-I(\lambda )d\lambda ,$

The reason for this is that when the frequency increases, the wavelength decreases, so the minus sign must be taken into account. —Preceding unsigned comment added by 152.78.121.124 (talk • contribs) 14:50 04 June 2007

Neither is correct. What is true is that integrating $I(\nu )d\nu$ over any given range of frequencies produces the same radiance within that range as integrating $I(\lambda )d\lambda ,$ over the corresponding range of wavelengths. It is customary when integrating to go from small to large. This convention automatically handles the sign issue by interchanging the lower and upper limits when passing from frequency to wavelength. I've rewritten the relevant portion of the lead accordingly, hope it makes sense. --Vaughan Pratt (talk) 00:22, 16 August 2008 (UTC)

Discussion merged from different talk pages

From User talk:66.167.26.233:

Regarding your edits to Planck's law: I think you have misunderstood the term radiance. I have undone your edits, as you appear to be using radiance and perhaps radiant intensity in a manner inconsistent with their formal definitions.--Srleffler (talk) 18:24, 21 January 2008 (UTC)

The symbol in the equations is I which stands for intensity and the units for I(lambda) are W/nm*steradian, so I expect that the accompanying discussion *should* be about differential(flux) per differential(solid-angle)and bandwidth. Flux within a solid-angle is .. well the definition of intensity, per the SI fundamental unit candela in photometry, among other applications of the word (see ANSI/IESNA RP-16, Nomenclature for Illumination Engineering.) I understand that I may have removed "radiance" when I should have been removing "radiant flux" - but quite simply the discussion is .. unclear at best! The symbol I and word intensity are misused (if one means radiance then do use R, and do not say intensity or use I) and most particularly, the units in the equation did not balance (no steradians in any of the elements listed for the right side of the equation, and the infinitesimal mentioned was *area*) and the confusing reference to pi (which "carrys" the steradian unit and is not just a numerical factor) all needed some refinement. Still does - I see one mention of infinitesimal solid angle (I recall adding it in two places) remains, but the infinitesimal area that is part of the definition of radiance is now missing entirely. Really the discussion could be about spectral hemispherical radiation, spectral intensity *or* spectral radiance - but right now it reads like an unfortunate hodge-podge of all three - and dimensional analysis indicated the discussion was (and apparently still is) flawed. I would rather correct the page once than correct all my students' homeworks and then the page as well - but I can now use it as an example of the errors found in plain sight when one uses dimensional analysis to check what is presented. —Preceding unsigned comment added by 66.167.26.233 (talk) 03:34, 23 January 2008 (UTC)

While there may be some confusion present in the article, jumping to conclusions and making changes based on assumptions is how articles end up that way. While I agree that it is odd to use the symbol I for radiance, you should certainly not have assumed it meant intensity. Symbols mean what they are defined to mean, and nothing else. The article specified the equation was for radiance, so that is the definition of I in this article. Further down the article did call it intensity, but in that case it explicitly gave the dimensions of I as energy/time/area/solid angle/frequency. I have corrected the text there from "intensity" to "radiance".

Some of the confusion arises from the fact that what we call "radiance" in optics is commonly called "intensity" in astrophysics, contrary to usage of that term in every other field I know of. The reference cited for the first two equations is an astrophysics text. I have checked the reference, and the equation is transcribed correctly, although the symbol I has been used in place of the text's B. From the accompanying graph in the textbook, B_ν has units erg·s^-1·cm^-2·Hz^-1·sr^-1, making it clearly radiance not intensity (in standard optics terminology).

Regarding dimensional analysis: it is an important tool, but do remember that steradians are dimensionless. They are not "carried by" the pi, although the pi may be required to make the equation correct depending on the system of units chosen. --Srleffler (talk) 05:23, 23 January 2008 (UTC)

Spectracalc link

I don't have that big a problem with the Spectracalc link. It is an informative link that also allows calculation of black body radiation on-line. The page that is linked allows back-button use, but when you go to the site, it throws up a new page, so no back button. I'm not sure there is a dark motive there. Yes, it is a commercial site, but why is that so offensive? Commercial sites are not prohibited, as long as they add knowledge and depth to the Wikipedia article. Spamming is definitely bad, but this is not a spam link. PAR (talk) 13:24, 20 March 2008 (UTC)

True, I've just tried it and back linked to here okay. --Michael C. Price ^talk 13:29, 20 March 2008 (UTC)

So, could you put it back in? I would do it myself, but I'd better leave it to you all to reinsert, since there was some concern about commercial sites and I have some affiliation with www.spectralcalc.com. Anyway, if you think it's OK, go ahead and reinsert it. mm (talk) 17:29, 31 March 2008 (UTC)

The curve of blackbody radiation

According to figure 2.4 page 57,Elementary Modern Physics, for Atam P.Arya,the curve you have used belongs to frequency (and not wavelength) versus intensity of radiation. The curve must be reversed

Quite right, Arya needs to reverse his curve to show the steep part on the left, not the right. Also his formula 8 pi h nu^3/c^3 doesn't agree with the article either for some reason. But his figure 2.5 refining figure 2.4 looks fine. --Vaughan Pratt (talk) 01:35, 17 August 2008 (UTC)

Where does Planck's law peak?

According to the article Planck's law peaks at hν = 2.82kT, citing Kittel. According to Wien's displacement law it peaks at 4.965114231744276kT. The 2.82 coefficient can't be right: sunlight peaks at 500 nm or ν = 600 terahertz, which would make the Sun's temperature 10200 K, about 75% higher than it actually is. What's Kittel's justification for 2.82? --Vaughan Pratt (talk) 17:47, 12 August 2008 (UTC)

Perhaps the difference is due to looking at the Planck distribution asa function of wavelength and frequency. If you look at the emitted power as a function of wavelength, you are considering the emitted power per unit wavelength interval. You could e.g. consider the power emitted in the range between lambda and lambda + 0.01 nm as a function of lambda. This is not equivalent to looking at the emitted power as a function of the frequency while fixing some frequency interval, because d lambda = -c/f^2 df.

Anyway, this is easy to check (but I don't have much time right now). Count Iblis (talk) 19:46, 12 August 2008 (UTC)

Right, I realized this myself just now. In fact Wien's displacement law gives both constants, which arise as values of x making x/(1-exp(-x)) = 3 (frequency) or 5 (wavelength). This difference should be pointed out in the article.

I suppose a more neutral notion of peak would be that of the function $\left({\frac {\nu }{\lambda }}\right)^{2}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$, namely where x/(1-exp(-x)) = 4, which would be 3.9206905. For the Sun this peak would be at 635 nm, counterintuitively red given that we see a more yellow sun. Perhaps the yellow appearance results from our picking up green more strongly than red. --Vaughan Pratt (talk) 21:04, 12 August 2008 (UTC)

Another neutral notion would be the median radiance, namely the frequency (and also wavelength) with equal radiance above and below. This would be at x = 3.50302, where the Sun is even redder at 710.6 nm, consistent with the fact that over half the Sun's radiation is infrared. Which peak is the applicable one is evidently heavily application dependent. --Vaughan Pratt (talk) 22:31, 12 August 2008 (UTC)

Amusing, but erroneous statistic

The cute bit about 1 photon/m^2 per thousand years for a room temp blackbody is, of course, wrong. Check with online calculator: http://www.spectralcalc.com/blackbody_calculator/blackbody.php using 0.38 um to 0.75 um for visible band, and you get 0.00388496 photons/s/m^2/sr, or 0.012205 photons/s/m^2, which works out to 1 photon every 82 seconds, nowhere close to 1000 years. —Preceding unsigned comment added by 74.10.115.66 (talk) 19:44, 25 August 2009 (UTC)

Utterly incomprehensible

No real encyclopedia would have such an abstruse article written solely by and for physicists. Even an educated and intelligent layman will get absolutely nothing from this "article." —Preceding unsigned comment added by 70.245.158.192 (talk) 15:13, 26 May 2008 (UTC)

The spectral energy density equations have different units - which is quite confusing. —Preceding unsigned comment added by 79.121.56.99 (talk) 12:17, 24 October 2009 (UTC)

$I(\nu )$ and $I(\lambda )$

There must be a mistake in Planck's law here. These are the two reasons:

1. Seconds cancle out => impossible for an equation determining a power
2. When drawn in Excel the peak doesn't correspond to Wien's displacement law

I didn't have the time to verify it all the way through but it seems as though it should be

$I(\nu )=2h\nu ^{5}/c^{3}/(e^{h\nu /kT}-1)\,$

This way it corresponds to equations I have found that stated lamda instead of the frequency and recalculating. It also corresponds to Wien's law like this. As I said, no guarantee for what I say here and I won't be able (or willing to invest the time) to redo the derivation.

Seems like there is work needed on that derivation. I hope someone does it correctly because it will look really odd, if I am going to change the equation without touching the derivation.

Thomas

Thomas - don't change it. You forgot to transform the differential $d\nu$. In other words, you don't want to say

$I(\nu )=I(\lambda )\qquad \mathbf {(WRONG)}$

you want to say:

$I(\nu )d\nu =I(\lambda )d\lambda \qquad \mathbf {(RIGHT)}$ (to within sign - PAR 2007-10-12)

Since $\lambda \nu =c$, you have $d\lambda =-cd\nu /\nu ^{2}$ and $d\nu =-cd\lambda /\lambda ^{2}$. If you redo the math, and realize that the signs on the differentials don't matter, you will see that everything is ok. PAR 16:23, 31 July 2006 (UTC)

Ok I need to register this request: The first two equations don't jive in terms of (frequency) and (wavelength) because they are connected by nu x lambda = c but one is ^3 and the other ^5. That could use some clearing up at the get-go I think. imho. imnvho.

Please read above discussion. I put a clarification in, I hope its clear. PAR 06:16, 15 March 2007 (UTC)

I got confused by the same thing. Thank's to PAR.

But I was wondering.. isn't it clearer if we put something like this?:

$dI(\nu ,T)={\frac {2h\nu ^{3}d\nu }{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}.$

$dI(\lambda ,T)={\frac {2hc^{2}d\lambda }{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}.$

In this way, dI represents an actual intensity, and its units are W/(m^2·s). And all we have to do to obtain the intensity of a frecuency (or wavelength) band is to integrate dI.

Skiel85 23:15, 11 November 2007 (UTC)

People deal with intensity, not the derivative of intensity. That is a useful equation, but it follows immediately when the relationship between ν and λ is understood. PAR 05:35, 12 November 2007 (UTC)

I think using I(ν,T)dν instead of I(ν,T) would be the best solution, since the inexperienced readers may easily forget about dν (as the one above). —Preceding unsigned comment added by 83.31.164.76 (talk) 19:35, 10 May 2009 (UTC)

Link to a better version of Planck's 1901 paper

I am about to change the link for this paper from http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html to http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf . The former has a number of problems, such as the HTML text using "q" or in one case "0" instead of theta. The latter is a PDF file and is based on the former, but has this and some other corrections, plus some German next not in the former. Robin Whittle (talk) 02:56, 27 February 2009 (UTC)

Unless the first link is really bad, I suggest leaving it in the article as well as adding the better link. --Michael C. Price ^talk 05:05, 27 February 2009 (UTC)

In the article here is a reference (8) to a paper by Helge Kragh summarising the doubtful history of Planck's law, the link http://www.physicsweb.org/articles/world/13/12/8/1 is dead, I think now here http://physicsworld.com/cws/article/print/373. I would change it myself but I am not sure how to do it since there is only a reference in curly brackets which looks a bit like this {){reflist})} with which I am not familiar (I guess it a link to some emebedded software). Will someone else make the change or show me how to, please?--Damorbel (talk) 13:29, 16 March 2009 (UTC)

Is the partition function defined correctly?

How can the partition function depend on n_1,n_2 and n_3? As far as I understand, the partition function is supposed to describe the entire system of photons, not any particular state? —Preceding unsigned comment added by 130.230.131.53 (talk) 22:54, 17 May 2009 (UTC)

Edit: Yes it is. I missunderstood. —Preceding unsigned comment added by 130.230.131.53 (talk) 23:09, 17 May 2009 (UTC)

"nu" is not "vee".

If this were USENET or some other ASCII-only medium, then this substitution would be understandable. But we have LaTeX here in Wikipedia. There is no need to, and there is potential confusion from, and it is aesthetically displeasing to make this substitution. If no one objects, in a couple of days I will replace the v with $\scriptstyle \nu$. 74.104.160.199 (talk) 05:40, 16 September 2009 (UTC)

I think it uses nu just about everywhere (and anywhere it doesn't, please go ahead and fix) - but in the default Wikipedia font, an italic v looks very much like an italic nu, and the latex nu looks like a v. (''v'', ''ν'', <math>\nu</math> gives v, ν, $\nu$). Djr32 (talk) 07:12, 16 September 2009 (UTC)

unclear text

"This function represents the emitted power per unit area of emitting surface, per unit solid angle, and per unit frequency."

I bet this means something to a physicist, but not to me. Does "per unit area" mean "per 1 square meter", "per unit solid angle" mean "per 1 steradian", and "per unit frequency" mean "per Hz" ?

75.226.155.234 (talk) 04:19, 21 November 2009 (UTC)

Specifying constants

This article writes down formulae without specifying, or giving values for the constants used in these formulae. Although, if one knows something of physics, one may make an educated guess as to the meaning of the formulae, the purpose of an encyclopedia article should be to explain things to non-physicists. Moreover, the fact of elementary mistakes of exposition does not inspire confidence in the competence of the author. Please would an expert edit the article. —Preceding unsigned comment added by 91.111.95.41 (talk) 11:30, 21 December 2009 (UTC)

Disambiguation: Factor of Pi, units, and Graph Scaling

In the lexicon of Radiometry, the equation I` at the top of this article (versus wavelength) is indeed called "Spectral Radiance", and should have a symbol of "R" (or perhaps "B" for brightness). The units (power per unit area, per unit solid angle, per unit wavelength) are indeed correct, but "per unit area" refers to the source, not an observing "detector". This is why the symbol I or I` (for intensity, or Irradiance) is a very confusing choice. In fact, "Intensity" is a subjective term, without universally accepted rigor.

The factor of Pi enters into the equation when one calculates Total spectral radiance into the forward hemisphere, assuming a Lambertian (cosine) distribution. Integral of cosine over the differential solid angle (which has a factor sine) from normal to 90 degrees is Pi. The units are then just power per unit (source) area, per unit wavelength. This is the form presented by Planck. Total power spectral density emitted from a blackbody surface.

The graph on the right of the article is supposed to be a graph of the energy spectral density. You can achieve the proper scaling by multiplying I` by 4Pi/c. However, the plotted units are incorrect. The author meant to show kJ/m^4, not kJ/nm. First of all, this is a silly way to present Planck's law of Blackbody radiation. Secondly, one should attempt to preserve the wavelength units independent of surface area, and not combine them into an SI factor. Thirdly, I think it's totally meaningless, as one should be talking about power density, not energy density. I think the author should have used I(nu) not I`(lambda) times 4pi/c, and then the time units would be preserved.

The proper graph should show spectral radiance of a blackbody radiator, as presented by Planck. That is, with the factor of Pi included. Astrophysicists don't seem to like this presentation form, as they try to compare stellar material to a black-body radiator which emits into all 4pi steradians... I've seen some texts just flat out omit the scaling and just show "arbitrary units". It's a bit incomplete for my taste. —Preceding unsigned comment added by Robertbuckles (talk • contribs) 17:02, 2 March 2010 (UTC)

Changing 280°K to 288°K in the percentiles table

Since the conventional temperature for Earth is 288°K or 15°C I'm going to warm up the percentiles table by 8°C to reflect this. For anyone who had come to rely on the 280°K table for whatever reason I've made a copy of it here. --Vaughan Pratt (talk) 22:57, 22 May 2010 (UTC)

Percentile	.01%	.1%	1%	10%	20%	25.0%	30%	40%	41.8%	50%	60%	64.6%	70%	80%	90%	99%	99.9%	99.99%
280°K planet $\lambda$ (µm)	3.25	3.95	5.2	7.85	9.55	10.3	11.1	12.8	13.1	14.7	16.9	18.2	20.0	24.5	33.5	81.7	184	405

More intuitive form

Plancks law:

$I(\nu ,T)={\frac {2hf^{3}}{c^{2}}}{\frac {1}{e^{\frac {hf}{kT}}-1}}$

This can be written $I(\nu ,T)={\frac {2kT}{\lambda ^{2}}}{\frac {hf}{kT}}{\frac {1}{e^{\frac {hf}{kT}}-1}}$ This seems more intuitive to me. λ2 would be the area that a single photon interacts with

For comparison we have the Maxwell–Boltzmann distribution: $f(v)={\sqrt {{\frac {2}{\pi }}{\frac {m}{kT}}}}\,{\frac {mv^{2}}{kT}}{\frac {1}{e^{\frac {-mv^{2}}{2kT}}}}$

Here is the Bose–Einstein distribution:

$n_{i}={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/kT}-1}}$

Just granpa (talk) 19:15, 30 November 2010 (UTC)

The power 2 is right, but it is a specific radiative intensity, not a flux (density)

The power 2 is right, according to Planck (at his equation 300 on page 176 of the translation) and other textbooks. But is is a specific radiative intensity, not a flux (density per unit area); the other arguments of the specific radiative intensity are not made explicit because the field is isotropic and homogeneous.Chjoaygame (talk) 12:33, 21 December 2010 (UTC)