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## Bound – Bound Transitions

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**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**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**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**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**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**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**Now we do Some Math**Solution of () for Bν Boltzman Equation Substitute for Nj* and note that Ni* falls out! Bound Bound Transitions**Onward!**Use this to multiply the top and bottom of the previous equation But this is just the Planck function so Bound Bound Transitions**A Revelation!**The Intersection of Thermodynamics and QM Bound Bound Transitions**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**The Equation of Transfer**Note the Following: Bound Bound Transitions**The Coefficients Are:**Line Absorption Coefficient usually denoted lν Departure Coefficient Gather the terms on I Boltzmann Equation giBij = gjBji Bound Bound Transitions**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**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**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**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**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**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**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**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**The Solution is**Damped Harmonic Oscillator • γ is called the classical damping constant Bound Bound Transitions**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**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**The Broadening Profile**Lorentz Profile Bound Bound Transitions**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 = ht ≈ Ah • This means that ν A γ Bound Bound Transitions**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**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**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**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**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**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**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**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**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