P Indelicato
email: paul.indelicato@spectro.jussieu.fr

J P Santos

S Boucard

J.-P Desclaux
email: jean-paul.desclaux@wanadoo.fr

QED and relativistic corrections in superheavy elements

Keywords: 31, 30, Jv -31, 25, Eb -31, 25, Jf -32, 70, Cs

In this paper we review the different relativistic and QED contributions to energies, ionic radii, transition probabilities and Landé g-factors in super-heavy elements, with the help of the MultiConfiguration Dirac-Fock method (MCDF). The effects of taking into account the Breit interaction to all orders by including it in the self-consistent field process are demonstrated. State of the art radiative corrections are included in the calculation and discussed. We also study the non-relativistic limit of MCDF calculation and find that the non-relativistic offset can be unexpectedly large.

Introduction

In the last decades, accelerator-based experiments at GSI and Dubna have lead to the discovery of super-heavy elements up to Z = 116 and 118 [1] (for a recent review see [2]). Considerable theoretical work has been done to predict the ground configuration and the chemical properties of those superheavy elements. Relativistic Hartree-Fock has been used to predict the ground configuration properties of superheavy elements up to Z = 184 in the early 70's [3,4]. The Multiconfiguration Dirac-Fock (MCDF) method was used to predict orbital properties of elements up to Z = 120 [5], electron binding energies up to Z = 118 [6,7], and K-shell and L-shell ionization potentials for the superheavy elements with Z = 112, 114, 116, and 118 [8]. Ionization potential and radii of neutral and ionized bohrium (Z = 107) and hassium (Z = 108) have been evaluated with large scale MCDF calculations [9]. Kaldor and coworkers have employed the relativistic coupled-cluster method to predict ground state configuration, ionization potential, electron affinity, binding energy of the negative ion of several elements with Z ≥ 100 [10,11,12,13,14,15,16,17].

Very recently, laser spectroscopy of several fermium (Z = 100) transitions has been performed, the spectroscopy of nobelium (Z = 102) is on the way [18,19,20], and large scale MCDF calculations of transition energies and rates have been performed by several authors for superheavy elements with Z = 100 [18,21], Z = 102 [21] and Z = 103 [22], which are in reasonable agreement with the fermium measurements.

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There are however many unanswered questions, that need to be addressed in order to assess the accuracy and the limit of current theoretical methods. For inner-shells, or highly ionized systems, QED effects must be very strong in superheavy elements, where the atomic number Z approaches the limit Zα → 1 (α = 1/137.036 is the fine structure constant), at which the point-nucleus Dirac equation eigen-energies become singular for all s 1/2 and p 1/2 states (the energy depends on j + 1 2 2 -(Zα) 2 , where j is the total angular momentum). This means that all QED calculations must be performed for finite nuclei, and to all orders in Zα. QED calculations for outer-shell are very difficult. At present, only the simplest one-electron, one-loop diagrams can be calculated, using model potentials to account for the presence of the other electrons. The evaluation of many-body effects, that remains large for neutral and quasi-neutral systems, is also made very difficult by the complex structure of these atoms, in which there may be several open shells. In that sense, methods based on Relativistic Many-Body Perturbation theory (RMBPT) and MCDF methods are complementary. The former one usually allowing for more accurate results, but limited to (relativistic) closed shell systems minus one or plus one or at most two electrons, while the latter is completely general but convergence becomes problematic for large size configuration set, particularly if one wants to optimize all orbitals.

The paper is organized as follows. In Sec. 2 we present briefly the MCDF method, with emphasis on the specific features of the code we have been using, and we described the QED corrections that have been used in the calcula-tions. In Sec. 3 we describes specific problems associated with the MCDF method (or more generally to all-order methods). We thus study the non-relativistic limit of the MCDF codes, and specific problems associated with high-Z. In Sec. 3.2 we study a number of systems from highlycharges ions to neutral atoms, for very large atomic numbers. The evaluation of atomic charge distribution size and Landé factors is performed in Sec. 5 and in Sec. 6 we state our conclusion.

Calculation of atomic wavefunctions and transition probabilities 2.1 The MCDF method

In this work, bound-states wavefunctions are calculated using the 2006 version of the Dirac-Fock program of J.-P. Desclaux and P. Indelicato, named mdfgme [START_REF] Indelicato | Mcdfgme, a multiconfiguration dirac fock and general matrix elements program[END_REF]. Details on the Hamiltonian and the processes used to build the wave-functions can be found elsewhere [START_REF] Desclaux | [END_REF][START_REF] Desclaux | Methods and Techniques in Computational Chemistry[END_REF][START_REF] Indelicato | [END_REF]27].

The total wavefunction is calculated with the help of the variational principle. The total energy of the atomic system is the eigenvalue of the equation H no pair Ψ Π,J,M (. . . , r i , . . .) = E Π,J,M Ψ Π,J,M (. . . , r i , . . .),

(1) where Π is the parity, J is the total angular momentum eigenvalue, and M is the eigenvalue of its projection on the z axis J z . Here,

H no pair = N i=1 H D (r i ) + i<j V(|r i -r j |), (2)
where H D is the one electron Dirac operator and V is an operator representing the electron-electron interaction of order one in α. The expression of V ij in Coulomb gauge, and in atomic units,is

V ij = 1 r ij (3a) - α i • α j r ij (3b)
- α i • α j r ij [cos ω ij r ij c -1] + c 2 (α i • ∇ i )(α j • ∇ j ) cos ωij rij c -1 ω 2 ij r ij , (3c)
where r ij = |r ir j | is the inter-electronic distance, ω ij is the energy of the exchanged photon between the two electrons, α i are the Dirac matrices and c is the speed of light.

We use the Coulomb gauge as it has been demonstrated that it provides energies free from spurious contributions at the ladder approximation level and must be used in many-body atomic structure calculations [28,29]. The term (3a) represents the Coulomb interaction, the term (3b) is the Gaunt (magnetic) interaction, and the last two terms (3c) stand for the retardation operator. In this expression the ∇ operators act only on r ij and not on the following wavefunctions.

By a series expansion of the operators in expressions (3b) and (3c) in powers of ω ij r ij /c ≪ 1 one obtains the Breit interaction, which includes the leading retardation contribution of order 1/c 2 . The Breit interaction is, then, the sum of the Gaunt interaction (3b) and the Breit retardation

B R ij = α i • α j 2r ij - (α i • r ij ) (α j • r ij ) 2r 3 ij . (4)
In the many-body part of the calculation the electronelectron interaction is described by the sum of the Coulomb and the Breit interactions. Higher orders in 1/c, deriving from the difference between expressions (3c) and ( 4) are treated here only as a first order perturbation. All calculations are done for finite nuclei using a Fermi distribution with a tickness parameter of 2.3 fm. The nuclear radii are taken or evaluated using formulas from reference [30].

The MCDF method is defined by the particular choice of a trial function to solve equation ( 1) as a linear combination of configuration state functions (CSF):

|Ψ Π ,J,M = n ν=1 c ν |ν, Π , J, M . (5)
The CSF are also eigenfunctions of the parity Π , the total angular momentum J 2 and its projection J z . The label ν stands for all other numbers (principal quantum number, ...) necessary to define unambiguously the CSF. The c ν are called the mixing coefficients and are obtained by diagonalization of the Hamiltonian matrix coming from the minimization of the energy in equation ( 1) with respect to the c ν . The CSF are antisymmetric products of oneelectron wavefunctions expressed as linear combination of Slater determinants of Dirac 4-spinors

|ν, Π , J, M = Nν i=1 d i ψ i 1 (r 1 ) • • • ψ i m (r 1 ) . . . . . . . . . ψ i 1 (r m ) • • • ψ i m (r m ) , (6)
where the ψ-s are the one-electron wavefunctions and the coefficients d i are determined by requiring that the CSF is an eigenstate of J 2 and J z . The one-electron wavefunctions are defined as

ψ (r) = χ µ κ (Ω)P (r) iχ µ -κ (Ω)Q(r) , (7)
where χ µ κ is a two-component spinor, and P and Q are respectively the large and small component of the wavefunction.

Application of the variational principle leads to a set of integro-differential equations, which determines the radial wavefunctions and a Hamiltonian matrix, which provides the mixing coefficients c ν by diagonalization. The mdfgme code provide the possibility to obtain wavefunctions and mixing coefficient with either only the Coulomb (3a) interaction used to obtain the differential equations and the Hamiltonian matrix that is diagonalized to obtain mixing coefficient or the full Breit operator (4). The convergence process is based on the self-consistent field process (SCF). For a given set of configurations, initial wavefunctions, obtained for example, with a Thomas-Fermi potential, are used to derive the Hamiltonian matrix and set of mixing coefficients. Direct and exchange potential are constructed for all orbitals, and the differential equations are solved. Then a new set of potentials is constructed and the whole process is repeated. Each time the largest variation of all wavefunction has been reduced by an order of magnitude, a new Hamiltonian matrix is build and diagonalized, and a new cycle is done.

The so-called Optimized Levels (OL) method was used to determine the wavefunction and energy for each state involved. This allow for a full relaxation of the initial and final states and provide much better energies and wavefunctions. However, in this method, spin-orbitals in the initial and final states are not orthogonal, since they have been optimized separately. The formalism to take in account the wavefunctions non-orthogonality in the transition probabilities calculation has been described by Löwdin [31] and Slater [START_REF] Slater | Quantum Theory of Molecules and Solids[END_REF]. The matrix element of a one-electron operator O between two determinants belonging to the initial and final states can be written as

νΠ JM | N i=1 O (r i ) |ν ′ Π ′ J ′ M ′ = × 1 N ! ψ 1 (r 1 ) • • • ψ m (r 1 ) . . . . . . . . . ψ 1 (r m ) • • • ψ m (r m ) × m i=1 O (r i ) φ 1 (r 1 ) • • • φ m (r 1 ) . . . . . . . . . φ 1 (r m ) • • • φ m (r m ) , (8)
where the ψ i belong to the initial state and the φ i and primes belong to the final state. If ψ = |nκµ and φ = |n ′ κ ′ µ ′ are orthogonal, i.e., nκµ|n ′ κ ′ µ ′ = δ n,n ′ δ κ,κ ′ δ µ,µ ′ , the matrix element (8) reduces to one term ψ i | O |φ i where i represents the only electron that does not have the same spin-orbital in the initial and final determinants.

Since O is a one-electron operator, only one spin-orbital can change, otherwise the matrix element is zero. In contrast, when the orthogonality between initial and final states is not enforced, one gets [31,[START_REF] Slater | Quantum Theory of Molecules and Solids[END_REF] νΠ JM

| N i=1 O (r i ) |ν ′ Π ′ J ′ M ′ = i,j ′ ψ i | O |φ j ′ ξ ij ′ D ij ′ , (9)
where D ij ′ is the minor determinant obtained by crossing out the ith row and j ′ th column from the determinant of dimension N × N , made of all possible overlaps ψ k |φ l ′ and ξ ij ′ = ±1 the associated phase factor.

The mdfgme code take into account non-orthogonality for all one-particle off-diagonal operators (hyperfine matrix elements, transition rates. . . ). The overlap matrix is build and stored, and minor determinants are constructed, and calculated using standard LU decomposition.

Evaluation of QED corrections

In superheavy elements, the influence of radiative corrections must be carefully studied. Obviously the status of the inner orbital and of the outer ones is very different. It is not possible for the time being, to do a full QED treatment. Here we use the one-electron self-energy obtained using the method developed by Mohr [33,34]. These calculations have been extended first to the n = 2 shell [35,36] and then to the n = 3, n = 4 and n = 5 shells, for |κ| ≤ 2 [37]. More recently, a new coordinate-space renormalization method has been developed by Indelicato and Mohr, that has allowed substantial gains in accuracy and ease of extension [38,39]. Of particular interest for the present work, is the extension of these calculation to arbitrary κ values and large principal quantum numbers [40]. All known values to date have been implemented in the 2006 version of the mdfgme code, including less accurate, inner shell ones, that covers the superheavy elements [41,42]. The self-energy of the 1s, 2s and 2p 1/2 states is corrected for finite nuclear size [43]. The self-energy screening is taken into account here by the Welton method [44,45], which reproduces very well other methods based on direct QED evaluation of the one-electron self-energy diagram with screened potentials [46,47,48]. Both methods however leave out reducible and vertex contributions. These two contributions, however, cancels out in the direct evaluation of the complete set of one-loop screened self-energy diagram with one photon exchange [49]. The advantages of screening method, on the other hand is that they go beyond one photon exchange, which may be important for the outer shells of neutral atoms. Recently special studies of outer-shell screening have been performed for alkali-like elements, using the multiple commutators method [50,51].

The comparison between the Welton model and the results from Ref. [50,51] is presented in Table 1. This table confirms comparison with earlier work at lower Z. It shows that the use of a simple scaling law, as incorporated in GRASP 92 and earlier version of mdfgme does not provide correct values. This scaling law is obtained by comparing the mean value of the radial coordinate over Dirac-Fock radial wave-function r DF to the hydrogenic one r (Z eff ) hydr. . This allow to derive an effective atomic number Z eff by solving r (Z eff ) hydr. = r DF . One then use Z eff to evaluate the self-energy screening from oneelectron self-energy calculations. The superiority of the Welton model can be easily explained by noticing that the range of QED corrections is the electron Compton wavelength Λ C = α a.u., while mean atomic orbital radii are dominated by contributions from the n 2 Z a.u. range, which is much larger.

When dealing with very heavy elements in the limit Zα → 1, one should consider if perturbative QED is still a valid model. Unfortunately, we still lack the tools to answer this fundamental question. However, the use of numerical all-order methods may gives some partial answers. In particular they allow to include the leading contribution to vacuum polarization to all orders by adding the Uehling potential [53] to the MCDF differential equations. This possibility has been implemented in the mdfgme code as described in [54], with the help of Ref. [55].

It also allows to calculate the effect of vacuum polarization in quantities other than energies, like hyperfine structure shifts [54], Landé g-factors [56], or transition rates. It can also provides some hints on the oder of magnitude of QED effects on atomic wavefunction, orbital or atom radii, and electronic densities. For high-Z, higher order QED corrections are also important. In the last decade, calculations have provided the complete set of values for two-loops, one-electron diagrams to all orders in Zα [57,58,59,60,61] and also low-Z expansions. All available data has been implemented in the mdfgme code. However, this data is limited to the n ≤ 2 shells.

3 Limitation of the MCDF method 3.1 Non-relativistic limit

The success of relativistic calculations in high-Z elements atomic structure is impressive. It has been shown many times, that only a fully relativistic formalism and the use of a fully relativistic electron-electron interaction can reproduce the correct level ordering and energy in heavy systems. As a non-exhaustive list of example, we can cite the case of the 1s2p 3 P 0 -3 P 1 level crossing for Z = 47 in heliumlike systems [62,63,64], and the prediction of the 1s 2 2s2p 3 P 0 -3 P 1 inversion in Be-like iron [65], both due to relativistic effects and Breit interaction. Relativistic effects determine also the structure of the ground configuration of many systems, as was recognized for example in the study of lawrencium which has a 7p 2 P in place of a 6d 2 D configuration [66,67,68].

There is one caveat that must be taken into consideration when performing such calculations, that has been recognized in low-Z systems, but never explored in the super-heavy elements region: it may sound rather paradoxical to investigate the nonrelativistic limit of MCDF and, more generally, of all-order calculations, when studying superheavy elements. Here we show, however, that there is a problem that has to be taken into account if one wants to obtain the correct fine structure splitting in all cases. We believe it is the first time this problem is recognized in the highly-relativistic limit.

This problem was first found many years ago, in systems like the fluorine isoelectronic sequence [69]: the nonrelativistic limit, obtained by doing c → ∞ in a MCDF code, is not properly recovered. States with LSJ label 2S+1 L J and identical L and S, and different J, which should have had the same energy in the non-relativistic limit do not. The energy difference between the levels of identical LS labels but different J is called the nonrelativistic (NR) offset. This offset leads to slightly incorrect fine structure in cases when a relativistic configuration has several non-relativistic parents (i.e., several states with one electron less, and different angular structure, that can recouple to give the same configuration). This effect can be large enough to affect comparison between theory and experiment. It should be noted that such a problem does not show if one works in the Extended Average Level (EAL) version of the MCDF. In this case a single wavefunction is used for all the members of a given multiplet, and the relaxation effects that are the source of the NR offset disappear, at the price of less accurate transition rates and energies, for a given configuration space.

Very recently, this non-relativistic problem was shown to be general to any all-order methods, in which subclasses of many-body diagrams are re-summed, without including all the diagrams relevant of a given order. In the MCDF method, in particular, one should add all configurations with single excitations of the kind nκ → n ′ κ, that in principle, should have no effect on the energy in a non-relativistic calculation, due to Brillouin's theorem [70]. In the iso-electronic sequence investigated up to now, this effects became less severe when going to higher Z, and it has thus never been considered in very heavy systems: in neutral, or quasi-neutral superheavy elements, the outershell structure can be complex, with several open shells, and thus many possible parents core configurations. It is then worthwhile to study if this problem could arise. We have studied a number of cases. As a first example we have studied neutral uranium. The ground state configuration is known to be [Rn]5f 3 6d7s 2 5 L 6 . We have calculated all levels of the ground configuration with J = 0 to J = 9, both in normal conditions, and taking the speed of light to infinity in the code. The results are shown on Table 2 and Fig. 1. There are two group of levels that can be affected by a NR offset: the 5f 3 6d7s 2 5 H J , J = 3, 4 and the 5f 3 6d7s 2 5 L J , J = 6 to J = 9. The non-relativistic offset is evaluated for both groups of levels as the difference between the energy of a configuration with a given J and the one for the configuration with the lower energy in the NR limit. The figure clearly shows a NR offset of 25 meV for the 5 H 4 and 5 H 5 , and one of less than 1 meV for the four 5 L J levels. While it can be significant compared to the accuracy of a laser measurement, it is probably negligible compared to the accuracy of realistic correlation calculations.

In order to assess the generality of this problem, we have investigated several other characteristic systems. Element 125, for example, is the first element with a populated 5g orbital [3,4]. We have calculated the NR offset for a configuration with 125 electrons [Rn]5f 14 6d 10 7s 2 7p 6 8s 2 5g6f 4 and Z = 125, 135 and 140. The results are presented on Fig. 2. We find three groups of levels with LSJ labels, four levels with label 6 F J (J = 1/2 to J = 7/2), three levels with label 6 K J (J = 9/2 to J = 13/2) and four levels with label 6 N J (J=15/2 to J = 21/2). The NR offset in each group is of the order of a few meV, while the (nonrelativistic) energy difference between the two first groups is 0.19 eV, and between the first and the last groups is around 0.25 eV. The figure also shows that the NR offset gets smaller when Z increases as expected.

Table 2. Total relativistic energy [Ener. (BSC)], including allorder Breit and QED corrections, total non-relativistic energy [ Ener. (NR)], and NR Offset for the ground configuration of uranium, relative to the 5f 3 6d7s 2 5 L6 energy, which is the lowest level of the configuration. Correlation has not been included. One can observe a splitting in the NR energy between the 5 H levels and the 5 L levels, which should be exactly degenerate. The fine-structure can be improved by subtract the NR offset from the relativistic energy. We have also investigated the lower excited states of a somewhat simpler system, element 118 (eka-radon). We have explored the [Rn]5f 14 6d 10 7s 2 7p 5 8s which should exhibit no NR offset (it has single parent states) and [Rn]-5f 14 6d 10 7s 2 7p 5 7d. The results are presented on Table 3. As expected, the 7p 5 8s 3 P J states do not exhibit an NR offset, within our numerical accuracy. The 7p 5 7d 3 L J configurations however do have a strong NR offset, up to 0.8 eV, much larger than we expected from the other results presented above, and the largest ever observed. Clearly, such a large offset would render any calculation of the fine structure splitting of eka-radon useless, unless the results are corrected for the NR offset.

Label Ener. (BSC) Ener. (NR) NR Offset

We would like to note that subtracting the NR offset is only a partial fix, since it was shown in Ref. [70] that not only the fine structure is affected, but also the level energy. The only possible solution are thus to do calcu- Fig. 1. Non-relativistic offset on the ground configuration of uranium. "Ener. (BSC off. sub)": total MCDF energy, with all QED corrections, using the full Breit operator in the SCF process, relative to the 5f 3 6d7s 2 5 L6 energy to which the nonrelativistic offset has been subtracted. "Ener. (NR off. sub)": Total non-relativistic energy, to which the non-relativistic offset for members of the same LS multiplet has been subtracted, relative to the 5f 3 6d7s 2 5 L6 energy. The two 5 HJ and the four 5 LJ levels have thus identical non-relativistic energy as it should be. Uncorrected energy and NR offset are displayed in Table 2 [6 lations with a large number of configurations, including all single excitations, or to use the EAL method. In view of the complexity of calculations on the superheavy elements, and of the extra convergence difficulties associated with the presence of Brillouin excitations, the first solution is probably not very universal, but the second one should always work. In conclusion, we have shown for the first time, that it is important to check for the non-relativistic Table 3. Non-relativistic offset on the lower excited configurations of eka-radon (Z=118). For each member of a multiplet, as, e.g., 7p 5 7d 3 D, we evaluate the difference between the member with the lowest non-relativistic energy and the others. Non-relativistically, all member of a multiplet with identical LS labels should have an energy independent of the total angular momentum J. limit in super-heavy elements and to correct for the nonrelativistic offset, as it can be large in some cases.

[3P0] [5D1] [5G2] [5H3] [5H4] [5K5] [5L6] [5L7] [5L8] [5L9]
F 1 /2 ] [6 F 3 /2 ] [6 F 5 /2 ] [6 F 7 /2 ] [6 K 9 /2 ] [6 K 1 1 /2 ] [6 K 1 3 /2 ] [6 N 1 5 /2 ] [6 N 1 7 /2 ] [6 N 1 9 /2
J 7p 5 7d 3 D J 7p 5 7d 3 F J 7p

Effect of the all-order Breit operator on simple systems

The use of different form of the electron-electron interaction in the self-consistent field process has a profound qualitative influence on the behavior of variational calculations that goes beyond changes in energy. In particular the mixing coefficients between configurations contributing to intermediate coupling are strongly affected (and thus the values of many operators would be likewise affected). For example, let us examine the very simple case of the 1s2p 3 P 1 state in two-electron system. In a MCDF calculation, intermediate coupling is taken care of by calculating

| 1s2p 3 P 1 = c 1 | 1s2p 1/2 J = 1 + c 2 | 1s2p 3/2 J = 1 .
The evolution as a function of the atomic number Z of the c 2 coefficient, is plotted on Fig. 3, with only the Coulomb interaction, or the full Breit interaction made self-consistent. The figure shows clearly that the inclusion of the Breit interaction in the SCF process lead to values of c 2 that are one order of magnitude lower at high-Z that when only the Coulomb interaction is included. It means that the JJ coupling limit is reached much faster. This has some influence even in the convergence of the calculation: as the exchange potential for the 2p 3/2 orbital is

proportional to c1 c2 2
, it becomes very large, and the calculation does not converge. This can be traced back to a negative energy continuum problem. If we use the method described in Ref. [START_REF] Indelicato | [END_REF] to solve for the 2p 3/2 , then convergence can be reached, provided the projection operator that suppress coupling between positive and negative energy solution of the Dirac equation is used. On Fig. 4, the different contributions to the 1s2p 3 P 1 level energy are plotted at Z = 40. The minimum, which corresponds to the level energy, is obtained for a mixing coefficient (habitually obtained by diagonalization of the Hamiltonian matrix), c 2 = 0.104. The shape of the magnetic and retardation energy contribution, as observed on the figure, shows clearly that the curve used to find the minimum (which represents the sum of the contribution of the mean values of the operators in Eqs. (3a), Eqs. (3b), and ( 4)) is shifted to the left compared to the pure Coulomb contribution. This explain why the mixing coefficients gets much smaller when using the Breit interaction in place of the Coulomb interaction in the SCF process. At high-Z this lead to an extra difficulty to achieve convergence: as can be seen in Figs. 5 and6, the minimum corresponding to the 1s2p 3 P 1 state (the one lower in energy) is very close to c 2 = 0. It thus sometimes happens during the convergence, that c 2 changes sign, leading to very tedious tuning of the convergence process. This is even worse for the 1s2p 1 P 1 state, because one is trying to reach the maximum energy. In that case the oscillation of the c 1 coefficient around zero are impossible to damp. Obviously, such problems will slowly disappear when going to neutral systems. For example, in neutral nobelium (Z = 102) c 2 changes only from 0.47250012 to 0.47316712. It is interesting to investigate simple many-body system, that can be calculated accurately, to see which kind of highly relativistic effects can be expected in the limit Zα → 1. In that sense the beryllium isoelectronic sequence is an interesting model case, as it exhibit a very strong intrashell coupling between the 1s 2 2s 2 J = 0 and 1s 2 2p 2 1/2 J = 0 configurations, which are almost degenerate in energy. We thus calculated all contributions to the energy, with and without including the vacuum polarization and Breit interaction in the SCF process. From this we could deduce the loop-after-loop Uehling contribution to the total energy, and the intrashell correlation. Most quantities contributing to the total energy do not exhibit any specific behavior when Zα → 1. However, a major changes in behavior of the system occurs around Z = 125 as shown in Fig. 7. One can see that the ground state, which is 1s 2 2s 2 J = 0 at lower Z, becomes 1s 2 2p 2 1/2 J = 0. This can be seen on the mixing coefficients as plotted on Fig. 7. This translates into a strong increase in the loop-afterloop vacuum polarization contribution. Obviously, if we were able to evaluate other second-order QED calculation than loop-after-loop vacuum polarization, including off-diagonal two-electron self-energy matrix elements for quasi-degenerate state, following recent work on heliumlike systems [71,72,73], there could be more unexpected effects to observe.

While not displaying such a feature, the total correlation energy increases strongly, reaching up to 3.6 keV. One can observe effects on other properties of the atom, like orbital energies and mean orbital radius. Figure 8 shows that, in the same atomic number range when the ground state changes of structure, the 2p 3/2 orbital radius and energy exhibit a very strong change. The behavior of the ground state must be connected to the fact that the small component of 2p 1/2 orbitals, as can be seen from Eq. ( 7), has a s behavior, and the ratio between small and large component is of order Zα. It is thus understandable that such effects could occurs when Zα → 1. It should be noted that this effects happens even with a pure Coulomb electron-electron interaction, which is one more proof it is only connected with the behavior of the oneelectron wavefunctions. We investigated the similar case of the magnesiumlike sequence, which exhibit strong intrashell coupling between the 3s, 3p and 3d orbitals, but we could not observe any effect on energies in this range of Z. However the [Ne]3p 2 1/2 mixing coefficient started to increase faster around Z = 128, but convergence problems prohibited us to investigate higher Z.

Relativistic correlations on the neon isoelectronic sequence

Calculating completely correlated energies can be performed only on relatively small systems. Neonlike ions, with 10 electrons are a small enough system that can be calculated with rather large basis sets. We have extended the calculation performed in Ref. [7] to superheavy elements (Z = 134). All 10 electrons are excited to all virtual orbitals up to a maximum nℓ, that is varied from 3d to 6h. The results are presented in Table 4 and plotted on Fig. 9.

The trend found up to Z = 94 in Ref. [7] extend smoothly to larger Z, but is enhanced. The total correlation energy becomes very large. It doubles when going from Z = 95 to Z = 134, the highest Z for which convergence could be reached. The speed of convergence as a function of nℓ does not change with increasing Z.

Transition energies and probabilities in nobelium and element 118 (eka-radon)

In this section we study the different contributions to the energy and transition probabilities of nobelium (Z = 102) and eka-radon (Z = 118). Nobelium is the next candidate for measurement of its first excited levels transition en- ergy by laser spectroscopy. Large scale calculations have been performed recently [21]. Here we examine a number of corrections not considered in Ref. [21]. For the 7s 2 1 S 0 ground state, we did a MCDF calculation taking into account all single and double excitations, except the one corresponding to the Brillouin theorem, from the 5f and 7s shells to the 7p, and 6d shells (48 jj configurations). For the excited states we included excitations from 5f , 7s and 7p shells to the 7p and 6d ones. This leads to 671 jj configurations for the 7s7p 3 P 1 and 7s7p 1 P 1 states and 981 for 7s7p 3 P 2 . The results of this calculation are presented on Table 5 for the transition energies and on Table 6 for transition probabilities. It is clear from these two tables that many contributions that are important for level energies are completely negligible for transition probabilities in this case. This is not true for highly charged ions, even at much lower Z. In particular two-loop QED corrections, even though many of them have not been calculated from n > 2, should remain completely negligible, since they will be of the same order of magnitude as the loop-after-loop and Källèn and Sabry contributions. The self-energy screening is almost exactly compensating the self-energy, but leaves a contribution that should be visible if one can calculate correlation well enough. At the present level of accuracy, calculation of the pure Coulomb correlation is the real challenge as it requires considerable effort on the size of configuration space.

The transition probabilities on Table 6 have been evaluated with different approximation, using theoretical energies. Here the choice of the wavefunction optimization technique has a sizeable effect. It remains small compared to correlation, but still at the level of 0.2%. We also investigated the effect of using fully relaxed orbitals on both initial and final state. In particular we checked at the Dirac-Fock level of approximation, what is the order of magnitude of taking into account non-orthogonality between initial and final state orbitals. We found 2.2% for the 3 P 1 → 1 S 0 transition, 0.014% for the 1 P 1 → 1 S 0 transition and -0.025% for the 3 P 2 → 1 S 0 . For calculations between correlated wavefunctions, it is anyway im- Orbital energy (eV)

c 2 OE(1s) OE(2s) OE(2p 1/2 ) OE(2p 3/2 ) r(1s) r(2s) r(2p 1/2 ) r(2p 3/2 )
Orbital radii (a.u.)

Fig. 8. Orbital radii (right axis) and one-electron energies (left axis) for Be-like ions portant to evaluate the matrix elements for off-diagonal operators, taking account non-orthogonalities between initial and final state orbitals, as it can change dramatically the contribution of a given CSF, since overlaps between correlations orbitals in initial and final states can be very different from either one or zero.

As another example we have also calculated several transition energies and rates for element 118, which are displayed in Table 7. For the most intense 7p 5 7d → 7s 2 7p 6 transitions and for the 7p 5 8s → 7s 2 7p 6 transitions we did a MCDF calculation, taking into account all single and double excitations, (except the ones corresponding to the Brillouin theorem) from the 7s and 7p shells to the 7d and 6f shells for the 7p 6 1 S 0 ground state (38 jj configurations) and for the 7p 5 7d excited states (657 jj configurations), and from the 7p and 8s shells to the 7d and 6f shells for the 7p 5 8s excited states (151 jj configurations).

The transition energy was calculated for some of these transitions, with and without including the vacuum polarization and Breit interaction in the SCF process, and we concluded that the transition energy is not significantly affected by the inclusion of these interactions in the SCF. This is illustrated on Fig. 10 that shows the transition energy values for 7p 5 7d 3 P 1 transition. Table 5. Transition energies for the lower energy levels of nobelium (eV). Coul.: DF Coulomb energy. Mag. (pert), Ret. (pert.), Higher order ret. (pert.): Contribution of the Magnetic, Breit retardation and higher-order retardation in first order of perturbation. All order Breit: effect of including the full Breit interaction in the SCF. Coul. Corr.: Coulomb correlation. Breit Corr.:Contribution of all Breit terms to the correlation energy. Self-energy (FS): self-energy and finite size correction. Self-energy screening: Welton approximation to self-energy screening. Vac. Pol. (Uehling): mean value of the Uehling potential (order α(Zα)). VP (muons, Uehling): vacuum polarization due to muon loops. VP Wichman and Kroll: correction to the Uehling potential (order α(Zα) 3 ). Loop-after-loop V11: iterated Uehling contribution to all orders. VP (Källèn et Sabry): twoloop contributions to vacuum polarization. Other 2nd order QED: sum of two-loop QED corrections not accounted for in the two previous one. Recoil: sum of lowest order recoil corrections (see, e.g., [74]). The number of digits presented in the table is not physically significant, but is necessary to show the size of some contributions. The physical accuracy is not better than 1 digits. 5 Relativistic and QED effects on Landé factors, atomic radii and electronic densities

Contribution 3 P1 → 1 S0 1 P1 → 1 S0 3 P2 → 1 S0
Although transition energies and probabilities, as well as ionization potential, are important quantities, it is interesting to study relativistic and QED effects on other atomic parameters, like Landé g-factors, atomic radii and electronic densities. Landé factors, which define the strength of the coupling of an atom to a magnetic field, can help characterize a level.

In some experiments concerning superheavy elements, singly ionized atoms are drifted in a gas cell, under the influence of an electric field. The drift speed can be related, in first approximation, to the charge distribution radius of the ion [75,76]. In this context, it can also be interesting to look at the atomic density, and see how it is affected by relativistic and QED effects. In that case, though, we can only get a feeling of this effect by comparing densities calculated with and without the Breit interaction self-consistent, or with and without the Uehling potential included in the differential equation. There is currently no formalism that would enable to account for changes in the wavefunction related to the self-energy.

We define the radial electronic density as

r 2 ρ (r) = dΩr 2 ρ (r, Ω) = i∈occ.orb. ̟ i P i (r) 2 + Q i (r) 2 , (10)
where P and Q are defined in Eq. (7), and the ̟ i are the orbital effective occupation numbers,

̟ i = j c 2 j n i j , (11)
where the sum extend over all configuration containing orbital i, and n i j is the occupation number of this orbital in the configuration j. The density is normalized to ∞ 0 drr 2 ρ (r) = N e , where N e is the number of electrons in the atom or ion.

The effect of the Breit interaction and vacuum polarization on the charge density of the ground state of Fm + ([Rn]5f 12 6s) is shown on Fig. 11. The inclusion of both contributions leads to local changes of around 1% in the charge density. It is rather unexpected that both contribution extend their effects way pass their range, in particular for the vacuum polarization potential, which has a very small contribution for r > 1/α = 0.0073 a.u. Fig. 11. Total electronic density of Fm + . The vertical lines represents, from left to right the mean radius, the mean spherical radius ("rms") and the mean radius of the outermost orbital (7s). "Density": total electronic density Eq. (10). "Diff (Bsc)": absolute value of the variation in the density due to the inclusion of the Breit interaction in the SCF. "Diff (Vsc)": absolute value of the variation in the density due to the inclusion of the Uehling potential in the differential equations (NB: the dip in the curve correspond to sign changes of the correction).

Charge distribution can be described in a number of ways, and the definition of atomic or ionic radius has evolved over the years. Slater noticed a long time ago the correlation between the value of the maximum charge density of the outermost core electron shell and the ionic radius of an atom [77]. Such a definition, however, does not provide a way to take into account mixing of outer shell in MCDF calculations. Other authors have chosen either the mean radius of a specific orbital r n,l,j [78] or weighted mean radius [79]. Results have been obtained for bohrium and hassium. It is not at all obvious either than what is true in crystals must be valid for singly charged ions drifting in a gas. Here we have evaluated four different quantities and their dependence in the Breit interaction and vacuum polarization. We have evaluated the position of the maximum density of the outermost orbital, the mean radius of the outermost orbital r n,l,j , the atom mean radius, and the atom mean spherical radius. The two first quantities have been tested in detail, but they cannot represent the ionic radius when one has a complex outer shell structure or when one calculate correlation. The mean radius of the atom is represented as

r (p) at. = 1 N e ∞ 0 drr p r 2 ρ (r) = 1 N e i∈occ.orb. ̟ i r (p) ni,li,ji (12)
for p = 1, and the mean spherical radius is obtained as r

at. . Both can be calculated in a MCDF model, as the ̟ i contain both the occupation numbers and the mixing coefficients. As an example, we have plotted the contribution to r ni,li,ji for the different orbitals. Contrary to the individual orbitals quantities or to the average performed only on the outer shell, like in Ref. [9], both quantities have sizeable contributions from several outer shells. The 7s, 6p, 5f and 6d contribute significantly to both the mean and the mean spherical atomic radius, but the contribution of the 6d orbitals is more dominant for the mean spherical radius. The results for singly charged ions with 57 ≤ Z ≤ 71 and 90 ≤ Z ≤ 108 are presented on Table 8 and plotted on Fig. 13. The comparison between mean radius and mean spherical radius with the Breit interaction in the SCF process with Coulomb values, shows changes around 0.04% to 0.08% depending on the atomic number. Self-consistent vacuum polarization has an effect 2 to 4 times smaller, depending on the element. We emphasize the fact that we used for the ion the configuration corresponding to the ground state of a Dirac-Fock calculation (i.e., without correlation), as given in [6]. For a few elements, the physical ground state configuration, as given for example on the NIST database, is different.

We have used the definition above to evaluate the effect of correlation on the ion radii. The calculation has been performed on neutral nobelium (Z = 102). The results are presented on Table 9. One can see that the radius of maximum density follows exactly the trend of the mean radius of the outer orbital. However the atomic mean radius and mean spherical radius follow a different trend.

There has been two experiments measuring drift time of singly ionized transuranic elements in gases. One mea-sured the relative velocity of Am + with respect to Pu + [75] and an other one measured the same quantity for Cf + and Fm + [18]. The results of these experiments are expressed as

∆r A-B = r A -r B r B . (13)
The experiments above provide ∆r Am + -Pu + =-3.1 ± 1.3% and ∆r Fm + -Cf + =-2% respectively, which shows a shrinkage of Am as compares to Pu and of Fm as compared to Cf. The comparison between these experiments and the calculation of singly charged ion radii from Table 8 is shown in Table 10. Although it is difficult to draw any firm conclusion from so few data, it seems that r

(2)

at. reproduces best the experiment, followed by r

(1) at. . We want to point out that the < r 7s > values for neutral atom, from Ref. [5], or ours, which are in good agreement gives a -1.5% change for Am/Pu and -3.0% for Fm/Cf, compare well with experiment. The 7s radius of the singly charged ions, as shown in Table 8, increases when going from Am to Pu, thus leading to a ratio of the wrong sign, while the 5f average radius, or the global atomic radii as defined here all show the right trend. The change of behavior for Am, is due to the relative diminution of the l(l + 1)/r 2 barrier compared to the Coulomb potential as a function of Z. The 5f radius reduces then strongly, leading to the change of the structure of the ground configuration, and to the change of the radius of the 7s orbital. to Sg+ of individual values of ̟i r (p) n i ,l i ,j i for the different orbitals and p = 1 and 2, together with the total values on the right. Stared orbital labels correspond to the orbital with j = l -1 2 .

QED corrections on Landé factors of one and three electron atoms have been studied in great details in the last few years, because of the increase in experimental accuracy associated with the use of Penning traps in a new series of experiments [80,81]. These experiments provided a strong incentives to evaluate very accurately QED corrections on Landé factors, beyond the one due to the anomalous magnetic moment of the electron [82,83,84, 85,86,87,88,89]. To our knowledge, there has been only one calculation dealing with QED corrections in heavy elements, in which the Feynman diagrams corresponding to self-energy and other corrections were evaluated. This calculation concerned alkali elements up to francium (Z = 87) [90]. Here we deal with much more complex system, with several open shells, and we calculate QED corrections due to the inclusion of the Breit interaction and of the vacuum polarization in the SCF, as well as the contribution from the electron anomalous moment. The coupling of an atom with a magnetic moment µ to a homogeneous magnetic field B gives an energy change

-Pu + [75] -3.1% (± 1.3%) < rAt. > -2.1% < r 2 At. > -3.1% < r 5f > -0.8% < r7s > 2.8% ∆r Fm + -Cf + [18] -2.0% < rAt. > -1.7% < r 2 At. > -2.4% < r 5f > -5.9% < r7s > -1.3%
∆E = -µ • B ( 14
)
where µ = -g J µ B J, g J being the Landé factor and µ B the Bohr magneton. The anomalous electron magnetic moment corrections can be written (see, e.g., [91] and reference therein) as As an example, we illustrate the different effects by evaluating the Landé g-factors for the ground configuration of neutral actinide and transactinide up to Z = 106, and of singly ionized lanthanide, actinide and transactinide up to Z = 106. The results are displayed on Table 11 for neutral atoms and on Table 12, for singly ionized atoms. The ground configuration of neutral atom comes from [START_REF] Dragoset | Periodic Table: Atomic properties of the elements[END_REF]. For singly ionized atoms, it has been taken from Refs. [6] for Z up to 106 and from [9] for Z = 107 and 108. Depending on the outer shell structures, the QED corrections can be dominated by either the g -2 correction or the Breit correction (order of 0.1% of the Landé factor). The vacuum polarization correction is at best one order of magnitude smaller at Z = 106.

∆g J = (g -2) J||βΣ||J J(J + 1)(2J + 1) (15)
In Table 13, we present correlation effects on the Landé g-factor of the lowest levels of No, evaluated with the same wavefunctions as in Sec. 4.3, with the Breit operator in the SCF process. The effect of correlation ranges from 0.6% to 0.02% depending on the level.

Conclusions

In this work we have evaluated the effects of self-consistent Breit interaction and vacuum polarization on level energies, transition energies and probabilities Landé g-factors on super-heavy elements. We have also studied their effect on orbitals and atomic charge distribution radii and found very large effects on highly charged ions. We have found some hints, when studying Be-like ions, of rather strong non-perturbative correlation effects for Z ≈ 128. For neutral or quasi-neutral systems, self-consistent Breit interaction and vacuum polarization have a small but noticeable effects on Landé factor and transition rates, which could be felt experimentally. Transition energies, on the other hand, are heavily dominated by Coulomb correlation. This is rather good news, since treating the Breit interaction self-consistently obliges to evaluate magnetic and retardation integrals during the SCF process, which are about one order of magnitude more numerous than Coulomb ones, leading to calculations that cannot fit on even on the largest computers available today, even with relatively small configuration space.

We have shown that very large non-relativistic offset may affect the fine structure separation of elements with several open shells. This should be carefully taken into account to avoid providing completely wrong results with all-order methods. We have also shown that the inclusion of the Breit interaction in the SCF process, because it get mixing coefficients closer to the jj limit, can noticeably complicate numerical convergence.

Fig. 2 .

 2 Fig.2. Non-relativistic offset for the 6 FJ , 6 KJ and 6 NJ LS configurations for the ground configuration of an atom with 125 electron. Z = 125, 130 and 1340 have been evaluated. This contribution should be negligible compared to correlation. For each LS group, the offset is evaluated by subtracting the lower energy of the group to the others.

Fig. 3 .

 3 Fig. 3. Variation with Z of mixing c2 coefficient for the 1s2p 3 P1 level of helium-like ions. BP: Only the Coulomb interaction is used in the SCF process. BSC: The full Breit interaction is used in the SCF process

Fig. 4 .

 4 Fig. 4. Variation of the different contributions to the energy as a function of the mixing coefficient c2, for the 1s2p 3 P1 level at Z = 40. The arrow indicates the position of the 1s2p 3 P1 energy, at the minimum around c2 = 0.104. Left axis: "Mag.": Magnetic energy, Eq. (3b). "Ret.": Breit retardation, Eq. (4). "H.O. Ret.": higher-order retardation. Right axis: "Coul.": Coulomb energy Eq. (3a). "Total e-e": sum of the 4 preceding contributions. " Total": total level energy including all QED corrections.

Fig. 5 .

 5 Fig. 5. Variation of the different contributions to the energy as a function of the mixing coefficient c2, for the 1s2p 3 P1 level at Z = 92. The arrow indicates the position of the 1s2p 3 P1 energy, at the minimum around c2 = 0.002. Left axis: "Mag.": Magnetic energy, Eq. (3b). "Ret.": Breit retardation, Eq. (4). "H.O. Ret.": higher-order retardation. Right axis: "Coul.": Coulomb energy Eq. (3a). "Total e-e": sum of the 4 preceding contributions. " Total": total level energy including all QED corrections.

Fig. 6 .

 6 Fig. 6. Variation of the different contributions to the energy as a function of the mixing coefficient c1, for the 1s2p 1 P1 level at Z = 92. The arrow indicates the position of the 1s2p 1 P1 energy, at the maximum around c1 = 0.002. Left axis: "Mag.": Magnetic energy, Eq. (3b). "Ret.": Breit retardation, Eq. (4). "H.O. Ret.": higher-order retardation. Right axis: "Coul.": Coulomb energy Eq. (3a). "Total e-e": sum of the 4 preceding contributions. " Total": total level energy including all QED corrections.

Fig. 7 .

 7 Fig.7. Loop-after-loop Uehling contribution to berylliumlike ions ground state energy (changed of sign), obtained by including the Uehling potential in the SCF. The intrashell correlation energy is also plotted (left axis), as well as the square of the mixing coefficients of the 1s 2 2s 2 J = 0, 1s 2 2p 2 1/2 J = 0 and 1s 2 2p 2 3/2 J = 0 configurations (right axis).

Fig. 9 .

 9 Fig.9. Total correlation energy for neonlike ions, with the Breit interaction included in the SCF, as a function of the most excited orbital included in the basis set.

Fig. 10 .

 10 Fig. 10. Changes in the 7p 5 8s 3 P1 → 7s 2 7p 6 1 S0 transition energy with respect to the approximation made, for element 118. BSC: Breit self-consistent. VPSC: Uelhing potential self consistent. BP: Breit in perturbation. VPP: Uelhing vacuum polarization in perturbation.

  Sg + on Fig. 12 of individual values of ̟ i r (p)

Fig. 13 .

 13 Fig. 13. Variation of < r >, √ < r 2 > (right scale), of the radius of the outermost orbital < rout > and of the maximum of the charge density of the outer orbital (Rmax. dens.) (left scale as a function of the atomic number Z.

where g = 2 1

 1 + α π + • • • is the electron magnetic moment, β and Σ are 4 × 4 matrices, β = I is a 2 × 2 identity matrix and σ are Pauli matrices.

Table 1 .

 1 Self-energy and self-energy screening for element 111

	Level	1s	7s
	SE (point nucl.)	848.23	3.627
	Welton screening Finite size Total SE (DF)	-18.80 -50.37 779.06	-3.283 -0.260 0.084
	Pyykkö et al. [51] a		0.087
	Pyykkö et al. [51] b		0.095
	r (GRASP) [52] c		0.018
	a screening calculated using Dirac-Fock potential
	b screening calculated using Dirac-Slater potential fitted to
	EDF
	c use hydrogenic values with Z eff obtained by solving
	r (Z eff ) hydr. = r (Z eff ) DF

Table 4 .

 4 Total correlation energy for neonlike ions, with the Breit interaction included in the SCF, as a function of the most excited orbital included in the basis set.

	Z 10 15 25 35 45 55 65 75 85 95 105 -13.205 all → 3d all → 4f all → 5g all → 6h -5.911 -8.306 -9.339 -9.709 -5.989 -8.712 -9.838 -10.280 -6.374 -9.310 -10.494 -10.967 -6.710 -9.850 -11.099 -11.609 -7.074 -10.482 -11.816 -12.372 -7.515 -11.269 -12.710 -13.322 -8.067 -12.260 -13.833 -14.511 -8.752 -13.375 -15.247 -16.007 -9.772 -15.119 -17.056 -17.916 -11.160 -17.231 -19.429 -20.415 -20.146 -22.700 -23.853 114 -15.975 -23.966 -26.974 -28.332 124 -21.222 -30.897 -34.694 -36.389 130 -26.714 -37.777 -42.304 -44.300 134 -32.248 -44.412 -49.598 -51.860

Table 6 .

 6 Effect of correlation, Breit interaction and all order vacuum polarization on transition probabilities of nobelium. DF:

		Coul.	1.69909	3.20051	2.19931
		Mag. (pert) Ret. (pert.) Higher order ret. (pert.) All order Breit (pert) Coul. Corr.	0.00070 -0.00048 -0.00119 0.00002 0.59560	-0.00420 0.00005 0.05048 -0.05877 0.60414	-0.00347 -0.00058 -0.00231 0.00003 0.55545
		Breit Corr. Self-energy (FS) Self-energy screening	-0.00208 -1.60884 1.59030	-0.00229 -1.72025 1.72940	-0.00194 -1.76085 1.74228
		Vac. Pol. (Uehling)	0.00610	0.00435	0.00648
		VP (muons. Uehling)	0.00000	0.00000	0.00000
		VP Wichman and Kroll Loop-after-loop Uehl.	-0.00030 0.00002	-0.00019 0.00012	-0.00033 0.00003
		VP (Källèn and Sabry)	0.00004	0.00002	0.00005
		Other 2nd order QED	0.00000	0.00000	0.00000
		Recoil Total	-0.00082 2.28	-0.00081 3.80	-0.00082 2.73
		Ref. [21] II	2.34	3.49
		Ref. [21] I	2.60	3.36
	Dirac-Fock. VP: vacuum polarization. ex.: excitation
	initial final 7s7p 3 P1 → 7s 2 1 S0 (E1) Breit Pert.; VP. Pert. Breit Pert.; VP SC Breit SC; VP Pert. Breit SC; VP SC 7s7p 1 P1 → 7s 2 1 S0 (E1) Breit Pert.; VP. Pert. Breit Pert.; VP SC Breit SC; VP Pert. Breit SC; VP SC 7s7p 3 P2 → 7s 2 1 S0 (M2) Breit Pert.; VP. Pert. Breit Pert.; VP SC Breit SC; VP Pert. Breit SC; VP SC	7s7p 2S+1 PJ DF → 7s 2 1 S0 DF 1.071 × 10 6 1.073 × 10 6 1.062 × 10 6 1.064 × 10 6 2.705 × 10 8 2.697 × 10 8 2.711 × 10 8 2.704 × 10 8 1.717 × 10 -4 1.713 × 10 -4 1.721 × 10 -4 1.717 × 10 -4	7s7p 2S+1 PJ ex. 7s, 7p → 7p, 6d 7s7p 2S+1 PJ ex. 5f , 7s, 7p → 7s, 7p, 6d → 7s 2 1 S0 ex. 7s → 7p, 6d → 7s 2 1 S0 ex. 7s → 7p, 6d 3.520 × 10 6 1.947 × 10 6 3.526 × 10 6 1.954 × 10 6 3.501 × 10 6 1.920 × 10 6 3.507 × 10 6 1.927 × 10 6 3.500 × 10 8 3.680 × 10 8 3.559 × 10 8 3.738 × 10 8 3.509 × 10 8 3.686 × 10 8 3.568 × 10 8 3.743 × 10 8 5.187 × 10 -4 5.492 × 10 -4 5.173 × 10 -4 5.477 × 10 -4 5.200 × 10 -4 5.513 × 10 -4 5.187 × 10 -4 5.499 × 10 -4

Table 7 .

 7 Transition energies and transition rates to the ground state 5f 15 6d 10 7s 2 7p 6 1 S0 of eka-radon (element 118). The transition values from states labeled with c were calculated with correlation up to 6f , as well as the ground state.

	Initial State			Transition Energy	Transition Rate
				(eV)	(s -1 )
	5f 15 6d 10 7s 2 7p 5 7d 3 D1 + 3 P1 c 3 P1 c 3 D1 + 1 P1 c 3 P2 3 F2 3 F3 1 D2 3 D3 3 F4 3 P0 c	10.43 6.81 7.2 6.08 16.95 6.15 6.25 6.22 6.01 6.64	9.86 × 10 8 5.53 × 10 7 9.98 × 10 6 1.41 × 10 -2 1.11 × 10 -2 1.30 × 10 -3 1.10 × 10 -3 7.52 × 10 -4 3.91 × 10 -14
	5f 15 6d 10 7s 2 7p 5 8s	1 P1 3 P2	c c	4.73 4.3	2.04 × 10 8 2.04 × 10 -3

Table 8 .

 8 Singly ionized atom radii (a.u.) for lanthanide 57 ≤ Z ≤ 71 and actinide and transactinide 90 ≤ Z ≤ 108

	Z	Config.	Label	< r >	√ < r 2 >	outer orb.
						lab.	Rmax	< 4f 5/2 >	< 4f 7/2 >	< 5d3/2 >	< 5d 7/2 >	< 6s 1/2 >
	57	5d 2	3 F2	0,6903	1,0365	5d	2,2820			2,8629	2,8900
	58	4f 5d 2	4 H 7/2	0,6789	1,0121	5d	2,2005	1,0862	1,1033	2,7568	2,7949
	59	4f 3 6s	5 I4	0,6750	1,0458	6s	3,8004	1,0589	1,0667			4,2924
	60	4f 4 6s	6 I 7/2	0,6646	1,0256	6s	3,7359	1,0054	1,0190			4,2252
	61	4f 5 6s	7 H2	0,6544	1,0062	6s	3,6741	0,9624	0,9796			4,1608
	62	4f 6 6s	8 F 1/2	0,6442	0,9874	6s	3,6171	0,9249	0,9392			4,1012
	63	4f 7 6s	9 S4	0,6342	0,9692	6s	3,5627	0,8920	0,8999			4,0438
	64	4f 7 5d6s	10 D 5/2	0,6357	0,9708	6s	3,3408	0,8218	0,8221	2,4547	2,4846	3,7930
	65	4f 9 6s	7 H8	0,6162	0,9373	6s	3,4711	0,8502	0,8541			3,9568
	66	4f 10 6s	6 I 17/2	0,6074	0,9219	6s	3,4289	0,8254	0,8345			3,9163
	67	4f 11 6s	5 I8	0,5987	0,9070	6s	3,3871	0,8022	0,8165			3,8758
	68	4f 12 6s	4 H 13/2	0,5903	0,8926	6s	3,3464	0,7804	0,7995			3,8359
	69	4f 13 6s	3 F4	0,5820	0,8786	6s	3,3070	0,7605	0,7811			3,7976
	70	4f 14 6s	2 S 1/2	0,5739	0,8649	6s	3,2699	0,7430	0,7613			3,7611
	71	4f 14 6s 2	1 S0	0,5876	0,9133	6s	2,9761	0,6922	0,7057			3,4229
								< 5f 5/2 >	< 5f 7/2 >	< 6d 3/2 >	< 6d 7/2 >	< 7s 1/2 >
	90	5f 2 6d	4 K 11/2	0.5873	0.9117	6d	2.5751	1.96004	1.95812	3.16141	3.18613
	91	5f 2 6d7s	5 K5	0.5977	0.9521	7s	3.5491	1.54181	1.57544	2.85704	2.9741	3.99617
	92	5f 3 6d7s	6 L 11/2	0.5926	0.9377	7s	3.4779	1.42706	1.44552	2.78857	2.86351	3.92351
	93	5f 4 6d7s	7 L5	0.5873	0.9239	7s	3.4116	1.3478	1.36699	2.72666	2.79424	3.85537
	94	5f 5 6d7s	8 K 7/2	0.5821	0.9107	7s	3.3498	1.28511	1.30541	2.67874	2.73928	3.7917
	95	5f 7 7s	9 S4	0.5699	0.8822	7s	3.4279	1.27013	1.29869			3.8992
	96	5f 7 7s 2	8 S 7/2	0.5805	0.9217	7s	3.2104	1.16578	1.18716			3.65864
	97	5f 8 6d7s	9 G7	0.5670	0.8761	7s	3.1840	1.14449	1.16614	2.62581	2.688	3.62195
	98	5f 10 7s	6 I 17/2	0.5553	0.8493	7s	3.2905	1.14462	1.18171			3.77066
	99	5f 11 7s	5 I8	0.5503	0.8385	7s	3.2461	1.10705	1.14777			3.72772
	100	5f 12 7s	2 H 11/2	0.5456	0.8293	7s	3.2294	1.07281	1.11537			3.72114
	101	5f 13 7s	3 F4	0.5403	0.8174	7s	3.1625	1.0417	1.08491			3.64662
	102	5f 14 7s	2 S 1/2	0.5353	0.8071	7s	3.1235	1.01352	1.05406			3.6087
	103	5f 14 7s 2	1 S0	0.5435	0.8385	7s	2.8837	0.95846	0.990419			3.32578
	104	5f 14 6d7s 2 2 D 3/2	0.5441	0.8392	7s	2.7464	0.917421	0.941578	2.33571		3.16377
	105	5f 14 6d 2 7s 2 3 F2	0.5436	0.8355	7s	2.6309	0.881333	0.900734	2.17264	2.25313	3.02942
	106	5f 14 6d 4 7s 6 D 1/2	0.5376	0.8127	7s	2.5150	0.850008	0.867203	2.11798	2.21807	2.87277
	107	5f 14 6d 4 7s 2 5 D0	0.5409	0.8240	7s	2.4351	0.820103	0.834045	1.95902	2.01018	2.80344
	108	5f 14 6d 5 7s 2 6 S 5/2	0.5396	0.8183	7s	2.3426	0.790744	0.807474	1.86004	1.95516	2.69468

Table 9 .

 9 Correlation effects on the mean radius and mean spherical radius of No.

		< r > (a.u.)	var.	√	< r 2 > (a.u.)	var.
	Ra5f 14 7s 2 1 S0	0.574391			0.932605
	+ excit. of 7s → 7p, 6d + excit. of 7s, 5f → 7p, 6d	0.573724 -0.12% 0.573309 -0.19%		0.928191 -0.47% 0.925313 -0.78%

Table 10 .

 10 Comparison between different definition of the ionic radii and experiment. We define the average radius of 5f orbitals as < r 5f >= (6 < r 5f 5/2 > +8 < r 5f 7/2 >)/14

	Exp.	Theo.
	∆r Am +

Table 13 .

 13 Correlation effect on the Landé g-factor of the first excited states of nobelium.

	Level	7s7p 3 P1 7s7p 1 P1 7s7p 3 P2
	DF	1.49073	1.00932	1.50111
	7, 7p exc. → 7p, 6d 7, 7p, 5f exc. → 7p, 6d	1.47691 1.48845	1.02829 1.01507	1.50027 1.50080

Table 11 .

 11 QED [Eq.(15)], Breit and Uehling corrections on Landé g-factors of neutral atoms with 90 ≤ Z ≤ 106

	Z	Conf.	Label	Landé (Coul.)	QED corr.	Breit contr.	Uehl. corr.	Total
	90 6d 2 7s 2	3 F2	0.68158912 -0.00073903 -0.00031059 -0.00002136 0.68051814
	91 5f 2 6d7s 2 92 5f 3 6d7s 2	4 K 11/2 5 L6	0.81847964 -0.00042068 -0.00020095 -0.00001531 0.81784271 0.73962784 -0.00060501 -0.00090733 -0.00001069 0.73810480
	93 5f 4 6d7s 2 94 5f 6 7s 2	6 L 11/2 7 F0	0.63735126 -0.00084251 -0.00108525 -0.00000780 0.63541570
	95 5f 7 7s 2 96 5f 7 6d7s 2	8 S 7/2 9 D2	1.96702591 2.60336698	0.00224893 0.00372915	0.00193804 0.00336317	0.00000967 1.97122254 0.00002669 2.61048600
	97 5f 9 7s 2 98 5f 10 7s 2	6 H 15/2 5 I8	1.29988703 1.22357731	0.00070147 0.00052300	0.00177562 0.00108516	0.00000642 1.30237054 0.00000375 1.22518921
	99 5f 11 7s 2 100 5f 12 7s 2	4 I 15/2 3 H6	1.18834647 1.16189476	0.00043997 0.00037811	0.00047015 0.00019838	0.00000159 1.18925817 0.00000062 1.16247187
	101 5f 13 7s 2 102 5f 14 7s 2	2 F 7/2 1 S0	1.14180915	0.00033132	0.00005605	0.00000020 1.14219672
	103 5f 14 7s 2 7p 104 5f 14 6d 2 7s 2	2 P 1/2 3 P0	0.66661695 -0.00077309	0.00003007	0.00000003 0.66587396
	104 5f 14 7s 2 7p 2	3 F2	0.69055158 -0.00071837 -0.00035975 -0.00006392 0.68940954
	105 5f 14 6d 3 7s 2 106 5f 14 6d 4 7s 2	4 F 3/2 5 D0	0.45467514 -0.00126763 -0.00122595 -0.00016406 0.45201750

Table 12 .

 12 QED [Eq.(15)], Breit and Uehling corrections on Landé g-factors of the ground configuration of singly ionized atoms with 57 ≤ Z ≤ 71 and 90 ≤ Z ≤ 108.

	Z	Conf.	Label	Landé (Coul.)	QED corr.	Breit contr.	Uehl. corr.	Total
	57 5d 2	3 F2	0.66973929 -0.00076609 -0.00012975 -0.00000075 0.66884270
	58 4f 5d 2 59 4f 3 6s	4 H 7/2 5 I4	0.71736392 -0.00065375 0.60262203 -0.00092068 -0.00031540 -0.00000026 0.60138569 0.00039406 -0.00000787 0.71709635
	60 4f 4 6s 61 4f 5 6s	6 I 7/2 7 H2	0.44690832 -0.00128150 -0.00029522 -0.00000023 0.44533137 0.00677005 -0.00230319 -0.00086739 -0.00000061 0.00359886
	62 4f 6 6s 63 4f 7 6s	8 F 1/2 9 S4	3.95749786 1.99480186	0.00687585 0.00230987	0.00701619 0.00035927	0.00001509 3.97140499 0.00000034 1.99747133
	64 4f 7 5d6s 65 4f 9 6s	10 D 5/2 7 H8	2.56089325 1.36838651	0.00362452 0.00085813	0.00067458 0.00067487	0.00000119 2.56519354 0.00000040 1.36991992
	66 4f 10 6s 67 4f 11 6s	6 I 17/2 5 I8	1.28727646 1.24630282	0.00067021 0.00057447	0.00068999 0.00033068	0.00000039 1.28863705 0.00000018 1.24720815
	68 4f 12 6s 69 4f 13 6s	4 H 13/2 3 F4	1.22865909 1.24882255	0.00053325 0.00057982	0.00015805 0.00006252	0.00000008 1.22935048 0.00000004 1.24946493
	70 4f 14 6s 71 4f 14 6d 2	2 S 1/2 1 S0	1.99992029	0.00231927 -0.00000949 -0.00000006 2.00223002
	90 5f 2 6d 91 5f 2 6d7s	4 K 11/2 5 K5	0.80868853 -0.00044409 -0.00040145 -0.00001738 0.80782561 0.69942415 -0.00069727 -0.00041498 -0.00003386 0.69827803
	92 5f 3 6d7s 93 5f 4 6d7s	6 L 11/2 7 L5	0.63834785 -0.00083965 -0.00081792 -0.00001647 0.63667382 0.52126261 -0.00111157 -0.00103625 -0.00001135 0.51910344
	94 5f 5 6d7s 95 5f 7 7s	8 K 7/2 9 S4	0.26613829 -0.00170714 -0.00271582 -0.00002210 0.26169324 1.97238902 0.00226041 0.00159104 0.00001036 1.97625083
	96 5f 7 7s 2 97 5f 8 6d7s	8 S 7/2 9 G7	1.95878066 1.48746928	0.00223118 0.00113601	0.00242861 0.00161739	0.00001187 1.96345232 0.00001838 1.49024105
	98 5f 10 7s 99 5f 11 7s	6 I 17/2 5 I8	1.27036131 1.23940637	0.00063117 0.00055819	0.00097651 0.00042388	0.00000581 1.27197480 0.00000222 1.24039065
	100 5f 12 7s 101 5f 13 7s	2 H 11/2 3 F4	1.09903182 1.24906758	0.00023261 0.00057982	0.00025286 0.00004304	0.00000531 1.09952260 0.00000013 1.24969057
	102 5f 14 7s 103 5f 14 7s 2	2 S 1/2 1 S0	1.99988521	0.00231927 -0.00002001 -0.00000035 2.00218413
	104 5f 14 6d7s 2 105 5f 14 6d 2 7s 2	2 D 3/2 3 F2	0.79977826 -0.00046385 0.70320927 -0.00068901 -0.00043211 -0.00005473 0.70203342 0.00005358 0.00000012 0.79936810
	106 5f 14 6d 4 7s 107 5f 14 6d 4 7s 2	6 D 1/2 5 D0	3.19780436	0.00510655	0.00311970	0.00051784 3.20654845
	108 5f 14 6d 5 7s 2	6 S 5/2	1.87088576	0.00203080	0.00399742	0.00034326 1.87725723

Acknowledgments

Laboratoire Kastler Brossel is Unité Mixte de Recherche du CNRS n • C8552. This research was partially supported by the FCT projects POCTI/FAT/44279/2002 and POCTI-/0303/2003 (Portugal), financed by the European Community Fund FEDER, and by the French-Portuguese collaboration (PESSOA Program, Contract n • 10721NF). One of us (P.I.) thanks M. Sewtz for several discussions on this subject. The 8-processors workstation used for the calculations has been provided by an "infrastructure" grant from the "Ministère de La Recherche et de l'Enseignement Supérieur".


S Gales

SPIRAL2 at GANIL: A world leading ISOL facility at the dawn of the next decade

SPIRAL2 at GANIL: A world leading ISOL facility at the dawn of the next decade

S.

Introduction

In the recent years, Rare Isotope Beams (RIB), i.e. beams of synthesized radioactive isotopes, have been recognized by the international scientific community as one of the most promising avenue for the development of fundamental nuclear physics and astrophysics, as well as in applications of nuclear science. The importance of the RIB's has been underlined by NuPECC (Nuclear Physics European Collaboration Committee -an expert committee of the European Science Foundation) which recommend the construction of two new-generation complementary RIB facilities in Europe. One based on the inflight fragmentation (IFF) as proposed for the FAIR facility at GSI (Darmstadt, Germany) and the other on isotope-separation on-line (ISOL) method. In ISOL method, a thick target is bombarded with a primary light or heavy-ion beam or with a secondary neutron beam in case of fission targets. The produced radioactive nuclei are post-accelerated to energies from keV to tens of MeV. The ISOL method provides high-intensity beams of high optical quality comparable to stable beams. The ultimategeneration RIB ISOL facility will be realised when the EURISOL facility will be built. However, a full engineering design study and necessary R&D programme lead to the expected beginning of operation of the EURISOL facility around 2020. Because of the time-line for EURISOL, NuPECC recommends the construction of intermediate-generation facilities that will benefit the EURISOL project in terms of R&D and that will give the community opportunities to perform research and applications with RIB. Among the proposed intermediate facilities SPIRAL2 at GANIL (Caen, France), meets in terms of physics potential, site and size of the investments, the criteria of European dimension.

In March 2005, the ESFRI (European Strategy Forum on Research Infrastructures) published the "List of Opportunities", which includes 23 Research Infrastructure projects corresponding to major needs of the European scientific community in the coming years. The FAIR and SPIRAL 2 are among the selected projects and were recommended as Research Infrastructures necessary to maintain Europe's position at the cutting-edge of world research.

The French Ministry of Research took a decision on the construction of SPIRAL 2 in the end of May 2005. Its construction cost, estimated to 130 Meuros (including personnel and contingency), will be shared by French founding agencies CNRS/IN2P3 and CEA/DSM, the Region of Basse Normandie and the European partners. The construction will last about six years and operation of the facility will cost 8.5 Meuros per year.

The investments costs amount to 95 Meuros and personnel costs to 35 Meuros, for the construction period 2006-2012. Funding from EU 7th framework and from others partnership countries are expected to contribute for about 20% to this budget.

The first beams are expected by the end of 2011. As an added bonus, the facility will also produce over 10 15 neutrons per second, making it the world's most powerful source of fast neutrons for several years.

The facility is an extension to the SPIRAL 1 device, which opened five years ago at GANIL. SPIRAL 2 will reinforce European leadership in the field of exotic nuclei and will serve a community of about 600 scientists.

Description of the project

The quest for Rare Isotope Beams (RIB), which are orders of magnitude more intense than those currently available, is the main motivation behind the SPIRAL 2 project.

The new SPIRAL2 facility is a major improvement of the SPIRAL1 facility. More than 20 laboratories from 10 countries from 2001 to 2005 have developed the conceptual and the technical design of SPIRAL2. More than 200 scientists and engineers have contributed to the scientific case and the design of the facility1).

The SPIRAL 2 facility (figure 1) is based on a high power, superconducting driver LINAC, which will deliver a high intensity, 40 MeV deuteron beam as well as a variety of heavy-ion beams with mass over charge ratio equals to 3 and energy up to 14.5 MeV/nucl.. Using a carbon converter, the 5 mA deuteron beam and a uranium carbide target , a fast-neutron induced fission is expected to reach a rate of up to 10 14 fissions/s. The RIB intensities in the mass range from A=60 to A=140 will surpass by one or two order of magnitude any existing facilities in the world. A direct irradiation of the UCx target with beams of deuterons, 3,4 He, 6,7 Li, or 12 C can be used if higher excitation energy leads to higher production rate for a nucleus of interest.

The neutron-rich fission RIB could be complemented by beams of nuclei near the proton drip line provided by fusion-evaporation or transfer reactions. For example, a production of up to 8x10 4 atoms of 80 Zr per second using a 200 µA 24 Mg 8+ beam on a 58 Ni target should be possible. Similarly, the heavyand light-ion beams from LINAG on different production targets can be used to produce high-intensity light RIB with ISOL technique.

The extracted RIB will be subsequently accelerated to energies of up to 20 MeV/nucl. (typically 6-7 MeV/nucl. for fission fragments) by the existing CIME cyclotron. Thus using different production mechanisms and techniques SPIRAL 2 would allow performing experiments in a wide range of neutronand proton-rich nuclei far from the line of stability (figure 2).

One of the important features of the future GANIL/SPIRAL/SPIRAL 2 facility will be a possibility to deliver up to three stable or radioactive beams simultaneously in the energy range from keV to several tens of MeV/nucl.. SPIRAL2 has also a remarkable potential for neutron-based research both for fundamental physics and various applications. In particular, in the neutron energy range from a few MeV to about 35 MeV this research would have a leading position for the next 10-15 years if compared to other neutron facilities in operation or under construction worldwide.

Scientific opportunities and performances

Very intense stable heavy ions beams will be available at the exit of the LINAG accelerator. These heavy ions (from He to Ca in a first step, then up to U), will reach intensities from 1pµA up to 1pmA using new PHOENIX ion sources in combination with A/Q=1/3 RFQ injector and later A/Q=1/6 for heavier ions (Kr to U) as illustrated in Fig. 3.

Reaching such beams intensities will open a wide range of physics experiments not accessible to day. Super Heavy Elements synthesis and spectroscopy, spectroscopy at or beyond the proton drip line, multinucleon transfer and deep inelastic scattering, study of mechanisms and reaction product distributions, production and study of isomers ,Ground state properties of rare nuclei.

A new device, S3 (for Super Separator Spectrometer) has been proposed for experiments with the very high intensity stable beams of SPIRAL2 coming from the LINAG accelerator1).

Most of the physics cases listed above can be seen as either the study of rare processes or as the study of a secondary reaction with exotic ions produced in a first step reaction.

S3 shall be designed to answer most experimental needs of these physics cases. The crucial point of the device is to cope with the very high intensity beams provided by the LINAG accelerator. That means:

-the targets should be able to cope with very high power dissipation, -the reaction products should be separated from the beam ions before they can be detected, -the full optics of the spectrometer should reduce the background coming from scattered ions and activation products, Meanwhile, the vast array of different physics that could be possible with S3 implies that it should be compatible with various detection settings .The basic design considered could be the following (see figure 3):

The primary beam will impinges on a primary target which should be made resistant to the high fluxes (rotating cooled targets, gas or liquid targets...)

The separation of the beam can proceed in two steps, the first being a high rejection device and an appropriate beam stopper.

After this first step, ions of interest could, if necessary, react on a secondary target. As they deal with moderate fluxes, this secondary target point could have gamma or particle prompt detection.

Next to this point, a secondary spectrometer helps with separation if very high purity is needed. It has some mass/momentum resolution and could rotate for reactions products at large angles.

The final focal plane can be equipped with the relevant detection system. The SPIRAL2 facility at GANIL will produce radioactive isotopes ranging from the lightest to very heavy elements beyond uranium. Different production mechanisms will be utilised: fission of 238 U to produce medium-mass, neutron-rich isotopes, fusion evaporation for medium-mass, neutron-deficient nuclei, and transfer and deep-inelastic reactions for light to heavy nuclei closer to the line of stability. Examples of reactions leading to the production of light neutron rich and N=Z exotic nuclei are listed in Table.1 below.

SPIRAL2 beam intensities for fission fragments after acceleration are shown in fig. 5. For SPIRAL2 the RIB intensities in the mass range from A=60 to A=140 will be of the order of 10 6 to 10 11 part./s surpassing by one or two order of magnitude any existing facilities in the world. For example, the intensities should reach, 10 9 part./s for 132 Sn and 10 10 pps for 92 Kr

The DESIR collaboration, formed after the SPIRAL2 workshop on low-energy physics at GANIL Table 2: Calculated production yields for light and N=Z exotic nuclei using transfer or fusion evaporation reactions in July 2005, proposes the construction of an experimental facility to exploit the low-energy "exotic" beams from SPIRAL2. Experiments performed at other ISOL facilities (e.g., ISOLDE or ISAC) show that a high degree of purity is necessary for the ions beams. The most efficient and universal way of achieving isotopically pure beams is a high-resolution mass separator. At SPIRAL2 and its low-energy beam facility DESIR, there is an opportunity of building three kinds of setup to try to cover the whole nuclear chart. A β -NMR setup will be suitable to study precisely the moments of key cases and, in combination with collinear hyperfine spectroscopy, get unambiguous spin assignments. With a collinear spectroscopy setup, we will have the possibility to study series of isotopes in the intermediate and heavy mass regions. For the heavy elements, a double laser + RF spectroscopy in a Paul trap will be used to study the hyperfine anomaly and higher-order moments up to very high precision.

Two main regions of the mass chart in the vicinity of the N=50, and 82 shell closures are of special interest 1) The N=50 region: One important issue is to determine how thick can be the neutron skin for nuclei far from stability? From the measurement of the charge radius, one has access to the neutron radius. A key element would be germanium which is produced in large quantities in neutron stars. So the measurements of the charge radius change of this element and its neighbours (Zn, As, Ga, Se) over the magic shell N=50 using collinear laser spectroscopy will answer this question.

Another typical example of an element that has to be studied is of course nickel as close as possible to 78 Ni. How can the charge radius evolve so far from stability? Is this doubly-magic nucleus still spherical?

2) The N=82 region: For all the studied elements between tellurium (Z=52) and samarium (Z=62), the charge radius change exhibit a kink at N=82 showing an increase in deformation beyond this magic number. The interesting feature is that the more one approaches tin (Z=50) the less pronounced is the kink. A lot of nuclear models predict that no kink is present at N=82 for tin, a fact which should be a signature of a neutron skin for the very neutron-rich isotopes. Studying tin very far from stability will be a very stringent test for these nuclear models. In contrast, indium (Z=49), iodine (Z=53) and cadmium (Z=48) should present interesting kinks at N=82. Silver isotopes (Z=48) have been studied until A=110. This element is the heaviest in this region to show a sizeable shape change at N=60. The heaviest one could have interesting shapes.

Nuclei were created during major events that made the evolution of the universe and are the witness of the origin of matter: while the lightest elements were created in the first minutes following the big bang, heavier ones are synthesized in stars. Understanding our world, its origin and its evolution is one goal of nuclear physics. However, these various nucleosynthesis processes involve radioactive nuclei. Due to their short life-time most of them have not survived long enough to be present on earth. To be studied those "exotic" nuclei must be synthesized in laboratories. SPIRAL 2 will abundantly produce some of them opening new investigation field.

The future SPIRAL2 facility will contribute prominently to several areas of active research in nuclear astrophysics, such as explosive hydrogen burning, s-process and r-process nucleosynthesis, which are linked to astrophysical observations (novae, X-ray bursts, type 2 supernovae,etc. . . ).

For example, radiative capture cross-sections can be determined from spectroscopic factors (or Asymptotic Normalization Coefficients) obtained in transfer reaction or elastic resonant scattering. From a general point of view, detailed spectroscopy of the nuclei of interest is crucial to insure efficient application of indirect methods, which can be obtained by using these two reactions.

Neutron capture cross-sections are often calculated in the framework of the statistical Hauser-Feschbach model, which assumes the presence of a high density of states above the neutron separation energy with various spin and parity values. The use of this statistical approach is not appropriate to calculate neutron capture cross sections for neutron-rich at closed shell, since the neutron-separation energy is small and the level density low. In such cases, the main contribution is obtained from direct captures on few bound states, mainly of s or p waves, or/and by resonance capture slightly above the neutron-energy threshold.

The (d,p) reaction can be used to simulate the (n,α) capture on medium-mass nuclei at SPIRAL2 near the r-process possible paths .

The way the energy of theses s or p states evolve from Z=50 down to Z=40 determines how neutron capture cross sections will evolve along the N=82 shell closure and how the r process synthesis of nuclei will be blocked at this shell closure. So again, the evolution of these p states and the underlying physics which makes them move with respect to other orbits is an essential ingredient to model the r process. The neutron rich isotopes of Sn and Cd nuclei are excellent test cases.

The neutron-capture through the soft Pigmy resonance could also play an important role if the energy of the soft pigmy vibration mode is peaked around the Sn value. So far some models predict this resonance to lie around 4 MeV, whereas recent experiment in the Sn isotopic chain show that this resonance is peaked at higher energy. The way this energy is shifted when moving from Sn to Cd isotopic is certainly an important issue to consider. Since the direct measurements of neutron-capture cross sections on very unstable nuclei is technically not feasible their determination should be provided by models, constrained by spectroscopic information (energy, spin, spectroscopic factors of the bound and unbound levels in the A+1 nucleus) obtained by (d,p) transfer reactions.

The scientific opportunities discussed above are just a few examples among many exciting avenues which can be investigated with SPIRAL2.

New exotic shapes and excitation modes, halo-like and molecular structures, single particle properties and new magic numbers, new modes of nuclear decay have been recently observed, while studies of fundamental symmetries, testing and refinement of the Standard Model and of fundamental interactions in very exotic nuclei have all a huge potential of discovery.

In addition, let us stress once more time that, thanks to the high energy and high intensity neutron flux available at SPIRAL2, the facility will offer a unique opportunity for material irradiations and cross-section measurements both for fission (notably accelerator driven systems (ADS) and Gen-IV fast reactors) and fusion related research, tests of various detection systems and of resistance of electronics components to irradiations, etc.

Construction of the facility and International Collaborations

Since September 2005, a project construction group has been formed and the whole project has been organized around three main sections -The driver accelerator -The secondary radioactive beams (pro- For the driver accelerator, most of the work packages are under the responsibility of IPN Orsay and DAPNIA/DSM/Saclay laboratories. The R&D on prototypes' for the RFQ, SC cavities and RF couplers are completed and order of driver components will be completed at the end of this year A SPIRAL2-France consortium had been set to coordinate the efforts of 9 French nuclear physics laboratories from CNRS and CEA .

Moreover SPIRAL2 facility will be built by a large European and International collaboration. Several MoU have been already signed or are in final negotiation stages with JINR Dubna, Israel Roumania, Isolde (CERN), Italy, UK ,Germany,Poland,Canada,Japan and US ,both for accelerator components (control system, beam loss monitors, beam lines) and/or construction of components of the RIB production units (neutron converter , beam lines, target-ions sources ,laser ion sources, highresolution magnetic separator ,. . . .).

At the EU research organisation level, SPIRAL2 as well as FAIR have been selected by the ESFRI Committee and a proposal for the construction phase in the framework of the 7th FPRD is under preparation.

The process of formation of international collaborations around the SPIRAL 2 research programme consists of: • Call for proposals in Spring-Summer 2007.

In terms of schedule for the new facility, the ambition is to start operation of the driver accelerator with intense stable beams before the end of 2011 and to complete the whole project including RIB production and post acceleration of fission fragments before the beginning of 2013.

Conclusion

Nuclear physics has been revolutionized by the recent development of the ability to produce accelerated beams of radioactive nuclei. For the first time it will be possible to study reactions to produce the 6000 to 7000 nuclei we believe exist rather than the 300 stable ones that nature provides. At the larger scale one wants to understand the limits of nuclear stability by producing exotic nuclei with vastly different numbers of neutrons and protons.

SPIRAL2 is major expansion of the SPIRAL facility at GANIL, which will help maintain European leadership in ISOL (Isotope separation on line) development, aiming at two orders of magnitude increase of the secondary beams available for nuclear physics studies.

The technical challenges of the acceleration, targetry and experimental equipment will provide essential knowledge on the road toward EURISOL; the most advanced nuclear physics research facility presently imaginable and based on the ISOL principle. It is expected, that the realisation of SPIRAL2 will substantially increase the know-how of technical solutions to be applied not only for EURISOL but also in a number of other European/ world projects. The scientific programme, proposes the investigation of the most challenging nuclear and astrophysics questions aiming at the deeper understanding of the nature of matter.

SPIRAL2 will contribute to the physics of nuclear fission and fusion based on the collection of unprecedented detailed basic nuclear data, to the production of rare radioisotopes for medicine, to radiobiology and to material science.

Figure 1 :

 1 Figure 1: Schematic layout of the SPIRAL 2 facility at GANIL

Figure 2 :

 2 Figure 2: Regions of the chart of nuclei accessible for research of nuclei far from stability at SPIRAL2.

Figure 3 :

 3 Figure 3: Expected intensities of stable heavy ion beams at the exit of the LINAG accelerator for various versions of PHOENIX ion sources.

Figure 4 :

 4 Figure 4: Basic scheme for S3 design

Figure 5 :

 5 Figure 5: Fission fragments beam intensities from SPIRAL2 after acceleration

Figure 6 :

 6 Figure 6: SPIRAL2 driver accelerator divided into main components

Table 1 :

 1

default RI Beam Reaction Production method Yield (min. -max.) pps 6 He 9 Be(n, α) 6 He ISOL 5x10 7 -10 12 1 1C 14N (p, α) 11 C ISOL 10 7 -3x10 11 1 5O 15 N(d, 2n) 15 O ISOL 3x10 7 -10 10 18 Ne 19 F (p, 2n) 18 Ne ISOL 6x10 6 -7x10 9 34 Ar 35 Cl(p, 2n) 34 Ar ISOL 2x10 6 -2x10 8 56 Ni 58 Ni(p, p2n) 56 Ni Batchmode 2x10 4 -10 8 58 Cu 58 Ni(p, n) 58 Cu Batchmode 10 4 -10 8 80 Zn 24 Mg + 58 Ni Inf light < 3x10 4