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PLANCK'S RADIATION LAW FOR FINITE
CAVITIES AND RELATED PROBLEMS
H. P. BALTES
Zentrale Forschung u. Entwicklung, LGZ Landis & Gyr Zug AG, Zug, Switzerland
{Received 28 August 1975)
Abstract—The fundamental thermal radiation laws are valid only if the dimensions of the cavity
are everywhere large compared to the wavelengths. Blackbody size and shape effects are reviewed
together with the statistical physics of other finite noninteracting systems based on the distribu-
tion of eigenfrequencies for the wave equation in a finite domain. Analytical results for the
refined density of states are emphasized. Quantum-size effects are discussed.
1. INTRODUCTION
The study of thermal radiation at the turn of the century*1_4) started a revolution
in thought that has dominated physics ever since. The current interest in this oldest
subject of modern physics has two main sources.
(I) Quantum optics and the theory of partial coherence investigate the statistical
properties of radiation and study the blackbody as the best known example of chaotic
fields.(5-10)
(II) Size, shape, and proximity effects and their impact on the venerable laws of
thermal radiation are studied in connection with the following problems.
(i) The Casimir effect for finite temperatures*11_14) and for cavities bounded in three
dimensions/15-17*
(ii) The radiative energy transfer between closely spaced bodies/18_20)
(iii) The spectral energy density of small blackbody cavities related to the problem
of far i.r. radiation standards/21-26*
(iv) The concept of the density of states and quantum-size effects in the statistical
mechanics of finite noninteracting systems.(24_29)
(v) The aspects of coherence and size effects are combined in the study of the correla-
tion of thermal radiation in finite cavities(30'31) and the quantum electrodynamic
proximity effects occurring in the vicinity of a material wall.(32'33)
In this paper, we are mainly interested in the items Il(iii) and II(iv) of the above
list. The blackbody size and shape effects are reviewed together with the statistical
physics of other perfect gases based on the distribution of eigenfrequencies for the wave
equation in a finite domain.
2. THE VALIDITY OF PLANCK'S LAW
Planck's radiation formula
l/ (v, T) dv = hv lQhv/KT -IT1 D (v) dv (1)
0 0
with the mode density or Jean's number
D (v) = Snc~3Vv2 (2)
0
gives the spectral density of the electromagnetic energy in an empty closed cavity of
volume V in thermal equilibrium with the walls at temperature T. The formula is valid
only for cavities with linear dimensions large compared with the wavelengths of the
radiation considered. Planck apparently was aware of this question as can be seen
from sparse references in his papers.
1
2 H. P. BALTES
(i) In an early remark antedating even the formula (1), the difficulty of defining black-
ness for arbitrary longwavelengths and the requirement of wavelengths small compared
with any radius of curvature are pointed out.(34)
(ii) In the early model, an expression for the radiative damping valid only at a large
distance from the oscillating dipole is used.(35) Thus the vicinity of the walls is
excluded.
(iii) The overall assumption of any linear dimension and radius of curvature being
large with respect to the wavelengths is made [Section 2 in reference (1)], and
the explicit conditions
ILv/O 1, 2Ldv/c> 1 (3)
are specified in the case of a cubical cavity with edge length L showing perfectly
reflecting walls [Section 169 in reference (1)]. Here, dv denotes the spectral resolu-
tion,
(iv) Planck(36) and independently Schäfer(37) discuss the analogous problem of elastic
vibrations in small crystals and predict a quantum-size-effect type deviation from
the Debye T3 law. These effects were demonstrated only recently (38'39) and explained
quantitatively along the line suggested by Planck and Schäfer.(40_42)
Optical blackbodies of usual laboratory size are characterized by Lv/c ^ 105 and
fulfill the first condition of (3), whereas far i.r. cavity standards and high-resolution
grating spectrometers rather obey Lv/c » 50 and Ldv/c ^ 1, and the validity of (1) can-
not be taken for granted. Furthermore, the thermal radiation in small cavities is expected
to be inhomogeneous, anisotropic, partially polarized, and to depend on the shape of
the cavity. Mathematically, the restricted validity of (1) and (2) is obvious from the
proof of WeyPs theorem(43) asserting that the mode density is independent of the shape
in the limit v —► oo.
3. WEYL'S PROBLEM AND THE STATISTICAL
PHYSICS OF FINITE SYSTEMS
By Weyl's problem we understand the problem of calculating the eigenvalue distribu-
tion for the wave equation in a finite domain with some boundary conditions. Weyl
studied three different types of boundary value problems, the scalar wave equation,
the electromagnetic vector wave equation (related to the blackbody), and the elastic
vibration problem/43} He concidered the behaviour of the mode number
N(v) = Σ 1 + Σ 1/2, (4)
vn < v v„ = v
i.e. the number of eigenfrequencies v not exceeding a certain value v, in the limit of
n
large v. He obtained the asymptotic behaviour
N(v)~N (v) = (Zn/3)V(v/c)3 (5)
0
for the electromagnetic mode number, the mode density (2) being the derivative of
N(v). Weyl assumed simple model boundary conditions, e.g. perfectly reflecting walls
0
for the cavity problem. Substantial progress in solving Weyl's problem was made in
the past decade on the mathematical side by both computational and analytical
methods.(44) This progress was paralleled by the applications to the statistical mechanics
of finite systems, in particular in the case of the scalar problem (Δ + k2) φ = 0 corre-
sponding to the time-independent Schrödinger equation for free particles of mass m
and kinetic energy e with the wavenumber k = (2m e)1/2/h. Here, the solution of Weyl's
problem provides the basic energy eigenvalue distribution. Similarly Weyl's electromag-
netic problem is fundamental for the statistical mechanics of the photon gas in a finite
enclosure, and the elastic problem leads to the frequency distribution of the low fre-
quency acoustic phonons in grains. The photon gas is a particularly important example
allowing the study of the pure boundary effects because of the absence of any interaction,
Planck's radiation law for finite cavities 3
whereas the perfect quantum gas is an approximation for other small systems, e.g. elec-
trons in metal grains or nucleons in heavy nuclei. It is the aim of this paper to (i)
report the progress achieved in Weyl's problem with particular attention to the thermal
radiation in finite cavities, and (ii) demonstrate that the blackbody problem is still
stimulating for the solution of formally similar problems in other fields. Both items
are extensive enough to fill a book(44) and only a few selected results are presented
in this paper. In particular, we restrict our presentation as follows. We dispense with
the full mathematical background available elsewhere.(44,45) We consider properties of
the whole system only (e.g. spectral distribution, total energy) rather than the—likewise
interesting—spatial variations.(32'33) We admit only simple boundary conditions, e.g.
Dirichlet or Neumann conditions for the scalar, and perfectly reflecting walls for the
cavity, problem. Although computational results are an indispensable heuristic
tool,(22'26'28'44) we shall emphasize analytical results.
4. QUASI-CONTINUUM AND QUANTUM-SIZE REGIMES
The following hierarchy of approximations is possibly helpful in order to overlook
both the progress in calculating the eigenvalue distribution and the corresponding appli-
cations (see Table 1). In the quasi-continuum approach, the finite distance between the
eigenlevels is smoothed out by some broad-band average procedure and an asymptotic
expansion, usually a polynom, is obtained for the mode number. In the quantum-size
regime the distance between adjacent eigenfrequencies is no longer negligible in compari-
son to the spectral resolution, the thermal energy XT, etc. The exact strongly oscillating
eigenvalue distribution or at least some level statistics must be taken into account.
There is still an intermediate approach where analytical formulae account for the main
features of the fluctuations of the mode number around the average value in the sense
of a narrow-band smoothing/24'46,47*
Table 1. The hierarchy of approximations for the eigenvalue distribution of the wave equation
regime sub-regime results/applications/remarks
quasi- infinite volume term, Weyl's theorem
continuum: space limit Planck's and Debye's formulae
analytical ideal quantum gas
mode number
asymptotic O-remainders improving the vo- ;
error esti- lume term/the mathematician's
mates approach
smoothed surface,edge,curvature, ...,
asymptotic terms/surface vibrational spe-
expansions, cific heat,surface and shape
integral tension of nuclei and perfect
transforms quantum gases
quantum summation specific heat and paramagne-
size: over exact tism of metal grains/level
finite levels or structure hidden in the
level stochastic summation
distance distributions
funny-hill nuclear shell correction in
regime:exact the mass formula. FIR absorp-
levels or tion of metal grains?/features
narrow-band of level structure appear
smoothing
4 H. P. BALTES
Some current investigations in the field of the spectra of finite systems are listed
below (the items indicated by * are presented in more detail).
THEORY
(i) Non-relativistic perfect quantum-gases
Thermodynamic equations of state.(48'49) Bose-Enstein condensation/50,5 υ
(ii) Photons
Spectral* and total* energy/24,27) Coherence and quantum electrodynamics.(30'33)
Radiative heat transfer/1 8,19) Casmir effect/11_17)
(iii) Solid grains
Electrons: level statistics/52'54) spherical modes,(55) cubical modes/56) Accoustical
phonons: quasi-continuum*,(57_59) quantum i >M36'37»40-42) Optical phonons: ionic
s ze
crystal cubes/60)
(iv) Nuclei
Surface and shape energy/61-63* Shell correction and fission barrier/46,47,64*
EXPERIMENT
(1) Photons
Fluorescence lifetime/65) Radiative heat transfer/19) Casimir effect/66,67*
(ii) Electrons in metal grains
Paramagnetism/68) Far i.r. absorption/69* Optical absorption/70*
(iii) Acoustical phonons
Vibrational specific heat*/38,39) Mössbauer/71)
(iv) Optical phonons
FIR absorption/72,73* FIR emission/74*
5. THE BLACKBODY: SMOOTHED EXPANSIONS FOR THE
MODE NUMBER
_ Let us recollect the analytical results for the smoothed electromagnetic mode number
N(k), k = In v/c, in the quasi-continuum regime. The complete asymptotic expansion
is known for the lossless cuboidal cavity with edge lengths L L , L (24,26)
l9 2 3
N(k) = ^ L, L L k3 - ^ (L + L + L )k + \ + Ö(k~r) (6)
2 3 x 2 3
with arbitrary r > 0. The bars over N and 0 indicate that an averaging procedure
suppressing the fluctuations of N(k) was used in deriving the result. No first-order correc-
tion proportional to the surface area of the cavity and to k2 appears in (6). The vanishing
of such a term can be shown for general cavity shapes(24_26) and is typical for the
electromagnetic problem: The TM and TE modes each give the same surface contribu-
tion, but of opposite sign. For the cuboid/55) these partial mode numbers read
1 . . . ,, 1 ,. . . . . . ,,, 1
N™{k) ^i?LiLlL^ " 8^(Ll Lz + L*L3 ~ Ll L*>*2 - in^ (Li+ +L *L- -L ^Lk )k+ +*
2 3 1
(7)
N (k) - ^L L L k3 + ^ (L L + L L - L, L )k2
TE x 2 3 x 3 2 3 2
-^(L +L + 3L )/c + | (8)
1 2 3
Planck's radiation law for finite cavities 5
and lead to N(k) = N + N as given in (6). The linear term in (6) accounts for
TM TE
the edges of the cavity, whereas the constants in (7) and (8) are related to the corners
and the connectivity.
For the sphere of radius R the linear term in N(k) reads -2Rkßn;{15) for a general
smooth cavity wall one has(25)
N(k) ~ Vk" - -i- f dS(Kr1 + Rix) (9)
2 2
3n Yin J
surface
where the R denote the principal radii of curvature. For the pill-box cavity (circular
{
cylinder of radius R and length L) the linear term reads —{R/2 + 2Lßn)k.(26) The
corresponding term is known for general cylinders with arbitrary cross section and
for spherical sectors (cones).(26'44) The corresponding partial mode numbers are available
as well.
_We compare these results with those for the scalar problem (where the complete
N(k) is known for any domain shape) and to the elastic problem [where only the volume
and surface terms are known].(57_59)
The smoothed mode number of the scalar Dirichlet or Neumann problem in a cuboidal
domain reads(75)
+ ^(L + L + L )k( + )i (10)
1 2 3
and does show a correction term proportional to k2 and to the surface of the domain.
For general smooth boundaries, the surface term of the mode number expansion reads
(+)Sk2/l6n (11)
with S denoting the surface area. The curvature term reads*75)
-i- f dS(R^ + R^) (12)
2
1Z7C J surface
with the positive sign for both the Dirichlet and the Neumann boundary condition.
The constant term of the scalar Dirichlet mode number is(76)
-±-J dS(R^-Ril). (13)
JlZ7l%/surface
Similar terms account for edges and corners in the case of non-smooth boundaries.
Six terms of the mode number expansion are known for the scalar Dirichlet problem
in the case of the spherical domain(76)
The expansion of the mode number corresponding to the elastic vector wave equation
with stress-free boundary starts with the volume term
-±- (2v;3 + vr3)Vk3 (14)
2
accounting for the two transverse branches and the longitudinal branch with sound
velocities v and v respectively. The surface term cannot be obtained as a sum of
t b
the terms corresponding to the three branches, but is proportional to(57)
2u4-3ü2i;2 + 3t;4 ,
02
vfvf(vf-vf) Sk' W
A systematic comparison of the electrodynamic cavity problem with the related scalar
and elastic problems for a variety of domain shapes is given in reference (44).
Let us now discuss the corrections of the radiation laws corresponding to (6). The
average mode density to be inserted in (1) reads D(v) = D (v) — A/c with
0
^cube = Li + L + L , /t = 4Ä/3, Runder = Lß + 7i#, etc. Planck's formula, the
2 3 sphere
Wien displacement law, the Stefan-Boltzmann law, the mean square fluctuation, and
6 H. P. BALTES
the Einstein coefficient of stimulated emission are modified accordingly. Only the ther-
modynamic relation E = 3 PV for the total energy E and the radiation pressure P
survives.
6. THE STEFAN-BOLTZMANN LAW FOR THE CUBE-SHAPED CAVITY
The total radiation energy E is known for arbitrary temperatures Tand edge lengths
L covering both the quasi-continuum and the quantum size regimes in the case of
the cube-shaped cavity/27'29'3 υ The result can be represented in terms of the asymptotic
expansion around 0 = KTL/hc-+ oo and reads up to a factor L/hc
— Ε(θ) ~—θ4-^θ2+$θ-α-^θ3 Σ' ν~' {(2πν0Γ 3 - cosh(27iv0)
flC lj T" ^ v1,v2,v3=— oo
[sin(27rv0)r3} + 12π0 Σ' \rn\~l [sinh(2n|m|ö)]-2,
(16)
m- — /
= (? + vi + vi)1/2.
v v
The primes indicate that the terms corresponding to v = 0 and m = 0 are omitted
in the summations. The constant term reads α = (π3 — Σ'ν"4)/16π2 = 0-0916... .Only the
first four terms of (16) are practicable for the numerical calculation of E and yield more
than 1% accuracy for LT> 0-3 cm K. In the quantum-size region (LT< 1 cm K),
however, one rather has to consult the corresponding expansion around 0—►O, i.e. the
original exact series defining E in statistical physics.(29) The analogous result for the
cuboid is known(77) as well.
7. VIBRATIONAL SPECIFIC HEAT IN GRAINS
The above blackbody quantum-size effect has never been investigated experimentally,
but the analogous effect in the elastic vibration of small free crystallites of approximately
spherical shape was observed/3 8'39) The measurements are at variance with the quasi-
continuum approach specific heat C = C + constant (RT)2 with R denoting the
vib Debye
radius of the grain,(57-59) but are compatible with the summation of the discrete scalar
Neumann spectrum for the spherical domain,
*™ (2/ + l)x2 exp(x ) _ twa'
s />s hs
vibX h [expfo..) - l]2 'Xl->—RT (17)
Here a denotes the 5th zero of the derivative j[ of the spherical Bessel function j
Us t
and v stands for an effective sound velocity.(40_42)
8. FUNNY HILLS
Although the overall effect of the discreteness of the spectrum is clearly observed
in the above examples, one cannot see the level structure because of the summations
involved. The fluctuations of the eigenvalue distribution appear, however, in the shell
correction of the mass formula established in terms of the independent particle model
of heavy nuclei related to the scalar Dirichlet problem/62_64) An improved density
of states is constructed using a moderate degree of smoothing that yields the average
D(e) including the volume, surface, and curvature terms and, in addition, slowly fluctuat-
ing terms related to the shell correction. The degree of smoothing is chosen large enough
in order to average over the levels of minor importance between the major shells, but
small enough in order to reveal the most relevant "hills" due to a bunching of eigen-
values appearing in D(e) — D(e).
As an example we mention the scalar Dirichlet problem for the sphere of unit radius
with the Lorentzian smoothing function.(63) Putting (2m)1/2/h = 1, the corresponding den-
sity of states is defined as
p(e) de = (y/n) £ (21 + 1) [(e - k2)2 + y2y1 *,./,(*,,,) = 0, k = k + ik = (e + iy)1^
y s r t
l,s
(18)
Planck's radiation law for finite cavities 7
where the bandwidth y controls the degree of smoothing. For not too small fc > 0-2, the
f
analytical expansion
Kß* ~ i[l " (2/^tang-1(/c//c,)] + (\ßn)K(k2 + kfy1
i r
?π / 1 \1/2
+ Y sin —( —βϊηπ/ρ Im{(-i)pkl/2 3lti/4e2',ffcsin^J (19)
e
^3,4 P \np )
p
describes the density of states. The three leading terms correspond to the volume, surface,
and curvature contributions. The fluctuations are characterized by two main oscillations
with periods 2π/3 ^3 « 1-21 and 2π/4 >/2 » 1-11 leading to a beating with the period
2π/(4 yjl — 3 >y3)« 13-4. A similar beat effect was observed in the far i.r. absorption
of small spherical aluminum particles(69) and can possibly be interpreted along this
line.
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