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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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