Egidio Landi: a Life in the Science and Teaching of Polarimetry

Home , Giorgio Abetti, Giulio Racah, Piero Calamandrei

Egidio Landi: A Life in the Science and Teaching of Polarimetry

R. Casini High Altitude Observatory, National Center for Atmospheric Research,1 P.O. Box 3000, Boulder, CO 80307-3000, U.S.A.

Abstract. This is inevitably a very personal and perhaps even biased account of the work of Prof. Egidio Landi Degl’Innocenti, during his nearly 45 years long scientific career, in the field of polarimetry as it applies to the investigation of solar processes, as well as in the broader context of astrophysics. Despite the breadth of Egidio’s contributions to scientific research and teaching, I will not be providing (nor would I have been able to) a complete account of his work. I instead made the choice to emphasize Egidio’s style in approaching new science challenges as well as revisiting older problems. This style is first of all a product of both personal discipline and deeply rooted curiosity, but also a legacy of the cultural and academic ambient where Egidio spent his formative years, that is, Florence and its University.

Egidio Landi Degl’Innocenti was born in Florence, on January 25th, 1945, only a few months after the liberation of the city from the occupation of the German army achieved by the local Resistance groups and the Anglo-American armed forces. Despite the dramatic pillaging and destruction of the city during one year of Nazi occupation, the cultural spirit of the place had remained unscathed, and this certainly contributed to a relatively speedy recovery in the years of the “reconstruction” that followed the end of World War II. Indeed, one could claim that those were the years of a second “renaissance” of Florence, when local intellectual and authority ﬁgures, motivated by a healthy measure of pride in the cultural and spiritual legacy of the city, helped maintain high the hopes and vision of the post-war years. Men of culture, politics, and religion, such as Pietro Annigoni, Piero Calamandrei, Elia Card. Dalla Costa, Luigi Dallapiccola, Fr. Giulio Facibeni, Eugenio Garin, Giorgio La Pira, Mario Luzi, Fioretta Mazzei, Vasco Pratolini, Vasco Ronchi, Gaetano Salvemini, Giuliano Toraldo di Francia, provided a deep and fertile intellectual soil and social fabric, which allowed the rebirth of Florence as a true center of “integral humanism”. In the mathematical and natural sciences, particularly in Physics and Astronomy, Florence had already witnessed an important revolution in the 1920s (which sadly also coincided with the advent of Fascism in Italy), while Antonio Garbasso was the director of the Institute of Physics, and Giorgio Abetti was the director of the Arcetri Observatory, both located on the hill of Arcetri, at the city’s south end. Garbasso was a physicist, who had also spent a period of study in Germany under Hertz and von Hel- moltz before returning to Italy at the turn of the 20th century, ﬁrst in Genoa, and then, in 1913, in Florence. Abetti was an astronomer, and the son of Antonio Abetti, who had in turn been the director of the Arcetri Observatory for almost thirty years, when

1The National Center for Atmospheric Research is sponsored by the National Science Foundation.

1 2

Figure 1. Fermi, Carrara, and Rasetti in Florence, posing in front of the “pozzo di Arcetri”, in the cloister of the Institute of Physics. Sitting in the back is Rita Brunetti, a research assistant of Garbasso along with Rasetti. in 1922 Giorgio inherited the incumbency of running the observatory after his father. Both Garbasso and Abetti were sensitive to the recent dramatic changes of the physical sciences—at a time when Einstein’s general theory of relativity had been given solid experimental evidence (by the results of Eddington’s 1919 solar eclipse expedition), and the new quantum view of the microscopic world was still being shaped—and they both took bold steps to secure a thriving scientific environment at their respective institutes in Florence. The enthusiasm of the times are probably best demonstrated by the array of “students” and “assistants” that Garbasso was able to gather around himself during the years of his tenure as the director of the Institute of Physics in Arcetri; names such as Franco Rasetti, Bruno Rossi, Giuseppe (Beppo) Occhialini, Vasco Ronchi, and Giulio Racah, each of whom would make important contributions to the scientific research of the 20th century (Rasetti in nuclear physics, Rossi and Occhialini in cosmic rays, Ronchi in optics, Racah in the quantum theory of spectroscopy). However, a most important turn of events happened near the end of 1924, when Garbasso succeeded in bringing Enrico Fermi to Florence. Fermi had earned his Physics degree two years earlier at the Scuola Normale Su- periore in Pisa, where he had been a fellow student with Rasetti and Nello Carrara (see Figure 1). The latter, despite being born in Florence, did not actively participate to the establishment of the early Florentine school of Physics, although he would later on become the founder and first director of the local Istituto di Ricerca sulle Onde Elet- tromagnetiche (IROE), which is today named after him. After his degree, Fermi had spent a brief period abroad, first in Göttingen with Max Born, and then in Leiden with 3

Figure 2. Mandò and Franchetti (on the right) discuss with I. I. Rabi (center) on the rooftop of the Institute of Physics. The Arcetri Observatory is in the background, on the top of the hill.

Paul Ehrenfest. He had returned to his native Rome, and briefly worked at the Physics institute directed by Orso Maria Corbino, when he got the opportunity to join the newly established Università degli Studi of Florence.2 The two years Fermi worked in Florence were extremely productive. He was charged with lecturing classes on Analytic Mechanics and Mathematical Physics, but at the same time he was also able to pursue his own scientific interests, both in theoretical research and experimental physics. While in Florence, he produced one of his most important works, which is the derivation of the properties of the Fermi-Dirac statistics describing identical particles with half-integer spin numbers (Fermi 1926). Quite rele- vant to the subject area of this polarization workshop is the theoretical and experimental work that Fermi and Rasetti pursued about the effects of an alternating magnetic field on the polarization of resonance radiation (Fermi & Rasetti 1925). Unfortunately, Fermi and Rasetti did not stay in Florence very long. Fermi went back to Rome in 1927, to fill the first Italian professorship in Theoretical Physics, while his childhood friend and colleague Enrico Persico moved in the opposite direction, from Rome to Florence, to fill the position left vacant by Fermi. Shortly after, Rasetti rejoined with Fermi in Rome, to fill the position left vacant by Persico. This amusing shuffling of personnel between the Florentine and Roman institutions was in fact instrumental in cementing the professional and personal interactions between the two research groups for years to come.

2October 1st, 1924 is the oﬃcial date of the institution of the modern university by royal decree. However, from a cultural standpoint, we can trace its origins back to the Studium generale et Universitas scholarium of Florence, which was founded in 1321, the same year of Dante Alighieri’s death. 4

Persico carried on the teaching of Analytic Mechanics and Mathematical Physics in Florence, and began establishing a strong school on the new “wave mechanics” within the Italian academia, first in Florence, and later in Turin, to where he moved in 1931. Two of his students were Racah in Florence and Ugo Fano in Turin (who also happened to be a first-degree cousin of Racah), who were to make remarkable and lasting contributions to the field of spectroscopy and polarimetry. In addition, some of Fermi’s students in Florence were there to stay (such as Giulio Calamai, who had attended Fermi’s course on Analytic Mechanics, and passed the exam with a very hon- orable 28/30 grade3), and, providentially, a few of them had also diligently transcribed the lecture notes of his courses. To some extent, this also helped carry on the baton of Fermi’s teaching and research styles at the University. The physicists that formed in Florence during those times, or who had been called to fill available positions there, such as Gilberto Bernardini, left a lasting legacy in the Florentine atheneum, characterized by the breadth, depth, and rigor of scientific research and teaching. As Manlio Mandò had put it (see Bonetti & Mazzoni 2007), the “echo” of the passage of Rasetti, Fermi, and Persico was still resounding loudly when in 1930 he started attending the university courses in Arcetri. Beppo Occhialini, who had been an assistant professor in Arcetri before moving to Cambridge to work with Blackett, likened the triad Garbasso–Abetti–Persico to a “Trinity”, or more figuratively to a “three-legged stool” upon which the Arcetri school of Physics had been solidly established, and was able to thrive in the years ahead. Abetti, in particular, had established a cycle of scientific “seminaries”, through which many rising personalities of modern physics (among those, Hans Bethe) were able to exchange their experiences and scientific accomplishments with their Florentine colleagues. Beside the people already mentioned, in the decade between 1930 and 1940, Arcetri produced several other graduates who will become stable and critically important figures of the local academic circle: Simone Franchetti, Guglielmo Righini, and Giuliano Toraldo di Francia. Sadly, Franchetti (visible in Figure 2 on the right, next to Mandò) was banned from teaching in 1938, as a consequence of the newly imposed national racial laws of the Fascist regime. However, he was able to return to the Uni- versity of Florence after the war, and eventually became the director of the Institute of Physics until 1977. Righini became the director of the Arcetri Observatory until his death in 1978, and was instrumental in solidifying the solar research directions that had been initiated by Abetti, recognizing the important problems of the science, and select- ing the right people to tackle them (among them, Egidio Landi). Toraldo di Francia was an eclectic mind, an avid scholar of Dante, a skillful chess player, a profound episte- mologist, but above all a formidable scientist and influential teacher. Not by chance, at the beginning, I included his name in that very diverse list of cultural personalities that made the post-war “renaissance” of Florence possible. At the beginning of the 1950s, the Istituto Nazionale di Fisica Nucleare (INFN) was founded. A Florentine section was established in Arcetri as early as 1952, and Mandò became its first director, leading it for most of 20 years. The increasingly theoretical slant of nuclear and particle physics made immediately manifest the need to create a program of post-graduate education to make up for the absence of a national

3Calamai ﬁlled various teaching positions, such as Calculus, Theoretical Physics, Experimental Physics, Optics, and eventually became astronomer at the Arcetri Observatory (see Casalbuoni, Frosali, & Pelosi 2014). 5

Figure 3. Egidio Landi at the blackboard, lecturing on the polarization scattering K matrix, whose expression involves the set of irreducible spherical tensors TQ for polarization, which he introduced in the early 1980s (Landi Degl’Innocenti 1984).

Ph.D. program.4 The objective was to address the preparation of the new generation of scientists for the highly technical challenges of those disciplines. Thus, the Scuola di Perfezionamento in Fisica was born. This came to Florence in 1963, and Raoul Gatto was brought in to become its ﬁrst director. In Florence, Gatto found a fertile ambient, and was able to gather a group of valuable young graduates, such as Marco Ademollo, Emilio Borchi, Claudio Chiuderi, Giorgio Longhi, and Giovanni Martucci, as well as Gatto’s own student Gabriele Veneziano (arguably the “father” of string theory, who laid several foundational ideas while in Florence). They represented the core of a thriving, and still existing, local School of Theoretical Physics. Many of these characters would become the teachers of Egidio Landi (shown in Figure 3 during a recent lecture on the polarization scattering matrix), when he enrolled in the Physics program of the University of Florence in the academic year 1963-64, at a time when the “spirit of Arcetri” was still alive and well. Egidio earned his Physics degree in 1971, with a thesis on the mechanisms that produce the UV solar spectrum (in English, the title of the thesis literally translates to Opacity Sources in the Ultravio- let Solar Spectrum). This work was performed under the guidance of Giancarlo Noci, who was then a professor at the University. Noci had studied with Mario Rigutti, who in turn had earned his degree with Guglielmo Righini, one of Giorgio Abetti’s pupils. Hence, Egidio’s scientiﬁc lineage directly roots back to the early days of the Institute of Physics, with its solid foundation on the Garbasso–Abetti–Persico triad. After attaining his MS in Physics, Egidio enrolled in the Scuola di Perfezionamento, attending a

4This was not established until the early 1980s. 6

k q ¢ ¡ p k p

Figure 4. Feynman diagrams for photon-atom processes of 1st order. p and q are the initial and ﬁnal atomic states, respectively. k is the incoming or outgoing photon.

k ′ q k q q k ′ q q

r r k r k ′ r k r

¤ ¦ § £ p

k p k ′ p k p p ¥

Figure 5. Feynmandiagrams for photon-atomprocesses of 2nd order. p and q are the initial and final atomic states, respectively, whereas r is an intermediate (virtual) state. k and k′ are the incoming or outgoing photons. A summation over all possible intermediate states, r, is implied by these diagrams. variety of specialized courses. However, in the end, he decided not to pursue the formal acquisition of the higher academic degree. Egidio’s thesis, and the corresponding paper that was published with Noci in Solar Physics (Landi Degl’Innocenti & Noci 1973), already display the main characteristics of Egidio’s approach to scientific research, which would mark all of his future work: the clarity of the physical assumptions, the rigor of the formal derivation of the results, and the reliance on the fundamental principles of Physics in interpreting his findings. However, the first appearance in the literature of Egidio’s scientific production was not the publication of his thesis work, but the treatment of a fundamental problem in radiative transfer (RT), which would set the path of his investigations for a lifetime. This work was conducted in collaboration with his brother Maurizio (Landi Degl’Innocenti & Landi Degl’Innocenti 1972), two years his senior, and had originally been proposed to them by Alberto Righini, the son of then Arcetri Observatory’s director Guglielmo Righini. The paper concerned the formulation of the general problem of spectral line formation in a magnetized plasma, and in the hands of the Landi Degl’Innocenti broth- ers, it aimed at deriving from first principles—within the realm of non-relativistic quantum electrodynamics and the density-matrix formalism (Fano 1957)—an expression for the transfer equation of polarized radiation, expressed in terms of its Stokes parameters. This was required in order to be able to describe the interaction of radiation with complex atomic structures that have no classical analog, as well as to provide a unified scheme for the description of radiation and atomic polarization. In order to lay down the bases of this new approach, Egidio and Maurizio considered at this stage the simpler case of a plasma satisfying the conditions of local thermodynamic equilibrium (LTE). The main idea behind the approach followed in that work was to consider the evolution equation for the statistical operator (or density matrix) describing the quantum system composed of the interacting ensembles of the atoms and the photons, d 1 ρ(t) = H(t), ρ(t) . (1) dt i~ 7

Here H(t) is the sum of the atomic Hamiltonian, HA, the radiation Hamiltonian, HR, and the interaction Hamiltonian, HI(t), the latter being responsible for all atom-photon processes. Equation (1) admits a formal integral solution that can be expressed as an infinite perturbation series, explicitly identifying the various orders of atom-photon processes. This is most clearly manifested by writing this solution in the form † ρ(t) = U(t, t0) ρ(t0) U (t, t0) , (2) where U(t, t0) is the evolution operator, from an initial time t0, for the system described by the total Hamiltonian, H(t). This operator satisfies Schrödinger’s equation, and its formal solution can be written in the form of the infinite perturbation expansion ∞ (i~)−n t t U(t, t ) = 1 + dt ··· dt T{H(t ) ··· H(t )} , (3) 0 n! Z n Z 1 n 1 Xn=1 t0 t0 where T{···} is Dyson’s time-ordered product (e.g., Mandl & Shaw 1984). It is well known that eq. (3) can be interpreted as a series expansion of Feynman diagrams describing all possible atom-photon processes. For example, Figs. 4 and 5 show all Feyn- man diagrams corresponding to processes of 1st and 2nd order, respectively. In their paper, Egidio and Maurizio only considered 1st-order processes, in order to be able to treat the mechanisms of radiation absorption and emission. Since the quantum statistical operator ρ(t) depends on both atomic and radiation variables, they performed a trace operation over the atomic observables, in order to “project” the evolution equation (1) onto the space of the radiation observables. After this operation, they were able to identify the resulting expression with the RT equation (in fact, a set of four coupled differential equations) for the polarized radiation field. The complementary operation of tracing eq. (1) over the radiation “bath” had already been exploited in previous work (e.g., Lamb & Ter Haar 1971), for the purpose of deriving the statistical-equilibrium (SE) equations of the atomic system. With their paper, Egidio and Maurizio demonstrated that the formal symmetry of the problem with respect to the “trace” operation could in fact be given a straightforward physical interpretation, in terms of the complementarity of the SE and RT equations. Their result also allowed to test the validity of prior heuristic derivations of the RT equation for polarized radiation, and to correct the corresponding expressions where necessary. Two further articles, published in the Nuovo Cimento in 1975 and 1976, aimed at clarifying the quantum-mechanical bases of the general problem of polarized line formation, as well as assessing the potential of the new formalism for treating the difficult case of departure from LTE. The first paper (Landi Degl’Innocenti & Landi Degl’Innocenti 1975), again by Egidio and Maurizio, can be considered a generalization to the non-LTE case of the RT equation derived in the original 1972 paper. The second paper (Landi Degl’Innocenti, Landolfi, & Landi Degl’Innocenti 1976) revisited the derivation of the SE equations for the atomic system, explicitly taking into account the effects of a magnetic field, as well as of the departure of the plasma conditions from LTE. It must be pointed out that the separation of the non-LTE line formation problem into two sets of equations that describe the evolution of the atomic and radiation sys- tems is a forceful but necessary approximation. It is essentially dictated by the need to attack the practical problem of polarized line formation by means of iterative numerical schemes, where one seeks the convergence to an equilibrium condition between the excitation state of the atomic system and the local radiation field, which interacts with the 8

Figure 6. The self-consistency loop of the non-LTE problem of line formation, breaking down the mutual interaction of the atomic system with the ambient radiation field into successive steps of an iterative numerical scheme. atom and is modified by it. The numerical implementation of this iterative scheme has been termed the self-consistency loop of the polarized line formation problem (Landi Degl’Innocenti & Landolfi 2004, see Figure 6). All along, the new formalism was being applied by Egidio and collaborators to tackle concrete problems of solar physics, and it would eventually become an important part of Egidio’s lecture course on Spectroscopy, which he held from 1982, when he was appointed as associate professor at the Institute for Astronomy (later to become the Department of Astronomy and Space Sciences) of the University of Florence, until his retirement in 2015. A further and ultimate generalization of this formalism became the content of a series of four papers published in Solar Physics between 1983 and 1985 (Landi Degl’Innocenti 1983a,b, 1984, 1985). In that work, who Egidio single-authored, he fully exploited the power of the description of physical symmetries and the represen- tation of the corresponding observables by means of irreducible tensorial sets, an approach that had been preeminently fostered by Ugo Fano and Giulio Racah (Fano 1957; Fano & Racah 1959). In particular, Egidio introduced a new set of polarization tensors (cf. Figure 3), which, in the following applications of his formalism to the investigation of line formation problems, has greatly facilitated the grasping of symmetries and the identification of allowed processes in the interaction of polarized radiation with an atomic system. Additionally, in this work, the atomic system was described under the most general excitation conditions, allowing in particular for the existence of quantum interference between atomic levels, and its modification due to the presence of an ambient magnetic (or electric) field, and to the possible presence of fine structure. The 9 problem of describing polarized line formation under these more general atomic conditions has been dubbed by Egidio himself the “non-LTE problem of the 2nd kind”. Its wide applicability to the modeling of polarized spectral lines has enabled the scientific community to explore the diagnostic potential of the Hanle effect and of level-crossing effects for the investigation of the magnetism of solar and astrophysical plasmas. A notable example is represented by the application of this formalism to the formation of the chromospheric line of He I at 587.6 nm, which has been an important diagnostic of the plasma and magnetic properties of solar prominences. In a seminal paper (Landi Degl’Innocenti 1982), Egidio showed how level-crossing effects were important in determining the shape of the circular polarization profiles of this line, for the typical magnetic field strengths of solar prominences, giving rise to very strong de- partures from the usually assumed profile shape due to the Zeeman effect. Even at a time when the Hanle-effect diagnostic for this type of investigations had already been pursued (House 1971; Bommier & Sahal-Bréchot 1978; Bommier 1980), the realiza- tion that important magnetic effects in circular polarization had to be expected, despite the relatively weak fields of prominences, was instrumental in providing critical con- straints for the interpretation of spectro-polarimetric observations of prominences (see, e.g. López Ariste & Casini 2003). These level-crossing effects, which are induced by the ambient magnetic field when the Zeeman-effect energy splitting becomes compa- rable with the fine structure separation of the atomic levels, give rise to the mixing between even (atomic alignment) and odd (atomic orientation) multipole orders of the irreducible spherical tensor representing the atomic density matrix (Kemp, Macek, & Nehring 1984; Landi Degl’Innocenti & Landolfi 2004), and this so-called alignment- to-orientation transfer mechanism is responsible for the net circular polarization that is often observed in the He I 587.6 nm emission from solar prominences. The theory was also able to explain the manifestation of peculiar linear polarization signatures in the chromospheric line of He I at 1083 nm observed on the disk, caused by the presence of atomic alignment in the ground state of ortho-helium (Trujillo Bueno et al. 2002). Other successes of the general formalism developed by Egidio for the study of polarized line formation problems, just to name a few, are: 1) the description of the full-Stokes polarization of forbidden coronal emission lines, in particular the effects of atomic alignment on the circular polarization signal (Casini & Judge 1999); 2) the modeling of the effects of simultaneous magnetic and electric fields on the scattering polarization from hydrogen-like atoms, and the prediction of strong net circular polarization (Favati, Landi Degl’Innocenti & Landolfi 1987; Casini 2005; Casini & Manso Sainz 2006); 3) the application of the formalism to the study of resonance scattering polarization and the Hanle effect in molecules (Landi Degl’Innocenti 2003). One notable limitation of the formalism developed by Egidio up to the mid 1980s is the fact that only first-order atom-photon processes (i.e., single-photon absorption and emission) are taken into account. As it was already pointed out above (see para- graph after eq. (3)), this restrictive hypothesis was embraced from the very beginning, since the seminal paper that Egidio and Maurizio published in 1972. Within that physical scheme, the scattering of radiation must be described as a temporal succession of single-photon absorption and re-emission. The connecting element between these two distinct processes, in their contribution to the overall phenomenon of radiation scattering, is represented by the solution of the statistical equilibrium for the atomic system illuminated by the incoming radiation: a “regime” solution for the atomic density ma- 10 trix, subject to the various first-order radiation (and possibly collisional) processes, is derived first, and it is next used in the first-order emissivity to model the scattered radiation. Because the first-order statistical equilibrium problem only contains spectral av- erages of the ambient radiation field, in such a description of radiation scattering the details of the spectral shape of the incident radiation are completely lost by the time that the photon re-emission occurs. Hence, the description of radiation scattering as a non- coherent sequence of single-photon absorption and re-emission is physically consistent only when the incident radiation has no spectral structure (flat-spectrum approximation), or if only the broadband intensity and polarization properties of the scattered radiation are of interest. Since in this picture the memory of the energy of the incident photon is completely lost in the re-emission process, the spectral distribution of the scattered radiation is fully determined by the natural shape of the probability distribution of the atomic population across the energy spectrum (e.g., Heitler 1954). For an atom at rest, this corresponds to the usual Lorentzian distribution with a full-width at half maximum given by the inverse lifetime of the energy level. Hence, the frequency of the incident radiation gets completely redistributed (CRD) by the re-emission process in accordance with the natural shape of the atomic level. On the other hand, radiation scattering is an intrinsically 2nd-order process (described by the first two diagrams of Fig. 5), where frequency (and polarization) correla- tions between the incoming and the outgoing photons are generally present. These are manifested in the form of a partial redistribution (PRD) of the frequency of the incident radiation across the emitted spectrum, which leads to important differences in the spectral shape of the scattered radiation with respect to the predictions of the CRD theory. Hence, all the atom-photon interaction processes where the spectral details of the scattered radiation are deemed to be important, and for which the fundamental physical assumptions of CRD are not guaranteed (e.g., in a collisionless plasma), cannot be described through a formalism where only first-order atom-photon interaction processes are considered. The inadequacy of the traditional first-order formalism in dealing with PRD effects was overcome by Egidio in 1995 (Landi Degl’Innocenti, Landi Degl’Innocenti, & Landolfi 1997), when he developed a new picture of the atom-photon interaction processes based on the so-called “metalevel” (or “internal state”) description of the atomic system (e.g., Woolley & Stibbs 1953). In that picture, the atomic density matrix of the first-order statistical equilibrium problem acquires an additional dimension, which is used to parametrize the probability distributions of the population and coherence of the atomic levels. In so doing, it becomes possible to circumvent the “memory loss” of the spectral details of the incident radiation, which in the ordinary picture is intrinsic to the first-order approximation of the atom-photon interaction. The metalevel theory has been successfully applied to a plethora of problems on the formation of chromospheric lines in the polarized solar spectrum. Egidio himself tackled the modeling of the “enigmatic” linear polarization of the Na I D-doublet at 590 nm as observed near the solar limb (Landi Degl’Innocenti 1998). Others have applied that formalism to predict the scattering polarization of transition region diagnostics such as H I Ly-α at 122 nm and the Mg II h–k doublet at 280 nm (Belluzzi, Trujillo Bueno, & Štepánˇ 2012; Belluzzi & Trujillo Bueno 2012). The main results of the metalevel approach, in the limit of collisionless plasmas, have also been confirmed by other theoretical developments, which relied instead on 11 describing the evolution of the interacting system of atoms and radiation (cf. eq. (3)) to perturbation orders larger than one (Bommier 1997a,b; Casini et al. 2014). In particular, the coherent scattering (2nd-order) term of the emissivity, which allows to include PRD effects in the formation problem for polarized spectral lines, for a Λ-type atomic transition l → u → f is given by (cf. Casini & Manso Sainz 2016, eqs. (2) and (3)),

4 ∗ ∗ e (rq)ul(rq′ ) ′ ′ (rp)u′ f (rp′ ) (2) ′ 4 0 4 q′+p′ u l u f ′ kˆ ′ εi (ωk , ) ≡ Nωk′ ρll (−1) 3 ~2c4 ǫ ′ + iω ′ Xll′ uuX′ f Xqq′ Xpp′ uu uu ′ ′ 1 1 K 1 1 K K′ ′ × (2K + 1)(2K + 1) T ′ (i, kˆ ) −q q′ −Q −p p′ −Q′ Q XKQ KX′Q′ p ∞ kˆ 3 d K × dω R(Ω , Ω ′ ; Ω , Ω ′ , Ω ; ω ,ω ′ , Θ) T ( j, kˆ) S (ω , kˆ) , (4) Z k u u l l f k k I 4π Q j k 0 Xj=0

′ where (ωk, kˆ ) and (ωk′ , kˆ ) are the incident and scattered photons, respectively, and where Ωa = ωa + iǫa is the “complex frequency” of the atomic level a, inclusive of its inverse lifetime. We note that the incoming and outgoing radiation fields are weighted in frequency by the redistribution function R(ωk,ωk′ , Θ), whereas their directional con- K kˆ tributions are weighted by the polarization tensors TQ (i, ) originally introduced by Egidio (Landi Degl’Innocenti 1984, see also Figure 3). Additionally, the entire 2nd- order emissivity is weighted by the atomic density matrix ρll′ of the initial (lower) state. We also recognize the presence of a “depolarizing” denominator in the expression (4), ′ which depends on the energy separation ωuu′ of the interfering levels u and u of the upper state, which originates from the possible presence of fine structure and/or exter- nally applied fields, and on the radiative and collisional lifetime ǫuu′ of the upper state. In particular, this denominator is responsible for the manifestation of the Hanle effect in the upper state. While the extension of the theory of polarized line formation to include PRD effects has recently succeeded in attaining realistic modelings of chromospheric magnetism diagnostics in the Hanle effect regime (e.g., del Pino Alemán, Casini, & Manso Sainz 2016, see also the contribution by R. Manso Sainz in these proceedings), the in- ner consistency of the formalism is still work in progress. Still to these days, Egidio himself is exploring new venues to formulate the general problem, as always relying on a clever mix of fundamental principles and physical intuition. I would like to conclude this account of Egidio’s work by commenting on some aspects of his teaching. As I mentioned earlier, Egidio held the only course on Spec- troscopy at the Department of Astronomy and Space Sciences since he became an associate professor at the University of Florence in 1982, until his retirement in 2015 as a full professor (an appointment that Egidio earned in 2000). I remember that I had started attending his lectures in the late 1980s, after the course had been enthusiasti- cally described to me by a fellow student a few years my senior, who had come to the conclusion that Egidio’s lectures offered the only opportunity in the Astronomy department to really see “in action” all the quantum mechanics machinery that we had been taught so beautifully by professors Ademollo and Longhi. Of course, Egidio’s lectures were always very carefully crafted and presented, and his hand written lecture notes provided the perfect companion to refresh and further deepen the concepts that had been presented in class. Recently, Egidio undertook the effort of parsing his lecture 12

Figure7. Egidio Landi, with some of his 1st and 2nd generation “pupils”: (left to right) the author (HAO), J. Štepánˇ (Astron. Inst. ASCR), J. Trujillo Bueno (IAC), E. Landi, S. Fineschi (OATO), S. Bagnulo (Armagh Obs.), R. Manso Sainz (MPS), A. Asensio Ramos (IAC), T. del Pino Alemán (HAO), E. Alsina Ballester (IAC), and L. Belluzzi (IRSOL). notes into a proper textbook, which is also available in English language for the greater benefit of the wider scientific community (Landi Degl’Innocenti 2014). It must be said that Egidio enjoyed a reputation for clarity of presentation and ex- actness of scientific arguments also among his peers at the University. I had originally considered pursuing my master thesis project with prof. Giovanni Godoli, since I had been very favorably impressed by his deep commitment to teaching and the overall ad- vancement of students. I had taken his course on Solar Physics, and I had also prepared with him my exam on Astrophysics (the course was officially held by Chiuderi), and during my many conversations in Godoli’s office, he had clearly come to the conclusion that my thirst for exact argumentations, backed up by unambiguous mathematical evidence, would have made my professional interaction with him a very frustrating ex- ercise. Thus, he encouraged me to talk to Egidio instead, and try to pursue a thesis project with him. For that, I still feel deeply indebted to prof. Godoli, who sadly passed away ten years ago. When Godoli retired near the end of the 1990s, Egidio relieved his course on Solar Physics. The subject was refocused closer to Egidio’s interests and the modern challenges of understanding the Sun’s spectrum (a significant part of Godoli’s course was on plasma physics and basic MHD), and a textbook containing the new set of lecture notes was published in 2007 (Landi Degl’Innocenti 2007, unfortunately, only published in Italian). Of course, under Egidio’s hands, the presentation of the subject attained a whole new level of sophistication. When finally Chiuderi retired from teaching, Egidio also inherited the course on Astrophysics. He also has spent time preparing a full set of lecture notes for that sub- 13 ject, which he is in the process of turning into a new textbook. As part of his teaching of introductory courses for Physics undergraduates, Egidio has also prepared (again, start- ing from very first principles, and with an incredibly low level of reliance on external sources), lecture notes on Fluidodynamics, Thermodynamics, and Statistical Mechan- ics. Unfortunately, for a series of different reasons, including internal politics, Egidio was not able to establish a local school of Polarimetry in Arcetri, and thus secure a succession for his role in the teaching of the subject at the University of Florence. All of his students in the field (some of them portrayed in Fig. 7), with the exception of Marco Landolfi (now retired), have in fact relocated elsewhere. Nonetheless, Egidio Landi’s legacy lives on, through his papers and books, and through the work of his students and collaborators, within the wider solar physics community. The actuality of his past and ongoing work is also poised to remain fruitful, as many more groundbreaking results can still be expected in the study of solar magnetism, through the theoretical, observational, and interpretational investigation of its polarized spectrum.

References

Belluzzi, L., & Trujillo Bueno, J. 2012, ApJ, 750, 11L Belluzzi, L., Trujillo Bueno, J., & Štepán,ˇ J. 2012, ApJ, 755, 2 Bommier, V. 1980, A&A, 87, 109 Bommier, V. 1997a, A&A, 328, 706 Bommier, V. 1997b, A&A, 328, 726 Bommier, V., & Sahal-Bréchot, S. 1978, A&A, 69, 57 Bonetti, A., & Mazzoni, M. 2007, L’Università degli Studi di Firenze nel centenario della nascita di Giuseppe Occhialini (1907-1993) (Florence: Firenze University Press) Casalbuoni, R., Frosali, G., & Pelosi, G. 2014, Enrico Fermi a Firenze (Florence: Firenze University Press) Casini, R. 2005, Phys.Rev.A, 71, 062505 Casini, R., & Judge, P. G. 1999, ApJ, 522, 524 Casini, R., & Manso Sainz, R. 2006, J. Phys. B, 39, 3241 Casini, R., & Manso Sainz, R. 2016, ApJ, 824, 135 del Pino Alemán, T., Casini, R., & Manso Sainz, R. 2016, ApJ, 830, 24L Casini, R., Landi Degl’Innocenti, M., Manso Sainz, R., Landi Degl’Innocenti, E., & Landolﬁ, M. 2014, ApJ, 791, 94 Fano, U. 1957, Rev. Mod. Phys., 29, 73 Fano, U., & Racah, G. 1959, Irreducible Tensorial Sets (New York: Academic Press) Favati, B., Landi Degl’Innocenti, E., & Landolﬁ, M. 1987, A&A, 179, 329 Fermi, E. 1926, Z. Physik, 36, 902 Fermi, E., & Rasetti, F. 1925, Nat, 115, 764 Heitler, W. 1954, Quantum Theory of Radiation (Oxford: Oxford University), 3 ed. House, L. L. 1971, in Solar Magnetic Fields, IAU Symposium 43, 130 Kemp, J. C., Macek, J. H., & Nehring, F. W. 1984, ApJ, 278, 863 Lamb, F. K., & Ter Haar, D. 1971, Phys. Reps., 2, 253 Landi Degl’Innocenti, E. 1982, Solar Phys., 79, 291 Landi Degl’Innocenti, E. 1983a, Solar Phys., 85, 3 Landi Degl’Innocenti, E. 1983b, Solar Phys., 85, 33 Landi Degl’Innocenti, E. 1984, Solar Phys., 91, 1 Landi Degl’Innocenti, E. 1985, Solar Phys., 102, 1 Landi Degl’Innocenti, E. 1998, Nat, 392, 256 Landi Degl’Innocenti, E. 2007, Fisica Solare (Florence: UniText) 14

Landi Degl’Innocenti, E. 2014, Atomic Spectroscopy and Radiative Processes (Milan: Springer) Landi Degl’Innocenti, E. 2003, in Solar Polarization Workshop 3, ASP Conf. Ser. 307, eds. J. Trujillo Bueno & J. Sanchez Almeida, 164 (2003) Landi Degl’Innocenti, E., & Landi Degl’Innocenti, M. 1972, Solar Phys., 27, 319 Landi Degl’Innocenti, E., & Noci, G. 1973, Solar Phys., 29, 287 Landi Degl’Innocenti, E., & Landi Degl’Innocenti, M. 1975, Nuovo Cim. B, 27, 134 Landi Degl’Innocenti, E., & Landolfi, M. 2004, Polarization in Spectral Lines (Dordrecht: Kluwer) Landi Degl’Innocenti, M., Landolfi, M., & Landi Degl’Innocenti, E. 1976, Nuovo Cim. B, 35, 117 Landi Degl’Innocenti, E., Landi Degl’Innocenti, M., & Landolfi, M. 1997, in Science with THÉMIS, Obs. de Paris-Meudon, 14-15 Nov. 1996, eds. P. Mein & S. Sahal-Bréchot, 59 López Ariste, A., & Casini, R. 2003, ApJ, 582, 51L Mandl, F., & Shaw, G. 1984, Quantum Field Theory (New York: Wiley) Trujillo Bueno, J., Landi Degl’Innocenti, E., Collados, M., Merenda, L., & Manso Sainz, R. 2002, Nat, 415, 403 Woolley, R. v. d. R., & Stibbs, D. W. N. 1953, The Outer Layers of a Star (London: Oxford Univ.)