Calculation of even-parity states in 16O

Nuclear Physics 82 (1966) 321--330; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

CALCULATION OF EVEN-PARITY STATES IN 160 JERZY BORYSOWICZ t

The Niels Bohr Institute, University of Copenhagen, Denmark Received 20 December 1965 Abstract: Energies and wave functions for even-parity states in x~O are calculated using the simplest

available shell-model configurations. A method given by Brink and Nash 1~) is developed and some simplification in comparison with standard shell-model calculations is obtained. Results and their discussion are given.

1. Introduction

The problem of positive parity states in 160 has been recognized some time ago. The difficulty to get low-lying energies of the first excited states was realized 1-s) and both the shell model 1,4 - 6, s) andthe alpha-cluster model 2, 3, 7) were used to explain the character of these states. The description in terms of rotational bands 3, 7) as well as of the breathing mode 2, 4, s ) has been assumed. Measurements by Gorodetzky 9) showed a strong E2 transition between the first excited 0 + and 2 + levels, and it became clear 10) that these levels are closely related to each other, and that they have collective character. In the meantime, the SU3 techniques 11) were applied 12) and the 0 ÷, 2 ÷, 4 ÷ levels at 6.06, 6.92, 9.84 and 10.36 MeV were assigned as the members o f K = 0 and K = 2 rotational bands with (42) SU3 symmetry. The possible existence of a band with (04) SU3 symmetry was pointed out. The assumption of (42) symmetry for low lying states in 160 also made it possible to obtain 13) good transition probabilities between these states. After recent experiments by Carter, Mitchell and Davis 14) the existence of the rotational band based on the 0 ÷ state at 6.06 MeV with members 2 ÷, 4 ÷, 6 ÷ at 6.92, 10.36 and 16.2 MeV was established and also further possible bands were recognized 15). The calculation presented here does not bring the spectrum of excited states low enough; however, because not only states with J = 0 but also many others were calculated, the spectrum is more complete and it is possible to note some common features with the experimental one 15). In sect. 2 the coupling scheme, the method of calculation and the forces used in the calculation are presented. Sect. 3 contains results and their description as well as a discussion of possible improvements. In the appendix some formulae and coefficients used in the calculation are given. t On leave from the Institute for Nuclear Research, Warsaw, Poland. Supported by a grant from the Ford Foundation. 321

322

J. BORYSOWICZ

2. Coupling Scheme and Calculation A shell-model type calculation was performed, i.e. a Hamiltonian ~'p2/2rn+ 1I, where V = ~p<~V(pq), was diagonalized in a certain subbasis of the spherical harmonic oscillator functions. The basis was restricted to the states with the lowest 2 hco excitation energy ineluding both two particle-two hole states and one particle-one hole states. In addition, only states with the highest [nAA/.] permutation symmetry for the orbital wave function were allowed. The assumption of [.n/z n.] symmetry implies that the total spin and the total isospin are zero 16), so that spin-orbit splitting and spin-orbit forces are neglected. Also, in the case of two particle-two hole states, the orbital symmetry of the particles must be the same as that of the holes 12). If i,j label orbital states of particles outside of the 160 core, k, l orbital states of holes in the 160 core, and spinisospin states of both particles and holes are labelled by ~, fl, ~ ( = nl", n$, pT, p$), then states with [A.A.A.4]permutation symmetry are: one particle-one hole states:

liTk) = Z a+ ak~,lO),

(1)

g

two particle,two hole states:

[i,j; k, l) = ~ a,,,ak~ajaalalO), + +

(2)

where 10) denotes the closed-shell configuration of 160. After normalization and symmetrization in particles and holes these states are

li, j; k, l)s = n'~l(i,j; k, l)(li, j; k, l ) + l j , i; k, l)), li, j; k, l)it = n~J(i,j; k, l)(li, j; k, l ) - I j ,

i; k, l)),

Ii; k) = ½1i; k), where

n2(i, j ; k, l) = 24(1 + 6ij)(1 + 6kl), 2

nit ` ~-

40.

(3)

In this form they provide a convenient orthogonal basis. Eq. (2) may be written in the form

[i,j; k, 1) = E a,,a~aj#aia+ + it#

x ~ • • a ~ . . . alp+ • • "1) = ~ ' - " closed shell

a,~+ • .. a j#+ "" • I),

(2)

~#

where I ) denotes t h e vacuum in the second quantization scheme. Then [A.A.A.n.] symmetry states can be obtained from the closed-shell configuration by replacing orbital states of the core corresponding to the holes k, I with orbital states corresponding to the particles i, j, and states formed in this way should be summed for all possible spin-isospins with equal weights. The construction of the states with more

EVEN-PARITY STATES IN leO

323

than two particles and two holes with [A/..A4.] symmetry is analogous. The matrix elements of a two-body potential (i,j; k, II Vli',j', k', l') are reduced to the standard two-body matrix elements ( p q l v ( p q ) l p q ) - ( p q l v ( p q ) l q p ) , using Wick 18) techniques for creation and annihilation operators. This is simplified due to the [4444] symmetry of the states (see appendix). The relative simplicity of the calculation of the matrix elements is obtained because states of the basis are not eigenstates of total angular momentum J. The size of a matrix for diagonalization is then considerably larger than in the case when matrices are built separately for different J values. However, it is possible'to choose the basis for yet unspecified states i, j, k, l so that the projection of angular momentum M is conserved. The compromise way of diagonalization in the basis with unspecified J, but with given M, seems to have the following advantages: the matrix elements are easy to calculate and the matrix itself is not too big. The suitable basis for single-particle orbital states was suggested by Brink 17) tp~ = In+ n_ no) =

J(,,+)!(,,.)!(.o)!

+n+ (a_) + " " (a0) + " ° Ih .o.), (a+)

(4)

where a+

-- ( 1 / x / 2 ) ( a

+ + ta,"+ ),

a_+ = (1/~/2)(a + --ia+), a~" = a z + ,

(5)

and

ax+ = (1/x/2-m)(moJx- ipx), a+ = (1/x/~)(mo~y-

ipy),

a + = (1/~/~)(mo~z-ipz)

(6)

are boson type creation operators for oscillator quanta in the x, y and z directions, and [h.o.) is the vacuum state of a two-dimensional spherical harmonic oscillator. The projection of angular momentum of states tp~ and ]i,j; k, l) is, respectively, rni = n + - n _

(7)

and

M = m~+mj-mk-m,. The Cartesian basis (4) seems to be preferable to the usual spherical basis Inlrn) for single-particle states 19), because coefficients of transformations to the relative and c.m. systems (see appendix) !~--~--] tp, \ ~/2 ] /

(8)

are simpler in the first. Also the relation to the SUa functions is more apparent in the Cartesian basis. The coefficients mentioned above and the coefficients of trans-

324

J. BORYSOWICZ

formation from the Cartesian to the spherical basis (n+n_no[nlm) (see appendix) are the only "coupling" coefficients used in the calculation. Energies of states consisting of core + one particle, and core + one hole, which are necessary in the calculation, have been calculated from a 17-particle Hamiltonian according to the formula 2o)

Eq, core =

C X (%~0,lvl~o,>,

(9)

where q denotes hole or particle and l runs over orbital states of the core. The energy o f the closed-shell configuration of 16 particles is assumed to be 0. The coefficient C is 1.5, 1.5, 0.5, 4.5 for singlet-even, triplet-even, singlet-odd and triplet-odd forces. Because phenomenological single-particle hole energies are not used, the Hamiltonian is translationally invariant and spurious states are eigenstates. Their kinetic energy is hog, and the potential energy for states with two spurious excitations is 0, and for states with one spurious excitation it is the potential energy of the corresponding single-particle hole state. This method of extracting spurious states has been used by Brink and Nash 12). Within the assumed symmetry there are 39 states with M = 0 and 11 of them are spurious 12). The diagonalization was performed for sets of states with M = 6, 5, 4, 3, 2, 1, 0 and angular momenta of states have been assigned successively. The SU3 wave functions were constructed by diagonalization of suitable two-body operators diagonal in SU3 20). In the case of the (42), (31) and (20) SU3 symmetries, different sets of functions within given (2, #) symmetry are probably not labelled by the Elliott 11) quantum number K, however, they are orthogonal. Matrices were constructed and diagonalized with the help of the electronic computer GIER. The exponential forces with hard core, acting in relative s-atates given by Kallio and Kolltveit 21) have been used. This potential has not the character of residual forces commonly used in shell-model calculations, and was not adjusted to fit the nuclear spectra, but rather to fit the scattering length in the s-state and the binding energy of the deuteron. The separation method of Moszkowsky and Scott 22) was applied to calculate the matrix element of the potential with a hard core, as was shown by KaUio and Kolltveit 21). 3. Results and Discussion Calculated energy levels together with experimental levels 14, 28) are shown in fig. I. Theoretical levels are shifted 17 M e V d o w n to fit the experimental 0 + at 6.06 MeV. All experimentally known levels up to 14.83 MeV with the exception of the 2 + level at 13.1 MeV and all calculated levels with energies smaller than 19.91 MeV in the scale o f the figure are given, with the exception of 0 +, 2 + and 2 + states at - 1.97 MeV (fig. scale) 11.76 and 17.57 MeV. The first one of them consists in 96 ~ of one particle-one hole states and contains a similar percentage of the breathing mode. The position of the breathing mode is determined by the curvature

EVEN-PARITY

STATES

I N 180

325

o f the saturation curve for the binding energy at the equilibrium point, and this is not expected to be in the right position when the separation method is used 23). The separation method gives a proper binding energy at the proper density, but the saturation curve does not have an extremum at this point. The 2 + level at 11.76 MeV contains 45 ~o of one particle-one hole states and seems to be partly a quadrupole mode. (4+ ) 20.44 20

4+

--"

lg.So 1+ 1 5 . 4 1

®

(4+) 16.$ 6+

1.I.

16.2

ccO

15

2 2+

-

U x LLI 4+ 10

4+

13.80

3+

11.07

2+

9.84

16.21

3+

15.25 14.72

5÷ (2+) 1+

2 ~.

11.51

O+

11.25

14 2 .... 13,6S

6+

4*

14.86

2+

14.33

13.5

io.3s

-

15.16

9.25

16.15

2+ 13.76 1+ . . . . 13.30 4+

11.73

3+

10.16

2+

g.54

2:

10.46 10.05

4+

t"UJ

5

2+

6.92

0+

6.06

(a)

-

2+

6.84

0+

5.06

(4 2)

(b)

(4 2)

(0 4)

(3 1)

Fig. 1. Experimental and calculated even-parity levels of 1~O. a) Experimental levels and spin assignments 14,~8). b) Theoretical levels are shifted 17 MeV down. The dominant SU3 configurations are given below the appropriate bands. Some of the levels are not shown, as explained in sect 2. TABLE l

hole-core interaction

1 = 0

--37.55

MeV

particle-core

1 = 0

--25.76

MeV

interaction

1 = 2

--23.28

MeV

As in previous calculations s, 12) the calculated energy levels are much too high. However, part o f the 17 M e V discrepancy is specific to this calculation. The calculated energies o f particle-core and hole-core interaction (table 1) give the unperturbed energy of two particle-two hole states at about 40 MeV. Usually this quantity is extracted from experimental data for the neighbouring nuclei and is about 6 MeV smaller 24). To get a stronger particle-core interaction from the calculation one should use a smaller hco for particles in a ls-0d shell 23-2s). Also the assumption

326

J. BORYSOWICZ

Of a correlated ground state rather than the closed-shell configuration should increase the particle-core interaction and decrease the hole-core interaction. Further improvement in the absolute position of the spectrum should be obtained with more realistic forces. Kallio-Kolltveit forces contain an interaction only in the relative sstate. The effect of interaction in other relative states on the energy of levels with (42) and (04) SU3 symmetry was given by Brink and Nash 12). According to this, inclusion of the interaction in relative 0p states shifts the spectrum down about 5 MeV per 1 MeV strength of matrix element in the 0p state. Similarly, forces with 1 MeV matrix element in relative 0d states give a 2-3 MeV shift in the spectrum. The total shift expected from changes in forces is about 4-5 MeV. The main part of TABLE 2 T h e overlaps o f calculated states with SUa states. L 0 0 2 4 2 0 3 2 4 2 1 6 2 2 4 5 3 2 1 4

excit. energy --1.97 6.06 6.84 9.25 9.54 10.05 10.16 10.46 11.73 11.75 13.30 13.50 13.76 14.33 14.86 15.16 16.15 17.57 18.41 19.80

(42) 1 --0.05 --0.95 --0.69 --0.44 0.21 0.17 0.80 --0.07 0.88 --0.63

(31) 2

(04)

1

2

(12)

--0.20 --0.60 0.58

0.22 0.28 --0.83

--0.69 --0.39 0.21

0.13 0.18 --0.22

--0.18 --0.18 0.25 --0.05 --0.16

0.04 0.61 --0.54

0.39

--0.53 --0.69 0.15 --0.15 0.92

0.97 --0.11 --0.22 --0.30

1

(20) 2

0.06 0.24

0.13 --0.07

3 0.99 --0.70

--0.44

0.09 0.11

0.05 0.52

0.05

--0.15

--0.15 --0.08

0.16

--0.18

0.13

0.66

--0.10 0.12 --0.42

--0.05 0.29

0.05 --0.11

--0.05 ×0.21

0.19

0.72 --0.30 --0.39 --0.62

--0.18 0.71

--0.06

--0.49

--0.12

1.00 0.07 --0.24 --0.07 1.00 0.60 --0.09 --0.17

--0.55

0.18 0.86 --0.68

0.35

--0.10

--0.15

0.65

0.92 --0.39

T h e calculated energy is given in t h e scale as in fig. 1. T h e labelling o f states within a given (2,/~) s y m m e t r y p r o b a b l y does n o t c o r r e s p o n d to the q u a n t u m n u m b e r K (reg. xt))

the remaining 7-8 MeV discrepancy in the calculated spectrum should vanish after configurations with 4ho~ excitation energy are taken into account. It is possible, however, in a basis more general than the spherical oscillator basis to get the spectrum low using configurations with only 2hto excitations. The simplest modification is to keep the oscillator basis, but with different oscillator constants for different directions and for different shells 23, 25). There is an interesting indication that both positive and negative parity bands can be projected from intrinsic states built on the Nilsson 26) basis 10). Also the magnitude of the potential matrix element between the 0 + at 6.06 MeV and the ground-state levels indicates the necessity of having other

327

EVEN-PARITY STATES IN leo

components in the basis. This matrix element has been estimated from the experimental E2 transition probabilities in 160 and 1So to be about 2-3 MeV 13,27). The value of about 2 MeV is obtained under the assumption that the 0 + level at 6.06 MeV would fit exactly into a rotational band based on it, if there were no interaction with the ground state. T h e present calculation gives for this matrix element about 8 MeV, using Kallio-Kolltveit forces, and 5 MeV using simple positive-parity, Wigner forces - 3 5 exp(-r2/1.72) MeV (in both cases the wave function obtained from diagonalization of KaUio-KoUtveit forces has been used). This matrix element can be cut down either by using the deformed basis for excited states 13) or having a considerable four particle-four hole component (which has no matrix element to the ground state) in this state. From the discussion above it follows that the amount of two particle-two hole configurations in the ground state cannot be obtained properly from the present calculation, and similarly the transition probabilities between the ground state and excited states. The calculated transition probability B(E2, 0 ÷ ~ 2 ÷) between levels at 6.06 and 6.92 MeV is 15 e 2 • fm 4 what is in reasonable agreement with the experimental value 11 e 2 • fm 4. The large calculated value of B(E2, 0 ÷ - 2 +) is not surprising if one notes the big overlaps of the 0 + and 2 + states with (42) SU3 states (table 2). In table 2 the overlaps of states with SU3 states are given. As mentioned before, the different sets of (42), (31) and (20) states are labelled rather arbitrarily. From the table it is possible to recognize the definite SU3 character of most of the states, but this is not strong enough to say that SUa gives a good quantum number for different bands. The large overlaps for high spin and unnatural-parity levels are not convincing, because in the two particle-two hole basis there are not many other possibilities for these states. It is necessary to note that the comparison of the overlaps with SU3 functions with experimental alpha widths 14, 2s) is in contradiction to the selection rule for decay of the excited states of 160. According to the selection rule 12), decay into the alpha particle and ground state of 12C should be allowed for states with (42), (31) and (20) SU3 symmetries and forbidden for (04) symmetry. However, not only states with largest (04) components have big alpha widths, as noted previously is), but also states with (42) and (31) symmetries as 2 ÷ at 11.51 MeV, 2 ÷ at 9.84 MeV, 4 ÷ at 10.36 MeV have very small alpha widths. The author is extremely grateful for the hospitality extended by the Niels Bohr Institute and for inspiration and guidance from Professors Bohr, Brown, Hecht and Mottelson and Dr. David Brink, which have essentially contributed to this research. In particular the author wishes to thank Prof. G. E. Brown for suggesting the calculation and Dr. J. Gunn for invaluable help during the computation.

Appendix A.1. THE MATRIX ELEMENT OF THE TWO-BODY POTENTIAL The matrix element between single particle-hole states

(i, kl Vii', k') is a sum of

328

S. BORY$OWICZ

more elementary matrix elements . Now, the state a~= + a~10> can be graphically represented as in fig. 2, where the full line denotes the Slater o ict

g0¢ Fig. 2.

determinant 10) and oi,,ik~indicates that in this determinant the state q~k=is replaced with state ~o~=. From fig. 3 it is seen that the Slater determinants ai+#a,,#[O) and + a~=[0) are different only in the two pairs of states encircled by the dotted lines. ai=

,=

= <~o (~ ~,,, (2~ l ~ ~2> ] ,j,,~ ~, ~,~ c2>> 11

•

kO~

k? ¢"

I /

0

s"

/

J

/

[:/o i'~

I"

I i: Fig. 3.

Only these states appear in the matrix element, the direct term of which is written in the figure. In the case of one particle-one hole states this result is rather trivial, for two particle-two hole states the graphical method seems to be easier than standard Wick is) reductions. An example of a matrix element between two particle-two hole states is given in fig. 4. The exchange term is skipped again. An additional minus

/

/

,/"

oil) /

o jB

/


o+

=-<,p~,~c~) ~oo ¢2)[v c~2>1 % c~> ,p~ ¢2)> k~p ,: ~ - \

oj p ~

"-.. -.

tp \

"-,

j'Bo "~---~_ _ I' Fig. 4.

sign comes in because states tpj# and q~j,# should be interchanged to get the state %,# in both Slater determinants in the same place. From the fact that matrix elements of spin and isospin operators #~ and ~ vanish between states with [dddd] symmetry (states with [4"] symmetry provide scalar representations in SU4 and the operators # and ¥ are not scalar in this group) follows that the ratios of matrix elements of forces with different exchange character are

329

EVEN-PARITY STATES IN leO

constant between states with [4444] symmetry 2o). These ratios are the ratios of scalar parts of corresponding exchange operators. For example for singlet-even, triplet-even, singlet-odd and triplet-odd forces, projection operators are 1-trlt723+J1J 2 4 4

3+trltr2 1 - J 1 J 2 4 4

1-orltr 2 1 - J i J 2 4 4

3+altr 23+J1J2 4 4

and the corresponding matrix elements are proportional to the numbers 3 , ~ , 14, 9 16"

A.2. COEFFICIENTS OF TRANSFORMATION TO THE RELATIVE AND c.m. SYSTEM In any Cartesian basis these coefficients are separable in three basic directions, in particular in the basis (4)

(n +i n_ 1nollIN+ N_ No> = (n_, n_2[n_ N_>, where In+>

_

__

IN+)

1 x/(_~)+).l\

l

-x/-~-

]

10+>,

(a+l+a+21N+ / Io+>

and similarly for N_, n_, No, no ; 10± o) denotes the vacuum for a one-dimensional nlq harmonic oscillator. The coefficients {nlnz[nN) = A~1~2 are easily calculated 17, 20). Some of them are given below.

Inl)lnz) = I0>10> = I0>11> = 10>12> = l l>ll> =

~., A~,~2In>IN>, ~N IO>lO>, 0/~/2)10>11>- (1/x/2)l 1>10>, ½10>12>- ½x/211>11> + ½12>10>, (I/x/2)10>12>- (1/x/2)12>10>.

A.3. SHORT TABLE OF COEFFICIENTS In+n-no>

=

E

n,.

Cn÷n_no[nlm)

[ooo) = IOOO) IOOl) = IOLO)

1010> = [01 - 1 ) 1002) = x/~ 1020)-x/JrllO0) lOll) - 1 0 2 - 1 ) 11020) = 102-2> l110) = x/]1020) + x/i[ 100).

330

J. BORYSOWICZ

T h e s e coefficients a r e u s e f u l to e x p r e s s t h e m a t r i x c l e m e n t (n+ n_nolvln'+ n'_ no) as a s u m o f the m a t r i x e l e m e n t s (nlm[v[n'lm). The l a t t e r c a n be c a l c u l a t e d as in ref. 19).

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28)

R. F. Christy and W. A. Fowler, Phys. Rev. 96 (1954) 851 D. M. Dennison, Phys. Rev. 96 (1954) 378 H. Morinaga, Phys. Rev. 101 (1955) 254 P. J, Redmond, Phys. Rev. 101 (1956) 751 R. A. Ferrell and W. M. Visscher, Phys. Rev. 102 (1956) 450 J. J. Griffin, Phys. Rev. 108 (1957) 328 R. K. Sheline and K. Wildermuth, Nuclear Physics 21 (1961) 196 W. Vinh-Mau and G. E. Brown, Phys. Lett. 1 (1962) 14 S. Gorodetzky et al., Phys. Lett. 1 (1962) 14 A. Bohr and B. Mottelson, private communication J. P. Elliott, Proc. Roy. Soc. 245 (1958) 128, 562 D. M. Brink and G. F. Nash, Nuclear Physics 40 (1963) 608 G. E. Brown, Proc. Int. Conf. on Nucl. Phys., Paris, 1964 E. B. Carter, G. E. Mitchell and R. H. Davis, Phys. Rev. 133 (1964) B1421; G. E. Mitchell, E. B. Carter and R. H. Davis, Phys. Rev. 133 (1964) B1434 J. Borysowicz and R. K. Sheline, Phys. Lett. 12 (1964) 219 M. Hammermesh, Group theory (Addison-Wesley, Reading, Mass. 1962) D. M. Brink, Private communication G. C. Wick, Phys. Rev. 80 (1950) 258 T. A. Brody and M. Moshinsky, Tables of transformation brackets (Monografias del Instituto de Fisica, Mexico, 1960) D. M. Brink, Nuclear Physics 40 (1963) 593 A. Kallio and Kolltveit, Nuclear Physics 53 (1964) 87 B. L. Scott and S. A. Moszkowsky, Nuclear Physics 29 (1962) 665 G. E. Brown, private communication G. E. Brown, L. Castillejo and J. A. Evans, Nuclear Physics 22 (1961) 1 A. Volkov, private communication S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, no. 16 (1955) T. Engeland, Nuclear Physics 72 (1965) 68 F. Ajzenberg-Selove and T. Lauritsen, Nuclear Physics I1 (i959) 1