






Theoretical framework 






The many body problem is solved in the framework of the mean field
approximation accounting for pairing correlations, the HartreeFockBogoliubov
(HFB) approximation. The selfconsistent HFB equations are solved using an iterative
method. They are deduced from the minimization of the total energy of the nucleus : 







δ ( < Φ 
H λ_{Z}Z
 λ_{N}N
μ_{2} Q_{20}
 Φ >) =0 










• Φ>_{} is the HFB wave function. 





• λ_{N} and
λ_{Z} are the
Lagrange parameters fixing the number of neutrons
N and protons Z 





• μ_{2}
is the Lagrange parameter to fix the quadrupole moment
q_{20} defined by 







q_{20} =
< ΦQ_{20} Φ > 
with the operator 
Q_{20}=
(16 π / 5)^{½}
r^{2} Y_{20} 








• H is the nuclear Hamiltonian which reads 







H = ∑_{i} T_{i}
+ ½
∑_{i≠j} V_{ij} 







where V_{ij}
is the
Gogny effective nucleonnucleon interaction and
T_{i} the kinetic energy term. 








The HFB equations are solved in an harmonic oscilator basis within the axial
symetry hypothesis. The size of the basis is defiend by the number
of major shells N_{0}
used to develop the HFB wave function. This number depends on the number of
nucleons contained in the nucleus. It is chosen so that the number of states
in the basis is 8 time the maximum number of occupied states. 








These states also depend on the nucleus constrained deformation and are
defined by the quantum numbers of the deformed harmonic oscillator
( n_{⊥},
m
and n_{z} )
which obey the inequation 







(2n_{⊥}+m+1) hω_{⊥}
+(n_{z}+½)
hω_{z} ≤
(N_{0}+2) hω_{0} 
with 
(hω_{0})^{3}=
(hω_{⊥})^{2}
hω_{z} 








where
ω_{⊥} and
ω_{z} are the two
parameters of the axial oscillator. 








These parameters should, in principle be determined minimizing the system total energy.
This criterion is applied to determine, for each deformation, the optimal value of
ω_{0} .
However, the ratio q=ω_{⊥}/
ω_{Z} is not optimized but rather
estimated thanks to the relation : 







q=exp [1.5 β cos(γ) / (2 β +1)]

with
γ=0
for β>0 and
γ=π
for β≤0









based on the approximate deformation
β obtained when using the
Liquid drop approximation of the nucleus. In the latter approach,
both the axial symetry and the parity are conserved. 








The treatment of oddA and oddodd nuclei is performed using the blocking
procedure without breaking the time reversal symetry. This procedure consists in
blocking the unpaired nucleon in a fixed orbital during the minimization procedure.
Several cases are thus required to determine the blocked quasiparticle yielding the
minimum energy. For oddA nuclei, 11 configurations have been tested for each
deformation, while 25 configurations are considered for oddodd nuclei. 










Technical details 






The potential energy surfaces show the nucleus HFB energy, namely 














They are plotted as functions of the deformation parameter
β : 







β=(5 π / 9)^{½} q_{20}
/ (A R_{0}^{2}) 







where A=N+Z is the nucleus mass,
R_{0}=1.2 A^{1/3}
its radius (expressed in fm) and
q_{20}
is the mass quadrupolar moment defined by 







q_{20} =
< ΦQ_{20} Φ > 
with the operator 
Q_{20}=
(16 π / 5)^{½}
r^{2} Y_{20} 








When plotting only potential energy surfaces, dashed lines have been added corresponding
to the approximate rotational energy correction for spins
I = 8, 16 et 24 . These are obtained adding
to the nucleus binding energy the rotational energy 







E_{rot} = [ I(I+1) ]
/ 2ℑ_{x} 







given by the simplest rotational model using however the moment of inertia
ℑ_{x} calculated
for every deformation

β

≥ 0.15. 








The theoretical binding energies (defined as the minimum close to β=0)
as well as the experimental masses taken form the Audi_{}Wapstra
[4] mass tables are also indicated. 












The chemical potentials are given by the Lagrange parameters
λ_{Z} and
λ_{N} for the
protons and neutrons respectively. 








The quadrupolar collective masses
M_{20}
and the moments of inertia
ℑ_{x}
have been calculated using the InglisBeliaev
[5] approximation.
The determination of the zero point energies (ZPE) is also described in the same paper. 








Concerning the proton and neutron pairing energies (p/n),
they are defined by 







E_{P}^{(p/n)} =
½ Tr ( Δ^{(p/n)}
κ^{(p/n)} ) 







where Δ^{(p/n)}
is the pairing field and
κ^{(p/n)}
the pairing tensor obtained from the solution of the HFB equations. 








The β_{2(p/n)} and
β_{4(p/n)} are deduced from
the multipole moments
q_{20}
and q_{40}
defined from the protons and neutrons distribution. More precisely, 







β_{4(p/n)} =
q_{40}^{(p/n)}
/(A R_{0}^{4})

with 
q_{40}^{(p/n)}=
< Φ^{(p/n)}
 r^{4} Y_{40} 
Φ^{(p/n)} >









Finally, the protons and neutrons radii are given by the square roots
of the mean square radii
<r^{2}>
_{(p/n)} defined by 







< r^{2} >_{(p/n)}=
< Φ^{(p/n)}
 r^{2} 
Φ^{(p/n)} > 




