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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

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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.
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, d σ . Planck considered the reflective nature of d σ to ascertain whether its reflection coe ffi cients 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, K ν (plain arrow), at an angle θ , contained in the conical section, d Ω , of the first medium (upper right quadrant) which is reflected by the bounding surface into the conical section d Ω in the upper left quadrant and refracted into the conical section d Ω ′ of the second medium, at an angle θ ′ , in the lower left quadrant. Note that in order to preserve the proper specific intensities, K ν , in the upper left quadrant, Planck must sum the reflected portion of the incident specific intensity of medium 1, ρ ν K ν , 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, η ν , by d σ . B) Schematic representation of the incident specific intensity, K ′ ν (plain arrow), at an angle θ ′ ,contained in the conical section, d Ω ′ , of the second medium (lower right quadrant) which
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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 Kirchho’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
Arming Kirchho’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 Kirchho’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 Kirchho’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 Kirchhos 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. Kirchho’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) Kirchho’s Law of thermal emission [ 6, 7], 2) the
irreversability of heat radiation, and 3) the adoption of dis-
crete states.He utilized Kirchho’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 Kirchho[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-
ho’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 Kirchho’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 Kirchho’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 Kirchho’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 Kirchho’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 Kirchho’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 Kirchho’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 Kirchho[6,7] must be discussed, especially as related to
his treatment of reflection. Then, finally, a detailed analysis
of Max Planck’s derivation of Kirchho’s Law, as outlined
in “The Theory of Heat Radiation” [5], will be presented. It
will be demonstrated that Planck’s derivation suers, 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 Kirchho’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 sucient 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 Kirchho: Physics from Theory Alone
Soon after Balfour Stewart formulated the Law of Equiva-
lence [1], Gustav Kirchhopublished his law of thermal
emission [6, 7]. Almost immediately, the work was translated
into English by F. Guthrie [7] and Kirchho’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
Kirchhoand Stewart.The problem centered on Kirchho’s
attempt to dismiss Stewart’s priority claims for the Law of
Equivalence. Kirchhodid so by arguing that Stewart had
not brought forth sucient 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, Kirchho’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,
Kirchhoformulated 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, Kirchho, 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 Kirchho’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] Kirchhopresented 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)
Kirchhoeventually set A=1 [7, §3]. In modern notation,
one could express Kirchho’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], Kirchhooutlined 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 Kirchhospeaks 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, Kirchho’s “E” has the same units as his “e” and neither is equal
to 1. Kirchho, stated that “e” was a universal function and believed that its
elucidation was a matter of great scientific importance.
Relative to Kirchho’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 Kirchholacked 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 Kirchho’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, Kirchhoutilized 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
Kirchhohad properly treated reflection, because the mirrors
did, in fact, reflect radiation. However, he had dismissed the
possibility that the plates considered could possess diering
surface reflection [16]. As shall be discovered below, Max
Planck committed the same error, when he attempted to for-
mulate Kirchho’s Law [5, §36–38]. In his second proof,
Kirchhounknowingly 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 Kirchho’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
Kirchho’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 coecients to arrive at Planck’s law, physics
could dispense with the proof of Kirchho’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 Kirchho. In truth, it appears that those con-
cerned with bringing forth a proper proof for Kirchho’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 Kirchho’s Law could no longer provide sucient inter-
est. The entire issue appears to have come to a slow death,
without proper resolution.
It is certain that all theoretical proofs of Kirchho’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 Kirchho’s Law
suer 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-
ho’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 Kirchho’s Law must al-
ways be the one outlined by Max Planck himself (see [5, §1
51]), forty years after Kirchho[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 Kirchho’s own deriva-
tions, as the theoretical validity of Kirchho’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 Kirchho’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 Kirchho’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 Kirchho’s Law: Part I
Throughout his derivation of Kirchho’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 Kirchho, 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 Kirchhoconsidered
a blackbody as absorbing over an infinitely thin element. He
stated [5, §10],
In defining a blackbody Kirchhoalso 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 Kirchho’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 Kirchho’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 Kirchhoand 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
dierent 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 eects 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 eect 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 Kirchho’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
coecient of emission (Emissionskoezienten) 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 coecient of absorption (Absorptionko-
ezienten), αν, he was truly referring to the dimensionless
absorptivity, as we know it today. Insucient 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]. Suce 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 Kirchhos Law in the following form [5,
Eq. 48],
E
A=I=dσcos θdKνdν,
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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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, d, was given by [5, Eq. 1],
dt dτddν2ǫν,(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 dτddν2(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 Kirchho, 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 dier from one another, since at thermal equilibrium
1ρν=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
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like Eq. 12. Then, consider Eq. 16. This relationship is im-
portant, because, like the form presented by Kirchho(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 Kirchho’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 Kirchho’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 Kirchho’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 sucient to counter all of the argu-
ments which Planck later utilized to arrive at Kirchho’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 Kνin 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 Kirchho’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 Kirchho’s Law, in §35–37, Planck
placed two dierent 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 Kirchho’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
eect 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
Kirchho’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
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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 dσto ascertain whether its reflection coecients 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, d, of the first medium (upper right quadrant) which is reflected by the bounding surface into the conical
section din the upper left quadrant and refracted into the conical section dof 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
νρ
ν)Kν, 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, Kν(plain arrow), at an angle θ,contained in the conical section, d, of the second medium (lower right quadrant) which
is reflected by the bounding surface into the conical section, d, in the lower left quadrant and refracted into the conical section, d, of
the first medium, at an angle θ, in the upper left quadrant. Note that, in order to preserve the proper specific intensities, Kν, in the lower
left quadrant, Planck must sum the reflected portion of the incident specific intensity of medium 2, ρ
νKν, with the refracted portion of the
incident specific intensity of medium 1, (1 ανρν)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=1ρν, 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=1ρναν. Again, this completely prevents
further progress towards Kirchho’s Law [5, Eq. 42].
Planck considered the reflected rays in the first medium,
of specific intensity Kνat incidence [5, Eq. 38],
ρνdt dσcos θdKνdν, (17)
which were augmented by rays of incident specific intensity
Kνrefracted from the second medium [5, Eq. 39],
1ρ
νdt dσcos θdKνdν. (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
q2=1ρ
ν
1ρν
,(19)
where qand qcorrespond 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
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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 coecient 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 coecients 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 ρcan 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 (90oθB), 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.
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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 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 oered 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 Kirchhos
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 Kirchho, 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 θdKνdν. (21)
The proper form of Eq. 18 [5, Eq. 39] must also include a term
for absorptivity of the second medium, α
ν,
1ρ
να
νdt dσcos θdKνdν. (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=1ρ
να
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
q2=1ρ
να
ν
1ρναν
.(23)
This equation can never lead to Kirchho’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
Kirchho’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 Kirchho’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)
T4V=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. Kirchho’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 Kirchho’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 Kirchho’s Law was true. The
cavities were either both empty (i.e. Kirchho’s Law was not
valid), or both filled with radiation (i.e. Kirchho’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 Kirchho’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
Kirchho’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 Kirchho’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 Kirchho’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 Kirchho’s Law. Unfor-
tunately, the conclusion that ρ=ρ[5, Eq. 40] constituted a
distortion of known physics and, by extension, so did Kirch-
ho’s formulation.
Without a proper proof of Kirchho’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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132 P.-M. Robitaille and S. Crothers. “The Theory of Heat Radiation” Revisited
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