Volume 11 (2015)

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“The Theory of Heat Radiation” Revisited: A Commentary on the Validity of Kirchhoﬀ’s Law of Thermal Emission
and Max Planck’s Claim of Universality
Pierre-Marie Robitaille1 and Stephen J. Crothers2
1Department of Radiology, The Ohio State University, 395 W. 12th Ave, Columbus, Ohio 43210, USA 2Queensland, Australia
E-mails: robitaille.1@osu.edu, steve@plasmaresources.com
Aﬃrming Kirchhoﬀ’s Law of thermal emission, Max Planck conferred upon his own equation and its constants, h and k, universal signiﬁcance. All arbitrary cavities were said to behave as blackbodies. They were thought to contain black, or normal radiation, which depended only upon temperature and frequency of observation, irrespective of the nature of the cavity walls. Today, laboratory blackbodies are specialized, heated devices whose interior walls are lined with highly absorptive surfaces, such as graphite, soot, or other sophisticated materials. Such evidence repeatedly calls into question Kirchhoﬀ’s Law, as nothing in the laboratory is independent of the nature of the walls. By focusing on Max Planck’s classic text, “The Theory of Heat Radiation’, it can be demonstrated that the German physicist was unable to properly justify Kirchhoﬀ’s Law. At every turn, he was confronted with the fact that materials possess frequency dependent reﬂectivity and absorptivity, but he often chose to sidestep these realities. He used polarized light to derive Kirchhoﬀ’s Law, when it is well known that blackbody radiation is never polarized. Through the use of an element, dσ, at the bounding surface between two media, he reached the untenable position that arbitrary materials have the same reﬂective properties. His Eq. 40 (ρ = ρ′), constituted a dismissal of experimental reality. It is evident that if one neglects reﬂection, then all cavities must be black. Unable to ensure that perfectly reﬂecting cavities can be ﬁlled with black radiation, Planck inserted a minute carbon particle, which he qualiﬁed as a “catalyst”. In fact, it was acting as a perfect absorber, fully able to provide, on its own, the radiation sought. In 1858, Balfour Stewart had outlined that the proper treatment of cavity radiation must include reﬂection. Yet, Max Planck did not cite the Scottish scientist. He also did not correctly address real materials, especially metals, from which reﬂectors would be constructed. These shortcomings led to universality, an incorrect conclusion. Arbitrary cavities do not contain black radiation. Kirchhoﬀ’s formulation is invalid. As a direct consequence, the constants h and k do not have fundamental meaning and along with “Planck length”, “Planck time”, “Planck mass”, and “Planck temperature”, lose the privileged position they once held in physics.

. . . That the absorption of a particle is equal to its radiation, and that for every description of heat.
Balfour Stewart, 1858 [1]
1 Introduction
Seldom does discovery bring forth scientiﬁc revolution [2]. In this regard, there can be no greater exception than Max Planck’s [3] introduction of the quantum of action, at the beginning of the twentieth century [4, 5]. Within “The Theory of Heat Radiation” [5] Planck outlined the ideas which gave life both to this revolution and to the concept that fundamental constants existed which had universal signiﬁcance throughout nature. The pillars which supported his ideas included: 1) Kirchhoﬀ’s Law of thermal emission [6, 7], 2) the irreversability of heat radiation, and 3) the adoption of dis-

crete states.∗ He utilized Kirchhoﬀ’s Law not only to assist in the derivation of his equation, but to infer universality. Max Planck concluded that all cavities, irrespective of experimental evidence, would eventually become ﬁlled with blackbody, or normal, radiation. He argued that, if a cavity did not contain black radiation, the cause was a lack of thermal equilibrium, which could be easily rectiﬁed by the introduction of a minute particle of carbon [8]. For Max Planck, as for his teacher Gustav Kirchhoﬀ [9], cavity radiation was independent of the nature of the enclosure. In reality, such ideas were not supported by experiment, as arbitrary cavities do not contain black, or normal, radiation. By applying his law to all cavities, the father of quantum theory detached his equation from physical reality itself. In truth, Planck’s equation was only valid for laboratory blackbodies constructed from highly
∗The Theory of Heat Radiation is readily available online [5].

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absorbing materials. As a direct consequence, Planck’s equation was never
linked to a particular physical process and he did not provide physics with a cause for thermal emission. In fact, Kirchhoﬀ’s Law prevented him from advancing such a link [8, 10]. The exact nature of the oscillators responsible for thermal radiation could not be identiﬁed. Planck emphasized that [5, § 111],
“. . . to attempt to draw conclusions concerning the special properties of the particles emitting the rays from the elementary vibrations in the rays of the normal spectrum would be a hopeless undertaking”.
Studying Planck’s classic text, the reader is eventually brought to the equation which governs spectral energy density u [5, Eq. 275],

8 π h ν3 1

u=

c3

e hν kT

, −1

(1)

wherein ν, c, h, k and T represent the frequency of interest, the speed of light,∗ Planck’s constant, Boltzmann’s constant, and absolute temperature, respectively. The validity of this equation appears to have been established for blackbodies; namely those specialized heated cavities whose interior is always lined with good absorbers over the frequency of interest, such as graphite, soot, carbon black, or other specialized materials (see [8] and references therein). Max Planck recognized that blackbodies were complex devices, as the data provided for his analysis had been obtained by some of the premier experimentalists in Germany [11–13].
He relied on the work of Rubens and Kurlbaum [11,13] to secure the data which led to Eq. 1. In this regard, it is important to note the elaborate experimental setup used [11, 13]. It was very far from a simple cavity. These results made use of “the method of residual rays”, a process which actually took place well beyond the conﬁnes of the cavity [11, 13]. Repeated reﬂections were supported by using crystals of quartz, ﬂuorite, rocksalt, and sylvine, each for a given frequency of interest [11, 13]. The desired data points could only be obtained with an apparatus used to select the frequency of interest at the proper intensity.
In themselves, such extreme experimental methods conﬁrmed that not all enclosures were ﬁlled with black radiation. Surely, if arbitrary cavities contained black radiation, there should have been no need for the use of these sophisticated approaches [13].
In this regard, it is also interesting to note that when faced with non-compliant experimental facts, scientists often invoke the inability to reach thermal equilibrium. This is especially true when cavities are constructed from materials with a low emissivity. Such arguments are not reasonable, given

∗The United Nations has declared that 2015 will be the “Year of Light”.

the speed of light and the relative ease of maintaining temperature equilibrium in metallic objects through conductive processes. Laboratory ﬁndings do not support Planck’s position relative to Kirchhoﬀ’s Law.
Clearly, real blackbodies were much more than simple arbitrary cavities [11–13]. Yet, Max Planck believed with certainty in the universality of Kirchhoﬀ’s Law. It is this aspect of Planck’s work which must be carefully considered. For if it holds true, then Eq. 1 continues to have far-reaching consequences. It can be applied to any thermal spectrum, whether on Earth in the laboratory, or within any astrophysical context, provided of course, that thermal equilibrium can be demonstrated.† However, if Kirchhoﬀ’s Law can be shown to be false, then Planck’s equation, while still valid for laboratory blackbodies, loses all universal signiﬁcance [8, 10, 14– 19].
It could no longer be used indiscriminately outside of the laboratory, at least if the observer could not ensure that the source of the observed spectrum originated from a known solid. Hence, all applications of Planck’s law in astronomy would very likely constitute violations of its required setting. In addition, the fundamental nature of Planck’s constant, Boltzmann’s constant, and of “Planck length”, “Planck time”, “Planck mass”, and “Planck temperature” would forever be lost. All would have ordinary signiﬁcance. They would be no more fundamental for physics than the mile versus the kilometer. Everything simply becomes a question of the scale physics chooses to select, rather than scales being imposed upon mankind by nature itself. Consequently, Max Planck’s conclusion that Eq. 1 could be applied to all arbitrary cavities had great implications.
It remains an experimental fact that good reﬂectors, such as silver, are never utilized to construct blackbodies, in direct contradiction to Kirchhoﬀ’s claim that cavity radiation is independent of the nature of the walls from which it is comprised. Silver walls would prefer to increase their temperature when confronted with an inﬂux of heat, such as that typically used to drive blackbodies in the laboratory (see [8] and references therein). They would not easily maintain their temperature while building a radiation ﬁeld within a cavity using reﬂection (see [19] for a discussion). It has also not been established that cavities constructed from walls of low emissivity can contain Lambertian emission. These are some of the reasons why Kirchhoﬀ’s Law fails.
As such, how could this law have survived for so long? In order to answer this question, it is important to revisit both the experimental and theoretical foundations which brought forth Kirchhoﬀ’s Law. For this exposition, the journey will begin with the experiments of Balfour Stewart [1] in keeping with the reality that experiments [10], not solely theory, govern the laws of physics. At this point, the work of Gus-
†There must be radiative equilibrium, no temperature changes, and no conduction or convection taking place in the system of interest.

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tav Kirchhoﬀ [6,7] must be discussed, especially as related to his treatment of reﬂection. Then, ﬁnally, a detailed analysis of Max Planck’s derivation of Kirchhoﬀ’s Law, as outlined in “The Theory of Heat Radiation” [5], will be presented. It will be demonstrated that Planck’s derivation suﬀers, not only with minor problems, but with signiﬁcant departures from experimental reality.
2 Balfour Stewart and the Law of Equivalence
Balfour Stewart was a Scottish physicist. In 1858, one year before Kirchhoﬀ’s Law was proposed [6, 7], Stewart published what can be considered one of the most important works in the history of thermal emission [1]. His analysis of radiation was entirely based on experimental grounds. Hence, he never claimed, as law, principles which could not be proven experimentally [1]. Using actual measurements with material plates made of various substances, Stewart formulated the Law of Equivalence, ﬁrst in §19 of his work [1],
“The absorption of a plate equals its radiation, and that for every description of heat”,
and then in §33 [1],
“That the absorption of a particle is equal to its radiation, and that for every description of heat”.
At the same time, he addressed cavity radiation, arriving at a general principle by considering a single theoretical argument. For Stewart, this principle did not rise to the level of a law, precisely because the conclusion had not been experimentally veriﬁed. He treated cavity radiation purely from a theoretical perspective and highlighted that the radiation which should come to ﬁll the cavity resulted from the radiation emitted, in addition to the radiation which had been built up by reﬂection. The arguments advanced, being theoretical and not experimental, prevented him from formally proposing a new law with respect to cavity radiation. Rather, he spoke of a general principle [1],
“Although we have considered only one particular case, yet this is quite suﬃcient to make the general principle plain. Let us suppose we have an enclosure whose walls are of any shape, or any variety of substances (all at a uniform temperature), the normal or statical condition will be, that the heat radiated and reﬂected together, which leaves any portion of the surface, shall be equal to the radiated heat which would have left that same portion of the surface, if it had been composed of lampblack. . . Let us suppose, for instance, that the walls of this enclosure were of polished metal, then only a very small quantity of heat would be radiated; but this heat would be bandied backwards and forwards between surfaces, until the total amount of radiated and re-

ﬂected heat together became equal to the radiation of lampblack”.
The problem is that good reﬂectors do not readily emit radiation. As such, in order to drive the reﬂection term, one must try to inject heat into the walls of these cavities, while hoping that additional photons will be produced. But, if one attempts to pump heat into their walls using conduction, for instance, the temperature of the walls can simply increase [18, 19]. Nothing dictates that new photons can become available for the buildup of the reﬂective term, while maintaining the cavity at the same temperature. One can infer that good reﬂectors can easily move away from the temperature of interest and fall out of thermal equilibrium. As a result, they cannot easily be ﬁlled with the desired radiation, even if theoretical arguments suggest otherwise. In the real world, nothing is independent of the nature of the materials utilized.
Stewart recognized that, if one could “drive the radiation” in a cavity made from arbitrary materials, by permitting the slow buildup of reﬂected radiation, the interior could eventually contain black radiation. The argument was true in theory, but not demonstrated in practice. Stewart remained constrained by experimental evidence. The situation could not be fully extended in the laboratory.
From Balfour Stewart, we gain three important lessons. First, he correctly supplied the Law of Equivalence: Given thermal equilibrium, the emission of an object is equal to its absorption. Second, he outlined the principle that cavity radiation can become black, in theory, in the event that the reﬂective term can be driven. Third, and most importantly, he did not advance a new law of physics without experimental conﬁrmation.
3 Gustav Kirchhoﬀ: Physics from Theory Alone
Soon after Balfour Stewart formulated the Law of Equivalence [1], Gustav Kirchhoﬀ published his law of thermal emission [6, 7]. Almost immediately, the work was translated into English by F. Guthrie [7] and Kirchhoﬀ’s paper was then re-published in the same journal where Stewart had presented his law the year before. At this point, a battle ensued between Kirchhoﬀ and Stewart.∗ The problem centered on Kirchhoﬀ’s attempt to dismiss Stewart’s priority claims for the Law of Equivalence. Kirchhoﬀ did so by arguing that Stewart had not brought forth suﬃcient theoretical support for his law. As for Stewart, he believed that the law had been experimentally proven, even if his mathematical treatment might have lacked sophistication.
In any event, Kirchhoﬀ’s paper went much beyond the Law of Equivalence. Thus, Stewart, who had outlined the principle that arbitrary cavities might come to hold black radiation, did not insist that this was always true [1]. Conversely, Kirchhoﬀ formulated this conclusion as a law of physics, but
∗An excellent treatment of this incident has already been published [20] and one of the authors has also addressed the issue [8].

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he did so without recourse to a single experiment. Both of his proofs were theoretical [6, 7].
To begin his investigation, Kirchhoﬀ, in the ﬁrst section of his text, deﬁned a blackbody as follows [7, § 1]:

“This investigation will be much simpliﬁed if we imagine the enclosure to be composed, wholly or in great part, of bodies which, for inﬁnitely small thickness, completely absorb all rays which fall upon them”.

Note the emphasis on the absorption by an element of inﬁnitely small thickness. The contrast between Kirchhoﬀ’s deﬁnition of a blackbody and that adopted by Max Planck was profound [5], as will be discovered below. In any event, in §3 of his classic paper [7] Kirchhoﬀ presented his law as follows,

“The ratio between the emissive power and the absorptive power is the same for all bodies at the same temperature”.

In § 13, he explicitly wrote the following form,

E

= e.

(2)

A

Kirchhoﬀ eventually set A = 1 [7, § 3]. In modern notation,∗

one could express Kirchhoﬀ’s Law as follows:

Eν = f (T, ν),

(3)

αν

where f (T , ν) corresponds to the right side of Eq. 1 above, as ﬁrst deﬁned by Max Planck [4, 5]. In §17 of his classic paper [7], Kirchhoﬀ outlined his law as follows,

“When a space is surrounded by bodies of the same temperature, and no rays can penetrate through these bodies, every pencil in the interior of the space is so constituted, with respect to its quality and intensity, as if it proceeded from a perfectly black body of the same temperature, and is therefore independent of the nature and form of the bodies, and only determined by the temperature. The truth of this statement is evident if we consider that a pencil of rays, which has the same form but the reverse direction to that chosen, is completely absorbed by the inﬁnite number of reﬂections which it successively experiences at the assumed bodies. In the interior of an opaque glowing hollow body of given temperature there is, consequently, always the same brightness whatever its nature may be in other respects.”

∗Though Kirchhoﬀ speaks of absorptive power, A, he was actually referring to the unitless absorptivity, αν. Conversely, when referring to emissive power, E, he was, in fact, referring to this quantity, even in modern terms. That is, Kirchhoﬀ’s “E” has the same units as his “e” and neither is equal to 1. Kirchhoﬀ, stated that “e” was a universal function and believed that its elucidation was a matter of great scientiﬁc importance.

Relative to Kirchhoﬀ’s formulation, three important concerns must be raised. First, the law becomes undeﬁned in the perfect reﬂector, as αν = 0 under that condition. Planck himself recognized this fact [5, § 48], but might not have exercised proper care relative to its consequences. Second, it is clear that Kirchhoﬀ lacked an accurate understanding of what was happening within his cavity, as an “inﬁnite number” of reﬂections will never amount to absorption. An “inﬁnite number” of reﬂections does not involve the exchange of energy. Conversely, when absorption occurs, energy is exchanged between the ﬁeld in the interior of the cavity and the walls. Third, and the most serious objection to Kirchhoﬀ’s Law, centers upon his improper treatment of reﬂection. One of the authors has previously addressed these problems in detail [16].
In brief, within his ﬁrst proof, Kirchhoﬀ utilized transmissive plates to accomplish the proof, even if blackbody cavities must always be opaque. He addressed transmission by positioning mirrors behind his plates. In so doing, it appeared that Kirchhoﬀ had properly treated reﬂection, because the mirrors did, in fact, reﬂect radiation. However, he had dismissed the possibility that the plates considered could possess diﬀering surface reﬂection [16]. As shall be discovered below, Max Planck committed the same error, when he attempted to formulate Kirchhoﬀ’s Law [5, § 36–38]. In his second proof, Kirchhoﬀ unknowingly permitted the cavity to fall out of thermal equilibrium, depending on the order in which operations were performed (see [16] for a detailed presentation).
It is evident that no valid theoretical proof of Kirchhoﬀ’s Law existed before Max Planck formulated his law of emission (see [21] for an excellent presentation). In fact, physicists continued to argue about a proper theoretical proof for Kirchhoﬀ’s Law until well after Planck’s ideas became accepted [21]. Thus, in search of a proof, those provided by Planck, Hilbert, or Pringsheim may be the most relevant [21]. Yet, the proofs provided by Pringsheim and Hilbert have their own shortcomings [21].† It has even been claimed that, by applying Einstein coeﬃcients to arrive at Planck’s law, physics could dispense with the proof of Kirchhoﬀ’s Law [21]. However, Einstein’s derivation utilized the energy density associated with a Wien radiation ﬁeld, something which could only be found within a blackbody. Surely, Wien had not dispensed with Kirchhoﬀ. In truth, it appears that those concerned with bringing forth a proper proof for Kirchhoﬀ’s Law were never able to reach their goal. The problem of ﬁnding a valid proof, seems to have simply been displaced by “more exciting physics”, as the long sought deﬁnitive formulation of Kirchhoﬀ’s Law could no longer provide suﬃcient interest. The entire issue appears to have come to a slow death, without proper resolution.
It is certain that all theoretical proofs of Kirchhoﬀ’s Law
†The authors have not been able to locate an analysis of the proof advanced by Max Planck within “The Theory of Heat Radiation”.

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will be found to contain signiﬁcant misapplications of experimental facts. The inability to provide a proper proof before the days of Planck [21], has not been easily overcome by some new insight into the nature of materials, after Planck. It remains true that all theoretical proofs of Kirchhoﬀ’s Law suﬀer from one or more of the following: 1) an improper treatment of reﬂection, absorption, or transmission; 2) the invocation of polarized light, when heat radiation is always unpolarized; 3) the use of transmissive materials, when Kirchhoﬀ’s Law refers to opaque enclosures; and 4) the existence of hypothetical objects which can have no place in the physical world.
However, the central proof of Kirchhoﬀ’s Law must always be the one outlined by Max Planck himself (see [5, § 1– 51]), forty years after Kirchhoﬀ [6,7]. For it is upon this proof (see [5, § 1–51]) that Eq. 1 was derived and through which Planck would ultimately attempt to lay the foundation for universality. Hence, it is best to forgo Kirchhoﬀ’s own derivations, as the theoretical validity of Kirchhoﬀ’s Law now rests with Max Planck [5, § 1–51].
4 Max Planck and Departure from Objective Reality
Having held such reverence for Max Planck over the years [3], it is with some regret that the following sections must be composed, outlining his sidestep of known experimental physics in the derivation of Kirchhoﬀ’s Law. Fortunately, in Planck’s case, the validity of his equation is preserved, but only within the strict conﬁnes of the laboratory blackbody. The quantum of action continues to hold an important place in physics. Yet, the loss of universality cannot be taken lightly, as this aspect of Planck’s work was the pinnacle of his career. In fact, above all else, it was universality which Planck sought, believing that he had discovered some great hidden treasure in nature [5, § 164],
“Hence it is quite conceivable that at some other time, under changed external conditions, every one of the systems of units which have so far been adopted for use might lose, in part or wholly, its original natural signiﬁcance. In contrast with this it might be of interest to note that, with the aid of the two constants h and k which appear in the universal law of radiation, we have the means of establishing units of length, mass, time, and temperature, which are independent of special bodies or substances, which necessarily retain their signiﬁcance for all times and for all environments, terrestrial and human or otherwise, and which may, therefore, be described as ‘natural units’ ”.
This was an illusion. With the collapse of Kirchhoﬀ’s Law, there are no “natural units” and all the constants of physics become a manifestation of the scales which the scientiﬁc community chooses.

4.1 Planck’s Derivation of Kirchhoﬀ’s Law: Part I
Throughout his derivation of Kirchhoﬀ’s Law (see [5, § 1– 51]), Max Planck sub-optimally addressed reﬂection, transmission, and absorption. This can be seen in the manner in which he redeﬁned a blackbody, in an array of quotations [5, § 4],
“Strictly speaking, the surface of a body never emits rays, but rather it allows part of the rays coming from the interior to pass through. The other part is reﬂected inward and according as the fraction transmitted is larger or smaller, the surface seems to emit more or less intense radiation”.
For Planck, photons were being released from an object, not because they were emitted by its surface, but simply because they managed to be transmitted throughout, or beyond, its interior. The blackbody became a sieve. Planck stated [5, § 10],
“A rough surface having the property of completely transmitting the incident radiation is described as ‘black’ ”.
Planck continued [5, § 12],
“Thus only material particles can absorb heat rays, not elements of surfaces, although sometimes for the sake of brevity, the expression absorbing surfaces is used.
Note the contrast, with Kirchhoﬀ, which can be repeated for convenience [7, § 1],
“This investigation will be much simpliﬁed if we imagine the enclosure to be composed, wholly or in great part, of bodies which, for inﬁnitely small thickness, completely absorb all rays which fall upon them”.
Planck acknowledged in a footnote that Kirchhoﬀ considered a blackbody as absorbing over an inﬁnitely thin element. He stated [5, § 10],
“In deﬁning a blackbody Kirchhoﬀ also assumes that the absorption of incident rays takes place in a layer ‘inﬁnitely thin’. We do not include this in our deﬁnition.”
With his words, Planck redeﬁned the meaning of a blackbody. The step, once again, was vital to his derivation of Kirchhoﬀ’s Law, as he relied on transmissive arguments to arrive at its proof. Yet, blackbody radiation relates to opaque objects and this is the ﬁrst indication that the proofs of Kirchhoﬀ’s Law must not be centered on arguments which rely upon transmission. Planck ignored that real surface elements must possess absorption, in apparent contrast with Kirchhoﬀ and without any experimental justiﬁcation. Planck would expand on his new concept for a blackbody with these words [5, § 10],

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“. . . the blackbody must have a certain minimum thickness depending on its absorbing power, in order to insure that the rays after passing into the body shall not be able to leave it again at a diﬀerent point of the surface. The more absorbing a body is, the smaller the value of this minimum thickness, while in the case of bodies with vanishingly small absorbing power only a layer of inﬁnite thickness may be regarded as black.”
Now, he explicitly stated that bodies which are poor absorbers can still be blackbodies. Yet, we do not make blackbodies from materials which have low absorptivities, because these objects have elevated reﬂectivities, not because they are not inﬁnite. Planck had neglected the important eﬀects of absorption and reﬂection when formulating his new deﬁnition for a blackbody. This may have consequences throughout physics and astronomy [8, 17, 22].
In the end, Planck’s surface elements must be composed of material particles. Since Planck was a theoretical physicist, he cannot work solely in the vacuum of a mathematical world. His derivations and conclusions must be related to physical reality. Yet, Planck’s treatment had moved away from laboratory experiments with thin plates. These experiments were vital to the development of blackbody radiation science from the days long before Balfour Stewart [1]. Planck stated that [5, § 12],
“Whenever absorption takes place, the heat ray passing through the medium under consideration is weakened by a certain fraction of its intensity for every element of path traversed.”
Clearly, Planck’s element at the “bounding surface”, as will soon be discovered, was an “element of path traversed”. He therefore cannot neglect its absorption. Planck was well aware of this fact [5, § 12]:
“We shall, however, consider only homogeneous isotropic substances, and shall therefore suppose that αν has the same value at all points and in all directions in the medium, and depends on nothing but the frequency ν, the temperature T , and the nature of the medium.”
and again [5, § 32],
“Consider then any ray coming from the surface of the medium and directed inward; it must have the same intensity as the opposite ray coming from the interior. A further immediate consequence of this is that the total state of radiation of the medium is the same on the surface as in the interior.”
Still, at every turn, he attempted to include the eﬀect of transmission, when it had no proper place in the treatment of blackbody radiation, as found in opaque bodies [5, § 14],

“Let dσ be an arbitrarily chosen, inﬁnitely small element of area in the interior of a medium through which radiation passes.”
Planck thereby included the transmissive properties of the element, dσ, though he should have avoided such an extension. In the end, his deﬁnition of a blackbody was opposed to all that was known in the laboratory. Blackbodies are opaque objects without transmission, by deﬁnition. By focusing on transmission, Planck prepared for his move to universality, as will now be discussed in detail.
4.2 Planck’s Derivation of Kirchhoﬀ’s Law: Part II
In the ﬁrst section of his text, leading to his Eq. 27, [5, Eq. 27], Planck chose to formally neglect reﬂection, even though the total energy of the system included those rays which are both emitted/absorbed and those which would have been maintained by driving reﬂection [18, 19]. Such an approach was suboptimal. Planck must have recognized that the reﬂective contributions could eventually be canceled. Perhaps, that is why he simply neglected these terms, but the consequence was that insight was lost. In addition, by adopting this approach, Max Planck explicitly prevented the newcomer to the ﬁeld of thermal radiation from appreciating the crucial importance of reﬂection within cavity radiation, as Balfour Stewart had well demonstrated [1, 18, 19].
In order to properly follow Planck’s work, it is important to recognize his unusual conventions with respect to symbols. Dimensional analysis reveals that even though he spoke of a coeﬃcient of emission (Emissionskoeﬃzienten) and utilized the symbol now reserved for emissivity, ǫν, he was not referring to the emissivity in this instance. Rather, he was invoking the emissive power, E, an entity with units. Conversely, when he spoke of the coeﬃcient of absorption (Absorptionkoeﬃzienten), αν, he was truly referring to the dimensionless absorptivity, as we know it today. Insuﬃcient attention relative to Planck’s notation has, in fact, caused one of the authors to revise some of his previous works [18, 19]. Suﬃce it to note for the time being that, in order to remain consistent with Planck’s notation, the following conventions will now be adopted: The symbol ǫν, will represent emissive power, E, and not emissivity. The symbols αν and ρν will retain their modern meaning and represent dimensionless absorptivity and reﬂectivity, respectively. This is in keeping with Planck’s notation. At the same time, we shall add the symbol ην, in order to deal with dimensionless emissivity, since Max Planck had already utilized the needed symbol when expressing emissive power.∗
∗In § 44, Planck presented Kirchhoﬀ’s Law in the following form [5, Eq. 48],
E A = I = dσ cos θ dΩ Kν dν, where A is actually the unitless absorptivity. Then, in § 45, Planck set A = 1. But, he also set, E = A. In so doing, he removed dimensionality from the emissive power, E.

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At the outset, Max Planck considered the radiation within the interior of an isotropic medium. Inside this material, the total energy emitted from a volume element, dτ, in frequency range of interest, ν + dν, and in time, dt, in the direction of a conical element, dΩ, was given by [5, Eq. 1],

dt dτ dΩ dν 2ǫν,

(4)

which led to [5, Eq. 27],

Kν

=

ǫν . αν

(12)

Note that in this expression, Planck, like Kirchhoﬀ, removed all consideration of reﬂection. Conversely, by combining Eqs. 8 and 10, we obtain that,

from which Planck immediately surmised, by integrating over all directions and frequencies, that the total energy emitted

∞

∞

dt v 8π (ǫν + ρνKν) dν = dt v 8π (αν + ρν) Kνdν. (13)

0

0

corresponded to [5, Eq. 2],

This expression leads to the following relation,

∞

dt dτ 8π ǫν dν.

(5)

0

ǫν + ρνKν = ανKν + ρνKν.

(14)

He then moved to present the same equation, in slightly modiﬁed form in § 25 as,

∞

dt v 8π ǫν dν,

(6)

0

where v now corresponded to the volume element.

But since this element was contained within the

medium of interest, it must also be reﬂecting radiation from

other elements within the medium. That is because, as Bal-

four Stewart correctly highlighted, the total radiated power

measured from a particle is to that portion which was emitted

by the particle itself and that portion which it reﬂected [1].

This reﬂective component corresponds to the reﬂection coef-

ﬁcient, ρν, multiplied by the speciﬁc intensity, Kν, of the radiation leaving the second element, dτ′, positioned at the end of

Planck’s conical section. The proper form of Eq. 4 [5, Eq. 1],

including all of the radiation which leaves the particle, be-

comes,

dt dτ dΩ dν 2 (ǫν + ρνKν) .

(7)

This expression, rather than leading to Eq. 6, results in,

∞

dt v 8π (ǫν + ρνKν) dν.

(8)

0

Similarly, Planck characterized the fate of the radiation which

strikes the volume element, by including only absorption [5,

Eq. 25],
∞

dt v 8π αν Kν dν.

(9)

0

If however, one considers that the radiation incident to the

volume element, v, can be either absorbed or reﬂected, then

Eq. 9 [5, Eq. 25] becomes,

∞

dt v 8π (αν + ρν) Kν dν.

(10)

0

Equating Eqs. 6 and 9, Planck obtained,

∞

∞

dt v 8π ǫν dν = dt v 8π αν Kν dν,

(11)

0

0

If one eliminates the terms involving reﬂection, this expres-

sion immediately leads to Eq. 12 [5, Eq. 27]. More impor-

tantly, since αν + ρν = 1 at thermal equilibrium, then a second

expression, which retains the importance of reﬂectivity, is ob-

tained,

ǫν = (1 − ρν) Kν.

(15)

Since Eq. 14 leads directly to Eq. 12, it now becomes clear why Max Planck chose to ignore the contribution of reﬂection in his derivation. He adopted a physically incomplete picture, but without mathematical consequence, at least in this instance. It could also be argued that Eq. 12 and Eq. 15 do not diﬀer from one another, since at thermal equilibrium 1 − ρν = αν. However, mathematically this is not the case. Eq. 12 becomes undeﬁned when the absorptivity, αν, is set to zero. This is precisely what happens in the perfect reﬂector. Conversely, Eq. 15 is never undeﬁned, as long as the reﬂective term is retained. As such, the prudent course of action for Max Planck might have been to adopt Eq. 15.
At this point, a trivial observation can be easily advanced. As mentioned above, given thermal equilibrium, then 1−ρν = αν. But at the same time, αν = ην. This is the Law of Equivalence, ﬁrst presented by Balfour Stewart [1]. As a result, it can be readily noted that Eq. 15 can be expressed as,

ǫν = ην Kν or Eν = ην Kν,

(16)

which is similar to Planck’s Eq. 26 [5, Eq. 26]. In this case, Kν is given by Planck [5, Eq. 300]. It corresponds to a Planck function multiplied by the square of the index of refraction of the medium. Note what Eq. 16 is stating: The emissive power of an arbitrary cavity at thermal equilibrium is equal to the emissivity of the material which makes up the cavity multiplied by a function. This constitutes a proper and direct contradiction of universality. The nature of the radiation within the cavity becomes dependent on the nature of the cavity itself.
Thus, if the derivation is accomplished while including reﬂection, additional insight is gained. If given the choice, a function which is never undeﬁned, like Eq. 15, must always take precedence over a function which can become undeﬁned,

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like Eq. 12. Then, consider Eq. 16. This relationship is important, because, like the form presented by Kirchhoﬀ (Eq. 2) and Planck (Eq. 12), it is devoid of the consideration of reﬂection. But, when confronted with Eq. 16, it is impossible to conclude that arbitrary cavities contain black radiation.
In this initial treatment, Planck had not yet formally introduced Kirchhoﬀ’s Law. In order to accomplish this feat, he had to explore more than one medium at a time. Nonetheless, in this initial exposition of Planck’s derivation, an important lesson has been learned: it is vital to recognize that the manner in which a result is presented can have a great deal of inﬂuence on its interpretation. Nowhere is this more applicable than in Planck’s formal presentation of Kirchhoﬀ’s Law, as he leads the reader from Eq. 27 to Eq. 42 [5, Eq. 27–42]. It is here that Planck sidestepped experimental reality.

spectrum, can never be interpreted as heat radiation.”
In order to arrive at Kirchhoﬀ’s Law, in § 35–37, Planck placed two diﬀerent homogeneous isotropic media in contact with one another, as illustrated in Figure 1. The whole system was “enclosed by a rigid cover impermeable to heat”. He then considered two arbitrary plane-polarized waves, one from each of the media, incident upon an element of area dσ at the bounding surface of the two media. It can be seen in § 38, that Planck initially endowed this element with differing reﬂectivities, depending on whether the incident rays approached from medium 1 or medium 2. For Planck, both waves underwent reﬂection and refraction. He sidestepped that the ray could be absorbed, a decision vital to his ability to derive Kirchhoﬀ’s law [5, § 9],

4.3 Planck’s Derivation of Kirchhoﬀ’s Law: Part III
Heat radiation is unpolarized, by deﬁnition [23, p. 450]. In § 4 of The Theory of Heat Radiation [5], Planck considered a homogeneous isotropic emitting substance. Any volume element of such a material necessarily emits heat radiation uniformly in all directions. In § 5 Planck admitted that homogeneous isotropic media emit only natural or normal, i.e. unpolarized, radiation [5, § 5]:
“Since the medium was assumed to be isotropic the emitted rays are unpolarized.”
This statement alone, was suﬃcient to counter all of the arguments which Planck later utilized to arrive at Kirchhoﬀ’s Law [5, Eq. 42]. That is because the important sections of Planck’s derivation, namely § 35–37 make use of plane-polarized light. These steps were detached from experimental reality, relative to heat radiation [5, § 35],
“Let the speciﬁc intensity of radiation of frequency ν polarized in an arbitrary plane be Kν in the ﬁrst substance . . . and K′ν in the second substance . . . ”
Planck also stated [5, § 36],
“. . . we have for the monochromatic planepolarized radiation. . . ”
As such, to prepare for his use of polarized light in later sections, Planck resolved, in § 17, the radiation into its two polarized components. However, note that he could have arrived at Eq. 12 [5, Eq. 27] without ever resolving the radiation into its components. Nonetheless, his proof for the universality of Kirchhoﬀ’s Law [5, Eqs. 27–42] depended upon the use of polarized light [5, § 35–37]. Planck utilized polarized light in an isotropic medium, even though he had already recognized in § 5, that such radiation must be unpolarized. He clearly remarked in § 107,
“For a plane wave, even though it be periodic with a wave lying within the optical or thermal

“. . . a discontinuous change in both the direction and the intensity of a ray occurs when it reaches the boundary of a medium and meets the surface of a second medium. The latter, like the former, will be assumed to be homogeneous and isotropic. In this case, the ray is in general partly reﬂected and partly transmitted.”
Planck invoked a small element of area dσ at the boundary of his two contiguous media. This element had no consistent meaning in Planck’s analysis. First, in § 36 and § 42 Planck placed this element in the bounding surface and, in so doing, allocated it properties characteristic of medium 1 on one half and medium 2 on the other. However, in § 43, he placed the element ﬁrmly within the surface of medium 2,
“. . . and falls on the surface element dσ of the second medium.”
Note that Planck had already introduced three causes for objection. First, what exactly was the location of dσ? In reality it must rest in one of the two media. Second, Planck neglected the fact that real materials can possess ﬁnite and differing absorptivities. While these can be ignored within the medium when treating propagation, because of the counter eﬀect of emissivity, they cannot be dismissed at the boundary. Third, the simplest means of nullifying the proof leading to Planck’s Eq. 42, is to use a perfect reﬂector as the second medium. In that case, a refractive wave could never enter the second medium and Planck’s proof fails. The same objection can be raised using any fully opaque material for the second medium (i.e. αν + ρν = 1), as for all of them, τν=0. This would include many materials typically used to construct real blackbodies in the laboratory. Consequently, for his proof of Kirchhoﬀ’s Law, Planck eliminated, by deﬁnition, virtually all materials of interest. In fact, he even excluded the perfect reﬂector, the very material he had chosen to consider throughout much of his text [5].
In § 36 Planck considered a monochromatic planepolarized ray of frequency ν, emitted in time dt. In order to

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Fig. 1: Expansion of Figure 3 in “The Theory of Heat Radiation” [5] depicting the full complement of rays involved in treating the
interaction between two media separated by a “bounding surface” which contained a hypothetical element of interest, dσ. Planck considered
the reﬂective nature of dσ to ascertain whether its reﬂection coeﬃcients were identical depending on whether the incident ray originated
from medium 1, (A), or medium 2, (B). A) Schematic representation of the incident speciﬁc intensity, Kν (plain arrow), at an angle θ,
contained in the conical section, dΩ, of the ﬁrst medium (upper right quadrant) which is reﬂected by the bounding surface into the conical section dΩ in the upper left quadrant and refracted into the conical section dΩ′ of the second medium, at an angle θ′, in the lower left
quadrant. Note that in order to preserve the proper speciﬁc intensities, Kν, in the upper left quadrant, Planck must sum the reﬂected portion of the incident speciﬁc intensity of medium 1, ρνKν, with the refracted portion of the incident speciﬁc intensity of medium 2, (1 − α′ν − ρ′ν)K′ν, depicted in B. This fact is represented by the feathered arrow. However, he neglected to include that part of the speciﬁc intensity in the upper left quadrant was being produced by emission in that direction, ην, by dσ. B) Schematic representation of the incident speciﬁc intensity, K′ν (plain arrow), at an angle θ′,contained in the conical section, dΩ′, of the second medium (lower right quadrant) which is reﬂected by the bounding surface into the conical section, dΩ′, in the lower left quadrant and refracted into the conical section, dΩ, of the ﬁrst medium, at an angle θ, in the upper left quadrant. Note that, in order to preserve the proper speciﬁc intensities, K′ν, in the lower left quadrant, Planck must sum the reﬂected portion of the incident speciﬁc intensity of medium 2, ρ′νK′ν, with the refracted portion of the incident speciﬁc intensity of medium 1, (1 − αν − ρν)Kν, as depicted in A. This fact is represented by the feathered arrow. However, he neglected to include that part of the speciﬁc intensity in the lower left quadrant was being produced by emission in that direction, η′ν, by dσ.

address absorption at the “bounding surface”, as mentioned under the second objection above, the total radiation which was both emitted and reﬂected by an element within the medium of interest (i.e. the incident ray) towards the “bounding surface” must be considered, as illustrated in Figure 2.
Note in this case, that the ray which is approaching the bounding surface will be transformed into three components: 1) that which will be absorbed at the “bounding surface” and then re-emitted in the direction of reﬂection; 2) that which will be reﬂected into the same medium; and 3) that which will be refracted into the other medium. The distinction is important, for Planck inferred that ρν + τν = 1, whereas the correct expression involves ρν+τν +αν = 1.∗ Planck permitted himself to state that τν = 1 − ρν, whereas he should have
∗Note that in §36 Planck referred to frequency dependent reﬂectivity, ρν, but chose to write it simply as ρ. In this case, since he was dealing with the frequency dependent value, the subscripted form will be utilized throughout the presentation which follows. As such, the equations presented by Max Planck will be modiﬁed such that ρ is replaced with ρν in accordance with his description that the term was frequency dependent.

obtained τν = 1 − ρν − αν. Again, this completely prevents further progress towards Kirchhoﬀ’s Law [5, Eq. 42].
Planck considered the reﬂected rays in the ﬁrst medium, of speciﬁc intensity Kν at incidence [5, Eq. 38],

ρν dt dσ cos θ dΩ Kν dν,

(17)

which were augmented by rays of incident speciﬁc intensity K′ν refracted from the second medium [5, Eq. 39],

1 − ρ′ν dt dσ cos θ′ dΩ′ K′ν dν.

(18)

In this setting, the resultant rays in medium 1 consist of com-

ponents from both media, the reﬂected and the refracted rays.

Planck then obtained the following equation, at the end of

his § 36,

Kν K′ν

·

q2 q′2

=

1 1

− ρ′ν , − ρν

(19)

where q and q′ correspond to speeds of light in ﬁrst and sec-

ond media, respectively. He rapidly moved to [5, Eq. 40],

ρν = ρ′ν,

(20)

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convention is to use the electric vector E [24, § 1.4.2]. Planck used the erstwhile magnetic vector convention.
The “angle of polarization” is Brewster’s angle [23, p. 450]. The angle between reﬂected and refracted rays resulting from a given incident ray is then 90o. The reﬂected wave is entirely plane-polarized∗, as shown in Figure 3,

Fig. 2: Schematic representation of the fate of an incident ray, 1, which strikes a bounding surface. The ray will be split into three components: 1) the reﬂected ray, ρν; 2) the refracted ray, τν; and 3) that portion of the ray which is ﬁrst absorbed, αν, then immediately re-emitted, ην, in order to preserve energy balance, in the direction of the reﬂected ray (αν = ην). Thus, it is possible to describe this problem mathematically as 1 = ρν + τν + αν.

The result was stunning. Max Planck had determined that the reﬂectivities of all arbitrary media were equal. Yet, he attempted to dismiss such a conclusion by stating relative to Eq. 20 [5, Eq. 40]:
“The ﬁrst of these two relations, which states that the coeﬃcient of reﬂection of the bounding surface is the same on both sides, is a special case of a general rule of reciprocity ﬁrst stated by Helmholtz.”
Planck provided for the element of the bounding surface two separate coeﬃcients of reﬂection. These must, in fact, correspond to those of the media utilized. Planck has already stated in § 35 that
“. . . let all quantities referring to the second substance be indicated by the addition of an accent.”
Consequently, ρ and ρ′ can only take meaning with respect to the media under consideration. Thus, how did Planck possibly reach the conclusion that these values must be equal? At the onset in Eq. 19 [5, § 35], Planck sought to force ρν = ρ′ν, in general, by ﬁrst making ρν = ρ′ν = 0, in particular. To accomplish this feat, he considered rays that were,
“polarized at right angles to the plane of incidence and strike the bounding surface at the angle of polarization” [5, § 37].
Again, such rays could never exist in the context of heat radiation [23, p. 450].
The “plane of incidence” is that containing the unit normal vector from the surface of incidence and the direction of the incident ray. There are two natural ways by which the orientation of an electromagnetic wave can be ﬁxed; by the electric vector E or the magnetic vector B. Contemporary

Fig. 3: Schematic representation of Brewster’s Law. The dots correspond to the electric vector perpendicular to the page, whereas the double-headed arrows represent the electric vector in the plane of the page. An unpolarized, or arbitrarily plane-polarized, incident ray (upper right quadrant), strikes a surface at an angle of incidence, θB, corresponding to the Brewster’s angle, or the angle of polarization. The reﬂected ray, depicted in the upper left quadrant will be entirely plane-polarized in such a way that it has no component of its electric vector in the plane of incidence. The transmitted ray produced at the angle of refraction, θ′B, depicted in the lower left quadrant, will be partially polarized. The angle between the reﬂected and refracted rays is 90o. The angles, θB and θ′B are complementary θ + θ′B = 90o . This process depends on the refractive indices of the two media involved, n1 and n2, such that the process is deﬁned by Snell’s Law, n1 sin θB = n2 sin (90o − θB), which in turn becomes n1 sin θB = n2 cos θB, or tan θB = n2/n1.
Planck’s medium 2 has a Brewster’s angle complementary to the Brewster’s angle of his medium 1 (θB + θ′B =90o). Brewster’s angle is deﬁned in terms of a reﬂected and a refracted beam. Unpolarized light, and plane-polarized light that is not “at right angles to the plane of incidence”, produce reﬂected and refracted beams, in accordance with Brewster’s Law. Planck invoked Brewster’s Law [23, p. 450] with the special condition that incident rays are orthogonal to the plane of incidence. In this case, there could be no reﬂection, but only refraction, in accordance with Snell’s Law. He simultaneously applied these same restricted conditions to medium 2.
“Now in the special case when the rays are polarized at right angles to the plane of incidence and strike the bounding surface at the angle of polarization, ρ = 0, and ρ′ = 0.”
∗The reﬂected ray has no E component in the plane of incidence.

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However, Planck’s two contiguous media were homogeneous and isotropic. They could only emit unpolarized light and not plane-polarized light. Since the entire system was enclosed by a barrier impermeable to heat, there was no external source of any incident plane-polarized rays. All incident rays considered must be unpolarized and all resultant composite rays, at best, partially polarized. This implied that the reﬂectivities of both media were never zero. Yet, Planck made all rays plane-polarized and, in this special case, orthogonal to the plane of incidence (magnetic vector convention). Since plane-polarized rays in both media were chosen orthogonal to their common plane of incidence, they had no components which could be reﬂected. The conclusion that the reﬂectivities were equal was therefore never properly tested, as Planck had oﬀered no possibility of any reﬂection taking place. Consequently, Planck’s conclusion, that ρν = 0, and ρ′ν = 0 cannot be true. Thus, Planck becomes unable to move to Kirchhoﬀ’s Law, as presented in his Eq. 42 [5, Eq. 42].
The situation was actually more complex, as Planck did not provide the proper form for Eqs. 17, 18, and 19. In reality, he neglected the contribution from emission or absorption in Eqs. 17 and 18. He had already redeﬁned the blackbody as possessing a purely transmissive surface, in contradiction to Kirchhoﬀ, as seen above. This was a critical error. The proper form of Eq. 17 [5, Eq. 38] must also include a term for emissivity, ην, in the direction of the conical element,

(ην + ρν) dt dσ cos θ dΩ Kν dν.

(21)

The proper form of Eq. 18 [5, Eq. 39] must also include a term for absorptivity of the second medium, α′ν,

1 − ρ′ν − α′ν dt dσ cos θ′ dΩ′ K′ν dν.

(22)

That is because the intensity of the ray from medium 2 which
is refracted into medium 1 corresponds to the transmissivity (τ′ν = 1 − ρ′ν − α′ν). Clearly, the intensity of the transmitted ray must account for the reduction of the incident ray
within medium 2 as a result of both reﬂection and absorption.
Planck cannot ignore the absorption of the surface. Conse-
quently, Eq. 19 should have included the emissivity of the ﬁrst medium, ην, and the absorptivity of the second medium, α′ν. If one considers that the emissivity of the ﬁrst medium, ην, is equal to its absorptivity, αν, then Eq. 19 becomes,

Kν K′ν

·

q2 q′2

=

1 − ρ′ν 1 − ρν

− α′ν . − αν

(23)

This equation can never lead to Kirchhoﬀ’s Law [5, Eq. 42]. As a consequence, it is readily apparent that Planck,
through Eqs. 17-20, adopted a presentation which selectively applied the rules of reﬂection and refraction to polarized rays, irrelevant to the discussion of heat radiation. Furthermore, he then arbitrarily chose the plane of polarization such that when the waves were incident at Brewster’s angle, there would be

no reﬂection. Nonetheless, if there could be no reﬂection, then Brewster’s angle, or the angle of polarization, could have no meaning. That is because such an angle depends on the reﬂected and refracted rays being at 90o to one another. But since Planck insisted that no reﬂection occurred, then clearly the reﬂected and refracted rays could not form a 90o angle. Importantly, not only did Planck advance Eq. 20 (i.e. Planck’s Eq. 40) by neglecting absorptivity and emissivity, he thereby selected materials which have little or no relevance to heat radiation. Planck could not neglect absorption and emission, treating only transmission and reﬂection, if he wished to have any relevance to actual blackbodies. In addition, he hypothesized a bounding surface without any true physical meaning. Given this array of shortcomings, this derivation of Kirchhoﬀ’s law can never be salvaged. Planck’s claims for universality were without proper theoretical conﬁrmation.
5 Planck’s Perfectly Reﬂecting Cavities and the Carbon Particle
Throughout “The Theory of Heat Radiation”, Planck had recourse to a perfectly reﬂecting cavity, in which he placed a minute carbon particle (see [8] for a detailed treatment). Obviously, cavities comprised solely of perfectly reﬂecting surfaces, can never contain black radiation, as such materials cannot emit photons [16]. Nonetheless, Planck believed that these cavities contained radiation. He was careful however, not to state that this radiation was black [5, § 51],
“. . . in a vacuum bounded by totally reﬂecting walls any state of radiation may persist.”
This statement, by itself, was a violation of Kirchhoﬀ’s Law. Nonetheless, Planck believed that he could transform the radiation contained in all cavities into the thermodynamically stable radiation by inserting a carbon particle [5, § 51],
“If the substance introduced is not diathermanous for any color, e.g., a piece of carbon however small, there exists at the stationary state in the whole vacuum for all colors the intensity Kν of black radiation corresponding to the temperature of the substance”.
and later [5, § 52],
“It is therefore possible to change a perfectly arbitrary radiation, which exists at the start in the evacuated cavity with perfectly reﬂecting walls under consideration, into black radiation by the introduction of a minute particle of carbon. The characteristic feature of this process is that the heat of the carbon particle may be just as small as we please, compared with the energy of radiation contained in the cavity of arbitrary magnitude. Hence, according to the principle of the conservation of energy, the total energy of radiation remains essentially constant during the

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change that takes place, because the changes in the heat of the carbon particle can be entirely neglected, even if its changes in temperature should be ﬁnite. Herein the carbon particle exerts only a releasing (auslo¨send) action” .
Recall however, that Stewart’s law insisted that [1],
“. . . That the absorption of a particle is equal to its radiation, and that for every description of heat.”
When Planck moved the carbon particle into the cavity, clearly the emissive ﬁeld of the particle also entered the cavity provided the former had some real temperature. However, if one assumes that the particle was at T=0K, then no radiation from the carbon particle could enter the cavity. At the same time, if the particle was allowed to come into physical contact with the walls of the cavity, then energy could ﬂow from the walls into the particle by conduction. Hence the particle, being perfectly emitting, would ﬁll the entire cavity with black radiation. Alternatively, if the carbon particle could be suspended within the cavity, with no thermal contact to its walls, then the only radiation entering the system, would be that which accompanied the carbon particle itself [16]. That is because the walls of the cavity would not be able to “drive” the carbon particle, since they could emit no radiation. In that case, the radiation density within the cavity would remain too low and characterized only by the carbon particle. Unlike what Planck believed, the carbon particle could never be a simple catalyst, as this would constitute a violation of Stewart’s law [1]. Catalysts cannot generate, by themselves, the product sought in a reaction. They require the reactants. Yet, the carbon particle was always able to produce black radiation, in accordance with Stewart’s ﬁndings [1]. This was evidence that it could not be treated as a catalyst.
6 Planck’s Treatment of Two Cavities
Planck’s suboptimal treatment of the laws of emission continued [5, § 69],
“Let us ﬁnally, as a further example, consider a simple case of a irreversible process. Let the cavity of volume V, which is everywhere enclosed by absolutely reﬂecting walls, be uniformly ﬁlled with black radiation. Now let us make a small hole through any part of the walls, e.g., by opening a stopcock, so that the radiation may escape into another completely evacuated space, which may also be surrounded by rigid, absolutely reﬂecting walls. The radiation will at ﬁrst be of a very irregular character; after some time, however, it will assume a stationary condition and will ﬁll both communicating spaces uniformly, its total volume being, say, V′. The presence of a carbon particle will cause all conditions of black radiation to be satisﬁed in the new state. Then,

since there is neither external work nor addition of heat from the outside, the energy of the new state is, according to the ﬁrst principle, equal to that of the original one, or U′ = U and hence from (78)
T ′4V′ = T 4V

T′ =

4

V

T

V′

which deﬁnes completely the new state of equilibrium. Since V′ > V the temperature of the
radiation has been lowered by the process.”

This thought experiment was unsound. First, both cavities were made of perfectly reﬂecting walls. As such, Planck could not assume that the second cavity contained no radiation. To do so, constituted a violation of the very law he wished to prove. Kirchhoﬀ’s Law stated that the second cavity could not be empty. Therefore, Planck could not surmise that the temperature had dropped.
If one accepted that Kirchhoﬀ’s Law was false, as has been demonstrated above, then both cavities must be viewed as empty, other than the minute contribution made by the carbon particle. Here again, Max Planck had moved beyond the conﬁnes of reality, for he advanced a result which could not be correct, whether or not Kirchhoﬀ’s Law was true. The cavities were either both empty (i.e. Kirchhoﬀ’s Law was not valid), or both ﬁlled with radiation (i.e. Kirchhoﬀ’s Law was valid). One could not be ﬁlled, while the other was empty. Planck’s equation, in the quote above, was incorrect.

7 Conclusion
Throughout “The Theory of Heat Radiation’ [5] Planck employed extreme measures to arrive at Kirchhoﬀ’s Law. First, he redeﬁned the nature of blackbodies, by adopting transmission as a central element of his derivation. Second, he neglected the role of absorption at the surface of such objects, in direct contradiction to experimental ﬁndings and Kirchhoﬀ’s understanding of blackbodies. While it could be argued that absorption does not take place entirely at the surface, Planck could not assume that no absorption took place in this region. He was bound to include its contribution, but failed to meet this requirement. Third, he sidestepped reﬂection, by neglecting its presence in arriving at Eq. 12 [5, Eq. 27]. Nonetheless, the energy of the system under investigation included both that which was involved in emission/ absorption and that associated with the reﬂection terms. Stewart has well highlighted that such terms are central to the nature of the radiation within arbitrary cavities [1] and the concept has recently been re-emphasized [18, 19]. Fourth, Planck had recourse to plane-polarized light, whereas blackbody radiation is never polarized.
In the end, Planck’s presentation of Kirchhoﬀ’s Law did not properly account for the behavior of nature. Arbitrary

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cavities are not always black and blackbodies are highly specialized heated objects. Planck’s characterization of the carbon particle as a simple “catalyst” constituted a dismissal of Stewart’s Law [1]:
“. . . That the absorption of a particle is equal to its radiation, and that for every description of heat.”
Planck could not transform a perfect absorber into a catalyst. Yet, without the carbon particle [8], the perfectly reﬂecting cavities, which he utilized throughout “The Theory of Heat Radiation” for the derivation of his famous Eq. 1 [4, 5], remained devoid of radiation. Perfectly reﬂecting cavities are incapable of producing radiation, precisely because their emissivity is 0 by deﬁnition. Planck can only properly arrive at Eq. 1 by having recourse to perfectly absorbing materials, a truth which he did not acknowledge. The presence of reﬂection must always be viewed as suboptimal to the creation of a blackbody, since signiﬁcant reﬂection acts as a hindrance to the generation of photons through emission. It is never clear that the reﬂection term can easily be driven to arrive at the desired radiation, since thermal equilibrium, under these circumstances, can easily be violated, as the temperature of the cavity increases.
Planck’s detachment from experimental ﬁndings relative to Kirchhoﬀ’s Law was evident in his presentation of Eq. 20 [5, Eq. 40]. His conclusion, with respect to the equivalence of the reﬂection in arbitrary materials, was false. Obviously, if reﬂection was always the same, then all opaque cavities would become identical. Eq. 20 [5, Eq. 40] became the vital result in Planck’s derivation of Kirchhoﬀ’s Law. Unfortunately, the conclusion that ρ=ρ′ [5, Eq. 40] constituted a distortion of known physics and, by extension, so did Kirchhoﬀ’s formulation.
Without a proper proof of Kirchhoﬀ’s Law, Planck’s claim for universality loses the role it plays in science. This has signiﬁcant consequences in both physics and astronomy [8, 17, 24]. The constants h and k do not have fundamental meaning. Along with “Planck length”, “Planck time”, “Planck mass”, and “Planck temperature”, they are to be relegated to the role of ordinary and arbitrary constants. Their value has been deﬁned by our own selection of scales, not by nature itself.
Dedication
This work is dedicated to the memory of Balfour Stewart [1].
Submitted on: January 24, 2015 / Accepted on: January 25, 2015 First published online on: January 28, 2015
References
1. Stewart B. An account of some experiments on radiant heat, involving an extension of Pre´vost’s theory of exchanges. Trans. Royal Soc. Edinburgh, 1858, v. 22, no. 1, 1–20.
2. Kuhn T.S. The Structure of Scientiﬁc Revolutions. University of Chicago Press, Chicago, Il, 1962.

3. Robitaille P.-M. Max Karl Ernst Ludwig Planck: (1858–1947). Progr. Phys., 2007, v. 4, 117–120.
4. Planck M. U¨ ber das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik, 1901, v. 4, 553–563.
5. Planck M. The theory of heat radiation. P. Blakiston’s Son & Co., Philadelphia, PA, 1914, http://gutenberg.org/ebooks/40030.
6. Kirchhoﬀ G. U¨ ber den Zusammenhang zwischen Emission und Absorption von Licht und. Wa¨rme. Monatsberichte der Akademie der Wissenschaften zu Berlin, sessions of Dec. 1859, 1860, 783–787.
7. Kirchhoﬀ G. U¨ ber das Verha¨ltnis zwischen dem Emissionsvermo¨gen und dem Absorptionsvermogen. der Ko¨rper fur Wa¨rme und Licht. Poggendorfs Annalen der Physik und Chemie, 1860, v. 109, 275–301.
8. Robitaille P.-M. Blackbody radiation and the carbon particle. Progr. Phys., 2008, v. 3, 36–55.
9. Agassi J. The Kirchhoﬀ–Planck radiation law. Science, 1967, v. 156(3771), 30–37.
10. Robitaille P.-M. Kirchhoﬀ’s Law of thermal emission: 150 Years. Progr. Phys., 2009, v. 4, 3–13.
11. Hoﬀmann D. On the experimental context of Planck’s foundation of quantum theory. In: Revisiting the Quantum Discontinuity, Max Planck Institute for the History of Science, Preprint 150, 2000, 47–68.
12. Lummer O. and Pringsheim E. Kritisches zur schwarzen Strahlung. Annalen der Physik, 1901, v. 6, 192–210.
13. Rubens H. and Kurlbaum F. Anwendung der Methode der Reststrahlen zur Pru¨fung der Strahlungsgesetzes. Annalen der Physik, 1901, v. 2, 649–666; Rubens H. and Kurlbaum F. On the heat radiation of long wave-length emitted by black bodies at diﬀerent temperatures. Astrophys. J., 1901, v. 74, 335–348.
14. Robitaille P.M. On the validity of Kirchhoﬀ’s Law of thermal emission. IEEE Trans. Plasma Sci., 2003, v. 31, no. 6, 1263–1267.
15. Robitaille P. M. L. An analysis of universality in blackbody radiation. Progr. Phys., 2006, v. 2, 22–23; arXiv: physics/0507007.
16. Robitaille P.-M. A critical analysis of universality and Kirchhoﬀ’s Law: A return to Stewart’s law of thermal emission. Progr. Phys., 2008, v. 3, 30–35 (also in arXiv: 0805.1625).
17. Robitaille P.-M. Blackbody Radiation and the Loss of Universality: Implications for Planck’s Formulation and Boltzmann’s Constant. Progr. Phys., 2009, v. 4, 14–16.
18. Robitaille P.-M. On the equation which governs cavity radiation. Progr. Phys., 2014, v. 10, no. 2, 126–127; (see also Errata – Notice of Revision. Progr. Phys., 2015, v. 11, no. 1, 88.)
19. Robitaille P.M. On the equation which governs cavity radiation II. Progr. Phys., 2014, v. 10, no. 3, 157-162; (see also Errata – Notice of Revision. Progr. Phys., 2015, v. 11, no. 1, 88.)
20. Siegel D.M. Balfour Stewart and Gustav Robert Kirchhoﬀ: Two independent approaches to Kirchhoﬀ’s Law. Isis, 1976, v. 67, no. 4, 565– 600.
21. Schirrmacher A. Experimenting theory: the proofs of Kirchhoﬀ’s radiation law before and after Planck. Hist. Stud. Phys. Biol. Sci., 2003, v. 33, 299–335.
22. Robitaille P.M. Forty lines of evidence for condensed matter – The Sun on trial: Liquid metallic hydrogen as a solar building block. Progr. Phys., 2013, v. 4, 90–142.
23. Jenkins F.A. and White H.E. Fundamentals of Optics (4th Edition), McGraw-Hill, Inc, New York, 1976.
24. Born M. and Wolf E. Principles of Optics (6th edition), Cambridge University Press, 1980.

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