Decay schemes of excited states in Mg24 and Si28 and the collective model

I.D.4: 1.E.l:3.A[

[ Nuclear Physics 29 (1962) 604---622; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

D E C A Y S C H E M E S O F E X C I T E D S T A T E S I N M g ~t A N D Si 2s A N D THE COLLECTIVE MODEL A. V. C O H E N a n d J. A. C O O K S O N

Atomic Weapons Research Establishment, Aldermaston, Berkshire R e c e i v e d 17 J u l y 1961 A b s t r a c t : G a m m a d e c a y m o d e s of t h e first six excited s t a t e s of Mg s4 a n d t h e first f o u r excited s t a t e s of Si 2s h a v e b e e n d e t e r m i n e d w i t h inelastic p r o t o n s c a t t e r i n g . ~M_gz~ h a s b e e n i n t e r p r e t e d b o t h o n t h e a x i a l l y s y m m e t r i c r o t a t i o n a l - v i b r a t i o n a l , a n d t h e r o t a t i n g t r i a x i a l ellipsoid collective m o d e l s . T h e d e c a y m o d e s require, o n t h e f o r m e r model, < 2 % a m p l i t u d e b a n d m i x i n g ; t h e y i m p l y 7 ~ 14° on t h e l a t t e r model, while t h e positions of t h e s p i n 2 levels s u g g e s t 7 = 22°. T h e h i g h e r Si z* levels c a n n o t be e x p l a i n e d o n e i t h e r m o d e l . T h u s Mg t4 is m o r e s u s c e p t i b l e to a d e f o r m e d n u c l e u s collective m o d e l i n t e r p r e t a t i o n t h a n Si2S; t h i s is correlated w i t h t h e k n o w n g r e a t e r n u c l e a r d e f o r m a t i o n of Mg ~4.

1. I n t r o d u c t i o n

A considerable b o d y of evidence has been built up in recent years in favour of the existence of collective behaviour in the 2s-ld shell. In particular, the odd-mass nuclei A12sand Mg 25, whose excited states have been studied in considerable detail 1), have several bands of rotational levels, each band being based on an independent particle level, which could be explained using the Nilsson model 2) of an axially symmetric, elliptically distorted nucleus. Collective behaviour has hitherto been less fully demonstrated for the even nuclei of this shell. R a k a v y 3) has used the Nilsson model for this shell, and predicts, for some of the even nuclei, deformations e comparable with those of the most distorted odd-mass nuclei. He compares these with the available experimental results, and notes that collective behaviour is most marked for those nuclei with predicted e greater than roughly 0.35. For Mg 24 and Si ~s, deformations of 0.47 and 0.23, respectively, are predicted. Some confirmation of this m a y be obtained 4) from the measured B(E2) values for Coulomb excitation of the first excited states of the two nuclei: 0.034 and 0.025 in units of e2× 10-4s cm 4, respectively. Assuming uniform charge distribution throughout both nuclei, these results m a y be used 5) to give values of e which m a y be compared with the above predictions: 0.51 for Mg 24 and 0.34 for Si 2s. It might therefore be expected that rotational collective behaviour is more strongly developed in the nucleus Mg a4 than in Si ~s. The level schemes of the two nuclei are summarised in fig. 1. Each nucleus has a ground-state 0 + , and first and second excited states 2 + and 4 + , respectively e), characteristic 604

EXCITED STATES IN Mg14 AND Siz8

605

but not necessarily indicative of a collective excitation. Higher excited states in both nuclei have been observed ~. s) and in many cases the spins and parities determined 6). The level energies in fig. 1 are those of refs. ~,s), with the exception of the value for the recently discovered e.9) sixth excited state of Mg 24, which was determined in the present work. EXCITATION ENERGY

Jr, K

(M,V)

EXCITATION ENERGY

jlr

(M,V)

6.432 6.005

0 4

0 2

5.224

3+ 2

4.2,52 4.122

2+ 2 4+ 0

1.368

2÷ 0

6.276

3 (+0

4.975 4.617

0 4 .4-

1.771

2+

0+ 0 Mq24

04. Si ztt

F i g . 1. L e v e l s c h e m e s of M g s* a n d Si sS.

It has been pointed out 3, e) that the third, fourth and fifth excited states of Mg ~4 are characteristic of a rotational band built up on the third excited state, identified with the single phonon n~ = 1, K = 2 mode. A similar level scheme can however, be qualitatively derived from the rotation of a triaxial ellipsoid using the methods of D a v y d o v and Filippov zo). The positions and spins of the third and fourth Si "s excited states bear no resemblance to these characteristically collective sequences. Further predictions of these collective models refer to the decay modes of excited states. In the Davydov-Filippov model, these are determined Zl) b y the degree of triaxiality and so are directly related to the positions of the higher levels. On the rotational-vibrational models, the predictions are not so detailed, b u t none-the-less, the partition of decays from one level, between the various members of a different band is accurately predictable both if K is a good quantum number 1,) and also in terms of a small degree of band-mixing zs). The smallness, and the degree of constancy of the required band-mixing coefficient within a pair of bands are clearly measures of the correctness of this model in a given nucleus. Other information m a y also be obtained from

606

A. V. COHEN AND J. A. COOKSON

decay modes; thus transitions within a K - b a n d should be favoured 13) over those between bands, and there is a selection rule 14) on Anp and An~. The decay of the 4.232 MeV level excited in Mg24(p, p') has been meas'~:ed b y Batchelor et al. 16), using a proton energy at which this level was prel~,c,tially excited, and found to be consistent with the levelbeing almost pure K --= 2, only a 3 % amplitude mixing of b ~..ds being necessary. A possible error in this technique, which would tend to over-estimate the proportion of bandmixing, would arise if a few per cent of the 4.122 MeVlevel, decaying exclusively by cascade, were excited, and its radiation unresolved from the cascade mode of decay of the 4.232 MeV level *. Decay schemes for other I~ . . . . of Mg 24 and Si 28 have been measured 16,1~) using resonant proton energies in Na 23 (p, 7) and A12~(p, 7); the inherent difficulties of this technique imply that only certain levels m a y be studied, and that decay modes are determined with a usuall 3 large quoted error. Many of the difficulties encountered b y previous workers are removed in measuring the decay modes of highly excited states, if one observes only those gamma rays in coincidence with the associated emitted charged particle. In the present experiment the reactions Mg*4(p, p') and Si28(p, p') have been investigated, the proton groups inelastical!y scattered at 90 ° to the incident proton beam b y nuclei of Mg ~4 and Si *s were magnetically resolved and a given group of inelastically scattered protons was used to gate a sodium iodide crystal, so that onl thosey gammas from the decay of the selected excited state were recorded. Results have been obtained for the decay schemes of the first six excited states of Mg 24 and the first four excited states of Si 28.

2. E x p e r i m e n t a l Method 2.1. A P P A R A T U S

In assessing the decay scheme of a level, itis necessary to allow for anisotropic gamma emission. Moreover, for inelastic scattering of 8-12 MeV protons b y Mg "4 and Si 2s, the reaction mechanism is quite complicated, as is evidenced b y the sharp peaks in the excitation curves is). In these circumstances, the only certain gamma symmetry is that of mirror reflection about the plane containing the incident and scattered protons and it is necessary to measure the gamma spectrum at as many positions as possible in the appropriate hemisphere. The design of apparatus is largely determined b y the neutron and gamma background detected in the sodium iodide crystal when the proton bearr, of incident energy up to 11 MeV, is stopped. Collimators must be few, of heavy material and as distant from the target as possible, while the Faraday cup measuring the beam must be remote from the NaI crystal and large enough to collect the major part of the beam, which diverges slightly after multiple * R. Batchelor, private communication

E X C I T E D STATES I N Mg 14 AND Si 18

607

scattering in the target. This requirement, and the need to observe the gamma spectrum over as much of a hemisphere as possible, makes the design of target chamber difficult unless the magnetic spectrometer is placed at a fixed azimuthal angle 0, which in this case is chosen to be 90 °. The proton beam from the A W R E T a n d e m Electrostatic Generator 19) was analysed b y a 90 ° beam-bending magnet, and focussed onto a target by two sets of magnetic quadrupole lenses, being defined by a 2.5 mm square gold slit assembly placed 85 cm before the target. The target itself (150 /~g/cm 2 of 99.9 °/o Mg 24 obtained from A E R E , Harwell and deposited on 100 #g/cm 2 gold or 100/~g/cm 2 natural silicon on 20 #g/cm 2 carbon) was set at 45 ° to the incident beam, in a holder at 0 = 90 ° and angle with horizontal ¢ ---- 45% The spherical target chamber was a brass shell, 10.3 cm diameter and 1.6 mm wall thickness. Tubes soldered into this contained the target holder, allowed observation of the target, enabled the reaction products emitted at 0 = 90 °, = 0 °, to be observed in the magnetic particle spectrometer, and allowed the beam to emerge at 0 = 0 °. The beam then passed into a series of brass tubes of increasing diameter, being finally stopped some 2 m from the target b y a 15 cm diameter gold-covered plate. The latter tube assembly was electrically insulated from the target chamber and used as a Faraday cup. The NaI crystal (11.4 cm long and 11.4 cm diameter) could be placed 11.4 cm from the target in nine distinct positions A-I, as listed in table 1, around the upper hemisphere, which was t he reby almost completely explored. The crystal was placed on a rotatable arm pivoting about the vertical axis, for positions with ~b = 0 ° and on suitable brackets resting on this arm for the remaining positions. TABLE Position Position 0 ~b

Magnet --90 ° 0

I

of the

1 NaI

crystal

A

B

C

D

E

F

G

+35 ° 0

+90 ° 0

+145 ° 0

180 ° 45 °

0° 45 °

90 °

--145 ° 0

H --35 ° 0

I +90 ° 45 °

The energy spectrum of protons emitted from the target at --90 ° in the horizontal plane was analysed in a 180 ° double-focussing magnetic particle spectrometer 2o) (61 cm radius, 0.007 sr solid angle, energy resolving power 850) b y means of which a selected group of protons could be detected in a CsI crystal, 2 cm diameter and 2 m m thick placed behind an adjustable slit at the spectrometer focus. The observed pulse-height spectrum in the crystal had a well-defined peak, due to the selected proton group, together with a background, due partly to protons scattering their way round the magnet and p a r t l y to gamma rays penetrating the lead shielding the CsI crystal. This background could be estimated b y altering the magnet current in suitable

608

A.V.

COHEN AND J. A. COOKSON

steps and observing the proton spectrum; it became relatively serious ( > 10 ~/o) for proton energies less than about 1.5 MeV. The gamma pulse-height spectrum from a D u Mont 6364 photomultiplier attached to the NaI crystal was amplified and observed with a 100-channel CDC kicksorter. Mumetal and iron shields round the photomultiplier reduced the effect of stray magnetic fields. Background in the NaI crystal, due to gamma rays coming from the beam collimating slits and from beam scattered onto the brass walls of the beam piping, was reduced very considerably b y filling all but a small region near the axis of the beam piping with lead, b y lining the entrance and exit tubes of the target chamber with gold and b y placing lead shielding near the beam collimating slits. Since over half the gamma counts then came from the target itself, this shielding was considered adequate, and no lead was put round the crystal. The beam was focussed to give minimum background for a given beam current, which was then adjusted so that the background greater than 0.3 MeV gamma energy was always below 106 per sec, at which level tests showed that the gamma energy resolution was not significantly reduced. Under typical conditions beam currents of between 0.15 and 0.25 /~A were used. The conventional fast-slow coincidence system 21) used had a fast coincidence unit of the type described b y Garg 23). The gamma ray slow discriminator was usually set to just exclude 0.51 MeV annihilation radiation. The fast coincidence time was set at 30 ns, the lowest time consistent with the time-spread in the magnet s0). Full fast-slow coincidences were used to operate a 1/,s gate on the slow gamma spectrum, which was then displayed on a 100-channel CDC kicksorter (channel A). Suitable delays could be inserted between the two fast lines, and the plateau of the delay curve was thus determined empirically for each group of inelastically scattered protons. To measure the random coincidonce spectrum (always less than 10 ~ of real counts) a duplicate coincidence unit was employed, with an extra 100 ns delay between the fast lines, and the gated slow gamma spectrum displayed on a second kick-sorter (channel B). The extra delay of 100 ns eliminates any effect of high-frequency beam intensity variation, known to be less than 10 ~/o, induced b y interference of the 20 MHz ~.F. ion source with the first Einzel lens of the Tandem injector. 2.2. P R O C E D U R E

The yield of inelastic proton scattering on Mg ~ and Si ~s with incident protons of 8-12 MeV is a strongly peaked function ! s) of energy, and so the energies at which maximum yield of a particular selected inelastic proton group occurred were determined, subject to the condition that the emergent proton energy exceeded 1.5 MeV, both to cut down proton counter background, and to achieve a reasonable emergent barrier penetrability. The lowest peak

EXCITED

STATES

IN

M g 24 A N D

Si 2s

609

energy above this value was usually selected, as the gamma and neutron background rises with incident beam energy. However, there were sometimes other considerations; thus the group corresponding to the 6.432 MeV level in Mg24 has as a close neighbour for 0 = 90 °, the group corresponding to inelastic scattering to the 6.135 MeV level in the O le contamination of the target, and so in this case the energy chosen was t h a t of the first maximum in the ratio of Mg.4 yield to O le yield at which clear resolution was possible. With the NaI crystal in each of the nine positions, the gamma spectra in both channels A and B were thus observed, for a standard number of proton counts, usually extending over 1½ to 2 h. For very complex spectra, where 100 kicksorter channels gave insufficient resolution, the experiment was performed at two kicksorter biases, and the two sets of spectra suitably combined. To assess the decay mode, all nine spectra were added together, moving peaks into alignment where necessary, since secular drift of electronics over several days of observation and stray magnetic field effects sometimes caused the gain to alter b y a few per cent. The slight difference between the gains of the two channels was allowed for before the total channel B curve (randoms) was subtracted from the total channel A curve (reals). The random coincidences were so few ( < 10 %) that any slight differences in the fast resolving times of the two channels added only a small error to the results. Thus the sum of the nine spectra after removal of background (the "summed spectrum") could be plotted as a function of gamma energy, after allowing for some nonlinearity revealed by energy calibration. After the peaks due to each gamma ray had been identified in each summed spectrum, attention was turned to each of the individual nine spectra A to I. The yield of each g a m m a . r a y in each of the nine positions was determined, using measurements ofa~ (where a~ is the effective solidangle as a fraction of a sphere and e the efficiency) determined as below (sect. 2.3). For this purpose, two definitions of efficiency were used: for gamma energies less than 2.75 MeV, the counts in the photo-peak and for higher gamma energies, the total counts corresponding to energies greater than 1 MeV below the peak energy. Wherever necessary, counts due to gamma rays of higher energy were allowed for. Errors were estimated and included those due to statistical effects, the difficulty of estimating the background of proton counts, the error in estimating o~, the error in subtracting the random coincidence curve and the effects of higher energy gamma rays, and the difficulty of precisely defining the region of kicksorter spectrum over which the total of gamma counts was estimated. For gamma rays of weak intensity ( < 2 0 %) it was necessary to estimate the yield, not from the individual spectra, but from the summed spectrum, with consequent slight ev4tra error. The gamma yield in the small portion of the hemisphere not covered by the nine positions A to I was estimated

610

^. v.

COHEN ^~D

J. A . C O O K S O ~

in three separate ways; one was an average over the nine positions and the others assumed symmetry about the 0 = 0 position and about the target nucleus recoil direction, respectively. A mean of these three was taken in estimating the gamma yield per proton; the spread of the three estimates indicated the error in this procedure, which was rarely as much as 10 %. In this manner an estimate was made of the yield of each gamma ray per inelastically scattered proton, for each excited state being studied. 2.3. C A L I B R A T I O N

Calibration determines the quantity coe as a function of gamma energy, and establishes an energy scale; for determination of efficiencies one is limited to ~ study of first states and other levels known to decay exclusively by one mode. The procedure outlined in sect. 2.2 was applied and cos determined for all the gamma rays listed in table 2. TABLE 2 G a m m a rays used for t h e d e t e r m i n a t i o n of efficiencies Target nucleus

Mg24

Mg24

Level (MeV)

1.368

4.122

Mg24 I

Si2S

Sia8

CX'

O16

5.224 ~) 1.771

4.617

4.433

6.135

1.771 2.846

4.433

6.135

b

G a m m a energies (MeV)

1.368

1.368 2.754

1.368 3.856

t 1.771 I

a) S h o w n in t h e p r e s e n t work to possess effectively a single decay mode.

This procedure has the advantage t h a t calibration methods are identical with those used for determining the more complex spectra of other levels. 1.5-

i L

1.0 ¢oc x uo =

2--

4

0.5

t I

L 2

I 3

L 4

1 S

I 6

I 7

GAMMA-RAY ENERGY (McV) Fig. 2. Efficiency of t h e sodium iodide crystal (11.4 c m diameter, 11.4 c m long, front face 11.4 c m from source). The energy dependence of co 8 (where co is the effective solid angle as a fraction of a sphere a n d 8 t h e efficiency) is given as d e t e r m i n e d from region 1 MeV below peak energy (upper curve), and as d e t e r m i n e d from area of p h o t o peak (lower curve).

EXCITED

STATES IN

M g ~4 A N D

Si ~

Bll

In this way, the curves shown in fig. 2 of we as a function of energy between 1.37 MeV and 6.14 MeV were.obtained. The shape of a gamma spectrum varies steadily with energy and could be constructed by interpolation at any intermediate energy; such synthetic curves were used in subtracting out the effects of high energy gamma rays from the superposed peaks corresponding to lower energy gamma rays. 3. R e s u l t s 3.1. T H E

M g 24 L E V E L S

First and second excited states (2~, 1.368 MeV and 4-[-, 4.122 MeV). The first and second excited states were used only for calibration, since it is wellknown that the second excited state decays exclusively by cascade. Third excited state (2~, 4.232 MeV). The working energy chosen was 9.46 MeV at a peak in the yield. The "summed spectrum" is shown in fig. 3. Threc

1.36~BMeV

[

P

u~ 5

r

°

'

, I 4.~'52

i'

2.804 M e i i

.

MeV .

=

I

.

p

.

l

, r

o

,J,

i r

~ ,

I

,

2

J

~ ,

3

4

%',

~'~"F+ .....

.5

E~,(M,vt F i g . 3. S u m m e d

s p e c t r u m of g a m m a

r a y s f r o m t h e t h i r d e x c i t e d s t a t e of M g ~¢ a t 4.282 MeV.

gamma rays are present: one (4.232 MeV) due to the crossover transition and the other two (2.864 MeV and 1.368 MeV) due to cascade radiation, the yield of the last two being the same within the limits of error. The ratio of crossover to cascade (using the procedure of sect. 2.2) is estimated to be 3.44-4-0.5, which is to be compared with the result of Batchelor et al. 15) of 2.95=E0.4. Fourth excited state (3+, 5.224 MeV). In principle, M3 decay is possible to the ground state a n d mixed E2 and M1 decay to all three intermediate excited states. Broude and Gove 6) detected the 1.368 MeV and 3.856 MeV gamma ays characteristic of cascade through the first excited state, other modes being less than 5 % of this, and showed that the 3.856 MeV radiation is pure E2.

612

A. V .

COHEN

AND

J.

A. C00KSON

We observed this level at an incident proton energy of 9.46 MeV, where there is a peak in the yield of width about 200 keV. The summed spectrum is shown in fig. 4; the predominant mode of decay is that observed b y Broude and Gove. The full curve in the diagram is a 3.856 MeV gamma ray. 1.36B McV

l

~6 o

T

==4 3.856 McV

z

÷

i,,+, 0

I

Z

3

4

$

6

7

r~ CM,v) Fig. 4. S u m m e d s p e c t r u m of g a m m a r a y s f r o m t h e f o u r t h excited s t a t e of IvIg14 a t 5.224 MeV.

Alternative modes of decay are: 5.224 MeV to the ground state (measured here to be < 5 %, compared with < 2 % quoted b y Glaudemans le)) and 1.102 MeV and 0.992 MeV to the second and third excited states, respectively, followed in each case b y the characteristic radiation of the level. Curve fitting in the region below the 1.368 MeV peak showed that there was less than 3 % of 1 MeV radiation. Thus decay is predominantly-through the first excited state ( > 92 %) justifying our use of this level for crystal efficiency determinations. Fifth excited state (J = 4, 6.005 MeV). The parity of this level is not known. D e c a y b y a 4.637 MeV gamma ray to the first excited state was observed b y Glaudemans le) and b y Broude and Gore e) who say it is quadrupole; decay to the ground state is improbable. Other possible decay modes of this level are: a ) t o the 5.224 MeV level, giving 0.781 MeV, 3.856 MeV and 1.368 MeV gamma rays; b) to the 4.232 MeV level giving 1.773 MeV gamma rays together with 77.5 % of 4.232 MeV gammas and 22.5 % of 2.864 MeV and 1.368 MeV g adnmas; c) to the 4.122 MeV level, giving 1.883 MeV, 2.754 MeV and 1.368 MeV gamma rays.

EXCITED

STATES

IN

Mg

24

AND

613

Si 18

The excitation curve of the reaction again had a peak at 9.46 MeV incident proton energy, which was selected for the work. The summed spectrum is shown in fig. 5, which demonstrates that the main decay mode is through the first excited state. The full curve shows a suitably normalised 4.637 MeV

1.368MeV

z6

g

*

u

4.637 McV

1.883MeV 2.7514MtV

2

,

0

,

i

i

,

I

r

i

p

i

2

i

i

~

i

S

I

~y(M,V)

%

,

,

~

4

5

T

,.

6

Fig. 5. S u m m e d s p e c t r u m of g a m m a r a y s f r o m t h e f i f t h e x c i t e d s t a t e of Mg 24 a t 6.005 MeV. N o t e t h e p e a k s n e a r 1.883 MeV a n d 2.754 MeV a n d t h e a b s e n c e of a p e a k n e a r 4.232 MeV.

gamma ray. In addition to this and the 1.368 MeV gamma ray, two other peaks are clearly present: one near 1.8 MeV, corresponding to ( 9 + 2 ) % of transitions, and the other near 2.8 MeV, corresponding to (5.5-4-3)% of transitions. The peak near 1.8 MeV implies that ( 9 + 2 ) % of transitions occur by modes b) and c). The peak near 2.8 MeV implies that about half of this occurs by mode c); this is consistent with there being no positive indication ( < 5 %) of 4.232 MeV radiation. These results are consistent with the observations of Glauderoans, that less than 15 % of decays occur by modes other than through the first excited state. Sixth excited state. This was first located by Gove at 6.4 MeV, decaying by cascade through the first excited state, the s y m m e t r y of the gamma correlations implying 6) a spin of 0. Hinds et al. 9) measured the energy of the state to be 6.44+0.02 MeV in the reaction C12(0 le, ,t)Mg24. We have observed protons from this level, with protons from inelastic

A. V. C 0 1 i E N A N D

614

J. X. C O O K . S O N

scattering b y target contaminant 018 as a close neighbour. Spectra were analysed at several proton energies in the range 8.8 to 9.8 MeV. In fig. 6(a) is shown the spectrum at 9.40 MeV; from the Hp of the peaks the incidel~t proton energy El, was deduced, since the excitation of the 018 level (6.1354'~00

2',0

,n zoo

o=

c~ 150 I00 I .50

I

I

2;)0

225

Z30

235

Z40

245

250

Hp, k G - c m

I00

75 -w

(b)

50

,<1 25

8.4

I.

I

I

I

I

1

~

8.6

8.8,

9.0

9.2

9.4

9.6

9.8

I

I0.0

P~O'tOX F.HE~C,'f ( H t ~ )

Fig. 6. T h e s i x t h e x c i t e d s t a t e of Mg 2¢. (a) O b s e r v e d p r o t o n s p e c t r u m a t 0 = 90 ° w i t h a n i n c i d e n t p r o t o n e n e r g y of 9.40 MeV. (b) Difference in e n e r g y A E a t 0 = 90 ° b e t w e e n p r o t o n s f r o m t h i s l e v e l a n d t h o s e f r o m t h e 6.135 MeV level in O t* ,as a f u n c t i o n of i n c i d e n t p r o t o n energy'.

0.010 MeV) is well-known 23). The difference AE in energy of the two groups was also deduced. Fig. 6(b) shows AE plotted as a function of Ep together with the best fit, dependence being linear at 90°; the best line corresponds to a nucleus of mass 24.5-+-2 with a difference in Q-values of the two inelastic processes of 2974-3 keV. Thus the energy of this level is 6.432~:0.010 MeV. For gamma decay observations, the lowest proton energy (9.37 MeV) consistent with a maximum yield of Mg ~4 relative to 018 peak was selected, and the gamma radiation was found to be isotropic within 4-5 %, confirming the spin assignment 0. In fig. 7, which shows the summed spectrum, some

EXCITED

STATES

IN

M g s4 A N D

Si t 8

615

6.135 MeV gamma radiation from O is is evident. The predominant gamma rays are 5.064 MeV and 1.368 MeV, corresponding to cascade through the first excited state. The full curve shows a suitably normalised 5.064 MeV gamma ray. There is in addition, a very clear peak near 2.200 MeV, corresponding to (174-3)% of decays.

I

o .ffi. z

,~,L 2

I '

2.200MeV

.

I

t

I

0t 6

2.864k,v I

0

I

2

5

E-},(MeV)

. ~.¢+A,.-¢~. L

5

6.135 McV

5

?

Fig. 7. S u m m e d s p e c t r u m of g a m m a r a y s f r o m the sixth excited state of Mg 24 a t 6.432 MeV. N o t e the peak at 2.200 MeV and the shape of the s p e c t r u m near 4.232 MeV.

If the level is of spin 0 + , there are only two likely transitions, both E2, via the spin 2 levels at 1.368 MeV and 4.232 MeV. If the level is 0--, E3 radiation to the 3 + level at 5.224 MeV is possible with yield comparable to the M2 radiation to other levels. The peak near 2.200 MeV can in either case arise only from decays to the 4.232 MeV level. This is confirmed b y the shape of the spectrum at 4.232 MeV near the expected associated gamma ray, and to a lesser extent, from the associated 2.864 MeV gamma ray. It is hard to decide if any decays occur through the 5.224 MeV level; if present, they are less than 5 % and would indeed be negligible on the assumption of positive parity for the 6.432 MeV level. The decay mode is then 83 ~o via the first excited state and 17 ~/o via the third excited state, the ratio of decay modes being 4.9+0.8. 3.2. T H E Si 28 L E V E L S

First and second excited states ( 2 + , 1.771 MeV and 4 + , 4.617 MeV, respectively). These were used for calibration; it was confirmed that, as expected, the second excited state decayed exclusively ( > 98 °/o ) b y cascade.

616

A. V. COHEN

AND

J.

A.

COOKSON

T h i r d excited state ( J = o, 4.975 MeV). The parity is unknown, decay b y emission of 0.358 MeV gamma radiation to the second excited state is unlikely and gamma-decay to the ground state is forbidden. These expectations are confirmed b y experiment. Protons from the 4.433 MeV level of C1~ form a close neighbour to those from this level, and the gamma spectrum was therefore I

1.771 HeY

1

°

*

z :m

o

4

+ z

4.505 HtV

2.84 MeV 2

++++ +kr+~'+ ,

0

1

~¢,1',0"

+

+!++. . . . . . I

2

5

4

5

G(H,v) Fig. 8. S u m m e d s p e c t r u m of g a m m a r a y s f r o m the f o u r t h excited s t a t e of Si ts a t 6.276 MeV. The small p e a k a t 2.846 MeV corresponds to ~ 8 % of decays t h o u g h the 4.617 level. The associated 1.659 MeV g a m m a r a y is p r e s u m a b l y m a s k e d b y the 1.771 MeV peak.

observed at an optimum incident proton energy for resolution and yield (7.93 MeV). Decay is as expected exclusively b y cascade through the first excited state, there being less than 10 % of the 2.846 MeV gamma radiation associated with the second excited state. Fourth excited state (J = 3, 6.276 MeV). The parity is probably positive, with a 2 % level of significance e). The gamma spectrum was observed at an incident proton energy of 9.18 MeV, where there is a maximum in the yield curve of width 300 keV. The summed spectrum is shown in fig. 8 and is consistent with 91.7 ~/o of decays via the first excited state and 8.3 ~/o via the second excited state, a ratio of (11-4- 5): 1. This confirms the proportion of 90 : 10 observed b y Endt and Heyligers 17). 4. D i s c u s s i o n 4.1. L E V E L P O S I T I O N S I N Mg 2~

The ground state and first two excited states of Mg 24 form a O+, 2W, 4 + sequence, not in itself necessarily proof of rotational excitation, since the

EXCITED STATES IN Mg~ AND Si t8

617

sequence might arise from properly symmetrised configuration excitation 24). A more sensitive, though still not definitive, test is the ratio of the excitation energies of the second and first excited states; this is 3.00 compared with an unperturbedrotational value of ~-. 3 Departure from this ratio m a y be explained b y band-mixing, the level energies being then given 14) in perturbation theory by El = A J ( J + I ) - - B ~ ( ] + I ) ] 2 , (1) with the empirical values A = 237 keV, B = 1.56 keV. These figures are related to the positions of the first/i- and 7- vibrational levels. The third excited state is 2 + , and its decay is characteristic 15) of a K = 2 level; it m a y well be the ~,-vibrational mode, n 7 = 1. If so, and accepting an empirical factor 14) derived from Pu ~88, the values of A and B imply that the first/t-vibrational level, 0 + , n p = 1, is very roughly at 20 MeV, b u t this estimate must be viewed cautiously. The fourth excited state, 3 + at 5.224 MeV, and the fifth excited state, J = 4 at 6.005 MeV, might well be part of a 2 + , 3 + , 4 + rotational band based on the third excited state. D a v y d o v and Filippov 10) have shown that a 2 + , 3 + , 4 + sequence can be derived from the rotation of a triaxial ellipsoid, without introducing any vibrational modes, the energies of this level sequence being determined b y an angle 7 which measures the degree of triaxiality. The position of the third excited state corresponds to an angle 7 = 22% An extension of this theory 25) allows for the presence of an n p = 1 level (the n~ = 1 level not being considered) which disturbs the positions of the levels, so that they are now determined not only b y 7, but also b y a parameter/~ related to the influence of the n p = 1 mode. The sixth excited state ] ---- 0 might be the n p = 1 level and would give p = 0.38. The positions of the fourth and fifth levels can now be independently predicted on all these models, while that of the second excited state is also determined on the two rotating triaxial ellipsoid models. The predictions are summarised in table 3. Thus none of these models predicts the level positions accurately, though TABLE 3 Predicted level positions (MeV) E x c i t a t i o n (MeV) J , ~

1) A s s u m i n g A and B required for the 0 + , 2 + , 4 + s e q u e n c e 2). D a v y d o v model w i t h o u t interactions (7 ~ 22°) 3) P e r t u r b e d D a v y d o v model (Y ~ 22°, P ~ 0.38)

4.122

5.224

6.005

4+

3+

J = 4

5.49

6.98

4.1

5.5

8.2

3.7

5.1

6.6

618

A. V. COHEN AND J. A. C00KSON

it is possible that more complex collective models might be more successful. An alternative shell model explanation has been proposed b y Banerjee 2e). An intermediate coupling calculation in the 2s-ld shell, using the SU 3 representation, predicts approximate level excitations up to the fourth excited state, though the second and third excited states are inverted. The wave-functions and Hamiltonian bear some resemblance to those of the collective model;, indeed Banerjee points out that the wave-functions have almost pure K-values, the mixing being sufficient, Elliott observes 3~), to explain the observed branching ratio of the third excited state. 4.2. D E C A Y

MODES

O F M g t4

On the axially symmetric rotational vibrational model, if a level of spin It belonging to a band K decays with multipolarity ~ to two levels of spins It and It', respectively, both belonging to the same band Kr = 0, then the ratio of the reduced decay widths is given 13) in terms of Clebsch-Gordan coefficients, b y the simple form

ItKt --->It Kt ItKt --~ I r ' K t

=

(It2Kt(Kr--Kt)[If Kt) 3 (It2Kt(Kr--Kt)[Ir, K t ) 3

.

(2)

If there is a small admixture of K-bands, then Mottelson has shown in ref. x3) that the above formula is modified b y a function ](It, It, I t, z), where z is a quantity which includes both the amplitude of band mixing and a factor describing the preferential emission of gamma rays within a band relative to that between bands. Thus measurement of decay modes m a y give an estimate of band purity. The Davydov-Filippov model 10.11) makes explicit predictions about decay modes as a function of ~,; in particular for 7 < 10° the predictions of this model are close to those of eq. (2). The Davydov-Chaban model 35) as yet has not been used to predict decay modes except in the Ifmit of weak coupling; the derived/~ value indicates that this limit is not appropriate. Third excited state (2+, 4.232 MeV). The ratio of cross-over radiation to the 0 + ground state to cascade radiation via the 2 + first excited state is 3.4440.5, compared with the 2.94-0.5 of Batchelor et al. 15), who also found that the ratio of the amplitude of E2 radiation to that of M1 in the cascade is (234-9): 1. This high proportion of E2 radiation would tend to support the assignment of K = 2 to this level, since for AK = 2, ZlL > 2 to the first order x3). Moreover, since the integrated cascade intensity is thus almost pure E2, while the crossover radiation must be pure E2, ratios of B (E2) values m a y be found b y simply extracting the E 5 term giving B(E2; 22 --->20)/B(E2; 22 ~ 00) = 2.474-0.35, compared with 2.954-0.5 of ref. 15). With pure K-bands, eq. (2) predicts a B(E2) ratio of 1.72. The observed ratios correspond to z = 0.0634-0.017 (present work) and 0.0934-0.025 (Batchelor et al.) with corresponding amplitude mixing of 2 °/o and 2.9 %

EXCITED

STATES

IN

Mg ~

AND

Siu

619

respectively, taking as a reasonable assumption, that transitions within bands are inherently ten times more probable t h a n those between bands. For the reasons mentioned in sect. 1, Batchelor's estimate of the degree of bandmixing might be expected to be slightly larger that, ours and this indeed is so, though the discrepancy is within the quote4 errors. These branching ratios are also consistent with intermediate coupling calculations in the shell model 26,27) using the SU a group. On the Davydov-Filippov model, the branching ratio corresponds to y---16.5°+2 ° inconsistent with the value ~ ~- 22 ° necessary to explain the level position. Fourth excited state (3 + , 5.224 MeV). The cascade radiation to the first excited state is pure E2 and more than 92 ~/o of total decays. Comparing transitions to this level and to the 4 + level at 4.122 MeV, both in the K = 0 band, and extracting an E 5 term (which assumes that only E2 transitions, if any, occur to the 4 + level) gives B(E2; 32 -+ 40)/B(E2; 32 -~ 20) ~ 12. This corresponds to z ~_ 0.39, consistent with the results for the 4.232 MeV level. The Davydov-Filippov model requires ? < 21 °. Filth excited state (J = 4, 6.005 MeV). On the assumption that this level is a member of the K ---- 2 band and therefore of positive parity, transitions both to the first and second excited states (91 °/o and 5.5-9 °/o, respectively) are expected to be predominantly electric quadrupole. The observed ratio of the two intensities after extraction of E 5 terms is then given b y B (E2; 42 -+ 40)/B (E2; 42 -+ 20) = 4.9-8.8, while the ratio for pure K-bands is 2.95. Thus z lies between 0.04 and 0.07 which is consistent with the figure 0.063 obtained from the decay of the third excited state. The proportion of decays going to the third excited state at 4.232 MeV is (3.5-4-3.5)%. On the assumption t h a t the fifth excited state is 4 + , this decay must be E2; after extracting the E 5 terms one has B(E2; 42 ~ 22)/B(E2; 42 -+ 20) ----4.3-+-4.3. Because of band mixing effects, it is difficult to use this figure to estimate the enhancement of quadrupole transitions within bands over those between bands. The Davydov-Filippov model accounts for the observed proportion of transitions to the 4 + state at 4.122 MeV, with a choice of 7 = 124-2°, inconsistent with the value of 22 ° necessary to explain the position of the third excited state. It is unable to explain the upper limit for transitions to the third excited state, since any choice of 7 below 22 °, implies a favouring greater than 15. Sixth excited state (J = 0, 6.432 MeV, K = 0). Whatever the parity of this state, decay with reasonable yield is possible only by quadrupole emission to the first and third excited states and is observed in the ratio 83 : 17. After extracting the E 5 terms, decay to the third state (K = 2) is favoured over decay to the first state (K = 0) b y a factor of 134-2 although the K-values

~0

A.v.

co~eN

A N D J. ^. COOKSO~

might lead one to expect favouring the other way. Eq. (2) cannot be applied to this case, but clearly an additional selection rule must be at work. A level of spin 0 in this region might conceivably be n p = 1, n r = 0. and it was so assumed in applying the Davydov-Chaban model in sect. 4.1. An alternative possibility for a spin 0 level is as part of an (rip ~---0, n,~ = 2) doublet ( J = 0 + , K = 0 and J --- 4 + , K = 4) expected on the simplest t heory to be degenerate at twice the energy of the n~ = 1 level, viz 8.46 MeV. The observed decay mode seems to favour this latter assumption. To a first order there is a selection rule 14) of 0 or + 1 on both Anp and An~. On the assumption np = 1, there is no reason in the weak coupling theory why transitions to the 9,+ level at 4.232 MeV should be so favoured; the alternative t hat this level is n r = 2, np = 0 seems consistent with the observed decay mode and implies a favouring of transitions for ~n~ = 1 over An~ = 2. . 4.3. L E V E L

STRUCTURE

O F Mg 24

'i

On the rotational-vibrational model of a deformed but axially symmetric nucleus, a broad picture of the level-structure is obtained, the 0 + , 2 + , 4 + sequence belonging to the K = 0 band and the 2 + , 3 + , 4 sequence belonging to a K = 2 band built on a vibrational state n r = 1, the mixing of bands being less th an about 2% in amplitude as is indicated b y the observed decay modes. The sixth excited state appears to be n~ = 2. However, the precise positions of the levels cannot be explained; the parameters required to describe the 0 + , 2 + , 4 + sequence do not adequately describe the positions of the higher levels. On the Davydov-Filippov model, the positions of the first three levels are consistent with 7 = 22° ,those of the fourth and fifth excited states are not v e ry well predicted and the decay modes of the third and fifth excited states require a smaller y. The Davydov-Chaban model predicts level positions better, assuming th at the sixth excited state has np = 1; decay mode predictions have not yet been made on this model. It would be v e r y interesting to compare theoretical predictions with these results, particularly for the sixth excited state, since the weak coupling rotational-vibrational model appears to favour a different origin for this level. 4.4. L E V E L S O F Si 2s

The excitation energies of the first and second excited states are in the ratio 2.61 and can be described with a choice A ~ 323 keV, B = 4.59 keV, corresponding to a rotational sequence with strong interference from othel states. The next two levels J = 0 and J = 3 are difficult to explain on a collective model of either the axially symmetric rotational-vibrational or the D a v y d o v type. It m a y be added t hat the observed decay of the fourth excited state (92 % 1 :

EXCITED STATES IN Mg24 AND Si28

621

to the first excited state and 8 % to the second excited state, of which the former is mainly dipole 6)), cannot be explained on a simple collective model. 5.

Conclusions

In broad outline, the positions and decay modes of the fir~t six excited states of Mg 24 suggest interpretation in terms of a collective model. However, the behaviour of the nucleus does not necessarily preclude a shell model, since Banerjee and Elliott were able to predict approximate positions of levels with an appropriately small amount of K-mixing; their technique predicts behaviour similar to that expected in a collective model. An elementary axially symmetric rotational-vibrational model gives a reasonable account of the order, but not the precise positions of the levels, while the decay modes correspond to fairly pure K-bands (z = 0.063-~-0.017). The sixth excited state on this model appears to be n~ = 2. The DavydovFilippov model requires a smaller ~ for the decay modes than for the positions of the spin 2 levels and does not account well for the positions of the higher levels, while the Davydov-Chaban model gives a somewhat better account of the level positions, but assumes that the sixth excited state is n B = 1. For this reason it would be of interest to extend the Davydov-Chaban model to predict decay modes. The higher levels of Si~8 do not appear to correspond to the predictions of either type of collective model. The deformations of the two nuclei, both as predicted theoretically, and as inferred from the measured B(E2) values, reinforce the conclusion that Mg24 is more susceptible to a deformed nucleus collective interpretation than is Si28. We are indebted to Dr. K. W. Allen for much helpful advice and encouragement during the course of this work. Thanks are also due to Dr. A. E. Litherland and Dr. N. MacDonald for m a n y helpful disscusions. The technical assistance of Mr. J. L. Wankling and Mr. M. H. Grant is gratefully acknowledged.

Note added in proo/: Recent work confirms that all the levels shown in fig. 1 have positive parity (J. A. Kuehner, A. E. Litherland, E. Almqvist and J. R. Evans, private communication). References 1) A . E . L i t h e r l a n d , H. M c M a n u s , E. ]3. P a u l , D. A. B r o m l e y a n d H. E. G o r e . Can. J. P h y s . 3 b (1958) 378 2) S. G. Nilsson, Mat. F y s . Medd. D a n . Vid. Selsk. 29, no. 16 (1955) 3) G. R a k a v y , N u c l e a r P h y s i c s 4 (1957) 375 4) H . E. C o v e a n d C. B r o u d e , P r o c . 1960 Conf. a t G a t l i n b u r g , T e n n e s s e e ( J o h n Wiley, N e w Y o r k 1960) p. 57

622

A.

V. C O H E N

AND

J. A. C O O K S O N

5) K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 6) H. E. Cove and C. Bronde, Nucl. Instr. Meth. l l (1961) 63; H. E. Cove, Proc. Int. Conf. on Nuclear Structure, Kingston, 1960 (North-Holland Publishing Co., Amsterdam, p. 471; (1960) private communication 7) S. Hinds and R. Middleton, Proc. Phys. Soc. 7b (1960) 553 8) S. Hinds and R. Middleton, Proc. Phys. Soc. 76 (1960) 545 9) S. Hinds, R. Middleton and A. E. Litherland, Proc. Phys. Soc. (In the press) 10) A. S. Davydov and G. F. Filippov, Nuclear Physics 8 (1958) 237; A. S. Davydov and V. S. Rostovsky, Nuclear Physics 12 (1959) 58 l l ) P. P. Day and C. A. Mailman, Argonne National Laboratory Report, ANL-6184 Argonne, Illinois, U.S.A. 12) G. Alaga, K. Alder, A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, no. 9 (1955)

13) P. Gregers Hansen, O. B. Nielsen and R. K. Sheline, Nuclear Physics 12 (1959) 389 14) R. K. Sheline, Rev. Mod. Phys. 32 (1960) 1 15) R. Batchelor, A. J. Ferguson, H. E. Cove and A. E. Litherland, Nuclear Physics 16 (1960) 38 16) P. W. M. Glaudemans (unpublished), reported b y P. M. E n d t ; Nuclear I n s t r u m e n t s and Methods l l (1961) 3 17) P. M. E n d t and A. Heyligers, Physiea 26 (1960) 230 18) A. V. Cohen and J. A. Cookson, Nuclear Physics 24 (1961) 529 19) K. W. Allen, F. A. Julian, W. D. Allen, A. E. P y r a h and J. Blears, Nature 184 (1959) 303 20) A. V. Cohen, J. A. Cookson and J. L. Wankling, Nuclear Instruments and Methods 10 (1961) 84 21) R. E. Bell, R. L. Graham and H. E. Perch, Can. J. Phys. 30 (1952) 35 22) J. B. Garg, Nuclear Instruments and Methods b (1960) 187 23) F. Ajzenberg-Selove and T. Lauritsen Nuclear Physics 11 (1959) 1 24) D. Kurath, Nuclear Spectroscopy, P a r t B, ed. b y F. Ajzenberg-Selove (Academic Press, New York, 1960) p. 983 25) A. S. Davydov and A. A. Chaban, Nuclear Physics 20 (1960) 499; E. D. Klema, C. A. Mailman and P. P. Day, Nuclear Physics 25 (1961) 266 26) M. K. Banerjee, Proc. Int. Conf. on Nuclear Structure, Kingston, 1960 (North-Holland Publishing Co., Amsterdam, 1960) p. 461 27) J. P. Elliott, Proc. Int. Conf. on Nuclear Structure, Kingston, 1960 (North-Holland Publishing Co., Amsterdam, 1960) p. 419