Nuclear Physics A153 (1970) 445 --459; ( ~ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

B O O T S T R A P T H E O R Y O F V I B R A T I O N S

(II). Extended particle-phonon coupling and R P A

in closed shell nuclear regions

AMIT GOSWAMI t and ORHAN NALCIOGLU t Institute of Theoretical Science and the Department of Physics, University of Oregon, Eugene, Oregon

and

ARNOLD I. SHERWOOD tt

Department of Physics, University of Arizona, Tucson, Arizona

Received 15 December 1969 (Revised 13 April 1970)

Abstract: The bootstrap theory of vibrations is developed for closed-shell nuclei where the particle and phonon modes of these nuclei are generated simultaneously and self-consistently. It is shown that in first order the phonon solutions of this theory correspond to the random-phase approximation (RPA). The properties of the particle modes is also studied in this order. It is found that particle (hole) states are pushed up in energy which can account for the discrepancy between the shell-model particle-hole energies and the experimental ones. The non-normal parity states (particle-phonon coupled mode) are also studied and it is found that some of these can be expected to lie close in energy to the normal parity (particle mode) states, as it is experimentally found. The possibility of a different approach where the calculation of even- and odd-mass nuclei is carried out separately, the extended-particle-phonon-coupling theory (EPPC without bootstrap) is also pointed out. A numerical example using the perturbation theoretical solution of EPPC theory is given for 39.40.41Ca"

1. Introduction

In the preceding paper t) it was shown how a boots t rap theory of phonons and qua-

siparticles could be obta ined in a self-consistent manner . In this paper we shall develop

an analogous theory for vibrat ions in closed shells using the particle representation.

I t has been realized for a long time that in order to generate the vibra t ional character

of the low-lying odd-par i ty states of closed shell nuclei, one has to take in to account

the correlat ions of the ground state, which is done by the R P A method 2). However,

unt i l recently, the effect of the g round state correlations on the single-particle (hole)

states was ignored. Very recently, one impor tan t effect of the ground state correla-

t ions on the particle (hole) modes was pointed out by refs. 3 ,4) t t t ; namely that the

t Work supported in part by US Atomic Energy Commission. tt Supported in part by US National Science Foundation.

*tt Ref. 3) only includes the ground state correlation effects due to the particle-particle T = 1 force. In contrast to this we shall consider the particle-hole force in the 3- T = 0 state and show that the effect of the latter one is also quite large.

445

446 A. GOSWAMI e t aL

observed splittings between particles and holes are considerably larger than their shell-model unperturbed values. This is because the ground state is depressed much more by the correlations than the single-particle (or hole) states due to the blocking effect of the extra particle (or hole). Other authors have pointed out the possibility of considerable loss of single-particle (or hole) strength due to the ground state correlations s, 6). The calculations of the ground state correlations mentioned above have been carried out only to the extent of a perturbation admixture of 2p-2h states in one case 3) and a diagonalization including only a severely truncated subset of the 2p-2h states in the other case 4). However, in the RPA theory (which works for the pho- non states) correlations of 4p-4h, 6p-6h, etc. type are also included in the ground state. The calculation of the ground state depression due to these additional configurations would be very difficult with the approach of refs. 3, 4). The calculations of refs. 3, 4), however make it clear that the inclusion of the ground state correlations, which is important for the phonons is also essential for generating the particle (or hole) modes for the closed shell plus (minus) one nucleon systems. The bootstrap theory of ref. 1), which is a self-consistent theory of particle and phonon modes and their coupling should be well-suited for this.

The coupling of particles and phonons in the closed shell region has been considered by other authors 7). The closest work to the present one is of refs. s, 9) which used: (i) the equation of motion method 10), and (ii) the RPA wave functions for the phonons. The works of refs. s, 9) still did not take the ground state correlations directly into account for the odd-mass nuclei. Moreover, the RPA theory t* for the phonons has recently been criticized for various reasons ~1). The direct account of ground state correlations is achieved by extending the equations of motion to include the coupling to the backward going amplitudes. We shall call this the extended-particle- phonon-coupling 13) (EPPC)***. In this way, the ground state correlations can be accounted for in a non-perturbative manner and higher order correlations are also treated as in RPA, in contrast to the work of refs. 3, 4). Furthermore, we shall formu- late the bootstrap theory which will enable us to generate the phonon modes simulta- neously and self-consistently with the particle (and hole) modes.

Additionally, we shall show that the perturbative solution of the bootstrap equa- tions lead to the RPA for the phonons. In the same order of perturbation, the particle (and hole) modes will be shown to be 'pushed up' in energy, thus accounting for the largeness of experimental particle-hole splitting compared to their shell-model value [in agreement with refs. 3, 4)]. This theory also enables us to calculate the loss of single-particle strength. The particle-phonon coupled mode will also be studied and it will be shown that some of these states can be expected to lie close in energy to the

t The first use of the equation of motion method in the present context was made by Klein and Kerman lo). For references to other related papers see Johnson e t al. lo).

*t The breakdown of RPA is evidenced in some heavy spherical nuclei by the discovery of large static quadrupole moments, see ref. ~). **t This is named in analogy to EQPC formulated in ref. 13).

BOOTSTRAP THEORY OF VIBRATIONS II 447

particle states in the odd-mass nuclei as observed experimentally. Numerical calcula- tions based on the perturbative solutions of bootstrap theory will be given for 3 9 , 4 o , 4 1 C a "

In sect. 2, we define the Hamiltonian H and discuss the Tamm-Dancoff (TD) and RPA methods for the treatment of vibrations in the closed shell nuclei. In sect. 3, we shall introduce EPPC and bootstrap theory. The discussion of the perturbative solu- tions of bootstrap theory will be presented in sect. 4. Sect. 5 contains a discussion of the EPPC theory without bootstrap. In sect. 6, we shall present a numerical example of the perturbative solution of the bootstrap theory. Finally, sect. 7 contains the summary and conclusion of the paper.

2. The Hamiltonian H, TD and RPA

In this section, we shall review the Tamm-Dancoff (TD) and the random-phase approximation (RPA) for the (odd parity T = 0) phonon states of the closed shell nuclei. In order to keep the formalism and numerical calculations simple, we shall assume the two-body interaction to be the separable multipole-multipole force. The method of this paper can, however, be readily extended to the case of a more general two-body interaction.

We start with the Hamiltonian t

Z = X e,c:c~-½ E ZxA¢~" M~" (1) ~t 2

Here e~ are the single-particle (hole) energies and Mx is the 2 a pole multiple operator Mg, t, and M~ = ~¢=~h x (¢ being the charge index)

M~ = V~ 2-~ E (-)~Mx(hp)[A~:°(hP)+(-)"A'a~°(hP)] , (2) hp

with A f ~ M T ( h p ) = 2 { ~jh--Mh+~--,h/"jhjp;t p½½T ,,+ ~.--] x'~-MhMplt ~'~--'thtpMT top Ch, (3 )

MhMp ~h~la

where C is a Clebsch-Gordan coefficient. The TD method consists of diagonalizing the Hamiltonian in the particle-hole basis

](h-*p);t#TMr) = A~m~(hp)]~p0>, (4)

where ](Po> is the unperturbed ground state of the closed shell core. Unfortunately, the TD method does not produce enough coherence to generate

the T = 0 odd-parity collective vibrations observed experimentally [see, for example, ref. 9)]. One way of introducing the necessary coherence is through RPA, which takes account of the correlations in the closed shell ground state.

t We explicitly consider the T = 0 particle-hole force, since our method does no t seem to be suited for the t reatment o f ground state correlations (it" any) due to T = 1 particle-hole force, which is repulsive. We also neglect the correlation effects due to the particle-particle force.

448 A. GOSWAMI e t al.

If we let [To> denote the actual correlated ground state and [Tn> the excited collec- tive vibrational state, then we can define the forward and backward going amplitudes, respectively, by

X(hpBT) = <TB[A~-~nMT(hp)[To>, (5a)

.~(hpBT) = Sn<TBlA~r~-M~(hp)l~eo>, SB = (--)~-~t~+r-ML (5b)

The A operators are treated as boson operators. Taking the commutator of these operators with the Hamiltonian, and retaining only the terms linear in the A and A + operators in the resulting equations, one obtains the following dispersion equation for 2 B pole T = 0 collective mode:

2B+ 1 _ vM2(hp)eph ~ , (6)

4ZB hp ~ph -- con

where

~ph ~ Ep--/~h"

The amplitudes X(hpBT = 0) and X (hpBT = 0) are given by

X(hpBT 0) ~/2 M•(hp) . . . . Zn - - (-)BMn(BO), (7a) 2B + 1 eph- con

X(hpBT = 0) - ~/ '2 XB MB(hP) MB(BO), (7b) 2B + 1 ep~ + cob

where MB(BO) is to be determined from the RPA normalization condition

X [X2(hP BT = O)-X2(hP BT = 0)3 = 1, (7e) hp

M~(BO) - 2B+4~B 1 . ~ { ~ V ~- 7 q - - - - ~ M2(hp)%ht - ~ " (7d) (~p~-co.) J

Eqs. (6, 7) determine the energy eigenvalues con of the states ITB) relative to the ground state ]To>, as well as the amplitudes X and X. The collective state ITn) can now be written as

l eB> + = 12B,~.l~o>, (8) where

Q+~. = E [X(hpBT = 0)A~oM"°(hp)-SB X(hpBT = 0)A~o~"°(hp)]. (9) hp

The Q~-~z, will be referred to as phonon operators and the states as one-phonon states.

The hermitian conjugate operator QBMn defines the ground state by

QnM.ITo> = 0. (10)

BOOTSTRAP THEORY OF VIBRATIONS II 449

It follows that the Q operators also satisfy boson commutation relations. Eqs. (8) and (10) determine the one-phonon and the ground states of the closed

shell core. We shall also assume the existence of core states of two or more phonons, although these might be difficult to identify experimentally, because of anharrnonie effects. In particular in RPA, the two-phonon state is given by:

1 - - CMBMB'MI QBMs QBM~,[ ~o). (11) I~x) x/2 mBmB,E nnx + +

In the subsequent sections, we shall consistently use I~n) for one-phonon states, [ ~ ) for two-phonon states.

3. Extended particle-phonon coupling (EPPC) and bootstrap theory

3.1. E Q U A T I O N O F M O T I O N M E T H O D

Let us consider the equation of motion of a single-particle operator t

Ep(~plc+[~o) = (~pl[H, c+]l~o)

= pl I vo) - Z x (g, pl FM , c -IM + vo)

-½ Z ;~(~pl[ M+z, [M~, c+]][Vo) • (12)

The double commutator term can be shown to cancel if one takes the Hamiltonian in the normal ordered form. Therefore, we shall ignore it. The second term on the r.h.s, of eq. (12) represents the coupling of the single amplitude to triple amplitudes. We re-express this term as a single amplitude times a double amplitude by employing a complete set expansion of the core states I~a) (which includes the ground state

2hB (13)

The amplitudes like (~kplC~l ~vn) are called backward-going amplitudes for obvi- ous reasons. The coupling of these forward- and backward-going amplitudes will be referred to as extended particle-phonon coupling (EPPC). In the bootstrap theory, the matrix elements of M~ between core states (e.g. (kVnlMn[~o) are to be regarded as self-consistent potentials to be evaluated in terms of the solutions of the coupled (EPPC) equations in a self-consistent manner. In practice, this procedure is numeri- cally quite complicated and one has to make the following simplifying assumptions: (i) use a truncated set of states I ~B) keeping only those states which are expected to give large matrix elements lo) (~B[Mnl~o) on the basis of experimental data. A fortiori justification of the truncation can be given using perturbation solutions. If

t The kets I ~ ) and I~P) will denote odd-mass and core states respectively.

4 5 0 A. GOSWAMI e t al.

there is a collective 2 8 pole vibration which exhaust the major part of the 2 ~ pole sum rule, this procedure is obviously justified. For closed-shell nuclei, only the lowest 3- T = 0 state satisfies this criterion and, therefore the octupole component of the force can be expected to be treated well just by including the collective octupole state. In contrast, the 5- vibrations do not satisfy the criterion too well and therefore one might have to include more than one state to treat the corresponding component of the interaction. There is no collectivity for any other J~, T = 0 state and the correlation of the ground state due to these other states can be ignored. The treatment of these non-collective states will be discussed in a later paper; (ii) The second approximation consist of cutting off the coupled amplitudes at some point. With this truncation only zero-, one- and two-phonon states of the core will be included in the complete set expansion in this paper. In contrast to negative-parity one-phonon states, these two- phonon states, it should be noted, have positive parity. Even though, two-phonon amplitudes will also be coupled to three-phonon amplitudes, inclusion of three-phonon states may not be too meaningful. Because of this truncation the problem is cast into the (self-consistent) solution of a finite hermitian matrix. With these approximations the EPPC-bootstrap equations are easily written down which is done in the next subsection. The equation of motion of the phonon coupled amplitude

= a.<~,Jc~+l~%> - ~ ;~x<jhIM~[jp><~pJC+I~K><~KIM~"[~8>, (14) IlK MK

where I~x>, which can be either I~UB.> or I~> , contains the excitation energy ~o~ of the state I ~B> of the core which must also be determined self-consistently. This is done by considering the commutator of H with Mx itself as shown in ref. ~).

3.2. ANGULAR MOMENTUM COUPLING AND EPPC-BOOTSTRAP EQUATIONS

In this paper, we shall be mainly concerned with closed shell nuclei like 4°Ca where particle and hole states have opposite parity. The equations are easily generalized to the cases where the above is not true.

3.2.1. Particle nuclei. Particle states (normal parity). The following amplitudes are coupled:

~ o ~ = <~lc~+l~o>6~p, (15) ~ox~,~ = ~ rlJ~'J. /,t, I~ + (16a) "~'M~rMp,U**N'F edt-p" l ~l>,

MIMp"

XBh= Z n jh ja + -- C,,,.MhM.<~0JCN I~B>. (16b) MBMh

3.2.2. Particle nuclei. Particle-phonon coupled states (non-normal parity). The for- ward going amplitude

q~np~ ~ rBJ~J. /,t, , ~+ ,~> (17) : X'~MBMpM~NtFaOt'p I MBMp

BOOTSTRAP THEORY OF VIBRATIONS I I 45 ]

is coupled to the backward going amplitudes

Zlhot 2 IjhJa + = CMIMIaM~<~¢ot[Ch [ ~ / ' > "

(18a)

(18b)

<Ipa[HlI' p' o:> = (% + cot)61r 6pp, + Z Zz Ma(p' p)Mx(I'I)(--)r + h,+ ~'W(IpI'p' ; a2), . . . . . (24~)

< IpalH[B'h' ct> = £ Xa M a(h'p)Ma(B'I)(- )"" + jp + J'W(IpB'h' ; a2), (24d)

<0aalHl0a~> = 8p¢~pat, <OaalHII' p' o:> = O,

(0~IHIB'h'~¢> = ZB, Mn,(h,a)Mw(B,O)(_)S'ap,,

[B'] = 2S '+ l , [c¢] = 2j~+l,

(23a)

(23b)

(23c)

(24a)

(24b)

MIMh

3.2.3. Hole nuclei. Hole states (normal parity). The forward going amplitudes

~Po~ = S~(~lcJ To>~h, (19a)

r~J,,:,, u ~ eh[ ~,>, (19b) cp,~ = y, Sh,-, , .- ,~:. .Z,/ , I MIMh

are coupled to the backward going amplitude

XBp, = ~ SpC~L"Mp-M~,<~,I%I~B>. (20) MBMp

3.2.4. Hole nuclei. Hole-phonon coupled states (non-normal parity). The forward going amplitude

Sh cM.-M~-M.<~0=IChl ~'B> (21) M•Mrt

is coupled to the backward going amplitudes

~o= = s=<~O=lc=l ~o>a=., (22a)

~rP at 2 ljpJa C = Sp CMI_Up_M.(~k.I pl ~I>- (22b) MzMp

The coupled equations between the amplitudes q~opp, ~Ptp~ and Xw.. for the particle states I~k.> is given as

E~ ¢Po= = <0~lHI0~>~Oo~, + ~ <0ctctlHlI'p'ct>(pr¢~ + ~ <0~ctlnlB'h'~>XB,h,,, I 'p ' B'h'

E~ Cpip~ = <Ip~lHl0a~>~o0~ + ~ <Ip~lHlI'p'cx>q~rp'~, + ~ <Ip~IHIB'h'cc>Zn,h'~, l'p" B'h"

E~ZBh~ = <Bh~lHl0~a>CPo..+ ~ <Bh~lnlI'P'a>q~rp'. + ~ <Bh°tlnlB'h'°t>Zn'w,,, l'p" B'h"

where

452 A. GOSWA~t et al.

(Bh°~lHlB'h'o~) = (th + ~0S)bSn' 6h~, + Z ZZ Mz(h'h)M,~(B'B)(-)s' +~*+~ ~, = evell

x W(BhB'h'; a2), (24e)

with the serf-consistent potentials given as

MB,(B'O) = 2 ( _ ) n ' ~ MB,(hp){x/~-~]q~oppZ,,h~,+(_)jh-h,X/-~]tp,,phZOhh}, a/2B' + 1 hp

(25a)

Mx(IB) = 2 Z [a](-)r+JP+JaM~(hp){W(IBph; 2a)q~tr~Xnh.+ W(IBhp; 2a)r.pBpaZlha}, hp

a ( 2 5 b )

Mz(B'B) = 2[ ~ [a](-)B+ih'+J~Mz(hh')W(B'Bh'h; 2a)ZBh~Za,h, , h ' h a

+ ~ [ct](-)~+J~+Y~+B'Mz(pp')W(B'Bpp'; 2a)~B¢~B,~], (26a) p'pa

Mz(I'I) = 2[ ~ [ct](-)I+ih'+J~Mz(hh')W(I'Ih'h; 2a)ZIh~Zrh'~ h ' h a

+ ~ [~](-)a+~"+J"+rMa(pp')W(I'Ipp'; 2a)~p,,~rp~]. (26b) p ' p a

The matrix elements in eq. (26) are not expected t o be large, but are given for the sake of completeness. It should be noted that the amplitude tpxp. is also coupled to three phonon-hole coupled amplitudes through a large matrix element which we have ignored. It should also be noted that in the bootstrap theory the states I tgi) are not given by by eq. (11), but can be anticipated to have similar properties. Furthermore, the coupling to the deformed 0 + state is neglected in this work. This can partially be remedied by considering the coupling to the quadrupole moment (due to admixture of deformed state) of the 2 z pole collective state. Such considerations are reserved for a future publication.

The coupled equation for ~PBp~, ~o~ and Z~h~ are given as

g a CPBP ~ "~" Z ( BPctIH]B' P' °~)qgn'p'~ + (BpctlHl0a~)Xo~, + ~ ( Bpo*lnlI'h' o~) Xrh, , , (27) B'p' I'h"

t t 0~ t t Ea Zo,, = ~', (0~ctIHIB p ~)q'n'p', + (0~IHI0~a)Xo,, + Z (0~ IHII h ~)Xrh',, (28a) B" p' l'h"

E,, X,h, = ~ (IhaiHiB'p'o~)tPn'p'~, + (IhalHi0~a)Zo,, + ~ (IhctlHlI'h'cz)Xrh,~,, (28b) B ' p ' I ' h '

where

(BpaIHIB'p'a) = (ep+COa)fsB, rpp,+ ~', zzM~(p' p)Ma(B'B) ~, = e v e n

(-)a'+J'+JPW(BpB'p'; a2), (29a)

(Bp~[HI0aa) = - X~ ~ / _ ~ / _ ~ M .( a p )M .( BO )( -- )B + j. - J" Sh. , [ ] E ]

(29b)

B O O T S T R A P T H E O R Y O F V I B R A T I O N S I I 453

(Bp~lHlI'h' ~) = Z zxMa(h'p)Ma(BI')(-)~'+J"+BW(Bp I'h' ; a2), (29c) 2

(0a~ lH[0~) -- eh6,,h, (0ct~lHll'h'~) = O, (30)

(Ihc~lHlI'h'~) = (eh + o~,)6n, 6h~,' + ~ Za Mx(h'h)M~(I'I)(-)J° + j~ + r W(IhI'h'; a2), = e v e n

(31)

where the potentials M;~(B'B) and MB(BO), M;~(BI') are given by eqs. (25a, b and 26a, b).

The equation of motion for the amplitudes of hole nuclei can easily be obtained by replacing p, h by h, p respectively and changing the sign of 5, and multiplying the off- diagonal elements by - ( - )x .

3.3. NORMALIZATION

The above EPPC-bootstrap coupled equations have been cast in the form of hermitian matrices and can be solved by a matrix diagonalization routine. This deter- mines the eigenvalues and the amplitudes except for the normalization. The normaliza- tion t of the amplitudes is chosen 1 o) in such a way that the Pauli principle relations taken between the core states are satisfied, i.e.

(~Kl{C,, c+}I~K) = 1, (32)

where K = 0, B, I and ~ = p, h. Performing the angular momentum summation, we obtain the following equations

1 ~ (~pgpp)2 + ~. 2 = 2 (;(Opp) , (33a) /1 B

1 Z (t~g~h)2 + ,, 2 = Z (ZO,h) , (33b) n B

(2B + 1)(2p + 1) = E (2b + 1)(~p#) 2 + ~ (2b + 1)(~;p#) 2, (33c) b , n b , n

( 2 B + 1 ) ( 2 h + l ) = Z ( 2 b + l ) ( t ~ p ) 2 + ~ ( 2 b + l ) ( z ~ p ) 2, (33d) b,n b,n

(2 I+1 ) (2p+1) = Z (2b+l)(a~pp)2+ E (2b+l) (~vp) 2, (33e) b, n b , rl

(21 + 1)(2h + 1) = Z (2b + 1)(q3~p) 2 + Z (2b + 1)(X~p) 2, (33f) b, n b , n

wher n denotes a state of odd-mass nucleus with same angular momentum fl but differ- ent energy. Since the eigenfunctions determined from EPPC matrix diagonalization yield the ratios of the amplitudes ¢p and Z, X can be substituted into eq. (33) to get coupled equations in terms of ~o. The solution of these coupled equations in terms of tp yield the normalized amplitudes. However, as pointed out in ref. 16), a determina-

, The normalization chosen here is somewhat different from that used in our earlier papers x. la). The present normalization gives the right perturbation limit namely RPA value for the singie-particle strengths.

454 A. GOSWAMI et al.

tion of the ~0 using only the above relations violate the homogeneous Pauli principle relations (those with (~KI {G, c~- }[~s) J # K and/or ~ # fl). It is therefore suggested in ref. ~6) that the determination of ~p be carried out by a least-squares procedure using all the Pauli principle relations.

3.4. THE oJ-EQUATION

The bootstrap is completed by adding further equations to evaluate the core ener- gies o9 B and fOz. In this section, we shall only give the fO-equation for fOB. The equation for fOz may be obtained easily by similar arguments to these we shall give in this section for foB. In what follows, we shall use the RPA value for foz, i.e. 2 foB. The fob equation follows by taking the matrix element of [H, M~] between I LgB) and I kUo)

c°BMB(BO) = x/~ E (ep-eh)Ms(hP)[(-)sX(hpB)-X(hpB)] hp

- ~ Z;t(~nIM~.ITK)(TKI[M~ ~', MMB]I~O)~;.BbuMB. (34) KMK

At*

The largest contribution to the summation comes when [ ~ K ) = I~o). But (~ol [ M+~, M~][~o) = 0. Therefore, the effect of the second term should be small, representing the effect of admixture of non-collective B-states on the energy of the collective state.

The EPPC-bootstrap method consists of the self-consistent solution of the coupled equations (23), (27, 28) subject to the normalization condition (33), calculation of the potentials equations (25, 26) and the evaluation of fob from eq. (34). In the next section, we shall consider the perturbation solutions.

4. Perturbative solutions of the EPPC-bootstrap equations and physical consequences

4.1. RPA FOR THE PHONONS

When the energy of the core phonon state I~PB) is much smaller than that of the particle-hole state:

fOB ~ /~ph~

then the corresponding coupled equations can be solved by perturbation approach. In this section, we assume that the phonon states included in the EPPC expansion satisfy the above criterion. To illustrate the perturbation procedure, consider the coupled equations for the normal parity states t:

Ep (Popp = gp (~Opp "~ E (0pplHlBhp)ZBhp, Bh

Ep (pi,p,p = (•'p q- 2fO~)tpi,p,p + E ( I ' p ' p ] H I B ' h ' p ) Z B , h , p , (23b) B'h'

Ep •B'h'p = (B'h'plHl0pp)~popp + ~ (B'h'plHlI'p'p)tPrp,p + (eL + fOB)ZBhp, (23c) l'p'

* In the perturbation limit 'small' core matrix elements, e.g. M;~(B'B), M;.(I'I) can easily be seen to vanish. Therefore, from now on we shall omit them and also assume cot = 2~o8.

BOOTSTRAP THEORY OF VIBRATIONS I I 4 5 5

putting Ep = ~p in the second and third equations yield

[ep-(eh + fo~)]ZB'h'p = <B'h' plnlOpp)~oo,,p + Z (n 'h ' plHII' p' p)q, rCp, (35a) l"p"

[ S p - - ( e ' p + 2 f o ~ ) ] t P i , p , p = (I'p'plHlB'h'p)Zv,h,p, ( 3 5 b )

with q~opp = 1 to the first order ~O~p,p and ZB'h'p becomes

q~opp = 1, q~tp,p = 0, (36a)

(B'h'plHl0pp) ZB'h'p = (36b) I t

~p -- /~h - - foB

In a similar manner, we obtain for the 'phonon' particle coupled amplitudes,

~0Bp ~ = 1, (37a)

(Bp~IH[0~) Z 0 h h - - (~th ' (37b)

8p + to B - - ~h

X.,~,~, - (Bp~lHllh' o~) , (37c) ¢

/~p - - ~h - - (0B

substituting in the expression for MB(BO) eq. (25a).

Me(B0) = 2 { ZB x/2B+ 1 (_)B ~ MB(hp)

hp 4 ~ B ~ 4 E p ~ ( S p - - / ~ h - - foB)

Ma(hp)MB(BO)(- ) n ~/-~

- - Ks (--)JP-Jh+BMB(hp)MB(BO)] +(-)Jn+Jp4[h] 4 ~ 4 [ - ~ ~ j . (38)

We get the well-known RPA secular equation (6). In the same order, the m-equation in the form eq. (34) also gives the RPA secular equation.

4.2. PARTICLE (HOLE) MODES

From eq. (23a), one can now find Ep in the second order, by substituting the value of ~nhp from eq. (36) into eq. (23a)

S "x~M~(BO) MnZ(hP) (39) = + foB)"

Similarly for the hole state one gets

x~MB2(BO) M~(hp) = E . (40)

v" [B]Eh] (% -- ~h + foB)

The particle-hole splitting (the observed quantity) is given in second order as

AE ---- E p + ~ h = gp - -Sh+ 2 Z~M~(B0) M~(hp)_. + y,z~M~(B0) M~(hp) (41)

which is larger than the shell-model value eph-

456 A. GOSWAMI e t aL

To calculate the escape of particle (hole) strength, we have to calculate tpopp (and q30hh) in second order using the normalization condition. It is easily seen that in second order the single-particle (hole) strength is

([B]~ ~2(hpB T = 0), (42a) \ [P]]

Sh = ~02hh = 1--½ ~.. [ [B]] X2(hpB T = 0). (42b) B. \ [h]!

4.3. PARTICLE-PHONON COUPLED MODES

The energy of the particle-phonon coupled modes can be obtained in second order as7

E~ = ep+oga+ 1 z2M~(hp)M2(BO) 6h ~

,,2 ~,t2rua~ X" M2( h'p)(2I + 1)WZ(Bplh'; aB) (43)

where we used RPA value for MB(BI ). Because of a Racah coefficient in the third term and the large energy denominator in the second, the pushing up of the phonon- like level is expected to be less than that of the particle level. Further the Mn(BI) is expected to be less than the RPA value used above. If ~ = h, one can calculate the hole spectroscopic factor for the state I~Oh) as,

Z2hh = ½ Z ([B]~ X2(hpB T = 0). (44) . , \[h]/

5. EPPC without bootstrap

The bootstrap method as outlined in sect. 3, is numerically difficult (although not formidable) to calculate with. An alternative method would be to separate the problem into two parts:

(i) Calculating the phonon states for the even core using modified RPA (MRPA) [ref. 15)] or boson expansion method 14).

(ii) Calculation of states in the odd-mass nuclei using the even core wave functions. This method has the advantage of not having the difficult self-consistency problem. A numerical comparison of this method with the bootstrap method is now underway.

6. Numerical calculation

In this section, we present the numerical calculation for 39, 40, 4tCa for the pertur- bation solutions of the EPPC-bootstrap equations. Only the octupole phonon is treated, since the perturbation theory is applicable for this case. The shell-model single-particle energies are taken from ref. 3) with a l f u l d ~ splitting of 5.4 MeV. The

BOOTSTRAP THEORY OF VIBRATIONS I I 457

splitting of other particle and hole levels relative to l ft and ld~ respectively are fixed from experimental data. These are given in fig. 1.

Eq. (6) yields for B = 3 with ~o3 = 3.73 MeV a coupling constant Z3 of 0.05 MeV. The reduced matrix element M3(30 ) is then calculated from eq. (7d). Also the value of IM3(30 ) protonl 2 is calculated to be 1.98 x 104 e 2. fm 6 in agreement with the experi- mental value of 1.78 x 104 e 2 • fm 6.

6.2_____.~5 . . . . .

4 . 2 5 - _ - - -

1.90 ~ ~ ~ - -

7.40 6.70 6 . 7 7 ~ f f 5 / 2

4 . 9 5 5.4________0__0 2p I /2 4 . 6 0

3 . 0 5 2 .72 ~ 2p 5 / 2

2 .40

1.15 1.15 1.15 If 7 / 2

> >~ > >

0 t"d tO 0 ~l" O~ tM N

- 5 . 4 0 " . . " - . ' -6.10 - 6 . 0 5

- 6 . 7 7

- 7 . 7 0 \ \ \

- 9 . 9 0

Id 3 / : 9

2s I / 2 - 8 . 5 5

\ \ - / 0 . 2 7 - 9 . 3 2 ~ . - - - 1 0 . 0 5 _ _ Id 5 / 2

-10 .56 -10 .55

INPUT TH. TH. EXP. (2 ND (4 TM

ORDER) ORDER)

Fig. 1. Particle and hole levels for 4°Ca (in MeV). The input from ref. a) is shown in the first column, the calculated values in the second and third columns, the experimental values in the last. The experi- mental spectrum is shifted so that the f} level appears at 1.15 MeV, the calculated value. Levels in all

the columns are in the order shown in the extreme right.

The 'pushed up' particle hole spectrum is also shown in fig. 1. The splitting of lf~-ld~ is found to be 7.9 MeV which is somewhat overestimated compared to the experimental value of 7.2 MeV but agrees within 25 % of the calculated value of ref. 4). The overestimation is partially due to the use of perturbation theory. Also, presumably a better fit would be obtained with a smaller value of the 'unperturbed' splitting. Furthermore, the energy of the 'particle' states are lowered somewhat by second order mixing with physical particle plus two phonon states. Perturbation calculation shows that the lf~-ld~ splitting is lowered to 7.25 MeV when this mixing is taken into account. These results are also shown in fig. 1.

4 .5 8 A. G O S W A M I et aL

The spectroscopic factor of the l f l state is found to be 0.97 in second order, in agreement with the RPA value (only 2p-2h correlations included) of ref. 17).

The 'phonon' states based on the f~ state are shown in fig. 2. Obviously, they are pushed up almost as much as the particle state. This is due to the RPA treatment of the two-phonon coupling which is overestimated. The hole spectroscopic factor of the 3 + level ( 3 - + f~ ) is found to be 0.015. This is in serious disagreement with the value of 0.04 extracted for the 2.15 MeV 3 + level in ref. 12). This perhaps indicates a different origin of this ~+ state, but may also mean that the extraction method for small spectroscopic factors is faulty.

4 . 8 5

5 .70 13/2 "f" 5 .67 1 /2 +

5 .20 9 / 2 ~ 5.18 / 3 ~5"-/2'~ , 'CT 4. 77 712 + 4 . 6 2 1112"

3 . 7 3 I12 +, 31P-.. +, 512+, 712 + , 912 ÷ , II12", 1312 +

1,15 I f 7 / 2

I f 7 / 2 (2 ND O R D E R )

Fig. 2. The phonon levels in 41Ca (in MeV) formed by coupling the f} particle to the 3- phonon state 4°Ca. The unperturbed situation is shown on the left.

To conclude this section, we have demonstrated that the correlation effects of the ground state in the calculation of particles and holes can be accounted for by EPPC- bootstrap method.

7. Conclusion

To summarize, we have shown that one can and should include 'backward going' amplitudes (correlations) in the particle representation for odd-mass nuclei by means of the EPPC-bootstrap theory, which provides a natural extension of RPA to odd- mass nuclei. Using EPPC-bootstrap theory one can calculate:

(i) the 'pushing-up effect' for particle states. It may also account for the nearness in energy of normal and non-normal parity states in odd-mass nuclei. A numerical

BOOTSTRAP THEORY OF VIBRATIONS II 459

demonstration of the importance of this effect was given in sect. 6 for the case of 39Ca and 41Ca.

(ii) the hole stripping probability from the closed-shell ground state to the non- normal parity states and modification of particle spectroscopic factors.

What remains to be done is the complete solution of the bootstrap equations particularly in those cases where considerable deviation from RPA has been found for the phonon (e.g. 2°spb and 12C). Moreover, the phonon in the particle-particle channel (namely pairing vibration) has to be brought in for the treatment of the particle-particle force 3).

We wish to thank Prof. J. McCullen for helpful discussions. Thanks are due to Mr. R. Trilling for pointing out an error in the original manuscript.

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