Atomic Physics Transition Probabilities

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					   Physics 80301 – Fall 2009




Atomic Physics
Transition Probabilities
  The First Computer (the ABC)?
  John Atanasoff, American electronics engineer. Atanasoff was … in the long list
  of scientists involved in “who discovered/invented” first. Professor Atanasoff
     d    d         d Clifford B        b il h        ld' first l      i di i l
  and graduate student Cliff d Berry built the world's fi electronic-digital
  computer at Iowa State University between 1939 and 1942. The Atanasoff-Berry
  Computer (the ABC) represented several innovations in computing, including a
                     arithmetic,        processing                 memory
  binary system of arithmetic parallel processing, regenerative memory, and a
  separation of memory and computing functions. However, Presper Eckert and
  John Mauchly (Atanasoff had actually sent his ideas to Mauchly for review)
                                               device               computer
  were the first to patent a digital computing device, the ENIAC computer. A
  patent infringement case (Sperry Rand Vs. Honeywell, 1973) voided the ENIAC
  patent as a derivative of John Atanasoff's invention. Eckert and Mauchly
  received most of the credit for inventing the first electronic-digital computer.
                                           g                       g        p
  Historians now believe that the Atanasoff-Berry computer (the ABC) was the
  first.

Clifford E. Berry (1918 - 1963) was born in Gladbrook, Iowa on 19 April 1918 to Fred Gordon
Berry and Grace Strohm…. http://www.scl.ameslab.gov/ABC/Biographies.html#Berry
         Review – detailed balance
               (Einstein) - a
Emission and absorption processes
in a 2-level system


  Rate into N1 = rate out of N2




     Then in equilibrium


  Principle of detailed balance gives for any states i and j:
                                                 the t     f b     ti ( i i )
                                   Pij (Pji) are th rates of absorption (emission)
                                   ωij = (Ej – Ei)/ħ
            Review – detailed balance
                  (Einstein)
                  (Ei t i ) - b
 With no radiation present, we can solve:
 Giving the lifetime tj in terms of transition
 rates Aji from upper level j to all lower
 levels i.



   Assuming the Boltzmann and Planck distributions for the states and the radiation
             l
   respectively:




We obtain the relationships
between the Einstein A and B
coefficients:
                            E-
                Atoms in an E-M field
In an electromagnetic field: E = -▼φ - ∂A/∂t         ; B = ▼ x A,
the Hamiltonian of the atom in the field becomes




  Which can be written as the zero order atomic part, plus the interaction part:




Assume the radiation field is a
superposition of planewaves (ω, k),
and φ=0.
Choosing a gauge ▼·A = 0 (waves are transverse), and neglecting the two photon term A2 ,
assuming only a weak field, yields the perturbation hamiltonian:

                     H’ = (e/m) A · p
              Semiclassical approach
  We assume only weak fields are present, so that the atomic state wavefunctions
  (stationary state eigenfunctions of H0 ) will be close to those of the atom in a vacuum.
  The external-field interactions will lead to time dependent changes of the distribution of
  states of the atom.




   Where cn(t) are the time-dependent occupation indices of the atomic states.

   Substituting theses states the time-dependent Schrodinger equation:




Since the wavefunctions are
eigenfunctions of H0
     Semiclassical approach, part 2

     Assuming that we begin in the i-th state: ci(0) = 1,
     and also that H’ is small so that all other cn remain close to zero,
     Then we can approximate the right –hand side to one term….




Familiar trick: multiply from left by the complex conjugate wavefunction ψj*, and
integrate:




   Gives the time dependence of cj – but note that H’ is also time-dependent…
      The Radiation Hamiltonian H’
                                H
For a plane wave, the vector potential is
 - the unit vector describes the polarization             and let



     Then:



  Integrating each term gives our answer: (the second term is the only one retained)




Near the absorption frequency ω ≈ ωji, the first term can be neglected..
(the rotating wave approx.)
g    g
giving…
                        B-
   Link to the Einstein B-coefficient
   We need to calculate the transition rate |cj(t) |2/t, and relate it to the B-
   coefficient – then the A coefficient (spontaneous emission rate) is also known
                                          important
   Assuming no phase relationships are important, we need the energy density of
   the plane waves:
       The energy in terms of A0 is

                Where A02 =

   N   integrate the transition rate over all “relevant” frequencies ω1 to ω2:
   Now i          h       ii               ll “ l      ”f        i




For times large compared with 1/ω, we can write the


Giving the transition rate:
                              Discussion
1. From the definition of the Einstein coefficient B: dN/dt = B.ρ(ω) , we get




 2.   We used 1st order perturbation theory (weak field only, not a laser present)
 3.   We have assumed non-degenerate levels (easy to fix with gi gj, etc)
 4.
 4    Only frequencies near the transition frequency (assumed infinite lifetime upper level)
 5.   Only one electron affected (would need to sum over all electrons)

 6. Evaluating the matrix element will lead to selection rules for the transitions…
    - electric and magnetic multipole transitions -
Electric dipole approximation, #1
    Now expand the matrix element in terms of ik·r,
                   |k| 2π/λ
    the wavevector |k|=2π/λ

                                                    1st term => the dipole term

     Size of 2nd term:

Hence, the first term dominates -- what is the difference for nuclear transitions?
Evaluation of the first term – use the commutator relations (eg for linearly polarized light)




     H0 commutes so that

  So that the transition
  probability is
    Electric dipole approximation, #2
          Hence for unpolarized radiation


        where

 Using the relations between Aij and Bij,
 we get the spontaneous emission rate:

    Q
    Questions:              g               yp             p
                      How big is Aji for a typical visible photon?
                      How big is Aji for a typical gamma ray?

For a set of non-degenerate lower levels, e.g. mi


Upper mj states all have the same lifetimes: thus


  And the transition rate can be written
  Transition rate parameters, #1
  Definition: The Line Strength S


Relation between S and A for a given transition



Energy loss rate P·ΔE:


  Brief review: a classical damped harmonic oscillator: - > exponential decay

  d2x/dt2 + γdx/dt + ω2x =0 => x = x0 e-γt/2 cos ωt

  Energy W = ½ m(dx/dt)2 -> -dW/dt = γW(t) = (1/4πε0).(e2/3c3).ω4x02

  having used γ = (1/4πε0).(2/3) (e2ω2/mc3) as the radiation damping force
 The transition oscillator strength
Define the oscillator strength
or f-value as


 In       f
 I terms of Bij

 In terms of Aji ..
              j       fij=Aji gi/gj

   (For completeness),
   in terms of the transition strength:
                                         ( )
      Selection rules for E1 transitions (m)
   We have found the basic
   matrix element as

Treat each x, y, z component separately:


                  t
        z-component




     x, y components




 These are the m-selection rules giving so-called π and σ polarizations – the electric
 vectors parallel and perpendicular to the z-axis
                                            ( )
         Selection rules for E1 transitions (L)
 Just from symmetry arguments – since r is odd, and the total integral must be
 even, then the two wavefunctions ψi and ψj must have opposite parity.
 Note l h i            h     h        l               f1 h
 N also that since a photon has angular momentum of 1, then Δl = ± 1

In detail, we can use the orthogonality of the associated legendre polynomials:




 => Calculate an emission rate R(ll’), where F(..) are the angular parts




     (do one example for homework)
 Higher order multipole transitions
These come from the higher order terms in the expansion of the interaction operator




   For example, just keeping the first 2 terms leads to magnetic dipole and
   electric quadrupole terms… terms of the type < p (k·r) >

           Example term (trick):

   First part gives
   The Magnetic dipole transition
                                                  =

  The second part can be written:



The Electric quadrupole
transition                                                     =