108 Chapter VII Negative-Parity States of the Alpha Particle § 1

108 Supplement of the Progress of Theoretical Physics, No. 56, 1974

Chapter VII

Negative-Parity States of the Alpha Particle

Motoki SAKAI, Yoshinori AKAISHI and Hajime TANAKA

Department of Physics~ Hokkaido Universz'tyJ Sapporo

(Received February 12, 1975)

Variational calculations are made by the use of the Ramada-Johnston potential for the T=O negative-parity states of the alpha particle on the basis of the ATMS method. Total energies obtained as the upper-bound are -6.4 MeV, -2.9 MeV and +2.1 MeV for the o-, 2- and 1- levels, respectively. The observed energy spectrum is reproduced r~asonably. The physical picture in A TMS is found to be valid not only in the ground state but in the excited states. The tensor force in the triplet-eve~). state plays an essential role in the binding mechanism of the excited states. The content of the energies has a close resemblance to that of the triton, which suggests a cluster-like structure such asp-3H or n-3He. The even-state forces as well as the odd-state ones contribute to the splitting among the o-, 1- and 2- states. This mechanism is different from that in the 1p-1h model where the splitting is caused only from the .odd-state forces.

§ 1. Introduction

The study of excited states of the alpha particle based on the realistic nuclear force is expected to reveal some aspects on the nuclear binding mecha

nism which do not appear in the ground state. There exists a nucleon in p orbit in the excited states as well as nucleons in s orbit. The p-nucleon ought

to play an important role ip a cluster-like structure of 4He* such as p-3H, n-3He and d-d. We have arrived at a stage that we can investigate the excited

stat~s by means. of the variational method called ATMS which is proposed

to the few nucleon problems. The method has been already applied to the three- and four-nucleon

ground states, 1> and upper-bound energies yielded by the Hamada-Johnston potential are -6.0 MeV for the triton and -20.6 MeV for the alpha particle

as is seen in Chapter II and Chapter III, respectively. The value for the triton is comparable to those reproduced by other conventional methods. 2>

Our treatment is the only one that succeeds in making the four-nucleon system bound as for the variational approach using realistic potentials. The analysis of the binding energies obtained has revealed the decisive role of the tensor force in triplet-even state that actual nuclei cannot be bound without it.

This success is brought about by a clear-cut picture that nucleons move

in a nucleus via multiple scattering processes mutually interacting with one

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N~gatz've-Parity States of the Alpha Particle I09

another according to the Bethe-Goldstone equation. A trial wave function

o,f the system can be constructed in three steps along the picture. Com

plicated nucleon-pair correlations, which are induced not only by the repulsive core and the tensor component of the realistic force but by the Pauli. principle,

are included into the trial function in a dynamical manner of the independent

pair picture. It is to be noted that objects of this prescription are not confined

to ground states, and that it is applicable also to excited states with keeping

the orthogonality of their wave functions to that of the ground state in trans-" parent way.

The purpose of the present chapter consists in an investigation of the

binding mechanism in three odd-parity states with T =0 as a first step of the

study, i.e., ]=O- (2l.I MeV), 2- (22.1MeV) and I- (31.0 MeV), 3> through

the analysis of variational energies for these states. The negative-parity states

of the alpha particle are regarded as members of the IS-dimensional S U1 supermultiplet in the one particle-one hole configuration from a viewpoint of the shell model since de-Shalit and Walecka. 4> These levels are to be

degenerate for the spin-independent force. As was shown in their calculation,

the observed level order of the T =0 states cannot be reproduced so far as the

spin-dependent central force plus a single-particle spin-orbit one is con

cerned. Shell model calculations with realistic forces were made in the framework of the lp-Ih model,5),6) and qualitative agreements were obtained with

respect to the level order. It was pointed out there that the splitting among O-, 2-, and I- levels is caused from a competition of the tensor component

with the spin-orbit one of the two-body force in the odd states. 5) Comparing

the present results with those of the Ip-Ih model, we will discuss the extent

to which the model possesses its validity and also make clear what cannot be

grasped by the model considerations. It goes without saying that the alpha particle is one of the most fun

damental constituents for the cluster-like structure of nuclei. The analysis

of the binding mechanism in the excited states ofthe alpha particle, combined

with knowledges on the ground state, would give us a clue to the understanding

on the mode of existence of the alpha particle in nuclei.

§ 2. The A TMS method in the excited states

We give an outline of the variational method in the excited states and

construct trial wave functions in this. section. Ritz's variational principle,

(I)

ts applicable to the excited state 1Jf(j1r, T) with a nuclear spin], an isospin

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110 M. Sakai, Y. Akaishi and H. Tanaka

T and a parity 7T, if we give lJ'(jrr, T) so as to keep its orthogonality to the ground state Pgr(O+, 0), where His the Hamiltonian of the four-nucleon system and Eo(]rr, T) is the eigenvalue of H. In the present approach of ATMS, 1J' (Jrr, T) is related to an initial state ~(]rr, T) as

(2)

F being solved in the same procedure as in Chapter III. Equation (2) makes it very easy to impose the orthogonality on lJ'(jrr, T). We can obtain P(jrr, T) orthogonal to lJ'gr(O+, 0) automatically, if only @(jrr, T) is kept orthogonal to @gr(O+, 0).

The correlation function of the ,total system F is represented in the same expression as that of the ground state in terms of the two-body correlations ~(zj) and '(zj) (see Eq. (Sa) in Chapter III). The state dependence of TJ(ij) should, however, be replaced by the following:

TJ(zj) = 1TJ s(rij)P ~1 + 3'r] .s(rtj) ·P ~1 + 3TJ D(rij) ·Sw P ~~

+ 17Jp(rij)·P~9 + 37Jp(r·ij),~P~9, (3)

where P~i(a=lE, 3E, 10, 30) is the projection operator onto the two-body state and sij is the tensor operator,

(4)

We prqceed to prepan~ the wave functions of the o-, 1- and 2- levels with T=O. The initial 'states @cJ) in Eq. (2) are given as coupled states of the orbital angular momentum L =I with the spin S = l correspondingly to the lp-lh model as follows:

3 @cJ)=@s :E [ui@{S=l, T=O}MA,i]U>, (]=0, l, 2) (5)

i=i

where

(6)

{S T} R .is the orthonormal spin-isospin basis, which belongs to the irreducible representation [R] of the symm.etric group S4, 7> and {ui, i=l, 2, 3} are the interpair vectors.?) An, abbreviation [j~L)@{ST} R]U> is, used to denote the state with a total angular momentum J generated by the coupling of a orbital one L with a spin S,

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Negat£ve-Par£ty States of the Alpha Part-icle 111

Then the tdal wave functions PU> are derived from Eq. (2), (]><.!) being multi

plied by F, and are expanded into the following components:

1TF(J)=1TI(cJ)+1TI(Tf), (/=0 1 2)' T T r , , (7a)

where 3

PC()= ~ [j~~Ml]>iQ9 {10} MA,i]U> i=l

3 + ~ [j~~M1J>iQ9 {10} MS,i]U> (7b)

i=l

and

PV)= , ~ v3C2J'+ I)· wc12;1; 1/') /'=1,2,3

3 J' J' X ~ {(/~,~s.t® {10} MA,i]U> +(f~,~A,iQ?) {10}Ms,i]U>}

i=l

+ ~ v3(2f'+I)· wc2211; 1J') /'=1,2,3 '

(7c)

' '

In Eq. (7c) W(abcd;ej) is the Racah coefficient, and explicit expressions of

the ~~~RD and JV.'A are given in the Appendix. PV) contains correlations caused mainly by the 3 E-tensor force, while

PC() is composed of correlations by 'Central forces. In usual, effects of the

tensor force are included in the effective interaction in a form of the effective

ce~tral force. Hence PV) ought not to ~a.~pear in the treatment of the 1p-1h model with realistic forces. -

it is a merit of the present wave function that we can get an internal

energy without the troublesome problem about the center-of-mass spuriosity,

because PU>· is described only by the relative coordinates rij· PU> has, of

course, the property that it vanishes if any nucleon pair comes close to each

other inside the hard core radius of the potential.

§ 3. Results and discussion ·

§ 3-1. The 1p-1h model

The T =0 negative parity states are investigated here in the 1p-1h model

for latter discussions. The energy differences among these levels are given in the model by

(8a)

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II2 M. Sakai, Y; Akaishi and H. Tanaka

(8b)

where the effective interaction in the triplet-odd state 3vj-) is composed of . 3v~--:-), · 3v~-) and 3v~-), which are the central, tensor and spin-orbit parts of the two-body force, respectively:

3vC-) -3vC-)_4·3vC-)_2·3vC-) · J=O- C T LS ' .

3vC-) - 3vC-) +2·3vC-) - 3vc~) J=l- C T LS'

2 3vC-) _ 3vC-) _ -·3vC -) + 3vC-) J=2- C 5 T LS ·

(9a)

(9b)

(9c)

The bracket <3vj-)) means the matrix element with respect to the two-body

relative P-state. The total energy E(J-) is given by

(IO)

where sis a constant energy for ] O-, I- and 2-. The forces in the even ) ·.

states do not contribute to the splitting as is seen fro~ Eqs. (8a) and (8b). The values of <3v~-)) and <3v~--]) are extracted from the experimental energy splitting such as

<3v~))=l.33 MeV,

<3v~--J)= -,--1.37 MeV.

(II a)

(Ilb)

Effects of the 30-LS and 30-tensor forces on the levels are seen from Fig. l. The experimental spectrum is shown in Fig. I(c). The contribution of the 30-tensor force is subtracted in Fig. I(b) from each level of Fig. I(c), where the value of (IIa) is used as the matrix element <3v~-)). Levels are degenerate without the 30-tensor and 30-LS forces [Fig. I(a)]. The experimental level order (O-, 2- and I-) cannot be reproduced by the central force and the spin-orbit force whose effect is equivalent to the single-particle spin-orbit one [Fig. I(b)J. The 30-tensor force makes the Q- state considerably lower [Fig. 1( c)]. The observed splitting is brought as a result of the competiti~n between the 30-LS and 30-tensor forces.

The tail part of the Ham ada-Johnston potential8) is adopted for the eff~ctive interaction. The separation distances are taken to be dr=0.90 fm and dLs=0.75 fm so that the values (IIa) and (IIb) are reproduced at liw= 22 MeV ,(which Barrett used). ·The hw-dependence of <3v~)) and <3vt]) are shown in Fig. 2. The values of the matrix elements are sensitive to the spread of the wave function. In the triton,

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Negative-Parity States of the Alpha Particle

(MeV) 32 T=O states of

4 He 1

30

0 :<.::2 u 6'\

22

20

,,.--/

0-/ ,--.I I \1 I II

! ft\ ! ,,----'-1 \

- - _/l \ ~I \

(a)

\ I \ I \ -·I

\ ___ ~ \ -'L_l_ \ o-l..___=:..

(b) (c)

Fig. 1. Effects of the odd-state forces .on the level splitting of the T=O states in the 1 p-lh model.

The spectrum (c) is the observed one in Ref. 3). The contribution of the 30-tensor force is subtracted in the spectrum (b) from the spectrum (c). The degenerate spectrum (a) is produced by subtracting the contribution of the 30-LS force from the spectrum (b). The values (lla) and (llb) are used as the matrix elements <3vt-)) and <3v~8)).

1.4

1.2 1.0 0.8 0.6 0.4 0.2

8 10 12 14 16 18 20 22 1lw (MeV)

Fig. 2. nw-dependence of <svt-)) and <av~"S)).

The plotted curves are calculated by the use of the tail parts of the H-J potential. The separation distances are chosen to be dr=0.9 fm and dLs=0.75 fm so as to reproduce the values (lla) and (llb) at nw=22 MeV.

fi2 I

liw= M X (rms)-2

=41 x(l.7 fm)- 2c---14 MeV.

113

Then (3v~-)) and ( 3vtd> are reduced by factors ~1/2 and ~1/3, respectively, compared with the values at liw=22 MeV.

In the 1p-1h model one cannot but use the same value of liw as the ground state because of the spuriosity problem of the center-of-mass motion. Hence the model deals with not the absolute value of the excitation energy but the splitting only.

§ 3-2. Total energies by the AT MS method

Variational calculations of Eq. (1), are performed by the use, of the Hamada-

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Johnston (H-J) potential. The on-shell correlations 17]s, 37]s and 37JD are solved from the Bethe-Goldstone equation. To the on-shell correlations in the relative odd-state and to the off-shell one, the following forms are assumed: ,

where rc is the hard core radius of the potential (rc=0.485 fm in H-J). Only the parameter liw (=f3n 2/M, M: nucleon mass) is varied (y is fixed 'at 0.6). The numerical integration for ErP over nine variables {ui, i=1, 2, 3} is carried out by the quasi-random number· method, 9) whose error is estimated in Chapter III to be 2 MeV at most for the total energy and 1 MeV at most

I

for each term of the energy. Results are summarized in Table I and Fi'g. 3, where the value nw= 10 MeV is taken. A contribution of the quadratic

' Table I. Upper-bound energies of the o-, 1- and 2- states and its details. The Ramada-Johnston potential is used, but contributions from

its quadratic spin-orbit forces are not included in this table. Results are obtained with hw=10 MeV. P(S) mea17-s a state probability for the state with the intrinsic spi~ S.

State I

J~O-I

J=1-I

I==2-

Total energy ECJ) (MeV) I

-6.4 I I

2.1 I

-2.9

Kinetic energy (MeV) I

74.8 I

76.0 I

75.4

,Potential energy (MeV) , I -81.2 I

-73.9 I

-78.3

> Central forces -34.1 -31.2

'

-32.6 v Vc(lE) -19.4 -16.2 -17.6 e Vc('6E) -14.9 -15.2 -15.1 513 Vc(lO) 0.31 0.35 0.33 1-< v .:: Vc(30) -0.15 -0.16 -0.14 v

] Tensor forces -47.5 -42.5 -44.6 ..... .:: v VT(3E) -45.2 --,-43.9 -44.3 ..... 0 p.. VT(30) ---'- 2. 3 1.4 -0.33 4-1 0

Spin-orbit forces ..$ 0.44 --0.16 -1.19 'Cj

VLs(3E) -0.73 -0.75 -0.67 ~(,

Q VLs(30} 1.16 0.59 -0.51

State probabilities (%)

P(S=1) 92.4 93.7 93.6 P(S=2) 7.6 6.3 6.4

Root-mean-square radius (fm) 2.26 2.29 2.28

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Negatz've-Parity States of the Alpha Partz'cle 115

spin-orbit force of the H-J potential is not included in the present results.

What is to be pointed out, .first of all, is that the total energies obtained

as the upper bound are -6.4 MeV, +2.1 MeV and -2.9 MeV for the O-, land 2- states, respectively. They ·are obtained not as the excitation energies

but as the total ones without any parametrization of the ground state energy. The result that the system is bound as for the o- and 2- states shows a validity

of the ATMS method, that is, it is powerful enough for the excited states.

The energy spectrum obtained is shown in Fig. 3 together with the observ

ed one. The correct level order of the o-, 2- and 1- states is reproduced in

the present calculation. It may be said that the fitting to the observed

spectrum is fairly reasonable. There is a possibility to attain a comprehensive

understanding of the T = 0 states in<;:luding the ground O+ state, if the three-body

force is taken account of. It is pointed out in Chapter V that its effect on the

ground state energy amounts to 8,.._,10 MeV attraction in the alpha particle,

while it yields only ""'1 MeV attraction in the triton. The contribution

from the .three-body force is expected to be ·reduced in the excited states to

the nearly same value as in the triton, if the cluster-like _structure such asp-

3H or n-3He appears as is mentioned in the following. Then the deficiencies

of the energies in the excited states as well as in the gr?und states will be

explained more reasonably by the three-body force. This energy shift between

5 (MeV)

o --'<:27.4 --r-------------------------

- 5

>. -10 (J)

L (lJ

c (lJ

0 -20 .,_.

0 .,_.

-25

-30

26.4 2-

1 I

I

i ,--

~~~. o-21.1 __ o~~~--

0 ... ,---1 I

I I

I t i

I I

I

o.o · o~i T=O T=O

experimental calculated · 4 He

Fig. 3. Energy spectrum of the negative-parity states of 4He. ·

The experimental spectrum is referred to Ref. 3). The calculated levels are the o-, 2- and 1- states with T=O. As for the ground O+ state, we have alieady obtained -20.6 MeV as the upper-bound energy in Chapter III.

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116 M. Sakai, Y. ~kaishi and H. Tanaka

the observed spectrum and the theoretical one may be regarded as a phenomenological data concerning the strength C P of the three-body force in Chapter V.

An attention should be paid to a fact that each term of the energies in Table I has a close resemblance to that of the triton. In the ground state the alpha particle has the kinetic energy of ,_,130 MeV and the potential energy of ,......., -150 MeV. The contribution from the 3 E-tensor force is ,_, _,_., 70 MeV (Chapter III). On the other hand, the binding energy of the triton is brought by the kinetic one of .-:..,65 MeV and the potential one of "-' -70 MeV, and the 3 E-tensor force gives ,_,33 MeV attraction (Chapter II). In the exCited states, the kinetic and potential energies are considerably reduced to ,_,75 MeV and ,._,:..__80·MeV, respectively, which are similar rather to the triton. The essential role of the 3 E-tensor force which is seen in the ground state of the alpha particle and in the, triton remains unchanged in the excited states. Namely, it gives a large contribution· to the interaction energy (e.g., --,---45.2 MeV in the O- state) which surpasses an amount of those from the central forces ( -34.1 MeV in the o- state). These facts seem to suggest a clusterlike structure in the excited states such as p-3H or n-3He. The existence of the nucleon in p orbit prevents the system from being bound tightly. In fact, the root-mean-square radius of ,_,2.3 fm indicates that the system is loosely bound rather like p-3H or n-3He.

§ 3-3. The splitting among the o-. 1- and 2- states

The 30-tensor and 30-spin-orbit (LS) forces have appreciable effects on the level splitting, although their contributions to the interaction energy are very small compared with the 3 E-tensor and central forces (Table I). The actual spectrum is arisen as a consequence of a competition among these forces. It is, however, noted that the competition between the 30-tensor and 30-LS forces is reduced by a factor 1/3,_,1/4 in comparison with the lp-lh model calculation, as is seen from Table II. The even-state forces, i.e. 1 the 3 E- · tensor and lE-central ones, recover the deficiency of the splitting energy

Table II. Comparison of the contributions from the odd-state forces between the present and the lp-lh model.

(unit in MeV)

~ Contribution from the 30-tensor force Contribution from the 30-LS force

State~ Ours a) I

1p-1h b) Ours a) I

1p-lh b) T=O,J-

o- -2.3 -7.98 1.16 4.11

1- 1.4 3.99 0.59 2.06

2- -0.33 -0.80 -0.51 -2.06

a) Values are taken 'from the Table I ~here liw=10 MeV. b) Values are given according to Eqs. (9a),-.,(9c) by the use of (lla) and (llb).

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Negative-Parity States of the Alpha Particle · II7

due to the reduction of the contribution from, the 3 0-state: forces. This strikingly contrasts with the situation in the lp.:.Ih model where only the odd-state forces are responsible for .the splitting.

In the Ip-Ih model, the spread parameter; of the wave function nw is taken to be nw=22 MeV common to the ground state in order to avoid the difficulty on the spuriosity of the center-of-mass motion. 4); 5) Then the odd-state forces .are compelled to. give too large contributions as is shown io Table II. The value nw=22 MeV is, however, unsuitable for the excited states, because the system ought to be loosely spread rather like the triton plus the proton as is suggested from the present results. The contributions of the odd-state

·forces are reduced by a factor I/2~I/3 at nw .I4 MeV which corresponds to the root-mean-square radius of the triton (§3-I). In fact, our results show such a reduction of the contribution from the odd-state forces, and the evenstate ones compensate the splitting for the deficiency of energy differences due to this reduction. That is to say, the observed spectrum can be understood reasonably by the present treatment without any restriction which is necessarily imposed on the Ip-Ih model, because the wave function lJ'U) is free from the center-of-mass motion. Thus the even-state forces as well as the odd-state ones play a role in the splitting among the o-, I- and 2- states.

The reason why such a mechanism appears lie~ in the wave function lJ'(.f). The correlations which cannot be described with the independentpair model (I. P A. model) are. included in lJ'U). That is, the correla,tions induced by the multiple scattering process are responsible for the role of the even-state forces in the splitting. If lJ'U) is approximated to the I. P A. model, it can be shown that the contribution from the even-state forces is just the same for the o-, I- and 2- states, and that only the odd-state ones contribute to the splitting. The deviation from the I. PA. model is rather small, which is seen from the fact that, for example, the 3 E-tensor force brings only the differences less than IO% among three levels; VT(3E) are -45.2 MeV

. and -43.9 MeV in the o- and I- states, respectively. Nevertheless, the deviation has an appreciable effect on the level· splitting. The effective intera.ction given by the I. PA. model may have room to be reexamined m the excited states in this respect.

§ 4. Concluding remarks

The ATMS method succeeds in treating the excited states of the alpha particle. The total energies yielded by the Hamada-Johnston potential are -6.4 MeV, -2.9 MeV and +2.1 MeV in the o-, 2- and I- states with T=O, respectively. They are obtained as the upper bound of the variational energy. The observed energy spectrum is reproduced reasonably. This shows that the A TMS method is applicable enough not only to the ground

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state but to the excited states .

. Some aspects are revealed in the excited states. The observed spectrum

will be understood more entirely, if the three-body force is taken into account

in addition to the two-body force. The essential role of the triplet-even tensor force remains unchanged in the binding mechanism of the excited states. The content of the total energies has a close resemblance to that

of the triton, which suggests a cluster-like structure such as p-3H or n-3He.

The binding mechanism which appears in the splitting of the levels is quite

different from the usual grasp by the lp-lh model where only the odd-state

forces contribute to the splitting. The present approach makes it possible to give the reasonable understanding of the spectrum without any restriction

imposed on the lp-lh ·model. It is concluded that the ATMS method is the firsLtreatment that succeeds

in attaining the comprehensive understanding of the total energies of the excited states as well as the ground state on the basis of the realistic nuclear

force.

Acknowledgements

The authors would like to express their thanks to Professor J. Hiura

and Professor H. Bando for valuablr and stimulating discussions. Numerical

calculations are made by HITAC 8800/8700 computer at the University of Tokyo and FACOM 230-75 computer at Hokkaido University.

Appendix

Spatial functions f~~R.v and JVJ. in Eqs. (7b) and (7c) are given in this

appendix explicitly. The,.operator F, which describes the correlations of the system, Is com

posed of two parts, F c and' Fr, common to the O-, 1- and 2- states.

F=Fc+Fr,

Fe= I:; I:; Cl.J~1·PT1 ll(ij)

and

where

and

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Negative-Parity States of the Alpha Particle 119

with

u(r)= 1- '(r).

Then_ PC/) and PiO are expanded into Eqs. (7b) and (7c), where PC/)=Fc'<PU> and PY)=Fr·l!>U>. Space partsf~~R.D and/YJ in Eqs. (7b) and .(7c) are given in the following :

M . d . ' b . (3d' ' 1) jCL-D jCf') zxe symmetrzc aszs . - 1menswna : c.Ms.t, r.MS.i.

f~~Ml],I = urll>s·[-5B+(1/2) {Q}/1 +ClJf4 +ClJf4 +ClJla +ClJ~i +ClJ~~

+ ClJf2 + ClJ~4 + C!Jf4 + C!Jqa}]

+u2·lPs·(1/2) { '-- (C!Jf4- C!Jfa) +(C!Jf4- C!Jqa)}

+ua·ll>s·(1/2) {- CC!Jf2- ClJ/4) +CC!Jf2- C!J~4)},

where

C!Jtt = (1/4)(3·CIJ~~ +C!J~~). I

C!J?i= (1/4)(CIJ~~ +3·C!J~~), B= [1 u(nJ).

( ii)

f~~Ml], 2 ={Permute particle 2 and particle 3 in /~~M:Y, 1 .},

f~~Ml],3 = {Permute particle 3 and particle 4 in/~~M:Y, 1 .},

JV.2ts.I = -v5/3·(1/4)·ll>s

X [u1@ {4}~~· T}~+e[J;-'i· T~2J+4J~IJ· T}?.J+ 4}~~· T~~

-4(e[J~~· T}~+e[J~-'i· T~?.J)}

+u2® { .-e[J~-'i· T}2J+4J~~· T~~} +u3@ { -ct}~~· T}~+4J~-'i-- T~2)} ]Cf'),

J>/.2ts,2 . {Permute particle 2 and particle 3 in/>/.2ts,1 .},

JV.2ts,3= {Permute particle 3 and particle 4 in/>/.'JJ5 , 1 .},

where T~)) is the irreducible tensorial set with .rank 2 defined by

and

e[J~lj =(3/r~J)·C!J~f.

Mixed basis (2-dimensionaJ): J>/.'Jt.i:

j>/,'Jt,1 =·(v15/2)-ll>s[u1@(4}~~· Ti~-4}~· T~?.J) - Ua@(CfJ~IJ · T}?.J- Q}~f· T~~)]Cf'),

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JV.'it,2= (vS /2)·@s[u10(CZJ~~· Ti'1i-cfJ~~· T~2J) +u3Q9(CZJ~~· Ti2d-cfJ~· T~~)

-2u2Q9(CZJ~i· Ti~~ cfJ;~· T~2J)]CJ').

Mixed antz'symrnetric basis (3-dimensional): f~~=tll,t, j>/,~A,i.

/cv=n . Y 3 ·@s[u

1 {- (OTEC-)_ a TOC-))- (O T EC-)_ oroC-))

C,M A ,1 8 '112 '112 ·, '134 '134 / + (Ci.Jfi-) '--- C!.Jf4-))- (Q}f3C-)- C!.J~~- ))}

+ U2 {- (CI.Jf4C-) + C!.Jf4-)) + CC!.Jf3C-) + C!.J~~-))}

+u3 { + CCIJ~c-) + C!.Jf~-)) + CC!.Jt4c-) + C!.J~4-))} ],

where

f~~A1lJ, 2 ----. (-){Permute particle 2 and particle 3 in /~~A1lJ, 1 .},

f~~A1lJ, 3 =(- ){Permute particle 3 and particle 4 inj~~A1lJ, 1 .},

f cf') __ y 5_·@s[Ulfx\(CfTaD.TC2)+0T3D,TC2) T,MA,l- 4 '<Y '112 12 '134 34

- e[J[i · Ti2d- cfJ~~ · T~1i) + u2®CCfJl~ · Ti2d- ClJ~~ · T~~) . + U3Q9(- CfJ~f · Ti~ + cq;~ · T~2J)]CJ'),

JV.'itA,2=(- ){Permute particle· 2 and particle 3 inJ>/.'itA;l·},

JV.'itA,3 =(-){Permute particle 3 and particle 4 inf>/.'itA, 1 .}.

References

1) Y. Akaishi, M. Sakai, J. Hiura and H. Tanaka, Prog. Theor. Phys. 51 (1974), 134. M. Sakai, I. Shimodaya, Y. Akaishi, J. Hiura and H. Tanaka, Prog. Theor. Phys. 51 (1974), 15.5.

2) L. M. Delves and M.A. Hennell, Nucl. Phys. A168 (1971), 347. A. D. Jackson, A. Lande and P. U. Sauer, Phys. Letters 35B (1971), 365. P. A. Malfliet and J. A. Tjon, Phys. Letters 35B (1971), 487. E. P. Harper, Y. E. Kim and A. Tubis, Phys. Rev. Letters 28 (1972), 1553. J. Bruinsma and R. van Wageningen, Phys. Letters 44B (1973), 221.

3) S. Fairman and W. E. Meyerhoff, Nucl. Phys. A206 (1973), 1. 4) A. de-Shalit and J, D. Walecka, Phys. Rev. 147 (1966) 1 763. 5) B. R. Barrett, Phys. Rev. 154 (1967), 955. 6) T. T. S. Kuo and J. B. McGrory, Nucl. ·Phys. A134 (1969), 633.

P. P. Szydlik, J. R. Borysowicz and R. F. 'Wagner, Phys. Rev. C6 (1972), 1902. 7) J. E. Beam, Phys. Rev. 15.8 (1967), 907. 8) T. Hamada and I. D. Johnston, Nucl. Phys, 34 (1962), 382. 9) V. B. Haselgrove, Math. Camp; 15 (1961), 323.

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108 Chapter VII Negative-Parity States of the Alpha Particle § 1

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Transcript of 108 Chapter VII Negative-Parity States of the Alpha Particle § 1