Electromagnetic properties of sd shell nuclei

1

l.D.1

1

NucZearPJzysiesA170 (1971) 273-308;

@ ~ortb-Ho~ln~d Publ~sbia~ Co., Amsterdam

Not to be reproducedby photoprintor microtilmwithoutwrittenpermissionfrom the publisher

A. ARIMA Depar@nepllof Physics, Universify of Tokyo, Tokyo M. SAKAKURA t.tf ~~titate for Nuclear Study, Uniuersity of Tokyo, Ta~~bi-shi,

Tokyo

and T. SEBE department

of Applied Physics, Hosei University, Ko@aBei, Tokyo

Received 24 October 1970 (Revised 1 March 1971) Ah&a& The electroma~~c properties of the nuclei 19*200, 1g*25*21F, 1g*20*zi*22Ne, 21*22Na and 2zMg are studied in a shell-model basis in which the valence nucIeons are restricted to the (Od, 1s) shell and the I60 core is assumed to be inert. Central two-body interactions are taken as the residual interactions. Beyond mass number A = 20, the SU(3) and SU(4) groups are used to truncate the shell-model space to a dimension less than sixty. Good a~ment with experiment is obtained, except for 22Na.

In this paper we study the electromagnetic properties of the nuclei 1‘?“0, I’, “* 21F, I’* ‘OS“* 22Ne, “, “Na and “Mg on the assumption that the 1‘%I core is inert and that the extra-core nucleons fill the &I+, Qd, and Is+ single-particle levels. The SU(3) and SU(4} groups are used to truncate shell-model states beyond mass number 20. The residul two-body interaction is assumed to be cent& and the exchange character is the same as that used in our previous papers L “1. Good agreement with experiment is generally found except for 22Na. The gyromagnetic ratios gs and gr are assumed to be the same as those for free m&eons. An isoscalar effective charge of t3.5e is assumed for quadrupole moments and E2 transition rates.

Although there is experimental evidence indicating that the I60 core cannot be considered inert 3, “> and th a t excitations of particles to the pf shel?.from the sd shell canrnot be neglected “> in treating the properties of the low-lying states of sd shell t This work is based on a thesis for the degree of Ph.D. of one of the authors (MS.) at The University of Tokyo, Tokyo. tr Present address: The Computer Centre, The University of Tokyo, Tokyo. 273

274

A. ARXMA ef aL

nuclei, it is highly possible that the main features of the electromagnetic properties are governed by nucleons in the sd shell orbits outside an inert 160 core. Our model Hamiltonian is assumed to have the form A = di,+Efs+

where JY, is the Hamiltonian

v,

of the I60 core and

which will be called the residual interaction. 170, we obtain

IJsing the observed level structures of

5 = 2.03 MeV,

E

=

1.15

MeV.

We fix 5 to be 2.03 MeV but change .afor each nucleus. It is assumed that the residual interaction Vij between pairs of nucleons outside the 160 core is central and of the form

where .FzT.+1,2S+1 is the projection operator for the state of isospin T and spin 5’ in which the strength of the residual interaction is expressed by 2T’ r, “’ ’ K The radial dependence,~(~~~), is taken to be a Gaussian exp( - (~~$~)‘). The range parameter L appearing in the matrix element of the residual interaction is given the value 0.7 (,? = P,J$, v = XPX~,&). The standard strengths used in this paper are: 13V = 60.0 MeV,

31V = 45.0 MeV,

llY = 0.0 MeV,

33V = -22.5 MeV..

A slightly weaker pot~~~al than that men~o~ed above gives better agreement with spectra for nuclei lighter than “Ne, while a slightl y stronger one gives better results beyond 20Ne. Definitions of the matrix elements of the electromagnetic operators with which we are mainly concerned in this paper are given in the following. In these matrix elements, Ji and JE stand for the spin of the initial and final states, respectively. (i) Magnetic properties The magnetic dipole operator is given by

where pP = 2.79 n.m. and lu, = - 1.91 n.m. I_Jsingthis operator, we can express the

sd SHELL NUCLEI

275

reduced Ml transition probabilities as B(M1, Ji + Jr) = The Weisskopf unit for B(M1) is 1.8 ,u: (1,~ = 1 n.m.). The magnetic dipole moments of states with spin J can be calculated by using the formula

(ii) Electric properties The electric quadrupole operator is defined by AI:) = &(l + Ser + 6eN) C I”:Yi2)(6i, +i) 1

-+e(l

+ 6% - 6eN) T T,(i)ff Y4(2)(Oi,+J,

where eGeP is an additional charge to a proton charge, induced by the core polarization, and e6eN is that to a neutron charge ‘* ‘) . These additional charges 6eP and Sen are assumed to be 0.5 in this calculation. The reduced E2 transition probabilities are given by this A&‘) as follows:

The Weisskopf unit for B(E2) is about 3.4 e 2 - fm4 in the mass-number region A M 20. The quadrupole moments of states with spin f are given by Q =

vf20’sf) J~
Calculated and observed reduced transition probabilities are summarized for comparison in table 1. The agreement between theoretical and observed values is reasonable except for some newly observed transitions from a highly excited state in lgF and for some transitions in 22Na where our calculation also fails to reproduce the level structure “). From table 1, transitions from the highly excited state in “I? are omitted. Discussions will be found in subsect. 3.1. 0ur calculated values agree excellently with those obtained by the Oak Ridge Group ‘), except in 22Na where their results both for the energy levels and transitions are better. Our residual interaction is phenomenological and central, while their interaction is based on the realistic two-body interaction which includes a non-central

TABLE I A Summary of EZ transition strengths (in W.U.) NucI.

A

Ji

2.40*0.03 24.5 ho.6

170

17

I’F

19

Exp.

Jf

‘90

2 0.37

‘9F

6.87-+0.13 6.8 1-0.7 7.7 51.5 5.0 *1.1 0.06{+“.‘8 0.06 3.0 {‘“,:Z

lgNe

21Na

2aNe %fg

B

6.51 6.44 6.71 4.46 0.022 0.84

0.83 6.31 6.31 6.61 4.55

C

l.O@

5 12

1.68

D

1.46

5.34

D

8.6

16.3 20.0 16.6 0.71

D

14.9 18.5 15.2 3.0

16.6 40.7

19.9

E

18 4.7 7.0 6.4 6.1 12.5 3.8 &

&4 11.7 22.9 A2.9 14.0 s5.5 +2.9 0.09

20.1 10.1 17.6 14.6 9.7 15.9 8.2 4.3

F

14

22.2

F

6

zzNa

A

c

29.5 &4.9 6.8 {“:I”, 20.7 f7.0 0.01 0.06

22

a

OX.

12.50

17.8 _il.7 15.4 11.4 27.8 16.0 3.8

21Ne

21

2.4 21.6 0.41 % 0.82

=*F 2oNe

Param.

12.5 f0.6

200 20

Calc.

0.8 2.7 10.9 42.0 0.04

18.0 A3.3 I 10.9 10.8 14.8 f3.3

16.3

28.3 ‘“2.:

20.3

19.9 23.3 10.0 14.9 13.2 8.9, 15.Q 7.6

27.9 6.9 24.9 0.2

G

16.4

15.0 H

19.S

H

We use the wave fmsctions which were calculated with the following parameters sets A-I,, for the c&c&ation of E2 transition strengths, with effective charge ese = 0.5e. A: s~l~p~cle estimate M(E2, s* -+ ds) for a suitable oscillator parameter. This is not dependent on E, E and potential parameters. B: c: D: E: F:

E= 5 = E= t = 6 =

2.03, 2.03, 2.03, 2.03, 2.03,

G: H: I: J:

E= E= 6 = 5 =

2.03, 2.03, 2.03, 2.03,

K:

L:

2.03, E = 2.03, E =

1.15 SF 5 0.0, E = 0.78, E = 1.27, E = 0.00, E = 0.78, E= s = s = E=

0.78, 1.15, 0.00, 0.40,

G = 1.15, 8 = 1.15,

=v =v -V =V =v =V =v l3V =v

= = = = = = = = =

60, 50, 60, 60, 70, 60, 70, IO, 70,

3lY = 45, siv = 37.5, 31v=45, 31V = 45. 3lV= 52, 3lV = 45, ‘lV = 52, Sll’c 52, 31V = 52,

=y_= llY= =v= ‘IV = llV =

0, 0, 0, 0, 0,

33 v = 33V = =v = =v== 33V=

1lV = IlV= llV= llV=

0, 0, 0, 0,

s3 v = -22s =Vz -26 =V, -26

13V = 60, =v = 50,

3”V=45, 31v = 37.5,

1lV=O, x11/= 0,

The calculated results of the Oak Ridge Group (0.R.) are taken from Hdbert nuclear physics, vol. 4, to be p~b~shed.

33y=

=v =V

-22.5 -18.75 -22.5 -22.5 -26

-26

= -22.5 = -18.75

et ai., Advances in

TABLE 1B Summary of Ml transition strengths (in W-u.) Nucl.

A

Ji

J;

Exp. 0.018~0.0006 & 1x10-4 0.20 hO.03 1.4 rtO.2

190 19

19F

0.30 0.024 0.019 0.61 5.0

=)F

20

21

0.033 &0.01X 0.072IfiO.022 0.12 jo.03 0.077,t,o.o3 > 0.017 > 0.02s 0.061 f0.011

21Ne

*‘Na 22Ne

2 0.02s (0.21 f0.04) x 10-s (0.2X&0.04)x 1O-3

22 2ZNa

Calc.

Param.

O.R.

0.036 0.16 0.21 0.63

K K I., 1,

< 0.006 0.04

0.039 0.13 8.3 0.078 6.5

D D D D D

0.03 0.05 2.8 0.26 2.4

0.0x1 0.094 0.29 0.29 0.39 1.5 0.074

F F F F F F I

0.039 0.16 0.19 0.19

0.0088 0.41 x10-3 0.84 x lo- 3

F G G

0.51 x 1o-3 0.89 x 10-S

See the caption of table IA. TAH.B 1C Summary of magnetic moments (in n.m.) A

17

.P

T

*

&+

Magnetic moments

Nucl.

1’0 l7l?

19

Isoscalar part of ,u

exp.

present talc.

-1.894 4.722

-1.913* 4.7934

2.629 -1.886

2.919 -2.058

2.87

3.705 -0.81s

3.55

1.959 0.544

1.92

l9F lgNe

-3.69 10.04 0.740~0.00s

!=F Z”Na

&0.36~$&)03

Q

21 5 +

2o

’

2it

21

k

Cl

21Ne 21Na

-0.662 2.386

22

0 0

3,+ 11 +

22Na 22Na

1.746 0.535~0.010

O.R.

-1.098 2.832

-0.90

1.826 0.787

See the caption of table IA. * Schmidt values of the dg state. For comparison follows:

exp.

Param.

present talc.

1.414

1.440*

0.372

0.431

B

1.475*0.05

1.443

B

1.230

1.252

D

0.862

0.867

I

1.82

A*

G G

we show the Schmidt values of s+ and d+ as

d+

& = 4.793&

flu, = -1.913Jk3

dt sB

0.124 2.793

1.148 1.913

= 1*440,&j ,%sos.z.tar 0.636 0,440

A. ARIMA

278

TABLE

Summary of qua~polo

et al. 1D

moments

(in e - fm2) Param. -

Oak Ridge

A

Nucl.

T

J=

Qexll

“9F lgNe “90

i

5 + i::

~11.0*2.0

19

- 9.61 - 12.73 $0.32 NN-0.12

C C B

z”Ne 2OF 20Na 200

0 1 1 2

-24.0f3.0

D D D B

-14.3 7.6

2%+

- 14.91 6.63 10.48 -f-O.23 E -3.24

) 4

& $1

22

tr 3%+ 2t * 21+

F F D

2ZNa 2we z2Mg

I 0 1 1

10.07 11.77 -11.00

10.3

21

T%e 21Na 2l.F

- 9.36 -15.49 -16.58

G J J

20

51

2r+ 21 + 21 +

9.3+1.0

-21.0*4.0

Qoak

-

9.2

-

0.1

-

4.6

-10.9 22.1

See the caption of table 1A.

part. It is very interesting to observe that level energies and transition probabilities are rather independent of the details of the residual interactions except for a few cases (especially for doubly odd nuclei). This is particularly true in the ground state band. Some transitions from the highly excited states are, however, sensitive to the residual interactions. There is a very iu~eresting disagreement between the calculations and experimental observations: this concerns the quadrupole moments of 2oNe and “Ne, although in both nuclei, B(E2, 2: -+ 0:) can be explained. 3.1. l9O, lgF AND

lsNe

We show the results of a calculation of energy levels for the “I! nucleus witb rather weak residual interactions (fig. 1). The agreement with experiment is very good up to about 7 MeV, except for the state at 3.91 MeV (‘“) in lpF. Only one Ml ~ra~si~on and one E2 transition have been observed in ‘% [ref. ‘)I (table 2). The &I1 transition from the first excited state to the ground state is very weak, which suggests that these two states have a nearly pure (d5)3 ~on~~ra~on or (d,, s+.)~conhguration with good generalized seniorities 13) in which the Ml transition cannot occur. The E2 transition between these levels is not enhanced, which is consistent with the above statement about the Ml transition. These facts indicate that “0 is spherical. The E2 transition probabilities between the lowest energy levels are remarkably well explained in lgF (table 3A). It is known that the level structure of “F can also be explained by the rotational model I’). Calculated transition probabilities with A? = 1 are very small, except for the transition from the ($)1 state to the ground state.

219

sd SHELL NUCLEI IO-

9-

B-

------7 -13 -7 -5

1 i

I

OL

-5 -7

z: EXP

-5 -1-l

a--a-51 E= 078

c=1.15 z--2.03

L

13V =5O

3’v = 37.5

“V

=V =-18.75

=

0

E =0.5 5 = 2.03 =v= 60 3’v=45 ‘IV = (y 3%~ z-255

-5

K-f-% Oak Ridge

Fig. 1. Level structure of 19F. Experimentally observed excitation energies and spin-parity assignments are taken from the references cited in the last reference of ref. *). In the shaded region above six MeV, the six positive-parity levels with 4 d J $3 were reported by Smotrich $‘) who used the reaction 15N(~, a’). Spins J are twice their values, e.g., 3 should mean J = 8. The state shown by the dotted line at 3.9 MeV has spin & but the parity is not known.

This is qualitatively consistent with the prediction given by the rotational model, but the observed transition probabilities from ($)1 to (a), and from (q)l to (s)1 deviate from those given by the rotational model and they arevery close to our calculated values. The pure rotational model predicts the ratio of B(E2, _“z”+ 3) to B(E2, 3 + 3) to be 1.57 but the observed value is at most 1. This is very interesting because small

280

A. ARIMA

eta!.

TABLE2A

JI

Ji

HMO

@02b

B(E2)

(e2 * fm4)

present

O.R.

present

O.R.

< 0.01 0.03 < 0.01 0.17

1.23” 7.61 4.10 1.26 0.21 0.80 2.16

2.6 8.6 3.0 0.6 0.1

& gz Qz & Q2

0.9

4,

*1

5. 81

Q1 81 %z

%l %1 %I

0.06** 0.05 0.04

$2

ri 21

0.08

%, 1e1

$1

Ql

2.69 0.28”“”

0.07

Ji

B(M1) -

f&02) B(E2) (e2*fin’)

present

O.R.

0.57 0.03 2.81

0.17 < 0.01

Jf

j& & 81

Qa $2

0.007 0.002

%2

present

O.R.

2.25 0.07 0.19 0.55 3.05 1.69 0.77

2.1 0.4 0.6 0.5 1.8

Present calculations: we use the wave functions calculated with thep~~e~rs .$ = 2.03, E = 1.15, 13Y = 60, 31 Y = 45, XIV = 0, 33Y = -22.5. O.R.: see ref. *). * B(E2, iI -+ tl),, & 1.11; ref. 9). ** B(M1, g1 -5 &), & 0.032+0.01; ref. 9). *** B(M1, 41 -+Ql)ex. & 0.0002; L. F. Chase et al., BuII. Am. Phys. Sot. 10 (1965) 426. TABLE2B CaIcuIated magnetic and quadrupole moments of IgO J

Mag. mom. &,) present talc. -1.57 -1.74 -1.72 +0.08 -1.58 +0.13 -0.78 +0.44 +0.97

Quad. mom.

(e - fm’)

present caIc.

O.R.

-0.12 -2.93 1-1.79 -0.17

-0.1

+3.00 -2.13 -2.61

-l-2.9

See the caption of tabIe 2A.

values for E2 ~a~sitio~s from a high angular rn~rne~t~ state indicate the termination of the rotational band at a certain angular momentum. Calculated Ml transition probab~i~es in 191F:from {& to (& from ($).)1to (s), and from (-“i_‘)lto ($)1 are much weaker than those from ($J1 to (& from {& to (g)1 and from (y)1 to (q)1 (table 3A). This can easily be explained by the L-S coupling scheme, although this scheme is to some extent broken by the spin-orbit interaction. In the L-S coupling limit the wave function of the (& state is given by 22S4, the (a), state by 22D3 and the (& by 22D, and so on. The Ml operator cannot change the orbital angular momentum L by two units and therefore cannot connect I’S), with (+),, but can connect (s)l with ($)1 in this L-5’ coupling limit. This is con-

sd SHELL NUCLEI

281

TABLE 3A B(M1, Ji -+ Jf) and B(E2, Ji + Jf) in 19F

B(E2, Ji + Jf) (e2 * fm4)

HML Ji -+ Jf> Cue’) Ji

Jf

exp.

talc.

exp.

present

O.R.

0.05 8.20

0.03 3.98

1

20.6910.39 “) 20.48h2.11 “)

21

*1 91 %1 *1 %l a1 t1 $1 $1 61

talc.

23.1914.52 “) 0.27 7.66 0.40

15.0 h3.4 0 * 18+0.540.18

b)

0.3610.05 ‘) 2.5 10.36 ‘)

5.14

9.0 “g.y.

c)

c1

present

O.R.

19.6 19.4 7.5 20.2 2.2 18.5 0.4 13.4 0.065 8.8 2.5

19.0 19.0 8.2 19.9 1.5 18.4 2.7 13.7 0.4 3.0

Present calculation: we use the wave functions calculated with the parameters 5 = 2.03, E = 0..78, 13V = 50, 31V = 37.5, llY = 0, 33V = -18.75, the level structure is shown in fig. 1. O.R.: see ref. 8). “) K. P. Jackson et al,, Phys. Lett. 30B (1969) 162. “) K. Bharuth-Ram et aZ., Nucl. Phys. Al37 (1969) 262. “) D. D. Tolbert, Ph.D. thesis, University of Kansas, 1968. TABLE 3B B(M1, Ji + Jf) and B(E2, Ji + Jf) in 19Ne

B(M1, Ji --f Jf) (po2) talc.

HE&

Ji -+ Jr)

(e’

* fm4)

exp.

talc.

37.711.7*

37.61 37.68 18.02 45.82 6.38 25.96 31.14 3.97 0.77 0.14 14.58

0.047 6.222 22.3: I:$** 0.273

6.076 0.442 4.577

See the caption of table 3A. * J. A. Becker et al., cited in S. J. Skorka et al., Nucl. Data 2 (1967) 347 ** R. D. Gill et al.; Nucl. Phys. Al52 (1970) 369

firmed experimentally since B(M1, (q)1 -+ ($)1) is very small while B(M1, (q-)-)1 --f (y),) is very large. For 19F, very large values have been recently reported for B(MI, (& + ($),),

282

A. ARIMA

3c

-hRLE

Calculated SfMl,.Ji

Ji

+ &> achy

Jf

et al.

ex&ed statesin lsF and l We ijn h2)

I9I?

-exp.

0.92i0.23

0.52&0.14

0.58zkO.22

2.0 *0.45

lpNe talc.

talc.

0.024 0.006 0.047 0.479 0.002 0.049 0.032 0.018 0.199 0.098 0.028 0.079 0.004 0.055 0.294 0,308 0.096 1.419 0.052 0.199 3.573 2.603 4.957 0.004 0.105 0.390 0.193

0.023 0.009 0.085 0.577 0.0004 0.050 0.035 0.020 0.250 0.099 0.021 0.074 0.0004 0.070 0.193 0.168 0.107 1.128 0.041 0.186 3.212 3.840 0.005 0.117 0.352 0.201

See the caption of table 3A.

B(ML $X -+ (+&), @ML (2”)~ + (S)1) and @ML & --, (-$)I) @able 3C). They are 0.92,0.52,0.58 and 2.0 in units of &. Our calculated values for all transitions are almost zero. furthermore, the observed B(E2, (& -+ (&) is very large, namely 41421 e2 =fm2, and the calculated value is about one fifteenth of this value. The observed B(E2, (& -+ (2j1) and S(E2, (& -+ (+&) values are also very large compared with the calculated values (table 3D). These facts suggest that we have insutbcient understanding of these highly excited states, although the nature of the ground state band is well understood. Benson introduced 2h-5p states, but he still has difficulties in explaining such large B(M1) values ‘l>. Serious ~screpan~i~s between theory and obse~a~on are found in the transitions from ($& at 5.47 MeV and from ($jZ at 3.91 MeV. However it is possible that the observed ($), at 5.5 MeV is a member of the ground state band. In that case the agreement with experiment becomes much better. For example, the strong E2 decay from (& to (-& is more reasonable.

sd SHELL

283

NUCLEI

TABLE 3D Calculated B(E2, Ji + Jf) from highly excited states in 191? and 19Ne (in ez * fm4)

Ji

Jf

19Ne

“9F exp.

< 8.6 59*34

41121 67+74 --42

talc.

CalC. present

O.R.

3.10 1.77 0.23 0.12 0.06 0.25 2.04 0.95 0.14 0.64 2.49 6.29 1.69 1.64 0.17 2.05 0.75 6.53 0.63 0.41 2.46 14.64 9.25 25.14 2.94 10.47

1.3

0.2 0.6

0.1 2.1 1.4

3.58 5.99 0.46 0.71 0.33 1.20 0.14 0.10 0.64 0.06 5.06 6.29 2.12 6.11 2.32 5.13 0.23 7.98 0.89 2.08 1.51 11.78 19.92 0.02 8.48 2.15

See the caption of table 3A.

The calculated magnetic moments and quadrupole moments of “F are in good agreement with the observed values (table 3E). Because of an increase in the proton number, the calculated B(E2) values and quadrupole moments in “Ne are about twice as large as those for lgF (tables 3B and D). Except for the magnitudes, the arguments given for lgF can be applied to ‘gNe. Our calculation gives reasonable values for magnetic moments, particularly for the first excited state (table 3F). 3.2. zoO, 20F AND

z”Ne

We have few data on transition probabilities in “0 and ‘OF. The available data consist of an upper limit on B(E2,2, -+ 0,) in 2o0 [ref. ‘“)I and a few B(M1) values and the magnetic moment of the 2+ state in 20F (table 4A and table 5A). The upper limit B(E2,2, + 0,) in 2o0 is seven times as large as our calculated value which agrees, however, with a result obtained by the Oak Ridge group “). Our

A. AROMA et ai.

284

TABLE 3E Magnetic and quadrupole moments of lgF

Magnetic moments Go)

Quadrupole moments

exv.

present talc.

O.R. talc.

2.63 “>

2.92 -1.52 3.10 -0.39 4.00 0.67 3.80 2.14 -0.72 -1.43 3.63 0.55 2.88 -0.71

2.87

3.69&-0.04 b,

3.55

exv.

111.012.0

(e - fm2)

vresent talc.

O.R. talc.

- 6.39 - 9.61 - 9.36 - 12.05 - 7.73 -12.82

-6.3 -9.2

‘)

-

2.90 7.00 3.97 2.45 x.09

See the caption of table 3A. “) Ref. 35). ““) R. M. Freedman, Nucl. Fhys. 26 (1961) 44.6. “) K. Sugimoto, A. ~~zobuchi and K. Nakai, Phys. Rev. 134 (1964)B539. TABLE 3F

Magnetic and quadrupolemomentsof “Ne J

/&AC

t-kw -1.886 “) -0.740~0.0077

b)

-2.06 2.83 -0.82 3.91 0.94 5.52 3.23 -1.88 1.52 2.78 - 0.75 2.39 1.00 5.01

e c&J - 8.86 -12.73 -11.41 - 16.02 --11.51 -16.22

-

5.02 9.42 6.09 0.47 4+39

Set the caption of table 3A. “) Ref. 35). b, J. Heck et aZ., Nucl. Phys. Al23 (1969) 65.

calculated values of B(E2) are not similar either to those given by the rotational model or to those by the ~bra~~o~al model, because B(E2, 41 -+ 2,) and B(E2,61 -+ 41) are much smaller than B(E2, 2r 4 0,) and further B(E2,2, + 2,) and B(E2,G + 21)

285

sd SHELL NUCLEI TABLE4A Calculated B(E2, Ji + Jf) in “0

Ji

Jf

21

01 01 21 21 21

22 41 22 02 42

21

B(E2) (e” +fm4)

Ji

present

O.R.

5.4* 0.096 0.13 7.0 0.74 4.7

4.1 0.6 1.9 0.6 0.0

B(E2) (e’ . fm4)

Jf

22 42 61 02 42

41 41 41 22 22 42

61

present

OX

4.7 4.5 0.21 3.8 0.003 2.0

1.8

6.1

We use the wave functions calculated with the parameters 6 = 2.03, E = 1.27, 13V = 60, = 45, llV = 0, 33V = -22.5. * Observed B(E2, 21 -> 0,) 5 38.7 e 2 * fm4; D. Eccleshall, private communication to H. Gove.

=V

TABLE4B Calculated B(M1, Ji + 4) in “0

Ji

Jf

22

21

42

41

Present talc. (,uo2)

O.R. talc. (,uO”)

0.016 0.022

0.03

See the caption of table 4A. TABLE4C Magnetic and quadrupole moments of 2oO J

kllc

a 21 41 61

22 42

-0.271 -0.914 -0.834 -0.690 -0.889

Q CalC

(PO21 b

a

b

O.R.

-0.322 -0.970 -0.828 -0.574 -0.761

+0.23 -4.48 -2.82 -0.53 -1.06

-1.13 -1.36 -2.79 +0.63 -1.92

-4.6 -1.7

of moments

are prepared

with the parameters

The wave functions used for the calculations shown in the caption of table 4A, except for F. “) E = 1.27; b, E = 0.78.

are smaller than B(E2,2, + 0,). This nature reflects the generalized seniority scheme in which the degeneracy of d* and s&is assumed. In this scheme, the four-particle system has just a half-closed shell for neutrons. While the E2 transitions between states with the same seniority are forbidden, those which change the seniority by one unit are enhanced 13). The observed B(M1,3, + 2,) for 2“F [ref. “)I is about eight times as large as the

f)(Ml,

2v

____. exp. 31 22 11 23 12 32

33 22 23

41 32

21 21 21 21 21 21 21

0.54 0.043 0.034”

3% 31 31 31

9.0

33

31

II

22

23

22

12

22

32

22

33

22

01

11

23

11

12

11

12

01

12

23

32

23

33

23

32

41

33

4i

33

32

TABLE 5A Jj -+ &) in ‘OF and ‘ONa (in &) present talc.

OX.

0.070 0.23 14.90 0.33 0.14 0.71 0.057 11.74 0.0016 0.15 0.022 0.053 0.104 0.042 0.0013 0.00001 0.0020 10.04 0.33 2.25 5.91 0.047 0.23 5.45 11.39 0.024 0.52

1.1

caIc.

2oNa present ealc.

0.06 0.09 5.06

0.106 0.22 13.38 0.39 0.014 0.79 0.061 9.84 0.0029 0.138 0.120 0.087 0.012 0.047 0.0012 0.0013 0.0009 10.74 0.34 1.89 5.33 0.052 0.25 5.89 9.75 0.10 0.45

0.46 0.05 4.30 0.10 < 0.01 0.23 0.01 0.03

2.6

The wave functions used for the present calculation are prepared with the parameters: *lV= 0, 33V= -22.5. E = 2.03, E = 1.27, I3 V = 60, 31V=45, * An assignment of l+ to the 984 keV state is not definite. TABLB SB Calculated B(E2, Xi -+ Jf) in ‘OF and ZffNa (in ez

Ji

Jf

B(E2, Ji + Jf) (ez

ZONa

20F

31 41 41 61 22 11 01

21 2, 31 41 21 21 2X

present

O.R.

17.20 9.65 18.94 17.03 8.14 8.19 5.89

27.2 11.4 21.2

0.2 5.4

12

21

3.88

32 33

21 21

4.55 7.07

11 12 32 33

31 31 31

4.10 15.17 0.55

31

4.96

See the caption of table 5A.

* fm”)

* I%?)

7.5 18.9

11.1 0.8 3.5

38.39 24.49 46.95 37.56 15.21 16.43 20.72 12.85

6.71 17.61 15.86 14.73 2.87 10.77

287

sd SHELL NUCLEI TABLE5C Magnetic and quadrupole moments of 20F and 20Na J

P

cue) 20Na

+1.96* +1.47 +2.15 $3.03 -1.26 -3.35 -2.28 +0.82 -4.14 +2.59

21 31 41 61 11 22 32 12 23 33

+0.54 ** +2.24 +2.49 +3.59 +2.31 +5.44 f7.91 +0.75 -2.05 +0.91

20F

Q(e *fm’) O.R.

+ 6.63 - 4.91 - 5.05 -10.30 + 4.03 - 2.67 - 2.93 - 0.26 - 7.74 - 5.74

20Na

7.6 -3.2 -5.0

+10.48 - 9.76 - .x20 - 14.45 + 6.53 - 6.52 - 4.69 - 1.06 - 10.75 - 5.68

s=o

O.R.

See the caption of table 5A. Observed values of the magnetic moments are as follows: * PObS = 2.092 po; ref. 3s). ** ,u.bs= 0.368 po; H. Schweikert,private communication. TABLE6A Calculated B(M1, Ji -+ Jf) in ‘ONe (in po2>

Ji 22

24 42 23 24 24 62 82

Jf

.? = 1.15

e = 0.78

E = 0.4

21

1.07 x 10-s 5.90 x 10-b 1.21 x10-5 1.08x10-~ 2.70 x 1O-3 1.69 x 1O-3 5.46~10-~ 1.69 x 1O-2

1.16~10-~ 7.43 x10-4 1.14x 10-6 9.70 x 10-s 3.49 x 10-S 1.28~10-~ 5.25 x 1O-3 1.69 x 1O-2

1.23 x 1O-5 8.55 x 1O-4 1.03 x10-5 8.75 x 1O-5 3.95 x10-3 9.57 x 10-a 5.10 x 10-S 1.69 x lo-*

21 41 22 22 23 61 81

1.30 x 10-S 9.40 x 10-d 8.61 x 1O-6 8.53 x lo- 5 4.17 x 1o-3 7.66 x 1O-4 5.00 x 1o-3 1.69 x lo-’

3.0x10-5 2.0x10-5

In *ONethe quantities B(Ml), B(E2), p and Q are calculated by the use of the wave functions with parameters: E = 2.03, 13V = 60, 31V = 45, 11 I.‘= 0, 33V = -22.5 and four values of E.

calculated value, but B(M1,2, -+ 3 1) is in good agreement with our calculated value. The previous calculation predicted around 500 keV the 2, state which corresponds to a level at 823 keV which is assigned 2’. This calculation predicted two low-lying 1’ states. The levels at 984 keV and 1054 keV are reported to be l+, but the former may not be a 1’ state because our calculation shows a possibility of a strong Ml transition from the 1 + state to the ground state while the observed one is very weak. Many data are available on “Ne (table 6). The excitation energies show an interesting deviation from the I(I+ 1) rule, which is well explained by previous calculations I*‘*1“). Some of the excited O+ states around 7 MeV should be the core-excited states. One of the authors (A.A.) and Strottman have taken into account the coreexcited states to explain these excited states around 7 MeV [ref. “)I.

TABLE 6B Calculated B(E2, Ji --f Jr) in ‘ONe (in e2 * fm4)

Ji

Jf

Exp.

21

01

41 6t 81

21 41

57.3 =tz 8.0a) 49.7 & 4.5 “) 89.8 & 19.2 “)

22

01

23

01

02

21 21

22

21

23

21

42

21 41 41 4% 41

23 42 62 22

02

23

02

22

03

23

03

23

22

42

22

42

23

42 02 62 62

61 62

6 0.970) 12.341

?

3

42

61 81

82

0.78

E =

52.6 64.5 53.4 32.8 8.4x1O-4 0.30 2.3 2.9 I.1 0.07 3.0 0.19 0.03 0.15 0.50 24.0 3.7 0.11 2.4 37.4 4.0 10.7 5.2 18.7 0.91 1.4 2.2

61

03

22

S = 1.15

51.7 63.3 52.3 32.7 7.6 x 1O-3 0.33 4.7 2.4 1.9 0.02 3.6 0.43 0.002 0.15 0.49 21.7 6.0 0.09 2.9 29.5 1.8 12.1 6.6 18.0 0.77 1.5 2.2

E = 0.4 50.7 61.9 51.0 32.5 3.2 x 1O-2 0.35 7.7 1.8 2.7 0.001 4.3 0.74 0.01 0.14 0.49 19.6 7.8 0.07 3.5 21.6 0.56 12.5 8.0 17.3 0.58 1.7 2.2

6=0 49.5 60.3 49.5 32.3 6.1 x 1O-2 0.36 10.8 1.3 3.8 0.007 5.0 1.1 0.07 0.10 0.50 18.1 9.2 0.05 4.2 15.7 0.06 12.1 9.5 16.8 0.37 1.9 2.2

OR. 48.1 59.7 48.8 31.2 0.1 9.8 2.6 0.0 2.8 0.7 0.7 0.0 16.2

12.5 0.1 x.1

See the caption of tabIe 6A. Observed values are taken from “) H. G. Evans et al., Can. J. Phys. 43 (1965) 82. b, E. A. Litherland et al., private comm. by H. E. Gove. “) J. D. Pearson and R. H. Spear, Nucl. Phys. 54 (1964) 434. d, C. van der Leun et al., Phys. Lett. 38 (1958) 134. TABLE 6C Ma~etic

41 61 81 2, 23 24

42 62 82

P Cud E = 0.78

& = 0.0

1.006 2.016 3.037 4.086 1.021 1.022 1.263 2.026 3.361 4.489

1.006 2.016 3.037 4.086 1.023 1.019 1.292 2.026 3.352 4.489

1.007 2.017 3.038 4.086 1.025 1.018 1.313 2.027 3.342 4.489

See the caption of table 6A. * Ref. 27).

moments of zONe Q (e * fmz)

E = 1.15

J

21

and qua~pole

E=1.15 -14.9 -18.8 -20.5 -20.7 2.6 - 3.1 9.3 -15.9 - 9.5 -11.7

.a=O.78

E = 0.0

O.R.

exp.

-14.8 -18.7 -20.4 -20.7 5.1 - 5.7 9.1 -15.8 - 9.0 -11.7

-14.4 -18.5 -20.1 -20.7 7.6 - 8.3 8.5 -15.2 - 8.1 -11.7

-14.3 -18.2 -39.6 -19.8

-24&3*

289

sd SMELL NUCLEI

B(M1, .& +. Jf) and B(E2, Xi + Jr) in zlP

BWl)

@02>

present talc.

B(E2) (e2 +fm4) present talc.

O.R.

68.54” 6.07 6.01 17.47 22.03 3.02 5.99 10.43 1.69 0.18 14.57 2.90 2.71 2.51 10.35 1.52 24.97 11.45

68.6

3.54 0.65 0.23 0.83 2.93 0.69 0.26

0.08

0.20 1.15

We use the watrefunctions for the calculation of the electro~a~etic properties in 2’F prepared with parameters E=2.03,~=0, 13Y=60,31V=45, 11Y=Q,33V= -22.5. * B(E2, $I --f &) = 57.1k2.5 e2 * fm4; ref. g).

Magnetic and quadrtrpob moments of 21F J

%1 %1 t1 3-1 91 +1 *z tz %z

Q (e * fmz)

Pcuo)

present talc. 3.77 3.23 3.45 3.30 4.23 2.82 -1.20 2.74 -1.64

present talc.

O.R.

-11.2 - 3.1 -13.8 - 4.6 -14.8

-10.9 - 4.4 -13.6 - 7.2

-

-

4.0 7.06 1.74

6.5

See the caption of table 7A.

The observed transition probability from the first excited state co the ground state of 20Ne agrees well with the calculated value. The calculated B(E2,4, + 2,) is slightly larger than the observed vake, while the calctiated B(E2,6, -+ &) is slightly smaller

290

A. ARIMA et crl.

than the observed one (table 6B). A new experiment performed at Chalk River seems, however, to reduce these discrepancies 16). A large transition probability has been measured from the excited 0: state at 6 MeV to the tist excited 2: state, If E is assumed to be the same as that in 17Q, the calculated value is only one seventh the measured value. Even if the core-excited state is admixed, B(E2) remains too small “). This transition probability is, however, sen-15,9 9

-15 T&/$ ..__A_

,J:y

-3

._*

________ T=%,J=%

-

-5

-

-9

-7

=-=+===A

-9

il 13

:

--

5

-5 ====1

9

t+) ==-=+-=s

3 5

-3 -5 -9

-5 -3

-3

ml=-==-9

scGizzG;

m9

-.----I m5mnr‘7’+’

===-5J=7

--m--7

n-EmRm7

-5

=--==5 -5 5==-==3-3-=-==-3=-=5=-=3=---wa3~3 &=i,27 &=a.78 Ew.

====-=7

=+==-5

-5

E = 0.0

RI=mwnz7

-5

E= 1.15

K + “0

3 = 2.03 5=2.03

*‘Ne

,

‘“V-70 “V=

0

‘v

“If=

52

( 3’V=_26

= 60

3’V = 4 5 “V3 0 =V =-22.5

Oak Ridge

Fig. 2. Level structureof 21Ne. Experimentallyobserved excitation energies and spin-parity assignments are taken from refs. 19*22) and the following references: M. B. Burbank et nl., Nucl. Phys. All9 (1968) 184; A. J. Howard eb nl., Phys. Rev. 184 (1969) 1095 Spin values are shown in the same maer as in fig. f .

TABLE8A B(M1, Ji + Jf) and B(E2, Ji + Jf) in the same rotational bands of 21Ne

BWL Ji -+ Jf) Cd>

B(E2, Ji + Jf) (2 . fm4)

exp.

present talc.

O.R.

exp.

present talc.

O.R.

0.06&0.02

0.145

0.07

0.13+0.04

0.169

0.28

0.22&0.06

0.527

0.35

0.14kO.05

0.525

0.35

63113 16+ 6 24110 22110 21*14 43119 13110

80.0 34.4 51.4 45.3 30.7 51.6 26.1

< 0.03

0.704

69.1 34.8 60.4 50.3 33.4 54.7 28.3 48.7 14.9 47.6 13.5 31.9 42.5 17.7 49.6 6.5 8.2

> 0.3

0.832 0.058 0.173 0.157 0.002 Present calculation: we use the wave functions prepared with parameters lay= 70, =V= 52, l’V=O, aaV= -26. The observed values of B(M1) and B(E2) are taken from refs. lg. 22).

E = 2.0, E = 0.78,

TABLESB

Calculated B(M1, Ji --f Jf) and B(E2, Ji -+ Jf) in 21Ne

Ji

Jf

O.R. 0.72

a1 21 a21

2.71’ 0.55 0.56 2.19

a21 h21 621 I21

0.09 2.96

i+l

%2 tz tz 32 121 %, $2

:I

a2

%l

%z % %z t2

81

42

$2 L22 a22 222 32 122 $52 82 31

B(Ml) Gut?) present talc.

61 %I HI $1 221 41

0.09 2.45 0.49 0.16 2.28

31 81

%I %z 32 ++l -82

3.62 0.15 0.08 2.75

0.41 0.13

B(E2) (e2 * fm4) present talc. O.R. 10.4 2.7 5.8 1.0 0.4 19.6** 19.3 0.8 2.0 0.5 1.1 5.8 4.1 0.9 5.0

1.5 0.1 2.7

0.4 3.5 4.6 2.3 38.3 2.7 0.1

See the caption of table 8A. * B(M1, t1 --f $1) 2 0.05 y,2; P. J. K. Smulders and T. EC.Alexander, Phys. Lett. (1966). ** B(E2, +I + +I) = 18.3~tl4.1 e2 * fm4; R. D. Rent et al., Third Conf. on reactions between complex nuclei, Gathnburg, 1963, p. 417.

A. AROMA et al.

292

TAIXS 8C Ml transition probabifities Ji

of ‘INa

compared with 21Ne

E = 1.27 bo2)

E = 0.0 (/Joy

21Na

21Ne

‘INa

2”Nc

ZINa

0.1334 0.383 0.529 0.885 0.840 1.167 0.095 0.016 0.158 0.044 3.066 0.306 0.583 0.258 3.408 0.025 4.911 0.345 3.492

2.660 0.526 2.283 0.186 0.520 0.887 0.058 2.81-7 0.973 0.398 0.057 1.716 0.994 0.074 0.322 0.326 0.01 I 0.297 0.291

3.433 0.711 3.309 0.279 Q.639 1.044 0.088 3.911 1.429 0.481 0.072 2.296 1.289 0.127 0.348 0.389 0.024 0.353 0.388

2.780 0.635 2.044 0.123 0.622 0.875 0.145 3.190 0.485 0.737 0.146 1.907 0.830 0.081 0.267 0.222 0.003 0.232 0.457

3.634 0.850 2.968 0.189 0.768 1.031 0.128 4.384 0.731 0.914 0.186 2.585 1.100 0.138 0.289 0.258 0.010 0.275 0.604

E = 1.27 (,uo’)

& = 0.0 (&)

“INe

21Na

21Ne

0.187 0.096 0.594 0.412 0.722 0.744 0.055 0.311 0.812 0.0001 2.526 0.104 0.558 0.139 2.178 0.021 3.612 0.104 2.832

0.247% 0.118 0.745 0.499 0.892 0.911 0.078 0.372 0.263 0.002 3.308 0.171 0.803 0.178 3.093 0.032 5.002 0.166 3.726

0.100 0.309 0.425 0.723 0.690 0.943 0.066 81.020 0.104 0.021 2.315 0.193 0.400 0.202 2.419 0.016 3.600 0.223 2.617

Ji

Jf

We use the same values of the residual interaction and 5 shown in the caption of table 8A, and used for 8 = 1.27, 0.78 and 0.0. * ObservedB(Ml, & -+ &) = 0~1~~~.02~*’ in 2”Na; ref. zz),

sitive to E [ref. I’)]; when E = 0.5, the calculated value can reach one-half of the experimental value. A calculated ~~adru~o~e moment of the first excited state is smaller than an observed value I*), even though B(E2,2: -+ 0:) is &ted to the observed one by adjusting an effective charge 5e (table 6C). This kiiad of discrepancy can be also found in “Ne, and possibly in 24Mg. Calculated magnetic moments of the rotational states belonging to the ground state band show an interesting behavior; they are given by g,l where g, = 0.5 n.m. and I is the spin of the state (table 6@). This coincides exactly with what the rotational model predicts. It is, however, easily understood in the following way. These states have [4]‘%, = L as the main components. Hn this L-S coupling scheme, ,uis nothing but +L. because all parts depending on z and c vanish. It is interesting to find that an admixture of [31] induces only small changes in these magnetic moments. 3.3 21F, 21Ne AND

zlNa

A calculated E2 ~ansi~io~ probability from the ($)1 state to the ground (g)i state in ‘IF is 20 W.U. which is in good agreement with the observed value and the value calculated by the Oak Ridge group (table 7A).

sd SHELL

293

NUCLEI

TABLE SD 332~r~ns~~jo~ proba~~~~~s of 2iNa compared with asNe (in e2 * fm4)

65.75 35.04 58.07 52.31 32.26 55.83 27.49 50.06 14.43 48.91 13.67 29.64 39.98 49.90 18.14 5.47 56.93 7.41 11.59 5.53 6.35 1.04 2.40 18.03 0.30 16.15 1.30 1.93 3.29 32.03

72.w 42.05 74.96 45.31 24.20 59.69 36.00 42.13 9.31 51.32 20.88 40.29 49.51 63-U 25.39 5.51 62.62 21.4f 21.94 11.65 1.88 7.19 0.04 llS6 0.39 5.71 0.29 4.16 0.03 361.70

72.01 34.23 6G35 47.00 34.23 52.60 29.05 46.20 15.26 45.16 13.00 33.89 44-33 48.98 16.02 8.11 56.48 9.10 8.80 0.30 4.63 0.91. 0.75 16.79 0‘43 12.68 2.42 3.08 1.27 32.82

80.54” 41.61 71.95 46.80 26.73 58.16 32.20 40.51 9.89 48.14 17.38 45.05 50.98 62.26 19.46 7.46 60.13 11.17 19.52 2.55 0.86 6.66 0.38 11.22 0.32 3.59 1.07 7.23 1.Q4 35.60

28.95 20.14 0.48 1.79 0.09 4.48 0.25 1.42 9.92 3.83 0.25 0.34 0.68 7.08 0.23 0.04 2.62 4.31 1.76 4.00 27.99 3.55 1.88 10.45 0.65 x.35 0.18 0.16 9.03 42.83

7.18 27.40 3.54 X2.25 0.34 0.07 0.001 0.63 0.69 4.66 6.16 0.87 0.16 9.61 0.97 2.73 5.13 5.82 5.82 0.20 31.94 8.15 4.48 2.53 0.31 3.44 0.89 0.46 x3.55 28.52

8.73 17.08 1.43 2.27 0.03 6.78 0.89 0.74 1.69 4.52 0.42 0.24 1.29 2.26 0.55 0.01 5.28 4.79 1.19 1.19 29.23 4.93 1.15 5.74 X.88 2.22 0.001 0.12 9.74 33.72

0.05 1x.17 6.40 13.06 0.68 0.55 0.22 0.15 1.57 5.69 6.54 0.65 0.04 4.01 0.31 2.56 7.45 8.66 4.85 1.89 31.98 10.64 2.87 0.57 1.53 4.30 0.18 0.49 34.57 20.21

See the caption ofta& SC. * Observed .+I&& -+ &) = 30-&X4 e2 * fm4 in aiNa; ref. 22).

We show here revised ca~c~la~o~sfor the level structure of 21i%e@g. 21, because we have found some mistakes in the input data for the ca~c~atiol~sreported in ref. “1. ~lculatio~s show very good agreement with ex~er~~e~t es~c~~~~yfor the ground band but not for excited bands, caky t~~si~o~ ~~obab~~~esin 21Ne have recendy beeTPmeasured for both I%1 and 332t~a~si~ons ’ “1. ~~ee~ent between the calculated and observed values is very ~easo~abKe(table 8A), although the rota~o~al model with. the band mixing gives s~~gh~ybetter a~ee~e~t I”). ft is worth ~entio~~~g that the I%?~~ans~tio~sfrom states with high angular mocotta in 21’Naare not much larger or even smaller t&m those in 2*Ne (table SD).

294

et al.

A. ARIMA

TABLE SE

Calculated rna~e~c J

E = 1.27

lf-kd

’ INe

alNa

moments of 2tNe aud ‘*Na E=0.78&o)

%I iz-i B e1

-1.12* -1.34 -0.52 -0.096

;: Q1

0.53 1.27 1.67

5.29 5.57 6.19

-1.11* -1.25 -0.52 -0.044 0.52 1.35 1.66

61 $2 s22 $2 a22

-1.23 1.77 0.25 2.33 1.68

1.93 -0.46 2.36 1.00 2.87

--I,16 1.77 0.23 2.31 1.69

$2 %a

0.15 1.81 0.43

0.41 -0.51 2.20

791

63

2.87** 4.19 4.33 4.94

2.86** 4.08 4.33 4.87 5.31 5.49 6.21

0.087 1.82 0.47

See the caption of table SC. * p,bs = -0.66 p. for 2iN($r +); ref. 35). The c&uIational result of ref. a) is -0.90 ** ,u.,,,~= 2.39 ,uuofor 21Na($$l+); ref. 23).

*lNe

2wa

-1.10* -1.10 -0.51 0.046 0.51 1.47 1.63

2.83”” 3.91 4.32 4.76 5.32 5.35 6.24

21Na

21Ne

0.0 (f.60)

E =

1.85 -0.46 2.38 1.03 2.86

-1.04 1.73 0.20 2.25 1.71

1.72 -0.41 2.42 1.10 2.83

0.48 -0.52 2.15

-0.017 1.86 0.57

0.60 -0.56 2.04

pe.

TABLE SF

Calculated quadrupole moments of z%e E = 1.27 (e+fm’>

E = 0.78 (e

ZlNe

21Na

$1 81 $2 $1 3-t %I %I

9.96* - 5.54 - 8.58 -16.02 --15..56 -20.08 -18.87

82 $2 92 32 $3 $3

J

and alNa e = 0.0 (e

* fm2)

* fma)

zlNe

*rNa

‘lNe

21Na

11.76 - 9.15 - 9.18 -20.20 -15.04 -23.45 -17.69

10.07* - 4.35 - 8.60 -15.33 -15.41 -19.68 -18.75

11.77 - 1.99 - 9.17 -19.44 -14.91 -22.71 -17.62

10.18” - 2.75 - 8.65 -14.17 -15.17 -19.07 -18.51

11.72 - 6.32 - 9.14 -18.07 - 14.68 -21.45 -17.47

- 9.39 - 5.47 -16.82 -13.85

-11.41 - 4.70 -20.00 -11.82

- 9.40 - 6.50 -16.19 -14.36

-11.37 - 5.73 -19.27 -12.68

- 9.39 - 7.47 -15.09 -15.08

-11.34 - 6.85 - 17.94 -14.13

-

- 10.09 11.06

-

- 10.02 10.66

-

-

9.83 8.74

See the caption of table SC. * &bs = 9.3hl.O e - fm2 for “‘Ne(&+);

9.88 8.34

ref. 35).

9.81 7.41

9.72 9.72

OX 2’Ne -.--. 10.3’ - 3.4 - 9.5 -14.7 -15.3 -19.0

sd SHELL

295

NUCLEI

TABLE SG Isoscalar parts of the magnetic moments of A = 19, 2’ = + and A = 21, 2’ = 4 (in ~0) A = 19

J

121

$1 r21 221 31 Q1 PI $1 61 %z Bz $2 %z 42 %3 -33

A = 21

exp.

E = 1.15

8 = 0.78

0.372

0.432 0.655 1.443 1.741 2.466 3.118 3.538

0.431 0.656 1.443 1.764 2.481 3.097 3.538

0.670 1.442 1.960 2.503 0.430 0.858 1.486

0.678 1.443 1.942 2.500 0.429 0.875 1.473

1.475 hO.025

exp.

E = 1.27

E = 0.78

& = 0.0

0.862

0.878 1.424 1.901 2.421 2.914 3.423 3.933 0.352 0.656 1.301 1.665 2.273 0.280 0.652 1.318

0.878 1.418 1.903 2.415 2.915 3.418 3.934 0.347 0.657 1.304 1.668 2.273 0.285 0.651 1.314

0.867 1.407 1.904 2.405 2.916 3.410 3.934 0.339 0.659 1.310 1.675 2.273 0.294 0.650 1.305

The calculation is done using the wave functions prepared with the same values of E and residual interactions shown in tha captions of table 319 and table 8A.

This seems to be strange because “Na has more protons than ‘lNe has. This can be partially explained by the fact that the main contribution to the E2 transition comes from the “Ne core. The same situation will be observed in 22Ne and “Mg. No shell-model calculation has succeeded in giving the correct excitation energy for the (a), state in ‘lNe. Wong and Harvey observed that the L-S interaction has a large effect in raising this level but the L-S part of the F&o-Brown interaction is still not large enough to shift this level up to the observed position *, ““). Transition probabilities from and to this level give important information about the nature of this level. Present calculations show a large transition probability from (+)1 to (a), and to the ground state (table SB), which indicates rather large K-mixing. The magnetic moment of the ground state of “Ne has been measured ‘I). The calculated value is larger by 0.4 n.m. than the observed value (table 8E), but it agrees better than does the Schmidt value (table 1C). It is very interesting to find a .rdependence for the calculated magnetic moments similar to that predicted by the rotational model. In particular, the magnetic moments of the states having a spin greater than J = 9 are predicted to change sign for the ground state. Hopefully this sign change will be tested experimentally. An observed Ml transition probability ““) from the first excited state to the ground state in 21Na is reproduced very well by this calculation (table SC), but an observed E2 transition probability 22) between these two states is about half the calculated value (table SD). This observed value of B(E2) is, however, considerably smaller than that in 21Ne and it is hard to be understood by a shell-model theory.

A. ARXMA et al.

296

The magnetic moment of the ground state of “Na is observed 23) and it deviates by 0.4 n.m. from the calculated value (table SE). It is, however, very interesting that an averaged value of two magnetic moments

3w*;

21Ne) -i-p@‘; 21Na)],

is predicted remarkably well by the present calculation (table XG). There is a similar situation in A = 19. These averaged values can be given as the expectation values of the isoscalar part of the magnetic dipole operator J&F). Thus the 0.4 n.m. discrepancy in A = 21 is due to an error in the estimation of the isovector part. This is very reasonable, because the exchange current modifies mainly this isovector part ““). g_s_

17 -..---0

-8

-7 -10 -0

-7

-7

-7 -9

-@i5 -.---61

-8 _7 7 6

=8 B -9 -7

6 III.zz?,

IIIz5

10 ? 10

-8 6 1

-7

-3 -22 =======2b -8 6-

-

-4

_--..-1

-6

!%?SzzSp

B

-4 -2 -3

=$

Xz;

-3

-3

n==Ea-4 ----2

-4 -2

Sw.m,4 -2

-2

-2

-2

-2

0 t -0

-0

-0 t: E0.78

-0 E10.4

-2 4m4

2t

EXP

22Ne

E = 1.15

\

5 = 2.03 3’V=52

‘3V’70 “V= 0

J%J = -26

4,2 -0 w 34 ====%6,3 -‘4, = $1 -2 -3

-2 -4 -3

%

-------2 -4

-2

-2

-0

n!mmXuo

&=O.O

5 = 2.03 13V~60 “‘V =45 ‘IV=

K+” 0

Oak

Ridge

0

=v s-225

Fig. 3. Level structure of ZzNe. Experimentally observed excitation energies and spin-parity assignments are taken from the following references (except the 8+ state at 11.15 MeV): W. Kutschera et al., Nucl. Fhys. AYH (1968) 529; 18. H. Wjide~thal and E. Mewman, Phys. Rev. 975 (1968) 1431; A. J. Howard and J. D. Pronko, in Contrib. to theInt. Conf.o~properties of nuc~ears~ates,Montreal, 1969, p, 259. The SC state is taken from ref. 25).

TABLE PA Calculated @MI, *We

Ji

Jf

11 22 23

01 21 21 21 21 41 41 22

31 11 31 51 23

JI -F .&) in nuclei with A = 22 and T = 1 (in po2)

0.053 0.016* 0.617 0.003 0.075 O.OOOl 0.003 0.084

94g

Ji

Jf

zzNe

=Mg

0.065 0.016 0.656 0.003 0.095 0.0002 0.003 0.090

31 11 31 11 42 42 61 11

22 22 2, 23 31 51 51 02

0.071 0.180 0.163 0.050 0.056 0.061 0.015 0.420

0.073 0.207 0.177 0.063 0,056 0.060 0.014 0.496

The calculation is done using the wave functions prepared with 5 = 2.03, E = 0.78, 13Y = 70, 0, 33V, e-26. 31Y= 52, =P= *

B(MI,

22 + 2,)

& 0.05 po2; ref. 2G). TABLE 9B

B(E2, J, + Jf) in nuclei with A = 22 and T = 1 (in e2 . fm’) Ji

2zNe

Jf ew.

21

01

41 61

21 4P

81

61

101 31 42 42 51 51

I

66 &I2 45 &3 54.2k11.7

“) b) =)

81

60.1

54.9

103.6123.9 44.4 4)

65.0 66.6 44.4 8.1

72.5

23.3 70.2 41.0 52.2

22 31 31 42

See the caption of table 9A. “) Ref. 27). “) Ref. zs).

present talc.

exp.

101.0

22

z2Mg

O.R.

present talc.

“) Ref. 26).

74.5 83.1 62.8 39.2 11.4 131.7 36.9 94.8 58.3 52.3

83.8 25.9 56.1

“) B. Povh, private communication.

TABLE PC Calculated B&2, Ji -+ J,) of inter-band transitions in zzNe and “Mg

Ji

* fm4>

Jf

==Ne

=Mg

&

Jf

=Ne

=Mg -

01

0.23 2.41 6.40 0.79 0.42 0.25 9.58 6.81 2.28 15.85 0.64 7.26 0.21

7.31 1.20 25.09 1.28 11.17 0.37 10.94 8.74 3.16 0.20 0.04 25.61 4.95

11 02 31

22 22 23

1.15 2.94 0.27 17.27 3.24 24.05 1.32 2.19 0.60 0.84 0.29 29.03

0.04 6.16 0.33 18.80 0.03 0.12 0.76 0.03 0.07 3.87 5.32 37.00

22 2s 22 2, 31

21

11

21

01

21 21

42 02 22 23 31 51 23

(in ez

21

21 41 41 41 41 22

See the caption of table 9A.

11

42 02 11 61 61 82 82 101

23 23

23 31 51 42 61 81 82

A. ARIMA

298

et al.

TABLE 9D Magnetic and quadrupo~e moments of 22Mg compared with 22Ne Mag. mom. i&,)

J 22Ne

isoscalar part of p

2ZNe

=Mg

1.228 2.982 3.987 4.455 6.185 1.827 2.426 3.987 3.903 5.038 1.201 1.968

1.014 2.039 3.053 4.059 5.108 1.016 1.517 2.028 2.527 4.060 0.536 1.023

- 15.4* -20.5 -23.5 -24.8 -19.0 15.4 1.0 - 5.4 -10.1 -17.9 5.0 --10.1

-16.4 -25.4 -30.2 -31.2 -14.8 16.1 - 0.17 - 5.3 -13.7 -16.3 4.1 -14.1

0.799 1.096 2.118 3.464 4.032 0.204 0.608 0.728 1.153 3.083 0.130 0.079

21

41 61 81 101 22 31 42 51 82 11 23

Quad. mom. (e . fm2)

2’Mg

See the caption of table 9A. * Observed Q(21f) = -21 At4 e * fm’; ref. 27). TABLE 9E

Intrinsic qua~~pole Z”Ne (8 = 1.15)

J

61 81 101

the captions

2zMg (e = 0.78)

54.0* 56.4 58.8 59.0 43.6

59.1 70.0 75.3 74.0 34.1

of table 6A. Table 9A and text.

* Observed Q, (21+) = [til::

3.4. zzNe, zzNa

“‘Ne (& = 0.78)

52.1” 51.7 51.1 49.2

21 41

See

moments of 20Ne and 22Ne (in e * fm’)

AND

z

: iz: E: :I;:.

22Mg

We have predicted in 22Ne another example of the irregularities in excitation energies “) which are well known in “Ne. Recently an 8’ state was observed at 11.15 MeV at Chalk River “). The predicted energy is 11.27 MeV. According to the calculation, this level comprises about 70 % of the (82) representation of SU(3) which consists of 57.6 y0 of K = 0 and 12.7 % of K = 2, This calculation predicts another X+ state at 12.12 MeV just above the 8: state, and a 10’ state at 15.8 MeV (a = 0.4). It is interesting to check this possibility (fig. 3). Two E2 transition probabilities have been measured in 22Ne [ref. 26)J. Agreement between the theoretical and observed values are very good (table 9B). Deviations of the calculated values from values predicted by the rotations model increase with increasing spin.

sd SHELL NUCLEI

299

The quadrupcale moment of the first excited state has been measured 27) but the observed value is larger than the calculated value by 30 % (table 9D). The same sit~a~o~ is already fuund ia “Ne. The quadrup ale murne~~s QI of other states ape calculated. Assuting the rotatiional model, we Carl calculate the intrinsic quadrupale rnorn~~t Qe, for each. state 1 which must be corset irrespective of I for pure rota= tional motion. This QO can be derived from Q, by use of the reMiiorm (K = 0 is assumed)

As discussed in ref. 2, and as seen in table BE, Q, ilz “‘Ne decreases as P increases, while Q. in ‘%Ie increases up to 6’. In a naive way we can say that the deformation of 2*Ne decreases but the deformation of “Pse increases as excitation energy increases. A spin de~e~de~~~ of calculated magnetic moments is very interesting because ~1 is not equal to gKI. This is very different from the case of ‘%e. However, ~~~~~~~~~~~g the isoscalar part &(,(I$ of the magnetic moment, i.e.

we I&d that gr,(P) is nearly equal to gR& where gR M 05 n.m.. This fact shows that the isovector part of the magnetic moment has larger contributions which are not ~~opor~o~a~ to the spin 1 (table 9D), The calculated E2 transition probabilities in zzMg are not very different from the corresponding values in “Ne. The Calculated E2 transition probabilities seem to be decisively deterrn~~~d by co~t~~b~~o~s from the “We core. The two nucleans outside 2oNe contribute very little to the transition rate (table 9B). In order to ccmti this statement, let us take the SUf?) limit, We have the m~ique S-state brat two D-states in [42](82). Atiyama’s wave f~~~t~o~s, which are not classified by HlioWs K quartz number, give the foll,owiag matrix elements of the MC’) operator:

where S and V starmdfor matrix elements of the isoscalar and isovector parts of the IZI operator. The quantum numbers a and j? are introduced to discriminate between the two D-states of the (82) representation. The upper sign must be taken for “Ne (LYz= -I-I) and the lower one for 22Mg (T, = - 1). The CC’) are th6: ~~~o~rn~~~~~d

300

A. ARIMA TABLE

et al. 10A

B(M1, $2 + .Ff) and B(E2, Jr -+ Jr) ofz2Na

Ji

Jf

41 42 42 51 52 52 51 52 52 11 11

13

31 3, 32 31 31 32 41 41 42 31 32 31 32 31 32 11 11

13

12

12 12 13 13 12

mv

(e2 . fm4)

exp.

O.R.

108.0

118

102.0

25.0

&::z

25.3

75.8

125.6

91.2

0.0371_<

1%

0.7

0.21&- ?

31 31 32

21 22 22

-.& = 0.78 3.07 17.24 0.0207 9.75 63.42 0.186 40.02 53.09 0.955 155.0 17.X 0.150 41.3 2.362 2.52 71.70 0.355 0.0011 15.5 0.6 4.3

mw

(PC3

.s= 0.78

0.00073” 0.00151 0.00267

0.00151** 0.00038 0.0154

0.018 0.000017 0.00570 0.0256 0.0107 0.0286

We use the wave functions prepared with the standard strength of residual interaction cited in text, and with E = 2.03 and a = 0.78. Experimentally observed vahtes of B(E2) are tafren from R. C. Haight, Ph.D. thesis, Princeton University, 1969. * @Ml, 41 -+ 3,) = (0.3710.08) x 10-3p02. ** B(M1, 51 -+ 4,) = (0.50&0.08)x 1O-3 po2. TABLE 1OB

Magnetic and quadrupole moments of 22Na f

31 32 33 11 12. 13 41 42 51

52 21 22

Magnetic moment (,u~> - exp. E = 0.78 1.746*

0.535~0*01**

1.828 1.787 1.357 0.787 0.390 0.719 2.312 2.168 2.853 2.825 1.119 1.112

See the caption of table 1OA. * L. Davis, Jr., et al., Phys. Rev. 76 (1949) 1068. ** A. W. Synyar et al., Phys. Rev. 151 (1966) 910.

Quadrupole moment E = 0.78 -

9.362 14.426 -14.356 - 4.724 - 0.720 - 1.031 6.312 - 14.417 - 17.337 - 7.311 - 8.122 14.949

(e +fin’)

sd SHELL

NUCLEI

301

spherical harmonics defined as [J(2A.-i- 1>/47r]Ycnt. The E2 matrix elements are given with the same sign convention as follows: (~42](82)~31D211M(z~i~~42](82)31So) ([42](82)~31D,\jiW(z~j~[42](82)31So) XC--

I: f5.3441 +Sep+GeN) &0.84(1 -I-Gtp- SeN))&. Y

Thus the E2 transitions from the rx-states do not display any dih?erence between “Ne and “Mg, except in phase. The @-state decays mainly through the scalar part of the A$@) operator, and the transition probabilities again are almost the same both in “Ne and “Mg. The actual wave function of the 2+ state is a mixture of these two D states and other basic states, but the p-state is still dominant. Thus we can conclude that this calculation gives a small difference between “Ne and “Mg. The lifetime of the first excited state in “P&g has been observed 36). The B(E2, 2: --+0;) (104*,, 44 e2 - fm”) derived from this lifetime seems to be larger than our calculated value (74 e2 * fm2>. It is very interesting to measure other E2 transitions in “Mg in order to study contributions from the isoscalar part and from the isovector part of the E2 operator separately. The level structure of 22Na was not well explained in onr previous calculation ‘1. The electromagnetic properties are not explained either, except for the quadrupole moment of the ground state (tables IOA and 10B). The 0ak Ridge group’s calculation shows much more satisfactory results concerning these properties together with the level structure in “2Na. Reasons why we failed to obtain reasonable results are being studied. 4. Electromagnetic

transitions with AT = 1 and j?-decay

The y-~ansitions with AT = 1 have recently been measured for A = 19 and 20 nuclei “, 3“). The Ml transitions from the analogue state for the ground state of the neighbouring nucleus are very interesting. These transitions will give important information for comparing the Ml transitions with the P-decays induced by GT interactions. The GT type P-decay does not break the SU(4) symmetries because the interaction is g - z, so that this decay occurs only through small admixtures of different symmetries in the daughter state, because the parent nucleus has a symmetry from that of the daughter in the L-S coupling limit. Although the Ml transitions with AT = I modify the above situation through the I- z interaction which breaks the symmetries, the CT- z part is still most important. We can thus expect that there must be great similarity between the P-decay and the Ml transition. This is really the case as found in our calculations.

302

A. ARIMA eb al. TABLE log

Initial state nucl.

111%

ft values of @-decay

Final. state

T

J’

Exp.

mlcl.

T

170

4

%1

19F

+ *

4l P;2

2QF

z

1%

3.73”

1 0 0 1 1 0

12 21

5.0

zONe 2oNe 2ONe zlNe

I

s” ;

P:

!

i

Param.

case 1 case 2

3.6

3.45 3.48 5.71 5.17 4.07 3.47 3.79

3.37 3.16 3.20 4.47 3.93 5.53 5.82 3.44 3.47 5.82 5.28 4.18 3.52 3.30

5.0

4.62 3.68

4.73 3.69

4.49

4.60

3.38 3.3

21

21 21 01

Present Cal.

3.31 3.08 3.11 4.36 3.82 5.42 5.71

3.48

4

A B C c c c f)

Observed log f$ values are taken from C. M. Lederer, Table of isotopes, 1968. Ila the column labelled param. we distinguish the wave function used in calculation of log ft values, A: d%-+ de single-particle estimate. 33-Y= -18.75. B: E = 2.03, 31v = 37.5, IXY = 0, 8 = 0.78, =Y = 50, 33v = -22.5. llY= 0, C: t = 2.03, 31Y = 45, E = 1.27, 13Y = 60, 33V = -26, IlV = 0, D: < = 2.03, 31v= 52, =v = 70, E = 1,27, *

H.

B.

Mak

et

al.,

Phys. Rev. C2 (1970) 1729

Thefi value far the allowed P-decays are giveieaas follows

element of Fermi iateractions a&i G is that of ~arno~-delver interactions. We use two sets of Iparameters of B and x because the values of the coupling constants are not precisely known, these are as follow:

where I; is a matrix

case 1 [refs. 3%q case 2 33)

B = 2500, B = 2840,

x =10.59, x = 0.52.

~a~~~~~~~~and observed Ilog& values are ~~~~a~~e~ fCpr~o~~ari~~~ in table I IA. Agreement between theoretical and observed values is very reasonable. Only the

calcu:ulatedlogft values for the decays from 2o0 to a OF and from ‘OF to “Ne deviate by about 0.5 from the experimental values. These perhaps come from the fact that our ~a~~~la~o~s s~rne~~at failed to reproduce the level str~~t~e of ‘OF [ref. “>I. The Ml trarPsitions from the analogue states of IgO to the levels with T = 3 in “33

sd SHELL NUCLEI

303

TABLE11B Ml and E2 reduced transition probabi~ties from 2’ = @,J = Q and -&in lqF, and log ft values of /G-decay of I90 Initial Ti

Ji

Final -Ii”f Jf

i: 2 ::

exp.

> 4.45 5.5 5.41

ii3

:‘l.2 -ii3

3.54

logft present talc. O.R. case 1 case 2

4.50 4.38 6.23 5.42

4.61 4.49 6.37 5.53

3.89

4.00

3.92 5.12 3.35

4.03 5.23 3.47

4.97

5.08

4.90 4.19 4.61 5.59 7.70 5.02 3.57

5.01 4.30 4.72 5.70 7.81 5.13 3.68

4.6 4.4 4.7 6.4

4.2 3.4

exp.

B(M1) W) present O.R. talc.

1.01 < 0.11 0.40 < 0.20 4.68 < 0.58

0.13 < 0.90 0.22 < 0.07 0.05 < 0.20

1.337 1.244 0.465 0.039 3.056 3.824 0.913 8.807

0.45 0.47 0.13 0.09

0.767 0.503 2.060 1.170 0.001 0.082 0.168 6.336

0.25 0.05

0.20 3.42

0.29 0.005 0.02 0.007

B(E2) (ea * fm4> present talc.

0.136 0.146 0.740 0.021 0.141 0.648 0.541 0.230 1.661 0.301 0.229 0.810 0.115 0.047 0.006 0.165 1.821 1.410 0.001 0.223 0.511 0.302 0.037

Observed values of log ft and B(M1) are taken from ref. 29). Calculated values of cited as O.R. are also taken from ref. 29). Present calculations: we use the wave functions of B in table 1 la.

and the P-decays from the ground state of “0 to the low-lying levels of ‘“F are listed in table 11B. Excellent agreement is obtained both for Ml transitions and j-decays, especially from the state T = 3, J” = 5’ to the members of the ground rotational band in lpF. However, the Ml transition strengths from the state T = 3, J” = 3’ to the members of the ground band show large values which are about six times as large as the observed values. The calculated energy distance of these two leveb with T = 4 is about 800 keV while the observed value is about 100 keV in fig. 1. This indicates that our wave function for the state b” = $+, T = 3 may not be very accurate. The comparison of Ml strengths with theft values shows that the contribution of the c-part in A@‘) is predominant. For example, the ratio of the cont~butions to the Ml strength from the t-part and cr-part is about one-fourth in the transitions from the state T = 3, J” = 3’ to the T = 3 levels in “F. This fact remains true for the mass-number region treated in this paper. Hanna et al. “) o b served only one strong cascade transition from the T = 2,

304

A. ARIMA etaf. TABLE1 IC Ml reduced transition probabilities with d T = 1 in 20Ne (in ~0’)

T = 1, J* = lr+ ~-i

1 2 3 4 5 6 7 8 9

-%.(MeV)

14.841 16.006 18.079 18.176 19.057 20.806 21.516 21.723 22.235

B(M1, T = 2, J=O,+T= l,J= 1,) & (T = 2, f = 0,) =

22.625 MeV

J=

B(M1,T = 1, B(M1,T =L-1, 11 3 1” = 0, J = 01) J = Ii,-+ T = 0, J- = 21) .z&(T = 0, J = 0,) = E,(T=O,J=21)= 0.0MeV 1.677 MeV

3.070 2.714 0.855 1.201 2.901 28.000 7.329 2.265 0.318

0.9045 0.0993 0.0058 0.0295 0.0234 0.0578 0.0013 0.0200 0.1447

0.5189 0.0204 0.0009 0.6487 0.0016 0.0002 0.0471 < 10-S 0.0185

We use the wave functions obtained with the parameters 6 = 2.03, E = 1.27, 31 V = 60,31V = 45, 11V = 0,33V = -22.5. E, (T,J*)orL?, express the calculated excitation energies.

Ml and E2 reduced transition ~robabi~ties

ew.

5.23f0.10 > 6.0 4,76&0.05

5.07hO.07

from T = p, J = 8 and + in “Ne, and logft of,&decay of =F log ft case 1

7.02 5.38

case 2

7.13 5.49

B(M1) ec2) present talc.

B(E2) (e2 * fm4) present talc.

0.027 0.039

0.269 0.057

10.1s 6.60 5.42 5.51

10.29 6.71 5.53 5.62

WO 0.148 0.030 0.074

0.010 0.159 0.177 0.034

5.85 5.81

5.96 5.92

0.035 0.255

0.057 0.068

6.45

6.56

0.028

0.096 0.170

0.002 0.220 5.24

5.35

0.010

6.39 5.08

5.19 6.50 5.21 6.63 7.68

0.049 0.016 0.736 0.004 0.012

5.10

6.52 7.57

0.214 0.802 0.174 0.789 0.002 0.393

observed values of log&+ are taken from ref. 3a). In our calculation we use the same wave functions as D in table I1 A.

305

sd SMELL NUCLEI TABLS 12A The dependence of B(EZ) on E in a2Ne (ez

4

4

21 41 61 81 101 31 42 42 51 51

0, 21 4:1 61

22

Qi

22

21

42 31

21 21

& = 1.15 60.1 59.7 66.3 44.1 6.4 100.6 21.6 66.9 40.2 47.9 0.2 7.5 14.1 0.3

81

22 22 31 3, 42

* fm4)

E = 0.78 60.1 65.0 66.6 44.4 8.1 101 .o 23.3 70.2 41.0 52.2 0.2 6.4 9.6 0.4

E = 0.4 60.1 69.1 66.9 45,o 10.5 101.2 24.7 72.9 41.7 55.5 0.3 5.7 6.2 0.5

We use the wave functions prepared with the same E and residual interactions as table 9A but different 8. TABLE12B The dependence of ,u and Q on E in Z2Ne s

$2 (e E = 1.15

21 41 61 81 1% 22

31 42 51 li

-15.3 -20.4 -23.6 -2S.l -19.0 15.2 0.9 - 5.8 -10.1 5.1

- fmz)

1” obo)

E = 0.78

E = 0.4

& = 1.15

E = 0.78

-15.4 -20.5 -23.5 -24.8 -19.0 15.4 1.0 - 5.4 -10.1 5.0

-15.5

0.80 0.96 2.07 3.52 4.01 0.17 0.58 0.73 1.09 -0.10

0.80

0.80

1.10 2.12 3.46 4.03 0.20 0.61 0.73 1.15 -0.13

1.22 2.16 3.39 4.06 0.24 0.63 0.76 1.22 - -0.17

-20.5 -23.4 -24.4 -18.9 15.5 1.1 - 5.1 -10.1 4.9

& = 0.4

See the caption of table 12A.

J” = 0” state in 2*I%e; a qualitative explanation was shown in ref. “). Our calculated values are listed in table 11C. Kurath 31) obtained similar results. The P-decays from the boded state of “F to the low-lying states of ‘INe have been observed 34). Comparison between calculated and observed ft values is given in table Ill3 together with calculated Ml transitions with AT = 1. 5. Sensitivity of the calculated vabes to several parameters Magnetic dipole and electric quadrupole transition probabilities for both the intraband and interband transitions of ‘lNe are not very sensitive to a change of E except

A. ARXMA

306

et al.

TABLB 12C The sensitivity of ,u and Q ou a change in the residual interactions for 21Ne and %Ie Nucl.

J

B

#I fi $1 $:

-1.05 -1.32 -0.46 -0.05 -0.64 1.37

-1.12 -1.34 -0.52 -0.10 0.53 1.27

9.99 - 5.44 - 8.51 -15.90 -19.78 -15.50

9.96 - 5.54 - 8.58 -16.02 -20.08 -15.56

$1 21 41 61 81 22 3, 42

1.86 0.76 0.85 1.96 344 0.11 0.55 0.65

1.67 0.80 0.96 2.07 3.52 0.17 0.58 0.73

-18.64 -15.25 -20.30 -23.48 -24.80 15.11 0.48 - 5.68

-18.87 -15.33 -20.40 -23.58 -25.11 15.21 0.91 - 5.81

$1 alNe

zzNe

Q (e . fl3P)

p6 @Jo> A

A

B

We use the two sets of the wave function obtained with the parameters: A:

< = 2.03,

E = 1.27,

=V=

B:

E = 2.03,

E = 1.27,

=v

60, = 70,

45,

IlV = 0,

33 v = -22.5.

e1v = 52,

eiv=

llV = 0,

33V= -26.

for the magnetic transitions of (-& -+ ($)r and (& + (& (tables 8C and SD). Magnetic dipoIe and electric quadrupole moments of “Ne are also very insensitive to a change of E except for the magnetic moment of (.g)1 and the quadrupole moment of (& (tables 8E and 8F). This insensitivity to a change of E is found elsewhere except for a few cases. Exceptions are, for example, some quadrupole moments of lgO and “0 and some transition probabilities in “0 and “0. This is easily understood because they are spherical nuclei and there is a competition between a few leading states, for example, (d+j3 and (d+)‘(s,) for J = 3 in IgO the energies of which are sensitive to E. Table 12 shows how these iroperties depend on a change in the strength of the residual interactions; insensitivity is again observed, particularly in the quadrnpole moments. The transition probabilities also proved to be insensitive to small modification of the residual interac~on. paretic moments are expected to be modified strongly by a change of j-r,which we did not study in this paper. The modification induced by non-central interactions should also be studied, but the reasonable agreement of our results with those of the Oak Ridge group indicate that they do not induce serious changes, at least where we have good agreement with these works. One of the authors (MS.) wishes to thank the Institute for Nuclear Study, Tokyo University, for giving him a fellowship to enable him to stay and continue to study this subject. The authors would like to thank Drs. C. Broude, W. 6. Davice and H.

sd SHELL NUCLEI

307

~ch~eickert for supplying their expedients results before ~ublicatiou. Our thanks are due to Dr. E. Halbert and her colleague for sending us their results before publication. East but not least, we express sincere thanks to Professor J. Ginocchio and Dr. K. Ogawa for a careful reading of our manuscript. This research was financially supported by The Institute for Nuclear Study, The University of Tokyo, Tanashi-shi, Tokyo. The computer, JSITAC! 502OE, in the Computer Centre of the University of Tokyo was used.

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