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Atomic Spectra in Astrophysics

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Atomic Spectra in Astrophysics Atomic Spectra in Astrophysics Lida Oskinova, Helge Todt Astrophysik Institut für Physik und Astronomie Universität Potsdam WiSe 2016/2017 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 1 / 142 The Hydrogen Atom L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 2 / 142 Contents importance of hydrogen, origin the hydrogen spectrum (brief) history of atom models quantum mechanics and solution of the central-force problem L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 3 / 142 Hydrogen discovered 1766 by Cavendish (metal + acid), and found as constituent of water by de Lavoisir (1787) ! hydrogen = generator of water simplest atom: proton & electron - mass 1:6738 × 10−27 kg Eion ≈ 13:6 eV isotopes: deuterium (1 neutron) and tritium (2 neutrons) origin: Big Bang; deuterium from primordial nucleosynthesis (1 min after BB at 60 MK ^= 80 keV); recombination at 378 000 yr (z = 1100) ! transparent universe fuel for stars (fusion) via proton-proton chain reaction or CNO cycle L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 4 / 142 The hydrogen spectrumI Spectrum of a Balmer lamp: ! low pressure gas-discharge tube (H. Geißler 1857) filled with hydrogen Ångström (1862): spectral lines of hydrogen in spectrum of sun L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 5 / 142 The hydrogen spectrumII Balmer (1885): spectral lines of hydrogen given by hm2 λ = (n = 2; m = 3; 4; 5;:::) (1) m2 − n2 with h = 3645:6 × 10−10 m and 10−10 m = 1 Å, typical size of an atom predicted lines for m > were found in A stars Rydberg (1888): generalization to other series 1 1 1 7 −1 = RH 2 − 2 ; RH = 1:096775854 × 10 m (2) λ n1 n2 generalization to H-like ions (e.g. He II, Li III): 1 2 1 1 = Z RX 2 − 2 (3) λ n1 n2 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 6 / 142 Atom modelsI Parmenides (500 v. Chr.): atoms (indivisible) as building blocks of the world J. Dalton (1803): chemical elements consist of atoms of different mass L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 7 / 142 Atom modelsII J.J. Thomson (1900): atoms contain negatively charged electrons in a positively charged continuum (cathode rays experiments) L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 8 / 142 Atom models III Rutherford (1910): positively charged nucleus is smaller (10−15 m) than atom (scattering of helium nuclei on gold foil) Problems of Rutherford model Why do electrons not fall into nucleus? ! circular orbit Why don’t they emit like electric dipole, what about spectral lines? ! Bohr model L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 9 / 142 Bohr modelI Bohr’s postulates (1913) to explain observations: Motion of electrons in atoms obeys quantum rules 1 Electrons orbit in atoms the nucleus on so-called stationary orbits with discrete energies En. (The angular momentum is restricted to integer multiples of a fixed unit L = n~ (n = 1; 2; 3;:::).) 2 Atoms can only gain or lose energy by the transition of an electron from one stationary orbit to another stationary orbit, this energy is discrete and given by ∆E = En − En+1 = hν for the involved photon of frequency ν. Postulate (1) can be also written as phase space integral 1 I pdq = n (n = 1; 2; 3;:::) (4) 2π ~ L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 10 / 142 Bohr modelII Application to the hydrogen atom: Electron orbits in a Coulomb potential (polar coordinates) with Ze2 F = − 2 (5) 4π"0rn with stationary radius rn, balanced by centrifugal force 2 vn 2 Fz = mr = mrrn!n (6) rn with reduced mass mr so that 0 is in nucleus with mass mc m2 me mr = = (7) 1 + m2 1 + me m1 mc Force balance yields 2 Ze 2 2 = mrrn!n (8) 4π"0rn L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 11 / 142 Bohr model III As vn = rn!n the momentum and position are pn = mrvn = mrrn!n; qn = rnφn (9) I Z 2π 1 1 2 2 pndqn = mrrn !n dφ = mrrn !n = n~ (10) 2π 2π 0 n~ ! !n = 2 (n = 1; 2; 3;:::) (11) mrrn Inserting force balance to elimnate !n: 2 2 4πn ~ "0 rn = 2 (n = 1; 2; 3;::: ) (12) Zmre Z 2m e4 ! = r (13) n 2 3 3 2 16π n ~ "0 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 12 / 142 Bohr modelIV This can be used to obtain the kinetic energy of the electron m m Z 2m e4 E = r v 2 = r r 2!2 = r (14) kin n n n 2 2 2 2 2 2 32π n ~ "0 Its potential energy is given by qV (r) = − R Fdr: Ze2 Z 2m e4 E = − = − r (15) pot 2 2 2 2 4π"0rn 16π n ~ "0 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 13 / 142 Bohr modelV both forces are conservative forces, ! conservation of energy and the energy levels are therefore Z 2m e4 E = E + E = − r (16) n kin pot 2 2 2 2 32π n ~ "0 2 4 Z me e = − (n = 1; 2; 3;:::) (17) 32π2n2 2"2 1 + me ~ 0 mc from 2nd Bohr’s postulate (∆E = hν) and by ν = c/λ: 1 Z 2m e4 1 1 = e − (18) m 2 2 λ 32π2c"2 1 + e n1 n2 0 mc L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 14 / 142 Bohr modelVI 94 nm Lyman series 95 nm 97 nm 103 nm 122 nm 656 nm486 nm 434 nm n = 1 410 nm Balmer series 1875 nm n = 2 1282 nm n = 3 1094 nm n = 4 n = 5 n = 6 −1 1 1 λ = 911:8 Å 2 − 2 n1 n2 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 15 / 142 Bohr model VII 2 2 2 comparison to Rydberg’s formula 1/λ = Z RX 1=n1 − 1=n2 for hydrogen-like atoms suggests: R R = 1 (19) X 1 + me mc m e4 R = e = 1:097373177 × 107m−1 (20) 1 2 2 32π c"0 Application: Pickering lines of He II ! blackboard note: R1 sometimes used for atomic energies L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 16 / 142 Bohr model VIII Problems and limits of Bohr model Experiments: H-like spectra of alkali metals need more than one quantum number for discription (e.g. Na D line ≈ 5890 Å transition 3s − 3p) no explanation of other spectra Theory: Heisenberg’s uncertainty principle contradicts electron orbits Street light using ! blackboard sodium lamp. Orange light from Na D line L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 17 / 142 The Schrödinger equationI wave equation, used for explanation of hydrogen spectrum Schrödinger (1926) @ { j (t)i = H^ j (t)i (21) ~@t @ 2 e.g. { (~r; t) = − ~ ∆ + V (~r; t) (~r; t) (22) ~@t 2m derived from de Broglie and dispersion relation: p = ~k and E = ~! describes unperturbed evolution in time of non-relativistic quantum systems, linear PDE of 2nd order with complex solutions linearity: superposition principle of solutions probability of presence (find particle at position x): j (~r; t)j2 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 18 / 142 The Schrödinger equationII Stationary Schrödinger Eq. (@=@t = 0 ), separation: (~r; t) = '(~r) f (t) ! blackboard H^ '(~r) = E'(~r) (23) −{ E t (x; t) = '(x) e ~ (24) with eigenvalues E (energy) of Hamilton operator: the probability density j (x; t)j2 = j (x)j2 (Why?) does not depend on t, same holds for expectation values of dynamic variables p2 as H(x; p) = 2m + V (x) is classical Hamiltonian ≡ total energy ! hHi = E (Why?) general solution is a linear combination of separable solutions: 1 X −{Ent=~ (x; t) = cn n(x)e (25) n=1 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 19 / 142 The Central-force problemI 3D: Potential is of form V (~r) = V (j~rj) The Laplace operator in 3D for spherical coordinates: 1 @ @ 1 @ @ 1 @2 ∆ = r 2 + sin θ + (26) r 2 @r @r r 2 sin θ @θ @θ r 2 sin2 θ @φ2 Use separation: '(r; θ; φ) = R(r) · Y (θ; φ), thus 2 Y @ @R 2 1 @ @Y 2 R @2Y − ~ r 2 − ~ R sin θ − ~ 2m r 2 @r @r 2m r 2 sin θ @θ @θ 2m r 2 sin2 θ @φ2 + V (r) RY = ERY (27) with eigenvalue E L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 20 / 142 The Angular EquationI 2 1 Multiply ·r , and RY , again: obtain another separation constant 2 1 @ @R 2 1 @ @Y − ~ r 2 + r 2V (r) − Er 2 = ~ sin θ (28) 2m R @r @r 2m Y sin θ @θ @θ 2 1 @2Y + ~ 2m Y sin2 θ @φ2 2 = ~ [−`(` + 1)] (29) 2m Next separation: Y (θ; φ) = Θ(θ) · Φ(φ) Φ @ @Θ Θ @2Φ sin θ + = −`(` + 1) (30) ΘΦ sin θ @θ @θ ΘΦ sin2 θ @φ2 L. Oskinova, H. Todt (UP) Atomic Spectra in Astrophysics WiSe 2016/2017 21 / 142 The Angular EquationII term Φ=Φ and Θ=Θ cancels out, multiply sin2 θ 1 @ @Θ 1 @2Φ sin θ sin θ + `(` + 1) sin2 θ = − = m2 (31) Θ @θ @θ Φ @φ2 with new separation constant m: @2Φ + m2Φ = 0 ) solution: Φ(φ) = e±{mφ (32) @φ2 as rotation around φ = 2π in space means same as original state: Φ(φ + 2π) =! Φ(φ) ) m = 0; ±1; ±2;::: (33) L.
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