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AN ESTIMATION OF THE RATES OF
(TWO-NEUTRINO) DOUBLE BETA DECAY AND RELATED PROCESSES
J. Abad, A. Morales, R. Núñez-Lagos, A. Pacheco
To cite this version:
J. Abad, A. Morales, R. Núñez-Lagos, A. Pacheco. AN ESTIMATION OF THE RATES OF (TWO-
NEUTRINO) DOUBLE BETA DECAY AND RELATED PROCESSES. Journal de Physique Collo-
ques, 1984, 45 (C3), pp.C3-147-C3-150. �10.1051/jphyscol:1984328�. �jpa-00224042�
AN ESTIMATION OF THE RATES OF (TWO-NEUTRINO) DOUBLE BETA DECAY AND RELATED PROCESSES
J.
Abad, A. Morales, R. ~ d f ; e z - ~ a ~ o s and A.F. Pachecom s t i t u t o de ~ & c a Nuclear y Altas Energiias, Universidad d e Zaragoza, Zaragoza, Spain
Rdsumd
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La d e s i n t d g r a t i o n double-bdta, l a c a p t u r e d l e c t r o n i q u e avec dmission de p o s i t r o n s e t l a c a p t u r e d l e c t r o n i q u e double ( t o u ' o u r s avec Bmission de d e w neu- t r i n o s ) o n t dtB c a l c u l d e s en supposant que l ' d t a t 1' l e p l u s bas du noyau intermd- d i a i r e e s t dominant. En u t i l i s a n t l e s v a l e u r s expdrimentales pour l e s Bldments de m a t r i c e de d g s i n t d g r a t i o n b Q t a simple e t l e s ddnominateurs d ' d n e r g i e , on o b t i e n t d e s r B s u l t a t s q u i ne ddpendent pas des c a l c u l s de physique n u c l s a i r e . Les r B s u l t a t s pour l a d s s i n t d g r a t i o n double-b6ta sont en accord avec l e s donndes gdochimiques rBcentes.D ' a u t r e s p r d d i c t i o n s s o n t 2 l a p o r t d e d e s p o s s i b i l i t G s a c t u e l l e s de d B t e c t i o n . A b s t r a c t
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The (two-neutrino) double b e t a decay, t h e e l e c t r o n c a p t u r e w i t h p o s i t r o n emission and t h e d o u b l e e l e c t r o n c a p t u r e decay r a t e s a r e e s t i m a t e d f o r se- v e r a l n u c l i d e s by assuming t h e dominance of t h e lowest 1+ s t a t e of t h e i n t e r m e d i a t e i s o b a r . By u s i n g single-8 decay m a t r i x elements and energy denominators a s e x p e r i - mental imputs we g e t v a l u e s independently of n u c l e a r physics c a l c u l a t i o n s . The re- s u l t s f o r double 8- decay a r e i n agreement w i t h t h e r e c e n t geochemical v a l u e s . Other p r e d i c t i o n s a r e w i t h i n t h e p r e s e n t experimental s e n s i t i v i t y .The r e c e n t y e a r s have w i t n e s s e d a g r e a t d e a l of work /1/ i n t h e f i e l d of double be- t a decay because of i t s relevance f o r t h e q u e s t i o n of the,(Plajorana) n e u t r i n o mass which might b e i n f e r r e d i f t h e l e p t o n number v i o l a t i n g n e u t r i n o l e s s double b e t a de- cay would t a k e p l a c e . From t h e t h e o r e t i c a l p o i n t of view t h e r e e x i s t s alarming d i s - c r e p a n c i e s /2/ between t h e conventional (28)-decay t h e o r y and t h e geochemical measu- rements / 3 , 4 , 5 / . Although i t i s commonly accepted t h a t t h e u n c e r t a i n t i e s i n t h e computations a r i s e mainly from t h e n u c l e a r m a t r i x elements i t i s t r o u b l i n g t o n o t e t h a t r e c e n t improved c a l c u l a t i o n s w i t h t h e s t a t e - o f - t h e - a r t s h e l l model techniques have exacerbated t h e s e d i s c r e p a n c i e s / 2 , 6 / . Moreover, t h e proper treatment / 2 , 7 / of t h e Coulomb c o r r e c t i o n amounts a l s o t o an i n c r e a s e of t h e a b s o l u t e r a t e s , i . e . , e a c h new refinement seems t o go i n t h e wrong d i r e c t i o n . I n t h e d i r e c t experiments t h e r e e x i s t s , s o f a r , one " p o s i t i v e " r e s u l t 181, i n t h e 8 2 ~ e c a s e , which i s i n c o n t r a d i c - t i o n with t h e corresponding geochemical r e s u l t 161. On t h e o t h e r hand, t h e r e a r e two geochemical experiments /3,4/ measuring t h e famous Tellurium r a t i o which give incompatible r e s u l t s of f a r r e a c h i n g consequences / 2 , 4 , 7 / .
Should t h e geochemical r e s u l t s / 4 , 5 / b e c o r r e c t one i s faced t o t h e uneasy s i t u a t i o n of having t h e o r e t i c a l a b s o l u t e r a t e s c o n s i d e r a b l y l a r g e r t h a n t h e experimental ones and, which i s worse, a l l t h e t h e o r e t i c a l improvements make t h e r a t e s even l a r g e r . One might g i v e up some of t h e approximations u s u a l l y made i n computing t h e r a t e s , l i k e f o r i n s t a n c e , t h e c l o s u r e approximation employed when performing t h e sum over t h e i n t e r m e d i a t e n u c l e a r s t a t e s / 6 , 9 / o r t o look f o r a l t e r n a t i v e mechanisms i n o r d e r t o reduce t h e t h e o r e t i c a l r a t e s
/ l o / .
I n Ref 6 , devoted t o t h e Te i s o t o p e s , t h e sum over t h e i n t e r m e d i a t e 1+ s t a t e s i s performed f o r e x c i t a t i o n e n e r g i e s up t o about 20 MeV, i n s e r t i n gLheahedXcdty
campcLted s i n g l e 8- and f.?+ decay amplitudes from t h e 0+ground s t a t e s t o t h e v a r i o u s I+ i n t e r m e d i a t e s t a t e s , i n s t e a d of using t h e (28)-decay amplitudes c o n v e n t i o n a l l y surmned up by c l o s u r e . With t h e n u c l e a r s t a t e s d e s c r i b e d i n a BCS p a i r i n g model they o b t a i n even h i g h e r r a t e s t h a n t h o s e p r e v i o u s l y computed
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984328
C3-148 JOURNAL DE PHYSIQUE
/2/. In Ref. 9 the sum over the intermediate states is saturated by keeping only
%he lowest-lying 1+ state, inserting the
expenimentae
matrix elements of the single 8- and 6+ decay amplitudes from the 1+ to the 0+ ground states. This very naive and drastic a roach, as noted also in Ref. 6, already yields the observed 26-decay rates in the E58Te, 1 3 0 ~ 8 and 82Se cases, as given by the geochemical data. These results may cast some doubts about the reliability of the geochemical measurements or the closure approximation, or both.In the present paper we briefly review our results /9/ and extend them to other two- neutrino processes related to the double beta decay, i.e. double positron emission, e- -+ e+ conversion and double electron capture Ill/.
In the conventional (26)* decay theory the (two-neutrino) decay rate in O+ + 0+
transitions is given by
~(68)
;,
= (96a7)- 1 (GFcos8) 4~4rn9 lMGT1
2 ~ $ (years)-1A=1.25 and S=<E,>-EI is the difference between the energy of the initial state and some averaged energy of the intermediate states. The Gamow-Teller matrix element is given as usual by
GT
It is straightforward to check that if one assumes the dominance of the 1' lowest- lying intermedia5e state,+one woukd obta&n, instead of
MGT
and S, respectively the quantities <OF1 1
u11
l><lll o1 1
01>= M and !?=E~+-E~~ which can be taken from the expe- rimental data. In fact, in the single-5 Gamow-Teller transitions l+ + 0+ one haswhere the indexes 1 and 2 refer to the two branches of decay from the intermediate state 1+ to the initial and final O+ ground states. The resulting half-life is
T~/~(BB)~, = (4.26~10'~)/(
]GI2
)'1 yearsThe phase space integrals I* have been computed numerically, with the Coulomb correc tion factors
F*
as suggested by Haxton et a1 121. Table 1 shows the half-life pre-- ---
'I,
Nuclide (log ft)l (log ft)2 s/m
laGTI2
I TlI2 (years)TABLE 1
Nuclide 64„ 3QZn 78Kr 36Kr 48Cd 48Cd 50Sn 120Te 52 130,, 56Ba 132B 56Ba 136Ce 58Le 138 Ce 58Ce 162Fr 68Er
log(ft)1 5.0 4.8 4.9 4.8 4.7 4.5 5.1
-
4.6-
5.0 log(ft)2 5.3 5.5 4.4 4.4 4.1 5.1 5.4-
5.5-
5.6 IMI
2 1 GT1 7.22xl0~3 7.22xl0~3 7.22xl0~3 9.08xl0-2 2.28x10"l 3.62xl0"2 4.55xl0"3 4.55xl0-3 1.14xl0-2 1.14xl0~2 3.61xl0~3s/m 2.13 2.35 1.39 4.23 2.29 2.92 1.86 3.50 1.92 3.04 1.56
T /m 0
-
1.64 1.44- - -
1.04-
0.70- -
(eV)
2vh -
0.835xl0~3 0.206xl0~3- - -
0.358xl0~5-
0.39xl0-7- -
T1/2(yr)
-
27 7.07x10 27 2.86x10- - -
30 2.61x10-
9.55xl031- -
T /m 0 0.13 3.61 3.39
-
1.70 1.26 2.97-
2.62-
1.50 (K6+)2vh
0.70xl0~9 12.0 16.0-
0.59x10" -1 0.40xl0-2 3.7-
1.4-
0.44xl0_1T1/2(yr) 6.4xl034 2.2xl024 22 6.8xlOzz
-
24 5.2xlOZ4 4.4xl026 3.0xl024-
2.8xl024-
1.7xl026T /m 0 2.11 5.58 5.34 0.43 3.64 3.19 4.90 1.48 4.54 1.23 3.39 (KK)2v
h
0.315 12.0 21.0 4.4xl0~5 2.42 0.93 10.35 2.2xl0~2 7.4 1.3xl0-2 3.3 x1/2(yr) 7.3xl026 6.4xl024 22 6.5x10 2.5xl028 1.4xl023 1.8xl024 23 8.3x10 3.9xl026 23 3.7x10 2.1xl026 l.lxlO24C3-150 JOURNAL DE PHYSIQUE
d i c t i o n s f o r (24)5v decay i n some n u c l i d e s where t h e 1+ intermediate-state-dominan- c e - h y p o t h e s i s i s a p p l i c a b l e . As i t was s a i d above t h e r e s u l t s a r e i n f a i r agreement w i t h t h e geochemical d a t a f o r t h e 1 2 8 ~ e , 130Te and 8 2 ~ e (28)- decay c a s e s .
I n much t h e same way t h e I+ dominance approach c a n b e a p p l i e d t o t h e (two n e u t r i n o ) e-+e+ c o n v e r s i o n p r o c e s s (Kf3)2v whose r a t e i s g i v e n c o n v e n t i o n a l l y 1121, by
1
r ( ~ B ) 2, = (24a5)- ( ~ c o s e ~ ) * A 4
I
Y (0)1 1 ~
2 m 6 ~ 2 (years)-'~ ~ 1
where To+m- Ee+
K ~ + L ~ + K L )
I
pe+ 14+dEe+lo
E:~E:~ dEvl( 2 2 2 2-
1K,= (s+B,-~+E v 1 ) - ' + ( S + E ~ + + E ~ ~ )
,
L ~ = ( s+B~-+E v2 ) - ' + ( S + E ~ + + E ~ l )-
1BK i s t h e b i n d i n g e n e r g y of t h e K e l e c t r o n and S h a s t h e same meaning a s above. By i n s e r t i n g t h e e x p e r i m e n t a l
3
and3
i n s t e a d o f MGT and S , one g e t s t h e e-+e+ h a l f - l i - f e from t h e e x p r e s s i o nT ~ , ~ ( K B + ) Z v = ( 3 . 5 3 ~ 1 0 2 1 ) /(a3Z3
[ % I
y e a r swhere t h e a t o m i c wave f u n c t i o n h a s b e e n approximated, a s u s u a l , by
1
Y ( o ) ~ ~ = ( ~ z ) ~ ~ ~ I T - ~ As f a r a s t h e (two-neutrino) d o u b l e e l e c t r o n c a p t u r e (KK)zV i s concerned, t h e ,&+ do-minance h y p o t h e s i s amounts, a s i n p r e v i o u s c a s e s , t o r e p l a c e t h e e x p e r i m e n t a l M and
3
i n s t e a d of and S i n t h e c o n v e n t i o n a l r a t e /13/r
(KK) 2 v = ( ~ T ~ ) - ~ ( G C O ~ ~ ~ ) 4 1 4 m 3I
Y (0)1 *
IllGT 121) (years.)-' where0 d ~ v l ~ : l ~ ~ 2 ( ~ ~ + ~ ? j + ~ 3 ~ 3 ) / 3
,
K3=(S-ni+BK+Evl)-1,
L ~ = ( s - ~ + B ~ + E ~ ~ ) - ~ The h a l f - l i f e is g i v e n byT1/2 ( ~ ~ ) ~ ~ = ( 1 . 8 2 x 1 0 ~ ~ ) / ( a ~ ~ ~ 12121~) y e a r s
The p a r a m e t e r s e n t e r i n g t h e above formulae and t h e p r e d i c t e d h a l f - l i v e s f o r proces- s e s ( ~ ~ ) $ ~ , ( ~ $ 9 2 , and (KK)2\, a r e g i v e n i n t a b l e 2.
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