Volume 134B, number 6 PHYSICS LETTERS 26 January 1984

REMOVAL OF REDUNDANCY FROM THE SINGLE PARTICLE EQUATION OF MOTION METHOD

FOR ODD MASS NUCLEI

Amit GOSWAMI and Peter ROLNICK Department of Physics and the b~stitute of Theoretical Science, University of Oregon, Eugene, OR 97403, USA

Received 30 August 1983

A method is developed for the removal of redundancy which is known to plague the calculation of low lying spectra of odd mass nuclei by the equation of motion method. The feasibility of the method is verified numerically.

Low lying excited states of near-spherical even nu- clei can be described microscopically as vibrational boson excitations (hereafter called phonons) made from pairs of fermions (as in tile theory of random phase approximation or RPA). One can also build a theory of even nuclei solely using boson field opera- tors as the recent success of the interacting boson model [ 1 ] indicates, although it is a bit puzzling as to why this theory works as well as it does. Considerable amount of work is now being devoted to discover tile fermion connection of the interacting boson model [2] ; however, the boson description is regarded by many as adequate, although the fermion description would be more satisfying from a microscopic point of view.

In describing odd mass nuclei, on the other hand, since one nucleon must remain unpaired, use of ex- plicit fermion operators is necessary. And if we are really to properly investigate the dynamical inter- play of the single particle modes of odd mass nuclei and the collective boson-likq excitations of the even core, we had better use a fermion description of the even core as well. Thus a proper extension of the earlier RPA theory (because it is a theory based on fermion operators) and its coupling to single particle modes has remained an important problem to solve in nuclear structure theory.

Kisslinger and Sorensen [3] describe near-spherical odd mass nuclei with some success as quasiparticle states (the eigenstates of the pairing hamiltonian) coupled to the ground and one phonon states of the

0.370-2693/84/S 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

even core (let us call this quasiparticle phonon cou- pling or QPC):

a+~lO>, [a~ 12>l~,

where a~ is a quasiparticle operator; [0) and 12) are the ground and one phonon states of the even core re- spectively; 12) = p t 10), where p t is given by RPA. [ ]a denotes angular momentum coupling t o J =/'a.

One basic assumption of QPC is that in spite of the ground state correlation as implied by RPA, the ground state remains an approximate quasiparticle vacuum:

alO)~ O.

But as the long range nuclear interactions get strong- er, and we start deviating from strict sphericity, as in- dicated by the discovery o f the static quadrupole mo- ment of the phonon state in some nuclei, the assump- tion al0) ~ 0 no longer holds for the treatment of the adjacent odd mass nuclei. And here is precisely where the QPC description breaks down as evidenced by the appearance of anomalous states in the low energy spectra of these odd mass nuclei. Goswami and col- laborators [4] corrected this deficiency of QPC by in- cluding unphysical ("backward going") states aal0} and [a~12)]~ in the basis for diagonalizing the hamil- tonian. (6 is the titne reversed counterpart of the la- bel a.) Let us call this theory extended quasiparticle phonon coupling or EQPC.

However, whereas the EQPC theory solved the problem of the anomalous states in the "transitional"

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Volume 134B, n u m b e r 6 PHYSICS L E T T E R S 26 Janua ry 1984

odd mass nuclei, it did so at the expense of calculat- ing an overcomplete set of states {IUn)} half of which are actually unphysical. It remained unclear how much new information comes from the inclusion of backward coupling and how much from double counting due to overcompleteness of the basis. Also any attempts to use the EQPC odd mass states to calculate the even nuclear properties such as the pho- non quadrupole moment were plagued by the fact that neither

mation that diagonalizes the normalization matrix of the overcomplete EQPC basis, and the second diagonalizes the hamiltonian in the resulting ortho- normal basis after truncation. Below we shall outline further details of the method and present some pre- liminary numerical calculations that indicate the va- lidity of the method.

We start with the pairing plus quadrupole hamilto- nian, perform a Bogoliubov transformation on it, and get

IU.><u,,I, ",ill s t a t e s

n o r

IUn><Unl, physica l

s t a t e s o n l y

is necessarily equal to 1. This overcompleteness of states due partly to the

inadequate treatment of the Pauli principle and partly to the additional redundancy introduced by the "backward coupling" has generally plagued all at- tempts to further develop a self consistent equation of motion method [5] for the simultaneous treat- ment of the low lying states of both odd mass and even nuclei in the near spherical region [6].

A hint as to how to solve the redundancy problem has come from an analysis of the BCS theory. The BCS theory can be looked upon as a diagonalization of the pairing hamiltonian in the extended particle basis:

c~10), c,~ 10).

But this too is a nonorthonormal and redundant ba- sis. However, the beauty of the BCS-Bogoliubov theo- ry is that the same transformation (Bogoliubov trans- formation) which diagonalizes the pairing hamilto- nian also diagonalizes the normalization (or overlap) matrix of the above nonorthonormal basis, as shown explicitly by Ghosh and Goswami [7].

Following the above broad hint, in this letter we present a simple method which orthonormalizes the EQPC basis for generating odd mass states while it still includes the effect of backward coupling. We ac- complish this by performing two transformations on tile quasiparticle hamiltonian: the first is a transfor-

H = ~l : '~a;ac~-3× ~ 0uQ+u, (1) c~ p,

where E,~ denotes quasiparticle energy, × is the qua- drupole force strength determined by fitting the ener- gy of the 2 + phonon of the core, and,(2 u is the qua- drupole operator written in the quasiparticle repre- sentation, the explicit form of which is omitted for brevity as is the explicit mention of neutron/proton indices.

As already mentioned the EQPC basis consists of the following:

.t

[a~ I0), [a~ [2)]~, a,~ I0), [aBi2)] a}" (2)

As the first step, we evaluate the hamihonian matrix in this basis, following the equation of motion meth- od, using the Klein-Kerman factorization along with the usual "saturation" assumption. The result is given in previous work by one of us [8]. The matrix ele- ments of H generally contain self-consistent poten- tials involving matrix elements of (~u between core states: however, in the first order perturbation limit that we shall use in this paper, the self-consistent po- tentials take on values as given by RPA [8]. (That the first order perturbation solution of tile equation of motion method gives tile RPA description for the phonons of the core is, of course, one of the initial triumphs of this theory [8].)

In order to calculate the normalization matrix N~. of the overcomplete basis {I ¥~>}

(N aX)nn, : (Xn iXn,) , (3)

we use a formal complete set expansion in terms of the (positive energy) eigenstates of H, denoted by

IWm>

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= C <XXIW,%><WmLXX.>. (4) ?r/

The quantities (XnlW~n) are given by the first order perturbation solution of EQPC as given in ref. [8] ex- cept for one important correction. Since N~. contains

a a terms in second order in the quantities (XnlW,n), we must normalize them up to second order. This nor- malization procedure is not straightforward for a non- orthonormal basis. We use the recipe that, while nor- malizing {IW~n)}, the positive energy states of the ba- sis (iX~')} have a large norm which we assume to be 1, and the negative energy states a small norm of 0. This completely determines the matrix elements of N~..

Now we diagonalize the matrix N~ by the transfor- mation A :

A- 1N~.A = / ~ y , (5)

'~ is diagonal and the Y-representation is defined Ny

I Yn,) = ~ (A-l)nn,IXn). (6) n

Since the original set was overcomplete by a factor of two, half the diagonal elements of N~ should be zero (or much smaller than the other half'). By throwing out the states [Y~,) for which (Y~,IY~,) ~" 0, we will be left with the orthogonal set (i Y~')truncated} which will consist of linear combinations of the original states IXff). Next we transform H~ to H~ - A- IH~A. Now we truncate H~ by throwing out all elements (YffIHIY~,) for which (YnlYn)or (Y~,IY~,)-~-0- that is, if

a =/0( b 0

Ny 0 0 0 '

0 0 0

then

H ~ = q 0 Ytruncated 0 0 "

0 0

I . ! l m i l ' H a = q r ' Y U O

y Z

This insures that all physical interactions are included only once when diagonalizing H.

The matrix/~Ytruncated is now diagonalized, giving a final set of eigenstates {IZn)), which includes im- portant effects of backward coupling and is nearly or- thonormal.

Because the diagonalization of N is not expected to lead to the exact form

000 0 0

0 0

the eigenstates IZ~) will deviate slightly from strict normalization. They can be left this way (slightly un- normalized), but we have to remember this fact while using them as a complete set for subsequent calcula- tion, that is,

C - a a a IZ, ,><z n I I<Z n IZn> = 1. (7)

/1

To calculate the fractional parentage coefficients of the odd mass states, we do indeed use these eigen- states IZ n) of H. This involves using inverse transfor- mations which cause the formal problem that the ma- trix P which diagonalizes the truncated H is a 2 × 2 matrix. However, the formal solution to this prob- lem is straightforward. We generalize P to the form of the supermatrix

Pexpanded = (~ ~ ) '

where each of the zeroes represent a 2 X 2 null ma- trix. Such a matrix does not have an inverse, but we can get by with its transpose

( 0) (p xrr = P - 1

expanded! 0 0 "

To test this method, we have done a preliminary numerical calculation on 107Ag, using the pairing data of ref. [3] for the quasiparticle states. We consider only the states obtained from coupling the 9/2 + qua- siparticle to one and zero phonon core states. Our set

0x /25) : {,,;/2+Jo>, * * [a9/2 +P2 +10)] 9/2,

[a9/2+e2+lO)] 9/2}" a97~+10) ' - - +

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The normalization matrix in the X-basis is now calcu- lated to be

1.00 0.00 --0.26 0 .52 \

0.00 1.00 0.25 0 .53[ N 912+ = / . (8) . A 0.26 0.25 0.13 0.00 /

/ 0.52 0.53 0.00 0 .54/

Most gratifyingly, upon diagonalization the diagonal form of N was found to be:

/1 .54 0.00 0.00 0.001

Ngy/2+ = 10.00 1.13 0.00 0.00~.

/0 .00 0.00 0.00 0 .00]

\0 .00 0.00 0.00 0 .00!

This form of N makes the resl of the procedure, trun- cation etc., unambiguous and straightforward.

The numerical results for the 9/2 ÷ states of 107Ag are presented in table 1, where the results are com- pared with the previous EQPC results of ref. [4]. We have also calculated the fractional parentage coeffi- cients

(9/2+ qplc~9/2+lO)/(9/2+qp19/2+qp)

Table 1 Calculated energies and fractional parentage coefficients. The energy (in MeV) of the quasiparticle-like eigenstate, the pho- non-like eigenstate, and the~ difference as calculated for 1°TAg by QPC, EQPC, and the present method. The experi- mental energy gap between these two states is shown for l°SAg. The normalization factor (Zn~Z ~) in the expression for the parentage coefficient is a reminder that our [Zn)'S are slightly unnormalized.

QPC EQPC This work Experi- ment

Eph 1.41 2.03 1.95 Eqp 0.84 1.33 1.03 AE = E p h - Eqp 0.57 0.70 0.92

(9/2+phlc~fg/210)

X (9/2+phL9/2÷ph) -l - -0.03 0.02

(9/2+qplc~/210)

X (9[2+qp19/2+qp) -1 - 0.62 -0.30

1.04 a)

a) For l°SAg.

and

(9 /2 +phlctg9/2*lO)/9/ 2 +ptq9/2 +ph) ,

for both the present method and EQPC which are given in table 1.

I f H ~ were never truncated then P would be of the same dhnensions as A to start with and p - I A - I (the transformation which defines Z a as a function of X '~) would be a unitary transformation. But since we are trying to connect 2 spaces of different dimen- sions, we need to use the truncate Hr/expand P pro- cedure. The price we pay is that APexpanded has no inverse. If we substitute (Pexpanded) TrA.-1 for the

(non-existent) (APexpanded)- 1 and evaluate

a =zzTr Tr -1 a N Z = (Pcxpanded) A N X APexpanded,

we find N~ has picked up small off diagonal ele- ments. The off diagonal elements are very small, how- ever, showing the validity of our procedures, and we treat them as zero for calculational purposes.

Explicitly,

/ 1 . 54 0.05 0.00 0.00 /

NZ9/2+ = {0.05 1.14 0.00 0 .00 ] .

0.00 0.00 0.00 0 . 0 0 |

\0.00 0.00 0.00 0.00/

1.04

,~ / -0"01

N~ = / -.0.27

\ 0.53

To further check the consistency of our method of evaluating N~, with our final results, we recalcu- lated the normalization matrix via a complete set ex- pansion in the states {LZn) ) [eq. (7)] and found

-0.01 -0 .27 0 .53 \

0.97 0.25 0 .51 | .

0.25 0.13 - 0 . 0 1 J

0.51 -0 .01 0 .54/

Notice this is very close to the original value of N~ given in eq. (8).

| t is most encouraging that the numerical results basically bear out the soundness of the present tech- nique as well as confirm that the importance of the backward coupling as emphasized in ref. [4] is not fortuitous. Indeed, the "pushing up effect" sug- gested in this earlier work as a solution to the prob- lem of anomalous positive parity states in the Tc--

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Ag region is found to be still present, only slightly augmented. However, the correction to the fraction- al parentage coefficients over EQPC is large. This is, of course, to be expected.

What remains to be done now is the extension of the present idea to the complete self-consistent solu- tion of the problem which will provide us with re- liable energy and fractional parentage coefficients for the odd mass nuclei as well as such core quanti-

ties as the phonon quadrupole moment. We have ar- gued elsewhere [9] that for this extension we will need three transformations instead of the present two. l lowever, obtaining these self consistent solu- tions is a major numerical undertaking, and the re-

sults will be reported when completed.

References

[2] See fur example. A. Klein. Algebraic methods for a di- rect calculus of observables in the theory of nuclear band structure, UPR-0191T (Univ. of Penn., 1982), preprint.

[3] I..S. Kisslinger and R.A. Sorensen, Rev. Mod. Phys. 35 (1963) 853.

[4] A.I. Sherwood and A. Goswami, Nucl. Phys. 89 (1966) 465; A. Goswami and O. Nalcioglu, Phys. Lett. 26B (1968) 363.

[51 R.M. Dreizler, A. Klein, C.S. Wu and G. I)o Dang, Phys. Rev. 156 (1967) 1169.

[6 ] G. Borse, W.C. tluang and L.S. Kisslinger, Nucl. Phys. A164 (1971) 422.

17] M.K. Ghosh and A. Goswami, Phys. Lett. 39B (1972) 433.

[8] A. Goswami, O. Nalcioglu and A.I. Sherwood, Nucl. Phys. A153 (1970) 433.

191 A. Goswami and P. Rolnick, Bull. Am. Phys. Soc. 28 (1983) 21.

[ 1 ] See for example, Interacting Bose-Fermi systems in nu- clear physics, ed. F. lachello (Plenum. New York, 1981).

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