Bound – Bound Transitions

1. Bound – Bound Transitions

2. Einstein Relation for Bound- Bound Transitions • Lower State i : gi = statistical weight • Upper State j : gj = statistical weight • For a bound-bound transition there are three direct processes to consider: • #1: Direct absorption: upward i → j • #2: Return Possibility 1: Spontaneous transition with emission of a photon • #3: Return Possibility 2: Emission induced by the radiation field Bound Bound Transitions

3. The Absorption Process Possibility 1: Upward i → j • Ni(ν) Rij dω/4π = Ni(ν) Bij Iν dω/4π • Rij is the rate at which the transition occurs • Ni(ν) is the number of absorbers cm-3 in state i which can absorb in the range (ν, ν + dν) • The equation defines the coefficient Bij • Transitions are not infinitely sharp so there is a spread of frequencies φν (the absorption profile) Bound Bound Transitions

4. Let Us Absorb Some More • If the total number of atoms is Ni then the number of atoms capable of absorbing at frequency ν is Ni(ν) = Niφν • In going from i to j the atom absorbs photon of energy hνij = Ej - Ei • The rate that energy is removed from the beam is: • aν is a macroscopic absorption coefficient such as Κν Bound Bound Transitions

5. The Return Process: j → i Going Down Once • Spontaneous transition with the emission of a photon • Probability of a spontaneous emission per unit time ≡ Aij • Emission energy rate: • ρjν (spontaneous) = Nj Aji (hνij/4π) ψν • The emission profile is normalized: ∫ψνdν = 1 Bound Bound Transitions

6. The Return Process: j → i Going Down Twice • Emission induced by the radiation field • ρjν(induced) = Nj Bjiψν Iν (hνij/4π) • Note that the induced emission profile has the same form as the spontaneous emission profile. • Spontaneous emission takes place isotropically. • Induced emission has the same angular distribution as Iν Bound Bound Transitions

7. Assume TE This gives us Iν = Bν and the ratio of the state populations! • Iν = Bν and the Boltzman Equation • Ni* Bij Bν = Nj* Aji + Nj* Bji Bν () • Upward = Downward • What did we do to get ()? • Integrate over ν and assume that Bν≠f(ν) over a line width. The two pieces that depend on ν are Bν (assumed a constant) and (φν,ψν) but the latter two integrate to 1! Bound Bound Transitions

8. Now we do Some Math Solution of () for Bν Boltzman Equation Substitute for Nj* and note that Ni* falls out! Bound Bound Transitions

9. Onward! Use this to multiply the top and bottom of the previous equation But this is just the Planck function so Bound Bound Transitions

10. A Revelation! The Intersection of Thermodynamics and QM Bound Bound Transitions

11. What Does this Mean? • We have used thermodynamic arguments but these equations must be related only to the properties of the atoms and be independent of the radiation field • ==> These are perfectly general equations • ==> These arguments predicted the existence of stimulated emission. • NB: Stimulated (induced) emission is not intuitively obvious. However, it is now an observed fact. Bound Bound Transitions

12. The Equation of Transfer Note the Following: Bound Bound Transitions

13. The Coefficients Are: Line Absorption Coefficient usually denoted lν Departure Coefficient Gather the terms on I Boltzmann Equation giBij = gjBji Bound Bound Transitions

14. The Line Source Function Source Function = j/ Divide by NjBjiψν Use relations between A & B’s The equation of transfer for a line! Bound Bound Transitions

15. Let Us Simplify Some • Assume φν = ψν (Not unreasonable) • If we assume LTE then bi = bj = 1 • Then lν = NiBijφν(1-e-hν/kT) • The term (1-e-hν/kT) is the correction for stimulated emission • Sν = Bν Bound Bound Transitions

16. An Alternate Definition With Respect to the Energy Density • Define Bi,j as • uνBij = Bij 8π/c3 (hν3/(ehν/kT-1)) • Bi,j is the probability that an atom in lower level i will be excited to upper level j by absorption of a photon of frequency ν = (Ej - Ei) / h • This B is defined with respect to the energy density U of the radiation field, not Iν (=Bν) as previously. • For this definition • Aji = (8πhν3/c3) Bji • giBij = gjBji Bound Bound Transitions

17. Now Let Us Try To Determine Aji, Bij, and Bji • Classical Oscillator and EM Field • Dimensionally correct but can be off by orders of magnitude • QM Atom and Classical EM Field • Correct Expression for Bij • QM Atom and a Quantized EM Field • Gives Bij, Bji, and Aji • However, note that Bij, Bji, and Aji are interrelated and if you know one you know them all! Bound Bound Transitions

18. A Classical Approach • The probability of an event generally depends on the number of ways that it can happen. Suppose photons in the range (ν, ν + dν) are involved. The statistical weight of a free particle: Position (x, x+dx) and momentum (p, p+dp) is dN/h3 = dxdydzdpxdpydpz/h3 • Then the statistical weight per unit volume of a particle with total angular momentum p, in direction dΩ, is p2dpdΩ/h3. Bound Bound Transitions

19. Free Particles • Momentum of a photon is p = hν/c so the statistical weight per unit volume of photons in (ν, ν+dν) is ν2dνdΩ/hc2. • This means high frequency transitions have higher transition probabilities than low frequency transitions. Bound Bound Transitions

20. Semiclassical Treatment of an Excited Atom • Oscillating Dipole (electric) in which the electron oscillates about the nucleus. • For the case of no energy loss the equation of motion is: • ν0 is the frequency of the oscillation • The electric dipole radiates (classically) and energy is lost due to the radiation Bound Bound Transitions

21. A Damped Oscillator • The energy loss leads to a further term in the equation of motion which slows the electron • The damping force is • If F is small then the motion can be considered to be near a simple harmonic r = r0 cos (2πν0t) Bound Bound Transitions

22. The Solution is Damped Harmonic Oscillator • γ is called the classical damping constant Bound Bound Transitions

23. Electric Field • The field set up by the electron is proportional to the electron displacement • At time t = (γ/2)-1 we get E(t) = E0/e which characterizes the time scale of the decay Bound Bound Transitions

24. A Result! • The decay time must be proportional to Aji -the probability of a spontaneous transition downwards • Aclassical = γ • For Hα, = 6563Å; ν0 = c/λ so γ  5(107) s-1 • The field decays exponentially so the frequency is not monochromatic. To get the frequency dependence do a Fourier analysis • The square of the spectrum is the intensity of the field! Bound Bound Transitions

25. The Broadening Profile Lorentz Profile Bound Bound Transitions

26. More On Broadening • In QM the natural broadening arises due to the uncertainty principle • The uncertainty in the time in which an atom is in a state is its lifetime in that state. • Average lifetime A-1 • Thus the uncertainty E = ht ≈ Ah • This means that ν A  γ Bound Bound Transitions

27. Oscillator Strengths • Since γ is of the same order as A we usually express exact values of A in terms of the classical value of γ. • Oscillator Strength nupper→ mlower • Anm ≡ 3 (gm/gn) fnmγ • = (gm/gn) (8π2e2ν2/mec3) fnm • = (gm/gn) 7.42(10-22) ν2 fnm • Bmn = (πe2/mehν) fnm Bound Bound Transitions

28. Oscillator Strengths Balmer Series of H This is known as Kramer’s Formula. GI is the Gaunt Factor which is order 0.1 - 10 Bound Bound Transitions

29. The Absorption Coefficient • We are now ready to determine the absorption coefficient aν per atom of an absorption line centered at ν0. • It is defined so that the probability of absorption per unit path length of a photon is Naν where N is the number density of atoms capable of absorption. • Consider a beam of intensity Iν traveling across a unit cross-sectional area. In time dt the photons have traveled cdt. Bound Bound Transitions

30. Energy Considerations • The energy removed from the beam in (ν, ν + dν) is Iν naν cdt. Alternately, the number of transitions is Iν naν cdt / hν. The volume occupied by the photons is cdt (unit cross sectional area). • The number of transitions per unit volume per unit time per unit frequency due to the beam is Iν naν / hν. • Integrating over solid angle and assuming thermal equilibrium (4πIν = cUν) the number of absorptions is cUν naν / hν. Bound Bound Transitions

31. Onward • We now integrate over frequency over the line profile only to obtain the number of transitions from lower state to upper state per unit volume per unit time • But this is equal to Uν Bnm N so Bound Bound Transitions

32. Line Absorption The nature of line absorption • Uν does not vary across the line • aν is small except near ν0: ν0 = (En-Em) / h • Uν/hν varies slowly across the line For ν = ν0 cm2 Hz Bound Bound Transitions

33. We are about done • The integral of aν over ν can be thought of as the total absorption cross section per atom initially in the lower state. We rewrite aν as • φν is the absorption profile and we expect it to follow the Lorentz profile Bound Bound Transitions

34. Combine the Profile • In principal fnm ~ γ but there are other broadening mechanisms. • Natural width (γ) is small compared to other mechanisms (Stark, Doppler, van der Waals, etc) • For ν = ν0: a0 = (πe2/mec) (fnm/γ) So Bound Bound Transitions

35. Stimulated Emission • We have yet to account for stimulated emission • We usually assume emission is isotropic but if it goes as BmnIν then it correlates with Iν • We treat this as a negative absorption and reduce the value of aν by the appropriate factor. • In our discussion of the Einstein coefficients we found the factor to be (1 - e-(hν/kT)) so the corrected aν is Bound Bound Transitions