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Nuclear Physics A l l 5 (1968) 79--96; ~0 North-Hoiland Publishing Co., Amsterdam

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T H E 4STi(~, ~ ' ) R E A C T I O N

A N D S Y S T E M A T I C S O F O C T U P O L E S T A T E S I N T H E Ti I S O T O P E S

A. M. BERNSTEIN, E. P. LIPPINCOTT *, G. T. SAMPLE and C. B. THORN

Department of Physics and Laboratory for Nuclear Science it,

M1T, Cambridge, Massachusetts, USA

Received 1 April 1968

Abstract: The 48Ti(ct, ~') reaction has been studied at 31 MeV with 100 keV resolution and high statistics. Spin and parity assignments have been made on the basis of the systematic shapes of the differential cross sections and by comparison with distorted wave Born approximation calculations. These assignments have been made up to an excitation energy of 6.4 MeV. The magni- tudes of the cross sections have been related to electromagnetic transition rates by use of the vibrational model. Based on the assumption of pure lf~ configurations, the lowest 4 + and the second 2 + states should be excited by two-step processes only. The differential cross sections for these states do not support this hypothesis. The most interesting feature of the results is the identification of seven 3- states in 48Ti ranging in excitation energy from 3.36 to 6.33 MeV which vary in strength from 1 to 3 Weisskopf units. Three 3- states of comparable strength have been previously located in 4eTi and five 3- states in b°Ti. The strongest of these states has 3.5 Weiss- kopf units in ~STi and 7 Weisskopf units in ~°Ti. The weakness of the octupole states in this region compared to the 26 Weisskopf units found for the lowest 3- state in 4°Ca is surprising and may be difficult to reconcile with our understanding of current microscopic descriptions of this mode.

E NUCLEAR REACTIONS 4STi(ct, ct'), E = 31 ; measured tr(E=,, 0). tSTi deduced levels, J, zr. Enriched target.

1. Introduct ion

As p a r t o f a s tudy o f the ine las t ic sca t te r ing o f et-particles c o n d u c t e d at the M I T

C y c l o t r o n 1,2), levels o f 48Ti have been inves t iga ted . P r ev ious ly 4aTi has been in-

ves t iga ted wi th the (~, 0~') [refs. 3,4)], (p, p , ) [ref. 5)], (d, p) [ref. 6)], (d, ~) [ref. 7)],

(p, d) [ref. 8)] and (t, p ) [ref. 9)] r eac t ions and by fl- and y-decay s tudies 10-12) .

C o m p a r e d to the p r ev ious (c~, C ) e x p e r i m e n t s 3,4) a t 43 and 44 M e V , the p re sen t

e x p e r i m e n t has be t t e r ene rgy reso lu t ion .

T h e low- ly ing pos i t i ve -pa r i t y levels can be c o m p a r e d wi th the she l l -mode l calcula-

t ions o f M c C u l l e n , B a y m a n and Z a m i c k ( M B Z ) 13). Se lec t ion rules based on this

m o d e l p red ic t t ha t the second 2 + and first 4 + levels shou ld be exci ted by s e c o n d - o r d e r

t r ans i t ions 14).

Present address: Battelle Memorial Institute, Pacific N.W. Laboratory, Richland, Wash. tt This work is supported in part through funds provided by the Atomic Energy Commission under

Contract AT(30-1)-2098.

79

80 A. M. BERNSTEIN et al.

500

550

600

650

700 6 z

c

750

800 1

850

900

'2C 4.43

E

f . - . ~

< - ~ 12 C

<~-- t60

~ x I/ioo

950 - -

I 0 500

Contarmnant

7,15 6.99 6.8~ 6.76

6.47 6.56 6.24 6.09

5.86

5.54 5.54 5.16

4.96

4.59

4.39 4.35

4.05

3.85

3.36 5.24

3.00

(3-) ( Y )

(3-)

3- (4") (4")

2*

3-

3-

3- 4*

2.42 2* 2.29

0.984 2*

I I I I000 1500 ~000

Counts

0.00 O*

Fig . 1. S p e c t r u m o f s c a t t e r e d s - p a r t i c l e s f r o m 4aTi a t a l a b a n g l e o f 30.8 °. T h e s t a t e s i n d i c a t e d b y d o t t e d l ines a r e t he o n e s o b s e r v e d in th i s e x p e r i m e n t .

48Ti(~z, off) REACTION 81

Another reason for interest in 48Ti was to continue the study of the systematics of

octupole states. In particular the octupole strength has been found to be fractionated in the Ca isotopes 1) with three states found in 4°Ca and 48Ca and six in 42Ca and 44Ca. It is interesting to examine the behavior of the octupole strength in 4STi where there are both neutrons and protons outside of closed shells.

2. Experimental procedure

The experimental apparatus has been described in a previous paper 1). Alpha particles of 31 MeV energy were scattered by a thin foil (about 1 mg/cm 2) of metallic 48Ti enriched to 99 %. Scattered alpha particles were detected with 500/~m silicon surface-barrier detectors, and the resulting energy spectra were recorded on a 1024- channel analyser. The beam intensity was typically 0.2/~A, and over-all resolution was approximately 100 keV.

Spectra were taken at 1.8 ° intervals from 15 ° to 60 °. A typical spectrum at an angle that favors negative-parity states is shown in fig. 1. The angular aperture was about 3o in the scattering plane. The relative angular accuracy is 0.2 °, whereas the absolute angular accuracy is +_0.4 °. The data were analysed by computer using a least-squares fitting procedure to give the cross section at each angle. Absolute cross sections were obtained by comparison to the previously measured 4°Ca cross section at the 30 ° maximum with an overall error of 15 o/ /O"

The Q-values of the excited states were determined by assuming the first, third and sixth excited states to lie at 0.984, 2.423 and 3.365 MeV, respectively. This calibra- tion leads to an error of about 10 keV for each MeV of excitation.

3. Elastic scattering

The elastic scattering angular distribution is shown in fig. 2. The solid line is an optical-model fit to the data. A four-parameter complex potential well of the form

U (r) = - ( V + i W ) [ 1 + e ('-R)/" ] -

was used and the parameters varied for the best least-squares fit. The parameters for the curve shown in fig. 2 are V = 56.3 MeV, W -- 13.0 MeV, R = 5.58 fm and a = 0.646 fm. These parameters were used to calculate the inelastic scattering cross sections. Other parameter sets that give equivalent fits were also obtained.

4. Inelastic scattering

4.1. SPIN AND PARITY ASSIGNMENTS

The inelastic scattering data obtained in this experiment were analysed using the DWBA in which the ingoing and outgoung s-particle waves were distorted by the

82 A, M, BERNSTEIN et al.

optical potential found from the elastic scattering t 5). The angular distributions are calculated by using the collective model in which the surface of the nucleus is vibrating

4~x 8Ti Elastic i03~ xx

, o -

IO --

x

i0 ~ -- i ~

Io-' I I I I I I0 2 0 30 4 0 50 60 70

Ocrn ( d e g )

Fig. 2. Elastic scatter ing cross section fo r 31 M e V 0c-particles f r o m 48Ti. The sol id l ine is an opt ical - model fit.

around a spherical equilibrium shape according to the formula (to first order)

R(0', ,p') = Ro[1 + Y~ %.r?(o', ~o')1. lm

The effect of this oscillation on the optical potential is to introduce non-spherical

4STi(~t, ~t') REACTION 83

terms which give rise to inelastic scattering. This theory has the advantage that the shape of the angular distribution is then determined without any additional parameters. Although this theory leans on the vibrational model, it can be shown that the shapes of the predicted cross sections are sensitive only to the angular momentum transfer and not to the specific nuclear model employed.

As has been shown in detail in a previous (e, e') experiment in the Ca isotopes 1), the DWBA theory is in excellent agreement with the shapes of the observed differential cross sections. These angular distributions have characteristic shapes for different angular momentum transfers so that spin and parity assignments can be made with confidence. The reliability of this procedure is indicated by the high degree of similari- ty of the shapes of the differential cross sections for states of the same final spin and parity.

The only free parameter in the DWBA theory is determined by the magnitude of the cross section. This parameter is (/~/)2, the root-mean square deformation of the ground state due to zero point oscillations. Assuming that neutrons and protons move in phase, as one expects for AT = 0 excitations, the value o f / ~ obtained from inelastic scattering data can then be used to calculate an electromagnetic transition rate. It is assumed that the amplitude of vibration of the charge distribution and optical potentials are equal so that (/3R)E ~ = (/~R)~, where EM indicates the appropriate electromagnetic value and e the quantity measured in e-particle scattering 16).

In the past it has been customary to calculate the inferred electromagnetic transition rate from the vibrations of a uniform charge distribution whose radius REM is 1.2 A ~ fm [ref. 1)]. This procedure is convenient because it leads to a simple formula. However the measured charge distributions ~7) in nuclei are not uniform but can be more accurately characterized by the Fermi shape

p(r) = p0[1 +e('-c)/~] -1,

where on the average c = 1.08 A 6 fm and a = 0.585 fm [ref. 17)]. (This gives an equivalent 10 ~o to 90 % skin thickness of 2.5 fro.) The multipole transition rates calculated from a uniform-charge distribution compared to that calculated from a Fermi-charge distribution will be underestimated, particularly for the high multipolarities, where the surface region is highly weighted ~s). In the case of 48Ti, using the parameters just indicated, the ratios of the inferred electromagnetic transitions rates calculated by the use of a Fermi-charge distribution to that of a uniform-charge distribution are 1.14, 1.43 and 2.0 for multipolarities l of 2, 3 and 4, respectively. Therefore, the inferred electromagnetic transition rates have been calculated using the vibrations of a Fermi charge distribution. These rates which are presented in tables 1 and 2 are given in terms of a single-particle (Weisskopf) unit 19)

Bs.p.(El),O~l)--2l+l(3 ) 2 47r ~ (REM)2l e2 fm2/"

84 A. M. BERNSTEIN e l al.

TABLE 1 Resul ts for 2 + and 4 ~ states in aSTi

j= E ( M e V ) /?,~. G~ ~) Error in G~ ( ~ )

2 + 0.984 0.2l 16 15 2.42 0.058 1.2 15 4.96 0.045 0.72 25

4 + 3.24 0.082 4.6 15 5.16 b) 0.036 0.9 25 5.34 ~') 0.051 1.8 30

a) The inferred e lectromagnet ic t ransi t ion rates in single-particle units, (For discuss ion see text.) The error in G z is based on exper imental uncertainties. ' ) Tentat ive ass ignment only.

TABLE 2 3 state in 46,48, 5UTi

f12

Nucleus E (MeV ) MIT Saclay Argonne Gz

5°Ti 4.38 0.11 r) 0.113 5.6 6.57 0.068 0.066 2.1 6.72 0.055 0.088 ~) 1.4 7.13 0.073 2.5 7.72 0.048 0.063 1.4

48Ti 3.36 n) 0.079 2.9 L 2 5 °o 3.85 0.056 0.083 0.063 1 . 4 ± 2 0 ~ 4.59 0.070 0.070 0.068 2.3 ± 2 0 o/ 5.54 0.054 1.4 ~-20 °o 5.86 h) 0.054 1 . 4 ~ 2 0 % 6.24 h) 0.051 1 . 2 4 - 2 0 ~ 6.26 h) 0.052 1.3~ 20°~o

a6Ti 3.05 0.067 0.070 2.2 3.54 0.075 0.080 2.8 4.14 0.083 0.091 3.5 5.87 0.073 2.5

a) The exci tat ion energies are taken f rom the Saclay exper iments for 4eTi and ~°Ti and f rom this exper iment for 48Ti. b) This experiment . e) The 44 MeV (~, co') exper iment , ref. 4). The energy resolut ion was approx imated 180 keV. a) The 43 MeV (c¢, c¢') experiment , ref. 3). The energy resolut ion varied between 135 and 175 keV. e) The inferred E3 electromagnet ic t ransi t ion rate in Wei s skopf units (see text for discussion). For 46Ti and ~°Ti, the average value of/?3 was used wherever more than one value is indicated. For 4*Ti, the values offl3 f rom the present exper iment were used. The error in G3 is presented for this exper iment only and is based on exper imental uncertainties. r) The Saclay da ta were analysed us ing the Austern-Bla i r model . The results were normal ized to the /?~ value o f the 4.38 MeV 5°Ti state found in the Argonne experiment . This procedure el iminates the known systemat ic discrepancy between the analyses based on the Austern-Blai r model and those based on D W B A . g) This exper iment reports a 3 - group at 5.29 MeV which appears to be resolved into two 3- levels in the Saclay experiment . ~) Tentat ive 3- ass ignment .

Z

I"1-1 I

+

m

K'-'I] I i

i I

IIII I

1 I

I I

]1111 I

I I

I ]Iril

i I

r ,

llVl, I

i T

I

CO ~

J ~"

I

-o %_

To To

Js UP

q,,, np

t- o

o

<= e~

o

o_

2 ~J

0

T o -

10 4 I ] I I - - - - T - - - 7 - -

48T i 5 -

Io ~ _ / / ~ ' , , ;

10 2 _ - u ~

I0' I- o-~. V °~ ° ~ o __

,o-' i :,_.~ ~ _

-

1(5' I I I I I I 0 I0 20 30 40 50 60 70

ec ru ( deg )

Fig . 4. C ros s s ec t i ons for 3 s ta tes in 48Ti. The so l id l ines were D W B A c a l c u l a t i o n s . A n a s t e r i s k i nd i ca t e s an a s s i g n m e n t wh ich is less cer ta in .

4s1"i(ct, ~t') REACTION 87

It would be interesting to compare the inferred electromagnetic transition rates obtained in this experiment with those obtained by electromagnetic methods. At the present time this can be done only for the lowest 2 + level for which the electromagnetic value 2o) of 13.6_+2.7 Weisskopf units is in good agreement with that obtained in this experiment.

101 f

i i i I I I

48 Ti 4+

o I0

b ~ lo-' =I=

x 3 . 2 4

5.16"

n ) "~ X

v /

-3 io L_

0 I0 20 30 40 50 60 70

e c r u (deg)

Fig. 5. Cross sections for 4 + states in 48Ti. The solid lines arc D W B A calculations. The asterisk indicates an assignment which is less certain.

4.2. RESULTS As was indicated in the previous section, the shapes of the observed angular distri-

butions were compared to DWBA calculations and to each other to make spin and

88 A.M. BERNSTEIN e t al.

p a r i t y a s s i g n m e n t s . T h e levels t h a t we re iden t i f i ed in t h i s m a n n e r a re s h o w n in figs.

3-6. F r o m the m a g n i t u d e s o f t he d i f f e r e n t i a l c ro s s s e c t i o n s fo r e a c h s ta te , t he va lues

o f fl~ a n d o f t he i n f e r r e d e l e c t r o m a g n e t i c t r a n s i t i o n r a t e were o b s e r v e d . T h e s e r e su l t s

I° I A

E I.E -

b ,%,~\ _

I I ] I I I ] 012

i0

1.o

b

B

h t = 2

- ', \

:37',. \ / \, \ f ~ ',. \ / \o

\ /~, / ', \ ,/ ~ , , . \ \/' ~\ ii ~', \\ / ' ",,,

\ / ', Y F',~ "-" / \ /

O.I \ i \ / , , . /

0.05 I I i I I I ! 15 20 75 50 35 40 45 50

@CM (de(])

Fig. 6. Cross section for the 3.36 MeV group in ~STi which is assigned to be a doublet consisting of a 3 - and a positive-parity state. (a) Data with DWBA calculations representing the sum of l = 3 and 1 = 4 excitations for three combinations offl3 and/74. Curve 1 is for/?s = 0.081, fl~ = 0.063; curve 2 for/?s = 0.079, f14 = 0.070; curve 3 for/?s = 0.077,/?4 ~ 0.077. (b) Data with the results of curve 2 of (a) in which the individual contributions of the l = 3 and l = 4 components are shown as well as

the sum.

4gTi(~t, ct') REACTION 89

are presented in tables 1 and 2. Several levels have been identified (see fig. 1) whose angular distributions do not show any structure. The most likely reason for this is that these levels are unresolved multiplets. No assignments were made for these levels and they will not be considered further.

The angular distributions for 2 + states are shown in fig. 3. The 0.98 and 2.42 MeV levels are known to be 2 + and the assignment to the 4.96 MeV level is new. On the basis of pure lf~ shell model, the 2.42 MeV level should not be excited by a first-order transition 14). We shall discuss this point further in subsect. 5.1.

Fig. 4 shows the angular distributions to six levels which have been identified as 3- . The 3.85 and 4.59 MeV levels have been previously identified 3,4) as 3- and these assignments are confirmed by this experiment. The other 3- assignments are new.

The state at 3.36 MeV has been previously identified as 3- by Matsuda in a (p, p ') experiment 5). The angular distribution for this state is presented in fig. 6, and it is evident that it does not have the same structure as the 3- states that are presented in fig. 4, even though the cross section for this state is larger than for any of the states presented in fig. 4. This state has been analysed as a resolvable doublet with the 3.24 MeV, 4 + state. The data analysis was carefully checked to be certain that there was no "cross talk". The well-defined structure in the angular distribution of the weaker 3.24 MeV state also indicates that the data analysis was properly done. The high- resolution 4VTi(d, p)48Ti experiment 6) indicates a positive-parity level at 3.361 MeV, which is 16 keV above the 3- level identified by Matsuda 2). Because it was considered important to determine the octupole strength in this nucleus, the data were analysed as a sum of a 3- and a 2 + or a 4 + state *. The best fit was obtained for a 3- and 4 + combination. In fig. 6a, we present the results for several combinations for the magni- tudes of the 3- and 4 + cross sections. In fig. 6b, the individual 3- and 4 + cross sections and their sums for combination which give the best fit to the data are presented. Although the theoretical curves are high at small angles, the agreement is sufficiently good to tentatively assign the 3.36 MeV group as a doublet consisting of 3- and positive-parity states. The results for this group presented by Yntema and Satchler 3) are consistent with this interpretation. The strength of the 3- component is listed in table 2. Because of the uncertainty involved in summing the theoretical curves, the estimated error in /33 has been increased for this state. A higher resolution (:~, ~') experiment would be desirable.

Three angular distributions which agree with a 4 + one-step excitation process are presented in fig. 5. The 3.24 MeV state has been previously measured lo) to be 4 +, whereas the assignments to the 5.16 and 5.34 (tentative assignments) MeV states are new. In fig. 7a, we present the differential cross sections for the 3.24 and 2.296 MeV 4 + states. As was indicated in fig. 5, the cross section for the 3.24 MeV state is in agreement with a one-step l = 4 transition and is included in fig. 7a for comparison. The state at 2.296 is known to be a 4 + state lo), but the shape of its differential cross sec-

t The 6 + state 1o) at 3.342 MeV was not taken into account as 6 + states are usually weakly excited in (ct, cd) experiments 1).

9 0 A . M . BERNSTEIN e l aL

I I I I I 1 I I r I i I i I I I I I I I

i i !

:;2 22 -2--'~ )~

2~

_ TO

I " ' ' '

I I l J ~ l

?o

L

0

I 2 . _ ~ J l l u I : : _ 3 ~ J : : : : l : I r t i i J J r

o T ' ~o o o -

Js ~ p qua ~p

0 I,.-

_o

o

- - 0

._o

o

0 tD

o

E

0 9 o

0

rr ~.

- '~ <+., 0

< ~ ' ~

,-, i <-

- <--,

0

o ~

o 0

4STi(~, ~¢') REACTION 91

tion (fig. 7a) is that of a one-step 1 = 1 transition. The nuclear structure aspects of this result will be treated in subsect. 5.1. For the present we confine our attention to the following question: To what extent does an apparent assignment of a single-step l = 1 transition to a known 4 + state throw doubt on the other assignments made in (c~, e ') experiments? As has been noted previously 1) there is a lower limit of the magnitude of the differential cross section for which J~ assignments can be made with confidence on the basis of (e, e') data alone. The reason is that the DWBA theory is based on single-step transitions, and for weak states there is possible interference with two step processes. For the case of 31 MeV alpha particles in the Ca isotopes, it was decided empirically that approximately 0.2 mb/sr in the vicinity of 30 ° (or approximately 0.4 mb/sr in the vicinity of 20 °) was this limit 1). For smaller values of the cross section, J= assignments could not be made reliably. In the present case, the magnitude of 0.4 mb/sr at 20 ° for the 2.296 MeV level is at the previously made esti- mate of the lower limit for reliable assignments. Because of this criterion, the 5.16 MeV level whose differential cross section near 20 ° is approximately 0.3 mb/sr is given a tentative assignment only. All J~ assignments made in this paper correspond to larger cross sections and are not likely to be in error because of the presence of two- step transitions.

5. Discussion of results

5.1. POSITIVE-PARITY STATES OF 48Ti

The energies and inferred transition rates found in this experiment for 2 + and 4 + states are presented in table 1. The known 6,10) positive-parity states of 48Ti up to an excitation energy of 4 MeV along with the levels predicted by MBZ 13) based on the lf~ shell model are presented in fig. 8. Since the MBZ calculation neglected p- states, this calculation should not be expected to be accurate at high excitation energies. As has been demonstrated in the 47Ti(d, p)48Ti reaction 6), p-wave neutron capture is significant starting at the 2.42 MeV state. For excitation energies above 4 MeV, f~ neutron capture is negligible. Therefore it only makes sense to compare the MBZ prediction 13) for excitation energies below 4 MeV where there is in fact a reasonable qualitative agreement between these predictions and experiment. The main disagreements are that the predicted 3 + state at 3 MeV has not been found, and the existence of a 0 + level 12) at 3.0 MeV has not been predicted. It is tempting to speculate that this 0 + state is a deformed state as was found in 4°Ca, 42Ca and pos- sibly in 44Ca [ref. 1)].

It is common for shell-model calculations to reproduce energy levels far better than dynamic quantities. For the 2+ state (second 2 + state) at 2.42 MeV, the pure lf~ shell model predicts that the excitation should proceed via a two-step excitation 14). However, based on the shape of the differential cross section (fig. 3), our data indicate that the excitation is predominantly via a single-step excitation. This "violation" should be expected on the basis of the 47Ti(d, p)48Ti reaction 6) which did not find

92 A.M. BERNSTEIN et al.

the In = 3 stripping strength predicted for this state but instead found that this level was entirely excited by a 1, -- 1 transition.

The 4 + state at 2.296 M e V is predicted by the lf~ shell m o d e l 13) to be excited by a

two-step transition, and the 4~ state at 3.23 MeV is predicted to be excited by a first- order transition 14). The differential cross sections for these 4 + states are shown in

3,- (._9

0c LLI

Z Ld

Z C'

I'-"

L) X tO

= + + 4 2+

I+ - - +

- - +

- - ( 6 l . 6+

- - + - - 6 6+ ~ ( 4 1 4+

t+

3 o 3 +

- - 4 + - - 2"

4

2+ 2

. 2 +

0 0 + . 0 +

E X P M B Z

Fig. 8. Energy levels of positive-parity states in 4STi to an excitation energy of 4 MeV. The theory is from MBZ, ref. 13). The experimental results T M ) that are shown do not include states at 3.71, 3.79 and 3.86 MeV for which no parity information exists or the 3- states at 3.36 MeV. The level density

above 4 MeV is high and includes positive-parity levels at 4.05 and 4.09 MeV.

fig. 7a. The 4 + state is in agreement with a one-step l = 4 excitation in agreement with the theoretical predictions. For the 4+ state, we find the surprising result that the observed angular distribution can be fitted with a one-step I = 1 transition. A calcula- t ion o f the differential cross section of this state using a combinat ion o f mult iple (M) and direct ( D ) excitat ion mechanisms 3) has been performed by Satchler 2z). The results are shown in fig. 7b for several combinat ions o f Mand D (M - D, M and M + D). It can be seen that these curves do not even qualitatively resemble the data. In particular, the peak near 20 ° is always much broader than is observed experimentally. Curves for several other combinat ions o f M and D (M_+ D/2 ) were also calculated and they all have the same broad m a x i m u m near 20 ° .

4STi(ot, 0t') REACTION 93

An attempt has been made in this laboratory to observe a ground state 7-ray branch from the 3.23 MeV level to the ground state to see whether there is an accidental de- generacy of a 4 + and a 1- state 23). No ground state 7-rays were seen, making this possibility quite unlikely. We believe that the state we have observed is an isolated 4 + level, and we have no explanation for the observed angular distribution.

Yntema and Satchler have excited these states with 43 MeV s-particles 3) and con- clude that the 4 + and 2~- states are excited by multiple excitations. The angular distributions observed in this experiment have much deeper minima than in this latter work. A possible explanation of this difference may be the better resolution of the present experiment. In the spectrum shown by Yntema and Satchler the 2 + and 4 + states are not resolved 3).

In their original paper on the shell-model selection rules in 48Ti, Garvey et al. ~4) also reported a 48Ti(p, p ') experiment with 17.5 MeV protons. They reported relative excitations of the + + 2~/22 states of 15 and 5, respectively, which 4 z/41 states and the + + they claimed as verification of these selection rules. The ratios of 13 and 4 found in the present experiment are in agreement with their results. On the other hand, we do not agree that these ratios necessarily prove the validity of the shell-model selection rules. The electromagnetic transition rates of 16 and 1.2 Weisskopf units, respectively, for the 2~- and 2 + states (table 1) indicate that the transition to the 2 + state is enhanced rather than that the transition to the 2~- state is retarded. In both 42Ca and 44Ca [ref. a)] as well as in most doubly even nuclei 24), the second 2 + state is approximately an order of magnitude weaker than the first 2 + state.

5.2. SYSTEMATICS OF 3- STATES IN THE Ti ISOTOPES

Table 2 lists the results for the 3- states that have been found to date in the even- mass Ti isotopes. There have been three (~, ~') experiments on the Ti isotopes; at Argonne with 135 to 170 keV resolution 3), at Saclay with approximately 180 keV resolution 4) and the present experiment with 100 keV resolution. For each nucleus we shall use the results with the best resolution in the final results which are plotted in fig. 9. The most striking feature is the large number of states found to date. An additional surprise is the relatively small magnitude of the transition strengths G 3 (single-particle units). The strongest 3 - state is the lowest state in 50Ti with 5.6 single- particle units. These numbers compare to the lowest (and strongest) 3 - states of 4°Ca and 48Ca whose strengths are 26 and 9 single-particle units, respectively. The rapid and surprising drop in the 3- strength in going from 4°Ca and 48Ca has already been discussed ~). As can be seen from the data presented in fig. 9, the relationship of 50Ti to 48Ca is similar to that of 42Ca to 4°Ca in that the addition of the two extra core particles gives rise to an additional fractionation of the observed octupole strength and reduction of that strength. For both 5°Ti and 42Ca, the energy of the lowest 3 - state is slightly lower than that of 48Ca and 4°Ca, respectively.

The situation in 46Ti and 48Ti appears to be more complex. First of all we note that the energy of the lowest 3 - states is approximately 3 MeV which is more charac-


teristic of the 4°Ca core than the 48Ca core. Second we note that the spacing between the first and second 3- states is smaller than in 4ZCa and 44Ca. Third we note the lack of a really collective 3- state in 41Ti and 48Ti. In fact, in 46Ti the lowest 3 - state is

not the strongest and the first three 3- states increase in strength as the excitation energy increases. Since microscopic calculations of the octupole strength have only been carried out for 4°Ca in this region, we do not know what this model will predict. It is clear that a simple vibrational picture of the 3- state that essentially is slowly varying with mass number is not correct. In order to reproduce the data, a calculation will have to take into account the couplings of the extra core nucleons.

8 '

7 ¸

6

1

o l I 4Oco

~ 5

o:: w4. z uJ

_g3.

I---

×2 w

I 42Ca 46Ti r i 4aCa

m B

5OTi 52Cr 54Fe ~Fe

I 58N i

Fig. 9. Systematics o f 3- states f rom 4°Ca to ferred electromagnetic transit ion strength in

The horizontal scale is

58Ni. The length of each line is proport ional to the in- single-particle units (see text, subsects. 4.1 and 5.2). 10 single-particle units per box.

I t is of interest to inquire whether the weakness of the octupole states observed in the Ti isotopes can be explained by a deformation of these nuclei. In the Gd and Sm isotopes, the strength of the lowest 3 - state has been observed 25) to drop rapidly as the neutron number becomes greater than 88. This drop in the E3 strength occurs at the onset of permanent deformation. However, it appears unlikely that this is what is happening in the Ti isotopes because no evidence of rotational bands is seen in 46Ti or 48Ti.

In order to discuss the systematics of the octupole strength in the Ca and Ti isotopes, it is useful to exhibit these systematics in the rest of the shell. In fig. 9, data are plotted for the octupole states of 52Cr [refs . 26 '27) ] , 54Fe [ref. 4 ) ] , 56Fe [ref. 28)] and 58Ni [ref. 24)]. For nuclei with 28 neutrons (48Ca, 5°Ti, 52Cr and 54Fe), the position and

48Ti(ct, ~') REACTION 95

strength of the lowest octupole state remain approximately constant. The octupole strength starts to increase in S SNi and increases to about the same strength as in 4°Ca in the Zn isotopes 3o). It appears that the rapid and dramatic drop in strength as nucleons are added to 4°Ca is associated with the fact that these nucleons are added to the If k shell. It can also be seen that it makes a large difference whether there are both lf~ neutrons and protons rather than neutrons alone or protons alone. When nucleons are added above the 1 f~ shell, then the octupole strength increases again 30). At the present time, there are no calculations which bear on this phenomenon. It re- mains to be seen whether this effect is within the scope of present models.

6. Conclusions

The findings of this experiment raise as many questions as are answered. For the 4 + state further theoretical work is required to understand the excitation mechanism. Further experimental work at higher alpha-particle energies might also be useful in this respect to see whether the anomalous shape of the differential cross section for the 4 + state persists.

The major result of this experiment concerns the fractionation of the octupole strength in 48Ti and the observation that none of the octupole states in 46Ti or 48Ti are very strong. The rapid and surprising drop in the octupole strength for nuclei in the lf~ shell merits a careful calculation to see whether this effect can be explained by current theories of the octupole state. It is our opinion that these data will prove quite difficult to fit with present models.

We would like to thank W. J. Kossler for communicating his experimental results before publication and for valuable criticisms of the manuscript. We would also like to thank D. R. Hendrie for communicating his results prior to publication and G. R. Satchler for his performance of coupled-channel calculations. The assistance of the cyclotron staff of E. F. White, F. Fay, W. Bucelewicz and J. Byrne was extremely valuable. The programming assistance of T. Provost is gratefully acknowledged. The assistance of Mrs. M. E. White and Mrs. K. H. Smith in preparing the manuscript is appreciated.

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