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Units, Constants and Useful Formulas
Blackbody Radiation
-------------------
Blackbody radiation is a type of electromagnetic
radiation emitted by a blackbody (a hypothetical
body that completely absorbs all radiant energy
falling upon it, reaches some equilibrium temperature,
and then reemits that energy as quickly as it absorbs
it). It can either be within or surround a body in
thermodynamic equilibrium with its environment.
It can also be emitted by an opaque and non-reflective
'black' body held at constant, uniform temperature.
The radiation has a specific spectrum and intensity
that depends only on the temperature of the body.
The radiated energy can be considered to be produced
by standing waves or resonant modes of the cavity
which is radiating. Consider electromagnetic radiation
inside a cavity with sides, L. The standing waves
in the cavity have to satisfy the wave equation:
∇^{2}E = (1/c^{2})∂^{2}/∂t^{2}. With solution:
E = E_{0}sin(2πx/λ)sin(2πy/λ)sin(2πz/λ)sin(2πct/λ)
This can be written as:
E = E_{0}sin(k_{x}x)sin(k_{y}y)sin(k_{z}z)sin(2πct/λ)
where k = 2π/λ
The solution must give zero amplitude at the walls,
since a non-zero value would dissipate energy and
violate the condition of equilibrium. This condition
can be met by:
L = nλ/2 ∴ λ = 2L/n
Now k = 2π/λ
= nπ/L
In 3D we can write:
k^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = (n_{x}^{2} + n_{y}^{2} + n_{z}^{2})π^{2}/L^{2}
_{ } = n^{2}π^{2}/L^{2}
The number of states, N, in n space occupies 1/8
of a sphere. The volume of a shell of radius n
and thickness dn is:
N(n)dn = (1/8)4πn^{2}dn
Now n^{2} = k^{2}L^{2}/π^{2} so n = kL/π and dn = Ldk/π
N(k)dk = (1/8)(4πk^{2}L^{2}/π^{2})(L/π)dk
= (1/2π^{2})L^{3}k^{2}dk
N(k) = (1/2π^{2})L^{3}∫k^{2}dk
= (1/6π^{2})L^{3}k^{3}
ω = ck gives:
N(ω) = (1/6π^{2})(L^{3}/c^{3})ω^{3}
ω = 2πν gives:
N(ν) = (4π/3)(L^{3}/c^{3})ν^{3}
k = 2π/λ gives:
N(λ) = 4πL^{3}/3λ^{3}
Now there are 2 polarizations allowed for the
photon. Therefore, the constant needs to be
multiplied by 2:
N(ω) = (1/3π^{2})(L^{3}/c^{3})ω^{3}
N(ν) = (8π/3)(L^{3}/c^{3})ν^{3}
N(λ) = 8πL^{3}/3λ^{3}
This is the total number of modes. To get the
distribution by ω, ν or λ it is necessary to
take the derivative of the number of modes with
respect to ω, ν or λ (i.e., the rates of change).
Therefore:
dN(ω)/dω = (1/π^{2})(L^{3}/c^{3})ω^{2}
dN(ν)/dν = (8πL^{3}/c^{3})ν^{2}
check: ω = 2πν and dω = 2πdν
∴ (1/2π)dN(ν)/dν = (1/π^{2})(L^{3}/c^{3})4π^{2}ν^{2}
∴ dN(ν)/dν = (8πL^{3}/c^{3})ν^{2}
dN(λ)/dλ = -8πL^{3}/λ^{4}
check: ν = c/λ and dν = -c/λ^{2}dλ
∴ (-λ^{2}/c)dN(λ)/dλ = (8π)(L^{3}/c^{3})c^{2}/λ^{2}
∴ dN(dλ)/dλ = -8πL^{3}/λ^{4}
In the latter case, the negative sign shows that
the number of modes decreases with increasing
wavelength.
Equipartion of Energy
---------------------
The theorem of equipartition of energy states
that each mode of radiation associated with a
classical harmonic oscillator has an energy equal
to K_{B}T. Therefore, the energy of the radiation
per unit ω, ν or λ per unit volume (≡ energy
density per unit ω, ν or λ) is:
E_{ω} = (ω^{2}/π^{2}c^{3})(K_{B}T)
or,
E_{ν} = (8πν^{2}/c^{3})(K_{B}T)
or,
E_{λ} = (8π/λ^{4})(K_{B}T))
Dimensional check:
LHS: [E/ωV] = [mL^{2}/t^{2}/(1/t)L^{3}] = [m/Lt]
RHS: [(mL^{2}/t^{2})(1/t^{2})/L^{3}/t^{3}] = [m/Lt]
This is the classical RAYLEIGH-JEANS law that
says the energy is proportional to the square
of the frequency. Since there was no reason to
believe that their existed a maximum frequency
inside the cavity, the energy would increase
towards infinity as the frequency is increased.
However, experimentally this did not happen. This
was referred to as the ULTRAVIOLET CATASTROPHE.
It wasn't until the advent of Quantum Mechanics
that this paradox was solved. Planck assumed
that the sources of radiation are atoms in a
state of oscillation and that the vibrational
energy of each oscillator may have any of a
series of discrete values but never any value
between. Planck further assumed that when an
oscillator changes from a higher state of energy
to a lower state of energy a discrete quantum of
radiation given by E = hν is emitted.
Planck was able to calculate the value of h from
experimental data. His result, 6.55 x 10^{-34} J/s,
is within 1.2% of the currently accepted value. He
was also able to make the first determination of
the Boltzmann constant K_{B} from the same data and
theory.
If we replace the classical oscillator with the
quantum oscillator we get:
K_{B}T -> hν/(exp(hν/K_{B}T) - 1)
Which results in:
E_{ω} = (ω^{2}/π^{2}c^{3})[hω/((exp(hω/K_{B}T) - 1)]
or,
E_{ν} = (8πν^{2}/c^{3})[hν/((exp(hν/K_{B}T) - 1)]
or,
E_{λ} = (8π/λ^{4})[hc/λ((exp(hc/K_{B}λT) - 1)]
This is PLANCK'S RADIATION LAW.
Note: For small ν, exp(x) - 1 = x so the formulae
reduces to the classical case.
A comparison between the Rayleigh-Jeans law and
Planck's law is shown below.
Stefan-Boltzmann Law
--------------------
The Stefan-Boltzmann law gives the total energy
radiated per unit surface area of a black body
across all wavelengths per unit time. It is the
radiative power per unit area.
To find the radiated power per unit area from
a surface it is necessary to multiply the energy
density by c/4. This comes about because in
thermal equilibrium the inward energy flow equals
the outward flow giving a factor of 1/2 for the
radiated power outward. Then it is required to
average over all angles, which gives another
factor of 1/2. Therefore, the radiated power
per unit area as a function of wavelength is:
P_{λ} = (c/4)(8π/λ^{4})[hc/λ((exp(hc/K_{B}λT) - 1)]
= (2πhc^{2}/λ^{5})[1/((exp(hc/K_{B}λT) - 1)]
If we make the substitution x = hc/K_{B}T and
dx = -(hc/λ^{2}K_{B}T)dλ, we can write:
P_{λ} = 2π(K_{B}T)^{4}/(h^{3}c^{2})∫x^{3}/(exp(x) - 1)dx
This is a standard integral equal to π^{4}/15. Thus,
we get:
P_{λ} = (2π^{5}K_{B}^{4}/15h^{3}c^{2})T^{4}
= σT^{4}
To get the total power in Watts it is
necessary to multiply the above equation by the
area, A.
For hot objects other than ideal radiators, the
law is expressed in the form:
P_{λ} = eAσT^{4}
where e is the emissivity of the object (e = 1
for an ideal radiator).
If the object is radiating/absorbing energy to
its surroundings at temperature T_{environment},
the net radiation loss/gain rate takes the form:
P_{λ} = eAσ(T_{object}^{4} - T_{environment}^{4})
If T_{object} > T_{environment} the object is radiating.
If T_{object} < T_{environment} the object is absorbing.
Wien's Displacement Law
-----------------------
Return to the Planck radiation law:
E_{λ} = (8πL^{3}/3λ^{4})hν/((exp(hc/K_{B}Tλ) - 1)
The maximum value of this function can be found by
differentiating w.r.t. λ, setting the result = 0
and solving for λ. This gives:
λT = (hc/K_{B})/5[1 - exp(-hc/K_{B}λT)]
Which must be solved numerically to give:
λ_{peak}T = 2.898 x 10^{-3} meter Kelvin
This is the WIEN DISPLACEMENT LAW. It says that
as the temperature of a blackbody radiator increases,
the overall radiated energy increases and the peak
of the radiation curve moves to shorter wavelengths.
When the maximum is evaluated from the Planck
radiation formula, the product of the peak wavelength
and the temperature is found to be a constant. This
is useful for the determining the temperatures of
objects whose temperature is far above that of their
surroundings, such as hot radiant stars, by looking
at their spectra.