X Mougeot

C Bisch

Consistent calculation of the screening and exchange effects in allowed β -transitions

The atomic exchange effect has previously been demonstrated to have a great influence at low energy in the 241 Pu β -transition. The screening effect has been given as a possible explanation for a remaining discrepancy. Improved calculations have been made to consistently evaluate these two atomic effects, compared here to the recent high precision measurements of 241 Pu and 63 Ni β spectra. In this paper, a new screening correction has been defined to account for the spatial extension of the electron wave functions. Excellent overall agreement of about 1% from 500 eV to the endpoint energy has been obtained for both β spectra, which demonstrates that a rather simple β decay model for allowed transitions, including atomic effects within an independent particles model, is sufficient to describe well the current most precise measurements.

I. INTRODUCTION

Beta emission probabilities from neutral atoms are highly influenced at low energy by atomic effects, as demonstrated in [1]. The sudden change of the nuclear charge can induce atomic excitations (shake-up) or internal ionizations (shake-off) because initial and final state orbitals are not strictly orthogonal. But the two major atomic effects are the screening and the exchange effects. The latter arises from the creation of a β electron in a bound orbital of the daughter atom corresponding to one which was occupied in the parent atom. An atomic electron from the bound orbital simultaneously makes a transition to a continuum orbital of the daughter atom. This process cannot be distinguished from the direct decay to a final state containing one continuum electron.

At present, precise knowledge of the shape of energy spectra from β transitions, coupled with well-established uncertainties, are sought by end users from the nuclear power industry, the medical care sector ( [2], [3]) or for ionizing radiation metrology( [4], [5]). These shapes have been little studied since the late 1970s. At that time, the knowledge of the spectral shape was thought to be appropriate. Following this demand, a program to calculate analytically the shape of β spectra for allowed and forbidden unique transitions has already been developed and described elsewhere [START_REF] Mougeot | LSC2010[END_REF]. This work improves it in calculating exchange and screening effects consistently.

A β spectrum is the product of (i ) a weak interaction coupling constant, (ii ) a statistical phase space factor pW q 2 which simply reflects the momentum distribution between the electron and the neutrino, (iii ) the so-called Fermi function F 0 L 0 which corrects for the Coulomb effects, and (iv ) a shape factor C(W ) which contains all the remaining energy dependencies, such as leptonic and nuclear matrix elements or corrections for atomic effects. Thus, following Behrens' formalism used throughout this * xavier.mougeot@cea.fr work [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF]

dN dW ∝ p W q 2 F 0 L 0 C(W ), (1)
with W = 1 + E/m e the total energy of the electron defined from its kinetic energy E and its rest mass m e , the corresponding momentum p = √ W 2 -1, and the neutrino momentum q = W 0 -W where W 0 = 1 + E max /m e . The Fermi function, usually denoted by F (Z, W ), is

F 0 L 0 = α 2 -1 + α 2 1 2p 2 , (2)
where the α κ are the Coulomb amplitudes of the electron radial wave functions, defined in Sec. II B. The parameter κ will be defined in Sec. II. Denoting (J i ,π i ) and (J f ,π f ) the initial and final nuclear spins and parities, β decays are classified according to ∆J = |J i -J f | and π i π f . For allowed and forbidden unique β transitions, the energy dependence of the nuclear matrix elements can be factored out. Introducing L = 1 if ∆J = 0 and L = ∆J otherwise, the shape factor of an allowed (L = 1) or an (L -1) th forbidden unique transition is

C(W ) = (2L -1)! L k=1 λ k p 2(k-1) q 2(L-k) (2k -1)![2(L -k) + 1]! (3) with k = |κ| and λ k = α 2 -k + α 2 k / α 2 -1 + α 2 1 .
These λ k parameters have to be calculated numerically and the procedure is not straightforward. For this reason, a usual but not justified assumption is to set λ k ≡ 1 in classical β spectra calculations, even in the most recent ones [START_REF] Huber | [END_REF].

In our previous study [1], the exchange effect was evaluated through Harston and Pyper's formalism [9] using the analytical electron wave functions given by Rose in [START_REF] Rose | Relativistic electron theory[END_REF]. A remaining discrepancy was highlighted, possibly due to the usual, but overly simple, screening correction [START_REF] Good | [END_REF]. In the present work, we propose a consistent calculation of these two atomic effects. The continuum and bound orbital electron wave functions have been determined consistently using Behrens' formalism [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], resolving numerically the Dirac equation within an independent particle model. The complete process is given in Sec. II, detailing the structure of the atomic potential for an extended nucleus including screened and exchange potentials. Atomic corrections for allowed β transitions are stated in Sec. III, with a new screening correction for the β spectrum. These calculations are compared to recent precise measurements of the 63 Ni and 241 Pu β spectra in Sec. IV.

Consistent with Behrens' formalism, natural units = m e = c = 1 are used throughout this work, except when explicitly specified. All numerical integrations were performed using a three point lagrangian interpolation.

II. ELECTRON WAVE FUNCTIONS

A β -decay changes a neutron into a proton, ejecting an electron and an antineutrino. The relativistic behaviour of the electron within the Coulomb field of the nucleus is described by the Dirac equation. The Coulomb potential is assumed to be scalar, static and spherically symmetric. The electron wave function can then be split into a radial part and an angular part as [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF]

Ψ( r) = S κ f κ (r)χ µ -κ g κ (r)χ µ κ , (4)
with the electron radial wave functions f κ (r) (small component) and g κ (r) (large component), the spin-angular functions χ µ κ , and the sign of κ S κ = κ/|κ| = κ/k. κ is the eigenvalue of the operator K = β( σ • L + 1) which appears by applying the theory of angular momentum to an electron in a Coulomb central field: β is the Dirac matrix, σ designates the Pauli matrices σ x,y,z and L is the orbital angular momentum operator. The usual spinangular functions χ µ κ are expanded into the orthonormal basis of the spherical harmonics

Y µ-m l χ µ κ = ı l m C(l1/2j; µ -m, m)Y µ-m l χ m , (5)
with the Clebsch-Gordan coefficients C(...) and the twocomponent spin eigenfunctions χ m . l = k and j = l -1/2 if κ > 0, or l = k -1 and j = l + 1/2 if κ < 0. Notice that the factor ı l in this definition of χ µ κ is specific to Behrens' formalism.

The Dirac equation therefore leads to the following system of coupled differential equations

         df κ dr = (κ -1) r f κ -[W -1 -V (r)]g κ dg κ dr = [W + 1 -V (r)]f κ - (κ + 1) r g κ (6)
with V (r) a central potential. Beyond free states, analytical solutions for f κ and g κ exist only for pure Coulomb potential, i.e. V (r) = -αZ/r. Here, Z is the atomic number and α the fine structure constant. These solutions can be found in [START_REF] Rose | Relativistic electron theory[END_REF].

Even a slightly more complex description of the nucleus requires a numerical solution. We have followed the method described in detail in [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], where the electron radial wave functions are expressed locally as power series expansions.

A. Atomic potential

The structure of the atomic potential is essential in the determination of the electron wave functions. Its construction is detailed here with our choices regarding the β spectra calculation in which we are interested.

Using Behrens' method [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], a power series expansion of the potential is also needed. Eqs. ( 6) exhibit two singular points, r = 0 and r = ∞. Assuming a local C ∞ function, the atomic potential can be expressed locally as a Taylor series, namely, with r 0 denoting an ordinary point,

V (r) = ∞ m=0 v m (0)r m , r ∼ 0 (7) V (r) = ∞ m=1 v m (∞)r -m , r ∼ ∞ (8) rV (r) = ∞ m=0 ṽm (r 0 )(r -r 0 ) m , r ∼ r 0 (9)
Notice that in β decay the initial system is a neutral atom and the final system is an ion plus an electron at infinity. The emitted electron thus feels the potential of one elementary charge. The general structure is the same as depicted in [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], but the required information to perform the calculation is very poorly documented in the literature. We give below all the necessary equations and our choices for the few parameters of this model.

Construction

To take into account the finite nuclear size effect in β spectra calculations, the nucleus is decribed as a uniformly charged sphere with nuclear radius R, leading to the usual quadratic behaviour of the potential inside the nucleus. Further, the influence of the atomic electrons has to be taken into account, e.g. through a screened potential. A simple model for a neutral atom has been adopted, well adapted to β decay [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF] V

(r) = - αZ r N i=1 a i e -βir = - αZ r φ(r), (10)
where a i and β i are fitted parameters for the considered atom. These parameters have been tabulated many times in the literature. The ones from [12] were chosen in this study because of their completeness and their assumption of independent particles. Using N = 3 is sufficient to represent a screened potential with good accuracy in most applications.

The structure of the atomic potential has to be consistent with the one described by Eqs. [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF] to (9) where the space is divided into three parts, and a quadratic behaviour inside the nucleus is required. This potential has to be continuous in any point. A point R 2 , sufficiently far away to ensure the convergence of the asymptotic solutions, was introduced. To deal with the exponential part in Eq. [START_REF] Rose | Relativistic electron theory[END_REF], a new space region has been defined by introducing a radius R 3 . Within R 3 ≤ r ≤ R 2 , the remaining charge, which comes from the screened potential, is spread substituting the exponential by a second order polynomial that ensures continuity at R 3 and R 2 .

For evaluating the bound states, we have also accounted for the fermion nature of the electrons, which is characterized by exchange symmetry within the manyparticle description. An exchange potential, adopted here, has already been introduced by Slater and depends only on the electron charge density by applying the free electron gas assumption [13]. Following Behrens' formalism, this exchange potential is

V ex (r) = K F ex φ 1/3 ex (r) with (11)
F ex = 3α 2 3Z 4π 2 1/3 , φ ex (r) = N i=1 a i β 2 i r e -βir . (12)
Some Hartree-Fock calculations include a parametrisation of Slater's exchange potential, the convergence process being performed also on these parameters [14]. This possibility has been added to our exchange potential through the adjustment of the constant K (see Sec. II B).

Notice that the charge density does not depend upon the atomic orbitals, which might be expected since each orbital does not have the same number of electrons and the same mean radius. The exchange potential can be therefore ajusted for each orbital.

To simplify further notations, we introduce V ex and φ as the first derivatives of V ex and φ, and V ex and φ as their second derivatives. All these considerations have led us to define the following atomic potential

V (r) =                                          t 1 + t 2 2 r 2 , 0 ≤ r ≤ R - αZ r Cφ(r) + δ + C αZ rV ex + δ ex , R ≤ r ≤ R 3 - αZ r 1 -C + C σt 2 3 + µt 3 3 + νt 4 3 , R 3 ≤ r ≤ R 2 - αZ r (1 -C), R 2 ≤ r ≤ ∞ (13)
where the following parameters have been defined

C = Z -1 Z , δ ex = - CR αZ [2V ex (R) + RV ex (R)] , η = i a i 1 -(1 + β i R)e -βiR , δ = 1 -C + Cη, t 1 = - 3αZ 2R 1 + 2 3 CRφ (R) + 2 3 δ ex , t 2 = αZ R 3 1 - 2CR αZ V ex (R) , t 3 = r -R 2 R 3 -R 2 , σ = D 2 2 -3D 1 + 6D 0 , µ = -D 2 + 5D 1 -8D 0 , ν = D 2 2 -2D 1 + 3D 0 , D 0 = φ(R 3 ) + η + R 3 αZ V ex (R 3 ) + δ ex C , D 1 R 3 -R 2 = φ (R 3 ) + V ex (R 3 ) + R 3 V ex (R 3 ) αZ , D 2 (R 3 -R 2 ) 2 = φ (R 3 ) + 2V ex (R 3 ) + R 3 V ex (R 3 ) αZ . (14)
The first derivative of V (r), also continuous, and the second derivative, have been established by straightforward calculations. The potential for continuum states is simply obtained by setting all the exchange terms to zero in Eqs. (13) and (14).

The parameters R 2 and R 3 influence the shape of the atomic potential. For r ≥ R 2 , an electron in a continuum state feels the potential of a unique charge. Thus, R 2 is a kind of atomic radius. The authors of [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF] do not provide any specific criterion to set R 2 and R 3 . Being inspired by [15], these parameters have been set to R 2 = 350(W/p) and R 3 = 150(W/p). No convergence problem of the asymptotic solutions was noticed in our study.

Validity of the asymptotic solutions of the bound states, beyond R 2 , can be closer than for the continuum states when considering inner orbitals. To speed up the calculation, these parameters have been set differently for the bound states. The radius R 2 has been determined as the smallest radius for which the asymptotic solutions of the considered orbital converges. The radius R 3 has been empirically set to R 3 ∼ R 2 /2 in order to smooth the reconnection to the asymptotic potential.

Usable form for calculation

Particular attention has to be paid when going from the potential of Eq. ( 10) to the potential defined in Eqs. [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF] to (9). For the continuum states, the atomic potential with screened potential but no exchange potential is needed. The following helpful relation was used

e -βr = e -βr0 ∞ m=0 1 m! (-β) m (r -r 0 ) m . ( 15
)
The v m coefficients have been established within the different space regions, in 0

≤ r ≤ R V (r) = 2 m=0 v m r m with v 0 = t 1 , v 1 = 0, v 2 = t 2 2 , ( 16
) in R ≤ r ≤ R 3 rV (r) = ∞ m=0 v m (r -r 0 ) m with v 0 = (-αZ) δ + C i a i e -βir0 , v m = (-αZ)C i a i e -βir0 (-β i ) m m! ∀m > 0, ( 17
) in R 3 ≤ r ≤ R 2 rV (r) = 4 m=0 v m (r -r 0 ) m with v 0 = (-αZ)[1 -C + CU 0 ], v m = (-αZ)CU m ∀m > 0, (18)
where

R d = r 0 -R 2 , t 4 = R d R 3 -R 2 , U 0 = σt 2 4 +µt 3 4 +νt 4 4 , U 1 R d = 2σt 2 4 + 3µt 3 4 + 4νt 4 4 , U 2 R 2 d = σt 2 4 + 3µt 3 4 + 6νt 4 4 , U 3 R 3 d = µt 3 4 + 4νt 4 4 , U 4 R 4 d = νt 4 4 , and in R 2 ≤ r ≤ ∞ V (r) = v 1 r -1 with v 1 = -α. (19)
For the bound states, the parameter

v 1 is identical in [R 2 , ∞[. The v m coefficients within [0, R] and [R 3 , R 2 ]
remain the same provided that the parameters t 1 , t 2 , D 0 , D 1 and D 2 are evaluated by taking the exchange potential into account.

In [R, R 3 ], a specific treatment is required because of the r -1/3 behaviour. In this study, an exact evaluation of the potential was implemented. Eq. ( 9) is just the Taylor series of any function f (r) in the vinicity of r 0 . Thus we can define

∞ m=0 f (m) (r 0 ) m! (r -r 0 ) m = ∞ m=0 ṽm (r 0 )(r -r 0 ) m . ( 20
)
The three first terms of this series are trivial. The residual can be calculated exactly since the exact values of the potential and its derivatives are known

Res = rV (r) - 2 m=0 f (m) (r 0 ) m! (r -r 0 ) m . (21)
A final third order term is therefore added, identifying Res ≡ v 3 (r 0 )(r -r 0 ) 3 . This cubic term of the series is not exact, but the calculated potential is exact and the polynomial exhibits an appropriate behaviour up to at least m = 2. Of course, this requires a dense enough r grid to have a small residual. In practice, this residual is about 10 -4 % and always less than 0.1%.

B. Radial wave functions

In Behrens' method [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], the electron radial wave functions are expressed locally as power series expansions, allowing Eqs. ( 6) to be solved using recurrence relations. Special treatment is needed for the solutions relative to the regular singular point r = 0, and for the asymptotic solutions relative to the irregular singular point r = ∞. Then, the calculation consists in evaluating the wave functions near r = 0, near r = ∞, and step by step between these two points in order to reconnect each solution with the appropriate renormalisation and phase shift. Coulomb amplitudes α κ are given, and defined, by the reconnection of the solutions near r = 0 with the ordinary solutions. They are linked to the values of the radial wave functions at the nuclear radius. Similarly, the phase shifts are given by the reconnection of the asymptotic solutions with the ordinary solutions.

In order to be sure of our calculations for the continuum wave functions, the tables from [START_REF] Behrens | New Series, Group I[END_REF] of various parameters used to calculate β spectra were recalculated. Unscreened Coulomb functions (Table II in [START_REF] Behrens | New Series, Group I[END_REF]) were perfectly reproduced, and the ratios of screened to unscreened Coulomb functions (Table III in [START_REF] Behrens | New Series, Group I[END_REF]) were in very good agreement for all Z. Incidently, this allows us to calculate these parameters at any desired energy, especially the λ k parameters involved in the shape factors of Eq. ( 3), avoiding any interpolation in these tables.

For the bound wave functions, the procedure is basically the same but the orbital energy T is not known in advance. An iterative procedure is needed which has to start with a relatively good initial energy to ensure a quick convergence toward the actual energy.

Behrens and Bühring have proposed a Newton iteration method [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], used in this study. Starting from the analytical energy T 0 given in [START_REF] Rose | Relativistic electron theory[END_REF], the zeros of the following estimator have to be found

D(T ) = g R (r M )f L (r M ) -g L (r M )f R (r M ) (22)
with r M = γ(γ + n )/αZ, using the recurrence relation

T n+1 = T n -D(T )/ Ḋ(T ), (23)
with Ḋ(T ) the first derivative of this estimator. Our calculation stops when |T n -T n-1 | < 10 -13 . This procedure is not sufficient because the power series expansions of the wave functions do not depend explicitly on the main quantum number n, but only on κ.

Thus, for the same value of κ, several convergence energies are possible. The number of nodes of f κ and g κ have therefore to be checked to select the desired orbital.

The orbital energy determines the oscillation frequency of the radial wave functions. As an overlap between the bound and continuum wave functions is needed for the exchange effect, good accuracy of the orbital energy is required. Significant discrepancies have been found for low lying orbitals using the procedure detailed just above. An additional procedure has thus been implemented in order to ensure good convergence. The one electron energies E Des of Desclaux in [START_REF] Desclaux | [END_REF] have been choosen for their completeness and for the quality of the calculations. The K constant parameter of the exchange potential in Eq. ( 11) is adujsted by dichotomy inside an empirical interval K min = -2 and K max = 2. The convergence is reached in about 10 iterations, when the orbital energy is close to E Des at the 0.01% level, with a minimum of 0.1 eV. Finally, notice that this method is generic in the sense that the tabulated energies from [START_REF] Desclaux | [END_REF] can be changed to experimental ones, for instance.

To verify our calculations, the parameters of Table V from [START_REF] Behrens | New Series, Group I[END_REF] for the bound wave functions, useful in electron capture calculations, were recalculated. Good agreement for all Z has been obtained. The Coulomb amplitudes of the bound wave functions given in Table IX by Bambynek et al. [18] were also recalculated and good agreement has also been achieved.

III. ATOMIC EFFECTS ON β SPECTRA A. Exchange effect

The formalism of the exchange effect has already been set out in detail in [19]. For allowed transitions, only β electrons created in an s bound orbital of the daughter atom can take part. This effect is expressed as a correction factor 1 + η T ex (E) which modifies the β spectrum defined in Eq. ( 1). The total exchange factor η T ex (E) depends on the β electron energy E and involves an overlap between the electron radial wave functions of the bound and continuum orbitals. It can be written in terms of the subshell contributions

η ns ex η T ex (E) = n η ns ex (E) + m,n (m =n) µ m µ n , (24)
and the exchange factor of the n-th subshell is given by

η ns ex (E) = f µ 2 n -2µ n . ( 25
)
The parameters µ n and f are expressed using the bound (f b n,κ , g b n,κ ) and continuum(f c κ , g c κ ) electron radial wave functions evaluated at the nuclear radius according to

µ n = Es |ns g b n,κ (R) g c κ (R) , f = g c κ (R) 2 g c κ (R) 2 + f c κ (R) 2 . ( 26
)
For s orbitals, κ = -1. Es |ns is the overlap between the initial bound orbital of the parent atom |ns and the final continuum orbital of the daughter atom Es | over the entire space.

In contrast with our previous study [1] and the study in [9], here the screened electron wave functions for both the continuum and bound states have been determined in a complete and consistent manner.

B. A new screening correction

When calculating a β spectrum, the screening effect is generally corrected for using a constant Thomas-Fermi potential which is subtracted from the total energy of the particle [START_REF] Good | [END_REF]. This method creates a non-physical discontinuity at the minimum energy defined by the Thomas-Fermi potential [START_REF] Mougeot | LSC2010[END_REF]. This minimum energy is 20 keV, hence the discontinuity does not generally affect the practical use of the spectrum. Physically, the influence of the atomic electrons is expected to be significant when the β wavelength is comparable to the atom size, i.e. at low energy. As it has been shown in [1], this simple method, which allows analytical wave functions to be used, cannot be reasonable for high Z and at low energy.

Beta spectra are classically calculated using the electron radial wave functions evaluated at the nuclear radius. To take the screening into account, it is not just sufficient to renormalise the wave functions and evaluate them at the nuclear radius because the screened potentials are very weak at this distance, and so, the modification is completely negligible over the entire range of the spectrum. For example, in the case of 241 Pu and using the parameters given in [12], one can evaluate [1 -φ(R)] = 7.24 • 10 -6 .

It is, in fact, necessary to take into account the spatial extension of the wave functions. As described in [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF], this can be done by evaluating the transition matrix of the corresponding β decay. However the calculations become very complicated because the electron wave functions are coupled with the nuclear matrix elements involved. To avoid this difficulty, we can benefit from the parameter f defined in Eq. ( 26) for the exchange effect. This parameter represents the fraction of β electrons which emerge in a continuum κ state if the exchange process is omitted. Instead of squared wave functions evaluated at the nucleus surface, the mean value of the squared radial wave functions over the entire space has been used

g 2 κ = 1 ∆R ∆R g 2 κ (r)dr, ( 27
)
where ∆R is the effective integration interval. The correction factor that takes the screening into account is

C sc = 1 - ∆R unsc ∆R sc • 1 - f sc f unsc , ( 28
)
where the subscript sc (unsc) means that the screened (unscreened) wave functions were used. In this way, the screening correction is physically meaningful because of the definition of the Fermi function in Eq. ( 2).

C. Numerical procedure

The calculation of the exchange effect is performed in two stages. First, the bound wave functions have to be calculated and stored. The minimal sampling that corresponds to the adaptive step given in [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF] is used for the r grid. The continuum wave functions are calculated at the desired energies in the final β spectrum, with their own adaptive r grids. Next, the overlap between the bound and continuum wave functions can be determined, and thus the contribution of each orbital and the total correction factor for the exchange effect.

To perform the calculation of the overlap, the bound and continuum r grids are concatenated and the wave functions are evaluated at these new points. However, if the common grid is not sufficiently dense, numerical fluctuations appear in the overlap calculation, and thus in the final β spectrum. In practice, we observed that a refined grid with just one more point between each point of the common grid is sufficient to ensure good convergence with a precision 10 -3 %.

IV. RESULTS

Accurate calculations, based on inclusion of the various effects covered above, have been performed for the allowed transition of 63 Ni and the first forbidden nonunique transition of 241 Pu. Radiative corrections have also been included but their influence on the β spectrum is negligible, as shown in [START_REF] Mougeot | LSC2010[END_REF]. 63 Ni decays entirely by β -emission to the ground state of 63 Cu. This allowed transition has an endpoint energy of E max = 66.980 (15) keV [START_REF] Bé | Table of Radionuclides -Monographie BIPM-5[END_REF]. 241 Pu decays mainly by β -emission to the ground state of 241 Am. This first forbidden non-unique transition, with E max = 20.8 (2) keV [START_REF] Bé | Table of Radionuclides -Monographie BIPM-5[END_REF], fulfills the assumption of the ξ approximation and can be calculated as allowed [START_REF] Schopper | Weak interactions and nuclear beta decay[END_REF], as it has been proved in [1].

The low maximum energies of these two transitions make them ideal cases for evaluating the influence of the atomic effects. The β spectra of 63 Ni [START_REF] Le-Bret | [END_REF] and 241 Pu [START_REF] Loidl | [END_REF] were recently measured using metallic magnetic calorimeters. Each source was enclosed in a gold absorber assuring a 4π solid angle and 100% detection efficiency. These spectra are first compared to classical β calculations, and then taking into account the screening and exchange effects. The theoretical spectra were normalized to the data by integration from 500 eV to E max for both 63 Ni and 241 Pu decays, for being as much as possible independent of the statistical fluctuations of the measurement.

A. A simple statistical analysis

To evaluate the global quality of our theoretical spectra, some simple elements of statistical analysis are proposed. For a set of n measurements {y i }, the fit quality of a model, leading to n predictions {y th i }, with respect to the data is given by the quantity

R 2 = 1 - var(ê i ) var(y i ) . ( 29
)
Valid for n 1, this definition is general, without any restriction on the type of model. This quantity falls in 0 ≤ R 2 ≤ 1, where R 2 = 1 corresponds to perfect predictions. The modeling error is êi = y i -y th i . Non biased estimators of the variances are

var(y i ) = n i=1 (y i -y) 2 n -1 , var(ê i ) = n i=1 (ê i ) 2 n -p -1 (30)
with y the simple mean of the measurements, and p the number of parameters of the model. For β spectra, p = 2 was considered, namely the maximum energy of the transition and the normalisation of the theoretical spectrum to the data. Therefore, the coefficient (1 -R 2 ) quantifies the general disagreement between the measurements and the predictions.

For the end user, this criterion does not provide any information about the uncertainty of the emission probability at a given energy. We propose first to look at the distribution of the standardized residuals

r i = êi var(ê i ) . ( 31
)
If these residuals are almost equally distributed around zero, it is appropriate to assume that they are distributed according to a Gaussian probability variance var(r i ). The standard deviation σ ri = var(r i ) then gives an estimate of the overall uncertainty of the calculated spectrum in the energy range where the measurements exist. Notice that this analysis is not relevant if the standardized residuals are not equally distributed.

Given the energy thresholds of the measured β spectra considered in this study, these statistical parameters have been calculated from 500 eV to the maximum energy of the transition for both 63 Ni and 241 Pu.

B. 63 Ni beta spectrum Fig. 1 presents the total exchange factor η tot for the 63 Ni decay, with the contribution of each orbital η ns . These results are consistent with those calculated in the framework described in [1] where relativistic analytical wave functions and relativistic effective nuclear charges had been used. Both in this previous framework and the present one, a partial negative contribution of one orbital was found, creating a maximum in the total exchange factor of 10.7% at 500 eV for this study.

Fig. 2 compares the experimental spectrum with three theoretical predictions: an analytical β calculation, without screening and exchange; a numerical calculation from this work, with screening but without exchange; a complete numerical calculation with screening and exchange. Notice that the endpoint energy from the measured spectrum is E max = 67.176 (173) keV [START_REF] Le-Bret | [END_REF]. This value disagrees with the lastest evaluated one [START_REF] Bé | Table of Radionuclides -Monographie BIPM-5[END_REF]. The latter value comes from mass measurements and is therefore well known. The value from [START_REF] Le-Bret | [END_REF] probably points out some small non linear effects of about 0.1% that were not taken into account within the data analysis. Nevertheless, this spectrum is the only one available of such high quality at low energy. These two endpoint energies were tested and the results did not change significantly. Eventually, an excellent agreement has been obtained with the experimental spectrum if the screening and exchange effects are taken into account. The standardized residuals are equally distributed around a mean value of r = 0.093%. The standard deviation of the residuals is σ ri = 1.03% and the global disagreement is (1 -R 2 ) = 0.028%. The discrepancy between 500 eV and the energy threshold of the measurement at 200 eV comes directly from the influence of the 3s orbital in our calculation, but it is difficult to know if it is due to the experiment or to the model. Mean energies are also interesting quantities: E an = 17.45 keV for the analytical calculation; E sc = 17.40 keV for the numerical calculation with screening; E scex = 17.14 keV for the complete calculation. As expected, the screening effect is weak for this low Z nucleus, but the exchange effect is of importance, as is confirmed experimentally.

FIG. 2. (Color online) Measured 63

Ni β spectrum compared to analytical calculation (green), numerical calculation with screening (red), and numerical calculation with screening and exchange (blue). Normalization to the data was done by integration from 500 eV to Emax. Standardized residuals between the data and the most complete calculation are also given. A zoom on the low energy region is also given.

C. 241 Pu beta spectrum Fig. 3 presents the total exchange factor η tot for 241 Pu decay, with the contribution of each orbital η ns . These results are consistent with those calculated in the framework described in [1]. The total exchange factor is more pronounced in this study, with a higher contribution of the inner orbitals. Fig. 4 compares the experimental spectrum with the same three theoretical predictions as for the 63 Ni spectrum. The fall of the experimental spectrum between 500 eV and the energy threshold of the measurement at 300 eV is most likely due to the data analysis. Excellent agreement has been obtained with the experimental spectrum if the screening and exchange effects are taken into account. The standardized residuals are equally distributed around a mean value of r = 0.0019%. The global disagreement is (1 -R 2 ) = 0.040% and the standard deviation of the residuals is σ ri = 0.99%. For the mean energies, we find E an = 5.24 keV, E sc = 5.18 keV and E scex = 5.03 keV. As expected, the screening effect is strong for this high Z nucleus, equivalent to the magnitude of the exchange effect, as is confirmed experimentally.

V. CONCLUSION

A consistent calculation of the screening and exchange effects in allowed β -transitions has been given in detail. The exchange effect has been determined with Harston and Pyper's formalism [9], but using more precise relativistic electron wave functions calculated within Behrens' formalism [START_REF] Behrens | Electron radial wave functions and nuclear beta decay[END_REF]. Special care has been taken to explain the details of these calculations because of the lack of information available in the literature.

This work has demonstrated that a rather simple calculation within an independent particles model is sufficient to obtain a fine understanding of the atomic effects that occur in β decays. A new screening correction that accounts for the spatial extension of the electron wave functions has had to be defined to reach the required precision due to the high quality of the measurements used for comparison. Excellent overall agreement of about 1% from 500 eV to the endpoint has been obtained for both the β spectra of 63 Ni and 241 Pu decays. Even though more complex calculations, such as multiconfigurational Dirac-Fock ones, can take into account chemical effects through ionised atomic states, they are not deemed nec-essary due to the current precision of β spectra measurements.

Finally, this work allows the exact calculation of the leptonic matrix elements within Behrens' formalism. The obvious next step, on which we are currently working, is to generalize the screening and exchange corrections to forbidden unique transitions, which can be performed without taking into account the nuclear matrix elements. A longer term goal is the generalization to forbidden nonunique transitions, which will inherently require the calculation of the nuclear matrix elements. All these new calculations will be compared to existing and available new measurements. (Color online) Measured 241 Pu β spectrum compared to analytical calculation (green), numerical calculation with screening (red), and numerical calculation with screening and exchange (blue). Normalization to the data was done by integration from 500 eV to Emax. Standardized residuals between the data and the most complete calculation are also given. A zoom on the low energy region is also given.

  FIG. 1.(Color online) Total exchange factor η tot for the 63 Ni decay, with the contribution of each orbital η ns .

  FIG. 3.(Color online) Total exchange factor η tot for the 241 Pu decay, with the contribution of each orbital η ns .

  FIG. 4.(Color online) Measured 241 Pu β spectrum compared to analytical calculation (green), numerical calculation with screening (red), and numerical calculation with screening and exchange (blue). Normalization to the data was done by integration from 500 eV to Emax. Standardized residuals between the data and the most complete calculation are also given. A zoom on the low energy region is also given.

ACKNOWLEDGMENTS

We warmly thank M. Loidl (CEA-Saclay, LNHB, France) for providing us the measured β spectra of 63 Ni and 241 Pu, B. Najjari (IPHC, Strasbourg, France) for fruitful discussions, and the referee for helping us to clarify the presentation of this work.


Sa Yao

Clément Falaise

Nathalie Leclerc

Catherine Roch-Marchal

Mohamed Haouas
email: mohamed.haouas@uvsq.fr

Emmanuel Cadot

Improvement of the Hydrolytic Stability of the Keggin Molybdo-and Tungsto-phosphate Anions by Cyclodextrins

Keggin-type molybdo-and tungsto-phosphate polyoxoanions are among the most popular polyoxometalates (POMs), but suffer from their limited stability at low pH in aqueous solution. Their superchaotropic properties generate strong supramolecular complexes with cyclodextrins (CDs) which significantly affect the hydrolytic stability of POM. This chaotropically-driven stabilization effect was systematically monitored by 31 P NMR spectroscopy covering a wide range of pH (from 0 to 8) and varying the nature of the CD (a-, b-, and g-form).

A shift of ca. two pH units of the stability domains of these POMs were found in the presence of two eq. g-CD compared to pure water, leading to keep intact the PW12O40 3-anion without any decomposition up to pH 3.5 (versus 1.5 in pure water) and pH 2.5 for PMo12O40 3-which begins to decompose even at pH 0 in pure water. The effect of the smaller CDs (a-and b-form) is much less pronounced (only 0.5 pH units shift of stability domain) confirming the importance of host-guest size matching to form a sandwich-type inclusion complex, and thus protect the POM structure against basic hydrolysis. Such complexation was further supported by 183 W and 1 H NMR spectroscopy. Finally, using quantitative 31 P NMR analyzes, the new speciation and formation constants of phosphomolybdates and phosphotungstates in the presence of cyclodextrins are determined and compared to those previously reported in pure water or in 50:50 water/1,4-dioxane mixture.

INTRODUCTION

The dodecamolybdophosphate and the dodecatungstophosphate are among the most known and studied polyoxometalates (POMs) due to their important properties valuable for many applications in various fields of research such as catalysis, 1,2 energy, [START_REF] Proust | Functionalization and Post-Functionalization: A Step towards Polyoxometalate-Based Materials[END_REF][START_REF] Genovese | Polyoxometalate Modified Inorganic-Organic Nanocomposite Materials for Energy Storage Applications: A Review[END_REF] biology, [START_REF] Rhule | Polyoxometalates in Medicine[END_REF] and environment. 6,[START_REF] D'cruz | Polyoxometalate-Based Materials for the Removal of Contaminants from Wastewater: A Review[END_REF] This class of highly soluble metal-oxo clusters was first discovered by Berzelius in the early 19 th century, [START_REF] Berzelius | Beitrag Zur Näheren Kenntniss Des Molybdäns[END_REF] but their structure was not determined until more than 100 years later by Keggin. [START_REF] Keggin | Structure of the Crystals of 12-Phosphotungstic Acid[END_REF] The heteropolyanion is composed of a metal-oxo shell containing twelve MO6 polyhedra (M = Mo VI or W VI ) connected together that defined a central tetrahedral cavity in which a phosphorus atom P V is located (Figure 1a), leading to a general chemical formula PM12O40 3-, hereafter abbreviated {PM12}. Due to their very low global charge, their stability in aqueous solution is weak and they only exist in strong acidic medium, 10,[START_REF] Gumerova | Polyoxometalates in Solution: Speciation under Spotlight[END_REF] that greatly limits their use at pH close to neutral or physiological medium. It is crucial that POM integrity should be demonstrated (e.g., by NMR) at pH 7-8 before performing any biomedical studies. There are several reports in the literature in which the POMs is certainly hydrolyzed at the pH conditions and concentrations employed, which raises the question: what is the actual speciation? [START_REF] Colovic | Polyoxometalates in Biomedicine: Update and Overview[END_REF] For instance, the use of Keggin-type polyoxotungstate and -molybdate, H3PW12O40 and H3PMo12O40, as disinfectants was reported by Bae et al., [START_REF] Bae | Photocatalytic Bacterial Inactivation by Polyoxometalates[END_REF] who found a high photocatalytic inactivation of two different bacterial strains (Escherichia coli and Bacillus subtilis). In another study, 14 in vitro inhibitory activity of 12-tungstophosphoric acid (H3PW12O40), on E-NTPDase activity was shown using rat synaptic plasma membranes as a model system. The IC50 value, calculated from concentration dependent sigmoidally-shaped inhibition curve, was in the micromolar range, 1.6 10 -6 mol/L, even though monolacunary Keggin anion was found as the main active molecular species present under physiological conditions (in the enzyme assays, pH 7.4). Hydrolysis of these anions leads to highly charged lacunary Keggin-type POMs like PM11O39 7-(M = W or Mo) or PMo9O34 9-which dominate the phospho-tungstate or -molybdate speciation, respectively. [START_REF] Gumerova | Polyoxometalates in Solution: Speciation under Spotlight[END_REF] Although such a degradation process can be avoided in organic solvents, [START_REF] Hori | Characterization of Alpha-Dodecamolybdophosphae and Beta-Dodecamolymdophosphae Complexes Formed in Aqueous-Solutions Including Acetonitrile, Acetone, or Dioxane[END_REF][START_REF] Fan | The Stabilizing Role of Cyclodextrins on Keggin Phosphotungstic Acid by Complexation Unveiled by Electrospray Mass Spectrometry[END_REF] their utilization in biological systems remains hampered. Therefore, finding chemical additives to stabilize the phosphorus-based Keggin structure in aqueous solution is rather demanding for basic and application purposes. POMs have shown a remarkable ability to interact with organic and hybrid moieties to form supramolecular assemblies with specific physicochemical properties. [START_REF] Stuckart | Polyoxometalates as Components of Supramolecular Assemblies[END_REF][START_REF] Cameron | Supramolecular Assemblies of Organo-Functionalised Hybrid Polyoxometalates: From Functional Building Blocks to Hierarchical Nanomaterials[END_REF][START_REF] Zhu | Constructing Chiral Polyoxometalate Assemblies via Supramolecular Approaches[END_REF] For instance, Keggin-type anions are capable of forming non-covalent complexes in aqueous solution with the toroidal cyclodextrin (CD) macrocycles. [START_REF] Fan | The Stabilizing Role of Cyclodextrins on Keggin Phosphotungstic Acid by Complexation Unveiled by Electrospray Mass Spectrometry[END_REF][START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF][START_REF] Zhang | Induced Chirality and Reversal of Phosphomolybdate Cluster via Modulating Its Interaction with Cyclodextrins[END_REF][START_REF] Su | Gas-Phase Fragmentation of Host-Guest Complexes of Cyclodextrins and Polyoxometalates[END_REF][START_REF] Jiang | A Nonconventional Host-Guest Cubic Assembly Based on Gamma-Cyclodextrin and a Keggin-Type Polyoxometalate[END_REF][START_REF] Yao | Redox-Responsive Host-Guest Association between Γ-Cyclodextrin and Mixed-Metal Keggin-Type Polyoxometalates[END_REF][START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF][START_REF] Pacaud | Hybrid Core-Shell Nanoparticles by "Plug and Play" Self-Assembly[END_REF] Some of these CD/Keggin hybrids have shown improved properties in sensor [START_REF] Hatami | Construction of Beta-Cyclodextrin-Phosphomolybdate Grafted Polypyrrole Composite: Application as a Disposable Electrochemical Sensor for Detection of Propylparaben[END_REF] , or in electrochemical energy storage, [START_REF] Yang | Polyoxometalate-Cyclodextrin Metal-Organic Frameworks: From Tunable Structure to Customized Storage Functionality[END_REF] highlighting the beneficial effects of these supramolecular composites. CDs are natural oligosaccharides made up of 6, 7, or 8 glucopyranose units leading to ring-shaped molecules of different sizes, denoted respectively by a-, b-, and g-CD (Figure 1b). However, host-guest complexation was only observed with the larger g-CD molecule which allows a perfect size matching between its inner macrocyclic cavity and the outer surface of the Keggin POM, leading to a 1:2 sandwich assembly with phosphomolybdate, {PMo12}@2g-CD, [START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF] or to a 1:1 complex with phosphotungstate, {PW12}@g-CD, [START_REF] Jiang | A Nonconventional Host-Guest Cubic Assembly Based on Gamma-Cyclodextrin and a Keggin-Type Polyoxometalate[END_REF] while a simple adduct arising from weak mutual interactions was obtained with the smaller a-CD molecule, {PW12}•a-CD. [START_REF] Yang | Polyoxometalate-Cyclodextrin Metal-Organic Frameworks: From Tunable Structure to Customized Storage Functionality[END_REF] The crystallographic structures shown in Figure 2 revealed various modes of intermolecular hydrogen bonding, involving not only the outer wall of the CD but also the protons of the internal cavity through the primary or secondary face of the conical cavitand, depending on the type of the host-guest association. Indeed, the POM interacts with the larger secondary face in the 1:1 complex {PW12}@g-CD, while in the case of the sandwich {PMo12}@2g-CD, the POM is connected to the CD through its smaller primary rim. In all cases, it is now accepted that the driving force of such supramolecular assemblies lies in the chaotropic effect which consists of a water structure recovery process resulting from the desolvation of the interacting components, namely the POM and organic cavitand. [START_REF] Assaf | The Chaotropic Effect as an Assembly Motif in Chemistry[END_REF] Likewise, such an effect has also been observed in other processes involving Keggin POMs where solvation comes into play such as adsorption on micelle surfaces, [START_REF] Buchecker | Polyoxometalates in the Hofmeister Series[END_REF][START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] on temperature-responsive polymer, [START_REF] Buchecker | Self-Assembly of Short Chain PoIy-N-Isopropylacrylamid Induced by Superchaotropic Keggin Polyoxometalates: From Globules to Sheets[END_REF] or aggregation with proteins. [START_REF] Sole-Daura | Structure-Activity Relationships for the Affinity of Chaotropic Polyoxometalate Anions towards Proteins[END_REF] It has been shown that the chaotropic character of Keggin POMs increases with the decrease in their overall charge according to the Hofmeister series. [START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF][START_REF] Assaf | The Chaotropic Effect as an Assembly Motif in Chemistry[END_REF][START_REF] Buchecker | Polyoxometalates in the Hofmeister Series[END_REF] Accordingly, host-guest stabilization of one of the less charged POMs, the Lindqvist anion M6O19 2-with M = Mo or W, was achieved in aqueous solution using g-CD. [START_REF] Falaise | Probing Dynamic Library of Metal-Oxo Building Blocks with Gamma-Cyclodextrin[END_REF] Thus, CDs can be used as trapping or stabilizing agents for elusive and unstable POMs. The supramolecular interactions between Keggin POMs and CDs in solution can be studied and followed, primarily by nuclear magnetic resonance (NMR) spectroscopy and mass spectrometry. [START_REF] Fan | The Stabilizing Role of Cyclodextrins on Keggin Phosphotungstic Acid by Complexation Unveiled by Electrospray Mass Spectrometry[END_REF][START_REF] Su | Gas-Phase Fragmentation of Host-Guest Complexes of Cyclodextrins and Polyoxometalates[END_REF][START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF][START_REF] Pacaud | Hybrid Core-Shell Nanoparticles by "Plug and Play" Self-Assembly[END_REF] However, these studies have rarely been conducted in POM stabilization context and no systematic pH study has been reported so far. The scope of this report will then focus on the impact of this crucial parameter, namely pH, for the POMs chemistry, hitherto ignored in previous studies about POM-CD systems. Indeed, the acidic polycondensation and basic hydrolysis of metal oxides are processes mainly governed by the pH. In this contribution, we provide a detailed 31 P NMR study on the CDs stabilizing effect of Keggin phospho-tungstate and -molybdate POMs over a wide range of pH. The phosphates speciation and the related formation constants are modeled in various systems studied, demonstrating the influence of CDs for expanding the pH domain of the PW12O40 3- and PMo12O40 3-anions.

EXPERIMENTAL SECTION

Chemicals and POM syntheses. All reagents were purchased from commercial sources and used without further purification. Solutions were prepared in Milli-Q water.

H3[PW12O40]•9H2O, 35 and H3[PMo12O40]•29H2O, 36 were synthesized and purified according to published procedures. The purity and chemical composition of the POMs were confirmed by EDS, TGA, multinuclear NMR ( 31 P and 183 W) and FT-IR spectroscopies. For H3[PW12O40]•9H2O: EDS, obs. W/P: 12/1.4. TGA showed a weight loss of 5.3% in the 30-600 °C temperature range corresponding to 9 water molecules (calculated 5.3%), due to both crystal lattice and hydration water. [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] P and 183 W NMR (90 mM in D2O), d31P -14.3 ppm, d183W -97.4 ppm, 2 JW-P 1.1 Hz. FT-IR (Diamond ATR, ATR correction applied) n (cm -1 ) 1074 (P-Oa), 974 (W=Ot), 899 (W-Ob-W), 779 (W-Oc-W). For H3[PMo12O40]•29H2O: EDS, obs. Mo/P: 12/1.1. TGA showed a weight loss of 22% in the 30-600 °C temperature range corresponding to 29 water molecules (calculated 22%). Physical methods. Fourier transform infrared (FT-IR) spectra were recorded on a 6700 FT-IR Nicolet spectrophotometer, using the diamond ATR technique. The spectra were recorded on non-diluted compounds and ATR correction was applied. Energy-dispersive X-ray spectroscopy (EDS) measurements were performed using a SEM-FEG (scanning electron microscope enhanced by a field emission gun) equipment (JSM 7001-F, Jeol). The measures were acquired with a SDD XMax 50 mm 2 detector and the Aztec (Oxford) system working at 15 kV and 10 mm working distance. The quantification is realized with the standard library provided by the constructor using Lα lines. To determine water contents, thermal gravimetric analysis (TGA) measurements were performed using a Mettler Toledo TGA/DSC 1, STAR e System apparatus under oxygen flow (50 mL min -1 ) at a heating rate of 5 °C min -1 up to 700 °C.

Nuclear magnetic resonance. NMR spectra were measured in D2O at 28 °C. 1 H and 31 P NMR spectra were recorded on a Bruker Avance 400 spectrometer at Larmor frequencies of 400.1 and 162.0 MHz, respectively, using 5 mm standard NMR tubes. The 183 W NMR spectra were obtained with 10 mm NMR tubes and a Bruker Avance 500 spectrometer operating at a Larmor frequency of 20.8 MHz. The 1 H NMR spectra were recorded with one pulse sequence at 30° flip angle (pulse duration 2.7 μs), using 0.1 s recycle delay, 3 s acquisition time, and 8 number of scans.

The 31 P spectra were recorded in quantitative mode with a 90° pulse of duration 16 μs and a length of free induction decay 1.6 s. A relaxation delay of at least 200 s was necessary to ensure a return to equilibrium of the magnetization. Depending on the concentration of the sample, the number of scans was adjusted and can vary from 8 to more than 1000. The duration of an experiment then varied from 30 min to more than 10 h. The 183 W spectra were performed with a Hahn echo sequence using a 90 ° pulse of 37 μs, a relaxation delay of 2 s and a scan number of over 30,000.

Chemical shifts are scaled with respect to standards (d = 0 ppm): Me4Si (1% CDCl3) for 1 H, 85 wt% H3PO4 for 31 P, and Na2WO4 (D2O, 1 M) for 183 W.

Samples preparation. The solutions were prepared in D2O. Each sample was prepared from stock solutions of POM (80-100 mM H3PW12O40 or H3PMo12O40) and each CD (a-, b-, or g-CD) which are mixed in known proportions to make 1-or 3-mL final solutions. The volume was adjusted with D2O to achieve the desired concentrations of POM and CD. Usually, the POM solution was added last. The pH of the solutions is measured systematically and adjusted with concentrated solutions of NaOH (6-12 M). The samples were then transferred into 5 mm (600 µL) NMR tubes for the 1 H and 31 P analyzes, or 10 mm (2.5 mL) tubes for the 183 W analyzes. The pH can change over time (especially for tungstate samples). pH measurements were then taken at each start and end of an NMR experiment, which can last several hours. Similar series were also prepared in pure D2O or in a 50:50 v/v D2O/1,4-dioxane mixture for comparison.

RESULTS AND DISCUSSION

Before studying the stabilizing role of CDs on PW12O40 3-and PMo12O40 3-, we must first understand their mutual interaction. Such an interaction should be strong and comparable owing to their similar low charge. [START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF][START_REF] Yao | Redox-Responsive Host-Guest Association between Γ-Cyclodextrin and Mixed-Metal Keggin-Type Polyoxometalates[END_REF] Indeed, the main factor affecting the strength of the attractive association of CDs with a given POM of comparable size and shape is its overall charge which is directly related to its solvation properties and its chaotropic character. [START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF] However the size of the cavitand should also be taken under consideration, as this has been reported in similar host-guest systems. [START_REF] Assaf | High-Affinity Host-Guest Chemistry of Large-Ring Cyclodextrins[END_REF][START_REF] Ivanov | Size-Exclusion Mechanism Driving Host-Guest Interactions between Octahedral Rhenium Clusters and Cyclodextrins[END_REF] In their pioneer work, Stoddart et al. had also shown significant differences in the behavior of PMo12O40 3-when it reacted with gor b-CD. [START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF] Here, we systematically reinvestigate the behavior of these two POMs in the presence of the three types of native CD.

Titration of CDs with POMs.

The study of the interaction of Keggin-type POMs with cyclodextrins was carried out by proton NMR through the variation of the chemical shifts of the six types of protons in the molecule, labelled from H1 to H6 (see Figure 1). These experiments consisted of the preparation of aqueous solutions of 4 mM CD to which is added a variable number of equivalents of PW12O40 3-or PMo12O40 3-anions. The pH in these solutions was not controlled and varied from ca. 0.5 to 2 depending on the proportion of the POM. The purpose of this titration is to identify the protons that will be most affected and to assess its magnitude as a result of the interaction between the POM and the CD. The anomeric position signal (H1) appears shielded (~5 ppm) and is usually the least sensitive.

Protons pointing towards the cavity (H3 and H5) and from the methoxy group (H6) are generally the most subject to interactions with guest molecules. The shape of the spectra and the effects of the presence of POM are different depending on the size of the CD. However, similar results were obtained with the two POMs, i.e., PW12O40 3-and PMo12O40 3- , confirming that the nature of the metal center does not affect the interaction with the CD.

The spectra of molybdenum-based systems are shown in Figure 3, while those of the tungsten-analogues are supplied in Figures S1-S3, in Supporting Information (SI). The interaction of γ-CD with {PW12} species has been extensively studied by our group. [START_REF] Yao | Redox-Responsive Host-Guest Association between Γ-Cyclodextrin and Mixed-Metal Keggin-Type Polyoxometalates[END_REF][START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF] It has been shown very strong interaction leading to inclusion complexes with association constants of the order of 10 5 M -1 . Figure 3 shows the evolution of the 1 H NMR spectra of a 4 mM γ-CD solution with gradual addition of increasing amount of {PMo12} up to 8 equivalents. The evolution of the spectra is very similar to that observed with {PW12} (see Figure S1, in SI), [START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF] but different from that reported by Stoddard et al., [START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF] probably because of the difference in experimental conditions (our solutions being 4 times more concentrated). As expected, H2 and H4 show negligible effect with only a slight chemical shift variation for H4. However, strong interactions are observed with H5 and H6, undergoing a clear deshielding with a slightly less effect on H5 in the case of {PMo12} compared to {PW12}. Nevertheless, H3 is the most interacting hydrogen, as shown by about 0.7 ppm variation for large POM/CD ratio. Thus, g-CD appears to interact with POM preferentially through H3 and H5 hydrogens when encapsulated in its cavity. The interaction with the primary face of γ-CD is then privileged as shown by the variations observed on H6. This confirms the mode of POM insertion into the γ-CD cavity from the primary rim.

As it has been demonstrated in previous studies, [START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF] the driving force behind such an association arises from the chaotropic nature of POM which facilitates its desolvation and thus promotes inclusion complexation.

The titration of 4 mM β-CD solution by {PMo12} is shown in Figure 3, and compares well with that obtained with {PW12}, shown in Figure S2, in SI. [START_REF] Pacaud | Hybrid Core-Shell Nanoparticles by "Plug and Play" Self-Assembly[END_REF] The six types of protons Monitoring hydrolytic stability by 31 P NMR. The stability of PW12O40 3-in pure water and in presence of two eq. of the three forms of CD (a-, b-, and g-CD) was studied over a pH range from 0 to 8.5 by 31 P NMR. The concentration of the POM was kept constant and the pH was adjusted using concentrated NaOH solutions. The POM concentration was fixed at 95 mM for the pure water series, and at 60 or 7.5 mM for two eq. g-CD (120 mM) or b-CD (15 mM) series, taking into account the solubility limits of each CD, respectively. In the case of a-CD series, we observed a white solid precipitate when mixing with phosphotungstic acid in a concentrated system. Such a precipitate would correspond to a CD-POM composite analogous to that reported by Khashab et al., [START_REF] Yang | Polyoxometalate-Cyclodextrin Metal-Organic Frameworks: From Tunable Structure to Customized Storage Functionality[END_REF] since IR spectroscopy showed the presence of both the characteristic vibration bands of a-CD and {PW12}. Then, we had to reduce the concentrations down to 7.5 mM POM and 15 mM a-CD in order to avoid any precipitations. Representative 31 P NMR spectra obtained for the pure water system, compared to those contained two equivalents of aand g-CD are shown in Figure 4. The overall spectra as well as those of the series with 2 eq. b-CD are given in Figures S5-S8.

The hydrolytic conversions of {PW12} anion have been extensively studied by Maksimovskaya using 31 P NMR, and more recently the previously published work in the field, including her research, has been carefully reviewed, providing the most comprehensive hydrolysis scheme, currently existing. 10,[START_REF] Maksimovskaya | Hydrolysis of Heteropoly Acid H(3)PW[END_REF] The present study allows identifying at least 4 main signals in addition to the signal of the parent Keggin POM {PW12}, labelled A, B, C, H and F according to the notation of Maksimovskaya and Maksimov. 10 The assignment of these [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] P NMR resonances is reported in Table 1, based on this review, 10 although the authors did not provide a clear assignment for the H and F signals. (species H), and P2W21O71(H2O)3 6-(species F). These two latter correspond to sandwich-mike POMs resulting from assemblies of two trilacunary Keggin PW9O34 9-linked by two or three W centers, have been isolated by Tourné et al., [START_REF] Tourné | Aquanonadecatungstodiphosphate(14-) Popyanion[END_REF][START_REF] Tourné | Triaquahenicosatungstodiphosphate(6-) Heteropolyanion[END_REF] and their 31 P NMR spectrum is consistent with such assignments.

Similar studies were performed in presence of two eq. α-and β-CDs. The 31 P NMR spectra corresponding to the a-CD system (see Figure 4), reveal that the first intermediate, species A (PW11O39•WO2 5-) appears at pH 2. This species continues to form accompanied by a second intermediate, the species C ((PW11)2WO2). Nevertheless, the most abundant species that forms corresponds to the monocavant {PW11} anion. Furthermore, it can be worth noting that the presence of α-or β-CD reduces the number of intermediates generated form the {PW12} degradation. Indeed, the species H (P2W20O70(H2O)2 10-) and F (P2W21O71(H2O)3 6-) do not appear here, while the species C ((PW11)2WO2) and A (PW11•WO2) persist over an extended pH range.

Similar distribution of species is also observed with b-CD (see Figure S7, in SI), indicating that aand b-CD produce the same effects upon the hydrolytic behavior of the {PW12} anion.

In the presence of two eq. g-CD, the chemical system becomes remarkably simple and restricted to the {PW12} anion and its hydrolysis product {PW11} (see Figure 4). For pH < 3.5, only {PW12} is present while for pH 4, its intensity ratio decreases in favor of that of the resonance attibuted to {PW11} (labelled B). The pH must be increased up to 6.7 to observe the complete disappearance of the {PW12} in favor of {PW11}. Then, no intermediates were observed in this system, and the stability of {PW12} was found to be extended at much higher pH.

As already shown, the {PW12} anion appears very sensitive to pH and hydrolysis equilibria take place as soon as pH = 1.7. Decomposition leads to numerous degradation products but the monovacant POM {PW11} becomes predominant from pH 3.5. In the presence of a-CD or b-CD, much fewer intermediate species were observed and only those based on the {PW11} fragment persist. However, the most dramatic effect is observed with g-CD which suppresses all intermediate species and improves the stability of {PW12} at significantly higher pH up to 6.

The stability of PMo12O40 3-in pure D2O and in the presence of two eq. of g-CD as well as in a 50:50 v/v D2O:1,4-dioxane mixture, was studied over a pH range of 0 to 7 by 31 P NMR. The concentration of POM was kept constant at 60 mM in all three systems and the pH was adjusted using concentrated NaOH solutions. Figure 5 shows representative 31 P NMR spectra obtained from these three systems, while spectra of full series are shown in Figures S9-S11, in SI. The degradation of Keggin POM {PMo12} in water with increasing pH have been reported in the literature, and the 31 P NMR spectra obtained in this study (see Figure 5) are comparable to those published. [START_REF] Mc Garvey | A Study of Solution Species Generated During the Formation of 12-Heteropoly Oxometalate Catalysts[END_REF][START_REF] Bajuk-Bogdanovic | A Study of 12-Tungstosilicic and 12-Molybdophosphoric Acids in Solution[END_REF][START_REF] Bajuk-Bogdanovic | Study of the Decomposition Pathway of 12-Molybdophosphoric Acid in Aqueous Solutions by Micro Raman Spectroscopy[END_REF][START_REF] Krukovsky | Optimization of Phosphomolybdic Acid Synthesis Using 31(p) NMR Data[END_REF] The four main identified POM species resulting from the {PMo12} decomposition in the pH range 0-7 are PMo11O39 2). In the presence of two eq. g-CD, the 31 P NMR spectra (Figure 5) show the presence of only {PMo12} for pH < 2.8, but its signal decreases in intensity to the detriment of the signal of {PMo11} when pH increases. The complete disappearance of {PMo12} occurs at approximately pH = 4.2. The formation of the {PMo11} anion occurs until pH = 6 associated with the presence of the A-and B-forms of the lacunar {PMo9}, as minor species. The stabilizing effect of g-CD on {PMo12} is evidenced by two observations. The first one lies on the fact that {PMo12} can be dissolved in water without any decomposition as in a mixed hydro-organic medium. The second observation arises from the pH domain of the {PMo12} presence which is extended to more than two pH units. However, no significant change in phosphate speciation could be observed for pH above 5, where highly charged {PMo11} converts into {P2Mo5} and free phosphates as observed in pure water. There is almost no action of the g-CD in this pH range from 5 to 7.

Finally, we also carried out the same experiments in a mixture 50:50 v/v D2O:1,4dioxane 50/50, known to stabilize the {PMo12} POM, with the purpose to compare the influence of the CD with those of the medium (Figure 5). Only the 31 P NMR {PMo12} signal is visible over a wide pH range from 0.5 to 4.5. It is remarkable to note the presence of the monovacant {PMo11} and trivacant {PMo9} anions only as minor products.

However, the fraction of {PMo12} decreases until disappearance at pH ca. 7 in favor of {P2Mo5} as the major end-product. The comparison clearly showed that g-CD acts as a very good specific stabilizer for the {PMo12} anion. Even if the hydro-organic mixture makes possible to expand the hydrolytic stability of the {PMo12} species until pH = 5-6, it should be worth noting that such a comparison requires activity corrections which were not taken into account in this study.

Phosphate speciation. The previous results showed that distribution of the P-containing species is dependent on the nature of the medium, which can shift the thermodynamic equilibria.

From these experimental measurements, the fraction of the phosphate species can be plotted as a function of the pH for the different media such as pure water (D2O), in the presence of two eq. CD (a-, b-, or g-form), and in 50:50 v/v D2O:1,4-dioxane, hydro-organic mixture (Figures 6 and7).

The curves are calculated to best fit our experimental data, making it possible to estimate the formation constants of the various POMs identified during the degradation process of the Keggintype anion. For this, we use the individual formation equilibria associated to the formation of each species (see section S3 in SI). The formation constants of phosphates, isopolyoxomolybdates, and isopolyoxotungstates are taken from the literature. [START_REF] Falaise | Probing Dynamic Library of Metal-Oxo Building Blocks with Gamma-Cyclodextrin[END_REF] For phospho-molybdates and -tungstates, their constants were optimized as a function of the medium to best fit the experimental data. Tables 3 and4 gather our results and compare them with data from literature when available.

We did not find in the literature any formation constants related to the formation of phosphotungstates species. Nevertheless, the constant values corresponding to the formation of {PW12} and {PW11} (Table 3) are 40 orders of magnitude higher than their molybdenum analogues, confirming that phosphotungstate POMs are thermodynamically more stable than their molybdenum equivalents. The presence of CDs led to a significant increase in the formation constant of {PW12} by 14 orders of magnitude with aand b-CD and by 25 orders of magnitude with g-CD. Such a stabilizing effect of the {PW12} anion takes place at the expense of other less stable POMs such as {P2W20} and {P2W21} anions. For phosphomolybdate speciation, our calculations compare well with published data (see Table 4). [START_REF] Pettersson | Speciation in the Aqueous H+-MoO42--HPO42-System as Deduced from a Combined EMF-P-31 NMR-Study[END_REF] The stability constant of the {PMo12} anion increases by 10 orders of magnitudes in the presence of g-CD, but also remains ca.

10 orders of magnitude lower than the value observed in the mixture 50:50 v/v D2O:1,4-dioxane.

The equilibrium shifts affect mainly the stability domain of {PMo11}, which disappears almost completely in aquo-organic medium but still persists in water with the presence of g-CD although considerably reduced (see Figure 7). These results confirm quantitatively the stabilizing influence of CDs. 3. 4. Evidence of the stabilizing role of host-guest complexation. Clearly, the presence of CDs, and particularly the g-form, improves the stability of the phosphorus-containing Keggin-type POMs. In this section, we will focus on the mechanism of such the stabilizing role. In particular, we will further demonstrate the protective role of supramolecular inclusion complexation against hydrolytic attack of water.

Figure 8 shows the 1 H NMR spectra of a 60 mM solution of {PW12} in the presence of γ-CD at 120 mM at various pH (the same solutions studied by [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] P NMR and shown in Figure 4). Similar results are also observed with the aand b-CD systems, where the shape of the 1 H NMR spectrum changes at the pH for which the Keggin POM {PW12} decomposes, around pH = 3 (See Figures S12 and13, in SI). The same is also obtained with molybdenum-containing POM {PMo12} in the presence of 2 eq. g-CD (see Figure S14). In this case, the shape change in the 1 H NMR spectra occurs at pH = 5 corresponding to the limit of stability for {PMo12} in this system.

All of these data clearly indicate that CDs interact exclusively with the starting saturated Keggin POM, while the less chaotropic vacant POM species show no specific interaction with any type of the three CDs. To provide further information on these interactions, a study by NMR 183 W was carried out. The starting {PW12} anion is characterized by a single sharp doublet ( 2 JW-P = 1.1 Hz) at -95.4 ppm corresponding to the twelve W equivalent nuclei of the POM structure (Figure 9).

With the addition of two eq. γ-CD, this signal broadens and shifts towards high fields of about 3 ppm (Figure 9). These effects clearly indicate the formation of the inclusion complex with a very strong association which provide a significant shielding on the W nuclei. The line broadening of the 183 W signal (guest) like that of the H3 proton (host) results from a strong host-guest contact.

These effects are not observed with the other smaller CDs (Figure 9), confirming the unique behavior of γ-CD highlighted by the formation of inclusion complex. All our results obtained so far are in agreement with the conclusion that the host-guest complexation of the Keggin-type anion is at the origin of its exceptional stabilization. However, it

is not yet clear whether such a stabilization is dynamic and then reversible. To address such a purpose, we compared the effect of the sequence of addition of γ-CD with respect to {PW12}. As previously shown, at pH = 3, aqueous solution of {PW12} anion is unstable giving many intermediate. This decomposition can be fully canceled when the {PW12} anion is dissolved in g-CD-containing aqueous solution. However, when g-CD is added to an aqueous solution of POM at pH = 3, the restoration of the {PW12} anion by shifting the chemical equilibria, is only partial (see Figure 10). Only the intermediate decomposition products exhibit reversible transformation while the most stable decomposition product {PW11} appears to be unaffected and retains almost the same fraction as that before the addition of the g-CD. This result clearly shows that the presence of CD makes it possible to protect {PW12} from basic hydrolysis, but also capable of restoring it quasi-quantitatively from the first decomposition products labelled A, C, H and F (see Figure 4).

However, it no longer becomes possible to restore {PW12} with the CD from the degradation product {PW11} anion. The high negative charge the latter makes it poorly chaotrope, and may explain its weak affinity towards the CD. These observations give also information about the equilibrium relationships that link {PW12} to its degradation products since these transformations are dynamic and partially reversible.

Highlighting stability domain enhancement of phosphorus-Keggin POMs.

Supramolecular associations between Keggin-type POMs and CDs alter significantly the properties of the POM in aqueous solution. Depending on the nature of these interactions, the stability domains of POM can be shifted to a large extent as shown in Figure 11. The stability of POM in the presence of α-or β-CD is only slightly improved by about one pH unit (Figure 11). These two CDs have shown to behave similarly with the Keggin anion. In fact, the interactions have been identified as non-specific, involving mainly the outer surface of the CDs, leading to very similar and comparable effects. However, γ-CD, acting as a true host molecule for inclusion complexation, its stabilizing effect is much larger. This CD is able to maintain the {PW12} anion intact up to near pH = 4, and to expand its existence in aqueous solution up to pH = 6, a value close to that of the physiological pH. In addition, no intermediate decomposition products are observed during the transformation of {PW12} into {PW11}. Similar behavior was also observed with the less stable {PMo12}, but with one pH unit lower, i.e., maintaining its structure intact up to pH = 3, and its existence up to pH = 5 (Figure 11).

Table of Contents Synopsis

The stabilizing role of cyclodextrins (CDs) on Keggin-type molybdo-and tungsto-phosphates, was investigated using 31 P NMR spectroscopy. The formation of supramolecular host-guest inclusion complexes observed with g-CD provides a protective shell on the POM surface against hydrolytic attack which extends the stability domains by more than three pH units. With the smaller CDs (a-, and b-type), the stabilization is only moderate, demonstrating the importance of size matching effect in host-guest complexation and its protective role.

Table of Contents Graphic

Figure 1 .

 1 Figure 1. Structural representations of molecular objects used in this study: a) the Keggin-type POM {PM12} (M = Mo or W), b) the macromolecular torus a-CD (C36H60O30), b-CD (C42H70O35), and g-CD (C48H80O40) resulting from condensation of 6, 7, and 8 glucopyranose units, respectively.

Figure 2 .

 2 Figure 2. Top and side views of most representative known examples of cyclodextrin-{PM12} Keggin-type complexes: a) 1:1 adduct {PW12}•a-CD, b) 1:1 host-guest complex {PW12}@g-CD, and c) 1:2 host-guest sandwich-type complex {PMo12}@2g-CD.

31 P

 31 NMR (80 mM in 50:50 D2O:1,4-dioxane + one drop HCl 6 M), d31P -3.2 ppm. FT-IR (Diamond ATR, ATR correction applied) n (cm -1 ) 1076 (P-Oa), 958 (Mo=Ot), 875 (Mo-Ob-Mo), 784 (Mo-Oc-Mo).

Figure 3 .

 3 Figure 3. Partial 1 H NMR spectra (from left to right) of 4 mM a-CD, b-CD, and g-CD after adding increasing amounts of PMo12O40 3-up to 8 equivalents.

(

  Figure S3, in SI, for {PW12}. The six proton types of a-CD are observed as before with a difference in the shape of the H6 signal which changes from a multiplet to a singlet during the titration. As in the case of b-CD, the signals of the hydrogen H2 and H3 exhibit only slight shifts.The resonances of the H4 and H5 hydrogens change by ca. 0.1 ppm and reach a plateau at around two equivalents of POM. Finally, the H6 hydrogen undergoes the largest shift of its signal from 3.8 to almost 4.0 ppm after adding about four equivalents of POM. Thus, the interactions between α-CD and POM are probably similar to those demonstrated with β-CD, indicating POM-CD interactions mostly occur at the exposed outer wall of the CD and its primary face. It also confirms that like β-CD, α-CD is too small to entirely accommodate POM in its cavity.In summary, the {PMo12} or {PW12} Keggin-type anions behave similarly toward CD (see the comparison of 1 H chemical shifts in FigureS4, in SI). However, the three CDs interact with the Keggin POM in different ways. The aand b-CD interact mostly with their outer surface, while γ-CD leads to the usual host-guest inclusion, involving internal

Figure 4 .

 4 Figure 4. Representative 31 P NMR spectra (from left to right) of 95 mM H3PW12O40 in D2O, 7.5 mM H3PW12O40 in presence of 15 mM a-CD in D2O, and 60 mM H3PW12O40 in presence of 120 mM g-CD in D2O, at various pH, adjusted with concentrated NaOH.

F41

  At very low pH, the -14.3 ppm resonance is attributed to {PW12}. As shown in FigureS5, the hydrolytic degradation of the {PW12} occurs as soon as pH > 1.5. From a pH 1.8 solution, many new signals in the [-10.5 ; -13.5 ppm] range arise due to the hydrolytic degradation of the {PW12} anion. These species (labelled A, B, C, H, and F) grow as the pH increases to the detriment of the {PW12} anion which disappears at pH 2.5. It is also possible that the monovacant PW11O397- species, labelled B and abbreviated {PW11} hereafter, begins to form in this pH range but remains as minor product. From pH = 3.9, the {PW11} anion becomes predominant, while the other intermediates have almost entirely disappeared. Furthermore, the signal attributed to {PW11} is very pH dependent and stabilizes around -9.7 ppm in the 4-8 pH range. Because the resulting transient cationic species WO2(H2O)4 2+ and WO(H2O)5 4+ arising from the {PW12} hydrolysis would be stabilized in acidic medium, they should be trapped by vacant POMs forming various complexes such as PW11O39•WO2 5-(species A), (PW11O39)2WO2 12-(species C), P2W20O70(H2O)210-

  7-({PMo11}), A-PMo9O31(H2O)3 3-(A-{PMo9}), B-H2PMo9O34 7- (B-{PMo9}), and P2Mo5O23 6-({P2Mo5}) (see Table

Figure 5 .

 5 Figure 5. Representative 31 P NMR spectra of 60 mM H3PMo12O40 (from left to right) in pure D2O, in presence of 120 mM g-CD in D2O, and in 50:50 v/v D2O:1,4-dioxane, at various pH, adjusted with concentrated NaOH. Asterisk indicates free phosphate H2PO4 -.

46 A47

 46 In pure D2O system (Figure5), the ca -3 ppm resonance is attributed to {PMo12} and the two other minor signals at -0.6 and 0.4 ppm are assigned to the A-{PMo9} species and to free phosphates H3PO4/H2PO4 -, respectively. These species are not synthetic impurities but identified as decomposition products since they are absent in the 50/50 v/v mixture of D2O/1,4-dioxane medium. From pH = 1.6, many new signals begin to appear between -1 and +3.4 ppm due to the decomposition species of {PMo12}. The amount of these species increases with pH to the detriment of {PMo12} which disappears at a pH approximately equal to 2. The {PMo11} anion is formed from pH = 1.6 and becomes predominant until pH = 4.5. For pH = 5, we observe the formation of the trivacant Keggintype anion B-{PMo9} identified by the resonance at +0.6 ppm. Above pH = 6, the solution is dominated by the presence of the {P2Mo5} species and solvated phosphates H2PO4 - /HPO4 2-observed respectively at +2.4 ppm and +3.4 ppm.

Figure 6 .

 6 Figure 6. Phosphate species distribution in a) 95 mM H3PW12O40 in D2O, b) 7.5 mM H3PW12O40 in presence of 15 mM a-CD in D2O,and c) 60 mM H3PW12O40 in presence of 120 mM g-CD in D2O, from 31 P NMR spectra. The computed curves correspond to the best fits of experimental data using formation constants reported in Table3.

Figure 7 .

 7 Figure 7. Phosphate species distribution of 60 mM H3PMo12O40 in a) D2O, b) presence of 120 mM g-CD in D2O, and c) 50:50 v/v D2O:1,4-dioxane, from 31 P NMR spectra. The computed curves correspond to the best fits of experimental data using formation constants reported in Table4.

Before addition of the base ( 6 M

 6 NaOH), the spectrum shows a characteristic feature of a solution in the POM/g-CD = 0.5 molar ratio observed previously in Figure3. Such a broadening is characteristic of strong host-guest inclusion complexation. When the pH increases up to 4 only slight shifts in the signals are observed due mostly to the pH effect. From pH 4, the spectrum in the region of H4/H2 hydrogens (3.4-3.6 ppm) begins to change. From pH 7, the triplet signal of hydrogen H3 at ca. 3.9 ppm narrows considerably and the spectra evolve back to the initial signature of free γ-CD. This indicates that cyclodextrin no longer interacts with POMs due to their degradation at neutral pH. According to the 31 P NMR, the Keggin {PW12} ion disappears completely at pH 6, which corresponds to the change of the 1 H NMR spectrum shape.

Figure 8 . 1 H

 81 Figure 8. 1 H NMR spectra of 60 mM H3PW12O40 in presence of 120 mM g-CD in D2O, at various pH. For comparison, the spectrum of 120 mM g-CD in D2O is shown above.

Figure 9 .

 9 Figure 9. 183 W NMR spectra of 7.5 mM H3PW12O40 (from top to bottom) in pure D2O, in the presence of 15 mM a-CD, b-CD, and g-CD. The pH was adjusted at ca. 0.5 each time.

Figure 10 shows the 31 P

 31 Figure 10 shows the 31 P NMR spectra of the mixture of γ-CD (120 mM) and {PW12} (60 mM) at pH = 3, before and after addition of the CD, for which the sequence of the POM-CD addition has been inverted.

Figure 10 .

 10 Figure10.[START_REF] Bae | Photocatalytic Bacterial Inactivation by Polyoxometalates[END_REF] P NMR spectra of 60 mM H3PW12O40 (bottom) before and (middle) after addition of 2 eq. g-CD in D2O, compared to (top) the spectrum of a mixture of 60 mM H3PW12O40 and 120 mM g-CD in D2O for which the POM is added the last. For all the experiments, the pH is adjusted to 3.

Figure 11 .

 11 Figure 11. Effect of medium on stability domains of Keggin POM {PW12} and {PMo12}.

Table 1 . Assignment of Main 31 P NMR Signals from Degradation of H3PW12O40.

 1

	Label	Species	31 P shift (ppm)	Reference
	{PW12} PW12O40 3-	-14.3 to -14.2	39
	A	PW11O39•WO2 5-	-12.3 to -12.0	39
	B	PW11O39 7-	-11.4 to -9.8	39
	C	(PW11O39)2WO2 12-	-11.3 to -11.2	10
	H	P2W20O70(H2O)2 10-	-12.6 to -12.5	40

Table 2 . Assignment of Main 31 P NMR Signals from Degradation of H3PMo12O40.

 2

	Label	Species	31 P shift (ppm)	Reference
	{PMo12}	PMo12O40 3-	-2.9 to -2.8	46
	{PMo11}	PMo11O39 7-	-0.9 to +0.2

Table 3 .

 3 Formation Constants (LogK) at 28 °C of Phosphotungstate Species Calculated from Degradation of H3PW12O40, in Pure Water (D2O), and in Presence of two eq. a-, b-, and g-CD.

	species	Protonation state	2 eq. g-CD in D2O 2 eq. b-CD in D2O 2 eq. a-CD in D2O D2O
	{PW12}	PW12O40 3-	208.0	196.6	196.6	183.1
	{PW11}	PW11O39 7-	175.0	170.2	170.5	160.2
	{(PW11)WO2}	PW11O39WO2(OH) 6-		187.7	187.7	176.0
	{(PW11)2WO2} (PW11O39)2WO2 12-		363.3	363.5	340.0
	{P2W20}	P2W20O70(OH)2 12-				299.3
	{P2W21}	P2W21O71(OH)3 9-				320.2

Table 4 .

 4 Formation Constants (LogK) at 28 °C of Phosphomolybdate Species Calculated from the degradation of H3PMo12O40, in Water (D2O), with two eq. g-CD or in 50:50 v/v D2O/1,4-Dioxane Mixture.

	species	Protonation state	50:50 D2O:Dioxane 2 eq. g-CD in D2O D2O	Literature a
	{PMo12}	PMo12O40 3-	159.7	152.3	142.7 139.7
	{PMo11}	PMo11O39 7-	127.0	125.2	125.3 118.7
	A-{PMo9} A-PMo9O31(H2O)3 3-			111.1 107.2
	A-{PMo9} A-PMo9O31(OH)(H2O)2 4-116.9	112.9		104.7
	B-{PMo9} B-H2PMo9O34 7-			101.1 104.9
	B-{PMo9} B-HPMo9O34 8-	96.6	95.5		102.0
	{P2Mo5}	P2Mo5O23 6-	86.1	83.4	84.7	62.0
	a) At 25 °C, ionic strength µ = 3 M (NaCl), from reference 46 .

ACKNOWLEDGMENTS

M. Lajoye and F. Nagbo are thanked for their technical assistance during their Master internships. Authors gratefully acknowledge financial supports from LabEx CHARMMMAT (grant number ANR-11-LBX-0039), University of Versailles Saint-Quentin, and the Paris Ile-de-France Region -DIM "Respore". S.Y. thanks the China Scholarship Council (201904910419) for her scholarship.

DEDICATION

In memory of Raisa I. Maksimovskaya.

CONCLUSIONS

The stabilizing role of cyclodextrins on PW12O40 3-and PMo12O40 3-has been studied by NMR spectroscopy. While degradation of POMs occurs at pH ca. 1.5 and becomes complete until pH = 3, we show that specific supramolecular interactions are able to expand the pH stability scale.

The aand b-CD showed similar interactions in solution, which were identified as non-specific due to small size of the cavities compared to that of the Keggin-type anion. Conversely, the larger g-CD exhibits specific recognitions with the POM, leading to a stable host-guest inclusion complex involving two γ-CDs and a POM. As a result, aand β-CD have a limited stabilizing effect on POMs by expanding their pH stability domain by only about one pH unit. Due to the formation of close host-guest assembly, γ-CD acts as a protecting shell for the POM against hydrolytic attack of hydroxide ions. Significant stability improvement has been demonstrated, up to pH = 5 for PMo12O40 3-and pH = 6 for PW12O40 3-.

It then clearly appears that the presence of CDs modify the thermodynamic properties of POMs in aqueous solutions. A previous report showed that such supramolecular interactions alters significantly their redox properties, [START_REF] Yao | Redox-Responsive Host-Guest Association between Γ-Cyclodextrin and Mixed-Metal Keggin-Type Polyoxometalates[END_REF][START_REF] Yao | Effect in the Keggin-Type Polyoxotungstate Series[END_REF] while in the same way, we highlight that these supramolecular interactions, driven by the chaotropic effect were strong enough to modify the speciation diagram of metalate ions or increase significantly the rate of the polycondensation processes. [START_REF] Falaise | Probing Dynamic Library of Metal-Oxo Building Blocks with Gamma-Cyclodextrin[END_REF][START_REF] Falaise | Host in Host" Supramolecular Core-Shell Type Systems Based on Giant Ring-Shaped Polyoxometalates[END_REF] This is confirmed by the present study since CDs modify the distribution of the intermediates resulting from the degradation of the Keggin-type anion. Moreover, such an interaction-type, directed by the chaotropic effect reveals important consequences about the solution properties of POMs, that could be one of the important keys for a rational understanding of the POM behavior in complex matrices, such as biological media or aqueous wastes.

ASSOCIATED CONTENT

Supporting Information

The Supporting Information is available free of charge.

Additional 1 H and 31 P NMR data.

AUTHOR INFORMATION

Corresponding Authors

mohamed.haouas@uvsq.fr