Alpha decay and the structure of the Kπ = 0+ states in the Th-U region

I

5

Nuclear Physics A130 (1969) 31 --.40; (~) North-tlolland Publishin9 Co., Amsterdam

Not to be reproduced by photoprintor microfilmwithout written permissionfi'omthe publisher

ALPHA DECAY AND THE STRUCTURE OF THE K "~ = 0 + S T A T E S I N T H E Th-U R E G I O N M. I. CRISTU Institute for Atomic Physics, Bucharest, Romania O. DUMITRESCU t, Chemistry l-'aculty, Bucharest University, Bucharest, Romania N. I. PYATOV Joint Institute Jbr Nuclear Research, Dubna, USSR and A. SANDULESCU Max-Planck-Institut fiir Chemic, Mainz, Bandesrepublik Deutschland Received 12 August 1968 Abstract: The influence of the long-range effective forces on alpha decay to first K~ - 0 + states of boundary nuclei 232U, 23°Th and 228Th is analysed. It is shown that only the model with pairing, quadrupole and spin-quadrupole effective forces can describe both the energies and the alpha hindrance factors for//-states (K n -- 0 a ) in this region. From the experimental data, the strength parameters for quadrupole and spin-quadrupole forces are determined.

1. Introduction The problem of the effective interactions in deformed heavy nuclei is a very complicated problem. One can solve this problem by l o o k i n g at the experimental data and guessing the principal c o m p o n e n t s of the effective forces which reproduce the experimental data. In this paper, we try to obtain some i n f o r m a t i o n c o n c e r n i n g these comp o n e n t s from the alpha intensities to the first K ~ = 0 + excited states, i.e. from a l p h a decay t o / ~ - v i b r a t i o n a l states. It is k n o w n that the pairing and q u a d r u p o l e effective force model 1) is very successful in explaining the general trends of the experimental data. O n the other hand, this model encounters significant difficulties in the description of //-vibrational states. F o r instance, such a model does not fit well the experimental//-energies a n d ~.-hindrance factors for/J-states in the T h - U region. Moreover, this model failed to explain the appearance of the second K ~" = 0 + excited state below the energy gap. Such states were found in a n u m b e r of deformed nuclei 2-6). Thus, there is a d e m a n d for the extension of the model of interaction. Recently, B~s a n d Broglia 7) developed the theory of pairing vibrations a s s u m i n g that the first K ~ = 0 + excited states have a pure pairing structure. In a previous p a p e r [ref. 8)], we have shown that the alpha intensities are strongly enhanced by the quat Permanent address: Institute for Atomic Physics, Bucharest, P.O. Box 35, Romania. 31

M. I. C R I S T U et al.

32

drupole forces and that for the nuclei in the middle of the transuranium region, only the pairing and quadrupole forces model can describe the alpha decay to fl-vibrational states. The collective states are described as a superposition of two-quasiparticle states. As a rule, the quasiparticle pairs occupying the levels near the Fermi sea have the largest amplitudes in the 0 + state wave function 9). The experimental data for [ref. 10)] and fl [rcf. 11)] transitions point out that these amplitudes are smaller than those predicted by the pairing and quadrupole effective forces model. Recently, Pyatov has shown 12-, 3) that a signiiicant decrease of the diagonal amplitudes may be due to the spin-quadrupoIe effective forces. Also the spin-quadrupole force may generate a new K ~ = 0 + state below the energy gap. This force was studied first by Kisslinger a4) in single-closed-shell spherical nuclei. In the present paper, the pairing, quadrupole and spin-quadrupole effective forces model was used for the description of the fl-vibrational states in the Th-U region. It is shown that this model is able to explain the energies and the ~.-hindrance factors for these states. We concluded that the fl-vibrational states in this region have a pairing, quadrupole and spin-quadrupole structure i.e. a complex structure.

2. Hindrance factor

The ratio of the actual half-life to the one calculated from the Geiger-Nuttal empirical law with the actual alpha energy for the ground-ground transition of the nearest doubly even nucleus is called the hindrance factor 10). For a doubly even nucleus, the hindrance factor is just the ratio of the reduced width of the ground-ground transition to that of the ground-excited state transition (HF)trrr,f = [ Z

k°z(B)?'L( 00+, 0 0 + ) ] 2 [ Z k~.(B')TL(O0+, lfKfnf)] -2,

L

(1)

L

where 7i.(Ii K i hi, If Kf gr) is the amplitude of the reduced width for the transition from the state Ii, Ki, n I to the state It, Kf, af with the :~-momcntum transfer L, I being the total spin of the state, K its projection on the nuclear Oz-axis and n the parity. The kT'L(B) are the eJements of the Fr6man 1s) matrix accounting for the anisotropic part of the barrier which scatters L-waves near the nuclear surface into outgoing channels outside the barrier. Here B' is equal ~ 0.9 for transuranium elements ,6). To compare the alpha intensities to the states of the rotational band of an intrinsic K ~ = U state and the alpha intensity to that intrinsic K ~ state, we use the reduced wave amplitude defined by the relation

b,f = [ ~_, kg{z(B')yL( 00+ , I f K f n 0 ] [ Z krfL(B rf ,)~'L(O0+ , Kf Kf ~f)]-a L

L

=

For fl-states Kf = 0, ;re = + and If = 0, 2, 4 . . . . .

(2)

~-DECAY

33

In the hypothesis that the superfluid properties of the initial and final nuclei are the same, the reduced width in [(~')(Z)(h2/MRo)]l units is given by the relations 17) for the ground-ground a-transition ":L(00+ . O0 . .+),., . = y~ A+Lo_, + _(vvlo9o9)¢~¢~,

(3)

vco

and for the ground-(/3-state) c~-transition ~L(00 + , 00+)a.~.

- 2 - } { Z AL°+-,+ -(vv'lc°t°)f~¢¢~,+ Z A~°--,. + _(vvloxo )Gf~,,o,}. vv" o~

v¢o¢:~"

(4) Here the A~e Is ' ,,,(vv , 1o909, ) coefficients are the reduced alpha-transfer amplitudes 18) associated with proton transfer from orbitals vr, v'z' and neutron transfer from ¢oa, co'tr'; v, v' and o~, co' denote the magnitude of the angular momenta components (f2) along the nuclear symmetry axis and the additional quantum numbers necessary for complete specification of the single-particle states; z, z' and o, a' are the signs off2. For the calculation of these coefficients, we have used the approximation formula given in ref. 19) based on the method/3 = ~, where/3 is the Gaussian size parameter of the ~-particle and 7 the harmonic-oscillator size parameter. This method gives values of the A coefficients of an accuracy better than 1 ~ . The superfluid weights for favoured and collective transitions are

G = GV~,

(5)

f~, = 2-'[w~,,(V, Vs,+ Us Us,)+O~,(V~V~,- U~Us,)].

(6)

The quantities Us and Vs are the amplitudes of the Bogoliubov-Valatin canonical transformation given by the relations

2 l + [(E • _~iZ+AZ3 ~ ,

V~ = 2

1 - [ ( E s - - 2 ) 2 + A2]~J ,

(7)

where E~ are the single-particle energies, 2 the chemical potential and A the energy gap. i and gss" i from relation (6) are given in terms of different comThe quantities wss, ponents of the effective forces. As mentioned before, we assumed three components of these forces ~3), e.g. pairing forces, quadrupole forces -- tCqr 2 r22 Y*.(I ) Y:.(2)

(S)

-- x, r z rZ[~ r t Yz(1)]*u[a2 Yz(2)]2u,

(9)

and spin-quadrupole forces

where ~Cqand tct are the quadrupole and spin-quadrupole coupling strength parameters, respectively, Y2u the spherical harmonics functions and tr i the Pauli spinmatrices.

34

M.I.

CRISTU

et al.

In RPA, the ground state of a doubly even nucleus Z is defined as a phononless state and the first excited states as p h o n o n states. Thus, we have Q~z = o,

(lO)

Q~y. = IK" = 0+>,

where Q~ and Q~+ are the absorption and the creation operators respectively of fl phonons for the K ~ = 0 + excited states where i denotes the order of the root of the secular equation ~) for the energies of the K ~ = 0 + states. T h e p h o n o n operators satisfy the boson c o m m u t a t i o n relations

[Q,, Qj] = [Q?, Q+I,j Ea,,

(11)

=

In terms of quasiparticle operators, they are given by Ql

f) (P)-L/~(n)

(12)

with Ol n°rp) = 2 -1 ;

* A(ss , ) - ~ ,~A + (ss)], , • [Oss'

(13)

A(ss') = 2 -~ L / . ~ s ' , a s - , .

(14)

/l

o~.,. = Vsas_.+#Vsa~ +,

(15)

+

where by as. (as.) and ~t~(~.) we have denoted the emission (absorption) o p e r a t o r for particles and for quasiparticles respectively. The amplitudes wL. and g:s' are connected with ~k~s. and q~*s."from relation (13) as follows: 2~b~. = gs~' ' + w~., ' •

' ' 2~s,' = gs~.w~.,

(16)

where 12-13) gI~' = ( 2 Z 7 ' ) ~ [q~,V~s.e.s, +D(E,a ' ~¢,) ...... . . . . . 2

.... S-2-

2

b s s . -2

2

8ss, - - E l #

[ ~'ss' - - E l #

8ss - -

(17)

E2p

w~s. = (2Z[-,)½ [qss, U.~,'E,a +D(E,a , ~: . tss,Lss, e~s.

[ ess,--Eitj

8ss, - - L'i#

_~,

.... 2EL&A~ •

F'_~_ ~ ,

e~(es~--E~) Yl 2

2

2A "

essEi~

(18) '

in which e,s, = e ~ + e s , ,

(19)

L,s, = Us V , , - U~, Vs,

(20)

U,,, = U,V,,+U,,V,.

(21)

~-DECAV

35

Here by qs," and t~, we have denoted the single-particle quadrupole and spin-quadrupole matrix elements and by Eta the ith root energy ofK" = 0 + state. The quantities D(E,~, r.t), Z, = Z,(E,B, ~c,), F~, ~' and ~' are the following: O(Eip, ~t) _

tq X(E,a)

(22)

1 - xt S(E~I~)'

2_A2Ei~ E F2(E'a) Zi(Eia, tq) = Y(Eip, xt)+ ?2(E,a ) , e,(e2_Ei~)2

4A2E'a - q'sr~(E'a) ~(~i~) ~ e,(e,~-E,a)2.... 2-~,'

(23)

where

}, Y(Eit~, Kt) = ~1 /dFo(Ei~) ( ~ + 2D(Eit~, tq) OX(E'p)+D2(EI,s,xt)d-~Sa(-E~) ?'Eta

(24)

F.,(Eia) = ¢(E,a) - 4(E, - ,;.)r/(E,a),

(26)

¢(E,~) = ~] q~"'[aA2- E$ + 4(E~- ),)(E,,- ;03 2 2 2 ~.~, ~ , ( ~ - 2 E,~)(~,~,E,~)

(27)

rl(Ei,~) = Z .Y(Eia)

=

~ SS'

q~(Es-E¢) 2 2

2

!h_.. 2 - -

•

E~ + 4(E~ - ;t)(E¢- 2), 2 e~, 2 ~,- E,,~) 2 Eilj)(

(28) (29)

~,ses,(e2s_

2

2

S(E,~) = 2 ~S~ e~.~.L.,.~,.t~s~ ~L,-E?~ ' X(E,,) = 2

E,a U~,_L~, %,_t_~s, . es2,-- El2p

(30) (31 )

From relation (14), we can see that the alpha particle can be formed from two like nucleons situated in different levels (s' -~ s) or situated in the same level (s' = s). The asymptotic behaviour of the collective superfluid weights f~, when the twoquasiparticle energy is far from the pole (e,~, >> Ea) and the single-particle energy is far from the Fermi surface (IE~-21 >> A) is C

{q,, Fsign(E~,-2)

sig_~_(#,,.~)l

+t~, IS i-gn.(E' .---2-) - s-i-g-n--(-E--~-;t)lt , L IE~-21 IE~,-21 JJ where C as well as C' and

C" in eq. (33), which follows, are positive constants.

(32)

N.I. CRISTUet al.

36

Thus, the superfluid weightsf~s, have two terms, one f~(~!) due to quadrupole forces and the other o n e f ~ ) due to spin-quadrupole forces. In the first case (s' ~ s), the ratio F.}.~!)/Gs"is positive for all combinations of levels lying both below the Fermi surface or with s-level below and s'-level above provided that ]E~-21 < ]Es-2[ or with s'-level below and s-level above provided that ]E~,-2[ > ]E~-2[. This ratio is otherwise negative. The ratiof,(2)/G, is positive when both levels are lying above the Fermi surface and [E,,-2] < [E~-).[ or below and ]E~,-21 > [E,-).] or when the s'-level is below and the s-level above. This ratio is otherwise negative. Thus, taking into account the magnitude and the signs of G~, and t.w matrix elements, we have essentially a cancellation effect for the non-diagonal terms. In the second case (s = s'), the second term f}]) due to spin-quadrupole forces cancels. The asymptotic behaviour keeps the form given in our previous paper s) when we have considered the conventional pairing and quadrupole effective force model L~ ~ - C '

1

IE~-;.I

C " sign ( E s - 2 ) IE _;.12 G,.

(33)

From relation (33), we can see that when the single-particle states are below the Fermi sea 2, to maxima in Gs correspond also maxima in J;~, whereas for energies above 2, to maxima in q~s correspond minima in f,.~. These rules are visualized in figs. 1 and 2, where the single-particle quadrupole matrix-elements q~ and the superfluid weightsf~ s are drawn as a function of the 42 single-particle energies. We can see also that the.f~ values are strongly enhanced by the quadrupole forces and strongly hindered by the spin-quadrupole forces.

3. Discussion

In the framework of the pairing, quadrupole and spin-quadrupole effective force models, we have computed the hindrance factors and the reduced wave amplitudes for three nuclei in the Th-U region for values of the quadrupole ~Cqand spin-quadrupole 1q strength parameters. The results are shown in tables 1 and 2. The first two cases for each nucleus from table 1 illustrate the pairing and quadrupole effective force model (~ct = 0). Choosing the ~c, parameter to fit the experimental fl-energies E~ (case I), the alpha hindrance factors appear to be .smaller than the experimental data by approximately an order of magnitude. Fitting the alpha hindrance factors (case 2), the fl-energies Ep are approximately twice as large as the experimental ones. Thus, the model fails to explain these experimental data. By comparing case 2 with case 1, we can see that the alpha hindrance factors increase strongly if the to, parameter decreases. This implies that the K" = 0 + states are not pure vibrational states 8).

,

/

0.1

02

.......

~''-"

.-J ...-'f

t

"~'/

/

.. ~\

/\\,

~G ~

o./+

i ..../

os

_/.~.--_~

/'-I L.....

o'~ '~°o'~ -f

~ 0'8" ..........

69

lo

11

........

"q-...

j/Am k.k~...L~es-:?-~--:---:-.......=:.:_

!/

/"

,/,,',

/:~

' < - I/ /~tT,J+.-£V ./--_~ _-1 . ~ ~ ! " ~ ' / / / _ S ,k _---'"." . . . . . . . - . . . . . . . .

/

S:-"/L4!

03

. .....

/

I\

I, ~

i

i ~.t

13

1/+

15

.... Z_-Q. . . . . . . . - .... --'--~-"-~-'-

....

EI~

512 000 0,123(] /+oo ooo o21~s /+00109501196 3 ~ 6 - - i o 9 5 o.12,o

l~q

232U a-'~22BTh

Proton System

n 12

~

2. ,

1

CQse

16

151

18

Ig

-~-,~==-~.~.-'-=-~ . . . . bmgle porticle energy

Fig. 1. The quadrupole m o m c n t s qvv a n d the superfluid weights ~vv for the a l p h a transition to the g r o u n d state and those fg, to the fl-vibrational state o f n S T h as a function o f the 42 single-proton energies E v for s o m e values o f the q u a d r u p o l c Kq a n d s p i n - q u a d r u p o l e Kt strength p a r a m e t e r .

-3-

>> 4. ~r *2-

o

+2-i.

00-

Oh

O3

---=O5

"~ 06.

, 07.

08

o

~h

K

03

o

I

~.'-.. . . . . . . .

. .

~-~

"'

\

~,

09C

110

"

120 :

L50 1,10

,,o

1LO

01075 01085

"1S0

160

........ // '. . .,.'., . . . . . '~/"

130

System

1035 ~35

ooo o2,,o

-,,,, Ea 6gs ooo O.tt~0

Neutron

3 z,

2

1

~Case ~

7"-~ .......

10.0

'~,,'-T ~' ~.~' \ '\ ', "~.

',,(1)

c~

\

/--'--;::-;". . . . . . . . .

/

\

Single p a r t i c l e

energy

Fig. 2. The quadrupole moments q,o,o and the superfluid weights ~o~o for the alpha transition to the ground state and thosef~oo~ to the/~-vibrational state of 23~U as a function of the 42 single-neutron energies Eo~ for some values of the quadrupole ~q and spin-quadrupole ~t strength parameter.

-~-

/

/

" o~o ''o3o ~,-o,~, ~-, ,o60 o,o ' " ~ . .

:

i

':~

,,,,./\

II

'', . I.: j] / / ,,) : /

I

Ol.

i ,

05

,'i (, ' , ,, : ,

06

08

oo,,:fLo,j %

~i

"9-

310.

311.

~2-

.N

(7

~r

:t- DECAY

39

TABLE 1 The experimental and calculated fl-energies and hindrance factors H F for 228Th, 2a°Th and za2U for some values o f the quadrupole rq and spin-quadrupole r t strength parameters Daughter nucleus

Case

1 2

228Th (E#) = 825 keV exp

23°Th (E#) = 636 keV exp

232 U

(E#) = 694 keV exp

rt

rq

0 0

5.12 4.00

E (keV) 825.33 1434.72

HF (B' = 0) 1.1 10.2

HF (B' = 0.9) 0.7 4.7

3

10.95

4.00

802.52

27.2

18.25

4 5

10.95 10.95

3.80 3.44

811.91 825.33

37.9 74.9

26.2 50.5

6.21 5.68

635.58 1097.21

0.6 3.5

0.4 2.3

5.68 3.82

635.58 635.58

4.1 105.6

2.6 66.0

6.95 5.30

693.86 1607.88

0.4 10.4

0.2 3.9

4.50 4.10

717.22 723.89

65.4 100.2

37.6 57.9

1 2

0 0

HF exp.

11 q- 4

40--10 3 4 1 2 3 4

10.50 10.58 0 0

.

10.35 10.35

12 ~ 4

The F r 6 m a n parameter B' for fl-states has been assumed zero or equal to the value 0.9 as for the ground state. TABLE 2

The experimental and calculated reduced wave amplitudes b2 for the first rotational level associated to fl-vibrational states of 22aTh and 232U Daughter nucleus

22RTh

232U

Case

1 2 3 4 5

b2 (calc)

bz (exp)

0.69 1.14 0.67 0.66 0.70

0.56

1

0.91

2 3 4

0.36 0.99 1.01

0.27

The next cases illustrate the pairing, quadrupole and spin-quadrupole effective force model. Taking the ~ct parameter constant and decreasing the ~q parameter, we can fit both the fl-energies E~ and the alpha hindrance factors HF. F r o m the experimental data, we evaluated the strength parameters for quadrupole (~q 228Th = 4.00, ~:q 23orb = 3.82, ~cq23~u = 4.56) and spin-quadrupole (x t ~28Th = 10.95, rCt 2~ox~= 10.58, tot 23~U = 10.35) forces. N o trials have been made to obtain more convergent strength parameters for this region.

40

M. 1. CRISTU et

al.

Th e i n t r o d u c t i o n o f the F r b m a n m a t r i x for the first K" = 0 + excited states will deci'ease the h i n d r a n c e factors by an insignificant factor a m o u n t i n g 1.78 for 23°U, 1.65 for 2 3 ° T h an d 1.47 for 228Th. T h e reduced wave amplitudes b2, defined in eq. (2), for K ~ = 0 + states are given in table 2. M o r e detailed analysis must be effectuated in o r d er to o b t a i n some i n f o r m a t i o n f r o m these data. O n e o f us (A.S.) wishes to t h a n k Professor H. K i i m m e l for his interest in this pap er an d for the kind hospitality at the M a x - P l a n c k - I n s t i t u t e Mainz.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

V. G. Soloviev, Nucl. Phys. 69 (1965) 1 R. Graetzer, G. B. Hageman, K. A. Hageman and B. Elbek, Nucl. Phys. 76 (1966) 1 J. H. Bjerregaard, O. Hansen, O. Nathan and S. Hinds, Nucl. Phys. 86 (1966) 145 O. LOnsj6 and G. B. Hageman, Nucl. Phys. 88 (1966) 624 J. Vrzal et al., lzv. Akad. Nauk SSSR (ser. fiz.) 31 (1967) 604 H. L. Nielsen, K. Wilsky, I. Zylicz and G. Sorensen, Nucl. Phys. A93 (1967) 385 D. R. B6s and R. A. Broglia, Nucl. Phys. 80 (1966) 289 A. S~ndulescu and O. Dumitrescu, Phys. Lett. 24B (1967) 212 K. M. Zheleznova et al., preprint D-2157 Dubna (1965) S. Bjornholm, Nuclear excitations in even isotgpes of the heaviest elements, thesis (Munksgaard, Copenhagen, 1965) Y. Yoshizawa, B. Elbek, B. Herskind and M. C. Olesen, Nucl. Phys. 73 (1965) 273 N. I. Pyatov, Proc. Lysekil Symp. (1966) M. I. Cherney, N. I. Pyatov and K. M. Zheleznova, Izv. Akad. Nauk SSSR (ser. fiz.)31 (1967) 550 L. S. Kisslinger, Nucl. Phys. 35 (1962) 114 P. O. FrOman, Mat. Fys. Medd. Dan. Vid. Selsk. 1, No. 3 (1957) J. K. Poggenburg, Ph.D. thesis UCRL 16187 (1965) A. Shndulescu and O. Dumitrescu, Phys. Lett. 19 (1965) 404 A. S~mdulescu, Nucl. Phys. 48 (1963) 345 A. Sgmdulescu, Proc. Liperi Summer School of Theoretical Physics, Helsinki (1967); J. dc Phys. 29 (1968) C1-35