Bull. Kyushu Inst. Tcch.(M. &!.S.) No. 12, 1965

GROUIM)•STATE CORRIilLATION IN EXCrlrED e+ STATES OF Oi6

Hiroyuki NAGAiDepartment of Physics, Kyushu Institutc of Tcchnology, Tobata, Kitakyushu

and

Mitsuo !TAyADepartment of Physics, Kyushu Industrial Univcrsity, Kashii, Fukuoka

"eceived Dccembcr 21, 1964)

The excited O' states have been calculated, including the effects of ground-state correlation, by the usual harmonic-oscillator shel!-model method and using

a Yukawa force with Rosenfeld mixture. A first-order ealculation of theamount of ground-state eorrelation indicates that this amount is eonsiderablein Oi6 and that ground-state correlations have important effeets on the properties

of excited O" states, Also the previous resu]ts obtained by numerieal calcula-tions on a desk computer have been eonfirmed by recaleulation on a electronie

digital computer.

gl. introduction

In the previous paperi), one of the present authors has investigated theproperties of the first excited state of Oi6 2) by the use of the harmonic oscil-

lator shell model, assurning the residual interaetion with a Yukawa shape andRosenfeld mixture. The spin-orbit interaction has been neglected. The firstexcited state has been assumed to be some admixture of possible configurationsWith an excitation energy of 2he above the simple shell-model ground stateWhieh had a wave function corresponding to the double closed shell configura-tion (ls)`(lp)i2. The lowest exeited O" state obtained by the usual shell modelCaleulation has been found to be at 7.00 MeV above the ground state. However,OUr shell-model wave funetions eorresponding to these excited configurationsinelude the spurious states in which the eenter-of-mass motions are excited telp and 2s states from the ground ls state. Thus it is neeessary to remove theseSPUrious states to avoid any error caused by the eenter-of-mass motion. Indeed,We have found that this lowest state corresponds to the spurious state wlththe 2s eenter-of-mass motion. These calculations were carried out on the hand-

Worked desk computer. rt is suggested that the second exeited O" state obtained by this usual shellMOdel ealculation eorrespond to the lowest excited state obtained by diagonaliz-

57

58 H. NAcAT and M, rTAyAing the energy mqtrix which is eonstrueted by use of only the admissible wavefunctions, removing the spurious states. But this suggestion has not beeneonfirmed in the previous work. Then, in S3, we carry out the matrix diago-nalization on the OKITAC-5090 computer to give eigenvalues and wave fune-tions for exeited O" states. There, we can confirm this suggestion and thewave functions obtained is used to estimate the monopole transition matrixelements between the ground and excited O" states, On the simple shell model, the ground state of OiG belongs to the configura-tion (ls)`(lp)i2. Residual interactions between nucleons are expected to eauseadmixtures of states of other configurations. The effects of such ground-statecorrelations on the exeitation energies of certain collective states of Oi5 and on

their transition matrix elements to the ground state have been considered byFerrel13) and Fallieros`) (monopole and quadrupole states) and by Brown andCoworker5•6) and Barker') (dipole and octupole states). However, in the previ-

ous work we have nat taken into account ground-state eorrelations. In g4, justthe same way as in S3, according to the usual shell model to calculate theeffeets of admixtures between the ground-state (ls)`(lp)i2 on the simple shellmodel and excited eonfigurations we construct the energy matrix of the residualinteraction with these states and then diagonalize it. The wave functions ob-tained there is used to calculate the monopole transition matrix elements. rng5, these results are compared with those obtained in S3.

S Z Wave functions and the residual interactions

We employ standard shell model techniques to calculate the energjes ofexcited O" states and their transition matrix elements to the ground state andthe effects of admixtures between the ground and excited states, and aisoterminology used here is just the same as in the previous worki). Their detailsare found in reference') (which we shall denote by I), so that heTe we give onlya brief description of states, wave functiens and residual interactions assumedby us. The Oi6 ground state on the simple harmonic-oscillator shell model belongSto the eonfiguration (ls)"(lp)'2. This is the double closed shell, so that theexcited O' states belong to the exeited configurations with an excitation energYof an even number of quanta. Woking only to the lowest erder the excitedeonfigurations eonsidered here are as follows:

(a) (ls)3(lp)'2(2s)=(ls)-i(2s), (b) (ls)` (lp)'i (2p) = (lp)-i (2p),

(c) (ls)`(lp)'O(ld)2==(lp)'2(ld)2, (1) (d) (ls)`(lp)'O(2s)2=(lp)-2(2s)2, (e) (ls)`(lp)'e(ld)(2s)=(!p)-2(ld)(2s),

Ground-Statc Correlation in Excitcd O+ States of ois 59

Fer simplieity, the spin-orbit interaetion is here neglected which is supposed tobe less important in the e' states. Therefore, all calculations are earried outin the L-S coupling scheme and only the states with T=S=L=O aTe taken intoconsideration.

The states belonging to the ground and one-hole configurations are unique,respectively, while the O" states of the two-hole configurations (c), (d) and (e)

can be elassified by the charge-spin multiplieity and the resultant orbitalangular momentum of the configuration (lp)iO. Consequently, in the L-S couplingrepresentation the simple ground state and twelve iiS states belonging to theexeited configurations (1) can be written as follows:

e ((ls)4(lp)'2) !!i IV'G(O) =- Åë ((a s.)),

Åë ((ls)-' (2s)) = Åë ((1, G)), Åë (( lp) "' (2p )) i!! Åë ((1, b)),

di({ap)-2 i3,3is, ad)2 i3,3is}iis) .. tu((1, c)i3,3is),

Åë({(lp)-2 i3•3iD, (ld)2 i3•3iD}iiS) =-. O ((1, ,)i3•3iD), (2)

di({(lp)-2 ii,33p, (ld)2 ii,33p}iis) =. o(a, ,)ii,33p),

di({(lp)-2 i3, 3is, (2,)! i3,3is}iis) =. e((1, d)i3,3is),

to({(lp)"2 i3•3iD, (ld) (2,)i3i3iD}iiS) =. e((1, ,)i3•3iD).

Except for the two "S states, e((1, es)), Åë((1, b)), these itS states do notbelong to any irreducible representation of symmetric group Si6. However,s]nee such a strong exchange interaetion as the Rosenfeld type favours themost symmetrie wave functions in the orbital space we may well take those iiSstates which are of the most symmetric type [4444]. They can be easilywritten down as follows:

91 o = Åë ((c. s.)),

gi i = O ((1, a)),

Åë,= e((1, b)),

vr,= (2)"t2 [Åë ((1, c)i3S)-Åë((1, c)3'S)], (3) ip4 = (2)-ii2 [e ((1, c)i3D ) - o ((1, c)3 iD )],

Åës = (10)-ii2[Åë ((1, c)iip) + 3e ((1, c)33p)],

Åë, = (2)-ii2[to ((1, d)i3S) ny Åë((z, d)3is)],

Åë7 = (2)-il2[Åë ((1, e)'3D) - O((1, e)3'D)].

Later by diagonalization of energy matrices we shall verify that this restrie-tiOn of the wave funetions (2) to the wave funetions(3) of the irreducible repre-

Sentation [4444] is a good approximation for the residual interaction with theRosenfeld admixture•

60 H. NAcAT and M. ITAyAHowever, any state belonging to excited configurations which contain two ermore unfi11ed shells includes an exeited eenter-of-mass motion in the oscillatorpotential.8) Our shell-model wave-functions for excited configurations ineludethe spurious states in which the center-of-mass motions are exeited to lp and2s states from the ground ls state. In order to avoid serious errers caused bythe center-of-mass motion, it is neeessary to take appropriate wave funetionswith the ls state of the center-of-mass motion which we shall hereafter referto as admissible states or wave functions. InI we have found that there arefive admissible states and two spurious states in the [4444]"S states, and there

these wave funetions are explicitly given as Iinear combinations of the wavefunetions (3) in the irredueible representation [4-l44]. [See Eqs. (12)-and(13) I]. pt6 eorresponds to the lp sta!e oS th-e ceNnter-of:mass. motion and Sif7 te

the 2s states. Five admissible states Wi, Yr2, Y3, V4 and Ys which are orthogonalto the spurious states pt6 and di7 are constructed by SehmidVs method. The shell-medel wave function (2) is angular momentum eoupled, antisym-metrized product of the single-particle harmonic-oseillator wave funetions withthe size parameter (1/p)l=(hlmo)'}. This y has been determined by Carlsonand Talmi") from the Coulomb energy differenee between the Oi5 and Ni5 groundstates, assuming the radii of the Oi6, O'5, and IVi5 to be equal. The experimental

value of v is O.349 x1026 em'2 which gives he=14.5 MeV and ro ==1.3Å~10-i3 em

The residual interaetion is taken as follows:

V(1, 2) =eTi, (ri•r2) {O.3+ O.7 (ai•c2)} exp (- ri21a)/(ri2/es), (4)

where ri2 :lri-r21, Vc=40 MeV, and a=1.37Å~10-i3cm. It has been success-fully used by Elliott and Flowers in the problems of the odd-parity states ofOi6 iO) and the low-lying states of the nuclei of mass 18 and 19.ii)

S 3. Results of prelirninary maehine calculations

In I, by the calculation on the desk computer without taking the effectE efground-state correlations into aceount we have found that the lowest exelted:,ta,:gw.il.t2zh,z",7"zL.ih,;'.L-,m.l:ge,kw.a,le,,s"2e./roefi.:ozx2sgo,gg,s,,zo,,kh%,slu,:g",g

states to the [4444] irredueible representation is a good approximation for.theresidual interaetion with the Rosenfeld admixture. It seems to us that JuStthe seeond excited state with the usual shell-model wave funetions correspondS

to the true lowest excited state. In the present work in order to eonfirm moreexactly these eonclusions the complete diagonalization of the same energYmatriees is carried out on the electronie digital computer OKITAC-5090 90give eigenvalues and eigenveetors, The results thus obtained, taking as a baSiS

Ground-State Cerrelation in Exeited O+ States er Oie 61

L6

3uueq:

liv.o e vv utctn.2 ..; .Egg -a.

vv o"u u.E a tr)g.9. "'1Sm ':'a = -ue "Vl --1 -Tl "gg ss:, •s-

'g•g a"

a2 gv= ai-e )- tn "odi .eg•g' i,

gg e.pae -fi8'; lg:" ego e

-str si-g

.pa Tm`ss"b

p-.l

mtsa=ptl

x$sptthca- .

oN8r9T

oRct).

9I

ii!

sN-tsse

q21

s!E2

-

98ci

1

Ef

g":Y

x•/,g•/.

./ S'

/

co ts e an co anoann co2- g$$ ge g- s :. gs ge sl66 e' o' ddddd666crl e c) et) a) ct) (N u') co m t) .-ggNI&Nggss ft s---t--------za x es ." en. gg=EggB8S at R st su 68e892------------l11 l,- ts ao"--o en o cu o: ge ggEsgk3ekk---e--------

di gS ax g88 ;tt ge B8Rat st su st sg2xggss6 ti dd e' d6d6d6dR9 za tR 5g Est g gg R El :8886gg8 ge 8888------------g:6gg gg gg za g gt Rgy g#g88 se E92XX--e-e----e--co ene an ea ts neN"ggk ge gsgg g- g- 'g' sdd e' d6dd66 o' e' df! 8gR as S; 8g8g ge RgB za Rx$gRsgsgd6 di o' 6d di ti dd e' 6g- g $ g- s : g g- : g- E gd ci ei 6d ci dd e' e' e' e'es: th. et) c" m ,-- cc) cn cc) r..t co mge :gg fi sE g- k ge kN--------t--- Il 1111 1th -" pt en a ts Kf - ts a-:- sEEggg ge g- E ge $6dd ci 6dddd666 lll Illl 1RA eA, eq"' G "A ac ac c',E. cA,)- gA Aa

,S- E- iis, ils` S6k ':lsL S'ls• ;ls` EtsL kA ;gL i'isL

uupt- -- H- -- -- -- -"n-" "'

62 H. NAGAJ and )vf. ITAyAb) With the usual shell-modcl wave functions (3) in the [`Vl44] irreduciblc rcprcsentation

=.i',gllEIIIEIill:..I -6.s22g -ii,i46o -ig.sso3 -g,s376 -2i.g"s -i6,o7sg -i6.3ns

":

Yts

yis

"4

yr,s

?la

lyr ;

O. 7304, O. 2381 -O. 2276 -O. 0429 -O. 1633 O. 48 89 O. 3oo7-O.4215 O.8396 O.041e -O.e682 O.O170 O.3116 -O.1172-O.OO17 O.O051 O.6053 -O.4210 -O.64,47 -O.0833 O.IS37 O.0621 O.1620 O,0642 O.8335 -O.4S17 -O.1816, -O.0779-- O. 0966 -O. 0626 O. 5408 O. 3316 O. 4278 O. 2372 O. 5874-O. O193 -O. 3tl41 O. 2783 O. ]03S -O. C614 O. 67 20 --- O. 58 12

-O.5247 •-O.2996 -O.4539 O.0302 -O.3724 O.3408 O,4162

c) Wilh the admissiblc wave functions (13) in I

-- 7. 2024 -11. 6548 -- 19. 9662 -9, 8552 -16. 7034

ip, e. 9266 -O. 2699 --- O. 0638 -O. 0906 O. 2372ip. e. 2155 O. 9385 -O. 0422 -O, 0927 O. 2500iP'3 -- O. 0478 O. 0234 O. B466 -O. 4581 O. 265Bip, O. 0491 -O. 1138 O. 4tl OO O. 8794 O. 1331iit, I -O. 3oo4r -O. 18]6 -O. 2B97 O. O035 o. sgocl

TablÅë Z The perccntage of spurious statcs in thc cigenvectors in Table 1, a)*

'Nx- x-he1gcnvectors spurioussta)IIE----SL-N---s-..

]p center-of-mass motion

2s ccntcref-mass motion

total percetage

istexcitcd 2nd 3th 4th Sth 6th 7thlevel

1.6 2.0 6.5 84.9 -2.5 O.1 2.375.7 9.8 5.6 1.9 3.5 O,O 3.377.3 11.8 12.1 86.8 6.0 O.1 5,6

* The result for thc cigcnvectors in Table 1, bÅr is just the same as thesc in Table 1, a).

usual shell-model wave funetions (2) and (3), and admissible wave functionSiPri, •••, IZ7s (See (13) I), are listed in Tables 1, a), b) and c) respeetively. For Vhe

sake of easy comparison a few low-lying O" levels are indieated in Fig. 1 withexperimental levels. Percentages of spurious states with lp and 2s center-Of'mass motions in the eigenvectors are given in Table 2. By reference to TableS1 and 2 and Fig. 1, we can conclude that in either case, in which the usualshe!1-model wave function is used as a basis, the first exeited iiS state correspondS

to .the spurious state with the 2s center-of-mass rnotion and the 4th excited "S

state to the spurious state with the lp eenter-of-mass motion and just the

second exeited state to the true first exeited O" state.

Ground-Statc Cerrelation in Excitcd O+ Statcs ef Oie 63

22.29 22.48 21,80

19. 07 19, 16 19. 14 17. sil 17- 85 lz 3-SFgz 12.90 12.92 12. 43o 12.30F IZ67 -T'- 12.69E5

X 9. 0B 9. 12 9. 03t

7.02 7.os 6. os

(a) (b) (c) EXPERIMENTAL CALCVLATED Fig. 1. Lcvcl splitting of O+ Statcs in Oie

Tablc 3. Calculated valucs of thc matrix clcments for the O"-)O+

transition in Oro (in 10-:e cm!)

Matrixcicmcnt l NO,.g,',O,Y.",gsS.t,"te Ground-statc correlation :

With the usual shell-modcl

wavc functions

Åq Fc l -O l ]F" ist År

Åq yrc I l? 1 vt,nd År

O. 43

O. 48

O. 76

3. 9S

With aclmissible ss,ave functions ÅqSP'cl-OllP'i,:År O- 46 O• 38

The eigenvectors obtained above are used to calculate the matrix elementOf the monopole operator g=S..7]r?• (1-ri,)/2 between the ground and excited O" iStates- In either case, the ealculated values of the monopole transition matrixeleMent are about 12.oot of the experimental value of 3.8x10-26 cm2.

From Table 1, a), it is eonfirmed that the restriction of the i'S wavefUnCtions (2) in the L-S coupling scheme to the wave funetiens (3) of symmetry

tYPe [4444] is a good approximation for the residual interaetion with theRosenfeld admixture.

64 I-l. NAcAi and M. ITAyA

S- Ground-statecorrelatiens

So far we have not taken "ground-state correlations" into aceount. How-ever, residual interactions between nucleons is expeeted to eause not onlymixing between the excited configurations (1) with an excitation energy of twoquanta, but also admixture between the simple ground state, (c. s.), and exeitedeonfigurations. Thus, in this seetion, just the same way as in S3, according tothe usual shell model we calculate the effeets of such ground-state correlationson the exeitation energies of excited O" states of O'6 and on their transition

matrix elements to the ground state. The matrix elements of the residual interaetion (4) between the simpleground state and twelve excited states in (2) ean easily be calculated by thesame standard shell-rnodel techniques as in 1. These elements are expressedin terms of the Slater integrals Rk. For the eonveneence of calculation, further,these S]ater integrals are expressed in terms of Talmi integrals li2). Theresults are given in the Appendix. To give eigenvalues and eigenveetors thesematrix elements are used for construetion of the energy matrix together withthose caleulated in 1, and then we diagonalize it by OKITAC-5090. The resultsare given in Table 4 and Fig. 2. As expected from large off-diagonal elements,

Ground-statc cerrelation - - - - No ground-state correlation i1. 9`l, lst 12- IO. 95 10.92 t g.os 2nd T g.12 2nd t g.o3 ------m 8-;I--8i7--"-lst . 8'29 lst V'------ 7. 02 7. 06 4-i]

o------- ------- --------4-

-8-

-l2 --

v --14•15 G v "- 13. 76 G

-10.20 G

Fig. 2. The effect of greuud-statc cerrclatiens on level spt{ttingu of O+ state in OiO

Grouncl-Statc Correlation in Excited O+ States ofOi6 65

ts

e:

I!)

lie,,o. E

v:to.9a-- -Hgti a.vs e.gg es1" "=9 vv= =.- .- -U: ,EeO .-.='= N" es vL..d - Mzg •s,

uv v': v =-- =es " -E as v:.2 l.

-: tgo .o- e tn ve. ge s8be' iÅr" v=" so.: m•ge•; g

ve" BE6 .:s

- ---i: l"vbo AN

.

evsae

caenptts

oov1

"NptFts

aaco

-e

R8di

1

eaR99

1

a"Nnts

-ogny

9l

aoaneng:1

if

sgMcatscodi

l

gll

di

"1

g'/-'ts!•2-

ln

fi 8- ee af g g. thg g'i gs opt. tg go - - o o .d N cu ts .. --. m en-------------se R se 88 ge g88 st eg MSB8 Es 9=E sc sg9 : a. R-------------E- sgR fi gg g- s$ e- g:

----+-----1--pt amo pt o eo th e co oe co$s$:e fi ggs2!gEci d ci d ti dddd666d66gkg su #g88 en. !., 8R : as ge e:RgÅíg8RRdd6dd ti 6d6d di ddggggzzg-zgszggci dd di di ddddd66 ei2 z s. g os g s ,Å} $ : 'k g $

dd ti 666 ti d666d6!- zs i.' i g- :gs g- gk k-

wa g, fi k$xg:s R, s:ddd6dd6dd ci 6ddg• !• Zsi pt: g- i oz. g- : g- :,

E :- : ge ssilkg$Ez

sE. gggg'tg E. sg. z.s

1 lll IIII 1RsE: ge gsg:g ,6 E -g6d6d gogoeo909 AA i-. AAAAA'il {l .'.N. Clg" 'ill t6 9-. }..-s ?-. Z'-h M?s Il.lv :-.

g g- g g- g .g g- .g g- .g .g g- g

66 H, NAGAfand M. ITAyAb) With the usual shell-modcl wave functions (S) in thc [44`l4] irreduciblc rcprescntation

cigenvaLue

bas{s

Vro

L;ri

VT:

Yts

Yll

Vs

Yte

"7

-42.7623 -6.7001 -2.0577 --20.7115 -10.0193 -tB.0558 -16.2682 -!4.ltl12

O.7993 -O.084•3 -O.53B5 O.0803 O.0409 O.1547 O.O120 -O.1784O.i642 O.7846 e.1146 -O.l852 -O,1721 O.4-el63 -O.0574 O.2791O.280l, -O.2386 O.6540 O.1436 -O.371tl, O.3101 O.25B7 -O.3392O.3776 O.O18B O.2025 -O.O`!-77 -O.4341 -O.6652 -O.2336 O.3582O.2918 O.2012 O.4156 -O.2823 O.7030 -O.2658 -O.O097 -O.238Se.0859 -O.0380 O.2074 O.7419 O.3090 O.1675 -O.4308 O.2978O. 0994 -O. 0315 O. O131 O. I681 O. 1925 -O. 07 12 O. 7969 O. 5326

O.1025 -O.5264 O.1110 -O.5273 O.1246 O.3667 -O.2333 O.4666

c) NVith thc admissiblc wave functions (13) in I

bll;ISilEE:EIELe

vr pEYc(e)

e, ijs iv-7is

ip4 iie-s

-38. 5910 -7. 23 67 -Il. 6775 -l6. 0413 -3. 7236 -17.11 18

O. 8459 O. 0306 -- O. OI 06 -O. 1!OO -O. 5083 O, 1133 O. D785 O. 90Bl - O. 1915 O. OL19B O. 2387 O. 2701-- O. 2646 O. 2704 O. 6728 O, 3747 - O.512e O. 0289 O. 3313 -O. 0527 -O. e636 O.8130 O. 2909 -O. 3712 O. 3129 -- O. 0486 O. 7077 - O. 2330 O. 5772 O. le70-O. 0228 -O,3100 -O. 0752 O. 3602 O, 0619 O. 8742

admixtures oÅí the states of exeited configurations to the ground state are veryimportant and amount to about 36iee' and 28o/o' in all for the caleulations withthe usual shell-model wave functions involving the spurious states and with theadmissible wave funetions, respectively. Here again, although more or lessobscure in comparison with the results obtained in g3, we can eonfirm that thefirst and fourth exeited states with the usual shell-model wave functionseorrespond to the spurious states with the 2s and lp center-of-mass motions,respeetively. The eigenvectors listed in Table 4, a), b) and c) are used to estimate thematrix element of the monopole operator 9 between the ground and excited O"states. The results are given in Table 3. The values with eigenveetors inTable 4, b) are practically the same as with corresponding eigenveetors in Table

4, a). Therefore, we do not list the results with eigenvectors in Table 4, b)•

g5. Discussion

On the simple shell model, the O'6 nueleus in the ground state has beensupposed to be a double-closed-shell nucleus and hence very rigid. Therefore, if we make an appropriate choiee for a model wave function

Ground-State Corrclation in Excitcd O+ States of OiS 67

and moreover a residual interaction taken by us matches with this model wavefunetion, admixtures of excited config"urations with considerable excitationenergies to the ground state are expected to be very small. AceoTdingly, theeffects of such ground-state eorre]ations on the excitation energies of O' excitedstates of Oi6 and on their transition matrix elements to the ground state should

be, also, not very important. On the contrary, on the view of Tables 4, 5 andFig. 2, our present calculation shows that these effeets of ground-state correla-tions are very important.

On the other hand, in I in order to understand the effeets of configrtrationmixing between degenerate configurations with the same excitation energy of2fito, we have compared the results obtained on the single configuration modelwith the results in configuration mixing. The comparison has shown that theresonance energy amounts to about 1 MeV at most. Contrary to our expecta-tion, by configuration mixing between degenerate exeited configurations wehave improved only a little on the numerical results for the exeitation energyof the O" excited state and this monopole transition matrix element to theground state. This is beeause of the smallness ef the matrix elements for sueheonfiguration mixing.

However, the present ealculation, taking ground-state correlations intoaccount, shows that the admixture of the, states of excited configurations at 29

MeV to the ground state amounts to about 30% and that the ground state isdepressed by a very large amount of about 13 MeV. The amounts and effectsof admixtures of the ground state to excited states are comparable to or morethan those in configuration mixing, mentioned above, between only excitedconfigurations (1). Indeed, the effeets of ground-state correlations on the low-lying excited states are much smaller than these effects on the ground state,still the former seem to be much too large in comparison with configurationMixing between degenerate excited eonfigurations (1) because of the largenessOf the unperturbed energy separation between the ground and excited configu-rations.

These results are quite ineonsistent with the simple shell-model supposi-tion about the O'G eore mentioned at the beginning of this seetion. On the otherhand this supposition has been supported by various authorst3) by means of

SPeCtroscopic ealculations on the shell model in which any form of the radialParts of the single-nueleon wave function and the residual interaction has notbeen chosen explieitly and the Oi6 core has been assumed to be very inert in(ld, 2s)-shell nuclei as a hard eore,

. The explanation of this diserepancy seems to be found in the greater partin the harmonic-oscillator wave funetion and its too Iarge shell spacing used inthe calculation. Instead of the harmonic-oscillator well of infinite depth and

68 H. ri"AGAi and M. rTAyAsteep ascent a more realistic well of finite depth and gentle slope gives wavefunctions with larger tails and smaller wave length so that the Bhell spacingsand the overlap integrals for the residual interaction are more deereased forthe higher level and radial matrix elements sueh as the monopele transitionmatrix element are increaged, for fixed nuclear mean-square radius. L The present work was partially finaneed by the Grant fer Seientifie'Researehes of the Ministry of Edueation.

Append ix

Eehe Slater integrals in Table 5 are as follovvs:

ek (n.lanblb) = I:I: Rn.tb(ri)Rnbtb (ri)Rn.tb(r2) Rnbib(r2)fk (ri, r2) aridr2,

and

Rk(nalanblb, nclcndld) = S:!:Rn.i. (ri)Rn.i.(ri)Rnbrb (r2)Rndid (r2)fk (ri, r2) dridr2•

Here fh is defined'by

Table 5. Thc matrix clements of the residual interaction (4) between thc simple gtound state (c. s.) and tsvelve excited :iS states in (2)

state 1 Matrix ciemcnt 'O((1, a))

O((1, b)År

o(o, c)ias)

Åë((1, c)sis)

Åë ÅqÅq1, c)':D)

o ((1, c)aiD)

e((1, c):ip)

Åë ((1, c)ssp År

O((1, d)i:S)

e(o, d)ats)

e ((1, e)iaD)

e ((1, e)iiD)

-g{RO(1s2s, 1s1s)+4Rt(2.s1p, lp1s)},

-gVt[itl][RD(ip2p, ipip) +i3Rs(:p2p, ipip)]- Bll[3Ri(is2p, ipts),

i- 7•t(sT{i4G'(ldlp)+9Ca(ldlp)),

3 3svrs- {14C'(1dlp) +gGsadlp)},

1- 7v3-s{49C:Åqldlp)ÅÄ9Gi(ldlp)},

3 3sv 7s (49e'(i dip) + 9Gs(ld lp)},

EiltTV it]{-7C:(1dlp) +3Gs(1dlp)},

-l7-VIIIIs {-7ciodipÅr+3cnodip)},

-R'(k2s, lplp)=--Gi(lpds),

3-'s'Ci (lp2s),

V72irRi(ldas, lplp),

.gvT2 R;(ld2s, lplp),

Ground-State Correlation in Exeited O+ States of Otfi 69

Vc exp (- ri2/a)/(ri2/a) = E]fk (ri, r2)Ph (cos w), kwhere Ph(cos to) is a Legendre polynomial of degree k for the angle to betweenthe veetoTs ri and r2.

In the fol!owing expressions, the Slater integrals neeessary in these cal-cu!ations are given in terms of li.

Ci(lp2s) == -ftlT (111o - 41Ii + 6512 - 35r3),

Ro(1,2s, lsls) == 1/ gt: (lo - ii),

1 RO (2slp, lslp) = 4sf6- (lo + 41i -- 5I2),

RO(lp2p, lplp)= ft l/it (3io'Ir+512-713),

Ro (ls2p, lslp) = 'pll- }/t2 (Ie - I2),

Ri (2slp, lpls) = -II' }/St (lo " 61i + 512)J

R' (ls2p, lpls) = i1/t2(3Io - 6ii + 3i2),

R' (ld2s, lplp) = 'k- J/-:-(-- Io + Ii -- 712 + 7I3),

R2(lp2p, lplp) == e1/llti (- Jr + I2)•

ReÅíerences1) H• NAoAr, Prog. Theor. Phys. 27(1962År,619•2) See, in detail, refcrence 1). V. GT-ET and N. V:NH MAu, Phys. Letters 1 (1962), 25-

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