G L Wilson

W N Catford
email: w.catford@surrey.ac.uk

N A Orr

C Aa

Diget

A Matta

G Hackman

S J Williams

I C Celik

N L Achouri

H Al Falou

R Ashley

R A E Austin

G C Ball

J C Blackmon

A J Boston

H C Boston

S M Brown

D S Cross

M Djongolov

T E Drake

U Hager

S P Fox

B R Fulton

N Galinski

A B Garnsworthy

D Jamieson

R Kanungo

K Leach

J N Orce

C J Pearson

M Porter-Peden

F Sarazin

E C Simpson

S Sjue

D Smalley

C Sumithrarachchi

S Triambak

C Unsworth

R Wadsworth

D F Geesaman

Shell evolution approaching the N = 20 island of inversion: Structure of 26 Na

Keywords:

γ -Ray transitions

Transfer reactions Shell migration

The levels in 26 Na with single particle character have been observed for the first time using the d( 25 Na, pγ ) reaction at 5 MeV/nucleon. The measured excitation energies and the deduced spectroscopic factors are in good overall agreement with (0 + 1)hω shell model calculations performed in a complete spsdf p basis and incorporating a reduction in the N = 20 gap. Notably, the 1p 3/2 neutron configuration was found to play an enhanced role in the structure of the low-lying negative parity states in 26 Na, compared to the isotone 28 Al. Thus, the lowering of the 1p 3/2 orbital relative to the 0 f 7/2 occurring in the neighbouring Z = 10 and 12 nuclei - 25,27 Ne and 27,29 Mg -is seen also to occur at Z = 11 and further strengthens the constraints on the modelling of the transition into the island of inversion.

© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP 3 .

The breakdown of the shell model magic number N = 20 for neutrons, in the case of neutron-rich isotopes near 31 Na and 32 Mg [1,2], has led to the concept of an "island of inversion", where neutrons preferentially occupy orbitals above the normal N = 20 gap, leaving vacancies below it [3,4]. The mechanism is now understood in terms of the shell model to involve specific valence nucleon interactions [3,5] as well as a monopole shift in the effective single particle energies [6]. In a wider picture, the monopole migration of levels is understood as arising from the tensor and three-body components of the nucleon-nucleon force between valence nucleons in partially filled orbitals [7,8].

The level schemes of nuclei with odd N, such as N = 15 and N = 17 (Fig. 1, provide important insight into the evolution of the single particle structure approaching the island of inversion. Between Z = 8 and 14, the proton d 5/2 orbital is filled, and this are derived from refs. [9][10][11][12][13] and compilations, whilst those for oxygen are from refs. [14,15]. The energy reference is chosen to be the 7/2 -level (see text) and the Z position of 26 Na is indicated.

affects the energies of the neutron orbitals. The reference energy is the 7/2 -level, as it is expected that the energy of the f 7/2 neutron orbital (a j > orbital, i.e. j = + 1/2) will depend relatively weakly on the filling of the proton d 5/2 orbital (also j > [7]). In the case of 21,23 O the J π = 7/2 -assignments are tentative [14,15] but the small width observed in 23 O [15] supports the 7/2 -assignment. The systematics and also the shell model [15] suggest a near degeneracy between the 3/2 + and 3/2 -states in oxygen. As may be seen, 26 Na falls in a key region where the shell closures at N = 20 and 16 are weakening and strengthening respectively.

As such, compared to the isotone 28 Al, the levels in 26 Na corresponding to a neutron in the 0d 3/2 orbital would be expected to lie higher in energy, whereas states with a neutron occupying the 1p 3/2 orbital may be degenerate with the 0 f 7/2 levels or even lie below them. 2 In order to explore this behaviour the present study has been undertaken to locate the corresponding levels using neutron transfer onto 25 Na, produced as a secondary beam. As will be shown, the detection of coincident gamma-rays has dramatically enhanced the excitation energy resolution and provided vital complementary information on the spins and parities of the populated levels.

Previous studies have identified the ground state of 26 Na as 16,17] and there are three 1 + states fed by the beta-decay of 26 Ne [18,19]. Apart from the low-lying quartet of states (3 + , 1 + , 2 + , 2 + ) below 407 keV [20], which have been very recently probed via Coulomb excitation [21], the many observed states [20,22,23,19] have a structure that is almost completely unknown. The (d, p) reaction employed here will selectively populate the levels with a predominantly single-particle structure, which are those most useful for testing shell model predictions.

having J π = 3 + [
The experiment was undertaken at the ISAC-II facility at TRI-UMF. A pure beam of 5.0 MeV/nucleon 25 Na ions of intensity 3 × 10 7 pps was employed to bombard a self-supporting (CD 2 ) n deuterated polythene foil of thickness 0.5 mg/cm 2 . The target was mounted at the centre of the SHARC silicon strip detector array [24] which was surrounded by 8 segmented germanium clover gamma-ray detectors of the TIGRESS array [25]. Four clovers were centred at 90 • in the laboratory and four at 135 • , giving a total absolute photopeak efficiency of some 3% at 1.33 MeV. In order to identify events in SHARC arising from fusion-evaporation reactions initiated on the carbon in the target (a source of significant background), an Al foil followed by a thin plastic scintillator detector (the TRIFOIL [26]) were installed 400 mm downstream of the target. The Al thickness of 30 μm was chosen to stop fusionevaporation residues whilst transmitting the beam and the products of direct reactions [27]. Further, the scintillator foil was thin enough (10 μm) to transmit the radioactive beam.

Events were recorded whenever a particle was registered in SHARC, including a logic signal to indicate whether the TRIFOIL had fired during the same beam bunch. The TRIFOIL allowed the events corresponding to the production of 26 Na via the (d, p) reaction to be highlighted in the analysis in order to optimise the software gating. In the final determination of the absolute differential cross sections of the (d, p) reaction, the TRIFOIL requirement was not imposed, however, owing to its efficiency being dependent on the position on the foil, and hence on the angle of the recoil proton [28]. The coincident gamma-rays were recorded in TIGRESS and their energies were corrected for Doppler shift (v/c ≈ 0.1) [28].

The excitation energy spectrum for states populated in 26 Na, reconstructed from the observed energy and angle of the protons, is shown in Fig. 2 where the resolution is 350 keV (FWHM). This figure illustrates how closely spaced states could be distinguished using the gamma-ray data. The FWHM resolution in gamma-ray energy at 1.8 MeV, after Doppler correction, was 18 keV at 135 • and 23 keV at 90 • . This resolution, rather than that of the protonderived excitation energy spectrum, defined the precision with which individual states could be selected. On the other hand, the proton-derived excitation energy was critical in determining the energy at which the 26 Na was populated in the (d, p) reaction (i.e. prior to gamma-decay of the states). In addition, gating on this excitation energy was employed in the determination of the gamma-ray branching ratios [28].

Elastically scattered deuterons from the target were prominent in the spectrum of particle energy versus angle and the corresponding differential cross section was extracted [28]. Optical model parameters for d + 26 Mg at an almost identical centre of mass energy [29] were employed to construct a theoretical angular distribution and this was found to reproduced well the form of the measured distribution (including the observed minimum). The spectrum for all events is shown (red), plus the results for coincidences with the known gamma-rays of 233 keV (blue) and 407 keV (black histogram) in 26 Na. The data are for proton laboratory angles backward of 90 • . A TRIFOIL requirement has been imposed in all cases (see text).

The best-fit normalisation thus gave the product of the deuteron target thickness and the total integrated beam flux. The uncertainty associated with this fitting was estimated to be 3%. An estimate of systematic uncertainties in this analysis was obtained by repeating the procedure with parameters taken from d + 28 Si scattering at the same energy [30]. The variation was less than 2%.

Fig. 3 shows the Doppler corrected gamma-ray energy (E γ ) plotted against the excitation energy in 26 Na (E x ) as derived from the proton energy and angle. The superior resolution of the gamma-ray energy is clearly apparent, and the states that overlap in E x can be distinguished using the gamma-ray energy. A number of levels with ground-state gamma-ray branches (E γ = E x ) can be seen, including many of the more strongly populated states. The neutron separation energy is S n = 5.57 MeV [31]. The ground state branch generally has a lower and better defined underlying background than any other peak in Fig. 3 and hence for the most part only these peaks were chosen for initial analysis 3 (Table 1). With suitable background subtraction [28], the yield of protons could be deduced for individual states in 26 Na as a function of laboratory angle. These distributions were converted to absolute differential cross sections by taking into account the geometry of SHARC, the elastic scattering normalisation, the measured gamma-ray branching ratios and the gamma-ray detection efficiency (corrected for Doppler and relativistic angular aberration effects). The overall systematic error associated with these effects was typically 5-6%.

Differential cross sections for states populated in 26 Na are shown in Fig. 4, plotted in terms of the laboratory scattering angle. These are compared with reaction calculations performed using the code TWOFNR [START_REF] Tostevin | University of Surrey version of the code TWOFNR[END_REF] employing the Adiabatic Distorted Wave Approximation (ADWA) of Johnson and Soper [START_REF] Johnson | [END_REF] with standard input parameters [35] including the Chapel-Hill (CH89) nucleonnucleus optical potential [36]. This formalism benefits inter alia from having no requirement for a deuteron-nucleus optical potential. The magnetic substate populations from TWOFNR were used to check that the gamma-ray coincidence requirement did not alter the shape of the differential cross sections by more than a few percent across the full range of proton angles [28]. Spectroscopic factors, S, were extracted by normalising the calculated differential 3 A full analysis including cascade decays and the more weakly populated states will follow. 26 Na states populated via the d( 25 Na, p) reaction, as deduced from the proton energy and angle, versus the energy of any coincident gamma-ray (after Doppler correction), E γ . Gamma-decays directly to the ground state lie along the diagonal E x = E γ , whilst cascade decays result in events lying above the diagonal. Data are for laboratory proton angles backward of 90 • . A TRIFOIL requirement has been imposed (see text). cross sections to the data using the full range of angles shown in Fig. 4. The transferred angular momentum, L, that best describes the shape of the measured angular distribution for each state is listed in Table 1.

States for which more than one value is possible for the transferred angular momentum have been analysed by fitting a linear combination of the calculated cross sections using the two possible values L 1 and L 2 (where L 1 = L). It is important to note that the contribution from the larger of L 1 and L 2 is suppressed (for a similar spectroscopic factor) due to poorer kinematic matching. As a result, the spectroscopic factor deduced for the higher L is seen generally to exhibit a larger statistical uncertainty. As may be seen in Table 1, the addition of L 2 to the fit does not significantly change the spectroscopic factor deduced using just L 1 , i.e. S 1 ≈ S. The overall uncertainties assigned to the spectroscopic factors are some 20% (dominated overwhelmingly by the reaction calculations) [35]. The inferred spin and parity assignments are discussed below.

The 25 Na projectile has a structure, in the simplest shell model picture, of three 0d 5/2 protons coupled to 5/2 + and a closed 0d 5/2 neutron orbital. The positive parity orbitals 1s 1/2 and 0d 3/2 , as well as the negative parity orbitals 0 f 7/2 , 1p 3/2 , 1p 1/2 . . . are thus available for neutron transfer. As shown in Fig. 2, the known low-lying positive parity states below 450 keV [16,20] are populated. As discussed below, the data also show clearly that there is population of negative parity states, as would be expected from the previous (d, p) studies of 25,27 Ne [12,13], 27 Mg [37] and 28 Al [38,37,39]. In order to interpret the results, shell model calculations have been performed including all 0hω sd-shell configurations for positive parity states and all 1hω excitations in an spsdp f basis for negative parity states. The program OXBASH [START_REF] Brown | Computer code OXBASH[END_REF] was used with the USD-A [32] and WBP-M [13] interactions respectively. The WBP-M interaction, which describes the neighbouring nuclei 25,27 Ne and 29 Mg in a consistent manner [13], is a modification of the WBP interaction [START_REF] Warburton | [END_REF] that shifts the energies of the f p-shell orbitals down by 0.7 MeV. It was previously noted [13] that for the neon and magnesium isotopes the strongly populated 3/2 -and 7/2 -states (S ≈ 0.4-0.6) appear to track fairly closely the shift in the energy

G.L. Wilson et al. / Physics Letters B ••• (••••) •••-•••

Table 1

Excitation energies (E x ) and proposed J π for 26 Na levels in the present work, with angular momentum transfers and deduced spectroscopic factors (±20%) §. The L-transfer dominating the yield is denoted by L, or L 1 where two L-values are possible; the second value is then denoted by L 2 . Values of S and S 1,2 were obtained by fitting cross sections using a single L and a sum of two L contributions, respectively, assuming orbitals (nlj). Shell model spectroscopic factors S SM and S SM 1,2 are shown for each fitted orbital (nlj) and, for completeness, its spin-orbit partner also. Numbering of states as in Fig. 4.

No. E x a) E SM x b) J π c) J π SM
Single L analysis Two L analysis (where applicable) a) Present work, excitation energy in MeV (±1 keV) deduced from Doppler-corrected gamma-ray energies.

L nlj S S SM L 1 n 1 l 1 j 1 S 1 S SM 1 L n 2 l 2 j 2 S 2 S SM 2 0 0 3 + 3 + 1 *
b) Excitation energy in MeV, calculated using the shell model (SM) interactions USD-A (π = +) [32] and WBP-M (π = -) [13] (see text).

c) Inferred in present work (see text) except g.s. [16] and first four excited states [20,18,19]. d) Excitation energy of this isomeric state deduced from gamma-decays feeding both this level and the g.s. (see Fig. 6). of the f p-shell orbitals. In fact, earlier work by Bender and coworkers also applied a similar interaction called WBP-a that improved the agreement with the shell model for states in nearby (and only slightly higher Z ) isotopes of Al [42] and P [43,44]. Excitation energies for the WBP-M and USD-A calculations were computed relative to the (positive parity) ground state calculated with the same interaction. The association of experimentally observed states with shell model states, indicated in Table 1 and Fig. 5, was based on the observed L-transfer, the excitation energy and the gammadecay selectivity (Fig. 6) as discussed below.

Above ∼2.5 MeV excitation, no states of positive parity are expected to be strongly populated, according to the shell model calculations. The states seen here at higher energies are therefore likely to have negative parity. Of these, the most easily identified is the strongly populated state at 3.511 MeV which is observed to have a yield dominated by L = 1 neutron transfer at large laboratory angles (Fig. 4). The only observed gamma-decay branch for this state is to the 3 + ground state, in line with the favoured decay pattern for the lowest 4 -state in the isotone 28 Al [38]. This state is thus assigned to be the 4 - 1 state in 26 Na. There are two strongly populated higher lying levels that gamma-decay via the 3.511 MeV level (Fig. 6). In terms of predicted levels in this energy range, the obvious candidates are the lowest 5 -and 6 -states (Table 1). These could reasonably be expected to gamma-decay via the 4 -state, according to the pattern observed [39] in 28 Al.

The decays of both the proposed 5 -and 6 -states proceed in part (20% and 31% respectively [28]) via the state at 2.116 MeV.

Given that this level is populated via an L = 2 transfer (Fig. 4) and that it is fed by these higher lying negative parity states, it has a likely J π assignment of greater than 4, and a comparison of possible spins with shell model calculations indicates an assignment of 5 + (supported by both the energy and the weak spectroscopic factor). Indeed, there is an analogous state at 2.581 MeV in 28 Al, perhaps weakly populated in the decay of the 6 -state [39], that has a 5 + assignment [45,46].

The two reasonably strong states seen just below the 4 -are most naturally associated with the 2 -and 3 -states predicted in the shell model (see Table 1). In particular, the identification of the 3 -at 3.135 MeV is confirmed by its gamma-decay branch to the lowest lying 1 + state at 1.507 MeV as occurs in 28 Al (Table 28.6 of ref. [46])). This is further supported by the mixed L 1 = 1 plus L 2 = 3 differential cross section (Fig. 4). The spectroscopic factors extracted from the proton differential cross sections are compared to the shell model predictions in Table 1 and Fig. 5. The 1s 1/2 strength that leads to 2 + states is concentrated in the two known states below 0.5 MeV and sums to approximately 0.45 in both theory and experiment. The tentative identifications of 3 + 2 and 4 + 2 states are based on a comparison with the shell model calculations as follows. The observed 0d 3/2 strength leading to J π = 3 + is concentrated in one state near 1.8 MeV with a spectroscopic strength in very good agreement with the predictions. The 0d 3/2 strength leading to the 4 + states has a comparable magnitude in theory and experiment and is located close to 2 MeV, so that the experiment is compatible with the strength being concentrated in the second 4 + state. We note that the USD-A interaction [32] predicts the excitation energies for positive parity states to within ∼150 keV, in line with typical shell model accuracy. When we instead used the WBP-M interaction to calculate the energies of states with a strong 0d 3/2 neutron character (namely the 3 + and 2 + states), then the predictions lay about 0.4 MeV lower than experiment. This is because the WBP-M incorporates the USD interaction [47] to compute the 0hω positive parity levels and this is known to underestimate 0d 3/2 neutron energies [12].

The spectroscopic factors deduced for the 4 -state indicate a comparable strength to that predicted for the 1p 3/2 transfer, albeit somewhat weaker. However, the shell model fails completely to reproduce the substantial 0 f 7/2 strength. The excitation energy is given accurately by the WBP-M calculation, as are those of the other negative parity states, within 100 keV ( J π = 2 -, 3 -, 5 -, 6 -). These latter states are all predicted to have a structure that overlaps substantially with that of a neutron in the 0 f 7/2 orbital coupled to the 0d 5/2 proton of the 25 Na ground state. It is worthwhile noting that the 3 -state is predicted also to have a similar spectroscopic strength for the coupling with a 1p 3/2 neutron. Because of a better kinematic matching, the L = 1 transfer dominates the yield; a similar situation is observed for the lowest 3 -state in 28 Al as populated in (d, p) [48,49,45]. In 28 Al, the 0 f 7/2 measured spectroscopic factor actually exceeds that of the 1p 3/2 orbital by a factor of three [48] (Table 2). Another 3 -state in 26 Na is predicted by the WBP-M calculations to have approximately equal mixing and to lie at 4.462 MeV, whilst a predominantly 1p 3/2 neutron state occurs at 4.774 MeV. These states may be populated but have not been identified in the present analysis.

In the case of the 4 -state in 26 Na, the deduced 1p 3/2 spectroscopic factor is twice as large as that reported for 28 Al whereas the 0 f 7/2 spectroscopic factor is effectively unchanged (Table 2). In contrast, for the 3 -state, the spectroscopic factors are three times and six times lower, respectively, than in 28 Al [48,46]. A clearer picture emerges from Table 2 if the relative magnitudes of the 1p 3/2 and 0 f 7/2 spectroscopic factors for each of the 26 Na states are compared with the behaviour in 28 Al. For both the 3 -and 4 - states in 26 Na, the 1p 3/2 spectroscopic factor is half the magnitude of that for 0 f 7/2 . In contrast, the analogous states in 28 Al exhibit 1p 3/2 spectroscopic factors that are three to five times smaller. This demonstrates an enhanced role emerging for the 1p 3/2 orbital in the structure of the low-lying negative parity states in 26 Na as compared to 28 Al. Indeed, in the case of 28 Al the spectroscopic factors (Table 2) indicate that the states with predominantly 1p 3/2 structure lie in the region of 4.8 MeV, significantly above the 0 f 7/2 -dominated states which are closer to 4.0 MeV (Table 2 of ref. [48]). It would be very interesting if the spectroscopic factors for the higher lying 2 -and 3 -states in 26 Na could be measured and compared. As is evident in Fig. 5, a theory-based comparison between the 1p 3/2 and 0 f 7/2 strengths in 26 Na indicates that the levels with a 1p 3/2 structure are on average around 1 MeV below those with 0 f 7/2 structure (in fact, 0.83 MeV when weighted by the spectroscopic factors), an inversion that is in accord with the systematics of Fig. 1.

In conclusion, as noted in the introduction and illustrated in Fig. 1, the ordering of levels in nuclei with A 25-30 evolves dramatically as they become more neutron-rich, driven largely by the interaction between protons in the 0d 5/2 orbital and the valence neutrons. Importantly in this context, the results presented here confirm that the evolution is also manifest in 26 Na. In particular, the low-lying 3 -and 4 -states in 26 Na are found to exhibit an enhanced influence of the 1p 3/2 neutron orbital, compared to the isotone 28 Al. In addition the WBP-M shell model calculations, in which the f p-shell orbitals are lowered to reduce the N = 20 gap by 0.7 MeV, succeed in reproducing the energies of the negative parity states in 26 Na as they do in the neighbouring neon and magnesium isotopes [12,13]. From a theoretical perspective, a less ad hoc description of the transition into the island of inversion represents an interesting and important challenge. Mixing between 1p 3/2 and 0f 7/2 configurations in 26 Na, compared with the isotone 28 Al. Values of (2J+1)S from [48] are converted to S using the confirmed spins [46]. 26 Na, present work 28 Al, ref. [48] J π E x a)

S 1 (1p

Fig. 1 .

 1 Fig. 1. (Colour online.) Energies of lowest levels in isotones N = 15 (dashed line) and N = 17 (full line). Energies for N = 13 are included for oxygen. The points for Z = 10-16

Fig. 2 .

 2 Fig.2. (Colour online.) Excitation energy spectrum for states in26 Na, reconstructed from the measured energy and angle of the protons from the d( 25 Na, p) reaction. The spectrum for all events is shown (red), plus the results for coincidences with the known gamma-rays of 233 keV (blue) and 407 keV (black histogram) in26 Na. The data are for proton laboratory angles backward of 90 • . A TRIFOIL requirement has been imposed in all cases (see text).

Fig. 3 .

 3 Fig. 3. (Colour online.) Plot of the excitation energy E x of26 Na states populated via the d( 25 Na, p) reaction, as deduced from the proton energy and angle, versus the energy of any coincident gamma-ray (after Doppler correction), E γ . Gamma-decays directly to the ground state lie along the diagonal E x = E γ , whilst cascade decays result in events lying above the diagonal. Data are for laboratory proton angles backward of 90 • . A TRIFOIL requirement has been imposed (see text).

Fig. 4 .

 4 Fig. 4. (Colour online.) Differential cross sections for the reaction d( 25 Na, p) at 5.0 MeV/nucleon. The results of ADWA reaction calculations are also shown, normalised to the data, for the angular momentum transfers indicated and listed in Table1(dashed line = fit with single angular momentum, labelled as L or as L 1 where another fit is also shown, full line = sum of two contributions L 1 and L 2 ).

Fig. 5 .

 5 Fig. 5. (Colour online.) Level scheme of 26 Na as deduced from the present work compared to the results of shell model calculations employing the USD-A and WBP-M interactions (see text). The lengths of the coloured lines correspond to the spectroscopic factors as follows: red s-wave, blue p-wave, green d-wave, orange f -wave.

Fig. 6 .

 6 Fig. 6. (Colour online.) Decay scheme of 26 Na as deduced from the present work, including the gamma-ray branching ratios.

  Extraction of these differential cross sections beyond scope of present analysis (see text).

							1 s 1/2		0.61	*	1s 1/2		0.61	*	0d 3/2		0.01
		0.082 d)	0.077	1 +	1 + 1	*	0d 3/2		0.29						0d 5/2		0.01
	(i)	0.232	0.149	2 +	2 + 1	0	0d 5/2 1 s 1/2	0.13	0.11 0.15	0	1s 1/2	0.10	0.15		0d 3/2	0.19 †	0.10
	(ii)	0.405	0.416	2 +	2 + 2	0	1 s 1/2	0.33	0.27	0	1s 1/2	0.30	0.27		0d 5/2 0d 5/2	0.13 †	0.09 0.03
		1.507	1.409	1 +	1 + 2	*	0d 3/2		0.09						0d 3/2		0.03
	(iii)	1.805	1.676	(3 + )	3 + 2	2	0d 5/2 0d 3/2	0.37	0.10 0.33	2	0d 3/2	0.33 †	0.33		1 s 1/2	0.01 ‡	0.00
	(iv) (v)	2.116 2.225	2.241 2.048	5 + (4 + )	5 + 1 4 + 2	2 2	0d 5/2 0 d 5/2 0d 3/2	0.16 0.43	0.02 0.08 0.51	2	0d 5/2		0.02
		2.843	2.936	(2 -)	2 -1	*	0d 5/2 0f 7/2		0.01 0.20	*	0f 7/2		0.20	*	1p 3/2		0.05
	(vi)	3.135	3.228	3 -	3 -1	1	0f 5/2 1p 3/2	0.07 †	0.00 0.15	1	0f 5/2 1p 3/2	0.06 †	0.00 0.15		1p 1/2 0f 7/2	0.10 ‡	0.04 0.13
	(vii)	3.511	3.513	4 -	4 -1	1	1p 1/2 1 p 3/2	0.30	0.02 0.44	1	1p 1/2 1p 3/2	0.25	0.02 0.44		0f 5/2 0f 7/2	0.51 †	0.00 0.00
		4.305	4.401	(5 -)	5 -1	*	0f 7/2		0.46						0f 5/2		0.00
		4.917 5.009	4.881	(6 -) (3 -,4 -)	6 -1	* *	0f 5/2 0 f 7/2		0.00 0.61
	§Dominated by reaction theory contribution; statistical errors typically several percent except where noted as 10% ( †) or 35% ( ‡).

*

  3/2 ) b) S 2 (0f 7/2 ) b) Excitation energy in MeV. b) Statistical error in italics. c) Beyond E x range of[48].

					E x	a)	S 1 (1p 3/2 )	S 2 (0f 7/2 )
	3 -	3.135	0.06 0.01	0.10 0.03	3.591	0.19	0.60
	4 -	3.511	0.25 0.01	0.51 0.05	3.465	0.11	0.62
	3 -				4.691	0.31	0.08
	2 -				4.766	0.41	0.15
	2 -				4.905	0.24	0.06
	3 -				5.134	c)
	a)

A multiplet of levels for each neutron configuration is expected, according to the coupling of the neutron with the unpaired proton in the 0d 5/2 orbital.

Acknowledgements

The assistance of our late colleague and friend R.M. Churchman in preparing the experiment is gratefully acknowledged. The efforts of the ISAC operations team in supplying the 25 Na beam are appreciated. Illuminating discussions with B.A. Brown are also acknowledged. We acknowledge support from the Science and Technologies Facility Council (UK) (grants ST/J000051/1, STFC-EP/D060575/1) and the Natural Sciences and Engineering Research Council of Canada; TRIUMF is funded via a contribution agreement with the National Research Council of Canada.


Adaptive Multi-Resolution

  L'archive ouverte pluridisciplinaire

INTRODUCTION

Recent developments in image coding have demonstrated that H.264 appears as the most efficient solution for both video and still image compression [START_REF] Côté | Recent Advances in Image Compression Standards[END_REF]. In particular, it can over-class Jpeg2000 at every bit-rates but at the depend of computation time [START_REF]Performance comparison: H.26L intra coding vs. JPEG2000[END_REF]. However, H.264 does not provide a scalable bitstream. The LAR (Locally Adaptive Resolution) method has been originally introduced as a fast method for high compression of gray-level images [3]. Extensions have permitted to address also middle and high quality encoding, automatic reconstruction of a region representation at the decoder allowing both Region Of Interest coding and region-level chromatic images encoding [START_REF] Déforges | Region of Interest Coding for Low Bit-Rate Image Transmission[END_REF]. This paper is organized as follows : section 2 recalls the outlines of the LAR method, while section 3 presents the new advances over the first high compression stage. Some comparative results are given in section 4, and we conclude in section 5.

LAR METHOD OVERVIEW

One of the main improvements introduced in H.264 is the adaptation to the image content by considering different sizes of block. For Intra-picture mode, flat areas are encoded using 16x16 blocks, whereas more active ones involve 4x4 blocks. Our coding scheme is based on the same principle but declined much further. Indeed, the block size fits the actual local resolution so that a meaningful low resolution image is provided. The LAR method is a two layers codec (Fig. 1) : a spatial and a spectral one. Low bit rate compression can be achieved when considering only the spatial coder. The second layer allows encoding the image texture.

Spatial Coder

The principle relies on a variable image resolution. The image is divided into macroblocks (16x16) possibly split up from a local activity estimation onto a quadtree data structure until 2x2 size. A low resolution image can be obtained when filling each block by its average intensity. Many works use such representation, but our approach differs by three essential points:

-the activity is estimated from a local morphological gradient. As a result, the variable block size representation constitutes a coarse edge-driven segmentation map (small blocks upon contours, large blocks in smooth areas).

-simple linear quantizations can be applied but based on psychovisual concepts : fine quantization for large blocks (strong sensitivity of human eye to brightness

REGIONS CODER

SPECTRAL CODER

U/V Blocks Error

GRAY-LEVEL CODER CHROMATIC CODER

Grid

ROI Selection

Chromatic control

Region Representation

Mask of Validated blocks

variations in uniform areas), coarse quantization for small blocks (lower sensitivity).

A classical quad-tree decomposition would lead to an expensive encoding of the block luminance value. Till now, the original LAR method was based on a one raster scan DPCM scheme. The next section will introduce a new multi-resolution decomposition providing both efficiency and scalability.

The output of the spatial coder is an image subsampled by two preserving edges (2x2 blocks transformed into single pixel). Perceptible block artifacts in homogeneous areas are easily removed by a simple efficient post-processing based on an adaptive bilinear interpolation (Fig. 2).

We can summarize the major features of this coder: -a very fast and efficient technique for high compression ratio, -a method simplifying the source image by removing the local texture but globally preserving the object boundaries. -a method naturally allowing a spatial scalability (CIF -> QCIF for instance), suitable for broadcast applications involving different kind of receivers [START_REF] Sun | Architectures for MPEG compressed domain bitstream[END_REF].

Spectral coder

To obtain higher image quality a second spectral coder can be added to encode the whole error image (texture image). We use a DCT adaptive block-size approach, but where both the size and the DC components are provided by the spatial coder. The use of adapted square sizes offers two major advantages :

-The dynamic of the block content is bounded (except for 2x2 blocks). So fewer AC coefficients are generated with lower energy than for standard fixed block size DCT.

-As the block sizes are coded independently the coding scheme is content-based scalable (small blocks -> edge enhancement, large blocks -> smooth area refinement).

This second coder can still be improved by including some H.264 techniques such as spatial AC prediction, two zigzag scans within 4x4 blocks, and context modeling together with the entropy coding.

self-extracting region representations d

Unlike traditional compression techniques, the low resolution LAR image does not introduce strong distortions which usually prevent from a reliable segmentation process. Indeed, the variable block size representation resulting from a local gradient distance can constitute a "split" step in a segmentation scheme. The original idea of the self-extracting region representation is to transmit the Y low resolution LAR image, and to continue the segmentation ("merge" step) at both the sender and receiver. As regions are built as union of blocks, two main advantages are introduced by the method : b -a free region representation (no cost overhead for the contours description), -contours regions and texture coding share the same grid of representation. A Region Of Interest can then be selected either at the coder or the decoder, and the spectral coder activated only for blocks included in this ROI. By adding a very low cost chromatic control during the segmentation process, reliable regions in terms of regions are generated, allowing very high compression rate coding of U and V images at the region levels [START_REF] Déforges | Region of Interest Coding for Low Bit-Rate Image Transmission[END_REF].Coding the error images at the block level (spatial coder) leads to very good quality chromatic images.

SPATIAL CODER IMPROVEMENT

Pyramidal predictive decomposition

To get a scalable low resolution image while maintaining a compression efficiency, we have introduced a new quadtree decomposition scheme. The previous spatial LAR coder encoded in one pass (raster scan) the intensity block values by means of the Gradient Adjusted Predictor (GAP) [START_REF] Wu | A context-based, adaptive, lossless/nearly-lossless coding scheme for continuoustone images[END_REF]. As for the new scalable DPCM scheme, it is based on the predictor described in [START_REF] Wu | Lossless compression of Continuous-Tone Images via Context Selection, Quantization and Modelling[END_REF] by Wu. For full resolution image, errors are coded by means of three interlaced sampling of the original image. By this way, we tend to obtain a spatial configuration of 360° type context surrounding a given pixel. The general principle is as follows (Fig. 3) : the first pass encodes through a classical DPCM system an uniform subsampled image formed by the average of two diagonally adjacent pixels within each 2x2 block. Then the second pass will provide the values of the two pixels We propose to extend the previous full resolution scanning scheme to get a pyramidal representation of the image.

Bottom-Up building

Starting from the initial LAR partition, four blocks N/2xN/2 are gathered into one block NxN valued by the average of the two blocks of the first diagonal (Fig. 4). By this way, the value of a NxN block is equal to the average over the whole diagonal in the finest resolution image.

The highest level (level 4) of the pyramid corresponds to the 16x16 blocks (largest block size in the LAR method).

Top-Down decomposition

Let the original image be of width W and height H, the first stage of our pyramid concerns the image of size W/16xH/16 (level 4). The first pass of the Wu algorithm (simple causal DPCM coding of the block values) is applied. Then, each square is split until its maximal size is reached. For a given level (block size) of the pyramid, the blocks of smaller or equal size are processed by successively passes 2 and 3. The values of higher size blocks are copied out in order to refine the context (Fig. 5). By this way, taking into account the variable resolution LAR image, a progressive content-based method is developed : if a restricted context is sufficient for the large blocks, a more accurate and informative context exists for the small blocks (objects boundaries).

The main property of this method is that the pyramidal decomposition does not introduce additional symbols when applying a minimal quantization : the number of encoded symbol is equal to the number of blocks of our LAR representation. Indeed, for a group of four blocks obtained by the split of a block from the above level, the diagonal mean is already known. The exact value of the second value on the diagonal can be deduced with one bit knowing the first value. This bit disappears with quantization by a minimum step of two. Figure 6 gives a representation of the decomposition. This method has also be extended to lossless compression, nearly achieving performances of state of the art CALIC [START_REF] Wu | Lossless compression of Continuous-Tone Images via Context Selection, Quantization and Modelling[END_REF], while offering a scalable coding scheme.

Interpolation

From our low resolution image sub-sampled by two, an interpolation stage can be envisaged in order to build a full resolution image.

We have then introduced the very efficient adaptive image interpolation described in [START_REF] Muresan | Optimal Recovery Approach to Image Interpolation[END_REF], based on the theory of optimal recovery. After determining locally, at a given location, the quadratic signal class from a set of training vectors, optimal recovery estimates the missing pixels. By this way, designing an interpolation filter is equivalent to minimize the integrated square error in the frequency domain. This new type of interpolation is essentially based on learning : the coarse scale image can be then used to improve the resulting image. Hence, we have applied this algorithm on the subsampled post-processed luminance image provided by the LAR spatial coder. Clearly, images obtained thanks to this method are of visually much better quality than traditional technique such as bi-cubic interpolation, especially on the contours (no more jagged effect). Computation time is nevertheless more prohibitive (about one second on a standard PC). Comparative results are given figure 8 between intra mode Mpeg-4 (Momusys), Jpeg2000, intra mode H.26l (most optimized mode) and the LAR approach. Jpeg2000 is a fully scalable coding method, the LAR is partially scalable (by layers), whereas Mpeg4 and H.26l are not scalable. While LAR's PSNR is about 1.5 db lower than H.261, the LAR method does not introduce the same kind of distortions as the other approaches, smoothes texture details, but preserves a pleasant visual aspect.

CONCLUSION AND PERSPECTIVES

We have introduced new improvements realized for the LAR codec. The main idea is to build a totally scalable algorithm, where the low resolution image is of good quality. Even if H26L uses also a decomposition with different but fewer block sizes, progressivity is not one of its properties. Visually one can observe the good performances of the LAR coding given thanks to the optimal recovery interpolation. Further works will concern the lowering of the cost of the AC coefficients at the spectral coder : H26L proposes efficient schemes in this domain. Thereby, LAR coder could be seen as a new functionality of this state-of-theart.

Figure 2 :

 2 Figure 2: Spatial codec, a) Source image (CIF), b) variable size grid : 0.0029 bpp, c) Var. bl. size image :0.123 bpp (QCIF), d) after postprocess.

  the previous image. At this stage, the prediction 360° degrees type context consists in the already known values of the current pass and the diagonal means coded by the first pass. Finally, the third pass encodes the remaining half of the original image. Once again, thanks to the reconstructed pixels, a completely spatially enclosing and adjacent context is available to predict the modeled pixel.

Figure 3 :

 3 Figure 3: Wu algorithm : 3 passes, prediction neighborhood.

Figure 4 :

 4 Figure 4: Bottom-up building

Figure 5 :

 5 Figure 5: Top-down decomposition

Figure 6 :

 6 Figure 6: Multiresolution representation : grid : 0.029 bpp, lev1 (sub 16) : 0.005 bpp, lev2 (sub 8) : 0.015 bpp, lev3 (sub 4): 0.051 bpp, lev4 (sub 2) : 0.094 bpp

Figure 7 :

 7 Figure 7: Full size (0.123 bpp), a) bi-cubic b) optimal recovery

2x2 Blocks Y Texture 16x16 Blocks Y Texture

	Y image			U/V images
	Nonuniform	Y Blocks
	Sub-sampling
	Blocks		Region
	Average		Segmentation
	Blocks		Regions
	Quantization		Average
	Post-		Regions	Blocks
	processing		Quantization	Average
	Low Resolution Y		Region-based U/V
	+	-		-	+
		Y Texture
	FDCT 2x2 blocks AC coef quant./coding	FDCT 16x16 blocks AC coef quant./coding	SPATIAL CODER	Blocks Quantization Post-processing
				Low Resolution U/V

SPATIAL CODER Figure 1: Global LAR codec scheme