- Email: [email protected]

Review

Shell model and nuclear structure Etienne Caurier ∗ Institut Pluridisciplinaire Hubert Curien (IN2P3-Universit´e Louis Pasteur), F-67037 Strasbourg Cedex 2, France

Abstract The present possibilities, problems and limitations of the description of the nuclear structure in the framework of the Shell Model are discussed. Some applications are presented, in particular to the calculation of the nuclear matrix elements for ββ decays. c 2007 Elsevier B.V. All rights reserved.

Keywords: Nuclear structure; Shell model; Double beta decay

1. Introduction The nuclear shell model (SM) was introduced [1,2] some 55 years ago to explain the regularities of the nuclear properties associated with magic numbers. Its authors proposed an independent particle model, assuming that the main effect of the two body nucleon–nucleon interaction was to generate a mean field. They modelled this mean field as the sum of a harmonic oscillator potential plus a spin–orbit force. 1 Es . mω2r 2 − C lE 2 The success of the independent particle model strongly suggests that the very singular free NN interaction can be regularized in the nuclear medium. Starting with a regularized interaction, the exact solution of the secular problem, in the (infinite) Hilbert space built on the mean field orbits, is approximated in the large scale shell model calculations by the solution of the Schr¨odinger equation in the valence space, using an effective interaction such that: h(r ) =

H Ψ = EΨ −→ Heff. Ψeff. = EΨeff. . ∗ Tel.: +33 03 88 10 64 84; fax: +33 03 88 10 62 02.

E-mail address: [email protected] c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2006.12.012

227

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

In general, effective operators have to be introduced to account for the restrictions of the Hilbert space hΨ |O|Ψ i = hΨeff. |Oeff. |Ψeff. i. The fundamental advantage of the SM is the theoretical possibility of describing simultaneously all the spectroscopic properties of the low-lying states – either of collective or single-particle nature – for a large domain of nuclei. The limitations come from the fact that sometimes the tractable valence spaces are too small to encompass the desired properties of a nucleus. To carry out an SM study it is necessary: (1) To define a valence space (inert core, active shells). (2) To derive an effective interaction. (3) To build and diagonalize the Hamiltonian matrix. The plan of this article is as follows; in Section 2 we discuss the problems related to the valence space, in Section 3 we deal with the problem of the effective interactions, in Section 4 we present the Lanczos method, in Section 5 we discuss the ways to solve the secular problem, and, finally, in Section 6 we present some results of recent large scale shell model calculations for ββ decays. 2. Valence space The choice of the valence space is strongly conditioned by the problem of the diagonalization of the energy matrix. The dimension of the matrices increases exponentially with the number of single particle states in the valence space and with the number of particles (holes) in it (valence particles). To grasp the orders of magnitude, let us examine a particular example. For the description of nuclei in the region 50 ≤ N , Z ≤ 82 a reasonable valence space is the one spanned by the shells 0g 7 , 1d 5 , 1d 3 , 2s 1 , 0h 11 . We call this valence space r4 h, r p being the 2 2 2 2 2 set of nl j orbits with 2n + l = p except the orbit j = (2 p + 1)/2 (for lighter nuclei with 28 ≤ N , Z ≤ 50 we have the similar valence space, r3 g r3 ≡ 0 f 5 , 1 p 3 , 1 p 1 , g ≡ 0g 9 ). In the 2

2

2

2

valence space r4 h 116 Sn is semi-magic and as a consequence the dimension of its secular matrix is small (Dim(Jz = 0) = 16 · 106 ). However, to describe correctly the B(E2) it is necessary to take into account the proton excitations from the 100 Sn core. If we introduce in the valence space 2 p–2h proton excitations from the 0g 9 shells in r4 space, the dimension becomes 12·1010 . 2 Such calculation is, at present, out of reach. However for spherical nuclei, like the semi-magic, a seniority truncation scheme is very efficient. Limiting the seniority to v ≤ 8, the dimensions become tractable (for 116 Sn Dim(Jz = 0) = 750 · 106 ) and, as we can see on Fig. 1, we are able to get a fair description of the B(E2) for Sn isotopes [3] and that with standard effective charges e p = 1.5 en = 0.5. Their description would need excitations from the g 9 to the h 11 2 2 shells, but these excitations generate strong center of mass excitations (spurious states). If, in the harmonic oscillator basis, all the components corresponding to an excitation until n h¯ ω are contained in the valence space, the spurious states can be exactly eliminated. If not, to limit the importance of these components, we must avoid excitations between shells with maximum j : 0 p 3 → 0d 5 → 0 f 7 → 0g 9 → 0h 11 · · · . 2 2 2 2 2 Another important remark is that a valence space can be adequate to describe some properties and completely inadequate for others. In Table 1 we exemplify this for the case of 48 Cr. Notice

228

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Fig. 1. Comparison of measured B(E2↑) with SM calculations that take into account 2 p–2h proton excitations. Table 1 Properties of 48 Cr in different valence spaces 48 Cr

( f 7 )8

( f 7 p 3 )8

0.0 0.63 1.94 77 0.90

−23.3 0.44 2.52 150 0.95

2

Q(2+ ) (e fm2 ) E(2+ ) (MeV) E(4+ )/E(2+ ) BE2(2+ → 0+ ) (e2 fm4 ) P B(GT )

2

2

( f p)8 −23.8 0.80 2.26 216 3.88

that for the quadrupole properties 0 f 7 1 p 3 is a rather good space whereas for magnetic and 2 2 Gamow–Teller processes the inclusion of the spin orbit partners is needed in order to get reasonable results. 2.1. Light nuclei Light nuclei have been extensively studied in the framework of the SM: • 2 ≤ N , Z ≤ 8 p shell nuclei [4] valence space: 0 p 3 , 0 p 1 2 2 • 8 ≤ N , Z ≤ 20 sd shell nuclei [5,6] valence space: 0d 5 , 0d 3 , 1s 1 . 2

2

2

The next step is the p f shell which has also been extensively studied [7] at least for nuclei with 20 ≤ N , Z ≤ 32. For heavier nuclei the influence of the 0g 9 orbit becomes non-negligible. 80 Zr,

2

the next harmonic oscillator double closure after 4 He,16 O and 40 Ca, is not doubly magic but well deformed instead. The strong spin–orbit interaction changes the magic numbers from the harmonic oscillator ones, for instance N , Z = 40, to N , Z = 50, 82, 126. The valence space r3 g (0 f 5 , 1 p 3 , 1 p 1 , 0g 9 ) looks adequate (and tractable) to describe the 2 2 2 2 region 28 ≤ N , Z ≤ 50 except for the strongly deformed nuclei around Z ' N ' 40. The description of prolate states (which coexist with oblate ones) would need the introduction of the 1d 5 shell in the valence space (the quadrupole partner of the 0g 9 shell). This valence space 2 2 is, for the moment, too large for standard SM calculations. Other methods, like the Vampir approach [8], have been successfully used to describe these nuclei. Notice that the shell closure Z = 40 reappears when N = 50; a good example of the evolution of shell structure due to the proton–neutron interaction.

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

229

Fig. 2. Dimension of the matrices for the valence space r4 h. The curve denotes the present computational limit.

2.2. Heavy nuclei The dimensions of the matrices in heavy nuclei may become huge. However the recent breakthroughs in SM calculations have made nuclei close to semi-magic tractable (in Fig. 2 we have depicted the situation for the r4 h valence space). And when the calculations are tractable, the results can be excellent. For the N = 126 isotones [9] a very good agreement (see Fig. 3) with experimental data can be achieved in the r5 i space. The only exceptions are the 3− states which are known to be of very collective nature, thus having important contributions from the excitations of the core. 2.3. Mixed spaces We can take different valence spaces for protons and neutrons. For light neutron rich nuclei A ∼ 40, protons and neutrons are in different harmonic oscillator shells, p f for neutrons and sd for protons [10]. For mid-mass systems, a mixed space composed by the p f shell for protons and r3 g for neutrons (48 Ca core) can be adequate. It will allow the description of heavy Ni or Zn isotopes. In this space the effect of the tensor force [11] appears clearly with the interplay between f → r3 (p) and r3 → g (n). A similar valence space may be built on a 90 Zr core, consisting of the gds-shell for the protons and r4 h for neutrons. This space is adapted to the description of the nuclei with 45 ≤ Z ≤ 50 and 50 ≤ N ≤ 82 like the ββ emitter 116 Cd. For smaller Z the 1 p 1 proton shell must be included in the space (88 Sr core).

2

2.4. No-core shell model Another approach is the ab-initio no-core shell model (NCSM) [12]. Here the valence space includes all the states in a harmonic oscillator basis until an excitation N h¯ ω. Starting from a realistic NN potential, the Lee–Suzuki method [13] is used to derive an effective interaction corresponding to the valence space. There are no free parameters, and no problems with the

230

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Fig. 3. Shell model and experimental level schemes for N = 126 isotones in the r5 i space.

center of mass since the full N h¯ ω space is taken in account. For evident technical reasons the studies have been achieved only in the lighter nuclei ( p shell nuclei). The theoretical results are in fair agreement with the experimental data with the noticeable exception of the intruder states. In 12 C and 16 O, even if we are able to deal with a valence space with N = 8, the NCSM misses the description of the first excited 0+ state. Another important result of the NCSM studies has been to show the effect of a genuine three-body force. With its introduction, the well known problem of the inversion of the states 1+ and 3+ in N = Z odd–odd nuclei has been solved for 10 B [14]. 2.5. Intruder states A SM description of the intruder states is not easy. The dimensions of the matrices explode, the center of mass excitations contaminate the calculations, etc. In spite of the difficulties, several studies have been successfully completed. • The excited states in 16 O [15] • The island of inversion around 32 Mg [16] • The intruder states in the 40 Ca region. We adopt a valence space r2 − p f which is tractable and in which the 28 Si core ensures that the most important spurious components are eliminated. In this space it has been possible

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

231

Fig. 4. Experimental spectra and SM calculations with a realistic interaction.

to describe deformed and superdeformed bands in 36 Ar and 40 Ca [17] within a spherical basis. In the same space, we have described the nuclear charge radii in the calcium isotopic chain. The parabolic shape (a mean field approach can produce only a linear evolution) and the strong odd–even staggering are qualitatively well reproduced [18]. 3. Effective interactions A natural approach would be to use realistic potentials, i.e. consistent with NN scattering data. It has been shown that these potentials produce two-body matrix elements (TBME) very similar to each other, sharing the same defects: • The spectroscopy deteriorates when the number of particles (holes) increases. An example is 131 given in Fig. 4. Starting from 101 50 Sn51 (extrapolated individual energies) we get for 50 Sn81 133 (one neutron hole) and 51 Sb82 (one proton particle with closed shell neutrons) a very bad spectroscopy. • The shell closures are not reproduced. As we show in Table 2, with all the realistic interactions, 48 Ca is not a double closed shell nucleus, and in 56 Ni the deformed state (excitation of 4 p–4h), experimentally at 5 MeV, is the ground state. • There are systematic spectroscopic defects like the inversion of 1+ and 3+ in N = Z odd–odd nuclei (10 B, 22 Na). An empirical approach is to consider the TBME as parameters and to fit them to the experimental data. For instance, the spectra of nuclei 20 ≤ N , Z ≤ 28 were long ago described with a valence space limited at the f 7 shell and TBME extracted from the spectrum of 42 Sc [19]. 2 The best known example of the empirical approach is the USD interaction, which gives a very good spectroscopy for the sd shell nuclei. The drawback is that some of the TBME are so different from those of the realistic interactions that USD cannot be considered “realistic compatible”. For instance the TBME Vi JjklT with i = 0d 5 , j = 0d 3 , k = 1s 1 , l = 0d 3 , J = 2, T = 0 has the 2 2 2 2 following values: V (Bonn C) = −1.30

V (Kuo) = −1.54

V (USD) = +0.28.

232

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Table 2 2+ energies of Ca isotopes with different G matrices Exp.

KB

KLS

Bonn A

Bonn B

Bonn C

1.16 1.35 3.83 1.03

1.45 1.45 1.80 1.41 0.468

1.43 1.42 1.60 1.35 0.381

1.31 1.26 1.23 1.27 0.214

1.25 1.22 1.30 1.10 0.345

1.26 1.23 1.41 1.17 0.437

h( f 7 )16 |ΨG S i

0.39 0.04

0.31 0.015

0.43 0.018

0.42 0.011

0.42 0.019

hn 3 i

4.5

5.2

5.7

5.2

5.0

2+ 1 excitation energy 44 Ca 46 Ca 48 Ca 50 Ca

h( f 7 )8 |ΨG S i 2

56 Ni model space ( f 56 Ni

7 2

p 3 )16 2

2.70

2

2

For this reason, following the pioneering work of Pasquini, Poves, and Zuker (see [20] and ref. therein) we will look for a minimal cure of the realistic interactions. From [21] we know that any effective interaction can be split into two parts: H = Hm (monopole) + HM (multipole). Hm contains all the terms responsible for the spherical mean field. This important property can be written as: hC S ± 1|H |C S ± 1i = hC S ± 1|Hm |C S ± 1i where C S ± 1 means closed shell plus or minus a particle. For all the realistic G-matrices Hm is wrong and has to be empirically corrected to reproduce the structure of the “simple” nuclei C S ± 1. Starting from the particle–particle representation for H : X JT J T 00 H= Vi JjklT [(ai+ a + j ) (a˜k a˜l ) ] JT

we can write H in the particle–hole representation: X λτ 00 H= Wikλτjl [(ai+ a˜k )λτ (a + j a˜l ) ] . λτ

Hm corresponds only to the terms λτ = 00 and 01 which implies that i = j and k = l and with the definition of the particle number n i = (ai+ a˜i )0 can be written as: X X Hm = n i i + n i .n j Vi j . i

i≤ j

Now we can understand what happens in 56 Ni; there is a competition between a closed shell state f 16 and a 4 p–4h state f 12r 4 ( f ≡ 0 f 7 and r ≡ 0 f 5 , 1 p 3 , 1 p 1 ). 2 2 2 2 The monopole energy of these two states reads: E C S = 16 f +

16 ∗ 15 Vf f 2

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

233

E 4 p4h = 12 f + 4r + 66V f f + 48V f r + 6Vrr and the difference 1E = 4( f − r ) + 48(V f f − V f r ) + 6(V f f − Vrr ). A small change of V f f − V f r will produce a spherical or deformed ground state for 56 Ni. It is possible to rewrite the monopole Hamiltonian separating a global term H0 (depending only on the total number of particles) from a linear term H1 and a quadratic term H2 [22,23] z(z − 1) n(n − 1) + Wππ + Wνπ nz H0 = E 0 + ν n + π z + Wνν 2 2 X Dν − n − 1 n−1 z H1 = Γi ηi + η¯ i + (qi − ηi ) Dν − 2 Dν − 2 Dπ i6=1 X Dπ − z − 1 z−1 n ηα + η¯ α + + Γα (qα − ηα ) Dπ − 2 Dπ − 2 Dν α6=1 X X X H2 = Γi j Wi j + Γαβ Wαβ + Γi Γα Wiα α6=β

i6= j

i,α

ni n¯ i n1 n¯ 1 with Γi = Di D −D = Di D −D = −Γ¯ i i 1 1 i (2) (2) n n 2n n D D Γi j = i2 j DiiD jj − i(2) − j(2) = Γ¯ i j = Γi j (n¯ i , n¯ j ) and the definitions: Di

Dj

i(α): neutron (proton) shells n i , n¯ i : number of particles, holes in the shell I X X n(n − 1) Di = 2 ji + 1 n(z) = n i(α) Dν(π) = Di(α) n (2) = 2 i(α) i(α) For closed shell states Γi = Γi j = 0. Only H0 contributes. For closed shell states with one particle (hole) Γi j = 0, H1 gives the spectrum in the example of the mass region 50 ≤ N , Z ≤ 82 with the r4 h valence space: • ηi determines the spectrum of 101 Sn • η¯ i determines the spectrum of 131 Sn • qi determines the spectrum of 133 Sb. These parameters produce a LINEAR evolution of the effective individual energies. This explains the importance of the experimental data on nuclei with closed shells ±1 particle. After the adjustment of the these parameters, only small modifications of H2 (quadratic monopoles) and of multipole terms (pairing, . . . ) can produce interactions still realistic compatible and however able to give a very satisfying spectroscopy. An illustration is given in Fig. 5. After this adjustment the obtained interaction for the r4 h space can be used reliably. 4. Lanczos method The Lanczos algorithm consists in the construction of an orthonormal basis in which H is tridiagonal. Starting from an initial vector |1i, we calculate H |1i and write H |1i = |Φi = E 11 |1i + E 12 |2i. E 11 = hΦ|H |1i E 12 and |2i are obtained by normalization of E 12 |2i = |Φi − E 11 |1i.

234

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Fig. 5. 2+ and 4+ excitations in Sn isotopes and N = 82 isotones with modified Bonn C interaction.

At the second iteration we will get H |2i = |Φi = E 21 |1i + E 22 |2i + E 23 |3i. The hermiticity of H implies E 21 = E 12 . E 22 = hΦ|H |2i. E 23 |3i = |Φi − E 12 |1i − E 22 |2i. At the third iteration: H |3i = |Φi = E 32 |2i + E 33 |3i + E 34 |4i. The hermiticity of H implies E 31 = E 13 = 0. At rank N, we will get: H |Ni = E N ,N −1 |N − 1i + E N ,N |Ni + E N ,N +1 |N + 1i. The iterative process continues until all the required eigenvalues are converged. 4.1. Lanczos strength functions Furthermore, the Lanczos method can be used to compute global properties of the spectra, even if approximate, via the calculation of strength distributions. The idea, due to Whitehead [24] is very simple. Suppose the ground state |Gi of a given nucleus has been obtained by the conventional Lanczos method. Let Ω λ denote the multipole operator we are interested in. Acting with Ω λ on |Gi we obtain the sum rule state. It is not a physical state if H does not commute with the multipole operator, but its norm is the sum rule or equivalently the total strength of the multipole operator in the ground state of the nucleus. Let us call its normalized version |Λi and

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

235

Fig. 6. Evolution of the GT strength function of 48 Ca with the number of Lanczos iterations.

develop it in energy eigenstates as: |Λi =

1 X S(E)|Ei N E

where N2 =

X

S 2 (E)

E

and S 2 (E) is the strength function. The equality hΛ|H n |Λi =

1 X 2 S (E)E n N2 E

shows that, taking |Λi as the pivot, each Lanczos iteration brings in the expectation value of two new powers of H on |Λi, therefore after N iterations we have 2N moments of the strength function S 2 (E). Diagonalizing the Lanczos matrix after N iterations, the overlaps of the eigenvectors with |Λi give an approximation to S 2 (E) which has the same lowest 2N moments. In other words, one starts with a single peak that contains all the strength located at the centroid. Iteration after iteration it goes on fragmenting until the number of iterations equals the dimension of the basis. An illustration is given in Fig. 6 with the GT strength function of 48 Ca. However, for a relatively small number of iterations we should have already a very good understanding of the global behaviour of the strength. Actually, if we smooth the spikes with gaussians having the

236

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Fig. 7. Convolution of the GT strength function with Gaussians.

experimental width, the shape of the resonance can be already obtained with a limited number of iterations as can be seen in Fig. 7. 5. Shell model codes Given a valence space, the optimal choice of the basis is related to the physics of the particular problem to be solved. Depending on the states or properties we want to describe and depending on the type of nucleus (deformed, spherical) different choices of the basis are favored. There are two main possibilities: the M-scheme, and the coupled (J or J T ) schemes. In the M-scheme the basis is composed of Slater determinants (SD) Y Ď Ď Ď |K i = ai |0i = ai1 · · · ai A |0i. i=nl jmτ

In the coupled scheme the states of n i particles in a given ji shell are defined as: |γi i = |( ji )ni vi Ji xi i, where vi is the seniority and xi any extra quantum number needed to specify uniquely the state. Next, the states of N particles distributed in several shells are obtained by successive angular momentum couplings of the one-shell basic states [25]: [[[|γ1 i|γ2 i]Γ2 |γ3 i]Γ3 · · · |γk i]Γk .

(1)

The advantages of the coupled scheme are: • The smaller dimensions of the basis: in the M scheme we have in the basis all the possible J (T ) states (only Jz and Tz are good quantum numbers). This is especially acute for J = 0 states (ground states of even–even nuclei). The dimensions for 60 Zn in the p f shell are the following: Dim(Jz = 0) = 2.3 ∗ 109

Dim(J = 0) = 3.1 ∗ 107

Dim(J = 4) = 2.1 ∗ 108 .

• The possibility to make seniority truncations (very useful for heavy spherical nuclei). • To avoid some numerical problems in the Lanczos process appearing in the M scheme when we need many Lanczos iterations. (J 2 is conserved by construction of the basis.) The advantages of the M scheme are: • The N -body matrix elements (NBME) are easy to calculate. They are the TBME in the decoupled basis (with a phase due to the permutation of the operators) (no need of c f ps or 9 j coefficients).

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

237

• The matrices are very sparse. In the case of 60 Zn: (H I J 6= 0, Jz = 0) = 2.2 ∗ 1012 (H I J 6= 0, J = 4) = 2.8 ∗ 1014 .

(H I J 6= 0, J = 0) = 7.5 ∗ 1012

This last argument is decisive and explains why the M scheme is most often used for standard SM calculations. Even with very sparse matrices and huge storage disks, we have to deal with giant matrices. It means that the number of non-zero matrix elements (NZME) is so large that it is not possible to precalculate and store them before diagonalization. They must be recalculated at each Lanczos iteration. The first breakthrough in the solution of this problem was due to Whitehead [26]. His idea is as clever as it is simple: with each Slater Determinant (SD) (M scheme) K is associated a number N (K ). With each individual state i = nl jmτ is associated a bit of this number N (K ). The bit value is 1 or 0 depending on the occupation of the state. These integer numbers N (K ) are ordered. Ď Ď The Hamiltonian is written in the decoupled basis Vi jkl ai a j ak al . At each Lanczos iteration the code works as follows: (1) (2) (3) (4) (5) (6)

Loop on K Ď Ď Loop on the operators ai a j ak al Check bit(k) = bit(l) = 1 and bit(i) = bit( j) = 0 if not continue (2) N0 = N (K ) − bit(k) − bit(l) + bit(i) + bit( j) With the bisection method identify N0 = N (J ) Calculate the phase (permutation of the operators) and get HK J = ±Vi jkl .

The shell model code ANTOINE1 [27] has kept this idea. Moreover, it takes advantage of the fact that the dimensions of the proton and neutron spaces are small compared with the full space dimension, with the obvious exception of semi-magic nuclei. For example, the 2 292 604 744 SD with M = 0 in 60 Zn are generated with only the 184 756 SD (corresponding to all the possible M values) in 50 Ca. The states of the basis are written as the product of two SDs, one for protons and one for neutrons: |I i = |i, αi. The Slater determinants i and α can be classified by their M values, M1 and M2 . The total M being fixed, the SDs of the two subspaces will be associated only if M1 + M2 = M. A pictorial example is given in Fig. 8. Before the diagonalization, all the calculations that involve only the proton or the neutron spaces separately are carried out and the results stored. The proton–proton (neutron–neutron) NZME in the full space are generated by a simple loop on the proton (neutron) states: pp

H I (i,α)J ( j,β) = h i j δ(α, β)

H Inn(i,α)J ( j,β) = h αβ δ(i, j).

For the proton–neutron interaction, we have to calculate only one-body operators. Defining Ď Ď O(k) = am an and O(γ ) = aλ aµ , the NZME will be generated with three integer additions. i, α → I

j, β → J

k, γ → K

H (I, J ) = V (K ).

1 A version of the code can be downloaded: http://sbgat194.in2p3.fr/ theory/antoine/main.html.

238

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Fig. 8. Schematic representation of the shell-model basis (M-scheme).

As long as two Lanczos vectors can be stored in the RAM memory of the computer, the calculations are straightforward. Until recently this was the fundamental limitation of the SM code ANTOINE. It is now possible to overcome it dividing the Lanczos vectors into segments: X (k) Ψf = Ψf . (2) k

The Hamiltonian matrix is also divided in blocks so that the action of the Hamiltonian during a Lanczos iteration can be expressed as: X (q) (k) Ψf = H (q,k) Ψi . (3) q

The k segments correspond to specific values of M. This technique allows the diagonalization of matrices with dimensions in the order of one billion. All the nuclei of the p f -shell can now be calculated without truncation. The coupled SM code NATHAN [27,28] keeps the fundamental idea of the code ANTOINE i.e. splitting the valence space into two parts and writing the full space basis as the product of states belonging to these two parts. Now, |ii and |αi are states with good angular momentum. The only difference with the M-scheme is that instead of having a one-to-one association (M1 + M2 = M), for a given J1 we now have all the possible J2 , Jmin ≤ J2 ≤ Jmax , with Jmin = |J0 − J1 | and Jmax = J0 + J1 . The generation of the proton–proton and neutron–neutron NZME proceeds exactly as in M-scheme. For the proton–neutron NZME the one-body operators Ď in each space can be written as O λp = (a j1 a j2 )λ . There exists a strict analogy between 1m in the M-scheme and λ in the coupled scheme. The NZME read now: hI |H |J i = hJ |H |I i = h i j · h αβ · W (K ), with h i j =

hi|O λp | ji, h αβ

=

i W (K ) ∝ V (K ) · j λ

hα|Oωλ |βi, α β λ

J J , 0

(4)

and

(5)

where V (K ) is a TBME. We need to perform – as in the M-scheme code – the three integer additions which generate I, J , and K , but, in addition, there are two floating point multiplications

239

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

to be done, since h i j and h αβ , which in the M-scheme were just phases, are now a product of c f ps and 9 j symbols (see formula 3.10 in [25]). The dimensions of the matrices being smaller, compared to the M scheme, the parallelization has been easy to do. Each processor has in its RAM the initial and a final vector. The calculation of the NZME is shared P between the different processors (each processor taking a piece of the Hamiltonian, H = k H (k) ) leading to different vectors that are added to obtain the full one: X (k) (k) Ψi . (6) Ψ f = H (k) Ψi , Ψf = k

6. ββ decay The double beta decay is the rarest nuclear weak process. It takes place between two even–even isobars, when the decay to the intermediate nucleus is energetically forbidden or hindered by the large spin difference between the parent ground state and the available states in the intermediate nuclei. It comes in two forms: The two-neutrino decay ββ2ν : A Z XN

−→ ZA+2 X N −2 + e1− + e2− + ν¯ 1 + ν¯ 2

is just a second order process in the weak interaction and has been observed in a few nuclei with 2ν ∼ 1020 years. T1/2 The second mode, the neutrinoless decay ββ0ν : A Z XN

−→ ZA+2 X N −2 + e1− + e2−

needs an extension of the standard model of electroweak interactions as it violates lepton number. It has not yet been experimentally observed. The double beta decay without emission of neutrinos would be the only way to sign the Majorana character of the neutrino and to distinguish between the different scenarios for the neutrino mass differences. The theoretical expression of the half-life of the 2ν mode can be written as: 2ν −1 ] [T1/2

=

2ν 2 G 2ν |MGT |

with

2ν MGT

=

+ σ t k0+ i X h0+f kE σ t− k1+ − i m ih1m kE m

Em + E0

.

G 2ν contains the phase space factors and the axial coupling constant g A . The SM calculation of 2ν proceeds as follows: MGT • Calculation of the initial 0i+ and final 0+f states • Generation of the doorway states σE t− |0i+ i and σE t+ |0+f i • Lanczos Strength Function on a doorway state. After N iterations we get N 1+ states in the intermediate nucleus, with excitation energies E m , then we calculate the overlap with the other doorway state and add up the N contributions with their energy denominators, until full convergence is reached. In Fig. 9 we give the contributions of the different intermediate states in the 2νββ decay of 48 Ca. Notice the interfering positive and negative contributions. The theoretical expression of the half-life of the 0ν mode can be written as: 0ν + [T1/2 (0 → 0+ )]−1 = G 0ν |M 0ν |2 hm ν i2

where hm ν i is the effective neutrino mass, G 0ν the kinematic phase factor.

240

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

Fig. 9. LSF for 48 Ca → 48 Ti 2ν decay. Table 3 ββ emitters with Q ββ > 2 MeV Transition

Q ββ (keV)

Ab.(232 Th=100)

110 Pd → 110 Cd

2013 2040 2288 2479 2533 2802 2995 3034 3350 3667 4271

12 8 6 9 34 7 9 10 3 6 0.2

76 Ge → 76 Se 124 Sn → 124 Te 136 Xe → 136 Ba 130 Te → 130 Xe 116 Cd → 116 Sn 82 Se → 82 Kr 100 Mo → 100 Ru 96 Zr → 96 Mo 150 Nd → 150 Sm 48 Ca → 48 Ti

The summation over all the intermediate states, a poor approximation in the two-neutrino mode, is here a good one, since the energy denominators are almost constant because the average energy of the virtual neutrino is typically of 100 MeV. We are then left with a “standard” nuclear structure problem X λ,K Ď Ď hΨ f kO(K ) kΨi i with O(K ) = Wi jkl [(ai a j )λ (a˜k a˜l )λ ]0 i jkl (0ν)

gV2

(0ν)

M(0ν) = MGT − 2 M F gA + + * * X X 2 g = 0+f h(σn .σm )tn− tm− 0i+ − V2 0+f htn− tm− 0i+ . n,m gA n,m Among the nuclei (see Table 3) which are the most interesting from the experimental point of view (Q ββ larger) only one, 150 Nd, is not already reachable by large scale SM calculations. The systematic theoretical studies aim to answer the fundamental – from the experimental point of view – question: what is (are) the best candidate(s) to observe 0νββ decay? The

241

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242 Table 4 0νββ matrix elements mν

(T 1 = 1025 y.)

48 Ca

0.85 0.90 0.42 0.45 1.92 0.35 0.41

2

76 Ge 82 Se 124 Sn 128 Te 130 Te 136 Xe

0ν MGT

J =0 MGT

J =2 MGT

2 v0i

v02 f

0.67 2.35 2.26 2.11 2.36 2.13 1.77

3.16 5.59 5.32 4.46 7.07 6.61 5.72

−1.96 −2.35 −2.11 −2.21 −3.08 −4.17 −2.55

0.98 0.43 0.50 0.95 0.70 0.78 0.97

0.58 0.26 0.44 0.60 0.37 0.49 0.71

2 v0i( f ) is the weight of the component of seniority 0 in the initial (final) state.

difficulty of this task is that we have to deal with different valence spaces, to tune the effective interactions, to check different spectroscopic properties of the nuclei in the region (and not only the 2νββ decay), etc. For instance before calculating the 0νββ decay of 100 Mo we must be able to describe the evolution of the molybdenum isotopes between N = 50 (spherical) and N = 60 (strongly deformed), because 100 Mo is just at the transition point. These studies have been already performed for some of the possible emitters: • • •

48 Ca

in the p f -shell. and 82 Se in the r3 g space. 124 Sn,128,130 Te and 136 Xe in the r h space. 4 76 Ge

Preliminary results are available for

116 Cd,

which needs the introduction of the 0g 9 proton

(90 Zr

2

shell core). The updated results are gathered in Table 4. The Nuclear Matrix Elements (0νNME) depend only weakly on • The effective interactions, provided they are compatible with the spectroscopy of the region. For A = 48 with KB3, GXPF1, FPD6 the MGT is respectively 0.67, 0.62 and 0.72 but drops to 0.34 with the initial G matrix Bonn C. • The nuclei, except when the seniority decomposition varies strongly between the two nuclei (one doubly magic nucleus), in that situation the J = 0 component of the MGT is strongly suppressed. For the fictitious transition 48 Ti → 48 Cr we get MGT = 1.30. 48 Ti is more similar to 48 Cr than to 48 Ca. We are carrying on systematic studies of the influence of the deformation of parent and granddaughter in the value of the 0νNME. The goal is to understand whether 150 Nd is or not a good candidate to observe the 0νββ decay. This nucleus (N = 90) is much strongly deformed than its descendant 150 Sm (N = 88). The excitation energies of the first 2+ state are respectively 130 and 334 keV, the ratio E(4+ )/E(2+ ) 2.92 and 2.21 and the B(E2)(0+ → 2+ ) 2.75e2 b2 and 1.35 e2 b2 , and we do not know yet to what extent the mismatch of deformation between the initial and final nuclei may quench the 0νNME. References [1] [2] [3] [4] [5] [6]

M. Mayer, Phys. Rev. 75 (1949) 1969. O. Axel, J.H.D. Jensen, H.E. Suess, Phys. Rev. 75 (1949) 1766. A. Banu, et al., Phys. Rev. C 72 (2005) 061305. S. Cohen, D. Kurath, Nuclear Phys. 73 (1965) 1. B.H. Wildenthal, Prog. Nucl. Phys. 11 (1984) 5. B.A. Brown, B.H. Wildenthal, Nuclear Phys. A 474 (1987) 290.

242 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242 E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, A.P. Zuker, Rev. Modern Phys. 77 (2005) 427. A. Petrovici, K.W. Schmid, A. Faessler, J.H. Hamilton, A.V. Ramayy, Prog. Part. Nucl. Phys. 43 (1999) 485. E. Caurier, M. Rejmund, H. Grawe, Phys. Rev. C 67 (2003) 054310. J. Retamosa, E. Caurier, F. Nowacki, A. Poves, Phys. Rev. C 55 (1997) 1266; Phys. Rev. C 58 (1998) 2033. T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, Y. Akaishi, Phys. Rev. Lett. 95 (2005) 232502. P. Navratil, B.R. Barrett, Phys. Rev. C 57 (1998) 3119. K. Suzuki, S.Y. Lee, Prog. Theoret. Phys. 64 (1980) 2091. P.M. Navratil, W.E. Ormand, Phys. Rev. C 68 (2003) 034305. A.P. Zuker, B. Buck, B. McGrory, Phys. Rev. Lett. 21 (1968) 39. A. Poves, J. Retamosa, Phys. Lett. B 184 (1987) 311; Nuclear Phys. A 571 (1994) 221. E. Caurier, F. Nowacki, A. Poves, Phys. Rev. Lett. 95 (2005) 042502. E. Caurier, K. Langanke, G. Martinez-Pinedo, F. Nowacki, P. Vogel, Phys. Lett. B 522 (2001) 240. J.B. Mc Cullen, B.F. Bayman, L. Zamick, Phys. Rev. 134 (1964) 515. A. Poves, A.P. Zuker, Phys. Rep. 70 (1981) 235. M. Dufour, A.P. Zuker, Phys. Rev. C 54 (1996) 1641. A. Abzouzi, E. Caurier, A.P. Zuker, Phys. Rev. Lett. 66 (1991) 1134. A.P. Zuker, Nuclear Phys. A 576 (1994) 65. R.R. Whitehead, in: B.J. Dalton, et al. (Eds.), Moment Methods in Many Fermion Systems, Plenum, New York, 1980. J.B. French, et al., Adv. Nucl. Phys. 3 (1969) 193. R.R. Whitehead, et al., Adv. Nucl. Phys. 9 (1977) 123. E. Caurier, F. Nowacki, Acta Phys. Polon. B30 (1999) 705. E. Caurier, G. Martinez-Pinedo, F. Nowacki, J. Retamosa, A. Poves, A.P. Zuker, Phys. Rev. C 59 (1999) 2033.