The Nuclear Shell Model Feats and Challenges

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					The Nuclear Shell Model:
 Feats and Challenges.


         Alfredo Poves
               o
Universidad Aut´noma de Madrid

    Bordeaux, October 2006




   Strasbourg-Madrid collaboration
          OUTLINE


Introduction
Extreme correlations
Exotic nuclei
Double beta decay
The three pillars of the shell
          model


The Effective Interaction
The Valence Space
The Algorithms and their Codes




                   ınez-Pinedo, F. Nowacki, A. Poves
E. Caurier, G. Mart´
and A. P. Zuker. “The Shell Model as a Unified View
of Nuclear Structure”, Reviews of Modern Physics, 77
(2005) 427-488
 The Effective Interaction: Key aspects


The evolution of the Spherical mean field in the valence
spaces. What is missing in the monople hamiltonian
derived from the realistic NN interactions, be it through
a G-matrix, Vlow−k or other options?

The multipole hamiltonian does not seem to demand
major changes with respect to the one derived from
the realistic nucleon-nucleon potentials

Do we really need three body forces? Would they be
reducible to simple monopole forms?
              The Valence Space(s)
Miscellanea of computationally accessible valence
spaces:
(note: in a major HO shell of principal quantum number p the
orbit j=p+1/2 is called intruder and the remaining ones are
denoted by rp)


• Classical 0¯ ω valence spaces are the p, sd and pf
             h
  shells

• r2-pf : intruders around N and/or Z=20
                     76         82
• r3-g9/2(d5/2):          Ge,        Se, and the region around
  80
     Zr

• r3-g9/2(d5/2) for the neutrons and pf for protons:
  neutron rich Cr, Fe, Ni, Zn

• r4-h11/2 for neutrons and p1/2 − g9/2-r4 for protons:
  96
     Zr, 100Mo,110Pd, 116Cd
                                             124
• r4-h11/2 for neutrons and protons:               Sn, 128−130Te,
  136
      Xe
      The Algorithms and the Codes


Algorithms include Direct Diagonalisation, Lanczos,
Monte Carlo Shell Model, Quantum Monte Carlo
Diagonalization, Projected Shell-Model, DMRG and
different extrapolation ansatzs

The Strasbourg-Madrid codes can deal with problems
involving basis of 1010 Slater determinants, using
relatively modest computational resources
              Some Physics Goals


Precision Spectroscopy towards larger masses

Changing Magic Numbers far from Stability: The
competing roles of spherical mean field and correlations

Double β decay, the key to the nature of the neutrinos,
the absolute scale of their masses and their hierarchy

No core shell model for light nuclei.         Ab initio
description of the low-lying intruder states and of the
origin of the Gamow-Teller quenching

Nuclear Structure and Nuclear Astrophysics
   Extreme Correlations. Extreme
Coexistence. Spherical, Deformed and
 Superdeformed states in the same
      nucleus; The case of 40Ca

   In the valence space of two major shells
            1f5/2

           2p1/2

           2p3/2

           1f7/2

                    pf -shell

           1d3/2

           2s1/2

           1d5/2

                    sd-shell
The relevant configurations are:

[sd]24 0p-0h in   40
                       Ca, SPHERICAL

[sd]20 [pf]4 4p-4h in     40
                               Ca, DEFORMED

[sd]16 [pf]8 8p-8h in     40
                               Ca, SUPERDEFORMED

                  1f5/2

                  2p1/2

                  2p3/2

                  1f7/2

                               pf -shell

                  1d3/2

                  2s1/2

                  1d5/2

                               sd-shell
       The superdeformed band: 8p-8h


 20
 18                       exp
 16                       sdpf-8p-8h

 14
 12
J 10
  8
  6
  4
  2
  0
   0        1000      2000             3000   4000
                   Eγ (in keV)
       The Superdeformed band: Mixed
                 calculation


 20
                          exp
 18
                          sdpf-full
 16
 14
 12
J 10
  8
  6
  4
  2
  0
   0        1000      2000            3000   4000
                   Eγ (in keV)
       Huge correlation energies!!!


-260
           Lowest Slater Determinant
           Lowest energy at fix np-nh
-265                    +
           The lowest 0 ’s after mixing


-270

-275

-280

-285

-290
       0            2             4           6   8   10
                                      np-nh
  The np-nh energies as a function of J


            +
-266       0
             +
           2
-268       4
             +
             +
-270       6
             +
           8
               +
-272       10
               +
           12
-274       14
               +
               +
-276       16

-278
-280
-282
-284
       0           2   4           6   8   10
                           np-nh
                                         Transition Quadrupole Moments


                                       250
Transition quadrupole moment (e fm )
2




                                       200




                                       150




                                       100


                                         0   2   4   6   8       10   12   14   16   18
                                                             J
Quasi-SU3 + Pseudo-SU3 interpretation


In the 4p-4h intrinsic state of 40Ca, the two neutrons
and two protons in the pf -shell can be placed in the
lowest K=1/2 quasi-SU3 level of the p=3 shell. This
gives a contribution Q0=25 b2. In the pseudo-sd shell.
p=1 we are left with eight particles, that contribute
with Q0=7 b2. For the 8p-8h state the values are
Q0=35 b2 and Q0=11 b2

Using the proper values of the oscillator length it
obtains:
40
     Ca 4p-4h band Q0=125 e fm2 (Q0=148 e fm2)
40
     Ca 8p-8h band Q0=180 e fm2 (Q0=226 e fm2)

In very good accord with the data. The values in blue
assume strict SU3 symmetry in both shells. The SM
results almost saturate the quasi-SU3 bounds. The
SU3 values are a 25% larger.
The geometry of the spherical mean field orbits giving
rise to deformed rotors pertains to variants of Elliott’s
SU3 (pseudo-SU3, quasi-SU3).

np-nh configurations across N=Z=20 produce
superdeformed shapes that can be explained in the
pseudo-SU3+quasi-SU3 scheme. This scheme will also
apply to other mass regions either protron rich as in
80
   Zr, or neutron rich as in 42S.
Shell Model at the limits of the stability


At the very neutron rich or very proton rich edges,
the T=0 and T=1 channels of the effective nuclear
interaction weight very differently than they do at the
stability line. Therefore the effective single particle
structure may suffer important changes, leading in
some cases to the vanishing of established shell closures
or to the appearance of new ones.
Shell Model at the limits of the stability


N=20 The region around 31Na provides a beautiful
example of intruder dominance in the ground states.
This has been known experimentally (Detraz, Thibault,
Guillemaud, Klotz, Walter) since long. Early mean
field (Campi) and shell model (Warburton, Retamosa)
interpretations pointed out the role of deformed
intruder configurations 2p-2h neutron excitations from
the sd to the pf -shell and started to study the
boundaries of the so called “island of inversion”
Recently there has been a renewal of interest in this
region, triggered by the advent of new radioactive ion
beam facilities that pave the way toward spectroscopic
studies far from stability.

The limits of the “island of inversion” depend crucially
of the effective spherical single particle energies
(ESPE’s)
The evolution of the spherical mean field
                at N=20


Strasbourg-Madrid
             10

                                                 2p3/2
              5
                                                 1f7/2
                                                 1d3/2
              0                                  2s1/2
                                                 1d5/2
ESPE (MeV)




              -5


             -10


             -15


             -20


             -25
                   28   30   32   34   36   38    40
                                  A
The evolution of the spherical mean field
                at N=20


Tokyo
             10

                                                 2p3/2
              5
                                                 1f7/2
                                                 2d3/2
              0                                  2s1/2
                                                 2d5/2
ESPE (MeV)




              -5


             -10


             -15


             -20


             -25
                   28   30   32   34   36   38   40
                                  A
The saga of the Calcium isotopes; How
          many are magic?


                             40             48
    The classics:                 Ca and         Ca

    The proton-rich challengers:
                36                34
                     Ca and            Ca
                34
   (why not?,        Si is doubly magic, isn’t it?)


 The very neutron rich ones: 52Ca,
     54
        Ca and, perhaps, 60Ca
The monopole drift in the Calcium chain


              0
ESPE (MeV)




              -5




             -10


                   20      28    32   34    40
                        Neutron number
KB3G-A (solid line) GXPF1-A (dashed line)

They show almost identical behavior at N=28 and
N=32. 52Ca is (weakly) magic.
                             54
                        Is        Ca doubly magic?


              0
ESPE (MeV)




              -5




             -10


                   20                 28    32   34   40
                                   Neutron number
The monopole drift caused by the filling of the orbit
1p3/2 is very different for both interactions. GXPF1-A
increases strongly the 1p3/2-0f5/2 gap,while KB3G-A
reduces it slightly

Therefore, N=34 should be magic with GXPF1-A and
not at all with KB3G-A
A word of caution on the monopole drifts




All these monopole drifts involve solely the neutron-
neutron interaction.    Therefore, once the single
particle spectrum of 41Ca is fixed, the magicity of
N=34 depends only of the neutron-neutron interaction.
Mutatis mutandis, the same applies to the N=14 and
N=16 closures in the Oxygen isotopes



Our present knowledge of the effective interactions
is not precise enough as to predict the details
of the monopole drifts far from stability. Some
phenomenological ingredients have to be extracted
from key experiments. The N=34 case is a good
example.
           54
      Is          Ca doubly magic? Exhibits from
                     Isolde and elsewhere


                                       52
                                              Ca
                                                           4+      6.48
                                      (4+)          5.95                  4+          5.93
4+            5.47  2+ 4+     5.32
2+                            5.18                         2+      5.09 2+
4+            5.30 4+         5.14                                 4.48               4.90
              5.22 2+                                      0+
                              4.43                       2+        4.39 + +
                                                                          0           4.28
 2+                                   (3-)                +             2
              4.11
                    0+                              3.99 4         4.34 +             3.94
 0+                           3.69                                                    3.90
              3.42                    (1+,2+)       3.15   1+      3.29 41+
                                                                                      3.08
1+         2.38              2.26     2+            2.56   2+      2.55 2+
2+                  2+                                                                2.35
           2.35    1+        2.25



 0+           0     0+            0   0+            0      0+      0      0+          0

      GXPF1              GXPF1A              Exp.          KB3GA               KB3G




F. Perrot, F. Marechal et al. (Isolde)
            54
       Is        Ca doubly magic? Exhibits from
                    Isolde and elsewhere


                                         53
                                               Ca
5/2-         2.85 5/2-         2.94
                                      (3/2-)        2.22 3/2-       2.12 3/2-
                    3/2-                                                           2.16
                               2.00
3/2-         1.46                                        5/2-       1.23 5/2-
                                                                                   1.03

1/2-         0      1/2-       0      1/2-          0    1/2-       0    1/2-      0

  GXPF1               GXPF1A                 Exp.           KB3GA           KB3G




F. Perrot, F. Marechal et al. (Isolde)
       54
Is          Ca doubly magic? Exhibits from
               Isolde and elsewhere


                              55
                                   Ti
5/2-            0.89


                                               1/2-
1/2-                   1/2-             0.04   5/2-
                                                             0.19
                       5/2-
       GXPF1A                 KB3GA                   KB3G
        54
   Is        Ca doubly magic? Exhibits from
                Isolde and elsewhere


                                          56
  8+
                                               Ti
             5.33
  6+         4.89
                                                        8+
                    8+        4.37
                                                                 4.51   8+
                    6+                           4.21   6+       4.17
                                                                                    4.31
                              4.09
                                                                        6+          3.85


  6+                                 6+
             3.04   6+                           2.98   6+       2.89   6+          2.87
                              2.86
  4+         2.53
                    4+               4+
                              2.27               2.29   4+       2.16   4+          2.00
  2+         1.51
                    2+        1.17   2+          1.13   2+       1.05
                                                                        2+          0.89


  0+         0      0+        0      0+          0      0+       0      0+          0

   GXPF1             GXPF1A               Exp.           KB3GA               KB3G




R. Janssens, B. Fornal et al. MSU
  Is 54Ca doubly magic? The strange
behavior of the B(E2)’s in the Titanium
                isotopes


                   800
                                         exp
                                         gxpf1
                   700                   kb3g
B(E2) (in e fm )
4




                   600
2




                   500

                   400

                   300

                   200
                     46   48   50   52   54      56   58
                                    A

isovector effective charge equal to zero
  Is 54Ca doubly magic? The strange
behavior of the B(E2)’s in the Titanium
                isotopes


                   800
                                         exp
                                         gxpf1
                   700                   kb3g
B(E2) (in e fm )
4




                   600
2




                   500

                   400

                   300

                   200
                     46   48   50   52   54      56   58
                                    A

isovector effective charge equal to 0.6e
                            54
     Predicted                   Ca low-lying spectrum


                                     54
 +
                                          Ca
2            3.83
                        +
                    2                2.95

                                            2+           1.77
                                                                2+             1.32


0+                  +
                    0                       0
                                             +
                                                                0
                                                                    +
                                                         0                     0

     GXPF1                  GXPF1A               KB3GA                  KB3G
                42
     N=28:           Si; doubly magic or well
                       deformed?


There have been recent claims of double magicity
for 42Si by Friedman et al. (Nature 2005) based
in indirect evidence from a two proton knock-out
experiment at MSU. Previous shell model calculations
in a model space comprising the full sd-shell for protons
and the full pf -shell for neutrons that reproduced
successfully the spectroscopy of the Sulfur isotopes,
predicted a heavily mixed 42Si, oblate, with Q0–
intrinsic quadrupole moment– equal to –56 e fm2,
a 2+ excitation energy of 1.49 MeV and a doubly
closed (N=28, Z=14) component that amounts only
to 28% [Caurier, Nowacki and Poves (2004)]. With
this interaction, the proton gap at Z=14 has a value of
6.2 MeV, and the N=28 closure vanishes below 46Ar,
due to the combined effect of a slight reduction of
the N=28 neutron gap when sd-protons are removed,
and the availability of valence protons that favor the
neutron excitations across N=28 via the quadrupole-
quadrupole neutron proton interaction
   42
        Si, an oblate, well deformed, rotor

In an experiment of in-beam γ-spectroscopy performed
at GANIL, S. Grevy et al. have measured the excitation
energy of the 2+ of 42Si, (770 keV). We have tried
to understand whether 42Si can have such a low-lying
2+, a feature associated to deformed rotors, without
a massive breaking of the Z=14 proton closure, as
implied by the MSU result. In order to lower the 2+
excitation energy we have reduced the pairing in the
pf -shell orbits 300 keV, to bring the 2+ excitation
energy of 36Si to its experimental position. Then we
have modified the d5/2–pf -shell monopoles equally, to
obtain a smaller value of the Z=14 gap, 5.8 MeV.
With this interaction, 42Si is more clearly an oblate
rotor; the 2+ excitation energy is now 810 keV,
and the intrinsic quadrupole moment –87 e fm2,
corresponding to β=–0.45. The doubly magic N=28,
Z=14 component represents only 20% of the ground
state wave-function. The ground state has (in average)
2.2 neutrons above N=28 and 1.1 protons above Z=14.
This means that the massive breaking of the N=28 shell
closure can indeed be achieved with a relatively modest
opening of the Z=14 one.
 N=28: from doubly magic 48Ca toward
  well deformed drip line 40Mg; The
            physical picture


As we remove protons from doubly magic 48Ca,
the N=28 neutron gap slowly shrinks. In 46Ar the
collectivity induced by the action of the four valence
protons in the quasi-degenerate quasi-spin doublet
1s1/2-0d3/2 is not enough to beat the N=28 closure.
46
   Ar comes out oblate non-collective. Notice that,
were the N=28 gap wiped out, it would be oblate and
collective. This can be easily understood assuming the
development of quasi-SU3 in the neutron orbits and
pseudo-SU3 in the proton orbits.

In 44S, the quadrupole collectivity has already set
in. The N=28 closure blows out and prolate and
oblate states coexist. The ground state and the first
excited 2+ form the germ of a prolate rotational band.
However, it dies out already at the 4+ state. No
regular band appears on top of the oblate 0+ isomer,
predicted by the shell model calculations and recently
found also at Ganil.
 N=28: from doubly magic 48Ca toward
  well deformed drip line 40Mg; The
            physical picture


The scene changes suddenly in 42Si, because the
pseudo-spin doublet is not relevant any more. The
proton collectivity can only develop promoting particles
through the Z=14 closure. As we have seen, even with
relatively large values of the gap it does. The collective
coupling scheme is now quasi-SU3, and the favored
shape oblate. The calculations produce a very regular
oblate band that resists up to J=8+. An excited 0+
state, prolate, predicted at 1.3 MeV, does not generate
any band.

The situation is even better in 40Mg –that should
be the heavier magnesium isotope according to some
calculations– because now the proton orbit 0d5/2 is
open. The calculations produce a very collective
prolate rotor, with a deformation similar in absolute
value to that of 42Si, a 2+ excitation energy even lower
(∼600 keV) and a very regular rotational band up to
J=8+.
 N=28: from doubly magic 48Ca toward
  well deformed drip line 40Mg; The
            physical picture


In conclusion, the combined effects of the erosion of
the semi-magic closures N=28 and Z=14, and the
action of the quadrupole interaction, produce a very
rich variety of behaviors and shapes in the even N=28
isotones; spherical 48Ca; oblate non-collective 46Ar;
coexistence in 44S, and two rotors, oblate 42Si and
prolate 40Mg.
 SHELL MODEL CALCULATIONS OF
THE NEUTRINOLESS DOUBLE BETA
           DECAY



The “crisis” of the calculations of the 0ν, ββ
nuclear matrix elements



The QRPA “explosion”

gpp, the miraculous factor

Does a good 2ν m.e. guarantee a good 0ν m.e.?

To quench or not to quench . . .
   The quest for better wave functions


Quality indicators

• Good spectroscopy for parent, daughter and grand-
  daughter, even better if its extend to a full mass
  region

• GT-strengths and strength functions, 2ν matrix
  elements, etc.

Large scale shell model calculations (LSSM) vs QRPA,
the pros and cons

• Interaction

• Valence space

• Pairing

• Deformation
             Update of the 0ν results

In the valence spaces r3-g9/2 (76Ge, 82Se) and r4-
h11/2 (124Sn, 128−130Te, 136Xe) we have obtained high
quality effective interactions by carrying out multi-
parametrical fits whose starting point is given by
realistic G-matrices. In the valence space proposed
for 96Zr, 100Mo, 110Pd and 116Cd, the results are still
subject to further improvement


        mν for T 1 = 1025 y.     MGT
                                  0ν    1-χF
                  2


        48
           Ca         0.85      0.67    1.14
        76
           Ge         0.90      2.35    1.10
        82
           Se         0.42      2.26    1.10
        (110Pd)       0.67      2.21    1.15
        (116Cd)       0.27      2.49    1.18
        124
            Sn        0.45      2.11    1.13
        128
            Te        1.92      2.36    1.13
        130
            Te        0.35      2.13    1.13
        136
            Xe        0.41      1.77    1.13
Dependence on the effective interaction


The results depend only weakly on the effective
interactions provided they are compatible with the
spectroscopy of the region.

For the lower pf shell we have three interactions
that work properly, KB3, FPD6 and GXPF1. Their
predictions for the 2ν and the neutrinoless modes are
quite close to each other


                    KB3     FPD6     GXPF1

        MGT (2ν)    0.083   0.104    0.107
        MGT (0ν)    0.667   0.726    0.621


Similarly, in the r3g and r4h spaces, the variations
among the predictions of spectroscopically tested
interactions is small (10-20%)
 Learning from the 48Ca →48Ti and the
     (fictitious) 48Ti →48Cr decays

The influence of deformation

Changing adequately the effective interaction we can
increase or decrease the deformation of parent, grand-
daughter or both, and so gauge its effect on the decays.
We have artificially changed the deformation of 48Ti
and 48Cr adding an extra λQ · Q term to the effective
interaction. A mismatch of deformation can reduce
the ββ matrix elements by factors 2-3. This exercise
shows that the effect of deformation is very important
and cannot be overlooked
 The influence of the spin-orbit partner
Similarly, we can increase artificially the excitation
energy of the spin-orbit partner of the intruder orbit.
Surprisingly enough, this affects very little the values
of the matrix elements, particularly in the neutrinoless
case. Even removing the spin-orbit partner completely
produces minor changes



                    48
                         Ca →48Ti   48
                                         Ti →48Cr

        MGT (2ν)            0.083           0.213
        MGT (0ν)            0.667           1.298


              Without spin-orbit partner

                    48
                         Ca →48Ti   48
                                         Ti →48Cr

        MGT (2ν)            0.049           0.274
        MGT (0ν)            0.518           1.386
   The contributions to the 0ν matrix
element as a function of the J of the of
       the decaying pair : A=82

            6
            5                                                           82
                                                                            Se
            4
            3
            2
            1
      MGT




            0
            -1
            -2
            -3
            -4
            -5
            -6
                 0+        1+        2+ - 3+ - 4+ - 5+ - 6+ - 7+ - 8+ - 9+
                      0-        1-     2    3    4    5    6    7    8


          10
                      82
            8              Se, (9 lev.)
                                         2ν       0ν
            6                  gpp=0.87: M =0.11, Μ =3.09
     0ν




            4
      M




            2


            0


          -2
                 0         1         2        3   4         5   6   7   8        9
                                                            J
   The contributions to the 0ν matrix
element as a function of the J of the of
      the decaying pair : A=130

            7
            6
                                                                                                            130
            5                                                                                                     Te
            4
            3
            2
            1
      MGT




            0
            -1
            -2
            -3
            -4
            -5
            -6
            -7
                 0+       1+       2+       3+        4+       5+       6+       7+       8+       9+     10+ 11+
                      -        -        -         -        -        -        -        -        -        -      -
                      0        1        2        3         4        5        6        7        8        9   10


          12
                      130
          10
                               Te, (13 lev.)
                                             2ν                0ν
            8                  gpp=0.84: M =0.017, Μ =2.12
     0ν




            6
      M




            4


            2


            0


          -2
                 0        1         2             3            4        5         6            7            8      9
                                                                        J
The multipole structure of the 0ν matrix
                element


The transformation of a two body interaction from the
p-p to the p-h form is highly non-unique

[(a† a† )J · (a1a2)J ]0 can go to
   1 2
   †
[(a1a1)λ · (a† a2)λ]0 or to
               2
   †           †
         γ
[(a1a2) · (a2a1)γ ]0

We make the choice of keeping all the orderings of the
matrix elements even if they are redundant. Notice
however that other choices may lead to different
decompositions, that have the same physical content

Our results differ markedly of those of the QRPA
calculations
 The multipole structure of the 0ν matrix
             element: A=76


        1

      0.8                                                    76
                                                                  Ge
      0.6

      0.4

      0.2
MGT




        0

      -0.2

      -0.4

      -0.6

      -0.8

       -1
             0+   -   1+   -   2+ - 3+ - 4+ - 5+ - 6+ - 7+ - 8+ - 9+
                  0        1     2    3    4    5    6    7    8
 The multipole structure of the 0ν matrix
             element: A=82


        1

      0.8                                                    82
                                                                  Se
      0.6

      0.4

      0.2
MGT




        0

      -0.2

      -0.4

      -0.6

      -0.8

       -1
             0+   -   1+   -   2+ - 3+ - 4+ - 5+ - 6+ - 7+ - 8+ - 9+
                  0        1     2    3    4    5    6    7    8
 The multipole structure of the 0ν matrix
            element: A=130


        1

      0.8                                                                                              130
                                                                                                             Te
      0.6

      0.4

      0.2
MGT




        0

      -0.2

      -0.4

      -0.6

      -0.8

       -1
             0+       1+       2+       3+       4+       5+       6+       7+       8+       9+     10+ 11+
                  -        -        -        -        -        -        -        -        -        -
                  0        1        2        3        4        5        6        7        8        9   10-
• Large scale shell model calculations with
  high quality effective interactions are
  available or will be in the immediate future
  for all but one of the neutrinoless double
  beta emitters

• The theoretical spread of the values of
  the nuclear matrix elements entering in
  the lifetime calculations is greatly reduced
  if the ingredients of each calculation are
  examined critically and only those fulfilling
  a set of quality criteria are retained

• A concerted effort of benchmarking
  between LSSM and QRPA practitioners
  would be of utmost importance to increase
  the reliability and precision of the nuclear
  structure input for the double beta decay
  processes

				
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posted:9/6/2011
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