microscopic foundation of the interacting boson model in thes d-shell nuclei

Z. Phys. A - Atomic Nuclei 331, 243-253 (1988) Zeitschrift fiJr Physik A

Atomic Nuclei �9 Springer-Verlag 1988

Microscopic Foundation of the Interacting Boson Model in the s d-Shell Nuclei

R. Kuchta*

Institute of Mechanical Engineering, Plzefi, Czechoslovakia

Received March 11, 1988; revised version June 20, 1988

Starting from an isospin invariant shell-model hamiltonian, we describe a method for deriving microscopically the IBM-hamiltonian appropriate to light sd-shell nuclei. The key ingredients of our approach are: a) the Belyaev-Zelevinsky-Marshalek (BZM) bosonization procedure; b) two successive unitary transformations that extract the "maximally

(s~, s~) and J decoupled" collective bosons with angular momenta J = 0 + s~+, + + + d+~(T= 0), d+~(T= 1)). The method is applied to obtain the low-energy spec- = 2(d,~,,, d~,

tra and the electron scattering form factors for the 0~-~2~- transitions in g~ and 24Mg. Good agreement with the exact shell-model results is achieved. The inclusion of proton-neutron bosons (s~+~, d~+~ (T= 1), d~+~(T=0)), as well as the renormalization of boson parameters due to the non-collective degrees of freedom, are shown to play a crucial role.

PACS: 21.60.Ev; 21.60.Cs

1. Introduction

The Interacting Boson Model (IBM) has achieved an impressive success in the phenomenological description of collective motion in the medium-mass and heavy even-even nuclei at low excitation energies [1]. The simplest version of the model (IBM-l) [2] treats an even-even nucleus with 2N valence nucleons as a system of N bosons carrying angular momentum J = 0 (s+-boson) and J = 2 (d+-boson). Soon after- wards, it has been realized that [3]

a) the basic ingredients of the IBM-1 (s--- and d+-bosons) have properties similar to those of correlated pairs of nucleons with angular momentum J- -0 (S § and J = 2 (D § respectively;

b) the interaction among only identical nucleons is not sufficient to cause nuclei to deform and conse- quently to produce rotational spectra;

c) in nuclei involving both valence protons and neutrons, the proton-neutron interaction plays a dominant role in the description of low-lying collective states.

* Present address: Laboratory of Theoretical Physics, JINR Dubna, PO Box 79, SU-101000 Moscow, USSR

These observations have led to a more elaborate version of the IBM, called IBM-2, which distinguishes neutron bosons + + (s~,dw) from the proton ones

+ + (s~, d~) and introduces an appropriate quadrupole- quadrupole interaction between them [4]. Despite this refinement of the model, the IBM-2 is not flexible enough to provide states with good isospin [5]. This drawback may not be of great importance in heavy nuclei where neutrons and protons occupy very different single-particle orbits. However, in light nuclei (such as those with 16<A<40), both protons and neutrons are filling the same valence shell (the s d- shell) and it is well known [6] that large errors arise if one does not take into account the isospin symme- try. That is why for applications to these nuclei, the IBM has to be put into an isospin invariant form. Two such versions, IBM-3 [5] and IBM-4 [7], have been considered so far. Their encouraging success (especially that of the IBM-4) [8] raises the important question of the microscopic foundation of the IBM in light nuclei. Although tremendous progress has been achieved in formulating the theoretical basis of the IBM in heavy nuclei [9], the same cannot as yet be said about light nuclei. The work which has been done in this field [10] is still at the beginning and

244 R. Kuchta: Interacting Boson-Model

further investigations are called for. The basic problem one encounters in the course of establishing a true microscopic foundation of the IBM consists in that there are non-negligible admixtures of the higher- multipole nucleon pairs in the low-lying collective states [11], while the phenomenological description of the same states does not indicate the need for an explicit inclusion of the corresponding higher-multipole bosons [1]. In other words, the higher-multipole degrees of freedom occur naturally in microscopy, but are not required in phenomenology. This is particu- larly pronounced in rotational nuclei. Since the sd- shell region 16 < A < 40 contains some experimentally well studied nuclei with clearly identified rotational character [e.g. 2~ 24Mg], the above inconsistency must be resolved before one can aim at a microscopic understanding of the success of the IBM-4.

In the present paper we show a possible way of achieving this goal. The starting point of our approach is the Belyaev-Zelevinsky-Marshalek (BZM) boson mapping [12] of an isospin invariant shell- model hamiltonian. The further step consists in two successive unitary transformations. The first of them determines the most collective J = 0(s +) and J = 2(d +) bosons as creation operators of the energetically low- est one-boson eigenstates of the boson hamiltonian. The second transformation then eliminates (to a certain degree) the two-body interaction of these collective s +- and d+-bosons with the rest of the boson space, thereby producing an IBM hamiltonian which only weakly couples the collective and non-collective subspaces.

The organization of the paper is as follows. In Sect. 2 we describe the relevant formalism and show how to treat the spurious states arising due to the violation of the Pauli principle in the boson space. Section 3 presents the results of an application of the method to calculating the energy spectra and the electron scattering form factors for the excitation of the 2~+-states in 2~ and 24Mg. Conclusions from our investigation are collected in Sect. 4.

2. The Formalism

2.1. The Microscopic Boson Hamiltonian

The starting point of our considerations is a "realistic" shell-model hamiltonian written in the (uncou- pled) second quantized form as

where ~+ ((I;~) is the creation (annihilation) operator for a nucleon in the single-particle (s.p.) shell-model

state ~ - (G l~j~ m, ~) with G = �89 z, = - �89 for a neutron and proton, respectively, e, stands for the corresponding s.p. energy and the quantities V~p~0 repre- sent the antisymmetrized (V~0=-Vp,~0=-V,p~7) matrix elements of an effective nucleon-nucleon (NN) interaction. By the notion "realistic" we mean that the s.p. energies and the effective NN interaction have been chosen so as to reproduce some important experimental properties of the nuclei considered. The hamiltonian (1) is taken to be hermitian, i.e. the matrix elements V~70 are required to satisfy V~p~= Vy~r Furthermore, it is assumed that HSM is rotationally and isospin invariant, so that it can be transcribed as a scalar in the JT-coupled form

H~= F. a ~.[r • G ] ~176 a

�88 Z vZ~(-)'+ ~([e2 • r [~o • G] "~) abcd

~ (2)

where

a = ( G , l~,j~),

[r , ~ + ~ ,x, ~-~b ] M M T

= ~ (j~m~jpmp[JM} l 1 + (7 G ~ z#[ T M r } C,+m~,, ~bm# z#, m~m B rezfl

(.. . [...) is the usual Clebsch-Gordan coefficient, the symbol ([ ]JT. [ ]JT) means a scalar product of tensor operators, ff;am,~ is defined as (-)J=+"=+~+~= l~a_,, _,~ and VSr e are the angular-momentum- and isospin- coupled versions of the matrix elements V~pra, appearing in (1). Using the relation

r r G r = 6B~ G + r G + G r r (31

we can rewrite the hamiltonian (1) in the form

ap 7

_1 Z v.,~r + G r r (4) eP76

which will be more convenient for our purposes. Now, we introduce the boson creation and annihi-

lation operators b+~ and b~p, which correspond to fermion pair operators II22 112; and (E~ 112,, respectively. These boson operators are assumed to satisfy the following antisymmetry and commutation relations:

b2~ = - b L , (Sa)

[b,p, b,,e, ] = [b 3 , b~,] =0, (5b)

[b~p, b~B, ] = 5~, 6p~,-- 6~, 5~,. (5c)

R. Kuchta: Interacting Boson-Model 245

We also define the boson vacuum I0)B by the condition

b~ e 10), = 0. (6)

In the Belyaev-Zelevinsky-Marshalek (BZM) approach [12], the boson images of the bifermion opera-

+ q- tors C~ C , , ~ , ~ and ~7~ + ~ are expressed as power series in the boson operators b+~, b=e and the unknown coefficients of these expansions are determined by requiring that the original fermion commutation relations remain valid in the boson space. In the case of the operators C + ~e this requirement leads to a very simple finite form

(r r = Z b+ be,, (7)

where the suffix B on the left-hand side denotes the boson image of the corresponding fermion operator. On the other hand, the operators (lE+lE~), and (IE e (E~), are in general represented by infinite expansions, thereby giving rise to the problem of conver- gence. Nevertheless, these complicated operators are not needed to find the boson image of the fermion shell-model hamiltonian if the latter is conveniently expressed in the form (4). In this case only the simple boson operators (7) are relevant. Replacing the bifermion operators lE~+ lEe in (4) by their boson images (lE~+ lE~)~ according to (7) and arranging the boson operators in the interaction term into the normal order with respect to the boson vacuum 10)~, we obtain an exact boson image H B of the shell-model hamiltonian HSM in the form

+ + + e~e,ob~pb,a- �88 ~ (8) V~e~o b~p be~ ba~ b~,

aflya aeyo where oe

1 e,e~o = e, 6~ 3ea + ~ V~ ~. (9)

Since the bosons associated with actual states of a nucleon system are expected to have good angular momentum and isospin, it is useful to introduce the following coupled boson operators

BfTMM r (a b)

= ~ (J~m~jemelJM) (~zt�89 TMT) b + - am~z=, bm3z 3 " trtc, m ~

~ (10)

In order to take into account some of the nucleon- nucleon correlations, we further define the collective boson operators

B + i J T M M T - - ~ E i Dab )~abST BfTMMT( a b), (11) ab

i where Dab=/1 q-6,b and {Z,bsr} represents a com- plete set of two-nucleon wave functions, i.e.

Z~bJr = (--)J~+Ja +J+ 7"Z~.jr, (12a)

2 i i' - 2 E DabZabJT)&blr - ~ii', (12b) ab

i i E )~abJT Za'b'JT i

=Dab2 {(~aa,(~bb, q-(--)J~+J~+J+ T (~a,b(~ab,}, (12c)

(~a -~- ~b) i I ; JT i ZabST -[- �89 t 2 Da'b' " a'b~ ab Za'b'JT

a'b"

, i (12d) = E Jr XabJr"

The condition (12a) reflects the Pauli principle for a nucleon pair [see (Sa)]. Equations (12b) and (12c) are the orthonormality and completeness relations, respectively, ensuring the unitarity of the transformation (11). Equation (12d) is the dynamical condition determining the i-th two-nucleon eigenstate of HSM with eigenenergy ESt. Of course, this is equivalent to demanding that the one-boson state created by (11) be an eigenstate of the boson hamiltonian (8). It is easily seen that (12b) guarantees that the operators (11) satisfy proper boson commutation relations

[BijTMM~, Br+j'T'M'M'T] = •ii' (~ JS" (~TT' (~MM' ~MTM" T" (13)

The main significance of (12c) consists in that it en- ables us to invert (11),

+ D ..i B + (14) BjTM~t~(a b)= ~ ~b,~abaT iJTMMT" i

Using (10), (12), (14) and the properties of the Clebsch- Gordan coefficients, we can express (8) in the form

HB=H~ + HiBnt, (15a)

H~ + - oo [Bisr x B~sr]oo, iJT

(15b) Y = ~ + l ,

B i s r v v ~ = ( - )J + r - M- M~ B ~ j r - M- vT;

Hint__ --JT + B f ] J T [~3 X ~4]JT), E Ew 2 (E< • �9 1234ST (15C)

l =(il J1Tt), 2=(i2 J2 T2), ...

where

WIJ2T34 : w/2T34 _}_(__ 1)J3 + J4-~J -I T 3 q- T 4 ac T v~JT1243, (16 a)


with

W1J2T4:�88 Z Z (--)Je+']'f+K+J3+d4+S§ abcdef J'T'KS

�9 J, J2 J3 J 4 Tt T 2 T3 T4 V.~ 'r'

�9 ( 2 J ' + 1) ( 2 K + 1) ( 2 T ' + 1) ( 2 S + 1)

j f j~ j f 1 1 1 1

D a e D b f D a e D c f i~ i2 'a i4 (16b) �9 ZaeJl T1 )~bfJ2 T2 ZdeJ3 T3 ZcfJ4 T4"

Having determined E~T , i Z,bST from (12), one can see from (15) and (1.6) that the parameters of Hn are completely specified in terms of the microscopic quantities e,, VSrc~. Using (16), it is also easy to verify that HB is hermitian and the interaction term HiB nt is properly symmetrized. It should be emphasized that up to this point no approximations have been made, i.e. as long as the sums in (15) extend over all possible values of the labels l = ( i t J t T O, 2=(i2JzT2), ..., HB is an exact boson image of the original shell-model hamiltonian HsM. We have thus arrived at a microscopically founded hermitian one- plus two-body boson ha- mittonian, which describes the full dynamics of the nucleon system considered in the language of interacting bosons. However, the essential physics we are in- terested in is contained mainly in the description of the low-lying collective states. Guided by the phenomenological success of the IBM, we would like to truncate HB tO the most collective bosons with angular momentum J = 0 and J = 2 in such a way that HB provides no serious coupling between the retained (collective) and the neglected (non-collective) bosons. We turn to this point now.

2.2. Derivation of an IBM-like Hamiltonian

First we specify in more detail the selected degrees of freedom which are intended as candidates for the IBM-bosons. If we arrange the indices i labelling different solutions of (12d) so that Eiar increases with i (for given J and T), the most collective bosons (11) will be those characterized by the smallest i, say i= 1. We then have the following types of bosons with J = 0 or J = 2:

s § ( i=1, J = 0 , T = I ) , T=I

d~=~ ( i = I , J = 2 , T = I ) , (17)

d + (i= 1, J = 2 , T=0). T=0

In the proton-neutron formalism, the isospin triplet s~= a corresponds to proton-proton, neutron-neutron

and proton-neutron bosons + § and + s~ , sw s~v, respectively. Similarly, the operators {d~= 1} denote three types of d-bosons with Mr = - l(d~+~), Mr = + 1 (d~+v), Mr=O(d+~v) and the isospin singlet d~= o represents an independent proton-neutron d-boson d~+~ with Mr = 0. It is interesting to notice that there exists no s+-boson with T= 0, in contrast to d~-=o. It is a consequence of (12a), because for J = 0 one has j~=jp=j (half integer), so that (12a) requires T to be odd. [-We have in mind the case of one major shell only, i.e. that j~ =j0 implies a = hi. In the following, the triplets of indices (ikJk Tk) which characterize the collective bosons (17) will be denoted by Ck, i.e.

ct = (it, Jr, r~); (18)

(it, Jk, Tk)E {(1, 0, 1); (1, 2, 1); (1, 2, 0)}.

The remaining indices which label all the other (non- collective) bosons will be denoted by nt, i.e.

nk=(ik, Jk, T~); (19)

(i~, at, T~)r {(1, o, 1); (1, 2, 1); (1, 2, 0)}.

With this specification we can define the collective boson subspace ~ spanned by the vectors

I{c}>8_-B+ n+ + c~ - c~ . . . Be, , t % , (20)

where N is the number of bosons (nucleon pairs) in the system, ]0)B is the boson vacuum defined by (6) and the symbol {c} stands for the set {q, c2 . . . . , cN}. Similarly, we can define the non-collective subspace N, generated by the vectors

]{n}>B = B + B+ n+ B+,, I%; k _ 1 (21) �9 " " I l k ~ c k + 1 " " "

i.e. N, contains states with at least one non-collective boson. For simplicity of notation, the angular momentum and isospin coupling is suppressed in (20) and (21)�9 The explicit separation of the boson space into the collective and non-collective subspaces en- ables us to decompose the hamiltonian (15) into 3 parts,

_ r4~ol l , r4 . . . . . 1~ ~_ u~o~pl (22) H B - - ~ . B v~,a B i . ~ B ,

where H~ ~ acts on the collective subspace (20), H~ ~176 operates on the non-collective subspace (21) and /v_/~oupt couples the two subspaces. For our purposes only the explicit forms of H~ ~ and H~ ~ will be needed;

H~ ~ = ~~176 + (i"0H~~ (23 a)

00 (~176 = ~, ~ E~, [B + x B j 0 o , (23b) C l


(int)HB ~ = 21- E E [ '~ ' T CLC2C3C4

CLC2C3C4 J T

�9 ([B + x B~,I Jr�9 [/~c3 x /~ j sw) , (23c)

where ~ l = ~ / ( 2 J ~ + l ) ( 2 T q + l ) , E ~ - ~1 - - E J1 T1 ,

H~ ~ -= (H~ 3 + h.c.) + (H22 + h.c.), (24 a)

<3 =1 E Z w2 .. ?11C2C3C 4 J T

�9 ([B + x Bs Jr. [/3c~ x J~C4-1Jz)

Cln2C3r 4 J T

�9 ([B + x B,+] "r. [/3~ x ~jSW), (24b)

H2 2 = 1 E E ~rnJlTz . . . .

nln2c3c4 .IT

�9 ([B. + x BL] "r" [/~, x/~j,r). (24c)

In the following we will write H~ ~ as

n ~ o u . l = x y ; Z w d 5 4 ( [ B ? • B ~ y ~. [~3 • ~ 4 Y ~ ) , 1234 Jr (25)

where the primed symbol ~ ' means that the summa- 1234

tion is restricted to the indices given explicitly in (24 a - c).

Now, we are at the stage which is crucial for our attempt to determine the IBM-bosons. These should be chosen so as to minimize H~ ~ In other words, the interaction between the collective and non-collective subspaces (20) and (21) should be made to vanish (in some approximation) by the choice of the IBM bosons. To this end, we introduce a unitary transformation U [13] which ensures that, up to a certain order, the transformed hamiltonian

~ = UH B U + (26)

leaves invariant the collective subspacc (20). In this case, the operators

B7 = u + B~ + u ;

c=(iJT)~{(1, O, 1); (1, 2, 1); (1, 2, 0)} (27)

can be regarded as creating the IBM-bosons. We write U as

U=e 2, 2 + = - 2 (28)

where 2 is an anti-hermitian, number conserving, rotationally and isospin invariant operator involving

both the collective and non-collective bosons. In this paper, 2 is assumed to have the form of H~ ~

2 - - 1 ~ ' ~ , z~2ra4(I-B( x B;] JT. [/~3 x/~43 Jr) 1234 J T

(29)

[cf. (25), (24)1�9 The unknown coefficients Jr Z1234 are determined by requiring that the transformed hamiltonian (26) contains no terms of the first order in the coupling matrix elements I7~2r34 given explicitly in (24 b, c). Using the formula

= e z H , e - ~ = y~ [ . . . [ U , , 2 ] , . . . , 21 k=0 "

k we get

(30)

~=HcB~ q- H~ ~176176 ~,*.'B - - L"B,rUO 21)+ ..., (31)

so that the condition for eliminating H ~ ~ to the first order in the interaction reads

[ H~ , 2,i = H~ ~ (32)

Substituting (15b), (25) and (29) into (32), we finally obtain

Jr ~Ir4 (33) Z1234 --

E 4 + E 3 - - E a - - E 1

with the indices (1234) implicitly assumed to take only the values from the set

I~ ~--- {(Yl 1 C 2 C 3 C4) , (C 1 tl 2 C 3 C4) , (C 1122 I//3 C4), (Cl C2 C3 n4),

(nl n2 c3 c4), ( q c2 n3 n4)}. (34)

Using (16a, b) and (33), it can easily be verified that

J T J T Z1234=--24321~

which means the antihermiticity of (29), and conse- quently, the unitarity of (28).

Once 2 has been fixed, we can construct the IBM- bosons (27),

B~ + = e - ~ B~ + e ~

=B~ + -- [2, n~ + ] + 1 [2, [2, B~+]] + .... Z. ~

(35)


A straightforward calculation gives

[2 , B~ +]

= Z ' Z ( - ) + +'' + �9

123 J K T S

� 9 1 ~ - 1 ( 2 J + 1)(2 T + 1)

�9 [ B i ~ x [ B ~ x ~3 ]"~] ~o~o, (36)

where L = 2 ~ i - , {iii} is the 6j-symbol and the

sum is restricted in such a way that JT is well de- Z123c fined, i.e. that the quadruplet of indices (123c) belongs to the set (34). Using (36), we can further calculate the multiple commutators in (35), e.g.

[2, [2, B+]]

= ~,' Z ~(a~ K L J R S T ; z) 12345 K L J R S T

�9 IBm- x [[B~ x/~3] K" x [B~ x B5]LS]JT] J~176 (37)

The explicit expression for t~]Ozs45(KLJRST;z) is rather lengthy and will be omitted here. From (35)- (37) one can see that the IBM-bosons B~ + are complicated many-body operators involving infinite expansions in terms of both B +, B~ and B +, B,. Hence a suitable approximation is required in practice�9 In this paper we follow the approximation suggested by Ot- suka and Ginocchio [14]. This approximation consists in

a) retaining only the J = 0, T= 0 term in (37), b) replacing the operator [[B~- x B3] KR X [B2

x B5]~R] ~176 by its expectation value in an appropriate state [~b), i.e.

[2, [2, B+]] ~%Bt, (38 a)

q~= ~ ' Z ~ 3 4 5 ( K K O R R O ; z ) 2345 K R

�9 <qS] [[B~- x B3] KR x [B + x B5]KR] ~176 I~b>. (38b)

Using (38 a), we have

[2, [2, .. �9 [2, B+]. . . ] ="~ gocm B~ +

2m

[2 , [2 , � 9 + ] � 9 1 4 9 m ~ + �9 �9 =q~ [Z, Bc ],

2m~+ 1

so that we can evaluate the sum in (35) to obtain

B + ~ ,=

cosh([/~c) + 1 . ^ + B c - - ~ s m h ( l / ~ ) [ Z , B ~ ] ;

~o~>__0

i sin (~f-~) [2, B + ];

~oc<0�9 (39)

In order to determine ~0 c, we have to specify the state Iq~) appearing in (38b). We take it in the form

1 [~b) = ~ (M +)N 10)B ' (40 a)

where N is the number of bosons (nucleon pairs) and

M+=~flaB+oo; 1-=(i~J1T,), MI=0 , Mr1=0;

1 (40b)

E 1fi112 = 1. (40c) 1

The boson state (40) is an example of the (number- projected) coherent state [15] which has proved to be very useful in treating the many-boson systems [16]. We assume that the system described by (40) is axially symmetric (angular momentum projection M = 0) and contains an equal number of valence protons and neutrons (isospin projection Mr=0). We also note that the indices 1 - ( i l J I T 0 in (40b) run over all admissible values, i.e. M + contains contribu- tions from both collective and non-collective bosons. The coefficients fl~ are determined variationally by minimizing the expectation value of (15) in the state (40). This leads to the following non-linear eigenvalue problem

E, fil + (N- i) ~ ,A,(o),12 f12 = gila, 2

Z l / ~ , I 2 = l , 1

(41 a)

where

= Z Z I - ) J + T r a G 4 34 J T

�9 <dl O J2 0]d O) <J3 O J4 0[J O)

�9 (T~0 T2 01T0> (T30 T401T0) fi3f14 (41 b)


with W1J2T4 given by (16). The solutions ilk(N) of (41) are obtained using standard iterative techniques�9 In terms of ilk(N), we obtain q~ (38b) in the final form

~oc(N)=N ~ ' Z ( - - ) s'+T~+J'+T,+K+R 2 3 4 5 K R

�9 r (KKORR0; z) il2(N) il5 (N)

yo ~ ~ ~ ~ 1/(2/(+1)(2R+1i ' F i 3 ' 4 J3J4 T3T40J2J5 T 2 T s ( ~ 2 ~ - l i ( 2 T ~ 2 ~ ~

1 + ( N - 1) il3 (N) il4(N)

( 2 K + 1)(2R + 1)

�9 <J2OJ3OIKO> < J 4 O J s O > l K O )

�9 (T20 T30IR 0> <T40 T50[R 0)~, (42) )

where we have marked the dependence of r on N. Now, approximating [2, B +] in (39) similarly as (38a), we get from (35)

B~ + ~ r~(N) B +, (43)

where

r ~ ( N ) =

r sinh ( ~ N ) ) ; c o s h ( ] f ~ ) - ~ , )

~o~(N)>0

cos (~-~o~ (N)) - r (N) sin ( V - (N)).

~o~(m) < o

(44)

with

~(N) = N Z ' Z ~ 6 ~ , ( - / ~+ ~+~+ ~ 2 3 JT

1 �9 (2J + 1) (2 T + 1)

(2./2 + 1) (2 T2 + 1) (2J~ + 1) (2 T~ + 1) JT �9 z~23c il2 (N) 133 (N). (45)

The IBM-hamiltonian

HIB M = e- ~ H ~ ~ e 2

then takes the form

H m M ~ d l ( , . ) E ~ : ( N ) [B +q • B~,]oo- oo Cl

+~ Z Z ~'w/~ ~ C1C2C3C 4 \~'] CLC2C3C4 JT

�9 ([B + • ~.~n+qJr �9 [Bc~ • Bc,,]ST), (46)

where

(~)Ec~(N) = rZ~(N) Ec 1

(~) FV s Y (N~ �9 " c 1 c 2 c 3 c 4 \ - - i

= r,x(N) r~(N) rc~ (N) r~,(N) ~JT~,c~c~, �9 (47)

We thus end up with a boson hamiltonian of type (23), but with renormalized parameters�9 The renor- realization comes from taking into account (in an averaged way) the effects of the non-collective bosons. It is worthwhile to notice that our microscopically determined parameters of HmM are number-depen- dent, which is consistent with the results of many other approaches [9], irrespective of the fact that the latter use completely different techniques�9 Having derived HmM in the form (46), we can simply diagonalize it on the boson space (20) to obtain the eigenenergies and eigenvectors. Before doing so, however, the important problem of eliminating unphysical (spurious) boson states must be briefly discussed�9

2.3. Unphysical (Spurious) Boson States

It is well known [17] that the treatment of many- fermion systems in terms of bosons is connected with the occurrence of unphysical (spurious) states. The latter have nothing to do with the underlying fermion system because they arise as a consequence of the overcompleteness of the boson basis with respect to the space available to fermions. Therefore, they must be removed before diagonalization of the boson hamiltonian. To this end, let us consider the following fermion states

[{C})F=Fc + Fc + ... Fc + ]O>v, (48)

where

+ _ 1 2 Dab i Fc + ~- FiJT -- 2 )~abJT [ ~ : X ~ : ] J T , ( 4 9 )

ab

(i, J, T)E {(1, 0, 1); (1, 2, 1); (1, 2, 0)},

Z~bJT are solutions of (12d) and 10>v is the fermion vacuum, ~ [ 0 ) F = 0 . As in (20), the angular momentum and isospin coupling in (48) is suppressed for the sake of simplicity�9 In contrast to the boson states (20), the fermion states (48) are not necessarily orthon- ormal. To construct orthonormalized basis vectors, we diagonalize the norm matrix:

E ~<{c} I {c'}>~ u~% = ~ % , ~c,~ (5o)

U(c ) U{c} (e}

250 R. Kuchta : Interacting Boson-Model

Recall that the symbol {c} represents the set {cl, c2, ..., cN}. Let us denote the zero-eigenvalue solutions of (50) by 0.= 0.o (JV,o =0). Using these, we can define orthonormalized basis vectors as

1 I0.))F = ~ ~ u{r [{C})F, 0.*0.o (51)

which satisfy the orthonormality relations

((0" [ a'))F = 6,~,,. (52)

Next, we construct the orthonormalized boson ana- logues of (51),

this form factor squared at momentum transfer q is given by [18]

4~z 1 '~ " "lri--*ftq)12 - - Z 2 (2,//+ 1)(2 T~+ 1)

�9 l(Yf T/l[ ~(q)IIIJ~ T~)I z (54)

where Z is the atomic number of the target nucleus, the transition operator ~ ( q ) has the second quantized form

~M(q) = e ~ dr ~(e [ JL(q r) YLM( 1 - t3)[fl) ~+ ff~p (55) 0 ~b'

[O-))B = ~ U;} [{C})~, 0-4:0" 0 (53) {c}

where [{c}>B are given by (20). The boson states (53) span the so-called physical

subspace of N~={[{c}>n}, in which every state has its counterpart in the original fermion space. There- fore, in order to obtain physically meaningful results, the IBM hamiltonian (46) must be diagonalized in the basis (53). The resulting eigenvalues produce the energy spectrum and the corresponding eigenvectors can be used to calculate the relevant spectroscopic quantities.

2.4. The Electron Scattering Form Factors

The electromagnetic interaction has long been recog- nized C18, 19J as one of the best tools for studying nuclear structure effects. Especially detailed information on nuclear states can be obtained from an analysis of the electron scattering form factors. A character- istic feature of these form factors is their dependence on both the radial structure of nuclear wavefunctions and an appropriate description of nuclear collectivity. As to the first point (i.e. the radial dependence), it might seem difficult to relate this kind of information to the abstract algebraic space in which the IBM is formulated. This is certainly a problem in a purely phenomenological version of the IBM. However, the explicit radial dependence can be identified quite naturally by resorting to the microscopic description of the bosons in terms of correlated pairs of nucleons with known radial wavefunctions. This is just the case we are dealing with in the present paper.

Our aim is to calculate the charge (Coulomb) electron scattering form factor for a transition from an initial nuclear state characterized by angular momentum Ji and isospin T~ to a final state with Jf and T I. In the plane-wave Born approximation (PWBA)

and the symbol (...[[[ ... [[1...) means that the matrix element is reduced both in the ordinary and isospin spaces�9 In (55), e represents the charge of a nucleon, JL(q r) and YLM stand for the spherical Bessel functions and the spherical harmonics, respectively, and t 3 is the operator of the third isospin component,

t31~>=-�89 t31v>=+�89 (56)

In the present boson formalism, the vectors [JITI), Idi T~) in (54) are taken to be the relevant eigenstates of the IBM-hamiltonian (46). The associated boson image of the transition operator (55) is obtained by first replacing I1~ + IE a by (C~ + 112p)B according to (7), then expressing b~+a in terms of B i J T M M : r + [see (10), (14)], truncating to the collective bosons Bc+~ [see (18)] and finally replacing Bc+~ by Bc+~ [see (43)]. The resulting expression reads

x/~ 1Lr (57a) E~M(q)]mM = 2 z., xT (,)~:Lr_c, c2, ~',(N" q) EBc + c2Jgo, clc2 T

where

(r)'rLT [[V" q) ~c 1 c 2 ~ ,

e

r.(N) l/ (2L +

�9 ~ dr (a[ I jL(qr ) YL[[b) J~l L j a ) l T ~ T � 8 9 0

1 1 (57b) " D a d D b d )(,adJc, Tcl )~adJc~Tc 2 "

Equation (57 a) thus represents an IBM-like transition operator for proton-neutron systems and (57b) determines microscopically its parameters. It is clearly seen from (57) how the two above-mentioned items (radial dependence and nuclear collectivity) enter into the structure of the IBM transition operator. Information concerning the spatial localization of the transition

R. Kuchta : Interacting Boson-Model 251

is completely embodied in the coefficients (r)'rLT ( I V " q) ~ C l C 2 \ ~ ' ,

through the matrix elements (alljL(qr)YL nb>, while information on nuclear collectivity is partly absorbed in the same coefficients (through the quantities r~(N) accounting for the renormalization effects from shell- model states outside the IBM space) and partly in the operator [B~ x / ~ j ~ t r (which is responsible for the angular momentum content of the IBM states). It was shown in [28] that the sensitivity of the final results on the concrete form of the radial nucleon single-particle wavefunctions le> is rather small. It appears that a much more important role is played by the nuclear collectivity effects. In this paper we therefore choose the simplest single-particle basis con- sisting of harmonic oscillator wavefunctions. The associated oscillator length parameter b is given in the next section.

3 . A p p l i c a t i o n t o 2 ~ a n d 2 4 M g

In order to see if our method for deriving microscopically the IBM in the sd-shell nuclei really works, we have applied it to calculate the energy spectra and the electron scattering form factors for the excitation of the first 2+-states in 2~ and 24Mg. The single- particle energies associated with the j-orbits of the sd-valence shell are taken to be [20] (both for protons and neutrons)

8(0 d5/2) = -4 .15 MeV, 8(1&/2)= -3 .28 MeV,

8(0 d3/2) = q- 0.93 MeV.

As for the nucleon-nucleon interaction, we adopt a form which is simple and yet reasonable [21]

V ( i , j ) = Vo 5 (r ,- rj) {1 -r /+r /~( i ) -a( j )}

�9 {1 - # + ~ ~(i). ,(j)}. (58)

The parameters Vo, q and p, as well as the oscillator length parameter b necessary for the evaluation of two-particle matrix elements V~p~o, have been determined so that the shell-model hamiltonian (1) provides the best fit to the experimental low-lying energy spectrum of a given nucleus. In this way we have obtained

a) for ZONe: Vo=68.9(MeVfm3), t/=0.276,

# = 0.490, b = 1.70 (fro),

b) for 2 4 M g : Vo=71.5(MeVfm3), ~/=0.257,

g = 0.511, b = 1.65 (fro).

In Figs. 1 and 2 we compare with experimental data the low-energy collective states of 2~ and 24Mg, respectively, calculated in different approxima-

8

7 - - 0 '

& 6

z 4

'!f ,,>I, - - 2 ~ _ _ _

E•

2ONe - - Z' f - - Z' o ~ _ _ 1++ - - 2' - - - - Z + . . . . . < < ~ 2 ~

O* . . . . . . . 0 ~"

4' .. . . r

--f 2- - - 2 ......

SM IBM BWR 8WI

2 +

Fig. 1. Excitation spectrum of 2~ Experimental values are taken from [24]. SM: shell-model results obtained within the (0ds/2, l s l / z , O d 3 / 2 ) configuration space using the nucleon-nucleon interaction (58). The energies of s.p. levels as well as the parameters of the interaction are specified in the text. IBM: results obtained by diagonalizing the microscopically derived IBM-hamil tonian (46) in the physical boson basis (53). BWR: results coming from the corresponding boson calculation wi thout renormalizat ion of the hamil tonian parameters - (23). BWI: results obtained by truncat ing the hamil tonian (23) to n 7c ( T = 1, M T = - 1) and v v ( r = 1, M T = -t- i) bosons - the boson calculation without isospin invariance

8 f- . . . . . 2' --0" 7 _ _ 0 t . . . . . O' 4t )

/

N 3+- . . . . 3t __ ) / " + ,z., ,- B / . .4- , 2 ) _ _ - - 2 ) . . ,"

oz ~ ~;=:-~::z 4 a . . . . ~++"

2+- . . . . . 2" . . . . . 2t'"

EXP SN l ~ BWR

24 l g

2 t

- -4 " r - - 3 t

- - 2 ~

8WI

Fig. 2. Excitation spectrum of 24Mg. Experimental values are from [25]. Fo r further details see caption to Fig. 1

tion schemes. In column 2 we show the results of an exact shell-model calculation (SM). The levels dis- played in column 3 (IBM) have been obtained by diagonalizing the renormalized IBM hamiltonian (46) in the basis (53). The fourth column contains the results of the corresponding boson calculation without renormalization (BWR), i.e. the hamiltonian (23) has been diagonalized. In order to get an idea on the importance of proton-neutron (re v) bosons in the description of s d-shell nuclei, we have also calculated the spectra with the hamiltonian (23) truncated to contain only rc rc (T= 1, M r = - 1 ) and v v (T= 1, M r = + I) bosons. The results of such a boson calculation


10 '

10 4

lOS

'/_./L

SM '., . . . . IBM . . . . . BWI ' \

...... 8W~

0:5 1:0 1.'5 210 2.5 q ( fm -+)

Fig. 3. Form factors for inelastic electron scattering to the first 2 +-state in 2~ Experimental data are from [26]. Various approximations are explained in the caption to Fig. 1

2+Mg . . . o ~ §

�9 o 0 ; + 2 ~

/ ~ o'

+ . i / + > r - . .

- - ~ . / .,

--,, +o-+ } SM

. . . . IBM lo + ..... ",,. ,/("-

..... B W I ' ,' ' . ," { V

o'.5 1;0 1;5 i.o 215 q (frn -4}

Fig. 4. The same as in Fig. 3 for 2r Experimental data are from [27]

without isospin invariance (BWI) are shown in the last column.

Several conclusions can be drawn from Figs. 1 and 2. First of all, one observes that the shell-model results are in a good agreement with experimental data (at least for the states considered). Since the boson results provide more or less accurate approximations to the shell-model ones, the quality of the former has direct relevance to physical reality. It is further seen that the BWI calculation produces very poor results. Hence, the explicit inclusion of rc v-bosons (isospin invariance) is crucial for the description of the energy spectra in sd-shell nuclei. The BWR calculation re- produces some qualitative features of the spectra, but it is not sufficient to provide a quantitative agreement. On the other hand, the renormalized IBM results are very close to the exact shell-model ones. This indicates that the effects of the non-collective degrees of freedom are essential and that they are properly~ taken into account by means of the transformation e z. Nev- ertheless, it should be noticed that the agreement between the IBM and SM results deteriorates when we move up to higher energies in the spectrum. This fact seems to have its origin in the neglect of higher-order boson terms in the transformed hamiltonian, i.e. in the mean-field approximation involved in our treatment [cf. (37), (38)]. This problem is now being inves- tigated by the author.

In Figs. 3 and 4 we show the electron scattering form factors for the excitation of the 2~--state in Z~ and 2+Mg, respectively, calculated in the same approximations as the energy spectra. A quantitative

agreement with experimental data is now somewhat worse (especially for larger momentum transfers), but the conclusions as to the importance of rcv-bosons and the role of renormalization remain valid. It is thus seen that the IBM-hamiltonian (46) provides a reasonable approximation to the shell-model hamiltonian (1) not only for the description of energy levels but also for the description of electron scattering to the 2~--states in some sd-shell nuclei. It should be emphasized that once the shell-model parameters e~, V~#~+ have been fixed, all the IBM parameters are calculated from them and no further adjustment is performed. In view of this fact, the agreement of the IBM results with the SM ones is indeed remarkable. The discrepancy between the SM and experimental form factors is, of course, out of the scope of the present method. The differences can most probably be traced back to limitations of the SM space. In fact, it has been shown at the phenomenological level [22] that the effects associated with the polarization of the 160 core in sd-shell nuclei play a non-negligible role in the correct reproduction of experimental data. A microscopic understanding of this observation is a very exciting topic which deserves further study. The main effect to be expected is the renormalization of the nucleon charge e to an "effective" value eel r known from the standard phenomenology [23].

4 . C o n c l u s i o n s

In this paper, we have proposed a method for deriving microscopically an extended IBM-like hamiltonian


which would be appropriate to account for the properties of the low-lying collective states in s d-shell nuclei. We have shown how the IBM picture emerges quite naturally from an isospin invariant shell-model hamiltonian by means of the BZM bosonization procedure and two successive unitary transformations. The first transformation isolates the most collective bosons with angular momentum J = 0 and J = 2, the second one minimizes the coupling between these collective and all other (non-collective) boson degrees of freedom. Because of the isospin invariance of the hamiltonian, proton-neutron (~ v) IBM-bosons with T= 1 and T= 0 have been introduced in addition to the ordinary 7z rc and v v bosons. Special care has been devoted to the elimination of unphysical (spurious) boson states arising from violation of the Pauli principle in the boson space.

The microscopically derived IBM hamiltonian has been applied to calculating the low-energy spectra and the electron scattering form factors for the excitation of the 2~ states in 2~ and 24Mg. The inclusion of rc v-bosons, as well as the renormalization of boson operators due to the neglected non-collective degrees of freedom, have turned out to be of crucial importance with respect to a reasonable reproduction of experimental data. The present method seems to be suitable for a microscopic analysis of the core polarization effects in sd-shell nuclei. Work on this subject is now in progress.

Fruitful discussions with Dr. M. Gmitro, who has attracted the author's attention to the study of electron scattering form factors, are gratefully acknowledged.

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R. Kuchta Laboratory of Theoretical Physics Joint Institut for Nuclear Research - JINR Dubna P.O. Box 79 SU-10 t000 Moscow USSR

microscopic foundation of the interacting boson model in thes d-shell nuclei

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