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8.2 Einstein Coefficients

m
n
m
The three coefficients α , β , and β , describing the probabilities of the
n
n
m
three possible transitions:
m
df sp = α dν dΩ dt,
n
n
df ab = β u ν (s) dν dΩ dt,
m
m
df st = β u ν (s) dν dΩ dt, (12)
n
are called the “Differential Einstein Coefficients”. They are usually isotropic,
i.e. they do not depend on the direction of propagation of radiation.
Einstein originally described the transition probabilities in the following
form, for the case of an isotropic radiation field, and with the particles at
rest:
m
dW sp = A dt,
n
n
dW ab = B U ν dt,
m
m
dW st = B U ν dt, (13)
n
−1
where U ν = H u ν (s) dΩ is the spectral energy density (in units of J m −3 Hz ).
n
m
m
The coefficients A , B , and B , are called the “Einstein Coefficients”. In
n
n
m
such a case, the spectral lines due to the transitions between discrete energy
levels must be monochromatic lines having no Doppler broadening, since the
particles are at rest. If we introduce f(ν) as the probability distribution of
frequency within a spectral line in a real interstellar medium in motion, the
two sets of coefficients are related to each other, as follows:
m
A f(ν)
α m = n ,
n
4π
n
n
β m = B f(ν),
m
m
β n m = B f(ν). (14)
n
8.3 Number Density of Photons
Let the number densities of particles (number of particles per unit volume) in
the states Z m and Z n be n m and n n , respectively. Then the number density of
photons emitted by the Z n → Z m transition into the solid angle dΩ around
the direction -s, within the bandwidth dν, during the time interval dt, is
equal to:
m
m
n n (df sp + df st ) = n n (α + β n u ν ) dν dΩ dt.
n

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