1
Chapter 18
BoseEinstein Gases (Part I)
18.1 BlackBody Radiation—a perfect quantal gas
18.1.1 Introduction
First, one may ask: What is blackbody radiation?
A perfectly black body radiation
absorbs all radiation falling on it.
This is very similar to the case that radiation enters a
small hole in the wall of an opaque enclosure: after many reflections nearly all radiation
is absorbed by the inner surface of the enclosure.
(Fig. 18.1)
All bodies emit electromagnetic (EM) radiation by virtue of their temperature but usually
this radiation is not in thermal equilibrium.
If we consider the radiation within an opague
enclosure whose wall is kept at a uniform temperature T, then radiation and walls reach a
thermal equilibrium state, in which the radiation has quite definite properties.
To study
this equilibriumstate radiation, one cuts a small hole in the wall of the enclosure.
Such a
hole, if sufficiently small, will not disturb the equilibrium in the cavity and the emitted
radiation will have the same properties as the cavity radiation.
The emitted radiation also
has the same properties as the radiation emitted by a perfectly black body at the same
temperature T as the enclosure.
For this reason, cavity radiation is also called
black
body radiation
.
To the end of 19
th
century, blackbody radiation was one of the most outstanding
problems in classical physics.
By that time, it had been known that
both heat radiation
and light radiation are essentially the same – EM wave
.
The nature of thermal EM
radiation was a subject of intense interest—a central issue in physics in that time!
In
these studies, a central physical quantity is the
density of radiation energy u(
)
:
u(
)d
is the radiation energy in the frequency range of
to
+d
.
In 1894, Wien found an empirical formula:
,
)
(
/
3
1
2
d
e
c
d
u
T
c
(18.1)
which agrees with the experimental data well at high frequency
.
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(Fig. 18.2)
In 1900, Rayleigh and Jeans applied classical EM theory and classical statistical physics
to treat blackbody radiation and obtained
,
8
)
(
2
3
d
kT
c
d
u
(18.2)
where c is the speed of light. This theoretical formula agrees with the experimental data
well at low frequency
.
As
, however, u(
)
.
This is the socalled ultraviolet
catastrophe—one of the biggest problems faced by classical physics
!
Then in 1900,
Max Plank
obtained an excellent empirical formula
.
1
)
(
3
/
1
2
d
e
c
d
u
T
c
(18.3)
This formula agrees very well with the experimental data in the whole frequency range,
both low and high
. It can be easily seen from Eq.(18.3) that as
, u(
)
0.
This
removed the socalled ultraviolet catastrophe!
At high
, Eq. (18.3) reduces to Eq.(18.1).
More remarkably, Plank did not stop there!
Two months later, he found that Eq.
(18.3) can be derived theoretically if one makes the following assumption: For EM
radiation with a frequency
, its energy can only be absorbed or emitted in the multiple of
h
, that is, E=nh
, where n is an integer.
In other words, the energy of EM radiation
in
emission and absorption
is quantized. This is the concept (assumption) of
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 Winter '10
 Dr.asdas
 Thermodynamics, Radiation, Black body

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