Macro-microscopic mass formulae and nuclear mass predictions

Nuclear Physics A 847 (2010) 24–41 www.elsevier.com/locate/nuclphysa

Macro-microscopic mass formulae and nuclear mass predictions G. Royer ∗ , M. Guilbaud, A. Onillon Laboratoire Subatech, UMR: IN2P3/CNRS-Université-Ecole des Mines, 4 rue A. Kastler, 44307 Nantes Cedex 03, France Received 26 April 2010; received in revised form 28 June 2010; accepted 30 June 2010 Available online 15 July 2010

Abstract Different mass formulae derived from the liquid drop model and the pairing and shell energies of the Thomas–Fermi model have been studied and compared. They include or not the diffuseness correction to the Coulomb energy, the charge exchange correction term, the curvature energy, different forms of the Wigner term and powers of the relative neutron excess I = (N − Z)/A. Their coefficients have been determined by a least square fitting procedure to 2027 experimental atomic masses (G. Audi et al. (2003) [1]). The Coulomb diffuseness correction Z 2 /A term or the charge exchange correction Z 4/3 /A1/3 term plays the main role to improve the accuracy of the mass formula. The Wigner term and the curvature energy can also be used separately but their coefficients are very unstable. The different fits lead to a surface energy coefficient of around 17–18 MeV. A large equivalent rms radius (r0 = 1.22–1.24 fm) or a shorter central radius may be used. An rms deviation of 0.54 MeV can be reached between the experimental and theoretical masses. The remaining differences come probably mainly from the determination of the shell and pairing energies. Mass predictions of selected expressions have been compared to 161 new experimental masses and the correct agreement allows to provide extrapolations to masses of 656 selected exotic nuclei. © 2010 Elsevier B.V. All rights reserved. Keywords: Mass formula; Liquid drop model; Wigner term; Curvature energy

* Corresponding author.

E-mail address: [email protected] (G. Royer). 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.06.014

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1. Introduction Predictions of masses of unknown nuclei far away from the β-stability valley may be tentatively performed from mass formulae working well for the known isotopes. Semi-macroscopic liquid drop models including the pairing effects have been firstly developed to reproduce the experimental nuclear masses [2,3]. Later on, these macroscopic–microscopic approaches have been developed, mainly the finite-range liquid drop model and the finite-range droplet model [4], to describe the masses and the fission, fusion and alpha [5] processes in taking into account the shell effects, the proximity energy and the nuclear deformations. Nuclear masses have also been reproduced accurately within the statistical Thomas–Fermi model with a Seyler–Blanchard effective interaction [6,7] and microscopic Hartree–Fock self-consistent theories using meanfields and Skyrme or Gogny forces and pairing correlations [8–12] as well as relativistic mean field theories [13] or Garvey–Kelson relations [14]. All these recent macroscopic–microscopic approaches aiming at reproducing the nuclear binding energy and then the nuclear mass contain the usual volume, surface, Coulomb energy terms and the shell and pairing energies and include or not the diffuseness correction to the Coulomb energy, the charge exchange correction term, the curvature energy, a constant term, different forms of the Wigner term and different powers of the relative neutron excess I = (N − Z)/A. The purpose of the present work is, firstly, to determine the most efficient mass formulae to reproduce the most precisely known masses given in the 2003 atomic mass evaluation [1]. Four possible radii are considered to calculate the Coulomb energy: the equivalent rms radius deduced from the experimental charge radius [15], a central radius used to determine the proximity energy [16], a radius free to evolve to minimise the difference between the theoretical and experimental masses and a fixed r0 value. Secondly, the mass predictions of the selected formulae are compared with 161 new known experimental masses. The agreement allows finally to provide extrapolated masses of 656 unknown realistic exotic nuclei and to compare with the data given in the 2003 atomic mass evaluation. 2. Macro-microscopic liquid drop model binding energy Different subsets of the following expansion of the nuclear binding energy in powers of A−1/3 and |I | have been studied:     2   1 3 e2 Z 2 B = a v 1 − kv I 2 A − a s 1 − k s I 2 A 3 − a k 1 − k k I 2 A 3 − 5 R0 4

Z2 Z3 + ac,exc 1 − Epair − Eshell − EWigner . + fp A A3

(1)

The volume energy corresponding to the saturated exchange force and infinite nuclear matter is given by the first term. I 2 A is the asymmetry energy of the Bethe–Weizsäcker mass formula. The second term is the surface energy. Its origin is the deficit of binding energy of the nucleons at the nuclear surface and corresponds to semi-infinite nuclear matter. The following term is the curvature energy. It results from non-uniform properties which correct the surface energy and depends on the mean local curvature. This term appears in the TF model [7] but vanishes in the FRLDM [4]. The decrease of binding energy due to the Coulomb repulsion is given by the fourth term. Different charge radii will be assumed. The Z 2 /A term is the diffuseness correction to the sharp radius Coulomb energy (called also the proton form-factor correction in [4]). The

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Fig. 1. Shell energies of the Thomas–Fermi model [7].

Fig. 2. Pairing energies of the Thomas–Fermi model [7].

Z 4/3 /A1/3 term is the charge exchange correction term. The pairing and shell energies of the recent Thomas–Fermi model [6,7] have been selected. Their dependence on the proximity of the proton and neutron magic numbers and on the parity is displayed in Figs. 1 and 2. The shell effects add 12.84 MeV to the binding energy of 208 Pb, for example. The Wigner energy appears in the counting of identical pairs in a nucleus and depends on I . Its effect is to decrease the binding energy when N = Z. Four versions of the Wigner term have been 2 considered here, namely: the original linear version W1 = kw1 |I |, W2 = kw2 |N − Z| × e−(A/50) , 2 W3 = kw3 |N − Z| × e−A/35 and W4 = kw4 e−80I . W2 and W4 have been proposed in [17] but with different coefficients. To obtain the coefficients of the selected expressions by a least square fitting procedure, the masses of the 2027 nuclei verifying the two conditions: N and Z higher than 7 and the one standard deviation uncertainty on the mass lower than or equal to 150 keV [1] have been used.

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Table 1 Coefficient values and root mean square deviation (in MeV) between the theoretical and experimental binding energies. The theoretical shell and pairing energies are taken into account. The Coulomb energy is determined by 0.6e2 Z 2 /(1.28A1/3 − 0.76 + 0.8A−1/3 ). av

kv

as

ks

fp

ac,exc

W1

W2

W3

W4

σ

15.8548 15.8427 15.8276 15.8328 16.038 15.5172 15.2508 15.6233 15.3989 15.5002 15.2312 15.5003 15.5389 15.2986 15.5899 15.3684 15.5868 15.3587 15.6096 15.3841

1.7281 1.7368 1.7681 1.8931 1.9801 1.7753 1.7840 1.8412 1.8492 1.7860 1.7949 1.8088 1.8585 1.8690 1.8840 1.8924 1.8602 1.8687 1.8543 1.8625

17.3228 17.2607 17.176 17.1077 18.4563 17.9474 17.9475 18.1709 18.1703 17.8829 17.8831 17.8136 17.7736 17.7739 17.9004 17.9003 18.0012 18.0007 18.1132 18.1127

0.8179 0.8727 1.0540 1.9361 2.2201 1.6575 1.6577 1.92097 2.0320 1.7290 1.7291 1.8401 2.1451 2.1444 2.2326 2.2316 2.0434 2.0427 2.0021 2.0014

– – – – – 2.1401 – 1.7987 – 2.1612 – 2.1032 1.9299 – 1.7779 – 1.8294 – 1.80856 –

– – – – – – 2.0195 – 1.6983 – 2.0393 – – 1.8212 – 1.6783 – 1.7272 – 1.7073

– – – 41.003 – – – – – – – – 21.437 21.401 19.554 19.528 – – – –

– 0.4083 – – – – – – – 0.4645 0.4641 – – – – – – – 0.4700 0.4696

– – 1.1872 – – – – – – – – 0.9951 – – – – 0.9428 0.9424 – –

– – – – −8.4670 – – −2.4136 −2.4059 – – – – – −1.2050 −1.2004 −1.9495 −1.9425 −2.4953 −2.4886

1.402 1.368 1.334 1.199 0.994 0.692 0.691 0.661 0.661 0.597 0.596 0.591 0.591 0.590 0.583 0.582 0.567 0.567 0.558 0.558

3. Dependence on the Coulomb radius In a previous study [18] it has been shown that, when the Coulomb reduced radius r0 is a priori free and adjusted on the experimental masses as the other coefficients of the mass formulae, the adjustment leads always to a large value of around 1.21–1.255 fm. To complete this study, the coefficients of different mass formulae have been adjusted, assuming a priori a specific expression for the Coulomb energy. Firstly, the Coulomb radius has been calculated using the expression R0 = 1.28A1/3 − 0.76 + 0.8A−1/3 (see Table 1). This formula proposed in Ref. [16] is derived from the droplet model and the proximity energy and simulates rather a central radius for which R0 /A1/3 increases slightly with the mass. This radius is much smaller than the equivalent rms radius for which the experimental value of R0 /A1/3 decreases slightly with the mass [15,19] and has an isospin dependence and a mean value of 1.2257 fm. It has been shown [5,20] that this selected more elaborated expression can also be used to reproduce accurately the fusion, fission and cluster and alpha decay data. The main result is that an rms deviation of only 0.56 MeV can be reached within seven adjustable parameters. In Table 2 the assumed charge radius is simply R0 = 1.16A1/3 fm. This often used mean value does not allow to reach an accuracy better than 0.72 MeV. The surface coefficient is generally higher. So it seems that a large equivalent rms radius or a very low central Coulomb radius may be used in the mass formulae but that an intermediate radius is less efficient. The values of the parameters and their correlations are discussed in the next section. 4. Correlations between the LDM parameters The correlations between the different terms of the semi-empirical LDM mass formulae have been deeply investigated by M.W. Kirson [21] using correlation and error matrix starting directly

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Table 2 Same as Table 1 but the Coulomb energy is determined by 0.6e2 Z 2 /(1.16A1/3 ). av

kv

as

ks

fp

ac,exc

W1

W2

W3

W4

σ

16.0975 16.1273 16.1388 16.1017 16.3374 15.7511 15.4781 16.0616 15.9131 15.7862 15.5185 15.7855 15.7174 15.4034 16.1454 15.9900 16.1478 16.0082 16.0889 15.9426

1.6145 1.5934 1.5552 1.5836 1.9404 1.6597 1.6665 1.8498 1.8552 1.6384 1.6457 1.5936 1.5313 1.5363 1.7460 1.7505 1.8064 1.8058 1.8245 1.8295

18.4883 18.642 18.711 18.5293 19.9732 19.1291 19.129 19.7833 19.7832 19.2616 19.2616 19.3991 19.3998 19.4002 20.4634 20.4634 20.1843 20.1842 19.8986 19.8984

0.3684 0.2466 0.0449 0.1728 2.0989 1.1916 1.1914 1.9150 1.9149 1.0584 1.0583 0.8600 0.5023 0.5014 1.2294 1.2289 1.6570 1.6568 1.7673 1.7670

– – – – – 2.1955 – 1.1963 – 2.1520 – 2.2700 2.5229 – 1.2486 – 1.1238 – 1.1767 –

– – – – – – 2.0710 – 1.1278 – 2.0302 – – 2.3807 – 1.1782 – 1.0597 – 1.1098

– – – −7.8076 – – – – – – – – −33.3855 −33.4319 −49.1756 −49.1935 – – – –

– −1.0105 – – – – – – – −0.9546 −0.9550 – – – – – – – −0.9394 −0.9397

– – −1.8053 – – – – – – – – −2.0089 – – – – −2.2277 −2.2280 – –

– – – – −11.0907 – – −7.0645 −7.0659 – – – – – −10.1039 −10.1026 −8.161 −8.1614 −6.9010 −6.9003

1.757 1.583 1.631 1.751 1.186 1.233 1.233 1.079 1.079 1.001 1.001 0.996 1.098 1.098 0.739 0.739 0.722 0.722 0.814 0.814

from the four basic terms of the Bethe–Weizsäcker mass formula (see also [22]). Here I 2 A is called the symmetry term and I 2 A2/3 the surface symmetry term. Let us first recall the main conclusions of this study [21]. There is not much correlation of the volume, Coulomb, pairing and shell correction terms with other terms. The shell correction term is the most important single term to be added to the initial Bethe–Weizsäcker mass formula followed by the surface symmetry contribution. The pairing term is remarkably stable and there is essentially no correlation between the pairing term and any other term. The surface symmetry term is always important but less so when a Wigner term is present. The inclusion of the curvature term may seem questionable. Significant correlation exists between symmetry and exchange Coulomb term (Z 4/3 /A1/3 term), between surface and curvature terms and between Wigner and surface symmetry terms. The drastically variable terms are the Wigner and curvature terms for which even their sign is not definite. It is tempting to rule out any mass formula which does not contain a surface symmetry term. The direct and exchange Coulomb terms are anti-correlated. There is no compelling reason to introduce a congruence term to forego the linear form of the Wigner term as there is no compelling reason to prefer a non-linear form of the surface symmetry term to the linear form. As recommended in this study, all the recent versions of the liquid drop model [4,20,23] contain a symmetry term, a surface symmetry term and shell and pairing corrections. The fact that the shell and pairing energies are uncorrelated with the other macroscopic terms justifies the possibility to select specific shell energies and formulae for the pairing corrections. The analyzes of the tables in Ref. [18] and of Tables 1 and 2 of the present work are in agreement with all the above mentioned conclusions. The shell corrections of the Thomas–Fermi

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model selected in our approach go beyond simple coarse-grained global analytical shell corrections. This leads to a large improved accuracy and allows to extract the following supplementary information from Tables 1 and 2 and from [18]. The correlation between the Coulomb diffuseness correction term and the surface energy term is the same as the one between the charge exchange correction term and the surface energy term. The diffuseness correction term (in Z 2 /A) has the advantage to be continuous during the transition from one-body to two-body shapes, at least in symmetric exit or entrance channels. The diffuseness correction term or the charge exchange correction term plays the main role to improve the accuracy of the mass formulae. The W2 and W3 terms are as efficient as the usual W1 term when the diffuseness correction term is added. The convergency of the coefficients of the W2 and W3 terms is better than the one of the W1 term. The introduction of a Coulomb diffuseness correction term divides W1 by two and vanishes the efficiency of the curvature correction term. The parameters of the LDM mass formula probably lie in the following range of values: av = 15.3–15.9 MeV, as = 16.5–18.5 MeV, kv = 1.7–1.9, ks = 1.4–2.8, r0 = 1.20–1.25 fm, fp = 0.9 − 2.2, W1 = 16–42 MeV, W2 = 0.45–0.5 MeV, W3 = 0.9–1.3 MeV and W4 = 1.2–2.5 MeV. Let us recall that it has been previously shown that an |I | or I 4 dependence in the surface and volume energies improves only slightly the efficiency of the expansion and that the introduction of a constant term is difficult since its value is highly fluctuating while a congruence term leads to high discontinuities at the scission point of fission or fusion barriers. The surface coefficient and the relative radius r0 are correlated since r0 diminishes when the surface coefficient increases. 5. Different possible mass formulae The following formulae (2)–(6) have been obtained assuming the proportionality of R0 with A1/3 but the reduced radius r0 is provided as the other coefficients by the adjustment to the experimental masses. The rms deviations are respectively: 0.633, 0.579, 0.610, 0.564 and 0.543 MeV. The Wigner terms are more efficient than the curvature term but they induce a high value of r0 . The combination of two Wigner terms allows to reach a very good accuracy. Nevertheless the formula (2) with only 6 parameters and without Wigner terms already leads to a correct precision.     2 B = 15.4122 1 − 1.7116I 2 A − 17.5371 1 − 1.4015I 2 A 3 − 0.6 Z2 − Epair − Eshell . A    2  B = 15.3529 1 − 1.8084I 2 A − 16.9834 1 − 1.9647I 2 A 3 − 0.6

e2 Z 2 1

1.2177A 3

+ 1.3736

Z2 − 21.3975|I | − Epair − Eshell . A    2  B = 15.4529 1 − 1.8722I 2 A − 18.0588 1 − 2.7716I 2 A 3 − 0.6

(2) e2 Z 2 1

1.2324A 3

+ 0.9675

+ 1.3712

  1 Z2 + 1.348 1 − 45.I 2 A 3 − Epair − Eshell . A

(3) e2 Z 2 1

1.2204A 3 (4)

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G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

.5.5 Fig. 3. Difference between the theoretical and experimental masses of the 2027 selected nuclei for the formulas (2)–(4), (6)–(8) from the top left to the bottom right as a function of the mass number.

   2  B = 15.2508 1 − 1.7489I 2 A − 16.8015 1 − 1.6077I 2 A 3 − 0.6 Z2 2 − 0.6806|N − Z| × e−(A/50) − Epair − Eshell . A    2  B = 15.4133 1 − 1.7962I 2 A − 17.3079 1 − 1.7858I 2 A 3 − 0.6

e2 Z 2 1

1.2426A 3

+ 1.0736

+ 0.8956

(5) e2 Z 2 1

1.2318A 3

Z2 2 2 − 0.4838|N − Z| × e−(A/50) + 2.2 × e−80I − Epair − Eshell . A

(6)

Experimentally for nuclei verifying N and Z higher than 7 the set of 782 ground state nuclear charge radii presented in Ref. [15] leads for the equivalent rms charge radius given by R0 = √ (5/3)r 2 1/2 to R0 = 1.22572 A1/3 fm and σ = 0.124 fm. This radius value is imposed in the formula (7). It allows also to obtain a good accuracy of 0.584 MeV.

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Fig. 4. Difference between the theoretical masses obtained with the formula (8) and the experimental masses of the 2027 selected nuclei.

   2  B = 15.3848 1 − 1.7837I 2 A − 17.1947 1 − 1.8204I 2 A 3 − 0.6

e2 Z 2 1

1.2257A 3

Z2 − 16.606|I | − Epair − Eshell . (7) A In the last formula (8) the radius is taken as the central radius previously used in Table 1 and σ = 0.558 MeV. So it is possible to obtain accurate mass formulae with a large constant reduced radius r0 or with a more sophisticated central radius corresponding to a smaller value of r0 increasing with the mass.     2 B = 15.6096 1 − 1.8543I 2 A − 18.1132 1 − 2.0021I 2 A 3 + 1.1035

− 0.6

e2 Z 2 1 3

− 13

1.28A − 0.76 + 0.8A

+ 1.8086

+ 2.4954 × e−80I − Epair − Eshell . 2

Z2 2 − 0.47|N − Z| × e−(A/50) A (8)

The Wigner energy terms are supposed to be approximately independent of the nuclear shape and they may induce discontinuities at the scission point of fission or fusion barriers. For example for 2 symmetric decay of 254 Fm, the expressions 20|I |, 0.4|N − Z| × e−(A/50) , |N − Z| × e−A/35 , 2 2e−80I and Econg = −10e−4.2|I | lead to discontinuities at the contact point of the nascent fragments of respectively 4.25, 0.03, 1.4, 0.05 and 4.1 MeV. The difference between the theoretical masses calculated with the formulae (2)–(4), (6)–(8) and the experimental masses of the 2027 selected nuclei are given from the top left to the bottom

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Fig. 5. Distribution of the 2027 nuclei in each error range.

right in Fig. 3 as a function of the mass number. The structure observed in the whole data are about the same for the different formulae. They come probably mainly from the assumed shell and pairing energies. The errors are more important for the lighter nuclei. The mass of the heaviest nuclei is better reproduced by the formulae (2), (4) and (7). The formula (6) is the best one for the light nuclei while the formula (8) is a good compromise. As an illustration, the difference between the theoretical masses obtained with the formula (8) and the experimental masses of the 2027 nuclei used for the adjustment of the coefficients is indicated in Fig. 4. The darker the colour, the higher the accuracy. The other formulae lead to similar pictures. The distribution of the nuclei in each error range is given explicitly in Fig. 5. The errors are slightly larger for the light nuclei. The same behaviour is encountered by all the mass models. Nevertheless the error is very rarely higher than 2 MeV. 6. Test of the predictability on 161 new experimental masses Since the last mass evaluation [1] other masses have been newly or more precisely obtained. The predictions given by the formulae (6)–(8) (not readjusted) are compared in Fig. 6 with 161 new experimental masses for which the one standard deviation uncertainty is lower than or equal to 800 keV [1,24]. The accuracy is correct in the whole mass range showing the predictability of such formulae. The rms deviations are respectively 0.937, 0.985 and 0.976 MeV.

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Fig. 6. Difference between the theoretical masses given by the formulas (6)–(8) and the experimental masses of 161 newly known nuclei.

Fig. 7. Difference between the theoretical masses obtained with the formula (8) and the 161 new experimental masses.

Nucleus 24 O 31 Na 37 Si 40 Si 44 S 52 Ca 59 V 62 Mn 64 Fe 66 As 67 Se 71 Ni 79 Y 81 Ga 83 Nb 85 As 88 Tc 91 Tc 93 Pd 95 Kr 100 Sn 112 I 128 Cd 141 I 146 Ho 152 Yb 160 W 171 Ho 194 At 225 Rn 230 Ac 242 Np

δE

19.07 12.65 −6.58 5.47 −9.12 −32.51 −44.19 −37.07 −48.04 −54.97 −51.5 −46.55 −55.2 −58.36 −57.98 −58.96 −63.24 −61.68 −75.98 −59.44 −56.16 −56.78 −67.06 −67.29 −60.3 −51.24 −46.31 −29.36 −54.52 −1.19 26.56 33.81 57.42

3.74 −0.88 −1.51 −2.2 −2.07 2.21 0.37 −0.94 0.02 −0.69 1.16 0.69 −1.07 0.07 −0.07 2.29 −0.02 1.07 −0.52 1.09 −0.71 −0.37 0.24 −0.65 1.15 −0.19 0.00 0.7 0.27 0.13 0.72 0.53 0.3

Nucleus 26 F 33 Na 37 Al 41 P 45 Cr 54 Sc 57 Ti 60 Cr 62 Cr 65 Mn 66 Co 68 Fe 71 Br 79 Zn 81 Zr 83 Ge 86 Mo 89 Ru 91 Ru 94 Kr 95 Pd 101 Rb 118 Pd 130 Cd 143 Gd 147 Tm 153 Lu 161 Re 172 Hg 195 Os 226 Rn 230 Fr

Eexp

δE

18.27 24.89 9.95 −5.28 −18.97 −34.22 −33.54 −46.5 −40.41 −40.67 −56.41 −43.13 −57.06 −53.44 −58.49 −60.8 −64.56 −58.99 −68.24 −61.35 −69.96 −43.6 −75.47 −61.57 −68.23 −35.97 −38.41 −20.88 −1.09 −29.69 28.74 39.5

2.32 −0.24 −1.34 −1.29 −2.39 2.88 −0.17 −0.94 −0.42 −0.05 −0.25 −2.1 0.49 −0.96 0.7 −0.15 1.86 0.65 0.13 −0.17 −0.07 −0.5 0.26 −1.03 0.62 0.29 −0.13 0.31 0.55 0.46 0.95 0.83

Nucleus 28 S 34 Mg 38 Al 42 P 46 Cl 55 Sc 58 V 60 V 63 Mn 65 Fe 66 Se 68 Co 71 Kr 80 Zn 81 Zn 83 As 86 As 89 Tc 92 Ru 94 Rh 97 Pd 107 Sb 119 Ba 138 Te 143 Xe 148 Cs 156 Nd 164 Os 186 Tl 202 Au 227 Rn 231 Ra

Eexp

δE

4.07 8.81 16.05 0.94 −14.71 −29.58 −40.21 −32.58 −46.35 −51.22 −41.83 −51.35 −46.92 −51.65 −46.2 −69.56 −58.96 −67.39 −74.3 −72.91 −77.8 −70.66 −64.59 −65.76 −60.25 −47.3 −60.53 −20.46 −20.19 −24.4 32.88 38.23

−0.25 −0.09 −1.11 −2.13 0.86 1.03 0.45 −0.56 −0.5 −0.91 0.6 −1.08 0.7 −1.09 −0.6 −0.2 0.57 0.26 −0.39 −0.15 −0.29 0.22 0.14 −0.09 0.58 0.23 0.18 0.68 0.13 0.56 0.79 0.75

Nucleus 29 Ne 35 Al 39 Si 43 S 50 K 56 Ti 58 Cr 61 Cr 63 Fe 65 As 67 Fe 69 Co 72 Ni 80 As 82 As 84 Zr 87 Mo 90 Tc 92 Rh 94 Pd 97 Ag 111 I 120 Ba 140 Tb 144 Ho 148 Tm 156 Hf 168 Pt 189 W 223 Rn 228 Rn 232 Ra

Eexp

δE

18.06 −0.13 1.93 −11.97 −25.35 −38.94 −51.83 −42.18 −55.63 −46.78 −45.69 −50.00 −53.94 −72.21 −70.32 −71.42 −67.69 −70.72 −63.00 −66.1 −70.82 −64.97 −68.89 −50.48 −44.61 −38.77 −37.85 −11.04 −35.48 20.4 35.25 40.5

0.32 −0.43 −1.87 −1.31 2.67 −0.02 −0.04 −0.12 0.00 1.11 −0.67 −1.33 −1.23 −0.53 −0.36 0.65 1.53 −0.09 0.5 −0.06 −0.35 0.6 0.09 −0.54 0.55 −0.19 0.17 0.95 −0.81 0.69 1.01 0.77

Nucleus 30 Ne 36 Al 40 Ti 43 Cl 52 Sc 56 V 59 Cr 61 Mn 64 Mn 66 Fe 67 Co 70 Ni 75 Sr 80 Y 82 Ge 85 Nb 87 As 90 Ru 93 Rh 94 Ag 100 In 112 Te 123 Ag 140 I 145 Ho 151 Yb 157 Ta 169 Dy 190 W 224 Rn 229 Rn 236 Pa

Eexp

δE

23.1 5.78 −8.85 −24.17 −40.36 −46.8 −47.89 −51.56 −42.62 −49.57 −55.06 −59.15 −46.62 −61.14 −65.58 −67.15 −55.62 −64.88 −69.01 −53.33 −64.17 −77.57 −69.38 −63.6 −49.12 −41.54 −29.63 −55.6 −34.3 22.44 39.36 45.35

−0.39 −0.88 −1.72 −0.76 1.87 1.74 −0.25 −0.13 −0.08 −1.73 −0.97 −1.21 −0.25 −0.3 −0.09 1.66 0.31 0.66 −0.26 1.3 −0.59 0.53 −0.12 0.9 0.28 0.53 −0.2 0.26 −0.39 0.97 1.08 0.43

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

57 V

Eexp

34

Table 3 Experimental mass excess (in MeV) and difference δE between the values predicted with the formula (8) and these new 161 experimental data.

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35

Fig. 8. Difference between the theoretical and AME extrapolated masses for the 656 nuclei and the formulas (6)–(8) from the top left to the bottom.

The location of these 161 nuclei as a function of their neutron and proton numbers is displayed in Fig. 7 as the difference between the theoretical masses calculated from the formula (8) and the experimental ones. These new data are well distributed around the better known masses. The experimental mass excess for these nuclei is given in Table 3 as well as the difference between the mass excess derived from the formula (8) and the experimental ones for these 161 new nuclei. 7. Mass predictions for 656 exotic nuclei Finally, the predictions for 656 other more exotic nuclei for which the mass is still unknown are compared to the extrapolations given in Ref. [1] with an assumed uncertainty often higher than 500 keV. Firstly, the predictions given by the formulae (6)–(8) (not readjusted) are compared in Fig. 8 with these new 656 extrapolated masses. The rms deviations are respectively 1.011, 1.067 and 1.065 MeV. The global behaviour is similar. For about 50% of nuclei the difference is lower than 0.5 MeV and for 80% of nuclei the difference is lower than 1 MeV. The formula (7) is slightly closer to the 2003 AME predictions for the heaviest nuclei but for which the uncertainties are larger. These comparisons seem to confirm that this is the microscopic part of the mass formulae which induces these structures in E. The location of these 656 nuclei around the known valley of isotopes and the difference between the theoretical masses calculated from the formula (8) and the 2003 AME ones is displayed in Fig. 9. Similar pictures are obtained using the formulae (6)–(7). The theoretical mass excesses predicted with the formula (8) and 2003 AME values are given and compared in Table 4. Without readjustment the formulae (6)–(8) lead

36

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

Fig. 9. Difference between the theoretical masses obtained with the formula (8) and 656 2003 AME extrapolated masses.

to σ = 0.704, 0.748 and 0.733 MeV for the 2844 nuclei. When the coefficients are readjusted on these 2844 data the errors are about the same, namely σ = 0.697, 0.726 and 0.725 MeV, which justifies the adjustment on the most known 2027 nuclei. 8. Summary and conclusion The coefficients of different macro-microscopic Liquid Drop Model mass formulae including the pairing and shell energies of the Thomas–Fermi model have been determined by an adjustment to 2027 experimental atomic masses. The Coulomb diffuseness correction Z 2 /A term or the charge exchange correction Z 4/3 /A1/3 term plays the main role to improve the accuracy of the mass formula. The Wigner term and the curvature energy can also be used separately but their coefficients are very unstable. The remaining differences come probably mainly from the determination of the shell and pairing energies. An rms deviation of 0.54 MeV can be reached between the experimental and theoretical masses. A large constant coefficient r0 = 1.22–1.23 fm or a small central radius increasing with the mass can be used. The different fits lead rather to a surface energy coefficient of around 17–18 MeV. Mass predictions of selected expressions have been compared to 161 new experimental masses and the correct agreement allows to provide extrapolations to masses of 656 selected exotic nuclei. Acknowledgements We thank K. Blaum, M. Block, S. Schwarz and B. Singh for providing us different nuclear mass data.

Table 4 Theoretical mass excess (in MeV) predicted with the formula (8) and 2003 AME values for the 656 selected exotic nuclei. Nucleus 40 Al 42 Si 43 Cr 45 P 46 Fe 47 Co 49 Ni 51 K 52 Ni 53 Ni 55 Ca 56 Zn 58 Sc 59 Ga 60 As 62 Ge 63 As 65 Cr 67 Mn 69 Fe 71 Fe 73 Co 74 Ni 77 Ni 78 Zr 82 Ga 83 Mo 84 Mo 86 Ga 87 Ru 89 Ru 91 Se

26.74 13.87 −4.29 18 −0.96 9.01 16.33 7.26 −19.03 −24.49 −31.36 −15.68 −26.03 −14.12 −33.88 −6.46 −41.37 −32.89 −28.51 −34.32 −40.13 −33.01 −38.81 −49.54 −38.14 −41.87 −53.22 −44.77 −51.32 −33.4 −43.83 −56.81 −50.06

EAME 29.3 18.43 −2.13 17.9 0.76 10.7 18.4 9 −22 −22.65 −29.37 −18.12 −25.73 −15.17 −34.12 −6.4 −42.24 −33.82 −27.8 −33.4 −38.4 −31 −37.04 −48.37 −36.75 −41.7 −53.1 −47.75 −55.81 −34.35 −47.34 −59.51 −50.34

Nucleus 40 V 42 V 44 Si 45 Mn 47 Si 48 Cl 49 Cl 50 Ar 51 Co 52 Cu 53 Cu 55 Cu 56 Ga 58 Ti 59 Ge 61 V 62 As 64 V 65 Se 67 Br 69 Kr 71 Rb 73 Ni 74 Sr 77 Cu 79 Cu 82 Zr 84 Ga 85 Ga 86 Ge 88 Ge 89 Rh 91 Rh

Eth 9.51 −9.98 30 −7 10.02 −1.27 3.38 −10.73 −28.76 −3.75 −15.04 −32.79 −4.51 −31.63 −16.77 −30.81 −24.27 −16.6 −32.75 −32.68 −32.77 −32.74 −51.1 −41.69 −49.76 −43.01 −61.74 −43.66 −39.65 −49.47 −39.7 −44.7 −57.83

EAME 10.33 −8.17 32.84 −5.11 8 −4.7 0.3 −14.5 −27.27 −2.63 −13.46 −31.62 −4.74 −30.77 −17 −29.36 −24.96 −15.4 −32.92 −32.8 −32.44 −32.3 −49.86 −40.7 −48.58 −42.33 −64.19 −44.11 −40.05 −49.84 −40.14 −47.66 −59.1

Nucleus 41 Al 42 Cr 44 P 45 Fe 47 Cl 48 Ar 49 Ar 50 Co 51 Ni 53 K 54 Ca 55 Zn 57 Sc 58 Ga 60 Ti 61 Ge 63 V 64 Cr 66 Cr 68 Mn 70 Fe 72 Fe 73 Rb 75 Co 77 Y 79 Zr 82 Nb 84 Ge 85 Ge 86 As 88 As 90 As 91 Pd

Eth 31.96 4.01 9.62 11.83 −7.79 −20.35 −14.07 −19.02 −13.47 −8.89 −21.84 −14.69 −19.94 −23.97 −23.55 −33.35 −22.99 −34.7 −26.12 −29.15 −38.16 −30.47 −46.26 −31.12 −47.24 −46.99 −51.24 −58.23 −52.63 −49.18 −50.05 −40.85 −44.84

EAME 35.7 5.99 12.1 13.58 −10.52 −23.72 −18.15 −17.2 −11.44 −12 −23.89 −14.92 −20.69 −23.99 −21.65 −33.73 −20.91 −33.15 −24.8 −28.6 −35.9 −28.3 −46.05 −29.5 −46.91 −47.36 −52.97 −58.25 −53.07 −53.21 −51.29 −41.45 −47.4

Nucleus 41 Ti 43 Si 44 Cr 46 Si 47 Mn 48 Fe 49 Fe 50 Ni 52 K 53 Ca 54 Cu 56 Sc 57 Zn 58 Ge 60 Ga 61 As 63 Cr 64 As 66 Mn 68 Br 70 Br 72 Co 73 Sr 75 Ni 78 Cu 80 Cu 83 Zn 84 As 85 Mo 87 Ge 88 Ru 90 Se 92 Se

Eth −17.15 22.58 −15.48 2.55 −23.73 −19.33 −26.14 −5.1 −13.08 −24.71 −23.16 −23.29 −32.94 −8.54 −39.01 −17.5 −35.73 −38.38 −36.06 −38.49 −50.93 −41.19 −32.6 −45.16 −45.6 −36.84 −36.29 −65.86 −55.51 −43.42 −52.23 −55.62 −46.93

EAME −15.7 26.7 −13.46 0.7 −22.26 −18.16 −24.58 −3.79 −16.2 −27.9 −21.69 −25.27 −32.8 −8.37 −40 −18.05 −35.53 −39.52 36.25 −38.64 −51.43 −39.3 −31.7 −43.9 −44.75 −36.45 −36.3 −66.08 −59.1 −44.24 −55.65 −55.93 −46.65

Nucleus 41 V 43 V 44 Mn 46 Mn 47 Fe 48 Co 49 Co 51 Ar 52 Co 53 Sc 54 Zn 56 Cu 57 Ga 59 Ti 60 Ge 62 V 63 Ge 65 V 67 Cr 69 Mn 70 Kr 72 Rb 74 Co 76 Y 78 Y 82 Zn 83 Ga 84 Nb 85 Tc 87 Tc 89 As 90 Rh 92 Pd

Eth

EAME

37

−1.96 −0.21 −19.75 −18.02 4.47 6.4 −14.25 −12.37 −8.5 −6.62 0.19 1.64 −10.92 −9.58 −3.78 −7.8 −35.27 −33.92 −36.06 −37.62 −6.76 −6.57 −39.38 −38.6 −15.86 −15.9 −26.04 −25.22 −27.56 −27.77 −25.84 −24.42 −45.63 −46.91 −13.2 −11.25 −20.28 −19.05 −26.25 −25.3 −41.57 −41.3 −38.38 −38.12 −33.99 −32.25 −39.23 −38.7 −52.39 −52.53 −42.87 −42.46 −49.29 −49.39 −59.24 −61.88 −43.65 −47.67 −56.27 −59.12 −46.5 −47.14 −50.51 −53.22 −53.43 −55.5 (continued on next page)

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

48 Ni

Eth

38

Table 4 (continued) Nucleus 93 Se 95 Br 96 Ag 97 In 99 In 101 Sn 103 Sb 106 Zr 107 Nb 108 Mo 109 Mo 110 I 112 Mo 114 Mo 115 Mo 116 Ru 117 Rh 118 Ba 119 La 120 Ce 122 Rh 122 Pr 124 Pd 125 Ag 126 Pr 127 Nd 128 Sm 129 Sm 131 Cd 132 Pm 133 Eu 135 In

EAME −40.72 −43.9 −64.57 −47 −61.27 −59.56 −56.18 −38.58 −59.7 −64.92 −71.3 −67.25 −60.32 −58.83 −51.31 −46.31 −64.45 −68.95 −62.37 −54.97 −49.71 −52.9 −44.89 −58.8 −64.8 −60.26 −55.42 −39.05 −42.25 −55.27 −61.71 −47.28 −47.2

Nucleus 93 Br 95 Ag 96 Cd 98 Kr 99 Sn 102 Rb 104 Sr 105 Y 106 Nb 107 Te 108 Sb 110 Zr 111 Nb 112 Cs 114 Tc 115 Tc 116 Cs 117 Ba 118 La 119 Ce 121 Pd 122 Pd 123 Pd 124 Ag 125 Ce 126 Nd 127 Pm 129 Ag 130 Ag 131 Pm 132 Sm 134 In 135 Sn

Eth −52.98 −59.59 −55.2 −46.07 −48.27 −38.75 −45.35 −51.87 −67.15 −60.23 −72.34 −44.37 −50.55 −45.39 −58.81 −56.16 −62.26 −57.18 −49.9 −43.89 −66.45 −64.62 −60.56 −66.11 −66.96 −53.41 −45.47 −53.19 −47.01 −60.3 −55.48 −52.04 −60.42

EAME −53.05 −60.1 −56.1 −44.8 −47.2 −38.31 −44.4 −51.35 −67.1 −60.54 −72.51 −43.9 −50.63 −46.29 −59.73 −57.11 −62.07 −57.29 −49.62 −44 −66.26 −64.69 −60.61 −66.47 −66.66 −52.89 −45.06 −52.45 −46.16 −59.74 −55.25 −52.02 −60.8

Nucleus 93 Ag 95 Cd 97 Br 98 In 100 Kr 103 Sr 104 Y 105 Zr 106 Sb 108 Y 108 I 110 Nb 111 Mo 113 Nb 114 Ru 115 Cs 116 Ba 117 La 119 Ru 120 Ru 121 La 122 Ag 123 La 124 Ce 125 Pr 126 Pm 128 Ag 129 Cd 130 Pm 131 Sm 132 Eu 134 Sm 135 Eu

Eth −45.89 −46.29 −35.65 −54.05 −36.59 −48.49 −54.93 −62.6 −66.36 −38.52 −51.61 −53.82 −61.29 −40.87 −70.36 −59.48 −54.23 −46.25 −53.11 −50.46 −62.36 −70.67 −68.76 −64.99 −58.4 −40.11 −55.78 −64.6 −56.22 −50.63 −42.9 −61.57 −54.65

EAME −46.78 −46.7 −34.65 −53.9 −36.2 −47.55 −54.91 −62.36 −66.33 −37.74 −52.65 −53.62 −61.1 −42.2 −70.53 −59.7 −54.6 −46.51 −53.24 −50.94 −62.4 −71.23 −68.71 −64.82 −57.91 −39.57 −54.8 −63.2 −55.47 −50.2 −42.5 −61.51 −54.19

Nucleus 94 Se 96 Br 97 Kr 99 Kr 100 Rb 103 Y 104 Zr 105 Te 107 Y 108 Zr 109 Zr 110 Mo 111 Xe 113 Mo 114 I 115 Ba 117 Tc 118 Ru 119 Rh 120 Rh 121 Ce 122 La 123 Ce 124 Pr 125 Nd 127 Ag 128 Nd 129 Nd 130 Sm 131 Eu 133 In 134 Eu 135 Gd

Eth −37.48 −39.51 −49.18 −40.27 −47.68 −59.43 −67.02 −51.73 −43.75 −52.89 −47.81 −65.62 −53.65 −52.99 −72.75 −48.2 −48.74 −57.59 −62.81 −58.98 −52.82 −64.78 −60.43 −53.77 −48.15 −59.22 −60.74 −62.84 −47.84 −39.4 −58.43 −50.54 −44.72

EAME −36.8 −38.63 −47.92 −39.5 −46.7 −58.94 −66.34 −52.5 −42.72 −52.2 −47.28 −65.46 −54.4 −54.14 −72.8 −49.03 −49.85 −57.92 −63.24 −59.23 −52.7 −64.54 −60.18 −53.13 −47.62 −58.9 −60.18 −62.24 −47.58 −39.35 −57.93 −49.83 −44.18

Nucleus 94 Br 96 Kr 97 Cd 99 Cd 101 In 103 Sn 104 Sb 106 Y 107 Zr 108 Nb 109 Nb 110 Sb 112 Nb 113 Tc 114 Cs 116 Tc 117 Ru 118 Rh 119 Pd 120 La 121 Pr 122 Ce 123 Pr 124 Nd 126 Ag 127 Pr 128 Pm 129 Pm 130 Eu 132 Cd 133 Sm 134 Gd 136 Sn

Eth −48.16 −54.29 −60.14 −70.19 −69.51 −68.27 −59.08 −47.08 −55.55 −60.77 −58.23 −77.33 −45.4 −63.04 −54.31 −51.75 −59.9 −65.02 −71.64 −57.91 −41.75 −57.94 −50.66 −44.97 −61.15 −65.01 −48.74 −53.51 −33.74 −51.97 −57.57 −41.9 −56

EAME −47.8 −53.03 −60.6 −69.85 −68.61 −66.97 −59.18 −46.77 −55.19 −60.7 −58.1 −77.54 −45.8 −63.72 −54.54 −52.75 −60.01 −65.14 −71.62 −57.69 −41.58 −57.84 −50.34 −44.5 −61.01 −64.43 −48.05 −52.95 −33.94 −50.72 −57.13 −41.57 −56.5

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

105 Sr

Eth −40.99 −44.78 −64.41 −47.8 −61.76 −60.55 −56.01 −39.86 −60.21 −64.86 −71.41 −67.46 −59.83 −58.7 −50.24 −45.21 −64.22 −68.74 −62.17 −55.06 −49.77 −52.46 −45.47 −58.67 −64.61 −60.96 −56.08 −39.02 −42.38 −56.48 −62.39 −47.85 −47.13

Table 4 (continued) Nucleus 136 Sb 137 Sb 138 Gd 139 Gd 140 Ho 142 I 143 Dy 146 Er 148 Er 149 Tm 150 Tm 151 La 152 Lu 153 Hf 155 La 155 Ta 157 Pr 158 Ta 159 W 160 Re 161 Ta 162 Re 163 Gd 164 Re 165 Os 166 Ir 167 Re 169 Gd 170 Dy 171 Hg 173 Dy 174 Er

−64.44 −60.13 −55.91 −57.62 −30.08 −53.75 −51.14 −36.59 −43.62 −51.07 −43.79 −46.42 −53.76 −33.18 −26.82 −37.71 −24.38 −48.29 −31.1 −24.99 −16.95 −38.7 −22.54 −61.17 −27.54 −21.43 −13.32 −34.75 −43.71 −53.49 3.57 −43.62 −52.09

EAME −64.88 −60.26 −55.78 −57.53 −29.31 −55.72 −52.32 −36.91 −44.71 −51.65 −44.04 −46.61 −54.29 −33.42 −27.3 −38.8 −23.67 −48.97 −31.02 −25.23 −16.66 −38.73 −22.35 −61.49 −27.64 −21.65 −13.21 −34.84 −43.9 −53.66 3.5 −43.78 −51.95

Nucleus 136 Eu 137 Eu 138 Tb 139 Tb 141 Te 142 Tb 143 Ho 145 Xe 146 Tm 148 Yb 149 Yb 150 Yb 151 Lu 153 Ba 154 La 155 Ce 156 Ce 157 Nd 158 W 160 Nd 161 Nd 161 W 162 Os 163 Os 164 Ir 165 Ir 166 Pt 167 Pt 169 Tb 170 Ir 172 Dy 173 Ho 174 Au

Eth −56.69 −60.19 −44.27 −48.5 −49.4 −55.88 −42.62 −51.08 −30.24 −29.51 −32.86 −38.32 −29.94 −36.48 −41.3 −47.65 −44.6 −56.36 −23.81 −46.91 −42.36 −30.08 −14.13 −15.9 −7.55 −11.32 −4.35 −6.17 −49.99 −23.4 −47.65 −49.11 −13.87

EAME −56.26 −60.02 −43.63 −48.17 −51.56 −57.06 −42.28 −52.1 −31.28 −30.35 −33.5 −38.73 −30.2 −37.62 −42.38 −48.4 −45.4 −56.79 −23.7 −47.42 −42.96 −30.41 −14.5 −16.12 −7.27 −11.63 −4.79 −6.54 −50.1 −23.32 −47.73 −49.1 −14.2

Nucleus 136 Gd 137 Gd 138 Dy 139 Dy 141 Dy 142 Dy 143 Er 145 Er 147 Xe 149 Cs 150 Cs 150 Lu 152 Ba 153 La 154 Ce 155 Pr 156 Pr 157 Hf 159 Pr 160 Pm 161 Pm 162 Pm 163 Pm 164 Sm 165 Eu 166 Eu 167 Eu 168 Gd 169 Pt 170 Au 172 Ho 173 Er 175 Ho

Eth −49.44 −51.59 −35.39 −38.31 −45.65 −50.13 −31.55 −38.24 −42.54 −43.41 −38.29 −24.45 −41.72 −46.02 −52.25 −55.37 −51.28 −38.54 −40.68 −52.73 −50.12 −46.07 −43.07 −48.36 −50.66 −46.68 −43.65 −48.17 −12.03 −3.7 −51.09 −53.54 −43.25

EAME −49.05 −51.21 −34.94 −37.69 −45.32 −49.96 −31.35 −39.69 −43.26 −43.85 −38.96 −24.94 −42.6 −46.93 −52.7 −55.78 −51.91 −38.75 −41.45 −53.1 −50.43 −46.31 −43.15 −48.18 −50.56 −46.6 −43.59 −48.1 −12.38 −3.61 −51.4 −53.65 −42.8

Nucleus 136 Tb 137 Tb 139 Sb 140 Te 141 Ho 142 Ho 144 I 145 Tm 147 Ba 149 Ba 150 Ba 151 Cs 152 La 153 Ce 154 Lu 155 Nd 156 Ta 158 Pr 159 Nd 160 Sm 161 Sm 162 Sm 163 Sm 164 Eu 165 Gd 166 Gd 167 Gd 168 Tb 169 Au 171 Tb 172 Ir 173 Hg 175 Er

Eth −36.76 −41.53 −49.13 −55.61 −34.92 −38.15 −44.58 −28.17 −60.26 −53.16 −50.12 −34.34 −49.34 −54.97 −39.88 −62.38 −26.2 −43.86 −49.6 −60.5 −56.88 −54.8 −50.83 −53.02 −56.41 −54.52 −50.67 −52.35 −1.71 −43.48 −27.48 −2.31 −48.73

EAME −35.97 −41 −50.32 −56.96 −34.37 −37.47 −46.58 −27.88 −60.6 −53.49 −50.6 −35.22 −50.07 −55.35 −39.57 −62.47 −25.8 −44.73 −50.22 −60.42 −56.98 −54.75 −50.9 −53.1 −56.47 −54.4 −50.7 −52.5 −1.79 −43.5 −27.52 −2.57 −48.65

Nucleus

Eth

EAME

137 Sn

−49.45 −50.31 −53.44 −55.15 139 Te −59.43 −60.8 140 Dy −43.22 −42.84 142 Te −45.55 −47.43 143 I −50.08 −51.64 144 Xe −56.46 −57.28 146 Xe −47.9 −48.67 147 Er −46.3 −47.05 149 La −60.66 −60.8 150 La −56.7 −57.04 151 Ba −45.15 −45.82 152 Ce −59.26 −59.11 153 Yb −47.09 −47.06 154 Hf −32.87 −32.73 155 Hf −34.18 −34.1 157 Ce −39.66 −40.67 158 Nd −53.94 −54.4 159 Pm −56.5 −56.85 160 Eu −63.14 −63.37 161 Eu −61.57 −61.78 162 Eu −58.43 −58.65 163 Eu −56.47 −56.63 164 Gd −59.76 −59.75 165 Tb −60.27 −60.66 166 Re −31.99 −31.85 167 Tb −55.69 −55.84 168 Ir −18.73 −18.74 170 Tb −46.13 −46.34 171 Dy −49.76 −50.11 172 Au −9.06 −9.28 174 Ho −45.65 −45.5 176 Er −46.89 −46.5 (continued on next page) 138 Sb

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

144 Er

Eth

39

40

Table 4 (continued) Nucleus 176 Au 178 Tl 180 Tl 184 Lu 187 Hf 190 Ta 194 Re 215 Pb 220 U 222 U 231 Fr 233 Ra 234 Am 236 Ac 237 Th 238 Th 239 Bk 240 Es 242 U 243 Cf 244 Fm 246 Md 247 Md 248 No 249 No 251 No 252 Lr 253 Rf 255 Cf 256 Es 257 Rf 258 Lr

EAME −18.54 −4.75 −9.4 −36.41 −32.98 −28.66 −27.55 −13.1 4.48 23.03 24.3 42.33 44.77 44.53 51.51 50.2 52.63 54.29 64.2 58.62 60.95 69.01 76.28 76.04 80.66 81.82 82.91 88.84 93.79 84.81 87.19 95.93 94.84

Nucleus 176 Tl 179 Tm 181 Yb 184 Bi 187 Ta 191 W 198 Ir 209 Hg 217 Bi 221 At 223 At 231 Am 233 Ac 235 Ac 236 Th 237 Am 238 Bk 239 Cf 241 U 242 Bk 243 Es 245 Es 247 Pu 248 Am 249 Am 250 Es 251 Lr 253 Bk 254 Bk 255 Lr 256 Lr 257 Db

Eth 0.87 −42.19 −41.58 1.88 −37.73 −31.72 −25.39 −8.54 9.78 17.4 24.39 42.09 41.91 48.09 46.8 46.79 54.27 58.24 56.49 57.91 64.44 65.82 69.11 70.43 73.26 72.57 86.76 80.96 84.74 88.81 90.89 98.94

EAME 0.55 −41.6 −40.85 1.05 −36.77 −31.11 −25.82 −8.35 8.82 16.81 23.46 42.44 41.5 47.72 46.45 46.57 54.29 58.15 56.2 57.74 64.78 66.44 69 70.56 73.1 73.23 87.9 80.93 84.39 90.06 91.87 100.34

Nucleus 177 Er 179 Yb 181 Lu 185 Hf 188 Hf 192 W 202 Pt 210 Hg 218 Bi 221 U 226 Np 232 Fr 233 Am 235 Am 236 Am 237 Cm 238 Cf 240 Pa 241 Bk 242 Es 243 Fm 245 Fm 247 Am 248 Bk 249 Es 250 Md 252 Cm 253 Md 254 Md 255 Rf 256 Db 258 Es

Eth −43.06 −46.88 −45.29 −39.06 −31.73 −29.98 −22.66 −5.14 14.28 24.05 32.44 46.99 43.11 44.62 46.25 49.36 57.05 57.27 56.12 64.44 68.88 69.49 66.85 67.76 70.62 77.84 79.3 80.42 82.89 93.03 99.14 92.54

EAME −42.8 −46.42 −44.74 −38.36 −30.88 −29.65 −22.6 −5.11 13.34 24.59 32.74 46.36 43.17 44.66 46.18 49.28 57.2 56.8 56.1 64.97 69.26 70.22 67.15 68.08 71.18 78.64 79.06 81.3 83.51 94.4 100.72 92.7

Nucleus 177 Tm 179 Pb 182 Lu 185 Bi 188 Ta 192 Re 204 Au 211 Tl 219 Po 222 At 228 Fr 232 Np 234 Ra 235 Cm 236 Cm 237 Bk 239 Pa 240 Bk 241 Cf 242 Fm 244 Np 245 Md 247 Es 248 Es 249 Fm 250 No 252 Bk 253 No 254 Lr 255 Db 257 Es 258 Fm

Eth −47.76 2.34 −42.24 −1.47 −34.59 −32.1 −20.5 −5.61 13.72 21.58 34.05 37.42 47.81 47.98 47.75 53.04 53.64 55.66 59.16 67.88 63.5 74.78 67.96 69.79 72.82 80.58 78.59 83.38 88.69 98.47 89.14 89.92

EAME −47.47 2 −41.88 −2.21 −33.81 −31.71 −20.75 −6.08 12.8 20.8 33.28 37.36 47.23 47.91 47.89 53.1 53.34 55.67 59.36 68.4 63.2 75.29 68.61 70.3 73.62 81.52 78.53 84.47 89.85 100.04 89.4 90.43

Nucleus 178 Tm 180 Yb 183 Lu 186 Hf 189 Ta 193 Re 205 Au 212 Tl 220 Po 222 Pa 228 Np 232 Am 234 Ac 235 Bk 236 Bk 237 cf 239 Cm 240 Cf 241 Es 243 Np 244 Es 246 Es 247 Fm 248 Md 249 Md 251 Md 252 Md 253 Lr 254 Rf 256 Cf 257 Lr 258 No

Eth −44.44 −45.15 −40.21 −37.44 −32.55 −30.48 −18.8 −0.84 16.51 21.27 33.5 43.16 45.49 52.44 53.32 57.73 51.4 57.85 63.39 59.98 65.53 67.39 70.89 76.37 76.46 78.15 79.67 87.48 92 87.11 91.64 90.83

EAME −44.12 −44.4 −39.52 −36.43 −31.83 −30.3 −18.75 −1.65 15.47 22.12 33.7 43.4 45.1 52.7 53.4 57.82 51.19 58.03 63.84 59.88 66.03 67.9 71.58 77.15 77.33 79.03 80.63 88.69 93.32 87.04 92.74 91.48

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

208 Hg

Eth −18.14 −4.38 −9.07 −37.01 −33.9 −29 −27.15 −13.62 5.67 22.6 24.17 43.04 45.18 44.43 51.75 50.42 52.88 54.28 63.9 58.8 60.72 68.45 75.71 75.27 79.81 80.9 81.91 87.72 92.61 84.81 87.01 94.86 93.83

G. Royer et al. / Nuclear Physics A 847 (2010) 24–41

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