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Fine-And Hyperfine-Structure Effects in Molecular Photoionization: I

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Fine-And Hyperfine-Structure Effects in Molecular Photoionization: I Fine- and hyperfine-structure effects in molecular photoionization: I. General theory and direct photoionization Matthias Germann1 and Stefan Willitsch1, a) Department of Chemistry, University of Basel, Klingelbergstrasse 80, 4056 Basel, Switzerland (Dated: 2 October 2018) We develop a model for predicting fine- and hyperfine intensities in the direct photoionization of molecules based on the separability of electron and nuclear spin states from vibrational-electronic states. Using spherical tensor algebra, we derive highly symmetrized forms of the squared photoionization dipole matrix elements from which which we derive the salient selection and propensity rules for fine- and hyperfine resolved photoionizing transitions. Our theoretical results are validated by the analysis of the fine-structure resolved photoelectron spectrum of O2 (reported by H. Palm and F. Merkt, Phys. Rev. Lett. 81, 1385 (1998)) and are used for predicting hyperfine populations of molecular ions produced by photoionization. I. INTRODUCTION dynamics.19–21 Thus, there is a growing need for theo- retical models capable of describing fine- and hyperfine Photoionization and photoelectron spectroscopy are effects in molecular photoionization. Whereas the hy- among the eminent experimental techniques to gain infor- perfine structure in Rydberg spectra has previously been mation on the electronic structure of molecules, on their treated within the framework of multichannel quantum- 14–18 photoionization dynamics and the structure and dynam- defect theory (MQDT), we are not aware of any pre- ics of molecular cations.1–3 Since the first photoelectron vious treatments of hyperfine intensities in direct pho- spectroscopic studies of molecules in the 1960s, the reso- toionization which can be applied to the interpretation lution of the technique has steadily been improved, such of line intensities in high-resolution photoelectron spec- that vibrational and thereon rotational structure were tra and to the prediction of level populations of molec- resolved during the following decades.3,4 Along with the ular cations produced by photoionization. The present experimental progress, the theoretical understanding of work aims at filling this gap by developing closed expres- molecular photoionization has been successively refined. sions for fine- and hyperfine-structure resolved intensities Vibrational structure in the spectra can often be modeled in molecular photoionization. The theory is developed in terms of the Franck-Condon principle (see, e.g., Ref. here for diatomic molecules in Hund’s coupling case (b), 2). Concerning rotational structure, Buckingham, Orr but can readily be extended to other coupling cases by 22 and Sichel (BOS) presented in a seminal paper5 in 1970 a suitable basis transformation of our final result or to a model to describe rotational line intensities in photo- symmetric- and asymmetric-top molecules by a suitable 4 electron spectra of diatomic molecules. Subsequently, ex- modification of the rotational basis functions. tended models, e.g., for resonance-enhanced multiphoton In the present paper, we develop the general theory for ionization6 or to describe the angular distribution of the fine- and hyperfine-resolved photoionization intensities photoelectrons,7,8 have been developed. Also, the BOS and apply our model to the analysis of the fine structure model has been rephrased in terms of spherical tensor of the photoelectron spectrum of O2 from Ref. 13 and of algebra9 and extended to asymmetric rotors.4 Moreover, hyperfine propensities in the photoionization of N2. In 23 dipole selection rules for photoionizing transitions have the subsequent companion paper, we extend our model been developed based on these intensity models10 as well to resonance-enhanced multiphoton-ionization processes as on general symmetry considerations.11,12 and address the problem of hyperfine-preparation of Over the last decades, fine (spin-rotational) struc- molecular cations. ture has been resolved in high-resolution photoelec- tron spectra.13 Hyperfine structure has been resolved in millimeter-wave spectra of high Rydberg states of rare II. GENERAL CONSIDERATIONS gas atoms such as Kr and diatomics such as H2 and its isotopomers, see, e.g., Refs. 14–18. Fine- and hyperfine- We consider the ionization of a molecule M yielding structure effects in molecular photoionization have also the molecular ion M+ by ejection of a photoelectron e− become of importance in precision spectroscopy and dy- through interaction with electromagnetic radiation via namics experiments with molecular ions produced by arXiv:1606.02990v1 [physics.chem-ph] 9 Jun 2016 the electric-dipole operator µ: photoionization in which the ionic hyperfine populations µ are governed by the underlying hyperfine photoionization M −−−−! M+ + e−: (1) The relevant transition matrix element is given by a)Electronic mail: [email protected] (h M+ j h e− j) µ j Mi ; (2) 2 with j Mi standing for the internal quantum state of the III. FINE-STRUCTURE RESOLVED PHOTOIONIZATION neutral molecule and j M+ ij e− i the product of the in- INTENSITIES ternal state of the molecular ion M+ and the state of the photoelectron e−. We start by developing the theory for fine-structure- resolved photoionization intensities which will form the basis for the subsequent inclusion of hyperfine struc- The squared magnitude of the transition matrix ele- ture in Sec. IV. We note that a similar result for fine- ment summed over all the quantum states contributing structure resolved photoionization has previously been to the observed ionization rate, derived by McKoy and co-workers.6,24 In Hund’s case (b), fine structure manifests itself in the coupling of the orbital-rotational angular momentum N (corresponding to the mechanical rotation of the molecule in Σ states) with the electron spin S to form J: J = N + S. Therefore, we evaluate Eq. (3) for a diatomic + X X X 2 P (M ! M ) = (h M+ j h e− j) µ j Mi ; molecule in Hund’s case (b) by expressing the quan- M M+ e− tum states of the neutral molecule j Mi and the molec- (3) ular ion j M+ i in the basis jnΛ; v; NΛSJMJ i and is proportional to the ionization probability per unit + + + + + + + + n Λ ; v ;N Λ S J MJ , respectively. The defini- time. Here, the sums over M and M+ include all de- tions of the relevant angular-momentum quantum num- generate (or spectroscopically unresolved) states of the bers are summarized in Tab. I. n (n+) and v (v+) denote neutral molecule and the molecular ion, respectively, in- the electronic and the vibrational quantum number in volved in the photoionizing transition. The sum over the neutral molecule (molecular ion). The state of the e− includes the orbital angular momentum and the spin photoelectron j e− i is expressed as a tensor product of state of the emitted photoelectron. We suppose that nei- its spin state and its partial wave js; msi jl; mli. ther the energy nor the angular distribution nor the spin The quantity P (J; J +) which is proportional to the polarization of the photoelectron is detected in the pho- ionization probability on the photoionizing transition toionization experiment. J ! J + may hence be written as + 1 l s J J 2 + X X X X X + + + + + + + + P (J; J ) = n Λ ; v ;N Λ S J MJ hs; msj hl; mlj µ jnΛ; v; NΛSJMJ i : + l=0 ml=−l ms=−s MJ =−J + MJ =−J (4) We follow the approach of Xie and Zare9 and identify the state of the photon. Cjm stands for a Clebsch- j1m1j2m2 electric-dipole operator µ and the photoelectron partial 25–27 k Gordan coefficient. The spherical tensor operator Tp wave jl; m i with the spherical tensors T1 and Tl , l µ0 −ml of Eq. (5) describes the combined effect of absorbing respectively. We then contract the product of these two electromagnetic radiation via the electric dipole opera- 22,25,26 spherical tensors according to tor and ejecting a photoelectron in the state jl; mli (with l = k ± 1). l+1 X Tl ⊗ T1 = Ckp Tk; (5) The term with k = l does not contribute to the sum −ml µ0 l−ml1µ0 p k=jl−1j in Eq. (4) because of parity selection rules and may be omitted.9,11 Ignoring proportionality constants, the ma- where p = −ml + µ0 and µ0 denotes the polarization trix element in Eq. (4) is thus expressed as + + + + + + + + n Λ ; v ;N Λ S J MJ hs; msj hl; mlj µ jnΛ; v; NΛSJMJ i X = Ckp n+Λ+; v+;N +Λ+S+J +M + hs; m j Tk jnΛ; v; NΛSJM i ; (6) l−ml1µ0 J s p J k=l±1 where negative values for k are to be excluded. To proceed, we decouple spin and orbital-rotational 3 TABLE I. Angular momentum quantum numbers relevant to the photoionization of diatomic molecules Magnitude Mol.-fixed Space-fixed quant. num. projection projection Description N Λ MN Orbital-rotational angular momentum of the neutral molecule S – MS Total electron spin of the neutral molecule J – MJ Total angular momentum of the neutral molecule excluding nuclear spin I – MI Nuclear spin of the neutral molecule F – MF Total angular momentum of the neutral molecule + + + N Λ MN Orbital-rotational angular momentum of the molecular ion + + S – MS Total electron spin of the molecular ion + + J – MJ Total angular momentum of the molecular ion excluding nuclear spin + + I – MI Nuclear spin of the molecular ion + + F – MF Total angular momentum of the molecular ion l – ml Orbital angular momentum (partial wave) of the photoelectron s – ms Spin of the photoelectron (s = 1=2) 1 – µ0 Angular momentum due to the electric-dipolar interaction with the electromagnetic field k q p Total orbital angular momentum transferred to/from the molecule in the ionization process (p = −ml + µ0) u – w Total angular momentum transferred to/from the molecule in the ionization process (w = −ms + p) angular momenta in the neutral state according to coordinates using Wigner rotation matrix elements k X k jnΛ; v; NΛSJMJ i Tk = [Dk ]∗T0 ; (10) X p pq q = CJMJ jnΛ; v; NΛM ; SM i ; (7) q=−k NMN SMS N S MN ;MS where the 0 denotes operators in molecule-fixed coordi- and analogously in the ionic state.
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