collective magnetic dipole transitions: dependence of the energies and rates on the nuclear...

Nuclear Physics A467 (1987) 29-43

North-Holland, Amsterdam

COLLECTIVE MAGNETIC DIPOLE TRANSITIONS:

Dependence of the energies and rates on the nuclear effective interaction

Huan LIU and Larry ZAMICK

Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA

Received 1 September 1986

(Revised 18 December 1986)

Abstract: The dependence on the properties of the effective interaction of the energies of and excitation

strengths to magnetic dipole states in open shell nuclei is studied. In particular the single j-shell for 48Ti is used as an example. In this case there are two l+ states with isospin T = 2 and one with

isospin T = 3. The conditions for having a strong low-lying collective l* state are examined. Focus

is also given on the excitation strength to the analog T = 3 state since this is of relevance also to

p’ Gamow-Teller reactions and to double beta decay. It is found that in the rotational limit there

is no strength to the T = 3 state and there is an overly strong low-lying l+ T = 2 state. This is almost

also true for a quadrupole-quadrupole interaction. At the other extreme, as shown by Halse, with

a pairing interaction all the strength goes to the T=3 state. Other interactions considered are

pairing plus quadrupole, spin-dependent delta and Kuo-Brown bare and renormalized, and matrix

elements taken from the spectrum of ‘%c. It is found that the p’ strength can be reduced either by making the two-body J = 2 T = 1 matrix element or J = 1 T = 0 matrix element more attractive,

just as was shown by others in heavier nuclei. However such parameter changes have effects on

other properties of the l+ spectrum, which can serve as indicators as to whether or not these

changes are justified.

1. Introduction

In previous works le3) ‘t 1 was suggested that relatively strong, mainly isovector

collective magnetic dipole excitations should occur in all nuclei in which there are

open shells of both neutrons and protons. The specific example of the even-even

titanium isotopes (together with the cross conjugate nuclei) was considered. The

single j-shell approximation (j = f,J was used in the calculation. Such states have

indeed been found in 46Ti and 48Ti both with high resolution electron scattering,

and with medium energy protons “).

In this work we address the question: what can we learn about the effective

interaction between nucleons in the nucleus by studying such states? How sensitive

are the energies and transition strengths to the parameters of the effective interaction?

Aside from the intrinsic interest of this problem in the f,,* region, we hope that

such a study might help to cast some light on what is happening in heavier nuclei,

in particular the deformed region in the neighborhood of 156Gd [ref. 5)], where such

collective Ml orbital modes were first found. In that region there are many calcula-

tions and many controversies 6-1o). In this pap&r the deformed rotational model and

0375-9474/87/$03.50 @ Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

30 H. Liu, L. Zamick / Magnetic dipole transitions

the pairing interaction model are first examined. Then a variety of effective interac-

tions are introduced and compared on the ground of producing the collective

magnetic dipole excitations. In the last section we study the changes in Ml matrix

element and energy levels in terms of certain component in two-body matrix elements

that were obtained from the spectrum of 42Sc.

The single j-shell model. We use 48Ti as an example to illustrate the single j-shell

calculations.

In the f7,,2 model the low-lying even-parity states of 48Ti arise from the configuration

(f:,J,(f;,$)_ i.e. two protons and two neutron holes. The two protons can couple

to angular momenta L, = 0, 2,4 or 6; Iikewise the two neutron holes. We can form

a general state of angular momentum J as follows:

* = ,C, ~“(~pL,)[(j2)L~(j-2)L~lJ, P n

where @(L,L,) is the probability amplitude that in the ath state of total angular

momentum J, the protons couple to angular momentum L, and the neutrons to L,.

To get the coefficients DJ* (L&J one needs to diagonalize the hamiltonian. To

this end we require eight two-body matrix elements

EJ = ((f~,2)Jfvt12)l(f:,2>J), J=O, 1, 2,... 7 (for even J the isospin is T = 1, for odd J it is T = 0).

In the original work of McCullen, Bayman and Zamick (MBZ) I’) the matrix

elements EJ were taken from the spectra of 42Ca and 42Sc. However some of the

odd J assignments were incorrect due to the sparsity of data at that time. We refer

to these matrix elements as E(MBZ0).

Subsequently, calculations were done with correct experimental matrix elements

by Kutshera, Brown and Ogawa “). We refer to the matrix element with the correct

42Sc spectrum E (42Sc). To make things clear we here show some detailed results for E(42S~) in table 1.

Note that the wave function of the l+ T = 3 state is independent of the effective

interaction. This is because the T = 3 analog state in Sc4* is a unique configuration

( f,+ f;/:,)‘=‘. In the above as well as in all the subsequent calculations the value

of g,-g, was taken to be the unquenched value 2.202. The expression used is

where for j= i+$, g = (l/j)gr+g,/2j.

We see that the lowest state is the most strongly excited [a point made pre-

viously ‘**I. In the single j-shell model the transition is necessarily isovector.

As a prelude to showing the sensitivity to parameters we note that with the original

MBZ matrix elements 11) (EJ =O.OOO, 1.03, 1.51, 2.25, 3.00, 1.96, 3.40 and 0.62 MeV

forJ=O,l,..., 7) the corresponding values of B(M1) are 0.94, 0.20 and 0.03 CL;.

For the lowest l+ state there is not too much changed by the change of interaction,

H. Liu, L. Zamick / Magnetic dipole transitions 31

TABLET

The values of B(M1) 0, + 1: using two-body matrix elements from the spectrum of ‘%

a) Matrix elements

J(T=l) E’ ( MeV) J(T=O) E’ ( MeV)

0 0.000 1 0.611 2 1.586 3 1.490 4 2.812 5 1.511 6 3.231 I 0.617

b) Ground-state waoe function (J = 0:)

D’(O0) = 0.9286, LP(22) = -0.3701 P(44) = -0.0252, P(66) = -0.0071

c) Wavefunctions of J = l+ states and B(M1) 0, + 1”

Energy

(MeV) Isospin D(22) D(44) D(66)

NMl) P’,

3.66 2 0.9297 -0.3683 -0.007 1 0.741

6.58 2 0.3098 0.7914 -0.5270 0.138

12.99 3 0.1992 0.4880 0.8298 0.087

but for the transition to the T = 3 J = If state there is a difference in B(M1) of

about a factor of 2.5 for the two sets of parameters. The AT = 1 Ml rate is proportional

to the j?’ rate in (n, p) reactions so the extreme sensitivity noted here must be taken

into account.

2. Deformed model versus the pairing interaction model

It is the purpose of his work to examine the dependence of the calculated Ml

strength on the nuclear effective interaction. That the results are very sensitive to

the effective interactions can be demonstrated rather dramatically by considering

two extremes: the deformed model and the pairing interaction calculation.

In the deformed model ‘) one can easily show that there will be no transition

from the J = O+ T = 2 ground state of 48Ti to the l+ T = 3 analog state. In contrast,

Halse 13) has shown that using a pairing interaction (action between J =0 T = 1

pairs only) the transition will go only to the J = 1 T = 3 states and will not to the

J = 1 T = 2 states.

The deformation argument goes as follows: for the K = 0 ground state band the

configuration, as shown in fig. la, is


/ K = 7/2

c.)

Fig. 1. (a) The intrinsic state for the K = 0 ground state band of 48Ti. (b) The “proton” K = 1 intrinsic

state. (c) The “neutron” K = 1 intrinsic state.

This will connect to either a K = 1 proton excitation K, = 2, + $K,, same as before

(fig. lb) or a K = 1 neutron excitation K, =$, -tK, =& -& i, -2, I, -3 (fig. lc). In

the “proton” excitation case we have 3 nucleons in K =t. They must have isospin

4. We have 3 nucleons in K =$ They must also have isospin 1. We also have 2

neutrons in K =$ with isospin 1. The total isospin is i+fi- 1. This cannot add up

to T=3.

For the neutron excitations we have 4 nucleons in K = 4. They must have isospin

zero. We then have 4 remaining neutrons which must have isospin 2. Again we

cannot have isospin 3.

For the pairing calculation of Halse 13), one has to be a bit careful because all 3

l+ states are degenerate. But one can construct explicitly the T = 3 J = 1 wave

function because it is the analog of the unique 48Sc state (f7,2V, f&)J=l. That is to

say, in 48Ti the state with J = 1 T = 3 is T_( f7,2rf&,). From this one obtains

DJ=“‘T=3)(L,L,)=J~(2L”+1) U(JAL,; L,j)

where j = f,,2, and U is the unitary Racah coefficient.

One then verifies his result that all the strength goes to l+ T = 3 and none to l+

T=2.

However if one looks at the “realistic” results using the spectrum of 42Sc as input,

(shown in the previous section) one sees that the answer lies closer to the deformed

limit than the pairing limit in the sense that the lowest l+ T =2 state is excited

much more strongly than the l+ T =3 state.

A further distinction should be made between the pairing model of Halse 13) in

which any pair of J = 0 T = 1 nucleons can interact, and a more simple but often

used model in which only like nucleons in a J = 0 state interact (i.e. one neglects


the neutron proton interaction in J = 0 T = 1). For this simpler model it turns out

that in the single j-shell space all magnetic multipole transitions from the ground

state will vanish so that these would be nothing to discuss here. This is because the

MA operator only connects states of the same seniority. With the like particle pairing

model the ground state of 48Ti would be a state of seniority zero in both the neutrons

and the protons. To reach a state of angular momentum A (A = 1, 3, 5 etc.) one

clearly has to break a pair.

What this section shows is that the presence of low-lying l+ collective states is

not inevitable but that it does depend on the nature of the interaction. This will be

explored in more detail in the next section where we will consider a variety of

interactions.

3. Pairing plus quadrupole interaction

3.1. THE QUADRUPOLE-QUADRUPOLE INTERACTION

We consider an interaction -xQ. Q and choose x the same as the choice made

by MBZ in their original paper. The results are given in table 2.

TABLE 2

The values of B(M1) 0, --f 1,’ using a quadrupole-quad-

rupole interaction

Matrix elements

J E’(T=l)(MeV) J E’ ( T = 0) ( MeV)

0 -2.151 1 -1.741

2 -1.004 3 -0.1012

4 0.717 5 1.127

6 0.717 7 -1.004

E (I+) MeV B(M1) /.&

4.44 2.648

8.99 0.005

10.78(T=3) 0.0001

These results should be compared with the rotational model in the limit of zero

deformation which was described in the previous section. In that model the results

are

B(M1) 0’T=2+1:T=2=5.534pN,

B(M1) O+T=2+1+T=3=0,

B(M1) O+T=2+1;=0,


We see that with the Q * Q interaction we do not achieve a perfect realization of

the rotational limit, but we do have the qualitative features that the first l+ state is

excited very strongly (much more than experiment) and that the excitation rates to

other l+ states, including the analog state l+ T = 3 are negligible. [ N.B. The result

for B( Ml) + l+ T = 3 with Q * Q is so small that the reader might suspect that the

answer should be exactly zero, with the value of 0.0001 coming from round off

errors on the computer. We have convinced ourselves that this is not the case. The

finite result is a real one.]

We next add a pairing term -2 MeV S,,, to the previous matrix elements of the

quadrupole-quadrupole interaction. That is to say

V= -2 MeV S,,-xQ* Q,

The results for the 0, + 1: transitions are given in table 3.

We see, as expected that relative to the pure Q * Q case (table 2) the strength to

the yrast l+ state is weaker and that to the T = 3, J = l+ analog state is stronger.

Roughly speaking we tend to get results which are closer to those obtained with

the spectrum of 42Sc [refs. 11*12)].

TABLE 3

The values of B(M1) 0, + 1: for a pairing

plus quadrupole interaction

E(l+) MeV B(M1) P&

7.53 1.059

12.08 0.029

13.86(T=3) 0.123

V=-AS,,,-~9.9 withA=2MeV.

At the other extreme we consider a pure pairing interaction. All three l+ states,

two of which have isospin T = 2 and one of which has isospin T = 3, are degenerate

in energy. If one removes the degeneracy ever so slightly by, say adding a very weak

quadrupole-quadrupole term, one verifies Halse’s result that all the strength goes

to the l+ T = 3 state. The value of the strength is B(M1) = 1.154~2,.

4. The delta interaction

We here consider the delta interaction -G( 1 + xPa)8( rl - r2), where P" is the

spin exchange operator f( 1 + (+r * ~~2). The delta interaction and the pairing interac-

tions are both short range so that one should expect similar results. (One often says

that the pairing interaction is of shorter range than the delta interaction.)

It is of interest to study the delta interaction because, even though it is not realistic,

it is often used, e.g. the Landau-Migdal interaction is a delta interaction and the

Ii. fiu, L. Zamick j Magnetic dipole transjtio~ 35

Skyrme interaction consist of delta interactions and derivatives thereof. We might

guess in advance that the delta interaction might give too little strength to the lowest

1’ state and too much strength to the li T = 3 state because of its similarity to a

pairing interaction.

However we do have the spin term xP” to play with. We will show in table 4 the

results of for several values of x.

TABLE 4

The values of @MI) Op lzwith the delta interaction -G(l+xP”)G(r, - r2), with G= 300 MeV * fm”

x=-l x=0 x = 0.3

E( I+) MeV B(M1) & E( 1’) MeV NM1) P: E( 1’) MeV B(M1) pu:

11.024 0.015 5.522 0.067 3.870 0.079

11.263 (T=3) 0.900 7.005 0.239 5.490 0.589

12.698 0.009 9.825 (T=3) 0.382 9.490 (T=3) 0.155

Considered as a simple G-matrix the best value of x is about 0.3. The results are

given for this value as well as for x = -1 and x = 0. For x = -1, the delta interaction

simulates a pairing interaction as close as possible because in this case all the T = 0

matrix elements are zero.

Indeed for x = -1 the T = 3 state is most strongly excited whereas there are two

T = 2 states one just below and one just above the T = 3 state which are excited

only very weakly. Thus for x = -1 the delta interaction looks something like a pairing

interaction. For x = 0.3, the “best value”, on the other hand we find that neither

the lowest energy state (E = 3.87 MeV, B( Ml) = 0.080 pu’,) or the highest-lying state

with T =3 (E =9.490 MeV, B(M1) =0.155 FL) is the most strongly excited, but

rather a state in the middle has most of the strength (E = 5.490 MeV, B(M1) =

0.591&k). Such a behavior is qualitatively different from that obtained from the

spectrum of 42Sc, and apparently disagree with experimental findings of Richters’

group “). The latter group find that the lowest 1’ state is the one most strongly excited.

We may therefore have found an operational method of discarding the delta

interaction as a suitable effective interaction in producing collective magnetic excita-

tions. The delta interaction, for all values of x leads to a prediction that the lowest

If state will be the one most weakly excited, but more reasonable interactions and

experiment indicate that the lowest 1’ state will be the one that is most strongly

excited.

5. The Kuo-Brown interaction

We show the results for the B(Ml)‘s and energies of l+ states in 48Ti for both

the bare and core-renormalized Kuo-Brown interactions 14) in table 5.

36

TABLE 5

The values of B(Ml)O, + 1,’ with the bare and with the core renormalized Kuo-Brown Interaction

Matrix elements Bare KB Renormalized KB

E’

J(T=l) 0 2 4

6

T=O

1

3

5 7

-0.869 -0.664 -0.297

-0.120

-0.230 -0.525

-0.211. -0.208 -0.604 -0.502 -2.185 -2.199

Energies of l+ states and B(M1)

-1.807 -0.785

-0.OS9 0.226

E(l+) (MeV)

1.117

3.604 7.280 (T = 3)

B(Ml) E(l+) B@fl) l-44 (MeW CLZN

2.211 2.657 1.013

0.03 1 5.772 0.151 0.0002 10.34s 0.044

Note that with the bare Kuo-Brawn interaction one obtains a very low-lying lf

state (E = 1.12 MeV) with a very strong B(M1) from ground (B(M1) =2.217 pi).

This should be compared with the results from the spectrum of 42S~ (E = 3,76 MeV,

B(M1) = 0.858 ,u&). The latter results are very close to experiment (E = 3.8 MeV,

B(M1) = 0.7 /A;), Note further that with the bare Kuo-Brown interaction 14) the

energy of the T = 3, J = If state comes much too low in energy E = 7.28 MeV as

compared with E = 13.82 MeV obtained with E(42S~). We know that the answer for

E(“%c) is very close to experiment because we can extract this by looking at the

energy of the analog state in 48Sc (J = 1’ T = 3) and calculating the Coulomb energy

difference between 48Sc and 48Ti.

When the renormalized Kuo-Brown interaction 14) is used results improve con-

siderably, although perhaps not enough. When the cure polarization effects are

turned an the energy of the lowest 1’ state changes from the bare value E =

1.117 MeV to 2.057 and B(M1) gets reduced from 2.217 F$ to 1.016 pk. The energy

of the J = If T = 3 analog state increases from 7.280 MeV to 10.34 MeV. T!5s is

certainly in the right direction but we are still far away from the correct energy of

about 13.8 MeV.

We also note a big change in the B(M1) to the J= l* T=3 states. The result is

negligibly small for the bare KB interaction @(Ml) = 0.0002 E_L~) but for the dressed

KB interaction it is 0.044 pk. Recall that with E(42S~) the value is 0.078 ph.


As a curiosity one notes that the results for the bare Kuo-Brown interaction 14)

are remarkably close to those of quadrupole-quadrupole interaction, in the sense

that B(M1) is very large to the lowest If state and is negligible to the J = l+ T = 3

analog state. Whereas the B(Ml)‘s in order of increasing energy of a bare KB are

(2.217, 0.031, 0.0002) JL~ while for Q. Q they are (2.656, 0.005, and 0.0001) J.L$.

The reason for this is that in the T = 1 channel the bare KB matrix elements look

remarkably like those of a Q. Q interaction, a point which we believe has not been

made before. This can be seen by adding a constant to all the bare KB matrix

elements so that EJ=‘- - 0, and do likewise to the Q - Q interaction. We arbitrarily

normalize the Q. Q interaction so that the matrix element ( EJc4 - EJco) is the same

as that for the bare Kuo-Brown. We then find that the T = 1 matrix elements for

the two interactions as well as for the dressed KB are

EJ J=O J=2 J=4 J=6

bare KB 0 0.205 0.572 0.749

Q*Q 0 0.229 0.572 0.572

renormalized KB 0 1.01 1.72 2.04

The key point here is that the bare Kuo-Brown interaction there is a large splitting

between EJz4 and EJc2. This large splitting leads to the characteristic behavior

such that the lowest J = l+ state becomes strongly excited and the analog J = l+

T = 3 state very weakly excited.

Whereas for the bare KB the ratio EJz4/EJz2 is 2.79, and for Q * Q it is 2.5, for

the dressed KB it is only 1.70. Evidently in the act of renormalization the T = 1

matrix elements pick up on effective pairing component.

6. Systematic study of the effects on the Ml matrix elements in ““ri of varying selected

two-body matrix element

In this section we start with the two-body matrix elements obtained from the

spectrum of 42Sc and we study the effects of changing certain key matrix elements.

The results are presented in three figures. In each figure the vertical axis corre-

sponds to the Ml matrix element. In fig. 2 for the horizontal axis (AMEJ,‘,Z,) we

have the deviation of the J =2 T = 1 two-body matrix element from the value

obtained from the spectrum of 42Sc, namely 1.586 MeV. All other matrix elements

are kept fixed to the values from E(42S~). In a similar vein in fig. 3 only the J = 1

T = 0 matrix element is varied and in fig. 4 only the J = 7 T = 0 matrix element is

varied.

In each figure three curves are drawn, corresponding to the three l+ states obtained

in the diagonalization for 48Ti. The Ml matrix element for the lowest or yrast state

(Yr) is drawn with a dashed line. The Ml matrix element for the J = l+ T = 3

isobaric analog state (IA) is plotted with a solid line; the Ml matrix element for

the third state, which like the yrast state has isospin T = 2, is drawn with a dot dash

line. This third state is usually but not always above the Yr state and below the IA


0.6

- 0.2

-

5 -“*4

s - 0.6

-0.8

AME::: ( MeV)

Fig. 2. Calculated Ml matrix elements from the three l+ states to the ground state versus the change of the two-body matrix element ME:“,: in the unit of MeV. In this figure as well as in figs. 3 and 4 the

value of (gp- g,) is taken to be 2.05.

state in energy. However for certain choices of the interaction it could lie above the IA state.

Note that in all three figures the results for A ME = 0 correspond to results obtained with E(42S~). These results for the three Ml matrix elements are:

Yr Ml = -0.86~~ IA Ml = --0.37/~~

third state Ml = -0.29j~~

These results for E(42S~) serve as a standard for comparison with results when the 2-body matrix elements are changed.

Before discussing the results further words on motivation are in order. Sometimes one is tempted to vary the interaction matrix elements to make one quantity come out correct, but such a variation will have effects on other properties as well. For


-2.0 0 2 .o

AME”,: k ( MeV 1

Fig. 3. Calculated Ml matrix elements from the three l+ states to the ground state versus the change of the two-body matrix element ME’,‘_‘, in the unit of MeV.

example the transition from J = 0 T = 2 to J = 1 T = 3 in 48Ti is closely connected with the transition J = 0 T = 2 in 4RTi to J = l+ T = 3 in 48Sc, the latter being realized in a (n, p) reaction. [There will however be no orbital contribution in (n, p).] Even further in the double beta decay of 48Ca+48Ti (ground state) the transition goes via virtual excitations of l+ states in 48Sc. In the severe models space truncation of f7,2 only the double beta decay matrix element of UT from .I = 0 T = 2 in 48Ti ground state to the f = l+ T = 3 state in 48Ti or 48Sc. From the work of Tsubai et af. 15) on double beta decay we see that truncation to a single j-shell is woefully inadequate. Nevertheless the virtual l+ state represented by f 7,2*f;,ZV does represent an important amplitude which adds coherently with other amplitudes corresponding to other virtual l+ states. Therefore we may ask whether if we change the two-body interaction matrix elements so as to get a certain desired value for the J = 0 T = 2 + J = l+ T = 1 transition matrix elements, being motivated by (p, n) or double beta decay experiments, what will be the consequences for other quantities, such as the transitions to J=l T=2 states?

Conversely, if we try to adjust the low-lying collective l+ state (Yr) what happens to the T = 3 analog state?

40 H. Liu L. Zamick /

> -0.6

3 * -0.8 ZE

-1.0

-1.2

/ - IA.state

Fig. 4. Calculated Ml matrix elements from the three It states to the ground state versus the change of

the two-body matrix element ME’;=‘c in the unit of MeV.

We now examine the behavior in the 3 figures. In fig. 2 where only the J = 2 T = 1 interaction matrix element is varied, we note that as we decrease E$Zt relative to the E(42S~) value i.e. as AMEJT”_2, becomes increasingly negative the Ml to the YR state becomes stronger and the Ml to the state IA and to the third state becomes weaker. This behavior can be understood as being due to the fact that making EC::

smaller effectively increases the attractive quadrupole-quadrupole part of the interaction. We are pulling EJ=* towards EJao and away from EJE4.

Note that there is an anticorrelation between what happens to the IA state and third state (Ml gets smaller) and what happens to the lowest state (Yr) as AMJ” becomes more negative.

In fig. 3 where only the J = 1 T = 0 matrix element is varied we find that again as we decrease EJ” the Ml for the T = 3 IA state decreases. In fact the behavior for the Ml for IA is reasonably similar to what happens in fig. 2 where only Efs2 is varied. Thus if our only motive is to decrease Ml for the T = 3 IA state we can do it in two completely different ways.

However the behavior for the other states is quite different. As we make E”=’ more attractive the Ml for the Yr state remains fairly constant but the magnitude of Ml for the third state increases a great deal. Thus if we make EJ=’ so small as to cause Ml T = 3 IA to vanish, we will have produced two states with large Ml’s, the Yr state and the third state. Thus we have a check as to whether making EJ”’ very attractive is correct or not. If we find two strong 1+ states the procedure might be correct, but if there is only one strong 1’ state then the procedure is questionable.

H. Liu, L. Zamick / Mugneiic dipole transitions 41

The behavior that we have found for the Ml rate for J = l+ T = 3 state in 48Ti is

qualitatively similar to what has been obtained in the context of double beta decay

by others.

For example, Klapdor and Grotz 16) have shown that by adding a quadrupole-

quadrupole interaction they could decrease double beta decay matrix elements in

medium mass nuclei. Vogel and Zirnbauer “) obtained the same behavior by adding

an attractive J = 1 T = 0 interaction. These results are qualitatively the same that

we obtain in our calculations.

Our calculation is too far away in mass region and model space to make any

definitive remarks about the validity of the suggestions of the two sets of authors

mentioned above. However we have shown that there are consequences for other

properties which will serve either to confirm or to repudiate these suggestions.

Thus, if like Klapdor and Grotz 16) we have a strong attractive quadrupole-

quadrupole interaction, motivated by a desire to decrease the J = 1 T = 3 Ml matrix

element, we will make the yrast J = I T = 2 matrix element very strong. If on the

other hand we achieve the same objective by introducing a strong J = 1 T = 0

attraction as Vogel and Zirnbauer I’) have done we will predict two strong Ml

collective states rather than just one.

From the experiments done thus far in 46Ti and 48Ti by Richter’s group “) it does

not appear that there are two strong Ml states of the (f7,2)” characters -just one.

There are however collective states of the form f;jzf5,2 at higher energies. To really

sort these out larger model space calculations will have to be carried out and indeed

are being done.

In fig. 4 we have considered the variation of the J = 7 T = 0 two-body interaction

matrix element, the reason for looking at this matrix element is that it is rather low

in the spectrum, essentially degenerate with the J = 1 T = 0 state at an excitation

energy of 0.61 MeV.

We see that as we increase the J = 7 T = 0 interaction the main thing that happens

is that the Ml strength to the Yr state decreases rapidly. This tells us that if the

J = 7 T = 0 matrix elements were not sufficiently attractive there would be no

low-lying collective Ml state.

Thus many different things come into play in producing a strong low-lying

collective state. In the T = 1 channel a large separation of the J = 4 from the J = 2

matrix element is conclusive to having a strong low-lying Ml and in the T = 0

channel having the J = 7 ‘stretched’ state come down low in energy is also important.

What about changes in the energy levels? In table 6 we list changes in calculated

excitation energies of the l+ states resulting from changes in the two-body matrix

elements. We note a remarkable insensitivity of the energies of the If states to

variations of the J = 1, T = 0 matrix element, this despite the fact that the correspond-

ing B(M 1) rates vary considerably. On the other hand when the J = 7, T = 0 matrix

element is decreased the energy of the higher isospinl+ state, with T = 3, increases

sharply. The energies of the other two l+ states show much less variation. In contrast


TABLE 6

The changes of energy levels (MeV) corresponding to changes of two-body matrix element (MeV)

-1.0 -0.5 0.0 0.5 1.0

a) AME<:;

IA state

3rd state yrast state

b) AME;:‘,

IA 3rd

Yr

c) AME;:;

IA 3rd Yr

13.90 13.85 13.82 13.79 13.77

6.99 6.96 6.95 6.96 7.07 3.85 3.80 3.76 3.72 3.64

14.22 13.93 13.82 13.82 13.91 7.59 7.18 6.95 6.89 7.07

2.76 3.18 3.76 4.41 5.07

16.76 15.29 13.82 12.35 10.91 8.46 7.69 6.95 6.26 5.63 4.19 3.97 2.76 3.53 3.28

to this, when the J = 2, T = 1 matrix element is decreased the energy of the T = 3

l+ state does not change significantly but the energy of lowest It state does decrease

somewhat.

7. Conclusions

In this work we have studied the sensitivities of the energies of l+ states to the

types of nuclear effective interactions that are used. We have shown, working in an

admittedly small model space - the singlej-shell, that one gets the opposite behavior

in two extremes. In one case we have the rotational model for which all the B(M1)

strength goes to the lowest T = 2, J = 1 state and in the other extreme the pairing

interaction model for which all the B(M1) strength goes to the T = 3, J = 1 state.

When discussing other interactions e.g. bare and renormalized Kuo-Brown, or

pairing plus quadrupole we can relate the results to how close we are to one extreme

or the other. We have also made the point that if we vary matrix elements with

particular motive in mind we will make other changes which may or may not be

desirable. For example if we make the J = 1 T = 0 two-body matrix element

sufficiently attractive to cause B(M1) = I ,, T_-2_J=l T=3 to vanish, then we will find _

that we will have two l+ T =2 states with almost equal collective Ml strength. On

the other hand if the same J = 1 T = 0 matrix element is obtained from the spectrum

of 42Sc only the lowest T = 2 l+ state will carry significant Ml strength. The energies

of the 1+ states on the other hand are scarcely affected by the changes in the J = 1

T = 0 matrix element. Different but equally significant changes occur when one

alters other other matrix elements such as J = 2 T = 1 and J = 7 T = 0. This indicates

that energies and B(M1) rates are sensitive to all the two-body matrix elements.


Therefore the results of magnetic dipole excitation energies and strengths will provide

important input on the nature of the effective interaction in a nucleus.

This work was supported in part by the US Department of Energy DE-FGOS-

86ER40299. One of us (L.Z.) currently has a US Senior Scientist Humboldt award

at T-H Darmstadt.

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