Cluster structure in the highly-excited states of 24Mg

Nuclear Physics A463 (1987) 399c - 404c North-Holland, Amsterdam

399c

CLUSTER STRUCTURE IN THE HIGHLY-EXCITED STATES OF 24Mg

Kiyoshi KATO, Hiroshi KAZAMA and Hajime TANAKA Department of Physics, Hokkaido U n i v e r s i t y , Sapporo 060, Japan

We propose a w - t r u n c a t i o n method f o r the m u l t i - c l u s t e r o r t h o g o n a l i t y c o n d i t i o n model (OCM) to reduce the large number ofgmbasis states. The c l u s t e r structures in the h i g h l y - e x c i t e d l 6 s t a t e s of L~Mg are discussed on the basis of s o l u t i o n s obtained by the O+2e OCM.

I.

INTRODUCTION From a molecular v i e w p o i n t of n u c l e i ,

8Be to

l o w - l y i n g states of l i g h t nuclei from

24Mg have been s u c c e s s f u l l y studied by

microscopic c l u s t e r models. 1

Since various kinds

w i t h increase of the e x c i t a t i o n energy, 2 i t

i n v e s t i g a t e the h i g h l y - e x c i t e d states by means of a

and

semi-

to solve the equation of motion of the m u l t i - c l u s t e r

of c l u s t e r s t r u c t u r e is very important

microscopic

model 3 which can describe many c l u s t e r c o n f i g u r a t i o n s . of

microscopic

Our i n t e r e s t is n a t u r a l l y extended to the c l u s t e r

s t r u c t u r e of h i g h l y - e x c i t e d states. are expected

using

to

multi-cluster

However, i t

is not easy

system, because the number

degrees of freedom to be solved or the number of basis states of the m u l t i -

c l u s t e r system becomes very large. In t h i s paper, we propose a powerful method, c a l l e d

w - t r u n c a t i o n method, to

truncate a large model space of the m u l t i - c l u s t e r o r t h o g o n a l i t y c o n d i t i o n model. The o r t h o g o n a l i t y c o n d i t i o n model (OCM), 4 being a semi-microscopic model, has 1 been shown to be very useful f o r study of l i g h t nuclei. In t h i s model, microscopic e f f e c t s

coming from the Pauli p r i n c i p l e are taken i n t o account

the c o n d i t i o n t h a t r e l a t i v e wave f u n c t i o n s between c l u s t e r s must be to the P a u l i - f o r b i d d e n states. 160+2~

The w - t r u n c a t i o n method w i l l

by

orthogonal

be applied to

the

OCM5 and the 3~ OCM6, and shown to be very powerful to reduce the large

number of basis states. We i n v e s t i g a t e c l u s t e r s t r u c t u r e in h i g h l y - e x c i t e d states of 24Mg by means of an 160+2~

model.

The 160+2~ model includes t w o - c l u s t e r c o n f i g u r a t i o n s such as

20Ne-~ and 160-8Be and t h r e e - c l u s t e r c o n f i g u r a t i o n s such as ~-160-~ . Present address: RCNP, Osaka U n i v e r s i t y ,

I b a r a k i , Osaka 567, Japan

0 3 7 5 - 9 4 7 4 / 8 7 ] $ 0 3 . 5 0 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Threshold

400c

K. KatO et al. / Cluster structure

energies of these c l u s t e r channels are observed at 9.31MeV (20Ne+~), 14.04 Mev (160+~+~) and 14.14 MeV (160+ 8Be), r e s p e c t i v e l y . 7 and

discussed

region

by Abe et a l . 8, large c l u s t e r i z a t i o n is expected at

from about the threshold energy to the energy of about

height. from

As pointed by Ikeda et al. 2 the

Coulomb

energy barrier

Therefore, we analyze the 160+2~ OCM s o l u t i o n s with e x c i t a t i o n energies

I0 MeV ( =20Ne~ threshold energy ) to 20MeV ( ~160+8Be threshold energy +

Coulomb

b a r r i e r height ) in d e t a i l through c a l c u l a t i o n of

~- and

8Be-reduced

widths. In

section 2,

we e x p l a i n the

and discuss the a p p l i c a b i l i t y we

H - t r u n c a t i o n method f o r the m u l t i - c l u s t e r OCM

to 160+2~ and 3~

s h o w r e s u l t s of the 160+2~

OCM and

e x c i t e d states with Ex=lO ~ 20 MeV.

systems.

In sections 3 and 4,

discuss c l u s t e r s t r u c t u r e

of

highly-

In section 5, conclusion is given.

2. p-TRUNCATION METHOD FOR MULTI-CLUSTER SYSTEMS The basis states of r e l a t i v e motions between c l u s t e r s are represented by Pauli-allowed

states which are c l a s s i f i e d by the harmonic o s c i l l a t o r

( X , u ) - l a b e l of the E l l i o t t number

the

quanta

SU(3) group and an a d d i t i o n a l quantum number ~.3

N, The

of the P a u l i - a l l o w e d states r a p i d l y increases with the harmonic o s c i l l a -

t o r quanta N.

Since the d e v e l o p e d - c l u s t e r states are described by the

relative

wave functions with higher o s c i l l a t o r

quanta, we must take a large value of N max to get e n e r g e t i c a l l y - c o n v e r g e d s o l u t i o n s in d i a g o n a l i z a t i o n of the OCM equation. In order to reduce the number ( dimension ) of basis states, we propose the

u-

t r u n c a t i o n method.

a)H-TruncationMethod

The H - t r u n c a t i o n method consists of two

E

steps;

z( ~

in step I, the

I

I \

\

!

....... ~im

zok

~2oo

!

OCM equation is solved

dim.

]::

w i t h i n the subspace of the f i x e d - u

b)Fu[[ Space (Me~ ~

,oo

l

states in

the P a u l i - a l l o w e d

]0

0

space with N < Nmax, and in step I I ,

the

OCM equation is again d i a g o n a l i z e d by using

-IC

20

22

24

the s o l u t i o n s with the

26

28 30 Nrnax

32

3~

-~0~

36

16

]8

energy E
Fig. 1

in step I f o r each H-

by the 160+2a OCM, Dotted l i n e

s t a t e as basis states.

(dimension) of b a s i s s t a t e s .

20

22

24 26 Nnna~

-0

2'8

3'0 3'2

Energy convergence of the 0+ s t a t e s o b t a i n e d shows

the number

401 c

K. K a t 6 et al. / Cluster structure

We apply the

~ - t r u n c a t i o n method to the 160+2~ OCM.5

energy of 0+ states converges f o r Nmax. space

c a l c u l a t i o n increases r a p i d l y w i t h Nmax,

while it

the ~ - t r u n c a t i o n method increases o n l y g r a d u a l l y . tion

method,

tions

for

energies The

in the c a l c u l a t i o n

Therefore, in the

by

~-trunca-

we can take large value of Nmax so as to get well-converged s o l u -

d e v e l o p e d - c l u s t e r states, for

almost f u l l y

Figure 1 shows how the

The number of basis states in the f u l l -

0+ states.

Figure 2 shows the

Emax-dependence of

They show a good convergence f o r Emax > I00 MeV and

reproduce the r e s u l t s of the f u l l - s p a c e c a l c u l a t i o n ,

~ - t r u n c a t i o n method is also applied to the 3~ OCM.6

The second 0+ s t a t e

E 10r

(7.66MeV) of 12C has been well known to have the developed cluster structure,

dim 5

6

I ".

"-.

. / / . "'

'°

~;~22Z2 .

.

.

.

.

For an i n s u f f i c i e n t

'..';'~o .

value of N , the max energy of the 0+ s t a t e is not obtained c o r r e -

. . .

2bo :,oo

]°

-5~

5F ",,

3T

io 20

30

40

~0 Nrmax

c t l y below the energy of the 3- s t a t e (9.6 MeV). 6

-10

To r e a s o n a b l y

reproduce the order of Energy convergence

Fig,2

N must be l a r g e r max than about 50 as shown

e n e r g i e s o f t h e 0+ s t a t e s

of the 0+ and

obtained by the 160+2a

obtained by

in Fig.3,

OCM.

3,

E -dependence of max

Fig,3

energies, the value of

3-states the 3~

OCM,

~-160-e LINEAR-CHAIN STRUCTURE We i n v e s t i g a t e the s t r u c t u r e of the 0+ states which are obtained by the

+2~ OCM c a l c u l a t i o n .

states f o r the 20Ne(l~)+(~ and 8Be(l~)+160 channels, i)

the

ii)

seen

that

X

+

the 07 s t a t e has large values of

the 20Ne-~

energy region of E

X

The

result

e-RW in both channels

configuration

F r o m i ) and i i ) ,

c l u s t e r s t r u c t u r e becomes dominant

> I0 MeV,

of i i i )

states

no s t a t e in E <20 MeV region has large values

p o s i t i v e - and n e g a t i v e - p a r i t y states of the 2ONe nucleus. is

0+

These r e s u l t s i n d i c a t e t h a t

states above about I0 MeV have large values of ~-RW, w h i l e the

below I0 MeV have small ones, of 8Be-RW, and i i i )

160

Figure 4 shows the reduced widths (RW'S) of obtained

in

the

of it

excited

+

suggests t h a t the 07 s t a t e has an

~-160-~ l i n e a r

chain

because such a c o n f i g u r a t i o n leads to the r e s i d u a l states of 2ONe

402c

K. KatO et al.

/

Cluster structure

having an ~-160 c o n f i g u r a t i o n with a p a r i t y violated intrinsic

s t a t e by removing one

"Ne(~,* )

cluster. Furthermore, t h i s conclusion t h a t + the 07 s t a t e has the e-160-~ l i n e a r chain

Ol

o,~%,i ~'Ne(2"I

+ I ) The wave f u n c t i o n of the 07

motion between two ~ c l u s t e r s has much higher quanta.

1

oi O2¸ "Ne ( 0" )

s t a t e has large amplitudes t h a t the r e l a t i v e oscillator

I_

o 02

s t r u c t u r e is confirmed by analyses of the wave f u n c t i o n :

a)

02

°.I

This means t h a t two

~'Ne( 5" )

c l u s t e r s s t a t e are separated from each other. + 2) The wave f u n c t i o n of the 07 s t a t e has a dominant amplitude ( =70% ) in the (~ .0)

r

io

,I

Ex

20 MeV

b)

"NeI 3" )

component of the SU(3) basis f u n c t i o n s . This + r e s u l t i n d i c a t e s t h a t the 07 s t a t e has a

-[~

, I

"Ne I I" )

stretched c o n f i g u r a t i o n . The 2+ and 4+ states which are assigned as

,o

members of a r o t a t i o n a l band with the ~-160-~ 'Be (~.')

l i n e a r chain c o n f i g u r a t i o n are obtained at

02

about 20 MeV. The moment of i n e r t i a is ~2/2~

oi

_L Jo!°~

¢ )

o

=0.106 MeV, and t h i s value is very consistent

02

=Be(2*)

with the value (0.107 MeV) estimated by assuming a r i g i d r o t o r o

the ~-160-~

I ,

o o2

configuration.

~Be(O')

ol

The

decay

widths of the

~-160-~

linear

o

io

chain states (0 +, 2+ and 4+ ) i n t o 20Ne(l~)+(~ channels are shown in Table I.

24Mg

\E x(MeV) l~c(fm) 0+ 2+ 4+ 6+ 1

36-

0+

2+

18.50

19.18

5.16 55KeV 218

6.00 99KeV 351

32 48 1.6xlO -20 4 . 9 x I 0 -20 14 3.0xlO -2 0

II 2 . 7 x i 0 -2 0

5.16

4+ 20.59 6.00

I O O K e V 130KeV 176

262

49 80 1.2xlO -7 2.7×10 -7 39

20 MeV

0+ states obtained by the 160+2~ OCM.

are s t r o n g l y desired.

J\J~

]

Fig. 4 e - and 8Be-RW's of the

Experimental

studies on the h i g h l y - e x c i t e d states of

,I Ex

24

1.5

1.3

0

0

6.16

6.00

57KeV

60KeV

2.5

7.1

89 62 5 . 2 x i 0 -3 1.9xlO 2 5.1 I0

2.9 5.6

2.6x10 -6 2.2×10 -6

Table I of

20Ne(1)+~widths

e-160-~ l i n e a r chain

states, radius.

a

C

is the channel

K. KatLi et al. /Cluster

4.

a-REDUCED

for

WIDTH

Fig.5.

In

we

OF

O+--6+ states.

region

of

Ex>lO

iii)

RW.

distribution

energies

energy

E,(I).

shifts

to higher

channel

spin

value

these results, 20 Ne-a cluster

we

can

highly-excited

states

structure

large

energy

values

cl-RW are

of

much

of each

structure of

of energy

widths RW's

appear

larger

channel

with

a peak

(RW's)

in the

than for

obtained

those

J7-states

at a certain

energy of 8Beand excita-

the with

From

conclude

that

is dominant

with

8Be-reduced

a-RW's

cluster.

also

and

of

a gross

shows

I of the

with

values of the

iv) The

403c

STATES of o-

states

ii) The

tion

the

results

i) The

excitation

peak

the

MeV.

The

HIGHLY-EXCITED

show

structure

10 MeV

the

in the

< Ex < 20

MeV. Figure of

6 shows

"Ne(I'=O+

channels states iii)

for with

and

channel

energy Ex=lO

iv),

at an energy

spins well

J =

and

model

4+)+o

diagram the

system.

the On

mix

a-RW the

from

the

values

are

other

5.

are

the

crossing

of o-Rw

in the

band

which

excited

for

the

of

the

is

band

20Ne(Of,

2+,

level

in Fig.6

then

among

state region

d).

In

channel

values many

of

levels.

at a distance has

dominant the

have

of a-RW

different and

a peak

contributing

schematic shown

in

each

with

channel

basis

for

strongly

large

channel

state

Fig.5 of the

o-

0+~6+

belongs.

and

states

160+2a

8Be-RW's obtained

of the by the

OCM.

CONCLUSION

In

conclusion,

of developed-cluster to

the

distributed

band

rotational

Ne(I')to

region,

hand,

the

states

The

crossing

amplitudes

of

(~))+a

As mentioned

a structure

on

of cr-RW's

4+

structure

(8CM)'

of tt.s BCM

band

20

the

interpreted

crossing

spin

The

Such

I.

and

distribution

a gross

of

(b)

and

E,(I).

peak

distribution

2+

Q 25 MeV.

the

shows

to the

the

(a),

reduce

the

it is essential states,

computational

and

to take then

difficulty

the

large

value

the u-truncation of the

large

of N

method dimension

in descripzion max is very powerful of

model

space.

404c

K. Kat6 et al. / Cluster structure

The

results

in the

Ne-a

we

O+Zo

highly-excited

distribution 20

16

of the

e-16O-o

model

mean

elastic,

the

basis

channels

of the

for

Ex=

can

of the

the

d) shows

2' and

2+-inelastic

Ne-e

and

cluster

band

show

schematic

system.

4'-inelastic

which

energy

Full, rotational

large of the

16 by the otzo 20 Ne(2+)+o and

,

diagram

dashed

has

states

obtained

*'Ne(O+)+a

of

Furthermore,

model.

negative-parity

% 25MeV) of

is dominant

characteristic

as a manifestation

crossing

and

a-RW's the

structure

also

at Ex 2 18 MeV,

(Ex=10

4+)to

They

interpreted

of positive-

c) show

Figure

20Ne(0t,

be

structure

Of Q 6+ states

and

20

the

lO'%?O MeV.

which

on the

b) OCM. Figures a), 20 Ne(4')+a channels. crossing

with

that

linear-chain

in both

The a-RW's

show

widths

structure

the

a-reduced widths 20 Ne nucleus.

Fig.6

states

of o-reduced

cluster

obtain

OCM

and

of the dotted

band lines

bands.

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Ikeda,

(1968).

et al.,

4) s. Saito,

Prog.

Prog.

and

6) H. Horiuchi,

H.

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Theor. Band;.

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Y. Abe,

Theor.

Phys.

Suppl.

No.68

H. Horiuchi,

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Theor.

Phys.

No.62

(1977).

Y. Kondo

9) T. Matsuse.

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Nucl.

and

Y. Abe

Phys.

Theor.

7) F. Ajzenberg-Selove, 8)

and

(1980). Phys.

29.

Suppl.

Extra

Number

464.

3) H. Horiuchi.

5) K. Kate

Prog.

N. Takigawa

4J

(1969),

Theor.

Phys. Phys.

T. Matsuse,

and

Suppl.

Y. Kondo.

Phys.

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62

(1974),

A300

90.

705. (1979), 1266:

(1978),

53

644. (1975),

447.

1.

Prog.

Theor.

Phys.

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Theor.

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303.