Matematica Open Source
– http://www.extrabyte.info Physics Papers – 2021The BlackBody Radiation
Marcello Colozzo
CONTENTS
Contents
1 Experimental features 2
1.1 Stefan-Boltzmann law. Wien’s law. Displacement law . . . 2
2 Theoretical setting 5
2.1 The Rayleigh-Jeans radiation formula . . . 5 2.2 Planck’s hypothesis . . . 6
3 Planck’s interpolation 8
4 Radiated power 10
5 The cosmic background radiation 11
1 EXPERIMENTAL FEATURES
1 Experimental features
1.1 Stefan-Boltzmann law. Wien’s law. Displacement law
A body C (solid or liquid) in thermodynamic equilibrium at temperature T , emits electro- magnetic radiation which, for T not excessively high, is in the infrared band. Let C be mathematizable through a domain of the ordinary space R3. The energy emitted by the surface element dσ in the cone with solid angle dΩ (fig.1) whose axis forms an angle θ with the unit vector n of the external normal to ∂C, in the unit of time and in the interval of frequencies [ν, ν + dν], can be written as:
e (ν, T ) = cos θdνdΩdσ (1)
where the non-negative quantity e (ν, T ) is theemissive power di C.
Figure 1: Energy emitted by the surface element dσ of solid angle dΩ.
If instead we irradiate the body, part of the incident energy is absorbed, and part is reflected. We then define the absorbing power:
a (ν, T ) = absorbed energy
incident energy ≤ 1, ∀ν, T
Here we mean “energy absorbed” relative to the frequency range [ν, ν + dν], so it is an energy spectral density.
Definition 1 C is a black body if
a (ν, T ) = 1, ∀ν, T (2)
Having stated this, let us consider a cavity whose walls are in equilibrium at the tempera- ture T . For the above, the walls emit electromagnetic energy; since it is a cavity, this energy cannot ”exit” so it is reabsorbed by the walls, until a state of equilibrium is reached (emis- sion=absorption). Through thermodynamic considerations it can be demonstrated that in such conditions the spectral energy density is a universal function u (ν, T ), in the sense that it does not depend on the point at which the electromagnetic field is determined nor on the direction of propagation of the field itself. In other words, at equilibrium the field inside the cavity is homogeneous and isotropic. Let us now fix our attention on the walls of the cavity (fig. 2).
1 EXPERIMENTAL FEATURES
Figure 2: Cavity.
Taking arbitrarily the surface element dσ of the wall of the cavity, we denote by n the unit vector of the internal normal. The energy that propagates in the unit of time in the cone of solid angle dΩ and of frequency in [ν, ν + dν] is
c
4πu (ν, T ) dΩdν (3)
where c is the speed of light. Part of this energy is absorbed by the aforementioned surface element. Precisely:
a (ν, T ) c
4πu (ν, T ) dΩdνdσ cos θ (4)
On the other hand, the energy emitted by dσ in the unit of time and in the same dν is
e (ν, T ) dΩdνdσ cos θ (5)
To balance
a (ν, T ) c
4πu (v, T ) dΩdνdσ cos θ = e (ν, T ) dΩdνdσ cos θ
so e (ν, T )
a (ν, T ) = c
4πu (ν, T ) (6)
In the particular case of a black body:
e (ν, T ) = c
4πu (ν, T ) (7)
It follows that the spectral density of energy inside a cavity whose walls are kept at the temperature T , is less than a multiplicative constant, the emissive power of the black body.
In other words, if we can determine the emissive power of the black body, we know the universal function u (ν, T ). For this purpose we use a cavity (oven) whose walls can be brought to a temperature T . The walls then have a hole of negligible size compared to the linear dimensions of the walls (so as not to disturb the state of equilibrium). In these conditions, the cavity is a black body since the incoming radiation remains “trapped” thanks to the infinitesimal dimensions of the hole. In correspondence with the latter we fix a system of cartesian axes Oxyz with origin in the hole and z axis oriented according to the external normal to the surface of the cavity (fig. 3).
Let’s take a vertex cone in O and infinitesimal half-aperture such as to identify the elementary solid angle dΩ, whose axis forms an angle θ with the unit vector n which, as
1 EXPERIMENTAL FEATURES
Figure 3: Blackbody scheme. The shape of the cavity is irrelevant.
stated, is the versor of the z axis. We orient the x-axis of the aforementioned system of axes so that θ is the colatitude of a polar reference with pole at O and the polar axis coincident with the z-axis. With this arrangement of the axes, dΩ is the surface element of the sphere with center O and unit radius, ie
dΩ = sin θdθdϕ
We recall that the emissive power e (ν, T ) is the energy emitted in dΩ in the unit of time and in the frequency interval dν. So
e (ν, T ) cos θdνdΩdσ = e (ν, T ) sin θ cos θdνdσdθdϕ, (8) where dσ is the surface element relative to the hole. To obtain the energy emitted by the hole (in the unit of time and in the frequency interval dν) we need to integrate the (8) for 0 ≤ θ ≤ π2, 0 ≤ ϕ ≤ 2π. If we refer to the unit of area:
W (ν, T ) = Z π/2
0
dθ sin θ cos θ Z 2π
0
dϕe (ν, T )
= 2πe (ν, T ) Z π/2
0
sin θd (sin θ) = 2πe (ν, T ) · 1
2 sin2θ
π/2 0
That is
W (ν, T ) = πe (ν, T ) (9)
which is the energy in the unit of time (power) and per unit interval of frequencies and per unit area. This quantity is therefore the spectral power density emitted by a black body in equilibrium at temperature T . To obtain the power emitted by the unit area, we must integrate over all frequencies:
Z +∞
0
W (ν, T ) dν
which is ageneralized integral. We have W (ν, T ) > 0, so this function is certainly integrable.
Physically, it must be summable:
Z +∞
0 W (ν, T ) dν < +∞
Experimentally (Stefan–Boltzmann law):
Z +∞
0
W (ν, T ) dν = σT4 (10)
where σ = 5.68 · 10−5erg · cm−2s−2. That is, the power emitted by the unit area of a black body is proportional to the fourth power of the equilibrium temperature.
2 THEORETICAL SETTING
Experimentally, we had then seen (displacement law):
λmaxT = costante (11)
where λmax is the point of absolute maximum of the function W (λ, T ), being λ = νc the wavelength. The (11) is a particular case of Wien’s Law (1893), expressed by
W (ν, T ) = ν3F ν T
(12) where F (x) is an unknown function. Indeed:
Z +∞
0
W (ν, T ) dν = Z +∞
0
ν3F ν T
dν =
x=Tν T4 Z +∞
0
x3F (x) dx For an F (x) such that x3F (x)is summable in [0, +∞)
Z +∞
0
W (ν, T ) dν = σT4, σ = Z +∞
0
x3F (x) dx
While (12) is mathematically acceptable, it is not physically so since the argument of the function F is constituted by the ratio of non-homogeneous quantities, so we redefinev
f
hν kBT
where kB is Boltzmann’s constant, while h > 0 is a new constant with the dimensions of an action (energy×time). Also, to ensure invariance under a change in units of measure, we need to insert a new universal constant, in addition to Boltzmann’s constant. Since u (ν, T ) is a universal function, the only universal constant available in the classical framework is the speed of light c. So
u (ν, T ) = ν2 c3kBT f
hν kBT
from which
W (ν, T ) = ν2 4c2kBT f
hν kBT
(13)
2 Theoretical setting
2.1 The Rayleigh-Jeans radiation formula
The cavity that creates the black body is home to an electromagnetic field of spectral density u (ν, T ). From Statistical Mechanics [1] we know that this field is equivalent to a system of uncoupled harmonic oscillators, of unitary mass, whose frequencies reproduce the spectrum of characteristic frequencies of the cavity. The mechanical energy of single oscillator is
E (ν, T ) = p2
2 + 2π2ν2q2, (14)
where (q, p) are the canonical variables in the Hamiltonian formalism, i.e. position and momentum. By the equipartition of energy theorem, the average energy is
E (ν, T ) =¯ p2
2 + 2π2ν2q2 = 1
2kBT + 1
2kBT = kBT (15)
2 THEORETICAL SETTING
The number of frequency oscillators in [ν, ν + dν] is [1]:
dN = 8π
c3V ν2dν (16)
where V is the volume of the cavity. So u (ν, T ) dν = dN
V E (ν, T ) =¯ 8π
c3 ν2dνkBT from which
W (ν, T ) = 8π
c3 kBT ν2 (17)
which is the Rayleigh-Jeans radiation formula. Comparing with (13):
f (x) = 8π,
which is manifestly not summable in [0, +∞). We conclude that the aforementioned formula is not acceptable, even if it faithfully reproduces the experimental data in the limit ν ≪ 1.
2.2 Planck’s hypothesis
Let’s go back to the equation
u (ν, T ) dν = dN
V E (ν, T )¯ which is correct. What is wrong, however, is the result:
E (ν, T ) = k¯ BT
deriving from the energy equipartition theorem and therefore, from the Boltzmann distribu- tion law. Let’s review the latter, writing the energy of a single oscillator in the form:
E (q, p; ν) = p2
2 + 2πν2q2 (18)
The oscillator system is in equilibrium at temperature T , and as such follows the Boltzmann statistic according to which the infinitesimal probability that the representative point (p, q) of the energy oscillator (18) lies in the≪volume≫ element dqdp of the phase space Γ, is
Ae−E(q,p;ν)kB T (19)
where
A = Z
Γ
dqdpe−E(q,p;ν)kB T (20)
We pass from the canonical variables (p, q) to the variables (ξ, η) ξ =√
2πνq, η = 1
√2p (21)
So
E (ξ, η) = ξ2+ η2 (22)
2 THEORETICAL SETTING
In the ξη plane we pass to polar coordinates
ξ = r cos ϕ, η = r sin ϕ (23)
πνdqdp = dξdη = rdrdϕ (24)
From (22)
dE = 2ξdξ + 2ηdη (25)
By differentiating the (23):
dE = 2rdr So the (24) becomes
πνdqdp = 1 2dEdϕ
which allows us to perform a variable change in (19).Precisely:
Ae−E(q,p;ν)kB T = A πν
1
2e−kB TE dEdϕ; A = Z +∞
0
dE Z 2π
0
dϕ 1
2πνe−kB TE = 1 ν
Z +∞
0
e−kB TE dE whence the probability that the oscillator of frequency ν has energy in[E, E + dE]
A
2πνe−kB TE dE Z 2π
0
dϕ = A
νe−kB TE dE Substituting the value of A
e−kB TE dE Z +∞
0
e−kB TE dE
(26)
which is independent of frequency. The average energy is
E =¯ Z +∞
0
Ee−kB TE dE Z +∞
0
e−kB TE dE
=
β= 1
kB T
− ∂
∂β ln Z +∞
0
e−kB TE dE = − ∂
∂β ln β−1 = 1 β,
that is
E = k¯ BT,
as we wanted to demonstrate. What did Planck do? We rewrite the (26) e−kB TE dE
Z +∞
0
e−kB TE dE
Here E varies continuously between 0 and +∞. on the other hand, assumed En= nε with n = 0, 1, 2, ..., e ε > 0 assigned quantity (for now unknown). That is, the energy of a single oscillator is quantized. With this mathematical trick the Boltzmann distribution is
e−kB Tnε
+∞
X
n=0
e−kB Tnε
(27)
3 PLANCK’S INTERPOLATION
So the average energy
E =¯
+∞
X
n=0
nεe−kB Tnε
+∞
X
n=0
e−kB Tnε
= − ∂
∂β ln
+∞
X
n=0
e−nεβ (28)
The argument of the logarithm is a geometric series:
x = e−kB Tε < 1 =⇒
+∞
X
n=0
e−kB Tnε =
+∞
X
n=0
xn = 1
1 − x = 1
1 − e−kB Tε It follows
E =¯ ε ekB Tε − 1
Planck put ε = hν, where h is the constant we introduced in the previous issue. So u (ν, T ) dν = dN
V E =¯ 8π
c3ν2dν hν ekB Thν − 1
Taking into account that W = 4cu, we finally obtain the Planck’s formula W (ν, T ) = 2π
c2ν2 hν ekB Thν − 1
(29) For hν ≪ kBT we can expand the exponential in Taylor series, stopping the expansion at first order:
W (ν, T ) = 2π
c2 ν2kBT,
i.e. the Rayleigh-Jeans law. Note that the constant h disappears in this limit. The latter is Planck’s constant.
h ≃ 6.266 · 10−27erg s (30)
The (29) reproduces the correct asymptotic behavior for ν → +∞. In fig. 4 we report the trend of W as a function of frequency, for different values of T , obtaining a family of Planckians.
The physical interpretation of Planck’s result is the following: the processes of emis- sion/absorption of radiation consist of a succession of elementary acts characterized by the same energy quantity of energy ε = hν.
3 Planck’s interpolation
As seen, Planck found the law:
W (ν, T ) = 2π
c2ν2 hν ekB Thν − 1
(31)
as a Rayleigh-Jeans interpolation formula
W (ν, T ) = 8π
c3 kBT ν2 (32)
3 PLANCK’S INTERPOLATION
10 20 30 40
h Ν k T 2. ´ 109
4. ´ 109 6. ´ 109 8. ´ 109 1. ´ 1010 c2 k3
W HΝ ,TL
Figure 4: Behavior of a family of Planckians.
valid in the limit of low frequencies, and of the Wien formula W (ν, T ) = ν2
4c2kBT f
hν kBT
, con f (x) = costante · e−x x
valid in the opposite limit. That being said, we dimensionless the independent variable:
x = hν
kBT =⇒ x ∈ [0, +∞) It follows
W (x, T ) = 2π c2
kBT h
3
g (x) (33)
where
g (x) = x3
ex− 1 (34)
This function has a discontinuity that can be eliminated at x = 0:
x→0lim+g (x) = 0
0 = lim
x→0+
x2
ex−1 x2
= 0
1 = 0 (35)
and is an infinitesimal of order infinitely large as x → +∞
x→+∞lim g (x) = 0 (36)
This guarantees convergence of the integral Z +∞
0
g (x) dx The first derivative is
g′(x) = x2[(3 − x) ex− 3]
(ex− 1)2 ,
from which g′(0) = 0, i.e. the graph of g(x) starts from x = 0 with horizontal tangent. To determine any extrema relative to the finite, we need to solve the equation
(3 − x) ex− 3 = 0,
4 RADIATED POWER
and this must be done numerically, obtaining xmax= 2.82 and it is easy to convince oneself that it is a relative maximum point. Given a right neighborhood of x = 0 with radius δ, we have by expanding the numerator exponential in Taylor series:
x ∈ (0, δ) =⇒ g (x) ≃ x3
1 + x − 1 = x2
It follows that in the aforementioned neighborhood the function behaves like x2. By restoring the old variable, we find the Rayleigh-Jeans formula. In the opposite limit:
x >> 1 =⇒ ex ≫ 1 =⇒ g (x) ≃ x3e−x as we see in the graph of fig. 5
xmax
x 0.5
1.0 1.5 y
y=gHxL y=x2
y=x3e-x
Figure 5: Function (34).
4 Radiated power
To determine the power radiated by a unit area of a black body, all we have to do is integrate the spectral power density. That is
W (T ) = Z +∞
0
W (ν, T ) = 2π c2h
Z +∞
0
ν3 ekB Thν − 1
(37) Performing the usual change of variable:
W (T ) = 2π h3kB4T4
Z +∞
0
x3dx
ex− 1 (38)
The integral appearing in this equation is a special case of Iα =
Z +∞
0
xα−1dx
ex− 1 = Γ (α) ζ (α) con α > 1
where Γ e ζ are respectively the Eulerian gamma function, and the Riemann zeta function.
In the case of even integer index: α = 2n Z +∞
0
x2n−1dx
ex− 1 = (2π)2nB2n
4n
5 THE COSMIC BACKGROUND RADIATION
where B2n are the Bernoulli numbers. In our case it is n = 3 Z +∞
0
x3dx
ex− 1 = (2π)4B4
8 = π4 15 We conclude
W (T ) = σT4 (39)
being
σ = 2π5kB4
15h3c2 (40)
the Stefan-Boltzmann constant.
5 The cosmic background radiation
The study of the black body emission spectrum is also fundamental in Cosmology, in the sense that our universe emits electromagnetic radiation having the spectrum of a black body in thermodynamic equilibrium at the temperature T0 = 2.7 K.
Let’s proceed in order.
In 1964, physicists Arno Penzias and Robert Woodrow Wilson were working for a tele- phone company, trying to implement a technology for using microwaves. In this circumstance the two encountered the reception of a signal dominated by a noise The cleaning of the an- tenna from the pigeon dung had no effect, so Pensias and Wilson established the existence of a signal coming not from an assigned direction in the celestial sphere , by virtue of the isotropy of the signal itself (rotating the antenna did not decrease the intensity). A more detailed analysis showed that it was the emission of a black body in equilibrium at ∼ 3 K, for which the maximum peak corresponded to a wavelength of the microwave spectrum. In this regard, we recall the relation established above:
λmaxT = costante ≃ 0.290 cm K
so knowing the equilibrium temperature we can determine the wavelength corresponding to the emission peak. More accurate measurements give as equilibrium temperature:
T0 = 2.736 ± 0.017 K
Then expressing the spectral power density emitted per unit area, as a function of the wavelength λ = c/ν
W (λ, T ) = 2πc λ3
h ekB T λhc − 1 whose graph is shown in Fig.6
In any model of an expanding universe, such a low temperature is equivalent to asserting the existence of an epoch in which the temperature was extremely higher (theoretically infinite). In terms of observational data, this constituted evidence for the existence of a
“hot” beginning of the universe (hot big-bang).
REFERENCES
0.1 0.2 0.3 0.4
Λ 2 ´ 10-16
5 ´ 10-16 1 ´ 10-15 2 ´ 10-15 5 ´ 10-15 1 ´ 10-14 W
Figure 6: Spectral density of the emission power (on a logarithmic scale) of a black body in equilibrium at temperature T = 2.7 K.
References
[1] Caldirola P. Prosperi G., Cirelli R. Introduzione alla fisica teorica UTET.
