P Granholm
email: patric.granholm@turkuamk.fi

T Lönnroth

J Suhonen

J Bergman

K-M Källman

J-O Lill

M Norrby

E Ydrefors

P Tikkanen

On the decay of the J π = 11/2 -, T 1/2 = 38.9 h, isomer in 133 Ba: Search for the E5 transition and verification of the EC branch

Keywords: numbers: 23.20.-g, 23.35.+g, 21.60.-n, 21.60.Cs, 23.40.-s

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Introduction

The isomeric 11/2 -level in 133 Ba has been known for several decades. It was first observed as a (40.0 ± 0.5) h isomer by Cork and Smith 1941 [1], although in that study the spin was wrongly assigned to 9/2. After this it has been studied by several others, see references [2,3,4,5,6]. The half-life of the isomeric state is T 1/2 = 38.9 h and the main M4 decay of the level proceeds to an excited 3/2 + state at 12 keV [7]. A weak branch to the 1/2 + ground state, of multipolarity E5, was reported in an Annual Report of Tokyo University [6]. Furthermore, the 11/2 -isomer has a weak EC decay branch to 133 Cs, I EC = 0.0096 (11), notably to an 11/2 + level at 633 keV. An earlier work claims 133 Ba illustrating the M4/EC decay of the T 1/2 =38.9 hours J π = 11/2 -isomer, and the possible E5 decay branch. The isomeric EC decay branch to the J π = 11/2 + state in 133 Cs has an intensity of 0.011%. The other branch, to the J π = 11/2 -state, is under dispute. The 10.7 yr EC decay of the 133 Ba ground state is included for completeness (its accompanying γ-rays will be discussed in the text).

The level scheme is based on [7,8].

decay to both this level and another almost degenerate 11/2 -level at 605 keV, cf. [5,8] but this latter claim is not supported in [6]. These two decay paths (γ-ray and EC) are marked with question marks in figure 1 and discussed in more detail in section 2. This paper presents an investigation on the competing E5 branch 11/2 -→ 1/2 + to the main 11/2 -→ 3/2 + M4 in 133 Ba. Furthermore, the study aims at settling the issue of the existence of the 11/2 -level in 133 Cs, suggested in reference [8]. This level should also be fed by the EC decay of the 38.9 h isomer in 133 Ba. The isomeric state in 133 Ba was populated using the reaction 133 Cs(p,n) 133 Ba and a CsI target.

In order to understand the transition rates related to the microscopic structures of the involved states, we have performed calculations within a QRPA (quasiparticle random-phase approximation)-based approach of the microscopic quasiparticle-phonon model (MQPM) [9,10]. These theoretical studies are used to shed light on the results and support the experimental findings on the two issues that have raised controversy:

1) Despite the high multipoles of the transitions 11/2 -→ 3/2 + and 11/2 -→ 1/2 + , the 11/2 -state decays predominantly by γ-ray emission [11]. The latter seems to be much more hindered than the former. We intend to study the existence of the E5 transition.

2) Does the 11/2 -level exist in the level structure of 133 Cs, and what are the EC decay strengths to the 11/2 -and/or 11/2 + levels?

Experimental method

The Tandem van de Graaff at the Accelerator Laboratory at Helsinki University was used to determine a suitable particle energy for the reaction 133 Cs(p,n) 133 Ba to populate the 11/2 -state in 133 Ba. The advantage of using the Tandem van de Graaff is that the energy can be changed rapidly but the disadvantage is that it does not reach sufficiently high in energy. In the experiment a proton beam with the average current of 50 nA and five different energies; 5.0 MeV, 6.2 MeV, 7.4 MeV, 8.6 MeV and 9.8 MeV was used. The proton irradiation of a thin CsI crystal target resulted in two different reactions; the desired reaction producing 133 Ba and the reaction 127 I(p,n) 127 Xe. Singles γ-rays were measured online with a Ge-detector perpendicular to the direction of the particle beam.

To avoid the relatively long decay time for the isomer 11/2 -in 133 Ba the production rate for the 13/2 -state at 902 keV in Ba was studied. This state decays solely with 613.6 keV gamma emission to the 11/2 -state and thus it gives a minimum of the production rate of the 11/2 -state. The experimental production rate was normalised by setting the production rate to 1 at the incident energy of 5.0 MeV, see figure 2. From the experimental values the excitation function for the 11/2 -state was estimated to have a maximum at an incident energy of about 12 MeV. This was confirmed by an excitation function calculated with the software package TALYS [START_REF] Koning | Proc. of the Int. Conf. on Nuclear Data for Science and Technology (Nice)[END_REF] giving a maximum at 10.5 MeV and by the results in a recent study by F. Tárkányi et al. [START_REF] Tárkányi | [END_REF], as well as in two older ones [14,15]. The fact that the accompanying reaction 127 I(p,n) 127 Xe shows a cross section maximum at the same energies [16] is not a problem as the halflife of the 9/2 -isomeric state in 127 Xe is 69.2 s, which is short compared to that of the the isomeric state in 133 Ba. Furthermore, the decay of the 127 Xe ground state through electron capture, T 1/2 = 36.4 d, does not have disturbing secondary γ-rays [17]. Figure 2 shows experimental data from this work and from references [START_REF] Tárkányi | [END_REF]15] as well as the calculated data.

The actual activation study of the 11/2 -state in 133 Ba was done at the Accelerator Laboratory at Åbo Akademi. The target, a (10 × 10 × 2) mm 3 CsI crystal, was placed in a water cooled target holder with a 25 µm stainless steel foil in front of it. The foil degrades the particle energy with 0.49 MeV. The target was irradiated with a 1 µA, 12 MeV proton beam for 3 hours. After the irradiation the target was removed from the holder and the dose rate of the target was measured to 4 mSv/h at a distance of 5 cm.

The decay of the activated CsI target was measured during a period of 11 days. Singles gamma-ray were recorded using one high-purity Ortec Ge detector of 19% efficiency. During the first 84 hours the source was placed 1 m from the detector to keep to a count rate between 1000 cps at the beginning of the measurement and 600 cps at the end. The low count rate reduces the dead time in the measurement system. Assuming a nonparalyzable model [START_REF] Knoll | Radiation Detection and Measurement[END_REF] for the dead time, the ratio between the dead-time-corrected count rate and the measured count rate was 1.0101 for the highest measured count rate of 1000 cps and a pulse width of 10 µs. On the average the ratio between the length of the measurement interval and the live time of the system was 1.0099 and thus the live time is well corrected for the dead time. After the half-life measurement the target was moved closer to the detector and the measurement was continued with a count rate of 7000 cps to get higher statistics. A spectrum collected during the first 12 hours after the activation is showed in figure 3. Strong peaks from the decay of the isomeric state in 133 Ba, 276 keV, and the decay of the ground state of 127 Xe can be seen in the upper panel. The peak origin from the decay of the 11/2 + state at 633 keV to the ground state in 133 Cs can be found in the lower panel.

The decay of the 11/2 -isomer

The half-lives of the a Purely statistical error, the systematic error is expected to be of the same order of magnitude or lower than the statistical one.

the two transitions to the ground state. The relative intensities for both transitions were given as 0.011 % and 0.013 % and the log f t values as 7.9 and 8.0, respectively. In another work by Kikuchi [5] both levels in 133 Cs have been assigned spin and parity. The assignments were based on angular distribution measurement of the radiation from the reaction 133 Cs(n, n' γ). The level at 633 keV was assigned the spin and parity 11/2 + and was considered to be based on the g 9/2 proton state coupled to the lowest 2 + phonon state in 132 Xe. The other level at 605 keV was assigned spin 11/2 -from the h 11/2 proton state. In order to study the 605 keV line the position of the fitted peak was locked to the correct position and the shape was obtained from near-by peaks. But the peak did not grow with increasing recording time in the same manner as the peak for the 633 keV transition which is known to be fed from the EC-decay of 133m Ba, see figure 5 and the insert in the figure. The peak at 605 keV grows, without significant deviations, linearly in time which indicates that it originates from a long-lived nucleus in the background. However, the source could not be identified. Furthermore, the 11/2 -→ 11/2 -transition should be faster since it corresponds to an allowed EC decay whereas the 11/2 -→ 11/2 + transition corresponds to a first-forbidden EC decay. Thus, if the unidentified peak at 605 keV would origin from the decay of a 11/2 -state in 133 Cs it should be larger than the peak at 633 keV. However, in this study the intensity ratio of the former to the latter was found to be smaller than 0.12.

Ando et al. report a weak E5 transition between the isomeric 11/2 -state and the 1/2 + ground state in 133 Ba. The relative intensity was found to be (3.6 ± 2.5) • 10 -4 of the 276 keV M4 transition [6]. In the present study spectra collected during the first 72 h after the activation were used in the analysis. With the relative intensity given above the peak area at 288 keV should be about 9.7 • 10 3 counts as there were 2.70 • 10 7 counts in the 276 keV peak. However, a clear peak cannot be found in the spectrum. To establish an upper limit for the intensity a peak, with the shape and background parameters locked, was forced at the position of the assumed 288 keV peak. Figure 6 shows the forced-peak fit at 288 keV and fits to the near-lying peaks at 295 keV and 303 keV. The same background and parameters for peak shape were used for all three peaks. The forced peak height 125 counts and the width 3.0 channels resulted in a peak area of 490 counts and a peak at this position would be smaller. As the internal conversion coefficients for both the M4 and the E5 transitions are of the same order [START_REF] Kibédi | [END_REF], 4.64 and 4.08 respectively, internal conversion can be neglected. This gives an upper limit of 1.8 • 10 -5 for the E5/M4 intensity ratio. The ratio between the calculated transition probabilities is 3.1 • 10 -6 , see table 2, in agreement with the experimental upper limit.

Theoretical calculations

In this paper the MQPM [10] was used for theoretical calculations of the wave functions and excitation energies of 133 Ba and 133 Cs. In the model the three-quasiparticle states of an odd-A nucleus are formed by coupling one-quasiparticle states to QRPA (quasiparticle random-phase approximation [START_REF] Ring | The Nuclear Many-Body Problem[END_REF]) phonons. The MQPM calculations were performed in four steps: First the single-particle energies were calculated from the Coulomb-corrected Woods-Saxon potential with the Bohr-Mottelson parametrization [START_REF] Bohr | Nuclear Structure[END_REF].

In the calculations a valence space containing 11 neutron orbitals from 0f 7/2 to 0h 11/2 and 9 proton orbitals from 1p 1/2 and 0h 11/2 was used. The calculated results could be improved by slightly adjusting the proton and neutron single-particle energies in the vicinity of the respective Fermi surfaces.

BCS calculations were then performed separately for neutrons and protons. The Bonn one-boson-exchange potential [START_REF] Holinde | [END_REF] was used as the two-body interaction. The experimental neutron and proton pairing gaps were calculated from the three-point formulae (see e.g. [START_REF] Suhonen | From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory[END_REF]). The monopole matrix elements were renormalized by adjusting the proton and neutron pairing strengths [START_REF] Suhonen | [END_REF] so that the lowest one-quasiparticle energy was approximately equal to the experimental pairing gap.

The QRPA model was subsequently used to calculate the excitation spectrum of the even-even reference nucleus ( 134 Ba). For each multipole J π the particle-particle (g pp ) and the particle-hole (g ph ) strengths were adjusted so that the experimental low-lying energy spectra were reproduced.

In the final step of the MQPM calculations the interaction Hamiltonian was diagonalized (see [10]) to obtain the MQPM wave functions and excitation energies. A phonon basis containing all QRPA states having angular momentum J ≤ 10 (π = +, -) and excitation energy E ω ≤ 3.5 MeV was used. The obtained energy spectra are compared with experimental data in figures 7 and 8 and the main configuration of selected states are given in table 2. As seen in the figures, the agreement is very good and the calculations does not support two 11/2 states at about 600 keV in 133 Cs.

Furthermore, electromagnetic transition probabilities and half-lives have been calculated for the γ-transitions shown in figure 1. The calculated partial half-life for the M4 transition, using free-nucleon gyromagnetic ratios, is 0.5027 • 10 5 s ≈ 14.0 h. However, effective gyromagnetic ratios of gν l = -0.07, gν s = -2.30 and gπ l = 1.13, gπ s = 3.546 from [START_REF] Castel | Modern Theories of Nuclear Moments[END_REF] for the neutrons and protons, respectively, gave T 1/2 = 0.1426 • 10 6 s ≈ 39.6 h, in excellent agreement with the experimental value. The calculated partial half-life of a possible E5 admixture (1.64 • 10 11 s) supports the findings in reference [4] that this transition is of pure M4 character.

For the suggested 288 keV E5 transition from the 11/2 -state to the ground state the partial half-life was calculated using effective charges e ν ef f = χe and e π ef f = (1 + χ)e with χ = Z/A [START_REF] Suhonen | From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory[END_REF] i.e. 0.421e and 1.421e for the neutron and proton respectively, resulting in T 1/2 = 4.563 • 10 10 s = 1.27 • 10 7 h and branching of 3.1 • 10 -6 as mentioned above. The effective gyromagnetic ratios and effective charges may origin from meson exchange currents and core polarizations, respectively, see reference [START_REF] Castel | Modern Theories of Nuclear Moments[END_REF] for details.

The calculated partial half-lives for the other transitions in 133 Cs and 133 Ba are in good agreement with the experimental half-lives of the states with some exceptions. The low energy transition of 12 keV between the 3/2 + and 1/2 + states in 133 Ba is an l-forbidden M1 transition with a low E2 admixture [4]. A similar argument explains the overestimated half-life for the 53 keV transition between the 1/2 + state and the 3/2 + state and the 81 keV transition between the low-lying 5/2 + state and the ground state, both l-forbidden M1 transitions [START_REF] Thun | [END_REF]. In these cases the calculations gives the realistic partial half-lives for the M1 transitions; 9.25 • 10 21 s, 3.00 • 10 27 s and 3.46 • 10 21 s, respectively. This is because the three-quasiparticle amplitudes of the MQPM states in these cases are very small and thus leading to an almost vanishing M1 matrix element. The results are summarized in table 2.

Conclusions

Both the EC-and the γ-decay branches of the isomeric 11/2 -state in 133 Ba have been studied. The proposed competing EC-decay branch to a 11/2 -level at 605 keV in 133 Cs was not found, even though it was possible to fit a small peak at the correct position in the obtained energy spectrum. However, the EC branch to the 11/2 + level at 633 keV was confirmed. The peak at 605 keV behaves as if it would originate from a long-lived source in the background. Furthermore, the MQPM calculation gives an 11/2 + state at 795 keV with the main configuration arising from a proton g 7/2 ⊗ 2 + phonon and, in addition a 11/2 -state at 952 keV with the main configuration h 11/2 . Corresponding levels are also found experimentally at 633 keV and 1071 keV, respectively [7]. Thus the existence of a 11/2 -state at 605 keV is not supported by experiment nor by MQPM calculations. Therefore, the only existing EC-decay from the isomeric state in 133 Ba goes to the level at 633 keV in 133 Cs.

The weak E5 γ-transition to the ground state in 133 Ba could not be detected. Therefore, only an upper limit for the intensity relative to the main γ decay branch at 276 keV could be established to 2.2 • 10 -5 . The theoretical calculation suggests that the E5 transition should be a single-particle transition h 11/2 → s 1/2 and the M4 transition should have the single-particle structure h 11/2 → d 3/2 . The suppression of the E5 transition relative to the M4 transition is thus simply the suppression arising from the higher multipolarity λ = 5 of the electric transition, as can be simply estimated by using the formalism described in [START_REF] Suhonen | From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory[END_REF].

  On the decay of the J = 11/2, T1/2 = 38.9 h isomer in 133Ba: search for the E5 transition and verification of the EC branch P Granholm, T Lönnroth, J Suhonen, J Bergman, K-M Källman, J-O Lill, M Norrby, E Ydrefors, P Tikkanen To cite this version: P Granholm, T Lönnroth, J Suhonen, J Bergman, K-M Källman, et al.. On the decay of the J = 11/2, T1/2 = 38.9 h isomer in 133Ba: search for the E5 transition and verification of the EC branch. Journal of Physics G: Nuclear and Particle Physics, 2011, 38 (1), pp.15101. 10.1088/0954-3899/38/1/015101. hal-00600868

Figure 2 .

 2 Figure 2. The figure shows a plotted experimental data from this work ( ) and the cross sections from F. Tárkányi et al. [13] (×), Sakamoto et al. [15] ( ), as well as the calculated data (----)

Figure 4 .

 4 Figure 4. The figure shows the decrease of the corrected activity corresponding to the decay 11/2 -state in 133 Ba as a function of time. The line is fitted to the data.

Figure 5 .

 5 Figure 5. A part of the spectrum showing that it is possible to fit a peak at 605 keV. The spectra were measured during 84 h. The inserted figure shows the growth of the peak at 605 keV (×) and 633 keV ( * ) as a function of time. The line is fitted to the data.

Figure 6 .

 6 Figure 6. A fit to the peaks close to the position of the 288 keV peak. The assumed peak at 288 keV was created by force with the peak and background shape fixed based on the analysis of the peak at 275 keV. The forced peak height was 125 counts and the width 3.0 channels

Figure 8 .

 8 Figure 8. Theoretical and experimental states of 133 Cs. The experimental data is obtained from [7].

Table 1 .

 1 11/2 -state in133 Ba have been reported by several authors, see table1. In this work the half-life was determined from the M4 γ-decay. The activity measured during time intervals of 12 h for the first 72 hours were used to determine the half-life. The total counts where divided with the live time of the measuring system and corrected to the beginning of the time interval. The half-life and the statistical error, T 1/2 = (38.92 ± 0.09) h, was obtained from an exponential fit to the values using gnuplot 4.2.0, see figure4. The systematical error is expected to be of the same order of magnitude or lower than the statistical one. The result clearly confirms the previous results by Wille et al. but the shorter half-life determined by Ando et al. drops outside the error limits.Sugiyama and Kikuchi[8] have reported electron-capture decays to two levels at 633 keV and 605 keV in133 Cs in addition to the 275.6 keV M4 transition in 133 Ba. These two EC-branches where identified by detection of gamma-rays corresponding to Reported half-lives for the isomeric 11/2 -state 133 Ba and intensity ratios E5/M4.

			Gamma-energy [keV]
		50	150	250	350
		1800
	]	1400
	Counts/Channel [ 10 3	600 1000
		200

Ac

Figure

3

. A spectrum collected during the first 12 hours after the activation. The strongest peak origins from the decay of the isomeric state in 133 Ba. Strong peaks originating from the decay of the ground state of 127 Xe can also be found.

  Theoretical and experimental states of 133 Ba. The experimental data is obtained from[7].

	1.0 2.0	3/2+ 1/2+ 5/2+ 3/2+ 5/2+ 7/2+ 7/2-9/2-3/2+ 15/2-7/2+ 11/2-		0.823 0.854 0.909 1.001 1.051 1.083 1.121 1.140 1.169 1.172 1.198 1.225	3/2+,5/2+ (3/2,5/2)+ 3/2+,5/2+ + (9/2)+ 3/2+,5/2+ 13/2 3/2+,5/2+ 15/2-3/2+,5/2+ 3/2+,5/2+,7/2+ 3/2+,5/2+		0.631 0.676 0.794 0.858 0.863 0.883 0.887 0.902 0.924 0.969 1.022 1.112 1.212
		7/2+		0.615	1/2+ (7/2)+		0.540 0.578
					3/2+		0.302
		5/2+		0.386	(5/2)+		0.291
		11/2-		0.302	11/2-		0.288
		3/2+		0.050	3/2+		0.012
	0.0	1/2+	MQPM	0.000	1/2+	133Ba exp.	0.000
	1.0 Figure 7. 0.0 7/2+ 5/2+ 3/2+ 5/2+ 1/2+ 3/2+ 7/2+ 3/2+ 1/2+ 11/2+ 9/2+ 5/2+ 7/2+ 11/2-9/2+ 3/2+ 2.0 5/2+	MQPM	0.000 0.165 0.557 0.606 0.638 0.681 0.721 0.749 0.788 0.795 0.806 0.837 0.921 0.952 0.967 1.106 1.158	7/2+ 5/2+ 5/2+ 3/2+ 1/2+ 11/2+ 3/2+ 9/2+ 9/2+ (5/2,7/2,9/2)+ 7/2+ (11/2-)	133Cs exp.	0.000 0.081 0.161 0.384 0.437 0.633 0.640 0.706 0.728 0.768 0.789 0.819 0.872 0.916 0.942 1.071 1.089 1.173 1.219

Table 2 .

 2 Summary of MQPM results for the interesting states in 133Cs and 133 Ba and comparison with experimental data from reference[7]. The experimental T 1/2 values are half-lives of the states whereas theoretical values, calculated using effective gyromagnetic ratios and charges, are the partial half-lives of the transitions. Only the main configurations used in the calculations are shown.

		th. multipole		E2	M1	M1	E2	M1	E2	M1	E2	M1	E2	E2	M1	E2	M1	E2	M1	E2	M1	M1	E2	E2	M1	M4	E5	E5
	Decay	E γ [keV] final state exp. multipole T th 1/2 [s]		81 g 7/2 M1+E2 8.37 • 10 -5	3.46 • 10 21	80 d 5/2 M1+E2 5.52 • 10 -6	7.51 • 10 -5	161 g 7/2 M1+E2 1.02 • 10 -7	1.99 • 10 -7	223 g 7/2 ⊗ 2 + M1+E2 1.34 • 10 -6	3.89 • 10 -6	303 d 5/2 M1+E2 1.73 • 10 -12	9.97 • 10 -9	384 g 7/2 E2 1.40 • 10 -9	2.61 • 10 21	53 d 3/2 M1+E2 8.83 • 10 -6	3.00 • 10 27	276 g 7/2 ⊗ 2 + E2 2.48 • 10 -6	3.70 • 10 -7	356 d 5/2 E2 5.59 • 10 -10	8.79 • 10 21	633 g 7/2 E2 2.06 • 10 -9	6.73 • 10 -10	12 s 1/2 M1 1.07	9.25 • 10 21	276 d 3/2 M4 1.43 • 10 5	1.64 • 10 11	288 s 1/2 E5 4.56•10 10
	Energy [keV]	theory exp. T 1/2 [s]	g.s. g.s.	165 81 6.28 • 10 -9		606 161 1.72 • 10 -10				577 384 2.1 • 10 -11						638 437 ≤ 1.50 • 10 -10						795 633 5.4 • 10 -12	g.s. g.s.	50 12 7.0 • 10 -9		302 288 1.40 • 10 5
						2 +																2 +
	Main	Conf	g 7/2	d 5/2		g 7/2 ⊗				d 3/2						s 1/2						g 7/2 ⊗	s 1/2	d 3/2		h 11/2
		State	Cs 7/2 +	5/2 +		5/2 +				3/2 +						1/2 +						11/2 +	Ba 1/2 +	3/2 +		11/2 -
			133																				133

Acknowledgments

Patric Granholm has been supported with a grant from Svenska Kulturfonden.


D Fraix-Burnet

The Fundamental Plane of Early-Type Galaxies as a Confounding Correlation

Keywords: galaxies, fundamental parameters -methods, statistical -galaxies, evolutiongalaxies, formation -

Early-type galaxies are characterized by many scaling relations. One of them, the so-called fundamental plane is a relatively tight correlation between three variables, and has resisted a clear physical understanding despite many years of intensive research. Here, we show that the correlation between the three variables of the fundamental plane can be the artifact of the effect of another parameter influencing all, so that the fundamental plane may be understood as a confounding correlation. Indeed, the complexity of the physics of galaxies and of their evolution suggests that the main confounding parameter must be related to the level of diversification reached by the galaxies. Consequently, many scaling relations for galaxies are probably evolutionary correlations.

INTRODUCTION

Galaxies are huge systems of stars, gas and dust, that have assembled through a complex history of collapses, internal perturbations, interactions, mergers and even ejections of some material. This complexity renders quite remarkable the scaling relations found between many observables and properties, like size, mass, surface brightness, velocity dispersion, magnitude, color, metallicity, black hole mass or spectral features. The fundamental plane is a famous correlation between three variables, the effective radius, the central velocity dispersion and the surface brightness within the effective radius (Dressler et al. 1987;Djorgovski & Davis 1987;Nieto et al. 1990). For more than 20 years of investigations, it has resisted a satisfactory physical understanding (Robertson et al. 2006;D'Onofrio et al. 2008;Gargiulo et al. 2009;Graves et al. 2009;Nigoche-Netro et al. 2009;Fraix-Burnet et al. 2010).

For (astro)physicists, such scaling relations reveal some underlying physical laws that are described by a set of equations. However, for statisticians, correlations are not always causal, as lurking or confounding parameters can do the trick. In general, these correlations are called "spurious" although this more specifically concerns acknowledged false correlations.

A famous example of a "spurious" correlation in statistics is the relation between ice cream sales and number of drownings. The confounding parameter is the high temperature which favours both the ice cream consumption and the number of people swimming, hence the probability of drownings. So even unrelated phenomena can appear correlated. Other examples, like the relation between the size of a children and his school level or between the distance ⋆ E-mail: fraix@obs.ujf-grenoble.fr of the Pioneer probe to the Earth and the average Earth temperature since its launch, have time or evolution as confounding factor.

Indeed the correlation between children's size and school level is statistical: the taller ones being more probably found at higher levels. This is reminiscent of a classical reasoning used in the determination of the cosmic distance ladder: for example, more luminous stars are more probably found in bigger galaxies. Simillarly, because of evolution, bigger galaxies are more probably more massive, more luminous, more metallic, with a higher velocity dispersion, etc. Hence statistically, a lot of properties must appear correlated, the confounding parameter being the level of diversification.

In this paper, we show that correlations between the three variables of the fundamental plane can easily be the artifact of the effect of another parameter influencing all. In other words, we show that the fundamental plane may be understood as a confounding correlation. We propose that the confounding parameter(s) is (are) related to evolution. Consequently, many scaling relations of galaxies are probably evolutionary correlations.

In Sect. 2 we derive the conditions for parametric relations to yield a planar correlation in a 3-variable space. We then dedicate a full section to the fundamental plane (Sect. 3) to show how easily these conditions can be fulfilled. We finally discuss why and how the evolution of galaxies is probably the confounding parameter (Sect. 4).

CONFOUNDING CORRELATIONS

Let us consider three variables Ω 1 , Ω 2 , Ω 3 that are all functions of a same generic parameter X. For sake of simplicity in this paper, let us assume that these functions are power laws:

c 2011 RAS L2 D. Fraix-Burnet            Ω 1 = A 1 X p Ω 2 = A 2 X s Ω 3 = A 3 X t (1)
with A 1 , A 2 and A 3 being constant. Any linear correlation of the form

log Ω 1 = a log Ω 2 + b log Ω 3 + c (2)
where a, b and c are constant, is thus expressed as

p log X + log A 1 = as log X + bt log X + a log A 2 + b log A 3 + c. (3)
This expression should be valid for all X, implying that

p = as + bt log A 1 = a log A 2 + b log A 3 + c. ( 4
)
This set of two linear equations generally yields solutions for a and b. The result is thus a plane in the 3-D space (Ω 1 , Ω 2 , Ω 3 ).

The generic parameter X can be a multivariate component, making the parametric dependence of Ω 1 , Ω 2 and Ω 3 mutlivariate. For instance, consider power laws with two parameters, X 1 and X 2 :

           Ω 1 = A 1 X p 1 X p ′ 2 Ω 2 = A 2 X s 1 X s ′ 2 Ω 3 = A 3 X t 1 X t ′ 2 (5)
Then it is easy to show that a correlation like equation (2) holds if:

         p = as + bt p ′ = as ′ + bt ′ log A 1 = a log A 2 + b log A 3 + c. (6)
This set of equations is more constraining than equation (4). But if for instance the second relation in equation ( 6) is not exactly fulfilled, then the parameter X 2 can be considered as noise adding a dispersion to the correlation defined by the two other relations. In other words, X 2 could generate a thickness to the plane defined by equation (2).

To be complete, we must discuss the physics in the parameter X. This parameter is supposed to influence all the three variables. We distinguish two possbilities: causality or evolution. In the first case, the relation between the variables and X is driven by direct causality, that is some parameter influences directly each of the three variables Ω 1 , Ω 2 , Ω 3 through physical laws. This is illustrated in Sect. 3.2. In the second case, the relation is statistical in the sense that each variable is bound to evolve so that even totally unrelated variables can show an apparent correlation. This is discussed in Sect. 3.3 and Sect. 4.

THE FUNDAMENTAL PLANE OF EARLY-TYPE GALAXIES

General constraint

Let us put Ω 1 = r e the effective radius, Ω 2 = σ the velocity dispersion, and Ω 3 = L the luminosity. equation ( 1) becomes:

           r e = A 1 X p σ = A 2 X s L = A 3 X t (7)
The surface brightness µ e can be expressed as

µ e = -2.5 log L/4πr 2 e + m = (-2.5t + 5p) log X + 2.5 log(4π) + m (8)
where m is a constant of normalisation. Any linear correlation of the form log r e = a log σ + bµ e + c (9) translates to

p log X + log A 1 = as log X + b (-2.5t + 5p) log X + a log A 2 + b 2.5 log 4πA 2 1 /A 3 + m + c. ( 10
)
This implies

p = sa + (-2.5t + 5p) b logA 1 = a log A 2 + b 2.5 log 4πA 2 1 /A 3 + m + c. ( 11
)
If a solution can be found for a and b from equation ( 11), then the equation of the fundamental plane equation ( 9) is obtained. There is no need of any further assumption to explain the fundamental plane. This demonstration is made here with a simple power-law assumption in equation ( 7). But this result is also true for more complex functions, with equation ( 11) being replaced by more complicated conditions.

To understand the origin equation ( 9), we need to solve equation (11). Since there are too many unknowns, additional input is required. Two approaches are possible: either input some a priori knowledge to determine the functions of X and derive coefficients a and b (Sect.3.2), or conversely use the observations to determine a and b and derive constraints on the functions of X (Sect.3.3).

A priori physical input

In this section, we consider the input of a priori knowledge believed to be relevant for the physics of galaxies. At first glance, it is quite logical to consider that mass, either dynamical, true or stellar, is somehow related to the radius, the velocity dispersion and the surface brightness, essentially because it influences the density of stars and their kinematics. So we consider X = M in equation ( 7).

One obvious way to link the kinematics to the mass is through the virial theorem. Hence, let us assume that the virial equation holds and that the ratio between the average squared velocity and σ 2 , and the ratio between r e and the gravitational radius, are constant. This gives

σ 2 r e ∝ M ⇒ 2s + p = 1 (12)
Using equation ( 11) and equation ( 12) we obtain

1 -2s = sa + (-2.5t + 5 (1 -2s)) b ⇒ s (a + 2 -10b) = 2.5tb -5b + 1 ⇒ 2.5tb -5b + 1 = 0 if a + 2 -10b = 0 s = (2.5tb -5b + 1) / (a + 2 -10b) otherwise. ( 13
)
Unfortunately, the brightness has no direct relation to mass, but it might be assumed that the mass is essentially due to the stars that are responsible for the luminosity. If we also assume that the ratio M/L is constant for a given population of galaxies, we must have:

L ∝ M ⇒ t = 1. ( 14
)
Replacing this value of t in equation ( 13), one obtains two solutions:

(i) 2.5b -5b + 1 = 0 and a + 2 -10b = 0 ⇒ b = 0.4 and a = 2 or (15) (ii) s = (2.5b -5b + 1) / (a + 2 -10b) ⇒ s = (1 -2.5b) / (a + 2 -10b)
The first solution is well known, being the so-called virial plane a = 2 and b = 0.4, a classical solution obtained with the assumptions made to derive equation ( 12) and equation ( 14). In this case the dependency of σ (or r e ) on the mass M is undetermined (s and p cannot be computed).

However, the second solution yields different values of the coefficients a and b with the same classical assumptions. In this case, the dependency of r e or σ on M can be computed. Hence the tilt of the observed fundamental plane with respect to the virial plane could simply be explained in this way whithout relaxing any of the assumptions made above. Taking typical observed values a ≃ 1.1 and b ≃ 0.3 (e.g. Fraix-Burnet et al. 2010), we find s ≃ 2.5 and p ≃ -4. This yields strong dependences of the variables on mass. In particular, the r e vs M relation is steeper than that found for instance in numerical simulations by Robertson et al. (2006).

As a conclusion, the two possible solutions implied by equation ( 16), leading to s and p either undefined or relatively high, can be debated, but it appears clearly that the assumptions made to derive the virial plane are a very restrictive case of more general conditions (equation ( 11)) that lead to a fundamental plane correlation. Interestingly, one usual modification to explain the apparent tilt of the observed fundamental plane with respect to the virial plane is to relax the assumption in equation ( 14) with t 1 and rely on the observations for this value. In the next section we show that we can go a step further and use the more general conditions in equation (11) with the sole observations as constraints without any a priori input from the virial hypothesis. This allows the exploration of wider physics: the virial conditions (equations ( 12) and ( 14)), if fulfilled, should be derived from the observations, not taken as an a priori.

Observational constraints

Henceforth, let us use the relations in equation ( 11) to derive the constraints on the functions in equation ( 7) from the observations. To derive the three exponents s, p and t, we take the values for a and b as provided by the observed fundamental plane correlations, and also other observed correlations.

As mentioned above, the observed fundamental plane yields typical values of a ≃ 1.1, b ≃ 0.3 and c = -8.5 (and m = 4.45 in the R band). However, these are known to depend on the sample, the wavelength, the redshift and more generally on the group of galaxies (e.g. Fraix-Burnet et al. 2010). This strongly indicates that the functions of X in equation ( 7) and X itself are not universal and depend on the population of galaxies and their assembly histories. This can be seen in the numerical simulations by Robertson et al. (2006) where the r e vs M relations change according to the sequences and nature of the merging events.

With the observed values for a, b and c, equation (11) becomes

           p ≃ 1.1s + (-2.5t + 5p) 0.3 log A 1 ≃ 1.1 log A 2 + 0.3 2.5 log 4πA 2 1 /A 3 + 4.45 -8.5 ⇒ p ≃ -2.2s + 1.5t A 1 ≃ 2 10 -13 A -2.2 2 A 1.5 3 . (16)
The above set of relations is independent of the actual meaning of X. The physics can here come to our help to consider different possibilities for the confounding parameter X, but here we still only use observations (of real objects or synthetic ones from numerical simulations) to determine the shape of the functions of X.

Let us consider the fig. 5 in Hopkins et al. (2008) that gives the dependence of r e and σ on the mass fraction f starburst of the starburst which is an indicator of the level of dissipation in passed merger. The dissipation has been found to be a key element to explain the very existence of the fundamental plane. In simple words, without dissipation, the merger product would not follow a plane, that is to say that the correlation of equation ( 9) would not exist. Hence if we assume that the relations given in their fig. 5 are observations (of synthetic populations of galaxies in this case), then we derive that p ≃ -1 and s ≃ 1. We then compute the exponent t from equation ( 16) and find t ≃ 2.1. We thus have

         r e ∝ f -1 starburst σ ∝ f 1 starburst L ∝ f 0.8 starburst (17)
These functions are in agreement with the expected role of dissipation: evacuates angular momentum, concentrates the matter in the bulge (r e diminishes), increases the surface brightness, and increases the velocity dispersion.

Let us consider another candidate for X: M BH the mass of the central black hole. It can be guessed that it could play a key role in the evolution of the properties of galaxies in the course of diversification. On the Fig. 4 in Hopkins et al. (2009), there is a clear dependence of r e and σ on M BH . The relations are not exactly linear, but let us assume that it is the case to a first approximation. We find that p ≃ 0.63 and s ≃ 0.28, implying from equation ( 16) that

t ≃ 0.83. Then          r e ∝ M 0.63 BH σ ∝ M 0.28 BH L ∝ M 0.83 BH (18)
These values compare very favorably to the results obtained by Gültekin et al. (2009) (σ ∝ M 0.25 BH and L ∝ M 0.9 BH ) and Marconi & Hunt (2003) (L ∝ M 0.83 BH ) for elliptical galaxies. We then conclude that the fundamental plane is explained by solely assuming that the black hole mass evolves in parallel with other properties of galaxies, without any direct physical modeling of each relation in equation ( 18), Finally, it is interesting to remark that we can combine the two examples above with X = ( f starburst , M BH ) like in equation ( 5) and obtain the conditions equation ( 6). We then obtain exactly a combination of equation ( 17) and equation ( 18):

         r e ∝ f -1 starburst M 0.63 BH σ ∝ f 1 starburst M 0.28 BH L ∝ f 2.1 starburst M 0.83 BH ( 19
)
This solution is given for illustration only because one might question the independence between f starburst and M BH (see below).

We have thus proven that the fundamental plane correlation equation ( 9) can be easily and plausibly obtained with confounding parameters like merger dissipation or central black hole mass, without any assumption linked to the virial equilibrium.

EVOLUTIONARY CORRELATIONS

In the course of diversification, many properties of galaxies change, and they tend to statistically change in a more or less monotonous way. For instance mass and metallicity are both bound to increase with the complexity of a galaxy assembly history, so that they appear to be statistically correlated. It seems difficult to avoid the evolution to act as a confounding factor.

We thus propose that the main confounding parameter is X = T with T an indicator of the level of diversification, being something like an evolutionary clock not necessarily easily related to time or redshift.

In the simple case of a single stellar population, the time since formation is naturally a good evolutionary clock for some parameters like luminosity L, color and metallicity. However this is less obvious for r e or σ. Time since formation can probably be as good for a homogeneous population of galaxies if they are not affected by significant transformation events (interactions, mergers...). However, galaxies from the same homogeneous population (identical properties) do not form at the same epoch in the Universe, so that the time since formation is not related to redshift. In addition, galaxies are much more complex than single stellar populations, so that the "age" of a galaxy unfortunately is only an average over the different stellar populations and does not characterize its complete evolutionary stage.

Considering now X = (1+ z), we take from Saglia et al. ( 2010) that p ≃ -0.5 and s ≃ 0.4. Note that these values are obtained at fixed mass, which probably biases significantly the derived evolutions of r e and σ. Anyhow if a tight fundamental plane correlation exists with the parameters previously used (those given in [START_REF] Saglia | A6+[END_REF], close to ours, do not change the main result here), then equation ( 16) yields :

         r e ∝ (1 + z) -0.5 σ ∝ (1 + z) 0.4 L ∝ (1 + z) 0.25 (20)
Our luminosity evolution is clearly weaker than the one estimated in [START_REF] Saglia | A6+[END_REF]: L ∝ (1 + z) 1 . This discrepancy could probably be explained by the various hypotheses they have to make to try disentangling all effects in such a multivariate and evolutionary context. In particular, the evolution of r e and σ are computed at fixed mass. possibly introducing interdependencies of variables through the M/L ratio. More importantly, the cosmological clock (1 + z) is not a good evolutionary clock for a mixture of different populations of galaxies since they do not evolve at the same time, at the same space and along the same paths. It is well known that the tilt of the fundamental plane depends on the sample (D'Onofrio et al. 2008) or on the group considered (Fraix-Burnet et al. 2010).

Anyhow, diversification cannot be summarized with only one simple property (like redshift or mass) because galaxies are too complex objects and do not evolve linearly in a unique way. In some diagrams, that is for some set of variables, a particular property could crudely depict the general trend of diversification. In the case of r e , σ and µ e , and to a first approximation, mass could well represent a satisfactory driving parameter for the fundamental plane correlation, but it is certainly not unique as shown in Sect. 3.3. Since it is only approximate, some dispersion is expected.

A lot of observables evolve with diversification, at least statistically, so that we should not be surprised by the many scaling relations found for galaxies and the difficulty to pinpoint the driving parameters and mechanisms. We also better understand why several characteristic parameters (mass, luminosity, metallicity...) and also the samples themselves have been found to influence the shape of the fundamental plane without providing a clearer picture of its origin.

This might also explain some of the observed dispersion is most scatter plots. For instance, it has been found that the disper-sion of the fundamental plane strongly depends on the evolutionary group (Fraix-Burnet et al. 2010), the correlation equation ( 9) even not holding in the least diversified groups. Also relations several parameters like in equation ( 19) are most often wrong because the evolutionary clock T makes most variables to be non independent. Hence, dispersion may be explained by the statistical (non-causal) nature of the correlation and the heterogeneity of the samples as far as diversification is concerned.

Indeed, the evolutionary clock, i.e. the factor X = T , can be hidden, not understandable analytically and not directly observable. It is more directly related to an evolutionary classification, and is a well-known problem of comparative methods in phylogeny (e.g. Felsenstein 1985).

CONCLUSION

The fundamental plane correlation and similarly scaling relations for galaxies can be formalized as confounding correlations. The confounding factor X is very probably related to the level of diversification and may be identified in some cases with some variables that trace the global evolution of galaxies. The dependence of the variables involved in the observed correlations on X has been assumed here to be power law functions for sake of simplicity but they could be more complicated without changing the result of the present paper. In particular, these functions can be multivariate, with several confounding factors.

The physics thus should not be invoked to explain causally each of the observed correlations, but rather to explain the confounding or evolutionary nature of these correlations. Since the galaxy assembly history and transformation processes are complex, it is quite improbable that a simple physical theory can yield X and the dependence of observables on this parameter. Indeed, these functions more probably come from the statistics in the course of galaxy diversification. In addition, X is probably hidden, not understandable analytically and not directly observable. It might also be different depending on the set of variables and the group of galaxies considered.

Gaining insights on the confounding parameter X requires several complementary statistical approaches. A first one is to combine several scaling relations and several galaxy samples in order to identify some common variables that could play the role of the confounding parameter. A second approach is to use numerical simulations to produce synthetic populations of galaxies with as many assembly configurations as possible (like in e.g. Robertson et al. 2006;Hopkins et al. 2008Hopkins et al. , 2009)). The great advantage here is to play with unobservable parameters. Finally, the third approach is to group galaxies according to their assembly history. Since the confounding parameter X is probably mainly linked to the level of diversification, the scaling relations necessarily depend on the evolutionary groups. Indeed, the very nature of X and the functions characterizing the dependence of the variables on X are expected to depend on the assembly history of galaxies.

Combining evolutionary classifications, numerical simulations, observed scaling relations and recognizing the latter as evolutionary correlations, will lead us toward a much better understanding of the history of galaxy formation. Interestingly, all astrophysical objects evolve, and evolutionary correlations could probably also explain scaling relations for other stellar systems such as globular clusters (e.g. Misgeld & Hilker 2011;Fraix-Burnet et al. 2009).

c 2011 RAS, MNRAS 000,[1][2][3][4][5]

This paper has been typeset from a T E X/ L A T E X file prepared by the author.c 2011 RAS, MNRAS 000,[1][2][3][4][5]

ACKNOWLEDGMENTS

I would like to thank my colleagues Asis Chattopadhyay, Tanuka Chattopadhyay, Emmanuel Davoust and Marc Thuillard for numerous and enlightning discussions.


Irving O Morales

A Frank

Improving nuclear mass predictions through the Garvey-Kelson relations

Keywords: numbers: 21, 10, Dr,21, 60, -n

We describe a procedure to improve the nuclear mass predictions obtained by means of the Garvey-Kelson (GK) iterative process. This is achieved by reducing the degrees of freedom involved in the process, expressing the Garvey-Kelson relations in terms of one-neutron and one-proton separation energies and using a theoretical estimation for them. This approach has the effect that the intrinsic error associated to the iterative process grows significantly more slowly. This leads to a better quality of the nuclear mass predictions and therefore extends the range for which the predictions are accurate. We test these ideas using different mass models.

I. INTRODUCTION

A precise knowledge of nuclear masses in regions away from measured values is an important topic in nuclear physics, due to their importance in astrophysical calculations [1], but a large amount of these masses remain unknown. The values of the unknown nuclear masses are obtained through theoretical predictions but, unfortunately, there is a lack of consensus and most predictions differ drastically from each other, specially in the region of large neutron excess [1,2]. There are a large diversity of models and techniques to predict nuclear masses [1] which can essentially be classified into global and local approximations.

Global models use the totality of currently available experimental information corresponding to the measured nuclear masses. Examples of this kind of models are the liquid drop model [3], the finite-range droplet model [4], the Duflo-Zuker model [5], the Hartree-Fock approach [6] and the CLEAN deconvolution algorithm [7]. These models assume that the current experimental knowledge is sufficient to determine the necessary parameters of the theory, so they can be used to describe the behavior of the masses outside the known region. The models thus focus on explaining the general characteristics and overall systematics of the mass surface on the N, Z plane. While it is true that it takes a great effort to capture the main properties of these global systematics, in order to predict such a large amount of information, small details and surface structures at smaller scales are usually not possible to describe. These "higher resolution" structures , however, are required to provide sufficient accuracy in the predictions, needed for most practical applications [1].

On the other hand, local approaches view the problem with a completely different philosophy. These techniques attempt to predict nuclear masses much more precisely, but in a limited region close to the experimentally known one. These models predict one nucleus at a time and consider only the information arising from the neighboring nuclei, i.e. the structures in the immediate vicinity and how they vary as a function of the number of neutrons and protons. Essentially there are two methods to predict nuclear masses in this fashion. The Audi-Wapstra extrapolation method [8], based on the systematics and smoothness of the mass surface and its derivatives, the one and two particle separation energies and the Q β and Q α decay values. The second one arises from the use of the Garvey-Kelson relations [9], and in this article we will concentrate on them. The GK relations are algebraic equations between the masses of six neighboring nuclei obtained by considering an extreme single-particle model, where the interactions between neutrons and protons vary smoothly along the (N,Z) plane. These relations are obtained by constructing combinations of neighboring masses in such a way that the residual interactions cancel out. The expressions obtained in this way are the following

M (N + 2, Z -2) -M (N, Z) +M (N, Z -1) -M (N + 1, Z -2) (1) +M (N + 1, Z) -M (N + 2, Z -1) = 0, M (N + 2, Z) -M (N, Z -2) +M (N + 1, Z -2) -M (N + 2, Z -1) (2) +M (N, Z -1) -M (N + 1, Z) = 0,
where M (N, Z) is the nuclear mass with N neutrons and Z protons. Two restrictions must be taken into account : none of the nuclei in the equations should have N = Z odd and, for relation 1 to be satisfied is necessary that N > Z, to secure a proper cancelation of the isospin dependence of the residual interactions [9]. Using either of these relationships it is possible to predict the value of a given nuclear mass if the other five are known. Varying the position of the mass to be predicted and using both GK relations, it is possible to obtain several predictions for the same mass, as long as the neighboring nuclear masses are known. This technique has been used previously in an iterative fashion to predict whole regions in the N,Z plane [10], using the predicted nuclear masses as known values for the next iterations. The main drawback with this procedure is that the GK relations are not satisfied exactly by the nuclear masses. The average error in the GK relations for the experimentally known nuclear masses is of about 100 keV. In fact, we have found no evidence of the error growing when we go further away from the region of stability [11]. The problem arises when the relations are used to predict in an iterative fashion since, as the number of iterations increases, the intrinsic error grows rapidly. After a few iterations the deviation on the predictions is larger than the desired precision. An extensive analysis of how the error on the prediction grows as a function of the iterations has been done in [10]. In this paper we propose a way to improve the predictions of the Garvey-Kelson relations, something we achieve by first modifying the iterative process to predict more nuclear masses per iteration, as we explain below.

II. THE GARVEY-KELSON RELATIONS AND THE SEPARATION ENERGIES

The Garvey-Kelson relations involve five degrees of freedom, which are not independent of each other. Through the iterative process each of these degrees of freedom accumulates error, leading to a growing innacuracy. It is possible, however, to reduce the error in the process by modifying the way in which the different degrees of freedom are related to each other. The first Garvey-Kelson relation, eq.1 can be expressed in two ways, in terms of the separation energies of neutrons and protons

S n (N + 2, Z -2) + S n (N + 1, Z) +M (N, Z -1) -M (N + 2, Z -1) = 0, (3)
S p (N + 2, Z -1) + S p (N, Z) -M (N + 1, Z) + M (N + 1, Z -2) = 0, (4)
where

S n (N, Z) = M (N, Z) -M (N -1, Z), (5)
S p (N, Z) = M (N, Z) -M (N, Z -1). (6)
In the same fashion, it is possible to express the second Garvey-Kelson relation as

S n (N + 2, Z) + S n (N + 1, Z -2) -M (N + 2, Z -1) + M (N, Z -1) = 0, (7)
S p (N + 2, Z) + S p (N, Z -1) +M (N + 1, Z -2) -M (N + 1, Z) = 0. ( 8
)
The former expressions relate one-nucleon separation energies with mass values. If we can obtain the values of the separation energies, then it is possible to reduce the number of degrees of freedom at play. Assuming that we know the value of the separation energies and using equations 3, 4, 7 and 8, there are 8 possible ways of estimating the value of a given nuclear mass (predicted mass) in terms only of the value of another mass (generator mass). The separation energies function as a backbone for the prediction, so we require these energies to use this technique. One may think that determining the separation energies is equivalent to knowing the nuclear masses, but this is far from being correct. The separation energies can be considered as the discrete derivatives of the mass surface, so in order to obtain the value of the actual surface it is necessary to integrate them and the corresponding integration constants are needed. Although it is possible to use the known experimental masses as integration constants, the error due to this process grows very rapidly.

Once the separation energies have been determined, the iteration procedure works in the same way as in the one described in [10]. In each iteration there are nuclei that meet 2 conditions, 1) the value of the nuclear mass to be determined is unknown and 2) we do know the value of a neighboring nuclear mass, involved in at least one of the relations expressed on eqs. 3, 4, 7 and 8. On every nucleus that meets these conditions, we calculate as many estimates as possible, averaging these values in order to obtain a prediction for the unknown mass of the nucleus. The mass values predicted on previous iterations are used as known values for the next iteration. It is important to note that the values used for the separation energies are defined from the beginning and do not change throughout the procedure, so the number of degrees of freedom which can accumulate error is reduced to one per relation. Having reliable predictions for the separation energies is thus crucial.

The new iterative procedure (IGK2), which uses equations 3, 4, 7 and 8, has several advantages over the old iterative process (IGK1), defined in terms of the original Garvey-Kelson relations. The main advantage is the smaller number of iterations necessary to reach a given nucleus. While IGK1 requires previous knowledge of at least 5 neighboring masses in order to predict a new one, we only need one mass in IGK2 (plus the information provided by the one-nucleon separation energies, of course). Since the condition of knowing one mass is satisfied more often than knowing 5 masses, the number of nuclei predicted is greater in the new process. It has been shown previously that the error on the predictions grows systematically on each iteration [10], which implies that if a given nucleus is reached in less iterations, the mass prediction of that nucleus will have a smaller accumulated error. This effect can be clearly seen in table I ,which compares the number of nuclei for which the mass is predicted per iteration for both processes. In both cases the iterative process starts from the masses reported on the AME03 compilation [8]. A graphical comparison of the two processes and the regions covered after the first 10 iterations is shown in figure 1. It is clear from the table and the figure that the IGK2 iterative process is significantly more effective. TABLE I: Number of nuclei predicted per iteration on the first 10 iterations of the process using the original Garvey-Kelson relations (IGK1) and the iterative process using the modified Garvey-Kelson relations of equations 3, 4, 7 and 8 (IGK2). An additional advantage provided by this technique arises from the relative position of the predicted and the generator mass. These two masses always belong to nuclei in the same set of isotopes or isotones, i.e., each of the equations relate two masses with the same number of neutrons or protons. This allows the predictions to advance directly towards the proton-or neutron-rich regions. The IGK1 procedure, in contrast, moves forward on the A =N+Z or T = (N-Z)/2 directions, making it less efficient. Finally, the nature of the error implies an additional advantage for IGK2, since part of the deviation comes form the separation energy estimation, which can be improved systematically with better analyses of these energies, using both global and local approaches.

III. PREDICTION TEST

The IGK1 iterative procedure has been tested in [10]. In order to quantitatively compare it to the new iterative process, we use the same set of data. The test consists in dividing the experimentally known masses into two subsets, one used to fit the model and the other to be compared to the predictions. Specifically, the reported masses on the AME03 compilation [8] have been divided into a subset conformed by the masses measured up to the year 1995, i.e. reported in the AME95 compilation [12], and a prediction subset comprised by those masses present on the AME03 compilation but absent on AME95. For this test we only consider nuclei with N ≥ 28 and Z ≥ 28.

The first step to apply the IGK2 process is to estimate the experimental separation energies. As should be evident, the quality of the mass predictions depends on an accurate description of them. We stress that the main reason for which a better accuracy is possible to achieve with this procedure, is that the (N,Z) surface produced by the separation energies has simpler systematics than the mass surface. In addition, the one-nucleon separation energies can be directly measured experimentally and do not require, as the masses, a subsequent analysis which may introduce further systematic errors. As a first approximation we take the simple parametrization described in [13] which assumes that the one-neutron separation energies can be parametrized as

S n (N, Z) = (an 1 + an 2 A 1/3 ) N Z -1 -an 3 + an p δ(N ) A 1/2 -an s µ(N ), (9)
where A = N + Z, the an p term models the effects of pairing, δ(N ) = 1 if the neutron number is even or δ(N ) = -1 if it is odd. The an s term is a correction for the nuclear shell effects which subtracts a constant energy on every shell, the value of µ depends on the shell, µ(N

) = 0 if 1 ≤ N ≤ 28, µ(N ) = 1 if 29 ≤ N ≤ 50, µ(N ) = 2 if 51 ≤ N ≤ 82, µ(N ) = 3 if 83 ≤ N ≤ 126 and µ(N ) = 4 if 127 ≤ N .
The one-proton separation energy parametrization is the same, but changing N by Z and adding an expression to model the energy due to the Coulomb interaction. Including this term the parametrization can be expressed as

S p (N, Z) = (ap 1 + ap 2 A 1/3 ) Z N -1 -ap 3 + ap p δ(Z) A 1/2 -ap s µ(Z) -ap c Z A 1/3 , (10)
where δ(Z) and µ(Z) has the same behavior as δ(N ) and µ(N ).

The validity of these equations to model separation energies has been quantified in [13]. In order to use these estimations on the iterative process the values of the parameters involved on equations 9 and 10 have been obtained fitting the models to the known separation energies, obtained through the experimentally known nuclear masses on the fitting subset. The values obtained for the parameters are shown on table II. Once the separation energies have been determined, the iterative procedure can be applied. TABLE II: Values of the parameters obtained for the one-neutron separation energies, eq. 9 and for the one-proton separation models, eq. 10.

an1 an2 an3 anp ans 5.178 3.549 5.354 11.169 1.566 ap1 ap2 ap3 app aps apc 11.120 1.054 1.043 12.473 1.474 1.111 On each iteration two quantities are calculated, the root mean square deviation for the nuclei predicted on that iteration σ and the root mean square deviation for the nuclei predicted up to that iteration Aσ. These quantities, for the i ′ th iteration, are defined as follows

σ i =   1 N i Ni j=1 (M GK i,j -M exp i,j ) 2   1/2 (11)
Aσ i =   1 N i ni i=1 Ni j=1 (M GK i,j -M exp i,j ) 2   1/2 (12)
where the index j goes over the N i nuclei predicted on iteration i. The nuclear mass value obtained by this method is M GK , while the experimental value for that nuclear mass is M exp .

Table III shows the performance of IGK2 for the nuclei predicted at each iteration in the prediction subset. The same table compares this prediction with the one obtained with IGK1 on the same nuclei. The number of nuclei predicted on the prediction subset is shown for each iteration as well as the number of nuclei predicted up to that iteration, shown in parenthesis. It is important to notice that after only two iterations the IGK2 process has covered almost all nuclei on the prediction subset, 319 of 371 nuclei. On each iteration both the root mean square deviation per iteration σ and the accumulated deviation Aσ are lower in IGK2. In the case of the first iteration which includes 227 nuclei, the root mean square deviation is lower by 50%. These results show that through the use of separation energy information in the Garvey-Kelson relations, it is possible to significantly improve the predictive power of the iterative process, thus improving the range for which the predictions are accurate. However, the model we have considered for the separation energies, eq. 9 and 10, is very simple. We now show that the iterative process can be further improved by introducing better estimations of the separation energies. As an example, we will use the separation energies obtained in terms of the predictions of the Duflo-Zuker mass model. We call this iterative process IGKDZ. In order to test the accuracy of this procedure we have again subjected it to the same test as before. Comparing the results on table III and table IV we find a very significant improvement in accuracy. Furthermore, the predictions obtained with the IGK2 methodology are much better than the ones using the original Garvey-Kelson relations. On the first four iterations we observe that the results are of high quality. According to table I these four iterations cover about 2000 nuclei where the predictions are improved in comparison with the predictions of the model used to generate the separation energies. In this fashion, the new iterative process can be thought as a procedure to improve the predictive power of existing nuclear mass models.

IV. CONCLUSION

A new way to use the Garvey-Kelson relations has been presented. This approach involves expressing the original Garvey-Kelson relations in terms of the one-neutron and one-proton separation energies. The modified GK relations are then used on an iterative fashion, helped by (either local or global) estimations for the separation energies, which are used as a backbone constraining the process. The inclusion of separation energies reduce the number of degrees of freedom involved on the iterative process and allows for a more efficient and accurate predictions. Comparisons of the original iterative process with the one proposed in this work are presented, using different estimations for the separation energies. It has been shown that the predictions using the new approach (IGK2) are more accurate and are competitive with the best predictions available [1], at least for the 2000 nuclei closest to the experimentally known region. Finally, this approach can be combined with existing nuclear mass models to significantly improve their predictive power, as demonstrated for the Duflo-Zuker model.

FIG. 1 :

 1 FIG. 1: (Color online) Nuclei predicted per iteration in the first 10 iterations of the process using (a) the original Garvey-Kelson relations (IGK1) and (b) the modified Garvey-Kelson relations defined on equations 3, 4, 7 and 8 (IGK2).

TABLE III :

 III Root mean square deviations σ i and Aσ i in MeVs for the nuclei predicted on each iteration of IGK2 and the predictions obtained with the IGK1 process.The second row shows the number of nuclei predicted on iteration i and in parenthesis the number of nuclei predicted up to iteration i. Both numbers are constrained to belong to the prediction subset. Aσ i σ i Aσ i σ i Aσ i σ i Aσ i

	Iteration 1 Iteration 2 Iteration 3 Iteration 4
	227 (227) 92 (319) 34 (353) 10 (363)
	σ i IGK2 0.445 0.445 0.824 0.580 1.129 0.653 1.300 0.679
	IGK1 1.002 1.002 0.925 0.981 1.536 1.047 2.517 1.108

  Table IV displays the results. The table shows the deviation σ per iteration as well as the accumulated deviation Aσ for the IGKDZ process. The same quantities have been computed for the predictions obtained directly with the Duflo-Zuker mass model on those nuclei, in order to compare both methodologies.

TABLE IV :

 IV Root mean square deviations σ i and Aσ i in MeVs for the nuclei predicted on each iteration of the iterative process IGKDZ and the predictions obtained directly with the Duflo-Zuker model. The second row shows the number of nuclei predicted on iteration i and in parenthesis the number of nuclei predicted up to iteration i. Both numbers are constrained to nuclei within the subset of prediction masses. Aσ i σ i Aσ i σ i Aσ i σ i Aσ i

	Iteration 1 Iteration 2 Iteration 3 Iteration 4
	227 (227) 92 (319) 34 (353) 10 (363)
	σ i IGKDZ 0.227 0.227 0.300 0.250 0.286 0.254 0.370 0.258
	DZ 0.326 0.326 0.335 0.329 0.319 0.328 0.396 0.330