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X- and γ -ray spectroscopy Carlo Ferrigno Department of astronomy - University of Geneva https://cms.unige.ch/isdc/ferrigno/ 2016 December 7 • Based on Rybick & LightMan Radiative Processes in astrophysics 1985 Wiley • Advanced material available in X-ray spectroscopy in astrophysics, 1997, Paradijs, Jan van, Bleeker, Johan A.M. (Eds.) Introduction Let’s pass from processes giving rise to a continuum radiation to processes producing lines: • Continuum Spectrum – – – – – Curvature Compton scattering Synchrotron Bremsstrahlung Pair productiona • Line spectrum – – – – – – – a Atomic structure Transition probabilities Absorption and emission coefficients Line shapes Collisional excitation Photoionization Conditions of plasma from lines Feature at 511 keV Introduction Emisison line spectrum in X-rays (Hitomi-Collaboration et al. 2016) Absorption line spectrum in X-rays (Pintore et al. 2016) 1 Principal quantum number Most of the properties can be derived from classical quantum mechanics whose basic principles are: • Action is quantized (Sommerfield): H pi qi dqi = ni h • Heisenberg indetermination principle: states with ∆pi ∆qi < h cannot be distinguished • Pauli exclusion principle: two Fermions cannot occupy the same cell in phase space. In case of electrons in atoms, they cannot have the same quantum numbers. Quantization of action in central potential of atom with central charge Ze leads to the first quantization number n for the energy levels of radial operator: n2~2 n2 n2 rn = = a0 ∼ 0.5Å 2 Zme Z Z (1) where a0 = ~2/me2 ∼ 0.5 Å is the first Bohr radius. Quantization of azimuthal momentum can be introduced for the second quantum number l in elliptical orbits. Also take into account the finite nucleus mass by changing electron m into µ = me M/(me + M). A more straightforward way of considering the angular momentum is the use of Schrodinger equation. Atomic structure 1 Lines from jump in shells - Order of magnitude Metalsa Hydrogen Line emission or absorption through transition between orbits with different quantum number n. Photon energy is: hν21 = e2Z 1 r1 − 1 r2 2 2 = Z e a0 1 n12 − 1 n22 where the constant is know as 1 Rydberg ' 13.6 eV. e2 1 Ry = 2a0 Atomic structure (2) Inner shell transitions for heavier atoms are (3) fluorescence “K", “L” ... lines. a astronomical 2 Digression Iron lines Iron has atomic number 26, the inner shell transitions are in the X-ray domain, quite isolated from other features and then "easy" to detect. Depending on the environment, Iron can be ionized at many different level. Neutral Iron fluorescence Kα is at 6.38 keV, while Fe XVI is at 6.7 keV. Emission lines for weakly ionized iron. On the left the Kα lines, on the right, the Kβ lines. Emission lines for strongly ionized Fe XXIV– Fe XXVI mostly. The best-ever X-ray spectrum of a Galaxy cluster. Atomic structure 3 Schrödinger equation For an Hamiltonian H , we can write the time-dependent Schrödinger equation i~ ∂ψ = Hψ ∂t (4) And then look for the stationary solutions by separating the time and space parts of the wavefunction ψ , which is possible if H is time independent ψ(t, r) = ψ(r)eiEi /~. We end with the equation: Hψ = Eψ (5) where E is the energy and ψ the eigenfunction of the corresponding energy state. For a nucleus of charge Ze, neglecting spin, relativistic effects and nuclear effects, the Hamiltonian is X 1 X e2 ~2 X 2 − + −E ψ=0 ∇j − Ze 2m r r j ij j i>j j | {z } | {z } | {z } 1 2 (6) 3 where the terms are 1) kinetic, 2) Coulomb, 3) electron self interaction. By expressing length as function of a0 and energy as e2/a0=27.2 eV, the equation can be written in a dimensionless form 1 2 Atomic structure X j 2 ∇j + Z X1 j X1 − − E ψ = 0 rj rij (7) i>j 4 Solution of Schrödinger equation in a central field. The central potential implies that angular momentum is constant and thus the corresponding operator commutes with the Hamiltonian. It is useful to consider one electron in a central filed also for high-Z atoms. Limiting cases: 1 • one electron is far away: he others screen nucleus: V (r ) → Z −N+ ,r →∞ r • one electron is close to the nucleus V (r ) → − Zr , r → 0. A solution can be found by separating variables and is: ψ(r , θ, φ) = r −1R(r )Y (θ, φ) (8) where the spherical harmonics are defined as (l − |m|)! 2l + 1 Ylm (θ, φ) = (l + |m|)! 4π 1/2 |m| (−1)(l+|m|)/2Pl (cos(θ))eimφ (9) where Plm is the associate Legendre function and l and m are integers. Atomic structure 5 The n, l an m quantum numbers: bound states. The spherical harmonics are an orthonormal base, solution of each central potential, and are eigenfunctions of the angular momentum operator L = r × p, expressed in units of ~ L2Ylm (θ, φ) = l(l + 1)Ylm (θ, φ) Lz Ylm (θ, φ) = mYlm (θ, φ) (10) (11) where l is integer and m = −l...0... + l The radial part of the eigenfunction is obtained by solving 2 1 d Rnl l(l + 1) + R − V (r ) − Rnl = 0 2 2 2 dr 2r (12) the radial eigenfunction depends on l , but not on m, the index n labels the energy states in increasing order of energy n = l + 1, l + 2, l + 3.... There is also a continuous set of eigenfunctions for unbound states. Atomic structure 6 Radial functions and orbitals The bound states are described by 1/2 Z (n − l − 1)! −p/2 l+1 2l+1 e p Ln+l (ρ) Rnl (r ) = − 3 2 n [(n + l)!] En = −Z 2/2n2(e2/a0) (13) (14) l+1 where ρ = 2Zr /n and L2n+l (ρ) are the associated Laguerre polynomials. Normally, the angular momentum quantum number is indicated with letters: l=0 l=1 l=2 l =3 l= 4 l= 5 l= 6 s p d f g h i = sharp = principal = diffuse = fundamental ... ... Atomic structure 7 Spin and many electron systems In non-relativistic approximation, the spin can be described with a doublet: 1 1 1 0 = − = 2 0 2 1 (15) (relativistic treatment requires the solution of Dirac’s equation). We have a set of N Fermions, each described by four quantum numbers n, l, m, ms , defining the orbitals. We need to form the product of the sort ua(1)ub (2) ... uk (N) , (16) where each subscript a, b...k represents the set of four quantum numbers (n.l, m, ms ), and the numbers 1...N represent the space and spin coordinates of each electron. These functions include the spacial plus angular parts ψ and the spin. Particle cannot be distinguished, we make a linear combination of all permutations of the states and take into account the Pauli exclusion principle. The wave function becomes (N!) −1/2 X p P(ua(1)ub (2) ... uk (N)) , (17) p where p = ±1, according if the permutation is even or odd. This prevents two electrons being in the same orbital. There is also complete antisymmetry of the wave function under interchange of electrons. Atomic structure 8 Atomic structure Recap on typical notations: l = 0 1 2 3 (subshell) s p d f n= 0 1 2 3 (shell) K L M N Order of filling is the following: 1s , 2s , 2p , 3s , 3p , 4s , 3d , 4p , 5s , 4d , 5p, 6s , 4f , 5p , 6p , 6s , 6d , ... Some shells are not in the expected numbering, because for some angular momenta the shells penetrate closer in the central potential. Electrons are subject to a quantum mechanics repulsion called exchange potential → electrons with the same spin are repulsed. Atomic structure 9 Spectroscopic notation I • Closed shell are spherically symmetric and thus interactions of these electrons with external fields is minimal • Non-equivalent electrons differ in either n or l • Equivalent electrons have same n and l . The notation is: two equivalent s electrons are: s 2. two non-equivalent s electrons are: s · s or ss 0. • Total spin and total angular momentum combine: S= P i si L= P i li • In the ground state, Hund’s rules hold – terms with larger S tend to lie lower in energy, because for the Pauli principle they are further apart. – among terms with a given spin configuration, those with largest L tend to lie lower in energy because they orbit further apart. • Due to exchange potential, electrons prefer to occupy the filled shells in equivalent states. E.g.: 1s 2 2s 2 2 p 2, rather than ... 2p · p . Atomic structure 10 Spectroscopic notation II Spectroscopic notation is used to specify how many combinations with different energy can assume a set of electrons in a given state. #ms (Total L) (18) • Two non-equivalent electrons in states 1s 2s (excited Helium) can have only L = 0, but the spin can be S = 0, 1 (1/2-1/2, 1/2+1/2). There can be two possible terms: 1S , for S = 0, called singlet and 3S , for S = 1, called triplet. • Two equivalent electrons in states s (same shell number) can have only opposite spin: 1S . • two non-equivalent p electrons, the possible L−S combinations are S = 0, 1, L = 0, 1, 2, and spectroscopic terms are 1S , 1P , 1D , 3S , 3P , 3D and 1 + 3 + 5 + 3 + 9 + 15 = 36 different states. • two equivalent p electrons (same shell) can have only 15 distinguishable states (Exercise). Atomic structure 11 LS coupling I Configuration of quantum numbers is not sufficient to determine the state of ions. For instance the different ml and ms states are all degenerate in a central potential which does not depend on spin. Total spin and total angular momentum combine: S= P i L= si P i li and then form the total angular momentum J=S+L The effect is to split each term into a set of levels, each labelled by the remaining quantum number J . The idea is that the central electric field is seen by the moving electron as a magnetic field. In the rest frame of the electron, we have: l dU B=− v×E= , (19) c mecr dr e which interacts with the electron’s magnetic moment µ = − mc s. And interact as U = − 12 µ · B. 1 Atomic structure 12 LS coupling II The spin-orbit Hamiltonian is often written as the sum of interactions of all electrons X X Hso = ξ Si · Li = ξS · L (20) i i We can find the splitting of a given term as function of the total angular momentum. J = (S + L) · (S + L) = S2 + L2 + 2S · L (21) so that 1 2 2 2 Hso = ξ J − S − L (22) 2 Since the operators are mutually commuting, the energy shifts are proportional to J(J + 1) 1 EJ+1 − EJ = C(J + 1) (23) 2 where C > 0 for shells less than half full (normal) and C < 0 for more than half full (inverted). Lande interval rule: the spacing between two consecutive levels of a term is proportional to the larger of the two J values involved. Atomic structure 13 LS coupling III The J level of a level is given as a subscript o the term symbol: 2S+1 LJ (24) e.g., 3P2, or 2S 1 2 Often the terms of allowed J are expressed as a list: e.g., 2P1/2,3/2. The number of J values in any term is given by the smaller of (2L + 1) and (2S + 1). Atomic structure 14 Classical oscillator • When electrons transit between levels, photons can be absorbed or emitted at energy hν = E2 − E1 , (25) where E1 and E2 are the energy associated with the two levels. • During transition the electron behaves as an oscillator with that resonance frequency. • We might perform semi-classical computation, but we will directly compute the “oscillator strength” derived from quantum mechanics to obtain correct results. • Line emission m → n is treated using the Einstein A coefficient. The power emitted per unit volume is: dP = Nm hνmn Amn (26) dV where Nm is the number density in level m and hνmn the energy of the transition. Radiation-Atom interaction 1 Electromagnetic Hamiltonian Hamiltonian of a particle with charge e in an external EM field 2 2 4 1/2 H = (cp − eA) + m c + eφ (27) In the non-relativistic limit ignoring the rest mass and using the Coulomb gauge ∇ · A = φ = 0 eA H= p− 2m c 1 2 p2 2 e2A2 + eφ = + eφ A·p+ 2 2m mc 2mc (28) if we compute the ration of the terms in A, the linear dominates on the quadratic, unless the phoe ton density is extremely high (n ∼ 1025 cm−3 )a We will use the term H 1 = (− mc 2 )A · p as a perturbation to the atomic Hamiltonian. H = H0 + H1 (29) where the H 0 is the atomic Hamiltonian, independent of time. We express the atomic eigenvalues and eigenfunction as Ek and φk , so that the zeroth-order time dependent wave functions are φk exp{−iEk t/~}. The actual wave function can be expanded in this complete set: ψ(t) = a X ak (t)φk exp{−iEk t/~} (30) CMB nγ ∼ 400 cm−3 ; Sun surface: nγ ∼ 1012 cm−3 Radiation-Atom interaction 2 Transition probability The probability per unit time for a transition from state |ii to state |f i is: RT 1 1 iωt 1 H (t)e dt H (ω) ≡ fi fi 2π 0 2 R ∗ 1 3 4π 1 2 1 wfi = Hfi (ωfi ) with Hfi (t) ≡ φf H φi d x ~T ω ≡ Ef −Ei fi (31) ~ and the perturbation is active only in the time interval [0, T ]. We assume that the perturbation is a wave A(r, t) = A(t)eik·r with A(t) limited in time, but large enough to host many waves; using pj → −i~∇j , we obtain X ie~ ik·r ∇j |ii Hfi (ωfi ) = A(ωfi ) · hf |e mc 1 The transition rate is then 4π 2e2 wfi = m2c 2T 2 X 2 ik·r |A(ωfi )| hf |e 1 · ∇j |ii (32) (33) where the unit vector 1 represents the polarization vector traveling in direction n ω2 and we can substitute the intensity per unit are and frequency as I = cT |A(ω)|2. Radiation-Atom interaction 3 Dipole approximation Transition probability has terms like hf |e ik·r 1· X Z ∇j |ii = φ∗j eik·r1 · X ∇φj d3x (34) 2 which we what to expand in k · r as eik·r = 1 + ik · r + 12 (ik · r) and this is appro∆E priate since k · r ∼ ka0 ∼ a0~c ∼ Z2α << 1 If we retain only the constant term, we get the electric dipole. If this is zero for symmetry reasons of the electron configurations, we need to climb the higher terms: magnetic dipole, electric quadrupole, octupole. For the dipole term, we can write Z φ∗j 1 · X ∇φj d3x = i~−1(1 · pj)fi (35) using commutation relations rj p2j − p2j rj = 2i~pj wfi = 4π 2 ~2c Radiation-Atom interaction 2 |(1 · d)fi | I(ωfi ) with d ≡ e X rj (36) j 4 Einstein coefficients A21 is the coefficient of spontaneous emission, B12 is for absorption, and B21 for stimulated emission. Multiplying the Eistein coefficients to the average flux density, we get the transition probability for each process. A21 = 3 2hν21 c2 B21 g1B12 = g2B21 (37) with gi state multiplicity. Let’s relate them to the transition probability. Radiation-Atom interaction 5 Oscillator strength I For the case of unpolarized radiation D 2 |(1 · d)fi | E = 1 |dfi |2 3 since hcos θi = 1/3. We can relate the previous computations to the Einstein B coefficient, after noting that u and l refer to the upper and lower states hwlu i = Blu Jνlu We note that Jνul = (4π)−1I(νul ) = 12 I(ωul ), so that hwlu i = 4π 2 1 2 ul ) = ul ) |d | I(ω B I(ω lu fi 3c~2 2 By developing the above relations, we can express the Einstein coefficients as (let’s consider nondegenerate states): 2 Blu = 2 Aul = 3 4ωul |dfi | Exercise: get the above Radiation-Atom interaction 3c 3~ ' 4c 3 a0 8π 2 |dfi | = Bul (38) α4Z 4 ∼ 109Z 4s−1 (39) 3c~2 6 Oscillator strength II It is convenient to define the absorption oscillator strength flu by the relations: 4π 2e2 classical flu ≡ Blu flu hνul mc X 2m 2 flu = − E |d | (E ) u l lu 3c~2e2 Blu = (40) (41) The classical B coefficient can be associated to a classical oscillator and expresses the amount of radiation extracted from a beam of radiation. Z 0 ∞ πe2 classical hνul σ(ν)dν = = Blu mc 4π Order of magnitude for a typical dipole transition rate is Aul ∼ 108 − 1012 s−1. Which means that exited states have a decay time of the order of the nanosecond or less. Therefore transitions are highly probable. Radiation-Atom interaction 7 Oscillator strength III In a system with Z active electrons, the oscillator strength obeys to the rules: X n<m fmn + X fmn = Z (42) n>m Known as Reiche-Thomas-Kuhn rule, it states that the total number of equivalent electrons cannot be more than the total number of electrons. For two levels, following Einstein rules, the oscillator strength has the property gm fmn = −gn fnm (43) where gm,n are the statistical weights. For a level l , g = 2l + 1, for the n-th shell of a H atom gn = 2n2. Values of the oscillator strengths are computed via numerical models and also measured empirically, they are available at the NIST atomic database at https: //www.nist.gov/pml/atomic-spectra-database Radiation-Atom interaction 8 Selection rules In general there is always some probability of transition between states, but in some cases, it can be exceedingly small. This happens when the transition probability is strictly zero for dipole, but non null for higher multipoles. Allowed lines: dipole selection rules: P • Laporte’s rule: there are no transitions between states with the same parity (−1) li , with li angular momentum numbers of the individual orbitals. • Configuration of electrons must change • Dipole operator is a vector, thus ∆l ± 1 and ∆m = 0, ±1. This rule applies to the moving electron, like in one electron atoms H I He II or the alkali metals • Rules for multi-electron atoms . They involve the total quantities L,S , and J . A general rule, which is true also for higher multipoles is that transitions from J = 0 to J = 0 are always forbidden because photon is a boson. We have then in L-S coupling: ∆S = 0 ∆L = 0, ±1 ∆J = 0, ±1 except J = 0 → J = 0 ∆MJ = 0, ±1 except J = 0 → J = 0 Radiation-Atom interaction (44) (45) (46) (47) 9 Selection rules Taking the higher terms of the expansion for the transition probability, we get the magnetic dipole and electric quadrupole. Forbidden lines: Magnetic dipole (k · r ) • The spontaneous emission coefficient can be written as Aul = 2 order of the Bohr magneton |µB | = 3 4ωul 3~c 3 |µul |2 where |µul |2 is of the e~ 2me c • This implies that Aul ∼ 104 s−1 and therefore the state is meta-stable, when Aul > 104 s−1, the state is semi-forbidden.a • There are many ways to express the magnetic dipole one is µ = L + 2S, in which we need to add spin with factor 2 from Dirac’s equation. • The most important difference from dipole is that parity is unchanged • No change in electronic configuration is needed • Spin flip is possible a Exercise: derive the rate from eq. (38) using the expansion in k · r, which is the suppression factor? Radiation-Atom interaction 10 Selection rules Taking the higher terms of the expansion for the transition probability, we get the magnetic dipole and electric quadrupole. Forbidden lines: electric quadrupole E2 (k · r ) • No change in parity • Term is quadratic in r: at least one change in nl ∆S = 0 ∆L = 0, ±1, ±2 ∆J = 0, ±1, ±2 except J = 0 → J = 0 (48) (49) (50) Two photon transitions • Transitions with J = 0 ↔ 0 are not allowed for nature of photons (1s2 − 1s2s 1S0) • In H-like atoms transitions are not allowed L = 0 ↔ 0 (1s − 2s) • Only emission of two photons can preserve angular momentum • Conservation of energy requires that the sum of energies is preserved • The most probable case is the emission of two dipole photons within the allowed time ∆t∆E ∼ ~a a Exercise: using the probability of dipole and the uncertainty principle, derive the suppression factor of the emission coefficient from the dipole case. Radiation-Atom interaction 11 Line width I - natural broadening • Uncertainty principle implies natural broadening of lines ∆E∆t ∼ ~ . The P 0 0 , where n spontaneous decay from a state n proceeds at rate γ = A 0 nn n are the states with lower energies. • If radiation is present, we should add the induced rate • The coefficient of the wave function for the perturbed Hamiltonian (eq. 30) is thus of the form e−γt/2 and leads to a decay of the electric field of the same factor (energy as e−γt ) • The Fourier transform of a decaying sinusoid is a Lorentzian function γ/4π 2 φ(ν) = (51) 2 2 (ν − ν0) + (γ/4π) where ν0 is the transition frequency and γ is the line width (Full width half maximum) • The above is valid strictly only for the ground state, since it has infinite live time, if both states are broadened (finite live time), γ = γl + γu . Radiation-Atom interaction 12 Line width II - Doppler broadening • Astronomical systems are hot: atoms are shacked and emit in a frame which is not the observer’s frame, here ν0 is the rest-frame frequency. ν0 v z ν − ν0 = (52) c • The number of atoms with the velocity in the range vz − vz + dz is proportional to the 2 m v 2 Maxwellian distribution exp − 2akTz dvz , with ma mass of the atom • By substituting eq. (52), we obtain the line profile, in the assumption that natural broadening is negligible φ(ν) = 1 2 2 √ e−(ν−ν0) /(∆νD ) ∆νD π (53) where the Doppler width is defined as ∆νD = ν0 c s 2kT ma • If microturbulence is present, stochastic motion << mean free path of atoms s ν0 2kT ∆νD = +ξ c ma with ξ rms measure of the turbulent velocity. Radiation-Atom interaction 13 Complication on Line • Fine structure lines are forbidden lines with ground state multiplet, usually seen in IR and cool gas • Hyperfine structure lines interaction with nuclear spin, e.g. electron spin flip produces the 21 cm line of neutral Hydrogen • Satellite lines Lines produced from excited atoms, in which inner shell electrons are missing – perturbed energy levels – slightly different energy levels from the lines of the ground state – They appear as satellite of the lines in the ground state Radiation-Atom interaction 14 What next In the next lecture, we will apply these basic notions to the explain some features in high-resolution X-ray spectra and derive the physical conditions of the emitting/absorbing regions. Summary 1 Summary Bibliography 0–30a [1607.07420v1] Pintore, F., Sanna, A., Di Salvo, T., et al. 2016, arXiv.org [1601.05215v1] Hitomi-Collaboration, Aharonian, F. A., Akamatsu, H., et al. 2016, arXiv.org 0