Atomic Physics ASTR 2110 Sarazin

Homework #5 Due Wednesday, October 4 due to fall break

Test #1 Monday, October 9, 11-11:50 am Ruffner G006 (classroom) You may not consult the text, your notes, or any other materials or any person Bring pencils, paper, calculator ~2/3 Quantitative Problems (like homework problems) ~1/3 Qualitative Questions Multiple Choice, Short Answer, Fill In the Blank questions No essay questions

Test #1 (Cont.) Equation/Formula Card: You may bring one 3x5 inch index card with equations and formulae written on both sides. DO NOT LIST pc, AU, Μ, L, R DO NOT INCLUDE ANY QUALITATIVE MATERIAL (text, etc.)

Test #1 (Cont.) No problem set week of October 2 9 to allow study for test Review Session: Discussion session Friday, October 6, 3-4 pm

Uncertainty Principle Heisenberg 1927: lumpiness of nature means we cannot make perfectly accurate measurements Newtonian physics: F = ma = dp/dt. If you knew position and momentum of every particle perfectly, you could exactly predict everything in the future. But, can t measure position and momentum accurately.

Uncertainty Principle Measure position with microscope. What kind of light to use? Measure position to accuracy of Δx, need λ Δx But, photon will bounce off particle, change particle momentum by Δp p γ = hν c = h λ h Δx Δx Δp more accurately ΔE Δt for energy

Relativity and Quantum Mechanics Spin-Statistics Theorem: Social properties of particles depend on spin Integer Spin (S = nħ) Bosons All want to be doing the same thing (in the same state) Example: photons, S = 1 ħ. Stimulated Emission (lasers) all photons in exactly the same state Half-Integer Spin [S = (n+1/2)ħ] Fermions Can NEVER be in exactly the same state Examples: electrons, protons, neutrons Exclusion Principle

Relativity and Quantum Mechanics Sizes of particles: Δp = mδv mc Δx Δp mc = size (radius r) of particle Note: m 0 r effects at distances Long range forces due to massless particles Photons è Electromagnetism Gravitons è Gravity Strong, Weak forces short range

Relativity and Quantum Mechanics Virtual Particles: Δt r c = # % $ ΔE Δt = & ( 1 mc ' c = shortest time 2 mc mc 2 = mc2 = rest mass energy Make virtual particles e - e - + e - e + e - + e - e + e - e + etc. Reason for field description of particles, number of particles is not fixed

Relativity and Quantum Mechanics Forces = Exchange of Virtual Particles: e - e + photon Electromagnetism = exchange of photons

Nucleus: Atomic Structure Strong force strong, short range p s & n s è nucleus (10-13 cm = 10-15 m) Electron bound to nucleus by electric force Strength of electrical forces PE(electron) e2 r e = e 2 m e c PE / m e c 2 = e2 c α 1 137 " % $ 'm e c 2 # c & = e2 fine structure constant Electrical forces are relatively weak e - e - + e - e + ~ α 2 ~ 10-4

Atomic Structure Electrons: Force from nucleus is F = q 1 q 2 /r 2 = -Ze 2 /r 2 just like gravity Newtonian Physics: elliptical orbits, etc. Problem: Electrons charged, accelerating èmust radiate (Maxwell s eqs.), would lose energy, atom would collapse in ~10-18 sec!! Note: nucleus actually 10 5 x smaller

Bohr Theory of Atom L quantized, orbital angular momentum L = m e vr = n Bound, Circular Orbits: Radius: m e v 2 r m 2 e v 2 r 2 = Ze 2 m e r = Ze2 r 2 multiply by m e r 3 n 2 2 = Ze 2 m e r r = n2 2 Ze 2 m e = n2 r e Zα Orbits quantized As n increases, r increases Smallest orbit, n=1 Bohr radius (Z=1) = r e /α =5.29 x 10-9 cm = 0.529 Å =0.0529 nm

Bohr Theory of Atom Bound, Circular Orbits: Velocity: m e vr = n plug in r, solve for v! v = Zα $ # &c nonrelativistic as long as Z <<137 " n % Energy: E = KE + PE = 1 2 m ev 2 Ze2 r E = Z 2 α 2 2n 2 m e c 2 E < 0 bound E(hydrogen) ~ ½ α 2 m e c 2 << m e c 2 n=1 most bound, n large nearly unbound m e èμ e = m e m nucl /(m e + m nucl )

Bohr Theory of Atom Unbound Electrons: m e v 2 Ze2 no energy quantization (energy levels) r r 2 E > 0 Why are atoms stable? 1. n =1 (ground state) has no where to go 2. Upper levels don t spiral in, E atom & E photon both quantized

Hydrogen Spectral Lines E photon = ΔE atom hν ab = E a E b ν ab = Z 2 α 2 m e c 2 2h # % $ 1 n b 2 1 n a 2 1 = ν ab λ ab c = R # % 1 2 n 1 2 $ b n a & ( ' & ( (for Z =1) ' R =109, 678 cm 1 Rydberg constant R 1 = 912 A o = 91.2 nm n b n a

Hydrogen Spectral Lines Label by lower n (roman letters) & Δn (greek letter) n 1 Lyman lines (UV) Lyα n = 2 n =1 λ Lyα =1216 A o =121.6 nm Lyβ n = 3 n =1 n 2 Balmer lines (optical) Hα n = 3 n = 2 Hβ n = 4 n = 2 λ Hα = 6563 A o = 656.3 nm (red) λ Hβ = 4861 A o = 486.1 nm

Red Hα Line in Planetary Nebula

More Complex Atoms: Energy Levels E>0 free Energy # States n=3 n=2 E<0 bound 0-1.5 ev -3.4 ev 18 8 }Periodic Table n=1-13.6 ev 2 1 ev = ΔE of electron moving through 1 volt 1 ev =1.6 10 12 erg =1.6 10 19 J α 2 2 m e c2 =13.6 ev Ionization potential of H Bound, quantized energy levels & Free continuous energies

Atomic Processes Transitions: radiative & collisional Bound-bound: ΔE = E 2 E 1 Excitation (é): E 2 E>0 free E Radiative = Absorption line ΔE E<0 bound hν = ΔE rate α intensity of light De-Excitation (ê) Radiative = Emission line E 1 Intensity emission line absorption line hν

Atomic Processes Bound-free: ΔE > IP ionization potential Ionization (é): Removes electron Radiative: hν > IP Absorption edge Recombination (ê): Adds an electron Emission Edge IP Intensity E>0 free E<0 bound emission edge absorption edge hν E

Atomic Processes Free-free: Only energy condition is ΔE < initial E of electron Continuous emission or absorption of radiation Intensity hν

Radiation Photons: Photons = lumps of light Light = waves of electric and magnetic field Electric field = wave function, gives probability of finding a photon E photon = hν Bulk Properties of Radiation: Natural Illumination: lots of photons, radiation not coherent, generally not polarized, etc. Main property = carrier of energy Intensity: I ν = energy / (time x area x solid angle x frequency) solid angle = direction from which radiation came (steradian)

Interaction of Radiation and Matter 1. Transmission 2. Absorption 3. Scattering 4. Emission 5. Stimulated emission Makes exact clone of an incoming photon. Result of Spin-Statistics Thm. Photons have S = 1, bosons. View scattering = absorption, then emission Stimulated emission = negative absorption exactly Define: Extinction = true absorption + scattering stimulated emission Two processes: emission & extinction 1 5 3 2 4

Radiative Transfer Emissivity: j ν = (energy emitted) / (time x volume x solid angle x freq.) Opacity: Think of each atom as target, area A, mass m n = # atoms / volume α = n A (units cm -1 ) κ = A / m (units cm 2 / gm) Equation of Radiative Transfer area A, mass m d I ν d x = α ν I ν + j ν absorption emission distance

Optical Depth: Radiative Transfer τ α dx ~ αx (dimensionless) τ <<1 little interaction of matter and radiation optically thin τ >> 1 lots of interaction optically thick

Radiative Transfer Realistic astrophysics Matter: Many elements, atoms, molecules, ions Each with many, many energy levels Many, many transitions between the levels Radiation: Need opacities, emissivities for all lines. Many frequencies, wavelengths covering EM spectrum Transfer equation for all frequencies, directions Big, messy problem = solve on computer

Thermodynamic Equilibrium Closed system Wait a long time Thermodynamic equilibrium: Unique state with Energy fixed E = constant Entropy maximum, S max Characterized by one number Temperature T Same value throughout box Ludwig Boltzmann

Thermodynamic Equilibrium Given all possible molecules, ions, energy levels, emission lines, photon energies, etc., how can there be one unique state of thermodynamic equilibrium? Detailed Balance: In TE, every process is exactly balanced by its inverse In TE, populations of states generally vary as population α exp( - E / kt), where E is energy of state

Thermodynamic Equilibrium Bound states: E>0 free n 2 = e ( E2 E1 )/kt = e ΔE/kT n 1 E 2 E kt << ΔE every thing in ground state kt >> ΔE many states populated Bound-free (ionization): Saha equation kt << IP lower ionization kt >> IP higher ionization ΔE IP E 1 E<0 bound

fully ionized neutral atom T (K)

Thermodynamic Equilibrium Free Particles: Maxwellian distribution <E> = <½ mv 2 > = 3/2 kt Hotter = faster moving atoms

Radiation in Thermodynamic Equilibrium Planck or Blackbody Spectrum: I ν = B ν 2hν 3 c 2 1 e hν /kt 1 Rayleigh-Jeans Limit: hν << kt I ν 2ν 2 kt c 2

Wien s Law: Radiation in Thermodynamic Equilibrium λ max = c ν max = 0.3 cm T (K) Hotter = shorter wavelength = bluer Stefan-Boltzmann Law: L = (area) σt 4 σ = 5.67 10 5 erg/(cm 2 sec K 4 )

Pressure Forces Newton s 3 rd Law è most forces cancel in bulk But, new force appears Pressure pressure = force / area

Pressure Forces p = momentum Force = dp dt = Δp atom Δp atom = 2 p = 2mv x x atoms time = nv (area) x n atoms / volume P = 2np x v x = 2nm v x 2 atoms time P = nkt = ρkt µm p ideal gas law atom wall

Hydrostatic Equilbrium Planets & Stars Gravity is balanced by Pressure: Hydrostatic Equilibrium Consider a spherically symmetric object F = G M(r)m r 2 ê r M(r) mass interior to r r m

Hydrostatic Equilbrium For mass m, take a small cylinder Area A Height Δr Filled with fluid with mass density ρ A Δr r Assume pressure P and density only depend on radius [P(r), ρ(r)], and that pressure decreases with radius.

Hydrostatic Equilbrium Pressure forces = P Area F pres = PAê r (P + ΔP)Aê r P + ΔP = (P P ΔP)Aê r Δr A = ΔPAê r F grav = G M(r)m r 2 ê r P r M(r) mass interior to r m = ρ volume = ρaδr

Hydrostatic Equilbrium F grav = G M(r)ρAΔr r 2 F tot = F grav + F pres = 0 ê r A P + ΔP F pres = F grav Δr ΔPAê r = G M(r)ρAΔr r 2 ΔP Δr = G M(r)ρ r 2 ê r P r dp dr = G M(r)ρ r 2

Stellar Spectra ASTR 2110 Sarazin Solar Spectrum

Theory of Stellar Atmospheres Divide stars into Atmosphere Narrow outer layer, 1 mean free path, τ 1 Makes light we see Interior Not directly observable

Theory of Stellar Atmospheres Assume 1. Very thin Δr << R Treat as flat plane 2. No energy sources L in = L out = L 3. Static Forces balance, hydrostatic equilibrium

Theory of Stellar Atmospheres dp dr = GM(r)ρ Hydrostatic equilibrium r 2 M(r) mass interior to r = M * ρ mass density = mass/volume r R * constant, thin atmosphere g * GM * R * 2 surface gravity, constant dp dr = g *ρ Hydrostatic equilibrium P = ρkt µm p Pressure, ideal gas law

Theory of Stellar Atmospheres L = constant, thin atmosphere, flux important L = 4π R *2 σt 4 eff, F = L / (4π R 2 * ) = σt 4 eff constant Equation of Radiative Transfer di ν dx = α I +ε ν ν ν absorption reduces, emission increases ε ν,α ν emissivity, opacity, depend on T,ρ, composition

Theory of Stellar Atmospheres dp dr = g *ρ Hydrostatic equilibrium P = ρkt µm p Pressure, ideal gas law F = L / (4π R 2 * ) = σt 4 eff constant di ν dx = α ν I ν +ε ν Eqn. Radiative Transfer

Theory of Stellar Atmospheres dp dr = g *ρ Hydrostatic equilibrium P = ρkt µm p Pressure, ideal gas law F = L / (4π R 2 * ) = σt 4 eff constant di ν dx = α νi ν +ε ν Eqn. Radiative Transfer Four unknowns: P, ρ, T, I ν vs. r Four equations solvable

Inputs: F = L / 4π R 2 * = σt 4 eff = constant g * α ν,ε ν,µ depend on ρ, T, composition R * but only scales the fluxes

Stellar Spectra and Stellar Atmospheres Determined by (in decreasing order of importance) 1. T eff 2. g * 3. Composition

Solar Spectrum

Stellar Spectra General result Stellar spectra = continuum emission + absorption lines ~blackbody hotter brighter, bluer hotter molecular lines atomic lines ions (more and more ionized)

Stellar Spectra hotter cooler

Stellar Spectra ions hotter molecules cooler

Stellar Spectra ions molecules hotter 50000 25000 10000 8000 6000 5000 4000 3000 Temperature cooler