Bound – Bound Transitions

Bound Bound Transitions2 Einstein Relation for Bound- Bound Transitions Lower State i : g i = statistical weight Lower State i : g i = statistical weight Upper State j : g j = statistical weight Upper State j : g j = statistical weight For a bound-bound transition there are three direct processes to consider: For a bound-bound transition there are three direct processes to consider: #1: Direct absorption: upward i → j #1: Direct absorption: upward i → j #2: Return Possibility 1: Spontaneous transition with emission of a photon #2: Return Possibility 1: Spontaneous transition with emission of a photon #3: Return Possibility 2: Emission induced by the radiation field #3: Return Possibility 2: Emission induced by the radiation field

Bound Bound Transitions3 The Absorption Process N i (ν) R ij dω/4π = N i (ν) B ij I ν dω/4π N i (ν) R ij dω/4π = N i (ν) B ij I ν dω/4π R ij is the rate at which the transition occurs R ij is the rate at which the transition occurs N i (ν) is the number of absorbers cm -3 in state i which can absorb in the range (ν, ν + dν) N i (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 B ij The equation defines the coefficient B ij Transitions are not infinitely sharp so there is a spread of frequencies φ ν (the absorption profile) Transitions are not infinitely sharp so there is a spread of frequencies φ ν (the absorption profile) Possibility 1: Upward i → j

Bound Bound Transitions4 Let Us Absorb Some More If the total number of atoms is N i then the number of atoms capable of absorbing at frequency ν is N i (ν) = N i φ ν If the total number of atoms is N i then the number of atoms capable of absorbing at frequency ν is N i (ν) = N i φ ν In going from i to j the atom absorbs photon of energy hν ij = E j - E i In going from i to j the atom absorbs photon of energy hν ij = E j - E i The rate that energy is removed from the beam is: The rate that energy is removed from the beam is: a ν is a macroscopic absorption coefficient such as Κ ν a ν is a macroscopic absorption coefficient such as Κ ν

Bound Bound Transitions5 The Return Process: j → i Spontaneous transition with the emission of a photon Spontaneous transition with the emission of a photon Probability of a spontaneous emission per unit time ≡ A ij Probability of a spontaneous emission per unit time ≡ A ij Emission energy rate: Emission energy rate: ρj ν (spontaneous) = N j A ji (hν ij /4π) ψ ν ρj ν (spontaneous) = N j A ji (hν ij /4π) ψ ν The emission profile is normalized: The emission profile is normalized: ∫ψ ν dν = 1 ∫ψ ν dν = 1 Going Down Once

Bound Bound Transitions6 The Return Process: j → i Emission induced by the radiation field Emission induced by the radiation field ρj ν (induced) = N j B ji ψ ν I ν (hν ij /4π) ρj ν (induced) = N j B ji ψ ν I ν (hν ij /4π) Note that the induced emission profile has the same form as the spontaneous emission profile. Note that the induced emission profile has the same form as the spontaneous emission profile. Spontaneous emission takes place isotropically. Spontaneous emission takes place isotropically. Induced emission has the same angular distribution as I ν Induced emission has the same angular distribution as I ν Going Down Twice

Bound Bound Transitions7 Assume TE I ν = B ν and the Boltzman Equation I ν = B ν and the Boltzman Equation N i * B ij B ν = N j * A ji + N j * B ji B ν (  ) N i * B ij B ν = N j * A ji + N j * B ji B ν (  ) –Upward = Downward What did we do to get (  )? What did we do to get (  )? –Integrate over ν and assume that B ν ≠ f(ν) over a line width. The two pieces that depend on ν are B ν (assumed a constant) and (φ ν,ψ ν ) but the latter two integrate to 1! This gives us I ν = B ν and the ratio of the state populations!

Bound Bound Transitions8 Now we do Some Math Solution of (  ) for B ν Boltzman Equation Substitute for N j * and note that N i * falls out!

Bound Bound Transitions9 Onward! Use this to multiply the top and bottom of the previous equation But this is just the Planck function so

Bound Bound Transitions10 A Revelation! The Intersection of Thermodynamics and QM

Bound Bound Transitions11 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 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 are perfectly general equations ==> These arguments predicted the existence of stimulated emission. ==> These arguments predicted the existence of stimulated emission. NB: Stimulated (induced) emission is not intuitively obvious. However, it is now an observed fact. NB: Stimulated (induced) emission is not intuitively obvious. However, it is now an observed fact.

Bound Bound Transitions12 The Equation of Transfer Note the Following:

Bound Bound Transitions13 The Coefficients Are: Departure Coefficient Gather the terms on I Boltzmann Equation g i B ij = gjB ji Line Absorption Coefficient usually denoted l ν

Bound Bound Transitions14 The Line Source Function Source Function = j/  Divide by N j B ji ψ ν Use relations between A & B’s The equation of transfer for a line!

Bound Bound Transitions15 Let Us Simplify Some Assume φ ν = ψ ν (Not unreasonable) Assume φ ν = ψ ν (Not unreasonable) If we assume LTE then b i = b j = 1 If we assume LTE then b i = b j = 1 Then = N i B ij φ ν (1-e -hν/kT ) Then l ν = N i B ij φ ν (1-e -hν/kT ) –The term (1-e -hν/kT ) is the correction for stimulated emission S ν = B ν S ν = B ν

Bound Bound Transitions16 An Alternate Definition Define B i,j as Define B i,j as u ν B ij = B ij 8π/c 3 (hν 3 /(e hν/kT -1)) u ν B ij = B ij 8π/c 3 (hν 3 /(e hν/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 = (E j - E i ) / h –This B is defined with respect to the energy density U of the radiation field, not I ν (=B ν ) as previously. For this definition For this definition –A ji = (8πhν 3 /c 3 ) B ji –g i B ij = g j B ji With Respect to the Energy Density

Bound Bound Transitions17 Now Let Us Try To Determine A ji, B ij, and B ji Classical Oscillator and EM Field Classical Oscillator and EM Field –Dimensionally correct but can be off by orders of magnitude QM Atom and Classical EM Field QM Atom and Classical EM Field –Correct Expression for B ij QM Atom and a Quantized EM Field QM Atom and a Quantized EM Field –Gives B ij, B ji, and A ji However, note that B ij, B ji, and A ji are interrelated and if you know one you know them all! However, note that B ij, B ji, and A ji are interrelated and if you know one you know them all!

Bound Bound Transitions18 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/h 3 = dxdydzdp x dp y dp z /h 3 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/h 3 = dxdydzdp x dp y dp z /h 3 Then the statistical weight per unit volume of a particle with total angular momentum p, in direction dΩ, is p 2 dpdΩ/h 3. Then the statistical weight per unit volume of a particle with total angular momentum p, in direction dΩ, is p 2 dpdΩ/h 3.

Bound Bound Transitions19 Free Particles Momentum of a photon is p = hν/c so the statistical weight per unit volume of photons in (ν, ν+dν) is ν 2 dνdΩ/hc 2. Momentum of a photon is p = hν/c so the statistical weight per unit volume of photons in (ν, ν+dν) is ν 2 dνdΩ/hc 2. This means high frequency transitions have higher transition probabilities than low frequency transitions. This means high frequency transitions have higher transition probabilities than low frequency transitions.

Bound Bound Transitions20 Semiclassical Treatment of an Excited Atom Oscillating Dipole (electric) in which the electron oscillates about the nucleus. Oscillating Dipole (electric) in which the electron oscillates about the nucleus. For the case of no energy loss the equation of motion is: For the case of no energy loss the equation of motion is: –ν 0 is the frequency of the oscillation The electric dipole radiates (classically) and energy is lost due to the radiation The electric dipole radiates (classically) and energy is lost due to the radiation

Bound Bound Transitions21 A Damped Oscillator The energy loss leads to a further term in the equation of motion which slows the electron The energy loss leads to a further term in the equation of motion which slows the electron The damping force is The damping force is If F is small then the motion can be considered to be near a simple harmonic r = r 0 cos (2πν 0 t) If F is small then the motion can be considered to be near a simple harmonic r = r 0 cos (2πν 0 t)

Bound Bound Transitions22 The Solution is γ is called the classical damping constant γ is called the classical damping constant Damped Harmonic Oscillator

Bound Bound Transitions23 Electric Field The field set up by the electron is proportional to the electron displacement The field set up by the electron is proportional to the electron displacement At time t = (γ/2) -1 we get E(t) = E 0 /e which characterizes the time scale of the decay At time t = (γ/2) -1 we get E(t) = E 0 /e which characterizes the time scale of the decay

Bound Bound Transitions24 A Result! The decay time must be proportional to A ji -the probability of a spontaneous transition downwards The decay time must be proportional to A ji -the probability of a spontaneous transition downwards A classical = γ A classical = γ –For Hα, = 6563Å; ν 0 = c/λ so γ  5(10 7 ) s -1 The field decays exponentially so the frequency is not monochromatic. To get the frequency dependence do a Fourier analysis 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! The square of the spectrum is the intensity of the field!

Bound Bound Transitions25 The Broadening Profile Lorentz Profile

Bound Bound Transitions26 More On Broadening In QM the natural broadening arises due to the uncertainty principle 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. The uncertainty in the time in which an atom is in a state is its lifetime in that state. Average lifetime A -1 Average lifetime A -1 Thus the uncertainty  E = h  t ≈ Ah Thus the uncertainty  E = h  t ≈ Ah This means that  ν  A  γ This means that  ν  A  γ

Bound Bound Transitions27 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. 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 n upper → m lower Oscillator Strength n upper → m lower A nm ≡ 3 (g m /g n ) f nm γ A nm ≡ 3 (g m /g n ) f nm γ = (g m /g n ) (8π 2 e 2 ν 2 /m e c 3 ) f nm = (g m /g n ) (8π 2 e 2 ν 2 /m e c 3 ) f nm = (g m /g n ) 7.42( ) ν 2 f nm = (g m /g n ) 7.42( ) ν 2 f nm B mn = (πe 2 /m e hν) f nm B mn = (πe 2 /m e hν) f nm

Bound Bound Transitions28 Oscillator Strengths This is known as Kramer’s Formula. G I is the Gaunt Factor which is order LineU-Lf HαHαHαHα HβHβHβHβ HγHγHγHγ HδHδHδHδ HεHεHεHε Balmer Series of H

Bound Bound Transitions29 The Absorption Coefficient We are now ready to determine the absorption coefficient a ν per atom of an absorption line centered at ν 0. We are now ready to determine the absorption coefficient a ν per atom of an absorption line centered at ν 0. It is defined so that the probability of absorption per unit path length of a photon is Na ν where N is the number density of atoms capable of absorption. It is defined so that the probability of absorption per unit path length of a photon is Na ν where N is the number density of atoms capable of absorption. Consider a beam of intensity I ν traveling across a unit cross-sectional area. In time dt the photons have traveled cdt. Consider a beam of intensity I ν traveling across a unit cross-sectional area. In time dt the photons have traveled cdt.

Bound Bound Transitions30 Energy Considerations The energy removed from the beam in (ν, ν + dν) is I ν na ν cdt. Alternately, the number of transitions is I ν na ν cdt / hν. The volume occupied by the photons is cdt (unit cross sectional area). The energy removed from the beam in (ν, ν + dν) is I ν na ν cdt. Alternately, the number of transitions is I ν na ν 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 I ν na ν / hν. The number of transitions per unit volume per unit time per unit frequency due to the beam is I ν na ν / hν. Integrating over solid angle and assuming thermal equilibrium (4πI ν = cU ν ) the number of absorptions is cU ν na ν / hν. Integrating over solid angle and assuming thermal equilibrium (4πI ν = cU ν ) the number of absorptions is cU ν na ν / hν.

Bound Bound Transitions31 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 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 U ν B nm N so But this is equal to U ν B nm N so

Bound Bound Transitions32 Line Absorption U ν does not vary across the line U ν does not vary across the line a ν is small except near ν 0 : ν 0 = (E n - E m ) / h a ν is small except near ν 0 : ν 0 = (E n - E m ) / h U ν /hν varies slowly across the line U ν /hν varies slowly across the line The nature of line absorption For ν = ν 0 cm 2 Hz

Bound Bound Transitions33 We are about done The integral of a ν over ν can be thought of as the total absorption cross section per atom initially in the lower state. We rewrite a ν as The integral of a ν over ν can be thought of as the total absorption cross section per atom initially in the lower state. We rewrite a ν as φ ν is the absorption profile and we expect it to follow the Lorentz profile φ ν is the absorption profile and we expect it to follow the Lorentz profile

Bound Bound Transitions34 Combine the Profile In principal f nm ~ γ but there are other broadening mechanisms. In principal f nm ~ γ but there are other broadening mechanisms. –Natural width (γ) is small compared to other mechanisms (Stark, Doppler, van der Waals, etc) For ν = ν 0 : a 0 = (πe 2 /m e c) (f nm /γ) For ν = ν 0 : a 0 = (πe 2 /m e c) (f nm /γ) So

Bound Bound Transitions35 Stimulated Emission We have yet to account for stimulated emission We have yet to account for stimulated emission We usually assume emission is isotropic but if it goes as B mn I ν then it correlates with I ν We usually assume emission is isotropic but if it goes as B mn I ν then it correlates with I ν We treat this as a negative absorption and reduce the value of a ν by the appropriate factor. We treat this as a negative absorption and reduce the value of a ν by the appropriate factor. In our discussion of the Einstein coefficients we found the factor to be (1 - e -(hν/kT) ) so the corrected a ν is In our discussion of the Einstein coefficients we found the factor to be (1 - e -(hν/kT) ) so the corrected a ν is