Bootstrap theory of quasirotational spectra

] I.D.2 ) Nuclear Physics AIM (1972) 6X-672; @ North-Holland Publishing Co., Amsterdam

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BOOTSTRAP THEORY OF QUASIROTATIO~AL SPECTRA

AMIT GOSWAMI f

Institute of Theoretical Science, and Department of Physics,

University of Oregon, Eugene, Oregon

and

ORHAN NALCIO&LU +

Department of Physics, University of California, Davis, California

Received 7 September 1970

(Revised 23 July 1971)

Abstract: In a previous paper we have developed a self-consistent (bootstrap) theory of vibrations which is particularly suited for the description of the ~nha~onicity of the even nuclear spectra. In this paper we combine the bootstrap idea with the group theoretical approach. We show that this theory enables us to understand the smooth transition from vibrational to rotational spectrum and in particular the quasirotational spectrum which is the intermediate situation. Applications of this theory is made for the quadrupole-quadrupole and the pairing ~amiltonians.The observation is made that the smooth transition from vibrational to rotational spectrum for both cases is very similar.

1. Introduction

The need for a uniform description of vibrational and rotational nuclei has been realized for a long time. The fact that one has to invoke two different models for these two limiting cases is not a very satisfactory one. At least, one needs to understand the connection between the two kinds of collective modes. Fu~hermore, the so-cahed “spherical” to deformed “phase transition” (as a function of N or 2) has been found to be a smooth one, passing through a phase where the spectrum is intermediate between vibrational and rotational. The latter has been called the quasirotational spectrum.

Refs. 1-3) give extensive experimental evidence for the existence of the quasirotational spectra. In particular, Ejiri et al. “) have shown that the energies of the o+, 2+, 4+ . . . sequence for the quasirotational spectra have the interesting angufar momentum Z-dependence of aZ+bZ(Zf 1) which can be interpreted as the sum of a vibrational part (proportional to I) and a rotational part, proportional to Z(Z+ 1). ObviousIy a unified theory of these modes of motion must be able to predict the major characteristics of all these types of spectrum. At the weak coupling limit one should obtain:

(i) a O+ ground state and 2+ one-phonon first excited state at energy o followed by a Of, 2+, 4+ almost-degenerate two-phonon triplet at an energy of 203.

t Supported in part by the US Atomic Energy Commission.

658

QUASIROTATIONAL SPECTRA 659

(ii) The static quadrupole moment of the 2+ one-phonon state (and the other phonon states as well) to be approximately equal to zero. The Z3(E2)2,0 w 3B(E2),,, with Z = (O’+, 2’+, 4+) and B(E2),,o w 0. These kinds of harmonic vibrations seem to be well described by the method of quasi-random phase approximation “) (QRPA).

In the limit of strong coupling on the other hand, one should be able to predict the characteristics of rotational nuclei, i.e. a ground rotational band with O+ ground state followed by 2+, 4” etc. rotational states with Z(Z+ 1) type spacing.

Successful description of rotational states has been given through symmetry breaking self-consistent field theories (Hartree-Fock and Hartree-Fock-Bogoliubov). An alternative method is due to Elliott “) which makes explicit use of the Q * Q nature of the two-nucleon effective interaction and the group properties of suitably general- ized quadrupole operators.

In the region of intermediate coupling, one should be able to predict the various observed features of anharmonic phonon spectra. Of these features the most important ones are the large static quadrupole moment of the one-phonon 2+ state observed in some nuclei and the quasirotational spectra. In this paper, we shall concern ourselves with the second feature of phonon anharmonicity; in particular we shall try to achieve an understanding of the transition from vibrational to rotational spectrum through the intermediate phase of quasirotation.

In this intermediate region, the deformed HF and HFB type theories cannot be expected to be appropriate since an HFB calculation would mostly predict a spherical equilibrium shape. Even if deformed equilibrium shapes are found, it is likely that the fluctuation of the order parameter characterizing the deformation would be as large as the order parameter itself. One method of approach under these circumstances is that of Kumar and Baranger 6), which however is complicated. Alternatively, one can approach the problem from the vibrational side.

In a previous paper ‘) (hereafter often referred to as I), we have developed a self- consistent (bootstrap) theory of vibrations which is particularly suited for this problem. This theory has already been shown ‘) to have the proper weak coupling limit, namely QRPA which corresponds to its perturbation solution. The applicability of the theory to cases of strong coupling has been demonstrated for the pairing-force Hamiltonian “). Furthermore, the large static quadrupole moment of the 2+ phonon in heavy spherical nuclei seems also to follow “) from this theory. The important dif- ference of our work ‘) from earlier work 7a) is that the self-consistent evaluation of the energies of the phonon states is achieved through a set of plausible equations, referred to as bootstrap energy equations. In this paper, we shall show how these energy equations lead to an understanding of the quasirotational spectra.

Marumori et al. ’ “) have presented a theory of quasirotational spectra for a single j-shell model using the quadrupole force for the quasiparticle interaction. Their method consists of evaluating the equation of motion of two-quasiparticle operators A&, and A,, (where ZJ is the z-component of the angular momentum J = 2). Further- more they made the essential observation that the commutator [A:,,, A,,] is propor-

660 A. GOSWAMI AND 0. NALCIOGLU

tional to the angular momentum operator for the singlej-shell case if the rank three

component of the commutator is neglected. They were then able to obtain an expres-

sion for the energies of the quasirotational band which clearly displays quasirotation-

al characteristics as well as those of the vibrational and rotational limits.

We shall first demonstrate the quasirotational properties of the bootstrap energy

equation using essentially the above ideas of Marumori er al. lo) for the single,j-case.

For the general case of many+shells, we shall make use of the Elliott model. We re-

place the ordinary quadrupole operator by Elliott’s velocity-dependent quadrupole

operator, the two being completely equivalent since we restrict ourselves to a single

major shell. This enables us to make use of the algebra of the Elliott quadrupole

operators to demonstrate the quasirotational feature of the bootstrapenergyequations.

An analogous situation to quasirotation occurs for the pairing case.

In this paper, using the pairing-force Hamiltonian, we shall show that the ground

state energies of neighboring doubly even nuclei can be expressed as E(N) =

aN+ bN ‘, where N is the number of particles in the nucleus. This energy equation

can also be viewed as a sum of vibrational contribution, proportional to N, and a

rotational part proportional to N2 (the term “rotational” here is used in analogy

with the quadrupole case). The first term is important when pairing excitations are

vibrational (weak coupling limit) and the second teim is important when BCS theory

holds (strong coupling limit).

In sect. 2, a review of the bootstrap energy equation is given. The single_+shell case

is demonstrated in sect. 3. The incorporation of the Elliott model into the bootstrap

theory is accomplished in sect. 4. The pairing case is treated and other possibilities

of application are pointed out in sect. 5. Finally a summary and the conclusions of

the paper are given in sect. 6.

2. The bootstrap energy equations

As in I, we shall start with the quadrupole-quadrupole force Hamiltonian in the

quasiparticle representation

H = CE,da,-+x~Q,Q~, tz P

(1)

Q, = 2+5 $ Q.,CU,,(Al(ac)+(-~~-_,(ac))+ v,c(A,“+(uc)+(-)“AO_,(ac))l, (2)

A,t(ac) = c c~$&7~, (3) mamy

A,O’(ac) = c SyC&!$,&+u,, m,my

(4)

u,, = u,vc+u,va, v,, = u,u,-vv,v,, (5)

Q,, = <.Lll~2&lljc>, sy = (-l)‘c-my.


We use units of 1 /b for the operator r where b is the harmonic oscillator constant and x is suitably defined. The definition of the reduced matrix element follows de-Shalit and Talmi I’).

The bootstrap method is based on the equation of motion method and a subsequent factorization based on the largeness of only a few matrix elements ‘). Equations of motion are written down for the amplitudes of excitation of odd-mass nuclear states in terms of large even-core matrix elements acting as potentials and even-core energies. These core matrix elements and the core energies are all evaluated self-consistently (bootstrapped) in terms of the solutions of the equations of motion for odd-mass nuclear states.

In this paper our main concern will be the bootstrap equations for the energies of the core states in the quasirotational limit, i.e. when the off-diagonal matrix elements of the quadrupole operator are large only for states that differ by two units of angular momentum e.g. 0+, 2+, 4+, . . . etc. (diagonal matrix elements of the quadrupole operator are also large, of course). This means that ( Y’N]Qp]YL) is large if L = iV, N&2.

In order to calculate the energies of the core states, we take the commutator of the quadrupole operator Q, with the Hamiltonian H

[H, Qvl = kQ,l-+X c [Q,Qt, QA P

where

= CT, QJ-3x c (CQ,, QvlQ;+Q,SQ;, QvII, (6) Ir

T = c E,ata,. (7) CL

The energy equation that will give the excitation energy wN of the core state ]YN) relative to oL of state 1 Yu,) is obtained by taking the matrix elements of eq. (6) between the core states ( Yy, 1 and I Y,), where I YN), I Y,) are consecutive members of the quasirotational band, i.e. N = L+2. We then insert a complete set expansion in terms of the core states I Y, ), cJ I Y, )( YJ I between Q,, and [QL, Q, ] and likewise in the last term of eq. (6). Hence

(~,ICH, Qvll Y,> = (0~ -&(YNIQ~IYL) = ~m<YivlQvlY~>

= <Y’,ICT Q,ll~u,>-tx 1 .lmnJ

x W’~VIQ,IWWJICQ~~ ;2dlW+<W~Q,> Qv1lY.J

x < %lQ;I Y&I. (8) In I, we have been able to show that the (Y, I [T, Q,]l Yu,> term alone on the right hand side leads to QRPA if perturbation solutions of the bootstrap equations are used for its evaluation ‘. Presumably, this is the origin of the vibrational term in Ejiri for-

t Note that this is a dynamical result, because solutions of the bootstrap equations (which include the interaction) are being employed.

662 A. GOSWAMI AND 0. NALCIOdLU

mula and the rotational term has to come from the commutator term of Q with the

Q-Q interaction. The key point here is that the quadrupole-quadrupole interaction

is an effective interaction and therefore operates only in a truncated shell-model space

which introduces a velocity dependence in it. This would make the commutator

[Q, Q] non-zero. In the next two sections we shall see how we can obtain quasirota-

tion with two different models for this commutator.

3. Single j-shell case

We will start with the quadrupole-quadrupole Hamiltonian In the quasiparticle

representation for a single j-shell:

where

H = C EjaJ”,aj,,,--hx 1 Q,QL, m P

(9)

Q,l = 215 Qj(Cijj[Al,+(-)“A,_,]+2~jjA~:3, (IO) \

At: = C ( -)j-m’C~!,,,,p ajm aj,,,, , mm’

(1 lb)

with

ujj = 2ujvj,

vjj = l+vf,

Qj = (.illr2Y~llj>.

By straightforward algebra, using eq. (IO), we obtain

(12a)

(12b)

(13)

[Q,, Q,] = cQj[l-( -)“]C$%V(jjL?; 2j)I/i”‘A~~, (14)

where

v!+’ = (2ujVj)2+ VA. J (15)

Terms like At and A do not occur in the above commutator because of the anti-

commutation relations of the quasiparticle operators. At this stage, following Maru-

mori et al., we make the following important assumptions:

(i) keep only th e 2” = 1 term in the commutator, eq. (14);

(ii) treat the quadrupole force in the quasi-random phase approximation “)

(QRPA), which amounts to neglecting the term Vfj in eq. (15).

Now let us see what happens to the first summation term of eq. (8). As we discussed

before in the quasirotational limit the matrix element (Y, IQ,1 YJ) is large only if

J = N, N+2, (16)


but since the states 1 Y,) and 1 YL) are consecutive members of the quasirotational band we also have L = iV+ 2. Using this result we can re-express J in eq. (8) in terms of L rather than N. Thus from eq. (16) the matrix element ( yiN IQ,] Y,) is large when

J = LF2,L. (17)

On the other hand from assumption (i) the commutator [QL, Q,] is proportional to a tensor operator of rank one and this restricts the value of J in the matrix element

O’,I[Qf, Q,lly,> to J= L, L+I. (18)

We see that in order to satisfy eqs. (17) and (18), simultaneously, we must have J = L. After using a similar argument for the second summation term in eq. (8) (in this case J = N) the J-summation in eq. (8) drops and we get

(%-- ~&u’~lQvl’J’u,> = <YdET Q,WL)

-4 ; <Y~~Q~l~~><Y~l[Q~ 3 Qvll~d

-3x; WNICQ,, Q,~~%XWQ:~~L)~ (19)

Since only the diagonal matrix elements of the first-rank tensor operator occur in eq. (19), we can replace them by the total angular-nlomentum operator as was done by Marumori et al. ’ “). After substituting the total angular-momentum operator and retaining only the QRPA part of Q, in [1,, Q,], the energy equation (19) becomes

CGN,Q,(NL) = L Qj Ej Uij[X,(NL)-X,(NL)] Vj5

+XQ~&W(jj12; 2j)W(NL12; 2~)~~+)Y~(~~)Q~(~L)

-XQTt/3W(jjl2; 2j)W(LN12; 2L)Vj(+‘Y,(LL)Q,(NL),

where

i- {~~lJll~> Ye = (y~ll~~+lly~> = 43 (jiiJlli) ’

X,(W = <~NII~~II~,)2

%(W = <~i,ll~*ll~A

Q,VL) = <Y,llQJl!J’,).

The Racah coefficients appearing in eq. (20) are

W(NL12; 2N) = N(Nfl)-L(Lfl)-t6

2~~~~(~~~)’

(20)

(21)

(224

(22b)

(231

(244


W(LN12; 2L) = L(L+l)--N(N+l)+6

_p . 2J&L(L+ 1)(2L+ 1)

Furthermore

(NI/JIIN) = &v(fV+ 1)(2N-i- 1).

By substituting eqs. (21) (24a, b, c) into eq. (20), we get

(24b)

(24c)

oNLQ~(NL) = -Ei Qj Ujj[Xz(NL)-X,(NL)] J*

+y-~~(2uv.)‘Zx,,Qz(NL) “5 JJ

> (25)

where

z= v’ --._-- 15 W(jjl2; 2j)

. Z lij(j +7(G) ' (26)

cxNL = N(N+l)-L&+1), (27)

It has already been shown in ref. ‘) that the first term on the right-hand side of eq. (25) gives rise to QRPA vibrations when the ~rturbative solutions of the bootstrap equations are employed for the amplitudes X,(NL), x, (NL) and the commutator term [Q,Qi, Q,*] was neglected. We can thus substitute the following equations for the amplitudes X, and x, (as shown in ref. ‘) and as also follows from QRPA)

X2(NL) =I: _!__ Qj UjjQz(NL) --- , 45 2Ej-Uo

X,(NL) = X QjujjQ,(NL) ,is'----' ZEj+wo

(284

Wb)

in the perturbation limit. In the above equation, w. is the solution of the QRPA secular equation

This equation can be solved to give

(2Ej)“-Wi -___...- 2Ej

= 4XQf(2Ujuj)2.

Upon substituting eqs. (%a, b) and (30) into eq. (25) and dividing 2Ej Qz (NL ) we get

rliVt = ylo [

x 2EjQ3(2UjVj)' - -- 5 [(2Q2_o;f 1 +(1-q )zaN,, ii

(30)

both sides by

(31)


where

VNL = $,

J

00 q0=-.

2Ej

t 324

(32b)

Since the quantity in the brackets is equal to one, as shown in eq. (29), eq. (31) be-

comes

I]NL = ?O+(l-?;)zaNL* (33)

This essentially gives the angular momentum dependence of the empirical energy

equation of Ejiri et al., E(N) = aN+bN(N+ 1) where a = $oo, b = Z(4ET -w$2Ej).

Therefore, by treating the first term of the bootstrap energy equation through per-

turbation theory (QRPA) and the commutator term through Marumori’s idea, one

obtains the quasirotational energy formula of Ejiri et al. “) at least for the single.j-

shell model. Note that the rotational term is zero only when q. + 1, i.e. x = 0.

Note also that our procedure has led to a slightly different formula than Marumori

et al., who get

VNL = 4(1-?~)ZaNL+Jt(1-r]~)ZZ2C1~L+tl~. (34)

This is due to their substitution of

x,(N,r) = J_ Qj ujjQ*(NL) JS 2Ej-WNL ’

(35a)

X,(NL) = J_ Qj UjjQdNL) JS 2Ej+ONL ’

(3%)

for the values of Xz(NL) and x, (NL) using the true value of oNL rather than o. in

the denominators. This is somewhat inconsistent since the expressions (35a) and (35b)

can be written in this form only in the QRPA method [which is equivalent to the first

order perturbation solution of the bootstrap theory ‘)I. Therefore the QRPA value

of wNL, that is oo, should be used in the denominators of eqs. (35a, b) as we have

done in eqs. (28a, b).

Figs. 1 and 2 compare our quasirotational energy equation with that of Marumori

et al. Obviously the two formulas are quite close.

4. Many j-shell case

4.1. THE ELLIOTT MODEL

If we confine ourselves to a single major shell, the quadrupole operator defined by

02, = +a 7 (rf Y2p(pi) + Pf ~jOi)l9 (36)

a = J&c,


- PRESENT WORK

- - MARIJMORI

0.2 0.4 0.6 0.8 1.0

Fig. 1. Curves of the ratio of the energies q& as function of q. forj = 8. Solid curve represents the present work, whereas the dotted curve that of ref. lo),

TL

\G

IO -

5-

- PRESENT WORK

- - MARUMORI

I 2 46 8 -!_

Fig. 2. Curves of the ratio q& as function of angutar momentum f. for j = g are shown for different values of Q,. Solid curves represent present work, dotted curves that of ref. lo).


is equivalent to the ordinary quadrupole operator Q2, = CirT YZP(Pi). However, the five components of the irreducible tensor operator oZP along with the 3 components of orbital angular momentum operator form the eight generators of the SU(3) group. Furthermore, the operator Q commutes with the harmonic oscillator Hamiltonian. We now note that the quadrupole-quadrupole force can be written as

-&* Q” = -Cf3L2,

where C is one of the Casimir operators of the SU(3) group.

(37)

The Q . Q force cannot couple different irreducible representations of SU(3). Also C has only diagonal matrix elements within an irreducible representation of SU(3). It is obvious that within a representation of SU(3) the eigenvalue of Q - Q is proportional to L(L+ 1). Also from eq. (37) the lowest SU(3) representation is the one that maximizes (C) and turns out to be the state of maximum orbital symmetry. For doubly even nuclei one further has spin S = 0 for all states of maximum orbital symmetry so that for these states L = J and we get the J(Jf 1) rotational interval rule.

In reality, the nuclear Hamiltonian has strong spin-orbit coupling and pairing forces (and also corrections to single-particle energies )which complicates the situation, so that in practice Elliott S&‘(3) model has been used only as a classification scheme for light nuclei.

4.2. QRPA AND THE ELLIOTT MODEL

We propose to combine bootstrap theory and the essential ingredient of the Elliott model, namely the algebra of the Q operator rather than using the group CIassi~cation which is impractical for heavy nuclei. This is done simply by replacing the Q-operator in eq. (1) by Elliott’s quadrupole operator and then substituting the commutation relation

for [Qe, Q,] in eq. (8). We then obtain

WNL(Y’NIQ~I y,> = <~NIIX Qvll YI.>

where a2 = Mn 5 *

The arguments in the previous section can now be repeated to show that the bootstrap assumptions would enable us to keep only states of J = L and J = N in the second and third terms on the right hand side. Since the states IY,) are eigenstates of total


angular momentum, we now write L= J-S.

We can argue that at least for the states of the ground quasirotational band the expectation value of S should be small because these states are collective excitations without spin-flip from a ground state which is basically composed of singlet s-wave pairs of particles. Besides one might invoke the argument of maximum orbital symmetry also in this case. After neglecting the S term, we get

In writing eq. (39) we have neglected a small non-RPA term as in ref. ‘). We now use the perturbation solution of the bootstrap equation to evaluate the first term on the right-hand side:

where o,, satisfies the QRPA equation

This term then becomes

The factor in the parentheses is equal to one and we finally get

coNLQ2(NL) = woQ,(A’L)+ ;t [N(N+l)-L(L+l)]Q,(NL).

The Ejiri formula follows:

OIy = UN + bN(N + 1), with

(42)

(43)

Note that the rotational term is zero when x + 0. Also for heavy nuclei b z 0.018 MeV. So by combining bootstrap theory with the Elliott model we can get some general idea of the behaviour of the solutions of the nuclear ~amiltonians usually


employed. When the single-particle energy part of the Hamiltonian is basically given

by the harmonic oscillator (e.g. light nuclei) the kinetic energy commutator term in

eq. (8) gives zero and we get J(J+ 1) spectra at least for the states ofmaximum orbital

symmetry. It is important to note that in light nuclei like “Ne pairing is not impor-

tant either. As one goes to heavier nuclei, the importance of the pairing force (and

1 . s force) leads to the Hamiltonian (I). The treatment of the quadrupole force then

naturally gives a vibrational term in the bootstrap approximation besides the rota-

tional term coming from the algebra of the quadrupole operator. Since the pairing-

energy gap is about 1 MeV and the vibrations have the energy 2 0.5 MeV in heavy

spherical nuclei, the rotational contribution of O.O18J(J+ 1) MeV is unimportant.

However, as we approach the quasirotational region o. becomes 0.3-0.4 MeV and

the rotational term is comparable. Finally when w. + 0 (corresponding to spherical

to deformed phase transition in the context of QRPA the rotational term dominates.

We should also point out that if the full bootstrap solutions are employed for

evaluating the single-quasiparticle energy commutator we obtain some deviation

from the vibrational behaviour even from this term.

Finally, we note that the derivation of the quasirotational equation vitally depends

on being able to write the Hamiltonian as a sum of single-quasiparticle energy terms

and the quadrupole-quadrupole interaction.

5. Algebraic method for other Hamiltonians

In this section we want to stress the other very important aspect of our work:

whether the bootstrap method can be combined with the algebra of operators for

approximate solutions of other types of Hamiltonians. To this extent we have to note

that most of effective interactions used in nuclear physics can be written as a scalar

product of two operators relevant for the problem and can be diagonalized by em-

ploying group theory when the relevant operators are the generators of the group in-

volved. Examples besides the quadrupole-quadrupole force are the pairing Hamil-

tonian and the charge-independent pairing Hamiltonian (where the groups are SU(2)

and O(8) respectively). The problem is the treatment of the remaining part of the

Hamiltonian (for example the single-particle part in the pairing theory.

The bootstrap theory enables one to treat the single-particle part of the Hamil-

tonian well provided that there exist large matrix elements of the relevant operator

between a band of doubly even nuclear states. The symmetry breaking part (causing

deviation from the “vibrational” band) can be explicitly included by using the algebra

of the relevant operators.

5.1. PAIRING HAMILTONIAN

The case of the pairing Hamiltonian is a good example of the general method we

discussed. The pairing Hamiltonian can be written as

H = c E,&,-)IGIS+ S-, (44) a

670

where

A. GOSWAMI AND 0. NALCIOCLU

(45)

(46)

% = Zj,+l.

The commutator of [S,, S-f is given as

[S, , s-1 = 2s,, (47) where

s, = c sq = c (2N,-L&J, (48)

Is:, s*-J : f4S,. (49)

The operators S+, S- and S, have the commutation rules of angular momentum and constitute the generators of the SU(2) quasi-spin group r3). The classification in terms of the irreducible representation of the quasi-spin group (commonly known as the seniority c~assi~cation) in which the pairing force is diagonal simplifies the problem to the extent that only the single-particle part has to be diagonalized. Our approach uses the bootstrap theory for the single-particle part. Consider the commutator [H, S,],

<N+2lw, S,llW = [~(N3_2)--wfN)J<Nt_2lS,IN)

= : c %<Nf2ESZ, S+IIW

-al&N+2ls+cs_ 9 S+]IN). (50)

Here IN) and IN+Z) denote the ground Of states of N and Nf2 nucleon doubly even systems and w(N+2) is the energy of ]N+2> state. Substituting the values of the commutators from eqs. (47) and (49) we obtain

rw(N+2)-w(N)I<N+2lS+IN)

= 2 c &,<N~2~S”,~N)+3G(N+2]S+S,1N>. (51) n

After making a complete set expansion CL/ L>(L( in terms of the core states IL), we get

= 2 c q7<N+2lS”,]N)+3G c <N+2lS,ILXWzIN>. (52) a L

Now, we argue as before that there exists a band of states for which the matrix eIe- ment of S+ is large, namely, the ground 0” states of theadjacent doubly even nuclei. We can thus truncate the complete set expansion and the only state that contributes


with a large matrix element occurs when 1 L} = IN). Since the doubly even ground states are approximate eigenstates of S, we can replace (NjS, IN) by 2(N) -~,S&, here (N) is the expectation value of the number operator. Eq. (52) then becomes

[o(N+2)-w(N)]d(N+2) = ICI c s,(N+21S:IN) II

where

-4Gld(N+2)C ~~+lG~N~(N+2), (53) *

4Nf2) = tlGl C (Nf2P”,IN), (54) a

is the potential of the bootstrap equations. These equations were first derived by Dang and Klein 14) and are given as

s,(N+l)u,(N) = [F+~&Y(N)fd(N+2)~,(N+2), (55)

.a,(N+l)o,(N+2) = d(Nf2)u,,(N)+[F-&]u,(N+2),

%(N) = <$a (N + Qlch INA (56) Q(N+~) = -<$,(N-t-O/w-AN>,

F = j&4V+2)+o(N)-+lGl,

& = &,-p-1,

with p = +IGl as the self-energy and 1 = +[w(l\i+ 2)- w(N)] as the chemical potential.

In order to solve the bootstrap equations self-consistently, we need to express d in terms of odd-mass amplitudes u and u. This is accomplished by inserting an odd- mass complete set and using the definition of u’s and u’s:

.4(N+2) = +/Cl c ~~~~~(~)u~(N+2) = c d,(N+2). u u

The perturbative solutions of eqs. (55) yield for

(57)

v,(N+2) = 1,

4N-k2) u,(N) = ~ ,

2&, - 00 (58)

where w. = 2J. and is also the energy of the pairing phonon in Tamm-Dancoff approximation (TIIA). Substituting these perturbative values for U, and u, gives d,(N+2); substituting into eq. (53) we get

+J++-dN) = WIQJO ; & + IGIN. @ 0 (59)


Since in TDA

we finally obtain

which implies

where

WI T & = 1, a 0

o(N+2)-w(N) = w,+lGIN,

o(N) = aN+6NZ,

a = ho-WI, b = +lGl.

(60)

(61)

(62)

Eq. (62) can be regarded as the quasirotational equation for pairing rotational bands.

This has been particularly stressed by Bohr II).

6. Summary and conclusions

We have shown that the bootstrap energy equations first formulated in I lead to an

understanding of the connection between vibrational and rotational behaviour and

also the intermediate phase of quasirotation. The demonstration of quasirotation is

carried out in the context of two different models: the single j-model of Marumori

et al. lo) and the Elliott “) model. The latter is used to the extent of the algebra of the

quadrupole operators only. The possibility of the general use of such operator alge-

bra in conjunction with bootstrap theory is pointed out. A second application is made

to the pairing Hamiltonian and a “quasirotational” equation is derived for this case.

The interplay of the interactions causing anharmonicity is surprisingly similar in both

cases, as particularly stressed in ref. ‘l).

References

1) R. K. Sheline, Rev. Mod. Phys. 52 (1960) 1 2) M. Sakai, Nucl. Phys. A104 (1967) 301 3) E. Ejiri, M. Ishihara, M. Sakai, K. Katori and T. Inamura, J. Phys. Sot. (Jap.) 24 (1968) 1189 4) See for example, M. Baranger, Phys. Rev. 120 (1960) 957 5) J. P. Elliott, Proc. Roy. Sot. 245 (1958) 128; 245 (1958) 562 6) M. Baranger and K. Kumar, Nucl. Phys. A92 (1967) 608 7) A. Goswami, 0. Nalcioglu and A. I. Sherwood, Nucl. Phys. A153 (1970) 433

7a) R. M. Dreizler, A. Klein, C. S. Wu and G. Do Dang, Phys. Rev. 156 (1967) 1169 8) A. Goswami and 0. Nalcioglu, Phys. Rev. C2 (1970) 1573 9) A. Goswami and 0. Nalcioglu, to be published

10) T. Marumori, Y. Shono, M. Yamamura, A. Tokunaga and Y. Miyanishi, Phys. Lett. 25B (1967) 249 Proc. Tokyo Conf. on nucl. structure 1967, p. 581

11) A. Bohr in Proc. of Int. Conf. on nuclear structure, Dubna 1968, p. 179 12) A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) 13) A. K. Kerman, Ann. of Phys. 12 (1961) 300 14) G. Do Dang and A. Klein, Phys. Rev. 143 (1966) 735

Documents

Bootstrap theory of quasirotational spectra