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Interacting boson model of collective states I. The vibrational limit
ANNALS
OF PHYSICS
99, 253-317 (1976)
interacting
Boson Model of Collective I. The Vibrational Limit
States
A. ARIMA Department
of Physics,
University
of Tokyo,
Tokyo,
Japan
AND
F. IACHELLO Kernfysisch
Versneller Instituut, University Groningen, The Netherlands
of Groningen,
Received March 2, 1976
We propose a unified description of collective nuclear states in terms interacting bosom. We show that within this model both the vibrational tional limit can be recovered. We study in detail the vibrational limit and to the possible existence of an unbroken SU(5) 3 O+(5) symmetry. We set of analytic relations for energies and electromagnetic transitions.
of a system of and the rotabring attention derive a large
1. INTRODUCTION In contrast with other many-body systems, the nuclear collective motion is characterized by vibrational and rotational frequencies of comparable order of magnitude, preventing a clear-cut distinction between the two types of motion. Moreover, other intermediate situations can occur in nuclei, corresponding for instance to asymmetric rotations and/or spectra which are neither vibrational nor rotational. In order to accomodate these facts we have recently proposed [l] a unified description of the collective nuclear motion in terms of a system of interacting bosom. Boson representations of the collective motion are not new. Following the suggestion of Belyaev and Zelevinski [2] and of Marumori et al. [3] many elaborate calculations have been performed and we refer the reader to somerecent papers by Sorensen [4], Kishimoto and Tamura [5], Marshalek [6], and Holzwarth and Lie [7] for a review of this work. Our approach is different from the above mentioned authors in that we do not attempt, at this stage, a derivation of the boson 253 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
254
IACHELLO
AND
ARIMA
Hamiltonian from the fermion basis, although some work in this direction has been done by one of us (F.I.) in collaboration with Feshbach [8]. Rather we are interested in (i) providing a unified description of collective nuclear states irrespective of their detailed nature (i.e., vibrational, rotational, etc.) and (ii) producing simple limiting situations which can be solved analytically. In relation to these simple situations, our main purpose is to point out the possible existence of unbroken boson symmetries (and to classify nuclei according to representations of the associated symmetry groups). By exploiting their group structure is then possible to obtain analytic relations to be tested experimentally. One of the main advantages of the model of [l] is that, under certain conditions to be described below, it can be shown to have analytic solutions corresponding to the vibrational and axially symmetric rotational limit. These limits are always defined with respect to the classical description of the collective quadrupole motion given by Bohr and Mottelson [9] which forms the basic framework for a discussion of all nuclear collective properties. The second advantage of the model described in [l] is that it reproduces, after slight rearrangement, the Hamiltonian derived by Janssen, Jolos, and Diinau [lo] using the Lie algebra of pair operators. We believe that [lo] is of considerable importance in the understanding of a microscopic theory of the collective nuclear motion since it produces a Hamiltonian of finite order in the boson operators and which, in addition, includes the effects of the Pauli principle through the cut-off factor. In the present paper, after a short review of the basic idea, we will consider in detail the vibrational limit. By exploiting the SU(5) I> O+(5) group structure of this limit and using standard group theoretical methods we will derive sets of analytic relations for energies and transition rates. In addition to the quadrupole excitation mode we will also discuss other excitation modes, in particular the collective octupole mode and two-quasiparticle modes. Again, under certain restrictive assumptions, we will derive analytic expressions for energies and transition rates. These relations will not be confined to the electric quadrupole (E2) operator but will include the Ml, EO,El, M2, and E3 operators. Contrary to what one might expect, these operators appear to be equally well described in terms of boson operators. This is somewhat surprising since, in contrast with the E2 case, the induced transitions are not enhanced but in fact largely retarded. Finally we will consider a small breaking of the SU(5) 1 O+(5) symmetry and derive expressions for some observable quantities in first-order perturbation theory. These relations are useful in practice, since a small breaking of the symmetry, even in the best of the cases, will always be present. A discussion of the boson SU(3) group and the sets of analytic relations which can be derived from its group structure in the axially symmetric rotational limit, will form the subject of a subsequent paper.
INTERACTING
2. THE INTERACTING
255
BOSON MODEL
BOSON MODEL:
QUADRUPOLE
STATES
In constructing a simple model of the collective nuclear motion we start by realizing the quadrupole (L = 2) character of the excitations. We also note that, because of the limited number of particles in each self-consistent shell, there will be a cut-off in the vibrational and rotational bands built on each shell. A model which has these properties has been proposed in [l]. Here a number of positive parity states are constructed as states of a system of N bosons having no intrinsic spin but able to occupy two levels, a ground state level with angular momentum L = 0 and an excited level with angular momentum L = 2, as shown in Fig. 1. For zero splitting between the two levels and in the absence of any interaction between the bosons, the five components of the L = 2 state, called d for convenience, and the single component of the L = 0 state, called s, span a linear vector space which provides a basis for the representation of the group SCJ(6). These representations are characterized by the symmetry properties of the wavefunctions. For bosons the only allowed representations are the totally symmetric ones, belonging to the partition [N] of SU(6). In the absenceof interaction and for zero splitting, all states belonging to [N] are degenerate. The energy difference E = cd - Edand the boson-boson interaction lift the degeneracy and give rise to a definite spectrum. The spectrum is defined by E, by the two-body matrix elements (d2L 1 V 1d2L). (L = 0, 2,4), (d20 I I’/ s20), (ds 2 I V I ds2), (ds 2 1 V 1d22), (s20 / V 1s20) and by the partition [N] of SU(6) to which it belongs. Thus in our approach, each nucleus is characterized by a set of nine parameters and the energy levels can be found by diagonalizing the model Hamiltonian H = E,S+S+ cdC dvn+d,,, +
c $(2L + 1)1/2cl[(d+d+)‘l’ (dd)(O](O) I. + Z’e[(d’d’)k (&)‘2’ + ;;:;+;M (dd)(2)](0) + (l/21/2) fio[(d+d+)‘o’(B)(O) + (s+s+)(~) (dd)(“‘](o’ + u2[(d+s+)‘2’ (ds)(2)](o)+ $u~[(s+s+)(~) (s,s)(~)](~)
(2.1)
where d+(d), S+(S)are the creation (annihilation operators for bosonsin the L = 2, L = 0 state and the parentheses denote angular momentum couplings. The ”
‘d
.I
”
d
L=2
s
L=O
t
E ES
b,,,,
FIG. 1. The configuration&PinthebosonSU(6)model.
256
IACHELLO
AND
ARIMA
parameters CL, CL, uL have been introduced for convenience and are related to the two-body matrix elements by cL = E2 = 5, = u2 = ug =
(d2L j V 1d2L) (ds 2 I V 1d22)(5/2)1/2 (d20 1 V 1s”O>(I/~)~~~ (ds 2 1 V 1ds 2)51/2 (s20 1 Y j s20).
(2.2)
The numerical coefficients appearing in Eq. (2.2) are the inverse of the matrix elements of the operators in square-brackets in Eq. (2.1) between two-boson states. To the extent that Eq. (2.1) describes collective quadrupole states, there are associated transition operators. The electric quadrupole operator is defined in terms of the two reduced matrix elements (d I/ Q iid) and (d/I Q 11S) and given by Ty’
= q”,[(dts)t’
+ (std)f’]
+ q,‘(dQ@.
(2.3)
All quadrupole transitions are thus characterized by two number which have been written for convenience as
= @II Q IIsW5Y” 92’ = (dll Q II 4. q2
(2.4)
Again the numerical coefficients in Eq. (2.4) are the inverse of the matrix elements of the operators in bracket in Eq. (2.3) between one boson states. It is interesting to note that Eqs. (2.1) and (2.3) reduce to the Hamiltonian and quadrupole operator derived by Janssen, Jolos, and Diinau [IO] using the Lie algebra of pair operators. Consider in fact the basis states 1snsdnd[N]xLM> where x is whatever quantum number is needed to specify uniquely the states, and n, = N - ngj. Operating with H on these states one can replace the 9 and s operators by c-number functions of nd H j sN+dnd[N] =
I
xLM)
E,(N - nd) + cdtzd + 1 Q(2L + 1Y2 c&dtd’)‘“’
(dd)‘=‘l’“’
L
+ u,(N - n,J n8/51f2 + u. i(N - nJ(N
- nd - 1) I sNpnd
1
+ d,{[(d+&)@’ d](O) (N - n,)li2 [ sN-nd-‘d”d[N + (N - nd + I)li2 [dt(dd)c2)](‘)
““EN1xLM>
- 11 xLM)
1sNendfldnd[N + 11 xLM)}
+ $o{(dtdt)(OJ ($(N - n,)(N - nd - I))“”
[ sN--nd-2dnd[N - 21 xLM)
+ (&(N - nd + l)(N - nd + 2))lj2 (dd)“’
1sNwnd’“dBd[N - 21 xLM)).
(2.5)
INTERACTING
Eliminating to give
BOSON
257
MODEL
the degrees of freedom of the s-boson one can rearrange this equation
H 1d’““XLM)
= p* + E’& + c 3(2L + 1)1/2 c,‘[(d+d+)‘l’
@d)(L)](O)
L
+ 02’[[(d+d+)(2) d](O) (N - nd)l12 + (N - nd + l)l/% [d+(dd)‘2’]‘o’]
+ vo’[(d+d+)‘o’((N - n,)(N - lid - l))‘/”
(ddpl! I d”dXLM)
+ ((N - lzd + l)(N - ?zd+ 2))“2
(2.6)
where ho = E,N + uogv(N - 1) E’ = E + ((u2/W2) - u&N - 1) CL ’ =
CL
+
(u,
-
(u2/W))
(2.7)
us’ u = 212 ilo’ = zo/21/2 which is identical to [lo, Eq. (13)]. Similarly, operating with TiF2’ on the basis states j PG?“~[N]xL.M~ one has T(E~) k
is N--nddnd[N] =
q2{dk+(N
XLM) -
np2
+ (IV - nd f + qJdtd)p’
j s
N--nd-ld”qv - l] XLM)
1)112(-)” / sN-“ddnd[N]
d-, 1 Pnd+ldnd[N
+ l] xLM))
xLM).
(2.8)
Eliminating again the degrees of freedom of the s-boson, one can rewrite this equation as Tp2’ j d’QLM) = {&[dkt(iv
(2.9) -
/I,$/~ + (A’ - nd + l)lj2 (-)”
d-k] + q,‘(d+d):‘}
1 dndxLM\.
We now show that for different choices of the parameters E,... the Hamiltonian and transition operators of Eqs. (2.1) and (2.3) produce both vibrational and rotational spectra. As these parameters change the SU(6) model spans the entire veriety of observed spectra. We begin by considering the case in which the energy difference E is much larger than the interaction terms cL , 6, , uL . In this casethe Hamiltonian is invariant under separate transformations among the five components of the L = 2 state. Thus the statesare characterized by the number of bosons occupying the L = 2 level, nd, and an (approximately) unbroken symmetry
258
IACHELLO
AND ARIMA
emerges from the decomposition SU(6) 1 SU(5) @ U(1). The quantum number n, plays no role in this case. The representations of SU(5) contained in [N] are all the symmetric representations [nd = 01, [nd = 11, [nd = 21 up to [IZ~ = Iv]; This is the limit which will be discussed in detail in this paper. The other limiting situation occurs when the energy, E is small and of the same order of magnitude of the two-body matrix elements cL , 6, , uL . In particular if both the energies E, , cd and the two-body matrix elements correspond to those of a quadrupole-quadrupole interaction Y = -K Ci,j Qi ’ Qj in a major oscillator shell, where K is the strength of the interaction and Qi the quadrupole moment of the ith boson, another approximate symmetry occurs [Ill. The related wavefunctions serve now as a representation space for the groups SU(6) r) SU(3) r) O+(3) and they are characterized by the quantum numbers ][N](X, p) KLM) where (h, CL)label the representations of SU(3) belonging to the partition [N] of SU(6). In this limit the energy levels are given by
WW,
pL)KLW = 4%-W + 1) - CC&p>l
(2.10)
where C(h, p) is the Casimir operator of SU(3), C(h, CL)= h2 + pz + hp + 3(h + p). The properties of the boson W(3) limit, which describes the axially symmetric rotor, will form the subject of a subsequent paper.
3. THE VIBRATIONAL LIMIT:
STATES OF THE dna CONFIGURATION
In the vibrational limit, states are characterized by the number of bosons occupying the L = 2 state, n&d nd configurations). As mentioned above the quantum number n, is not needed in this case and it will be omitted from here on. As it is well known [12], states of nd d-bosons form the basis for the totally sym metric irreducible representations of the group SU(5) and of its subgroup O+(5). Five quantum numbers are needed to classify uniquely the states [13]. Three of them are trivial, the d-boson number nd , the angular momentum L and its third component M. The fourth is the boson seniority Y. Instead of v one can introduce another quantum number no which counts boson pairs coupled to zero angular momentum. n, is related to u by v = nd - 2n, .
(3.1)
Finally one can introduce a fifth quantum number nd which counts boson triplets coupled to zero angular momentum. The total number na is partitioned by na and nA as nd = 2nB + 3nA + h (3.2)
INTERACTING
BOSON
and the values of the total angular momentum of h by
259
MODEL
for each nd , n, , ~2~are given in terms
L = A, x + 1, h + 2)...) 2h - 2, 2h
(3.3)
where the conspicuous absence of L = 2h - 1 should be noted. The vibrational limit in the conventional sense of the word, is described, in the boson represen‘tation, by the Bohr Hamiltonian [9] H = Ec d,+d,,,
(3.4)
111
and transition
operator TLE2) = q2 (dk + + (-)” h
d-,)
(3.5)
where the boson number nd can take on any value 0, 1, 2,..., co. This Hamiltonian and transition operator can be derived from the SU(6) model of the previous section in the following way. First rewrite Eqs. (2.6) and (2.9) as H = EC drn+dv,+ C &(2L + 1)lj2 cL[(d+d+)‘L’(dd)(L)](o) +w~~2{[(d+d+)(2)~](oJ (1 - (r~,/N))l/~ + (1 - ((nd - 1)/N))1/2 [d+(dd)‘2’]‘0’} + uoKd+d+)(0) ((1 - (ndW(l + ((1 - ((n, - l>/NMl
- ((nd + l)lNWi”
- (h
- 2)lW)Y12 WY09
(3.6)
and ,p,
= qJd+(l - (t~d/N))l’~ + (1 - ((nd - l)/N))l’”
(-)”
de,] + q,‘(d+d)‘,)
(3.7)
where the prime has been dropped on E and cL , ho = 0, 1’2= v2’(N)lj2, v. = zlo’N and q2 = g2(N)l12. Now let N -+ co (no cut-off) and neglect all terms in Eqs. (3.6) and (3.7) except the first one. This reproduces Eqs. (3.4) and (3.5) respectively. Nuclei away from closed-shells and regions of large deformations are indeed often characterized by vibrational spectra [14] but with large splittings between members of the two-phonon (nd = 2) triplet O+, 2+, 4+, relatively large quadrupole moments of the first excited 2,+ state and rather small values of the transition matrix elements for the forbidden An, = 2 transitions. Motivated by this observation, we prefer [15] to introduce another vibrational limit described by the Hamiltonian H = Ec dm+dln+ c 4(2L+ 1)lj2 cL[(d+d+)tL’(dd)(L)](o) (3.8) nt
L
and by the transition operator T;F’ = q2(dk++ (-)”
hk) + q2’(d+d$?.
(3.9)
260
IACHELLO
AND ARlMA
This limit can be obtained from Eqs. (3.6) and (3.7) by neglecting the d-bosonnumber changing terms v2 and v0 and by letting N -+ 00. In this limit the observed large anharmonicities are caused by the boson-boson interaction, the quadrupole moment of the first excited 2,+ state is due to the second term in Eq. (3.9) and the dnd = 2 transitions are strictly forbidden. This model, which we call the interacting d-boson model, is related to other models previously discussed in the literature. In particular it is identical to that of Das, Dreizler, and Klein [16] if one drops the terms d, , czl , a, in [16, Eqs. (8) and (9)] and to that of Brink, Kerman and de Toledo Piza [17] if one assumes that the d-boson-number changing terms are small from the beginning and therefore no unitarity transformation is needed to bring the Hamiltonian in the form of Eq. (3.8). This is a necessary phenomenological condition since otherwise the terms eliminated from the Hamiltonian will reappear in the transition operator thus giving rise to An, = 2 transitions which are observed to be small. The advantage of the d-boson limit is that the additional terms in Eq. (3.8) split but do not admix the different representations of SU(5) r) O+(5). Thus using standard group theoretical methods we are able to obtain the eigenvalues and transition rates in analytic form. It is this symmetry (also proposed in a slightly less general form by Ferreiro, Alcaras, and Navarro [18]) and its experimental verification that we want to emphasize.
4. SOLUTION OF THE d-BosoN MODEL:
THE BOSON QUASI-SPIN
GROUP
In this section we discuss an analytic solution to the eigenvalue problem H 1ndvnALM) = E 1ndvnALM)
(4.1)
where His given by Eq. (3.8), and nd , v, nA , L, A4 are the five quantum numbers which label the totally symmetric irreducible representations of SU(5). The method of solution is due to Racah l-191. In the course of our discussion we will also review some known properties of the boson quasispin group SU(1, 1) which is used in the solution. We begin by noting that two d-bosom can only have three values of the total angular momentum L = 0,2 and 4. Therefore three coefficients c,, , c2, c, completely describe the boson-boson interaction. Tnstead of the second quantized form of Eq. (3.8) we may use the first quantized form. Since there are only three coefficients which describe it, we rewrite the interaction Vij between the ith and jth boson in terms of three operators, which we choose as the unit operator Iif, the pairing operator Bij and the operator L?~~= 21i . Ii , where Ii , lj are the angular momenta of the ith and jth boson (4.2)
INTERACTING
BOSON
MODEL
261
The parameters (Y,/3, y are related to the parameters c0 , ca , c, by the relations Eq. (4.22) below. The expectation value of the operator I = c lij
(4.3)
i
in the state j ndunJA4)
is trivially given by (1) = g?&d - 1).
(4.4)
9 = 1 Pij
(4.5)
That of the operator i
is also easily obtained, since 9 is related to the total angular momentum Lz=p;+z.
L = xi Ii (4.6)
Thus the expectation value of 9 is (8)
= L(L + 1) - 6n, .
(4.7)
In order to find the expectation value of the pairing operator we introduce three operators S, = 4 C (-)” m
dmtd:,
S- = 4 C (-)” ?n
d,d-,
(4.8)
So = $ C (dm+dm + d,d,+). m
These operators are the generators of the boson quasi-spin group SU(1, 1) [20, 211. This group is noncompact contrary to the fermion case in which the quasispin group is the compact SU(2). Using the commutation relations of the boson operators d+(d) it is easy to show that [S+ ) s-1 = -2s,
Kl ?S+l = s+ [S, ) s-1 = -s-
(4.9) .
In Cartesian coordinates s, = &.(S+ + s-) s, = (1/2i)(S+ - SJ s, = s, 595/99/2-3
(4.10)
262
IACHELLO
the commutation
AND
ARIMA
relations are [S, , S,] = -8,
[S, , &I = is, LX , &I = is, to be compared with ordinary W(2) group
the commutation
relations
(4.11)
among the generators
L-L, -&I = i-G [L, , Ll = i.L
of the
(4.12)
[L, , L,] = iL, . Similarly the Casimir operator of SU(1, 1) can be written
as
c = $2 - S,” - s,2 = s,2 - (S+S- + S,) = s,2 - s, - s+sto be compared with the Casimir operator
(4.13)
of SU(2)
c = L,2 + L,? + L,2
(4.14)
The eigenvalues of the pairing operator .P now can be easily found since 9 = 4&S-
(4.15)
and therefore B = 4[S&s, -
1) - S(S -
I)]
(4.16)
where we have used the fact that c = S(S -
1).
(4.17)
The eigenvalues of S,, can be obtained from its definition Eq. (4.8). By commuting the last two operators we have s, = (l/2) c a,+LL + (l/4) c 1 = (%/2) + 5/4. ?n m
(4.18)
The eigenvalues of S can be obtained in a similar fashion [21]. They are given by s = (U/2) + 5/4
(4.19)
Inserting Eqs. (4.18) and (4.19) in Eq. (4.16) we obtain (9)
= (Hd - v>(n, + 2’ + 3)
(4.20)
INTERACTING
and the eigenvalues of the interacting
BOSON
263
MODEL
d-boson Hamiltonian
E(nd, 0, HA, L, M) = End + &7&,
- 1) + &,
[15,22]
- v)(n, + v + 3)
+ y[L(L + 1) - 64 which do not depend on nd and M. The parameters c2 , cq by considering states with nd = 2. One has
(4.21) 01,/3, y can be related to cO ,
cp = CY+ 8y c, == cx- 6y
(4.22)
ro = a + lop - 12y and the inverse relations 01= (l/14)(&,
+ 8c,)
P = (3/7(J) Ca- (l/7) cz + (l/IO) co
(4.23)
Y = (l/14)($ - ce). The matrix elementsof the operators I, 9, LY have also been obtained in a slightly different way by Kishimoto and Tamura [5]. In addition to providing the eigenvalues of the pairing interaction 9, the group reduction SU,(l, 1) @ O,+(3) is also useful in other respects. For later use we list some of its properties [21]. All states with same o but different nd can be generated from the maximum seniority states by means of the relation I S, S, , nd , L, M\ = ( I/JV(S, Sz))(S.k)sz-s/ S, S, = S, nd , L, M)
(4.24)
where the normalization factor is obtained from the recurrence relation Jv(S, S,) = (S, - S)(S, + s - 1) J-(S, s, - 1)
(4.25)
Jv”(S, s, = S) = 1
(4.26)
with Thus 1s, sz , ?lA , L, M)
= ( (S,- &I$
- I)! !
l/2
(S+)sz-s I S, S, = S, nd , L, Mj.
(4.27)
Replacing S and S, by Eqs. (4.18) and (4.19) we have 112 nv + Pm (l/Z) h-o) I nd , 4 11 A ’ Ly M, = ((hd - u)/2)! T(v + (S/2) + (nd - e)/2) 1 @+I X j nd = c, c, nA , L, M) (4.28)
264
IACHELLO
AND
ARlMA
where r(n + 1) = n!. The matrix elements of & in the seniority representation can be obtained from those in the quasispin representation a &’ I s+ I s, &> = ~,~,Ss,,((~z + S)(S, - s + lW2
= &s,~,S,-1KSz - SK% + s - l))‘i”.
(4.29)
They are
6.,f,n,+2W2)
+ ((nd+ WW
+ Knd- WWi2
+ ((nd + W)Mnd - d2)W
(4.30)
The interacting d-boson Hamiltonian generates spectra, Eq. (4.21) with strong regularities. An example is shown in Fig. 2. For reasons which will become apparent in the following Section 5 we have arranged the different states into “bands.” We have named the important bands Y, X, Z, X’, Z’, p, d and defined them as follows Y-band
/ nd , nd , 0, L = 2nd 2 m
X-band
1nd , nd , 0, L = 2nd - 2, M)
Z-band
j nd, nd, 0, L = 2nd- 3, M)
r-band
1nd , nd, 0, L = 2nd- 4, M)
Z’-band
/ nd , nd , 0, L = 2nd- 5, M)
(4.31)
1nd , nd - 2,0, L = 2nd- 4, M) 1nd, nd, 1, L = 2nd- 6, M).
p-band d-band
Some of these bands are shown in Fig. 2. Our classification coincides to some extent with that of Sakai [23] who following a suggestion by Sheline [24] introduced the concept of quasi-band in vibrational nuclei. For later purposes, it is interesting to plot the energy differences dE = E(n, + 1) - E(n,) as given by Eq. (4.21) for the various bands (Figs. 3). Here the different bands appear as a set of parallel linear trajectories, with equal slope c4 . The corresponding equations can easily be derived from Eq. (4.21). For the important bands they are
A& AE, AE, AE,
= = = =
E+ E+ E+ E+
cqnd cqnd- 8y cand- 12y cqnd+ 48 - 16~
nd 3 nd 3 nd 3 nd >
0 2
3 4.
(4.32)
A 3-i
‘i
i
y
p I
g&y
>-
,Ol
4+*,ol r9(4.4,01 5+&4,0) ,+J43(4.4;0)
r” w 2-
;I+&,01
2”4L4,1,
g&3,0) ++B.O) 4+&?,o)
z+fi
3+(3.3,01
,Ol
o+(3.3,1) ,+~,01
2+m,o)
I-
+ (4.0,O) o-
*t&O)
0 1,+(0,0,01 FIG. 2. A typical spectrum in the d-boson limit. The numbers in parenthesis are the SU(5) quantum numbers (na , u, Q). The angular momentum quantum number is explicitly written to the left of each level. The parameters used are 6 = 579 keV, c, = 39.4 keV, ca = -95.3 keV, co = -27.4 keV. 1000
r
750
soo5 0
2
4
8
8 “d
FIG. 3. The energy differences AE = E(n, + 1) - E(nd) given by Eq. (4.3i) as a function of nd . The parameters d, c,, , c, , cd are the same as in Fig. 2.
266
IACHELLO
AND
ARIMA Y
FIG 4. The energy E as a function (a) c, < c2 < c,, ; (b) c, > c2 > c, .
of the angular momentum
L for the various bands.
In Fig. 4 we plot the energy E as a function of the angular momentum L for the various bands. In case (a) cq < c2 < c, the bands never meet. In cast (b) c, > c2 > cOthe bands start crossing for a certain value L,, of the angular momentum L, becoming in turn yrast bands. In the interacting d-boson model all energies are expressed in terms of four parameters E, c, , c2 , c4 which can be determined by a fit to the observed energies. Moreover, the energies of the members of the Y-band depend only on E and cp , while the X, Z, X’, Z’ bands depend only on E, cq and c2 . Thus having determined E and cq by a fit to the Y-band, one can then use the X and Z bands to extract the third parameter c2. A knowledge of some members of the p-band is however necessary to obtain c,, . Incidentally we note that the energies of the members of the Y-band can be rewritten as E(nd , nd , 0, L = 2nd, M) = i(4e - 3~~)L + &L(L
+ 1)
(4.33)
which is the formula proposed by Ejiri [25] in his fits to the ground state bands of rotational and transitional nuclei.
INTERACTING 5. ELECTRIC
267
BOSON MODEL
QUADRUPOLE
TRANSITIONS
Another advantage of the interacting d-boson model is that the matrix elements of the electric quadrupole operator, Eq. (3.9), can be obtained in compact form. Since the boson-boson interaction does not admix states of different boson number, the results of this section apply to the harmonic quadrupole vibrator as well. The reduced matrix elements of the E2 operator of Eq. (3.9) can be obtained numerically by means of the usual coefficient of fractional parentage technique. Here (na + 1, x’, I’ /I d+ I/ /?a , x, 1) = (nd + l)i” il[~P(xI)
dI’ I} d”d+‘,y’I’]
(5.1)
and 0% , x, III dll nd + 1, x’, 0
= (-)“-‘
+ 1, x’, I’ II d+ II nd , x, 0.
(5.2)
In these equations x is whatever quantum number is needed to specify uniquely the states, f = (21+ l)lj2 and the bracketed symbol is a boson cfp. Similarly one can construct the reduced matrix elements of the operator U;’
= ( 1/5”2)(d+d)l”’
(5.3)
as (nd , x, 111U’) Ij nd , x’, I’> = nd ~J(-)r+Jii’[d”“-l(tcJ)
dZ I>dndxI]
. [d”d-l(tcJ) dl’ I} d”“x’I’] 1; ‘J ‘;I
(5.4)
where the symbol in curly bracket is a Wigner 6 - j coefficient. Since tables of boson cfp are available [26] the numerical calculation of the reduced matrix elements can be easily carried on. However, we are interested in obtaining analytic expressions for these matrix elements and we will therefore describe in this section an alternative approach to obtain matrix elements in algebraic form. We restrict ourselves to the matrix elements of the d+(d) operators since those of the U2) operator can be obtained in a similar fashion. We begin with matrix elements among states of maximum seniority. The basis states can be written as 1nd , n, = 0, nA = 0; L, M) = A$,,,
C (2,2n, - 2, m, M, j L, M) d,+ 77040
x 1nd - 1, n, = 0, nA = 0; 2nd - 2, M,,).
(5.5)
268
IACHELLO
AND
ARIMA
From here up to Eq. (5.16) we will drop the unnecessary indices ns , nd . The normalization constant in Eq. (5.5) is determined from the condition
1 = Jci.L ,,C,
M P,2nd-22,
m’, MO’ I L, M)(2, 2nd - 2, m, MO I LM)
* #J*1. 0’
*(r&j--
1;2nd-2,M,‘jd,,d,+Ind-
1;2n,-2,M,)
which can be evaluated by using the boson commutation
(5.6) relations
d,,,jd,+ = dmtd,l + a,,,
the Wigner-Eckart
(5.7)
theorem, and the basic relation + 1)li2
(nd ; L = 2nd 11d+ 11nd - 1; L’ = 2n, - 2) = (r1#/~(2L
(5.8)
The result is A”
-
nd.L -
112 l)(4na - 3) W(2n, - 2, L, 2nd - 4,2nd - 2; 2,2) ) . (5.9)
1
t 1 + (-)”
(nd -
Here and in the following formulas we have preferred to use the Racah coefficient W(f, , I2 , 11’, I,‘; J, k) = (-)z1+1~+z1’+z2’ i, i
f,
1
k”l
(5.10)
rather than the 6 - j symbol, because of notational convenience. Incidentally, one recovers the result mentioned in Section 3, that L = 2nd - 1 is not allowed, since in this case W = l/(nd - 1)(4n, - 3) and Jy;ld,L --+ co. In order to calculate the reduced matrix elements of the transition operator we write
(nd+ l;L’,M’IdmtInd;L,M)
* (2, 2nd - 2, m”, Mi j L, M) x (nd 3.2n, 3 MO’ I d,,d m‘d;” 1nd-
1;2n a - 2,
Ml).
(5.11) ,
Then making use of the relation (5.12)
INTERACTING
of the Wigner-Eckart
BOSON
269
MODEL
theorem and of the relations (5.8) and (5.9) we obtain
= (2L’ + 1)1/S (n&1/2 [(-)“’
S,,,,, + (-)L ((2L + 1)(4n, + l))“/”
x W(L, 2~ - 2, L’, 2n, ; 2, 2)(1 + (-)”
(nd - 1)(4n, - 3)
x W(2nd - 2, L, 2nd - 4, 2nd - 2; 2,2))] /(l + (-)“’ x (1 + (-)”
nd(4n, + I) W(2n,,
L’, 2nd - 2, 2nd ; 2, 2))1/2
1)(4n, - 3) W(2n, - 2, L, 2nd - 4,2n, - 2; 2,2))‘/“.
(nd -
(5.13)
The corresponding matrix elements of d can be obtained from the phase relation (5.2). Eq. (5.13) can be specialized to the various cases. Denoting by F the reduced matrix elements of d
F= (nd;LIIdllnd+
1,L’)
(5.14)
we have for transitions within the bands of Eq. (4.31) Y-band
L’ = 2n, + 2 + L = 2n,
F= h
X-band
L’ = 2nd
F=
Z-band
L’ = 2nd -
-+L=2n,-2
1 -+ L = 2n, - 3
+ 1)(4nd + 5)Y
+ 3)(4n, + 1) lj2 > (4nd - 1) dnct - 2)(2n, + 1)(4n, - 1) l/2 (nd - l)Ch - 1) (5.15) (nd -
F =
1X%
and for intraband transitions X+Y
L’=2n,
Z+Y
L’=2nd-
Z-+X
L’=2nd-l+L=2n,-1
(4n, + 2)(4nd + 1) v2 (4nd - 1)
-+L = 2n, l+L=2n,
F= (
t2% - 2)(4nd + ‘> l” (2nd - 1)
(5 16)
v2 ’
These expressions are summarized in Fig. 5. The B(E2) values can be obtained from Eqs. (5.15) and (5.16) in the usual way B(E2 Ii -+ I,) = (l/(21, + 1)) \(I-- I/ TcE2’ I/ Ii>12.
(5.17)
The “bands” defined in the previous section appear then to be connected by transitions with large B(E2) values. For each band, except the Y-band, an increasing
270
IACHELLO
AND
ARIMA
L-ad-3
FIG.
5. Reduced matrix elements of d between states of maximum
seniority.
fraction of the decay goes into the Y-band as the spin decreases. However, for large spin values, the lateral bands, X, Z,... are very stable, in the sense that the major part of their decay proceeds within the band. The fraction which branches out behaves as I/Q (Fig. 5). Thus for large spins, an anharmonic vibrator whose anharmonicity is only due to the boson-boson interaction gives rise to a set of “channels.” These channels do not “communicate” to each other until low enough values of the spins are reached. Analytic expressions for the matrix elements of the transition operator can also be obtained for members of the p-band. It is convenient here to make use of the boson quasi-spin group SU(1, 1). The basis states of the p-band can be written as /nd,nB=
1,n,=O;L=2nd-4,M)
= (2/(2n, + 1))lj2 S+ 1nd - 2, nB = 0, nA = 0; L = 2nd - 4, M).
(5.18)
In order to calculate the reduced matrix elements of d+(d) between states of the form (5.18) we observe that d+ commutes with S, [d,+, S+] = 0.
(5.19)
Using this relation, the first of Eq. (4.9) and the phase relation Eq. (5.2) we then obtain the reduced matrix elements of d for transitions within the B-band
@band
L’ = 2n, - 2 + L = 2,~~- 4 F =
end- l):22nnd,++:):4nd - 3) )1’2a (5.20)
INTERACTING
The intraband
BOSON
271
MODEL
matrix elements can be obtained making use of the relation [S- , d,+] = (-)”
(5.21)
d-,
and the phase relation Eq. (5.2). They are given by F =
B-Y
L’=2nd-2+L=2n,
8-X
L’=2n,-2+L=22n,-2
F=
B-Z
L’=2nd-2+L=2n,-3
F=-
2k(4~ + 1) 1’2 ( (2nd + 3) ) (
;;z;;;;-si’
(
For nd 2 4 the p-band can also decay to the X-band. can be obtained using the sum rule
)1’2
T;;;;;f;-r;))1’2.
The matrix elements /3 - X’
+ 1)) C I
(1/W.
(5.22)
(5.23)
L’
and they are given by B-X’
L’ = 2nd - 2 ---f L = 2nd - 4 Wd
- Wd
112
- 1)
1 .
(5.24)
The results (5.20), (5.22), (5.24) are summarized in Fig. 6. Before concluding this
Lz:r,~‘l
FIG. 6. Reduced matrix elements of d between the j%band and the states of maximum
seniority
272
IACHELLO AND ARIMA
section we remark that the algebraic forms of Eqs. (5.13), (5.20), (5.22), and (5.24) can be used to calculate the boson cfp from the relation Eq. (5.1) Vd(XZ) aT II dnd+lX’J’l = (& : 1)1,2 (ZI’ +y 1)1,2 0% + 1, x’, 1’ II cl+II % , x, 0. (5.25) These algebraic relations may therefore be useful in other contexts, as for instance within the framework of the Greiner and Gneuss [27] method.
6. THE VIBRATIONAL
REGIONS: A SURVEY
We now consider the regions where a description in terms of a SU(5) 3 O+(5) symmetry may be appropriate. A necessary but not sufficient condition is that the forbidden dn, = 2 transitions be small, and that a vibration-like two-phonon pattern is present with Of, 2+, 4f states in the neighborhood of twice the energy of the first excited 2+ state. The regions in the periodic table which exhibit these properties are shown in Fig. 7. They are characterized by having both neutrons
N
FIG. 7. Possible vibrational regions (dashed areas). The even-even nuclei shown in the chart are those for which the energy of the first excited 2+ state is known.
and protons outside the closed-shell. The most favorable conditions seem to be when N (or 2) is only 4-6 particles away from the closed-shell and 2 (or N) is S-10 particles away from it. For larger particle number a rotational pattern is exhibited. The best candidates appear to be the Pd(Z = 46) and Xe(Z = 54) isotopes (Regions II and III) and the Pt(Z = 78) isotopes (Region V). From the predictions of the quadrupole d-boson model without phonon mixing obtained in analytic form in Section 4 and 5, it is now easy to see whether they
INTERACTING
BOSON MODEL
213
provide a first-order approximation to energy levels and transition rates. In Figs. 8-10 we show the comparison between observed and calculated levels in lo2Ru, l°Cd, and l**Pt. The cases of lo2Ru and “OCd illustrate some difficulties which arise in comparison with experiments. Since vibrational nuclei exhibit other excitation modes such as two-quasiparticles and collective octupole modes, special care must be taken in comparing with experiment. The level scheme alone is not sufficient to determine whether or not a state belongs to the quadrupole mode. Detailed knowledge of their electromagnetic decays is also necessary. In the case of “OCd this information is available (see Section 7) and one is in a position to compare the observed levels with the collective quadrupole mode prodictions. In general, states of the octupole mode are easily identified by their negative parity. Instead, for states of positive parity, it is more difficult to distinguish between collective states and states of two-quasiparticle character. The latter will appear in the spectrum at excitation energies E > 24 (where A is the pairing gap) as shown in Fig. 11. Unless the mixing with the collective quadrupole states is small (as it appears to be in ll°Cd) these states will admix with the quadrupole states washing out the collective structure. Recent results of (heavy ion, xny), (LX,xny), and (p, xny) experiments, a technique
OL-,*
-o+
8. Comparison betweenexperimental and theoretical spectrum in loaRu. The experimental levels 4+ at 1872 keV and Of at 1837 keV are taken from [28], the 3+ level at 1837 keV from [29], and all others from [30] and the nuclear data sheets. The parameters in the theoretical spectrum are E = 481 keV, c, = -18 keV, cz = 141 keV, c, = 144 keV. FIG.
- i 8.--
8’
FIG. 9. Comparison between experimental and theoretical spectrum in “OCd. The experimental levels are taken from [30] and the nuclear data sheets. The parameters in the theoretical spectrum are E = 722 keV, c0 = 29 keV, c2 = -42 keV, cq = 98 keV.
Other
Stcies
‘B8P+ 78 110
FIG. 10. Comparison between mental levels 4+ at 1084 keV and [30] and the nuclear data sheets. co = 148 keV, c, = 30 keV, cq =
experimental and theoretical spectrum in IsaPt. The experi5+ at 1442 keV are taken from [31], all the other levels from The parameters in the theoretical spectrum are E = 281 keV, 110 keV.
INTERACTING
,
Two
C
BOSON
Collective
-
&adruDoie
States
s+
-
2+
3+ -
-
‘I
275
MODEL
0’
0’
"2A
0
E
-
2+
-
0’
FIG. 11. Schematic representation
of the energy spectrum in even-even vibrational
nuclei.
E (MeV)
FIG. 12. Comparison between experimental [33] and theoretical energies of the Y and X bands in loOPd. The parameters used are E = 680 keV, ce = - 160 keV, cg = 45 keV. The average deviation between experimental and theoreGca1 energies is 10 keV for the Y-band and 36 keV for the X-band.
276
IACHELLO
AND
ARIMA
pionered by Morinaga and Gugelot [32], have provided much additional information. These reactions favor the population of collective states and act as a sort of filter allowing an easier identification of the collective states. An example is shown in Fig. 12. Here comparison with the observed levels is straightforward but a precise identification of the various bands remains difficult since a careful study of their decay properties cannot be performed. (For example, in the case of l”OPd it is not clear whether the observed band to the right of the Y-band is the X-band or rather one of the bands discussed in Sect. 11). Figs. 8-10 and 12 show that the d-boson model without phonon mixing provides a reasonable first order approximation to the energy spectrum. Similar conclusions can be drawn if one considers the electromagnetic decay properties of the collective quadrupole states. In Table I we show a collection of B(E2) branching ratios in the Xe-isotopes. We observe that all forbidden dn, = 2 transitions appear to be small, TABLE
I
B(E2) Branching Ratios in the Xe-isotopes” lszXe 54 78
;;I z 2:
/
31+ + --------.( 31+ -+ 3t+ + 31+ +
21+ 2,+ 41* 2,+
lsoXe 54 78
lesXe 54 14
lzeXe 54
1.01
1.48-1.99
12
lz4Xe 54
PO
d-boson limit
(x 10-S)
0.17
0.57-0.64
x 1o-3
1.03
1.41-1.54
0.51
0.24-0.25
-
3.11-3.42
-
1.28
-
-
0.95-1.05
-
0.94
0.90
10 - = 0.909 11 0
5t+ --L 4,+ (x10-2) 5,+ + 31+ 51+ - 4,+ 5lf + 31+ 5r+ --f 6,+ 51’ -+ 31+
1.99 0.46-0.72
1.86 - 3.89 2.99
0
0.16
2 -=0.400 5
-
3.38
-
4.97
3.90
-
0.46
-
1.26
0.98
-
0.64
-
-
-
0
5 - = 0.454 11 104 - = 0.450 231
o The experimental values are from the nuclear ,data sheets. When the experimental results disagree. with each other, both the smaller and the larger value is quoted. All transitions are assumed to be pure E2. Note that forbidden An, = 2 transitions are given in units of 10-e.
1NTERACTING
BOSON MODEL
277
about 1O-2 of the allowed transitions, and that the allowed transitions appear to follow the values expected for the d-boson model without mixing. We recall that the expected values are just ratios of integers (Section 5). By looking at the first line in Table I one can observe that there is a general tendency for the forbidden transitions to increase toward the middle of the shell. This tendency supports the idea of Sakai [23] that there is a smooth change from vibration-like to rotation-like spectra. The data of Table I also illustrate another difficulty which arises in comparing with theoretical predictions. The branching ratios quoted have been obtained by assuming that all transitions are pure E2 transitions. In order to make more detailed comparisons with experiments the amount of E2/Ml mixing has to be known. In few cases the mixing ratios are known and we will discuss one of these cases in Section 7. Further, before doing so, we consider the effect on energies and transition rates of a breaking of the SU(5) I) O+(5) symmetry. A small breaking will be always present, even in the best of the cases, since the boson symmetries are not symmetries of the underlying fermion Hamiltonian, but rather of the boson Hamiltonian. However, the breaking of the SU(5) r) O+(5) symmetry can still be treated within the framework of the unified interacting boson model, Eq. (2.1), to which we now return for a proper treatment of the energy spectrum and transition rates.
7. SYMMETRY BREAKING
IN PERTURBATION: ENERGIES AND QUADRUPOLE TRANSITIONS
In regions where the boson SU(5) r) O+(5) symmetry is only slightly broken one can use the results of the previous sections to perform first-order perturbation calculations based on the vibrational limit. In the Hamiltonian of Eq (3.6) the interacting d-boson model diagonalizes the first two terms and we now treat the remaining two terms as the perturbation. (This eigenvalue problem has been solved numerically by Janssen, Jolos, and Donau [IO]). However, if th.e additional terms 2,2 , o,, are small, we prefer to use perturbation theory, in view of the transparent results. We also note that the effects of the one- and two-phonon changing terms have been investigated by Sorensen [34], Kishimoto and Tamura [35], and Holzwarth and Lie [7], although without cut-off. We find the inclusion of the cutoff essential in producing results which are in agreement with experiment. Moreover, it is the presence of the cut-off in Eq. (3.6) which provides the SU(6) group structure necessary to produce rotational spectra. To demonstrate the importance of this term we consider the simple Hamiltonian (without cut-off) H = EC d,+d, - 60(51/2/4)[(d+d+)(0) + (dd)“‘]. m
595/9912-4
(7.1)
278
IACHELLO
AND
ARIMA
This Hamiltonian can be diagonalized by introducing group, Eq. (4:8),
the generators of the SU( 1, 1)
H = ~(2s~ - 5/2) - Go& .
By performing
(7.2)
a complex rotation in the space of the group So + ch @$, f sh @S, (7.3)
S, --f sh @$, + ch @S, and choosing the angle @ such that th@ = 6,126
(7.4)
one can bring H to diagonal form H = -~(5/2)
+ 2(9 - (6,2/4))“2
$,
(7.5)
with eigenvalues E = -(5/2)
li = 0, l)...) co.
E + (5/2)(c2 - (6,2/4))lj2 + (e2 - (6,2/4))‘/2 ri
(7.6)
Thus the addition of the term [(LP~+)~~)+ (dd)(O)] without cut-off is ineffective in changing the structure of the spectrum, since it only displaces and compresses it. Returning to Eq. (3.6) we introduce the notation + (1 - ((n, - 1)/N))1/2 CM)
H = i& + i&(C+(l - (n,/N))‘l”
+ 2fio,(~+Kl - hdmu
- (42 + ww2
+ ((1 - @d - l)lWl
- ((&I - ww1’2
s-1
(7.7)
where f& = E c d,+d, + c g(2L + 1)1/Z c&I+d+)‘l’
nl
C+ = (5/2)lj2 [d+d+)‘2’ d](O)
c- = c++
S+ = (51/2/2)(d+d+)(0)
s- = s++
co = vo/5112
(@(S](O)
L
C2 = (2/5)‘/”
(7.8)
v2 .
We denote the unperturbed solutions of ir, by [ nd, L, v, nA) and display v and nd only when necessary. To construct the perturbed wavefunctions, denoted by 1nd , L, v, nd>, we need the matrix elements of the operators Ch, S, in the basis defined by ir, . The matrix elements of Si have already been given in Eq. (4.30).
INTERACTING
BOSON
MODEL
279
TABLE II Boson Wavefunctions in First-order Perturbation
Theory
izd= 0 1o,o>> = I 0, o> + (y
(yy
f I 2,O)
726= 1 11,2))=
; 1,2) + (9,“”
~12.2>+(;)l’z(~)*‘~(~)1’z~,3,2)
Fld = 2 2, 4)) = I 2, 4) + (y(yy2 j 2,2)) = - (Jgy
E’ I 3,4> + & I’ I 1,2> + I 2,2) + Jg
+&(~)“‘(~)l”6,4,2,v 1 2,0> = - (;)li2
(~)l’p(~)l’*
6 ( 4, 4, 5 = 2)
(yys
r ) 3,2)
=2>
(v)l”
f I 0, 0) + I 2,0> + 61ia (%)I”
6’ 13,o)
+71/a(~)1’a(~)“2~,4,0> ??d= 3 I3,6)) = I3,6)+3
(;)l”
I 3,4)) = - (ye
(qy
f’ I 2,4) + I 3,4> - g.g
(F)lla
5’ I4,4, o=2)+(;)1’z
+ 30’/” 4(’ Iv*
1 3,3)) = I 3,3) + (y ] 3,2)) = - (;y
(v)lil
5’ / 4,6)+(;)“’
(Jglla
(2Lg
(J!y)1’2 (L!gL)“”
+ 13,2) + y
(=)I”
(?)I” (Jgli2
P-3”’
fJ ) 4,4, v = 4) 515,4, v=3>
5 ( 5, 3)
f ) 1,2> - 3g
(~)“”
p , 2,2)
6’ 1 4,2, v = 4) + -?-. (N--l)“’
I’ I 2, o> + I 3, o> + q
+ (!31’y~)1’8(231’~~15,0)
t/5,6, u=3>
(~)1’2(~)1’p
+3(“,‘).‘2(~~‘2f,5,2,v=3) ) 3,0> = --6l/*
(y)1’2
71’2 (=y
N f’ j 4,O)
N
5’ 1 4 9 2 , tr zz 2)
TABLE
III
Matrix Elements of the Quadrupole Operator of Eq. (3.7) in First-Order Perturbation
Theory”
N---l{ Cl, 2 II T’EZ’ II 1,2> = Y’2 lY*f + 9.2SY2(2)“2 N) <(I, 2 (1TfE2’ /I 2,4> = 3 ;qp 21’2 (y)‘:’ 3(110)“2 <2, 4 II T’E2’ I]2 >4)) = ___ 7 <<2,2 II T’E2’ ll2,4> ((1 \ > 211 p21 [j2,2 >,=
I = 7l2 (d
(1 + t y)
+ q;t’ ; (?)“‘I
N-2 + qze wY’” - N 1 N-l - q2P’ ___ 7 ~2(2)‘la ( N ( p
5112jq,2111
(!g”‘(l
I 15N-2 -
+ &q)
:q;*qy)“l
g2 x , 211 T’EZ’ II 2,2> = 51’ 2 (i - 37 q2’ - q2f’ &?)I/2 (!$A N-l (7
<(2, 0 11TIE*’ 112,2> = [2q?’ - q$sI (2Y go, 0 II T’EZ’ 112,2> = w <(1,21IT’E2’)\2,0>=
1-q2g- (!!$)“’
/l T’-’
113,4>
+ qz’,$21/z(!!$)“‘/
[Bz2’!9(~)1’z(l
<<2,2\\T’E2’(\3,4>=
= 3 /q2 (G)‘!’
-S;tEfy)
(y)l”
+ qa’( (y
+4m(~)y
+[y)
+q;
(1 + fy)
3 /q2(;)1’z(~)1’p(1
<(l, 2 )I z-‘EZ’ 113,4> = 3 1 -q2r (y
- q;<‘s
(~)““l
+
(T)‘” (N--g
C2,4,,T’EP’,,3,3)_7ii~j~~(~~‘d(~)1’a(1 <2I , 2 11T@’
+ ; !!y)/
I - p-2___ N 1\
<2,4~~~‘~~‘~~3,f$)=,3V/qE3112(~~‘2(l <2,4
)I
1) 3, 3)) = 71/a 1 - q2(;)1’z(E$2)l,l(l
(yy (N-g/ +[y) +*!!z$)
-q;+$(~)l’rl +q;~~(~)“P~
~l,21,T1Bzll13,3)~71,P~~qI~(~~‘e(~~’~(~~’a + qz’# (q!)‘/’
(!yy*
(!g”“1
0 Some matrix elements involving states with nd = 4 are included.
Table
continued
TABLE
III
(continued)
282
IACHELLO
AND
ARIMA
Those of C+ are given in terms of the reduced matrix elements of the operator of Eqs. (5.3) and (5.4) as
U@)
0% 7x, L I c- I % + 1, x’, L) = (S/2)“” (nd + 1)1’2 1 (i/E)[dnd(tc’J’) K’.J’
dL I> dnd+lx’L]
II n dY tc’, J’).
x (4t 5 x 3 L II W’
(7.9)
Since they involve only cfp’s, these matrix elements can be obtained in algebraic form using the methods of Section 5. We now list in Table II the perturbed wavefunctions for states up to nd = 3. The notation is e = i&,/e, e = &/E. Using these TABLE
IV
Energies of the Boson States in Perturbation
Theory
nd = 0 E(O.0)
= E(0, 0) -
5y
g%
a!?(& 2) = E(l, 2) - 7 y y
8% - N+
I’%
nd = 2 &2,4)
E(2, 4) - 9 y
7
I”< -
,??(2,2) = E(2, 2) - 9 y
y
g”c + (y
=
k(2,O) = E(2,O) + (5 y
Tld=
-
y
!!!I$
gf2< - f y)
g-e
14N-2N-3 N-x-
)
3
&3,6)
= E(3,6)
- 11
&3,4)
=
E(3,4)
- 11N+ !$? 52~+ (; !?$ - g N+) (‘zE
&3, 3)
=
E(3, 3)
- 11N+ y
gee
INTERACTING
BOSON
283
MODEL
wave functions and the results of Section 5, we can calculate the reduced matrix elements of the quadrupole transition operator of Eq. (3.7) in first-order perturbation theory. Some of them are given in Table III. Similarly one can calculate the effect of the perturbations on the energies. This is shown in Table IV, where -&a 2 L) denote the unperturbed energies given by Eq. (4.21) and &nd, L) the perturbed energies. For sake of completeness we have also calculated wavefunctions and transition matrix elements in second-order perturbation theory up to nd = 2. They are given in Tables V and VI. To this order one has to take into account the effect of the boson-boson scattering terms cL . The notation in Tables V and VI is 74 = cq/c, qz = c,le, 70 = C,/C We now return to Table III in order to study the effect of the various symmetry breaking terms. To examine the effect of the cut-off we set 6 = c’ = 0. In Fig. 13 B(Ez,L’=2nd-+
L=2nd-2)
E(E2,L4=2-L=O)
Vibrational
Model
/
3
5
6
10
7
14
9
18
II
“d
22
L’
FIG. 13. Effect of the cut-off on the B(E2) ratios.
we show the ratios B(E2, L’ = 2n, + L = 2~ - 2)/B(E2,2 -+ 0) as a function of nd . One can see that these ratios deviate more and more from the pure vibrational limit as one approaches the top of the band, na = N. However, the cut-off does not affect B(E2) ratios for decays of states belonging to the same nd . For instance B(E2, 31+-+ 4,+)/B(E2, 3,f - 2,+) = 2/5 which is the same as in Table I. The effect of the .$ term in conjunction with the cut-off modifies B(E2) ratios even between members of the same multiplet. For the two-phonon triplet, it lowers the O,+ --t 2,+ transition relative to the 41+ + 2,f and 2,+ + 2,f. Finally the 5 term further modifies the B(E2) ratios. The combined effect of the symmetry breaking terms is shown schematically in Fig. 14.
TABLE
V
Boson Wavefunctions in Second-Order Perturbation
tld= 0 IO,O>= (1 - ;y + (;jl’a
f;) j o,o>+(;j”‘(~j”2f(l (!yj1’2
(!yj”’
Theory
-$jl2,0> 51’ 1 3, 0)
rid = 1 I1,2>=
7N-1
(1 -yfyNNfaj + (yy
[fv
-
+ (!yj1’2
(!!g2
N-2
/ 1,2)
72) + 2y [(;j1’2
ffj
12, 2)
f ( 1-~~~-f~~-~~~)+(~)“‘~‘]I3,2>
rtd = 2
1%4>=
(1 -? 11!X$p~-$?$2C!$*~)~2,4) + (y’
(y”*
[ 5’ (1 - ; ‘la - ;
Q)
+ 27
ffq
+(~j”‘(~~‘*[~f~-~~~-~Z)1-~~4)+~~~],4,4,~=2) _(~~‘z(~j1’*5(~)1’z~l,4,4,a
12,2>=
=4) -;E+;yyf2),2,2)
(1 -;yp -(~)li*[t’(l-%~+4~ff$l,2) + (yj1’2
[G
1.5 N-3 + -1411e N
ff
I’ (1 -
N-l 141/a -
g Q + ; 112 - 5 To) EE’] I3,2>
+(~~‘z(~~‘a,~f(l-~~~-t~~-~~~j+~~~]14,2,u=2> +(~~‘a(~)l’a~~~,4,2,u=4) I2,0>=
(1 -;ypr
7N-2N-3
7
p -
3y
c”)
( 2, o>
-(gyy~‘*f(l-+):o,o) +6qgy(l
-331,+?,)+2yffq
13,O)
1 3,4>
INTERACTING
Matrix Elements of the Quadrupole
Cl, 2 II FEZ’ II 17 2) = y/a qa -N-lN I
2(2F
lON-1 + q2’ (1 - czy7
[
TABLE VI Operator in Second-Order Perturbation
N-2 C(l - 72) + 51’3 -yN--l + I”,,
1
N-2 )I
=3~q1(~~‘*2’/“[1+g(l-~~?,-~~,-~so)~-g~(~-~~)
IN-lN-2
285
BOSON MODEL
9N-2N-3
Theory
286
IACHELLO B(E2.L’-L
AND
ARIMA
)
B(E2.2,
-
0,)
~Conventional ,JVibrationaI
(2*-2,)‘... 2
i
---------
tl-
oi
0.1
0.2
L’
i Model
0.3
14. Combined effect on the B(E2) ratios of the symmetry breaking terms as a function of 5 for rqz’;= 0.7 6 q2/W2and N = 6. The experimental data are from [36]. FIG.
1542.4 -1475.7 ‘1473.2
0+-
FIG.
15. Decay scheme llomAg -+ WZd [37].
INTERACTING
BOSON
287
MODEL
A test of the d-boson model with perturbations is meaningful only when many of the matrix elements of Table III are known experimentally. Presently there are few such cases. The decay of the high spin 6+ state in llomAg (Fig. 15) provides an almost unique example since it populates only high spin states in “OCd thus acting as a filter in the selection of the collective states (see Fig. 8 for comparison). Krane and Steffen [37] have carefully analyzed E2/M1 mixing ratios in this decay. From their data it is possible to extract 8 B(E2) ratios. Moreover, Milner et al. [38] have measured by Coulomb excitation 3 more B(E2) values, and the quadrupole moment of the 2,’ state is also known 1391. Thus in total there are 12 matrix elements known to be fitted with the 5 parameters .$, [‘, q2 , q2’ and N. We have done the fit by arbitrarily assuming N = 6, 5 = -[‘, q2’ = -(0.7/5112) q2 and the corresponding values are shown in Table VII. The picture which emerges from TABLE
Comparison
between Experimental
Transition B(E2),,r,(ezfm4)
B(E2),,,,/B(E2)w
and Calculated B(E2) Values in ll°CdG Ratio to
Ratio talc
Ratio exp
1.63 1.30 1.25 1.1 x 10-Z
(1.53 + 0.19) (1.08 f 0.29) (5.5 rt 0.9) x 10-a
31-+2, 31-+2,
0.31 2.0 x 10-Z
(0.47 f 0.20) (1.7 * 1.0) x 10-a
1533 1222 1171 14 989 304 20 831
30 49 39 37 0.45 32 10 0.64 27
592
19
42 - 2,
0.71
15 762 292 300 26
0.48 24 9 10 0.84
4, - 2,
1.8 x 10-Z
’ - 0.23 (0.7 rt 0.1) x 10-a
51 + 3, 5, + 31 51- 31
0.38 0.40 3.4 x 10-z
(0.39 f 0.23) (0.54 f 0.10) (0.7 zt 0.6) x lo-*
938
Calc
Q, (e
VII
fm9
B(E2)2, -+ Ol(e2 fm4)
-42 938* (fixed)
o
23
+
0.38
EXP
-31 * 9 or -55 l 8 934 f 38
a The data for the 21 -+ O1 , 4, --f 2, , 22 + 2, transitions are from [38], the QZ value from [39], and the remaining data have been extracted from the measurements of [37]. The errors shown are our estimates. The parameters used are 5 = -F = 0.3, q2 = 24.5 e fm*, qz’ = -7.7 e fme. B(E2)w in column 3 is the Weisskopf single particle estimate.
288
IACHELLO
AND
ARIMA
Table VII is that the transition rates in “OCd are not very different from those of a pure vibrator. The forbidden And = 2 transitions are all systematically small, of the order of single particle units or less. The allowed transitions, with the possible exception of the 4, --f 4, , are also close to the pure vibrational limits 0.40(3, + 4,), 0.91(4, -+ 4,), 0.45(5, --+ 6,), and 0.45(5, -+ 4& despite the large anharmonicities observed in the energies. This supports the suggestion that the large anharmonicities in the energies are due to the boson-boson interaction and that the symmetry breaking terms are small and give rise only to small deviations from the vibrational limit in the transition rates. We have also explored the sensitivity of the calculated B(E2) values to the parameters .$ and [‘, and found that equally good fits can be obtained for e and 5’ in the range 0.1 < t, -F < 0.3 by appropriately choosing q2 and cd. The symmetry breaking terms also affect the energies. In Fig. 16 we compare the experimental and calculated level sequence. In addition to the levels observed by Krane and Steffen [37] we have added two more levels which we believe to be the remaining members of the 3-phonon quintuplet. Comparing Fig. 16 with Fig. 8 ,tmc305a
6t312462) 4+-(2302)
4+J&
(15l81
*+~(I4431
2+~(2288) 3+-(2202)
ty~(2002)
(-j+J5g1473)
2t-(704)
0+2 FIG. 16. Quadrupole states in “OCd. The calculated energies are in parenthesis. In addition to f = -5 = 0.3 the parameters used are E = 740 keV, c, = 100 keV, c, = -120 keV, co = 30 keV.
one can observe that no substantial improvement in the calculated energies is obtained by introducing the symmetry breaking terms. This can be understood since these terms act in second order in the energies (Table VI), while the bosoninteraction acts in first order. Moreover, because of the cut-off, their matrix elements decrease as nd increases, while the matrix elements of the boson-boson interaction increase with nd . Once more we stress the fact that it is the combined
INTERACTING
BOSON
MODEL
289
effect of the cut-off and of the symmetry breaking terms which makes them unimportant. Before concluding this section we remark that, according to Table III, all quadrupole transitions in the d-boson limit with perturbations are connected by a set of simple relations. Thus they should show systematic behavior across a set of isotopes. This systematic behaviour has been observed experimentally [36] and it is shown in Figs. 17 and 18 for the Cd and Pd isotopes. In cadmium the
56
60
64
68
56
60
64
68
56
60
64
68
2.0 i I
0
LI~L.--...LI!~i
FIG. 17. Systematics of the Cd isotopes. The B(E2) data are from [38] and the quadrupole moment data are from the nuclear data sheets. The curves are drawn to guide the eye.
290
IACHELLO
0.5
t
I 56
'
AND ARIMA
I .--U.-L 60
64
Pd Z=46
-50
2 0” “2--
0 -25
ii;:-i
25w 56
FIG. 18. Systematics of the Pd isotopes. The B(E2) data are from [36] and the quadrupole moments from the nuclear data sheets. The curves are drawn to guide the eye.
effect of a partial shell closure at N = 64 is clearly observed [40]. From Fig. 17 it appearsthat the symmetry breaking terms which give rise to the B(E2), 2, --t 0, , and to deviations from the vibrational limit in the other transitions, are smallest for 112Cdand increaseas we move away from this nucleus. We have not attempted a fit to the other Cd isotopes similar to that of “OCd becausethere are no available
INTERACTING
291
BOSON MODEL
data comparable in detail to those of Krane and Steffen. Such data, especially the measurement of E2/Ml mixing ratios, would be very helpful in determining the nature of the symmetry breaking terms.
8. OTHER ELECTROMAGNETIC
TRANSITIONS:
EO AND Ml
Since the interacting d-boson model appears to provide a good description of the observed energiesand electric quadrupole transitions in the vibrational limit, we inquire whether an equally good description of other electromagnetic transitions is possible within the sameframework. The basic one-body operators in the theory are (d+s)@), (d+d)c2) and (s+s)co).Thus the first natural extension of Eq. (2.3) is to consider electromagnetic operators of the form
T:' =
c?$,,[(d+s)~)
+ (s+d):‘]
+ &(d+d):’
+ yE6108mo(s+s)~).
(8.1)
The additional terms now contain all electromagnetic operators EO, Ml,... up to 1 = 4. We begin by considering the Ml operator TiM1’ which we rewrite, for notational convenience, TCM1) = m,(d+d)F) k
(8.2)
In a microscopic description of the boson model [8] one can show that the coefficient m, is expected to be of the order of single particle units. However, the operator of Eq. (8.2) is proportional to the boson angular momentum operator L[41] and therefore in the boson basisit has only diagonal matrix elementscontributing to the g-factor of the excited states. This is true even if the wave functions are the perturbed wave functions of Table III sincethe angular momentum remains diagonal in this representation. The matrix elements of the operator of Eq. (8.2) are given by
(nd , x, L II TtM1’ II nd’, x’, L’) = m, (
L(L
+ 1)(2L + 1) lj2 6
10
1
nJL&’
*
(8.3)
In order to have nonzero matrix elements of the Ml operator we must therefore go to next order and form an effective two body operator from the product (d+s)(2) @ (d+d)fl). The most general operator of this form is CITl[(d+s)(2)(d+d)(z) + (d+d)(z)(s+d)(2)](1’. Ho wever, by recoufiling the operators d+d+d, one can see that there is only one independent term in the sum. Thus the most general Ml operator of this form can be written as TiM1) = m,(d+d)f)
+ fil’[(d+s)(2)
(d+d)“)
+ (d+d)(‘) (s+d)‘2’]t’.
(8.4)
292
IACHELLO
AND
ARIMA
The coefficient 6,’ is presumably small since it can come, in microscopic description [8] only from the Pauli principle in the fermion basis. As a consequence, Ml transitions are expected to be largely retarded in vibrational nuclei, a conclusion which is borne out by the experimental results. Within the boson basis j s”@~[N] xLA4) the operator of Eq. (8.4) is equivalent to @‘fl) = m,(dtd)(,) + ml’{[dt(dtd)“y’
(1 - @2,/N))“” + (1 - ((fld - 1)/N))“”
where m,’ = fi,‘(N)liz. The vibrational N + co in Eq. (8.5) and it gives Tf’fl) = m,(dtd)p
[(&Q(l) d]?‘} (8.5)
limit with no cut-off, is obtained by letting
+ m,‘[dt(dtd)(‘)
+ (dfd)(l) d]!‘.
G3.6)
The matrix elements of this operator can be easily evaluated and yield +d , x, L II ‘P’l’
II G + 1, x’, L’)
= ml’(-)L+L’+l
3112 L(L + 1)(2L + 1) 112 ( 10 1 (8.7)
where use of Eq. (8.3) has been made in evaluating the matrix elements of (#d)(l). Since the matrix elements of d have been obtained in algebraic form, Section 5, it is easy to construct those of TtM1) given in Table VIII. All matrix elements are given in terms of the single parameter ml’. Using Table VIII and the matrix elements of Section 5 one can calculate E2/M1 mixing ratios in the pure vibrational limit. TABLE Some Reduced Matrix Elements of the Ml
VIII
Operator in the d-boson Limit without Cut-off
II2,2) /I 3,4> (13, 3) 113, 3) I) 3,2> II4,6>
= = = = = =
--m,’ (21/10)‘la -m,’ 3(1 1/10)1~* ml’ 3(2/5)‘/* -m,’ 3(3/10)1/” --ml’ (3/5)1/S --ml’ (273/1O)‘/2
114, 5) = -ml’
(2731fa/5)
)I4, S> = m,’ (3/5) 21*j2
INTERACTING
BOSON
293
MODEL
However, because of the simple structure of Eq. (8.7), it is also possible to obtain these mixing ratios in analytic form. The mixing ratios ~WP/Ml'
=
(
nay
x,
L
II TtE2'
II nd
-t
1, x',
L'>/(nd,
x,
L
II F"~'
II nd
+
1, x',
L')
are given by L’ = L
A’E2IMl’ = --A(10/((2L
L’=L+l
A@/fifl’
L’=L-1
A(E’LIMl’ = --A(10/(3(L
- 1)(2L + 3))‘/2)
= -A(10/(3L(L
+ 2))‘/2)
(8.8)
- l)(L + 1))1/2)
where A = q2/m,‘. Some of these are shown in Table IX. In comparing the mixing ratios given in Table IX with those observed experimentally we remark that quite often the value of 8(E2/M1’= (P2’/PM1’)1/2 is quoted instead of d(E21M1’, where r(E2’(r(M1’) are the partial radiative widths for E2(Ml) radiations. SfE2jM1’ is related to dtE21M1’by a(=/Ml’ = 0.832 x 10-2J$d(E2Ibfl’
(8.9)
where E, is in MeV and AtE21M1) in e fm2/pn , TABLE
IX
E2/M1 mixing Ratiosin the VibrationalLimit without Cut-off (2, 2 1)d’EelM1) 113, 3) <2,411 @'IM1'II 3, 3) <2, 2 II LI’~‘+‘~) 113, 2> (3,6 II dfEZIM1) 114,6> (3,611 d'EP'M1)II 4,5> <3,41t~(E2’M”II
4,5>
= = = = = = = =
-A -A -A -A -A -A -A -A
(lo/219 (lo/7719 (5/69
(2/3)51ie (10/2119
2(5/33)'je 2 (5/21)‘/2 (5/3(2)'/*)
We next study the effect of the symmetry breaking terms by evaluating matrix elements of the operator of Eq. (8.5) in the perturbed basis of Table II. These matrix elements are given in Table X where we have also included the diagonal matrix elements ((1, 2 // TfM1’ II I, 2)), ((2,4 // TIM1’ II 2, 4)), ((2, 2 11TtM1’ jl 2, 2)) which are related to the g-factors of the excited states. It is interesting to note that the ratio (2, 2 /I TtM1’ I/ 2, 2)) g2+2 _ ___ <:1, 2 11T’M1’ 11I, 2) g2t1 595i99h
(8.10)
294
IACHELLO
AND
TABLE Matrix
Elements
of the Ml
X
Operator
in First-Order
\‘> I, 2 jj T’M”
111, 2> = 311% m, - In,’ I’
x:.2,4 11I+‘”
jl2,4))
<\I 7 2 [I FM”
112, 2>, =
<2,2
11TfM1’ II2,2>>
= 3(2)“’
(C2,41/ ‘,I,2
TcM1’ II 3,3>
\<2 > 2 /I T’M”
+ *!y]
m, + m,' F
L
3,3> 113,2>
= m,’ 3 (3”’ j
= --ml'3
14 II2 N-
ii
3
-m,’
@‘i”(~j-
(w-2)~'2 [f
($!-j"' (!yj"'
27:'!2(N;3)'!'
(3 7 4 jl TfM1' 114, 5> = m,': 21112 (Fg" 11TfM1' II4,5?) = -m,'c
- 4 (;)‘“!7)]
[1 + (Kg]
(N+l)'i'
<3,6/j TcM1)/I 4,6> = --ml' fz?)"' q3, 6 11TcM1) 114,5;, = -ml'
1
[1 + (!y]
(3lo j':"f!!$,"'
/I, 2 /I TIM" (I 3,2> = ml'(' (i)"'
~“2, 4
N
1
+ fy]
(!yj’i’
(2)'i' = ml' e3 3 =
Theory
m, - ml’ I’ 11 ,s i2j1’%
-m~3($'"(~)':'[1
II TtM1j I/ 3,3>
11T’M1l[l
Perturbation
(?Lj’:“fy)“‘[I
-m,’
= 31/2
q2,411T’M1)j/3,4>= <2,2
[
ARIMA
f 66'12
-
((;!y
- yg)]
(y)*'* [I + f!y] [1 + *N-+] [I + *!$A]
(N-+2)"'
(Ng
which is 1 in the vibrational limit can be considerably affected by the perturbation. Recent experimental results obtained by Katayama, Morinobu, and Ikegami (private communication to Arima) show that this ratio is indeed different from one in ls2Pt and lg4Pt. An application of the formulas given in Table X is shown in Table XI. Once more we observe the surprisingly good agreement with experiment. In particular the sign is correctly predicted to be the same for all transitions. Thus it appears that the d-boson model is useful not only in correlating the enhanced E2 transitions
INTERACTING
BOSON
TABLE E2/Ml Transition
A tale
295
MODEL
XI
Mixing Ratios in *l°Cd [37]
(vibrational limit)
A,,l,
(with 5 = - 5’ = 0.3)
A exp
- I50* (fixed)
- 150* (fixed)
-140
-142
-102
-91
-175 & 22 -190 + 140 -70 -153 + 95
-78
-75
-44 rt 36
-67
-67
-120
-81
-82
-170 f 50
forbidden
-85
-44 do 8
forbidden
-60
-32 + 3
5 50
a All values are in e fme/pN .
but also the retarded Ml transitions. We remark that the mixing ratio d for the 2, -+ 2, transition corresponds to a B(M1) of approximately 1O-2 Weisskopf units. We also note that a treatment similar to the one presented here has been recently given by Maruhn-Rezwani, Greiner, and Maruhn [42]. In an analogous fashion we can calculate matrix elements of the EO operator T’Eo’ = qO(s+s)‘O’ + &‘(d+d)‘O’.
(8.11)
These matrix elementsare trivial since the two terms in Eq. (8.11) are proportional to the s- and d-boson number operators n, and nd . Within the boson basis 1Pdna[N] xLM) the operator of Eq. (8.11) is equivalent to PO) = qoN + q0’n,/W2
(8.12)
where qo’ = 4; - 51i2qo.
(8.13)
The first term in Eq. (8.12) contributes only to the root mean square radius, while the second term gives rise to EO transitions. Using the perturbed wavefunctions of Table JI, we can calculate the matrix elementsshown in Table XII. EO transition matrix elements vanish in the vibrational limit with no perturbation. Again all matrix elements are given in terms of the number qo’. It would be interesting to seeto what extent these relations are fulfilled in practice.
296
IACHELLO AND ARIMA
EO
TABLE XII Matrix Elements in First-Order Perturbation
((0, 0 II z-0’
II 2, o> = 40’5
w2 ((iv
II 2, 2> = 40’2’ (1/5)1’2
<(2, 4 11P”’
1) 3,4>
= q,‘f’
-
((1, 2 11T’EO’ (1 3, 2> = q(f (14/5)“2 I/ 3, 2)) = qa’f’ 2(2/35)“’ II 3,OB
(3,
II 4, 6)) = q,‘f’(3/W21
6 11P”’
9. OCTUPOLE
= q;f’
(6/5V
((N
l>/w’s - 2NV)“2
((Iv - l)/lv)l’”
<<2, 2 /I T’“’ <2, 0 11P”
1)/N)“’
((N
(22/35)“’
Theory
(W
-
((A’ -
2YNY
((N
3h’W’2
-
((N - 2)lw’2
2)/Wiz
STATES. ENERGIES AND TRANSITIONS
In addition to the quadrupole d-boson, other excitation modes are expected to play an important role in the description of the nuclear collective motion. Of particular importance is the collective 3- mode, hereafter referred as jlboson. In order to construct states of octupole character we consider a system of two different kinds of bosons, N quadrupole d-bosons able to occupy a L = 2 and a L = 0 level and N’ octupole f-bosons able to occupy a L = 3 and L = 0 level, as shown in Fig. 19. The most general Hamiltonian describing this system is H = Hd + Hf + V,,
(9.1)
where Hd is the Hamiltonian describing the quadrupole the Hamiltonian describing the octupole mode H, = ~,t~Jsf+ + Cf ~Qfi,,
Ef
+
dof~(fTfl(o:;s,sf)(o)
+
&f[(f+Sf+p
+ lY2 CfLKf’f’Y” wY”‘1’“’
+ ,=g 4 o 4w
. *>
+ (fif)'"']'"'
f
(Sf+Sf+p +
mode, Eq. (2.1) Hf is
(ff)'"']'O'
t.Tof[(sf+.sf+)'O'
(9.2)
(sfsfpp
L’3
fd Sf L=o
Es
FIG. 19. The boson configuration
s5ds @ $fl.
d
L=2
5
L=O
INTERACTING
BOSON
and V,, represents the octupole-quadrupole Vfd
1
=
x‘[(d+f+)‘“’
297
MODEL
interaction
(dfyLq(o)
+
zq(d+s
+
s+dy2)
(9.3)
(f+f)‘“‘]‘“‘.
L=1.2.3,4.5
Here f+(f), Sf+(.sf)are the creation (annihilation) operators for octupole bosons in the excited L = 3 level and in the ground L = 0 level. The basis states are now of the product form 1s”YP~[N] xdL, ; s,““~f”~[N’] xfLf ; LM). The Hamiltonian of Eq. (9.1) is identical to a shell-model Hamiltonian with two kinds of particles able to occupy two levels each. It can be solved using usual shell-model techniques. In analogy with the quadrupole case, we expect the boson-boson interaction terms cfL , xL to be the dominant terms in the vibrational limit [43] and we shall therefore treat all the remaining terms in perturbation theory. We thus first consider the simpler Hamiltonian H = edz d,+d, + c &2L + l)'/' ~~[(d+d+)'~' (dd)(L)](o)+ Ef zfmlfm+
nz
L
?n
+ c $(2L + 1)1/2CfL[(f+f+)(L)
nl
(ff)'"']'"'
+ 1 XL(d+f+)(L)(d!)(L). (9.4) L
Rather than constructing explicit solutions of the Hamiltonian of Eq. (9.4) in the most general case, we shall restrict ourselves to the special case n, = 1. The diagonal matrix elements of Eq. (9.4) within basis states of the form / dnydLd ; f; LM) are
;h LW
+ of + n&L,
+ 1) C V-‘(XJ) XJ
dLd I} dndxdL$
while the off-diagonal matrix elements are given by (dndxdLd ; f; LM I H [ dndxdfLd’;f;
LM)
= rl&(2Ld + 1)1/Z (2Ld’ + 1)1/Z * ,cJ [dnd-l(xJ)
dL, I} d”dxdLd][dnd-l(xJ)
dL,’ [} d4xd’Ld’]
P-6) In these expressions, E(dny,L,) represents the energy of the dnd configuration as given by Eq. (4.21). These expressions simplify further in the case in which the
298
IACHELLO
AND
ARIMA
states of the d”a-configuration to which the f-boson is coupled belong to the ground state band L, = 2nd . Then Eq. (9.5) becomes
L, = 2~) + l f + n,(2L, + 1) c (25’ + 1) XI’ IL’; Ir,
2 ;
$j2. (9.7)
N
Y
I
/
7-___ / / I II? 5-
---_
/
;/ /
N’
__ 4
3-
/ 2+
-f -o+ FIG. 20. A typical octupole spectrum. The parameters in the Hamiltonian are q = 400 keV, q = 50 keV, .q = 1100 keV, x5 = -50 keV, A, = -50 keV. The heavy lines represent E2 transitions, the thin lines E2 + Ml transitions, the broken lines El transitions and the dotted lines Ml transitions.
INTERACTING
BOSON
299
MODEL
In the case in which the total spin L corresponds to the totally alligned value, L = 2nd + 3, or to the totally alligned value minus one, L = 2nd + 2, even simpler expressions can be obtained. For these states, the Hamiltonian of Eq. (9.4) is already diagonal and its eigenvalues are given by E(d”“, Ld = 2n, ;f; L = 2n, + 3) = E(dnd, L, = 2~) + cf + ntiy5
(9.8)
and E(dnd, Ld = 2n, ;f; L = 2n, + 2) = E(d”“, L = 2n,) + q + ndx5 -t ((24
+ 3)/5) A4
(9.9)
where (9.10)
A, = x4 -x5.
In Fig. 20 we show the corresponding spectrum. In this figure we have arranged the negative parity states into “bands,” defined as follows N-band
/ dnd, Ld = 2n, ;,f; L = 2nd + 3, M)
N/-band
1dnd, Ld = 2nd ; f; L = 2nd + 2, M).
(9.11)
The other negative parity states which arise from the coupling dnd @f do not form a band structure since they are admixed to other states, thus loosing their collective character and systematic pattern. For example the state I d”a, L, = 2nd ;f; L = 2nd + 1) will be admixed to the state 1dna, Ld = 2n, - 2;fi L = 2nd + 1) since both states have the same zeroth-order energy and they are coupled by the large matrix elements Eq. (9.6). For this reason other bands are not shown in Fig. 20. It is instructive to plot the energy differences AE = E(n, + 1) - E(n,) as a function of nd , as in Fig. 3. This plot (Fig. 21) shows that the coupled bands N‘ and N’ appear as straight lines A&
=
Ed +
c4%
+
x5
AEN,
=
Ed +
c&f
+
X5
(9.12) +
(z/5)
A,.
In a similar fashion one can derive transition matrix elements. The most general expressions for the E2 transition operator is now ,f2)
= &(d+s + s+d)t) + g,‘(dtd)‘,)
+ q;(f+f)~‘.
(9.13)
The term ( f’f)(2) which describes the quadrupole moment of the octupole 3- state is expected to be small and it will be neglected in the following. The transition operator of Eq. (9.13) then becomes, in the vibrational limit with no cut-off, T
(E21 k
=
q,(d,+
+
(->”
d-k)
+
qAd+d$’
300
IACHELLO
250
AND
ARIMA
t-
I
I
0
I
I
I
1 6
4
2
“d
FIG. 21. The energy differences AE = E(nd + 1) - I?(&) as a function of nd for the coupled bands N and N’. The parameters are the same as in Fig. 20.
as before, Eq. (3.9). The matrix elements of this operator are easily constructed within the basis states dnd @f. For example, (nd
+
1, Xd’,
Ld’;f;
L’
11 d+
11 ‘Id,
Xd
, Ld
ifi L)
= (-)L?‘+3+L (2L + 1y2 (2L’ + 1)“2 12’ x
hd
+
1, Xd’,
L,’
11 d+
11 nd,
Xd
;:
;I (9.15)
, Ld).
This expression can be specialized to the various cases. Denoting by F the reduced matrix elements of d F =
(9.16)
L’)
we have for transitions within the bands N-band
L’ = 2nd + 5 -+ L = 2nd + 3 F = ((n, + 1)(4&j + ll))‘/”
N’-band
L’ = 2nd + 4 + L = 2n, + 2
F =
d2%
+
5)(4nd
(2nd
+
+
(9.17)
g,
3)
The intraband transitions N’ -+ N are given by N’-+N
L’ = 2n, -j- 4 --f L = 2n, + 3
F = - ( ;fn;+‘3;)
l”.
(9.18)
1NTERACTlNG
BOSON
301
MODEL
The second term in Eq. (9.14) does not contribute to the nd + 1 -+ nd transitions in the vibrational limit. In addition to E2, Ml, EO ,..., other electromagnetic transition operators can now be constructed by combining quadrupole and octupole operators. Of particular importance are the El, M2, and E3 operators. We begin by considering the El operator Tp’
= cjJ(d?~)‘~’ (sr’f)‘“’ + (ftsJc3) (sV)‘~‘];’
+ q,‘[(~Pd)‘~’ (sr’f + f’~~)‘~‘]t) (9.19)
which in the vibrational Tp’
limit with no cut-off N --f co, N’ ---f co becomes
= q,(d+j-+ fV)t’
+ q,‘[(dtd)‘2) (f + ft)‘“‘];‘.
(9.20)
The reduced matrix elements of the operator (dtf)o) between basis states of the form dnd @fare 0% + 1, xd , Ld ;fnfco;
L II (d+j)'l'
II nd , xd', Ld’;fnfsl;
L’)
= (nd + 1)1’2 [dRd(xd’Ld’)dL, I} dnd+lxdLd] L,?‘(-)‘d+r,’
,:,,
This expression can be specialized to various cases. Denoting matrix elements of (dtf)cl) one has for interband transitions N-+Y
L’=2nd+3+L=2nd+2
N’-tY
L’=2n,$2-t~=2n&2
F’ = (
2,
21.
by F’ the reduced
(nd + 1)(4n, + 7) 1/Z 7 1
F’=-(
(9.21)
(9.22)
nd(4n, + 5) l/2 21 ) .
The second term in Eq. (9.20) does not contribute to these transitions in the vibrational limit. Combining Eqs. (9.17), (9.18) and (9.22) it is possible to write the B(El)/B(E2) branching ratios for the decays of the N- and N’-bands in the form N+Y
B(El,L’=2nd+3+L=2n,+2) =- nd+lC 7nd B(E2, L’ = 2nd + 3 --f L = 2n, + 1)
N-+N N’+
N’+
Y
N’
N’-+ Y N’+N
B(E1, L’ = 2n, + 2 -+ L = 2n, + 2) B(E2, L’ = 2n, -j- 2 + L = 2nJ =
%Pci + 1) 1)(2n, + 3) ’
21(n, -
L’ = 2nd + 2 --f L = 2n, + 2) = n,(2n, + 1) B(E2,L’=2n,+2+L=2nd+1) 63 c B(E1,
where C = (q1/q.J2is a constant for each nucleus.
(9.23)
302
IACHELLO AND ARIMA
The relations Eqs. (9.8) and (9.23) are compared in Figs. 22 and 23 and Table XIII with recent experimental results. However, this comparison must be taken with care since the nuclei 150Srnand 152Gd are at the beginning of the transitional region and therefore the symmetry breaking terms may be important. The comparison is shown here to point out that a first-order approximation seems already to provide reasonable results. Together with the El operator one may also consider the M2 operator Ti’-
= C~~[(d~s)(~) ($f)‘“’
+ (f@)
(sV)‘~‘]~’ + m,‘[(~Pd)(~)(s/j’+
f?~$~‘];’
(9.24) which in the vibrational
limit becomes
TtM2) = wz,(&f + fV)f’ k
+ WZ,‘[(&)(~) (f + ft)‘2’]~‘.
Y I
6 Y P w
(9.25)
N EXP
I
Th
195 17-
Th
EXP 4
14+
15-
I
1312+ 3
I
IIlo+
,
92 a+
6+ I
7-~ 5--3----
i
1
4+ 2+ 0
i
‘3m*13
0
FIG. 22. Comparison between experimental [44] and theoretical spectrum in IsOSm. The parameters in the theoretical spectrum are Q = 356 keV, c1 = 66 keV, q = 1077 keV, x5 = -47 keV.
INTERACTING
Y I
N
16’ 15‘ ;I If
5
I
EXP
14’
2
14
17.
T 17.
15‘
15.
13.
13.
14’
12’
12+
II-
iI-
IO’
IO’
9‘
9.
8’
8’
7‘-
c-
6+ 4’
+
2’ 0
17‘l-
16’
3
Th
Th
17‘
4
303
BOSON MODEL
if 0’
5-
6+
3-
-
7-
ir_ __
-
5. II
3-
4’ 2’
0’
FIG. 23. Comparison between experimental [45] and theoretical spectrum in lKeGd. The parameters in the theoretical spectrum are 6 = 365 keV, cq = 45.3 keV, 4 = 1123 keV, x5 = -10 keV. Comparison
between Experimental
TABLE XIII B(El)/B(EZ) Branching Eq. (9.23)
Ratios” and those Predicted by
15*Gd
150Sm
zi -+ z, 5- +4+ 5-*37-+6f 7- -f 59- + a+ 9- + lll-+10+ ll-+913---t 12+ 13----t lla In units of lo-’ e-l fm-2.
Exp t441
Th
ExP 1451
Th
20
2
15
1.5
21 14
13.3
2.2 & 0.4
1.33
10 & 3
12.5
2.6 f 0.4
1.25
12 & 2
12
0.8 i 0.2
1.2
304
IACHELLO
AND
ARIMA
The matrix elements of the operator (dtf)@) between basis states of the form d”d @f are
-L ;fnf=‘; L II (d+f’f)(2)II lid , xd’, L,‘; f = (nd + 1)1’2 [d”Qd’Ld’)
nf=l;
L’)
dL, I} dnd+‘xdLd] x?e’(-)l+rd+Ld’
I
Ll,
L,2
Lj. 3,
(9.26)
This expression can be specialized to the various cases. Denoting by F” the reduced matrix elements of (d+f )(2) one has for interband transitions N-tY N'+
L’ = 2n, f 3 ---f L = 2n, + 2
Y
L’ = 2n, + 2 -
F” = ,(
(nd + 2)(4n, + 7) l/2 14 ) nd(4nd + 7>(4n, + 5) v2
L = 2n, f 2
F” = - (
14(4n, + 3)
i
.
(9.27)
From Eqs. (9.22) and (9.27) one can calculate the mixing (L 11TfM2) 11L’)/(L 11TfE1) 1)L’) as N+Y N’+
~m42/El)
=
D
(4 !
A(MB/El)
Y
=
2(nd
+ +
2)
1’2
1)
(9.28)
3(4&j + 7) 1’2 2(% + 3)
D
where D = m,/q, is a constant for each nucleus. The mixing related to the ratio 8(M2/E1) = (PM2)/P1))1/2 by pfwu
ratios dCM21E1)=
ratio A(M21E1) is
= 0.921 x 10-4~~,~Mw)
(9.29)
when E, is in MeV and A(M2/E1) in t+Je. Next we consider E3 transitions. In first order, the related operator is (9.30)
+E3)
=
k
which in the vibrational
q&f
+q
+
limit with no cut-off becomes T!?)
=
dfk
+
f nf=“; L’ 1)TcE3)IInd,xd,Ld;f
(9.31)
(-)3-kf-k).
The only matrix elements of some importance
gf>$’
in the present context are
nf=l; L) = (-p+L+B
q3e8LdLd~sXdXd~ . (9.32)
INTERACTING
BOSON
305
MODEL
Finally we note that Ml transitions are also possible between members of the N and N’ bands. We only quote here the transitions N + N’ which are expected to be large since they are induced by the first term in the Ml transition operator of Eq. (8.4). The general expression (4
, Xd
2 Ld
;fnf=l;
L
L&L’ =
(1)
II W)
‘Ld L’
LL,
c-1
II G',
L Ld’
I
&';
Xd',
p=1;
L')
3 (nd
1
, xd
, Ld
11 (d+d)“’
‘I”
= -
11 nd’,
xd’,
Ld’)
(9.33)
yields in this case 3nd(4nd + 7))1/2. (9.34) 5(2n, + 3)
N-tN’
L’ = 2nd f 3 + L = 2nd + 2
Combining
this with Eq. (9.17) one can obtain B(Ml)/B(E2) N-N’
B(M1,2nd+3+2nd+2)= B(E2, 2n, + 3 --f 2n, + I)
N-N
G
branching ratios 3 Wnd + 3)
(9.35)
where G = (nzl/q# is a constant for each nucleus.
10. SYMMETRY BREAKING IN PERTURBATION: OCTUPOLE STATES The symmetry breaking will change the zeroth-order predictions of the previous sections. However, if the breaking is small, it can once more be treated in perturbation. The perturbed Hamiltonian is H = l d C dm+dnl+ C 3(2L + l)ljz cL[(d+d+)‘L’(dd)‘L’]‘o) L
+ ~~([(d+d+)‘z’d](O)(1 - (nd/N))1/2 + (1 - ((nd - 1)/N))1/2 [d+(dd)‘2’]c0’} f uo[(d+d+)‘*’((1 - (n,/N))(l + ((1 - 0,
- l)lN))(l
+
+
Ef cfnltfm
c
H2L
- ((% + l)/N)))1’2
- ((nd - 2YNNYi2 +
1F
cfLKftf+YL’
WY*)1 WY”‘1’“’
L +
Vafhf3(*)
+
((1
-
((1 (01~
-
-
(nf/N’NU
1)lN’NU
-
((nf
(h-f -
+
lMW)>1~2
2>/W>Y/”
(ffY*‘l
+ ; xL[(dtft)‘L’ (df)‘L’]‘o’ + ~,{[(ftf)(~)
d+](O)(1 - (n,/iV))‘i2 f (1 - ((nd - 1)/N))‘/” [d(f+j)‘2’]‘o’} (10. 1)
306
IACHELLO
AND
TABLE Octupole
States
ARIMA
XIV
in First-Order
Pertubation
Theory
n, = 0 IO, 0; 3-B = I 0,o;
3-j
+ (;j”’
(~j”’
f 1 2,0;
3-)
+ (;jLi2
e / I) 2; 3->
F7,j = 1 / 1,2;
5-B =
/ 1,2;
5-> + (T)liS
+ ; (y
f’ 1 2, 2; 5->
(!yB
6 / 2,4;
+ @liZ
5-> + ; (y
(Jgy
(fgy
f j 3,2;
(y)‘;’
e / 2,2;
5-)
(qy
e I 3,4;
7->
n, = 2 1 2,4;
7-p
= 1 2,4;
7-)
+ &
+ (y’
(Tj”’
+ (Jg
(y)1’2 (q+y
5’ i 3,4; f / 4,4,
(!!I$,“”
e 1 3, 5; 7->
7-)
v = 2; 7-) + (g)‘”
rtd = 3 I3,6;
9-B
= I 3,6;
9->
+ (;j*‘* + (;j”’
+ 3 (;j”’
(y)“’
5’ 1 4,6;
(y)*”
(!$.!)*‘2
f / 5, 6, 0 zz 3; 9-)
(y)“’
B 1 4,8;
9->
+ i&j”’
9->
(y)“’
fl 1 4,6;
n, = 4 14,8;11-))=
~4,8;11-)+2(~)1’*(~)1’a~,5,8;ll-> + (;j’”
(y)“’
5 +- 13311S (yjl”
(y)*” 6’ 15, 10; ll->
f ! 6, 8, v = 4; ll-> + 2 (&jl’*
0 I 5, 8; ll->
9->
5->
INTERACTING
BOSON MODEL
307
and the three perturbing terms of interest here are Pi , L:,,and w2. Introducing the notation Q/E = .$‘, Q,/E = 5 (as before) and W,/C = 13one can study the effect of the various terms. Some perturbed wave functions are given in Table XIV. In Tables XV-XVIII we give the matrix elements of the E2, El, and A42 operators between the basis states of Table XIV. These formulas can be combined together to give B(El)/B(E2) branching ratios and B(M2)/B(El) mixing ratios. We also give in Table XVIII the energies of some negative parity states. It should be noted that the coupling of the octupole and quadrupole bands in transitional nuclei has been recently studied by Nomura [46] and Zolnowski et al. [45] using the coupling term w.J(d+ + d)~z~(f’f)~z~]~o~(no cut-off). The effect of this term in perturbation cannot be distinguished from that of the boson-boson interaction term a-, . This can be easily seen by letting N 4 co in Table XVIII. The perturbation correction to the energies Al? then becomes
A Eh , 2nd ; 2n,
+ 3) = -(n,/42) TABLE
e2e+ A&O, 0; 3)
(10.2)
XV
Matrix Elements of the E2 Operator of Eq. (9.14) between the Basis States of Table XIV
,! I, 2; 5- 11FEZ’ II2,4;
7-B
= 15112 1iq” 2 ‘2(!g2(1 1 5112 + q;e 7(3)1/Z
+ [!y)
(N;
:‘2, 4; 7- /I T’-) 113, 6; 9-B = 19112 q2 3112 (EgZ(l
i 3, 6; 9-
1)Fe’ 114,8; 11-B = 231/2 q2 2
l,“‘/
+ *!?I$)
+q;q!r$)l’e
308
IACHELLO
AND
TABLE Matrix
Elements
of the El
Operator
XVI
of Eq. (9.20)
N-l 1 + 5 7)
<2,4
<3,6
/I T’E2’ II 1,2;
II F*’ II 1,2;
<4, 8 1) P2’
<5, 10 II P”
II 2,4;
II 3,6;
5-B
]
6”yJg)y 91 -y-
+ oq,‘;
(fq+j’“j
= 1 lljZ
5-B
= 11”’
7->
= 151’2
9-B = 19”’
04,
I
341j2 35(21)“2
eq, &
between
(N;‘)‘;’
the Basis
States
+ eq , 6CW2 l--El
+ $?!$)
N-1 fT)
I(h&
ARIMA
l/P
N-2 (-Nj
(yj”‘/
+q;(2!E(K;)1’2
l/e) i
of Table
XIV
INTERACTING
BOSON
TABLE
309
MODEL
XVII
Matrix Elements of the M2 Operator of Eq. (9.24) between the Basis States of Table XIV
311% + eIn,f____ 7(70)1/Z
<2,4 11T’M2’ I/ 1,2; 5-> = 1 II/* 1m, x&gy
+ *!y)
+ ~m;?gqy)l’n
(!y/ + em, -.-L98(5)‘/2
+ [!y)
<<3,6 11FM*’ I/2,4; 7-> = 15 3112 (N; 49(5)“2
--em,
2)‘“/
<4,8 iJ FM21 I/3,6; 9-> = 19’j2 f mz
i
39l/* (N; em2 7(110)‘/2
l)‘/’
3(17)‘/* (N;2)1/2 <4,8 jl TtM2) ;I 2,4; 7-B = 15112 em, 49(5)‘/2 1
<5, 10 I] FM2) II 3, 6; 9-B = 191’2
em,
551je (N; 7(57)1/2
(N-2)li:
(N+3,‘:‘\
3)“z (N-4r”/
+ *~m;f(!g)I’P
310
IACHELLO
AND
TABLE
ARIMA
XVIII
Energies of some Negative Parity States in First-Order Perturbation l?(O, 0; 39 l?(l, 2; 5-) x??(2,4; 7-) g(3, 6; 9-j .R(4, 8; 1I-)
= E(0, 0) = &I, 2) = @2,4) = &3,6) = &(4, 8) -
Theory”
(l/7) 8% (1/6)((N - 1)/N) 8% (4/21)((N - 2)/N) t-e (3/14)((N - 3)/N) 6% (S/21)(@ - 4)/N) e*E
a The energies &n, , L) are given in Table IV.
i.e., Al? is linear in nd as in Eq. (9.12). However, a careful study of the electromagnetic transitions can distinguish between the two coupling schemes. We believe the boson-boson coupling scheme to be the appropriate scheme for vibrational nuclei and that the importance of the w2 term increases as one moves from the vibrational to the rotational region. In this respect lsoSm and 152Gd are presumably in an intermediate situation and both terms should be included to reproduce accurately the experimental data. No such a detailed analysis has been performed so far.
11.
TWO-QUASIPARTICLE
MODES
As we have mentioned in the introduction and schematically shown in Fig. 11, two quasiparticle states will begin to appear in the spectrum at about twice the energy gap A and will be present with increasing density from there one. These states are largely filtered out in (heavy ion, xnr) reactions since their E2 transitions are considerably retarded with respect to the collective ones. There is however one case in which two-quasiparticle states can be observed in (hi., xnr) and (01,xnr) reactions. This is the case in which the two quasiparticle state (j&) f has high spin and yet lies below the collective state of the same spin. The two-quasiparticle state is then the yrast state. Although in general the coupling of particle states to the collective excitations of the core require a much more detailed analysis, which we hope to be able to present in the near future, we consider here the question whether or not a simple description of the two-quasiparticle states is possible in terms of elementary excitation modes [43]. In this drastic approximation (in which the two-quasiparticle mode is treated as a boson mode), one can take as Hamiltonian for the coupled system, that given in [43] H = Hd + Hz + V,,
(11.1)
INTERACTING
where now 1 labels two-quasiparticle the boson operators are
BOSON MODEL
311
states with angular momentum
1. For these
(11.2) where a+(a) are the creation (annihiliation) operators for quasiparticles. Using techniques similar to those of Section 9 one can derive a set of relations for the energies of the states dnd @ (jJ,)Z. If the interaction V,, is of the form V,, = 1 y,[(d+b+)‘L’ (db)(L’]‘o)
(11.3)
L
QF
5
QP’
2 sw 4
3
2
I
2+ 0
FIG. 24. Coupling
0+ i-
of a two-quasiparticle
state with spin 8+ to the ground state band.
312
IACHELLO
AND ARIMA
the totally alligned, L = 2nd + 1, and totally alligned minus one, L = 2nd + I - 1, states again give rise to bands whose energies are given by E[d”“, L, = 24 ; (j,j,)Z; L = 2n, + I] = E(d”“, L, = 24) + cl + n,yl+?
(11.4)
and E[dnd, Ld = 2n, ; (j,j,)Z; L = 2n, + I - l] = E(dnd, Ld = 2nd + l z + n,y,+,
+ ((24 + Ml+
2)) 4+, .
(11.5)
Here ez represents the energy of the two-quasiparticle state, the coefficients y, parametrize the interaction between the quasiparticles and the quadrupole d-boson and (11.6) AZ-1 = Yz-1 - Yz * The resulting spectrum is shown in Fig. 24 where the two new bands have been called QP and QP’. It is, once more, instructive to consider the energy differences LIE. However, it is more convenient in this case to plot AE = E(Z + 2) - E(Z) as a function of the angular momentum I as shown in Fig. 25. The equations defining the two lines in Fig. 25 are “EOP
=
0
Ed +
W'4
+
I
I
I
2
4
6
Yz+e
I
8
=
Ed +
((I-
0/2)C,
+
I
I
I
I
I
10
12
14
16
18
.J'z+2
(11.7)
I I
FIG. 25. The energy differences AE = E(Z+ 2) - E(Z)as a function of the angular momentum I.
INTERACTING
BOSON
313
MODEL
and A.E,,~ = cz + w4 + ~z+z + = Q + ((I- z + W)
(2/U
~4
+ 2)) A,+, + yz+z + (2/U
+
2))
A,,,
(11.8)
and they appear again to be parallel to the Y-band. Since in this case the decays of members of the QP and QP’ bands to members of the ground state Y-band require a large change in phonon number, these transitions are highly forbidden. The two quasiparticle bands QP and QP’ will therefore preferentially decay within the bands until they reach the band-head. The final transition from the band-head to the ground state band should be retarded relative to transitions within the band. The presence of a two-quasiparticle state with high spin below the corresponding state in the ground state band will give rise to a dramatic backbending effect. However, if the two quasiparticles couple to the quadrupole mode as in Eq. (11.3) the slope of the QP-band in the AE versus Z plot should be identical to that of the Y-band. In Fig. 26 we show some recent experimental data [47] in 156Er which is the isotone to 150Sm and lj2Gd. Finally in Fig. 27 we compare the available data in lh6Er with the predictions in Sections 4 (Y-band), 9 (N-band), and 11 (QP-bands). The energy level systematics are fairly well reproduced by the simple model of [43]. However, in view of the transitional character of 156Er definitive statement about the nature of the lateral bands will have to await the results of a more detailed study of the role of the symmetry breaking terms.
FIG. 26. AE versus Z plot for the yrast band in IseEr. The experimental from [47]. The broken line is the theoretical prediction, Eq. (11.7).
data (full line) are
314
IACHELLO
s
I
E~P
i
.Th
-
23--r-
Exp
AND
i
Th
ARIMA
,,,“I’
Th
E,,“i’“Th
“‘+T"+T
FIG. 27. Probable band structure of IseEr. In the absence of detailed information on the decay properties of the lateral bands, the assignment QP= lo+ 0 dn, QP” = 3+ 0 d”must be regarded as tentative. The parameters in the theoretical spectrum are c = 365 keV, c, = 78 keV, q = 1647 keV, x5 = -162 keV, cl,,+ = 2965 keV, yIz = 32 keV, c3+ = 1728 keV yS = -189 keV.
12. CONCLUSIONS Here and in [I] we have presented a model for the collective states of nuclei which pictures them as states of a system of interacting bosons. After formulating a general framework in which vibrational and rotational nuclei appear as limiting cases, we have explored in detail the vibrational limit and presented algebraic expressions for energies and transition matrix elements. Our main purpose here has been that of providing complete and, in some case, simple expressions to be checked experimentally, rather than that of performing detailed numerical calculations. In the one case which we have studied in detail, llOCd, it appears that a detailed and accurate description can be achieved within the framework of the vibrational limit with small perturbations. This points to the possible existence, in certain regions of the periodic table, of approximate W(5) 1 O+(5) boson symmetries. Moreover, the boson representation seems to provide a good description not only of the enhanced E2 transitions but also of the retarded Ml transitions.
INTERACTING
315
BOSON MODEL
We hope that the simple algebraic relations presented here will stimulate experimentalists to perform more detailed analysis of data on other nuclei. For tests of this and similar models the best set of experiments appears to be that of the Krane-Steffen [32] type, in which energy levels, E2 and Ml transitions are studied simultaneously. It would also be interesting to compare with experiment the predictions of EO transitions, and to study the systematic behavior of the various parameters cO, c2, cq , etc. which appear in our formulas. We stress once more the phenomenological aspect of our approach in comparison with the more elaborate boson expansion methods or with the calculations of the Kumar-Baranger [48] and Greiner-Gneuss [27] type. Although a generalization of the formal results presented here to odd-A nuclei is straightforward, we have postponed it until we will have completed a detailed analysis of some specific examples. ACKNOWLEDGMENTS We wish to thank H. Feshbach, A. Kerman, D. Kurath, A. Lande, R. D. Lawson, M. Macfarlane, M. Peshkin, J. Schiffer, R. H. Siemssen, and I. Talmi for interesting discussions and 0. Scholten for performing some of the numerical calculations presented here and Mrs. M. Beijerman for her expert typing. We are also grateful for the interest shown in our work by many experimental groups, especially by Z. Sujkowski, P. P. Singh, 0. Schult, R. D. Smither, P. von Brentano, J. F. W. Jansen, A. Gelberg, R. F. Casten and M. J. A. de Voigt. One of us (A. A.) wishes to thank R. H. Siemssen for the hospitality extended to him at the K.V.I. where part of this work was done. This work was performed as part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie” (F.O.M.) with financial support from the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek” (Z.W.O.).
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Mat. Fys. Me&. Dan. Vid. Selsk. 27, No. 16 (1953). 10. D. JANSSEN,R. V. JOLOS, AND F. D~NAU, Nucl. Phys. A224 (1974),
93.
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11. J. P. ELLIOTT, Proc. Roy. Sot. A245 (1958), 128; 245 (1958), 562. 12. M. HAMERMESK, “Group Theory and its Applications to Physical Problems,” AddisonWesley, Reading, Mass., 1962. 13. N. KEMMER, D. L. PURSEY, AM) S. A. WILLIAMS, J. &f&z. Phys. 9 (1968), 1224; S. A. WILLIAMS AND D. L. PURSEY, J. Math. Phys. 9 (1968), 1230; H. J. WEBER, M. G. HAKER, ANJJ W. GREINER, 2. Physik 190 (1966), 25. 14. G. SCHARFF-GOLDHABER AND J. WENESER, Phys. Rev. 98 (1955), 212. 15. F. IACHELL~ AND A. ARIMA, Phys. Len. 53B (1974), 309. 16. T. K. DAS, R. M. DREIZLER, AND A. KLEIN, Pbys. Rev. C2 (1970), 632. 17. D. M. BRINK, A. F. R. DE TOLEDO PEA, AND A. K. KERMAN, Phys. Lett. 19 (1965), 413. 18. P. LEAL, FERREIRA, J. A. CASTILHO ALCARAS AND V. C. AGUILERA NAVARRO, Phys. Rev. 136 (1964), B 1243. 19. G. RACAH, Group theory and spectroscopy, Princeton lectures 7 and 8, 1951. 20. H. J. LIPKIN, “Lie Groups for Pedestrians,” North-Holland, Amsterdam, 1965. 21. H. UI, Ann. Phys. (N.Y.) 49 (1968), 69. 22. F. IACHELLO, Proc. Int. Conf. on Nucl. Phys. and Spectroscopy, Vol. I, p. 163. Amsterdam, 1974. 23. M. SAKAI, Proc. Topical Conf. on Vibrational Nuclei, Zagreb, 1974; J. Phys. Sot. Jupun 24 (1968), 576. 24. R. K. SHELINE, Rev. Mod. Phys. 32 (1960), 1. 25. H. Enm, Institute for Nuclear Studies (Tokyo), Rep. Nos. INSJ 101 (1966) and 104 (1967), unpublished; H. EJIRI, M. ISHIHARA, M. S-1, K. KATORI AND T. IMAMURA, J. Phys. Sot. Japan 24 (1968), 1189. 26. B. BAYMAN AND A. LANDE, Nucl. Phys. 77 (1966), 1. 27. G. GNEUSS, V. MOSEL, AND W. GREINER, Phys. Lett. 30B (1969), 397; 31B (1970), 209; 32B (1970), 161; G. GNEU~.~ AND W. GREINER, Nucl. Phys. A171 (1971), 449. 28. H. A. M. HUSSEIN, 2. Physik 230 (1970), 358. 29. H. W. TAYLOR, A. H. KIJKOC, AND B. SINGH, Nucl. Phys. A141 (1970), 641. 30. M. SAKAI, Nucl. Data Tables 15 (1975), 513. 31. T. NUMAO, Y. YOSHIKAWA, Y. &DA, AND M. SAKAI, Bull. Phys. Sot. Japan, Oct. 13a-H-12 (1974). 32. H. MORINAGA AND P. C. GUGELOT, Nucl. Phys. 46 (1963), 210. 33. G. SCHARFF-GOLDHABER, M. MCKEOWN, A. H. LUMPKIN, AND W. F. PIEL, Phys. Lett. 44B (1973), 416. 34. B. SORENSEN, J. Phys. Sot. Japan Supplement 34 (1973), 399. 35. T. I~HIMOTO AND T. TAMURA, J. Phys. Sot. Japan Supplement 34 (1973), 393. 36. A. L. ROBINSON, F. K. MCGOWAN, P. H. STELSON, W. T. MILNER, AND R. 0. SAYER, Nucl. Phys. Al24 (1969), 553. 37. K. S. KRANE AND R. M. STEFFEN, Phys. Rev. C2 (1970), 724. 38. W. T. MILNER, F. K. MCGOWAN, P. H. STELSON, R. L. ROBINSON, AND R. 0. SAYER, Nucl. Phys. Al29 (1969), 687. 39. R. P. HARPER, A. CHRISTY, I. HALL, I. M. NAGIB, AND B. WAKE~~LD, Nucl. Phys. Al62 (1971), 161. 40. S. G. STEADMAN, A. M. KLEINFELD, G. G. SEAMAN, J. DE BOER, AND D. WARD, Nucl. Phys. Al55 (1970), 1. 41. J. M. EISENBERG AND W. GREINER, “Nuclear Theory,” Vol. I. North-Holland, Amsterdam, 1970. 42. V. MARUHN-REZWAMI, W. GREINER, AND J. A. MARHUN, Phys. Lett. 57B (1975), 109. 43. A. ARIMA AND F. IACHELLO, Phys. Left. 57B (1975), 39.
INTERACTING
BOSON
MODEL
317
44. M. J. A. DE VOIGT, Z. SUJKOWSKI, D. CHMIELEWSKA, J. F. W. JANSEN, J. VAN KLINKEN, AND S. J. FEENSTRA, Phys. Lett. 59B (1975), 137. 45. D. R. ZQLNOWSKI, T. KISHIMOTO, Y. GONO, AND T. T. SUGIHARA, Phys. Lett. 55B (1975), 453. 46. M. NOMURA, Phys. Lett. 55B (1975), 357. 47. A. SIJNYAR, private communication. 48. M. BARANGER AND K. KUMAR, Nucl. Phys. A62 (1965), 113; All0 (1968), 490; Al22 (1968), 241; K. KLJMAR AND M. BARANGER, Nucl. Phys. A97 (1967), 1; A119 (1968), 65; Al42 (1970), 411.
OF PHYSICS
99, 253-317 (1976)
interacting
Boson Model of Collective I. The Vibrational Limit
States
A. ARIMA Department
of Physics,
University
of Tokyo,
Tokyo,
Japan
AND
F. IACHELLO Kernfysisch
Versneller Instituut, University Groningen, The Netherlands
of Groningen,
Received March 2, 1976
We propose a unified description of collective nuclear states in terms interacting bosom. We show that within this model both the vibrational tional limit can be recovered. We study in detail the vibrational limit and to the possible existence of an unbroken SU(5) 3 O+(5) symmetry. We set of analytic relations for energies and electromagnetic transitions.
of a system of and the rotabring attention derive a large
1. INTRODUCTION In contrast with other many-body systems, the nuclear collective motion is characterized by vibrational and rotational frequencies of comparable order of magnitude, preventing a clear-cut distinction between the two types of motion. Moreover, other intermediate situations can occur in nuclei, corresponding for instance to asymmetric rotations and/or spectra which are neither vibrational nor rotational. In order to accomodate these facts we have recently proposed [l] a unified description of the collective nuclear motion in terms of a system of interacting bosom. Boson representations of the collective motion are not new. Following the suggestion of Belyaev and Zelevinski [2] and of Marumori et al. [3] many elaborate calculations have been performed and we refer the reader to somerecent papers by Sorensen [4], Kishimoto and Tamura [5], Marshalek [6], and Holzwarth and Lie [7] for a review of this work. Our approach is different from the above mentioned authors in that we do not attempt, at this stage, a derivation of the boson 253 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
254
IACHELLO
AND
ARIMA
Hamiltonian from the fermion basis, although some work in this direction has been done by one of us (F.I.) in collaboration with Feshbach [8]. Rather we are interested in (i) providing a unified description of collective nuclear states irrespective of their detailed nature (i.e., vibrational, rotational, etc.) and (ii) producing simple limiting situations which can be solved analytically. In relation to these simple situations, our main purpose is to point out the possible existence of unbroken boson symmetries (and to classify nuclei according to representations of the associated symmetry groups). By exploiting their group structure is then possible to obtain analytic relations to be tested experimentally. One of the main advantages of the model of [l] is that, under certain conditions to be described below, it can be shown to have analytic solutions corresponding to the vibrational and axially symmetric rotational limit. These limits are always defined with respect to the classical description of the collective quadrupole motion given by Bohr and Mottelson [9] which forms the basic framework for a discussion of all nuclear collective properties. The second advantage of the model described in [l] is that it reproduces, after slight rearrangement, the Hamiltonian derived by Janssen, Jolos, and Diinau [lo] using the Lie algebra of pair operators. We believe that [lo] is of considerable importance in the understanding of a microscopic theory of the collective nuclear motion since it produces a Hamiltonian of finite order in the boson operators and which, in addition, includes the effects of the Pauli principle through the cut-off factor. In the present paper, after a short review of the basic idea, we will consider in detail the vibrational limit. By exploiting the SU(5) I> O+(5) group structure of this limit and using standard group theoretical methods we will derive sets of analytic relations for energies and transition rates. In addition to the quadrupole excitation mode we will also discuss other excitation modes, in particular the collective octupole mode and two-quasiparticle modes. Again, under certain restrictive assumptions, we will derive analytic expressions for energies and transition rates. These relations will not be confined to the electric quadrupole (E2) operator but will include the Ml, EO,El, M2, and E3 operators. Contrary to what one might expect, these operators appear to be equally well described in terms of boson operators. This is somewhat surprising since, in contrast with the E2 case, the induced transitions are not enhanced but in fact largely retarded. Finally we will consider a small breaking of the SU(5) 1 O+(5) symmetry and derive expressions for some observable quantities in first-order perturbation theory. These relations are useful in practice, since a small breaking of the symmetry, even in the best of the cases, will always be present. A discussion of the boson SU(3) group and the sets of analytic relations which can be derived from its group structure in the axially symmetric rotational limit, will form the subject of a subsequent paper.
INTERACTING
2. THE INTERACTING
255
BOSON MODEL
BOSON MODEL:
QUADRUPOLE
STATES
In constructing a simple model of the collective nuclear motion we start by realizing the quadrupole (L = 2) character of the excitations. We also note that, because of the limited number of particles in each self-consistent shell, there will be a cut-off in the vibrational and rotational bands built on each shell. A model which has these properties has been proposed in [l]. Here a number of positive parity states are constructed as states of a system of N bosons having no intrinsic spin but able to occupy two levels, a ground state level with angular momentum L = 0 and an excited level with angular momentum L = 2, as shown in Fig. 1. For zero splitting between the two levels and in the absence of any interaction between the bosons, the five components of the L = 2 state, called d for convenience, and the single component of the L = 0 state, called s, span a linear vector space which provides a basis for the representation of the group SCJ(6). These representations are characterized by the symmetry properties of the wavefunctions. For bosons the only allowed representations are the totally symmetric ones, belonging to the partition [N] of SU(6). In the absenceof interaction and for zero splitting, all states belonging to [N] are degenerate. The energy difference E = cd - Edand the boson-boson interaction lift the degeneracy and give rise to a definite spectrum. The spectrum is defined by E, by the two-body matrix elements (d2L 1 V 1d2L). (L = 0, 2,4), (d20 I I’/ s20), (ds 2 I V I ds2), (ds 2 1 V 1d22), (s20 / V 1s20) and by the partition [N] of SU(6) to which it belongs. Thus in our approach, each nucleus is characterized by a set of nine parameters and the energy levels can be found by diagonalizing the model Hamiltonian H = E,S+S+ cdC dvn+d,,, +
c $(2L + 1)1/2cl[(d+d+)‘l’ (dd)(O](O) I. + Z’e[(d’d’)k (&)‘2’ + ;;:;+;M (dd)(2)](0) + (l/21/2) fio[(d+d+)‘o’(B)(O) + (s+s+)(~) (dd)(“‘](o’ + u2[(d+s+)‘2’ (ds)(2)](o)+ $u~[(s+s+)(~) (s,s)(~)](~)
(2.1)
where d+(d), S+(S)are the creation (annihilation operators for bosonsin the L = 2, L = 0 state and the parentheses denote angular momentum couplings. The ”
‘d
.I
”
d
L=2
s
L=O
t
E ES
b,,,,
FIG. 1. The configuration&PinthebosonSU(6)model.
256
IACHELLO
AND
ARIMA
parameters CL, CL, uL have been introduced for convenience and are related to the two-body matrix elements by cL = E2 = 5, = u2 = ug =
(d2L j V 1d2L) (ds 2 I V 1d22)(5/2)1/2 (d20 1 V 1s”O>(I/~)~~~ (ds 2 1 V 1ds 2)51/2 (s20 1 Y j s20).
(2.2)
The numerical coefficients appearing in Eq. (2.2) are the inverse of the matrix elements of the operators in square-brackets in Eq. (2.1) between two-boson states. To the extent that Eq. (2.1) describes collective quadrupole states, there are associated transition operators. The electric quadrupole operator is defined in terms of the two reduced matrix elements (d I/ Q iid) and (d/I Q 11S) and given by Ty’
= q”,[(dts)t’
+ (std)f’]
+ q,‘(dQ@.
(2.3)
All quadrupole transitions are thus characterized by two number which have been written for convenience as
= @II Q IIsW5Y” 92’ = (dll Q II 4. q2
(2.4)
Again the numerical coefficients in Eq. (2.4) are the inverse of the matrix elements of the operators in bracket in Eq. (2.3) between one boson states. It is interesting to note that Eqs. (2.1) and (2.3) reduce to the Hamiltonian and quadrupole operator derived by Janssen, Jolos, and Diinau [IO] using the Lie algebra of pair operators. Consider in fact the basis states 1snsdnd[N]xLM> where x is whatever quantum number is needed to specify uniquely the states, and n, = N - ngj. Operating with H on these states one can replace the 9 and s operators by c-number functions of nd H j sN+dnd[N] =
I
xLM)
E,(N - nd) + cdtzd + 1 Q(2L + 1Y2 c&dtd’)‘“’
(dd)‘=‘l’“’
L
+ u,(N - n,J n8/51f2 + u. i(N - nJ(N
- nd - 1) I sNpnd
1
+ d,{[(d+&)@’ d](O) (N - n,)li2 [ sN-nd-‘d”d[N + (N - nd + I)li2 [dt(dd)c2)](‘)
““EN1xLM>
- 11 xLM)
1sNendfldnd[N + 11 xLM)}
+ $o{(dtdt)(OJ ($(N - n,)(N - nd - I))“”
[ sN--nd-2dnd[N - 21 xLM)
+ (&(N - nd + l)(N - nd + 2))lj2 (dd)“’
1sNwnd’“dBd[N - 21 xLM)).
(2.5)
INTERACTING
Eliminating to give
BOSON
257
MODEL
the degrees of freedom of the s-boson one can rearrange this equation
H 1d’““XLM)
= p* + E’& + c 3(2L + 1)1/2 c,‘[(d+d+)‘l’
@d)(L)](O)
L
+ 02’[[(d+d+)(2) d](O) (N - nd)l12 + (N - nd + l)l/% [d+(dd)‘2’]‘o’]
+ vo’[(d+d+)‘o’((N - n,)(N - lid - l))‘/”
(ddpl! I d”dXLM)
+ ((N - lzd + l)(N - ?zd+ 2))“2
(2.6)
where ho = E,N + uogv(N - 1) E’ = E + ((u2/W2) - u&N - 1) CL ’ =
CL
+
(u,
-
(u2/W))
(2.7)
us’ u = 212 ilo’ = zo/21/2 which is identical to [lo, Eq. (13)]. Similarly, operating with TiF2’ on the basis states j PG?“~[N]xL.M~ one has T(E~) k
is N--nddnd[N] =
q2{dk+(N
XLM) -
np2
+ (IV - nd f + qJdtd)p’
j s
N--nd-ld”qv - l] XLM)
1)112(-)” / sN-“ddnd[N]
d-, 1 Pnd+ldnd[N
+ l] xLM))
xLM).
(2.8)
Eliminating again the degrees of freedom of the s-boson, one can rewrite this equation as Tp2’ j d’QLM) = {&[dkt(iv
(2.9) -
/I,$/~ + (A’ - nd + l)lj2 (-)”
d-k] + q,‘(d+d):‘}
1 dndxLM\.
We now show that for different choices of the parameters E,... the Hamiltonian and transition operators of Eqs. (2.1) and (2.3) produce both vibrational and rotational spectra. As these parameters change the SU(6) model spans the entire veriety of observed spectra. We begin by considering the case in which the energy difference E is much larger than the interaction terms cL , 6, , uL . In this casethe Hamiltonian is invariant under separate transformations among the five components of the L = 2 state. Thus the statesare characterized by the number of bosons occupying the L = 2 level, nd, and an (approximately) unbroken symmetry
258
IACHELLO
AND ARIMA
emerges from the decomposition SU(6) 1 SU(5) @ U(1). The quantum number n, plays no role in this case. The representations of SU(5) contained in [N] are all the symmetric representations [nd = 01, [nd = 11, [nd = 21 up to [IZ~ = Iv]; This is the limit which will be discussed in detail in this paper. The other limiting situation occurs when the energy, E is small and of the same order of magnitude of the two-body matrix elements cL , 6, , uL . In particular if both the energies E, , cd and the two-body matrix elements correspond to those of a quadrupole-quadrupole interaction Y = -K Ci,j Qi ’ Qj in a major oscillator shell, where K is the strength of the interaction and Qi the quadrupole moment of the ith boson, another approximate symmetry occurs [Ill. The related wavefunctions serve now as a representation space for the groups SU(6) r) SU(3) r) O+(3) and they are characterized by the quantum numbers ][N](X, p) KLM) where (h, CL)label the representations of SU(3) belonging to the partition [N] of SU(6). In this limit the energy levels are given by
WW,
pL)KLW = 4%-W + 1) - CC&p>l
(2.10)
where C(h, p) is the Casimir operator of SU(3), C(h, CL)= h2 + pz + hp + 3(h + p). The properties of the boson W(3) limit, which describes the axially symmetric rotor, will form the subject of a subsequent paper.
3. THE VIBRATIONAL LIMIT:
STATES OF THE dna CONFIGURATION
In the vibrational limit, states are characterized by the number of bosons occupying the L = 2 state, n&d nd configurations). As mentioned above the quantum number n, is not needed in this case and it will be omitted from here on. As it is well known [12], states of nd d-bosons form the basis for the totally sym metric irreducible representations of the group SU(5) and of its subgroup O+(5). Five quantum numbers are needed to classify uniquely the states [13]. Three of them are trivial, the d-boson number nd , the angular momentum L and its third component M. The fourth is the boson seniority Y. Instead of v one can introduce another quantum number no which counts boson pairs coupled to zero angular momentum. n, is related to u by v = nd - 2n, .
(3.1)
Finally one can introduce a fifth quantum number nd which counts boson triplets coupled to zero angular momentum. The total number na is partitioned by na and nA as nd = 2nB + 3nA + h (3.2)
INTERACTING
BOSON
and the values of the total angular momentum of h by
259
MODEL
for each nd , n, , ~2~are given in terms
L = A, x + 1, h + 2)...) 2h - 2, 2h
(3.3)
where the conspicuous absence of L = 2h - 1 should be noted. The vibrational limit in the conventional sense of the word, is described, in the boson represen‘tation, by the Bohr Hamiltonian [9] H = Ec d,+d,,,
(3.4)
111
and transition
operator TLE2) = q2 (dk + + (-)” h
d-,)
(3.5)
where the boson number nd can take on any value 0, 1, 2,..., co. This Hamiltonian and transition operator can be derived from the SU(6) model of the previous section in the following way. First rewrite Eqs. (2.6) and (2.9) as H = EC drn+dv,+ C &(2L + 1)lj2 cL[(d+d+)‘L’(dd)(L)](o) +w~~2{[(d+d+)(2)~](oJ (1 - (r~,/N))l/~ + (1 - ((nd - 1)/N))1/2 [d+(dd)‘2’]‘0’} + uoKd+d+)(0) ((1 - (ndW(l + ((1 - ((n, - l>/NMl
- ((nd + l)lNWi”
- (h
- 2)lW)Y12 WY09
(3.6)
and ,p,
= qJd+(l - (t~d/N))l’~ + (1 - ((nd - l)/N))l’”
(-)”
de,] + q,‘(d+d)‘,)
(3.7)
where the prime has been dropped on E and cL , ho = 0, 1’2= v2’(N)lj2, v. = zlo’N and q2 = g2(N)l12. Now let N -+ co (no cut-off) and neglect all terms in Eqs. (3.6) and (3.7) except the first one. This reproduces Eqs. (3.4) and (3.5) respectively. Nuclei away from closed-shells and regions of large deformations are indeed often characterized by vibrational spectra [14] but with large splittings between members of the two-phonon (nd = 2) triplet O+, 2+, 4+, relatively large quadrupole moments of the first excited 2,+ state and rather small values of the transition matrix elements for the forbidden An, = 2 transitions. Motivated by this observation, we prefer [15] to introduce another vibrational limit described by the Hamiltonian H = Ec dm+dln+ c 4(2L+ 1)lj2 cL[(d+d+)tL’(dd)(L)](o) (3.8) nt
L
and by the transition operator T;F’ = q2(dk++ (-)”
hk) + q2’(d+d$?.
(3.9)
260
IACHELLO
AND ARlMA
This limit can be obtained from Eqs. (3.6) and (3.7) by neglecting the d-bosonnumber changing terms v2 and v0 and by letting N -+ 00. In this limit the observed large anharmonicities are caused by the boson-boson interaction, the quadrupole moment of the first excited 2,+ state is due to the second term in Eq. (3.9) and the dnd = 2 transitions are strictly forbidden. This model, which we call the interacting d-boson model, is related to other models previously discussed in the literature. In particular it is identical to that of Das, Dreizler, and Klein [16] if one drops the terms d, , czl , a, in [16, Eqs. (8) and (9)] and to that of Brink, Kerman and de Toledo Piza [17] if one assumes that the d-boson-number changing terms are small from the beginning and therefore no unitarity transformation is needed to bring the Hamiltonian in the form of Eq. (3.8). This is a necessary phenomenological condition since otherwise the terms eliminated from the Hamiltonian will reappear in the transition operator thus giving rise to An, = 2 transitions which are observed to be small. The advantage of the d-boson limit is that the additional terms in Eq. (3.8) split but do not admix the different representations of SU(5) r) O+(5). Thus using standard group theoretical methods we are able to obtain the eigenvalues and transition rates in analytic form. It is this symmetry (also proposed in a slightly less general form by Ferreiro, Alcaras, and Navarro [18]) and its experimental verification that we want to emphasize.
4. SOLUTION OF THE d-BosoN MODEL:
THE BOSON QUASI-SPIN
GROUP
In this section we discuss an analytic solution to the eigenvalue problem H 1ndvnALM) = E 1ndvnALM)
(4.1)
where His given by Eq. (3.8), and nd , v, nA , L, A4 are the five quantum numbers which label the totally symmetric irreducible representations of SU(5). The method of solution is due to Racah l-191. In the course of our discussion we will also review some known properties of the boson quasispin group SU(1, 1) which is used in the solution. We begin by noting that two d-bosom can only have three values of the total angular momentum L = 0,2 and 4. Therefore three coefficients c,, , c2, c, completely describe the boson-boson interaction. Tnstead of the second quantized form of Eq. (3.8) we may use the first quantized form. Since there are only three coefficients which describe it, we rewrite the interaction Vij between the ith and jth boson in terms of three operators, which we choose as the unit operator Iif, the pairing operator Bij and the operator L?~~= 21i . Ii , where Ii , lj are the angular momenta of the ith and jth boson (4.2)
INTERACTING
BOSON
MODEL
261
The parameters (Y,/3, y are related to the parameters c0 , ca , c, by the relations Eq. (4.22) below. The expectation value of the operator I = c lij
(4.3)
i
in the state j ndunJA4)
is trivially given by (1) = g?&d - 1).
(4.4)
9 = 1 Pij
(4.5)
That of the operator i
is also easily obtained, since 9 is related to the total angular momentum Lz=p;+z.
L = xi Ii (4.6)
Thus the expectation value of 9 is (8)
= L(L + 1) - 6n, .
(4.7)
In order to find the expectation value of the pairing operator we introduce three operators S, = 4 C (-)” m
dmtd:,
S- = 4 C (-)” ?n
d,d-,
(4.8)
So = $ C (dm+dm + d,d,+). m
These operators are the generators of the boson quasi-spin group SU(1, 1) [20, 211. This group is noncompact contrary to the fermion case in which the quasispin group is the compact SU(2). Using the commutation relations of the boson operators d+(d) it is easy to show that [S+ ) s-1 = -2s,
Kl ?S+l = s+ [S, ) s-1 = -s-
(4.9) .
In Cartesian coordinates s, = &.(S+ + s-) s, = (1/2i)(S+ - SJ s, = s, 595/99/2-3
(4.10)
262
IACHELLO
the commutation
AND
ARIMA
relations are [S, , S,] = -8,
[S, , &I = is, LX , &I = is, to be compared with ordinary W(2) group
the commutation
relations
(4.11)
among the generators
L-L, -&I = i-G [L, , Ll = i.L
of the
(4.12)
[L, , L,] = iL, . Similarly the Casimir operator of SU(1, 1) can be written
as
c = $2 - S,” - s,2 = s,2 - (S+S- + S,) = s,2 - s, - s+sto be compared with the Casimir operator
(4.13)
of SU(2)
c = L,2 + L,? + L,2
(4.14)
The eigenvalues of the pairing operator .P now can be easily found since 9 = 4&S-
(4.15)
and therefore B = 4[S&s, -
1) - S(S -
I)]
(4.16)
where we have used the fact that c = S(S -
1).
(4.17)
The eigenvalues of S,, can be obtained from its definition Eq. (4.8). By commuting the last two operators we have s, = (l/2) c a,+LL + (l/4) c 1 = (%/2) + 5/4. ?n m
(4.18)
The eigenvalues of S can be obtained in a similar fashion [21]. They are given by s = (U/2) + 5/4
(4.19)
Inserting Eqs. (4.18) and (4.19) in Eq. (4.16) we obtain (9)
= (Hd - v>(n, + 2’ + 3)
(4.20)
INTERACTING
and the eigenvalues of the interacting
BOSON
263
MODEL
d-boson Hamiltonian
E(nd, 0, HA, L, M) = End + &7&,
- 1) + &,
[15,22]
- v)(n, + v + 3)
+ y[L(L + 1) - 64 which do not depend on nd and M. The parameters c2 , cq by considering states with nd = 2. One has
(4.21) 01,/3, y can be related to cO ,
cp = CY+ 8y c, == cx- 6y
(4.22)
ro = a + lop - 12y and the inverse relations 01= (l/14)(&,
+ 8c,)
P = (3/7(J) Ca- (l/7) cz + (l/IO) co
(4.23)
Y = (l/14)($ - ce). The matrix elementsof the operators I, 9, LY have also been obtained in a slightly different way by Kishimoto and Tamura [5]. In addition to providing the eigenvalues of the pairing interaction 9, the group reduction SU,(l, 1) @ O,+(3) is also useful in other respects. For later use we list some of its properties [21]. All states with same o but different nd can be generated from the maximum seniority states by means of the relation I S, S, , nd , L, M\ = ( I/JV(S, Sz))(S.k)sz-s/ S, S, = S, nd , L, M)
(4.24)
where the normalization factor is obtained from the recurrence relation Jv(S, S,) = (S, - S)(S, + s - 1) J-(S, s, - 1)
(4.25)
Jv”(S, s, = S) = 1
(4.26)
with Thus 1s, sz , ?lA , L, M)
= ( (S,- &I$
- I)! !
l/2
(S+)sz-s I S, S, = S, nd , L, Mj.
(4.27)
Replacing S and S, by Eqs. (4.18) and (4.19) we have 112 nv + Pm (l/Z) h-o) I nd , 4 11 A ’ Ly M, = ((hd - u)/2)! T(v + (S/2) + (nd - e)/2) 1 @+I X j nd = c, c, nA , L, M) (4.28)
264
IACHELLO
AND
ARlMA
where r(n + 1) = n!. The matrix elements of & in the seniority representation can be obtained from those in the quasispin representation a &’ I s+ I s, &> = ~,~,Ss,,((~z + S)(S, - s + lW2
= &s,~,S,-1KSz - SK% + s - l))‘i”.
(4.29)
They are
6.,f,n,+2W2)
+ ((nd+ WW
+ Knd- WWi2
+ ((nd + W)Mnd - d2)W
(4.30)
The interacting d-boson Hamiltonian generates spectra, Eq. (4.21) with strong regularities. An example is shown in Fig. 2. For reasons which will become apparent in the following Section 5 we have arranged the different states into “bands.” We have named the important bands Y, X, Z, X’, Z’, p, d and defined them as follows Y-band
/ nd , nd , 0, L = 2nd 2 m
X-band
1nd , nd , 0, L = 2nd - 2, M)
Z-band
j nd, nd, 0, L = 2nd- 3, M)
r-band
1nd , nd, 0, L = 2nd- 4, M)
Z’-band
/ nd , nd , 0, L = 2nd- 5, M)
(4.31)
1nd , nd - 2,0, L = 2nd- 4, M) 1nd, nd, 1, L = 2nd- 6, M).
p-band d-band
Some of these bands are shown in Fig. 2. Our classification coincides to some extent with that of Sakai [23] who following a suggestion by Sheline [24] introduced the concept of quasi-band in vibrational nuclei. For later purposes, it is interesting to plot the energy differences dE = E(n, + 1) - E(n,) as given by Eq. (4.21) for the various bands (Figs. 3). Here the different bands appear as a set of parallel linear trajectories, with equal slope c4 . The corresponding equations can easily be derived from Eq. (4.21). For the important bands they are
A& AE, AE, AE,
= = = =
E+ E+ E+ E+
cqnd cqnd- 8y cand- 12y cqnd+ 48 - 16~
nd 3 nd 3 nd 3 nd >
0 2
3 4.
(4.32)
A 3-i
‘i
i
y
p I
g&y
>-
,Ol
4+*,ol r9(4.4,01 5+&4,0) ,+J43(4.4;0)
r” w 2-
;I+&,01
2”4L4,1,
g&3,0) ++B.O) 4+&?,o)
z+fi
3+(3.3,01
,Ol
o+(3.3,1) ,+~,01
2+m,o)
I-
+ (4.0,O) o-
*t&O)
0 1,+(0,0,01 FIG. 2. A typical spectrum in the d-boson limit. The numbers in parenthesis are the SU(5) quantum numbers (na , u, Q). The angular momentum quantum number is explicitly written to the left of each level. The parameters used are 6 = 579 keV, c, = 39.4 keV, ca = -95.3 keV, co = -27.4 keV. 1000
r
750
soo5 0
2
4
8
8 “d
FIG. 3. The energy differences AE = E(n, + 1) - E(nd) given by Eq. (4.3i) as a function of nd . The parameters d, c,, , c, , cd are the same as in Fig. 2.
266
IACHELLO
AND
ARIMA Y
FIG 4. The energy E as a function (a) c, < c2 < c,, ; (b) c, > c2 > c, .
of the angular momentum
L for the various bands.
In Fig. 4 we plot the energy E as a function of the angular momentum L for the various bands. In case (a) cq < c2 < c, the bands never meet. In cast (b) c, > c2 > cOthe bands start crossing for a certain value L,, of the angular momentum L, becoming in turn yrast bands. In the interacting d-boson model all energies are expressed in terms of four parameters E, c, , c2 , c4 which can be determined by a fit to the observed energies. Moreover, the energies of the members of the Y-band depend only on E and cp , while the X, Z, X’, Z’ bands depend only on E, cq and c2 . Thus having determined E and cq by a fit to the Y-band, one can then use the X and Z bands to extract the third parameter c2. A knowledge of some members of the p-band is however necessary to obtain c,, . Incidentally we note that the energies of the members of the Y-band can be rewritten as E(nd , nd , 0, L = 2nd, M) = i(4e - 3~~)L + &L(L
+ 1)
(4.33)
which is the formula proposed by Ejiri [25] in his fits to the ground state bands of rotational and transitional nuclei.
INTERACTING 5. ELECTRIC
267
BOSON MODEL
QUADRUPOLE
TRANSITIONS
Another advantage of the interacting d-boson model is that the matrix elements of the electric quadrupole operator, Eq. (3.9), can be obtained in compact form. Since the boson-boson interaction does not admix states of different boson number, the results of this section apply to the harmonic quadrupole vibrator as well. The reduced matrix elements of the E2 operator of Eq. (3.9) can be obtained numerically by means of the usual coefficient of fractional parentage technique. Here (na + 1, x’, I’ /I d+ I/ /?a , x, 1) = (nd + l)i” il[~P(xI)
dI’ I} d”d+‘,y’I’]
(5.1)
and 0% , x, III dll nd + 1, x’, 0
= (-)“-‘
+ 1, x’, I’ II d+ II nd , x, 0.
(5.2)
In these equations x is whatever quantum number is needed to specify uniquely the states, f = (21+ l)lj2 and the bracketed symbol is a boson cfp. Similarly one can construct the reduced matrix elements of the operator U;’
= ( 1/5”2)(d+d)l”’
(5.3)
as (nd , x, 111U’) Ij nd , x’, I’> = nd ~J(-)r+Jii’[d”“-l(tcJ)
dZ I>dndxI]
. [d”d-l(tcJ) dl’ I} d”“x’I’] 1; ‘J ‘;I
(5.4)
where the symbol in curly bracket is a Wigner 6 - j coefficient. Since tables of boson cfp are available [26] the numerical calculation of the reduced matrix elements can be easily carried on. However, we are interested in obtaining analytic expressions for these matrix elements and we will therefore describe in this section an alternative approach to obtain matrix elements in algebraic form. We restrict ourselves to the matrix elements of the d+(d) operators since those of the U2) operator can be obtained in a similar fashion. We begin with matrix elements among states of maximum seniority. The basis states can be written as 1nd , n, = 0, nA = 0; L, M) = A$,,,
C (2,2n, - 2, m, M, j L, M) d,+ 77040
x 1nd - 1, n, = 0, nA = 0; 2nd - 2, M,,).
(5.5)
268
IACHELLO
AND
ARIMA
From here up to Eq. (5.16) we will drop the unnecessary indices ns , nd . The normalization constant in Eq. (5.5) is determined from the condition
1 = Jci.L ,,C,
M P,2nd-22,
m’, MO’ I L, M)(2, 2nd - 2, m, MO I LM)
* #J*1. 0’
*(r&j--
1;2nd-2,M,‘jd,,d,+Ind-
1;2n,-2,M,)
which can be evaluated by using the boson commutation
(5.6) relations
d,,,jd,+ = dmtd,l + a,,,
the Wigner-Eckart
(5.7)
theorem, and the basic relation + 1)li2
(nd ; L = 2nd 11d+ 11nd - 1; L’ = 2n, - 2) = (r1#/~(2L
(5.8)
The result is A”
-
nd.L -
112 l)(4na - 3) W(2n, - 2, L, 2nd - 4,2nd - 2; 2,2) ) . (5.9)
1
t 1 + (-)”
(nd -
Here and in the following formulas we have preferred to use the Racah coefficient W(f, , I2 , 11’, I,‘; J, k) = (-)z1+1~+z1’+z2’ i, i
f,
1
k”l
(5.10)
rather than the 6 - j symbol, because of notational convenience. Incidentally, one recovers the result mentioned in Section 3, that L = 2nd - 1 is not allowed, since in this case W = l/(nd - 1)(4n, - 3) and Jy;ld,L --+ co. In order to calculate the reduced matrix elements of the transition operator we write
(nd+ l;L’,M’IdmtInd;L,M)
* (2, 2nd - 2, m”, Mi j L, M) x (nd 3.2n, 3 MO’ I d,,d m‘d;” 1nd-
1;2n a - 2,
Ml).
(5.11) ,
Then making use of the relation (5.12)
INTERACTING
of the Wigner-Eckart
BOSON
269
MODEL
theorem and of the relations (5.8) and (5.9) we obtain
= (2L’ + 1)1/S (n&1/2 [(-)“’
S,,,,, + (-)L ((2L + 1)(4n, + l))“/”
x W(L, 2~ - 2, L’, 2n, ; 2, 2)(1 + (-)”
(nd - 1)(4n, - 3)
x W(2nd - 2, L, 2nd - 4, 2nd - 2; 2,2))] /(l + (-)“’ x (1 + (-)”
nd(4n, + I) W(2n,,
L’, 2nd - 2, 2nd ; 2, 2))1/2
1)(4n, - 3) W(2n, - 2, L, 2nd - 4,2n, - 2; 2,2))‘/“.
(nd -
(5.13)
The corresponding matrix elements of d can be obtained from the phase relation (5.2). Eq. (5.13) can be specialized to the various cases. Denoting by F the reduced matrix elements of d
F= (nd;LIIdllnd+
1,L’)
(5.14)
we have for transitions within the bands of Eq. (4.31) Y-band
L’ = 2n, + 2 + L = 2n,
F= h
X-band
L’ = 2nd
F=
Z-band
L’ = 2nd -
-+L=2n,-2
1 -+ L = 2n, - 3
+ 1)(4nd + 5)Y
+ 3)(4n, + 1) lj2 > (4nd - 1) dnct - 2)(2n, + 1)(4n, - 1) l/2 (nd - l)Ch - 1) (5.15) (nd -
F =
1X%
and for intraband transitions X+Y
L’=2n,
Z+Y
L’=2nd-
Z-+X
L’=2nd-l+L=2n,-1
(4n, + 2)(4nd + 1) v2 (4nd - 1)
-+L = 2n, l+L=2n,
F= (
t2% - 2)(4nd + ‘> l” (2nd - 1)
(5 16)
v2 ’
These expressions are summarized in Fig. 5. The B(E2) values can be obtained from Eqs. (5.15) and (5.16) in the usual way B(E2 Ii -+ I,) = (l/(21, + 1)) \(I-- I/ TcE2’ I/ Ii>12.
(5.17)
The “bands” defined in the previous section appear then to be connected by transitions with large B(E2) values. For each band, except the Y-band, an increasing
270
IACHELLO
AND
ARIMA
L-ad-3
FIG.
5. Reduced matrix elements of d between states of maximum
seniority.
fraction of the decay goes into the Y-band as the spin decreases. However, for large spin values, the lateral bands, X, Z,... are very stable, in the sense that the major part of their decay proceeds within the band. The fraction which branches out behaves as I/Q (Fig. 5). Thus for large spins, an anharmonic vibrator whose anharmonicity is only due to the boson-boson interaction gives rise to a set of “channels.” These channels do not “communicate” to each other until low enough values of the spins are reached. Analytic expressions for the matrix elements of the transition operator can also be obtained for members of the p-band. It is convenient here to make use of the boson quasi-spin group SU(1, 1). The basis states of the p-band can be written as /nd,nB=
1,n,=O;L=2nd-4,M)
= (2/(2n, + 1))lj2 S+ 1nd - 2, nB = 0, nA = 0; L = 2nd - 4, M).
(5.18)
In order to calculate the reduced matrix elements of d+(d) between states of the form (5.18) we observe that d+ commutes with S, [d,+, S+] = 0.
(5.19)
Using this relation, the first of Eq. (4.9) and the phase relation Eq. (5.2) we then obtain the reduced matrix elements of d for transitions within the B-band
@band
L’ = 2n, - 2 + L = 2,~~- 4 F =
end- l):22nnd,++:):4nd - 3) )1’2a (5.20)
INTERACTING
The intraband
BOSON
271
MODEL
matrix elements can be obtained making use of the relation [S- , d,+] = (-)”
(5.21)
d-,
and the phase relation Eq. (5.2). They are given by F =
B-Y
L’=2nd-2+L=2n,
8-X
L’=2n,-2+L=22n,-2
F=
B-Z
L’=2nd-2+L=2n,-3
F=-
2k(4~ + 1) 1’2 ( (2nd + 3) ) (
;;z;;;;-si’
(
For nd 2 4 the p-band can also decay to the X-band. can be obtained using the sum rule
)1’2
T;;;;;f;-r;))1’2.
The matrix elements /3 - X’
+ 1)) C I
(1/W.
(5.22)
(5.23)
L’
and they are given by B-X’
L’ = 2nd - 2 ---f L = 2nd - 4 Wd
- Wd
112
- 1)
1 .
(5.24)
The results (5.20), (5.22), (5.24) are summarized in Fig. 6. Before concluding this
Lz:r,~‘l
FIG. 6. Reduced matrix elements of d between the j%band and the states of maximum
seniority
272
IACHELLO AND ARIMA
section we remark that the algebraic forms of Eqs. (5.13), (5.20), (5.22), and (5.24) can be used to calculate the boson cfp from the relation Eq. (5.1) Vd(XZ) aT II dnd+lX’J’l = (& : 1)1,2 (ZI’ +y 1)1,2 0% + 1, x’, 1’ II cl+II % , x, 0. (5.25) These algebraic relations may therefore be useful in other contexts, as for instance within the framework of the Greiner and Gneuss [27] method.
6. THE VIBRATIONAL
REGIONS: A SURVEY
We now consider the regions where a description in terms of a SU(5) 3 O+(5) symmetry may be appropriate. A necessary but not sufficient condition is that the forbidden dn, = 2 transitions be small, and that a vibration-like two-phonon pattern is present with Of, 2+, 4f states in the neighborhood of twice the energy of the first excited 2+ state. The regions in the periodic table which exhibit these properties are shown in Fig. 7. They are characterized by having both neutrons
N
FIG. 7. Possible vibrational regions (dashed areas). The even-even nuclei shown in the chart are those for which the energy of the first excited 2+ state is known.
and protons outside the closed-shell. The most favorable conditions seem to be when N (or 2) is only 4-6 particles away from the closed-shell and 2 (or N) is S-10 particles away from it. For larger particle number a rotational pattern is exhibited. The best candidates appear to be the Pd(Z = 46) and Xe(Z = 54) isotopes (Regions II and III) and the Pt(Z = 78) isotopes (Region V). From the predictions of the quadrupole d-boson model without phonon mixing obtained in analytic form in Section 4 and 5, it is now easy to see whether they
INTERACTING
BOSON MODEL
213
provide a first-order approximation to energy levels and transition rates. In Figs. 8-10 we show the comparison between observed and calculated levels in lo2Ru, l°Cd, and l**Pt. The cases of lo2Ru and “OCd illustrate some difficulties which arise in comparison with experiments. Since vibrational nuclei exhibit other excitation modes such as two-quasiparticles and collective octupole modes, special care must be taken in comparing with experiment. The level scheme alone is not sufficient to determine whether or not a state belongs to the quadrupole mode. Detailed knowledge of their electromagnetic decays is also necessary. In the case of “OCd this information is available (see Section 7) and one is in a position to compare the observed levels with the collective quadrupole mode prodictions. In general, states of the octupole mode are easily identified by their negative parity. Instead, for states of positive parity, it is more difficult to distinguish between collective states and states of two-quasiparticle character. The latter will appear in the spectrum at excitation energies E > 24 (where A is the pairing gap) as shown in Fig. 11. Unless the mixing with the collective quadrupole states is small (as it appears to be in ll°Cd) these states will admix with the quadrupole states washing out the collective structure. Recent results of (heavy ion, xny), (LX,xny), and (p, xny) experiments, a technique
OL-,*
-o+
8. Comparison betweenexperimental and theoretical spectrum in loaRu. The experimental levels 4+ at 1872 keV and Of at 1837 keV are taken from [28], the 3+ level at 1837 keV from [29], and all others from [30] and the nuclear data sheets. The parameters in the theoretical spectrum are E = 481 keV, c, = -18 keV, cz = 141 keV, c, = 144 keV. FIG.
- i 8.--
8’
FIG. 9. Comparison between experimental and theoretical spectrum in “OCd. The experimental levels are taken from [30] and the nuclear data sheets. The parameters in the theoretical spectrum are E = 722 keV, c0 = 29 keV, c2 = -42 keV, cq = 98 keV.
Other
Stcies
‘B8P+ 78 110
FIG. 10. Comparison between mental levels 4+ at 1084 keV and [30] and the nuclear data sheets. co = 148 keV, c, = 30 keV, cq =
experimental and theoretical spectrum in IsaPt. The experi5+ at 1442 keV are taken from [31], all the other levels from The parameters in the theoretical spectrum are E = 281 keV, 110 keV.
INTERACTING
,
Two
C
BOSON
Collective
-
&adruDoie
States
s+
-
2+
3+ -
-
‘I
275
MODEL
0’
0’
"2A
0
E
-
2+
-
0’
FIG. 11. Schematic representation
of the energy spectrum in even-even vibrational
nuclei.
E (MeV)
FIG. 12. Comparison between experimental [33] and theoretical energies of the Y and X bands in loOPd. The parameters used are E = 680 keV, ce = - 160 keV, cg = 45 keV. The average deviation between experimental and theoreGca1 energies is 10 keV for the Y-band and 36 keV for the X-band.
276
IACHELLO
AND
ARIMA
pionered by Morinaga and Gugelot [32], have provided much additional information. These reactions favor the population of collective states and act as a sort of filter allowing an easier identification of the collective states. An example is shown in Fig. 12. Here comparison with the observed levels is straightforward but a precise identification of the various bands remains difficult since a careful study of their decay properties cannot be performed. (For example, in the case of l”OPd it is not clear whether the observed band to the right of the Y-band is the X-band or rather one of the bands discussed in Sect. 11). Figs. 8-10 and 12 show that the d-boson model without phonon mixing provides a reasonable first order approximation to the energy spectrum. Similar conclusions can be drawn if one considers the electromagnetic decay properties of the collective quadrupole states. In Table I we show a collection of B(E2) branching ratios in the Xe-isotopes. We observe that all forbidden dn, = 2 transitions appear to be small, TABLE
I
B(E2) Branching Ratios in the Xe-isotopes” lszXe 54 78
;;I z 2:
/
31+ + --------.( 31+ -+ 3t+ + 31+ +
21+ 2,+ 41* 2,+
lsoXe 54 78
lesXe 54 14
lzeXe 54
1.01
1.48-1.99
12
lz4Xe 54
PO
d-boson limit
(x 10-S)
0.17
0.57-0.64
x 1o-3
1.03
1.41-1.54
0.51
0.24-0.25
-
3.11-3.42
-
1.28
-
-
0.95-1.05
-
0.94
0.90
10 - = 0.909 11 0
5t+ --L 4,+ (x10-2) 5,+ + 31+ 51+ - 4,+ 5lf + 31+ 5r+ --f 6,+ 51’ -+ 31+
1.99 0.46-0.72
1.86 - 3.89 2.99
0
0.16
2 -=0.400 5
-
3.38
-
4.97
3.90
-
0.46
-
1.26
0.98
-
0.64
-
-
-
0
5 - = 0.454 11 104 - = 0.450 231
o The experimental values are from the nuclear ,data sheets. When the experimental results disagree. with each other, both the smaller and the larger value is quoted. All transitions are assumed to be pure E2. Note that forbidden An, = 2 transitions are given in units of 10-e.
1NTERACTING
BOSON MODEL
277
about 1O-2 of the allowed transitions, and that the allowed transitions appear to follow the values expected for the d-boson model without mixing. We recall that the expected values are just ratios of integers (Section 5). By looking at the first line in Table I one can observe that there is a general tendency for the forbidden transitions to increase toward the middle of the shell. This tendency supports the idea of Sakai [23] that there is a smooth change from vibration-like to rotation-like spectra. The data of Table I also illustrate another difficulty which arises in comparing with theoretical predictions. The branching ratios quoted have been obtained by assuming that all transitions are pure E2 transitions. In order to make more detailed comparisons with experiments the amount of E2/Ml mixing has to be known. In few cases the mixing ratios are known and we will discuss one of these cases in Section 7. Further, before doing so, we consider the effect on energies and transition rates of a breaking of the SU(5) I) O+(5) symmetry. A small breaking will be always present, even in the best of the cases, since the boson symmetries are not symmetries of the underlying fermion Hamiltonian, but rather of the boson Hamiltonian. However, the breaking of the SU(5) r) O+(5) symmetry can still be treated within the framework of the unified interacting boson model, Eq. (2.1), to which we now return for a proper treatment of the energy spectrum and transition rates.
7. SYMMETRY BREAKING
IN PERTURBATION: ENERGIES AND QUADRUPOLE TRANSITIONS
In regions where the boson SU(5) r) O+(5) symmetry is only slightly broken one can use the results of the previous sections to perform first-order perturbation calculations based on the vibrational limit. In the Hamiltonian of Eq (3.6) the interacting d-boson model diagonalizes the first two terms and we now treat the remaining two terms as the perturbation. (This eigenvalue problem has been solved numerically by Janssen, Jolos, and Donau [IO]). However, if th.e additional terms 2,2 , o,, are small, we prefer to use perturbation theory, in view of the transparent results. We also note that the effects of the one- and two-phonon changing terms have been investigated by Sorensen [34], Kishimoto and Tamura [35], and Holzwarth and Lie [7], although without cut-off. We find the inclusion of the cutoff essential in producing results which are in agreement with experiment. Moreover, it is the presence of the cut-off in Eq. (3.6) which provides the SU(6) group structure necessary to produce rotational spectra. To demonstrate the importance of this term we consider the simple Hamiltonian (without cut-off) H = EC d,+d, - 60(51/2/4)[(d+d+)(0) + (dd)“‘]. m
595/9912-4
(7.1)
278
IACHELLO
AND
ARIMA
This Hamiltonian can be diagonalized by introducing group, Eq. (4:8),
the generators of the SU( 1, 1)
H = ~(2s~ - 5/2) - Go& .
By performing
(7.2)
a complex rotation in the space of the group So + ch @$, f sh @S, (7.3)
S, --f sh @$, + ch @S, and choosing the angle @ such that th@ = 6,126
(7.4)
one can bring H to diagonal form H = -~(5/2)
+ 2(9 - (6,2/4))“2
$,
(7.5)
with eigenvalues E = -(5/2)
li = 0, l)...) co.
E + (5/2)(c2 - (6,2/4))lj2 + (e2 - (6,2/4))‘/2 ri
(7.6)
Thus the addition of the term [(LP~+)~~)+ (dd)(O)] without cut-off is ineffective in changing the structure of the spectrum, since it only displaces and compresses it. Returning to Eq. (3.6) we introduce the notation + (1 - ((n, - 1)/N))1/2 CM)
H = i& + i&(C+(l - (n,/N))‘l”
+ 2fio,(~+Kl - hdmu
- (42 + ww2
+ ((1 - @d - l)lWl
- ((&I - ww1’2
s-1
(7.7)
where f& = E c d,+d, + c g(2L + 1)1/Z c&I+d+)‘l’
nl
C+ = (5/2)lj2 [d+d+)‘2’ d](O)
c- = c++
S+ = (51/2/2)(d+d+)(0)
s- = s++
co = vo/5112
(@(S](O)
L
C2 = (2/5)‘/”
(7.8)
v2 .
We denote the unperturbed solutions of ir, by [ nd, L, v, nA) and display v and nd only when necessary. To construct the perturbed wavefunctions, denoted by 1nd , L, v, nd>, we need the matrix elements of the operators Ch, S, in the basis defined by ir, . The matrix elements of Si have already been given in Eq. (4.30).
INTERACTING
BOSON
MODEL
279
TABLE II Boson Wavefunctions in First-order Perturbation
Theory
izd= 0 1o,o>> = I 0, o> + (y
(yy
f I 2,O)
726= 1 11,2))=
; 1,2) + (9,“”
~12.2>+(;)l’z(~)*‘~(~)1’z~,3,2)
Fld = 2 2, 4)) = I 2, 4) + (y(yy2 j 2,2)) = - (Jgy
E’ I 3,4> + & I’ I 1,2> + I 2,2) + Jg
+&(~)“‘(~)l”6,4,2,v 1 2,0> = - (;)li2
(~)l’p(~)l’*
6 ( 4, 4, 5 = 2)
(yys
r ) 3,2)
=2>
(v)l”
f I 0, 0) + I 2,0> + 61ia (%)I”
6’ 13,o)
+71/a(~)1’a(~)“2~,4,0> ??d= 3 I3,6)) = I3,6)+3
(;)l”
I 3,4)) = - (ye
(qy
f’ I 2,4) + I 3,4> - g.g
(F)lla
5’ I4,4, o=2)+(;)1’z
+ 30’/” 4(’ Iv*
1 3,3)) = I 3,3) + (y ] 3,2)) = - (;y
(v)lil
5’ / 4,6)+(;)“’
(Jglla
(2Lg
(J!y)1’2 (L!gL)“”
+ 13,2) + y
(=)I”
(?)I” (Jgli2
P-3”’
fJ ) 4,4, v = 4) 515,4, v=3>
5 ( 5, 3)
f ) 1,2> - 3g
(~)“”
p , 2,2)
6’ 1 4,2, v = 4) + -?-. (N--l)“’
I’ I 2, o> + I 3, o> + q
+ (!31’y~)1’8(231’~~15,0)
t/5,6, u=3>
(~)1’2(~)1’p
+3(“,‘).‘2(~~‘2f,5,2,v=3) ) 3,0> = --6l/*
(y)1’2
71’2 (=y
N f’ j 4,O)
N
5’ 1 4 9 2 , tr zz 2)
TABLE
III
Matrix Elements of the Quadrupole Operator of Eq. (3.7) in First-Order Perturbation
Theory”
N---l{ Cl, 2 II T’EZ’ II 1,2> = Y’2 lY*f + 9.2SY2(2)“2 N) <(I, 2 (1TfE2’ /I 2,4> = 3 ;qp 21’2 (y)‘:’ 3(110)“2 <2, 4 II T’E2’ I]2 >4)) = ___ 7 <<2,2 II T’E2’ ll2,4> ((1 \ > 211 p21 [j2,2 >,=
I = 7l2 (d
(1 + t y)
+ q;t’ ; (?)“‘I
N-2 + qze wY’” - N 1 N-l - q2P’ ___ 7 ~2(2)‘la ( N ( p
5112jq,2111
(!g”‘(l
I 15N-2 -
+ &q)
:q;*qy)“l
g2 x , 211 T’EZ’ II 2,2> = 51’ 2 (i - 37 q2’ - q2f’ &?)I/2 (!$A N-l (7
<(2, 0 11TIE*’ 112,2> = [2q?’ - q$sI (2Y go, 0 II T’EZ’ 112,2> = w <(1,21IT’E2’)\2,0>=
1-q2g- (!!$)“’
/l T’-’
113,4>
+ qz’,$21/z(!!$)“‘/
[Bz2’!9(~)1’z(l
<<2,2\\T’E2’(\3,4>=
= 3 /q2 (G)‘!’
-S;tEfy)
(y)l”
+ qa’( (y
+4m(~)y
+[y)
+q;
(1 + fy)
3 /q2(;)1’z(~)1’p(1
<(l, 2 )I z-‘EZ’ 113,4> = 3 1 -q2r (y
- q;<‘s
(~)““l
+
(T)‘” (N--g
C2,4,,T’EP’,,3,3)_7ii~j~~(~~‘d(~)1’a(1 <2I , 2 11T@’
+ ; !!y)/
I - p-2___ N 1\
<2,4~~~‘~~‘~~3,f$)=,3V/qE3112(~~‘2(l <2,4
)I
1) 3, 3)) = 71/a 1 - q2(;)1’z(E$2)l,l(l
(yy (N-g/ +[y) +*!!z$)
-q;+$(~)l’rl +q;~~(~)“P~
~l,21,T1Bzll13,3)~71,P~~qI~(~~‘e(~~’~(~~’a + qz’# (q!)‘/’
(!yy*
(!g”“1
0 Some matrix elements involving states with nd = 4 are included.
Table
continued
TABLE
III
(continued)
282
IACHELLO
AND
ARIMA
Those of C+ are given in terms of the reduced matrix elements of the operator of Eqs. (5.3) and (5.4) as
U@)
0% 7x, L I c- I % + 1, x’, L) = (S/2)“” (nd + 1)1’2 1 (i/E)[dnd(tc’J’) K’.J’
dL I> dnd+lx’L]
II n dY tc’, J’).
x (4t 5 x 3 L II W’
(7.9)
Since they involve only cfp’s, these matrix elements can be obtained in algebraic form using the methods of Section 5. We now list in Table II the perturbed wavefunctions for states up to nd = 3. The notation is e = i&,/e, e = &/E. Using these TABLE
IV
Energies of the Boson States in Perturbation
Theory
nd = 0 E(O.0)
= E(0, 0) -
5y
g%
a!?(& 2) = E(l, 2) - 7 y y
8% - N+
I’%
nd = 2 &2,4)
E(2, 4) - 9 y
7
I”< -
,??(2,2) = E(2, 2) - 9 y
y
g”c + (y
=
k(2,O) = E(2,O) + (5 y
Tld=
-
y
!!!I$
gf2< - f y)
g-e
14N-2N-3 N-x-
)
3
&3,6)
= E(3,6)
- 11
&3,4)
=
E(3,4)
- 11N+ !$? 52~+ (; !?$ - g N+) (‘zE
&3, 3)
=
E(3, 3)
- 11N+ y
gee
INTERACTING
BOSON
283
MODEL
wave functions and the results of Section 5, we can calculate the reduced matrix elements of the quadrupole transition operator of Eq. (3.7) in first-order perturbation theory. Some of them are given in Table III. Similarly one can calculate the effect of the perturbations on the energies. This is shown in Table IV, where -&a 2 L) denote the unperturbed energies given by Eq. (4.21) and &nd, L) the perturbed energies. For sake of completeness we have also calculated wavefunctions and transition matrix elements in second-order perturbation theory up to nd = 2. They are given in Tables V and VI. To this order one has to take into account the effect of the boson-boson scattering terms cL . The notation in Tables V and VI is 74 = cq/c, qz = c,le, 70 = C,/C We now return to Table III in order to study the effect of the various symmetry breaking terms. To examine the effect of the cut-off we set 6 = c’ = 0. In Fig. 13 B(Ez,L’=2nd-+
L=2nd-2)
E(E2,L4=2-L=O)
Vibrational
Model
/
3
5
6
10
7
14
9
18
II
“d
22
L’
FIG. 13. Effect of the cut-off on the B(E2) ratios.
we show the ratios B(E2, L’ = 2n, + L = 2~ - 2)/B(E2,2 -+ 0) as a function of nd . One can see that these ratios deviate more and more from the pure vibrational limit as one approaches the top of the band, na = N. However, the cut-off does not affect B(E2) ratios for decays of states belonging to the same nd . For instance B(E2, 31+-+ 4,+)/B(E2, 3,f - 2,+) = 2/5 which is the same as in Table I. The effect of the .$ term in conjunction with the cut-off modifies B(E2) ratios even between members of the same multiplet. For the two-phonon triplet, it lowers the O,+ --t 2,+ transition relative to the 41+ + 2,f and 2,+ + 2,f. Finally the 5 term further modifies the B(E2) ratios. The combined effect of the symmetry breaking terms is shown schematically in Fig. 14.
TABLE
V
Boson Wavefunctions in Second-Order Perturbation
tld= 0 IO,O>= (1 - ;y + (;jl’a
f;) j o,o>+(;j”‘(~j”2f(l (!yj1’2
(!yj”’
Theory
-$jl2,0> 51’ 1 3, 0)
rid = 1 I1,2>=
7N-1
(1 -yfyNNfaj + (yy
[fv
-
+ (!yj1’2
(!!g2
N-2
/ 1,2)
72) + 2y [(;j1’2
ffj
12, 2)
f ( 1-~~~-f~~-~~~)+(~)“‘~‘]I3,2>
rtd = 2
1%4>=
(1 -? 11!X$p~-$?$2C!$*~)~2,4) + (y’
(y”*
[ 5’ (1 - ; ‘la - ;
Q)
+ 27
ffq
+(~j”‘(~~‘*[~f~-~~~-~Z)1-~~4)+~~~],4,4,~=2) _(~~‘z(~j1’*5(~)1’z~l,4,4,a
12,2>=
=4) -;E+;yyf2),2,2)
(1 -;yp -(~)li*[t’(l-%~+4~ff$l,2) + (yj1’2
[G
1.5 N-3 + -1411e N
ff
I’ (1 -
N-l 141/a -
g Q + ; 112 - 5 To) EE’] I3,2>
+(~~‘z(~~‘a,~f(l-~~~-t~~-~~~j+~~~]14,2,u=2> +(~~‘a(~)l’a~~~,4,2,u=4) I2,0>=
(1 -;ypr
7N-2N-3
7
p -
3y
c”)
( 2, o>
-(gyy~‘*f(l-+):o,o) +6qgy(l
-331,+?,)+2yffq
13,O)
1 3,4>
INTERACTING
Matrix Elements of the Quadrupole
Cl, 2 II FEZ’ II 17 2) = y/a qa -N-lN I
2(2F
lON-1 + q2’ (1 - czy7
[
TABLE VI Operator in Second-Order Perturbation
N-2 C(l - 72) + 51’3 -yN--l + I”,,
1
N-2 )I
IN-lN-2
285
BOSON MODEL
9N-2N-3
Theory
286
IACHELLO B(E2.L’-L
AND
ARIMA
)
B(E2.2,
-
0,)
~Conventional ,JVibrationaI
(2*-2,)‘... 2
i
---------
tl-
oi
0.1
0.2
L’
i Model
0.3
14. Combined effect on the B(E2) ratios of the symmetry breaking terms as a function of 5 for rqz’;= 0.7 6 q2/W2and N = 6. The experimental data are from [36]. FIG.
1542.4 -1475.7 ‘1473.2
0+-
FIG.
15. Decay scheme llomAg -+ WZd [37].
INTERACTING
BOSON
287
MODEL
A test of the d-boson model with perturbations is meaningful only when many of the matrix elements of Table III are known experimentally. Presently there are few such cases. The decay of the high spin 6+ state in llomAg (Fig. 15) provides an almost unique example since it populates only high spin states in “OCd thus acting as a filter in the selection of the collective states (see Fig. 8 for comparison). Krane and Steffen [37] have carefully analyzed E2/M1 mixing ratios in this decay. From their data it is possible to extract 8 B(E2) ratios. Moreover, Milner et al. [38] have measured by Coulomb excitation 3 more B(E2) values, and the quadrupole moment of the 2,’ state is also known 1391. Thus in total there are 12 matrix elements known to be fitted with the 5 parameters .$, [‘, q2 , q2’ and N. We have done the fit by arbitrarily assuming N = 6, 5 = -[‘, q2’ = -(0.7/5112) q2 and the corresponding values are shown in Table VII. The picture which emerges from TABLE
Comparison
between Experimental
Transition B(E2),,r,(ezfm4)
B(E2),,,,/B(E2)w
and Calculated B(E2) Values in ll°CdG Ratio to
Ratio talc
Ratio exp
1.63 1.30 1.25 1.1 x 10-Z
(1.53 + 0.19) (1.08 f 0.29) (5.5 rt 0.9) x 10-a
31-+2, 31-+2,
0.31 2.0 x 10-Z
(0.47 f 0.20) (1.7 * 1.0) x 10-a
1533 1222 1171 14 989 304 20 831
30 49 39 37 0.45 32 10 0.64 27
592
19
42 - 2,
0.71
15 762 292 300 26
0.48 24 9 10 0.84
4, - 2,
1.8 x 10-Z
’ - 0.23 (0.7 rt 0.1) x 10-a
51 + 3, 5, + 31 51- 31
0.38 0.40 3.4 x 10-z
(0.39 f 0.23) (0.54 f 0.10) (0.7 zt 0.6) x lo-*
938
Calc
Q, (e
VII
fm9
B(E2)2, -+ Ol(e2 fm4)
-42 938* (fixed)
o
23
+
0.38
EXP
-31 * 9 or -55 l 8 934 f 38
a The data for the 21 -+ O1 , 4, --f 2, , 22 + 2, transitions are from [38], the QZ value from [39], and the remaining data have been extracted from the measurements of [37]. The errors shown are our estimates. The parameters used are 5 = -F = 0.3, q2 = 24.5 e fm*, qz’ = -7.7 e fme. B(E2)w in column 3 is the Weisskopf single particle estimate.
288
IACHELLO
AND
ARIMA
Table VII is that the transition rates in “OCd are not very different from those of a pure vibrator. The forbidden And = 2 transitions are all systematically small, of the order of single particle units or less. The allowed transitions, with the possible exception of the 4, --f 4, , are also close to the pure vibrational limits 0.40(3, + 4,), 0.91(4, -+ 4,), 0.45(5, --+ 6,), and 0.45(5, -+ 4& despite the large anharmonicities observed in the energies. This supports the suggestion that the large anharmonicities in the energies are due to the boson-boson interaction and that the symmetry breaking terms are small and give rise only to small deviations from the vibrational limit in the transition rates. We have also explored the sensitivity of the calculated B(E2) values to the parameters .$ and [‘, and found that equally good fits can be obtained for e and 5’ in the range 0.1 < t, -F < 0.3 by appropriately choosing q2 and cd. The symmetry breaking terms also affect the energies. In Fig. 16 we compare the experimental and calculated level sequence. In addition to the levels observed by Krane and Steffen [37] we have added two more levels which we believe to be the remaining members of the 3-phonon quintuplet. Comparing Fig. 16 with Fig. 8 ,tmc305a
6t312462) 4+-(2302)
4+J&
(15l81
*+~(I4431
2+~(2288) 3+-(2202)
ty~(2002)
(-j+J5g1473)
2t-(704)
0+2 FIG. 16. Quadrupole states in “OCd. The calculated energies are in parenthesis. In addition to f = -5 = 0.3 the parameters used are E = 740 keV, c, = 100 keV, c, = -120 keV, co = 30 keV.
one can observe that no substantial improvement in the calculated energies is obtained by introducing the symmetry breaking terms. This can be understood since these terms act in second order in the energies (Table VI), while the bosoninteraction acts in first order. Moreover, because of the cut-off, their matrix elements decrease as nd increases, while the matrix elements of the boson-boson interaction increase with nd . Once more we stress the fact that it is the combined
INTERACTING
BOSON
MODEL
289
effect of the cut-off and of the symmetry breaking terms which makes them unimportant. Before concluding this section we remark that, according to Table III, all quadrupole transitions in the d-boson limit with perturbations are connected by a set of simple relations. Thus they should show systematic behavior across a set of isotopes. This systematic behaviour has been observed experimentally [36] and it is shown in Figs. 17 and 18 for the Cd and Pd isotopes. In cadmium the
56
60
64
68
56
60
64
68
56
60
64
68
2.0 i I
0
LI~L.--...LI!~i
FIG. 17. Systematics of the Cd isotopes. The B(E2) data are from [38] and the quadrupole moment data are from the nuclear data sheets. The curves are drawn to guide the eye.
290
IACHELLO
0.5
t
I 56
'
AND ARIMA
I .--U.-L 60
64
Pd Z=46
-50
2 0” “2--
0 -25
ii;:-i
25w 56
FIG. 18. Systematics of the Pd isotopes. The B(E2) data are from [36] and the quadrupole moments from the nuclear data sheets. The curves are drawn to guide the eye.
effect of a partial shell closure at N = 64 is clearly observed [40]. From Fig. 17 it appearsthat the symmetry breaking terms which give rise to the B(E2), 2, --t 0, , and to deviations from the vibrational limit in the other transitions, are smallest for 112Cdand increaseas we move away from this nucleus. We have not attempted a fit to the other Cd isotopes similar to that of “OCd becausethere are no available
INTERACTING
291
BOSON MODEL
data comparable in detail to those of Krane and Steffen. Such data, especially the measurement of E2/Ml mixing ratios, would be very helpful in determining the nature of the symmetry breaking terms.
8. OTHER ELECTROMAGNETIC
TRANSITIONS:
EO AND Ml
Since the interacting d-boson model appears to provide a good description of the observed energiesand electric quadrupole transitions in the vibrational limit, we inquire whether an equally good description of other electromagnetic transitions is possible within the sameframework. The basic one-body operators in the theory are (d+s)@), (d+d)c2) and (s+s)co).Thus the first natural extension of Eq. (2.3) is to consider electromagnetic operators of the form
T:' =
c?$,,[(d+s)~)
+ (s+d):‘]
+ &(d+d):’
+ yE6108mo(s+s)~).
(8.1)
The additional terms now contain all electromagnetic operators EO, Ml,... up to 1 = 4. We begin by considering the Ml operator TiM1’ which we rewrite, for notational convenience, TCM1) = m,(d+d)F) k
(8.2)
In a microscopic description of the boson model [8] one can show that the coefficient m, is expected to be of the order of single particle units. However, the operator of Eq. (8.2) is proportional to the boson angular momentum operator L[41] and therefore in the boson basisit has only diagonal matrix elementscontributing to the g-factor of the excited states. This is true even if the wave functions are the perturbed wave functions of Table III sincethe angular momentum remains diagonal in this representation. The matrix elements of the operator of Eq. (8.2) are given by
(nd , x, L II TtM1’ II nd’, x’, L’) = m, (
L(L
+ 1)(2L + 1) lj2 6
10
1
nJL&’
*
(8.3)
In order to have nonzero matrix elements of the Ml operator we must therefore go to next order and form an effective two body operator from the product (d+s)(2) @ (d+d)fl). The most general operator of this form is CITl[(d+s)(2)(d+d)(z) + (d+d)(z)(s+d)(2)](1’. Ho wever, by recoufiling the operators d+d+d, one can see that there is only one independent term in the sum. Thus the most general Ml operator of this form can be written as TiM1) = m,(d+d)f)
+ fil’[(d+s)(2)
(d+d)“)
+ (d+d)(‘) (s+d)‘2’]t’.
(8.4)
292
IACHELLO
AND
ARIMA
The coefficient 6,’ is presumably small since it can come, in microscopic description [8] only from the Pauli principle in the fermion basis. As a consequence, Ml transitions are expected to be largely retarded in vibrational nuclei, a conclusion which is borne out by the experimental results. Within the boson basis j s”@~[N] xLA4) the operator of Eq. (8.4) is equivalent to @‘fl) = m,(dtd)(,) + ml’{[dt(dtd)“y’
(1 - @2,/N))“” + (1 - ((fld - 1)/N))“”
where m,’ = fi,‘(N)liz. The vibrational N + co in Eq. (8.5) and it gives Tf’fl) = m,(dtd)p
[(&Q(l) d]?‘} (8.5)
limit with no cut-off, is obtained by letting
+ m,‘[dt(dtd)(‘)
+ (dfd)(l) d]!‘.
G3.6)
The matrix elements of this operator can be easily evaluated and yield +d , x, L II ‘P’l’
II G + 1, x’, L’)
= ml’(-)L+L’+l
3112 L(L + 1)(2L + 1) 112 ( 10 1 (8.7)
where use of Eq. (8.3) has been made in evaluating the matrix elements of (#d)(l). Since the matrix elements of d have been obtained in algebraic form, Section 5, it is easy to construct those of TtM1) given in Table VIII. All matrix elements are given in terms of the single parameter ml’. Using Table VIII and the matrix elements of Section 5 one can calculate E2/M1 mixing ratios in the pure vibrational limit. TABLE Some Reduced Matrix Elements of the Ml
VIII
Operator in the d-boson Limit without Cut-off
II2,2) /I 3,4> (13, 3) 113, 3) I) 3,2> II4,6>
= = = = = =
--m,’ (21/10)‘la -m,’ 3(1 1/10)1~* ml’ 3(2/5)‘/* -m,’ 3(3/10)1/” --ml’ (3/5)1/S --ml’ (273/1O)‘/2
114, 5) = -ml’
(2731fa/5)
)I4, S> = m,’ (3/5) 21*j2
INTERACTING
BOSON
293
MODEL
However, because of the simple structure of Eq. (8.7), it is also possible to obtain these mixing ratios in analytic form. The mixing ratios ~WP/Ml'
=
(
nay
x,
L
II TtE2'
II nd
-t
1, x',
L'>/(nd,
x,
L
II F"~'
II nd
+
1, x',
L')
are given by L’ = L
A’E2IMl’ = --A(10/((2L
L’=L+l
A@/fifl’
L’=L-1
A(E’LIMl’ = --A(10/(3(L
- 1)(2L + 3))‘/2)
= -A(10/(3L(L
+ 2))‘/2)
(8.8)
- l)(L + 1))1/2)
where A = q2/m,‘. Some of these are shown in Table IX. In comparing the mixing ratios given in Table IX with those observed experimentally we remark that quite often the value of 8(E2/M1’= (P2’/PM1’)1/2 is quoted instead of d(E21M1’, where r(E2’(r(M1’) are the partial radiative widths for E2(Ml) radiations. SfE2jM1’ is related to dtE21M1’by a(=/Ml’ = 0.832 x 10-2J$d(E2Ibfl’
(8.9)
where E, is in MeV and AtE21M1) in e fm2/pn , TABLE
IX
E2/M1 mixing Ratios
4,5>
= = = = = = = =
-A -A -A -A -A -A -A -A
(lo/219 (lo/7719 (5/69
(2/3)51ie (10/2119
2(5/33)'je 2 (5/21)‘/2 (5/3(2)'/*)
We next study the effect of the symmetry breaking terms by evaluating matrix elements of the operator of Eq. (8.5) in the perturbed basis of Table II. These matrix elements are given in Table X where we have also included the diagonal matrix elements ((1, 2 // TfM1’ II I, 2)), ((2,4 // TIM1’ II 2, 4)), ((2, 2 11TtM1’ jl 2, 2)) which are related to the g-factors of the excited states. It is interesting to note that the ratio (2, 2 /I TtM1’ I/ 2, 2)) g2+2 _ ___ <:1, 2 11T’M1’ 11I, 2) g2t1 595i99h
(8.10)
294
IACHELLO
AND
TABLE Matrix
Elements
of the Ml
X
Operator
in First-Order
\‘> I, 2 jj T’M”
111, 2> = 311% m, - In,’ I’
x:.2,4 11I+‘”
jl2,4))
<\I 7 2 [I FM”
112, 2>, =
<2,2
11TfM1’ II2,2>>
= 3(2)“’
(C2,41/ ‘,I,2
TcM1’ II 3,3>
\<2 > 2 /I T’M”
+ *!y]
m, + m,' F
L
3,3> 113,2>
= m,’ 3 (3”’ j
= --ml'3
14 II2 N-
ii
3
-m,’
@‘i”(~j-
(w-2)~'2 [f
($!-j"' (!yj"'
27:'!2(N;3)'!'
(3 7 4 jl TfM1' 114, 5> = m,': 21112 (Fg" 11TfM1' II4,5?) = -m,'c
- 4 (;)‘“!7)]
[1 + (Kg]
(N+l)'i'
<3,6/j TcM1)/I 4,6> = --ml' fz?)"' q3, 6 11TcM1) 114,5;, = -ml'
1
[1 + (!y]
(3lo j':"f!!$,"'
/I, 2 /I TIM" (I 3,2> = ml'(' (i)"'
~“2, 4
N
1
+ fy]
(!yj’i’
(2)'i' = ml' e3 3 =
Theory
m, - ml’ I’ 11 ,s i2j1’%
-m~3($'"(~)':'[1
II TtM1j I/ 3,3>
11T’M1l[l
Perturbation
(?Lj’:“fy)“‘[I
-m,’
= 31/2
q2,411T’M1)j/3,4>= <2,2
[
ARIMA
f 66'12
-
((;!y
- yg)]
(y)*'* [I + f!y] [1 + *N-+] [I + *!$A]
(N-+2)"'
(Ng
which is 1 in the vibrational limit can be considerably affected by the perturbation. Recent experimental results obtained by Katayama, Morinobu, and Ikegami (private communication to Arima) show that this ratio is indeed different from one in ls2Pt and lg4Pt. An application of the formulas given in Table X is shown in Table XI. Once more we observe the surprisingly good agreement with experiment. In particular the sign is correctly predicted to be the same for all transitions. Thus it appears that the d-boson model is useful not only in correlating the enhanced E2 transitions
INTERACTING
BOSON
TABLE E2/Ml Transition
A tale
295
MODEL
XI
Mixing Ratios in *l°Cd [37]
(vibrational limit)
A,,l,
(with 5 = - 5’ = 0.3)
A exp
- I50* (fixed)
- 150* (fixed)
-140
-142
-102
-91
-175 & 22 -190 + 140 -70 -153 + 95
-78
-75
-44 rt 36
-67
-67
-120
-81
-82
-170 f 50
forbidden
-85
-44 do 8
forbidden
-60
-32 + 3
5 50
a All values are in e fme/pN .
but also the retarded Ml transitions. We remark that the mixing ratio d for the 2, -+ 2, transition corresponds to a B(M1) of approximately 1O-2 Weisskopf units. We also note that a treatment similar to the one presented here has been recently given by Maruhn-Rezwani, Greiner, and Maruhn [42]. In an analogous fashion we can calculate matrix elements of the EO operator T’Eo’ = qO(s+s)‘O’ + &‘(d+d)‘O’.
(8.11)
These matrix elementsare trivial since the two terms in Eq. (8.11) are proportional to the s- and d-boson number operators n, and nd . Within the boson basis 1Pdna[N] xLM) the operator of Eq. (8.11) is equivalent to PO) = qoN + q0’n,/W2
(8.12)
where qo’ = 4; - 51i2qo.
(8.13)
The first term in Eq. (8.12) contributes only to the root mean square radius, while the second term gives rise to EO transitions. Using the perturbed wavefunctions of Table JI, we can calculate the matrix elementsshown in Table XII. EO transition matrix elements vanish in the vibrational limit with no perturbation. Again all matrix elements are given in terms of the number qo’. It would be interesting to seeto what extent these relations are fulfilled in practice.
296
IACHELLO AND ARIMA
EO
TABLE XII Matrix Elements in First-Order Perturbation
((0, 0 II z-0’
II 2, o> = 40’5
w2 ((iv
II 2, 2> = 40’2’ (1/5)1’2
<(2, 4 11P”’
1) 3,4>
= q,‘f’
-
((1, 2 11T’EO’ (1 3, 2> = q(f (14/5)“2 I/ 3, 2)) = qa’f’ 2(2/35)“’ II 3,OB
(3,
II 4, 6)) = q,‘f’(3/W21
6 11P”’
9. OCTUPOLE
= q;f’
(6/5V
((N
l>/w’s - 2NV)“2
((Iv - l)/lv)l’”
<<2, 2 /I T’“’ <2, 0 11P”
1)/N)“’
((N
(22/35)“’
Theory
(W
-
((A’ -
2YNY
((N
3h’W’2
-
((N - 2)lw’2
2)/Wiz
STATES. ENERGIES AND TRANSITIONS
In addition to the quadrupole d-boson, other excitation modes are expected to play an important role in the description of the nuclear collective motion. Of particular importance is the collective 3- mode, hereafter referred as jlboson. In order to construct states of octupole character we consider a system of two different kinds of bosons, N quadrupole d-bosons able to occupy a L = 2 and a L = 0 level and N’ octupole f-bosons able to occupy a L = 3 and L = 0 level, as shown in Fig. 19. The most general Hamiltonian describing this system is H = Hd + Hf + V,,
(9.1)
where Hd is the Hamiltonian describing the quadrupole the Hamiltonian describing the octupole mode H, = ~,t~Jsf+ + Cf ~Qfi,,
Ef
+
dof~(fTfl(o:;s,sf)(o)
+
&f[(f+Sf+p
+ lY2 CfLKf’f’Y” wY”‘1’“’
+ ,=g 4 o 4w
. *>
+ (fif)'"']'"'
f
(Sf+Sf+p +
mode, Eq. (2.1) Hf is
(ff)'"']'O'
t.Tof[(sf+.sf+)'O'
(9.2)
(sfsfpp
L’3
fd Sf L=o
Es
FIG. 19. The boson configuration
s5ds @ $fl.
d
L=2
5
L=O
INTERACTING
BOSON
and V,, represents the octupole-quadrupole Vfd
1
=
x‘[(d+f+)‘“’
297
MODEL
interaction
(dfyLq(o)
+
zq(d+s
+
s+dy2)
(9.3)
(f+f)‘“‘]‘“‘.
L=1.2.3,4.5
Here f+(f), Sf+(.sf)are the creation (annihilation) operators for octupole bosons in the excited L = 3 level and in the ground L = 0 level. The basis states are now of the product form 1s”YP~[N] xdL, ; s,““~f”~[N’] xfLf ; LM). The Hamiltonian of Eq. (9.1) is identical to a shell-model Hamiltonian with two kinds of particles able to occupy two levels each. It can be solved using usual shell-model techniques. In analogy with the quadrupole case, we expect the boson-boson interaction terms cfL , xL to be the dominant terms in the vibrational limit [43] and we shall therefore treat all the remaining terms in perturbation theory. We thus first consider the simpler Hamiltonian H = edz d,+d, + c &2L + l)'/' ~~[(d+d+)'~' (dd)(L)](o)+ Ef zfmlfm+
nz
L
?n
+ c $(2L + 1)1/2CfL[(f+f+)(L)
nl
(ff)'"']'"'
+ 1 XL(d+f+)(L)(d!)(L). (9.4) L
Rather than constructing explicit solutions of the Hamiltonian of Eq. (9.4) in the most general case, we shall restrict ourselves to the special case n, = 1. The diagonal matrix elements of Eq. (9.4) within basis states of the form / dnydLd ; f; LM) are
;h LW
+ of + n&L,
+ 1) C V-‘(XJ) XJ
dLd I} dndxdL$
while the off-diagonal matrix elements are given by (dndxdLd ; f; LM I H [ dndxdfLd’;f;
LM)
= rl&(2Ld + 1)1/Z (2Ld’ + 1)1/Z * ,cJ [dnd-l(xJ)
dL, I} d”dxdLd][dnd-l(xJ)
dL,’ [} d4xd’Ld’]
P-6) In these expressions, E(dny,L,) represents the energy of the dnd configuration as given by Eq. (4.21). These expressions simplify further in the case in which the
298
IACHELLO
AND
ARIMA
states of the d”a-configuration to which the f-boson is coupled belong to the ground state band L, = 2nd . Then Eq. (9.5) becomes
L, = 2~) + l f + n,(2L, + 1) c (25’ + 1) XI’ IL’; Ir,
2 ;
$j2. (9.7)
N
Y
I
/
7-___ / / I II? 5-
---_
/
;/ /
N’
__ 4
3-
/ 2+
-f -o+ FIG. 20. A typical octupole spectrum. The parameters in the Hamiltonian are q = 400 keV, q = 50 keV, .q = 1100 keV, x5 = -50 keV, A, = -50 keV. The heavy lines represent E2 transitions, the thin lines E2 + Ml transitions, the broken lines El transitions and the dotted lines Ml transitions.
INTERACTING
BOSON
299
MODEL
In the case in which the total spin L corresponds to the totally alligned value, L = 2nd + 3, or to the totally alligned value minus one, L = 2nd + 2, even simpler expressions can be obtained. For these states, the Hamiltonian of Eq. (9.4) is already diagonal and its eigenvalues are given by E(d”“, Ld = 2n, ;f; L = 2n, + 3) = E(dnd, L, = 2~) + cf + ntiy5
(9.8)
and E(dnd, Ld = 2n, ;f; L = 2n, + 2) = E(d”“, L = 2n,) + q + ndx5 -t ((24
+ 3)/5) A4
(9.9)
where (9.10)
A, = x4 -x5.
In Fig. 20 we show the corresponding spectrum. In this figure we have arranged the negative parity states into “bands,” defined as follows N-band
/ dnd, Ld = 2n, ;,f; L = 2nd + 3, M)
N/-band
1dnd, Ld = 2nd ; f; L = 2nd + 2, M).
(9.11)
The other negative parity states which arise from the coupling dnd @f do not form a band structure since they are admixed to other states, thus loosing their collective character and systematic pattern. For example the state I d”a, L, = 2nd ;f; L = 2nd + 1) will be admixed to the state 1dna, Ld = 2n, - 2;fi L = 2nd + 1) since both states have the same zeroth-order energy and they are coupled by the large matrix elements Eq. (9.6). For this reason other bands are not shown in Fig. 20. It is instructive to plot the energy differences AE = E(n, + 1) - E(n,) as a function of nd , as in Fig. 3. This plot (Fig. 21) shows that the coupled bands N‘ and N’ appear as straight lines A&
=
Ed +
c4%
+
x5
AEN,
=
Ed +
c&f
+
X5
(9.12) +
(z/5)
A,.
In a similar fashion one can derive transition matrix elements. The most general expressions for the E2 transition operator is now ,f2)
= &(d+s + s+d)t) + g,‘(dtd)‘,)
+ q;(f+f)~‘.
(9.13)
The term ( f’f)(2) which describes the quadrupole moment of the octupole 3- state is expected to be small and it will be neglected in the following. The transition operator of Eq. (9.13) then becomes, in the vibrational limit with no cut-off, T
(E21 k
=
q,(d,+
+
(->”
d-k)
+
qAd+d$’
300
IACHELLO
250
AND
ARIMA
t-
I
I
0
I
I
I
1 6
4
2
“d
FIG. 21. The energy differences AE = E(nd + 1) - I?(&) as a function of nd for the coupled bands N and N’. The parameters are the same as in Fig. 20.
as before, Eq. (3.9). The matrix elements of this operator are easily constructed within the basis states dnd @f. For example, (nd
+
1, Xd’,
Ld’;f;
L’
11 d+
11 ‘Id,
Xd
, Ld
ifi L)
= (-)L?‘+3+L (2L + 1y2 (2L’ + 1)“2 12’ x
hd
+
1, Xd’,
L,’
11 d+
11 nd,
Xd
;:
;I (9.15)
, Ld).
This expression can be specialized to the various cases. Denoting by F the reduced matrix elements of d F =
(9.16)
L’)
we have for transitions within the bands N-band
L’ = 2nd + 5 -+ L = 2nd + 3 F = ((n, + 1)(4&j + ll))‘/”
N’-band
L’ = 2nd + 4 + L = 2n, + 2
F =
d2%
+
5)(4nd
(2nd
+
+
(9.17)
g,
3)
The intraband transitions N’ -+ N are given by N’-+N
L’ = 2n, -j- 4 --f L = 2n, + 3
F = - ( ;fn;+‘3;)
l”.
(9.18)
1NTERACTlNG
BOSON
301
MODEL
The second term in Eq. (9.14) does not contribute to the nd + 1 -+ nd transitions in the vibrational limit. In addition to E2, Ml, EO ,..., other electromagnetic transition operators can now be constructed by combining quadrupole and octupole operators. Of particular importance are the El, M2, and E3 operators. We begin by considering the El operator Tp’
= cjJ(d?~)‘~’ (sr’f)‘“’ + (ftsJc3) (sV)‘~‘];’
+ q,‘[(~Pd)‘~’ (sr’f + f’~~)‘~‘]t) (9.19)
which in the vibrational Tp’
limit with no cut-off N --f co, N’ ---f co becomes
= q,(d+j-+ fV)t’
+ q,‘[(dtd)‘2) (f + ft)‘“‘];‘.
(9.20)
The reduced matrix elements of the operator (dtf)o) between basis states of the form dnd @fare 0% + 1, xd , Ld ;fnfco;
L II (d+j)'l'
II nd , xd', Ld’;fnfsl;
L’)
= (nd + 1)1’2 [dRd(xd’Ld’)dL, I} dnd+lxdLd] L,?‘(-)‘d+r,’
,:,,
This expression can be specialized to various cases. Denoting matrix elements of (dtf)cl) one has for interband transitions N-+Y
L’=2nd+3+L=2nd+2
N’-tY
L’=2n,$2-t~=2n&2
F’ = (
2,
21.
by F’ the reduced
(nd + 1)(4n, + 7) 1/Z 7 1
F’=-(
(9.21)
(9.22)
nd(4n, + 5) l/2 21 ) .
The second term in Eq. (9.20) does not contribute to these transitions in the vibrational limit. Combining Eqs. (9.17), (9.18) and (9.22) it is possible to write the B(El)/B(E2) branching ratios for the decays of the N- and N’-bands in the form N+Y
B(El,L’=2nd+3+L=2n,+2) =- nd+lC 7nd B(E2, L’ = 2nd + 3 --f L = 2n, + 1)
N-+N N’+
N’+
Y
N’
N’-+ Y N’+N
B(E1, L’ = 2n, + 2 -+ L = 2n, + 2) B(E2, L’ = 2n, -j- 2 + L = 2nJ =
%Pci + 1) 1)(2n, + 3) ’
21(n, -
L’ = 2nd + 2 --f L = 2n, + 2) = n,(2n, + 1) B(E2,L’=2n,+2+L=2nd+1) 63 c B(E1,
where C = (q1/q.J2is a constant for each nucleus.
(9.23)
302
IACHELLO AND ARIMA
The relations Eqs. (9.8) and (9.23) are compared in Figs. 22 and 23 and Table XIII with recent experimental results. However, this comparison must be taken with care since the nuclei 150Srnand 152Gd are at the beginning of the transitional region and therefore the symmetry breaking terms may be important. The comparison is shown here to point out that a first-order approximation seems already to provide reasonable results. Together with the El operator one may also consider the M2 operator Ti’-
= C~~[(d~s)(~) ($f)‘“’
+ (f@)
(sV)‘~‘]~’ + m,‘[(~Pd)(~)(s/j’+
f?~$~‘];’
(9.24) which in the vibrational
limit becomes
TtM2) = wz,(&f + fV)f’ k
+ WZ,‘[(&)(~) (f + ft)‘2’]~‘.
Y I
6 Y P w
(9.25)
N EXP
I
Th
195 17-
Th
EXP 4
14+
15-
I
1312+ 3
I
IIlo+
,
92 a+
6+ I
7-~ 5--3----
i
1
4+ 2+ 0
i
‘3m*13
0
FIG. 22. Comparison between experimental [44] and theoretical spectrum in IsOSm. The parameters in the theoretical spectrum are Q = 356 keV, c1 = 66 keV, q = 1077 keV, x5 = -47 keV.
INTERACTING
Y I
N
16’ 15‘ ;I If
5
I
EXP
14’
2
14
17.
T 17.
15‘
15.
13.
13.
14’
12’
12+
II-
iI-
IO’
IO’
9‘
9.
8’
8’
7‘-
c-
6+ 4’
+
2’ 0
17‘l-
16’
3
Th
Th
17‘
4
303
BOSON MODEL
if 0’
5-
6+
3-
-
7-
ir_ __
-
5. II
3-
4’ 2’
0’
FIG. 23. Comparison between experimental [45] and theoretical spectrum in lKeGd. The parameters in the theoretical spectrum are 6 = 365 keV, cq = 45.3 keV, 4 = 1123 keV, x5 = -10 keV. Comparison
between Experimental
TABLE XIII B(El)/B(EZ) Branching Eq. (9.23)
Ratios” and those Predicted by
15*Gd
150Sm
zi -+ z, 5- +4+ 5-*37-+6f 7- -f 59- + a+ 9- + lll-+10+ ll-+913---t 12+ 13----t lla In units of lo-’ e-l fm-2.
Exp t441
Th
ExP 1451
Th
20
2
15
1.5
21 14
13.3
2.2 & 0.4
1.33
10 & 3
12.5
2.6 f 0.4
1.25
12 & 2
12
0.8 i 0.2
1.2
304
IACHELLO
AND
ARIMA
The matrix elements of the operator (dtf)@) between basis states of the form d”d @f are
nf=l;
L’)
dL, I} dnd+‘xdLd] x?e’(-)l+rd+Ld’
I
Ll,
L,2
Lj. 3,
(9.26)
This expression can be specialized to the various cases. Denoting by F” the reduced matrix elements of (d+f )(2) one has for interband transitions N-tY N'+
L’ = 2n, f 3 ---f L = 2n, + 2
Y
L’ = 2n, + 2 -
F” = ,(
(nd + 2)(4n, + 7) l/2 14 ) nd(4nd + 7>(4n, + 5) v2
L = 2n, f 2
F” = - (
14(4n, + 3)
i
.
(9.27)
From Eqs. (9.22) and (9.27) one can calculate the mixing (L 11TfM2) 11L’)/(L 11TfE1) 1)L’) as N+Y N’+
~m42/El)
=
D
(4 !
A(MB/El)
Y
=
2(nd
+ +
2)
1’2
1)
(9.28)
3(4&j + 7) 1’2 2(% + 3)
D
where D = m,/q, is a constant for each nucleus. The mixing related to the ratio 8(M2/E1) = (PM2)/P1))1/2 by pfwu
ratios dCM21E1)=
ratio A(M21E1) is
= 0.921 x 10-4~~,~Mw)
(9.29)
when E, is in MeV and A(M2/E1) in t+Je. Next we consider E3 transitions. In first order, the related operator is (9.30)
+E3)
=
k
which in the vibrational
q&f
+q
+
limit with no cut-off becomes T!?)
=
dfk
+
f nf=“; L’ 1)TcE3)IInd,xd,Ld;f
(9.31)
(-)3-kf-k).
The only matrix elements of some importance
gf>$’
in the present context are
nf=l; L) = (-p+L+B
q3e8LdLd~sXdXd~ . (9.32)
INTERACTING
BOSON
305
MODEL
Finally we note that Ml transitions are also possible between members of the N and N’ bands. We only quote here the transitions N + N’ which are expected to be large since they are induced by the first term in the Ml transition operator of Eq. (8.4). The general expression (4
, Xd
2 Ld
;fnf=l;
L
L&L’ =
(1)
II W)
‘Ld L’
LL,
c-1
II G',
L Ld’
I
&';
Xd',
p=1;
L')
3 (nd
1
, xd
, Ld
11 (d+d)“’
‘I”
= -
11 nd’,
xd’,
Ld’)
(9.33)
yields in this case 3nd(4nd + 7))1/2. (9.34) 5(2n, + 3)
N-tN’
L’ = 2nd f 3 + L = 2nd + 2
Combining
this with Eq. (9.17) one can obtain B(Ml)/B(E2) N-N’
B(M1,2nd+3+2nd+2)= B(E2, 2n, + 3 --f 2n, + I)
N-N
G
branching ratios 3 Wnd + 3)
(9.35)
where G = (nzl/q# is a constant for each nucleus.
10. SYMMETRY BREAKING IN PERTURBATION: OCTUPOLE STATES The symmetry breaking will change the zeroth-order predictions of the previous sections. However, if the breaking is small, it can once more be treated in perturbation. The perturbed Hamiltonian is H = l d C dm+dnl+ C 3(2L + l)ljz cL[(d+d+)‘L’(dd)‘L’]‘o) L
+ ~~([(d+d+)‘z’d](O)(1 - (nd/N))1/2 + (1 - ((nd - 1)/N))1/2 [d+(dd)‘2’]c0’} f uo[(d+d+)‘*’((1 - (n,/N))(l + ((1 - 0,
- l)lN))(l
+
+
Ef cfnltfm
c
H2L
- ((% + l)/N)))1’2
- ((nd - 2YNNYi2 +
1F
cfLKftf+YL’
WY*)1 WY”‘1’“’
L +
Vafhf3(*)
+
((1
-
((1 (01~
-
-
(nf/N’NU
1)lN’NU
-
((nf
(h-f -
+
lMW)>1~2
2>/W>Y/”
(ffY*‘l
+ ; xL[(dtft)‘L’ (df)‘L’]‘o’ + ~,{[(ftf)(~)
d+](O)(1 - (n,/iV))‘i2 f (1 - ((nd - 1)/N))‘/” [d(f+j)‘2’]‘o’} (10. 1)
306
IACHELLO
AND
TABLE Octupole
States
ARIMA
XIV
in First-Order
Pertubation
Theory
n, = 0 IO, 0; 3-B = I 0,o;
3-j
+ (;j”’
(~j”’
f 1 2,0;
3-)
+ (;jLi2
e / I) 2; 3->
F7,j = 1 / 1,2;
5-B =
/ 1,2;
5-> + (T)liS
+ ; (y
f’ 1 2, 2; 5->
(!yB
6 / 2,4;
+ @liZ
5-> + ; (y
(Jgy
(fgy
f j 3,2;
(y)‘;’
e / 2,2;
5-)
(qy
e I 3,4;
7->
n, = 2 1 2,4;
7-p
= 1 2,4;
7-)
+ &
+ (y’
(Tj”’
+ (Jg
(y)1’2 (q+y
5’ i 3,4; f / 4,4,
(!!I$,“”
e 1 3, 5; 7->
7-)
v = 2; 7-) + (g)‘”
rtd = 3 I3,6;
9-B
= I 3,6;
9->
+ (;j*‘* + (;j”’
+ 3 (;j”’
(y)“’
5’ 1 4,6;
(y)*”
(!$.!)*‘2
f / 5, 6, 0 zz 3; 9-)
(y)“’
B 1 4,8;
9->
+ i&j”’
9->
(y)“’
fl 1 4,6;
n, = 4 14,8;11-))=
~4,8;11-)+2(~)1’*(~)1’a~,5,8;ll-> + (;j’”
(y)“’
5 +- 13311S (yjl”
(y)*” 6’ 15, 10; ll->
f ! 6, 8, v = 4; ll-> + 2 (&jl’*
0 I 5, 8; ll->
9->
5->
INTERACTING
BOSON MODEL
307
and the three perturbing terms of interest here are Pi , L:,,and w2. Introducing the notation Q/E = .$‘, Q,/E = 5 (as before) and W,/C = 13one can study the effect of the various terms. Some perturbed wave functions are given in Table XIV. In Tables XV-XVIII we give the matrix elements of the E2, El, and A42 operators between the basis states of Table XIV. These formulas can be combined together to give B(El)/B(E2) branching ratios and B(M2)/B(El) mixing ratios. We also give in Table XVIII the energies of some negative parity states. It should be noted that the coupling of the octupole and quadrupole bands in transitional nuclei has been recently studied by Nomura [46] and Zolnowski et al. [45] using the coupling term w.J(d+ + d)~z~(f’f)~z~]~o~(no cut-off). The effect of this term in perturbation cannot be distinguished from that of the boson-boson interaction term a-, . This can be easily seen by letting N 4 co in Table XVIII. The perturbation correction to the energies Al? then becomes
A Eh , 2nd ; 2n,
+ 3) = -(n,/42) TABLE
e2e+ A&O, 0; 3)
(10.2)
XV
Matrix Elements of the E2 Operator of Eq. (9.14) between the Basis States of Table XIV
,! I, 2; 5- 11FEZ’ II2,4;
7-B
= 15112 1iq” 2 ‘2(!g2(1 1 5112 + q;e 7(3)1/Z
+ [!y)
(N;
:‘2, 4; 7- /I T’-) 113, 6; 9-B = 19112 q2 3112 (EgZ(l
i 3, 6; 9-
1)Fe’ 114,8; 11-B = 231/2 q2 2
l,“‘/
+ *!?I$)
+q;q!r$)l’e
308
IACHELLO
AND
TABLE Matrix
Elements
of the El
Operator
XVI
of Eq. (9.20)
N-l 1 + 5 7)
<2,4
<3,6
/I T’E2’ II 1,2;
II F*’ II 1,2;
<4, 8 1) P2’
<5, 10 II P”
II 2,4;
II 3,6;
5-B
]
6”yJg)y 91 -y-
+ oq,‘;
(fq+j’“j
= 1 lljZ
5-B
= 11”’
7->
= 151’2
9-B = 19”’
04,
I
341j2 35(21)“2
eq, &
between
(N;‘)‘;’
the Basis
States
+ eq , 6CW2 l--El
+ $?!$)
N-1 fT)
I(h&
ARIMA
l/P
N-2 (-Nj
(yj”‘/
+q;(2!E(K;)1’2
l/e) i
of Table
XIV
INTERACTING
BOSON
TABLE
309
MODEL
XVII
Matrix Elements of the M2 Operator of Eq. (9.24) between the Basis States of Table XIV
311% + eIn,f____ 7(70)1/Z
<2,4 11T’M2’ I/ 1,2; 5-> = 1 II/* 1m, x&gy
+ *!y)
+ ~m;?gqy)l’n
(!y/ + em, -.-L98(5)‘/2
+ [!y)
<<3,6 11FM*’ I/2,4; 7-> = 15 3112 (N; 49(5)“2
--em,
2)‘“/
<4,8 iJ FM21 I/3,6; 9-> = 19’j2 f mz
i
39l/* (N; em2 7(110)‘/2
l)‘/’
3(17)‘/* (N;2)1/2 <4,8 jl TtM2) ;I 2,4; 7-B = 15112 em, 49(5)‘/2 1
<5, 10 I] FM2) II 3, 6; 9-B = 191’2
em,
551je (N; 7(57)1/2
(N-2)li:
(N+3,‘:‘\
3)“z (N-4r”/
+ *~m;f(!g)I’P
310
IACHELLO
AND
TABLE
ARIMA
XVIII
Energies of some Negative Parity States in First-Order Perturbation l?(O, 0; 39 l?(l, 2; 5-) x??(2,4; 7-) g(3, 6; 9-j .R(4, 8; 1I-)
= E(0, 0) = &I, 2) = @2,4) = &3,6) = &(4, 8) -
Theory”
(l/7) 8% (1/6)((N - 1)/N) 8% (4/21)((N - 2)/N) t-e (3/14)((N - 3)/N) 6% (S/21)(@ - 4)/N) e*E
a The energies &n, , L) are given in Table IV.
i.e., Al? is linear in nd as in Eq. (9.12). However, a careful study of the electromagnetic transitions can distinguish between the two coupling schemes. We believe the boson-boson coupling scheme to be the appropriate scheme for vibrational nuclei and that the importance of the w2 term increases as one moves from the vibrational to the rotational region. In this respect lsoSm and 152Gd are presumably in an intermediate situation and both terms should be included to reproduce accurately the experimental data. No such a detailed analysis has been performed so far.
11.
TWO-QUASIPARTICLE
MODES
As we have mentioned in the introduction and schematically shown in Fig. 11, two quasiparticle states will begin to appear in the spectrum at about twice the energy gap A and will be present with increasing density from there one. These states are largely filtered out in (heavy ion, xnr) reactions since their E2 transitions are considerably retarded with respect to the collective ones. There is however one case in which two-quasiparticle states can be observed in (hi., xnr) and (01,xnr) reactions. This is the case in which the two quasiparticle state (j&) f has high spin and yet lies below the collective state of the same spin. The two-quasiparticle state is then the yrast state. Although in general the coupling of particle states to the collective excitations of the core require a much more detailed analysis, which we hope to be able to present in the near future, we consider here the question whether or not a simple description of the two-quasiparticle states is possible in terms of elementary excitation modes [43]. In this drastic approximation (in which the two-quasiparticle mode is treated as a boson mode), one can take as Hamiltonian for the coupled system, that given in [43] H = Hd + Hz + V,,
(11.1)
INTERACTING
where now 1 labels two-quasiparticle the boson operators are
BOSON MODEL
311
states with angular momentum
1. For these
(11.2) where a+(a) are the creation (annihiliation) operators for quasiparticles. Using techniques similar to those of Section 9 one can derive a set of relations for the energies of the states dnd @ (jJ,)Z. If the interaction V,, is of the form V,, = 1 y,[(d+b+)‘L’ (db)(L’]‘o)
(11.3)
L
QF
5
QP’
2 sw 4
3
2
I
2+ 0
FIG. 24. Coupling
0+ i-
of a two-quasiparticle
state with spin 8+ to the ground state band.
312
IACHELLO
AND ARIMA
the totally alligned, L = 2nd + 1, and totally alligned minus one, L = 2nd + I - 1, states again give rise to bands whose energies are given by E[d”“, L, = 24 ; (j,j,)Z; L = 2n, + I] = E(d”“, L, = 24) + cl + n,yl+?
(11.4)
and E[dnd, Ld = 2n, ; (j,j,)Z; L = 2n, + I - l] = E(dnd, Ld = 2nd + l z + n,y,+,
+ ((24 + Ml+
2)) 4+, .
(11.5)
Here ez represents the energy of the two-quasiparticle state, the coefficients y, parametrize the interaction between the quasiparticles and the quadrupole d-boson and (11.6) AZ-1 = Yz-1 - Yz * The resulting spectrum is shown in Fig. 24 where the two new bands have been called QP and QP’. It is, once more, instructive to consider the energy differences LIE. However, it is more convenient in this case to plot AE = E(Z + 2) - E(Z) as a function of the angular momentum I as shown in Fig. 25. The equations defining the two lines in Fig. 25 are “EOP
=
0
Ed +
W'4
+
I
I
I
2
4
6
Yz+e
I
8
=
Ed +
((I-
0/2)C,
+
I
I
I
I
I
10
12
14
16
18
.J'z+2
(11.7)
I I
FIG. 25. The energy differences AE = E(Z+ 2) - E(Z)as a function of the angular momentum I.
INTERACTING
BOSON
313
MODEL
and A.E,,~ = cz + w4 + ~z+z + = Q + ((I- z + W)
(2/U
~4
+ 2)) A,+, + yz+z + (2/U
+
2))
A,,,
(11.8)
and they appear again to be parallel to the Y-band. Since in this case the decays of members of the QP and QP’ bands to members of the ground state Y-band require a large change in phonon number, these transitions are highly forbidden. The two quasiparticle bands QP and QP’ will therefore preferentially decay within the bands until they reach the band-head. The final transition from the band-head to the ground state band should be retarded relative to transitions within the band. The presence of a two-quasiparticle state with high spin below the corresponding state in the ground state band will give rise to a dramatic backbending effect. However, if the two quasiparticles couple to the quadrupole mode as in Eq. (11.3) the slope of the QP-band in the AE versus Z plot should be identical to that of the Y-band. In Fig. 26 we show some recent experimental data [47] in 156Er which is the isotone to 150Sm and lj2Gd. Finally in Fig. 27 we compare the available data in lh6Er with the predictions in Sections 4 (Y-band), 9 (N-band), and 11 (QP-bands). The energy level systematics are fairly well reproduced by the simple model of [43]. However, in view of the transitional character of 156Er definitive statement about the nature of the lateral bands will have to await the results of a more detailed study of the role of the symmetry breaking terms.
FIG. 26. AE versus Z plot for the yrast band in IseEr. The experimental from [47]. The broken line is the theoretical prediction, Eq. (11.7).
data (full line) are
314
IACHELLO
s
I
E~P
i
.Th
-
23--r-
Exp
AND
i
Th
ARIMA
,,,“I’
Th
E,,“i’“Th
“‘+T"+T
FIG. 27. Probable band structure of IseEr. In the absence of detailed information on the decay properties of the lateral bands, the assignment QP= lo+ 0 dn, QP” = 3+ 0 d”must be regarded as tentative. The parameters in the theoretical spectrum are c = 365 keV, c, = 78 keV, q = 1647 keV, x5 = -162 keV, cl,,+ = 2965 keV, yIz = 32 keV, c3+ = 1728 keV yS = -189 keV.
12. CONCLUSIONS Here and in [I] we have presented a model for the collective states of nuclei which pictures them as states of a system of interacting bosons. After formulating a general framework in which vibrational and rotational nuclei appear as limiting cases, we have explored in detail the vibrational limit and presented algebraic expressions for energies and transition matrix elements. Our main purpose here has been that of providing complete and, in some case, simple expressions to be checked experimentally, rather than that of performing detailed numerical calculations. In the one case which we have studied in detail, llOCd, it appears that a detailed and accurate description can be achieved within the framework of the vibrational limit with small perturbations. This points to the possible existence, in certain regions of the periodic table, of approximate W(5) 1 O+(5) boson symmetries. Moreover, the boson representation seems to provide a good description not only of the enhanced E2 transitions but also of the retarded Ml transitions.
INTERACTING
315
BOSON MODEL
We hope that the simple algebraic relations presented here will stimulate experimentalists to perform more detailed analysis of data on other nuclei. For tests of this and similar models the best set of experiments appears to be that of the Krane-Steffen [32] type, in which energy levels, E2 and Ml transitions are studied simultaneously. It would also be interesting to compare with experiment the predictions of EO transitions, and to study the systematic behavior of the various parameters cO, c2, cq , etc. which appear in our formulas. We stress once more the phenomenological aspect of our approach in comparison with the more elaborate boson expansion methods or with the calculations of the Kumar-Baranger [48] and Greiner-Gneuss [27] type. Although a generalization of the formal results presented here to odd-A nuclei is straightforward, we have postponed it until we will have completed a detailed analysis of some specific examples. ACKNOWLEDGMENTS We wish to thank H. Feshbach, A. Kerman, D. Kurath, A. Lande, R. D. Lawson, M. Macfarlane, M. Peshkin, J. Schiffer, R. H. Siemssen, and I. Talmi for interesting discussions and 0. Scholten for performing some of the numerical calculations presented here and Mrs. M. Beijerman for her expert typing. We are also grateful for the interest shown in our work by many experimental groups, especially by Z. Sujkowski, P. P. Singh, 0. Schult, R. D. Smither, P. von Brentano, J. F. W. Jansen, A. Gelberg, R. F. Casten and M. J. A. de Voigt. One of us (A. A.) wishes to thank R. H. Siemssen for the hospitality extended to him at the K.V.I. where part of this work was done. This work was performed as part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie” (F.O.M.) with financial support from the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek” (Z.W.O.).
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