Revista Mexicana de Física 39, Suplemento 2 (1999) 29-96 Double beta decay in heavy deformed nuclei* O. CASTAÑOS Instituto de Ciencias Nucleares Universidad Nacional A utónoma de México Apartado postal 70-543, 04510 México, D.F., México J.G. HIRSCH Departamento de Física Centro de Investigación y Estudios Avanzados del IPN Apartado postal 14-740, 07000 México, D.F., México P.O. HESSt Institut ¡ür Theoretische Physik Justus-Liebig- Universitiit Heinrich-Buff-Ring 16, D-6300 Ciessen, Cermany ABSTRACT. \Ve estimate the double beta half-life of several heavy deforrned nuclei in the two neutrino mode using the pseudo SU(3)shellmodel scheme,whichis quite successfulin describing rotationa1 nuclei. In this first approach by using the closure approximation and neglecting the pairing interaction, we found forbidden decays for five potential double beta ernitters but finite values of the half life for other six nuclei. These forbidden decays represent a possible test of our model and the other predictions are in good agreement with tite scarce available experimental data. Details of the procedure are given for the evaluation of the half life of 23'U. RESU~tEN. Estimarnos la vida media del decaimiento beta doble de varios núcleos pesados y deformadosenelmododedosneutrinosusandoelesquemapseudoSU(3)delmodelodecapas,que tiene hastante éxito en describir los núcleos rotacionales. En este trabajo encontrarnos decaimientos prohibidos para cinco posibles emisores beta doble y valores finitos de las vidas medias de otros seis núcleos, usando la aproximación de cerradura y no tomando en cuenta la interacción de apareamiento. Estos decaimientos prohibidos representan una posible prueba de nuestro modelo. Las otras predicciones estan en buen acuerdo con los escasos datos experimentales disponibles. Damos los detalles del procedimiento para la evaluación de la vida media de 238U. PACS:23.40.Hc;21.60.Fw;27.90.+b A double beta (f3f3) half-Jife of 2.0 :l:0.6 X 10 21 Y for 238U was recently reported [1 J, preliminary results were given for 150Nd[2},and an active search is in progress for 244pu [3). Taking 238Uas a representative case, ten years ago Haxton et al. [41used a Nilsson plus pairing model and they got a result one order of magnitude smaller than the experimental -Work supported in part by CONACYT 1570-E9208,UNAM-DGAPAIN-103091and DFG. tpennancnt Address: Instituto de Ciencia.s Nucleares, UNAM.
Transcript
Revista Mexicana de Física 39, Suplemento 2 (1999) 29-96
Double beta decay in heavy deformed nuclei*
O. CASTAÑOSInstituto de Ciencias Nucleares
Universidad Nacional A utónoma de MéxicoApartado postal 70-543, 04510 México, D.F., México
J.G. HIRSCHDepartamento de Física
Centro de Investigación y Estudios Avanzados del IPNApartado postal 14-740, 07000 México, D.F., México
ABSTRACT. \Ve estimate the double beta half-life of several heavy deforrned nuclei in the twoneutrino mode using the pseudo SU(3) shell model scheme, which is quite successful in describingrotationa1 nuclei. In this first approach by using the closure approximation and neglecting thepairing interaction, we found forbidden decays for five potential double beta ernitters but finitevalues of the half life for other six nuclei. These forbidden decays represent a possible test of ourmodel and the other predictions are in good agreement with tite scarce available experimentaldata. Details of the procedure are given for the evaluation of the half life of 23'U.
RESU~tEN. Estimarnos la vida media del decaimiento beta doble de varios núcleos pesados ydeformados en el modo de dos neutrinos usando el esquema pseudo SU(3) del modelo de capas, quetiene hastante éxito en describir los núcleos rotacionales. En este trabajo encontrarnos decaimientosprohibidos para cinco posibles emisores beta doble y valores finitos de las vidas medias de otrosseis núcleos, usando la aproximación de cerradura y no tomando en cuenta la interacción deapareamiento. Estos decaimientos prohibidos representan una posible prueba de nuestro modelo.Las otras predicciones estan en buen acuerdo con los escasos datos experimentales disponibles.Damos los detalles del procedimiento para la evaluación de la vida media de 238U.
PACS: 23.40.Hc; 21.60.Fw; 27.90.+b
A double beta (f3f3) half-Jife of 2.0 :l: 0.6 X 1021 Y for 238U was recently reported [1J,preliminary results were given for 150Nd[2},and an active search is in progress for 244pu [3).Taking 238U as a representative case, ten years ago Haxton et al. [41used a Nilsson pluspairing model and they got a result one order of magnitude smaller than the experimental
-Work supported in part by CONACYT 1570-E9208,UNAM-DGAPA IN-103091and DFG.tpennancnt Address: Instituto de Ciencia.s Nucleares, UNAM.
30 O. CASTAÑOS ET AL.
va1ue, which was measured following the procedure indicated in that work. Later Vogelused the QRPA with a schematic spin-isospin interaction and obtained a similar result [5],while the latest Klapdor's results with schematic and realistic interactions [6] show widerdispersion.Thus we were looking for a theoretical framework which can describe rotational heavy
nuelei and incorporate therefore the strong neutron proton residual interaction. This isprovided by the pseudo 5U(3) shell model and its pseudo symplectic extensions and wewant to propose them as an alternative to deal with the double beta decay [7J.The Nilsson Hamiltonian yields, for heavier nuelei, the formation of new shells arranged
by the orbitals j = {1/2, 3/2, ... N - 1/2} called as normal parity plus a single partielestate with j = N + 3/2 of a different parity, where N denotes the harmonic oscillatorshell. This result implies that the normal oscillator symmetry is badly broken by thespin-orbit interaction, however the normal parity orbitals can be identified with orbitalsof an harmonic oscillator of one quanta less Ñ = N - 1 [8]. This set of orbitals, withi. = j .=!+ S, pseudo spin s = 1/2, and pseudo angular momentum taking the values1 = N, N - 2, ... , 1 or O define the so called pseudo space and only recently it hasbeen found an analytic expression for the transformation that take us from the normalparity orbitals to the pseudo space [9J. Applying this transformation to the sphericalNilsson Hamiltonian it can be shown explicitly that the strength of the pseudo spin orbitinteraction is almost zero and the orbitals j = i::1:~ are nearly degenerate doublets [9J.We are going to describe, through an example, how the pseudo 5U(3) model can be
used. We consider 238U,which has 10 protons and 20 neutrons in the 82-126 and 126-184shells, repectively. These partieles are distributed between the orbitals in the normal andabnormal spa.ces, i.e.
(Sl/2; d3/2d5/2; 97/299/2)K; (iI3/2)K,
(jí¡/2P3/2; i5/2i7/2; i'9/2hl1/2)V; (j15/2)v.
From all the possible choices for the number of nucleons that satisfy n~ + 1l~= 10, andn{Y + n~ = 20, only the most probable values are considered. These are determined fromthe appropriate Nilsson diagrams by selecting a reasonable deformation, f3 - 0.25, andfilling each level with a pair of partieles in order of increasing energy, obtaining
n': = 6, n; = 4, n~= 12, (1)
The rnany particle states of no nucleons in a given shell TIa, with the subindex a denotingprotons (,,) or neutrons (v), can be defined by the totally antisymrnetric irreduciblerepresentations {ln~'} and {ln~}; with 710 = n~+ n~, of unitary groups of dimensionsn[t = (Tia + 1)(Tia + 2) and n~= 2T10 + 4, respectively. A complete elassification of thestates can be defined by the following chains of groups:
w::'} {jo}U(n~) J U(n[t /2)
{jo} 'Yo (>'0,1'0)x U(2) J 5U(3)
jNo (2a)
x 5U(2) J 50(3) x 5U(2) J 5UJ(2),
DOUBLE BETA DECAY IN HEAVY DEFORMED NUCLEI 31
Va f3a J:~ USp(n:) ~ SU(2)'
(2b)
where aboye each group the quantum numbers that characterize its irreducible represen-tations (irreps) are given and the 'Ya, f3a and Ka are multiplicity labels of the indicatedreductions. For this example the dimensions of the unitary groups are given by: n: = 14,n~ = 30, n~ = 16, and n~ = 42 and its corresponding irreps are defined by (1). Forthe normal parity spaces (2a) the pseudo LS coupling scheme is used and the followingrelations between its quantum numbers are satisfied
while for the abnormal parity spaces (2b) the seniority configurations Va are appropriate.For even-even heavy nuclei, it has been shown that if the residual neutron-proton
interaction is of the quadrupole type, independently of the interaction in the pro tonand neutron spaces, the yrast states below the backbending region are found under thefollowing assumptions [101:i) The most important normal parity configurations are thosewith highest spatial symmetry and ii) Only seniority zero configurations, v. = Vv = O,are taken into account. Then, in our example according the expressions (1) and (2) wehave that
(3a, b)
which imply that the yrast states have pseudo spin S. = Sv = O.Next, we have to determine the pseudo SU(3) irreps, (Aa, /La), which are contained in
{ia}, this can be done through the Littlewood procedure or by considering,a method whichexploits the simplicity in decomposing Young diagrams in terms of Kronecker productsof their completely symmetric or antisymmetric components, and is adequate to realizenumerical calculations (11). Then we get
Prom the sets (4) we consider only the irreps with maximum eigenvalue of the Casimiroperator, (C2)a, that is (18,0). and (36,0)v. To get the complete wave function we usethe strong coupled lirnit that is SU.(3) x SUv(3) ~ SU.(3) [101,and from the Kroneckerproduct
(18,0). x (36,0)v = {(54,0), (50,2), (46,4), ... },
we chose the SU.(3) irrep with rnaximum eigenvalue of the Casimir operator, C2, and sothe irrep (54, O) will dominate the low-Iying energy structure because we are consideringa quadrupole-quadrupole interaction QC . QC, which in a given shell can be replaced by
32 O. CASTAÑOS ET AL.
4C2 - 3L2 anú QC = Q~ + Q~. Then the groundstate band of 238U is characterized bythe following product of states
where the first term indicates the normal part and the second the abnormal one.An extension of the pseudo SU(3) coupling scheme has been proposed recently [12] in
which 2hw multiple monopole, Dto and quadrupole Dtm excitations are included acting ontop of the normal parity part of (5). Using this approach the excitation energies and DE2values for 238Uwere calculated by essentially considering a QC . QC interaction. As it wasshown in [12] the agreement between the theoretical calculations and the experimentaldata is very good.We now come to the explicit calculation of the half life for {3{3-decay:In the standard
approximation, the inverse half life of the two neutrino mode of the {3{3-decay,({3{3hv,can be given as a product of a integrated kinematical factor G2v times the square of theabsolute value of the nuclear matrix element, M2v, that is
(6)
The M2v depends strongly on the conside~ed nuclear model and can be written in theclosure approximation as follows [7,13]
Mcio, = 2.(o+lr. rlo:t')2v E f I(7)
where E is the average excitation energy of the intermediate virtual 1+ states and thelabels i and f refer to the ini tial and final states. The r is the Gamow-Teller operatorwhich is defined by the expression
m=1,0,-1. (8)
We want to evaluate the nuclear matrix element (7) for the {3{3decay of ~~8Uto ~~8pu.The IOn state is given in (5), with Ks = 1, and Ls = Ms = O,and to determine lO/) wefollow the procedure indicated previously. For this case the most probable occupancies ofthe valence neutrons are the following two choices: i) 1l~ = 8, 1l~ = 10, and ii) 1l~ = 6,1l~ = 12; which together with 1l~ = 6 and 1l~ = 6 yields two possible ground stateconfigurations:
(9a)
(9b)
DOUBLE BETA DECAY 1:-1HEAVY DEFORME!) NUCLEI 33
From these two possible states, the 10th state was seleded by using the following eriteria:eheeking the last filled orbitallooking at the ground state spin of the neighboring even-oddnucleus, ~Fpu, whieh has to be eonsistent with the spin of the last filled Nilsson level. Ina near future we plan to propose for the ground-state of 238pu a linear eombination ofthe states (9) and find the appropriate mixing by adding to the hamiltonian the pairinginteraction.We express now the one body Gamow-Teller operator in the seeond quantization for-
malism
rm = ¿(-l)/+H' )2(2j' + 1) W(Bj'j; 11)Tlljj'
(10)
where the matrix element of the spin operator 17m was evaluated and ("It-Iv) = 1. Wemake now the transformation to the pseudo space [141 and keep all the possible eontribu-tions, tbus we have formally the expression
r - rNN + rN,! + rA!>! + rA,t.m - 0'1 m 0'2 m 0'3 m 0'4 m , 7n=1,O,-I,
where the superindiees N N, N A, ... are indieating the space of tbe fermion ereation andannihilation operators, respedi vely.Afterwards we eonstruet r2 and tbe only part whieb give a non-zero eontribution is
indieated here
(lla)
witb 1/v = 6 and iív = 5. The tensor operators T. and Tv are defined by tbe expressions
(llb)
(lle)
where tbe a t denotes a ereation proton operator in tbe j-j eouplillg soheme with j = 1/v+ ~
and ii an annihilatioll neutron operator in the pseudo L-S sebeme with i= iív. Thuseffectively r2 creates a pair of protons coupled tú total angular rHornentum zero in theorbit (;13/2)' and annihilates two neutrons of the normal parity space coupled to pseudoorbital angular momentum and pseudo spin equal to zero.
where we substitute the expression (Ua) for r2, uncoupled the 5U,(3) states in its neutronand proton parts, and use orthonormality properties of the different subspaces.The evaluation of the remaining reduced matrix elements of the last expression can
be obtained by using 5U(3) Racah ca!Culus in the normal parity part and the quasispinformalism [15] for the abnormal parity sector. The results are given by
5ubstituting (13) into the expression (12) ami evaluating the triple bar reduced matrixelement by using the Wiguer Eekart theorern and explicit fermionie expressions for theneutron normal parity part of the wave fundions we get [7,16]
(14)
The value for the integrated kinematical factor, G2v = 8.78 X 10-20 MeV2/y wasobtained following the procedure indicated by Doi el al. [13], with g,1/ gv = 1.0. For Ewe have used two extreme limits E = ECT = 1.12Al/2 [17) and Erro;" = ~EGT, which areindicated in the fourth column of Table 1. For these values we ealculated the ,I3,13-halflifeof 238Uand the results are given in the fifth column of Table 1.Following the method indicated in this work we also estimate the ,I3,13-halflife of several
heavy deformed uuclei which are possible ,13,13emitters. Its correspondiug results are sum-rnarized in Table 1. Besides of the transitions indicated in Table 1, we predict, according
DOUBLEBETADECAYIN HEAVYDEFOR~tEDNUCLEI 35
TABLE I. Theoretical estimates for the half-life fJfJ-deeay in the 21/mode for several hcavy de-formed nuclei are given and comparcd with the available experimental data.
146N'd -t 146Sm -0.97 5.6 x 10-31 7(14) 9.3 (37.2) x 101114'Nd ~ 1'.Sm 1.31 1.28 x lO-lo 7(14) 2.2 (8.9)150Nd -t 150Sm 1.31 1.44 x 10-17 7(14) .02 (.08)1'.W ~ ".Os 0.258 1.53 x 10-" 7.5(15) 5.5 (22) x 1031920s ~ 10'pt 0.38 5.92 x 10-" 7.5(15) 6.6 (26) x 103
"'U ~ ".Pu 1.51 8.71 x 10-'. 9(18) 4 (16) 20 :l:6
to the assumptions of the model, that the double beta transitions:
154Sm --l 154Cd
160Cd --l 160Dy
176Yb --l 176Hf
232Th --l 232U
244pu --l 244Cm
are forbidden. It must be emphasized that the inelusion of pairing will not allow the /3/3decay in these nuelei. Only if pairing dominates over the Q . Q interaetion, mixing twoneighbour shells of protons or neutrons, these decays will become allowed. But then, thedescription of them as heavy deformed nuelei will collapse. By the other hand, the matrixelements of the six nuelei given in the Table 1 could be reduced by the pairing mixing ofsorne SU(3) irreps, and therefore giving longer (never shorter) /3/3 half lives.In summary, we think that the pseudo SU(3) formalism is a very promisory scheme
for evaluating /3/3 half lives of heavy deformed nuelei. Its simplest version, using theelosure approximation and neglecting pairing, predicts forbidden decay for five potential/3/3 emiters, and finite values, consistent with the present experimental evidence, for othersix.We aeknowledge encouraging discussions with S. Pittel and P. Voge!.
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