Gohar Hovhannesyan

Vincent Boudon

Maxence Lepers

Transition intensities of trivalent lanthanide ions in solids: Extending the Judd-Ofelt theory

Keywords:

We present a modified version of the Judd-Ofelt theory, which describes the intensities of f-f transitions for trivalent lanthanide ions (Ln 3+ ) in solids. In our model, the properties of the dopant are calculated with well-established atomic-structure techniques, while the influence of the crystal-field potential is described by three adjustable parameters. By applying our model to europium (Eu 3+ ), wellknown to challenge the standard Judd-Ofelt theory, we are able to give a physical insight into all the transitions within the ground electronic configuration, and also to reproduce quantitatively experimental absorption oscillator strengths. Our model opens the possibility to interpret polarizedlight transitions between individual levels of the ion-crystal system.

Introduction

The Judd-Ofelt (JO) theory has been successfully applied since almost 60 years, to interpret the intensities of absorption and emission lines of crystals and glasses doped with trivalent lanthanide ions (Ln 3+ ) [START_REF] Judd | Optical absorption intensities of rare-earth ions[END_REF][START_REF] Ofelt | Intensities of crystal spectra of rare-earth ions[END_REF][START_REF] Hehlen | 50th anniversary of the judd-ofelt theory: An experimentalist's view of the formalism and its application[END_REF]. Despite its remarkable efficiency, this standard JO theory cannot reproduce some of the observed transitions, because of its strong selection rules. It is especially the case for europium (Eu 3+ ) [START_REF] Tanner | Some misconceptions concerning the electronic spectra of tri-positive europium and cerium[END_REF][START_REF] Binnemans | Interpretation of europium (III) spectra[END_REF], well known to challenge the standard JO theory [START_REF] Walsh | Judd-Ofelt theory: principles and practices[END_REF]. Many extensions of the original model have been proposed to overcome this drawback [START_REF] Smentek | Theoretical description of the spectroscopic properties of rare earth ions in crystals[END_REF], including e.g. J-mixing [START_REF] Tanaka | Contribution of mixing to the 5 0 -7 0 transition of Eu 3+ ions in several host matrices[END_REF][START_REF] Kushida | Transition mechanisms and spectral shapes of the 5 0 -7 0 line of Eu 3+ and Sm 2+ in solids[END_REF][START_REF] Kushida | Spectral shape of the 5d0-7f0 line of Eu 3+ and Sm 2+ in glass[END_REF], the Wybourne-Downer mechanism [START_REF] Downer | A new contribution to spinforbidden rare earth optical transition intensities: Gd 3+ and Eu 3+[END_REF][START_REF] Burdick | A new contribution to spinforbidden rare earth optical transition intensities: Analysis of all trivalent lanthanides[END_REF], velocitygauge expression of the electric-dipole (ED) operator [START_REF] Smentek | Theoretical description of 0 ↔ 0 and 0 ↔ 1 transitions in the Eu 3+ ion in hosts with 2 symmetry[END_REF], relativistic or configuration-interaction (CI) effects [START_REF] Smentek | Relativistic f↔f transitions in crystal fields[END_REF][START_REF] Smentek | Relativistic f↔f transitions in crystal fields: Ii. beyond the single-configuration approximation[END_REF][START_REF] Wybourne | Relativistic effects in lanthanides and actinides[END_REF][START_REF] Ogasawara | Optical spectra of trivalent lanthanides in LiYF 4 crystal[END_REF][START_REF] Dunina | Modified theory of f-f transition intensities and crystal field for systems with anomalously strong configuration interaction[END_REF], purely ab initio intensity calculations [START_REF] Wen | Ab-initio calculations of Judd-Ofelt intensity parameters for transitions between crystal-field levels[END_REF]. In this respect, Smentek and coworkers were able to reproduce experimental absorption oscillator strengths with a very high accuracy, with up to 17 adjustable parameters [START_REF] Ke ¸dziorski | Extended parametrization scheme of f-spectra[END_REF]. But in spite of all these improvements, even the most recent experimental studies use the standard version of the JO theory [START_REF] Ćirić | JOES: An application software for Judd-Ofelt analysis from Eu 3+ emission spectra[END_REF][START_REF] Ćirić | An extension of the Judd-Ofelt theory to the field of lanthanide thermometry[END_REF].

In the standard JO theory, the line strength characterizing a given transition is a linear combinations of three parameters Ω (with = 2, 4 and 6), which are functions of both the properties of the Ln 3+ ion and the crystal-field parameters [START_REF] Hehlen | 50th anniversary of the judd-ofelt theory: An experimentalist's view of the formalism and its application[END_REF][START_REF] Walsh | Judd-Ofelt theory: principles and practices[END_REF]. Since the Ω -parameters are adjusted by least-square fitting, those two types of contributions cannot be separated. However, the properties of the impurity can be investigated by means of free-ion spectroscopy. In this respect, recent joint experimental and theoretical investigations have provided a detailed knowledge of some free-Ln 3+ ion structure [START_REF] Meftah | Observation and interpretation of the Tm 3+ free ion spectrum[END_REF][START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF][START_REF] Meftah | Analysis of the free ion spectrum of Er 3+ (Er IV)[END_REF][START_REF] Arab | Observation and interpretation of the 5 5 4 3 5 core-excited configuration in triply ionized neodymium Nd 3+ (Nd IV)[END_REF]. Although such a study has not been made with Eu 3+ , the continuity of the atomic properties along the lanthanide series opens the possibility to compute the Eu 3+ spectrum using a semi-empirical method, based on adjusted parameters of neighboring elements [START_REF] Wyart | On the interpretation of complex atomic spectra by means of the parametric Racah-Slater method and Cowan codes[END_REF].

gohar_hovhannesyan@etu.u-bourgogne.fr (G. Hovhannesyan); maxence.lepers@u-bourgogne.fr (M. Lepers) In this article, we present a modified version of the JO theory in which the properties of the free Ln 3+ ion, i.e. energies and transition integrals, are computed using a combination of ab initio and least-square fitting procedures available in Cowan's suite of codes [START_REF] Cowan | The theory of atomic structure and spectra[END_REF][START_REF] Kramida | Cowan code: 50 years of growing impact on atomic physics[END_REF]. This allows us to relax some of the strong assumptions of the JO theory, for instance the strict application of the closure relation. The line strengths appear as linear combinations of three adjustable parameters which are only functions of the crystal-field potential, giving access to the local environment around the ion. We account for the spin-orbit (SO) interaction responsible of spin-changing transitions by calculating the line strengths at the third order of perturbation theory. Our results on Eu 3+ suggest that the spin-mixing transitions are mainly due to the SO mixing within the ground electronic configuration, in contradiction with the Wybourne-Downer mechanism described in Ref. [START_REF] Downer | A new contribution to spinforbidden rare earth optical transition intensities: Gd 3+ and Eu 3+[END_REF][START_REF] Burdick | A new contribution to spinforbidden rare earth optical transition intensities: Analysis of all trivalent lanthanides[END_REF]. In addition, our model gives a simple physical interpretation of the transitions that are forbidden in the framework of the standard JO theory, including 7 0 ↔ 5 0 , 7 0 ↔ 5 or 7 ↔ 5 0 with odd. To benchmark our model, we reproduce quantitatively the set of experimental absorption oscillator strengths of Babu et al. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF], although we overestimate the strength of 7 0 ↔ 5 0 transition. The paper is organized as follows. Section 2 contains our analytical developments resulting in the ED line strengths, which then allow for calculating oscillator strengths and Einstein coefficients. Our model is based on the timeindependent perturbation theory, up to second and third orders (see Subsections 2.1 and 2.2 respectively). Then in Section 3, we apply our model to the case of europium, describing first the free-ion properties required for our model in Subsections 3.1-3.4, and then the f-f transitions within the ground configuration in Subsection 3.5. Section 4 contains conclusions and prospects.

Electric-dipole line strengths

The aim of the present section is to derive analytical expressions for the electric-dipole (ED) line strengths, which enable to characterize absorption and emission intensities of Ln 3+ -doped solids. Unlike the magnetic-dipole (MD) and electric-quadrupole (EQ) transitions [START_REF] Dodson | Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: Calculated emission rates and oscillator strengths[END_REF], the ED ones are activated by the presence of the host material, which relaxes the free-space selection rules. We use similar hypotheses as in the original JO model [START_REF] Judd | Optical absorption intensities of rare-earth ions[END_REF][START_REF] Ofelt | Intensities of crystal spectra of rare-earth ions[END_REF]: the crystal-field (CF) potential slightly admixes the levels of the ground configuration [Xe] [START_REF] Tanner | Some misconceptions concerning the electronic spectra of tri-positive europium and cerium[END_REF] and those of the first excited configuration [Xe]4 -1 5 , where [Xe] denotes the ground configuration of xenon, dropped in the rest of the article. In the resulting perturbative expression of the ED line strength, we assume that all the levels of the excited configuration have the same energy. However, we relax some of the original hypothesis, by accounting for the energies of the ground-configuration levels, and by applying the closure relation less strictly. Unlike the standard and most common extensions of the JO model, we do not introduce effective operators, like the so-called unit-tensor operator ( ) [START_REF] Cowan | The theory of atomic structure and spectra[END_REF], but rather work on the matrix elements of the CF or ED operators. To calculate the line strength, we firstly use the second-order perturbation theory (see subsection 2.1) and then the thirdorder perturbation theory (see subsection 2.2), for which the free-ion spin-orbit operator is within the perturbation.

The common starting point of those two calculations is the multipolar expansion of the crystal-field potential,

CF = ∑ ∑ =1 ( ) ( , ) ≡ ∑ ( ) (1)
where is a non-negative integer and = -, -+ 1, ..., + , ( , , ) are the spherical coordinates of the -th ( = 1 to ) electron in the referential frame centered on the nucleus of the Ln 3+ ion, and ( ) are the Racah spherical harmonics of rank and component , related to the usual spherical harmonics by ( ) ( , ) = √ 4 ∕(2 + 1) × ( , ), see for example Chap. 5 of Ref. [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF]. In Eq. ( 1), the quantities ( ) represent the electric multipole moment as defined in Chaps. 14 and 15 of Ref. [START_REF] Cowan | The theory of atomic structure and spectra[END_REF]. The simplest way of calculating the CF parameters is to assume that they are due to distributed charges inside the host material. More elaborate models can be used, like distributed dipoles resulting in the so-called dynamical coupling [START_REF] Smentek | Theoretical description of the spectroscopic properties of rare earth ions in crystals[END_REF], or the vibration of the ion center-of-mass. This would affect the physical origin of the coefficients, but not the validity of the forthcoming results [START_REF] Judd | Optical absorption intensities of rare-earth ions[END_REF].

Second-order correction

In the theory of light-matter interaction, the ED approximation arises at the first order of perturbation theory. Furthermore, the f-f transitions in Ln 3+ -doped solids are only possible if the free-ion levels are perturbed by the CF potential. Therefore, using the first-order correction on the ion levels to calculate the matrix element of the ED operator gives in total a second-order correction.

We call |Ψ ⟩ the eigenvectors associated with the ion+crystal system (without electromagnetic field). In the framework of perturbation theory, we express them as |Ψ ⟩ = ∑ |Ψ ⟩, where denotes the order of the perturbative expansion. In this subsection, we consider that the 0-th, i.e. unperturbed, eigenvectors |Ψ 0 ⟩ are the free-ion levels. Those belonging to the ground configuration (with = 4 for Ln 3+ ions) are written in intermediate coupling scheme [START_REF] Cowan | The theory of atomic structure and spectra[END_REF]

|Ψ 0 ⟩ = ∑ | | ⟩ (2)
where , and are the quantum numbers associated with the orbital, spin and total electronic angular momentum respectively, while is associated with the -projection of the latter. The free-ion levels of energy 0 ≡ are degenerate in . Finally in Eq. ( 2), is a generic notation containing additional information like the seniority number [START_REF] Cowan | The theory of atomic structure and spectra[END_REF]. In the 4 configuration of Ln 3+ ions, the energy levels are usually well described in the LS coupling scheme (see Table 2).

In the first excited configuration -1 ′ ′ , with ( , ′ ′ ) = (4 , 5 ) for Ln 3+ ions, we consider free-ion levels in pure LS coupling,

|Ψ 0 ⟩ = | -1 , ′ ′ ′ ′ ′ ′ ⟩, (3)
where the overlined quantum numbers characterize the -1 subshell alone. As Table 4 shows, the LS coupling is not appropriate for the energy levels of the excited configuration. But since, in our ED matrix element calculation, we will assume that all the levels of the excited configuration have the same energy, the choice of coupling scheme is arbitrary, and so we take the simplest one. Now we express the ED transition amplitude 12 between eigenvectors

|Ψ 0 ⟩ + |Ψ 1 ⟩ ( = 1, 2)
, perturbed by the CF potential up to the first order,

⎤ ⎥ ⎥ ⎦ , ( 4
)
where the index = 0 denotes light polarization, and = ±1 denote ± polarizations. We recall that ⟨Ψ 0 1 | (1) |Ψ 0 2 ⟩ = 0, because in free space, there is no ED transition between levels of the same electronic configuration. In what follows, we assume that all the energies of the excited configuration are equal, ≈ ′ ′ . Rather than the center-of-gravity energy of the excited configuration, can be regarded as the mean energy for which the coupling with both levels 1 and 2 is significant (see Fig. 2).

Equation ( 4) contains matrix elements of (1) and CF , itself function of ( ) as Eq. ( 1) shows. Being irreducible tensor operators, the matrix elements of ( ) satisfy the Wigner-Eckart theorem [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF]

⟨ Ψ 0 | | | ( ) | | | Ψ 0 ⟩ = ′ ′ √ 2 + 1 ⟨ Ψ 0 ‖ ‖ ‖ ( ) ‖ ‖ ‖ Ψ 0 ⟩ (5)
where

′ ′ = ⟨ ′ ′ | ′
⟩ is a Clebsch-Gordan (CG) coefficient, and ⟨Ψ 0 ‖ ( ) ‖Ψ 0 ⟩ the reduced matrix element given in Eq. [START_REF] Martin | Atomic energy levels-the rareearth elements. the spectra of lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium[END_REF], which is independent from , ′ and . By contrast, the products of the kind

⟨Ψ 0 | ( ) |Ψ 0 ⟩ ×⟨Ψ 0 | (1)
|Ψ 0 2 ⟩ are not irreducible tensors; still we overcome this problem by expanding the product of two CG coefficients given in [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF], which yields

⟨ Ψ 0 1 | | | ( ) | | | Ψ 0 ⟩ ⟨ Ψ 0 | | | (1) | | | Ψ 0 2 ⟩ = ∑ (-1) 1 + 2 - √ 2 + 1 2 1 + 1 1 1 1 2 2 × 1 2 1 ′ ⟨ Ψ 0 1 ‖ ‖ ‖ ( ) ‖ ‖ ‖ Ψ 0 ⟩ ⟨ Ψ 0 ‖ ‖ ‖ (1) ‖ ‖ ‖ Ψ 0 2 ⟩ (6)
where the quantity between curly brackets is a Wigner 6j symbol. Equation ( 6) is interesting because the only dependence on quantum numbers is in the CG coefficient Applying the same reasoning for the third line of Eq. ( 4), we obtain the same result as Eq. ( 6) except the permutations of the couples of indexes ( , ) and (1, ). Using the symmetry relation of CG coefficients 1 = (-1) 1+ - 1 , we get to the final expression for the transition amplitude

12 = ∑ 1 1 1 1 1 1 ∑ 2 2 2 2 2 2 ∑ ∑ (-1) 1 + 2 - √ 2 + 1 2 1 + 1 1 1 1 2 2 × ∑ ′ 1 2 1 ′  ( 1) 12, ′ + (-1) 1+ - 1 2 1 ′  (1 ) 12, ′ , (7)
where we have introduced the quantities

 ( 1) 12, ′ = 1 1 -′ ′ ∑ , ′ ′ ′ ⟨ 1 1 1 1 ‖ ‖ ‖ ( ) ‖ ‖ ‖ , ′ ′ ′ ⟩ ⟨ , ′ ′ ′ ‖ ‖ ‖ (1) ‖ ‖ ‖ 2 2 2 2 ⟩ (8)
 (1 ) 12, ′ = 1 2 -′ ′ ∑ , ′ ′ ′ ⟨ 1 1 1 1 ‖ ‖ ‖ (1) ‖ ‖ ‖ , ′ ′ ′ ⟩ ⟨ , ′ ′ ′ ‖ ‖ ‖ ( ) ‖ ‖ ‖ 2 2 2 2 ⟩ (9) in which | , ′ ′ ′ ⟩ is a condensed representation of |Ψ 0 ⟩, see Eq. ( 3
). The superscripts ( 1) and (1 ) correspond to the order in which the tensor operators ( ) and (1) are written.

For eigenvectors |Ψ 0 1,2 ⟩ belonging to the ground configuration and |Ψ 0 ⟩ belonging to the first excited configuration, the 3-j symbol of Eq. ( 35) imposes that the CF potential matrix elements are non-zero for = 1, 3 and 5, which, according to Eq. ( 6), imposes = 0, 1, ..., 6. By contrast, in the standard version of the JO theory, = + 1 = 2, 4 and 6. The = 0 contribution in Eq. ( 6) comes from the dipolar term = 1 of the CF potential; it is the only non-zero contribution when 1 = 2 = 0, for instance the 5 D 0 ↔ 7 F 0 transition in Eu 3+ . Our odd-contributions are responsible for the transitions like 5 D 0 ↔ 7 F 3,5 and 5 D 3 ↔ 7 F 0 ; they arise because we consider distinct energies for levels 1 and 2, 1 ≠ 2 , unlike the standard JO theory. But since the energy difference | 2 -1 | is significantly smaller (although not negligible) compared to ′ ′ -1,2 , those transitions are weak. Finally, since the operators ( ) do not couple different spin states, the spin-changing transitions are only due to the mixing of different spin states within the ground-configuration levels |Ψ 0 1,2 ⟩, see Eq. [START_REF] Ofelt | Intensities of crystal spectra of rare-earth ions[END_REF]. In other words, Eq. ( 7) does not account for the so-called Downer-Wybourne mechanism [START_REF] Downer | A new contribution to spinforbidden rare earth optical transition intensities: Gd 3+ and Eu 3+[END_REF].

At present, we calculate the ED line strength  ED = ∑ 1 2 ( 12 ) 2 . Expressing Eq. ( 7) twice gives many sums: in particular on , 1 , 2 , , , and ′ , but also ′ , ′ , ′ and ′′ (coming from the second expansion of 12 ). Focusing on the sum involving CG coefficients, we have

∑ ′ ′ 1 ′ ′ ′ ′ 1 ∑ 1 2 1 1 2 2 1 1 2 2 ′ ′ 2 1 + 1 = ∑ ′ ′ 1 ′ ′ ′ ′ 1 ′ ′ 2 + 1 = ′ 2 + 1 ∑ ′ 1 ′ ′ 1 = ′ ′ ′ 2 + 1 , ( 10
)
where the Kronecker symbols come from the orthonormalization relation of CG coefficients. Plugging Eq. ( 10) into the line strength gives

 ED = ∑ 1 1 1 1 1 1 ∑ 2 2 2 2 2 2 ∑ 1 1 1 1 1 1 ∑ 2 2 2 2 2 2 ∑ | | 2 2 + 1 × ∑ (2 + 1) ∑ ′ 1 2 1 ′  ( 1) 1 ,2 , ′ + (-1) 1+ - 1 2 1 ′  (1 ) 1 ,2 , ′ × ∑ ′′ 1 2 1 ′′  ( 1) 1 ,2 , ′′ + (-1) 1+ - 1 2 1 ′′  (1 ) 1 ,2 , ′′ . ( 11
)
When expanded, the last two lines contain four terms: two of the kind

∑ (2 + 1) 1 2 2 1 ′ 1 2 2 1 ′′  ( 1 2 ) 1 ,2 , ′  ( 1 2 ) 1 ,2 , ′′ = ′ ′′ 2 ′ + 1  ( 1 2 ) 1 ,2 , ′  ( 1 2 ) 1 ,2 , ′ (12)
where the sum on is actually the orthonormalization relation of 6-j symbols; and two terms of the kind

∑ (-1) 1 + 2 -(2 + 1) 1 2 2 1 ′ 2 1 2 1 ′′  ( 1 2 ) 1 ,2 , ′  ( 2 1 ) 1 ,2 , ′′ = (-1) 1 + 2 + ′ + ′′ 1 1 ′ 2 2 ′′  ( 1 2 ) 1 ,2 , ′  ( 2 1 ) 1 ,2 , ′′ (13)
where we use some properties of 6-j symbols (see Ref. [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF], p. 305). The final expression of the line strength is

 ED = ∑ 1 1 1 1 1 1 ∑ 2 2 2 2 2 2 ∑ 1 1 1 1 1 1 ∑ 2 2 2 2 2 2 ∑ | | 2 2 + 1 × ∑ ′ ⎡ ⎢ ⎢ ⎣  ( 1) 1 ,2 , ′  ( 1) 1 ,2 , ′ +  (1 ) 1 ,2 , ′  (1 ) 1 ,2 , ′ 2 ′ + 1 + ∑ ′′ (-1) 1+ + ′ + ′′ × 1 ′ 1 2 ′′  ( 1) 1 ,2 , ′  (1 ) 1 ,2 , ′′ + 1 1 ′ 2 ′′  (1 ) 1 ,2 , ′  ( 1) 1 ,2 , ′′ ⎤ ⎥ ⎥ ⎦ . ( 14
)
Equation ( 14) looks very different from the standard JO line strength

 ED = ∑ Ω ⟨Ψ 0 1 ‖ ( ) ‖Ψ 0 2
⟩, especially because it does not depend on , but depends on ′ and ′′ (which are by contrast eliminated in the standard case). The index is still relevant in the ED transition amplitude 12 , see Eq. ( 7), because it allows for deriving the selection rules, but it disappears in the line strength, where we consider unpolarized light and ions (that is to say sums on , 1 and 2 ). In Eq. ( 14), the influence of the CF potential are only contained in the three parameters

= (2 + 1) -1 ∑ | | 2
, for = 1, 3 and 5, which are -averages of the square of . In what follows, they will be treated as adjustable parameters, whereas all the atomic properties will be computed using atomic-structure methods.

Third-order correction

In this section, we address the influence of spin-orbit (SO) mixing in the excited configuration on spin-changing f-f transitions. Contrary to the ground configuration, the LS coupling scheme is by far not appropriate to interpret the levels of the 4 -1 5 configuration (see Table 4), because the electrostatic energy between 4 and 5 electrons and the SO energy of the 5 electron are comparable. Therefore one can expect these excited levels to play a significant role in the spin-changing transitions.

To check this hypothesis, we will investigate the effect of the SO Hamiltonian of the ion SO using perturbation theory. Namely we define a perturbation operator containing SO and CF interactions,

= SO + CF . ( 15
)
In consequence, the new unperturbed eigenvectors related to the ground configuration are called manifolds, i.e. atomic levels for which the SO energy is set to 0. Those manifolds | Ψ0 ⟩, of energy ̃ are degenerate in as previously, but also in , and they are characterized by one and one quantum number,

| Ψ0 ⟩ = ∑ ̃ | | ⟩ . ( 16
)
Some manifolds, like the lowest 5 D one in Eu 3+ , are linear combination of different terms having the same and but different seniority numbers, hence the sum on in Eq. ( 16).

For the excited configuration, the unperturbed eigenvectors are those given in Eq. ( 3).

The selection rules associated with SO and CF are very different. In particular, SO couples unperturbed eigenvectors of the same configuration, whereas the odd terms of CF couple configurations of opposite parities. Therefore, the influence of both SO and CF potentials appears as products of matrix elements like ⟨ Ψ0

1 | SO | Ψ0 ⟩⟨ Ψ0 | CF |Ψ 0
⟩, and we need to go to the third order of perturbation theory to calculate the transition amplitude,

12 = ⟨ Ψ1 1 | | | (1) | | | Ψ1 2 ⟩ + ⟨ Ψ2 1 | | | (1) | | | Ψ0 2 ⟩ + ⟨ Ψ0 1 | | | (1) | | | Ψ2 2 ⟩ ( 17
)
where the second-order correction of eigenvectors is given in Eq. [START_REF] Kunz | Changes in fluorescence lifetimes induced by variable optical environments[END_REF].

By expanding Eq. ( 17), we get six terms corresponding to the six possible products of matrix element of SO , (1) and ( ) . Since SO couples states of the same configuration, unlike (1) and ( ) , we distinguish two kinds of terms:

• ⟨ Ψ0 1 | ( ) |Ψ 0 ⟩ × ⟨Ψ 0 | SO |Ψ 0 ⟩ × ⟨Ψ 0 | (1) | Ψ0 2 ⟩ and ⟨ Ψ0 1 | (1) |Ψ 0 ⟩ × ⟨Ψ 0 | SO |Ψ 0 ⟩ × ⟨Ψ 0 | ( ) | Ψ0 2 ⟩
, for which the SO interaction mixes levels of the excited configuration, for example quintets and septets in Eu 3+ ;

• ⟨ Ψ0 1 | SO | Ψ0 ′ 1 ⟩ × ⟨ Ψ0 ′ 1 | ( ) |Ψ 0 ⟩ × ⟨Ψ 0 | (1) | Ψ0 2 ⟩, ⟨ Ψ0 1 | SO | Ψ0 ′ 1 ⟩ × ⟨ Ψ0 ′ 1 | (1) |Ψ 0 ⟩ × ⟨Ψ 0 | ( ) | Ψ0 2 ⟩, ⟨ Ψ0 1 | ( ) |Ψ 0 ⟩ × ⟨Ψ 0 | (1) | Ψ0 ′ 2 ⟩ × ⟨ Ψ0 ′ 2 | SO | Ψ0 2 ⟩ and ⟨ Ψ0 1 | (1) |Ψ 0 ⟩ × ⟨Ψ 0 | ( ) | Ψ0 ′ 2 ⟩ × ⟨ Ψ0 ′ 2 | SO | Ψ0 2 ⟩.
In those cases, the SO interaction mixes manifolds of the ground configuration, for example in Eu 3+ , 7 F with 5 D, 5 F and 5 G.

Because SO is a scalar, i.e. a tensor operator of rank 0, the application of the Wigner-Eckart theorem gives a CG coefficients ′ ′ 00 = ′ ′ . So, applying the Wigner-Eckart theorem to ( ) and (1) as in Eq. ( 5), the product of three matrix elements can be expanded in a similar way to Eq [START_REF] Walsh | Judd-Ofelt theory: principles and practices[END_REF]. For example,

⟩ = ∑ (-1) 1 + 2 - √ 2 + 1 2 1 + 1 1 1 1 2 2 1 2 1 ′ × ⟨ Ψ 0 1 ‖ ‖ ‖ ( ) ‖ ‖ ‖ Ψ 0 ⟩ ⟨ Ψ 0 | | | SO | | | Ψ 0 ⟩ ⟨ Ψ 0 ‖ ‖ ‖ (1) ‖ ‖ ‖ Ψ 0 2 ⟩ ( 18
)
where |Ψ 0 , ⟩ are two eigenvectors of the excited configuration with the same total angular momentum ′ . The other products give similar results: the order of reduced matrix elements in the last line is of course the same as the order of matrix elements in the first line; if (1) appears before ( ) , the 1 and are interchanged in the CG coefficients and 6-j symbols, like in Eq. [START_REF] Smentek | Theoretical description of the spectroscopic properties of rare earth ions in crystals[END_REF].

Gathering the six matrix-element products, we can write the ED transition amplitude as

12 = ∑ 1 2 ̃ 1 ̃ 2 ∑ ∑ (-1) 1 + 2 - √ 2 + 1 2 1 + 1 1 1 1 2 2 ∑ ′ 1 2 1 ′  (0 1) 12, ′ +  ( 01) 12, ′ +  ( 10) 12, ′ + (-1) 1+ - 1 2 1 ′  (01 ) 12, ′ +  (10 ) 12, ′ +  (1 0) 12, ′ , ( 19
)
where the terms 

( 1 2 3 ) 12, ′
are built in analogy to Eqs. ( 8) and ( 9): the order of the superscripts is the one in which the matrix element of operators appear (the "0" standing for SO ). Firstly, the 

(0 2 3 )
12, ′ are such that

 (0 2 3 ) 12, ′ = 1 Δ 2 3 ∑ ′ 1 ′ 1 ′ 1 ′ 1 ̃ ′ 1 ̃ ′ 1 ̃ 1 -̃ ′ 1 × ⟨ 1 1 1 1 | | SO | | ′ 1 ′ 1 2 1 ⟩ × ∑ , ′ 2 ⟨ ′ 1 ′ 1 2 1 ‖ ‖ ‖ ( 2 ) ‖ ‖ ‖ , ′ 2 2 ′ ⟩ ⟨ , ′ 2 2 ′ ‖ ‖ ‖ ( 3 ) ‖ ‖ ‖ 2 2 2 2 ⟩ , ( 20
)
with

Δ = ̃ 1 -′ ′ for ( , ) = ( , 1) ̃ 2 -′ ′ for ( , ) = (1, ). (21)
In Eu 3+ for example, for | Ψ0 1 ⟩ in the lowest 5 D manifold and | Ψ0 1 ⟩ in the 7 F manifold, SO couples | Ψ0 1 ⟩ to the 7 ′ 1 manifolds on the ground configuration (actually there is only one: 7 F). The quantum numbers ( , ′ 2 ) characterize the septet levels ( 2 = 3) of the excited configuration. Similarly,

 ( 1 2 0) 12, ′ = 1 Δ 1 2 ∑ , ′ 1 ⟨ 1 1 1 1 ‖ ‖ ‖ ( 1 ) ‖ ‖ ‖ , ′ 1 1 ′ ⟩ × ∑ ′ 2 ′ 2 ′ 2 ′ 2 ̃ ′ 2 ̃ ′ 2 ̃ 2 -̃ ′ 2 × ⟨ , ′ 1 1 ′ ‖ ‖ ‖ ( 2 ) ‖ ‖ ‖ ′ 2 ′ 2 1 2 ⟩ ⟨ ′ 2 ′ 2 1 2 | | SO | | 2 2 2 2 ⟩ , ( 22
)
where Δ 1 2 is given by Eq. ( 21). Here the SO Hamiltonian couples the 7 F manifold to the various quintet manifolds, for instance 5 D, 5 F and 5 G manifolds, since ′ 2 -2 = 0, ±1. Finally, the terms 

( 1 0 3 )
12, ′ correspond to the Wybourne-Downer mechanism [START_REF] Downer | A new contribution to spinforbidden rare earth optical transition intensities: Gd 3+ and Eu 3+[END_REF] where SO couples the quintet and septet levels of the excited configuration. Namely,

 ( 1 0 3 ) 12, ′ = 1 Δ 2 1 3 ∑ ∑ ′ 1 ′ 2 ⟨ 1 1 1 1 ‖ ‖ ‖ ( 1 ) ‖ ‖ ‖ , ′ 1 1 ′ ⟩ × ⟨ , ′ 1 1 ′ | | | SO | | | , ′ 2 2 ′ ⟩ ⟨ , ′ 2 2 ′ ‖ ‖ ‖ ( 3 ) ‖ ‖ ‖ 2 2 2 2 ⟩ , ( 23
)
where Δ 1 3 is given by Eq. ( 21). If we assume that in Equations ( 20), ( 22) and ( 23), the spin-orbit interactions are of the same order of magnitude (see Table 1), the main difference between them comes from the energy denominator. The quantity Δ is on the order of several tens of thousands of cm -1 , while the differences

̃ 1 -̃ ′ 1 and ̃ 2 -̃ ′ 2
, which are the energies between different manifolds of the ground configuration, are on the order of several thousand cm -1 . This means that Eq. ( 23) is, roughly speaking, one order of magnitude smaller than Eqs. ( 20) and ( 22). This fact is really a precious information that brings the third order correction.

Combining Eqs. ( 14) and ( 19), we can see that the ED line strength  ED now contains 36 terms, containing products of the kind 

( 1 2 3 ) 1 ,2 , ′ ×  ( 1 2 3 )
1 ,2 , ′′ . For the 18 terms in which and 1 appear in the same order in ( 1 2 3 ) and ( 1 2 3 ), we have the same prefactor as the second line of Eq. ( 14), that is (2 ′ + 1) -1 . For the 18 other terms in which and 1 appear in different orders, we have the prefactors with the 6-j symbols as in the second and third lines of Eq. ( 14). Namely, we can write the line strength as

 ED = ∑ 1 1 ̃ 1 ̃ 1 ∑ 2 2 ̃ 2 ̃ 2 ∑ | | 2 2 + 1 ∑ ∑ ′ ̃ ,( 1) ̃ ,( 1) + ̃ ,(1 ) ̃ ,(1 ) 2 ′ + 1  ( ) 1 ,2 , ′  ( ) 1 ,2 , ′ + ∑ ′′ (-1) 1+ + ′ + ′′ 1 ′ 1 2 ′′ ̃ ,( 1) ̃ ,(1 ) + 1 1 ′ 2 ′′ ̃ ,(1 ) ̃ ,( 1)  ( ) 1 ,2 , ′  ( ) 1 ,2 , ′′ (24)
where = ( 1 2 3 ) and = ( 1 2 3 ) designate the possible combinations of indices , 1 and 0. The quantity ̃ ,( 1) = 1 if is a combination in which appears firstly and 1 secondly, namely = ( 10), ( 01), (0 1), and 0 otherwise.

The quantity ̃ ,(1 ) corresponds to the inverse situation. Similarly to Eq. ( 14), the line strength ( 24) depends on the CF potential through the three parameters = (2 + 1) -1 ∑ | | 2 , which will be treated as adjustable in the next section.

Application to Europium

In this section, we aim at benchmarking our model with experimental data. To this end, we have chosen the measurements of absorption oscillator strengths by Babu et al. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF], who performed a thorough spectroscopic study of Eu 3+ -doped lithium borate and lithium fluoroborate glasses. Because our model relies on free-ion properties, we start with studying the free-ion energies of the two lowest Eu 3+ electronic configurations 4 6 , of even parity, and 4 5 5 , of odd parity, and the free-space transitions between them.

Calculation of free-ion energy parameters

The calculations of the Eu 3+ free-ion spectrum are performed with the semi-empirical technique provided by Robert Cowan's atomic-structure suite of codes [START_REF] Kramida | Cowan code: 50 years of growing impact on atomic physics[END_REF], whose theoretical background is presented in Ref. [START_REF] Cowan | The theory of atomic structure and spectra[END_REF]. In a first step, ab initio radial wave functions for all the subshells of the considered configurations are computed with the relativistic Hartree-Fock (HFR) method. Those wave functions are used to calculate energy parameters, for instance center-of-gravity configuration energies av , direct and exchange electrostatic integrals, or spin-orbit integrals , that are the building blocks of the atomic Hamiltonian. In a second step, the latter are treated as adjustable parameters of a least-square fitting calculation, in order to find the best possible agreement between the Hamiltonian eigenvalues and the experimental energies. To make some comparisons between different elements and ionization stages, one often defines the scaling factor (SF) = f it ∕ HFR between the fitted and the HFR value of a given parameter .

In an attempt to improve the quality of the fit (and therefore, the accuracy of the resulting eigenvectors), a variety of "effective-operator" parameters, called , and and "illegal"-k , have been introduced, representing corrections to both the electrostatic and the magnetic singleconfiguration effects [START_REF] Cowan | The theory of atomic structure and spectra[END_REF]. "Illegal"-k means that these are the values of for which + + ′ is odd. These effective parameters, unlike other parameters, can not be calculated ab initio. By contrast, we do not include the , and parameters that are sometimes used in Ln 3+ -ion ground configuration. The general methodology for our fitting calculations is as follows: (a) fitting the parameters with an ab initio values while effective parameters are forced to be zero; (b) fixing the parameters resulting from step (a) and fitting the effective parameters; (c) using the final values of (b), fitting all the parameters together.

Our fitting calculations require experimental energies. For the Eu 3+ ground configuration 4 6 , we find them on the NIST ASD database [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.7.1)[END_REF]. However, no experimental level has been reported for the 4 5 5 configuration. Because the 4 configurations (with 2 ≤ ≤ 12) and the 4 -1 5 ones (with 3 ≤ ≤ 13) possess the same energy parameters, we perform a least-square fitting calculation of some 4 -1 5 configurations for which experimental levels are known, namely for Nd 3+ ( = 3) and Er 3+ ( = 11) [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF][START_REF] Meftah | Analysis of the free ion spectrum of Er 3+ (Er IV)[END_REF][START_REF] Arab | Observation and interpretation of the 5 5 4 3 5 core-excited configuration in triply ionized neodymium Nd 3+ (Nd IV)[END_REF]. Then, relying on the regularities of the scaling factors along the lanthanide series, we multiply the obtained scaling factors given in Table 1 by the HFR parameters for Eu 3+ to compute the energies of 4 5 5 .

The interpretation of Nd 3+ and Er 3+ spectra show that, because CI mixing is very low, a one-configuration approximation can safely be applied in both parities, which is done here. For Nd 3+ , experimentally known levels are taken from the article of Wyart et al. [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF]. There are 41 levels for 4 3 configuration and 111 for 4 2 5 configuration. For Er 3+ , 38 experimental levels of the configuration 4 11 and 58 of 4 10 5 are taken from Meftah et al. [START_REF] Meftah | Analysis of the free ion spectrum of Er 3+ (Er IV)[END_REF]. For the 4 6 configuration of Eu 3+ , the NIST database gives 12 levels [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.7.1)[END_REF].

Table 1 shows a comparison of the final SFs (for ab initio parameters) or the fitted values (for effective parameters), for the two lowest configurations of the above mentioned ions. It also shows the parameter values used in the Eu 3+ spectrum calculations of the next subsections. In the 4 configurations, the least-square fitting calculations, performed for each element, illustrates the regularities of SFs for and parameters. Regarding effective parameters, the negative values of are usual, while the small values of and of Er 3+ are not. The regularities are also visible between 4 2 5 and 4 10 5 configurations of Nd 3+ and Er 3+ respectively. Therefore, we calculate our Eu 3+ parameters by multiplying the HFR values by the average SF obtained for Nd 3+ and Er 3+ . The effective parameters are those obtained for Nd 3+ , and the center-of-gravity energy of 4 5 5 is calculated by assuming that the difference av (4 -1 5 )av (4 ) increases linearly with .

Figure 1 shows the levels computed with the parameters of Table 1, whose energies are between 0 and 100000 cm -1 , for the 4 6 and 4 5 5 configurations. They will be analyzed in details in the next two paragraphs.

Energy levels of the ground configuration 4 6

For the 4 6 configuration, values from the NIST database [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.7.1)[END_REF] were taken as the experimentally known energy levels. Because the free ion has not been analyzed yet, the energies was determined by interpolation or extrapolation of known experimental values or by semi-empirical calculation [START_REF] Martin | Atomic energy levels-the rareearth elements. the spectra of lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium[END_REF]. Table 2 shows a good agreement between these experimental values, our computed values and the theoretical values calculated by Freidzon and coworkers [START_REF] Freidzon | Ab initio calculation of energy levels of trivalent lanthanide ions[END_REF]. Our values are closer to the experimental ones in the 5 D manifold. Note that a direct comparison with the article of Ogasawara and coworkers [START_REF] Ogasawara | Optical spectra of trivalent lanthanides in LiYF 4 crystal[END_REF] is difficult, as the authors do not give tables of energy levels for Eu 3+ . In total, the 4 6 configuration contains 296 levels with values ranging from 0 to 12.

Table 2 also illustrates that the ground-configuration levels are well described by the LS coupling scheme. Some levels are mainly characterized by a single term, like 7 F or 5 L, but others are shared between several terms with the same and quantum numbers, but different seniority

Table 2

Comparison between the experimental, computed and other theory [START_REF] Freidzon | Ab initio calculation of energy levels of trivalent lanthanide ions[END_REF] values for the levels of 4 6 configuration of Eu 3+ , with total angular momenta from = 0 to 6 and energies up to 30000 cm -1 , as well as first three LS-coupling eigenvectors with their percentages. All energy values are in cm -1 . ). The small deviations from LS coupling are due to the SO interaction, for example, a small 5 D component in the 7 F levels. The terms coupled by SO are such that Δ = 0, ±1 and Δ = 0, ±1 in agreement with Eq. ( 36). Finally, Table 3 contains the energy value and eigenvector of the manifolds with = 2 and 3, calculated by setting to 0 the spin-orbit parameter of Table 1. This information is necessary to build our third-order theory, see Eq. ( 16). Note that the first excited manifold is a superposition of 5 D3, 5 D1 and 5 D2 terms. But due to its strong importance in Eu 3+ spectroscopic studies, it will be denoted 5 D in the rest of the paper.

Exp. This work

Energy levels of the first excited configuration

4 5 5

This subsection is devoted to the energy levels of the first excited configuration 4 5 5 . The parameters necessary for the calculations are given in Table 1. The 4 5 5 configuration contains 1878 levels with -values from 0 to 14, and according to our calculations, with energies from 74438 to 243060 cm -1 . The dominant eigenvector of the 74438-cm -1 level is 4 5 ( 6 H ) 5 ( 7 K 4 ) with 93.8 %. As examples, Table 4 shows the 20 lowest energy levels with = 0, 1 and 2, along with their three dominant eigenvectors.

Table 4 shows that the levels of the 4 5 5 configuration do not possess a strongly dominant eigenvector (or a group of eigenvectors) characterized by the same and quantum numbers. This means that, unlike the ground configuration, see Table 2, the LS coupling scheme is not appropriate for the excited configuration. It can be shown that the coupling scheme is not appropriate either, because the spin-orbit energy of the 5 electron is of the same order of magnitude as the electrostatic energy between 4 and 5 electrons. The eigenvectors are therefore written in pair coupling, i.e. linear combination of LS-coupling states.

In a given energy level, the and quantum numbers, which characterize the parent term of the 4 5 subshell, are common to the majority of the eigenvectors. With increasing energy, the levels mainly possess 6 H , 6 F and 6 P characters; then come the quartet and doublet parent terms. Indeed the SO interaction within the 4 5 subshell is too small to significantly mix different and of the 4 5 subshell. By contrast, the total and quantum numbers of the LS states differ at most by one unity. For example, we notice the pairs 7 H-5 G (Δ = 1 and Δ = 1), 7 G-7 F (Δ = 0 and Δ = 1) and 5 F-7 F (Δ = 1 and Δ = 0) for the level at 78744, 79541 and 80396 cm -1 respectively. In consequence, the mixing between quintet and septet states of Eu 3+ is mainly due to the SO interaction of the 5 electron. That is why we ignore the influence of the 4 electrons to account for the Wybourne-Downer mechanism (see Subsection 2.2 and Eq. ( 37)).

Free-ion transitions between the two configurations

Equations ( 14) and [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF] show that our f-f transition line strengths require the reduced multipole moments of some free-ion transitions which only occur between levels of the ground and excited configurations. In this subsection, we focus on the electric-dipole (ED) free-ion transitions ( = 1), that are the most intense. A widely used quantity for the discussion of spectral lines and transitions is the absorption oscillator strength 12,ED , which is related to the ED line strength ED through the expression

12,ED = 2 2 0 ( 2 -1 ) 3ℏ 2 (2 1 + 1)  ED , ( 25
)
where 1 (2) denotes the lower (upper) levels of energy 1

( 2 ) and total angular momentum 1 ( 2 ), ℏ is the reduced Planck constant, electron mass, 0 = 4 0 ℏ 2 ∕ 2 the Bohr radius, 0 the vacuum permitivity and the electron charge. In Eq. ( 25), the ED line strength is in atomic units (units of 2 2 0 ). Because the oscillator strength for stimulated emission is defined as 21 = - +1 12 , the so-called weighted oscillator strength

2 1 +1 2 2
ED = (2 1 + 1) 12,ED = -(2 2 + 1) 21,ED (26)
does not depend on the nature of the transition. For ED freeion transitions, the line strength of Eq. ( 25) is the square of the reduced ED matrix element,

 ED = |⟨Ψ 1 ‖ (1)
‖Ψ 2 ⟩| 2 . In the rest of the article, we will focus on the absorption oscillator strengths, and so will drop the "12" subscripts.

Figure 2 shows the dependence of the logarithm of the weighted oscillator strengths given by Eq. ( 26) on the energy of the excited-configuration level, for transitions involving two levels of the ground configuration. It shows that the energy band with strong transitions is rather narrow and lies in the range of 80000-100000 cm -1 , while for larger excitedlevel energies, the values of log( ) for the level 7 F 1 (blue dots) decrease faster than those for 5 D 1 (red dots). Indeed, the total spin of 4 5 5 levels tends to decrease with energy (see Table 4), the coupling with levels of the 4 6 7 F manifold drops faster than the coupling with levels of the quintet manifolds. Therefore, in the framework of the JO theory, the excited-configuration energy ′ ′ appearing in the denominators of the line strengths, see Eqs. ( 14) and [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF], is not the center-of-gravity energy of the excited-configuration, but rather the strong-coupling window between 80000 and 100000 cm -1 : in practice, we take ′ ′ = 90000 cm -1 .

In addition to the free-ion ED reduced matrix elements, the JO theory requires those for = 3 (octupole) and = 5, which depend on the radial transition integral

⟨ ′ ′ | | ⟩ = ∫ ∞ 0 ′ ′ ( ) ( )
, where = 4 and ′ ′ = 5 . We have calculated those integrals with a home-made Octave code, reading the HFR radial wave functions 4 and 5 computed by Cowan's code RCN. We obtain 1.130629 0 , -3.221348 3 0 and 21.727152 5 0 for = 1, = 3 and = 5, respectively, while the = 1 value calculated by Cowan is 1.130618 0 .

f-f transitions in Eu 3+ -doped solids

Now that we have all the necessary information about the free-ion spectrum, in this subsection, we aim to validate our model with experimental data. To that end, we have chosen the thorough investigation of Babu et al. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF], who measured absorption oscillator strengths and interpreted them with the standard JO theory. Their study deals with transitions within the ground manifold 7 F and between the ground and first excited manifold 5 D for Eu 3+ -doped lithium fluoroborate Table 4 First 20 energy levels for 4 5 5 configuration of Eu 3+ with total angular momentum = 0, 1 and 2, as well as first three eigenvectors with their percentages. As an example, ( 6 H) 7 H is used for brevity for 4 5 ( 6 H )5 7 H , where the superscript "o" indicates odd parity. All energy values are in cm -1 .

Energy

First three eigenvectors and percentages 78744 2 ( 6 H) 7.6 % ( 6 F) 5 P 2.0 % 91979 0 ( 6 F) 5 D 61.5 % ( 6 F) 7 F 7.9 % ( 6 P) 5 D 5.9 % 93243 2 ( 6 F) 5 D 47.0 % ( 6 F) 5 G 16.9 % ( 6 F) 5 F 5.6 % glass. In the latter case, the transitions involve a change in spin, well known to challenge the standard JO theory.

Description of our calculations

We have written a FORTRAN program which firstly reads the energies and four leading eigenvectors of the ground-configuration free-ion levels (see Table 2) and manifolds (see Table 3). Then, the code performs a linear leastsquare fitting of experimental line strengths exp and the ED part of the theoretical ones ED given by Eqs. ( 14) and [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF], with the free adjustable parameters

= + ∑ =- | | 2 2 + 1 (27)
for = 1, 3 and 5, which describes the electrostatic environment at the ion position. During the least-square step, we seek to minimize the standard deviation on line strengths

= ⎡ ⎢ ⎢ ⎣ ∑ tr =1 exp, -ED, 2 tr -par ⎤ ⎥ ⎥ ⎦ 1 2 , ( 28
)
where tr is the number of transitions included in the calculation and par = 3 is the number of adjustable parameters. The experimental line strengths are extracted from the absorption oscillator strengths exp by inverted Eq. ( 25),

exp = 3(2 1 + 1)ℏ 2 2 2 0 ( 2 -1 ) ED exp (29)
where is the host refractive index, and ED = ( 2 + 2)∕9 the local-field correction in the virtual-cavity model (see for example Ref. [START_REF] Aubret | Understanding local-field correction factors in the framework of the Onsager-Bottcher model[END_REF]). In contrast with the free-ion case, Eq. ( 29) takes into account the host material through its refractive index ; for lithium fluoroborate, = 1.57 is assumed wavelength-independent. Note that our code can also apply the fitting procedure to Einstein coefficients, as the latter are transformed in line strengths.

After the fitting, using these optimal parameters, we can predict line strengths, oscillator strengths and Einstein coefficients, for other transitions. Of course, that procedure only involves transitions with a predominant ED character; magnetic-dipole (MD) transitions like 5 D 0 ↔ 7 F 1 and 5 D 1 ↔ 7 F 0 are therefore excluded from the fit. For them, the MD line strength  MD , oscillator strengths and Einstein coefficients can be calculated from the free-ion eigenvectors (see Table 2) [START_REF] Dodson | Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: Calculated emission rates and oscillator strengths[END_REF].

Results of the least-square fitting

We have included 9 out of the 14 transitions measured with the so-called L6BE glass in Table 3 of Ref. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF]. We have excluded three predominant MD transitions, 5 D 0 ↔ 7 F 1 , 5 D 1 ↔ 7 F 0 and 5 D 2 ↔ 7 F 1 , as well as the 5 G 4 ↔ 7 F 0 and 5 D 0 ↔ 7 F 0 for which we observe large deviations between theory and experiment. They are probably due to the fact that the 5 D 0 and 5 G 4 are further from LS coupling than the other levels. In particular, the four leading components represent 88.4 and 86.9 % of the total eigenvectors respectively.

Table 5 shows the results of our least-square calculations with the second-and third-order theory, in comparison Table 5 Transition labels, experimental oscillator strengths (× 10 -6 ) [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF] and ratios between theoretical and experimental line strength for the standard Judd-Ofelt theory ( 0 ), as well as for our second-order ( 1 ) and third-order ( 2 ) corrections of our theory. The last lines present the absolute and relative standard deviations for each model (see text with the standard JO theory used in Ref. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF]. For each transition, the table contains the experimental values of the oscillator strength (×10 -6 ) [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF] and the ratios between the theoretical and experimental oscillator strengths, where 0 is the ratio for the standard Judd-Ofelt theory, and the 1 and 2 are the ratios, respectively, for second and third order corrections of theory (see subsection 2.1 and 2.2). For each model, we present the absolute and relative standard deviations, taken by dividing Eq. ( 28) by the largest experimental line strength  max = 3.057 × 10 -4 for the 7 F 6 ↔ 7 F 1 transition. Figure 3 gives a visual insight into the results of Table 5, with histograms of the experimental absorption oscillator strengths, and those resulting from the standard JO theory and our third-order correction, plotted as functions of the transition wavelength. Globally, the three methods have similar performances. That shows that the SO interaction in the excited configuration has little effect, since it is included in the thirdorder correction and not in the second-order one. Our thirdorder corrections better describes transitions between 7 F and 5 D manifolds. However, it predicts the smallest oscillator strength for 5 G 2 ↔ 7 F 0 , owing to the proximity between the 5 G3 and 5 H1 manifolds (see Table 3), which puts into question the use of SO interaction as a perturbation. On the other hand, the second-order correction fails to describe the 5 D 4 ↔ 7 F 0 transition. The three methods tend to underestimate the oscillator strengths for high-energy transitions, where the refractive index is larger than 1.57.

The final fitting parameters are given in Table 6 for the standard JO calculation of Ref. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF] (see Set B of Table 4), as well as our second-order correction ( 14) and thirdorder correction [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF]. The orders of magnitude of the are the same for the two corrections. The parameter 3 are the largest, then the 1 are roughly one order of magnitude smaller than the 3 , and the 5 are roughly two orders of magnitude smaller than 3 . It is hard to make direct comparisons with the standard JO parameters given in Table 4 of Ref. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF] (data set B); but we see that that the Ω 6 parameter, responsible of the 7 F 6 ↔ 7 F 0,1 and 5 L 6 ↔ 7 F 0,1 transitions just like 5 is respectively 9 and 6 times smaller than Ω 2 and Ω 4 . To give more insight values of the parameter, we notice that the quantities √ × ⟨ | | ′ ′ ⟩ is the order-of-magnitude energy of the ion-field interaction: in the third-order correction, they are respectively equal to 298, 2226 and 1207 cm -1 for = 1, 3 and 5.

Transitions with a MD character

Now that we have the parameters, we can calculate oscillator strengths for transitions not present in the fit. In particular, we can predict the percentage of ED and MD characters for the transitions having both characters [START_REF] Freed | Multiple nature of elementary sources of radiation-wide-angle interference[END_REF][START_REF] Kunz | Changes in fluorescence lifetimes induced by variable optical environments[END_REF][START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF][START_REF] Chacon | Measuring the magnetic dipole transition of single nanorods by spectroscopy and Fourier microscopy[END_REF], assuming that the total oscillator strength is equal to the sum ED + MD . The ED part can be calculated by inverting Eq. ( 29) and replacing the subscripts "exp" by "ED", while

where the MD line strength is written in units of 2 2 0 [28]

 MD = 2 4 | | ⟨Ψ 1 ‖ ‖ + ‖ ‖ Ψ 2 ⟩ | | 2 (31)
with the fine-structure constant and the electronic-spin g-factor. Because the orbital and spin angular momenta are even-parity tensors of rank one, MD transitions can occur in free space or in solids, between levels of the same configuration and with Δ ≤ 1 except ( 1 , 2 ) = (0, 0). Table 7 presents experimental and theoretical absorption oscillator strengths for the transitions having in principle an ED and a MD character. The MD oscillator strengths are calculated by multiplying the free-ion one computed with Cowan's code by the host refractive index , see Eq. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF]. The table clearly shows that the 5 D 1 ↔ 7 F 1 transition is purely electric (at least 99.9 %), hence its inclusion in the fit. The 5 D 2 ↔ 7 F 1 is also mainly electric, but to a lesser extent, roughly at 95 %. The two others are mostly magnetic, but the experimental and theoretical MD oscillator strengths significantly differ from each other. Still, the ED character looks larger for the 5 D 1 ↔ 7 F 0 transition (4-9 %) than for the 5 D 0 ↔ 7 F 1 one (1-2 %).

The

5 D 0 ↔ 7 F 0 transition
Since the 5 D 0 ↔ 7 F 0 transition is forbidden by the selection rules of the standard JO model, it has attracted a lot of attention (see Ref. [START_REF] Binnemans | Interpretation of europium (III) spectra[END_REF] and references therein), in order to understand its origin. Even though it is not forbidden in our model, we had to exclude it in the fit, because of a strong discrepancy between the experimental and our computed oscillator strength. With our optimal parameters , we obtain an oscillator strength 1.25×10 -7 , that is 7.8 times larger than the experimental value. In this paragraph, we investigate in closer details the possible origin of that discrepancy and how to reduce it.

Firstly, as mentioned in Subsection 2.2, the sums in Eqs. ( 19) and ( 24) involves quintet and septet manifolds of the ground configuration. But a closer look at the eigenvectors shows that the 5 D 0 level contains 6.7 % of 3 P6 character, see Table 2, as well as 5.1 % of 3 P3, while 7 F 0 contains 0.1 % of 3 P6 character. These small components are likely to contribute to the transition amplitude, and so they need to be accounted for in a future work, through a complete description of the free-ion eigenvectors.

The selection rules associated with Eq. ( 19) show merely the terms with = 1 of the CF potential can induce a transition of the kind ( 1 , 2 ) = (0, 0). This result seems consistent because: (i) those terms are stronger in sites with low symmetries, and (ii) observing the 5 D 0 ↔ 7 F 0 transition is an indication of , or point groups at the ion site [START_REF] Nieuwpoort | Linear crystal-field terms and the 5do-7fo transition of the eu3+ ion[END_REF][START_REF] Binnemans | Application of the Eu 3+ ion for site symmetry determination[END_REF][START_REF] Binnemans | Influence of dipicolinate ligands on the spectroscopic properties of europium (III) in solution[END_REF].

Another frequently invoked mechanism to explain the 5 D 0 ↔ 7 F 0 transition is -mixing [START_REF] Tanaka | Contribution of mixing to the 5 0 -7 0 transition of Eu 3+ ions in several host matrices[END_REF][START_REF] Kushida | Transition mechanisms and spectral shapes of the 5 0 -7 0 line of Eu 3+ and Sm 2+ in solids[END_REF][START_REF] Kushida | Spectral shape of the 5d0-7f0 line of Eu 3+ and Sm 2+ in glass[END_REF], especially between levels of the lowest manifold 7 F. However, because this mixing is limited to 10 %, it cannot explain the strongest 0-0 transitions listed in Ref. [START_REF] Chen | The standard and anomalous crystal-field spectra of eu3+[END_REF]. Charge-transfer states are also likely to play a role in the 0-0 transition, especially in hosts with oxygen-compensating sites around by which the CF tends to be strongly deformed [START_REF] Karbowiak | Site-selective emission spectra of Eu 3+ :Ca 5 (PO 4 ) 3 F[END_REF]. However, those two mechanisms are not present in our model.

Radiative lifetime of the 5 D 0 level

In addition to absorption oscillator strengths, our model also makes it possible to calculate the ED Einstein coefficient for the spontaneous emission from level 2 to 1, ED = 2 2 0 ( 2 -1 ) 3 3 0 ℏ 4 3 (2 2 + 1) ED  ED , [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF] where is the speed of light and  ED is given by Eq. [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF]. We can also compute the MD Einstein coefficients MD , by multiplying the free-ion value calculated with Cowan by 3 ; namely MD = 2 2 0 ( 2 -1 ) 3 3 0 ℏ 4 3 (2 2 + 1) 3  MD [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.7.1)[END_REF] where  MD is given by Eq. [START_REF] Dodson | Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: Calculated emission rates and oscillator strengths[END_REF].

From them, we can deduce the radiative lifetime of a given level. In particular for the 5 D 0 level, it reads . (34) Transitions 5 0 ↔ 7 , where = 1 to 6, are not included in our fit, and so are considered as additional transitions, for which our program calculated line strengths and Einstein coefficients. For the transition 5 0 ↔ 7 1 , the total Einstein coefficient is the sum of the electric and the magnetic parts, calculated using Cowan code. The latter is found to be MD ( 5 D 0 , 7 F 1 ) = 53.44 s -1 . The sum of Einstein coefficients for all other transitions, including the electric part of transition 5 0 ↔ 7 1 , is 500.529 s -1 . That sum includes the transition 5 0 ↔ 7 0 , whose Einstein coefficient [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF] is calculated using the line strength deduced from the experimental oscillator strength following Eq. ( 29). This yields the very small value of 0.029 s -1 . The resulting radiative lifetime is ( 5 D 0 ) = 1805 s, which is close to the experimental value of 1920 s reported in Ref. [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF]. In principle, the relaxation limiting the lifetime is due to radiative as well as nonradiative processes; however the latter are expected to be unlikely for the 5 D 0 level [START_REF] Rabouw | Europium-doped NaYF 4 nanocrystals as probes for the electric and magnetic local density of optical states throughout the visible spectral range[END_REF], due to the large gap between the 5 D 0 and 7 F 6 levels, see Table 2.

Conclusion

In this article, we have developed an extension of the Judd-Ofelt model enabling to calculate intensities of absorption and emission transitions for Ln 3+ -doped solids. In our model, the properties of the Ln 3+ impurity are fixed parameters calculated with free-ion spectroscopy, while the crystal-field ones are adjusted by least-square fitting. In particular, the line strengths, oscillator strengths and Einstein coefficients are functions of three least-square fitted crystalfield parameters.

We have benchmarked our model with a detailed spectroscopic study of europium-doped lithium borate glasses. Not only our model allows for giving a simple physical insight into the transitions which are not described by the standard Judd-Ofelt theory, but it also reproduces measured oscillator strengths with a similar accuracy to the standard theory [START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF]. Moreover, we demonstrate that the spin-changing transitions in Eu 3+ mainly result from the spin-orbit mixing within the ground electronic configuration, even if its levels are well described by the LS-coupling scheme.

In consequence, our model may be improved in the future, by taking into account all the eigenvector components of the free-ion levels, while the four leading ones are taken into account in the current study. We expect this improvement to give more a precise calculation of the 7 F 0 ↔ 5 D 0 intensity. We also plan to account for the wavelengthdependence of the refractive index of the host material. Finally, the fact of separating the dopant and crystal-field parameters opens the possibility to interpret transitions between individual crystal-field levels or involving polarized light, which is especially relevant for nanometer-scale host materials.

In contrast, spectroscopic studies of free Ln 3+ ions indicate that configuration-interaction mixing, between the configurations 4 and 4 -1 6 on the one hand, 4 -1 5 and 4 -1 6 on the other hand, does not have a strong role in the energy spectrum [START_REF] Wyart | Analysis of the free ion Nd 3+ spectrum (Nd IV)[END_REF][START_REF] Meftah | Analysis of the free ion spectrum of Er 3+ (Er IV)[END_REF], and so shall not be included in our model. However, in the case of Er 3+ [START_REF] Arab | Observation and interpretation of the 5 5 4 3 5 core-excited configuration in triply ionized neodymium Nd 3+ (Nd IV)[END_REF], the lowest core-excited configuration of opposite parity compared to the ground one, 5 5 4 3 5 starts at 182000 cm -1 . Assuming a similar order of magnitude for the 5 5 4 6 5 configuration of Eu 3+ , and taking the relativistic Hartree-Fock value of the radial integral ⟨5 | |5 ⟩ = 1.62 0 , we can expect the excitation of the 5 core electrons toward the 5 orbital, to have a sizeable effect on the crystal-field coupling to opposite-parity configurations.

√

( + 1) (2 + 1) ( + 1) (2 + 1) 2 1 + 1 2 2 + 1 × √ 2 1 + 1 2 2 + 1 ∑ (-1) + + + + 1 + 1 1

1 2 1 1 2 ×( -1 |} 1 1 1 ) × ( -1 |} 2 2 2 ) (36)

2 ,

 2 while ′ is absent. The equation appears as a sum of irreducible tensors of rank and component coupling directly |Ψ 0 1 ⟩ and |Ψ 0 2 ⟩. The selection rules governing this coupling are Δ = | 2 -1 | ≤ ≤ 1 + 2 and 1 = 2 + . Moreover, the triangle rule associated with 1 imposes = , ± 1, = + and -≤ ≤ + .

Figure 1 :

 1 Figure 1: Energy levels of the 4 6 (blue) and 4 5 5 (red) configurations of Eu 3+ as functions of the electronic angular momentum . The plot is restricted to energy values between 0 and 100000 cm -1 .

Figure 2 :

 2 Figure 2: Logarithm of the weighted ED oscillator strengths, see Eq. (26), as functions of the energy of the excitedconfiguration levels, for transitions implying the 7 F 1 (blue) and 5 D 1 (red) levels of the ground configuration.

Figure 3 :

 3 Figure 3: Comparison between experimental[START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF] and theoretical oscillator strengths of absorption, plotted as function of the transition wavelength (not at scale). The transitions are labeled with the LS-term quantum numbers of the Eu 3+ free ion.

  0 , 7 F ) + MD ( 5 D 0 , 7 F 1 ) -1

Table 1

 1 Scaling factors (SFs) of ab initio parameters (see text), values of av and fitted effective parameters f it (in cm -1 ) for two lowest configurations of Nd 3+ , Er 3+ and Eu 3+ , as well as all the fitted parameters for Eu 3+ . Here, for brevity, we use ( ) as (4 , 5 ), as 4 and as 5 . Nd 3+ ) SF(Er 3+ ) SF(Eu 3+ ) value (Eu 3+ ) SF(Nd 3+ ) SF(Er 3+ ) SF(Eu 3+ ) value (Eu 3+ )

	Param.		4				4 -1 5
	name
	SF(av 24898.0	35577.1		65609.0	88430.0 133432.5		137500.0
	2 ( )	0.738	0.754	0.781	88732.0	0.759	0.756	0.758	91269.9
	4 ( )	0.825	0.919	0.950	67778.0	0.909	0.988	0.949	72029.8
	6 ( )	0.773	0.898	0.800	41060.7	0.870	0.798	0.834	45638.6
		19.1	-0.2		22.6	22.6	32.4		22.6
		-558.5	-204.7		-605.0	-605.0	-668.4		-605.0
		1690.5	55.8		292.2	292.2	1409.8		292.2
		0.930	0.979	0.928	1313.6	0.947	0.995	0.971	1481.1
						0.972	0.916	0.945	1290.0
	1 ( )					1025.3	1370.6		0
	2 ( )					0.726	0.776	0.751	23046.8
	3 ( )					111.5	2330.4		0
	4 ( )					1.128	1.124	1.126	16815.7
	1 ( )					0.762	0.653	0.707	9113.2
	2 ( )					2199	411.1		0
	3 ( )					1.005	0.838	0.922	10103.2
	4 ( )					2016.0	0		0
	5 ( )					0.874	0.680	0.778	6603.8

Table 3

 3 Manifolds of quintet ( = 2) and septet ( = 3) multiplicities of 4 6 ground configuration of Eu 3+ . All energy values are in cm -1 .

	Energy						Eigenvector
	3895						| 7 F⟩
	24561	√	0.576 | 5 D3⟩ -	√ 0.406 | 5 D1⟩ -	√	0.019 | 5 D2⟩
	28212						| 5 L⟩
	28268	√ 0.408 | 5 G3⟩ -√ 0.803 | 5 D3 ⟩ -	√	0.165 | 5 D1⟩ -	√	0.031 | 5 D2⟩
	45888						| 5 P⟩
	47553			√	0.652 | 5 H2⟩ -	√	0.348 | 5 H1⟩
	57251						| 5 S⟩
	62588			√ 0.970 | 5 I1⟩ -	√	0.030 | 5 I2⟩
	64164			√	0.755 | 5 F1⟩ -	√	0.245 | 5 F2⟩
	75177	√	0.513 | 5 G1⟩ -	√	0.483 | 5 G3⟩ -	√ 0.004 | 5 G2⟩
	76112	√	0.430 | 5 D1⟩ -	√ 0.392 | 5 D3⟩ -	√	0.177 | 5 D2⟩

  7 H 57.6 % ( 6 H) 5 G 14.1 % ( 6 F)7 H 14.0 % 79541 1 ( 6 H) 5 F 52.9 % ( 6 H) 7 G 20.7 % ( 6 H) 7 F 10.1 % 80396 2 ( 6 H) 5 F 41.2 % ( 6 H) 7 F 23.7 % ( 6 H) 7 G 12.0 % 81171 0 ( 6 H) 7 F 91.5 % ( 4 G) 5 D 3.1 % ( 4 G) 5 D 2.0 % 81493 1 ( 6 H) 7 F 65.0 % ( 6 H) 7 G 23.1 % ( 6 F) 7 G 4.3 % 82105 2 ( 6 H) 7 G 48.6 % ( 6 H) 7 F 34.1 % ( 6 F) 7 G 9.5 % 83096 1 ( 6 H) 7 G 32.1 % ( 6 H) 5 F 26.6 % ( 6 H) 7 F 17.4 % 83849 2 ( 6 H) 7 F 31.6 % ( 6 H) 5 F 18.1 % ( 6 H) 5 G 14.7 % 84398 1 ( 6 F) 7 G 73.0 % ( 6 H) 7 G 18.5 % ( 6 F) F) 7 D 81.8 % ( 6 F) 7 P

			5 F	3.9 %
	84785 2 ( 6 F) 7 G	75.1 % ( 6 H) 7 G 14.8 % ( 6 F) 5 F	2.4 %
	85060 2 ( 6 F) H 47.4 % ( 6 H) 5 G 18.3 % ( 6 H) 5 F 10.9 %
	86736 0 ( 6 F) 7 F	81.5 % ( 6 F) 5 D	7.4 % ( 6 H) 7 F	2.6 %
	87056 1 ( 6 F) 7 F	82.2 % ( 6 F) 5 D	6.7 % ( 6 H) 7 F	2.4 %
	87134 2 ( 6 H) 5 G 37.2 % ( 6 H) 7 H 28.1 % ( 6 F) 7 H 18.1 %
	87679 2 ( 6 F) 7 F	80.9 % ( 6 F) 5 D	5.9 % ( 6 P) 7 F	2.2 %
	89165 1 ( 6 F) 7 D 84.2 % ( 6 F) 5 P	7.8 % ( 4 F) 5 P	1.8 %
	89220 2 ( 6 F) 7 P	78.1 % ( 6 F) 7 D	6.0 % ( 6 F) 5 P	5.0 %
	90024 2 ( 6

  ).

	Tr. label	Exp. o.s.	0		1	2
	5 D 4 ↔ 7 F 0	0.489	1.14	0.07	0.80
	5 G 2 ↔ 7 F 0	0.523	0.82	0.59	0.28
	5 L 6 ↔ 7 F 0	3.338	0.38	0.31	0.29
	5 L 6 ↔ 7 F 1	1.383	0.17	0.33	0.29
	5 D 3 ↔ 7 F 1	0.302	0.87	0.12	1.10
	5 D 2 ↔ 7 F 0	0.333	1.41	1.66	0.88
	5 D 1 ↔ 7 F 1	0.450	0.99	1.00	1.00
	7 F 6 ↔ 7 F 0	1.232	1.83	1.81	1.83
	7 F 6 ↔ 7 F 1	1.983	0.93	0.95	0.94
					0.266	0.252	0.258
	∕ max			8.71 % 8.24 % 8.45 %
	3.5
					Experiment
					Our theory (3rd order correction)
	3				Standard Judd-Ofelt theory
				7 F 0 -5 L 6
	2.5							7 F 0 -7 F 6
	2							7 F 1 -7 F 6
	1.5				7 F 1 -5 L 6
	0.5 1	7 F 0 -5 D 4	7 F 0 -5 G 2		7 F 1 -5 D 3 7 F 0 -5 D 2 7 F 1 -5 D 1
	0
		361	382	394	400	414	466	534 2087 2202

Table 6

 6 Fitted parameters from the absorption oscillator strengths of Ref.[START_REF] Babu | Optical spectroscopy of Eu 3+ ions in lithium borate and lithium fluoroborate glasses[END_REF]. The second column gives standard JO parameters Ω 2,4,6 ; the third and fourth ones give obtained with Eqs. (14) and (24), respectively.

		Std. JO	2nd-order	3rd-order
		Ω +1 (10 -20 cm 2 )	(a.u.)	(a.u.)
	1	17.93 9.424 × 10 -7 1.441 × 10 -6
	3	11.92 2.330 × 10 -5 9.916 × 10 -6
	5	2.13 7.187 × 10 -8 6.406 × 10 -8

Table 7

 7 Experimental absorption oscillator strengths, as well as ED and MD theoretical ones, for transitions having both an ED and a MD character (numbers in brackets are the powers of 10).

	Tr. label	exp	th,ED	th,MD
	5 D 1 ↔ 7 F 0 7.8(-8)	2.74(-9)	2.67(-8)
	5 D 0 ↔ 7 F 1 5.7(-8)	7.70(-10) 3.93(-8)
	5 D 1 ↔ 7 F 1 4.50(-7) 4.50(-7)	1.92(-11)
	5 D 2 ↔ 7 F 1 2.48(-7) 1.74(-7)	4.96(-9)
	the MD part reads [31]
	MD =	2	2 0 ( 2 -1 ) 3(2 1 + 1)ℏ 2	 MD

J o u r n a l P r e -p r o o f

Acknowledgements

We would like to thank Reinaldo Chacón, Gérard Colas des Francs and Aymeric Leray for fruitful discussions. We acknowledge support from the NeoDip project (ANR-19-CE30-0018-01 from "Agence Nationale de la Recherche"). M.L. also acknowledges the financial support of "Région Bourgogne Franche Comté" under the projet 2018Y.07063 "ThéCUP". Calculations have been performed using HPC resources from DNUM CCUB (Centre de Calcul de l'Université de Bourgogne).

Appendix: Useful relations

In the appendix, the LS-coupling basis functions of the ground and excited configurations are respectively written | ⟩ and | , ′ ′ ′ ⟩. The reduced matrix element of the electric-multipole operator ( ) are given by [START_REF] Cowan | The theory of atomic structure and spectra[END_REF] ⟨

where

|}

) is a coefficient of fractional parentage introduced by Racah [START_REF] Racah | Theory of complex spectra. III[END_REF]. The matrix element of the spin-orbit operator within the ground configuration is

J o u r n a l P r e -p r o o f

Extending the Judd-Ofelt theory

In the excited configuration, we assume that the off-diagonal matrix elements are only due to the outermost ′ ′ = 5 electron,

The second-order correction on the eigenvector

where

J o u r n a l P r e -p r o o f

Highlights  An extension of the Judd-Ofelt theory is developed.

 All transition intensities in europium-doped (Eu 3+ ) solids are accounted for.

 The model is based on accurate calculations of fee-ion properties.

 Eu 3+ transitions are from spin-orbit mixing in the ground electronic configuration.

J o u r n a l P r e -p r o o f

Declaration of interests

X The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:


Kyriaki Dariva

Pedro Jaramillo

Reconstruction of trajectories-Challenge AMIES

Given a set of successive images which contain the positions of people in a certain space and at a certain time, we propose a method to reconstruct their trajectories. A surveillance 3D-camera takes a photo of the place every a certain fraction of the second and provides us with the data, after processing each picture and retrieving the position of the people in its view. We connect points from successive images by considering that people's position evolves with optimal transport . Our approach is mainly implemented by using a linear programming function, already available in Python. As the camera produces imperfect measurements, we define a framework to classify data and filter out noise. After presenting the problem of optimal transport of finite points as a linear programming problem, we provide some details of our modeling procedure. Finally, we show some of the results obtained with our method while we also suggest ideas to further improve our work.

Introduction

The problem of trajectory reconstruction was one of the challenges in the virtual event

"Challenge mathématiques et entreprises" organized by the Agence pour les Mathématiques en Interaction avec l'Entreprise (AMIES). It was proposed by the French company EU-RECAM whose main activity is to design and construct sensors that count the number of people inside a certain region. They are currently interested in determining the flux of people entering and leaving an area. This could provide valuable information on their customers as it could describe their foot traffic behavior or help improve security measures without needing someone to actually review hours of footage.

A 3D-Camera is set to oversee a certain place. It takes photos of the scene with a certain frequency and then for each photo it provides a set of positions in space while it also 1 identifies the position of the head. Therefore, the data we are working with are successive images, let's say 1, 2, 3, ... to put them in order. In each image i there is a set of points and each point is described as a tuple with 5 coordinates:

(i, x, y, z, h) ∈ N × R 2 × R 2 + , ( 1
)
where (x, y) is the position in the horizontal plane, z is the distance of the person's feet to the camera (in the coordinate system of the sensor) and h is the height measured from the top of the person's head to the floor. The z coordinate is only used to plot the data, it plays no role in our modeling and can be ignored. We do include it in all our definitions just to be consistent with our dataset. Images are taken in very short time intervals, one after the other (more than two images are produced each second). Therefore we consider linking points between two consecutive images with line segments as a first approximation. As we dispose many images, we can construct trajectories using straight lines and still be accurate enough. Thus, it suffices to determine how to link successive images together to see how trajectories evolve in time. Before giving some implementation details of how to create, delete, and track trajectories, we first describe the association process between two images.

Optimal transport and linear programming

Optimal transport theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimise her total effort, quantified for instance as the total cost, distance or time spent carrying shovelfuls of sand.

In our case the pile of sand pile represents points from one image and the sand castle represents the points in the next image. Therefore we wish to minimize the distances that would be traveled by everybody. Hence the cost function should take into account the distance between two points in different images. The cost function should take into account other quantities that should be minimized as well, but we will give more details and all the justifications in the next section. For now we only consider a general cost function which we denote by c. Let us now translate our optimal transport problem to a linear programming problem.

Let (x i ) i∈I denote the set of points in image k and (y j ) j∈J the set of points in the next image k + 1.

Firstly assume that |I| = |J|. For each pair (x i , y j ) i,j we can calculate the cost of linking the point x i with y j . We thus obtain a cost matrix, also denoted by c, of size I × J. We now define the matrix p which is of the same size as c. For each i ∈ I and j ∈ J, there are two possibilities for the entries of p: p i,j ∈ {0, 1}. If p i,j = 1 then the point x i is linked to the point y j , whereas if p i,j = 0 then those two points are not coupled. Thus, for any coupling matrix p we obtain a linking between the points in those two images. Matrix p can have a very general form but as we are only interested in trajectory reconstruction, each point in image k can be associated to only one point in image k + 1. Hence, we impose the restriction:

j p i,j = 1, ∀i ∈ I.
(2)

Similarly two points in image k cannot be link to the same point of the image k + 1 as two different people cannot occupy the same position. Thus, we also consider the following constraint

i p i,j ≤ 1, ∀j ∈ J. ( 3
)
Finally, the quantity that we want to minimize is the following function of p:

i,j p ij c(x i , y j ) (4)
subject to the constraints (2) and (3). As (4) is a linear function of p and the constraints are also linear, our case is a linear programming problem. For more information on linear programming we refer to [START_REF] Matousek | Understanding and Using Linear Programming[END_REF]. We reformulate this to put it in the standard form used by the linprog library in Python.

    

min p {c T • p}, such that A eq p = b eq , and

A ub p ≤ b ub , (LP)
where A eq and A ub are the conditions (2) and (3) in matrix form, c and p are the cost and the coupling matrix respectively, in vector form, and b eq and b ub are column vectors filled with ones. The values in b eq and b ub correspond to the right hand side of (2) and (3).

In the general case |I| ̸ = |J|: This is possible since people can enter or leave the region tracked by the camera. Let n k denote the number of points in image k and n k+1 the number of points in image k + 1. There are three possibilities:

• n k = n k+1 . In that case every point in image k is connected to exactly one point in image k + 1 and every point in image k + 1 receives exactly one point from image k. It is the case of (LP ) described above with (3) becoming an equality (LP is written using only the matrix A eq ).

• n k > n k+1 . Every point in image k + 1 receives exactly one point from image k and every point in image k is linked to at most one point from the image k + 1.

• n k < n k+1 . Every point in image k is connected to exactly one point in image k + 1 and every point in image k + 1 can receive at most one point from image k.

The solution to (LP) is the coupling matrix p which gives the linking of points between two images by minimizing the total cost. Remark 1. We would like to mention here that in a more general framework the vectors b eq , b ub can have other entries besides 1 and 0. For example, if we consider b ub and b eq to be column vectors consisted of twos, that would mean (respectively) that a point in image k has two possible choices in image k + 1 and that each point in image k + 1 could be connected with two points from the previous image. If that scenario was taken into consideration, it would improve the robustness of our solution, but remains to be treated in some future work.

Remark 2. We chose the revised simplex method for linprog since it is known to be computationally efficient. For n k points in image k and n k+1 points in image k + 1 the revised simplex method takes about n k n k+1 computations in practice. This means that we could easily consider hundreds of people and still manage to keep the computational cost relatively low.

Modeling

In this section we will describe some of the difficulties that appear when we are working with real data. We will present our approach to overcome some of these hurdles, the assumptions that we make and some of the issues that we could not solve. Let us first introduce the notations and give some basic definitions.

In what follows the term frame will be used interchangeably to denote the image taken by the camera and the set of data points that are associated to that image.

Preliminaries

Let Ω ⊂ R 2 × R + be the smallest rectangular set that contains all the data points recorded by the camera. We will call this Ω the domain of observation. We now detail what exactly we mean by a trajectory: Definition 1. A trajectory is an ordered list of states [S 1 , S 2 , S 3 ....., S current ]. States are described with six coordinates:

S := (f, x, y, z, h, v) ( 5
)
where f is the number of the picture (frame) where the data point (f, x, y, z, h) was recorded and v = (v x , v y ) is the velocity at this point of the person that follows this trajectory. The point associated to state S i goes to the point of state S i+1 with optimal transport. Definition 2. We will say that a trajectory dies at frame f 0 if its last state is inside Ω and it is expected to be outside Ω before the next frame f 0 + 1 is recorded. The set of dead trajectories at frame f 0 will be the trajectories that have died before or at frame f 0 . We say a trajectory is active at frame f 0 if it is not dead and if its last state is in frame f 0 .

Observing the given dataset helped us to come up with a solution. An algorithm was constructed based on the behavioral trends of the data that we have to describe. Apart from false measurements taken by the sensor, the fact that the camera tracks real people makes the flow very heterogeneous. Hence, wrong measurements can be attributed to different factors.

Observations

A careful study of the videos provided by the camera leads us to the following remarks:

• The rate of frames varies in time. Additionally, it may not always be high enough to have an accurate approximation by line segments.

• Measurements are noisy, meaning that either there are points missing (people that are not recognized by the camera at all) or that there are points that do not correspond to human beings. Indeed, a big part of the noise is due to objects in the scene which are captured as people crossing the region.

• Another important source of noise is the multiple measurements of the same person in one frame. For example, it is not uncommon to see that in one frame, the head, the knees and the feet of a person are recorded by the camera as five different people that are really close to each other.

Directly applying the approach described in section 2 will not work. That is because, even without all the above phenomena, we still need to appropriately handle the people that leave and enter the domain of observation. We will divide our solution in 3 steps.

1. For each frame, we will first filter the currently active trajectories to remove noise and dead trajectories.

2. Next, we will filter noise points from the next frame.

3. Lastly, we will link currently active trajectories to points from the next frame. Based on the number of active trajectories in the current frame and the number of data points in the next frame, we will either neglect data points, create new trajectories or extend trajectories.

We consider the following setup to model this problem. The state of the system at frame f 0 will be described by: the set of active trajectories at frame f 0 , the set of dead trajectories at frame f 0 and the set of noise points until the frame f 0 . The idea behind this, is to be able to record the past trajectories and the noise registered by the camera, and to get an idea of the type our domain (for example if it is a street, a plaza or an elevator). That way we can more accurately eliminate noise points from frames and classify trajectories as time passes and more data is collected.

Filtering trajectories

For each frame we first filter noisy trajectories and then dead trajectories. The order in which this is done is important (if done differently some noisy trajectories could be classified as real trajectories).

Noisy trajectories

When looking at the footage and the data provided by EURECAM we notice that for a considerable amount of the time the domain of observation had nobody inside it.

In that case all the measurements were noise, and as most of the noise is linked to static objects in the scene (for example a table), the trajectories that were created just moved arbitrarily in one part of the domain. This gave us the idea to classify a trajectory as noise if it ever lived too long or if it changed directions abruptly too many times. Definition 3. We say that a trajectory changes direction abruptly at a frame f 0 if its current velocity v 0 and its velocity v 1 at in the next frame verify

v 0 • v 1 < 0. ( 6
)
Let n vc ∈ N be a fixed parameter denoting the upper bound of the number of changes in velocity direction that we allow. If a trajectory abruptly changes direction more than n vc times, then we consider all of its states as noisy data points, we remove it from the set of active trajectories and we add all of the points in the noise set.

Choosing the value of n vc parameter correctly may need to take into consideration the frame rate and the general speed expected for people traversing the scene. For instance, in a hall inside a building one would generally expect people to abruptly change their direction at most once. In that case we could pick n vc = 1 or n vc = 0.

Another possibility of a noisy trajectory is when points tracked by the camera generate trajectories that turn around in circles (or that do not move much) without abruptly changing directions many times. For that scenario we define the lifetime of trajectories.

Definition 4. We call lifetime of a trajectory the difference between the frame number of its last state and the frame number of its next state.

We provide a threshold for a very long lifetime. Let n lt ∈ N be a fixed parameter. We consider that a trajectory has lived too long if its lifetime is greater than n lt .

Dead trajectories

It is necessary to determine which people are leaving from Ω at frame f 0 , so that we stop considering their trajectories as active in the next frame.

As we cannot actually measure data points outside Ω, we use the last state of a trajectory to predict if it will exit the scene or not. To do so, we take into account the last position of a trajectory at the current frame and the velocity with which it is moving.

Let δ x , δ y > 0 be two fixed parameters and let x min , x max , y min , y max ∈ R define the border of Ω. We expect a trajectory to leave Ω if its last state (f 0 , x, y, z, h, (v x , v y )) verifies:

|x -x min | < δ x , or (7) |x -x max | < δ x , or (8) |y -y min | < δ y , or (9) |y -y max | < δ y , ( 10
)
and

(x + v x , y + v y ) ̸ ∈ Ω. ( 11
)
Ideally, the last condition would suffice but we add the first set of conditions just to avoid any unforeseen behavior attributed to noise. Once a trajectory dies we remove it from the active trajectories list and add it to the list of dead trajectories.

Filtering data points

We distinguish two sorts of noise:

• Points from static objects in the scene.

• Multiple measurements for the same person.

Static objects in the scene

This is one of the simplest tests that we had to implement. The basic idea is that if there is an object in the scene, we do not expect to find measurements of actual people near it. This is reasonable as people tend to avoid obstacles when walking. In order to give a sense of what near means we consider the usual euclidean distance:

d((x 0 , y 0 , h 0 ), (x 1 , y 1 , h 1 )) = |x 0 -x 1 | 2 + |y 0 -y 1 | 2 + |h 0 -h 1 | 2 .
Let δ n > 0 be a fixed parameter. For every point in a new frame, we check if its euclidean distance to any of the points in the noise set is less than δ n . If it is the case, then we classify it as noise and we neglect it (we put it in the noise set). Otherwise, we continue to observe it. This is a crude test but it was a good enough start for testing the real data from EURECAM.

Duplicate measurements

What we call duplicate points is when we have two points near each other (in the xy-plane) but with considerably different heights. For example, what tends to happen

most often is that the camera registers the head of the people along with both of their feet as if they were three different people.

A first approach to deal with this imperfect measurements could be to just remove all data points with height less than a certain value. We did not choose to do it because regularly only a person's foot is registered by the camera. Nevertheless, our approach remains simplistic but quick to implement and iterate on. More precisely, for each data point in the frame we look if it is near to another point in the frame. If this is the case, we ignore the data point and keep it otherwise.

Let δ d , δ h > 0 be fixed parameters. To give a meaning to the notion of near, we introduce the following function

d h ((x 0 , y 0 , h 0 ), (x 1 , y 1 , h 1 )) = |x 0 -x 1 | 2 + |y 0 -y 1 | 2 +δ h min(|h 0 -h 1 |, min(|h 0 |, |h 1 |)).
We consider two data points (f, x 0 , y 0 , z 0 , h 0 ), (f, x 1 , y 1 , z 1 , h 1 ) to be near each other if they verify

d h ((x 0 , y 0 , h 0 ), (x 1 , y 1 , h 1 )) < δ d ( 12
)
Points classified as multiple measurements of one person are just removed from a frame and are not considered when linking points. Now that we have filtered the duplicates we proceed with the final step of our method, that is to link the current active trajectories to the (clean) data points from the next frame.

Linking data points to trajectories

From now on we will consider that all the data points represent real people in the scene.

As described in section 2 we will associate trajectories to data points by minimizing the cost function. In the simplest case one would consider the sum of the euclidean distances between the last position of all active trajectories at the current frame and the data points from the next frame. Even in a framework without noise, this solution would not be robust. Changes in frame rate, missing data and differences in height of data points are all factors that should be taken into consideration in the cost function. Instead of using the euclidean distance we propose the following function:

d OT ((f 0 , x 0 , y 0 , z 0 , h 0 , (v x , v y )), (f 1 , x 1 , y 1 , z 1 , h 1 )) = cost h + cost v + cost d ,
where

cost h = (1 -C align )|h 0 -h 1 | + C align min(|h 0 -h 1 |, min(|h 0 |, |h 1 |)), ( 13
)
C align =    vx |(vx,vy)| (x 1 -x 0 ) |(x 1 -x 0 ,y 1 -y 0 )| + vy |(vx,vy)| (y 1 -y 0 ) |(x 1 -x 0 ,y 1 -y 0 )| , if |v||(x 1 -x 0 , y 1 -y 0 )| > 0, 1, otherwise cost v = |x 1 -x 0 -(f 1 -f 0 )v x | 2 + |y 1 -y 0 -(f 1 -f 0 )v y | 2 , and (14
)

cost d = |x 0 -x 1 | 2 + |y 0 -y 1 | 2 . ( 15
)
The cost h term helps differentiate people by their height without penalizing the distance of their head to their feet. In that way, if at some frame the position of a person's foot is registered by the 3D-camera, and at the next frame what is registered is the same person's head, these two points should be relatively close according to the distance function. We do emphasize that the term cost h recognizes some height differences, for example it can differentiate adult from children fairly well.

We also considered an alignment coefficient C align inside cost h . As the feet are usually either in front or behind the person itself, we wanted to take that into account in our cost function. To do so, we use the velocity with which the person was moving. The less aligned a point is to a trajectory the more we penalize errors in height. This prevents us from linking one person's head with another person's feet if they happen to walk near each other.

The second term takes into consideration the inertia of a person. In short, we expect a person to continue moving with their current velocity between the current frame and the next one. We thus calculate the distance not only to the current position of the trajectory but also to its expected position. This is just a way to add some robustness to our model, as we know that people need time to change direction.

Finally, the third term is the usual 2D euclidean distance one would expect to use. We can now calculate the cost function as in section 2 and obtain a pairing between trajectory and data points. We next proceed to describe the different possible cases we could encounter at this stage.

Same number of trajectories and data points

Let X ∈ N be the number of active trajectories at the current frame f and Y ∈ N the number of data points in the next frame. If X = Y we consider that nobody has entered the scene in view of the camera and thus we expect that each active trajectory just continues in the next frame. This means that we optimize the distance introduced in the last section and then we add each point to the end of the trajectory it is paired with. Before extending the trajectory, we transform the new point into a trajectory state by adding the velocity component.

Let a trajectory's last state be (f 0 , x 0 , y 0 , z 0 , h 0 , (v x , v y )) and the point it is paired with be (f 1 , x 1 , y 1 , z 1 , h 1 ). The next state of the trajectory is then given by

f 1 , x 1 , y 1 , z 1 , h 1 , x 1 -x 0 f 1 -f 0 , y 1 -y 0 f 1 -f 0 ( 16
)
We now consider the case of more data points in the next frame than active trajectories in the current frame. This is not very different from what we already did but it is still worth detailing.

More data points than trajectories

The only possibility in that case is that people have entered the scene. We will therefore add them as new trajectories to the set of active trajectories. First, we pair trajectories as described above. Then, the unpaired points will be transformed into a list each with only one element. If (f,x,y,z,h) is such a data point we will consider the new trajectory that starts at this point: traj = [(f, x, y, z, h, (0, 0))], using Python code notation.

(

) 17
The choice of the velocity vector (0, 0) is arbitrary. There are probably better options, but we chose it like this just to avoid adding additional cost to this state when linking it to another point later on.

More trajectories than data points

Lastly we see what happens when there are more active trajectories in the current frame than data points in the next frame. As dead trajectories have already been taken into account, this can only mean that some people have not been recognized by the camera. It could also mean that we were maybe too strict when eliminating noise from the frame, but we will not consider this case.

We have a few ways to handle this scenario. A first option would be to join trajectories as usual and then extend the unpaired trajectory by just considering that their actual position is their next expected position. This is a sensible idea but in practice it makes it hard to filter noise trajectories. What happens is that static objects create new trajectories and before they can be recognized as noise they exit the scene and are thus registered as people.

Another choice would be to keep unpaired trajectories as they are. This can also cause problems. Indeed, it is possible that such a trajectory stays inside the domain when the person corresponding to it exits the domain. Consequently this trajectory could be linked to the data point of another trajectory or eventually be considered as noise and removed .

We chose the second approach, as we prioritized to have fewer but true dead trajectories. It is however easy to imagine some kind of hybrid approach where for example we use the second option until the noise set is big enough and then change our approach to the first option.

Results and conclusions

We now show some of the results when we tested our code with the data provided by EURECAM. We cannot include the actual data in this work but we do provide our code and a video that shows our results [2]. For the most part in this section, we just show some images which reveal the behavior of the proposed solution.

We first show the case of two people walking together across the scene in a straight line. We can see in figure 1 two successive frames of the video recorded by the camera. We can see in figure 1a that the one person has produced multiple measurements and the other has only one, which is a foot. We count these data points as if it were only one person. This can be due to the size of the parameter δ d that we are currently using in combination with lack of measurements in certain frames, as can been seen in figure 1b. In that case the last point of the trajectory (in blue) and the red cross are close enough according to our test. They are therefore considered to represent the same person. As a result, the red cross is not taken into account when linking trajectories.

We would like to note that we did not have many criteria to chose the parameters the way we did. Accuracy can easily be improved by optimizing the choice of parameters. Figure 2 shows a common problem that we had when people leave the domain. What tends to happen, is that when someone exits the domain, their head may leave from the current frame but the foot, which is behind the head, is tracked by the camera in the next frame. As such, the trajectory turns back as in figure 2b. Hence the person has left the domain but the trajectory representing that person still remains. It can then infect some future trajectories as in figure 3. This could be easily fixed with a small change in the part of our code which filters out duplicate measurements. We could, for example, consider removing points that are duplicate and keep the one that is the farthest away from the floor.

Figure 3: Someone is leaving the are while someone else is entering it. We can see that if the trajectory of a person leaving is not treated correctly it can infect future trajectories.

The same infection phenomenon can be seen in 4 below. We remark that in that case we do prevent some of the noise coming from the static objects (right part of the image) as can be seen in figure 4c. The red crosses are data points that have been picked up by the camera but we manage to correctly identify them as noise and do not infect the other trajectories. Figure 6: Our method can even handle people that are moving relatively fast. These two frames belong to a person running across the area. Her trajectory is accurately registered and classified as dead when she is outside the domain of observation.

Figure 7: Image of a frame when a lot of people are passing through the domain as a group. We see most of the problems described above happening simultaneously

In figure 7 we see why our method should still be improved.

Iterating on our code

Our method can be improved with relatively simple changes. Most of these can be easily implemented in the code because most of the solution comes down to the cost function that we choose. One small change could be to make the parameters introduced above depend on the dead trajectories and the noise set. That way, ideally, the approach would improve over time. Moreover this could help us determine the possible trends in the behavior of the people inside Ω. If for instance the observed domain is a busy street, then we could calculate the mean velocity of people to filter out noise more accurately.

If we wish to avoid mistakes when linking trajectories to points, we could also take into account the value of the cost function and compare it to the mean value of the total cost from previous frames. Even a simple enough test could provide some insight to help detect if we should create a new trajectory from a data point to the next frame or not. This could help in the case where a person enters Ω and another person who is already inside is not detected by the camera.

Other simple changes we could consider would be to better define velocities and velocities of a trajectory. As implemented in our code right now we consider the frame rate of the camera to be constant. A better definition would actually use time intervals. We could also consider improving upon our noise detection by consider more subtle tests. Right now we just consider a data point to be noise if it is near something that has already been classified as a noise. It would be better if we took into account the number of noise points that are close to it, for example.

We remark that even though all these changes can improve the accuracy of our solution, in the end the best choice is probably to deal with more than one frames at a time. That way our solution would be more robust against noise and lack of measurements. This combination is one of the major flaws in our approach. The reason we cannot manage it more appropriately is because, as described so far, we consider one frame at a time when linking trajectories to points. If we make a mistake at some frame we cannot later go back and correct it. In order to get a significant improvement, more frames need to be taken into account, however that requires a subtle analysis.

Annexe

Plotting

The images are extrapolated from the videos animated with the simple transformation of a 3D object (x, y, z) in cm to a 2D object (ix, iy) of the image in pixels: ix = 1 2 ( xf oc z + cx) iy = 1 2 ( yf oc z + cy) where f oc is the focal in pixels and (cx, cy) are the coordinates of the optical center.

  (a) First frame (b) Next frame

Figure 1 :

 1 Figure 1: Two successive frames where only two people are traversing the scene in a relatively straight line.

Figure 2 :

 2 Figure 2: Two successive frames where one person is leaving the field of view of the camera.

Figure 4 :Figure 5 :

 45 Figure 4: Noise coming from static objects can infect people that are far away if we blindly apply optimal transport without considering the actual value of the cost function.

Acknowledgements

We would like to thank Sébastien Tran Tien for his contribution to the development of the code, the video animation and his constructive discussions and suggestions.