Alpha decay of light nuclei

Nuclaar Phyaltx A272 (1976) 303-316 ; © North-Holland Publiahiny Co., Mutsrdam Not to be reproduced by photoprlnt or microfilm without written permlwon from tha pnbWha

ALPHA DECAY OF LIGHT NUCLEI B . APAGYI Institute oJPhysics, Technical Unitxrsity, Budapest

and G . FÁI and J . NÉMETH Institutejor Theoretiml Physics, Roland Eôtt+ós Unioersity, Budapest

Received 3 December 1975 (Revised 6 April 1976) Abstract : Theoretical a spectroscopic amplitudes are calculated in,jjwupling and isotopic spin formalism . The formulae obtained are appliod for calculating the a reduced width amplitudes of the lowest lying 2+ levels of' 60 using the shell-model wave fundiona of Zuker, Buck and McGrory.

l. Introduction The a spectroscopic amplitude (ASA) can be determined in different models . In the late 6fües Mang 1) expressed it in the n-p formalism which is applicable for heavy nuclei . The ASA has also been derived by Ichimura et al. Z) in the LS coupling SU(3) scheme which is, ofcourse, suitable for SU(3) shell-model (SM) wave functions. Recently, Kurath and Towner a) have calculated it in the .v coupling harmonic oscillator SM for the case of a proton and neutron pair decomposition. Using wave functions which are the results of a general SM calculation in .,ij coupling and isospin formalism, one has to derive ASA accordingly. The formulae obtained in the present work contain the basis of the SM calculation in a general manner, and therefore they retain their applicability for every SM calculation of the above type. Sincethe a reduced width amplitude (ARWA) can be expressed in terms oftheASA, such a calculation is necessary for example in determining the parity violating adecay width of the 8.87 MeV 2~ level of 160. Here, ARWA of all the opposite parity 2,+ states contribute ~) to the decay width, even the 2i level which lies below the a-decay threshold energy of 7.16 MeV. The ARWA of this state cannot be taken from a-decay experiments, and therefore we are forced to calculate it theoretically. In sect . 2, following the notation of ref. Z), the definition of the ASA and its relation to the ARWA are presented for SM wave functions. In sect. 3 the ASA required is derived and, as an application, numerical results are given for the lowest lying 2,+ levels of 160calculated with the Zuker, Huck and McGrory (ZHM)wave functions s). 303

304

B . APAGYI et al.

In sect . 4 we use a method for extracting a reduced widths (ARW defined as ARWA squared) from experiment, and the experimental and theoreticalwidths arecomparod. Sect . 5 gives a short summary and conclusion . 2. I)eßoitioo of ASA for SM wave fmctioos To calculate the width of an a-decay we have to determine the overlap integral FJMáNr:J'T'J "T,(r~) -

AI T' ~(SA'f~a " T""(~a/~Y~Ir~~TYr(SAh

dSA~Sa~~ ~4/~ J

(1 )

where the square bracket, as usual, means vector coupling ; ~xT~T represents the antisymmetric internal wave funáion of the nucleus X, in the JMTMT state ; ~x refers to its internal'ooordinates ; A, A' and a denote the parent,,daughter and a~ nucleus, respectively ; finally, r",,' is the vector connecting the c.m . vectors of the fragment nuclei . The overlap integral F can be expanded in terms of any orthonormal complete set of fmctions in the r",,. space {9~,u(r~A')} [at least in a finite range if Fcannot be normalized ~)]. One refers to the expansion coefficients as the "four-particle spectroswpic amplitudes". If we restrict the four-particle state ~;" x "r"~ r T" to be an aparticle ground state by choosing Ja = T~ = 0 in eq. (1), We get the expansion coefficients called ;`a spectroscopic amplitudes" (ASA). In the following we shall consider such a-decays of the doubly even nuclei where the daughter nucleus is in its ground state. This means that we choose the quantum numbers T, T' and J' equal to zero in eq. (1). With these restrictions F can be written as FMáo: oooo(r~,) _ ~ , INL(A, A~xtrÁr w'~ N where the ASA is defined as ANL(~+A7 =

~4~

/ d~A~~a`~sA'~A' ~U \~A'l'ri~00~~a1Y'NLY~raA'l~A ~~(SAl"

With the aid of eq . (3) the ARWA can be expressed as

where RNL(r) is the radial part of the wave function 9~Nrar(r) of«l . (2~ ra is the channel radius and uA', is the reduced mass of A and a. The calculation of ANL using SM wave functions ~Px, instead of the internal ~x is descaibed in detail in ref. ~). It was shown 2) that in this case the ASA can be

a-DECAY

writteII a8 A ~

,q

305

N+ta a x

f

d~a~á(~a~reer(Pa ; vo)~a(Ca~

where harmonic oscillator ~ wave functions {rpr(r; vo)} with siu parameter vo = mocu/fl are used fo_r the expansion (2) ; SPa denotes a complete set of four-body SM wave functions; Ca means the four-particle SM coordinates ; pa = 2Ra, where Ra is the vector directed from the center of the SM well to the c.m . of the a-particle ; <~w" ~`al}~,,% is a four-particle coefficient of fractional parentage; thefactor (A/A-4~+tL is a correction due to the separation of the centers of mass of the nuclei participating in the decay. In eq. (5) and throughout this paper we use the following sets of coordinates ya = lYh P2+ P3, 1Qt, Ti}+ 1 - 1, . . ., 4),

The connections between the two systems and the infinitesimal volume elements are as follows :

Pa = z(rl+rz - ra -ra~

Pa = 31r1+rz+r3+r~,

a a dSa = ~ drt dQtdTt = - ~ dpt~tdTt = -d~adPa, t=1 t-1 where rt, Q t and r, stand for the spatial, spin and isospin coordinates of the ith particle, respectively . Using SM wave functions which have an inert 1 zC core and specifying the decaying nucleus to be 160 we get for the ASA lz~ ( `4NL(160, yI e

}i: ~, ~

J d~a`+l~a~á(yaWNL~(lt+~a ; v0)~a(~a~ a where the coefficients c~ are given by the SM.

3. Derivation of the ASA and some featores of the 2+ states of 1`O

In every SM calculation of the type "inert core+more than or equal to four particles", the four-body state can be described as follows : 7Ya(A, A~ ; Sa) = ha(A, A~ ; b, kltl, kztz ; .laTa) tPa(klth kztz ; JaTa ; yah

306

B. APAGYI et al .

where SYa has a special order of coupling which can be represented by the following ket ~a(klty kzt2 ; JaTa ; ba) = 112(kltl)34(kzts~aTa ; {rt; rnTth i = 1, . . ., 4),

(8)

that is two pairs of particles are first formed, then the resultants k 1 t 1 and k2 t 2 are coupled to the total four-particle angular momentum Ja and isospin Ta. Additional quantum numbers are denoted by 8, and the symbol ha contains fractional parentage and Racah coefilcients as well as instructions for possible summations. In the appendix the ha are calculated for the 2 states of 160 using the ZBM shell model s) . From eqs. (5}{8) the ASA can be written as A~(A, A7 =

A lx+fir.

CA -4I

~ L~BT.oar i ~lha(~ A~ ; ó, klty k2tz ; JaTal"xL(klth k2í1 ; Jal+ a

where axL(klty kztl ;

Ja) =

J

dba`Wa~á(yalY'NLAIV'a+ VO)~a(kltl+ k2t2 ; Ja~; ~~~ (1~)

The wave function ~a(~a) standing for the ground state of the a-particle consists of a symmetrical spatial part ~ and an antisymmetrical spin-isospin part ~, namely ~a(~a) = y a~a b(Pv Pz~ P3)Xa({QtTt}+ i = 1, . . , 4~ where b _ ~,, VOaW000~2+ ~~,, r^ ~~,,~3+ v0 J~ ~POOOWl+ VOa/Y'000

Xa

= -

1/61p34p23 -p23+ 1)X00(Q1Q2)X00(Q3Q411 1 +p13p24)Xll(T1T2)XI-1(Z3T41+

( 12) (13 )

where Ply exchange the Q,, Q~ as well as the Tt, T~ coordinates ; Xoo means the singlet spin function, while Xh corresponds to the triplet one. Separating the spin-isospfn part in eq. (8), the wave function ~a can be written as ~a(klty kztl ; Ja~+ `a) _ ~

C(k1m1k2m21JaMa)

~l~l 1

1

XJ1J2J3.Îa ~ ^'1^~2"1~2 .TL43 xix3

x

1

.

2

1

~2

1

`S 1 kl

l3

~4

~2

J3

!4

k2

S2

~ C(~1P1S1Ms,Iklm1~(Z2h2S2Msl~k2m2) AIS,YS3 pt1~3

x ~ill2 ; ~1P1i~ 13 1a ; ~21~2iXs,x s,(~l~z)Xs3urs,(Q3~a) X ~Xt,wk,(T1T2)Xt,-~,(T3T4/1-111 ~~~th

(14)

a-DECAY

307

where A =_ (2x+ 1)}, the symbol C( ~), means a Clebsch-Gordan coefficient, and the curly bracket represents the usual 9j coefficients appearing due to the transformation from the,~j coupling to the LS scheme. Taking into account eqs. (11}-(14), the integration over the spin and isospin coordinates ({Qli,}, i = 1, . . ., 4) can be immediately carried out in eq. (10) : 4

~~1

d~IdTÜAa ({QiTi} + 1 = 1, ' ' ', 4)XStMgI(Q1Q2)XSaAlgs(Q3Q4) x

E Xt,iwtt(TlZ2)Xt,-nttl(2 3T411 - lJ t- ~tt~tl ~t, _ ~(_ 1 ) i+ ~ssas s,b~s,-xsz(as=iatto+V ~s~oat i)' t t

(15)

Using this result, summing over S l and Msl , and introducing the notations Sz =_ S, =_ Ms, the integral of eq . (10) can be written in the form

Ms2

aNC(kiti~kztl ;J,~ _ ~ ~ C(k1n+lkzmzlJall~IJ

x~l.%2J3J4

~ Sx,xl

"'1" .zJ2

h

Iz

~

~

ji

jz

.l i

l3

la

.l z

ki

.ls ja

kz

S

2

~

S

x(asiat,o+~asoatti) ~ ( -1~sC(~iPiS-Mslkimi) ~aPtps rr°ár x C(.1zuzSMs~kzmz)r ro~{vr
(16)

where IH,.e, contains integration over the spatial coordinates only INKY°~{vrlrZKutt} ~ i = 1, . . ., 4; K

= 1, 2)

a

~.dPi~ b*(PiPzPa ; vo~Hr.~r(Pa ; vo)wihvzlz+~iPiilvslsvala~ ~zPzi' != i .

(1~

(Note the dif%rence between v~ and vo!) To proceed we have now to choose a special representation for the two-body nuclear state wlv'l' ; ~.Pi . We use the harmonic oscillator representation of the size parameter vo because of its very advantageous applicability in the analytical calculation. Moreover, the structure of light nuclei can be well described by using harmonic oscillator wave functions z). Using this representation and applying the Brody-Moshinsky transformation technique 6), one can perform the integration over {pi ; i = 1, . . ., 4} in eq . (1~ giving the ßnal expression for the integral aNL(kltl~ kzti~ JJ

= a~.a

~6

( y +v o ~ oa \J~/~

a. nPnc~rl a ~r .

3os

xptlzß.ia~t~z~ z ~ ~t~z~z(-1)~1+k,+L x (astSrlo+~asoarlt) x

kt {~, z

S

zt kz

L~

jt ~~

lz

~t

l3

~1

.Í2

kl

~3 ~4

~

s

~ wtjt(rt)vzlz(rzutlnt0(Pt)Nt~i(~/~(ri+rz)~i,,is t

j4

~z s

kz

"u~

x ws13(r3)vaia(ra~zlnz0(Pz)Nz~z(~{r3+r~uzii~ x ~NtAi(~(rt+rz)NzZz(~~(rs+r~~InsO(Ps)NL(P4~is>r

~

x s

C

(n~+~)~ l~lvoa "~ nil , voa+v -vJ

(18)

where { : : :} denotes the 6j coefficient, the brackets marked with the subscript BM stand for the Brody-Moshinsky coefficients 6), and the summations over n t and nz are restricted by the energy conservation : 2(vt +v z)+ h+l z

2(N t +Nz)+ .lt

= 2(nt+Nt)+At,

+ .l z =

2(n3+N)+L.

Substituting eq. (18) into eq . (9) one gets the ASA in a general way. As an application, we have calculated the ASA for the different 2+ states of' 6 0 using ZBM wave functions'). ['The reason we examine just these states is that they have a special importance in the parity-violating a-decay of this nucleus 4).] Results are shown in fig. 1, where the ASA are drawn as a function of the nuclear oscillator parameter v o ranging from 0.19 to 0.59 frn -z ; in this manner we cover the whole T~a~.e 1 Calculated a spedroeoopic amplitudes for the different 2+ states of'60 with the two vo as dmaibed in the text ~+ A°i Af ~ A°~ A~ vo (fm-=)

ro) vo ~ vos

(a) vo - vol 24 0.0081 0.0028 -0 .1540 -0 .0597 0.3938

23

2z

-0 .0020 -0 .0285 0.0841 0.0429 0.0130 -0 .1870 0.0767 0.0102 0.3938

0.3938

21

2s

23

2Z

21

0.0084 0.0274 -0 .0000 -0.0555 0.0001 -0 .0877 -0 .0094 0.0092 0.0533 -0 .0074 0.4530 -0 .0649 0.0101 -0 .0955 0.2610 -0 .3560 -0.0416 0.0863 0.0060 -0.4000 0.3938

0.244

0.594

0.194

0.594

309

a-DECAY v o+ .+

0

O

C~f

II II Z Z

II Z

N II Z

Of M O

N lC!

~n d

in

o

d .~

n z

M

u z

ó

1

o u z

n

vrn M ó

N ~n d

O

a

~

óe ó

~ff

d

MO II II 11 Z Z

I

N II Z

~y +

N ~+ 4.

W

}N CY _

R O

O O

d

O1 O

v N n z

+ ..

d

O n z

0

ó

.

r ~., I

~ u z

ó M

n z

In

ó

ó

n

u

310

B . APAGYI et al.

region of physical interest . The size parameter v~ of the a-particle was fixed at 0.625 fm - ~. The results do predict strong 4p-4h and 2p-2h mixing for the 2i state and definite 2p-2h structure for the 22 and 24 states within the whole range ofvo. By this method the ZBM wave function does not predict any general character for the 2; state within the v o range oonsiderod . In table 1 the numerical values of the ASA are shown for the four 2+ states of 160 in two different cases. In case (a) the ratio vo/vim is chosen to be equal to (6)~, while _ in case (b) v~ is fixed at 0.625 fm s and the nuclear size parameter v o varies in the range 0.19-0.59 fin- 2 to have a value vo = X02 which provides good ARW compared to experiment (see fig. 2 of sect. 5). The 4p-4h character and the considerable 2p-2h mixing in the 2i state of 160 are well establishod'~ 14). From table 1 and fig. 1 it can be seen that, using ZBM wave functions s), this state has the largest A~ amplitude compared to that of the other 2 states taken into account. 4. Res~ilts for the a-deay widths of 160

the best expression for the total width I'a of the a-decay has been derived by Arima and Yoshida a) who based their a-decay theory on the single-channel singlelevel approximation of the R-matrix theory. In their theory I'Q can be expressed as 2 (19) ra - [P/yZ+SP-P(S-B)]l[PZ+(s-B)2 ] -~' where the dot means derivatives with respect to the energy ; P is the penetrability defined by P = ~, J(Fz+rr~~ (20) where k is the wave number of the relative motion of the a-particle and the daughter nucleus, r~ is the channel radius parameter ; S is the shift function where f = (1/k~f/âr ; energy shift

S = P(~~'+~~'~ B

(21)

is the boundary condition parameter chosen to have zero B = S -P tan ~,

and ~ is the hard sphere scattering phase shift given by the relation tan ~ _ ~/~.

(22) (23)

All the quantities defined by eqs. (20}{23) depend on the modified Coulomb funotions ~ F and ~ which have to be taken at the resonance energy E, of the a-nucleus system and at the channel radius r o. It has been shown in ref. e) that the value of l', does not depend on the choice of

a-DECAY

31 I

r~ when it is calculated from eq. (I9) . This means that we can express ARW (yZ) using eq. (19) with experimental values for the total width l'a and the resonance energy E~. The ARW thus derived can be called "experimental ARW' (Y~~. On the other hand, using the theoretical ASA and eq. (4), we can calculate the ARW called "theoretical ARW' (ymeor) . For modifying the Coulomb functions we have used a Woods-Saxon potential with the fixed parameters ro = 4.3 1m and a = 0.5 fm given in ref. 10~, while the depth parameter Vo has been chosen for a given resonance energy El to have a resonance solution and the desired number of nodes N for r; inside the barrier. In table 2 the values of - Vo can be seen for the dit%rent N and four E, corresponding to three 2+ and one 2 - levels of 160 above the a-decay threshold energy . In the calculation the Vo with N = 2 were used for the 2á and 2Z states and the Vo with ly = 3 were accepted for the 23 and 2- states . T~ate 2

Resonance depth parameter - Va for the Woods-Saxon potentials modifying the Coulomb field as a function of the number of nodes N 2; 24 23 2~ 2i

E, (MeV) 5.86 4.36 2.69 1 .71

N=

0

1

2

3

4

6.55 7.92 10.71 12 .44

20.71 21 .47 24.41 26 .56

4(1.40 40 .33 43 .38 46 .02

65 .81 64 .72 67 .74 70 .89

96 .96 94 .68 97 .53 101 .20

In fig. 2 the theoretical and experimental ARWA (yt~~r and yep can be seen for the four lowest lying 2 + states of 160 as a function of the channel radius r~ in the range 2 .1 < r~ < 6.1 fin. Since only the absolute value of the experimental ARWA can be determined from eq. (19), its sign was chosen according to the theoretical curve. In each case we have drawn two theoretical curves, one of which corresponds to the ASA calculated with vo = vol, where vol/vim _ (tb)~ holds. The second theoretical curve has been evaluated with the ASA with siu parameter vo = vo2, for which good fitting could be obtained to the experimental values in the range r~ = 3-b fin (cf: table 1). In the latter case vo ~ has been fixed at 0.625 fm -2 . For the 24 and 23 states we found a good fit at about vo = 0.244 and 0.594 fm -2, respectively. For the 2? state we did not find any ASA set which approximates the experimental curve reasonably well in the range 3 < r~ < 6 fin. The possible accord could have bcen reached outside the v-range considered atabout vo x 0.144 fines 2. For illustration we have drawn the curve obtained with vo2 = 0 .194 fm-2 . In these three cases above threshold we have used Woods-Saxon wave functions for the functions RNL of eq. (4) to improve the asymptotical behaviour of y~~. The Woods-Saxon potentials are the same as used for deriving the modified Coulomb functions. In the case of the first 2 + state of 16 0 below the a-decay threshold one cannot

o~ ~

~~~ ~~ ~~~~

a-DECAY

31 3

derive experimental ARW from a~lecay experiments. [T'here have been some attempts 14) to extract ASA for this state from a-transfer reactions.] We have seen in the preceding suction that this 2i state has the largest ASA compared to the other 2 + states . In this way this state has the largest ARWA inside the well (r~ _ 4.3 fm). On the other hand, this 2i state lies near the 8.87 MeV 21 level, the parityviolating a-decay of which has been measured recently ' 1). As it was shown in ref. 4), the parity-violating ARWA of this 2~ state can be expressed in terms of the ARWA of the 2 states as Y~(21) = E I(~+ 12~ hd~+ ~

(~)

n

where jis the mixing amplitude treated in detail and denoted by F, in ref. °). It is just thej(2i 12 1) ~PUtude which has the largest magnitude compared to the other j(2 121) values 4). From these two facts it follows that we can write e4 . (24) to good approximation in the form (25) Y~(2 1) ^" Î(21 121)Y(21)" Using oscillator wave functions for calculating yls~(2 ; ), this result gives an opportunity to fmd an approximate vo2 value for the ARWA of the 2i state, since one can doduce experimental yß(2 1) values from the measured quantity Cam" 10 - ib MeV [ref. 11 )] . This .yß(21) curve is also shown in fig. 2. [The sign was deduced from yl~(2 i ) because f(2i 121) is Positive 4).] Comparing the maximum, zero and minimum positions of the yl~~r(2 i ) curve to that of the y one, we found the value vo = vo2 = 0.594 fm -2 for the 2; state. TABLE 3 Experimental and calculated a-widths for the 2+ stater of 160 at the channel ridü ro = 5.6, 4.3 and 3.6 fm

vo3 2+ (fm _ :) 0.244 0.594 0.194 0.594

2~ 2; 2= 2i ro (fm)

Y~e..r(~o:) (MeV' ~')

IY..,I (MeV'«)

rwQ(~o~ (Me~

rnv (MeV)

0.1060 -0.0527 -0 .0852 -0.0919 0.0295 0.1280 0.0129 -0.0705 -0.0838 0.3800 -0 .0740 -0 .1530

0.0550 0.0960 0.0870 0.1130 0.0270 0.1370 0.0155 0.0251 0 .0323

0.453 0.053 0.141 0.074 0.326 0.094 0.0006 0.0097 0.0072

0.160 0.110 0.0010

3.6

3.6

3.6

4.3

5.6

4.3

5.6

4.3

5.6

Table 3 contains the experimental and theoretical values of the ARWA as well as the total widths at three r~ values (5.6, 4.3 and 3 .6 fm). Results at ra = 5.6 fm are includod to show the good agreement of experiment and theory in the asymptotic region. Results at rQ = 3.6 fm are, however, more suitable for calculating yp`'(21) because of the wrong asymptotic behaviour of the harmonic oscillator functions in Yln.a(2 i)"

B. APAGYI et al .

314

5. S®mary sod cooclasion The theoretical ASA (a spectroscopic amplitudes) were calculated in the framework of the ZBM shell model s) for the four lower lying 2,+ levels of 16 0. It has been found that the 2i wave function reproduces well the 4p-4h and 2p-2h mixing in the 2i state of 160 (fig . 1 and table 1). The 22 and 24 levels have a general 2p-Zh character due to the overall dominance ofA~ in the whole domain of size parameters ranging from 0.19 to 0.56 fin-z . The 23 state is the most sensitive for choosing the value of vo because in this case there is no A~ which predominates throughout the total range of vo considerod . The calculated ASA were then applied for determining the ARWA (a reduced width amplitudes) of the first four 2 states of 160. Using the a-decay theory of Arima and Yoshida s) the absolute value of the ARWA was extracted from the a-decay experiments for the above-threshold states and even for the 2i state at 8.87 MeV which decays through parity violation 11) . By choosing the size parameter vo suitably, the theoretical ARWA could be fitted to the experimental values, except in the case of the 22 state where this is possible only for an extremely small value of wo at about 0.144 fin-z . Nevertheless, the a-decay of this state is hindered compared to the other 2+ states by a fator of about one hundered (sce table 3), and this hindrance comes almost entirely from the effect of the nuclear structure. From the examinations of the ASA in sect . 3, we conclude that the ZBM wave functions of the 2; states of 160 contain the essential information about the aprocesses. Analysing the ARWA in sect. 4 we have found that the 2i level gives themost significant contribution to the parity-violating a-decay of the 2i state of 160. The authors are indebted to Dr. A. Zuker for supplying them with the 160 wave functions. They express their thanks to 1?r. G. Ripka for valuable discussions and to the ICTP Trieste (G.F.) and CEN Saclay (J.N.), respectively, for kind hospitality. Appendix

CALCULATION OF h~ FOR THE 2+ STATES OF ' 6 0 IN THE ZBM SHELL MODEL

The ZBM shell model ') has an inert core 1 zC, and therefore any basis vector tP~ of the 1~, T = 2+ , 0 levels of the 160 nucleus is just an antisymmetrised four-particle state .The calculation ofASA [see eqs. (9) and (18)] needs a coupling of the four particles dit%rent from that of ref. '), namely

I1

?(kltl)~(kztz) ;

2 +Ui~

(A.1)

The coupling of !Pe of ref. ') can be expressed in terms of (A.1) as follows [see eq. (7) and the explanation there] SP~

=

hp('6G~ 1zC~

b~ klth kztz~

~)112(kltl)~(kztz)~ 2 + Ui~

(A .2)

a-DECAY

31 5

According to the maximal number n of particles in one orbit (d~., s} or p.~), there are three dif%rent forms for ha (a) n = 2. From the positive parity requirement it follows that the two coupling orders are the same . This means that ha = 1 ; (b) n = 3. From parity considerations the particles can be distributed only in the d~. and,s~ orbits . In the very simple case when the three particles are in the s~ orbit ha is equal to -1. In the other case, using

x (2k'+ 1 x2kz + 1 ) jk' a

js k~ ~ 112(k t 34(k t 20~, t t) z z); 2 kz

(A.3)

and employing Racah recoupling as well as coefficient of fractional parentage (c.f.p.) techniques 1z), one gets for ha hv = ~ Ljz(vikiti)lz; k't'I}j3(v'x~~7 v~ttkl

x ~ (_ k2t=

1~t+1+~,g

,~

tzt, 8 t

+

cz +

kt 4

j 2

k' kz}'

(A.4)

where j = i, j4 = 2, vl and v' are the seniorities of the jz andj s systems, respectively, and x distinguishes the states having the same v'k't' quantum numbers s). The three-particle c.f.p. [jz . . . 1}j3 . . .] with non-maximal isospin t' = i in the j = i shell are given in ref. t3) . (c) n = 4. Due to the Pauli principle, this case is realised only ifthe four particles are in the d~. shell. Similarly to the former case one gets ~ ~.lz(viktti).t~+ k't'I}J3(vxk't~] he = u'k't' ~ [j3(v'xk't')jZ ; 201}I~vy~)~ ntk,tt

x

~(_ l~t+z~S t t k,t,

_' br}

+

z+

~kt .1 j 2

~~ ~ kz

(A .5)

where the four-particle c.f p. in the j = i shell with zero isospin are given in ref. 13), and y distinguishes states having the same seniority v, spin J = 2 and isospin T = 0.

1) 2) 3) 4)

References

H. J. Mang, Z. Phys. 148 (1957) 582 M. Ichimura et al., Nucl . Phys. A204 (1973) 225 D. Kurach and I. S. Towns, Nud. Phys . A222 (1974) l M. Gari, Phys. Ittports 6C (1973) 317; B. Apagyi, G. Fái and J. Németh, Nucl . Phys . A272 (1976) 317 ~ A. P. Zuker, B. Buck and J. H. McGrory, Informal Report, PD-99, BNL 14085, October (1969) 6) T. A. Brolly and M. Moshinsky, Tables of transformation bracrets (Gordon and Breach, New Yorly 1967)

31 6 7) 8) 9) 10) 11) 12) 13) 14)

B. APAGYI et al. G. E. Brown and A. M Grew, Nucl . Phys. 7S (1966) 401 A. Arima and S. Yoshida, Nud. Phys . A219 (1974) 475 G. Midraad, L. Scherf and E. Vogt, Phys. Rev. Cl (1970) 864 E. B. Carter, G. E. Mitchell and R. 'H . Davis, Phys. Rev. 133 (1964) B1421 K. Neoheft, H. Schóber and H. Witfust, Phys. Rev. C10 (1974) 920 A. de-Shalitand I. Talmi, Nuclear shell theory (Academic Prees, New Yort, 1%3) I. S. Tawner and J. C. Hardy, Nud. Data Tables A6 (1969) 153 H. M. Laebenstein et al., Nud. Phys . A91 (I%7) 481 ; F. Pilhlhofer et al., Nud . Phys. A147 (1970) 258