A P Severyukhin

J Margueron

I N Borzov

N Van Giai

Sensitivity of β-decay rates to the radial dependence of the nucleon effective mass

Keywords: numbers: 23.40.-s, 26.30.-k, 21.10.Re

We analyze the sensitivity of β-decay rates in 78 Ni and 100,132 Sn to a correction term in Skyrme energy-density functionals (EDF) which modifies the radial shape of the nucleon effective mass. This correction is added on top of several Skyrme parametrizations which are selected from their effective mass properties and predictions about the stability properties of 132 Sn. The impact of the correction on high-energy collective modes is shown to be moderate. From the comparison of the effects induced by the surface-peaked effective mass in the three doubly magic nuclei, it is found that 132 Sn is largely impacted by the correction, while 78 Ni and 100 Sn are only moderately affected. We conclude that β-decay rates in these nuclei can be used as a test of different parts of the nuclear EDF: 78 Ni and 100 Sn are mostly sensitive to the particle-hole interaction through the B(GT) values, while 132 Sn is sensitive to the radial shape of the effective mass. Possible improvements of these different parts could therefore be better constrained in the future.

I. INTRODUCTION

Weak processes such as β-decay rates, electron capture, neutrino scattering and absorption play an important role during the late evolution of massive stars [1]. They are largely responsible for the electron fraction in the core during the core-collapse phase and they continue to play a determinant role in the nuclear synthesis r-process [2]. Because of their great importance in astrophysical applications, weak processes were extensively investigated within various approaches. The large-scale shell model Monte Carlo (SMMC) method was, for instance, applied to compute β ± decay rates for stellar conditions for more than 100 nuclei in the mass range A = 45-65 [START_REF] Langanke | At. Data Nucl. Data Tables[END_REF][START_REF] Martínez-Pinedo | Astrophys[END_REF]. Recently, mean-field based models have been used for the prediction of electron-capture cross sections and rates. Finite-temperature charge-exchange RPA (CERPA) models based on Skyrme or relativistic functionals have been applied to predict electron-capture cross sections using different interactions [5][6][7]. Meanfield predictions around the Fermi energy are, however, known to suffer from their underestimation of the density of states. In this work, we explore a small correction to the mean field models which increases the density of states around the Fermi energy [8]. Here, we compare the predictions of this model to known experimental values such as β half-lives or collective modes, as a first step before using it for astrophysical applications.

Since the pioneering work of Brown et al. [9] it is known that the level density around the Fermi energy in stable nuclei indicates that the in-medium nucleon effective mass is close to the bare mass. The description of giant resonances such as the giant dipole resonance requires, on the other hand, that the nucleon effective mass in the nuclear medium should be reduced as compared to its value in vacuum [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitation[END_REF]. Analysis of the momentum dependence of the nuclear optical potential also favors an in-medium effective mass lower than in vacuum [11,12].

These apparently diverging properties of the inmedium effective mass m * can be reconciled by considering the two contributions to m * : the k-mass which is also called the non-locality mass, and the ω-mass which is induced by dynamical correlations such as particle-phonon coupling [13][14][15][16][17]. The coupling of the collective modes to the single-particle (s.p.) motion is, however, difficult to perform in a self-consistent approach. One of the main problems is coming from the fragmentation of s.p. strength which increases exponentially at each iteration of the self-consistent method. It has therefore been tried to include these correlation directly in the mean field, either at the level of the interaction with density-dependent gradient terms [18], or, loosing the relation with an interaction, at the level of the nuclear energy density functional (EDF) so as to produce a surface-peaked effective mass (SPEM) which, at the same time, does not strongly modify the mean-field [8]. In this study, we will explore the second approach.

Predictions of β-decay rates throughout the nuclear chart within a consistent microscopic nuclear model are difficult. Tuning of models according to the system under study is usually performed, and the description of βdecay rates through a unique microscopic nuclear model does not exist. Since β-decay rates are known to depend strongly on the fine structure around the Fermi level, the difficulties to have a general description could be related to the common issue with mean-field models that the s.p. level density around the Fermi level is too low. The increase of the level density, by using for instance a model producing a SPEM could, in principle, lead to a better description of β-decay rates throughout the nuclear chart.

In microscopic approaches, calculations of nuclear β -decay rates are rather complex. Due to phase-space amplification effects, the calculated β-decay rates are sensitive to both nuclear binding energies and β-strength functions. In an appropriate β-decay model, the correct amount of the integral β-strength should be placed within the properly calculated Q β -window provided that the spectral distribution is also close to the "true" β-strength function. Furthermore, for consistency the model has to yield correct positions and strengths of the Gamow-Teller (GT) and first-forbidden resonances in the continuum [19]. Another complication is related with the largescale predictions of nuclear β-decay rates. Such a program is a compromise between accurate results and the necessity to cover extended regions of the nuclear chart including deformed nuclei or even the region with triple prolate-oblate-spherical shape coexistence scenario. In this work, we shall consider only the case of spherical nuclei. A plausible way to detect a change of the β-decay strength function profile due to higher-order corrections could be the analysis of the integral characteristics of βdecay. The half-life is one of such characteristics, being sensitive enough to the β-strength distribution [19]. It is worth to analyze first the doubly-magic β ± -unstable nuclides, such as 100,132 Sn, since one can use the simpler CERPA model. Also, we compare to the most neutronrich ((N -Z)/A = 0.28) doubly-magic nucleus 78 Ni which is an important waiting point in the r-process [20]. The next step would then be to study the delayed neutron and especially delayed multi-neutron emission [21]. This is a more difficult task since the delayed neutron emission probability (P n -value) [22] puts an additional constraint on the β-strength distribution in the near-threshold region. This paper is organized as follows. In Sec. II we briefly present the modifications to the nuclear EDF which produce a SPEM, and we describe the protocol used to adjust the strength of this correction. In Sec. III, we analyze the results of the calculations of β-decay rates in 78 Ni and 100,132 Sn and the properties of the giant quadrupole resonance (GQR) and Gamow-Teller resonance (GTR) of 208 Pb. Conclusions are drawn in Sec. IV.

II. THE MEAN FIELD MODELS

Skyrme-type EDF are known to give an accurate description of masses and charge radii over the whole nuclear chart, from Z = 8 up to heavy elements [23]. As most of the mean-field approaches, they however lead to a s.p. level density around the Fermi surface which is lower than the experimental one [9]. Here, we introduce a correction term to the Skyrme EDF which leads to a SPEM and increases the average s.p. level density [8]. We hereafter present this correction term and then briefly describe the calculations of β-decay rates carried out consistently in the framework of Hartree-Fock-CERPA approach.

A. The standard Skyrme functional

The standard Skyrme functional for the time-even energy density is expressed as [23] H sky (r) =

2 2m τ 0 + t=0,1 C ρ t (ρ 0 )ρ 2 t + C ∆ρ t ρ t ∆ρ t +C τ t ρ t τ t + 1 2 C J t J 2 t + C ∇J t ρ t ∇ • J t , (1)
where the indices t = 0, 1 stand for the isoscalar and isovector part of the corresponding densities, respectively. For instance, the nucleonic densities ρ 0 and ρ 1 are defined as,

ρ 0 (r) = ρ n (r) + ρ p (r), ρ 1 (r) = ρ n (r) -ρ p (r), (2)
where the densities ρ q (q = n, p) are expressed in terms of the s.p. wave functions ϕ q i as

ρ q (r) = i |ϕ q i (r)| 2 . ( 3
)
The kinetic energy and spin-current densities, τ t and J t , are defined similarly [23]. The coefficients C j i in Eq. ( 1) are constants (see, e.g., Ref. [23]) except for the coefficient C ρ t which depends of the isoscalar density ρ 0 as:

C ρ t (ρ 0 ) = C ρ t (0) + (C ρ t (ρ 0,sat ) -C ρ t (0)) ρ 0 ρ 0,sat α , (4)
where ρ 0,sat is the saturation density in infinite nuclear matter.

The standard Skyrme functional can be separated into neutron and proton channels, and neutron and proton effective masses are introduced:

m m * q = 1 + 2m 2 [(C τ 0 + C τ 1 )ρ q + (C τ 0 -C τ 1 )ρ q ], (5)
Then, neutron and proton mean fields can be obtained (see appendix A). Among the large number of Skyrme parametrizations, we have selected 6 of them based on the following requirements:

• First, the Skyrme EDF should predict 132 Sn as a β-unstable nucleus at the mean field level. This is based on the common expectation that the Landau parameter G ′ 0 in the spin-isospin channel is repulsive and will shift up the GT strength.

• Second, we wish to explore different values of effective mass in the bulk, and different isospin splittings of the effective mass.

In 132 Sn the first condition can be related to the s.p. energy difference between the π2d 5 2 and ν2d 3 2 states (which contribute mostly to the GT transition towards , where the last term stands for the mass difference between the neutron and the hydrogen atom, ∆M n-H = 0.782 MeV. If this transition energy is positive at the mean field level -hereafter called the HF transition energy -the system is βstable since the CERPA correlations could only push it up, while it is expected to be actually β-unstable. Anticipating the discussion of the results in Sec. III we observe that models having positive HF transition energies predict β-decay half-lives which are too large in 132 Sn. We therefore consider only models having a HF energy difference ǫ π2d 5 2 -ǫ ν2d 3 2 < 0. This condition is indeed quite drastic, and we found that an appreciable number of well established Skyrme models do not fulfill it. Among these are SIII [24], BSK14-17 [25], SKM * [24], SLy4-5 [24], SKO [26]. In addition, the models which predict that the HF transition energy is larger than 0.782 MeV are: RATP [24], SGII [27], LNS [28], LNS1, LNS5 [29], SKI1-5 [30], SAMi [31]. These models have therefore not been used here.

For the few remaining models, we restrict ourselves to the parametrizations SLyIII0.7, SLyIII0.8 and SLyIII0.9 [32] which predict in the bulk nuclear matter the effective mass values m * /m=0.7, 0.8 and 0.9, respectively. We have also considered the f -, f 0 and f + [33] models which predict an effective mass of 0.7 in symmetric matter, with either a positive, zero, or negative isospin splitting of the effective mass (ISEM) in neutron matter, defined as m * n /m -m * p /m. Notice that m * p /m could be calculated in neutron matter without any ambiguity: the proton density shall simply be set to zero, see for instance Eq. ( 5).

The bulk properties of the selected interactions are given in Table I. It is observed that the saturation density ρ 0 , the energy per particle at saturation E 0 , and the symmetry energy J sym are very similar for these interactions. The slope of the symmetry energy L sym is varying between 24.75 and 43.79 MeV, which is a rather wide range, but these models are still considered as iso-soft ones. The incompressibility of SLyIII0.7, SLyIII0.8 and SLyIII0.9 is quite large. However, this does not affect the processes explored in this work. The main difference among these models comes from their effective masses and ISEM. The models SLyIII0.7, SLyIII0.8, SLyIII0.9 have a different effective mass in symmetric matter, and a positive ISEM. The models f -, f 0 and f + have the same effective mass in symmetric matter and different signs for the ISEM. The models SLyIII0.7, SLyIII0.8, SLyIII0.9 and f + have a positive ISEM, as expected from microscopic BHF and DBHF calculations [34], while f 0 has no splitting and f -has a negative ISEM. Finally, the values of the Landau parameter for the models SLyIII0.7, SLyIII0.8 and SLyIII0.9 is given by

G ′ 0 = -N 0 1 4 t 0 + 1 24 t 3 ρ α3 + 1 8 k 2 F (t 1 -t 2 ) (6)
and for the models f -, f 0 and f + , 2 2 is the level density, with k F being the Fermi momentum and m * the nucleon effective mass. In Eq.( 7), the parameter t 4 is coming from an additional density-dependent term besides the usual densitydependent t 3 term, and the coefficient in front of the density dependent terms have been modified w.r.t. the standard notations [33]. The values of the Landau parameter G ′ 0 are given in the last column of Table I. At saturation density (ρ = ρ 0 ), the models SLyIII0.7, SLyIII0.8, SLyIII0.9 predict rather large values for G ′ 0 ≈ 0.3 -0.35, while the models f -, f 0 and f + predict smaller value with G ′ 0 ≈ 0. The forces f -, f 0 and f + clearly predict not enough positive G ′ 0 values [19]. In addition to the different effective masses we therefore expect to observe substantial differences between these two sets of models in the charge-exchange channel.

G ′ 0 = -N 0 1 4 t 0 + 1 4 t 3 ρ α3 + 1 4 t 4 ρ α4 + 1 8 k 2 F (t 1 -t 2 ) (7) where N 0 = 2k F m * /π

B. Surface-peaked effective mass correction

In Ref. [8], a surface-peaked effective mass correction to the Skyrme-type Hamiltonian was proposed with the following form, and the new functional can be written as H = H sky +∆H.

△H = C τ (∇ρ) 2 0 τ (∇ρ) 2 + C ρ 2 (∇ρ) 2 0 ρ 2 (∇ρ) 2 , (8)
The first term of Eq.( 8) is designed to modify the effective mass profile at the nuclear surface, while the second term is introduced in order to compensate the effects of the first term in the nuclear mean-field. Without the second term, the effects of the first term on the mean field are too large and drastically limit the possible values for the strength of the SPEM, as in Ref. [35]. The compensation was found to be optimal for intermediate mass and heavy nuclei if one uses the following constant relation between the two new parameters [8]:

C ρ 2 (∇ρ) 2 0 = -10 fm C τ (∇ρ) 2 0 . ( 9
)
One can expect an impact of the SPEM on the properties of the lowest quadrupole excitation if the isoscalar terms (8) are taken into account. On the other hand, the energy-weighted sum rule (EWSR) is an integral characteristic and it is particularly sensitive to the giantresonance properties which can be described by the EDF without the terms (8). In the present work, the values of C τ (∇ρ) 2 0 =-210 and -420 MeV fm 10 are fixed so that the isoscalar quadrupole EWSR in 208 Pb is modified by 1% and 2%, respectively. Consequently, a change of less than 0.04 of the neutron and proton effective masses at the nuclear surface of 208 Pb is predicted, see Fig. 1. This procedure is slightly different from that used in Ref. [8], and it leads to a SPEM less strongly peaked at the surface. We have added the terms (8) without refitting the existing standard parameterizations. Using this perturbative approach, we observe a small change of the binding energies which is larger than the tolerance of the protocol for the parameter fitting. In particular, in 208 Pb the binding energy changes by 0.35% for the SLyIII0.9 set, 0.37% for the SLyIII0.8 set, 0.38% for the SLyIII0.7 set and 0.45% for the f 0 , f -and f + sets. A fine tuning of other parameters in order to compensate for these energy changes has still to be done.

In Fig. 1 are shown the effective mass profiles in 208 Pb for the f 0 , f -and f + models where we have considered different values of the parameter governing the strength of the SPEM, C τ (∇ρ) 2 0 =0, -210 and -420 MeV fm 10 . We remind that the differences between the models f 0 , f - and f + are mostly the ISEM in asymmetric matter: f + has m * n > m * p in neutron rich matter, while f -has m * n < m * p , and f 0 has m * n = m * p in the same conditions of isospin asymmetry. The effect of the sign difference of the effective mass splitting can also be observed on panels (a) and (d) (without SPEM): Since 208 Pb is a neutron rich nucleus, the neutron effective mass is larger than the proton one for the f + model, an opposite effect is found for f -, and no effect is observed for f 0 . Additionally, it is observed in panels (b), (c), (e) and (f) that the SPEM correction is almost unaffected by the effective mass splitting, since the correction is isoscalar.

C. Calculations of β-decay rates

We describe the collective modes in the chargeexchange random phase approximation (CERPA) using the same Skyrme interactions as above. Making use of the finite-rank separable approximation (FRSA) [36][37][38] for the p-h interaction enables us to perform CERPA calculations in very large configuration spaces. Although it is well known that the tensor interaction influences also the description of the β --decay half-lives [39], in the present study the tensor force is neglected in order to focus on the impact of the SPEM.

The experimentally known values of the half-lives put an indirect constraint on the calculated GT strength distributions within the Q β -window. To calculate the halflives an approximation worked out in Ref. [40] is used. It allows one to avoid an implicit calculation of the nuclear masses and Q β -values. However, one should realize that the related uncertainty in constraining the parent nucleus ground state calculated with the chosen Skyrme interaction is transferred to the values of the neutron and proton chemical potentials. In the allowed GT approximation, the β ± -decay rate is expressed by summing the probabilities of the energetically allowed transitions (in units of G 2 A /4π) weighted with the integrated Fermi function. For the β --decay case we have:

T β - 1/2 = D GA GV 2 k f 0 (Z + 1, A, E i -E 1 + k )B(GT ) - k , (10)
E i -E 1 + k ≈ ∆M n-H + µ n -µ p -E k (11)
while for the β + -decay case this becomes:

T β + 1/2 = D GA GV 2 k f 0 (-Z + 1, A, E i -E 1 + k )B(GT ) + k , (12)
E i -E 1 + k ≈ -∆M n-H -2m e -µ n + µ p -E k . ( 13
)
Here, D=6147 [START_REF] Suhonen | From Nucleons to Nucleus[END_REF] is a constant, G A /G V =1.25 [START_REF] Suhonen | From Nucleons to Nucleus[END_REF] is the ratio of the weak axial-vector and vector coupling constants, m e is the positron mass; µ n and µ p are the neutron and proton chemical potentials. E i is the ground state energy of the parent nucleus (Z, A) and E 1 + k denotes a state of the daughter nucleus. The E k are the 1 + eigenvalues of the CERPA equations. The CERPA wave functions allow us to determine GT transitions whose operator is Ô± = i,m t ± (i)σ m (i).

B(GT

) ± k = N ± 1, Z ∓ 1; 1 + k | Ô± |N, Z; 0 + gs 2 . (14)
Expressions ( 10)-( 14) will be used in the next section to calculate the β-decay rates and the collective modes. All the calculations are performed without any quenching factor.

III. RESULTS FOR COLLECTIVE MODES AND β-DECAY RATES

We now analyze first the results of the β-decay rates which are sensitive to the low-energy part of the CERPA strength, and then the GT collective modes. The effects of the SPEM will be discussed.

The p-h interaction in the spin-isospin channel is assumed of the following form:

V (r 1 , r 2 ) = N -1 0 G ′ 0 (r 1 )σ (1) •σ (2) τ (1) •τ (2) δ(r 1 -r 2 ) ( 15
)
where σ (i) and τ (i) are the spin and isospin operators. As expected, the largest contribution to the calculated β ± -decay half-life comes from the 1 + 1 state, the structure of which is dominated by one unperturbed configuration. They are the 1p-1h configurations {π2d 5 2 , ν2d 3 2 }, {ν1g 7 2 , π1g 9 2 } and {π2p 3 2 , ν2p 1 2 } of 132 Sn, 100 Sn and 78 Ni, respectively. In other words, the 1 + 1 state is non-collective and, therefore, the β-decay is related to the lowest unperturbed 1 + energy. We first examine the s.p. energy differences given in Table II for the selected Skyrme models and for various strength of the SPEM parameter C τ (∇ρ) 2 0 . They are small (about 1 MeV) in 132 Sn but rather large in 78 Ni and 100 Sn (5 to 8 MeV). In 100 Sn, the energy differences without SPEM are mostly sensitive to the Coulomb component of the EDF, with a small additional effect due to the effective mass (the larger effective mass, the smaller the energy difference). In 132 Sn, the energy difference is related mostly to the symmetry energy: the larger the symmetry energy, going from SLyIII0.9 to SLyIII0.7 for instance, the larger the energy difference. In addition, the increase of the effective mass also contributes, with a smaller impact, to the decrease the energy difference, as it can be deduced from the comparison of the energy difference for the forces f -, f 0 , and f + which has increasing effective mass in neutron rich matter, see Fig. 1. It can be seen (cf. Table II) that the shifts in the energy differences between the cases without, and with maximal SPEM (C τ (∇ρ) 2 0 =-420 MeV fm 10 ) are almost constant and independent of the models considered. It varies by about 0.3 MeV in 132 Sn and in 78 Ni. From Table II, we can anticipate that the SPEM will have a larger impact on the calculation of the β half-life of 132 Sn and a weaker one in the case of 78 Ni and 100 Sn. In 132 Sn, the experimental value is -1.305 MeV [START_REF] Mezilev | [END_REF]. No data exist for 78 Ni and 100 Sn.

The

E i -E 1 + 1
energies, the B(GT ) - 1 values and β -decay half-lives of 132 Sn and 78 Ni are given in Tables III and IV, the β + -decay properties of 100 Sn in Table V. The evolution of the transition energies and the B(GT ) ± 1 values is reflected in the half-life behaviour, see Eqs. [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitation[END_REF] and (12). As in Table II, the results shown in Tables III, IV and V correspond to the selected interactions with and without the SPEM represented by the value of the parameter C τ (∇ρ) 2 0 . In the case of 132 Sn, the model SLyIII0.7 predicts positive energy differences for the dominant transition of the β-decay half-lives (see Table II), and it leads to half-lives which are much larger than the experimental value, as anticipated.

One can see from Tables III, IV and V that the β-decay half-lives are much more sensitive to the effective mass distribution in the case of the low-Q β nucleus 132 Sn than in 100 Sn and 78 Ni. For 132 Sn, a strong decrease of the half-life can be directly correlated to either the increase of the effective mass in symmetric matter, or to the increase of the SPEM, while in 78 Ni and 100 Sn, the correlation, while still being present, is much less pronounced. This can be easily understood from the energy difference of the most important transition given in Table II: The energy differences are much smaller in the case of 132 Sn than in the case of 100 Sn and 78 Ni, which makes the β-decay half-lives more sensitive to a small modification of the s.p. energies induced by the SPEM. Let us examine whether the SPEM could improve the agreement between the model predictions and the experimental values. As one can see from Tables III, IV and V the inclusion of the terms (8) leads to minor effects on the B(GT ) ± 1 values. We find that the SPEM induces an increase of the transition energies and it results in a decrease of the half-lives. We first concentrate on the models SLyIII0.7, SLyIII0.8 and SLyIII0.9, which correspond to different values of the effective mass in symmetric matter. From the comparison of the theoretical predictions with the experimental half-lives shown in Tables III, IV and V, it is difficult to conclude which model is better: For 132 Sn, the model SLyIII0.8 is preferred, for 78 Ni and 100 Sn, it is SLyIII0.7. Now, if we concentrate on the models f + , f 0 and f -, it is f + which always comes the closest to the experimental value. This indicates that, in addition to the effective mass, the residual interaction is very important. It was already anticipated that the value of the Landau parameter G ′ 0 for the selected models (cf. Table I) could have an impact on charge-exchange related observables. For the models f + , f 0 and f -the values of G ′ 0 are too small. Since the impact of the SPEM on the β-decay rates in 78 Ni and 100 Sn is quite small, these nuclei could be, in the future, used to calibrate the residual interaction almost independently from the profile of the effective mass. The modification of G ′ 0 could be obtained from a refitting of the Skyrme functional with a different value of the strength of the SPEM. One could increase G ′ 0 by about 0.1-0.2 by introducing the spin-density dependent extension of the Skyrme model [49,50]. This will be left for future investigations.

Up to this point, we have mostly focused on the relation between the SPEM and the low energy part of the strength, since it represents the main contribution to the β-decay rates. We now turn to the higher energy part and show in Figs. 2 and3 the effects of the SPEM on the properties of the GQR and GTR in 208 Pb. In the figures, the calculated strength distributions are folded out with a Lorentzian distribution of 1 MeV width. The excitation energies refer to the ground state of the parent nucleus 208 Pb. The arrows indicate the maxima of the strength distributions corresponding to the case of the f + model and C τ (∇ρ) 2 0 =0 MeV fm 10 . Since the isoscalar quadrupole EWSR is changed by only about 1% by the SPEM, we expect the collective modes at higher energy to be only marginally impacted.

The GQR strength distribution consists mostly of a main peak. Comparing the cases without SPEM and with maximal SPEM, we find that the peak is shifted down by about 500 keV. One can notice that the GQR strength distribution is almost identical for the three models f 0 , f + and f -. As can be seen from Table VI, while the 2 + 1 energy is practically unaffected by the SPEM, the B(E2) value is decreasing as the SPEM increases. Some overestimate of the experimental energy indicates that there is room for the two-phonon effects [51].

The GTR is much more fragmented than the GQR, as seen in Fig. 3. The strength distribution is globally shifted up as the isospin splitting is going from positive (f + ) to negative values (f -). As in the case of the GQR, the high energy peaks of the strength distribution are shifted to lower energies (by about 500 keV) as the SPEM gets larger. This is an effect of the slight increase of the level density induced by the SPEM.

IV. CONCLUSIONS

Starting from different Skyrme EDF which predict 132 Sn β-unstable, we have studied the effects of introducing a surface-peaked effective mass on top of existing Skyrme models. The main effect of this additional term is a compression of the s.p. level spacing around the Fermi level, or equivalently, an increase of the level density. This systematically increases the β-decay rates (i.e., decreases the half-lives). The collective modes at higher energy are only slightly impacted by the SPEM.

This work is a first step towards improving Skyrme functionals by adding extra terms to the energy functional. Our motivation is based on both having a better agreement with nuclear data, and also predicting weak transition rates for astrophysical applications. The results of our analysis allow for a better understanding of the effects at play. The β-decay rates in doubly-magic unstable nuclei ( 100,132 Sn, 78 Ni) are indeed very sensitive both to the s.p. energies and residual interactions, and none of the Skyrme models selected in this work are fully satisfactory in this respect. From our analysis, we have however identified two nuclei ( 100 Sn, 78 Ni) where the β-decay half-lives are only weakly impacted by the SPEM. They can be considered as good benchmark nuclei since they potentially offer the possibility to calibrate the residual interaction, with a weak influence of the effective mass. In a complementary approach, 132 Sn could be used to test different strengths of the SPEM, for a fixed residual interaction.

The tensor force has not been considered in this work, although it can affect the neutron-proton s.p. energies in some cases. We have aimed at understanding just the contribution of the SPEM to the β-decay and GT mode in order to disentangle the respective roles of the effective mass and the residual interaction. An additional modification of the Skyrme functional was proposed earlier in order to stabilize the nuclear matter equation of state [49,50]. It has been recently used in nuclei and, since it brings an additional repulsive term to the G ′ 0 Landau parameter, it was shown to shift the centroids of the GT collective mode to higher energies by a few hundreds keV up to 1 MeV [52]. In the future, we plan to explore the predictions of a general mean field model including all these ingredients, and to compare them to known experimental data, as done in this work. These calibration processes are important to set-up boundaries for the additional parameters before making predictions for astrophysical cases.

FIG. 1 :

 1 FIG. 1: (Color online) Panels (a) and (d): Proton and neutron effective masses in 208 Pb as functions of radial distance; panels (b), (c), (e) and (f): effective-mass difference between the results of the HF calculation with the surface peaked terms (C τ (∇ρ) 2 0=-210 MeV fm 10 and -420 MeV fm 10 ), and without. Solid, dashed and dotted lines correspond to HF calculations with f0, f-and f+ forces, respectively.

FIG. 2 :

 2 FIG. 2: (Color online)The quadrupole strength distribution of 208 Pb. Solid, dashed and dotted lines correspond to RPA calculations with f0, f-and f+ models, respectively. The experimental centroid of the GQR is at 10.89±0.30 MeV[46].

FIG. 3 :

 3 FIG.3:(Color online) Same as Fig.2, for the GT strength distribution obtained within the CERPA. The experimental centroids of the GTR is at 19.2 MeV[47,48].

TABLE I :

 I Bulk properties of the selected interactions. + state of 132 Sb). The lowest unperturbed transition energy is ǫ π2d5 2 -ǫ ν2d 3 2 -∆M n-H

	Skyrme	ρ0,sat	E0	K0	Jsym	Lsym	m * s /m	∆m * /m	G ′ 0
		(fm -3 )	(MeV)	(MeV)	(MeV)	(MeV)	(MeV)
	SLyIII0.7 [32]	0.153	-16.33	361.4	31.98	30.78	0.7	0.18	0.30
	SLyIII0.8 [32]	0.153	-16.32	368.8	31.69	28.24	0.8	0.29	0.33
	SLyIII0.9 [32]	0.153	-16.31	374.5	31.44	24.75	0.9	0.38	0.34
	f+ [33]	0.162	-16.04	230.0	32.00	41.53	0.7	0.17	0.08
	f0 [33]	0.162	-16.03	230.0	32.00	42.42	0.7	0	-0.01
	f-[33]	0.162	-16.02	230.0	32.00	43.79	0.7	-0.28	-0.13
	the 1

TABLE II :

 II Energy differences between the dominant s.p. states in 132 Sn and 100 Sn. For each Skyrme parameterization, the energy difference is calculated with the surface peaked term or without (C

			τ (∇ρ) 2 0	= 0). See text for more details.
				132 Sn	100 Sn	78 Ni
	Skyrme	C	τ (∇ρ) 2 0	ǫ π2d 5 2	-	ǫ ν1g 7 2	-	2 ǫ π2p 3	-
				ǫ ν2d 3 2	ǫ π1g 9 2	ǫ ν2p 1 2
		(MeV fm 10 )	(MeV)	(MeV)	(MeV)
	SLyIII0.7		0	0.3		-7.2	-5.1
	SLyIII0.7		-210	0.2		-7.3	-5.2
	SLyIII0.7		-420	0.1		-7.3	-5.3
	SLyIII0.8		0	-0.6		-7.5	-6.2
	SLyIII0.8		-210	-0.7		-7.5	-6.3
	SLyIII0.8		-420	-0.9		-7.6	-6.4
	SLyIII0.9		0	-1.3		-7.7	-7.0
	SLyIII0.9		-210	-1.4		-7.7	-7.2
	SLyIII0.9		-420	-1.6		-7.8	-7.3
	f+		0	-0.6		-5.9	-5.8
	f+		-210	-0.7		-6.0	-5.9
	f+		-420	-0.8		-6.2	-6.0
	f0		0	-0.5		-6.0	-5.6
	f0		-210	-0.6		-6.1	-5.7
	f0		-420	-0.7		-6.2	-5.8
	f-		0	-0.4		-6.2	-5.4
	f-		-210	-0.5		-6.3	-5.5
	f-		-420	-0.6		-6.5	-5.6

TABLE III :

 III SPEM effects on β --decay properties of 132 Sn. Data are from Ref. [43].

	Skyrme	C	τ (∇ρ) 2 0	Ei -E 1 + 1	B(GT ) -1	T 1/2
		(MeV fm 10 )	(MeV)		(s)
	SLyIII0.7		0	0.07	2.6	389400
	SLyIII0.7		-210	0.21	2.7	9840
	SLyIII0.7		-420	0.34	2.7	1930
	SLyIII0.8		0	0.97	2.5	57
	SLyIII0.8		-210	1.11	2.5	33
	SLyIII0.8		-420	1.26	2.6	21
	SLyIII0.9		0	1.70	2.4	6.7
	SLyIII0.9		-210	1.84	2.5	4.7
	SLyIII0.9		-420	2.01	2.6	3.3
	f+		0	1.12	4.6	18
	f+		-210	1.25	4.6	12
	f+		-420	1.36	4.6	8.5
	f0		0	1.14	5.9	13
	f0		-210	1.27	5.8	8.8
	f0		-420	1.37	5.8	6.5
	f-		0	1.12	8.8	6.4
	f-		-210	1.25	8.7	5.0
	f-		-420	1.36	8.6	3.6
	Expt.			1.794±0.009		39.7±0.8

TABLE IV :

 IV SPEM effects on β --decay properties of 78 Ni. Data are from Refs.[44].

	Skyrme	C	τ (∇ρ) 2 0	Ei -E 1 + 1	B(GT ) -1	T 1/2
		(MeV fm 10 ) (MeV)		(s)
	SLyIII0.7		0	5.49	1.0	0.157
	SLyIII0.7		-210	5.61	1.1	0.140
	SLyIII0.7		-420	5.74	1.1	0.121
	SLyIII0.8		0	6.60	1.0	0.057
	SLyIII0.8		-210	6.73	1.0	0.051
	SLyIII0.8		-420	6.87	1.0	0.045
	SLyIII0.9		0	7.48	1.0	0.025
	SLyIII0.9		-210	7.61	1.0	0.023
	SLyIII0.9		-420	7.79	1.0	0.020
	f+		0	6.40	1.9	0.031
	f+		-210	6.51	1.9	0.028
	f+		-420	6.60	1.9	0.027
	f0		0	6.33	2.6	0.020
	f0		-210	6.44	2.5	0.019
	f0		-420	6.52	2.5	0.018
	f-		0	6.33	3.9	0.010
	f-		-210	6.42	3.9	0.009
	f-		-420	6.50	3.8	0.009
	Expt.					0.1222±0.0051

TABLE V :

 V SPEM effects on β + -decay properties of 100 Sn. Data are from Ref.[45].

	Skyrme	C	τ (∇ρ) 2 0	Ei -E 1 + 1	B(GT ) + 1	T 1/2
		(MeV fm 10 )	(MeV)		(s)
	SLyIII0.7		0	4.26	15.2	0.232
	SLyIII0.7		-210	4.38	15.2	0.221
	SLyIII0.7		-420	4.42	15.2	0.213
	SLyIII0.8		0	4.60	15.1	0.178
	SLyIII0.8		-210	4.64	15.1	0.172
	SLyIII0.8		-420	4.67	15.0	0.167
	SLyIII0.9		0	4.86	15.1	0.138
	SLyIII0.9		-210	4.90	15.0	0.134
	SLyIII0.9		-420	4.92	15.0	0.131
	f+		0	3.47	16.3	0.593
	f+		-210	3.62	16.3	0.492
	f+		-420	3.72	16.2	0.433
	f0		0	3.80	16.8	0.381
	f0		-210	3.94	16.8	0.323
	f0		-420	4.04	16.7	0.290
	f-		0	4.38	17.6	0.190
	f-		-210	4.51	17.5	0.168
	f-		-420	4.60	17.5	0.154
	Expt.			3.08±0.34		1.16±0.20

TABLE VI :

 VI SPEM effects on the energy and B(E2)-value for the up-transition to the first 2 + state in 208 Pb. Data are from Ref. [43].

	Skyrme	C	τ (∇ρ) 2 0	Energy	B(E2; 0 + gs → 2 + 1 )
		(MeV fm 10 )	(MeV)	(e 2 fm 4 )
	f+		0	5.12	3130
	f+		-210	5.09	2530
	f+		-420	5.09	2180
	f0		0	5.13	3250
	f0		-210	5.09	2650
	f0		-420	5.09	2310
	f-		0	5.09	3440
	f-		-210	5.06	2850
	f-		-420	5.06	2500
	Expt.			4.09	3000±300

Acknowledgments

A.P.S. and I.N.B. thank the hospitality of IPN-Orsay and IPN-Lyon where a part of this work was done. This work is partly supported by the IN2P3-JINR agreement and the ANR project "SN2NS: supernova explosions, from stellar core-collapse to neutron stars and black holes".

Appendix A: Decomposition of the Skyrme functional into neutron and proton channels Here, the Skyrme functional is expressed in terms of the neutron and proton densities instead of the isoscalar and isovector densities,

where the different terms of the energy density are

and the effective mass factor f Sky q = m/m * q is defined as

By functional derivation the one-body Hamiltonian H q is obtained as,

where the central potential is given by

Here, the central-density potential is given by:

the central-gradient potential by:

and the central-J potential by:

The spin-orbit potential is:

Appendix B: Modification of the mean-field equations induced by the SPEM

The kinetic energy correction induced by the effective mass in Eq. (A2) is now given by f q = f Sky q + f corr q where

and the mean field central potential (A7) reads:

where V Sky q (r) is the mean field deduced from the Skyrme interaction, e.g. Eq. (A7), and V corr (r) is the correction term induced by Eq. ( 8):

V corr (r) = -2C τ (∇ρ) 2 0 τ (r)∇ 2 ρ(r) + ∇τ (r)∇ρ(r) -2C ρ 2 (∇ρ) 2 0 ρ(r)(∇ρ(r)) 2 + ρ(r) 2 ∇ 2 ρ(r) .

(B3)


Marc Fourmigué

Julien Lieffrig

Organizing Radical Species in the Solid State with Halogen Bonding

Keywords: radical, halogen bond, magnetism, spin crossover, chirality, organic conductors, charge transfer, nitronyl, tetrathiafulvalene, dithiolene Contents

The electronic properties (conductivity, magnetism) of radical systems in the solid state essentially depends on (i) the extend of delocalization of the spin density in the molecule and (ii) the intermolecular interactions between radicals. Halogen bond has proven very efficient to engineer such magnetic or conducting structures and recent advances along these lines are reviewed here. Three situations are considered, (i) halogenated radical species acting as halogen bond donors, as found in iodotetrathiafulvalene-based chiral conductors or bilayer systems, and in spin crossover (SCO) complexes with halogenated ligands, (ii) radical species acting as halogen bond acceptors, such as neutral nitronyl species or anionic, mixed-valence dithiolene complexes, interacting with closed-shell halogen bond donors (iodoperfluoro alkanes and arenes, iodo-or bromo pyridinium cations) and finally (iii) charge transfer salts where both halogen bond donor and acceptor are radical species.

Introduction

Radical species are particularly investigated in the solid state for their magnetic (and eventually conductive) properties, which intimately depend on the interaction pathways that establish between them [START_REF] Fourmigué | The importance of π-interactions in crystal engineering[END_REF]2]. In that respect, intermolecular interactions are expected to play two strikingly different roles. They can introduce some control elements on the solid state organization of the radical species, and accordingly contribute to the specific magnetic and/or conductive properties of the material induced by its crystal structure. However, this structural role can be further complemented by an added functionality, i.e. the introduction of a new electronic interaction path for radical interactions or delocalization (electronic control) through the non-bonding interaction, provided some spin density or orbital overlap takes place. Such effects have been reported for hydrogen bonding interactions [START_REF] Otsuka | Crystal structures and magnetic properties of acid-base molecular complexes, (p-pyridyl nitronylnitroxide)2X (X=hydroquinone, fumaric acid and squaric acid)[END_REF][START_REF] Romero | Evidence for Transmission of Ferromagnetic Interactions through Hydrogen Bonds in Alkyne-Substituted Nitroxide Radicals: Magnetostructural Correlations and Polarized Neutron Diffraction Studies[END_REF][START_REF] Maspoch | Radical para-Benzoic Acid Derivatives: Transmission of Ferromagnetic Interactions through Hydrogen Bonds at Long Distances[END_REF] and the question arises if they could be also investigated with halogen bonding interactions, as they have proven to be very good and predictable crystal engineering tools [6,[START_REF] Meyer | Halogen bonding at work: recent applications in synthetic chemistry and materials science[END_REF]. The halogen bond "structural" approach to control the solid state associations of radical species has been investigated in the last twenty years and several reviews are available until 2008 which describes the use of halogen bonding in molecular conductors [START_REF] Fourmigué | Activation of Hydrogen-and Halogen-Bonding Interactions in Tetrathiafulvalene-Based Crystalline Molecular Conductors[END_REF], in conducting or magnetic systems [START_REF] Fourmigué | Halogen bonding in Conducting or Magnetic Molecular Materials[END_REF][START_REF] Metrangolo | Halogen Bonding in Supramolecular Chemistry[END_REF]. In this chapter, we want to complement these reviews by more recent results on this topic, but with a special highlight on those situations where radical interactions or delocalization (electronic control) through the halogen bond were investigated and eventually identified. This contribution will be organized into three main sections, the first one dedicated to radical molecules or complexes acting as halogen bond donors, the second one describing radical molecules or complexes acting as halogen bond acceptors, and finally a third section reporting on charge transfer salts where both halogen bond donor and acceptor are also radical species, a rare situation which provides an opportunity to indeed evaluate the electronic interactions between radical species. Note that in the following, intermolecular distances will be compared with the sum of van der Waals radii, as defined by Bondi [START_REF] Bondi | van der Waals Volumes and Radii[END_REF], with H: 1.20; N: 1.55; O: 1.52; F: 1.47; S: 1.80; Cl: 1.75; Br: 1.85 and I: 1.98 Å.

Radicals as Halogen Bond Donors

In order to act as halogen bond donors, molecules need to be not only in their radical form but also need to bear halogen atoms as substituents. Furthermore, as for any halogen bond interactions (see other chapters in this book), the strongest effects are anticipated with the most polarizable bromine and iodine atoms. Such molecules are actually extremely rare among neutral radicals, either C-centered or N-O centered ones. In the latter, halogen bonding interactions within halogenated neutral nitroxides have been indeed identified [START_REF] Fourmigué | Halogen bonding in Conducting or Magnetic Molecular Materials[END_REF], the main information being that the oxygen atom in N-O• based radical species is able to act as an efficient halogen bond acceptor. Such situation will be detailed in Section 3.1.

Also, many neutral organic radicals are based on heavily chlorinated aryl derivatives (Fig. 1) such as the perchlorophenalenyl or the polychlorotriphenyl radicals [START_REF] Veciana | Stable Radicals[END_REF][START_REF] Koutentis | Perchlorophenalenyl Radical[END_REF]. When described, the X-ray crystal structures of such compounds show inevitably intermolecular Cl•••Cl contacts, without identifying particularly short ones. In perchlorophenalenyl for example, the steric interactions between the Cl atoms in peri position in the molecules leads to a strong distortion from planarity which induces a ruffled molecular structure [START_REF] Koutentis | Perchlorophenalenyl Radical[END_REF]. As a consequence, the shortest intermolecular Cl•••Cl distances within the stacks amount to 3.78 Å, above the sum of van der Waals radii (3.50 Å). The bromo or iodo analogs of such complexes were not described, probably for stability reasons. In that respect, the brominated trioxytriangulene (Fig. 2a) described below in Section 2.1 is an exception which deserves special mention. Besides, the other efficient source of stable radical species acting as halogen bond donors remains the cation radical of iodinated tetrathiafulvalene derivatives (Fig. 2b). It has been shown indeed that these radical species are not only stable, but able to act as efficient halogen bond donors, thanks to the iodine activation by the cationic state of the TTF core. In the last years, the halogen bonding interaction with iodinated TTFs has been purposely used to favor very specific solid state structures, such as chiral, non-centrosymmetric or hexagonal organizations, as detailed in Section 2.2. A third possibility is associated with cationic and paramagnetic coordination complexes, with halogenated organic ligands (Fig. 2c). The solid state organization of such complexes was shown to be controlled to some extend by the halogen bonding interactions, an important issue for the engineering of magnetic properties, particularly when associated with spin crossover (SCO) systems.

Halogenated Trioxytriangulene Radical

Following earlier work on the 6-oxophenalenoxyl neutral radical for its use as cathode-active materials for battery applications [START_REF] Morita | Stable Radicals[END_REF], Morita et al. recently introduced the trioxytriangulene motif (noted TOT in the following), characterized by a SOMO and two degenerate LUMOs, which arise from the topological symmetry of the π-system [START_REF] Morita | Organic tailored batteries materials using stable open-shell molecules with degenerate frontier orbitals[END_REF]. Among these series, the tribromo derivative, noted Br3TOT, forms a one-dimensional columnar structure stabilized by strong π-π stacking. The outside of these columns is covered by bromine and hydrogen atoms susceptible to act as halogen (respect. hydrogen) bond donors, and bromine atoms. The molecule is located on a mirror plane incorporating one bromine, Br(1) and one oxygen O(2) atoms, while Br(2) and O(1) are in general position. As shown in Fig. 3, Br(2) atoms from two coplanar molecules are halogen bonded to the oxygen O(2) atom located on a mirror plane, while the other bromine atom is essentially non-bonded. The Br(2)•••O(2) distance is not very short, that is 95% the sum of van des Waals radii. The other oxygen atom O(1) is engaged in a cyclic R2 2 (10) hydrogen bonded motif with a C-H group. Magnetic properties of this compound were not described, but their comparison with the known tris(tertbutyl) analog would be particularly interesting to investigate the role of the intermolecular interactions taking place between the columns. Another attractive perspective would be the preparation of the triodo analog.

TTF Radical Cations Salts: Toward Chiral Conductors

Halogen-bonded TTF salts were first described by Kato in 1998 [START_REF] Imakubo | Novel radical cation salts of organic π-donors containing iodine atom(s): the first application of strong intermolecular-I•••X-(X = CN, halogen atom) interaction to molecular conductors[END_REF], from the electrocrystallization of EDT-TTF-I in the presence of counter ions acting as halogen bond acceptors, as detailed in Fig. 4. This topic has been reviewed in 2004 [START_REF] Fourmigué | Activation of Hydrogen-and Halogen-Bonding Interactions in Tetrathiafulvalene-Based Crystalline Molecular Conductors[END_REF], and more recent results can be found in more general reviews [START_REF] Fourmigué | Halogen bonding in Conducting or Magnetic Molecular Materials[END_REF][START_REF] Metrangolo | Halogen Bonding in Supramolecular Chemistry[END_REF]. Many different anions were used as halogen bond acceptors, from the simple halides [START_REF] Imakubo | Novel radical cation salts of organic π-donors containing iodine atom(s): the first application of strong intermolecular-I•••X-(X = CN, halogen atom) interaction to molecular conductors[END_REF][START_REF] Imakubo | Supramolecular organic conductors based on diiodo-TTFs and spherical halide ion X-(X = Cl, Br)[END_REF] and polyhalides (I3 -, IBr2 -, ...) [START_REF] Domercq | Hal•••Hal interactions in a series of three isostructural salts of halogenated tetrathiafulvalenes. Contribution of the halogen atoms to the HOMO-HOMO overlap interactions[END_REF], polyhalometallates, polycyanometallates, polythiocyanatometallates [START_REF] Hervé | Iodine Substituted Tetrathiafulvalene Radical Cation Salts with [M(isoq)2(NCS)4] -Anions where M = Cr III , Ga III : Role of I•••S and S•••S Contacts on Structural and Magnetic Properties[END_REF] to cluster anions such as [Re6Se6(CN)6] 4- [START_REF] Ranganathan | Balancing framework densification with charged, halogen-bonded-π-conjugated linkages[END_REF][START_REF] Barrès | The 8:1:1 ternary hybrid framework in the system[END_REF] or [Mo3S7Cl6] 2- [START_REF] Alberola | Halogen bonding interactions with the [Mo3S7Cl6] 2cluster anion in the mixed valence salt [EDT-TTFI2]4[Mo3S7Cl6]•CH3CN[END_REF] or polymeric anionic networks such as [PbI3]∞ - [START_REF] Devic | EDT-TTF-I2)2PbI3•H2O: an ambient pressure metal with a β′ donor slab topology[END_REF], the layered [Pb5/6□1/6I2] -system with a PbI2 structure [START_REF] Devic | Single Crystalline Commensurate Metallic Assemblages of π-slabs and CdI2-Type Layers: Synthesis and Properties of β-(EDT-TTF-I2)2[Pb5/6□1/6I2]3 and β-(EDT-TTF-I2)2[Pb2/3+xAg1/3-2x□xI2]3, x = 0.05[END_REF] or complex cadmium thiocyanate lacunar networks such as [Cd3(NCS)8] 2-and [Cd5(SCN)14] 4- [START_REF] Fourmigué | Anionic Layered Networks Reconstructed from [Cd(SCN)3]∞ -Chains in Pseudo One-Dimensional Conducting Salts of Halogenated Tetrathiafulvalenes[END_REF]. In essentially all cases, the halogen bonding interactions were shown to coexist with the stacking of partially oxidized TTF derivatives, affording highly conducting salts. It was also demonstrated that the halogen atoms (Br, I) directly linked to the TTF core as in EDT-TTFI or EDT-TTFI2 also contribute to the band dispersion as the HOMO of such TTFs exhibits no-zero coefficients on the halogen atoms. Note also that in most cases, the counter ion acting as Lewis based is centrosymmetric, affording halogen bonded systems with high symmetry. An illustrative counter example is obtained with the tetrahedral ClO4 -anion, as detailed below in section 2.2.1.

More recently, the XB interaction has been also used to tentatively favor the formation of chiral conductors. Indeed, as described by Rikken et al. [START_REF] Rikken | Electrical Magnetochiral Anisotropy[END_REF], such conductors are expected to exhibit a weak magneto-chiral effect on the conductivity, which can be tentatively described as an added contribution to the magnetoresistance of a compound, only present in chiral compounds. This effect, first observed in chiral nanotubes [START_REF] Martino | Nonlinear Magnetotransport in Interacting Chiral Nanotubes[END_REF] or in Bismuth twisted wires [START_REF] Krstic | Magneto-chiral anisotropy of the free electron on a helix[END_REF], has been a strong incentive in the domain of molecular conductors where the large chiral pool of organic molecules should allow for an easy functionalization with chiral substituents, either of the TTF core itself [START_REF] Avarvari | Strategies towards chiral molecular conductors[END_REF][START_REF] Griffiths | Synthetic strategies to chiral organosulfur donors related to bis(ethylenedithio)tetrathiafulvalene[END_REF], or within chiral counter ions [START_REF] Galan-Mascaros | A Chiral Ferromagnetic Molecular Metal[END_REF]. This ambitious objective was recently reached by Avarvari et al. [START_REF] Pop | Electrical magnetochiral anisotropy in a bulk chiral molecular conductorNature[END_REF], in a cation radical salt of a chiral BEDT-TTF derivative. In order to further enhance the structuring effect of a localized chiral center on the whole crystalline structures, intermolecular interactions such as hydrogen bonding [START_REF] Griffiths | Synthetic strategies to chiral organosulfur donors related to bis(ethylenedithio)tetrathiafulvalene[END_REF] or halogen bonding can be also considered, as detailed below in sections 2.2.2 and 2.2.3.

IodoTTFs with Non-Centrosymmetric ClO4 -Anion

Both EDT-TTFI2 [START_REF] Shin | Strong Iodine•••Oxygen Interactions in Molecular Conductors Incorporating Sulfonate Anions[END_REF] and tTTF-I [START_REF] Shin | Inter-layer charge disproportionation in the dual-layer organic metal (tTTF-I)2ClO4 with unsymmetrical I•••O halogen bond interactions[END_REF] were reported to form halogen bonded salts with the ClO4 -anion (Fig. 5). In the 1:1 (EDT-TTFI2)(ClO4) salt, the EDT-TTFI2 •+ radical cations are associated two-by-two into face-to face dyads allowing for a strong σ overlap of the frontier orbitals [START_REF] Fourmigué | The importance of π-interactions in crystal engineering[END_REF]. A particularly interesting 2:1 salt was also isolated with tTTF-I, formulated as (tTTF-I)2ClO4, where two oxygen atoms of the ClO4 -anion are engaged in I•••O interactions, each of them however with a crystallographically independent tTTF-I molecule (Fig. 6) [START_REF] Shin | Inter-layer charge disproportionation in the dual-layer organic metal (tTTF-I)2ClO4 with unsymmetrical I•••O halogen bond interactions[END_REF]. This asymmetry induced by the differential halogen bonding interaction is transferred to the whole salt, as each crystallographically independent molecule segregates into two crystallographically independent slabs. Band structure calculations for each of them gives a different Fermi level for an hypothetical common +0.5 charge for both tTTF-I molecules. Since there is only one Fermi level associated with the chemical potential of the salt, a charge transfer has necessarily to occur between both stacks, implying a deviation from a common +0.5 charge for each donor molecule.

IodoTTFs with Halogen Bonding to Chiral Anions

The electrocrystallization of iodoTTF derivatives such as EDT-TTF-I in the presence of chiral counter ions appears as a powerful way to introduce chiral center into halogen-bonded organic conductors, provided one has at hand chiral anionic species able to act as XB acceptors. We have shown that sulfonate anions are indeed efficient XB acceptors and various EDT-TTFI2 salts were reported with 1,5-naphtalene-bis(sulfonate), 2,6-naphtalene-bis(sulfonate) or 2,6-anthracenebis(sulfonate) [START_REF] Shin | Strong Iodine•••Oxygen Interactions in Molecular Conductors Incorporating Sulfonate Anions[END_REF]. Camphorsulfonate is a chiral anion, available both as enantiopure compound and racemic mixture. The electro-crystallization of EDT-TTFI2 in the presence of the D-isomer afforded indeed a 2:1 salt with metallic conductivity and two-dimensional Fermi surface [START_REF] Brezgunova | Combining halogen bonding and chirality in a two-dimensional organic metal (EDT-TTF-I2)2(Dcamphorsulfonate)•H2O[END_REF]. Interestingly, the two crystallographically independent EDT-TTFI2 molecules exhibit a different XB pattern. Molecule A is halogen bonded both to the sulfonate and carbonyl moieties of Dcamphorsulfonate anions while molecule B is only interacting with the oxygen atom of a water molecule (see Fig. 7 for structural XB characteristics). This diffe-rentiation is associated to a departure from the averaged +0.5 charge on the two EDT-TTFI2 molecules. Indeed, based on the comparison of intramolecular bond lengths, the compound is tentatively formulated as [A +0.75 B +0.25 ][Dcamphorsulfonate], i.e. with the most oxidized molecule A involved in the strongest intermolecular halogen-bond interactions, an observation to be correlated to the electrostatic contribution to halogen bond. It should be stressed also that the racemic salt obtained with DLcamphorsulfonate is no isostructural with its enantiopure counterpart [START_REF] Shin | Strong Iodine•••Oxygen Interactions in Molecular Conductors Incorporating Sulfonate Anions[END_REF]. Albeit a mixed-valence 2:1 salt was also obtained, it crystallizes with solvent inclusion, giving rise to a one-dimensional electronic structure and semi-conducting character. Halogen bonding interactions with the EDT-TTFI2 +0.5 molecule is still present with the shortest O•••I distances above 2.81Å, which is 80% the sum of van der Waals radii.

Chiral iodoTTF salts

Another approach to chiral conductors exhibiting halogen bonding in the solid state relies on the preparation of halogenated and chiral tetrathiafulvalene derivatives. Only one example has been reported to date, where the ethylene group of EDT-TTFI2 is replaced by a 1,3-dimethyl-1,3-propylene substituent in Me2PDT-TTFI2 [START_REF] Lieffrig | Toward chiral conductors: combining halogen bonding ability and chirality within a single tetrathiafulvalene molecule[END_REF]. Its electrocrystallization with halide anions afforded a 2:1 mixed valence salt, whose conductivity is limited by the steric constraints imposed by the bulky dimethylpropylene moiety.

Fig. 8 The chiral Me2PDT-TTFI2 and the structural motif observed in its chloride salt.

To conclude this section on chiral conductors, it appears that the introduction of low symmetry counter ions systematically favor the presence of multiple crystallographically independent molecules, with then the possibility of different degrees of charge transfer, and competition with charge localization [START_REF] Lieffrig | Chiral Conducting Salts of Nickel Dithiolene Complexes[END_REF].

Paramagnetic Metal Complexes with Halogenated Ligands

Halopyridine complexes of metal halides

The ability of halometallates [START_REF] Libri | Metal Fluorides Form Strong Hydrogen Bonds and Halogen Bonds: Measuring Interaction Enthalpies and Entropies in Solution[END_REF][START_REF] Espallargas | Noncovalent Interactions under Extreme Conditions: High-Pressure and Low-Temperature Diffraction Studies of the Isostructural Metal-Organic Networks (4-Chloropyridinium)2[CoX4] (X = Cl, Br)[END_REF][START_REF] Awwadi | The Analogy of C-Br•••Br-C, C-Br•••Br-Fe, and Fe-Br•••Br-Fe Contacts: Crystal Structures of (26DAPH)FeBr4 and (26DA35DBPH)2FeBr4[END_REF] (as well as cyano [START_REF] Derossi | Halogen Bonded Supramolecular Assemblies of [Ru(bipy)(CN)4] 2-Anions and N-Methyl-Halopyridinium Cations in the Solid State and in Solution[END_REF][START_REF] Ormond-Prout | Halogen Bonded Supramolecular Assemblies of [Ru(bipy)(CN)4] 2-Anions and N-Methyl-Halopyridinium Cations in the Solid State and in Solution[END_REF] and thiocyanato [START_REF] Hervé | Iodine Substituted Tetrathiafulvalene Radical Cation Salts with [M(isoq)2(NCS)4] -Anions where M = Cr III , Ga III : Role of I•••S and S•••S Contacts on Structural and Magnetic Properties[END_REF] analogs) to act as halogen bond acceptors has been extensively explored [START_REF] Bertani | Halogen bonding in metal-organic-supramolecular networks[END_REF], particularly by Brammer et al., and several series of salts with halo-pyridinium cations as halogen bond donors were reported.

Besides, neutral halopyridines (Fig. 9) were also used with a twin role, as they act as Lewis bases for metal coordination through the nitrogen atom and, simultaneously as halogen bond donor [START_REF] Zordan | M-X•••X'-C Halogen-Bonded Network Formation in MX2(4halopyridine)2 Complexes (M = Pd, Pt; X = Cl, I; X' = Cl, Br, I)[END_REF] On the other hand, the analogous 2,5-dibromopyridine derivatives, formulated as [CuCl2(2,5-Br2-py)2] and [CuBr2(2,5-Br2-py)2], are not isostructural and exhibit antiferro-or ferromagnetic interactions respectively [START_REF] Awwadi | The Electrostatic Nature of Aryl-Bromine-Halide Synthons: The Role of Aryl-Bromine-Halide Synthons in the Crystal Structures of the trans-bis(2-bromopyridine)dihalocopper(II) and trans-bis(3bromopyridine)dihalocopper(II) Complexes[END_REF][START_REF] Awwadi | Copper-halide bonds as magnetic tunnels; structural, magnetic and theoretical studies of trans-bis(2,5dibromopyridine)dihalo copper(II) and trans-bis(2-bromopyridine)dibromo copper(II)[END_REF]. The antiferromagnetic interactions in the copper chloride salt have been attributed, in spite of a relative long Cl•••Cl distance (4.16 Å) to this two-halide exchange pathway, in accordance with a non-negligible spin density on the chloride ions (0.109). The copper bromide salt, [CuCl2(2,5-Br2-py)2], adopts a structure comparable with that of [CuBr2(2-Br-py)2] (Fig. 10). Since the bromide are too far apart from each other, the proposed rationale for the observed ferromagnetic interaction is based on the orthogonal interaction between the very small spin density on the organic bromine (0.005), located along the C-Br bond, and the spin density of the bromide along the Cu-Br bond. Similar ferromagnetic interactions mediated by the halogen bonds were already reported in similar copper chloride and bromide with 2chloro-3-methylpyridine and 2-bromo-3-methylpyridine [START_REF] Herringer | Copper(II) complexes of 2halo-3-methylpyridine: synthesis, structure, and magnetic behaviour of Cu(2-X-3-CH3py)2X′2 [X, X′ = chlorine or bromine; py = pyridine][END_REF].

The structures of trans-[CoCl2(3,5-X2py)2], X = Cl, Br, are characterized (Fig. 11) by short XB interactions which develop in the ab plane, giving rise to original tetrameric motifs [START_REF] Coronado | Effects of halogen bonding in ferromagnetic chains based on Co(II) coordination polymers[END_REF]. Magnetic interactions between cobalt ions are characteristic of linear cobalt chloride chain compounds, albeit the XB interactions limit here the magnetic interactions between the chains. In conclusion of this section, it appears that spin density on the halogen atoms of coordinated halogenated pyridines is extremely weak and that, as a consequence, magnetic interactions through the halogen bond cannot be expected to play an important role.

Spin Crossover cooperativity and halogen bonding

As illustrated above, the XB interactions introduce structural constraints which can play an important role in the solid state organization of magnetic systems. This can be particularly important in spin crossover systems (SCO) where the high spin/low spin (HS/LS) conversion can be turned into an abrupt structural transition in the presence of cooperative intermolecular interactions [START_REF] Halcrow | Structure:function relationships in molecular spin-crossover complexes[END_REF]. Only a few examples of complexes showing the involvement of halogen bonding in SCO have been described to date and involve either Fe II or Fe III complexes (Fig. 12). The Fe II [Fe(phen)2(NCS)2] complex is regarded as the prototypical mononuclear SCO system, between a high temperature HS state (S = 2) to a low temperature LS state (S = 0) [START_REF] Gütlich | Spin-Crossover Materials: Properties and Applications[END_REF]. The observed abrupt transition has been extensively investigated and can be tentatively attributed to the presence of intermolecular π-π interactions. Among the various modifications of the phenanthroline ligand explored to date, the 3-bromophenanthroline (Fig. 12a) afforded a series of solvated iron thiocyanate complexes, formulated as [Fe(3-Br-phen)2(NCS)2]•S (S = 0.5 MeOH, 2 CH2Cl2, 0.5 Me2CO) [START_REF] Naik | Relevance of supramolecular interactions, texture and lattice occupancy in the designer iron(II) spin crossover complexes[END_REF], which exhibit magnetic features atypical of the classic [Fe(phen)2(NCS)2] [51]. In the solid state (Fig. 13), a characteristic halogen bond interaction takes places between the bromine atom and the sulfur atom of the thiocyanate ligand. As already observed with non-coordinated NCS -anion [START_REF] Cauliez | The thiocyanate anion as a polydentate halogen bond acceptor[END_REF], the halogen bond donor points toward the sulfur atom, perpendicular to the C-S bond, with a Br•••S distance at 91% the sum of the rvdW. Another example of SCO system involves the Fe III [Fe(Iqsal)2] + species, as [Ni(dmit)2] -salt (dmit = 1,3-dithiole-3-thione-4,5-dimercapto) (Fig. 12b). This salt could have been described in the Section 4 dedicated to halogen bonded systems between two radical species as the [Ni(dmit)2] -is also paramagnetic, but is shown here in this Section on SCO compounds for completeness. As shown in Fig. 14, a directional I•••S interaction takes place between cationic and anionic moieties [START_REF] Fukuroi | Synergistic Spin Transition between Spin Crossover and Spin-Peierls-like Singlet Formation in the Halogen-Bonded Molecular Hybrid System[END_REF]. The [Ni(dmit)2] -radical anions are organized into face-to-face dyads, in a way that parallel the one already observed with radical cations. A cooperative spin crossover occurs at T½↓ = 150 K, characterized with an abrupt decrease of the χΜ T product. One remarkable feature is that the I•••S distance found at 3.690 (2) at 293 K and 3.634(3) Å at 175 K, above T½↓, actually increases abruptly up to 3.796(3) Å at 130 K, indicating that the halogen-bond interactions disappeared when crossing the SCO process. These observations along with the spin-singlet formation in the [Ni(dmit)2] dimer suggest that the halogen-bond interactions between the cation and the anion compete with the π-stacking interactions between the face-to-face [Ni(dmit)2] anions. The last example with the complex ligand L shown in Fig. 12c [55] exhibits a short Cl•••F interaction between the strongly activated chlorine atom of the triazine ring and one fluorine atom of the pentofluorophenyloxy moiety. The Cl•••F distance varies between, 2.969(4) Å and 3.013(4) Å in the LS and HS phase respectively, both notably shorter than the sum of van der Waals radii (3.22 Å). Together with other intermolecular interactions, this Cl•••F contact is believed to contribute to the strongly cooperative SCO, characterized by an abrupt transition with hysteresis loop. These three rare examples demonstrate that halogen bonding interactions can indeed provide an efficient and powerful tool to modify and eventually control cooperativity in SCO systems. The extreme sensitivity to such complexes to minute modifications of their structural environment makes however difficult to fully rationalize the observed effects.

Radicals as Halogen Bond Acceptors

The association of radical molecules acting as halogen bond acceptors is based on two conditions: (i) the presence of halogen bond donor moieties in the radical molecules and (ii) the availability of strong halogen bond donor molecules. This second point is essentially fulfilled with perfluorinated iodoalkanes and iodoarenes which have been used extensively in crystal engineering [6,[START_REF] Meyer | Halogen bonding at work: recent applications in synthetic chemistry and materials science[END_REF][START_REF] Metrangolo | Halogen Bonding in Supramolecular Chemistry[END_REF], and specifically here also with radical molecules. The first point gives us a way to classify the radical halogen bond acceptors. The interest for the solid state structure of radical molecules in the solid state is associated to their magnetic properties, which are intimately related to the spatial arrangement of the spin carriers in the crystal lattice. The introduction of XB interactions involving radical species might modify the way they interact with each other and accordingly their magnetic and/or transport properties. Three main classes of radical species acting as halogen bond acceptors are identified, and developed in the three following sections, nitronyl radicals, paramagnetic metallic complexes, and metal complexes with noninnocent ligands.

Nitronyl radicals

As shown from theoretical [START_REF] Cimino | Halogen Bonds between 2,2,6,6-Tetramethylpiperidine-N-oxyl Radical and C=HyFzI Species: DFT Calculations of Physicochemical Properties and Comparison with Hydrogen Bonded Adducts[END_REF][START_REF] Zhao | Halogen Bonding or Hydrogen Bonding between 2,2,6,6-Tetramethylpiperidine-noxyl Radical and Trihalomethanes CHX3 (X=CI, Br, I)[END_REF] and temperature-dependent EPR studies [START_REF] Mugnaini | Noncovalent paramagnetic complexes: detection of halogen bonding in solution by ESR spectroscopy[END_REF], nitronyl radicals act as very good halogen bond donors and their cocrystallization with 1,4-diiodotetrafluorobenzene provided the very first example of an halogen bonded TEMPO radical, namely a trimeric entity obtained with 4amino-TEMPO (Fig. 15a) [START_REF] Boubekeur | Self-assembly of nitroxide radicals via halogen bonding directional NO•••I interactions[END_REF]. This work was extended by Micallef to isoindoline nitroxide [START_REF] Hanson | Halogen Bonding between an Isoindoline Nitroxide and 1,4-Diiodotetrafluorobenzene: New Tools and Tectons for Self-Assembling Organic Spin Systems[END_REF][START_REF] Davy | Vapour phase assembly of a halogen bonded complex of an isoindoline nitroxide and 1,2diiodotetrafluorobenzene[END_REF]: a similar trimeric entity was crystallized (Fig. 15b) while EPR spectroscopy performed in solution with pentafluoroiodobenzene indicated that halogen bonding induces indeed an increase in electron density at the nitroxide nitrogen nucleus and an increase in the nitroxide rotational correlation time. With 1,2-diiodotetrafluorobenzene [START_REF] Davy | Vapour phase assembly of a halogen bonded complex of an isoindoline nitroxide and 1,2diiodotetrafluorobenzene[END_REF]62], tetrameric motifs were isolated (Fig. 15c), with the NO group acting as simple or bifurcated halogen bond acceptor, allowing for a modulation of the magnetic properties, when compared with the radicals alone. Nitronyl nitroxyde radicals can also act as good halogen bonds acceptors and can act as bidendate donors, with a spin density delocalized on both NO groups [START_REF] Hosokoshi | Magnetic properties and crystal structures of 2-hydro and 2-halo nitronyl nitroxide radical crystals[END_REF]. Their co-crystallization with 1,4-diiodotetrafluorobenzene or the extended 4,4'-diiodooctafluoro-biphenyl (Fig. 16a,b) afforded indeed magnetic chains with very short I•••O distances, below 80% the sum of van der Waals radii [START_REF] Espallargas | Onedimensional organization of free radicals via halogen bonding[END_REF]. The lack of magnetic exchange through halogen bonds has been confirmed by EPR spectroscopy, and the magnetic interactions rely only on direct weak overlap between the radical species.

Metal complexes with essentially metallic spin density

Paramagnetic metal complexes can act efficiently as halogen bond acceptors, and attractive interactions dominated by electrostatics were indeed described by Brammer et al. in a series of systematic studies on C-X•••X'-M [START_REF] Zordan | Supramolecular Chemistry of Halogens: Complementary Features of Inorganic (M-X) and Organic (C-X') Halogens Applied to M-X•••X'-C Halogen Bond Formation[END_REF][START_REF] Espallargas | Designing Intermolecular Interactions between Halogenated Peripheries of Inorganic and Organic Molecules: Electrostatically Directed M-X⋅⋅⋅X'-C Halogen Bonds[END_REF][START_REF] Brammer | Combining metals with halogen bonds[END_REF]. This work was also recently extended to cyanometallates (C-X•••N≡C-M) as halogen bond donors, in a series of halopyridinium salts of hexacyanometallates [M(CN)6] 3-(M = Cr, Fe, Co) [START_REF] Ormond-Prout | Cyanometallates as Halogen Bond Acceptors[END_REF]. The geometry of interaction of the halogen bond donor (C-X) with the cyanide ligand is either a shorter interaction predominantly with the exo lone pair of the nitrogen atom (C≡N•••X > 145°) or a longer interaction with the C≡N π-bond (C≡N•••X < 105°). These studies provide important structural informations which can be further used when associating such paramagnetic complexes with radical species, particularly cation radical derived from tetrathiafulvalenes [START_REF] Fourmigué | Activation of Hydrogen-and Halogen-Bonding Interactions in Tetrathiafulvalene-Based Crystalline Molecular Conductors[END_REF]. One can cite in this context halometallates such the S = 5/2 Fe III FeCl4 - [START_REF] Nishijo | d-Electron-Induced Negative Magnetoresistance of a π-d Interaction System Based on a Brominated-TTF Donor[END_REF][START_REF] Shirahata | New organic conductors based on dibromo-and diiodo-TSeFs with magnetic and non-magnetic MX4 counter anions (M = Fe, Ga; X = Cl, Br)[END_REF], cyanometallates such as [Cr(CN)6] 3- [START_REF] Thoyon | Conducting Materials Containing Paramagnetic Hexacyanometallate [Cr(CN)6] 3and Iodine Substituted Organic Donor [DIETS[END_REF] or isothiocyanatometallates such as the S = 3/2 Cr III [Cr(isoq)2(NCS)4] -complexes [START_REF] Hervé | Iodine Substituted Tetrathiafulvalene Radical Cation Salts with [M(isoq)2(NCS)4] -Anions where M = Cr III , Ga III : Role of I•••S and S•••S Contacts on Structural and Magnetic Properties[END_REF], all acting as halogen bond acceptors vs. iodinated tetrathiafulvalenium cations. In all reported examples however, the halogen bond was found to provide a very poor channel for super-exchange magnetic interactions between the paramagnetic complex and the S = ½ iodotetrathiafulvalenium cation.

Metal complexes with non-innocent ligands

Dithiolene complexes such as [M(mnt)2] -• and [M(dmit)2] -• (M = Ni, Pd, Pt) are paramagnetic species (S = ½) with a spin density strongly delocalized on the two dithiolene moieties. They have been acting as halogen bond acceptors, either through the CN group in [Ni(mnt)2] -• [START_REF] Nishijo | Molecular metals with ferromagnetic interaction between localized magnetic moments[END_REF][START_REF] Devic | Cyano-Halogen Interactions in[END_REF] or through the sulfur atoms in [Pd(dmit)2] -• , [START_REF] Imakubo | Synthesis and crystal structure of the molecular metal based on iodine-bonded π-donor, (IEDT)[Pd(dmit)2][END_REF] particularly when engaged in salt with iodinated tetrathiafulvalenium cations [START_REF] Fourmigué | Halogen bonding in Conducting or Magnetic Molecular Materials[END_REF]. More recently, Kato et al. investigated mixed valence salts of [Ni(dmit)2] -• with non-magnetic halogenated cations such as N-Me-3,5dihalopyridinium, N-Me-2,5-dihalopyridinium or N-ethyl-4-halothiazolium [START_REF] Kato | Development of π-Electron systems Based on [M(dmit)2] (M = Ni and Pd); dmit: 1,3-dithiole-2-thione-4,5-dithiolate) Anion Radicals[END_REF]. 18), with a localized state in Layer I and a twodimensional closed Fermi surface in layer II. 13 C NMR showed evidence for the coexistence of localized spins (Layer I) and conduction π electrons (layer II) [START_REF] Fujiyama | Suppression of electronic susceptibility in metal-Mott-insulator alternating material (Me-3,5-DIP)[Ni(dmit)[END_REF], while Shubnikov-de Haas and angular-dependent magnetoresistance oscillations clearly show that there exists a two-dimensional Fermi surface in layer II, whose spins are strongly coupled with the localized spins in layer I [START_REF] Hazama | Fermi surface and interlayer transport in the two-dimensional magnetic organic conductor (Me-3,5-DIP)[Ni(dmit)2]2[END_REF]. It is believed that the asymmetric environment mediated by the halogen (and weak hydrogen) bond interactions favors here the formation of such bilayer systems, as already mentioned above in Section 2.2.2 for (tTTF-I)2ClO4 [START_REF] Shin | Inter-layer charge disproportionation in the dual-layer organic metal (tTTF-I)2ClO4 with unsymmetrical I•••O halogen bond interactions[END_REF]. [START_REF] Kosaka | Supramolecular Ni(dmit)2 salts with halopyridinium cations -development of multifunctional molecular conductors with the use of competing supramolecular interactions[END_REF]. Another series was developed with the 2,5-dihalopyridium cations [START_REF] Kusamoto | Bilayer Mott System Based on Ni(dmit)2 (dmit = 1,3-dithiole-2-thione-4,5-dithiolate) Anion Radicals: Two Isostructural Salts Exhibit Contrasting Magnetic Behavior[END_REF], as well as with N-ethyl-4-halothiazolium cations (see Fig. 17) [START_REF] Kusamoto | Utilization of σ-Holes on Sulfur and Halogen Atoms for Supramolecular Cation•••Anion Interactions in Bilayer Ni(dmit)2 Anion Radical Salts[END_REF].

Halogen-Bonded Charge Transfer Salts

This last section is dedicated to systems where both the halogen bond donor and the halogen bond acceptor are radical species. Such a rare situation is actually found in a few charge-transfer salts between electron donor and acceptor molecules. We have detailed above in Section 3 the possibility for radical species to act as halogen bond acceptors. In that respect, it is surprising that radical anion species such as TCNQ -• were not reported until recently to form salts with halogenated cations where XB could be observed. Actually, neither its neutral complex with para-diiodobenzene [START_REF] Allen | Molecular recognition via iodo•••nitro and iodo•••cyano interactions: crystal structures of the 1:1 complexes of 1,4diiodobenzene with 1,4-dinitrobenzene and 7,7,8,8-tetracyanoquinodimethane (TCNQ)[END_REF], nor various pyridinium salts [START_REF] Chen | Organic radical molecular solids based on [(TCNQ)n] -(n = 1 or 2): Syntheses, crystal structures, magnetic properties and DFT analyses[END_REF][START_REF] Liu | Four novel ion-pair compounds consisting of 7,7,8,8tetracyanoquinodimethane (TCNQ) radical: Syntheses, crystal structures and properties[END_REF], exhibit any halogen bond interaction with the nitrile substituents of TCNQ. Perhaps the first mention is to be found in a charge-transfer complex with an iodinated TTF (Fig. 19), formulated as (EDO-TTF-I)2(TCNQ) [START_REF] Iyoda | Syntheses, Structure and Conducting Properties of Halogenated Ethylene-dioxytetrathiafulvalenes[END_REF].

In this compound, a short and directional I•••N interaction is found at 3.07 Å, between the centrosymmetric TCNQ molecule and the EDO-TTF-I electron donor (Fig. 20). A close inspection of the intramolecular bond lengths within both the TTF and TCNQ cores, shows however that an electron transfer didn't take place upon complex formation. In other words, we are in presence of a neutral, so-called charge-transfer complex (and not salt). This behavior finds its origin in the relative redox potentials of both molecules. EDO-TTF-I oxidizes indeed at Eox = +0.47 V vs. SCE, a potential to high when compared with the reduction potential of TCNQ (Ered = +0.17 V). A very similar complex is obtained indeed with EDT-TTF-I (Eox = +0.46 V) [START_REF] Lieffrig | Competition between the C-H•••N Hydrogen Bond and C-I•••N Halogen Bond in TCNQFn (n = 0, 2, 4) Salts with Variable Charge Transfer[END_REF], with a comparable I•••N contact at 3.08 Å. These two examples demonstrate that the electron transfer, and ionic species, is not a necessary condition to observe the halogen bonding interaction between the iodine of the TTF derivative and the nitrile of TCNQ. The question then arises if this interaction is actually strengthened in the salts. For that purpose, TCNQ was replaced by the more oxidizing TCNQF2 (Ered = +0.36 V), affording indeed an ionic salt formulated as (EDT-TTF-I +0.5 )2(TCNQF2 -) with a slightly shorter N•••I interaction at 3.04 Å [START_REF] Lieffrig | Competition between the C-H•••N Hydrogen Bond and C-I•••N Halogen Bond in TCNQFn (n = 0, 2, 4) Salts with Variable Charge Transfer[END_REF]. The situation is however not so simple since moving from TCNQF2 to the even more oxidizing TCNQF4 afforded a 1:1 ionic salt, formulated as (EDT-TTF-I +• )(TCNQF4 -• ), where cations and anions are actually linked by a strong C-H•••N hydrogen bond rather than the expected C-I•••N halogen bond [START_REF] Lieffrig | Competition between the C-H•••N Hydrogen Bond and C-I•••N Halogen Bond in TCNQFn (n = 0, 2, 4) Salts with Variable Charge Transfer[END_REF].

A similar series of compounds was obtained with the diiodo TTF derivative, EDT-TTFI2 [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF][START_REF] Lieffrig | C-I•••NC halogen bonding in two polymorphs of the mixed-valence 2:1 charge-transfer salt (EDT-TTF-I2)2(TCNQF4), with segregated versus alternated stacks[END_REF], where the competition with C-H•••N hydrogen bond cannot take place. In order to establish a correlation between the halogen bond length and the degree of charge transfer, a complete series was crystallized from the less oxidizing TCNQ to the most oxidizing TCNQF4. All four compounds exhibit the very same halogen bonded trimeric motif (EDT-TTFI2)(TCNQFn)(EDT-TTFI2), n = 0, 1, 2, 4, similar to the one shown in Fig. 20. Three important conclusions were drawn from the structural, electronic and spectroscopic investigations of these series (Table 1, Fig. 21 and22):

(i) the degree of electron transfer is directly correlated to the redox difference ∆E = Ered(TCNQFn) -Eox(EDT-TTFI2), with ionic salts even when ∆E is slightly negative, an indication of the coulombic stabilization of ionic forms in the solid state, (ii) the degree of electron transfer can be modified with the temperature, with a neutral-to-ionic conversion observed upon cooling in the intermediate (EDT-TTFI2)2(TCNQF), (iii) the degree of electron transfer is correlated to the length of the I•••N halogen bond. In other words, shorter halogen bonds are found with the most ionic compounds, indicated unambiguously an added electrostatic contribution to the halogen bond interaction. A first very active research period, started at the end of the nineties, and aimed at investigating halogen bonding interactions essentially in organic conductors and radical nitronyl systems, has provided numerous examples of radical systems with solid state structures controlled to a large extend by halogen•••Lewis base interactions. While the contribution of halogen atoms to the band dispersion of organic metals was demonstrated, the preliminary assumptions that this intermolecular interaction could also work as a channel for magnetic interactions were not really fulfilled, essentially because the halogen atoms in radical halogen bond donors most often bear a very small part of the spin density. On the other hand, the predictability of halogen bonding associated to its strong linearity allowed for an efficient structural control of the solid state associations of radical species. We believe that this structural control can still play an important role in the tuning of intermolecular interactions, particularly in systems where cooperativity and delocalization play a crucial role. This is particularly true in molecular conductors and recent examples based on mixed valence salts, halogen bonded to noncentrosymmetric counter ions, provide novel bilayer conducting systems. Similarly, cooperativity effects are particularly involved in the nature of the spin crossover (SCO) between high spin and low spin states. Halogen bonding in the crystal structure of such SCO complexes opens the way to new structural organizations and new behaviors.

Fig. 1

 1 Fig. 1 Examples of heavily chlorinated carbon-based radicals. The crystal structure of the triazahexachlorophenalenyl is not known.

Fig. 2

 2 Fig. 2 Structures of radical, halogen bond donors described in Section 2.

Fig. 3

 3 Fig. 3 Projection view of the crystal structure of Br3TOT, perpendicular to the molecular plane, showing the C-Br(2)•••O(2) halogen bond [Br(2)•••O(2): 3.21 Å; C-Br(2)•••O(2): 156.4°; Br(2)•••O(2)=C: 144.6°] and C-H(2)•••O(1) [H(2)•••O(1): 2.44 Å; C-H(2)•••O(1): 150.7°: H(2)•••O(1)=C: 153.4°] hydrogen bonds

Fig. 4

 4 Fig. 4 Examples of halogen bonded TTF salts.

Fig. 5

 5 Fig. 5 Detail of the 1:1 ClO4 -salts with EDT-TTFI2 (left) and tTTF-I (right) showing the I•••O halogen bonds and the face-to-face interaction between radical species

Fig. 6

 6 Fig. 6 Detail of the two crystallographically independent conducting slabs in (tTTF-I)2ClO4

Fig. 7

 7 Fig. 7 Detail of the asymmetric unit in (EDT-TTFI2)2(D-camphorsulfonate). Halogen bonds with red dotted lines: I(1)•••O(4) 3.02 Å, I(2)•••O(1A) 2.68 Å, I(3)•••O(5) 2.84 Å. Hydrogen bond between water molecule and sulfonate moiety as black dotted line.

  . It is found for example in trans-[MCl2(4-X'py)2] (M = Pd, Pt; X' = Cl, Br) and trans-[PdI2(4-I-py)2]. While no specific magnetic interactions can be expected from these low spin d 8 complexes, other examples were reported incorporating S = ½ ions, in d 7 Co 2+ or d 9 Cu 2+ species.

Fig. 9

 9 Fig. 9 Mono-and dihalopyridines used in the coordination of metal halides (X, X' = Cl, Br, I) Several CuX2 complexes with halogenated pyridine ligands were reported, as in trans-[CuCl2(3-Cl-py)2] [45], or [CuX2(2-Br-py)2] and -[CuX2(3-Br-py)2] [46]. The data indicates that the halogen bond interaction taking place between the coordinated chloro or bromo pyridine are much weaker than those observed in the cationic bromo or chloropyridinium salts, a clear indication of electrostatic activation in the salts. Furthermore, the bromine-halide distances are almost equal to the

Fig. 10

 10 Fig. 10 View of the halogen bonded chains in [CuBr2(2-Br-py)2]. The Br•••Br distance, 3.46 Å, is to be compared with the sum of van der Waals radii, namely 3.70 Å.

Fig. 11

 11 Fig. 11 Views of trans-[CoCl2(3,5-Br2py)2] with, (a) projection along the c axis showing the tetrameric halogen bonded motif and, (b) the CoCl2 chain running along c.

Fig. 12 .

 12 Fig. 12. Metal complexes with halogenated ligands exhibiting spin crossover conversion

Fig. 13

 13 Fig. 13 Br•••S interactions in [Fe(3-Br-phen)2(NCS)2]•2 CH2Cl2

Fig. 14 I

 14 Fig. 14 I•••S interactions in the RT structure of [Fe(Iqsal)2][Ni(dmit)2]•CH3CN• H2O.

Fig. 15

 15 Fig. 15 Trimeric or tetrameric motifs observed in halogen bonded nitronyl radicals

Fig. 16

 16 Fig. 16 Tetrameric motifs observed in halogen bonded nitronyl nitroxyde radicals

Fig. 17

 17 Fig. 17 Anionic dithiolene complexes and their halogenated cations

Fig. 18

 18 Fig. 18 Detail of the crystal structure of (3,5-I2py)[Ni(dmit)2]2 Varying the electrocrystallization conditions or the nature of the halogens in the analogous 3,5-dibromo and 3,5-bromoiodopyridinium salts afforded a series of salts formulated as (3,5-Br2Py)[Ni(dmit)2]2, (3,5-I2py)[Ni(dmit)2]3, α-(3,5-I2py)[Ni(dmit)2]2, β-(3,5-I2py)[Ni(dmit)2]2 and (3,5-BrIpy)[Ni(dmit)2]2 [79]. Another series was developed with the 2,5-dihalopyridium cations [80], as well as with N-ethyl-4-halothiazolium cations (see Fig.17) [81].

Fig. 19

 19 Fig. 19 Structures of halogenated TTFs and different TCNQs involved in charge transfer complexes and salts

Fig. 21

 21 Fig. 21 Evolutions of the I•••N distance with charge and temperature within the (EDT-TTFI2)2(TCNQFn) trimers, n = 0-2


Noémie Villard

Mélanie Pélégrini-Issac

Pierre-François Pradat

Stephane Lehéricy

Habib Benali

Mohamed-Mounir El

Symmetry-based reorientation algorithm for spinal cord 3T MR images

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