Bull• Kyushu lnst. Tech• ' , '

Bull• Kyushu lnst. Tech• ' , ' (M. & N.S) No.1, :955 ' Properties of Light Nuclei, with Harmonic Oscillator Wave .limctions I

Hitoyuki NAGAI and NTobuo ]ELAYANO

Department of Physics, Kyushu lnstitutc of Technology . , and

- . Yoshimi,YAMAJI

` . • (Received November' 15, 1954)

Abstract: We investigate tlie properties of light nuclei (Z=9 to 20) on the' bnsis ' of the individual particle model with harmonic oscillator wave functions. In this paper we Consider the configurations Åq3d{}-)S and (3d", )2 (2s",)i in the jJ:coupling, and tlie configurations (3d)S and (3d)2 (2s)' in the LS and in"tetmediaJte cotipling.

In order to caTculate the energy matrix elements of the tenso'r interaction and '

.

the mutual Spin-orbit interaction introduced by Case and Pais, the Talmi method is '

extensively used in every coupling scheme. In the jj-doupling, by using'nucleon•

-

t.

nucleon interaetions hitherto proposed by various authors to exTplain nuclear two- -

bedy and three•bedy data, we cannot explain the occun'ence of the ground state with J= -g- for NeM and Naas, and the (3d.E)2 (2sJ.)' configuratien for F'"

LÅr - .-

tr

gives us tihe level scheme svhich .qualitative)y coincides with the observed trend,, but doe9 not give any 'explaDation for'the deviation from the Schmidt line. If '

we assunie tliat the strong spin-orbit Coupling required by Mayer's shell mode!

'

be entirely attributed fo the Case'&'Pais' mutual spin-orbit interaction,'it seems that for thesie nuclei the interinetliate coupling is more probable than the jJ'-coupling, because in the'LS coupling the contribution from central iriteraction '

within the unfilled shell is considerably Iarge compared with tlie contribunon'from '

int'eraction between the unfilled shell and closed shells. The inagnetic rnoment of F.i9'calculated in the intermediate coup]ing is in cxcellent agreement with the -

' 6b9erved Value. From a rougli estimation it seems that the occurrence of t}ie' "

ground state svith J=-g- for ii"e2i ana Na23 is ex'plttinecl by the intermediate'

--i -

coupling. And sve calculate the niagnetic'and guadrupole moments of Naes and obtain the faiir results.

Introduction

/ 1J h, 1

The'Strong spin-orbit couplingittshell model" proposed by Mayert) and Hax'el-Jensen- Suess2) from; a. consideration of the pronounced stability of cez'tain nuclear species, succeeded in exp1aining manY properties of nuclei, particularly the prediction oÅí spins of ground nuclear states te a suprisingly good approximation in spite of the extreme simplicity iof the model. 'Namely, the distinct division of magnetic rnotnents of odd-neutron or odd proton nuelei into-,two gtoups has been excellently correlated sVith the ,extreme single•

partidle Schmidt line, Further, the C"degrees of forbiddenness" of the beta decay of odd

23

24 H• NAGAI nndNHAYANO and Y• YAMAJI ' " ^{. '} '' ^,.

nuclei, and also matrix elemcnts of some ' gamma transitions calculated on the assttmptaon that the single odd nucleon alone takes part in the transition are generally in atiqreement

isrith experimental data within an order of magnitude. However, at the same time a number of features have become apparent which seemed to contradict such a single particle model, such as the large values founa for the m,atrix element' s of a nuinber of E 2 transitions, the sign and magnitude of quadrupole moments, and the considerable deviation of.nuclear magnetic moments to the inside region between two Schmiclt lines. The last deviation has been considered to suggest that the orbital•angular rnomentum is nota good quantum nnmber. rt was once suggested that the direction of deviation of the magnetic, monients be exTplained by a mixture of an l==j-+-l, state svith an l==jÅ}l state. But it' has been soon pointed out that tlris interpretation should be rejected since these two states have oppostte parity and canllot combine. Next, the contribution of exchange currents to the observed mabonetic rnoments wus examined in order to explain this deviation, but diis contribution was found to be not s.ufficiently large,'and can give rise to the deviation to the out$ide Tegion as well as to the inside of the two Schmidt limits. At present it may fairly be said that it is still questionable how important their contribution is. And also a quenching

mechanism of the intrinsic magnetic mement of the odd nucleon has bqen propoEed by Miyazawa3), etc.

The existence of the proten forbidden zone between the two Schmidt lines which has as its lirnits the Dirae line$ seems to supporL the use of this quenching mechanism to explain the deviations from the Schmidt lines, but the occurrence of a forbidden zene for the neutrons raises the question whether this coinoidence is not accidental, Therefore, it seems that a complete neglect of the structure of the {tcore" of an odd nucleus which consists ef the rest of the nucleus except an odd nucleon and its consideration only as the "can'ier"

of the required central nuclear field is an oversimplification of the problem. We qan divid these ways into two groups by which effects due to a "core structure" are taken into account. In the first way we take account of only the nucleons in the unfilled shell (so- calledLextra nucleons) and consider that completely closed shells do not make any contri- bution-to properties of nuclei and aTeFregarded as t"a spherical core", what is called the individual parijcle model. In the secend ene we consider that the cor-e is deformable liquid drop deformed by coupling with the odd particle (so-called collective rnodel)`). And then the combination between two models has been considered by various authorsS). The collective model has been proposed to explain too large values of the nuclear quadrupole moments ef heavier nuelei. At present it is likely that this model may be appropriate for heavy nuclei with more than eighty nucleens Tather than light nuclei. But in this paper we intend to investigate properties of light nuclei (Z==9 to 20). Therefore we wil1 perforTn

all caleulations based on the individual particle model. - L

At any rate the essential feature which is common to all types of the shell medel is the

aEsignment of the CCconfiguration"' of extra-nucleons. If this assignment of C'the configuration

of nucleons" represents the physical feature of nuclear structurei we may work with the

i roperties of'Ligllt ,Ny.clpi,,}vith, Harmpniq P-scillator Wave Functions I 25

methods of atomi.c spectrescopy in,the predictiop.of,energies qnd spips of Iower excited ,s.tates pf nuclei. Such tTeatmenls of nuclei have heen already carried opt by.various authorse). It was pointed out thatl for heavier nuclei did the jj-coupling take place, but -in the'liglltest. puclei the, L-S collplirtg treat:nent might predominate over•the jj-coupling.

Moreover the ,rnagic, numbe]'s Z er N--8, 20 can be explained not only, by the streng ,jj-coup]ing,.but also by the L-S conpling. , Therefore, we can suppose that the transitiop froma.the L-S coupling to the ,jicoupling takes place in the region where ssre will investigate proper-ties:,of-nuclei. So we carry out the calculations not only in the jj-covplipg,• but also

in the'L7S coupling and the intermediate coupling. • • i '

'

-. .As pointed.out by TalrniD, the Slater metl}od which is very avai1ahje for the atomic spectroscopy is not always advantageous.in the nuclear spectroscepy, because the form ,of the nuclear interaction potential and its exchange character aTe not sufficiently kno.wn, and there exists a lack of sufficient experimental 'inforrnatien about nuclear spectra-to determine '

the several paraineters used to describe the nuclear Ievels and• for non--central interactions which must here be taken into account this inethod looses even its formal $implicity and becomes complicated. However, we ,choose as the individua.1 particle "'casre functions the isotropic three-dimensional harmonic oscillator wave functions, so sve can adopt the Talmi met}ibd to ctilctilate variouS ihteraction matrix elements and avoid disadvaniages of the

'Slater'me]thiOd. ' Our choice'of wave function for individual nticleons may be re'asonable i

i

' fro'Th 'the physical situatiori i'n tlie light nuclei. (Chtipi 2) '

L '

'Iri this'P'relitninary Papel' ",e have taken up F'", Ne?' out bf nuclei with Z = 9 to 20.

The configuration of the extra-preten in Fre is (2si)l, and that of the iexLta-neutrons in

L 4p.

Ne?: is.(3d.E")eN by Klinkenberg and eteS). We supiose that tbe configurations of even extra o

nucleons in Fi9 and NePi are (3d-,s,.)3'N aucl (2sx)t, respectively. For extra-neutt-ons of FiS, there

.' • o,- ,

is the possibility of the conf]'gur'ation (2sk)?i, too. But for this configuration the protpns "

cpup,4e.tQgetber to .the.cqrrlple,tely clofied sliel{, so the magnetic moment of F'g must'be

tF J. rlt t

/t

"

p,qu{il to ,that of the proton only. Further, onr assignment is in a.frreement with the Ievel

/t

o,r,der give.n, by iSi.fgyer:, For extTa protons of Ne2'i there is the possibility of t})e configuration (3d-l,)r',• Acco.rdin.rr,to, tbe level order given by Mayer, this assignment should be taken

t. Tq,th,e.r flian purs., Froip the fact that the configurations for.odd-pucleens are different in

-1,t Lt

t.t

1 F.'9 ,andiNe?l, qg me,ntioned above and that the eonfiguration for.Nar' is {Åq3dgÅr' (2s-)2},

-1 {Åq3d,.s.),i.(2s+)?}Nl.Iit.is likely,that this Mayer's level order is ngt always rigorous, iand -.qie.se" two l,e"v,els is very .close. As one of, reasons for an occasional reversal of thQse,two ,levels jp "protQn an,d neutron shell sc,hen)es,xs'e could probably reekon "ivith the consideratian prgpgsediby lnglisP) ,tp explain the diffet'ence ,ef cloublet splitting in ,LiT and BeT.,our assign, ,mept oS the,configuTation (2sl):, for Ne2' makes the proton shell completely closed

an4 therefore, thq mathematical treiatment becornes very easy. . . .

In Cliap. 3 we discuss the c6hfigurations,{(2sl,)l, (3d.i-)2N} and {(2s,!):• (3de)S.•} witlt

sonie phenomenological two-body nuclear interaction potentials introdiced bSr various authorsiO) to explain many- two-body experimental data and sometimes t/hree-body data.

I

26 H• NAGAI and N. HAYANO and Y• YAMAn

The contributions of eentral interactions, tensor- and mutual spin-orbit interactions to every levels are calculated by the sum-method, If on]y the configuration ef equivalent nucleons is considered, of course, the results for l e2i are suitable for Na23.

In Chap. 4 the same configurations as that in Chap. 3 except the omiltecl j-values are tTeated in the L-S c:oupling. The sum method is also used to obtajn the central contrih- ution foT the term energies in the L-S coupling, but is not applicable to the calculation of the contributions of the non-central interactions to the ]eyels. At the first place we

get SL eigenfunctions by the angulaT momentum opeEator-method and then using these eigenfullctiens we coinpute the matrix elements for (J =S+L,M=J,S,L)- states and finally

-- have the contributions of mutual Case-Pais spin-orbit interHctions and tensor interactions by employing the Ltnde interval rule and the Racah coefficients, respectively.

In Chap 5, we calculate the non-diagonal matrix elements for non-central interactions by the same method in Chap. 4 and discuss the intermediate coupling for the configurations {(2s)i (3d):}p and Åq3d)k and apply these results te the nuclei Fi", Ne?', Cl3: and Nare.

2. Method of calculation

Since the shell models are in fact the former Hartree approximation, to the zeroth-order the wave functions of equivalent nucleons are antisymmetr'ized linear combinations of pro- ducts of the wave functions of the single Ducleon in the averaged central field. The zeroth- order wave function of N nucleons in the state charaeterized by quantum number A which consists of the inclividual sets ai, a2, ••- aN is :

il,'A(•fti, x2, d-• xN) == 71-Nr2y (-1)P ua:(xi)ua2(x2)•-• uaN(xN) (1) (for example, in the L-S coupling A are Mn and Ms, at's are ml and ml, where The' zeroth oraer` energy of the nucleus is' degenerate, since it is the same for al1 the states of the configuration. If the mutual interactions of the nucleons are taken into account, this degeneracy is partly removed. The first order energies are the diagonal matrix elements of the interaction energy with the cofrect zeroth-order wave functions belonging to the states characterized by L and S in the Russel-Saunders coupling or J in the jj-coupling.

The correet zeroth order wave functions are definite Iinear combinations of the wave functions ytA Eq. (1) having the same quanturn numbers which determine the total system (e.g. the nucleus). Therefore the matrix element of the interaction energy is a sum of matrix elements (AIVlB). Here the interaction energy V has the form iAli,, V(ij) where V(ij) deseribes the mutual interaction between the i-th and j-th nucleons. so the matrix elements (AIViB) are zero except for the t}iree eases, those in which B differs from A by two individual sets (case a), by ene (case b), by none (case c) For these three cases, (AlV1B) is given by the following formula;

(AIVIB)=Å}[ffuak'(xi) uat'(xj) l7(ii) ubk(xi) ubt(-t'j) d7i dV

'L:'reperties of Light Nuclei with Hannonic Oscil]ator Wave Functio ts I 27

---ffuak'(xi) uat*(.ftj) J7'(ij) ub,(xi) ubk(.vJ') d7'i, di7"Li], (2,a) ' (AI JZIB)---Å} {i,l[ffuak'(•ri) uat'(xl') V(ii) tr•bk(-ri) uaL(xl') dTi (l7'j

--ffUak'(Xi) uat'(xli) V(ti) ua,(xi) u"bk(xj) d7i dlj], (2,b) (Al l"IB)==,.Z7-,[ffttak'(xi) uat'(.rj) V(ii) itak(xi) uat(-x'i') dTi clT7'

, HfJuak'(xi) itat'(x7') V(ij) ua,(.xi) uak(x'i') clTi tITi ]. ' (2,c)

Consequently the matrix elements of V in the zeroth order scheme are the sunis of matrix

elements of the foIlowing type: '

.II"l,ffua'(7],ai) ub'(1,a?) v(1,2) u'c(i,a,) ud(tL,a2) dir] d3r2. (3)

The coeMcients of these sums depend on the operatoTs commuting svi!h the interaction energy and are independent of the exact form of the wave function. These coefficients for various cases have been given by Condon and Shortley'i), Racah]2) and the othersi:).

'

tt/

gl The S]eter methodt`)

In this section we shall describe briefly the fetiture of the Slater method in the case of the rnatrix elements for the central interaetion energy. Iri this ease the matrix element to be calculated is:

ffv.'(t) "b'(7,) v(J7,--r• 1) thc(7,) yrd(-,,) "r, d3r,. (4)

--

V(lr,-r2 i) can be expanded in the series ef Legendre polynornials

- -+ ec

J77(lr,-r,1)=,]X.,77lr(ri,ra) Pk(cos tu:,), (s)

fk(r,,r,)= 2k2+1 ftl pT(I7,-:. i) pk(cos a,,2) "d(cos (D:t),

-- where toip is the angle between ri and r2. The single-nucleon eigenfunction is given by the following:

.1

Yra(rT)= ,,-R,(nala) e,(lamta) q),(mta). ' ', (6)

when we snhstitute Eqs. (5) and (6), the matrix element (4) becomes ' •

,X"-,focoJooofk(r:,r:) R,(nala) R2(nblb) Ri(ncge) R2(ndld) dr, di,

Å~forrJo2"fo"fo2"pk(cos(o,,) e,(lamta) 0?(lbmtb) (E):(lcmia) @,(ldnztd) (I)r"(mta) Åë2'(,nlb) (I),(nnc) Åqb2(mld) sin e, sin e2 dei da), de, (l{p,. (7) Further, by using the Hddition theorem

Pk(cos toi?)== 2k4+rr1 .;II"t-, ei(km+) (E)a(km) q)i(,n)(I)?'(nt), ..

28 H• NAGAI and N. HAYANO and Y• Ymm

the second factor, involving only the angular integrations, can be expressed by the product of two integrals of the following type:

ck(irni,"mr)==1/T5/i,-tTl]fiie(k,mi-rni') e(bni) e(i'Tii,i) sin e de.

We can easily obtain these value by using the Gaunt's formula.

Finally, the matrjx element (4) becomes

:l ] ck(lamla,lcmtc) 'ck(ldnud,lt"mlb) Rk(nala,nblb,nclc,ndld), (8)

lt=1ru1

where Rk(nala,nblb,nclc,ndtd)=S:S,"fk(r,,r2) R,(nala) R2(nblb) Rt(nclc) R2(ndld) dridr2 (9) In the case of the atemic spectroscopy where there are a great many ef experimental data [g,r.,m.a"g,e,".i'.g.y,le,,ei%tS,ek",d,,i,1'.,1"le,grg,s5.'2e:d,3.oli.i".f:cl•,9e.,c,o.Tp:ke.d,;a:.S,a,r:

nuclear spectroscopy availttble data aTe very few, so we cannot employ such a treatment and must iperform tlie calculations of all Rk's. But even fer usual eentral forces• fk's ate so complicated that the Slater method is •impractical and still more for complicated inte•

'

ractions, such as tensor forces and mutual spin-orbit interaction.

E2. The Talmi methodD

In tJie matrix elements (4) which we must calculate to determine nuclear state ene-tgi-ts tlLLe mutual nuclear interactions of any type depend only on the relative co6rdinate 'r=r?-ri spacially, and on the other hand remaining factors are the products of individual nucleon wave functions each of which depends only on the single•nucleon ceordinate. Thergfore.if we can express these preducts as products of two functions of ==1- 7, and fi= riSr2, respectively, by a comparatively simple way, our labors for calculations are consideTably reduced. Our hopeLs easil-y, satisfied provided that w!;en we transform the individual nucleon coerdinates r, and r? te the relative coordinate r and the coordinate of the center of gravity i{ the Hamiltonian of the uncoupled two-nucleon systern is separable with respect to'i; and 7. That is to say, th'is is the case if the petential of -t-h,e cen!Eal aveTaged field, which is the sum V(r?)+V(r;'), is separable in the eoordinates r and R.

Then we try to find the condition for such,a deeompQsition. . .

- --- ---

' Sinee R=l(ri+rf) and r=r7-ri,

re -. .

x= rt",= R7+ 4 -- R.r =g -- n, .- "

P -..

y=",= R2+ 4 +R.r=g+",

wherei.

- -,-

g = R7 + m4--, ny==Rer.

If we assume that the central averaged field V(x) is a regular function of x(;llO), we can

Propenies of Light ]Nuclei vrith Harmonic Oscillator Wave Funetions I 29

expand V(x) in a power series of x;

n

V(.r)=:i] avxv, , (9)

v=a

so

Pr(r";,)+ JZ(rt)= J7(.r) + V( }r) =,=ec., ay(.rv +- )t") =.2M., av[(g-v)p+ (g+ ol)"]•

This equation is separable in R andrif and anly if for p=O, 1 av's are not zero and remaining ay's are alI zero. As the term forv==O is the constant contTibution to the energy, it is generalJy omitted. Finally, we have

V(r?')+J7(r.) =a,rf+a,n"2=2a,(R'+ f )== l,(4a,R?+a,"). (10)

Therefore the central averaged field is the potential of a Eingle nucleon bound harmonically, and we shall employ the harmonic oscillator wave fiinctions in erder to ]earn the behaviour of the energy levels and other propenies ef light nuclei, On the other hand, se far'as the light nuclei are coneerned, cempared with other model wave functions it seems that,

physicElly, those of the harmonic oscillator are quite good, as the ]evel order in the shell model is in agreement with that of an oscillator potential well.

The eigen-functions for the Schrodinger wave equation with tlie Hamiltonian of the Harmonic oscillator:

,lil== 21. ([P2+m'(e?r') .

are given by the following: .

Yrnim(r,e,g) = R",iSL')- yil"(e,rp), Rnl (r) == N,it e-t" rl + i v,,1 (r) ,

Lt

where v= W ^iM / , Nnt is a riormalization factor, and vnl is.an associated Laguerre polyno- rnia1. The first few functions which wil1 be used in the following are; ilt

'

' for n=:O

Ri(r) =:Nie--;"'Tt+i and N:t;=i/-ii-il.'3'if.?/i7i+i)•

for n==1

vit=i-2t2+"3" and'tv't:ifll-;l12,t3+l.((221i++3i))- for n=2

v2t=-i-2i`."3r"+(2i.34)"i2i.s)r` and N,?=";ig"tlll'iilT3(--2.I;?3i(.2ii+)52-.

Of course,

f,coR2ni(r)di= 1, J,"f:f'l Ytm(e,q))Fsine de dgp=i.

As the zeroth-order Htmiiltonian of two nucleons moving in the field of the oscillator

potentia1 we take Hb:

30 H• NAGAI and N. HAYANO and Y- YAMAJI

HTo`= 21. (P?i+m2o'Bi)+ i. (P+m2to'O. '

when we make now the cenonical coordinate transformation:

- -- --År --) -

-

' r=:r2--r], p..?' ^SPL-L

--)' i

ft.,,Lr?--+i-r.!--, F,=pt.,'. i

we have, by substitution ai=lmo: in Eq•(10),

H,= 21ii (Pe'+M'ck)'R2)+ iA, (p' -l-lk'e)lr2), , ,

where .

M:2m, p==Sm•

This is also the Hamiltonian of two harmonic osciilators with masses M tmd p. Therefore, the calculations of matrix elements (4) are reduced to the cemputations of Integrals of the forms

int =- .f gCR'.i(r) V(r)dr

and

lnkn'r=Se'eRni(r)Rn'l'(r) J7T(r) dr. . -

These -integTals with n, n' =F O can be easily expressed as sums of integrals It•t which we shall write simply as It in the following.

From the fact verified in the previoug paragraph, we can'express the wave function yr:,Z(T:,) Vt !la(72) of two nucleons with delinite quantum oumbers niltm, and n24nts, as a linear combination of products '""ilifL(X) ftI";:A(Tir) of eigenfunctions qf harmonic oscillators with mafises M and p respectively, where ni, n2, n, aDd N are the number of nodes whic,l characterize these wave functions, and li, 4, L, A and mi,m2,M,m are the angular momenta and these z-cornponents respectively. Concerning what values of N,L,M and n,A,m should appear in such a expansion, we have the four restrictions: The conservation law of the z-component of the orbitaJ angular momentum (1), of the energy (2), and of the parity (3) and the symmetTy reguirement (4). From these fgur resnictions, when 4 Fnd op are ' both even or both odd, we haVe. •

r,',t!'f---:1ap,:,1,},(7,)dg,•;f.,i,(Til,T)+r•,r)rz,l,(TI,T)y•L:,l,i,(';T,)l

=XÅqnili,ni,n2e2,n2 I NL M,nAntÅr I,;nSfL(il)ab;ll.(T;) +(- 1)iii;rll3L(',)TlnT.(il)], (]1a) .

M+m=mi - me

-

2(N+n) +L+A=2(ni+n2)+li +4 '

A==even ,-: '' '- :

Propertiee of Light Nuclei with Harmonie Oscillator Wave T"unctions I 31

,/i, Ialn,m,,},(7,)y.;;,:,(';,I) -ytzl,},( r•)ftie ,",2,,(T:,")l

=2Åqnilimi•,i2A.nt2 1 IV]LM•n A mÅr [v.ilIII.(Iii)v,-;'r.( r)T +(-i)iTv,ril[IL(7)ik•1".(IiilL)] (iib)

M+m :=mt+m2

2(N+ p) +L+A= 2(ni+n2)+l; + Li

A=odd

lf one .of l: and 4 is even and the other is odd, we have .,/}-2-[v,:lj,(':i)ynlll,i,(';l•)+th,",',},(1•)"r;;IE,(7,)l '

=2Åqn,limi,n24nlelIVLntf,niimÅrYrlil3Ll (Xi)Vr?,.(T]lt). (12) M+m==mi + m2

2(N+ n) +L+A= 2(ni+n2)+lt +4 '

A= even

In the same case, the expansion of the antisymmetric wave funetion is easily obtained from

- -ÅÄ

Eq. (12) by interchangiDg R with r and multiplying the coefficients by (-1)h. In order to find the coeflrricients of the expansions, we write down the Ieft side of Eq. (11) or (12) in the case gi=g2=ca=cb, and express them as functions of R and r,and finally compare these expressions with a linear combination of the admissible wave functions V,NIXIL(R) Y.I,IA(r) according to the above conditions. But we cannot obtain the coefficients by such a way in the cases of antisymmetric wave functions with ni==n2, IT=4, and m,=-rn2. In such eases the expansien is easily obtained by operating the angular momentum operator (llx+l2=)Å}i(lly+l2y)=(L=+Ax)Å}i(Ly+Apt) on already obtained antisymmetric wave '

'

'

functions. The results which we need are given in the Appendix I.

'

-3, The jj-coupling

gl The low states of Ne2: and Nace -

r,

i

Since both in Ne2' and Nan the odd nucleons are in the dj -state according to the 2 Mayer's level scheme, by her single nucleon model both nuclei are expected to have spin

=- 25 in the ground state. However, it is confirmed by many experiments that for both nuelei the J= S- state is the ground state. So the configuration of equivalent nucleons to the odd nucleon, i, e., (diÅrS, has been considered by various aut]iorsi)G)7). In the

2 - • long range approximation Racah'5) showed that for any n in the j" (of course, n is odd)

configuration the' gr6und state has the 'minimum spin allowed by the Pauli principle, namely

3/2 in the case of three h'ucleons (or holes) in a shell, and l in the cases of more than

three nucleons, On the other hand Mayer:) showed that in the case of the S-type intera-

ction the ground qtate spin is J==j. Therefore, iÅí one passes from one limit to the other,

a 6ross-ovhr of these twQ•levels oCcursi Kuraih and Talmi huve found that this cross-

over occuts in the case of the Gaussian potential at X=1•32 (where X=rol/-v"-7; ro==force

32 H• NAGAI aad N. HAYANO and Y• YAMAJI ,

range), namely

at re :nuelear raClius R. Further, Talmi has pointed out that the value ef X at which the crosg.-over occurs becomes too big, especially if we assurne that the potential is a "deep hole" potential (e.g. Yukawa potential), rather than the "Ccut ofP' potential (e.g. Gaussian or rectangular well). On the other hand, as far as scattering experiments are coneerned, it is pointed out tliat the "Cdeep hole" potential is better than the "cut--ofP potentiali"). It

seems that in the nuclear spectroscopy the t"short range" approximation is more reasonable than the rtlong range" approximations. The results outlined above are obtained for the bentral interactions of the Wigner and Majorana type. In this section we calculate the energy levels belongillg to the configuration (d-g-)3 with the various two-nucleon interactions introd-

2r uced by many authqrs te explain two-nucleon data and sometirnes three-nucleon problems (H3 and He3), and discuss influences of exchange types and tensor and mutual spin-orbit interactions on nuclear levels. We use the following nuclear interactions:

V=(--67.8 Mev)}(1+P.x)S[(1+ny)+(1-ny)Ql ei",'- , (13.a)

where

p" =1.18xlO-"cm, n=O.686.

V=l(1+P.)(-46.1 Me.)[ e'ju"3' +O.54 S,, eZit,' ], (13.b)

where

ILn' i -•-- 1.1B x IO-i3cnz, iLri = 1.69 x 10'i3cm.

-- --

s12= 3(o]-r) -(c2.r) -:-(:l.;2)

V= (-25.3 Mev)[}(1+Ppt).Ii(1--")+(1-v)Ql e-pa"i,' +v(a+bPx)S;rgZffi;-t,' ], (13.c)

where, a=O.37, b=:O.65, and for p•.-i==p,-`'=1.35Å~10-'3cm, "=1.4034 7=L9 and for pts'=1.35x10-'3cm End ibg"==1.82x10-'3cm, o=O.877, 7==O.45.

--

v=(-- 46,g6 Mev)("5T2)[[i- ?g+ lg(r;,.a',)] e'p',',e' +s. eZ '], (i3.d)

where g=1.985, 7=O.5085, pat-'==1.7Å~10-'3cm, pc/pt==1.44; • V=(-49.35 Mev)l(1+Pec) e'p",;' +( -.22 Mev)s. e)"tt,'- (l3.e)

where,

pa." =1.14 x lO'tsern, ";i=L25 x lO-'3cm.

Further, we consider the effect en'levels of the intreductien of the mutual spin-orbit interaetion proposed by Case and Pais to explain nuclear,scattgring'experments, on thg bases of the charge-independent nuclear forces:

r

Vcp==(24 Mev)[(ti-72)Å~ (p-'i-R'2)](ll,+lll)(aPx+B) 'C,P d(,3,,p)(e,/,',O; ), (14)

where rep == 1.l x lO-i3cn}.

Properties ef Light Nuclei with Harrnonic Oscillator Wave Iinnctions 1 33

In the first place the energies of the various states of the configuration (d.E-)3 with ce"trel forces are calculated by the sum method frem the diagonal matrix eletnenis of the (ji,j:,

m]' i,mi'n.)-seheme. The wave functions of a single nucleon with .criven n, l,i--l+a-, and mj is given by

u (nij=l+lmj t i)= ]/tLIt llil]v'."'-u(,ilnij - id, p)xl (o,) + v/i '-2'-;•i'iit (ittm7 +l, l i)xi.'"' (ff,),

(15a)

or briefly

rs(nijmj i 1) =fue(nlmj --bi 1) )C//(cr,)+gu,(nb7i+ ll 1)Xil(o',). (15b)

We define the direct integral J and the exchange integral K of any interaction V(12) in

the (jij2mi'imjs)-scheme by: '

J(nlimi',n'l'i'm'j) ==.]ii, ).,ffu"(nijnut 1)u'(n'l'i'm'j l 2) V( 12)u(nijmJ' [, i)u(n't'i'm'j ! 2) Å~

d!riaSr!,

K(nl]'mj,n'l)"m'i')=.:,il.,f ftt*(nll'mj 1 1)u•'(n'l'i'nt'l' [ 2) V(12)u(nlinlj i 2)u(n'l'i'm.'j l 1) x

d3rid3r!. L

When the summation over the spin coordinates is carred o"t, we liave finally a sum of matrix elements of type (4) in the (l,"ml,m-l')-scheme, the coefficients of which are prod- ucts of f,g (of the mj-function) and of f',g' (of the mj').

From the spin dependency of the wave function (15) it is easily shown that JB(mj,mj')==KM(mi',mj') and KB(,nj,mj')==.IM(mj,mj')

JH(mj,mj')=KVIC(mj,mj') and KH(nlj,wf)==JM7(mj,wf),

and that JM(mi",ml') and KM(mj,mj') are expressed by expansions with the same coeMci- ents as those of JW and KW respectively, but these matrix elements of type (4) have last two arguments reversed in order. In our case (l= 2, J----g ) the matrix elements for ordinary (Wigner) 'forces in the Q-i,is,mj,mj')-scheme are given as follows;

J( 2 , 2 )= s J(2,1) + s J(2,2), J( 2 , 2 )= -s-J(2,O) +--s--J(2,1), J( g, S)., i3 J(i,o)+ ,8, J(i,i)+ ,3, J(2,o)+ ,2, J(2,i),

JÅq-:-, - S )= is J(1,-1)+ S; J(1,O) + 22s J(2, -- 1)+ 23s J(2,o)

J(-g-, --,3-)= g J(2,-2)+gJ(2,-i), J(g, -S)==gJ(2,-i)+gJ(2,o)

J( g , - g )=- ,`, J(i,--2)+-,'il?-J(i,-i)+-i-, J(2,--2)+ ,`, J(2,-i) J( S ,- i )--,6BJ(o,-i)+ ,9, J(o,o)+e,-J(i,-i)+ ,6, Ja,o)

KÅq-8.-, .g)=gK(2,i), K(.-,fi-. -i-)..-g---K(2,O), •

K( g ,- g )= ,i K(2,-2), K(Jg-,- S ) --2,-K(2,-i)

(16)

34 H- NAGAT and N. HAYANO and Y. YAmm

K( g , -S-År=.5.Z-K(i,o)+415. i.li!L' J(i,i,o,2År+ ,2, K(2,i)

K(--32--, -- i )= 28s K(1,--1)+4-i2/sJ(i-J(1,o;---1,2)+-23EgK(2,o)

KÅq 9t ', --' g )=24igTK(1,-2)+ 28s J(1,-1;-2,2)+ 24s K(2,--1) KÅq--S , - i )== 26-s-K(O, -- 1)+ SZ J(O,O;-1,1)+r2r6s K(1,o).

The rnatrix elements in the (ml,ml')-scheme are easily calculated in terms of Ii by directly using the wave funetions of the dd-configuration in Appendix I, or by using the Slater method and transforming the Slater integrals Fk to the Talmi integral Il. The forrner method is rnore convenient than the Iatter, because we can obtain the exchange integral from direct inte.qrals by changiDg signs of Ii with odd rs (i.e., which arise from functions antisymmetrie in the space coordinates of two nucleons).

In order to employ the sum method to calculate the energy levels, we svTite down the Table of complete set for the (di)3 configuration classified by MJ:

2

Diagonalmatrixelements MI

---

Statesinthe(mji,ml'2,mj3)-scheme

sJr--"--T ^exchange

531

li•ii•:r 535131mnv---rri

2'22'22'2

53INr2-,-2-'--E2`1

li•1;ET•-li-2,li

2

5 533511-2-•-2•--2-27•'ii•--J2U

'2'

L

i•r2el-2r•--l;17'H'if

515111--P.---liF,'2-2,22'2

3-ll 535513--ff--M-rrt---

2'2'22'2'2

535535---r2'•ir'2,22'2

g•e)g•-ge•-g313111ir•illi•-iil•-l;

Thus we see that there are 3 independent states, namely those with J=-g-,J=-il- and J=lj3-. Only one state belongs to eEch of the total spin J=g, , -:t- and -g-• Therefore the

sum" method can be straightforwardly e4nied out to calculate the contribution of various central forces to the levels, For the configuration of equivalent nucleons the diagonal matrix element are of type (2c), and the results are given in the Iast column of the

above table.

The inteTactien energies for nuclear Btates J=-2-, -;t- and Åí- from central inteTaction

Properties of Light Nuclei with HarmonSc Oscillater Wave Functipng I 35

potentials of various exchange types are given in Tal)le I:

TABLE I Central interaetion Energy fer (3d-E)' in j-j coupling 2

a) Slater integral J

9 2 5 2 3

- -

12 L

312 30 3FO- s,.7, F?--

2- p-Fi '

3PD

]

9 2 5 2 60

51

13

3F '-- ' g2.7iF- 21ffrv 2

MAJORAINTA -- 36 Iiv+ 72.Fe+ 1329-Fti

25 5.7` 52• 441

4 56 651

-- sr•FO+ s.7, FO+ s.4.41 F"

..--

g-- Fa- ,i.2,, Fe+-,-9Z-Zi i`

Ewigner= '- EHeisenberg EMajorana== -- EBartiett

b) Talmi integral J

9 2 5 2 3

2

WIGNER

iiils l il167(Io +L) + i29(ii + i3) -- Ss}il!I? l

g i 22 (i,+k)+ Z (i,+i,)+2-,9i,]

-} [ -?lig- (I, + I,) + -{liQ- (I, + I,) - t-s-6I, i

J! lMAJORANA

--SsEI, 9v iis+ti167(Io-I,)H-l-2-5(I,+I,.)+21gt{i5I,

+2- ^{s9I, 5i} g+%t(I,+k)-{t(I,+I,)+Vt7I,

-t-s-6I, 3

7 -g-+.?.g-(I,+I,)--54-4,(I,+I,)+1-s5--6.I,)s -

Evvrigner=: -EHeisenberg EMajorana=-EBartlett

From the definition of the Fk and the Il it is shown tliHt in the short range approxim- ation 1 Id Åq l Io l (lÅr o) ahd in the lo ng range approxim ation 1 Fk 1 Åq i Fs k w]iile in the long range limit all Il are equal and in the short range limit Fk---(2k+1) Fo. Therefoie by reference to Table I immediately we see that in the long range approximation the ground

state has J=3!2 and the first excited state has J=5/2 and, on the other hand, in the short range approximation the ground state has J=5/2 and the first excited state has J--3/2 in both Majorana and Wigner forces. As in t]ie cuses of nuclear interactions (13a), (13b), Åq13c) and (13e) the central potentiuls contain the Serber's exchange operater

l(1+Px) as a factor, central forces act only on the singlet states of two nucleons. On the other hand both teusor and mutual spiri-orbit interactions act only on the triplet states.

Hence vatiations of force constants in these non-central interactions have not any influence

on the central interactions, The contribution's to levels of these non-centraL interactions

and' those of eentral interactions are completely additive and, therefore, are discussed

independently.

36 . H• NAGAI and N: HAYANO and Y• YAMAJT

For the Yukawa potelltial, the Il are calculated by Talmi. CSee reference '(7) p. 205) Here we ean fix :he value of v by using Lhe formu]a for the nuclear radius

-

R2 =-i'=:Ni[tlSo'C'e-vmp21+4dr:::=:(21+3)/2v -

and i

Rt-vl.5 Å~A-I'i- Å~ 10'L]" cm.

For A= 21, 23, we have R =4.2Å~10-iScrn. For the value of R and various force ranges, the beheviour of It is shown in Fig. I. From this figure we see that the short range Fis. I. Bettavtours of Taemt Irue2Tals

L,• !s lz T,

-.o

2.o

to

re Z, rz 13 1-

-o.1

-O.2

-a3

-o.-

-- O, 5

approximation is reasonahle for these physically interesting force--ranges. In the case of nuclear interactions (13a), (13b), (13c) and (13e), from above-mentioned circumstances it turns eut that only Wigner and Majorana interactions are mixed with equal percentage.

Immediately we can suppose that the state with J==5/2 is lowesq abov!; IEt, lies the state with J=3/2. In the cuse of (13d) owing to the like-nucleon-system Ti.r?=1, and in order to obtain the expreseion with a linear combination of exchange operatoTs, we replace . g by }(1-n). Owing to gÅr1, the weight for the Wigner interaction becomes naganve

and the weight for Bartlett positive. Theiefore in the case cf (13d) the ground state

has J==9/2 and the state with J=3/2 is next to it. Consequently we may cqnclude that

)L ropertiee of Light Nuclei with Harmonic Oscillator Wave Functions I 37

the occurrence of the ground state with J= 3/2 in both Ne2' and Nats canpot be explained by using central parts of two-nueleon interactions with the Yukaxva potential introduced by various authors to explain many two-body and three-body experimental results. Splittings of nllclear Ievels obtained by using resu]ts given in Table 1 and nuclear potentials (13) (only central part) are shown in Fig. 2.

Es.2 CelztraL Ener8yLtvets vte.sus 'ro -tl

XfO CPL

-as IP' IS 2e "m-'To

Mev

J=g --•.- 2

.. - J={ .

s J- :;

'År'-"hN...--.----.-.h--.

Thus, we investigate the effects of non-central interactions on energy levels. In the first place we calculate the contributions to levels of the tensor intei'actions which has been introduced to explain the quadrupole moment of the deuteron and has been used very extensively in the analysis of high eneTgy nucleon-nucleon scatterings. For the sake of convenience ef calculations we use one-twelfth ef the usual expression of the tensor

lnterqctaon ;

-- -" ---

J77,,= S'.• P'1(r) =(( S, .r) .(`Si•r) /" --- S• (S, tS2)). Vl(r). (17)

The operator Si,2 of L17) are re-written in terms of spherical harmonics with 1=2 and .the operaters S+, S- and Sz, where S+=Sx+iSy, S--=S=---iSy. And then we have the

following expression for the matrix element of the tensor interaction in the (jij2 mjT mj?)- scheme.

,

38 H• NAGAI and N. HAYANO and Y• YAMAJI

Tensor interaetion J(nlintj, ntltjim)-i)

= 1/'JI-ilFs'--- -4i- f2f'!Sfit'(nlmj-l I 1)ttr'(n'd'md' -- l, l 2) v(r) yp.tt(nlmj -- -} : i)

u(n'l' nlj' - l,: 12)dTidT2

+ J/'z8s- -il-i- g"g"SSii'(nlmj + :.).` 1 1)u'(n'l' nij'+ ,'i 1 2) I7TÅqr) YGu(nlntj + i!t l 1)it(nl'ntj '+ :-, [ 2)

- }/-411s- 7ili-(f2g'2I i it'Åqnlmi' - li]1)u"Åqnil'mLi' + -l,12) li(r) Ye.tt(nlmti -- }. 11)u(n, l' nej, + .i-, i2)

+g`?f' 2 S i u'(nlm7+ l, l 1)u'(n'l'rni'-l• i2) V(r) Y9,u(nlntJ+ }, i 1)u(n'l'm7' -}, 1 2År)

- Vt-is J-ifg'gf"(ffu'(nlmj- .k, l 1)u'Åqn't'mi' +;.• 2ÅrIi(r)Y?.rt(nlmj+ .r-, 1 1)u(n, l, m,i, -,, 1 2År

+SSu"(nlmi + l• 1 1)u'(4'l' m] '- /, 12) V( r) Y8tt (nimj- !.,- : 1)uÅqn'l' mv' + }, i2)]

-I;illliiTL; l'-f?g'f' I ! it'(i71m '-;J.• l 1)u'(n• 't'm,' '+ l: l 2) Ii(r) Vl,ti(nimv- 't, n)u(n'tt mj' - ,, 1 2)

+i71i .g -l• b,2g'f' SSu*(nlnij + }• l 1)u'(n 'l' mj' + lr 1 2) JiÅqr) IVI,rt(nlm7' + '.",- 1 1)u,(ti 'l' mj' •-- }, I 2)

"- i-/!rs =Z, gff'2! S u'(nlmj + !7!1)a'(n 't',nj' -•- 1-• i2) V(r) Yl,ttÅqnbni-- },l1)u,(n,l' mj' - .i-, [2)

+ ttl-1-s ,i-• tgfg'2J" fts'(nlmj +,}• 1 1)u'(n 't'mj' + }• l 2) V(r) Yl.u(nlmi- :-, l 1)u(n'l, mj' - .i-., I2)

+lt1--s- l•if'f'g' ! S w'(nlmj --- }. 1/ 1)u*(n`l'mj'-•}- i 2)V(r) Y.r]te(nlnlj-l. s1)u,(n'l' mj' +El 2)

"H i/li :s i2 g`lf' g' ! ! u' (n lop + dl) i 1) u' (u 'l' mi ' - }• 1 2) VÅq rÅr Y.: 'u (n ln ej + l, I 1)nÅqtt ' l' mi• ' + ;-, l 2)

+ ir/11s l• fgf'2i S tt'(nlmj - l• I1)ut'(,i 't' mj' - ,!i l2) VÅqr) Yliiu.(nlm7' + /li 1)u(n'l'ntj' --- l, t2)

-i/lirs }fgg'2 ! l u'(nlmj-l} l 1)u'(n(l'n}j' -t- t, 1 2) J7T(r) Y,- 'u(nlmj+ t,- l 1)u(n,lt nzj' +M 2)

+ i]ilrs l, gfg 'f' S ! u' (nlm]' + -i:• I 1) u" (n 'l' mj ' + l• i 2) V( rÅr IY;u (n lmj - i..., p) u(n r l, mj r - }, l 2)

fi',Ii,'`1.'XlfES't.S,1."l`1'IMt':•li":tutt'i8ue'v'e:il(eiiteVE(IYvl12iSn"tros$li'le':(ln"A:ie11Ki2

:.'nle:.gya:,g::ciiEelella"'iZeXS;3.fe#i,;,t//lzae/j'G,,,:Oznji:"i•::•;"niTii•,:iol,xesk[.l,:T"/SelS:iOc.',rni:.,i,::.ÅÄt2g,r,t:ei,grkli,I•a

sum rnethod we calcu]ate the first oTder contribtitipns to levels of the tensor mterRction

and have the same results given by Talmi'a):

Properties of Light Nuclei with HarmoDic Oscil]ator Wave 1;unctions I 39

gii- 52 i2+ ,3, i3 for J---2,-

-g--i,-2i, +-g. i, nJ= --g gI, -- 172 I,+:I, nJ=-23-

(18)

The results for a potential which is multiplied by t]ie space exchange operator Ps is derived from those for an ordinary potential by changing all signs. In the case of (13b) because of the exchange type of the tensor interaction, it has no effect on levels. In other cases it ]ias no magnitude so as to change the order of ]esrels. Only in the case of (13c) tlie level spacing between the state with J==312 and the ground state diniinishes. These results are shown in Fig. 3.

Finally we consider the eontribution to Ievels of the mutual spin-orbit interaction• Case and PaisiO) introduced phenomenelogically the strong spin-orbit force (14) in order to preserve the charge independent theory of the nuclear forces in analyzing high eneTgy nucleon-nucleon scattering and fiirther shewed that this interaction could give a doulet splitting of the right order of magnitude necessary for the sliell model. In the atomic spech'oseopy the Thomas interaLtion imd the rnflgnetic spin-ei-bit interaction have already been used very extensively as the spin-orbit interactions ancl on the other hand in the

nuclear spectrescopy these are too small to account for the wide doublet separation. There- fore in the first order approximation xve need not take into account these two spin-orbit interactions. For the sake of convenience of calculation (14) is s"ritten in the forni;

M(r)( }. (sC ) +s9' )A- + l. (sC-t' +s(..?))A. + (sE') + s`,2))A.)

where

- -- - --

lt .tL =(r2 -- rl) Å~ (p2 -P,)

A-l-=Ax+iAy, A.-=A,x-iA.s

Thus the djrect integral fer the niutual spin-orbit interaction in the (mj, m'j)-scheme is:

Spin-orbit interaction

Vi2= V( r){/, (s: +s!. )A-.- +l, (sL +s!- )A. + (si+sE)A,} .

J(mj, mj[) l:.?i, ].,f .( {fu(mj - ;., i1)ai + gu(nij + t.J ti)B,}' {f' u(m,'j- i!. !, 2)ct'.. +g'u(nt.i' + :tt, [ 1)B,}* Ii,.-

Å~ {fit,(m•J'- l•- l 1Årcvi+gtL(mi' +"• 1 1)Bi}

{f' u,(mj' - }, i2)a, -L g' u,(mj' - :,1`P" ÅrB?}dT,dT2 ., }, fgf'2 J(m,1- ---b, mf' -- yl,; A-mj+i, ntj'-S) . +Sfgg" J(,nj-;"S, nzj' +l,i A-nlj+l, m]''+h)

+ J}, f' g'f2 J( nij - .i, , nij ' --- ;1-; iN -,itl' --- l,, n,1" + i: )

+-Sf'g',g2 J(mj+S, nz,''--l; A-mj+S, nt•i" +l•) ' +5 gff'2 J(nTj+}, nij' --- l,; A.nti'- },, mj' --- -!,)

+l• gfg'2 J(md+5, mj'+S; A+nij-l, mj' -+ l)

40 H• NAGAI aud N. HAYANO and Y• YAMAJI

+l. g'f'f2 1(nlj---l,, mj'+l.,; A.mj-l,, ntj' --l) +,t, g'f'g`' J(mJ'+-l•, ,,tJ" +S; A+nij+S, mj'-l) +f'f': J(md--l, mj' -- l-; Aznlj•-- +•, nof -l•) -g9g'2 J(mi+", mti'+}, A,nlj+, nlj'+}).

In this case the integrals can be calculated by the same way as in the case of the tensor interaction. The exchange inte.qTals K(mj,mtj) are easily obtained from the diTect integrals J(m,1' m'j) by changing the sign of Il with odd. Because, just as the operator S'zar-. tige

-- - - operator (Si +S2).A is diatiaonal with respect to the magnitude of the resultant spin S (S

-- ==Si+S2) and its eigenvalue in the singlet state is zere. We calculate the connibution to levels of the mutual spin orbit interaction introduced by Case and Pais (a=O,B==1) and

the result is;

,3 ,i:+tbu3i3+,g,i]]= -5-2,-ii"-29-s-i?+-8-g-is forJ--Z-)

sl o lT+ Zg I3-N 21s Iii =--22-s-Ii + l I?'F-7s-I3 for J=-;- (19)•

i3 o li+ :: I3+ 26s lit== ?o I] '- g I2 + ; I3 for J= {l-

The results fer ct=:1, B==O is easiry derived from (19) by changing the sign of al1 terms,.

Here the Il's differ ftom the Ilts fnr the central and tensor interactions. For the Case- Pais potential, V'(r)==VD II -d-dx(-e-iX-);(.fv= ;o ), the Il are calculated as follows.

it == lv; .( iie p'(r)e-vr2 r""dr= - ,i,Vl!'-,P-:; ,).L;i J,co( ll + i)(---fi- -)-' ,-e2--i'gm+i dij

=-t.IVm7'-iJUr,O ,X.,.e,t'f: ex'( Åí +1År( k )-'g'+i dx

= - i/ -2•lfi- !vl,--. VoN3. ept'(2/-JoOOe-x'(x -- pa)2' + f1 'e-x2.(x-pa)?i-i dx]

wheTe -

x=i/riJro,g=i/-;r, /i= 21N and x==}+ ft . .

This gives for- the Talmi integrals which sve need : ii=r(;'ii[-,/t,2.r.rT2t2.IL-l!L-(i--.Åë(p))e#2]

I2=-IZ-g-[.T/2,:- 4P` +48pP,2--2--- (2pa'+ s)a-(b(pÅr) ept2)

I, .. iltlftos.[i/4- 7 g-t.Z.e+26t'4`i.+, 24rf-4-• (3s+ 2sP2 + 4/i`)(i -- (l)(p))e,t'] (2o)

I4= gK?s[rt8-.-'4paS+5r2pae+1.ii,pa.St9=6.ud2=-1ny2 ,

-- (31s + 37sl-2 -t- 84-L4 + 8rf')(1 - ,sP(F)) ep:)

gF.(x)=-.i/2,--.--Sge'Rbt• . .

Properties of Light Nuclei with Harmonic Oscillator Wave Functions I 41

SinCe P==.ovilit+- -3, for R'N'4•2Å~10-t3 crn, roe-vl.1xlO-'" cm, and l=3, the It decreases very rapidly with lowing to the singular preperty of V(r), Assuming that ct==O and i?=1 in (14), we have as the contril)utions to Ievels

EF,:O==-O.098 Mev, ES,-O==+O.O04 Mev, EEO :--O.167 Mev.

- These eontribution$ cannot, also, cliange the order of levels. See Fig. 3. Therefore any nuclear potential given in (13) and (14) cannot exp]ain the occurrence of the ground state with J =-Il- in the jj-coupling•

-

However, as a resutt of our calculation the conti'ibution to level spiittings of the tensor inter:Lction is coniparatively insensitive to the value of ro(-to the value of p) and on the other hand that of the mutual spiTi-erbit interaction introduced by Case and Pais is very sensitive to the value of pa There exist g.ome possibilities in the point of view.

1, 5-

Mev

1,O

O,5

,- .-. .. •--- XVtthLecct C•ase2RLi$'

g2- LezePL'ng lnieraclion

--- Witk ,,

---- ---

--- "---

Slir rEi,---L--{k -:.'l':-:-.-iiz2'

5Er

i-,.,tt:.-,T-;h-T:[Z

-"--- --- ---

O:/3-.l-'`!ÅíZ:,:SI'",, ll,(/i3ii'i'tS-".=li/lll5illgcs),,,;.i.l].-:-:1[-:-u,,(,,2f'(T,-,-)-;Si;f

cm cTn. Fig.5 Enersy.eeve{s of Ne2ifar vaTio"s 7zcu tear b'nteractlores

S2. The Fta nucleus

As mentioned in iritroduction, we assume that the extra-n"cleong. in F'9 ]iave the config- uration (3d-})2N(2s,!)ip. In eontradiction to our expectation, since the wave function for the state with J==iis

y•(J- -S•• M,-,lt)-= -,,i, [(iS• -S; -5-) -- ( g • -g; g) +(g-• -g; -,-i-) i•

the magnetie moment of Fi9 is equal to that of the sing]e proton. '"Ve shall pass this

42 HNAGAI and N. HAYANO and Y. YAMAn

prol)lem nn the intermediate co"pling (Chap. 5). In this g.ection we caleiilate nuclear Ievels in the same way as in the previous section,

In order to employ the sum method to caleulate the energy levels, we wrrite down the Table of complete set for the {(3dL. ){y( 2gk)l,} configuration classified by MJ:

2 ..). ,

Mjl (m h,rn j..;m ]3)-scheme din

i

l states in the i Diagonai matt' elements

: exchange . or ary e-- r-.-U-"

9

"2'-

7 2

5 2

(g•g;s)

( :,-, g, ,i-.) e, , g, ; -S)

(:-' -rir` S)(-3'-' -5"; -n2mi )

(3,-,-};S)

(g•g)

(g• s)(g• g-)

tT -. tTtTrTr l

(-g.-,.l)(g,-+,) Åqg•-})

31

-2-

1 2

(g, -gi e)(g, --s; -- -}) (g,,i-,--s-)(g•-S;S)

(g, nvrÅrs., S-)(g, -- g; S)

ÅqS• -S; -5-)(g• - g; -t) (g• -s; --s-)

(g•-g)(g•-})

(g• i)(g• -e)

(g, - -52mÅr(g, - -32.")

Åqg• - -i--)(E,-. -g)

(g•-s)

(g• s)(-;--• sÅr

(g• S)(S• -5- )• (g, - ,i--)

(-;'-' '-'El-)

(g, -l-T)(- rr5T, S) (m5,.T, .- ,i-)

(-li• -,l-')('S2' -})( 5-" S)

(s,-• g)(-g• s)(g• - i,-)

(-s• -s)(g• -s)

(s• -s)(g• s-)(-i• -b-)

r (-g--, S)(--l' • S)(;-• 2i--)

(-g'S)('S"-''5')(-'5'-•',r') (g• -eÅr(- -3,--• -3-) (-}• -S)(-li-• - 5)

Thus we see that for this configuration there are 5 independent states, namely thote with J==g, -ill-. -5i-, g and -5--. Here, also, only one state belongs to each of the total spin and we can straightforwardly use tlie sum method to calculate the contributions to levels ef eentral and non-central interactions. Central interaction energies for var'ious exchange types aTe given in Table n.

Next, we calculate the contibution to leve}s of a erdinary tensoT internction, The Tesult

Prepenies of Light Nuctei with Harmonic Oscillator Wave Functions I 43

E[/ == --g- -I,-- --g--I, -- --liq-Md, ,

E.1;-,l =-- }- - I, --- sl -I3 + 4Mas ,

E[l/l ---- -} -I, 3 I,+ :g I.,- -g- -Md, ,

Et/i,- g-i,- g i,+ :g i,,+ i,2 Md', , Etll,i--g--i,-2i, + g i,

(21)

wheTe Mds= 2i4,ie ----s-o5-4-i7-is 7roi2;]e+ 2goiit + Eilii ii+-2is--i!+-ig-o }/' il ism

Here, the terms with the factor Mde arise from the neutron-proton interactions. Remaining terms derive frem the neutron-neutron interaction. The contributions to levels of the tensor interaction with space exchange are easily derived from (21) by changing the sign of aft terms originating from the n-n intei'action and of those inte.(rrals in Mds which arise from antisyinmetric wave functions (odd A, like Il', I3;n', etc.)

TABLE fi Central interaction energy for (2si)b(3dhsT)k ' .p o

--

a) Slater integral xiiXX

9 2

7

2 5

i- 2'-

3 2

1

2

28 35

2FO(d,,)+ FO--s: 7, F2- s.21,F` 9

2

28 35 2rv(d,s) + rv - Irs- py 5• 7-2 5,212 28

525

2FO(d,s) + F' + s, . 72 F2 - g2v;-2'in2FS

: 7 2

28 525

2Fe(d,s) + IP + s2.7? F2 -s-2.212rv

5 2

280 1050

2FC(d,s) + IO+ s, .7, F2 + s-,. 212F'

3

". 2"-

1 2

Majorana

...

g.-G7(d,,) .-- --s3- Fo + s3.67, Fe + s?27i,F4

g G:(d,,) -- -g-- Fe+ s3.67,, r2+ s9Si2p

- l-i- G2(d,s) -- -i--s Fo -- s.ti-g, I;y? + l.ll, il;91 ,F4

IOO 1389

-- s -G2(d,s).- Li}g- )t -- s:.72 pe + S7t-2I-2ps

Z G2(d,s)+ Å} Fe+,l.43F2+,4S9,p

44 H NAGAI and N. HAYANO and Y• YAMAJI

'L J "

x

"--")-y

T-9

2

7

2

5 -- 2'-"

3 -2-

1

"2-

Bartlett

-g-rv(d,,) . gipu --s3t7F - ,\5,,rv

"3s-rFQ --s3-/-7-,In -- s?:1:rv

f Fo(d,s) + t-i,ip + si,.9g,F2 -- si,3. :?,P

grv(d,s)+,-i,Fe+gi..gO,-,in-s-',tL:?,rv

x Heisenberg Jx

;sG2(d,s) -IF"+ -s-2;87-..F2 +-gli!l31iil72P

---

rv+ -Iil.i}l87,1pu + s\gI2rv

SliG2(d,s)-FO-t2..lll77,F'+t5/-,5,,F'

/'s'G2(d•s)-I'-g;,\}{.87?F2+go5.r;2121r,F`

2 in(d,,) -ri}Fo --gl,fll9,F2 -- ge..4g?,rv m} gG2(d,,) -rv-s2..e',rv -,l.?g?,rv

b) Talmi integral

xXNJX

Wigner

xX,JX

Majorana

97 2A+-

}ItiIl6/([o+i4)+2-(ir+is)+'gi21 -7-

97

-tSB+g[fe(i.+i,)-i(i,ÅÄi,)+gi,l

7i 2A+li-lii,T(k+LÅr+2(i,+i,)+-Z-i,l 7Y -Z-B+-}-[ii6Ii6(in+i4)--2-(ii+i3År+-2,-i2l

5v 2A+- ll[:Il,T(i,+i,)+ti6'(i,+i,)-•?gi,i 5i -i?B+-}I,9-,-(k+k)-5",g.(i,+i,)+fll51i,]

3'2- 2A"Tli`[9o(io+i4)+iil-/(ii+ii)"-Il'ILi!] ^3'ii- gB+-li-[tilli(I,+I,)-ii-/(I,+I,)+EIitLI,]

1i

rr

!T-rl2---

-2

s-B+- li-iIlÅÄg(k+I,)-2-49(r,+I,)+!gtl!II,]

Properties of Light Nucrei with Harmonic Oscillator Wave Functions l 45

J Bartlett

-

J Heisenberg

/.---.--

9-2' -:-A--:'166(Io+I,)-n:i,-(Ii+I,)+-2sJJ? ^9T, ;s-B-gt/-6(Io+I.,)+-48-(I,\-IM "--"---.- ,r4itil--

-t-tt

"L.--

r--v-n-"----""-t-J

z2

--

g-1-66(I,+k)--2--(I,+I,)+gl,M

77 -gP,(k+i,)+ij(i,+i,)+gi,

-il}.2

rllll'A-rg'`ill6(To+i4)-?'65'(io+is)ÅÄiilitLT2)s

ISIiB--ll-l5(I,+I,)+iil67(I,+I,)-IIii61I,

-g--A-gfa(I,+I,)--fil-o5(I,+I,)+t5I,

{lllB-Tg-fl,T(ie+ii)+ili(i:+i3)-ll+6i?

-i2-

-2A---g--Ii63(i,+i,)-2,?(i,.i,).igiitS

nrr"

l2 gB-gtfo'(i,+i,)-t'(i,+i,)-k'i,

where

A = Fo(d,g. )= -352-i,+ i52 i, +--4ig-i,- i72 is+-Zlli-i4

B .. G2(d,,)=.g-:.LI, - ;2 I, + l5s-5I,- 127il, + !30-i5I,

Finally "re take into account the m"tual spin--orbit interaction (14). The first order contributions to the energy Ievels can be cdleulated with the help of the sllm method. The resiTIts for an ordinary potential are

ETS.e O=TtlbT(I,+11 I3) +'s4'A,

ELg:1`'= '11`oH(Ii+ii Ig)•

"t .

s--o 1

E-g,L ="-lo'ITh23s3o'I3+rl'12-4s'i]i+ il A, (22)

.

Elgil,-O=•-ilo--I,-23-s3o-I,+ili2-4s-I,,+r-g--A,

S..O 7

1

E-L == - -2s I,- -s--I,, + A, 2

"Tliere

A== ' -i-4- Ii + -254 I2+ "l-s"Is+ 43o ITt +L27'4'Ii:•

The terins with the factor A and the otliers arise from the n-p and n-n interactions respective]y. The contribusiens of the potential with the gpace exchange operator differ frem (22) by the sign of the integrals arising from antisymmetric wave functions (odd A, like I3, Iii, etc•)

We can eesily transform the expressions (21) and (22) to the expressions which contain

oniy the Il's, by using the following formula:

46 H• NAGAI and N. HAYANO and Y• YAMAJI

Iu= --2-l-+2 "3 Il ••-- (21+3) Il.1 + --2-(2+- -5-rl+2 ,

i,i,io==VZ(-"3,-i,-gi,), i3,ii==V-;u(gi,--5--i,).

By means of those results which are obtained for central, tensor and mutual spin--orbit interaction energies we cslculate the energy levels of the eonfigiiration {(3dJ:-)?N (2sÅÄ)"}- In order to obtain numerical results "Te have to evaluate the Il. Since in the case 6f the harmonic oscillator wave functions the (3d) and (2s) wave fllnetions give the same exp- ressions for the average nuclear radius as a function of v, if we assume that in the first approximation tlie nuclear Tadius for protons is equal to that for neutrens it is justified that hitherto we use the same v in the (3d) and (2s) wave functions. We use the game value of v as that for )AIXe2'. It may be a slight underestimate, but in viesv of accuracy of our treatment this estimation is good enough for the calculation of energy levels. The numerical results for the interaction types (13a), (13b), and (13e) with (14) are illust- rated in Fig. 4. Here in the case of (13e) which has betin proposed by Christian and Noyes on the basis of the charge dependent theory ".e use (i3c) between (3d)--neutrongL and (13e) between (3d)-ncutrons and a (2s)-proten. TheF,e results is guaiitatively in

Mev zo

t5

1.0

as

o

d6- Coupi!ins wn iio ";.{ilili/lgltl n' ---" WtittL "

%•

d." -.% ====7E d--- -

- d-- ---

{ii51

%

-e-

%

--

%

.='

.'

i-i-k

-3 :d:I:,,-..,:.. -k ==-==-% z

i9iE)

-- ---d--- --- -dt-p

--- ---v --" ^•IE

" (isa) Åqla'b) ]Ill.ijg.C,1,s.t,sii,,,Åítij.C,,iB,io-,,,.,. (tae) - '

F7i8 (t Eneray Eevees of fi'9 lbrvarioas ntcceear

`htteractions

Prepenies of Light Nuclei with Harmon{c Oseillater Wsve Functieos r 47

agreement with the level diagram reported by Ajzenberg and Lauritzeni'), and the others:t) in which the lst and 2nd exeited states form one group and the 3rd and 4th excited states form another group. Ouz' results aTe ascribed to the faet that the configuration (d-:-)?N form three levels J==O,2 and 4, and the coupling of these ]evels with the (2s-!.)' proton gives rise to the doublet splittings of levels with J =2,4 throttgh small Bartlett 2

and Heisenberg central interacLions, and non-central interactions. (Refer to Table II and (21)

and (22).) Spins of low-excited states have not yet been determined by experiments•

However, according to our calculations the state "rith J==7i- is lowest, above it lies J=-Il-

and the state with J= --E-is next to it. " -

.-

4. The L S coupling

In the previous chaptev we see that if we restrict tlie nucleen-nucleon interactien pote•

ntial within potentiaJs introduced by various authors to explain two-body and three-body experimental data, we cannot explain the occurrence of the ground state with J=-il- for the configuration (3d-l-)a aTid the deviation of t]ie magnetic moment of Fi9 fro'

m the

Schmidt value and, however, the assignment of the configuration {(3d-gi)?N (2sl){•} is very probable in view of the energy values of lower excited states of F'g. in this "chapter, we investig.ate these difficnrties by means of the LS coupling scheme. The tahle of complete sett for the configuration (3d)3 classified by Mi. and Ms. values is given hy Table III.

TABLE III Configuration for (3d)3 in I-S coupling

Ms

MI. ^3'IT

2"

2H 5 ^(2,2,1) +-+

:G Tm 4 (2,1,1)(2,2,O) +---l.ÅÄ--+--L."

4FSF ^3;-rt (2",1",O') -(2+,O'-.1-i' )(2+,1- ^,o"' )(1',2- ,o' ^)(2'iJ ,2H ,-1+ ₎

TT--L-"

2Da:DP

[-.

4P!P

2 (2,1,-I,) (2+,--IM ,1+ )(2+,o- ,o+ xl+,1',oLi' )(2+,1' ,-1+

1 (2,O,-1)

(2+,1",-2't' ₎

(2",2P ^{,i'- )(2',-i' ,o" )(i"'} ,o- ,o" )(2",o-- ,-i+

o ^(1,O,-1) ++-F

(2+,O",-2+)

(2",--2d ,O" )(1".-1-- ,O+ )(2",-1' ,-1' )(1',O- ,-1+

From this table we see that there are eight tenns for this configuration: i. e., llI, ?G, 4F,

aF, -P and 2P occur once and gD twice. Tenn energies for 2H, VG, 'F, !F, 4P, and :P are

48 H• NACAI and N. HAYAJr"O and Y• YAMAJI

easily ebtained by the diagonal-sum melhod, but for tsvo 2D-terms this rnethod cannot determined their separate energies and can merely give their average vallle. These

Testtlts are :

Central ,interaction energy

1

Wigner - 2H i 3Fo-fi46

g-p--- ^i,42-iP

1-•--- -

i

2G i 3FOH-a-gl"IT2- 414ft Fi

4F I/ 3Fci --- S,g-F2 - -- ii.lil4?r F;

i

'F l 3Fe+9F:- 87 iig 49 441

l :

;

l'

l

.D ' 3Fe+ 49g F?,• i41 F`

Majorana m.1.-8".-F,+p15 rv

49 44•1

13 F+ 40 '

49 441 F

- 3Fn+ 15 rv + 72 rv

49 441 - 3 pe+ 120 rv

49 441

1

4p I 3)n ---l-t7urv

i 441

[

2p i 2)e.6 F2 H"-1"2mrv 49 441

13 pa+ 285 2Å~ 49 441

-3Fe + 147F

441

[

+ 3rv+ 9 F2 -m4.7--I,

49 441 ' E""Vigner =-EHeisenberg EMajorana=-EBartIett

Two 2D-terms can be separated by finding the complete rnatrix of central inteTactiens if we kno"r a set of LS coupling eigenfunctions. By the angu]ar-moinentum--nperator rnethod we can obtain the 2D-state wave functions Yt(ID) and Vr(gD) with the seniority number 1 and 3, respectively. (See Appendix II wave functions for the configuration (3d)e in the (SLJM)-scheme.)

We find the second-order matrix connecting tlie two ?D states. This matiix fer an ordinary central interaction has been given bÅrr Ufford and Shortleyi").

?D,l,2 3Fe+7F2+63F4 31/7t (F2-5F4)

1r F

g•D,l,,2I 31/2Mi (F,-5F,) I 3F,+3F,-57F,,

' 1 ⁽²⁴⁾

where

F,==F', F".=-t•g--- and F4=-4pm4=r'.

The matrix for'an Majorana interaetion is found to be

Properties of Light Nuclei with Harmonie Oscillator Wave Functione I 49

:D,.llH,2 35 -17/ ^1- glT(1sF,--115F,)

:D,-ll-,2

--+

!'i:'

2-

lg(1sF,--11sF,)

---

ll-(9F,-255F,)

1

(25)

If we assume the nuclear interaction to be (13c), the energy matrix to be diagonalized is

g{(24)+(25)}. (26)

which has its eigenvalues

E =3F,+ 2i F2 +Zl-iF, Å}-S- 1/s6s,Fzi4-643sF,,n--+-r29'9ii75F,? (27) The eigenfunetions for these terms (for Ms=:-}, MI.=2) can be easily obtained•

ctftlr(?D) +pvtÅqgD), (2s)

where

"=:

[3I),'i-itlilfli{,3"iF..':`/6,ig"i;,lii!i2+i'i'

49 441

B=-3E ^,' /;i(223F')lii;6iit"E-,-)Ea

'

(29)

The table of complete sets for the configuration (3d)2 (2s)' classified by Mr. and Ms is TABLE IV Configuration for (3d)'. (2s)" in L-S coupling

2G 4FiF

sD 4P2P

2S

ML

4 3 2

1

o

Ms

2

++ ^{(2 ,1 ;O+)}

+++ ^{(2 ,}

o ;o )

ÅÄ+

+ (1 ,O ;O ) (2', -1";O+) (1+, -l+; O+)

+ ++

Åq2 ,--2 ;O )

2 (2+,2- ;O" ₎

(2 +, 1-; O+)(2d , l+ ; Od )(2+, 1+; O- ) (2", o- ^{; o'} )(2- ^{, o'} ; o" )(2', o+; o' )a", 1'; o," ) (1', O' ^{; o"} )(1- ^{, o+ ; o'} x2 ", . 1' ^{; o F} ₎ (2-, -i+; o')(i", o"; o- )(2', -i -F; o-

) (1 ', - 1- ^{; O+} )(1-, -1 "; O" )(2 ", -2- ; O' )

(2" , -2" ;O" )(O",O- ;O' )Åq.1", --1';O- ) (2", '-"-2"; O-

)

From this table we see that there exist seven temns for this cenfigurntion: 2S, 4P, 2P, SD,

?F, qF and 2C, Term energies fnr these terms are obtained by the sum method. The result

is

50 H• NAGAI and N. HAYANO and Y• YAMAJI

Central interaction energy X..

Term .

tt-- ' ?s -l :P

4P 2D eF 4F 2G

Wigner

Fi(d,d ) + 14-F2 + 126Fg + 2F,( d,s)

Fo(d,d) + 7F, - 84F4 + 2Fo( d,s) FD(d,d)+ 7F2-- 84F4+2Fo(d,s) Fo(d,d)- 3F?+ 36F4+2Fe(d,s) Fo(d,d)- 8F2- 9F,+2Fo(d,s) FD(d,d)- 8F2-- 9F4+2Fo(d,s) Fe(d,d)+ 4F!+ F4+2Fn(d,S)

Majorana

F,(d,s) + 14F2 + 126F4 + 2G2(d,s )

-- Fo (d,d) - 7F2 + 84F4 +2G2(d,s)

-Fo(d,d)- 7F2+ 84F4+2G2(d,s) i Fo(d,d)- 3Fi+ 36F4+2G?(d,s)

- Fo(d,d) + 8F2 + 9F" + 2G2(- d,s)

-F,(d,d)+ 8F,+ 9F,+2G,(d,s) Fo(d,d)+ 8F2+ 9F4+2G2(d,s)

ii31iiiiiixx.

Bartlett Heisenberg.-L-.

T2S -Fo(fi,d)-14F2-126F,+Fe(d,s)! -Fo(d,d)-14F2-126F`+G2(d,s)

?P Fo(d,d)+7F2'84F4"2Fo(d,s) -Fo(d,dÅr'7F2+84F,mG2(d,s)

,P F,(d,d)+7F2-84F4+2FD(d,s) -Fo(d,d)-7F:+84F,+2G;(d,s)

2D _"- Fo(d,d)+3F2-'36Fa+Fo(d,s) --- FoÅqd,d)+3F2'-36F4+G!(d,s)

?F Fa(d,d)-8F2-9F`-Fo(d,s) :-Fo(d,d)+8F2+9F4-G2(d,s)

4F Fo(d,d)-L8F2'-9F4+2Fo(d,8) -FD(d,d)+8F2+9Fg+2G?(d,s)

2G _-- Fe(d,d)-4F?-Fg+Fv(d,s) -Fo(d,d)'-4F2-F`+G2(d,s)

F,=:FO, F,= tg' , F,--- 44F"1, Fz,=F2,,(d,d) G2

C,(d,s)=

5.

Now we calculate the connibutions of the tensor and rnutuaR spin-orbit interactions. In the jj-coupling we have obtained these contributiops to energy.levels by the sum method.

On the other hand, in the LS coupling we cannot straightforwardly use thig method to evaluate these contributions. With Tables M and IV we find the wave functions in the

(SLJM)-scheme hy the angular-mementum-operator method. In order to obtain the con- tributions of the tensor interaction, at first by using these wave functions we calculate the

diagonal matrix element of the tensor interaction energy ITTr-lll"ij Sij V(rij) for the (S,L,J

=S+L,M=:.1)--state and then obtain the contribution to the levels with different J's and the same S and L by means of the forn}ula:

(ln a SUM l X •SejV(rij) I lna'S'L'JM) ==( --1)S+L'-rU7(SLS'L';J2)

`'j x(lnaSL II T(22) ll lna'S'L') (30)