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Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

“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 Robitaille1and 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, hand 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 polar-

ized. 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 prop-

erties. 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 Stew-

art 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 con-

tain black radiation. Kirchhoﬀ’s formulation is invalid. As a direct consequence, the

constants hand kdo 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 be-

ginning of the twentieth century [4, 5]. Within “The The-

ory of Heat Radiation” [5] Planck outlined the ideas which

gave life both to this revolution and to the concept that fun-

damental constants existed which had universal signiﬁcance

throughout nature. The pillars which supported his ideas in-

cluded: 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 experimen-

tal evidence, would eventually become ﬁlled with blackbody,

or normal, radiation. He argued that, if a cavity did not con-

tain black radiation, the cause was a lack of thermal equilib-

rium, 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 indepen-

dent of the nature of the enclosure. In reality, such ideas were

not supported by experiment, as arbitrary cavities do not con-

tain 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].

120 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited

Issue 2 (April) PROGRESS IN PHYSICS Volume 11 (2015)

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, Kirch-

hoﬀ’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 conc lusions 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 speciﬁc intensity Kν

[5, Eq. 300],

Kν=hν3

c2

1

ehν

kT −1

,(1)

wherein ν,c,h,kand Trepresent 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 al-

ways lined with good absorbers over the frequency of inter-

est, such as graphite, soot, carbon black, or other specialized

materials (see [8] and references therein). Max Planck rec-

ognized 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 impor-

tant to note the elaborate experimental setup used [11, 1 3]. 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]. Re-

peated 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 ob-

tained with an apparatus used to select the frequency of inter-

est at the proper intensity.

In themselves, such extreme experimental methods con-

ﬁrmed 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 in-

voke the inability to reach thermal equilibrium. This is espe-

cially 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 tem-

perature equilibrium in metallic objects through conductive

processes. Laboratory ﬁndings do not support Planck’s posi-

tion relative to Kirchhoﬀ’s Law.

Clearly, real blackbodies were much more than simple ar-

bitrary cavities [11–13]. Yet, Max Planck believed with cer-

tainty in the universality of Kirchhoﬀ’s Law. It is this as-

pect 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 astrophysi-

cal 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 set-

ting. 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 kilo-

meter. 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 di-

rect 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 tem-

perature 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 keep-

ing 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.

P.-M. Robitaille and S . Crothers. “The Theory of Heat Radiation” Revisited 121

Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

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 ex-

perimental 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 pub-

lished what can be considered one of the most important

works in the history of thermal emission [1]. His analysis of

radiation was entirely based o n 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 formu-

lated 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 ar-

gument. For Stewart, this principle did not rise to the level of

a law, precisely because the conclusion had not been exper-

imentally 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 radia-

tion 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 partic-

ular 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 tem-

perature), 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 u s suppose, fo r in-

stance, 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 sur-

faces, until the total amount of radiated and re-

ﬂected heat together became equal to the radia-

tion of lampblack”.

The problem is that good reﬂectors do not readily emit radia-

tion. 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 cav-

ity 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 eventu-

ally contain black radiation. The argument was true in the-

ory, but not demonstrated in practice. Stewart remained con-

strained 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 ra-

diation can become black, in theory, in the event that the re-

ﬂective 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 Equiva-

lence [1], Gustav Kirchhoﬀpublished 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

Kirchhoﬀand Stewart.∗The problem centered on Kirchhoﬀ’s

attempt to dismiss Stewart’s priority claims for the Law of

Equivalence. Kirchhoﬀdid 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 radi-

ation, did not insist that this was always true [1]. Conversely,

Kirchhoﬀformulated 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].

122 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited

Issue 2 (April) PROGRESS IN PHYSICS Volume 11 (2015)

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 in-

ﬁnitely 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] Kirchhoﬀpresented 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

A=e.(2)

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

one could express Kirchhoﬀ’s Law as follows:

Eν

αν

=f(T, ν),(3)

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], Kirchhoﬀoutlined 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 inte-

rior 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 evi-

dent 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 inﬁ-

nite number of reﬂections which it successively

experiences at the assumed bodies. In the inte-

rior 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 Kirchhoﬀspeaks of absorptive power, A, he was actually refer-

ring 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 per-

fect reﬂector, as αν=0 under that condition. Planck him-

self recognized this fact [5, §48], but might not have ex-

ercised proper care relative to its consequences. Second, it

is clear that Kirchhoﬀlacked an accurate understanding of

what was happening within his cavity, as an “inﬁnite num-

ber” of reﬂections will never amount to absorption. An “in-

ﬁnite number” of reﬂections does not involve the exchange

of energy. Conversely, when absorption occurs, energy is ex-

changed 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 de-

tail [16].

In brief, within his ﬁrst proof, Kirchhoﬀutilized transmis-

sive plates to accomplish the proof, even if blackbody cavities

must always be opaque. He addressed transmission by posi-

tioning mirrors behind his plates. In so doing, it appeared that

Kirchhoﬀhad 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 for-

mulate Kirchhoﬀ’s Law [5, §36–38]. In his second proof,

Kirchhoﬀunknowingly permitted the cavity to fall out of ther-

mal 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 emis-

sion (see [21] for an excellent presentation). In fact, physi-

cists continued to argue about a proper theoretical proof for

Kirchhoﬀ’s Law until well after Planck’s ideas became ac-

cepted [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 ap-

plying Einstein coeﬃcients to arrive at Planck’s law, physics

could dispense with the proof of Kirchhoﬀ’s Law [21]. How-

ever, Einstein’s derivation utilized the energy density asso-

ciated with a Wien radiation ﬁeld, something which could

only be found within a blackbody. Surely, Wien had not dis-

pensed with Kirchhoﬀ. In truth, it appears that those con-

cerned 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 inter-

est. 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 ad-

vanced by Max Planck within “The Theory of Heat Radiation”.

P.-M. Robitaille and S . Crothers. “The Theory of Heat Radiation” Revisited 123

Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

will be found to contain signiﬁcant misapplications of exper-

imental 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 in-

vocation of polarized light, when heat radiation is always un-

polarized; 3) the use of transmissive materials, when Kirch-

hoﬀ’s Law refers to opaque enclosures; and 4) the existence

of hypothetical objects which can have no place in the physi-

cal world.

However, the central proof of Kirchhoﬀ’s Law must al-

ways 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 uni-

versality. Hence, it is best to forgo Kirchhoﬀ’s own deriva-

tions, 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 ca-

reer. 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 spe-

cial bodies or substances, which necessarily re-

tain their signiﬁcance for all times and for all en-

vironments, terrestrial and human or otherwise,

and which may, therefore, be described as ‘natu-

ral 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, trans-

mission, 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 radi-

ation”.

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 in-

terior. The blackbody became a sieve. Planck stated

[5, §10],

“A rough surface having the property of com-

pletely transmitting the incident radiation is de-

scribed as ‘black’ ”.

Planck continued [5, §12],

“Thus only material particles can absorb heat

rays, not elements of surfaces, although some-

times for the sake of brevity, the expression ab-

sorbing 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 Kirchhoﬀconsidered

a blackbody as absorbing over an inﬁnitely thin element. He

stated [5, §10],

“In deﬁning a blackbody Kirchhoﬀalso 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 transmis-

sion. Planck ignored that real surface elements must possess

absorption, in apparent contrast with Kirchhoﬀand without

any experimental justiﬁcation. Planck would expand on his

new concept for a blackbody with these words [5, §10],

124 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited

Issue 2 (April) PROGRESS IN PHYSICS Volume 11 (2015)

“. . . the blackbody mu st 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 absorb-

ing a body is, the smaller the value of this min-

imum 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 absorp-

tion and reﬂection when formulating his new deﬁnition for a

blackbody. This may have consequences throughout physics

and astronomy [8, 17, 2 2].

In the end, Planck’s surface elements must be composed

of material particles. Since Planck was a theoretical physi-

cist, he cannot work solely in the vacuum of a mathemati-

cal 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 exper-

iments 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 ανhas the same value at all points and in all

directions in the medium, and depends on noth-

ing 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 conse-

quence 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 trans-

mission, when it had no proper place in the treatment of

blackbody radiation, as found in opaque bodies [5, §14],

“Let dσbe 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 el-

ement, 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 main-

tained 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 ap-

proach, Max Planck explicitly prevented the newcomer to the

ﬁeld of thermal radiation from appreciating the crucial impor-

tance 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 refer-

ring to the emissivity in this instance. Rather, he was invok-

ing the emissive power, E, an entity with units. Conversely,

when he spoke of the coeﬃcient of absorption (Absorptionko-

eﬃzienten), αν, he was truly referring to the dimensionless

absorptivity, as we know it today. Insuﬃcient attention rel-

ative to Planck’s notation has, in fact, caused one of the au-

thors 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 ανand ρνwill retain

their modern meaning and represent dimensionless absorp-

tivity 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 express-

ing emissive power.∗

∗In §44, Planck presented Kirchhoﬀ’s Law in the following form [5,

Eq. 48],

E

A=I=dσcos θdΩKνdν,

where Ais 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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Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

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 dτdΩdν2ǫν,(4)

from which Planck immediately surmised, by integrating over

all directions and frequencies, that the total energy emitted

corresponded to [5, Eq. 2],

dt dτ8πZ∞

0

ǫνdν. (5)

He then moved to present the same equation, in slightly mod-

iﬁed form in §25 as,

dt v8πZ∞

0

ǫνdν, (6)

where vnow 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 radi-

ation 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 dτdΩdν2(ǫν+ρνKν).(7)

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

dt v8πZ∞

0

(ǫν+ρνKν)dν. (8)

Similarly, Planck characterized the fate of the radiation which

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

Eq. 25],

dt v8πZ∞

0

ανKνdν. (9)

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 v8πZ∞

0

(αν+ρν)Kνdν. (10)

Equating Eqs. 6 and 9, Planck obtained,

dt v8πZ∞

0

ǫνdν=dt v8πZ∞

0

ανKνdν, (11)

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 com bining Eqs.

8 and 10, we obtain that,

dt v8π

Z∞

0

(ǫν+ρνKν)dν=dt v8π

Z∞

0

(αν+ρν)Kνdν. (13)

This expression leads to the following relation,

ǫν+ρνKν=ανKν+ρνKν.(14)

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 reﬂec-

tion 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 reﬂec-

tive 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 Equiv-

alence, ﬁrst presented by Balfour Stewart [1]. As a result, it

can be readily noted that Eq. 15 can be expressed as,

ǫν=ηνKνor Eν=ηνKν,(16)

which is similar to Planck’s Eq. 26 [5, Eq. 2 6]. In this case,

Kνis 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 di-

rect contradiction of universality. The nature of the radiation

within the cavity becomes dependent on the nature of the cav-

ity 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,

126 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited

Issue 2 (April) PROGRESS IN PHYSICS Volume 11 (2015)

like Eq. 12. Then, consider Eq. 16. This relationship is im-

portant, because, like the form presented by Kirchhoﬀ(Eq. 2)

and Planck (Eq. 1 2), it is d evoid o f the consideratio n of re-

ﬂection. 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 intro-

duced 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 man-

ner in which a result is presented can have a great deal of

inﬂuence on its interpretation. Nowhere is this more applica-

ble 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.

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 ho-

mogeneous 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 argu-

ments 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 νpolarized in an arbitrary plane be

Kνin the ﬁrst substan ce . . . and K′νin the sec-

ond substance . . . ”

Planck also stated [5, §36],

“. . . we have for the monochromatic plane-

polarized radiation. . . ”

As such, to prepare for his use of polarized light in later sec-

tions, Planck resolved, in §17, the radiation into its two po-

larized 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

spectrum, can never be interpreted as heat radi-

ation.”

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 sys-

tem 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 dif-

fering 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],

“. . . a discontinuous change in b oth the direction

and the intensity of a ray occurs when it reaches

the boundary of a medium and meets the sur-

face 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 dσat 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 th e surface element d σof the

second medium.”

Note that Planck had already introduced three causes for

objection. First, what exactly was the location of dσ? In re-

ality it must rest in one of the two media. Second, Planck ne-

glected the fact that real materials can possess ﬁnite and dif-

fering 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 bound-

ary. 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 through-

out much of his text [5].

In §36 Planck considered a monochromatic plane-

polarized ray of frequency ν, emitted in time dt. In order to

P.-M. Robitaille and S . Crothers. “The Theory of Heat Radiation” Revisited 127

Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

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 dσto 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 dΩin 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 “bound-

ing 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 ρis replaced with ρνin 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 Kνat incidence [5, Eq. 38],

ρνdt dσcos θdΩKνdν, (17)

which were augmented by rays of incident speciﬁc intensity

K′νrefracted from the second medium [5, Eq. 39],

1−ρ′

νdt dσcos θ′dΩ′K′νdν. (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−ρ′

ν

1−ρν

,(19)

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

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

ρν=ρ′

ν,(20)

128 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited

Issue 2 (April) PROGRESS IN PHYSICS Volume 11 (2015)

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 sur-

face is the same on both sides, is a special case of

a general rule of reciprocity ﬁrst stated by Helm-

holtz.”

Planck provided for the element of the bounding surface two

separate coeﬃcients of reﬂection. These must, in fact, cor-

respond to those of the media utilized. Planck has already

stated in §35 that

“. . . let a ll quantities referring to the secon d sub-

stance be indicated by the addition of an accent.”

Consequently, ρand ρ′can only take meaning with respect to

the media under consideration. Thus, how did Planck possi-

bly 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 inci-

dence and strike the bounding surface at the an-

gle of polarization” [5, §37].

Again, such rays could never exist in the context of heat radi-

ation [23, p. 450].

The “plane of incidence” is that containing the unit nor-

mal 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 Eor the magnetic vector B. Contemporary

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. 3: Schematic representation of Brewster’s Law. The dots cor-

respond 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 polariza-

tion. 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 pro-

duced at the angle of refraction, θ′

B, depicted in the lower left quad-

rant, will be partially polarized. The angle between the reﬂected

and refracted rays is 90o. The angles, θBand θ′

Bare complementary

θ+θ′

B=90o. This process depends on the refractive indices of the

two media involved, n1and n2, such that the process is deﬁned by

Snell’s Law, n1sin θB=n2sin (90o−θB), which in turn becomes

n1sin θB=n2cos θB, or tan θB=n2/n1.

Planck’s medium 2 has a Brewster’s angle complemen-

tary 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 re-

fracted 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 simulta-

neously applied these same restricted conditions to medium 2.

“Now in the special case when the rays are po-

larized 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 Ecomponent in the plane of incidence.

P.-M. Robitaille and S . Crothers. “The Theory of Heat Radiation” Revisited 129

Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

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 en-

closed 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 reﬂec-

tivities 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 reﬂectivi-

ties were equal was therefore never properly tested, as Planck

had oﬀered no possibility of any reﬂection taking place. Con-

sequently, 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 dσcos θdΩKνdν. (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′

νdt dσcos θ′dΩ′K′νdν. (22)

That is because the intensity of the ray from medium 2 which

is refracted into medium 1 corresponds to the transmissiv-

ity (τ′

ν=1−ρ′

ν−α′

ν). Clearly, the intensity of the trans-

mitted 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−α′

ν

1−ρν−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 90oto one another. But

since Planck insisted that no reﬂection occurred, then clearly

the reﬂected and refracted rays could not form a 90oangle.

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 emis-

sion, 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 re-

course to a perfectly reﬂecting cavity, in which he placed a

minute carbon particle (see [8] for a detailed treatment). Ob-

viously, cavities comprised solely of perfectly reﬂecting sur-

faces, 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 v acuum 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 ra-

diation 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 car-

bon however small, there exists at the stationary

state in the whole vacuum for all colors the inten-

sity Kνof black radiation corresponding to the

temperature of the substance”.

and later [5, §52],

“It is therefore possible to change a perfectly ar-

bitrary 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 radi-

ation contained in the cavity of arbitrary magni-

tude. Hence, according to the principle of the

conservation of energy, the total energy of ra-

diation remains essentially constant during the

130 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited

Issue 2 (April) PROGRESS IN PHYSICS Volume 11 (2015)

change that takes place, because the changes in

the heat of the carbon particle can be entirely ne-

glected, even if its changes in temperature should

be ﬁnite. Herein the carbon particle exerts only

a releasing (ausl ¨osend) 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 con-

tact with the walls of the cavity, then energy could ﬂow from

the walls into the particle by conduction. Hence the parti-

cle, 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 re-

main 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 ra-

diation, 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 con-

tinued [5, §69],

“Let us ﬁnally, as a further example, consider a

simple case of a irreversible process. Let the cav-

ity 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 open-

ing a stopcock, so that the radiation may escape

into another completely evacuated space, which

may also be surrounded by rigid, absolutely re-

ﬂecting walls. The radiation will at ﬁrst be of a

very irregular character; after some time, how-

ever, 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′=T4V

T′

T=4

rV

V′

which deﬁnes completely the new state of equi-

librium. 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 radi-

ation. To do so, constituted a violation of the very law he

wished to prove. Kirchhoﬀ’s Law stated that the second cav-

ity 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 car-

bon 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 em-

ployed 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 sur-

face, 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 re-

ﬂection, by neglecting its presence in arriving at Eq. 12 [5,

Eq. 27]. Nonetheless, the energy of the system under investi-

gation included both that which was involved in emission/ab-

sorption 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 b een r e-emphasized [18, 19]. Fourth, Planck had

recourse to plane-polarized light, whereas blackbody radia-

tion is never polarized.

In the end, Planck’s presentation of Kirchhoﬀ’s Law did

not properly account for the behavior of nature. Arbitrary

P.-M. Robitaille and S . Crothers. “The Theory of Heat Radiation” Revisited 131

Volume 11 (2015) PROGRESS IN PHYSICS Issue 2 (April)

cavities are not always black and blackbodies are highly spe-

cialized heated objects. Planck’s characterization of the car-

bon 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 cata-

lyst. Yet, without the carbon particle [8], the perfectly re-

ﬂecting cavities, which he utilized throughout “The Theory of

Heat Radiation” for the d erivation of his famous E q. 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 reﬂec-

tion 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 cir-

cumstances, 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 vi-

tal result in Planck’s derivation of Kirchhoﬀ’s Law. Unfor-

tunately, the conclusion that ρ=ρ′[5, Eq. 40] constituted a

distortion of known physics and, by extension, so did Kirch-

hoﬀ’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 hand kdo not have fundamen-

tal meaning. Along with “Planck length”, “Planck time”,

“Planck mass”, and “Planck temperature”, they are to be rel-

egated 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

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