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A self-consistent description of systems with many interacting bosons
Nuclear Physics @ North-Holland
A425 (1984) 93-119 Publishing Company
A SELF-CONSISTENT DESCRIPTION OF SYSTEMS WITH MANY INTERACTING BOSONS J. DUKELSKY, Department0
G.G.
DUSSEL*,
de Fisica, Comisih
National
R.P.J.
PERAZZO*,
S.L. REICH
and
de Energia ArcSmica, Au. de1 Libertador Aires, Argentina
Received
14 December
H.M.
SOFiA
8.250, 1429 Buenos
1983
Abstract: A self-consistent description for the ground state of many interacting bosons is discussed. The excited bandheads are obtained by solving the TDA and RPA equations. The latter can be used to isolate the spurious states that arise as a consequence of broken symmetries. The rotational moment of inertia is obtained in different approximations. The interband and intraband electromagnetic decay modes are also analyzed. The use of this framework is exemplified describing systems containing bosom with angular momenta I = 0,2 and 4.
1. Introduction The collective low-energy nuclear spectrum has been described as arising from a system of many interacting bosons. Within the collective model ‘) the phonon degrees of freedom are associated to the quantized vibrations of the nuclear surface in which a large number of nucleons participate. These shape oscillations are described with the Bohr collective hamiltonian. In this approach the (intrinsic) dynamical variables are associated to the geometric parametrization of the distorted nuclear This has been worked out in all the necessary detail for the important quadrupole deformations 2*3) and within different restricting assumptions case of octupole fields “).
surface. case of for the
More recently the interacting boson approximation (IBA or IBA-1) ‘) considers that all the nucleons that participate in the collective motion are pairwise coupled and each (coherent) pair is treated as a boson. The possible angular momenta of these is restricted to be 0 or 2 and are allowed to interact via a two-body residual interaction 576). This “shell-model” point of view for dealing with nuclear boson excitation is able to cast each of the important types of collective motion into distinct coupling schemes that are in turn associated to various group chains that can be used to classify wave functions and energy eigenvalues. Both points of view have been related ‘) to each other allowing one to draw a correspondence between the different kinds of potential-energy surfaces and the different group classification schemes. Through this correspondence it is also possible * Member of the CONICET,
Argentina. 93
94
J. Dukelsky et al. / Se!i2onsistenr description
to identify momenta
which
power
is associated
The exact solution of an IBA hamiltonian,
in the intrinsic
deformation
with the two-body-effective
of the Bohr hamiltonian,
parameters
many-boson
or generalized
interaction.
on the one hand, or the diagonalization
on the other, can only be extended
beyond
the well-known
case of quadrupole (and octupole) distortions at the expense of overcoming great difficulties. The geometrical parametrization of high multipole distortions becomes increasingly complicated leading to the solution of (2h + 1)-coupled differential equations for multipole distortions of order A, still not counting the coupling of different modes. The IBA approach is instead limited by the same sort of problem that face any shell-model approach. These are the rapid increase of the dimension of the space of many-boson wave functions when bosons with high A are included, and the complications to calculate two- (and many-) body matrix elements. Higher (>2) multipole distortions appear as relevant in several nuclear physics problems “) and have been dealt with under very drastic simplifications ‘). To mention only a few we recall the description of the collective motion preceding fission, the appearance of low-lying bandheads with large K-values lo), the structural changes at high spins I’), etc. The proper inclusion of high multipole bosonic degrees of freedom is also a relevant I’) conceptual problem for testing bosonic hamiltonians that result from the mapping of a microscopic many-fermion problem through various kinds of boson expansions. One possible way out is to turn to approximate self-consistent methods that are based upon few relevant physical ingredients. These ideas have been extensively applied in nuclear physics 13) picturing the nucleus as a (finite) interacting Fermi liquid. As regards the bosonic excitations a similar approach is possible “*‘5) using similar ideas to the ones that have been introduced for extended systems as early as in refs. 16). Th ese self-consistent methods assume that a stable minimum occurs in the deformation parameters. The stability of it is measured by the fluctuations that can be proved, to be of 0( l/ N), N being the (large) number of bosons present in the system. In the present paper we present this approach and explore its application to model situations. The overall plan of the work is the following. In sect. 2 we work out the average (distorted) one-body hamiltonian together with an intrinsic state, minimizing the g.s. energy. In sect. 3 we construct the one-boson intrinsic excited states by considering TDA and RPA contributions that include all the corrections of 0(1/N) in the bandhead energies and allow one to isolate the spurious states that arise due to the broken symmetries of the self-consistent hamiltonian. In sect. 4 we discuss the symmetry properties of the total many-boson wave functions that involve both intrinsic and rotational degrees of freedom. Both sets of coordinates are assumed to remain decoupled. In sect. 5 we make use of this to discuss the moment of inertia obtained with the Inglis prescription I’). We also take a further step indicating how it is possible to perform a self-consistent cranking calculation in which an external rotational frequency is introduced and the expecta-
J. Dukelsky
tion value momentum
of H is varied
situations
with the subsidiary
has a given value.
the last section
et al. / Selj-consistent
we discuss
condition
In sect. 6 we evaluate
some applications
95
description
that the average
transition
matrix
of the above
angular
elements.
framework
In
to model
in a space of s-, d- and g-bosons.
2. The intrinsic ground state and the average hamiltonian Let us consider a system consisting of a large but finite number N of interacting bosons. Each of these can either be pictured as representing a pair of nucleons or a particle-hole excitation of a many-fermion system. The dynamical behaviour of this system can be described in the lowest possible order as arising from independent (dressed) bosons moving in an average self-consistent field. To emphasize the difference from the many-fermion situation we shall refer to this as the Hartree-Bose problem (HB). Under the assumption that the N HB bosons are non-interacting the wave function for the ground state is that of a condensate in which all the bosons are placed in a single state - labelled 0 - of lowest energy. We may thus write it as
Ic#I~)=(N!)-"'(~;)"(O). In (2.1) rl stands for bare vacuum. If [4+,) of it can only lead to most general variation to it, can be obtained
the creation operator of a (dressed) HB boson and IO) for the corresponds to a solution of the HB problem any variation an increase in the expectation value of the g.s. energy. The of a many-body permanent of the type (2.1), not orthogonal as: MIN)= JVJ
0 =c
(2.1)
c&T,
N=(
e@l&>,
(2.2) (p#O).
1 +y,q’)“*
(2.3)
P
In (2.3) the real coefficients individual
boson
C, are arbitrary
states that form together
numbers with rl
and
p
(ZO) label the remaining
the complete
that (2.2) is also a condensate of the type (2.1) is simple define an operator ?b’ that is the transform of rz as f,+=~T’e~~o+e-@=X-’
z-:+x
c,r; P
With this definition
the wave function
(&)=(N!)-“*[.h-
j4k)
can be written
‘e”r,‘e-Q]NIO)=(N!)-“2(f~)N10),
HB set. The proof
if we note that we can
> .
(2.4)
as
(2.5)
and the proof of our statement is complete. The variational arguments to define the HB hamiltonian can be worked out in close parallelism with the fermion (HF) case i8). We first calculate the expectation
J. Duke/sky et al. / Self-consistent description
96
value of the many-boson hamiltonian with the varied wave function I+‘&)up to and including second-order terms in 0. We thus get (##+#&J
= ~-2N~(~~l~l~~}
+(&Vl@+H +H@IcbN)
+(&,@+ff@ +$+‘ff
+fH@*)(b,) +
’ ’ *}
.
(2.6)
To fix ideas the hamiltonian appearing in (2.6) is assumed to have at most two-body residual interaction terms. Interactions involving three or more bodies may be of great importance in many-boson systems, for instance to account for triaxial deformations ‘). The extension of the present treatment to include them can be worked out similarly and is omitted here. We write for H:
ij
(2.7)
ijkl
In (2.7) we have used a different notation for the boson creation (annihilation) operators 7: (y,) to stress the fact that (2.7) is written in some standard reference basis labelled
The condition that the value (fbNIHl~N) re p resents a stationary value for the g.s. energy is equivalent to imposing that the linear terms in 0 appearing in (2.6) vanish identically: (f$Nl@+H+HOI&)=O.
(2.9)
The quadratic terms instead help to impose the stability condition of the HB minimum and lead to RPA equations, We turn to this point in sect. 3. To impose (2.9) we use as variational parameters the transformation coefficients 7 that relate the self-consistent HB basis with the reference basis in which the many-boson hamiltonian (2.7) has been written:
G=CI ;
The orthogonality
77:J?v,r
=44*,
I$ d,i%.j
(2.1 Ob)
(P#O),
rl$Ypf
=
St,,
l%v=o,P).
(2.11)
equations (2.11) follow immediately from the condition that the
J. Duke/sky
et al. / Se&consistent descripfion
transformation (2.10) must preserve the commutation (2.10) the hamiltonian (2.7) is written as
97
relations. With the aid of
(2.12) (2.13a) lJkf
rk
Ho*
=
H&
=
(2.13b)
C P
(2.13~) (2.13d) pq &kl
(2.13e) (2.13f) In eqs. (2.13) all the summations are extended to states p, q, . . . f 0. We immediately see that the stationary condition (2.9) only involves the terms HI0 and Ho, (eq. (2.13b)). Introducing these in eq. (2.9) we notice that the CP’s being arbitrary, this vanishes if (2.14) In (2.14) we have introduced
hj = tj
+w - 1)c vIpqo,kq&,
(2.15)
kl
The matrix h, corresponds to the one-body, average distorted field generated self-consistently by the N particles of the system. The parallelism with the fermion case is clear if we realize that in the present case there is only one state that is occupied and can therefore be considered as “belonging to the Fermi sea”. It is therefore natural that eqs. (2.14), (2.15) and (2.16) involve only the q-coefficients that are needed to define the “occupied” state 0 through eq. (2.10). The value E. defined in (2.16) can be interpreted as the energy needed to add one more boson to a condensate of N particles. This should not be confused with the energy of the g.s. of N bosons. This is defined by the Ho0 term in (2.11) and turns out to be (2.17)
98
J. Dukeisky et al. / Self-consistent description
that differs from NE, in the rearrangement terms due to the two-body interaction. The source of this difference is clear if we remind that the structure of the boson that builds the g.s. condensate changes for each value of N. Thus the g.s. energy 8(N) should be regarded as the result of adding N di$erent values of E,,, each one resulting from calculations performed for systems with 1,2,. . . N bosons. The sets of eqs. (2.14)-(2.16) can be solved without any refrerence to the coefhcients 77p,,.The standard procedure is in fact to guess an initial value for the Q, calculate h,,, and E,, and use (2.14) to obtain new values for the no,,. With this method the set of coefficients np,, have remained undetermined except for the fact that they must fulfill the orthogonality conditions (2.11). This indetermination is a well-known fact. In the fermion problem the HF self-consistent states are usually chosen such as to cancel the particle-hole hamiltonian matrix elements. Once this is achieved any further transformation within the single-particle basis, (that is obtained combining the particle and the hole states among themselves but without combining one kind with the other) is also a self-consistent solution. Similarly in the HB problem there is only one “hole” level and all the other single-boson states of higher energy (“particle” levels) remain undetermined except for the fact that all must be orthogonal to that.
3. The one-boson excitations To define the single-boson excited states we assume that a set of the states p, orthogonal to 0 has been determined diugonalizing the average one-body hamiltonian (2.15). In this procedure the creation operator r,’ are expressed as in (2.10b) with np,i defined through the eigenvectors of the matrix h, (with p # 0). In doing so we have defined new single-boson states {p} that are orthogonal to the state 0. The corresponding eigenvalues .I$,of the matrix h, can thus be interpreted as the energies required to add one more boson - different from that of the condensate -to the g-s. of (N - 1) bosons. The basis (0, p} obtained in this fashion can conceptually be assimilated to that obtained with a Nilsson scheme for the many-fermion nuclear problem. In both situations some symmetry is broken to allow for a lower g.s. energy and a set of independent-particle states is constructed in this distorted frame of reference. In the many-boson problem the energies Ep obtained in this fashion are, however, wrong in the O(N) as a consequence of the fact that terms of that order in the hamiltonian have not beeen considered in h, These terms can be considered making use of the uncertainties in the definition of the excited single-boson states to change from the basis {p} to a different basis {(w}using the stability conditions of the HB equations. These amount to demand that the second-order terms in the expansion (2.6) correspond to a positive definite quadratic form in the (arbitrary) coefficients C, and lead to the well-known non-hermitian eigenvalue problem of the RPA. This
99
J. Dukelsky et al. / Self-consistent description
is also equivalent
to linearizing
l3,‘=+
the equations
x;r;r’)-
of motion
Y;r;r,,
fi,
of new operators
~(x;)'-(Y;)2=
1,
(3-l)
P
that are supposed to act on the (correlated) g.s. of the N-boson system. It is worth noting that if one neglects terms 16)of 0( 1/N), eqn. (3.1) becomes formally equivalent to a Bogoliubov transformation defining quasi-particles. We turn to this point at the end of this section. To obtain the RPA equations we assume that the hamiltonian (2.12) has been diagonalized in the HB approximation. Thus H,,, and H,, vanish and the one-body parts of HZ0 and Ho2 are included in the eigenenergies E,,. We consequently retain only the two-body part of the hamiltonian. Since we only study the excitation energies of “one-boson” states we take the zero of the single-boson energy scale at E. and therefore define
ip=Ep-~,. The linearization
of the equations
(3.2)
of motion [H, B:] = Q&
lead to the well-known
non-hermitian
eigenvalue
(3.3) problem:
(_:*_;*)(;:)=a(;:), where the matrices
(3.4)
A and B are defined
(3Sa)
(3Sb) In calculating the commutators of eqs. (3.5) neither H,, nor H2, give any contribution and only those due to H,,, EZ,, and H,,, survive. Within the TDA the wave functions of these states is very simple I*). This approximation corresponds to cancelling the B-matrices or equivalently to neglecting the backward amplitudes YF in (3.1). Thus
Ih+J = 1 x;[(N - iyl-V;(r;y-‘(0) P
=
rmv - 1)!]-‘/2(r,t)N-‘10)p
r:l+,_,) .
(3.6)
100
.I. Duke/sky
et al. / Self-consistent
description
In this approximation all the “one-boson” excited states are determined by the H, , part of the hamiltonian. Within the assumptions made hitherto, the eigenstates (3.6) or the ones obtained from the RPA equations describe different configurations that can be assumed to remain essentially unchanged when the system performs collective rotations. We thus regard each of the states I4,+) as a band-head on top of which a rotational band can be built. Among the TDA solutions there are some that are spurious as a consequence of the introduction of a (self-consistent, average) distorted field. These spurious states correspond to the motion of the many-body system as a whole without a true change in the internal arrangement of its particles. In the case of axial symmetry this happens with a K = +I pair of states that are obtained through the action of the operators J, and Jy on the g.s. condensate 19). Such states correspond to a rotation of the system as a whole and therefore should not be considered as a genuine, different intrinsic configuration that can act as a bandhead. In a general situation, the spurious states that arise as a consequence of broken symmetries do not appear decoupled from the remaining true physical states with the same quantum numbers and require a proper technique to separate them out. This is achieved through the use of the full RPA equations. The situation can be stated in full generality*~,*‘) as follows: Let J& be a one-body operator that generates a transformation that leaves the hamiltonian H invariant. Let us further assume that this symmetry is broken in the Hartree approximation. The operator JU then fulfills EY4=0,
4 f 0*
[Kfet
(3.7)
In eq. (3.7) we have written HHB for the hamiltonian truncated to contain only Ho,,, W,, and H,, (HB approximations). Associated with JU there exists a spurious state. Its wave-function can be written as the action of % on the HB ground state: Is)l-le= 4#iV) -
(3.8)
The state Is)~~ is set at zero excitation energy in the RPA. To see this we note that for any one-body operator .4 the following equations hold: (g.s.(RPA)([&, [H, &]]]g.s.(RPA)) =2x
m
~(gs.(RPA)(.4+(RPA))~*~~
(3.9) In (3.9) we have written M for the (column) matrix with matrix elements M,,O,where I]g.s.(RPA))
= C Mp,,F;&]g.s.(RPA)) w (3.10)
101
J. Dukelsky et al. / Self-consistent description
and
Q
and
immediately
M’
for its transposed
and
hermitian
conjugate,
see that since [H, &/III= 0, then from (3.8) it follows
respectively.
We
that
]&PA = Jll]g.s.(RPA))
(3.11)
represents an eigenstate of the RPA matrix (3.4) with eigenvalue zero. We can further conclude from (3.9) that none of the excited states ]a(RPA)) are connected with the g.s. through the one-body operator JII and therefore 1~)~~~ collects all the “.Mstrength”. The algebra leading to eq. (3.9) can be used to extend to many boson systems the well-known Thouless-theorem ‘“) on the conservation of the sum rule. It is easy to prove that (g.s.(RPA)ILK
[K
4lbW’AN
= kf+vIW,[H, -41hb~) .
(3.12)
In the case of axial symmetry the HB states remain labelled by K, the matrices (3.5) reduce to block diagonal form, each block labelled by a different value of K. The matrices (3.5) are then always of the same dimension as those of (2.15) that in most practical cases are very small. For the sake of completeness we discuss here briefly the structure of the g.s. wave function within the RPA. The expression (2.1) has to be changed to include the correlations that make possible to have a non-vanishing result when acting with the bakcward components of Bz. The wave function can be obtained as in the manyfermion
problem
by setting “) (3.13) \P4
The expression (3.13) is formally equivalent to the wave function of the g.s. of uncoupled harmonic oscillators. The constant floN’ is a normalization and the matrix Gpq has to be determined with the condition that (3.13) is the vacuum of the annihilation
operators
B,. Thus
B,lg.s.(RPA))=O=fioN’exp (3.14) To derive (3.14) we have made follows from the commutation vanishes identically if
use of the fact that Gpq is a symmetric matrix as properties of ri and ri. The expression (3.14)
YX
(3.15)
= 6,, .
(3.16)
G,, = t C {(X-‘X 01 with c (x-‘);x: 01
102
J. Dukelsky et al. / pelf-consistent
deseripti~~
The norm tioN’ can be found to be
I
AloN’= 1tt nl(NN_!2n)l[2tr(~z)ln}~“2;;;;_exp[tr(G2)1.
(3.17)
It is worth remarking that the exponential in (3.13) or the sum in (3.17) truly represent finite polynomic expressions because N is finite. If we let N +OO and keep only the leading terms in l/N then the Bogoliubov prescription 16) (3.18)
r%+=rOkbN)=~N(~N)
can be used. In this case eq. (3.1) can be regarded as a quasiparticle transformation in which the number of bosons is no longer preserved and (3.13) becomes equivalent to the wave function of the BCS vacuum. 4. Wave functions and symmetry properties The many-body wave functions (2.1), (3.69, etc., correspond to intrinsic states in which the many-body system is described using a single-boson basis for which there are privileged orientations in space. The existence of these is a direct consequence of the occurrence of distortions in the self-consistent one-body field. The complete wave function of the system has therefore to involve not only the intrinsic degrees of freedom but also rotational variables specifying the orientation of the body-fixed, privileged frame of reference with respect to a lab system. The existence of a stable, distorted minimum in the g.s. energy allows one, as usual, to assume that the total wave function can be factorized into two parts that depend respectively on the intrinsic coordinates (x) and orientation variables (0): w N.raM=
(4-l)
~N,a(X)&,d~).
The standard procedure 22V23) for discussing the effects of a transfo~ation 9 on (4.1) is to work out two realizations of it, one acting on the intrinsic, 9(x), and the other on the collective coordinates, 5(w). If this can be done the effect of 5 on one set of degrees of freedom can be corrected by Y-’ on the complementary set leaving the wave function unchanged. Thus a wave function that remains invariant under .Y can be constructed within the separability assumption (4.1) as &W&t = ‘J&l + ~(X)~-‘(~))WN,loM(X,
0)
-
(4.2)
For the impo~ant case of axial symmetry the intrinsic wave functions can be labelled by the projection K of the angular momentum along the body-fixed z-axis that is taken to be along the symmetry axis. Using the argument 23) leading to (4.2) and taking for Y a rotation of 180” around an axis perpendicular to the intrinsic z-axis, it is possible to show that the two intrinsic solutions &,K and #NV2= Y(X)
J. Lhkelskv et al. / Self-consistent description [c$~.~]
combine
In a general
case without
can be expanded +.N,IKM
6,
w)
into a single
set of rotational
axial symmetry
in eigenstates
103
states as
the intrinsic
part of the wave function
with good K thus the more general
expression
for
is
%,,I&, ml = cK g: The special considerations of parity and time reversal them here.
%V,,K&,
WI.
23) for the K = 0 case and for the additional symmetries also hold in the present situation and we do not discuss
5. Rotational spectrum Each of the intrinsic states I+N,n) is a head for a rotational band. As long as the intrinsic degrees of freedom remain decoupled from the collective motion, the rotational spectrum will follow an Z(Z + 1) law in which the moment of inertia is the only parameter that enters in the description of the collective band. To calculate this we shall assume for the sake of concreteness that the system is axially symmetric and that the intrinsic states can thus be labelled by K and some extra quantum number LY:I+N,K,a). If the intrinsic states are described within the RPA, the spurious state appears at zero energy.
By separating
tum that is associated the inertial prescription
the canonically
to this Goldstone
conjugate (zero-energy)
parameter associated to the rotations. that yields the following expression
pair of coordinate boson,
it is possible
and momento evaluate
This is the Thouless-Valatin for the moments of inertia:
24,2’)
where (A + B)K,u*,K,rolts = AK,a,,K,,a,, + BKra,,Krsao. In (5.1) A and B are the matrices prescription:
(3.5a,b).
$1 = 2 c K’a’
Eq. (5.1) reduces
~(~NK$&N)~’ K’CY’
to the sell-known
Inglis
(5.2)
in the limit in which the terms due to the two-body residual interaction V are neglected. For such a case the excitation energies g.K’a’ in (5.2), are given by (3.2), i.e. as differences of the eigenvalues of the self-consistent field h, (eq. (2.15)).
104
J. Dukelsky
If axial symmetry
is preserved
et al. / Self-consistent
description
then the transformation
(2.10) can be written
as (5.3)
Using (5.2) and (5.3) to obtain
the moment
of inertia
of the g.s. rotational
band we
get
It is easy to see that the only intrinsic state I+N,K,a,) that is involved in (5.2) is the spurious state (see eq. (3.8)) as in the Thouless-Valatin prescription. We can take a further step to investigate the structure of the rotational bands by performing a self-consistent cranking calculation 13). This consists in minimizing a cranked hamiltonian 14) with the subsidiary condition that the expectation value of the angular
momentum
has the required ~((&,,(w)lH
value.
Namely,
- WJXI4N,&)))
= 0
(5.5)
G#Q4,awIJx14N,a(~)) = [HI + I)-- w'*.
(5.6)
with the condition
Upon an initial guess of w, the variational parameters qo,, are determined and a TDA (or RPA) calculation of the intrinsic state can’be performed. Finally the condition (5.6) can be imposed adjusting w with the required expectation value. This procedure can be followed without heavy numerical work because the size of the matrices is very small even for quite ambitious assumptions upon the single-boson space. 6. Electromagnetic
transitions
All the transition matrix elements can be calculated making use of the separation (4.3) of the total wave function into intrinsic and collective part. Each intrinsic state is assumed to present a (large) static multipole moment. The rapidly changing electromagnetic field that is produced as a consequence of the collective (rotational) motion of the intrinsic structure induces enhanced electromagnetic transitions between the different rotational states. This physical picture is the same as the one of the well-known “unified model” ‘*25) with the only change that the intrinsic part of the wave functions is of bosonic structure. Any multipole operators of order L in the lab and in the intrinsic system are related through Tb(lab)
= C 9h,,Th(intr). M’
In (6.1) Tb(intr)
has a large expectation
value
in any of the (deformed)
(6.1)
intrinsic
J. Lhkelsky
states. The Wigner
9-function
et al. / Self-consistent
that accounts
for the transformation
fixed to the lab systems
acts on the collective
(4.3). The matrix
of (6.1) can immediately
(@,,,
elements
(rotational)
11TL(lab)ll !@N,SKV) = ?? 1 M’
+
105
descriptron
part of the wave function
be worked f,
K
from the body-
out to be
(~NKollT~,(intr)J~NK,~,) >
I
I’
L
-K
M’
MI-K’
>
In calculating (6.2) we have taken for the matrix the following choice of phases 23):
=(-I
elements
among
intrinsic
states
(6.3)
“+“+“‘(~NKaI~4MI~N~).
In (6.3) the transformation .Y is - as in sect. 4 - a rotation of 180” around an axis perpendicular to the symmetry axis. The intrinsic matrix elements in (6.2) are calculated assuming that the multipole transition of order L proceeds via a one-boson operator. This is defined through its matrix elements c$,~” in a spherical (reference) basis: TL,(intr)
= C q!:“‘)Y+y,. ‘I
(6.4)
The labels i and i include the single-boson angular momentum and its projections and the tensor character (L, M’) is guaranteed by vector coupling coefficients that we take to be contained in the q!,FTM”. For the sake of simplicity we shall assume that the intrinsic states are defined within the TDA and we neglect the corrections due to the self-consistent change of the moment of inertia for increasing angular momenta. With this assumptions we can write
(6.5) using the definitions of r: used in eq. (3.6). Using (6.5) in (6.4) we obtain T&(intr)
=
t,“,“‘r,‘r, +c t;r’(r:rO+ro+ra) + 1 t::‘ryaf, n
aa’
(6.6)
1. Rukelsky et ai./ Self-consistent
106
description
with (6.7a)
(6.7b)
In (6.6) and (6.7) we have assumed that 7 and $ are real. while ton Out of the three contributions (6.7) too and t,, are related to intraband and f,,, (LYf a’) are relevant to interband transitions. Within the present approximations the &EL) or &ML) values that depend upon these matrix elements are completely defined by the intrinsic structure. Thus as expected from the separability (4.3), the decay intensities within a band are in the ratio of Clebsch-Gordan coefficients
(see eq. (6.2)) exactly in the same fashion
as within
the unified
model.
7. Calcutations 7.1. THE MODEL
In this section we discuss the methods already presented at the light of the numerical results obtained in different model situations. We consider a system of N bosons with angular momenta I = 0,2,4. All the calculations were performed for N = 30. The hamiltonian is H=KQ*
Q
(7.1)
with
+JW3&d%): We omit the L - L term because
(7.2)
+(r:~~)2:}-X443~(~41j4):. its effect is only to renormalize
the moment
of
inertia 8;. The free coefficients XZ2, X,, and X,, allow one to reproduce a variety of different dynamical situations, very much in the same fashion as in the Warner and Casten 26) model. For X,, = X,, = X,, = 1 the five operators 0: together with the three components of the angular momentum form the eight generators of the algebra of SU(3) built “) with bosons with angular momenta I= 0,2 and 4 @U(3) [sdg]). Taking X, = X,, = 0 and X,, = &v%$ the situation is again that of SU(3) 5, but with bosons of angular momenta I = 0 and 2 (SU(3)[sd)), leaving the 1= 4 modes as inert spectators. Finally, within the space spanned by s- and d-bosons and letting XZZ+ 0, it is also possible to approach the O(6) scheme 5, (0(6)[sd]).
107
J. Dukelsky et al. / Self-consistent description
To discuss the numerical results we recall that the self-consistent methods are approximate solutions of the many-boson problem taking as a (small) expansion parameter the inverse of the (large) number of bosons N. We can thus regard any physical observable P of order r in N (O(r)) expanded as P=Z’,N’+P,N’-‘+..a.
(7.3)
This expansion can also be performed with the exact (group theoretical) results. In the SU(3) sheme the energy eigenvalues are E Uhf --K{~L(L+l)-(A~+~z+h/.l -
In the lowest representations in the system 5@*9)
the label A is proportional
(A,~~=~2N,O),(2N-4,2),(2N-8,4~,~2N-6,0) (A,
-t-3(A +jAu)>}.
(7.4)
to the number of bosons
,...,
SWXsdl
,
. . . , SU(S)[sdg].
p)=(4N,0),(4N-4,2),(4N-6,3),(4N-8,4)*,(4N-6,0),
(7.5) We immediately see that the leading term in the g.s. energy is of O(N*). The excitation energies of the rotational bandheads are of O(N) and that the splitting within a rotational band -given by the L* term - is of 0( 1). THE g.s. ENERGY
7.2. O(N*):
The g.s. energies are shown in fig. 1 in units of KN* and compared with the exact SU(3) limit given by (7.4). The dimensionless value of .E,,./KN~ decreases as X,, + 0, correspondingly Eg s increases (for an attractive interaction). This is because the smaller influence of the 2= 2 boson causes a smaller deformation of the system. This can also be seen in the structure of the boson r,’ with which the g.s. condensate is built (fig. 2). The difference in the g.s. energies for the SU(3)[sdg] and the SU(3)[sd] models is a measure of the renormalization of the intensity between both models. The SU(3)[sd] and 0(6)[sd] results can be compared against those of ref. ‘) in which also an expansion in powers of l/N is possible. If only I = 0 and 2 bosons are considered (see appendix) we can write r,’ = %,0oY&+ %,20Y:o
(7.6)
with ‘) 770.00 =
J( ;
1-$-++O(N-2)
TO.20
,
=
WNsdl
, .
(7.7a) ,
To,20=
; J(
I+&+O(N-‘1
,
WXsdl
(7.7b)
108
1. Dukelsky
0.
.2
et al. / Seljbnsistent
.4
.6
.8
description
t.
Fig. 1. Ground-stateenergy in
units of KN~ as a function of the parameters operator (see eq. (7.2)) for different models.
defining
the quadrupole
The numerical results are seen to reproduce properly the rather drastic change in the structure of the boson of the g.s. condensate. Similar changes can be observed in a model involving bosons of I= 0,2 and 4 although no analytic expressions such as (7.7) have been worked out for this case. The g.s. energy can be obtained with the use of eq. (7.7) to leading order in IV. Using eq. (2.17) we get -$&=~(10&0
+%?3,20 +(N - 1w210>2)
+ 4 1 +& (
+O(W2)
>
=--&
&,,(SU(3)[sd])
+O(N-‘)
.
(7.8) In (7.8) we have used (see eq. 6.5)) (ff IQIP)=C
Q&*,*6%
‘I as the matrix elements of the quadrupole
(7il3,*=
rl0.d
operator in the HB-TDA basis.
(7.9)
109
.i. Dukelsky et al. / Seff-consistent descripiion I.0 ‘
0. Fig. 2. Structure
I
.2
of the boson
1
I
4
.6
I’: as a function
I
i
.8
1.
I
of X in the same model
situation
as fig. I.
In eq. (7.8) still a contribution of (3/4N) is missing inside the parenthesis to get the exact result. The RPA contributions to the g.s. energy 2’) contain 15)this correction thus reproducing the exact results up to and including terms of O(N) (see appendix).
7.3. O(N):
THE
To study
EXCITATION
the excitation
ENERGIES
energies
OF INTRINSIC
of the intrinsic
STATES
states
we perform
a TDA (or
RPA) calculation. Except when otherwise stated we restrict our discussion to TDA results. In general, the corrections introduced with the RPA in the present calculations are very small and can hardly be distinguished in the plots. However, there are a few special circumstances in which these become conceptually impo~ant and these are commented on in detail. To compare the approximate results with the exact ones we use (7.4) setting h = mN - m’ (m = 2 for SU(3)[sd] and m = 4 for SU(3)[sdg]). We thus get
J%hP>
-=(2m’-p)--&{m’2-(p mI+J
+3)(m’-p)-$(L+l)}.
(7.10)
J. Dukelsky et al. / Self-consistent description
110
! &“i(mN ,,_~~~Z”’ 1
-3
(4N-6.3) --t?Y
J&LO)
i
01 --rio EXACT
-I’S’
-
-0
-
SU(3)(sdg]
Jxl
+N> PkN>
@
’ ‘NJ’
TDA
SU(3)(sd]
E/)(mN t
-2
-0
5
,iI
Lj:
fZN,O) 0
Fig. 3. Correspondence of the exact W(3) limit. The level scheme in the labels different states with the same corresponding SU(3) representation following
r;‘---
-1’S’
-00
-
-
Jxl
“, >
MN>
and approximate (HB-TDA) bandhead excitation energies in the right-hand side is labelled by I< and in the left by I. A superscript K-value. An (s) denotes the spurious state. The (A, p) value of the are shown between parenthesis. The value of x in this and the figures is taken as a positive quantity.
Eq. (7.10) clearly displays the fact that the excitation energy of the intrinsic states is a O(N) effect. The values obtained with (7.10) are compared with the TDA results in fig. 3. These (or even those obtained with the RPA) agree with (7.10) except for differences of O(1). Among these small corrections the exact results contain the rotational kinetic energy term that is absent in TDA (or RPA) whose eigenstates remain Iabelled by K and not by L as the exact ones. The lowest SU(3) representations correspond to intrinsic states that within the TDA can be pictured as the promotion of only one of the bosons of the condensate
J. Dukelsky
111
et al. / Se/$-consistent description
to one of the unoccupied single-boson states of higher energy. Excited states’of a higher energy within HB-TDA are constructed promoting two bosons. Correspondingly its excitation
energy
of rn’ and p correspond
is the sum of the one-boson in fact to this harmonic
energies.
pattern
The lowest values
(see fig. 3). To construct
the K-values of the two boson states we remind that both the positive and negative values of K must be added up in agreement with what was said in sect. 4. This is schematically indicated with the creation operators in fig. 3. In figs. 3 and 4 low-energy K = 1 states appear that are almost insensitive to the free parameters of the residual interaction. These are the spurious states that are obtained by the action of the operators J,, on the g.s. wave function. In a general situation J,lg.s.) appear distributed among all the K = 1 states. To elliminate these states the full RPA equations must be used. For the case of an [sd] space and a quadrupole-quadrupole interaction the RPA equations can be worked out analytically. We first note that the boson rl of the condensate appears at an energy E. given by (a?lO)
wlQlo>
no,00 and the TDA excitation &(TDA)=
energies
E
,)mo WlQlo) ______
+K(N_
ho.00 2
&=&O+K(N-l)-p=
770.20
(7.12)
l)((olQIO)(KIQIK)+~(OIQIK)12)-Eo.
In eqs. (7.11) and (7.12) e. and .s2 are the single-boson
I
energies
(B)
(4 N.0)
I
jxz2.x
x*4=x&&=0 .2
4
6
that arise from the
I
(4
O/ 0
(7.11)
’
of the states with K = 0 is
&2+2K(N-
I
a770.20
.E
1.
1.2
_I
X,,.l
[
0
0
.2
.4
.6
.B
I.
x;;.x4,=x 1.2
Fig. 4. Excitation energy of the rotational bandheads [sd](A) and [sdg](B) models as a function of the internal parameters of the quadrupole operator (see eq. (7.2)). The exact SU(3) and O(6) limits are indicated in the figure. The superscript (s) indicates the spurious state. The numbers close to each curve indicate the K-value of the bandhead. A subscript labels the different states with the same value of K.
112
J. Dukelsky
contracted part of the two-body two by two and the eigenvalues
et al. / leaf-consislen!
residual interaction. For K # 0 the RPA matrix can readily be found to be
flK =(&(TDA)-4&V-This equation
description
can be used together
l)2~(O~Q~K)~4)“2.
is
(7.13)
with (7.11) and (7.12) to check that the RPA
frequency of the K = 1 state is zero. The structure of the intrinsic states changes
with the structure
of the Q-operator.
The transition from the SU(3)[sd] to the 0(6)[sd] scheme is shown in fig. 4(A). For small values of X,, there appear strong instabilities and a lack of convergence in the self-consistent procedure. These features indicate that a situation is approached in which there is not a sharp, deformed minimum in the g.s. energy. Correspondingly the RPA corrections become increasingly important. This is clear from fig. 4(A). Using eq. (7.11) it is simple to prove that in the 0(6)[sd] limit the frequency of the K = 2 state is zero as befits a situation of unstable (infinite-amplitude) oscillations. In fig. 4(B) it is also possible to see how the self-consistent procedure is able to reproduce the drastic changes from the SU(3)[sdg] situation in which the f = 4 boson is strongly coupled to one in which it is an inert spectator in the presence of a perturbed SU(3)[sd] scheme, giving rise to a degenerate multiplet of bandheads with K = 0,f 1, ;t2, *3 and &4. A few examples of the changes in the wave functions of the K = 0 bandheads are shown in fig. 5. In this figure it can be clearly seen how the K = 0’ state becomes a pure 1= 4 and the other becomes an o~hogonal tion to the g.s. (fig. 2) that only contains I = 0 and 2 components.
7.4. O(1): THE ROTATIONAL The
belong
rotational
kinetic
combina-
MOTION
energy term causes a (small) splitting
to the same representation
among
(h, 1~). This can be obs&ved
the states that
in the left part of
..
0 Fig. 5. Structure
.2
.4
.6
.a
1.x'x24=X4&
of the excited K = 0 bandheads in an sdg model (fig. 4(B)). The dotted correspond to the lowest (highest) R = 0 excited state (0,).
(full) lines
113
J. Dukelsky et al. / Self-consistent description
fig, 3. Since no angular momentum projection has been made for the TDA eigenvalues that are plotted in the right part, these appear in degenerate multiplets regardless the value of K. This rotational motion is described by a single parameter that is the moment of inertia 9. The SU(3) value for 2 obtained from (7.4) or (7.10) is
-=3 1 2&f~f
(7.14)
4K*
The Thouless-Valatin and Inglis prescriptions (5.1) and (5.2) to calculate the moment of inertia can be worked out analytically to leading order in N in the SU(3) limit. Using eqs. (5.4) and the energy of the Is: = 1 state obtained in the appendix (A.9) and (A-14) we get $r=6N7&o(E~B=1)-‘= 8;” = 6N&,(A,,
-$c
(7.15a)
+O(N-‘),
+ B, J-1 = 6Nn&(2E’TDA)-’
c- -$K +O(N-‘)
.
(7.15b) The results (7.15a) are seen to be very close to (7. ISb) that in turn agree with the exact ones up to the leading order in 1/N. A partial inclusion of the O(N) terms of the hamiltonian contained in HII and neglecting Ho2 and Z-ZzO, (TDA) would have badly spoiled the result (7.15a). The O(6) case can not be discussed in a straightforward manner since in this case the symmetry that is broken is of a more complex nature. In this case the moment of inertia also has contributions coming from the dynamics that is involved with the seniority-dependent terms. The moment of inertia depends on the residual interaction and on the single-boson space that is active. This can be checked through the examples in fig. 6.
7.5. MULTIPOLE
MOMENTS
AND
TRANSITION
The last dynamical ingredients that we study are the matrix elements for quadrupole electromagnetic transitions (eg. (6.2)). The expansion in powers of N can be obtained directly from eq. (6.6) using the wave functions (3.6). The corresponding matrix elements are (~~~~~,(intr)~~~)
(7.16a)
= NW/Q/O),
(#N,K,nIQ~,(intr)I~N,K,ol) = (N - l)(OlQlO>+(KIQlK)
.
(7.16b)
The values (7.16) give a measure of the static quadrupole deformation of the intrinsic states. This is clearly a collective O(N) effect. The difference between the ground and excited states is instead of O(l), as befits a situation in which only one of N bosons of the system has changed its configuration.
J. Dukelsky et al. / Self-consistent
114
description
.6 -
Fig. 6. The Thouless-Valatin moment of inertia of the ground-state rotational bands as a function internal parameters of the quadrupole operator for two different model situations.
of the
These features can also be found 27V28)in the Elliott SU(3) model. Within this framework the many-boson wave functions in a Cartesian basis can be organized in decreasing values of the numbers N, and N, of oscillation quanta in the z- and x-directions. The state with the biggest N, and N,, (P~:,~,, being called the leading state. The labels (A, CL) and N, and N, for this state are related through h=N,-N,;
/_L=N~-N,.
(7.17)
We have seen in subsect. (7.3) how different values of (A, CL)correspond to different sets of excited bandheads. We can thus regard the states (P~;,~, as intrinsic, in the same way as in the HB-TDA language. The expectation value 27) of the quadrupole operator in the leading state is ((PN,,N,10~,(i~r)l~~~,~,)(Y(2h
+~)=2mN+(~-24,
(7.18)
which agrees with (7.13a,b) as regards the order of N that is involved. In fig. 7 we show how the different matrix elements of the quadrupole operator change as a function of the parameters of the model. From the numbers in fig. 7 one can easily reconstruct the values (7.18) in the SU(3) limit. In most practical cases the actual value of (KIQIK) is irrelevant since it amounts to a correction of O(1) in the matrix element.
J. Lhkelsky et al. / Self-consistent
description
115
Fig. 7. Matrix elements of the intrinsic quadrupole operator for two [sdg] model situations. The full lines correspond to X = X,, = X,, = X, and the dotted lines to X = X,, = X, with X,, = I .O. Ste fig. 3 for the identification of the different states.
The transitions between bands SU(3) limit because the quadrupole
labelled by different (A, CL) are forbidden in the operator cannot connect wave functions belong-
ing to different SU(3) representations. element is in general of O(N”‘),
In spite of this fact the transition
(~N,K,~IQ~M’(intr)I~N,K,,a~ > = (KIQIK?fi
3
matrix
(7.19)
as follows from eq. (6.6). This feature causes that even quite small departures from the SU(3) limit turn this transitions into measurable lo) quantities since the corresponding (&EL)) value is magnified by a factor of N.
8. Conclusions In the preceding sections we have presented an application of self-consistent procedures to systems of many interacting bosons. The g.s. is described as a condensate in which the N bosons of the system are in the lowest single-particle state. The structure of the elementary excitation can be obtained within a TDA. Within this framework each of these excited states can be pictured as the promotion of one of the bosons of the condensate to a single-boson state of a higher energy.
116
J. Dukelsky et al. / Sel$consistent description
Among these a spurious state occurs as a consequence of a broken symmetry that corresponds to the motion of the N-body system as a whole. This state can be eliminated performing an RPA that brings it to zero energy. Each of this intrinsic states can act as a bandhead. The collective rotational motion that can be constructed on top of each of these states can be described by a single inertial parameter, i.e. the moment of inertia. This can be evaluated using the cranking-model prescription. A more elaborate approach - a self-consistent cranking - is also possible. The method can be applied successfully to situations in which a stable, (deformed) minimum occurs in the g.s. expectation value of the hamiltonian. However, we have proved that we can encompass a wide variety of dynamical situations. When the approximate results are compared to exact-group theoretical -values, they turn to be accurate except for terms of higher order to power l/N, N being the number of bosons in the system. This error is corrected if all the RPA contributions are included, although this study may be avoided because N is in most practical cases a large number. All these features hold even if the single-boson space includes, for instance, several bosons of the same 1 and many different possible Z’s The numerical work is kept under reasonable limits even for quite ambitious choices of the single-boson basis. With these advantages the method can be used to test a wide variety of many-boson hamiltonians obtained, for instance, by mapping of a many-fermion problem or in a study of the interplay of different multipole degrees of freedom. Both of these problems rapidly become intractable by other methods. The study of more complicated potential-energy surfaces in terms of shape-deformation parameters requires, however, the proper consideration of many-boson residual-interaction terms. Appendix ANALYTIC
EXPRESSION
FOR [sd] MODELS
In this appendix we summarize the expressions that can be derived for the HB, TDA and RPA equations for the case in which only s- and d-bosons are involved and axial symmet~ is assumed. (i) 7’he reference and HB basis. The six different boson creation operators in the spherical basis remain labelled by the angular momentum I( =0,2) and the projection m(=-I,..., I). The distorted self-consistent basis is in many respects similar to the fermion Nilsson scheme. The transformation between both bases (eq. (2.10)) reserves the z-projection and can be written
(A-1) In an [sd] model it is possible to simplify the notation dropping all the redundant
3. Dukelsky et al. j Seljkansisient description
117
subindices. We thus write I-;= %Yzo + 772Y2f0, G
= A*,
G = 77*Yiho- %do , r:,
,
= d&2.
(A.3
(A.3)
The 0’ state is commonly referred to as the &vibration. We assume real coefficients n. The further corrections that are introduced through the TDA do not change the two q-coefficients since the corresponding matrices do not connect different K-values and the matrices therefore are of dimension 1. We thus have r) = ;i (eq. (6.5)). (ii) The quadrupole operator. The quadrupole written as
operator in the spherical basis is
Q, =(ro’u~~,+r:,~~)+x(~:~*)~=~~~.
(A-4)
for the SU(3) and 0 for the O(6) The internal parameter x takes the values symmetries. Using (A.2) and (A.3) Q can be written in the HB basis. A few relevant matrix elements are
(OlQ10) = bwr~~xr~:, W]QIO’) = -2rlow&I;,
+J:xwz,
(O[Q\W = 7: - d
(lIQ/l>=J~~=(-1101-1>, OtQ12) = -4x
= t-4Q1-2,
(4 010) = &rtz
+ TO,
(1(Qk%= vn’%xrl~.
(A.3
Using eqs. (7.7) and ref. ‘) these matrix elements can in turn be expressed as an expansion in powers of (I/ IV). (iii) Tke hamiltonian in the H3 approximation.
quadrupole-quadrupole
We only discuss the case of a
interaction: H=@
0.
(A.@
The contracted terms give rise to a single-boson energy term 6):
E~=K~i(~llQll~)I~&=
5’&,o+(l +X2)‘+.
The symmetrized two-body interaction terms that are left can be written as
(A.7)
J. Lhkelsky et al. / Self-consistent description
118
Thus the HB single-boson hamiltonian (2.15) reads ~IK,I,K,=&K,{G%,+ GN - 1) (O\QiOX~~iQi~‘K’)
I
. (A-9) iI12 With this matrix, the energy E,, that is needed to add one more boson to the condensate of N - 1 is
+C (~KIQKI~,O)(~,OIQ-K(-)~~~‘K)~,,~~~,
E,=
=
(OlQb
E,,+K(N-
l)-
++K(N-
I)&
a ~(01 QkV -$QlO)
772
E,& + ~~77; +~K(N - l)l(OlQ10)12.
=
(A.10)
Within the HB approximation the energy required to add a boson p # 0, 0’ to the condensate is just the diagonal term in (A.9). Each of the states is twofold degenerate. Each pair *K combine into a single rotational state. The pair K = *l is spurious. The g.s. energy of N bosons is 8,,(N)
= N(s071; + ~217:)
+ KN(N
-
l)(OiQb)’ .
(A.1 1)
If the TJ coefficients for the g.s. of N or N - 1 bosons are assumed to be the same (thus neglecting O(l/ N) differences) it is easy to check that 8&N)-
8,,(N
- 1) = E,, .
(A.12)
(iu) The Q(N) corrections (TDA and RPA). Eq. (3Sa) that gives the TDA equations amounts to introducing a correction ;(N -
1)
C V~%J~J, JI JzJ3-14
= K(N
-
~o,~,rlq,~~~o,~~
~K’lQ~K,)~2) V, = W .
l)S,,((OlQ(O)(K,(Q(K,)-
(A.13)
The TDA excitation energy of the spurious state is (eq. (3.2)) “TDA
El
=
Ei-DA
I
-Eo=
K(N-
@(Q(O) a 2(0~Q(0)(1~Q~1)+2~(0~Q~1)~*---ii,),(OiQ/O))
1)
772
= -
~K(N
-
(A.14)
I)(( 1(Q(O)\’.
The RPA backward correction (3.5b) is also a one-by-one matrix. BKK’=
Consequently
K(N- 1)~KK~((OlQ10)(~lQl~)+l(~l~l~)12) I
the RPA eigenvalue problem is (K # 0) (~~-f12:)+B~k=0, OK = *{&
-‘~K~(N-
1)‘~(0~Q~K)~“}“‘.
(A.15)
J. Dukelsky
This is seen to cancel
et al. / SelJconsistent
for K = 1 and to give a minor
states. This result is needed
for the calculation
2&*(N)
= ?&J(N)
119
description
(O(N-‘))
correction
of the g.s. energy
to all other
in the RPA as
+c f
References I) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 4 (1952) 2) G.G. Dussel and D.R. Bes, Nucl. Phys. Al43 (1970) I 3) K. Kumar and M. Baranger. Nucl. Phys. A92 (1967) 608; J.M. Eisenberg and W. Greiner, Nuclear models, vol. I (North-Holland, Amsterdam, 1970) ch. 5 4) P.O. Lipas, Ph.D thesis, Rensselaer Polytechnic Institut (1961); P.O. Lipas and J.P. Davidson, Nucl. Phys. 26 (1961) 80 5) A. Arima and F. Iachello, Ann. of Phys. 99 (1976) 253; 111 (1978) 201; 123 (1979) 468 6) 0. Scholten, The interacting boson approximation model and applications, Ph.D. thesis, University of Groningen, (1980). and references therein 7) J.N. Ginocchio and M.W. Kirson, Nucl. Phys. A350 (1980) 31: Phys. Rev. Lett. 44 (1980) 1744 8) P. Van Isacker, K. Heyde, M. Waroquier and G. Wanes, Nucl. Phys. A380 (1982) 383; Phys. Lett. 1048 (1981) 5 9) Wu Hua-Chuan, Phys. Lett. 1lOB (1982) 1 10) D.D. Warner, R.F. Casten and W.F. Davidson, Phys. Rev. C24 (1981) 1713 1I) K. Sugawara-Tanabe and A. Arima, Phys. Lett. 1lOB (1982) 87 12) T. Otsuka, Nucl. Phys, A368 (1981) 244; T. Otsuka, A. Arima and N. Yoshinaga, Phys. Rev. Lett. 48 (1982) 387; D.R. Bes, R.A. Broglia, E. Maglione and A. Vitturi, Phys. Rev. Lett. 48 (1982) 1001 13) H.J. Mang. Phys. Reports 18C (1975) 325, and references therein 14) J. Dukelsky, G.G. Dussel, R.P.J. Perazzo and H.M. Sofia, Phys. Lett. 13BB (1983) 123 15) J. Dukelsky, G.G. Dussel, R.P.J. Perazzo and H.M. Sofia, Phys. Lett. 129B (1983) 1 16) N.N. Bogoliubov, J. Phys. (USSR) 11 (1947) 23; S. Beliaev, JETP (Sov. Phys.) 7 (1958) 289 17) D.R. Inglis, Phys. Rev. 96 (1954) 1059 18) A.E.L. Dieperink and 0. Scholten, Nucl. Phys. A346 (1980) 125 19) A. Bohr and B.R. Mottelson, Phys. Scripta 25 (1982) 28 20) D.J. Thouless, The quantum mechanics of many body systems (Academic Press, NY, 1972) ch. III 21) P. Ring and P. Schuck, The nuclear many-body problem (Springer, New York, 1980) ch. 8 22) G.G. Dussel, R.P.J. Perazzo, D.R. Bes and R.A. Broglia, Nucl. Phys. Al75 (1971) 513 23) A. Bohr and B.R. Mottelson, Nuclear structure, vol. 2 (Benjamin, NY, 1975) ch. 6 24) D.J. Thouless and J.G. Valatin, Nucl. Phys. 31 (1962) 211 25) A. Bohr and B.R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 26) D.D. Warner and R.F. Casten, Phys. Rev. Lett. 48 (1982) 1385 27) J.P. Elliott, Proc. Roy. Sot. A245 (1958) 128, 562; Group theory and the nuclear shell model, Notes of the Escuela Latinoamericana de Fisica, Mexico, 1962 28) Hsi-Tseng Chen and A. Arima, A cylindrical sp-boson for large deformed nuclei, Univ. of Tokyo, preprint; Hsi-Tseng Chen, Proc. Workshop on nuclear collective states, Suzhou, China, September 1983, p. 30
A425 (1984) 93-119 Publishing Company
A SELF-CONSISTENT DESCRIPTION OF SYSTEMS WITH MANY INTERACTING BOSONS J. DUKELSKY, Department0
G.G.
DUSSEL*,
de Fisica, Comisih
National
R.P.J.
PERAZZO*,
S.L. REICH
and
de Energia ArcSmica, Au. de1 Libertador Aires, Argentina
Received
14 December
H.M.
SOFiA
8.250, 1429 Buenos
1983
Abstract: A self-consistent description for the ground state of many interacting bosons is discussed. The excited bandheads are obtained by solving the TDA and RPA equations. The latter can be used to isolate the spurious states that arise as a consequence of broken symmetries. The rotational moment of inertia is obtained in different approximations. The interband and intraband electromagnetic decay modes are also analyzed. The use of this framework is exemplified describing systems containing bosom with angular momenta I = 0,2 and 4.
1. Introduction The collective low-energy nuclear spectrum has been described as arising from a system of many interacting bosons. Within the collective model ‘) the phonon degrees of freedom are associated to the quantized vibrations of the nuclear surface in which a large number of nucleons participate. These shape oscillations are described with the Bohr collective hamiltonian. In this approach the (intrinsic) dynamical variables are associated to the geometric parametrization of the distorted nuclear This has been worked out in all the necessary detail for the important quadrupole deformations 2*3) and within different restricting assumptions case of octupole fields “).
surface. case of for the
More recently the interacting boson approximation (IBA or IBA-1) ‘) considers that all the nucleons that participate in the collective motion are pairwise coupled and each (coherent) pair is treated as a boson. The possible angular momenta of these is restricted to be 0 or 2 and are allowed to interact via a two-body residual interaction 576). This “shell-model” point of view for dealing with nuclear boson excitation is able to cast each of the important types of collective motion into distinct coupling schemes that are in turn associated to various group chains that can be used to classify wave functions and energy eigenvalues. Both points of view have been related ‘) to each other allowing one to draw a correspondence between the different kinds of potential-energy surfaces and the different group classification schemes. Through this correspondence it is also possible * Member of the CONICET,
Argentina. 93
94
J. Dukelsky et al. / Se!i2onsistenr description
to identify momenta
which
power
is associated
The exact solution of an IBA hamiltonian,
in the intrinsic
deformation
with the two-body-effective
of the Bohr hamiltonian,
parameters
many-boson
or generalized
interaction.
on the one hand, or the diagonalization
on the other, can only be extended
beyond
the well-known
case of quadrupole (and octupole) distortions at the expense of overcoming great difficulties. The geometrical parametrization of high multipole distortions becomes increasingly complicated leading to the solution of (2h + 1)-coupled differential equations for multipole distortions of order A, still not counting the coupling of different modes. The IBA approach is instead limited by the same sort of problem that face any shell-model approach. These are the rapid increase of the dimension of the space of many-boson wave functions when bosons with high A are included, and the complications to calculate two- (and many-) body matrix elements. Higher (>2) multipole distortions appear as relevant in several nuclear physics problems “) and have been dealt with under very drastic simplifications ‘). To mention only a few we recall the description of the collective motion preceding fission, the appearance of low-lying bandheads with large K-values lo), the structural changes at high spins I’), etc. The proper inclusion of high multipole bosonic degrees of freedom is also a relevant I’) conceptual problem for testing bosonic hamiltonians that result from the mapping of a microscopic many-fermion problem through various kinds of boson expansions. One possible way out is to turn to approximate self-consistent methods that are based upon few relevant physical ingredients. These ideas have been extensively applied in nuclear physics 13) picturing the nucleus as a (finite) interacting Fermi liquid. As regards the bosonic excitations a similar approach is possible “*‘5) using similar ideas to the ones that have been introduced for extended systems as early as in refs. 16). Th ese self-consistent methods assume that a stable minimum occurs in the deformation parameters. The stability of it is measured by the fluctuations that can be proved, to be of 0( l/ N), N being the (large) number of bosons present in the system. In the present paper we present this approach and explore its application to model situations. The overall plan of the work is the following. In sect. 2 we work out the average (distorted) one-body hamiltonian together with an intrinsic state, minimizing the g.s. energy. In sect. 3 we construct the one-boson intrinsic excited states by considering TDA and RPA contributions that include all the corrections of 0(1/N) in the bandhead energies and allow one to isolate the spurious states that arise due to the broken symmetries of the self-consistent hamiltonian. In sect. 4 we discuss the symmetry properties of the total many-boson wave functions that involve both intrinsic and rotational degrees of freedom. Both sets of coordinates are assumed to remain decoupled. In sect. 5 we make use of this to discuss the moment of inertia obtained with the Inglis prescription I’). We also take a further step indicating how it is possible to perform a self-consistent cranking calculation in which an external rotational frequency is introduced and the expecta-
J. Dukelsky
tion value momentum
of H is varied
situations
with the subsidiary
has a given value.
the last section
et al. / Selj-consistent
we discuss
condition
In sect. 6 we evaluate
some applications
95
description
that the average
transition
matrix
of the above
angular
elements.
framework
In
to model
in a space of s-, d- and g-bosons.
2. The intrinsic ground state and the average hamiltonian Let us consider a system consisting of a large but finite number N of interacting bosons. Each of these can either be pictured as representing a pair of nucleons or a particle-hole excitation of a many-fermion system. The dynamical behaviour of this system can be described in the lowest possible order as arising from independent (dressed) bosons moving in an average self-consistent field. To emphasize the difference from the many-fermion situation we shall refer to this as the Hartree-Bose problem (HB). Under the assumption that the N HB bosons are non-interacting the wave function for the ground state is that of a condensate in which all the bosons are placed in a single state - labelled 0 - of lowest energy. We may thus write it as
Ic#I~)=(N!)-"'(~;)"(O). In (2.1) rl stands for bare vacuum. If [4+,) of it can only lead to most general variation to it, can be obtained
the creation operator of a (dressed) HB boson and IO) for the corresponds to a solution of the HB problem any variation an increase in the expectation value of the g.s. energy. The of a many-body permanent of the type (2.1), not orthogonal as: MIN)= JVJ
0 =c
(2.1)
c&T,
N=(
e@l&>,
(2.2) (p#O).
1 +y,q’)“*
(2.3)
P
In (2.3) the real coefficients individual
boson
C, are arbitrary
states that form together
numbers with rl
and
p
(ZO) label the remaining
the complete
that (2.2) is also a condensate of the type (2.1) is simple define an operator ?b’ that is the transform of rz as f,+=~T’e~~o+e-@=X-’
z-:+x
c,r; P
With this definition
the wave function
(&)=(N!)-“*[.h-
j4k)
can be written
‘e”r,‘e-Q]NIO)=(N!)-“2(f~)N10),
HB set. The proof
if we note that we can
> .
(2.4)
as
(2.5)
and the proof of our statement is complete. The variational arguments to define the HB hamiltonian can be worked out in close parallelism with the fermion (HF) case i8). We first calculate the expectation
J. Duke/sky et al. / Self-consistent description
96
value of the many-boson hamiltonian with the varied wave function I+‘&)up to and including second-order terms in 0. We thus get (##+#&J
= ~-2N~(~~l~l~~}
+(&Vl@+H +H@IcbN)
+(&,@+ff@ +$+‘ff
+fH@*)(b,) +
’ ’ *}
.
(2.6)
To fix ideas the hamiltonian appearing in (2.6) is assumed to have at most two-body residual interaction terms. Interactions involving three or more bodies may be of great importance in many-boson systems, for instance to account for triaxial deformations ‘). The extension of the present treatment to include them can be worked out similarly and is omitted here. We write for H:
ij
(2.7)
ijkl
In (2.7) we have used a different notation for the boson creation (annihilation) operators 7: (y,) to stress the fact that (2.7) is written in some standard reference basis labelled
The condition that the value (fbNIHl~N) re p resents a stationary value for the g.s. energy is equivalent to imposing that the linear terms in 0 appearing in (2.6) vanish identically: (f$Nl@+H+HOI&)=O.
(2.9)
The quadratic terms instead help to impose the stability condition of the HB minimum and lead to RPA equations, We turn to this point in sect. 3. To impose (2.9) we use as variational parameters the transformation coefficients 7 that relate the self-consistent HB basis with the reference basis in which the many-boson hamiltonian (2.7) has been written:
G=CI ;
The orthogonality
77:J?v,r
=44*,
I$ d,i%.j
(2.1 Ob)
(P#O),
rl$Ypf
=
St,,
l%v=o,P).
(2.11)
equations (2.11) follow immediately from the condition that the
J. Duke/sky
et al. / Se&consistent descripfion
transformation (2.10) must preserve the commutation (2.10) the hamiltonian (2.7) is written as
97
relations. With the aid of
(2.12) (2.13a) lJkf
rk
Ho*
=
H&
=
(2.13b)
C P
(2.13~) (2.13d) pq &kl
(2.13e) (2.13f) In eqs. (2.13) all the summations are extended to states p, q, . . . f 0. We immediately see that the stationary condition (2.9) only involves the terms HI0 and Ho, (eq. (2.13b)). Introducing these in eq. (2.9) we notice that the CP’s being arbitrary, this vanishes if (2.14) In (2.14) we have introduced
hj = tj
+w - 1)c vIpqo,kq&,
(2.15)
kl
The matrix h, corresponds to the one-body, average distorted field generated self-consistently by the N particles of the system. The parallelism with the fermion case is clear if we realize that in the present case there is only one state that is occupied and can therefore be considered as “belonging to the Fermi sea”. It is therefore natural that eqs. (2.14), (2.15) and (2.16) involve only the q-coefficients that are needed to define the “occupied” state 0 through eq. (2.10). The value E. defined in (2.16) can be interpreted as the energy needed to add one more boson to a condensate of N particles. This should not be confused with the energy of the g.s. of N bosons. This is defined by the Ho0 term in (2.11) and turns out to be (2.17)
98
J. Dukeisky et al. / Self-consistent description
that differs from NE, in the rearrangement terms due to the two-body interaction. The source of this difference is clear if we remind that the structure of the boson that builds the g.s. condensate changes for each value of N. Thus the g.s. energy 8(N) should be regarded as the result of adding N di$erent values of E,,, each one resulting from calculations performed for systems with 1,2,. . . N bosons. The sets of eqs. (2.14)-(2.16) can be solved without any refrerence to the coefhcients 77p,,.The standard procedure is in fact to guess an initial value for the Q, calculate h,,, and E,, and use (2.14) to obtain new values for the no,,. With this method the set of coefficients np,, have remained undetermined except for the fact that they must fulfill the orthogonality conditions (2.11). This indetermination is a well-known fact. In the fermion problem the HF self-consistent states are usually chosen such as to cancel the particle-hole hamiltonian matrix elements. Once this is achieved any further transformation within the single-particle basis, (that is obtained combining the particle and the hole states among themselves but without combining one kind with the other) is also a self-consistent solution. Similarly in the HB problem there is only one “hole” level and all the other single-boson states of higher energy (“particle” levels) remain undetermined except for the fact that all must be orthogonal to that.
3. The one-boson excitations To define the single-boson excited states we assume that a set of the states p, orthogonal to 0 has been determined diugonalizing the average one-body hamiltonian (2.15). In this procedure the creation operator r,’ are expressed as in (2.10b) with np,i defined through the eigenvectors of the matrix h, (with p # 0). In doing so we have defined new single-boson states {p} that are orthogonal to the state 0. The corresponding eigenvalues .I$,of the matrix h, can thus be interpreted as the energies required to add one more boson - different from that of the condensate -to the g-s. of (N - 1) bosons. The basis (0, p} obtained in this fashion can conceptually be assimilated to that obtained with a Nilsson scheme for the many-fermion nuclear problem. In both situations some symmetry is broken to allow for a lower g.s. energy and a set of independent-particle states is constructed in this distorted frame of reference. In the many-boson problem the energies Ep obtained in this fashion are, however, wrong in the O(N) as a consequence of the fact that terms of that order in the hamiltonian have not beeen considered in h, These terms can be considered making use of the uncertainties in the definition of the excited single-boson states to change from the basis {p} to a different basis {(w}using the stability conditions of the HB equations. These amount to demand that the second-order terms in the expansion (2.6) correspond to a positive definite quadratic form in the (arbitrary) coefficients C, and lead to the well-known non-hermitian eigenvalue problem of the RPA. This
99
J. Dukelsky et al. / Self-consistent description
is also equivalent
to linearizing
l3,‘=+
the equations
x;r;r’)-
of motion
Y;r;r,,
fi,
of new operators
~(x;)'-(Y;)2=
1,
(3-l)
P
that are supposed to act on the (correlated) g.s. of the N-boson system. It is worth noting that if one neglects terms 16)of 0( 1/N), eqn. (3.1) becomes formally equivalent to a Bogoliubov transformation defining quasi-particles. We turn to this point at the end of this section. To obtain the RPA equations we assume that the hamiltonian (2.12) has been diagonalized in the HB approximation. Thus H,,, and H,, vanish and the one-body parts of HZ0 and Ho2 are included in the eigenenergies E,,. We consequently retain only the two-body part of the hamiltonian. Since we only study the excitation energies of “one-boson” states we take the zero of the single-boson energy scale at E. and therefore define
ip=Ep-~,. The linearization
of the equations
(3.2)
of motion [H, B:] = Q&
lead to the well-known
non-hermitian
eigenvalue
(3.3) problem:
(_:*_;*)(;:)=a(;:), where the matrices
(3.4)
A and B are defined
(3Sa)
(3Sb) In calculating the commutators of eqs. (3.5) neither H,, nor H2, give any contribution and only those due to H,,, EZ,, and H,,, survive. Within the TDA the wave functions of these states is very simple I*). This approximation corresponds to cancelling the B-matrices or equivalently to neglecting the backward amplitudes YF in (3.1). Thus
Ih+J = 1 x;[(N - iyl-V;(r;y-‘(0) P
=
rmv - 1)!]-‘/2(r,t)N-‘10)p
r:l+,_,) .
(3.6)
100
.I. Duke/sky
et al. / Self-consistent
description
In this approximation all the “one-boson” excited states are determined by the H, , part of the hamiltonian. Within the assumptions made hitherto, the eigenstates (3.6) or the ones obtained from the RPA equations describe different configurations that can be assumed to remain essentially unchanged when the system performs collective rotations. We thus regard each of the states I4,+) as a band-head on top of which a rotational band can be built. Among the TDA solutions there are some that are spurious as a consequence of the introduction of a (self-consistent, average) distorted field. These spurious states correspond to the motion of the many-body system as a whole without a true change in the internal arrangement of its particles. In the case of axial symmetry this happens with a K = +I pair of states that are obtained through the action of the operators J, and Jy on the g.s. condensate 19). Such states correspond to a rotation of the system as a whole and therefore should not be considered as a genuine, different intrinsic configuration that can act as a bandhead. In a general situation, the spurious states that arise as a consequence of broken symmetries do not appear decoupled from the remaining true physical states with the same quantum numbers and require a proper technique to separate them out. This is achieved through the use of the full RPA equations. The situation can be stated in full generality*~,*‘) as follows: Let J& be a one-body operator that generates a transformation that leaves the hamiltonian H invariant. Let us further assume that this symmetry is broken in the Hartree approximation. The operator JU then fulfills EY4=0,
4 f 0*
[Kfet
(3.7)
In eq. (3.7) we have written HHB for the hamiltonian truncated to contain only Ho,,, W,, and H,, (HB approximations). Associated with JU there exists a spurious state. Its wave-function can be written as the action of % on the HB ground state: Is)l-le= 4#iV) -
(3.8)
The state Is)~~ is set at zero excitation energy in the RPA. To see this we note that for any one-body operator .4 the following equations hold: (g.s.(RPA)([&, [H, &]]]g.s.(RPA)) =2x
m
~(gs.(RPA)(.4+(RPA))~*~~
(3.9) In (3.9) we have written M for the (column) matrix with matrix elements M,,O,where I]g.s.(RPA))
= C Mp,,F;&]g.s.(RPA)) w (3.10)
101
J. Dukelsky et al. / Self-consistent description
and
Q
and
immediately
M’
for its transposed
and
hermitian
conjugate,
see that since [H, &/III= 0, then from (3.8) it follows
respectively.
We
that
]&PA = Jll]g.s.(RPA))
(3.11)
represents an eigenstate of the RPA matrix (3.4) with eigenvalue zero. We can further conclude from (3.9) that none of the excited states ]a(RPA)) are connected with the g.s. through the one-body operator JII and therefore 1~)~~~ collects all the “.Mstrength”. The algebra leading to eq. (3.9) can be used to extend to many boson systems the well-known Thouless-theorem ‘“) on the conservation of the sum rule. It is easy to prove that (g.s.(RPA)ILK
[K
4lbW’AN
= kf+vIW,[H, -41hb~) .
(3.12)
In the case of axial symmetry the HB states remain labelled by K, the matrices (3.5) reduce to block diagonal form, each block labelled by a different value of K. The matrices (3.5) are then always of the same dimension as those of (2.15) that in most practical cases are very small. For the sake of completeness we discuss here briefly the structure of the g.s. wave function within the RPA. The expression (2.1) has to be changed to include the correlations that make possible to have a non-vanishing result when acting with the bakcward components of Bz. The wave function can be obtained as in the manyfermion
problem
by setting “) (3.13) \P4
The expression (3.13) is formally equivalent to the wave function of the g.s. of uncoupled harmonic oscillators. The constant floN’ is a normalization and the matrix Gpq has to be determined with the condition that (3.13) is the vacuum of the annihilation
operators
B,. Thus
B,lg.s.(RPA))=O=fioN’exp (3.14) To derive (3.14) we have made follows from the commutation vanishes identically if
use of the fact that Gpq is a symmetric matrix as properties of ri and ri. The expression (3.14)
YX
(3.15)
= 6,, .
(3.16)
G,, = t C {(X-‘X 01 with c (x-‘);x: 01
102
J. Dukelsky et al. / pelf-consistent
deseripti~~
The norm tioN’ can be found to be
I
AloN’= 1tt nl(NN_!2n)l[2tr(~z)ln}~“2;;;;_exp[tr(G2)1.
(3.17)
It is worth remarking that the exponential in (3.13) or the sum in (3.17) truly represent finite polynomic expressions because N is finite. If we let N +OO and keep only the leading terms in l/N then the Bogoliubov prescription 16) (3.18)
r%+=rOkbN)=~N(~N)
can be used. In this case eq. (3.1) can be regarded as a quasiparticle transformation in which the number of bosons is no longer preserved and (3.13) becomes equivalent to the wave function of the BCS vacuum. 4. Wave functions and symmetry properties The many-body wave functions (2.1), (3.69, etc., correspond to intrinsic states in which the many-body system is described using a single-boson basis for which there are privileged orientations in space. The existence of these is a direct consequence of the occurrence of distortions in the self-consistent one-body field. The complete wave function of the system has therefore to involve not only the intrinsic degrees of freedom but also rotational variables specifying the orientation of the body-fixed, privileged frame of reference with respect to a lab system. The existence of a stable, distorted minimum in the g.s. energy allows one, as usual, to assume that the total wave function can be factorized into two parts that depend respectively on the intrinsic coordinates (x) and orientation variables (0): w N.raM=
(4-l)
~N,a(X)&,d~).
The standard procedure 22V23) for discussing the effects of a transfo~ation 9 on (4.1) is to work out two realizations of it, one acting on the intrinsic, 9(x), and the other on the collective coordinates, 5(w). If this can be done the effect of 5 on one set of degrees of freedom can be corrected by Y-’ on the complementary set leaving the wave function unchanged. Thus a wave function that remains invariant under .Y can be constructed within the separability assumption (4.1) as &W&t = ‘J&l + ~(X)~-‘(~))WN,loM(X,
0)
-
(4.2)
For the impo~ant case of axial symmetry the intrinsic wave functions can be labelled by the projection K of the angular momentum along the body-fixed z-axis that is taken to be along the symmetry axis. Using the argument 23) leading to (4.2) and taking for Y a rotation of 180” around an axis perpendicular to the intrinsic z-axis, it is possible to show that the two intrinsic solutions &,K and #NV2= Y(X)
J. Lhkelskv et al. / Self-consistent description [c$~.~]
combine
In a general
case without
can be expanded +.N,IKM
6,
w)
into a single
set of rotational
axial symmetry
in eigenstates
103
states as
the intrinsic
part of the wave function
with good K thus the more general
expression
for
is
%,,I&, ml = cK g: The special considerations of parity and time reversal them here.
%V,,K&,
WI.
23) for the K = 0 case and for the additional symmetries also hold in the present situation and we do not discuss
5. Rotational spectrum Each of the intrinsic states I+N,n) is a head for a rotational band. As long as the intrinsic degrees of freedom remain decoupled from the collective motion, the rotational spectrum will follow an Z(Z + 1) law in which the moment of inertia is the only parameter that enters in the description of the collective band. To calculate this we shall assume for the sake of concreteness that the system is axially symmetric and that the intrinsic states can thus be labelled by K and some extra quantum number LY:I+N,K,a). If the intrinsic states are described within the RPA, the spurious state appears at zero energy.
By separating
tum that is associated the inertial prescription
the canonically
to this Goldstone
conjugate (zero-energy)
parameter associated to the rotations. that yields the following expression
pair of coordinate boson,
it is possible
and momento evaluate
This is the Thouless-Valatin for the moments of inertia:
24,2’)
where (A + B)K,u*,K,rolts = AK,a,,K,,a,, + BKra,,Krsao. In (5.1) A and B are the matrices prescription:
(3.5a,b).
$1 = 2 c K’a’
Eq. (5.1) reduces
~(~NK$&N)~’ K’CY’
to the sell-known
Inglis
(5.2)
in the limit in which the terms due to the two-body residual interaction V are neglected. For such a case the excitation energies g.K’a’ in (5.2), are given by (3.2), i.e. as differences of the eigenvalues of the self-consistent field h, (eq. (2.15)).
104
J. Dukelsky
If axial symmetry
is preserved
et al. / Self-consistent
description
then the transformation
(2.10) can be written
as (5.3)
Using (5.2) and (5.3) to obtain
the moment
of inertia
of the g.s. rotational
band we
get
It is easy to see that the only intrinsic state I+N,K,a,) that is involved in (5.2) is the spurious state (see eq. (3.8)) as in the Thouless-Valatin prescription. We can take a further step to investigate the structure of the rotational bands by performing a self-consistent cranking calculation 13). This consists in minimizing a cranked hamiltonian 14) with the subsidiary condition that the expectation value of the angular
momentum
has the required ~((&,,(w)lH
value.
Namely,
- WJXI4N,&)))
= 0
(5.5)
G#Q4,awIJx14N,a(~)) = [HI + I)-- w'*.
(5.6)
with the condition
Upon an initial guess of w, the variational parameters qo,, are determined and a TDA (or RPA) calculation of the intrinsic state can’be performed. Finally the condition (5.6) can be imposed adjusting w with the required expectation value. This procedure can be followed without heavy numerical work because the size of the matrices is very small even for quite ambitious assumptions upon the single-boson space. 6. Electromagnetic
transitions
All the transition matrix elements can be calculated making use of the separation (4.3) of the total wave function into intrinsic and collective part. Each intrinsic state is assumed to present a (large) static multipole moment. The rapidly changing electromagnetic field that is produced as a consequence of the collective (rotational) motion of the intrinsic structure induces enhanced electromagnetic transitions between the different rotational states. This physical picture is the same as the one of the well-known “unified model” ‘*25) with the only change that the intrinsic part of the wave functions is of bosonic structure. Any multipole operators of order L in the lab and in the intrinsic system are related through Tb(lab)
= C 9h,,Th(intr). M’
In (6.1) Tb(intr)
has a large expectation
value
in any of the (deformed)
(6.1)
intrinsic
J. Lhkelsky
states. The Wigner
9-function
et al. / Self-consistent
that accounts
for the transformation
fixed to the lab systems
acts on the collective
(4.3). The matrix
of (6.1) can immediately
(@,,,
elements
(rotational)
11TL(lab)ll !@N,SKV) = ?? 1 M’
+
105
descriptron
part of the wave function
be worked f,
K
from the body-
out to be
(~NKollT~,(intr)J~NK,~,) >
I
I’
L
-K
M’
MI-K’
>
In calculating (6.2) we have taken for the matrix the following choice of phases 23):
=(-I
elements
among
intrinsic
states
(6.3)
“+“+“‘(~NKaI~4MI~N~).
In (6.3) the transformation .Y is - as in sect. 4 - a rotation of 180” around an axis perpendicular to the symmetry axis. The intrinsic matrix elements in (6.2) are calculated assuming that the multipole transition of order L proceeds via a one-boson operator. This is defined through its matrix elements c$,~” in a spherical (reference) basis: TL,(intr)
= C q!:“‘)Y+y,. ‘I
(6.4)
The labels i and i include the single-boson angular momentum and its projections and the tensor character (L, M’) is guaranteed by vector coupling coefficients that we take to be contained in the q!,FTM”. For the sake of simplicity we shall assume that the intrinsic states are defined within the TDA and we neglect the corrections due to the self-consistent change of the moment of inertia for increasing angular momenta. With this assumptions we can write
(6.5) using the definitions of r: used in eq. (3.6). Using (6.5) in (6.4) we obtain T&(intr)
=
t,“,“‘r,‘r, +c t;r’(r:rO+ro+ra) + 1 t::‘ryaf, n
aa’
(6.6)
1. Rukelsky et ai./ Self-consistent
106
description
with (6.7a)
(6.7b)
In (6.6) and (6.7) we have assumed that 7 and $ are real. while ton Out of the three contributions (6.7) too and t,, are related to intraband and f,,, (LYf a’) are relevant to interband transitions. Within the present approximations the &EL) or &ML) values that depend upon these matrix elements are completely defined by the intrinsic structure. Thus as expected from the separability (4.3), the decay intensities within a band are in the ratio of Clebsch-Gordan coefficients
(see eq. (6.2)) exactly in the same fashion
as within
the unified
model.
7. Calcutations 7.1. THE MODEL
In this section we discuss the methods already presented at the light of the numerical results obtained in different model situations. We consider a system of N bosons with angular momenta I = 0,2,4. All the calculations were performed for N = 30. The hamiltonian is H=KQ*
Q
(7.1)
with
+JW3&d%): We omit the L - L term because
(7.2)
+(r:~~)2:}-X443~(~41j4):. its effect is only to renormalize
the moment
of
inertia 8;. The free coefficients XZ2, X,, and X,, allow one to reproduce a variety of different dynamical situations, very much in the same fashion as in the Warner and Casten 26) model. For X,, = X,, = X,, = 1 the five operators 0: together with the three components of the angular momentum form the eight generators of the algebra of SU(3) built “) with bosons with angular momenta I= 0,2 and 4 @U(3) [sdg]). Taking X, = X,, = 0 and X,, = &v%$ the situation is again that of SU(3) 5, but with bosons of angular momenta I = 0 and 2 (SU(3)[sd)), leaving the 1= 4 modes as inert spectators. Finally, within the space spanned by s- and d-bosons and letting XZZ+ 0, it is also possible to approach the O(6) scheme 5, (0(6)[sd]).
107
J. Dukelsky et al. / Self-consistent description
To discuss the numerical results we recall that the self-consistent methods are approximate solutions of the many-boson problem taking as a (small) expansion parameter the inverse of the (large) number of bosons N. We can thus regard any physical observable P of order r in N (O(r)) expanded as P=Z’,N’+P,N’-‘+..a.
(7.3)
This expansion can also be performed with the exact (group theoretical) results. In the SU(3) sheme the energy eigenvalues are E Uhf --K{~L(L+l)-(A~+~z+h/.l -
In the lowest representations in the system 5@*9)
the label A is proportional
(A,~~=~2N,O),(2N-4,2),(2N-8,4~,~2N-6,0) (A,
-t-3(A +jAu)>}.
(7.4)
to the number of bosons
,...,
SWXsdl
,
. . . , SU(S)[sdg].
p)=(4N,0),(4N-4,2),(4N-6,3),(4N-8,4)*,(4N-6,0),
(7.5) We immediately see that the leading term in the g.s. energy is of O(N*). The excitation energies of the rotational bandheads are of O(N) and that the splitting within a rotational band -given by the L* term - is of 0( 1). THE g.s. ENERGY
7.2. O(N*):
The g.s. energies are shown in fig. 1 in units of KN* and compared with the exact SU(3) limit given by (7.4). The dimensionless value of .E,,./KN~ decreases as X,, + 0, correspondingly Eg s increases (for an attractive interaction). This is because the smaller influence of the 2= 2 boson causes a smaller deformation of the system. This can also be seen in the structure of the boson r,’ with which the g.s. condensate is built (fig. 2). The difference in the g.s. energies for the SU(3)[sdg] and the SU(3)[sd] models is a measure of the renormalization of the intensity between both models. The SU(3)[sd] and 0(6)[sd] results can be compared against those of ref. ‘) in which also an expansion in powers of l/N is possible. If only I = 0 and 2 bosons are considered (see appendix) we can write r,’ = %,0oY&+ %,20Y:o
(7.6)
with ‘) 770.00 =
J( ;
1-$-++O(N-2)
TO.20
,
=
WNsdl
, .
(7.7a) ,
To,20=
; J(
I+&+O(N-‘1
,
WXsdl
(7.7b)
108
1. Dukelsky
0.
.2
et al. / Seljbnsistent
.4
.6
.8
description
t.
Fig. 1. Ground-stateenergy in
units of KN~ as a function of the parameters operator (see eq. (7.2)) for different models.
defining
the quadrupole
The numerical results are seen to reproduce properly the rather drastic change in the structure of the boson of the g.s. condensate. Similar changes can be observed in a model involving bosons of I= 0,2 and 4 although no analytic expressions such as (7.7) have been worked out for this case. The g.s. energy can be obtained with the use of eq. (7.7) to leading order in IV. Using eq. (2.17) we get -$&=~(10&0
+%?3,20 +(N - 1w210>2)
+ 4 1 +& (
+O(W2)
>
=--&
&,,(SU(3)[sd])
+O(N-‘)
.
(7.8) In (7.8) we have used (see eq. 6.5)) (ff IQIP)=C
Q&*,*6%
‘I as the matrix elements of the quadrupole
(7il3,*=
rl0.d
operator in the HB-TDA basis.
(7.9)
109
.i. Dukelsky et al. / Seff-consistent descripiion I.0 ‘
0. Fig. 2. Structure
I
.2
of the boson
1
I
4
.6
I’: as a function
I
i
.8
1.
I
of X in the same model
situation
as fig. I.
In eq. (7.8) still a contribution of (3/4N) is missing inside the parenthesis to get the exact result. The RPA contributions to the g.s. energy 2’) contain 15)this correction thus reproducing the exact results up to and including terms of O(N) (see appendix).
7.3. O(N):
THE
To study
EXCITATION
the excitation
ENERGIES
energies
OF INTRINSIC
of the intrinsic
STATES
states
we perform
a TDA (or
RPA) calculation. Except when otherwise stated we restrict our discussion to TDA results. In general, the corrections introduced with the RPA in the present calculations are very small and can hardly be distinguished in the plots. However, there are a few special circumstances in which these become conceptually impo~ant and these are commented on in detail. To compare the approximate results with the exact ones we use (7.4) setting h = mN - m’ (m = 2 for SU(3)[sd] and m = 4 for SU(3)[sdg]). We thus get
J%hP>
-=(2m’-p)--&{m’2-(p mI+J
+3)(m’-p)-$(L+l)}.
(7.10)
J. Dukelsky et al. / Self-consistent description
110
! &“i(mN ,,_~~~Z”’ 1
-3
(4N-6.3) --t?Y
J&LO)
i
01 --rio EXACT
-I’S’
-
-0
-
SU(3)(sdg]
Jxl
+N> PkN>
@
’ ‘NJ’
TDA
SU(3)(sd]
E/)(mN t
-2
-0
5
,iI
Lj:
fZN,O) 0
Fig. 3. Correspondence of the exact W(3) limit. The level scheme in the labels different states with the same corresponding SU(3) representation following
r;‘---
-1’S’
-00
-
-
Jxl
“, >
MN>
and approximate (HB-TDA) bandhead excitation energies in the right-hand side is labelled by I< and in the left by I. A superscript K-value. An (s) denotes the spurious state. The (A, p) value of the are shown between parenthesis. The value of x in this and the figures is taken as a positive quantity.
Eq. (7.10) clearly displays the fact that the excitation energy of the intrinsic states is a O(N) effect. The values obtained with (7.10) are compared with the TDA results in fig. 3. These (or even those obtained with the RPA) agree with (7.10) except for differences of O(1). Among these small corrections the exact results contain the rotational kinetic energy term that is absent in TDA (or RPA) whose eigenstates remain Iabelled by K and not by L as the exact ones. The lowest SU(3) representations correspond to intrinsic states that within the TDA can be pictured as the promotion of only one of the bosons of the condensate
J. Dukelsky
111
et al. / Se/$-consistent description
to one of the unoccupied single-boson states of higher energy. Excited states’of a higher energy within HB-TDA are constructed promoting two bosons. Correspondingly its excitation
energy
of rn’ and p correspond
is the sum of the one-boson in fact to this harmonic
energies.
pattern
The lowest values
(see fig. 3). To construct
the K-values of the two boson states we remind that both the positive and negative values of K must be added up in agreement with what was said in sect. 4. This is schematically indicated with the creation operators in fig. 3. In figs. 3 and 4 low-energy K = 1 states appear that are almost insensitive to the free parameters of the residual interaction. These are the spurious states that are obtained by the action of the operators J,, on the g.s. wave function. In a general situation J,lg.s.) appear distributed among all the K = 1 states. To elliminate these states the full RPA equations must be used. For the case of an [sd] space and a quadrupole-quadrupole interaction the RPA equations can be worked out analytically. We first note that the boson rl of the condensate appears at an energy E. given by (a?lO)
wlQlo>
no,00 and the TDA excitation &(TDA)=
energies
E
,)mo WlQlo) ______
+K(N_
ho.00 2
&=&O+K(N-l)-p=
770.20
(7.12)
l)((olQIO)(KIQIK)+~(OIQIK)12)-Eo.
In eqs. (7.11) and (7.12) e. and .s2 are the single-boson
I
energies
(B)
(4 N.0)
I
jxz2.x
x*4=x&&=0 .2
4
6
that arise from the
I
(4
O/ 0
(7.11)
’
of the states with K = 0 is
&2+2K(N-
I
a770.20
.E
1.
1.2
_I
X,,.l
[
0
0
.2
.4
.6
.B
I.
x;;.x4,=x 1.2
Fig. 4. Excitation energy of the rotational bandheads [sd](A) and [sdg](B) models as a function of the internal parameters of the quadrupole operator (see eq. (7.2)). The exact SU(3) and O(6) limits are indicated in the figure. The superscript (s) indicates the spurious state. The numbers close to each curve indicate the K-value of the bandhead. A subscript labels the different states with the same value of K.
112
J. Dukelsky
contracted part of the two-body two by two and the eigenvalues
et al. / leaf-consislen!
residual interaction. For K # 0 the RPA matrix can readily be found to be
flK =(&(TDA)-4&V-This equation
description
can be used together
l)2~(O~Q~K)~4)“2.
is
(7.13)
with (7.11) and (7.12) to check that the RPA
frequency of the K = 1 state is zero. The structure of the intrinsic states changes
with the structure
of the Q-operator.
The transition from the SU(3)[sd] to the 0(6)[sd] scheme is shown in fig. 4(A). For small values of X,, there appear strong instabilities and a lack of convergence in the self-consistent procedure. These features indicate that a situation is approached in which there is not a sharp, deformed minimum in the g.s. energy. Correspondingly the RPA corrections become increasingly important. This is clear from fig. 4(A). Using eq. (7.11) it is simple to prove that in the 0(6)[sd] limit the frequency of the K = 2 state is zero as befits a situation of unstable (infinite-amplitude) oscillations. In fig. 4(B) it is also possible to see how the self-consistent procedure is able to reproduce the drastic changes from the SU(3)[sdg] situation in which the f = 4 boson is strongly coupled to one in which it is an inert spectator in the presence of a perturbed SU(3)[sd] scheme, giving rise to a degenerate multiplet of bandheads with K = 0,f 1, ;t2, *3 and &4. A few examples of the changes in the wave functions of the K = 0 bandheads are shown in fig. 5. In this figure it can be clearly seen how the K = 0’ state becomes a pure 1= 4 and the other becomes an o~hogonal tion to the g.s. (fig. 2) that only contains I = 0 and 2 components.
7.4. O(1): THE ROTATIONAL The
belong
rotational
kinetic
combina-
MOTION
energy term causes a (small) splitting
to the same representation
among
(h, 1~). This can be obs&ved
the states that
in the left part of
..
0 Fig. 5. Structure
.2
.4
.6
.a
1.x'x24=X4&
of the excited K = 0 bandheads in an sdg model (fig. 4(B)). The dotted correspond to the lowest (highest) R = 0 excited state (0,).
(full) lines
113
J. Dukelsky et al. / Self-consistent description
fig, 3. Since no angular momentum projection has been made for the TDA eigenvalues that are plotted in the right part, these appear in degenerate multiplets regardless the value of K. This rotational motion is described by a single parameter that is the moment of inertia 9. The SU(3) value for 2 obtained from (7.4) or (7.10) is
-=3 1 2&f~f
(7.14)
4K*
The Thouless-Valatin and Inglis prescriptions (5.1) and (5.2) to calculate the moment of inertia can be worked out analytically to leading order in N in the SU(3) limit. Using eqs. (5.4) and the energy of the Is: = 1 state obtained in the appendix (A.9) and (A-14) we get $r=6N7&o(E~B=1)-‘= 8;” = 6N&,(A,,
-$c
(7.15a)
+O(N-‘),
+ B, J-1 = 6Nn&(2E’TDA)-’
c- -$K +O(N-‘)
.
(7.15b) The results (7.15a) are seen to be very close to (7. ISb) that in turn agree with the exact ones up to the leading order in 1/N. A partial inclusion of the O(N) terms of the hamiltonian contained in HII and neglecting Ho2 and Z-ZzO, (TDA) would have badly spoiled the result (7.15a). The O(6) case can not be discussed in a straightforward manner since in this case the symmetry that is broken is of a more complex nature. In this case the moment of inertia also has contributions coming from the dynamics that is involved with the seniority-dependent terms. The moment of inertia depends on the residual interaction and on the single-boson space that is active. This can be checked through the examples in fig. 6.
7.5. MULTIPOLE
MOMENTS
AND
TRANSITION
The last dynamical ingredients that we study are the matrix elements for quadrupole electromagnetic transitions (eg. (6.2)). The expansion in powers of N can be obtained directly from eq. (6.6) using the wave functions (3.6). The corresponding matrix elements are (~~~~~,(intr)~~~)
(7.16a)
= NW/Q/O),
(#N,K,nIQ~,(intr)I~N,K,ol) = (N - l)(OlQlO>+(KIQlK)
.
(7.16b)
The values (7.16) give a measure of the static quadrupole deformation of the intrinsic states. This is clearly a collective O(N) effect. The difference between the ground and excited states is instead of O(l), as befits a situation in which only one of N bosons of the system has changed its configuration.
J. Dukelsky et al. / Self-consistent
114
description
.6 -
Fig. 6. The Thouless-Valatin moment of inertia of the ground-state rotational bands as a function internal parameters of the quadrupole operator for two different model situations.
of the
These features can also be found 27V28)in the Elliott SU(3) model. Within this framework the many-boson wave functions in a Cartesian basis can be organized in decreasing values of the numbers N, and N, of oscillation quanta in the z- and x-directions. The state with the biggest N, and N,, (P~:,~,, being called the leading state. The labels (A, CL) and N, and N, for this state are related through h=N,-N,;
/_L=N~-N,.
(7.17)
We have seen in subsect. (7.3) how different values of (A, CL)correspond to different sets of excited bandheads. We can thus regard the states (P~;,~, as intrinsic, in the same way as in the HB-TDA language. The expectation value 27) of the quadrupole operator in the leading state is ((PN,,N,10~,(i~r)l~~~,~,)(Y(2h
+~)=2mN+(~-24,
(7.18)
which agrees with (7.13a,b) as regards the order of N that is involved. In fig. 7 we show how the different matrix elements of the quadrupole operator change as a function of the parameters of the model. From the numbers in fig. 7 one can easily reconstruct the values (7.18) in the SU(3) limit. In most practical cases the actual value of (KIQIK) is irrelevant since it amounts to a correction of O(1) in the matrix element.
J. Lhkelsky et al. / Self-consistent
description
115
Fig. 7. Matrix elements of the intrinsic quadrupole operator for two [sdg] model situations. The full lines correspond to X = X,, = X,, = X, and the dotted lines to X = X,, = X, with X,, = I .O. Ste fig. 3 for the identification of the different states.
The transitions between bands SU(3) limit because the quadrupole
labelled by different (A, CL) are forbidden in the operator cannot connect wave functions belong-
ing to different SU(3) representations. element is in general of O(N”‘),
In spite of this fact the transition
(~N,K,~IQ~M’(intr)I~N,K,,a~ > = (KIQIK?fi
3
matrix
(7.19)
as follows from eq. (6.6). This feature causes that even quite small departures from the SU(3) limit turn this transitions into measurable lo) quantities since the corresponding (&EL)) value is magnified by a factor of N.
8. Conclusions In the preceding sections we have presented an application of self-consistent procedures to systems of many interacting bosons. The g.s. is described as a condensate in which the N bosons of the system are in the lowest single-particle state. The structure of the elementary excitation can be obtained within a TDA. Within this framework each of these excited states can be pictured as the promotion of one of the bosons of the condensate to a single-boson state of a higher energy.
116
J. Dukelsky et al. / Sel$consistent description
Among these a spurious state occurs as a consequence of a broken symmetry that corresponds to the motion of the N-body system as a whole. This state can be eliminated performing an RPA that brings it to zero energy. Each of this intrinsic states can act as a bandhead. The collective rotational motion that can be constructed on top of each of these states can be described by a single inertial parameter, i.e. the moment of inertia. This can be evaluated using the cranking-model prescription. A more elaborate approach - a self-consistent cranking - is also possible. The method can be applied successfully to situations in which a stable, (deformed) minimum occurs in the g.s. expectation value of the hamiltonian. However, we have proved that we can encompass a wide variety of dynamical situations. When the approximate results are compared to exact-group theoretical -values, they turn to be accurate except for terms of higher order to power l/N, N being the number of bosons in the system. This error is corrected if all the RPA contributions are included, although this study may be avoided because N is in most practical cases a large number. All these features hold even if the single-boson space includes, for instance, several bosons of the same 1 and many different possible Z’s The numerical work is kept under reasonable limits even for quite ambitious choices of the single-boson basis. With these advantages the method can be used to test a wide variety of many-boson hamiltonians obtained, for instance, by mapping of a many-fermion problem or in a study of the interplay of different multipole degrees of freedom. Both of these problems rapidly become intractable by other methods. The study of more complicated potential-energy surfaces in terms of shape-deformation parameters requires, however, the proper consideration of many-boson residual-interaction terms. Appendix ANALYTIC
EXPRESSION
FOR [sd] MODELS
In this appendix we summarize the expressions that can be derived for the HB, TDA and RPA equations for the case in which only s- and d-bosons are involved and axial symmet~ is assumed. (i) 7’he reference and HB basis. The six different boson creation operators in the spherical basis remain labelled by the angular momentum I( =0,2) and the projection m(=-I,..., I). The distorted self-consistent basis is in many respects similar to the fermion Nilsson scheme. The transformation between both bases (eq. (2.10)) reserves the z-projection and can be written
(A-1) In an [sd] model it is possible to simplify the notation dropping all the redundant
3. Dukelsky et al. j Seljkansisient description
117
subindices. We thus write I-;= %Yzo + 772Y2f0, G
= A*,
G = 77*Yiho- %do , r:,
,
= d&2.
(A.3
(A.3)
The 0’ state is commonly referred to as the &vibration. We assume real coefficients n. The further corrections that are introduced through the TDA do not change the two q-coefficients since the corresponding matrices do not connect different K-values and the matrices therefore are of dimension 1. We thus have r) = ;i (eq. (6.5)). (ii) The quadrupole operator. The quadrupole written as
operator in the spherical basis is
Q, =(ro’u~~,+r:,~~)+x(~:~*)~=~~~.
(A-4)
for the SU(3) and 0 for the O(6) The internal parameter x takes the values symmetries. Using (A.2) and (A.3) Q can be written in the HB basis. A few relevant matrix elements are
(OlQ10) = bwr~~xr~:, W]QIO’) = -2rlow&I;,
+J:xwz,
(O[Q\W = 7: - d
(lIQ/l>=J~~=(-1101-1>, OtQ12) = -4x
= t-4Q1-2,
(4 010) = &rtz
+ TO,
(1(Qk%= vn’%xrl~.
(A.3
Using eqs. (7.7) and ref. ‘) these matrix elements can in turn be expressed as an expansion in powers of (I/ IV). (iii) Tke hamiltonian in the H3 approximation.
quadrupole-quadrupole
We only discuss the case of a
interaction: H=@
0.
(A.@
The contracted terms give rise to a single-boson energy term 6):
E~=K~i(~llQll~)I~&=
5’&,o+(l +X2)‘+.
The symmetrized two-body interaction terms that are left can be written as
(A.7)
J. Lhkelsky et al. / Self-consistent description
118
Thus the HB single-boson hamiltonian (2.15) reads ~IK,I,K,=&K,{G%,+ GN - 1) (O\QiOX~~iQi~‘K’)
I
. (A-9) iI12 With this matrix, the energy E,, that is needed to add one more boson to the condensate of N - 1 is
+C (~KIQKI~,O)(~,OIQ-K(-)~~~‘K)~,,~~~,
E,=
=
(OlQb
E,,+K(N-
l)-
++K(N-
I)&
a ~(01 QkV -$QlO)
772
E,& + ~~77; +~K(N - l)l(OlQ10)12.
=
(A.10)
Within the HB approximation the energy required to add a boson p # 0, 0’ to the condensate is just the diagonal term in (A.9). Each of the states is twofold degenerate. Each pair *K combine into a single rotational state. The pair K = *l is spurious. The g.s. energy of N bosons is 8,,(N)
= N(s071; + ~217:)
+ KN(N
-
l)(OiQb)’ .
(A.1 1)
If the TJ coefficients for the g.s. of N or N - 1 bosons are assumed to be the same (thus neglecting O(l/ N) differences) it is easy to check that 8&N)-
8,,(N
- 1) = E,, .
(A.12)
(iu) The Q(N) corrections (TDA and RPA). Eq. (3Sa) that gives the TDA equations amounts to introducing a correction ;(N -
1)
C V~%J~J, JI JzJ3-14
= K(N
-
~o,~,rlq,~~~o,~~
~K’lQ~K,)~2) V, = W .
l)S,,((OlQ(O)(K,(Q(K,)-
(A.13)
The TDA excitation energy of the spurious state is (eq. (3.2)) “TDA
El
=
Ei-DA
I
-Eo=
K(N-
@(Q(O) a 2(0~Q(0)(1~Q~1)+2~(0~Q~1)~*---ii,),(OiQ/O))
1)
772
= -
~K(N
-
(A.14)
I)(( 1(Q(O)\’.
The RPA backward correction (3.5b) is also a one-by-one matrix. BKK’=
Consequently
K(N- 1)~KK~((OlQ10)(~lQl~)+l(~l~l~)12) I
the RPA eigenvalue problem is (K # 0) (~~-f12:)+B~k=0, OK = *{&
-‘~K~(N-
1)‘~(0~Q~K)~“}“‘.
(A.15)
J. Dukelsky
This is seen to cancel
et al. / SelJconsistent
for K = 1 and to give a minor
states. This result is needed
for the calculation
2&*(N)
= ?&J(N)
119
description
(O(N-‘))
correction
of the g.s. energy
to all other
in the RPA as
+c f
References I) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 4 (1952) 2) G.G. Dussel and D.R. Bes, Nucl. Phys. Al43 (1970) I 3) K. Kumar and M. Baranger. Nucl. Phys. A92 (1967) 608; J.M. Eisenberg and W. Greiner, Nuclear models, vol. I (North-Holland, Amsterdam, 1970) ch. 5 4) P.O. Lipas, Ph.D thesis, Rensselaer Polytechnic Institut (1961); P.O. Lipas and J.P. Davidson, Nucl. Phys. 26 (1961) 80 5) A. Arima and F. Iachello, Ann. of Phys. 99 (1976) 253; 111 (1978) 201; 123 (1979) 468 6) 0. Scholten, The interacting boson approximation model and applications, Ph.D. thesis, University of Groningen, (1980). and references therein 7) J.N. Ginocchio and M.W. Kirson, Nucl. Phys. A350 (1980) 31: Phys. Rev. Lett. 44 (1980) 1744 8) P. Van Isacker, K. Heyde, M. Waroquier and G. Wanes, Nucl. Phys. A380 (1982) 383; Phys. Lett. 1048 (1981) 5 9) Wu Hua-Chuan, Phys. Lett. 1lOB (1982) 1 10) D.D. Warner, R.F. Casten and W.F. Davidson, Phys. Rev. C24 (1981) 1713 1I) K. Sugawara-Tanabe and A. Arima, Phys. Lett. 1lOB (1982) 87 12) T. Otsuka, Nucl. Phys, A368 (1981) 244; T. Otsuka, A. Arima and N. Yoshinaga, Phys. Rev. Lett. 48 (1982) 387; D.R. Bes, R.A. Broglia, E. Maglione and A. Vitturi, Phys. Rev. Lett. 48 (1982) 1001 13) H.J. Mang. Phys. Reports 18C (1975) 325, and references therein 14) J. Dukelsky, G.G. Dussel, R.P.J. Perazzo and H.M. Sofia, Phys. Lett. 13BB (1983) 123 15) J. Dukelsky, G.G. Dussel, R.P.J. Perazzo and H.M. Sofia, Phys. Lett. 129B (1983) 1 16) N.N. Bogoliubov, J. Phys. (USSR) 11 (1947) 23; S. Beliaev, JETP (Sov. Phys.) 7 (1958) 289 17) D.R. Inglis, Phys. Rev. 96 (1954) 1059 18) A.E.L. Dieperink and 0. Scholten, Nucl. Phys. A346 (1980) 125 19) A. Bohr and B.R. Mottelson, Phys. Scripta 25 (1982) 28 20) D.J. Thouless, The quantum mechanics of many body systems (Academic Press, NY, 1972) ch. III 21) P. Ring and P. Schuck, The nuclear many-body problem (Springer, New York, 1980) ch. 8 22) G.G. Dussel, R.P.J. Perazzo, D.R. Bes and R.A. Broglia, Nucl. Phys. Al75 (1971) 513 23) A. Bohr and B.R. Mottelson, Nuclear structure, vol. 2 (Benjamin, NY, 1975) ch. 6 24) D.J. Thouless and J.G. Valatin, Nucl. Phys. 31 (1962) 211 25) A. Bohr and B.R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 26) D.D. Warner and R.F. Casten, Phys. Rev. Lett. 48 (1982) 1385 27) J.P. Elliott, Proc. Roy. Sot. A245 (1958) 128, 562; Group theory and the nuclear shell model, Notes of the Escuela Latinoamericana de Fisica, Mexico, 1962 28) Hsi-Tseng Chen and A. Arima, A cylindrical sp-boson for large deformed nuclei, Univ. of Tokyo, preprint; Hsi-Tseng Chen, Proc. Workshop on nuclear collective states, Suzhou, China, September 1983, p. 30